"Stark
+.6.>"resonance
+.6.>"van der Waals
+.6.>"natural
+.6.>"Doppler
+30.2XH (P/T)
+1.8P/To.7
+3.1 X 10-5
+
+--- Page 525 ---
+510
+Plasma Diagnostics
+8.5.4.3 Reonance broadening
+Resonance broadening is caused by collisions between 'like' particles (e.g.
+two hydrogen atoms) where the perturber's initial state is connected by an
+allowed transition to the upper or lower state of the radiative transition
+under consideration. Typically, the three perturbing transitions that must
+be considered are g ---+ I, g ---+ U, and I ---+ U, where g stands for the ground
+electronic state, and I and U for the lower and upper states of the radiative
+transition. Using the expression given by Griem (1964, p 97), we obtain
+3e2
+.6.·\esonance = 16 2
+2
+7r comec
+'-v--"
+6.72 x 1O-16 m-2
+Using
+the
+constants
+of Wiese
+et
+al (1966)
+(Aul = 486.132nm,
+Alg = 121.567 nm, Aug = 97.2537 nm, gu = 32, gg = 2, gl = 8, fgl = 0.4162,
+fgu = 0.02899, fiu = 0.1193), we obtain the resonance HWHM listed in
+table 8.5.1.
+8.5.4.4
+Van der Waals broadening
+Van der Waals broadening is caused by collisions with neutral perturbers
+that do not share a resonant transition with the radiating particle. Griem
+(1964, p 99) gives the following expression for a radiating species r colliding
+with a perturber p:
+~ ul
+a
+3/5
+A2 (97r1i5 R2 )2/5_
+.6.Avan derWaals ~ 2c
+16m~E]
+Vrp Np
+(2)
+where vrp is the relative speed of the radiating atom and the perturber, Ep is
+the energy of the first excited state of the perturber connected with its ground
+state by an allowed transition, Np is the number density of the perturber, and
+the matrix element R~ is equal to
+2" ~ 1
+EH
+[
+z2 EH
+]
+Ra ~ 2. E _ E
+5 E
+_ E + 1 - 31a (ta + 1) .
+00
+a
+00
+a
+(3)
+In equation (3), EH and Eoo are the ionization energies of the hydrogen atom
+and of the radiating atom, respectively, Ea is the term energy of the upper
+state of the line, la its orbital quantum number, and z is the number of
+effective charges (z = 1 for a neutral emitter, z = 2 for a singly ionized
+emitter, ... ). For H(3, we have EH = Eoo = 13.6eV, Ea = 12.75eV, and
+z = 1. The H(3 transition is a multiplet of seven lines (see table 8.5.2)
+
+--- Page 526 ---
+Plasma Emission Spectroscopy
+511
+Table 8.5.2. Components of the H(J transition multiplet and their properties.
+Wavelength
+Aul
+Upper level
+Lower level
+gu
+gl
+Relative
+air (nm)
+(S-I)
+configuration
+configuration
+intensity
+(% of total
+H(J emission)
+486.12785
+1.718 x 107
+4d2D3/2
+2p 2 P?/2
+4
+2
+25.5
+486.12869
+9.668 x 106
+4p211/2
+2s2 SI/2
+4
+2
+14.4
+486.12883
+8.593 x 105
+4s 2 SI/2
+2 20
+p PI /2
+2
+2
+0.6
+486.12977
+9.668 x 106
+4p 2 P?/2
+2S2 SI/2
+2
+2
+7.2
+486.13614
+2.062 x 107
+4d2D5/2
+2p211/2
+6
+4
+45.9
+486.13650
+3.437 x 106
+4d2D3/2
+2p211/2
+4
+4
+5.1
+486.13748
+1.719 x 106
+4s 2 SI/2
+2p211/2
+2
+4
+1.3
+originating from upper states 4s, 4p, and 4d of orbital angular momenta
+la = 0, 1, and 2. For la = 0, 1, and 2, (R~)2/5 takes the values 13.3, 12.9,
+and 12.0, respectively. As listed in table 8.5.2, the components issued from
+the 4s, 4p, and 4d states represent 1.9, 21.6, and 76.5% of the total H{3
+emission, respectively. We use ~ese percentages as weighting factors to
+determine an average value of (R;)2/5 = 12.2.
+The relative velocity term v;P of equation can be related to the mean
+speed as follows:
+v;j,5 = (4/1f)2/1Or(9/5)(vrp)3/5 9:! 0.98(vrp )3/5 = 0.98(8kT /1fm;p)3/1O
+(4)
+where m;p is the reduced mass of the radiating species and its perturber.
+Summing over all perturbers present in the plasma, and introducing the
+mole fraction Xp of perturber p, equation becomes
+,2 (9 1052 )2/5
+[
+X
+]
+~
+Aut
+1fn Ra
+3/10 P
+p
+.6.Avan derWaals ~ 0.98 2c
+16m3 E2
+(8kT /1f)
+kT L
+4/5(. )3/10 .
+e p
+p
+Ep
+mrp
+(5)
+In air plasmas, 0, N, N2, °2, and NO represent 98% of the chemical
+equilibrium composition for temperatures up to 10 000 K. We computed
+the equilibrium mole fractions of these five species up to 10 000 K and
+combined them with the Ep and m;p values listed in table 8.5.3 in order to
+evaluate the summation term in equation (5). The value of this term is
+found to be approximately constant over the entire temperature range and
+equal to 0.151 ± 0.007. The final expression for the van der Waals HWHM
+of H{3 in air plasmas with a small amount of hydrogen added is given in
+table 8.5.1.
+
+--- Page 527 ---
+512
+Plasma Diagnostics
+Table 8.5.3. Constants needed in equation (5) when the radiating species is a hydrogen atom.
+Perturber
+M;p
+Transition issued from the first
+Ep
+M;p -0.3 E;;0.8
+(g/mole)
+excited state optically connected to
+(eY)
+(g/mol)-0.3 ey-O.8
+the ground state
+0
+0.94
+3sO _
+3p
+9.5
+0.17
+N
+0.93
+4p _
+4sO
+10.3
+0.16
+O2
+0.97
+B3z:,;; -
+X 3z:,i (Schumann-Runge)
+6.2
+0.23
+N2
+0.97
+bIng _
+X lz:,; (Birge-Hopfield I)
+12.6
+0.13
+NO
+0.97
+A 2z:,+ _
+X 2n (gamma)
+5.5
+0.26
+8.5.4.5 Doppler broadening
+For a collection of emitters with a Maxwellian velocity distribution
+(characterized by a temperature Th ), Doppler broadening results in a
+Gaussian lineshape with HWHM given by Griem (1964, p. 101):
+1
+D.ADoppler = "2 AUl
+The Doppler HWHM of H(3 is given in table 8.5.1.
+8.5.4.6 Natural broadening
+Natural broadening gives a Lorentzian line profile of HWHM:
+D.Anatural = :;~ (L Aun + LAin)
+n
+E
+-
+-8
+iii
+C
+C)
+en
+-6
+en c
+-4
+a:: o
+-2
+10
+15
+20
+25
+30
+Time (J.IS)
+Figure 8.6.7. Experimental ring-down traces with the laser tuned to the Nt absorption
+bandhead (inset) with the high-voltage pulse (solid line) and without the high-voltage
+pulse (dashed line).
+uncertainty in the electrical measurement (10%) is primarily from uncer-
+tainties in the momentum transfer cross-section (5 %), the discharge area
+(4%), and the average gas temperature (8%). Column 4 of table 8.6.l
+shows that the electron number densities found from optical and electrical
+measurements overlap within their error bars. This excellent agreement
+gives us confidence in our results for the electron number density.
+8.6.3.6
+Temporal profiles of Nj concentration and electron number density
+Figure 8.6.7 shows ring-down traces obtained with and without firing the
+high voltage pulse, and with the laser tuned to the Nt B-X (0,0) bandhead.
+In the absence of the high-voltage pulse (dashed line) the absorption losses
+are constant in time, and the signal decays as a single-exponential. In the
+trace with the pulse (solid line), the light decays more steeply after the
+pulse, reflecting an increased concentration of Nt. The spike in the latter
+trace coincides with the firing of the pulse, and is caused by rf interference
+generated by the pulser. To verify that we are observing changes in the Nt
+concentration, we examine the analogous traces but with the laser de tuned
+from the absorption band (see figure 8.6.8). These traces confirm that the
+only effect of the high voltage pulse on the ring-down system is to generate
+the interference spike. We analyze these traces to determine over what
+region the interference spike affects the data. We vary the delay of the
+high-voltage pulse relative to the laser shot so that we can obtain ion concen-
+trations at different times.
+
+--- Page 544 ---
+Ion Concentration Measurements
+529
+-12
+-
+-10
+>
+E -
+-8
+C; c
+CJ
+-6
+en
+tn
+C
+-4
+a::
+0
+-2
+IHV
+-.......... -
+- -, Pulse
+o
+5
+10
+15
+20
+25
+30
+Time (JJS)
+Figure 8.6.8. Experimental ring-down traces with the laser tuned away from the Nt
+absorption (inset). We slightly scale «5%) the amplitude of the traces for visual clarity.
+The detuned trace (dashed line) is offset by 0.2 m V to make it more visible.
+We quantify the time-varying Nt concentration using equation (4) with
+a 1 ~s window. This time interval represents a good compromise in making
+the window short compared to the timescale of the process studied yet
+affording an acceptable signal-to-noise level. The empty-cavity losses
+(mirror reflectivity) are found from the ring-down signals with the laser
+detuned, and these losses are subtracted in the analysis. Using tabulated
+line strengths and the discharge dimensions, we find the absolute Nt center-
+line concentrations as a function of time. Figure 8.6.9 presents the time-
+varying concentrations (symbols). The error bars reflect uncertainties in
+the population fractions, as well as uncertainty associated with a possible
+change in shape of the concentration profile. The latter uncertainty is
+estimated by chemical kinetic considerations (see Yalin et aI2002). One micro-
+second after the pulse, the Nt concentration is '" 1.5 x 1013 cm -3, and then Nt
+recombines to the dc level in about 1 0 ~s. The dc level is found by analyzing the
+pulsed data at sufficiently long time delays after the pulse, and its value is
+consistent with that found in the dc plasma without the pulser.
+For the pulsed discharge, we also determine the electron concentration
+by measuring the electrical conductivity. The temporally resolved electron
+concentrations are shown with a swath in figure 8.6.9. The uncertainty in
+the dc electron concentration reflects uncertainties in the profile shape, the
+momentum transfer cross-section, and the gas temperature. The colli-
+sional-radiative model predicts that Nt is the dominant ion produced by
+the pulse. Thus, the agreement between the time-dependent electron and
+Nt concentrations during plasma recombination verifies the temporally
+
+--- Page 545 ---
+530
+Plasma Diagnostics
+16 -
+'1
+E
+(,) 12
+... ... o
+"I:""
+-
+8
+....
++
+~ 4
+S' -4,,.----------,
+til
+'-" -5
+00-6
+o
+0:::
+~ -7
+------------"_._----
+5 -8h10-.."...I!--~=--.,..4.-,8,-1
+O+-~~_r~~~~_y~~~~~
+o
+2
+4
+6
+8
+10
+12
+14
+Time after Pulse (~)
+16 -
+'1
+12 E
+(,)
+...
+"'0
+8 ~
+CD
+C
+Figure 8.6.9. CRDS measurements of Nt concentrations (circles) and conductivity
+measurements of electron densities (swath) versus time following the firing of a high-
+voltage pulse in an atmospheric pressure nitrogen dc plasma. The dc level of Nt concen-
+tration found by CRDS is shown with a hatched bar. The inset shows the ring-down signals
+(plotted on a semi-log scale) with the HV pulse (solid), and without the HV pulse (dotted).
+resolved CRDS measurement. The measured recombination time is consis-
+tent with reported (Park 1989) dissociative recombination rate coefficients
+for Nt (approximately 5 x 10-8 cm3/s).
+8.6.3.7 Non-equilibrium discharge
+To have a measure of the degree of non-equilibrium in the dc discharges, we
+examine the ratio of the measured electron number density (at the radial half-
+maximum) to the LTE electron number density at the corresponding
+gas temperature. These ratios are given in column 3 of table 8.6.2 for
+the four conditions studied in the dc discharge. The measured ion and
+electron concentrations in the discharge are significantly higher than those
+Table 8.6.2. Ratio of the measured dc electron number density
+to the concentration corresponding to a L TE
+plasma at the same gas temperature.
+i (rnA)
+Tg (K)
+ne-CRDS/ne-LTE
+52
+3100
+2.8 x 104
+97
+3600
+980
+142
+4200
+48
+187
+4700
+5.6
+
+--- Page 546 ---
+Ion Concentration Measurements
+531
+corresponding to LTE conditions at the same gas temperature. The results
+quantify the degree of ionization non-equilibrium in the discharges. At
+higher values of discharge current the LTE concentration of charged species
+rises steeply, so that the ratio of measured concentration to LTE concentra-
+tion reduces. Related work in our laboratory has shown that by more rapidly
+flowing the gas, comparable electron densities may be achieved with lower
+gas temperatures. Clearly, additional non-equilibrium is generated in the
+pulsed discharge. The high voltage pulse has a negligible effect on the gas
+temperature (and hence corresponding LTE number density) yet the
+measured electron number density in the discharge increases by a factor of
+at least 4 immediately following the high voltage pulse.
+8.6.4 NO+ measuremeuts
+8.6.4.1
+RF air plasma
+The experimental set-up is shown schematically in figure 8.6.10. Atmospheric
+pressure air plasmas are generated with a 50 kW rf inductively coupled
+plasma torch operating at a frequency of 4 MHz. The torch is operated
+with a voltage of 8.9 kV and a current of 4.6 A. The torch has been
+extensively characterized at similar conditions, and the plasma is known to
+be near LTE with a temperature of about 7000 K (Laux 1993).
+8.6.4.2 CRDS measurements
+Unlike the Nt ion, the NO+ ion does not have optically accessible electronic
+transitions. To perform CRDS measurements, the ion must be probed by
+accessing its infrared vibrational transitions. The strongest vibrational tran-
+sitions are the fundamental bands, and for these transitions one finds that the
+Nozzle
+(7 em diameter)
+Quartz
+Tube
+Power and
+___
+Cooling Water .........
+Coil
+Plasma Exit Velocity: -10 mls
+'t:tlow (5 em) = -5 ms
+'t:cllemistry < I ms
+Gas Injectors:
+• Radial
+• Swirl
+• Axial
+Figure 8.6.lO. Schematic cross-section of torch head with 7 cm diameter nozzle.
+
+--- Page 547 ---
+532
+Plasma Diagnostics
+9.0><10.5
+8.0><10.5
+70x10·5
+6. 0x10 5
+fl 5. 0x10 5
+r:::
+til -e 4.0><10.5
+0
+UI
+.c « 3.0><10.5
+2.0x10·5
+1.0><10.5
+0
+3.8
+4.0
+4.2
+4.4
+4.6
+4.8
+A. (Il-m)
+Figure 8.6.11. Modeled absorbance of the air plasma at LTE temperature of 7000K over
+pathlength of 5 cm. Absorption by NO, OH, and NO+ are included. Rotationally resolved
+lines of the vibrational transitions are shown.
+absorbance per NO+ ion is about 20000 times less than that of the electronic
+transitions of the Nt ion. Figure 8.6.11 shows the modeled absorbance, as a
+function of wavelength, for the air plasma at the conditions used. The simu-
+lation is performed with SPECAIR and assumes a pathlength of 5 cm, and
+LTE conditions at a temperature of 7000 K (Tg = Tr = Tv = Telectronic =
+7000 K). The simulation includes the infrared absorption features of NO,
+OH, and NO+. The absorption by NO and OH is relatively weak, while
+the various fundamental bands ofNO+ have stronger predicted absorbances.
+It is evident that the NO+ absorption begins at a wavelength of about
+3950 nm, and is a maximum at about 4lO0nm. Accessing these infrared
+wavelengths is challenging in terms of available laser sources. The current
+measurements have been performed using a Continuum-Mirage OPO
+system. The Mirage laser is designed to operate at a maximum wavelength
+of 4000 nm; however, we optimized the alignment in a manner that enabled
+operation in the vicinity of 4lO0 nm, in order to be nearer to the peak NO+
+absorption. Ring-down cavity alignment at these wavelengths is challenging,
+since the beam (and its back-reflections) are not readily observable. The ring-
+down cavity was aligned using a combination of LCD (liquid crystal display)
+paper to locate the beam, and a helium-neon laser to act as a reference. With
+the plasma off, ring-down times of about 1.2 jlS were obtained, corresponding
+to mirror reflectivities of about 0.998 (approximately an order of magnitude
+worse than the mirrors used for the Nt experiments).
+
+--- Page 548 ---
+Ion Concentration Measurements
+533
+Our initial attempts to perform CRDS measurements in the plasma
+torch used the same cavity-geometry as was used in the Nt experiments-
+a g-parameter of 0.5. With the plasma off, this geometry yielded excellent
+stability in the ring-down times: 1 % standard deviation in ring-down time
+for single shot ring-down signals. However, with the plasma on, the beam
+steering reduced the stability significantly. In the rf plasma, as compared
+to the smaller nitrogen plasma, the cavity-geometry considerations are
+different. In the smaller nitrogen plasma, we wanted to minimize simulta-
+neously the cavity beam-waist and the beam-walk, leading to a g-parameter
+of -0.5 (see discussion above). On the other hand in the rf plasma, the
+plasma dimension (about 5 cm) is significantly larger than the beam dimen-
+sion (about 1 mm). Therefore, the exact beam dimension is not critical,
+and the cavity-geometry may be selected solely to minimize beam-walk.
+The numerical modeling of Spuler and Linne (2002) indicates that mini-
+mizing the beam-walk may be accomplished with a g-parameter of about
+0.25, which we implemented by using a cavity of length 75 cm, and mirrors
+of radius-of-curvature of 1 m. This geometry did indeed reduce the beam-
+walk and enabled improved stability (about 2% standard deviation in
+empty cavity ring-down times).
+As will be discussed, the identification of spectral lines in the analysis of
+the air plasma spectra is challenging. In order to assist in identifying NO+
+spectral features, we also collected CRDS spectra with the plasma running
+with argon and nitrogen (as opposed to air), conditions that are not expected
+to have any significant NO+ concentration.
+8.6.4.3
+Results and discussion
+Figure 8.6.12 shows a measured absorbance spectrum along the centerline of
+the air plasma. The experimental data were obtained by averaging 16 laser
+shots at each spectral position. The plotted CRDS data have been converted
+to absorbance, and fitted with a peak-fitting program. (Fitted peaks are
+shown in black, while raw data are shown with blue symbols.) Also shown
+is the modeled NO absorbance assuming the expected plasma conditions
+of path length 5 cm, and L TE at 7000 K. The modeled contributions from
+OH and NO absorption are negligible on this scale. Comparing the CRD
+spectrum in the air plasma to the CRD spectrum in the argon/nitrogen
+plasma provides information as to line identities. The largest spectral feature
+(at "-'4127.7nm) is present in both spectra, and therefore is presumed not to
+be NO+. Comparing the other observed spectral features with the model does
+not yield good agreement. To the best of our knowledge, the spectroscopic
+constants used in our modeling are the most recent and accurate ones
+available (Jarvis et aI1999). The exact locations of the rotationally resolved
+lines are largely determined by the rotational constants B, which have a
+quoted uncertainty of ±0.005 cm- 1 (or about 0.25%). Based on the quoted
+
+--- Page 549 ---
+534
+Plasma Diagnostics
+0.0004 .,------------"11""""-----------,
+~ 0.0002
+Data~
+s::: as
+.
+.0
+.
+...
+0
+I/)
+.0 «
+0.0000
+-0.0002 -I--r---.--..,.--..----,---T""-,---.---,--.--,-----.---i
+41220
+41240
+41260
+41280
+41300
+41320
+41340
+~(A)
+Figure 8.6.12. Experimental and modeled absorbance spectrum from the air plasma near
+4100 nm. Raw data (blue symbols) as well as fitted peaks (top black line) are shown, as well
+as the modeled NO+ lines (plotted negative for visual clarity). The precision of the spectro-
+scopic constants used in the model is insufficient to predict accurately the locations of the
+rotational lines.
+uncertainty we performed an uncertainty analysis, and found that with this
+level of precision it is not possible to accurately predict the locations of
+the rotational lines. Therefore, any match between the experimental data
+and model would be fortuitous. Our experimental features are repeatable
+(to within experimental uncertainty) and have approximately the correct
+integrated area, so we do believe they belong to NO+.
+8.6.5 Conclusions
+Spatial and temporal profiles of Nt concentration have been measured in dc
+and pulsed atmospheric pressure nitrogen glow discharges by cavity ring-
+down spectroscopy. Special care in the selection of cavity geometry is
+needed in the atmospheric pressure plasma environment. Sub-millimeter
+spatial resolution, microsecond temporal resolution, and sub-ppm concen-
+tration sensitivity have been achieved. The signal-to-noise ratio suggests a
+dc detection limit of about 7 x 1010 cm-3 for Nt ions at our experimental
+conditions (corresponding to an uncertainty in column density of about
+1.4 x 1010 cm -2). Using a collisional-radiative model we infer electron
+number densities from the measured ion profiles. The values of electron
+number density found in this way are consistent with those found from
+
+--- Page 550 ---
+Ion Concentration Measurements
+535
+spatially integrated electrical conductivity measurements. The spectroscopic
+technique is clearly favorable, because it offers spatial resolution and does
+not require knowledge of other discharge parameters. Furthermore, the
+spectroscopic technique enables measurements of the speciation of the ion
+density, information not available from direct electrical measurements.
+Measurements of the NO+ ion in air plasmas have also been demon-
+strated. The accessible spectral features of NO+ are vibrational transitions,
+considerably weaker than the ultraviolet electronic transitions used to
+probe Nt. Nevertheless, CRDS data from air plasmas were obtained, and
+spectral features attributed to NO+ were observed. This technique shows
+promise for the measurement of NO+ concentrations once more accurate
+spectroscopic constants of NO+ become available.
+References
+Aldener M, Lindgren B, Pettersson A and Sassenberg U 2000 'Cavity ringdown laser
+absorption spectroscopy: nitrogen cation' Physica Scripta 61(1) 62-65
+Berden G, Peeters R and Meijer G 2000 'Cavity ring-down spectroscopy: experimental
+schemes and applications' Int. Rev. Phys. Chern. 19(4) 565-607
+Booth J P, Cunge G, Biennier L, Romanini D and Kachanov A 2000 'Ultraviolet cavity
+ring-down spectroscopy of free radicals in etching plasmas' Chern. Phys. Lett.
+317(6) 631-636
+Brown S S, Ravishankra A R and Stark H 2000 'Simultaneous kinetics and ring-down:
+rate coefficients from single cavity loss temporal profiles' J. Chern. Phys. A 104
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+Busch K Wand Busch A M (eds) 1999 Cavity-Ringdown Spectroscopy (acS Symposium
+Series) (Oxford: Oxford University Press)
+Grangeon F, Monard C, Dorier J-L, Howling A A, HoUenstein C, Romanini D and
+Sadeghi N 1999 'Applications of the cavity ring-down technique to a large-area
+RF-plasma reactor' Plasrna Sources Sci. Technol. 8448-456
+Jarvis G K, Evans M, Ng C Y and Mitsuke K 1999 'Rotational-resolved pulsed field
+ionization photoelectron study of NO+ X lI;+, v+ = 0-32) in the energy range of
+9.24-16.80eV' JCP 111(7) 3058-3069
+Kessels W M M, Leroux A, Boogaarts M G H, Hoefnagels J P M, van de Sanden M C M
+and Schram D C 2001 'Cavity ring down detection ofSiH3 in a remote SiH4 plasma
+and comparison with model calculations and mass spectrometry' 1. Vac. Sci.
+Technol. A 19(2) 467-476
+Kotterer M, Conceicao J and Maier J P 1996 'Cavity ringdown spectroscopy of molecular
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+Laux C 0 1993 'Optical diagnostics and radiative emission of air plasmas' Mechanical
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+Laux C 0, Gessman R J, Kruger C H, Roux F, Michaud F and Davis S P 2001 'Rotational
+temperature measurements in air and nitrogen plasmas using the first negative
+system of Nt JQSRT 68(4) 473-482
+Michaud F, Roux F, Davis S P, Nguyen A-D and Laux C 0 2000 'High resolution Fourier
+spectrometry of the 14Nt ion' J. Molec. Spectrosc. 203 1-8
+
+--- Page 551 ---
+536
+Plasma Diagnostics
+Park C 1989 Nonequilibrium Hypersonic Aerothermodynamics (New York: Wiley)
+Pierrot L, Yu L, Gessman R J, Laux C 0 and Kruger C H 1999 'Collisional-radiative
+modeling of non-equilibrium effects in nitrogen plasmas' in 30th AIAA Plasma-
+dynamics and Lasers Conference, Norfolk, VA
+Quandt E, Kraemer I and Dobele H F 1999 'Measurements of Negative-Ion Densities by
+Cavity Ringdown Spectroscopy' Europhysics Lett. 45 32-37
+Schwabedissen A, Brockhaus A, Georg A and Engemann J 2001 'Determination of the
+gas-phase Si atom density in radio frequency discharges by means of cavity ring-
+down spectroscopy' J. Phys. D: Appl. Phys. 34(7) 1116-1121
+Shkarofsky I P, Johnston T Wand Bachynski M P 1966 The Particle Kinetics of Plasmas
+(Addison-Wesley)
+Siegman A E 1986 Lasers (Mill Valley: University Science Books)
+Spuler S and Linne M 2002 'Numerical analysis of beam propagation in pulsed cavity ring-
+down spectroscopy' Appl. Optics 41(15) 2858-2868
+Yalin, A P and Zare R N 2002 'Effect of laser lineshape on the quantitative analysis of
+cavity ring-down signals' Laser Physics 12(8) 1065-1072
+Yalin A P, Zare R N, Laux C 0 and Kruger C H 2002 'Temporally resolved cavity ring-
+down spectroscopy in a pulsed nitrogen plasma' Appl. Phys. Lett. 81(8) 1408-1410
+Zalicki P and Zare R N 1995 'Cavity ring-down spectroscopy for quantitative absorption
+measurements' J. Chem. Phys. 102(7) 2708-2717
+
+--- Page 552 ---
+Chapter 9
+Current Applications of Atmospheric
+Pressure Air Plasmas
+M Laroussi, K H Schoenbach, U Kogelschatz, R J Vidmar, S Kuo,
+M Schmidt, J F Behnke, K Yukimura and E Stoffels
+9.1
+Introduction
+High-pressure non-equilibrium plasmas possess unique features and charac-
+teristics which have provided the basis for a host of applications. Being
+non-equilibrium, these plasmas exhibit electron energies much higher than
+that of the ions and the neutral species. The energetic electrons enter into
+collision with the background gas causing enhanced level of dissociation,
+excitation and ionization. Unlike the case of thermal plasmas, these reactions
+occur without an increase in the gas enthalpy. Because the ions and the
+neutrals remain relatively cold, the plasma does not cause any thermal
+damage to articles they may come in contact with. This characteristic
+opens up the possibility of using these plasmas for the treatment of heat-
+sensitive materials including biological tissues. In addition, operation in
+the high-pressure regime lends itself to the utilization of three-body processes
+to generate useful species such as ozone and excimers (excited dimers and
+trimers).
+Low-temperature high-pressure non-equilibrium plasmas are already
+routinely used in material processing applications. Etching and deposition,
+where low-pressure plasmas have historically been dominant, are examples
+of such applications. In the past two decades, non-equilibrium high-pressure
+plasmas have also played an enabling role in the development of excimer
+VUV and ultraviolet sources (Elias son and Kogelschatz 1991, EI-Habachi
+and Schoenbach 1998), plasma-based surface treatment devices (Dorai and
+Kushner 2003), and in environmental technology such as air pollution
+control (Smulders et at 1998). More recently, research on the biological
+and medical applications of these types of plasmas have witnessed a great
+537
+
+--- Page 553 ---
+538
+Current Applications of Atmospheric Pressure Air Plasmas
+interest from the plasma and medical research communities. This is due to
+newly found applications in promising medical research such as electro-
+surgery (Stoffels et al 2003, Stalder 2003), tissue engineering (Blakely et al
+2002), surface modification of bio-compatible materials (Sanchez-Estrada
+et al 2002), and the sterilization of heat-sensitive medical instruments
+(Laroussi 2002). These exciting applications would not have been possible
+were it not for the extensive basic research on the generation and sustainment
+of relatively large volumes of 'cold' plasmas at high pressures and with rela-
+tively small input power. However, as seen in the previous chapters of this
+book, in the case of air several challenges still remain to be overcome to
+arrive at an optimal generation scheme that is capable of producing large
+volume of air plasmas without a prohibitive level of applied power. Nonethe-
+less, as will be shown in this chapter, success in this research endeavor will
+potentially bring with it substantial economical and societal benefits. In
+particular, the semiconductor industry, chemical industry, food industry,
+and health and environmental industries, as well as the military stand to
+be great beneficiaries from the novel applications of 'cold' air plasmas.
+In this chapter, several applications of non-equilibrium air plasma are
+covered in details by experts who have extensively contributed to this
+research. The selected applications are of the kind that have had or poten-
+tially will have a significant impact on industrial, health, environmental, or
+military sectors. The first two sections (9.2 and 9.3) discuss electrostatic pre-
+cipitation and ozone generation. This choice is motivated by the fact that
+historically these two applications of electrical discharges were the first to
+have been applied on a large industrial scale: electrostatic precipitation for
+the cleaning of air from fumes and particulates, and ozone generation for
+the disinfection of water supplies. Section 9.4 discusses the reflection and
+absorption of electromagnetic waves by air plasmas. This has direct applica-
+tions in military radar communications, and opens the possibility of using
+plasmas as a protective shield from radar and high power microwave
+weapons. Section 9.5 introduces the concept of using air plasmas to mitigate
+the effects of shock waves in supersonic/hypersonic flights. Plasma has been
+shown to reduce drag, which leads to lower thermal loading and higher fuel
+efficiency. Section 9.6 discusses the use of air plasma to enhance combustion.
+Ignition delays can be reduced and the combustion of hydrocarbon fuels can
+be increased by the presence of radicals generated by the plasma. Section 9.7
+gives an extensive coverage of material processing by high-pressure non-
+equilibrium plasmas. The cleaning of surfaces, functionalization (such as
+for better adhesion), etching, and deposition of films are discussed and prac-
+tical examples are presented. Section 9.8 explores on the use of plasma
+discharges for the decomposition of NOx and VOCs. All practical aspects
+of the decomposition processes are discussed in detail. Sections 9.9 and
+9.10 introduce the reader to the biological and medical applications of
+'cold' plasmas. The emphasis of section 9.9 is on the use of air plasma to
+
+--- Page 554 ---
+Electrostatic Precipitation
+539
+inactivate bacteria efficiently and rapidly. The sterilization of heat-sensitive
+medical tools and food packaging and the decontamination of biologically
+contaminated surfaces are particularly attractive applications. The emphasis
+of section 9.10 is the use of 'bio-compatible' plasmas for in vivo treatment
+such as in electrosurgery. Cell detachment without damage using the
+'plasma needle' is discussed. Wound healing is one example where 'bio-
+compatible' plasma sources can be used.
+Research on non-equilibrium air plasmas has been to a large extent
+application-driven. Inter-disciplinary and cross-disciplinary efforts are
+necessary to drive plasma-based technology forward and into new fields
+and applications where air plasma has not been traditionally a component,
+but its use can substantially improve the established conventional processes.
+References
+Blakely E A, Bjornstad K A, Galvin J E, Montero 0 R and Brown I G 2002 'Selective
+neutron growth on ion implanted and plasma deposited surfaces' in Proc. IEEE
+Int. Conf. Plasma Sci., Banff, Canada, p 253
+Dorai R and Kushner M 2003 'A model for plasma modification of polypropylene using
+atmospheric pressure discharges', J. Phys. D: Appl. Phys. 36 666
+EI-Habachi A and Schoenbach K H 1998 'Emission of excimer radiation from direct
+current, high pressure hollow cathode discharges' Appl. Phys. Lett. 72 22
+Eliasson Band Kogelschatz U 1991 'Non-equilibrium volume plasma processing' IEEE
+Trans. Plasma Sci. 19(6) 1063
+Laroussi M 2002 'Non-thermal decontamination of biological media by atmospheric
+pressure plasmas: Review, analysis and prospects', IEEE Trans. Plasma Sci. 30(4)
+1409, 1415
+Sanchez-Estrada F S, Qiu H and Timmons R B 2002 'Molecular tailoring of surfaces via rf
+pulsed plasma polymerizations: Biochemical and other applications' in Proc. IEEE
+Int. Conf Plasma Sci., Banff, Canada, p 254,
+Smulders E H W M, Van Heesch B E J M and Van Paasen B S V B 1998 'Pulsed power
+corona discharges for air pollution control' IEEE Trans. Plasma Sci. 26(5) 1476
+Stadler K 2003 'Plasma characteristics of electro surgical discharges' in Proc. Gaseous Elec-
+tronics Conf, San Fransisco, CA, p 16
+Stoffels E, Kieft I E and Sladek R E J 2003 'Superficial treatment of mammalian cells using
+plasma needle' J. Phys. D: Appl. Phys. 36 1908
+9.2 Electrostatic Precipitation
+9.2.1
+Historical development and current applications
+The influence of electric discharges on smoke, fumes and suspended particles
+was described by William Gilbert as early as 1600. Gilbert acted as the
+
+--- Page 555 ---
+540
+Current Applications of Atmospheric Pressure Air Plasmas
+president of the British Royal College of Physicians and also as physician to
+Queen Elizabeth I of England. His famous work De M agnete (on the magnet)
+was a comprehensive review of what was then known about electrical and
+magnetic phenomena. In 1824 Hohlfeld in Leipzig reported an experiment
+of clearing smoke in a jar by applying a high voltage to a corona wire
+electrode. Similar experiments were later repeated in Britain by Guitard in
+1850 and by Lodge in 1884. Sir Oliver Lodge was the first to systematically
+investigate this effect and to put it to test on large scale in lead smelters at
+Bagillt in Flintshire, UK, to suppress the white lead fume escaping from
+the chimney (Hutchings 1885, Lodge 1886). To supply the corona current
+special electrostatic induction machines of the Wimshurst type were
+designed, with rotating glass plates of 1.5 m diameter. This can be considered
+the first, although not totally successful, commercial application of electro-
+static precipitation for pollution control. The importance of this new
+'electrical process of condensation for a possible purification of the atmos-
+phere' was clearly recognized, and international patent coverage was
+obtained (Walker 1884). Practically simultaneously and independently a
+German patent was issued for a cylindrical precipitator (Moller 1884).
+A number of important industrial applications followed the pioneering
+work of Frederick Gardner Cottrell, a professor of physical chemistry at the
+University of California-Berkeley. Starting in 1906 he conducted research on
+air pollution control, responding to growing nuisance caused by factories in
+his native San Francisco. The result was an improved precipitator, an elec-
+trical device, which could collect dusts and fumes as well as acid mists and
+fogs. Cottrell was the first to realize that for precipitation the negative
+corona discharge was superior to the positive corona, and who took advan-
+tage of the newly developed synchronous mechanical rectifier (Lemp 1904)
+and better high voltage step-up transformers. Within a few years commercial
+applications evolved for collecting sulfuric acid mists, for zinc and lead
+fumes, for cement kiln dust, for gold and silver recovery from electrolytic
+copper slimes, and for alkali salt recovery from waste liquors in paper-
+pulp plants (Cottrell 1911). In 1923 the first use of electrostatic precipitators
+(ESPs) collecting fly ash from a pulverized coal-fired power plant was
+reported. This process became by far the largest single application of
+ESPs. The fine wire corona discharge electrode, as it is used in many precipi-
+tators today, one of the most important advances in precipitator technology,
+was introduced and patented W A Schmidt (1920), a former student of
+Cottrell. In the following years investigations by Deutsch (1922, 1925a,b)
+and Seeliger (1926) brought new insight in the physical processes involved
+in electrostatic precipitation and a first quantitative formulation of precipi-
+tator performance. The Deutsch equation has been used ever since for
+sizing precipitators. For further details the reader is referred to the classical
+comprehensive treatment of industrial electrostatic precipitation by H J
+White (1963), to some more recent books (Oglesby and Nicholls 1978,
+
+--- Page 556 ---
+Electrostatic Precipitation
+541
+Cross 1987, Parker 1997) and to well written review articles (White 1957,
+1977/781984, McLean 1988, Lawless et a11995, Lawless and Altman 1999).
+The main advantages of electrostatic precipitators are that various
+types of dust, mist, droplets etc. can be collected under both dry and wet
+conditions, and also that submicron size particles can be collected with
+high efficiency. ESPs can handle very large air or flue gas streams, typically
+at atmospheric pressure, with low power consumption and low pressure
+drop.
+These properties have led to a number of large-scale commercial appli-
+cations in the following industries: steel mills, non-ferrous metal processing,
+cement kilns, pulp/paper plants, power plants and waste incinerators,
+sulfuric acid plants, and in petroleum refineries for powder catalyst recovery.
+Much smaller ESPs of different design are used for indoor air cleaning in
+homes and offices.
+9.2.2
+Main physical processes involved in electrostatic precipitation
+Electrostatic precipitation is a physical process in which particles suspended
+in a gas flow are charged electrically by ions produced in a corona discharge,
+are separated from the gas stream under the influence of an electric field, and
+are driven to collecting plates, from which they can be removed periodically
+by mechanical rapping (dry ESP) or continuously by washing (wet ESP).
+Typical configurations are corona wires centered in cylinders or wires
+mounted at the center plane between parallel plates forming ducts (figure
+9.2.1).
+The discharge electrodes can be simple weighted wires, barbed wires,
+helical wires, or rods, serrated strips and many other kinds. They all have
+in common that they have parts with a small radius of curvature or sharp
+edges to facilitate corona formation (see also chapters 2 and 6). The particle
+laden gas flow is channeled to pass through many cylinders or ducts either in
+___ Negative High Voltage
+~~
+~
+Discharge
+Electrodes
+--- Weights----
+~~jIff'JI¥,I'-
+Collecting
+Plates
+at Ground
+Potential
+Figure 9.2.1. Cylindrical and planar precipitator configurations with weighted wire corona
+discharge electrodes.
+
+--- Page 557 ---
+542
+Current Applications of Atmospheric Pressure Air Plasmas
+the vertical (cylinders) or horizontal direction (ducts). In large precipitators
+negative coronas are used almost exclusively because they have a larger
+stability range and can be operated at higher voltages. For these devices
+electrode plate distances of O.2-O.4m and voltages in the range 50-110kV
+are common. Small ESPs for indoor air cleaning normally use positive
+coronas, because they produce less ozone, a matter of great concern for
+indoor applications.
+9.2.2.1
+Generation of electrons and ions
+The active corona region in which electrons as well as positive and negative
+ions are generated is restricted to a very thin layer around the corona elec-
+trodes. Typically ionization occurs only in a layer extending a fraction of
+1 mm into the gas volume. Positive ions travel only a short distance to the
+negative electrode, while electrons and negative ions start moving towards
+the collecting surface at ground potential. In air or flue gas mixtures at
+atmospheric pressure electrons rapidly attach to 02> CO2 or H20 molecules,
+thus forming negative ions. As a consequence, most of the space in the duct is
+filled with negative ions. They are utilized to charge dust particles so that
+these can be subjected to electrical forces in order to separate the dust
+from the gas stream. With modern computational tools it is possible to calcu-
+late the ion charge density distribution for complicated electrode structures.
+An example is given in figure 9.2.2 for one helical electrode (left part) and for
+three helical electrodes in a duct formed by specially shaped collecting plates
+(right part).
+It is interesting to note that practically no ions are produced on the inner
+side of the helical discharge electrode ( dark zone) because of shielding effects.
+The shape and orientation of the ion clouds in the duct depends very much on
+Figure 9.2.2. Ion charge density on a helical corona electrode and in three different hori-
+zontal planes of an ESP duct formed by specially shaped collecting plates (maximum
+charge density: 10-4 As m -3).
+
+--- Page 558 ---
+Electrostatic Precipitation
+543
+0.5
+1
+1.5
+(mAIm:)
+Figure 9.2.3. Current density on collecting plates and ion-induced secondary flow in an
+ESP duct with helical corona electrodes and specially formed collecting plates.
+where the horizontal plane used in the visualization cuts the helix as well as
+on the location and shape of the closest collecting plane and on the distance
+to the neighboring electrodes. The complicated ion flow leads to a very
+inhomogeneous current density distribution on the collecting plates
+including zones of zero current density (figure 9.2.3). Such inhomogeneous
+current distributions were measured as well. They also show up in the
+deposited dust patterns.
+9.2.2.2 Space charge limitations and saturation current
+For practical purposes the active corona layer where ionization takes place
+can be regarded as very thin and as a copious source of charge carriers, in
+this case negative ions. The amount of current that is drawn depends on
+the characteristics of the ion drift region, which again depends on the applied
+voltage. The maximum current scales linearly with the ion mobility p, and
+with U2, when U is the applied voltage. The current is limited by the space
+charge accumulated in the duct. A unipolar ion drift region can be described
+by the following set of equations:
+E = - V = -grad
+V2 = divgrad = -pleo
+j = pp,E
+V . j = div j = o.
+(9.2.1 )
+(9.2.2)
+(9.2.3)
+(9.2.4)
+In these equations E stands for the electric field, for the potential, p for the
+ion space charge density, eo for the vacuum permittivity (8.85 x 10-12 As/
+V m), and j for the current density. Poisson's equation (9.2.2) enforces a
+strong coupling between the ion space charge and the electric field. Adequate
+
+--- Page 559 ---
+544
+Current Applications of Atmospheric Pressure Air Plasmas
+boundary conditions have to be formulated at the rim of the active corona
+region and at the collecting plane.
+Because of this strong dependence on the voltage, ESPs operate at the
+maximum possible voltage stable corona discharge operation will allow.
+Since the highest possible voltage is beneficial both for charging and precipi-
+tation, ESPs are automatically controlled to run close to the sparking limit by
+allowing a certain number of sparks per unit of time to occur (up to 60 sparks
+per minute). Modern ESPs utilize all-solid-state high voltage rectifiers and
+microcomputer controls.
+9.2.2.3
+Main gas flow and electric wind
+Ions, traveling in the duct at a speed of the order 100 mis, move perpendicu-
+larly to the gas stream flowing at a speed of about 1 m/s. Since they have
+practically the same mass as the neutral components of the gas flow there
+is an efficient collisional momentum transfer. As a result strong secondary
+flows are induced. This phenomenon, referred to as the ion wind or electric
+wind, has been known for a long time and has been reviewed by Robinson
+(1962). At high applied voltages the magnitude of the ion-induced secondary
+flow component in an ESP becomes comparable to the main flow velocity. In
+a complicated electrode duct geometry like the helical discharge electrodes
+discussed earlier, this leads to stationary or oscillating vortex structures
+(Egli et al 1997), as demonstrated in the right-hand part of figure 9.2.3.
+The computed cross flow velocity distribution is shown in a vertical plane
+perpendicular to the main flow, located between the second and third helical
+discharge electrode.
+As already suspected by Ladenburg and Tietze (1930) the electric wind
+can have a major adverse influence on particle collection. Recent 3D compu-
+tations of corona charging, particle transport in the flow field and particle
+collection show that this is indeed the dominating effect at certain operating
+conditions (Egli et a11997, Lowke et aI1998).
+9.2.2.4 Particle charging
+The physical processes involved in corona charging of powders and droplets
+have been studied in great detail. Apart from precipitators these phenomena
+are utilized in electrophotography (Crowley 1998), copying machines,
+printers, liquid spray guns, and in powder coating (Mazumder 1998). Solid
+particles or droplets entering a precipitator pass many corona zones, undergo
+collisions with ions resulting in charge accumulating, and are subjected to
+Coulomb forces in the electric field and to drag forces in the viscous flow.
+The charging process of solid particles or droplets has two main contri-
+butions, the relative importance of which depends on particle size. Field
+charging is the dominating process for particles of diameter of about 2 Ilm
+
+--- Page 560 ---
+Electrostatic Precipitation
+545
+or more. It is described by the following differential equation:
+dqf = p7rr2 ILpE (1 _ qf)2
+dt
+P
+qs
+(9.2.5)
+in which qf is the accumulated particle charge due to field charging,
+p = 3cr/(2 + cr ), rp is the particle radius, and qs is the saturation charge.
+The parameter p depends on the relative dielectric constant Cr of the particle
+and varies only moderately between the value p = 1 for Cr = 1 and p = 3 for a
+metallic particle (cr = (0). Charging stops when the saturation charge qs is
+reached. At this point additional approaching ions will be deflected in the
+electric field of the previously accumulated charges on the particle and will
+no longer be able to impact.
+(9.2.6)
+At the ion densities and electric fields encountered in ESPs, field charging is a
+fast process. Its rate is proportional to the ion density, the cross section of the
+particle and to the electric field strength. Also the maximum attainable
+charge is proportional to the particle cross section and the electric field.
+Under typical precipitator conditions a 5)lm particle may accumulate several
+thousand elementary charges.
+For very small particles with r p :::; 1 )lm, field charging gets very slow and
+another charging process depending on the Brownian motion of ions takes
+over (Fuchs 1964). This process is referred to as diffusion charging and
+follows a different law:
+ILP
+qd
+co exp (
+qd· e
+) _ 1
+47rcorpkT
+(9.2.7)
+where qd is the particle charge accumulated due to diffusion charging, e is the
+elementary charge, k is the Boltzmann constant (1.38 x 10-23 J/K), and Tis
+the gas temperature.
+Diffusion charging is a much slower process than field charging. It
+does not depend on the electric field and does not reach a saturation
+charge. At the exit of a precipitator, after 10-15 s transit time, a 0.3)lm
+particle has accumulated about 100 elementary charges. The theoretical
+limit is reached (if ever) when the field at the particle surface has reached a
+value where gas breakdown is initiated. In the intermediate particle size
+rage O.I)lm < r p < lO)lm both charging mechanisms are of comparable
+speed and occur simultaneously. The charging equations (9.2.5) and (9.2.7)
+have to be integrated along the particle trajectories, simultaneously with
+solving the coupled codes describing the corona discharge and the fluid
+phenomena (Choi and Fletcher 1997, Egli et a11997, Meroth 1997, Gallim-
+berti 1998, Medlin et aI1998). Instead of integrating (9.2.7) often a useful
+
+--- Page 561 ---
+546
+Current Applications of Atmospheric Pressure Air Plasmas
+approximate relation for the charge qd reached at time t is used:
+3r kT
+qd(t) = _P -
+In(AIl,pt).
+e
+(9.2.8)
+In this relation, suggested by Kirsch and Zagnit'ko (1990), A is a constant. It
+shows that the charge obtained by diffusion charging is proportional to the
+gas temperature and that it grows with the logarithm of the time t.
+9.2.3 Large industrial electrostatic precipitators
+Industrial precipitators can be very large installations. As an example the
+precipitator at the exit of a pulverized-coal fired utility boiler of a 500 MW
+power plant is described. Coal consumption is about 200 tons per hour
+resulting in fly ash quantities of 20-80 tons per hour, depending on the
+origin and quality of coal. Fly ash particles range from 0.1 to 10/lm size.
+At the exit of the boiler they are dispersed in a flue gas stream of about 2.5
+million m3 per hour with a mass concentration of about 20 g/m3. To meet
+tolerable output concentrations of 20 mg/m3 the precipitator has to reach a
+weight collection efficiency of 99.9%. With modern technology this can be
+achieved. In extreme cases even 99.99% efficiency has been obtained. It is
+one of the major achievements of modern precipitator technology that
+these goals can be reached with an almost negligible power consumption
+of 0.1 % of the generated power and a pressure drop of only 1 mbar.
+9.2.3.1
+Structural design
+To handle such a large gas flow the flue gas is slowed down to about 1 m/s,
+channeled into many parallel ducts of 15 m height, up to 15 m length, and
+0.3-0.4 m width. Such large ESPs are subdivided into fields of about 5 m
+length. About 11 0-150 such ducts add up to a total width of 45 m, being
+typically sectionalized into 3 x 15 m. In total 60000 m2 of collecting area
+are provided. At the center plane of each duct the discharge electrodes are
+mounted. (See figure 9.2.4.) The helical electrodes shown in figures 9.2.2
+and 9.2.3 have the advantage that, mounted under tension in metal frames
+at the center plane of each duct, they are always self centered. In addition,
+rapping of the metal frames induces vibration of the discharge electrodes,
+thus efficiently cleaning them of deposited fly ash. The charged particles
+impinging on the collecting plates, usually made of mild steel, and kept at
+ground potential, form a dust cake, which is held in position by electric
+forces. It is removed periodically by mechanical rapping using either side-
+or top-mounted hammers. Upon rapping the collected material is dislodged
+and slides down into hoppers at the bottom from where it is removed by
+conveyor belts. The special shape of the collecting plates indicated in figures
+9.2.2 and 9.2.3 is chosen to give them mechanical strength and to reduce
+rapping-induced re-entrainment of already collected material.
+
+--- Page 562 ---
+Electrostatic Precipitation
+547
+High Voltage Supplies ~~:::::::~ .........
+Screens for Gas Deceleration
+and Distribution
+Flue Gas with Fly Ash
+coming from Boiler
+Hoppeffif~ ~
+Dust Collection
+Figure 9.2.4. Structure of a large precipitator behind a coal-fired utility boiler (Flakt
+design).
+9.2.3.2 Numerical modeling
+For many years ESPs have been sized according to the Deutsch equation
+which was derived in 1922 and which, for the first time, established a quan-
+titative relation between the collection efficiency TJ of a precipitator and some
+operational and geometry parameters:
+TJ= 1- (Cexit/CO) = 1-exp(-wA/Q).
+(9.2.9)
+The quantities Cexit and Co are the dust concentrations at the exit and entrance
+of the precipitator, respectively, A is the total collection area and Q is
+the volumetric gas flow. The parameter w has the dimension of a velocity
+and is called the migration velocity. For ESP sizing this parameter was
+determined empirically and contained all the pertinent information about
+precipitator design, dust properties and corona operation.
+With a better understanding of all the physical processes involved, and
+taking advantage of fast computers and advanced computational tools,
+individual particle paths can now be followed through a large industrial
+precipitator. This approach requires that sufficiently accurate computational
+models are available for the field distribution and ion production, the
+charging process, the flow field and the particle motion. Since there is a
+strong interaction between the different processes involved the differential
+equations describing the different processes have to be solved simultaneously
+with appropriate boundary conditions. As an example some results are given
+of numerical studies in which individual particle paths where followed
+through a 12 m long ESP duct in which they passed 45 helical corona
+
+--- Page 563 ---
+548
+Current Applications of Atmospheric Pressure Air Plasmas
+a i
+0.1
+~
+0.1
+0.1
+l.
+lIkII:1ricwlnd
+)
+~
+0.01
+0.01
+~
+(lOOl,._u.'
+0.001
+0.001
+0.01
+0.1
+10
+001
+0.1
+10
+0.01
+0.1
+10
+Particle Diameter (fJITl)
+Figure 9.2.5. Fractional particle penetration curves demonstrating the influence of
+different parameters.
+electrodes (Kogelschatz et al 1999). For each size class 2000 particles with
+different initial positions at the entrance were traced.
+The plots, referred to as penetration curves, show the fraction of particles
+that are able to pass the whole precipitator without getting collected, as a
+function of particle size. The left-hand part of figure 9.2.5 demonstrates the
+overwhelming influence of the electric wind. If it were not present, collection
+would improve by more than 2 orders of magnitude. In the model computa-
+tion this was simulated by switching off the electric volume forces on the
+flow. These computations were performed for the specially formed collecting
+plates (figures 9.2.2, 9.2.3), a O.4m duct, an initial flow velocity of 1 mls and
+a corona voltage of 56kV. The middle graph of figure 9.2.5 shows results
+for different flow velocities at a fixed voltage of 56kV in a O.4m duct with
+planar walls. Clearly, slower transport velocity, and consequently longer resi-
+dence time, results in better particle collection. The right-hand part shows the
+influence of the applied voltage for a fixed initial flow velocity of 1 m/s. All
+computations show that there is a particle size range between 0.1 and lllm
+diameter that is difficult to collect. Larger particles are more efficiently
+collected because they accumulate sufficient charge in the corona zones and
+are subjected to strong electric forces. Very small particles are also easily
+collected despite the reduced electric forces. The reason is that they experience
+less flow resistance when particle diameters approach the mean free path of the
+gas molecules (Cunningham slip). Measurements of particle size distributions
+at the entrance and exit oflarge industrial precipitators yield the same form of
+the penetration curves. Such numerical simulations, based on the fundamental
+physical processes and validated in real situations, have become a powerful
+tool for optimizing ESP performance.
+9.2.3.4 Limitations by corona quenching and dust cake resistivity
+The practical performance of electrostatic precipitators can be limited by
+additional effects not mentioned so far. If large amounts of fine dust enter
+
+--- Page 564 ---
+Electrostatic Precipitation
+549
+the precipitator, the corona current in the entrance sections can drop to a
+small fraction of what it had been without dust. This very pronounced
+effect is called corona quenching. The reason is that the properties of the
+corona discharge that were originally determined by ion mobility and ion
+space charge are now determined by the much smaller dust mobility and
+the dust space charge. Fortunately, after collecting most of this fine dust,
+the corona recovers to its original current density, typically after a few
+meters in the duct.
+The collected material on the collecting plates can also pose limitations
+on electrostatic precipitation. If particles have a very low electrical resistivity,
+for example metal particles, they do not adhere to the collecting plates, thus
+preventing collection. On the other hand, if dust resistivity is very high, one
+might expect that the deposited dust layer would finally limit the current flow
+and stop the corona. Normally a different phenomenon, called back corona,
+occurs instead. Since the deposited dust forms a porous layer of growing
+thickness and voltage drop, gas breakdown in interstices and on particle
+surfaces can occur. When this happens, the corona current suddenly
+increases and collection is severely effected. Now positive ions, generated
+by back corona inside the dust cake, travel towards the center electrodes
+and counteract the charging process with negative ions. This results in
+what is called a bipolar corona. Obviously, for optimum charging conditions
+we depend on a unipolar ion flow.
+Back corona is observed in precipitators serving boilers using low sulfur
+coal and also in powder coating, where high resistivity polymer particles and
+pigments are deposited. It was first observed by Eschholz in 1919. The
+described effects limit the useful range of electrostatic precipitators to
+material with resistivity in the range of about 108 n·cm to less than
+1013 n·cm. The resistivity range for optimum ESP performance is 108 to
+1010 n·cm. In many cases high dust resistivity can be reduced by raising the
+temperature or by conditioning, which means by using additives like H20
+or S03. The cohesive properties of the dust cake can be influenced by
+adding NH3 to the gas stream. It is also possible to detect malfunctioning
+of a precipitator section as a consequence of corona quenching or back
+corona and counteract by modifying the electrical feeding of the corona.
+9.2.4 Intermittent and pulsed energization
+In many cases pronounced improvement of ESP performance has been
+obtained by abandoning the classical dc high voltage on the discharge elec-
+trodes. Microprocessor control of the supply voltage allows simple variations
+in the way the corona discharge in ESPs is fed. Intermittent energization can
+be achieved by suppressing voltage half cycles or even several cycles in the
+rectifier circuit. This way, peak voltages higher than those achievable with
+dc energization, and lower average voltages and average currents are
+
+--- Page 565 ---
+550
+Current Applications of Atmospheric Pressure Air Plasmas
+obtained. In addition to energy savings this can result in improved perfor-
+mance if back corona is a problem.
+Even better results can be obtained if pulsed energization is used. This
+technique originated about 1950 following pioneering research and develop-
+ment by Hall and White (Hall 1990). We speak of a pulsed corona if the
+duration of the applied voltage pulse is shorter than the ion transit time
+from the discharge electrode to the collecting plate. In a large ESP this is
+typically of the order 1 ms. Using this technique, periodic short high-voltage
+pulses are superimposed on a dc high voltage. Typical pulse widths of < IllS
+to about 300 IlS and repetition rates of about 30 to 300 per second are used.
+Pulsed energization introduces a number of new parameters that can be
+optimized: pulse duration, pulse repetition frequency, base dc voltage. It
+increases the uniformity of the corona along the discharge electrodes and
+on the collecting plates. It helps to suppress back corona in the collection
+of high resistivity dust. Experience shows that application of short HV
+pulses to high resistivity dusts of 1010_10 13 O'cm results in significant perfor-
+mance improvement over that achievable with dc energization.
+In conclusion it can be stated that electrostatic precipitation is the
+leading and most versatile procedure for high-efficiency collection of solid
+particles, fumes and mists escaping from industrial processes. It presents
+by far the most important application of industrial air pollution control.
+About one hundred years of practical experience with various kinds of
+dust, a growing understanding of the physical processes involved, and
+more recently, the use of advanced computational tools simulating the
+whole particle charging, particle motion and collection process have led to
+its present supremacy.
+References
+Choi B S and Fletcher C A J 1997 J. Electrost. 40/41 413--418
+Cottrell F C 1911 J. Ind. Eng. Chern. 3 542-550
+Cross J A 1987 Electrostatics: Principles, Problems and Applications (Bristol: Adam Hilger)
+Crowley J M 1998 'Electrophotography' in Wiley Encyclopedia of Electrical and
+Electronic Engineering Webster J G (ed) (New York: Wiley-Interscience) vol 6,
+pp 719-734
+Deutsch W 1922 Ann. Phys. 68335-344
+Deutsch W 1925a Z. Techn. Phys. 6423--437
+Deutsch W 1925b Ann. Phys. 76 729-736
+EgJi W, Kogelschatz U, Gerteisen E A and Gruber R 1997 J. Electrostat. 40/41 425--439
+Eschholz 0 H 1919 Trans. Am. Inst. Mining Metall. Eng. LX 243-279
+Fuchs N A 1964 The Mechanics of Aerosols (Oxford: Pergamon)
+Gallimberti I 1998 J. Electrostat. 43 219-247
+Gilbert W 1600 Tractatus, sive Physiologia de Magnete, Magnetisque corporibus magno
+Magnete tellure, sex libris comprehensus (London: Excudebat Petrus Short)
+Guitard C F 1850 Mech. Mag. (London) 53346
+
+--- Page 566 ---
+Ozone Generation
+551
+Hall H J 1990 J. Electrostat. 25 1-22
+Hohlfeld M 1824 Arch.f d. ges. Naturl. 2205-206
+Hutchings W M 1885 Berg- u Hiittenmiinn Zeitschr. 44 253-254
+Kirsch A A and Zagnit'ko A V 1990 Aerosol Sci. Technol. 12465--470
+Kogelschatz U, Egli Wand Gerteisen E A 1999 ABB Rev. 4/1999 33--42
+Ladenburg R and Tietze W 1930 Ann. Phys. 6 581-621
+Lawless P A and Altman R F 1999 'Electrostatic precipitators' in Wiley Encyclopedia of
+Electrical and Electronic Engineering, Webster J G (ed) (New York: Wiley-
+Interscience) vol 7 pp 1-15
+Lawless P A, Yamamoto T and Oshani 1995 'Modeling of electrostatic precipitators and
+filters' in Handbook of Electrostatic Processes, Chang J S, Kelly A J and Crowley
+J M (eds) (New York: Marcel Dekker) pp 481-507
+Lemp H 1904 Alternating current selector, US Pat No. 774,090
+Lodge 0 J 1886 J. Soc. Chem. Ind. 5 572-576
+Lowke J J, Morrow R and Medlin A J 1998 Proc. 7th Int. Con! on Electrostatic Precipita-
+tion (ICESP VII), Kyonju, Korea 1998, pp 69-75
+Mazumder M K 1999 'Electrostatic processes' in Wiley Encyclopedia of Electrical and
+Electronic Engineering, Webster J G (ed) (New York: Wiley-Interscience) vol 7
+pp 15-39
+McLean K J 1988 lEE Proc. 135347-361
+Medlin A J, Fletcher C A J and Morrow R 1998 J. Electrostat. 43 39--60
+Meroth A M 1997 Numerical Electrohydrodynamics in Electrostatic Precipitators (Berlin:
+Logos-Verlag)
+Moller K 1884 Rohrenformiges Gas und DampjJilter, German Pat. No. 31911
+Oglesby S and Nichols G 1978 Electrostatic Precipitation (New York: Decker)
+Parker K R (ed) 1997 Applied Electrostatic Precipitation (London: Blackie)
+Robinson M 1962 Am. J. Phys. 30 366--372
+Schmidt W A 1920 Means for separating suspended matter from gases, US Pat. No.
+1,343,285
+Seeliger R 1926 Z. Techn. Phys. 7 49-71
+Walker A 0 1884 A process for separating and collecting particles of metals or metallic
+compounds applicable for condensing fumes from smelting furnaces and for other
+purposes, Brit Pat No. 11,120
+White H J 1957 J. Air Poll. Contr. Ass. 7167-177
+White H J 1963 Industrial Electrostatic Precipitation (Reading: Addison-Wesley)
+White H J 1977/78 J. Electrostat. 4 1-34
+White H J 1984 J. Air Poll. Contr. Ass. 34 1163-1167
+9.3 Ozone Generation
+9.3.1
+Introduction: Historical development
+In 1785 the natural scientist Martinus van Marum described a characteristic
+odor forming close to an electrostatic machine, and in 1801 Cruikshank,
+
+--- Page 567 ---
+552
+Current Applications of Atmospheric Pressure Air Plasmas
+performing water electrolysis, noticed the same odor at the anode. Only in
+1839 Schonbein, professor at the University of Basel, also working on elec-
+trolysis, established that this very pronounced smell was due to a new
+chemical compound which he named ozone after the Greek word OSElV for
+to reek or smell. It took another 25 years of scientific vehement dispute
+before J L Soret could establish in 1865 that this new compound was made
+up of three oxygen atoms.
+Industrial ozone generation is the classical application of non-equilibrium
+air plasmas at atmospheric pressure. Low temperature is mandatory because
+ozone molecules decay fast at elevated temperature. At the same time a
+relatively high pressure is required because ozone formation is a three-
+body reaction involving an oxygen atom, an O2 molecule and a third collision
+partner, O2 or N2 • The dielectric barrier discharge (silent discharge) origin-
+ally proposed by Siemens (1857) for 'ozonizing air' is ideally suited for this
+purpose. Siemens' invention came at the right time. The foundations of
+bacteriology had been laid through the work of the French microbiologist
+Louis Pasteur and the German district surgeon Robert Koch. It had been
+established that infectious diseases like cholera and typhoid fever were
+caused by living micro-organisms, which were dispersed by contaminated
+drinking water, food and clothing. Cholera epidemics like the ones reported
+in Hamburg (1892) and in St Petersburg (1908) caused hundreds of casualties
+per day. Occasional typhoid fever epidemics were common in many cities.
+Ozone is an extremely effective oxidant, surpassed in its oxidizing power
+only by fluorine or radicals like OH or 0 atoms. Siemens succeeded in
+persuading Ohlmiiller, professor at the Imperial Prussian Department of
+Health, to test the effect of ozone exposure on cholera, typhus and coli
+bacteria. The result was complete sterility after ozone treatment. Soon after
+the first official documentation of these bactericidal properties (Ohlmiiller
+1891), industrial ozone production started for applications in small water
+treatment plants in Oudshoorn, Holland (1893) and in Wiesbaden and Pader-
+born, Germany (1901/2). Within the following years major drinking water
+plants using ozone disinfection were built in Russia (St Petersburg 1905), in
+France (Nice 1907, Chartres 1908, Paris 1909) and in Spain (Madrid 1910).
+The water works at St Petersburg already treated 50000 m3 of drinking
+water per day with ozone, those of Paris 90000 m3. Thus, historically speaking,
+ozonation was the first successful attempt of disinfecting drinking water on a
+large scale. Ever since, ozone generating technology has been closely linked to
+the development of water purification processes. In many countries ozonation
+in water treatment was later replaced by more cost-effective processes using
+chlorine or chlorine compounds, which are not only cheaper but also more
+soluble in water than ozone. Recent concerns about potentially harmful disin-
+fection by-products have reversed this, tending towards the use of ozone again.
+Many European cities and some Canadian cities have abandoned chlorination
+in favor of ozone technology to disinfect water. Water works in the US as well
+
+--- Page 568 ---
+Ozone Generation
+553
+as in Japan are increasingly turning to ozone, in order to be able to meet more
+stringent legislation about disinfection by-products like trihalomethanes
+(THMs) and haloacetic acids. These compounds can be formed when chlorine
+is added to the raw water containing organic water pollutants or humic
+materials. Some THMs are suspected to cause cancer. For this reason many
+experts consider ozone treatment the technology of choice for potable water
+treatment. In the United States more than 250 operating plants use ozone.
+For many years the Los Angeles Aqueduct Filtration Plant treating two
+million m3 jday (600mgd) of drinking water with ozone generating capacity
+of close to 10000kg per day, was the largest US plant. Very recently larger
+ozone generating facilities have been installed at the Alfred Merrit Water
+Treatment Plant in Las Vegas, the East Side Water Treatment Plant in
+Dallas, Texas, and the Metropolitan Water District in Southern California.
+In Europe, more than 3000 cities use ozone to disinfect their municipal
+water supplies.
+9.3.2 Ozone properties and ozone applications
+0 3 is a triangle shaped molecule with a bond angle of 117° and equal bond
+lengths of 0.128 nm. Ozone is a practically colorless gas with a characteristic
+pungent odor (Horvath et a11985, Wojtowicz 1996). At -112°C it condenses
+to an indigo blue liquid which is highly explosive. Below -193°C ozone
+forms a deep blue-violet solid. Because of explosion hazards ozone is used
+only in diluted form in gas or water streams. Its solubility is about 1 kg per
+m3 of water. Due to its oxidizing power it finds applications as a potent
+germicide and viricide as well as a bleaching agent. In many applications
+ozone is increasingly used to replace other oxidants such as chlorine that
+present more environmental problems and safety hazards. Strong oxidants
+are chemically active species. Their storage, handling and transportation
+involve substantial hazards. An important issue is also the question of
+residues and side reactions. In all respects ozone represents a superior
+choice due to its innocuous side product, oxygen. As a consequence of its
+inherent instability ozone is neither stored nor shipped. It is always generated
+on the site at a rate controlled by its consumption in the process.
+The most important application of ozone is still for the treatment of
+water. It is capable of oxidizing many organic and inorganic compounds in
+water. Ozone chemistry in water involves the generation of hydroxyl free
+radicals, very reactive species approaching diffusion controlled reaction
+rates for many solutes such as aromatic hydrocarbons, unsaturated
+compounds, aliphatic alcohols, and formic acid (Glaze and Kang 1988,
+Hoigne 1998). Besides applications in drinking water, ultra-pure process
+water, swimming pools, and cooling towers, ozone also finds applications
+in municipal waste water treatment plants and in industrial processes. Very
+large amounts of ozone are also used for pulp bleaching.
+
+--- Page 569 ---
+554
+Current Applications of Atmospheric Pressure Air Plasmas
+9.3.3 Ozone formation in electrical discharges
+Ozone can be generated in different types of gas discharges in which the
+electron energy is high enough to dissociate O2 molecules and in which the
+gas temperature can be kept low enough for the 0 3 molecules to survive
+without undergoing thermal decomposition. Mainly non-equilibrium
+discharges can meet these requirements, above all corona discharges and
+dielectric barrier discharges.
+9.3.3.1
+Ozone formation in corona discharges
+Ozone formation in both positive and negative corona discharges has been
+extensively investigated and is reasonably well understood. Ozone formation
+is restricted to the thin active corona region where ionization takes place.
+Since it is rarely used on an industrial scale it will not be treated in detail.
+The reader is referred to the following references: Peyrous (1986, 1990),
+Peyrous et al (1989), Boelter and Davidsen (1997), Held and Peyrous
+(1999), Yehia et al (2000), Chen (2002), and Chen and Davidson (2002,
+2003a,b).
+9.3.3.2
+Ozone formation in dielectric barrier discharges
+The preferred discharge type for technical ozone generators has always been
+the dielectric barrier discharge (silent discharge) as originally proposed by
+Siemens. In recent years industrial ozone generation profited substantially
+from a better understanding of the discharge properties and of the ozone
+formation process (Filippov et a11987, Kogelschatz 1988, 1999, Samoilovich
+et al 1989, Braun et al 1991, Kogelschatz and Eliasson 1995, Pietsch and
+Gibalov 1998). Operating in air or oxygen at pressures between 1 and 3
+bar, at frequencies between 0.5 kHz and 5 kHz, and using gap spacings in
+the mm range the discharge is always of the filamentary type. Major improve-
+ments were obtained by tailoring microdischarge properties in air or in
+oxygen in such a way that recombination of oxygen atoms is mimimized
+and ozone formation is optimized. This can be achieved by adjusting the
+width of the discharge space, the operating pressure, the properties of the
+dielectric barrier, and the temperature of the cooling medium. Changing
+the operating frequency has little influence on individual microdischarge
+properties. The power dissipated in the discharge is determined by the ampli-
+tude and frequency of the operating voltage. In connection with the cooling
+circuit, it determines the average temperature in the discharge gap. Cylind-
+rical as well as planar electrode configurations have been used. The majority
+of commercial ozone generators use cylindrical electrodes forming narrow
+annular discharge spaces of 0.5-1 mm radial width. The outer electrode is
+normally a stainless steel tube, which is at ground potential and which is
+
+--- Page 570 ---
+Ozone Generation
+555
+Discharge Gap
+Outer Steel
+Cooling Water Flow
+Fuses
+Wiring
+Figure 9.3.1. Configuration of water-cooled discharge tubes in an ozone generator.
+water-cooled. These tubes have a length of 1--4 m. The coaxial inner electrode
+is a glass or ceramic tube, closed at one side, and having an inner metal
+coating as a high voltage electrode (figure 9.3.1), or a closed steel cylinder
+which is covered by a dielectric layer (ceramic, enamel). The feed gas is
+streaming in the axial direction through the annular discharge region
+between the inner and outer tube. Each volume element of the flowing gas
+is subjected to the action of many microdischarges and leaves enriched
+with ozone.
+9.3.4 Kinetics of ozone and nitrogen oxide formation
+Any electric discharge in air or oxygen causes chemical changes induced by
+reactions electrons or ions with N2, O2 or trace elements like H20 and
+CO2 and subsequent free radical reactions. Extensive lists of possible reac-
+tions have been collected, and reliable sets of rate coefficients have been
+established (Krivosonova et a11991, Kossyi et a11992, Herron 1999, 2001,
+Herron and Green 2001, Sieck et al 2001). As far as ozone formation is
+concerned, extensive reaction schemes also exist (Yagi and Tanaka 1979,
+Samoilovich and Gibalov 1986, Eliasson and Kogelschatz 1986a,b, Eliasson
+et a11987, Braun et a11988, Peyrous 1990, Kitayama and Kuzumoto 1997,
+1999). It turns out that ion reactions play only a minor role and that the
+main trends can be described by tracing the reactions of the atoms generated
+by electron impact dissociation of O2 and N2 and those of a few excited
+molecular states.
+9.3.4.1
+Ozone/ormation in oxygen
+In pure oxygen, which is actually used in many large ozone generation
+facilities, ozone formation is a fairly straightforward process. Ozone
+always originates from a three body reaction of oxygen atoms reacting
+
+--- Page 571 ---
+556
+Current Applications of Atmospheric Pressure Air Plasmas
+with 202 molecules:
+0+ O2 + O2 -
+0 3 + O2 -
+0 3 + O2
+(9.3.1)
+where 0 3 stands for a transient excited state in which the ozone molecule is
+initially formed after the reaction of an 0 atom with an O2 molecule. The
+time scale for ozone formation in atmospheric pressure oxygen is a few
+microseconds.
+o is formed in reaction of electrons with O2 after excitation to the A 3~~
+state with an energy threshold of about 6 eV and via excitation of the B 3~~
+state starting at 8.4eV.
+Fast side reactions, also using 0 atoms or destroying 0 3 molecules,
+compete with ozone formation.
+0+0+02 -
+202
+o + 0 3 + O2 -
+302
+OeD) + 0 3 -
+202
+o + 0 3 + O2 -
+302 .
+(9.3.2)
+(9.3.3)
+(9.3.4)
+(9.3.5)
+The undesired side reactions (9.3.2)-(9.3.5) pose an upper limit on the atom
+concentration, or the degree of dissociation, tolerable in the microdischarges.
+Since equation (9.3.2) is quadratic in atom concentration while the ozone
+formation equation (9.3.1) is linear one would expect that extremely low
+atom concentrations are preferable. Computations with large reactions
+schemes show that complete conversion of 0 to 0 3 can only be expected if
+the relative atom concentration [0]/[02] stays below 10-4 . There are other
+considerations, however, that exclude the use of extremely weak micro-
+discharges. If the energy density in a micro discharge and consequently also
+the degree of dissociation is too low, a considerable fraction of the deposited
+energy is dissipated by ions (up to 50%). Since ions do not appreciably
+contribute to ozone formation this situation has to be avoided. A reasonable
+compromise between excessive energy losses due to ions and best use of 0
+atoms for ozone formation is found when the relative oxygen atom concen-
+tration in a microdischarge reaches about 2 x 10-3 in the micro discharge
+channel. This concentration can be obtained at an energy density of about
+20mJ/cm-3 (Eliasson and Kogelschatz 1987). In this case energy losses to
+ions are negligible and 80% of the oxygen atoms are utilized for ozone
+formation. At zero ozone background concentration this leads to a
+maximum energy efficiency of ozone formation corresponding to roughly
+25%. The efficiency of ozone formation is normally related to the enthalpy
+of formation, which is 1.48 eV /03 molecule or 0.82 kWh/kg. Thus 100%
+efficiency corresponds to the formation of 0.6803 molecules per eV or
+1.22 kg ozone per kWh. The indicated reaction paths requiring dissociation
+of O2 first (dissociation energy: 5.16eV) puts an upper limit at 0.7 kg/kWh.
+
+--- Page 572 ---
+Ozone Generation
+557
+,:) .. ~----..... ---------,
+to"
+10-'
+T_lsl
+Figure 9.3.2. Evolution of particle species after a short current pulse: with zero ozone
+background concentration (left) and at the saturation limit (right) (p = 1 bar, T = 300 K).
+If the electron energy distribution in oxygen is considered, and the combined
+actual dissociation processes at 6 and 8.4eV, this value is further reduced to
+0.4 kg/kWh. The best experimental laboratory values obtained at vanishing
+0 3 background concentration are in the range 0.25-0.3 kg/kWh.
+The ozone concentration in the gas stream passing through the ozone
+generator is built up due to the accumulated action of a large number of
+microdischarges. With increasing ozone concentration back reactions gain
+importance. In addition to the already mentioned reactions equations
+(9.3.2)-(9.3.5), 0 3 reactions with electrons and excited O2 molecules have
+to be considered. This finally leads to a situation where each additional
+microdischarge destroys as much ozone as it generates (figure 9.3.2, right-
+hand section). The attainable saturation concentration defined by this
+equilibrium depends strongly on pressure and on gas temperature.
+9.3.4.2
+Ozone formation in dry air
+In air the situation is more complicated. The presence of nitrogen atoms and
+excited atomic and molecular species as well as the nitrogen ions N+, Nt, Nt
+add to the complexity of the reaction system. Again, ions are of minor
+importance for ozone formation. Excitation and dissociation of nitrogen
+molecules, however, lead to a number of additional reaction paths involving
+nitrogen atoms and the excited molecular states N 2(A 3~~) and N 2(B 3IIg),
+that can produce additional oxygen atoms for ozone generation.
+N +02 -
+NO+O
+N+NO----N2+O
+N +N02 -
+N20+O
+N 2(A,B) +02 ---- N2 +20
+N2(A) + O2 -
+N 20 + O.
+(9.3.6)
+(9.3.7)
+(9.3.8)
+(9.3.9)
+(9.3.10)
+
+--- Page 573 ---
+558
+Current Applications of Atmospheric Pressure Air Plasmas
+lime (s)
+Figure 9.3.3. Evolution of particle species after a short current pulse in a mixture of 80%
+N2 and 20% O2 simulating dry air (p = I bar, T = 300 K).
+These oxygen atoms, generated in addition to those obtained from direct
+electron impact dissociation of 02, contribute about 50% of the ozone
+formed in air, which now takes longer, roughly about 1001lS. The result is
+that a substantial fraction of the electron energy initially lost in collisions
+with nitrogen molecules can be recovered and utilized for ozone generation
+through reactions (9.3.6)-(9.3.10). In addition to ozone a variety of nitrogen
+oxide species are generated: NO, N 20, N02, N03, and N 20 5. All these
+species have been measured at realistic ozone generating conditions (Elias son
+and Kogelschatz 1987, Kogelschatz and Baessler 1987). In the presence of
+ozone only the highest oxidation stage N20 5 is detected in addition to the
+rather stable molecule N20 (nitrous oxide, laughing gas). Figure 9.3.3
+shows results of a numerical simulation using a fairly extended reaction
+scheme in dry air (20% 02, 80% N2). The formation of ozone and different
+NOx species due to a single short discharge pulse is followed for a reasonably
+long time.
+A few results demonstrating special characteristics of ozone generation
+in air are added. The maximum attainable energy efficiency is reduced to
+about to 0.2 kg/kWh and it shifted to higher reduced electric field values
+(200-300 Td). This has to be expected because dissociation of N2 requires
+higher electron energies.
+The maximum attainable ozone concentration is lower and, surprisingly
+enough, no saturation concentration exists. When the power is increased or
+the air flow is reduced, the ozone concentration passes through a maximum
+
+--- Page 574 ---
+Ozone Generation
+559
+and then decreases again until it drops to zero. This effect, referred to as
+discharge poisoning, was reported by Andrews and Tait (1860), only a few
+years after Siemens had presented his ozone discharge tube. The poisoning
+effect was correctly associated with the presence of nitrogen oxides. Today
+we know that catalytic processes involving the presence of NO and N02
+can use up 0 atoms at a fast rate thus preventing 03 formation and can
+also destroy already formed ozone. This is a phenomenon that involves
+only fast chemical reactions between neutral particles and has little influence
+on electrical discharge parameters. Addition of 0.1 % NO or N02 to the feed
+gas of an ozone generator can completely suppress ozone formation. In the
+absence of ozone only NO, N02 and N20 can be detected at the exit. In
+dry air the catalytic reactions leading to enhanced removal of 0 and 03
+are as follows:
+0+ NO + M -
+N02 + M
+(9.3.11)
+0+N02
+-NO+02
+(9.3.12)
+0+0
+-02
+(9.3.13)
+and
+0+ N03 -
+N02 + O2
+(9.3.14)
+0+N02 -
+NO+02
+(9.3.15)
+0+03 -202
+(9.3.16)
+These NOy reactions also playa dominant role in atmospheric chemistry
+(Crutzen 1970, Johnston 1992).
+9.3.4.3 Ozone formation in humid oxygen and air
+The situation is further complicated if water vapor is present in the feed gas.
+Even traces of humidity drastically change the surface conductivity of the
+dielectric. At the same electrical operating conditions fewer and more intense
+microdischarges result. In addition, a strong influence on major reaction
+paths results from the presence of OH and H02 • The hydroxyl radical OH
+is formed by electron impact dissociation of H20 and, in most cases more
+importantly, by fast reactions of electronically excited oxygen atoms and
+nitrogen molecules:
+e + H20 -
+e + OH + H
+OeD) + H20 -
+20H
+N2(A 3~~) + H20 -
+N2 + OH + H.
+H02 is then formed in a reaction of OH radicals with ozone:
+OH+03 -
+H02 +02·
+(9.3.17)
+(9.3.18)
+(9.3.19)
+(9.3.20)
+
+--- Page 575 ---
+560
+Current Applications of Atmospheric Pressure Air Plasmas
+The presence of OH and H02 can limit ozone production in oxygen by intro-
+ducing a further catalytic ozone destruction cycle:
+OH +03 -- H02 +02
+H02 + 0 3 -- OH + 202
+In air an additional fast NO oxidation reaction occurs:
+NO + H02 -- N02 + OH.
+(9.3.21)
+(9.3.22)
+(9.3.23)
+(9.3.24)
+The main paths for NO removal in wet air are oxidation to N02 and fast
+conversion to HN02 and HN03.
+NO + OH + M -- HN02 + M
+N02 +OH+M -- HN03 +M.
+9.3.5 Technical aspects of large ozone generators
+(9.3.25)
+(9.3.26)
+Large ozone generators use several hundred discharge tubes and now
+produce up to 100 kg ozone per hour. In most water works several ozone
+generators are installed. Figure 9.3.4 shows a photograph of the entrance
+section of a large ozone generator. One can see the glass tubes mounted in
+slightly wider steel tubes, the high voltage fuses at the center of each tube
+Figure 9.3.4. Large ozone generator at the Los Angeles Aqueduct Filtration Plant.
+
+--- Page 576 ---
+Ozone Generation
+561
+and the electric wires connecting them. Depending on the feed gas, ozone
+concentration up 5wt% (from air) or up to 18wt% (from oxygen) can be
+obtained. Advanced water treatment processes utilize ozone at concentra-
+tions up to 12 wt%. Depending on the desired ozone concentration the
+energy required to produce 1 kg of 03 ranges from 7.5 to 10 kWh in
+oxygen and from about 15 to 20 kWh in air. Information on the technical
+aspects of ozone generation and ozone applications can be found in Rice
+and Netzer (1982, 1984) or in Wojtowicz (1996).
+9.3.5.1
+Design aspects and tolerances
+To obtain such performance several design criteria and operating conditions
+have to be met. The desired small width of the discharge gap in the range 0.5-
+1 mm puts severe tolerance limits on the diameters and on the straightness of
+the cylindrical dielectric and steel tubes. It is essential that the inner dielectric
+tube is perfectly centered inside the outer steel tube. Even a small displace-
+ment results in a drastic drop of performance. Microdischarge efficiency,
+heat removal and axial flow velocity depend strongly on the width of the
+discharge gap, which must be kept in tight tolerances. Also the pressure
+has to be kept close to the design value, about 2 bar in O2 and closer to
+3 bar in air. For a given dielectric tube its optimum value depends on the
+desired ozone concentration, the gap width, the temperature of the cooling
+fluid, and the power density the ozone generator is operated at.
+9.3.5.2
+Feed gas preparation
+The feed gas for most ozone generators is air or oxygen. In large installations
+operating at high ozone concentrations and power density also O2 with a
+small admixture of N2 is used. It is essential that the feed gas contains
+only a few ppm H20 (dew point below -60 QC). As mentioned above,
+humidity has a strong influence on the surface conductivity of the dielectric
+and on the properties of the microdischarges. In addition, we observe the
+changes in the chemical reaction scheme as described in section 9.3.4.3.
+Also traces of other impurities like Hb NOx and hydrocarbons have an
+adverse influence on ozone formation. Some of them lead to a catalytically
+enhanced recombination of 0 atoms, others to catalytic ozone destruction
+cycles.
+These requirements necessitate a feed gas preparation unit to remove
+humidity even if air is used. For this reason many large ozone installations
+use oxygen as a feed gas. If cryogenic oxygen is used one has to be aware
+of the fact that in polluted areas hydrocarbons may accumulate in the
+liquid oxygen. Oxygen prepared by pressure swing or vacuum swing
+adsortion-desorption techniques, on the other hand, is practically free of
+hydrocarbons « 1 ppm).
+
+--- Page 577 ---
+562
+Current Applications of Atmospheric Pressure Air Plasmas
+9.3.5.3
+Heat balance and cooling circuit
+The ozone formation efficiency and the stability of the 0 3 molecule deterio-
+rate at elevated temperature. As a consequence only non-equilibrium
+discharges are suited for ozone generation and efficient cooling of the
+discharge gap is mandatory. This is the reason why ozone generators are
+essentially built like heat exchangers. The average temperature increase
+due to discharge heating in the narrow annular discharge gap can be approxi-
+mated by a simple formula. After a few cm of entrance length stationary
+radial profiles of velocity and temperature are established. The radial
+temperature profile is a half parabola with its maximum at the inner
+uncooled dielectric tube if a uniform power deposition in the discharge is
+assumed. The average temperature increase in the gap /~.Tg is then deter-
+mined by the power dissipated in the discharge and the heat removed through
+the cooled steel electrode and kept at the wall temperature Tw. Unfor-
+tunately, only a minor fraction of the energy is used for ozone formation
+(efficiency: 'f)).
+(9.3.26)
+In this formula d is the gap width, ). is the heat conductivity of the feed gas
+(discharge plasma) and P / F is the power density referred to the electrode
+area F. For efficient ozone generation, especially at higher 0 3 concentrations,
+the temperature has to kept as low as possible, definitely below 100 oe. If a
+second cooling circuit is used to additionally cool the inner tube, the average
+temperature increase t::..Tg is reduced by a factor of four. This allows for a
+considerable increase of power density. However, it is rarely done in
+commercial ozone generators, because it requires cooling of the high voltage
+electrodes and introduces additional sealing problems.
+9.3.5.4
+Power supply units
+Originally ozone generators were run at line frequency or were fed by motor
+generators operating at rather low frequencies. Step-up transformers are
+required to reach the desired voltage level. To achieve reasonable power
+densities, high voltages (up to 50 kV) had to be used. Dielectric failure was
+a common problem. Since all tubes are connected in parallel, high voltage
+fuses were used to disconnect faulty elements. Modern high-power ozone
+generators take advantage of solid state power semiconductors. They utilize
+thyristor or transistor controlled frequency converters to impress square-
+wave currents or special pulse trains in the frequency range 500 Hz to
+5 kHz. Using this technology, applied voltages can be reduced to the range
+of about 5 kV. Dielectric failure is no longer a problem. With large ozone
+generators power factor compensation has become an important issue.
+
+--- Page 578 ---
+Ozone Generation
+563
+Typical power densities now reach 1-lOkW/m2 of electrode area. Using
+semiconductors at higher frequencies brought several advantages: increased
+power at lower voltage, fast shut off and improved process control.
+9.3.6 Future prospects of industrial ozone generation
+A better understanding of microdischarge properties in non-equilibrium
+dielectric barrier discharges and advances in power semiconductors resulted
+in improved performance and reliability of industrial ozone generation in
+recent years. Raised ozone generating efficiency and drastically reduced
+size of the ozone generators helped to lower the cost. Today, ozone can be
+produced at a total cost of about 2 US$/kg. Further progress can be
+expected. Engineering efforts for superior dielectric properties, better flow
+control and improved thermal management will continue. Rapid advances
+in power semiconductor design resulting in improved GTOs (gate turnoff
+thyristors) and IGBTs (insulated gate bipolar transistors) will have a
+major impact. Encapsulated IGBT modules now switch 1000 A at 5 kV. It
+is foreseeable that soon bulky step-up transformers will be no longer required
+and that almost arbitrary wave forms can be generated. Investigations into
+homogeneous self-sustained volume discharges may even lead to more
+favorable plasma condition for ozone formation (Zakharov et al 1988,
+Kogoma and Okazaki 1994, Nilsson and Eninger 1997).
+References
+Andrews T and Tait P G 1860 Phi. Trans. Roy. Soc. (London) 150 113
+Boelter K and Davidsen J H 1997 Aerosol Sci. Techno!. 27689-708
+Braun D, Kuchler U and Pietsch G 1988 Pure Appl. Chem. 60741-746
+Braun D, Kuchler U and Pietsch G 1991 J. Phys. D: Appl. Phys. 24 564-572
+Chen J 2002 Direct current corona-enhanced chemical reactions PhD thesis, Minneapolis,
+University of Minnesota
+Chen J and Davidson J H 2002 Plasma Chem. Plasma Process 22 199-224
+Chen J and Davidson J H 2003a Plasma Chem. Plasma Process 2383-102
+Chen J and Davidson J H 2003b Plasma Chem. Plasma Process 23501-518
+Crutzen P J 1970 Quart. J. Roy. Meteor. Soc. 96 320-325
+Eliasson Band Kogelschatz U 1986a J. Chim. Phys. 83279-282
+Eliasson Band Kogelschatz U 1986b J. Phys. B: At. Mol. Phys. 19 1241-1247
+Eliasson Band Kogelschatz U 1987 Proc 8th Int Symp on Plasma Chemistry (ISPC-8),
+Tokyo 1987, vol 2, pp 736-741
+Eliasson B, Hirth M and Kogelschatz U 1987 J. Phys. D: Appl. Phys. 20 1421-1437
+Filippov Yu V, Boblikova V A and Panteleev V I 1987 Electrosynthesis of Ozone (in
+Russian), (Moscow: Moscow State University Press).
+Glaze W Hand Kang J W 1988 J. A WWA 88 57-63
+Held Band Peyrous R 1999 Eur. Phys. J AP 7 151-166
+Herron J T 1999 J. Phys. Chem. Ref Data 281453-1483
+
+--- Page 579 ---
+564
+Current Applications of Atmospheric Pressure Air Plasmas
+Herron J T 2001 Plasma Chern. Plasma Proc. 21 581-609
+Herron J T and Green D S 2001 Plasma Chern. Plasma Process 21459-481
+Hoigne J 1998 'Chemistry of aqueous ozone and transformation of pollutants by ozona-
+tion and advanced oxidation processes' in Handbook of Environmental Chemistry,
+Vol 5, Part C: Quality and Treatment of Drinking Water II, Hrubec J (ed)
+(Berlin: Springer) pp 83-141
+Horvath M, Bilitzky L and Huttner J 1985 Ozone (New York: Elsevier Science Publishing)
+Johnston H S 1992 Ann. Rev. Phys. Chern. 43 1-32
+Kitayama J and Kuzumoto M 1997 J. Phys. D: Appl. Phys. 302453-2461
+Kitayama J and Kuzumoto M 1999 J. Phys. D: Appl. Phys. 323032-3040
+Kogelschatz U 1988 'Advanced ozone generation' in Process Technologiesfor Water Treat-
+ment Stucki S ed (New York: Plenum Press) pp 87-120
+Kogelschatz U and Baessler P 1987 Ozone Sc. Eng. 9 195-206
+Kogelschatz U 1999 Proc. Int. Ozone Symp., Basel, pp 253-265
+Kogelschatz U 2000 'Ozone generation and dust collection' in Electrical Discharges for
+Environmental Purposes: Fundamentals and Applications van Veldhuizen E M (ed)
+(Huntington, NY: Nova Science Publishers) pp 315-344
+Kogelschatz U and Eliasson B 1995 'Ozone generation and applications' in Handbook of
+Electrostatic Processes, Chang J S, Kelly A J and Crowley J M (eds) (New York:
+Marcel Dekker) pp 581-605
+Kogoma M and Okazaki S 1994 J. Phys. D: Appl. Phys. 27 1985-1987
+Kossyi I A, Kostinsky A Yu, Matveyev A A and Silakov V P 1992 Plasma Sources Sci.
+Technol. 1 207-220
+Krivosonova 0 E, Losev S A, Nalivaiko V P, Mukoseev Yu K and Shatolov 0 P 1991
+'Recommended data on the rate constants of chemical reactions among molecules
+consisting of Nand 0 atoms' in Reviews of Plasma Chemistry, Smirnov B M Ed
+(New York: Consultants Bureau) vol I, 1-29
+Nilsson J 0 and Eninger J E 1997 IEEE Trans. Plasma Sci. 25 73-82
+Ohlmuller W 1891 Ueber die Einwirkung des Ozons auf Bakterien (Berlin: Springer)
+Peyrous R 1986 Simulation de ['evolution temporelle de diverses especes gazeuses creees par
+['impact d'une impulsion etectronique dans ['oxygene ou de ['air, sec ou humide PhD
+Thesis, Universite de Pau
+Peyrous R 1990 Ozone Sci. Eng. 12 19-64
+Peyrous R, Pignolet P and Held B 1989 J. Phys. D: Appl. Phys. 22 1658-1667
+Pietsch G and Gibalov V 11998 Pure Appl. Chern. 70 1169-1174.
+Rice R G and Netzer A 1982 and 1984 (eds) Handbook of Ozone Technology and Applica-
+tions volland 2 (Ann Arbor: Ann Arbor Science Publishers)
+Samoilovich V G and Gibalov V I 1986 Russ. J. Phys. Chern. 60 1107-1116
+Samoilovich V G, Gibalov V I and Kozlov K V 1989 Physical Chemistry of the Barrier
+Discharge (in Russian) (Moscow: Moscow State University Press) (English transla-
+tion: Dusseldorf: DVS-Verlag 1997, Conrads J P F and Leipold F (eds»
+Sch6nbein C F 1840 Compt. Rend. Hebd. Seances Acad. Sci. 10706-710
+Sieck L W, Herron J T and Green D S 2001 Plasma Chern. Plasma Process 20 235-258
+Siemens W 1857 Poggendorfs Ann. Phys. Chern. 10266-122
+Soret J L 1865 Ann. Chim. Phys. (Paris) 7 113-118
+Wojtowicz J A 1996 'Ozone' in Kirk-Othmer Encyclopedia of Chemical Technology, (John
+Wiley) 4th edition, vol 17, pp 953-994
+Yagi S and Tanaka M 1979 J. Phys. D: Appl. Phys. 12 1509-1520
+
+--- Page 580 ---
+Electromagnetic Reflection, Absorption, and Phase Shift
+565
+Yehia A, Abdel-Salam M and Mizuno A 2000 1. Phys. D: Appl. Phys. 33 831-835
+Zakharov A I, Klopovskii K S, Opsipov A P, Popov A M, Popovicheva 0 B, Rakhimova
+TV, Samarodov V A and Sokolov A P 1988 Sov. J. Plasma Phy.\'. 14 191-195
+9.4 Electromagnetic Reflection, Absorption, and Phase Shift
+9.4.1
+Introduction
+The effect of plasma on electromagnetic (EM), wave propagation in the
+ionosphere is well known and documented by Budden (1985) and Gurevich
+(1978). A particularly striking example of plasma in air is the EM black out
+and fluctuation of radar cross section (ReS), associated with re-entry
+vehicles reported by Gunar and Mennella (1965) and discussed by Ruck
+et al (1970, pp 874--875). A shock wave and resulting plasma develop
+around a vehicle because of the increasing gas pressure and friction as it
+descends from space. At an altitude of 200000 ft (60.9 km) and higher, a
+5 GHz radar frequency is greater than the plasma frequency and the
+momentum-transfer collision rate between electrons and the bulk gas, the
+ReS corresponds to the bare skin value. At ",180000ft (55km), however,
+the plasma frequency increases to approximately the radar frequency and
+the ReS decreases up to 10 dB because of refraction from the plasma
+enclosing the re-entry vehicle. At 150000ft (45.7km) the plasma frequency
+is significantly greater than the radar frequency and an enhanced reflection
+produces a net increase in ReS of 5-10 dB. At 60000 ft (18.3 km) the
+atmosphere is significantly thicker, and the momentum-transfer collision
+rate is ",9 x 109 s-1, which is roughly equal to the plasma frequency with
+both exceeding the radar frequency. In this collision dominated plasma,
+absorption dominates and the Res decreases approximately 15 dB. At
+lower altitudes the re-entry vehicle slows, the plasma dissipates, and the
+ReS returns to its bare skin value.
+Another example is an artificial ionospheric mirror. Borisov and Gure-
+vich (1980) and Gurevich (1980) suggest that a reflective plasma layer below
+the D-layer could be generated at the intersection of two high-power EM
+pulses. The utility of such a mirror is the ability to reflect radio waves at
+frequencies above those supported by the ionosphere to great distances.
+This would permit long range high-frequency point-to-point communication
+and may even permit some radar to extend their range by bouncing their
+signals off such mirrors.
+In this section, EM effects based on a cold collisional plasma with a
+spatially varying plasma density are discussed. The dispersion relation and
+density profile theory is quantified, summary formulas for reflection, trans-
+mission, absorption, and phase shift provided, air-plasma characteristics
+
+--- Page 581 ---
+566
+Current Applications of Atmospheric Pressure Air Plasmas
+quantified, electron-beam produced plasmas discussed, and typical applica-
+tions described.
+9.4.2 Electromagnetic theory
+The theory of an EM wave propagating in air plasma is that of a wave propa-
+gation in a cold collisional plasma. In this approximation ions are assumed to
+be at rest compared to electrons. In the presence of a strong electric field it is
+possible for a non-equilibrium system to develop with an electron, ion, and
+bulk gas temperatures that are all different. The following material describes
+a cold system where the contribution to electrical conductivity by ions is
+small and has been neglected.
+9.4.2.1
+Cold collisional dispersion relationship
+For wave propagation in an air plasma the effect of collisions between
+electrons and the bulk gas is important. The Langevin equation of motion
+for electrons includes the damping of electron motion due to momentum-
+transfer collisions (Tanenbaum 1967),
+du
+me dt = -e(E + u X B) - mevu
+(9.4.1)
+where me is electron mass, u is electron velocity, e is electron charge, E is
+electric field strength, B is magnetic field density, v is momentum-transfer
+collision rate, and MKS units are used throughout. For propagation of
+a transverse EM wave at frequency f through a collisional plasma, the
+dispersion relation provides a succinct relation between angular frequency
+w = 21Tf and a complex wavenumber k,
+w
+k(w) =-c
+w2
+1 -
+p
+w(w - iv)
+(9.4.2)
+where c is the speed of light, wp = (nei /come)1/2 is the plasma frequency,
+i = +(_1)1/2, ne is electron density, e is electron charge, and co is the free-
+space permittivity. Wave propagation is proportional to exp[+i(wt - kz)]'
+where t is time and z is distance. For v = 0 in (9.4.2) the dispersion relation
+reduces to a cold lossless dispersion relation with a cutoff frequency at w = wp.
+For a lightly ionized collisional plasma with
+Iw~/w(w - iv)1 « 1
+equation (9.4.2) can be expanded and factored into real and complex parts,
+W
+wp
+[
+2]
+kr(w) = C 1 - 2(J + v 2) ,
+(9.4.3)
+The value of kr is directly proportional to frequency. The leading term of kr
+can be interpreted as ko = w/ c, which is the free-space wavenumber, but the
+
+--- Page 582 ---
+Electromagnetic Reflection, Absorption, and Phase Shift
+567
+first-order plasma term is proportional to both wand ne. Consequently, an
+EM wave will encounter an impedance that depends on ne. If ne exhibits a
+step-like change in number density there will be a coherent reflection. If
+the change in ne is smooth and extends over several free-space wavelengths,
+reflections along the smooth profile add incoherently and can be quite small.
+The value of ki for w < v is effectively independent of frequency and so
+implies that a collisional plasma is a broadband EM wave absorber. These
+two processes of reflection and absorption are present in collisional plasmas
+with the dominant effect depending on the profile for ne and the frequency of
+observation.
+9.4.2.3
+Electron density profiles
+The exact profile for ne depends on the plasma source and the intended
+application. Large changes in ne over a distance of less than one free-space
+wavelength generally result in a strong coherent reflection (often modeled
+as a slab discontinuity), whereas the same change in ne over several wave-
+lengths produces an incoherent reflection. Ruck et al (1970, pp 473--484)
+describe a layered-media matrix approach that takes internal reflections
+into account and can be applied to an arbitrary plasma distribution. The
+values of ne and the momentum-transfer collision rate can change for each
+layer and equation (9.4.2) is used to generate a complex wavenumber for
+each frequency of interest. A few distribution functions for ne yield analytic
+results. Budden (1985) provides analytic expressions for the reflection and
+transmission coefficients for linear, piecewise linear, parabolic, Epstein,
+and sech2 electron distributions. The Epstein distribution is used to model
+a variety of plasma sources that generates a high electron density near the
+source, which diminishes with distance from the source. The Epstein distribu-
+tion is particularly useful in modeling the plasmas generated by a high-energy
+electron beam or beta rays and photo processes that adhere to the Beer-
+Lambert law such as photo-ionization.
+10.3.2.3 Epstein distributions
+Epstein (1930) discussed a general electron density distribution with three
+arbitrary constants and wave propagation in absorbing media. Specific
+wave solutions are discussed by Budden (1985). Vidmar (1990) adapt the
+Epstein distribution to one suitable for modeling ionization sources. The
+electron number density utilized is
+no
+n (z) - ----'------:--:-
+- 1 + exp( -z/zo)
+(9.4.4)
+where Zo is a dimensional scale factor and no is the maximum electron
+concentration for z ----* +00. Equation (9.4.4) varies from n(z = -(0) = no
+
+--- Page 583 ---
+568
+Current Applications of Atmospheric Pressure Air Plasmas
+to n(z = +00) = O. For a source that deposits energy over a finite distance,
+it is possible to match n(z) at the 95% (z/zo = +2.944), 50% (z/zo = 0),
+and 5% (z/zo = -2.944) values and so determine an approximate value
+for zoo
+9.4.2.3
+Epstein's power reflection and transmission coefficients
+Using the Epstein distribution in (9.4.4) for a wave incident at an angle e,
+where e = 0 implies backscatter and e = 90° implies grazing incidence. The
+power reflection, R, and transmission, T, coefficients are
+R = IC - q121r[1 + ikozo(q + C)]1 4
+C + q r[l + ikozo(q - C)]
+4C2 I
+r2[1 + ikozo(q + C)]
+12
+T = IC + ql2 r[l + 2ikozoq]r[1 + 2ikozoCJ exp[+2Im(koq)z]
+2
+2
+2
+wp
+q = C -
+---,----.!:.----:-
+w(w - iv)
+(9.4.5)
+(9.4.6)
+(9.4.7)
+where C = cos e and q is a solution of the Booker quartic. For some atmos-
+pheric plasma the arguments of the gamma functions become large, complex,
+and produce an overflow condition. Lanczos (1964) provides an asymptotic
+expansion for r and evaluation of In r avoids overflow.
+9.4.2.4
+Attenuation and phase-shift coefficients for an Epstein profile
+In some applications, such as those relating to radar, the effects on signal
+attenuation and phase path-length for round trip propagation through
+plasma with reflection from a good conductor are of interest. Analytic
+expressions are evaluated using the approximate values for k in (9.4.3) and
+evaluating exp( +2i f kdz), where the integral is from z = -00 to the
+reflective surface. For reference the integration of w~ is proportional to ne,
+(9.4.4), and the integral of ne from z = -x to z = +x is nox. By noting
+that the ionization source plasma was modeled by (9.4.4) from the 95-5%
+values, the integration of f kdz is from free space for z = -2.944zo to
+the conductive body and ionization source at z = +2.944zo. For
+Iw~/w(w - iv) « 11 the round trip attenuation, A in dB, and the net phase
+change, ~ on no, h,f, and //.
+9.4.3 Air plasma characteristics
+The air chemistry for a plasma depends on many factors such as air density
+determined from altitude, moisture content, electron density, present
+populations of excited states, electron temperature, bulk gas temperature,
+magnitude of electric field, and method of ionization. For production of
+plasma without any external wire electrodes, a high-energy electron-beam
+source is proposed. A 250 kV electron beam source, for example, is capable
+of producing a plasma cloud that extends 1.5 m from its source at 30000 ft
+(",9.l4km) altitude. Macheret et al (2001) investigated electron beam
+produced air plasmas and quantified a return current from free space to
+the source, due to charge transport by fast electrons. Their electric field
+varies spatially, being most intense near the source. Consequently, the
+plasma generated by an electron beam varies spatially in electron concentra-
+tion, electric field, and electron temperature. The air chemistry production-
+deionization solution must also treat these variations. Analytic air-chemistry
+approaches are tedious due to the complexity and nonlinear aspects of the air
+chemistry. Numerical approaches can easily involve hundreds of reactions to
+model the air chemistry but provide useful estimates of plasma lifetime for
+pulsed systems and estimates of power expenditure with curves of species
+as a function of time for a variety of excitation waveforms.
+9.4.3.1
+Momentum-transfer collision rate
+For an electron beam source an electric field may be present with sufficient
+magnitude to elevate the electron temperature above thermal. Lowke
+(1992) has investigated free electrons in air as a function of water-vapor
+content and the reduced electric field E/N, where N is the bulk gas density.
+The curves Lowke generated explicitly treat the effects of N2, O2, CO2, and
+H20 as a gas mixture on the electron energy as a function of E / N. The
+momentum-transfer collision rates in table 9.4.1 were deduced from Lowke
+and appear as a function of altitude from sea level to 300000 ft
+(",91.4 km). Atmospheric parameters of pressure, bulk gas density, and
+temperature appear below each altitude.
+9.4.3.2 Major attachment mechanisms
+Electrons attach primarily to oxygen molecules in a three-body process,
+Bortner and Baurer (1979) and Vidmar and Stalder (2003) for E/N
+
+--- Page 585 ---
+Table 9.4.1. Momentum transfer collision rate and atmospheric parameters.t
+E/N
+Momentum transfer collision rate (S-l)
+Sea level
+30 000 ft
+60 000 ft
+(9.14km)
+(18.3 km)
+764 torr
+228 torr
+54.8 torr
+2.55 x 1019cm-3
+9.58 x 1018
+2.43 X 1018
+V -cm-2
+288.1 K
+228.8K
+216.6K
+0.0
+9.53 X IO lD S-l
+3.58 X IO lD
+9.09 X 109
+1.0 X 10-19
+9.53
+3.58
+9.09
+5.0 x 10-19
+1.31 X lOll
+4.92
+1.25 x IO lD
+1.0 X 10-18
+1.75
+6.60
+1.67
+1.5 X 10-18
+6.75
+1.91 x 1011
+4.90
+1.0 X 10-17
+7.25
+2.63
+6.72
+1.5 X 10-17
+1.11 X 1012
+5.60
+1.42 x 1011
+1.0 X 10-16
+1.94
+8.41
+2.13
+1.5 X 10-16
+3.31
+1.24 x 1012
+3.16
+1.0 x 10-15
+3.92
+1.47
+3.74
+t 1962 US standard atmosphere
+100 000 ft
+200 000 ft
+(30.5km)
+(60.9km)
+8.45 torr
+149 x 10-3 torr
+3.58 X 1017
+5.86 X 1015
+226.9K
+244.6K
+1.34 X 109
+2.19x107
+1.34
+2.19
+1.84
+3.01
+2.47
+4.04
+7.18
+1.11 X 108
+9.86
+1.54
+2.09 X IO lD
+3.41
+3.14
+5.14
+4.66
+7.62
+5.52
+9.03
+300000 ft
+(91.4 km)
+1.31 x 10-3 torr
+5.91 X 1013
+214.2K
+2.21 X lOS
+2.21
+3.03
+4.07
+1.19 x 106
+1.63
+3.45
+5.18
+7.68
+9.10 x 107
+VI
+-.l
+o
+()
+;:::
+....
+....
+~
+;:: -
+:.t..
+~
+:::-:
+'"'
+!:)
+5·
+;::
+c.,
+~
+:.t..
+§'
+<::>
+~
+;::..
+~
+....
+;::;.
+"tI
+....
+~
+c.,
+c.,
+;:::
+....
+~
+:.t..
+::;;.
+"tI
+is"'
+~
+!:)
+c.,
+
+--- Page 586 ---
+Electromagnetic Reflection, Absorption, and Phase Shift
+571
+dependencies. The resulting O2 ion undergoes numerous charge-transfer
+reactions, hydration, and eventually becomes N03 and N03·H20 prior to
+negative-ion/positive-ion recombination. The rate for three-body attachment
+of electrons to O2 depends on the altitude-dependent O2 concentration and
+the E/N-dependent electron temperature. The extent to which O2 or N03
+is the dominant ion depends on how long the plasma is generated. A typical
+time scale for generation and deionization for an aircraft flying near the
+speed of sound that generates then flies through a plasma cloud"", 1.5 m in
+extent is "",5 ms. A time scale of several hundred microseconds to several
+milliseconds typifies many plasma applications for aircraft.
+9.4.3.3
+lie plasma lifetime
+The 1/ e plasma lifetime is the time for plasma that has been suddenly ionized
+to an electron density of no to deionize to a value of no/ e. A set of curves that
+quantifies plasma lifetime as a function of altitude and electron density
+appears in Vidmar (1990) and quantifies electron densities, where the
+dominant process for electron loss is three-body attachment to O2 with an
+electron as the third body, three-body attachment with O2 as the third
+body, and electron-positive ion recombination. These curves have been
+extended to include an E / N dependency in Vidmar and Stalder (2003).
+Plasma lifetime is shown to increase by approximately an order of magnitude
+for 10-17 V cm -2 < E / N < 10-16 V cm -2. The increase in lifetime corresponds
+to a decrease in the rate of three-body attachment for E / N > 10-17 V cm-2
+predicted by Aleksandrov (1993). This trend towards longer lifetime reverses
+for E / N ~ 10-16 V cm -2, when the reaction rate for dissociative attachment
+to oxygen increases significantly and dominates the attachment process.
+9.4.4 Plasma power
+The energy deposited by an electron beam to generate an electron-ion pair in
+dry air, Ej , is 33.7 eV. For a pulsed source a lower estimate of the power per
+unit volume, P / V, is approximated by using Ej , the electron number density,
+and the plasma lifetime:
+P
+nOEj
+V
+T
+(9.4.10)
+where no is the peak electron concentration and T is plasma lifetime. The
+value of T as a function of altitude is quantified in Vidmar (1990) and
+Vidmar and Stalder (2003). For example, an electron density of 1010 cm-3
+at 30000ft (9.l4km) with E/N = 0 has a plasma lifetime of l57ns with
+P / V = 343 m W Icm3 or 343 kW 1m3• Plasma lifetime is effectively indepen-
+dent of electron number density below 1010 cm -3, because the dominant
+electron loss mechanism is three-body attachment to O2, which is linear
+
+--- Page 587 ---
+572
+Current Applications of Atmospheric Pressure Air Plasmas
+with respect to electron concentration. Consequently, power is proportional to
+no, and the total power is the integral of P / V over the electron distribution.
+For plasma generated by an electron beam and sustained by an electric
+field, the expression for power includes a term to account for louIe heating,
+P = noEj +J.E
+V
+T
+(9.4.11)
+where J = aE is current density and (J is the plasma conductivity. Vidmar
+and Stalder (2003) calculated plasma lifetime as a function of E / N for a
+continuous electric field and quantified total power at 30000 ft (9.14 km).
+Although louIe heating increases as the square of electric field strength,
+the increase in plasma lifetime for 1O-17 Vcm-2 < E/N < 1O-l6 Vcm-2
+results in a net decrease in total power from 343 to 230 m W jcm3 for a
+plasma density of 1010 cm -3. This decrease in net power is also accompanied
+by an increase in excited states with Oil ~g) reaching 8 x 109 cm-3 .
+Additional research on power in air plasma involves continuous and
+pulsed ionization to quantify the concentrations and effect of excited states
+as a function of time. Because the energy deposited in plasma eventually
+heats the bulk gas, the concentration of all species will decrease due to volu-
+metric expansion. Over short intervals such as those for an aircraft in flight,
+the generation of excited states under some conditions can significantly
+reduce the concentration of ground state species. These two effects slow
+the attachment process. The reaction rates for all the excited states on the
+major attachment, detachment, and deionization processes are not well
+known. Consequently, additional research, both theoretical and experi-
+mental, is necessary to quantify total power deposition in air plasma as a
+function of electron concentration, E / N, and altitude.
+9.4.5 Applications
+The application of collisional plasma for reflection, absorption, and phase
+shift has been motivated by early investigations of the ionosphere (Epstein
+1930). Reflection from plasma slabs with sharp discontinuities is well
+understood and application to a surface radar for beam steering has been
+investigated (Manheimer 1991). Reflections from an ionospheric mirror
+have been advanced by Borisov and Gurevich (1980) and Gurevich (1980).
+A set of curves that apply to an ionospheric mirror at 230000 ft (70.1 km)
+appears in Vidmar (1990) based on the Epstein distribution and the profile
+for n(z) in equation (9.4.4). These curves quantify the power reflection
+coefficient at a shallow angle of 75° off broadside for an electron density of
+107 cm-3 and v = 7.4 X 107 s-l. It was found that the power reflection
+coefficient was 0.80 or greater for frequencies below 100 MHz and
+Zo < 10 m. At higher frequencies or for Zo > 10 m the power reflection
+coefficient decreased substantially. In terms of the profile in (9.4.4) the
+
+--- Page 588 ---
+Electromagnetic Reflection, Absorption, and Phase Shift
+573
+value of Zo = 10m implies the means of ionization must transition the air at
+230000ft (70.1 km) from 5-95% of the maximum electron concentration
+over a distance of h = 5.888zo = 58.88 m.
+The use of microwave absorption as a diagnostic technique to determine
+electron concentration is well known. Spencer et al (1987) experimentally
+measured the amplitude and phase in a microwave cavity to quantify the
+plasma lifetime, complex conductivity, and momentum-transfer collision
+rate of an electron-beam generated plasma.
+The application of the Epstein distribution to model collisional plasma
+as a broadband absorber by Vidmar (1990) has curves of absorption versus
+frequency and zoo These curves quantify total reduction, which refers to the
+sum of the reflected power, R in equation (9.4.5), the round-trip absorption,
+A in equation (9.4.9), and points out the power advantage of generating
+plasma in a noble gas rather than air. The total reduction curves that
+appear in Vidmar (1990) imply 10-40 dB signal reduction at frequencies
+that extend from how> c/(4zo) and extends to fhigh < v/5. Physically, the
+broadband reduction requires approximately five collisions per cycle and
+the 5-95% gradient of the Epstein distribution, h = 5.888zo must be one to
+two wavelengths at the lowest frequency. The total reduction noted transfers
+of EM energy from a wave to heat via momentum-transfer collisions with the
+bulk gas. This reduction in reflected power reduces the RCS for the surface
+directly behind the plasma. The results of Santoru and Gregoire (1993)
+provide an experimental link between the Epstein theory for reflection and
+absorption with laboratory measurements.
+Some radar systems utilize coherent integration over many cycles to
+improve their signal-to-noise ratio. For such radars a sudden change in
+phase interferes with the coherent integration and so degrades radar perfor-
+mance. The phase change ~ in (9.4.9) can be used to quantify such effects in
+terms of radar frequency, electron number density, collision rate, and Epstein
+gradient.
+For all of these applications the EM effects of plasma on reflectivity and
+RCS are approximated by the Epstein distribution and the derived expressions
+for reflectivity, transmission, absorption, and phase shift. The means to
+achieve a man-made Epstein distribution in air all require power. The
+means of plasma generation for a particular application that minimizes net
+power required is not known at this time. Electron-beam generated air
+plasma is a candidate system for some applications because it has a unique
+excited-state air chemistry, the advantage that no wires are necessary in
+the plasma, and that the beam energy controls the Epstein gradient. A
+detractor on the use of electron beams is the problem of window heating
+that limits beam current and duty cycle. This problem is addressed by
+liquid cooling around the window or within the window (Vidmar and
+Barker 1998), or by propagation from vacuum to air through a small
+opening. Additional research on power required as a function of a
+
+--- Page 589 ---
+574
+Current Applications of Atmospheric Pressure Air Plasmas
+continuous or pulsed source, altitude, and electron concentration is neces-
+sary to prove the utility of the electron beam approach.
+References
+Aleksandrov N L 1993 Chern. Phys. Lett. 212 409--412
+Borisov N D and Gurevich A V 1980 Geomagn. Aeronomy 20587-591
+Bortner M Hand Baurer T 1979 Defense Nuclear Agency Reaction Rate Handbook, 2nd
+edition, NTIS AD-763699 ch 22
+Budden K G 1985 The Propagation of Radio Waves, The Theory of Radio Waves of Low
+Power in the Ionosphere and Magnetosphere (New York: Cambridge University
+Press) 438--479
+Epstein P S 1930 Proc. Nat. A cad. Sci. 16627-637
+Gunar M and Mennella R 1965 Proceedings of the 2nd Space Congress-New Dimensions
+in Space Technology, Canaveral Council of Technical Societies 515-548
+Gurevich A V 1978 Nonlinear Phenomena in the Ionosphere, Physics and Chemistry in Space
+vol 10 (New York: Springer) p 370
+Gurevich A V 1980 Sov. Phy. Usp. 23862-865
+Lanczos C 1964 J. SIAM Numer. Anal. Ser. B 1 86-96
+Lowke J J 1992 J. Phys D: Appl. Phys. 25202-210
+Macheret S 0, Shneider M N and Miles R B 2001 Physics of Plasmas 81518-1528
+Manheimer W M 1991 IEEE Trans. Plasma Sci. PS-19 1228-1234
+Ruck G T, Barrick D E, Stuart W D and Krichbaum C K 1970 Radar Cross Section Hand-
+book vol 2 (New York: Plenum) 473--484 and 874-875
+Santoru J and Gregoire D J 1993 J. Appl. Phys. 74 3736-3743
+Spencer M N, Dickinson J S and Eckstrom D J 1987 J. Phys D: Appl. Phys. 20923-932
+Tanenbaum B S 1967 Plasma Physics (New York: McGraw-Hill) 62-86
+Vidmar R J 1990 IEEE Trans. Plasma Sci. PS-18 733-741
+Vidmar R J and Barker R J 1998 IEEE Trans. Plasma Sci. PS-26 1031-1043
+Vidmar R J and Stalder K R 2003 AIAA 2003-1189
+9.5
+Plasma Torch for Enhancing Hydrocarbon-Air Combustion
+in the Scramjet Engine
+9.5.1 Introduction
+The development of the scramjet propulsion system [1-3] is an essential part
+of the development of hypersonic aircraft and long-range (greater than 750
+miles (1207 km)) scram jet-powered air-to-surface missiles with Mach-8
+cruise capability [4]. This propulsion system has a simple structure as
+required by the hypersonic aerodynamics. Basically, the combustor has the
+shape of a flat rectangular box with both sides open. Air taken in through
+the frontal opening mixes with fuel for combustion and the heated exhaust
+
+--- Page 590 ---
+Plasma Torchfor Enhancing Hydrocarbon-Air Combustion
+575
+gas at the open end is ejected through a MGD accelerator and a nozzle to
+produce the engine thrust.
+For the hydrocarbon-fueled scramjet in a typical startup scenario, cold
+liquid JP-7 is injected into a Mach-2 air crossflow (having a static tem-
+perature of ",500 K); under these conditions, the fuel-air mixture will not
+auto-ignite. Instead, some ignition aid-for example a cavity flameholder
+in conjunction with some mechanism to achieve a downstream pressure
+rise--is necessary to initiate main-duct combustion. With sufficient down-
+stream pressure rise, a shock front will propagate upstream of the region
+for heat release. The heat release from combustion will maintain the pre-
+combustion shock front, while subsonic conditions in the mixing and
+combustion region favor stable combustion and flameholding.
+Of course, even though the device operates as a ramjet under startup
+conditions (i.e. subsonic flow downstream of the pre-combustion shock)
+the residence time through the combustion region is short, of order 1 ms.
+Within scramjet test facilities, the typical mechanisms for achieving the
+required downstream pressure rise (and stable combustion) are the so-
+called aero-throttle, where a 'slug' of gas is injected in the downstream
+region, and the heat is released from the pyrophoric gas silane (SiH4).
+Indeed, silane injection into the combustor is the current mechanism by
+which the X43A scramjet vehicle is started. Both of these approaches,
+however, have their disadvantages: for example, the aero-throttle approach
+may not allow re-lighting attempts and silane poses obvious safety risks.
+Thus, an alternative approach is desired.
+For the purpose of developing techniques to reduce the ignition delay
+time and increase the rate of combustion of hydrocarbon fuels, Williams
+et al [5] have carried out kinetics computations to study the effect of
+ionization on hydrocarbon-air combustion chemistry. The models being
+developed-which include both the normal neutral-neutral reactions and
+ion-neutral reactions-focus primarily on the development of plasma-
+based ignition and combustion enhancement techniques for scramjet
+combustors. The results computed over the 900-1500 K temperature range
+show that the ignition delay time can be reduced significantly (three order
+of magnitude over the 900-1500 K temperature range) by increasing the
+initial temperature of fuel-air mixture.
+Moreover, detailed kinetics modeling also shows a significant decrease
+in ignition delay in the presence of initial ionization-in the form of a
+H30+ INO+ Ie ~ plasma-at levels of ionization mole fractions greater than
+1O~6. The ignition delay time is decreased most significantly at low tempera-
+tures. Indeed, the computational results suggest that even larger effects may
+be observed at the low temperatures encountered under engine startup.
+Plasma torches can deliver enough heat to replace silane for ignition
+purpose. Moreover, use of a torch as a fuel injector also introduces an initial
+ionization in the fuel. The significant decrease in the ignition delay time and
+
+--- Page 591 ---
+576
+Current Applications of Atmospheric Pressure Air Plasmas
+Figure 9.5.1. A photo of the plasma torch module. (Copyright 2004 by IEEE.)
+the initial energy carried by plasma may elevate the heat release from
+combustion to exceed a threshold level for flameholding. These are the
+primary reasons that plasma torches [6-8] are being developed for the appli-
+cation.
+Nevertheless, to make use of the high-temperature torch effluent, which
+may include quantities of radicals, ions, and electrons, it is necessary to
+project this gas into the engine in such a way that it readily mixes with a
+fuel-air stream. Poor penetration of the torch plume into the combustor,
+and/or improper placement of each torch-that is, more than one torch
+may be required-will limit its effectiveness. Shown in figure 9.5.1 is a
+photo of a plasma torch module, which is developed [9-10] in the present
+effort for the generation of torch plasma. The unique features of this
+plasma torch make it well suited for the purpose of ignition in a scramjet
+engine. These features include the following.
+1. The compact size. It can be easily mounted to the combustor wall and
+requires no water cooling.
+2. Flexible design. It can deliver high peak powers (and pulse/cycle energy)
+in 60 Hz or pulsed modes. Furthermore, it can deliver high mass flow rates
+due to the large annular flow area.
+3. High mass flow operation. It can be configured to deliver 10 g offeedstock
+(which can be the fuel) per second.
+4. Durability. It can be run for long periods with an air feedstock.
+5. High-voltage operation. Rather than running at high current, the torch
+runs at high voltage, which allows greater penetration of the arc into
+the combustor and reduces the power loss to the electrodes (leading to
+
+--- Page 592 ---
+Plasma Torch for Enhancing Hydrocarbon-Air Combustion
+577
+longer electrode life); higher E / N also enhances dissociations in fuel and
+air by direct electron impact.
+9.5.2
+Plasma for combustion enhancement
+In the combustion, fuel-air mixing is critical. Without oxygen, fuel will not
+burn by itself. The hydrocarbon fuel provides hydrogen and carbon to
+react with oxygen in the combustion process. The reaction rate increases
+with the temperature of the mixture, which changes the ratios of the compo-
+nents in the composition of the mixture. In low temperature, the gas mixture
+contains mainly neutral molecules, and neutral-neutral reactions are often
+immeasurably slow. For example, the rate coefficient for the reaction
+between H2 and O2 is 6 X 10-23 cm3 S-I. As temperature increases, some radi-
+cals such as atomic species are produced. Neutral-radical reactions have
+rates in the range of 10-16_10- 11 cm3 S-I. For example, the reaction between
+Hand O2 has a rate coefficient equal to 1 x 10-13 cm3 S-I. Reactions also
+occur between radicals, which in fact have higher rates in the range 10-13_
+10-10 cm3 S-I. Hence, the combustion rate is increased as the percentage of
+radicals in the mixture becomes significant by the temperature increase. If
+the temperature of the mixture is high enough to cause significant ioniza-
+tions, the combustion rate is further enhanced. This is because ion-neutral
+and ion-electron reactions have rates larger than 10-9 and 10-7 cm3 S-I,
+respectively. For instance, the reaction Hi + O2 has a rate coefficient of
+8 x 10-9 cm3 S-I. It turns out only long-range ion-electron and ion-dipole
+reactions are fast enough to react on hypersonic flow time scales in the micro-
+second range. Therefore, it is desirable to use energy to heat the mixture and
+also to introduce ionized species to the mixture. Usually, thermal plasma is
+not very energy efficient to introduce ionized species to the mixture. Non-
+equilibrium plasmas produced by corona, streamer, pulsed glow and micro-
+wave discharges have been suggested, as alternatives to the torch plasma, for
+aiding the ignition. These discharges run at high E / N can potentially
+enhance dissociations in fuel and air by direct electron impact [11], where
+E is the electric field and N is the gas density. However, the practical issue
+of the research efforts is the combustion efficiency, rather than the energy
+efficiency of the igniter. The combustion efficiency depends not only on the
+chemical processes but also on the spatial distribution of the plasma
+energy, in particular, in a supersonic combustor. If the igniter can only
+start the ignition locally, for instance, near the wall, a considerable percen-
+tage of injected fuel will not be ignited before exiting the combustor. The
+plasma torch presented in the following demonstrates that it can produce
+high enthalpy supersonic plasma jet to penetrate the supersonic cross flow,
+as required to be a practical igniter of a supersonic combustor.
+Two types of power supply are applied to operate the torch module
+shown in figure 9.5.1. One is a 60 Hz source, which sustains the discharge
+
+--- Page 593 ---
+578
+Current Applications of Atmospheric Pressure Air Plasmas
+periodically. Such produced plasma will be termed '60 Hz torch plasma' in
+the following. This power source [12] includes (1) a power transformer
+with a turn ratio of 1: 25 to step up the line voltage of 120 V from a wall
+outlet to 3 kV, (2) capacitors of C = 3 IlF in series with the electrodes, and
+(3) a serially connected diode (made of four diodes, connected in parallel
+and each having 15kV and 750 rnA rating) and resistor (R = 4kO) placed
+in parallel to the electrodes to further step up the peak voltage. The series
+resistor is used to protect the diode by preventing the charging current of
+the capacitor from exceeding the specification (750 rnA) of each diode and
+to regulate the time constant of discharge. In one half cycle when the
+diode is forward biased, the capacitor is charging, which reduces the avail-
+able voltage for the discharge in the torch module. However, since the time
+constant RC = 12 ms is longer than the half period 8.5 ms of the ac input,
+the discharge can still be initiated during this half cycle (even though the
+discharge has lower current and voltage than the corresponding ones in
+the other half cycle). During this other half period, the diode is reversed
+biased and the charged capacitor increases considerably the available voltage
+and current for the discharge in the torch module. The torch energy (i.e. the
+thermal energy carried by torch plasma) in each cycle varies with the gas
+supply pressure Po. The dependence measured in the pressure range from
+1.36 to 7.82 atm is presented in figure 9.5.2(a).
+As shown, the dependence has a maximum at the gas supply pressure
+Po = 6.12 atm, where the plasma energy is 25.6J. The increasing dependence
+of the plasma energy on the flow rate in the region of low gas supply pressure
+(i.e. Po < 6.12 atm) is realizable because the supplied gas flow works to
+increase the transit time of charge particles by keeping the discharge away
+from the shortest (direct) path between two electrodes. As the flow rate
+increases, the transit time loss of charge particles is reduced and thus the
+plasma energy increases. However, when the flow rate becomes too high
+(i.e. Po > 6.12 atm), the mobilities of charge particles crossing the flow
+becomes significantly affected by the flow. In such a way that the torch
+energy decreases with increasing pressure. It is noted in figure 9.5.2(a) that
+there is a significant plasma energy drop at Po = 4.08 atm. This unexpected
+result may be explained as follows. Schlieren images indicate that a transition
+from subsonic to supersonic flow at the exit of the module occurs near
+Po = 3.4 atm, which was identified by the sudden appearance of the shock
+structure at the exit of the torch nozzle in the schlieren image of the flowfield.
+After the transition, the flow becomes underexpanded. At Po = 4.08 atm, the
+low pressure region in the flow that favors gas breakdown is narrow in the
+flow direction and close to the exit of the module. Thus the discharge channel
+is narrow and the transit times of charge particles are small. Consequently,
+the plasma energy is reduced. As the pressure is further increased, this low-
+pressure region extends rapidly outward from the exit of the module so
+that the discharge can again appear in a larger region.
+
+--- Page 594 ---
+Plasma Torchfor Enhancing Hydrocarbon-Air Combustion
+579
+30
+f-O-i
+.......
+20
+.......
+~
+.....
+>+l
+W
+10
+0
+(a)
+pO(atrr$
+20
+E
+E
+15
+10
+5
+5
+(b)
+mm
+Figure 9.5.2. (a) Dependence of the plasma energy in one cycle on the gas supply pressure
+and (b) a planar image of torch plasma taken by an ultra-fast CCD camera with lOns
+exposure to laser-induced fluorescence from NO molecules. (Copyright 2004 by IEEE.)
+As a consequence of the high-voltage nature of the discharge, the arc
+loop can be many times the distance between the anode and cathode. The
+arc loop structure is illustrated in the image (typical of those recorded)
+shown in figure 9.5.2(b), which was recorded through a 239nm interference
+filter, 10 nm FWHM, with an intensified CCD camera (Roper Scientific
+PIMAX) set for an 80 ns exposure time. The current loop is coincident
+with the thin, intense emission loop shown in the figure. For this measure-
+ment, pure nitrogen with a pressure of 1.7 atm was supplied to the torch
+module. The horizontal extent of the arc loop is ca. 3.2 mm, whereas the
+vertical extent is about 2.5 cm. Such an extended arc loop increases the
+path length of the charged particles in the discharge by more than 15 times
+the direct path length from the cathode to the anode. Also shown in figure
+
+--- Page 595 ---
+580
+Current Applications of Atmospheric Pressure Air Plasmas
+9.5.2(b) is laser-induced fluorescence (LIF) from nitric oxide, NO, obtained
+using a Nd:YAG-pumped dye laser system to generate laser radiation at
+226 nm probing the overlapped QI (12.5) and Q2(l9.5) transitions in the
+8(0,0) band of NO. The LIF image appears as the diffuse, less intense
+background and is best seen on the left-hand side of the figure towards the
+outer portion of the arc loop. NO is produced within the torch plume in
+the region where the hot torch gas (pure N2), i.e. the gas near the arc,
+mixes with quiescent laboratory air. Thus, NO is formed primarily near
+the outer portion of the arc loop.
+The extended arc loop structure produced with this torch module has
+several distinct advantages. For instance, such images indicate that high
+temperature, dissociated, and ionized air extends well above the surface of
+the torch module, which is important for ignition applications. The long
+electrode lifetime may in part be due to extended arc length since the charged
+particles' kinetic energy is reduced before hitting electrodes. Furthermore,
+the conversion of electrical energy to plasma energy may be enhanced due
+to the longer interaction region. Images such as that shown in figure 9.5.2(b)
+indicate that the length of the arc loop is not strongly sensitive to the flow
+rate, but the width of the loop becomes narrower as the flow rate increases,
+which is consistent with the change in the flowfield structure as the jet becomes
+underexpanded and supersonic with increased supply pressure.
+The other power supply applied is a dc pulsed discharge source, which
+uses a RC circuit for charging and discharging, where a 281lF capacitor is
+used. A very energetic torch plasma, albeit one with a low repetition rate,
+can be generated. In the circuit, a ballasting resistor R2 is connected in
+series with the torch to regulate the discharging current and adjust the
+pulse duration. Shown in figure 9.5.3(a) is a power function obtained by
+connecting a resistor of R2 = 26 n in series with the torch. This power func-
+tion has a peak of about 300 kW and a pulse length of about 800 IlS, which is
+very close to the time constant R2C = 728Ils. The difference is accountable
+from the effective resistance of the discharge. As R2 is increased to 250 n,
+now the power function shown in figure 9.5.3(b) consists of two parts: an
+initial part with a large peak of about 20 kW for the ignition of the discharge
+and a subsequent near-constant low-power part keeping at about 2.5 kW for
+10 ms, which maintains the discharge. The energy contained in the pulse is
+about 50J.
+Because torch plasma delivers adequate energy, it can be an ignition aid
+and combustion enhancer within a scramjet engine.
+9.5.3 Plasma torch for the application
+The performances of plasmas produced by the torch module in a Mach-2.5
+supersonic crossflow are discussed in the following. Measurements consist
+of video images of the torch emission and of the flowfield schlieren. We
+
+--- Page 596 ---
+Plasma Torch/or Enhancing Hydrocarbon-Air Combustion
+581
+400
+! 200
+~
+Go
+01-_"""",1 '---__________ _
+(a)
+25
+20
+_ 15
+1,0
+5
+-0.5
+o
+0.5
+1
+t(ms)
+1.5
+2
+o
+~-~5--~~~0~~~~5~~~~~,0~~~~,5
+(b)
+t(ms)
+2.5
+Figure 9.5.3. Power functions of pulsed dc discharges with no flow in the background; gas
+supply pressure of the torch module is 2.72 atm. (a) R2 = 26 r! and (b) R2 = 250 r!.
+note that due to the limited framing rate, 30 frames per second, these images
+represent a temporal average during the frame time. Thus, one does not
+freeze the arc-loop structure as was done with the intensified CCD (figure
+9.5.2(b)). This is true regardless of whether one is viewing the 60 Hz or
+pulsed discharge.
+Experiments [13, 14] were conducted in the test section, measuring
+38 cm x 38 cm, of a supersonic blow-down wind tunnel. The upstream flow
+had a flow speed of 570 mis, a static temperature TI = 135 K, and a pressure
+PI = 1.8 X 104 N/m2 (about 0.20 atm). These conditions approximate the
+scramjet startup conditions listed earlier, though the temperature and
+pressure are somewhat low (e.g. the static temperature for engine startup is
+about 500 K). The torch plume is injected normally into the supersonic
+flow, and the performance of torch plasma in terms of its height and shape
+in the supersonic flow is studied. In experiments, the air supply pressure is
+varied from 1.7 to 9.2 atm.
+We first investigate the 60 Hz torch plasma in the wind tunnel. Presented
+in figure 9.5.4(a) is an airglow image of the plasma torch produced in the
+Mach-2.5 crossflow with 4.1 atm of air pressure supplied to the gas
+chamber of the torch module. This image shows the typical shape of the
+plasma torch in each half cycle; clearly, the supersonic flow causes significant
+deformation in the shape of the plasma torch. The penetration height of
+
+--- Page 597 ---
+582
+Current Applications of Atmospheric Pressure Air Plasmas
+E
+E
+o
+20
+90
+(a)
+mm
+(b)
+Figure 9.5.4. (a) Sideview of the airglow image of ac torch plasma in each half cycle in the
+Mach-2.5 crossflow. The gas supply pressure of the torch module is 4.1 atm. In the insert,
+d, = d)' = 11.4 mm define the horizontal and vertical scales of the image. (b) Shadow image
+of the flow; an oblique shock wave is generated in front of the torch. (c) Airglow image of
+pulsed dc torch plasma in a supersonic crossflow (about 10° off the sideview line); the field
+of view is estimated to be 9.5 cm x 6 cm; the gas supply pressure of the torch module is
+2.72 atm. (d) Schlieren image of pulsed dc torch plasma; the backpressure of the torch is
+9.2 atm. (Copyright 2004 by IEEE.)
+
+--- Page 598 ---
+Plasma Torch lor Enhancing Hydrocarbon-Air Combustion
+583
+e e
+o
+10
+20
+30
+Figure 9.5.4. (Continued)
+(c)
+(d)
+the torch is reduced significantly as the plume is swept downstream by the
+high-speed flow; nevertheless, the torch plume can still penetrate into the
+supersonic crossflow by more than I cm and also extends downstream
+about 1 cm, based on these emission images. A bow shock wave is also
+generated in front of the torch (since the torch acts as an obstruction to
+
+--- Page 599 ---
+584
+Current Applications of Atmospheric Pressure Air Plasmas
+the oncoming flow), as observed by the image presented in figure 9.5.4(b).
+This, of course, is typical behavior for a jet injected normally in a supersonic
+crossflow.
+We next study the torch operation in the supersonic flow using the high-
+power pulsed power supply. Shown in figure 9.5.4(c) is an airglow image of
+the torch plasma in the supersonic crossflow; the supply pressure was 2.7 atm.
+As shown in the figure, the (penetration) height of the torch is again reduced
+considerably by the wind tunnel crossflow. Comparing with that shown in
+figure 9.5.4(a), obtained in the case of higher gas supply pressure but lower
+power, the one shown in figure 9.5.4(c) extends about five times as far in
+the downstream direction and has a slightly larger penetration depth into
+the crossflow.
+Clearly, the increased discharge power produces larger volume plasma,
+which is evident in comparing figures 9.5.4(a) and 9.5.4(c). To increase torch
+penetration height in the wind tunnel, the air supply pressure was increased
+to 9.2 atm. The resulting schlieren image is shown in figure 9.5.4(d). An
+oblique shock wave is also generated in front of the torch as shown in this
+schlieren image. The voltage and current of the discharge as well as the
+shape and dimension of torch plasma vary with the torch flow rate and
+the crossflow condition. The results show that in addition to increasing the
+flow rate, one can increase the torch power to improve the penetration of
+the plasma into the crossflow.
+Initial evaluation of plasma-assisted ignition of hydrocarbon fuel was
+conducted in a supersonic, Mach-2 flow facility, at Wright-Patterson Air
+Force Research Laboratory, with heated air at a total temperature and
+pressure of 590 K and 5.4 atm, respectively. The resulting static temperature
+was thus ",330 K, still a relatively low value insofar as ignition is concerned.
+This facility allows testing of an individual concept with both gaseous and
+liquid hydrocarbon fuels without a cavity based flame-holder. In the tested
+configuration, a 15.2 cm x 30.5 cm test section floor plate fits into a simulated
+scram jet combustor duct with an initial duct height of 5.1 cm. At the
+upstream edge of the test section insert, the simulated combustor section
+diverges on the injector side by 2S. This particular hardware was intention-
+ally designed not to study main-duct combustion (ignition of the entire duct),
+but to reduce the chance of causing main-duct combustion by limiting the
+equivalence ratio of the tunnel below 0.1. In particular, this was accom-
+plished by placing the fuel injector at the centerline of the tunnel and not
+adding any flame-holding mechanisms such as a cavity or backwards-
+facing step. This approach allows the interactions of the fuel plume with
+the plasma torch to be studied by itself, and any flame produced is strictly
+created by this interaction, hence decoupling the ignition and flameholding
+problems as much as possible from the combustor geometry. Tests have
+been conducted using gaseous ethylene fuel, with the 15° downstream-
+angled single hole.
+
+--- Page 600 ---
+Plasma Torchfor Enhancing Hydrocarbon-Air Combustion
+585
+Figure 9.5.5. Flame plume ignited by 60 Hz torch plasma with fuel injected by a single-hole
+injector. (Copyright 2004 by IEEE.)
+The 60 Hz plasma torch module was evaluated and was found to
+produce a substantial flame plume as observed both from flame chemi-
+luminescence and OH planar laser-induced fluorescence [14]. The flame
+chemiluminescence (blue emission in the tail of the plume) is illustrated in
+figure 9.5.5, which shows a single frame taken from video recordings of a
+flame plume ignited by the 60 Hz plasma torch in operation 5 cm downstream
+of the ethylene-fueled single-hole injector. Several feedstock flowrates
+were tried over the torch module operational range and a flowrate of
+",500 SLPM was determined to produce the largest visible flame for the
+current electrode configuration. Air produced a larger flame when compared
+to nitrogen as the torch feedstock. This difference in flame size indicates that
+this type of flame is very sensitive to the local equivalence ratio and coupling
+of the ignition source with the mixture.
+Shown in figure 9.5.6 is a schematic of a conceptual Ajax vehicle and its
+engine. The engine is located at the bottom of the vehicle. Plasma torch
+modules are installed on the top wall of the box-shaped combustor right
+Power Demanding ...... Excess energy
+Payload
+-
+Power Conditioning
+Plasma Generation and...----
+Systems
+Control Systems
+Aerodynamic heat
+Masnetoplasmochemlcal engine
+~
+Thrust
+Figure 9.5.6. Schematic of a conceptual Ajax vehicle and the engine.
+
+--- Page 601 ---
+586
+Current Applications of Atmospheric Pressure Air Plasmas
+behind the fuel injectors to work as igniters. The torch modules can also be
+used as injectors to directly introduce ionizations and heat in the fuel for
+reducing ignition delay. It is worth pointing out that shock waves generated
+in front of torch plasma can help for holding flame and increasing its spread
+to achieve thorough combustion.
+References
+[I] Gruber M, Jackson, K Mathur T, Jackson T and Billig F 1998 'A cavity-based fuel
+injector/flameholder for scramjet applications' 35th JANNAF Airbreathing
+Propulsion Subcommittee and Combustion Subcommittee Meeting, Tucson, AZ,
+p 383
+[2] Mathur T, Streby G, Gruber M, Jackson K, Donbar J, Donaldson W, Jackson T,
+Smith C and Billig F 1999 'Supersonic combustion experiments with a cavity-
+based fuel injector' AIAA Paper 99-2102, American Institute of Aeronautics and
+Astronautics, Washington, DC, June 1999
+[3] Gruber M, Jackson K, Mathur T and Billig F 1999 'Experiments with a cavity-based
+fuel injector for scramjet application' ISABE Paper IS-7154
+[4] Mercier R A and Weber J W 1998 'Status of the US Air Force Hypersonic
+Technology Program' 35th JANNAF Airbreathing Propulsion subcommittee and
+Combustion Subcommittee Meeting, Tucson, AZ, p 17
+[5] Williams S, Bench P M, Midey A J, Arnold S T, Viggiano A A, Morris R A, Maurice
+L Q and Carter C D 2000 Detailed Ion Kinetic Mechanisms For Hydrocarbon/Air
+Combustion Chemistry, AFRL report 2000, Hanscom AFB, MA 01731-3010, pi
+[6] Wagner T, O'Brien W, Northam G and Eggers J 1989 'Plasma torch igniter for
+scramjets' J. Propulsion and Power 5(5)
+[7] Masuya G, Kudou K, Komuro T, Tani K, Kanda T, Wakamatsu Y, Chinzei N,
+Sayama M, Ohwaki K and Kimura I 1993 'Some governing parameters of
+plasma torch igniter/flameholder in a scramjet combustor' J. Propulsion and
+Power 9(2) 176-181
+[8] Jacobsen L S, Carter C D and Jackson T A 2003 'Toward plasma-assisted ignition in
+scramjets' AIAA Paper 2003--0871, American Institute of Aeronautics and
+Astronautics, Washington, DC
+[9] Kuo S P, Koretzky E and Orlick L 1999 'Design and electrical characteristics of a
+modular plasma torch' IEEE Trans. Plasma Sci. 27(3) 752
+[10] Kuo S P, Koretzky E and Orlick L 2001 Methods and Apparatus for Generating a
+Plasma Torch (United States Patent No. US 6329628 BI)
+[II] Parish J and Ganguly B 2004 'Absolute H atom density measurement in short pulse
+methane discharge' AIAA Paper 2004--0182, American Institute of Aeronautics and
+Astronautics, Washington, DC
+[12] Koretzky E and Kuo S P 1998 'Characterization of an atmospheric pressure plasma
+generated by a plasma torch array' Phys. Plasmas 5(10) 3774
+[13] Kuo S P, Bivolaru D, Carter C D, Jacobsen L S and Williams S 2003 'Operational
+Characteristics of a Plasma Torch in a Supersonic Cross Flow', AIAA Paper
+2003-1190, American Institute of Aeronautics and Astronautics, Washington, DC
+[14] Kuo S P, Bivolaru D, Carter C D, Jacobsen L S and Williams S 2004 'Operational
+characteristics of a periodic plasma torch', IEEE Trans. Plasma Sci., February issue
+
+--- Page 602 ---
+Plasma Mitigation of the Shock Waves
+587
+9.6 The Plasma Mitigation of the Shock Waves in
+Supersonic /Hypersonic Flights
+9.6.1
+Introduction
+A flying object agitates the background air; the produced disturbances
+propagate, through molecule collisions, at the speed of sound. When the
+object flight approaches the speed of sound (roughly 760mph in level
+flight), those disturbances deflected forward from the object move too
+slowly to get away from the object and form a sound barrier in front of
+the flying object. Ever since Chuck Yeager and his Bell X-I first broke the
+sound barrier in 1947, aircraft designers have dreamed of building a
+passenger airplane that is supersonic, fuel efficient and economical. However,
+the agitated flow disturbances by the flying object at supersonic/hypersonic
+speed coalesce into a shock appearing in front of the object. The shock
+wave appears in the form of a steep pressure gradient. It introduces a
+discontinuity in the flow properties at the shock front location, at the
+reachable edge of the flow perturbations made by the object. The back-
+ground pressure behind the shock front increases considerably, leading to
+significant enhancement of the flow drag and friction on the object.
+Shock waves have been a detriment to the development of supersonic/
+hypersonic aircraft, which have to overcome high wave drag and surface
+heating from the additional friction. The design of high-speed aircraft
+tends to choose slender shapes to reduce the drag and cooling requirements.
+While that profile is fine for fighter planes and missiles, it has long dampened
+dreams to build a wide-bodied airplane capable of carrying hundreds of
+people at speeds exceeding 760 mph. This is an engineering tradeoff between
+volumetric and fuel consumption efficiencies and this tradeoff significantly
+increases the operating cost of commercial supersonic aircraft. Moreover,
+shock wave produces a sonic boom on the ground. This occurs when flight
+conditions change, making the shock wave unstable. The faster the aircraft
+flies, the louder the boom. The noise issue raises environmental concerns,
+which have precluded for, example, the Concorde supersonic jetliner from
+flying overland at supersonic speeds.
+A physical spike [1] is currently used in the supersonic/hypersonic object
+to move the original bow shock upstream from the blunt-body nose location
+to its tip location in the new form of a conical oblique shock. It improves the
+body aspect ratio of a blunt-body and significantly reduces the wave drag.
+However, the additional frictional drag occurring on the spike structure
+and related cooling requirements limit the performance of a physical spike.
+Also another drawback of a physical spike is its sensitivity to off-design
+operation of the vehicle, i.e. flight Mach number and vehicle angle of
+attack. A failure regime at aspect ratios less than one also prohibits the
+practical uses of these physical spikes alone for shock wave modification.
+
+--- Page 603 ---
+588
+Current Applications of Atmospheric Pressure Air Plasmas
+Therefore, the development of new technologies for the attenuation or
+ideal elimination of shock wave formation around a supersonic/hypersonic
+vehicle has attracted considerable attention. The anticipated results of
+reduced fuel consumption and having smaller propulsion system require-
+ments, for the same cruise speed, will lead to the obvious commercial gains
+that include larger payloads at smaller take-off gross weights and broadband
+shock noise suppression during supersonic/hypersonic flight. These gains can
+make commercial supersonic flight a reality for the average traveler.
+9.6.2 Methods for flow control
+Considerable theoretical and experimental efforts have been devoted to the
+understanding of shock waves in supersonic/hypersonic flows. Various
+approaches to develop wave drag-reduction technologies have been explored
+since the beginning of high-speed aerodynamics. In the following, a few of
+these are discussed.
+Buseman [2] suggested that geometrical destructive interference of shock
+waves and expansion waves from two different bodies could work to reduce
+the wave drag. However, the interference approach is effective only for one
+Mach number and one angle of attack, which makes the design for practical
+implementation difficult.
+Using electromagnetic forces for the boundary layer flow control have
+been suggested as possible means to ease the negative effect of shock wave
+formation upon flight [3]. However, an ionized component in the flow has
+to be generated so that the fluid motion can be controlled by, for instance,
+an introduced j x B force density, where j and B are the applied current
+density and magnetic field in the flow.
+Thermal energy deposition in front of the flying body to perturb the
+incoming flow and shock wave formation has been studied numerically [4, 5].
+Heating of the supersonic incoming flow results in a local reduction of the
+Mach number. This in turn causes the shock front to move upstream and
+thus in this process the stronger bow shock is modified to a weaker oblique
+shock with significantly lower wave drag to the object and much less shock
+noise. Although this heating effect is an effectual means of reducing the
+wave drag and shock noise in supersonic and hypersonic flows, it requires a
+large power density to significantly elevate the gas temperature [5]. It is
+known that using the thermal effect to achieve drag reduction in supersonic
+and hypersonic flight does not, in general, lead to energy gain in the overall
+process. Thus this is not an efficient approach for drag reduction purposes,
+but it can be a relatively easy approach for sonic boom attenuation.
+Direct energy approaches have also been applied to explore the non-
+thermal/non-local effect on shock waves. Katzen and Kaattari [6] investi-
+gated aerodynamic effects arising from gas injection from the subsonic
+region of the shock layer around a blunt body in a hypersonic flow. In one
+
+--- Page 604 ---
+Plasma Mitigation of the Shock Waves
+589
+particular case, when helium was injected at supersonic speed, the injected
+flow penetrated the central area of the bow shock front, modifying the
+shock front in that area to a conical shape with the vertex much farther
+from the body (at about one body diameter). Laser pulses [7, 8] could
+easily deposit energy in front of a flying object. However, plasma generated
+at a focal point in front of the model had a bow radius much smaller than the
+size of the shock layer around the model, and its non-local effect on the flow
+was found to be insignificant.
+Plasma can effectively convert electrical energy to thermal energy for gas
+heating. Moreover, it has the potential to possibly offer a non-thermal
+modification effect on the structure of shock waves. The results from early
+and recent experiments conducted in shock tubes exhibited an increased
+velocity and dispersion on shock waves propagating in the glow discharge
+region [9, 10]. Measurements using laser beam photo deflection concluded
+that the dispersion and velocity increase of shock wave were attributed to
+the inhomogeneous plasma heating by the local electric field [11]. Plasma
+experiments were also conducted in wind tunnels. When plasma was gener-
+ated ahead of a model either by the off-board or on-board electrical discharge
+[12-15] or microwave pulses [16, 17] the experimental results showed that the
+shock front had increased dispersion in its structure as well as increased
+standoff distance from the model. One of the non-thermal plasma effects
+was evidenced by an experiment [18] investigating the relaxation time of
+the shock structure modification in decaying discharge plasma. The observed
+long-lasting effect on the shock structure was attributed to the existence of
+long-lived excited states of atoms and molecules in the gas.
+The study of the plasma effect on shock waves was further inspired by a
+wind tunnel experiment conducted by Gordeev et al [19]. High-pressure
+metal vapor (high Z) plasma, produced inside the chamber of a cone-
+cylinder model by exploding wire by electrical short circuit, is injected into
+the supersonic flow through a nozzle. A significant drag reduction was
+measured [19]. A brief history of the development in this subject area was
+reported in an article published in lane's Defence Weekly [20].
+The research in plasma mitigation of the shock waves has two primary
+goals:
+1. to improve the effective aerodynamic shape of an aircraft, but without the
+cooling requirements of a physical spike, and
+2. to reduce the shock noise and possibly make net energy savings.
+9.6.3 Plasma spikes for the mitigation of shock waves: experiments
+and results
+To further study plasma effects on shock waves, Kuo et al [21] have carried
+out experiments in a Mach-2.5 wind tunnel. A cone-shaped model having a
+
+--- Page 605 ---
+590
+Current Applications of Atmospheric Pressure Air Plasmas
+Figure 9.6.1. Plasma produced in front of the model, which is moving around the tip in
+spray-like forms. (Copyright 2000 by AlP.)
+60° cone angle was placed in the test section of the wind tunnel. The tip and
+the body of the model were designed as two electrodes with the tip of the
+model designated as the cathode for gaseous discharge. A 60 Hz power
+supply was used in the discharge for plasma generation. The peak and
+average powers of the discharge during the wind tunnel runs were measured
+to be about 1.2kW and lOOW, respectively. Shown in figure 9.6.1 is the
+airglow image of a spray-like plasma generated by the 60 Hz self-sustained
+diffused arc discharge, at the nose region of the model, where the usual
+attached conical shock is formed in the supersonic flow. The plasma density
+and temperature of the discharge were not measured. However, the electrode
+arrangement and the power supply were similar to those used in producing a
+60 Hz torch plasma, which was measured [22] to have peak electron density
+and temperature exceeding 1013 electrons/cm3 and 5000 K (time averaged
+temperature [23] is less than 2000 K), respectively. During the run, the back-
+ground pressure drops, thus the plasma density is expected to increase
+slightly. On the other hand, the electron plasma is cooled considerably by
+the supersonic flow. The produced spray-like plasma acted as a spatially
+distributed spike, which could deflect the incoming flow before the flow
+reached the original shock front location. The effect of this plasma spike
+on the shock wave formation was explored by examining a sequence of
+shadowgraphs taken during typical wind tunnel runs.
+The shadowgraph technique is briefly described as follows. A uniform
+collimated light beam is introduced to illuminate the flow. The second deriva-
+tive of the flow density deflects the light rays to a direction perpendicular to
+the light beam, which results in light intensity variation on a projection
+
+--- Page 606 ---
+Plasma Mitigation of the Shock Waves
+591
+(a)
+(c)
+(b)
+(d)
+Figure 9.6.2. A sequence of shadowgraphs taken during a wind tunnel run at Mach-2.S in
+the presence of plasma. (a) At the instant close to initiating plasma, (b) at a later time
+during the run, (c) at a later time during the same run, and (d) at the time when the
+discharge is around the peak and the shock wave is eliminated. (Copyright 2000 by AlP.)
+screen showing the shadow image of the flow field. Thus the location of a
+stationary shock front in the flow, where the second derivative of the
+density distribution is very large, is revealed in the shadowgraph as a dark
+curve because the light transmitted through that region is reduced to a
+mInImum.
+In the shadowgraphs shown in figure 9.6.2 the flow is from left to right.
+The upstream flow has a flow speed v = 570m/s, temperature T J = 135K,
+and a pressure PI = 0.175 atm. Figure 9.6.2(a) is a snapshot of the flow at
+the instant close to initiating the plasma. As shown, an undisturbed conical
+shock is formed in front of the plasma-producing model. To further examine
+the flow structure, a Pitot tube was installed in the tunnel, which can be seen
+on the top portion of the shadowgraph with its usual detached shock front.
+Figure 9 .6.2(b) taken at a later time during the run, on the other hand, clearly
+demonstrates the pronounced influence of plasma on the shock structure.
+Comparison of figures 9.6.2(a) and (b) clearly indicates an upstream
+displacement of the shock front along with a larger shock angle, indicating
+
+--- Page 607 ---
+592
+Current Applications of Atmospheric Pressure Air Plasmas
+a transformation of the shock from a well defined attached shock into a
+classic highly curved bow shock structure. It is also interesting to note that
+the shock in front of the Pitot probe, which is placed at a distance above
+the plasma-producing model, has been noticeably altered as is evident
+from the larger shock angle. A highly diffused detached shock front is
+observed in figure 9.6.2(c) taken at a later time during the same run. The
+diffused form of the shock front could be the result of less spatial coherency
+in the flow perturbations introduced by the spatially distributed plasma; it
+could also be ascribed to a visual effect from an asymmetric shock front
+caused by the non-uniformity of the generated plasma, a well-known
+integration effect inherent in the shadowgraph technique when visualizing
+a three-dimensional flow field. This phenomenon is commonly observed
+when the spatial extent of the region leading to the shock is small compared
+to the test section dimensions.
+Closer examination of figure 9.6.2(c) demonstrates a further upstream
+propagation of the bow shock, having an even more dispersed shape and a
+larger shock angle. It is also interesting to note that the shock wave in
+front of the Pitot probe has also moved upstream and some evidence of
+flow expansion may be seen near the tip of the probe. This is an interesting
+result indicating that the effect of plasma is not confined to the vicinity of
+the plasma-generating model but rather influences a large region of the
+flow field. As a final example, figure 9.6.2(d) demonstrates the effectiveness
+of the plasma in eliminating the shock near the model, an encouraging
+result, which may have significant consequences in the effectiveness of this
+scheme in minimizing wave drag and shock noise at supersonic speeds.
+In summary, the experimental results represented by the shadowgraphs
+(figures 9.6.2(b)-(d)) of the flowfield show that the spray-like plasma has
+strong effect on the structure of the shock wave. It causes the shock front
+to move upstream toward the plasma front and to become more and more
+dispersed in the process (figures 9.6.2(b) and (c)). A shock-free state (figure
+9.6.2(d)) is observed as the discharge is intensified.
+A follow up experiment by Bivolaru and Kuo [24] further demonstrated
+the plasma effect on shock wave mitigation. The experiment used a similar
+truncated cone model except that the nose of the model has a 9 mm
+protruding central spike, which also served as the discharge cathode. More-
+over, the power supply was a dc pulse discharge source using RC circuits for
+charging (Re = 10 kO) and discharging (Rd = 1500 to ballast the dischar-
+ging current) and a 5 kV/400mA dc power supply to charge the capacitor
+(C = 150IlF). It produced very energetic plasma with a low repetition rate.
+The peak power exceeded 40 kW and the energy in each discharge pulse
+was about 150J. Again, the plasma density and temperature were not
+measured during the runs. However, from the current measurement, the
+peak electron density is estimated to exceed 1014 electrons/cm3 . Without
+the spike, a detached curved shock would be generated in front of the
+
+--- Page 608 ---
+Plasma Mitigation of the Shock Waves
+593
+(a)
+Figure 9.6.3. (a) A baseline schlieren image of a Mach-2.5 flow over 60° truncated cone
+(pin hole knife-edge of 0.2mm in diameter); the aspect ratio of the spike length I to the
+spike diameter d, lid = 6, (b) video graph of the plasma airglow showing a cone-shaped
+plasma around the spike of the model; and (c) schlieren image of the flowfield modified
+by the cone-shaped plasma shown in (b). (Copyright 2002 by AlP.)
+truncated cone model. The added spike with the selected length modified the
+structure of the curved shock (which is the one intended to be modified by the
+plasma) only in the central region around the spike, where the shock front
+becomes conical and attached to the tip. This is seen in figure 9.6.3(a), a base-
+line schlieren image of the flow field around the spike and the nose of the
+cone; the flow is from left to right. The use of this design facilitates the
+discharge (starting at the base of the truncated cone model) to move
+upstream through the subsonic region of the boundary layer, along the
+spike/electrode surface, so that plasma can always be generated in the
+region upstream of the curved shock front (but it will appear behind the
+oblique part of the shock front as shown later).
+In the schlieren method, again, a uniform collimated light beam is
+introduced to illuminate the flow. In addition, an obstruction (i.e. a light
+ray selecting device) is introduced in the light path (e.g. a knife-edge placed
+at the focal point of the image-forming lens). It uniformly decreases the
+image illumination in the absence of any disturbance; however, when a
+density gradient exists in the flow, only some rays will pass the obstruction
+with a specific variation in the image illumination. The contrast of the
+image will be proportional to the density gradient in the flow. When rays
+
+--- Page 609 ---
+594
+Current Applications of Atmospheric Pressure Air Plasmas
+are deflected toward the knife-edge, the image field becomes darker (negative
+contrast) and vice-versa. The images can be recorded directly by a CCD
+camera, without going through an image projection screen. It is noted that
+if too many rays are stopped, the image quality will deteriorate. Therefore,
+the knife-edge must be adjusted with a compromise between image quality
+and contrast.
+Much more energetic plasma was generated by this pulsed dc discharge
+than that generated by 60 Hz discharge in the other experiment. This spike
+also guided the pulsed electrical discharge to move upstream such that
+plasma was easily generated in the region upstream of the curved shock
+front. As plasma was generated, it was found that the schlieren image of
+the flowfield became quite different from that shown in figure 9.6.3(a). The
+discharge was symmetric; it produced a cone-shaped plasma around the
+spike of the model, as shown by the video graph in figure 9.6.3(b).
+Comparing the corresponding schlieren image of the flowfield presented in
+figure 9.6.3(c), again the flow is from left to right, with the baseline schlieren
+image shown in figure 9.6.3(a), it is found that the original curved shock
+structure in front of the truncated cone is not there any more. The
+complicated shock structure in figure 9.6.3(a) is now modified to a simple
+one displaying a single attached conical (oblique) shock similar to the one
+generated by a perfect cone in the absence of plasma. In other words, it
+seems that plasma has reinstated the model to a perfect cone configuration.
+The wave drag to the model caused by oblique shock is much smaller than
+that caused by the original bow shock.
+This experiment has demonstrated that the performance of a small
+physical spike on the body aerodynamics can be greatly improved by
+generating plasma around it to form a plasma aero-spike, without increasing
+the cooling requirement to that for a large physical spike. A change of the
+shock wave pattern from bow shock dominated structure to oblique shock
+structure is equivalent to an effective increase in the body aspect ratio
+(fineness), from L/ D = 0.5 (blunt conical body) to L/ D = 0.85 (conical
+body), by 1.7 times (70%). Although the modification on the shock wave
+structure by this plasma aero-spike is characteristically different from that
+by a spread-shaped plasma that causes the shock front to have increased
+dispersion in its structure as well as standoff distance from the model, both
+are effective in the mitigation of shock waves. Moreover, it was found, in
+both experiments, that significant plasma effect on the shock wave was
+observed only when two criteria were met: (1) plasma is generated in the
+region upstream of the baseline shock front and (2) plasma has a symmetrical
+spatial distribution with respect to the axis of the model.
+Although experiments have clearly demonstrated that plasmas can
+significantly modify the shock structure and reduce the wave drag to the
+object, neither the physical mechanism nor a net energy saving from the
+drag reduction were confirmed. More experiments are needed to resolve
+
+--- Page 610 ---
+Plasma Mitigation of the Shock Waves
+595
+these issues. Some of the facts deduced from the experimental results,
+however, suggest that deflection of the incoming flow by a symmetrically
+distributed plasma spike in front of the shock may prove to be a useful
+process against shock formation.
+The effect of plasma aerodynamics on the shock wave observed in
+experiments may be understood physically. A shock wave is formed by
+coherent aggregation of flow perturbations from an object. In the steady
+state, a sharp shock front signified by a step pressure jump is formed to
+separate the flow into regions of distinct entropies. The shock wave angle
+(3 depends on the Mach number M and the deflection angle () of the flow
+through a ()-~M relation, where (3 increases with (). Since the shock front
+is at the far reachable edge of the flow perturbations deflected forward
+from an object, flow is unperturbed before reaching the shock front. In
+order to move the shock wave upstream, the flow perturbations have to
+move upstream beyond the original shock front. An easy way is to start
+the flow perturbation in front of the location of the original one by, for
+instance, introducing a longer physical spike. The added plasma spike
+serves the same purpose; it encounters the flow in the region upstream of
+the location of the original shock front. It increases the deflection angle ()
+of the incoming flow as well as the oblique angle (3 of the tip-attached
+shock. As the discharge is intensified, the induced flow perturbations from
+the plasma spike can be large enough to coalesce into a new shock front,
+which replaces the original one located behind it. This is also realized by
+the ()-(3-M relation. When the deflection angle of the flow exceeds the
+maximum deflection angle in the ()-(3-M relation, then the oblique shock
+in this region does not exist any more. Instead, the shock structure in this
+region becomes curved and detached (figure 9.6.2(c)). The deflection
+mechanism is also applicable for explaining the plasma effect shown in
+figure 9.6.3(c). As shown in figure 9.6.3(b), on-board generated plasma
+filled the truncated part of the model. It deflected the incoming flow as
+effectively as a perfect cone. Because much less flow could reach and be
+deflected by the frontal surface of the truncated cone, the original bow
+shock was replaced by an oblique shock attached to the tip of this 'virtually
+perfect cone'.
+The shock front is also expected to appear in a dispersed form because
+the effective plasma spike is distributed spatially and is not as rigid as the
+tip of the model or a physical spike. In other words, the flow perturbations
+by the plasma spike are less coherent as they coalesce into a shock and
+consequently form a weaker new shock.
+References
+[I] Chang P K 1970 Separation of Flow (Pergamon Press)
+[2] Buseman A 1935 'Atti del V Convegna "Volta'" Reale Accademia d'italia, Rome
+
+--- Page 611 ---
+596
+Current Applications of Atmospheric Pressure Air Plasmas
+[3] Kantrowitz A 1960 Flight Magnetohydrodynamics (Addison-Wesley) pp 221-232
+[4] Levin V A and Taranteva LV 1993 'Supersonic flow over cone with heat release in the
+neighborhood of the apex' Fluid Dynamics 28(2) 244-247
+[5] Riggins D, Nelson H F and Johnson E 1999 'Blunt-body wave drag reduction using
+focused energy deposition' AIAA J. 37(4)
+[6] Katzen E D and Kaattari G E 1965 'Inviscid hypersonic flow around blunt bodies'
+AIAA J. 3(7) 1230-1237
+[7] Myrabo L Nand Raizer Yu P 1994 'Laser induced air-spike for advanced trans-
+atmospheric vehicles' AIAA Paper 94-2451, 25th AIAA Plasmadynamics and
+Laser Conference, Colorado Springs, CO, June
+[8] Manucci MAS, Toro P G P, Chanes Jr J B, Ramos A G, Pereira A L, Nagamatsu
+H T and Myrabo L N 2000 'Experimental investigation of a laser-supported
+directed-energy air spike in hypersonic flow' 7th International Workshop on
+Shock Tube Technology, hosted by GASL, Inc., Port Jefferson, New York,
+September
+[9] Klimov A N, Koblov A N, Mishin G I, Serov Yu L, Khodataev K V and Yavov I P
+1982 'Shock wave propagation in a decaying plasma' Sov. Tech. Phys. Lett. 8
+240
+[10] Voinovich P A, Ershov A P, Ponomareva S E and Shibkov V M 1990 'Propagation of
+weak shock waves in plasma oflongitudinal flow discharge in air' High Temp. 29(3)
+468-475
+[11] Bletzinger P, Ganguly B Nand Garscadden A 2000 'Electric field and plasma
+emission responses in a low pressure positive column discharge exposed to a low
+Mach number shock wave' Phys. Plasmas 7(7) 4341-4346
+[12] Mishin G I, Serov Yu. Land Yavor I P 1991 Sov. Tech. Phys. Lett. 17413
+[13] Bedin A P and Mishin,G I 1995 Sov. Tech. Phys. Lett. 21 14
+[14] Serov Yu Land Yavor I P 1995 Sov. Tech. Phys. 40248
+[15] Kuo S P and Bivolaru D 2001 'Plasma effect on shock waves in a supersonic flow'
+Phys. Plasmas 8(7) 3258-3264
+[16] Beaulieu W, Brovkin V, Goldberg I et al 1998 'Microwave plasma influence on
+aerodynamic characteristics of body in airflow' in Proceedings of the 2nd
+Workshop on Weakly Ionized Gases, American Institute of Aeronautics and
+Astronautics, Washington, DC, p 193
+[17] Exton R J 1997 'On-board generation of a "precursor" microwave plasma at Mach 6:
+experiment design' in Proceedings of the 1st Workshop on Weakly Ionized Gases, vol
+2, pp EE3-12, Wright Lab. Aero Propulsion and Power Directorate, Wright-
+Patterson AFB, OH
+[18] Baryshnikov A S, Basargin I V, Dubinina E V and Fedotov D A 1997
+'Rearrangement of the shock wave structure in a decaying discharge plasma'
+Tech. Phys. Lett. 23(4) 259-260
+[19] Gordeev V P, Krasilnikov A V, Lagutin V I and Otmennikov V N 1996 'Plasma
+technology for reduction of flying vehicle drag' Fluid Dynamics 31(2) 313
+[20] 'Drag Factor' 1998 Jane's Defence Weekly (ISSN 0265-3818) 29(24) 23-26
+[21] Kuo S P, Kalkhoran I M, Bivolaru D and Orlick L 2000 'Observation of shock wave
+elimination by a plasma in a Mach 2.5 flow' Phys. Plasmas 7(5) 1345
+[22] Kuo S P, Bivolaru D and Orlick L 2003 'A magnetized torch module for plasma
+generation and plasma diagnostic with microwave', AIAA Paper 2003-135,
+American Institute of Aeronautics and Astronautics, Washington, DC
+
+--- Page 612 ---
+Surface Treatment
+597
+[23] Kuo S P, Koretzky E and Vidmar R J 1999 'Temperature measurement of an
+atmospheric-pressure plasma torch' Rev. Sci. Instruments 70(7) 3032-3034
+[24] Bivolaru D and Kuo S P 2002 'Observation of supersonic shock wave mitigation by a
+plasma aero-spike' Phys. Plasmas 9(2) 721-723
+9.7 Surface Treatment
+9.7.1
+Introduction
+Low-temperature non-equilibrium plasmas are effective tools for the surface
+treatment of various materials in micro-electronics, manufacturing and other
+industrial applications. The application of atmospheric pressure discharges
+presents advantages such as plasma treatment with cheap gas mixtures,
+low specific energy consumption and short processing time. Plasma pro-
+cedures in chemically reactive gases are easy to control and, as dry processes
+with low material insert, they are environmentally friendly.
+The interaction of plasmas with surfaces can be systematized according
+to the following definitions:
+1. Etching means the removal of bulk material. The process includes
+chemical reactions which produce volatile compounds containing atoms
+of the bulk material. Sputtering is a physical process which removes
+bulk atoms by collisions of energetic ions with the surface. Applications
+are, for example, structuring in micro-electronics and micro-mechanics.
+These processes are connected with a loss of a weighable amount of the
+bulk substance.
+2. Cleaning is the removal of material located on the surface which is not
+necessarily connected with the removal of bulk material. This process is
+applied, for example, in assembly lines as a preparation step for sub-
+sequent procedures.
+3. Functionalization leads to the formation of functional groups and/or of
+cross links on the surface by chemical reactions between gas-phase species
+and surface species/reactive sites and/or surface species (Chan 1994).
+Grafting is a surface reaction between gas phase and polymer material.
+The mass yield or loss in these processes is very small. Functionalization
+changes, but mostly improves the wettability, the adhesion, lamination to
+other films, the printability, and other coating applications. Biological
+properties may be influenced too, for example, the probability of settle-
+ments of cells or bacteria.
+4. Interstitial modifications occur, for example, by ion implantation for the
+hardening of metal surfaces.
+5. Deposition of films of non-substrate material change the mechanical
+(tribology), chemical (corrosion protection), and optical (reflecting and
+
+--- Page 613 ---
+598
+Current Applications of Atmospheric Pressure Air Plasmas
+Table 9.7.1. Plasma components and their efficiency in surface treatment (Meichsner
+2001).
+Plasma
+Kinetic
+component
+energy
+Ions, neutrals
+~lOeV
+Electrons
+5-10eV
+Reactive neutrals
+Thermal
+O.OSeV
+Photons
+>SeV (VUV)
+<5eV (UV)
+Processes and effects in the
+material
+Adsorbate sputtering, chemical
+reactions
+Inelastic collisions, surface
+dissociation, surface ionization
+Adsorption, chemical surface
+reactions, formation of functional
+groups, low molecular (volatile)
+products
+Diffusion and chemical reactions
+Photochemical processes
+Secondary processes
+Depth of
+interaction
+Monolayer
+~lnm
+Monolayer
+Bulk
+100SOnm
+11m range
+decorative) properties of materials. For films that are not too thin the
+mass yield is weighable. Systems of thin films with different electrical
+properties are the basic essentials of micro electronics.
+6. The depth scale of the different processes are as follows: etching 10-
+100nm, functionalization 1 nm, coating 1O-1000nm (Behnisch 1994).
+In reality these different processes are not strongly separated, e.g. cleaning may
+include sputtering or functionalization. The efficiency of the various plasma
+components in surface treatment is presented in table 9.7.1 (Meichsner 2001).
+The dielectric barrier discharge (DBD) seems to be the most promising
+plasma source for a plasma-assisted treatment of both large-area metallic
+and polymer surfaces at atmospheric pressure. Investigations of the homoge-
+neous DBD commonly known as 'atmospheric pressure glow discharge'
+(APGD) (Kogoma et al 1998), and of the filamentary or disperse DBD
+(Behnke 1996, Schmidt-Szalowski et a12000, Massines et a12000, Sonnenfeld
+2001b) proved the applicability ofDBD for surface treatment techniques.
+Special applications of DBD under atmospheric pressure exist in the
+modification of large-area surfaces for the purpose of the corrosion
+protection of metals and of an improvement of e.g. the wetting behavior of
+polymers.
+This modification of surfaces usually consists of three steps:
+1. the cleaning of the bulk material of hydrocarbon containing lubricants
+and other fatty contaminants,
+2. especially for metals, the deposition of a stable oxide layer of a thickness
+of some 10 nm as a diffusion barrier of the metallic bulk material, and
+
+--- Page 614 ---
+Surface Treatment
+599
+3. the deposition of a surface protecting thin layer (thickness of some
+hundreds of nm) with a good adhesive characteristics of a primer coating.
+The surface functionalization of polymers takes place after the cleaning
+procedure.
+The advantage of the surface treatment of metals by means of the DBD
+plasma consists in the fact that all three sub-processes can run off successively
+in the same plasma equipment (Behnke et al 2002).
+The effect of plasma treatment depends on the energy input into the
+process. For the energy flow on the mostly moving substrate, the dosage D
+is used (Softal Report 151 E Part 2/3)
+D = :v [~2]
+where P is the power introduced into the discharge [W], s is the electrode
+width [m], and v is the substrate velocity [m/s].
+The power density L in the discharge volume is given by
+P
+L=Ej=-
+[W/m3]
+Aa
+where E is the averaged voltage gradient inside the plasma [VIm], j is the
+current density [A/m2], A is the electrode surface [m2] , and a is the gap
+distance of the discharge [m].
+The power density 0 on the electrode surface is defined by
+O=~
+[W/m2].
+A
+D is an important parameter to achieve desired surface properties, L charac-
+terizes the plasma properties, 0 is a measure of the electrode strain. For a
+resting substrate the dosage is given by the product of 0 and the treatment
+time.
+This section is organized as follows: it first deals with experimental
+questions mainly oriented to the dielectric barrier discharge. The next part
+is devoted to cleaning by atmospheric pressure discharges. Then oxidation
+and functionalization are discussed, followed by plasma etching. The final
+topic deals with coating of substrates by deposition of a thin film. Closing
+remarks outline the advantages and limits of surface treatment by atmos-
+pheric pressure discharges in air.
+9.7.2 Experimental
+Here are presented special investigations with typical parameters which are
+used for surface cleaning, oxidation and thin film deposition (Behnke
+2002). The DBD apparatus consists of two dielectric high-voltage electrodes
+of rectangular cross section. The ceramic shell (AI20 3) of this hollow block is
+
+--- Page 615 ---
+600
+Current Applications of Atmospheric Pressure Air Plasmas
+gas flow
+:
+:
+U {O ... 20kV)'
+b
+PTFEblock
+.~.p .
+(gas flow and electrode support)
+banier profile
+substrate
+1,111'1 II :UI! .rrl;I}llllllllltI mill III [Ill II :Iilll !111, 11111; Ill: II r:IIIIIHIIIIIlIIllllldll] III 1:1
+movable substrate electrode
+c
+Figure 9.7.1. Scheme of the DBD equipment for surface treatment with a dynamic
+electrode arrangement.
+about 0.1 cm thick, 2cm wide and l5-50cm long, and coated inside with a
+silver layer for the electrical contacts.
+The DBD operates within the region between the electrodes and the
+substrate (grounded electrode) with a gap of 0.05-0.1 cm. The electrodes
+are moved periodically along the substrate by a step motor. The effective
+treatment time tp depends on the relative speed between the substrate and
+dielectric electrodes vs, the length b and the number n of the electrodes and
+the number of the moving periods p during the plasma process
+tp = pnb/vs• The slit between the rectangular profiles is used to introduce a
+laminar flow of the process gas mixture (air, vapors of silicon organic
+compounds as hexamethyldisiloxane (HMDSO, (CH3hSiOSi(CH3h) and
+tetraethoxysilane (TEOS, (CH3CH20)4Si)) into the discharge zone. To
+reduce excess heating the electrode system as well as the substrate holder
+are cooled by a flowing liquid. The DBD is driven by a sinusoidal voltage
+of some 10 kV in a continuous or pulsed mode of frequencies between 5
+and 50 kHz. For characterization of the experimental conditions the elec-
+trical power absorbed in the discharge is measured.
+A schematic view of the experimental set-up is given in figure 9.7.1. The
+typical operating conditions during plasma treatment are represented in table
+9.7.2. The cleaning and coating experiments are carried out with aluminum
+plates (80 mm x 150 mm) and Si wafers for ellipsometric measurements of
+the layer properties.
+For the investigation of the cleaning process the substrates were covered
+with defined quantities of oil (80-300 nm). For the deposition experiments
+the substrates are chemically pre-cleaned and cleaned in the DBD in air
+under atmospheric pressure with effective treatment times of about 100 s.
+
+--- Page 616 ---
+Surface Treatment
+601
+Table 9.7.2. Typical operation conditions during DBD-plasma treatment.
+Cleaning
+Oxidation
+Deposition
+Functionalization
+Frequency (kHz)
+10-25
+10-25
+6.6
+0.050-125
+Voltage (kV)
+<15
+<15
+<15
+3-50
+Power (yV)
+60-80
+80
+45
+Power density (W cm-2)
+2.2-3.0
+3
+1-1.6
+Volume power density
+20-60
+30-60
+10-30
+(yVcm-3)
+Dosage (Jjcm2)
+5-10
+5-10
+50-80
+1-300
+Discharge gap (mm)
+0.5-1.0
+0.5-1.0
+0.5-1.0
+1-5
+Process gas
+dry air
+dry air
+N2 or dry air
+Air
+Reactive gas
+0 220%
+0 220%
+TEOS 0.1 %
+HMDSOO.l%
+Gasfiow (slm)
+1.6
+1.6
+1
+1-10
+Effect. treatment time (s)
+<120
+<600
+<90
+10-100
+Mean residence time (s)
+0.06
+0.06
+0.1
+The time dependence of the oil removal and of the mass increase during the
+oxidation phase as well as the deposition of SiOxCyHz coatings are measured
+gravimetrically by weighing the samples with a micro-scale. The contami-
+nated and cleaned substrates are quasi in-situ characterized ellipsometrically
+by a spectroscopic polarization modulation ellipsometer (633 nm). The
+thickness of the deposited Si organic layer is also measured gravimetrically.
+The chemical composition of the substrate surface before and after
+plasma treatment is studied by x-ray photoelectron spectroscopy (XPS)
+and Fourier transform infrared (FTIR) spectroscopy. The surface morpho-
+logical properties are investigated by scanning electron microscopy (SEM)
+and contact angle measurements.
+9.7.3 Cleaning
+Metal surfaces are frequently covered with fats and oils in order to protect
+them temporarily against corrosion and to improve their manufacturing
+properties. For the following surface treatments this contamination must
+be removed by wet-chemical cleaning procedures or by vapor cleaning tech-
+niques using chlorinated and chloro-fluoro compounds. These processes are
+critically estimated to be environmentally undesirable. A plasma-assisted
+treatment operating at atmospheric pressure without greenhouse gases
+represents an environmentally friendly economical alternative. Since for
+such procedures no vacuum equipment is needed, they can be easily
+integrated in process lines (Klages 2002).
+Non-thermal atmospheric pressure air plasmas generate reactive oxygen
+atoms and ozone, which easily react with organic compounds and produce
+
+--- Page 617 ---
+602
+Current Applications of Atmospheric Pressure Air Plasmas
+40 ~~
+____ r-__ ~
+____ ~
+____ ~
+____ r-__ ~
+____ -r __
+~1BO
+D
+35
+30
+25
+2D
+15
+10
+.oaOD--.o~
+P /
+----0---...0
+/
+o;;;;owing
+-0- A
+,
+cilln Ilmple
+\ I
+y~""m;" ... "
+-A-Y
+~.A6A
+____ AA-__ ~6~ ___ 6a----6----A
+t[s]
+160
+140
+120
+100
+BD
+flO
+I>
+Figure 9.7.2. \IT and Ll during a whole cleaning process (633nm, PDBD = 80W) in
+dependence on the effective treatment time in seconds.
+volatile reaction products like CO, CO2 and H20. Air plasmas have been
+tested for surface cleaning, especially of contaminated metal.
+In order to understand the cleaning procedure in a DBD in air, the
+erosion of oil contamination on silicon surfaces was investigated by ellipso-
+metry and fluorescence microscopy (Behnke et aI1996a,b, Thyen et a12000,
+Behnke et al 2002).
+Figure 9.7.2 shows a typical plot of the ellipsometric angles \II and Do
+versus treatment time, which was monitored during the whole cleaning
+procedure (A = 633nm, DBD power 80W) of a contaminated Si wafer.
+The ellipsometric angles were measured before and after the oil contamina-
+tion (Wisura Akamin) (Behnke et al 2002).
+The angles \II (decreases) and Do (increases) change considerably during
+the surface treatment. In a short time they approach the values of pure
+silicon. That means the purification process runs very fast « 10 s). However,
+the initial values before the contamination are not reached, because the Si
+surface properties were changed by oxidation.
+More information about cleaning and the following oxidation process
+is elucidated by spectroscopic ellipsometrical investigations. The layer
+thickness d(t) and therefore the etching rate ret) are also evaluated from
+the ellipsometrical data of the wavelengths between 1.5 and 4.5 eV by
+means of the dispersion formula of Cauchy using model approximations.
+The contamination thickness and etching rate decrease nearly exponentially.
+
+--- Page 618 ---
+100
+80
+~
+60
+&:: tj
+:2 -
+40
+'0
+20
+0
+o
+Surface Treatment
+603
+•
+oil thickness
+0
+etching rate
+--model
+d(t) = do *exp(-tlt)
+r(t) = d d(t)/d t .. dJ't*exp(-tlt)
+do = 105 nm
+T = 2.64 5
+5
+treatment time [s]
+•
+10
+40
+30
+CD g:
+:j"
+20 CQ
+ii1 it
+'S'
+10~
+o
+Figure 9.7.3. Contamination thickness d and etching rate r versus treatment time for the
+discharge power of SOW. Substrate: Si wafer.
+The etching rate reaches values up to 40 nm s ~ 1• It decreases linearly with the
+contamination thickness. An example for the exponential decay of thickness
+and etching rate is given in figure 9.7.3. The following relations are valid:
+d(t) = doe~t/T
+r(t) = ladtl = dt
+at
+T
+d(t) =! = const
+r(t)
+T
+where T is a time constant which characterizes the cleaning process in
+dependence on the discharge power and of the initial contamination do.
+The same functional correlation is described by (Thyen et al 2000) for the
+cleaning of contaminated Si wafers. A similar exponential temporal behavior
+of the erosion of the contamination was determined from gravimetric
+measurements on aluminum substrates (Behnke et al 2002) as well as from
+fluorescence microscopic measurements on steel substrates (Thyen et al
+2000).
+In contrast to these results, cleaning investigations in rf oxygen low-
+pressure discharges show a linear reduction of the contamination thickness
+
+--- Page 619 ---
+604
+Current Applications of Atmospheric Pressure Air Plasmas
+and thus a constant etching rate during the entire plasma process. Hence it
+follows that in low-pressure discharges each sub-layer of the contamination
+is removed with a constant rate.
+One reason for the exponential behavior may be the statistical character
+of the cleaning process. A single filament removes nearly all the contami-
+nation from the sample within the relevant area. The temporal sequence of
+the filaments is statistically distributed on the substrate. That means that
+removed mass dm in the time interval dt is proportional to the mass m of
+the contamination.
+dt
+dm=-m-.
+T
+The second reason is the polymerization of the lubricant for higher initial
+thickness. That is clearly seen from the increase of the optical constants n
+and k of the layer which is related to higher layer density. Also Thyen et al
+(2000) explained the exponential decline of d(t) by initiation of polymeriza-
+tion reactions of the oil.
+An improved understanding will be achieved by studying the etching
+process in the remote plasma outside the DBD. There the contaminated
+metallic plate is not touched by filaments. Etching takes place only due to
+active species which are produced by the discharge. Under these conditions
+the etching rates are much lower and the process stops if the contamination
+reaches about 20% of the initial thickness. That means the filaments are
+essential for the cleaning process. Without filaments the polymerization of
+the lubricant becomes the most preferred mechanism. In case of small
+contamination thickness (l00-150nm) substrates can be completely cleaned
+using any tested values of power. The time constants for the removal of the
+contamination decrease approximately linearly with discharge power.
+Contamination above 6 g m -2 could not be removed by a barrier discharge.
+The cleaning rate r depends strongly on the oxygen content in the
+process gas. Thyen et al (2000) found that in pure nitrogen the rate is over
+ten times lower than in the air mixture. An admixture of 0.5% oxygen to
+the process gas raises the rate in relation to that in pure nitrogen by a
+factor around 3, but in pure oxygen this factor again decreases to 1.3. On
+the other hand the removal rate increases in dependence on the gas flow. A
+saturation is reached at a gas throughput of around 5 slm (Thyen et al
+2000). With increasing flow rate more dismantling products of the hydro-
+carbons in the exhaust gas stream are removed, because a higher flow
+counteracts a reassembly of these products on the surface. The saturation
+of the rate is achieved if the flux of broken hydrocarbon chains equals the
+products removed by the gas flow (Behnke 1996b). Concerning the chemical
+reactions of an air plasma with hydrocarbons the reader will be referred to
+the discussion of the plasma-functionalization of polypropylene as an
+example of hydrocarbons in section 9.7.5.
+
+--- Page 620 ---
+Surface Treatment
+605
+Becker and coworkers (Korfiatis et a12002, Moskwinski et a12002) have
+been using a non-thermal atmospheric-pressure plasma generated in a
+capillary plasma electrode configuration (Kunhardt 2000; see also chapter
+2 of this book) to clean Al surfaces contaminated with hydrocarbons.
+Efficient hydrocarbon removal of essentially 100% of the contaminants in
+this discharge type was reported for plasma exposure times of only a few
+seconds and contaminant films of up to 300 nm. Specifically, these
+researchers have studied the utility of a plasma-based cleaning process in
+removing oils and grease from Al surfaces both during manufacturing and
+prior to the use of the Al in a specific application.
+All these experimental investigations show that hydrocarbons can be
+removed completely from metallic substrates by using an atmospheric
+plasma in air. From the ellipsometric measurements on a silicon wafer it
+was found that the residual contamination is in the order of one atomic
+layer.
+One important parameter for the characterization of the surface
+cleanness is the specific surface energy, which is determined by means of
+contact angle measurements of several liquids (Owen plot). After the
+plasma cleaning procedure the total surface tension (67 mN/m) is very
+high. For further treatment procedures the time behavior of the surface
+tension is important. While the dispersive fraction does not change
+(27 mN/m) the polar fraction decreases exponentially in time (time constant:
+166 h). A high wettability of the cleaned surface remains stable for 24 h if the
+energy dosage of the DBD plasma process is between 50 and 100Jcm-2 •
+9.7.4 Oxidation
+Metallic substrates (e.g. AI, Si, eu) are usually covered with a native, mostly
+fragile oxide coating with a thickness of some nm during long storage in air.
+This layer must be conventionally chemically eliminated in order to treat the
+surface for corrosion protection. Afterwards the deposition of a stable
+thicker oxide coating follows (e.g. Al20 3 on aluminum surfaces) which is
+produced conventionally by a galvanic anodization. The plasma-supported
+treatment will also win extra relevance in the future because of the polluting
+disposal of galvanic baths.
+In the example given in figure 9.7.2 the values of the initial ellipsometric
+angles IT! and ~ of a silicon wafer without contamination cannot be reached
+completely after the air plasma cleaning in a DBD. Moreover ~ decreases
+again after reaching a maximum. The main reason for this is the oxide
+growth on the substrate. This result is also confirmed by the XPS measure-
+ments. The XPS spectra of an Al layer were measured before and after the
+plasma treatment. Before treatment the intensity of the Al 2p peak reaches
+20% of the oxide peak. After the treatment the oxide peak remarkably increases
+and the Al 2p peak almost disappears (figure 9.7.4). An increase of the oxide
+
+--- Page 621 ---
+606
+Current Applications of Atmospheric Pressure Air Plasmas
+1170
+1175
+1180
+1185
+1500
+before plasma
+treatment
+!'~\
+aluminium
+'3'
+/ \l·
+1000
+\/
+.!. 500
+aluminium --... lr
+... :
+lit
+oxyde
+..
+.
+F
+/
+......
+h-
+~ 0 f-.L."'_"'f,,-~_
+•• 1,,-J;4_· 'oIM
+........ "..-'-~ ....... I._
+......... _
+......... ---1---'
+.. _ ............
+~-'-fr_
+... L-t
+1500
+1000
+500
+o
+after plasma
+treatment
+1170
+1175
+1180
+~n(eV)
+1185
+Figure 9.7.4. XPS spectra of an aluminum layer deposited on a silicon wafer before and
+after the air plasma treatment.
+thickness from 3.2 to 8.6 nm is shown by angle resolved measurements. Figure
+9.7.5 shows the increase of the weight of an AI-substrate in dependence on the
+plasma treatment time (Behnke et aI2002). In both cases the thickness of the
+oxide increases approximately proportional to 0.
+Therefore oxide growth of the oxide is diffusion determined. Diffusion
+coefficients of about 2-7 x 10-16 cm2 S-1 are estimated. These are typical
+0.8
+~mox' •• t. = mo(t - to)o .•
+plasma treatment time t.,.,
+C>
+E 0.6
+-
+~
+I/)
+I/)
+m
+0.4
+E 0.2
+0.0
+2
+4
+6
+8
+10
+12
+time I min
+0
+t •• ,:84s
+• t.,., : 42 s
+6
+t.,., : 10 s
+14
+16
+18
+20
+Figure 9.7.5. Increase of the weight after treatment of an aluminum surface with a DBD
+(P = 80 W), parameter: plasma treatment time.
+
+--- Page 622 ---
+Surface Treatment
+607
+values for grain boundary diffusion (Wulff and Steffen 2001). The quality of
+this oxide depends on the treatment time. If the samples are treated con-
+tinuously for some minutes the oxide layer is rough. If the samples are treated
+intermittently only for some seconds with breaks, no roughness can be
+observed. For aluminum samples the thickness of the oxide reaches about
+10 nm after some minutes.
+The formation of an oxide layer (AI20 3, Si02) starts if the DBD is
+filamented. The high local energy input by the individual filaments leads to
+a restructuring of the natural oxide coating and to a local evaporation of
+the bulk material (AI, Si).
+The evaporated aluminum or silicon atoms are oxidized by the oxygen
+atoms inside the DBD plasma and deposited as oxide on the surface. The
+high current densities between 102 and 103 Acm-2 of an individual micro-
+discharge causes a compaction of the deposited oxide coating. The local
+evaporation of the bulk atoms is prevented by increasing oxide thickness
+and the layer growth is finished. The oxide coating in filamentary air
+discharge reaches a layer thickness of up to 10-20 nm. This process was
+monitored by the time-dependent measurement of the aluminum resonance
+line in a ferro-electrical barrier discharge. The relative line intensity
+decreased exponentially with the treatment time (Behnke et at 1996b). In
+summary it can be asserted that the DBD supported oxide coating is of a
+high quality. It has a high density with small roughness.
+9.7.5 Functionalization
+One important task of functionalization is the improvement of adhesion
+properties, e.g. for better printing and easier coating. Plastic foils, fibers
+and other polymer materials are mostly characterized by non-polar
+chemically inert surfaces with surface energies in the 20-40 mN/m range
+(polyamide 43.0 mN/m, polyethylene 31.0 mN/m, polytetrafluorethylene
+18.5mN/m). In general polymers are wetted by liquids when the surface
+energy of the polymer exceeds the surface energy of the liquid. The surface
+energy of common organic solvents is lower (toluene 28.4mN/m, carbon
+tetrachloride 27mN/m, ethanol 22.1 mN/m) than that of the polymers,
+therefore paint and inks based on organic solvents are successfully applied
+to polymers. Environmental requirements call for a replacement by water-
+based paints, inks, or bonding agents. Because of the high surface strength
+of water (72.1 mN/m) a treatment of polymer surfaces is necessary to
+improve their surface energy (Softal Report 102 E).
+On the one hand low surface energy impedes surface contamination and
+allows easy cleaning, but on the other hand it complicates printing, coating,
+sticking, etc. The surface properties are determined by a thin layer of
+molecular dimensions and can be changed without influencing the bulk
+properties of the polymer. Various processes have been developed for surface
+
+--- Page 623 ---
+608
+Current Applications of Atmospheric Pressure Air Plasmas
+treatment to enhance adhesion, such as mechanical treatment, wet-chemical
+treatments, exposure to flames, and plasma treatments in corona and glow
+discharge plasmas. What is meant by corona discharge is explained in
+chapter 6. In most cases the corona discharge for the polymer treatment is
+a dielectric barrier discharge because the non-conductive, dielectric plastic
+film inside the discharge gap is the barrier. Corona treatment is a well estab-
+lished method. High-capacity systems have been developed and offered by
+various manufacturers, and are applied to various synthetics. The principles
+of the action of an air plasma on a polymeric material will be exemplified by
+the case of polypropylene (PP). After this some characteristic examples for
+recent activities in surface functionalization will be presented.
+Dorai and Kushner (2002a,b, 2003) investigated in detail the processes
+associated with surface functionalization of an isotactic polypropylene film
+(0.05 mm thick) in an atmospheric pressure discharge in humid air. Industrial
+equipment (Pillar Technologies, Hartland, WI) was used for the corona
+treatment. The discharge is operated at a frequency of 9.6 kHz between a
+ceramic coated steel ground roll and stainless steel 'shoes' as the powered
+electrode, separated by a gap of 1.5 mm. The corona energy varied from
+0.1 to 17 W s/cm2 . The relative humidity of the air flow in the discharge
+region was either 2-5% or 95-100% at 25°C. The treated surface was
+analyzed to determine its chemical composition by ESCA, its surface
+energy by contact-angle measurements and its topology by AFM. Addition-
+ally the molecular weight of water-soluble low-molecular-weight oxidized
+material (LMWOM) was investigated. These materials can be separated by
+washing of the surface in polar solvents like water and alcohols.
+The untreated polypropylene surface is free of oxygen. The oxygen
+content grows with increasing discharge energy. A significant decrease of
+oxygen is observed after washing. A careful investigation of the LMWOM
+shows an averaged molecular weight of 400 amu. These oligomers originate
+from cleavage of the PP chain and contain oxidized groups such as COOH,
+CHO, or CH20H. The molecular weight is independent on the discharge
+energy and the humidity of air. Agglomerates of LMWOM are visible by
+AFM.
+The increase of the discharge energy is associated with a decrease as well
+as of the advancing and receding water contact angle, that means increasing
+wettability. The decrease of the advanced contact angle is much smaller for
+washed samples than for unwashed.
+For the treatment of PP in humid air plasma a model was developed
+(Dorai and Kushner 2003). It includes gas phase chemistry with the forma-
+tion of 0, H, OH radicals and 0 3 as important active species. Excited O2
+molecules, N atoms and H02 need not to be taken into account because of
+their lower reactivity towards PP. The reactivity of radicals with the PP is
+different for the position of the C atom where the reaction occurs. Primary
+C atoms are bound with only one C atom, secondary with two and tertiary
+
+--- Page 624 ---
+Surface Treatment
+609
+with three C atoms inside the polymer. The reaction probability is maximum
+for the primary C atoms, decreases for secondary and is minimum for tertiary
+C atoms. The surface reactions can be classified in analogy to polymerization
+processes in initiation, propagation, and termination.
+The initiation reaction is the abstraction of an H atom from the
+polypropylene surface by an 0 radical
+O(g) +
+H
+I
+- CH2 C-CH2 -
+I
+CH3
+or by an OH radical
+H
+I
+OH(g) + - CH2- y-CH2 -
+CH3
+-
+-CH-C-CH -
+2
+I
+2
++
+OH(g)
+CH3
+-CH-C-CH -
+2
+I
+2
+CH3
+associated with the generation of an alkyl radical.
+The propagation leads to peroxy radicals on the PP surface in a reaction
+of the alkyl radical with O2:
+O2 + -CH-C-CH -
+2
+I
+2
+CH3
+Alkoxy radicals are formed by the reaction of 0 atoms with the PP alkyl
+radicals:
+Also reaction with ozone results in alkoxy radical formation:
+o·
+I
+-CH-C-CH -
+2
+I
+2
+- CH2y-CH2 -
++
+02(g)
+CH3
+CH3
+The abstraction of a neighboring H atom of the PP surface by a peroxy
+radical produces hydroperoxide:
+O·
+0'
+I
+-CH-C-CH -
+2
+I
+2
+CH3
+H
+I
++ -CH-C-CH-
+2
+I
+2
+CH3
+O-H
+0'
+I
+-CH-C-CH - +
+2
+I
+2
+CH3
+The reaction of the alkyl radical with O2 may generate, as shown, new peroxy
+radicals.
+
+--- Page 625 ---
+610
+Current Applications of Atmospheric Pressure Air Plasmas
+A scission of the carbon chain occurs via alkoxy radicals and leads to the
+formation of ketones
+-{
+O·
+I
+- CH:zC-CH2-
+I
+CH3
+or aldehydes:
+H
+o· H
+I
+I
+I
+-CH-C-C-C-CH -
+-
+2
+I
+I
+I
+2
+CH3 H CH3
+-CH-C-CH -
+2 II
+2
+0
+/CH3
+-CH-C
+2
+~
+0
+H
+I
+-CH-C·
++
+2
+I
+CH3
++
++
+CH3
+• CH2-
+o H
+II
+I
+C-C-CH -
+I
+I
+2
+H CH3
+Alcohol groups are formed m reactions of alkoxy radicals with the
+polypropylene:
+O·
+I
+-CH-C-CH -
+2
+I
+2
+CH3
+H
+I
++ -CH-C-CH-
+2
+I
+2
+CH3
+OH
+I
+-CH-C-CH -
+2
+I
+2
+CH3
++ -CH2"y-CH2-
+CH3
+Alkoxy radicals are generated by reactions of 0 and OH radicals:
+OH
+I
+o{g) + - CH:zy-CH2-
+CH3
+OH
+I
+OH{g) + - CH:zy-CH2-
+CH3
+Termination reactions are
+H{g) +
+- CH:zC-CH2-
+I
+CH3
+OH{g) + - CH:zC-CH2-
+I
+CH3
+H I
+OH{g) +
+-CH-C-C=O
+2 I
+CH3
+0-
+I
+- CH:zy-CH2-
++ OH{g)
+CH3
+O·
+I
+- CH:zC-CH2-
+I
+CH3
+H I
+-CH2-C-CH2-
+I
+CH3
+OH
+I
+- CH:zC-CH2-
+I
+CH3
+H OH
+I
+I
+-
+-CH-C-C=O
+2 I
+CH3
+
+--- Page 626 ---
+Surface Treatment
+611
+The reactions with OH result in the formation of alcohols and acids,
+respectively.
+These reactions illustrate some possibilities of radical production by
+plasma reactions with a polypropylene surface. Reactions leading to cross
+linking of the polypropylene matrix must also be taken into account in a
+detailed description of the plasma-polymer interaction. The probabilities
+of surface reactions of ultraviolet radiation and ions are supposed to be
+small.
+The surface reaction processes together with the reaction probabilities
+or reaction rate coefficients are listed in table 9.7.3 (Dorai and Kushner
+2003). The calculated values for the percentage coverage of the polypropy-
+lene surface by alcohol (-C-OH), peroxy (-C-OO) and acid (-COOH)
+groups accord well with experimental results (O'Hare et at 2002). This
+successful approach indicates that in spite of the complexity the essential
+processes of this plasma-surface interaction were comprehensible.
+Table 9.7.3. Surface reaction mechanism for polypropylene (Dorai and Kushner 2003).
+Reaction"
+Probabilities or reaction rate
+coefficientsb
+Initiation
+Og + PP-H -
+PP* + OHg
+10-3, 10-4, 10-5
+OHg + PP-H -
+PP* + H20 g
+0.25, 0.05, 0.0025
+Propagation
+PP* + Og -
+PP-O*
+10-1, 10-2, 10-2
+pp* + 02,g -
+PP-OO*
+1.0 X 10-3, 2.3 X 10-4, 5.0 X 10-4
+PP* + 03.g -
+PP-O* + 02,g
+1.0, 0.5, 0.5
+PP-OO* + PP-H -
+PP-OOH + PP* 5.5 X 10- 16 cm2 S-1
+PP-O* -
+aldehydes + PP*
+10 S-1
+PP-O* -
+ketones + PP*
+500 S-1
+Og + PP=O -
+OHg + * PP=O
+0.04
+OHg + PP=O -
+H20 g +* PP=O
+0.4
+Og +* PP=O -
+CO2,g + PP-H
+0.4
+OHg +* PP=O -
+(OH)PP=O
+0.12
+PP-O* + PP-H -
+PP-OH + PP*
+8.0 X 10-14 cm2 S-1
+Og + PP-OH -
+PP-O + OHg
+7.5 x 10-4
+OHg + PP-OH -
+PP-O + H20 g
+9.2 X 10-3
+Termination
+Hg + pp* -
+PP-H
+0.2, 0.2, 0.2
+OHg + PP* -
+PP-OH
+0.2, 0.2, 0.2
+"Subscript g denotes gas phase species, PP-H denotes PP.
+b Those coefficients without units are reaction probabilities.
+CommentC
+C
+C
+C
+C
+C
+C
+C
+c C = reaction probabilities for tertiary, secondary, and primary radicals, respectively.
+
+--- Page 627 ---
+612
+Current Applications of Atmospheric Pressure Air Plasmas
+The atmospheric plasma surface treatment of polypropylene was a
+subject of various studies.
+A comparison of the action of a homogenous N 2 barrier discharge and a
+filamentary air discharge (Guimond et al 2002) shows that the maximum
+surface energy 'Y is higher in the first than in the second one (N2:
+'Y = 57 mN/m, E: 2.8 W s/cm2, air: 'Y = 39 mN/m, E: 0.6 W s/cm2), but
+requires a higher specific energy input E. A rapid decrease of the surface
+energy is observed during the first week of storage, but then the surface
+energy is fairly stable for more than three months (N2: 'Y = 49 mN/m,
+untreated film: 'Y = 27 mN/m).
+The action of homogenous and filamentary DBD in various gases,
+including air, on polypropylene was studied by (Mas sines et al 2001). Cui
+and Brown (2002) studied the chemical composition of a polypropylene
+surface during the air plasma treatment. Changes appear to terminate after
+about 25% of the surface carbon is oxidized. Oxidation produces polar
+groups like acetals, ketones and carboxyl groups which enhance the surface
+energy.
+A comparison of the treatment of several hydrocarbon polymers (poly-
+ethylene PE, polypropylene PP, polystyrene PS and polyisobutylene PIB)
+by air plasmas at atmospheric pressure of a silent or dielectric barrier
+discharge and at low pressure (0.2 torr) of an inductively coupled
+13.56 MHz discharge was presented by Greenwood et al (1995). The dielec-
+tric barrier discharge between two plane Al electrodes with a gap of 3 mm
+was driven by an operating voltage of 11 kV at 3 kHz. The samples on the
+lower grounded electrode were treated for 30 s and investigated by x-ray
+photoelectron spectroscopy and atomic force microscopy. Carbon singly
+bonded to oxygen was found to be the predominant oxidized carbon func-
+tionality for all polymers and discharges. The maximum amount of oxygen
+is incorporated into polystyrene with its 7r bonds. DBD modification
+increases the surface roughness of PP, PIB, and PS more than the low
+pressure discharge. For PE a smoothing is observed. Atmospheric pressure
+plasma treatment of polyethylene was studied also by Lynch et al (1998)
+and Akishev et al (2002). The latter compare the results with polypropylene
+and polyethylene terephthalate. The surface properties of polypropylene and
+tetrafluoroethylene perfluorovinyl ether copolymer were investigated after
+treatment in an atmospheric plasma pretreatment system with a discharge
+distance of up to 40 cm, which is suitable for a large plastic molding, e.g.
+an automobile bumper (Tsuchiya et al 1998). The increase of the water
+contact angle with storage time after plasma treatment is explained by a
+migration of oxygen from a very thin surface area into the inner layer.
+Polyimide is an interesting material in the electronics industry for
+flexible chip carriers. It is characterized by low costs, outstanding properties
+such as flame resistance, high upper working temperature (250-320 0q, high
+tensile strength (70-150 MPa), and high dielectric strength (22 kV /cm). The
+
+--- Page 628 ---
+Surface Treatment
+613
+application as a chip carrier demands a metallization with copper. The low
+surface energy must be enhanced to improve the adhesion of copper. The
+modification of po1yimide surface in a DBD in air is studied by Seeb6ck
+et al (2000, 2001) and Charbonnier et al (2001). The DBD operates at
+125 kHz between two plane copper or stainless steel electrodes which have
+diameters between 0.6 and 2 cm and are separated by a gap of 0.1 mm.
+There, the dielectric barrier is the polyimide film (thickness 50 or 38/lm).
+The dielectric barrier discharge with a specific energy input of 3 x 103 W sf
+cm2 leads to an increase of the surface roughness. For a polyimide foil
+filled with small alumina grains (to improve thermal conductivity) a rough-
+ness between 50 and 100 nm is measured. Microscopic inspection shows an
+increasing number of alumina grains visible at the surface as a consequence
+of the etching of the polymer. On the surface of the plasma-treated pure poly-
+imide foil, crater-like structures are observed. The DBD in air at atmospheric
+pressure is filamentary with ignition of the filaments at random spatial pos-
+itions. The crater formation is assumed as a consequence of repeated ignition
+of a filament at the same site. This surface roughness enables a metallization
+with good adhesion (SeebOck et al 2001). An obvious enhancement of the
+surface energy is observed after air plasma treatment. This is caused by the
+formation of oxygen containing polar groups at the polyimide surface
+(Seeb6ck et al 2000). XPS investigations demonstrate the increase of
+oxygen concentration at the surface and show the opening of the aromatic
+ring under the action of the plasma (Charbonnier et al 2001). This bond
+scission in the imide rings is an important step in the plasma surface reaction
+with aromatic polymers. For aliphatic polymers H atom abstraction is an
+essential reaction step, as has been discussed for polypropylene above.
+An example for air plasma treatment of a natural material refers to the
+felt-resistant finishing of wool. By means of an atmospheric pressure barrier
+discharge in air the content of carboxyl-, hydroxyl- and primary amino-
+groups on the wool surface is increased. The resulting improved adhesion
+to special resins enables a uniform and complete coating that leads to a
+felt-resistance comparable with the results of the environmentally polluting
+traditional procedures (VDI-TZ 2001, Rott et al 1999, Jansen et al 1999,
+Softal Report 152 E).
+Non-woven fabrics of synthetic material were successfully treated to
+increase the surface energy by an air plasma at atmospheric pressure (Roth
+et al 2001a). The treatment of metals was also reported. The removal of
+mono-layers of contaminants is supposed to be the dominant process of
+surface energy improvement (Roth et al2001 b).
+9.7.6 Etching
+Concerning the chemical processes, etching is closely related to cleaning,
+especially if the removal of hydrocarbons or similar materials is studied.
+
+--- Page 629 ---
+614
+Current Applications of Atmospheric Pressure Air Plasmas
+Here examples will be presented of the plasma etching of photo-resists
+supplemented by one example of plasma etching of Si-based materials and
+the decomposition of soot in the diesel engine exhaust.
+The etch rate of photo-resist on a silicon wafer in a He/02 mixture
+placed on the powered electrode is investigated in an atmospheric pressure
+dielectric barrier discharge (20-100 kHz, air gap 5-l5mm) (Lee et aI2001).
+Both electrodes are coated with 50 11m polyimide. The grounded electrode
+is additionally covered with a dielectric plate (thickness 8 mm) furnished
+with capillaries to induce glow discharges. For a He/02 mixture (2.5 or
+0.2 slm) 20.7 kHz, 10 mm air gap, and an aspect ratio of 10 average etch
+rates up to 200 nm/min were obtained. In front of the capillaries an etch
+rate >3 11m/min was observed.
+The photo-resist etching in a dielectric barrier discharge in pure oxygen
+is studied in dependence on the specific energy input (J/cm2 and J/cm3) with
+the result that the DBD at atmospheric pressure is an alternative to low-
+pressure plasma processing (Falkenstein and Coogan 1997).
+To overcome the difficulties in surface treatment of thick samples or
+samples with a complicated shape, spray-type reactors were developed
+(Tanaka et aI1999). In a reaction gas Ar/02 (100: 1) ashing rates of organic
+photo-resist of up to 111m/min were achieved.
+The application of a barrier discharge in air (5-7 kHz, 8.5-11 kV, gap
+width up to 1.5 cm) leads to etching rates of 270 nm/min (Roth et al
+2001b). The appearance of vertical etching structures under such conditions
+is observed.
+The remote and active plasma generated in a pulsed corona (400 Hz
+20 ns rise time, 30 kV) is tested for etching of a photo-resist coating on a
+silicon wafer (air plasma, remote, 9 nm/min) and the removal of organic
+films. Etching of the latter is more effective in the active plasma than
+under remote conditions (Yamamoto et aI1995).
+An increase of the etch rate of Si-based materials (Si02: 111m/min;
+SiN: 211m/min; poly Si: 211m/min) by more than one order of magnitude
+in relation to low-pressure plasma etching is observed in an atmospheric
+pressure of 40.68 MHz discharge in an 02/CF4 (up to 1: 1) mixture (Kataoka
+et aI2000).
+An interesting application of plasma etching in an air discharge
+concerns the soot decomposition in diesel engine exhaust (Muller et al
+2000). The reactor operates with a dielectric barrier discharge (lOkVpp ,
+'" 10 kHz, power on/power off: 3: 7, 1: 1, 3: 7) with an outer tube like
+porous SiC ceramics electrode (width of the honeycomb channel 5.6 mm)
+and an inner dielectric barrier electrode (4.2 mm diameter). The flue gas
+from the diesel engine flows across the discharge gap and is afterwards
+filtered by the porous outer electrode, leaving the soot particles on its surface.
+They were decomposed either in the continuous mode or by a regeneration
+procedure from time to time. More than 95% of the soot particles are
+
+--- Page 630 ---
+Surface Treatment
+615
+removed by the reactor and due to the soot decomposition on the surface a
+continuous gas flow is achieved across the reactor.
+9.7.7
+Deposition
+Investigations about plasma deposition with DBD have been performed on a
+broad variety of films in the past ten years. The spectrum ranges from coat-
+ings on plastic materials (e.g. polypropylene) for the improvement of the
+long-term behavior of the wetting ability (Meiners et at 1998, Massines et at
+2000) and hard carbon-based films (Klages et at 2003) up to layer systems for
+the corrosion protection on metal surfaces (Behnke et at 2002, 2003, 2004,
+Foest et at 2003, 2004). The kind of precursor used determines the function-
+ality of the deposited layer. The precursors hexamethyldisiloxane (HMDSO,
+(CH3)3SiOSi(CH3h) and tetraethoxysilane (TEOS, (CH3CH20)4Si) are
+frequently studied in atmospheric plasmas concerning their applicability
+for plasma-supported chemical vapor deposition of silicon-organic thin
+films (Sonnenfeld et at 2001 b, Schmidt-Szalowski et at 2000, Behnke et at
+2002, Klages et at 2003).
+The decomposition of HMDSO and TEOS in the plasma of DBD is
+controlled by electron impacts (Sonnenfeld et at 2001a,b, Basner et at
+2000). The electron impact induced scission of Si-CH3 and/or the Si-O
+bond of the HMDSO monomer is important for the layer deposition via
+this precursor. The cleavage of the Si-O bond is the main reaction path of
+the plasma chemical conversion of TEOS with the separation of
+CH3-CH2-O- radicals. In further reaction sequences ethanol and water
+are produced.
+The silicon-organic polymer film is mostly deposited from nitrogen or
+air DBD with an admixture of the silicon-organic precursor in the order
+of 0.1% (see table 9.7.2).
+The deposition occurs on the basis of small fragments of the silicon-
+organic precursor. These radicals are adsorbed on the substrate surface.
+For high energy dosage the gas phase reactions of the precursor and the inter-
+action of the plasma with the surface leads to highly cross-linked films. The
+films have good adhesion to the substrate surface, they are visually uniform,
+and transparent. The films are chemically resistant and protect the substrate
+against corrosive liquids (e.g. NaOH, NaCl, water). SEM images show that
+damages of the substrate surface «350 m) are uniformly covered by the
+films.
+The thickness of the deposited silicon-organic polymer films are esti-
+mated by gravimetric measurements under the assumption of a film mass
+density of 1 g cm -3, also by XPS, SEM and interferometric measurements.
+The average deposition rate strongly depends on the discharge power density
+and on the structure of the DBD plasma. One example is presented in
+figure 9.7.6. Up to the maximum of the deposition rate the DBD appears
+
+--- Page 631 ---
+616
+Current Applications of Atmospheric Pressure Air Plasmas
+Power density, W/cm2
+340
+1.0
+1.2
+1.4
+1.6
+1.8
+3.6
+320 L~
+• i3.4 ~
+300
+13.2 ~
+~ 280
+iii
+•
+3.0 ~
+:II 260
+7~'
+1""
+c:
+~ 240
+film strudure
+2.6~
+:E
+0
+I- 220
+2.4 5r
+200
+-.- deposition for 92 sec.
+'2.2°
+•
+at speed of O.037cmlsec
+~2.0
+180
+25
+30
+35
+40
+45
+50
+Power, W
+Figure 9.7.6. Thickness and deposition rate ofSiO, polymer films versus discharge power,
+effective deposition time 92 s, N2 DBD with admixture of 0.1 % TEOS.
+quasi-homogeneous: in this discharge range the deposition is quasi-
+homogeneously dispersed across the substrate. As long as the discharge
+changes to the mode of stronger filamentation with higher power densities
+the deposition rate describes a minimum. The film morphology alters to a
+stripe-shaped structure on the substrate, possibly due to some turbulent
+convection processes connected with the non-homogeneity of the discharge.
+FTIR measurements were carried out on the substrate after plasma
+treatment. Figure 9.7.7 shows spectra of films produced by an air plasma,
+S
+I::
+~
+'E
+II)
+I:: g
+!E
+C
+1,1
+1,0
+0,9
+0.8
+0,7
+0,6
+----.. --.-.. --............. ---•...... -..... ~-----............ -.... .
+.... .. _.' ........... _4 ... --_ ...... -....... - ........... -....... ... _...
+.
+.. ~ .... , : ..........
+...... 0.1 % TEeS air
+---'0.1 %TEeSAr
+--0.1 % TEeS N2
+,
+I
+~~¢
+G"
+Q ii5 .-
+0,5 .h--~..;~~;;;:;:::::::::;::==~~cn~-~cnu
+3500
+3000
+2500
+2000
+1500
+1000
+wave number I cm'1
+Figure 9.7.7. FTIR spectra of SiO, polymer films deposited in air, N2 and Ar DBD with
+precursor admixture of 0.1 % TEOS (P = 50 W).
+
+--- Page 632 ---
+Surface Treatment
+617
+a N2 plasma and an Ar plasma with 0.1 % TEOS admixture. The spectra are
+dominated by a broad peak in the region of 1000-1250cm-1 which denotes a
+macromolecular structure of the form (Si-Ox)n- The feature is more
+broadened for the films produced with air- and Nz-containing plasmas,
+indicating a slightly enhanced cross linking as compared to the Ar-based
+film. The specific energy per precursor molecule is comparable in all three
+cases, hence the effect is presumably caused by increased oxidation of the
+film.
+The (Si-OJn structure is overlapped by the prominent SiOx peak at
+l240cm- l . Both features along with the very low carbon content (e.g. CH3
+at 2950 cm -I) reveal the pronounced inorganic chemical nature of the film
+indicative for rather high specific energies per precursor molecule. With
+increasing specific energy the inorganic character of the film increases-a
+common effect proven for several silicon-organic precursors, such as
+HMDSO (Behnke et at 2002).
+Different technical test procedures for the estimation of the adhesion of
+a primer on the polymer layer and the determination of the corrosion protec-
+tion properties of the coating show a sufficient effect only for substrate
+temperatures above 40°C. With the dissociation of TEOS in the DBD,
+ethanol and water are formed, which are linked into the layer without a
+chemical bonding. The stoichiometric relationship of SiOx (x >::;j 2) thereby
+is never reached and the layer does not become leak-proof. The water
+stored in the layer withdraws with time and the residual hole-like laminated
+structure decreases both the adhesion and the anti-corrosion properties of
+the coating. The water entering the layer is avoided if the coating process
+is performed at higher substrate temperatures. Layers, which are deposited
+in a filamentary air DBD plasma, show a better adhesion and corrosion
+protection effect in contrast to those which are coated by quasi-homogeneous
+nitrogen DBDs. There will be also an improvement of these layer character-
+istics, if the layer is deposited only by a one-cycle procedure as a 'mono' layer
+in relation to a deposition in a multi-cycles procedure (Behnke et at 2003).
+9.7.8 Conclusions
+Atmospheric pressure plasmas are successfully implemented for various
+surface treatment tasks. When comparing atmospheric-pressure plasma
+processing with the well established low-pressure plasma processes, one
+has to consider that the latter methods have been continuously developed
+for more than 50 years. In contrast, the study of plasma processing at
+atmospheric pressure on a broader scale has just begun.
+The main advantage of atmospheric pressure plasma processing is that it
+requires much lower investment costs, because no vacuum devices are
+needed-in the case of ambient air, not even a housing. Hence, the implemen-
+tation of devices into assembly lines with renouncement of batch procedures
+
+--- Page 633 ---
+618
+Current Applications of Atmospheric Pressure Air Plasmas
+is greatly facilitated. The majority of atmospheric plasmas such as DBD and
+corona discharges are easily scaled up.
+The low level of maturity is one of the disadvantages of atmospheric
+pressure plasma processing in our day. Tailored plasma diagnostic tech-
+niques have to be developed for an effective process control.
+The state of the art atmospheric pressure plasma technology is holding
+promising prospects from the economical and environmental point of view.
+Therefore it is encouraging further research and development activities.
+Acknowledgments
+The financial support of our activities in the field of atmospheric pressure
+discharges by the BMBF of Germany Project no. 13N7350/0 and
+l3N7351/0 is gratefully acknowledged.
+References
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+thermal plasma sources for cold surface treatment of polymer films and fabrics at
+atmospheric pressure' Plasma and Polymers 7 261-289
+Basner R, Schmidt M, Becker K and Deutsch H 2000 'Electron impact ionization of
+organic silicon compounds' Adv. Atomic, Molecular and Optical Phys. 43 147-185
+Behnisch J 1994 'P1asmachemische Modifizierung von Cellulose-Moglichkeiten und
+Grenzen', Das Papier no. 12780--783
+Behnke J F, Lange H, Michel P, Opalinski T, Steffen H and Wagner H-E 1996b 'The
+cleaning process of metallic surfaces in barrier discharges' Proc. 5th Int. Symp.
+on High Pressure Low Temperature Plasma Chemistry (HAKONE V) Janca J
+et al (eds) Milovy/Czech Rep. pp 138-142
+Behnke J F, Sonnenfeld A, Ivanova 0, Hippler R, To R T X H, Pham G V, Vu K 0 and
+Nguyen T D 2003 'Study of corrosion protection of aluminium by siliconoxide-
+polymer coatings deposited by a dielectric barrier discharge under atmospheric
+pressure' 56th Gaseous Electronics Conference, 21-24 October 2003, San Francisco,
+CA. Poster GTP.015 http://www.aps.org/meet/GEC03/baps/abs/Sll0015.html
+Behnke J F, Sonnenfeld A, Ivanova 0, To T X H, Pham G V, Vu K 0, Nguyen T D, Foest R,
+Schmidt M and Hippler R 2004 'Study of corrosion protection of aluminium by sili-
+conoxide-polymer coatings deposited by a dielectric barrier discharge at atmospheric
+pressure' Proc. 9th Int. Symp. on High Pressure Low Temperature Plasma Chemistry
+(HAKONE IX) M. Rea et al (eds) 23-26 August 2004, Padova (Italy) in print
+Behnke J F, Steffen H and Lange H 1996a 'Elipsometric investigations during plasma
+cleaning: Comparison between low pressure rf-plasma and barrier discharge at
+atmospheric pressure' Proc. 5th Int. Symp. on High Pressure Low Temperature
+Plasma Chemistry (HAKONE V) Janca J et al (eds) Milovy/Czech Rep. pp 133-137
+Behnke J F, Steffen H, Sonnenfeld A, Foest R, Lebedev V and Hippler R 2002 'Surface
+modification of aluminium by dielectric barrier discharges under atmospheric pres-
+sure' Proc. 8th Int. Symp. on High Pressure Low Temperature Plasma Chemistry
+(HAKONE VIII) Haljaste A and Planck T (eds), Tartu/Estonia 2 410
+
+--- Page 634 ---
+Surface Treatment
+619
+Chan C-M 1994 Polymer Surface Modification and Characterization (Munich: Carl
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+Charbonnier M, Romand M, Esrom Hand Seebock R 2001 'Functionalization of polymer
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+Cui N-Y and Brown N M D 2002 'Modification of the surface properties of a propylene
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+Dorai R and Kushner M 2002b 'Plasma surface modification of polymers using atmos-
+pheric pressure discharges' 29th ICOPS Banff, Canada
+Dorai R and Kushner M 2003 'A model for plasma modification of polypropylene using
+atmospheric pressure discharges' J. Phys. D: Appl. Phys. 36 666-685
+Falkenstein Z and Coogan J J 1997 'Photoresist etching with dielectric barrier discharges in
+oxygen' J. Appl. Phys. 82 6273-6280
+Foest R, Adler F, Sigeneger F and Schmidt M 2003 'Study of an atmospheric pressure
+glow discharge (APG) for thin film deposition' Surf Coat. Technol. 163/164 323-
+330
+Foest R, Schmidt M and Behnke J 2004 'Plasma polymerization in an atmospheric
+pressure dielectric barrier discharge in a flowing gas' in Gaseous Dielectrics vol X,
+ed. Christophorou L G (New York: Kluwer Academic/Plenum Publisher) in print
+Greenwood 0 D, Boyd R D, Hopkins J and Badyal J P S 1995 'Atmospheric silent
+discharge versus low pressure plasma treatment of polyethylene, polypropylene,
+polyisobutylene, and polystyrene' J. Adhesion Sci. Technol. 9 311-326
+Guimond S, Radu I, Czeremuszkin G, Carlsson D J and Wertheimer M R 2002 'Biaxially
+orientated polypropylene (BOPP) surface modification by nitrogen atmospheric
+pressure glow discharge (APGD) and by air corona' Plasma and Polymers 7 71-88
+Jansen B, Kummeler F, Muller H B and Thomas H 1999 'EinfluB der Plasma- und
+Harzbehandlung auf die Eigenschaften der Wolle' Proc. Workshop Plasmaanwen-
+dungen in der Textilindustrie Stuttgart, Germany, 17-23
+Kataoka Y, Kanoh M, Makino N, Suzuki K, Saitoh S, Miyajima Hand Mori Y 2000 'Dry
+etching characteristics of Si-based materials used CF4/02 atmospheric-pressure
+glow discharge plasmas' Jpn. J. Appl. Phys. 39 294-298
+Kersten H, Behnke J F and Eggs C 1994 'Investigations on plasma-assisted surface
+cleaning of aluminium in an oxygen glow-discharge' Contr. Plasma Phys. 34 563
+Klages C P and Eichler M 2002 'Coating and cleaning of surfaces with atmospheric
+pressure plasmas' (in German) Vakuum in Forschung und Praxis 14149-155
+Klages C P, Eichler M and Thyen R 2003 'Atmospheric pressure PA-CVD of silicon- and
+carbon-based coatings using dielectric barrier discharges' New Diamond Front C
+Tee 13175-189
+Kogoma M, Okazaki S, Tanaka K and Inomata T 1998 'Surface treatment of powder in
+atmospheric pressure glow plasma using ultra-sonic dispersal technique' Proc. 6th
+Int. Symp. on High Pressure Low Temperature Plasma Chemistry (HAKONE VI),
+Cork, Ireland, 83-87
+Korfiatis G, Moskwinski L, Abramzon N, Becker K, Christodoulatos C, Kunhardt E,
+Crowe Rand Wieserman L 2002 'Investigation of Al surface cleaning using a
+novel capillary non-thermal ambient-pressure plasma' in Atomic and Surface
+Processes eds Scheier P and Mark T D, University of Innsbruck Press (2002)
+
+--- Page 635 ---
+620
+Current Applications of Atmospheric Pressure Air Plasmas
+Kunhardt E E 2000 'Generation of large-volume atmospheric-pressure, non-equilibrium
+plasmas' IEEE Trans. Plasma Sci. 28 189-200
+Lee Y-H, Yi C-H, Chung M-J and Yeom G-Y 2001 'Characteristics of He/02 atmospheric
+pressure glow discharge and its dry etching properties of organic materials' Surface
+and Coatings Technology 146/147 474-479
+Lynch J B, Spence P D, Baker D E and Postlethwaite T A 1999 'Atmospheric pressure
+plasma treatment of polyethylene via a pulse dielectric barrier discharge: Com-
+parison using various gas composition versus corona discharge in air' J. Appl.
+Polym. Sci. 71319-331
+Massines F, Gherardi N and Sommer F 2000 'Silane based coatings on propylene. Depos-
+ited by atmospheric pressure glow discharge' Plasmas and Polymers 5151-172
+Massines F, Gouda G, Gherardi N, Duran M and Croquesel E 2001 'The role of dielectric
+barrier discharge atmosphere and physics on polypropylene surface treatment'
+Plasma and Polymers 6 35-49
+Meichsner J 2001 'Low-temperature plasmas for polymer surface modification' in Low
+Temperature Plasma Physics Hippler R, Pfau S, Schmidt M and Schonbach K
+(eds) (Berlin: Wiley-VCH) 453-472
+Meiners S, Salge J G H, Prinz E and Foerster F 1998 'Surface modifications of polymer
+materials by transient gas discharges at atmospheric pressure' Surf Coat. Technol.
+98 1112-1127
+Moskwinski L, Ricatto P J, Babko-Malyi S, Crowe R, Abramzon N, Christodoulatos C
+and Becker K 2002 'AI surface cleaning using a novel capillary plasma electrode
+discharge' GEC 2002, Minneapolis, MN (USA), Bull. APS 47(7) 67
+Muller S, Conrads J and Best W 2000 'Reactor for decomposing soot and other harmful
+substances contained in flue gas' International Symposium on High Pressure
+Low Temperature Plasma Chemistry, (Hakone VII), Greifswald, Germany,
+Contr. Papers 2 340-344
+O'Hare L A, Leadley Sand Parbhoo B 2002 'Surface physicochemistry of corona-
+discharge-treated polypropylene film' Surface and Interface Analysis 33335-342
+Roth J R, Chen Z, Sherman D M, Karakaya F, Tsai P P-Y, Kelly-Wintenberg K and Montie
+T C 200la 'Increasing the surface energy and sterilization of nonwoven fabrics by
+exposure to a one atmosphere uniform glow discharge plasma (OAUGDP), Int.
+Nonwoven J. 1034-47
+Roth J R, Chen Z Y and Tsai P P-Y 200 I b 'Treatment of metals, polymer films, and fabrics
+with a one atmosphere uniform glow discharge plasma (OAUGDP) for increased
+surface energy and directional etching' Acta Metallurgica Sinica (English Letters)
+14391-407
+Rott U, Muller-Reich C, Prinz E, Salge J, WolfM and Zahn R-J 1999 'Plasmagestutzte
+Antifilzausrustung von Wolle-Auf der Suche nach einer umweltfreundlichen'
+Alternative Proc. Workshop Plasmaanwendungen in der Textilindustrie Stuttgart,
+Germany, 7-16
+Schmidt-Szalowski K, Rzanek-Boroch Z, Sentek J, Rymuza Z, Kusznierewicz Z and
+Misiak M 2000 'Thin film deposition from hexamethyldisiloxane and hexamethyl-
+disilazane under dielectric barrier discharge (DB D) conditions' Plasmas and
+Polymers 5 173
+Seebock R, Esrom H, Char bonnier M and Romand M 2000 'Modification of polyimide in
+barrier discharge air-plasma: Chemical and morphological effects' Plasma and
+Polymers 5 103-118
+
+--- Page 636 ---
+Chemical Decontamination
+621
+Seebi:ick R, Esrom H, Charbonnier M, Romand M and Kogelschatz U 2001 'Modification
+of polyimide using dielectric barrier discharge treatment' Surf Coating Technol.
+142/144455-459
+Softal Report 102 E 'Corona pretreatment to obtain wettability and adhesion' Softal
+Electronic GmbH, D21107 Hamburg, Germany
+Softal Report 151 E Part 2/3 'New trends in corona technology for stable adhesion' Softal
+Electronic GmbH, D21107 Hamburg, Germany
+Softal Report 152 E Part 3/3 'New trends in corona technology for stable adhesion' Softal
+Electronic GmbH, D21107 Hamburg, Germany
+Sonnenfeld A, Kozlov KV and Behnke J F 2001a 'Influence of noble gas on the reaction of
+plasma chemical decomposition of silicon organic compounds in the dielectric
+barrier discharge' Proc. 15th Int. Symp. on Plasma Chern. Contr. Orleans/France
+9-13 July 2001 Bouchoule A et al (eds) vol 5, pp 1829-1834
+Sonnenfeld A, Tun T M, Zajickova L, Wagner H-E, Behnke J F and Hippler R 2001 The
+deposition process based on silicon organic compounds in two different types of an
+atmospheric barrier discharge' in Proc.15th Int. Symp. on Plasma Chern. Contr.
+Orleans/France 9-13 July 2001, Bouchoule A et al (eds) vol 5, pp 1835-1840
+Sonnenfeld A, Tun T M, Zajickova M, Kozlov K V, Wagner H E, Behnke J F and Hippler
+R 2001b 'Deposition process based organosilicon precursors in dielectric barrier
+discharges at atmospheric pressure' Plasma and Polymers 6 237
+Steffen H, SchwarzJ, Kersten H, Behnke J F and Eggs C 1996 'Process control ofrfplasma
+assisted surface cleaning' Thin Solid Films 283 158
+Tanaka K, Inomata T and Kogoma M 1999 'Ashing of organic compounds with spray-
+type plasma reactor at atmospheric pressure' Plasma and Polymers 4 269-281
+Thyen R, Hi:ipfner K, Kliike N, and Klages C-P 2000 'Cleaning of silicon and steel surfaces
+using dielectric barrier discharges' Plasma and Polymers 5 91-102
+Tsuchiya Y, Akutu K and Iwata A 1998 'Surface modification of polymeric materials by
+atmospheric plasma treatment' Progress in Organic Coatings 34 100-107
+VDI-TZ Physikalische Technologien, Dusseldorf, Germany (Ed.) 2001 Plasmagestutzte
+Filzausrustung von Wolle Info. Phys. Tech. No. 32
+Wulff H and Steffen H 2001 'Characterization of thin solid films' in Low Temperature
+Plasma Physics Hippler R, Pfau S, Schmidt M and Schoenbach K H (eds)
+(Wiley-VCH)
+Yamamoto T, Newsome J R and Ensor D S 1995 'Modification of surface energy, dry
+etching, and organic film removal using atmospheric-pressure pulsed-corona
+plasma' IEEE Transactions Ind. Applications 31 494-495
+9.8 Chemical Decontamination
+9.8.1 Introduction
+NOx gases are emitted from coal burning electric power plant, boilers in
+factories, co-generation system and diesel vehicles. Some liquids and gases
+such as trichloroethylene, acetone and fluorocarbon are useful for clean-up
+of materials used in the semiconductor industry, for refrigerants, and so
+
+--- Page 637 ---
+622
+Current Applications of Atmospheric Pressure Air Plasmas
+on. However, recently, it has been noticed that these are harmful to human
+health. These must be processed for global environmental problems.
+Concerning NOx processing, selective catalytic reductions (SCRs)
+have been used. Soot and S02 exhausted from diesel engines prevent the
+conventional SCR from removing NOr Non-thermal plasmas (NTP) are
+attractive for decomposing these gases because the majority of the electrical
+energy goes into the production of energetic electrons with kinetic energies
+much higher than those of the ions or molecules. Energetic electron impact
+brings about the decomposition of the harmful gases or induced radicals
+facilitate the decompositions.
+In this section, removal of the harmful gases by NTPs is discussed. In
+sections 9.8.2-9.8.4, mainly de-NOx processes and kinetics, instrumentation
+and influencing parameters for de-NOy will be treated. In section 9.8.5,
+processing of environmentally harmful gases such as halogen gases, hydro-
+carbons, and chlorofluorocarbon removed by NTPs will be presented.
+9.8.2 de-NO x process
+Decomposition of NOx to their molecular elements (N2 and O2) is the most
+attractive method. However, it is seen that the major mechanism of NOx
+removal is oxidation to convert NO into N02 as shown in figure 9.8.1 for
+NO/N2/02 without water vapor. First, N2 and O2 collide with energetic
+electrons in the NTP to generate ions, excited species and radicals, in
+which oxygen related species such as 0, O2 and 0 3 mainly contribute to
+convert NO into N02. In the case of exhaust gases, including air with
+water vapor, not only oxygen related radicals but also hydroxyl radicals
+(OH radicals) are produced and contribute to oxidize NO to N02. However,
+in these systems, NO is only oxidized to N02, directly or indirectly, by these
+radicals. As a result, the net reduction of NOx (NO + N02) remains
+unchanged. Gases such as ammonia, H20 2, hydrocarbon, N2H4, hydrogen
+and catalyst as additives are used to dissolve N02. The case that ammonia
+is added into the NO stream field is shown in figure 9.8.2. NO is converted
+into N02 by hydroxyl and peroxy radicals as well as oxygen radicals. N02
+Figure 9.S.1. NO/N2/02 system without H20.
+
+--- Page 638 ---
+Chemical Decontamination
+623
+I RNO I
+NO
+oa~~tHj
+Figure 9.8.2. NOjN2j02 system with H20 and NH3 as an additive.
+reacts with OR to form RN03 and, further, NR4N03 is produced by the
+reaction between RN03 and ammonia. When ammonia is subjected to elec-
+tron impact in NTP, ammonia radicals are generated. This reaction scheme is
+shown in figure 9.S.3. NO reacts with ammonia radicals (NR3, NR2 and NR)
+Figure 9.8.3. NOjNH3 system.
+
+--- Page 639 ---
+624
+Current Applications of Atmospheric Pressure Air Plasmas
+I CH41
+e
+0
+NO
+O2
+~
+NO
+NO
+Figure 9.8.4. Hydrocarbon system.
+produced by electron impact. NH2 radicals are a major contributor to
+oxidize NO to N02, through which NH4N03 that is used for fertilizer is
+produced.
+NO decomposition by hydrocarbons is shown schematically in figure
+9.8.4. When hydrocarbons are added, the reaction by peroxy radicals
+(R-OO) is a major pathway to decompose NO [1-4], although the reactions
+are complicated. CHi (i = 1-3) radicals (CH3, CH2, CH etc.) are also
+produced by electron impact in NTPs to decompose NO through HCN,
+NCO and HCO radicals [5].
+There are many kinds of hydrocarbons such as CH4, C2H2, C2H4, C3H6
+and C3HS' However, reactions generated are commonly used to produce
+peroxy radicals R-OO. H02 is an example of a peroxy radical [3], i.e.
+R+O+O+M -
+R-OO+M.
+(9.8.2.1)
+R-OO strongly oxidizes NO into N02 as shown in equation (9.8.2.2) [6].
+R-OO + NO -
+R-O + N02 •
+(9.8.2.2)
+The detailed R -00 species of C3H6 is described in references [2] and [6] and
+C3Hs in reference [6].
+N02 reacts with OH radicals to make HN03. A part ofN02 is changed
+into CO2, where N02 is reacted with deposited soot at the proper tempera-
+ture. Oxygen radicals preferably react with hydrocarbon molecules thereby
+initiating a reaction chain forming several oxidizing radicals [7].
+
+--- Page 640 ---
+Chemical Decontamination
+625
+Carbon dioxide, CO2, is also included in exhaust gases [8]. CO2 hardly
+contributes to the decomposition of NO, because the majority of the energy
+deposited from the non-thermal plasma may be lost to the vibrational and
+rotational excitations of CO2 • Although it is thought that electrons impact
+CO2 to make CO, NO can be reduced only at very high temperatures as
+shown in equations (9.8.2.3) and (9.8.2.4) [9].
+e+C02 -
+CO+O+e
+CO + NO -
+CO2 +!N2 •
+(9.8.2.3)
+(9.8.2.4)
+NO is reproduced by the reaction between CO2 and nitrogen radicals as
+shown in equation (9.8.2.5) [10].
+N + CO2 -
+NO + CO.
+(9.8.2.5)
+NOs are reproduced by N02 reduction by oxygen and hydrogen radicals,
+and reactions between nitrogen and OH radicals as shown in equations
+(9.8.2.6}-(9.8.2.8).
+N02 +0 -
+NO+02
+N02 + H -
+NO + OH
+N +OH -
+NO+H.
+(9.8.2.6)
+(9.8.2.7)
+(9.8.2.8)
+In summary, NO is converted into final products through the production of
+N02 by additives in a NO stream field. The energetic electron impact is the
+origin of these reactions. Electrons directly impact to NO or produce radicals
+to convert NO into N02. N02 further changes to NH4N03 when ammonia is
+added. NO is also reproduced by oxygen and hydroxyl radicals.
+9.S.3 Non-thermal plasmas for de-NOx
+Plasma reactors that have been utilized for NOx remediation are: (1) di-
+electric barrier discharge, (2) corona discharge, (3) surface discharge, (4)
+glow discharge and (5) microwave discharge. Reactor groups are subdivided
+according to their power source: dc and pulsed. Electrode configurations in
+corona discharge and dielectric barrier discharge are (1) plate, (2) needle or
+multi-needle, (3) thin wire and (4) nozzle. A grounded electrode is placed
+in parallel or coaxial form near these electrodes.
+Hybrid systems combining plasma with electron beam [11, 12] or catalysts
+were also developed [13-15]. As indirect decomposition systems, radical shower
+systems were developed using ammonia gases [16,17] and methane gases [18].
+9.8.3.1
+Efficiency
+The efficiency of NOx reduction using pulsed or stationary NTPs is a
+complex function of parameters that include pulse width, pulse polarity,
+
+--- Page 641 ---
+626
+Current Applications of Atmospheric Pressure Air Plasmas
+current density, repetition rate and reactor size. For de-NOx , removal
+efficiency TlNO, and energy efficiency TIE are often used to evaluate the decom-
+position system. These are defined as equations (9.8.3.1) and (9.8.3.2).
+TlNo, = [NO]before - [NO]after x 100
+(%)
+(9.8.3.1)
+.
+[NO] before
+- L
+[NO] before
+TlNo, x ~
+x ~
+TIE -
+X
+106
+X 100
+22.4
+P
+(gjkWh)
+(9.8.3.2)
+where [NO]before and [NO]after are NO concentrations before and after the
+process in units of ppm. L is NO flow rate in units of l/min, the molecular
+weight of NO is 30 g, and P is consumed energy in units of kWh. The elec-
+trical conversion efficiency that refers to the efficiency for converting wall
+plug electrical power into the plasma is important in the evaluation of the
+total efficiency for the decomposition of NO".
+9.8.3.2
+Plasma reactors
+Figures 9.8.S(a)-(f) show schematics offundamental plasma reactors for NO
+decomposition. Figure 9.8.S(a) shows a DBD reactor. The electrode is coated
+with dielectric materials. To prevent charging-up of the dielectric materials,
+the power source is ac or burst ac signals with a frequency of 50 Hz to several
+tens to hundreds of kHz. For the electrode arrangement, parallel plate, multi-
+point [19] and coaxial types [16] are used. A series of filamentary discharges
+are produced at the gap. Figure 9.8.S(b) shows a coaxial electrode configura-
+tion [20] for generating corona discharge. The central electrode consists of a
+thin wire. By applying a high voltage, corona discharges are produced
+around the wire by stationary (ac and dc) and pulsed discharges [21-24].
+For dc corona discharge, a polar effect appears (positive and negative
+corona discharges). The electrode configurations are a wire [20], pipe and
+Electrode
+Dielectric
+Figure 9.8.5. (a) Dielectric barrier discharge reactor.
+
+--- Page 642 ---
+Gas flow
+t
+Electrode
+Discharge
+Wire
+Figure 9.8.5. (b) Corona discharge reactor.
+Chemical Decontamination
+627
+r~
+Gas flow
+nozzle electrodes [17]. For generating pulsed corona discharges, there are
+several types of electrode arrangement, i.e. point-to-plate [25], wire-to-
+plate [26, 27], wire-to-cylinder [28, 29], nozzle-to-plate [30] and pin-to-plate
+[31, 32]. For power sources, dc/ac superimposed source [33] and bi-polar
+polarity of pulsed source [28, 34] are also used. Streamer corona discharge,
+which is generated with a voltage rise time of 10-50 ns and a duration of
+50-500ns FWHM (full-width at half-maximum), can decompose pollutant
+gases.
+The catalyst coated-electrode configuration to facilitate de-NOx is
+shown in figure 9.8.5(c). NOx gases flow in the plasma and the catalyst to
+undergo decomposition.
+Figure 9.8.5(d) shows a tubular packed-bed corona reactor. The pellets
+of dielectric materials are coated with or without catalyst. The catalyst is
+activated by energetic particles, i.e. electrons, photons, excited molecules,
+ions etc. [14]. By applying a high ac voltage to pellets filled in a chamber,
+Gas flow
+Catalyst
+Discharge
+Wire
+Figure 9.8.5. (c) Corona discharge--catalyst reactor.
+Gas flow
+
+--- Page 643 ---
+628
+Current Applications of Atmospheric Pressure Air Plasmas
+Gas flow
+Discharge
+Wire
+Figure 9.S.S. (d) Packed-bed corona discharge reactor.
+micro-discharges in the gap and/or on the surface are generated. This is
+called a packed bed discharge, which is also expected to have a catalytic
+effect at the surface of pellets [35].
+Figure 9.8.5(e) shows a radical injection NTP system: a pipe electrode
+with nozzle pipes from which gas additives flow, that are spouted to generate
+Gas now
+Electrode
+,
+(ACIDC)
+R~
+NOx
+Figure 9.S.S. (e) Radical injection reactor.
+
+--- Page 644 ---
+Induction
+Electrode
+Outer
+Figure 9.S.5. (f) Surface discharge reactor.
+Chemical Decontamination
+629
+Discharge
+Electrode
+(grounded)
+streamer corona discharges in the NOx stream field. Thus, the NOx is directly
+exposed to the corona discharge [30]. On the other hand, radicals are
+supplied to the NOx stream field by DBD generated in a separate chamber
+from the NOx stream field. In this case, NOx is not exposed to the plasma.
+DBD is generated by an intermittent power source so as to control the
+discharge power. Ammonia radicals are injected into the NO stream field
+[16]. Remediation by radical shower systems is achieved using dielectric
+barrier discharges and corona discharges. Plasma-induced radicals from
+ammonia [16, 17, 36, 37], methane [18, 36] and hydrogen [36], are injected
+into the NOx stream region or via the corona zone.
+Figure 9.8.5(f) shows a reactor of surface discharge. One of the elec-
+trodes is inside the ceramics. By applying a high ac voltage, surface discharge
+(a kind of dielectric barrier discharge) is generated at a surface of the inner
+ceramics [38].
+Microwave discharges at atmospheric pressure are also used for NOx
+removal [39, 40] and are effective to decompose N2/NO and N2/02/NO
+mixtures [40]. Because the gas temperature becomes high when operating
+stationary discharges, a pulsed mode operation is employed [39]. NO is
+also decomposed into N2 and O2 by a microwave discharge in a NO/He
+mixture [41]. Micro-structured electrode arrays allow generation of a large-
+area glow discharge, which removes two nitrogen oxides (NO and N20).
+DC or rf power is applied to the arrays [42].
+A hybrid system using NTP and an electron beam is effective in simul-
+taneous removal of NO and S02 [12]. An electron beam is used together with
+a corona discharge ammonia radical injection system.
+
+--- Page 645 ---
+630
+Current Applications of Atmospheric Pressure Air Plasmas
+9.8.4 Parametric investigation for de-NOx
+In the de-NOx process by NTPs, optimization of the following parameters is
+desired: (1) energy efficiency, (2) removal efficiency, (3) process cost, (4)
+controllability, (5) by-products and (6) lifetime of the system and maintenance.
+These parameters are directly influenced by: (1) power source (output voltage,
+pulse width and polarity etc.), (2) electrode configuration, (3) catalyst, (4)
+radical species, (5) additives, (6) reactor size etc.
+In addition to the conventional electrode configurations mentioned
+above, pyramid [19, 43] and multi-needle geometry [44] have been employed
+to lower the operating voltage. In the pyramid type, tip angle and height were
+varied [19]. In the multi-needle type, gap length was varied [44]. These
+parameters of gap length and height have a close relationship to the
+plasma initiation voltage leading to the reduced electric field strength and
+the consumed energy in the plasma. When the angle of the tip point becomes
+small, energy efficiency decreases due to larger energy consumption. The
+lower reduced electric field strength was obtained for a shorter gap length
+to lead to a lower rate of ozone production for the multi-needle type. As a
+result, the de-NOx rate becomes low.
+The influence of height of the pyramid-shaped electrode was also inves-
+tigated [43]. It was shown that NO removal rate increases with decreasing
+heights, in other words, depth of the groove, at the same gas residence
+time. This change of the removal rate may be related to the change of the
+discharge modes in DBD and surface discharge.
+A heated wire is used for corona discharge generation and energetic
+electrons are emitted [20]. A heated corona wire is able to produce energetic
+electrons and activate the oxidation by the generated ozone. It was shown
+that the average corona currents increased and the corona starting voltages
+decreased with an increase in the wire temperature. The relation between de-
+NOx rate and wire temperature was investigated. For generating corona
+discharge, metallic wires are often used to make a high electric field. The
+dependence of de-NOx rate on the wire materials, tungsten and copper,
+was examined by a pulsed corona discharge with a wire-to-plate electrode
+system. A higher de-NOx rate is obtained by tungsten wire covered with
+W03 because a streamer corona discharge is easily generated, while a dc
+stationary corona is only generated in the case of copper wire [26].
+A pair of reticulated vitreous carbon (10 pores per inch) is used for
+generating streamer corona discharge to convert NO into N02. This elec-
+trode configuration is advantageous in scaling-up the system and gives rise
+to large total NOx removal. At the surface of the carbon electrodes, N02
+oxidizes carbon surfaces and finally nitrous acid is formed [9].
+Reactor size and power sources are also parameters that influence the
+de-NOy characteristics. Instead of the conventional ac and dc power sources
+to generate corona discharges, a high voltage (60 kV) and large current
+
+--- Page 646 ---
+Chemical Decontamination
+631
+(approximately 200 A) pulsed power unit was used to generate a lOO ns-
+duration streamer corona discharge. The output voltage is from a Blumlein
+line generator. The short-duration pulsed power produces high-energy elec-
+trons while the temperature of the ions and the neutrals remains unchanged,
+and thus the energy consumed is reduced. The maximum energy efficiency
+was 62.4 g/kWh [45]. A similar test is carried out using the Blumlein line
+system with an output voltage of 40 kV and a current of 170 A [23]. Actual
+flue gas from a thermal power plant was used. It was shown that about
+90% of the NO was removed at a flow rate of 0.8 liters/min and a repetition
+rate of 7 pps [23]. Using a traveling wave transmission in a coaxial cable, a
+series of alternative discharge pulses generate pulsed corona discharge. Fila-
+ment streamer discharges were generated at an applied reciprocal voltage
+with an output of 40 kV. The NO gas with a concentration of 170 ppm was
+reduced to one fourth of the original concentration in a time of 0.6 s [46].
+The influence of the reactor diameter for pulsed positive corona discharges
+on the de-NOx rate is discussed for a concentric coaxial cylindrical configura-
+tion of the electrode. As a result, the increase of inner diameter of the reactor
+from 10 to 22 mm could be a way to minimize energy losses in the process of
+NOx removal from flue gas [47]. Generally, the current through the plasma
+increases with increasing an applied voltage. In an ammonia radical injection
+system, the corona current shows a hysteresis characteristic against the
+applied voltage. This might be based on the NH4N03 aerosol production.
+The deposition of aerosol particles also affects the NOx removal rate [30].
+The main pathway for NOx removal in catalyst-based technology is
+reduction. Selective catalytic reduction (SCR) has been studied using either
+ammonia (NH3) or hydrocarbons (HCs) as additional reducing agents.
+The combination of NTP, catalyst and the additives are effective to signifi-
+cantly reduce nitric oxides (NO and N02) synergistically to molecular
+nitrogen. For example, NOx is converted into N2 and H20 through electron
+impact in NTP, gas-phase oxidation and catalytic reduction as shown in
+equations (9.8.4.1}-(9.8.4.3). This is called plasma-enhanced NHrSCR
+[48]. When HCs are used, this is called HC-SCR.
+NTP:
+e + O2 -
+e + 20
+(9.8.4.1)
+Gas phase oxidation:
+0 + NO + M -
+N02 + M
+(9.8.4.2)
+Catalytic reduction:
+NO + N02 + 2NH3 -
+2N2 + 3H20.
+(9.8.4.3)
+As catalysts, Pd-AI20 3, Ti02, aluminosilicate, Ag/mordenite, ')'-A1203 and
+Zr02 were examined for plasma-enhanced HC-SCR [48, 49].
+The pulsed corona plasma reactor was followed by a Co-ZSM5 catalyst
+bed of honeycomb type [14]. NO is converted into N02 in the plasma reactor
+and then N02 is reduced in the Co-ZSM5 catalyst bed. No formation of
+NH4N03 occurs. In the plasma-enhaced SCR system, plasma-treated N02
+was reduced effectively with NH3 over the Co-ZSM catalyst at a relatively
+
+--- Page 647 ---
+632
+Current Applications of Atmospheric Pressure Air Plasmas
+low temperature of 150°C [14]. Ti02 [50] as catalyst is also effective to de-
+NO". NTP improves the de-NO" rate with an appropriate content of water
+vapor and Na-ZSM-5 catalyst at any temperature [13].
+9.8.5 Pilot plant and on-site tests
+The de-NOx exhausted from pilot plants and diesel engines can be directly
+processed by NTP. A diesel engine exhaust of a vehicle with a 3 liter exhaust
+output is used as a stationary NOx source with the engine speed set at 1200
+rpm, where the plasma reactor consisting of a coaxial DBD with a screw-type
+electrodes is mounted on the vehicle [51]. The DBD deNOx system is applied
+to an actual vehicle with an exhaust output of 2.5 liters and the oxidation of
+hydrocarbon is recognized, where geometric and electric parameters such as
+dielectric surface roughness and gap width of the coaxial reactor are investi-
+gated [52]. A pulsed corona discharge process is applied to simultaneously
+remove S02 and NOx from industrial flue gas of an iron-ore sintering
+plant. The corona reactor is connected to the power source consisting of a
+magnetic pulse compression modulator with a system supplying chemical
+additives such as ammonia and propylene. The problem regarding the life-
+time of the closing switch can be solved by using magnetic pulse compression
+technology [53]. Propylene used as the chemical additive was very effective in
+the enhancement of NOx removal. The increase in C3H6 concentration gives
+rise to an enhancement of NOx [53].
+NOx and S02 from coal burning boiler flue gases are simultaneously
+removed by dc corona discharge ammonia radical shower systems in pilot
+scale tests, where multiple-nozzle electrodes are utilized for generating a
+corona discharge. Tests were conducted for the flue gas rate from 1000 to
+1500Nm3/h, the gas temperature from 62 to 80°C, the ammonia-to-total
+acid gas molecule ratio from 0.88 to 1.3, applied voltage from 0 to 25 kV
+and NO initial concentration from 53 to 93 ppm for a fixed S02 of
+800 ppm. As a result, approximately 125 g of NOx was removed by 1 kWh
+of energy input with 75% of removal efficiency [54]. A plasma/catalyst
+continuously regenerative hybrid system is introduced to reduce diesel parti-
+culate matter (DPM), NOx , Co etc., contained in diesel exhaust gas from a
+passenger diesel car (2500 cm\ A corona discharge is generated in front 'of
+a nozzle-type hollow electrode, where ammonia, hydrocarbon, steam,
+oxygen, nitrogen etc. are injected. The hybrid system test shows thatIiJPM
+and CO were almost removed and NOx reduced to 30% simultaneously by
+the system [25].
+9.8.6 Effects of gas mixtures
+It is known that, in addition to NOx, exhaust systems also release varying
+concentrations of N2, O2, CO2, H20 etc. In coal burning electric plant,
+
+--- Page 648 ---
+Chemical Decontamination
+633
+sulfur oxide (S02) and fly ash are also contained. In diesel exhaust gas, soot is
+included. One must consider the effect of these mixtures with NOx . These are
+molecules and therefore, when present together with NOx in a plasma, the
+plasma energy is partly consumed in these mixtures and is expended as
+vibrational and rotational energies. This energy expense may not contribute
+to the reaction. Thus, de-NOx efficiency can be enhanced using chemicals like
+H20, H20 2, 0 3, NH3, or hydrocarbons that are introduced into NTPs as an
+additive. As a result, NO and S02 are finally converted into NH3N04 and
+(NH4hS04, respectively, where ammonia is used as an additive.
+9.8.6.1
+Particulate matter, soot, andfly ash
+Fly ash is contained in the exhaust gas from coal-burning thermal electrical
+power plants. Diesel particulate matter, NOx , CO2, etc., contained in diesel
+exhaust gas emitted from a passenger car, were reduced using a dc corona
+discharge plasma/catalyst regenerative hybrid system. The effects of
+repetitive pulses and soot chemistry on the plasma remediation of NOx are
+computationally investigated [55]. It was pointed out that N02 reacts with
+deposited soot in the plasma reactor at the proper temperature [25]. An
+outer porous electrode made of SiC ceramics is used for decomposition of
+soot-containing exhaust gas and acts as both electrode for dielectric barrier
+discharge and particulate filter. Toxic and soot containing harmful
+substances from exhaust gas are subjected to plasma processing. The flue
+gas is let out through the porous electrode which is gas-permeable but filters
+hold back the soot particles. Reaction products were CO and CO2. The soot
+decomposition was achieved by a cold oxidation process. Thus, the soot is
+constantly oxidized during all engine operating conditions [56].
+Fly ash including NOx gas was removed using pulsed streamer
+discharges, generated by the configuration of wire and cylinder electrodes.
+Fly ash with particle sizes from 0.08 to 3000ilm was injected into the
+discharge region. The removal rate of NO and NOx including the fly ash
+was increased in the presence of moisture. It was explained that the presence
+of H20 generates the OH radicals by dissociation [57].
+9.8.6.2 S02
+S02 is often processed using ammonia as an additional gas. The reaction is
+shown as
+2S02 + 40H -
+2H2S04
+H2S04 + 2NH3 -
+(NH4hS04·
+(9.8.6.1 )
+(9.8.6.2)
+When S02 reacts with oxygen atoms to form S03, S03 is converted into
+H2S04 as
+(9.8.6.3)
+
+--- Page 649 ---
+634
+Current Applications of Atmospheric Pressure Air Plasmas
+S02 was simultaneously removed with NOx using dc corona discharge
+ammonia radical shower systems as pilot plant tests. Both removal and
+energy efficiencies for S02 decomposition increase with increasing
+ammonia-to-acid gas ratio and decrease with increasing flue gas temperature.
+The maximum removal efficiency exists at an applied power of about 300 W.
+Approximately 9 kg of S02 were removed by an energy input of 1 kWh with
+99% of S02 removal [54].
+S02 and NOx from industrial flue gas of iron-ore sintering plant were
+processed using pilot-scale pulsed streamer corona discharges generated by
+magnetic pulse compression technology. The sulfuric acid was neutralized
+by ammonia in the discharges to finally obtain ammonia sulfate. The
+removal of S02 was greatly enhanced when ammonia was added to the
+flue gas. The high removal efficiency may be caused by chemical reaction
+between S02 and NH3 in the presence of water vapor as well as the hetero-
+geneous chemical reaction among S02, NH3 and H20 [53].
+Flue gas from a heavy oil-fired boiler contains 200-1000 ppm of S02 and
+about 50-200 ppm ofNOx . When processed at a hybrid gas cleaning test plant
+using a corona discharge-electron beam hybrid system, up to 5-22% of NOx
+and 90-99% of S02 could be removed by operating the corona discharge with
+an ammonia radical injection system. It was found that total NOy and S02
+reduction rates increase non-monotonically with increasing applied voltage,
+hence, corona current or discharge input power [12].
+9.8.6.3
+O2
+When oxygen molecules are mixed with a mixture of N2 and NO, oxygen
+atoms are generated by electron impact, followed by formation of ozone
+by a reaction with oxygen molecules as shown in equations (9.8.6.4) and
+(9.8.6.5),
+e+02 -
+O+O+e
+0+02 +M -
+0 3 +M.
+Qzone oxidizes NO to form N02 as shown in equation (9.8.6.6),
+NO+03 -
+N02 +02.
+(9.8.6.4)
+(9.8.6.5)
+(9.8.6.6)
+When the N02 with ammonia as additive is used, NH4N03 is formed as
+shown in figure 9.8.3. However, because of the excessive concentration of
+oxygen molecules, N02 is reduced to NO. In this case, oxygen atoms do
+not contribute to remove NOn but reproduce NO as shown in equation
+(9.8.6.7),
+(9.8.6.7)
+Using dielectric barrier discharge with multipoint electrodes [44], NO
+removal was carried out. NO removal rate and NO conversion into N02
+
+--- Page 650 ---
+Chemical Decontamination
+635
+were discussed in NO/N2/02 mixed gas, where the oxygen concentration was
+varied from I to 4%. Removal rates of NO and NOx increase with increasing
+concentration of O2 in gas mixture, but conversion into N03 via N02 from
+NO is limited in low NO concentration.
+9.8.6.6 H20
+Water vapor H20 leads to production of OH and H02 radicals. As H20
+vapor concentration increases, more OH and H02 radicals can be generated
+to oxide NO to form N02 and further HN03 [7]. Therefore, NO and NOx
+(NO + N02) are removed with increasing H20 vapor concentration being
+in a range of 1100-32000 ppm [5]. Increase in the de-NO" rate was also
+seen in humid (10% H20) gas mixture [58], and in dc corona discharge
+over a water surface [59].
+9.8.6.5
+Hydrocarbon radical injection
+Hydrocarbons were used as an additive. NO/NOx is removed with acetylene
+(C2H2) as an additive using a coaxial wire-tube reactor with dielectric barrier
+discharge, where the feeding gases include N2, O2, NO and C2H2. The effect
+of oxygen with concentrations of 0-10% is discussed for de-NOr The rate of
+NO converted into N02 increases with increasing oxygen concentration.
+Thus, NO to N02 oxidation is largely enhanced as the amount of hydro-
+carbon increases. The hydrocarbon acts as a getter of 0 and OH radicals,
+with the products reacting with O2 to yield peroxy radicals (H02) which
+efficiently convert NO to N02. The conversion of NO into N2 by NH and
+N radicals produced via HCN, NCO and HCO radicals is shown in figure
+9.8.4. The de-NOx rate decreases with increasing the oxygen concentration
+from 2.5-10%. This is due to the oxidation to CO or C03 by the reaction
+between CHx and oxygen radicals. In low oxygen concentration, acetylene
+C2H2 reacts with oxygen radicals to form hydrocarbon radicals that facilitate
+to form HCN, NCO and HCO radicals. Thus, oxygen strongly influences the
+de-NOx process [5].
+9.8.6.6 Ammonia radical injection
+An ammonia radical injection system for converting NO into harmless
+products was developed [60], where the radicals are generated in a separate
+chamber from the NO stream chamber. NO gas is not in the plasma. In
+order to confirm the energy efficiency of de-NOx using an intermittent one-
+cycle sinusoidal source for generating DBDs, the NO concentration is
+increased to 3000 ppm by varying the oxygen concentration from 2-5.6%.
+For containing oxygen gas in the NO stream field, lower NO temperature
+operation is possible to obtain a higher de-NOx rate. At an applied voltage
+
+--- Page 651 ---
+636
+Current Applications of Atmospheric Pressure Air Plasmas
+slightly higher than the threshold voltage for plasma initiation, the removal
+amount of NO reaches maximum, presenting maximum energy efficiency. In
+particular, for an oxygen concentration of 5.6% and a duty cycle of 5-10%,
+a high energy efficiency is obtained to be 98 g/kWh. This means that the
+appropriate electrical power is deposited in the DBD plasma at this duty
+cycle. In the system, NO is mainly reduced by NH2 radicals for NO to
+convert into NH4N03 through H02 radicals as shown in figure 9.8.3.
+9.8.7 Environmentally harmful gas treatments
+Volatile organic compounds (VOCs) are converted into CO2 and H20 and
+other by-products (e.g. HCl and H2) in the desired reaction stoichiometry
+by oxygen and hydroxyl radicals. This stoichiometry is difficult to achieve
+by NTPs, because other intermediate products are produced. According to
+the process conditions, not only CO and nitric oxide such as N20 but also
+phosgene (COC12) may be produced, which may require a second-stage
+treatment. The end products include poisonous materials such as phosgene
+which must be separated from the gas stream and/or be processed in a
+second-stage treatment [61].
+The mechanism of decomposition is based on the electron impact on the
+harmful gases [62, 63]. Therefore, the simulation model includes a solution of
+Boltzmann's equation for the electron energy distribution [61]. It was
+reported that more N20 was generated for higher concentration of water
+vapor and decomposition energy efficiency. Power sources with frequencies
+such as 50 and 60 Hz are often used. In this case, the metal catalyst is
+contained in the dielectric barrier discharge to remove the by-product by
+facilitating the decomposition of the harmful gases. NTPs are effective to
+decompose VOCs and the increase of the decomposition rate is desirable
+for a practical flue gas process system.
+The parameters influencing their decompositions are (1) electrical char-
+acteristics of plasmas (power, energy, applied voltage, frequency, repetition
+rates and rise time), (2) water, (3) carrier gases and flow rate, (4) ionization
+potential of the target gases, and (5) gas temperatures. These parameters
+are closely related to bring high selectivity of the target products [64].
+A parametric study for decomposing VOC will be introduced below.
+9.8.7.1
+Plasma sources
+Plasma chemical processes have been known to be highly effective in
+promoting oxidation, enhancing molecular dissociation, and producing
+free radicals to enhance chemical reactions [65]. VOCs are also processed
+using NTPs, in the same way as NO is used. Four types of plasma reactor
+have been mainly used for the application of VOC destruction: surface
+discharge [66], dielectric barrier discharge [67], ferroelectric packed-bed
+
+--- Page 652 ---
+Chemical Decontamination
+637
+discharge [68], and pulsed corona discharge. Most of the power source
+frequency is 50-60 Hz [62, 68, 69]. The destruction is also carried out by dc
+discharge [65], capillary tube discharge [65] and microwave discharge
+processes as well as electron beam. In order to improve energy efficiency
+and control of undesirable by-products, hybrid systems in which NTPs are
+combined with catalysts are used [67]. Synergetic effects are expected.
+Deposition of by-products is not desirable during the process. Pevovskite
+oxides such as barium titanate (BaTi03) act as a highly dielectric compound
+[68]. The perovskite oxides can be catalytically activated by free radicals of
+ultraviolet irradiation from the plasma [68].
+Uniform generation of the corona discharge contribute to reduce
+toluene. The higher destruction efficiency of toluene is attributed to more
+uniform corona-induced plasma activities throughout the reactor volume.
+The size of the pellets contributes to the plasma uniformity [62].
+9.8.7.2 Processes
+Halogen gases such as chlorine and fluorine are finally converted into CO2
+and halogenated hydrogen, respectively. It was found that the destruction
+efficiency decreases in the order of toluene, methylene chloride and tri-
+chlorotrifluoroethane (CFC-1l3: CF2CICFCI2). CFCl13 has the strongest
+bonding and is stable [62]. Toluene (C6HSCH3) is reduced by a dielectric
+barrier discharge, where the reactor consists of a coaxial cylindrical electrode
+system. Packed Ti02 pellets or coated Ti02 on the inner electrode surface are
+used. Ti02 as catalyst is activated using plasma with coaxial electrodes. The
+energy efficiency is improved due to synergetic effects between plasma and
+activated catalyst [67]. The mechanism of toluene destruction involves not
+only plasma-induced destruction in the gas phase but also the adsorption/
+desorption of toluene on the Ti02 as well as catalytic reaction [67].
+Abatement of CFC-l13 (which is one of the fluorocarbons) was first
+reported using ferroelectric packed bed discharge [62] and surface discharge
+[66]. In the surface discharge case [66], CFC-1l3 with a concentration of
+1000 ppm was processed at a destruction rate of 98 % for a discharge
+power of 70W. Recently, CHF3 gas was reduced in H20/He plasma
+(13.56 MHz) and disappeared at 700W. The by-products were CHF3, CF4,
+H20, CO2 and SiF4 [70]. In CF4 destruction under identical experimental
+conditions as in the CH3 case, the maximum destruction efficiency using
+Hr 0 2/He as a carrier gas is higher by a factor of approximately 2 than
+that using 02/He gas. Hydrogen atoms contribute to the CF4 destruction.
+The by-products were CO2, HF and H20 [70]. Ar diluted CF4 as per fluor-
+ocarbon was abated using atmospheric pressure microwave plasma (2.45
+GHz) with TMolO mode. 10 sccm CF4 with 100 sccm Ar in 2 lpm O2 and
+10 lpm N2 flow was treated. CO2, COF2, H20 and NO were identified as
+the by-products [71].
+
+--- Page 653 ---
+638
+Current Applications of Atmospheric Pressure Air Plasmas
+The principal processes of the destruction of toluene are electron and
+radical dissociation in the discharges, although charge transfer of toluene
+with ions and recombination of toluene ions may also be responsible. Ti02
+activated by plasma may induce various reactions on the surface of the
+Ti02, resulting in an enhanced toluene destruction. Ti02 plays a role to
+enhance the destruction efficiency based on the following reactions: (1)
+photocatalyst process by ultraviolet light emission from plasma [67], (2)
+direct activation by fast energetic electrons and active species, (3) oxidation
+by oxygen radicals produced by the destruction of 0 3 on Ti02 catalyst [72]
+and (4) chemical reactions by OH and H02 radicals [72]. Toluene was
+mostly reduced to CO, COb H20 by OH radicals, 0 3 and ° [62, 65, 67,
+73]. Ozone generation is dependent on the heat by the gas discharge. In the
+presence of air or nitrogen, nitrogen atoms are produced in the direct and/
+or sensitized cleavage of nitrogen molecules and produce N20, NO and
+N02 [68]. N20 concentration is significant [68]. In air, triplet oxygen mole-
+cules are the most reactive oxygen source in the presence or absence of
+water, and carbon balance can be improved with suppression of by-products
+due to promoted autoxidation processes [68].
+The principal processes of the VOC destruction are electron and radical
+impact dissociation of molecules. For toluene, the reaction of toluene with
+OH radicals is effective to make H20 as a final product [65] and water can
+be reduced in NTP to give OH radicals and hydrogen atoms. The effect of
+water was discussed in the destruction of butane. In low voltage application,
+higher destruction efficiencies were obtained under wet conditions compared
+with dry conditions. However, at higher voltages, water had almost no or
+some negative effect on butane destruction efficiency [68]. This is much
+different from NO destruction. Benzene was reduced using alumina-hybrid
+and catalyst-hybrid plasma reactors. It was found that Ag-, Cu-, Mo-, Ni-
+supported Al20 3 can suppress the N20 formation [74].
+Carbon tetrachloride (CCI4) was reduced using catalysis-assisted plasma
+technology. Catalysts such as Co, Cu, Cr, Ni and V were coated on 1 mm
+diameter BaTi03 pellets. For high frequency operation at 18 kHz, the best
+CCl4 destruction was achieved with the Ni catalyst although the destruction
+ofCCl4 is based on the direct electron impact and short-lived reactive species
+[63, 75]. That is,
+e + CCl4 ---- Cl- + CCI3 .
+(9.8.7.1)
+CCl4 is reproduced by three-body reaction through CCI3,
+CI + CCl3 + M ---- CCl4 + M.
+(9.8.7.2)
+On the other hand, O2 scavenges the CCl3 through the reaction
+CCl3 + O2 ---- CCI30 2 .
+(9.8.7.3)
+Methylene chloride (CH2CI2) was destroyed by a packed bed plasma rector.
+Because the chlorine in methylene chloride is strongly bonded with carbon, it
+
+--- Page 654 ---
+Chemical Decontamination
+639
+is much more stable chemically than toluene, and it is expected that higher
+electron energies are necessary to reduce methylene chloride [62]. Tri-
+chloroethylene (C2HCI3, or TCE) was reduced in DBD [61] and in a capillary
+discharge [65]. The majority of the CI from TCE was converted into HCl, C12,
+and COC12 [61] and CO2, CO, N02 are also identified [65]. The destruction
+efficiency of TCE is smaller in humid mixtures compared to dry mixtures due
+to interception of reactive intermediates by OH radicals [61]. The reaction to
+form COCl2 is as follows:
+C2HC13 + OH -
+C2Cl3 + H20
+(9.8.7.4)
+C2HCl3 + CI -
+C2Cl3 + HCI
+(9.8.7.5)
+(9.8.7.6)
+TCE reacts with hydroxyl radicals, but the rate coefficient is no larger than
+that with 0 atoms. There are intermediates such as CHOCl, CCl2 and CIO
+due to 0 and OH radicals produced by electron impact dissociation of O2
+and H20. The ClO radical is attributed with an important role in oxidizing
+TCE [61, 76]. TCE can be dissociated or ionized by a direct electron
+impact to form C2C13, C2HCI2, C2HClj etc. It was pointed out that negative
+ions such as Cl- and C-might play an important role in the destruction
+process [65]. These form terminal species such as CO, CO2, HCI and
+COCl2 [61]. N02 is also produced after the process [65].
+9.8.8 Conclusion
+Processing of exhaust gases emitted from motor vehicle and different
+factories and harmful gases emitted from various industries is increasingly
+necessary to preserve our earth environment, thus improving our living
+conditions. For practical use of the NTP system, we must make greater
+effort to increase the process efficiency and reduce unit cost. In order to
+realize an easy handling unit, not only modification of the conventional
+process is needed but also development of new systems, in particular new
+plasma sources, is very important. Combinations of different systems are
+effective in bringing fruitful processing results.
+References
+[1] Filmonova E A, Amirov R H, Hong S H, Kim Y H and Song Y H 2002 Proc.
+HAKONE VIII, International Symposium on High Pressure Low Temperature
+Plasma Chemistry, 337-341
+[2] Filmonova E A, Kim Y H, Hong SHand Song Y H 2002 J. Phys. D: Appl. Phys. 35
+2795-2807
+[3] Kudrjashov S V, Sirotkina E E and Loos D 2000 Proc. HAKONE VII, International
+Symposium on High Pressure Low Temperature Plasma Chemistry, 257-261
+[4] Dorai R and Kushner M J 2002 J. Phys. D: Appl. Phys. 35 2954-2968
+
+--- Page 655 ---
+640
+Current Applications of Atmospheric Pressure Air Plasmas
+[5] Chang M B and Yang S C 2001 Environmental and Energy Engineering 471226-1233
+[6] Dorai R and Kushner M J 2001 J. Phys. D: Appl. Phys. 34 574-583
+[7] Hammer Th 2000 Proc. HAKONE VII, International Symposium on High Pressure
+Low Temperature Plasma Chemistry, 234-241
+[8] Aritoshi K, Fujiwara M and Ishida M 2002 Jpn. J. Appl. Phys. 41 7522-7528
+[9] Kirkpatrick M, Finney W C and Locke B R 2000 IEEE Trans. Industry Applications
+36500-509
+[10] Eichwald 0, Yousfi M, Hennad A and Benabdessadok M D 1997 J. Appl. Phys. 82
+4781-4794
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+Ghiuta V 2000 Radiation Phys. Chern. 57 501-505
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+Trans. Industry Applications 32 131-137
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+Applications Conference 31865-1870
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+604-613
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+Electrical Insulation and Dielectric Phenomena, 828-833
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+1998 Combust. Sci. Tech. 13393-105
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+[21] Mok Y S and Ham S W 1998 Chern. Engineering Sci. 53 1667-1678
+[22] Puchkarev V 2002 Conference record of 25th International Power Modulator
+Symposium and 2002 High-Voltage Workshop, 161-164
+[23] Tsukamoto S, Namihira T, Wang D, Katsuki S, Akiyama H, Nakashima E, Sato A,
+Uchida Y and Koike M 2001 Electrical Engineering in Japan 134 28-35
+[24] Takaki K, Sasaki T, Kato S, Mukaigawa S and Fujiwara T 2002 Conference record of
+25th International Power Modulator Symposium and 2002 High-Voltage
+Workshop, 575-578
+[25] Chae J 0, Hwang J W, Jung J Y, Han J H, Hwang H J, Kim Sand Demidiouk V I
+2001 Phys. Plasmas 8 1403-1410
+[26] Gasprik R, Gasprikova M, Yamabe C, Satoh Sand Ihara S 1998 Jpn. J. Appl. Phys.
+374186-4187
+[27] Penghui G, Hyashi N, Ihara S, Satoh Sand Yamabe C 2002 Proc. HAKONE VIII,
+International Symposium on High Pressure Low Temperature Plasma Chemistry,
+347-350
+[28] Yan K, Hui H, Cui M, Miao J, Wu X, Bao C and Li R 1998 J. Electrostatics 44
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+[29] Mutaf-Yardimci 0, Kennedy L A, Nester S A, Saveliev A V and Fridman A A 1998
+Proc. 1998 SAE International Fall Fuels and Lubricants Meeting, Plasma Exhaust
+Aftertreatment SP-1395 1--6
+[30] Moon J-D, Lee G-T and Geum S-T 2000 J. Electrostatics 50 1-15
+
+--- Page 656 ---
+Chemical Decontamination
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+International
+Fall
+Fuels
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+Aftertreatment SP-1395, 93-99
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+[33] Yan K, Higashi D, Kanazawa S, Ohkubo T, Nomoto Y and Chang J S 1998 Trans.
+lEE Japan 118-A 948-953
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+Asia-Pacific International Symposium on the Basis and Application of Plasma
+Technology, 39--44
+[35] Kawasaki T, Kanazawa S, Ohkubo T, Mizeraczyk J and Nomoto Y 2001 Thin Solid
+Films 386177-182
+[36] Boyle J, Russell A, Yao S C, Zhou Q, Ekmann J, Fu Y and Mathur M 1993 Fuel 72
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+Technol. 4 1--4
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+Trans. Industry Applications 29 781-786
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+Source Sci. Technol. 11 1-9
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+Phys. 39(2) L933-L935
+[42] Scheffler P, Gellner C and Gericke K-H 2000 Proc. HAKONE VII, International
+Symposium on High Pressure Low Temperature Plasma Chemistry, 407--411
+[43] Yamada M, Ehara Y and Ito T 2000 Proc. HAKONE VII, International Symposium
+on High Pressure Low Temperature Plasma Chemistry, 370-374
+[44] Toda K, Takaki K, Kato S and Fujiwara T 2001 J. Phys. D: Appl. Phys. 342032-
+2036
+[45] Namihira T, Tsukamoto S, Wang D, Katsuki S, Hackam R, Akiyama H and Uchida
+Y 2000 IEEE Trans. Plasma Sci. 28434--442
+[46] Kadowaki K, Nishimoto Sand Kitani I 2003 Jpn. J. Appl. Phys. 42 L688-L690
+[47] Dors M and Mizeraczyk J 2000 Proc. HAKONE VII, International Symposium on
+High Pressure Low Temperature Plasma Chemistry, 375-378
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+330
+[49] Miessner H, Francke K P and Rudolph R 2002 Appl. Catalysis B: Environmental 36
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+[50] Ogawa S, Nomura T, Ehara Y, Kishida H and Ito T 2000 Proc. HAKONE VII,
+International Symposium on High Pressure
+Low Temperature Plasma
+Chemistry, 365-369
+[51] Higashi M and Fujii K 1997 Electrical Engineering in Japan 120 1-7 [1996 Trans.
+IEEJ 116-A 868-872]
+[52] LepperhoffG, Hentschel K, Wolters P, NeffW, Pochner K and Trompeter F-J 1998
+Proc. 1998 SAE International Fall Fuels and Lubricants Meeting, Plasma Exhaust
+Aftertreatment SP-1395, 79-86
+[53] Mok Y Sand Nam I-S 1999 IEEE Trans Plasma Science 27 1188-1196
+[54] ChangJ S, Urashima K, TongYX, Liu WP, Wei H Y, Yang FM and LiuXJ 2003 J.
+Electrostatics 57 313-323
+
+--- Page 657 ---
+642
+Current Applications of Atmospheric Pressure Air Plasmas
+[55] Dorai R, Hassouni K and Kushner M J 2000 J. Appl. Phys. 886060-6071
+[56] MUller S, Conrads J and Best W 2000 Proc. HAKONE VII, International Symposium
+on High Pressure Low Temperature Plasma Chemistry, 340-344
+[57] Tsukamoto S, Namihira T, Wang D, Katsuki S, Hackam R, Akiyama H, Sato A,
+Uchida Y and Koike M 2001 IEEE Trans Plasma Science 29 29-36
+[58] Khacef A, Nikravech M, Motret 0, Lefaucheux P, Viladrosa R, Pouvesle J M and
+Cormier J M 2000 Proc. HAKONE VII, International Symposium on High
+Pressure Low Temperature Plasma Chemistry, 360-364
+[59] Fujii T, Aoki Y, Yoshioka N 'and Rea M 2001 J. Electrostatics 51/52 8-14
+[60] Nishida M, Yukimura K, Kambara S and Maruyama T 2001 Jpn. J. Appl. Phys. 40
+1114-1117
+[61] Evans D, Rosocha L A, Anderson G K, Coogan J J and Kushner M J 1993 J. Appl.
+Phys. 74 5378-5386
+[62] Yamamoto T, Ramanathan K, Lawless P A, Ensor D S, Newsome J R, Plaks Nand
+Ramsey G H 1992 IEEE Trans. Industry Applications 28 528-534
+[63] Yamamoto T, Mizuno K, Tamori I, Ogata A, Nifuku M, Michalska M and Prieto G
+1996 IEEE Trans. Industry Applications 32100-105
+[64] Kozlov K V, Michel P and Wagner H-E 2000 Proc. HAKONE VII, International
+Symposium on High Pressure Low Temperature Plasma Chemistry, 262-266
+[65] Kohno H, Berezin A A, Chang J S, Tamura M, Yamamoto T, Shibuya A and Honda
+S 1998 IEEE Trans. Industry Applications 34 953-966
+[66] Oda T, Takahashi T, Nakano H and Masuda S 1993 IEEE Trans. Industry
+Applications 29 787-792
+[67] Kanazawa S, Li D, Akamine S, Ohkubo T and Nomoto Y 2000 Trans. Institute of
+Fluid-Flow Machinery, No 107,65-74
+[68] Futamura S, Zhang A, Prieto G and Yamamoto T 1998 IEEE Trans. Industry
+Applications 34 967-974
+[69] Proeto G, Prieto 0, Gay C R and Yamamoto T 2003 IEEE Trans. Industry
+Applications 39 72-78
+[70] Kogoma M, Abe T and Tanaka K 2002 Proc. HAKONE VIII, International
+Symposium on High Pressure Low Temperature Plasma Chemistry, 303-
+307
+[71] Hong J, Kim S, Lee K, Lee K, Choi J J and Kim Y-K 2002 Proc. HAKONE VIII,
+International Symposium on High Pressure Low Temperature Plasma
+Chemistry, 360-363
+[72] Kim H H, Tsunoda K, Katsura S and Mizuno A 1999 IEEE Trans. Industry
+Applications 35 1306-1310
+[73] Ponizovsky A Z, Ponizovsky L Z, Kryutchkov S P, Starobinsky V Ya, Battleson D,
+Joyce J, Montgomery J, Babko S, Harris G and Shvedchikov A P 2000 Proc.
+HAKONE VII, International Symposium on High Pressure Low Temperature
+Plasma Chemistry, 345-349
+[74] Ogata A, Yamanouchi K, Mizuno K, Kushiyama S and Yamamoto T 1999 IEEE
+Trans. Industry Applications 35 1289-1295
+[75] Penetrante B M, Bardsley J N and Hsiao M C 1997 Jpn. J. Appl. Phys. 36 5007-
+5017
+[76] Vertriest R, Morent R, Dewulf J, Leys C and Langenhove H V 2002 Proc. HAKONE
+VIII, International Symposium on High Pressure Low Temperature Plasma
+Chemistry, 342-346
+
+--- Page 658 ---
+Biological Decontamination
+643
+9.9 Biological Decontamination by Non-equilibrium
+Atmospheric Pressure Plasmas
+In this section, a review of various works on the germicidal effects of atmos-
+pheric pressure non-equilibrium plasmas is presented. First, a few of the
+variety of plasma sources, which have been used by various research
+groups, will be briefly presented. In-depth discussion of these sources and
+others can be found in chapter 6. Analysis of the inactivation kinetics for
+various bacteria seeded in (or on) various media and exposed to the
+plasma generated by these devices is then outlined. Three basic types of
+survivor curves have been shown to exist, depending on the type of microor-
+ganism, the type of medium, and the type of exposure (direct versus remote)
+(Laroussi 2002). Lastly, insights into the roles of ultraviolet radiation, active
+species, heat, and charged particles are presented. The most recent results
+show that it is the chemically reactive species, such as free radicals, that
+play the most important role in the inactivation process by atmospheric
+pressure air plasmas.
+It is important to stress to the reader that only experiments carried out at
+pressures around 1 atm are the subjects of this presentation. For comprehen-
+sive studies conducted at low pressures, the reader is referred to Moreau et al
+(2000) and Moisan et al (2001). In addition, works that used etching-type gas
+mixtures, such as 02/eF 4, or which used plasmas only as a secondary
+mechanism to assist a chemical-based sterilization method will not be
+covered. To learn about these, the reader is referred to Lerouge et al
+(2000), Boucher (1980) and Jacobs and Lin (1987).
+9.9.1
+Non-equilibrium, high pressure plasma generators
+Here, a few methods that have been used to generate relatively large volumes
+of non-equilibrium plasmas, at or near atmospheric pressure (sometimes
+referred to as 'high' pressure) are briefly presented. This is far from being
+a comprehensive list of all existing methods. The devices presented here
+were chosen mainly because they have been used extensively to study the
+germicidal effects of low-temperature high-pressure plasmas. More detailed
+analysis of the physics of these devices can be found in chapter 6 of this book.
+9.9.1.1
+DBD-based diffuse plasma source
+One of the early developments of diffuse glow discharge plasma at atmos-
+pheric pressure was reported by Donohoe (1976). Donohoe used a large
+gap (cm) pulsed barrier discharge in a mixture of helium and ethylene to
+polymerize ethylene (Donohoe and Wydeven 1979). Later, Kanazawa et al
+(1988) reported their development of a stable glow discharge at atmospheric
+pressure by using a dielectric barrier discharge (DBD). The most common
+
+--- Page 659 ---
+644
+Current Applications of Atmospheric Pressure Air Plasmas
+configuration of the DBD uses two parallel plate electrodes separated by a
+variable gap. The experimental set-up of a DBD is shown in chapter 6
+(section 6.6, figure 6.4.1). At least one of the two electrodes has to be covered
+by a dielectric material. After the ignition of the discharge, charged particles
+are collected on the surface of the dielectric. This charge build-up creates a
+voltage drop, which counteracts the applied voltage, and greatly decreases
+the voltage across the gap. The discharge subsequently extinguishes. As the
+applied voltage increases again (at the second half cycle of the applied
+voltage) the discharge re-ignites.
+Laroussi (1995, 1996) reported the use of the DBD-based glow discharge
+at atmospheric pressure to destroy cells of Pseudomonasfluorecens. He used
+suspensions of the bacteria in Petri dishes placed on a dielectric-covered
+lower electrode. The electrodes were placed within a chamber containing
+helium with an admixture of air. He obtained full destruction of concentra-
+tions of 4 x 106 jml in less than 10 min. Subsequently, gram-negative bacteria
+such as Escherichia coli, and gram-positive bacteria such as Bacillus subtilis
+were inactivated successfully by many researchers using various types of
+high pressure glow discharges (Kelly-Wintenberg et a11998, Herrmann et al
+1999, Laroussi et a11999, Kuzmichev et aI2001).
+9.9.1.2
+The atmospheric pressure plasma jet
+The atmospheric pressure plasma jet (APPJ) (Scutze et a11998) is a capaci-
+tively coupled device consisting of two co-axial electrodes between which a
+gas flows at high rates. Figure 9.9.1 is a schematic of the APPJ. The outer
+electrode is grounded while the central electrode is excited by rf power at
+13.56 MHz. The free electrons are accelerated by the rf field and enter into
+collisions with the molecules of the background gas. These inelastic collisions
+produce various reactive species (excited atoms and molecules, free radicals,
+etc.) which exit the nozzle at high velocity. The reactive species can therefore
+react with a contaminated surface placed in the proximity (cm) of the nozzle
+1
+Feed gas inlet
+Effluent
+RF electrode
+Ground Electrode
+Figure 9.9.1. The atmospheric pressure plasma jet (Scutze et aI1998).
+
+--- Page 660 ---
+Biological Decontamination
+645
+(Herrmann et aI1999). As in the case of the diffuse DBD, the stability of the
+APPJ plasma (as well as its non-thermal characteristic) depends on using
+helium as a carrier gas. Herrmann used the APPJ to inactivate spores of
+Bacillus globigii, a simulant to anthrax (Bacillus anthracis) (Herrmann et al
+1999). They reported the reduction of seven orders of magnitude of the
+original concentration of B. globigii in about 30 s.
+9.9.1.3
+The resistive barrier discharge
+The concept of the resistive barrier discharge (RBD) is based on the DBD
+configuration. However, instead of a dielectric material, a high resistivity
+sheet is used to cover at least one of the electrodes (see section 6.4, figure
+6.4.7). The high resistivity layer plays the role of a distributed ballast
+which limits the discharge current and therefore prevents arcing. The advan-
+tage of the RBD over the DBD is the possibility to use dc power (or low
+frequency ac, 60 Hz) to drive the discharge. Using helium, large volume
+diffuse cold plasma at atmospheric pressure can be generated (Laroussi
+et aI2002a).
+Using the RBD, up to four orders of magnitude reduction in the original
+concentration of vegetative B. subtilis cells in about 10 min was reported
+(Richardson et al 2000). Endospores of B. subtilis were also inactivated,
+but not as effectively as the vegetative cells. In these experiments, a gas
+mixture of helium: oxygen 97: 3 % was used.
+9.9.2 Inactivation kinetics
+The concept of inactivation or destruction of a population of microorgan-
+isms is not an absolute one. This is because it is impossible to determine if
+and when all microorganisms in a treated sample are destroyed (Block
+1992). It is also impossible to provide the ideal conditions, which inactivate
+all microorganisms: some cells can always survive under otherwise lethal
+conditions. Therefore, experimental investigation of the kinetics of cell
+inactivation is paramount in providing a reliable temporal measure of
+microbial destruction.
+9.9.2.1
+Survivor curves and D-value
+Survivor curves are plots of the number of colony forming units (CFU s) per
+unit volume versus treatment time. They are plotted on a semi-logarithmic
+scale with the CFUs on the logarithmic vertical scale and time on the
+linear horizontal scale. Figure 9.9.2 shows an example of a survivor curve
+obtained by exposing a culture of E. coli to an atmospheric pressure glow
+discharge in a helium/air mixture (Laroussi and Alexeff 1999). A line,
+such as shown in figure 9.9.2, indicates that the relationship between the
+
+--- Page 661 ---
+646
+Current Applications of Atmospheric Pressure Air Plasmas
+1e+7
+1ei6
+1e+5
+1e+4
+E
+en 1e+3
+::::>
+u.
+0
+1e+2
+1e+1
+1e+O
+1e-1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Treatrrent TirTe (mnutes)
+Figure 9.9.2. Survivor curve of E. coli exposed to DBD plasma.
+concentration of survivors and time is given by
+10g[N(t)/Nol = -kt
+3.0
+3.5
+4.0
+where No is the initial concentration and k is the 'death rate' constant.
+One kinetics measurement parameter, which has been used extensively
+by researchers studying sterilization by plasma, is what is referred to as the
+'D' (decimal) value. This parameter was borrowed from studies on heat
+sterilization. The D-value is the time required to reduce an original concen-
+tration of microorganisms by 90%. Since survivor curves are plotted on
+semi-logarithmic scales, the D-value is determined as the time for a 10gIO
+reduction. Sometimes the D-value is referred to as the 'log reduction time'
+(Block 1992) and expressed as follows:
+Dv = t/(logNo -logNs)
+where t is the time to destroy 90% of the initial population, No is the initial
+population, and Ns is the surviving population (Block 1992).
+Another parameter, which is of great importance for practical systems,
+is the inactivation factor (IF). The IF is the percentage kill of a microbial
+population by a particular treatment (Block 1992). The IF is generally deter-
+mined for spores (highly resistant microorganisms), by taking the ratio of the
+initial count to the final extrapolated count (Block 1992). Since the IF
+depends on the initial count (before treatment, what is referred to as the
+
+--- Page 662 ---
+Biological Decontamination
+647
+'bioburden'), its value reveals the expected number of viable microorganisms
+after the treatment. Therefore, the IF of a treatment method directly reflects
+its sterilizing effectiveness, given a certain bioburden.
+9.9.2.2
+Survivor curves of plasma-based inactivation processes
+To date, the experimental work on the germicidal effects of cold, atmospheric
+pressure plasmas has shown that survivor curves take different shapes
+depending on the type of microorganism, the type of the medium supporting
+the microorganisms, and the method of exposure (direct exposure: samples
+are placed in direct contact with the plasma; remote exposure: samples are
+placed away from the discharge volume or in a second chamber. The reactive
+species from the plasma, but not the plasma itself, are allowed to diffuse and
+come in contact with the samples) (Laroussi 2002).
+Herrmann (APPJ, remote exposure), Laroussi (diffuse DBD-type
+discharge, direct exposure), and Yamamoto (corona discharge with H20 2,
+remote exposure) reported a 'single slope' survivor curve (one-line curve)
+for B. globigii on glass coupons (dry samples), for E. coli in suspension,
+and for E. coli on glass, respectively (Herrmann et al 1999, Laroussi et al
+2000, Yamamoto et al 2001). The D-values ranged from 4.5 s for the B.
+globigii on glass (APPJ), to 15 s for E. coli on glass (Corona with H20 2
+plasma), to 5 min for E. coli in liquid suspensions (DBD-type plasma).
+Two-slope survivor curves (two consecutive lines with different
+slopes) were reported by Kelly-Wintenberg (DBD-type, direct exposure)
+for S. aureus and E. coli on polypropylene samples, and by Laroussi for
+Pseudomonas aeruginosa in liquid suspension (Kelly-Wintenberg et a11998,
+Laroussi et al 2000). The curves show that the D-value of the second line
+(D2) was smaller (shorter time) than the D-value of the first line (Dd.
+Montie also reported the same type of survivor curve for E. coli and B.
+subtilis on glass, agar, and polypropelene (all under direct exposure to a
+DBD-type discharge) (Montie et al 2000). Montie claimed that D J was
+dependent on the species being treated and that D2 was dependent on the
+type of surface (or medium) supporting the microorganisms (Montie et al
+2000). A given explanation of the 'bi-phasic' nature of the survivor curve
+was the following. During the first phase, the active species in the plasma
+react with the outer membrane of the cells, inducing damaging alterations.
+After this process is advanced enough, the reactive species can then quickly
+cause cell death, resulting in a rapid second phase (Kelly-Win ten berg et al
+1998).
+Multi-slope survivor curves were also reported for E. coli and P. aerugi-
+nosa on nitrocellulose filter (diffuse DBD-type, direct exposure) and for B.
+stearothermophilus on stainless steel strips (pulsed barrier discharge,
+remote exposure) (Laroussi et al 2000, Kuzmichev et al 2001). Each line
+has a different D-value. Similar survivor curves (three phases) were reported
+
+--- Page 663 ---
+648
+Current Applications of Atmospheric Pressure Air Plasmas
+in low pressure studies (Moreau et al 2000, Moisan et al 2001). Moisan
+explains that the first phase, which exhibits the shortest D-value, is mainly
+due to the action of ultraviolet radiation on isolated spores or on the first
+layer of stacked spores. The second phase, which has the slowest kinetics,
+is attributed to a slow erosion process by active species. Finally the third
+phase comes into action after spores and debris have been cleared by
+phase 2, hence allowing ultraviolet to hit the genetic material of the still
+living spores. The D-value of this phase was observed to be close to the D-
+value of the first phase. It is important to note that the explanation given
+above would not apply to the case of atmospheric pressure air plasmas,
+which generate a negligible ultraviolet power output at the germicidal wave-
+lengths (200-300 nm).
+9.9.3 Analysis of the inactivation factors
+This section presents a discussion on the contributions of the various agents
+emanating from non-equilibrium air plasmas to the killing process. These are
+the heat, ultraviolet radiation, reactive species, and charged particles. Note
+that in general various gas mixtures can be used to optimize the generation
+of one inactivation agent or another and ultimately to optimize the killing
+efficiency. The following results and discussions, however, are limited to
+the case of atmospheric pressure air (containing some degree of humidity).
+As a plasma generation device, a DBD is used.
+9.9.3.1
+Heat and its potential effect
+High temperatures can have deleterious effects on the cells of microorgan-
+isms. A substantial increase in the temperature of a biological sample can
+lead to the inactivation of bacterial cells. Therefore, heat-based sterilization
+techniques were developed and commercially used for applications that do
+not require medium preservation. In heat-based conventional sterilization
+methods, both moist heat and dry heat are used. In the case of moist heat,
+such as in an autoclave, a temperature of 121°C at a pressure of 15 psi is
+used. Dry heat sterilization requires temperatures close to 170 °C and treat-
+ment times of about 1 h.
+To assess if heat plays a role in the case of decontamination by an air
+plasma, a thermocouple probe was used to measure the temperature increase
+in a biological sample under plasma exposure. In addition, the gas tempera-
+ture in the discharge can be measured by evaluating the rotational band of
+the 0-0 transition of the second positive system of nitrogen. Figure 9.9.3
+shows that the gas temperature and the sample temperatures in a DBD air
+plasma undergo only a small increase above room temperature (Laroussi
+and Leipold 2003). Based on these measurements no substantial thermal
+effects are expected.
+
+--- Page 664 ---
+Biological Decontamination
+649
+350
+340
+0
+0
+Gas Temperature
+~
+.II.
+Sample Temperatura
+!!! 330
+.II.
+0
+.II.
+:::l -
+.II.
+.II.
+I!! 2t 320
+0
+E
+~ 310
+0
+rn
+to
+(!)
+300
+0
+290
+0
+2
+4
+6
+8
+10
+12
+Flow Rate [I/min]
+Figure 9.9.3. Gas and sample temperature versus air flow rate at a power of 10 w.
+9.9.3.2
+Ultraviolet radiation and its potential effect
+Among ultraviolet effects on cells of bacteria is the dimerization of thymine
+bases in their DNA strands. This inhibits the bacteria's ability to replicate
+properly. Wavelengths in the 220-280 nm range and doses of several
+mW s/cm2 are known to have the optimum effect. Figure 9.9.4 shows the
+emission spectrum between 200 and 300 nm from a DBD air plasma
+0.25
+I
+I
+I
+I
+-
+0.20 I-
+::i
+.!!!.
+iii
+c: 0.15 I-
+0)
+U5 ...
+. ~ 0.10 -
+0.
+:2
+:::l
+E
+0
+0.05 -
+15
+.s::.
+a..
+J
+0.00
+.II
+I
+I
+I
+200
+220
+240
+260
+280
+300
+Wavelength [nm]
+Figure 9.9.4. Emission spectrum of an air plasma in the ultraviolet region.
+
+--- Page 665 ---
+650
+Current Applications of Atmospheric Pressure Air Plasmas
+(Laroussi and Leipold 2003). ultraviolet emission at wavelengths greater
+than 300 nm was also detected. The spectrum is dominated by N2 rotational
+bands 0-0 transition (337nm) and NO,6 transition around 304nm. Measure-
+ments of the ultraviolet power density by a calibrated ultraviolet detector, in
+the 200-31Onm band, showed that less than 1 mW/cm2 was emitted, under
+various plasma operating conditions. Therefore, according to these measure-
+ments, the ultraviolet radiation has no significant direct influence on the
+decontamination process of low temperature air plasmas. This is consistent
+with the results of several investigators (Laroussi 1996, Herrmann et al
+1999, Kuzmichev et aI2001).
+9.9.3.3
+Charged particles and their potential effects
+Mendis suggested that charged particles may playa very significant role in
+the rupture of the outer membrane of bacterial cells. By using a simplified
+model of a cell, they showed that the electrostatic force caused by charge
+accumulation on the outer surface of the cell membrane could overcome
+the tensile strength of the membrane and cause its rupture (Mendis et al
+2000, Laroussi et al 2003). They claim that this scenario is more likely to
+occur for gram-negative bacteria, the membrane of which possesses an
+irregular surface. Experimental work by Laroussi and others has indeed
+shown that cell lysis is one outcome of the exposure of gram-negative
+bacteria to plasma under direct exposure (Laroussi et al 2002b). However,
+it is not clear if the rupture of the outer membrane is the result of the charging
+mechanism or a purely chemical effect. Figure 9.9.5 shows SEM micrographs
+of controls and plasma-treated E coli cells (Laroussi et aI2002b). The micro-
+graph of the plasma-treated cells shows gross morphological damage.
+9.9.3.4
+Reactive species and their inactivation role
+In high-pressure non-equilibrium discharges, reactive species are generated
+through electron impact excitation and dissociation. They play an important
+(a)
+(b)
+Figure 9.9.5. SEM micrographs of controls (a) and plasma-treated bacteria (b) E. coli cells.
+The plasma-treated cells show gross morphological damage.
+
+--- Page 666 ---
+Biological Decontamination
+651
+role in all plasma-surface interactions. Among the radicals generated in air
+plasmas, oxygen-based and nitrogen-based species such as atomic oxygen,
+ozone (03), NO, N02, and the hydroxyl radical (OR) have direct impact
+on the cells of microorganisms, especially when they come in contact with
+their outer structures such as the outer membrane. Membranes are made of
+lipid bilayers, an important component of which is unsaturated fatty acids.
+The unsaturated fatty acids give the membrane a gel-like nature. This allows
+the transport of the biochemical by-products across the membrane. Since
+unsaturated fatty acids are susceptible to attacks by hydroxyl radical (OR)
+(Montie et al 2000), the presence of this radical can therefore compromise
+the function of the membrane lipids. This will ultimately affect their vital
+role as a barrier against the transport of ions and polar compounds in and
+out of the cells (Bettelheim and March 1995). Imbedded in the lipid bilayer
+are protein molecules, which also control the passage of various compounds.
+Proteins are basically linear chains of aminoacids. Aminoacids are also
+susceptible to oxidation when placed in the radical-rich environment of the
+plasma. Therefore, oxygen-based and nitrogen-based species are expected to
+playa crucial role in the inactivation process.
+The following are measurements of nitrogen dioxide (N02), hydroxyl
+(OR), and ozone (03) obtained from a DBD operated in atmospheric
+pressure air (Laroussi and Leipold 2003). Figure 9.9.6 shows the concentra-
+tion of N02 in the DBD, as measured by a calibrated gas detection system.
+The presence of OR was measured by means of emission spectroscopy,
+looking for the rotational spectrum of OR A-X (0--0) transition. This
+molecular band has a branch at about 306.6 nm (R branch) and another
+one at 309.2nm (P branch). Figure 9.9.7 shows the emission spectrum in
+900
+I
+800
+..
+E 700
+Co
+..
+.3,
+N
+600
+0
+•
+z
+500
+c::
+•
+..
+..
+..
+0
+•
+:;:::
+400
+!!!
+•
+•
+..
+.....
+c:
+300
+..
+lOW
+Q)
+•
+0
+• 5W
+c:
+200
+0
+1.5W
+0
+•
+U
+0
+100
+0
+0
+0
+0
+0
+0
+0
+8
+10
+12
+14
+Gas Flow [11m in]
+Figure 9.9.6. Concentration of nitrogen dioxide versus air flow rate, for different powers.
+
+--- Page 667 ---
+652
+Current Applications of Atmospheric Pressure Air Plasmas
+8
+OH R-Branch
+307
+308
+Wavelength [nm]
+OH P-Branch
+309
+Figure 9.9.7. Emission spectra from a humid air discharge showing OH lines.
+the range between 306 and 310 nm and it indicates the OR band heads.
+Figure 9.9.8 shows the relative concentration of OR in the discharge as a
+function of power and air flow rate. Ozone concentration produced by the
+DBD in atmospheric air was measured for varying flow rate and at various
+Figure 9.9.8. Relative concentration of OH versus power and air flow rate.
+
+--- Page 668 ---
+Biological Decontamination
+653
+power levels by ultraviolet absorption spectroscopy and by a chemical titra-
+tion method. Concentrations up to 2000 ppm could be obtained. Ozone
+germicidal effects are caused by its interference with cellular respiration.
+9.9.4 Conclusions
+Research on the interaction of both low-pressure and high-pressure non-
+equilibrium plasmas with biological media has reached a stage of maturity,
+which indicates that this emerging field promises to yield valuable technolo-
+gical novelty. In the medical field, the use of plasma to sterilize heat-sensitive
+re-usable tools in a rapid, safe, and effective way is bound to replace the
+present method which relies on the use of ethylene oxide, a toxic gas. In
+the food industry, the use of plasmas to sterilize packaging will lead to
+safer food with a longer shelf life. In space applications, plasma is considered
+as a potential method to decontaminate spacecraft on planetary missions.
+The goal in this application is to avoid transporting microorganisms from
+Earth to the destination planet (or moon). Air plasma is also a potential tech-
+nology that can be used for the destruction of biological warfare agents.
+Extensive research on the use of high-pressure low-temperature plasmas
+to inactivate microorganisms is a relatively recent event. There are still a lot
+of basic issues that need more in depth investigations. Among these are the
+effects of plasma on the biochemical pathways of bacteria. A clear under-
+standing of these will lead to new applications other than sterilization/decon-
+tamination. However, for practical devices intended for the destruction of
+pathogens, all the available results indicate that non-equilibrium plasmas
+generated in atmospheric pressure air offer a very efficient decontamination
+method. This is mainly due to the efficient production of oxygen-based and
+nitrogen-based reactive species, which interact directly with the cells and
+can cause them irreversible damage.
+References
+Bettleheim F A and March J. 1995 Introduction to General, Organic, and Biochemistry 4th
+edition (Saunders College Pub.)
+Block S S 1992 'Sterilization' in Encyclopedia of Microbiology, vol4, pp 87-103 (Academic
+Press)
+Boucher (Gut) R M 1980 'Seeded gas plasma sterilization method' US Patent 4,207,286
+Donohoe K G 1976 'The development and characterization of an atmospheric pressure
+non-equilibrium plasma chemical reactor' PhD Thesis, California Institute of Tech-
+nology, Pasadena, CA
+Donohoe K G and Wydeven T 1979 'Plasma polymerization of ethylene in an atmospheric
+pressure discharge' J. Appl. Polymer Sci. 232591-2601
+Herrmann H W, Henins I, Park J and Selwyn G S 1999 'Decontamination of chemical and
+biological warfare (CBW) agents using an atmospheric pressure plasma jet' Phys.
+Plasmas. 6(5) 2284-2289
+
+--- Page 669 ---
+654
+Current Applications of Atmospheric Pressure Air Plasmas
+Jacobs P T and Lin S M 1987 'Hydrogen peroxide plasma sterilization system' US Patent
+4,643,876
+Kanazawa S, Kogoma M, Moriwaki T and Okazaki S 1988 'Stable glow plasma at atmos-
+pheric pressure' J. Appl. Phys. D: Appl. Phys. 21 838-840
+Kelly-Wintenberg K, Montie T C, Brickman C, Roth J R, Carr A K, Sorge K, Wadworth
+L C and Tsai P P Y 1998 'Room temperature sterilization of surfaces and fabrics
+with a one atmosphere uniform glow discharge plasma' J. Industrial Microbiology
+and Biotechnology 2 69-74
+Kuzmichev A I, Soloshenko I A, Tsiolko V V, Kryzhanovsky V I, Bazhenov V Yu,
+Mikhno I Land Khomich V A 2001 'Feature of sterilization by different type of
+atmospheric pressure discharges' in Proc. Int. Symp. High Pressure Low Tempera-
+ture Plasma Chem. (HAKONE VII), pp. 402-406, Greifswald, Germany
+Laroussi M 1995 'Sterilization of tools and infectious waste by plasmas' Bull. Amer. Phys.
+Soc. Div. Plasma Phys. 40(11) 1685-1686
+Laroussi M 1996 'Sterilization of contaminated matter with an atmospheric pressure
+plasma' IEEE Trans. Plasma Sci. 24(3) 1188-1191
+Laroussi M 2002 'Non-thermal decontamination of biological media by atmospheric pressure
+plasmas: review, analysis and prospects' IEEE Trans. Plasma Sci. 30(4) 1409-1415
+Laroussi M and AlexeffI 1999 'Decontamination by non-equilibrium plasmas' in Proc. Int.
+Symp. Plasma Chem., pp 2697-2702, Prague, Czech Rep., August
+Laroussi M and Leipold F 2003 'Mechanisms of inactivation of bacteria by an air plasma'
+in Proc. Int. Colloq. Plasma Processing, Juan les Pins, France, June
+Laroussi M, Alexeff I and Kang W 2000 'Biological decontamination by non-thermal
+plasmas' IEEE Trans. Plasma Sci. 28(1) pp. 184-188
+Laroussi M, AlexeffI, Richardson J P and Dyer F F 2002a 'The resistive barrier discharge'
+IEEE Trans. Plasma Sci. 30(1) 158-159
+Laroussi M, Mendis D A and Rosenberg M 2003 'Plasma interaction with microbes' New
+Journal of Physics 5 41.1-41.10
+Laroussi M, Richardson J P and Dobbs F C 2002b 'Effects of non-equilibrium atmos-
+pheric pressure plasmas on the heterotrophic pathways of bacteria and on their
+cell morphology' Appl. Phys. Lett. 81(4) 772-774
+Laroussi M, Sayler G S, Galscock B B, McCurdy B, Pearce M, Bright N and Malott C
+1999 'Images of biological samples undergoing sterilization by a glow discharge
+at atmospheric pressure' IEEE Trans. Plasma Sci. 27(1) 34-35
+Lerouge S, Werthheimer M R, Marchand R, Tabrizian M and Yahia L'H 2000 'Effects of
+gas composition on spore mortality and etching during low-pressure plasma steri-
+lization' J. Biomed. Mater. Res. 51 128-135
+Mendis D A, Rosenberg M and Azam F 2000 'A note on the possible electrostatic disrup-
+tion of bacteria' IEEE Trans. Plasma Sci. 28(4) 1304-1306
+Moisan M, Barbeau J, Moreau S, Pelletier J, Tabrizian M and Yahia L'H 2001 'Low
+temperature sterilization using gas plasmas: a review of the experiments, and an
+analysis of the inactivation mechanisms' Int. J. Pharmaceutics 226 1-21
+Montie T C, Kelly-Wintenberg K and Roth J R 2000 'An overview of research using the
+one atmosphere uniform glow discharge plasma (OAUGDP) for sterilization of
+surfaces and materials' IEEE Trans. Plasma Sci. 28(1) 41-50
+Moreau S, Moisan M, Barbeau J, Pelletier J, Ricard A 2000 'Using the flowing afterglow of
+a plasma to inactivate Bacillus subtilis spores: influence of the operating conditions'
+J. Appl. Phys. 881166-1174
+
+--- Page 670 ---
+Medical Applications of Atmospheric Plasmas
+655
+Richardson J P, Dyer F F, Dobbs F C, Alexeff I and Laroussi M 2000 'On the use of the
+resistive barrier discharge to kill bacteria: recent results' in Proc. IEEE Int. Can!
+Plasma Science, New Orleans, LA, p 109
+Scutze A, Jeong J Y, Babyan S E, Park J, Selwyn G S and Hicks R F 1998 The
+atmospheric pressure plasma jet: a review and comparison to other plasma sources'
+IEEE Trans. Plasma Sci. 26(6) 1685-1694
+Yamamoto M, Nishioka M and Sadakata M 2001 'Sterilization using a corona discharge
+with H20 2 droplets and examination of effective species' in Proc. 15th Int. Symp.
+Plasma Chem., Orleans, France, vol II, pp 743-751
+9.10 Medical Applications of Atmospheric Plasmas
+This section concludes the chapter devoted to practical aspects of atmospheric
+plasmas. At this point, the reader is provided with state of the art information
+on available plasma sources and their applications in inorganic/material
+technology, gas cleaning, combustion, etc. The remaining issue is the role of
+plasma in health care.
+Several biomedical applications of plasmas have been already identified,
+including surface functionalization of scaffolds, deposition of bio-compatible
+coatings, and bacterial decontamination. For in vivo treatment, plasma-
+based devices have been successfully used in wound sealing and non-specific
+tissue removal. Since the modern plasma sources have become quite friendly
+and 'bio-compatible', the area of applications is expanding rapidly and many
+novel medical techniques are under preparation. The most recent develop-
+ment is in vivo bacterial sterilization and tissue modification at the cellular
+level. All these techniques will be described in this section.
+9.10.1
+A bio-compatible plasma source
+A plasma can be considered 'bio-compatible' when it combines therapeutic
+action with minimum damage to the living tissue. In non-specific tissue
+removal, the penetration depth and the degree of devitalization must be
+controllable. In refined/selective tissue modification there are more restrictions
+on the thermal, electrical and chemical properties of the plasma. In this
+paragraph the necessary safety requirements will be briefly discussed.
+9.10.1.1
+Thermal properties of a non-equilibrium plasma
+Surface processing of materials usually involves non-thermal plasmas. 'Non-
+thermal' does not imply that such plasmas cannot inflict thermal damage; it
+means that they are non-equilibrium systems with electron temperature 100
+to 1000 times higher than neutral gas temperature. In table 9.10.1.1 typical
+
+--- Page 671 ---
+656
+Current Applications of Atmospheric Pressure Air Plasmas
+Table 9.10.1.1.
+Plasma source
+Type
+Gas
+T(K)
+Ref.
+Atmospheric
+RF capacitively
+Helium, argon
+400
+Park et at (2002)
+pressure plasma
+coupled
+jet (APPJ)
+Atmospheric
+AC/DC glow
+Air
+800-1500 Lu and Laroussi
+glow
+above water
+(2003)
+Cold arc-plasma
+AC 10-40kHz
+Air, N2, O2
+520
+Toshifuji et at
+jet
+(2003)
+Microwave torch 2.45GHz
+Argon + O2
+2200
+Moon and Choe
+(2003)
+AC plasma
+AC
+Helium+02
+800-900
+Moon and Choe
+(2003)
+DBD
+Dielectric barrier N2 +02 +NO
+300
+Baeva et at (1999)
+Pulsed DBD
+Dielectric barrier Argon+H20
+350-450
+Motret et at (2000)
+Atmospheric
+DC glow with
+Air
+2000
+Mohamed et at
+glow
+micro-hollow
+(2002)
+cathode electrode
+Plasma needle
+RF capacitively
+Helium+N2
+350-700
+Stoffels et at (2002)
+coupled, mm size
+RF micro-plasma Helium (+H2O)
+300
+Stoffels et at (2003)
+gas temperatures in several types of non-thermal plasmas are given. Most of
+these results have been obtained using spectroscopic methods: optical emis-
+sion and CARS (Baeva et aI1999). Moon and Choe (2003) have calibrated
+optical emission spectroscopy against thermocouples. Stoffels et al (2002,
+2003) has also used both methods; some details are given in section 9.10.3
+where the plasma needle is characterized.
+Most of these sources can be used for non-specific treatment, like burning
+and coagulation (see section 9.10.2). For this purpose the temperature may be
+quite high as long as there is no carbonization or deep damage. In other appli-
+cations, like specific treatment without tissue devitalization, temperature is an
+essential issue. The tissue may be warmed up to at most a few degrees above
+the ambient temperature, and exposure time must be limited to several
+minutes. Discharges suitable for this kind of treatment are the micro-plasmas
+(plasma needle) and possibly some kinds of DBDs.
+9.10.1.2
+The influence of electricity
+The influence of electric fields on living cells and tissues has been elaborately
+studied in relation to electrosurgery and related techniques. High electric
+
+--- Page 672 ---
+Medical Applications of Atmospheric Plasmas
+657
+fields are surely a matter of concern for the health of the patient, because they
+may interact with the nervous system, disturb the heartbeat, and cause
+damage to the individual cells.
+Much attention has been given to alternating (high-frequency) currents
+passing through the body. For detailed data the reader should refer to works
+like Gabriel et al (1996) (dielectric properties and conductivity of tissues),
+Reilly (1992) (nerve and muscle stimulation) and Polk and Postow (1995)
+(electroporation and other field-induced effects). These studies have revealed
+that the sensitivity of nerves and muscles decreases with increasing ac
+frequency. The threshold current that causes irritation is as high as 0.1 A
+at 100 kHz. It implies that for medical applications high-frequency sources
+should be employed. At present, most of the electro surgical equipment
+operates at 300 kHz or higher; the plasma needle is sustained by rf excitation.
+Under these conditions no undesired effects are induced.
+9.10.1.3
+Toxicity
+Plasma is a rich source of radicals and other active species. Reactive oxygen
+species (ROS) (0, OH and H02, peroxide anions O2 and H02, ozone and
+hydrogen peroxide) may cause severe cell and tissue damage, known under
+a common name of oxidative stress. On the cellular level, several effects
+leading to cell injury have been identified: lipid peroxidation (damage to
+the membrane), DNA damage, and protein oxidation (decrease in the
+enzyme activity). On the other hand, free radicals have various important
+functions, so they are also produced by the body. For example, macrophages
+generate ROS to destroy the invading bacteria, and endothelial cells (inner
+artery wall) produce nitric oxide (NO) to regulate the artery dilation. The
+natural level of radical concentration lies in the J.lM range (Coolen 2000).
+The density of radical species in the plasma can be determined using
+a variety of plasma diagnostics. However, for applications in biology/
+medicine, standard gas-phase plasma characterization is not very relevant.
+Instead, one has to identify radical species that penetrate the solution and
+enter the cell. Biochemists have some standard methods for radical detection,
+e.g. laser-induced fluorescence in combination with confocal microscopy.
+Special organic probes are used, which become fluorescent after reaction
+with free radicals. This yields detection limits below 0.01 J.lM in a solution,
+and allows three-dimensional profiling with a resolution of about 0.2 J.lm.
+9.10.2
+In vivo treatment using electric and plasma methods
+9.10.2.1
+Electrosurgery
+From very early times it was believed that electricity might have some healing
+properties. In the 17th century some cases of improving the heart function,
+
+--- Page 673 ---
+658
+Current Applications of Atmospheric Pressure Air Plasmas
+waking up from swoon, etc. were reported. About 200 years later the
+technology of artificial generation of electricity was ready for advanced
+medical applications. In 1893, d'Arsonval discovered that high-frequency
+current passing through the body does not cause nerve and muscle stimula-
+tion (d'Arsonval1893). Soon after, high-frequency devices were introduced
+for cutting of tissues.
+At present, electrosurgery has a solid, established name in medicine: the
+electrical cutting device replaces the scalpel in virtually all kinds of surgery. A
+detailed list of applications can be found in the database of ERBE (http://
+www.erbe-med.de). a leading company producing equipment for electric,
+cryogenic and plasma surgery. The electrosurgical tools manufactured by
+ERBE are powered by high-frequency generators, either at 330 kHz or at
+1 MHz. The reason for using these frequencies has been already discussed
+in the previous section: they are well above 100 kHz, the lower limit for
+electric safety. The devices can supply reasonably high powers-up to 200
+or 450 W, dependent on the type and application. The power can be (auto-
+matically) regulated during the operation, to obtain the desired depth of
+the incision. Various electrode designs and configurations are used: a
+monopolar high-frequency powered pin (in this case the current is flowing
+through the patient's body), a bipolar coaxial head, and a tweezers-like
+design (see figure 9.10.1). In the latter case the arms of the tweezers have
+opposite polarities, and the distance between their tips can be varied. The
+quality of cuts for all these configurations is about the same.
+The features that have made electric devices so successful and desired
+are: good cutting reproducibility, high precision, good control of depth,
+A
+c
+B ~
+c
+D
+Figure 9.10.1. Electrosurgery devices and techniques developed by ERBE (http://
+www.erbe-med.de/): (a) a monopolar cutting device, (b) bipolar cutting/coagulation
+tweezers, (c) tissue cutting using coaxial bipolar device, (d) tissue coagulation using bipolar
+tweezers.
+
+--- Page 674 ---
+Medical Applications of Atmospheric Plasmas
+659
+and the possibility of local coagulation. The latter is especially important in
+achieving hemostasis and thus preventing blood loss, formation of thrombus,
+and contamination of tissues during surgery. Electrical coagulation is also
+used on its own, when no incision is necessary-for this purpose a bipolar
+tweezers-like device is used (see figures 9.1O.1b,d). The current flowing
+through the tissue induces ohmic heating that allows for fast and superficial
+coagulation. This method is often used to seal small blood vessels.
+9.10.2.2 Argon plasma coagulation
+The step from electric to plasma surgery is readily made. The electric
+methods discussed above are based on local tissue heating. Devitalization
+by heat is a rather unsophisticated effect, which can be achieved by exposure
+to any heat source. Atmospheric plasma generated by a high-power electric
+discharge is one of the options. Needless to say, for these applications it is not
+required that the gas temperature in the plasma be low. On the contrary,
+controlled burning of the diseased tissue is an essential part of the therapy.
+The aim of the treatment is coagulation and stopping the bleeding, and some-
+times even total desiccation and devitalization of the tissue.
+An adequate discharge has been developed by ERBE, and the corre-
+sponding surgical technique is called argon plasma coagulation (APC). The
+design of the APC source resembles somewhat the APPJ (Park et al 2002),
+because the latter is also a plasma generated in a tube with flowing argon.
+The APC source has not been characterized, but considering the parameters
+(frequency of 350 kHz, operating voltage of several kV and power input of
+50 W) it seems to be a classical ac atmospheric jet. The gas temperature
+within the plasma can easily reach several hundreds of degrees Celsius.
+A schematic view of an APC device (figure 9.10.2) shows a tube through
+which argon is supplied. The flow rate is adjustable between 0.1 and 0.91/min.
+The powered electrode is placed coaxially inside the tube (monopolar
+Figure 9.10.2. An argon plasma coagulation device, developed by ERBE. Argon flow is
+blown through the tube, in which the high frequency electrode is placed. The plasma
+flame stretches out of the tube.
+
+--- Page 675 ---
+660
+Current Applications of Atmospheric Pressure Air Plasmas
+configuration). Like in monopolar electrosurgery, the patient is placed on a
+conducting sheet and the high-frequency current flows through the body. The
+APe electrode generates argon plasma, which stretches about 2-10 mm from
+the tip. Since the plasma is conductive, the current can flow to the tissue, but
+the electrode does not touch it. This is one of the most important advantages
+of APe: the energy is transferred in a non-contact way, so the problems with
+tissue sticking to the metal device, heavy burning and tearing can be avoided.
+Another unique feature of APe is its self-limiting character. Since the
+desiccated tissues have a lower electrical conductivity than the bleeding
+ones, the plasma beam will turn away from already coagulated spots
+toward bleeding or still inadequately coagulated tissue in the area receiving
+treatment. The argon plasma beam acts not only in a straight line (axially)
+along the axis of the electrode, but also laterally and radially and 'around
+the corner' as it seeks conductive bleeding surfaces. This automatically
+results in evenly applied, uniform surface coagulation. The tissues are not
+subjected to surface carbonization and deep damage, and the penetration
+depth is at most 3-4 mm. It should be mentioned that the action 'around
+the corner' is typical for all plasmas, but it cannot be achieved in e.g. laser
+surgery. Superficial scanning of irregular surfaces, small penetration
+depths, and low equipment costs, make plasma devices competitive with
+lasers.
+It is not entirely clear what causes the coagulation of the treated tissue. It
+may be the heat transferred directly from the hot gas as well as the heat gener-
+ated within the tissue by ohmic heating. It is also plausible that argon ions
+bombarding the tissue contribute to desiccation.
+Although the exact physical mechanism of coagulation is not yet
+completely understood, the APe device has been successfully applied in
+many kinds of surgery. The most obvious application is open surgery-
+promoting hemostasis in wounds and bleeding ulcers. Treatment of various
+skin diseases has been discussed by Brand et al (1998). Devitalization of
+mucosal lesions in the oral cavity (e.g. leucoplakia) has been also performed.
+However, the most obvious techniques are not necessarily the most
+frequently applied ones. Since ERBE has developed a flexible endoscopic
+probe, the way to minimally invasive internal surgery has been opened.
+The area of interest is enormous, and most of the APe applications involve
+endoscopy. In gastroenterology there are many situations where large
+bleeding areas must be devitalized. APe treatment has been used to destroy
+gastric and colon carcinoma or to remove their remains after conventional
+surgery, to reduce tissue ingrowth into supporting metal stents (e.g. stents
+placed in the esophagus), to treat watermelon stomach and colitis. APe
+techniques are also frequently used for various operations in the tracheo-
+bronchial system-removal of tumors, opening of various blockages
+(stenoses) in the respiratory tract (e.g. scar stenoses), etc. In the nasal
+cavity, APe can reduce hyperplasia of nasal concha (which causes
+
+--- Page 676 ---
+Medical Applications of Atmospheric Plasmas
+661
+respiratory problems) and hemorrhaging. More examples and detailed
+information about the medical procedures can be found on the website of
+ERBE. In all mentioned cases, the physicians are positive about the
+immediate body reaction and post-treatment behavior. Of course, during
+the operation the surgeon has to be careful not to cause membrane/tissue
+perforation by applying high powers and/or prolonging the treatment too
+much. When the treatment is performed correctly, the devitalized (necrotic)
+tissue dissolves and the healing proceeds without complications.
+9.10.2.3
+Spark erosion and related techniques
+Spark erosion is a special and unconventional application of plasma in
+surgery. It is remarkable for two reasons: first, as an attempt to treat athero-
+sclerosis, a complex cardiovascular disease that plagues most of the Western
+world, and second, as an example to show that a quite powerful discharge
+can be induced in vulnerable places, like blood vessels. In the following
+passage a brief description of atherosclerosis, its pathogenesis and current
+treatment methods will be given, followed by a discussion of the spark
+erosion technique.
+Atherosclerosis is a chronic inflammatory disease, where lipid-rich
+plaque accumulates in arteries. The consequences are plaque rupture and/
+or obstruction of the arteries. The occluded artery cannot supply blood to
+a tissue. This results in ischemic damage and infarct (necrosis). For example,
+direct obstruction of a coronary artery causes irreversible damage to a part of
+the heart muscle, and a myocardial infarct (heart attack). Plaque rupture
+produces thrombus that can cause vascular embolization and infarct far
+away from the actual site of plaque. Complications include stroke and
+gangrene of extremities. At present it is the principal cause of death in the
+Western world (Ross 1999).
+Atherosclerotic obstructions are usually removed surgically (Guyton
+and Hall 2000), by inflating and stretching the artery (balloon angioplasty).
+In severe cases an additional blood vessel must be inserted (bypass
+operation). However, there is no universal cure, because restenosis after
+balloon angioplasty occurs within six months in 30-40% of treated cases,
+and the bypasses are less stable than original arteries.
+In surgical treatment the plaque must be removed, but in a way that
+causes least damage to the artery, so as to minimize restenosis. Recently,
+laser methods have been applied with reasonable success. However, as
+mentioned earlier, lasers cannot act 'around the corner', which in this case
+is essential. In 1985 Slager presented a new concept, which lies between
+electrosurgery and plasma treatment (Slager et al 1985). This technique,
+called spark erosion, is based on plaque vaporization by electric heating.
+The tool developed by Slager is similar to the monopolar device used in
+APC, but no feed gas is used. Instead, the electrode is immersed directly in
+
+--- Page 677 ---
+662
+Current Applications of Atmospheric Pressure Air Plasmas
+Figure 9.10.3. A crater in the atherosclerotic plaque, produced by tissue ablation using the
+spark erosion technique (Slager et aI1985).
+the blood stream and directed towards the diseased area. Alternating current
+(250 kHz) is applied to the electrode tip in a pulsed way, with a pulse duration
+of lOms. The voltages are up to 1.2kV. Under these conditions, the tissue is
+rapidly heated and vaporized. The produced vapor isolates the electrode
+from the tissue, so that further treatment is performed in a non-contact
+way. After vaporization, electric breakdown in the vapor occurs and a
+small « Imm) spark is formed. Spark erosion allows removing substantial
+amounts of plaque--craters produced can have dimensions of up to
+1.7mm. The crater edges are smooth and the coagulation layer does not
+exceed 0.I-O.2mm (see figure 9.10.3).
+It is not yet clear whether spark erosion will become competitive with
+lasers and mechanical methods in treatment of atherosclerosis. One possible
+problem is formation of vapor bubbles, which may lead to vascular
+embolization. Nevertheless, the spark-producing electrode can be used in
+open-heart operations, e.g. in surgical treatment of hypertrophic obstructive
+cardiomyopathy (Maat et al 1994). The cutting performance is similar to
+electrosurgery but, as in plasma techniques, the treatment is essentially
+non-contact.
+Compared to argon plasma coagulation, thermal effects in spark surgery
+are minor. The spark plasma is much smaller than the argon plasma, so that
+heating is more local. Since there is no gas flow, no heat is transferred by
+convection, and pulsed operation suppresses the thermal load. The physical
+characterization of spark-like discharges was performed by Stalder et al
+(2001) and Woloszko et al (2002). The spark generated by these authors
+was similar to the discharge employed by Slager, but they focused on the
+plasma interactions with electrolyte solution. The electron density in such
+
+--- Page 678 ---
+Medical Applications of Atmospheric Plasmas
+663
+plasmas is in the order of 1018 m -3, and the electron temperature is about
+4eV. The gas temperature is about 100°C above the ambient.
+9.10.3 Plasma needle and its properties
+In the medical techniques described above the action of plasma is not
+refined-it is based on local burning/vaporization of the tissue. Using the
+analogy to material science, APC and spark erosion can be compared to
+cutting and welding. However, plasmas are capable of much more sophisti-
+cated surface treatment than mere thermal processing. If the analogy to
+material science holds, it is expected that fine tissue modification can be
+achieved using advanced plasma techniques.
+However, the construction of non-thermal and atmospheric plasma
+sources suitable for fine tissue treatment is not trivial. Moreover, most
+plasmas must be confined in reactors, so they cannot be applied directly
+and with high precision to a diseased area. In the following section another
+approach will be presented: a flexible and non-destructive micro-plasma for
+direct and specific treatment of living tissues.
+9.10.3.1
+Plasma needle
+Small-sized atmospheric plasmas are usually non-thermal. This is simply a
+consequence of their low volume to surface ratio. Energy transfer from
+electrons to gas atoms/molecules occurs in the volume, and the resulting
+heat is lost by conduction through the plasma boundary surface. A simple
+balance between electron-impact heating and thermal losses can be made
+for a spherical glow with a radius L:
+me
+4
+3
+b.T
+2
+ma VeanekBTe 3' 7fL = '" L 47fL
+where me a is the electron/atomic mass, Vea is the electron-atom collision
+frequency and '" is the thermal conductivity of the gas. This allows estimation
+of a typical plasma size:
+L=
+ma
+3",b.T
+me VeanekBTe'
+Dependent on the plasma conditions, the typical length scales of non-thermal
+plasmas with b.T < 10° C are of the order of 1 mm.
+A plasma needle (Stoffels et al 2002) fulfills the requirements of being
+small, precise in operation, flexible and absolutely non-thermal. This is a
+capacitively coupled rf (13.56 MHz) discharge created at the tip of a sharp
+needle. The experimental scheme, including a photograph of the flexible
+hand-held plasma torch, is shown in figure 9.10.4. Like most atmospheric
+
+--- Page 679 ---
+664
+Current Applications of Atmospheric Pressure Air Plasmas
+waveform
+RF amplifier
+power
+meter
+Figure 9.10.4. A schematic view of the plasma needle set-up. In the photograph of the
+flexible torch: rf voltage (right throughput) is supplied to the electrode (needle), confined
+in a plastic tube, through which helium is blown (bottom throughput).
+discharges, the needle operates most readily in helium: the voltage needed for
+ignition is only 200 V peak-to-peak. In fact, using helium as a carrier gas has
+other advantages. The thermal conductivity (144 W/m/K) is very high, and
+consequently the plasma temperature can be maintained low. Moreover,
+helium is light and inert, and possible tissue damage due to ion bombardment
+and toxic chemicals can be thus excluded. The therapeutic working of the
+plasma depends on the additives. As said in section 9.10.1, small doses of
+active species may be beneficial, while large doses inflict damage. In case of
+a plasma needle, the amount of active species is easy to regulate. The right
+dose can be administered by adjusting the plasma power, distance to the
+tissue, treatment time and gas composition. So far, helium plasmas with
+about 1 % of air have been used.
+The glow can be applied directly to the tissues. In figure 9.10.5 one can
+see how the plasma interacts with human skin: it spreads over the surface
+without causing any damage or discomfort.
+Prior to tests with living cells and tissues the needle has been character-
+ized in terms of electrical properties, temperature and thermal fluxes. In
+figure 9.1O.6(a) the temperature versus plasma power is shown for a needle
+with 1 mm diameter: the power lies in the range of several watts and the
+temperatures rise far above the tolerance limits for biological materials.
+For a thinner needle (0.3mm) the power dissipation is only 10-100mW
+and the temperature increase is at most a couple of degrees (figure
+9.1O.6(b). Thus, the needle geometry is important for its operation.
+The flux of radicals emanated by the plasma into a liquid sample has been
+determined using a fluorescent probe (see section 9.10.1). In figure 9.10.7 the
+
+--- Page 680 ---
+Medical Applications of Atmospheric Plasmas
+665
+Figure 9.10.5. Plasma generated in the flexible torch stretches out to reach the skin.
+550
+Q' 500
+...-'
+.... -
+';' 450 ~
+~ 400
+.,
+~ 350
+.,
+... 300
+250
+0
+2
+4
+6
+8
+10
+(a)
+power(W)
+30
+g 28
+!:!
+~ 26
+!:i
+0.. B 24
+0.15 W
+•
+22
+3
+5
+7
+9
+(b)
+distance to needle (mm)
+Figure 9.10.6. (a) Temperature of the plasma determined using a spectroscopic method for
+a 1 mm thick needle. (b) Temperature of the surface (thermocouple) as a function of the
+distance between the needle and the thermocouple for a 0.3 mm thick needle.
+
+--- Page 681 ---
+666
+Current Applications of Atmospheric Pressure Air Plasmas
+9
+8
+7
+:i 6
+'::5
+.!!
+. ~ 4
+"g
+I! 3
+2
+o •
+o
+•
+.-
+•
+2
+3
+•
+•
+•
+•
+•
+4
+5
+6
+7
+8
+time (min)
+9
+Figure 9.10.7. Active radical concentration in a 400 ~l water sample treated with the
+plasma needle, as a function of exposure time. The plasma power is about 50mW, the
+needle-to-surface distance is 1.5 mm.
+concentration of ROS as a function of exposure time is shown for a helium
+plasma with 1 % air. The estimated radical density in the gas phase is
+1019 m -3. The ROS concentration in the liquid lies in the 11M range. This
+amount can trigger cell reactions, but it is too low to cause tissue damage.
+9.10.4 Plasma interactions with living objects
+Interactions of non-thermal plasmas with living objects are an entirely new
+area of research. Of course, the ultimate goal of this research is introducing
+plasma treatment as a novel medical therapy. However, living organisms are
+so complicated that one has to begin with a relatively simple and predictable
+model system, like a culture of cells. In the following section it will be shown
+that even the simplest biological models can exhibit complex reactions when
+exposed to an unknown medium.
+9.10.4.1
+Apoptosis versus necrosis
+The essential difference between the non-thermal plasma needle and APC or
+spark erosion lies in the manner in which the cells are affected. In fine surgery
+cell damage should be minimal. Cell death should be induced only when
+necessary, and then it should fit in the natural pathway, in which the body
+renews and repairs its tissues.
+Cell death is the consequence of irreversible cell injury. It can be
+classified in two types described below.
+• Necrosis, or accidental cell death. Necrosis is defined as the consequence of
+a catastrophic injury to the mechanisms that maintain the integrity of the
+
+--- Page 682 ---
+Medical Applications of Atmospheric Plasmas
+667
+cell. There are many factors that cause necrosis: cell swelling and rupture
+due to electrolyte imbalance, mechanical stress, heating or freezing, and
+contact with aggressive chemicals (e.g. acids, formaldehyde, alcohols). In
+necrotic cells the membrane is damaged, and the cytoplasm leaks to the
+outside. Since the content of the cell is harmful to the tissue, the organism
+uses its immune reaction to dispose of the dangerous matter, and an
+inflammatory reaction is induced. In surgery, mechanical, thermal or
+laser methods always cause severe injury and necrosis. The necrotic
+tissue is eventually removed by the organism, but the inflammation slows
+down the healing and may cause complications, the most common being
+restenosis and scar formation.
+• Apoptosis, or programmed cell death. Apoptosis is an internal mechanism
+of self-destruction, which is activated under various circumstances. This
+kind of 'cell suicide' is committed by cells which are damaged, dangerous
+to the tissue, or simply no longer functional. Thus, apoptosis takes place
+in developmental morphogenesis, in natural renewal of tissues, in DNA-
+damaged, virus-infected or cancer cells, etc. Presumably, any moderate
+yet irreversible cell damage can also activate apoptosis. Known factors
+are ultraviolet exposure, oxidative stress (section 9.10.1) and specific
+chemicals. The role of radicals and ultraviolet has given rise to the
+hypothesis, that plasma treatment may also induce apoptosis.
+Since the intracellular mechanism of apoptosis is rather complex, no
+details will be given here. The reader may refer to textbooks on cell biology
+(Alberts 1994) or more specific articles (Cohen 1997). The morphological
+changes in the cell during apoptosis are easy to recognize. In early apoptosis,
+the DNA in the nucleus undergoes condensation and fragmentation and the
+cell membrane displays blebs. Later, the cell is fragmented in membrane-
+bound elements (apoptotic bodies). Note that the membrane retains its integ-
+rity, so no cytoplasm leakage and no inflammatory reaction occur. The apop-
+totic bodies are engulfed by macrophages or neighboring cells and the cell
+vanishes in a neat manner.
+It is clear that apoptosis is preferred to necrosis. Selective induction of
+apoptosis can make a pathological tissue disappear virtually without a
+trace. Such refined surgery is the least destructive therapeutic intervention.
+No inflammation, no complications in healing and no scar formation/
+stenosis is expected. In the next paragraph plasma induction of apoptosis
+and other cell reactions (without necrosis) will be discussed.
+9.10.4.2 Plasma needle and cell reactions
+A fundamental study on a model system is necessary to identify and classify
+the possible ways in which the plasma can affect mammalian cells. Stoffels
+
+--- Page 683 ---
+668
+Current Applications of Atmospheric Pressure Air Plasmas
+et al (2003) used two model systems: the Chinese hamster ovarian cells
+(CHO-KI) and the human cells MR65. CHO-KI cells are fibroblasts, a
+basal cell type that can differentiate in other cells, like muscle cells, chondro-
+cytes, adipocytes, etc. Fibroblasts are sturdy and easy to culture, which
+makes them a good model at the beginning of a new study. They are also
+actively involved in wound repair, so their reactions to plasma treatment
+may be of interest in plasma-aided wound healing. The MR65 cells are
+human epithelial cells, originating from non-small cell lung carcinoma
+(NSCLC). The NSCLC is one of the most chemically resistant tumors.
+The usage of MR65 has a twofold advantage: (a) information on epithelial
+cells brings one closer to medical applications, like healing of skin ailments,
+and (b) induction of apoptosis in tumor cells is anyway one of the major
+objectives of plasma treatment. Cells were treated using the plasma needle
+under various conditions and observed using phase contrast microscopy or
+fluorescent staining in combination with confocal microscopy. Initially,
+basic viability staining was used: propidium iodide (PI) and cell tracker
+green (CTG). Propidium iodide stains the DNA of necrotic cells red, while
+cell tracker green stains the cytoplasm of viable cells green. Apoptosis in
+tumor cells was assayed using the M30 antibody. Antibody assays are very
+specific. M30 recognizes a molecule, which is a product of enzymatic reaction
+that occurs solely in apoptosis-a caspase-cleaved cytoskeletal protein.
+When M30 binds to this product, a fluorescent complex is formed. The diag-
+nosis is unambiguous. Next to specific antibody assays, cells were observed to
+detect morphological changes characteristic for apoptosis. Various cell reac-
+tions are briefly described below.
+Plasma treatment ofliving cells can have many consequences. Naturally,
+a high dose leads to accidental cell death (necrosis). Typically, necrosis
+occurs when the plasma power is higher than 0.2 Wand the exposure time
+is longer than 10 s (per treated spot). In terms of energy dose, this
+corresponds to 20J/cm2, which is very high. However, even upon such
+harsh treatment the cells are not disintegrated, but they retain their shape
+and internal structure. A typical necrotic spot in a CHO-Kl sample is
+shown in figure 9.10.8. Note that the dead cells (red stained) are separated
+from the living cells (green) by a characteristic void. This void is ascribed
+to local loss of cell adhesion.
+A moderate cell damage can activate the apoptotic pathway. In MR65
+apoptosis occurs under the threshold dose for necrosis. Simultaneously, cell
+adhesion is disturbed. Typical images of plasma-treated cells are shown in
+figure 9.10.9. The whole cytoplasm of the cell is stained using the M30
+antibody, which detects the enzymatic activity that is displayed during
+apoptosis. The percentage of apoptosis after treatment is up to 10%; the
+plasma conditions still have to be optimized.
+When the power and treatment time is substantially reduced (to 50mW
+and I s per spot), neither necrosis nor apoptosis occur. Instead, the
+
+--- Page 684 ---
+Medical Applications of Atmospheric Plasmas
+669
+Figure 9.10.8. A sample of CHO-KI cells after plasma treatment: a necrotic zone (red
+stained with PI), an empty space and the viable zone (green stained with CTG).
+cells round up and (partly) detach from the sample surface: voids like in
+figure 9.10.8 (but without necrotic zone) are created in the sheet of cells.
+The cells remain unharmed and after 2-4 h the attachment is restored. It
+seems that plasma treatment induces a temporary disturbance in the cell
+(a)
+(b)
+Figure 9.10.9. Apoptosis induced in MR65 cells by plasma treatment, assayed by the M30
+antibody method: (a) early apoptosis (caspase activity in the cytoplasma, first changes in
+the cell shape), (b) late apoptosis (formation of apoptotic bodies).
+
+--- Page 685 ---
+670
+Current Applications of Atmospheric Pressure Air Plasmas
+metabolism, which is expressed (among others) by loss of adhesion. Further
+discussion of possible causes is given elsewhere (Stoffels et al 2003).
+Cell detachment without severe damage is a refined way of cell manip-
+ulation. The loosened cells can be removed (peeled) from a tissue but, as
+they are still alive, no inflammatory response can be induced. The area of
+plasma action is always well defined: the influenced cells are strictly localized
+and the borders between affected and unaffected zones are very sharp. Thus,
+plasma treatment can be performed locally and with high precision.
+The last but very important feature of plasma treatment is related to
+plasma sterilization. The latter is a well-known effect, demonstrated by
+many authors (Moisan et a1200l, Laroussi 2002) and even implemented in
+practice. Parallel to plasma-cell interactions, bacterial decontamination
+using a plasma needle was studied. It appeared that bacteria are much
+more vulnerable to plasma exposure than eukaryotic cells. Bacterial inactiva-
+tion to 10-4 of the original population can be achieved in 1-2 min at plasma
+power lower than lOmW, while under the same conditions the mammalian
+cells remain uninfluenced. This demonstrates the ability of a non-thermal
+plasma to selectively sterilize infected tissues.
+9.10.4.3
+Motivationfor the future
+Minimal destructive surgery using non-thermal plasmas is still in its infancy.
+So far several potentially useful cell reactions have been identified, but the
+way to clinical implementation will probably be long and painstaking.
+However, one thing can be stated for sure-non-thermal plasma can be
+used for controlled, high-precision cell removal without necrosis, be it by
+apoptosis, inhibiting proliferation or cell detachment. There are strong
+indications that no inflammatory reaction will be induced. After the
+necessary tests are completed, an enormous area of applications will open.
+Removal of cancer and other pathological tissues, cosmetic surgery, aiding
+wound healing, in vivo sterilization and preparation of dental cavities without
+drilling are just a few examples. The plasma needle can be also operated in a
+catheter (like in APC) and used endoscopically. An enormous effort must be
+invested in developing all these therapies, but considering the benefit for
+human health, it is certainly rewarding.
+References
+Alberts B 1994 Molecular Biology o/the Cell (New York: Garlands Publishing)
+Baeva M, Dogan A, Ehlbeck J, Pott A and Uhlenbusch J 1999 'CARS diagnostic
+and modeling of a dielectric barrier discharge' Plasma Chern. Plasma Proc. 19(4)
+445-466
+Brand C U, Blum A, Schlegel A, Farin G and Garbe C 1998 'Application of argon plasma
+coagulation in skin surgery' Dermatology 197 152-157
+
+--- Page 686 ---
+References
+671
+Cohen G M 1997 'Caspases: the executioners of apoptosis' Biochem. J. 326 1-16
+Coolen S 2000 'Antipirine hydroxylates as indicators for oxidative damage' PhD Thesis,
+Eindhoven University of Technology
+D'Arsonval A 1893 'Action physiologique des courants altematifs a grand frequence'
+Archives Physiol. Norm. Path. 5401-408
+Gabriel S, Lau R Wand Gabriel C 1996 'The dielectric properties of biological tissues. 2.
+Measurements in the frequency range 10 Hz to 20 GHz' Phys. Med. Bioi. 41(11)
+2251-2269
+Guyton A C and Hall J E 2000 Textbook of Medical Physiology (W B Saunders Company)
+Laroussi M 2002 'Non-thermal decontamination of biological media by atmospheric
+pressure plasmas: review, analysis, and prospects' IEEE Trans. Plasma Sci. 30(4)
+1409-1415
+Lu X P and Laroussi M 2003 'Ignition phase and steady-state structures of a non-thermal
+air plasma' J. Phys. D: Appl. Phys. 36(6) 661-665
+Maat L P W M, Slager C J, Van Herwerden L A, Schuurbiers J C H, Van Suylen
+R J, Koffiard MJM, Ten Cate FJ and Bos E 1994 'Spark erosion myectomy
+in hypertrophic obstructive cardiomyopathy' Annals Thoracic Surgery 58(2)
+536-540
+Mohamed A A H, Block Rand Schoen bach K H 2002 'Direct current glow discharges in
+atmospheric air' IEEE Trans. Plasma Sci. 30(1) 182-183
+Moisan M, Barbeau J, Moreau S, Pelletier J, Tabrizian M and Yahia L'H 2001 'Low
+temperature sterilization using gas plasmas: a review of the experiments, and an
+analysis of the inactivation mechanisms' Int. J. Pharmaceutics 226 1-21
+Moon S Y and Choe W 2003 'A comparative study of rotational temperatures using
+diatomic OH, O2 and Ni molecular spectra emitted from atmospheric plasmas'
+Spectrochimica Acta B: Atomic Spectroscopy 58(2/3) 249-257
+Motret 0, Hibert C, Pellerin Sand Pouvesle J M 2000 'Rotational temperature measure-
+ments in atmospheric pulsed dielectric barrier discharge-gas temperature and
+molecular fraction effects' J. Phys. D: Appl. Phys. 33(12) 1493-1498
+Park J, Henins I, Herrmann H W, Selwyn G S and Hicks R F 2001 'Discharge phenomena
+of an atmospheric pressure radio-frequency capacitive plasma source' J. Appl.
+Phys. 89(1) 20-28
+Polk C and Postow E (eds) 1995 Handbook of Biological Effects of Electromagnetic Fields
+(Boca Raton: CRC Press)
+Reilly J P 1992 Electrical Stimulation and Electropathology (Cambridge: Cambridge
+University Press)
+Ross R 1999 'Atherosclerosis-an inflammatory disease' New England J. Med. 340(2) 115-
+126
+Slager C J, Essed C E, Schuurbiers J C H, Born N, Serruys P Wand Meester G T 1985
+'Vaporization of atherosclerotic plaques by spark erosion' J. American College of
+Cardiology 5(6) 1382-1386
+Stalder K R, Woloszko J, Brown I G and Smith C D 2001 'Repetitive plasma discharges in
+saline solutions' Appl. Phys. Lett. 79 4503-4505
+Stoffels E, Flikweert A J, Stoffels W Wand Kroesen G M W 2002 'Plasma needle: a non-
+destructive atmospheric plasma source for fine surface treatment of (bio )materials'
+Plasma Sources Sci. Technol. 11 383-388
+Stoffels E, Kieft I E and Sladek R E J 2003 'Superficial treatment of mammalian cells using
+plasma needle' J. Phys. D: Appl. Phys. 36 2908-2913
+
+--- Page 687 ---
+672
+Current Applications of Atmospheric Pressure Air Plasmas
+Toshifuji J, Katsumata T, Takikawa H, Sakakibara T and Shimizu I 2003 'Cold arc-
+plasma jet under atmospheric pressure for surface modification' Surface and
+Coatings Technology 171(1-3) 302-306
+Woloszko J, Stalder K R and Brown I G 2002 'Plasma characteristics of repetitively-pulsed
+electrical discharges in saline solutions used for surgical procedures' IEEE Trans.
+Plasma Sci. 30 1376-1383
+
+--- Page 688 ---
+Appendix
+This Appendix contains three sections with results pertaining to section 5.3.3
+which were inadvertently omitted from the manuscript. They have been
+added in the proof stage as an Appendix.
+( C)
+Vibrational distribution of N2 ground state
+The V-T, V-V and V-V' rates of the foregoing section were implemented
+in the model and the vibrational distribution of the N2 ground and
+excited electronic states was determined by solving a system of kinetic
+equations at steady state in which the vibrational levels of the N2 ground
+and excited electronic states are the unknowns. The total concentration of
+N2 was determined with the two-temperature kinetic [12] model and fixed
+by replacing the vibrational level v = 0 of the ground electronic state by
+the mass conservation equation. The total populations of the other species
+were fixed and determined with the two-temperature kinetic model, and
+their internal distribution was calculated according to a Boltzmann distri-
+bution at the vibrational temperature Tv = Tg and at the electronic
+temperature Tel = Te. We now present our calculations of the vibrational
+distribution of the N2 ground state at Tg = 2000 K and for different electron
+temperatures.
+For electron temperatures Te lower than 6000 K, the vibrational distri-
+bution is very close to a Boltzmann distribution at the gas temperature
+Tg = 2000 K. Figures A.l and A.2 show the calculated vibrational distribu-
+tions for a gas temperature of 2000 K and an electron temperature of 9000 K
+and 16000K respectively. The Boltzmann distributions at Tv = Tg and
+Tv = Te are also shown on these figures.
+For Te = 9000 K, the vibrational excitation introduced by VE transfer is
+mainly redistributed via V-T relaxation of N2 by collision with N2, and via
+Nr N 2 V-V exchange. The N 2-02 and NrNO V-V' processes do not signif-
+icantly affect the populations of N2 levels. We checked that this conclusion
+remains valid if we assume a different internal distribution for the O2 and
+NO molecules.
+673
+
+--- Page 689 ---
+674
+Appendix
+1019
+-- calculated distribution
+1017
+1015
+1013
+?
+\,--___
+---- Boltzmann at Tv=T;
+,
+--__ --- Boltzmann at Ty=T.
+,
+--
+,
+--
+,
+--
+,
+--
+,
+--
+,
+--------
+~
+,
+,
+5 1011
+, , ,
+.!:
+c:
+109
+.2
+1U
+107
+"S
+C.
+0 C. 105
+103
+101
+10-1
+0
+, , , , , , , , , , , , , , , , , , , , , , , , , ,
+., •••• ~ •••• ~ •••• ~ ••• t ••• 1 ..•.... , .... ~ ....•...... ~ ....•... I. ~ ... .l ..... ,. A .. ~ ••• ~ •••• ~ •••••••• A •••• A L ., .... ~.~.,.., .. A ••••••••• ~ ••• ,. , • •• L..A ••• ~ •• L.~ .... l .... t
+10
+20
+30
+40
+vibrational level v
+Figure A.I. N2(X,V) vibrational distribution function at Tg = 2000K and Te = 9000K,
+p= I atm.
+h
+1017
+f· \
+----
+,
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\
+\ ,
+\
+--
+,
+109
+\
+I
+\\
+. , I
+•
+,
+! , .
+-- calculated distribution
+---- Boltzmann at Tv=Tg
+--- Boltzmann at Tv=T.
+.j
+-----.... ----------- ..... . j
+"
+~
+107 L
+\
+i
+\,
+!
+105 L, .... ,
+... ,
+.... ,
+........ ,
+.... , ....... , .. ,
+... , .. ,
+........ ,
+.... ,
+.... ,
+... , ....... ,
+.. ~~ •...... , ............ ,
+.... ,
+.... , . , ... 1...., •.• , .•.• , ...••... , .•.••.•.••... , .... , 1..., .... , ... , ... , .... , .... , ... 1
+o
+10
+20
+30
+40
+vibrational level v
+Figure A.2. N2(X, v) vibrational distribution function at Tg = 2000K and Te = 16 OOOK,
+p= I atm.
+
+--- Page 690 ---
+Appendix
+675
+Indeed, the rates of NrNO exchange are faster than those of N2-N2
+exchange above v = 3 (see figure 5.3.11 in section 5.3.3), but the total concen-
+tration of NO is two orders of magnitude lower than the concentration ofN2
+and the rates for Nr 0 2 v-v exchange are fast for v> 20 but the population
+of those levels is mainly governed by V - T transfer processes. The vibrational
+distribution at Te = 9000 K lies between the Boltzmann distributions at
+Tv = Tg and Tv = Te, but remains closer to a distribution at Tv = Tg.
+For Te = 16 000 K, almost 25% of the O2 molecules are dissociated and
+the vibrational excitation is mainly redistributed by V-T relaxation ofN2 by
+collision with 0 atoms and by Nr N2 v-v exchange. At this electron
+temperature, the vibrational distribution of the first 15 levels is close to the
+Boltzmann distribution at Tv = Te.
+( D )
+Inelastic electron energy losses in air plasmas
+Electron inelastic energy losses can now be calculated by summing the contri-
+butions of all electron impact collisional processes
+Qinel= L
+[LL~{(Ej-Ei)]
+processes
+I
+j
+(A.l)
+where Ej - Ei represents the internal energy gained by heavy species during
+the collision (Ej must be greater than Ei) and dnj/dt is the net volumetric
+rate of production of heavy species in the final energy level f. In an atmos-
+pheric pressure air plasma characterized by a gas temperature between
+1000 and 3000 K and electron temperatures up to 17 000 K, the dominant
+contribution to electron inelastic energy losses is the electron-impact
+vibrational excitation of N2 ground state. The electron impact vibrational
+excitation cross-sections of O2 and NO ground states are two orders of
+magnitude lower than those of N2, and therefore the contribution of these
+molecules is negligible.
+The total rate of energy loss can be expressed as
+(A.2)
+where VI and V2 are the initial and final vibrational levels of the transition,
+and where the elementary rate QVIV2 is written as
+QVIV2 = (kVIV2 [N2(X, vdl- kV2V1 [N2(X, v2)])ne~Ev2Vl·
+(A.3)
+In equation (A.3), ne is the concentration of electrons, kV1V2 and kV2V1 are the
+excitation and de-excitation rate coefficients and ~EV2Vl stands ~or the differ-
+ence of energy between the two vibrational levels V2 and VI· QVIV2 depends
+strongly on the N2 ground state internal distribution. Vibrational population
+distributions calculated with the method presented in the foregoing section
+are used in equation (A.3) to determine the electron inelastic energy losses.
+
+--- Page 691 ---
+676
+Appendix
+106
+105
+.. ~
+E
+104
+0
+~ 103
+E--K Tg=1800K
+*"-* Tg=2000K
+-Tg=2900K
+...
+1000
+~
+VI
+VI
+..Q
+>.
+!?
+Ql
+I:
+Ql
+500
+9000
+14000
+electron temperature T. (K)
+Figure A.4. Energy loss factor De at Tg = 1800,2000 and 2900 K, as a function of Te.
+heavy species (equal to Tg), and Deh represents the average frequency of
+collisions between the electrons and heavy particle h. Deh can be expressed
+in terms of the number density of neutral species nh, the electron velocity
+ge = J8kTe/7fme and the average elastic collision cross-section Q~h:
+(A.6)
+Figure A.4 shows the calculated electron energy loss factor as a function
+of the electron temperature for two values of the gas temperature, Tg = 1800
+and 2900 K. As can be seen from this figure, the inelastic loss factor is a rela-
+tively weak function of the gas temperature. It increases up to Te = 8000 K as
+the net rate of production of N2 molecules in vibrational level V2 > VI
+increases with Te, and then decreases due to the transition Tv ~ Tg to
+Tv ~ Te. When Tv becomes close to Te, the forward and reverse rates are
+practically balanced and the net rate of energy lost by VE transfer
+approaches zero.
+(E)
+Predicted DC discharge characteristics in atmospheric pressure air
+The results of the previous subsections enable us to convert the 'S-shaped'
+curve of ne vs. Te into electric field vs. current density discharge characteristics.
+This result is obtained by combining Ohm's law and the electron energy equa-
+tion. The latter incorporates the results of the collisional-radiative model to
+account for non-elastic energy losses from the free electrons to the molecular
+species. The predicted discharge characteristics for atmospheric pressure air at
+
+--- Page 693 ---
+678
+Appendix
+2000
+1800
+1012
+-1
+13
+~ 1
+1600
+nj'10 em
+- 1400
+..
+l
+E
+0
+~ 1200
+w
+1
+,; 1000
+1
+Q)
+u:::
+800
+I
+~g
+i
+~ 600
+400
+1
+200
+j
+~0-4
+10~
+10-2
+10-1
+10°
+101
+102
+Current Density. j (A.em-2)
+Figure A.S. Predicted discharge characteristics for atmospheric pressure air at 2000 K,
+2000 K are shown in figure A,S, These discharge characteristics exhibit
+variations that reflect both the S-shaped dependence of electron number
+density versus Te, and the dependence of the inelastic energy loss factor on
+the electron temperature and number density, We have used these predicted
+characteristics as a starting point to design the DC glow discharge experi-
+ments presented in section 5.2, If these predictions are correct, the produc-
+tion of 1013 electron/cm3 requires an electric field of rv 1.35 kV/cm, and a
+current density of rvl0.4A/cm2. Thus the power required to produce
+1013 electrons/cc in air at 2000 K is approximately 14 kW /cm3 .
+
+--- Page 694 ---
+Index
+AC corona 60, 61, 62
+AC torch 276, 350
+Active zone 48,49,50,51
+Aerodynamics 3
+Afterglow 137
+Air chemistry 5, 6, 124-182
+Anharmonicity effects 455-458
+Anode layer 51, 52, 53, 54
+Anti-Stokes scattering 455
+Arc discharge 17, 18, 35
+Arrhenius plot, 125
+Atmospheric layers 4, 5
+Atmospheric-pressure glow discharge
+(APGD) 255-257
+Attachment (dissociative) 99, 127,
+201
+Attachment coefficient 32, 33
+Ball lightning 8, 9
+Barrier corona 61, 62, 63
+Barrier discharge 276-278, 280, 283, 286,
+287, 291, 293, 294, 299, 300, 307,
+316,321
+Bio-compatibility 655
+Biological decontamination 643-653
+Boltzmann (Maxwell-Boltzmann)
+distribution 86-88, 128, 139, 184,
+200,376,450
+Breakdown 17,26,29,30,31,32,33,35,
+36,37, 38, 39, 63, 68, 69, 71, 185,
+247,262-274,279,281,298,300,
+303, 304, 307, 348, 354, 359
+Brillouin scattering 477
+Burst corona 42, 54
+Capture 100
+Cathode boundary layer (CBL)
+discharge 319
+CARS (coherent anti-Stokes Raman
+spectroscopy) 462, 471
+Cathode fall 34, 279, 281,304,307-310,
+316-319,324
+Cathode layer 34, 38, 48, 49, 50, 51, 54,
+282, 308, 329, 336
+Cavity ring down spectroscopy (CRDS)
+517-535
+CBL discharge (cathode boundary layer)
+307, 319, 320
+Cell reaction 667-670
+Charge transfer, 127, 144
+Chemical decontamination 621-639
+CHEMKIN 205,210
+Cleaning 597, 601-605
+Cold plasma 19, 21
+Collision 13
+cross section 190
+energy 138
+frequency 212
+inelastic 199
+one-body 94, 95
+two-body 96-103, 130
+term 106
+three-body 130
+Collisional-radiative model, 201
+Combustion enhancement 577-580
+Computer modeling, 183
+Corona discharge 12,14,17,41,47,54,
+60, 63, 64, 329, 338
+Corona-to-spark transition 52, 53
+679
+
+--- Page 695 ---
+680
+Index
+CPE discharge (capillary plasma
+electrode) 306, 307, 321-324
+Cross section 97-100, 125, 127
+Current density (electrons) 192,211,225,
+243
+Current-voltage characteristic 295-297,
+300 ,308,311-313
+D-value 645, 647
+DC corona 42, 47, 54, 61
+DC glow discharge (see glow discharge)
+Debye length 89, 213
+Decay rate 96
+Decontamination 3, 14
+De-NOx process 622-633
+Deposition 597, 615-617
+Detachment 55,59, 148
+Detailed balance 203
+Diagnostics 10, 14
+Dicke narrowing 461
+Dielectric-barrier discharge (DBD) 12,
+14, 17, 68, 184, 277, 260, 245-260
+Diffuse discharge 284,297, 301
+Diffusion 192
+Dispersion relation 566
+Dissociation (electron impact) 99, 126,
+201,207
+Dissociation (heavy particle) 100
+Distribution function 79-85
+Doppler broadening 447, 448, 461, 469,
+512
+Drift tube 140
+Efficiency (of plasma generation) 6, 7
+Electric field 227, 239
+Electric potential 241
+Electrical conductivity 191, 240
+Electromagnetic absorption 565-574
+Electromagnetic reflection 565-574
+Electromagnetic theory 566-569
+Electron 77, 124
+Electron-beam sustained plasma 427
+Electron density 488-500, 517,
+525-528
+Electron-driven reactions 99, 100,
+127-129
+Electron energy distribution function
+(EEDF) 125, 184,447,448
+Electron impact excitation 99
+Electron impact ionization 99
+Electron-ion recombination 13,418
+Electron lifetime 7
+Electron loss reduction 428
+Electron temperature (see temperature,
+electrons)
+Electrosurgery 657-663
+Electrostatic precipitation 539-551
+Emission bands 447
+H2 (Fu1chur band) 447,501-509
+N2 (second positive band) 212,
+505-509
+Nt (first negative band) 447, 501-509
+NO 506-508
+OH 221
+0 2 505
+Emission spectroscopy 390, 501- 516
+Epstein distribution 567
+Equilibrium 124, 139
+Equivalent circuit 72
+Etching 597, 613-615
+Excitation 99, 100, 125, 126
+electronic 125
+vibrational 127, 148-152, 161
+rotational 139
+Fine structure effects 512
+Flow control 588-589
+Functionalization (surface) 597, 607-613
+Glow corona 42, 43, 44, 54, 55, 56, 57
+Glow discharge 2, 18, 22, 23, 34, 38, 45,
+50,59, 184, 199,218,229,245-269,
+277,279,245,282-284,286-288,
+290,291,298,299,304,307,308,318
+319, 324, 328, 329, 334-344, 346
+Glow-to-arc transition 295, 318, 319
+Guided ion beam 159
+Heavy particles 76
+Heavy particle reactions 100-102
+Heavy particle ionization 201
+Heterodyne interferometry 488-500
+High temperature flowing afterglow
+(HTFA) 138-140, 145-148
+Hollow cathode discharge 276, 307,
+309-311, 313-315, 31~ 318
+
+--- Page 696 ---
+Homogeneous barrier discharge 277,
+286,293-305
+Humidity 4, 6
+Hydrocarbon-air combustion 574-586
+Hydrogen Balmer lines 403
+Inactivation factors 648-652
+Inactivation kinetics 645-648
+Instabilities 446
+Interferometry 482-488
+In-vivo treatment 657-662
+Ion 124
+Ion concentration 517-535
+Ionization 99, 100, 124, 126
+direct 124
+step-wise 124
+Ionization coefficient 30, 32, 33, 38, 46,
+48,49
+Ionization instability 57, 58
+Ion-molecule reactions 136, 140-178
+Ion-pair production 99
+Ionosphere 7, 138
+I - V characteristic 290, 307, 314, 342-345
+Kinetic equation 105-117
+Kinetic theory 78
+Laser ionization 364
+Laser pumping 364
+Laser scattering 450-481
+Laser-sustained plasma 365
+Life time 127
+Line width 448-450
+Gaussian 448-450
+Lorentzian 448-450
+Natural 509
+Resonance 509, 510
+Van der Waals 509, 510
+Voigt 448-450
+Lightning 3, 8
+Liquid crystal display (LCD, active
+matrix LCD) 262-263
+Local thermodynamic equilibrium 221,
+400, 501
+MATLAB 210
+Maxwell's equations 90
+Medical application 655-670
+Index
+681
+MHC discharge (microhollow cathode)
+230,276,307,309-311,313-315,
+317,318,321
+Microdischarge 69, 70, 71,72, 184,
+258-259,276-279,281,297,
+280-283,307,309-318,324,493
+Microstructured electrode arrays 309
+Microwave absorption 3, 14
+Microwave plasma 395
+Millimeter wave interferometry 482-488
+Modeling 1, 10
+Monte Carlo simulation 185, 255,
+266-268
+Multidimensional modeling 233
+Navier-Stokes equations 186
+Neutral particle (neutrals) 137
+Nonequilibrium air plasma chemistry
+154-167
+Number density,
+electrons 195, 196, 199
+ions 241
+Ohm's law 211
+Ozone 128,276,277,278,280,282,287,
+289,290,291,297,316,551-563
+Oxidation 605-607
+Particle-in-a cell model 185,255,
+266-268
+Paschen curve 30, 32, 34
+Penning ionization 100
+Phase shift 565-574
+Photo-excitation 102
+Photo-detachment 102
+Photo-dissociation 102
+Photo-ionization 14, 102
+Photon 78
+Pin-to-plane corona 233
+Plasma combustion 3, 14
+Plasma display panels 253-255, 263,
+265
+Plasma needle 663-666,667-670
+Plasma mitigation 587-597
+Plasma parameters 446
+Plasma processing 2, 3, 14
+Plasma spikes 589-594
+Plasma torch 350-361,395,574-586
+
+--- Page 697 ---
+682
+Index
+Poisson equation 189, 236, 238
+Pollution control 3, 14
+Power factor 73, 74
+Proton transfer reaction 164
+Pulsed breakdown 38
+Pulsed streamer corona 63
+Radiation-driven processes 102-103
+Raman scattering 451-455, 459, 469
+Raman spectroscopy 374
+Rate coefficient 125, 130-135, 200, 214
+Rayleigh scattering 459, 469
+Recombination 99,127,168-175
+Refractive index (index of refraction)
+488, 490
+Replenishment criterion 45, 48
+Replenishment integral 56
+Resistive barrier discharge 276, 293, 299,
+300
+RF discharge 14, 19,21,22
+RF plasma torch 362
+Runaway electrons 38, 39
+Saturation current 543
+Scramjet propulsion 574-586
+Shock waves 587-597
+Space charge 543
+Spark formation 51, 59, 60, 64, 329, 341
+Spark transition 47,52,53,58
+SPECAIR 222
+Sputtering 598
+Stark broadening 401, 509
+Stefan-Boltzmann law 88
+Sterilization 3, 14
+Stokes scattering 455
+Streamer 26,35,42, 54, 56, 63, 281, 297,
+298, 304, 324, 348
+Streamer breakdown 35, 44, 63,
+247-248,276,287,290,291,338
+Streamer corona 42, 44, 56, 63
+Streamer-to-spark transition 58
+Sub-breakdown 386
+Supersonic flight 587-597
+Surface dissociation 598
+Surface ionization 598
+Surface treatment 276,287,290,291,
+338, 597-618
+Temperature,
+electron 124, 183, 200
+gas 203, 217, 221
+ion 124
+neutral 124
+rotational 124, 200, 221
+translational 196
+vibrational 124, 136-144, 200
+Thermal conductivity 190
+Thermal plasma 19,21, 35,42, 124
+Thomson scattering 451-455, 459, 469
+Torch plasma 351, 354-358
+Townsend breakdown 29, 33, 35,
+247-248
+Townsend criterion 32
+Townsend discharge 30, 34, 44, 45, 46,
+283,284
+Townsend mechanism 35, 36, 281, 348
+Trichel pulse 43, 47, 48,50,51, 184,233,
+239,243,329,330-335,340
+Two-temperature model 200
+Ultraviolet radiation 279, 316, 650
+UV flash tube 362
+Velocity distribution 125, 200
+Vibrational distribution functions
+465-469
+Vibrational enhancement 149
+Viscosity 190
+Volt -ampere characteristic 44, 46
+Voltage-charge Lissajous figure 71
+Voltage-current characteristic 313, 320,
+321, 333, 335, 337, 338
+
+--- Page 701 ---
+11111111111111111111111111
+9 780750 309622
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+=== PAGE 2 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+collision rates between charged and neutral particles, and low degree of ionization (see, e.g., Becker et al.,
+2004). Streamers, for instance, only partially meet the three standard conditions that traditionally define a
+plasma (Bittencourt, 2004; pp. 6–11). These criteria define the plasma's ability to shield short-range electro-
+static interactions between individual particles, remain quasi-neutral, and respond collectively to long-range
+electromagnetic forces. The three conditions can be estimated for typical streamer properties at atmospheric
+pressure, that is, electron temperature of 23,000 K, or ∼2 eV, and electron density of 1018–1020 m−3 (Raizer,
+1991; section 12.3). First, the Debye length is ∼1–10 μm, which is relatively smaller than the streamer radius,
+∼0.1–1 mm (Naidis, 2009). Second, there are many electrons in a Debye sphere, ∼500–5,000. Third, the
+electron-neutral collision frequency is ∼1012 s−1, which is higher than the frequency of relevant processes,
+including the plasma frequency. Therefore, it can be said that the first two conditions are approximately
+met, but not the third one. On the other hand, it is easy to show that all three conditions are met in the
+return stroke channel. Therefore, even though the formal definition of a plasma is not always met within the
+many elements of a lightning flash, we refer to its constituting ionized gas as a “plasma,” because it remains
+quasi-neutral and responds collectively to applied electric fields.
+The aforementioned collective behavior in lightning is evidenced in the many types of ionization waves
+(e.g., streamer front, leader front, dart leaders, and return strokes), its ability to shield itself from exter-
+nally applied electric fields, and its negative differential resistance, which in its turn map into several
+phenomenological features, including its fractal structure, the contrasting behavior of positively and neg-
+atively charged extremities, and the fact that leader channels are enveloped by streamer zones and corona
+sheaths. This manuscript focus on perhaps the most important feature attributed to the plasma nature of
+lightning—its nonlinear resistance. A correct description of the channel resistance is required to better
+characterize lightning electromagnetic emissions, to correctly predict its deleterious effects in man-made
+structures, to quantify the impacts of lightning in atmospheric chemistry, and to address fundamental open
+questions regarding lightning initiation, propagation, and polarity asymmetries. The nonlinear plasma resis-
+tance is in its turn dependent on the history of energy deposition and losses in the channel and cannot be
+accurately determined without properly tracking the evolution of all other channel properties, including
+electric field, electron density, temperature, and radius.
+Efforts to characterize the nonlinear resistance and overall plasma properties of the lightning channel can
+be classified into three categories: (1) LTE gas-dynamic models (Aleksandrov et al., 2000; Chemartin et al.,
+2009; Hill, 1971; Paxton et al., 1986; Plooster, 1971; Ripoll et al., 2014a), (2) streamer-to-leader transition
+models (Aleksandrov et al., 2001; Bazelyan et al., 2007; da Silva & Pasko, 2013; da Silva, 2015; Gallimberti,
+1979; Gallimberti et al., 2002; Popov, 2003; 2009), and (3) semiempirical resistance models (Baker, 1990;
+De Conti et al., 2008; Koshak et al., 2015; Mattos & Christopoulos, 1990; Theethayi & Cooray, 2005). The
+three categories are described in the upcoming paragraphs. Instructive discussions and additional references
+regarding each of the three categories can also be found in sections 2.5, 2.3, and 4.4, respectively, of Bazelyan
+and Raizer's (2000) textbook. On a separate note, the literature concerning the resistance of short spark
+discharges in the laboratory is very rich and has provided many insights into building the models cited above
+(see, e.g., Engel et al., 1989; Kushner et al., 1985; Marode et al., 1979; Naidis, 1999; Riousset et al., 2010;
+Takaki & Akiyama, 2001). It is outside of our scope to provide a detailed review of these investigations, but
+it can easily be found elsewhere (da Silva & Pasko, 2013; Engel et al., 1989; Montano et al., 2006).
+The first group of investigations evaluates the resistance of a lightning channel under the assumption that
+the plasma is in LTE. In this framework, the electrical conductivity is only a function of temperature, that
+is, 𝜎= 𝜎(T), which is valid for atmospheric-pressure arcs at temperatures higher than ∼10,000 K, where T
+or simply the word “temperature” here and in the remainder of this manuscript corresponds to the tem-
+perature of the neutral gas. (The 10,000-K threshold is a rough estimate; see section 2.2 for justifications.)
+Following the return stroke simulations performed by Plooster (1971), these models describe how Joule
+heating deposition in the channel core heats the air and causes rapid hydrodynamic expansion. They solve
+a system of three equations accounting for conservation of mass, momentum, and energy (or enthalpy) of
+the neutral gas (air). They are often solved in a 1-D radial domain, with the exception being the work of
+Chemartin et al. (2009) where efforts are made to capture the 3-D tortuosity of a plasma arc. A few of these
+models also present a detailed description of the plasma radiative transfer (see, e.g., Paxton et al., 1986;
+Ripoll et al., 2014a).
+DA SILVA ET AL.
+9443
+
+=== PAGE 3 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+The second class is dedicated to a detailed description of the streamer-to-leader transition process, which
+takes place during the discharge onset or at the tip of a growing channel. Streamer-to-leader transition is the
+name given to the sequence of processes converting cold and low-conductivity plasma channels (streamers)
+into hot and highly conducting ones (leaders), a condition required to allow lightning channels to propagate
+for several kilometers in the atmosphere before decaying (Bazelyan & Raizer, 2000, p. 59). These models
+account for the hydrodynamic expansion of the neutral gas, such as the ones described in the first category.
+However, following in the footsteps of the seminal monograph by Gallimberti (1979), they also account for
+a non-LTE plasma conductivity arising from the detailed kinetic balance of an air plasma. The more recent
+models describe in detail the energy exchange between charged and neutral particles accounting for the
+partitioning of electronic power between elastic collisions, and excitation of vibrational and electronic states,
+and also delayed vibrational energy relaxation of nitrogen molecules (see, e.g., da Silva & Pasko, 2013). The
+non-LTE conductivity regime encompasses temperatures lower than ∼10,000 K. The models in this category
+(cited in this paragraph) do not account for photoionization, which is important at the high temperatures
+present in the return stroke channel.
+The third category groups investigations where a semiempirical expression for the channel resistance (per
+unit length) as a function of time, R(t), has been employed in return stroke simulations. The reasoning
+behind such approach is that it is impractical to use the self-consistent gas-dynamic simulations to calculate
+the resistance of a channel that is 5 (or more) orders of magnitude longer than wider. Therefore, a paramet-
+ric dependence for R(t) facilitates the implementation of a height-dependent, transmission-line-like return
+stroke model. These investigations use expressions for R(t) derived by Barannik et al. (1975), Kushner et al.
+(1985), and others, as reviewed by De Conti et al. (2008). To the best of our knowledge, only Liang et al.
+(2014) present an effort to couple a self-consistent resistance calculation with a transmission-line-like return
+stroke model. These authors use a two-temperature plasma model to infer the electronic conductivity. The
+model does not account for channel expansion or plasma chemistry, and it is unclear how well it compares
+to the conventional gas-dynamic return stroke simulations. Nonetheless, investigations such as done by De
+Conti et al. (2008) and Liang et al. (2014) raise the need for accurate and computationally efficient models
+for R(t).
+The objective of this work is to fill a gap in the peer-reviewed literature by introducing a comprehensive—yet
+simple—model that can exemplify the plasma nature of lightning channels (section 2.1). We describe
+a series of parameterizations that allow us to capture both the low-temperature/non-LTE and the
+high-temperature/LTE regimes, account for radial expansion, and include negative-ion chemistry, at little
+computational cost (section 2.2). The model is first tested by calculating the time scale for streamer-to-leader
+transition (section 3.1), it is then validated against experimental data on the steady-state negative differential
+resistance of plasma arcs (section 3.2), and finally, compared to well-established gas-dynamic return stroke
+simulations (section 3.3). As an application of the model, we simulate optical emissions of rocket-triggered
+lightning and compare to the experimental findings of Quick and Krider (2017) (section 3.4).
+2. Model Formulation
+2.1. Basic Equations
+In this work we describe the minimal model to qualitatively capture the consequences of the plasma nature
+of lightning channels. The key simplification here is to solve a set of zero-dimensional equations (i.e., with
+zero spatial dimensions) that describe the temporal dynamics of the plasma in a given cross section of the
+channel. Starting from a general 3-D problem, we can progressively reduce the dimensionality of the system.
+A schematical representation of the model is given in Figure 1a. It can be assumed that the lightning channel
+is a long cylinder. The axial symmetry indicates that the plasma conditions do not depend on the polar
+coordinate. Furthermore, the 2-D long cylinder geometry can be reduced to a 1-D radial one, by noting that
+variations along the channel have significantly larger length scales than along the radial direction. Thus,
+the change in plasma properties are driven by the conduction current created by the overall lightning tree
+dynamics and merely imposed in that channel section. Finally, the 1-D radial dynamics can be averaged
+over to produce self-similar solutions of average channel properties. The minimal set of equations can be
+written as follows:
+E = RI =
+I
+𝜎𝜋r2
+c
+=
+I
+e𝜇ene𝜋r2
+c
+(1)
+DA SILVA ET AL.
+9444
+
+=== PAGE 4 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 1. (a) Schematical representation of how the model simulates a cross sectional area of the lightning channel,
+provided only the current passing through that region I(t) and the channel initial conditions. (b) Current waveforms
+adopted in this study: constant current versus four-parameter pulsed profile. (c) Radial temperature profile and
+corresponding channel expansion. Lightning leader channels are surrounded by streamer zones and corona sheaths,
+which are not depicted in panel (a).
+𝜌mcp
+dT
+dt = 𝜂T𝜎E2 −4𝜅T
+r2
+g
+(T −Tamb
+) −4𝜋𝜖
+(2)
+dne
+dt = (𝜈i −𝜈a2 −𝜈a3
+) ne + 𝜈dnn + kepn2
+LTE −kepne
+(ne + nn
+)
+(3)
+dnn
+dt =
+(
+𝜈a2 + 𝜈a3
+)
+ne −𝜈dnn −knpnn
+(
+ne + nn
+)
+(4)
+dr2
+c
+dt = 4Da
+(5)
+dr2
+g
+dt = 4𝜅T
+𝜌mcp
+(6)
+Equation (1) is the Ohm's law applied to the channel's cross section, which relates the axial electric field E
+to the electrical current I, via the resistance per unit channel length R = 1∕𝜎𝜋r2
+c, where 𝜎is the electrical
+conductivity and rc is the plasma channel or current-carrying radius. (For the remainder of this manuscript,
+we refer to the resistance per unit channel length R as simply the resistance.) The electrical conductivity
+is given by 𝜎= e𝜇ene under the assumption that only the electron contribution is important, where e is the
+electronic charge, 𝜇e is the electron mobility, and ne is the electron density. This is a reasonable approxima-
+tion because the ion mobility is of the order of 10−4 m2·V−1·s−1 (at 1 atm), while the electron mobility is 2–4
+orders of magnitude larger in the range of typical electric fields present in electrical discharges (see, e.g.,
+Figure 3a).
+Equation (2) describes the rate of change of air temperature T, where 𝜌m is the air mass density and cp is
+the specific heat at constant pressure. The first term on the right-hand side is the rate of Joule heating of air,
+where 𝜂T ≃10% is the fraction of electron Joule heating power contributing to air heating. The second term
+represents cooling due to heat conduction, where rg is the thermal radius (delimiting the hot air region), 𝜅T
+is the thermal conductivity, and Tamb = 300 K is the ambient air temperature. The third term corresponds to
+DA SILVA ET AL.
+9445
+
+=== PAGE 5 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 2. The solid lines show the local-thermodynamic equilibrium (LTE) properties of air as a function of temperature used in the present paper. (a) Mass
+density 𝜌m, (b) specific heat at constant pressure cp, (c) the product 𝜌mcp, (d) thermal conductivity 𝜅T, (e) electrical conductivity 𝜎LTE, and (f) net emission
+coefficient 𝜖for an optically thin plasma. The red dashed line in panels (a) and (c) show the ideal gas law trend 𝜌m ∝1∕T. In the original references, data are
+only available for temperatures to the left of the vertical dash-dotted line. For higher temperatures, we perform an analytical extrapolation using the data in the
+range highlighted in green. The air-plasma properties shown in the figure are taken from Boulos et al. (1994, pp. 413–417), unless otherwise noted.
+energy loss due to radiative emission, where 𝜖is the net radiation emission coefficient. Equation (2) assumes
+isobaric air heating and neglects cooling by convection.
+Equation (3) describes the change in electron density ne. The first term on the right-hand side describes
+the rate of change due to field-induced, electron-impact processes, where 𝜈i, 𝜈a2, and 𝜈a3 are the ionization,
+two-, and three-body attachment frequencies, respectively. The second term describes electron detachment
+from negative ions, where 𝜈d is the detachment frequency and nn is the negative-ion density. The third term
+describes the effective rate of thermal ionization, where kep is the rate coefficient for electron-positive ion
+recombination, and nLTE is the electron density in local thermodynamical equilibrium (LTE), defined as
+nLTE = 𝜎LTE∕e𝜇e. The LTE conductivity 𝜎LTE is only a function of temperature (see, e.g., Figure 2e). The fourth
+DA SILVA ET AL.
+9446
+
+=== PAGE 6 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+term represents plasma decay due to electron-positive ion recombination. Charge neutrality is assumed;
+thus, the positive-ion density is equal to ne + nn.
+Equation (4) describes the evolution of an effective or generic negative-ion density nn. This quantity rep-
+resents O−and O−
+2 , the dominant negative ions in atmospheric discharges. These species are created by
+two- (𝜈a2) three-body attachment (𝜈a3), respectively. The last term in equation (4) represents a sink of
+negative ions due to negative-positive ion recombination, where knp is the corresponding rate coefficient.
+Equations (5) and (6) describe the rate of expansion of the current-carrying radius rc and of the thermal
+radius rg, respectively, where Da is the ambipolar diffusion coefficient. For all purposes, rc represents the dis-
+charge channel radius, because it enters in the calculation of Joule heating power deposited in the channel
+via equation (1). The parameter rg is best interpreted as a measure of the curvature of the radial temper-
+ature profile, and its only contribution in the system of equations is in the thermal conduction cooling in
+equation (2).
+The set of six equations (1)–(6) is solved to obtain the temporal dynamics of six unknowns E, T, ne, nn, rg,
+and rc, respectively. The input parameters are the source current dynamics I(t) and the initial conditions for
+the five state variables (T, ne, nn, rg, and rc), as shown in Figure 1a. The initial value of the electric field is
+given directly from equation (1).
+In order to solve equations (1)–(6), several coefficients are required. These coefficients are a function of E∕𝛿,
+T, or both. The quantity E∕𝛿is the so-called reduced electric field, where 𝛿is the reduction of air density
+in comparison to the sea level, room temperature value, defined precisely as 𝛿= 𝜌m(h, T)∕𝜌m(h = 0km, T =
+300 K); h here corresponds to the altitude above mean sea level. Figure 2 shows all LTE plasma coefficients
+used: (a) 𝜌m, (b) cp, (c) 𝜌mcp, (d) 𝜅T, (e) 𝜎LTE, and (f) 𝜖. The LTE parameters are, by definition, only function
+of temperature. Note that the assumption of isobaric heating combined with the ideal gas law would lead to
+a dependence 𝜌m ∝1∕T between mass density and temperature. This trend is shown in Figures 2a and 2c
+as a red dashed line. However, in the present work, we use the full equilibrium calculations given by Boulos
+et al. (1994), shown as blue solid lines in the figure.
+Figure 3 shows the field-dependent coefficients: (a) 𝜇e, (b) effective frequencies of electron production and
+loss processes, (c, d) recombination coefficients, and (e, f) Da. The conventional breakdown threshold is
+defined by the equality between electron-impact ionization (𝜈i) and two-body attachment (𝜈a2) in Figure 3b.
+For the coefficients used here its numerical value is Ek∕𝛿= 28.4 kV/cm. Figures 3c and 3d show both
+the electron-positive ion (kep) and negative-positive ion (knp) recombination coefficients, as a function of
+the reduced electric field and temperature, respectively. Similarly, Figures 3e and 3f show the ambipolar
+diffusion as a function of electric field and temperature, respectively.
+The coefficients have been obtained from the following references: 𝜌m, cp, and 𝜅T (Boulos et al., 1994); 𝜎LTE
+(Boulos et al., 1994; Yos, 1963); 𝜖(Naghizadeh-Kashani et al., 2002); 𝜇e (Cho & Rycroft, 1998); 𝜈i and 𝜈a2
+(Benilov & Naidis, 2003); 𝜈a3 (Morrow & Lowke, 1997); 𝜈d (Luque & Gordillo-Vázquez, 2012); kep and knp
+(Kossyi et al., 1992); and Da is defined by the Einstein relation (Raizer, 1991, p. 20). Both kep and Da effectively
+depend on the electron temperature Te. The expression for Te(E∕𝛿, T) is taken from Vidal et al. (2002). The
+rate coefficients are given for an air composition of 80% N2 and 20% O2. All rate coefficients used in this
+manuscript have been summarized in the form of two Matlab functions and made publicly available online
+(da Silva, 2019a).
+2.2. Key Assumptions
+1. Externally driven electrical current. A key assumption of the model is that the electrical current is gener-
+ated by the overall lightning discharge electrodynamics and merely imposed to the channel cross section
+of interest. This allows one to calculate the channel properties for a given constant or pulsed current
+waveform. Here we use two types of waveforms: a constant current (in sections 3.1 and 3.2) and a
+four-parameter pulsed current waveform (in sections 3.3 and 3.4). The pulsed current waveform quali-
+tatively captures most impulsive processes taking place in the lightning channel, and it is given by the
+following mathematical expression:
+I(t) =
+{ Ip t∕𝜏r
+if
+t ≤𝜏r
+(Ip −Icc) exp(−t∕𝜏f) + Icc if
+t > 𝜏r
+(7)
+DA SILVA ET AL.
+9447
+
+=== PAGE 7 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 3. Electric-field-dependent coefficients used in this investigation. (a) Electron mobility 𝜇e. (b) Effective frequencies of electron production and loss
+processes 𝜈i, 𝜈a2, 𝜈a3, and 𝜈d, from equation (3). (c, d) Recombination coefficients kep and knp. (e, f) Ambipolar diffusion coefficient Da. Panel (c) shows the
+recombination coefficients as a function of E∕𝛿for two different temperature values. Contrastingly, panel (d) shows the same coefficients as a function of T for
+two values of E∕𝛿. The same strategy is used to display Da in panels (e) and (f). Panel (d) also shows the rate coefficient for three-body electron-positive-ion
+recombination (electrons are the third body), or more precisely kep3ne, with ne = 1020 m−3. This process is not included in the model, and the coefficient is just
+shown for comparison with the two-body rate. Expressions for the rate coefficients shown in this figure are given by da Silva and Pasko (2013); see text for
+references.
+The four parameters in the waveform are peak current Ip, rise time 𝜏r, fall time 𝜏f, and continuing current
+Icc. These four parameters can be adjusted to represent a first or subsequent return stroke with or without
+continuing current. They can also be adjusted to allow the model to simulate the surge current injected
+in the leader channel following the stepping process (see, e.g., Winn et al., 2011), a dart leader reioniza-
+tion wave, or ICC pulses happening during the initial continuous current (ICC) stage of a rocket-triggered
+lightning flash. A schematical representation of this waveform is given in Figure 1b. It should be noted
+that several different analytical functions have been used to simulate the current waveform propagating
+through the lightning channel, such as the Heidler function (Heidler, 1985; Rakov & Uman, 1998), the dou-
+ble exponential (Bruce & Golde, 1941), or the asymmetric Gaussian (e.g., da Silva et al., 2016). The model
+DA SILVA ET AL.
+9448
+
+=== PAGE 8 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+can handle any of them as input; equation (7) is chosen for its simplicity and to facilitate the comparison
+with the work of Plooster (1971) and Paxton et al. (1986) in section 3.3 below.
+The overall strategy of prescribing I(t) and calculating the channel properties has been success-
+fully employed by a number of researchers to investigate the dynamics of streamer-to-leader and
+streamer-to-spark transition (Aleksandrov et al., 2001; da Silva & Pasko, 2012; Gallimberti et al., 2002;
+Popov, 2003) and to simulate the channel decay following a return stroke (Aleksandrov et al., 2000; Hill,
+1971; Paxton et al., 1986; Plooster, 1971). Although insightful, this strategy does not reveal the full lightning
+electrodynamics, because changes in the plasma conductivity should feedback into how much current is
+flowing in the channel. However, the approach used here allows us to provide a detailed characterization
+of the plasma-channel nonlinear resistance R(t) for a given current I(t). This manuscript should be seen
+as an initial effort toward quantifying the effects of the nonlinear plasma resistance into the overall elec-
+trodynamics of lightning leaders. Future investigations can leverage this model by replacing equation (1)
+with lumped or distributed circuit equations that describe the lightning discharge tree.
+2. Averaged radial dynamics. The radial profile of temperature is assumed to follow a step function so that
+T(r) = T for r ≤rg and T(r) = Tamb for r > rg. The radial expansion is given by an increase of rg at a rate
+given by equation (6). It is assumed here that the expansion rate is determined by thermal conduction
+or, in other words, the radial temperature profile follows the equation 𝜕T∕𝜕t = k∇2T, where k = 𝜅T∕𝜌mcp.
+The solution for this equation under a delta function initial condition is T(r, t) = exp(−r2∕4kt)∕
+√
+4𝜋kt. The
+solution is a Gaussian function with half-width rg =
+√
+4kt. Taking the time derivative of this expression,
+one obtains the expansion rate of the thermal radius in equation (6).
+The second term in the right-hand side (rhs) of equation (2) is the spatially averaged Laplacian of temper-
+ature, that is, the rhs of the heat conduction equation. The method for evaluating that term is illustrated
+in Figure 1c. It is assumed that the thermal conduction-driven expansion conserves the area under the
+curve in Figure 1c, or the quantity A = (T −Tamb)𝜋r2
+g. Therefore, 𝜕T∕𝜕t|thermal
+conduction is determined from set-
+ting 𝜕A∕𝜕t = 0. This is a rather robust assumption since it is virtually equivalent to enforcing energy
+conservation. However, in reality, the shape of the profile is not preserved as assumed here.
+Similar results are obtained by assuming that the plasma distribution expands with ambipolar diffusion,
+leading to the expansion rate given in equation (5). In this case, the conserved quantity is A = ne𝜋r2
+c,
+or simply the number of electrons per unit channel length. Conservation of A in this case is equivalent
+to conservation of mass. This analysis also yields a radially averaged ambipolar diffusion sink term in
+equation (3). However, this loss process is negligible in comparison to chemically driven losses and, there-
+fore, it is not included in equation (3). Our considerations here are similar to Braginskii's (1958), where
+the plasma channel boundary is assumed to behave as a moving piston that “snowplows” the ambient gas.
+Both models yield a channel radius expansion as rc ∝
+√
+t, but Braginskii's expansion rate is not deter-
+mined by ambipolar diffusion. In a comparison between several semiempirical models of the lightning
+return stroke resistance, De Conti et al. (2008) concluded that the model accounting for channel expan-
+sion rc ∝
+√
+t effects in the resistance yielded the most robust return stroke radiated electromagnetic field
+signatures.
+3. Thermal ionization rate. At temperatures of several thousand Kelvin, the plasma-channel composition is
+roughly made of equal parts electrons and NO+ ions (Aleksandrov et al., 1997; da Silva & Pasko, 2013;
+Popov, 2003). The NO+ ions are formed by associative ionization of N and O atoms at a rate F = kassocnOnN.
+The plasma density is dictated by a balance between associative ionization and electron-positive ion recom-
+bination, that is, by F = kepnenNO+ ≈kepn2
+e. Without knowing the precise rate F, we know that at high
+temperatures this equation should yield the LTE conductivity given in Figure 2e, or the corresponding elec-
+tron density nLTE = 𝜎LTE∕e𝜇e. This can be achieved by setting the rate of thermal (associative) ionization
+to be equal to F = kepn2
+LTE, as done in equation (3).
+Therefore, equation (3) is designed to essentially have two different modes of operation. At low
+(near-ambient) temperatures, the plasma population balance is driven by electron-impact ionization,
+attachment, and detachment, that is, the typical chemistry considered in the streamer breakdown of short
+air gaps (da Silva & Pasko, 2013; Flitti & Pancheshnyi, 2009; Liu & Pasko, 2004; Naidis, 2005; Pancheshnyi
+et al., 2005). However, at high temperatures (≳10,000 K) the equation yields the LTE conductivity 𝜎LTE(T),
+in alignment with the typical approach used for the simulation of free-burning arcs (Chemartin et al.,
+2009; Lowke et al., 1992) or used in gas-dynamic return stroke simulations (Aleksandrov et al., 2000;
+Paxton et al., 1986; Plooster, 1971). It is not possible to state exactly what is the minimum temperature at
+which the assumption of LTE regime yields accurate calculations. Both T and Te depend on the history of
+DA SILVA ET AL.
+9449
+
+=== PAGE 9 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+energy deposition and losses in the channel, which in its turn depend on the electric field and the elec-
+tron density. In this manuscript, we loosely give the value of 10,000 K as an estimate. This is the value at
+which the electron temperature is only 5% and 50% larger than the neutral gas one for electric fields of 10
+and 1000 V/m, respectively. In the present work, the electron temperature is obtained under the assump-
+tion that the electron energy balance equation is in steady state. Therefore, yielding the simple relation
+Te = T + f(E∕𝛿), where the function f(E∕𝛿) ∝(E∕𝛿)0.46 is taken from Vidal et al. (2002). Essentially, this
+equation asserts that the non-equilibrium results from the presence of an electric field in the discharge
+plasma and that equilibrium is only achieved when E = 0.
+In some types of plasmas the high-temperature density is given by a balance between electron impact ion-
+ization (driven by high T and not high E∕𝛿) and three-body electron-positive ion recombination (electrons
+are the third body). One such example are Argon arc discharges at atmospheric pressure (see, e.g., Sanson-
+nens et al., 2000; Tanaka et al., 2003). In this case, plasma losses would happen at a rate ≈kep3n3
+e, and using
+the assumptions discussed in the last two paragraphs, the plasma production rate would be ≈kep3n3
+LTE,
+where kep3 is the three-body electron-positive ion recombination rate coefficient given in units of m6/s.
+Owing to the cubic power law dependence, three-body electron-positive ion recombination is important
+when the plasma density is high. In this work, we assume that the high-temperature balance is given by the
+two-body processes, because they are the dominant ones in the temperature range between 2,000–9,000 K
+(i.e., in the transition to LTE regime), as discussed by Bazelyan and Raizer (2000, pp. 75–80) and Aleksan-
+drov et al. (2001). To verify that this assumption is true, we first plot the rate coefficients kep ∝T−1.5
+e
+and
+kep3 ∝T−4.5
+e
+in Figure 3d with rate coefficients taken from Kossyi et al. (1992) for an air plasma. Figure 3d
+actually shows kep3ne so that the units match, with ne = 1020 m−3, a typically large value in our simula-
+tions. It can be seen that due to the weaker fall off with temperature, two-body recombination increasingly
+dominates over three-body in the temperature range of interest. Second, we show later in section 3.3 quan-
+titative comparisons between the two rates for specific simulation results obtained with our model, further
+justifying our use of two-body process rates.
+4. Negative ion chemistry. Equation (4) describes the evolution of an effective or generic negative-ion density
+nn, representing O−(created by two-body attachment) and O−
+2 (created by three-body attachment), the
+dominant negative ions in ambient-temperature discharges. In the hot lightning channel, negative ions
+disappear, and the plasma composition is given by a balance of positive ions and electrons. By comparing
+equations (3) and (4), we can see that attachment works as a sink in the former, but as a source in the lat-
+ter. Detachment plays the opposite role. Therefore, the attachment-detachment cycle does not represent
+a true plasma loss. Effectively, electrons can be thought to be temporarily stored in negative ions to be
+released at a later time, after substantial accumulation. It is assumed here that O−
+2 created by three-body
+attachment quickly converts into O−in collisions with atomic oxygen favored by elevated temperatures
+in the lightning channel (da Silva & Pasko, 2013, Figure 11a). Therefore, detachment is dominantly
+driven by collisions between O−and N2 (Luque & Gordillo-Vázquez, 2012; Rayment & Moruzzi, 1978).
+These assumptions allow us to account for effects of negative-ion chemistry in a simple yet reasonably
+accurate manner.
+5. Fast air heating. The coefficient 𝜂T in the first term on the rhs of equation (2) is the fraction of elec-
+tronic power (or Joule heating rate 𝜎E2) that is directly transferred into random translational kinetic
+energy of neutrals and, thus, contributes to air heating. This quantity has been calculated to be 𝜂T ≃0.1
+at near-ambient temperatures (da Silva & Pasko, 2013; da Silva, 2015), largely arising from surplus energy
+from the quenching of excited electronic states and molecular (electron-impact) dissociation, which
+consist the so-called fast air heating mechanism (Popov, 2001, 2011; da Silva & Pasko, 2014).
+Most of the remainder electronic power is spent into the excitation of vibrational energy levels of nitrogen
+molecules. However, as temperature increases, rates of vibrational-translational energy relaxation quickly
+accelerate, effectively making 𝜂T
+≈1 for temperatures of 2,000 K and above (provided that radiative
+losses are treated in a separate sink term in the rhs of the energy balance equation). This delayed vibra-
+tional energy relaxation is typically described with an extra equation for the total vibrational energy of N2
+molecules. In the present work, we capture this phenomenology, without the need for an extra equation,
+by adopting a parametric dependence of 𝜂T on temperature, given by 𝜂T = 0.1+0.9[tanh(T∕Tamb−4)+1]∕2.
+The added second term in this expression simulates the acceleration of vibrational energy relaxation,
+yielding 𝜂T = 1 for T >2,000 K with a smooth ramp transition between 1,000–1,500 K.
+DA SILVA ET AL.
+9450
+
+=== PAGE 10 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+3. Results and Discussion
+3.1. Streamer-to-Leader Transition
+The most fundamental step in the formation of a lightning channel is the streamer-to-leader transition.
+Streamers are the precursor stage. They are thin filamentary discharge channels that propagate as a non-
+linear electron-impact ionization wave, self-enhancing the electric field at its tips. Their conductivity is of
+the order of 0.1–1 S/m. They require electric fields higher than 17% of the conventional breakdown thresh-
+old for stable propagation. Streamer lifetimes are rather short, approximately tens of microseconds, limited
+by attachment to oxygen molecules. Leaders are a necessity for the breakdown of air gaps longer than one
+meter (Bazelyan & Raizer, 2000, p. 59). It takes several milliseconds for a leader to come from the cloud to the
+ground. The only way to keep the leader channel conductive for so long is by substantially heating the air. In
+the hot air plasma, attachment loses its importance; instead, the electron density decays via electron-positive
+ion recombination, which is substantially slower. The transition between streamer and leader happens in
+a region in space called stem, a converging point where several streamers in a streamer corona are rooted.
+In this region the small current carried by individual streamers can add up to values ≳1 A to produce air
+heating and create a leader channel.
+da Silva and Pasko (2013) developed a first-principles model to investigate the dynamics of streamer-to-
+leader transition. It consists of four main blocks: (1) a set of fully nonlinear gas-dynamic equations that
+described the heating and radial expansion of the neutral gas; (2) a detailed kinetic scheme accounting for
+the most important processes in an air discharge plasma; (3) energy exchange between charged and neu-
+tral particles accounting for the partitioning of electronic power between elastic collisions, and excitation
+of vibrational and electronic states; and (4) delayed vibrational energy relaxation of nitrogen molecules. da
+Silva and Pasko's (2013) model was validated against streamer-to-spark transition time scales measured in
+centimeter-long laboratory discharges ( ˇCernák et al., 1995; Larsson, 1998). That model was also applied to
+simulation of leader speeds at reduced air densities and for interpretation of the phenomenology of gigan-
+tic jets (da Silva & Pasko, 2012), as well as to study the mechanism of infrasound emissions in sprites
+(da Silva & Pasko, 2014). Figure 4a shows, as discontinuous traces, the air heating rate calculated with da
+Silva and Pasko's (2013) model with an assumed Gaussian initial distribution of electron density in the
+streamer channel. The peak ne value is 2 ×1020 m−3 and the e-folding spatial scale is rc = 0.3, 0.5, and 1
+mm, respectively. The streamer-to-leader transition time scale 𝜏h is defined as the time required to heat the
+channel up to 2000 K; the heating rate shown in the figure is simply 1∕𝜏h. The 2000-K threshold is chosen
+because when the channel reaches this temperature level a thermal-ionizational plasma instability is trig-
+gered: vibrational relaxation is accelerated, temperature raises very sharply, N + O associative ionization
+starts to take place, and transition to leader mode is unavoidable.
+The present work's goal is to propose the minimal physical model to describe the dynamics of the leader
+plasma. As discussed in section 2.2, the model uses a simplified plasma chemistry and parameterized radial
+dynamics. As a means of validation, in Figure 4 we compare the present model with the simulations of
+da Silva and Pasko (2013). Figure 4a uses the same initial conditions as the previous work and an ini-
+tial current-carrying radius rc = 0.5 mm. The figure shows order-of-magnitude agreement between the two
+models. However, there is an inherently different slope between the two curves, attributed to the multiple
+parameterizations and simplifications introduced in this paper. The other three panels in the figure show
+the effects of the initial conditions in the air heating rate: ne (b), rc (c), and rg (d). The current-carrying
+radius is the parameter that has the largest influence on the heating rate (Figure 4c). The thermal radius rg
+has no effect on the heating rate at all (Figure 4d), because this quantity is exclusively related to the cooling
+rate of the channel (see equation (2)), which is negligible in submicrosecond time scales. The dependence
+on initial electron density is slightly more complicated. The heating rate is ∝∫
+𝜏h
+0
+𝜎E2dt which, according
+to equation (1) is also ∝∫
+𝜏h
+0
+I2∕nedt. The inverse 1∕ne dependence can be qualitatively seen when com-
+paring the 1020- and 1022-m−3 cases. But reducing the initial electron density tends to increase the electric
+field according to Ohm's law. If the electric field goes beyond Ek, ionization increases ne until the field drops
+down to the Ek level. This self-regulatory mechanism imposes a maximum heating rate given by the 1018-
+to 1020-m−3 curves in Figure 4b.
+For the sake of comparison, we have repeated the calculations shown here with a full LTE version of the
+model. This is done by replacing equations (3) an (4) with 𝜎= 𝜎LTE and by setting 𝜂T = 1. The calculated
+air heating rate is in the range of 1012–1015 s−1 for currents between 1 and 100 A. They are not shown in
+Figure 4 because they lie completely outside of the vertical-axis limits. This result indicates that a full-LTE
+DA SILVA ET AL.
+9451
+
+=== PAGE 11 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 4. Calculated heating rate (1∕𝜏h) leading to the conversion of a streamer into a leader channel. The title in the
+four panels list the initial conditions for electron density ne (ne in the figure), current-carrying radius rc (rc), and
+thermal radius rg (rg) used in the simulations. The ambient neutral temperature is 300 K, and there are no negative
+ions initially. Panel (a) shows as discontinuous traces the calculation of da Silva and Pasko (2013) for the same initial
+conditions, but three different values of rc. The gray shaded area delimiting the calculations of da Silva and Pasko
+(2013) is repeated in all four panels for comparison. Panels (b)–(d) emphasize the effect of changing the initial
+conditions for ne (b), rc (c), and rg (d).
+model completely overestimates the air heating rate, and cannot capture the finite streamer-to-leader (or
+to-spark) transition time scale, well known from laboratory studies to be a fraction of 1 μs ( ˇCernák et al.,
+1995; Larsson, 1998). The reason for the unreasonably high air heating rate of a full-LTE model lies in the
+fact that the LTE conductivity at 300 K is substantially lower than the typical conductivity in a streamer
+channel (see Figure 2e). Since conductivity is lower, the resistance per unit length R is larger, and so is the
+Joule heating rate RI2, which is the same argument presented when discussing Figure 4b.
+In summary, the present model compares very well to a first-principles theoretical simulation that has been
+validated with spark data from laboratory discharges. The proposed computer-simulation tool is able to
+account for the finite time scale of streamer-to-leader transition, something that a full-LTE model cannot.
+The following input parameters are used as initial conditions in all simulations below, unless otherwise
+noted: ne = 1020 m−3, rc = 0.5 mm, rg = 5 mm, nn = 0, T = 300 K.
+DA SILVA ET AL.
+9452
+
+=== PAGE 12 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 5. (a) Temporal dynamics of resistance in a discharge channel for several current values. Solid and dashed lines
+show the contrast between full model versus suppressed channel expansion, respectively. The figure also shows the
+data by Tanaka et al. (2000) as a solid black line, with the gray shaded area marking ±50% variability. (b–d) Resistance
+value at 10 ms as a function of current. Panel (b) also shows the data from Tanaka et al. (2000) at 10 ms (square with
+±50% error bar), as well as, the steady-state arc resistance measured by King (1961) (black solid line with ±50% gray
+shaded band). Panels (b)–(d) emphasize the effect of changing the initial conditions for ne (b), rc (c), and rg (d), with
+the initial conditions being listed in the panel titles and legends. The gray shaded band marking the results from King
+(1961) are repeated in panels (b-d) for comparison with our simulations.
+3.2. Steady-State Negative Differential Resistance
+The behavior of the steady-state resistance of arc channels has been used to discuss the phenomenology
+of lightning channels (Hare et al., 2019; Heckman, 1992; Krehbiel et al., 1979; Mazur & Ruhnke, 2014;
+Williams, 2006; Williams & Heckman, 2012; Williams & Montanyà, 2019). Steady-state plasma arcs exhibit
+the so-called negative differential resistance, that is, the resistance decreases with increasing electrical cur-
+rent. Such behavior is reproduced in our simulations and shown in Figure 5. Figure 5a shows the temporal
+evolution of resistance in the discharge channel for several values of electrical current between 1 A and
+10 kA. It is easy to see that, owing to channel expansion, there is no true steady-state resistance. A con-
+stant value for the steady-state resistance can only be obtained if channel expansion is suppressed (compare
+the solid and dashed lines). At low currents (see the 1-A curve), one can start to see the channel recovery
+DA SILVA ET AL.
+9453
+
+=== PAGE 13 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Table 1
+Fit Parameters for the Resistance per Unit Channel Length Formula R = A∕Ib
+Reference
+Current range (A)
+Time scale (s)
+A (Ω Ab/m)
+b
+Mean fit error (%)
+This Work
+100–104
+10−2
+4.27×103
+1.18
+35
+This Work
+100–104
+1
+4.81×103
+1.37
+74
+This Work: Region I
+100–101
+10−2
+1.24×104
+1.84
+9
+This Work: Region II
+101–103
+10−2
+2.82×103
+1.16
+4
+This Work: Region III
+103–104
+10−2
+0.18×103
+0.75
+1
+King (1961)
+100–104
+—
+2.87×103
+1.16
+25
+Bazelyan and Raizer (1998)
+—
+—
+3×104
+2
+—
+starting as early as 0.1 ms. The recovery in this case is due to the fact that the channel cools down to a suffi-
+cient level that three-body attachment becomes important, accelerating the rate of plasma density depletion.
+For currents higher than 10 A, the resistance is still decreasing at the 0.1 s mark; in some cases after a partial
+recovery. In Figure 5a we also show data from Tanaka et al. (2000) used by Chemartin et al. (2009) to validate
+their 3-D free burning arc simulations. Tanaka et al. (2000) report on 1.6-m-long arcs with 100-A current.
+Their measurements are shown in Figure 17 of Chemartin et al. (2009). We obtain a good agreement between
+our simulations and the measurements despite the fact that the 3-D tortuous nature of the arc channel is
+neglected in the present work.
+For the purpose of evaluating the negative differential resistance behavior predicted by our simulations, we
+evaluate the resistance (per unit length) at 10 ms for several different values of electrical current. The results
+are shown in Figure 5b alongside measurements from King (1961). We chose to compare our simulations to
+King's measurements because this work has been featured in a number of manuscripts in lightning-research
+literature (e.g., Heckman, 1992; Mazur & Ruhnke, 2014; Williams, 2006; Williams & Heckman, 2012). The
+data from King (1961) is shown as a black solid line with a ±50% variability gray shaded band. The gray
+band is repeated in panels (b)–(d) for comparison with our simulations. The time instant of 10 ms is chosen
+because it is when the time-dependent data from Tanaka et al. (2000) (shown as a square with ±50% error
+bar) best aligns with King's curve. Our calculations in Figure 5a show good agreement with King's curve;
+the average difference between the two is 40%. Figures 5b–5d show the effects of the initial conditions in the
+steady-state resistance: ne (b), rc (c), and rg (d). It can be seen that changes in the initial conditions have very
+little impact on the resistance in the 10-ms time scale. It is as if the channel “forgets” the initial conditions
+(Aleksandrov et al., 2001). Given the uncertainty in determining the initial conditions of the channel, this
+result lends robustness to the resistance calculations shown hereafter. However, in shorter time scales the
+resistance R does depend on the initial conditions. Similarly to the discussion in section 3.1, the dependence
+on ne and rg is weak, but the dependence on rc can be more noticeable. The dependence on the initial channel
+radius becomes weaker and weaker at higher currents. As an example, at the 10-μs mark, we find that the
+ratio R(rc=2 mm)/R(rc=0.5 mm) is of the order of 700 for a constant current of 10 A. The same ratio is only
+0.63 for a current of 1,000 A.
+The dependence of the resistance on electrical current can be approximated by the analytical formula
+R = A∕Ib, where A and b are positive constants. It is easy to see that with this dependence dR∕dI < 0 always,
+in accordance with the terminology “negative differential resistance.” The limiting case b = 1 corresponds
+to a constant steady-state electric field inside the channel (with numerical value equal to A). We have eval-
+uated the fit parameters that best match our model for the standard set of initial conditions (shown in the
+title of Figure 5a). The results are shown in Table 1 alongside the fit parameters for the King's curve and also
+values given by Bazelyan and Raizer (1998). It can be seen that the exponent b that best fits both the present
+work and King (1961) are very close to each other (b = 1.16–1.18). The empirical trend given by Bazelyan
+and Raizer (1998) has a substantially steeper slope (b = 2). If we run the simulation for a longer time, up to
+1 s, the power law index increases from 1.18 to 1.37 (see second row in Table 1). However, the mean fit error
+doubles indicating that the curve deviates further from the power law approximation.
+It can be seen from Table 1 that fitting the power law dependence to a four-decade current range produces
+errors of 35–74%. A better fit can be produced by braking down the current range in three regions: (I) 100–101
+A, (II) 101–103 A, and (III) 103–104 A. The three regions are marked in Figure 5c. It can be seen in Table 1
+DA SILVA ET AL.
+9454
+
+=== PAGE 14 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+that the three regions have different power law indexes, progressively lower as current increases. Detailed
+analysis of the temporal evolution of energy deposition in the channel reveals that the steady state is given by
+different mechanisms in the three regions. In Region I the steady state is given by a balance of Joule heating
+and heat conduction, that is, between the first and second terms in the right-hand side of equation (2).
+Meanwhile, In Region III the steady state is given by a balance with radiative emission, that is, between
+the first and third terms in the right-hand side of equation (2). Region II is marked by a comparable role
+between the two loss processes; radiative emission is important in the submillisecond time scale, while heat
+conduction is significant at later stages.
+3.3. Energy Deposition in Return Strokes
+The return stroke follows the attachment of lightning leader channels to ground structures. In the case of a
+negative cloud-to-ground discharge, the return stroke effectively lowers several coulombs of negative charge
+originally deposited along the downward propagating stepped leader. The high-current return stroke wave
+(with typically tens of kiloamperes) rapidly heats the channel to peak temperatures of the order of 30,000
+K, emitting intense optical radiation, and creating a channel expansion shock wave (that produces audible
+thunder). According to Rakov and Uman (1998), models that describe the lightning return stroke can be
+divided into four categories: gas-dynamic or physical, electromagnetic, distributed-circuit, and engineering
+models. The basic set of equations described in this manuscript fits into the first category, where the cur-
+rent flowing through the channel is an input parameter and all other channel properties can be calculated
+from first principles. Some of the most well-accepted investigations within this framework are the papers
+by Plooster (1971) and Paxton et al. (1986). These authors solve the hydrodynamic equations of motion for
+atmospheric-pressure air in a Lagrangian frame of reference. A description of this simulation approach,
+which shows contemporary versions of the pertinent equations, is given by Aleksandrov et al. (2000). The
+model resolves the 1-D radial profiles of all state variables and captures the shock wave expansion as driven
+by ohmic heating. The plasma is assumed to be in LTE and the conductivity is simply 𝜎= 𝜎LTE(T). These
+models also describe the radial transport of radiation, and primarily differ by its implementation and com-
+prehensiveness. Plooster (1971) used a single temperature-independent opacity to obtain radiation loss and
+absorption in each radial grid point, while Paxton et al. (1986) used a detailed multigroup radiative transport
+algorithm using a diffusion approximation. A detailed discussion on plasma radiative transport is given by
+Ripoll et al. (2014a).
+In Figure 6a we present a comparison between our model's results and the seminal works of Plooster (1971)
+and Paxton et al. (1986). The current waveform has the qualitative shape depicted in Figure 1b, with a rise
+time of 5 μs and a fall time of 50 μs (or simply written as 5/50 μs). The peak current is 20 kA, a typical value
+for first return strokes, and no continuing current is incorporated. The current waveform is the same one
+used in the two papers for the simulation case shown in Figure 8 of Paxton et al. (1986). We generate initial
+conditions by starting the simulation with the standard streamer-like channel parameters used in section 3.1
+and running a constant 10-A current through the channel during 4 μs. This strategy ensures that the channel
+has the properties of a leader discharge prior to the return stroke. These initial conditions are rc = 1 mm, rg
+= 1 cm, ne = 9×1017 m−3, and T = 5000 K. Additionally, instead of using the value of 𝜌m(T = 5000K) for
+the air mass density, the ambient value 𝜌m(T = 300K) = 0.7 kg/m3 is used. These initial conditions are very
+similar to the ones used in the aforementioned references. Note that even a steady current as low as 10 A can
+produce a leader with temperature of ∼5,000 K. This value is within the estimate for the predart and postdart
+leader channel temperatures provided by Rakov (1998), which are 3,000 K and 20,000 K, respectively. It
+can be seen from Figure 6a that our model compares very well with simulation results of Plooster (1971),
+predicting a peak temperature of 36,000 K. The mean difference between the two curves is 3%.
+Both curves (Plooster's and ours) deviate from the results of Paxton et al. (1986). It can be seen from Figure 6a
+that a better agreement with Paxton et al. (1986) can be found by simply multiplying the radiative emis-
+sion coefficient (last term in equation (2)) by a factor of 10. This fact can be better understood by looking
+at the energy deposition in the return stroke channel, depicted in Figure 6b. The figure shows (in order)
+the four terms in the energy equation (2): the internal energy is given by 𝜌mcpT𝜋r2
+c, the Joule heating by
+∫𝜂T𝜎E2𝜋r2
+cdt, the thermal conduction by ∫
+4𝜅T
+r2
+g
+(
+T −Tamb
+)
+𝜋r2
+cdt, and the radiative emission by ∫4𝜋𝜖𝜋r2
+cdt.
+It can be seen that the channel's temperature is dictated by a balance between Joule heating and cooling by
+radiative emission. Therefore, simply increasing the rate of channel cooling by radiation can lower the peak
+temperature and provide a better agreement with Paxton's results. As mentioned above, the models pre-
+DA SILVA ET AL.
+9455
+
+=== PAGE 15 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 6. (a) Evolution of temperature in a 20-kA return stroke channel: comparison between the present investigation and established results (Paxton et al.,
+1986; Plooster, 1971). (b) Energy deposition in the return stroke channel. The four lines, in the order listed in the figure legend, correspond to the four terms in
+the energy balance equation (2). Panel (c) is a zoom-in into the gray shaded rectangle in panel (b). Panels (d)–(f) show the radius, resistance per unit length,
+and rates of electron-positive ion recombination, respectively. Panel (f) justifies a posteriori neglecting the three-body process in equation (3).
+sented by Plooster (1971) and Paxton et al. (1986) are essentially the same and only differ by the treatment
+of radiative emission, lending further credence to the idea that peak temperatures are dictated by radiative
+emission.
+An important conclusion to be drawn here is that the effective representation of the radiative emission
+through a net emission coefficient (𝜖in Figure 2f) produces a proper description of the channel temperature
+dynamics, especially because all four curves in Figure 6a have similar qualitative shape and rate of cooling
+after the peak. Moreover, at 35 μs the total deposited energy in our simulations of 5.6 kJ/m compares well
+to the estimates of 2 and 3.8 kJ/m by Plooster (1971) and Paxton et al. (1986), respectively (see also Rakov &
+Uman, 1998, Table I). The state of the art in lightning spectroscopy is the recent investigations by Walker and
+Christian (2017, 2019). From the ratio of several atomic spectral lines recorded with 1-μs temporal resolution,
+these authors report peak temperatures ranging between 32 and 42 kK for five rocket-triggered lightning
+strikes with peak currents varying between 8.1 and 17.3 kA (Walker & Christian, 2019, Figure 4). There is
+not a clear linear correlation between peak current and peak temperature in their dataset and the average
+peak temperature between the five strikes is ≈36±4 kK. Remarkably, our work and Plooster's do a better job
+reproducing the measured peak temperatures than Paxton's. Further work is required to explain the highest
+value registered by Walker and Christian (2019), in excess of 42,000 K.
+DA SILVA ET AL.
+9456
+
+=== PAGE 16 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Figure 6d shows the channel radius as a function of time. We have verified that the proposed averaged
+radial dynamics qualitatively captures the radial expansion and also provides order-of-magnitude quantita-
+tive agreement with previous investigations alike (Braginskii, 1958; Koshak et al., 2015; Plooster, 1971). All
+of these models (including ours) predict an initial rapid channel expansion rate, leveling off when the chan-
+nel is cooling down. During the initial return stroke stage (0.5–5 μs), our calculated radius is 8–42% smaller
+than the results obtained by Braginskii (1958) and Plooster (1971), shown in Table II of Plooster (1971).
+Koshak et al. (2015) improved on the channel radial expansion rate derived by Braginskii (1958) and found a
+good agreement with Plooster (1971) at the 35-μs mark. Both investigations yielded a 1.5-cm radius at 35 μs,
+while our simulations yielded a value 57% lower. Generally, the results are in good agreement with previous
+investigations. However, it should be noted that our peak channel expansion rate is ∼500 m/s, which is a
+factor of 4 lower than in Koshak et al. (2015).
+Figure 6e presents the resistance (per unit length) as a function of time. It can be seen that the resistance
+drops by more than two orders of magnitude while the current is rising, illustrating how negative differential
+resistance works for a current changing over time. After that, while the current is decreasing exponentially
+in time, the resistance achieves a stable value between 0.6–1 Ω/m. This leveling off is in agreement with
+the trend seen in measurements (Jayakumar et al., 2006, Figure 4). Jayakumar et al. (2006) measured the
+electrical current to ground and the vertical electric field in close vicinity to a series of rocket-triggered
+lightning strikes in Florida. At the instant of peak power, these authors found resistance values between
+0.67 and 31 Ω/m in eight different strikes. In our calculations, we obtain R = 0.6 Ω/m, which is close to the
+lowest resistance value in their dataset. This value is closer to the measurements than the early estimate
+of 0.035 Ω/m by Rakov (1998). Additionally, Jayakumar et al. (2006) registered input electrical energies
+between 0.9–6.4 kJ/m, also in range with our calculations.
+Figure 6f shows the rates of electron-positive ion recombination. The figure shows a comparison between
+the rate of two- and three-body recombination with coefficients taken from Kossyi et al. (1992). The
+figure is included here to justify the model design assumptions discussed in section 2.2 (item #3). In
+the regime studied here and with the rate coefficients for an air plasma available in the literature, the
+three-body recombination rate is substantially slower than the two-body counterpart, justifying neglecting
+it in equation (3).
+3.4. Behavior of Light Emission in Return Strokes
+The net emission coefficient 𝜖describes the radiative emission in all bands of the optical spectrum, encom-
+passing the infrared, visible, and ultraviolet (Naghizadeh-Kashani et al., 2002). Most of the radiation
+escaping the plasma is in the vacuum ultraviolet range (wavelengths lower than 200 nm) and is caused by
+atomic emissions. However, this band is not easily detected because the radiation is readily absorbed by
+atmospheric-pressure air surrounding the plasma discharge (Cressault et al., 2015). Spectroscopic measure-
+ments of rocket-triggered lightning strikes show characteristic line emissions associated with neutral, singly,
+and doubly ionized nitrogen and oxygen, neutral argon, neutral hydrogen, and neutral copper (from the
+triggering wire) and present no detected molecular emissions (Walker & Christian, 2017).
+For the purposes of comparing our simulations with observations, we estimate the power (per unit channel
+length) emitted in the visible range as 𝜂vis4𝜋𝜖𝜋r2
+c, where 𝜂vis is the fraction of optical radiation emitted in
+the visible range (380–780 nm). We use a constant fraction 𝜂vis = 3% for the sake of simplicity. In reality 𝜂vis
+depends on the radial distribution of the plasma temperature and the cumulative balance of emission and
+absorption. Table 2 shows seven estimates of 𝜂vis based on different references and techniques. Perhaps the
+most pertinent is estimate #2, which is calculated by taking the ratio of 𝜖vis in the visible range calculated
+by Cressault et al. (2011, Figure 2) to 𝜖in the total optical range calculated by (Naghizadeh-Kashani et al.,
+2002, Figure 13) for an optically thin plasma. This strategy places 𝜂vis between 0.1% and 10% in the tempera-
+ture range between 3,000 and 30,000 K. Within this range, we adopt the value of 3% because it yields a good
+agreement with experimental data from Quick and Krider (2017) discussed below.
+Figure 7 shows properties of return stroke light emission and comparison to rocket-triggered lightning data
+collected by Quick & Krider (2017, Figures 15 and 16). From a 200-m distance to the lightning striking point,
+Quick and Krider (2017) recorded the luminosity of a 62-m-long channel segment near the ground. The
+radiometers used had an approximately flat spectral response in the 400- to 1,000-nm range. Figure 7a shows
+the simulated temporal dynamics of visible power and electrical current in the channel, for conditions that
+resemble the aforementioned observations. The waveform is 0.5/50 μs with a 12-kA peak current, similar to
+DA SILVA ET AL.
+9457
+
+=== PAGE 17 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+Table 2
+Fraction of Optical Power Radiated in the Visible Range by an Air Plasma
+#
+Estimation method and reference
+𝜂vis (%)
+1
+Black-body spectral radiance (Siegel, 2001, p. 22) (3,000–30,000 K)
+5.3–49
+2
+Visible 𝜖vis calculated by Cressault et al. (2011) (3,000–30,000 K)
+0.1–10
+3
+Visible radiance calculated by Cressault et al. (2015) (8,000–30,000 K)
+0.2–0.6
+4
+20-kJ/m hot air shock (Ripoll et al., 2014a, Figure 9 and section 3.1 )
+14.3
+5
+Several simulations in Table 1 of Ripoll et al. (2014a)
+4–30
+6
+Section 4.2 of Ripoll et al. (2014a)
+5.3–21.7
+7
+A 12-kA discharge (Ripoll et al., 2014b, Figures 9b and 10b)
+30
+Empirical (this work)
+3
+the median case in the data set (Quick & Krider, 2017, Table 1). Figure 7b shows the first 3 μs of light emis-
+sion, evidencing a 0.1-μs delay between the rise of current and optical emissions in the channel. Figure 7c
+shows the effects of increasing peak current, which lead to higher emitted power and longer duration of the
+light emission.
+The delay shown in Figure 7b is evaluated at the 20% of peak level. The 0.1-μs value is in excellent agreement
+with experimental results by Carvalho et al. (2014, 2015) and Quick and Krider (2017) who found delays of
+Figure 7. (a) Temporal evolution of power per unit channel length emitted by a return stroke in the visible range (left-hand side axis) and electrical current
+(right-hand side). Panel (b) is a zoom-in into the gray shaded rectangle in panel (a). (c) Visible power emitted for several different peak current values.
+(d) Visible peak power versus peak current for four different current waveforms. (e) Energy emitted in the visible range versus charge transferred to the ground
+(the integration time is 2 ms). Panels (d) and (e) show a comparison with the experimental data from Quick and Krider (2017). The big crosses indicate the
+average ± standard deviation in the dataset. The data were collected during a study conducted by the University of Arizona at the International Center for
+Lightning Research and Testing, in Camp Blanding, FL, in 2012.
+DA SILVA ET AL.
+9458
+
+=== PAGE 18 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+0.09 ± 0.05 and 0.09 ± 0.06 μs, respectively. Differently than Quick and Krider (2017), Carvalho et al. (2014)
+recorded luminosity from a 3-m-long channel segment near the ground. From such a short segment, the
+luminosity rise time is not masked by the geometrical growth of the return stroke in the field of view. The
+fact that both experimental investigations observing different channel lengths (62 and 3 m) yielded similar
+results lends robustness to the ∼0.1 μs measured delay. Furthermore, analysis of different types of pulses
+occurring in the return stroke channel (Zhou et al., 2014) and of several channel segments at different heights
+(Carvalho et al., 2015) have led to the general conclusion that current and luminosity have similar rise times
+and the delay between the two has the same order of magnitude as such time scales. More precisely, Carvalho
+et al. (2015) found that the delay is approximately linearly dependent on the current rise time according to
+the following fit formula: delay = 0.35 𝜏1.03
+rise , where 𝜏rise is the 10–90% current rise time given in microseconds.
+The fit comprises rise times between ∼0.1 μs (for return strokes) and ∼100 μs (for M components). Using
+this formula, we obtain a delay of 0.14 μs for the simulation shown in Figure 7b, once more indicating good
+agreement between simulation and measurements.
+In our simulations the delay between the rise of current and optical emissions highlighted in Figure 7b has
+a clear interpretation. It is attributed to the finite time scale of channel heating and expansion. Since the
+initial channel temperature for the simulations shown in this section is 5000 K, non-LTE effects play a minor
+role here. From equations (1) and (2), the air heating rate thus is 𝜕T∕𝜕t ≃(I2∕𝜎LTE𝜋2r4
+c −4𝜋𝜖)∕𝜌mcp. What
+determines the finite 0.1-μs delay, in a return stroke with 0.5-μs rise time, are the coefficients 𝜌m, cp, 𝜎LTE,
+and 𝜖, as well as the channel expansion rc(t). A comparison with a full-LTE version of the simulation code
+yielded a similar time delay between current and optical emissions, but the full-LTE model overestimated
+the peak optical power by a factor of 3–4.
+Figures 7d and 7e show the peak visible power versus peak current and total energy versus charge, respec-
+tively. The integration time for the charge and energy is 2 ms. The figures show simulations for different
+current waveforms and comparison with light emitted by rocket-triggered lightning. The data correspond
+to optical irradiance from 55 rocket-triggered lightning strikes (with currents and charges ranging between
+3–20 kA and 0.3–3 C, respectively) observed in Florida by Quick and Krider (2017) in 2012. The irradi-
+ance data is converted to power per unit channel length according to equation (2) in the original reference.
+The simulations use the same initial conditions as in Figure 6, and the results indicate a direct relationship
+between current and power and between total energy and charge. Additionally, the calculations (under the
+𝜂vis = 3% assumption) present good agreement with the observational data, especially near the average val-
+ues (the big crosses in the figures). The peak visible power shows little dependence on the current waveform
+parameters in the range used (𝜏r = 0.5 and 5 μs, and 𝜏f = 50 and 150 μs). The rise time also does not affect the
+relationship between energy emitted and charge transferred to the ground, shown in Figure 7e. The same
+figure also shows that strokes with a narrower current pulse (i.e., with shorter fall time) are more efficient
+in converting electrical energy into optical.
+There are two important issues that must be noted about the comparison made in Figures 7d and 7e.
+First, the radiometers used by Quick and Krider (2017) have a flat spectral response in the 400-1,000
+nm range. According to Ripoll et al. (2014a, 2014b), about twice as much energy is emitted in this range
+than in the visible, because it includes part of the infrared spectrum. Second, Quick and Krider (2017)
+state that rocket-triggered lightning strikes radiate around half as much energy as first strikes in natural
+cloud-to-ground flashes. But the simulations use initial conditions that best resemble first strikes in natu-
+ral lightning, similarly to the works by Plooster (1971) and Paxton et al. (1986). Therefore, if we attempt
+to scale the numerical results to correspond to optical power emitted in the 400-1,000 nm range (×2) by
+rocket-triggered lightning (×1/2), the factors of two cancel and the curves would stay in the same place in
+Figures 7d and 7e, which lends further credence to the comparison. Nonetheless, it should be noted that
+our numerical investigations did not capture the approximate quadratic scaling between peak luminosity
+and peak current, that is, luminosity ∝I2
+p, seen in observations (Carvalho et al., 2015; Quick & Krider, 2017;
+Zhou et al., 2014). Further work is required to explain all experimentally inferred relationships between
+current and luminosity derived from close-by observations of rocket-triggered lightning.
+When analyzing the light emission of return strokes, two additional factors must be noted. First, in
+rocket-triggered lightning there is a nonnegligible amount of copper emission within the visible spectrum,
+DA SILVA ET AL.
+9459
+
+=== PAGE 19 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+arising from the vaporization of the copper wire that connects the rocket to the ground (Walker &
+Christian, 2017). Second, there is a geometric growth effect of the optical emission within the field of view
+of the detector. For the sake of simplicity, these two effects are neglected in the simulations by assuming that
+the fraction of total energy radiated by neutral copper is small in comparison to all other emissions from
+the air plasma, and by assuming that within the narrow field of view of the detector (only 62 m of chan-
+nel length) the return stroke current amplitude does not change considerably. All these uncertainties are
+encapsulated within the parameter 𝜂vis, adjusted within reason to fit the measurements.
+In all simulations shown in Figure 7, the total energy deposited in the channel by Joule heating ranges
+between 10 J/m and 18 kJ/m. At the instant of peak electrical power, the channel resistance varies between
+0.6–130 Ω/m within all simulation cases presented in this section. For peak currents larger than 5 kA, this
+quantity shows little dependence on the current rise time and fall time values used, and can be fitted by
+the following formula R = A∕Ip, where A = 13 kA Ω/m (the mean error between fit and simulation results
+is lower than 3%). From this formula it is easy to see that in the range of peak currents between 10 and 20
+kA, the channel resistance per unit length at the instant of peak electrical power reduces from 1.3 to 0.65
+Ω/m. Once again these values are in good agreement with the experimental findings of Jayakumar et al.
+(2006, Table 2).
+4. Summary and Conclusions
+In summary, in this manuscript we introduced, validated, and used a physics-based computational tool to
+calculate the lightning channel's nonlinear plasma resistance. A model that bridges an existing gap in the
+literature, by providing a self-consistent evaluation of the plasma properties at little computational cost
+(i.e., at the cost of solving five ordinary differential equations). In this paper, we showed how the proposed
+computer-simulation tool can perform well in a wide range of current values, from 1 to 104 A. It can capture
+well the non-LTE plasma regime, by reproducing the finite time scale for streamer-to-leader transition with
+reasonable accuracy. Furthermore, in the high-current/full-LTE regime, the model can capture well the
+temporal evolution of the neutral-gas temperature and the estimated energy deposition by a return stroke,
+in good agreement with the work of Plooster (1971) and Paxton et al. (1986).
+The model also describes well the negative differential resistance behavior of steady-state arc discharges, in
+good agreement with the experimental findings of King (1961) and Tanaka et al. (2000). The steady-state
+resistance in the millisecond time scale has an inverse power law dependence on the current, that is,
+R = A∕Ib, where A and b are fitting constants. We found that the power law index b decreases with increas-
+ing current, because at different current regimes the steady state is dictated by distinct physical processes.
+At low currents (I < 10 A) the steady state is given by a balance of Joule heating and heat conduction, while
+at high currents (I > 1 kA) the steady state is given by a balance with radiative losses. The intermediate cur-
+rent range is marked by a comparable role between the two loss processes, with radiative emission being
+important in the submillisecond time scale, while heat conduction being significant at later stages.
+We presented a detailed description of the light emission in a return stroke. We showed that the proposed
+model can reproduce the experimentally inferred direct relationship between peak current and peak radi-
+ated power and between charge transferred to ground and total energy radiated, as experimentally inferred
+by Quick and Krider (2017). The caveat is that the quadratic power law relationship between the two remains
+unexplained. The model also captures the 0.1-μs delay between the rise of current and optical emissions in
+rocket-triggered lightning return strokes, as measured with high precision by Carvalho et al. (2014, 2015).
+It has been suggested that the negative differential resistance behavior of lightning channels plays an impor-
+tant role in the mechanism of current cutoff, which in its turn makes some flashes transfer charge to ground
+by a series of (discrete) return strokes, while others by a single stroke followed by a long continuing current
+(Krehbiel et al., 1979; Hare et al., 2019; Heckman, 1992; Mazur et al., 1995). Recent review articles argue that
+the role of negative differential resistance in the channel cutoff remains to be quantified (Mazur & Ruhnke,
+2014; Williams, 2006; Williams & Heckman, 2012; Williams & Montanyà, 2019). The model described in
+this manuscript can be applied for simulating multiple return strokes in a flash and other types of processes
+taking place in the lightning channel, such as dart-leader ionization waves and M components, provided
+that the current waveform is given (see Figure 1b). Suggestions of future work include coupling this tool to
+DA SILVA ET AL.
+9460
+
+=== PAGE 20 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
+distributed circuit models of the lightning return stroke, or to fractal models of the growing lightning-leader
+network. We speculate that this strategy will provide important insights into the physics of lightning channel
+cutoff.
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+Acknowledgments
+This research was supported by a
+Research Infrastructure Development
+award from New Mexico NASA
+EPSCoR. We have made the simulation
+data output (da Silva, 2019b) and rate
+coefficients (da Silva, 2019a) shown in
+this manuscript publicly available
+online.
+DA SILVA ET AL.
+9461
+
+=== PAGE 21 ===
+Journal of Geophysical Research: Atmospheres
+10.1029/2019JD030693
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+9463
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+=== PAGE 1 ===
+Abstract. Physical processes determining the ability of light-
+ning to change its trajectory by choosing high constructions to
+strike are discussed. The leader mechanism of lightning propa-
+gation is explained. The criterion for a viable ascending (up-
+ward) leader to originate from a construction is established. The
+mechanism of the weak long-distance interaction between the
+ascending counter leader originating from a grounded construc-
+tion and the descending (downward) leader from a cloud is
+analyzed. Current problems concerning lightning protection
+and lightning triggering by a laser spark are discussed, the
+latter being of special interest owing to a recent successful
+experiment along this line.
+1. Introduction
+Experiments to initiate a high-voltage discharge employing a
+laser-produced plasma and to direct the discharge along the
+channel of a long laser spark [1 ± 12] as well as the advent of
+lasers appropriate for this purpose have lent impetus to
+attempts to control lightning with lasers. Research in this
+field, which is being pursued in the USA, Japan, Canada, and
+Russia [13 ±31], until recently did not go beyond the scope of
+laboratory investigations, though goal-seeking. In recent
+years, however, a start was made on natural experiments in
+Japan. As a result of repeated attempts, two events of
+successful lightning triggering with the aid of a laser plasma
+produced near the summit of a tall tower were recorded in
+1997 [17, 18, 21]. These undeniably impressive results raised
+the expectations of many that the dawn of an era of laser
+techniques in lightning protection is near. Of prime impor-
+tance in this connection is a clear understanding of the
+lightning processes and a statement of what is definitely
+known about the basic lightning mechanisms and what
+invites elucidation or comprehensive investigation. This will
+facilitate the search for efficient ways of controlling lightning
+by laser action in an effort to promote both research and
+lightning protection. At the same time, this will guard against
+excessively optimistic expectations, especially where engi-
+neering practice is involved.
+Below we will consider some key physical mechanisms of
+the lightning process, discuss the potential of laser triggering
+of lightning and the requirements on the control laser spark,
+and highlight the currently topical problems of lightning and
+lightning protection physics that might be solved with the aid
+of lasers.
+2. How the lightning leader works
+Of prime interest for both lightning physics and practical
+lightning protection is descending lightning which originates
+in a cloud and propagates towards the ground. In conse-
+quence of the lightning ± ground contact, the cloud or part of
+it (a charged cell) eventually discharge. Usually, a lightning
+flash consists of several sequential components spaced at tens
+of milliseconds, which travel through a common channel (and
+EÂ M Bazelyan G M Krzhizhanovski|¯ Power Engineering Institute,
+Leninski|¯ prosp. 19, 117927 Moscow, Russian Federation
+Tel.: (7-095) 955-31 39; Fax: (7-095) 954-42 50
+Yu P Ra|¯zer Institute for Problems of Mechanics,
+Russian Academy of Sciences,
+pros. Vernadskogo 101, 117526 Moscow, Russian Federation
+Tel.: (7-095) 434-01 94; Fax: (7-095) 938-20 48
+E-mail: raizer@ipm.msk.ru
+Received 23 March 2000; revised 19 April 2000
+Uspekhi Fizicheskikh Nauk 170 (7) 753 ± 769 (2000)
+Translated by E N Ragozin; edited by A Radzig
+PHYSICS OF OUR DAY
+PACS numbers: 52.80. ± s, 52.80.Mg, 51.50.+v, 52.90.+z
+The mechanism of lightning attraction and the problem
+of lightning initiation by lasers
+EÂ M Bazelyan, Yu P Ra|¯zer
+DOI: 10.1070/PU2000v043n07ABEH000768
+Contents
+1. Introduction
+701
+2. How the lightning leader works
+701
+3. Initiation of descending lightning in a cloud
+704
+4. Build up of the leader of descending lightning and potential delivered to the ground
+705
+5. Attraction of lightning. Ascending counter leader
+707
+6. Physical mechanism for the attraction of lightning
+708
+7. Adverse effect of the corona on the initiation of ascending and counter leaders and the possibilities
+to overcome it
+709
+8. Demands for, capabilities of, and modern trends in lightning protection
+711
+9. Laser triggering of lightning
+712
+10. Requirements on a laser-produced channel
+714
+11. Conclusions
+715
+References
+716
+Physics ± Uspekhi 43 (7) 701 ± 716 (2000)
+#2000 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences
+
+=== PAGE 2 ===
+sometimes through different ones). The overall flash duration
+may be as long as a second; sometimes the `component' flicker
+of a channel is discernible to the human eye. The first
+component, which makes its way through the unperturbed
+air, is similar in nature to the laboratory spark leader which
+breaks down the long gap, say, between a high-voltage rod
+and a grounded plane.
+The electric field in this gap is strongly nonuniform. It
+focuses near the small-radius rod tip. The air in this region
+begins to ionize, which requires a field E > Ei 30 kV cmÿ1,
+with the effect that under specific conditions there arises a
+thin plasma channel growing towards the plane. Despite the
+fact that the channel soon enters the domain of a very weak
+external field not nearly strong enough to ionize air, it
+continues to grow. Due to the still high conduction of the
+channel, the high electrode potential U is transferred without
+significant losses to the front end of the channel Ð the tip of
+small radius r. The tip is a source of a strong field Em U=r,
+and the adjacent air ionizes. As soon as the new volume of air
+acquires a high conduction, the high potential is transferred
+to it, and this volume becomes the new tip. The length of the
+plasma channel therewith increases. The ionization process in
+the vicinity of the tip is inherently the propagation of an
+ionization wave. The structureless plasma channel thereby
+produced is referred to as a streamer (Fig. 1).
+The theory of streamers is in an advanced stage of
+development and permits estimation of the main parameters
+in agreement with experiment [32]. In air, for a voltage of 10 ±
+1000 kV, the streamer travels at a speed vs 107ÿ109 cm sÿ1
+and produces, immediately behind the tip, a plasma with an
+electron density up to 1014 cmÿ3 in a channel of radius
+r 0:1 ± 1 cm. But in cool air electrons attach themselves to
+oxygen molecules in 10ÿ7 s and also recombine rapidly with
+the resultant complex O
+4 ions. That is why a cool plasma
+channel does not live long and does not grow to very great
+lengths. As shown by experiments, in cool normal-density air,
+a positive (moving towards the cathode) streamer grows for
+as long as the average external field over its length exceeds
+Ecr 4:5ÿ5 kV cmÿ1, while Ecr 10ÿ12 kV cmÿ1 for a
+negative streamer. Hence, for U 5 MV Ð a nearly limiting
+voltage for laboratory experiments Ð a negative streamer can
+grow no longer than U=Ecr 5 m. Meanwhile, spark
+discharges longer than 100 m have been obtained at this
+voltage in the laboratory (to be more specific, at outdoor
+high-voltage test benches), whereas lightning ranges into
+kilometers for an average external field of only 100 ±
+200 V cmÿ1.
+The only way to prevent an air plasma from decaying in so
+weak a field is to heat the gas to a high temperature. For
+T 5 5000 K, the electron losses due to their attachment are
+virtually nonexistent, the electron recombination is moder-
+ated owing to the decay of complex ions, and the electron loss
+is compensated for by associative ionization involving O and
+N atoms, which does not require an electric field. But the
+radius of the channel which may be heated is sharply limited,
+for only a limited amount of energy can be expended for this
+purpose. As is well known, in charging a capacitor with
+capacitance C to a voltage U, an energy CU 2=2 dissipates,
+which is equal to the electric energy to be stored. About the
+same is the case with a growing long line with distributed
+parameters, typified by the channel [32]. The capacitance of a
+unit length of the channel of radius r and length L 4 r is
+approximately equal to
+C1
+2pe0
+ln
L=r 0:555
+ln
L=r pF cmÿ1 :
+
1
+The capacitance of a unit length of its tip, if it is taken to be
+a hemisphere, C1t 2pe0r=r 2p0e0 is ln
L=r times larger
+and does not depend on the radius at all. No more energy than
+C1tU2=2 pe0U2 can be spent to form a unit length of the
+channel, including its heating. For instance, 28 kJ cmÿ1 if
+U 10 MV, which is typical of weak lightning. This energy
+can heat an air column of radius r 1 cm to 5000 K (at a
+pressure of 1 atm, the specific enthalpy is equal to 12 kJ gÿ1).
+In laboratory conditions for U 1 MV, r 1 mm.
+However, a prodigious field U=r 106ÿ107 V cmÿ1
+would have been induced near the channel tip for so small a
+radius. The electric field around the cylindrical channel,
+E Ur ln
L=rÿ1,
+would
+also
+be
+very
+strong
+(ln
L=r 10). An extremely strong ionization wave would
+travel through the air surrounding the tip and the channel,
+which would immediately increase their radius. But in this
+case the amount of energy would fall short of the gas heating.
+Being cool, the channel would rapidly lose conductivity and
+the electrical link to the voltage source. It would cease to
+grow. We arrive at a vicious circle. The voltage should be
+augmented to increase the energy deposited into the channel,
+but simultaneously the volume of the conducting (and
+therefore heated) gas increases owing to the ionization
+expansion, with the effect that the specific energy deposition
+does not rise. This is precisely the reason why a long
+laboratory spark and lightning cannot constitute a structure-
+less plasma channel akin to a streamer. They propagate
+employing the leader mechanism.
+The leader is structurally much more complex. The thin
+plasma channel of a leader is embedded in a shell of space
+charge (termed a cover) of the same sign as the channel
+potential U. The cover radius RL 4 r. The potential U now
+falls off at a radial distance of the order of RL rather than r, as
++
++
++
++
+x
+a
+E
+x
+ncr
+n ÿ ne
+ne
x
+x
+b
+Ecr
+Em
+E
x
+Figure 1. Schematic of the front part of a positive streamer (a), and
+qualitative distributions of the electron density ne, the difference between
+the densities of positive ions and electrons, n ÿ ne, which determines the
+space charge density, and of the field E along the axis in the vicinity of the
+tip (b).
+702
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 3 ===
+was the case with a streamer. That is why the electric fields at
+the channel surface and near the leader tip prove to be
+moderate even for a very high voltage Ð ranging into tens
+of megavolts, as for lightning. Nevertheless, the field around
+the tip is high enough to initiate streamers, Et 30 ±
+50 kV cmÿ1. The tip serves as a source of a diverging bundle
+of numerous streamers which make up a continuous sequence
+starting from the tip as from a high-voltage electrode. On
+travelling a distance of the order of Rs U=Ecr, the streamers
+come to a halt. For a negative leader for U 10 MV,
+Rs 10 m. A streamer zone is thereby formed in front of the
+leader tip (Fig. 2). It is occupied with moving streamers and
+those already dead. The charge introduced by the streamers
+becomes the cover charge. Penetrating into the streamer zone
+preformed, the growing leader channel pulls on a cover of
+radius RL Rs.
+The channel tip moves to a new position, adding a new
+portion to the channel, when the current of many `young', just
+emitted and still well conducting streamers is concentrated in
+a thin column to heat it to a high temperature providing
+retention of the conductivity. This is the most important
+phase of the leader process Ð the current contraction to a thin
+filament is akin to the effect of constriction in a glow
+discharge and is associated with the action of an ionization-
+overheating (thermal) instability [33]. The scale for the leader
+velocity vL is supposedly the ratio between the length of the
+streamers that retain a good conductivity, l vs=na (vs is the
+velocity of streamers in the immediate neighborhood of the
+leader tip, and na is the electron attachment frequency), and
+the characteristic instability build-up time tins. The bundle of
+conducting streamers nearly in contact with each other, in
+which the electron density is still relatively high, supposedly
+forms what appears in the photographs as a bright leader tip.
+The tip radius r is therefore about the same as l. For the values
+vs 107 cm sÿ1 and na 107 sÿ1 typical of the streamer zone
+of laboratory leaders, one finds l 1 cm. The instability
+build-up time in this case is, according to calculations [32], of
+the
+order
+of
+tins 10ÿ6
+s.
+Hence
+it
+follows
+that
+vL l=tins 106 cm sÿ1. Estimated values of r and l agree,
+in order of magnitude, with those given by experiments. The
+lightning leader velocity vL is higher by an order of
+magnitude, since the tip voltage is 1 ± 2 orders of magnitude
+higher and all the processes are more intense. The effects and
+the processes in the leader tip and in the streamer region are so
+complicated that the dependence of the leader velocity on
+external factors is hard to represent in the form of a reliable
+and physically transparent formula. Neither an adequate
+theory, nor adequate numerical calculations exist at present.
+The understanding of the phenomena which determine the
+leader velocity does not, even qualitatively, go far beyond the
+scope of what was just stated. This issue is discussed some-
+what more fully in Ref. [32]. One can find there a numerical
+simulation of the instability development that is responsible
+for the contraction of the current in the leader tip to a thin
+filament, thus allowing the plasma heating up to a high
+temperature.
+In a leader, the ionization-overheating instability builds
+up in a somewhat different manner than in the contraction of
+a glow discharge. In the latter, the process proceeds for a fixed
+voltage, while in a leader for a fixed current. The source of this
+current is the streamer zone which possesses an extremely
+high resistance. It is as if this region served as a current
+generator, and no processes in the leader tip (including
+contraction of the currents of many streamers to a thin
+pinch) can alter this current.
+Progress toward understanding lightning processes is
+impossible without prescribing some reasonable dependence
+of the leader velocity on external parameters. Having no
+theoretical dependence at our disposal, subsequently (see
+Section 4) we will invoke an empirical relationship and,
+naturally, provide a physical substantiation of which of the
+external parameters is the controlling one as regards the
+velocity. We note that constructing a good leader theory is a
+topical problem for the future, if we are seriously interested in
+the processes underlying the development of long sparks and
+lightning. Determination of the leader velocity should be one
+of the outcomes of this theory.
+The situation with the theory of a leader channel is little
+better (from the quantitative standpoint). Without this
+theory, it is also hard to make advances in the description of
+the lightning processes. The voltage drop across the channel
+and, hence, the potential of the leader tip responsible for the
+leader movement depend on the intensity of the longitudinal
+field in the leader channel. The leader channel resembles the
+channel of an arc. The quasi-stationary state with a non-
+decaying quasi-equilibrium plasma with an electron density
+ne 1014 cmÿ3 is sustained in a leader channel and an arc by a
+relatively weak field. The state in an arc channel is determined
+by the current flowing through the arc. The plasma
+temperature and the longitudinal field depend on the
+current. For a relatively high current i 100 A, the plasma
+is quasi-equilibrium in the sense that the temperature of the
+electron gas Te and that of the gas of heavy particles T,
+including ions, are close to each other (Te T 10; 000 K),
+and the degree of ionization corresponds to this temperature
+according to the laws of thermodynamic equilibrium. For
+i 100 A, the plasma of an arc channel is sustained by electric
+fields of several volts per centimeter. Indeed, such are the
+leader currents in lightning. In a laboratory leader, the
+current is lower, i 1 A, and the electric field in the channel
+is stronger Ð according to different estimates, several
+hundred volts per centimeter ( 1 ± 5 kV cmÿ1 immediately
+after the initiation of a new portion of the channel). In an air
++ +
++ +
++ +
++ +
++ +
++ +
++ +
++ +
++ +
++ +
++ +
++ +
+U
++
++
++
++
++
++
++
++
++
++
++
+Anode
+Cover
+Channel
+Tip
+Streamer
+zone
+Streamer
+zone
+Channel
+Figure 2. Photograph (made in a laboratory) and schematic representation
+of a positive leader.
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+703
+
+=== PAGE 4 ===
+arc at atmospheric pressure for so low a current, the field is
+weaker though also close to 100 V cmÿ1. In low-current arcs,
+the gas temperature is distinctly lower than 104 K and the
+temperatures are appreciably different, viz. Te > T. It seems
+likely that the situation is also the same in the leader channel
+of a laboratory spark. Since the theory of the leader channel is
+also far from completion Ð and knowing the electric field in
+the channel and its dependence on the leader current is
+indispensable to an understanding of many lightning pro-
+cesses Ð in the subsequent discussion we will take advantage
+of the following approximation formula
+i b
+E ;
+b 300 V A cmÿ1 ;
+
2
+which describes in a crude way the calculated and experi-
+mental results relating to the volt ± ampere characteristic of
+an air arc at atmospheric pressure for moderate currents
+i 1ÿ100 A [33]. The leader and arc channels are compared
+more fully elsewhere [32].
+3. Initiation of descending lightning in a cloud
+On the average, about 90% of descending lightning carries a
+negative charge to the ground, the start being made from the
+lower, negatively charged part of the cloud dipole (Fig. 3).
+The initiation of descending lightning in a cloud is literally
+shrouded in mist. Nobody ever saw or recorded it. One may
+conjecture the initiation mechanism, but one thing is clear. A
+cloud is not a conductor and cannot be likened to an electrode
+of large radius connected to a high-voltage generator. The
+negative charge of the cloud resides in hydrometeors
+(droplets, snow flakes) Ð small low-mobile macroscopic
+particles separated by a dielectric air medium. In the short
+time it takes the lightning leader to propagate to the ground
+and the cloud to discharge, the carriers of the cloud charge
+have no time, so to say, to move out of the positions.
+The average electric field in the cloud cell (of the order of
+several kV cmÿ1) is not nearly strong enough to ionize the air,
+which requires at least 20 kV cmÿ1 at an altitude of 3 km. The
+initial ionization, without which a leader cannot originate,
+occurs owing to a chance field strengthening in a small
+volume. It is conceivable that a local accumulation (a
+vortex) of charged hydrometeors is responsible for this. By
+the way, even near uncharged hydrometeors the maximum
+field is at least three times stronger than the average, because a
+water droplet with a relative permittivity e1 80 polarizes
+almost like a metal conductor. For a spherical droplet, the
+polarization charge suffices to triple the electric field; for
+droplets elongated along the field, the effect is even stronger.
+It was hypothesized that the initial track of ionization is
+produced by a high-energy particle being a constituent of
+cosmic rays. Nobody knows this with certainty. It is beyond
+question that the lightning leader should originate from some
+ionized conducting plasma object extended along the vector
+of the cloud field E0. Owing to the polarization of a conductor
+of length l 4 r (Fig. 4), the field at its ends strengthens as
+Em E0 DU
+r
+ E0
+
+1 l
+2r
+
+:
+
3
+The tip of the initiator conductor serves as the source of
+streamers in the bundle of which there originates a leader [32].
+In this respect, both ends are equivalent, and therefore two
+leaders emerge. The twin leaders move in opposite directions.
+One, being negative, moves primarily down to the ground (if
+the leaders originated in a negatively charged cloud cell, as is
+shown in Fig. 5) while the other, the positive one, moves
+upwards. The leaders are electrically linked to each other and
+are therefore interdependent: as they grow, the charge flows
+from one to the other. In this case, the charge cloud remains in
+place. During their development, the leaders can bypass the
+charged regions altogether if they originated outside the
+charged cell. As the descending leader grows, it is supplied
+6
+km
+4
+2
+0
+Rc
+Qc
+ÿQc
+D
+H
+Figure 3. Charge distribution in a cloud and model of the equivalent cloud
+dipole. Sometimes beneath the negative-charge domain there resides a
+small positive charge, which is disregarded by the dipole model. Typical
+geometric and electric scales are: H D 3 km; Rc 0:5 km; Qc 10 C.
+Taking into account the mirror charge reflection by the perfectly
+conducting ground, the potential at the center of the negatively charged
+cell is U ÿ290 MV relative to the ground; the potential at the lower edge
+of the negatively charged sphere is ±180 MV.
+l
+x
+2r
+U
x
+U ÿE0x
+E0
+E0
+E
+E0
+Em
+Em
+t
+ÿ
+
+Figure 4. Cause of the field multiplication at the ends of a conducting rod
+embedded in and aligned with a uniform electric field E0. The diagram
+shows the distributions of the potential U (the dashed line corresponds to
+the absence of the rod), the field E, and the charge t of a unit length of the
+rod. The potential changes at the rod ends with respect to the external one
+are DU E0l=2.
+704
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 5 ===
+with negative charge not from the cloud. It takes the charge
+away from its twin, leaving it positive. The role of the cloud
+charge reduces exclusively to inducing the electric field which
+initiates and drives the leader process by supplying it with its
+electric energy.
+Naturally, the leaders are more likely to originate where
+the average cloud field is strongest. When we are dealing with
+a negative descending leader, this is the lower edge of the
+negatively charged cloud cell. At the center of the cell, the field
+is close to zero; outside the charged region, it falls off as we
+recede from this region. It is pertinent to note that the
+origination of twin leaders is observed in laboratory condi-
+tions by placing a polarizable metallic rod in the electric field,
+for instance, between plane electrodes (Fig. 6). Concerning
+lightning, this idea was apparently first stated by Kazemir
+[34]. We came across his forgotten, uncited, and inherently
+qualitative paper when we were quantitatively developing a
+similar notion in our monograph on lightning [35].
+In a similar manner, the twin leaders originate at and grow
+from the ends of an extended metallic body insulated from the
+ground when its long dimension is aligned with the electric
+field vector of a thundercloud, even though it may not be fully
+mature. This is the main reason why large-sized aircraft and
+rockets are struck by lightning. They suffer from lightning
+which they induce themselves rather than from accidental
+encounters with descending or intercloud leaders. Running
+tip, we note that it is possible, in principle, to provoke the
+origination of lightning in exactly the same way with a long
+laser spark. It is desirable to produce its conducting channel
+as close as possible to the lower cloud edge but within
+visibility range and, so far as possible, parallel to the vector
+of the local external field. It would then be possible to
+observe, with preparations made in advance, the origination
+and the subsequent growth of the descending leader. It is
+precisely this type of experiment that would hold greatest
+interest for lightning science.
+4. Build up of the leader of descending lightning
+and potential delivered to the ground
+The leader velocity vL is determined ultimately by the excess
+of the leader tip potential Ut over the external potential U0
x
+at the point of tip location x, DUt Ut ÿ U0. The quantity vL
+may equally be thought of as being dependent on the current
+iL which flows to the leader tip and feeds it:
+iL tvL;
+t C1
Ut ÿ U0;
+C1
+2pe0
+ln
L=RL ;
+
4
+where t is the charge, and C1 the capacitance of a unit length
+of the leader. The latter obeys the above formula (1), with the
+reservation that the channel radius r should be replaced with
+the effective cover radius RL that harbors the bulk of the
+leader charge. The velocity cannot depend directly on the
+external field E0
x ÿHU0 at the point of tip location. The
+mechanism of leader advance is indeed associated with the
+action of overwhelmingly stronger inherent fields induced by
+intrinsic charges. In the streamer region of a negative leader,
+Es 10 kV cmÿ1. This field determines the radius of the
+region and, hence, the radius of the charge cover around the
+channel: RL DUt=Es. In the proximity of the leader tip, the
+field is even stronger (Ei 50 kV cmÿ1) to initiate streamers.
+In the region of current contraction during the action of the
+instability, the field was calculated to be as high as 20 kV cmÿ1
+[32]. Meanwhile, the leader quite often propagates in the
+external field E0 100 V cmÿ1, which is weaker even than
+random variations of the intrinsic one.
+Not engaging in speculations as to the vL
DUt depen-
+dence, we take advantage of the empirical relationship
+vL
DUt1=2 established in laboratory experiments with
+positive leaders. Unlike a positive leader which moves in a
+near-continuous manner, a negative one propagates (both in
+a laboratory and with lightning) in a clearly defined
+intermittent, jump-like manner. A leader of this kind is
+termed stepped. The nature of the stepping is not completely
+understood; it is discussed in Refs [32, 35]. However,
+U
+U
t
+U0
x
+U
t 0 U00
+x
+Figure 5. Schematic of the initiation and the propagation of twin leaders
+which started near the lower edge of the lower cloud charge at the instant
+of time t 0. The potential distribution of the cloud dipole U0
x (taking
+into account the mirror reflection) along the x-coordinate is measured
+from the ground upwards. The leader channel is assumed to be perfectly
+conducting, so that its potential U is everywhere the same but changes with
+time.
+ÿU
+Streamer zone
+Streamer zone of
+the descending leader
+Metal electrode
+Tip of the ascending
+leader
+Tip of
+the descending leader
+10
+20
+30
+40 ms
+Figure 6. Time scan of the twin leaders which started from a 0.5-m long
+metal rod embedded in a uniform field in a 3-m long gap. The interdepen-
+dence of their development is evident.
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+705
+
+=== PAGE 6 ===
+experiments with sparks hundred meters long exhibited no
+fundamental differences between the average velocities of the
+positive and negative leaders. The same is also true of positive
+(continuous) and negative (stepped) lightning leaders. In the
+consideration of the growth of leaders of either sign, in what
+follows it is therefore assumed that
+vL a
+
+jUt ÿ U0j
+p
+;
+a 1500 cm sÿ1 Vÿ1=2 :
+
5
+Generally speaking, the potential distribution along the
+leader should be calculated in the context of the theory of a
+distributed-parameter long line. However, for a typical
+current of the lightning leader i 100 A and the field in the
+channel estimated using formula (2), the voltage drop across
+the channel is found to be relatively low in comparison with
+DUt. Hence, the entire channel formed by twin leaders (the
+descending and ascending ones) in the first approximation
+may be thought of as carrying a common potential U at every
+point in time, like a perfect conductor. Then, the growth of
+the leaders is described by the elementary equations
+dx1
+dt ÿa
+
+jU ÿ U0
x1j
+p
+;
+dx2
+dt a
+
+jU ÿ U0
x2j
+p
+;
6
+where x1 and x2 are the tip coordinates of the descending and
+ascending leaders (the leader axis is measured from the
+ground upwards). In this case, the instantaneous value of
+the channel potential U
t is determined by the condition that
+the total charge distributed along the combined channel of the
+leaders with a linear capacitance C1 is equal to zero:
+
x2
+x1
+t dx 0 ;
+t C1
U ÿ U0
x ;
+U
+1
+x2 ÿ x1
+
x2
+x1
+U0 dx :
+
7
+The calculation of the growth of a lightning leader is
+exemplified in Fig. 7. The leading role is played by the
+descending leader which hardly decelerates as it travels in
+the direction of the electric force of the external field and
+which feeds the ascending one with its current. Before long,
+the latter (leader) begins to decelerate, for it finds itself in the
+domain of a steeply rising cloud potential. In this case, the
+ascending leader travels in the direction opposite to the
+electric force (see Fig. 5) and grows so far as the charge is
+delivered to it from the considerably faster descending one.
+When the descending leader reaches the ground and stops, the
+charge ceases to be delivered to the channel for a moment.
+The ascending leader also comes to a halt. Immediately after
+this, a wave travels upwards through the channel to carry the
+zero ground potential and the highest lightning current, the
+wave velocity being only a few times lower than the speed of
+light. However, this is an entirely different stage of the
+lightning process. This stage is termed the principal, or
+return stroke, and we will not enlarge on this subject (it is
+considered in detail in the monograph [35]). Formally,
+according to Eqns (6), the ascending leader comes to a halt
+when the voltage change on a tip U ÿ U0
x2 0 but actually
+when this difference falls off to a relatively low value
+DUt min 0:4 MV 5 U, U0
x2. Such is the limit below
+which the leader cannot grow at all, as shown by laboratory
+experiments and calculations [32]. Therefore, the potential Ui
+which the descending leader delivers to the ground can be
+estimated even without considering the evolution of the
+leaders, employing only equalities (7) and putting simulta-
+neously U Ui U0
x2 and x1 0, which corresponds to
+cessation of motion of both leaders. Geometrically, this
+4
+3
+2
+1
+0
+5
+10
+15
+x2
+x0
+x1
+Altitude, km
+Time, ms
+a
+2
+1
+0
+ÿ1
+ÿ2
+ÿ3
+1
+2
+3
+4
+x, km
+t, mC mÿ1
+b
+2.0
+1.5
+1.0
+0.5
+200
+180
+160
+140
+120
+100
+0
+5
+10
+15
+Time, ms
+Velocity of the descending leader, 105 msÿ1
+ÿU, MV
+vL
+U
+c
+Figure 7. Simulation of the development of a pair of leaders that start from
+the
+lower
+boundary
+of
+the
+negative
+charge
+of
+a
+cloud
+dipole
+(H D 3 km, Rc 0:5 km, Qc 12:5 C): (a) positions of the tips of
+the negative descending (x1) and twin positive ascending (x2) leaders, and
+also of the point of zero potential difference U ÿ U0
x0 0; (b) distribu-
+tion of the linear charge along the leader axis at t 16 ms (calculated using
+an advanced model); (c) potential and velocity of a descending leader.
+706
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 7 ===
+corresponds to equality of the two figure areas enclosed by the
+U0
x curve and the U const straight line in Fig. 8. 1
+The potential Ui which the descending leader delivers to
+the ground is far lower in magnitude than the cloud potential
+U00 at its point of origin. Despite the widespread belief, this is
+not owing to the voltage drop across the channel, which is
+neglected in the above calculation altogether. The potential of
+a perfectly conducting channel which had its origin in a
+nonconducting space with an electric field need not necessa-
+rily coincide all the time with the potential of this field at the
+point of origin. This would be the case if the channel were
+connected to a voltage source having zero internal resistance
+or with a plate of a charged capacitor of unlimited
+capacitance. In the case under consideration, the potential
+assumes a value obtained by averaging the U0
x function
+over a length x2 ÿ x1, strongly asymmetric relative to the
+point of the channel origin. As the channel grows, jUj
+becomes progressively lower in comparison with jU00j. The
+reason is that the U0
x curve is strongly extended towards
+the ground from the point of leader origin, whereas it has the
+shape of a narrow deep well in the opposite direction (see
+Fig. 8). In the case of an unbranched vertical channel, as in
+Figs 7 and 8, about half the potential is delivered to the
+ground (Ui ÿ105 MV instead of U00 ÿ185 MV at the
+starting point of the lightning). The numerous branchings and
+path curvature usually inherent in lightning significantly
+reduce Ui, actually several-fold further.
+The magnitude of the potential delivered to the ground is
+the most important lightning parameter. The destructive
+lightning current upon leader ± ground contact is propor-
+tional to the delivered potential: I Ui=Z, where Z 500 O
+is the wave impedance of a long line formed by the leader
+channel. It is not inconceivable that record-high lightning
+currents of 200 kA correspond to those rare occasions
+when the descending leader develops nearly along a vertical
+line and without branching rather than to record-high
+charged thunderclouds. The magnitude of the potential
+delivered to the ground is significant in one more respect.
+The `force of attraction' of lightning for a tall grounded object
+depends on this potential, as discussed immediately below.
+The higher jUtj, the earlier the lightning sets off for the object
+and the greater the range of attraction.
+5. Attraction of lightning.
+Ascending counter leader
+It has been known for a long time that lightning exhibits
+selectivity, striking primarily tall objects. It is as if the tall
+grounded conductors attract it. This underlies the operation
+of lightning rods. As a rule, a cloud-to-grounded-object strike
+is preceded by the excitation of a counter leader from its
+summit. The descending and counter leaders grow, attracting
+each other. Their joining connects the descending lightning to
+the ground via the conducting object. There may be several
+counter leaders in a group of grounded objects (for instance,
+they can start from the summits of the lightning rod and the
+object under its protection). The earlier the counter leader
+originates and the more intense its development, the better
+the chance that it intercepts the lightning. The ascending
+leader may also originate in the absence of descending
+lightning, under the action of the field of the thundercloud
+alone (if the object is tall enough and the cloud field is strong).
+This is the way so-called triggered lightning is organized
+artificially: a small rocket is launched into a cloud, pulling a
+thin (0.2 ± 0.3 mm in diameter) grounded wire behind it [37].
+The ascending lightning starts when the rocket reaches an
+altitude of about 200 m. In experiments [17, 18] on laser
+triggering of lightning, the leader was also excited to ascend
+from a tall tower.
+The cause of the origination of the ascending leader is
+simple. If the charges of the descending lightning and (or) the
+cloud induce a vertical field E0 in the region of a grounded
+conductor of height h, the difference between the zero
+potential of the conductor summit and the potential of the
+external field at the point of its location is DU E0h. This
+gives rise to a region of local field strengthening near the
+summit. This field and DU may turn out to be sufficient to
+ionize the air and generate the leader (DU >DUt min
+0:4 MV). However, the lightning is affected only by that
+counter leader which is capable of travelling a distance L at
+least comparable with the object height, i.e. several tens to a
+hundred meters. Only then will the `gain' in an object height
+owing to the conducting leader channel become significant.
+For this to happen, the potential change near the tip of the
+counter leader DUt
E0 ÿ ELL DU0, where EL is the
+field strength in its channel, should not lessen in comparison
+with DU0 (Fig. 9). The condition for viability of the counter
+leader, E0 > EL, proves to be more rigorous than its
+origination condition, E0 > DUt min=h.
+According to formula (2), the current in the channel of a
+viable leader exceeds imin b=E0, where the current is given
+by expression (4). The requirement i > imin imposes condi-
+tions on the initial potential change DU E0h at the object
+summit and its height for a given external field or on the
+minimal intensity of the external field for an object of a given
+height:
+DUmin
+ b ln
L=RL
+2pe0a
+2=3
+1
+E 2=3
+0
+;
+
8
+hmin
+ b ln
L=RL
+2pe0a
+2=3
+1
+E 5=3
+0
+:
+Taking the values of b and a from formulas (2) and (5), and
+putting L=RL 10 (the dependence on this not-too-well
+determined parameter is very weak), we find for E0
+150 V cmÿ1 that DUmin 3:2 MV and hmin 210 m. Much
+Ui U0
x2
+U0
x
+x
+x2
+x0
+H
+U
++
+ÿ
+Figure 8. Employing the area equality condition to determine the electric
+potential delivered to the ground by a negative leader.
+1 Curiously, a similar condition for the equality of areas in the correspond-
+ing coordinates describes the static equilibrium (co-existence) of a great
+diversity of states in physics, e.g., the current and currentless regions in
+discharges, the burned and initial mixtures at the moment of a combustion
+flame stopping, and many others [36].
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+707
+
+=== PAGE 8 ===
+the same field at the ground is produced by a cloud dipole
+with a lower charge jQcj 10 C at an altitude H 3 km.
+These parameter values are quite moderate for thunder-
+clouds, and the triggering of lightning for a 200-m elevation
+of a rocket with a wire is a wholly realistic situation. From
+buildings of typical height, say, h 50 m, the counter leader
+is, according to formula (8), excited for an external field
+E0 350 V cmÿ1. Over plains, the thunderstorm field from a
+cloud charge is rarely, if ever, that strong (in the mountains,
+sometimes it is). The leader of the descending lightning should
+add about 200 V cmÿ1 by its charge. This may happen, for
+instance, when a leader carrying a potential U 37 MV to the
+ground descends along a vertical path to an altitude
+H0 5h 250 m at a horizontal distance R 3h 150 m
+from the object. The main contribution to the leader field at
+the ground is made by the charge localized in the portion of
+the leader of length H0 immediately behind the tip. Here, the
+linear charge density is t C1U 4:4 mC cmÿ1
+for
+U 37 MV. It may be said that the lightning trajectory
+deviates `purposefully' from the vertical at a point with
+coordinates H0 and R and rushes to the object instead of
+striking the ground a distance R away. The calculated figures
+given above are in reasonable accord with observations.
+6. Physical mechanism for the attraction
+of lightning
+Clearly the attraction of lightning for a tall building and
+most often for its extension Ð a counter leader Ð is
+attributable to the electric field produced by the charges
+induced in these bodies by the charges of the cloud and the
+developing lightning. But this commonplace statement is
+void of content unless what this field acts upon is specified
+and unless the specific physical mechanism of the interac-
+tion of two leaders is elucidated. For, while the leader tips
+are hundreds of meters apart, each of them is subject to the
+field of the other leader, which is little stronger than the
+cloud field. It is as weak as hundreds of volts per centimeter
+and they do not exert a noticeable effect on the magnitude
+of the leader velocity. This was explained in Section 4 and is
+inherent in formula (5). What is the mechanism of mutual
+attraction of the leaders?
+We allow ourselves to propose a hypothesis. The weak
+external field E0, which has no effect on the leader velocity vL
+determined by the magnitude of the potential change jDUtj at
+its tip, affects the leader acceleration:
+dvL
+dt
+
+dvL
+djDUtj
+ dU
+dt ÿ HU0
+dx
+dt
+
+
+
+dvL
+djDUtj
+ dU
+dt
E0vL
+
+:
+
9
+Here, the upper sign refers to the negative leader, and the
+lower sign to the positive one. The first factor in Eqn (9) is
+independent of E0 and is always positive, the second consists
+of two terms comparable in absolute value. The term dU= dt
+related to the charge redistribution along the growing light-
+ning channel is most often favorable to the moderation of the
+growth of the descending leader. The term
E0vL charac-
+terizes the direct dependence of the leader acceleration on the
+external field. The higher E0 and the smaller the angle
+between the vectors of the leader velocity and the `electric
+force' E0, the higher the acceleration, all other factors being
+equal. Hence, the leader will get to the ground or a grounded
+conductor sooner if it moves in the direction of the vector of
+the electric force.
+In reality, the growth of the descending leader involves
+inherently statistical factors. As revealed by frame-by-frame
+photography of a laboratory leader with an exposure time of
+the order of 10ÿ7 s, a growing leader always exhibits several
+leader tips. They are connected to the main channel by short,
+randomly oriented leader `branches' (Fig. 10). Of all these
+tips, the one whose branch grows closest to the direction of
+the external electric force has the highest probability of
+survival. More often than not the remaining tips soon die
+off, because the tip which grows along the E0 vector and
+thereby keeps ahead of them hinders the growth of those
+lagging behind through the repulsive action of the intrinsic
+charge. The infrequent survival of two tips initiates a
+U0 ÿE0x
+3
+1
+2
+DU
+U
+Figure 9. Viability criterion for the leader ascending from a grounded
+structure of height h in an external field E0: 1 leader is capable of
+developing and accelerating; 2 decelerating nonviable leader; 3 leader on
+the verge of viability.
+Channel
+Tips
+10 cm
+Figure 10. Photograph of a leader with several tips; the exposure time is
+0.3 ms.
+708
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 9 ===
+`macroscopic' leader branching clearly visible in photographs
+of lightning and sometimes of long sparks. The chance
+survival of a tip deflected from the direction of the external
+field causes the lightning trajectory to bend. However, the
+latter event becomes a rarity when the external field builds up
+in magnitude along some of the directions of the descending
+leader growth. The route to the counter leader is precisely the
+one.
+An assumption can be made as to the cause of the random
+origination of new tips. The surface of the equipotential
+plasma channel conductor is unstable. An accidental sharp
+spike induces a field enhanced along the spike direction.
+Under its action, the spike begins to grow. Growth is possible
+in any direction, including that at a significant angle to the
+weak external field.
+All of the aforesaid, we believe, provides a qualitative
+explanation why the leader on the average adheres in its
+motion to the external field line but does not necessarily
+follow it rigorously. By and large the descending leader is
+headed to the ground. But it is more likely to deviate from its
+principal direction as the cloud field is combined with a
+differently directed field of comparable intensity induced by
+some other source, for instance, by the charge carried by the
+counter leader. Naturally, the qualitative reasoning outlined
+above calls for a more rigorous theoretical substantiation
+and, which is desirable, numerical simulations employing,
+e.g., the Monte Carlo technique.
+7. Adverse effect of the corona on the initiation
+of ascending and counter leaders and the
+possibilities to overcome it
+It is well known, all other factors being the same, that the
+ascending leader is far less frequently excited from a
+stationary building than from a rocket with a grounded wire
+moving fast upwards. The reason lies with accumulation of
+the corona space charge nearby the summit of a grounded
+building, whereas this charge does not have time to form in
+front of a rocket flying with a velocity of 100 m sÿ1. The
+electric field near the summit of the building becomes weaker
+owing to the space charge, with the effect that a stronger
+external field E0, which is induced by the thundercloud alone
+or in combination with the leader of the descending lightning,
+is required to excite the ascending or counter leaders. We are
+dealing now with a `quiet' stationary corona, which is
+sometimes termed an ultracorona. It develops for a relatively
+slow rise of the voltage across the discharge gap. In the case
+under consideration, the field builds up with repeated
+accumulation of the charge of the cloud cell after each
+lightning discharge or as the thundery front approaches the
+location of the grounded building. Hence, times of no shorter
+than a second are the case in point.
+In a thin layer near the surface of the structure's summit,
+where the field is maximum 2, ionization of the air occurs. If
+the thundercloud is negative, as is the case in 90% of
+instances, the grounded electrode (the grounded structure) is
+positively charged. The electrons being produced enter it and
+the positive ions drift from the summit to the cloud. In an
+ultracorona, the electric field near the summit of the electrode
+is sustained close to what is defined by the condition for
+discharge self-maintenance, Ecor [33]. For a summit radius of
+several centimeters, the latter is nearly coincident with the
+ionization threshold, Ecor Ei 30 kV cmÿ1. The field is
+controlled automatically. If for some reason it is enhanced,
+the ionization speeds up and more positive charge is
+introduced into the space, which induces a negative charge
+at the summit to attenuate the field. If the field becomes
+weaker than Ecor, the corona is extinguished for some short
+time, the previously produced positive ions recede from the
+electrode, their action becomes weaker, and the field at the
+summit builds up to resume the ionization. Such is the case
+only for relatively slow voltage variations, because the
+controlling mechanism is based on the ion motion whose
+mobility is low. For a sharp rise of the voltage at the summit
+of the electrode, the space charge required for the stabiliza-
+tion has no time to form and the field rises there significantly
+to generate ionization waves Ð streamers. A streamer flash (it
+is referred to as a pulsed corona) may trigger the leader
+process. This is precisely how the counter leader originates,
+when the channel of descending lightning approaches the
+object with a velocity of 107 cm sÿ1. Figure 11 gives the
+results of numerical simulation of the ultracorona at the
+summit of a grounded rod embedded in the external field.
+The model, elaborated in cooperation with N L Aleksandrov,
+takes full account of the effect of all the charges on the corona
+field distribution, including those induced over the whole
+length of the rod.
+While the corona protects buildings from lightning to
+some extent by hindering the origination of a counter leader,
+it is detrimental to efficient operation of the lightning rod, for
+its task is the opposite Ð to emit the counter leader as early as
+possible and to intercept the descending lightning by itself. In
+principle, the performance of this function could be promoted
+by shooting, in due time, a `harpoon' with a metallic marline
+tied to the summit of the lightning rod in order to transport
+the conductor tip beyond the ion cloud. It is not improbable
+that the main role of a laser-produced spark in the experiment
+to trigger the ascending leader from a tower (reported in Refs
+[17, 18]) reduced precisely to the transfer of the conductor
+outside the corona cloud nearby the tower summit (see
+Section 9).
+We will consider the simplest corona model to gain an idea
+of how far and with what velocity the `extender' of the
+lightning rod should be ejected upwards. Let a corona be
+displayed by an immobile spherical electrode of radius r0 to
+which a voltage U
t is applied (r0 corresponds to the radius
+of the summit of a lightning rod of height h, and U E0
th is
+the potential difference of the summit and the growing
+external field E0
t at the point of summit location). Let us
+assume, and there are grounds for doing so, that the state of
+the ultracorona formed is quasi-stationary in the sense that
+the radial distributions of the field E
r and the space charge
+r
r closely follow the corona current i
t which varies
+relatively slowly in time. At every point in time, they
+correspond to the instantaneous value of i
t as if the current
+were invariable. In this case, the current through all the
+spherical sections of the charge cloud at a given moment is
+the same, i.e. a new portion of charge i dt introduced into the
+corona goes exclusively to expand the ion cloud, into an
+increment dRf of its front radius Rf
t. Under this assump-
+tion, the electrostatic and charge conservation equations
+1
+r2
+d
+dr r2E r
+e0
+;
+r
+e0
+
+i
+4pr2e0miE
+
10
+with a typical boundary condition for an ultracorona,
+E
r0 Ecor const; are easily integrated (mi is the ion
+2 In the absence of a corona, it may be estimated by formula (3).
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+709
+
+=== PAGE 10 ===
+mobility). Not writing out the somewhat unwieldy complete
+formulas, we give only the compact asymptotic expressions
+valid away from the electrode in the stage when the cloud has
+strongly expanded and Rf 4 r0, while the space charge in the
+gap, Q 4pe0R2
+f E
Rf, is much larger than the electrode
+charge qcor 4pe0r2
+0Ecor which does not vary during the
+corona discharge:
+E
r
+
+i
+6pe0mir
+s
+;
+r
r 1
+r
+
+3e0i
+8pmi
+s
+:
+
11
+More precisely, these formulas are appropriate where the
+electric field of the space charge exceeds the field of the
+electrode charge, Ecor
r0=r2.
+The electrode potential is calculated employing one of the
+equivalent expressions
+U
+
Rf
+r0
+E dr EfRf Ecorr0
+
Rf
+r0
+rr dr
+e0
+ 3EfRf ;
+
12
+where Ef E
Rf. The radius of the ion cloud and the current
+are found by integrating the equation vf _Rf miEf with
+expression (12) and a given function U
t. The latter is
+governed by the external conditions Ð for an atmospheric
+field, by the charge accumulation rate in the thundercloud. In
+particular, for U at, one finds
+Rf t
+
+mia
+3
+r
+;
+vf
+
+mia
+3
+r
+;
+i 2pe0at
+
+mia
+3
+r
+:
+
13
+For instance, let the cloud field attain a value E0 100 V cmÿ1
+one second after the commencement of growth, h 100 m,
+and mi 1:5 cm2 (V s)ÿ1. Then, a 106 V sÿ1, and at the
+point in time t 1 s we have U 1 MV, i 390 mA, Rf 7:1
+m, Ef 470 V cmÿ1, and vf 7:1 m sÿ1. These estimative
+figures are in reasonable agreement with numerical calcula-
+tions.
+If the corona-displaying electrode could move fast to
+travel through the preformed ion cloud with a velocity v far
+higher than vf, in a short time it would be ahead of the
+previously produced peripheral ions and the new peripheral
+part of the ion cloud formed in the course of motion would
+now be unable to be ahead of the electrode. In other words,
+the corona charge would cease to accumulate in front of the
+electrode.
+In
+the
+radial
+distribution
+of
+ion
+velocities
+vi miE
r given by the first of equalities (11), there exists a
+section rc such that vi < v for r > rc, and vi > v for r < rc.
+Roughly speaking, the region from rc to Rf is nonexistent in
+the new cloud. The contribution of the charge corresponding
+to this region to the U potential also vanishes. Since U
+remains unchanged, being given by an external source, this
+loss should be cancelled out by an increase in the electrode
+charge q 4pe0r2
+0E
r0 and the corresponding enhancement
+of the field E
r0 at its surface. Formulating these qualitative
+notions in the context of the spherical model, we can write a
+conditional equality which replaces the second of expressions
+(12):
+U E
r0r0
+
rc
+r0
+rr dr
+e0
+:
+
14
+Let the electrode velocity ensure the field strengthening
+from the previous value Ecor
+to half the maximum,
+Em U=r0, which would take place in the absence of the
+5
+10
+15
+0
+0.2
+0.4
+0.6
+With corona
+Without corona
+x, m
+Electric éeld, kV cmÿ1
+a
+0
+10
+20
+30
+0
+10
+20
+30
+40
+50
+60
+With corona
+Without corona
+x, cm
+Electric éeld, kV cmÿ1
+b
+15
+10
+5
+0
+1
+2
+3
+4
+Time, s
+c
+Charge front radius, m
+0
+5
+10
+15
+103
+104
+105
+106
+107
+Ion density, cmÿ3
+x, m
+d
+Figure 11. Results of numerical simulations of the corona in proximity to
+the hemispherical top of a grounded 30-m tall rod 3 cm in radius embedded
+in the external field; the average ion mobility is 1.5 cm2 (V s)ÿ1. The field
+builds up linearly with time up to 100 V cmÿ1 for t 1 s and is thereafter
+held constant. (a) Field distributions along the x-axis, reckoned from the
+rod upwards, for the instant of time t 5 s with and without the corona.
+(b) The same on an enlarged scale in proximity to the top. (c) Radius of the
+front of the ion cloud. (d) Ion density distribution at the moment t 5 s.
+710
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 11 ===
+corona. In the numerical example given above for r0 3 cm,
+Em 333 kV cmÿ1 and a field half as strong would suffice to
+excite the leader. Bearing in mind that E
r0 Em=2 4 Ecor
+and U 4 Ecorr0, we estimate rc from the condition which
+follows from expression (14):
+
rc
+r0
+rr
+e0
+dr U
+2 1
+2
+
Rf
+r0
+rr
+e0
+dr :
+
15
+Employing formulas (11), we find that rc=Rf 1=4,
+rc 1:8 m, and vc vi
rc 2vf 14:2 m sÿ1. These
+figures give an idea of the scale of the quantities. To
+eliminate the action of the corona, a conductor connected
+to the lightning rod is to be fired upwards from its top to a
+distance l of several meters (l > rc) with a velocity of several
+tens of meters per second (v > vc). Solving the two-
+dimensional axially symmetric problem of the field and
+space-charge-density distributions in the discharge of a
+spherical electrode in a gas flow would aid to refine these
+results. For a flow velocity v exceeding some value vc, the
+solution with Ecor const would cease to exist. The flow
+with a velocity v miEcor 450 m sÿ1 would indeed blow
+away all the ions completely. In this case, the potential
+U 4 Ecorr0 is to be induced only by the increased electrode
+charge. The critical value vc arrived at will indicate the
+lower velocity bound for firing the extender of the lightning
+rod. Also note that the numerical solution of the problem
+on corona discharge of a rapidly growing electrode
+encounters no difficulties.
+8. Demands for, capabilities of, and modern
+trends in lightning protection
+Half a century ago, the main goal of lightning protection was
+to eliminate fire arising from the contact of the lightning
+channel with combustible materials and to guard power
+transmission lines against storm overvoltages induced by
+the current and the strong electromagnetic field of lightning.
+Lightning rods cope with this `coarse task' easily. To solve
+this problem, it will suffice to divert lightning from a fire
+hazardous or dangerously explosive area. Power transmis-
+sion lines are safely protected by lightning protection wires.
+Suspended above the lines, they serve the function of an
+extended lightning rod by intercepting the lightning channel.
+So-called induced overvoltages turned out to be the first
+truly serious indication that the lightning protection is
+inadequate. Induced by the lightning current from a
+distance of several hundred meters, they bring a threat to
+relatively low-voltage power distribution networks (up to 10
+kV). It was recognized that the lightning hazard becomes
+more severe as the operating voltage in electric devices is
+lowered. Regrettably, this prediction was amply borne out
+with the advent of the microelectronic era, when electronic
+devices with operating voltages of tens-to-several volts came
+into being and became indispensable. Aeroplanes, space
+vehicles, communication and information processing facil-
+ities are literally stuffed with microelectronics. Here, the
+`long-range action' of lightning reveals itself in full measure.
+Damage may be caused not only by a direct lightning strike
+to an object, but also by quite remote discharges. Their
+electromagnetic fields may be extremely strong, for the
+lightning current build-up rate may exceed 1011 A sÿ1. We
+are forced to provide screening devices, quite often heavy
+and bulky, or to protect the object from any lightning,
+including remote lightning.
+No better is the situation concerning highly inflammable
+fuels, explosives, and gaseous exhaust into the atmosphere,
+produced in the operation of some technical facilities. All of
+these are an integral part of many present-day devices.
+Explosives have long ceased to be exclusively a means of
+destruction. Many compact one-time actuating mechanisms
+employ explosives. The explosion does not destroy but
+performs a specific, previously planned action. Lightning-
+induced actuation of such a pyrotechnic device cannot be
+tolerated, which it can well do by remotely exciting current in
+the electric ignition circuit. Nor need the lightning channel
+necessarily strike an inflammable gas mixture to set it on fire.
+Counter discharges discussed in the foregoing and all kinds of
+sparking due to electromagnetic noise can easily do the job. A
+home piezoelectric igniter sets fire to the gas in the kitchen
+with an incommensurably weaker electric spark.
+Experts in lightning protection have never abandoned the
+dream of diverting lightning to a safe place, far from the
+critical object. Nor have they abandoned the idea of finding a
+means for provoking lightning to discharge thunderclouds in
+uninhabited vacant areas, where the lightning would cause no
+damage. There is no question that this is basically possible.
+But when the question is raised as to the use of new means in
+lightning protection, issues of technical substantiation,
+reliability, and cost come to the forefront. These factors are
+intimately related. For instance, it is beyond reason to
+increase the power or the energy capacity of a complex and
+therefore expensive device in an attempt to attain a 100%
+efficiency of lightning interception with the use of this device
+if the device itself cannot ensure the controlling action with a
+reliability of over 0.9. A primitive and inexpensive metal
+lightning rod would easily ensure at least one more nine after
+the decimal point in a reliability index.
+Of course, there may be circumstances in which tradi-
+tional lightning rods are basically incompatible with the
+technological functions of an object. A lightning rod cannot
+be mounted within the field of vision of a large-scale radar
+antenna. A lightning rod of many meters high should not
+tower on the launching site of a space vehicle. It constitutes a
+real life hazard in the actuation of the astronauts rescue
+system, for an ejected capsule may collide with the metal
+frame of the lightning rod. Present-day technology rapidly
+multiplies the list of these examples, sending us in search for
+unconventional protection devices.
+It is not always possible to devise an electronic unit
+capable of withstanding the electromagnetic field of light-
+ning by the application of metal screens or pulsed overvoltage
+limiters. For the most critical and easily vulnerable objects, it
+is desirable to arrange protection in such a way as to prevent
+the lightning discharges from occurring anywhere near the
+object whatsoever. But it is hardly realistic to construct a
+fencing of lightning rods at the distant approaches to the
+object, the more so as this does not ensure that lightning will
+not break through. In principle, the problem could be solved
+by a mobile laser facility capable of discharging a thunder-
+cloud in a safe place. To do this, the laser should `shoot'
+kilometers upwards to provoke descending lightning by a
+plasma trace appropriate in length and other characteristics
+(see below). This would be an inestimable aid to investigators
+pursuing descending-lightning research. They would not have
+to set hopes upon good fortune and wait for a successful
+discharge within the field of sight of the short-run recording
+instruments. During a thunderstorm, it would be possible to
+excite lightning in the required place and ensure timing down
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+711
+
+=== PAGE 12 ===
+to a microsecond. In the same way it would be possible to
+solve the problem of modelling situations characteristic for
+the initiation of lightning from bulky aircraft. This would
+hold the great interest for both lightning science and practical
+lightning protection.
+The laser technique of exciting ascending lightning is
+much simpler but less expedient from the practical stand-
+point. First, a tall structure (`extended' by a laser) is required,
+because producing a very long laser spark (of the order of
+200 m, for the electric field at the ground is too weak) with
+appropriate conductive properties would require prodigious
+laser energy and power. Second, this technique nevertheless
+does not ensure perfect protection. Ascending leaders are
+quite often excited from the summit of the 540-m high
+Ostankino television tower in Moscow. However, they do
+not discharge the clouds completely. Though the density of
+descending lightning in the neighborhood of the tower is
+lower than usual, it is far from zero, and not all of the
+lightning strikes the tower. Furthermore, it is well known
+that subsequent lightning components do not always follow
+the same path. Nearly half of them do not take the path of the
+primary channel [38]. Hence, there persists a real danger that
+one of the components of the lightning provoked would strike
+the nearby protected object rather than the construction
+intended for the purpose. Of course, this does not diminish
+the significance of the experiment performed, which is the first
+real step toward laser control over lightning.
+It should be admitted that alternate, non-laser-based
+techniques of initiating and controlling lightning are also
+possible, some of them being technically simpler. The
+excitation
+of
+artificially
+triggered
+ascending
+lightning
+referred to in the foregoing text has been practiced since the
+70s, though for the purposes of research. A well-heated gas jet
+ejected from the top of a stationary lightning rod can be used
+to `extend' it and improve its efficiency. The lowering of gas
+density arising from the heating lowers the counter-discharge
+ionization and excitation thresholds. It is well known that the
+long wake of hot gas jets from aircraft and rocket engines
+facilitates the initiation of lightning from them. It is not
+unusual that combustion products are partly ionized; there
+also exist special techniques to produce plasma jets, which
+may, in principle, have an effect similar to that of a laser-
+produced spark.
+Controlling lightning is also possible by applying a high
+voltage to an object. In this case, there are several options.
+With a voltage of the same polarity as the descending
+lightning, the latter should be repelled from the object (in
+principle, this is a way to protect a structure). For an opposite
+polarity, the lightning is attracted, and this is a way to
+improve the efficiency of a lightning rod. However, from the
+technical standpoint it is clear that applying megavolt
+voltages at the necessary times with the required repetition
+rate is a complicated task. Lower voltages are out of the
+question, which was shown in the estimation of the excitation
+conditions for counter and ascending leaders. The problem of
+action of high voltage on lightning arose inevitably in the
+construction a 1150-kV power transmission line. The ampli-
+tude of the alternating voltage at its conductors relative to the
+ground is close to 1 MV, which is commensurable with the
+potential of the lightning leader. This gives rise to quite
+tangible difficulties in the design of a reliable lightning
+protection for the power transmission line. The feasibility of
+overcomingtheactionofthecoronawasdiscussedinSection7.
+The same effect may be attained if a voltage of polarity
+opposite to that of the cloud is applied to the electrode. The
+case in point are quite moderate voltages of the order of E0h,
+where h is the electrode height, and E0 100 V cmÿ1.
+There is no question that the above-listed methods of
+affecting lightning and similar methods are the right subject
+of discussion from the viewpoint of investigations, but they
+do not attract considerable attention when it comes to
+practical lightning protection. Pragmatic considerations
+underlie the skepticism of engineers Ð is the game worth
+the candle? We repeat: the reliability of lightning protection
+is primarily determined by the reliability of actuation of the
+entire sequence of complex technical devices that form the
+controlling action on the lightning rather than by the
+efficiency of the controlling action itself. One is forced to
+take into account the possibility of interruption of the
+power supply to the controlling devices caused by a
+thunderstorm, the operational lifetime, maintenance expen-
+diture, etc. The use of conventional lightning rods is not
+associated with these problems, and therefore dilettante
+inventors, and sometimes even solid companies, address
+themselves to precisely these rods, proposing inexpensive
+and allegedly efficient means to improve the reliability and
+extend the protection radius. As an example we refer to
+radioactive and piezoelectric attachments. In the view of
+their manufacturers, both ionize the air to prepare the
+easiest route for the lightning channel. In reality their effect
+is akin to the action of an ultracorona. The effect, if any, is
+the opposite of that expected. But even that is in fact
+nonexistent. A weak radioactive source, the more so a
+piezoelectric cell, cannot compete with a corona. The
+action of radioactive sources of safe intensity has been
+repeatedly verified in the laboratories. They have no effect
+on the origination and development of a long spark.
+9. Laser triggering of lightning
+Two schemes of producing a laser plasma for controlling
+lightning are now under development. One of them has roots
+stretching back 30 years, when a long laser spark was
+produced [7, 39 ± 43]. It is produced employing neodymium
+or CO2 lasers, in record-breaking versions with an energy of
+2 kJ or even 5 kJ [31] and a duration of the main part of the
+pulse of 50 ns. The respective threshold intensities for the
+breakdown of the pure and aerosol-containing air are
+109 W cmÿ2 and 107 ± 108 W cmÿ2, respectively. The virtue
+of this scheme involving a CO2 laser is that the channel can be
+heated to several thousands of degrees. Reducing the gas
+density N by an order of magnitude promotes the collisional
+ionization by electrons, whose rate constant is determined by
+the reduced field E=N. For a temperature above 4000 K, the
+associative ionization N O ! e NO, which does not
+depend on the field at all, becomes appreciable. Heating also
+strongly suppresses the electron losses due to their attachment
+and recombination. But the laser spark proves to be
+continuous only when it is not too long, no longer than
+several meters for the energy specified above. When the
+radiation is focused to a distance of tens or hundreds of
+meters, spark production does occur, but the resultant spark
+consists of separate plasma centers. The longer the focal
+distance, the greater their spacing. The discontinuity of the
+conductor hinders its polarization as of an entity in the
+external field and does not permit using it as an efficient
+`extender' of the lightning rod or for the triggering of
+lightning in the open atmosphere.
+712
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 13 ===
+The other scheme pursued in Refs [14, 16, 20, 22] is free
+from this drawback. It is suggested that a short and extremely
+intense pulse of ultraviolet radiation be employed to accom-
+plish the three-photon ionization of the O2 molecules and the
+four-photon ionization of N2. A longer pulse of visible
+radiation complements the short one to release the electrons
+from negative ions. In this case, far less energy goes to ionize
+the air as compared with the breakdown by a CO2 laser,
+because the energy is in fact not expended on anything else.
+The objective is to produce a long thin ionized channel in the
+open atmosphere. It will be polarized under the cloud field,
+and leaders will be excited from its ends.
+In laboratory experiments involving these laser pulses, the
+gap exhibited a lowering of the breakdown voltage and the
+spark discharge was observed to make its way through the
+laser-produced channel [14, 20]. A multistage laser system
+produced ultraviolet radiation with a wavelength l 248 nm
+starting from the fourth harmonic of a neodymium laser, with
+final amplification by an excimer KrF laser. The output was a
+10-ps long pulse with an energy of 10 mJ (1 GW in power).
+This pulse was superimposed on an alexandrite-laser pulse
+with a wavelength l 750 nm, an energy of 0.21 J, and a
+length of 2 ms. The authors are designing a system to provide a
+l 248-nm pulse with an energy of 50 mJ and a length of
+200 fs (250 GW in power), and also a l 750-nm pulse
+several joules in energy and tens of microseconds in length.
+They carried out a numerical simulation of the initial stage of
+the evolution of a thin channel several tens of meters long
+ionized by the laser radiation at a small altitude in the open
+atmosphere. A gradual field multiplication was seen at the
+ends (the calculations indicated a two-fold multiplication).
+However, the controlling parameter Ð the external field
+E0 6:5 kV cmÿ1 Ð adopted in the calculations seems to be
+unrealistically overrated. This supposedly led the authors to
+make an unjustifiably optimistic prediction that low-energy
+laser pulses would be sufficient. Real storm fields at the
+ground are weaker by a factor of several tens; even at an
+altitude of 2 km they are still 2 ± 3 times weaker than those
+adopted in the model.
+Experiments [17, 18] were carried out to model lightning
+with a laser on the shore of the Sea of Japan in the period of
+intense winter low-cloudage thunderstorms typical of this
+region (Fig. 12). In this case, the electric field at sea level is
+usually close to 100 V cmÿ1. To trigger the ascending leader, a
+tower with a height h 50 m (the authors do not give the
+magnitude of the h parameter most critical for the analysis;
+the figure was borrowed from an entirely different source [23])
+was constructed on a 200-m high hill. Data on the electric field
+profile in the neighborhood of the tower are not given, either.
+However, there are grounds to believe that the field was
+significantly more intense (in the classical problem of a
+conductive hemisphere on a grounded plane in a uniform
+field cited in textbooks of physics, the maximum field at the
+top of the hemisphere is three times stronger than the external
+one).
+Stationed on the ground were two CO2 lasers delivering
+50-ns pulses with an energy of 1 kJ. One laser beam was
+focused with a mirror on a dielectric target at the tower
+summit to produce the initial plasma. The other beam, also
+focused with a mirror, produced a two-meter-long laser spark
+from the tower summit. In addition, an ultraviolet laser was
+employed (like in the second scheme outlined above) for
+producing a weakly ionized channel to direct the leader to
+the cloud, which was slightly offset from the tower.
+The experimenters believed that the selection of the
+instant of laser actuation was one of the most critical
+elements of the operation. Should it be done too early,
+nothing would be accomplished owing to the smallness of
+E0. Should it be done too late, spontaneous descending
+lightning might originate in the cloud to strike the structure
+beneath. Special-purpose microwave instrumentation traced
+the state of the cloud, and the lasers were actuated at the
+instant of the onset of the cloud discharge, which may be
+considered as the precursor of the descending lightning. In the
+authors' opinion, among the many attempts made two were
+successful; the lightning thus provoked was synchronized
+with the laser pulses. The authors state that an ascending
+leader went off the tower upwards. As a consequence, the
+nearby cloud region measuring about 2 km discharged 3 C
+into the tower with a current of 35 kA typical of lightning.
+It is safe to assume that the cloud field E0 near the tower
+was so strong that the natural potential change DU E0h was
+on the verge of provoking an ascending leader, were it not for
+the screening corona action. Of course, we cannot expect the
+numerical value of DU to literally satisfy the estimative
+formula (8), which relies on the not-too-dependable relation-
+ships (2) and (5). Furthermore, it is highly improbable that
+condition (8) was not satisfied without a laser spark and came
+to be satisfied when the 50-m high tower became two meters
+longer. The entire experience of experimental investigation of
+long spark discharges suggests that the statistical scatter of
+their threshold values is much larger. It may well be that the
+function of the lasers was as follows: a moderately long and
+therefore continuous laser spark `shot through' (perforated)
+the corona to instantly bring the conductor summit beyond
+some portion of the ion cloud, which was responsible for the
+origination of the ascending leader. Upon its penetration into
+the thundercloud or in consequence of the interception of a
+travelling descending leader, there followed a completion of
+the lightning discharge. It is conceivable that the discharge
+was multicomponent and comprised its return strokes, for
+which the current with an amplitude of 35 kA measured is
+Ascending leader
+Structure under
+a cloud
+Laser
+Laser
+spark
+Mirror
+Tower
+h 50 m
+Hill
+Figure 12. Schematic diagram of the experiment on the laser triggering of
+lightning [17, 18].
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+713
+
+=== PAGE 14 ===
+quite typical. As regards the interpretation of the experi-
+mental results, there are some indications that preference
+should be given to the interception of the descending leader.
+Be it as it may, the current oscilloscope trace given in the
+paper does not exhibit a long-duration build-up of the current
+pulse up to several hundreds of amperes typical for ascending
+lightning.
+10. Requirements on a laser-produced channel
+In our opinion, the capability of triggering lightning high in
+the sky would hold the greatest interest for lightning science
+and lightning protection, in particular, for modelling the
+origination of lightning from aircraft. Let us see what the
+parameters of a channel between the cloud and the ground
+should be to permit the excitation of viable leaders from its
+ends. The channel should work as a good conductor. Hence,
+the electric field should be largely suppressed inside it but
+multiplied at the ends. Given this, a unit length will harbor a
+charge t 2pe0E0x= ln
L=r, where E0 is the external field
+parallel to the channel, L is its length, r its radius, and x the
+coordinate reckoned from the middle. This is explained by
+Fig. 4 and formula (1). The potential difference DU E0L=2
+originating at the ends of the initial conductor should ensure
+viability of the leaders. The requisite length L is defined by
+formula (8):
+Lmin 2
+ b ln
L=RL
+2pe0a
+2=3
+1
+E 5=3
+0
+:
+For instance, in order to excite lightning for E0 1 kV cmÿ1
+(say, at an altitude of 2 km, 1 km below the center of a cloud
+charge of 10 C), a length Lmin 20 m (DU 1 MV) is
+required. To polarize the plasma conductor, a charge
+Q pe0
+E0L2
+4 ln
L=r 90 m C
+should flow from its one half to the other. On the verge of
+possibility, it is afforded by a length-averaged ionization
+Ne min 2Q=
eL 5:5 1011 electrons cmÿ1. For the elec-
+trons to flow from one half of the conductor to the other
+before they recombine, the current i should be provided with a
+sufficiently large section. The magnitude of the electron
+density ne Ne=
pr2 has only a small effect on this, because
+the charge transfer time tp Q=i nÿ1
+e
+and the characteristic
+recombination time trec
bneÿ1 vary similarly in propor-
+tion to nÿ1
+e
+(b is the recombination coefficient). The time of
+charge transfer and significant attenuation of the electric field
+inside the plasma conductor is approximately
+tp
+Q
+pr2emeneE0
+
+1
+ln
L=r
+ L
+2r
+2
+tM ;
+where
+tM e0=
emene is
+the
+Maxwellian
+time,
+and
+me 600 cm2 (V s)ÿ1 the electron mobility. Unlike a plasma
+volume equally extended in all directions (L=2r 1) where
+the times of space-charge relaxation and field attenuation are
+close (tp tM), for an extended thin conductor tp 4 tM.
+The requirement tp < trec defines the lower permissible
+bound for the radius of the initial plasma channel
+rmin L
+2
+
+e0b
+eme ln
L=r
+s
+ 3:8 cm :
+The numerical value of rmin corresponds to the value
+b 10ÿ7 cm3 sÿ1 inherent in cold air. It is impossible to get
+by with a smaller radius in a scheme involving multiphoton
+ionization. However, it may be that a longer channel will
+prove to be hard to produce as far as radiation focusing is
+concerned, but this is quite a different matter. A long CO2-
+laser-produced spark, if it is continuous, usually proves to be
+heated. This circumstance is beneficial because a high
+temperature significantly suppresses both electron recombi-
+nation and attachment. However, considerably higher expen-
+ditures of laser energy are the price that has to be paid.
+We revert to the scheme involving multiphoton ioniza-
+tion. To induce the needed voltage change DU provided by the
+transfer of a charge Q, a very low ionization would suffice:
+ne min Ne min=
pr2
+min 1:2 1010 cmÿ3. But for so low an
+electron density the current would be too weak, i 0:1 A
+(even for an electric field still retaining the initial level, 1 kV
+cmÿ1), and the charge transfer time would be tp 1000 ms.
+For at least this time, electrons would have to be released
+from negative ions with the aid of a laser. The case in point
+now is a real laser with a pulse length t 10 ms. For the
+charge transfer to be accomplished during this time, a current
+i 10 A and an initial electron density ne 1012 cmÿ3 are
+required (for a field of the order of the initial one). There is
+little point in producing orders of magnitude higher electron
+densities employing an ultraviolet laser, because the density
+will inevitably lower to the 1012 cmÿ3 level owing to
+recombination
+during
+the
+same
+period
+of
+time
+trec
10ÿ7neÿ1 10 ms. To ionize a column of air of length
+L 20 m and radius r 3:8 cm to a level ne 1012 cmÿ3
+takes an ultraviolet radiation energy W pr2LneI 200 mJ
+(I 15 eV is the ionization potential).
+However, the above list of difficulties is not exhaustive.
+Until now, we have been dealing with the preparation of
+conditions for forming a potential change and a strong field
+multiplication at the ends of a long artificial conductor.
+However, it also takes time for the leaders to develop. This
+time is hard to estimate but, according to laboratory
+experiments, it runs into the tens of microseconds. Hence,
+negative ions will have to be destroyed for a longer period of
+time, though this will not exclude recombination. But most
+important of all, the leader process, namely, the propagation
+of two leaders in opposite directions, will require an
+uninterrupted charge transfer from one channel to the other,
+i.e. characteristic leader currents of 1 ± 100 A flowing through
+a conductor initially produced by artificial means. For the
+leader to commence unimpeded propagation and provoke
+real lightning, the laser-produced channel should acquire the
+properties of a true leader channel, i.e. become thin and
+strongly heated, like an arc, and additional ionization should
+proceed in it. In the leader tip, all this takes place through the
+action of the ionization-overheating instability. However,
+this process in the leader tip begins with a far thinner channel
+in a stronger electric field and for a higher electron density
+ne 1014 cmÿ3, which cannot persist in our case without
+heating for more than trec 10ÿ7 s. In essence, the question
+which we now are dealing with is the same as the glow-to-arc
+discharge transformation, the question of contraction or
+arcing in a weakly ionized cold plasma (the terms are many),
+which is still a long way from being solved [33].
+An alternate scenario for the course of events is also
+possible. If the conductivity in the cold laser-produced
+channel is somehow maintained for a time period such that
+the leader develops and travels a distance L, at least one (if the
+714
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
+
+=== PAGE 15 ===
+leaders of opposite polarity behave in a different way) viable
+conductor of the same length L will result. Subsequently, if
+the laser-produced channel decays, this new conductor will be
+polarized in the external field and the development of leaders
+from
+its
+ends
+will
+continue.
+For
+a
+leader
+velocity
+vL 2 106 cm sÿ1 and L 20 m, the time taken for this is
+about L=vL 100 ms. The time it takes the contraction to
+develop also runs into the tens of microseconds (according to
+our calculations [32] referring to the formation of the leader
+channel in the leader tip, where the conditions are, we repeat,
+more favorable, this proceeds faster Ð in a time t 1 ms).
+That is why the ionized state of the cold laser-produced
+channel will have to be artificially maintained for at least
+tens of microseconds. Which of the scenarios outlined above
+will be realized, if at all, will be revealed by a close theoretical
+treatment and numerical computations probably supported
+by a dedicated experiment Ð which presents a real challenge.
+It is conceivable that it will not be possible to dispense
+with the initial artificial heating of the primary channel
+altogether, and then preference will be given to the long
+laser spark produced by a CO2 laser. This will require a
+higher laser energy because the same 20-m long channel (for
+an external field of 1 kV cmÿ1) is to be made continuous. To
+make it clear what kind of energy expenditure will be dealt
+with, we point out that a 20-m long column of cool air 1 cm in
+diameter harbors, when heated to 4000 K at pressure 1 atm
+(to which there corresponds an equilibrium electron density
+ne 7 1012 cmÿ3), 16 kJ of energy. At present, CO2-laser
+pulses with an energy of 2 ± 5 kJ have been realized.
+In brief, it seems likely that the problem of lightning
+triggering at high altitudes is still a long way from receiving a
+final solution, despite the fact that there appear to be no
+fundamental obstacles. The reason is that the natural source
+for the origination of lightning is, we believe, the same kind of
+cool plasma object that we are dealing with. Here, we do not
+discuss the problem of focusing and transportation of high-
+power laser radiation to a high altitude provided that it does
+not induce air breakdown and is not absorbed on its path.
+When it comes to moderate altitudes, this problem does not
+generate skepticism among enthusiasts of laser triggering of
+lightning [16]. But, as the altitude decreases, the requirements
+on the length of the initial channel L and the laser energy
+become more stringent owing to weakening of the cloud field:
+L E ÿ5=3
+0
+. Conversely, the difficulties associated with trans-
+portation and focusing of the radiation become more severe
+with increasing altitude. One can see that the conditions for
+selecting the appropriate altitude are contradictory. There-
+fore, future work should proceed not only on the develop-
+ment of laser pulses of higher energy and power. It should
+search for ways of unimpeded transportation of the radiation
+to as high an altitude as possible.
+We emphasize once again that the very possibility of
+exciting twin leaders from an isolated conductor embedded
+in an external field is beyond question. This is precisely how
+lightning originates from airplanes, and experiments of this
+kind on metal rods of moderate length have been repeatedly
+staged in laboratories (see Fig. 6). The question arises of how
+to gain the `right' behavior of a plasma conductor, which
+possesses a far lower initial conductivity and is prone to lose
+it. This issue may and should be purposefully studied in a
+laboratory, as applied to the problem of triggering lightning.
+In doing this, emphasis should not be placed on lowering the
+breakdown voltage in a long gap or the use of a laser spark to
+direct the high-voltage spark, as have primarily been done
+until now. For simplicity, solid rods with a conduction well
+below that of metals are perhaps worth trying as the initiators.
+We point to the experimental fact which may be pertinent
+to the behavior of a discontinuous (broken) long spark. It is
+well known that a high-voltage discharge can propagate
+along a path in which small metal rods are placed at
+intervals. As the leader approaches, each of the small rods is
+polarized in the enhanced external field supposedly to emit a
+pair of leaders: one toward and the other in the same direction
+as the principal leader, and that is the way the spark
+propagates. It is significant that only a negative spark, and
+not a positive one, propagates in this way, which is clearly
+associated with the fact that the leader process is inherently
+stepwise in the former and void of steps in the latter.
+11. Conclusions
+So, in the foregoing we showed how and why lightning that
+propagates from a cloud to the earth opts to strike a tall
+structure, even though it may have to depart from its initial
+path. Under the action of the electric field induced by the
+charges of the lightning leader, electric charges are induced on
+the grounded structure and the electric field is multiplied at its
+summit; and the higher the structure, the greater the multi-
+plication. This is responsible for the origination of a leader
+ascending from the summit, the leader behaving like a high-
+voltage electrode. The criterion for viability of the counter
+leader imposes a constraint on the minimal structure height or
+the combined field of the charges of the lightning and the
+cloud acting on the structure. The mutual attraction of the
+descending and counter leaders, when they are widely
+separated (by over a hundred meters) and interact via weak
+fields, is determined by a subtle nontrivial mechanism which
+affects the acceleration. In this case, the absolute values of the
+leader velocities, which are determined by intrinsic fields in
+the proximity of the tips that are several orders of magnitude
+stronger, are virtually invariable.
+The joining of the leaders attracted to one another results
+in the closing of the electric cloud ± ground circuit. During the
+subsequent (not discussed in this paper) return stroke, the
+plasma channel between the structure summit and the cloud
+recharges acquiring the potential of the ground, with the
+result that an extremely high current flows through the
+structure. To protect buildings, recourse is made to lightning
+rods which are raised in the neighborhood of the object under
+protection but are made even higher in order for the counter
+leader to be excited from the lightning rod rather than from
+the object.
+In the quest to improve the reliability of protection of
+especially
+vulnerable
+and
+critical
+objects,
+different
+approaches to controlling lightning are basically possible.
+Attempts are being made to use lasers for this purpose as well.
+The laser triggering of lightning involves the production of an
+ionized air channel by employing laser radiation. Two major
+schemes are conceivable on this route. In one of them, the
+plasma channel is produced by a laser at the summit of a tall
+tower to promote the earlier excitation of an ascending leader,
+which intercepts the lightning. It is precisely this effect that
+was recently observed in Japan as a result of extensive
+preparatory work and after many unsuccessful attempts. It
+is conceivable that the role of the laser-produced plasma
+reduced to the extension of the top of the grounded conductor
+beyond the corona charge layer which was prohibitive to
+leader excitation.
+July, 2000
+The mechanism of lightning attraction and the problem of lightning initiation by lasers
+715
+
+=== PAGE 16 ===
+The other scheme under development involves laser-
+assisted production of a plasma channel in the open atmo-
+sphere so as to have lightning-provoking leaders excited at its
+ends, much as large airplanes do. The condition for the
+excitation of viable leaders from a plasma conductor is the
+same as for a grounded structure. It also defines the minimal
+conductor length. This approach to laser triggering of
+lightning is much more complicated but is of greater interest
+for both lightning science and, potentially, lightning protec-
+tion. That would be the way to excite descending lightning in
+the required place and time, timing the recording instruments
+to a fraction of a millisecond and, on the other hand, to
+discharge the cloud in a safe place. Many basic and practical
+difficulties will be encountered in reaching this goal, but a
+start has been made on this research and the scope of work
+will most likely expand. One of the major problems is to focus
+the laser radiation at as high an altitude as possible and in
+doing this to eliminate the breakdown of air over the path of
+radiation transportation. The higher the altitude of the
+plasma channel produced to excite the leaders, the shorter it
+may be, because the cloud field at a high altitude is stronger. A
+shorter laser-produced spark would require less laser energy.
+The laser radiation is easier to focus near to the earth, but in
+this case the requisite length of the initial laser-produced
+channel and the laser energy rise steeply.
+References
+1.
+Akhmatov A G, Rivlin L A, Shil'dyaev V S Pis'ma Zh. Eksp. Teor.
+Fiz. 8 417 (1968) [JETP Lett. 8 258 (1968)]
+2.
+Wili S R, Tidman D A Appl. Phys. Lett. 17 20 (1970)
+3.
+Norinski|¯ L V Kvantovaya Elektron. 5 108 (1971) [Sov. J. Quantum
+Electron. 1 519 (1971)]
+4.
+Koopman D W, Wilkerson T D J. Phys. D 42 1883 (1971)
+5.
+Norinski|¯ L V, Pryadein V A, Rivlin L A Zh. Eksp. Teor. Fiz. 63
+1649 (1972) [Sov. Phys. JETP 36 872 (1973)]
+6.
+Danilov O B, Tul'ski|¯ S A Zh. Tekh. Fiz. 48 2040 (1978) [Sov. Phys.
+Tech. Phys. 23 1164 (1978)]
+7.
+Zvorykin V D et al. Fiz. Plazmy 5 1140 (1979) [Sov. J. Plasma Phys. 5
+638 (1979)]
+8.
+Aleksandrov G N et al. Zh. Tekh. Fiz. 47 (10) 2122 (1977) [Sov. Phys.
+Tech. Phys. 22 1233 (1977)]
+9.
+Guenter A H, Bettis J R J. Phys. D 11 1577 (1978)
+10.
+Asinovski|¯ E I, Vasilyak L M, Nesterkin O P Pis'ma Zh. Tekh. Fiz.
+13 249 (1987) [Sov. Tech. Phys. Lett. 13 102 (1987)]; Teplofiz. Vys.
+Temp. 25 447 (1987)
+11.
+Vasilyak L M et al. Usp. Fiz. Nauk 164 (3) 263 (1994) [Phys. Usp. 37
+247 (1994)]
+12.
+Vasilyak L M ``Napravlyaemye Lazerom Elektricheskie Razryady''
+(Laser-Directed Electric Discharges), in Plasma, XX. Proceedings of
+the All-Russian Scientific-Educational Olympiad Comprising the
+Presentations to the FNTP-98 Conference and the Lectures at the
+School of Young Scientists, Petrozavodsk, 1998 (Ed. A D Khakhaev)
+Part 2 (Petrozavodsk: Izd. Petrozav. Univ., 1998) pp. 135 ± 156
+13.
+Ball L M Appl. Opt. 13 2292 (1974)
+14.
+Zhao X, Diels J-C, Cai Yi Wang, Elizondo J IEEE J. Quantum
+Electron. QE-31 599 (1995)
+15.
+Wang D et al. J. Geophys. Res. D 99 16907 (1994); J. Atm. Terrestrial
+Phys. 55 459 (1995)
+16.
+Diels J et al. Sci. Am. 277 50 (1997)
+17.
+Yasuda H et al. Pris. CHEO Pacific Rim. Optical Soc. Am. Paper
+PDI. 14 (1997)
+18.
+Uchida S et al. Opt. Zh. 66 (3) 36 (1999)
+19.
+Miki M, Uada A, Shindo T Opt. Zh. 66 (3) 25 (1999)
+20.
+Rambo P et al. Opt. Zh. 60 (3) 30 (1999)
+21.
+Uchida S et al., in Intern. Forum on Advanced High-Power Lasers and
+Appl. AHPLA'99, Osaka 1 ± 5 Nov. 1999. Paper 3886-22
+22.
+Diels J et al. AHPLA'99 Paper 3886-23
+23.
+Uchiumi M et al. AHPLA'99 Paper 3886-24
+24.
+Rezunkov Y, Borisov M, Gromovenko V, Lapshin V AHPLA'99
+Paper 3886-25; Borisov M F Opt. Zh. 60 (3) 40 (1999)
+25.
+La Fontaine et al. AHPLA'99 Paper 3886-26
+26.
+Apollonov V, Kazakov K, Pletnyev N, Sorochenko V AHPLA'99
+Paper 3886-27
+27.
+Yamauro M, Ihara S, Satoh C, Yamabe C AHPLA'99 Paper
+3886-28
+28.
+Shimada Y, Uchida S, Yamanaka C, Ogata A AHPLA'99 Paper
+3886-93
+29.
+Miki M, Wad A, Shindo T AHPLA'99 Paper 3886-95
+30.
+Aleksandrov G, Daineko G, Lekomtsev AHPLA'99 Paper 3886-104
+31.
+Apollonov V et al. AHPLA'99 Paper 3886-34
+32.
+Bazelyan E M, Raizer Yu P Iskrovo|¯ Razryad (Spark Discharge)
+(Moscow: Izd-vo MFTI, 1997) [Translated into English (Boca
+Raton, FL: CRC Press, 1998)]
+33.
+Raizer Yu P Fizika Gazovogo Razryada 2nd ed. (Gas Discharge
+Physics) (Moscow: Nauka, 1992) [Translated into English (Berlin:
+Springer-Verlag, 1991)]
+34.
+Kazemir H J. Geophys. Res. 65 1873 (1960)
+35.
+Bazelyan E M, Raizer Yu P Lightning Physics and Lightning
+Protection (Bristol, Philadelphia: IOP Publishing, 2000)
+36.
+Raizer Yu P Lasernaya Iskra: Rasprostranenie Razryadov (Laser-
+Induced Discharge Phenomena (Moscow: Nauka, 1974) [Trans-
+lated into English (New York: Consultants Bureau, 1977)]
+37.
+Uman M A The Lightning Discharge (Orlando: Academic Press,
+1987)
+38.
+Rakov V, Uman M, Thottappillil R J. Geophys. Res. 99 10745
+(1994)
+39.
+Basov N G et al. Dokl. Akad. Nauk SSSR 173 538 (1967)
+40.
+Hagen W F J. Appl. Phys. 40 511 (1969)
+41.
+Parfenov V N et al. Pis'ma Zh. Tekh. Fiz. 2 731 (1976) [Sov. Tech.
+Phys. Lett. 2 286 (1976)]
+42.
+Caressa J P et al. J. Appl. Phys. 50 6822 (1979); Smith D C,
+Meyerand R G, in Principles of Laser Plasmas (Ed. G Bekefi)
+(New York: Wiley, 1976)
+716
+EÂ M Bazelyan, Yu P Ra|¯zer
+Physics ± Uspekhi 43 (7)
diff --git a/run.bat b/run.bat
deleted file mode 100644
index 13a5935..0000000
--- a/run.bat
+++ /dev/null
@@ -1,59 +0,0 @@
-@echo off
-REM Tesla Coil Spark Course - Launch Script
-REM Creates virtual environment and runs PyQt5 application
-
-echo ========================================
-echo Tesla Coil Spark Physics Course
-echo ========================================
-echo.
-
-REM Navigate to spark-lessons directory
-cd spark-lessons
-
-REM Check if virtual environment exists
-if not exist venv (
- echo [*] Creating virtual environment...
- python -m venv venv
- if errorlevel 1 (
- echo [ERROR] Failed to create virtual environment!
- echo Please ensure Python 3.8+ is installed and in PATH.
- pause
- exit /b 1
- )
- echo [OK] Virtual environment created
-)
-
-REM Activate virtual environment
-echo [*] Activating virtual environment...
-call venv\Scripts\activate.bat
-
-REM Check if dependencies are installed
-if not exist venv\installed.flag (
- echo [*] Installing dependencies...
- python -m pip install --upgrade pip
- pip install -r requirements.txt
- if errorlevel 1 (
- echo [ERROR] Failed to install dependencies!
- pause
- exit /b 1
- )
- echo. > venv\installed.flag
- echo [OK] Dependencies installed
-)
-
-REM Run the application
-echo [*] Launching Tesla Coil Spark Course...
-echo.
-python app/main.py
-
-REM Capture exit code
-set EXIT_CODE=%ERRORLEVEL%
-
-REM Deactivate virtual environment
-call deactivate
-
-REM Return to original directory
-cd ..
-
-REM Exit with application's exit code
-exit /b %EXIT_CODE%
diff --git a/spark-lesson.txt b/spark-lesson.txt
deleted file mode 100644
index e03705a..0000000
--- a/spark-lesson.txt
+++ /dev/null
@@ -1,7327 +0,0 @@
-# Tesla Coil Spark Modeling - Complete Lesson Plan Index
-
-## Overview
-This lesson plan is designed to take someone from basic circuit concepts through advanced Tesla coil spark modeling. Each part builds progressively, with worked examples, visual aids descriptions, and practice problems.
-
----
-
-## **Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior**
-*Target: 2-3 hours of study*
-
-### Module 1.1: AC Circuit Fundamentals Review
-- Peak vs RMS values (why we use peak)
-- Complex numbers and phasor notation (j, magnitude, phase)
-- Resistance (R), Reactance (X), Impedance (Z)
-- Conductance (G), Susceptance (B), Admittance (Y)
-- Power in AC circuits: P = 0.5 × Re{V × I*}
-- **Worked Example 1.1:** Calculate power with peak phasors
-- **Practice Problems:** 3 problems on complex impedance calculations
-
-### Module 1.2: Capacitance in Tesla Coils
-- What is capacitance physically?
-- Self-capacitance vs mutual capacitance
-- Capacitance to ground (shunt capacitance)
-- The 2 pF/foot empirical rule
-- **Worked Example 1.2:** Estimate C_sh for a 2-meter spark
-- **Visual Aid:** Diagram showing field lines for C_mut and C_sh
-- **Practice Problems:** 2 problems on capacitance estimation
-
-### Module 1.3: The Basic Spark Circuit Topology
-- Why spark has TWO capacitances (C_mut and C_sh)
-- Drawing the circuit: parallel R||C_mut in series with C_sh
-- Where is "ground" in a Tesla coil?
-- The topload port (measurement reference)
-- **Worked Example 1.3:** Draw circuit for given geometry
-- **Visual Aid:** 3D geometry → circuit schematic translation
-- **Practice Problems:** 2 problems on circuit topology
-
-### Module 1.4: Admittance Analysis of the Spark Circuit
-- Why use admittance (Y) instead of impedance (Z)?
-- Parallel combinations are easy in Y
-- Deriving Y_total = ((G+jB₁)·jB₂)/(G+j(B₁+B₂))
-- Real and imaginary parts
-- Converting back to impedance
-- **Worked Example 1.4:** Calculate Y and Z for specific values
-- **Visual Aid:** Complex plane plots showing Y and Z
-- **Practice Problems:** 3 problems on admittance calculations
-
-### Module 1.5: Phase Angles and What They Mean
-- Impedance phase φ_Z vs admittance phase θ_Y
-- Why φ_Z = -θ_Y
-- The "famous -45°" myth
-- Physical meaning: how much does load look resistive?
-- **Worked Example 1.5:** Calculate φ_Z from given R, C_mut, C_sh
-- **Visual Aid:** Phase angle on complex plane
-- **Practice Problems:** 2 problems on phase angle interpretation
-
-### Module 1.6: Introduction to Spark Physics
-- What is a spark? (brief non-mathematical overview)
-- Streamers vs leaders (qualitative)
-- Why sparks need voltage AND power
-- The "hungry streamer" principle (conceptual introduction)
-- **Visual Aid:** Photos/diagrams of streamers vs leaders
-- **Discussion Questions:** 3 conceptual questions
-
-### Part 1 Summary & Integration
-- Checkpoint quiz (10 questions, multiple choice + short answer)
-- Concept map connecting all Module 1 topics
-- Preview of Part 2
-
-**Estimated Token Count: ~15,000-18,000**
-
----
-
-## **Part 2: Optimization and Power Transfer - Making Sparks Efficient**
-*Target: 2-3 hours of study*
-
-### Module 2.1: The Topological Phase Constraint
-- What is a topological constraint?
-- Deriving φ_Z,min = -atan(2√(r(1+r)))
-- Why r = C_mut/C_sh matters
-- The critical value r = 0.207
-- When is -45° impossible?
-- **Worked Example 2.1:** Calculate φ_Z,min for typical geometries
-- **Visual Aid:** Graph of φ_Z,min vs r
-- **Practice Problems:** 3 problems on phase constraints
-
-### Module 2.2: The Two Critical Resistances
-- R_opt_power: maximum power transfer
-- R_opt_phase: closest to resistive
-- Why R_opt_power < R_opt_phase always
-- Deriving R_opt_power = 1/(ω(C_mut + C_sh))
-- Deriving R_opt_phase = 1/(ω√(C_mut(C_mut + C_sh)))
-- **Worked Example 2.2:** Calculate both for f=200 kHz, various capacitances
-- **Visual Aid:** Power vs R curves showing optima
-- **Practice Problems:** 4 problems on optimal resistances
-
-### Module 2.3: The "Hungry Streamer" - Self-Optimization
-- How plasma conductivity changes with power
-- Temperature → ionization → conductivity loop
-- Why sparks naturally seek R_opt_power
-- Constraints that prevent optimization
-- Physical limits: R_min and R_max
-- **Worked Example 2.3:** Trace through optimization process
-- **Visual Aid:** Flowchart of self-optimization mechanism
-- **Discussion Questions:** 3 questions on optimization limits
-
-### Module 2.4: Power Calculations
-- Power to a load: P = 0.5|V|²Re{Z_load}/|Z_th+Z_load|²
-- Why V_top/I_base is wrong
-- Displacement current problem
-- Correct measurement at topload port
-- **Worked Example 2.4:** Calculate power with correct vs incorrect method
-- **Visual Aid:** Current flow diagram showing displacement currents
-- **Practice Problems:** 3 problems on power calculations
-
-### Module 2.5: Thévenin Equivalent Method (Part A)
-- What is a Thévenin equivalent?
-- Why it separates coil from load
-- Measuring Z_th (output impedance)
-- Step-by-step procedure
-- **Worked Example 2.5A:** Extract Z_th from simulation
-- **Visual Aid:** Circuit diagrams for measurement setup
-- **Practice Problems:** 2 problems on Z_th measurement
-
-### Module 2.6: Thévenin Equivalent Method (Part B)
-- Measuring V_th (open-circuit voltage)
-- Using Z_th and V_th to predict any load
-- Theoretical maximum power (conjugate match)
-- Why actual spark power is less
-- **Worked Example 2.6:** Complete Thévenin analysis
-- **Visual Aid:** Load line analysis
-- **Practice Problems:** 3 problems on load power prediction
-
-### Module 2.7: Quality Factor and Ringdown Measurements
-- What is Q? (energy storage vs loss)
-- Q₀ (unloaded) vs Q_L (loaded)
-- Measuring Q from ringdown waveform
-- Extracting spark admittance from Q_L, f_L measurements
-- **Worked Example 2.7:** Q measurement from oscilloscope capture
-- **Visual Aid:** Annotated ringdown waveform
-- **Practice Problems:** 3 problems on Q measurements
-
-### Part 2 Summary & Integration
-- Checkpoint quiz (12 questions)
-- Worked example combining all of Part 2
-- Design exercise: optimize R for a given coil
-- Preview of Part 3
-
-**Estimated Token Count: ~18,000-20,000**
-
----
-
-## **Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From**
-*Target: 3-4 hours of study*
-
-### Module 3.1: Electric Fields and Breakdown
-- Electric field basics (V/m)
-- Field concentration at sharp points
-- E_inception: initial breakdown (~2-3 MV/m)
-- E_propagation: sustained growth (~0.4-1.0 MV/m)
-- Tip enhancement factor κ
-- **Worked Example 3.1:** Calculate E_tip for given voltage and geometry
-- **Visual Aid:** Field line diagram with enhancement
-- **Practice Problems:** 3 problems on field calculations
-
-### Module 3.2: Energy Requirements for Growth
-- Energy per meter (ε) concept
-- Why different operating modes have different ε
-- QCW: 5-15 J/m (efficient)
-- Burst: 30-100 J/m (inefficient)
-- Physical mechanisms behind ε
-- **Worked Example 3.2:** Calculate energy needed for target length
-- **Visual Aid:** Energy budget breakdown
-- **Practice Problems:** 2 problems on energy requirements
-
-### Module 3.3: Growth Rate Equation
-- dL/dt = P_stream/ε (when E_tip > E_propagation)
-- Voltage limit vs power limit
-- When does growth stall?
-- Time to reach target length
-- **Worked Example 3.3:** Predict growth time for QCW ramp
-- **Visual Aid:** Length vs time curves for different modes
-- **Practice Problems:** 3 problems on growth dynamics
-
-### Module 3.4: Thermal Physics of Plasma Channels
-- Temperature in streamers vs leaders
-- Thermal diffusion time constant τ_thermal = d²/(4α)
-- Why observed persistence is longer
-- Convection and ionization memory
-- QCW advantage: maintaining hot channels
-- **Worked Example 3.4:** Calculate thermal time constants
-- **Visual Aid:** Temperature profile cross-section
-- **Practice Problems:** 2 problems on thermal dynamics
-
-### Module 3.5: The Capacitive Divider Problem
-- How V_tip < V_topload due to C_sh
-- V_tip = V_topload × C_mut/(C_mut+C_sh) (open circuit)
-- Effect of finite R
-- As spark grows, C_sh grows, V_tip drops
-- Why length scales sub-linearly with energy
-- **Worked Example 3.5:** Calculate V_tip for growing spark
-- **Visual Aid:** Equivalent circuit with divider highlighted
-- **Practice Problems:** 3 problems on voltage division
-
-### Module 3.6: Introduction to FEMM
-- What is FEMM? (Finite Element Method Magnetics)
-- Electrostatic analysis for capacitances
-- Setting up a problem: geometry, boundaries, materials
-- Meshing and solving
-- Extracting results
-- **Worked Example 3.6:** Step-by-step FEMM tutorial (simple geometry)
-- **Visual Aid:** Screenshots of FEMM interface
-- **Practice Problems:** 1 guided FEMM exercise
-
-### Module 3.7: Extracting Capacitances from FEMM
-- The Maxwell capacitance matrix [C]
-- Diagonal elements: self-capacitances (positive)
-- Off-diagonal: mutual capacitances (negative)
-- For 2-body problem: C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|
-- Validation: C_sh ≈ 2 pF/foot check
-- **Worked Example 3.7:** Extract values from FEMM output
-- **Visual Aid:** Annotated capacitance matrix
-- **Practice Problems:** 2 problems on matrix interpretation
-
-### Module 3.8: Building the Lumped Spark Model
-- Using FEMM capacitances in circuit
-- Choosing R = R_opt_power
-- Clipping to physical bounds (R_min, R_max)
-- Implementing in SPICE
-- Running AC analysis
-- **Worked Example 3.8:** Complete lumped model simulation
-- **Visual Aid:** Flowchart from FEMM to SPICE
-- **Practice Problems:** 1 complete modeling exercise
-
-### Part 3 Summary & Integration
-- Checkpoint quiz (15 questions)
-- Complete design project: predict spark length for given coil
-- Comparison exercise: simulation vs empirical rules
-- Preview of Part 4
-
-**Estimated Token Count: ~20,000-22,000**
-
----
-
-## **Part 4: Advanced Topics - Distributed Models and Real-World Application**
-*Target: 3-4 hours of study*
-
-### Module 4.1: Why Distributed Models?
-- Limitations of lumped model
-- Current distribution along spark
-- Tip vs base differences
-- When is distributed model necessary?
-- **Visual Aid:** Comparison showing where lumped fails
-- **Discussion Questions:** 3 questions on model selection
-
-### Module 4.2: nth-Order Model Structure
-- Dividing spark into n segments (typically n=10)
-- Circuit topology with multiple segments
-- Capacitance matrix grows to (n+1)×(n+1)
-- Including all segment-to-segment couplings
-- Optional: inductance matrix
-- **Worked Example 4.2:** Draw 3-segment distributed model
-- **Visual Aid:** Progressive complexity (n=1, 3, 5, 10)
-- **Practice Problems:** 2 problems on model structure
-
-### Module 4.3: FEMM for Distributed Models
-- Multi-body electrostatic analysis
-- Defining n cylindrical segments
-- Extracting large capacitance matrix
-- Matrix properties: symmetric, semi-definite
-- Numerical stability and passivity
-- **Worked Example 4.3:** FEMM setup for n=5 model
-- **Visual Aid:** FEMM geometry with labeled segments
-- **Practice Problems:** 1 FEMM exercise with multiple bodies
-
-### Module 4.4: Implementing Capacitance Matrices in SPICE
-- Challenge: negative off-diagonal elements
-- Solution 1: Partial capacitance transformation
-- Solution 2: Controlled sources (MNA approach)
-- Solution 3: Nearest-neighbor approximation
-- Validation and stability
-- **Worked Example 4.4:** Convert 3×3 Maxwell to SPICE
-- **Visual Aid:** Circuit comparison of methods
-- **Practice Problems:** 2 problems on matrix implementation
-
-### Module 4.5: Resistance Optimization - Iterative Method
-- Initialization: tapered R profile
-- Iterative power maximization algorithm
-- Damping for stability (α_damp ≈ 0.3-0.5)
-- Position-dependent bounds: R_min[i], R_max[i]
-- Convergence criteria
-- **Worked Example 4.5:** Hand-trace 3 iterations for small model
-- **Visual Aid:** Flowchart of optimization algorithm
-- **Pseudo-code:** Python-style implementation
-- **Practice Problems:** 2 problems on optimization
-
-### Module 4.6: Resistance Optimization - Simplified Method
-- Circuit-determined resistance: R[i] = 1/(ω×C_total[i])
-- Weak diameter dependence (logarithmic)
-- When is this good enough?
-- Comparison with iterative method
-- **Worked Example 4.6:** Calculate R distribution for n=10 model
-- **Visual Aid:** Comparison plot: iterative vs simplified
-- **Practice Problems:** 2 problems on simplified method
-
-### Module 4.7: Diameter and Self-Consistency
-- Nominal diameter choice (1 mm burst, 3 mm QCW)
-- Back-calculating implied diameter from R
-- Self-consistency iteration (usually 1-2 steps)
-- Why it matters (and when it doesn't)
-- **Worked Example 4.7:** Self-consistency check
-- **Visual Aid:** Iteration convergence diagram
-- **Practice Problems:** 1 problem on diameter calculation
-
-### Module 4.8: Complete Simulation Workflow
-- Step 1: FEMM electrostatic analysis
-- Step 2: Extract capacitance matrix
-- Step 3: Choose/optimize resistances
-- Step 4: Build SPICE model
-- Step 5: Run analysis (AC or transient)
-- Step 6: Validate results
-- **Worked Example 4.8:** End-to-end simulation project
-- **Visual Aid:** Comprehensive workflow diagram
-- **Practice Problems:** 1 complete simulation exercise
-
-### Module 4.9: Validation and Physical Checks
-- Power balance: P_in = P_spark + P_losses
-- Total R in expected range (5-300 kΩ at 200 kHz)
-- R distribution: base < tip
-- C_sh validation: 2 pF/foot rule
-- Convergence tests: n=5 vs n=10 vs n=20
-- **Worked Example 4.9:** Validate a questionable simulation
-- **Visual Aid:** Checklist with pass/fail criteria
-- **Practice Problems:** 2 validation exercises
-
-### Module 4.10: Calibration from Real Measurements
-- Measuring ε: known drive, measure final length
-- Measuring E_propagation: V_top and L at stall
-- Using ringdown for Y_spark
-- Iterative refinement of model parameters
-- Building a calibration database
-- **Worked Example 4.10:** Calibrate ε from test data
-- **Visual Aid:** Calibration workflow
-- **Practice Problems:** 2 calibration problems
-
-### Module 4.11: Advanced Topics Preview
-- Frequency tracking during growth
-- Branching models (power division)
-- Strike event simulation (R collapse)
-- 3D FEA for complex geometries
-- Monte Carlo for stochastic effects
-- **Visual Aid:** Gallery of advanced scenarios
-- **Further Reading:** Resources for each topic
-
-### Module 4.12: Complete Design Case Study
-- Given: Coil specifications (f, L_secondary, C_topload, etc.)
-- Goal: Predict spark length for QCW operation
-- Work through entire process step-by-step
-- Compare prediction to empirical rules
-- Discuss uncertainties and limitations
-- **Comprehensive Example:** Full documentation
-- **Visual Aid:** Annotated results presentation
-
-### Part 4 Summary & Final Integration
-- Comprehensive final quiz (20 questions)
-- Capstone project: Design and simulate your own coil
-- Troubleshooting guide: Common errors and fixes
-- Resources for continued learning
-- Community and collaboration suggestions
-
-**Estimated Token Count: ~22,000-25,000**
-
----
-
-## Appendices (Reference Material - Brief)
-*Can be included at end of Part 4 or as separate quick-reference*
-
-### Appendix A: Complete Variable Reference Table
-- All variables with units and definitions (condensed)
-
-### Appendix B: Formula Quick Reference
-- All key equations organized by topic
-
-### Appendix C: Physical Constants
-- Standard values for air properties, field thresholds, etc.
-
-### Appendix D: SPICE Component Reference
-- How to implement various elements
-
-### Appendix E: FEMM Quick Start Guide
-- Installation, basic navigation, common tasks
-
-### Appendix F: Troubleshooting Guide
-- Common problems and solutions organized by symptom
-
-**Estimated Token Count: ~5,000-6,000**
-
----
-
-## Teaching Philosophy Embedded in This Plan
-
-1. **Spiral learning:** Concepts introduced simply, then revisited with more depth
-2. **Worked examples:** Every mathematical concept has at least one complete example
-3. **Visual aids:** Descriptions provided so you can create diagrams/graphs
-4. **Practice problems:** Incremental difficulty, answers can be provided separately
-5. **Checkpoints:** Regular assessment to ensure understanding before proceeding
-6. **Real-world connection:** Every module ties back to actual Tesla coil behavior
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior
-
----
-
-## Module 1.1: AC Circuit Fundamentals Review
-
-### Peak vs RMS Values
-
-In AC circuits, voltage and current vary sinusoidally with time. We can express them in two ways:
-
-**Time domain:**
-```
-v(t) = V_peak × cos(ωt + φ)
-```
-
-**Two amplitude conventions:**
-- **Peak value:** The maximum value reached (V_peak)
-- **RMS value:** Root-Mean-Square, V_RMS = V_peak/√2 ≈ 0.707 × V_peak
-
-**For this entire framework, we use PEAK VALUES exclusively.**
-
-**Why peak values?**
-1. Tesla coils are concerned with maximum voltage (breakdown, field stress)
-2. Consistent with phasor notation in engineering
-3. Power formula becomes: P = 0.5 × V_peak × I_peak × cos(θ)
-
-**Example:** If your oscilloscope shows a 100 kV peak-to-peak waveform:
-- V_peak-to-peak = 100 kV
-- V_peak = 50 kV (one-sided amplitude)
-- V_RMS = 50 kV / √2 ≈ 35.4 kV
-
-### Complex Numbers and Phasors
-
-AC circuit analysis uses complex numbers to represent magnitude and phase simultaneously.
-
-**Rectangular form:**
-```
-Z = R + jX
-where j = √(-1) (imaginary unit, engineers use 'j' instead of 'i')
-R = real part (resistance)
-X = imaginary part (reactance)
-```
-
-**Polar form:**
-```
-Z = |Z| ∠φ = |Z| × e^(jφ)
-where |Z| = √(R² + X²) (magnitude)
- φ = atan(X/R) (phase angle)
-```
-
-**Conversion:**
-```
-R = |Z| × cos(φ)
-X = |Z| × sin(φ)
-```
-
-**Phasor notation:** A complex number representing sinusoidal amplitude and phase:
-```
-V = V_peak ∠φ_v
-I = I_peak ∠φ_i
-```
-
-**Complex conjugate:** Used in power calculations
-```
-If I = a + jb, then I* = a - jb (flip sign of imaginary part)
-```
-
-### Resistance, Reactance, Impedance
-
-**Resistance (R):** Opposition to current that dissipates energy as heat
-- Units: Ω (ohms)
-- Always real and positive
-- V = I × R (Ohm's law)
-
-**Reactance (X):** Opposition to current that stores energy (no dissipation)
-- Units: Ω (ohms)
-- Can be positive (inductive) or negative (capacitive)
-- **Capacitive reactance:** X_C = -1/(ωC) where ω = 2πf
-- **Inductive reactance:** X_L = ωL
-
-**Impedance (Z):** Total opposition to AC current
-```
-Z = R + jX (complex)
-|Z| = √(R² + X²)
-φ_Z = atan(X/R)
-```
-
-**Sign conventions:**
-- X > 0: inductive (current lags voltage)
-- X < 0: capacitive (current leads voltage)
-- φ_Z > 0: inductive
-- φ_Z < 0: capacitive
-
-### Conductance, Susceptance, Admittance
-
-For parallel circuits, **admittance (Y)** is more convenient than impedance.
-
-**Conductance (G):** Inverse of resistance
-```
-G = 1/R
-Units: S (siemens)
-```
-
-**Susceptance (B):** Inverse of reactance (BUT with opposite sign convention!)
-```
-For capacitor: B_C = ωC (positive!)
-For inductor: B_L = -1/(ωL) (negative)
-```
-
-**Important:** Susceptance sign convention is OPPOSITE of reactance:
-- Capacitor: X_C < 0, but B_C > 0
-- Inductor: X_L > 0, but B_L < 0
-
-**Admittance (Y):** Inverse of impedance
-```
-Y = G + jB = 1/Z
-|Y| = 1/|Z|
-φ_Y = -φ_Z (opposite sign!)
-```
-
-**Conversion between Z and Y:**
-```
-Y = 1/Z = 1/(R + jX) = R/(R² + X²) - jX/(R² + X²)
-
-Therefore:
-G = R/(R² + X²)
-B = -X/(R² + X²)
-```
-
-### Power in AC Circuits
-
-**Using peak phasors:**
-```
-P = 0.5 × Re{V × I*}
-
-where V and I are complex peak phasors
- I* is the complex conjugate of I
- Re{·} means "real part of"
-```
-
-**Why the 0.5 factor?**
-- Average power over a full AC cycle
-- Comes from time-averaging cos²(ωt), which equals 0.5
-- If you used RMS values, formula would be P = V_RMS × I_RMS × cos(θ), NO 0.5
-
-**Expanded form:**
-```
-If V = V_peak ∠φ_v and I = I_peak ∠φ_i, then:
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-```
-
-The angle difference (φ_v - φ_i) is the power factor angle.
-
----
-
-### WORKED EXAMPLE 1.1: Power Calculation with Peak Phasors
-
-**Given:**
-- Voltage: V = 50 kV ∠0° (peak, using 0° as reference)
-- Impedance: Z = 100 kΩ ∠-60° (capacitive load)
-
-**Find:** Real power dissipated
-
-**Solution:**
-
-Step 1: Calculate current using Ohm's law
-```
-I = V/Z = (50 kV ∠0°)/(100 kΩ ∠-60°)
-I = 0.5 A ∠(0° - (-60°)) = 0.5 A ∠60°
-```
-
-Step 2: Calculate power
-```
-P = 0.5 × Re{V × I*}
-P = 0.5 × Re{(50 kV ∠0°) × (0.5 A ∠-60°)}
-P = 0.5 × Re{25 kW ∠-60°}
-```
-
-Step 3: Convert to rectangular to get real part
-```
-25 kW ∠-60° = 25 kW × (cos(-60°) + j×sin(-60°))
- = 25 kW × (0.5 - j×0.866)
- = 12.5 kW - j×21.65 kW
-```
-
-Step 4: Extract real part and apply 0.5 factor
-```
-P = 0.5 × 12.5 kW = 6.25 kW
-```
-
-**Alternative method:** Using power factor angle
-```
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-P = 0.5 × 50 kV × 0.5 A × cos(0° - 60°)
-P = 0.5 × 25 kW × cos(-60°)
-P = 0.5 × 25 kW × 0.5
-P = 6.25 kW
-```
-
----
-
-### PRACTICE PROBLEMS 1.1
-
-**Problem 1:** A capacitor has reactance X_C = -80 kΩ at 200 kHz. What is its capacitance? What is its susceptance?
-
-**Problem 2:** An impedance Z = 50 kΩ - j75 kΩ has current I = 0.2 A ∠30° (peak). Calculate: (a) Voltage magnitude and phase, (b) Real power
-
-**Problem 3:** An admittance Y = 0.00001 + j0.00002 S. Convert to impedance Z = R + jX.
-
----
-
-## Module 1.2: Capacitance in Tesla Coils
-
-### What is Capacitance Physically?
-
-**Definition:** Capacitance (C) is the ability to store electric charge for a given voltage:
-```
-Q = C × V
-Units: Farads (F), typically pF (10⁻¹² F) for Tesla coils
-```
-
-**Physical picture:**
-- Electric field between two conductors stores energy
-- Higher field → more stored energy → more capacitance
-- Capacitance depends on geometry, NOT on voltage
-
-**For parallel plates:**
-```
-C = ε₀ × A / d
-
-where ε₀ = 8.854×10⁻¹² F/m (permittivity of free space)
- A = plate area (m²)
- d = separation distance (m)
-```
-
-**Key insight:** Capacitance increases with:
-- Larger conductor area (more field lines)
-- Smaller separation (stronger field concentration)
-
-### Self-Capacitance vs Mutual Capacitance
-
-**Self-capacitance:** Capacitance of a single conductor to infinity (or ground)
-- Topload has self-capacitance to ground
-- Depends on size and shape
-- Toroid: C ≈ 4πε₀√(D×d) where D = major diameter, d = minor diameter
-
-**Mutual capacitance:** Capacitance between two conductors
-- Energy stored in field between them
-- Both conductors at different potentials
-- Can be positive or negative in matrix formulation
-
-**For Tesla coils with sparks:**
-- **C_mut:** mutual capacitance between topload and spark channel
-- **C_sh:** capacitance from spark to ground (shunt capacitance)
-
-### Capacitance to Ground (Shunt Capacitance)
-
-Any conductor elevated above ground has capacitance to ground.
-
-**For vertical wire above ground plane:**
-```
-C ≈ 2πε₀L / ln(2h/d)
-
-where L = wire length
- h = height above ground
- d = wire diameter
-```
-
-**For Tesla coil sparks:** Empirical rule based on community measurements:
-```
-C_sh ≈ 2 pF per foot of spark length
-
-Examples:
-1 foot (0.3 m) spark: C_sh ≈ 2 pF
-3 feet (0.9 m) spark: C_sh ≈ 6 pF
-6 feet (1.8 m) spark: C_sh ≈ 12 pF
-```
-
-This rule is surprisingly accurate (±30%) for typical Tesla coil geometries.
-
----
-
-### WORKED EXAMPLE 1.2: Estimating C_sh for a Spark
-
-**Given:** A 2-meter (6.6 foot) spark
-
-**Find:** Estimated shunt capacitance
-
-**Solution:**
-```
-C_sh ≈ 2 pF/foot × 6.6 feet
-C_sh ≈ 13.2 pF
-```
-
-**Refined estimate using cylinder formula:**
-
-Assume spark is vertical cylinder:
-- Length L = 2 m
-- Diameter d = 2 mm (typical for bright spark)
-- Height above ground h = L/2 = 1 m (average height)
-
-```
-C ≈ 2πε₀L / ln(2h/d)
-C ≈ 2π × 8.854×10⁻¹² × 2 / ln(2×1/0.002)
-C ≈ 1.112×10⁻¹⁰ / ln(1000)
-C ≈ 1.112×10⁻¹⁰ / 6.91
-C ≈ 16 pF
-```
-
-The empirical rule (13 pF) and formula (16 pF) agree reasonably well.
-
----
-
-### VISUAL AID 1.2: Field Lines for C_mut and C_sh
-
-```
-[Describe for drawing:]
-
-Side view of Tesla coil with spark:
-
- Spark tip (pointed)
- |
- | C_sh field lines radiate from
- | spark to ground plane horizontally
- Spark | (curved lines going left/right to ground)
- body |
- |
- |
- Topload (toroid)
- |
- Secondary
-
-C_mut field lines: Connect topload surface to spark channel
- - Start on topload outer surface
- - End on spark channel surface
- - Concentrated near base of spark
- - These store mutual electric field energy
-
-C_sh field lines: Connect spark to remote ground
- - Start on spark surface
- - Radiate outward to walls, floor, ceiling
- - Distributed along entire spark length
- - These store shunt field energy
-
-Key observation: Same spark channel participates in BOTH capacitances!
-This is why we need parallel C_mut || R, then series C_sh
-```
-
----
-
-### PRACTICE PROBLEMS 1.2
-
-**Problem 1:** A 4-foot spark is formed. Estimate C_sh using the empirical rule. If the topload has C_topload = 30 pF unloaded, what is the total system capacitance with the spark?
-
-**Problem 2:** Using the cylinder formula, calculate C_sh for a spark with: L = 1.5 m, d = 3 mm, average height h = 0.75 m. Compare to the empirical rule.
-
----
-
-## Module 1.3: The Basic Spark Circuit Topology
-
-### Why Sparks Have TWO Capacitances
-
-A spark channel is a conductor in space with:
-1. **Proximity to the topload** → mutual capacitance C_mut
-2. **Proximity to ground/environment** → shunt capacitance C_sh
-
-**Both exist simultaneously** because the spark interacts with multiple conductors.
-
-**Analogy:** A wire near two metal plates
-- Capacitance to plate 1: C₁
-- Capacitance to plate 2: C₂
-- Both must be included in the circuit model
-
-### The Correct Circuit Topology
-
-```
- Topload (measurement reference)
- |
- [C_mut] ← Mutual capacitance between topload and spark
- |
- +---------+--------- Node_spark
- | |
- [R] [C_sh] ← Shunt capacitance spark-to-ground
- | |
- GND ------------ GND
-```
-
-**Equivalent description:**
-- C_mut and R in parallel
-- That parallel combination in series with C_sh
-- All connected between topload and ground
-
-**Why this topology?**
-1. C_mut couples topload voltage to spark
-2. R represents plasma resistance (where power is dissipated)
-3. C_sh provides current return path to ground
-4. Current through R must also flow through either C_mut or C_sh (series connection)
-
-### Where is "Ground" in a Tesla Coil?
-
-**Earth ground:** Actual connection to soil/building ground
-**Circuit ground (reference):** Arbitrary 0V reference point
-
-**For Tesla coils:**
-- Primary circuit: Chassis/mains ground is reference
-- Secondary base: Usually connected to primary ground via RF ground
-- **Practical ground:** Floor, walls, nearby objects, you standing nearby
-- **Measurement ground:** Choose ONE point as 0V reference (usually secondary base)
-
-**Important:** "Ground" in spark model means "remote return path" - could be walls, floor, strike ring, or actual earth.
-
-### The Topload Port
-
-**Definition:** The two-terminal measurement point between topload and ground where we characterize impedance and power.
-
-```
-Port definition:
- Terminal 1: Topload terminal (high voltage)
- Terminal 2: Ground reference (0V)
-```
-
-**All impedance measurements reference this port:**
-- Z_spark: impedance looking into spark from topload
-- Z_th: Thévenin impedance of coil at this port
-- V_th: Open-circuit voltage at this port
-
-**Not the same as:**
-- V_top / I_base (includes displacement currents from entire secondary)
-- Any two-point measurement along the secondary winding
-
----
-
-### WORKED EXAMPLE 1.3: Drawing the Circuit
-
-**Given:**
-- Spark is 3 feet long
-- FEMM analysis gives C_mut = 8 pF (between topload and spark)
-- Estimate C_sh using empirical rule
-- Assume R = 100 kΩ
-
-**Task:** Draw complete circuit diagram
-
-**Solution:**
-
-Step 1: Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 3 feet = 6 pF
-```
-
-Step 2: Draw topology
-```
- Topload (V_top)
- |
- [C_mut = 8 pF]
- |
- +-------- Node_spark
- | |
- [R = 100 kΩ] [C_sh = 6 pF]
- | |
- GND -------- GND
-```
-
-Step 3: Simplify to show parallel/series structure
-```
-Topload
- |
- +---- [C_mut = 8 pF] ----+
- | |
- +---- [R = 100 kΩ] ------+ Node_spark
- |
- [C_sh = 6 pF]
- |
- GND
-```
-
-This is the basic lumped model for a Tesla coil spark.
-
----
-
-### VISUAL AID 1.3: 3D Geometry → Circuit Schematic
-
-```
-[Describe for drawing:]
-
-Panel 1: Physical 3D view
-- Toroidal topload at top (labeled "Topload")
-- Vertical spark channel extending downward (labeled "Spark, length L")
-- Ground plane at bottom (labeled "Ground")
-- Dashed lines showing C_mut field (topload to spark)
-- Dotted lines showing C_sh field (spark to ground)
-
-Panel 2: Conceptual extraction
-- Topload → single node
-- Spark → two elements: resistance R and capacitances
-- Ground → common reference
-- Arrows showing "Extract C_mut from field between topload and spark"
-- Arrows showing "Extract C_sh from field between spark and ground"
-
-Panel 3: Circuit schematic (as drawn above)
-- Proper circuit symbols
-- Component values labeled
-- Ground symbol at bottom
-- Clear port definition marked
-
-Annotation: "Same physics, different representations"
-```
-
----
-
-### PRACTICE PROBLEMS 1.3
-
-**Problem 1:** Draw the circuit for a spark with: L = 5 feet, C_mut = 12 pF (from FEMM), R = 50 kΩ. Label all component values.
-
-**Problem 2:** A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark length using the empirical rule?
-
----
-
-## Module 1.4: Admittance Analysis of the Spark Circuit
-
-### Why Use Admittance?
-
-For the spark circuit topology (parallel R||C_mut, in series with C_sh), admittance simplifies calculations.
-
-**Parallel elements:** Add admittances directly
-```
-Y_total = Y₁ + Y₂ + Y₃ + ...
-vs impedances: 1/Z_total = 1/Z₁ + 1/Z₂ + ... (messy!)
-```
-
-**Our circuit:**
-```
-Y_mut_R = Y_Cmut + Y_R (parallel: C_mut || R)
-Then series with C_sh requires impedance: Z = Z_mut_R + Z_Csh
-Then convert back: Y_total = 1/Z_total
-```
-
-### Deriving the Total Admittance Formula
-
-**Step 1:** Admittance of R and C_mut in parallel
-
-```
-Y_R = G = 1/R
-Y_Cmut = jωC_mut = jB₁ (where B₁ = ωC_mut)
-
-Y_mut_R = G + jB₁
-```
-
-**Step 2:** Convert to impedance for series combination
-```
-Z_mut_R = 1/(G + jB₁)
-```
-
-**Step 3:** Add impedance of C_sh in series
-```
-Z_Csh = 1/(jωC_sh) = -j/(ωC_sh) = 1/(jB₂) (where B₂ = ωC_sh)
-
-Z_total = Z_mut_R + Z_Csh
-Z_total = 1/(G + jB₁) + 1/(jB₂)
-```
-
-**Step 4:** Find common denominator
-```
-Z_total = [jB₂ + (G + jB₁)] / [(G + jB₁) × jB₂]
-Z_total = [G + j(B₁ + B₂)] / [jB₂(G + jB₁)]
-```
-
-**Step 5:** Invert to get admittance
-```
-Y_total = 1/Z_total = [jB₂(G + jB₁)] / [G + j(B₁ + B₂)]
-
-Y_total = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-```
-
-This is the **fundamental admittance equation** for the spark circuit.
-
-### Extracting Real and Imaginary Parts
-
-Multiply numerator:
-```
-(G + jB₁) × jB₂ = jGB₂ + j²B₁B₂ = jGB₂ - B₁B₂
- = -B₁B₂ + jGB₂
-```
-
-So:
-```
-Y = [-B₁B₂ + jGB₂] / [G + j(B₁ + B₂)]
-```
-
-To separate real and imaginary parts, multiply numerator and denominator by complex conjugate of denominator:
-
-```
-Denominator conjugate: G - j(B₁ + B₂)
-Denominator magnitude squared: G² + (B₁ + B₂)²
-```
-
-After algebra (multiply out and simplify):
-
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-```
-
-These are the **working formulas** for calculating admittance from R, C_mut, C_sh.
-
-### Converting to Impedance
-
-From Y = G_total + jB_total:
-
-```
-Z = 1/Y = 1/(G_total + jB_total)
-
-Multiply by conjugate:
-Z = (G_total - jB_total) / (G_total² + B_total²)
-
-R_total = G_total / (G_total² + B_total²)
-X_total = -B_total / (G_total² + B_total²)
-
-Or directly:
-|Z| = 1/|Y|
-φ_Z = -φ_Y (opposite sign!)
-```
-
----
-
-### WORKED EXAMPLE 1.4: Complete Y and Z Calculation
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 2π × 200×10³ = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-- R = 100 kΩ = 10⁵ Ω
-
-**Find:** Y_total (rectangular), Z_total (rectangular and polar)
-
-**Solution:**
-
-Step 1: Calculate component values
-```
-G = 1/R = 1/(10⁵) = 10⁻⁵ S = 10 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06×10⁻⁶ S = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54×10⁻⁶ S = 7.54 μS
-```
-
-Step 2: Calculate Re{Y}
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Numerator: 10 × (7.54)² = 10 × 56.85 = 568.5 μS²
-Denominator: (10)² + (10.06 + 7.54)² = 100 + (17.6)² = 100 + 309.8 = 409.8 μS²
-
-Re{Y} = 568.5 / 409.8 = 1.387 μS
-```
-
-Step 3: Calculate Im{Y}
-```
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-
-Numerator inner: G² + B₁(B₁ + B₂) = 100 + 10.06×17.6 = 100 + 177.1 = 277.1 μS²
-Numerator: 7.54 × 277.1 = 2089.3 μS³
-Denominator: 409.8 μS² (same as before)
-
-Im{Y} = 2089.3 / 409.8 = 5.10 μS
-```
-
-Step 4: Admittance result
-```
-Y_total = 1.387 + j5.10 μS
-|Y| = √(1.387² + 5.10²) = √(1.92 + 26.01) = √27.93 = 5.28 μS
-φ_Y = atan(5.10/1.387) = atan(3.68) = 74.8°
-```
-
-Step 5: Convert to impedance
-```
-|Z| = 1/|Y| = 1/(5.28×10⁻⁶) = 189 kΩ
-φ_Z = -φ_Y = -74.8°
-
-In rectangular:
-R_total = |Z| × cos(φ_Z) = 189 × cos(-74.8°) = 189 × 0.263 = 49.7 kΩ
-X_total = |Z| × sin(φ_Z) = 189 × sin(-74.8°) = 189 × (-0.965) = -182 kΩ
-
-Z_total = 49.7 - j182 kΩ = 189 kΩ ∠-74.8°
-```
-
-**Interpretation:**
-- Impedance is strongly capacitive (φ_Z = -74.8°)
-- Equivalent resistance ≈ 50 kΩ (half of actual R due to capacitive divider)
-- Large capacitive reactance dominates
-
----
-
-### VISUAL AID 1.4: Complex Plane Plots
-
-```
-[Describe for drawing:]
-
-Two plots side-by-side:
-
-LEFT: Admittance plane (Y = G + jB)
-- Horizontal axis: G (conductance, μS), 0 to 2
-- Vertical axis: B (susceptance, μS), 0 to 6
-- Plot point at (1.387, 5.10) labeled "Y_total"
-- Vector from origin to point
-- Angle φ_Y = 74.8° marked from horizontal
-- Length |Y| = 5.28 μS labeled
-- Note: "Positive B means capacitive in admittance"
-
-RIGHT: Impedance plane (Z = R + jX)
-- Horizontal axis: R (kΩ), 0 to 60
-- Vertical axis: X (kΩ), -200 to 0
-- Plot point at (49.7, -182) labeled "Z_total"
-- Vector from origin to point
-- Angle φ_Z = -74.8° marked from horizontal (below axis)
-- Length |Z| = 189 kΩ labeled
-- Note: "Negative X means capacitive in impedance"
-
-Connection between plots:
-- Arrow showing "Invert Y → Z"
-- Note: "Angles are opposite: φ_Z = -φ_Y"
-- Note: "Magnitude inverts: |Z| = 1/|Y|"
-```
-
----
-
-### PRACTICE PROBLEMS 1.4
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF, R = 80 kΩ, calculate Y_total (real and imaginary parts).
-
-**Problem 2:** An admittance Y = 2.0 + j4.5 μS. Convert to impedance Z in both rectangular and polar forms.
-
-**Problem 3:** Show algebraically that if R → ∞ (open circuit), the formula reduces to Y = jωC_mut × C_sh/(C_mut + C_sh), which is two capacitors in series.
-
----
-
-## Module 1.5: Phase Angles and What They Mean
-
-### Impedance Phase vs Admittance Phase
-
-**Impedance phase angle φ_Z:**
-```
-φ_Z = atan(X/R) = atan(Im{Z}/Re{Z})
-
-Interpretation:
-φ_Z > 0: inductive (current lags voltage)
-φ_Z = 0: purely resistive (in phase)
-φ_Z < 0: capacitive (current leads voltage)
-```
-
-**Admittance phase angle θ_Y:**
-```
-θ_Y = atan(B/G) = atan(Im{Y}/Re{Y})
-
-Relationship: θ_Y = -φ_Z (OPPOSITE SIGNS!)
-```
-
-**Why opposite?** Because Y = 1/Z, so angles subtract:
-```
-If Z = |Z|∠φ_Z, then Y = (1/|Z|)∠(-φ_Z)
-```
-
-**Convention in this framework:** We primarily discuss **impedance phase φ_Z** because that's what measurements typically report.
-
-### The "Famous -45°" and Why It's Special (Sort Of)
-
-In power electronics, a load with φ_Z = -45° is sometimes called "well-matched" because:
-- Equal resistive and capacitive components: |R| = |X_C|
-- Power factor = cos(-45°) = 0.707 (reasonable power transfer)
-- Not maximum power transfer, but balanced
-
-**Formula:** For φ_Z = -45°:
-```
-tan(-45°) = -1 = X/R
-Therefore: R = |X| = 1/(ωC) for capacitive load
-Or: R ≈ |X_c| = 1/(ωC_total) approximately
-```
-
-This is why you'll see "spark resistance should equal capacitive reactance" in old Tesla coil literature.
-
-**BUT:** As we'll see in Part 2, achieving exactly -45° is **impossible** for many Tesla coil geometries due to topological constraints!
-
-### Physical Meaning of Phase Angle
-
-**φ_Z = 0° (purely resistive):**
-- All power dissipated
-- No energy storage/return
-- Voltage and current in phase
-
-**φ_Z = -90° (purely capacitive):**
-- No power dissipated
-- All energy stored and returned each cycle
-- Current leads voltage by 90°
-
-**φ_Z = -45° (mixed):**
-- Some power dissipated (cos(-45°) ≈ 71% of |V||I|)
-- Some energy stored
-- Current leads voltage by 45°
-
-**For Tesla coil sparks:** Typical φ_Z = -55° to -75°
-- Significant capacitive component (energy storage in C_mut, C_sh)
-- Moderate power dissipation (plasma heating)
-- More capacitive than the "ideal" -45°
-
----
-
-### WORKED EXAMPLE 1.5: Calculating Phase Angle
-
-**Given:** (from Example 1.4)
-- Z_total = 49.7 - j182 kΩ
-
-**Find:** φ_Z and interpret
-
-**Solution:**
-
-Step 1: Calculate phase angle
-```
-φ_Z = atan(X/R) = atan(-182/49.7)
-φ_Z = atan(-3.66) = -74.8°
-```
-
-Step 2: Verify with magnitude and components
-```
-|Z| = √(49.7² + 182²) = √(2470 + 33124) = √35594 = 189 kΩ ✓
-
-cos(φ_Z) = R/|Z| = 49.7/189 = 0.263
-φ_Z = arccos(0.263) = 74.8°, but X is negative, so φ_Z = -74.8° ✓
-```
-
-Step 3: Interpret
-- **Strongly capacitive:** |φ_Z| = 74.8° is much larger than 45°
-- **Comparison:** |R| = 49.7 kΩ, but |X| = 182 kΩ
- - Capacitive reactance is 3.66× larger than resistance
- - Far from "balanced" -45° condition
-- **Power factor:** cos(-74.8°) = 0.263
- - Only 26.3% of |V||I| is real power
- - Most current is reactive (charging/discharging capacitances)
-
-This is typical for Tesla coil sparks: strongly capacitive impedance.
-
----
-
-### VISUAL AID 1.5: Phase Angle on Complex Plane
-
-```
-[Describe for drawing:]
-
-Impedance plane (Z = R + jX):
-- Horizontal axis: R (resistance, kΩ), 0 to 100
-- Vertical axis: X (reactance, kΩ), -200 to +200
-
-Three vectors from origin:
-
-1. Resistive (φ_Z = 0°):
- - Point at (50, 0)
- - Horizontal vector, angle = 0°
- - Label: "Pure resistance, φ_Z = 0°"
-
-2. Balanced (φ_Z = -45°):
- - Point at (50, -50)
- - Vector at -45° angle
- - Dashed line showing equal R and |X|
- - Label: "Balanced, φ_Z = -45°, R = |X|"
-
-3. Typical spark (φ_Z = -75°):
- - Point at (50, -186)
- - Vector at -75° angle
- - Label: "Typical spark, φ_Z = -75°"
- - Annotation: "Strongly capacitive, |X| >> R"
-
-Additional marks:
-- φ_Z = -90° line (vertical downward): "Pure capacitor"
-- Shaded region between -45° and -90°: "Typical Tesla coil spark range"
-- Note: "More negative φ_Z = more capacitive"
-```
-
----
-
-### PRACTICE PROBLEMS 1.5
-
-**Problem 1:** An impedance Z = 60 + j40 kΩ. Calculate φ_Z. Is this inductive or capacitive?
-
-**Problem 2:** A spark has φ_Z = -60°. If |Z| = 150 kΩ, find R and X. Calculate the power factor.
-
----
-
-## Module 1.6: Introduction to Spark Physics
-
-### What is a Spark? (Qualitative)
-
-**Definition:** A spark is a transient electrical breakdown of air, creating a conducting plasma channel between two electrodes.
-
-**Basic process:**
-1. High electric field ionizes air molecules (electrons stripped from atoms)
-2. Free electrons accelerate, collide with more atoms → avalanche
-3. Plasma forms: mixture of electrons, ions, neutral atoms
-4. Plasma conducts electricity (lower resistance than air)
-5. Current heats plasma → thermal ionization → sustained conduction
-6. When voltage removed, plasma cools and recombines
-
-**Key point:** Plasma is not a simple resistor! Its properties change dynamically:
-- Temperature: 1000 K (cool streamers) to 20,000 K (hot leaders)
-- Conductivity: varies with temperature and ionization
-- Geometry: diameter, length change during growth
-
-### Streamers vs Leaders (Qualitative)
-
-**Streamers:**
-- **Thin:** 10-100 μm diameter (thinner than human hair)
-- **Fast:** Propagate at ~10⁶ m/s (1% speed of light!)
-- **Cold:** Low temperature, weakly ionized
-- **Mechanism:** Photoionization (UV from excited atoms ionizes ahead)
-- **Appearance:** Purple/blue, highly branched, brief flashes
-- **Resistance:** High (MΩ range)
-- **Energy inefficient:** Much energy → light/heat, little → length
-
-**Leaders:**
-- **Thick:** mm to cm diameter (visible as bright core)
-- **Slower:** Propagate at ~10³ m/s (walking speed to car speed)
-- **Hot:** 5,000-20,000 K, fully ionized plasma
-- **Mechanism:** Thermal ionization (Joule heating)
-- **Appearance:** White/orange, straighter, persistent glow
-- **Resistance:** Low (kΩ range)
-- **Energy efficient:** More energy → length extension
-
-**Transition:** Streamers can become leaders if sufficient current flows → heating → thermal ionization. This requires power and time.
-
-### Why Sparks Need Voltage AND Power
-
-**Voltage requirement (field threshold):**
-```
-E_tip > E_propagation ≈ 0.4-1.0 MV/m
-
-For spark to grow, tip field must exceed threshold
-If E_tip drops below threshold, growth stalls
-```
-
-**Power requirement (energy per meter):**
-```
-To extend spark by ΔL, need energy: ΔE ≈ ε × ΔL
-where ε ≈ 5-100 J/m depending on mode
-
-Power determines growth rate: dL/dt ≈ P/ε
-```
-
-**Analogy:** Starting a fire
-- Voltage = temperature of match (need minimum to ignite)
-- Power = fuel supply rate (determines how fast fire spreads)
-- Both are necessary: hot match but no fuel → small flame dies
-- Lots of fuel but no ignition heat → no fire
-
-**For Tesla coils:**
-- Insufficient voltage → spark won't start or grows slowly
-- Insufficient power → spark stalls before reaching potential length
-- **Both must be adequate** for target spark length
-
-### The "Hungry Streamer" Principle (Conceptual)
-
-**Key insight:** Plasma is not passive! It actively adjusts its properties to maximize power extraction from the circuit.
-
-**Mechanism (simplified):**
-1. More power → more Joule heating (I²R)
-2. Higher temperature → more ionization
-3. More ionization → higher conductivity → lower R
-4. Changed geometry → modified capacitances
-5. Circuit has new optimal R for max power transfer
-6. Plasma conductivity adjusts toward this new optimal R
-7. Equilibrium when R_actual ≈ R_optimal_for_max_power
-
-**Physical limits:**
-- R cannot be infinite (some conductivity always present)
-- R cannot be zero (finite electron mobility)
-- Source has limited voltage/current
-- Takes time to adjust (thermal time constants)
-
-**Result:** In steady state, plasma R tends toward the value that maximizes power transfer, within physical constraints.
-
-**Why this matters:** We can model spark as "choosing" R = R_opt_power without detailed plasma chemistry! The physics self-optimizes.
-
----
-
-### VISUAL AID 1.6: Streamers vs Leaders
-
-```
-[Describe for photo/diagram annotations:]
-
-Two-panel comparison:
-
-LEFT PANEL: Streamer
-- Photo/drawing of thin, branched, purple discharge
-- Annotations:
- * Diameter: 10-100 μm (draw scale bar)
- * Temperature: ~1000 K
- * Speed: ~1,000,000 m/s
- * Color: Purple/blue (label spectrum)
- * Structure: Highly branched (mark branching points)
- * Duration: <1 μs per event
- * Resistance: High (MΩ)
-
-RIGHT PANEL: Leader
-- Photo/drawing of thick, straight, white discharge
-- Annotations:
- * Diameter: 1-10 mm (draw scale bar)
- * Temperature: 5,000-20,000 K
- * Speed: ~1,000 m/s
- * Color: White/orange (label spectrum)
- * Structure: Straighter channel (mark path)
- * Duration: Seconds with sustained power
- * Resistance: Low (kΩ)
-
-BOTTOM: Transition diagram
-- Timeline showing streamer → leader conversion
-- Labels: "Initial: streamers form at tip"
- "Current flows → Joule heating"
- "Channel heats → thermal ionization"
- "Leader forms from base, grows toward tip"
- "Leader tip launches new streamers"
- "Cycle repeats for continued growth"
-```
-
----
-
-### DISCUSSION QUESTIONS 1.6
-
-**Question 1:** If a Tesla coil produces high voltage but very low current, would you expect long streamers or short leaders? Why?
-
-**Question 2:** A coil generates 500 kV but only 100 mA. Another generates 200 kV but 1 A. Which is more likely to produce longer sparks? (Consider both voltage and power requirements.)
-
-**Question 3:** Explain in your own words why the spark plasma can be modeled as a resistance that "optimizes itself" rather than as a fixed resistance value.
-
----
-
-## Part 1 Summary: Concepts Checklist
-
-Before proceeding to Part 2, ensure you understand:
-
-### Circuit Fundamentals
-- [ ] Difference between peak and RMS values
-- [ ] Complex number representation: rectangular (R+jX) and polar (|Z|∠φ)
-- [ ] Power calculation: P = 0.5 × Re{V × I*} with peak phasors
-- [ ] Impedance Z = R + jX and admittance Y = G + jB
-- [ ] Relationship: Y = 1/Z, and φ_Y = -φ_Z
-
-### Capacitances
-- [ ] Physical meaning of capacitance (charge storage)
-- [ ] Self-capacitance vs mutual capacitance
-- [ ] Shunt capacitance C_sh ≈ 2 pF/foot for sparks
-- [ ] Both C_mut and C_sh exist simultaneously
-
-### Circuit Topology
-- [ ] Spark circuit: (R || C_mut) in series with C_sh
-- [ ] Topload port as measurement reference (topload-to-ground)
-- [ ] Why V_top/I_base is incorrect
-
-### Admittance Analysis
-- [ ] Advantages of Y for parallel circuits
-- [ ] Formula: Y = [(G+jB₁)×jB₂]/[G+j(B₁+B₂)]
-- [ ] Extracting Re{Y} and Im{Y}
-- [ ] Converting Y ↔ Z
-
-### Phase Angles
-- [ ] φ_Z = atan(X/R) for impedance
-- [ ] Negative φ_Z means capacitive
-- [ ] The -45° "balanced" condition: R = |X|
-- [ ] Typical sparks: φ_Z ≈ -55° to -75° (more capacitive than -45°)
-
-### Spark Physics (Qualitative)
-- [ ] Streamers: thin, fast, cold, high R, branched
-- [ ] Leaders: thick, slower, hot, low R, straighter
-- [ ] Need both voltage (E-field) and power (energy/time)
-- [ ] "Hungry streamer": plasma self-optimizes R
-
----
-
-## Integration Exercise: Putting It All Together
-
-**Scenario:** You have a Tesla coil operating at 180 kHz with a 2-foot spark.
-
-**Given data:**
-- C_mut = 7 pF (from FEMM)
-- Assume R = 75 kΩ (plasma resistance)
-- Estimate C_sh using empirical rule
-
-**Tasks:**
-1. Calculate ω, B₁, B₂, G
-2. Calculate Y_total (real and imaginary parts)
-3. Convert to Z_total (magnitude and phase)
-4. Calculate φ_Z and interpret (is it more or less capacitive than -45°?)
-5. If V_top = 300 kV peak, calculate power dissipated
-
-**Work through this problem completely before checking the solution below.**
-
----
-
-### Integration Exercise Solution
-
-**Step 1:** Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 2 feet = 4 pF
-```
-
-**Step 2:** Calculate ω and component values
-```
-ω = 2πf = 2π × 180×10³ = 1.131×10⁶ rad/s
-
-G = 1/R = 1/(75×10³) = 13.33 μS
-B₁ = ωC_mut = 1.131×10⁶ × 7×10⁻¹² = 7.92 μS
-B₂ = ωC_sh = 1.131×10⁶ × 4×10⁻¹² = 4.52 μS
-```
-
-**Step 3:** Calculate Y_total
-```
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 13.33 × (4.52)² / [13.33² + (7.92+4.52)²]
- = 13.33 × 20.43 / [177.7 + 154.4]
- = 272.3 / 332.1
- = 0.82 μS
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
- = 4.52 × [177.7 + 7.92×12.44] / 332.1
- = 4.52 × [177.7 + 98.5] / 332.1
- = 4.52 × 276.2 / 332.1
- = 3.76 μS
-
-Y_total = 0.82 + j3.76 μS
-```
-
-**Step 4:** Convert to impedance
-```
-|Y| = √(0.82² + 3.76²) = √(0.67 + 14.14) = √14.81 = 3.85 μS
-
-|Z| = 1/|Y| = 1/(3.85×10⁻⁶) = 260 kΩ
-
-φ_Y = atan(3.76/0.82) = atan(4.59) = 77.7°
-φ_Z = -φ_Y = -77.7°
-
-Z_total = 260 kΩ ∠-77.7°
-
-In rectangular:
-R_eq = 260 × cos(-77.7°) = 260 × 0.213 = 55.4 kΩ
-X_eq = 260 × sin(-77.7°) = 260 × (-0.977) = -254 kΩ
-
-Z_total = 55.4 - j254 kΩ
-```
-
-**Step 5:** Interpret phase
-```
-φ_Z = -77.7° is more capacitive than -45° (larger magnitude)
-Ratio: |X|/R = 254/55.4 = 4.6
-Capacitive reactance is 4.6× the resistance
-Very capacitive load!
-```
-
-**Step 6:** Calculate power
-```
-Current: I = V/Z = (300 kV)/(260 kΩ) = 1.15 A peak
-
-Power: P = 0.5 × V × I × cos(φ_Z)
- = 0.5 × 300×10³ × 1.15 × cos(-77.7°)
- = 0.5 × 345×10³ × 0.213
- = 36.7 kW
-
-Alternative: P = 0.5 × I² × R_eq
- = 0.5 × 1.15² × 55.4×10³
- = 0.5 × 1.32 × 55.4×10³
- = 36.6 kW ✓ (checks!)
-```
-
-**Result:** 36.7 kW dissipated in the spark plasma.
-
----
-
-## Preview of Part 2
-
-In Part 2, we'll discover:
-
-- **Why -45° is often impossible:** The topological phase constraint
-- **Two critical resistances:** R_opt_power and R_opt_phase
-- **Thévenin method:** Properly characterizing the Tesla coil
-- **Power optimization:** How the "hungry streamer" finds R_opt_power
-- **Measurements:** Extracting spark parameters from real coils
-
-These concepts build directly on the circuit analysis and phase relationships you've learned in Part 1.
-
----
-
-## CHECKPOINT QUIZ - Part 1
-
-Answer these questions to verify your understanding:
-
-1. What is the relationship between peak and RMS voltage? If V_peak = 100 kV, what is V_RMS?
-
-2. Write the power formula using peak phasors. Why is there a factor of 0.5?
-
-3. For a capacitor, why is X negative but B positive?
-
-4. Draw the circuit topology for a spark (show C_mut, R, C_sh).
-
-5. What is the empirical rule for C_sh? If a spark is 4 feet long, estimate C_sh.
-
-6. The admittance phase angle θ_Y = +60°. What is the impedance phase angle φ_Z?
-
-7. An impedance has φ_Z = -30°. Is this inductive or capacitive?
-
-8. Why is V_top/I_base not the correct impedance measurement?
-
-9. Describe the difference between streamers and leaders (two key differences).
-
-10. Explain the "hungry streamer" concept in one sentence.
-
----
-
-**END OF PART 1**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 2: Optimization and Power Transfer - Making Sparks Efficient
-
----
-
-## Module 2.1: The Topological Phase Constraint
-
-### What is a Topological Constraint?
-
-**Definition:** A limitation imposed by the **structure** of the circuit itself, independent of component values.
-
-**Example:** Series RLC circuit
-- Can only have impedance phase between -90° (pure C) and +90° (pure L)
-- Cannot have φ_Z = +120° no matter what component values you choose
-- This is a topological constraint
-
-**For spark circuits:** The specific arrangement (R||C_mut) in series with C_sh creates a fundamental limit on how resistive the impedance can appear.
-
-### Deriving the Minimum Phase Angle
-
-From Part 1, we have:
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh
-```
-
-The impedance phase is:
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Question:** For fixed C_mut and C_sh, which R value minimizes |φ_Z| (makes most resistive)?
-
-**Mathematical result:** Taking derivative ∂φ_Z/∂G = 0 and solving:
-```
-G_opt = ω√[C_mut(C_mut + C_sh)]
-
-Therefore:
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-At this resistance, the phase angle magnitude is minimized to:
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-
-where r = C_mut/C_sh (capacitance ratio)
-```
-
-### The Critical Ratio r = 0.207
-
-Let's find when φ_Z,min = -45° is achievable:
-```
--45° = -atan(2√[r(1 + r)])
-tan(45°) = 1 = 2√[r(1 + r)]
-0.5 = √[r(1 + r)]
-0.25 = r(1 + r) = r + r²
-r² + r - 0.25 = 0
-
-Using quadratic formula:
-r = [-1 ± √(1 + 1)] / 2 = [-1 ± √2] / 2
-
-Taking positive root:
-r = (√2 - 1) / 2 ≈ 0.207
-```
-
-**Critical insight:**
-- If r < 0.207: Can achieve φ_Z = -45° (with appropriate R)
-- If r > 0.207: **Cannot achieve φ_Z = -45° no matter what R you choose!**
-- If r ≥ 0.207: φ_Z,min is more negative than -45°
-
-### Typical Tesla Coil Values
-
-**Large topload, short spark:**
-```
-C_mut = 10 pF, C_sh = 4 pF (2 feet)
-r = 10/4 = 2.5
-
-φ_Z,min = -atan(2√[2.5 × 3.5]) = -atan(2 × 2.96) = -atan(5.92) = -80.4°
-```
-
-**Small topload, long spark:**
-```
-C_mut = 6 pF, C_sh = 12 pF (6 feet)
-r = 6/12 = 0.5
-
-φ_Z,min = -atan(2√[0.5 × 1.5]) = -atan(2 × 0.866) = -atan(1.732) = -60.0°
-```
-
-**Common range:** r = 0.5 to 2.0, giving φ_Z,min ≈ -60° to -80°
-
-**Conclusion:** For most Tesla coil geometries, -45° is **mathematically impossible**!
-
----
-
-### WORKED EXAMPLE 2.1: Calculate Minimum Phase Angle
-
-**Given:**
-- Frequency: f = 200 kHz
-- C_mut = 8 pF
-- C_sh = 6 pF
-
-**Find:**
-(a) Capacitance ratio r
-(b) Minimum achievable phase angle φ_Z,min
-(c) R_opt_phase that achieves this angle
-
-**Solution:**
-
-**Part (a):** Capacitance ratio
-```
-r = C_mut / C_sh = 8 / 6 = 1.333
-```
-
-**Part (b):** Minimum phase angle
-```
-φ_Z,min = -atan(2√[r(1 + r)])
- = -atan(2√[1.333 × 2.333])
- = -atan(2√3.11)
- = -atan(2 × 1.764)
- = -atan(3.528)
- = -74.2°
-```
-
-**Part (c):** Resistance for minimum phase
-```
-ω = 2πf = 2π × 200×10³ = 1.257×10⁶ rad/s
-
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 1 / [1.257×10⁶ × √(8×10⁻¹² × 14×10⁻¹²)]
- = 1 / [1.257×10⁶ × √(112×10⁻²⁴)]
- = 1 / [1.257×10⁶ × 10.58×10⁻¹²]
- = 1 / (13.30×10⁻⁶)
- = 75.2 kΩ
-```
-
-**Interpretation:**
-- With r = 1.333, cannot achieve -45°
-- Best possible is -74.2° (much more capacitive)
-- This requires R = 75.2 kΩ
-- Any other R value gives |φ_Z| > 74.2°
-
----
-
-### VISUAL AID 2.1: Graph of φ_Z,min vs r
-
-```
-[Describe for plotting:]
-
-Graph with:
-- X-axis: r = C_mut/C_sh (log scale), range 0.1 to 10
-- Y-axis: φ_Z,min (degrees), range -90° to -40°
-
-Plot curve: φ_Z,min = -atan(2√[r(1+r)])
-
-Key points marked:
-- r = 0.207, φ_Z,min = -45° (mark with horizontal dashed line)
-- Shaded region r < 0.207: "Can achieve -45°"
-- Shaded region r > 0.207: "Cannot achieve -45°"
-- Typical Tesla coil range r = 0.5 to 2.0 highlighted
-- Example points:
- * r = 0.5, φ_Z = -60°
- * r = 1.0, φ_Z = -70.5°
- * r = 2.0, φ_Z = -79.7°
-
-Annotations:
-- "Larger r → more capacitive minimum"
-- "Large topload + short spark → high r"
-- "Small topload + long spark → low r"
-```
-
----
-
-### PRACTICE PROBLEMS 2.1
-
-**Problem 1:** For C_mut = 12 pF, C_sh = 8 pF at f = 180 kHz:
-(a) Calculate r
-(b) Find φ_Z,min
-(c) Can this circuit achieve -45°?
-
-**Problem 2:** A designer wants φ_Z,min = -50°. What maximum value of r is allowed? If C_sh = 10 pF, what is the maximum C_mut?
-
-**Problem 3:** Explain physically why larger r (more C_mut relative to C_sh) makes the impedance more capacitive.
-
----
-
-## Module 2.2: The Two Critical Resistances
-
-### R_opt_phase: Closest to Resistive (Revisited)
-
-From Module 2.1:
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-**Purpose:** Minimizes |φ_Z| to achieve φ_Z,min
-
-**Use case:** If you want the "most resistive-looking" impedance possible
-
-### R_opt_power: Maximum Power Transfer
-
-**Different question:** Which R maximizes real power delivered to the spark for a given topload voltage?
-
-**Setup:** Fixed voltage source V_top, variable load resistance R
-
-**Power to load:**
-```
-P = 0.5 × |V_top|² × Re{Y(R)}
-```
-
-where Y(R) depends on R through G = 1/R.
-
-**Mathematical derivation:** Take ∂P/∂G = 0, solve for G:
-
-After calculus (see framework document for full derivation):
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-```
-
-**Simpler formula!** Just total capacitance, not geometric mean.
-
-### Comparing the Two
-
-**Relationship:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Since √(C_mut(C_mut + C_sh)) < (C_mut + C_sh):
-
-R_opt_power < R_opt_phase ALWAYS
-```
-
-**Numerical relationship:** For typical r = 0.5 to 2:
-```
-R_opt_power ≈ (0.5 to 0.7) × R_opt_phase
-```
-
-**Phase angle at R_opt_power:**
-- Always more negative than φ_Z,min
-- Typically φ_Z ≈ -55° to -75° at R_opt_power
-- More capacitive than R_opt_phase, but delivers more power
-
----
-
-### WORKED EXAMPLE 2.2: Calculating Both Critical Resistances
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-
-**Find:** R_opt_phase, R_opt_power, and compare
-
-**Solution:**
-
-**Part 1:** R_opt_phase (from Example 2.1)
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 75.2 kΩ
-```
-
-**Part 2:** R_opt_power
-```
-C_total = C_mut + C_sh = 8 + 6 = 14 pF = 14×10⁻¹² F
-
-R_opt_power = 1 / (ωC_total)
- = 1 / (1.257×10⁶ × 14×10⁻¹²)
- = 1 / (17.60×10⁻⁶)
- = 56.8 kΩ
-```
-
-**Part 3:** Comparison
-```
-Ratio: R_opt_power / R_opt_phase = 56.8 / 75.2 = 0.755
-
-R_opt_power is 75.5% of R_opt_phase
-```
-
-**Part 4:** Phase angle at R_opt_power
-
-Calculate admittance with R = 56.8 kΩ:
-```
-G = 1/56800 = 17.61 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54 μS
-
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 17.61 × 56.85 / [310 + 309.8]
- = 1001.2 / 619.8
- = 1.615 μS
-
-Im{Y} = 7.54[310 + 176.9] / 619.8
- = 7.54 × 486.9 / 619.8
- = 5.928 μS
-
-φ_Y = atan(5.928/1.615) = atan(3.67) = 74.7°
-φ_Z = -74.7°
-```
-
-**Summary:**
-- R_opt_phase = 75.2 kΩ gives φ_Z = -74.2° (minimum)
-- R_opt_power = 56.8 kΩ gives φ_Z = -74.7° (slightly more capacitive)
-- Power is maximized at R_opt_power despite not having minimum phase
-- Difference is small: both are strongly capacitive
-
----
-
-### VISUAL AID 2.2: Power vs Resistance Curves
-
-```
-[Describe for plotting:]
-
-Two overlaid plots sharing X-axis:
-
-X-axis: R (kΩ), range 20 to 150, log scale
-
-TOP PLOT - Power:
-Y-axis: P (kW), normalized to max = 1.0
-Curve: Bell-shaped, peaks at R_opt_power
-- Peak marked at 56.8 kΩ, height = 1.0
-- Label: "R_opt_power = 56.8 kΩ"
-- Width shows power drops to 0.5 at ±50% R
-- Annotation: "Maximum power transfer"
-
-BOTTOM PLOT - Phase angle:
-Y-axis: φ_Z (degrees), range -90° to -40°
-Curve: Rises from -90° (R→0), peaks at R_opt_phase, falls back
-- Peak (least negative) marked at 75.2 kΩ, φ_Z = -74.2°
-- Label: "R_opt_phase = 75.2 kΩ, φ_Z,min = -74.2°"
-- -45° reference line (dashed)
-- Annotation: "Most resistive phase"
-
-Vertical lines:
-- At R_opt_power (56.8 kΩ): shows φ_Z = -74.7° on bottom plot
-- At R_opt_phase (75.2 kΩ): shows lower power on top plot
-
-Key insight box: "R_opt_power ≠ R_opt_phase"
- "R_opt_power delivers more power but is more capacitive"
-```
-
----
-
-### PRACTICE PROBLEMS 2.2
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF:
-Calculate R_opt_power and R_opt_phase.
-
-**Problem 2:** At 200 kHz, a spark has C_total = 12 pF. What is R_opt_power? If V_top = 400 kV, estimate the maximum deliverable power.
-
-**Problem 3:** Prove algebraically that R_opt_power < R_opt_phase always (hint: compare 1/(C_mut+C_sh) with 1/√(C_mut(C_mut+C_sh))).
-
-**Problem 4:** A measurement shows φ_Z = -68° at the operating point. Is R likely above or below R_opt_phase? Above or below R_opt_power?
-
----
-
-## Module 2.3: The "Hungry Streamer" - Self-Optimization
-
-### The Feedback Loop
-
-Plasma conductivity changes dynamically with power:
-
-**1. More power → Joule heating**
-```
-Heating rate: dT/dt ∝ I²R
-Higher current → faster heating
-```
-
-**2. Higher temperature → ionization**
-```
-Thermal ionization: fraction ∝ exp(-E_ionization / kT)
-Hotter plasma → more free electrons
-```
-
-**3. More electrons → higher conductivity**
-```
-σ = n_e × e × μ_e
-where n_e = electron density, μ_e = electron mobility
-σ ∝ n_e ∝ exp(-E_ionization / kT)
-```
-
-**4. Higher conductivity → lower R**
-```
-R = ρL/A = L/(σA)
-σ increases → R decreases
-```
-
-**5. Changed R → new circuit behavior**
-```
-New R changes Y_spark, power transfer changes
-If R < R_opt_power: reducing R further decreases power
-If R > R_opt_power: reducing R increases power
-```
-
-**6. Stable equilibrium at R ≈ R_opt_power**
-```
-When R approaches R_opt_power:
-- Small decrease → power decreases → cooling → R rises
-- Small increase → power increases → heating → R falls
-- Negative feedback stabilizes at R_opt_power
-```
-
-### Time Scales
-
-**Thermal response:** ~0.1-1 ms for thin channels
-- Heat diffusion time: τ = d²/(4α) ≈ 0.1 ms for d = 100 μm
-- Fast enough to track AC envelope (kHz modulation)
-- Too slow to track RF oscillation (hundreds of kHz)
-
-**Ionization response:** ~μs to ms
-- Recombination time varies with density and temperature
-- Can follow slower modulation
-
-**Result:** Plasma adjusts R on timescales of 0.1-10 ms, tracking power delivery changes.
-
-### Physical Constraints
-
-**Lower bound R_min:**
-- Maximum conductivity limited by electron-ion collision frequency
-- Typical: R_min ≈ 1-10 kΩ for hot, dense leaders
-- If R_opt_power < R_min: plasma stuck at R_min (can't optimize)
-
-**Upper bound R_max:**
-- Minimum conductivity of partially ionized gas
-- Typical: R_max ≈ 100 kΩ to 100 MΩ for cool streamers
-- If R_opt_power > R_max: plasma stuck at R_max
-
-**Source limitations:**
-- Insufficient voltage: spark won't form at all
-- Insufficient current: can't heat enough to reach R_opt_power
-- Power supply impedance: limits available power
-
-**When optimization fails:**
-- Source too weak: spark operates at whatever R it can sustain
-- Thermal time too long: can't adjust fast enough (burst mode)
-- Branching: power divides, none optimizes well
-
----
-
-### WORKED EXAMPLE 2.3: Tracing Optimization Process
-
-**Scenario:** Spark initially forms with R = 200 kΩ (cold streamer). Circuit has R_opt_power = 60 kΩ.
-
-**Trace the evolution:**
-
-**Initial state (t = 0):**
-```
-R = 200 kΩ >> R_opt_power
-Power delivered: P_initial (suboptimal, low)
-Temperature: T_initial (cool)
-```
-
-**Early phase (0 < t < 1 ms):**
-```
-Current flows → Joule heating: dT/dt = I²R/c_p
-R is high → voltage division favorable → some heating occurs
-Temperature rises → ionization begins → n_e increases
-Conductivity σ ∝ n_e increases → R decreases
-R drops toward 150 kΩ
-```
-
-**Middle phase (1 ms < t < 5 ms):**
-```
-R approaches 100 kΩ range
-Now closer to R_opt_power → power transfer improves
-More power → faster heating → faster ionization
-Positive feedback: lower R → more power → lower R
-R drops rapidly: 100 kΩ → 80 kΩ → 70 kΩ → 65 kΩ
-```
-
-**Approach to equilibrium (5 ms < t < 10 ms):**
-```
-R approaches R_opt_power = 60 kΩ
-Power maximized at this R
-If R < 60 kΩ: power would decrease → cooling → R rises
-If R > 60 kΩ: power would increase → heating → R falls
-Negative feedback stabilizes around R ≈ 60 kΩ
-```
-
-**Steady state (t > 10 ms):**
-```
-R oscillates around 60 kΩ ± 10%
-Temperature stable at equilibrium
-Power maximized and stable
-Spark is "optimized"
-```
-
-**If constraints active:**
-```
-If R_opt_power = 30 kΩ but R_min = 50 kΩ:
- Plasma can only reach R = 50 kΩ (not optimal)
- Power is less than theoretical maximum
- Spark is "starved" - wants more current than physics allows
-```
-
----
-
-### DISCUSSION QUESTIONS 2.3
-
-**Question 1:** Why does the optimization work? Why doesn't the plasma just pick a random R value?
-
-**Question 2:** In burst mode (short pulses, <100 μs), thermal time constants are longer than pulse duration. Would you expect the plasma to reach R_opt_power? Why or why not?
-
-**Question 3:** A coil produces sparks with measured R ≈ 20 kΩ, but calculations show R_opt_power = 80 kΩ. What might explain this discrepancy?
-
----
-
-## Module 2.4: Power Calculations and Common Errors
-
-### Correct Power Formula
-
-For AC circuit with peak phasors:
-```
-P = 0.5 × Re{V × I*}
-
-Expanded:
-P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
-
-For impedance Z:
-I = V/Z
-P = 0.5 × |V|² × Re{1/Z} = 0.5 × |V|² × Re{Y}
-```
-
-Or using impedance directly:
-```
-P = 0.5 × |I|² × Re{Z} = 0.5 × I² × R
-```
-
-### Why V_top/I_base is Wrong
-
-**The problem:** Current at secondary base (I_base) includes ALL return currents:
-
-1. **Capacitance to ground** along entire secondary
- - Each turn has C to ground
- - AC current: I_C = jωC × V
- - Sum of all displacement currents
-
-2. **Primary-to-secondary coupling**
- - Displacement current through C_ps
- - Part of transformer action
-
-3. **Strike ring/environment coupling**
- - Any nearby grounded object
-
-4. **The spark current** (what we actually want)
-
-**Result:**
-```
-I_base = I_spark + I_displacement_secondary + I_primary_coupling + I_environment
-
-V_top/I_base = wrong because denominator includes parasitic currents!
-```
-
-**Measured impedance is too low** (I_base too high).
-
-### Correct Measurement Port
-
-**Definition:** Topload-to-ground is the correct measurement port.
-
-**Current measurement:** Only the current **through the spark path** from topload.
-
-**Methods:**
-1. Measure I_spark return current separately (Rogowski/CT on spark ground return)
-2. Use circuit analysis (know V_top, calculate I_spark from model)
-3. Thévenin extraction (next modules)
-
----
-
-### WORKED EXAMPLE 2.4: Correct vs Incorrect Power Calculation
-
-**Given:**
-- V_top = 300 kV peak
-- I_base (measured at secondary base) = 5 A peak
-- I_spark (actual spark current) = 1.5 A peak
-- Spark impedance phase: φ_Z = -70°
-
-**Find:** Power using incorrect method, power using correct method
-
-**Solution:**
-
-**Incorrect method:** Using V_top/I_base
-```
-Z_apparent = V_top / I_base = 300 kV / 5 A = 60 kΩ
-
-This is NOT the spark impedance!
-
-If we naively calculated power:
-P_wrong = 0.5 × 300 kV × 5 A × cos(-70°)
- = 0.5 × 1500 kW × 0.342
- = 257 kW
-
-This is way too high!
-```
-
-**Correct method:** Using actual spark current
-```
-I_spark = 1.5 A peak
-
-Real spark impedance:
-Z_spark = V_top / I_spark = 300 kV / 1.5 A = 200 kΩ
-
-Power:
-P_correct = 0.5 × V_top × I_spark × cos(φ_Z)
- = 0.5 × 300 kV × 1.5 A × cos(-70°)
- = 0.5 × 450 kW × 0.342
- = 77 kW
-
-Or using resistance directly:
-R = |Z| × cos(φ_Z) = 200 kΩ × 0.342 = 68.4 kΩ
-P = 0.5 × I² × R = 0.5 × 1.5² × 68.4 kΩ = 77 kW ✓
-```
-
-**Error analysis:**
-```
-P_wrong / P_correct = 257 / 77 = 3.3×
-
-The incorrect method overestimates power by 330%!
-```
-
----
-
-### VISUAL AID 2.4: Current Flow Diagram
-
-```
-[Describe for drawing:]
-
-Side view of Tesla coil showing current paths:
-
-PRIMARY:
-- Primary coil at bottom (multi-turn)
-- Current I_primary flowing
-- Capacitor C_primary
-- Ground connection
-
-SECONDARY:
-- Tall helical coil
-- Multiple current paths illustrated with arrows:
-
-Path 1 (RED): Spark current
- - Flows from topload through spark to remote ground
- - Returns through earth/floor to secondary base
- - Labeled: "I_spark" (what we want to measure)
-
-Path 2 (BLUE): Displacement currents along secondary
- - From each turn to ground
- - Many small arrows radiating outward
- - Labeled: "I_displacement = Σ(jωC_turn × V_turn)"
-
-Path 3 (GREEN): Primary-secondary coupling
- - From primary through C_ps to secondary
- - Labeled: "I_coupling"
-
-Path 4 (YELLOW): Environmental coupling
- - To nearby objects, walls, strike ring
- - Labeled: "I_environment"
-
-AT SECONDARY BASE:
-- Large arrow labeled "I_base = I_spark + I_displacement + I_coupling + I_environment"
-- RED path continues to ground separately
-
-Key insight box: "I_base ≠ I_spark! Cannot use V_top/I_base for spark impedance!"
-```
-
----
-
-### PRACTICE PROBLEMS 2.4
-
-**Problem 1:** A simulation shows V_top = 250 kV, I_base = 3.5 A, but the spark circuit model predicts Z_spark = 180 kΩ. Calculate the actual spark current and power.
-
-**Problem 2:** Explain why displacement current is proportional to frequency (ω). If frequency doubles, what happens to I_displacement?
-
-**Problem 3:** An experimenter measures I_base = 4 A and calculates Z = V_top/I_base = 75 kΩ. Another measurement with a Rogowski coil on the spark return path shows I_spark = 1.2 A. What is the true spark impedance?
-
----
-
-## Module 2.5: Thévenin Equivalent Method - Part A (Measuring Z_th)
-
-### What is a Thévenin Equivalent?
-
-**Thévenin's Theorem:** Any linear two-terminal network can be replaced by:
-- A voltage source V_th (open-circuit voltage)
-- In series with an impedance Z_th (output impedance)
-
-```
-[Complex network] ≡ [V_th]---[Z_th]---o Output
- |
- GND
-```
-
-**Advantage:** Characterize the coil **once**, then predict behavior with **any load** instantly.
-
-### Measuring Z_th: Output Impedance
-
-**Procedure:**
-
-**Step 1:** Turn OFF primary drive
-- Set drive voltage to 0V (AC short circuit)
-- Keep all tank components in place (MMC, L_primary, damping resistors)
-- Tank circuit still present, just not driven
-
-**Step 2:** Apply test source
-- Apply 1V AC at operating frequency to topload-to-ground port
-- Use small-signal AC source (simulation or actual)
-
-**Step 3:** Measure current
-```
-I_test = current into topload port with 1V applied
-```
-
-**Step 4:** Calculate Z_th
-```
-Z_th = V_test / I_test = 1V / I_test
-
-Z_th = R_th + jX_th (complex impedance)
-```
-
-**Physical meaning:**
-- R_th: resistive losses (secondary winding, topload, damping)
-- X_th: reactive component (usually capacitive from topload)
-
-**Typical values at 200 kHz:**
-- R_th: 10-100 Ω (depends on Q and coil size)
-- X_th: -500 to -3000 Ω (capacitive)
-- |Z_th|: 500-3000 Ω
-
----
-
-### WORKED EXAMPLE 2.5A: Extracting Z_th from Simulation
-
-**Simulation setup:**
-- DRSSTC at f = 185 kHz
-- Primary drive set to 0V
-- All components remain (L_primary, C_MMC, secondary, topload)
-- AC test source: 1V ∠0° at topload-to-ground
-
-**Simulation results:**
-- I_test = 0.000412 ∠87.3° A = 0.412 mA ∠87.3°
-
-**Calculate Z_th:**
-
-**Step 1:** Impedance magnitude
-```
-|Z_th| = |V| / |I| = 1 V / 0.000412 A = 2427 Ω
-```
-
-**Step 2:** Impedance phase
-```
-φ_Z_th = φ_V - φ_I = 0° - 87.3° = -87.3°
-```
-
-**Step 3:** Polar form
-```
-Z_th = 2427 Ω ∠-87.3°
-```
-
-**Step 4:** Convert to rectangular
-```
-R_th = |Z_th| × cos(φ_Z_th) = 2427 × cos(-87.3°) = 2427 × 0.0471 = 114 Ω
-X_th = |Z_th| × sin(φ_Z_th) = 2427 × sin(-87.3°) = 2427 × (-0.9989) = -2424 Ω
-
-Z_th = 114 - j2424 Ω
-```
-
-**Interpretation:**
-- **R_th = 114 Ω:** Secondary losses (winding resistance, dielectric losses)
-- **X_th = -2424 Ω:** Strongly capacitive (topload dominates)
-- **Phase ≈ -87°:** Nearly pure capacitor with small series resistance
-- **Quality factor estimate:** Q ≈ |X_th|/R_th = 2424/114 ≈ 21
-
----
-
-### VISUAL AID 2.5A: Thévenin Measurement Setup
-
-```
-[Describe for drawing:]
-
-Two circuit diagrams side-by-side:
-
-LEFT: Full Tesla coil circuit (complex)
-- Primary side: Driver → L_primary → C_MMC → Ground
-- Magnetic coupling to secondary
-- Secondary: Base grounded, many turns, topload at top
-- All parasitics shown (C to ground, etc.)
-- Output port marked at topload
-- Label: "Complex original circuit"
-
-RIGHT: Thévenin equivalent (simple)
-- Just two components:
- * Voltage source V_th
- * Series impedance Z_th = 114 - j2424 Ω
-- Output port (same as left)
-- Label: "Thévenin equivalent"
-
-Arrow between them: "Extraction process"
-
-BOTTOM: Measurement configuration
-- Primary drive: OFF (0V symbol)
-- Test source: 1V AC at topload
-- Ammeter measuring I_test
-- Calculation: Z_th = 1V / I_test
-- Note: "All tank components remain in circuit"
-```
-
----
-
-### PRACTICE PROBLEMS 2.5A
-
-**Problem 1:** A test measurement gives I_test = 0.00035 ∠82° A for V_test = 1 ∠0° V at f = 200 kHz. Calculate Z_th in rectangular form.
-
-**Problem 2:** If Z_th = 85 - j1800 Ω, what is the unloaded Q of the secondary circuit?
-
----
-
-## Module 2.6: Thévenin Equivalent Method - Part B (Using V_th and Z_th)
-
-### Measuring V_th: Open-Circuit Voltage
-
-**Procedure:**
-
-**Step 1:** Remove load
-- Disconnect spark (or set spark to not break out)
-- Topload is open-circuit
-
-**Step 2:** Turn ON primary drive
-- Normal operating frequency and amplitude
-- Drive as you would for spark operation
-
-**Step 3:** Measure topload voltage
-```
-V_th = V(topload) with no load (complex magnitude and phase)
-```
-
-**Typical:** V_th = 200-500 kV peak for medium coils
-
-### Predicting Power to Any Load
-
-With Z_th and V_th known, calculate power to any load impedance Z_load:
-
-**Circuit with load:**
-```
-[V_th] --- [Z_th] --- [Z_load] --- GND
-
-Total impedance: Z_total = Z_th + Z_load
-Current: I = V_th / (Z_th + Z_load)
-Voltage across load: V_load = I × Z_load
-Power in load: P_load = 0.5 × |I|² × Re{Z_load}
-```
-
-**Direct formula:**
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**No re-simulation needed!** Just plug in different Z_load values.
-
-### Theoretical Maximum Power
-
-**Conjugate match condition:** Maximum power transfer occurs when:
-```
-Z_load = Z_th* (complex conjugate)
-
-If Z_th = R_th + jX_th, then Z_load = R_th - jX_th
-```
-
-**Maximum power:**
-```
-P_max = |V_th|² / (8 × R_th)
-```
-
-**BUT:** For spark loads, conjugate match is usually not achievable due to topological constraints (Module 2.1).
-
----
-
-### WORKED EXAMPLE 2.6: Complete Thévenin Analysis
-
-**Given:**
-- Z_th = 114 - j2424 Ω (from Example 2.5A)
-- V_th = 350 kV ∠0° (measured with drive on, no load)
-- Candidate spark load: Z_spark = 60 kΩ - j160 kΩ (from lumped model)
-
-**Find:**
-(a) Current through spark
-(b) Voltage across spark
-(c) Power dissipated in spark
-(d) Theoretical maximum power (conjugate match)
-
-**Solution:**
-
-**Part (a):** Current
-```
-Z_total = Z_th + Z_spark
- = (114 - j2424) + (60000 - j160000)
- = (60114 - j162424) Ω
-
-|Z_total| = √(60114² + 162424²) = √(3.614×10⁹ + 2.638×10¹⁰) = √3.00×10¹⁰ = 173 kΩ
-
-I = V_th / Z_total = (350 kV) / (173 kΩ) = 2.02 A peak
-```
-
-**Part (b):** Voltage across spark
-```
-Voltage divider:
-V_spark = V_th × [Z_spark / (Z_th + Z_spark)]
-
-|V_spark| = 350 kV × (170 kΩ / 173 kΩ) = 350 kV × 0.983 = 344 kV
-
-Most voltage appears across spark (Z_spark >> Z_th)
-```
-
-**Part (c):** Power in spark
-```
-P_spark = 0.5 × I² × Re{Z_spark}
- = 0.5 × (2.02)² × 60000
- = 0.5 × 4.08 × 60000
- = 122 kW
-```
-
-**Part (d):** Theoretical maximum
-```
-Conjugate match: Z_load = Z_th* = 114 + j2424 Ω
-
-P_max = |V_th|² / (8 × R_th)
- = (350×10³)² / (8 × 114)
- = 1.225×10¹¹ / 912
- = 134 MW
-
-Wait, this seems way too high! Let me recalculate...
-
-P_max = 0.5 × |V_th|² / (4 × R_th) [Correct formula]
- = 0.5 × (350×10³)² / (4 × 114)
- = 0.5 × 1.225×10¹¹ / 456
- = 134 MW
-
-This is still huge because R_th is so small (114 Ω).
-```
-
-**Reality check:**
-- Actual spark power: 122 kW
-- Theoretical maximum: 134 MW
-- Spark extracts: 122/134000 = 0.09% of theoretical maximum
-
-**Why such a huge difference?**
-- Conjugate match would require Z_load = 114 + j2424 Ω (very low resistance!)
-- Actual spark: Z_spark = 60000 - j160000 Ω (much higher resistance, wrong phase)
-- Topological constraints prevent achieving conjugate match
-- This is normal for Tesla coils!
-
----
-
-### PRACTICE PROBLEMS 2.6
-
-**Problem 1:** Given Z_th = 95 - j1850 Ω, V_th = 280 kV, and a spark model with Z_spark = 50 kΩ - j140 kΩ:
-(a) Calculate power delivered to spark
-(b) What percentage of theoretical maximum is this?
-
-**Problem 2:** A load Z_load = 200 + j200 Ω is connected. If Z_th = 100 - j2000 Ω and V_th = 300 kV, calculate the power. Is this inductive or capacitive load?
-
----
-
-## Module 2.7: Quality Factor and Ringdown Measurements
-
-### What is Quality Factor (Q)?
-
-**Definition:** Ratio of energy stored to energy dissipated per cycle:
-```
-Q = 2π × (Energy stored) / (Energy dissipated per cycle)
-
-For series RLC: Q = ωL/R = 1/(ωRC)
-For parallel RLC at resonance: Q = R/(ωL) = ωRC
-```
-
-**Physical meaning:**
-- High Q: oscillation persists many cycles (low damping)
-- Low Q: oscillation decays quickly (high damping)
-
-### Measuring Q from Ringdown
-
-**Procedure:**
-1. Excite coil (burst of AC at resonance)
-2. Turn off drive
-3. Measure voltage decay
-
-**Exponential envelope:**
-```
-V(t) = V₀ × exp(-t/τ) × cos(ωt)
-
-where τ = 2Q/ω = decay time constant
-```
-
-**From consecutive peaks:**
-```
-Ratio of amplitudes n cycles apart:
-A(t + nT) / A(t) = exp(-nT/τ) = exp(-nπ/Q)
-
-Solving for Q:
-Q = nπ / ln[A(t) / A(t + nT)]
-```
-
-**Practical:** Measure peak-to-peak over several cycles:
-```
-Q ≈ πf × Δt / ln(A₁/A₂)
-
-where Δt = time between measured peaks
-```
-
-### Extracting Spark Parameters from Q Measurements
-
-**Unloaded (no spark):**
-- Measure f₀, Q₀
-- Represents coil losses only
-
-**Loaded (with spark):**
-- Measure f_L, Q_L
-- Spark adds resistance and capacitance
-
-**At resonance:**
-```
-Q_L = ω_L × C_eq × R_p
-
-where R_p = equivalent parallel resistance at resonance
- C_eq = total capacitance = C₀ + ΔC
-```
-
-**Solving for conductance:**
-```
-G_total = 1/R_p = ω_L × C_eq / Q_L
-
-Spark contribution:
-G_spark ≈ G_total - G_0 = ω_L C_eq / Q_L - ω₀ C₀ / Q₀
-```
-
-**Capacitance from frequency shift:**
-```
-Frequency ratio: f₀/f_L = √(C_eq/C₀)
-
-Therefore: C_eq = C₀ × (f₀/f_L)²
-
-Spark capacitance: ΔC = C_eq - C₀
-```
-
-**Spark admittance:**
-```
-Y_spark ≈ G_spark + jω_L ΔC
-```
-
----
-
-### WORKED EXAMPLE 2.7: Q Measurement and Spark Extraction
-
-**Given measurements:**
-
-**Unloaded:**
-- f₀ = 200 kHz
-- Q₀ = 80 (from ringdown)
-- C₀ = 28 pF (calculated from geometry)
-
-**With spark:**
-- f_L = 185 kHz (frequency dropped)
-- Q_L = 25 (from ringdown with spark)
-
-**Find:** Spark admittance Y_spark
-
-**Solution:**
-
-**Step 1:** Calculate loaded capacitance
-```
-C_eq = C₀ × (f₀/f_L)²
- = 28 pF × (200/185)²
- = 28 pF × (1.081)²
- = 28 pF × 1.169
- = 32.7 pF
-
-ΔC = C_eq - C₀ = 32.7 - 28 = 4.7 pF
-```
-
-**Step 2:** Calculate conductances
-```
-ω₀ = 2π × 200×10³ = 1.257×10⁶ rad/s
-ω_L = 2π × 185×10³ = 1.162×10⁶ rad/s
-
-G₀ = ω₀ C₀ / Q₀
- = 1.257×10⁶ × 28×10⁻¹² / 80
- = 35.2×10⁻⁶ / 80
- = 0.44 μS
-
-G_total = ω_L C_eq / Q_L
- = 1.162×10⁶ × 32.7×10⁻¹² / 25
- = 38.0×10⁻⁶ / 25
- = 1.52 μS
-
-G_spark = G_total - G₀ = 1.52 - 0.44 = 1.08 μS
-```
-
-**Step 3:** Construct spark admittance
-```
-B_spark = ω_L ΔC = 1.162×10⁶ × 4.7×10⁻¹² = 5.46 μS
-
-Y_spark = G_spark + jB_spark
- = 1.08 + j5.46 μS
-```
-
-**Step 4:** Convert to impedance
-```
-|Y_spark| = √(1.08² + 5.46²) = √(1.17 + 29.8) = 5.56 μS
-
-Z_spark = 1/Y_spark
-|Z_spark| = 1/(5.56×10⁻⁶) = 180 kΩ
-
-φ_Y = atan(5.46/1.08) = atan(5.06) = 78.8°
-φ_Z = -78.8°
-
-Z_spark = 180 kΩ ∠-78.8°
-
-In rectangular:
-R = 180 × cos(-78.8°) = 180 × 0.194 = 35 kΩ
-X = 180 × sin(-78.8°) = 180 × (-0.981) = -177 kΩ
-
-Z_spark = 35 - j177 kΩ
-```
-
-**Interpretation:**
-- Spark added 4.7 pF capacitance (consistent with ~2.4 foot spark)
-- R ≈ 35 kΩ at 185 kHz
-- Strongly capacitive: φ_Z = -78.8°
-- Q dropped from 80 to 25 (spark loading dominates)
-
----
-
-### PRACTICE PROBLEMS 2.7
-
-**Problem 1:** A ringdown shows voltage dropping from 100 kV to 50 kV in 8 cycles at f = 195 kHz. Calculate Q.
-
-**Problem 2:** Measurements show: f₀ = 210 kHz, Q₀ = 65, f_L = 198 kHz (with spark), Q_L = 30. If C₀ = 25 pF, calculate the spark's added capacitance and equivalent resistance.
-
-**Problem 3:** Why does frequency decrease when a spark forms? Explain in terms of capacitance.
-
----
-
-## Part 2 Summary & Integration
-
-### Key Concepts Checklist
-
-- [ ] **Topological phase constraint:** φ_Z,min = -atan(2√[r(1+r)])
-- [ ] **Critical ratio:** r ≥ 0.207 makes φ_Z = -45° impossible
-- [ ] **R_opt_phase:** Minimizes |φ_Z|, gives φ_Z,min
-- [ ] **R_opt_power:** Maximizes power transfer to load
-- [ ] **Relationship:** R_opt_power < R_opt_phase always
-- [ ] **Hungry streamer:** Plasma self-adjusts toward R_opt_power
-- [ ] **Physical limits:** R_min (hot plasma) to R_max (cold plasma)
-- [ ] **Why V_top/I_base fails:** Includes displacement currents
-- [ ] **Correct port:** Topload-to-ground
-- [ ] **Thévenin Z_th:** Output impedance (drive off, test on)
-- [ ] **Thévenin V_th:** Open-circuit voltage (drive on, no load)
-- [ ] **Power formula:** P = 0.5|V_th|²Re{Z_load}/|Z_th+Z_load|²
-- [ ] **Conjugate match:** Usually unachievable due to constraints
-- [ ] **Q from ringdown:** Q = πfΔt/ln(A₁/A₂)
-- [ ] **Extract Y_spark:** From frequency shift and Q change
-
----
-
-## Comprehensive Design Exercise
-
-**Scenario:** Design matching for a DRSSTC
-
-**Given:**
-- Operating frequency: f = 190 kHz
-- Topload: C_topload = 30 pF
-- Target spark: 3 feet (estimate C_sh)
-- FEMM analysis: C_mut = 9 pF for 3-foot spark
-- Thévenin equivalent (measured): Z_th = 105 - j2100 Ω, V_th = 320 kV
-
-**Tasks:**
-
-1. **Calculate capacitance ratio and phase constraint:**
- - Find r = C_mut/C_sh
- - Calculate φ_Z,min
- - Can this achieve -45°?
-
-2. **Determine optimal resistances:**
- - Calculate R_opt_power
- - Calculate R_opt_phase
- - What is typical φ_Z at R_opt_power?
-
-3. **Build lumped spark model:**
- - Draw circuit with C_mut, R, C_sh
- - Use R = R_opt_power
- - Calculate Y_spark
-
-4. **Predict performance with Thévenin:**
- - Calculate Z_spark from Y_spark
- - Find total impedance Z_th + Z_spark
- - Calculate spark current
- - Calculate power delivered to spark
-
-5. **Compare to theoretical maximum:**
- - Calculate P_max (conjugate match)
- - What percentage is actually delivered?
- - Explain the difference
-
-**Work through this completely, then check solutions in appendix.**
-
----
-
-**END OF PART 2**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From
-
----
-
-## Module 3.1: Electric Fields and Breakdown
-
-### Electric Field Basics
-
-**Definition:** Electric field E is force per unit charge:
-```
-E = F/q [units: N/C or V/m]
-
-Related to voltage:
-E = -dV/dx (field is voltage gradient)
-
-For uniform field:
-E ≈ V/d (voltage divided by distance)
-```
-
-**Field at spark tip is NOT uniform** - concentrated by geometry.
-
-### Breakdown Field Thresholds
-
-**E_inception:** Field required to initiate breakdown from smooth electrode
-```
-E_inception ≈ 2-3 MV/m (at sea level, dry air)
-
-Physical process:
-- Natural cosmic rays create seed electrons
-- Strong field accelerates electrons
-- Collisions ionize more atoms
-- Avalanche breakdown begins
-```
-
-**E_propagation:** Field required to sustain spark growth
-```
-E_propagation ≈ 0.4-1.0 MV/m (for leader propagation)
-
-Lower than inception because:
-- Channel already partially ionized
-- Hot gas easier to ionize
-- Photoionization helps (UV from plasma)
-```
-
-**Altitude/humidity effects:**
-- Lower air density (altitude) → lower E_threshold (±20-30%)
-- Humidity adds water vapor → changes breakdown (~10%)
-- Temperature affects density → small effect
-
-### Tip Enhancement Factor κ
-
-Sharp tips concentrate field:
-
-```
-E_tip = κ × E_average
-
-where E_average = V/L (voltage divided by length)
- κ = enhancement factor ≈ 2-5 typical
-```
-
-**Physical origin:**
-- Charge accumulates at sharp points
-- Field lines concentrate at high curvature
-- Smaller radius → higher κ
-
-**FEMM calculates E_tip directly** from geometry and voltage.
-
-### Growth Criterion
-
-Spark continues growing when:
-```
-E_tip > E_propagation
-
-If E_tip drops below E_propagation:
-- Growth stalls
-- Spark cannot extend further
-- "Voltage-limited"
-```
-
----
-
-### WORKED EXAMPLE 3.1: Field Calculation
-
-**Given:**
-- Spark length: L = 1.5 m
-- Topload voltage: V_top = 400 kV
-- Tip enhancement: κ = 3.5 (from FEMM or estimate)
-
-**Find:**
-(a) Average field
-(b) Tip field
-(c) Can spark grow if E_propagation = 0.6 MV/m?
-
-**Solution:**
-
-**Part (a):** Average field
-```
-E_average = V_top / L
- = 400×10³ V / 1.5 m
- = 267 kV/m
- = 0.267 MV/m
-```
-
-**Part (b):** Tip field
-```
-E_tip = κ × E_average
- = 3.5 × 0.267 MV/m
- = 0.93 MV/m
-```
-
-**Part (c):** Compare to threshold
-```
-E_tip = 0.93 MV/m
-E_propagation = 0.6 MV/m
-
-E_tip > E_propagation ✓
-
-Yes, spark can continue growing.
-Margin: 0.93/0.6 = 1.55× above threshold
-```
-
-**If voltage drops to 300 kV:**
-```
-E_average = 300 kV / 1.5 m = 0.2 MV/m
-E_tip = 3.5 × 0.2 = 0.7 MV/m
-
-Still above 0.6 MV/m, but margin reduced to 1.17×
-```
-
-**If voltage drops to 250 kV:**
-```
-E_average = 250 kV / 1.5 m = 0.167 MV/m
-E_tip = 3.5 × 0.167 = 0.58 MV/m
-
-Below 0.6 MV/m - growth stalls!
-```
-
----
-
-### VISUAL AID 3.1: Field Enhancement
-
-```
-[Describe for drawing:]
-
-Two panels side-by-side:
-
-LEFT: Uniform field (parallel plates)
-- Two flat plates, voltage V between them
-- Evenly spaced field lines (vertical)
-- Formula: E = V/d (constant everywhere)
-- Label: "No enhancement, κ = 1"
-
-RIGHT: Point-to-plane (spark geometry)
-- Spherical topload at top (voltage V)
-- Sharp spark tip pointing down
-- Ground plane at bottom
-- Field lines:
- * Sparse near topload (low density)
- * Dense at tip (concentrated)
- * Spread out below tip
-- Color gradient showing field strength:
- * Blue (low) far from tip
- * Red (high) at tip
-- Annotations:
- * E_average = V/L marked along spark
- * E_tip at very tip (red zone)
- * "Enhancement: E_tip = κ × E_average, κ = 2-5"
-
-Inset graph: E vs distance from tip
-- Sharp peak at tip (E_tip)
-- Drops rapidly with distance
-- Approaches E_average far from tip
-```
-
----
-
-### PRACTICE PROBLEMS 3.1
-
-**Problem 1:** A 0.8 m spark has V_top = 280 kV, κ = 4. Calculate E_tip. If E_propagation = 0.5 MV/m, can it grow?
-
-**Problem 2:** A spark stalls at 2.0 m length with V_top = 500 kV and κ = 3. Estimate E_propagation for these conditions.
-
-**Problem 3:** Why is E_inception > E_propagation? Explain the physical difference.
-
----
-
-## Module 3.2: Energy Requirements for Growth
-
-### Energy Per Meter (ε)
-
-**Concept:** Extending spark by 1 meter requires approximately constant energy:
-
-```
-Energy to grow from L₁ to L₂:
-ΔE ≈ ε × (L₂ - L₁)
-
-where ε [J/m] depends on operating mode
-```
-
-**Not just ionization energy** - includes:
-1. Initial ionization (breaking molecular bonds)
-2. Heating to operating temperature
-3. Work against pressure (channel expansion)
-4. Radiation losses (light, UV, RF)
-5. Branching (wasted energy in short branches)
-6. Inefficiency (non-productive heating)
-
-### Typical ε Values by Operating Mode
-
-**QCW (Quasi-Continuous Wave):**
-```
-ε ≈ 5-15 J/m
-
-Characteristics:
-- Long ramp times (5-20 ms)
-- Channel stays hot throughout growth
-- Efficient leader formation
-- Minimal re-ionization
-```
-
-**Hybrid DRSSTC (moderate duty cycle):**
-```
-ε ≈ 20-40 J/m
-
-Characteristics:
-- Medium pulses (1-5 ms)
-- Mix of streamers and leaders
-- Some thermal accumulation
-- Moderate efficiency
-```
-
-**Burst mode (hard-pulsed):**
-```
-ε ≈ 30-100+ J/m
-
-Characteristics:
-- Short pulses (<500 μs)
-- Channel cools between pulses
-- Mostly streamers, bright but short
-- Must re-ionize repeatedly
-- Poor length efficiency
-```
-
-### Why Different Modes Have Different ε
-
-**QCW efficiency (low ε):**
-- Continuous power → channel stays ionized
-- Thermal ionization maintained
-- Leaders form efficiently
-- Each Joule goes into extension
-
-**Burst inefficiency (high ε):**
-- Peak power → brightening, branching
-- Channel cools between bursts
-- Energy into light, heat, not length
-- Must restart from cold each time
-
-**Analogy:** Boiling water
-- Low ε: Keep burner on, maintain simmer (efficient)
-- High ε: Pulse burner on/off, water cools (inefficient)
-
-### Theoretical Minimum Energy
-
-**Just ionization:**
-```
-Ionization energy per molecule ≈ 15 eV
-Air density ≈ 2.5×10²⁵ molecules/m³
-Channel volume ≈ π(d/2)² × L
-
-For d = 1 mm, L = 1 m:
-E_ionize = 15 eV × 2.5×10²⁵ × π×(0.5×10⁻³)² × 1
- ≈ 0.3 J/m (theoretical minimum)
-```
-
-**Why ε >> 0.3 J/m?**
-- Heating to 5000-20000 K (thermal energy)
-- Radiation (visible light, UV, IR)
-- Expansion work (push air aside)
-- Branching losses (many failed attempts)
-- Inefficiencies (not all current goes to useful ionization)
-
-**Result:** Real ε is 20-300× theoretical minimum.
-
----
-
-### WORKED EXAMPLE 3.2: Energy Budget
-
-**Given:**
-- Target spark: L = 2 m
-- Operating mode: QCW with ε = 10 J/m
-- Growth time: T = 12 ms
-
-**Find:**
-(a) Total energy required
-(b) Average power required
-(c) If only 80 kW available, what happens?
-
-**Solution:**
-
-**Part (a):** Total energy
-```
-E_total = ε × L
- = 10 J/m × 2 m
- = 20 J
-```
-
-**Part (b):** Average power
-```
-P_avg = E_total / T
- = 20 J / 0.012 s
- = 1667 W
- ≈ 1.7 kW
-```
-
-**Part (c):** With limited power
-```
-Available: P = 80 kW (much more than needed!)
-
-This is 80/1.7 = 47× the required power.
-
-Options:
-1. Grow much faster: T = 20 J / 80 kW = 0.25 ms (burst-like)
-2. Grow to longer length: L = P × T / ε
- For same 12 ms: L = 80 kW × 0.012 s / 10 J/m = 96 m (unrealistic!)
-
-Reality: Voltage limit kicks in first
- - Cannot maintain E_tip > E_propagation for 96 m
- - Spark stalls at voltage-limited length
-```
-
-**Key insight:** Need BOTH adequate power AND adequate voltage!
-
----
-
-### PRACTICE PROBLEMS 3.2
-
-**Problem 1:** A burst-mode coil has ε = 60 J/m. To reach 1.5 m in a 200 μs pulse, what power is required?
-
-**Problem 2:** Two coils both deliver 50 kW. Coil A (QCW, ε = 8 J/m) vs Coil B (burst, ε = 50 J/m). For 10 ms operation, which produces longer sparks?
-
----
-
-## Module 3.3: Growth Rate and Stalling
-
-### The Growth Rate Equation
-
-When field threshold is met:
-```
-dL/dt = P_stream / ε [units: m/s]
-
-where P_stream = power delivered to spark [W]
- ε = energy per meter [J/m]
-```
-
-**Physical meaning:**
-- More power → faster growth
-- Higher ε (inefficiency) → slower growth
-
-**When growth stops:**
-```
-If E_tip < E_propagation:
- dL/dt = 0 (stalled)
-
-Cannot grow regardless of available power
-```
-
-### Voltage-Limited vs Power-Limited
-
-**Voltage-limited:**
-```
-E_tip < E_propagation
-- Field too weak at tip
-- Spark cannot extend
-- More power doesn't help (without more voltage)
-- Common for small topload, long target
-```
-
-**Power-limited:**
-```
-E_tip > E_propagation, but P_stream < ε × (dL/dt)_desired
-- Field adequate, but not enough energy
-- Spark grows slowly or stalls before reaching potential
-- More voltage doesn't help (without more power)
-- Common for high-Q coils, weak drive
-```
-
-### Predicting Growth Time
-
-For constant power during ramp:
-```
-L(t) = (P_stream / ε) × t
-
-Time to reach L_target:
-T = ε × L_target / P_stream
-```
-
-**More realistic:** Power changes as spark grows (loading changes)
-```
-T = ∫₀^L_target (ε / P_stream(L)) dL
-
-Requires simulation or numerical integration
-```
-
----
-
-### WORKED EXAMPLE 3.3: Growth Prediction
-
-**Given:**
-- QCW coil, ε = 12 J/m
-- Target: L = 1.8 m
-- Power profile: P_stream = 100 kW (constant during ramp)
-- κ = 3.2, E_propagation = 0.7 MV/m
-- V_top ramps linearly: V(t) = 50 kV/ms × t
-
-**Find:**
-(a) Growth time if power-limited
-(b) Growth time if voltage-limited
-(c) Actual growth (considering both limits)
-
-**Solution:**
-
-**Part (a):** Power-limited case (assume infinite voltage)
-```
-T_power = ε × L / P_stream
- = 12 J/m × 1.8 m / 100000 W
- = 21.6 J / 100000 W
- = 0.000216 s
- = 0.216 ms
-```
-
-**Part (b):** Voltage-limited case
-
-At length L, need E_tip > E_propagation:
-```
-E_tip = κ × V(t) / L > E_propagation
-V(t) > E_propagation × L / κ
-
-For L = 1.8 m:
-V_required > 0.7×10⁶ × 1.8 / 3.2
-V_required > 0.394 MV = 394 kV
-
-With ramp V(t) = 50 kV/ms × t:
-T_voltage = 394 kV / (50 kV/ms) = 7.88 ms
-```
-
-**Part (c):** Actual growth (limited by slowest)
-```
-T_power = 0.216 ms (very fast if voltage available)
-T_voltage = 7.88 ms (slower, limited by ramp rate)
-
-Actual: T ≈ 7.88 ms (voltage-limited)
-
-The spark grows as fast as voltage ramps allow.
-Power is MORE than sufficient (100 kW available, only need ~2.7 kW)
-```
-
-**Verification of power requirement:**
-```
-P_needed = ε × L / T_actual
- = 12 × 1.8 / 0.00788
- = 2.74 kW
-
-100 kW available >> 2.74 kW needed ✓
-Confirms voltage-limited, not power-limited
-```
-
----
-
-### VISUAL AID 3.3: Growth Curves
-
-```
-[Describe for plotting:]
-
-Graph: Spark length L vs time t
-
-Three curves:
-
-CURVE 1 (Blue): Power-limited
-- Linear growth: L(t) = (P/ε) × t
-- Steep slope (fast growth)
-- Reaches target quickly (0.2 ms)
-- Label: "Power-limited: unlimited voltage"
-
-CURVE 2 (Red): Voltage-limited
-- Curved growth: L(t) must satisfy E_tip(V(t),L) > E_prop
-- Slower, follows voltage ramp capability
-- Reaches target at 7.88 ms
-- Label: "Voltage-limited: slow ramp"
-
-CURVE 3 (Green): Actual (realistic)
-- Follows faster curve initially
-- Transitions to limiting constraint
-- Usually voltage-limited for Tesla coils
-- Label: "Actual: limited by slowest constraint"
-
-Shaded regions:
-- Below curves: "Achieved length"
-- Above: "Not yet reached"
-
-Annotations:
-- "QCW: usually voltage-limited"
-- "Burst: can be power-limited"
-- "Need both P and V adequate"
-```
-
----
-
-### PRACTICE PROBLEMS 3.3
-
-**Problem 1:** A spark grows at 2 m/s when P = 40 kW and ε = 20 J/m. Verify this is consistent with dL/dt = P/ε.
-
-**Problem 2:** If E_propagation = 0.5 MV/m, κ = 3, and voltage is fixed at V = 300 kV, what is the maximum length the spark can reach (voltage-limited)?
-
-**Problem 3:** A coil delivers 30 kW to a spark with ε = 15 J/m. How long to reach 2.5 m? If this time is longer than the voltage ramp allows, which limit dominates?
-
----
-
-## Module 3.4: Thermal Physics of Plasma Channels
-
-### Temperature Regimes
-
-**Streamers (cold):**
-```
-T ≈ 1000-3000 K
-- Weakly ionized
-- Mostly neutral gas with some ions/electrons
-- Purple/blue color (N₂ emission)
-```
-
-**Leaders (hot):**
-```
-T ≈ 5000-20000 K
-- Fully ionized plasma
-- White/orange color (blackbody + line emission)
-- Approaching temperatures of stellar photospheres!
-```
-
-### Thermal Diffusion Time
-
-Heat diffuses radially from hot channel core:
-```
-τ_thermal = d² / (4α_thermal)
-
-where d = channel diameter
- α_thermal ≈ 2×10⁻⁵ m²/s for air
-```
-
-**Examples:**
-```
-Thin streamer (d = 100 μm):
-τ = (100×10⁻⁶)² / (4 × 2×10⁻⁵)
- = 10⁻⁸ / (8×10⁻⁵)
- = 0.125 ms
-
-Thick leader (d = 5 mm):
-τ = (5×10⁻³)² / (4 × 2×10⁻⁵)
- = 25×10⁻⁶ / (8×10⁻⁵)
- = 312 ms
-```
-
-### Why Observed Persistence is Longer
-
-**Pure thermal diffusion** predicts cooling in 0.1-300 ms, but channels persist longer due to:
-
-**1. Convection (buoyancy):**
-```
-Hot gas rises: v ≈ √(g × d × ΔT/T_amb)
-
-For d = 2 mm, ΔT = 10000 K:
-v ≈ √(9.8 × 0.002 × 10000/300)
- ≈ √(0.65) ≈ 0.8 m/s
-
-Rising column remains hot longer than conduction alone
-```
-
-**2. Ionization memory:**
-```
-Recombination time: τ_recomb = 1/(α_recomb × n_e)
-Can be 10 μs to 10 ms depending on density
-Ions/electrons persist after thermal cooling begins
-```
-
-**Effective persistence:**
-```
-Streamers: ~1-5 ms (convection + ionization)
-Leaders: seconds (buoyant column maintained)
-```
-
-### QCW Advantage
-
-**QCW ramp times (5-20 ms) exploit channel persistence:**
-```
-1. Initial streamers form (t = 0)
-2. Power heats channel → leader begins (t = 1 ms)
-3. Leader maintained by continuous power (t = 1-20 ms)
-4. Channel stays hot entire time
-5. New growth builds on existing ionization
-6. Efficient energy use
-```
-
-**Burst mode problem:**
-```
-1. Pulse creates bright streamer (t = 0-0.1 ms)
-2. Pulse ends, channel cools (t = 0.1-1 ms)
-3. Next pulse must re-ionize cold gas (t = 1 ms)
-4. Energy wasted re-heating
-5. Inefficient (high ε)
-```
-
----
-
-### WORKED EXAMPLE 3.4: Thermal Time Constants
-
-**Given:**
-- Channel diameter: d = 2 mm (typical leader)
-- Air thermal diffusivity: α = 2×10⁻⁵ m²/s
-
-**Find:**
-(a) Pure thermal diffusion time
-(b) Estimate convection velocity if ΔT = 8000 K
-(c) QCW ramp time recommendation
-
-**Solution:**
-
-**Part (a):** Thermal diffusion
-```
-τ_thermal = d² / (4α)
- = (2×10⁻³)² / (4 × 2×10⁻⁵)
- = 4×10⁻⁶ / (8×10⁻⁵)
- = 0.05 s
- = 50 ms
-```
-
-**Part (b):** Convection velocity
-```
-v ≈ √(g × d × ΔT/T_amb)
- ≈ √(9.8 × 0.002 × 8000/300)
- ≈ √(0.523)
- ≈ 0.72 m/s
-
-Upward velocity helps maintain hot column
-```
-
-**Part (c):** QCW ramp recommendation
-```
-τ_thermal = 50 ms
-
-Good QCW ramp: T_ramp << τ_thermal (finish before significant cooling)
-Reasonable: T_ramp = 5-20 ms (10-40% of τ)
-
-If T_ramp >> τ_thermal:
- - Channel cools during ramp
- - Must reheat repeatedly
- - Loses QCW efficiency advantage
-```
-
----
-
-### PRACTICE PROBLEMS 3.4
-
-**Problem 1:** A streamer has d = 150 μm. Calculate τ_thermal. If burst pulse is 500 μs, does channel cool significantly during pulse?
-
-**Problem 2:** Why do thick leaders persist longer than thin streamers? Give two physical reasons.
-
----
-
-## Module 3.5: The Capacitive Divider Problem
-
-### Voltage Division Along Spark
-
-From Part 1, spark circuit:
-```
- [C_mut]
-Topload ----||---- Spark
- |
- [R]
- |
- [C_sh]
- |
- GND
-```
-
-**Voltage divider:** V_tip depends on impedance ratio:
-```
-V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
-
-where Z_mut = (1/jωC_mut) || R (parallel combination)
- Z_sh = 1/(jωC_sh)
-```
-
-### Open-Circuit Limit (No Current)
-
-When R → ∞ (no conduction), only capacitances matter:
-```
-V_tip = V_topload × C_mut / (C_mut + C_sh)
-```
-
-**Problem:** As spark grows, C_sh increases (∝ length):
-```
-C_sh ≈ 2 pF/foot × L
-
-As L increases → C_sh increases → V_tip decreases!
-```
-
-**Example:**
-```
-V_topload = 400 kV (constant)
-C_mut = 8 pF (approximately constant)
-
-Short spark (1 ft): C_sh = 2 pF
-V_tip = 400 × 8/(8+2) = 320 kV (80%)
-
-Medium spark (3 ft): C_sh = 6 pF
-V_tip = 400 × 8/(8+6) = 229 kV (57%)
-
-Long spark (6 ft): C_sh = 12 pF
-V_tip = 400 × 8/(8+12) = 160 kV (40%)
-```
-
-**Tip voltage drops to 40% even with constant topload voltage!**
-
-### With Finite Resistance
-
-Real case with R = R_opt_power ≈ 1/(ω(C_mut+C_sh)):
-
-```
-Z_mut = R || (1/jωC_mut) ≈ complex value
-V_tip is lower and phase-shifted
-
-Effect is similar but worse:
-- Magnitude division (as above)
-- Plus current-dependent voltage drop across R
-- V_tip drops faster than capacitive case alone
-```
-
-### Impact on Growth
-
-```
-E_tip = κ × V_tip / L
-
-As L increases:
-- Numerator (V_tip) decreases (capacitive division)
-- Denominator (L) increases (geometry)
-- E_tip decreases as L²
-
-Growth becomes progressively harder!
-```
-
-**Why sub-linear scaling:**
-```
-If energy scales as E ∝ L², but division effect makes
-V_tip ∝ 1/L, then achievable length L ∝ √E
-
-This explains Freau's empirical observation: L ∝ √E for burst mode
-```
-
----
-
-### WORKED EXAMPLE 3.5: Voltage Division
-
-**Given:**
-- V_topload = 350 kV (maintained constant)
-- C_mut = 10 pF
-- Spark grows from 0 to 4 feet
-
-**Find:** V_tip at L = 1, 2, 3, 4 feet (open-circuit approximation)
-
-**Solution:**
-
-**At L = 1 ft:**
-```
-C_sh = 2 pF/ft × 1 ft = 2 pF
-
-V_tip = 350 kV × 10/(10+2)
- = 350 × 10/12
- = 292 kV (83% of V_topload)
-```
-
-**At L = 2 ft:**
-```
-C_sh = 4 pF
-
-V_tip = 350 × 10/14
- = 250 kV (71%)
-```
-
-**At L = 3 ft:**
-```
-C_sh = 6 pF
-
-V_tip = 350 × 10/16
- = 219 kV (63%)
-```
-
-**At L = 4 ft:**
-```
-C_sh = 8 pF
-
-V_tip = 350 × 10/18
- = 194 kV (55%)
-```
-
-**Summary table:**
-
-| Length | C_sh | V_tip | % of V_top |
-|--------|------|-------|------------|
-| 1 ft | 2 pF | 292 kV| 83% |
-| 2 ft | 4 pF | 250 kV| 71% |
-| 3 ft | 6 pF | 219 kV| 63% |
-| 4 ft | 8 pF | 194 kV| 55% |
-
-**Voltage drops almost linearly with length, making further extension difficult.**
-
----
-
-### PRACTICE PROBLEMS 3.5
-
-**Problem 1:** V_top = 300 kV, C_mut = 12 pF. Calculate V_tip for L = 2 ft and L = 5 ft. What percentage is lost?
-
-**Problem 2:** If E_propagation = 0.6 MV/m and κ = 3, what V_tip is needed for 2 m spark? Using C_mut = 8 pF, what V_topload is required?
-
----
-
-## Module 3.6: Introduction to FEMM
-
-### What is FEMM?
-
-**FEMM = Finite Element Method Magnetics**
-- Free, open-source electromagnetic FEA software
-- 2D planar and axisymmetric problems
-- Electrostatic, magnetostatic, AC magnetic, thermal analysis
-
-**For Tesla coils:** Use electrostatic solver to extract capacitances
-
-**Download:** www.femm.info
-
-### Basic Workflow
-
-**1. Define geometry:**
-- Draw conductors (spark, topload, ground)
-- Define materials (air, metal)
-- Set boundaries (Dirichlet, Neumann)
-
-**2. Assign properties:**
-- Conductor potentials (voltages)
-- Material properties (permittivity)
-- Boundary conditions
-
-**3. Mesh:**
-- Automatic triangulation
-- Refinement near conductors
-
-**4. Solve:**
-- Numerical solution of Laplace's equation
-- ∇²V = 0 in free space
-
-**5. Post-process:**
-- Extract capacitance matrix
-- Calculate electric fields
-- Visualize field lines
-
-### Problem Setup for Spark
-
-**Geometry:**
-```
-- Toroidal topload (axisymmetric)
-- Cylindrical spark channel (vertical)
-- Ground plane (large boundary)
-- Air region (surrounds everything)
-```
-
-**Materials:**
-```
-- Air: ε_r = 1.0
-- Conductors: Set potentials, not material
-```
-
-**Boundaries:**
-```
-- Outer boundary: V = 0 (grounded, far from coil)
-- Axisymmetric boundary: special condition (mirror)
-```
-
-**Potentials:**
-```
-- Topload: 1 V (arbitrary, will scale)
-- Spark: floating (capacitance extraction)
-- Ground: 0 V
-```
-
----
-
-### WORKED EXAMPLE 3.6: FEMM Tutorial (Conceptual)
-
-**Task:** Extract C_mut and C_sh for 1 m spark from 30 cm toroid
-
-**Step 1: Geometry (axisymmetric)**
-```
-r-z coordinates (cylindrical)
-- Toroid: major radius 15 cm, minor radius 5 cm, center at z = 0
-- Spark: cylinder radius 1 mm, extends from z = -5 cm to z = -105 cm
-- Ground plane: z = -120 cm (large disk)
-- Outer boundary: r = 200 cm, z = ±150 cm (large region)
-```
-
-**Step 2: Materials**
-```
-- Everything is "Air" (ε_r = 1)
-- Will assign potentials, not conductivities
-```
-
-**Step 3: Boundaries**
-```
-- r = 0: Axisymmetric boundary (axis of symmetry)
-- Outer box: V = 0 (Dirichlet)
-```
-
-**Step 4: Conductors**
-```
-Create 3 conductor groups:
-- Conductor 1: Topload surface, V = 1V
-- Conductor 2: Spark surface, floating (no fixed potential)
-- Conductor 3: Ground plane, V = 0V
-```
-
-**Step 5: Mesh and solve**
-```
-- Auto mesh: ~5000 elements typical
-- Solve electrostatic problem
-- Convergence <0.001%
-```
-
-**Step 6: Extract capacitance matrix**
-```
-FEMM outputs 3×3 Maxwell capacitance matrix [C]:
-
- Top Spark Ground
-Top [ 30 -8 -22 ] pF
-Spark [ -8 14 -6 ] pF
-Ground[ -22 -6 28 ] pF
-
-(Values are example)
-```
-
-**Step 7: Calculate C_mut and C_sh**
-```
-C_mut = |C[Top, Spark]| = |-8| = 8 pF
-
-C_sh = C[Spark, Spark] + C[Spark, Top]
- = 14 + (-8)
- = 6 pF
-
-Validation: 6 pF ≈ 2 pF/ft × 3.3 ft ✓
-```
-
----
-
-### VISUAL AID 3.6: FEMM Interface
-
-```
-[Describe for screenshot annotation:]
-
-FEMM main window with four panels:
-
-UPPER LEFT: Geometry editor
-- Drawing tools (point, line, arc, circle)
-- Coordinate display (r, z in cm)
-- Toroid drawn as rotated circle
-- Spark as vertical line segment
-- Ground as horizontal line
-- All in r-z plane (axisymmetric)
-
-UPPER RIGHT: Problem definition
-- Properties: Frequency = 0 (electrostatic)
-- Length units: centimeters
-- Problem type: Axisymmetric
-- Precision: 1e-8
-
-LOWER LEFT: Mesh view
-- Triangle mesh covering domain
-- Refined near conductors (smaller triangles)
-- Coarse far away (larger triangles)
-- Color = element size
-
-LOWER RIGHT: Solution view
-- Filled contours: equipotential lines (V)
-- Field vectors: E field (arrows)
-- Concentrated at topload and spark tip
-- Circuit property window showing capacitances
-```
-
----
-
-### PRACTICE PROBLEMS 3.6
-
-**Problem 1:** Why do we use V = 1 V instead of actual voltage (400 kV)? (Hint: electrostatics is linear)
-
-**Problem 2:** A FEMM simulation with 2 m spark gives C_sh = 14 pF. Does this match the empirical 2 pF/ft rule? (Show calculation)
-
----
-
-## Module 3.7: Extracting Capacitances from FEMM
-
-### The Maxwell Capacitance Matrix
-
-FEMM outputs matrix [C] where:
-```
-[Q] = [C] × [V]
-
-Q_i = charge on conductor i
-V_i = potential of conductor i
-
-Matrix properties:
-- Symmetric: C_ij = C_ji
-- Diagonal positive: C_ii > 0
-- Off-diagonal negative: C_ij < 0 for i≠j
-- Row sums to zero: Σ_j C_ij = 0
-```
-
-**Physical meaning:**
-- C_ii: self-capacitance (conductor i to infinity)
-- C_ij (i≠j): mutual capacitance (coupling between i and j, negative)
-
-### Two-Body System (Topload + Spark)
-
-Matrix for topload (1), spark (2), ground (implicit):
-```
- [1] [2]
-[1] [ C₁₁ C₁₂ ]
-[2] [ C₂₁ C₂₂ ]
-
-Example values:
- [Top] [Spark]
-[Top] [ 30 -8 ] pF
-[Spark][ -8 14 ] pF
-```
-
-### Extraction Formulas
-
-**C_mut (mutual capacitance):**
-```
-C_mut = |C₁₂| = |C₂₁|
-
-Take absolute value of off-diagonal element
-```
-
-**C_sh (spark to ground):**
-
-Method 1 - From row sum:
-```
-Ground capacitance = -(C₂₁ + C₂₂)
-But we want spark-to-ground only: C_sh
-
-C_sh = C₂₂ + C₂₁
- = C₂₂ - |C₁₂| (since C₂₁ = C₁₂ < 0)
-```
-
-Method 2 - Direct measurement:
-```
-Run second simulation with topload grounded
-Measure spark capacitance to ground directly
-```
-
-**Validation check:**
-```
-C_sh ≈ 2 pF/foot × L_spark
-
-If ratio is 1.5-2.5 pF/foot: good
-If significantly different: check geometry/mesh
-```
-
----
-
-### WORKED EXAMPLE 3.7: Matrix Interpretation
-
-**Given FEMM output:**
-```
-Conductor properties:
-Conductor 1 (Topload): 35.2 pF to ground
-Conductor 2 (Spark): 16.8 pF to ground
-
-Circuit properties:
-C[1,1] = 35.2 pF
-C[1,2] = -10.5 pF
-C[2,1] = -10.5 pF (symmetry)
-C[2,2] = 16.8 pF
-
-Spark length: 1.8 m = 5.9 ft
-```
-
-**Extract:**
-(a) C_mut
-(b) C_sh
-(c) Validate against empirical rule
-
-**Solution:**
-
-**Part (a):** Mutual capacitance
-```
-C_mut = |C[1,2]| = |-10.5| = 10.5 pF
-```
-
-**Part (b):** Shunt capacitance
-```
-C_sh = C[2,2] + C[2,1]
- = 16.8 + (-10.5)
- = 6.3 pF
-```
-
-**Part (c):** Validation
-```
-Empirical prediction:
-C_sh_predicted = 2 pF/ft × 5.9 ft = 11.8 pF
-
-FEMM result:
-C_sh_FEMM = 6.3 pF
-
-Ratio: 6.3 / 11.8 = 0.53
-
-This is LOWER than expected (by factor ~2)
-```
-
-**Possible explanations:**
-```
-1. Empirical rule assumes straight vertical spark
- - If spark is angled or curved, less capacitance
-
-2. Empirical rule from community measurements
- - May include some C_mut in "measured" value
- - Pure C_sh might be lower
-
-3. Ground plane distance matters
- - FEMM has specific ground geometry
- - Empirical rule assumes "typical" room
-
-4. Diameter assumption
- - Thinner diameter → lower C_sh (logarithmic)
-
-For modeling: Use FEMM value (more accurate for specific geometry)
-```
-
----
-
-### VISUAL AID 3.7: Capacitance Matrix Interpretation
-
-```
-[Describe for diagram:]
-
-Left: Physical picture
-- Topload (labeled "1")
-- Spark channel (labeled "2")
-- Ground plane (labeled "0" or implicit)
-- Field lines showing:
- * C₁₁: Topload to infinity (self)
- * C₂₂: Spark to infinity (self)
- * C₁₂: Topload to spark (mutual, shown in green)
-
-Center: Matrix representation
-```
-[C] = [ 35.2 -10.5 ]
- [-10.5 16.8 ]
-```
-- Diagonal highlighted (positive)
-- Off-diagonal highlighted (negative)
-- Symmetry shown with arrows
-
-Right: Circuit extraction
-- C_mut = |C₁₂| = 10.5 pF (between topload and spark)
-- C_sh = C₂₂ - |C₁₂| = 6.3 pF (spark to ground)
-- Circuit diagram showing extracted values
-
-Bottom: Key points
-- "Off-diagonal → mutual capacitance"
-- "Diagonal - mutual → shunt capacitance"
-- "Always check symmetry: C₁₂ = C₂₁"
-```
-
----
-
-### PRACTICE PROBLEMS 3.7
-
-**Problem 1:** FEMM gives C[1,1]=40 pF, C[1,2]=-12 pF, C[2,2]=20 pF for a 2 m spark. Extract C_mut and C_sh. Does C_sh match the empirical rule?
-
-**Problem 2:** Why are off-diagonal elements negative in the Maxwell matrix? What would happen if they were positive?
-
----
-
-## Module 3.8: Building the Lumped Spark Model
-
-### Complete Workflow
-
-**Step 1: FEMM electrostatic analysis**
-```
-- Geometry: topload + spark + ground
-- Axisymmetric 2D
-- Solve at frequency = 0 (electrostatic)
-- Extract [C] matrix
-```
-
-**Step 2: Calculate circuit elements**
-```
-C_mut = |C₁₂| from matrix
-C_sh = C₂₂ - |C₁₂| from matrix
-R = R_opt_power = 1/(ω(C_mut + C_sh))
-Clip to physical bounds: R = clip(R, R_min, R_max)
-```
-
-**Step 3: Build SPICE netlist**
-```
-* Lumped spark model
-.param freq=200k
-.param omega={2*pi*freq}
-
-V_topload topload 0 AC 1 ; 1V test source
-
-C_mut topload spark_node {C_mut}
-R_spark spark_node spark_r {R}
-C_sh spark_r 0 {C_sh}
-
-.ac lin 1 {freq} {freq}
-.print ac v(topload) i(V_topload)
-.end
-```
-
-**Step 4: Run AC analysis**
-```
-- Calculate Y = I/V at topload port
-- Extract Re{Y}, Im{Y}
-- Convert to Z if needed
-- Calculate power: P = 0.5 × |V|² × Re{Y}
-```
-
-**Step 5: Validate**
-```
-- Check φ_Z in expected range (-55° to -75°)
-- Check R in physical range (kΩ to hundreds of kΩ)
-- Check C_sh ≈ 2 pF/ft ± factor of 2
-- Compare to measurements if available
-```
-
-### Integration with Full Coil Model
-
-```
-[Primary circuit] → [Coupled transformer] → [Secondary] → [Topload] → [Spark model]
-
-Spark model appears as:
-- Load impedance at topload port
-- Affects loaded Q, resonant frequency
-- Extracts power from secondary
-```
-
----
-
-### WORKED EXAMPLE 3.8: Complete Lumped Model
-
-**Given:**
-- Frequency: f = 190 kHz
-- FEMM results: C_mut = 9.5 pF, C_sh = 7.2 pF
-- Physical bounds: R_min = 5 kΩ, R_max = 500 kΩ
-
-**Build and analyze model:**
-
-**Step 1:** Calculate R_opt_power
-```
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-C_total = C_mut + C_sh = 9.5 + 7.2 = 16.7 pF
-
-R_opt_power = 1/(ω × C_total)
- = 1/(1.194×10⁶ × 16.7×10⁻¹²)
- = 1/(19.94×10⁻⁶)
- = 50.2 kΩ
-```
-
-**Step 2:** Check bounds
-```
-R_min = 5 kΩ
-R_opt = 50.2 kΩ
-R_max = 500 kΩ
-
-5 < 50.2 < 500 ✓
-
-Use R = 50.2 kΩ
-```
-
-**Step 3:** Build SPICE model
-```
-* Spark lumped model - 190 kHz
-V_test topload 0 AC 1V
-C_mut topload n1 9.5p
-R_spark n1 n2 50.2k
-C_sh n2 0 7.2p
-
-.ac lin 1 190k 190k
-.print ac v(topload) i(V_test) vp(topload) ip(V_test)
-.end
-```
-
-**Step 4:** Simulate and extract (example results)
-```
-Simulation output:
-V(topload) = 1.000 V ∠0°
-I(V_test) = 5.23×10⁻⁶ A ∠74.5°
-
-Y = I/V = 5.23 μS ∠74.5°
-
-Re{Y} = 5.23 × cos(74.5°) = 1.39 μS
-Im{Y} = 5.23 × sin(74.5°) = 5.04 μS
-
-Convert to Z:
-|Z| = 1/5.23×10⁻⁶ = 191 kΩ
-φ_Z = -74.5°
-
-R_eq = 191 × cos(-74.5°) = 51 kΩ
-X_eq = 191 × sin(-74.5°) = -184 kΩ
-```
-
-**Step 5:** Validate
-```
-φ_Z = -74.5° : In expected range (-55° to -75°) ✓
-R_eq ≈ 51 kΩ : Close to R_opt = 50.2 kΩ ✓
-Physical: Between 5-500 kΩ ✓
-
-C_sh validation:
-L ≈ 7.2 pF / 2 pF/ft = 3.6 ft ≈ 1.1 m
-Reasonable for medium spark ✓
-```
-
-**Step 6:** Power calculation (if V_topload = 320 kV actual)
-```
-P = 0.5 × |V|² × Re{Y}
- = 0.5 × (320×10³)² × 1.39×10⁻⁶
- = 0.5 × 1.024×10¹¹ × 1.39×10⁻⁶
- = 71.2 kW
-```
-
-Model is complete and ready for coil integration!
-
----
-
-### PRACTICE PROBLEMS 3.8
-
-**Problem 1:** Build lumped model for: f=200 kHz, C_mut=11 pF, C_sh=9 pF. Calculate all component values and expected φ_Z.
-
-**Problem 2:** If SPICE simulation gives φ_Z=-85° (more capacitive than expected), what might be wrong with the model?
-
----
-
-## Part 3 Summary & Integration
-
-### Key Concepts Checklist
-
-- [ ] **E_inception:** ~2-3 MV/m to start breakdown
-- [ ] **E_propagation:** ~0.4-1.0 MV/m to sustain growth
-- [ ] **Tip enhancement:** E_tip = κ × E_avg, κ ≈ 2-5
-- [ ] **Growth criterion:** E_tip > E_propagation required
-- [ ] **Energy per meter ε:** 5-15 (QCW), 30-100 (burst) J/m
-- [ ] **Growth rate:** dL/dt = P/ε when field adequate
-- [ ] **Voltage vs power limited:** Both constraints exist
-- [ ] **Thermal time:** τ = d²/(4α), but persistence longer
-- [ ] **QCW advantage:** Maintains hot channel (low ε)
-- [ ] **Capacitive divider:** V_tip drops as C_sh grows
-- [ ] **Sub-linear scaling:** L ∝ √E for voltage-limited
-- [ ] **FEMM workflow:** Geometry → solve → extract [C]
-- [ ] **Maxwell matrix:** Diagonal positive, off-diagonal negative
-- [ ] **C_mut extraction:** |C₁₂| from off-diagonal
-- [ ] **C_sh extraction:** C₂₂ - |C₁₂|
-- [ ] **Validation:** C_sh ≈ 2 pF/ft ± factor 2
-- [ ] **Lumped model:** (R||C_mut) + C_sh
-- [ ] **R = R_opt_power:** For hungry streamer assumption
-
----
-
-## Final Integration Exercise
-
-**Complete design challenge:**
-
-**Given:**
-- DRSSTC at 185 kHz
-- Toroid: 40 cm major diameter, 10 cm minor
-- Target: 2 m spark
-- Thévenin: Z_th = 120 - j2200 Ω, V_th = 380 kV
-
-**Tasks:**
-
-1. **FEMM analysis (describe setup):**
- - Draw geometry for 2 m spark
- - What boundaries to use?
- - Expected C_sh range?
-
-2. **Assume FEMM gives:** C_mut = 11 pF, C_sh = 13 pF
- - Validate C_sh (empirical rule)
- - Calculate R_opt_power at 185 kHz
- - Is R within 5-500 kΩ bounds?
-
-3. **Build lumped model:**
- - Calculate Y_spark
- - Convert to Z_spark
- - What is φ_Z?
-
-4. **Predict performance:**
- - Calculate Z_total = Z_th + Z_spark
- - Find current I
- - Calculate power to spark
- - Compare to theoretical max (conjugate match)
-
-5. **Growth analysis:**
- - Assume QCW, ε = 10 J/m
- - How long to reach 2 m?
- - Check voltage requirement: E_prop = 0.6 MV/m, κ = 3.5
- - Is growth voltage-limited or power-limited?
-
-**This exercise integrates all of Part 3!**
-
----
-
-**END OF PART 3**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 4: Advanced Topics - Distributed Models and Real-World Application
-
----
-
-## Module 4.1: Why Distributed Models?
-
-### Limitations of Lumped Models
-
-**Lumped model treats entire spark as single R, C_mut, C_sh:**
-
-**Works well for:**
-- Short sparks (<1 m)
-- Impedance matching studies
-- Quick optimization
-- First-order power estimates
-
-**Fails to capture:**
-```
-1. Current distribution along spark
- - Base carries full current
- - Tip may have much less (capacitive shunting)
-
-2. Voltage distribution
- - Not linear drop from top to tip
- - Capacitive divider effects at each point
-
-3. Tip vs base differences
- - Base: hot, well-coupled, low R
- - Tip: cool, weakly-coupled, high R
-
-4. Streamer/leader transitions
- - Base forms leader (low R)
- - Tip remains streamer (high R)
- - Lumped model averages this out
-
-5. Very long sparks (>3 m)
- - Distributed effects dominate
- - Single lumped R is poor approximation
-```
-
-### When to Use Distributed Model
-
-**Use distributed when:**
-- Spark length > 1-2 meters
-- Need current distribution (for measurements)
-- Studying leader/streamer physics
-- Validating against detailed measurements
-- Research/publication quality results
-
-**Stick with lumped when:**
-- Quick design iterations
-- Coil-level optimization (matching)
-- Spark length < 1 meter
-- Engineering estimates sufficient
-
-**Computational cost:**
-- Lumped: <1 second
-- Distributed (n=10): ~10-30 seconds
-- Distributed (n=20): ~1-5 minutes
-
----
-
-### VISUAL AID 4.1: Lumped vs Distributed Comparison
-
-```
-[Describe for diagram:]
-
-Two-panel comparison:
-
-LEFT: Lumped model
-- Single box representing entire spark
-- Three components: C_mut, R, C_sh
-- Simple circuit
-- One current value
-- One voltage drop
-- Label: "Good for <1m, fast computation"
-
-RIGHT: Distributed model (n=5 shown)
-- Spark divided into 5 segments
-- Each segment has: C_mutual[i], R[i], C_shunt[i]
-- Coupling between segments shown
-- Current arrows varying in size (large at base, small at tip)
-- Voltage nodes at each junction
-- Gradient showing R: low (blue) at base, high (red) at tip
-- Label: "Captures physics, slower computation"
-
-BOTTOM: Feature comparison table
-| Feature | Lumped | Distributed |
-|----------------------|--------|-------------|
-| Setup time | Fast | Slow |
-| Computation | <1s | 10s-min |
-| Current distribution| No | Yes |
-| Tip/base difference | No | Yes |
-| Accuracy <1m | Good | Excellent |
-| Accuracy >3m | Poor | Good |
-```
-
----
-
-### DISCUSSION QUESTIONS 4.1
-
-**Question 1:** A 0.5 m spark shows good agreement between lumped model and measurements. A 3 m spark shows poor agreement. Why?
-
-**Question 2:** If you only care about total power delivered to spark (not distribution), when would distributed model still be necessary?
-
-**Question 3:** In what situation might even a distributed model fail? (Hint: think about branching)
-
----
-
-## Module 4.2: nth-Order Model Structure
-
-### Segmentation Strategy
-
-**Divide spark into n equal-length segments:**
-```
-n = number of segments (typically 5-20)
-L_segment = L_total / n
-
-Segment numbering:
-i = 1: Base (connected to topload)
-i = 2, 3, ..., n-1: Middle sections
-i = n: Tip (furthest from topload)
-```
-
-**Why equal lengths?**
-- Simplifies FEMM geometry
-- Uniform discretization
-- Easy to implement
-- Non-uniform possible but more complex
-
-### Circuit Topology
-
-**Each segment i has:**
-```
-1. Resistance R[i]
- - Plasma resistance of that segment
- - Variable, to be optimized
-
-2. Mutual capacitances C[i,j]
- - Coupling to all other segments j≠i
- - And to topload (j=0)
- - Extracted from FEMM
-
-3. Shunt capacitance to ground
- - Included in capacitance matrix
- - Not a separate component
-```
-
-**Full network:**
-```
-Topload (node 0)
- |
- +-- C[0,1] -- Node 1 (base segment)
- | |
- | R[1]
- | |
- +-- C[0,2] ----+-- Node 2
- | |
- | R[2]
- | |
- ...
- |
- +-- C[0,n] ----+-- Node n (tip segment)
- |
- R[n]
- |
-
-Plus C[i,j] between all segment pairs
-Plus C[i,ground] for each segment to ground
-```
-
-**Complexity:** For n segments + topload:
-- (n+1)×(n+1) capacitance matrix
-- n resistance values
-- Total unknowns: n (resistances)
-
----
-
-### WORKED EXAMPLE 4.2: Draw 3-Segment Model
-
-**Given:**
-- Total spark: 1.5 m
-- Divide into n = 3 equal segments
-- Each segment: 0.5 m
-
-**Task:** Draw circuit topology (conceptual)
-
-**Solution:**
-
-```
-Topload (V_top, node 0)
- |
- +---[C[0,1]]---+---[C[0,2]]---+---[C[0,3]]---+
- | | | |
- | | | |
-Node 1 -------[R[1]]-------------|--------------|
-(base) | | |
- [C[1,2]] [C[1,3]] |
- | | |
- Node 2 -----------[R[2]]--------[C[2,3]]
- (middle) | |
- [C_sh,2] |
- | |
- Node 3 --------[R[3]]
- (tip) |
- [C_sh,3]
- |
- GND
-
-Where:
-- C[i,j] = mutual capacitance between segments
-- C_sh[i] = shunt capacitance segment i to ground
-- R[i] = resistance of segment i
-```
-
-**Note:** This is conceptual. Actual implementation uses full (n+1)×(n+1) matrix.
-
-**Typical values (estimated):**
-```
-Segment 1 (base): R[1] = 10 kΩ (hot, well-coupled)
-Segment 2 (mid): R[2] = 30 kΩ (moderate)
-Segment 3 (tip): R[3] = 100 kΩ (cool, weak coupling)
-
-C[0,1] > C[0,2] > C[0,3] (coupling decreases with distance)
-```
-
----
-
-### PRACTICE PROBLEMS 4.2
-
-**Problem 1:** A 2.4 m spark is divided into n=6 segments. What is the length of each segment? Number them from base to tip.
-
-**Problem 2:** For n=10 segments, how many capacitance matrix elements are there? (Count all C[i,j] including diagonal)
-
-**Problem 3:** Why might R[1] (base) be much smaller than R[10] (tip)? Give two physical reasons.
-
----
-
-## Module 4.3: FEMM for Distributed Models
-
-### Multi-Body Electrostatic Setup
-
-**Geometry definition:**
-```
-For n segments + topload → (n+1) conductors
-
-Example n=5:
-- Body 0: Toroid topload
-- Body 1: Cylinder, length L/5, base at topload
-- Body 2: Cylinder, length L/5, above body 1
-- Body 3: Cylinder, length L/5, above body 2
-- Body 4: Cylinder, length L/5, above body 3
-- Body 5: Cylinder, length L/5, top segment (tip)
-- Ground plane at bottom
-```
-
-**Axisymmetric setup:**
-```
-r-z coordinates
-All bodies as cylindrical sections
-Diameter: 1-3 mm typical (uniform for simplicity)
-Spacing: slight gap (~0.1 mm) between segments for FEMM
-```
-
-**Conductor properties:**
-```
-Group each body as separate conductor:
-- Conductor 0: Topload, V = 1V
-- Conductors 1-n: Spark segments, floating potential
-- Ground: V = 0V (boundary condition)
-```
-
-### Solving and Extraction
-
-**Mesh requirements:**
-```
-- Finer mesh near conductors
-- Refinement at segment junctions
-- Typical: 10,000-50,000 elements for n=10
-- Convergence: <0.01% error
-```
-
-**Capacitance matrix output:**
-```
-FEMM circuit properties → Capacitance matrix
-
-(n+1)×(n+1) symmetric matrix [C]:
-
- [0] [1] [2] ... [n]
-[0] [ C₀₀ C₀₁ C₀₂ ... C₀ₙ ]
-[1] [ C₁₀ C₁₁ C₁₂ ... C₁ₙ ]
-[2] [ C₂₀ C₂₁ C₂₂ ... C₂ₙ ]
-...
-[n] [ Cₙ₀ Cₙ₁ Cₙ₂ ... Cₙₙ ]
-
-Properties:
-- Symmetric: Cᵢⱼ = Cⱼᵢ
-- Diagonal positive: Cᵢᵢ > 0
-- Off-diagonal negative: Cᵢⱼ < 0 for i≠j
-- Row sum = 0: Σⱼ Cᵢⱼ = 0
-```
-
-### Matrix Validation
-
-**Check 1: Symmetry**
-```
-|C[i,j] - C[j,i]| / |C[i,j]| < 0.01
-If not symmetric: numerical error, refine mesh
-```
-
-**Check 2: Positive definite**
-```
-All eigenvalues should be ≥ 0
-One eigenvalue = 0 (ground reference freedom)
-Rest positive
-```
-
-**Check 3: Physical values**
-```
-Nearby segments: larger |C[i,j]|
-Distant segments: smaller |C[i,j]|
-Base segments: larger C[i,0] (topload coupling)
-Tip segments: smaller C[n,0]
-```
-
-**Check 4: Total shunt capacitance**
-```
-C_sh_total = Σᵢ (Cᵢᵢ - |Cᵢ₀|) for all spark segments
-
-Should be approximately:
-C_sh_total ≈ 2 pF/foot × L_total
-
-Within factor of 2 is reasonable
-```
-
----
-
-### WORKED EXAMPLE 4.3: FEMM Setup for n=5
-
-**Given:**
-- Spark length: 2.0 m = 6.56 feet
-- Diameter: 2 mm
-- n = 5 segments → each 0.4 m long
-- Topload: 30 cm toroid
-
-**FEMM procedure:**
-
-**Step 1: Geometry (r-z coordinates)**
-```
-Topload:
-- Major radius: 15 cm, minor radius: 5 cm
-- Center at z = 0
-- Lowest point: z = -5 cm
-
-Segment 1 (base):
-- r = 1 mm (0.1 cm)
-- z from -5 cm to -45 cm
-- Length: 40 cm
-
-Segment 2:
-- z from -45 cm to -85 cm
-
-Segment 3:
-- z from -85 cm to -125 cm
-
-Segment 4:
-- z from -125 cm to -165 cm
-
-Segment 5 (tip):
-- z from -165 cm to -205 cm
-
-Ground plane:
-- z = -220 cm (15 cm below tip)
-- r = 0 to 300 cm (large)
-
-Outer boundary:
-- r = 300 cm, z = ±250 cm
-```
-
-**Step 2: Materials and conductors**
-```
-All regions: Air (ε_r = 1)
-
-Define 6 conductor groups:
-Group 0: Topload surface, V = 1V
-Groups 1-5: Segment surfaces, floating
-Ground: Boundary at z = -220 cm, V = 0V
-```
-
-**Step 3: Meshing**
-```
-Auto mesh with refinement:
-- Triangle size near conductors: 0.5 mm
-- Triangle size at boundaries: 50 mm
-- ~25,000 elements total
-```
-
-**Step 4: Solve**
-```
-Problem type: Electrostatic, axisymmetric
-Frequency: 0 Hz
-Precision: 1e-8
-```
-
-**Step 5: Extract matrix (example results)**
-```
-Matrix [C] in pF:
-
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
-
-(Values are illustrative)
-```
-
-**Step 6: Validate**
-```
-Symmetry check: C[1,2] = C[2,1] = -2.8 ✓
-
-Total shunt capacitance (approximate):
-C_sh ≈ Σᵢ₌₁⁵ (Cᵢᵢ - |Cᵢ₀|)
- = (14.8-9.2) + (10.4-3.1) + (8.6-1.2) + (7.4-0.6) + (5.8-0.3)
- = 5.6 + 7.3 + 7.4 + 6.8 + 5.5
- = 32.6 pF
-
-Expected: 2 pF/ft × 6.56 ft = 13.1 pF
-
-Ratio: 32.6/13.1 = 2.5
-
-Higher than expected, but within factor of 2-3 (acceptable)
-Difference due to matrix interpretation method
-```
-
----
-
-### PRACTICE PROBLEMS 4.3
-
-**Problem 1:** For n=10 segments, 3 m total, what is each segment length? What is the z-coordinate range for segment 5 if topload bottom is at z=0?
-
-**Problem 2:** A capacitance matrix shows C[3,7] = -0.4 pF and C[3,4] = -2.1 pF. Which segments are closer to segment 3? Does this make physical sense?
-
----
-
-## Module 4.4: Implementing Capacitance Matrices in SPICE
-
-### The Challenge
-
-**Maxwell matrix has negative off-diagonals:**
-```
-Literal SPICE capacitor implementation:
-C_12 node1 node2 10p ← OK, positive value
-C_12 node1 node2 -10p ← ERROR! Negative capacitance unphysical
-```
-
-**Problem:** Cannot directly use C[i,j] < 0 as SPICE capacitors
-
-### Solution 1: Partial Capacitance Transformation
-
-**Convert Maxwell → Partial (all-positive):**
-
-**Partial capacitance:** Capacitance with all other nodes grounded
-
-```
-For node i:
-C_partial[i,j] = -C_Maxwell[i,j] for i≠j (flip sign!)
-C_partial[i,i] = Σⱼ |C_Maxwell[i,j]| (sum of magnitudes)
-
-All C_partial > 0 → can implement as SPICE capacitors
-```
-
-**SPICE implementation:**
-```
-* Partial capacitance method
-* Between every node pair i,j (i 1 (distant segments)
-```
-
-**When acceptable:**
-- Large n (>10): distant couplings small
-- Quick estimates
-- Weak segment-to-segment coupling
-
-**Validation:** Compare full vs approximate impedance
-
----
-
-### WORKED EXAMPLE 4.4: Partial Capacitance Conversion (3×3)
-
-**Given Maxwell matrix (topload + 2 segments):**
-```
- [0] [1] [2]
-[0] [ 30.0 -8.0 -2.0 ] pF
-[1] [ -8.0 14.0 -3.0 ] pF
-[2] [ -2.0 -3.0 9.0 ] pF
-```
-
-**Convert to partial (all-positive) for SPICE:**
-
-**Step 1:** Between-node capacitances (flip signs)
-```
-C_partial[0,1] = -C_Maxwell[0,1] = -(-8.0) = 8.0 pF
-C_partial[0,2] = -C_Maxwell[0,2] = -(-2.0) = 2.0 pF
-C_partial[1,2] = -C_Maxwell[1,2] = -(-3.0) = 3.0 pF
-```
-
-**Step 2:** Ground capacitances
-
-For each node, start with diagonal, subtract partial caps:
-
-**Node 0:**
-```
-C[0,0] = 30.0 pF
-Sum of partials leaving node 0: 8.0 + 2.0 = 10.0 pF
-C_partial[0,gnd] = 30.0 - 10.0 = 20.0 pF
-```
-
-**Node 1:**
-```
-C[1,1] = 14.0 pF
-Partials: 8.0 (to 0) + 3.0 (to 2) = 11.0 pF
-C_partial[1,gnd] = 14.0 - 11.0 = 3.0 pF
-```
-
-**Node 2:**
-```
-C[2,2] = 9.0 pF
-Partials: 2.0 (to 0) + 3.0 (to 1) = 5.0 pF
-C_partial[2,gnd] = 9.0 - 5.0 = 4.0 pF
-```
-
-**Step 3:** SPICE netlist
-```
-* Partial capacitance implementation
-* Between nodes
-C_0_1 node0 node1 8.0p
-C_0_2 node0 node2 2.0p
-C_1_2 node1 node2 3.0p
-
-* To ground
-C_0_gnd node0 0 20.0p
-C_1_gnd node1 0 3.0p
-C_2_gnd node2 0 4.0p
-
-* Resistances (to be determined)
-R1 node1 node1_r {R1_value}
-R2 node2 node2_r {R2_value}
-```
-
-**Validation:** Verify total capacitance node0→gnd matches:
-```
-With node1, node2 grounded:
-C_total = C_0_gnd + C_0_1 || C_1_gnd + C_0_2 || C_2_gnd
-
-Should equal approximately 30 pF (check numerically)
-```
-
----
-
-### PRACTICE PROBLEMS 4.4
-
-**Problem 1:** Given C_Maxwell = [25, -6; -6, 10] pF (2×2), convert to partial capacitances. Draw the SPICE circuit.
-
-**Problem 2:** Why can't we just use "negative capacitors" in SPICE? What would it physically mean?
-
-**Problem 3:** In nearest-neighbor approximation for n=10, how many capacitances are kept vs full matrix? Calculate percentage reduction.
-
----
-
-## Module 4.5: Resistance Optimization - Iterative Method
-
-### Algorithm Overview
-
-**Goal:** Find R[i] for each segment that maximizes total power
-
-**Challenge:** R[i] values are coupled (changing one affects power in others)
-
-**Solution:** Iterative optimization with damping
-
-### Initialization: Tapered Profile
-
-**Physical expectation:**
-- Base: hot, well-coupled → low R
-- Tip: cool, weakly-coupled → high R
-
-**Initialize with gradient:**
-```
-For i = 1 to n:
- position = (i-1)/(n-1) # 0 at base, 1 at tip
- R[i] = R_base + (R_tip - R_base) × position^2
-
-Typical starting values:
- R_base = 10 kΩ
- R_tip = 1 MΩ
-
-Quadratic taper gives smooth transition
-```
-
-### Iterative Optimization Loop
-
-```
-iteration = 0
-converged = False
-
-While not converged and iteration < max_iterations:
-
- For i = 1 to n:
- # Sweep R[i] while keeping other R[j] fixed
- R_test = logspace(R_min[i], R_max[i], 20 points)
-
- For each R_test_value:
- Set R[i] = R_test_value
- Run AC analysis
- Calculate P[i] = power in segment i
-
- Find R_optimal[i] = R_test that maximizes P[i]
-
- # Apply damping for stability
- R_new[i] = α * R_optimal[i] + (1-α) * R_old[i]
-
- # Clip to physical bounds
- R[i] = clip(R_new[i], R_min[i], R_max[i])
-
- # Check convergence
- max_change = max(|R_new[i] - R_old[i]| / R_old[i])
- If max_change < 0.01: # 1% threshold
- converged = True
-
- iteration = iteration + 1
-```
-
-**Damping factor α:**
-```
-α = 0.3 to 0.5 typical
-- Lower α: more stable, slower convergence
-- Higher α: faster, may oscillate
-- Start with α=0.3 for safety
-```
-
-### Position-Dependent Bounds
-
-**Physical limits vary with position:**
-```
-position = (i-1)/(n-1)
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
- = 1 kΩ at base → 10 kΩ at tip
-
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position^2
- = 100 kΩ at base → 100 MΩ at tip
-```
-
-**Rationale:**
-- Base can achieve very low R (hot leader)
-- Tip unlikely to reach low R (cool, weak coupling)
-- Prevents unphysical solutions
-
-### Convergence Behavior
-
-**Well-coupled base segments:**
-- Sharp power peak at optimal R
-- Fast convergence (2-3 iterations)
-- Stable solution
-
-**Weakly-coupled tip segments:**
-- Flat power curve (many R values similar power)
-- Slow/no convergence to unique value
-- May stay at high R (physical - streamer regime)
-
-**Expected result:**
-```
-R[1] ≈ 5-20 kΩ (base leader)
-R[2] ≈ 10-40 kΩ
-...
-R[n-1] ≈ 50-200 kΩ
-R[n] ≈ 100 kΩ - 10 MΩ (tip streamer)
-
-Total: Σ R[i] should be in expected range (5-300 kΩ at 200 kHz)
-```
-
----
-
-### WORKED EXAMPLE 4.5: Iterative Optimization (n=3, simplified)
-
-**Given:**
-- 3 segments, f = 200 kHz
-- Capacitance matrix (from FEMM, simplified)
-- Initial: R[1]=50k, R[2]=100k, R[3]=500k
-
-**Iteration 1:**
-
-**Optimize R[1] (keeping R[2], R[3] fixed):**
-```
-Sweep R[1] = [10k, 20k, 30k, 40k, 50k, 60k, 80k, 100k]
-
-Results (example):
-R[1]=10k → P[1]=5.2 kW
-R[1]=20k → P[1]=8.1 kW
-R[1]=30k → P[1]=9.4 kW ← maximum
-R[1]=40k → P[1]=8.9 kW
-R[1]=50k → P[1]=7.8 kW (current value)
-...
-
-R_optimal[1] = 30 kΩ
-```
-
-**Apply damping (α=0.4):**
-```
-R_new[1] = 0.4 × 30k + 0.6 × 50k
- = 12k + 30k
- = 42 kΩ
-```
-
-**Optimize R[2]:**
-```
-With R[1]=42k (updated), R[3]=500k (fixed)
-
-Sweep R[2], find R_optimal[2] = 60 kΩ
-Current: R[2] = 100 kΩ
-
-R_new[2] = 0.4 × 60k + 0.6 × 100k
- = 24k + 60k
- = 84 kΩ
-```
-
-**Optimize R[3]:**
-```
-With R[1]=42k, R[2]=84k
-
-Sweep R[3], power curve is FLAT:
-R[3]=200k → P[3]=0.8 kW
-R[3]=500k → P[3]=0.85 kW
-R[3]=1M → P[3]=0.83 kW
-
-Weakly coupled! Peak not well-defined.
-Keep at R[3] = 500 kΩ (within bounds, acceptable)
-```
-
-**After iteration 1:**
-```
-R[1]: 50k → 42k (change = -16%)
-R[2]: 100k → 84k (change = -16%)
-R[3]: 500k → 500k (change = 0%)
-
-Max change = 16% > 1% → not converged, continue
-```
-
-**Iteration 2:**
-
-Repeat process with new R values...
-(typically 3-5 iterations to converge for base/middle segments)
-
-**Final converged result (example):**
-```
-R[1] = 35 kΩ (leader, base)
-R[2] = 75 kΩ (transition)
-R[3] = 500 kΩ (streamer, tip - weakly determined)
-
-Total: 610 kΩ at 200 kHz
-Check: Within expected range ✓
-```
-
----
-
-### PRACTICE PROBLEMS 4.5
-
-**Problem 1:** Initial R=[100k, 200k], optimal found R=[60k, 150k]. With α=0.3, what are the damped updates?
-
-**Problem 2:** Why use damping factor α<1 instead of just setting R=R_optimal directly? What could go wrong?
-
-**Problem 3:** After 10 iterations, base segment converged (0.5% change) but tip segment still changing 5% per iteration. What should you do?
-
----
-
-## Module 4.6: Resistance Optimization - Simplified Method
-
-### Circuit-Determined Resistance
-
-**Key insight:** If plasma always seeks R_opt_power, and C depends weakly on diameter:
-
-```
-For each segment i:
- C_total[i] = sum of all capacitances involving segment i
- R[i] = 1 / (ω × C_total[i])
- R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Extracting C_total from matrix:**
-```
-C_total[i] = |C[i,0]| + Σⱼ₌₁ⁿ |C[i,j]| (sum of absolute values)
-
-This is total capacitance "seen" by segment i
-```
-
-### Why This Works
-
-**Physical argument:**
-
-1. Hungry streamer seeks R = 1/(ωC_total) for max power
-2. C depends on diameter: C ∝ 1/ln(h/d)
-3. Logarithmic dependence: 2× diameter → ~10% capacitance change
-4. R_opt also changes ~10% for diameter change
-5. Diameter adjusts to match R_opt (self-consistent)
-6. Error from fixed C is comparable to other uncertainties
-
-**Typical uncertainties:**
-```
-FEMM extraction: ±5-10%
-Plasma physics (ε, E_prop): ±30-50%
-Empirical calibration: ±20-30%
-
-Diameter approximation: ±10-15%
-
-Diameter error is SMALL compared to physics uncertainties!
-```
-
-### When to Use
-
-**Good for:**
-- Standard cases (typical geometries, frequencies)
-- First-pass analysis
-- Quick evaluation of many designs
-- Educational purposes
-
-**Use iterative when:**
-- Research/validation
-- Extreme parameters (very long, very short, very low frequency)
-- Measurement comparison requires highest accuracy
-- Publishing results
-
-**Computational savings:**
-```
-Iterative: 5-10 iterations × 20 R-sweep points × n segments = 1000-2000 AC analyses
-Simplified: 1 AC analysis
-
-Speedup: 1000-2000× faster!
-```
-
----
-
-### WORKED EXAMPLE 4.6: Simplified R Calculation (n=5)
-
-**Given:**
-- f = 190 kHz, ω = 1.194×10⁶ rad/s
-- Capacitance matrix from Example 4.3 (repeated):
-
-```
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
-```
-
-**Calculate R[i] for each segment:**
-
-**Segment 1 (base):**
-```
-C_total[1] = |C[1,0]| + |C[1,2]| + |C[1,3]| + |C[1,4]| + |C[1,5]|
- = 9.2 + 2.8 + 0.9 + 0.4 + 0.2
- = 13.5 pF
-
-R[1] = 1 / (ω × C_total[1])
- = 1 / (1.194×10⁶ × 13.5×10⁻¹²)
- = 1 / (16.12×10⁻⁶)
- = 62.0 kΩ
-```
-
-**Segment 2:**
-```
-C_total[2] = |C[2,0]| + |C[2,1]| + |C[2,3]| + |C[2,4]| + |C[2,5]|
- = 3.1 + 2.8 + 2.1 + 0.7 + 0.3
- = 9.0 pF
-
-R[2] = 1 / (1.194×10⁶ × 9.0×10⁻¹²)
- = 93.0 kΩ
-```
-
-**Segment 3:**
-```
-C_total[3] = 1.2 + 0.9 + 2.1 + 1.8 + 0.5
- = 6.5 pF
-
-R[3] = 1 / (1.194×10⁶ × 6.5×10⁻¹²)
- = 129 kΩ
-```
-
-**Segment 4:**
-```
-C_total[4] = 0.6 + 0.4 + 0.7 + 1.8 + 1.4
- = 4.9 pF
-
-R[4] = 1 / (1.194×10⁶ × 4.9×10⁻¹²)
- = 171 kΩ
-```
-
-**Segment 5 (tip):**
-```
-C_total[5] = 0.3 + 0.2 + 0.3 + 0.5 + 1.4
- = 2.7 pF
-
-R[5] = 1 / (1.194×10⁶ × 2.7×10⁻¹²)
- = 310 kΩ
-```
-
-**Summary:**
-```
-R[1] = 62 kΩ (base - lowest)
-R[2] = 93 kΩ
-R[3] = 129 kΩ
-R[4] = 171 kΩ
-R[5] = 310 kΩ (tip - highest)
-
-Total: R_total = 765 kΩ
-```
-
-**Validation:**
-```
-At 190 kHz for 2 m spark:
-Expected total: 50-300 kΩ (from Part 2 guidelines)
-
-765 kΩ is higher than typical.
-
-Possible reasons:
-- Long spark (2 m), distributed effects significant
-- Tip resistance (310k) is high (streamer-dominated)
-- If measured, could be lower (iterative optimization might find lower R)
-
-Within factor of 2-3 of expectations - acceptable for first pass
-```
-
----
-
-### PRACTICE PROBLEMS 4.6
-
-**Problem 1:** Given C_total[i] = [15, 10, 8, 6, 4] pF for n=5 at f=200 kHz, calculate R[i] for all segments.
-
-**Problem 2:** Compare simplified method: one calculation (1 second) vs iterative: 10 iterations × 20 points × 5 segments = 1000 AC analyses (~100 seconds). For engineering design, which is more appropriate?
-
----
-
-## Module 4.7: Quick Validation Checks
-
-### Power Balance
-
-**Energy conservation:**
-```
-P_input = P_spark + P_secondary_losses + P_corona + P_radiation + P_other
-
-Check: P_spark should be 30-70% of P_input for typical coil
-```
-
-**If P_spark > 90% of P_input:**
-- Secondary losses too low (unrealistic Q)
-- Check winding resistance, dielectric losses
-
-**If P_spark < 20% of P_input:**
-- Excessive secondary losses
-- Or spark model R too high (not optimized)
-
-### Total Resistance Range Check
-
-**Expected at 200 kHz for 1-3 m sparks:**
-```
-Burst/streamer-dominated: 50-300 kΩ
-QCW/leader-dominated: 5-50 kΩ
-Very low frequency (<100 kHz) or very long: 1-10 kΩ
-
-R_total = Σ R[i] should fall in expected range
-```
-
-**If outside range:**
-- Check frequency (R ∝ 1/f)
-- Check optimization convergence
-- Verify capacitance matrix extraction
-- Consider if mode is truly different (all-leader vs all-streamer)
-
-### Resistance Distribution Check
-
-**Physical expectation:**
-```
-R[1] < R[2] < R[3] < ... < R[n]
-
-Base should be lowest (hot, coupled)
-Tip should be highest (cool, weakly coupled)
-
-Monotonic increase expected
-```
-
-**If non-monotonic:**
-- Check capacitance matrix (may have errors)
-- Verify optimization didn't get stuck
-- Physical interpretation: local heating/cooling variation
-
-### Phase Angle Check
-
-**Total impedance phase:**
-```
-Calculate Z_total at topload port
-φ_Z should be -55° to -75° typical
-
-If φ_Z > -45°: Too resistive (check if topological constraint violated)
-If φ_Z < -85°: Too capacitive (R values too high, not optimized)
-```
-
-### Convergence Check
-
-**For distributed models with n=5, 10, 20:**
-```
-Run same problem with different n:
-- n=5 → Z_total, P_spark
-- n=10 → Z_total, P_spark
-- n=20 → Z_total, P_spark
-
-Should converge: changes <10% from n=10 to n=20
-
-If still changing >20%: need finer discretization
-```
-
----
-
-### WORKED EXAMPLE 4.7: Validation Exercise
-
-**Given simulation results:**
-```
-Coil: DRSSTC at 185 kHz
-P_primary_input = 150 kW
-P_spark = 105 kW (from distributed model n=10)
-Spark: 2.5 m
-
-Distributed R values [kΩ]:
-[18, 25, 35, 48, 65, 88, 120, 165, 230, 320]
-
-Z_total = 185 kΩ ∠-68°
-```
-
-**Validate:**
-
-**Check 1: Power balance**
-```
-P_spark / P_input = 105 / 150 = 0.70 = 70%
-
-Expected: 30-70% typical ✓
-Reasonable - some secondary losses, but spark dominates
-```
-
-**Check 2: Total resistance**
-```
-R_total = Σ R[i] = 18+25+35+48+65+88+120+165+230+320
- = 1114 kΩ
-
-At 185 kHz, expected: 50-300 kΩ for typical
-Actual: 1114 kΩ
-
-High, but this is 2.5 m spark (long)
-Factor of 3-4× over typical
-Could indicate:
-- Very streamer-dominated (burst mode?)
-- Or optimization not fully converged
-- Or long spark genuinely has higher R
-
-Flag for investigation, but not necessarily wrong ✓?
-```
-
-**Check 3: Resistance distribution**
-```
-R[1]=18 < R[2]=25 < R[3]=35 < ... < R[10]=320
-
-Monotonic increasing ✓
-Expected pattern (base lower, tip higher) ✓
-```
-
-**Check 4: Phase angle**
-```
-φ_Z = -68°
-
-Expected range: -55° to -75°
-Actual: -68°
-
-Right in the middle ✓
-Indicates reasonable capacitive loading
-```
-
-**Check 5: Compare to lumped model**
-```
-Lumped model (from earlier): R ≈ 600 kΩ at similar conditions
-
-Distributed: R_total = 1114 kΩ
-
-Distributed is higher (factor ~2)
-This can happen:
-- Distributed captures tip streamer high-R better
-- Lumped averages to middle value
-- For long sparks, distributed more accurate
-
-Consistent with expectations ✓
-```
-
-**Overall assessment:**
-- Most checks pass
-- Total R is high but potentially physical for long streamer spark
-- Recommend: compare to measurement if available
-- Model is usable for predictions
-
----
-
-### PRACTICE PROBLEMS 4.7
-
-**Problem 1:** Simulation shows P_spark = 180 kW but P_input = 150 kW. What's wrong?
-
-**Problem 2:** Distributed model gives R = [50, 45, 40, 35, 30] kΩ (decreasing from base to tip). Is this physical? What might be wrong?
-
-**Problem 3:** At 150 kHz, 1.8 m spark, you get R_total = 2 kΩ. Check against expected range. Is this reasonable?
-
----
-
-## Module 4.8: Complete Simulation Summary
-
-### Workflow Checklist
-
-**Phase 1: Geometry and FEMM**
-- [ ] Define spark length L_total
-- [ ] Choose n segments (typically 10)
-- [ ] Create FEMM geometry (axisymmetric)
-- [ ] Set up conductors (topload + n segments)
-- [ ] Mesh and solve electrostatic
-- [ ] Extract (n+1)×(n+1) capacitance matrix [C]
-- [ ] Validate: symmetry, positive definite, C_sh ≈ 2 pF/ft
-
-**Phase 2: Resistance Determination**
-- [ ] Choose method: iterative or simplified
-- [ ] If simplified: R[i] = 1/(ω × C_total[i])
-- [ ] If iterative: initialize R[i], run optimization loop
-- [ ] Apply position-dependent bounds R_min[i], R_max[i]
-- [ ] Check convergence (<1% change)
-- [ ] Validate: R distribution monotonic, total in expected range
-
-**Phase 3: SPICE Implementation**
-- [ ] Convert [C] matrix to SPICE-compatible form (partial or controlled sources)
-- [ ] Add resistance elements R[i]
-- [ ] Define topload voltage source (or integrate with full coil model)
-- [ ] Set up AC analysis at operating frequency
-
-**Phase 4: Analysis**
-- [ ] Run AC simulation
-- [ ] Extract V, I at each node
-- [ ] Calculate P[i] in each segment: P[i] = 0.5 × I[i]² × R[i]
-- [ ] Calculate total P_spark = Σ P[i]
-- [ ] Calculate Y_spark or Z_spark at topload port
-
-**Phase 5: Validation**
-- [ ] Power balance: P_spark reasonable fraction of P_input
-- [ ] Total R in expected range for frequency and length
-- [ ] Phase angle φ_Z in typical range
-- [ ] Resistance distribution physical (increasing base→tip)
-- [ ] Compare to lumped model (should be similar order of magnitude)
-- [ ] Compare to measurements if available
-
-**Phase 6: Iteration (if needed)**
-- [ ] If validation fails, identify issue
-- [ ] Adjust and re-run
-- [ ] Document assumptions and uncertainties
-
----
-
-## Module 4.9: Calibration and Measurement Integration
-
-### Calibrating ε (Energy Per Meter)
-
-**Procedure:**
-
-**Step 1: Controlled test**
-```
-Run coil with known drive conditions
-Measure final spark length L_measured
-```
-
-**Step 2: Simulation**
-```
-Simulate same conditions
-Calculate E_delivered = ∫ P_spark dt over growth time
-```
-
-**Step 3: Extract ε**
-```
-ε_calibrated = E_delivered / L_measured
-
-Example:
-E_delivered = 18 J (from simulation)
-L_measured = 1.5 m (from photograph/measurement)
-
-ε = 18 J / 1.5 m = 12 J/m
-```
-
-**Step 4: Build database**
-```
-Repeat for different operating modes:
-- QCW long ramp: ε_QCW
-- Burst mode: ε_burst
-- Intermediate: ε_hybrid
-
-Use appropriate ε for future predictions
-```
-
-### Calibrating E_propagation
-
-**Procedure:**
-
-**Step 1: Measure stall condition**
-```
-Ramp voltage slowly
-Observe maximum length L_max when growth stops
-Measure V_topload at stall
-```
-
-**Step 2: FEMM field analysis**
-```
-Set up geometry with spark length = L_max
-Apply V = V_topload
-Calculate E_tip at tip using FEMM
-```
-
-**Step 3: Extract threshold**
-```
-E_propagation ≈ E_tip at stall
-
-Typical: 0.4-1.0 MV/m
-Calibrate for your specific conditions (altitude, humidity, geometry)
-```
-
-### Using Measurements to Refine Model
-
-**Ringdown method (from Part 2):**
-```
-1. Measure f₀, Q₀ (unloaded)
-2. Measure f_L, Q_L (with spark)
-3. Extract Y_spark from frequency shift and Q change
-4. Compare to model prediction
-5. Adjust R values if significant discrepancy (>factor of 2)
-```
-
-**Direct impedance measurement:**
-```
-If you have:
-- Calibrated E-field probe (V_topload)
-- Calibrated current probe on spark return path (I_spark, not I_base!)
-
-Then:
-Z_measured = V_topload / I_spark
-
-Compare to model Z_spark
-Adjust R values to match
-```
-
-**Iterative refinement:**
-```
-1. Initial model from FEMM + simplified R
-2. Simulate → predict Z_spark, power
-3. Measure actual Z_spark, power
-4. Adjust R distribution (proportionally) to match measured total R
-5. Validate that distribution shape is still physical
-6. Use refined model for future predictions
-```
-
----
-
-### WORKED EXAMPLE 4.9: Calibrating ε
-
-**Measurement:**
-```
-QCW coil, 12 ms ramp
-Final spark length: L = 2.2 m
-```
-
-**Simulation:**
-```
-Full model with distributed spark
-Calculate power to spark over time:
-P_spark(t) varies from 20 kW to 80 kW during ramp
-
-Total energy:
-E_delivered = ∫₀^0.012 P_spark(t) dt
- = 26 J (numerical integration)
-```
-
-**Calibration:**
-```
-ε = E_delivered / L_measured
- = 26 J / 2.2 m
- = 11.8 J/m
-```
-
-**Interpretation:**
-```
-This is at low end of QCW range (5-15 J/m)
-Indicates efficient leader formation
-Consistent with long ramp time (12 ms)
-
-Use ε = 12 J/m for future predictions with this coil in QCW mode
-```
-
-**Validation:**
-```
-Predict different condition:
-New ramp: 8 ms, available energy: E = 30 J
-
-Expected length: L = E/ε = 30/12 = 2.5 m
-
-Run test, measure actual length, compare
-If within ±20%: calibration good
-If >30% error: investigate (different mode? voltage limited?)
-```
-
----
-
-### PRACTICE PROBLEMS 4.9
-
-**Problem 1:** Simulation shows E = 40 J delivered, measurement shows L = 2.8 m. Calculate ε. Is this more consistent with QCW or burst mode?
-
-**Problem 2:** A calibration at sea level gives E_propagation = 0.5 MV/m. At 2000 m altitude (air density ~80% of sea level), estimate new E_propagation.
-
----
-
-## Part 4 Conclusion: Practical Guidelines
-
-### Decision Tree: Which Model to Use?
-
-```
-START
- |
- └─ Spark length < 1 m?
- ├─ YES → Use LUMPED model
- | * Fast, accurate enough
- | * R = R_opt_power
- |
- └─ NO → Spark length < 3 m?
- ├─ YES → Choice:
- | * Quick answer: LUMPED
- | * Best accuracy: DISTRIBUTED (n=10)
- |
- └─ NO (>3 m) → Use DISTRIBUTED (n=15-20)
- * Essential for accuracy
- * Captures tip/base differences
-
-Research/validation? → Always use DISTRIBUTED
-```
-
-### Typical Simulation Times
-
-```
-Lumped model:
-- FEMM: 2 min (single geometry)
-- SPICE: <1 sec
-- Total: ~3 minutes
-
-Distributed (n=10), simplified R:
-- FEMM: 5 min (multi-body)
-- SPICE: 1 sec (one analysis)
-- Total: ~6 minutes
-
-Distributed (n=10), iterative R:
-- FEMM: 5 min
-- SPICE: 100 sec (100 iterations × 1 sec)
-- Total: ~7 minutes
-
-Distributed (n=20), iterative R:
-- FEMM: 10 min (larger matrix)
-- SPICE: 300 sec (more elements)
-- Total: ~15 minutes
-```
-
-### Accuracy Expectations
-
-```
-Lumped model:
-- Impedance: ±20%
-- Power: ±30%
-- Good enough for: matching studies, coil optimization
-
-Distributed (simplified R):
-- Impedance: ±15%
-- Power: ±25%
-- Current distribution: ±30%
-
-Distributed (iterative R):
-- Impedance: ±10%
-- Power: ±20%
-- Current distribution: ±20%
-- Best available without plasma modeling
-
-Measurement comparison:
-- ±20-50% agreement is GOOD (plasma variability)
-- ±factor of 2: acceptable (many unknowns)
-- Better than factor of 2: excellent!
-```
-
-### Final Recommendations
-
-**For hobbyist design:**
-- Use lumped model
-- Calibrate ε from one measurement
-- Predict new conditions
-
-**For research:**
-- Use distributed model (n=10-15)
-- Iterative optimization
-- Document all assumptions
-- Compare to measurements
-- Report uncertainties
-
-**For publications:**
-- Distributed model required
-- Validation against measurements
-- Sensitivity analysis
-- Clear methodology section
-
----
-
-## Final Comprehensive Problem
-
-**Design Challenge: Predict Performance of New Coil**
-
-**Given:**
-- DRSSTC, f = 195 kHz
-- Topload: 35 cm toroid (major diameter)
-- Target: 2 m spark, QCW mode (10 ms ramp)
-- Primary input: P_input = 120 kW
-- Thévenin: Z_th = 110 - j2300 Ω, V_th = 340 kV
-
-**Required:**
-
-**Part 1: Distributed Model Setup**
-- Choose n (justify)
-- Describe FEMM geometry
-- What validation checks after extracting [C]?
-
-**Part 2: Resistance Calculation**
-- Choose method (iterative or simplified, justify)
-- Estimate expected R_total range
-- What bounds for R[i]?
-
-**Part 3: Performance Prediction**
-- Calculate Z_spark
-- Find current and power
-- What % of theoretical max?
-
-**Part 4: Growth Analysis**
-- Assume ε = 12 J/m (from calibration)
-- Can 2 m be reached in 10 ms with available power?
-- Check voltage: κ = 3.2, E_prop = 0.7 MV/m
-- Is growth voltage-limited or power-limited?
-
-**Part 5: Validation Plan**
-- What measurements would you take?
-- How would you refine the model?
-- What accuracy do you expect?
-
-**This problem integrates all four parts of the course!**
-
----
-
-## Course Summary: Master Checklist
-
-### Part 1 Concepts
-- [ ] Peak vs RMS phasor convention
-- [ ] Complex impedance and admittance
-- [ ] Power formula: P = 0.5 × Re{V × I*}
-- [ ] C_mut and C_sh in spark circuit
-- [ ] Circuit topology: (R||C_mut) + C_sh
-- [ ] Phase angles and capacitive loading
-
-### Part 2 Concepts
-- [ ] Topological phase constraint φ_Z,min
-- [ ] R_opt_power maximizes power transfer
-- [ ] Hungry streamer self-optimization
-- [ ] Why V_top/I_base is wrong
-- [ ] Thévenin equivalent extraction and use
-- [ ] Q measurement and ringdown analysis
-
-### Part 3 Concepts
-- [ ] E_inception and E_propagation thresholds
-- [ ] Energy per meter ε by mode
-- [ ] Growth rate dL/dt = P/ε
-- [ ] Thermal time constants and persistence
-- [ ] Capacitive divider problem
-- [ ] FEMM electrostatic analysis
-- [ ] Maxwell capacitance matrix extraction
-- [ ] Lumped model construction
-
-### Part 4 Concepts
-- [ ] When distributed models needed
-- [ ] nth-order segmentation
-- [ ] Multi-body FEMM analysis
-- [ ] Capacitance matrix in SPICE (partial capacitance)
-- [ ] Iterative R optimization with damping
-- [ ] Simplified R = 1/(ωC_total) method
-- [ ] Validation checks (power balance, R range, distribution)
-- [ ] Calibration from measurements (ε, E_prop)
-
----
-
-## Resources for Continued Learning
-
-**Software:**
-- FEMM: www.femm.info (free)
-- LTSpice: www.analog.com/ltspice (free)
-- Python + NumPy/SciPy for automation
-
-**Tesla Coil Communities:**
-- 4hv.org forums (active community)
-- highvoltageforum.net
-- teslamap.com (coil database)
-
-**Further Reading:**
-- "The Spark Gap" magazine (archived)
-- Lightning physics textbooks (Uman, Rakov)
-- Plasma physics introductions (Chen)
-- High voltage engineering (Kuffel)
-
-**This framework:**
-- Original document for full mathematical details
-- Implement in stages (lumped → distributed)
-- Calibrate to YOUR coil
-- Share results with community!
-
----
-
-**END OF PART 4**
-
-**END OF COMPLETE LESSON PLAN**
-
----
-
-**Congratulations!** You now have a complete framework to:
-1. Understand Tesla coil spark physics
-2. Extract parameters from FEMM
-3. Build circuit models (lumped and distributed)
-4. Predict performance
-5. Validate against measurements
-6. Iterate and improve
-
-**Next steps:**
-- Work through practice problems
-- Build your first model
-- Compare to real coil
-- Refine and calibrate
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Appendices: Quick Reference Materials
-
----
-
-## Appendix A: Complete Variable Reference Table
-
-### Circuit Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **C_mut** | F (pF) | Mutual capacitance between topload and spark | 5-15 pF |
-| **C_sh** | F (pF) | Shunt capacitance spark-to-ground | 2 pF/foot × length |
-| **C_total** | F (pF) | Total capacitance: C_mut + C_sh | 10-30 pF |
-| **C_eq** | F (pF) | Equivalent loaded capacitance | Calculated from f shift |
-| **R** | Ω (kΩ) | Spark plasma resistance | 5-500 kΩ @ 200 kHz |
-| **R_opt_power** | Ω | Resistance for maximum power transfer | 1/(ω(C_mut+C_sh)) |
-| **R_opt_phase** | Ω | Resistance for minimum phase angle | 1/(ω√(C_mut(C_mut+C_sh))) |
-| **R_min** | Ω | Minimum physical resistance (hot leader) | 1-10 kΩ |
-| **R_max** | Ω | Maximum physical resistance (cold streamer) | 100 kΩ - 100 MΩ |
-| **G** | S (μS) | Conductance: 1/R | 1-100 μS typical |
-| **B₁** | S (μS) | Susceptance of C_mut: ωC_mut | Positive (capacitive) |
-| **B₂** | S (μS) | Susceptance of C_sh: ωC_sh | Positive (capacitive) |
-| **Y** | S (μS) | Complex admittance: G + jB | - |
-| **Z** | Ω (kΩ) | Complex impedance: R + jX | - |
-| **Z_th** | Ω | Thévenin output impedance | 100-200 Ω + j(-2000 to -3000 Ω) |
-| **V_th** | V (kV) | Thévenin open-circuit voltage | 200-500 kV |
-| **φ_Z** | ° or rad | Impedance phase angle | -55° to -75° typical |
-| **φ_Z,min** | ° or rad | Minimum achievable phase: -atan(2√(r(1+r))) | More negative than -45° usually |
-| **r** | - | Capacitance ratio: C_mut/C_sh | 0.5-2.0 typical |
-
-### Frequency and Quality Factor
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **f** | Hz (kHz) | Operating frequency | 100-400 kHz |
-| **f₀** | Hz | Unloaded resonant frequency | - |
-| **f_L** | Hz | Loaded resonant frequency (with spark) | Lower than f₀ |
-| **ω** | rad/s | Angular frequency: 2πf | 6.28×10⁵ - 2.5×10⁶ |
-| **Q₀** | - | Unloaded quality factor | 50-200 typical |
-| **Q_L** | - | Loaded quality factor (with spark) | 20-80 typical |
-| **τ** | s (ms) | Time constant for decay | τ = 2Q/ω |
-
-### Power and Energy
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **P** | W (kW) | Real (average) power | - |
-| **P_spark** | W (kW) | Power dissipated in spark | 10-200 kW |
-| **P_avg** | W (kW) | Average power over time | - |
-| **P_max** | W (kW) | Theoretical maximum (conjugate match) | Usually unachievable |
-| **E** | J | Energy | - |
-| **E_total** | J | Total energy to grow spark | ε × L |
-| **ε** (epsilon) | J/m | Energy per meter for growth | 5-15 (QCW), 30-100 (burst) |
-| **ε₀** | J/m | Initial energy per meter | Before thermal accumulation |
-
-### Electric Fields
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **E** | V/m (MV/m) | Electric field strength | - |
-| **E_tip** | V/m (MV/m) | Field at spark tip | κ × V_top/L |
-| **E_average** | V/m (MV/m) | Average field: V_top/L | - |
-| **E_inception** | V/m (MV/m) | Field for initial breakdown | 2-3 MV/m |
-| **E_propagation** | V/m (MV/m) | Field for sustained growth | 0.4-1.0 MV/m |
-| **κ** (kappa) | - | Tip enhancement factor | 2-5 typical |
-
-### Geometric Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **L** | m | Spark length | 0.3-6 m typical |
-| **L_target** | m | Target design length | - |
-| **L_segment** | m | Length of one segment (distributed model) | L_total/n |
-| **d** | m (mm) | Spark channel diameter | 0.1-5 mm (streamers-leaders) |
-| **d_nominal** | m (mm) | Assumed diameter for FEMM | 1 mm (burst), 3 mm (QCW) |
-| **n** | - | Number of segments (distributed model) | 5-20, typically 10 |
-| **i** | - | Segment index (1 to n) | 1=base, n=tip |
-
-### Thermal Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **T** | K | Temperature | 1000 K (streamer) - 20000 K (leader) |
-| **ΔT** | K | Temperature rise above ambient | - |
-| **τ_thermal** | s (ms) | Thermal diffusion time: d²/(4α) | 0.1 ms (thin) - 300 ms (thick) |
-| **τ_effective** | s (ms) | Observed persistence time | Longer than τ_thermal |
-| **α_thermal** | m²/s | Thermal diffusivity of air | ~2×10⁻⁵ m²/s |
-
-### Matrix and Optimization
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **[C]** | F (pF) | Maxwell capacitance matrix (n+1)×(n+1) | - |
-| **C[i,j]** | F (pF) | Matrix element i,j | Diagonal >0, off-diagonal <0 |
-| **R[i]** | Ω (kΩ) | Resistance of segment i | Increases from base to tip |
-| **α_damp** | - | Damping factor for iteration | 0.3-0.5 |
-| **position** | - | Normalized position: (i-1)/(n-1) | 0=base, 1=tip |
-
-### Measurement Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **V_top** | V (kV) | Voltage at topload (peak) | 200-600 kV |
-| **V_tip** | V (kV) | Voltage at spark tip | V_top × C_mut/(C_mut+C_sh) |
-| **I_spark** | A | Current through spark | 0.5-3 A |
-| **I_base** | A | Current at secondary base (WRONG for spark) | Includes displacement currents |
-| **A₁, A₂** | V, A | Consecutive peak amplitudes in ringdown | - |
-
----
-
-## Appendix B: Formula Quick Reference
-
-### Basic Circuit Analysis
-
-**Admittance of spark circuit:**
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where: G = 1/R
- B₁ = ωC_mut
- B₂ = ωC_sh
-```
-
-**Real and imaginary parts:**
-```
-Re{Y} = GB₂² / [G² + (B₁+B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
-```
-
-**Impedance phase:**
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Power calculation:**
-```
-P = 0.5 × Re{V × I*} (with peak phasors)
-P = 0.5 × |V|² × Re{Y}
-P = 0.5 × |I|² × Re{Z}
-P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
-```
-
-### Optimal Resistances
-
-**Maximum power transfer:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-
-Example: f=200 kHz, C_total=12 pF
-R_opt_power = 1/(2π×200×10³×12×10⁻¹²) ≈ 66 kΩ
-```
-
-**Minimum phase angle magnitude:**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Always: R_opt_power < R_opt_phase
-```
-
-**Minimum phase angle:**
-```
-φ_Z,min = -atan(2√[r(1+r)])
-
-where r = C_mut/C_sh
-
-Critical value: r = 0.207 gives φ_Z,min = -45°
-If r > 0.207: cannot achieve -45°
-```
-
-### Thévenin Equivalent
-
-**Measuring Z_th (drive off, test source on):**
-```
-Z_th = V_test / I_test = 1V / I_test
-
-Apply 1V AC at topload-to-ground
-Measure current I_test
-```
-
-**Measuring V_th (drive on, no load):**
-```
-V_th = V(topload) with spark removed
-```
-
-**Power to any load:**
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**Theoretical maximum (conjugate match):**
-```
-Z_load = Z_th* (complex conjugate)
-P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
-
-Usually unachievable due to topological constraints
-```
-
-### Ringdown Method
-
-**Quality factor from decay:**
-```
-Q = πf × Δt / ln(A₁/A₂)
-
-where Δt = time between peaks
- A₁, A₂ = consecutive peak amplitudes
-```
-
-**At loaded resonance:**
-```
-Q_L = ω_L C_eq R_p = R_p/(ω_L L)
-
-Therefore:
-R_p = Q_L/(ω_L C_eq) = Q_L ω_L L
-G_total = ω_L C_eq/Q_L = 1/(Q_L ω_L L)
-```
-
-**Capacitance from frequency shift:**
-```
-C_eq = C₀(f₀/f_L)²
-ΔC = C_eq - C₀
-```
-
-**Spark admittance approximation:**
-```
-Y_spark ≈ (G_total - G_0) + jω_L ΔC
-```
-
-### Spark Growth Physics
-
-**Growth rate equation:**
-```
-dL/dt = P_stream/ε (when E_tip > E_propagation)
-dL/dt = 0 (when E_tip ≤ E_propagation, stalled)
-```
-
-**Time to reach target length (constant power):**
-```
-T = ε × L_target / P_stream
-```
-
-**Total energy required:**
-```
-E_total = ε × L_target
-```
-
-**Energy per meter with thermal accumulation:**
-```
-ε(t) = ε₀ / (1 + α∫P dt)
-
-where α has units [1/J]
-```
-
-**Field thresholds:**
-```
-E_inception ≈ 2-3 MV/m (initial breakdown)
-E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
-E_tip = κ × E_average = κ × V_top/L
-```
-
-### Thermal Time Constants
-
-**Pure thermal diffusion:**
-```
-τ_thermal = d² / (4α)
-
-where α ≈ 2×10⁻⁵ m²/s for air
-
-Examples:
-d = 100 μm → τ ≈ 0.125 ms
-d = 5 mm → τ ≈ 312 ms
-```
-
-**Convection velocity (buoyancy):**
-```
-v ≈ √(g × d × ΔT/T_amb)
-
-where g = 9.8 m/s²
-```
-
-### Capacitive Divider
-
-**Open-circuit voltage division:**
-```
-V_tip = V_topload × C_mut/(C_mut + C_sh)
-
-As spark grows: C_sh increases → V_tip decreases
-```
-
-**With finite resistance (more complex):**
-```
-V_tip = V_topload × Z_mut/(Z_mut + Z_sh)
-
-where Z_mut = (1/jωC_mut) || R
- Z_sh = 1/(jωC_sh)
-```
-
-### FEMM Capacitance Extraction
-
-**For 2-body system (topload + spark):**
-```
-Maxwell matrix:
- [Top] [Spark]
-[Top] C₁₁ C₁₂
-[Spark] C₂₁ C₂₂
-
-Extraction:
-C_mut = |C₁₂| = |C₂₁| (absolute value)
-C_sh = C₂₂ - |C₁₂|
-
-Validation: C_sh ≈ 2 pF/foot × L_spark
-```
-
-### Distributed Model
-
-**Simplified resistance calculation:**
-```
-For each segment i:
-C_total[i] = Σⱼ |C[i,j]| (sum of absolute values)
-R[i] = 1/(ω × C_total[i])
-R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Position-dependent bounds:**
-```
-position = (i-1)/(n-1) (0 at base, 1 at tip)
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position²
-```
-
-**Iterative optimization (damped update):**
-```
-R_new[i] = α × R_optimal[i] + (1-α) × R_old[i]
-
-where α = 0.3-0.5 (damping factor)
-```
-
----
-
-## Appendix C: Physical Constants and Typical Values
-
-### Universal Constants
-
-| Constant | Symbol | Value | Units |
-|----------|--------|-------|-------|
-| Permittivity of free space | ε₀ | 8.854×10⁻¹² | F/m |
-| Pi | π | 3.14159... | - |
-| Gravitational acceleration | g | 9.81 | m/s² |
-| Electron charge | e | 1.602×10⁻¹⁹ | C |
-
-### Air Properties (Sea Level, 20°C)
-
-| Property | Symbol | Value | Units |
-|----------|--------|-------|-------|
-| Density | ρ_air | 1.2 | kg/m³ |
-| Thermal diffusivity | α | 2×10⁻⁵ | m²/s |
-| Thermal conductivity | k | 0.026 | W/(m·K) |
-| Specific heat | c_p | 1005 | J/(kg·K) |
-| Molecular density | n | 2.5×10²⁵ | molecules/m³ |
-| Ionization energy | E_ion | ~15 | eV/molecule |
-
-### Field Thresholds (Dry Air, Sea Level)
-
-| Parameter | Value | Units | Notes |
-|-----------|-------|-------|-------|
-| E_inception | 2-3 | MV/m | Initial breakdown, smooth electrode |
-| E_propagation | 0.4-1.0 | MV/m | Sustained leader growth |
-| Altitude correction | -20 to -30 | %/1000m | Lower air density → lower threshold |
-| Humidity effect | ±10 | % | Variable, depends on conditions |
-
-### Energy per Meter by Mode
-
-| Operating Mode | ε Range | Units | Characteristics |
-|----------------|---------|-------|-----------------|
-| QCW (5-20 ms ramp) | 5-15 | J/m | Efficient, leader-dominated |
-| Hybrid DRSSTC | 20-40 | J/m | Mixed streamers/leaders |
-| Burst mode (<1 ms) | 30-100+ | J/m | Inefficient, streamer-dominated |
-| Single-shot burst | 50-150 | J/m | Very inefficient, bright but short |
-
-### Typical Spark Resistance (@ 200 kHz)
-
-| Spark Type | Length | Total R | Notes |
-|------------|--------|---------|-------|
-| Short burst | 0.5-1 m | 100-300 kΩ | Streamer-dominated |
-| Medium burst | 1-2 m | 150-400 kΩ | Mixed |
-| Long burst | 2-3 m | 200-500 kΩ | Difficult, high R |
-| QCW (short) | 0.5-1 m | 20-80 kΩ | Leader-dominated |
-| QCW (medium) | 1-2 m | 30-120 kΩ | Efficient |
-| QCW (long) | 2-4 m | 40-200 kΩ | Best mode for length |
-
-### Frequency Dependence
-
-| Frequency | R_typical | C_sh (per meter) | Notes |
-|-----------|-----------|------------------|-------|
-| 100 kHz | 5-50 kΩ | ~6 pF | Low frequency, low R |
-| 150 kHz | 10-100 kΩ | ~6 pF | Typical small coils |
-| 200 kHz | 20-200 kΩ | ~6 pF | Common frequency |
-| 300 kHz | 30-300 kΩ | ~6 pF | Higher frequency |
-| 400 kHz | 40-400 kΩ | ~6 pF | Very high, smaller coils |
-
-**Note:** R ∝ 1/f approximately, C_sh relatively constant
-
-### Thermal Time Constants
-
-| Channel Type | Diameter | τ_thermal | Persistence | Notes |
-|--------------|----------|-----------|-------------|-------|
-| Thin streamer | 50-100 μm | 0.05-0.2 ms | 1-5 ms | Convection extends |
-| Medium streamer | 200-500 μm | 0.2-1.5 ms | 2-10 ms | Mixed |
-| Thin leader | 1-2 mm | 6-25 ms | 50-500 ms | Buoyancy significant |
-| Thick leader | 5-10 mm | 150-600 ms | Seconds | Persistent column |
-
-### Tesla Coil Typical Parameters
-
-| Parameter | Small Coil | Medium Coil | Large Coil | Units |
-|-----------|------------|-------------|------------|-------|
-| Frequency | 300-500 | 150-250 | 80-150 | kHz |
-| Topload C₀ | 15-25 | 25-40 | 40-80 | pF |
-| Secondary Q₀ | 100-200 | 80-150 | 50-120 | - |
-| Spark length | 0.3-1.0 | 1.0-2.5 | 2.0-4.0 | m |
-| Power | 1-10 | 10-100 | 50-300 | kW |
-| Z_th magnitude | 1-3 | 0.5-2 | 0.3-1 | kΩ |
-| Z_th phase | -85 to -88 | -86 to -89 | -87 to -89 | degrees |
-
----
-
-## Appendix D: SPICE Component Reference
-
-### Basic Elements
-
-**Resistor:**
-```
-R node1 node2
-Example: R1 topload spark 50k
- R2 n1 n2 {R_value} ; parameterized
-```
-
-**Capacitor:**
-```
-C node1 node2
-Example: C_mut topload spark 10p
- C_sh spark 0 6p
-```
-
-**Voltage source:**
-```
-V node+ node-
-Example: V1 topload 0 AC 1V
- V2 drive 0 AC 100k ; 100 kV
-```
-
-**Current source:**
-```
-I node+ node-
-Example: I1 topload 0 AC 1m
-```
-
-### Parameterized Components
-
-**Define parameters:**
-```
-.param freq=200k
-.param omega={2*pi*freq}
-.param C_mut=10p
-.param C_sh=6p
-.param R={1/(omega*(C_mut+C_sh))}
-```
-
-**Use in components:**
-```
-C1 n1 n2 {C_mut}
-R1 n2 n3 {R}
-```
-
-### Controlled Sources (for capacitance matrix)
-
-**Voltage-controlled current source:**
-```
-G node+ node- ctrl+ ctrl-
-Example: G1 n1 0 n2 0 {j*omega*C[1,2]}
-```
-
-**Behavioral source:**
-```
-B node+ node- V={expression}
-Example: B1 n1 0 V={j*omega*C_mut*V(n2)}
-```
-
-### Analysis Commands
-
-**AC analysis:**
-```
-.ac lin
-Example: .ac lin 1 200k 200k ; single frequency
- .ac lin 100 180k 220k ; sweep 100 points
-```
-
-**Transient analysis:**
-```
-.tran
-Example: .tran 0.1u 10m ; 0.1 μs steps, 10 ms total
-```
-
-**Print/plot:**
-```
-.print ac v(topload) i(V1) vp(topload) ip(V1)
-.plot ac vdb(topload) ; dB magnitude
-```
-
-### Mutual Inductance (for transformer)
-
-**Inductors with coupling:**
-```
-L1 n1 n2
-L2 n3 n4
-K1 L1 L2
-
-Example:
-Lpri drive n1 100u
-Lsec n2 base 10m
-K_couple Lpri Lsec 0.15 ; k=0.15
-```
-
-### Subcircuits (for modular models)
-
-**Define subcircuit:**
-```
-.subckt spark_model topload ground
-+ params: C_mut=10p C_sh=6p R=50k
-C1 topload n1 {C_mut}
-R1 n1 n2 {R}
-C2 n2 ground {C_sh}
-.ends
-```
-
-**Use subcircuit:**
-```
-X1 topload 0 spark_model params: C_mut=12p C_sh=8p R=60k
-```
-
-### Example: Complete Lumped Model
-
-```
-* Tesla Coil Spark Lumped Model
-* Frequency: 200 kHz
-
-.param freq=200k
-.param omega={2*pi*freq}
-
-* Spark parameters from FEMM
-.param C_mut=10p
-.param C_sh=6p
-.param R_opt={1/(omega*(C_mut+C_sh))}
-
-* Clip to physical bounds
-.param R_min=5k
-.param R_max=500k
-.param R={min(max(R_opt,R_min),R_max)}
-
-* Circuit
-V_topload topload 0 AC 1V
-C_mut topload n1 {C_mut}
-R_spark n1 n2 {R}
-C_sh n2 0 {C_sh}
-
-* Analysis
-.ac lin 1 {freq} {freq}
-.print ac v(topload) i(V_topload) vp(topload) ip(V_topload)
-
-* Calculate admittance in post-processing:
-* Y = I/V, extract real and imaginary parts
-* Power = 0.5 * |V|^2 * Re{Y}
-
-.end
-```
-
----
-
-## Appendix E: FEMM Quick Start Guide
-
-### Installation
-
-1. **Download:** Visit www.femm.info
-2. **Install:** Run installer (Windows), or use Wine (Linux/Mac)
-3. **Launch:** Open FEMM 4.2 (main application)
-
-### Basic Interface
-
-**Main window sections:**
-- **Toolbar:** Problem type, zoom, view controls
-- **Drawing area:** Geometry creation
-- **Status bar:** Coordinates, snap mode
-- **Menus:** File, Edit, View, Problem, Mesh, Analysis
-
-### Creating Electrostatic Problem
-
-**Step 1: New document**
-```
-File → New
-Select: Electrostatics Problem
-Frequency: 0 (electrostatic)
-Length units: Centimeters (or your preference)
-Problem type: Axisymmetric
-Precision: 1e-8
-```
-
-**Step 2: Define materials**
-```
-Problem → Materials Library
-Select: Air (ε_r = 1.0)
-Add to model
-
-If needed, define custom materials:
-Problem → Materials → Add Property
-Name: Custom
-Permittivity: (relative value)
-```
-
-**Step 3: Draw geometry**
-```
-Use toolbar buttons:
-- Draw nodes (points): Click to place
-- Draw lines: Select two nodes
-- Draw arcs: Select two nodes, define angle
-- Draw circles: Center + radius
-
-For axisymmetric:
-- Draw in r-z plane (r ≥ 0)
-- r = 0 is axis of symmetry
-```
-
-### Tesla Coil Spark Geometry Example
-
-**Toroid (topload):**
-```
-1. Draw circle (minor diameter) at z=0, r=15 cm
-2. Use circular rotation: Operations → Mirror/Rotate
-3. Create toroidal surface
-```
-
-**Spark (cylinder):**
-```
-1. Draw vertical line from topload base to tip
- Example: r=0.1 cm, z=-5 to z=-105 cm (1 m spark)
-2. This represents axis of cylinder
-3. For multiple segments: Draw each as separate line
-```
-
-**Ground plane:**
-```
-1. Draw large circle or line at z = (below spark)
-2. Large enough to approximate "infinity"
-```
-
-**Outer boundary:**
-```
-1. Draw rectangle enclosing entire problem
-2. Far from coil (5-10× max dimension)
-```
-
-### Assigning Properties
-
-**Step 4: Define conductors**
-```
-Problem → Conductors
-Add conductor groups:
-- Conductor 1: Name "Topload", Voltage = 1V
-- Conductor 2: Name "Spark1", Floating
-- Conductor 3: Name "Spark2", Floating
-...
-- Conductor n+1: Name "Ground", Voltage = 0V
-```
-
-**Step 5: Assign to geometry**
-```
-Select line/arc/circle
-Right-click → Set Boundary
-Choose conductor group
-
-All segments of spark: Assign to separate conductors
-Topload surface: Assign to topload conductor
-Ground: Assign to ground conductor
-```
-
-**Step 6: Assign materials**
-```
-Select region (click inside enclosed area)
-Right-click → Set Block Property
-Material: Air
-Mesh size: Auto or specify
-```
-
-**Step 7: Boundary conditions**
-```
-Problem → Boundaries
-- Outer boundary: V=0 (Dirichlet)
-- r=0: Axisymmetric boundary
-- Others: Default (Neumann, E field normal)
-```
-
-### Meshing and Solving
-
-**Step 8: Create mesh**
-```
-Mesh → Create Mesh
-Wait for triangulation (seconds to minutes)
-Check mesh quality: Zoom in near conductors
-```
-
-**Step 9: Solve**
-```
-Analysis → Run
-Wait for solution (seconds to minutes)
-Look for convergence message
-```
-
-### Post-Processing
-
-**Step 10: View results**
-```
-File → Open Postprocessor
-(or automatically opens after solve)
-
-View field:
-- View → Contour Plot → V (voltage)
-- View → Vector Plot → E (field)
-- View → Density Plot → Field magnitude
-```
-
-**Step 11: Extract capacitance matrix**
-```
-Circuit Properties window (usually visible)
-If not: View → Circuit Properties
-
-Shows capacitance matrix [C]
-Copy values to spreadsheet/text file
-
-Format:
- [1] [2] [3] ...
-[1] C₁₁ C₁₂ C₁₃
-[2] C₂₁ C₂₂ C₂₃
-...
-```
-
-**Step 12: Calculate electric field at point**
-```
-Click on specific point
-View → Point Values
-Shows: V, E_r, E_z, |E| at that location
-
-For tip field: Click at spark tip
-```
-
-### Tips and Tricks
-
-**Efficient meshing:**
-```
-- Finer mesh near conductors (small triangle size)
-- Coarse mesh far away (large triangles)
-- Specify manually: Set Block Property → Mesh size
-```
-
-**Symmetry exploitation:**
-```
-- Use axisymmetric for cylindrical symmetry (2D → 3D)
-- Use planar for 2D problems
-- Reduces element count by 10-100×
-```
-
-**Convergence issues:**
-```
-- Increase precision (Problem → Precision: 1e-10)
-- Refine mesh near conductors
-- Enlarge outer boundary
-- Check for geometry errors (gaps, overlaps)
-```
-
-**Large matrix extraction:**
-```
-For n=20 segments → 21×21 matrix
-Circuit Properties window may be small
-Resize window or copy values programmatically
-Consider exporting to CSV
-```
-
-### Automation with Lua Scripting
-
-**FEMM supports Lua scripts for automation:**
-```lua
--- Example: Create spark segment
-newdocument(0) -- Electrostatics
-for i=1,10 do
- z_start = -i*10
- z_end = -(i+1)*10
- addnode(0.1, z_start)
- addnode(0.1, z_end)
- addsegment(0.1, z_start, 0.1, z_end)
- selectsegment(0.1, (z_start+z_end)/2)
- setconductor("Spark"..i, 0) -- Floating
-end
-```
-
-**Useful for:**
-- Parametric sweeps (vary length, diameter)
-- Batch processing multiple geometries
-- Extracting results programmatically
-
----
-
-## Appendix F: Troubleshooting Guide
-
-### Problem: Negative Phase Angle Too Large (φ_Z < -80°)
-
-**Symptoms:**
-- Impedance phase more negative than -80°
-- Very capacitive
-- Low power transfer
-
-**Possible causes:**
-1. R too high (not optimized)
-2. Capacitances overestimated
-3. Frequency too high for given R
-
-**Solutions:**
-- Run iterative R optimization
-- Verify FEMM capacitance extraction
-- Check R bounds (R_max too high?)
-- Recalculate R_opt_power
-
----
-
-### Problem: Power Balance Doesn't Close
-
-**Symptoms:**
-- P_spark > P_input (violates conservation)
-- Or P_spark << P_input (most energy missing)
-
-**Possible causes:**
-1. Incorrect power calculation (missing 0.5 factor?)
-2. Using RMS instead of peak values inconsistently
-3. Missing loss terms
-4. Measuring wrong current (I_base instead of I_spark)
-
-**Solutions:**
-- Verify formula: P = 0.5 × Re{V × I*} with peak
-- Check all quantities are peak (or all RMS, consistently)
-- Account for secondary losses separately
-- Measure I_spark on return path, not I_base
-
----
-
-### Problem: FEMM Capacitance Matrix Not Symmetric
-
-**Symptoms:**
-- C[i,j] ≠ C[j,i]
-- Non-physical
-
-**Possible causes:**
-1. Numerical error (insufficient precision)
-2. Mesh quality poor
-3. Geometry errors (overlaps, gaps)
-
-**Solutions:**
-- Increase precision: Problem → Precision: 1e-10
-- Refine mesh near conductors
-- Check geometry for errors (zoom in, look for gaps)
-- Ensure proper boundary conditions
-
----
-
-### Problem: Distributed Model Doesn't Converge
-
-**Symptoms:**
-- Iterative optimization oscillates
-- R values jumping around
-- No stable solution after many iterations
-
-**Possible causes:**
-1. Damping factor α too high
-2. Weakly coupled segments (tip)
-3. R bounds too restrictive
-4. Power curve very flat
-
-**Solutions:**
-- Reduce α to 0.2-0.3 (more damping)
-- Accept tip segments not converging (physical)
-- Widen R_max bounds for tip segments
-- Use simplified method if iterative fails
-
----
-
-### Problem: Simulation Predicts Too Short Spark
-
-**Symptoms:**
-- Predicted length << measured
-- Model underestimates performance
-
-**Possible causes:**
-1. ε too high (overestimating energy needed)
-2. E_propagation set too high
-3. Power transfer underestimated (R not optimized)
-4. Capacitances wrong (affects R_opt)
-
-**Solutions:**
-- Calibrate ε from measurements
-- Check E_propagation threshold
-- Verify R optimization ran correctly
-- Re-check FEMM extraction
-
----
-
-### Problem: Simulation Predicts Too Long Spark
-
-**Symptoms:**
-- Predicted length >> measured
-- Model overestimates performance
-
-**Possible causes:**
-1. ε too low (underestimating energy needed)
-2. E_propagation set too low
-3. Not accounting for capacitive divider voltage drop
-4. Using burst-mode ε for QCW (or vice versa)
-
-**Solutions:**
-- Increase ε (burst needs higher value)
-- Verify field threshold appropriate for conditions
-- Check V_tip calculation (capacitive division)
-- Use correct ε for operating mode
-
----
-
-### Problem: R_total Outside Expected Range
-
-**Symptoms:**
-- Total resistance 10× too high or too low
-- Doesn't match measurements or expectations
-
-**Possible causes:**
-1. Wrong frequency
-2. Capacitance extraction error
-3. Optimization failure
-4. Physical bounds too restrictive
-
-**Solutions:**
-- Verify frequency used in R calculation
-- Re-check capacitance matrix from FEMM
-- Try simplified R method as sanity check
-- Compare segment-by-segment to expected profile
-
----
-
-### Problem: SPICE Simulation Gives Nonsense Results
-
-**Symptoms:**
-- Negative resistance calculated
-- Infinite impedance
-- Convergence errors
-
-**Possible causes:**
-1. Capacitance matrix implementation wrong
-2. Negative capacitor values
-3. Ground reference missing
-4. Parameter syntax error
-
-**Solutions:**
-- Use partial capacitance transformation (all positive)
-- Verify every capacitor value >0
-- Ensure at least one node grounded
-- Check .param syntax (use {expression} for calculations)
-
----
-
-### Problem: Measured vs Simulated Impedance Differs by Factor >2
-
-**Symptoms:**
-- Model predicts Z = 200 kΩ
-- Measurement shows Z = 450 kΩ (or 90 kΩ)
-
-**Possible causes:**
-1. Measurement method wrong (V_top/I_base)
-2. Spark branching in measurement (not modeled)
-3. Operating mode different (burst vs QCW)
-4. Frequency shift not accounted for
-
-**Solutions:**
-- Use correct measurement port (topload-to-ground)
-- Model cannot capture branching (expected discrepancy)
-- Ensure ε appropriate for actual mode
-- Remeasure at loaded resonance frequency
-
----
-
-### Problem: Growth Stalls Before Target Length
-
-**Symptoms:**
-- Spark stops growing
-- More power doesn't help
-
-**Possible causes:**
-1. Voltage-limited (E_tip < E_propagation)
-2. Capacitive divider drops V_tip too much
-3. E_propagation higher than assumed
-4. Topload too small for target length
-
-**Solutions:**
-- Check E_tip calculation at stall length
-- Consider ramping voltage higher
-- Increase topload capacitance (less voltage division)
-- Reduce target length (be realistic)
-
----
-
-### Problem: QCW Gives Same Length as Burst (Expected Longer)
-
-**Symptoms:**
-- QCW and burst same performance
-- Not seeing efficiency advantage
-
-**Possible causes:**
-1. Using same ε for both (should be different)
-2. QCW ramp too short (not exploiting thermal memory)
-3. Insufficient power for QCW
-4. Leader formation not occurring
-
-**Solutions:**
-- Use ε_QCW = 8-15 J/m, ε_burst = 40-80 J/m
-- Lengthen ramp time (10-20 ms)
-- Increase average power
-- Check current sufficient for leader (>0.5 A)
-
----
-
-### Quick Diagnostic Flowchart
-
-```
-Problem occurs
- |
- ├─ Unreasonable value (negative, infinite, 1000× off)
- | → Check units, formula, syntax
- | → Verify all inputs are correct quantities
- |
- ├─ Non-convergence (oscillation, no stable solution)
- | → Reduce damping factor α
- | → Check if problem has solution (bounds?)
- | → Try simpler model first
- |
- ├─ Mismatch with measurement (factor 2-5)
- | → Verify measurement method
- | → Check operating mode matches
- | → Calibrate ε, E_propagation from data
- |
- └─ Physical impossibility (violates conservation, etc.)
- → Review assumptions
- → Check for double-counting or missing terms
- → Verify reference frames consistent
-```
-
----
-
-## Appendix G: Worked Solutions to Comprehensive Problems
-
-### Part 2 Comprehensive Design Exercise (Solution)
-
-**Given:**
-- f = 190 kHz
-- C_topload = 30 pF
-- Target spark: 3 feet (estimate C_sh)
-- C_mut = 9 pF (from FEMM)
-- Z_th = 105 - j2100 Ω, V_th = 320 kV
-
----
-
-**Task 1: Calculate capacitance ratio and phase constraint**
-
-```
-C_sh = 2 pF/ft × 3 ft = 6 pF
-
-r = C_mut/C_sh = 9/6 = 1.5
-
-φ_Z,min = -atan(2√[r(1+r)])
- = -atan(2√[1.5×2.5])
- = -atan(2√3.75)
- = -atan(2×1.936)
- = -atan(3.872)
- = -75.5°
-
-Cannot achieve -45° (r = 1.5 > 0.207) ✓
-```
-
----
-
-**Task 2: Determine optimal resistances**
-
-```
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-R_opt_power = 1/(ω(C_mut + C_sh))
- = 1/(1.194×10⁶ × 15×10⁻¹²)
- = 1/(17.91×10⁻⁶)
- = 55.8 kΩ
-
-R_opt_phase = 1/(ω√(C_mut(C_mut+C_sh)))
- = 1/(1.194×10⁶ × √(9×10⁻¹² × 15×10⁻¹²))
- = 1/(1.194×10⁶ × 11.62×10⁻¹²)
- = 1/(13.87×10⁻⁶)
- = 72.1 kΩ
-
-R_opt_power < R_opt_phase ✓ (55.8 < 72.1)
-
-At R_opt_power, expect φ_Z ≈ -76° (slightly more capacitive than minimum)
-```
-
----
-
-**Task 3: Build lumped spark model**
-
-```
-Circuit:
- Topload ---[C_mut=9pF]---+--- [C_sh=6pF]---GND
- |
- [R=55.8kΩ]
-
-Calculate Y_spark:
-G = 1/R = 1/55800 = 17.92 μS
-B₁ = ωC_mut = 1.194×10⁶ × 9×10⁻¹² = 10.75 μS
-B₂ = ωC_sh = 1.194×10⁶ × 6×10⁻¹² = 7.16 μS
-
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 17.92 × 51.27 / [321.1 + 319.7]
- = 918.8 / 640.8
- = 1.434 μS
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
- = 7.16 × [321.1 + 191.7] / 640.8
- = 7.16 × 512.8 / 640.8
- = 5.73 μS
-
-Y_spark = 1.434 + j5.73 μS
-```
-
----
-
-**Task 4: Predict performance with Thévenin**
-
-```
-Convert Y_spark to Z_spark:
-|Y_spark| = √(1.434² + 5.73²) = 5.91 μS
-|Z_spark| = 1/5.91×10⁻⁶ = 169 kΩ
-
-φ_Y = atan(5.73/1.434) = 76.0°
-φ_Z = -76.0°
-
-Z_spark = 169 kΩ ∠-76.0°
- = 169 × cos(-76°) + j × 169 × sin(-76°)
- = 41 - j164 kΩ
-
-Total impedance:
-Z_total = Z_th + Z_spark
- = (105 - j2100) + (41000 - j164000)
- = (41105 - j166100) Ω
- = 41.1 - j166.1 kΩ
-
-|Z_total| = √(41.1² + 166.1²) = 171 kΩ
-
-Current:
-I = V_th/Z_total = 320 kV / 171 kΩ = 1.87 A
-
-Power to spark:
-P_spark = 0.5 × I² × Re{Z_spark}
- = 0.5 × 1.87² × 41000
- = 0.5 × 3.50 × 41000
- = 71.7 kW
-```
-
----
-
-**Task 5: Compare to theoretical maximum**
-
-```
-For conjugate match: Z_load = Z_th* = 105 + j2100 Ω
-
-P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
- = 0.5 × (320×10³)² / (4 × 105)
- = 0.5 × 1.024×10¹¹ / 420
- = 122 MW
-
-Actual percentage:
-71.7 kW / 122000 kW = 0.0588%
-
-Spark extracts only 0.06% of theoretical maximum!
-
-Why such huge difference?
-- Conjugate match needs Z_load = 105 + j2100 Ω (very low R, inductive)
-- Actual spark: Z_spark = 41000 - j164000 Ω (high R, capacitive)
-- Topological constraints prevent achieving conjugate match
-- This is NORMAL for Tesla coils
-- The 71.7 kW is still significant useful power
-```
-
----
-
-### Part 4 Final Comprehensive Problem (Partial Solution)
-
-**Given:**
-- f = 195 kHz, 2 m target, QCW 10 ms
-- Topload 35 cm, P_input = 120 kW
-- Z_th = 110 - j2300 Ω, V_th = 340 kV
-
----
-
-**Part 1: Distributed model setup**
-
-```
-Choose n = 10 (good balance accuracy/speed)
-
-FEMM geometry (axisymmetric r-z):
-- Toroid: major R=17.5 cm, minor r=5 cm, center z=0
-- Segments: 10 cylinders, each 20 cm long
- Segment 1: r=0.15 cm, z=-5 to -25 cm
- Segment 2: z=-25 to -45 cm
- ...
- Segment 10: z=-185 to -205 cm
-- Ground plane: z=-220 cm, r=0 to 400 cm
-- Outer boundary: r=400 cm, z=±300 cm
-
-Validation checks after [C] extraction:
-1. Symmetry: C[i,j] = C[j,i] within 0.1%
-2. All diagonal positive
-3. All off-diagonal negative
-4. C_sh_total ≈ 2 pF/ft × 6.56 ft ≈ 13 pF
- (Sum across segments)
-```
-
----
-
-**Part 2: Resistance calculation (simplified method)**
-
-```
-ω = 2π × 195×10³ = 1.225×10⁶ rad/s
-
-Assume FEMM gives C_total[i] = [14, 11, 9, 7.5, 6.5, 5.5, 4.5, 3.5, 2.8, 2.0] pF
-
-R[i] = 1/(ω × C_total[i]):
-
-R[1] = 1/(1.225×10⁶ × 14×10⁻¹²) = 58.3 kΩ
-R[2] = 1/(1.225×10⁶ × 11×10⁻¹²) = 74.6 kΩ
-R[3] = 92.1 kΩ
-R[4] = 110 kΩ
-R[5] = 127 kΩ
-R[6] = 150 kΩ
-R[7] = 184 kΩ
-R[8] = 236 kΩ
-R[9] = 294 kΩ
-R[10] = 408 kΩ
-
-R_total = 1734 kΩ
-
-Expected range at 195 kHz for 2m QCW: 30-120 kΩ
-Actual: 1734 kΩ (high, but long spark distributed can be higher)
-
-Bounds check: All R[i] between 5 kΩ and 500 kΩ ✓
-Distribution: Monotonically increasing ✓
-```
-
----
-
-**Part 3: Performance prediction (abbreviated)**
-
-```
-Build SPICE with [C] matrix and R[i] values
-Run AC analysis at 195 kHz
-
-Expected results (estimated):
-Z_spark ≈ 600 kΩ ∠-72°
-I ≈ 0.5 A
-P_spark ≈ 40 kW
-
-Percentage of theoretical max: <0.1% (typical)
-```
-
----
-
-**Part 4: Growth analysis**
-
-```
-Power available: 40 kW (from part 3)
-ε = 12 J/m (QCW calibrated)
-Target: L = 2 m, Time: T = 10 ms
-
-Energy needed: E = ε × L = 12 × 2 = 24 J
-
-Power needed: P = E/T = 24/0.010 = 2.4 kW
-
-Available: 40 kW >> 2.4 kW needed ✓
-Power is MORE than sufficient
-
-Voltage check:
-V_top = 340 kV (from V_th, approximately)
-κ = 3.2, E_prop = 0.7 MV/m
-E_tip = κ × V_top/L = 3.2 × 340 kV / 2 m
- = 3.2 × 170 kV/m = 544 kV/m = 0.544 MV/m
-
-E_tip = 0.544 MV/m < E_prop = 0.7 MV/m ✗
-
-Growth is VOLTAGE-LIMITED!
-Cannot reach 2 m with 340 kV
-
-Required voltage:
-V_required = E_prop × L / κ = 0.7×10⁶ × 2 / 3.2
- = 437.5 kV
-
-Need to ramp to 438 kV to sustain growth to 2 m
-With 340 kV, maximum length ≈ 340/438 × 2 = 1.55 m
-
-Conclusion: Voltage limited, not power limited
-Need higher voltage ramp or accept shorter spark
-```
-
----
-
-**Part 5: Validation plan**
-
-```
-Measurements to take:
-1. Ringdown: f₀, Q₀ (unloaded); f_L, Q_L (loaded)
- → Extract Y_spark, compare to model
-2. High-speed video: Growth rate dL/dt
- → Validate power/ε relationship
-3. V_top with E-field probe (calibrated)
- → Check voltage predictions
-4. Final spark length with ruler/laser
- → Validate growth model
-
-Refinement process:
-1. If measured length > predicted:
- - Reduce ε (more efficient than assumed)
- - Check E_prop (may be lower)
-2. If measured length < predicted:
- - Increase ε
- - Check for branching (wastes energy)
-3. Adjust R distribution if impedance mismatch
-
-Expected accuracy:
-- Length: ±30% (good agreement)
-- Power: ±40% (acceptable)
-- Impedance: ±25% (reasonable)
-
-Better than factor of 2 on all parameters = success!
-```
-
----
-
-## Appendix H: Further Resources
-
-### Online Communities
-
-**4hv.org Forums**
-- Active Tesla coil community
-- Design sharing and troubleshooting
-- DRSSTC, QCW, SGTC sections
-- Measurement techniques
-
-**High Voltage Forum (highvoltageforum.net)**
-- International community
-- Advanced projects
-- Safety discussions
-
-### Software Tools
-
-**FEMM (femm.info)**
-- Free 2D electromagnetic FEA
-- This framework's primary tool
-- Active development and support
-
-**LTSpice (analog.com/ltspice)**
-- Free SPICE simulator
-- Excellent for circuit analysis
-- Large component library
-
-**Python Scientific Stack**
-- NumPy: Matrix operations
-- SciPy: Optimization algorithms
-- Matplotlib: Plotting
-- Free and powerful
-
-### Books and Papers
-
-**Lightning Physics:**
-- Uman, M.A. "The Lightning Discharge" (comprehensive)
-- Rakov & Uman "Lightning: Physics and Effects" (modern)
-
-**Plasma Physics:**
-- Chen, F.F. "Introduction to Plasma Physics" (accessible)
-- Raizer, Y.P. "Gas Discharge Physics" (detailed)
-
-**High Voltage Engineering:**
-- Kuffel, Zaengl, Kuffel "High Voltage Engineering Fundamentals"
-- Wadhwa, C.L. "High Voltage Engineering"
-
-**Tesla Coil Specific:**
-- "The Spark Gap" magazine archives (historical)
-- Tesla coil design guides (various online)
-
-### Academic Resources
-
-**IEEE Xplore**
-- Search: "spark discharge modeling"
-- "Tesla transformer"
-- "resonant transformer"
-
-**arXiv.org**
-- Physics preprints
-- Some Tesla coil research
-
-### Safety Resources
-
-**ALWAYS prioritize safety:**
-- High voltage safety guidelines
-- Grounding and bonding practices
-- First aid for electrical injuries
-- Equipment safety ratings
-
-**Key principle:** If you're not sure, DON'T DO IT.
-
----
-
-## Closing Remarks
-
-**You now have:**
-- Complete theoretical framework
-- Practical implementation guide
-- Worked examples throughout
-- Troubleshooting resources
-- Validation methodologies
-
-**Next steps:**
-1. Start with lumped model (simple coil)
-2. Calibrate ε from one measurement
-3. Predict new operating point
-4. Progress to distributed model
-5. Share results with community
-
-**Remember:**
-- All models are approximations
-- Plasma physics has uncertainties
-- ±20-50% agreement is GOOD
-- Document your assumptions
-- Compare to measurements
-- Iterate and improve
-
-**Most importantly:**
-- Stay safe
-- Have fun
-- Learn continuously
-- Contribute back to community
-
-**This framework is a starting point, not the final word. As you gain experience, you'll develop intuition and may improve upon these methods. That's the goal!**
-
----
-
-**END OF APPENDICES**
-
-**END OF COMPLETE TESLA COIL SPARK MODELING LESSON PLAN**
-
----
-
-**Total lesson plan:**
-- Part 1: ~18,000 tokens (Foundation)
-- Part 2: ~17,500 tokens (Optimization)
-- Part 3: ~17,800 tokens (Growth Physics & FEMM)
-- Part 4: ~17,900 tokens (Distributed Models)
-- Appendices: ~14,500 tokens (Reference)
-- **Grand Total: ~85,700 tokens**
-
-**Ready for teaching Tesla coil spark modeling from beginner to advanced!**
-
-
diff --git a/spark-lessons/CIRCUIT-SPECIFICATIONS.md b/spark-lessons/CIRCUIT-SPECIFICATIONS.md
deleted file mode 100644
index 93452ff..0000000
--- a/spark-lessons/CIRCUIT-SPECIFICATIONS.md
+++ /dev/null
@@ -1,470 +0,0 @@
-# Circuit Diagram Specifications
-
-This document provides **exact specifications** for creating 7 circuit diagrams that require manual attention for professional quality.
-
-**Recommended tools:** LTspice, CircuitLab, Inkscape, KiCad schematic editor, or professional drawing software.
-
-**Format:** PNG, 150 DPI minimum, white background
-
----
-
-## Circuit 1: Geometry to Circuit Translation
-
-**Filename:** `lessons/01-fundamentals/assets/geometry-to-circuit.png`
-**Size:** 1000 x 600 px
-**Referenced in:** fund-02 (Basic Circuit Model)
-
-### Description
-Side-by-side diagram showing physical geometry on left, equivalent circuit on right.
-
-### Left Side: 3D Visualization (Conceptual)
-```
-[Sketch/photo showing:]
-- Toroidal topload (or spherical)
-- Cylindrical spark channel extending downward
-- Ground plane at bottom
-- Arrows/labels indicating:
- * C_mut (coupling between topload and spark)
- * C_sh (spark to ground)
-```
-
-**Note:** Can use simplified 2D side-view sketch if 3D is difficult.
-
-### Right Side: Circuit Schematic
-
-**Topology (CRITICAL - verify this is correct):**
-
-```
-Topload node
- |
- +----[C_mut]----+
- | |
- +----[R]--------+
- |
- (Spark tip node)
- |
- [C_sh]
- |
- GND
-```
-
-**Component values to show:**
-- R: Variable (or "R_spark")
-- C_mut: "~8 pF" (typical)
-- C_sh: "~6 pF" (typical)
-
-**Layout guidelines:**
-- Vertical orientation
-- Clear node labels: "Topload", "Spark Tip", "GND"
-- R and C_mut in parallel (side-by-side, same start/end nodes)
-- C_sh in series below the parallel combination
-
-**Alternative if parallel is hard:** Show as impedance block "Z_mut = R || C_mut"
-
----
-
-## Circuit 2: Current Paths Diagram
-
-**Filename:** `lessons/01-fundamentals/assets/current-paths-diagram.png`
-**Size:** 1000 x 1200 px (vertical)
-**Referenced in:** fund-07 (Measurement Port)
-
-### Description
-Complete Tesla coil schematic showing **all** current return paths.
-
-### Schematic Components
-
-**Primary circuit (left side):**
-```
-[AC Source] -→ [IGBT/Switch] -→ [C_pri] -→ [L_pri] -→ GND
-```
-
-**Secondary circuit (right side, magnetically coupled):**
-```
-L_sec (coil symbol, coupled to L_pri via k = 0.1-0.2)
- |
- +-- [C_topload] --|
- | |
- +-- [Spark] |
- | |
- +-- [C_stray] ----+
- |
- GND
-```
-
-**Current paths to label (USE DIFFERENT COLORS):**
-1. **I_spark** (RED): Through spark resistance
-2. **I_displacement** (BLUE): Through C_topload to ground
-3. **I_coupling** (GREEN): Primary-to-secondary capacitive coupling
-4. **I_secondary** (PURPLE): Distributed capacitance along secondary
-5. **I_base** (BLACK, THICK): Total current at secondary base
-
-**Key annotation:**
-```
-I_base = I_spark + I_displacement + I_coupling + I_secondary + ...
-```
-
-**Mark measurement points:**
-- Correct: "Measure here" at topload-to-ground (V_top / I_spark)
-- Incorrect: "NOT here" with X at base (V_top / I_base)
-
-### Layout Guidelines
-- Primary on left, secondary on right
-- Clear coupling indicator (dashed lines or k = 0.1-0.2)
-- Use arrows for current directions
-- Color code or use different line styles for each current path
-- Legend showing which color = which current
-
----
-
-## Circuit 3: Thévenin Equivalent Circuit
-
-**Filename:** `lessons/02-optimization/assets/thevenin-equivalent-circuit.png`
-**Size:** 800 x 600 px
-**Referenced in:** opt-04 (Thévenin Calculations)
-
-### Description
-Simple Thévenin equivalent driving a spark load.
-
-### Schematic
-
-```
- +-------[R_th]-----[jX_th]------+
- | |
-[V_th source] [Z_spark load]
- | |
- +--------------------------------+
-```
-
-**More detailed Z_spark:**
-```
-Z_spark can be shown as:
- [R_spark] in series with [jX_spark]
- OR
- [(R || C_mut) in series with C_sh]
-```
-
-**Component labels:**
-- V_th: "350 kV" (typical value)
-- R_th: "114 Ω" (typical)
-- X_th: "-j2424 Ω" (typical, capacitive)
-- Z_spark: "Variable"
-
-**Annotations:**
-- "Thévenin Equivalent" label on left side
-- "Spark Load" label on right side
-- Formula below: **P = 0.5|V_th|² Re{Z_spark} / |Z_th + Z_spark|²**
-
-### Layout Guidelines
-- Horizontal orientation, left to right
-- V_th source on left
-- R_th and X_th clearly in series
-- Load impedance on right
-- Clean, minimal style
-
----
-
-## Circuit 4: Capacitive Divider Circuit
-
-**Filename:** `lessons/03-spark-physics/assets/capacitive-divider-circuit.png`
-**Size:** 600 x 800 px (vertical)
-**Referenced in:** phys-07 (Capacitive Divider)
-
-### Description
-Shows voltage division across C_mut and C_sh.
-
-### Schematic
-
-```
-V_topload (source)
- |
- +----[C_mut]----+
- | |
- +----[R]--------+
- |
- V_tip (measurement point) ← mark this clearly
- |
- [C_sh]
- |
- GND
-```
-
-**Component labels:**
-- V_topload: "Input"
-- C_mut: "~10 pF"
-- C_sh: "~6.6 L (pF)" where L is in meters
-- R: "R_spark"
-- V_tip: Mark with voltmeter symbol or arrow
-
-**Key formula (below circuit):**
-```
-V_tip = V_topload × [C_mut / (C_mut + C_sh)]
-
-C_sh grows with spark length: ~6.6 pF/m
-```
-
-### Layout Guidelines
-- Vertical orientation
-- Show V_tip measurement clearly (voltmeter symbol or highlighted node)
-- Annotate that C_sh increases with length
-- Clean parallel R||C_mut representation
-
----
-
-## Circuit 5: Lumped Model Schematic
-
-**Filename:** `lessons/04-advanced-modeling/assets/lumped-model-schematic.png`
-**Size:** 800 x 600 px
-**Referenced in:** model-01 (Lumped Model)
-
-### Description
-Clean, professional lumped spark model circuit.
-
-### Schematic (Same topology as Circuit 1, but cleaner)
-
-```
-Port (Topload connection)
- |
- +----[R]--------+
- | |
- +----[C_mut]----+
- |
- (Spark tip - internal node)
- |
- [C_sh]
- |
- GND
-```
-
-**Component values:**
-- R: "50 kΩ (typical)"
-- C_mut: "8 pF (typical)"
-- C_sh: "6 pF (typical)"
-
-**Annotations:**
-- "Port" or "Topload Connection" at top
-- "Internal Node" at spark tip
-- Box or note: "Typical values at 200 kHz for 3-foot spark"
-
-### Layout Guidelines
-- Very clean, professional appearance
-- Grid-aligned components
-- Perfect parallel alignment for R || C_mut
-- Clear port indication (terminal symbols)
-- Minimal, uncluttered
-
----
-
-## Circuit 6: Distributed Model Structure
-
-**Filename:** `lessons/04-advanced-modeling/assets/distributed-model-structure.png`
-**Size:** 1200 x 600 px (horizontal)
-**Referenced in:** model-03 (Distributed Model)
-
-### Description
-Shows n-segment distributed model with proper transmission-line style layout.
-
-### Schematic
-
-**Horizontal cascade layout (recommended):**
-
-```
-Topload --[C_01]-- Node1 --[C_12]-- Node2 -- ... --[C_n-1,n]-- Node_n
- | | |
- [R_1] [R_2] [R_n]
- | | |
- [C_1,gnd] [C_2,gnd] [C_n,gnd]
- | | |
- GND GND GND
-```
-
-**Alternative vertical cascade** (if horizontal too wide):
-```
-Topload
- |
-[C_01]
- |
-Node 1 --[R_1]--
- | |
-[C_1,gnd] (parallel)
- |
-[C_12]
- |
-Node 2 --[R_2]--
- | |
-[C_2,gnd] (parallel)
- |
- ...
-```
-
-**Component labeling:**
-- Show first 2 segments explicitly
-- Use "..." for middle segments
-- Show last segment (segment n)
-- Label: "n = 5 to 20 segments (typically n = 10)"
-
-**Capacitance matrix note:**
-- Annotation: "(n+1) × (n+1) capacitance matrix"
-- "Extracted from FEMM electrostatic analysis"
-
-### Layout Guidelines
-- Clear repeating pattern
-- Ellipsis (...) to indicate continuation
-- Symmetric, professional appearance
-- Not too cluttered
-
----
-
-## Circuit 7: Tesla Coil System Overview
-
-**Filename:** `assets/shared/tesla-coil-system-overview.png`
-**Size:** 1400 x 1000 px
-**Referenced in:** Multiple lessons
-
-### Description
-Complete DRSSTC system diagram showing all major components.
-
-### Schematic Components
-
-**Primary tank circuit:**
-```
-[DC Bus] → [H-Bridge / IGBT switches] → [C_pri (MMC)] → [L_pri] → GND
- ↑
- [Gate Driver]
- ↑
- [Feedback/Control]
-```
-
-**Secondary resonator:**
-```
-L_sec (large coil symbol, coupled to L_pri via k)
- |
-[C_topload]
- |
-[Spark gap or streamer symbol]
- |
-[Strike point / GND]
-```
-
-**Annotations:**
-- Coupling coefficient: "k = 0.1 to 0.2"
-- Primary frequency: "f_pri = f_resonant"
-- Secondary resonance: "f_sec = 1/(2π√(L_sec × C_top))"
-- Power flow arrows
-- "DRSSTC" or "Double-Resonant Solid State Tesla Coil" title
-
-**Components to show:**
-- DC power supply
-- Full bridge (4 IGBTs/MOSFETs) or half bridge
-- MMC (multiple capacitors in series-parallel)
-- Primary coil (few turns, heavy wire)
-- Secondary coil (many turns, fine wire)
-- Topload (toroid or sphere symbol)
-- Spark/streamer
-- Feedback path (CT or antenna back to controller)
-- Ground connections
-
-### Layout Guidelines
-- Primary on left or bottom
-- Secondary on right or top
-- Clear separation of power vs signal paths
-- Coupling indicated (dashed lines, double-headed arrow, or k annotation)
-- Professional, complete system view
-- Include legend if needed
-
----
-
-## General Guidelines for All Circuits
-
-### Style
-- **Clean, professional appearance**
-- Grid-aligned components
-- Consistent component symbols (IEEE or European standard)
-- Clear, readable labels (minimum 10pt font)
-- No overlapping text or components
-- White background
-
-### Components Symbols
-- Resistor: Standard zigzag (IEEE) or rectangle (IEC)
-- Capacitor: Two parallel lines
-- Inductor: Coil/loops
-- Ground: Standard ground symbol
-- AC source: Sine wave in circle
-- Voltage source: Circle with +/- or V label
-
-### Colors (if used)
-- Use sparingly, only for clarity
-- Current paths: different colors
-- Otherwise: black on white for print compatibility
-
-### Verification
-**CRITICAL:** Before finalizing any circuit:
-1. Verify topology matches spark-physics.txt equations
-2. Check that parallel vs series connections are correct
-3. Ensure component values are realistic (refer to physical-bounds.md)
-4. Review against worked examples for consistency
-
----
-
-## Priority Order
-
-**High Priority (needed for core lessons):**
-1. Circuit 5: Lumped Model Schematic
-2. Circuit 4: Capacitive Divider
-3. Circuit 3: Thévenin Equivalent
-
-**Medium Priority:**
-4. Circuit 1: Geometry to Circuit
-5. Circuit 6: Distributed Model
-
-**Low Priority (nice-to-have):**
-6. Circuit 2: Current Paths (complex, can use text description initially)
-7. Circuit 7: System Overview (general reference, not lesson-critical)
-
----
-
-## Tools Recommendations
-
-**Easy (recommended for quick creation):**
-- **CircuitLab** (web-based, clean output)
-- **LTspice** (free, professional, can export schematics)
-- **Falstad Circuit Simulator** (web-based, can screenshot)
-
-**Professional (for publication quality):**
-- **KiCad Schematic Editor** (free, excellent output)
-- **Inkscape** (manual drawing with circuit symbols)
-- **Adobe Illustrator / Affinity Designer** (professional vector graphics)
-
-**Advanced (if familiar with LaTeX):**
-- **CircuiTikZ** + LaTeX (publication-quality output)
-
----
-
-## Validation Checklist
-
-Before considering a circuit "done":
-
-- [ ] Topology verified against spark-physics.txt
-- [ ] Component values realistic and labeled
-- [ ] No overlapping elements
-- [ ] Grid-aligned, professional appearance
-- [ ] Clear node labels where needed
-- [ ] Formula or key annotation included
-- [ ] 150 DPI or vector format (scalable)
-- [ ] White background, high contrast
-- [ ] Filename matches specification
-- [ ] Placed in correct assets directory
-
----
-
-## Notes
-
-- These specifications are based on analysis of spark-physics.txt
-- Some topologies (especially parallel R||C_mut) are tricky - verify carefully
-- When in doubt, consult reference physics document
-- Can simplify complex parallel combinations as impedance blocks (Z = R||C) if clearer
-- Professional quality > programmatic generation
-
-**Created:** 2025-10-10
-**Status:** Awaiting manual creation
-**Current:** 0/7 circuits completed
diff --git a/spark-lessons/PyQt_PROGRESS.md b/spark-lessons/PyQt_PROGRESS.md
deleted file mode 100644
index 772bcd5..0000000
--- a/spark-lessons/PyQt_PROGRESS.md
+++ /dev/null
@@ -1,282 +0,0 @@
-# PyQt5 Application Development Progress
-
-**Project:** Tesla Coil Spark Physics Course - Interactive Desktop Application
-**Started:** 2025-10-10
-**Current Status:** Phase 2 - Main Window Complete ✅
-
----
-
-## Phase 1: Core Setup & Infrastructure (COMPLETED)
-
-### ✅ Completed Files
-
-**1. Environment & Launch**
-- ✅ `run.bat` - Launch script with virtual environment management
-- ✅ `requirements.txt` - PyQt5 and all dependencies
-
-**2. Database**
-- ✅ `resources/database/schema.sql` - Complete SQLite schema (8 tables)
-- ✅ `app/database.py` - Database manager with convenience methods
-
-**3. Configuration**
-- ✅ `app/config.py` - All paths, constants, colors, settings
-
-**4. Course Model**
-- ✅ `app/models/course_model.py` - Complete course structure loader
- - Course, Part, Section, Lesson, LearningPath classes
- - Fast lesson lookup by ID
- - Navigation (next/prev lesson)
- - Search by title/tag
- - Learning path filtering
-
-**5. Application Entry**
-- ✅ `app/main.py` - Basic application launcher
-- ✅ `app/__init__.py` - Package initialization
-- ✅ `app/models/__init__.py` - Models package
-
-### Database Schema
-
-**Tables Created:**
-1. **users** - User profiles and preferences
-2. **lesson_progress** - Lesson completion tracking
-3. **exercise_attempts** - All exercise attempts
-4. **exercise_completion** - Best scores per exercise
-5. **study_sessions** - Daily session tracking
-6. **achievements** - Badge system
-7. **bookmarks** - Saved lessons/notes
-8. **learning_path_progress** - Path-specific progress
-
-### Course Model Features
-
-**Loaded from course.json:**
-- 4 Parts with 30 Lessons
-- 18 Exercises (525 points)
-- 4 Learning Paths
-- Reference materials
-- Worked examples
-- Tags and metadata
-
-**Navigation Methods:**
-- `get_lesson(id)` - Fast O(1) lookup
-- `get_next_lesson(id)` - Sequential navigation
-- `get_prev_lesson(id)` - Sequential navigation
-- `get_lesson_by_index(i)` - Access by position (0-29)
-- `search_lessons(query)` - Search by title
-- `get_lessons_for_path(path_id)` - Filter by learning path
-- `get_lessons_by_tag(tag)` - Filter by tag
-
-### Testing the Setup
-
-**To test current progress:**
-```batch
-cd C:\git\spark-lesson
-run.bat
-```
-
-**Expected Behavior:**
-1. Creates virtual environment (first run)
-2. Installs PyQt5 and dependencies
-3. Connects to SQLite database (~/.tesla_spark_course/progress.db)
-4. Loads course.json (30 lessons, 4 parts)
-5. Validates lesson files exist
-6. Shows success dialog with course info
-
-**Current Output:**
-```
-Tesla Coil Spark Physics Course v1.0.0
-[*] Initializing database...
-[OK] Database ready: C:\Users\...\progress.db
-[*] Loading course structure...
-[OK] Course loaded: Tesla Coil Spark Physics: Complete Course
-[*] Validating lesson files...
-[OK] All lesson files found
-[*] Application setup complete
-```
-
----
-
-## Phase 2: Main Window & UI (COMPLETED)
-
-### ✅ Completed Components
-
-**Priority 1: Main Window Layout**
-- ✅ `app/views/main_window.py` - QMainWindow with 3-panel QSplitter
-- ✅ `app/views/navigation_panel.py` - Left sidebar (QTreeWidget)
-- ✅ `app/views/content_viewer.py` - Center (QWebEngineView)
-- ✅ `app/views/progress_panel.py` - Right sidebar (QScrollArea)
-- ✅ `app/views/__init__.py` - Views package
-
-**Priority 2: Navigation Tree** ✅
-- ✅ Tree structure showing 4 parts, 30 lessons
-- ✅ Status icons (✓ ⊙ ○ 🔒)
-- ✅ Learning path selector dropdown
-- ✅ Search functionality
-- ✅ Double-click to open lessons
-- ✅ Continue Learning button
-
-**Priority 3: Content Viewer** ✅
-- ✅ Markdown rendering (python-markdown + pymdownx)
-- ✅ MathJax equation rendering (CDN)
-- ✅ Image loading from assets/
-- ✅ Custom tag parsing ({exercise:id}, {image:file})
-- ✅ Styled HTML output with syntax highlighting
-- ⏳ Auto-scroll restoration (placeholder)
-
-**Priority 4: Progress Panel** ✅
-- ✅ Overall progress bar
-- ✅ Part-by-part progress (4 parts)
-- ✅ Current lesson info
-- ✅ Quick stats (points, time, streak)
-- ✅ Level system display
-- ✅ Exercise completion tracking
-
----
-
-## Architecture Overview
-
-```
-spark-lessons/
-├── run.bat ✅ DONE
-├── requirements.txt ✅ DONE
-├── app/
-│ ├── __init__.py ✅ DONE
-│ ├── main.py ✅ DONE
-│ ├── config.py ✅ DONE
-│ ├── database.py ✅ DONE
-│ ├── models/
-│ │ ├── __init__.py ✅ DONE
-│ │ ├── course_model.py ✅ DONE
-│ │ ├── progress_model.py 🔄 TODO (optional)
-│ │ └── user_model.py 🔄 TODO (optional)
-│ ├── views/ ✅ DONE (all)
-│ │ ├── __init__.py ✅ DONE
-│ │ ├── main_window.py ✅ DONE
-│ │ ├── navigation_panel.py ✅ DONE
-│ │ ├── content_viewer.py ✅ DONE
-│ │ └── progress_panel.py ✅ DONE
-│ ├── controllers/ 🔄 TODO (optional)
-│ │ ├── navigation_controller.py
-│ │ └── progress_controller.py
-│ └── utils/ 🔄 TODO (optional)
-│ ├── markdown_renderer.py
-│ └── icon_provider.py
-└── resources/
- ├── database/
- │ └── schema.sql ✅ DONE
- ├── styles/ 🔄 TODO
- │ └── main.qss
- └── icons/ 🔄 TODO
- └── status/
-```
-
----
-
-## Technical Stack
-
-**Core:**
-- Python 3.8+
-- PyQt5 5.15.0+
-- SQLite3
-
-**Content Rendering:**
-- python-markdown 3.5.0+
-- pymdown-extensions 10.5.0+ (for equations, syntax highlighting)
-- PyQt5-WebEngine (for rendering HTML/MathJax)
-
-**Data:**
-- PyYAML 6.0.1+ (for exercises)
-- JSON (for course structure)
-
----
-
-## Phase 3: Enhancements & Polish (NEXT)
-
-1. **Exercise System** (4-6 hours)
- - Create exercise YAML files
- - Exercise widget components
- - Answer validation
- - Hints system
- - Score tracking
-
-2. **Keyboard Navigation** (2-3 hours)
- - Next/prev lesson shortcuts
- - Search hotkey
- - Quick navigation
- - Lesson completion shortcut
-
-3. **Additional Features** (3-4 hours)
- - Bookmarking system
- - Notes editor
- - Export progress report
- - Print lesson content
-
-4. **Polish & UX** (2-3 hours)
- - Smooth scrolling
- - Loading indicators
- - Error handling improvements
- - Tooltips and help text
-
-**Estimated Time:** 11-16 hours for Phase 3
-
----
-
-## Known Issues / Notes
-
-1. **Lesson File Paths**: The course_model currently constructs paths by string manipulation. Works for current structure but may need refinement.
-
-2. **Exercise Files**: Exercise YAML files don't exist yet in the exercises/ directory. Need to create them or handle gracefully.
-
-3. **Images**: 22 images generated, 15 placeholders exist. Circuit diagrams (7) need manual creation.
-
-4. **MathJax CDN**: Currently points to CDN. For offline use, may want to bundle MathJax locally.
-
-5. **Single User**: Database designed for single-user desktop app. Multi-user would need authentication layer.
-
----
-
-## Success Metrics for Phase 2
-
-- ✅ Main window opens without errors
-- ✅ Navigation tree shows all 30 lessons with proper structure
-- ✅ Click lesson → content loads and displays
-- ✅ Markdown renders correctly with MathJax
-- ⏳ Images display from assets/ (when files exist)
-- ✅ Progress panel shows basic stats
-- ✅ Learning path filter works
-- ✅ Search functionality works
-- ✅ Progress tracking in database
-- ✅ Auto-save every 10 seconds
-- ✅ Menu bar with File/View/Help
-- ✅ 3-panel splitter layout
-
-**Phase 2 Complete!** All core UI components implemented and functional.
-
----
-
-**Last Updated:** 2025-10-10
-**Status:** Phase 2 complete - Full application UI working!
-
-## Testing the Application
-
-**To run the application:**
-```batch
-cd C:\git\spark-lesson\spark-lessons
-run.bat
-```
-
-**Current Functionality:**
-1. ✅ Browse all 30 lessons in tree structure
-2. ✅ Double-click lessons to view content
-3. ✅ Markdown content with equations renders properly
-4. ✅ Progress tracking automatically saves
-5. ✅ Filter by learning path
-6. ✅ Search lessons by title
-7. ✅ View overall and per-part progress
-8. ✅ Points and level system
-9. ✅ Study statistics (time, streak, exercises)
-
-**Known Limitations:**
-- Exercise widgets not yet interactive (placeholders only)
-- Scroll position restoration not implemented
-- Some lesson images need to be created
-- No keyboard shortcuts yet
diff --git a/spark-lessons/README.md b/spark-lessons/README.md
deleted file mode 100644
index 77f827f..0000000
--- a/spark-lessons/README.md
+++ /dev/null
@@ -1,465 +0,0 @@
-# Tesla Coil Spark Physics: Interactive Course
-
-Complete educational course teaching the physics, mathematics, and simulation techniques for understanding and modeling Tesla coil sparks. From basic circuit theory to advanced distributed modeling with FEMM.
-
-**Version:** 1.0.0
-**Created:** 2025-10-10
-**Format:** Structured markdown lessons with YAML metadata
-
----
-
-## 📚 Course Overview
-
-### What You'll Learn
-
-This course provides comprehensive coverage of:
-- Circuit fundamentals and admittance analysis
-- Topological phase constraints and optimization
-- Thévenin equivalent analysis and power calculations
-- Spark growth physics and energy requirements
-- Thermal dynamics and streamer-to-leader transitions
-- FEMM-based capacitance extraction
-- Lumped and distributed spark modeling
-- Resistance optimization algorithms
-
-### Prerequisites
-
-**Required:**
-- Basic AC circuit analysis (impedance, phasors)
-- Complex number arithmetic
-- Basic calculus (derivatives, integrals)
-- Familiarity with SPICE circuit simulation
-
-**Recommended:**
-- Electromagnetic field theory basics
-- Experience with FEMM or similar FEA software
-- Tesla coil operating experience
-
-### Course Statistics
-
-- **30 lessons** across 4 parts
-- **18 exercises** (525 total points)
-- **~14 hours** estimated completion time
-- **5 comprehensive worked examples**
-- **3 reference documents** (equations, bounds, glossary)
-- **45+ images** needed (specifications provided)
-
----
-
-## 📂 Directory Structure
-
-```
-spark-lessons/
-├── course.json # Course structure and navigation
-├── lessons/ # All lesson content
-│ ├── 01-fundamentals/ # Part 1: Circuit Fundamentals (8 lessons)
-│ ├── 02-optimization/ # Part 2: Optimization & Simulation (7 lessons)
-│ ├── 03-spark-physics/ # Part 3: Spark Growth Physics (9 lessons)
-│ └── 04-advanced-modeling/ # Part 4: Advanced Modeling (6 lessons)
-├── exercises/ # Practice problems in YAML format
-│ ├── 01-fundamentals/ # 10 exercises
-│ ├── 02-optimization/ # 3 exercises
-│ ├── 03-spark-physics/ # 4 exercises
-│ └── 04-advanced-modeling/ # 1 exercise
-├── worked-examples/ # Complete worked examples
-│ ├── calculating-ropt.md
-│ ├── thevenin-extraction.md
-│ ├── spark-growth-timeline.md
-│ ├── femm-lumped-extraction.md
-│ └── distributed-model-complete.md
-├── reference/ # Quick reference materials
-│ ├── equation-sheet.md # All key formulas
-│ ├── physical-bounds.md # Validation ranges
-│ └── glossary.yaml # 64 technical terms
-├── assets/ # Images and media
-│ ├── shared/ # Shared images
-│ └── IMAGE-REQUIREMENTS.md # Specifications for 45+ images
-└── _originals/ # Backup of source files
- ├── spark-lesson.txt
- └── spark-physics.txt
-```
-
----
-
-## 🎓 Course Structure
-
-### Part 1: Circuit Fundamentals (200 min)
-**Lessons 01-08** | Beginner to Intermediate
-
-Learn the foundational circuit theory for spark modeling:
-- AC circuit review and complex analysis
-- Basic spark circuit model (C_mut, C_sh)
-- Admittance analysis of parallel networks
-- Phase angles and topological constraints
-- Why -45° is often mathematically impossible
-- Correct measurement port determination
-
-**Key Outcomes:** Understand spark impedance, phase constraints, and measurement techniques.
-
----
-
-### Part 2: Optimization & Simulation (280 min)
-**Lessons 01-07** | Intermediate to Advanced
-
-Master power optimization and simulation methods:
-- R_opt_power vs R_opt_phase (two critical resistances)
-- The "hungry streamer" self-optimization principle
-- Thévenin equivalent extraction and analysis
-- Power calculations for any load impedance
-- **Frequency tracking and loaded poles** (critical!)
-- DRSSTC operating modes comparison
-
-**Key Outcomes:** Perform Thévenin analysis, optimize power transfer, understand frequency tracking importance.
-
----
-
-### Part 3: Spark Growth Physics (260 min)
-**Lessons 01-09** | Intermediate to Advanced
-
-Understand the physics of spark formation and growth:
-- Electric field thresholds (E_inception, E_propagation)
-- Voltage-limited vs power-limited operation
-- Energy per meter (ε) concept and calibration
-- Thermal time constants and channel persistence
-- Streamers vs leaders (transition mechanisms)
-- Capacitive divider problem
-- Freau's empirical scaling relationships
-
-**Key Outcomes:** Model spark growth, estimate energy requirements, understand operating mode differences.
-
----
-
-### Part 4: Advanced Modeling (285 min)
-**Lessons 01-06** | Advanced
-
-Build sophisticated spark models using FEMM:
-- Lumped model theory and workflow
-- FEMM electrostatic extraction for lumped models
-- Distributed nth-order model theory
-- FEMM extraction for distributed models (capacitance matrices)
-- Resistance optimization (iterative and circuit-determined methods)
-- Complete modeling project with validation
-
-**Key Outcomes:** Extract capacitance matrices from FEMM, build lumped and distributed models, optimize resistance distribution.
-
----
-
-## 🎯 Learning Paths
-
-### Beginner Path (~8 hours)
-Focus on fundamentals and basic simulation:
-- Part 1: All lessons (fund-01 through fund-08)
-- Part 2: Lessons 01, 03, 04 (skip hungry streamer details)
-- Part 3: Lessons 01-03, 08 (basic physics and scaling)
-- Part 4: Skip (or just lesson 01 for overview)
-
-### Complete Course (~14 hours)
-Full curriculum for comprehensive understanding:
-- All 30 lessons in sequence
-- All 18 exercises
-- All 5 worked examples
-
-### Simulation Focus (~10 hours)
-For those primarily interested in modeling:
-- Part 1: Lessons 01-03, 05, 08
-- Part 2: All lessons (especially 06!)
-- Part 3: Lessons 01-04
-- Part 4: All lessons
-
-### Physics Focus (~9 hours)
-For those primarily interested in spark physics:
-- Part 1: Lessons 01-03 (circuit basics only)
-- Part 2: Lessons 01-02 (optimization principles)
-- Part 3: All lessons (complete physics coverage)
-
----
-
-## 📖 Lesson Format
-
-Each lesson file includes:
-
-```markdown
----
-id: fund-01 # Unique identifier
-title: "Lesson Title"
-section: "Fundamentals"
-difficulty: "beginner" # beginner | intermediate | advanced
-estimated_time: 20 # minutes
-prerequisites: [] # List of required prior lessons
-objectives: # Learning goals
- - Objective 1
- - Objective 2
-tags: ["circuit-theory", ...] # Topic tags
----
-
-# Lesson Title
-
-## Introduction
-[Lesson content...]
-
-## Key Takeaways
-- Bullet point 1
-- Bullet point 2
-
-## Practice
-{exercise:fund-ex-01}
-
----
-**Next Lesson:** [Next Title](next-file.md)
-```
-
----
-
-## 📝 Exercise Format
-
-Practice problems are stored as YAML files:
-
-```yaml
-id: fund-ex-01
-type: calculation # calculation | conceptual | design | multi-part
-difficulty: easy # easy | medium | hard
-points: 10
-related_lesson: fund-02
-question: |
- [Full question text]
-
-hints:
- - "Hint 1"
- - "Hint 2"
-
-solution:
- steps:
- - "Step 1 description"
- - "Step 2 description"
- answer: "66.3"
- unit: "kΩ"
- tolerance: 2.0 # percentage
-
-explanation: |
- [Why this matters]
-
-related_concepts: ["concept1", "concept2"]
-```
-
----
-
-## 🔧 Using This Course
-
-### For Self-Study
-
-1. Start with `course.json` to see overall structure
-2. Follow your chosen learning path (see above)
-3. Read lessons in order (prerequisites specified in frontmatter)
-4. Complete exercises to reinforce learning
-5. Refer to worked examples when stuck
-6. Use reference materials (equation sheet, glossary) as needed
-
-### For Interactive App Development
-
-This course is **designed for PyQt application** development:
-
-1. **Parse `course.json`** for navigation structure
-2. **Render markdown lessons** with proper equation support (MathJax)
-3. **Load exercise YAML** for interactive practice
-4. **Track progress** using lesson IDs
-5. **Implement custom tags:**
- - `{exercise:ex-id}` → Load and display exercise
- - `{image:filename}` → Display image from assets/
- - `{interactive:type}` → Launch interactive element
-
-### For PDF Generation
-
-Compile to PDF using Pandoc:
-
-```bash
-# All lessons
-pandoc lessons/**/*.md -o tesla-coil-spark-course.pdf \
- --toc --number-sections --pdf-engine=xelatex
-
-# Single part
-pandoc lessons/01-fundamentals/*.md -o part1-fundamentals.pdf \
- --toc --pdf-engine=xelatex
-```
-
----
-
-## 📊 Reference Materials
-
-### Equation Sheet
-`reference/equation-sheet.md`
-
-45+ key formulas organized by category:
-- Circuit analysis (Y, Z, φ)
-- Optimization (R_opt_power, R_opt_phase)
-- Thévenin equivalent
-- Spark growth (ε, E_threshold, dL/dt)
-- Thermal physics
-- And more...
-
-### Physical Bounds
-`reference/physical-bounds.md`
-
-Validation ranges and typical values:
-- Resistance bounds (1 kΩ to 100 MΩ)
-- Capacitance values (2 pF/foot rule)
-- Field thresholds (0.4-3.0 MV/m)
-- Energy per meter (5-100 J/m by mode)
-- Phase angles (-55° to -75° typical)
-- And more...
-
-### Glossary
-`reference/glossary.yaml`
-
-64 technical terms with:
-- Full definitions
-- Units and typical ranges
-- Related concepts
-- Related lessons
-
----
-
-## 🖼️ Images
-
-**Status:** Specifications provided, images not yet created
-
-See `assets/IMAGE-REQUIREMENTS.md` for complete specifications of 45+ needed images:
-- Circuit diagrams
-- Field visualizations
-- Graphs and charts
-- FEMM screenshots
-- High-speed photography
-- Process flowcharts
-
-**Priority:**
-- **High priority:** Images 1-6, 9-11, 16-19, 28-30 (core concepts)
-- **Medium priority:** Images 7-8, 12-15, 20-27, 31-37 (supporting)
-- **Low priority:** Images 38-45 (nice-to-have)
-
----
-
-## 🎯 Key Concepts
-
-### Circuit Theory
-- **C_mut** (mutual capacitance): Coupling between spark and topload
-- **C_sh** (shunt capacitance): Spark to ground, ~2 pF/foot
-- **Admittance analysis**: Essential for parallel networks
-- **Topological phase constraint**: φ_Z,min = -atan(2√[r(1+r)])
-
-### Optimization
-- **R_opt_power**: Maximizes power transfer = 1/(ω(C_mut+C_sh))
-- **R_opt_phase**: Minimizes phase magnitude
-- **Hungry streamer**: Self-optimization toward R_opt_power
-- **Thévenin equivalent**: Z_th, V_th extraction for any load analysis
-
-### Spark Physics
-- **E_inception**: 2-3 MV/m (initial breakdown)
-- **E_propagation**: 0.4-1.0 MV/m (sustained growth)
-- **Energy per meter (ε)**: 5-15 J/m (QCW) to 30-100 J/m (burst)
-- **Thermal time constant**: τ = d²/(4α)
-- **Streamers**: Thin, fast, high-resistance, purple/blue
-- **Leaders**: Thick, slower, low-resistance, white/orange
-
-### Advanced Modeling
-- **Lumped model**: Single R, C_mut, C_sh (fast, <10 foot sparks)
-- **Distributed model**: n segments (slow, accurate, any length)
-- **Maxwell capacitance matrix**: Extract from FEMM electrostatics
-- **Resistance optimization**: Iterative power maximization
-
----
-
-## ⚠️ Important Notes
-
-### Frequency Tracking
-**Critical concept often overlooked!**
-
-When simulating with different R values, you MUST retune to the loaded pole frequency for each case. Comparing at fixed frequency measures detuning, not inherent matching quality.
-
-See: `lessons/02-optimization/06-frequency-tracking.md`
-
-### C_sh Validation
-For distributed models, extracted C_sh may differ from the 2 pF/foot rule by factor 2-3. This is **normal** - the matrix method includes all segment couplings differently. Use FEMM values.
-
-### Sign Conventions
-Maxwell capacitance matrices have **negative off-diagonal elements**. When extracting:
-- C_mut = |C_12| (take absolute value!)
-- C_sh = C_22 - |C_12| (subtract the absolute value)
-
----
-
-## 🚀 Next Steps
-
-### To Use This Course:
-
-1. **Review** `course.json` to understand structure
-2. **Choose** a learning path (beginner/complete/simulation/physics)
-3. **Start** with Part 1, Lesson 01
-4. **Complete** exercises as you go
-5. **Reference** equation sheet and glossary as needed
-
-### To Build Interactive App:
-
-1. **Parse** course.json for navigation
-2. **Implement** markdown renderer with MathJax
-3. **Load** YAML exercises
-4. **Track** user progress by lesson ID
-5. **Add** interactive elements for {exercise:}, {interactive:} tags
-
-### To Create Images:
-
-1. **Review** `assets/IMAGE-REQUIREMENTS.md`
-2. **Prioritize** high-priority images first
-3. **Create** using tools specified (Inkscape, matplotlib, FEMM, etc.)
-4. **Place** in appropriate assets/ subdirectories
-5. **Update** lesson markdown with actual filenames
-
----
-
-## 📄 License
-
-Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)
-
-You are free to:
-- Share: Copy and redistribute
-- Adapt: Remix, transform, and build upon
-
-Under these terms:
-- Attribution: Give appropriate credit
-- ShareAlike: Distribute under same license
-
----
-
-## 🙏 Acknowledgments
-
-Based on comprehensive Tesla coil spark modeling research from the community, including:
-- Steve Conner's "hungry streamer" principle
-- Empirical observations from builders worldwide
-- FEMM electromagnetic analysis techniques
-- Circuit-theoretical foundations
-
----
-
-## 📞 Support
-
-For questions or contributions:
-- **Repository:** [GitHub link to be added]
-- **Issues:** [GitHub issues link]
-- **Community:** [Tesla coil community forum]
-
----
-
-## 📅 Version History
-
-### Version 1.0.0 (2025-10-10)
-- Initial release
-- 30 lessons across 4 parts
-- 18 exercises in YAML format
-- 5 comprehensive worked examples
-- 3 reference documents
-- Complete image specifications
-- Course navigation structure
-
----
-
-**Ready to learn Tesla coil spark physics? Start with Part 1, Lesson 01!**
-
-`lessons/01-fundamentals/01-introduction.md`
diff --git a/spark-lessons/_originals/spark-lesson.txt b/spark-lessons/_originals/spark-lesson.txt
deleted file mode 100644
index e03705a..0000000
--- a/spark-lessons/_originals/spark-lesson.txt
+++ /dev/null
@@ -1,7327 +0,0 @@
-# Tesla Coil Spark Modeling - Complete Lesson Plan Index
-
-## Overview
-This lesson plan is designed to take someone from basic circuit concepts through advanced Tesla coil spark modeling. Each part builds progressively, with worked examples, visual aids descriptions, and practice problems.
-
----
-
-## **Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior**
-*Target: 2-3 hours of study*
-
-### Module 1.1: AC Circuit Fundamentals Review
-- Peak vs RMS values (why we use peak)
-- Complex numbers and phasor notation (j, magnitude, phase)
-- Resistance (R), Reactance (X), Impedance (Z)
-- Conductance (G), Susceptance (B), Admittance (Y)
-- Power in AC circuits: P = 0.5 × Re{V × I*}
-- **Worked Example 1.1:** Calculate power with peak phasors
-- **Practice Problems:** 3 problems on complex impedance calculations
-
-### Module 1.2: Capacitance in Tesla Coils
-- What is capacitance physically?
-- Self-capacitance vs mutual capacitance
-- Capacitance to ground (shunt capacitance)
-- The 2 pF/foot empirical rule
-- **Worked Example 1.2:** Estimate C_sh for a 2-meter spark
-- **Visual Aid:** Diagram showing field lines for C_mut and C_sh
-- **Practice Problems:** 2 problems on capacitance estimation
-
-### Module 1.3: The Basic Spark Circuit Topology
-- Why spark has TWO capacitances (C_mut and C_sh)
-- Drawing the circuit: parallel R||C_mut in series with C_sh
-- Where is "ground" in a Tesla coil?
-- The topload port (measurement reference)
-- **Worked Example 1.3:** Draw circuit for given geometry
-- **Visual Aid:** 3D geometry → circuit schematic translation
-- **Practice Problems:** 2 problems on circuit topology
-
-### Module 1.4: Admittance Analysis of the Spark Circuit
-- Why use admittance (Y) instead of impedance (Z)?
-- Parallel combinations are easy in Y
-- Deriving Y_total = ((G+jB₁)·jB₂)/(G+j(B₁+B₂))
-- Real and imaginary parts
-- Converting back to impedance
-- **Worked Example 1.4:** Calculate Y and Z for specific values
-- **Visual Aid:** Complex plane plots showing Y and Z
-- **Practice Problems:** 3 problems on admittance calculations
-
-### Module 1.5: Phase Angles and What They Mean
-- Impedance phase φ_Z vs admittance phase θ_Y
-- Why φ_Z = -θ_Y
-- The "famous -45°" myth
-- Physical meaning: how much does load look resistive?
-- **Worked Example 1.5:** Calculate φ_Z from given R, C_mut, C_sh
-- **Visual Aid:** Phase angle on complex plane
-- **Practice Problems:** 2 problems on phase angle interpretation
-
-### Module 1.6: Introduction to Spark Physics
-- What is a spark? (brief non-mathematical overview)
-- Streamers vs leaders (qualitative)
-- Why sparks need voltage AND power
-- The "hungry streamer" principle (conceptual introduction)
-- **Visual Aid:** Photos/diagrams of streamers vs leaders
-- **Discussion Questions:** 3 conceptual questions
-
-### Part 1 Summary & Integration
-- Checkpoint quiz (10 questions, multiple choice + short answer)
-- Concept map connecting all Module 1 topics
-- Preview of Part 2
-
-**Estimated Token Count: ~15,000-18,000**
-
----
-
-## **Part 2: Optimization and Power Transfer - Making Sparks Efficient**
-*Target: 2-3 hours of study*
-
-### Module 2.1: The Topological Phase Constraint
-- What is a topological constraint?
-- Deriving φ_Z,min = -atan(2√(r(1+r)))
-- Why r = C_mut/C_sh matters
-- The critical value r = 0.207
-- When is -45° impossible?
-- **Worked Example 2.1:** Calculate φ_Z,min for typical geometries
-- **Visual Aid:** Graph of φ_Z,min vs r
-- **Practice Problems:** 3 problems on phase constraints
-
-### Module 2.2: The Two Critical Resistances
-- R_opt_power: maximum power transfer
-- R_opt_phase: closest to resistive
-- Why R_opt_power < R_opt_phase always
-- Deriving R_opt_power = 1/(ω(C_mut + C_sh))
-- Deriving R_opt_phase = 1/(ω√(C_mut(C_mut + C_sh)))
-- **Worked Example 2.2:** Calculate both for f=200 kHz, various capacitances
-- **Visual Aid:** Power vs R curves showing optima
-- **Practice Problems:** 4 problems on optimal resistances
-
-### Module 2.3: The "Hungry Streamer" - Self-Optimization
-- How plasma conductivity changes with power
-- Temperature → ionization → conductivity loop
-- Why sparks naturally seek R_opt_power
-- Constraints that prevent optimization
-- Physical limits: R_min and R_max
-- **Worked Example 2.3:** Trace through optimization process
-- **Visual Aid:** Flowchart of self-optimization mechanism
-- **Discussion Questions:** 3 questions on optimization limits
-
-### Module 2.4: Power Calculations
-- Power to a load: P = 0.5|V|²Re{Z_load}/|Z_th+Z_load|²
-- Why V_top/I_base is wrong
-- Displacement current problem
-- Correct measurement at topload port
-- **Worked Example 2.4:** Calculate power with correct vs incorrect method
-- **Visual Aid:** Current flow diagram showing displacement currents
-- **Practice Problems:** 3 problems on power calculations
-
-### Module 2.5: Thévenin Equivalent Method (Part A)
-- What is a Thévenin equivalent?
-- Why it separates coil from load
-- Measuring Z_th (output impedance)
-- Step-by-step procedure
-- **Worked Example 2.5A:** Extract Z_th from simulation
-- **Visual Aid:** Circuit diagrams for measurement setup
-- **Practice Problems:** 2 problems on Z_th measurement
-
-### Module 2.6: Thévenin Equivalent Method (Part B)
-- Measuring V_th (open-circuit voltage)
-- Using Z_th and V_th to predict any load
-- Theoretical maximum power (conjugate match)
-- Why actual spark power is less
-- **Worked Example 2.6:** Complete Thévenin analysis
-- **Visual Aid:** Load line analysis
-- **Practice Problems:** 3 problems on load power prediction
-
-### Module 2.7: Quality Factor and Ringdown Measurements
-- What is Q? (energy storage vs loss)
-- Q₀ (unloaded) vs Q_L (loaded)
-- Measuring Q from ringdown waveform
-- Extracting spark admittance from Q_L, f_L measurements
-- **Worked Example 2.7:** Q measurement from oscilloscope capture
-- **Visual Aid:** Annotated ringdown waveform
-- **Practice Problems:** 3 problems on Q measurements
-
-### Part 2 Summary & Integration
-- Checkpoint quiz (12 questions)
-- Worked example combining all of Part 2
-- Design exercise: optimize R for a given coil
-- Preview of Part 3
-
-**Estimated Token Count: ~18,000-20,000**
-
----
-
-## **Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From**
-*Target: 3-4 hours of study*
-
-### Module 3.1: Electric Fields and Breakdown
-- Electric field basics (V/m)
-- Field concentration at sharp points
-- E_inception: initial breakdown (~2-3 MV/m)
-- E_propagation: sustained growth (~0.4-1.0 MV/m)
-- Tip enhancement factor κ
-- **Worked Example 3.1:** Calculate E_tip for given voltage and geometry
-- **Visual Aid:** Field line diagram with enhancement
-- **Practice Problems:** 3 problems on field calculations
-
-### Module 3.2: Energy Requirements for Growth
-- Energy per meter (ε) concept
-- Why different operating modes have different ε
-- QCW: 5-15 J/m (efficient)
-- Burst: 30-100 J/m (inefficient)
-- Physical mechanisms behind ε
-- **Worked Example 3.2:** Calculate energy needed for target length
-- **Visual Aid:** Energy budget breakdown
-- **Practice Problems:** 2 problems on energy requirements
-
-### Module 3.3: Growth Rate Equation
-- dL/dt = P_stream/ε (when E_tip > E_propagation)
-- Voltage limit vs power limit
-- When does growth stall?
-- Time to reach target length
-- **Worked Example 3.3:** Predict growth time for QCW ramp
-- **Visual Aid:** Length vs time curves for different modes
-- **Practice Problems:** 3 problems on growth dynamics
-
-### Module 3.4: Thermal Physics of Plasma Channels
-- Temperature in streamers vs leaders
-- Thermal diffusion time constant τ_thermal = d²/(4α)
-- Why observed persistence is longer
-- Convection and ionization memory
-- QCW advantage: maintaining hot channels
-- **Worked Example 3.4:** Calculate thermal time constants
-- **Visual Aid:** Temperature profile cross-section
-- **Practice Problems:** 2 problems on thermal dynamics
-
-### Module 3.5: The Capacitive Divider Problem
-- How V_tip < V_topload due to C_sh
-- V_tip = V_topload × C_mut/(C_mut+C_sh) (open circuit)
-- Effect of finite R
-- As spark grows, C_sh grows, V_tip drops
-- Why length scales sub-linearly with energy
-- **Worked Example 3.5:** Calculate V_tip for growing spark
-- **Visual Aid:** Equivalent circuit with divider highlighted
-- **Practice Problems:** 3 problems on voltage division
-
-### Module 3.6: Introduction to FEMM
-- What is FEMM? (Finite Element Method Magnetics)
-- Electrostatic analysis for capacitances
-- Setting up a problem: geometry, boundaries, materials
-- Meshing and solving
-- Extracting results
-- **Worked Example 3.6:** Step-by-step FEMM tutorial (simple geometry)
-- **Visual Aid:** Screenshots of FEMM interface
-- **Practice Problems:** 1 guided FEMM exercise
-
-### Module 3.7: Extracting Capacitances from FEMM
-- The Maxwell capacitance matrix [C]
-- Diagonal elements: self-capacitances (positive)
-- Off-diagonal: mutual capacitances (negative)
-- For 2-body problem: C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|
-- Validation: C_sh ≈ 2 pF/foot check
-- **Worked Example 3.7:** Extract values from FEMM output
-- **Visual Aid:** Annotated capacitance matrix
-- **Practice Problems:** 2 problems on matrix interpretation
-
-### Module 3.8: Building the Lumped Spark Model
-- Using FEMM capacitances in circuit
-- Choosing R = R_opt_power
-- Clipping to physical bounds (R_min, R_max)
-- Implementing in SPICE
-- Running AC analysis
-- **Worked Example 3.8:** Complete lumped model simulation
-- **Visual Aid:** Flowchart from FEMM to SPICE
-- **Practice Problems:** 1 complete modeling exercise
-
-### Part 3 Summary & Integration
-- Checkpoint quiz (15 questions)
-- Complete design project: predict spark length for given coil
-- Comparison exercise: simulation vs empirical rules
-- Preview of Part 4
-
-**Estimated Token Count: ~20,000-22,000**
-
----
-
-## **Part 4: Advanced Topics - Distributed Models and Real-World Application**
-*Target: 3-4 hours of study*
-
-### Module 4.1: Why Distributed Models?
-- Limitations of lumped model
-- Current distribution along spark
-- Tip vs base differences
-- When is distributed model necessary?
-- **Visual Aid:** Comparison showing where lumped fails
-- **Discussion Questions:** 3 questions on model selection
-
-### Module 4.2: nth-Order Model Structure
-- Dividing spark into n segments (typically n=10)
-- Circuit topology with multiple segments
-- Capacitance matrix grows to (n+1)×(n+1)
-- Including all segment-to-segment couplings
-- Optional: inductance matrix
-- **Worked Example 4.2:** Draw 3-segment distributed model
-- **Visual Aid:** Progressive complexity (n=1, 3, 5, 10)
-- **Practice Problems:** 2 problems on model structure
-
-### Module 4.3: FEMM for Distributed Models
-- Multi-body electrostatic analysis
-- Defining n cylindrical segments
-- Extracting large capacitance matrix
-- Matrix properties: symmetric, semi-definite
-- Numerical stability and passivity
-- **Worked Example 4.3:** FEMM setup for n=5 model
-- **Visual Aid:** FEMM geometry with labeled segments
-- **Practice Problems:** 1 FEMM exercise with multiple bodies
-
-### Module 4.4: Implementing Capacitance Matrices in SPICE
-- Challenge: negative off-diagonal elements
-- Solution 1: Partial capacitance transformation
-- Solution 2: Controlled sources (MNA approach)
-- Solution 3: Nearest-neighbor approximation
-- Validation and stability
-- **Worked Example 4.4:** Convert 3×3 Maxwell to SPICE
-- **Visual Aid:** Circuit comparison of methods
-- **Practice Problems:** 2 problems on matrix implementation
-
-### Module 4.5: Resistance Optimization - Iterative Method
-- Initialization: tapered R profile
-- Iterative power maximization algorithm
-- Damping for stability (α_damp ≈ 0.3-0.5)
-- Position-dependent bounds: R_min[i], R_max[i]
-- Convergence criteria
-- **Worked Example 4.5:** Hand-trace 3 iterations for small model
-- **Visual Aid:** Flowchart of optimization algorithm
-- **Pseudo-code:** Python-style implementation
-- **Practice Problems:** 2 problems on optimization
-
-### Module 4.6: Resistance Optimization - Simplified Method
-- Circuit-determined resistance: R[i] = 1/(ω×C_total[i])
-- Weak diameter dependence (logarithmic)
-- When is this good enough?
-- Comparison with iterative method
-- **Worked Example 4.6:** Calculate R distribution for n=10 model
-- **Visual Aid:** Comparison plot: iterative vs simplified
-- **Practice Problems:** 2 problems on simplified method
-
-### Module 4.7: Diameter and Self-Consistency
-- Nominal diameter choice (1 mm burst, 3 mm QCW)
-- Back-calculating implied diameter from R
-- Self-consistency iteration (usually 1-2 steps)
-- Why it matters (and when it doesn't)
-- **Worked Example 4.7:** Self-consistency check
-- **Visual Aid:** Iteration convergence diagram
-- **Practice Problems:** 1 problem on diameter calculation
-
-### Module 4.8: Complete Simulation Workflow
-- Step 1: FEMM electrostatic analysis
-- Step 2: Extract capacitance matrix
-- Step 3: Choose/optimize resistances
-- Step 4: Build SPICE model
-- Step 5: Run analysis (AC or transient)
-- Step 6: Validate results
-- **Worked Example 4.8:** End-to-end simulation project
-- **Visual Aid:** Comprehensive workflow diagram
-- **Practice Problems:** 1 complete simulation exercise
-
-### Module 4.9: Validation and Physical Checks
-- Power balance: P_in = P_spark + P_losses
-- Total R in expected range (5-300 kΩ at 200 kHz)
-- R distribution: base < tip
-- C_sh validation: 2 pF/foot rule
-- Convergence tests: n=5 vs n=10 vs n=20
-- **Worked Example 4.9:** Validate a questionable simulation
-- **Visual Aid:** Checklist with pass/fail criteria
-- **Practice Problems:** 2 validation exercises
-
-### Module 4.10: Calibration from Real Measurements
-- Measuring ε: known drive, measure final length
-- Measuring E_propagation: V_top and L at stall
-- Using ringdown for Y_spark
-- Iterative refinement of model parameters
-- Building a calibration database
-- **Worked Example 4.10:** Calibrate ε from test data
-- **Visual Aid:** Calibration workflow
-- **Practice Problems:** 2 calibration problems
-
-### Module 4.11: Advanced Topics Preview
-- Frequency tracking during growth
-- Branching models (power division)
-- Strike event simulation (R collapse)
-- 3D FEA for complex geometries
-- Monte Carlo for stochastic effects
-- **Visual Aid:** Gallery of advanced scenarios
-- **Further Reading:** Resources for each topic
-
-### Module 4.12: Complete Design Case Study
-- Given: Coil specifications (f, L_secondary, C_topload, etc.)
-- Goal: Predict spark length for QCW operation
-- Work through entire process step-by-step
-- Compare prediction to empirical rules
-- Discuss uncertainties and limitations
-- **Comprehensive Example:** Full documentation
-- **Visual Aid:** Annotated results presentation
-
-### Part 4 Summary & Final Integration
-- Comprehensive final quiz (20 questions)
-- Capstone project: Design and simulate your own coil
-- Troubleshooting guide: Common errors and fixes
-- Resources for continued learning
-- Community and collaboration suggestions
-
-**Estimated Token Count: ~22,000-25,000**
-
----
-
-## Appendices (Reference Material - Brief)
-*Can be included at end of Part 4 or as separate quick-reference*
-
-### Appendix A: Complete Variable Reference Table
-- All variables with units and definitions (condensed)
-
-### Appendix B: Formula Quick Reference
-- All key equations organized by topic
-
-### Appendix C: Physical Constants
-- Standard values for air properties, field thresholds, etc.
-
-### Appendix D: SPICE Component Reference
-- How to implement various elements
-
-### Appendix E: FEMM Quick Start Guide
-- Installation, basic navigation, common tasks
-
-### Appendix F: Troubleshooting Guide
-- Common problems and solutions organized by symptom
-
-**Estimated Token Count: ~5,000-6,000**
-
----
-
-## Teaching Philosophy Embedded in This Plan
-
-1. **Spiral learning:** Concepts introduced simply, then revisited with more depth
-2. **Worked examples:** Every mathematical concept has at least one complete example
-3. **Visual aids:** Descriptions provided so you can create diagrams/graphs
-4. **Practice problems:** Incremental difficulty, answers can be provided separately
-5. **Checkpoints:** Regular assessment to ensure understanding before proceeding
-6. **Real-world connection:** Every module ties back to actual Tesla coil behavior
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior
-
----
-
-## Module 1.1: AC Circuit Fundamentals Review
-
-### Peak vs RMS Values
-
-In AC circuits, voltage and current vary sinusoidally with time. We can express them in two ways:
-
-**Time domain:**
-```
-v(t) = V_peak × cos(ωt + φ)
-```
-
-**Two amplitude conventions:**
-- **Peak value:** The maximum value reached (V_peak)
-- **RMS value:** Root-Mean-Square, V_RMS = V_peak/√2 ≈ 0.707 × V_peak
-
-**For this entire framework, we use PEAK VALUES exclusively.**
-
-**Why peak values?**
-1. Tesla coils are concerned with maximum voltage (breakdown, field stress)
-2. Consistent with phasor notation in engineering
-3. Power formula becomes: P = 0.5 × V_peak × I_peak × cos(θ)
-
-**Example:** If your oscilloscope shows a 100 kV peak-to-peak waveform:
-- V_peak-to-peak = 100 kV
-- V_peak = 50 kV (one-sided amplitude)
-- V_RMS = 50 kV / √2 ≈ 35.4 kV
-
-### Complex Numbers and Phasors
-
-AC circuit analysis uses complex numbers to represent magnitude and phase simultaneously.
-
-**Rectangular form:**
-```
-Z = R + jX
-where j = √(-1) (imaginary unit, engineers use 'j' instead of 'i')
-R = real part (resistance)
-X = imaginary part (reactance)
-```
-
-**Polar form:**
-```
-Z = |Z| ∠φ = |Z| × e^(jφ)
-where |Z| = √(R² + X²) (magnitude)
- φ = atan(X/R) (phase angle)
-```
-
-**Conversion:**
-```
-R = |Z| × cos(φ)
-X = |Z| × sin(φ)
-```
-
-**Phasor notation:** A complex number representing sinusoidal amplitude and phase:
-```
-V = V_peak ∠φ_v
-I = I_peak ∠φ_i
-```
-
-**Complex conjugate:** Used in power calculations
-```
-If I = a + jb, then I* = a - jb (flip sign of imaginary part)
-```
-
-### Resistance, Reactance, Impedance
-
-**Resistance (R):** Opposition to current that dissipates energy as heat
-- Units: Ω (ohms)
-- Always real and positive
-- V = I × R (Ohm's law)
-
-**Reactance (X):** Opposition to current that stores energy (no dissipation)
-- Units: Ω (ohms)
-- Can be positive (inductive) or negative (capacitive)
-- **Capacitive reactance:** X_C = -1/(ωC) where ω = 2πf
-- **Inductive reactance:** X_L = ωL
-
-**Impedance (Z):** Total opposition to AC current
-```
-Z = R + jX (complex)
-|Z| = √(R² + X²)
-φ_Z = atan(X/R)
-```
-
-**Sign conventions:**
-- X > 0: inductive (current lags voltage)
-- X < 0: capacitive (current leads voltage)
-- φ_Z > 0: inductive
-- φ_Z < 0: capacitive
-
-### Conductance, Susceptance, Admittance
-
-For parallel circuits, **admittance (Y)** is more convenient than impedance.
-
-**Conductance (G):** Inverse of resistance
-```
-G = 1/R
-Units: S (siemens)
-```
-
-**Susceptance (B):** Inverse of reactance (BUT with opposite sign convention!)
-```
-For capacitor: B_C = ωC (positive!)
-For inductor: B_L = -1/(ωL) (negative)
-```
-
-**Important:** Susceptance sign convention is OPPOSITE of reactance:
-- Capacitor: X_C < 0, but B_C > 0
-- Inductor: X_L > 0, but B_L < 0
-
-**Admittance (Y):** Inverse of impedance
-```
-Y = G + jB = 1/Z
-|Y| = 1/|Z|
-φ_Y = -φ_Z (opposite sign!)
-```
-
-**Conversion between Z and Y:**
-```
-Y = 1/Z = 1/(R + jX) = R/(R² + X²) - jX/(R² + X²)
-
-Therefore:
-G = R/(R² + X²)
-B = -X/(R² + X²)
-```
-
-### Power in AC Circuits
-
-**Using peak phasors:**
-```
-P = 0.5 × Re{V × I*}
-
-where V and I are complex peak phasors
- I* is the complex conjugate of I
- Re{·} means "real part of"
-```
-
-**Why the 0.5 factor?**
-- Average power over a full AC cycle
-- Comes from time-averaging cos²(ωt), which equals 0.5
-- If you used RMS values, formula would be P = V_RMS × I_RMS × cos(θ), NO 0.5
-
-**Expanded form:**
-```
-If V = V_peak ∠φ_v and I = I_peak ∠φ_i, then:
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-```
-
-The angle difference (φ_v - φ_i) is the power factor angle.
-
----
-
-### WORKED EXAMPLE 1.1: Power Calculation with Peak Phasors
-
-**Given:**
-- Voltage: V = 50 kV ∠0° (peak, using 0° as reference)
-- Impedance: Z = 100 kΩ ∠-60° (capacitive load)
-
-**Find:** Real power dissipated
-
-**Solution:**
-
-Step 1: Calculate current using Ohm's law
-```
-I = V/Z = (50 kV ∠0°)/(100 kΩ ∠-60°)
-I = 0.5 A ∠(0° - (-60°)) = 0.5 A ∠60°
-```
-
-Step 2: Calculate power
-```
-P = 0.5 × Re{V × I*}
-P = 0.5 × Re{(50 kV ∠0°) × (0.5 A ∠-60°)}
-P = 0.5 × Re{25 kW ∠-60°}
-```
-
-Step 3: Convert to rectangular to get real part
-```
-25 kW ∠-60° = 25 kW × (cos(-60°) + j×sin(-60°))
- = 25 kW × (0.5 - j×0.866)
- = 12.5 kW - j×21.65 kW
-```
-
-Step 4: Extract real part and apply 0.5 factor
-```
-P = 0.5 × 12.5 kW = 6.25 kW
-```
-
-**Alternative method:** Using power factor angle
-```
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-P = 0.5 × 50 kV × 0.5 A × cos(0° - 60°)
-P = 0.5 × 25 kW × cos(-60°)
-P = 0.5 × 25 kW × 0.5
-P = 6.25 kW
-```
-
----
-
-### PRACTICE PROBLEMS 1.1
-
-**Problem 1:** A capacitor has reactance X_C = -80 kΩ at 200 kHz. What is its capacitance? What is its susceptance?
-
-**Problem 2:** An impedance Z = 50 kΩ - j75 kΩ has current I = 0.2 A ∠30° (peak). Calculate: (a) Voltage magnitude and phase, (b) Real power
-
-**Problem 3:** An admittance Y = 0.00001 + j0.00002 S. Convert to impedance Z = R + jX.
-
----
-
-## Module 1.2: Capacitance in Tesla Coils
-
-### What is Capacitance Physically?
-
-**Definition:** Capacitance (C) is the ability to store electric charge for a given voltage:
-```
-Q = C × V
-Units: Farads (F), typically pF (10⁻¹² F) for Tesla coils
-```
-
-**Physical picture:**
-- Electric field between two conductors stores energy
-- Higher field → more stored energy → more capacitance
-- Capacitance depends on geometry, NOT on voltage
-
-**For parallel plates:**
-```
-C = ε₀ × A / d
-
-where ε₀ = 8.854×10⁻¹² F/m (permittivity of free space)
- A = plate area (m²)
- d = separation distance (m)
-```
-
-**Key insight:** Capacitance increases with:
-- Larger conductor area (more field lines)
-- Smaller separation (stronger field concentration)
-
-### Self-Capacitance vs Mutual Capacitance
-
-**Self-capacitance:** Capacitance of a single conductor to infinity (or ground)
-- Topload has self-capacitance to ground
-- Depends on size and shape
-- Toroid: C ≈ 4πε₀√(D×d) where D = major diameter, d = minor diameter
-
-**Mutual capacitance:** Capacitance between two conductors
-- Energy stored in field between them
-- Both conductors at different potentials
-- Can be positive or negative in matrix formulation
-
-**For Tesla coils with sparks:**
-- **C_mut:** mutual capacitance between topload and spark channel
-- **C_sh:** capacitance from spark to ground (shunt capacitance)
-
-### Capacitance to Ground (Shunt Capacitance)
-
-Any conductor elevated above ground has capacitance to ground.
-
-**For vertical wire above ground plane:**
-```
-C ≈ 2πε₀L / ln(2h/d)
-
-where L = wire length
- h = height above ground
- d = wire diameter
-```
-
-**For Tesla coil sparks:** Empirical rule based on community measurements:
-```
-C_sh ≈ 2 pF per foot of spark length
-
-Examples:
-1 foot (0.3 m) spark: C_sh ≈ 2 pF
-3 feet (0.9 m) spark: C_sh ≈ 6 pF
-6 feet (1.8 m) spark: C_sh ≈ 12 pF
-```
-
-This rule is surprisingly accurate (±30%) for typical Tesla coil geometries.
-
----
-
-### WORKED EXAMPLE 1.2: Estimating C_sh for a Spark
-
-**Given:** A 2-meter (6.6 foot) spark
-
-**Find:** Estimated shunt capacitance
-
-**Solution:**
-```
-C_sh ≈ 2 pF/foot × 6.6 feet
-C_sh ≈ 13.2 pF
-```
-
-**Refined estimate using cylinder formula:**
-
-Assume spark is vertical cylinder:
-- Length L = 2 m
-- Diameter d = 2 mm (typical for bright spark)
-- Height above ground h = L/2 = 1 m (average height)
-
-```
-C ≈ 2πε₀L / ln(2h/d)
-C ≈ 2π × 8.854×10⁻¹² × 2 / ln(2×1/0.002)
-C ≈ 1.112×10⁻¹⁰ / ln(1000)
-C ≈ 1.112×10⁻¹⁰ / 6.91
-C ≈ 16 pF
-```
-
-The empirical rule (13 pF) and formula (16 pF) agree reasonably well.
-
----
-
-### VISUAL AID 1.2: Field Lines for C_mut and C_sh
-
-```
-[Describe for drawing:]
-
-Side view of Tesla coil with spark:
-
- Spark tip (pointed)
- |
- | C_sh field lines radiate from
- | spark to ground plane horizontally
- Spark | (curved lines going left/right to ground)
- body |
- |
- |
- Topload (toroid)
- |
- Secondary
-
-C_mut field lines: Connect topload surface to spark channel
- - Start on topload outer surface
- - End on spark channel surface
- - Concentrated near base of spark
- - These store mutual electric field energy
-
-C_sh field lines: Connect spark to remote ground
- - Start on spark surface
- - Radiate outward to walls, floor, ceiling
- - Distributed along entire spark length
- - These store shunt field energy
-
-Key observation: Same spark channel participates in BOTH capacitances!
-This is why we need parallel C_mut || R, then series C_sh
-```
-
----
-
-### PRACTICE PROBLEMS 1.2
-
-**Problem 1:** A 4-foot spark is formed. Estimate C_sh using the empirical rule. If the topload has C_topload = 30 pF unloaded, what is the total system capacitance with the spark?
-
-**Problem 2:** Using the cylinder formula, calculate C_sh for a spark with: L = 1.5 m, d = 3 mm, average height h = 0.75 m. Compare to the empirical rule.
-
----
-
-## Module 1.3: The Basic Spark Circuit Topology
-
-### Why Sparks Have TWO Capacitances
-
-A spark channel is a conductor in space with:
-1. **Proximity to the topload** → mutual capacitance C_mut
-2. **Proximity to ground/environment** → shunt capacitance C_sh
-
-**Both exist simultaneously** because the spark interacts with multiple conductors.
-
-**Analogy:** A wire near two metal plates
-- Capacitance to plate 1: C₁
-- Capacitance to plate 2: C₂
-- Both must be included in the circuit model
-
-### The Correct Circuit Topology
-
-```
- Topload (measurement reference)
- |
- [C_mut] ← Mutual capacitance between topload and spark
- |
- +---------+--------- Node_spark
- | |
- [R] [C_sh] ← Shunt capacitance spark-to-ground
- | |
- GND ------------ GND
-```
-
-**Equivalent description:**
-- C_mut and R in parallel
-- That parallel combination in series with C_sh
-- All connected between topload and ground
-
-**Why this topology?**
-1. C_mut couples topload voltage to spark
-2. R represents plasma resistance (where power is dissipated)
-3. C_sh provides current return path to ground
-4. Current through R must also flow through either C_mut or C_sh (series connection)
-
-### Where is "Ground" in a Tesla Coil?
-
-**Earth ground:** Actual connection to soil/building ground
-**Circuit ground (reference):** Arbitrary 0V reference point
-
-**For Tesla coils:**
-- Primary circuit: Chassis/mains ground is reference
-- Secondary base: Usually connected to primary ground via RF ground
-- **Practical ground:** Floor, walls, nearby objects, you standing nearby
-- **Measurement ground:** Choose ONE point as 0V reference (usually secondary base)
-
-**Important:** "Ground" in spark model means "remote return path" - could be walls, floor, strike ring, or actual earth.
-
-### The Topload Port
-
-**Definition:** The two-terminal measurement point between topload and ground where we characterize impedance and power.
-
-```
-Port definition:
- Terminal 1: Topload terminal (high voltage)
- Terminal 2: Ground reference (0V)
-```
-
-**All impedance measurements reference this port:**
-- Z_spark: impedance looking into spark from topload
-- Z_th: Thévenin impedance of coil at this port
-- V_th: Open-circuit voltage at this port
-
-**Not the same as:**
-- V_top / I_base (includes displacement currents from entire secondary)
-- Any two-point measurement along the secondary winding
-
----
-
-### WORKED EXAMPLE 1.3: Drawing the Circuit
-
-**Given:**
-- Spark is 3 feet long
-- FEMM analysis gives C_mut = 8 pF (between topload and spark)
-- Estimate C_sh using empirical rule
-- Assume R = 100 kΩ
-
-**Task:** Draw complete circuit diagram
-
-**Solution:**
-
-Step 1: Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 3 feet = 6 pF
-```
-
-Step 2: Draw topology
-```
- Topload (V_top)
- |
- [C_mut = 8 pF]
- |
- +-------- Node_spark
- | |
- [R = 100 kΩ] [C_sh = 6 pF]
- | |
- GND -------- GND
-```
-
-Step 3: Simplify to show parallel/series structure
-```
-Topload
- |
- +---- [C_mut = 8 pF] ----+
- | |
- +---- [R = 100 kΩ] ------+ Node_spark
- |
- [C_sh = 6 pF]
- |
- GND
-```
-
-This is the basic lumped model for a Tesla coil spark.
-
----
-
-### VISUAL AID 1.3: 3D Geometry → Circuit Schematic
-
-```
-[Describe for drawing:]
-
-Panel 1: Physical 3D view
-- Toroidal topload at top (labeled "Topload")
-- Vertical spark channel extending downward (labeled "Spark, length L")
-- Ground plane at bottom (labeled "Ground")
-- Dashed lines showing C_mut field (topload to spark)
-- Dotted lines showing C_sh field (spark to ground)
-
-Panel 2: Conceptual extraction
-- Topload → single node
-- Spark → two elements: resistance R and capacitances
-- Ground → common reference
-- Arrows showing "Extract C_mut from field between topload and spark"
-- Arrows showing "Extract C_sh from field between spark and ground"
-
-Panel 3: Circuit schematic (as drawn above)
-- Proper circuit symbols
-- Component values labeled
-- Ground symbol at bottom
-- Clear port definition marked
-
-Annotation: "Same physics, different representations"
-```
-
----
-
-### PRACTICE PROBLEMS 1.3
-
-**Problem 1:** Draw the circuit for a spark with: L = 5 feet, C_mut = 12 pF (from FEMM), R = 50 kΩ. Label all component values.
-
-**Problem 2:** A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark length using the empirical rule?
-
----
-
-## Module 1.4: Admittance Analysis of the Spark Circuit
-
-### Why Use Admittance?
-
-For the spark circuit topology (parallel R||C_mut, in series with C_sh), admittance simplifies calculations.
-
-**Parallel elements:** Add admittances directly
-```
-Y_total = Y₁ + Y₂ + Y₃ + ...
-vs impedances: 1/Z_total = 1/Z₁ + 1/Z₂ + ... (messy!)
-```
-
-**Our circuit:**
-```
-Y_mut_R = Y_Cmut + Y_R (parallel: C_mut || R)
-Then series with C_sh requires impedance: Z = Z_mut_R + Z_Csh
-Then convert back: Y_total = 1/Z_total
-```
-
-### Deriving the Total Admittance Formula
-
-**Step 1:** Admittance of R and C_mut in parallel
-
-```
-Y_R = G = 1/R
-Y_Cmut = jωC_mut = jB₁ (where B₁ = ωC_mut)
-
-Y_mut_R = G + jB₁
-```
-
-**Step 2:** Convert to impedance for series combination
-```
-Z_mut_R = 1/(G + jB₁)
-```
-
-**Step 3:** Add impedance of C_sh in series
-```
-Z_Csh = 1/(jωC_sh) = -j/(ωC_sh) = 1/(jB₂) (where B₂ = ωC_sh)
-
-Z_total = Z_mut_R + Z_Csh
-Z_total = 1/(G + jB₁) + 1/(jB₂)
-```
-
-**Step 4:** Find common denominator
-```
-Z_total = [jB₂ + (G + jB₁)] / [(G + jB₁) × jB₂]
-Z_total = [G + j(B₁ + B₂)] / [jB₂(G + jB₁)]
-```
-
-**Step 5:** Invert to get admittance
-```
-Y_total = 1/Z_total = [jB₂(G + jB₁)] / [G + j(B₁ + B₂)]
-
-Y_total = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-```
-
-This is the **fundamental admittance equation** for the spark circuit.
-
-### Extracting Real and Imaginary Parts
-
-Multiply numerator:
-```
-(G + jB₁) × jB₂ = jGB₂ + j²B₁B₂ = jGB₂ - B₁B₂
- = -B₁B₂ + jGB₂
-```
-
-So:
-```
-Y = [-B₁B₂ + jGB₂] / [G + j(B₁ + B₂)]
-```
-
-To separate real and imaginary parts, multiply numerator and denominator by complex conjugate of denominator:
-
-```
-Denominator conjugate: G - j(B₁ + B₂)
-Denominator magnitude squared: G² + (B₁ + B₂)²
-```
-
-After algebra (multiply out and simplify):
-
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-```
-
-These are the **working formulas** for calculating admittance from R, C_mut, C_sh.
-
-### Converting to Impedance
-
-From Y = G_total + jB_total:
-
-```
-Z = 1/Y = 1/(G_total + jB_total)
-
-Multiply by conjugate:
-Z = (G_total - jB_total) / (G_total² + B_total²)
-
-R_total = G_total / (G_total² + B_total²)
-X_total = -B_total / (G_total² + B_total²)
-
-Or directly:
-|Z| = 1/|Y|
-φ_Z = -φ_Y (opposite sign!)
-```
-
----
-
-### WORKED EXAMPLE 1.4: Complete Y and Z Calculation
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 2π × 200×10³ = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-- R = 100 kΩ = 10⁵ Ω
-
-**Find:** Y_total (rectangular), Z_total (rectangular and polar)
-
-**Solution:**
-
-Step 1: Calculate component values
-```
-G = 1/R = 1/(10⁵) = 10⁻⁵ S = 10 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06×10⁻⁶ S = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54×10⁻⁶ S = 7.54 μS
-```
-
-Step 2: Calculate Re{Y}
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Numerator: 10 × (7.54)² = 10 × 56.85 = 568.5 μS²
-Denominator: (10)² + (10.06 + 7.54)² = 100 + (17.6)² = 100 + 309.8 = 409.8 μS²
-
-Re{Y} = 568.5 / 409.8 = 1.387 μS
-```
-
-Step 3: Calculate Im{Y}
-```
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-
-Numerator inner: G² + B₁(B₁ + B₂) = 100 + 10.06×17.6 = 100 + 177.1 = 277.1 μS²
-Numerator: 7.54 × 277.1 = 2089.3 μS³
-Denominator: 409.8 μS² (same as before)
-
-Im{Y} = 2089.3 / 409.8 = 5.10 μS
-```
-
-Step 4: Admittance result
-```
-Y_total = 1.387 + j5.10 μS
-|Y| = √(1.387² + 5.10²) = √(1.92 + 26.01) = √27.93 = 5.28 μS
-φ_Y = atan(5.10/1.387) = atan(3.68) = 74.8°
-```
-
-Step 5: Convert to impedance
-```
-|Z| = 1/|Y| = 1/(5.28×10⁻⁶) = 189 kΩ
-φ_Z = -φ_Y = -74.8°
-
-In rectangular:
-R_total = |Z| × cos(φ_Z) = 189 × cos(-74.8°) = 189 × 0.263 = 49.7 kΩ
-X_total = |Z| × sin(φ_Z) = 189 × sin(-74.8°) = 189 × (-0.965) = -182 kΩ
-
-Z_total = 49.7 - j182 kΩ = 189 kΩ ∠-74.8°
-```
-
-**Interpretation:**
-- Impedance is strongly capacitive (φ_Z = -74.8°)
-- Equivalent resistance ≈ 50 kΩ (half of actual R due to capacitive divider)
-- Large capacitive reactance dominates
-
----
-
-### VISUAL AID 1.4: Complex Plane Plots
-
-```
-[Describe for drawing:]
-
-Two plots side-by-side:
-
-LEFT: Admittance plane (Y = G + jB)
-- Horizontal axis: G (conductance, μS), 0 to 2
-- Vertical axis: B (susceptance, μS), 0 to 6
-- Plot point at (1.387, 5.10) labeled "Y_total"
-- Vector from origin to point
-- Angle φ_Y = 74.8° marked from horizontal
-- Length |Y| = 5.28 μS labeled
-- Note: "Positive B means capacitive in admittance"
-
-RIGHT: Impedance plane (Z = R + jX)
-- Horizontal axis: R (kΩ), 0 to 60
-- Vertical axis: X (kΩ), -200 to 0
-- Plot point at (49.7, -182) labeled "Z_total"
-- Vector from origin to point
-- Angle φ_Z = -74.8° marked from horizontal (below axis)
-- Length |Z| = 189 kΩ labeled
-- Note: "Negative X means capacitive in impedance"
-
-Connection between plots:
-- Arrow showing "Invert Y → Z"
-- Note: "Angles are opposite: φ_Z = -φ_Y"
-- Note: "Magnitude inverts: |Z| = 1/|Y|"
-```
-
----
-
-### PRACTICE PROBLEMS 1.4
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF, R = 80 kΩ, calculate Y_total (real and imaginary parts).
-
-**Problem 2:** An admittance Y = 2.0 + j4.5 μS. Convert to impedance Z in both rectangular and polar forms.
-
-**Problem 3:** Show algebraically that if R → ∞ (open circuit), the formula reduces to Y = jωC_mut × C_sh/(C_mut + C_sh), which is two capacitors in series.
-
----
-
-## Module 1.5: Phase Angles and What They Mean
-
-### Impedance Phase vs Admittance Phase
-
-**Impedance phase angle φ_Z:**
-```
-φ_Z = atan(X/R) = atan(Im{Z}/Re{Z})
-
-Interpretation:
-φ_Z > 0: inductive (current lags voltage)
-φ_Z = 0: purely resistive (in phase)
-φ_Z < 0: capacitive (current leads voltage)
-```
-
-**Admittance phase angle θ_Y:**
-```
-θ_Y = atan(B/G) = atan(Im{Y}/Re{Y})
-
-Relationship: θ_Y = -φ_Z (OPPOSITE SIGNS!)
-```
-
-**Why opposite?** Because Y = 1/Z, so angles subtract:
-```
-If Z = |Z|∠φ_Z, then Y = (1/|Z|)∠(-φ_Z)
-```
-
-**Convention in this framework:** We primarily discuss **impedance phase φ_Z** because that's what measurements typically report.
-
-### The "Famous -45°" and Why It's Special (Sort Of)
-
-In power electronics, a load with φ_Z = -45° is sometimes called "well-matched" because:
-- Equal resistive and capacitive components: |R| = |X_C|
-- Power factor = cos(-45°) = 0.707 (reasonable power transfer)
-- Not maximum power transfer, but balanced
-
-**Formula:** For φ_Z = -45°:
-```
-tan(-45°) = -1 = X/R
-Therefore: R = |X| = 1/(ωC) for capacitive load
-Or: R ≈ |X_c| = 1/(ωC_total) approximately
-```
-
-This is why you'll see "spark resistance should equal capacitive reactance" in old Tesla coil literature.
-
-**BUT:** As we'll see in Part 2, achieving exactly -45° is **impossible** for many Tesla coil geometries due to topological constraints!
-
-### Physical Meaning of Phase Angle
-
-**φ_Z = 0° (purely resistive):**
-- All power dissipated
-- No energy storage/return
-- Voltage and current in phase
-
-**φ_Z = -90° (purely capacitive):**
-- No power dissipated
-- All energy stored and returned each cycle
-- Current leads voltage by 90°
-
-**φ_Z = -45° (mixed):**
-- Some power dissipated (cos(-45°) ≈ 71% of |V||I|)
-- Some energy stored
-- Current leads voltage by 45°
-
-**For Tesla coil sparks:** Typical φ_Z = -55° to -75°
-- Significant capacitive component (energy storage in C_mut, C_sh)
-- Moderate power dissipation (plasma heating)
-- More capacitive than the "ideal" -45°
-
----
-
-### WORKED EXAMPLE 1.5: Calculating Phase Angle
-
-**Given:** (from Example 1.4)
-- Z_total = 49.7 - j182 kΩ
-
-**Find:** φ_Z and interpret
-
-**Solution:**
-
-Step 1: Calculate phase angle
-```
-φ_Z = atan(X/R) = atan(-182/49.7)
-φ_Z = atan(-3.66) = -74.8°
-```
-
-Step 2: Verify with magnitude and components
-```
-|Z| = √(49.7² + 182²) = √(2470 + 33124) = √35594 = 189 kΩ ✓
-
-cos(φ_Z) = R/|Z| = 49.7/189 = 0.263
-φ_Z = arccos(0.263) = 74.8°, but X is negative, so φ_Z = -74.8° ✓
-```
-
-Step 3: Interpret
-- **Strongly capacitive:** |φ_Z| = 74.8° is much larger than 45°
-- **Comparison:** |R| = 49.7 kΩ, but |X| = 182 kΩ
- - Capacitive reactance is 3.66× larger than resistance
- - Far from "balanced" -45° condition
-- **Power factor:** cos(-74.8°) = 0.263
- - Only 26.3% of |V||I| is real power
- - Most current is reactive (charging/discharging capacitances)
-
-This is typical for Tesla coil sparks: strongly capacitive impedance.
-
----
-
-### VISUAL AID 1.5: Phase Angle on Complex Plane
-
-```
-[Describe for drawing:]
-
-Impedance plane (Z = R + jX):
-- Horizontal axis: R (resistance, kΩ), 0 to 100
-- Vertical axis: X (reactance, kΩ), -200 to +200
-
-Three vectors from origin:
-
-1. Resistive (φ_Z = 0°):
- - Point at (50, 0)
- - Horizontal vector, angle = 0°
- - Label: "Pure resistance, φ_Z = 0°"
-
-2. Balanced (φ_Z = -45°):
- - Point at (50, -50)
- - Vector at -45° angle
- - Dashed line showing equal R and |X|
- - Label: "Balanced, φ_Z = -45°, R = |X|"
-
-3. Typical spark (φ_Z = -75°):
- - Point at (50, -186)
- - Vector at -75° angle
- - Label: "Typical spark, φ_Z = -75°"
- - Annotation: "Strongly capacitive, |X| >> R"
-
-Additional marks:
-- φ_Z = -90° line (vertical downward): "Pure capacitor"
-- Shaded region between -45° and -90°: "Typical Tesla coil spark range"
-- Note: "More negative φ_Z = more capacitive"
-```
-
----
-
-### PRACTICE PROBLEMS 1.5
-
-**Problem 1:** An impedance Z = 60 + j40 kΩ. Calculate φ_Z. Is this inductive or capacitive?
-
-**Problem 2:** A spark has φ_Z = -60°. If |Z| = 150 kΩ, find R and X. Calculate the power factor.
-
----
-
-## Module 1.6: Introduction to Spark Physics
-
-### What is a Spark? (Qualitative)
-
-**Definition:** A spark is a transient electrical breakdown of air, creating a conducting plasma channel between two electrodes.
-
-**Basic process:**
-1. High electric field ionizes air molecules (electrons stripped from atoms)
-2. Free electrons accelerate, collide with more atoms → avalanche
-3. Plasma forms: mixture of electrons, ions, neutral atoms
-4. Plasma conducts electricity (lower resistance than air)
-5. Current heats plasma → thermal ionization → sustained conduction
-6. When voltage removed, plasma cools and recombines
-
-**Key point:** Plasma is not a simple resistor! Its properties change dynamically:
-- Temperature: 1000 K (cool streamers) to 20,000 K (hot leaders)
-- Conductivity: varies with temperature and ionization
-- Geometry: diameter, length change during growth
-
-### Streamers vs Leaders (Qualitative)
-
-**Streamers:**
-- **Thin:** 10-100 μm diameter (thinner than human hair)
-- **Fast:** Propagate at ~10⁶ m/s (1% speed of light!)
-- **Cold:** Low temperature, weakly ionized
-- **Mechanism:** Photoionization (UV from excited atoms ionizes ahead)
-- **Appearance:** Purple/blue, highly branched, brief flashes
-- **Resistance:** High (MΩ range)
-- **Energy inefficient:** Much energy → light/heat, little → length
-
-**Leaders:**
-- **Thick:** mm to cm diameter (visible as bright core)
-- **Slower:** Propagate at ~10³ m/s (walking speed to car speed)
-- **Hot:** 5,000-20,000 K, fully ionized plasma
-- **Mechanism:** Thermal ionization (Joule heating)
-- **Appearance:** White/orange, straighter, persistent glow
-- **Resistance:** Low (kΩ range)
-- **Energy efficient:** More energy → length extension
-
-**Transition:** Streamers can become leaders if sufficient current flows → heating → thermal ionization. This requires power and time.
-
-### Why Sparks Need Voltage AND Power
-
-**Voltage requirement (field threshold):**
-```
-E_tip > E_propagation ≈ 0.4-1.0 MV/m
-
-For spark to grow, tip field must exceed threshold
-If E_tip drops below threshold, growth stalls
-```
-
-**Power requirement (energy per meter):**
-```
-To extend spark by ΔL, need energy: ΔE ≈ ε × ΔL
-where ε ≈ 5-100 J/m depending on mode
-
-Power determines growth rate: dL/dt ≈ P/ε
-```
-
-**Analogy:** Starting a fire
-- Voltage = temperature of match (need minimum to ignite)
-- Power = fuel supply rate (determines how fast fire spreads)
-- Both are necessary: hot match but no fuel → small flame dies
-- Lots of fuel but no ignition heat → no fire
-
-**For Tesla coils:**
-- Insufficient voltage → spark won't start or grows slowly
-- Insufficient power → spark stalls before reaching potential length
-- **Both must be adequate** for target spark length
-
-### The "Hungry Streamer" Principle (Conceptual)
-
-**Key insight:** Plasma is not passive! It actively adjusts its properties to maximize power extraction from the circuit.
-
-**Mechanism (simplified):**
-1. More power → more Joule heating (I²R)
-2. Higher temperature → more ionization
-3. More ionization → higher conductivity → lower R
-4. Changed geometry → modified capacitances
-5. Circuit has new optimal R for max power transfer
-6. Plasma conductivity adjusts toward this new optimal R
-7. Equilibrium when R_actual ≈ R_optimal_for_max_power
-
-**Physical limits:**
-- R cannot be infinite (some conductivity always present)
-- R cannot be zero (finite electron mobility)
-- Source has limited voltage/current
-- Takes time to adjust (thermal time constants)
-
-**Result:** In steady state, plasma R tends toward the value that maximizes power transfer, within physical constraints.
-
-**Why this matters:** We can model spark as "choosing" R = R_opt_power without detailed plasma chemistry! The physics self-optimizes.
-
----
-
-### VISUAL AID 1.6: Streamers vs Leaders
-
-```
-[Describe for photo/diagram annotations:]
-
-Two-panel comparison:
-
-LEFT PANEL: Streamer
-- Photo/drawing of thin, branched, purple discharge
-- Annotations:
- * Diameter: 10-100 μm (draw scale bar)
- * Temperature: ~1000 K
- * Speed: ~1,000,000 m/s
- * Color: Purple/blue (label spectrum)
- * Structure: Highly branched (mark branching points)
- * Duration: <1 μs per event
- * Resistance: High (MΩ)
-
-RIGHT PANEL: Leader
-- Photo/drawing of thick, straight, white discharge
-- Annotations:
- * Diameter: 1-10 mm (draw scale bar)
- * Temperature: 5,000-20,000 K
- * Speed: ~1,000 m/s
- * Color: White/orange (label spectrum)
- * Structure: Straighter channel (mark path)
- * Duration: Seconds with sustained power
- * Resistance: Low (kΩ)
-
-BOTTOM: Transition diagram
-- Timeline showing streamer → leader conversion
-- Labels: "Initial: streamers form at tip"
- "Current flows → Joule heating"
- "Channel heats → thermal ionization"
- "Leader forms from base, grows toward tip"
- "Leader tip launches new streamers"
- "Cycle repeats for continued growth"
-```
-
----
-
-### DISCUSSION QUESTIONS 1.6
-
-**Question 1:** If a Tesla coil produces high voltage but very low current, would you expect long streamers or short leaders? Why?
-
-**Question 2:** A coil generates 500 kV but only 100 mA. Another generates 200 kV but 1 A. Which is more likely to produce longer sparks? (Consider both voltage and power requirements.)
-
-**Question 3:** Explain in your own words why the spark plasma can be modeled as a resistance that "optimizes itself" rather than as a fixed resistance value.
-
----
-
-## Part 1 Summary: Concepts Checklist
-
-Before proceeding to Part 2, ensure you understand:
-
-### Circuit Fundamentals
-- [ ] Difference between peak and RMS values
-- [ ] Complex number representation: rectangular (R+jX) and polar (|Z|∠φ)
-- [ ] Power calculation: P = 0.5 × Re{V × I*} with peak phasors
-- [ ] Impedance Z = R + jX and admittance Y = G + jB
-- [ ] Relationship: Y = 1/Z, and φ_Y = -φ_Z
-
-### Capacitances
-- [ ] Physical meaning of capacitance (charge storage)
-- [ ] Self-capacitance vs mutual capacitance
-- [ ] Shunt capacitance C_sh ≈ 2 pF/foot for sparks
-- [ ] Both C_mut and C_sh exist simultaneously
-
-### Circuit Topology
-- [ ] Spark circuit: (R || C_mut) in series with C_sh
-- [ ] Topload port as measurement reference (topload-to-ground)
-- [ ] Why V_top/I_base is incorrect
-
-### Admittance Analysis
-- [ ] Advantages of Y for parallel circuits
-- [ ] Formula: Y = [(G+jB₁)×jB₂]/[G+j(B₁+B₂)]
-- [ ] Extracting Re{Y} and Im{Y}
-- [ ] Converting Y ↔ Z
-
-### Phase Angles
-- [ ] φ_Z = atan(X/R) for impedance
-- [ ] Negative φ_Z means capacitive
-- [ ] The -45° "balanced" condition: R = |X|
-- [ ] Typical sparks: φ_Z ≈ -55° to -75° (more capacitive than -45°)
-
-### Spark Physics (Qualitative)
-- [ ] Streamers: thin, fast, cold, high R, branched
-- [ ] Leaders: thick, slower, hot, low R, straighter
-- [ ] Need both voltage (E-field) and power (energy/time)
-- [ ] "Hungry streamer": plasma self-optimizes R
-
----
-
-## Integration Exercise: Putting It All Together
-
-**Scenario:** You have a Tesla coil operating at 180 kHz with a 2-foot spark.
-
-**Given data:**
-- C_mut = 7 pF (from FEMM)
-- Assume R = 75 kΩ (plasma resistance)
-- Estimate C_sh using empirical rule
-
-**Tasks:**
-1. Calculate ω, B₁, B₂, G
-2. Calculate Y_total (real and imaginary parts)
-3. Convert to Z_total (magnitude and phase)
-4. Calculate φ_Z and interpret (is it more or less capacitive than -45°?)
-5. If V_top = 300 kV peak, calculate power dissipated
-
-**Work through this problem completely before checking the solution below.**
-
----
-
-### Integration Exercise Solution
-
-**Step 1:** Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 2 feet = 4 pF
-```
-
-**Step 2:** Calculate ω and component values
-```
-ω = 2πf = 2π × 180×10³ = 1.131×10⁶ rad/s
-
-G = 1/R = 1/(75×10³) = 13.33 μS
-B₁ = ωC_mut = 1.131×10⁶ × 7×10⁻¹² = 7.92 μS
-B₂ = ωC_sh = 1.131×10⁶ × 4×10⁻¹² = 4.52 μS
-```
-
-**Step 3:** Calculate Y_total
-```
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 13.33 × (4.52)² / [13.33² + (7.92+4.52)²]
- = 13.33 × 20.43 / [177.7 + 154.4]
- = 272.3 / 332.1
- = 0.82 μS
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
- = 4.52 × [177.7 + 7.92×12.44] / 332.1
- = 4.52 × [177.7 + 98.5] / 332.1
- = 4.52 × 276.2 / 332.1
- = 3.76 μS
-
-Y_total = 0.82 + j3.76 μS
-```
-
-**Step 4:** Convert to impedance
-```
-|Y| = √(0.82² + 3.76²) = √(0.67 + 14.14) = √14.81 = 3.85 μS
-
-|Z| = 1/|Y| = 1/(3.85×10⁻⁶) = 260 kΩ
-
-φ_Y = atan(3.76/0.82) = atan(4.59) = 77.7°
-φ_Z = -φ_Y = -77.7°
-
-Z_total = 260 kΩ ∠-77.7°
-
-In rectangular:
-R_eq = 260 × cos(-77.7°) = 260 × 0.213 = 55.4 kΩ
-X_eq = 260 × sin(-77.7°) = 260 × (-0.977) = -254 kΩ
-
-Z_total = 55.4 - j254 kΩ
-```
-
-**Step 5:** Interpret phase
-```
-φ_Z = -77.7° is more capacitive than -45° (larger magnitude)
-Ratio: |X|/R = 254/55.4 = 4.6
-Capacitive reactance is 4.6× the resistance
-Very capacitive load!
-```
-
-**Step 6:** Calculate power
-```
-Current: I = V/Z = (300 kV)/(260 kΩ) = 1.15 A peak
-
-Power: P = 0.5 × V × I × cos(φ_Z)
- = 0.5 × 300×10³ × 1.15 × cos(-77.7°)
- = 0.5 × 345×10³ × 0.213
- = 36.7 kW
-
-Alternative: P = 0.5 × I² × R_eq
- = 0.5 × 1.15² × 55.4×10³
- = 0.5 × 1.32 × 55.4×10³
- = 36.6 kW ✓ (checks!)
-```
-
-**Result:** 36.7 kW dissipated in the spark plasma.
-
----
-
-## Preview of Part 2
-
-In Part 2, we'll discover:
-
-- **Why -45° is often impossible:** The topological phase constraint
-- **Two critical resistances:** R_opt_power and R_opt_phase
-- **Thévenin method:** Properly characterizing the Tesla coil
-- **Power optimization:** How the "hungry streamer" finds R_opt_power
-- **Measurements:** Extracting spark parameters from real coils
-
-These concepts build directly on the circuit analysis and phase relationships you've learned in Part 1.
-
----
-
-## CHECKPOINT QUIZ - Part 1
-
-Answer these questions to verify your understanding:
-
-1. What is the relationship between peak and RMS voltage? If V_peak = 100 kV, what is V_RMS?
-
-2. Write the power formula using peak phasors. Why is there a factor of 0.5?
-
-3. For a capacitor, why is X negative but B positive?
-
-4. Draw the circuit topology for a spark (show C_mut, R, C_sh).
-
-5. What is the empirical rule for C_sh? If a spark is 4 feet long, estimate C_sh.
-
-6. The admittance phase angle θ_Y = +60°. What is the impedance phase angle φ_Z?
-
-7. An impedance has φ_Z = -30°. Is this inductive or capacitive?
-
-8. Why is V_top/I_base not the correct impedance measurement?
-
-9. Describe the difference between streamers and leaders (two key differences).
-
-10. Explain the "hungry streamer" concept in one sentence.
-
----
-
-**END OF PART 1**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 2: Optimization and Power Transfer - Making Sparks Efficient
-
----
-
-## Module 2.1: The Topological Phase Constraint
-
-### What is a Topological Constraint?
-
-**Definition:** A limitation imposed by the **structure** of the circuit itself, independent of component values.
-
-**Example:** Series RLC circuit
-- Can only have impedance phase between -90° (pure C) and +90° (pure L)
-- Cannot have φ_Z = +120° no matter what component values you choose
-- This is a topological constraint
-
-**For spark circuits:** The specific arrangement (R||C_mut) in series with C_sh creates a fundamental limit on how resistive the impedance can appear.
-
-### Deriving the Minimum Phase Angle
-
-From Part 1, we have:
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh
-```
-
-The impedance phase is:
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Question:** For fixed C_mut and C_sh, which R value minimizes |φ_Z| (makes most resistive)?
-
-**Mathematical result:** Taking derivative ∂φ_Z/∂G = 0 and solving:
-```
-G_opt = ω√[C_mut(C_mut + C_sh)]
-
-Therefore:
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-At this resistance, the phase angle magnitude is minimized to:
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-
-where r = C_mut/C_sh (capacitance ratio)
-```
-
-### The Critical Ratio r = 0.207
-
-Let's find when φ_Z,min = -45° is achievable:
-```
--45° = -atan(2√[r(1 + r)])
-tan(45°) = 1 = 2√[r(1 + r)]
-0.5 = √[r(1 + r)]
-0.25 = r(1 + r) = r + r²
-r² + r - 0.25 = 0
-
-Using quadratic formula:
-r = [-1 ± √(1 + 1)] / 2 = [-1 ± √2] / 2
-
-Taking positive root:
-r = (√2 - 1) / 2 ≈ 0.207
-```
-
-**Critical insight:**
-- If r < 0.207: Can achieve φ_Z = -45° (with appropriate R)
-- If r > 0.207: **Cannot achieve φ_Z = -45° no matter what R you choose!**
-- If r ≥ 0.207: φ_Z,min is more negative than -45°
-
-### Typical Tesla Coil Values
-
-**Large topload, short spark:**
-```
-C_mut = 10 pF, C_sh = 4 pF (2 feet)
-r = 10/4 = 2.5
-
-φ_Z,min = -atan(2√[2.5 × 3.5]) = -atan(2 × 2.96) = -atan(5.92) = -80.4°
-```
-
-**Small topload, long spark:**
-```
-C_mut = 6 pF, C_sh = 12 pF (6 feet)
-r = 6/12 = 0.5
-
-φ_Z,min = -atan(2√[0.5 × 1.5]) = -atan(2 × 0.866) = -atan(1.732) = -60.0°
-```
-
-**Common range:** r = 0.5 to 2.0, giving φ_Z,min ≈ -60° to -80°
-
-**Conclusion:** For most Tesla coil geometries, -45° is **mathematically impossible**!
-
----
-
-### WORKED EXAMPLE 2.1: Calculate Minimum Phase Angle
-
-**Given:**
-- Frequency: f = 200 kHz
-- C_mut = 8 pF
-- C_sh = 6 pF
-
-**Find:**
-(a) Capacitance ratio r
-(b) Minimum achievable phase angle φ_Z,min
-(c) R_opt_phase that achieves this angle
-
-**Solution:**
-
-**Part (a):** Capacitance ratio
-```
-r = C_mut / C_sh = 8 / 6 = 1.333
-```
-
-**Part (b):** Minimum phase angle
-```
-φ_Z,min = -atan(2√[r(1 + r)])
- = -atan(2√[1.333 × 2.333])
- = -atan(2√3.11)
- = -atan(2 × 1.764)
- = -atan(3.528)
- = -74.2°
-```
-
-**Part (c):** Resistance for minimum phase
-```
-ω = 2πf = 2π × 200×10³ = 1.257×10⁶ rad/s
-
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 1 / [1.257×10⁶ × √(8×10⁻¹² × 14×10⁻¹²)]
- = 1 / [1.257×10⁶ × √(112×10⁻²⁴)]
- = 1 / [1.257×10⁶ × 10.58×10⁻¹²]
- = 1 / (13.30×10⁻⁶)
- = 75.2 kΩ
-```
-
-**Interpretation:**
-- With r = 1.333, cannot achieve -45°
-- Best possible is -74.2° (much more capacitive)
-- This requires R = 75.2 kΩ
-- Any other R value gives |φ_Z| > 74.2°
-
----
-
-### VISUAL AID 2.1: Graph of φ_Z,min vs r
-
-```
-[Describe for plotting:]
-
-Graph with:
-- X-axis: r = C_mut/C_sh (log scale), range 0.1 to 10
-- Y-axis: φ_Z,min (degrees), range -90° to -40°
-
-Plot curve: φ_Z,min = -atan(2√[r(1+r)])
-
-Key points marked:
-- r = 0.207, φ_Z,min = -45° (mark with horizontal dashed line)
-- Shaded region r < 0.207: "Can achieve -45°"
-- Shaded region r > 0.207: "Cannot achieve -45°"
-- Typical Tesla coil range r = 0.5 to 2.0 highlighted
-- Example points:
- * r = 0.5, φ_Z = -60°
- * r = 1.0, φ_Z = -70.5°
- * r = 2.0, φ_Z = -79.7°
-
-Annotations:
-- "Larger r → more capacitive minimum"
-- "Large topload + short spark → high r"
-- "Small topload + long spark → low r"
-```
-
----
-
-### PRACTICE PROBLEMS 2.1
-
-**Problem 1:** For C_mut = 12 pF, C_sh = 8 pF at f = 180 kHz:
-(a) Calculate r
-(b) Find φ_Z,min
-(c) Can this circuit achieve -45°?
-
-**Problem 2:** A designer wants φ_Z,min = -50°. What maximum value of r is allowed? If C_sh = 10 pF, what is the maximum C_mut?
-
-**Problem 3:** Explain physically why larger r (more C_mut relative to C_sh) makes the impedance more capacitive.
-
----
-
-## Module 2.2: The Two Critical Resistances
-
-### R_opt_phase: Closest to Resistive (Revisited)
-
-From Module 2.1:
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-**Purpose:** Minimizes |φ_Z| to achieve φ_Z,min
-
-**Use case:** If you want the "most resistive-looking" impedance possible
-
-### R_opt_power: Maximum Power Transfer
-
-**Different question:** Which R maximizes real power delivered to the spark for a given topload voltage?
-
-**Setup:** Fixed voltage source V_top, variable load resistance R
-
-**Power to load:**
-```
-P = 0.5 × |V_top|² × Re{Y(R)}
-```
-
-where Y(R) depends on R through G = 1/R.
-
-**Mathematical derivation:** Take ∂P/∂G = 0, solve for G:
-
-After calculus (see framework document for full derivation):
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-```
-
-**Simpler formula!** Just total capacitance, not geometric mean.
-
-### Comparing the Two
-
-**Relationship:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Since √(C_mut(C_mut + C_sh)) < (C_mut + C_sh):
-
-R_opt_power < R_opt_phase ALWAYS
-```
-
-**Numerical relationship:** For typical r = 0.5 to 2:
-```
-R_opt_power ≈ (0.5 to 0.7) × R_opt_phase
-```
-
-**Phase angle at R_opt_power:**
-- Always more negative than φ_Z,min
-- Typically φ_Z ≈ -55° to -75° at R_opt_power
-- More capacitive than R_opt_phase, but delivers more power
-
----
-
-### WORKED EXAMPLE 2.2: Calculating Both Critical Resistances
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-
-**Find:** R_opt_phase, R_opt_power, and compare
-
-**Solution:**
-
-**Part 1:** R_opt_phase (from Example 2.1)
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 75.2 kΩ
-```
-
-**Part 2:** R_opt_power
-```
-C_total = C_mut + C_sh = 8 + 6 = 14 pF = 14×10⁻¹² F
-
-R_opt_power = 1 / (ωC_total)
- = 1 / (1.257×10⁶ × 14×10⁻¹²)
- = 1 / (17.60×10⁻⁶)
- = 56.8 kΩ
-```
-
-**Part 3:** Comparison
-```
-Ratio: R_opt_power / R_opt_phase = 56.8 / 75.2 = 0.755
-
-R_opt_power is 75.5% of R_opt_phase
-```
-
-**Part 4:** Phase angle at R_opt_power
-
-Calculate admittance with R = 56.8 kΩ:
-```
-G = 1/56800 = 17.61 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54 μS
-
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 17.61 × 56.85 / [310 + 309.8]
- = 1001.2 / 619.8
- = 1.615 μS
-
-Im{Y} = 7.54[310 + 176.9] / 619.8
- = 7.54 × 486.9 / 619.8
- = 5.928 μS
-
-φ_Y = atan(5.928/1.615) = atan(3.67) = 74.7°
-φ_Z = -74.7°
-```
-
-**Summary:**
-- R_opt_phase = 75.2 kΩ gives φ_Z = -74.2° (minimum)
-- R_opt_power = 56.8 kΩ gives φ_Z = -74.7° (slightly more capacitive)
-- Power is maximized at R_opt_power despite not having minimum phase
-- Difference is small: both are strongly capacitive
-
----
-
-### VISUAL AID 2.2: Power vs Resistance Curves
-
-```
-[Describe for plotting:]
-
-Two overlaid plots sharing X-axis:
-
-X-axis: R (kΩ), range 20 to 150, log scale
-
-TOP PLOT - Power:
-Y-axis: P (kW), normalized to max = 1.0
-Curve: Bell-shaped, peaks at R_opt_power
-- Peak marked at 56.8 kΩ, height = 1.0
-- Label: "R_opt_power = 56.8 kΩ"
-- Width shows power drops to 0.5 at ±50% R
-- Annotation: "Maximum power transfer"
-
-BOTTOM PLOT - Phase angle:
-Y-axis: φ_Z (degrees), range -90° to -40°
-Curve: Rises from -90° (R→0), peaks at R_opt_phase, falls back
-- Peak (least negative) marked at 75.2 kΩ, φ_Z = -74.2°
-- Label: "R_opt_phase = 75.2 kΩ, φ_Z,min = -74.2°"
-- -45° reference line (dashed)
-- Annotation: "Most resistive phase"
-
-Vertical lines:
-- At R_opt_power (56.8 kΩ): shows φ_Z = -74.7° on bottom plot
-- At R_opt_phase (75.2 kΩ): shows lower power on top plot
-
-Key insight box: "R_opt_power ≠ R_opt_phase"
- "R_opt_power delivers more power but is more capacitive"
-```
-
----
-
-### PRACTICE PROBLEMS 2.2
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF:
-Calculate R_opt_power and R_opt_phase.
-
-**Problem 2:** At 200 kHz, a spark has C_total = 12 pF. What is R_opt_power? If V_top = 400 kV, estimate the maximum deliverable power.
-
-**Problem 3:** Prove algebraically that R_opt_power < R_opt_phase always (hint: compare 1/(C_mut+C_sh) with 1/√(C_mut(C_mut+C_sh))).
-
-**Problem 4:** A measurement shows φ_Z = -68° at the operating point. Is R likely above or below R_opt_phase? Above or below R_opt_power?
-
----
-
-## Module 2.3: The "Hungry Streamer" - Self-Optimization
-
-### The Feedback Loop
-
-Plasma conductivity changes dynamically with power:
-
-**1. More power → Joule heating**
-```
-Heating rate: dT/dt ∝ I²R
-Higher current → faster heating
-```
-
-**2. Higher temperature → ionization**
-```
-Thermal ionization: fraction ∝ exp(-E_ionization / kT)
-Hotter plasma → more free electrons
-```
-
-**3. More electrons → higher conductivity**
-```
-σ = n_e × e × μ_e
-where n_e = electron density, μ_e = electron mobility
-σ ∝ n_e ∝ exp(-E_ionization / kT)
-```
-
-**4. Higher conductivity → lower R**
-```
-R = ρL/A = L/(σA)
-σ increases → R decreases
-```
-
-**5. Changed R → new circuit behavior**
-```
-New R changes Y_spark, power transfer changes
-If R < R_opt_power: reducing R further decreases power
-If R > R_opt_power: reducing R increases power
-```
-
-**6. Stable equilibrium at R ≈ R_opt_power**
-```
-When R approaches R_opt_power:
-- Small decrease → power decreases → cooling → R rises
-- Small increase → power increases → heating → R falls
-- Negative feedback stabilizes at R_opt_power
-```
-
-### Time Scales
-
-**Thermal response:** ~0.1-1 ms for thin channels
-- Heat diffusion time: τ = d²/(4α) ≈ 0.1 ms for d = 100 μm
-- Fast enough to track AC envelope (kHz modulation)
-- Too slow to track RF oscillation (hundreds of kHz)
-
-**Ionization response:** ~μs to ms
-- Recombination time varies with density and temperature
-- Can follow slower modulation
-
-**Result:** Plasma adjusts R on timescales of 0.1-10 ms, tracking power delivery changes.
-
-### Physical Constraints
-
-**Lower bound R_min:**
-- Maximum conductivity limited by electron-ion collision frequency
-- Typical: R_min ≈ 1-10 kΩ for hot, dense leaders
-- If R_opt_power < R_min: plasma stuck at R_min (can't optimize)
-
-**Upper bound R_max:**
-- Minimum conductivity of partially ionized gas
-- Typical: R_max ≈ 100 kΩ to 100 MΩ for cool streamers
-- If R_opt_power > R_max: plasma stuck at R_max
-
-**Source limitations:**
-- Insufficient voltage: spark won't form at all
-- Insufficient current: can't heat enough to reach R_opt_power
-- Power supply impedance: limits available power
-
-**When optimization fails:**
-- Source too weak: spark operates at whatever R it can sustain
-- Thermal time too long: can't adjust fast enough (burst mode)
-- Branching: power divides, none optimizes well
-
----
-
-### WORKED EXAMPLE 2.3: Tracing Optimization Process
-
-**Scenario:** Spark initially forms with R = 200 kΩ (cold streamer). Circuit has R_opt_power = 60 kΩ.
-
-**Trace the evolution:**
-
-**Initial state (t = 0):**
-```
-R = 200 kΩ >> R_opt_power
-Power delivered: P_initial (suboptimal, low)
-Temperature: T_initial (cool)
-```
-
-**Early phase (0 < t < 1 ms):**
-```
-Current flows → Joule heating: dT/dt = I²R/c_p
-R is high → voltage division favorable → some heating occurs
-Temperature rises → ionization begins → n_e increases
-Conductivity σ ∝ n_e increases → R decreases
-R drops toward 150 kΩ
-```
-
-**Middle phase (1 ms < t < 5 ms):**
-```
-R approaches 100 kΩ range
-Now closer to R_opt_power → power transfer improves
-More power → faster heating → faster ionization
-Positive feedback: lower R → more power → lower R
-R drops rapidly: 100 kΩ → 80 kΩ → 70 kΩ → 65 kΩ
-```
-
-**Approach to equilibrium (5 ms < t < 10 ms):**
-```
-R approaches R_opt_power = 60 kΩ
-Power maximized at this R
-If R < 60 kΩ: power would decrease → cooling → R rises
-If R > 60 kΩ: power would increase → heating → R falls
-Negative feedback stabilizes around R ≈ 60 kΩ
-```
-
-**Steady state (t > 10 ms):**
-```
-R oscillates around 60 kΩ ± 10%
-Temperature stable at equilibrium
-Power maximized and stable
-Spark is "optimized"
-```
-
-**If constraints active:**
-```
-If R_opt_power = 30 kΩ but R_min = 50 kΩ:
- Plasma can only reach R = 50 kΩ (not optimal)
- Power is less than theoretical maximum
- Spark is "starved" - wants more current than physics allows
-```
-
----
-
-### DISCUSSION QUESTIONS 2.3
-
-**Question 1:** Why does the optimization work? Why doesn't the plasma just pick a random R value?
-
-**Question 2:** In burst mode (short pulses, <100 μs), thermal time constants are longer than pulse duration. Would you expect the plasma to reach R_opt_power? Why or why not?
-
-**Question 3:** A coil produces sparks with measured R ≈ 20 kΩ, but calculations show R_opt_power = 80 kΩ. What might explain this discrepancy?
-
----
-
-## Module 2.4: Power Calculations and Common Errors
-
-### Correct Power Formula
-
-For AC circuit with peak phasors:
-```
-P = 0.5 × Re{V × I*}
-
-Expanded:
-P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
-
-For impedance Z:
-I = V/Z
-P = 0.5 × |V|² × Re{1/Z} = 0.5 × |V|² × Re{Y}
-```
-
-Or using impedance directly:
-```
-P = 0.5 × |I|² × Re{Z} = 0.5 × I² × R
-```
-
-### Why V_top/I_base is Wrong
-
-**The problem:** Current at secondary base (I_base) includes ALL return currents:
-
-1. **Capacitance to ground** along entire secondary
- - Each turn has C to ground
- - AC current: I_C = jωC × V
- - Sum of all displacement currents
-
-2. **Primary-to-secondary coupling**
- - Displacement current through C_ps
- - Part of transformer action
-
-3. **Strike ring/environment coupling**
- - Any nearby grounded object
-
-4. **The spark current** (what we actually want)
-
-**Result:**
-```
-I_base = I_spark + I_displacement_secondary + I_primary_coupling + I_environment
-
-V_top/I_base = wrong because denominator includes parasitic currents!
-```
-
-**Measured impedance is too low** (I_base too high).
-
-### Correct Measurement Port
-
-**Definition:** Topload-to-ground is the correct measurement port.
-
-**Current measurement:** Only the current **through the spark path** from topload.
-
-**Methods:**
-1. Measure I_spark return current separately (Rogowski/CT on spark ground return)
-2. Use circuit analysis (know V_top, calculate I_spark from model)
-3. Thévenin extraction (next modules)
-
----
-
-### WORKED EXAMPLE 2.4: Correct vs Incorrect Power Calculation
-
-**Given:**
-- V_top = 300 kV peak
-- I_base (measured at secondary base) = 5 A peak
-- I_spark (actual spark current) = 1.5 A peak
-- Spark impedance phase: φ_Z = -70°
-
-**Find:** Power using incorrect method, power using correct method
-
-**Solution:**
-
-**Incorrect method:** Using V_top/I_base
-```
-Z_apparent = V_top / I_base = 300 kV / 5 A = 60 kΩ
-
-This is NOT the spark impedance!
-
-If we naively calculated power:
-P_wrong = 0.5 × 300 kV × 5 A × cos(-70°)
- = 0.5 × 1500 kW × 0.342
- = 257 kW
-
-This is way too high!
-```
-
-**Correct method:** Using actual spark current
-```
-I_spark = 1.5 A peak
-
-Real spark impedance:
-Z_spark = V_top / I_spark = 300 kV / 1.5 A = 200 kΩ
-
-Power:
-P_correct = 0.5 × V_top × I_spark × cos(φ_Z)
- = 0.5 × 300 kV × 1.5 A × cos(-70°)
- = 0.5 × 450 kW × 0.342
- = 77 kW
-
-Or using resistance directly:
-R = |Z| × cos(φ_Z) = 200 kΩ × 0.342 = 68.4 kΩ
-P = 0.5 × I² × R = 0.5 × 1.5² × 68.4 kΩ = 77 kW ✓
-```
-
-**Error analysis:**
-```
-P_wrong / P_correct = 257 / 77 = 3.3×
-
-The incorrect method overestimates power by 330%!
-```
-
----
-
-### VISUAL AID 2.4: Current Flow Diagram
-
-```
-[Describe for drawing:]
-
-Side view of Tesla coil showing current paths:
-
-PRIMARY:
-- Primary coil at bottom (multi-turn)
-- Current I_primary flowing
-- Capacitor C_primary
-- Ground connection
-
-SECONDARY:
-- Tall helical coil
-- Multiple current paths illustrated with arrows:
-
-Path 1 (RED): Spark current
- - Flows from topload through spark to remote ground
- - Returns through earth/floor to secondary base
- - Labeled: "I_spark" (what we want to measure)
-
-Path 2 (BLUE): Displacement currents along secondary
- - From each turn to ground
- - Many small arrows radiating outward
- - Labeled: "I_displacement = Σ(jωC_turn × V_turn)"
-
-Path 3 (GREEN): Primary-secondary coupling
- - From primary through C_ps to secondary
- - Labeled: "I_coupling"
-
-Path 4 (YELLOW): Environmental coupling
- - To nearby objects, walls, strike ring
- - Labeled: "I_environment"
-
-AT SECONDARY BASE:
-- Large arrow labeled "I_base = I_spark + I_displacement + I_coupling + I_environment"
-- RED path continues to ground separately
-
-Key insight box: "I_base ≠ I_spark! Cannot use V_top/I_base for spark impedance!"
-```
-
----
-
-### PRACTICE PROBLEMS 2.4
-
-**Problem 1:** A simulation shows V_top = 250 kV, I_base = 3.5 A, but the spark circuit model predicts Z_spark = 180 kΩ. Calculate the actual spark current and power.
-
-**Problem 2:** Explain why displacement current is proportional to frequency (ω). If frequency doubles, what happens to I_displacement?
-
-**Problem 3:** An experimenter measures I_base = 4 A and calculates Z = V_top/I_base = 75 kΩ. Another measurement with a Rogowski coil on the spark return path shows I_spark = 1.2 A. What is the true spark impedance?
-
----
-
-## Module 2.5: Thévenin Equivalent Method - Part A (Measuring Z_th)
-
-### What is a Thévenin Equivalent?
-
-**Thévenin's Theorem:** Any linear two-terminal network can be replaced by:
-- A voltage source V_th (open-circuit voltage)
-- In series with an impedance Z_th (output impedance)
-
-```
-[Complex network] ≡ [V_th]---[Z_th]---o Output
- |
- GND
-```
-
-**Advantage:** Characterize the coil **once**, then predict behavior with **any load** instantly.
-
-### Measuring Z_th: Output Impedance
-
-**Procedure:**
-
-**Step 1:** Turn OFF primary drive
-- Set drive voltage to 0V (AC short circuit)
-- Keep all tank components in place (MMC, L_primary, damping resistors)
-- Tank circuit still present, just not driven
-
-**Step 2:** Apply test source
-- Apply 1V AC at operating frequency to topload-to-ground port
-- Use small-signal AC source (simulation or actual)
-
-**Step 3:** Measure current
-```
-I_test = current into topload port with 1V applied
-```
-
-**Step 4:** Calculate Z_th
-```
-Z_th = V_test / I_test = 1V / I_test
-
-Z_th = R_th + jX_th (complex impedance)
-```
-
-**Physical meaning:**
-- R_th: resistive losses (secondary winding, topload, damping)
-- X_th: reactive component (usually capacitive from topload)
-
-**Typical values at 200 kHz:**
-- R_th: 10-100 Ω (depends on Q and coil size)
-- X_th: -500 to -3000 Ω (capacitive)
-- |Z_th|: 500-3000 Ω
-
----
-
-### WORKED EXAMPLE 2.5A: Extracting Z_th from Simulation
-
-**Simulation setup:**
-- DRSSTC at f = 185 kHz
-- Primary drive set to 0V
-- All components remain (L_primary, C_MMC, secondary, topload)
-- AC test source: 1V ∠0° at topload-to-ground
-
-**Simulation results:**
-- I_test = 0.000412 ∠87.3° A = 0.412 mA ∠87.3°
-
-**Calculate Z_th:**
-
-**Step 1:** Impedance magnitude
-```
-|Z_th| = |V| / |I| = 1 V / 0.000412 A = 2427 Ω
-```
-
-**Step 2:** Impedance phase
-```
-φ_Z_th = φ_V - φ_I = 0° - 87.3° = -87.3°
-```
-
-**Step 3:** Polar form
-```
-Z_th = 2427 Ω ∠-87.3°
-```
-
-**Step 4:** Convert to rectangular
-```
-R_th = |Z_th| × cos(φ_Z_th) = 2427 × cos(-87.3°) = 2427 × 0.0471 = 114 Ω
-X_th = |Z_th| × sin(φ_Z_th) = 2427 × sin(-87.3°) = 2427 × (-0.9989) = -2424 Ω
-
-Z_th = 114 - j2424 Ω
-```
-
-**Interpretation:**
-- **R_th = 114 Ω:** Secondary losses (winding resistance, dielectric losses)
-- **X_th = -2424 Ω:** Strongly capacitive (topload dominates)
-- **Phase ≈ -87°:** Nearly pure capacitor with small series resistance
-- **Quality factor estimate:** Q ≈ |X_th|/R_th = 2424/114 ≈ 21
-
----
-
-### VISUAL AID 2.5A: Thévenin Measurement Setup
-
-```
-[Describe for drawing:]
-
-Two circuit diagrams side-by-side:
-
-LEFT: Full Tesla coil circuit (complex)
-- Primary side: Driver → L_primary → C_MMC → Ground
-- Magnetic coupling to secondary
-- Secondary: Base grounded, many turns, topload at top
-- All parasitics shown (C to ground, etc.)
-- Output port marked at topload
-- Label: "Complex original circuit"
-
-RIGHT: Thévenin equivalent (simple)
-- Just two components:
- * Voltage source V_th
- * Series impedance Z_th = 114 - j2424 Ω
-- Output port (same as left)
-- Label: "Thévenin equivalent"
-
-Arrow between them: "Extraction process"
-
-BOTTOM: Measurement configuration
-- Primary drive: OFF (0V symbol)
-- Test source: 1V AC at topload
-- Ammeter measuring I_test
-- Calculation: Z_th = 1V / I_test
-- Note: "All tank components remain in circuit"
-```
-
----
-
-### PRACTICE PROBLEMS 2.5A
-
-**Problem 1:** A test measurement gives I_test = 0.00035 ∠82° A for V_test = 1 ∠0° V at f = 200 kHz. Calculate Z_th in rectangular form.
-
-**Problem 2:** If Z_th = 85 - j1800 Ω, what is the unloaded Q of the secondary circuit?
-
----
-
-## Module 2.6: Thévenin Equivalent Method - Part B (Using V_th and Z_th)
-
-### Measuring V_th: Open-Circuit Voltage
-
-**Procedure:**
-
-**Step 1:** Remove load
-- Disconnect spark (or set spark to not break out)
-- Topload is open-circuit
-
-**Step 2:** Turn ON primary drive
-- Normal operating frequency and amplitude
-- Drive as you would for spark operation
-
-**Step 3:** Measure topload voltage
-```
-V_th = V(topload) with no load (complex magnitude and phase)
-```
-
-**Typical:** V_th = 200-500 kV peak for medium coils
-
-### Predicting Power to Any Load
-
-With Z_th and V_th known, calculate power to any load impedance Z_load:
-
-**Circuit with load:**
-```
-[V_th] --- [Z_th] --- [Z_load] --- GND
-
-Total impedance: Z_total = Z_th + Z_load
-Current: I = V_th / (Z_th + Z_load)
-Voltage across load: V_load = I × Z_load
-Power in load: P_load = 0.5 × |I|² × Re{Z_load}
-```
-
-**Direct formula:**
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**No re-simulation needed!** Just plug in different Z_load values.
-
-### Theoretical Maximum Power
-
-**Conjugate match condition:** Maximum power transfer occurs when:
-```
-Z_load = Z_th* (complex conjugate)
-
-If Z_th = R_th + jX_th, then Z_load = R_th - jX_th
-```
-
-**Maximum power:**
-```
-P_max = |V_th|² / (8 × R_th)
-```
-
-**BUT:** For spark loads, conjugate match is usually not achievable due to topological constraints (Module 2.1).
-
----
-
-### WORKED EXAMPLE 2.6: Complete Thévenin Analysis
-
-**Given:**
-- Z_th = 114 - j2424 Ω (from Example 2.5A)
-- V_th = 350 kV ∠0° (measured with drive on, no load)
-- Candidate spark load: Z_spark = 60 kΩ - j160 kΩ (from lumped model)
-
-**Find:**
-(a) Current through spark
-(b) Voltage across spark
-(c) Power dissipated in spark
-(d) Theoretical maximum power (conjugate match)
-
-**Solution:**
-
-**Part (a):** Current
-```
-Z_total = Z_th + Z_spark
- = (114 - j2424) + (60000 - j160000)
- = (60114 - j162424) Ω
-
-|Z_total| = √(60114² + 162424²) = √(3.614×10⁹ + 2.638×10¹⁰) = √3.00×10¹⁰ = 173 kΩ
-
-I = V_th / Z_total = (350 kV) / (173 kΩ) = 2.02 A peak
-```
-
-**Part (b):** Voltage across spark
-```
-Voltage divider:
-V_spark = V_th × [Z_spark / (Z_th + Z_spark)]
-
-|V_spark| = 350 kV × (170 kΩ / 173 kΩ) = 350 kV × 0.983 = 344 kV
-
-Most voltage appears across spark (Z_spark >> Z_th)
-```
-
-**Part (c):** Power in spark
-```
-P_spark = 0.5 × I² × Re{Z_spark}
- = 0.5 × (2.02)² × 60000
- = 0.5 × 4.08 × 60000
- = 122 kW
-```
-
-**Part (d):** Theoretical maximum
-```
-Conjugate match: Z_load = Z_th* = 114 + j2424 Ω
-
-P_max = |V_th|² / (8 × R_th)
- = (350×10³)² / (8 × 114)
- = 1.225×10¹¹ / 912
- = 134 MW
-
-Wait, this seems way too high! Let me recalculate...
-
-P_max = 0.5 × |V_th|² / (4 × R_th) [Correct formula]
- = 0.5 × (350×10³)² / (4 × 114)
- = 0.5 × 1.225×10¹¹ / 456
- = 134 MW
-
-This is still huge because R_th is so small (114 Ω).
-```
-
-**Reality check:**
-- Actual spark power: 122 kW
-- Theoretical maximum: 134 MW
-- Spark extracts: 122/134000 = 0.09% of theoretical maximum
-
-**Why such a huge difference?**
-- Conjugate match would require Z_load = 114 + j2424 Ω (very low resistance!)
-- Actual spark: Z_spark = 60000 - j160000 Ω (much higher resistance, wrong phase)
-- Topological constraints prevent achieving conjugate match
-- This is normal for Tesla coils!
-
----
-
-### PRACTICE PROBLEMS 2.6
-
-**Problem 1:** Given Z_th = 95 - j1850 Ω, V_th = 280 kV, and a spark model with Z_spark = 50 kΩ - j140 kΩ:
-(a) Calculate power delivered to spark
-(b) What percentage of theoretical maximum is this?
-
-**Problem 2:** A load Z_load = 200 + j200 Ω is connected. If Z_th = 100 - j2000 Ω and V_th = 300 kV, calculate the power. Is this inductive or capacitive load?
-
----
-
-## Module 2.7: Quality Factor and Ringdown Measurements
-
-### What is Quality Factor (Q)?
-
-**Definition:** Ratio of energy stored to energy dissipated per cycle:
-```
-Q = 2π × (Energy stored) / (Energy dissipated per cycle)
-
-For series RLC: Q = ωL/R = 1/(ωRC)
-For parallel RLC at resonance: Q = R/(ωL) = ωRC
-```
-
-**Physical meaning:**
-- High Q: oscillation persists many cycles (low damping)
-- Low Q: oscillation decays quickly (high damping)
-
-### Measuring Q from Ringdown
-
-**Procedure:**
-1. Excite coil (burst of AC at resonance)
-2. Turn off drive
-3. Measure voltage decay
-
-**Exponential envelope:**
-```
-V(t) = V₀ × exp(-t/τ) × cos(ωt)
-
-where τ = 2Q/ω = decay time constant
-```
-
-**From consecutive peaks:**
-```
-Ratio of amplitudes n cycles apart:
-A(t + nT) / A(t) = exp(-nT/τ) = exp(-nπ/Q)
-
-Solving for Q:
-Q = nπ / ln[A(t) / A(t + nT)]
-```
-
-**Practical:** Measure peak-to-peak over several cycles:
-```
-Q ≈ πf × Δt / ln(A₁/A₂)
-
-where Δt = time between measured peaks
-```
-
-### Extracting Spark Parameters from Q Measurements
-
-**Unloaded (no spark):**
-- Measure f₀, Q₀
-- Represents coil losses only
-
-**Loaded (with spark):**
-- Measure f_L, Q_L
-- Spark adds resistance and capacitance
-
-**At resonance:**
-```
-Q_L = ω_L × C_eq × R_p
-
-where R_p = equivalent parallel resistance at resonance
- C_eq = total capacitance = C₀ + ΔC
-```
-
-**Solving for conductance:**
-```
-G_total = 1/R_p = ω_L × C_eq / Q_L
-
-Spark contribution:
-G_spark ≈ G_total - G_0 = ω_L C_eq / Q_L - ω₀ C₀ / Q₀
-```
-
-**Capacitance from frequency shift:**
-```
-Frequency ratio: f₀/f_L = √(C_eq/C₀)
-
-Therefore: C_eq = C₀ × (f₀/f_L)²
-
-Spark capacitance: ΔC = C_eq - C₀
-```
-
-**Spark admittance:**
-```
-Y_spark ≈ G_spark + jω_L ΔC
-```
-
----
-
-### WORKED EXAMPLE 2.7: Q Measurement and Spark Extraction
-
-**Given measurements:**
-
-**Unloaded:**
-- f₀ = 200 kHz
-- Q₀ = 80 (from ringdown)
-- C₀ = 28 pF (calculated from geometry)
-
-**With spark:**
-- f_L = 185 kHz (frequency dropped)
-- Q_L = 25 (from ringdown with spark)
-
-**Find:** Spark admittance Y_spark
-
-**Solution:**
-
-**Step 1:** Calculate loaded capacitance
-```
-C_eq = C₀ × (f₀/f_L)²
- = 28 pF × (200/185)²
- = 28 pF × (1.081)²
- = 28 pF × 1.169
- = 32.7 pF
-
-ΔC = C_eq - C₀ = 32.7 - 28 = 4.7 pF
-```
-
-**Step 2:** Calculate conductances
-```
-ω₀ = 2π × 200×10³ = 1.257×10⁶ rad/s
-ω_L = 2π × 185×10³ = 1.162×10⁶ rad/s
-
-G₀ = ω₀ C₀ / Q₀
- = 1.257×10⁶ × 28×10⁻¹² / 80
- = 35.2×10⁻⁶ / 80
- = 0.44 μS
-
-G_total = ω_L C_eq / Q_L
- = 1.162×10⁶ × 32.7×10⁻¹² / 25
- = 38.0×10⁻⁶ / 25
- = 1.52 μS
-
-G_spark = G_total - G₀ = 1.52 - 0.44 = 1.08 μS
-```
-
-**Step 3:** Construct spark admittance
-```
-B_spark = ω_L ΔC = 1.162×10⁶ × 4.7×10⁻¹² = 5.46 μS
-
-Y_spark = G_spark + jB_spark
- = 1.08 + j5.46 μS
-```
-
-**Step 4:** Convert to impedance
-```
-|Y_spark| = √(1.08² + 5.46²) = √(1.17 + 29.8) = 5.56 μS
-
-Z_spark = 1/Y_spark
-|Z_spark| = 1/(5.56×10⁻⁶) = 180 kΩ
-
-φ_Y = atan(5.46/1.08) = atan(5.06) = 78.8°
-φ_Z = -78.8°
-
-Z_spark = 180 kΩ ∠-78.8°
-
-In rectangular:
-R = 180 × cos(-78.8°) = 180 × 0.194 = 35 kΩ
-X = 180 × sin(-78.8°) = 180 × (-0.981) = -177 kΩ
-
-Z_spark = 35 - j177 kΩ
-```
-
-**Interpretation:**
-- Spark added 4.7 pF capacitance (consistent with ~2.4 foot spark)
-- R ≈ 35 kΩ at 185 kHz
-- Strongly capacitive: φ_Z = -78.8°
-- Q dropped from 80 to 25 (spark loading dominates)
-
----
-
-### PRACTICE PROBLEMS 2.7
-
-**Problem 1:** A ringdown shows voltage dropping from 100 kV to 50 kV in 8 cycles at f = 195 kHz. Calculate Q.
-
-**Problem 2:** Measurements show: f₀ = 210 kHz, Q₀ = 65, f_L = 198 kHz (with spark), Q_L = 30. If C₀ = 25 pF, calculate the spark's added capacitance and equivalent resistance.
-
-**Problem 3:** Why does frequency decrease when a spark forms? Explain in terms of capacitance.
-
----
-
-## Part 2 Summary & Integration
-
-### Key Concepts Checklist
-
-- [ ] **Topological phase constraint:** φ_Z,min = -atan(2√[r(1+r)])
-- [ ] **Critical ratio:** r ≥ 0.207 makes φ_Z = -45° impossible
-- [ ] **R_opt_phase:** Minimizes |φ_Z|, gives φ_Z,min
-- [ ] **R_opt_power:** Maximizes power transfer to load
-- [ ] **Relationship:** R_opt_power < R_opt_phase always
-- [ ] **Hungry streamer:** Plasma self-adjusts toward R_opt_power
-- [ ] **Physical limits:** R_min (hot plasma) to R_max (cold plasma)
-- [ ] **Why V_top/I_base fails:** Includes displacement currents
-- [ ] **Correct port:** Topload-to-ground
-- [ ] **Thévenin Z_th:** Output impedance (drive off, test on)
-- [ ] **Thévenin V_th:** Open-circuit voltage (drive on, no load)
-- [ ] **Power formula:** P = 0.5|V_th|²Re{Z_load}/|Z_th+Z_load|²
-- [ ] **Conjugate match:** Usually unachievable due to constraints
-- [ ] **Q from ringdown:** Q = πfΔt/ln(A₁/A₂)
-- [ ] **Extract Y_spark:** From frequency shift and Q change
-
----
-
-## Comprehensive Design Exercise
-
-**Scenario:** Design matching for a DRSSTC
-
-**Given:**
-- Operating frequency: f = 190 kHz
-- Topload: C_topload = 30 pF
-- Target spark: 3 feet (estimate C_sh)
-- FEMM analysis: C_mut = 9 pF for 3-foot spark
-- Thévenin equivalent (measured): Z_th = 105 - j2100 Ω, V_th = 320 kV
-
-**Tasks:**
-
-1. **Calculate capacitance ratio and phase constraint:**
- - Find r = C_mut/C_sh
- - Calculate φ_Z,min
- - Can this achieve -45°?
-
-2. **Determine optimal resistances:**
- - Calculate R_opt_power
- - Calculate R_opt_phase
- - What is typical φ_Z at R_opt_power?
-
-3. **Build lumped spark model:**
- - Draw circuit with C_mut, R, C_sh
- - Use R = R_opt_power
- - Calculate Y_spark
-
-4. **Predict performance with Thévenin:**
- - Calculate Z_spark from Y_spark
- - Find total impedance Z_th + Z_spark
- - Calculate spark current
- - Calculate power delivered to spark
-
-5. **Compare to theoretical maximum:**
- - Calculate P_max (conjugate match)
- - What percentage is actually delivered?
- - Explain the difference
-
-**Work through this completely, then check solutions in appendix.**
-
----
-
-**END OF PART 2**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From
-
----
-
-## Module 3.1: Electric Fields and Breakdown
-
-### Electric Field Basics
-
-**Definition:** Electric field E is force per unit charge:
-```
-E = F/q [units: N/C or V/m]
-
-Related to voltage:
-E = -dV/dx (field is voltage gradient)
-
-For uniform field:
-E ≈ V/d (voltage divided by distance)
-```
-
-**Field at spark tip is NOT uniform** - concentrated by geometry.
-
-### Breakdown Field Thresholds
-
-**E_inception:** Field required to initiate breakdown from smooth electrode
-```
-E_inception ≈ 2-3 MV/m (at sea level, dry air)
-
-Physical process:
-- Natural cosmic rays create seed electrons
-- Strong field accelerates electrons
-- Collisions ionize more atoms
-- Avalanche breakdown begins
-```
-
-**E_propagation:** Field required to sustain spark growth
-```
-E_propagation ≈ 0.4-1.0 MV/m (for leader propagation)
-
-Lower than inception because:
-- Channel already partially ionized
-- Hot gas easier to ionize
-- Photoionization helps (UV from plasma)
-```
-
-**Altitude/humidity effects:**
-- Lower air density (altitude) → lower E_threshold (±20-30%)
-- Humidity adds water vapor → changes breakdown (~10%)
-- Temperature affects density → small effect
-
-### Tip Enhancement Factor κ
-
-Sharp tips concentrate field:
-
-```
-E_tip = κ × E_average
-
-where E_average = V/L (voltage divided by length)
- κ = enhancement factor ≈ 2-5 typical
-```
-
-**Physical origin:**
-- Charge accumulates at sharp points
-- Field lines concentrate at high curvature
-- Smaller radius → higher κ
-
-**FEMM calculates E_tip directly** from geometry and voltage.
-
-### Growth Criterion
-
-Spark continues growing when:
-```
-E_tip > E_propagation
-
-If E_tip drops below E_propagation:
-- Growth stalls
-- Spark cannot extend further
-- "Voltage-limited"
-```
-
----
-
-### WORKED EXAMPLE 3.1: Field Calculation
-
-**Given:**
-- Spark length: L = 1.5 m
-- Topload voltage: V_top = 400 kV
-- Tip enhancement: κ = 3.5 (from FEMM or estimate)
-
-**Find:**
-(a) Average field
-(b) Tip field
-(c) Can spark grow if E_propagation = 0.6 MV/m?
-
-**Solution:**
-
-**Part (a):** Average field
-```
-E_average = V_top / L
- = 400×10³ V / 1.5 m
- = 267 kV/m
- = 0.267 MV/m
-```
-
-**Part (b):** Tip field
-```
-E_tip = κ × E_average
- = 3.5 × 0.267 MV/m
- = 0.93 MV/m
-```
-
-**Part (c):** Compare to threshold
-```
-E_tip = 0.93 MV/m
-E_propagation = 0.6 MV/m
-
-E_tip > E_propagation ✓
-
-Yes, spark can continue growing.
-Margin: 0.93/0.6 = 1.55× above threshold
-```
-
-**If voltage drops to 300 kV:**
-```
-E_average = 300 kV / 1.5 m = 0.2 MV/m
-E_tip = 3.5 × 0.2 = 0.7 MV/m
-
-Still above 0.6 MV/m, but margin reduced to 1.17×
-```
-
-**If voltage drops to 250 kV:**
-```
-E_average = 250 kV / 1.5 m = 0.167 MV/m
-E_tip = 3.5 × 0.167 = 0.58 MV/m
-
-Below 0.6 MV/m - growth stalls!
-```
-
----
-
-### VISUAL AID 3.1: Field Enhancement
-
-```
-[Describe for drawing:]
-
-Two panels side-by-side:
-
-LEFT: Uniform field (parallel plates)
-- Two flat plates, voltage V between them
-- Evenly spaced field lines (vertical)
-- Formula: E = V/d (constant everywhere)
-- Label: "No enhancement, κ = 1"
-
-RIGHT: Point-to-plane (spark geometry)
-- Spherical topload at top (voltage V)
-- Sharp spark tip pointing down
-- Ground plane at bottom
-- Field lines:
- * Sparse near topload (low density)
- * Dense at tip (concentrated)
- * Spread out below tip
-- Color gradient showing field strength:
- * Blue (low) far from tip
- * Red (high) at tip
-- Annotations:
- * E_average = V/L marked along spark
- * E_tip at very tip (red zone)
- * "Enhancement: E_tip = κ × E_average, κ = 2-5"
-
-Inset graph: E vs distance from tip
-- Sharp peak at tip (E_tip)
-- Drops rapidly with distance
-- Approaches E_average far from tip
-```
-
----
-
-### PRACTICE PROBLEMS 3.1
-
-**Problem 1:** A 0.8 m spark has V_top = 280 kV, κ = 4. Calculate E_tip. If E_propagation = 0.5 MV/m, can it grow?
-
-**Problem 2:** A spark stalls at 2.0 m length with V_top = 500 kV and κ = 3. Estimate E_propagation for these conditions.
-
-**Problem 3:** Why is E_inception > E_propagation? Explain the physical difference.
-
----
-
-## Module 3.2: Energy Requirements for Growth
-
-### Energy Per Meter (ε)
-
-**Concept:** Extending spark by 1 meter requires approximately constant energy:
-
-```
-Energy to grow from L₁ to L₂:
-ΔE ≈ ε × (L₂ - L₁)
-
-where ε [J/m] depends on operating mode
-```
-
-**Not just ionization energy** - includes:
-1. Initial ionization (breaking molecular bonds)
-2. Heating to operating temperature
-3. Work against pressure (channel expansion)
-4. Radiation losses (light, UV, RF)
-5. Branching (wasted energy in short branches)
-6. Inefficiency (non-productive heating)
-
-### Typical ε Values by Operating Mode
-
-**QCW (Quasi-Continuous Wave):**
-```
-ε ≈ 5-15 J/m
-
-Characteristics:
-- Long ramp times (5-20 ms)
-- Channel stays hot throughout growth
-- Efficient leader formation
-- Minimal re-ionization
-```
-
-**Hybrid DRSSTC (moderate duty cycle):**
-```
-ε ≈ 20-40 J/m
-
-Characteristics:
-- Medium pulses (1-5 ms)
-- Mix of streamers and leaders
-- Some thermal accumulation
-- Moderate efficiency
-```
-
-**Burst mode (hard-pulsed):**
-```
-ε ≈ 30-100+ J/m
-
-Characteristics:
-- Short pulses (<500 μs)
-- Channel cools between pulses
-- Mostly streamers, bright but short
-- Must re-ionize repeatedly
-- Poor length efficiency
-```
-
-### Why Different Modes Have Different ε
-
-**QCW efficiency (low ε):**
-- Continuous power → channel stays ionized
-- Thermal ionization maintained
-- Leaders form efficiently
-- Each Joule goes into extension
-
-**Burst inefficiency (high ε):**
-- Peak power → brightening, branching
-- Channel cools between bursts
-- Energy into light, heat, not length
-- Must restart from cold each time
-
-**Analogy:** Boiling water
-- Low ε: Keep burner on, maintain simmer (efficient)
-- High ε: Pulse burner on/off, water cools (inefficient)
-
-### Theoretical Minimum Energy
-
-**Just ionization:**
-```
-Ionization energy per molecule ≈ 15 eV
-Air density ≈ 2.5×10²⁵ molecules/m³
-Channel volume ≈ π(d/2)² × L
-
-For d = 1 mm, L = 1 m:
-E_ionize = 15 eV × 2.5×10²⁵ × π×(0.5×10⁻³)² × 1
- ≈ 0.3 J/m (theoretical minimum)
-```
-
-**Why ε >> 0.3 J/m?**
-- Heating to 5000-20000 K (thermal energy)
-- Radiation (visible light, UV, IR)
-- Expansion work (push air aside)
-- Branching losses (many failed attempts)
-- Inefficiencies (not all current goes to useful ionization)
-
-**Result:** Real ε is 20-300× theoretical minimum.
-
----
-
-### WORKED EXAMPLE 3.2: Energy Budget
-
-**Given:**
-- Target spark: L = 2 m
-- Operating mode: QCW with ε = 10 J/m
-- Growth time: T = 12 ms
-
-**Find:**
-(a) Total energy required
-(b) Average power required
-(c) If only 80 kW available, what happens?
-
-**Solution:**
-
-**Part (a):** Total energy
-```
-E_total = ε × L
- = 10 J/m × 2 m
- = 20 J
-```
-
-**Part (b):** Average power
-```
-P_avg = E_total / T
- = 20 J / 0.012 s
- = 1667 W
- ≈ 1.7 kW
-```
-
-**Part (c):** With limited power
-```
-Available: P = 80 kW (much more than needed!)
-
-This is 80/1.7 = 47× the required power.
-
-Options:
-1. Grow much faster: T = 20 J / 80 kW = 0.25 ms (burst-like)
-2. Grow to longer length: L = P × T / ε
- For same 12 ms: L = 80 kW × 0.012 s / 10 J/m = 96 m (unrealistic!)
-
-Reality: Voltage limit kicks in first
- - Cannot maintain E_tip > E_propagation for 96 m
- - Spark stalls at voltage-limited length
-```
-
-**Key insight:** Need BOTH adequate power AND adequate voltage!
-
----
-
-### PRACTICE PROBLEMS 3.2
-
-**Problem 1:** A burst-mode coil has ε = 60 J/m. To reach 1.5 m in a 200 μs pulse, what power is required?
-
-**Problem 2:** Two coils both deliver 50 kW. Coil A (QCW, ε = 8 J/m) vs Coil B (burst, ε = 50 J/m). For 10 ms operation, which produces longer sparks?
-
----
-
-## Module 3.3: Growth Rate and Stalling
-
-### The Growth Rate Equation
-
-When field threshold is met:
-```
-dL/dt = P_stream / ε [units: m/s]
-
-where P_stream = power delivered to spark [W]
- ε = energy per meter [J/m]
-```
-
-**Physical meaning:**
-- More power → faster growth
-- Higher ε (inefficiency) → slower growth
-
-**When growth stops:**
-```
-If E_tip < E_propagation:
- dL/dt = 0 (stalled)
-
-Cannot grow regardless of available power
-```
-
-### Voltage-Limited vs Power-Limited
-
-**Voltage-limited:**
-```
-E_tip < E_propagation
-- Field too weak at tip
-- Spark cannot extend
-- More power doesn't help (without more voltage)
-- Common for small topload, long target
-```
-
-**Power-limited:**
-```
-E_tip > E_propagation, but P_stream < ε × (dL/dt)_desired
-- Field adequate, but not enough energy
-- Spark grows slowly or stalls before reaching potential
-- More voltage doesn't help (without more power)
-- Common for high-Q coils, weak drive
-```
-
-### Predicting Growth Time
-
-For constant power during ramp:
-```
-L(t) = (P_stream / ε) × t
-
-Time to reach L_target:
-T = ε × L_target / P_stream
-```
-
-**More realistic:** Power changes as spark grows (loading changes)
-```
-T = ∫₀^L_target (ε / P_stream(L)) dL
-
-Requires simulation or numerical integration
-```
-
----
-
-### WORKED EXAMPLE 3.3: Growth Prediction
-
-**Given:**
-- QCW coil, ε = 12 J/m
-- Target: L = 1.8 m
-- Power profile: P_stream = 100 kW (constant during ramp)
-- κ = 3.2, E_propagation = 0.7 MV/m
-- V_top ramps linearly: V(t) = 50 kV/ms × t
-
-**Find:**
-(a) Growth time if power-limited
-(b) Growth time if voltage-limited
-(c) Actual growth (considering both limits)
-
-**Solution:**
-
-**Part (a):** Power-limited case (assume infinite voltage)
-```
-T_power = ε × L / P_stream
- = 12 J/m × 1.8 m / 100000 W
- = 21.6 J / 100000 W
- = 0.000216 s
- = 0.216 ms
-```
-
-**Part (b):** Voltage-limited case
-
-At length L, need E_tip > E_propagation:
-```
-E_tip = κ × V(t) / L > E_propagation
-V(t) > E_propagation × L / κ
-
-For L = 1.8 m:
-V_required > 0.7×10⁶ × 1.8 / 3.2
-V_required > 0.394 MV = 394 kV
-
-With ramp V(t) = 50 kV/ms × t:
-T_voltage = 394 kV / (50 kV/ms) = 7.88 ms
-```
-
-**Part (c):** Actual growth (limited by slowest)
-```
-T_power = 0.216 ms (very fast if voltage available)
-T_voltage = 7.88 ms (slower, limited by ramp rate)
-
-Actual: T ≈ 7.88 ms (voltage-limited)
-
-The spark grows as fast as voltage ramps allow.
-Power is MORE than sufficient (100 kW available, only need ~2.7 kW)
-```
-
-**Verification of power requirement:**
-```
-P_needed = ε × L / T_actual
- = 12 × 1.8 / 0.00788
- = 2.74 kW
-
-100 kW available >> 2.74 kW needed ✓
-Confirms voltage-limited, not power-limited
-```
-
----
-
-### VISUAL AID 3.3: Growth Curves
-
-```
-[Describe for plotting:]
-
-Graph: Spark length L vs time t
-
-Three curves:
-
-CURVE 1 (Blue): Power-limited
-- Linear growth: L(t) = (P/ε) × t
-- Steep slope (fast growth)
-- Reaches target quickly (0.2 ms)
-- Label: "Power-limited: unlimited voltage"
-
-CURVE 2 (Red): Voltage-limited
-- Curved growth: L(t) must satisfy E_tip(V(t),L) > E_prop
-- Slower, follows voltage ramp capability
-- Reaches target at 7.88 ms
-- Label: "Voltage-limited: slow ramp"
-
-CURVE 3 (Green): Actual (realistic)
-- Follows faster curve initially
-- Transitions to limiting constraint
-- Usually voltage-limited for Tesla coils
-- Label: "Actual: limited by slowest constraint"
-
-Shaded regions:
-- Below curves: "Achieved length"
-- Above: "Not yet reached"
-
-Annotations:
-- "QCW: usually voltage-limited"
-- "Burst: can be power-limited"
-- "Need both P and V adequate"
-```
-
----
-
-### PRACTICE PROBLEMS 3.3
-
-**Problem 1:** A spark grows at 2 m/s when P = 40 kW and ε = 20 J/m. Verify this is consistent with dL/dt = P/ε.
-
-**Problem 2:** If E_propagation = 0.5 MV/m, κ = 3, and voltage is fixed at V = 300 kV, what is the maximum length the spark can reach (voltage-limited)?
-
-**Problem 3:** A coil delivers 30 kW to a spark with ε = 15 J/m. How long to reach 2.5 m? If this time is longer than the voltage ramp allows, which limit dominates?
-
----
-
-## Module 3.4: Thermal Physics of Plasma Channels
-
-### Temperature Regimes
-
-**Streamers (cold):**
-```
-T ≈ 1000-3000 K
-- Weakly ionized
-- Mostly neutral gas with some ions/electrons
-- Purple/blue color (N₂ emission)
-```
-
-**Leaders (hot):**
-```
-T ≈ 5000-20000 K
-- Fully ionized plasma
-- White/orange color (blackbody + line emission)
-- Approaching temperatures of stellar photospheres!
-```
-
-### Thermal Diffusion Time
-
-Heat diffuses radially from hot channel core:
-```
-τ_thermal = d² / (4α_thermal)
-
-where d = channel diameter
- α_thermal ≈ 2×10⁻⁵ m²/s for air
-```
-
-**Examples:**
-```
-Thin streamer (d = 100 μm):
-τ = (100×10⁻⁶)² / (4 × 2×10⁻⁵)
- = 10⁻⁸ / (8×10⁻⁵)
- = 0.125 ms
-
-Thick leader (d = 5 mm):
-τ = (5×10⁻³)² / (4 × 2×10⁻⁵)
- = 25×10⁻⁶ / (8×10⁻⁵)
- = 312 ms
-```
-
-### Why Observed Persistence is Longer
-
-**Pure thermal diffusion** predicts cooling in 0.1-300 ms, but channels persist longer due to:
-
-**1. Convection (buoyancy):**
-```
-Hot gas rises: v ≈ √(g × d × ΔT/T_amb)
-
-For d = 2 mm, ΔT = 10000 K:
-v ≈ √(9.8 × 0.002 × 10000/300)
- ≈ √(0.65) ≈ 0.8 m/s
-
-Rising column remains hot longer than conduction alone
-```
-
-**2. Ionization memory:**
-```
-Recombination time: τ_recomb = 1/(α_recomb × n_e)
-Can be 10 μs to 10 ms depending on density
-Ions/electrons persist after thermal cooling begins
-```
-
-**Effective persistence:**
-```
-Streamers: ~1-5 ms (convection + ionization)
-Leaders: seconds (buoyant column maintained)
-```
-
-### QCW Advantage
-
-**QCW ramp times (5-20 ms) exploit channel persistence:**
-```
-1. Initial streamers form (t = 0)
-2. Power heats channel → leader begins (t = 1 ms)
-3. Leader maintained by continuous power (t = 1-20 ms)
-4. Channel stays hot entire time
-5. New growth builds on existing ionization
-6. Efficient energy use
-```
-
-**Burst mode problem:**
-```
-1. Pulse creates bright streamer (t = 0-0.1 ms)
-2. Pulse ends, channel cools (t = 0.1-1 ms)
-3. Next pulse must re-ionize cold gas (t = 1 ms)
-4. Energy wasted re-heating
-5. Inefficient (high ε)
-```
-
----
-
-### WORKED EXAMPLE 3.4: Thermal Time Constants
-
-**Given:**
-- Channel diameter: d = 2 mm (typical leader)
-- Air thermal diffusivity: α = 2×10⁻⁵ m²/s
-
-**Find:**
-(a) Pure thermal diffusion time
-(b) Estimate convection velocity if ΔT = 8000 K
-(c) QCW ramp time recommendation
-
-**Solution:**
-
-**Part (a):** Thermal diffusion
-```
-τ_thermal = d² / (4α)
- = (2×10⁻³)² / (4 × 2×10⁻⁵)
- = 4×10⁻⁶ / (8×10⁻⁵)
- = 0.05 s
- = 50 ms
-```
-
-**Part (b):** Convection velocity
-```
-v ≈ √(g × d × ΔT/T_amb)
- ≈ √(9.8 × 0.002 × 8000/300)
- ≈ √(0.523)
- ≈ 0.72 m/s
-
-Upward velocity helps maintain hot column
-```
-
-**Part (c):** QCW ramp recommendation
-```
-τ_thermal = 50 ms
-
-Good QCW ramp: T_ramp << τ_thermal (finish before significant cooling)
-Reasonable: T_ramp = 5-20 ms (10-40% of τ)
-
-If T_ramp >> τ_thermal:
- - Channel cools during ramp
- - Must reheat repeatedly
- - Loses QCW efficiency advantage
-```
-
----
-
-### PRACTICE PROBLEMS 3.4
-
-**Problem 1:** A streamer has d = 150 μm. Calculate τ_thermal. If burst pulse is 500 μs, does channel cool significantly during pulse?
-
-**Problem 2:** Why do thick leaders persist longer than thin streamers? Give two physical reasons.
-
----
-
-## Module 3.5: The Capacitive Divider Problem
-
-### Voltage Division Along Spark
-
-From Part 1, spark circuit:
-```
- [C_mut]
-Topload ----||---- Spark
- |
- [R]
- |
- [C_sh]
- |
- GND
-```
-
-**Voltage divider:** V_tip depends on impedance ratio:
-```
-V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
-
-where Z_mut = (1/jωC_mut) || R (parallel combination)
- Z_sh = 1/(jωC_sh)
-```
-
-### Open-Circuit Limit (No Current)
-
-When R → ∞ (no conduction), only capacitances matter:
-```
-V_tip = V_topload × C_mut / (C_mut + C_sh)
-```
-
-**Problem:** As spark grows, C_sh increases (∝ length):
-```
-C_sh ≈ 2 pF/foot × L
-
-As L increases → C_sh increases → V_tip decreases!
-```
-
-**Example:**
-```
-V_topload = 400 kV (constant)
-C_mut = 8 pF (approximately constant)
-
-Short spark (1 ft): C_sh = 2 pF
-V_tip = 400 × 8/(8+2) = 320 kV (80%)
-
-Medium spark (3 ft): C_sh = 6 pF
-V_tip = 400 × 8/(8+6) = 229 kV (57%)
-
-Long spark (6 ft): C_sh = 12 pF
-V_tip = 400 × 8/(8+12) = 160 kV (40%)
-```
-
-**Tip voltage drops to 40% even with constant topload voltage!**
-
-### With Finite Resistance
-
-Real case with R = R_opt_power ≈ 1/(ω(C_mut+C_sh)):
-
-```
-Z_mut = R || (1/jωC_mut) ≈ complex value
-V_tip is lower and phase-shifted
-
-Effect is similar but worse:
-- Magnitude division (as above)
-- Plus current-dependent voltage drop across R
-- V_tip drops faster than capacitive case alone
-```
-
-### Impact on Growth
-
-```
-E_tip = κ × V_tip / L
-
-As L increases:
-- Numerator (V_tip) decreases (capacitive division)
-- Denominator (L) increases (geometry)
-- E_tip decreases as L²
-
-Growth becomes progressively harder!
-```
-
-**Why sub-linear scaling:**
-```
-If energy scales as E ∝ L², but division effect makes
-V_tip ∝ 1/L, then achievable length L ∝ √E
-
-This explains Freau's empirical observation: L ∝ √E for burst mode
-```
-
----
-
-### WORKED EXAMPLE 3.5: Voltage Division
-
-**Given:**
-- V_topload = 350 kV (maintained constant)
-- C_mut = 10 pF
-- Spark grows from 0 to 4 feet
-
-**Find:** V_tip at L = 1, 2, 3, 4 feet (open-circuit approximation)
-
-**Solution:**
-
-**At L = 1 ft:**
-```
-C_sh = 2 pF/ft × 1 ft = 2 pF
-
-V_tip = 350 kV × 10/(10+2)
- = 350 × 10/12
- = 292 kV (83% of V_topload)
-```
-
-**At L = 2 ft:**
-```
-C_sh = 4 pF
-
-V_tip = 350 × 10/14
- = 250 kV (71%)
-```
-
-**At L = 3 ft:**
-```
-C_sh = 6 pF
-
-V_tip = 350 × 10/16
- = 219 kV (63%)
-```
-
-**At L = 4 ft:**
-```
-C_sh = 8 pF
-
-V_tip = 350 × 10/18
- = 194 kV (55%)
-```
-
-**Summary table:**
-
-| Length | C_sh | V_tip | % of V_top |
-|--------|------|-------|------------|
-| 1 ft | 2 pF | 292 kV| 83% |
-| 2 ft | 4 pF | 250 kV| 71% |
-| 3 ft | 6 pF | 219 kV| 63% |
-| 4 ft | 8 pF | 194 kV| 55% |
-
-**Voltage drops almost linearly with length, making further extension difficult.**
-
----
-
-### PRACTICE PROBLEMS 3.5
-
-**Problem 1:** V_top = 300 kV, C_mut = 12 pF. Calculate V_tip for L = 2 ft and L = 5 ft. What percentage is lost?
-
-**Problem 2:** If E_propagation = 0.6 MV/m and κ = 3, what V_tip is needed for 2 m spark? Using C_mut = 8 pF, what V_topload is required?
-
----
-
-## Module 3.6: Introduction to FEMM
-
-### What is FEMM?
-
-**FEMM = Finite Element Method Magnetics**
-- Free, open-source electromagnetic FEA software
-- 2D planar and axisymmetric problems
-- Electrostatic, magnetostatic, AC magnetic, thermal analysis
-
-**For Tesla coils:** Use electrostatic solver to extract capacitances
-
-**Download:** www.femm.info
-
-### Basic Workflow
-
-**1. Define geometry:**
-- Draw conductors (spark, topload, ground)
-- Define materials (air, metal)
-- Set boundaries (Dirichlet, Neumann)
-
-**2. Assign properties:**
-- Conductor potentials (voltages)
-- Material properties (permittivity)
-- Boundary conditions
-
-**3. Mesh:**
-- Automatic triangulation
-- Refinement near conductors
-
-**4. Solve:**
-- Numerical solution of Laplace's equation
-- ∇²V = 0 in free space
-
-**5. Post-process:**
-- Extract capacitance matrix
-- Calculate electric fields
-- Visualize field lines
-
-### Problem Setup for Spark
-
-**Geometry:**
-```
-- Toroidal topload (axisymmetric)
-- Cylindrical spark channel (vertical)
-- Ground plane (large boundary)
-- Air region (surrounds everything)
-```
-
-**Materials:**
-```
-- Air: ε_r = 1.0
-- Conductors: Set potentials, not material
-```
-
-**Boundaries:**
-```
-- Outer boundary: V = 0 (grounded, far from coil)
-- Axisymmetric boundary: special condition (mirror)
-```
-
-**Potentials:**
-```
-- Topload: 1 V (arbitrary, will scale)
-- Spark: floating (capacitance extraction)
-- Ground: 0 V
-```
-
----
-
-### WORKED EXAMPLE 3.6: FEMM Tutorial (Conceptual)
-
-**Task:** Extract C_mut and C_sh for 1 m spark from 30 cm toroid
-
-**Step 1: Geometry (axisymmetric)**
-```
-r-z coordinates (cylindrical)
-- Toroid: major radius 15 cm, minor radius 5 cm, center at z = 0
-- Spark: cylinder radius 1 mm, extends from z = -5 cm to z = -105 cm
-- Ground plane: z = -120 cm (large disk)
-- Outer boundary: r = 200 cm, z = ±150 cm (large region)
-```
-
-**Step 2: Materials**
-```
-- Everything is "Air" (ε_r = 1)
-- Will assign potentials, not conductivities
-```
-
-**Step 3: Boundaries**
-```
-- r = 0: Axisymmetric boundary (axis of symmetry)
-- Outer box: V = 0 (Dirichlet)
-```
-
-**Step 4: Conductors**
-```
-Create 3 conductor groups:
-- Conductor 1: Topload surface, V = 1V
-- Conductor 2: Spark surface, floating (no fixed potential)
-- Conductor 3: Ground plane, V = 0V
-```
-
-**Step 5: Mesh and solve**
-```
-- Auto mesh: ~5000 elements typical
-- Solve electrostatic problem
-- Convergence <0.001%
-```
-
-**Step 6: Extract capacitance matrix**
-```
-FEMM outputs 3×3 Maxwell capacitance matrix [C]:
-
- Top Spark Ground
-Top [ 30 -8 -22 ] pF
-Spark [ -8 14 -6 ] pF
-Ground[ -22 -6 28 ] pF
-
-(Values are example)
-```
-
-**Step 7: Calculate C_mut and C_sh**
-```
-C_mut = |C[Top, Spark]| = |-8| = 8 pF
-
-C_sh = C[Spark, Spark] + C[Spark, Top]
- = 14 + (-8)
- = 6 pF
-
-Validation: 6 pF ≈ 2 pF/ft × 3.3 ft ✓
-```
-
----
-
-### VISUAL AID 3.6: FEMM Interface
-
-```
-[Describe for screenshot annotation:]
-
-FEMM main window with four panels:
-
-UPPER LEFT: Geometry editor
-- Drawing tools (point, line, arc, circle)
-- Coordinate display (r, z in cm)
-- Toroid drawn as rotated circle
-- Spark as vertical line segment
-- Ground as horizontal line
-- All in r-z plane (axisymmetric)
-
-UPPER RIGHT: Problem definition
-- Properties: Frequency = 0 (electrostatic)
-- Length units: centimeters
-- Problem type: Axisymmetric
-- Precision: 1e-8
-
-LOWER LEFT: Mesh view
-- Triangle mesh covering domain
-- Refined near conductors (smaller triangles)
-- Coarse far away (larger triangles)
-- Color = element size
-
-LOWER RIGHT: Solution view
-- Filled contours: equipotential lines (V)
-- Field vectors: E field (arrows)
-- Concentrated at topload and spark tip
-- Circuit property window showing capacitances
-```
-
----
-
-### PRACTICE PROBLEMS 3.6
-
-**Problem 1:** Why do we use V = 1 V instead of actual voltage (400 kV)? (Hint: electrostatics is linear)
-
-**Problem 2:** A FEMM simulation with 2 m spark gives C_sh = 14 pF. Does this match the empirical 2 pF/ft rule? (Show calculation)
-
----
-
-## Module 3.7: Extracting Capacitances from FEMM
-
-### The Maxwell Capacitance Matrix
-
-FEMM outputs matrix [C] where:
-```
-[Q] = [C] × [V]
-
-Q_i = charge on conductor i
-V_i = potential of conductor i
-
-Matrix properties:
-- Symmetric: C_ij = C_ji
-- Diagonal positive: C_ii > 0
-- Off-diagonal negative: C_ij < 0 for i≠j
-- Row sums to zero: Σ_j C_ij = 0
-```
-
-**Physical meaning:**
-- C_ii: self-capacitance (conductor i to infinity)
-- C_ij (i≠j): mutual capacitance (coupling between i and j, negative)
-
-### Two-Body System (Topload + Spark)
-
-Matrix for topload (1), spark (2), ground (implicit):
-```
- [1] [2]
-[1] [ C₁₁ C₁₂ ]
-[2] [ C₂₁ C₂₂ ]
-
-Example values:
- [Top] [Spark]
-[Top] [ 30 -8 ] pF
-[Spark][ -8 14 ] pF
-```
-
-### Extraction Formulas
-
-**C_mut (mutual capacitance):**
-```
-C_mut = |C₁₂| = |C₂₁|
-
-Take absolute value of off-diagonal element
-```
-
-**C_sh (spark to ground):**
-
-Method 1 - From row sum:
-```
-Ground capacitance = -(C₂₁ + C₂₂)
-But we want spark-to-ground only: C_sh
-
-C_sh = C₂₂ + C₂₁
- = C₂₂ - |C₁₂| (since C₂₁ = C₁₂ < 0)
-```
-
-Method 2 - Direct measurement:
-```
-Run second simulation with topload grounded
-Measure spark capacitance to ground directly
-```
-
-**Validation check:**
-```
-C_sh ≈ 2 pF/foot × L_spark
-
-If ratio is 1.5-2.5 pF/foot: good
-If significantly different: check geometry/mesh
-```
-
----
-
-### WORKED EXAMPLE 3.7: Matrix Interpretation
-
-**Given FEMM output:**
-```
-Conductor properties:
-Conductor 1 (Topload): 35.2 pF to ground
-Conductor 2 (Spark): 16.8 pF to ground
-
-Circuit properties:
-C[1,1] = 35.2 pF
-C[1,2] = -10.5 pF
-C[2,1] = -10.5 pF (symmetry)
-C[2,2] = 16.8 pF
-
-Spark length: 1.8 m = 5.9 ft
-```
-
-**Extract:**
-(a) C_mut
-(b) C_sh
-(c) Validate against empirical rule
-
-**Solution:**
-
-**Part (a):** Mutual capacitance
-```
-C_mut = |C[1,2]| = |-10.5| = 10.5 pF
-```
-
-**Part (b):** Shunt capacitance
-```
-C_sh = C[2,2] + C[2,1]
- = 16.8 + (-10.5)
- = 6.3 pF
-```
-
-**Part (c):** Validation
-```
-Empirical prediction:
-C_sh_predicted = 2 pF/ft × 5.9 ft = 11.8 pF
-
-FEMM result:
-C_sh_FEMM = 6.3 pF
-
-Ratio: 6.3 / 11.8 = 0.53
-
-This is LOWER than expected (by factor ~2)
-```
-
-**Possible explanations:**
-```
-1. Empirical rule assumes straight vertical spark
- - If spark is angled or curved, less capacitance
-
-2. Empirical rule from community measurements
- - May include some C_mut in "measured" value
- - Pure C_sh might be lower
-
-3. Ground plane distance matters
- - FEMM has specific ground geometry
- - Empirical rule assumes "typical" room
-
-4. Diameter assumption
- - Thinner diameter → lower C_sh (logarithmic)
-
-For modeling: Use FEMM value (more accurate for specific geometry)
-```
-
----
-
-### VISUAL AID 3.7: Capacitance Matrix Interpretation
-
-```
-[Describe for diagram:]
-
-Left: Physical picture
-- Topload (labeled "1")
-- Spark channel (labeled "2")
-- Ground plane (labeled "0" or implicit)
-- Field lines showing:
- * C₁₁: Topload to infinity (self)
- * C₂₂: Spark to infinity (self)
- * C₁₂: Topload to spark (mutual, shown in green)
-
-Center: Matrix representation
-```
-[C] = [ 35.2 -10.5 ]
- [-10.5 16.8 ]
-```
-- Diagonal highlighted (positive)
-- Off-diagonal highlighted (negative)
-- Symmetry shown with arrows
-
-Right: Circuit extraction
-- C_mut = |C₁₂| = 10.5 pF (between topload and spark)
-- C_sh = C₂₂ - |C₁₂| = 6.3 pF (spark to ground)
-- Circuit diagram showing extracted values
-
-Bottom: Key points
-- "Off-diagonal → mutual capacitance"
-- "Diagonal - mutual → shunt capacitance"
-- "Always check symmetry: C₁₂ = C₂₁"
-```
-
----
-
-### PRACTICE PROBLEMS 3.7
-
-**Problem 1:** FEMM gives C[1,1]=40 pF, C[1,2]=-12 pF, C[2,2]=20 pF for a 2 m spark. Extract C_mut and C_sh. Does C_sh match the empirical rule?
-
-**Problem 2:** Why are off-diagonal elements negative in the Maxwell matrix? What would happen if they were positive?
-
----
-
-## Module 3.8: Building the Lumped Spark Model
-
-### Complete Workflow
-
-**Step 1: FEMM electrostatic analysis**
-```
-- Geometry: topload + spark + ground
-- Axisymmetric 2D
-- Solve at frequency = 0 (electrostatic)
-- Extract [C] matrix
-```
-
-**Step 2: Calculate circuit elements**
-```
-C_mut = |C₁₂| from matrix
-C_sh = C₂₂ - |C₁₂| from matrix
-R = R_opt_power = 1/(ω(C_mut + C_sh))
-Clip to physical bounds: R = clip(R, R_min, R_max)
-```
-
-**Step 3: Build SPICE netlist**
-```
-* Lumped spark model
-.param freq=200k
-.param omega={2*pi*freq}
-
-V_topload topload 0 AC 1 ; 1V test source
-
-C_mut topload spark_node {C_mut}
-R_spark spark_node spark_r {R}
-C_sh spark_r 0 {C_sh}
-
-.ac lin 1 {freq} {freq}
-.print ac v(topload) i(V_topload)
-.end
-```
-
-**Step 4: Run AC analysis**
-```
-- Calculate Y = I/V at topload port
-- Extract Re{Y}, Im{Y}
-- Convert to Z if needed
-- Calculate power: P = 0.5 × |V|² × Re{Y}
-```
-
-**Step 5: Validate**
-```
-- Check φ_Z in expected range (-55° to -75°)
-- Check R in physical range (kΩ to hundreds of kΩ)
-- Check C_sh ≈ 2 pF/ft ± factor of 2
-- Compare to measurements if available
-```
-
-### Integration with Full Coil Model
-
-```
-[Primary circuit] → [Coupled transformer] → [Secondary] → [Topload] → [Spark model]
-
-Spark model appears as:
-- Load impedance at topload port
-- Affects loaded Q, resonant frequency
-- Extracts power from secondary
-```
-
----
-
-### WORKED EXAMPLE 3.8: Complete Lumped Model
-
-**Given:**
-- Frequency: f = 190 kHz
-- FEMM results: C_mut = 9.5 pF, C_sh = 7.2 pF
-- Physical bounds: R_min = 5 kΩ, R_max = 500 kΩ
-
-**Build and analyze model:**
-
-**Step 1:** Calculate R_opt_power
-```
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-C_total = C_mut + C_sh = 9.5 + 7.2 = 16.7 pF
-
-R_opt_power = 1/(ω × C_total)
- = 1/(1.194×10⁶ × 16.7×10⁻¹²)
- = 1/(19.94×10⁻⁶)
- = 50.2 kΩ
-```
-
-**Step 2:** Check bounds
-```
-R_min = 5 kΩ
-R_opt = 50.2 kΩ
-R_max = 500 kΩ
-
-5 < 50.2 < 500 ✓
-
-Use R = 50.2 kΩ
-```
-
-**Step 3:** Build SPICE model
-```
-* Spark lumped model - 190 kHz
-V_test topload 0 AC 1V
-C_mut topload n1 9.5p
-R_spark n1 n2 50.2k
-C_sh n2 0 7.2p
-
-.ac lin 1 190k 190k
-.print ac v(topload) i(V_test) vp(topload) ip(V_test)
-.end
-```
-
-**Step 4:** Simulate and extract (example results)
-```
-Simulation output:
-V(topload) = 1.000 V ∠0°
-I(V_test) = 5.23×10⁻⁶ A ∠74.5°
-
-Y = I/V = 5.23 μS ∠74.5°
-
-Re{Y} = 5.23 × cos(74.5°) = 1.39 μS
-Im{Y} = 5.23 × sin(74.5°) = 5.04 μS
-
-Convert to Z:
-|Z| = 1/5.23×10⁻⁶ = 191 kΩ
-φ_Z = -74.5°
-
-R_eq = 191 × cos(-74.5°) = 51 kΩ
-X_eq = 191 × sin(-74.5°) = -184 kΩ
-```
-
-**Step 5:** Validate
-```
-φ_Z = -74.5° : In expected range (-55° to -75°) ✓
-R_eq ≈ 51 kΩ : Close to R_opt = 50.2 kΩ ✓
-Physical: Between 5-500 kΩ ✓
-
-C_sh validation:
-L ≈ 7.2 pF / 2 pF/ft = 3.6 ft ≈ 1.1 m
-Reasonable for medium spark ✓
-```
-
-**Step 6:** Power calculation (if V_topload = 320 kV actual)
-```
-P = 0.5 × |V|² × Re{Y}
- = 0.5 × (320×10³)² × 1.39×10⁻⁶
- = 0.5 × 1.024×10¹¹ × 1.39×10⁻⁶
- = 71.2 kW
-```
-
-Model is complete and ready for coil integration!
-
----
-
-### PRACTICE PROBLEMS 3.8
-
-**Problem 1:** Build lumped model for: f=200 kHz, C_mut=11 pF, C_sh=9 pF. Calculate all component values and expected φ_Z.
-
-**Problem 2:** If SPICE simulation gives φ_Z=-85° (more capacitive than expected), what might be wrong with the model?
-
----
-
-## Part 3 Summary & Integration
-
-### Key Concepts Checklist
-
-- [ ] **E_inception:** ~2-3 MV/m to start breakdown
-- [ ] **E_propagation:** ~0.4-1.0 MV/m to sustain growth
-- [ ] **Tip enhancement:** E_tip = κ × E_avg, κ ≈ 2-5
-- [ ] **Growth criterion:** E_tip > E_propagation required
-- [ ] **Energy per meter ε:** 5-15 (QCW), 30-100 (burst) J/m
-- [ ] **Growth rate:** dL/dt = P/ε when field adequate
-- [ ] **Voltage vs power limited:** Both constraints exist
-- [ ] **Thermal time:** τ = d²/(4α), but persistence longer
-- [ ] **QCW advantage:** Maintains hot channel (low ε)
-- [ ] **Capacitive divider:** V_tip drops as C_sh grows
-- [ ] **Sub-linear scaling:** L ∝ √E for voltage-limited
-- [ ] **FEMM workflow:** Geometry → solve → extract [C]
-- [ ] **Maxwell matrix:** Diagonal positive, off-diagonal negative
-- [ ] **C_mut extraction:** |C₁₂| from off-diagonal
-- [ ] **C_sh extraction:** C₂₂ - |C₁₂|
-- [ ] **Validation:** C_sh ≈ 2 pF/ft ± factor 2
-- [ ] **Lumped model:** (R||C_mut) + C_sh
-- [ ] **R = R_opt_power:** For hungry streamer assumption
-
----
-
-## Final Integration Exercise
-
-**Complete design challenge:**
-
-**Given:**
-- DRSSTC at 185 kHz
-- Toroid: 40 cm major diameter, 10 cm minor
-- Target: 2 m spark
-- Thévenin: Z_th = 120 - j2200 Ω, V_th = 380 kV
-
-**Tasks:**
-
-1. **FEMM analysis (describe setup):**
- - Draw geometry for 2 m spark
- - What boundaries to use?
- - Expected C_sh range?
-
-2. **Assume FEMM gives:** C_mut = 11 pF, C_sh = 13 pF
- - Validate C_sh (empirical rule)
- - Calculate R_opt_power at 185 kHz
- - Is R within 5-500 kΩ bounds?
-
-3. **Build lumped model:**
- - Calculate Y_spark
- - Convert to Z_spark
- - What is φ_Z?
-
-4. **Predict performance:**
- - Calculate Z_total = Z_th + Z_spark
- - Find current I
- - Calculate power to spark
- - Compare to theoretical max (conjugate match)
-
-5. **Growth analysis:**
- - Assume QCW, ε = 10 J/m
- - How long to reach 2 m?
- - Check voltage requirement: E_prop = 0.6 MV/m, κ = 3.5
- - Is growth voltage-limited or power-limited?
-
-**This exercise integrates all of Part 3!**
-
----
-
-**END OF PART 3**
-
----
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Part 4: Advanced Topics - Distributed Models and Real-World Application
-
----
-
-## Module 4.1: Why Distributed Models?
-
-### Limitations of Lumped Models
-
-**Lumped model treats entire spark as single R, C_mut, C_sh:**
-
-**Works well for:**
-- Short sparks (<1 m)
-- Impedance matching studies
-- Quick optimization
-- First-order power estimates
-
-**Fails to capture:**
-```
-1. Current distribution along spark
- - Base carries full current
- - Tip may have much less (capacitive shunting)
-
-2. Voltage distribution
- - Not linear drop from top to tip
- - Capacitive divider effects at each point
-
-3. Tip vs base differences
- - Base: hot, well-coupled, low R
- - Tip: cool, weakly-coupled, high R
-
-4. Streamer/leader transitions
- - Base forms leader (low R)
- - Tip remains streamer (high R)
- - Lumped model averages this out
-
-5. Very long sparks (>3 m)
- - Distributed effects dominate
- - Single lumped R is poor approximation
-```
-
-### When to Use Distributed Model
-
-**Use distributed when:**
-- Spark length > 1-2 meters
-- Need current distribution (for measurements)
-- Studying leader/streamer physics
-- Validating against detailed measurements
-- Research/publication quality results
-
-**Stick with lumped when:**
-- Quick design iterations
-- Coil-level optimization (matching)
-- Spark length < 1 meter
-- Engineering estimates sufficient
-
-**Computational cost:**
-- Lumped: <1 second
-- Distributed (n=10): ~10-30 seconds
-- Distributed (n=20): ~1-5 minutes
-
----
-
-### VISUAL AID 4.1: Lumped vs Distributed Comparison
-
-```
-[Describe for diagram:]
-
-Two-panel comparison:
-
-LEFT: Lumped model
-- Single box representing entire spark
-- Three components: C_mut, R, C_sh
-- Simple circuit
-- One current value
-- One voltage drop
-- Label: "Good for <1m, fast computation"
-
-RIGHT: Distributed model (n=5 shown)
-- Spark divided into 5 segments
-- Each segment has: C_mutual[i], R[i], C_shunt[i]
-- Coupling between segments shown
-- Current arrows varying in size (large at base, small at tip)
-- Voltage nodes at each junction
-- Gradient showing R: low (blue) at base, high (red) at tip
-- Label: "Captures physics, slower computation"
-
-BOTTOM: Feature comparison table
-| Feature | Lumped | Distributed |
-|----------------------|--------|-------------|
-| Setup time | Fast | Slow |
-| Computation | <1s | 10s-min |
-| Current distribution| No | Yes |
-| Tip/base difference | No | Yes |
-| Accuracy <1m | Good | Excellent |
-| Accuracy >3m | Poor | Good |
-```
-
----
-
-### DISCUSSION QUESTIONS 4.1
-
-**Question 1:** A 0.5 m spark shows good agreement between lumped model and measurements. A 3 m spark shows poor agreement. Why?
-
-**Question 2:** If you only care about total power delivered to spark (not distribution), when would distributed model still be necessary?
-
-**Question 3:** In what situation might even a distributed model fail? (Hint: think about branching)
-
----
-
-## Module 4.2: nth-Order Model Structure
-
-### Segmentation Strategy
-
-**Divide spark into n equal-length segments:**
-```
-n = number of segments (typically 5-20)
-L_segment = L_total / n
-
-Segment numbering:
-i = 1: Base (connected to topload)
-i = 2, 3, ..., n-1: Middle sections
-i = n: Tip (furthest from topload)
-```
-
-**Why equal lengths?**
-- Simplifies FEMM geometry
-- Uniform discretization
-- Easy to implement
-- Non-uniform possible but more complex
-
-### Circuit Topology
-
-**Each segment i has:**
-```
-1. Resistance R[i]
- - Plasma resistance of that segment
- - Variable, to be optimized
-
-2. Mutual capacitances C[i,j]
- - Coupling to all other segments j≠i
- - And to topload (j=0)
- - Extracted from FEMM
-
-3. Shunt capacitance to ground
- - Included in capacitance matrix
- - Not a separate component
-```
-
-**Full network:**
-```
-Topload (node 0)
- |
- +-- C[0,1] -- Node 1 (base segment)
- | |
- | R[1]
- | |
- +-- C[0,2] ----+-- Node 2
- | |
- | R[2]
- | |
- ...
- |
- +-- C[0,n] ----+-- Node n (tip segment)
- |
- R[n]
- |
-
-Plus C[i,j] between all segment pairs
-Plus C[i,ground] for each segment to ground
-```
-
-**Complexity:** For n segments + topload:
-- (n+1)×(n+1) capacitance matrix
-- n resistance values
-- Total unknowns: n (resistances)
-
----
-
-### WORKED EXAMPLE 4.2: Draw 3-Segment Model
-
-**Given:**
-- Total spark: 1.5 m
-- Divide into n = 3 equal segments
-- Each segment: 0.5 m
-
-**Task:** Draw circuit topology (conceptual)
-
-**Solution:**
-
-```
-Topload (V_top, node 0)
- |
- +---[C[0,1]]---+---[C[0,2]]---+---[C[0,3]]---+
- | | | |
- | | | |
-Node 1 -------[R[1]]-------------|--------------|
-(base) | | |
- [C[1,2]] [C[1,3]] |
- | | |
- Node 2 -----------[R[2]]--------[C[2,3]]
- (middle) | |
- [C_sh,2] |
- | |
- Node 3 --------[R[3]]
- (tip) |
- [C_sh,3]
- |
- GND
-
-Where:
-- C[i,j] = mutual capacitance between segments
-- C_sh[i] = shunt capacitance segment i to ground
-- R[i] = resistance of segment i
-```
-
-**Note:** This is conceptual. Actual implementation uses full (n+1)×(n+1) matrix.
-
-**Typical values (estimated):**
-```
-Segment 1 (base): R[1] = 10 kΩ (hot, well-coupled)
-Segment 2 (mid): R[2] = 30 kΩ (moderate)
-Segment 3 (tip): R[3] = 100 kΩ (cool, weak coupling)
-
-C[0,1] > C[0,2] > C[0,3] (coupling decreases with distance)
-```
-
----
-
-### PRACTICE PROBLEMS 4.2
-
-**Problem 1:** A 2.4 m spark is divided into n=6 segments. What is the length of each segment? Number them from base to tip.
-
-**Problem 2:** For n=10 segments, how many capacitance matrix elements are there? (Count all C[i,j] including diagonal)
-
-**Problem 3:** Why might R[1] (base) be much smaller than R[10] (tip)? Give two physical reasons.
-
----
-
-## Module 4.3: FEMM for Distributed Models
-
-### Multi-Body Electrostatic Setup
-
-**Geometry definition:**
-```
-For n segments + topload → (n+1) conductors
-
-Example n=5:
-- Body 0: Toroid topload
-- Body 1: Cylinder, length L/5, base at topload
-- Body 2: Cylinder, length L/5, above body 1
-- Body 3: Cylinder, length L/5, above body 2
-- Body 4: Cylinder, length L/5, above body 3
-- Body 5: Cylinder, length L/5, top segment (tip)
-- Ground plane at bottom
-```
-
-**Axisymmetric setup:**
-```
-r-z coordinates
-All bodies as cylindrical sections
-Diameter: 1-3 mm typical (uniform for simplicity)
-Spacing: slight gap (~0.1 mm) between segments for FEMM
-```
-
-**Conductor properties:**
-```
-Group each body as separate conductor:
-- Conductor 0: Topload, V = 1V
-- Conductors 1-n: Spark segments, floating potential
-- Ground: V = 0V (boundary condition)
-```
-
-### Solving and Extraction
-
-**Mesh requirements:**
-```
-- Finer mesh near conductors
-- Refinement at segment junctions
-- Typical: 10,000-50,000 elements for n=10
-- Convergence: <0.01% error
-```
-
-**Capacitance matrix output:**
-```
-FEMM circuit properties → Capacitance matrix
-
-(n+1)×(n+1) symmetric matrix [C]:
-
- [0] [1] [2] ... [n]
-[0] [ C₀₀ C₀₁ C₀₂ ... C₀ₙ ]
-[1] [ C₁₀ C₁₁ C₁₂ ... C₁ₙ ]
-[2] [ C₂₀ C₂₁ C₂₂ ... C₂ₙ ]
-...
-[n] [ Cₙ₀ Cₙ₁ Cₙ₂ ... Cₙₙ ]
-
-Properties:
-- Symmetric: Cᵢⱼ = Cⱼᵢ
-- Diagonal positive: Cᵢᵢ > 0
-- Off-diagonal negative: Cᵢⱼ < 0 for i≠j
-- Row sum = 0: Σⱼ Cᵢⱼ = 0
-```
-
-### Matrix Validation
-
-**Check 1: Symmetry**
-```
-|C[i,j] - C[j,i]| / |C[i,j]| < 0.01
-If not symmetric: numerical error, refine mesh
-```
-
-**Check 2: Positive definite**
-```
-All eigenvalues should be ≥ 0
-One eigenvalue = 0 (ground reference freedom)
-Rest positive
-```
-
-**Check 3: Physical values**
-```
-Nearby segments: larger |C[i,j]|
-Distant segments: smaller |C[i,j]|
-Base segments: larger C[i,0] (topload coupling)
-Tip segments: smaller C[n,0]
-```
-
-**Check 4: Total shunt capacitance**
-```
-C_sh_total = Σᵢ (Cᵢᵢ - |Cᵢ₀|) for all spark segments
-
-Should be approximately:
-C_sh_total ≈ 2 pF/foot × L_total
-
-Within factor of 2 is reasonable
-```
-
----
-
-### WORKED EXAMPLE 4.3: FEMM Setup for n=5
-
-**Given:**
-- Spark length: 2.0 m = 6.56 feet
-- Diameter: 2 mm
-- n = 5 segments → each 0.4 m long
-- Topload: 30 cm toroid
-
-**FEMM procedure:**
-
-**Step 1: Geometry (r-z coordinates)**
-```
-Topload:
-- Major radius: 15 cm, minor radius: 5 cm
-- Center at z = 0
-- Lowest point: z = -5 cm
-
-Segment 1 (base):
-- r = 1 mm (0.1 cm)
-- z from -5 cm to -45 cm
-- Length: 40 cm
-
-Segment 2:
-- z from -45 cm to -85 cm
-
-Segment 3:
-- z from -85 cm to -125 cm
-
-Segment 4:
-- z from -125 cm to -165 cm
-
-Segment 5 (tip):
-- z from -165 cm to -205 cm
-
-Ground plane:
-- z = -220 cm (15 cm below tip)
-- r = 0 to 300 cm (large)
-
-Outer boundary:
-- r = 300 cm, z = ±250 cm
-```
-
-**Step 2: Materials and conductors**
-```
-All regions: Air (ε_r = 1)
-
-Define 6 conductor groups:
-Group 0: Topload surface, V = 1V
-Groups 1-5: Segment surfaces, floating
-Ground: Boundary at z = -220 cm, V = 0V
-```
-
-**Step 3: Meshing**
-```
-Auto mesh with refinement:
-- Triangle size near conductors: 0.5 mm
-- Triangle size at boundaries: 50 mm
-- ~25,000 elements total
-```
-
-**Step 4: Solve**
-```
-Problem type: Electrostatic, axisymmetric
-Frequency: 0 Hz
-Precision: 1e-8
-```
-
-**Step 5: Extract matrix (example results)**
-```
-Matrix [C] in pF:
-
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
-
-(Values are illustrative)
-```
-
-**Step 6: Validate**
-```
-Symmetry check: C[1,2] = C[2,1] = -2.8 ✓
-
-Total shunt capacitance (approximate):
-C_sh ≈ Σᵢ₌₁⁵ (Cᵢᵢ - |Cᵢ₀|)
- = (14.8-9.2) + (10.4-3.1) + (8.6-1.2) + (7.4-0.6) + (5.8-0.3)
- = 5.6 + 7.3 + 7.4 + 6.8 + 5.5
- = 32.6 pF
-
-Expected: 2 pF/ft × 6.56 ft = 13.1 pF
-
-Ratio: 32.6/13.1 = 2.5
-
-Higher than expected, but within factor of 2-3 (acceptable)
-Difference due to matrix interpretation method
-```
-
----
-
-### PRACTICE PROBLEMS 4.3
-
-**Problem 1:** For n=10 segments, 3 m total, what is each segment length? What is the z-coordinate range for segment 5 if topload bottom is at z=0?
-
-**Problem 2:** A capacitance matrix shows C[3,7] = -0.4 pF and C[3,4] = -2.1 pF. Which segments are closer to segment 3? Does this make physical sense?
-
----
-
-## Module 4.4: Implementing Capacitance Matrices in SPICE
-
-### The Challenge
-
-**Maxwell matrix has negative off-diagonals:**
-```
-Literal SPICE capacitor implementation:
-C_12 node1 node2 10p ← OK, positive value
-C_12 node1 node2 -10p ← ERROR! Negative capacitance unphysical
-```
-
-**Problem:** Cannot directly use C[i,j] < 0 as SPICE capacitors
-
-### Solution 1: Partial Capacitance Transformation
-
-**Convert Maxwell → Partial (all-positive):**
-
-**Partial capacitance:** Capacitance with all other nodes grounded
-
-```
-For node i:
-C_partial[i,j] = -C_Maxwell[i,j] for i≠j (flip sign!)
-C_partial[i,i] = Σⱼ |C_Maxwell[i,j]| (sum of magnitudes)
-
-All C_partial > 0 → can implement as SPICE capacitors
-```
-
-**SPICE implementation:**
-```
-* Partial capacitance method
-* Between every node pair i,j (i 1 (distant segments)
-```
-
-**When acceptable:**
-- Large n (>10): distant couplings small
-- Quick estimates
-- Weak segment-to-segment coupling
-
-**Validation:** Compare full vs approximate impedance
-
----
-
-### WORKED EXAMPLE 4.4: Partial Capacitance Conversion (3×3)
-
-**Given Maxwell matrix (topload + 2 segments):**
-```
- [0] [1] [2]
-[0] [ 30.0 -8.0 -2.0 ] pF
-[1] [ -8.0 14.0 -3.0 ] pF
-[2] [ -2.0 -3.0 9.0 ] pF
-```
-
-**Convert to partial (all-positive) for SPICE:**
-
-**Step 1:** Between-node capacitances (flip signs)
-```
-C_partial[0,1] = -C_Maxwell[0,1] = -(-8.0) = 8.0 pF
-C_partial[0,2] = -C_Maxwell[0,2] = -(-2.0) = 2.0 pF
-C_partial[1,2] = -C_Maxwell[1,2] = -(-3.0) = 3.0 pF
-```
-
-**Step 2:** Ground capacitances
-
-For each node, start with diagonal, subtract partial caps:
-
-**Node 0:**
-```
-C[0,0] = 30.0 pF
-Sum of partials leaving node 0: 8.0 + 2.0 = 10.0 pF
-C_partial[0,gnd] = 30.0 - 10.0 = 20.0 pF
-```
-
-**Node 1:**
-```
-C[1,1] = 14.0 pF
-Partials: 8.0 (to 0) + 3.0 (to 2) = 11.0 pF
-C_partial[1,gnd] = 14.0 - 11.0 = 3.0 pF
-```
-
-**Node 2:**
-```
-C[2,2] = 9.0 pF
-Partials: 2.0 (to 0) + 3.0 (to 1) = 5.0 pF
-C_partial[2,gnd] = 9.0 - 5.0 = 4.0 pF
-```
-
-**Step 3:** SPICE netlist
-```
-* Partial capacitance implementation
-* Between nodes
-C_0_1 node0 node1 8.0p
-C_0_2 node0 node2 2.0p
-C_1_2 node1 node2 3.0p
-
-* To ground
-C_0_gnd node0 0 20.0p
-C_1_gnd node1 0 3.0p
-C_2_gnd node2 0 4.0p
-
-* Resistances (to be determined)
-R1 node1 node1_r {R1_value}
-R2 node2 node2_r {R2_value}
-```
-
-**Validation:** Verify total capacitance node0→gnd matches:
-```
-With node1, node2 grounded:
-C_total = C_0_gnd + C_0_1 || C_1_gnd + C_0_2 || C_2_gnd
-
-Should equal approximately 30 pF (check numerically)
-```
-
----
-
-### PRACTICE PROBLEMS 4.4
-
-**Problem 1:** Given C_Maxwell = [25, -6; -6, 10] pF (2×2), convert to partial capacitances. Draw the SPICE circuit.
-
-**Problem 2:** Why can't we just use "negative capacitors" in SPICE? What would it physically mean?
-
-**Problem 3:** In nearest-neighbor approximation for n=10, how many capacitances are kept vs full matrix? Calculate percentage reduction.
-
----
-
-## Module 4.5: Resistance Optimization - Iterative Method
-
-### Algorithm Overview
-
-**Goal:** Find R[i] for each segment that maximizes total power
-
-**Challenge:** R[i] values are coupled (changing one affects power in others)
-
-**Solution:** Iterative optimization with damping
-
-### Initialization: Tapered Profile
-
-**Physical expectation:**
-- Base: hot, well-coupled → low R
-- Tip: cool, weakly-coupled → high R
-
-**Initialize with gradient:**
-```
-For i = 1 to n:
- position = (i-1)/(n-1) # 0 at base, 1 at tip
- R[i] = R_base + (R_tip - R_base) × position^2
-
-Typical starting values:
- R_base = 10 kΩ
- R_tip = 1 MΩ
-
-Quadratic taper gives smooth transition
-```
-
-### Iterative Optimization Loop
-
-```
-iteration = 0
-converged = False
-
-While not converged and iteration < max_iterations:
-
- For i = 1 to n:
- # Sweep R[i] while keeping other R[j] fixed
- R_test = logspace(R_min[i], R_max[i], 20 points)
-
- For each R_test_value:
- Set R[i] = R_test_value
- Run AC analysis
- Calculate P[i] = power in segment i
-
- Find R_optimal[i] = R_test that maximizes P[i]
-
- # Apply damping for stability
- R_new[i] = α * R_optimal[i] + (1-α) * R_old[i]
-
- # Clip to physical bounds
- R[i] = clip(R_new[i], R_min[i], R_max[i])
-
- # Check convergence
- max_change = max(|R_new[i] - R_old[i]| / R_old[i])
- If max_change < 0.01: # 1% threshold
- converged = True
-
- iteration = iteration + 1
-```
-
-**Damping factor α:**
-```
-α = 0.3 to 0.5 typical
-- Lower α: more stable, slower convergence
-- Higher α: faster, may oscillate
-- Start with α=0.3 for safety
-```
-
-### Position-Dependent Bounds
-
-**Physical limits vary with position:**
-```
-position = (i-1)/(n-1)
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
- = 1 kΩ at base → 10 kΩ at tip
-
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position^2
- = 100 kΩ at base → 100 MΩ at tip
-```
-
-**Rationale:**
-- Base can achieve very low R (hot leader)
-- Tip unlikely to reach low R (cool, weak coupling)
-- Prevents unphysical solutions
-
-### Convergence Behavior
-
-**Well-coupled base segments:**
-- Sharp power peak at optimal R
-- Fast convergence (2-3 iterations)
-- Stable solution
-
-**Weakly-coupled tip segments:**
-- Flat power curve (many R values similar power)
-- Slow/no convergence to unique value
-- May stay at high R (physical - streamer regime)
-
-**Expected result:**
-```
-R[1] ≈ 5-20 kΩ (base leader)
-R[2] ≈ 10-40 kΩ
-...
-R[n-1] ≈ 50-200 kΩ
-R[n] ≈ 100 kΩ - 10 MΩ (tip streamer)
-
-Total: Σ R[i] should be in expected range (5-300 kΩ at 200 kHz)
-```
-
----
-
-### WORKED EXAMPLE 4.5: Iterative Optimization (n=3, simplified)
-
-**Given:**
-- 3 segments, f = 200 kHz
-- Capacitance matrix (from FEMM, simplified)
-- Initial: R[1]=50k, R[2]=100k, R[3]=500k
-
-**Iteration 1:**
-
-**Optimize R[1] (keeping R[2], R[3] fixed):**
-```
-Sweep R[1] = [10k, 20k, 30k, 40k, 50k, 60k, 80k, 100k]
-
-Results (example):
-R[1]=10k → P[1]=5.2 kW
-R[1]=20k → P[1]=8.1 kW
-R[1]=30k → P[1]=9.4 kW ← maximum
-R[1]=40k → P[1]=8.9 kW
-R[1]=50k → P[1]=7.8 kW (current value)
-...
-
-R_optimal[1] = 30 kΩ
-```
-
-**Apply damping (α=0.4):**
-```
-R_new[1] = 0.4 × 30k + 0.6 × 50k
- = 12k + 30k
- = 42 kΩ
-```
-
-**Optimize R[2]:**
-```
-With R[1]=42k (updated), R[3]=500k (fixed)
-
-Sweep R[2], find R_optimal[2] = 60 kΩ
-Current: R[2] = 100 kΩ
-
-R_new[2] = 0.4 × 60k + 0.6 × 100k
- = 24k + 60k
- = 84 kΩ
-```
-
-**Optimize R[3]:**
-```
-With R[1]=42k, R[2]=84k
-
-Sweep R[3], power curve is FLAT:
-R[3]=200k → P[3]=0.8 kW
-R[3]=500k → P[3]=0.85 kW
-R[3]=1M → P[3]=0.83 kW
-
-Weakly coupled! Peak not well-defined.
-Keep at R[3] = 500 kΩ (within bounds, acceptable)
-```
-
-**After iteration 1:**
-```
-R[1]: 50k → 42k (change = -16%)
-R[2]: 100k → 84k (change = -16%)
-R[3]: 500k → 500k (change = 0%)
-
-Max change = 16% > 1% → not converged, continue
-```
-
-**Iteration 2:**
-
-Repeat process with new R values...
-(typically 3-5 iterations to converge for base/middle segments)
-
-**Final converged result (example):**
-```
-R[1] = 35 kΩ (leader, base)
-R[2] = 75 kΩ (transition)
-R[3] = 500 kΩ (streamer, tip - weakly determined)
-
-Total: 610 kΩ at 200 kHz
-Check: Within expected range ✓
-```
-
----
-
-### PRACTICE PROBLEMS 4.5
-
-**Problem 1:** Initial R=[100k, 200k], optimal found R=[60k, 150k]. With α=0.3, what are the damped updates?
-
-**Problem 2:** Why use damping factor α<1 instead of just setting R=R_optimal directly? What could go wrong?
-
-**Problem 3:** After 10 iterations, base segment converged (0.5% change) but tip segment still changing 5% per iteration. What should you do?
-
----
-
-## Module 4.6: Resistance Optimization - Simplified Method
-
-### Circuit-Determined Resistance
-
-**Key insight:** If plasma always seeks R_opt_power, and C depends weakly on diameter:
-
-```
-For each segment i:
- C_total[i] = sum of all capacitances involving segment i
- R[i] = 1 / (ω × C_total[i])
- R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Extracting C_total from matrix:**
-```
-C_total[i] = |C[i,0]| + Σⱼ₌₁ⁿ |C[i,j]| (sum of absolute values)
-
-This is total capacitance "seen" by segment i
-```
-
-### Why This Works
-
-**Physical argument:**
-
-1. Hungry streamer seeks R = 1/(ωC_total) for max power
-2. C depends on diameter: C ∝ 1/ln(h/d)
-3. Logarithmic dependence: 2× diameter → ~10% capacitance change
-4. R_opt also changes ~10% for diameter change
-5. Diameter adjusts to match R_opt (self-consistent)
-6. Error from fixed C is comparable to other uncertainties
-
-**Typical uncertainties:**
-```
-FEMM extraction: ±5-10%
-Plasma physics (ε, E_prop): ±30-50%
-Empirical calibration: ±20-30%
-
-Diameter approximation: ±10-15%
-
-Diameter error is SMALL compared to physics uncertainties!
-```
-
-### When to Use
-
-**Good for:**
-- Standard cases (typical geometries, frequencies)
-- First-pass analysis
-- Quick evaluation of many designs
-- Educational purposes
-
-**Use iterative when:**
-- Research/validation
-- Extreme parameters (very long, very short, very low frequency)
-- Measurement comparison requires highest accuracy
-- Publishing results
-
-**Computational savings:**
-```
-Iterative: 5-10 iterations × 20 R-sweep points × n segments = 1000-2000 AC analyses
-Simplified: 1 AC analysis
-
-Speedup: 1000-2000× faster!
-```
-
----
-
-### WORKED EXAMPLE 4.6: Simplified R Calculation (n=5)
-
-**Given:**
-- f = 190 kHz, ω = 1.194×10⁶ rad/s
-- Capacitance matrix from Example 4.3 (repeated):
-
-```
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
-```
-
-**Calculate R[i] for each segment:**
-
-**Segment 1 (base):**
-```
-C_total[1] = |C[1,0]| + |C[1,2]| + |C[1,3]| + |C[1,4]| + |C[1,5]|
- = 9.2 + 2.8 + 0.9 + 0.4 + 0.2
- = 13.5 pF
-
-R[1] = 1 / (ω × C_total[1])
- = 1 / (1.194×10⁶ × 13.5×10⁻¹²)
- = 1 / (16.12×10⁻⁶)
- = 62.0 kΩ
-```
-
-**Segment 2:**
-```
-C_total[2] = |C[2,0]| + |C[2,1]| + |C[2,3]| + |C[2,4]| + |C[2,5]|
- = 3.1 + 2.8 + 2.1 + 0.7 + 0.3
- = 9.0 pF
-
-R[2] = 1 / (1.194×10⁶ × 9.0×10⁻¹²)
- = 93.0 kΩ
-```
-
-**Segment 3:**
-```
-C_total[3] = 1.2 + 0.9 + 2.1 + 1.8 + 0.5
- = 6.5 pF
-
-R[3] = 1 / (1.194×10⁶ × 6.5×10⁻¹²)
- = 129 kΩ
-```
-
-**Segment 4:**
-```
-C_total[4] = 0.6 + 0.4 + 0.7 + 1.8 + 1.4
- = 4.9 pF
-
-R[4] = 1 / (1.194×10⁶ × 4.9×10⁻¹²)
- = 171 kΩ
-```
-
-**Segment 5 (tip):**
-```
-C_total[5] = 0.3 + 0.2 + 0.3 + 0.5 + 1.4
- = 2.7 pF
-
-R[5] = 1 / (1.194×10⁶ × 2.7×10⁻¹²)
- = 310 kΩ
-```
-
-**Summary:**
-```
-R[1] = 62 kΩ (base - lowest)
-R[2] = 93 kΩ
-R[3] = 129 kΩ
-R[4] = 171 kΩ
-R[5] = 310 kΩ (tip - highest)
-
-Total: R_total = 765 kΩ
-```
-
-**Validation:**
-```
-At 190 kHz for 2 m spark:
-Expected total: 50-300 kΩ (from Part 2 guidelines)
-
-765 kΩ is higher than typical.
-
-Possible reasons:
-- Long spark (2 m), distributed effects significant
-- Tip resistance (310k) is high (streamer-dominated)
-- If measured, could be lower (iterative optimization might find lower R)
-
-Within factor of 2-3 of expectations - acceptable for first pass
-```
-
----
-
-### PRACTICE PROBLEMS 4.6
-
-**Problem 1:** Given C_total[i] = [15, 10, 8, 6, 4] pF for n=5 at f=200 kHz, calculate R[i] for all segments.
-
-**Problem 2:** Compare simplified method: one calculation (1 second) vs iterative: 10 iterations × 20 points × 5 segments = 1000 AC analyses (~100 seconds). For engineering design, which is more appropriate?
-
----
-
-## Module 4.7: Quick Validation Checks
-
-### Power Balance
-
-**Energy conservation:**
-```
-P_input = P_spark + P_secondary_losses + P_corona + P_radiation + P_other
-
-Check: P_spark should be 30-70% of P_input for typical coil
-```
-
-**If P_spark > 90% of P_input:**
-- Secondary losses too low (unrealistic Q)
-- Check winding resistance, dielectric losses
-
-**If P_spark < 20% of P_input:**
-- Excessive secondary losses
-- Or spark model R too high (not optimized)
-
-### Total Resistance Range Check
-
-**Expected at 200 kHz for 1-3 m sparks:**
-```
-Burst/streamer-dominated: 50-300 kΩ
-QCW/leader-dominated: 5-50 kΩ
-Very low frequency (<100 kHz) or very long: 1-10 kΩ
-
-R_total = Σ R[i] should fall in expected range
-```
-
-**If outside range:**
-- Check frequency (R ∝ 1/f)
-- Check optimization convergence
-- Verify capacitance matrix extraction
-- Consider if mode is truly different (all-leader vs all-streamer)
-
-### Resistance Distribution Check
-
-**Physical expectation:**
-```
-R[1] < R[2] < R[3] < ... < R[n]
-
-Base should be lowest (hot, coupled)
-Tip should be highest (cool, weakly coupled)
-
-Monotonic increase expected
-```
-
-**If non-monotonic:**
-- Check capacitance matrix (may have errors)
-- Verify optimization didn't get stuck
-- Physical interpretation: local heating/cooling variation
-
-### Phase Angle Check
-
-**Total impedance phase:**
-```
-Calculate Z_total at topload port
-φ_Z should be -55° to -75° typical
-
-If φ_Z > -45°: Too resistive (check if topological constraint violated)
-If φ_Z < -85°: Too capacitive (R values too high, not optimized)
-```
-
-### Convergence Check
-
-**For distributed models with n=5, 10, 20:**
-```
-Run same problem with different n:
-- n=5 → Z_total, P_spark
-- n=10 → Z_total, P_spark
-- n=20 → Z_total, P_spark
-
-Should converge: changes <10% from n=10 to n=20
-
-If still changing >20%: need finer discretization
-```
-
----
-
-### WORKED EXAMPLE 4.7: Validation Exercise
-
-**Given simulation results:**
-```
-Coil: DRSSTC at 185 kHz
-P_primary_input = 150 kW
-P_spark = 105 kW (from distributed model n=10)
-Spark: 2.5 m
-
-Distributed R values [kΩ]:
-[18, 25, 35, 48, 65, 88, 120, 165, 230, 320]
-
-Z_total = 185 kΩ ∠-68°
-```
-
-**Validate:**
-
-**Check 1: Power balance**
-```
-P_spark / P_input = 105 / 150 = 0.70 = 70%
-
-Expected: 30-70% typical ✓
-Reasonable - some secondary losses, but spark dominates
-```
-
-**Check 2: Total resistance**
-```
-R_total = Σ R[i] = 18+25+35+48+65+88+120+165+230+320
- = 1114 kΩ
-
-At 185 kHz, expected: 50-300 kΩ for typical
-Actual: 1114 kΩ
-
-High, but this is 2.5 m spark (long)
-Factor of 3-4× over typical
-Could indicate:
-- Very streamer-dominated (burst mode?)
-- Or optimization not fully converged
-- Or long spark genuinely has higher R
-
-Flag for investigation, but not necessarily wrong ✓?
-```
-
-**Check 3: Resistance distribution**
-```
-R[1]=18 < R[2]=25 < R[3]=35 < ... < R[10]=320
-
-Monotonic increasing ✓
-Expected pattern (base lower, tip higher) ✓
-```
-
-**Check 4: Phase angle**
-```
-φ_Z = -68°
-
-Expected range: -55° to -75°
-Actual: -68°
-
-Right in the middle ✓
-Indicates reasonable capacitive loading
-```
-
-**Check 5: Compare to lumped model**
-```
-Lumped model (from earlier): R ≈ 600 kΩ at similar conditions
-
-Distributed: R_total = 1114 kΩ
-
-Distributed is higher (factor ~2)
-This can happen:
-- Distributed captures tip streamer high-R better
-- Lumped averages to middle value
-- For long sparks, distributed more accurate
-
-Consistent with expectations ✓
-```
-
-**Overall assessment:**
-- Most checks pass
-- Total R is high but potentially physical for long streamer spark
-- Recommend: compare to measurement if available
-- Model is usable for predictions
-
----
-
-### PRACTICE PROBLEMS 4.7
-
-**Problem 1:** Simulation shows P_spark = 180 kW but P_input = 150 kW. What's wrong?
-
-**Problem 2:** Distributed model gives R = [50, 45, 40, 35, 30] kΩ (decreasing from base to tip). Is this physical? What might be wrong?
-
-**Problem 3:** At 150 kHz, 1.8 m spark, you get R_total = 2 kΩ. Check against expected range. Is this reasonable?
-
----
-
-## Module 4.8: Complete Simulation Summary
-
-### Workflow Checklist
-
-**Phase 1: Geometry and FEMM**
-- [ ] Define spark length L_total
-- [ ] Choose n segments (typically 10)
-- [ ] Create FEMM geometry (axisymmetric)
-- [ ] Set up conductors (topload + n segments)
-- [ ] Mesh and solve electrostatic
-- [ ] Extract (n+1)×(n+1) capacitance matrix [C]
-- [ ] Validate: symmetry, positive definite, C_sh ≈ 2 pF/ft
-
-**Phase 2: Resistance Determination**
-- [ ] Choose method: iterative or simplified
-- [ ] If simplified: R[i] = 1/(ω × C_total[i])
-- [ ] If iterative: initialize R[i], run optimization loop
-- [ ] Apply position-dependent bounds R_min[i], R_max[i]
-- [ ] Check convergence (<1% change)
-- [ ] Validate: R distribution monotonic, total in expected range
-
-**Phase 3: SPICE Implementation**
-- [ ] Convert [C] matrix to SPICE-compatible form (partial or controlled sources)
-- [ ] Add resistance elements R[i]
-- [ ] Define topload voltage source (or integrate with full coil model)
-- [ ] Set up AC analysis at operating frequency
-
-**Phase 4: Analysis**
-- [ ] Run AC simulation
-- [ ] Extract V, I at each node
-- [ ] Calculate P[i] in each segment: P[i] = 0.5 × I[i]² × R[i]
-- [ ] Calculate total P_spark = Σ P[i]
-- [ ] Calculate Y_spark or Z_spark at topload port
-
-**Phase 5: Validation**
-- [ ] Power balance: P_spark reasonable fraction of P_input
-- [ ] Total R in expected range for frequency and length
-- [ ] Phase angle φ_Z in typical range
-- [ ] Resistance distribution physical (increasing base→tip)
-- [ ] Compare to lumped model (should be similar order of magnitude)
-- [ ] Compare to measurements if available
-
-**Phase 6: Iteration (if needed)**
-- [ ] If validation fails, identify issue
-- [ ] Adjust and re-run
-- [ ] Document assumptions and uncertainties
-
----
-
-## Module 4.9: Calibration and Measurement Integration
-
-### Calibrating ε (Energy Per Meter)
-
-**Procedure:**
-
-**Step 1: Controlled test**
-```
-Run coil with known drive conditions
-Measure final spark length L_measured
-```
-
-**Step 2: Simulation**
-```
-Simulate same conditions
-Calculate E_delivered = ∫ P_spark dt over growth time
-```
-
-**Step 3: Extract ε**
-```
-ε_calibrated = E_delivered / L_measured
-
-Example:
-E_delivered = 18 J (from simulation)
-L_measured = 1.5 m (from photograph/measurement)
-
-ε = 18 J / 1.5 m = 12 J/m
-```
-
-**Step 4: Build database**
-```
-Repeat for different operating modes:
-- QCW long ramp: ε_QCW
-- Burst mode: ε_burst
-- Intermediate: ε_hybrid
-
-Use appropriate ε for future predictions
-```
-
-### Calibrating E_propagation
-
-**Procedure:**
-
-**Step 1: Measure stall condition**
-```
-Ramp voltage slowly
-Observe maximum length L_max when growth stops
-Measure V_topload at stall
-```
-
-**Step 2: FEMM field analysis**
-```
-Set up geometry with spark length = L_max
-Apply V = V_topload
-Calculate E_tip at tip using FEMM
-```
-
-**Step 3: Extract threshold**
-```
-E_propagation ≈ E_tip at stall
-
-Typical: 0.4-1.0 MV/m
-Calibrate for your specific conditions (altitude, humidity, geometry)
-```
-
-### Using Measurements to Refine Model
-
-**Ringdown method (from Part 2):**
-```
-1. Measure f₀, Q₀ (unloaded)
-2. Measure f_L, Q_L (with spark)
-3. Extract Y_spark from frequency shift and Q change
-4. Compare to model prediction
-5. Adjust R values if significant discrepancy (>factor of 2)
-```
-
-**Direct impedance measurement:**
-```
-If you have:
-- Calibrated E-field probe (V_topload)
-- Calibrated current probe on spark return path (I_spark, not I_base!)
-
-Then:
-Z_measured = V_topload / I_spark
-
-Compare to model Z_spark
-Adjust R values to match
-```
-
-**Iterative refinement:**
-```
-1. Initial model from FEMM + simplified R
-2. Simulate → predict Z_spark, power
-3. Measure actual Z_spark, power
-4. Adjust R distribution (proportionally) to match measured total R
-5. Validate that distribution shape is still physical
-6. Use refined model for future predictions
-```
-
----
-
-### WORKED EXAMPLE 4.9: Calibrating ε
-
-**Measurement:**
-```
-QCW coil, 12 ms ramp
-Final spark length: L = 2.2 m
-```
-
-**Simulation:**
-```
-Full model with distributed spark
-Calculate power to spark over time:
-P_spark(t) varies from 20 kW to 80 kW during ramp
-
-Total energy:
-E_delivered = ∫₀^0.012 P_spark(t) dt
- = 26 J (numerical integration)
-```
-
-**Calibration:**
-```
-ε = E_delivered / L_measured
- = 26 J / 2.2 m
- = 11.8 J/m
-```
-
-**Interpretation:**
-```
-This is at low end of QCW range (5-15 J/m)
-Indicates efficient leader formation
-Consistent with long ramp time (12 ms)
-
-Use ε = 12 J/m for future predictions with this coil in QCW mode
-```
-
-**Validation:**
-```
-Predict different condition:
-New ramp: 8 ms, available energy: E = 30 J
-
-Expected length: L = E/ε = 30/12 = 2.5 m
-
-Run test, measure actual length, compare
-If within ±20%: calibration good
-If >30% error: investigate (different mode? voltage limited?)
-```
-
----
-
-### PRACTICE PROBLEMS 4.9
-
-**Problem 1:** Simulation shows E = 40 J delivered, measurement shows L = 2.8 m. Calculate ε. Is this more consistent with QCW or burst mode?
-
-**Problem 2:** A calibration at sea level gives E_propagation = 0.5 MV/m. At 2000 m altitude (air density ~80% of sea level), estimate new E_propagation.
-
----
-
-## Part 4 Conclusion: Practical Guidelines
-
-### Decision Tree: Which Model to Use?
-
-```
-START
- |
- └─ Spark length < 1 m?
- ├─ YES → Use LUMPED model
- | * Fast, accurate enough
- | * R = R_opt_power
- |
- └─ NO → Spark length < 3 m?
- ├─ YES → Choice:
- | * Quick answer: LUMPED
- | * Best accuracy: DISTRIBUTED (n=10)
- |
- └─ NO (>3 m) → Use DISTRIBUTED (n=15-20)
- * Essential for accuracy
- * Captures tip/base differences
-
-Research/validation? → Always use DISTRIBUTED
-```
-
-### Typical Simulation Times
-
-```
-Lumped model:
-- FEMM: 2 min (single geometry)
-- SPICE: <1 sec
-- Total: ~3 minutes
-
-Distributed (n=10), simplified R:
-- FEMM: 5 min (multi-body)
-- SPICE: 1 sec (one analysis)
-- Total: ~6 minutes
-
-Distributed (n=10), iterative R:
-- FEMM: 5 min
-- SPICE: 100 sec (100 iterations × 1 sec)
-- Total: ~7 minutes
-
-Distributed (n=20), iterative R:
-- FEMM: 10 min (larger matrix)
-- SPICE: 300 sec (more elements)
-- Total: ~15 minutes
-```
-
-### Accuracy Expectations
-
-```
-Lumped model:
-- Impedance: ±20%
-- Power: ±30%
-- Good enough for: matching studies, coil optimization
-
-Distributed (simplified R):
-- Impedance: ±15%
-- Power: ±25%
-- Current distribution: ±30%
-
-Distributed (iterative R):
-- Impedance: ±10%
-- Power: ±20%
-- Current distribution: ±20%
-- Best available without plasma modeling
-
-Measurement comparison:
-- ±20-50% agreement is GOOD (plasma variability)
-- ±factor of 2: acceptable (many unknowns)
-- Better than factor of 2: excellent!
-```
-
-### Final Recommendations
-
-**For hobbyist design:**
-- Use lumped model
-- Calibrate ε from one measurement
-- Predict new conditions
-
-**For research:**
-- Use distributed model (n=10-15)
-- Iterative optimization
-- Document all assumptions
-- Compare to measurements
-- Report uncertainties
-
-**For publications:**
-- Distributed model required
-- Validation against measurements
-- Sensitivity analysis
-- Clear methodology section
-
----
-
-## Final Comprehensive Problem
-
-**Design Challenge: Predict Performance of New Coil**
-
-**Given:**
-- DRSSTC, f = 195 kHz
-- Topload: 35 cm toroid (major diameter)
-- Target: 2 m spark, QCW mode (10 ms ramp)
-- Primary input: P_input = 120 kW
-- Thévenin: Z_th = 110 - j2300 Ω, V_th = 340 kV
-
-**Required:**
-
-**Part 1: Distributed Model Setup**
-- Choose n (justify)
-- Describe FEMM geometry
-- What validation checks after extracting [C]?
-
-**Part 2: Resistance Calculation**
-- Choose method (iterative or simplified, justify)
-- Estimate expected R_total range
-- What bounds for R[i]?
-
-**Part 3: Performance Prediction**
-- Calculate Z_spark
-- Find current and power
-- What % of theoretical max?
-
-**Part 4: Growth Analysis**
-- Assume ε = 12 J/m (from calibration)
-- Can 2 m be reached in 10 ms with available power?
-- Check voltage: κ = 3.2, E_prop = 0.7 MV/m
-- Is growth voltage-limited or power-limited?
-
-**Part 5: Validation Plan**
-- What measurements would you take?
-- How would you refine the model?
-- What accuracy do you expect?
-
-**This problem integrates all four parts of the course!**
-
----
-
-## Course Summary: Master Checklist
-
-### Part 1 Concepts
-- [ ] Peak vs RMS phasor convention
-- [ ] Complex impedance and admittance
-- [ ] Power formula: P = 0.5 × Re{V × I*}
-- [ ] C_mut and C_sh in spark circuit
-- [ ] Circuit topology: (R||C_mut) + C_sh
-- [ ] Phase angles and capacitive loading
-
-### Part 2 Concepts
-- [ ] Topological phase constraint φ_Z,min
-- [ ] R_opt_power maximizes power transfer
-- [ ] Hungry streamer self-optimization
-- [ ] Why V_top/I_base is wrong
-- [ ] Thévenin equivalent extraction and use
-- [ ] Q measurement and ringdown analysis
-
-### Part 3 Concepts
-- [ ] E_inception and E_propagation thresholds
-- [ ] Energy per meter ε by mode
-- [ ] Growth rate dL/dt = P/ε
-- [ ] Thermal time constants and persistence
-- [ ] Capacitive divider problem
-- [ ] FEMM electrostatic analysis
-- [ ] Maxwell capacitance matrix extraction
-- [ ] Lumped model construction
-
-### Part 4 Concepts
-- [ ] When distributed models needed
-- [ ] nth-order segmentation
-- [ ] Multi-body FEMM analysis
-- [ ] Capacitance matrix in SPICE (partial capacitance)
-- [ ] Iterative R optimization with damping
-- [ ] Simplified R = 1/(ωC_total) method
-- [ ] Validation checks (power balance, R range, distribution)
-- [ ] Calibration from measurements (ε, E_prop)
-
----
-
-## Resources for Continued Learning
-
-**Software:**
-- FEMM: www.femm.info (free)
-- LTSpice: www.analog.com/ltspice (free)
-- Python + NumPy/SciPy for automation
-
-**Tesla Coil Communities:**
-- 4hv.org forums (active community)
-- highvoltageforum.net
-- teslamap.com (coil database)
-
-**Further Reading:**
-- "The Spark Gap" magazine (archived)
-- Lightning physics textbooks (Uman, Rakov)
-- Plasma physics introductions (Chen)
-- High voltage engineering (Kuffel)
-
-**This framework:**
-- Original document for full mathematical details
-- Implement in stages (lumped → distributed)
-- Calibrate to YOUR coil
-- Share results with community!
-
----
-
-**END OF PART 4**
-
-**END OF COMPLETE LESSON PLAN**
-
----
-
-**Congratulations!** You now have a complete framework to:
-1. Understand Tesla coil spark physics
-2. Extract parameters from FEMM
-3. Build circuit models (lumped and distributed)
-4. Predict performance
-5. Validate against measurements
-6. Iterate and improve
-
-**Next steps:**
-- Work through practice problems
-- Build your first model
-- Compare to real coil
-- Refine and calibrate
-
-# Tesla Coil Spark Modeling - Complete Lesson Plan
-## Appendices: Quick Reference Materials
-
----
-
-## Appendix A: Complete Variable Reference Table
-
-### Circuit Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **C_mut** | F (pF) | Mutual capacitance between topload and spark | 5-15 pF |
-| **C_sh** | F (pF) | Shunt capacitance spark-to-ground | 2 pF/foot × length |
-| **C_total** | F (pF) | Total capacitance: C_mut + C_sh | 10-30 pF |
-| **C_eq** | F (pF) | Equivalent loaded capacitance | Calculated from f shift |
-| **R** | Ω (kΩ) | Spark plasma resistance | 5-500 kΩ @ 200 kHz |
-| **R_opt_power** | Ω | Resistance for maximum power transfer | 1/(ω(C_mut+C_sh)) |
-| **R_opt_phase** | Ω | Resistance for minimum phase angle | 1/(ω√(C_mut(C_mut+C_sh))) |
-| **R_min** | Ω | Minimum physical resistance (hot leader) | 1-10 kΩ |
-| **R_max** | Ω | Maximum physical resistance (cold streamer) | 100 kΩ - 100 MΩ |
-| **G** | S (μS) | Conductance: 1/R | 1-100 μS typical |
-| **B₁** | S (μS) | Susceptance of C_mut: ωC_mut | Positive (capacitive) |
-| **B₂** | S (μS) | Susceptance of C_sh: ωC_sh | Positive (capacitive) |
-| **Y** | S (μS) | Complex admittance: G + jB | - |
-| **Z** | Ω (kΩ) | Complex impedance: R + jX | - |
-| **Z_th** | Ω | Thévenin output impedance | 100-200 Ω + j(-2000 to -3000 Ω) |
-| **V_th** | V (kV) | Thévenin open-circuit voltage | 200-500 kV |
-| **φ_Z** | ° or rad | Impedance phase angle | -55° to -75° typical |
-| **φ_Z,min** | ° or rad | Minimum achievable phase: -atan(2√(r(1+r))) | More negative than -45° usually |
-| **r** | - | Capacitance ratio: C_mut/C_sh | 0.5-2.0 typical |
-
-### Frequency and Quality Factor
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **f** | Hz (kHz) | Operating frequency | 100-400 kHz |
-| **f₀** | Hz | Unloaded resonant frequency | - |
-| **f_L** | Hz | Loaded resonant frequency (with spark) | Lower than f₀ |
-| **ω** | rad/s | Angular frequency: 2πf | 6.28×10⁵ - 2.5×10⁶ |
-| **Q₀** | - | Unloaded quality factor | 50-200 typical |
-| **Q_L** | - | Loaded quality factor (with spark) | 20-80 typical |
-| **τ** | s (ms) | Time constant for decay | τ = 2Q/ω |
-
-### Power and Energy
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **P** | W (kW) | Real (average) power | - |
-| **P_spark** | W (kW) | Power dissipated in spark | 10-200 kW |
-| **P_avg** | W (kW) | Average power over time | - |
-| **P_max** | W (kW) | Theoretical maximum (conjugate match) | Usually unachievable |
-| **E** | J | Energy | - |
-| **E_total** | J | Total energy to grow spark | ε × L |
-| **ε** (epsilon) | J/m | Energy per meter for growth | 5-15 (QCW), 30-100 (burst) |
-| **ε₀** | J/m | Initial energy per meter | Before thermal accumulation |
-
-### Electric Fields
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **E** | V/m (MV/m) | Electric field strength | - |
-| **E_tip** | V/m (MV/m) | Field at spark tip | κ × V_top/L |
-| **E_average** | V/m (MV/m) | Average field: V_top/L | - |
-| **E_inception** | V/m (MV/m) | Field for initial breakdown | 2-3 MV/m |
-| **E_propagation** | V/m (MV/m) | Field for sustained growth | 0.4-1.0 MV/m |
-| **κ** (kappa) | - | Tip enhancement factor | 2-5 typical |
-
-### Geometric Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **L** | m | Spark length | 0.3-6 m typical |
-| **L_target** | m | Target design length | - |
-| **L_segment** | m | Length of one segment (distributed model) | L_total/n |
-| **d** | m (mm) | Spark channel diameter | 0.1-5 mm (streamers-leaders) |
-| **d_nominal** | m (mm) | Assumed diameter for FEMM | 1 mm (burst), 3 mm (QCW) |
-| **n** | - | Number of segments (distributed model) | 5-20, typically 10 |
-| **i** | - | Segment index (1 to n) | 1=base, n=tip |
-
-### Thermal Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **T** | K | Temperature | 1000 K (streamer) - 20000 K (leader) |
-| **ΔT** | K | Temperature rise above ambient | - |
-| **τ_thermal** | s (ms) | Thermal diffusion time: d²/(4α) | 0.1 ms (thin) - 300 ms (thick) |
-| **τ_effective** | s (ms) | Observed persistence time | Longer than τ_thermal |
-| **α_thermal** | m²/s | Thermal diffusivity of air | ~2×10⁻⁵ m²/s |
-
-### Matrix and Optimization
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **[C]** | F (pF) | Maxwell capacitance matrix (n+1)×(n+1) | - |
-| **C[i,j]** | F (pF) | Matrix element i,j | Diagonal >0, off-diagonal <0 |
-| **R[i]** | Ω (kΩ) | Resistance of segment i | Increases from base to tip |
-| **α_damp** | - | Damping factor for iteration | 0.3-0.5 |
-| **position** | - | Normalized position: (i-1)/(n-1) | 0=base, 1=tip |
-
-### Measurement Variables
-
-| Variable | Units | Definition | Typical Values |
-|----------|-------|------------|----------------|
-| **V_top** | V (kV) | Voltage at topload (peak) | 200-600 kV |
-| **V_tip** | V (kV) | Voltage at spark tip | V_top × C_mut/(C_mut+C_sh) |
-| **I_spark** | A | Current through spark | 0.5-3 A |
-| **I_base** | A | Current at secondary base (WRONG for spark) | Includes displacement currents |
-| **A₁, A₂** | V, A | Consecutive peak amplitudes in ringdown | - |
-
----
-
-## Appendix B: Formula Quick Reference
-
-### Basic Circuit Analysis
-
-**Admittance of spark circuit:**
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where: G = 1/R
- B₁ = ωC_mut
- B₂ = ωC_sh
-```
-
-**Real and imaginary parts:**
-```
-Re{Y} = GB₂² / [G² + (B₁+B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
-```
-
-**Impedance phase:**
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Power calculation:**
-```
-P = 0.5 × Re{V × I*} (with peak phasors)
-P = 0.5 × |V|² × Re{Y}
-P = 0.5 × |I|² × Re{Z}
-P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
-```
-
-### Optimal Resistances
-
-**Maximum power transfer:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-
-Example: f=200 kHz, C_total=12 pF
-R_opt_power = 1/(2π×200×10³×12×10⁻¹²) ≈ 66 kΩ
-```
-
-**Minimum phase angle magnitude:**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Always: R_opt_power < R_opt_phase
-```
-
-**Minimum phase angle:**
-```
-φ_Z,min = -atan(2√[r(1+r)])
-
-where r = C_mut/C_sh
-
-Critical value: r = 0.207 gives φ_Z,min = -45°
-If r > 0.207: cannot achieve -45°
-```
-
-### Thévenin Equivalent
-
-**Measuring Z_th (drive off, test source on):**
-```
-Z_th = V_test / I_test = 1V / I_test
-
-Apply 1V AC at topload-to-ground
-Measure current I_test
-```
-
-**Measuring V_th (drive on, no load):**
-```
-V_th = V(topload) with spark removed
-```
-
-**Power to any load:**
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**Theoretical maximum (conjugate match):**
-```
-Z_load = Z_th* (complex conjugate)
-P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
-
-Usually unachievable due to topological constraints
-```
-
-### Ringdown Method
-
-**Quality factor from decay:**
-```
-Q = πf × Δt / ln(A₁/A₂)
-
-where Δt = time between peaks
- A₁, A₂ = consecutive peak amplitudes
-```
-
-**At loaded resonance:**
-```
-Q_L = ω_L C_eq R_p = R_p/(ω_L L)
-
-Therefore:
-R_p = Q_L/(ω_L C_eq) = Q_L ω_L L
-G_total = ω_L C_eq/Q_L = 1/(Q_L ω_L L)
-```
-
-**Capacitance from frequency shift:**
-```
-C_eq = C₀(f₀/f_L)²
-ΔC = C_eq - C₀
-```
-
-**Spark admittance approximation:**
-```
-Y_spark ≈ (G_total - G_0) + jω_L ΔC
-```
-
-### Spark Growth Physics
-
-**Growth rate equation:**
-```
-dL/dt = P_stream/ε (when E_tip > E_propagation)
-dL/dt = 0 (when E_tip ≤ E_propagation, stalled)
-```
-
-**Time to reach target length (constant power):**
-```
-T = ε × L_target / P_stream
-```
-
-**Total energy required:**
-```
-E_total = ε × L_target
-```
-
-**Energy per meter with thermal accumulation:**
-```
-ε(t) = ε₀ / (1 + α∫P dt)
-
-where α has units [1/J]
-```
-
-**Field thresholds:**
-```
-E_inception ≈ 2-3 MV/m (initial breakdown)
-E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
-E_tip = κ × E_average = κ × V_top/L
-```
-
-### Thermal Time Constants
-
-**Pure thermal diffusion:**
-```
-τ_thermal = d² / (4α)
-
-where α ≈ 2×10⁻⁵ m²/s for air
-
-Examples:
-d = 100 μm → τ ≈ 0.125 ms
-d = 5 mm → τ ≈ 312 ms
-```
-
-**Convection velocity (buoyancy):**
-```
-v ≈ √(g × d × ΔT/T_amb)
-
-where g = 9.8 m/s²
-```
-
-### Capacitive Divider
-
-**Open-circuit voltage division:**
-```
-V_tip = V_topload × C_mut/(C_mut + C_sh)
-
-As spark grows: C_sh increases → V_tip decreases
-```
-
-**With finite resistance (more complex):**
-```
-V_tip = V_topload × Z_mut/(Z_mut + Z_sh)
-
-where Z_mut = (1/jωC_mut) || R
- Z_sh = 1/(jωC_sh)
-```
-
-### FEMM Capacitance Extraction
-
-**For 2-body system (topload + spark):**
-```
-Maxwell matrix:
- [Top] [Spark]
-[Top] C₁₁ C₁₂
-[Spark] C₂₁ C₂₂
-
-Extraction:
-C_mut = |C₁₂| = |C₂₁| (absolute value)
-C_sh = C₂₂ - |C₁₂|
-
-Validation: C_sh ≈ 2 pF/foot × L_spark
-```
-
-### Distributed Model
-
-**Simplified resistance calculation:**
-```
-For each segment i:
-C_total[i] = Σⱼ |C[i,j]| (sum of absolute values)
-R[i] = 1/(ω × C_total[i])
-R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Position-dependent bounds:**
-```
-position = (i-1)/(n-1) (0 at base, 1 at tip)
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position²
-```
-
-**Iterative optimization (damped update):**
-```
-R_new[i] = α × R_optimal[i] + (1-α) × R_old[i]
-
-where α = 0.3-0.5 (damping factor)
-```
-
----
-
-## Appendix C: Physical Constants and Typical Values
-
-### Universal Constants
-
-| Constant | Symbol | Value | Units |
-|----------|--------|-------|-------|
-| Permittivity of free space | ε₀ | 8.854×10⁻¹² | F/m |
-| Pi | π | 3.14159... | - |
-| Gravitational acceleration | g | 9.81 | m/s² |
-| Electron charge | e | 1.602×10⁻¹⁹ | C |
-
-### Air Properties (Sea Level, 20°C)
-
-| Property | Symbol | Value | Units |
-|----------|--------|-------|-------|
-| Density | ρ_air | 1.2 | kg/m³ |
-| Thermal diffusivity | α | 2×10⁻⁵ | m²/s |
-| Thermal conductivity | k | 0.026 | W/(m·K) |
-| Specific heat | c_p | 1005 | J/(kg·K) |
-| Molecular density | n | 2.5×10²⁵ | molecules/m³ |
-| Ionization energy | E_ion | ~15 | eV/molecule |
-
-### Field Thresholds (Dry Air, Sea Level)
-
-| Parameter | Value | Units | Notes |
-|-----------|-------|-------|-------|
-| E_inception | 2-3 | MV/m | Initial breakdown, smooth electrode |
-| E_propagation | 0.4-1.0 | MV/m | Sustained leader growth |
-| Altitude correction | -20 to -30 | %/1000m | Lower air density → lower threshold |
-| Humidity effect | ±10 | % | Variable, depends on conditions |
-
-### Energy per Meter by Mode
-
-| Operating Mode | ε Range | Units | Characteristics |
-|----------------|---------|-------|-----------------|
-| QCW (5-20 ms ramp) | 5-15 | J/m | Efficient, leader-dominated |
-| Hybrid DRSSTC | 20-40 | J/m | Mixed streamers/leaders |
-| Burst mode (<1 ms) | 30-100+ | J/m | Inefficient, streamer-dominated |
-| Single-shot burst | 50-150 | J/m | Very inefficient, bright but short |
-
-### Typical Spark Resistance (@ 200 kHz)
-
-| Spark Type | Length | Total R | Notes |
-|------------|--------|---------|-------|
-| Short burst | 0.5-1 m | 100-300 kΩ | Streamer-dominated |
-| Medium burst | 1-2 m | 150-400 kΩ | Mixed |
-| Long burst | 2-3 m | 200-500 kΩ | Difficult, high R |
-| QCW (short) | 0.5-1 m | 20-80 kΩ | Leader-dominated |
-| QCW (medium) | 1-2 m | 30-120 kΩ | Efficient |
-| QCW (long) | 2-4 m | 40-200 kΩ | Best mode for length |
-
-### Frequency Dependence
-
-| Frequency | R_typical | C_sh (per meter) | Notes |
-|-----------|-----------|------------------|-------|
-| 100 kHz | 5-50 kΩ | ~6 pF | Low frequency, low R |
-| 150 kHz | 10-100 kΩ | ~6 pF | Typical small coils |
-| 200 kHz | 20-200 kΩ | ~6 pF | Common frequency |
-| 300 kHz | 30-300 kΩ | ~6 pF | Higher frequency |
-| 400 kHz | 40-400 kΩ | ~6 pF | Very high, smaller coils |
-
-**Note:** R ∝ 1/f approximately, C_sh relatively constant
-
-### Thermal Time Constants
-
-| Channel Type | Diameter | τ_thermal | Persistence | Notes |
-|--------------|----------|-----------|-------------|-------|
-| Thin streamer | 50-100 μm | 0.05-0.2 ms | 1-5 ms | Convection extends |
-| Medium streamer | 200-500 μm | 0.2-1.5 ms | 2-10 ms | Mixed |
-| Thin leader | 1-2 mm | 6-25 ms | 50-500 ms | Buoyancy significant |
-| Thick leader | 5-10 mm | 150-600 ms | Seconds | Persistent column |
-
-### Tesla Coil Typical Parameters
-
-| Parameter | Small Coil | Medium Coil | Large Coil | Units |
-|-----------|------------|-------------|------------|-------|
-| Frequency | 300-500 | 150-250 | 80-150 | kHz |
-| Topload C₀ | 15-25 | 25-40 | 40-80 | pF |
-| Secondary Q₀ | 100-200 | 80-150 | 50-120 | - |
-| Spark length | 0.3-1.0 | 1.0-2.5 | 2.0-4.0 | m |
-| Power | 1-10 | 10-100 | 50-300 | kW |
-| Z_th magnitude | 1-3 | 0.5-2 | 0.3-1 | kΩ |
-| Z_th phase | -85 to -88 | -86 to -89 | -87 to -89 | degrees |
-
----
-
-## Appendix D: SPICE Component Reference
-
-### Basic Elements
-
-**Resistor:**
-```
-R node1 node2
-Example: R1 topload spark 50k
- R2 n1 n2 {R_value} ; parameterized
-```
-
-**Capacitor:**
-```
-C node1 node2
-Example: C_mut topload spark 10p
- C_sh spark 0 6p
-```
-
-**Voltage source:**
-```
-V node+ node-
-Example: V1 topload 0 AC 1V
- V2 drive 0 AC 100k ; 100 kV
-```
-
-**Current source:**
-```
-I node+ node-
-Example: I1 topload 0 AC 1m
-```
-
-### Parameterized Components
-
-**Define parameters:**
-```
-.param freq=200k
-.param omega={2*pi*freq}
-.param C_mut=10p
-.param C_sh=6p
-.param R={1/(omega*(C_mut+C_sh))}
-```
-
-**Use in components:**
-```
-C1 n1 n2 {C_mut}
-R1 n2 n3 {R}
-```
-
-### Controlled Sources (for capacitance matrix)
-
-**Voltage-controlled current source:**
-```
-G node+ node- ctrl+ ctrl-
-Example: G1 n1 0 n2 0 {j*omega*C[1,2]}
-```
-
-**Behavioral source:**
-```
-B node+ node- V={expression}
-Example: B1 n1 0 V={j*omega*C_mut*V(n2)}
-```
-
-### Analysis Commands
-
-**AC analysis:**
-```
-.ac lin
-Example: .ac lin 1 200k 200k ; single frequency
- .ac lin 100 180k 220k ; sweep 100 points
-```
-
-**Transient analysis:**
-```
-.tran
-Example: .tran 0.1u 10m ; 0.1 μs steps, 10 ms total
-```
-
-**Print/plot:**
-```
-.print ac v(topload) i(V1) vp(topload) ip(V1)
-.plot ac vdb(topload) ; dB magnitude
-```
-
-### Mutual Inductance (for transformer)
-
-**Inductors with coupling:**
-```
-L1 n1 n2
-L2 n3 n4
-K1 L1 L2
-
-Example:
-Lpri drive n1 100u
-Lsec n2 base 10m
-K_couple Lpri Lsec 0.15 ; k=0.15
-```
-
-### Subcircuits (for modular models)
-
-**Define subcircuit:**
-```
-.subckt spark_model topload ground
-+ params: C_mut=10p C_sh=6p R=50k
-C1 topload n1 {C_mut}
-R1 n1 n2 {R}
-C2 n2 ground {C_sh}
-.ends
-```
-
-**Use subcircuit:**
-```
-X1 topload 0 spark_model params: C_mut=12p C_sh=8p R=60k
-```
-
-### Example: Complete Lumped Model
-
-```
-* Tesla Coil Spark Lumped Model
-* Frequency: 200 kHz
-
-.param freq=200k
-.param omega={2*pi*freq}
-
-* Spark parameters from FEMM
-.param C_mut=10p
-.param C_sh=6p
-.param R_opt={1/(omega*(C_mut+C_sh))}
-
-* Clip to physical bounds
-.param R_min=5k
-.param R_max=500k
-.param R={min(max(R_opt,R_min),R_max)}
-
-* Circuit
-V_topload topload 0 AC 1V
-C_mut topload n1 {C_mut}
-R_spark n1 n2 {R}
-C_sh n2 0 {C_sh}
-
-* Analysis
-.ac lin 1 {freq} {freq}
-.print ac v(topload) i(V_topload) vp(topload) ip(V_topload)
-
-* Calculate admittance in post-processing:
-* Y = I/V, extract real and imaginary parts
-* Power = 0.5 * |V|^2 * Re{Y}
-
-.end
-```
-
----
-
-## Appendix E: FEMM Quick Start Guide
-
-### Installation
-
-1. **Download:** Visit www.femm.info
-2. **Install:** Run installer (Windows), or use Wine (Linux/Mac)
-3. **Launch:** Open FEMM 4.2 (main application)
-
-### Basic Interface
-
-**Main window sections:**
-- **Toolbar:** Problem type, zoom, view controls
-- **Drawing area:** Geometry creation
-- **Status bar:** Coordinates, snap mode
-- **Menus:** File, Edit, View, Problem, Mesh, Analysis
-
-### Creating Electrostatic Problem
-
-**Step 1: New document**
-```
-File → New
-Select: Electrostatics Problem
-Frequency: 0 (electrostatic)
-Length units: Centimeters (or your preference)
-Problem type: Axisymmetric
-Precision: 1e-8
-```
-
-**Step 2: Define materials**
-```
-Problem → Materials Library
-Select: Air (ε_r = 1.0)
-Add to model
-
-If needed, define custom materials:
-Problem → Materials → Add Property
-Name: Custom
-Permittivity: (relative value)
-```
-
-**Step 3: Draw geometry**
-```
-Use toolbar buttons:
-- Draw nodes (points): Click to place
-- Draw lines: Select two nodes
-- Draw arcs: Select two nodes, define angle
-- Draw circles: Center + radius
-
-For axisymmetric:
-- Draw in r-z plane (r ≥ 0)
-- r = 0 is axis of symmetry
-```
-
-### Tesla Coil Spark Geometry Example
-
-**Toroid (topload):**
-```
-1. Draw circle (minor diameter) at z=0, r=15 cm
-2. Use circular rotation: Operations → Mirror/Rotate
-3. Create toroidal surface
-```
-
-**Spark (cylinder):**
-```
-1. Draw vertical line from topload base to tip
- Example: r=0.1 cm, z=-5 to z=-105 cm (1 m spark)
-2. This represents axis of cylinder
-3. For multiple segments: Draw each as separate line
-```
-
-**Ground plane:**
-```
-1. Draw large circle or line at z = (below spark)
-2. Large enough to approximate "infinity"
-```
-
-**Outer boundary:**
-```
-1. Draw rectangle enclosing entire problem
-2. Far from coil (5-10× max dimension)
-```
-
-### Assigning Properties
-
-**Step 4: Define conductors**
-```
-Problem → Conductors
-Add conductor groups:
-- Conductor 1: Name "Topload", Voltage = 1V
-- Conductor 2: Name "Spark1", Floating
-- Conductor 3: Name "Spark2", Floating
-...
-- Conductor n+1: Name "Ground", Voltage = 0V
-```
-
-**Step 5: Assign to geometry**
-```
-Select line/arc/circle
-Right-click → Set Boundary
-Choose conductor group
-
-All segments of spark: Assign to separate conductors
-Topload surface: Assign to topload conductor
-Ground: Assign to ground conductor
-```
-
-**Step 6: Assign materials**
-```
-Select region (click inside enclosed area)
-Right-click → Set Block Property
-Material: Air
-Mesh size: Auto or specify
-```
-
-**Step 7: Boundary conditions**
-```
-Problem → Boundaries
-- Outer boundary: V=0 (Dirichlet)
-- r=0: Axisymmetric boundary
-- Others: Default (Neumann, E field normal)
-```
-
-### Meshing and Solving
-
-**Step 8: Create mesh**
-```
-Mesh → Create Mesh
-Wait for triangulation (seconds to minutes)
-Check mesh quality: Zoom in near conductors
-```
-
-**Step 9: Solve**
-```
-Analysis → Run
-Wait for solution (seconds to minutes)
-Look for convergence message
-```
-
-### Post-Processing
-
-**Step 10: View results**
-```
-File → Open Postprocessor
-(or automatically opens after solve)
-
-View field:
-- View → Contour Plot → V (voltage)
-- View → Vector Plot → E (field)
-- View → Density Plot → Field magnitude
-```
-
-**Step 11: Extract capacitance matrix**
-```
-Circuit Properties window (usually visible)
-If not: View → Circuit Properties
-
-Shows capacitance matrix [C]
-Copy values to spreadsheet/text file
-
-Format:
- [1] [2] [3] ...
-[1] C₁₁ C₁₂ C₁₃
-[2] C₂₁ C₂₂ C₂₃
-...
-```
-
-**Step 12: Calculate electric field at point**
-```
-Click on specific point
-View → Point Values
-Shows: V, E_r, E_z, |E| at that location
-
-For tip field: Click at spark tip
-```
-
-### Tips and Tricks
-
-**Efficient meshing:**
-```
-- Finer mesh near conductors (small triangle size)
-- Coarse mesh far away (large triangles)
-- Specify manually: Set Block Property → Mesh size
-```
-
-**Symmetry exploitation:**
-```
-- Use axisymmetric for cylindrical symmetry (2D → 3D)
-- Use planar for 2D problems
-- Reduces element count by 10-100×
-```
-
-**Convergence issues:**
-```
-- Increase precision (Problem → Precision: 1e-10)
-- Refine mesh near conductors
-- Enlarge outer boundary
-- Check for geometry errors (gaps, overlaps)
-```
-
-**Large matrix extraction:**
-```
-For n=20 segments → 21×21 matrix
-Circuit Properties window may be small
-Resize window or copy values programmatically
-Consider exporting to CSV
-```
-
-### Automation with Lua Scripting
-
-**FEMM supports Lua scripts for automation:**
-```lua
--- Example: Create spark segment
-newdocument(0) -- Electrostatics
-for i=1,10 do
- z_start = -i*10
- z_end = -(i+1)*10
- addnode(0.1, z_start)
- addnode(0.1, z_end)
- addsegment(0.1, z_start, 0.1, z_end)
- selectsegment(0.1, (z_start+z_end)/2)
- setconductor("Spark"..i, 0) -- Floating
-end
-```
-
-**Useful for:**
-- Parametric sweeps (vary length, diameter)
-- Batch processing multiple geometries
-- Extracting results programmatically
-
----
-
-## Appendix F: Troubleshooting Guide
-
-### Problem: Negative Phase Angle Too Large (φ_Z < -80°)
-
-**Symptoms:**
-- Impedance phase more negative than -80°
-- Very capacitive
-- Low power transfer
-
-**Possible causes:**
-1. R too high (not optimized)
-2. Capacitances overestimated
-3. Frequency too high for given R
-
-**Solutions:**
-- Run iterative R optimization
-- Verify FEMM capacitance extraction
-- Check R bounds (R_max too high?)
-- Recalculate R_opt_power
-
----
-
-### Problem: Power Balance Doesn't Close
-
-**Symptoms:**
-- P_spark > P_input (violates conservation)
-- Or P_spark << P_input (most energy missing)
-
-**Possible causes:**
-1. Incorrect power calculation (missing 0.5 factor?)
-2. Using RMS instead of peak values inconsistently
-3. Missing loss terms
-4. Measuring wrong current (I_base instead of I_spark)
-
-**Solutions:**
-- Verify formula: P = 0.5 × Re{V × I*} with peak
-- Check all quantities are peak (or all RMS, consistently)
-- Account for secondary losses separately
-- Measure I_spark on return path, not I_base
-
----
-
-### Problem: FEMM Capacitance Matrix Not Symmetric
-
-**Symptoms:**
-- C[i,j] ≠ C[j,i]
-- Non-physical
-
-**Possible causes:**
-1. Numerical error (insufficient precision)
-2. Mesh quality poor
-3. Geometry errors (overlaps, gaps)
-
-**Solutions:**
-- Increase precision: Problem → Precision: 1e-10
-- Refine mesh near conductors
-- Check geometry for errors (zoom in, look for gaps)
-- Ensure proper boundary conditions
-
----
-
-### Problem: Distributed Model Doesn't Converge
-
-**Symptoms:**
-- Iterative optimization oscillates
-- R values jumping around
-- No stable solution after many iterations
-
-**Possible causes:**
-1. Damping factor α too high
-2. Weakly coupled segments (tip)
-3. R bounds too restrictive
-4. Power curve very flat
-
-**Solutions:**
-- Reduce α to 0.2-0.3 (more damping)
-- Accept tip segments not converging (physical)
-- Widen R_max bounds for tip segments
-- Use simplified method if iterative fails
-
----
-
-### Problem: Simulation Predicts Too Short Spark
-
-**Symptoms:**
-- Predicted length << measured
-- Model underestimates performance
-
-**Possible causes:**
-1. ε too high (overestimating energy needed)
-2. E_propagation set too high
-3. Power transfer underestimated (R not optimized)
-4. Capacitances wrong (affects R_opt)
-
-**Solutions:**
-- Calibrate ε from measurements
-- Check E_propagation threshold
-- Verify R optimization ran correctly
-- Re-check FEMM extraction
-
----
-
-### Problem: Simulation Predicts Too Long Spark
-
-**Symptoms:**
-- Predicted length >> measured
-- Model overestimates performance
-
-**Possible causes:**
-1. ε too low (underestimating energy needed)
-2. E_propagation set too low
-3. Not accounting for capacitive divider voltage drop
-4. Using burst-mode ε for QCW (or vice versa)
-
-**Solutions:**
-- Increase ε (burst needs higher value)
-- Verify field threshold appropriate for conditions
-- Check V_tip calculation (capacitive division)
-- Use correct ε for operating mode
-
----
-
-### Problem: R_total Outside Expected Range
-
-**Symptoms:**
-- Total resistance 10× too high or too low
-- Doesn't match measurements or expectations
-
-**Possible causes:**
-1. Wrong frequency
-2. Capacitance extraction error
-3. Optimization failure
-4. Physical bounds too restrictive
-
-**Solutions:**
-- Verify frequency used in R calculation
-- Re-check capacitance matrix from FEMM
-- Try simplified R method as sanity check
-- Compare segment-by-segment to expected profile
-
----
-
-### Problem: SPICE Simulation Gives Nonsense Results
-
-**Symptoms:**
-- Negative resistance calculated
-- Infinite impedance
-- Convergence errors
-
-**Possible causes:**
-1. Capacitance matrix implementation wrong
-2. Negative capacitor values
-3. Ground reference missing
-4. Parameter syntax error
-
-**Solutions:**
-- Use partial capacitance transformation (all positive)
-- Verify every capacitor value >0
-- Ensure at least one node grounded
-- Check .param syntax (use {expression} for calculations)
-
----
-
-### Problem: Measured vs Simulated Impedance Differs by Factor >2
-
-**Symptoms:**
-- Model predicts Z = 200 kΩ
-- Measurement shows Z = 450 kΩ (or 90 kΩ)
-
-**Possible causes:**
-1. Measurement method wrong (V_top/I_base)
-2. Spark branching in measurement (not modeled)
-3. Operating mode different (burst vs QCW)
-4. Frequency shift not accounted for
-
-**Solutions:**
-- Use correct measurement port (topload-to-ground)
-- Model cannot capture branching (expected discrepancy)
-- Ensure ε appropriate for actual mode
-- Remeasure at loaded resonance frequency
-
----
-
-### Problem: Growth Stalls Before Target Length
-
-**Symptoms:**
-- Spark stops growing
-- More power doesn't help
-
-**Possible causes:**
-1. Voltage-limited (E_tip < E_propagation)
-2. Capacitive divider drops V_tip too much
-3. E_propagation higher than assumed
-4. Topload too small for target length
-
-**Solutions:**
-- Check E_tip calculation at stall length
-- Consider ramping voltage higher
-- Increase topload capacitance (less voltage division)
-- Reduce target length (be realistic)
-
----
-
-### Problem: QCW Gives Same Length as Burst (Expected Longer)
-
-**Symptoms:**
-- QCW and burst same performance
-- Not seeing efficiency advantage
-
-**Possible causes:**
-1. Using same ε for both (should be different)
-2. QCW ramp too short (not exploiting thermal memory)
-3. Insufficient power for QCW
-4. Leader formation not occurring
-
-**Solutions:**
-- Use ε_QCW = 8-15 J/m, ε_burst = 40-80 J/m
-- Lengthen ramp time (10-20 ms)
-- Increase average power
-- Check current sufficient for leader (>0.5 A)
-
----
-
-### Quick Diagnostic Flowchart
-
-```
-Problem occurs
- |
- ├─ Unreasonable value (negative, infinite, 1000× off)
- | → Check units, formula, syntax
- | → Verify all inputs are correct quantities
- |
- ├─ Non-convergence (oscillation, no stable solution)
- | → Reduce damping factor α
- | → Check if problem has solution (bounds?)
- | → Try simpler model first
- |
- ├─ Mismatch with measurement (factor 2-5)
- | → Verify measurement method
- | → Check operating mode matches
- | → Calibrate ε, E_propagation from data
- |
- └─ Physical impossibility (violates conservation, etc.)
- → Review assumptions
- → Check for double-counting or missing terms
- → Verify reference frames consistent
-```
-
----
-
-## Appendix G: Worked Solutions to Comprehensive Problems
-
-### Part 2 Comprehensive Design Exercise (Solution)
-
-**Given:**
-- f = 190 kHz
-- C_topload = 30 pF
-- Target spark: 3 feet (estimate C_sh)
-- C_mut = 9 pF (from FEMM)
-- Z_th = 105 - j2100 Ω, V_th = 320 kV
-
----
-
-**Task 1: Calculate capacitance ratio and phase constraint**
-
-```
-C_sh = 2 pF/ft × 3 ft = 6 pF
-
-r = C_mut/C_sh = 9/6 = 1.5
-
-φ_Z,min = -atan(2√[r(1+r)])
- = -atan(2√[1.5×2.5])
- = -atan(2√3.75)
- = -atan(2×1.936)
- = -atan(3.872)
- = -75.5°
-
-Cannot achieve -45° (r = 1.5 > 0.207) ✓
-```
-
----
-
-**Task 2: Determine optimal resistances**
-
-```
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-R_opt_power = 1/(ω(C_mut + C_sh))
- = 1/(1.194×10⁶ × 15×10⁻¹²)
- = 1/(17.91×10⁻⁶)
- = 55.8 kΩ
-
-R_opt_phase = 1/(ω√(C_mut(C_mut+C_sh)))
- = 1/(1.194×10⁶ × √(9×10⁻¹² × 15×10⁻¹²))
- = 1/(1.194×10⁶ × 11.62×10⁻¹²)
- = 1/(13.87×10⁻⁶)
- = 72.1 kΩ
-
-R_opt_power < R_opt_phase ✓ (55.8 < 72.1)
-
-At R_opt_power, expect φ_Z ≈ -76° (slightly more capacitive than minimum)
-```
-
----
-
-**Task 3: Build lumped spark model**
-
-```
-Circuit:
- Topload ---[C_mut=9pF]---+--- [C_sh=6pF]---GND
- |
- [R=55.8kΩ]
-
-Calculate Y_spark:
-G = 1/R = 1/55800 = 17.92 μS
-B₁ = ωC_mut = 1.194×10⁶ × 9×10⁻¹² = 10.75 μS
-B₂ = ωC_sh = 1.194×10⁶ × 6×10⁻¹² = 7.16 μS
-
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 17.92 × 51.27 / [321.1 + 319.7]
- = 918.8 / 640.8
- = 1.434 μS
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
- = 7.16 × [321.1 + 191.7] / 640.8
- = 7.16 × 512.8 / 640.8
- = 5.73 μS
-
-Y_spark = 1.434 + j5.73 μS
-```
-
----
-
-**Task 4: Predict performance with Thévenin**
-
-```
-Convert Y_spark to Z_spark:
-|Y_spark| = √(1.434² + 5.73²) = 5.91 μS
-|Z_spark| = 1/5.91×10⁻⁶ = 169 kΩ
-
-φ_Y = atan(5.73/1.434) = 76.0°
-φ_Z = -76.0°
-
-Z_spark = 169 kΩ ∠-76.0°
- = 169 × cos(-76°) + j × 169 × sin(-76°)
- = 41 - j164 kΩ
-
-Total impedance:
-Z_total = Z_th + Z_spark
- = (105 - j2100) + (41000 - j164000)
- = (41105 - j166100) Ω
- = 41.1 - j166.1 kΩ
-
-|Z_total| = √(41.1² + 166.1²) = 171 kΩ
-
-Current:
-I = V_th/Z_total = 320 kV / 171 kΩ = 1.87 A
-
-Power to spark:
-P_spark = 0.5 × I² × Re{Z_spark}
- = 0.5 × 1.87² × 41000
- = 0.5 × 3.50 × 41000
- = 71.7 kW
-```
-
----
-
-**Task 5: Compare to theoretical maximum**
-
-```
-For conjugate match: Z_load = Z_th* = 105 + j2100 Ω
-
-P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
- = 0.5 × (320×10³)² / (4 × 105)
- = 0.5 × 1.024×10¹¹ / 420
- = 122 MW
-
-Actual percentage:
-71.7 kW / 122000 kW = 0.0588%
-
-Spark extracts only 0.06% of theoretical maximum!
-
-Why such huge difference?
-- Conjugate match needs Z_load = 105 + j2100 Ω (very low R, inductive)
-- Actual spark: Z_spark = 41000 - j164000 Ω (high R, capacitive)
-- Topological constraints prevent achieving conjugate match
-- This is NORMAL for Tesla coils
-- The 71.7 kW is still significant useful power
-```
-
----
-
-### Part 4 Final Comprehensive Problem (Partial Solution)
-
-**Given:**
-- f = 195 kHz, 2 m target, QCW 10 ms
-- Topload 35 cm, P_input = 120 kW
-- Z_th = 110 - j2300 Ω, V_th = 340 kV
-
----
-
-**Part 1: Distributed model setup**
-
-```
-Choose n = 10 (good balance accuracy/speed)
-
-FEMM geometry (axisymmetric r-z):
-- Toroid: major R=17.5 cm, minor r=5 cm, center z=0
-- Segments: 10 cylinders, each 20 cm long
- Segment 1: r=0.15 cm, z=-5 to -25 cm
- Segment 2: z=-25 to -45 cm
- ...
- Segment 10: z=-185 to -205 cm
-- Ground plane: z=-220 cm, r=0 to 400 cm
-- Outer boundary: r=400 cm, z=±300 cm
-
-Validation checks after [C] extraction:
-1. Symmetry: C[i,j] = C[j,i] within 0.1%
-2. All diagonal positive
-3. All off-diagonal negative
-4. C_sh_total ≈ 2 pF/ft × 6.56 ft ≈ 13 pF
- (Sum across segments)
-```
-
----
-
-**Part 2: Resistance calculation (simplified method)**
-
-```
-ω = 2π × 195×10³ = 1.225×10⁶ rad/s
-
-Assume FEMM gives C_total[i] = [14, 11, 9, 7.5, 6.5, 5.5, 4.5, 3.5, 2.8, 2.0] pF
-
-R[i] = 1/(ω × C_total[i]):
-
-R[1] = 1/(1.225×10⁶ × 14×10⁻¹²) = 58.3 kΩ
-R[2] = 1/(1.225×10⁶ × 11×10⁻¹²) = 74.6 kΩ
-R[3] = 92.1 kΩ
-R[4] = 110 kΩ
-R[5] = 127 kΩ
-R[6] = 150 kΩ
-R[7] = 184 kΩ
-R[8] = 236 kΩ
-R[9] = 294 kΩ
-R[10] = 408 kΩ
-
-R_total = 1734 kΩ
-
-Expected range at 195 kHz for 2m QCW: 30-120 kΩ
-Actual: 1734 kΩ (high, but long spark distributed can be higher)
-
-Bounds check: All R[i] between 5 kΩ and 500 kΩ ✓
-Distribution: Monotonically increasing ✓
-```
-
----
-
-**Part 3: Performance prediction (abbreviated)**
-
-```
-Build SPICE with [C] matrix and R[i] values
-Run AC analysis at 195 kHz
-
-Expected results (estimated):
-Z_spark ≈ 600 kΩ ∠-72°
-I ≈ 0.5 A
-P_spark ≈ 40 kW
-
-Percentage of theoretical max: <0.1% (typical)
-```
-
----
-
-**Part 4: Growth analysis**
-
-```
-Power available: 40 kW (from part 3)
-ε = 12 J/m (QCW calibrated)
-Target: L = 2 m, Time: T = 10 ms
-
-Energy needed: E = ε × L = 12 × 2 = 24 J
-
-Power needed: P = E/T = 24/0.010 = 2.4 kW
-
-Available: 40 kW >> 2.4 kW needed ✓
-Power is MORE than sufficient
-
-Voltage check:
-V_top = 340 kV (from V_th, approximately)
-κ = 3.2, E_prop = 0.7 MV/m
-E_tip = κ × V_top/L = 3.2 × 340 kV / 2 m
- = 3.2 × 170 kV/m = 544 kV/m = 0.544 MV/m
-
-E_tip = 0.544 MV/m < E_prop = 0.7 MV/m ✗
-
-Growth is VOLTAGE-LIMITED!
-Cannot reach 2 m with 340 kV
-
-Required voltage:
-V_required = E_prop × L / κ = 0.7×10⁶ × 2 / 3.2
- = 437.5 kV
-
-Need to ramp to 438 kV to sustain growth to 2 m
-With 340 kV, maximum length ≈ 340/438 × 2 = 1.55 m
-
-Conclusion: Voltage limited, not power limited
-Need higher voltage ramp or accept shorter spark
-```
-
----
-
-**Part 5: Validation plan**
-
-```
-Measurements to take:
-1. Ringdown: f₀, Q₀ (unloaded); f_L, Q_L (loaded)
- → Extract Y_spark, compare to model
-2. High-speed video: Growth rate dL/dt
- → Validate power/ε relationship
-3. V_top with E-field probe (calibrated)
- → Check voltage predictions
-4. Final spark length with ruler/laser
- → Validate growth model
-
-Refinement process:
-1. If measured length > predicted:
- - Reduce ε (more efficient than assumed)
- - Check E_prop (may be lower)
-2. If measured length < predicted:
- - Increase ε
- - Check for branching (wastes energy)
-3. Adjust R distribution if impedance mismatch
-
-Expected accuracy:
-- Length: ±30% (good agreement)
-- Power: ±40% (acceptable)
-- Impedance: ±25% (reasonable)
-
-Better than factor of 2 on all parameters = success!
-```
-
----
-
-## Appendix H: Further Resources
-
-### Online Communities
-
-**4hv.org Forums**
-- Active Tesla coil community
-- Design sharing and troubleshooting
-- DRSSTC, QCW, SGTC sections
-- Measurement techniques
-
-**High Voltage Forum (highvoltageforum.net)**
-- International community
-- Advanced projects
-- Safety discussions
-
-### Software Tools
-
-**FEMM (femm.info)**
-- Free 2D electromagnetic FEA
-- This framework's primary tool
-- Active development and support
-
-**LTSpice (analog.com/ltspice)**
-- Free SPICE simulator
-- Excellent for circuit analysis
-- Large component library
-
-**Python Scientific Stack**
-- NumPy: Matrix operations
-- SciPy: Optimization algorithms
-- Matplotlib: Plotting
-- Free and powerful
-
-### Books and Papers
-
-**Lightning Physics:**
-- Uman, M.A. "The Lightning Discharge" (comprehensive)
-- Rakov & Uman "Lightning: Physics and Effects" (modern)
-
-**Plasma Physics:**
-- Chen, F.F. "Introduction to Plasma Physics" (accessible)
-- Raizer, Y.P. "Gas Discharge Physics" (detailed)
-
-**High Voltage Engineering:**
-- Kuffel, Zaengl, Kuffel "High Voltage Engineering Fundamentals"
-- Wadhwa, C.L. "High Voltage Engineering"
-
-**Tesla Coil Specific:**
-- "The Spark Gap" magazine archives (historical)
-- Tesla coil design guides (various online)
-
-### Academic Resources
-
-**IEEE Xplore**
-- Search: "spark discharge modeling"
-- "Tesla transformer"
-- "resonant transformer"
-
-**arXiv.org**
-- Physics preprints
-- Some Tesla coil research
-
-### Safety Resources
-
-**ALWAYS prioritize safety:**
-- High voltage safety guidelines
-- Grounding and bonding practices
-- First aid for electrical injuries
-- Equipment safety ratings
-
-**Key principle:** If you're not sure, DON'T DO IT.
-
----
-
-## Closing Remarks
-
-**You now have:**
-- Complete theoretical framework
-- Practical implementation guide
-- Worked examples throughout
-- Troubleshooting resources
-- Validation methodologies
-
-**Next steps:**
-1. Start with lumped model (simple coil)
-2. Calibrate ε from one measurement
-3. Predict new operating point
-4. Progress to distributed model
-5. Share results with community
-
-**Remember:**
-- All models are approximations
-- Plasma physics has uncertainties
-- ±20-50% agreement is GOOD
-- Document your assumptions
-- Compare to measurements
-- Iterate and improve
-
-**Most importantly:**
-- Stay safe
-- Have fun
-- Learn continuously
-- Contribute back to community
-
-**This framework is a starting point, not the final word. As you gain experience, you'll develop intuition and may improve upon these methods. That's the goal!**
-
----
-
-**END OF APPENDICES**
-
-**END OF COMPLETE TESLA COIL SPARK MODELING LESSON PLAN**
-
----
-
-**Total lesson plan:**
-- Part 1: ~18,000 tokens (Foundation)
-- Part 2: ~17,500 tokens (Optimization)
-- Part 3: ~17,800 tokens (Growth Physics & FEMM)
-- Part 4: ~17,900 tokens (Distributed Models)
-- Appendices: ~14,500 tokens (Reference)
-- **Grand Total: ~85,700 tokens**
-
-**Ready for teaching Tesla coil spark modeling from beginner to advanced!**
-
-
diff --git a/spark-lessons/_originals/spark-physics.txt b/spark-lessons/_originals/spark-physics.txt
deleted file mode 100644
index f49c8ad..0000000
--- a/spark-lessons/_originals/spark-physics.txt
+++ /dev/null
@@ -1,856 +0,0 @@
-# Tesla Coil Spark Modeling and Simulation Framework - Final Corrected Edition
-
-## Executive Summary
-
-This document presents a complete framework for modeling Tesla coil sparks using circuit analysis combined with electromagnetic field simulation (FEMM). The key insight is that spark plasma self-optimizes to maximize power transfer within circuit constraints, allowing accurate simulation without detailed plasma physics modeling. Two modeling approaches are presented: a simplified lumped model and a sophisticated nth-order distributed model.
-
-**Convention:** All phasor quantities use **peak values** (not RMS). Power formulas include the 0.5 factor: P = 0.5×Re{V×I*}.
-
----
-
-## Part 1: Fundamental Circuit Topology and Constraints
-
-### 1.1 Basic Spark Circuit Model
-
-Tesla coil sparks exhibit two capacitances revealed by FEMM electrostatic analysis:
-- **Mutual capacitance (C_mut)**: Coupling between spark and topload
-- **Shunt capacitance (C_sh)**: Spark-to-ground capacitance (~2 pF/foot empirically)
-
-The actual topology at the topload connection point is:
-```
-Topload ---[C_mut || R]--- Spark tip
- | |
- | [C_sh]
- | |
- GND ---------------------- GND
-```
-
-### 1.2 Admittance Analysis
-
-At angular frequency ω, with G = 1/R, B₁ = ωC_mut (positive susceptance), B₂ = ωC_sh (positive susceptance):
-
-**Input admittance at topload (looking into spark):**
-```
-Y = ((G + jB₁)·jB₂) / (G + j(B₁ + B₂))
-
-Re{Y} = GB₂² / (G² + (B₁ + B₂)²)
-
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / (G² + (B₁ + B₂)²)
-```
-
-**Admittance phase angle:**
-```
-θ_Y = atan(Im{Y}/Re{Y})
-```
-
-**Impedance phase angle (what we typically measure):**
-```
-φ_Z = -θ_Y = atan(-Im{Y}/Re{Y})
-```
-
-**Important:** When discussing impedance phase, we reference φ_Z. The common "-45°" refers to impedance phase, not admittance phase.
-
-### 1.3 Fundamental Phase Constraint
-
-The circuit topology imposes a **minimum achievable impedance phase angle**:
-
-```
-φ_Z,min = -atan(2√(r(1+r)))
-
-where r = C_mut/C_sh
-```
-
-**Critical insight:** When r ≥ 0.207, achieving φ_Z = -45° (traditionally considered "matched") becomes **mathematically impossible** regardless of R value. This is a topological constraint, not a plasma limitation.
-
-For typical Tesla coil geometries:
-- Large topload, short spark: r = 0.5 to 2.0
-- Resulting φ_Z,min ≈ -50° to -70°
-
-**Note:** Secondary losses add parallel conductance on the source side but don't change the spark's fundamental phase constraint.
-
-The commonly cited "R ≈ |X_c|" relationship emerges because power optimization within topological constraints naturally produces this approximate relationship, not because -45° is achievable.
-
----
-
-## Part 2: Two Critical Resistance Values
-
-### 2.1 R_opt_phase: Closest to Resistive
-
-Minimizes impedance phase magnitude to achieve φ_Z,min:
-```
-R_opt_phase = 1 / (ω√(C_mut(C_mut + C_sh)))
-```
-
-This represents the "most resistive-looking" impedance the circuit can present.
-
-### 2.2 R_opt_power: Maximum Power Transfer
-
-Maximizes real power delivered to the load for fixed topload voltage:
-```
-R_opt_power = 1 / (ω(C_mut + C_sh))
-```
-
-**Numeric example:** At f = 200 kHz with C_mut + C_sh = 12 pF:
-```
-R_opt_power = 1/(2π × 200×10³ × 12×10⁻¹²) ≈ 66 kΩ
-```
-
-**Key relationship:**
-```
-R_opt_power < R_opt_phase always
-
-R_opt_power typically gives phase angles of -55° to -75°
-```
-
-### 2.3 The "Hungry Streamer" Principle
-
-**Steve Conner's insight:** Streamers actively optimize their impedance to maximize power extraction. The plasma adjusts its properties (temperature, ionization, diameter, conductivity) to extract maximum available power from the resonant circuit.
-
-**Physical mechanism:**
-- More power → Joule heating (I²R) → increased temperature
-- Higher temperature → thermal ionization → increased n_e
-- Increased conductivity → R decreases
-- Changed geometry/expansion → modified C_mut, C_sh
-- Modified capacitances → new R_opt_power
-- Plasma conductivity adjusts toward new R_opt_power
-- **Stable equilibrium achieved when R_actual ≈ R_opt_power**
-
-**Constraints on optimization:**
-- Insufficient source current/voltage (primary limited)
-- Inception field not achieved (spark doesn't form)
-- Physical conductivity limits (R_min, R_max)
-- Thermal time constants (can't adjust faster than ~ms)
-
-When constraints prevent reaching R_opt_power, the spark operates sub-optimally or stalls.
-
----
-
-## Part 3: Impedance Measurement at Topload Port
-
-### 3.1 Why V_top/I_base is Wrong
-
-Measuring "impedance" as V_top/I_base is incorrect because I_base includes **all** displacement currents returning to ground:
-- Every secondary section's capacitance to ground
-- Strike ring coupling
-- Primary-to-secondary capacitance
-- **AND** the spark current
-
-This mixes the spark load with all parasitic return paths.
-
-### 3.2 Correct Measurement Port
-
-**The measurement port is topload-to-ground** where the spark physically connects. All impedance and power calculations reference this port.
-
-### 3.3 Thévenin Equivalent Extraction (Recommended)
-
-This method separates Tesla coil characterization from load analysis.
-
-**Step 1: Measure Z_th (output impedance with drive off)**
-- Set primary drive source to AC 0V (short voltage source)
-- Keep all tank components (MMC, L_primary, damping resistors) in circuit
-- Apply 1V AC test source at topload-to-ground
-- Measure current: I_test
-- Calculate: **Z_th = 1V / I_test = R_th + jX_th**
-
-**Step 2: Measure V_th (open-circuit voltage with drive on)**
-- Remove test source
-- Turn primary drive source ON at operating frequency
-- Remove spark load (open-circuit topload)
-- Measure: **V_th = V(topload)** (complex magnitude and phase)
-
-**Step 3: Calculate power to any load**
-For candidate load impedance Z_load:
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**Theoretical maximum power (sanity check):**
-If conjugate match were achievable (Z_load = Z_th*):
-```
-P_max = 0.5 × |V_th|² / (4×Re{Z_th})
-```
-Actual spark power will be less than this due to topological constraints.
-
-**Advantages:**
-- Characterize coil once, evaluate many loads instantly
-- No re-simulation for different spark parameters
-- Separates "coil behavior" (Z_th) from "drive conditions" (V_th)
-
-**Enhancement:** Measure Z_th(ω) and V_th(ω) over a frequency band (±10% of operating frequency) to account for frequency tracking as spark loads the system.
-
-### 3.4 Direct Power Measurement (Alternative)
-
-Keep full coupled model with spark load present:
-- Drive primary at operating frequency and amplitude
-- Run AC analysis
-- Measure power in spark: P = 0.5 × Re{V(top) × conj(I(spark))}
-- Step R to find maximum
-- **Critical:** For each R, retune to loaded pole frequency (resonance shifts with loading)
-
----
-
-## Part 4: DRSSTC Operating Modes and Pole Frequencies
-
-### 4.1 Coupled System Poles
-
-A Tesla coil is a coupled resonant system. Even without a spark, coupling between primary and secondary creates two resonant modes (eigenfrequencies):
-- **Lower pole:** Below the geometric mean
-- **Upper pole:** Above the geometric mean
-
-The spark modifies both pole **frequency and damping**, not just frequency.
-
-### 4.2 Frequency Shift with Loading
-
-As spark grows:
-- C_sh increases (~2 pF/foot)
-- Both poles shift and become more damped
-- Comparing different R values at fixed frequency measures detuning, not inherent matching quality
-
-**Best practice:** For each R value, sweep frequency to find loaded pole (max |V_top|), then measure power at that frequency. This gives true matched performance.
-
----
-
-## Part 5: Spark Growth Physics and Energy Requirements
-
-### 5.1 Voltage Limit: Field Threshold
-
-A spark continues to grow while the electric field at its tip exceeds a threshold.
-
-**Field requirements (at sea level, standard conditions):**
-```
-E_inception ≈ 2-3 MV/m (initial breakdown from smooth topload)
-E_propagation ≈ 0.4-1.0 MV/m (sustained leader growth)
-E_tip = κ × E_average (tip enhancement factor κ ≈ 2-5)
-```
-
-**Maximum voltage-limited length:**
-Solve: E_tip(V_top_peak, L) = E_propagation
-
-Use FEMM to compute E_tip for given V_top and length L. As spark grows, E_tip decreases due to:
-- Increased distance from topload
-- Geometric field dilution
-- Capacitive voltage division (see below)
-
-**Note:** E_propagation varies with altitude and humidity by ±20-30%.
-
-### 5.2 Power Limit: Energy per Meter
-
-Growth consumes approximately constant energy per unit length ε [J/m]:
-
-**Growth rate equation:**
-```
-dL/dt = P_stream / ε (when E_tip > E_propagation)
-dL/dt ≈ 0 (when E_tip < E_propagation, stalled)
-```
-
-**Over time T to reach length L:**
-```
-E_total ≈ ε × L
-P_avg ≈ ε × L / T
-```
-
-### 5.3 Empirical Energy per Meter Values
-
-Requires calibration per coil. Starting values:
-
-**QCW-style growth:**
-- ε ≈ 5-15 J/m
-- Long ramp times (5-20 ms)
-- Leader-dominated channels
-- Energy efficiently extends length
-
-**High duty cycle DRSSTC:**
-- ε ≈ 20-40 J/m
-- Hybrid streamer/leader formation
-- Some thermal accumulation
-- Moderate efficiency
-
-**Hard-pulsed DRSSTC (burst mode):**
-- ε ≈ 30-100+ J/m (single-shot)
-- Short pulses, mostly streamers
-- Much energy → brightening/branching
-- Poor length efficiency
-
-**Advanced refinement:** ε decreases during heating due to thermal accumulation:
-```
-ε(t) = ε₀ / (1 + α∫P_stream dt)
-
-where α has units [1/J] and ∫P_stream dt is accumulated energy
-```
-
-### 5.4 Thermal Memory and Operating Regimes
-
-**Pure thermal diffusion time constant:**
-```
-τ_thermal = d² / (4α)
-
-where α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air
-
-For thin streamers (d ~ 100 μm): τ ~ 0.1-0.2 ms
-For thick leaders (d ~ 5 mm): τ ~ 300-600 ms
-```
-
-**Observed channel persistence is longer than pure thermal diffusion** due to:
-- Buoyancy and convection maintaining hot gas column
-- Ionization memory (recombination slower than thermal diffusion)
-- Broadened effective channel diameter
-
-**Effective persistence times:**
-- Thin streamers: ~1-5 ms (convection/ionization dominated)
-- Thick leaders: seconds (buoyancy maintains hot column)
-
-**QCW advantage:**
-- Ramps of 5-20 ms exploit ionization/convection persistence
-- Channel stays hot throughout growth
-- Continuous energy injection maintains E_tip
-- Transitions streamers → leaders efficiently
-
-**Burst mode characteristics:**
-- Widely spaced bursts: channel cools/deionizes between pulses
-- Must re-ionize repeatedly
-- High peak current → bright, thick but short
-- Voltage collapse limits length before leader formation
-
-### 5.5 Streamers vs Leaders
-
-**Streamers:**
-- Thin (10-100 μm), fast (~10⁶ m/s), low current (mA)
-- Photoionization propagation
-- High resistance, short-lived (μs thermal time)
-- Purple/blue, highly branched
-- High ε (inefficient)
-
-**Leaders:**
-- Thick (mm-cm), slower (~10³ m/s), high current (A)
-- Thermally ionized (5000-20000 K)
-- Low resistance, persistent (seconds with convection)
-- White/orange, straighter
-- Low ε (efficient)
-
-**Transition sequence:**
-1. High E-field creates streamers
-2. Sufficient current → Joule heating
-3. Heated channel → thermal ionization → leader
-4. Leader grows from base
-5. Leader tip launches new streamers
-6. Fed streamers convert to leader
-
-### 5.6 The Capacitive Divider Problem
-
-As spark grows, voltage division limits tip voltage:
-
-```
-V_tip = V_topload × Z_mut/(Z_mut + Z_sh)
-
-where Z_mut = (1/jωC_mut) || R (complex)
- Z_sh = 1/jωC_sh
-```
-
-**Open-circuit limit (R → ∞):**
-```
-V_tip ≈ V_topload × C_mut/(C_mut + C_sh)
-```
-
-**With finite R ≈ R_opt_power:** V_tip is lower and complex. Since C_sh ∝ L:
-- As spark grows, C_sh increases
-- V_tip decreases even if V_topload maintained
-- E_tip decreases
-- Growth becomes harder
-
-This creates sub-linear scaling of length with energy.
-
-### 5.7 Freau's Empirical Relationship
-
-Community observations suggest:
-```
-Single-shot burst: L ∝ √(bang energy)
-Repetitive operation: L ∝ P_avg^(0.3 to 0.5)
-```
-
-**The single-shot √E relationship** applies when there's no thermal accumulation between events - each spark starts cold.
-
-**The repetitive power scaling** applies when thermal/ionization memory carries over between pulses.
-
-**Physical explanation for voltage-limited burst mode:**
-```
-E_field ≈ V_top/L
-Need: V_top > E_propagation × L
-Power to maintain voltage: P ∝ V_top²/Z_spark
-If Z_spark ∝ L, then: L ∝ √P
-```
-
-**QCW shows different scaling** (closer to linear, maybe L ∝ E^0.6-0.8) because:
-- Active voltage ramping compensates for divider
-- Leader formation more energy-efficient
-- Still fights capacitive divider but with mitigation
-
----
-
-## Part 6: Practical Simulation Workflow
-
-### 6.1 Calibration Procedure
-
-**Required measurements (one-time per coil type):**
-
-1. **Energy per meter (ε):**
- - Run coil with known drive, measure final spark length L
- - From SPICE, compute E_delivered = ∫P_spark dt
- - Calculate: ε = E_delivered/L
-
-2. **Field threshold (E_propagation):**
- - Use FEMM to compute E_tip for measured V_top and final L
- - E_propagation ≈ E_tip at stall point
- - Typical: 0.4-1.0 MV/m
-
-### 6.2 Prediction Workflow
-
-**Step 1: Voltage capability check**
-- Simulate to determine V_top(t)
-- Use FEMM: E_tip(V_top, L_target) ≥ E_propagation?
-- If not, target length is voltage-limited
-
-**Step 2: Power/energy requirement**
-- Choose growth time T (e.g., 10 ms for QCW)
-- Required: P_avg ≈ ε × L_target/T
-- Required: E_total ≈ ε × L_target
-
-**Step 3: Verify in SPICE**
-- Verify delivered P_stream meets requirement
-- Check coil stays near loaded pole
-
-**Step 4: Power balance validation**
-```
-P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation
-
-Check: P_spark / P_primary_input = expected efficiency
-```
-
-### 6.3 Growth Simulation (Advanced)
-
-For each time step dt:
-```
-1. Check: E_tip(V_top(t), L) ≥ E_propagation?
-2. If yes: dL/dt = P_stream(t)/ε(L,t)
-3. If no: dL/dt = 0 (stalled)
-4. Update: L = L + (dL/dt)×dt
-5. Update spark model parameters for new L
-6. Optionally track frequency to follow loaded pole
-```
-
----
-
-## Part 7: Lumped Spark Model Theory
-
-### 7.1 Model Structure
-
-Single lumped element:
-```
- C_mut
-Topload ----||---- Node_spark
- |
- [R]
- |
- [C_sh]
- |
- GND
-```
-
-### 7.2 FEMM Extraction
-
-**Electrostatic simulation:**
-- Topload at potential V
-- Spark as cylindrical conductor
-- Ground plane/boundaries
-- Solve for 2×2 capacitance matrix
-
-**Extract values from Maxwell capacitance matrix:**
-
-The Maxwell matrix has C_ii > 0 (self-capacitance) and C_ij < 0 for i≠j (mutual capacitance, negative).
-
-```
-C_mut = -C[topload, spark] = |C_12| (take absolute value of negative off-diagonal)
-C_sh = C[spark, spark] + C[spark, topload] = C_22 - |C_12| (total to ground)
-```
-
-**Sign convention note:** We're using the Maxwell capacitance matrix convention. If using partial capacitances, the extraction differs.
-
-**Typical validation:** C_sh ≈ 2 pF per foot confirms model accuracy.
-
-### 7.3 Determining R
-
-**Default (recommended):**
-```
-R = R_opt_power = 1/(ω(C_mut + C_sh))
-```
-
-**Physical bounds:**
-```
-R_min ≈ 1 kΩ (very hot, thick leader plasma)
-R_max ≈ 100 MΩ (cold, thin streamer plasma)
-R_actual = clip(R_opt_power, R_min, R_max)
-```
-
-If clipping occurs, check if source can provide required power/voltage for this impedance.
-
-### 7.4 User Measurement Integration
-
-**Ringdown method (improved):**
-
-For a parallel RLC equivalent at the loaded resonance ω_L:
-```
-Q_L = ω_L C_eq R_p = R_p/(ω_L L)
-
-Therefore: R_p = Q_L/(ω_L C_eq) or equivalently R_p = Q_L ω_L L
-
-And: G_total = 1/R_p = ω_L C_eq/Q_L or equivalently G_total = 1/(Q_L ω_L L)
-```
-
-**Measurement procedure:**
-1. Measure unloaded: f₀, Q₀, C₀ (from geometry or separate measurement)
-2. Measure with spark: f_L, Q_L
-3. Calculate equivalent capacitance: C_eq = C₀(f₀/f_L)²
-4. Calculate capacitance change: ΔC = C_eq - C₀
-5. Calculate total conductance: G_total = ω_L C_eq/Q_L (using either form above)
-6. Calculate unloaded conductance: G_0 = ω₀ C₀/Q₀
-7. Spark admittance: Y_spark ≈ (G_total - G_0) + jω_L ΔC
-
-**Note:** This method is sensitive to primary coupling effects. The Thévenin port method (Section 3.3) is more robust.
-
-**Direct measurement:**
-- Use E-field probe for V_top (isolated, calibrated)
-- Use Rogowski/CT for I_spark return current (not I_base)
-- Calculate: Y = I/V, extract R from circuit model
-- Low-level option: VNA with capacitive pickup (no spark) to verify Z_th
-
-### 7.5 Limitations
-
-**Good for:**
-- Impedance matching studies
-- Fast simulation
-- Coil design optimization
-
-**Cannot capture:**
-- Current distribution along spark
-- Tip vs. base differences
-- Streamer/leader transitions
-- Very long sparks (>10 feet)
-
----
-
-## Part 8: nth-Order Distributed Spark Model
-
-### 8.1 Model Structure
-
-Divide spark into n segments (typically n=10):
-```
-Topload
- |
-[C_01][R_1][C_1,gnd]
- |
-[C_12][R_2][C_2,gnd]
- |
- ...
- |
-[C_n-1,n][R_n][C_n,gnd]
-```
-
-Each segment: mutual capacitances, shunt capacitance, resistance. Optional: inductances if magnetic effects significant.
-
-### 8.2 FEMM Extraction
-
-**Electrostatic:**
-- n cylindrical segments + topload + environment
-- Solve for (n+1)×(n+1) capacitance matrix
-- Includes all segment-to-segment and segment-to-environment couplings
-
-**SPICE implementation challenge:**
-Maxwell C-matrix has negative off-diagonals (C_ij < 0 for i≠j). Direct implementation as literal capacitors problematic. Solutions:
-1. **Partial-capacitance matrix:** Use capacitances to ground with all others grounded (positive definite)
-2. **Controlled sources:** Implement via MNA: I_i = Σ_j C_ij dV_j/dt
-3. **Nearest-neighbor approximation:** Approximate with local couplings, validate against full matrix
-
-**Passivity check:** Ensure C-matrix is symmetric positive semi-definite (SPD). If numerical noise creates slight non-passivity, add small diagonal term (+0.1 pF) or small series R for numerical stability.
-
-### 8.3 Resistance Optimization: Iterative Power Maximization
-
-**Initialization (tapered, recommended):**
-```
-position = i/(n-1) # 0 at base, 1 at tip
-R[i] = R_base + (R_tip - R_base)×position²
-R_base = 10 kΩ, R_tip = 1 MΩ
-```
-
-**Iterative algorithm with damping:**
-```
-Iterate until convergence:
- For each segment i:
- Sweep R[i] to find value maximizing P[i]
- Apply damping: R_new[i] = α×R_optimal[i] + (1-α)×R_old[i]
- where α ≈ 0.3-0.5 for stability
- Clip to bounds: R[i] = clip(R_new[i], R_min[i], R_max[i])
- Check convergence: max relative change < 1%
-
-If poles shifted >5%, re-optimize at new frequency
-```
-
-**Physical bounds (position-dependent):**
-```
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ)×position
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ)×position
-```
-
-**Convergence behavior:**
-- Well-coupled base segments: sharp power peak, fast convergence to low R
-- Poorly-coupled tip segments: flat power curve, may not converge to unique value, stays at high R
-- This naturally produces leader (base) + streamer (tip) distribution
-
-**Typical total resistance validation:**
-
-At 200 kHz for 1-3 meter sparks:
-- **Streamer-dominated (burst mode):** Total R ≈ 50-300 kΩ
-- **Leader-dominated (QCW):** Total R ≈ 5-50 kΩ (hot, thick channels)
-- **Very low frequency (<100 kHz) or very long sparks:** Can approach 1-10 kΩ
-
-Calculate total: R_total = Σ R[i]
-
-Flag if significantly outside these ranges for your frequency and length.
-
-### 8.4 Circuit-Determined Resistance (Simplified Alternative)
-
-If plasma always adjusts to R_opt_power and C depends weakly on diameter (logarithmically):
-
-```
-For each segment:
- C_total[i] = C_shunt[i] + sum(C_mutual[i,:])
- R[i] = 1/(ω × C_total[i])
- R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Justification:**
-- C ∝ 1/ln(h/d): weak diameter dependence
-- R_opt ∝ 1/C: also weak diameter dependence
-- 2× diameter → ~10-15% change in C, R
-- Error acceptable given other uncertainties (FEMM ~10%, plasma variability ~50%)
-
-**When to use:** Standard cases within typical parameter ranges.
-**When to iterate:** Edge cases, validation studies, highest accuracy needs.
-
-### 8.5 Diameter Considerations
-
-**Circuit-first view (recommended):**
-1. Use nominal diameter in FEMM (e.g., 1 mm for burst, 3 mm for QCW)
-2. Calculate C matrices
-3. Calculate R_opt from C
-4. Plasma adjusts properties to match R_opt
-5. Diameter is dependent variable
-
-**Self-consistency check (optional):**
-```
-d_nominal = 1e-3 m # 1 mm starting guess
-C_mut, C_sh = FEMM(d_nominal)
-R_opt = 1/(ω(C_mut + C_sh))
-
-# Back-calculate implied diameter (typical partially ionized plasma):
-ρ_typical = 10 Ω·m
-L_segment = L_total/n_segments
-d_implied = sqrt(4×ρ_typical×L_segment / (π×R_opt))
-
-# If d_implied ≈ d_nominal (within factor of 2), self-consistent
-# If not, iterate once with d = (d_nominal + d_implied)/2
-```
-
-Because dependence is logarithmic, typically converges in 1-2 iterations if needed.
-
----
-
-## Part 9: Impedance Matching for Target Spark Length
-
-### 9.1 QCW Matching Strategy
-
-During QCW, spark grows from 0 to target length. Impedance changes dramatically.
-
-**Recommendation: Match at 50-70% of target length**
-
-**Reasoning:**
-- Decent power transfer throughout ramp
-- Spark grows fastest in middle phase
-- Frequency tracking compensates for mismatch
-
-**Rule of thumb: Match at 60% for first design iteration**
-
-### 9.2 Optimization Approach
-
-Minimize total energy over growth:
-```
-E_total = ∫₀ᵀ [ε × L(t)/η(t)] dt
-η(t) = power transfer efficiency
-```
-
-**Procedure:**
-1. Simulate growth with match points at 0%, 30%, 50%, 70%, 100%
-2. Calculate E_total to reach target for each
-3. Choose match point minimizing E_total
-
-### 9.3 Burst Mode Matching
-
-For non-ramping burst:
-- Match to final spark length (100%)
-- Coil rings up quickly
-- Steady-state matching more important
-
----
-
-## Part 10: Implementation Summary
-
-### 10.1 Lumped Model Workflow
-
-1. FEMM electrostatic: topload + single spark cylinder
-2. Extract C_mut = |C_12|, C_sh = C_22 - |C_12| from Maxwell matrix
-3. Calculate R = 1/(ω(C_mut + C_sh)), clip to bounds
-4. Build SPICE: (C_mut||R) in series with C_sh at topload port
-5. AC analysis: Thévenin equivalent or direct power measurement
-6. Use for matching optimization and performance prediction
-
-### 10.2 nth-Order Workflow
-
-1. FEMM: n segments + environment → full C-matrix
-2. Optional: magnetic analysis → L-matrix
-3. Initialize R with tapered profile
-4. Choose approach:
- - Full iterative optimization with damping (highest accuracy)
- - Simplified R = 1/(ωC_total) (good for typical cases)
-5. Export to SPICE with proper C-matrix handling (partial capacitances or controlled sources)
-6. AC analysis or transient simulation
-7. Validate: power balance, total R in expected range, R distribution physical
-
-### 10.3 Validation Strategy
-
-**Tests:**
-- Lumped vs. 1-segment nth-order (should match exactly)
-- Convergence: n=5 vs. n=10 vs. n=20 (diminishing changes)
-- Measurements: compare impedance, power, length to real coil
-- Self-consistency: R distribution shows base < tip, total R reasonable
-
----
-
-## Part 11: Key Equations Reference
-
-### Circuit Analysis
-```
-R_opt_power = 1/(ω(C_mut + C_sh))
-Example: f=200 kHz, C_total=12 pF → R_opt ≈ 66 kΩ
-
-R_opt_phase = 1/(ω√(C_mut(C_mut + C_sh)))
-
-φ_Z,min = -atan(2√(r(1+r))), r = C_mut/C_sh
-
-Y = ((G+jB₁)·jB₂)/(G+j(B₁+B₂))
-where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh (positive susceptances)
-
-φ_Z = -atan(Im{Y}/Re{Y}) (impedance phase)
-```
-
-### Thévenin Equivalent
-```
-Z_th = 1V/I_test (drive off, test source on)
-V_th = V(topload) (drive on, no spark)
-P_load = 0.5×|V_th|²×Re{Z_load}/|Z_th+Z_load|²
-
-Theoretical maximum (conjugate match):
-P_max = 0.5×|V_th|²/(4×Re{Z_th})
-```
-
-### Spark Growth
-```
-E_inception ≈ 2-3 MV/m (initial breakdown)
-E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
-
-dL/dt = P_stream/ε (when E_tip > E_propagation)
-
-ε ≈ 5-15 J/m (QCW), 20-40 J/m (hybrid), 30-100 J/m (burst)
-ε(t) = ε₀/(1 + α∫P dt), where [α] = 1/J
-
-V_tip ≈ V_topload×C_mut/(C_mut+C_sh) (open-circuit limit)
-
-τ_thermal = d²/(4α), α ≈ 2×10⁻⁵ m²/s for air
-d=100 μm → τ~0.1 ms; d=5 mm → τ~300 ms
-(Observed persistence longer due to convection/ionization)
-```
-
-### Physical Bounds
-```
-R_min ≈ 1-10 kΩ (hot leader plasma, position-dependent)
-R_max ≈ 100 kΩ - 100 MΩ (cold streamer, position-dependent)
-
-Typical total spark resistance at 200 kHz for 1-3 m:
-- Burst/streamer: 50-300 kΩ
-- QCW/leader: 5-50 kΩ
-- Low frequency/very long: can approach 1-10 kΩ
-
-Typical impedance phase: -55° to -75°
-```
-
-### Ringdown Method
-```
-At loaded resonance ω_L:
-Q_L = ω_L C_eq R_p = R_p/(ω_L L)
-
-R_p = Q_L/(ω_L C_eq) = Q_L ω_L L
-G_total = ω_L C_eq/Q_L = 1/(Q_L ω_L L)
-
-C_eq = C₀(f₀/f_L)²
-Y_spark ≈ (G_total - G_0) + jω_L(C_eq - C_0)
-```
-
----
-
-## Part 12: Open Questions and Future Work
-
-### 12.1 Remaining Uncertainties
-
-- ε variability with current density, frequency, ambient conditions
-- E_propagation dependence on geometry, humidity, altitude
-- Full thermal evolution including convection and radiation
-- Branching: power division among multiple channels
-
-### 12.2 Future Enhancements
-
-**Advanced physics:**
-- Dynamic capacitance: d_eff(E) = d₀×(1 + β×ln(E/E_threshold))
-- Radial temperature profiles: hot core, cool edges
-- Time-dependent ε with thermal memory
-- Branching models: I_branch ∝ d_branch^1.5
-
-**Simulation improvements:**
-- Full transient with L(t) evolution
-- 3D FEA for complex geometries
-- Monte Carlo for stochastic breakout/branching
-- Strike detection: R → few ohms when contact occurs
-
-**Validation needs:**
-- Systematic measurements across coil types, frequencies, power levels
-- High-speed photography for growth rate validation
-- RF current distribution measurements at multiple points
-- Database correlating spark parameters to operating conditions
-
----
-
-## Conclusion
-
-This framework provides practical, implementable Tesla coil spark modeling:
-
-**Core principles:**
-1. Circuit topology imposes fundamental phase constraints
-2. Plasma self-optimizes within constraints (hungry streamer)
-3. R_opt_power maximizes power transfer
-4. Capacitances depend weakly (logarithmically) on diameter
-5. Circuit determines R; plasma adjusts to match
-6. Growth requires E_tip > E_propagation AND sufficient energy (ε×L)
-
-**For basic use:** Lumped model with R = R_opt_power
-
-**For advanced use:** nth-order distributed model with iterative (highest accuracy) or simplified (good for typical cases) R optimization
-
-**Critical:** Calibrate ε and E_propagation from measurements, then predict new operating conditions with validated power balance.
-
-The framework balances theoretical rigor with practical implementation, acknowledging where empirical calibration fills gaps in complex plasma physics while maintaining solid circuit-theoretical foundations.
\ No newline at end of file
diff --git a/spark-lessons/app/__init__.py b/spark-lessons/app/__init__.py
deleted file mode 100644
index 0b43d6f..0000000
--- a/spark-lessons/app/__init__.py
+++ /dev/null
@@ -1,6 +0,0 @@
-"""
-Tesla Coil Spark Physics Course - PyQt5 Application
-"""
-
-__version__ = '1.0.0'
-__author__ = 'Tesla Coil Community'
diff --git a/spark-lessons/app/config.py b/spark-lessons/app/config.py
deleted file mode 100644
index 972ef34..0000000
--- a/spark-lessons/app/config.py
+++ /dev/null
@@ -1,259 +0,0 @@
-"""
-Configuration and constants for Tesla Coil Spark Course application
-"""
-
-from pathlib import Path
-
-# ============================================================================
-# Paths
-# ============================================================================
-
-# Base directory (spark-lessons/)
-BASE_DIR = Path(__file__).parent.parent
-
-# Content directories
-LESSONS_DIR = BASE_DIR / 'lessons'
-EXERCISES_DIR = BASE_DIR / 'exercises'
-REFERENCE_DIR = BASE_DIR / 'reference'
-WORKED_EXAMPLES_DIR = BASE_DIR / 'worked-examples'
-ASSETS_DIR = BASE_DIR / 'assets'
-
-# Course structure
-COURSE_JSON = BASE_DIR / 'course.json'
-
-# Resources
-RESOURCES_DIR = BASE_DIR / 'resources'
-STYLES_DIR = RESOURCES_DIR / 'styles'
-ICONS_DIR = RESOURCES_DIR / 'icons'
-DATABASE_DIR = RESOURCES_DIR / 'database'
-SYMBOLS_JSON = RESOURCES_DIR / 'symbols_definitions.json'
-IMAGES_DIR = ASSETS_DIR / 'images'
-
-# User data (created in user's home directory)
-USER_HOME = Path.home()
-USER_DATA_DIR = USER_HOME / '.tesla_spark_course'
-USER_DATA_DIR.mkdir(exist_ok=True)
-
-DATABASE_PATH = USER_DATA_DIR / 'progress.db'
-USER_NOTES_DIR = USER_DATA_DIR / 'notes'
-USER_NOTES_DIR.mkdir(exist_ok=True)
-
-# ============================================================================
-# Application Constants
-# ============================================================================
-
-APP_NAME = "Tesla Coil Spark Physics Course"
-APP_VERSION = "1.0.0"
-APP_AUTHOR = "Tesla Coil Community"
-
-# ============================================================================
-# UI Constants
-# ============================================================================
-
-# Window dimensions
-DEFAULT_WINDOW_WIDTH = 1400
-DEFAULT_WINDOW_HEIGHT = 900
-MIN_WINDOW_WIDTH = 1000
-MIN_WINDOW_HEIGHT = 600
-
-# Panel sizes
-NAVIGATION_PANEL_MIN_WIDTH = 250
-NAVIGATION_PANEL_DEFAULT_WIDTH = 300
-PROGRESS_PANEL_MIN_WIDTH = 280
-PROGRESS_PANEL_DEFAULT_WIDTH = 320
-CONTENT_PANEL_MIN_WIDTH = 600
-
-# Font sizes
-FONT_SIZE_SMALL = 10
-FONT_SIZE_NORMAL = 12
-FONT_SIZE_LARGE = 14
-FONT_SIZE_TITLE = 16
-
-# Colors (light theme)
-COLOR_PRIMARY = "#3498db" # Blue
-COLOR_SECONDARY = "#9b59b6" # Purple
-COLOR_SUCCESS = "#27ae60" # Green
-COLOR_WARNING = "#f39c12" # Orange
-COLOR_DANGER = "#e74c3c" # Red
-COLOR_ERROR = "#e74c3c" # Red (alias)
-COLOR_INFO = "#2ecc71" # Light green
-COLOR_BACKGROUND = "#ffffff" # White
-COLOR_PANEL_BACKGROUND = "#f8f9fa" # Light gray
-COLOR_TEXT = "#2c3e50" # Dark blue-gray
-COLOR_TEXT_SECONDARY = "#7f8c8d" # Gray
-COLOR_BORDER = "#dee2e6" # Light border
-COLOR_HIGHLIGHT = "#e3f2fd" # Light blue
-
-# Status colors
-COLOR_STATUS_COMPLETE = "#27ae60" # Green
-COLOR_STATUS_IN_PROGRESS = "#f39c12" # Orange
-COLOR_STATUS_NOT_STARTED = "#95a5a6" # Gray
-COLOR_STATUS_LOCKED = "#bdc3c7" # Light gray
-
-# Progress bar colors
-COLOR_PROGRESS_BG = "#ecf0f1"
-COLOR_PROGRESS_FG = "#3498db"
-COLOR_PROGRESS_COMPLETE = "#2ecc71"
-
-# ============================================================================
-# Course Constants
-# ============================================================================
-
-TOTAL_LESSONS = 30
-TOTAL_EXERCISES = 18
-TOTAL_POINTS = 525
-TOTAL_PARTS = 4
-
-# Difficulty levels
-DIFFICULTY_BEGINNER = "beginner"
-DIFFICULTY_INTERMEDIATE = "intermediate"
-DIFFICULTY_ADVANCED = "advanced"
-
-DIFFICULTY_COLORS = {
- DIFFICULTY_BEGINNER: "#2ecc71", # Green
- DIFFICULTY_INTERMEDIATE: "#f39c12", # Orange
- DIFFICULTY_ADVANCED: "#e74c3c" # Red
-}
-
-# Lesson status
-STATUS_NOT_STARTED = "not_started"
-STATUS_IN_PROGRESS = "in_progress"
-STATUS_COMPLETED = "completed"
-
-# Status icons (Unicode)
-ICON_COMPLETE = "✓"
-ICON_IN_PROGRESS = "⊙"
-ICON_NOT_STARTED = "○"
-ICON_LOCKED = "🔒"
-ICON_EXERCISE = "⚡"
-ICON_BOOKMARK = "⭐"
-
-# ============================================================================
-# Progress & Gamification Constants
-# ============================================================================
-
-# Level thresholds (points)
-LEVELS = [
- (0, "Novice", "Circuit Curious"),
- (100, "Learner", "Circuit Explorer"),
- (250, "Practitioner", "Circuit Master"),
- (400, "Expert", "Tesla Scholar"),
-]
-
-# Achievement definitions
-ACHIEVEMENTS = {
- 'quick_learner': {
- 'name': 'Quick Learner',
- 'description': 'Complete first lesson in under 15 minutes',
- 'icon': '🏆',
- 'condition': 'first_lesson_under_15min'
- },
- 'accuracy_master': {
- 'name': 'Accuracy Master',
- 'description': 'Maintain 85%+ average on exercises',
- 'icon': '🎯',
- 'condition': 'exercise_avg_85_percent'
- },
- 'bookworm': {
- 'name': 'Bookworm',
- 'description': 'Complete Part 1 in under 3 hours',
- 'icon': '📚',
- 'condition': 'part1_under_3hours'
- },
- 'streak_master': {
- 'name': 'Streak Master',
- 'description': 'Study for 7 consecutive days',
- 'icon': '🔥',
- 'condition': 'streak_7_days'
- },
- 'lab_rat': {
- 'name': 'Lab Rat',
- 'description': 'Complete 5 exercises with perfect scores',
- 'icon': '🧪',
- 'condition': 'perfect_5_exercises'
- },
- 'insight': {
- 'name': 'Insight',
- 'description': 'Average fewer than 2 hints per exercise',
- 'icon': '💡',
- 'condition': 'avg_hints_under_2'
- },
- 'power_user': {
- 'name': 'Power User',
- 'description': 'Use 10+ keyboard shortcuts',
- 'icon': '⚡',
- 'condition': 'shortcuts_10_plus'
- },
- 'graduate': {
- 'name': 'Graduate',
- 'description': 'Complete all 30 lessons',
- 'icon': '🎓',
- 'condition': 'all_lessons_complete'
- },
-}
-
-# ============================================================================
-# Auto-save Settings
-# ============================================================================
-
-AUTO_SAVE_INTERVAL = 10000 # milliseconds (10 seconds)
-PROGRESS_UPDATE_INTERVAL = 1000 # milliseconds (1 second for time tracking)
-
-# ============================================================================
-# Markdown Rendering
-# ============================================================================
-
-# MathJax CDN
-MATHJAX_CDN = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
-
-# Markdown extensions
-MARKDOWN_EXTENSIONS = [
- 'extra',
- 'codehilite',
- 'tables',
- 'toc',
- 'pymdownx.arithmatex',
- 'pymdownx.superfences',
- 'pymdownx.highlight',
-]
-
-# ============================================================================
-# Keyboard Shortcuts
-# ============================================================================
-
-SHORTCUTS = {
- 'next_lesson': 'Ctrl+Right',
- 'prev_lesson': 'Ctrl+Left',
- 'first_lesson': 'Ctrl+Home',
- 'last_lesson': 'Ctrl+End',
- 'search': 'Ctrl+F',
- 'bookmark': 'Ctrl+B',
- 'mark_complete': 'Ctrl+M',
- 'exercises': 'Ctrl+E',
- 'references': 'Ctrl+R',
- 'notes': 'Ctrl+N',
- 'dashboard': 'Ctrl+P',
- 'fullscreen': 'F11',
- 'quit': 'Ctrl+Q',
-}
-
-# ============================================================================
-# Default User Settings
-# ============================================================================
-
-DEFAULT_SETTINGS = {
- 'theme': 'light',
- 'font_size': FONT_SIZE_NORMAL,
- 'auto_save': True,
- 'show_hints': True,
- 'learning_path': 'intermediate',
- 'auto_mark_complete': True, # Auto-mark at 95% scroll
- 'sound_effects': False,
-}
-
-# ============================================================================
-# Debug Mode
-# ============================================================================
-
-DEBUG = True # Set to False for production
-VERBOSE_LOGGING = DEBUG
diff --git a/spark-lessons/app/database.py b/spark-lessons/app/database.py
deleted file mode 100644
index 0b0a221..0000000
--- a/spark-lessons/app/database.py
+++ /dev/null
@@ -1,332 +0,0 @@
-"""
-Database connection manager for Tesla Coil Spark Course
-Handles SQLite connections, schema creation, and queries
-"""
-
-import sqlite3
-import os
-from pathlib import Path
-from datetime import datetime
-
-
-class Database:
- """SQLite database manager"""
-
- def __init__(self, db_path=None):
- """
- Initialize database connection
-
- Args:
- db_path: Path to SQLite database file. If None, uses default location.
- """
- if db_path is None:
- # Default location: user's home directory
- home = Path.home()
- data_dir = home / '.tesla_spark_course'
- data_dir.mkdir(exist_ok=True)
- db_path = data_dir / 'progress.db'
-
- self.db_path = db_path
- self.connection = None
- self._connect()
- self._initialize_schema()
-
- def _connect(self):
- """Establish database connection"""
- try:
- self.connection = sqlite3.connect(
- self.db_path,
- check_same_thread=False # Allow usage from multiple threads
- )
- self.connection.row_factory = sqlite3.Row # Access columns by name
- print(f"[DB] Connected to database: {self.db_path}")
- except sqlite3.Error as e:
- print(f"[DB ERROR] Failed to connect: {e}")
- raise
-
- def _initialize_schema(self):
- """Create tables if they don't exist"""
- schema_file = Path(__file__).parent.parent / 'resources' / 'database' / 'schema.sql'
-
- if not schema_file.exists():
- print(f"[DB WARNING] Schema file not found: {schema_file}")
- return
-
- try:
- with open(schema_file, 'r') as f:
- schema_sql = f.read()
-
- cursor = self.connection.cursor()
- cursor.executescript(schema_sql)
- self.connection.commit()
- print("[DB] Schema initialized successfully")
- except sqlite3.Error as e:
- print(f"[DB ERROR] Failed to initialize schema: {e}")
- raise
-
- def execute(self, query, params=None):
- """
- Execute a query and return cursor
-
- Args:
- query: SQL query string
- params: Query parameters (tuple or dict)
-
- Returns:
- sqlite3.Cursor
- """
- try:
- cursor = self.connection.cursor()
- if params:
- cursor.execute(query, params)
- else:
- cursor.execute(query)
- return cursor
- except sqlite3.Error as e:
- print(f"[DB ERROR] Query failed: {e}")
- print(f"[DB ERROR] Query: {query}")
- raise
-
- def fetch_one(self, query, params=None):
- """Execute query and fetch one result"""
- cursor = self.execute(query, params)
- return cursor.fetchone()
-
- def fetch_all(self, query, params=None):
- """Execute query and fetch all results"""
- cursor = self.execute(query, params)
- return cursor.fetchall()
-
- def commit(self):
- """Commit transaction"""
- self.connection.commit()
-
- def close(self):
- """Close database connection"""
- if self.connection:
- self.connection.close()
- print("[DB] Connection closed")
-
- # =========================================================================
- # Convenience methods for common operations
- # =========================================================================
-
- def get_user(self, user_id=1):
- """Get user by ID (default user is ID 1)"""
- return self.fetch_one(
- "SELECT * FROM users WHERE user_id = ?",
- (user_id,)
- )
-
- def get_lesson_progress(self, user_id, lesson_id):
- """Get progress for a specific lesson"""
- return self.fetch_one(
- "SELECT * FROM lesson_progress WHERE user_id = ? AND lesson_id = ?",
- (user_id, lesson_id)
- )
-
- def update_lesson_progress(self, user_id, lesson_id, **kwargs):
- """
- Update lesson progress
-
- Args:
- user_id: User ID
- lesson_id: Lesson ID
- **kwargs: Fields to update (status, scroll_position, time_spent, etc.)
- """
- # First, ensure record exists
- existing = self.get_lesson_progress(user_id, lesson_id)
-
- if existing is None:
- # Create new record
- self.execute(
- """INSERT INTO lesson_progress
- (user_id, lesson_id, first_opened, last_accessed)
- VALUES (?, ?, ?, ?)""",
- (user_id, lesson_id, datetime.now(), datetime.now())
- )
-
- # Update fields
- if kwargs:
- # Add last_accessed to every update
- kwargs['last_accessed'] = datetime.now()
-
- set_clause = ', '.join([f"{key} = ?" for key in kwargs.keys()])
- values = list(kwargs.values()) + [user_id, lesson_id]
-
- query = f"""UPDATE lesson_progress
- SET {set_clause}
- WHERE user_id = ? AND lesson_id = ?"""
- self.execute(query, values)
- self.commit()
-
- def mark_lesson_complete(self, user_id, lesson_id):
- """Mark a lesson as completed"""
- self.update_lesson_progress(
- user_id, lesson_id,
- status='completed',
- completion_percentage=100,
- completed_at=datetime.now()
- )
-
- def get_all_lesson_progress(self, user_id):
- """Get progress for all lessons"""
- return self.fetch_all(
- "SELECT * FROM lesson_progress WHERE user_id = ?",
- (user_id,)
- )
-
- def record_exercise_attempt(self, user_id, exercise_id, user_answer,
- is_correct, points_earned, points_possible,
- hints_used=0, time_taken=0, lesson_id=None):
- """Record an exercise attempt"""
- # Get attempt number
- cursor = self.execute(
- """SELECT COALESCE(MAX(attempt_number), 0) + 1 as next_attempt
- FROM exercise_attempts
- WHERE user_id = ? AND exercise_id = ?""",
- (user_id, exercise_id)
- )
- attempt_number = cursor.fetchone()['next_attempt']
-
- # Insert attempt
- self.execute(
- """INSERT INTO exercise_attempts
- (user_id, exercise_id, lesson_id, attempt_number, user_answer,
- is_correct, points_earned, points_possible, hints_used, time_taken)
- VALUES (?, ?, ?, ?, ?, ?, ?, ?, ?, ?)""",
- (user_id, exercise_id, lesson_id, attempt_number, user_answer,
- is_correct, points_earned, points_possible, hints_used, time_taken)
- )
-
- # Update or create completion record
- existing = self.fetch_one(
- "SELECT * FROM exercise_completion WHERE user_id = ? AND exercise_id = ?",
- (user_id, exercise_id)
- )
-
- if existing is None:
- # First attempt
- self.execute(
- """INSERT INTO exercise_completion
- (user_id, exercise_id, best_score, max_possible, total_attempts,
- first_attempted, first_completed, last_attempted)
- VALUES (?, ?, ?, ?, ?, ?, ?, ?)""",
- (user_id, exercise_id, points_earned, points_possible, 1,
- datetime.now(), datetime.now() if is_correct else None, datetime.now())
- )
- else:
- # Update existing
- best_score = max(existing['best_score'], points_earned)
- first_completed = existing['first_completed']
- if is_correct and first_completed is None:
- first_completed = datetime.now()
-
- self.execute(
- """UPDATE exercise_completion
- SET best_score = ?, total_attempts = total_attempts + 1,
- first_completed = ?, last_attempted = ?
- WHERE user_id = ? AND exercise_id = ?""",
- (best_score, first_completed, datetime.now(), user_id, exercise_id)
- )
-
- self.commit()
-
- def get_overall_progress(self, user_id):
- """Get overall progress statistics"""
- # Total points earned
- points_result = self.fetch_one(
- """SELECT SUM(best_score) as total_points
- FROM exercise_completion
- WHERE user_id = ?""",
- (user_id,)
- )
- total_points = points_result['total_points'] or 0
-
- # Lessons completed
- lessons_result = self.fetch_one(
- """SELECT COUNT(*) as completed
- FROM lesson_progress
- WHERE user_id = ? AND status = 'completed'""",
- (user_id,)
- )
- lessons_completed = lessons_result['completed'] or 0
-
- # Total study time
- time_result = self.fetch_one(
- """SELECT SUM(time_spent) as total_time
- FROM lesson_progress
- WHERE user_id = ?""",
- (user_id,)
- )
- total_time = time_result['total_time'] or 0
-
- return {
- 'total_points': total_points,
- 'lessons_completed': lessons_completed,
- 'total_time': total_time,
- 'percentage': (lessons_completed / 30.0) * 100 # 30 total lessons
- }
-
- def update_study_session(self, user_id):
- """Update or create today's study session"""
- today = datetime.now().date()
-
- existing = self.fetch_one(
- "SELECT * FROM study_sessions WHERE user_id = ? AND session_date = ?",
- (user_id, today)
- )
-
- if existing is None:
- self.execute(
- """INSERT INTO study_sessions
- (user_id, session_date, session_start)
- VALUES (?, ?, ?)""",
- (user_id, today, datetime.now())
- )
- else:
- self.execute(
- """UPDATE study_sessions
- SET session_end = ?
- WHERE user_id = ? AND session_date = ?""",
- (datetime.now(), user_id, today)
- )
-
- self.commit()
-
- def get_study_streak(self, user_id):
- """Calculate current study streak (consecutive days)"""
- sessions = self.fetch_all(
- """SELECT session_date FROM study_sessions
- WHERE user_id = ?
- ORDER BY session_date DESC""",
- (user_id,)
- )
-
- if not sessions:
- return 0
-
- from datetime import timedelta
- streak = 0
- expected_date = datetime.now().date()
-
- for session in sessions:
- session_date = datetime.strptime(session['session_date'], '%Y-%m-%d').date()
- if session_date == expected_date:
- streak += 1
- expected_date -= timedelta(days=1)
- else:
- break
-
- return streak
-
-
-# Global database instance
-_db_instance = None
-
-def get_database():
- """Get global database instance"""
- global _db_instance
- if _db_instance is None:
- _db_instance = Database()
- return _db_instance
diff --git a/spark-lessons/app/main.py b/spark-lessons/app/main.py
deleted file mode 100644
index 4cbe21b..0000000
--- a/spark-lessons/app/main.py
+++ /dev/null
@@ -1,85 +0,0 @@
-"""
-Tesla Coil Spark Physics Course - Main Application Entry Point
-PyQt5 Desktop Application
-"""
-
-import sys
-from pathlib import Path
-
-# Add parent directory to path so we can import app package
-sys.path.insert(0, str(Path(__file__).parent.parent))
-
-from PyQt5.QtWidgets import QApplication, QMessageBox
-from PyQt5.QtCore import Qt
-
-# Import configuration and models
-from app import config
-from app.database import get_database
-from app.models import get_course
-from app.views import MainWindow
-
-
-def main():
- """Main application entry point"""
-
- # Create QApplication
- app = QApplication(sys.argv)
- app.setApplicationName(config.APP_NAME)
- app.setApplicationVersion(config.APP_VERSION)
- app.setOrganizationName(config.APP_AUTHOR)
-
- # Enable high DPI scaling
- app.setAttribute(Qt.AA_EnableHighDpiScaling, True)
- app.setAttribute(Qt.AA_UseHighDpiPixmaps, True)
-
- print("="*60)
- print(f"{config.APP_NAME} v{config.APP_VERSION}")
- print("="*60)
-
- try:
- # Initialize database
- print("[*] Initializing database...")
- db = get_database()
- print(f"[OK] Database ready: {db.db_path}")
-
- # Load course structure
- print("[*] Loading course structure...")
- course = get_course()
- print(f"[OK] Course loaded: {course.title}")
-
- # Validate lesson files
- print("[*] Validating lesson files...")
- if course.validate():
- print("[OK] All lesson files found")
- else:
- print("[WARN] Some lesson files missing (see above)")
-
- # Create and show main window
- print("[*] Creating main window...")
- window = MainWindow()
- window.show()
-
- print("[OK] Application ready!\n")
-
- # Run application event loop
- return app.exec_()
-
- except Exception as e:
- # Show error dialog
- print(f"\n[ERROR] {e}")
- import traceback
- traceback.print_exc()
-
- error_dialog = QMessageBox()
- error_dialog.setIcon(QMessageBox.Critical)
- error_dialog.setWindowTitle("Error")
- error_dialog.setText("Failed to initialize application")
- error_dialog.setInformativeText(str(e))
- error_dialog.setDetailedText(traceback.format_exc())
- error_dialog.exec_()
-
- return 1
-
-
-if __name__ == '__main__':
- sys.exit(main())
diff --git a/spark-lessons/app/models/__init__.py b/spark-lessons/app/models/__init__.py
deleted file mode 100644
index 86bc635..0000000
--- a/spark-lessons/app/models/__init__.py
+++ /dev/null
@@ -1,7 +0,0 @@
-"""
-Models package for Tesla Coil Spark Course
-"""
-
-from .course_model import Course, Lesson, Section, Part, LearningPath, get_course
-
-__all__ = ['Course', 'Lesson', 'Section', 'Part', 'LearningPath', 'get_course']
diff --git a/spark-lessons/app/models/course_model.py b/spark-lessons/app/models/course_model.py
deleted file mode 100644
index 25a5aa2..0000000
--- a/spark-lessons/app/models/course_model.py
+++ /dev/null
@@ -1,320 +0,0 @@
-"""
-Course Model - Loads and manages course structure from course.json
-"""
-
-import json
-from pathlib import Path
-from typing import Dict, List, Optional
-from app.config import COURSE_JSON, LESSONS_DIR
-
-
-class Lesson:
- """Represents a single lesson"""
-
- def __init__(self, data: dict, part_id: str, section_id: str, order: int = 0):
- self.id = data['id']
- self.filename = data['filename']
- self.title = data['title']
- self.estimated_time = data['estimated_time']
- self.difficulty = data['difficulty']
- self.part_id = part_id
- self.section_id = section_id
- self.order = order # Sequential order in course (1-30)
- self.points = data.get('points', 0) # Points for completion
-
- # Construct full path to lesson file
- section_path = LESSONS_DIR / section_id.replace('-', '_').replace('fundamentals', '01-fundamentals').replace('optimization', '02-optimization').replace('spark_physics', '03-spark-physics').replace('advanced_modeling', '04-advanced-modeling')
- self.file_path = section_path / self.filename
-
- def __repr__(self):
- return f""
-
-
-class Section:
- """Represents a course section (e.g., 'fundamentals')"""
-
- def __init__(self, data: dict, part_id: str, lesson_order_start: int = 1):
- self.id = data['id']
- self.title = data['title']
- self.path = data['path']
- self.description = data['description']
- self.part_id = part_id
-
- # Load lessons with sequential ordering
- self.lessons = []
- for i, lesson_data in enumerate(data['lessons']):
- lesson = Lesson(lesson_data, part_id, self.id, lesson_order_start + i)
- self.lessons.append(lesson)
-
- self.exercises = data.get('exercises', [])
- self.key_concepts = data.get('key_concepts', [])
-
- def get_lesson(self, lesson_id: str) -> Optional[Lesson]:
- """Get lesson by ID"""
- for lesson in self.lessons:
- if lesson.id == lesson_id:
- return lesson
- return None
-
- def __repr__(self):
- return f""
-
-
-class Part:
- """Represents a course part (e.g., 'Part 1: Fundamentals')"""
-
- def __init__(self, data: dict, part_number: int, lesson_order_start: int = 1):
- self.id = data['id']
- self.title = data['title']
- self.description = data['description']
- self.estimated_time = data['estimated_time']
- self.number = part_number # Part number (1-4)
-
- # Load sections with sequential lesson ordering
- self.sections = []
- current_order = lesson_order_start
- for section_data in data['sections']:
- section = Section(section_data, self.id, current_order)
- self.sections.append(section)
- current_order += len(section.lessons)
-
- # Create a convenience property for accessing all lessons in this part
- self.lessons = self.get_all_lessons()
-
- def get_lesson(self, lesson_id: str) -> Optional[Lesson]:
- """Get lesson by ID from any section in this part"""
- for section in self.sections:
- lesson = section.get_lesson(lesson_id)
- if lesson:
- return lesson
- return None
-
- def get_all_lessons(self) -> List[Lesson]:
- """Get all lessons in this part"""
- lessons = []
- for section in self.sections:
- lessons.extend(section.lessons)
- return lessons
-
- def __repr__(self):
- return f""
-
-
-class LearningPath:
- """Represents a learning path (e.g., 'beginner', 'complete')"""
-
- def __init__(self, data: dict):
- self.id = data['id']
- self.title = data['title']
- self.description = data['description']
- self.lessons = data.get('lessons', [])
- self.skip = data.get('skip', [])
-
- def includes_lesson(self, lesson_id: str) -> bool:
- """Check if lesson is included in this path"""
- if self.lessons == 'all':
- return lesson_id not in self.skip
- return lesson_id in self.lessons
-
- def __repr__(self):
- return f""
-
-
-class Course:
- """Main course model - loads and manages entire course structure"""
-
- def __init__(self, course_json_path: Path = None):
- """
- Load course from course.json
-
- Args:
- course_json_path: Path to course.json file (default: from config)
- """
- if course_json_path is None:
- course_json_path = COURSE_JSON
-
- self.json_path = course_json_path
- self._load_course()
-
- def _load_course(self):
- """Load and parse course.json"""
- try:
- with open(self.json_path, 'r', encoding='utf-8') as f:
- data = json.load(f)
-
- # Course metadata
- self.title = data['title']
- self.version = data['version']
- self.author = data['author']
- self.description = data['description']
- self.estimated_total_time = data['estimated_total_time']
- self.total_lessons = data['total_lessons']
- self.total_exercises = data['total_exercises']
- self.total_points = data['total_points']
-
- # Prerequisites
- self.prerequisites_required = data['prerequisites']['required']
- self.prerequisites_recommended = data['prerequisites']['recommended']
-
- # Load course structure (4 parts) with sequential numbering
- self.parts = []
- current_order = 1
- for i, part_data in enumerate(data['structure']):
- part = Part(part_data, i + 1, current_order)
- self.parts.append(part)
- current_order += len(part.get_all_lessons())
-
- # Reference materials
- self.reference_materials = data['reference_materials']
-
- # Worked examples
- self.worked_examples = data['worked_examples']
-
- # Learning paths
- self.learning_paths = [
- LearningPath(path_data)
- for path_data in data['learning_paths']
- ]
-
- # Tags
- self.tags = data.get('tags', {})
-
- # Metadata
- self.metadata = data.get('metadata', {})
-
- # Build lesson index for quick lookup
- self._build_lesson_index()
-
- print(f"[Course] Loaded: {self.title}")
- print(f"[Course] {self.total_lessons} lessons across {len(self.parts)} parts")
-
- except FileNotFoundError:
- print(f"[Course ERROR] course.json not found: {self.json_path}")
- raise
- except json.JSONDecodeError as e:
- print(f"[Course ERROR] Invalid JSON: {e}")
- raise
- except KeyError as e:
- print(f"[Course ERROR] Missing required field: {e}")
- raise
-
- def _build_lesson_index(self):
- """Build index for fast lesson lookup by ID"""
- self._lesson_index = {}
- for part in self.parts:
- for section in part.sections:
- for lesson in section.lessons:
- self._lesson_index[lesson.id] = lesson
-
- def get_lesson(self, lesson_id: str) -> Optional[Lesson]:
- """Get lesson by ID (fast lookup)"""
- return self._lesson_index.get(lesson_id)
-
- def get_all_lessons(self) -> List[Lesson]:
- """Get all lessons in course order"""
- lessons = []
- for part in self.parts:
- lessons.extend(part.get_all_lessons())
- return lessons
-
- def get_lesson_by_index(self, index: int) -> Optional[Lesson]:
- """Get lesson by sequential index (0-29)"""
- all_lessons = self.get_all_lessons()
- if 0 <= index < len(all_lessons):
- return all_lessons[index]
- return None
-
- def get_lesson_index(self, lesson_id: str) -> Optional[int]:
- """Get sequential index of a lesson (0-29)"""
- all_lessons = self.get_all_lessons()
- for i, lesson in enumerate(all_lessons):
- if lesson.id == lesson_id:
- return i
- return None
-
- def get_next_lesson(self, lesson_id: str) -> Optional[Lesson]:
- """Get next lesson in sequence"""
- index = self.get_lesson_index(lesson_id)
- if index is not None:
- return self.get_lesson_by_index(index + 1)
- return None
-
- def get_prev_lesson(self, lesson_id: str) -> Optional[Lesson]:
- """Get previous lesson in sequence"""
- index = self.get_lesson_index(lesson_id)
- if index is not None and index > 0:
- return self.get_lesson_by_index(index - 1)
- return None
-
- def get_part(self, part_id: str) -> Optional[Part]:
- """Get part by ID"""
- for part in self.parts:
- if part.id == part_id:
- return part
- return None
-
- def get_learning_path(self, path_id: str) -> Optional[LearningPath]:
- """Get learning path by ID"""
- for path in self.learning_paths:
- if path.id == path_id:
- return path
- return None
-
- def get_lessons_for_path(self, path_id: str) -> List[Lesson]:
- """Get all lessons for a specific learning path"""
- path = self.get_learning_path(path_id)
- if not path:
- return []
-
- all_lessons = self.get_all_lessons()
- if path.lessons == 'all':
- return [l for l in all_lessons if l.id not in path.skip]
- else:
- return [l for l in all_lessons if l.id in path.lessons]
-
- def get_lessons_by_tag(self, tag: str) -> List[Lesson]:
- """Get all lessons with a specific tag"""
- if tag not in self.tags:
- return []
-
- lesson_ids = self.tags[tag]
- return [self.get_lesson(lid) for lid in lesson_ids if self.get_lesson(lid)]
-
- def get_part_for_lesson(self, lesson_id: str) -> Optional[Part]:
- """Get the part that contains a lesson"""
- for part in self.parts:
- if part.get_lesson(lesson_id):
- return part
- return None
-
- def search_lessons(self, query: str) -> List[Lesson]:
- """Simple search by lesson title"""
- query = query.lower()
- results = []
- for lesson in self.get_all_lessons():
- if query in lesson.title.lower() or query in lesson.id.lower():
- results.append(lesson)
- return results
-
- def validate(self) -> bool:
- """Validate that all lesson files exist"""
- all_valid = True
- for lesson in self.get_all_lessons():
- if not lesson.file_path.exists():
- print(f"[Course WARN] Missing lesson file: {lesson.file_path}")
- all_valid = False
- return all_valid
-
- def __repr__(self):
- return f""
-
-
-# Global course instance
-_course_instance = None
-
-def get_course() -> Course:
- """Get global course instance (singleton)"""
- global _course_instance
- if _course_instance is None:
- _course_instance = Course()
- return _course_instance
diff --git a/spark-lessons/app/utils/__init__.py b/spark-lessons/app/utils/__init__.py
deleted file mode 100644
index 6dcfe34..0000000
--- a/spark-lessons/app/utils/__init__.py
+++ /dev/null
@@ -1,8 +0,0 @@
-"""
-Utilities package for Tesla Coil Spark Course
-"""
-
-from .symbol_loader import get_symbol_definitions, SymbolDefinitions
-from .variable_wrapper import VariableWrapper
-
-__all__ = ['get_symbol_definitions', 'SymbolDefinitions', 'VariableWrapper']
diff --git a/spark-lessons/app/utils/symbol_loader.py b/spark-lessons/app/utils/symbol_loader.py
deleted file mode 100644
index 38a178c..0000000
--- a/spark-lessons/app/utils/symbol_loader.py
+++ /dev/null
@@ -1,107 +0,0 @@
-"""
-Symbol Definitions Loader
-Loads and manages symbol/variable definitions for tooltips
-"""
-
-import json
-from pathlib import Path
-from typing import Dict, Optional
-
-
-class SymbolDefinitions:
- """Manages symbol definitions for variable tooltips"""
-
- def __init__(self, json_path: Optional[Path] = None):
- """
- Initialize symbol definitions
-
- Args:
- json_path: Path to symbols JSON file. If None, uses default from config.
- """
- if json_path is None:
- from app import config
- json_path = config.RESOURCES_DIR / 'symbols_definitions.json'
-
- self.json_path = json_path
- self.symbols = self._load_symbols()
-
- def _load_symbols(self) -> Dict:
- """Load symbols from JSON file"""
- try:
- with open(self.json_path, 'r', encoding='utf-8') as f:
- data = json.load(f)
-
- symbols = data.get('variables', {})
- print(f"[Symbols] Loaded {len(symbols)} symbol definitions")
- return symbols
-
- except FileNotFoundError:
- print(f"[Symbols WARNING] Symbol definitions file not found: {self.json_path}")
- return {}
- except json.JSONDecodeError as e:
- print(f"[Symbols ERROR] Invalid JSON in symbols file: {e}")
- return {}
-
- def get_tooltip(self, symbol: str) -> Optional[str]:
- """
- Get plain text tooltip content for a symbol (no HTML)
-
- Args:
- symbol: The symbol/variable name (e.g., "ω", "C_mut")
-
- Returns:
- Plain text string for tooltip, or None if symbol not defined
- """
- if symbol not in self.symbols:
- return None
-
- s = self.symbols[symbol]
-
- # Build tooltip as plain text with line breaks
- tooltip_parts = []
-
- # Symbol name
- tooltip_parts.append(f"{symbol}")
-
- # Add pronunciation/name if different from symbol
- if 'name' in s and s['name'] != symbol:
- tooltip_parts.append(f" ({s['name']})")
-
- # Definition
- if 'definition' in s:
- tooltip_parts.append(f"\n{s['definition']}")
-
- # Formula
- if 'formula' in s:
- tooltip_parts.append(f"\nFormula: {s['formula']}")
-
- # Units
- if 'units' in s:
- tooltip_parts.append(f"\nUnits: {s['units']}")
-
- return ''.join(tooltip_parts)
-
- def has_symbol(self, symbol: str) -> bool:
- """Check if a symbol is defined"""
- return symbol in self.symbols
-
- def get_all_symbols(self) -> list:
- """Get list of all defined symbols"""
- return list(self.symbols.keys())
-
-
-# Global singleton instance
-_symbol_defs_instance = None
-
-
-def get_symbol_definitions() -> SymbolDefinitions:
- """
- Get global SymbolDefinitions instance (singleton pattern)
-
- Returns:
- SymbolDefinitions instance
- """
- global _symbol_defs_instance
- if _symbol_defs_instance is None:
- _symbol_defs_instance = SymbolDefinitions()
- return _symbol_defs_instance
diff --git a/spark-lessons/app/utils/variable_wrapper.py b/spark-lessons/app/utils/variable_wrapper.py
deleted file mode 100644
index 11c57b5..0000000
--- a/spark-lessons/app/utils/variable_wrapper.py
+++ /dev/null
@@ -1,159 +0,0 @@
-"""
-Variable Wrapper Utility
-Automatically wraps variables in HTML content with tooltip spans
-"""
-
-import re
-import html
-from typing import List, Tuple
-from .symbol_loader import get_symbol_definitions
-
-
-class VariableWrapper:
- """Wraps known variables in HTML content with tooltip markup"""
-
- def __init__(self):
- """Initialize variable wrapper with symbol definitions"""
- self.symbols = get_symbol_definitions()
- self._build_patterns()
-
- def _build_patterns(self) -> None:
- """Build regex patterns for all known symbols"""
- # Get all symbols and sort by length (longest first) to avoid partial matches
- symbols_list = sorted(
- self.symbols.get_all_symbols(),
- key=len,
- reverse=True
- )
-
- # Single letters that commonly appear in regular text
- # Only match these in specific mathematical contexts
- common_words = {'A', 'I', 'V', 'P', 'Q', 'R', 'L', 'C', 'E', 'B', 'G', 'X', 'Y', 'Z', 'f', 'd', 'h'}
-
- # Very common English words that need extra-strict matching
- very_common = {'A', 'I'}
-
- self.patterns: List[Tuple[str, str]] = []
- self.context_patterns: List[Tuple[str, str]] = [] # Patterns requiring context
-
- for symbol in symbols_list:
- # Escape special regex characters
- escaped = re.escape(symbol)
-
- # For single-letter variables, only match in formula/code contexts
- if symbol in common_words:
- if symbol in very_common:
- # Extra restrictive for A, I - only in clear math context
- # Must be preceded by =, ×, +, -, /, ( with optional single space
- # Multiple patterns to handle both "=A" and "= A" cases
- # Use alternation to avoid variable-width lookbehind
- pattern = f'(?<=[=×+\\-/\\(])\\s?({escaped})(?=[\\s=+\\-*/()\\[\\]])'
- self.context_patterns.append((pattern, symbol))
- else:
- # More restrictive pattern - requires mathematical context
- # Match if preceded by: =, mathematical operators, but NOT punctuation
- pattern = f'(?<=[=])\\s?({escaped})(?=[\\s=+\\-*/()\\[\\],;<>])|(?<=\\s)({escaped})(?=[\\s=+\\-*/()\\[\\],;<>])'
- self.context_patterns.append((pattern, symbol))
- else:
- # Normal pattern for multi-character symbols
- # Use word boundaries but allow underscores and subscripts
- pattern = f'(? str:
- """
- Wrap known variables in HTML content with tooltip spans
-
- Args:
- html_content: HTML content to process
-
- Returns:
- HTML content with variables wrapped in tooltip spans
- """
- # Track which variables were found (for debugging)
- wrapped_vars = set()
-
- # Process normal patterns
- all_patterns = self.patterns + self.context_patterns
-
- for pattern, symbol in all_patterns:
- tooltip_text = self.symbols.get_tooltip(symbol)
- if not tooltip_text:
- continue
-
- # Escape for HTML attribute (newlines become
)
- tooltip_escaped = html.escape(tooltip_text, quote=True).replace('\n', '
')
-
- # Create replacement span with tooltip
- replacement = (
- f''
- f'\\1' # Captured group (the symbol itself)
- f''
- )
-
- # Count matches before replacement
- matches = list(re.finditer(pattern, html_content))
-
- if matches:
- wrapped_vars.add(symbol)
-
- # Replace pattern with wrapped version
- # Use negative lookahead to avoid wrapping already-wrapped variables
- pattern_with_check = f'(?)(?)'
- html_content = re.sub(
- pattern_with_check,
- replacement,
- html_content
- )
-
- if wrapped_vars:
- print(f"[VariableWrapper] Wrapped {len(wrapped_vars)} unique variables: {', '.join(sorted(wrapped_vars)[:10])}...")
-
- return html_content
-
- def wrap_in_context(self, html_content: str) -> str:
- """
- More sophisticated wrapping that parses HTML structure
- to avoid wrapping in code blocks, headings, etc.
-
- Args:
- html_content: HTML content to process
-
- Returns:
- HTML content with variables wrapped (context-aware)
- """
- # For now, use simple wrapping
- # TODO: Implement HTML parsing to be more selective
- # (e.g., skip , , - tags)
-
- # Simple exclusion: Don't process content inside or
- code_blocks = []
-
- def preserve_code(match):
- """Preserve code blocks and replace with placeholder"""
- code_blocks.append(match.group(0))
- return f"___CODE_BLOCK_{len(code_blocks) - 1}___"
-
- # Temporarily remove code blocks
- html_content = re.sub(
- r'<(code|pre)>(.*?)',
- preserve_code,
- html_content,
- flags=re.DOTALL
- )
-
- # Wrap variables
- html_content = self.wrap_variables(html_content)
-
- # Restore code blocks
- for i, code_block in enumerate(code_blocks):
- html_content = html_content.replace(
- f"___CODE_BLOCK_{i}___",
- code_block
- )
-
- return html_content
diff --git a/spark-lessons/app/views/__init__.py b/spark-lessons/app/views/__init__.py
deleted file mode 100644
index 79c0421..0000000
--- a/spark-lessons/app/views/__init__.py
+++ /dev/null
@@ -1,10 +0,0 @@
-"""
-Views package for Tesla Coil Spark Course
-"""
-
-from .main_window import MainWindow
-from .navigation_panel import NavigationPanel
-from .content_viewer import ContentViewer
-from .progress_panel import ProgressPanel
-
-__all__ = ['MainWindow', 'NavigationPanel', 'ContentViewer', 'ProgressPanel']
diff --git a/spark-lessons/app/views/content_viewer.py b/spark-lessons/app/views/content_viewer.py
deleted file mode 100644
index 5c1199e..0000000
--- a/spark-lessons/app/views/content_viewer.py
+++ /dev/null
@@ -1,432 +0,0 @@
-"""
-Content Viewer - Center panel for displaying lesson content
-"""
-
-from PyQt5.QtWidgets import QWidget, QVBoxLayout, QLabel
-from PyQt5.QtWebEngineWidgets import QWebEngineView, QWebEnginePage
-from PyQt5.QtCore import Qt, pyqtSignal, QUrl
-from pathlib import Path
-import markdown
-from pymdownx import superfences, arithmatex
-
-from app import config
-from app.models import Lesson
-from app.utils import VariableWrapper
-
-
-class ContentViewer(QWidget):
- """Center panel for displaying lesson content with markdown and MathJax"""
-
- # Signals
- scroll_position_changed = pyqtSignal(float) # For auto-save
-
- def __init__(self, parent=None):
- super().__init__(parent)
- self.current_lesson = None
- self.markdown_converter = self._init_markdown()
- self.variable_wrapper = VariableWrapper()
-
- self.init_ui()
-
- def init_ui(self):
- """Initialize the UI components"""
- layout = QVBoxLayout(self)
- layout.setContentsMargins(0, 0, 0, 0)
-
- # Lesson title bar
- self.title_label = QLabel("No lesson selected")
- self.title_label.setStyleSheet(f"""
- background-color: {config.COLOR_PRIMARY};
- color: white;
- font-size: 16pt;
- font-weight: bold;
- padding: 12px;
- """)
- self.title_label.setWordWrap(True)
- layout.addWidget(self.title_label)
-
- # Web view for content
- self.web_view = QWebEngineView()
- self.web_view.setPage(QWebEnginePage(self.web_view))
- layout.addWidget(self.web_view, 1)
-
- # Load welcome page
- self.show_welcome()
-
- def _init_markdown(self):
- """Initialize markdown converter with extensions"""
- return markdown.Markdown(
- extensions=[
- 'extra',
- 'codehilite',
- 'tables',
- 'toc',
- 'pymdownx.arithmatex',
- 'pymdownx.superfences',
- 'pymdownx.highlight',
- 'pymdownx.inlinehilite',
- ],
- extension_configs={
- 'pymdownx.arithmatex': {
- 'generic': True
- },
- 'codehilite': {
- 'css_class': 'highlight',
- 'linenums': False
- }
- }
- )
-
- def show_welcome(self):
- """Display welcome message"""
- html = self._wrap_html("""
-
-
Welcome to Tesla Coil Spark Physics Course
-
- Select a lesson from the navigation panel to begin learning.
-
-
- ⚡ Explore the fascinating world of Tesla coils and electromagnetic theory ⚡
-
-
-
📝 Exercise: {exercise_id}
-
Interactive exercise will be loaded here
-
'
- html = re.sub(r'\{image:([^}]+)\}', replace_image, html)
-
- return html
-
- def _wrap_html(self, content: str, title: str) -> str:
- """Wrap content in full HTML document with styling and MathJax"""
- return f"""
-
-
-
-
- {title}
-
-
-
-
-
-
-
-
-
-
- {content}
-
-
- """
-
- def show_error(self, message: str):
- """Display an error message"""
- html = self._wrap_html(f"""
-
- """, "Error")
- self.web_view.setHtml(html)
-
- def get_scroll_position(self) -> float:
- """Get current scroll position (0.0 to 1.0)"""
- # This would require JavaScript execution in QWebEngineView
- # For now, return 0.0 - can be implemented later
- return 0.0
-
- def set_scroll_position(self, position: float):
- """Set scroll position (0.0 to 1.0)"""
- # This would require JavaScript execution in QWebEngineView
- # For now, do nothing - can be implemented later
- pass
diff --git a/spark-lessons/app/views/main_window.py b/spark-lessons/app/views/main_window.py
deleted file mode 100644
index 97177e8..0000000
--- a/spark-lessons/app/views/main_window.py
+++ /dev/null
@@ -1,292 +0,0 @@
-"""
-Main Window - Primary application window with 3-panel layout
-"""
-
-from PyQt5.QtWidgets import (
- QMainWindow, QSplitter, QStatusBar, QMenuBar, QMenu,
- QAction, QMessageBox, QApplication
-)
-from PyQt5.QtCore import Qt, QTimer
-from PyQt5.QtGui import QKeySequence
-
-from app import config
-from app.models import Course, get_course
-from app.database import Database, get_database
-from .navigation_panel import NavigationPanel
-from .content_viewer import ContentViewer
-from .progress_panel import ProgressPanel
-
-
-class MainWindow(QMainWindow):
- """Main application window with 3-panel layout"""
-
- def __init__(self):
- super().__init__()
-
- # Load course and database
- self.course = get_course()
- self.db = get_database()
-
- # Get or create default user
- self.user_id = self._get_or_create_user()
-
- # Current state
- self.current_lesson_id = None
-
- # Initialize UI
- self.init_ui()
- self.create_menus()
-
- # Connect signals
- self.connect_signals()
-
- # Auto-save timer
- self.auto_save_timer = QTimer(self)
- self.auto_save_timer.timeout.connect(self.auto_save)
- self.auto_save_timer.start(config.AUTO_SAVE_INTERVAL * 1000) # Convert to ms
-
- # Load progress and restore state
- self.load_initial_state()
-
- def init_ui(self):
- """Initialize the user interface"""
- self.setWindowTitle(f"{config.APP_NAME} v{config.APP_VERSION}")
- self.setGeometry(100, 100, config.DEFAULT_WINDOW_WIDTH, config.DEFAULT_WINDOW_HEIGHT)
-
- # Create 3-panel splitter layout
- self.splitter = QSplitter(Qt.Horizontal)
-
- # Create panels
- self.navigation_panel = NavigationPanel(self.course, self)
- self.content_viewer = ContentViewer(self)
- self.progress_panel = ProgressPanel(self.course, self)
-
- # Add panels to splitter
- self.splitter.addWidget(self.navigation_panel)
- self.splitter.addWidget(self.content_viewer)
- self.splitter.addWidget(self.progress_panel)
-
- # Set initial splitter sizes
- self.splitter.setSizes([
- config.NAVIGATION_PANEL_DEFAULT_WIDTH,
- config.DEFAULT_WINDOW_WIDTH - config.NAVIGATION_PANEL_DEFAULT_WIDTH - config.PROGRESS_PANEL_DEFAULT_WIDTH,
- config.PROGRESS_PANEL_DEFAULT_WIDTH
- ])
-
- # Set as central widget
- self.setCentralWidget(self.splitter)
-
- # Create status bar
- self.status_bar = QStatusBar()
- self.setStatusBar(self.status_bar)
- self.status_bar.showMessage("Ready")
-
- def create_menus(self):
- """Create menu bar"""
- menubar = self.menuBar()
-
- # File Menu
- file_menu = menubar.addMenu("&File")
-
- exit_action = QAction("E&xit", self)
- exit_action.setShortcut(QKeySequence.Quit)
- exit_action.triggered.connect(self.close)
- file_menu.addAction(exit_action)
-
- # View Menu
- view_menu = menubar.addMenu("&View")
-
- toggle_nav_action = QAction("Toggle &Navigation Panel", self)
- toggle_nav_action.setShortcut("Ctrl+1")
- toggle_nav_action.triggered.connect(lambda: self.navigation_panel.setVisible(not self.navigation_panel.isVisible()))
- view_menu.addAction(toggle_nav_action)
-
- toggle_progress_action = QAction("Toggle &Progress Panel", self)
- toggle_progress_action.setShortcut("Ctrl+2")
- toggle_progress_action.triggered.connect(lambda: self.progress_panel.setVisible(not self.progress_panel.isVisible()))
- view_menu.addAction(toggle_progress_action)
-
- view_menu.addSeparator()
-
- reset_layout_action = QAction("&Reset Layout", self)
- reset_layout_action.triggered.connect(self.reset_layout)
- view_menu.addAction(reset_layout_action)
-
- # Help Menu
- help_menu = menubar.addMenu("&Help")
-
- about_action = QAction("&About", self)
- about_action.triggered.connect(self.show_about)
- help_menu.addAction(about_action)
-
- def connect_signals(self):
- """Connect signals between components"""
- self.navigation_panel.lesson_selected.connect(self.on_lesson_selected)
-
- def _get_or_create_user(self) -> int:
- """Get or create default user"""
- # Check if user exists
- row = self.db.fetch_one("SELECT user_id FROM users LIMIT 1")
-
- if row:
- return row[0]
-
- # Create default user
- cursor = self.db.execute("""
- INSERT INTO users (username, created_at)
- VALUES (?, datetime('now'))
- """, ("default",))
- self.db.commit()
-
- return cursor.lastrowid
-
- def load_initial_state(self):
- """Load progress and restore application state"""
- # Get overall progress
- progress = self.db.get_overall_progress(self.user_id)
-
- # Update progress panel
- completed_lessons = progress.get('lessons_completed', 0)
- total_points = progress.get('total_points', 0)
- total_time = progress.get('total_time', 0)
-
- self.progress_panel.update_progress(completed_lessons, total_points, total_time)
-
- # Update part progress
- for part in self.course.parts:
- part_completed = 0
- for lesson in part.lessons:
- lesson_prog = self.db.get_lesson_progress(self.user_id, lesson.id)
- if lesson_prog and lesson_prog['status'] == 'completed':
- part_completed += 1
- part_total = len(part.lessons)
- self.progress_panel.update_part_progress(part.number, part_completed, part_total)
-
- # Update study streak
- streak = self.db.get_study_streak(self.user_id) if hasattr(self.db, 'get_study_streak') else 0
- self.progress_panel.update_streak(streak)
-
- # Update exercises
- if hasattr(self.db, 'get_exercise_progress'):
- exercise_progress = self.db.get_exercise_progress(self.user_id)
- if exercise_progress:
- self.progress_panel.update_exercises(
- exercise_progress.get('completed', 0),
- self.course.total_exercises
- )
- else:
- self.progress_panel.update_exercises(0, self.course.total_exercises)
-
- # Update lesson statuses in navigation
- for lesson in self.course.get_all_lessons():
- lesson_progress = self.db.get_lesson_progress(self.user_id, lesson.id)
- status = lesson_progress['status'] if lesson_progress else 'not_started'
- self.navigation_panel.update_lesson_status(lesson.id, status)
-
- # Get last viewed lesson
- row = self.db.fetch_one("""
- SELECT lesson_id FROM lesson_progress
- WHERE user_id = ?
- ORDER BY last_accessed DESC
- LIMIT 1
- """, (self.user_id,))
-
- if row:
- last_lesson_id = row[0]
- # Don't auto-load, just highlight it
- self.navigation_panel.set_current_lesson(last_lesson_id)
-
- def on_lesson_selected(self, lesson_id: str):
- """Handle lesson selection from navigation"""
- lesson = self.course.get_lesson(lesson_id)
- if not lesson:
- return
-
- self.current_lesson_id = lesson_id
-
- # Update navigation highlight
- self.navigation_panel.set_current_lesson(lesson_id)
-
- # Load lesson content
- self.content_viewer.load_lesson(lesson)
-
- # Update progress panel
- self.progress_panel.update_current_lesson(
- lesson.title,
- lesson.points,
- lesson.estimated_time
- )
-
- # Update database (mark as in_progress if not already completed)
- lesson_progress = self.db.get_lesson_progress(self.user_id, lesson_id)
- current_status = lesson_progress['status'] if lesson_progress else 'not_started'
- if current_status == 'not_started':
- self.db.update_lesson_progress(
- self.user_id,
- lesson_id,
- status='in_progress'
- )
- self.navigation_panel.update_lesson_status(lesson_id, 'in_progress')
-
- # Update last accessed
- self.db.update_lesson_progress(
- self.user_id,
- lesson_id,
- last_accessed=True
- )
-
- # Update status bar
- self.status_bar.showMessage(f"Lesson {lesson.order}: {lesson.title}")
-
- def auto_save(self):
- """Auto-save progress"""
- if not self.current_lesson_id:
- return
-
- # Get scroll position
- scroll_pos = self.content_viewer.get_scroll_position()
-
- # Update in database
- self.db.update_lesson_progress(
- self.user_id,
- self.current_lesson_id,
- scroll_position=scroll_pos,
- time_spent_increment=config.AUTO_SAVE_INTERVAL
- )
-
- def reset_layout(self):
- """Reset window layout to defaults"""
- self.splitter.setSizes([
- config.NAVIGATION_PANEL_DEFAULT_WIDTH,
- config.DEFAULT_WINDOW_WIDTH - config.NAVIGATION_PANEL_DEFAULT_WIDTH - config.PROGRESS_PANEL_DEFAULT_WIDTH,
- config.PROGRESS_PANEL_DEFAULT_WIDTH
- ])
- self.navigation_panel.setVisible(True)
- self.progress_panel.setVisible(True)
-
- def show_about(self):
- """Show about dialog"""
- QMessageBox.about(self, "About", f"""
- {config.APP_NAME}
- Version {config.APP_VERSION}
- By {config.APP_AUTHOR}
-
- An interactive desktop application for learning about Tesla coils
- and electromagnetic theory.
-
- Course Statistics:
-
- - {self.course.total_lessons} Lessons
- - {self.course.total_exercises} Exercises
- - {self.course.total_points} Total Points
- - {len(self.course.parts)} Parts
-
- """)
-
- def closeEvent(self, event):
- """Handle window close event"""
- # Final auto-save
- self.auto_save()
-
- # Accept close
- event.accept()
diff --git a/spark-lessons/app/views/navigation_panel.py b/spark-lessons/app/views/navigation_panel.py
deleted file mode 100644
index 9f3c910..0000000
--- a/spark-lessons/app/views/navigation_panel.py
+++ /dev/null
@@ -1,211 +0,0 @@
-"""
-Navigation Panel - Left sidebar with course tree and navigation
-"""
-
-from PyQt5.QtWidgets import (
- QWidget, QVBoxLayout, QTreeWidget, QTreeWidgetItem,
- QLabel, QComboBox, QPushButton, QLineEdit, QHBoxLayout
-)
-from PyQt5.QtCore import Qt, pyqtSignal
-from PyQt5.QtGui import QIcon, QColor, QBrush
-
-from app import config
-from app.models import Course, Lesson, Part, Section
-
-
-class NavigationPanel(QWidget):
- """Left sidebar panel with course navigation tree"""
-
- # Signals
- lesson_selected = pyqtSignal(str) # lesson_id
-
- def __init__(self, course: Course, parent=None):
- super().__init__(parent)
- self.course = course
- self.current_lesson_id = None
- self.lesson_items = {} # lesson_id -> QTreeWidgetItem mapping
-
- self.init_ui()
- self.populate_tree()
-
- def init_ui(self):
- """Initialize the UI components"""
- layout = QVBoxLayout(self)
- layout.setContentsMargins(10, 10, 10, 10)
- layout.setSpacing(10)
-
- # Title
- title = QLabel("Course Navigation")
- title.setStyleSheet(f"font-size: 14pt; font-weight: bold; color: {config.COLOR_PRIMARY};")
- layout.addWidget(title)
-
- # Learning Path Filter (optional for Phase 2+)
- path_layout = QHBoxLayout()
- path_label = QLabel("Path:")
- self.path_combo = QComboBox()
- self.path_combo.addItem("All Lessons", None)
- for path in self.course.learning_paths:
- self.path_combo.addItem(path.title, path.id)
- self.path_combo.currentIndexChanged.connect(self.on_path_filter_changed)
- path_layout.addWidget(path_label)
- path_layout.addWidget(self.path_combo, 1)
- layout.addLayout(path_layout)
-
- # Search box
- search_layout = QHBoxLayout()
- self.search_box = QLineEdit()
- self.search_box.setPlaceholderText("Search lessons...")
- self.search_box.textChanged.connect(self.on_search_changed)
- search_layout.addWidget(self.search_box)
- layout.addLayout(search_layout)
-
- # Course tree
- self.tree = QTreeWidget()
- self.tree.setHeaderHidden(True)
- self.tree.setIndentation(20)
- self.tree.itemDoubleClicked.connect(self.on_item_double_clicked)
- layout.addWidget(self.tree, 1) # Expand to fill space
-
- # Quick actions
- btn_layout = QVBoxLayout()
- self.btn_continue = QPushButton("Continue Learning")
- self.btn_continue.setStyleSheet(f"background-color: {config.COLOR_SUCCESS}; color: white; font-weight: bold; padding: 8px;")
- self.btn_continue.clicked.connect(self.on_continue_learning)
- btn_layout.addWidget(self.btn_continue)
- layout.addLayout(btn_layout)
-
- self.setMinimumWidth(config.NAVIGATION_PANEL_MIN_WIDTH)
-
- def populate_tree(self):
- """Populate the tree with course structure"""
- self.tree.clear()
- self.lesson_items.clear()
-
- # Add course title as root
- root = QTreeWidgetItem(self.tree)
- root.setText(0, self.course.title)
- root.setExpanded(True)
- root.setFlags(root.flags() & ~Qt.ItemIsSelectable)
-
- # Add parts
- for part in self.course.parts:
- part_item = QTreeWidgetItem(root)
- part_item.setText(0, f"Part {part.number}: {part.title}")
- part_item.setExpanded(True)
- part_item.setFlags(part_item.flags() & ~Qt.ItemIsSelectable)
- part_item.setForeground(0, QBrush(QColor(config.COLOR_PRIMARY)))
-
- # Add sections (if any)
- if part.sections:
- for section in part.sections:
- section_item = QTreeWidgetItem(part_item)
- section_item.setText(0, section.title)
- section_item.setExpanded(True)
- section_item.setFlags(section_item.flags() & ~Qt.ItemIsSelectable)
-
- # Add lessons in section
- for lesson in section.lessons:
- self._add_lesson_item(section_item, lesson)
- else:
- # Add lessons directly to part
- for lesson in part.lessons:
- self._add_lesson_item(part_item, lesson)
-
- def _add_lesson_item(self, parent_item: QTreeWidgetItem, lesson: Lesson):
- """Add a lesson item to the tree"""
- lesson_item = QTreeWidgetItem(parent_item)
- lesson_item.setText(0, f"{lesson.order}. {lesson.title}")
- lesson_item.setData(0, Qt.UserRole, lesson.id) # Store lesson_id
-
- # Store reference for quick lookup
- self.lesson_items[lesson.id] = lesson_item
-
- # Add status icon (default: not started)
- self.update_lesson_status(lesson.id, 'not_started')
-
- def update_lesson_status(self, lesson_id: str, status: str):
- """Update the visual status of a lesson"""
- if lesson_id not in self.lesson_items:
- return
-
- item = self.lesson_items[lesson_id]
- lesson = self.course.get_lesson(lesson_id)
-
- # Status icons
- icon_map = {
- 'completed': '✓',
- 'in_progress': '⊙',
- 'not_started': '○',
- 'locked': '🔒'
- }
-
- icon = icon_map.get(status, '○')
- item.setText(0, f"{icon} {lesson.order}. {lesson.title}")
-
- # Color coding
- if status == 'completed':
- item.setForeground(0, QBrush(QColor(config.COLOR_SUCCESS)))
- elif status == 'in_progress':
- item.setForeground(0, QBrush(QColor(config.COLOR_WARNING)))
- else:
- item.setForeground(0, QBrush(QColor(config.COLOR_TEXT)))
-
- def set_current_lesson(self, lesson_id: str):
- """Highlight the current lesson"""
- # Clear previous selection
- if self.current_lesson_id and self.current_lesson_id in self.lesson_items:
- prev_item = self.lesson_items[self.current_lesson_id]
- prev_item.setBackground(0, QBrush(Qt.transparent))
-
- # Set new selection
- self.current_lesson_id = lesson_id
- if lesson_id in self.lesson_items:
- item = self.lesson_items[lesson_id]
- item.setBackground(0, QBrush(QColor(config.COLOR_HIGHLIGHT)))
- self.tree.scrollToItem(item)
-
- def on_item_double_clicked(self, item: QTreeWidgetItem, column: int):
- """Handle double-click on tree item"""
- lesson_id = item.data(0, Qt.UserRole)
- if lesson_id:
- self.lesson_selected.emit(lesson_id)
-
- def on_continue_learning(self):
- """Handle 'Continue Learning' button click"""
- # TODO: Get the next incomplete lesson from database
- # For now, just select the first lesson
- if self.course.lessons:
- first_lesson = self.course.lessons[0]
- self.lesson_selected.emit(first_lesson.id)
-
- def on_path_filter_changed(self, index: int):
- """Handle learning path filter change"""
- path_id = self.path_combo.itemData(index)
- if path_id:
- # Filter tree to show only lessons in this path
- lessons_in_path = self.course.get_lessons_for_path(path_id)
- path_lesson_ids = {lesson.id for lesson in lessons_in_path}
-
- # Hide/show items
- for lesson_id, item in self.lesson_items.items():
- item.setHidden(lesson_id not in path_lesson_ids)
- else:
- # Show all
- for item in self.lesson_items.values():
- item.setHidden(False)
-
- def on_search_changed(self, text: str):
- """Handle search text change"""
- if not text:
- # Show all
- for item in self.lesson_items.values():
- item.setHidden(False)
- return
-
- # Search lessons
- results = self.course.search_lessons(text)
- result_ids = {lesson.id for lesson in results}
-
- # Hide/show items
- for lesson_id, item in self.lesson_items.items():
- item.setHidden(lesson_id not in result_ids)
diff --git a/spark-lessons/app/views/progress_panel.py b/spark-lessons/app/views/progress_panel.py
deleted file mode 100644
index 53988ef..0000000
--- a/spark-lessons/app/views/progress_panel.py
+++ /dev/null
@@ -1,299 +0,0 @@
-"""
-Progress Panel - Right sidebar with progress tracking and statistics
-"""
-
-from PyQt5.QtWidgets import (
- QWidget, QVBoxLayout, QHBoxLayout, QLabel,
- QProgressBar, QPushButton, QFrame, QScrollArea
-)
-from PyQt5.QtCore import Qt
-from PyQt5.QtGui import QFont
-
-from app import config
-from app.models import Course
-
-
-class ProgressPanel(QWidget):
- """Right sidebar panel with progress statistics and tracking"""
-
- def __init__(self, course: Course, parent=None):
- super().__init__(parent)
- self.course = course
-
- self.init_ui()
-
- def init_ui(self):
- """Initialize the UI components"""
- # Main layout
- main_layout = QVBoxLayout(self)
- main_layout.setContentsMargins(10, 10, 10, 10)
- main_layout.setSpacing(10)
-
- # Title
- title = QLabel("Your Progress")
- title.setStyleSheet(f"font-size: 14pt; font-weight: bold; color: {config.COLOR_PRIMARY};")
- main_layout.addWidget(title)
-
- # Scroll area for content
- scroll = QScrollArea()
- scroll.setWidgetResizable(True)
- scroll.setHorizontalScrollBarPolicy(Qt.ScrollBarAlwaysOff)
- scroll.setFrameShape(QFrame.NoFrame)
-
- # Content widget
- content_widget = QWidget()
- layout = QVBoxLayout(content_widget)
- layout.setContentsMargins(0, 0, 5, 0)
- layout.setSpacing(15)
-
- # === Overall Progress Section ===
- layout.addWidget(self._create_section_header("Overall Progress"))
-
- self.overall_progress_bar = QProgressBar()
- self.overall_progress_bar.setStyleSheet(f"""
- QProgressBar {{
- border: 2px solid {config.COLOR_PRIMARY};
- border-radius: 5px;
- text-align: center;
- height: 25px;
- }}
- QProgressBar::chunk {{
- background-color: {config.COLOR_SUCCESS};
- }}
- """)
- layout.addWidget(self.overall_progress_bar)
-
- self.overall_stats_label = QLabel("0 / 30 lessons completed")
- self.overall_stats_label.setStyleSheet("font-size: 10pt; color: #666;")
- layout.addWidget(self.overall_stats_label)
-
- layout.addWidget(self._create_separator())
-
- # === Points and Level Section ===
- layout.addWidget(self._create_section_header("Points & Level"))
-
- points_layout = QHBoxLayout()
- self.points_label = QLabel("0 pts")
- self.points_label.setStyleSheet(f"font-size: 24pt; font-weight: bold; color: {config.COLOR_WARNING};")
- points_layout.addWidget(self.points_label)
- points_layout.addStretch()
- layout.addLayout(points_layout)
-
- self.level_label = QLabel("Level 1: Novice")
- self.level_label.setStyleSheet("font-size: 11pt; color: #666;")
- layout.addWidget(self.level_label)
-
- self.level_progress_bar = QProgressBar()
- self.level_progress_bar.setStyleSheet(f"""
- QProgressBar {{
- border: 1px solid #ccc;
- border-radius: 3px;
- text-align: center;
- height: 15px;
- }}
- QProgressBar::chunk {{
- background-color: {config.COLOR_WARNING};
- }}
- """)
- layout.addWidget(self.level_progress_bar)
-
- layout.addWidget(self._create_separator())
-
- # === Part Progress Section ===
- layout.addWidget(self._create_section_header("Progress by Part"))
-
- self.part_progress_widgets = []
- for part in self.course.parts:
- part_widget = self._create_part_progress(part.number, part.title, 0)
- self.part_progress_widgets.append(part_widget)
- layout.addWidget(part_widget)
-
- layout.addWidget(self._create_separator())
-
- # === Study Stats Section ===
- layout.addWidget(self._create_section_header("Study Statistics"))
-
- stats_grid = QVBoxLayout()
- stats_grid.setSpacing(8)
-
- self.time_stat = self._create_stat_row("⏱", "Total Time", "0 min")
- self.streak_stat = self._create_stat_row("🔥", "Streak", "0 days")
- self.exercises_stat = self._create_stat_row("📝", "Exercises", "0 / 18")
-
- stats_grid.addWidget(self.time_stat)
- stats_grid.addWidget(self.streak_stat)
- stats_grid.addWidget(self.exercises_stat)
-
- layout.addLayout(stats_grid)
-
- layout.addWidget(self._create_separator())
-
- # === Current Lesson Section ===
- layout.addWidget(self._create_section_header("Current Lesson"))
-
- self.current_lesson_label = QLabel("No lesson selected")
- self.current_lesson_label.setStyleSheet("font-size: 10pt; color: #666; padding: 10px;")
- self.current_lesson_label.setWordWrap(True)
- layout.addWidget(self.current_lesson_label)
-
- # Push everything to top
- layout.addStretch()
-
- # Set content widget to scroll area
- scroll.setWidget(content_widget)
- main_layout.addWidget(scroll, 1)
-
- # Initialize with default values
- self.update_progress(0, 0, 0)
-
- self.setMinimumWidth(config.PROGRESS_PANEL_MIN_WIDTH)
-
- def _create_section_header(self, text: str) -> QLabel:
- """Create a section header label"""
- label = QLabel(text)
- label.setStyleSheet(f"font-size: 11pt; font-weight: bold; color: {config.COLOR_SECONDARY};")
- return label
-
- def _create_separator(self) -> QFrame:
- """Create a horizontal separator line"""
- line = QFrame()
- line.setFrameShape(QFrame.HLine)
- line.setFrameShadow(QFrame.Sunken)
- line.setStyleSheet("color: #ddd;")
- return line
-
- def _create_part_progress(self, part_number: int, part_title: str, progress: int) -> QWidget:
- """Create a part progress widget"""
- widget = QWidget()
- layout = QVBoxLayout(widget)
- layout.setContentsMargins(0, 5, 0, 5)
- layout.setSpacing(5)
-
- # Part title
- title_label = QLabel(f"Part {part_number}: {part_title[:30]}...")
- title_label.setStyleSheet("font-size: 9pt; font-weight: bold;")
- layout.addWidget(title_label)
-
- # Progress bar
- progress_bar = QProgressBar()
- progress_bar.setValue(progress)
- progress_bar.setStyleSheet(f"""
- QProgressBar {{
- border: 1px solid #ccc;
- border-radius: 3px;
- text-align: center;
- height: 12px;
- font-size: 8pt;
- }}
- QProgressBar::chunk {{
- background-color: {config.COLOR_PRIMARY};
- }}
- """)
- layout.addWidget(progress_bar)
-
- # Store reference for updates
- widget.progress_bar = progress_bar
-
- return widget
-
- def _create_stat_row(self, icon: str, label: str, value: str) -> QWidget:
- """Create a statistics row"""
- widget = QWidget()
- layout = QHBoxLayout(widget)
- layout.setContentsMargins(5, 5, 5, 5)
- layout.setSpacing(10)
-
- # Icon
- icon_label = QLabel(icon)
- icon_label.setStyleSheet("font-size: 16pt;")
- layout.addWidget(icon_label)
-
- # Label
- text_label = QLabel(label)
- text_label.setStyleSheet("font-size: 10pt; color: #666;")
- layout.addWidget(text_label)
-
- layout.addStretch()
-
- # Value
- value_label = QLabel(value)
- value_label.setStyleSheet("font-size: 10pt; font-weight: bold;")
- layout.addWidget(value_label)
-
- # Store reference for updates
- widget.value_label = value_label
-
- return widget
-
- def update_progress(self, completed_lessons: int, total_points: int, total_time_minutes: int):
- """Update overall progress display"""
- # Overall progress
- total_lessons = self.course.total_lessons
- progress_percent = int((completed_lessons / total_lessons) * 100) if total_lessons > 0 else 0
- self.overall_progress_bar.setValue(progress_percent)
- self.overall_stats_label.setText(f"{completed_lessons} / {total_lessons} lessons completed")
-
- # Points
- self.points_label.setText(f"{total_points} pts")
-
- # Level
- level_info = self._get_level_info(total_points)
- self.level_label.setText(f"Level {level_info['level']}: {level_info['title']}")
- self.level_progress_bar.setValue(level_info['progress'])
-
- # Time
- if total_time_minutes < 60:
- time_str = f"{total_time_minutes} min"
- else:
- hours = total_time_minutes // 60
- minutes = total_time_minutes % 60
- time_str = f"{hours}h {minutes}m"
- self.time_stat.value_label.setText(time_str)
-
- def update_part_progress(self, part_number: int, completed: int, total: int):
- """Update progress for a specific part"""
- if 0 < part_number <= len(self.part_progress_widgets):
- widget = self.part_progress_widgets[part_number - 1]
- progress = int((completed / total) * 100) if total > 0 else 0
- widget.progress_bar.setValue(progress)
- widget.progress_bar.setFormat(f"{completed}/{total} ({progress}%)")
-
- def update_streak(self, days: int):
- """Update study streak"""
- self.streak_stat.value_label.setText(f"{days} days")
-
- def update_exercises(self, completed: int, total: int):
- """Update exercise completion"""
- self.exercises_stat.value_label.setText(f"{completed} / {total}")
-
- def update_current_lesson(self, lesson_title: str, lesson_points: int, estimated_time: int):
- """Update current lesson information"""
- text = f"""
- {lesson_title}
- Points: {lesson_points} | Est. time: {estimated_time} min
- """
- self.current_lesson_label.setText(text)
-
- def _get_level_info(self, points: int) -> dict:
- """Get level information based on points"""
- for i, (threshold, title, subtitle) in enumerate(config.LEVELS):
- if i < len(config.LEVELS) - 1:
- next_threshold = config.LEVELS[i + 1][0]
- if points < next_threshold:
- progress = int(((points - threshold) / (next_threshold - threshold)) * 100)
- return {
- 'level': i + 1,
- 'title': title,
- 'subtitle': subtitle,
- 'progress': progress,
- 'next_threshold': next_threshold
- }
-
- # Max level
- return {
- 'level': len(config.LEVELS),
- 'title': config.LEVELS[-1][1],
- 'subtitle': config.LEVELS[-1][2],
- 'progress': 100,
- 'next_threshold': config.LEVELS[-1][0]
- }
diff --git a/spark-lessons/course.json b/spark-lessons/course.json
deleted file mode 100644
index a07e73c..0000000
--- a/spark-lessons/course.json
+++ /dev/null
@@ -1,446 +0,0 @@
-{
- "title": "Tesla Coil Spark Physics: Complete Course",
- "version": "1.0.0",
- "author": "Tesla Coil Community",
- "description": "A comprehensive course teaching the physics, mathematics, and simulation techniques required to understand and model Tesla coil sparks. From basic circuit theory to advanced distributed modeling with FEMM.",
- "estimated_total_time": 840,
- "total_lessons": 30,
- "total_exercises": 18,
- "total_points": 525,
-
- "prerequisites": {
- "required": [
- "Basic AC circuit analysis (impedance, phasors)",
- "Complex number arithmetic",
- "Basic calculus (derivatives, integrals)",
- "Familiarity with SPICE circuit simulation"
- ],
- "recommended": [
- "Electromagnetic field theory basics",
- "Experience with FEMM or similar FEA software",
- "Tesla coil operating experience"
- ]
- },
-
- "structure": [
- {
- "id": "part-1",
- "title": "Part 1: Circuit Fundamentals",
- "description": "Foundation concepts for understanding spark impedance, admittance analysis, and topological constraints",
- "estimated_time": 200,
- "sections": [
- {
- "id": "fundamentals",
- "title": "Circuit Fundamentals",
- "path": "lessons/01-fundamentals",
- "description": "Learn the basic circuit model, admittance analysis, phase constraints, and measurement techniques",
- "lessons": [
- {
- "id": "fund-01",
- "filename": "01-introduction.md",
- "title": "Introduction and AC Circuit Review",
- "estimated_time": 20,
- "difficulty": "beginner"
- },
- {
- "id": "fund-02",
- "filename": "02-basic-circuit-model.md",
- "title": "Basic Spark Circuit Model",
- "estimated_time": 25,
- "difficulty": "beginner"
- },
- {
- "id": "fund-03",
- "filename": "03-admittance-analysis.md",
- "title": "Admittance Analysis of Parallel Circuits",
- "estimated_time": 30,
- "difficulty": "intermediate"
- },
- {
- "id": "fund-04",
- "filename": "04-phase-angles.md",
- "title": "Understanding Phase Angles",
- "estimated_time": 20,
- "difficulty": "intermediate"
- },
- {
- "id": "fund-05",
- "filename": "05-phase-constraint.md",
- "title": "Topological Phase Constraint",
- "estimated_time": 25,
- "difficulty": "intermediate"
- },
- {
- "id": "fund-06",
- "filename": "06-why-not-45-degrees.md",
- "title": "Why Not -45 Degrees?",
- "estimated_time": 15,
- "difficulty": "intermediate"
- },
- {
- "id": "fund-07",
- "filename": "07-measurement-port.md",
- "title": "Correct Measurement Port",
- "estimated_time": 20,
- "difficulty": "intermediate"
- },
- {
- "id": "fund-08",
- "filename": "08-review-exercises.md",
- "title": "Part 1 Review and Integration",
- "estimated_time": 45,
- "difficulty": "intermediate"
- }
- ],
- "exercises": [
- "fund-ex-02a", "fund-ex-02b", "fund-ex-02c",
- "fund-ex-03a", "fund-ex-03b",
- "fund-ex-04a", "fund-ex-04b",
- "fund-ex-05a",
- "fund-ex-08-comprehensive",
- "fund-ex-checkpoint-quiz"
- ],
- "key_concepts": [
- "mutual_capacitance",
- "shunt_capacitance",
- "admittance_analysis",
- "phase_constraint",
- "measurement_port",
- "topological_limits"
- ]
- }
- ]
- },
- {
- "id": "part-2",
- "title": "Part 2: Optimization & Simulation",
- "description": "Learn optimization principles, Thévenin analysis, and simulation techniques for Tesla coil sparks",
- "estimated_time": 280,
- "sections": [
- {
- "id": "optimization",
- "title": "Optimization & Simulation",
- "path": "lessons/02-optimization",
- "description": "Master power optimization, self-adjustment mechanisms, and Thévenin equivalent analysis",
- "lessons": [
- {
- "id": "opt-01",
- "filename": "01-two-resistances.md",
- "title": "Two Critical Resistances",
- "estimated_time": 35,
- "difficulty": "intermediate"
- },
- {
- "id": "opt-02",
- "filename": "02-hungry-streamer.md",
- "title": "The Hungry Streamer Principle",
- "estimated_time": 30,
- "difficulty": "advanced"
- },
- {
- "id": "opt-03",
- "filename": "03-thevenin-method.md",
- "title": "Thévenin Equivalent Extraction",
- "estimated_time": 40,
- "difficulty": "intermediate"
- },
- {
- "id": "opt-04",
- "filename": "04-thevenin-calculations.md",
- "title": "Power Calculations with Thévenin",
- "estimated_time": 45,
- "difficulty": "intermediate"
- },
- {
- "id": "opt-05",
- "filename": "05-direct-measurement.md",
- "title": "Direct Power Measurement",
- "estimated_time": 25,
- "difficulty": "intermediate"
- },
- {
- "id": "opt-06",
- "filename": "06-frequency-tracking.md",
- "title": "Frequency Tracking and Loaded Poles",
- "estimated_time": 45,
- "difficulty": "advanced"
- },
- {
- "id": "opt-07",
- "filename": "07-review-exercises.md",
- "title": "Part 2 Review and Design Challenge",
- "estimated_time": 60,
- "difficulty": "intermediate"
- }
- ],
- "exercises": [
- "opt-ex-01a",
- "opt-ex-01b",
- "opt-ex-thevenin-complete"
- ],
- "key_concepts": [
- "R_opt_power",
- "R_opt_phase",
- "hungry_streamer",
- "thevenin_equivalent",
- "frequency_tracking",
- "loaded_poles",
- "power_optimization"
- ]
- }
- ]
- },
- {
- "id": "part-3",
- "title": "Part 3: Spark Growth Physics",
- "description": "Understand the physics of spark formation, growth, and energy requirements",
- "estimated_time": 260,
- "sections": [
- {
- "id": "spark-physics",
- "title": "Spark Growth Physics",
- "path": "lessons/03-spark-physics",
- "description": "Master electric field thresholds, energy per meter, thermal dynamics, and streamer-to-leader transitions",
- "lessons": [
- {
- "id": "phys-01",
- "filename": "01-field-thresholds.md",
- "title": "Electric Field Thresholds",
- "estimated_time": 20,
- "difficulty": "intermediate"
- },
- {
- "id": "phys-02",
- "filename": "02-voltage-limits.md",
- "title": "Voltage-Limited Spark Length",
- "estimated_time": 25,
- "difficulty": "intermediate"
- },
- {
- "id": "phys-03",
- "filename": "03-energy-per-meter.md",
- "title": "Energy Per Meter Concept",
- "estimated_time": 30,
- "difficulty": "intermediate"
- },
- {
- "id": "phys-04",
- "filename": "04-empirical-epsilon.md",
- "title": "Empirical ε Values by Mode",
- "estimated_time": 35,
- "difficulty": "advanced"
- },
- {
- "id": "phys-05",
- "filename": "05-thermal-memory.md",
- "title": "Thermal Memory and Persistence",
- "estimated_time": 40,
- "difficulty": "advanced"
- },
- {
- "id": "phys-06",
- "filename": "06-streamers-vs-leaders.md",
- "title": "Streamers vs Leaders",
- "estimated_time": 35,
- "difficulty": "advanced"
- },
- {
- "id": "phys-07",
- "filename": "07-capacitive-divider.md",
- "title": "The Capacitive Divider Problem",
- "estimated_time": 30,
- "difficulty": "advanced"
- },
- {
- "id": "phys-08",
- "filename": "08-freau-relationship.md",
- "title": "Freau's Empirical Scaling",
- "estimated_time": 25,
- "difficulty": "intermediate"
- },
- {
- "id": "phys-09",
- "filename": "09-review-exercises.md",
- "title": "Part 3 Review and QCW Design",
- "estimated_time": 20,
- "difficulty": "advanced"
- }
- ],
- "exercises": [
- "phys-ex-01a",
- "phys-ex-03a",
- "phys-ex-comprehensive",
- "phys-ex-conceptual-limits"
- ],
- "key_concepts": [
- "E_inception",
- "E_propagation",
- "energy_per_meter",
- "epsilon_calibration",
- "thermal_diffusion",
- "streamers",
- "leaders",
- "capacitive_divider",
- "voltage_limited",
- "power_limited"
- ]
- }
- ]
- },
- {
- "id": "part-4",
- "title": "Part 4: Advanced Modeling",
- "description": "Learn FEMM extraction techniques and build lumped and distributed spark models",
- "estimated_time": 285,
- "sections": [
- {
- "id": "advanced-modeling",
- "title": "Advanced Modeling Techniques",
- "path": "lessons/04-advanced-modeling",
- "description": "Master FEMM capacitance extraction, lumped models, distributed models, and resistance optimization",
- "lessons": [
- {
- "id": "model-01",
- "filename": "01-lumped-model.md",
- "title": "Lumped Spark Model Theory",
- "estimated_time": 35,
- "difficulty": "advanced"
- },
- {
- "id": "model-02",
- "filename": "02-femm-extraction-lumped.md",
- "title": "FEMM Extraction for Lumped Model",
- "estimated_time": 50,
- "difficulty": "advanced"
- },
- {
- "id": "model-03",
- "filename": "03-distributed-model.md",
- "title": "Distributed Model Introduction",
- "estimated_time": 40,
- "difficulty": "advanced"
- },
- {
- "id": "model-04",
- "filename": "04-femm-extraction-distributed.md",
- "title": "FEMM Extraction for Distributed Model",
- "estimated_time": 55,
- "difficulty": "advanced"
- },
- {
- "id": "model-05",
- "filename": "05-resistance-optimization.md",
- "title": "Resistance Optimization Algorithms",
- "estimated_time": 55,
- "difficulty": "advanced"
- },
- {
- "id": "model-06",
- "filename": "06-review-exercises.md",
- "title": "Part 4 Review and Complete Project",
- "estimated_time": 50,
- "difficulty": "advanced"
- }
- ],
- "exercises": [
- "model-ex-lumped-complete"
- ],
- "key_concepts": [
- "lumped_model",
- "distributed_model",
- "FEMM_extraction",
- "Maxwell_capacitance_matrix",
- "partial_capacitance",
- "resistance_optimization",
- "iterative_algorithm",
- "circuit_determined_R",
- "passivity_check",
- "matrix_validation"
- ]
- }
- ]
- }
- ],
-
- "reference_materials": {
- "equation_sheet": "reference/equation-sheet.md",
- "physical_bounds": "reference/physical-bounds.md",
- "glossary": "reference/glossary.yaml"
- },
-
- "worked_examples": {
- "path": "worked-examples",
- "examples": [
- "calculating-ropt.md",
- "thevenin-extraction.md",
- "spark-growth-timeline.md",
- "femm-lumped-extraction.md",
- "distributed-model-complete.md"
- ]
- },
-
- "learning_paths": [
- {
- "id": "beginner",
- "title": "Beginner Path",
- "description": "For those new to Tesla coils or RF circuit analysis",
- "lessons": [
- "fund-01", "fund-02", "fund-03", "fund-04", "fund-06", "fund-07", "fund-08",
- "opt-01", "opt-03", "opt-04",
- "phys-01", "phys-02", "phys-03", "phys-08"
- ],
- "skip": ["opt-02", "opt-06", "phys-04", "phys-05", "phys-06", "phys-07", "part-4"]
- },
- {
- "id": "intermediate",
- "title": "Complete Course",
- "description": "Full course for comprehensive understanding",
- "lessons": "all"
- },
- {
- "id": "simulation-focus",
- "title": "Simulation Focus",
- "description": "For those primarily interested in modeling and simulation",
- "lessons": [
- "fund-01", "fund-02", "fund-03", "fund-05", "fund-08",
- "opt-01", "opt-03", "opt-04", "opt-05", "opt-06",
- "phys-01", "phys-02", "phys-03", "phys-04",
- "model-01", "model-02", "model-03", "model-04", "model-05"
- ]
- },
- {
- "id": "physics-focus",
- "title": "Physics Focus",
- "description": "For those primarily interested in spark physics",
- "lessons": [
- "fund-01", "fund-02", "fund-03",
- "opt-01", "opt-02",
- "phys-01", "phys-02", "phys-03", "phys-04", "phys-05", "phys-06", "phys-07", "phys-08", "phys-09"
- ]
- }
- ],
-
- "tags": {
- "circuit-theory": ["fund-01", "fund-02", "fund-03", "fund-04", "fund-05", "fund-07"],
- "admittance": ["fund-03", "fund-04", "opt-01", "opt-03"],
- "optimization": ["opt-01", "opt-02", "opt-03", "opt-04", "opt-05", "opt-06"],
- "thevenin": ["opt-03", "opt-04"],
- "frequency-tracking": ["opt-06"],
- "field-theory": ["phys-01", "phys-02", "phys-07"],
- "energy-budget": ["phys-03", "phys-04", "phys-08"],
- "thermal-physics": ["phys-05", "phys-06"],
- "plasma-physics": ["phys-06"],
- "FEMM": ["model-02", "model-04"],
- "modeling": ["model-01", "model-02", "model-03", "model-04", "model-05"],
- "SPICE": ["opt-05", "model-01", "model-04", "model-05"],
- "advanced": ["opt-02", "opt-06", "phys-04", "phys-05", "phys-06", "phys-07", "model-01", "model-02", "model-03", "model-04", "model-05"]
- },
-
- "metadata": {
- "created": "2025-10-10",
- "last_updated": "2025-10-10",
- "format_version": "1.0",
- "license": "Creative Commons Attribution-ShareAlike 4.0",
- "repository": "https://github.com/your-repo/spark-lessons"
- }
-}
diff --git a/spark-lessons/generate_circuits.py b/spark-lessons/generate_circuits.py
deleted file mode 100644
index 9c35dc8..0000000
--- a/spark-lessons/generate_circuits.py
+++ /dev/null
@@ -1,382 +0,0 @@
-"""
-Tesla Coil Spark Course - Circuit Diagram Generation
-
-Generates circuit schematics using schemdraw.
-Run from spark-lessons directory.
-
-Usage: python generate_circuits.py
-"""
-
-import schemdraw
-import schemdraw.elements as elm
-from pathlib import Path
-import matplotlib.pyplot as plt
-
-# Directories
-BASE_DIR = Path(__file__).parent
-ASSETS_DIRS = {
- 'fundamentals': BASE_DIR / 'lessons' / '01-fundamentals' / 'assets',
- 'optimization': BASE_DIR / 'lessons' / '02-optimization' / 'assets',
- 'spark-physics': BASE_DIR / 'lessons' / '03-spark-physics' / 'assets',
- 'advanced-modeling': BASE_DIR / 'lessons' / '04-advanced-modeling' / 'assets',
- 'shared': BASE_DIR / 'assets' / 'shared',
-}
-
-def save_circuit(drawing, filename, directory='fundamentals'):
- """Save circuit diagram"""
- filepath = ASSETS_DIRS[directory] / filename
- drawing.save(str(filepath), dpi=150)
- print(f"[OK] Generated: {filepath}")
-
-
-# ============================================================================
-# PART 1: FUNDAMENTALS CIRCUITS
-# ============================================================================
-
-def generate_geometry_to_circuit():
- """Image 2: Geometry to circuit schematic translation"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=12, font='sans-serif')
-
- # Draw the circuit on the right side
- # Topload node at top
- d += elm.Line().right(1).label('Topload', loc='top')
- d.push()
-
- # Parallel R and C_mut
- d += elm.Line().down(0.5)
- d.push()
- d += elm.Resistor().down(1.5).label('R', loc='right')
- d.pop()
- d += elm.Capacitor().down(1.5).label('C_mut', loc='right').at((1, d.here[1]))
- d += elm.Line().left(1)
-
- # Series point (spark tip node)
- d += elm.Dot().label('Spark Tip', loc='right', ofst=0.3)
- d += elm.Line().down(0.5)
-
- # C_sh to ground
- d += elm.Capacitor().down(1.5).label('C_sh', loc='right')
- d += elm.Ground()
-
- # Title is implicit in context - no annotation needed
-
- save_circuit(d, 'geometry-to-circuit.png', 'fundamentals')
-
-
-def generate_current_paths_diagram():
- """Image 6: Tesla coil showing all current paths"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=10, font='sans-serif')
-
- # Primary circuit (left)
- d += elm.SourceSin().label('Drive')
- d += elm.Capacitor().right(1.5).label('C_pri')
- d += elm.Inductor().down(2).label('L_pri')
- d += elm.Line().left(1.5)
- d += elm.Ground()
-
- # Coupling to secondary
- d.move(2, 1.5)
- d += elm.Inductor().up(3).label('L_sec', loc='right')
- d.push()
-
- # Topload capacitance
- d += elm.Line().right(0.5)
- d += elm.Capacitor().right(1).label('C_top')
- d += elm.Line().down(0.5)
-
- # Spark circuit
- d.push()
- d += elm.Capacitor().down(1).label('C_mut', loc='right')
- d += elm.Line().down(0.5)
- d += elm.Capacitor().down(1).label('C_sh', loc='right')
- d += elm.Ground()
- d.pop()
-
- # Ground path
- d += elm.Line().right(1.5)
- d += elm.Ground()
-
- # Add current labels
- d.here = (0, -2.5)
- d += elm.Annotate().label('I_base', fontsize=10, color='red')
-
- save_circuit(d, 'current-paths-diagram.png', 'fundamentals')
-
-
-# ============================================================================
-# PART 2: OPTIMIZATION CIRCUITS
-# ============================================================================
-
-def generate_thevenin_equivalent_circuit():
- """Image 12: Thévenin equivalent with spark load"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=12, font='sans-serif')
-
- # Thévenin source
- d += elm.SourceV().label('V_th')
- d.push()
-
- # Z_th (impedance)
- d += elm.Resistor().right(1.5).label('R_th')
- d += elm.Capacitor().right(1.5).label('X_th', loc='bottom')
-
- # Connection point
- d += elm.Dot()
- d.push()
-
- # Load (spark)
- d += elm.Line().down(0.5)
- d += elm.Resistor().down(1.5).label('R_spark', loc='right')
- d += elm.Capacitor().down(1.5).label('X_spark', loc='right')
- d += elm.Ground()
-
- # Close circuit
- d.pop()
- d += elm.Line().down(4.5)
- d += elm.Line().left(3)
-
- # Add formula annotation
- d.here = (1, -5.5)
- d += elm.Annotate(ofst=(0, -0.5)).label(
- 'P = 0.5|V_th|² Re{Z_spark} / |Z_th+Z_spark|²',
- fontsize=11
- )
-
- save_circuit(d, 'thevenin-equivalent-circuit.png', 'optimization')
-
-
-# ============================================================================
-# PART 3: SPARK PHYSICS CIRCUITS
-# ============================================================================
-
-def generate_capacitive_divider_circuit():
- """Image 25: Capacitive divider circuit"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=12, font='sans-serif')
-
- # Voltage source (topload)
- d += elm.Line().right(1).label('V_topload', loc='top')
- d += elm.Dot()
- d.push()
-
- # Parallel R and C_mut
- d += elm.Line().down(0.5)
- d.push()
- d += elm.Resistor().down(1.5).label('R')
- d.pop()
- d += elm.Capacitor().right(1.5).down(1.5).label('C_mut')
- d += elm.Line().left(1.5)
-
- # V_tip measurement point
- d += elm.Dot().label('V_tip', loc='right', ofst=0.3)
- d += elm.Line().down(0.5)
-
- # C_sh to ground
- d += elm.Capacitor().down(1.5).label('C_sh = L×6.6pF/m', loc='right')
- d += elm.Ground()
-
- # Add formula
- d.here = (0, -5)
- d += elm.Annotate().label(
- 'V_tip = V_topload × C_mut/(C_mut + C_sh)',
- fontsize=11
- )
-
- save_circuit(d, 'capacitive-divider-circuit.png', 'spark-physics')
-
-
-# ============================================================================
-# PART 4: ADVANCED MODELING CIRCUITS
-# ============================================================================
-
-def generate_lumped_model_schematic():
- """Image 28: Lumped model circuit schematic"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=11, font='sans-serif')
-
- # Topload connection
- d += elm.Line().right(1).label('Topload', loc='top')
- d += elm.Dot().label('Port')
- d.push()
-
- # Parallel combination
- d += elm.Line().down(0.3)
- d.push()
-
- # R branch
- d += elm.Resistor().down(2).label('R', loc='left')
-
- # C_mut branch
- d.pop()
- d += elm.Capacitor().right(2).down(2).label('C_mut', loc='right')
- d += elm.Line().left(2)
-
- # Spark tip node
- d += elm.Dot().label('Spark Tip', loc='right', ofst=0.3)
-
- # C_sh to ground
- d += elm.Line().down(0.3)
- d += elm.Capacitor().down(1.5).label('C_sh', loc='right')
- d += elm.Ground()
-
- # Add typical values
- d.here = (0, -5)
- d += elm.Annotate().label(
- 'Typical: R=50kΩ, C_mut=8pF, C_sh=6pF',
- fontsize=10
- )
-
- save_circuit(d, 'lumped-model-schematic.png', 'advanced-modeling')
-
-
-def generate_distributed_model_structure():
- """Image 32: nth-order distributed model structure"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=9, font='sans-serif')
-
- # Topload
- d += elm.Line().right(0.5).label('Topload', loc='top')
- d += elm.Dot().label('Node 0')
-
- # Segment 1
- d.push()
- d += elm.Capacitor().down(1.2).label('C_01', loc='left', ofst=-0.2)
- d += elm.Dot().label('Node 1', loc='right', ofst=0.2)
- d.push()
- d += elm.Resistor().right(1.5).label('R_1', loc='top')
- d.pop()
- d += elm.Capacitor().down(1.2).label('C_1,gnd', loc='left')
- d += elm.Ground()
-
- # Segment 2
- d.pop()
- d += elm.Line().right(3)
- d.push()
- d += elm.Capacitor().down(1.2).label('C_12', loc='left', ofst=-0.2)
- d += elm.Dot().label('Node 2', loc='right', ofst=0.2)
- d.push()
- d += elm.Resistor().right(1.5).label('R_2', loc='top')
- d.pop()
- d += elm.Capacitor().down(1.2).label('C_2,gnd', loc='left')
- d += elm.Ground()
-
- # Ellipsis
- d.pop()
- d += elm.Line().right(1.5)
- d += elm.Dot()
- d += elm.Line().right(0.3).linestyle('dotted')
- d += elm.Line().right(0.3)
- d += elm.Dot()
- d += elm.Line().right(1.5)
-
- # Segment n
- d.push()
- d += elm.Capacitor().down(1.2).label('C_n-1,n', loc='left', ofst=-0.2)
- d += elm.Dot().label('Node n', loc='right', ofst=0.2)
- d.push()
- d += elm.Resistor().right(1.5).label('R_n', loc='top')
- d.pop()
- d += elm.Capacitor().down(1.2).label('C_n,gnd', loc='left')
- d += elm.Ground()
-
- # Add note
- d.here = (3, -3.5)
- d += elm.Annotate().label(
- 'n = 5-20 segments\n(n+1)×(n+1) capacitance matrix',
- fontsize=9
- )
-
- save_circuit(d, 'distributed-model-structure.png', 'advanced-modeling')
-
-
-# ============================================================================
-# SHARED CIRCUITS
-# ============================================================================
-
-def generate_tesla_coil_system_overview():
- """Image 44: Complete Tesla coil system diagram"""
- with schemdraw.Drawing(show=False) as d:
- d.config(fontsize=10, font='sans-serif')
-
- # Primary side
- d += elm.SourceSin().label('Drive\nSource')
- d += elm.Line().right(0.5)
- d += elm.Switch().label('IGBT/FET')
- d += elm.Line().right(0.5)
- d += elm.Capacitor().right(1.5).label('MMC\n(C_pri)')
- d += elm.Inductor().down(3).label('L_primary', loc='bottom')
- d += elm.Line().left(3.5)
- d += elm.Ground()
-
- # Secondary side (coupled)
- d.move(4, 2)
- d += elm.Inductor().up(4).label('L_secondary', loc='right')
- d += elm.Line().up(0.5)
-
- # Topload
- d += elm.Capacitor().right(1.5).label('C_topload')
- d.push()
-
- # Spark
- d += elm.Line().down(1)
- d += elm.Gap().down(2).label('Spark\nGap')
- d += elm.Line().down(1)
- d += elm.Ground().label('Strike\nPoint')
-
- # Ground return
- d.pop()
- d += elm.Line().right(2)
- d += elm.Line().down(5.5)
- d += elm.Ground()
-
- # Add coupling annotation
- d.here = (2, 0)
- d += elm.Annotate(ofst=(0, 2)).label('k = 0.1-0.2', fontsize=10)
-
- # Add title
- d.here = (0, 7)
- d += elm.Annotate().label(
- 'Double-Resonant Solid State Tesla Coil (DRSSTC)',
- fontsize=12
- )
-
- save_circuit(d, 'tesla-coil-system-overview.png', 'shared')
-
-
-# ============================================================================
-# MAIN
-# ============================================================================
-
-def main():
- print("\n" + "="*60)
- print("TESLA COIL SPARK COURSE - CIRCUIT DIAGRAM GENERATION")
- print("="*60)
-
- print("\nGenerating Part 1 circuits...")
- generate_geometry_to_circuit()
- generate_current_paths_diagram()
-
- print("\nGenerating Part 2 circuits...")
- generate_thevenin_equivalent_circuit()
-
- print("\nGenerating Part 3 circuits...")
- generate_capacitive_divider_circuit()
-
- print("\nGenerating Part 4 circuits...")
- generate_lumped_model_schematic()
- generate_distributed_model_structure()
-
- print("\nGenerating shared circuits...")
- generate_tesla_coil_system_overview()
-
- print("\n" + "="*60)
- print("CIRCUIT GENERATION COMPLETE!")
- print("="*60)
- print(f"\nTotal circuit diagrams generated: 7")
- print("="*60 + "\n")
-
-
-if __name__ == '__main__':
- main()
diff --git a/spark-lessons/lessons/01-fundamentals/01-introduction.md b/spark-lessons/lessons/01-fundamentals/01-introduction.md
deleted file mode 100644
index 479e2c0..0000000
--- a/spark-lessons/lessons/01-fundamentals/01-introduction.md
+++ /dev/null
@@ -1,248 +0,0 @@
----
-id: fund-01
-title: "Introduction to Tesla Coil Spark Modeling"
-section: "Fundamentals"
-difficulty: "beginner"
-estimated_time: 20
-prerequisites: []
-objectives:
- - Understand the scope and goals of Tesla coil spark modeling
- - Review essential AC circuit fundamentals including peak vs RMS values
- - Master complex number notation and phasor representation
- - Learn power calculations using peak phasors
- - Understand impedance and admittance concepts
-tags: ["introduction", "ac-circuits", "phasors", "complex-numbers", "power"]
----
-
-# Introduction to Tesla Coil Spark Modeling
-
-## Overview
-
-This lesson plan is designed to take you from basic circuit concepts through advanced Tesla coil spark modeling. Tesla coil sparks are complex plasma phenomena that require understanding of AC circuits, electromagnetic fields, and plasma physics. By the end of this series, you'll be able to predict spark behavior and optimize coil performance.
-
-### What You'll Learn
-
-The complete course is divided into four parts:
-
-1. **Part 1: Fundamentals** - Circuits, impedance, and basic spark behavior
-2. **Part 2: Optimization** - Power transfer and efficiency
-3. **Part 3: Growth Physics** - FEMM modeling and energy requirements
-4. **Part 4: Advanced Topics** - Distributed models and real-world application
-
-This lesson begins Part 1 by establishing the circuit theory foundation you'll need throughout.
-
-## AC Circuit Fundamentals Review
-
-### Peak vs RMS Values
-
-In AC circuits, voltage and current vary sinusoidally with time:
-
-**Time domain:**
-```
-v(t) = V_peak × cos(ωt + φ)
-```
-
-**Two amplitude conventions:**
-- **Peak value:** The maximum value reached (V_peak)
-- **RMS value:** Root-Mean-Square, V_RMS = V_peak/√2 ≈ 0.707 × V_peak
-
-**For this entire framework, we use PEAK VALUES exclusively.**
-
-**Why peak values?**
-1. Tesla coils are concerned with maximum voltage (breakdown, field stress)
-2. Consistent with phasor notation in engineering
-3. Power formula becomes: P = 0.5 × V_peak × I_peak × cos(θ)
-
-**Example:** If your oscilloscope shows a 100 kV peak-to-peak waveform:
-- V_peak-to-peak = 100 kV
-- V_peak = 50 kV (one-sided amplitude)
-- V_RMS = 50 kV / √2 ≈ 35.4 kV
-
-### Complex Numbers and Phasors
-
-AC circuit analysis uses complex numbers to represent magnitude and phase simultaneously.
-
-**Rectangular form:**
-```
-Z = R + jX
-where j = √(-1) (imaginary unit, engineers use 'j' instead of 'i')
-R = real part (resistance)
-X = imaginary part (reactance)
-```
-
-**Polar form:**
-```
-Z = |Z| ∠φ = |Z| × e^(jφ)
-where |Z| = √(R² + X²) (magnitude)
- φ = atan(X/R) (phase angle)
-```
-
-**Conversion:**
-```
-R = |Z| × cos(φ)
-X = |Z| × sin(φ)
-```
-
-**Phasor notation:** A complex number representing sinusoidal amplitude and phase:
-```
-V = V_peak ∠φ_v
-I = I_peak ∠φ_i
-```
-
-**Complex conjugate:** Used in power calculations
-```
-If I = a + jb, then I* = a - jb (flip sign of imaginary part)
-```
-
-### Resistance, Reactance, Impedance
-
-**Resistance (R):** Opposition to current that dissipates energy as heat
-- Units: Ω (ohms)
-- Always real and positive
-- V = I × R (Ohm's law)
-
-**Reactance (X):** Opposition to current that stores energy (no dissipation)
-- Units: Ω (ohms)
-- Can be positive (inductive) or negative (capacitive)
-- **Capacitive reactance:** X_C = -1/(ωC) where ω = 2πf
-- **Inductive reactance:** X_L = ωL
-
-**Impedance (Z):** Total opposition to AC current
-```
-Z = R + jX (complex)
-|Z| = √(R² + X²)
-φ_Z = atan(X/R)
-```
-
-**Sign conventions:**
-- X > 0: inductive (current lags voltage)
-- X < 0: capacitive (current leads voltage)
-- φ_Z > 0: inductive
-- φ_Z < 0: capacitive
-
-### Conductance, Susceptance, Admittance
-
-For parallel circuits, **admittance (Y)** is more convenient than impedance.
-
-**Conductance (G):** Inverse of resistance
-```
-G = 1/R
-Units: S (siemens)
-```
-
-**Susceptance (B):** Inverse of reactance (BUT with opposite sign convention!)
-```
-For capacitor: B_C = ωC (positive!)
-For inductor: B_L = -1/(ωL) (negative)
-```
-
-**Important:** Susceptance sign convention is OPPOSITE of reactance:
-- Capacitor: X_C < 0, but B_C > 0
-- Inductor: X_L > 0, but B_L < 0
-
-**Admittance (Y):** Inverse of impedance
-```
-Y = G + jB = 1/Z
-|Y| = 1/|Z|
-φ_Y = -φ_Z (opposite sign!)
-```
-
-**Conversion between Z and Y:**
-```
-Y = 1/Z = 1/(R + jX) = R/(R² + X²) - jX/(R² + X²)
-
-Therefore:
-G = R/(R² + X²)
-B = -X/(R² + X²)
-```
-
-### Power in AC Circuits
-
-**Using peak phasors:**
-```
-P = 0.5 × Re{V × I*}
-
-where V and I are complex peak phasors
- I* is the complex conjugate of I
- Re{·} means "real part of"
-```
-
-**Why the 0.5 factor?**
-- Average power over a full AC cycle
-- Comes from time-averaging cos²(ωt), which equals 0.5
-- If you used RMS values, formula would be P = V_RMS × I_RMS × cos(θ), NO 0.5
-
-**Expanded form:**
-```
-If V = V_peak ∠φ_v and I = I_peak ∠φ_i, then:
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-```
-
-The angle difference (φ_v - φ_i) is the power factor angle.
-
-## Worked Example: Power Calculation with Peak Phasors
-
-**Given:**
-- Voltage: V = 50 kV ∠0° (peak, using 0° as reference)
-- Impedance: Z = 100 kΩ ∠-60° (capacitive load)
-
-**Find:** Real power dissipated
-
-**Solution:**
-
-Step 1: Calculate current using Ohm's law
-```
-I = V/Z = (50 kV ∠0°)/(100 kΩ ∠-60°)
-I = 0.5 A ∠(0° - (-60°)) = 0.5 A ∠60°
-```
-
-Step 2: Calculate power
-```
-P = 0.5 × Re{V × I*}
-P = 0.5 × Re{(50 kV ∠0°) × (0.5 A ∠-60°)}
-P = 0.5 × Re{25 kW ∠-60°}
-```
-
-Step 3: Convert to rectangular to get real part
-```
-25 kW ∠-60° = 25 kW × (cos(-60°) + j×sin(-60°))
- = 25 kW × (0.5 - j×0.866)
- = 12.5 kW - j×21.65 kW
-```
-
-Step 4: Extract real part and apply 0.5 factor
-```
-P = 0.5 × 12.5 kW = 6.25 kW
-```
-
-**Alternative method:** Using power factor angle
-```
-P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
-P = 0.5 × 50 kV × 0.5 A × cos(0° - 60°)
-P = 0.5 × 25 kW × cos(-60°)
-P = 0.5 × 25 kW × 0.5
-P = 6.25 kW
-```
-
-## Key Takeaways
-
-- Always use **peak values** for Tesla coil analysis
-- Complex numbers combine magnitude and phase: Z = R + jX = |Z|∠φ
-- Power calculation: **P = 0.5 × Re{V × I*}** with peak phasors
-- Admittance (Y = G + jB) is the inverse of impedance
-- **Sign convention critical:** X < 0 for capacitors, but B > 0
-- Phase angles are opposite: φ_Y = -φ_Z
-
-## Practice
-
-{exercise:fund-ex-01}
-
-**Problem 1:** A capacitor has reactance X_C = -80 kΩ at 200 kHz. What is its capacitance? What is its susceptance?
-
-**Problem 2:** An impedance Z = 50 kΩ - j75 kΩ has current I = 0.2 A ∠30° (peak). Calculate: (a) Voltage magnitude and phase, (b) Real power
-
-**Problem 3:** An admittance Y = 0.00001 + j0.00002 S. Convert to impedance Z = R + jX.
-
----
-
-**Next Lesson:** [Basic Circuit Model](02-basic-circuit-model.md)
diff --git a/spark-lessons/lessons/01-fundamentals/02-basic-circuit-model.md b/spark-lessons/lessons/01-fundamentals/02-basic-circuit-model.md
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----
-id: fund-02
-title: "The Basic Spark Circuit Model"
-section: "Fundamentals"
-difficulty: "beginner"
-estimated_time: 25
-prerequisites: ["fund-01"]
-objectives:
- - Understand what capacitance represents physically
- - Distinguish between mutual capacitance (C_mut) and shunt capacitance (C_sh)
- - Learn the empirical 2 pF/foot rule for spark capacitance
- - Draw the correct circuit topology for a Tesla coil spark
- - Identify the topload port as the measurement reference
-tags: ["capacitance", "circuit-topology", "C_mut", "C_sh", "measurement"]
----
-
-# The Basic Spark Circuit Model
-
-## Introduction
-
-A spark isn't just a resistor - it's a complex structure with multiple electrical properties. Understanding how to model a spark as a circuit with the correct topology is essential for analyzing Tesla coil performance.
-
-## What is Capacitance Physically?
-
-**Definition:** Capacitance (C) is the ability to store electric charge for a given voltage:
-```
-Q = C × V
-Units: Farads (F), typically pF (10⁻¹² F) for Tesla coils
-```
-
-**Physical picture:**
-- Electric field between two conductors stores energy
-- Higher field → more stored energy → more capacitance
-- Capacitance depends on geometry, NOT on voltage
-
-**For parallel plates:**
-```
-C = ε₀ × A / d
-
-where ε₀ = 8.854×10⁻¹² F/m (permittivity of free space)
- A = plate area (m²)
- d = separation distance (m)
-```
-
-**Key insight:** Capacitance increases with:
-- Larger conductor area (more field lines)
-- Smaller separation (stronger field concentration)
-
-## Self-Capacitance vs Mutual Capacitance
-
-**Self-capacitance:** Capacitance of a single conductor to infinity (or ground)
-- Topload has self-capacitance to ground
-- Depends on size and shape
-- Toroid: C ≈ 4πε₀√(D×d) where D = major diameter, d = minor diameter
-
-**Mutual capacitance:** Capacitance between two conductors
-- Energy stored in field between them
-- Both conductors at different potentials
-- Can be positive or negative in matrix formulation
-
-**For Tesla coils with sparks:**
-- **C_mut:** mutual capacitance between topload and spark channel
-- **C_sh:** capacitance from spark to ground (shunt capacitance)
-
-## Shunt Capacitance and the 2 pF/Foot Rule
-
-Any conductor elevated above ground has capacitance to ground.
-
-**For vertical wire above ground plane:**
-```
-C ≈ 2πε₀L / ln(2h/d)
-
-where L = wire length
- h = height above ground
- d = wire diameter
-```
-
-**For Tesla coil sparks:** Empirical rule based on community measurements:
-```
-C_sh ≈ 2 pF per foot of spark length
-
-Examples:
-1 foot (0.3 m) spark: C_sh ≈ 2 pF
-3 feet (0.9 m) spark: C_sh ≈ 6 pF
-6 feet (1.8 m) spark: C_sh ≈ 12 pF
-```
-
-This rule is surprisingly accurate (±30%) for typical Tesla coil geometries.
-
-### Worked Example: Estimating C_sh
-
-**Given:** A 2-meter (6.6 foot) spark
-
-**Find:** Estimated shunt capacitance
-
-**Solution:**
-```
-C_sh ≈ 2 pF/foot × 6.6 feet
-C_sh ≈ 13.2 pF
-```
-
-**Refined estimate using cylinder formula:**
-
-Assume spark is vertical cylinder:
-- Length L = 2 m
-- Diameter d = 2 mm (typical for bright spark)
-- Height above ground h = L/2 = 1 m (average height)
-
-```
-C ≈ 2πε₀L / ln(2h/d)
-C ≈ 2π × 8.854×10⁻¹² × 2 / ln(2×1/0.002)
-C ≈ 1.112×10⁻¹⁰ / ln(1000)
-C ≈ 1.112×10⁻¹⁰ / 6.91
-C ≈ 16 pF
-```
-
-The empirical rule (13 pF) and formula (16 pF) agree reasonably well.
-
-## Why Sparks Have TWO Capacitances
-
-A spark channel is a conductor in space with:
-1. **Proximity to the topload** → mutual capacitance C_mut
-2. **Proximity to ground/environment** → shunt capacitance C_sh
-
-**Both exist simultaneously** because the spark interacts with multiple conductors.
-
-**Analogy:** A wire near two metal plates
-- Capacitance to plate 1: C₁
-- Capacitance to plate 2: C₂
-- Both must be included in the circuit model
-
-
-
-**Field line visualization:**
-- **C_mut field lines:** Connect topload surface to spark channel
- - Start on topload outer surface
- - End on spark channel surface
- - Concentrated near base of spark
- - These store mutual electric field energy
-
-- **C_sh field lines:** Connect spark to remote ground
- - Start on spark surface
- - Radiate outward to walls, floor, ceiling
- - Distributed along entire spark length
- - These store shunt field energy
-
-**Key observation:** The same spark channel participates in BOTH capacitances! This is why we need a specific circuit topology.
-
-## The Correct Circuit Topology
-
-```
- Topload (measurement reference)
- |
- [C_mut] ← Mutual capacitance between topload and spark
- |
- +---------+--------- Node_spark
- | |
- [R] [C_sh] ← Shunt capacitance spark-to-ground
- | |
- GND ------------ GND
-```
-
-**Equivalent description:**
-- C_mut and R in parallel
-- That parallel combination in series with C_sh
-- All connected between topload and ground
-
-**Why this topology?**
-1. C_mut couples topload voltage to spark
-2. R represents plasma resistance (where power is dissipated)
-3. C_sh provides current return path to ground
-4. Current through R must also flow through either C_mut or C_sh (series connection)
-
-## Where is "Ground" in a Tesla Coil?
-
-**Earth ground:** Actual connection to soil/building ground
-**Circuit ground (reference):** Arbitrary 0V reference point
-
-**For Tesla coils:**
-- Primary circuit: Chassis/mains ground is reference
-- Secondary base: Usually connected to primary ground via RF ground
-- **Practical ground:** Floor, walls, nearby objects, you standing nearby
-- **Measurement ground:** Choose ONE point as 0V reference (usually secondary base)
-
-**Important:** "Ground" in spark model means "remote return path" - could be walls, floor, strike ring, or actual earth.
-
-## The Topload Port
-
-**Definition:** The two-terminal measurement point between topload and ground where we characterize impedance and power.
-
-```
-Port definition:
- Terminal 1: Topload terminal (high voltage)
- Terminal 2: Ground reference (0V)
-```
-
-**All impedance measurements reference this port:**
-- Z_spark: impedance looking into spark from topload
-- Z_th: Thévenin impedance of coil at this port
-- V_th: Open-circuit voltage at this port
-
-**Not the same as:**
-- V_top / I_base (includes displacement currents from entire secondary)
-- Any two-point measurement along the secondary winding
-
-We'll explore why V_top/I_base is incorrect in a later lesson.
-
-## Worked Example: Drawing the Complete Circuit
-
-**Given:**
-- Spark is 3 feet long
-- FEMM analysis gives C_mut = 8 pF (between topload and spark)
-- Assume R = 100 kΩ
-- Estimate C_sh using empirical rule
-
-**Task:** Draw complete circuit diagram
-
-**Solution:**
-
-Step 1: Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 3 feet = 6 pF
-```
-
-Step 2: Draw topology
-```
- Topload (V_top)
- |
- [C_mut = 8 pF]
- |
- +-------- Node_spark
- | |
- [R = 100 kΩ] [C_sh = 6 pF]
- | |
- GND -------- GND
-```
-
-Step 3: Alternative representation showing parallel/series structure
-```
-Topload
- |
- +---- [C_mut = 8 pF] ----+
- | |
- +---- [R = 100 kΩ] ------+ Node_spark
- |
- [C_sh = 6 pF]
- |
- GND
-```
-
-This is the basic lumped model for a Tesla coil spark.
-
-
-
-## Key Takeaways
-
-- Capacitance stores energy in electric fields, depends on geometry
-- **C_mut:** mutual capacitance between topload and spark
-- **C_sh:** shunt capacitance from spark to ground, approximately **2 pF/foot**
-- Both capacitances exist simultaneously on the same conductor
-- **Correct topology:** (R || C_mut) in series with C_sh
-- **Topload port:** measurement reference between topload and ground
-- Ground means "remote return path" in this context
-
-## Practice
-
-{exercise:fund-ex-02}
-
-**Problem 1:** Draw the circuit for a spark with: L = 5 feet, C_mut = 12 pF (from FEMM), R = 50 kΩ. Label all component values.
-
-**Problem 2:** A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark length using the empirical rule?
-
-**Problem 3:** A 4-foot spark is formed. Estimate C_sh using the empirical rule. If the topload has C_topload = 30 pF unloaded, what is the total system capacitance with the spark? (Hint: Consider how C_mut and C_sh combine in the circuit.)
-
----
-
-**Next Lesson:** [Admittance Analysis](03-admittance-analysis.md)
diff --git a/spark-lessons/lessons/01-fundamentals/03-admittance-analysis.md b/spark-lessons/lessons/01-fundamentals/03-admittance-analysis.md
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----
-id: fund-03
-title: "Admittance Analysis of the Spark Circuit"
-section: "Fundamentals"
-difficulty: "intermediate"
-estimated_time: 30
-prerequisites: ["fund-01", "fund-02"]
-objectives:
- - Understand why admittance is preferred over impedance for parallel circuits
- - Derive the total admittance formula for the spark circuit
- - Calculate real and imaginary parts of admittance
- - Convert between admittance and impedance representations
- - Apply formulas to practical Tesla coil examples
-tags: ["admittance", "circuit-analysis", "complex-algebra", "formulas"]
----
-
-# Admittance Analysis of the Spark Circuit
-
-## Introduction
-
-The spark circuit topology (R || C_mut in series with C_sh) requires careful analysis. While we could work entirely with impedances, using admittance simplifies the parallel combination and provides clearer insight into circuit behavior.
-
-## Why Use Admittance?
-
-For the spark circuit topology (parallel R||C_mut, in series with C_sh), admittance simplifies calculations.
-
-**Parallel elements:** Add admittances directly
-```
-Y_total = Y₁ + Y₂ + Y₃ + ...
-vs impedances: 1/Z_total = 1/Z₁ + 1/Z₂ + ... (messy!)
-```
-
-**Our circuit:**
-```
-Y_mut_R = Y_Cmut + Y_R (parallel: C_mut || R)
-Then series with C_sh requires impedance: Z = Z_mut_R + Z_Csh
-Then convert back: Y_total = 1/Z_total
-```
-
-Admittance makes the first step (parallel combination) trivial, and we only need to handle the series combination once.
-
-## Deriving the Total Admittance Formula
-
-Let's work through the complete derivation step by step.
-
-**Step 1:** Admittance of R and C_mut in parallel
-
-```
-Y_R = G = 1/R
-Y_Cmut = jωC_mut = jB₁ (where B₁ = ωC_mut)
-
-Y_mut_R = G + jB₁
-```
-
-**Step 2:** Convert to impedance for series combination
-
-```
-Z_mut_R = 1/(G + jB₁)
-```
-
-**Step 3:** Add impedance of C_sh in series
-
-```
-Z_Csh = 1/(jωC_sh) = -j/(ωC_sh) = 1/(jB₂) (where B₂ = ωC_sh)
-
-Z_total = Z_mut_R + Z_Csh
-Z_total = 1/(G + jB₁) + 1/(jB₂)
-```
-
-**Step 4:** Find common denominator
-
-```
-Z_total = [jB₂ + (G + jB₁)] / [(G + jB₁) × jB₂]
-Z_total = [G + j(B₁ + B₂)] / [jB₂(G + jB₁)]
-```
-
-**Step 5:** Invert to get admittance
-
-```
-Y_total = 1/Z_total = [jB₂(G + jB₁)] / [G + j(B₁ + B₂)]
-
-Y_total = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-```
-
-This is the **fundamental admittance equation** for the spark circuit.
-
-## Extracting Real and Imaginary Parts
-
-To use this formula, we need to separate it into Re{Y} and Im{Y}.
-
-Multiply numerator:
-```
-(G + jB₁) × jB₂ = jGB₂ + j²B₁B₂ = jGB₂ - B₁B₂
- = -B₁B₂ + jGB₂
-```
-
-So:
-```
-Y = [-B₁B₂ + jGB₂] / [G + j(B₁ + B₂)]
-```
-
-To separate real and imaginary parts, multiply numerator and denominator by complex conjugate of denominator:
-
-```
-Denominator conjugate: G - j(B₁ + B₂)
-Denominator magnitude squared: G² + (B₁ + B₂)²
-```
-
-After algebra (multiply out and simplify):
-
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-```
-
-These are the **working formulas** for calculating admittance from R, C_mut, C_sh.
-
-### Formula Summary
-
-Given R, C_mut, C_sh, and frequency f:
-
-**Step 1:** Calculate component values
-```
-ω = 2πf
-G = 1/R
-B₁ = ωC_mut
-B₂ = ωC_sh
-```
-
-**Step 2:** Calculate admittance
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-
-Y = Re{Y} + j×Im{Y}
-```
-
-**Step 3:** Magnitude and phase
-```
-|Y| = √[Re{Y}² + Im{Y}²]
-φ_Y = atan(Im{Y}/Re{Y})
-```
-
-## Converting to Impedance
-
-From Y = G_total + jB_total:
-
-```
-Z = 1/Y = 1/(G_total + jB_total)
-
-Multiply by conjugate:
-Z = (G_total - jB_total) / (G_total² + B_total²)
-
-R_total = G_total / (G_total² + B_total²)
-X_total = -B_total / (G_total² + B_total²)
-
-Or directly:
-|Z| = 1/|Y|
-φ_Z = -φ_Y (opposite sign!)
-```
-
-## Worked Example: Complete Y and Z Calculation
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 2π × 200×10³ = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-- R = 100 kΩ = 10⁵ Ω
-
-**Find:** Y_total (rectangular), Z_total (rectangular and polar)
-
-**Solution:**
-
-Step 1: Calculate component values
-```
-G = 1/R = 1/(10⁵) = 10⁻⁵ S = 10 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06×10⁻⁶ S = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54×10⁻⁶ S = 7.54 μS
-```
-
-Step 2: Calculate Re{Y}
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-
-Numerator: 10 × (7.54)² = 10 × 56.85 = 568.5 μS²
-Denominator: (10)² + (10.06 + 7.54)² = 100 + (17.6)² = 100 + 309.8 = 409.8 μS²
-
-Re{Y} = 568.5 / 409.8 = 1.387 μS
-```
-
-Step 3: Calculate Im{Y}
-```
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-
-Numerator inner: G² + B₁(B₁ + B₂) = 100 + 10.06×17.6 = 100 + 177.1 = 277.1 μS²
-Numerator: 7.54 × 277.1 = 2089.3 μS³
-Denominator: 409.8 μS² (same as before)
-
-Im{Y} = 2089.3 / 409.8 = 5.10 μS
-```
-
-Step 4: Admittance result
-```
-Y_total = 1.387 + j5.10 μS
-|Y| = √(1.387² + 5.10²) = √(1.92 + 26.01) = √27.93 = 5.28 μS
-φ_Y = atan(5.10/1.387) = atan(3.68) = 74.8°
-```
-
-Step 5: Convert to impedance
-```
-|Z| = 1/|Y| = 1/(5.28×10⁻⁶) = 189 kΩ
-φ_Z = -φ_Y = -74.8°
-
-In rectangular:
-R_total = |Z| × cos(φ_Z) = 189 × cos(-74.8°) = 189 × 0.263 = 49.7 kΩ
-X_total = |Z| × sin(φ_Z) = 189 × sin(-74.8°) = 189 × (-0.965) = -182 kΩ
-
-Z_total = 49.7 - j182 kΩ = 189 kΩ ∠-74.8°
-```
-
-**Interpretation:**
-- Impedance is strongly capacitive (φ_Z = -74.8°)
-- Equivalent resistance ≈ 50 kΩ (half of actual R due to capacitive divider)
-- Large capacitive reactance dominates
-
-
-
-**Visualization notes:**
-- LEFT: Admittance plane (Y = G + jB)
- - Point at (1.387, 5.10) μS
- - Angle φ_Y = 74.8° from horizontal
- - Positive B means capacitive in admittance
-
-- RIGHT: Impedance plane (Z = R + jX)
- - Point at (49.7, -182) kΩ
- - Angle φ_Z = -74.8° below horizontal
- - Negative X means capacitive in impedance
-
-- Connection: Angles are opposite (φ_Z = -φ_Y), magnitudes invert (|Z| = 1/|Y|)
-
-## Key Takeaways
-
-- **Admittance simplifies parallel combinations:** Y_parallel = Y₁ + Y₂ + ...
-- **Fundamental formula:** Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-- **Working formulas:**
- - Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
- - Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-- **Conversion:** |Z| = 1/|Y| and φ_Z = -φ_Y
-- Typical spark: strongly capacitive with large |Im{Y}| compared to Re{Y}
-
-## Practice
-
-{exercise:fund-ex-03}
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF, R = 80 kΩ, calculate Y_total (real and imaginary parts).
-
-**Problem 2:** An admittance Y = 2.0 + j4.5 μS. Convert to impedance Z in both rectangular and polar forms.
-
-**Problem 3:** Show algebraically that if R → ∞ (open circuit), the formula reduces to Y = jωC_mut × C_sh/(C_mut + C_sh), which is two capacitors in series.
-
----
-
-**Next Lesson:** [Phase Angles and Their Meaning](04-phase-angles.md)
diff --git a/spark-lessons/lessons/01-fundamentals/04-phase-angles.md b/spark-lessons/lessons/01-fundamentals/04-phase-angles.md
deleted file mode 100644
index df5fa90..0000000
--- a/spark-lessons/lessons/01-fundamentals/04-phase-angles.md
+++ /dev/null
@@ -1,203 +0,0 @@
----
-id: fund-04
-title: "Phase Angles and What They Mean"
-section: "Fundamentals"
-difficulty: "beginner"
-estimated_time: 20
-prerequisites: ["fund-01", "fund-02", "fund-03"]
-objectives:
- - Distinguish between impedance phase φ_Z and admittance phase φ_Y
- - Understand the relationship φ_Z = -φ_Y
- - Interpret the physical meaning of different phase angles
- - Learn why -45° is considered "balanced"
- - Recognize typical phase angles for Tesla coil sparks
-tags: ["phase-angle", "impedance", "admittance", "power-factor"]
----
-
-# Phase Angles and What They Mean
-
-## Introduction
-
-Phase angles tell us about the balance between resistive and reactive components in our circuit. Understanding what different phase angles mean physically helps us interpret circuit behavior and optimize performance.
-
-## Impedance Phase vs Admittance Phase
-
-**Impedance phase angle φ_Z:**
-```
-φ_Z = atan(X/R) = atan(Im{Z}/Re{Z})
-
-Interpretation:
-φ_Z > 0: inductive (current lags voltage)
-φ_Z = 0: purely resistive (in phase)
-φ_Z < 0: capacitive (current leads voltage)
-```
-
-**Admittance phase angle θ_Y:**
-```
-θ_Y = atan(B/G) = atan(Im{Y}/Re{Y})
-
-Relationship: θ_Y = -φ_Z (OPPOSITE SIGNS!)
-```
-
-**Why opposite?** Because Y = 1/Z, so angles subtract:
-```
-If Z = |Z|∠φ_Z, then Y = (1/|Z|)∠(-φ_Z)
-```
-
-**Convention in this framework:** We primarily discuss **impedance phase φ_Z** because that's what measurements typically report.
-
-## The "Famous -45°" and Why It's Special
-
-In power electronics, a load with φ_Z = -45° is sometimes called "well-matched" because:
-- Equal resistive and capacitive components: |R| = |X_C|
-- Power factor = cos(-45°) = 0.707 (reasonable power transfer)
-- Not maximum power transfer, but balanced
-
-**Formula:** For φ_Z = -45°:
-```
-tan(-45°) = -1 = X/R
-Therefore: R = |X| = 1/(ωC) for capacitive load
-Or: R ≈ |X_C| = 1/(ωC_total) approximately
-```
-
-This is why you'll see "spark resistance should equal capacitive reactance" in old Tesla coil literature.
-
-**BUT:** As we'll see in the next lesson, achieving exactly -45° is **impossible** for many Tesla coil geometries due to topological constraints!
-
-## Physical Meaning of Phase Angle
-
-Let's explore what different phase angles mean for circuit behavior.
-
-**φ_Z = 0° (purely resistive):**
-- All power dissipated
-- No energy storage/return
-- Voltage and current in phase
-- Power factor = cos(0°) = 1.0 (100%)
-
-**φ_Z = -45° (mixed):**
-- Some power dissipated (cos(-45°) ≈ 71% of |V||I|)
-- Some energy stored
-- Current leads voltage by 45°
-- Equal R and |X|: balanced condition
-
-**φ_Z = -90° (purely capacitive):**
-- No power dissipated
-- All energy stored and returned each cycle
-- Current leads voltage by 90°
-- Power factor = cos(-90°) = 0 (no real power)
-
-**For Tesla coil sparks:** Typical φ_Z = -55° to -75°
-- Significant capacitive component (energy storage in C_mut, C_sh)
-- Moderate power dissipation (plasma heating)
-- More capacitive than the "ideal" -45°
-
-## Worked Example: Calculating and Interpreting Phase Angle
-
-**Given:** (from previous lesson)
-- Z_total = 49.7 - j182 kΩ
-
-**Find:** φ_Z and interpret
-
-**Solution:**
-
-Step 1: Calculate phase angle
-```
-φ_Z = atan(X/R) = atan(-182/49.7)
-φ_Z = atan(-3.66) = -74.8°
-```
-
-Step 2: Verify with magnitude and components
-```
-|Z| = √(49.7² + 182²) = √(2470 + 33124) = √35594 = 189 kΩ ✓
-
-cos(φ_Z) = R/|Z| = 49.7/189 = 0.263
-φ_Z = arccos(0.263) = 74.8°, but X is negative, so φ_Z = -74.8° ✓
-```
-
-Step 3: Interpret
-- **Strongly capacitive:** |φ_Z| = 74.8° is much larger than 45°
-- **Comparison:** |R| = 49.7 kΩ, but |X| = 182 kΩ
- - Capacitive reactance is 3.66× larger than resistance
- - Far from "balanced" -45° condition
-- **Power factor:** cos(-74.8°) = 0.263
- - Only 26.3% of |V||I| is real power
- - Most current is reactive (charging/discharging capacitances)
-
-This is typical for Tesla coil sparks: strongly capacitive impedance.
-
-## Visualizing Phase Angles
-
-
-
-**Impedance plane (Z = R + jX):**
-
-Three key vectors from origin:
-
-1. **Resistive (φ_Z = 0°):**
- - Horizontal vector along R axis
- - Pure resistance, no reactance
- - All power dissipated
-
-2. **Balanced (φ_Z = -45°):**
- - Vector at -45° angle
- - Equal R and |X|
- - Traditional "well-matched" condition
-
-3. **Typical spark (φ_Z = -75°):**
- - Vector at -75° angle
- - Strongly capacitive
- - |X| >> R
-
-**Key regions:**
-- φ_Z = 0°: Pure resistance (horizontal axis)
-- φ_Z = -45°: Balanced point
-- -45° to -90°: Typical Tesla coil spark range (shaded region)
-- φ_Z = -90°: Pure capacitor (vertical downward)
-
-**Note:** More negative φ_Z means more capacitive behavior
-
-## Relationship to Power Factor
-
-The power factor relates phase angle to real power delivery:
-
-```
-Power Factor = cos(φ_Z)
-
-Real Power: P = 0.5 × |V| × |I| × cos(φ_Z)
-Reactive Power: Q = 0.5 × |V| × |I| × sin(φ_Z)
-```
-
-**Examples:**
-| φ_Z | Power Factor | % of Maximum Power |
-|-----|--------------|-------------------|
-| 0° | 1.00 | 100% |
-| -30° | 0.866 | 86.6% |
-| -45° | 0.707 | 70.7% |
-| -60° | 0.500 | 50.0% |
-| -75° | 0.259 | 25.9% |
-| -90° | 0.000 | 0% |
-
-Tesla coil sparks typically operate at 25-50% power factor - much energy is reactive (stored and returned each cycle) rather than dissipated in the plasma.
-
-## Key Takeaways
-
-- **Phase relationship:** φ_Z = -φ_Y (opposite signs)
-- **Negative φ_Z:** means capacitive (current leads voltage)
-- **φ_Z = -45°:** balanced condition with R = |X|
-- **Typical sparks:** φ_Z ≈ -55° to -75° (strongly capacitive)
-- **Power factor:** cos(φ_Z) determines fraction of power dissipated
-- More capacitive → lower power factor → less efficient power transfer
-
-## Practice
-
-{exercise:fund-ex-04}
-
-**Problem 1:** An impedance Z = 60 + j40 kΩ. Calculate φ_Z. Is this inductive or capacitive?
-
-**Problem 2:** A spark has φ_Z = -60°. If |Z| = 150 kΩ, find R and X. Calculate the power factor.
-
-**Problem 3:** Two sparks have the same |Z| = 200 kΩ. Spark A has φ_Z = -50°, Spark B has φ_Z = -70°. Which dissipates more power for the same applied voltage? By what factor?
-
----
-
-**Next Lesson:** [The Phase Constraint](05-phase-constraint.md)
diff --git a/spark-lessons/lessons/01-fundamentals/05-phase-constraint.md b/spark-lessons/lessons/01-fundamentals/05-phase-constraint.md
deleted file mode 100644
index b7cef9d..0000000
--- a/spark-lessons/lessons/01-fundamentals/05-phase-constraint.md
+++ /dev/null
@@ -1,235 +0,0 @@
----
-id: fund-05
-title: "The Topological Phase Constraint"
-section: "Fundamentals"
-difficulty: "intermediate"
-estimated_time: 25
-prerequisites: ["fund-01", "fund-02", "fund-03", "fund-04"]
-objectives:
- - Understand what a topological constraint is
- - Derive the minimum achievable phase angle φ_Z,min
- - Learn the critical capacitance ratio r = C_mut/C_sh
- - Calculate φ_Z,min for typical Tesla coil geometries
- - Understand R_opt_phase that achieves minimum phase
-tags: ["topology", "phase-constraint", "optimization", "mathematical-limit"]
----
-
-# The Topological Phase Constraint
-
-## Introduction
-
-Can we make a spark look purely resistive (φ_Z = 0°)? Can we at least achieve the "balanced" -45° condition? Surprisingly, the circuit topology itself imposes fundamental limits on what phase angles are achievable, regardless of component values.
-
-## What is a Topological Constraint?
-
-**Definition:** A limitation imposed by the **structure** of the circuit itself, independent of component values.
-
-**Example:** Series RLC circuit
-- Can only have impedance phase between -90° (pure C) and +90° (pure L)
-- Cannot have φ_Z = +120° no matter what component values you choose
-- This is a topological constraint
-
-**For spark circuits:** The specific arrangement (R||C_mut) in series with C_sh creates a fundamental limit on how resistive the impedance can appear.
-
-## Deriving the Minimum Phase Angle
-
-From our previous lesson, we have:
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh
-```
-
-The impedance phase is:
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Question:** For fixed C_mut and C_sh, which R value minimizes |φ_Z| (makes most resistive)?
-
-**Mathematical result:** Taking derivative ∂φ_Z/∂G = 0 and solving:
-```
-G_opt = ω√[C_mut(C_mut + C_sh)]
-
-Therefore:
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-At this resistance, the phase angle magnitude is minimized to:
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-
-where r = C_mut/C_sh (capacitance ratio)
-```
-
-**Key insight:** φ_Z,min depends only on the ratio r, not on absolute capacitance values or frequency!
-
-## The Critical Ratio r = 0.207
-
-Let's find when φ_Z,min = -45° is achievable:
-```
--45° = -atan(2√[r(1 + r)])
-tan(45°) = 1 = 2√[r(1 + r)]
-0.5 = √[r(1 + r)]
-0.25 = r(1 + r) = r + r²
-r² + r - 0.25 = 0
-
-Using quadratic formula:
-r = [-1 ± √(1 + 1)] / 2 = [-1 ± √2] / 2
-
-Taking positive root:
-r = (√2 - 1) / 2 ≈ 0.207
-```
-
-**Critical insight:**
-- If **r < 0.207:** Can achieve φ_Z = -45° (with appropriate R)
-- If **r = 0.207:** Minimum achievable phase is exactly -45°
-- If **r > 0.207:** **Cannot achieve φ_Z = -45° no matter what R you choose!**
-- If r ≥ 0.207: φ_Z,min is more negative than -45°
-
-## Typical Tesla Coil Values
-
-Let's examine realistic scenarios:
-
-**Large topload, short spark:**
-```
-C_mut = 10 pF, C_sh = 4 pF (2 feet)
-r = 10/4 = 2.5
-
-φ_Z,min = -atan(2√[2.5 × 3.5]) = -atan(2 × 2.96) = -atan(5.92) = -80.4°
-```
-
-**Medium configuration:**
-```
-C_mut = 8 pF, C_sh = 6 pF (3 feet)
-r = 8/6 = 1.33
-
-φ_Z,min = -atan(2√[1.33 × 2.33]) = -atan(2 × 1.76) = -atan(3.53) = -74.2°
-```
-
-**Small topload, long spark:**
-```
-C_mut = 6 pF, C_sh = 12 pF (6 feet)
-r = 6/12 = 0.5
-
-φ_Z,min = -atan(2√[0.5 × 1.5]) = -atan(2 × 0.866) = -atan(1.732) = -60.0°
-```
-
-**Common range:** r = 0.5 to 2.0, giving φ_Z,min ≈ -60° to -80°
-
-**Conclusion:** For most Tesla coil geometries, -45° is **mathematically impossible**!
-
-## Worked Example: Calculate Minimum Phase Angle
-
-**Given:**
-- Frequency: f = 200 kHz
-- C_mut = 8 pF
-- C_sh = 6 pF
-
-**Find:**
-(a) Capacitance ratio r
-(b) Minimum achievable phase angle φ_Z,min
-(c) R_opt_phase that achieves this angle
-
-**Solution:**
-
-**Part (a):** Capacitance ratio
-```
-r = C_mut / C_sh = 8 / 6 = 1.333
-```
-
-**Part (b):** Minimum phase angle
-```
-φ_Z,min = -atan(2√[r(1 + r)])
- = -atan(2√[1.333 × 2.333])
- = -atan(2√3.11)
- = -atan(2 × 1.764)
- = -atan(3.528)
- = -74.2°
-```
-
-**Part (c):** Resistance for minimum phase
-```
-ω = 2πf = 2π × 200×10³ = 1.257×10⁶ rad/s
-
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 1 / [1.257×10⁶ × √(8×10⁻¹² × 14×10⁻¹²)]
- = 1 / [1.257×10⁶ × √(112×10⁻²⁴)]
- = 1 / [1.257×10⁶ × 10.58×10⁻¹²]
- = 1 / (13.30×10⁻⁶)
- = 75.2 kΩ
-```
-
-**Interpretation:**
-- With r = 1.333, cannot achieve -45°
-- Best possible is -74.2° (much more capacitive)
-- This requires R = 75.2 kΩ
-- Any other R value gives |φ_Z| > 74.2°
-
-## Understanding the Constraint Graphically
-
-
-
-**Graph characteristics:**
-- X-axis: r = C_mut/C_sh (log scale), range 0.1 to 10
-- Y-axis: φ_Z,min (degrees), range -90° to -40°
-- Curve: φ_Z,min = -atan(2√[r(1+r)])
-
-**Key features:**
-- r = 0.207 marked: φ_Z,min = -45° (horizontal dashed line)
-- Region r < 0.207 (shaded): "Can achieve -45°"
-- Region r > 0.207 (different shade): "Cannot achieve -45°"
-- Typical Tesla coil range r = 0.5 to 2.0 highlighted
-
-**Example points:**
-- r = 0.1: φ_Z,min ≈ -35°
-- r = 0.207: φ_Z,min = -45° (critical point)
-- r = 0.5: φ_Z,min = -60°
-- r = 1.0: φ_Z,min = -70.5°
-- r = 2.0: φ_Z,min = -79.7°
-- r = 5.0: φ_Z,min = -84.5°
-
-**Trends:**
-- Larger r → more capacitive minimum
-- Large topload + short spark → high r → very capacitive
-- Small topload + long spark → low r → less capacitive (but still > -45° usually)
-
-## Physical Interpretation
-
-**Why does this constraint exist?**
-
-The series connection of C_sh means current must flow through it to reach ground. This creates a capacitive voltage drop that can never be completely eliminated, no matter how you adjust R.
-
-**Analogy:** Trying to make water flow uphill
-- C_sh is like a mandatory uphill section in your pipe
-- R adjusts resistance elsewhere, but can't remove the uphill section
-- The uphill section imposes a minimum "difficulty" for flow
-
-**Engineering implications:**
-1. Can't achieve purely resistive load (φ_Z = 0°)
-2. Usually can't achieve "balanced" -45° condition
-3. Must work with more capacitive phase angles
-4. Power transfer is inherently less efficient than with purely resistive load
-
-## Key Takeaways
-
-- **Topological constraint:** Circuit structure limits achievable phase angles
-- **Minimum phase:** φ_Z,min = -atan(2√[r(1 + r)]) where r = C_mut/C_sh
-- **Critical ratio:** r = 0.207 allows exactly -45°
-- **Typical range:** r = 0.5 to 2.0 → φ_Z,min ≈ -60° to -80°
-- **Optimal resistance:** R_opt_phase = 1/[ω√(C_mut(C_mut + C_sh))]
-- Most Tesla coils **cannot achieve -45°** due to geometry
-
-## Practice
-
-{exercise:fund-ex-05}
-
-**Problem 1:** Calculate r, φ_Z,min, and R_opt_phase for: f = 150 kHz, C_mut = 12 pF, C_sh = 8 pF.
-
-**Problem 2:** A coil designer wants to achieve φ_Z = -45°. If C_sh = 10 pF (5-foot spark), what maximum C_mut is allowed?
-
-**Problem 3:** Two coils have the same frequency and total capacitance (C_mut + C_sh = 20 pF). Coil A has r = 0.5, Coil B has r = 2.0. Which can achieve a more resistive phase angle? Calculate φ_Z,min for both.
-
----
-
-**Next Lesson:** [Why Not -45 Degrees?](06-why-not-45-degrees.md)
diff --git a/spark-lessons/lessons/01-fundamentals/06-why-not-45-degrees.md b/spark-lessons/lessons/01-fundamentals/06-why-not-45-degrees.md
deleted file mode 100644
index 4b4fe38..0000000
--- a/spark-lessons/lessons/01-fundamentals/06-why-not-45-degrees.md
+++ /dev/null
@@ -1,238 +0,0 @@
----
-id: fund-06
-title: "Why Not -45 Degrees?"
-section: "Fundamentals"
-difficulty: "beginner"
-estimated_time: 15
-prerequisites: ["fund-04", "fund-05"]
-objectives:
- - Understand the historical origin of the -45° target
- - Recognize why -45° is often impossible for Tesla coils
- - Distinguish between R_opt_phase and R_opt_power
- - Learn what resistance values are actually optimal
-tags: ["misconceptions", "optimization", "history", "phase-angle"]
----
-
-# Why Not -45 Degrees?
-
-## Introduction
-
-If you've read Tesla coil literature or online discussions, you've probably encountered the advice: "Make the spark resistance equal to the capacitive reactance for -45° phase angle." This lesson explains where this comes from, why it's often impossible, and what you should actually target instead.
-
-## The Historical -45° Target
-
-### Where Did This Come From?
-
-In power electronics and RF engineering, a load with φ_Z = -45° has some appealing properties:
-
-**Mathematical simplicity:**
-```
-φ_Z = -45° means tan(-45°) = -1
-Therefore: X/R = -1
-So: R = |X|
-```
-
-For a capacitive load: R = 1/(ωC_total)
-
-**Balanced characteristics:**
-- Equal resistive and reactive components
-- Power factor = cos(-45°) ≈ 0.707
-- Reasonable compromise between power delivery and energy storage
-
-**Easy to remember:** "Make resistance equal to reactance"
-
-### Why It Became Popular in Tesla Coil Literature
-
-Early Tesla coil experimenters borrowed concepts from radio engineering, where matching impedances for -45° was a common practice. The simple rule "R should equal capacitive reactance" was easy to communicate and remember.
-
-**The problem:** This advice doesn't account for the specific topology of the spark circuit!
-
-## The Reality: Why -45° is Often Impossible
-
-### The Topological Constraint
-
-As we learned in the previous lesson, the minimum achievable phase angle is:
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-
-where r = C_mut/C_sh
-```
-
-**For -45° to be achievable:** r must be ≤ 0.207
-
-**What this means:**
-```
-C_mut/C_sh ≤ 0.207
-C_mut ≤ 0.207 × C_sh
-```
-
-### Realistic Tesla Coil Scenarios
-
-Let's check if typical geometries can achieve -45°:
-
-**Scenario 1: 3-foot spark, medium topload**
-```
-C_sh ≈ 2 pF/foot × 3 = 6 pF
-C_mut ≈ 8 pF (from FEMM)
-r = 8/6 = 1.33
-
-Required for -45°: r ≤ 0.207
-Actual: r = 1.33
-
-1.33 > 0.207 → Cannot achieve -45°!
-φ_Z,min = -74.2° (actual minimum)
-```
-
-**Scenario 2: 5-foot spark, large topload**
-```
-C_sh ≈ 2 pF/foot × 5 = 10 pF
-C_mut ≈ 12 pF (larger topload)
-r = 12/10 = 1.2
-
-1.2 > 0.207 → Cannot achieve -45°!
-φ_Z,min = -71.6° (actual minimum)
-```
-
-**Scenario 3: 6-foot spark, small topload**
-```
-C_sh ≈ 2 pF/foot × 6 = 12 pF
-C_mut ≈ 6 pF (minimal topload)
-r = 6/12 = 0.5
-
-0.5 > 0.207 → Still cannot achieve -45°!
-φ_Z,min = -60° (actual minimum)
-```
-
-**The pattern:** Typical Tesla coils have r = 0.5 to 2.5, all well above the critical 0.207 threshold.
-
-### When CAN You Achieve -45°?
-
-You would need an extremely unusual geometry:
-```
-If C_sh = 10 pF (5-foot spark)
-Required: C_mut ≤ 0.207 × 10 = 2.07 pF
-
-This implies an extremely small topload with a very long spark!
-```
-
-Such configurations are rare because:
-1. Small topload = lower voltage capability
-2. Lower voltage = harder to initiate long sparks
-3. Contradictory requirements for practical operation
-
-## What Should You Target Instead?
-
-### Two Different Optimal Resistances
-
-There are actually **two** different optimal resistance values with different purposes:
-
-**1. R_opt_phase:** Minimizes |φ_Z| (most resistive phase angle)
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Achieves: φ_Z = φ_Z,min = -atan(2√[r(1+r)])
-```
-
-**2. R_opt_power:** Maximizes power transfer to the load
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-
-Achieves: Maximum real power dissipation
-```
-
-**Important relationship:**
-```
-R_opt_power < R_opt_phase (always!)
-
-Specifically: R_opt_power = R_opt_phase / √(1 + r)
-```
-
-### Which One Should You Use?
-
-**For Tesla coil sparks: Use R_opt_power!**
-
-**Why?**
-1. Sparks need **power** to grow (energy per meter)
-2. Maximum power = fastest growth = longest sparks
-3. The "hungry streamer" naturally seeks R_opt_power
-4. Phase angle is a consequence, not a goal
-
-**The -45° target is a red herring!** It doesn't maximize spark length or performance.
-
-## Worked Example: Comparing the Two Optima
-
-**Given:**
-- f = 200 kHz → ω = 1.257×10⁶ rad/s
-- C_mut = 8 pF
-- C_sh = 6 pF
-- r = 8/6 = 1.333
-
-**Calculate both optimal resistances:**
-
-**R_opt_power:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
- = 1 / [1.257×10⁶ × (8 + 6)×10⁻¹²]
- = 1 / [1.257×10⁶ × 14×10⁻¹²]
- = 1 / (17.60×10⁻⁶)
- = 56.8 kΩ
-```
-
-**R_opt_phase:**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 1 / [1.257×10⁶ × √(8 × 14)×10⁻¹²]
- = 1 / [1.257×10⁶ × 10.58×10⁻¹²]
- = 1 / (13.30×10⁻⁶)
- = 75.2 kΩ
-```
-
-**Comparison:**
-```
-R_opt_power = 56.8 kΩ → Maximizes power transfer
-R_opt_phase = 75.2 kΩ → Minimizes |φ_Z| (= -74.2°)
-
-Ratio: R_opt_phase / R_opt_power = 75.2 / 56.8 = 1.32 = √(1 + r) ✓
-```
-
-**What phase angle at R_opt_power?**
-Using the admittance formulas with R = 56.8 kΩ would give φ_Z ≈ -78° (slightly more capacitive than the minimum -74.2°, but delivers more power!)
-
-## The Bottom Line
-
-**Common misconception:**
-"Spark resistance should equal capacitive reactance for -45° phase angle."
-
-**Why it's wrong:**
-1. **Topology prevents it:** r > 0.207 for typical geometries
-2. **Wrong optimization target:** Should maximize power, not minimize |φ_Z|
-3. **Ignores self-optimization:** Plasma adjusts to R_opt_power naturally
-
-**What to do instead:**
-1. Calculate R_opt_power = 1/[ω(C_mut + C_sh)]
-2. Expect φ_Z ≈ -60° to -80° (more capacitive than -45°)
-3. Accept this is optimal for spark growth
-4. Don't worry about achieving -45°!
-
-## Key Takeaways
-
-- **-45° target:** Historical artifact from RF engineering
-- **Usually impossible:** Requires r ≤ 0.207, but typical coils have r = 0.5 to 2.5
-- **Two optima:** R_opt_phase (most resistive) vs R_opt_power (maximum power)
-- **Use R_opt_power:** Maximizes spark growth and length
-- **Expect highly capacitive:** φ_Z ≈ -60° to -80° is normal and optimal
-- **Don't chase -45°:** It's neither achievable nor desirable for most coils
-
-## Practice
-
-{exercise:fund-ex-06}
-
-**Problem 1:** For a coil with C_mut = 10 pF, C_sh = 8 pF, f = 180 kHz, calculate both R_opt_power and R_opt_phase. What is their ratio?
-
-**Problem 2:** A coil has r = 1.5. Can it achieve -45°? If not, what is φ_Z,min? Calculate the ratio R_opt_phase / R_opt_power and verify it equals √(1+r).
-
-**Problem 3:** Someone claims they achieved -45° on their Tesla coil. They measured C_sh = 8 pF for a 4-foot spark. What is the maximum C_mut their topload could have if this claim is true? Is this realistic?
-
----
-
-**Next Lesson:** [The Measurement Port](07-measurement-port.md)
diff --git a/spark-lessons/lessons/01-fundamentals/07-measurement-port.md b/spark-lessons/lessons/01-fundamentals/07-measurement-port.md
deleted file mode 100644
index 5640809..0000000
--- a/spark-lessons/lessons/01-fundamentals/07-measurement-port.md
+++ /dev/null
@@ -1,258 +0,0 @@
----
-id: fund-07
-title: "The Measurement Port and Why V_top/I_base is Wrong"
-section: "Fundamentals"
-difficulty: "intermediate"
-estimated_time: 20
-prerequisites: ["fund-01", "fund-02"]
-objectives:
- - Understand what displacement current is and why it matters
- - Recognize why V_top/I_base gives incorrect impedance
- - Identify all current paths in a Tesla coil system
- - Learn the correct measurement port definition
- - Calculate power using the correct method
-tags: ["measurement", "displacement-current", "power", "troubleshooting"]
----
-
-# The Measurement Port and Why V_top/I_base is Wrong
-
-## Introduction
-
-One of the most common mistakes in Tesla coil analysis is using V_top/I_base to calculate spark impedance. This seems logical - measure the voltage at the top and the current at the base - but it gives completely wrong results. This lesson explains why and shows the correct approach.
-
-## The Displacement Current Problem
-
-### What is Displacement Current?
-
-**Displacement current** flows through capacitances, not through physical conductors. It's given by:
-```
-I_displacement = jωC × V
-```
-
-**Key insight:** At AC, capacitors conduct current even though no charge physically crosses the dielectric!
-
-**For Tesla coils:**
-- Every turn of the secondary has capacitance to ground
-- Higher frequency → larger displacement current (proportional to ω)
-- These currents return to ground through the secondary base
-
-### Multiple Current Paths in a Tesla Coil
-
-A Tesla coil has **many** current paths returning to ground:
-
-**1. Spark current** (what we want to measure)
-```
-I_spark: From topload → through spark → remote ground → back to secondary base
-```
-
-**2. Displacement currents along secondary**
-```
-I_displacement: From each turn → through C_turn_to_ground → to ground → base
-Sum of all displacement currents: I_displacement = Σ(jωC_turn × V_turn)
-```
-
-**3. Primary-secondary coupling**
-```
-I_coupling: Displacement current through C_ps (primary-to-secondary capacitance)
-Part of transformer action
-```
-
-**4. Environmental coupling**
-```
-I_environment: Displacement currents to nearby objects, walls, strike ring
-Any grounded conductor near the secondary
-```
-
-**Total current at secondary base:**
-```
-I_base = I_spark + I_displacement + I_coupling + I_environment
-```
-
-**The problem:** Only I_spark goes through the spark! The other currents are parasitic paths that don't tell us about spark behavior.
-
-### Why V_top/I_base is Wrong
-
-```
-Z_apparent = V_top / I_base
-
-But I_base >> I_spark (often 3-5× larger!)
-
-Therefore: Z_apparent << Z_spark (impedance appears much lower than actual)
-```
-
-**Consequences:**
-- **Underestimate impedance:** Think load is more resistive than it is
-- **Overestimate power:** Calculate far too much power to spark
-- **Wrong optimization:** Make decisions based on incorrect data
-- **Model mismatch:** Can't reconcile measurements with theory
-
-
-
-**Diagram description:**
-- **RED path:** Spark current (I_spark) - the one we want
-- **BLUE paths:** Displacement currents along secondary (I_displacement)
-- **GREEN path:** Primary-secondary coupling current (I_coupling)
-- **YELLOW paths:** Environmental coupling currents (I_environment)
-- **At base:** All paths converge: I_base = sum of all currents
-
-**Key insight box:** "I_base ≠ I_spark! Cannot use V_top/I_base for spark impedance!"
-
-## The Correct Measurement Port
-
-**Definition:** The **topload port** is the two-terminal reference between topload and ground.
-
-```
-Port definition:
- Terminal 1: Topload (high voltage)
- Terminal 2: Ground reference (0V)
-```
-
-**Correct impedance:**
-```
-Z_spark = V_top / I_spark
-
-where I_spark is the current ONLY through the spark path
-```
-
-**Correct power:**
-```
-P = 0.5 × Re{V_top × I_spark*}
-P = 0.5 × |V_top| × |I_spark| × cos(φ_Z)
-```
-
-### Methods to Measure I_spark Correctly
-
-**Method 1: Separate return path measurement**
-- Run spark ground return through isolated conductor
-- Measure current with Rogowski coil or current transformer
-- Only captures I_spark, excludes parasitic currents
-
-**Method 2: Circuit modeling**
-- Know V_top (measure with voltage probe/antenna)
-- Calculate I_spark from circuit model using component values
-- Use admittance formulas from Lesson 3
-
-**Method 3: Thévenin extraction**
-- Characterize coil as Thévenin equivalent (covered in Part 2)
-- Predict load current from Z_th and V_th
-- Most accurate for design work
-
-## Worked Example: Correct vs Incorrect Power Calculation
-
-**Given:**
-- V_top = 300 kV peak
-- I_base (measured at secondary base) = 5 A peak
-- I_spark (actual spark current) = 1.5 A peak
-- Spark impedance phase: φ_Z = -70°
-
-**Find:** Power using incorrect method, power using correct method
-
-**Solution:**
-
-### Incorrect Method: Using V_top/I_base
-
-```
-Z_apparent = V_top / I_base = 300 kV / 5 A = 60 kΩ
-
-This is NOT the spark impedance!
-
-If we naively calculated power:
-P_wrong = 0.5 × 300 kV × 5 A × cos(-70°)
- = 0.5 × 1500 kW × 0.342
- = 257 kW
-
-This is way too high!
-```
-
-### Correct Method: Using Actual Spark Current
-
-```
-I_spark = 1.5 A peak
-
-Real spark impedance:
-Z_spark = V_top / I_spark = 300 kV / 1.5 A = 200 kΩ
-
-Power:
-P_correct = 0.5 × V_top × I_spark × cos(φ_Z)
- = 0.5 × 300 kV × 1.5 A × cos(-70°)
- = 0.5 × 450 kW × 0.342
- = 77 kW
-
-Or using resistance directly:
-R = |Z| × cos(φ_Z) = 200 kΩ × 0.342 = 68.4 kΩ
-P = 0.5 × I² × R = 0.5 × 1.5² × 68.4 kΩ = 77 kW ✓
-```
-
-### Error Analysis
-
-```
-P_wrong / P_correct = 257 / 77 = 3.3×
-
-The incorrect method overestimates power by 330%!
-```
-
-**Impedance error:**
-```
-Z_apparent = 60 kΩ (wrong)
-Z_spark = 200 kΩ (correct)
-
-Ratio: 200/60 = 3.3× (impedance underestimated)
-```
-
-**Why the same ratio?** Because I_base/I_spark = 5/1.5 = 3.3× - the displacement currents are 3.3× larger than the spark current in this example!
-
-## Why Displacement Current Increases with Frequency
-
-From the capacitor current equation:
-```
-I_C = jωC × V
-
-|I_C| = ω × C × |V| = 2πf × C × |V|
-```
-
-**Implication:** If frequency doubles, displacement current doubles!
-
-**For Tesla coils:**
-- Higher frequency operation → larger displacement currents
-- I_base becomes increasingly dominated by parasitics
-- V_top/I_base becomes even more wrong at high frequency
-- 200 kHz vs 400 kHz: displacement current 2× larger at 400 kHz
-
-**This is why measurement port definition is critical for comparison across different coils.**
-
-## Common Symptoms of Using I_base
-
-If you're using I_base incorrectly, you'll see:
-
-1. **Impedance too low:** Calculate 30-60 kΩ when should be 150-250 kΩ
-2. **Power too high:** Predict hundreds of kW when actual is tens of kW
-3. **Can't match models:** Circuit simulations disagree with "measurements"
-4. **Phase angle confusion:** Measured phase doesn't match expected
-5. **Efficiency paradox:** Calculate >100% efficiency (impossible!)
-
-**If you see these symptoms, check your measurement method!**
-
-## Key Takeaways
-
-- **I_base includes multiple current paths:** spark + displacement + coupling + environment
-- **Displacement current:** I = jωC×V, proportional to frequency
-- **V_top/I_base is wrong:** Gives impedance too low, power too high
-- **Correct port:** Topload-to-ground with I_spark only
-- **Typical error:** 3-5× underestimate of impedance
-- **Frequency dependence:** Displacement current ∝ ω, problem worse at high frequency
-
-## Practice
-
-{exercise:fund-ex-07}
-
-**Problem 1:** A simulation shows V_top = 250 kV, I_base = 3.5 A, but the spark circuit model predicts Z_spark = 180 kΩ. Calculate the actual spark current and power (assume φ_Z = -72°).
-
-**Problem 2:** Explain why displacement current is proportional to frequency (ω). If frequency doubles from 200 kHz to 400 kHz, what happens to I_displacement?
-
-**Problem 3:** An experimenter measures I_base = 4 A and calculates Z = V_top/I_base = 75 kΩ. Another measurement with a Rogowski coil on the spark return path shows I_spark = 1.2 A. What is the true spark impedance? What fraction of I_base is parasitic displacement current?
-
-**Problem 4:** A coil operates at 300 kV with Z_spark = 200 kΩ, φ_Z = -68°. Calculate the correct spark power. If someone incorrectly uses I_base = 4 A instead of the correct I_spark, what power would they calculate? What is the percentage error?
-
----
-
-**Next Lesson:** [Review and Exercises](08-review-exercises.md)
diff --git a/spark-lessons/lessons/01-fundamentals/08-review-exercises.md b/spark-lessons/lessons/01-fundamentals/08-review-exercises.md
deleted file mode 100644
index 03c012d..0000000
--- a/spark-lessons/lessons/01-fundamentals/08-review-exercises.md
+++ /dev/null
@@ -1,334 +0,0 @@
----
-id: fund-08
-title: "Part 1 Review and Integration"
-section: "Fundamentals"
-difficulty: "intermediate"
-estimated_time: 45
-prerequisites: ["fund-01", "fund-02", "fund-03", "fund-04", "fund-05", "fund-06", "fund-07"]
-objectives:
- - Review all fundamental concepts from Part 1
- - Apply concepts in an integrated example problem
- - Verify understanding through checkpoint quiz
- - Prepare for Part 2 optimization topics
-tags: ["review", "integration", "checkpoint", "summary"]
----
-
-# Part 1 Review and Integration
-
-## Introduction
-
-Congratulations on completing the fundamentals! This lesson reviews key concepts, provides an integration exercise that combines everything you've learned, and includes a checkpoint quiz to verify your understanding before moving to Part 2.
-
-## Concepts Checklist
-
-Before proceeding to Part 2, ensure you understand:
-
-### Circuit Fundamentals
-- [ ] Difference between peak and RMS values
-- [ ] Complex number representation: rectangular (R+jX) and polar (|Z|∠φ)
-- [ ] Power calculation: P = 0.5 × Re{V × I*} with peak phasors
-- [ ] Impedance Z = R + jX and admittance Y = G + jB
-- [ ] Relationship: Y = 1/Z, and φ_Y = -φ_Z
-
-### Capacitances
-- [ ] Physical meaning of capacitance (charge storage)
-- [ ] Self-capacitance vs mutual capacitance
-- [ ] Shunt capacitance C_sh ≈ 2 pF/foot for sparks
-- [ ] Both C_mut and C_sh exist simultaneously
-
-### Circuit Topology
-- [ ] Spark circuit: (R || C_mut) in series with C_sh
-- [ ] Topload port as measurement reference (topload-to-ground)
-- [ ] Why V_top/I_base is incorrect
-
-### Admittance Analysis
-- [ ] Advantages of Y for parallel circuits
-- [ ] Formula: Y = [(G+jB₁)×jB₂]/[G+j(B₁+B₂)]
-- [ ] Extracting Re{Y} and Im{Y}
-- [ ] Converting Y ↔ Z
-
-### Phase Angles
-- [ ] φ_Z = atan(X/R) for impedance
-- [ ] Negative φ_Z means capacitive
-- [ ] The -45° "balanced" condition: R = |X|
-- [ ] Typical sparks: φ_Z ≈ -55° to -75° (more capacitive than -45°)
-
-### Topological Constraints
-- [ ] φ_Z,min = -atan(2√[r(1+r)]) where r = C_mut/C_sh
-- [ ] Critical ratio r = 0.207 for -45°
-- [ ] Most Tesla coils cannot achieve -45°
-- [ ] R_opt_phase minimizes |φ_Z|, R_opt_power maximizes power
-
-### Measurement
-- [ ] Displacement current in secondary
-- [ ] I_base = I_spark + I_displacement + I_coupling + I_environment
-- [ ] V_top/I_base gives wrong impedance (too low)
-- [ ] Correct port: topload-to-ground with I_spark only
-
-### Spark Physics (Qualitative)
-- [ ] Streamers: thin, fast, cold, high R, branched
-- [ ] Leaders: thick, slower, hot, low R, straighter
-- [ ] Need both voltage (E-field) and power (energy/time)
-- [ ] "Hungry streamer": plasma self-optimizes R
-
-## Integration Exercise: Putting It All Together
-
-**Scenario:** You have a Tesla coil operating at 180 kHz with a 2-foot spark.
-
-**Given data:**
-- C_mut = 7 pF (from FEMM)
-- Assume R = 75 kΩ (plasma resistance)
-- Estimate C_sh using empirical rule
-
-**Tasks:**
-1. Calculate ω, B₁, B₂, G
-2. Calculate Y_total (real and imaginary parts)
-3. Convert to Z_total (magnitude and phase)
-4. Calculate φ_Z and interpret (is it more or less capacitive than -45°?)
-5. If V_top = 300 kV peak, calculate power dissipated
-
-**Work through this problem completely before checking the solution below.**
-
----
-
-### Integration Exercise Solution
-
-**Step 1:** Calculate C_sh
-```
-C_sh ≈ 2 pF/foot × 2 feet = 4 pF
-```
-
-**Step 2:** Calculate ω and component values
-```
-ω = 2πf = 2π × 180×10³ = 1.131×10⁶ rad/s
-
-G = 1/R = 1/(75×10³) = 13.33 μS
-B₁ = ωC_mut = 1.131×10⁶ × 7×10⁻¹² = 7.92 μS
-B₂ = ωC_sh = 1.131×10⁶ × 4×10⁻¹² = 4.52 μS
-```
-
-**Step 3:** Calculate Y_total
-```
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 13.33 × (4.52)² / [13.33² + (7.92+4.52)²]
- = 13.33 × 20.43 / [177.7 + 154.4]
- = 272.3 / 332.1
- = 0.82 μS
-
-Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
- = 4.52 × [177.7 + 7.92×12.44] / 332.1
- = 4.52 × [177.7 + 98.5] / 332.1
- = 4.52 × 276.2 / 332.1
- = 3.76 μS
-
-Y_total = 0.82 + j3.76 μS
-```
-
-**Step 4:** Convert to impedance
-```
-|Y| = √(0.82² + 3.76²) = √(0.67 + 14.14) = √14.81 = 3.85 μS
-
-|Z| = 1/|Y| = 1/(3.85×10⁻⁶) = 260 kΩ
-
-φ_Y = atan(3.76/0.82) = atan(4.59) = 77.7°
-φ_Z = -φ_Y = -77.7°
-
-Z_total = 260 kΩ ∠-77.7°
-
-In rectangular:
-R_eq = 260 × cos(-77.7°) = 260 × 0.213 = 55.4 kΩ
-X_eq = 260 × sin(-77.7°) = 260 × (-0.977) = -254 kΩ
-
-Z_total = 55.4 - j254 kΩ
-```
-
-**Step 5:** Interpret phase
-```
-φ_Z = -77.7° is more capacitive than -45° (larger magnitude)
-Ratio: |X|/R = 254/55.4 = 4.6
-Capacitive reactance is 4.6× the resistance
-Very capacitive load!
-```
-
-**Step 6:** Calculate power
-```
-Current: I = V/Z = (300 kV)/(260 kΩ) = 1.15 A peak
-
-Power: P = 0.5 × V × I × cos(φ_Z)
- = 0.5 × 300×10³ × 1.15 × cos(-77.7°)
- = 0.5 × 345×10³ × 0.213
- = 36.7 kW
-
-Alternative: P = 0.5 × I² × R_eq
- = 0.5 × 1.15² × 55.4×10³
- = 0.5 × 1.32 × 55.4×10³
- = 36.6 kW ✓ (checks!)
-```
-
-**Result:** 36.7 kW dissipated in the spark plasma.
-
-## Checkpoint Quiz
-
-Answer these questions to verify your understanding:
-
-**Question 1:** What is the relationship between peak and RMS voltage? If V_peak = 100 kV, what is V_RMS?
-
-**Question 2:** Write the power formula using peak phasors. Why is there a factor of 0.5?
-
-**Question 3:** For a capacitor, why is X negative but B positive?
-
-**Question 4:** Draw the circuit topology for a spark (show C_mut, R, C_sh).
-
-**Question 5:** What is the empirical rule for C_sh? If a spark is 4 feet long, estimate C_sh.
-
-**Question 6:** The admittance phase angle θ_Y = +60°. What is the impedance phase angle φ_Z?
-
-**Question 7:** An impedance has φ_Z = -30°. Is this inductive or capacitive?
-
-**Question 8:** Why is V_top/I_base not the correct impedance measurement?
-
-**Question 9:** Describe the difference between streamers and leaders (two key differences).
-
-**Question 10:** Explain the "hungry streamer" concept in one sentence.
-
-### Quiz Answers
-
-
-Click to reveal answers
-
-**Answer 1:** V_RMS = V_peak/√2. For V_peak = 100 kV, V_RMS = 100/√2 ≈ 70.7 kV
-
-**Answer 2:** P = 0.5 × Re{V × I*}. The 0.5 factor comes from time-averaging cos²(ωt) over a full cycle.
-
-**Answer 3:** For capacitors, reactance X_C = -1/(ωC) is negative, but susceptance B_C = ωC is positive. The sign conventions are opposite for impedance vs admittance.
-
-**Answer 4:**
-```
- Topload
- |
- [C_mut]
- |
- +----+----+
- | |
- [R] [C_sh]
- | |
- GND------GND
-```
-
-**Answer 5:** C_sh ≈ 2 pF/foot. For 4 feet: C_sh ≈ 8 pF.
-
-**Answer 6:** φ_Z = -θ_Y = -60°
-
-**Answer 7:** Capacitive (negative φ_Z indicates capacitive behavior)
-
-**Answer 8:** I_base includes displacement currents from the entire secondary, plus coupling currents and environmental currents. Only I_spark flows through the spark. V_top/I_base underestimates impedance because I_base > I_spark.
-
-**Answer 9:** (Any two of these)
-- Streamers: thin (10-100 μm), fast (~10⁶ m/s), cold (~1000 K), high R, branched
-- Leaders: thick (mm-cm), slower (~10³ m/s), hot (5000-20000 K), low R, straighter
-
-**Answer 10:** Plasma actively adjusts its conductivity to maximize power extraction from the circuit, naturally seeking R ≈ R_opt_power.
-
-
-
-## Key Formulas Summary
-
-**Admittance components:**
-```
-G = 1/R
-B₁ = ωC_mut
-B₂ = ωC_sh
-```
-
-**Total admittance:**
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-```
-
-**Conversion to impedance:**
-```
-|Z| = 1/|Y|
-φ_Z = -φ_Y
-```
-
-**Topological constraint:**
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-where r = C_mut/C_sh
-```
-
-**Optimal resistances:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-**Power:**
-```
-P = 0.5 × Re{V × I*}
-P = 0.5 × |V| × |I| × cos(φ_Z)
-P = 0.5 × |I|² × R
-```
-
-**Empirical rule:**
-```
-C_sh ≈ 2 pF/foot
-```
-
-## Common Mistakes to Avoid
-
-1. **Using RMS instead of peak values** - Always use peak for this framework
-2. **Using V_top/I_base** - Includes displacement currents, gives wrong Z
-3. **Expecting -45°** - Usually impossible due to topological constraint
-4. **Confusing R_opt_power and R_opt_phase** - Use R_opt_power for spark growth
-5. **Forgetting sign conventions** - X < 0 but B > 0 for capacitors
-6. **Ignoring phase in power calculations** - Must include cos(φ_Z) factor
-
-## Preview of Part 2
-
-In Part 2: Optimization and Power Transfer, we'll explore:
-
-- **Two critical resistances:** Detailed derivation and comparison of R_opt_power and R_opt_phase
-- **Thévenin method:** Properly characterizing the Tesla coil as V_th and Z_th
-- **Power optimization:** How the "hungry streamer" finds R_opt_power
-- **Measurements:** Extracting spark parameters from real coils using Q and ringdown
-- **Load line analysis:** Predicting performance with any load
-
-These concepts build directly on the circuit analysis and phase relationships you've mastered in Part 1.
-
-## Practice Problems
-
-{exercise:fund-ex-08}
-
-**Comprehensive Problem 1:**
-A Tesla coil operates at 220 kHz with a 3.5-foot spark. FEMM analysis gives C_mut = 9 pF. Assume R = 60 kΩ.
-- (a) Calculate C_sh, ω, G, B₁, B₂
-- (b) Calculate Y_total and Z_total
-- (c) Find φ_Z and compare to -45°
-- (d) Calculate r and φ_Z,min
-- (e) If V_top = 350 kV, find power dissipated
-
-**Comprehensive Problem 2:**
-Two coils have identical frequency (200 kHz) and total capacitance (C_mut + C_sh = 15 pF).
-- Coil A: C_mut = 10 pF, C_sh = 5 pF
-- Coil B: C_mut = 5 pF, C_sh = 10 pF
-- (a) Calculate r for both coils
-- (b) Calculate φ_Z,min for both
-- (c) Which can achieve more resistive phase?
-- (d) Calculate R_opt_power and R_opt_phase for both
-
-**Measurement Problem:**
-An experimenter measures V_top = 280 kV and I_base = 4.2 A. A separate measurement with a current probe on the spark return path shows I_spark = 1.3 A. The spark is 4 feet long.
-- (a) What is the true spark impedance?
-- (b) What would they calculate using V_top/I_base (incorrect)?
-- (c) What percentage of I_base is parasitic displacement current?
-- (d) Calculate the correct spark power (assume φ_Z = -68°)
-
----
-
-**Congratulations on completing Part 1: Fundamentals!**
-
-You now have a solid foundation in Tesla coil spark circuit modeling. You understand the topology, can calculate impedances, recognize the phase constraints, and know how to measure correctly. You're ready to move on to optimization and power transfer in Part 2.
-
-**Next:** [Part 2: Optimization and Power Transfer](../../02-optimization/01-introduction.md)
diff --git a/spark-lessons/lessons/01-fundamentals/README.md b/spark-lessons/lessons/01-fundamentals/README.md
deleted file mode 100644
index 1944eec..0000000
--- a/spark-lessons/lessons/01-fundamentals/README.md
+++ /dev/null
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-# Part 1: Fundamentals
-
-## Overview
-
-This section provides the foundational knowledge for Tesla coil spark modeling. You'll learn the circuit theory, analysis techniques, and key concepts needed to understand and predict spark behavior.
-
-## Lessons
-
-1. **[Introduction to Tesla Coil Spark Modeling](01-introduction.md)** (20 min)
- - AC circuit fundamentals review
- - Peak vs RMS values
- - Complex numbers and phasors
- - Power calculations with peak phasors
-
-2. **[The Basic Spark Circuit Model](02-basic-circuit-model.md)** (25 min)
- - Physical meaning of capacitance
- - Mutual capacitance (C_mut) vs shunt capacitance (C_sh)
- - The 2 pF/foot empirical rule
- - Correct circuit topology: (R || C_mut) in series with C_sh
-
-3. **[Admittance Analysis](03-admittance-analysis.md)** (30 min)
- - Why use admittance for parallel circuits
- - Deriving the total admittance formula
- - Calculating Re{Y} and Im{Y}
- - Converting between Y and Z
-
-4. **[Phase Angles and Their Meaning](04-phase-angles.md)** (20 min)
- - Impedance phase φ_Z vs admittance phase φ_Y
- - Physical interpretation of phase angles
- - The "famous -45°" and why it's special
- - Typical spark phase angles: -55° to -75°
-
-5. **[The Topological Phase Constraint](05-phase-constraint.md)** (25 min)
- - What is a topological constraint?
- - Deriving φ_Z,min = -atan(2√[r(1+r)])
- - The critical ratio r = 0.207
- - Why -45° is usually impossible
-
-6. **[Why Not -45 Degrees?](06-why-not-45-degrees.md)** (15 min)
- - Historical origin of the -45° target
- - Why it's often impossible for Tesla coils
- - R_opt_phase vs R_opt_power
- - What to target instead
-
-7. **[The Measurement Port](07-measurement-port.md)** (20 min)
- - Understanding displacement current
- - Why V_top/I_base gives wrong impedance
- - Multiple current paths in a Tesla coil
- - Correct measurement methods
-
-8. **[Review and Integration](08-review-exercises.md)** (45 min)
- - Complete concepts checklist
- - Integration exercise combining all topics
- - Checkpoint quiz
- - Preview of Part 2
-
-## Total Time
-
-Approximately 3-4 hours for complete mastery
-
-## Learning Outcomes
-
-After completing Part 1, you will be able to:
-
-- Use peak values and phasor notation correctly
-- Model a spark with proper circuit topology
-- Calculate impedance using admittance formulas
-- Understand phase angle constraints and their physical meaning
-- Recognize why -45° is rarely achievable
-- Measure spark impedance correctly
-- Avoid common measurement pitfalls
-- Apply integrated circuit analysis to real Tesla coil scenarios
-
-## Prerequisites
-
-- Basic algebra and trigonometry
-- Familiarity with sine waves and AC circuits (helpful but not required)
-- Scientific calculator or Python/MATLAB for calculations
-
-## Key Formulas
-
-**Admittance:**
-```
-Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
-where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh
-```
-
-**Topological constraint:**
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-where r = C_mut/C_sh
-```
-
-**Empirical rule:**
-```
-C_sh ≈ 2 pF/foot
-```
-
-**Power:**
-```
-P = 0.5 × Re{V × I*}
-```
-
-## Image Placeholders
-
-The following images should be created for the assets folder:
-
-1. `field-lines-capacitances.png` - C_mut and C_sh field lines
-2. `geometry-to-circuit.png` - 3D geometry to circuit schematic
-3. `complex-plane-admittance.png` - Y and Z on complex planes
-4. `phase-angle-visualization.png` - Phase angles on impedance plane
-5. `phase-constraint-graph.png` - φ_Z,min vs r graph
-6. `current-paths-diagram.png` - Multiple current paths in Tesla coil
-
-## Next Steps
-
-After mastering Part 1, proceed to:
-
-**[Part 2: Optimization and Power Transfer](../02-optimization/README.md)**
-
-Topics include:
-- R_opt_power and R_opt_phase derivations
-- Thévenin equivalent method
-- The "hungry streamer" self-optimization
-- Q measurements and ringdown analysis
diff --git a/spark-lessons/lessons/02-optimization/01-two-resistances.md b/spark-lessons/lessons/02-optimization/01-two-resistances.md
deleted file mode 100644
index 58fbad0..0000000
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----
-id: opt-01
-title: "The Two Critical Resistances"
-section: "Optimization & Simulation"
-difficulty: "intermediate"
-estimated_time: 35
-prerequisites: ["fund-08"]
-objectives:
- - Derive and understand R_opt_phase for minimum phase angle
- - Derive and understand R_opt_power for maximum power transfer
- - Compare the two resistances and their physical meanings
- - Calculate phase angles at different operating points
-tags: ["optimization", "impedance", "power-transfer", "phase-angle"]
----
-
-# The Two Critical Resistances
-
-In spark gap modeling, we encounter two fundamentally different optimization criteria that lead to two different "optimal" resistance values. Understanding the distinction between these is critical for both analysis and practical coil operation.
-
-## The Topological Phase Constraint
-
-Before we dive into the two resistances, we need to understand a fundamental limitation imposed by circuit topology.
-
-### What is a Topological Constraint?
-
-**Definition:** A limitation imposed by the **structure** of the circuit itself, independent of component values.
-
-**Example:** A series RLC circuit can only have impedance phase between -90° (pure capacitive) and +90° (pure inductive). You cannot achieve φ_Z = +120° no matter what component values you choose. This is a topological constraint.
-
-**For spark circuits:** The specific arrangement (R||C_mut) in series with C_sh creates a fundamental limit on how resistive the impedance can appear.
-
-### Deriving the Minimum Phase Angle
-
-From Part 1 fundamentals, we have the spark admittance:
-
-```
-Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
-
-where:
- G = 1/R (conductance)
- B₁ = ωC_mut (mutual capacitance susceptance)
- B₂ = ωC_sh (sheath capacitance susceptance)
-```
-
-The impedance phase is:
-```
-φ_Z = atan(-Im{Y}/Re{Y})
-```
-
-**Question:** For fixed C_mut and C_sh, which R value minimizes |φ_Z| (makes the impedance most resistive)?
-
-**Mathematical result:** Taking the derivative ∂φ_Z/∂G = 0 and solving:
-
-```
-G_opt = ω√[C_mut(C_mut + C_sh)]
-
-Therefore:
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-At this resistance, the phase angle magnitude is minimized to:
-
-```
-φ_Z,min = -atan(2√[r(1 + r)])
-
-where r = C_mut/C_sh (capacitance ratio)
-```
-
-### The Critical Ratio r = 0.207
-
-Let's find when φ_Z,min = -45° is achievable:
-
-```
--45° = -atan(2√[r(1 + r)])
-tan(45°) = 1 = 2√[r(1 + r)]
-0.5 = √[r(1 + r)]
-0.25 = r(1 + r) = r + r²
-r² + r - 0.25 = 0
-
-Using quadratic formula:
-r = [-1 ± √(1 + 1)] / 2 = [-1 ± √2] / 2
-
-Taking positive root:
-r = (√2 - 1) / 2 ≈ 0.207
-```
-
-**Critical insight:**
-- If r < 0.207: Can achieve φ_Z = -45° (with appropriate R)
-- If r > 0.207: **Cannot achieve φ_Z = -45° no matter what R you choose!**
-- If r ≥ 0.207: φ_Z,min is more negative than -45°
-
-### Typical Tesla Coil Values
-
-**Large topload, short spark:**
-```
-C_mut = 10 pF, C_sh = 4 pF (2 feet)
-r = 10/4 = 2.5
-
-φ_Z,min = -atan(2√[2.5 × 3.5]) = -atan(2 × 2.96) = -atan(5.92) = -80.4°
-```
-
-**Small topload, long spark:**
-```
-C_mut = 6 pF, C_sh = 12 pF (6 feet)
-r = 6/12 = 0.5
-
-φ_Z,min = -atan(2√[0.5 × 1.5]) = -atan(2 × 0.866) = -atan(1.732) = -60.0°
-```
-
-**Common range:** r = 0.5 to 2.0, giving φ_Z,min ≈ -60° to -80°
-
-**Conclusion:** For most Tesla coil geometries, achieving -45° is **mathematically impossible**!
-
-## R_opt_phase: Closest to Resistive
-
-**Formula:**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-
-**Purpose:** Minimizes |φ_Z| to achieve φ_Z,min
-
-**Use case:** If you want the "most resistive-looking" impedance possible for your given capacitances.
-
-**Physical meaning:** This is the geometric mean of the capacitive reactances, representing the resistance that balances the phase contributions from C_mut and C_sh.
-
-## R_opt_power: Maximum Power Transfer
-
-**Different question:** Which R maximizes real power delivered to the spark for a given topload voltage?
-
-**Setup:** Fixed voltage source V_top, variable load resistance R
-
-**Power to load:**
-```
-P = 0.5 × |V_top|² × Re{Y(R)}
-```
-
-where Y(R) depends on R through G = 1/R.
-
-**Mathematical derivation:** Take ∂P/∂G = 0 and solve for G:
-
-After applying calculus (expanding Re{Y} and differentiating):
-
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-```
-
-**Simpler formula!** Just the total capacitance reactance, not a geometric mean.
-
-## Comparing the Two Resistances
-
-### Relationship
-
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-
-Since √(C_mut(C_mut + C_sh)) < (C_mut + C_sh):
-
-R_opt_power < R_opt_phase ALWAYS
-```
-
-**Numerical relationship:** For typical r = 0.5 to 2:
-```
-R_opt_power ≈ (0.5 to 0.7) × R_opt_phase
-```
-
-### Phase Angle at R_opt_power
-
-- Always more negative (more capacitive) than φ_Z,min
-- Typically φ_Z ≈ -55° to -75° at R_opt_power
-- More capacitive than R_opt_phase, but delivers more power
-
-**Key insight:** The impedance that transfers maximum power is NOT the same as the impedance with minimum phase angle!
-
-## Worked Example: Calculating Both Critical Resistances
-
-**Given:**
-- Frequency: f = 200 kHz → ω = 1.257×10⁶ rad/s
-- C_mut = 8 pF = 8×10⁻¹² F
-- C_sh = 6 pF = 6×10⁻¹² F
-
-**Find:** R_opt_phase, R_opt_power, and compare
-
-### Solution
-
-**Part 1: R_opt_phase**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
- = 1 / [1.257×10⁶ × √(8×10⁻¹² × 14×10⁻¹²)]
- = 1 / [1.257×10⁶ × √(112×10⁻²⁴)]
- = 1 / [1.257×10⁶ × 10.58×10⁻¹²]
- = 1 / (13.30×10⁻⁶)
- = 75.2 kΩ
-```
-
-**Part 2: R_opt_power**
-```
-C_total = C_mut + C_sh = 8 + 6 = 14 pF = 14×10⁻¹² F
-
-R_opt_power = 1 / (ωC_total)
- = 1 / (1.257×10⁶ × 14×10⁻¹²)
- = 1 / (17.60×10⁻⁶)
- = 56.8 kΩ
-```
-
-**Part 3: Comparison**
-```
-Ratio: R_opt_power / R_opt_phase = 56.8 / 75.2 = 0.755
-
-R_opt_power is 75.5% of R_opt_phase
-```
-
-**Part 4: Phase angle at R_opt_power**
-
-Calculate admittance with R = 56.8 kΩ:
-```
-G = 1/56800 = 17.61 μS
-B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06 μS
-B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54 μS
-
-Re{Y} = GB₂²/[G² + (B₁+B₂)²]
- = 17.61 × 56.85 / [310 + 309.8]
- = 1001.2 / 619.8
- = 1.615 μS
-
-Im{Y} = 7.54[310 + 176.9] / 619.8
- = 7.54 × 486.9 / 619.8
- = 5.928 μS
-
-φ_Y = atan(5.928/1.615) = atan(3.67) = 74.7°
-φ_Z = -74.7°
-```
-
-**Summary:**
-- R_opt_phase = 75.2 kΩ gives φ_Z = -74.2° (minimum)
-- R_opt_power = 56.8 kΩ gives φ_Z = -74.7° (slightly more capacitive)
-- Power is maximized at R_opt_power despite not having minimum phase
-- Difference is small: both are strongly capacitive
-
-## Visual Aid: Power vs Resistance Curves
-
-
-
-*Image shows two overlaid plots:*
-- *Top: Power vs R (bell curve peaking at R_opt_power = 56.8 kΩ)*
-- *Bottom: Phase angle vs R (minimum at R_opt_phase = 75.2 kΩ)*
-- *Key insight: The two optimal points do not coincide*
-
-**Key features:**
-- X-axis: R (kΩ), range 20 to 150, log scale
-- Power curve: Bell-shaped, peaks at R_opt_power
-- Phase curve: Rises from -90° (R→0), peaks at R_opt_phase, falls back
-- Vertical lines show the two different optimum points
-
-## Key Takeaways
-
-- **R_opt_phase = 1/[ω√(C_mut(C_mut + C_sh))]** minimizes phase angle magnitude
-- **R_opt_power = 1/[ω(C_mut + C_sh)]** maximizes power transfer
-- **R_opt_power < R_opt_phase** always (typically 50-75% of R_opt_phase)
-- Most Tesla coils operate with r > 0.207, making φ_Z = -45° impossible
-- The impedance must be strongly capacitive due to topological constraints
-- Power optimization and phase optimization are different goals with different solutions
-
-## Practice
-
-{exercise:opt-ex-01}
-
-**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF:
-Calculate R_opt_power and R_opt_phase.
-
-**Problem 2:** At 200 kHz, a spark has C_total = 12 pF. What is R_opt_power? If V_top = 400 kV, estimate the maximum deliverable power (assume R at optimal value).
-
-**Problem 3:** Prove algebraically that R_opt_power < R_opt_phase always (hint: compare 1/(C_mut+C_sh) with 1/√(C_mut(C_mut+C_sh))).
-
-**Problem 4:** A measurement shows φ_Z = -68° at the operating point. Is R likely above or below R_opt_phase? Above or below R_opt_power? Explain your reasoning.
-
-**Problem 5:** Calculate the capacitance ratio r and minimum achievable phase angle φ_Z,min for:
-(a) C_mut = 12 pF, C_sh = 8 pF
-(b) Can this circuit achieve -45°?
-
----
-**Next Lesson:** [The Hungry Streamer - Self-Optimization](02-hungry-streamer.md)
diff --git a/spark-lessons/lessons/02-optimization/02-hungry-streamer.md b/spark-lessons/lessons/02-optimization/02-hungry-streamer.md
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----
-id: opt-02
-title: "The Hungry Streamer - Self-Optimization"
-section: "Optimization & Simulation"
-difficulty: "advanced"
-estimated_time: 30
-prerequisites: ["opt-01", "fund-06"]
-objectives:
- - Understand the physical feedback loop between power and plasma conductivity
- - Trace the thermal-electrical evolution of a spark
- - Recognize when and why plasma self-optimizes to R_opt_power
- - Identify physical constraints that prevent optimization
-tags: ["plasma-physics", "self-optimization", "thermal-dynamics", "feedback"]
----
-
-# The Hungry Streamer - Self-Optimization
-
-One of the most remarkable features of spark plasmas is their ability to **self-adjust** their resistance to maximize power extraction from the coil. This phenomenon, often described by Steve Conner's principle of the "hungry streamer," is a consequence of fundamental plasma physics and thermal dynamics.
-
-## The Physical Feedback Loop
-
-Plasma conductivity changes dynamically with the power it receives, creating a feedback mechanism:
-
-### Step 1: More Power → Joule Heating
-
-```
-Heating rate: dT/dt ∝ I²R
-
-Higher current → faster heating
-```
-
-The plasma channel experiences resistive heating (Joule heating) from the current flowing through it. The heating rate is proportional to I²R, so higher currents lead to faster temperature rise.
-
-### Step 2: Higher Temperature → Ionization
-
-```
-Thermal ionization: fraction ∝ exp(-E_ionization / kT)
-
-Hotter plasma → more free electrons
-```
-
-As temperature increases, more air molecules have sufficient thermal energy to ionize. The ionization fraction follows a Boltzmann-like distribution, increasing exponentially with temperature once the thermal energy approaches the ionization energy (~13.6 eV for many atmospheric species).
-
-### Step 3: More Electrons → Higher Conductivity
-
-```
-σ = n_e × e × μ_e
-
-where:
- n_e = electron density
- μ_e = electron mobility
- e = elementary charge
-
-σ ∝ n_e ∝ exp(-E_ionization / kT)
-```
-
-Electrical conductivity is directly proportional to the free electron density. More ionization means more free charge carriers, which means higher conductivity.
-
-### Step 4: Higher Conductivity → Lower R
-
-```
-R = ρL/A = L/(σA)
-
-σ increases → R decreases
-```
-
-The resistance of the plasma channel is inversely proportional to conductivity. As the plasma heats up and becomes more conductive, its resistance drops.
-
-### Step 5: Changed R → New Circuit Behavior
-
-```
-New R changes Y_spark, power transfer changes:
-
-If R < R_opt_power: reducing R further DECREASES power
-If R > R_opt_power: reducing R INCREASES power
-```
-
-This is the crucial step. The circuit's power transfer characteristics depend on the load resistance. From our previous lesson, we know that power is maximized at R_opt_power.
-
-### Step 6: Stable Equilibrium at R ≈ R_opt_power
-
-```
-When R approaches R_opt_power:
-- Small decrease → power decreases → cooling → R rises
-- Small increase → power increases → heating → R falls
-- Negative feedback stabilizes at R_opt_power
-```
-
-**This creates a stable operating point!** The system naturally seeks the resistance value that maximizes power transfer through negative feedback.
-
-## Time Scales
-
-Understanding the time scales involved is critical to predicting when self-optimization occurs.
-
-### Thermal Response: ~0.1-1 ms for Thin Channels
-
-**Heat diffusion time:**
-```
-τ = d²/(4α)
-
-where:
- d = channel diameter
- α = thermal diffusivity ≈ 2×10⁻⁵ m²/s for air
-
-For d = 100 μm (thin streamer): τ ≈ 0.1 ms
-For d = 5 mm (thick leader): τ ≈ 300 ms
-```
-
-**Implications:**
-- Fast enough to track AC envelope (kHz modulation in QCW/burst mode)
-- Too slow to track RF oscillation (hundreds of kHz carrier)
-- The plasma "sees" the RMS or average power, not instantaneous RF cycles
-
-### Ionization Response: ~μs to ms
-
-**Recombination time varies with:**
-- Electron density (higher density → faster recombination)
-- Temperature (higher temperature → slower recombination)
-- Gas composition (different species have different rates)
-
-**Typical:** ~1-10 ms for atmospheric pressure air plasmas
-
-### Result: 0.1-10 ms Adjustment Time
-
-The plasma can adjust its resistance on timescales of 0.1-10 ms, allowing it to:
-- Track power delivery changes in burst mode or QCW operation
-- Respond to voltage variations
-- Seek optimal operating conditions dynamically
-
-## Physical Constraints
-
-While the feedback mechanism drives the plasma toward R_opt_power, physical limitations can prevent this optimization:
-
-### Lower Bound: R_min
-
-**Physical limit:**
-- Maximum conductivity limited by electron-ion collision frequency
-- Even fully ionized plasma has finite conductivity
-- Typical: R_min ≈ 1-10 kΩ for hot, dense leader channels
-
-**If R_opt_power < R_min:**
-- Plasma stuck at R_min (cannot achieve lower resistance)
-- Power transfer is suboptimal
-- Spark cannot extract as much power as theoretically possible
-
-### Upper Bound: R_max
-
-**Physical limit:**
-- Minimum conductivity of partially ionized gas
-- Cool plasma or weak ionization
-- Typical: R_max ≈ 100 kΩ to 100 MΩ for cool streamers
-
-**If R_opt_power > R_max:**
-- Plasma stuck at R_max (cannot achieve higher resistance)
-- Usually not the limiting factor in Tesla coils
-- More common with very weak discharges
-
-### Source Limitations
-
-**Insufficient voltage:**
-- Spark won't form at all if V_top < V_breakdown
-- No optimization possible without a spark
-
-**Insufficient current:**
-- Cannot heat plasma enough to reach R_opt_power
-- Spark remains in cool streamer regime
-- High resistance, low power transfer
-
-**Power supply impedance:**
-- If Z_source >> Z_spark, source impedance limits available power
-- The "hungry streamer" is starved by a weak source
-
-## When Optimization Fails
-
-Several scenarios prevent the plasma from reaching R_opt_power:
-
-### Source Too Weak
-
-**Scenario:** Available power insufficient to heat plasma
-
-**Result:**
-- Spark operates at whatever R it can sustain
-- Typically remains at high R (cool streamers)
-- Low power transfer, short sparks
-
-### Thermal Time Too Long
-
-**Scenario:** Burst mode with pulse width << thermal time constant
-
-**Example:** 50 μs pulses with τ_thermal = 0.5 ms
-
-**Result:**
-- Plasma cannot respond fast enough
-- Operates in transient regime
-- Does not reach steady-state R_opt_power
-
-### Branching
-
-**Scenario:** Multiple discharge paths from topload
-
-**Result:**
-- Available power divides among branches
-- No single branch gets enough power to optimize
-- Multiple weak streamers rather than one strong leader
-
-## Worked Example: Tracing Optimization Process
-
-**Scenario:** Spark initially forms with R = 200 kΩ (cold streamer). Circuit has R_opt_power = 60 kΩ. Let's trace the thermal-electrical evolution:
-
-### Initial State (t = 0)
-
-```
-R = 200 kΩ >> R_opt_power
-Power delivered: P_initial (suboptimal, low)
-Temperature: T_initial (cool, ~1000 K)
-Current: I_initial ≈ V_top / Z_total (low)
-```
-
-The spark has just formed. It's essentially a weakly ionized streamer with high resistance.
-
-### Early Phase (0 < t < 1 ms)
-
-```
-Current flows → Joule heating: dT/dt = I²R/c_p
-R is high → voltage division favorable → some heating occurs
-Temperature rises → ionization begins → n_e increases
-Conductivity σ ∝ n_e increases → R decreases
-R drops toward 150 kΩ
-```
-
-**What's happening:**
-- Even though R is far from optimal, some power flows
-- Joule heating warms the plasma channel
-- Thermal ionization begins to create more free electrons
-- Resistance starts to drop
-
-### Middle Phase (1 ms < t < 5 ms)
-
-```
-R approaches 100 kΩ range
-Now closer to R_opt_power → power transfer improves
-More power → faster heating → faster ionization
-Positive feedback: lower R → more power → lower R
-R drops rapidly: 100 kΩ → 80 kΩ → 70 kΩ → 65 kΩ
-```
-
-**What's happening:**
-- As R approaches R_opt_power, power transfer increases
-- Positive feedback accelerates the process
-- This is the "hungry" phase - the plasma eagerly draws more power
-- Temperature may reach 5000-10000 K (transition to leader)
-
-### Approach to Equilibrium (5 ms < t < 10 ms)
-
-```
-R approaches R_opt_power = 60 kΩ
-Power maximized at this R
-
-If R < 60 kΩ: power would decrease → cooling → R rises
-If R > 60 kΩ: power would increase → heating → R falls
-
-Negative feedback stabilizes around R ≈ 60 kΩ
-```
-
-**What's happening:**
-- Feedback changes from positive to negative near R_opt_power
-- System naturally seeks the stable equilibrium point
-- Small perturbations are self-correcting
-
-### Steady State (t > 10 ms)
-
-```
-R oscillates around 60 kΩ ± 10%
-Temperature stable at equilibrium (~8000-15000 K for leaders)
-Power maximized and stable
-Spark is "optimized"
-```
-
-**What's happening:**
-- Plasma has reached thermal and electrical equilibrium
-- Continuous power input balances radiative/convective losses
-- The spark maintains maximum power extraction
-
-## What If Physical Limits Intervene?
-
-**Example with R_min constraint:**
-
-```
-If R_opt_power = 30 kΩ but R_min = 50 kΩ (plasma physics limit):
- Plasma can only reach R = 50 kΩ (not optimal)
- Power is less than theoretical maximum
- Spark is "starved" - wants more current than physics allows
-```
-
-This can happen with very hot, dense plasmas where even full ionization cannot achieve the low resistance needed for optimization.
-
-## Steve Conner's Principle
-
-**The "Hungry Streamer" Concept:**
-
-A spark will adjust its resistance to extract maximum power from the source, subject to physical constraints. The plasma behaves as if it is "hungry" for energy and actively optimizes its impedance to feed that hunger.
-
-**Why this matters:**
-- Explains why measured spark resistance tends to cluster around R_opt_power
-- Justifies using R_opt_power as a design target
-- Helps predict spark behavior in different operating modes
-- Guides optimization of coil parameters
-
-## Key Takeaways
-
-- Plasma resistance is not fixed - it dynamically adjusts based on power
-- **Feedback loop:** Power → Heating → Ionization → Conductivity → R changes → Power changes
-- **Stable equilibrium at R ≈ R_opt_power** due to negative feedback
-- Time scales: 0.1-10 ms for thermal/ionization response
-- Physical constraints: R_min (hot plasma limit), R_max (cool plasma limit), source limitations
-- Burst mode with short pulses may not reach equilibrium
-- The "hungry streamer" actively seeks maximum power extraction
-
-## Practice
-
-{exercise:opt-ex-02}
-
-**Question 1:** Why does the optimization work? Why doesn't the plasma just pick a random R value and stay there?
-
-**Question 2:** In burst mode (short pulses, <100 μs), thermal time constants are longer than pulse duration. Would you expect the plasma to reach R_opt_power? Why or why not?
-
-**Question 3:** A coil produces sparks with measured R ≈ 20 kΩ, but calculations show R_opt_power = 80 kΩ. What might explain this discrepancy? (Hint: Consider multiple possibilities)
-
-**Question 4:** Sketch the time evolution of R, T, and P for a spark that starts at R = 150 kΩ with R_opt_power = 50 kΩ. Label key phases.
-
-**Question 5:** Why might a branched spark (multiple discharge paths) fail to optimize? Explain in terms of power distribution.
-
----
-**Next Lesson:** [Thévenin Equivalent Method - Extraction](03-thevenin-method.md)
diff --git a/spark-lessons/lessons/02-optimization/03-thevenin-method.md b/spark-lessons/lessons/02-optimization/03-thevenin-method.md
deleted file mode 100644
index 7c943ef..0000000
--- a/spark-lessons/lessons/02-optimization/03-thevenin-method.md
+++ /dev/null
@@ -1,329 +0,0 @@
----
-id: opt-03
-title: "Thévenin Equivalent Method - Extraction"
-section: "Optimization & Simulation"
-difficulty: "intermediate"
-estimated_time: 40
-prerequisites: ["opt-01", "fund-08"]
-objectives:
- - Understand Thévenin's theorem applied to Tesla coils
- - Extract output impedance Z_th through test measurements
- - Extract open-circuit voltage V_th
- - Interpret Z_th components physically
-tags: ["thevenin", "impedance-measurement", "circuit-analysis", "simulation"]
----
-
-# Thévenin Equivalent Method - Extraction
-
-The Thévenin equivalent method is a powerful technique that allows us to characterize a Tesla coil **once** and then predict its behavior with **any load** without re-running full simulations. This dramatically simplifies optimization and design work.
-
-## What is a Thévenin Equivalent?
-
-### Thévenin's Theorem
-
-**Statement:** Any linear two-terminal network can be replaced by:
-- A voltage source **V_th** (the open-circuit voltage)
-- In series with an impedance **Z_th** (the output impedance)
-
-```
-┌─────────────┐ ┌────┐
-│ Complex │ │V_th├───[Z_th]───o Output
-│ Network │──o Output ≡ └────┘ |
-│ │ | GND
-└─────────────┘ GND
-```
-
-**Key advantage:** The Thévenin equivalent completely characterizes the network's behavior at the output terminals. Once extracted, you can predict performance with any load by simple circuit analysis.
-
-### Application to Tesla Coils
-
-For a Tesla coil, the "complex network" includes:
-- Primary tank circuit (L_primary, C_MMC)
-- Primary drive (inverter or spark gap)
-- Magnetic coupling
-- Secondary coil with all its distributed properties
-- Topload capacitance
-- All parasitic elements
-
-The **output port** is the topload-to-ground connection, where we connect the spark load.
-
-**Thévenin parameters:**
-- **V_th:** The voltage that appears at the topload with no spark (open circuit)
-- **Z_th:** The impedance "looking into" the topload terminal with the drive turned off
-
-## Step 1: Measuring Z_th (Output Impedance)
-
-The output impedance tells us how the coil "pushes back" against a load. It represents all the losses and reactive elements as seen from the topload.
-
-### Procedure
-
-**Step 1.1: Turn OFF primary drive**
-- Set drive voltage to 0V (AC short circuit)
-- Keep all tank components in place (MMC, L_primary, damping resistors)
-- The tank circuit is still present, just not energized
-- This "deactivates" all voltage sources in the network
-
-**Step 1.2: Apply test source**
-- Apply 1V AC at operating frequency to topload-to-ground port
-- Use small-signal AC source (in simulation or actual test equipment)
-- Frequency should match your intended operating frequency
-
-**Step 1.3: Measure current**
-```
-I_test = current flowing into topload port with 1V applied
-```
-
-In SPICE/simulation:
-- Place 1V AC source between topload and ground
-- Run AC analysis at operating frequency
-- Read current magnitude and phase
-
-**Step 1.4: Calculate Z_th**
-```
-Z_th = V_test / I_test = 1V / I_test
-
-Z_th = R_th + jX_th (complex impedance)
-```
-
-### Physical Meaning of Components
-
-**R_th (Resistance):**
-- Secondary winding resistance (copper losses)
-- Dielectric losses in the coil form
-- Damping resistors in primary circuit
-- Core losses (if any)
-- Typical: 10-100 Ω for medium coils at RF frequencies
-
-**X_th (Reactance):**
-- Usually negative (capacitive) due to topload
-- Includes reflected impedances from coupling
-- May include inductive component from coil
-- Typical: -500 to -3000 Ω (strongly capacitive)
-
-**Magnitude |Z_th|:**
-- Total opposition to current
-- Typical: 500-3000 Ω for Tesla coils at 100-400 kHz
-
-**Phase φ_Z_th:**
-- Usually -85° to -88° (nearly pure capacitive)
-- Small R_th compared to |X_th| gives phase close to -90°
-
-### Quality Factor from Z_th
-
-The quality factor Q represents how "lossy" the coil is:
-
-```
-Q = |X_th| / R_th
-
-Higher Q → lower losses → more efficient
-```
-
-Typical values:
-- Small coils: Q = 50-150
-- Medium coils: Q = 100-300
-- Large coils: Q = 200-500
-
-## Step 2: Measuring V_th (Open-Circuit Voltage)
-
-The open-circuit voltage tells us what voltage the coil produces with no load attached.
-
-### Procedure
-
-**Step 2.1: Remove load**
-- Disconnect spark (or ensure spark won't break out)
-- Topload is in open-circuit condition
-- No current flows to external loads
-
-**Step 2.2: Turn ON primary drive**
-- Normal operating frequency and amplitude
-- Drive the coil exactly as you would for spark operation
-- Primary current flows, secondary is excited
-
-**Step 2.3: Measure topload voltage**
-```
-V_th = V(topload) with no load
-
-Record both magnitude and phase (complex phasor)
-```
-
-In simulation:
-- Run AC analysis with drive on
-- Read voltage at topload node
-- This is your V_th
-
-In practice:
-- Use high-impedance voltage probe
-- Capacitive divider for high voltages
-- Or measure primary current and use coupling theory
-
-**Typical values:**
-- Small coils (few hundred watts): V_th = 100-300 kV
-- Medium coils (1-3 kW): V_th = 200-500 kV
-- Large coils (5-10+ kW): V_th = 500 kV - 1 MV+
-
-### Important Notes
-
-**Frequency dependence:**
-- Both Z_th and V_th depend on frequency
-- Extract at your operating frequency
-- Near resonance, small frequency changes cause large V_th changes
-
-**Linearity assumption:**
-- Thévenin theorem assumes linear network
-- Valid for small-signal analysis
-- For large sparks, nonlinear effects may require iterative refinement
-
-**Enhancement for frequency tracking:**
-- Measure Z_th(ω) and V_th(ω) over frequency band (±10%)
-- Accounts for resonance shift when spark loads the coil
-- Enables accurate predictions with different loads
-
-## Worked Example: Extracting Z_th from Simulation
-
-**Simulation setup:**
-- DRSSTC at f = 185 kHz
-- Primary drive set to 0V (AC short)
-- All components remain (L_primary, C_MMC, secondary, topload)
-- AC test source: 1V ∠0° at topload-to-ground
-
-**Simulation results:**
-```
-I_test = 0.000412 ∠87.3° A = 0.412 mA ∠87.3°
-```
-
-### Calculate Z_th
-
-**Step 1: Impedance magnitude**
-```
-|Z_th| = |V| / |I| = 1 V / 0.000412 A = 2427 Ω
-```
-
-**Step 2: Impedance phase**
-```
-φ_Z_th = φ_V - φ_I = 0° - 87.3° = -87.3°
-```
-
-**Step 3: Polar form**
-```
-Z_th = 2427 Ω ∠-87.3°
-```
-
-**Step 4: Convert to rectangular form**
-```
-R_th = |Z_th| × cos(φ_Z_th) = 2427 × cos(-87.3°) = 2427 × 0.0471 = 114 Ω
-
-X_th = |Z_th| × sin(φ_Z_th) = 2427 × sin(-87.3°) = 2427 × (-0.9989) = -2424 Ω
-
-Z_th = 114 - j2424 Ω
-```
-
-### Interpretation
-
-**R_th = 114 Ω:**
-- Represents all resistive losses in the system
-- Includes secondary winding resistance
-- Includes reflected primary losses
-- This is the "cost" of extracting power from the coil
-
-**X_th = -2424 Ω:**
-- Strongly capacitive (negative reactance)
-- Topload capacitance dominates
-- At 185 kHz: C_equivalent ≈ 1/(ω|X_th|) ≈ 35 pF
-
-**Phase ≈ -87°:**
-- Nearly pure capacitor (ideal would be -90°)
-- Small resistive component (R_th << |X_th|)
-- Typical for well-designed Tesla coils
-
-**Quality factor:**
-```
-Q = |X_th| / R_th = 2424 / 114 ≈ 21
-```
-
-This Q is relatively low, likely because:
-- Measurement includes all system damping
-- Primary circuit losses are reflected
-- This is the "loaded" Q of the coupled system
-
-## Visual Aid: Thévenin Measurement Setup
-
-
-
-*Image shows comparison between:*
-- *Left: Full Tesla coil circuit (complex, many components)*
-- *Right: Thévenin equivalent (simple: V_th in series with Z_th)*
-- *Bottom: Measurement configuration for Z_th extraction*
-
-**Key elements:**
-- Primary drive: OFF (0V) for Z_th measurement
-- Test source: 1V AC at topload for Z_th
-- All tank components remain in circuit
-- Ammeter measures test current I_test
-- Calculation: Z_th = 1V / I_test
-
-## Common Pitfalls
-
-### Pitfall 1: Removing Tank Components
-
-**Wrong:** Disconnecting C_MMC or shorting L_primary
-
-**Right:** Keep all components, just set drive to 0V
-
-**Why:** The tank circuit affects the output impedance. Removing components gives incorrect Z_th.
-
-### Pitfall 2: Wrong Frequency
-
-**Wrong:** Extracting Z_th at one frequency, using at another
-
-**Right:** Extract at operating frequency, or measure Z_th(ω) over range
-
-**Why:** Impedance is highly frequency-dependent near resonance
-
-### Pitfall 3: Ignoring Phase
-
-**Wrong:** Using only |Z_th| without phase information
-
-**Right:** Keep full complex impedance Z_th = R_th + jX_th
-
-**Why:** Phase affects power calculations and matching
-
-### Pitfall 4: Using I_base Instead of Port Current
-
-**Wrong:** Measuring current at secondary base for Z_th test
-
-**Right:** Measure current through test source at topload port
-
-**Why:** Base current includes displacement currents (see Module 2.4)
-
-## Key Takeaways
-
-- **Thévenin equivalent** reduces complex coil to simple V_th and Z_th
-- **Z_th extraction:** Drive OFF, apply 1V test, measure current, Z_th = 1V/I_test
-- **V_th extraction:** Drive ON, no load, measure topload voltage
-- **Z_th components:** R_th (losses), X_th (reactance, usually capacitive)
-- **Typical values:** R_th = 10-100 Ω, X_th = -500 to -3000 Ω, |Z_th| = 500-3000 Ω
-- **Quality factor:** Q = |X_th|/R_th indicates coil efficiency
-- **Frequency matters:** Extract at operating frequency or measure Z_th(ω)
-
-## Practice
-
-{exercise:opt-ex-03}
-
-**Problem 1:** A test measurement gives I_test = 0.00035 ∠82° A for V_test = 1 ∠0° V at f = 200 kHz. Calculate:
-(a) Z_th in polar form
-(b) Z_th in rectangular form (R_th + jX_th)
-(c) Quality factor Q
-
-**Problem 2:** If Z_th = 85 - j1800 Ω, what is the equivalent capacitance at f = 180 kHz?
-
-**Problem 3:** A coil has Z_th = 120 - j2100 Ω. Calculate:
-(a) Impedance magnitude and phase
-(b) Quality factor
-(c) Would you describe this as "high Q" or "low Q"?
-
-**Problem 4:** Explain why we short the drive voltage source (set to 0V) when measuring Z_th, but keep all passive components in place.
-
-**Problem 5:** Two coils have the same |Z_th| = 2000 Ω but different phases: Coil A has φ = -88°, Coil B has φ = -75°. Which coil has lower losses (higher Q)? Calculate Q for both.
-
----
-**Next Lesson:** [Thévenin Calculations - Using the Equivalent](04-thevenin-calculations.md)
diff --git a/spark-lessons/lessons/02-optimization/04-thevenin-calculations.md b/spark-lessons/lessons/02-optimization/04-thevenin-calculations.md
deleted file mode 100644
index bc898ce..0000000
--- a/spark-lessons/lessons/02-optimization/04-thevenin-calculations.md
+++ /dev/null
@@ -1,397 +0,0 @@
----
-id: opt-04
-title: "Using the Thévenin Equivalent - Power Calculations"
-section: "Optimization & Simulation"
-difficulty: "intermediate"
-estimated_time: 45
-prerequisites: ["opt-03", "opt-01"]
-objectives:
- - Calculate load voltage and current using Thévenin equivalent
- - Compute power delivered to arbitrary loads
- - Determine maximum theoretical power (conjugate match)
- - Understand why conjugate match is usually unachievable
-tags: ["thevenin", "power-calculation", "impedance-matching", "circuit-analysis"]
----
-
-# Using the Thévenin Equivalent - Power Calculations
-
-Now that we've extracted the Thévenin equivalent (V_th and Z_th), we can use it to predict coil performance with any load without re-running full simulations. This lesson shows how to perform these calculations and interpret the results.
-
-## Predicting Behavior with Any Load
-
-Once you have V_th and Z_th, the Tesla coil looks like this simple circuit:
-
-```
- ┌────┐
- │V_th├───[Z_th]───┬─── Output
- └────┘ │
- [Z_load]
- │
- GND
-```
-
-This is just a voltage divider! We can apply basic circuit analysis.
-
-### Voltage Across Load
-
-Using voltage divider rule:
-
-```
-V_load = V_th × [Z_load / (Z_th + Z_load)]
-```
-
-**Complex arithmetic:** Both numerator and denominator are complex numbers, so you need to handle magnitude and phase carefully.
-
-### Current Through Load
-
-Using Ohm's law on the series circuit:
-
-```
-I = V_th / (Z_th + Z_load)
-```
-
-This current flows through both Z_th and Z_load since they're in series.
-
-### Power Delivered to Load
-
-Power dissipated in the load (real power only):
-
-```
-P_load = 0.5 × |I|² × Re{Z_load}
-```
-
-Or equivalently:
-
-```
-P_load = 0.5 × Re{V_load × I*}
-```
-
-where I* is the complex conjugate of I.
-
-**Direct formula combining everything:**
-
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-This formula is gold! It lets you sweep different Z_load values and calculate power without any additional simulation.
-
-## Step-by-Step Calculation Process
-
-### Given Information
-- V_th (complex voltage phasor)
-- Z_th = R_th + jX_th (complex impedance)
-- Z_load = R_load + jX_load (spark impedance from model)
-
-### Step 1: Calculate Total Impedance
-
-```
-Z_total = Z_th + Z_load
- = (R_th + R_load) + j(X_th + X_load)
-
-R_total = R_th + R_load
-X_total = X_th + X_load
-
-|Z_total| = √(R_total² + X_total²)
-```
-
-### Step 2: Calculate Current
-
-```
-I = V_th / Z_total
-
-|I| = |V_th| / |Z_total|
-
-φ_I = φ_V_th - φ_Z_total
-```
-
-where φ_Z_total = atan(X_total / R_total)
-
-### Step 3: Calculate Load Voltage
-
-```
-V_load = I × Z_load
-
-|V_load| = |I| × |Z_load|
-
-φ_V_load = φ_I + φ_Z_load
-```
-
-Or use voltage divider directly (often simpler):
-
-```
-|V_load| = |V_th| × |Z_load| / |Z_total|
-```
-
-### Step 4: Calculate Power in Load
-
-```
-P_load = 0.5 × |I|² × R_load
-
-P_load = 0.5 × |I|² × Re{Z_load}
-```
-
-The factor of 0.5 accounts for peak phasor to RMS conversion in AC power.
-
-## Worked Example: Complete Thévenin Analysis
-
-**Given:**
-- Z_th = 114 - j2424 Ω (from previous lesson)
-- V_th = 350 kV ∠0° (measured with drive on, no load)
-- Spark load: Z_spark = 60 kΩ - j160 kΩ (from lumped model)
-
-**Find:**
-(a) Current through spark
-(b) Voltage across spark
-(c) Power dissipated in spark
-(d) Theoretical maximum power (conjugate match)
-
-### Part (a): Current Through Spark
-
-**Calculate total impedance:**
-```
-Z_total = Z_th + Z_spark
- = (114 - j2424) + (60000 - j160000)
- = (60114 - j162424) Ω
-
-R_total = 60114 Ω
-X_total = -162424 Ω
-
-|Z_total| = √(60114² + 162424²)
- = √(3.614×10⁹ + 2.638×10¹⁰)
- = √(3.000×10¹⁰)
- = 173.2 kΩ
-```
-
-**Calculate current:**
-```
-I = V_th / Z_total
-|I| = 350 kV / 173.2 kΩ = 2.02 A peak
-```
-
-### Part (b): Voltage Across Spark
-
-**Method 1: Voltage divider**
-```
-|Z_spark| = √(60000² + 160000²)
- = √(3.6×10⁹ + 2.56×10¹⁰)
- = √(2.92×10¹⁰)
- = 171 kΩ
-
-|V_spark| = |V_th| × |Z_spark| / |Z_total|
- = 350 kV × (171 kΩ / 173.2 kΩ)
- = 350 kV × 0.987
- = 345 kV
-```
-
-**Method 2: Using current**
-```
-|V_spark| = |I| × |Z_spark|
- = 2.02 A × 171 kΩ
- = 345 kV
-```
-
-**Observation:** Most voltage appears across the spark! This makes sense because Z_spark >> Z_th.
-
-### Part (c): Power in Spark
-
-```
-P_spark = 0.5 × |I|² × Re{Z_spark}
- = 0.5 × (2.02)² × 60000
- = 0.5 × 4.08 × 60000
- = 122 kW
-```
-
-This is the real power dissipated in heating, ionization, radiation, and sound in the spark.
-
-### Part (d): Theoretical Maximum Power
-
-The maximum power transfer theorem states that power is maximized when the load impedance is the **complex conjugate** of the source impedance.
-
-**Conjugate match condition:**
-```
-Z_load = Z_th* (complex conjugate)
-
-If Z_th = R_th + jX_th
-Then Z_load = R_th - jX_th
-
-For our case:
-Z_th = 114 - j2424 Ω
-Z_load_optimal = 114 + j2424 Ω
-```
-
-**Why this maximizes power:**
-- Reactive components cancel: Z_total = Z_th + Z_th* = 2R_th (purely real)
-- No reactive power circulation
-- All delivered power is real
-
-**Maximum power formula:**
-```
-P_max = |V_th|² / (8 × R_th)
-```
-
-**Calculate:**
-```
-P_max = (350×10³)² / (8 × 114)
- = 1.225×10¹¹ / 912
- = 134.3 MW
-```
-
-**Wait, this seems enormous!**
-
-Let's double-check:
-```
-With Z_load = 114 + j2424 Ω:
-
-Z_total = (114 - j2424) + (114 + j2424) = 228 Ω (purely resistive!)
-
-I = 350 kV / 228 Ω = 1535 A
-
-P = 0.5 × (1535)² × 114 = 134.3 MW ✓
-```
-
-### Part (e): Reality Check - Why Such a Huge Difference?
-
-**Actual spark power:** 122 kW
-**Theoretical maximum:** 134.3 MW
-**Efficiency:** 122 / 134,300 = 0.09% of theoretical maximum
-
-**Why such a huge discrepancy?**
-
-1. **Conjugate match requires Z_load = 114 + j2424 Ω**
- - This means R_load = 114 Ω (extremely low!)
- - This means X_load = +2424 Ω (inductive, not capacitive)
-
-2. **Actual spark: Z_spark = 60 kΩ - j160 kΩ**
- - R_spark = 60 kΩ (525× too high!)
- - X_spark = -160 kΩ (capacitive, wrong sign, 66× too large)
-
-3. **Topological constraints prevent achieving conjugate match:**
- - Spark structure (R||C_mut in series with C_sh) is inherently capacitive
- - Cannot produce positive (inductive) reactance
- - Cannot achieve R_load as low as 114 Ω with realistic plasma
-
-**This is normal for Tesla coils!** The impedance mismatch is fundamental to the physics of spark discharges. We cannot achieve conjugate match in practice.
-
-## Understanding Efficiency
-
-### What Does 0.09% Mean?
-
-It does NOT mean the coil is "inefficient" in the usual sense. Rather:
-
-- The coil has very low output impedance (114 Ω)
-- The spark has very high impedance (171 kΩ)
-- This is a 1500:1 impedance mismatch
-- The voltage divider heavily favors the spark (good!)
-- Most voltage appears at the spark, but current is limited
-
-### Voltage Transfer Efficiency
-
-```
-Voltage across spark / Total voltage:
-345 kV / 350 kV = 98.6%
-```
-
-We achieve excellent voltage transfer! This is what matters for spark length (field at tip).
-
-### Why Not Match Impedances?
-
-**In conventional circuits:** Match impedances for maximum power transfer
-
-**In Tesla coils:** We WANT high spark impedance because:
-- High voltage at spark tip drives field
-- High resistance means controlled current (safety)
-- Mismatch is unavoidable due to plasma physics
-- Optimization focuses on maximizing power given the constraints
-
-## Practical Use: Sweeping Spark Parameters
-
-The real power of Thévenin analysis is rapid parameter sweeps:
-
-**Given:** V_th = 350 kV, Z_th = 114 - j2424 Ω
-
-**Sweep:** Spark resistance R from 10 kΩ to 200 kΩ
-
-**For each R value:**
-1. Construct Z_spark from R and capacitances (using lumped model)
-2. Calculate Z_total = Z_th + Z_spark
-3. Calculate I = V_th / Z_total
-4. Calculate P = 0.5 × |I|² × R
-5. Plot P vs R
-
-**Result:** You find P_max at R ≈ R_opt_power without any new simulations!
-
-## When Thévenin Analysis Fails
-
-### Nonlinearity
-
-**Assumption:** Coil behaves linearly (impedances don't change with voltage/current)
-
-**Breaks down when:**
-- Magnetic cores saturate
-- Component heating changes parameters
-- Very large sparks significantly load the coil
-
-**Solution:** Iterate - use results to update model, re-extract Thévenin
-
-### Frequency Dependence
-
-**Assumption:** Operating at a single frequency
-
-**Breaks down when:**
-- Spark loading shifts resonant frequency
-- Comparing different loads at fixed frequency (detuning varies)
-
-**Solution:** Extract Z_th(ω) and V_th(ω), account for frequency shift (next lessons)
-
-### Coupled Modes
-
-**Assumption:** Single-mode operation
-
-**Breaks down when:**
-- Operating between two coupled poles
-- Mode hopping as spark changes loading
-
-**Solution:** Full coupled-mode analysis or stay clearly in one mode
-
-## Key Takeaways
-
-- **Thévenin circuit:** Simple series combination of V_th and Z_th
-- **Load voltage:** V_load = V_th × Z_load/(Z_th + Z_load)
-- **Load current:** I = V_th / (Z_th + Z_load)
-- **Load power:** P = 0.5 × |I|² × Re{Z_load} or P = 0.5 × |V_th|² × Re{Z_load}/|Z_th + Z_load|²
-- **Maximum power:** Requires conjugate match Z_load = Z_th*
-- **P_max = |V_th|²/(8R_th)** but usually unachievable
-- **Tesla coils operate far from conjugate match** due to physics constraints
-- **High voltage transfer efficiency** matters more than impedance matching
-- **Parameter sweeps** become trivial with Thévenin equivalent
-
-## Practice
-
-{exercise:opt-ex-04}
-
-**Problem 1:** Given Z_th = 95 - j1850 Ω, V_th = 280 kV, and a spark model with Z_spark = 50 kΩ - j140 kΩ:
-(a) Calculate total impedance
-(b) Calculate current through spark
-(c) Calculate power delivered to spark
-(d) Calculate theoretical maximum power (conjugate match)
-(e) What percentage of theoretical maximum is achieved?
-
-**Problem 2:** A load Z_load = 200 + j200 Ω is connected to a coil with Z_th = 100 - j2000 Ω and V_th = 300 kV.
-(a) Calculate the load voltage
-(b) Calculate power delivered
-(c) Is this load inductive or capacitive?
-(d) Is this load closer to conjugate match than a typical spark?
-
-**Problem 3:** For Z_th = 120 - j2200 Ω:
-(a) What load impedance gives conjugate match?
-(b) Calculate P_max if V_th = 400 kV
-(c) If actual spark has R = 70 kΩ, X = -180 kΩ, calculate actual power
-(d) Calculate the power transfer efficiency ratio
-
-**Problem 4:** A coil has V_th = 350 kV and Z_th = 110 - j2500 Ω. You want to deliver 100 kW to a purely resistive load. What resistance value is required? (Hint: Set P = 100 kW in power formula and solve for R)
-
-**Problem 5:** Explain physically why Tesla coils operate so far from conjugate match. Why can't we just add inductance to the spark to cancel its capacitive reactance?
-
----
-**Next Lesson:** [Direct Power Measurement Method](05-direct-measurement.md)
diff --git a/spark-lessons/lessons/02-optimization/05-direct-measurement.md b/spark-lessons/lessons/02-optimization/05-direct-measurement.md
deleted file mode 100644
index 4236575..0000000
--- a/spark-lessons/lessons/02-optimization/05-direct-measurement.md
+++ /dev/null
@@ -1,337 +0,0 @@
----
-id: opt-05
-title: "Direct Power Measurement Method"
-section: "Optimization & Simulation"
-difficulty: "intermediate"
-estimated_time: 25
-prerequisites: ["opt-04", "opt-01"]
-objectives:
- - Understand the direct measurement alternative to Thévenin
- - Set up simulations for direct power measurement
- - Extract spark resistance through power optimization
- - Compare advantages and disadvantages of each method
-tags: ["power-measurement", "simulation", "optimization", "methodology"]
----
-
-# Direct Power Measurement Method
-
-While the Thévenin equivalent method is powerful and elegant, there's an alternative approach: directly measure power delivered to the spark in a full simulation. Each method has advantages and trade-offs.
-
-## The Direct Measurement Approach
-
-### Concept
-
-Instead of extracting a simplified equivalent circuit, keep the **full coupled model** with the spark load present and directly measure power flow.
-
-**Setup:**
-1. Build complete simulation (primary, secondary, coupling, spark load)
-2. Drive primary at operating frequency and amplitude
-3. Run AC analysis (or transient with post-processing)
-4. Measure power dissipated in spark resistance
-5. Repeat for different spark resistance values
-
-**Goal:** Find the spark resistance R that maximizes measured power
-
-### Procedure
-
-**Step 1: Build Full Model**
-- Primary tank circuit (L_primary, C_MMC)
-- Secondary coil (distributed or lumped model)
-- Topload capacitance
-- Magnetic coupling k
-- **Spark load** modeled as R||C_mut in series with C_sh
-
-**Step 2: Set Operating Point**
-- Drive frequency: f_drive (initially at unloaded resonance)
-- Drive amplitude: V_drive or I_drive
-- Spark parameters: Choose initial R, C_mut, C_sh
-
-**Step 3: Run AC Analysis**
-- Solve circuit at drive frequency
-- Extract voltage and current at spark resistor
-- Calculate power: P = 0.5 × Re{V_spark × I_spark*}
-
-Or more directly:
-```
-P = 0.5 × |I_R|² × R
-
-where I_R is current through the resistance R
-```
-
-**Step 4: Sweep R Values**
-- Vary R from 10 kΩ to 200 kΩ (typical range)
-- For each R, measure P
-- Plot P vs R
-- Find R that gives maximum P → this is R_opt_power
-
-**Step 5: Validate**
-- Compare numerical R_opt_power to analytical formula
-- Check that it matches: R_opt = 1/[ω(C_mut + C_sh)]
-
-## Power Measurement in SPICE
-
-### Method 1: Using Current Through Resistor
-
-```
-.param Rspark = 50k
-Rspark topload node2 {Rspark}
-Cmut node2 0 8p
-Csh topload 0 6p
-
-.ac lin 1 185k 185k
-.step param Rspark list 10k 30k 50k 70k 100k 150k
-
-.meas ac Ispark_mag find mag(I(Rspark))
-.meas ac Pspark param '0.5 * Ispark_mag^2 * Rspark'
-```
-
-This sweeps Rspark and calculates power for each value.
-
-### Method 2: Direct Power Function
-
-Some SPICE variants support direct power measurement:
-
-```
-.meas ac Pspark_real find Re(V(topload)*conj(I(Rspark)))
-```
-
-This directly computes complex power and extracts the real part.
-
-### Method 3: Voltage and Current
-
-```
-.meas ac Vtop_mag find mag(V(topload))
-.meas ac Ispark_mag find mag(I(Rspark))
-.meas ac phase_diff param 'ph(V(topload)) - ph(I(Rspark))'
-.meas ac Pspark param '0.5 * Vtop_mag * Ispark_mag * cos(phase_diff)'
-```
-
-This accounts for phase difference in power calculation.
-
-## Worked Example: Direct Optimization
-
-**Given:**
-- DRSSTC simulation at f = 185 kHz
-- Primary drive: V_drive produces V_top ≈ 350 kV (unloaded)
-- Spark model: C_mut = 8 pF, C_sh = 6 pF, R = variable
-
-**Goal:** Find R_opt_power
-
-### Analytical Prediction
-
-First, predict what we should find:
-
-```
-C_total = C_mut + C_sh = 8 + 6 = 14 pF
-ω = 2π × 185×10³ = 1.162×10⁶ rad/s
-
-R_opt_power = 1/(ωC_total)
- = 1/(1.162×10⁶ × 14×10⁻¹²)
- = 61.5 kΩ
-```
-
-We expect maximum power near 61.5 kΩ.
-
-### Simulation Sweep
-
-**Run AC analysis with R values:**
-- R = 20 kΩ → P = 85 kW
-- R = 40 kΩ → P = 115 kW
-- R = 60 kΩ → P = 125 kW ← **Maximum**
-- R = 80 kΩ → P = 118 kW
-- R = 100 kΩ → P = 105 kW
-
-**Result:** Maximum power at R ≈ 60 kΩ
-
-**Validation:** Simulation (60 kΩ) matches theory (61.5 kΩ) within rounding!
-
-## Advantages of Direct Measurement
-
-### 1. No Approximations
-
-- Full coupled model captures all interactions
-- No linearization assumptions
-- Includes all nonlinear effects (if using transient analysis)
-
-### 2. Intuitive
-
-- Directly see what you care about: power to spark
-- No intermediate steps
-- Easy to visualize results
-
-### 3. Flexibility
-
-- Can use any circuit simulator
-- Works with complex topologies
-- Easy to add additional elements (damping, protection, etc.)
-
-### 4. Transient Capability
-
-- Can extend to time-domain (transient) analysis
-- Capture burst mode, ramping, dynamics
-- See energy transfer over time
-
-## Disadvantages of Direct Measurement
-
-### 1. Computational Cost
-
-- Must re-run full simulation for each R value
-- Sweep of 20 points = 20 full simulations
-- Slow for large parameter spaces
-
-### 2. Limited Insight
-
-- Doesn't reveal underlying equivalent circuit
-- Harder to understand why maximum occurs where it does
-- Less portable to different load types
-
-### 3. Frequency Coupling
-
-- Operating frequency may need adjustment for each R (see next lesson!)
-- Fixed-frequency comparison can be misleading
-- Must account for resonance shift
-
-### 4. Sensitivity to Setup
-
-- Results depend on drive amplitude, frequency, damping
-- Harder to isolate spark effects from system effects
-
-## Comparison: Thévenin vs Direct
-
-| Aspect | Thévenin Method | Direct Method |
-|--------|----------------|---------------|
-| **Speed** | Fast (single extraction + algebra) | Slow (simulation per R value) |
-| **Insight** | High (reveals equivalent circuit) | Moderate |
-| **Accuracy** | Excellent (if linear) | Excellent (includes nonlinearities) |
-| **Flexibility** | Any load instantly | One load per simulation |
-| **Complexity** | Requires understanding of method | Straightforward |
-| **Best for** | Sweeps, optimization, understanding | Validation, nonlinear cases |
-
-## When to Use Each Method
-
-### Use Thévenin When:
-- Exploring many different load configurations
-- Optimizing spark parameters
-- Building intuition about matching
-- Preparing design curves
-- Speed is important
-
-### Use Direct Measurement When:
-- Validating Thévenin results
-- Dealing with significant nonlinearities
-- Need transient/time-domain behavior
-- Checking specific operating points
-- Learning circuit behavior
-
-### Best Practice: Use Both
-
-1. **Start with Thévenin:** Fast exploration, find optimal regions
-2. **Validate with Direct:** Confirm key points, check assumptions
-3. **Iterate:** If discrepancies exist, understand why
-
-## Accounting for Displacement Currents
-
-Both methods can fall victim to the "I_base error" discussed in Module 2.4.
-
-### The Problem
-
-**Wrong:** Measuring total current returning through secondary base
-
-**Right:** Measuring current specifically through spark resistance
-
-### Why It Matters
-
-Total base current includes:
-- Spark current (what we want)
-- Displacement currents from secondary to ground
-- Coupling currents to primary
-- Environmental coupling
-
-**In SPICE:** This isn't usually a problem because you can measure specific branch currents. Use I(Rspark) not I(V_secondary_base).
-
-**In physical measurements:** You must use current probes on the spark return path, not the coil base.
-
-## Implementation Tips
-
-### Tip 1: Automate Sweeps
-
-Use SPICE .STEP or scripting:
-
-```
-.step param Rspark 10k 200k 5k
-```
-
-This automatically sweeps from 10 kΩ to 200 kΩ in 5 kΩ steps.
-
-### Tip 2: Log Scale for Wide Ranges
-
-Spark resistance varies over decades (10 kΩ to 1 MΩ). Use logarithmic stepping:
-
-```
-.step param Rspark list 10k 20k 50k 100k 200k 500k
-```
-
-### Tip 3: Extract Peak Directly
-
-Use .MEAS to find maximum automatically:
-
-```
-.meas ac Pmax MAX Pspark
-.meas ac Ropt WHEN Pspark=Pmax
-```
-
-### Tip 4: Verify Power Components
-
-Separately measure real and reactive power:
-
-```
-P_real = Re{V × I*}
-Q_reactive = Im{V × I*}
-S_apparent = |V × I*|
-```
-
-Check that Q >> P (highly reactive, as expected).
-
-## Key Takeaways
-
-- **Direct measurement:** Keep full model, measure power in spark, sweep R
-- **Advantages:** Intuitive, no approximations, handles nonlinearity
-- **Disadvantages:** Slow, less insight, multiple simulations required
-- **Power formula:** P = 0.5 × |I_R|² × R or P = 0.5 × Re{V × I*}
-- **Find R_opt:** Sweep R, plot P vs R, identify maximum
-- **Validation:** Should match analytical R_opt = 1/[ω(C_mut + C_sh)]
-- **Best practice:** Use Thévenin for exploration, direct measurement for validation
-- **Beware:** Measure spark current, not base current (displacement current issue)
-
-## Practice
-
-{exercise:opt-ex-05}
-
-**Problem 1:** You run simulations with the following results:
-
-| R (kΩ) | P (kW) |
-|--------|--------|
-| 30 | 92 |
-| 50 | 118 |
-| 70 | 128 |
-| 90 | 125 |
-| 110 | 115 |
-
-(a) Estimate R_opt_power from this data
-(b) If C_total = 12 pF and f = 200 kHz, what does theory predict?
-(c) Do they match?
-
-**Problem 2:** A simulation reports I_R = 2.1 A (peak) through R = 55 kΩ. Calculate the power dissipated.
-
-**Problem 3:** You measure V_topload = 340 kV ∠0° and I_spark = 1.8 A ∠-72°.
-(a) Calculate apparent power S = V × I*
-(b) Extract real power P = Re{S}
-(c) Extract reactive power Q = Im{S}
-(d) Is the spark more resistive or reactive?
-
-**Problem 4:** List two scenarios where direct measurement would be preferred over Thévenin extraction.
-
-**Problem 5:** Why is it important to measure I(Rspark) rather than I(V_secondary_base) when calculating power? Sketch the circuit showing both current paths.
-
----
-**Next Lesson:** [Frequency Tracking and Loaded Poles](06-frequency-tracking.md)
diff --git a/spark-lessons/lessons/02-optimization/06-frequency-tracking.md b/spark-lessons/lessons/02-optimization/06-frequency-tracking.md
deleted file mode 100644
index 7db0de1..0000000
--- a/spark-lessons/lessons/02-optimization/06-frequency-tracking.md
+++ /dev/null
@@ -1,485 +0,0 @@
----
-id: opt-06
-title: "Frequency Tracking and Loaded Poles"
-section: "Optimization & Simulation"
-difficulty: "advanced"
-estimated_time: 45
-prerequisites: ["opt-05", "opt-01", "fund-08"]
-objectives:
- - Understand coupled system poles and eigenfrequencies
- - Recognize frequency shift with loading
- - Implement proper frequency tracking in measurements
- - Avoid common detuning errors in optimization
- - Apply frequency tracking to DRSSTC operating modes
-tags: ["frequency-tracking", "coupled-resonators", "detuning", "poles", "DRSSTC"]
----
-
-# Frequency Tracking and Loaded Poles
-
-**This is one of the most commonly overlooked aspects of Tesla coil optimization.** Failing to account for frequency tracking leads to misleading power measurements and incorrect conclusions about optimal operating points.
-
-## The Critical Problem: Fixed-Frequency Comparison
-
-### Common Mistake
-
-**Scenario:** You want to find R_opt_power by measuring power delivered to different spark resistances.
-
-**Wrong approach:**
-1. Set drive frequency to f = 200 kHz (unloaded resonance)
-2. Measure power with R = 30 kΩ → P₁ = 95 kW
-3. Measure power with R = 60 kΩ → P₂ = 110 kW
-4. Measure power with R = 90 kΩ → P₃ = 105 kW
-5. Conclude: R_opt ≈ 60 kΩ
-
-**What's wrong?** Each different R value changes the system's resonant frequency. By staying at fixed f = 200 kHz, you're comparing:
-- R = 30 kΩ at **Δf = +8 kHz detuned**
-- R = 60 kΩ at **Δf = +3 kHz detuned**
-- R = 90 kΩ at **Δf = -2 kHz detuned**
-
-**You're not measuring inherent matching quality - you're measuring a combination of matching AND detuning!**
-
-### Right Approach
-
-**Correct procedure:**
-1. Set R = 30 kΩ
-2. **Sweep frequency to find loaded resonance** → f₁ = 192 kHz
-3. Measure power at f₁ → P₁ = 108 kW
-4. Set R = 60 kΩ
-5. **Sweep frequency to find new loaded resonance** → f₂ = 188 kHz
-6. Measure power at f₂ → P₂ = 125 kW
-7. Set R = 90 kΩ
-8. **Sweep frequency to find new loaded resonance** → f₃ = 185 kHz
-9. Measure power at f₃ → P₃ = 118 kW
-10. Conclude: R_opt ≈ 60 kΩ **(and each was measured at its optimal frequency)**
-
-**Key principle: For each R value, retune to the loaded pole frequency.**
-
-## Why Does Loading Change Frequency?
-
-### Capacitance Changes Resonance
-
-When you change the spark, you change its sheath capacitance C_sh:
-
-**Unloaded:**
-```
-C_total,0 = C_topload + C_secondary_stray ≈ 28 pF
-f₀ = 1/(2π√(L_sec × C_total,0)) = 200 kHz
-```
-
-**With spark (R = 60 kΩ, 3-foot leader):**
-```
-C_sh ≈ 2 pF/foot × 3 feet = 6 pF
-C_total,1 = C_total,0 + C_sh = 28 + 6 = 34 pF
-
-f₁ = f₀ × √(C_total,0 / C_total,1)
- = 200 × √(28/34)
- = 200 × 0.907
- = 181 kHz
-```
-
-**Frequency dropped by 19 kHz!** This is not a small shift.
-
-### Different Sparks → Different Frequencies
-
-| Spark Length | C_sh | C_total | f_loaded | Δf |
-|--------------|------|---------|----------|-----|
-| No spark | 0 pF | 28 pF | 200 kHz | 0 |
-| 2 feet | 4 pF | 32 pF | 187 kHz | -13 kHz |
-| 4 feet | 8 pF | 36 pF | 176 kHz | -24 kHz |
-| 6 feet | 12 pF | 40 pF | 167 kHz | -33 kHz |
-
-**Even for the same length, changing R changes the effective loading!**
-
-## Coupled System Poles
-
-Tesla coils are **coupled resonant systems**. Even without a spark, the primary-secondary coupling creates two resonant modes.
-
-### The Two Poles
-
-For coupled resonators with coupling coefficient k:
-
-**Lower pole (f₁):**
-```
-f₁ = f₀ / √(1 + k) < f₀
-```
-
-**Upper pole (f₂):**
-```
-f₂ = f₀ / √(1 - k) > f₀
-```
-
-where f₀ = √(f_primary × f_secondary) is the geometric mean.
-
-**Example with k = 0.15:**
-```
-f₀ = 200 kHz (geometric mean)
-f₁ = 200 / √(1.15) = 186.5 kHz (lower pole)
-f₂ = 200 / √(0.85) = 217.0 kHz (upper pole)
-```
-
-### Loading Modifies Both Poles
-
-When a spark loads the secondary:
-- **Both pole frequencies shift** (usually downward)
-- **Both pole damping increases** (Q decreases)
-- **Pole separation changes**
-
-The spark doesn't just add capacitance - it adds a complex load that couples into both modes.
-
-### Which Pole Should You Use?
-
-**For DRSSTC operation:**
-- Most coils operate on the **lower pole** (more stable)
-- Some operate between poles (dual-resonance mode)
-- Upper pole is rarely used (harder to control)
-
-**The loaded pole frequency is where voltage gain is maximized.**
-
-## DRSSTC Operating Modes
-
-Different DRSSTC drive strategies interact with frequency tracking differently.
-
-### Mode 1: Fixed Frequency (No Tracking)
-
-**Strategy:** Drive at fixed frequency (e.g., 200 kHz) regardless of loading
-
-**Advantages:**
-- Simple control electronics
-- No frequency sensing required
-- Predictable timing
-
-**Disadvantages:**
-- **Detunes as spark grows**
-- Voltage gain drops with larger sparks
-- Suboptimal power transfer
-- Risk of operating off-resonance
-
-**When acceptable:**
-- Very short bursts (spark doesn't grow much)
-- Controlled environments with consistent sparks
-- Systems designed with wide bandwidth
-
-### Mode 2: Frequency Tracking (PLL or Feedback)
-
-**Strategy:** Continuously adjust drive frequency to match loaded pole
-
-**Implementation:**
-- Phase-locked loop (PLL) tracks zero-crossing
-- Feedback from antenna or current sensor
-- Drive frequency follows resonance in real-time
-
-**Advantages:**
-- **Always at optimal frequency**
-- Maximum voltage gain throughout growth
-- Efficient power transfer
-- Adapts to varying sparks
-
-**Disadvantages:**
-- More complex electronics
-- Requires feedback sensing
-- Can be unstable if poorly tuned
-- Frequency limits needed for safety
-
-**This is the gold standard for QCW and high-performance DRSSTCs.**
-
-### Mode 3: Pre-Programmed Sweep
-
-**Strategy:** Drive frequency ramps down over time (anticipating C_sh increase)
-
-**Implementation:**
-- Start at f₀ (unloaded resonance)
-- Linearly or exponentially decrease frequency
-- End at f_target (expected loaded resonance)
-
-**Advantages:**
-- Simpler than PLL
-- No feedback required
-- Can be optimized per coil
-
-**Disadvantages:**
-- Not adaptive (doesn't match actual spark)
-- Requires characterization/tuning
-- Mismatch if spark growth differs from expectation
-
-**When useful:**
-- QCW with consistent spark growth patterns
-- Transition from no-spark to steady spark
-- Combined with current limiting
-
-## Frequency Response and Bandwidth
-
-### Quality Factor Limits Bandwidth
-
-The resonance has finite width determined by Q:
-
-```
-Δf_3dB = f₀ / Q (3 dB bandwidth)
-
-For Q = 100 at f₀ = 200 kHz:
-Δf_3dB = 200 kHz / 100 = 2 kHz
-```
-
-**Within ±1 kHz:** Still >70% of peak voltage (acceptable detuning)
-**Beyond ±5 kHz:** Down to ~30% of peak voltage (severe detuning)
-
-### High Q vs Low Q
-
-**High Q (narrow bandwidth):**
-- Sharper resonance peak
-- More sensitive to detuning
-- **Frequency tracking more critical**
-- Better efficiency when matched
-
-**Low Q (wide bandwidth):**
-- Broader resonance peak
-- More forgiving of detuning
-- Frequency tracking less critical
-- Lower peak voltage gain
-
-### Loaded Q vs Unloaded Q
-
-**Unloaded Q₀:**
-- No spark, only coil losses
-- Typically Q₀ = 100-300
-
-**Loaded Q_L:**
-- With spark, additional damping
-- Spark resistance adds loss
-- Typically Q_L = 20-80
-
-**Effect on bandwidth:**
-```
-Unloaded: Δf₀ = 200 kHz / 200 = 1 kHz (narrow!)
-Loaded: Δf_L = 185 kHz / 50 = 3.7 kHz (wider)
-```
-
-**Ironically, the spark broadens the resonance, making detuning slightly less critical. But the frequency shift is still large enough that you must track it.**
-
-## Implementing Frequency Tracking in Measurements
-
-### Simulation Approach
-
-**For each R value:**
-
-```python
-# Pseudocode for proper frequency tracking
-for R in [10k, 20k, 30k, ..., 200k]:
- set_spark_resistance(R)
-
- # Sweep frequency to find loaded pole
- for f in range(150k, 220k, 1k):
- run_AC_analysis(frequency=f)
- V_top[f] = measure_topload_voltage()
-
- # Find frequency with maximum voltage
- f_loaded = frequency_at_max(V_top)
-
- # Measure power at loaded frequency
- run_AC_analysis(frequency=f_loaded)
- P[R] = measure_spark_power()
-
- # Store results
- results[R] = {
- 'f_loaded': f_loaded,
- 'V_top': V_top[f_loaded],
- 'P': P[R]
- }
-
-# Now P[R] represents true matching, not detuning!
-R_opt = R_at_max(P)
-```
-
-### SPICE Implementation
-
-```spice
-* Sweep R and frequency together
-.param Rspark = 60k
-
-* First find loaded frequency for this R
-.ac dec 100 150k 220k
-.meas ac f_loaded WHEN mag(V(topload))=MAX(mag(V(topload)))
-
-* Then measure power at that frequency
-.ac lin 1 {f_loaded} {f_loaded}
-.meas ac Pspark param '0.5 * mag(I(Rspark))^2 * Rspark'
-
-* Repeat for each R value
-.step param Rspark list 10k 30k 50k 70k 90k 110k 150k 200k
-```
-
-**Challenge:** SPICE doesn't easily allow nested sweeps where inner result affects outer analysis. You may need to:
-- Run multiple simulations
-- Use scripting (Python + PySpice, MATLAB, etc.)
-- Manually extract f_loaded for key R values
-
-## Worked Example: Impact of Tracking vs Not Tracking
-
-**System:**
-- Unloaded: f₀ = 200 kHz, Q₀ = 150
-- V_th = 350 kV (at resonance)
-- Z_th = 110 - j2400 Ω (at 200 kHz)
-
-**Spark configurations:**
-
-| R | C_sh | C_total | f_loaded | Shift |
-|---|------|---------|----------|-------|
-| 40k | 5 pF | 33 pF | 188 kHz | -12 kHz |
-| 60k | 6 pF | 34 pF | 185 kHz | -15 kHz |
-| 80k | 7 pF | 35 pF | 183 kHz | -17 kHz |
-
-### Without Tracking (Fixed f = 200 kHz)
-
-**R = 40 kΩ:**
-```
-Detuning: Δf = +12 kHz
-Voltage penalty: V_actual / V_max ≈ 0.65
-Z_spark = 40k - j140k → |Z| = 146 kΩ
-I ≈ 350 kV × 0.65 / 146 kΩ = 1.56 A
-P = 0.5 × 1.56² × 40k = 48.6 kW
-```
-
-**R = 60 kΩ:**
-```
-Detuning: Δf = +15 kHz
-Voltage penalty: ≈ 0.55
-Z_spark = 60k - j160k → |Z| = 171 kΩ
-I ≈ 350 kV × 0.55 / 171 kΩ = 1.13 A
-P = 0.5 × 1.13² × 60k = 38.3 kW (WORSE despite higher R!)
-```
-
-**R = 80 kΩ:**
-```
-Detuning: Δf = +17 kHz
-Voltage penalty: ≈ 0.48
-Z_spark = 80k - j180k → |Z| = 197 kΩ
-I ≈ 350 kV × 0.48 / 197 kΩ = 0.85 A
-P = 0.5 × 0.85² × 80k = 28.9 kW
-```
-
-**Conclusion from fixed-frequency:** R_opt ≈ 40 kΩ (WRONG!)
-
-### With Tracking (Tune to f_loaded for each R)
-
-**R = 40 kΩ at f = 188 kHz:**
-```
-Detuning: 0 (by definition - we tuned to loaded pole)
-Voltage penalty: 1.0 (at resonance)
-I ≈ 350 kV / 146 kΩ = 2.40 A
-P = 0.5 × 2.40² × 40k = 115 kW
-```
-
-**R = 60 kΩ at f = 185 kHz:**
-```
-Detuning: 0
-Voltage penalty: 1.0
-I ≈ 350 kV / 171 kΩ = 2.05 A
-P = 0.5 × 2.05² × 60k = 126 kW (MAXIMUM!)
-```
-
-**R = 80 kΩ at f = 183 kHz:**
-```
-Detuning: 0
-Voltage penalty: 1.0
-I ≈ 350 kV / 197 kΩ = 1.78 A
-P = 0.5 × 1.78² × 80k = 127 kW (close!)
-```
-
-**Conclusion with tracking:** R_opt ≈ 60 kΩ (CORRECT!)
-
-**Power improvement with tracking:**
-- At R = 60 kΩ: 126 kW vs 38 kW = **3.3× more power!**
-- At R = 80 kΩ: 127 kW vs 29 kW = **4.4× more power!**
-
-**This is not a small effect. Frequency tracking is critical.**
-
-## Practical Implications
-
-### For Simulation Studies
-
-**Always:**
-- Report frequency used for each measurement
-- Either track frequency or clearly state fixed-frequency limitations
-- Specify whether results assume optimal tuning
-
-**When comparing:**
-- Ensure fair comparison (same tracking strategy)
-- Document detuning if fixed-frequency is used
-
-### For Physical Coils
-
-**DRSSTC with PLL:**
-- Tracks automatically - excellent
-- Monitor actual operating frequency
-- Check frequency stays within safe limits
-
-**DRSSTC with fixed frequency:**
-- Accept voltage/power reduction as spark grows
-- Consider pre-tuning to expected loaded frequency
-- Wider-bandwidth design helps (lower Q)
-
-**SGTC (Spark Gap):**
-- Frequency self-adjusts with loading (inherent tracking)
-- Spark gap firing adapts to LC resonance
-- Less of an issue for spark gap coils
-
-### For Optimization
-
-**When finding R_opt_power:**
-1. Use frequency tracking (simulation or actual)
-2. Report f_loaded for each R tested
-3. Verify analytical formula matches
-
-**When designing:**
-1. Choose f₀ based on unloaded resonance
-2. Expect f_operating ≈ f₀ - 10 to 30 kHz with sparks
-3. Ensure drive can operate over this range
-
-## Key Takeaways
-
-- **Critical principle:** For each R value, retune to loaded pole frequency
-- **Why it matters:** Loading changes C_sh, which shifts resonance by 10-30+ kHz
-- **Fixed-frequency comparison is misleading:** Measures detuning, not matching quality
-- **Coupled system has two poles:** Lower and upper, both shift with loading
-- **DRSSTC modes:** Fixed frequency (simple), PLL tracking (optimal), programmed sweep (compromise)
-- **Q affects sensitivity:** Higher Q = narrower bandwidth = more critical tracking
-- **Power difference:** Can be 3-5× between tracked and non-tracked measurements
-- **Simulation best practice:** Sweep frequency for each load to find f_loaded
-- **Physical coils:** PLL tracking gives best performance, fixed frequency is acceptable for short bursts
-
-## Practice
-
-{exercise:opt-ex-06}
-
-**Problem 1:** A coil has f₀ = 195 kHz unloaded with C_total,0 = 30 pF. A 4-foot spark adds C_sh = 8 pF.
-(a) Calculate the loaded capacitance
-(b) Calculate the loaded frequency
-(c) What is Δf (frequency shift)?
-
-**Problem 2:** You measure power at fixed f = 200 kHz:
-- R = 50 kΩ, f_loaded = 188 kHz → P₁ = 85 kW
-- R = 70 kΩ, f_loaded = 185 kHz → P₂ = 95 kW
-
-If Q = 80, estimate the voltage penalty factor for each case and calculate what power would be measured if you had tracked frequency.
-
-**Problem 3:** Explain why frequency tracking is MORE critical for high-Q coils than low-Q coils.
-
-**Problem 4:** A DRSSTC operates with fixed frequency drive. As the spark grows from 2 feet to 5 feet, what happens to:
-(a) Loaded resonant frequency
-(b) Detuning (if drive frequency is fixed)
-(c) Voltage gain
-(d) Power delivered
-
-**Problem 5:** For coupled resonators with k = 0.18 and f₀ = 210 kHz:
-(a) Calculate the lower pole frequency
-(b) Calculate the upper pole frequency
-(c) Which pole is typically used for DRSSTC operation?
-
-**Problem 6:** Sketch V_top vs frequency for three cases:
-(a) No spark (unloaded)
-(b) R = 60 kΩ spark (lightly loaded)
-(c) R = 30 kΩ spark (heavily loaded)
-
-Label the peak frequencies and relative peak heights. Explain how tracking helps maintain peak operation.
-
----
-**Next Lesson:** [Part 2 Review and Comprehensive Exercises](07-review-exercises.md)
diff --git a/spark-lessons/lessons/02-optimization/07-review-exercises.md b/spark-lessons/lessons/02-optimization/07-review-exercises.md
deleted file mode 100644
index fc16ffa..0000000
--- a/spark-lessons/lessons/02-optimization/07-review-exercises.md
+++ /dev/null
@@ -1,464 +0,0 @@
----
-id: opt-07
-title: "Part 2 Review - Optimization & Power Transfer"
-section: "Optimization & Simulation"
-difficulty: "intermediate"
-estimated_time: 60
-prerequisites: ["opt-01", "opt-02", "opt-03", "opt-04", "opt-05", "opt-06"]
-objectives:
- - Synthesize concepts from all optimization lessons
- - Apply multiple techniques to comprehensive design problems
- - Troubleshoot common optimization errors
- - Build complete optimization workflow
-tags: ["review", "comprehensive", "integration", "design"]
----
-
-# Part 2 Review - Optimization & Power Transfer
-
-This lesson integrates all concepts from Part 2, providing comprehensive exercises that require applying multiple techniques together.
-
-## Part 2 Summary: Key Concepts
-
-### Lesson 1: The Two Critical Resistances
-
-**R_opt_phase:**
-```
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-```
-- Minimizes impedance phase angle magnitude
-- Achieves φ_Z,min = -atan(2√[r(1+r)])
-- Makes impedance "most resistive" possible
-
-**R_opt_power:**
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-```
-- Maximizes real power transfer to load
-- Always smaller than R_opt_phase
-- Typical ratio: R_opt_power ≈ 0.5-0.7 × R_opt_phase
-
-**Topological constraint:**
-```
-If r = C_mut/C_sh > 0.207:
- Cannot achieve φ_Z = -45° (inherently capacitive)
-
-Most Tesla coils: r = 0.5 to 2.0 → φ_Z,min = -60° to -80°
-```
-
-### Lesson 2: The Hungry Streamer
-
-**Self-optimization mechanism:**
-1. Power → Joule heating
-2. Temperature → Ionization (exp(-E_i/kT))
-3. Ionization → Conductivity (σ ∝ n_e)
-4. Conductivity → Resistance (R = L/σA)
-5. Resistance → Circuit power
-6. **Feedback stabilizes at R ≈ R_opt_power**
-
-**Time scales:**
-- Thermal response: 0.1-1 ms (thin channels)
-- Ionization response: μs to ms
-- Can track kHz modulation, not RF cycles
-
-**Physical limits:**
-- R_min ≈ 1-10 kΩ (maximum conductivity)
-- R_max ≈ 100 kΩ to 100 MΩ (minimum conductivity)
-- Source limitations prevent optimization if insufficient power
-
-### Lesson 3-4: Thévenin Equivalent
-
-**Extraction:**
-```
-Z_th: Drive OFF, apply 1V test, measure I_test
- Z_th = 1V / I_test = R_th + jX_th
-
-V_th: Drive ON, no load, measure V_topload
-```
-
-**Using the equivalent:**
-```
-I = V_th / (Z_th + Z_load)
-V_load = V_th × Z_load / (Z_th + Z_load)
-P_load = 0.5 × |I|² × Re{Z_load}
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**Maximum power (conjugate match):**
-```
-Z_load = Z_th* → P_max = |V_th|² / (8 R_th)
-
-Usually unachievable due to topological constraints!
-```
-
-### Lesson 5: Direct Measurement
-
-**Alternative to Thévenin:**
-- Keep full coupled model
-- Measure power in spark directly
-- Sweep R, find maximum
-- Slower but handles nonlinearity
-
-**Best practice:**
-- Use Thévenin for exploration
-- Validate with direct measurement
-
-### Lesson 6: Frequency Tracking
-
-**Critical principle:**
-```
-For each R value, retune to loaded pole frequency!
-```
-
-**Why:**
-- Loading changes C_sh → shifts resonance
-- Typical shift: 10-30 kHz for medium sparks
-- Fixed-frequency comparison measures detuning, not matching
-
-**Loaded frequency:**
-```
-f_loaded = f₀ × √(C_total,0 / C_total,loaded)
-
-C_total,loaded = C_total,0 + C_sh
-```
-
-**DRSSTC modes:**
-- Fixed frequency: Simple, but detunes with loading
-- PLL tracking: Optimal, adapts in real-time
-- Programmed sweep: Compromise
-
-## Comprehensive Design Exercise
-
-**Scenario:** You're optimizing a medium DRSSTC for a 3-foot spark target.
-
-**Given System Parameters:**
-- Operating frequency: f ≈ 190 kHz (to be refined)
-- Topload: C_topload = 30 pF (measured)
-- Target spark: 3 feet
-- FEMM analysis gives: C_mut = 9 pF for 3-foot spark
-- Secondary stray capacitance: C_stray = 5 pF
-- Thévenin measurement (unloaded): Z_th = 105 - j2100 Ω at 200 kHz, V_th = 320 kV
-
-**Your tasks:** Work through the complete optimization workflow.
-
----
-
-### Task 1: Estimate Spark Capacitance
-
-Using the 2 pF/foot rule:
-
-**Question 1a:** What is C_sh for a 3-foot spark?
-
-**Question 1b:** What is the total secondary capacitance (unloaded)?
-
-**Question 1c:** What is the total capacitance with the 3-foot spark?
-
----
-
-### Task 2: Calculate Loaded Frequency
-
-**Question 2a:** If unloaded resonance is f₀ = 200 kHz, calculate the loaded resonance frequency with the 3-foot spark.
-
-**Question 2b:** What is the frequency shift Δf?
-
-**Question 2c:** If you operated at fixed f = 200 kHz (unloaded resonance), how detuned would you be? Express as a percentage of the original frequency.
-
----
-
-### Task 3: Determine Optimal Resistances
-
-**Question 3a:** Calculate R_opt_power at the loaded frequency (use result from Task 2).
-
-**Question 3b:** Calculate R_opt_phase at the loaded frequency.
-
-**Question 3c:** What is the ratio R_opt_power / R_opt_phase?
-
-**Question 3d:** Calculate the capacitance ratio r = C_mut / C_sh.
-
-**Question 3e:** Calculate the minimum achievable phase angle φ_Z,min. Can this system achieve -45°?
-
----
-
-### Task 4: Build Lumped Spark Model
-
-**Question 4a:** Draw the lumped spark circuit showing R, C_mut, and C_sh. Label all component values, using R = R_opt_power from Task 3a.
-
-**Question 4b:** Calculate the spark admittance Y_spark at the loaded frequency. Express in rectangular form (G + jB).
-
-**Question 4c:** Convert Y_spark to impedance Z_spark. Express in both polar and rectangular forms.
-
-**Question 4d:** Verify that the phase angle matches expectations from the topological constraint.
-
----
-
-### Task 5: Predict Performance with Thévenin
-
-Now use the Thévenin equivalent to predict performance. Adjust Z_th for the loaded frequency:
-
-**Note:** Z_th changes with frequency. For this exercise, assume:
-- Z_th ≈ 108 - j2050 Ω at f_loaded (slightly adjusted from 200 kHz value)
-- V_th ≈ 320 kV (approximately constant near resonance)
-
-**Question 5a:** Calculate the total impedance Z_total = Z_th + Z_spark.
-
-**Question 5b:** Calculate the current through the spark.
-
-**Question 5c:** Calculate the voltage across the spark.
-
-**Question 5d:** Calculate the real power dissipated in the spark.
-
-**Question 5e:** What percentage of V_th appears across the spark? Why is this ratio so high?
-
----
-
-### Task 6: Compare to Theoretical Maximum
-
-**Question 6a:** What load impedance would give conjugate match?
-
-**Question 6b:** Calculate P_max (maximum theoretical power with conjugate match).
-
-**Question 6c:** What percentage of P_max is actually delivered to the spark (from Task 5d)?
-
-**Question 6d:** Explain physically why the actual power is so much less than P_max. Why can't we achieve conjugate match?
-
----
-
-### Task 7: Frequency Tracking Impact
-
-Suppose you made a mistake and measured power at fixed f = 200 kHz instead of the loaded frequency.
-
-**Question 7a:** Estimate the voltage penalty factor. Assume Q_loaded ≈ 40 and use:
-```
-Voltage_ratio ≈ 1 / √[1 + (2Q × Δf/f)²]
-```
-
-**Question 7b:** How much would the measured power differ from the correctly tracked measurement?
-
-**Question 7c:** If you compared three different spark resistances at fixed f = 200 kHz, would you correctly identify R_opt_power? Why or why not?
-
----
-
-### Task 8: Self-Optimization Analysis
-
-**Question 8a:** Suppose the spark initially forms with R = 150 kΩ (cold streamer). Describe qualitatively what happens over the next 5-10 ms as the plasma heats up. Include R, T, σ, and P in your description.
-
-**Question 8b:** Why does the plasma naturally evolve toward R ≈ R_opt_power?
-
-**Question 8c:** If the calculated R_opt_power = 55 kΩ but physical limits give R_min = 80 kΩ, what would happen? Would the plasma reach R_opt_power?
-
-**Question 8d:** In burst mode with 50 μs pulses, would you expect the plasma to reach R_opt_power? Explain using thermal time constants.
-
----
-
-### Task 9: Alternative Measurement Validation
-
-You decide to validate your Thévenin results with direct power measurement.
-
-**Question 9a:** Describe the simulation setup for direct measurement. What components are included? What is varied?
-
-**Question 9b:** You sweep R from 20 kΩ to 120 kΩ. For each R value, should you:
-- (A) Measure at fixed f = 200 kHz?
-- (B) Sweep frequency to find loaded pole, then measure?
-
-Explain your choice.
-
-**Question 9c:** The direct measurement gives P_max at R = 58 kΩ, while your calculation gave R_opt_power = 55 kΩ. Is this agreement acceptable? What might explain the small difference?
-
----
-
-### Task 10: Design Recommendations
-
-Based on your analysis, provide design recommendations:
-
-**Question 10a:** What operating frequency should the DRSSTC use when driving this spark?
-
-**Question 10b:** Should the drive use fixed frequency or frequency tracking? Justify your recommendation.
-
-**Question 10c:** If using fixed frequency, what single frequency would you choose to balance unloaded and loaded operation?
-
-**Question 10d:** What power level should the primary tank be designed to deliver (approximately)?
-
-**Question 10e:** If you wanted a 4-foot spark instead, qualitatively describe how C_sh, f_loaded, R_opt_power, and delivered power would change.
-
----
-
-## Troubleshooting Common Errors
-
-### Error 1: "My calculated R_opt doesn't match simulation!"
-
-**Possible causes:**
-- Forgot to account for loaded frequency (used unloaded f₀)
-- Used wrong capacitance values (forgot C_stray or miscounted C_sh)
-- Simulation measured at wrong port (I_base instead of I_spark)
-- Simulation didn't converge properly
-
-**How to check:**
-- Verify C_total = C_topload + C_stray + C_sh
-- Verify ω = 2πf_loaded (not f₀!)
-- Plot power vs R to visually confirm peak location
-- Check simulation settings and convergence
-
-### Error 2: "Power is much lower than expected!"
-
-**Possible causes:**
-- Operating at wrong frequency (detuned)
-- High losses in simulation (R_th too large)
-- Incorrect power measurement (forgot factor of 0.5, or using wrong current)
-- Displacement currents included in measurement
-
-**How to check:**
-- Verify frequency matches loaded pole
-- Check Z_th extraction (is R_th reasonable? 10-100 Ω typical)
-- Verify power formula: P = 0.5 × I² × R for peak phasors
-- Measure current through R specifically, not total base current
-
-### Error 3: "Phase angle doesn't match theory!"
-
-**Possible causes:**
-- Using unloaded frequency instead of loaded
-- Incorrect capacitance ratio calculation
-- Measurement includes other components (not just spark)
-- Non-ideal behavior (resistance not purely in parallel with C_mut)
-
-**How to check:**
-- Recalculate r = C_mut/C_sh carefully
-- Verify φ_Z,min = -atan(2√[r(1+r)])
-- Check measurement port (topload to ground, not base)
-- Consider more complex model if simple lumped model doesn't fit
-
-### Error 4: "Conjugate match power is impossibly high!"
-
-**This is normal!** For Tesla coils:
-- Z_th has low R_th (10-100 Ω)
-- P_max = V_th²/(8R_th) can be tens or hundreds of MW
-- Sparks cannot achieve conjugate match (topological constraints)
-- Actual power is typically 0.01% to 1% of P_max
-
-**Not an error** - just shows extreme impedance mismatch is fundamental to Tesla coil operation.
-
-## Key Formulas Reference
-
-### Optimal Resistances
-```
-R_opt_power = 1 / [ω(C_mut + C_sh)]
-R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
-φ_Z,min = -atan(2√[r(1+r)]) where r = C_mut/C_sh
-```
-
-### Thévenin Equivalent
-```
-Z_th = 1V / I_test (drive OFF, 1V test source)
-V_th = V_topload (drive ON, no load)
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-P_max = |V_th|² / (8 R_th)
-```
-
-### Frequency Tracking
-```
-C_total,loaded = C_total,0 + C_sh
-f_loaded = f₀ √(C_total,0 / C_total,loaded)
-C_sh ≈ 2 pF/foot for typical sparks
-```
-
-### Lumped Model
-```
-Y_spark = [(G + jωC_mut) × jωC_sh] / [G + jω(C_mut + C_sh)]
-where G = 1/R
-```
-
-### Power Measurement
-```
-P = 0.5 × |I|² × Re{Z} (peak phasors)
-P = 0.5 × Re{V × I*} (complex power)
-```
-
-## Practice Problems - Solutions in Appendix
-
-### Problem Set A: Quick Calculations
-
-**A1.** Calculate R_opt_power for f = 180 kHz, C_mut = 7 pF, C_sh = 9 pF.
-
-**A2.** A spark has r = 1.5. Calculate φ_Z,min. Can it achieve -45°?
-
-**A3.** Z_th = 92 - j1950 Ω, V_th = 290 kV. Calculate P_max.
-
-**A4.** Unloaded f₀ = 205 kHz, C₀ = 32 pF. A 3.5-foot spark appears. Calculate f_loaded.
-
-**A5.** At f = 190 kHz with Q = 60, you're detuned by Δf = +8 kHz. Estimate the voltage penalty.
-
-### Problem Set B: Integration Problems
-
-**B1.** Complete Thévenin analysis:
-- Z_th = 115 - j2300 Ω, V_th = 340 kV
-- Spark: C_mut = 8 pF, C_sh = 5 pF, R = 65 kΩ, f = 188 kHz
-- Find: Current, voltage, power, compare to R_opt_power
-
-**B2.** Optimization with tracking:
-- f₀ = 198 kHz unloaded, C₀ = 28 pF
-- Test R = 40k, 60k, 80k with C_sh = 6 pF, C_mut = 9 pF
-- Calculate f_loaded for each R
-- Which R is closest to R_opt_power?
-
-**B3.** Self-optimization timeline:
-- R_opt_power = 70 kΩ, spark forms at R = 200 kΩ
-- Sketch R(t), P(t), T(t) vs time from t = 0 to 15 ms
-- Label key phases: initial, runaway, approach, equilibrium
-
-### Problem Set C: Design Challenges
-
-**C1.** Design matching for 4-foot target:
-- Given: f = 185 kHz, C_topload = 35 pF, C_stray = 6 pF
-- Determine: C_sh, C_total, f_loaded, R_opt_power, R_opt_phase
-- Build lumped model and calculate Z_spark
-
-**C2.** Frequency tracking implementation:
-- Coil operates 170-210 kHz range
-- Sparks vary from 2 to 5 feet
-- Calculate frequency range needed
-- Recommend: fixed frequency, PLL, or sweep?
-
-**C3.** Troubleshooting:
-- Simulation shows maximum power at R = 45 kΩ
-- Analytical R_opt_power = 62 kΩ
-- What could explain the discrepancy? List 3 possible causes and how to verify each.
-
----
-
-## Transition to Part 3
-
-You now have a complete toolkit for optimization and power transfer analysis:
-- Understanding the two critical resistances
-- Physical self-optimization mechanism
-- Thévenin equivalent extraction and use
-- Direct measurement validation
-- Frequency tracking principles
-
-**Part 3** builds on this foundation to explore:
-- Spark growth physics and field requirements
-- FEMM modeling for capacitance extraction
-- Energy budgets and growth rates
-- Voltage vs power limits
-- Complete growth simulations
-
-The optimization techniques from Part 2 combine with the growth physics of Part 3 to enable **full spark length prediction**.
-
----
-
-## Key Takeaways
-
-- **Two optimizations:** R_opt_power (max power) and R_opt_phase (min phase) are different
-- **Self-optimization:** Plasma naturally seeks R ≈ R_opt_power via thermal feedback
-- **Thévenin method:** Extract once, predict any load instantly
-- **Direct measurement:** Slower but handles nonlinearity, good for validation
-- **Frequency tracking is critical:** Must retune for each load to avoid detuning errors
-- **Topological constraints:** Most Tesla coils cannot achieve -45°, inherently capacitive
-- **Conjugate match unachievable:** Sparks operate far from theoretical maximum power
-- **Complete workflow:** Capacitance → frequency → R_opt → lumped model → power prediction
-
-## Practice
-
-{exercise:opt-ex-07}
-
-Work through the Comprehensive Design Exercise (Tasks 1-10) completely. Show all calculations and reasoning. Compare your results with the solutions appendix.
-
----
-**Next Section:** [Part 3: Spark Growth Physics and FEMM Modeling](../../03-spark-physics/01-electric-fields.md)
diff --git a/spark-lessons/lessons/03-spark-physics/01-field-thresholds.md b/spark-lessons/lessons/03-spark-physics/01-field-thresholds.md
deleted file mode 100644
index 3140b0a..0000000
--- a/spark-lessons/lessons/03-spark-physics/01-field-thresholds.md
+++ /dev/null
@@ -1,263 +0,0 @@
----
-id: phys-01
-title: "Electric Field Thresholds for Breakdown"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 35
-prerequisites: ["fund-07", "opt-07"]
-objectives:
- - Understand the electric field requirements for air breakdown
- - Calculate average and tip electric fields from voltage and geometry
- - Apply tip enhancement factors to predict spark inception
- - Determine when sparks can continue growing vs when they stall
-tags: ["electric-field", "breakdown", "tip-enhancement", "E-field", "threshold"]
----
-
-# Electric Field Thresholds for Breakdown
-
-Understanding electric fields is fundamental to predicting spark behavior. A spark will only initiate and grow when the electric field strength exceeds specific thresholds. This lesson covers the critical field values and how to calculate them.
-
-## Electric Field Basics
-
-**Definition:** The electric field E is force per unit charge:
-
-```
-E = F/q [units: N/C or V/m]
-```
-
-The electric field is related to voltage through the gradient:
-
-```
-E = -dV/dx (field is voltage gradient)
-```
-
-For a uniform field between parallel plates:
-
-```
-E ≈ V/d (voltage divided by distance)
-```
-
-**Critical insight:** The field at a spark tip is NOT uniform - it is concentrated by the sharp geometry.
-
-## Breakdown Field Thresholds
-
-Two key field thresholds govern spark behavior:
-
-### E_inception: Initial Breakdown Field
-
-**E_inception** is the field required to initiate breakdown from a smooth electrode:
-
-```
-E_inception ≈ 2-3 MV/m (at sea level, dry air)
-```
-
-**Physical process:**
-1. Natural cosmic rays create seed electrons
-2. Strong field accelerates these electrons
-3. High-energy electrons collide with air molecules
-4. Collisions ionize more atoms (avalanche breakdown)
-5. Breakdown begins when ionization exceeds losses
-
-### E_propagation: Sustained Growth Field
-
-**E_propagation** is the field required to sustain spark growth after initiation:
-
-```
-E_propagation ≈ 0.4-1.0 MV/m (for leader propagation)
-```
-
-**Why is E_propagation < E_inception?**
-- The channel is already partially ionized
-- Hot gas is easier to ionize than cold air
-- Photoionization helps (UV from plasma creates seed electrons ahead)
-- Thermal effects reduce the energy barrier
-
-### Environmental Effects
-
-Field thresholds vary with atmospheric conditions:
-
-**Altitude effects:**
-- Lower air density → lower E_threshold
-- Variation: ±20-30% from sea level to moderate altitude
-- Higher altitude → easier breakdown (less air to ionize)
-
-**Humidity effects:**
-- Water vapor changes breakdown characteristics
-- Typical variation: ~10%
-- Complex effects: water molecules have different ionization energy
-
-**Temperature effects:**
-- Affects air density
-- Small effect compared to altitude/humidity
-
-## Tip Enhancement Factor (κ)
-
-Sharp tips concentrate the electric field dramatically. The **tip enhancement factor** κ quantifies this concentration:
-
-```
-E_tip = κ × E_average
-
-where:
- E_average = V/L (voltage divided by spark length)
- κ = enhancement factor ≈ 2-5 typical
-```
-
-### Physical Origin of Enhancement
-
-**Why do tips concentrate field?**
-1. Charge accumulates at sharp points (boundary condition)
-2. Field lines must be perpendicular to conductor surfaces
-3. Closer spacing of equipotential lines near high curvature
-4. Smaller radius of curvature → higher κ
-
-**Typical values:**
-- Smooth sphere: κ ≈ 1.0 (no enhancement)
-- Mild tip (radius ~cm): κ ≈ 2-3
-- Sharp tip (radius ~mm): κ ≈ 3-5
-- Very sharp needle: κ ≈ 5-10
-
-**FEMM calculates E_tip directly** from geometry and voltage, eliminating the need to estimate κ.
-
-## Growth Criterion
-
-A spark continues growing when:
-
-```
-E_tip > E_propagation
-```
-
-**When growth stalls:**
-
-```
-If E_tip < E_propagation:
- - Growth stalls
- - Spark cannot extend further
- - System is "voltage-limited"
- - More power doesn't help without more voltage
-```
-
-**Practical implications:**
-- Small topload → lower voltage → shorter maximum length
-- Long target spark requires higher voltage to maintain E_tip
-- Enhancement factor κ helps by concentrating field at tip
-- But κ decreases as tip becomes less sharp
-
----
-
-## WORKED EXAMPLE 3.1: Field Calculation
-
-**Given:**
-- Spark length: L = 1.5 m
-- Topload voltage: V_top = 400 kV
-- Tip enhancement: κ = 3.5 (from FEMM or estimate)
-
-**Find:**
-(a) Average field
-(b) Tip field
-(c) Can spark grow if E_propagation = 0.6 MV/m?
-
-### Solution
-
-**Part (a): Average field**
-
-```
-E_average = V_top / L
- = 400×10³ V / 1.5 m
- = 267 kV/m
- = 0.267 MV/m
-```
-
-**Part (b): Tip field**
-
-```
-E_tip = κ × E_average
- = 3.5 × 0.267 MV/m
- = 0.93 MV/m
-```
-
-**Part (c): Compare to threshold**
-
-```
-E_tip = 0.93 MV/m
-E_propagation = 0.6 MV/m
-
-E_tip > E_propagation ✓
-
-Yes, spark can continue growing.
-Safety margin: 0.93/0.6 = 1.55× above threshold
-```
-
-**If voltage drops to 300 kV:**
-
-```
-E_average = 300 kV / 1.5 m = 0.2 MV/m
-E_tip = 3.5 × 0.2 = 0.7 MV/m
-
-Still above 0.6 MV/m, but margin reduced to 1.17×
-```
-
-**If voltage drops to 250 kV:**
-
-```
-E_average = 250 kV / 1.5 m = 0.167 MV/m
-E_tip = 3.5 × 0.167 = 0.58 MV/m
-
-Below 0.6 MV/m - growth stalls!
-```
-
-**Key insight:** Even moderate voltage reduction can cause growth to stall. Maintaining adequate voltage throughout the ramp is critical for long sparks.
-
----
-
-## Visual Understanding: Field Enhancement
-
-Imagine two scenarios:
-
-**LEFT: Uniform field (parallel plates)**
-- Two flat plates with voltage V between them
-- Evenly spaced field lines (vertical)
-- Formula: E = V/d (constant everywhere)
-- No enhancement: κ = 1
-
-**RIGHT: Point-to-plane (spark geometry)**
-- Spherical topload at top (voltage V)
-- Sharp spark tip pointing down
-- Ground plane at bottom
-- Field lines:
- - Sparse near topload (low field density)
- - Highly concentrated at tip (high field density)
- - Spread out below tip
-- Color gradient showing field strength:
- - Blue (low field) far from tip
- - Red (high field) at tip
-- E_average = V/L along spark
-- E_tip at very tip (red zone)
-- Enhancement: E_tip = κ × E_average, κ = 2-5
-
-**Field vs distance from tip:** Sharp peak at tip, drops rapidly with distance, approaches E_average far from tip.
-
-{image:field-enhancement-comparison}
-
----
-
-## Key Takeaways
-
-- **E_inception ≈ 2-3 MV/m**: Required to start breakdown from smooth surface
-- **E_propagation ≈ 0.4-1.0 MV/m**: Required to sustain spark growth (lower than inception)
-- **Tip enhancement**: E_tip = κ × E_average, where κ ≈ 2-5 for typical geometries
-- **Growth criterion**: Spark grows when E_tip > E_propagation, stalls when E_tip < E_propagation
-- **Environmental effects**: Altitude and humidity affect thresholds by ±20-30%
-- **FEMM advantage**: Directly computes E_tip from geometry, no need to estimate κ
-
-## Practice
-
-{exercise:phys-ex-01}
-
-**Problem 1:** A 0.8 m spark has V_top = 280 kV and κ = 4. Calculate E_tip. If E_propagation = 0.5 MV/m, can it grow?
-
-**Problem 2:** A spark stalls at 2.0 m length with V_top = 500 kV and κ = 3. Estimate E_propagation for these conditions.
-
-**Problem 3:** Why is E_inception > E_propagation? Explain the physical difference in 2-3 sentences.
-
----
-**Next Lesson:** [Voltage-Limited Length](02-voltage-limits.md)
diff --git a/spark-lessons/lessons/03-spark-physics/02-voltage-limits.md b/spark-lessons/lessons/03-spark-physics/02-voltage-limits.md
deleted file mode 100644
index 8eadf3a..0000000
--- a/spark-lessons/lessons/03-spark-physics/02-voltage-limits.md
+++ /dev/null
@@ -1,275 +0,0 @@
----
-id: phys-02
-title: "Maximum Voltage-Limited Length"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 30
-prerequisites: ["phys-01", "opt-07"]
-objectives:
- - Understand what causes voltage-limited spark growth
- - Calculate maximum achievable spark length for given voltage
- - Use FEMM to compute tip fields for realistic geometries
- - Recognize when more power cannot help extend sparks
-tags: ["voltage-limit", "FEMM", "E-field", "maximum-length", "altitude"]
----
-
-# Maximum Voltage-Limited Length
-
-Even with unlimited power, a spark cannot grow indefinitely. The maximum length is determined by the **voltage-limited condition**: when the tip field drops below the propagation threshold, growth stalls regardless of available power.
-
-## The Voltage-Limited Condition
-
-A spark is **voltage-limited** when:
-
-```
-E_tip < E_propagation
-```
-
-Under this condition:
-- Field at tip is too weak to sustain ionization
-- Spark cannot extend further
-- Growth rate: dL/dt = 0 (stalled)
-- More power doesn't help (without more voltage)
-- Common scenario: small topload, long target length
-
-**Contrast with power-limited:**
-- E_tip > E_propagation (field is adequate)
-- But P_stream < ε × (dL/dt)_desired
-- Spark grows slowly or stalls before reaching potential
-- More voltage doesn't help (without more power)
-- Common scenario: high-Q coils, weak drive
-
-## Calculating Maximum Length
-
-The maximum voltage-limited length L_max occurs when:
-
-```
-E_tip(V_top, L_max) = E_propagation
-```
-
-Using the tip enhancement approximation:
-
-```
-κ × (V_top / L_max) = E_propagation
-
-Solving for L_max:
-L_max = κ × V_top / E_propagation
-```
-
-**Important caveats:**
-- This assumes κ remains constant (simplification)
-- Reality: κ decreases as spark grows and tip becomes less sharp
-- Capacitive voltage division reduces V_tip (covered in Lesson 07)
-- Best accuracy: use FEMM to compute E_tip(V_top, L) iteratively
-
-### FEMM Field Computation
-
-**Finite Element Method Magnetics (FEMM)** provides accurate field calculations:
-
-**Workflow:**
-1. Define geometry (topload, spark channel, ground)
-2. Set boundary conditions (V_top on topload, 0V on ground)
-3. Mesh and solve Laplace's equation (∇²V = 0)
-4. Extract E_tip at spark endpoint
-5. Check: E_tip ≥ E_propagation?
-
-**Advantages over analytical formulas:**
-- Accounts for realistic topload geometry (toroids, spheres)
-- Includes ground plane proximity effects
-- Automatically computes κ from geometry
-- Handles multiple conductors and complex shapes
-
-**Iterative approach for L_max:**
-```
-1. Start with initial guess: L = L_guess
-2. Run FEMM with topload at V_top and spark length L
-3. Extract E_tip from FEMM results
-4. Compare E_tip to E_propagation:
- - If E_tip > E_propagation: try longer L
- - If E_tip < E_propagation: try shorter L
-5. Repeat until E_tip ≈ E_propagation (within tolerance)
-6. Result: L_max
-```
-
-## Altitude and Environmental Effects
-
-The propagation threshold E_propagation varies with environmental conditions:
-
-### Altitude Effects
-
-**Lower air density at higher altitude:**
-
-```
-ρ_air ∝ exp(-h/H) where H ≈ 8.5 km (scale height)
-
-E_propagation ∝ ρ_air
-
-Typical variation: ±20-30% from sea level to moderate altitude
-```
-
-**Practical implications:**
-- At 1500 m elevation: E_propagation reduced by ~15%
-- Same voltage produces ~15% longer sparks
-- Important for coilers at altitude to adjust expectations
-
-**Example:**
-```
-Sea level (ρ = 1.0): E_propagation = 0.6 MV/m
-1500 m (ρ ≈ 0.85): E_propagation ≈ 0.51 MV/m
-
-For V_top = 400 kV, κ = 3:
-Sea level: L_max = 3 × 400 kV / 0.6 MV/m = 2.0 m
-1500 m: L_max = 3 × 400 kV / 0.51 MV/m = 2.35 m (17% longer)
-```
-
-### Humidity Effects
-
-**Water vapor changes breakdown characteristics:**
-- Typical variation: ~10%
-- Less significant than altitude
-- Complex dependency on partial pressure
-
-### Temperature Effects
-
-**Affects air density:**
-- ρ_air ∝ 1/T (ideal gas law)
-- Small effect: ~10-15% from winter to summer
-- Usually overshadowed by altitude effects
-
-## Common Misconceptions
-
-**Misconception 1:** "More power always makes longer sparks"
-
-**Reality:** If voltage-limited, adding power just makes the spark brighter/hotter but not longer. Both adequate voltage AND adequate power are required.
-
-**Misconception 2:** "κ is constant for a given coil"
-
-**Reality:** κ changes as the spark grows. Initial sharp tip has high κ, but as spark extends and tip becomes less defined, κ decreases. This further limits maximum length.
-
-**Misconception 3:** "Small topload is fine if I have enough power"
-
-**Reality:** Small topload limits maximum voltage capability. Even unlimited power cannot overcome voltage limitation from inadequate topload capacitance.
-
----
-
-## WORKED EXAMPLE: Maximum Length Calculation
-
-**Given:**
-- Topload voltage capability: V_top_max = 500 kV
-- Tip enhancement factor: κ = 3.2 (estimated for this geometry)
-- Propagation threshold: E_propagation = 0.7 MV/m (sea level)
-- Same coil operated at 1500 m altitude
-
-**Find:**
-(a) Maximum spark length at sea level
-(b) Maximum spark length at 1500 m (assume E_propagation reduced by 15%)
-(c) Voltage required for 3 m spark at sea level
-
-### Solution
-
-**Part (a): Sea level maximum length**
-
-```
-L_max = κ × V_top_max / E_propagation
- = 3.2 × 500 kV / 0.7 MV/m
- = 3.2 × 500×10³ V / (0.7×10⁶ V/m)
- = 1600 kV / 700 kV/m
- = 2.29 m
-
-Maximum spark length ≈ 2.3 m
-```
-
-**Part (b): 1500 m altitude**
-
-At altitude, E_propagation reduced by 15%:
-
-```
-E_propagation(1500m) = 0.7 MV/m × 0.85 = 0.595 MV/m
-
-L_max = 3.2 × 500 kV / 0.595 MV/m
- = 1600 kV / 595 kV/m
- = 2.69 m
-
-Maximum spark length ≈ 2.7 m (17% longer than sea level)
-```
-
-**Part (c): Voltage for 3 m at sea level**
-
-Rearrange the equation:
-
-```
-V_required = E_propagation × L_target / κ
- = 0.7 MV/m × 3 m / 3.2
- = 2.1 MV / 3.2
- = 0.656 MV
- = 656 kV
-
-Need 656 kV to reach 3 m at sea level
-This exceeds V_top_max = 500 kV
-Therefore 3 m is not achievable with current topload
-```
-
-**Conclusion:** To reach 3 m at sea level, need to:
-- Increase topload size (higher voltage capability), OR
-- Operate at altitude (lower E_propagation), OR
-- Improve tip enhancement (sharper geometry, higher κ)
-
----
-
-## FEMM Tutorial Concept
-
-While detailed FEMM usage is beyond this lesson, here's the conceptual workflow:
-
-**Problem setup (axisymmetric):**
-```
-Geometry in r-z coordinates:
-- Toroid: major radius 20 cm, minor radius 7 cm, center at z = 0
-- Spark: cylinder radius 1 mm, extends from toroid to length L
-- Ground plane: large disk at z = -L - 30 cm
-- Outer boundary: large box (r = 150 cm, z = ±200 cm)
-
-Materials:
-- Air everywhere (ε_r = 1.0)
-
-Boundaries:
-- r = 0: Axisymmetric boundary (symmetry axis)
-- Outer box: V = 0 V (Dirichlet, grounded far field)
-- Topload surface: V = V_top
-- Ground plane: V = 0 V
-
-Solve:
-- Laplace equation: ∇²V = 0
-- Extract E_tip at spark endpoint
-```
-
-**Reading results:**
-- FEMM displays field magnitude |E| as color contours
-- Highest concentration (red) at spark tip
-- Extract numerical value at tip location
-- Compare to E_propagation threshold
-
-{image:femm-field-plot-example}
-
----
-
-## Key Takeaways
-
-- **Voltage-limited**: Growth stalls when E_tip < E_propagation, regardless of available power
-- **Maximum length**: L_max ≈ κ × V_top / E_propagation (simplified formula)
-- **FEMM accuracy**: Finite element analysis accounts for realistic geometry and provides E_tip directly
-- **Altitude benefit**: Lower air density reduces E_propagation by ~20-30%, enabling longer sparks
-- **Design implication**: Both adequate voltage AND adequate power are necessary for target length
-- **κ is not constant**: Tip enhancement decreases as spark grows, further limiting length
-
-## Practice
-
-{exercise:phys-ex-02}
-
-**Problem 1:** A coil has V_top = 350 kV, κ = 3.5, and E_propagation = 0.6 MV/m. Calculate L_max. If operating at 2000 m altitude (E_propagation reduced 20%), what is the new L_max?
-
-**Problem 2:** FEMM simulation shows E_tip = 0.55 MV/m for a 2.5 m spark at V_top = 450 kV. If E_propagation = 0.6 MV/m, what happens? Estimate the maximum length this voltage can support if κ ≈ 3.
-
-**Problem 3:** Explain why having 100 kW of available power doesn't guarantee a 3 m spark if the topload can only reach 400 kV. Use the concepts of voltage-limited vs power-limited growth.
-
----
-**Next Lesson:** [Energy Per Meter Concept](03-energy-per-meter.md)
diff --git a/spark-lessons/lessons/03-spark-physics/03-energy-per-meter.md b/spark-lessons/lessons/03-spark-physics/03-energy-per-meter.md
deleted file mode 100644
index d734e39..0000000
--- a/spark-lessons/lessons/03-spark-physics/03-energy-per-meter.md
+++ /dev/null
@@ -1,359 +0,0 @@
----
-id: phys-03
-title: "Energy Per Meter Concept"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 40
-prerequisites: ["phys-01", "phys-02"]
-objectives:
- - Understand the concept of energy per meter (ε) for spark growth
- - Apply the growth rate equation dL/dt = P/ε
- - Calculate total energy and average power for target spark length
- - Recognize the difference between theoretical minimum and practical ε values
-tags: ["energy-per-meter", "epsilon", "growth-rate", "power", "ionization"]
----
-
-# Energy Per Meter Concept
-
-Extending a spark requires energy. Surprisingly, the energy needed is approximately **constant per unit length**, regardless of how long the spark already is. This fundamental concept enables practical spark growth modeling.
-
-## The Energy Per Meter Parameter (ε)
-
-**Definition:** ε (epsilon) is the energy required to extend a spark by one meter.
-
-```
-Energy to grow from L₁ to L₂:
-ΔE ≈ ε × (L₂ - L₁) [Joules]
-
-where ε has units [J/m]
-```
-
-**Key characteristics:**
-- Approximately constant for a given operating mode
-- Independent of current spark length (first-order approximation)
-- Depends strongly on operating regime (QCW vs burst)
-- Empirical parameter that must be calibrated per coil
-
-**Why is this useful?**
-- Simple relationship: energy scales linearly with length
-- Easy to calculate power requirements
-- Enables growth rate predictions
-- Separates voltage limit (field) from power limit (energy)
-
-## What Does ε Include?
-
-The energy per meter is **NOT** just the ionization energy. It includes all energy processes:
-
-### 1. Initial Ionization
-Breaking molecular bonds to create ions and free electrons:
-```
-E_ionize ≈ 15 eV per molecule
-```
-
-### 2. Heating to Operating Temperature
-Raising channel temperature from ambient to 5,000-20,000 K:
-```
-E_thermal = m × c_p × ΔT
-```
-
-### 3. Work Against Pressure
-Expanding the channel against atmospheric pressure:
-```
-E_expansion = P × ΔV
-```
-
-### 4. Radiation Losses
-Emitted light, UV, infrared, and RF:
-```
-E_radiation = ∫ σ T⁴ dA dt (blackbody + line emission)
-```
-
-### 5. Branching Losses
-Energy wasted in short branches that don't contribute to main channel:
-```
-E_branching = ε × L_branches (failed growth attempts)
-```
-
-### 6. General Inefficiencies
-Non-productive heating, turbulence, and other losses:
-```
-E_losses = various mechanisms
-```
-
-**Result:** Practical ε is 20-300× larger than theoretical ionization minimum!
-
-## Theoretical Minimum Energy
-
-Let's estimate the absolute minimum energy needed for ionization alone:
-
-**Given:**
-- Ionization energy per molecule: ~15 eV
-- Air density: n ≈ 2.5×10²⁵ molecules/m³
-- Channel diameter: d = 1 mm (typical)
-- Length increment: ΔL = 1 m
-
-**Calculation:**
-
-```
-Volume of 1 m channel:
-V = π(d/2)² × L = π(0.5×10⁻³)² × 1 = 7.85×10⁻⁷ m³
-
-Number of molecules:
-N = n × V = 2.5×10²⁵ × 7.85×10⁻⁷ = 1.96×10¹⁹ molecules
-
-Energy to ionize:
-E_min = N × 15 eV × (1.6×10⁻¹⁹ J/eV)
- = 1.96×10¹⁹ × 15 × 1.6×10⁻¹⁹
- = 0.47 J/m
-
-Theoretical minimum: ε_theory ≈ 0.3-0.5 J/m
-```
-
-**Why is practical ε so much higher?**
-
-Compare to real values:
-- QCW: ε ≈ 5-15 J/m (10-30× theoretical)
-- Burst mode: ε ≈ 30-100 J/m (60-200× theoretical)
-
-The difference accounts for:
-- Heating to high temperature (major contribution)
-- Radiation losses (visible light alone is significant)
-- Expansion work (pushing air aside)
-- Branching inefficiency (many failed paths)
-- Re-ionization (especially in pulsed modes)
-
-## The Growth Rate Equation
-
-When the field threshold is met (E_tip > E_propagation), the growth rate is determined by power:
-
-```
-dL/dt = P_stream / ε [m/s]
-
-where:
- P_stream = power delivered to spark [W]
- ε = energy per meter [J/m]
-```
-
-**Physical interpretation:**
-- More power → faster growth
-- Higher ε (inefficiency) → slower growth for same power
-- Linear relationship: double power → double growth rate
-
-**When growth stops:**
-
-```
-If E_tip < E_propagation:
- dL/dt = 0 (stalled)
-
-Cannot grow regardless of available power
-(voltage-limited condition)
-```
-
-### Predicting Growth Time
-
-For constant power during ramp:
-
-```
-Growth rate: dL/dt = P_stream / ε
-
-Integrating: L(t) = (P_stream / ε) × t
-
-Time to reach target length:
-T = ε × L_target / P_stream
-```
-
-**More realistic scenario:** Power changes as spark grows (loading changes):
-
-```
-T = ∫₀^L_target (ε / P_stream(L)) dL
-
-Requires simulation or numerical integration
-```
-
----
-
-## WORKED EXAMPLE 3.2: Energy Budget
-
-**Given:**
-- Target spark length: L = 2 m
-- Operating mode: QCW with ε = 10 J/m
-- Growth time: T = 12 ms
-
-**Find:**
-(a) Total energy required
-(b) Average power required
-(c) If 80 kW is available, what changes?
-
-### Solution
-
-**Part (a): Total energy**
-
-```
-E_total = ε × L
- = 10 J/m × 2 m
- = 20 J
-```
-
-Remarkably modest! Only 20 J to create a 2 m spark.
-
-**Part (b): Average power**
-
-```
-P_avg = E_total / T
- = 20 J / 0.012 s
- = 1,667 W
- ≈ 1.7 kW
-```
-
-For 12 ms growth, need ~1.7 kW average power.
-
-**Part (c): With 80 kW available**
-
-Available power is 80 kW, but only need 1.7 kW!
-
-```
-Power ratio: 80 kW / 1.7 kW = 47× more than needed
-```
-
-**Option 1: Grow much faster**
-```
-T_min = E_total / P_available
- = 20 J / 80,000 W
- = 0.00025 s
- = 0.25 ms (burst-like growth)
-```
-
-**Option 2: Grow to longer length (in same 12 ms)**
-```
-L_max_power = P_available × T / ε
- = 80,000 W × 0.012 s / 10 J/m
- = 960 J / 10 J/m
- = 96 m (!!)
-```
-
-**Reality check:** 96 m is absurd! What limits this?
-
-**Voltage limit kicks in first:**
-- Cannot maintain E_tip > E_propagation for 96 m
-- Spark stalls at voltage-limited length
-- Typical: L_max ≈ 2-4 m for practical topload voltages
-
-**Key insight:** Tesla coils are almost always **voltage-limited**, not power-limited. Excess power goes into brightening, heating, and branching rather than length.
-
----
-
-## WORKED EXAMPLE 3.3: Comparing Operating Modes
-
-**Given:**
-- Two coils both deliver P = 50 kW average
-- Coil A: QCW mode, ε_A = 8 J/m
-- Coil B: Burst mode, ε_B = 50 J/m
-- Both operate for T = 10 ms
-
-**Find:** Which produces longer sparks?
-
-### Solution
-
-**Coil A (QCW):**
-
-```
-L_A = P × T / ε_A
- = 50,000 W × 0.010 s / 8 J/m
- = 500 J / 8 J/m
- = 62.5 m (voltage-limited in practice)
-```
-
-**Coil B (Burst):**
-
-```
-L_B = P × T / ε_B
- = 50,000 W × 0.010 s / 50 J/m
- = 500 J / 50 J/m
- = 10 m (still voltage-limited in practice)
-```
-
-**Comparison:**
-
-```
-Ratio: L_A / L_B = ε_B / ε_A = 50/8 = 6.25×
-
-QCW coil produces 6.25× longer sparks for same power!
-```
-
-**Practical reality:**
-- Both limited by voltage before reaching these lengths
-- But ratio still applies: QCW gives much better length efficiency
-- Coil A might reach 2.5 m while Coil B reaches 0.4 m
-- Burst mode wastes energy on brightness and branching
-
-**Why choose burst mode then?**
-- Spectacular brightness and branches (visual appeal)
-- Higher peak current (electromagnetic effects)
-- Simpler drive electronics
-- Better for musical/modulated output
-- Different aesthetic goals than pure length
-
----
-
-## Power-Limited vs Voltage-Limited
-
-Understanding the interplay between power and voltage limits:
-
-### Voltage-Limited Condition
-```
-E_tip < E_propagation
-- Field too weak at tip
-- Spark cannot extend
-- More power → brighter/hotter, not longer
-- Common for Tesla coils
-```
-
-### Power-Limited Condition
-```
-E_tip > E_propagation, but P_stream insufficient
-- Field adequate but not enough energy
-- Spark grows slowly or stalls before reaching potential
-- More voltage doesn't help without more power
-- Less common for Tesla coils (usually have excess power)
-```
-
-### Practical Implications
-
-**For most Tesla coils:**
-1. Design for adequate voltage (large topload, high primary voltage)
-2. Ensure sufficient power (but don't need enormous amounts)
-3. Optimize ε by choosing appropriate operating mode
-4. Accept that voltage limit dominates final length
-
-**Rule of thumb:**
-- If P × T / ε >> L_actual, you're voltage-limited
-- If P × T / ε ≈ L_actual, you might be power-limited
-- Most coils fall in first category (voltage-limited)
-
----
-
-## Key Takeaways
-
-- **ε definition**: Energy per meter [J/m], approximately constant for a given mode
-- **Growth rate**: dL/dt = P/ε when field threshold is met
-- **Total energy**: E_total ≈ ε × L (linear scaling)
-- **Theoretical minimum**: ε_theory ≈ 0.3-0.5 J/m (ionization only)
-- **Practical values**: 10-300× higher than theoretical (includes heating, radiation, losses)
-- **Operating mode matters**: QCW has low ε (efficient), burst has high ε (inefficient)
-- **Voltage limit dominates**: Most Tesla coils have more than enough power, limited by voltage
-
-## Practice
-
-{exercise:phys-ex-03}
-
-**Problem 1:** A burst-mode coil has ε = 60 J/m. To reach L = 1.5 m in a 200 μs pulse, what power is required? Is this realistic?
-
-**Problem 2:** A QCW coil delivers 30 kW average power for 15 ms with ε = 12 J/m. Calculate:
-(a) Total energy delivered
-(b) Maximum length if power-limited
-(c) If actual length is only 1.8 m, what does this tell you?
-
-**Problem 3:** Explain why practical ε is 50-100× larger than the theoretical ionization minimum. List at least three major energy sinks.
-
----
-**Next Lesson:** [Empirical ε Values](04-empirical-epsilon.md)
diff --git a/spark-lessons/lessons/03-spark-physics/04-empirical-epsilon.md b/spark-lessons/lessons/03-spark-physics/04-empirical-epsilon.md
deleted file mode 100644
index 6141087..0000000
--- a/spark-lessons/lessons/03-spark-physics/04-empirical-epsilon.md
+++ /dev/null
@@ -1,404 +0,0 @@
----
-id: phys-04
-title: "Empirical ε Values and Calibration"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 35
-prerequisites: ["phys-03"]
-objectives:
- - Learn typical ε values for different operating modes
- - Understand why QCW, DRSSTC, and burst modes have different ε
- - Calibrate ε from experimental measurements
- - Apply thermal accumulation effects to refine ε predictions
-tags: ["epsilon", "calibration", "QCW", "DRSSTC", "burst-mode", "thermal-accumulation"]
----
-
-# Empirical ε Values and Calibration
-
-The energy per meter (ε) is not a universal constant - it depends strongly on the operating mode. Understanding typical values and calibration methods is essential for accurate spark growth modeling.
-
-## Typical ε Values by Operating Mode
-
-### QCW (Quasi-Continuous Wave)
-
-**ε ≈ 5-15 J/m**
-
-**Characteristics:**
-- Long ramp times: 5-20 ms
-- Channel stays hot throughout growth
-- Efficient leader formation
-- Minimal re-ionization needed
-- Each joule efficiently extends length
-
-**Why low ε (efficient)?**
-- Continuous power maintains channel ionization
-- Thermal ionization kept active
-- Leaders form and persist
-- Minimal energy wasted on re-starting
-
-**Typical coil parameters:**
-- Medium-high power: 10-100 kW
-- Moderate duty cycle: 1-10%
-- Linear voltage ramp
-- Long sparks: 2-5+ m
-
-### Hybrid DRSSTC (Moderate Duty Cycle)
-
-**ε ≈ 20-40 J/m**
-
-**Characteristics:**
-- Medium pulse lengths: 1-5 ms
-- Mix of streamers and leaders
-- Some thermal accumulation between pulses
-- Moderate efficiency
-
-**Why moderate ε?**
-- Not quite continuous like QCW
-- Some cooling between bursts
-- Partial re-ionization required
-- Both streamer and leader mechanisms active
-
-**Typical coil parameters:**
-- High power: 50-200 kW peak
-- Moderate duty cycle: 5-15%
-- Partial interrupter control
-- Good balance: length and brightness
-
-### Burst Mode (Hard-Pulsed)
-
-**ε ≈ 30-100+ J/m**
-
-**Characteristics:**
-- Short pulses: <500 μs typical
-- Channel cools between pulses
-- Mostly streamers, bright but short
-- Must re-ionize repeatedly
-- Poor length efficiency
-
-**Why high ε (inefficient)?**
-- Peak power → intense brightening and branching
-- Channel cools between bursts (ms timescale)
-- Energy dumped into light and heat, not length
-- Must restart from cold each time
-- High ionization overhead
-
-**Typical coil parameters:**
-- Very high peak power: 100-500+ kW
-- Low duty cycle: 0.1-2%
-- Bang energy: 10-100+ J per burst
-- Short sparks: 0.5-2 m despite high energy
-
-### Single-Shot Impulse
-
-**ε ≈ 50-150+ J/m**
-
-**Characteristics:**
-- One-time discharge (capacitor bank)
-- No thermal memory from previous events
-- All energy must come from single pulse
-- Very high ε due to complete inefficiency
-
-**Why very high ε?**
-- Starting from completely cold air
-- No accumulated ionization
-- Transient streamer formation
-- Most energy into flash and noise
-
-## Physical Explanation for ε Differences
-
-### QCW Efficiency (Low ε)
-
-**Energy flow:**
-```
-1. Initial streamers form (t = 0)
-2. Current flows → Joule heating (t = 0-1 ms)
-3. Channel heats → thermal ionization (t = 1-2 ms)
-4. Leader forms from base (t = 2-5 ms)
-5. Leader maintained by continuous power (t = 5-20 ms)
-6. New growth builds on existing hot ionization
-7. Minimal wasted energy
-```
-
-**Result:** Each joule goes into extending the channel, not re-creating what already exists.
-
-### Burst Inefficiency (High ε)
-
-**Energy flow:**
-```
-1. Pulse creates bright streamer (t = 0-100 μs)
-2. Pulse ends, no more power (t = 100 μs)
-3. Channel begins cooling (t = 0.1-1 ms)
-4. Thermal diffusion and convection cool channel
-5. Ionization recombines
-6. Next pulse must re-ionize cold gas (t = 1-10 ms)
-7. Energy wasted heating the same air repeatedly
-```
-
-**Result:** Energy into brightening and repeated ionization overhead, not cumulative length.
-
-### Analogy: Boiling Water
-
-**Low ε (QCW):**
-- Keep burner on continuously
-- Maintain simmer (steady state)
-- Efficient: minimal energy to maintain temperature
-
-**High ε (Burst):**
-- Pulse burner on/off repeatedly
-- Water cools between pulses
-- Inefficient: must reheat repeatedly
-
-## Calibration Procedure
-
-To calibrate ε for your specific coil:
-
-### Step 1: Measure Delivered Energy
-
-**From SPICE simulation:**
-```
-E_delivered = ∫ P_spark(t) dt
-
-where P_spark = instantaneous power to spark
-Integration from t = 0 to end of ramp
-```
-
-**From measurements (if available):**
-```
-E_delivered ≈ E_capacitor - E_losses
-
-where E_capacitor = ½ C_primary V_primary²
- E_losses = resistive, core, switching losses
-```
-
-### Step 2: Measure Final Spark Length
-
-**Direct measurement:**
-- Photograph spark with scale reference
-- Measure from topload to tip
-- Average over multiple runs (sparks vary!)
-- Use median or typical length, not maximum outlier
-
-**Typical measurement uncertainty:**
-- ±10-20% due to spark variability
-- Branching makes "length" ambiguous
-- Use main channel length
-
-### Step 3: Calculate ε
-
-```
-ε = E_delivered / L_final [J/m]
-
-Example:
-E_delivered = 45 J (from SPICE)
-L_final = 1.8 m (measured)
-
-ε = 45 J / 1.8 m = 25 J/m
-```
-
-### Step 4: Verify and Refine
-
-**Repeat for different power levels:**
-- Change primary voltage or pulse width
-- Measure new E_delivered and L_final
-- Calculate ε for each run
-- Average to get robust estimate
-
-**Check for consistency:**
-- ε should be approximately constant (±30%)
-- Large variations indicate:
- - Voltage-limited at some power levels
- - Thermal accumulation effects
- - Operating mode changes
-
-## Thermal Accumulation Effects
-
-For more advanced modeling, ε can decrease during long ramps due to thermal accumulation:
-
-```
-ε(t) = ε₀ / (1 + α × ∫P_stream dt)
-
-where:
- ε₀ = initial energy per meter [J/m]
- α = thermal accumulation factor [1/J]
- ∫P_stream dt = accumulated energy [J]
-```
-
-**Physical meaning:**
-- As channel heats up, ionization becomes easier
-- Less energy needed per meter as temperature rises
-- ε decreases with accumulated heating
-
-**Typical values:**
-- ε₀ ≈ 15 J/m (initial, cold start)
-- α ≈ 0.01-0.05 [1/J]
-- After 50 J accumulated: ε ≈ 15/(1 + 0.03×50) = 6 J/m
-
-**When to use:**
-- Long QCW ramps (>10 ms)
-- High accumulated energy (>30 J)
-- For short bursts: ε ≈ ε₀ (constant)
-
-**Simplified model:**
-Most practitioners use constant ε for simplicity:
-- Choose ε representing average over ramp
-- Simpler and usually adequate
-- Advanced users can implement ε(t) in simulation
-
----
-
-## WORKED EXAMPLE: Calibration from Data
-
-**Given:**
-Three experimental runs on a QCW coil:
-
-| Run | V_primary | E_delivered | L_measured |
-|-----|-----------|-------------|------------|
-| 1 | 200 V | 25 J | 2.2 m |
-| 2 | 250 V | 38 J | 3.1 m |
-| 3 | 300 V | 55 J | 4.5 m |
-
-**Find:**
-(a) Calculate ε for each run
-(b) Average ε for this coil
-(c) Assess consistency
-
-### Solution
-
-**Part (a): ε for each run**
-
-```
-Run 1: ε₁ = E₁ / L₁ = 25 J / 2.2 m = 11.4 J/m
-Run 2: ε₂ = E₂ / L₂ = 38 J / 3.1 m = 12.3 J/m
-Run 3: ε₃ = E₃ / L₃ = 55 J / 4.5 m = 12.2 J/m
-```
-
-**Part (b): Average ε**
-
-```
-ε_avg = (ε₁ + ε₂ + ε₃) / 3
- = (11.4 + 12.3 + 12.2) / 3
- = 12.0 J/m
-
-Recommended value: ε ≈ 12 J/m
-```
-
-**Part (c): Consistency assessment**
-
-```
-Standard deviation: σ ≈ 0.5 J/m
-Coefficient of variation: CV = σ/μ = 0.5/12 = 4.2%
-
-Excellent consistency! (<5% variation)
-```
-
-**Interpretation:**
-- ε is nearly constant across power range
-- Coil is NOT voltage-limited in this range
-- Pure power-limited growth (field threshold always met)
-- Can confidently use ε = 12 J/m for predictions
-
-**If we saw large variation:**
-```
-Example: ε₁ = 10 J/m, ε₂ = 15 J/m, ε₃ = 30 J/m
-
-This would indicate:
-- Run 3 hitting voltage limit (inefficient growth)
-- Possible mode transition (streamers vs leaders)
-- Need to reassess model assumptions
-```
-
----
-
-## WORKED EXAMPLE: Predicting Performance Change
-
-**Given:**
-- Current coil: Burst mode, ε = 65 J/m, E_bang = 80 J, L_typical = 1.2 m
-- Proposed upgrade: Convert to QCW with ε = 12 J/m, same E_total = 80 J
-
-**Find:**
-(a) Predicted length after QCW conversion
-(b) Percentage improvement
-(c) Required power for 10 ms ramp
-
-### Solution
-
-**Part (a): Predicted QCW length**
-
-```
-L_QCW = E_total / ε_QCW
- = 80 J / 12 J/m
- = 6.67 m
-
-Predicted length ≈ 6.7 m
-```
-
-**Part (b): Improvement**
-
-```
-Improvement = (L_QCW - L_burst) / L_burst × 100%
- = (6.67 - 1.2) / 1.2 × 100%
- = 456% increase in length!
-
-Or: 6.67/1.2 = 5.6× longer sparks
-```
-
-**Part (c): Required power**
-
-```
-For 10 ms ramp:
-P_avg = E_total / T_ramp
- = 80 J / 0.010 s
- = 8,000 W
- = 8 kW average
-
-Peak power higher (depends on waveform)
-Typical: P_peak ≈ 1.5-2 × P_avg ≈ 12-16 kW
-```
-
-**Reality check:**
-- 6.7 m prediction assumes NOT voltage-limited
-- Actual length limited by topload voltage capability
-- Still expect major improvement over burst mode
-- Might achieve 3-4 m instead of 6.7 m (voltage limit)
-
----
-
-## Summary Table: ε by Operating Mode
-
-| Mode | ε Range [J/m] | Characteristics | Best For |
-|------|---------------|-----------------|----------|
-| **QCW** | 5-15 | Efficient leaders, long ramps | Maximum length |
-| **DRSSTC Hybrid** | 20-40 | Mixed streamers/leaders | Balanced length & brightness |
-| **Burst Mode** | 30-100+ | Bright streamers, short pulses | Visual spectacle, music |
-| **Single-Shot** | 50-150+ | One-time discharge | Impulse testing, demonstrations |
-
-**Choosing operating mode:**
-- **Goal: Length** → QCW (low ε)
-- **Goal: Brightness** → Burst (high peak power)
-- **Goal: Music/modulation** → Burst (rapid on/off)
-- **Goal: Efficiency** → QCW (low ε, lower losses)
-
----
-
-## Key Takeaways
-
-- **QCW: ε ≈ 5-15 J/m** - Most efficient, maintains hot channel
-- **Hybrid DRSSTC: ε ≈ 20-40 J/m** - Moderate efficiency, mixed mechanisms
-- **Burst mode: ε ≈ 30-100+ J/m** - Least efficient, repeated re-ionization
-- **Calibration**: ε = E_delivered / L_measured from experimental runs
-- **Consistency check**: ε should be approximately constant if power-limited
-- **Thermal accumulation**: Advanced models use ε(t) decreasing with heating
-- **Operating mode choice**: Trades off length efficiency vs brightness/aesthetics
-
-## Practice
-
-{exercise:phys-ex-04}
-
-**Problem 1:** A coil delivers 60 J in burst mode and produces 0.9 m sparks. Calculate ε. If converted to QCW with same energy, estimate new length assuming ε = 10 J/m.
-
-**Problem 2:** Calibration runs give: ε₁ = 14 J/m (25 J delivered), ε₂ = 13 J/m (40 J), ε₃ = 28 J/m (90 J). What does the sudden increase in ε₃ suggest?
-
-**Problem 3:** Explain why burst mode has higher ε than QCW despite delivering the same total energy. What happens to the "wasted" energy?
-
----
-**Next Lesson:** [Thermal Memory Effects](05-thermal-memory.md)
diff --git a/spark-lessons/lessons/03-spark-physics/05-thermal-memory.md b/spark-lessons/lessons/03-spark-physics/05-thermal-memory.md
deleted file mode 100644
index 5845485..0000000
--- a/spark-lessons/lessons/03-spark-physics/05-thermal-memory.md
+++ /dev/null
@@ -1,460 +0,0 @@
----
-id: phys-05
-title: "Thermal Memory and Channel Persistence"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 40
-prerequisites: ["phys-03", "phys-04"]
-objectives:
- - Understand thermal diffusion time constants for plasma channels
- - Calculate channel persistence times for different diameters
- - Recognize the role of convection in extending channel lifetime
- - Apply thermal memory concepts to QCW vs burst mode operation
-tags: ["thermal-diffusion", "convection", "channel-persistence", "time-constants", "ionization-memory"]
----
-
-# Thermal Memory and Channel Persistence
-
-Once formed, a plasma channel doesn't instantly disappear. It has **thermal memory** - the channel stays hot and partially ionized for some time after power is removed. Understanding these timescales is crucial for optimizing operating modes.
-
-## Temperature Regimes
-
-Plasma channels exist in different temperature regimes depending on current and power density:
-
-### Streamers (Cold Plasma)
-
-```
-Temperature: T ≈ 1000-3000 K
-- Weakly ionized (few % ionization)
-- Mostly neutral gas with some ions/electrons
-- Purple/blue color (N₂ molecular emission)
-- Low conductivity
-```
-
-### Leaders (Hot Plasma)
-
-```
-Temperature: T ≈ 5000-20,000 K
-- Fully ionized plasma
-- White/orange color (blackbody + line emission)
-- High conductivity
-- Approaching temperatures of stellar photospheres!
-```
-
-**Temperature comparison:**
-- Room temperature: 300 K
-- Candle flame: 1500 K
-- Thin streamers: 1000-3000 K
-- Thick leaders: 5000-20,000 K
-- Sun's photosphere: 5800 K
-
-Leaders are literally as hot as the surface of the Sun!
-
-## Thermal Diffusion Time
-
-Heat diffuses radially outward from the hot channel core according to:
-
-```
-τ_thermal = d² / (4α_thermal)
-
-where:
- d = channel diameter [m]
- α_thermal ≈ 2×10⁻⁵ m²/s (thermal diffusivity of air)
-```
-
-**Physical meaning:** Time for heat to diffuse a distance d through air by conduction.
-
-### Examples for Different Channel Sizes
-
-**Thin streamer (d = 100 μm):**
-
-```
-τ = (100×10⁻⁶)² / (4 × 2×10⁻⁵)
- = 10⁻⁸ m² / (8×10⁻⁵ m²/s)
- = 1.25×10⁻⁴ s
- = 0.125 ms
- ≈ 0.1-0.2 ms
-```
-
-**Medium channel (d = 2 mm):**
-
-```
-τ = (2×10⁻³)² / (4 × 2×10⁻⁵)
- = 4×10⁻⁶ m² / (8×10⁻⁵ m²/s)
- = 0.05 s
- = 50 ms
-```
-
-**Thick leader (d = 5 mm):**
-
-```
-τ = (5×10⁻³)² / (4 × 2×10⁻⁵)
- = 25×10⁻⁶ m² / (8×10⁻⁵ m²/s)
- = 0.3125 s
- = 312 ms
- ≈ 0.3-0.6 s
-```
-
-**Key insight:** Thermal diffusion time scales as d² - thicker channels persist much longer!
-
-## Why Observed Persistence is Longer
-
-Pure thermal diffusion predicts cooling in 0.1-300 ms, but channels persist longer due to additional effects:
-
-### 1. Convection (Buoyancy)
-
-Hot gas is less dense and rises:
-
-```
-Buoyancy velocity: v ≈ √(g × d × ΔT/T_amb)
-
-where:
- g = 9.8 m/s² (gravity)
- d = channel diameter
- ΔT = temperature excess above ambient
- T_amb = ambient temperature (≈300 K)
-```
-
-**Example: 2 mm channel at ΔT = 10,000 K**
-
-```
-v ≈ √(9.8 × 0.002 × 10000/300)
- ≈ √(9.8 × 0.002 × 33.3)
- ≈ √(0.653)
- ≈ 0.81 m/s
-```
-
-The hot channel rises at ~0.8 m/s, creating a continuously renewing hot column!
-
-**Effect on persistence:**
-- Rising column remains coherent (doesn't diffuse sideways as fast)
-- Maintains hot gas path for seconds
-- Why Tesla coil sparks leave visible "smoke trails"
-- Enhances thermal memory significantly
-
-### 2. Ionization Memory
-
-Even after thermal cooling begins, ions and electrons persist:
-
-```
-Recombination time: τ_recomb = 1/(α_recomb × n_e)
-
-where:
- α_recomb ≈ 10⁻¹³ m³/s (recombination coefficient)
- n_e = electron density [m⁻³]
-
-Typical: τ_recomb ≈ 10 μs to 10 ms
-```
-
-**Effect on persistence:**
-- Channel remains partially ionized after cooling
-- Lower resistance than cold air
-- Easier to re-ionize than virgin air
-- "Memory" of previous discharge path
-
-### 3. Broadened Effective Diameter
-
-Turbulence and mixing increase effective channel size:
-
-```
-d_effective > d_initial (due to turbulence)
-
-Larger diameter → longer τ_thermal
-```
-
-## Effective Persistence Times
-
-Combining all effects:
-
-**Thin streamers:**
-```
-Pure thermal: ~0.1-0.2 ms
-With convection: ~1-5 ms
-Ionization memory: ~0.1-1 ms
-Effective persistence: ~1-5 ms
-```
-
-**Thick leaders:**
-```
-Pure thermal: ~50-300 ms
-With convection: seconds (buoyant column maintained)
-Ionization memory: ~1-10 ms
-Effective persistence: seconds
-```
-
-**Visual evidence:** High-speed photography shows spark channels glowing and rising for seconds after power is removed.
-
-{image:spark-channel-persistence-sequence}
-
-## QCW Advantage
-
-QCW ramp times (5-20 ms) are designed to exploit channel persistence:
-
-### Timeline of QCW Growth
-
-```
-t = 0 ms:
- - Initial streamers form from topload
- - Thin, fast, purple channels
- - Temperature: ~2000 K
-
-t = 0.5-1 ms:
- - Current begins flowing through streamers
- - Joule heating: P = I²R
- - Temperature rising
-
-t = 1-2 ms:
- - Channel heats to 5000+ K
- - Thermal ionization becomes dominant
- - Leader formation begins at base
-
-t = 2-5 ms:
- - Leader established and growing
- - Hot channel maintained by continuous power
- - New growth builds on existing ionization
- - Temperature: 10,000-20,000 K
-
-t = 5-20 ms:
- - Leader continues extending
- - Persistence time >> growth time
- - Channel stays hot entire duration
- - Efficient energy use: no re-ionization needed
-
-t > 20 ms (after ramp ends):
- - Power removed
- - Channel begins cooling
- - Buoyancy carries hot gas upward
- - Visible glow for seconds
-```
-
-**Key advantage:** The ramp duration (5-20 ms) is shorter than thermal diffusion time (50+ ms for leaders), so the channel NEVER cools during growth!
-
-### Energy Efficiency Mechanism
-
-**QCW flow:**
-```
-Energy → Initial ionization (startup cost)
- → Heating to leader temperature
- → Maintaining hot channel (low cost)
- → Extending length (efficient)
-
-Result: Most energy after startup goes into extension
-ε_QCW ≈ 5-15 J/m (low, efficient)
-```
-
-## Burst Mode Problem
-
-Burst mode pulses are short (50-500 μs) with long gaps (ms):
-
-### Timeline of Burst Mode
-
-```
-t = 0 μs:
- - High voltage, cold air
- - Streamer inception
-
-t = 0-100 μs:
- - First pulse (high peak power)
- - Bright streamers form
- - Some heating but limited
- - Temperature reaches ~3000-5000 K
-
-t = 100 μs (pulse ends):
- - Power removed
- - Channel begins cooling immediately
- - Thermal diffusion time ~0.1-0.5 ms for thin channels
-
-t = 0.1-1 ms:
- - Channel cools significantly
- - Temperature drops to ~1000 K
- - Ionization recombines
- - Channel approaching cold air
-
-t = 1-10 ms (between pulses):
- - Next pulse arrives
- - Must re-ionize mostly cold gas
- - Energy wasted on re-heating
- - Little thermal memory remains
-
-Result: Each pulse restarts from nearly cold conditions!
-```
-
-**Energy inefficiency mechanism:**
-
-```
-Energy → Initial ionization (EVERY pulse)
- → Heating (REPEATED)
- → Brief brightening
- → Cooling (wasted)
- → Re-ionization overhead (high)
-
-Result: Energy into repeated startup, not cumulative growth
-ε_burst ≈ 30-100+ J/m (high, inefficient)
-```
-
-### Analogy: Boiling Water
-
-**QCW (efficient):**
-```
-Turn stove on and keep it on
-Water heats up once
-Maintain boiling continuously
-Minimal energy to sustain
-```
-
-**Burst (inefficient):**
-```
-Pulse stove on/off rapidly
-Water heats briefly
-Water cools between pulses
-Must reheat repeatedly
-High energy for little sustained boiling
-```
-
----
-
-## WORKED EXAMPLE: Thermal Time Constants
-
-**Given:**
-- Channel diameter: d = 2 mm (typical leader)
-- Air thermal diffusivity: α = 2×10⁻⁵ m²/s
-- Temperature excess: ΔT = 8000 K
-- Ambient temperature: T_amb = 300 K
-
-**Find:**
-(a) Pure thermal diffusion time
-(b) Convection velocity
-(c) QCW ramp time recommendation
-
-### Solution
-
-**Part (a): Thermal diffusion time**
-
-```
-τ_thermal = d² / (4α)
- = (2×10⁻³)² / (4 × 2×10⁻⁵)
- = 4×10⁻⁶ m² / (8×10⁻⁵ m²/s)
- = 0.05 s
- = 50 ms
-```
-
-**Part (b): Convection velocity**
-
-```
-v ≈ √(g × d × ΔT/T_amb)
- ≈ √(9.8 × 0.002 × 8000/300)
- ≈ √(9.8 × 0.002 × 26.67)
- ≈ √(0.523)
- ≈ 0.72 m/s
-```
-
-Upward velocity of ~0.7 m/s helps maintain hot column.
-
-**Part (c): QCW ramp recommendation**
-
-```
-τ_thermal = 50 ms
-
-For efficient QCW operation:
-T_ramp << τ_thermal (finish before significant cooling)
-
-Recommended: T_ramp = 0.1 × τ to 0.4 × τ
- = 5-20 ms
-
-Sweet spot: ~10 ms (20% of τ_thermal)
-```
-
-**Reasoning:**
-- If T_ramp >> τ_thermal (e.g., 200 ms):
- - Channel cools during ramp
- - Must reheat repeatedly
- - Loses QCW efficiency advantage
-
-- If T_ramp << τ_thermal (e.g., 1 ms):
- - May not form thick leaders
- - Closer to burst behavior
- - Doesn't exploit full persistence
-
-- Optimal: T_ramp ≈ 10-20 ms
- - Channel stays hot throughout
- - Leaders form and persist
- - Maximum efficiency
-
----
-
-## WORKED EXAMPLE: Burst vs QCW Timing
-
-**Given:**
-- Burst pulse: 200 μs every 5 ms (5 ms period)
-- QCW ramp: 15 ms continuous
-- Both use same average power
-
-**Find:**
-(a) Why burst is inefficient for thin channels (d = 100 μm)
-(b) Why QCW is efficient for thick channels (d = 3 mm)
-
-### Solution
-
-**Part (a): Burst with thin streamers**
-
-```
-Channel diameter: d = 100 μm
-Thermal time: τ = (100×10⁻⁶)² / (8×10⁻⁵) = 0.125 ms
-
-Timeline:
-t = 0: Pulse starts, channel forms
-t = 200 μs: Pulse ends (0.2 ms)
- Channel cooling for: 0.125 ms ≈ τ/1
-t = 5 ms: Next pulse
- Channel has cooled for: 5 ms = 40 × τ
- COMPLETELY COLD
-
-Result: Each pulse re-ionizes from scratch
- High ε (inefficient)
-```
-
-**Part (b): QCW with thick leaders**
-
-```
-Channel diameter: d = 3 mm
-Thermal time: τ = (3×10⁻³)² / (8×10⁻⁵) = 112 ms
-
-Timeline:
-t = 0: Ramp starts, initial streamers
-t = 2 ms: Heating → leader formation begins
-t = 5 ms: Leader well-established (hot)
-t = 15 ms: Ramp ends
- Total time elapsed: 15 ms = 0.13 × τ
-
-Cooling fraction: exp(-15/112) ≈ exp(-0.13) ≈ 0.88
-
-Result: Channel stays at 88% of peak temperature!
- Leader persists throughout ramp
- Low ε (efficient)
-```
-
----
-
-## Key Takeaways
-
-- **Thermal diffusion time**: τ = d²/(4α), scales quadratically with diameter
-- **Thin streamers**: τ ≈ 0.1-0.2 ms (fast cooling)
-- **Thick leaders**: τ ≈ 50-600 ms (slow cooling)
-- **Convection**: Hot gas rises at ~0.5-1 m/s, maintains hot column for seconds
-- **Ionization memory**: Partial ionization persists 0.1-10 ms after thermal cooling
-- **Effective persistence**: 1-5 ms for streamers, seconds for leaders
-- **QCW advantage**: Ramp time (5-20 ms) << leader thermal time (~50+ ms)
-- **Burst problem**: Gap between pulses (ms) >> streamer thermal time (~0.1 ms)
-
-## Practice
-
-{exercise:phys-ex-05}
-
-**Problem 1:** A streamer has d = 150 μm. Calculate τ_thermal. If burst pulse width is 500 μs with 10 ms between pulses, does the channel cool significantly?
-
-**Problem 2:** Why do thick leaders persist longer than thin streamers? Give two physical reasons with approximate timescales.
-
-**Problem 3:** A QCW coil uses 25 ms ramps. For a 3 mm diameter leader (τ ≈ 100 ms), estimate the fraction of peak temperature remaining at end of ramp (use exponential cooling approximation).
-
----
-**Next Lesson:** [Streamers vs Leaders](06-streamers-vs-leaders.md)
diff --git a/spark-lessons/lessons/03-spark-physics/06-streamers-vs-leaders.md b/spark-lessons/lessons/03-spark-physics/06-streamers-vs-leaders.md
deleted file mode 100644
index 1d8bbc4..0000000
--- a/spark-lessons/lessons/03-spark-physics/06-streamers-vs-leaders.md
+++ /dev/null
@@ -1,441 +0,0 @@
----
-id: phys-06
-title: "Streamers vs Leaders: Transition Sequence"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 45
-prerequisites: ["phys-05"]
-objectives:
- - Distinguish between streamer and leader discharge mechanisms
- - Understand the 6-step streamer-to-leader transition sequence
- - Recognize the efficiency differences between streamer and leader growth
- - Apply this knowledge to optimize coil operating modes
-tags: ["streamers", "leaders", "photoionization", "thermal-ionization", "transition", "mechanisms"]
----
-
-# Streamers vs Leaders: Transition Sequence
-
-Not all sparks are created equal. Two fundamentally different propagation mechanisms exist: **streamers** and **leaders**. Understanding the differences and transition between them is crucial for optimizing Tesla coil performance.
-
-## Streamer Characteristics
-
-**Streamers** are thin, fast, cold plasma channels:
-
-### Physical Properties
-
-```
-Diameter: 10-100 μm (thinner than human hair)
-Velocity: ~10⁶ m/s (1% speed of light!)
-Temperature: 1000-3000 K (weakly ionized)
-Current: mA to tens of mA (low)
-Resistance: MΩ range (high)
-Thermal time: ~0.1-0.2 ms (fast cooling)
-```
-
-### Propagation Mechanism: Photoionization
-
-**How streamers propagate:**
-
-1. **Electric field accelerates electrons** in partially ionized tip region
-2. **Energetic electrons collide** with neutral molecules, creating excited states
-3. **Excited molecules emit UV photons** (de-excitation radiation)
-4. **UV photons travel ahead** of the streamer tip (speed of light)
-5. **UV ionizes neutral air ahead** (photoelectric effect), creating seed electrons
-6. **Seed electrons avalanche** in high field at tip
-7. **New ionized region forms** ahead of previous tip
-8. **Process repeats** → rapid propagation
-
-**Key insight:** Propagation driven by photons (electromagnetic radiation), not thermal effects. This is why streamers are FAST - limited only by ionization avalanche time, not thermal diffusion.
-
-### Visual Appearance
-
-```
-Color: Purple/blue (N₂ molecular emission lines)
-Structure: Highly branched, tree-like
-Persistence: Brief flashes (<1 ms visible)
-Brightness: Moderate (low current)
-Pattern: Random, fractal-like branching
-```
-
-### Energy Efficiency
-
-```
-ε_streamer ≈ 50-150+ J/m (high, inefficient)
-
-Energy distribution:
-- Ionization: ~1%
-- Radiation (UV, visible): ~30-50%
-- Heating: ~20-40%
-- Branching losses: ~20-40%
-- Extension: ~5-10% (poor efficiency!)
-```
-
-**Why inefficient?**
-- Energy dumped into radiation (bright UV and visible light)
-- Massive branching (many failed paths)
-- Low current → high resistance → voltage drop limits length
-- No thermal memory between events
-
-## Leader Characteristics
-
-**Leaders** are thick, slower, hot plasma channels:
-
-### Physical Properties
-
-```
-Diameter: 1-10 mm (visible as bright core)
-Velocity: ~10³ m/s (walking speed to car speed)
-Temperature: 5000-20,000 K (fully ionized)
-Current: 100 mA to several A (high)
-Resistance: kΩ range (low)
-Thermal time: ~50-600 ms (slow cooling)
-```
-
-### Propagation Mechanism: Thermal Ionization
-
-**How leaders propagate:**
-
-1. **High current flows** through existing channel
-2. **Joule heating** (I²R) raises channel temperature
-3. **Thermal ionization** occurs as temperature exceeds ~5000 K
- - Collisional ionization from thermal energy
- - Lower resistance as more ions/electrons created
-4. **Hot channel tip** heats adjacent air by conduction/radiation
-5. **Adjacent air ionizes** thermally
-6. **Leader extends** into newly ionized region
-7. **Process repeats** → steady growth
-
-**Key insight:** Propagation driven by heat transfer (thermal effects), much slower than photoionization. But more efficient energy use - heat stays in channel.
-
-### Visual Appearance
-
-```
-Color: White/orange (blackbody + line emission)
-Structure: Straighter, fewer branches
-Persistence: Seconds with sustained power (or buoyant rise)
-Brightness: Very bright (high current)
-Pattern: More directed, follows field lines
-```
-
-### Energy Efficiency
-
-```
-ε_leader ≈ 5-20 J/m (low, efficient)
-
-Energy distribution:
-- Ionization: ~5-10%
-- Heating to operating T: ~30-50%
-- Extension work: ~20-40%
-- Radiation: ~10-20%
-- Branching: ~5-10% (minimal)
-```
-
-**Why efficient?**
-- Heat stays in channel (thermal memory)
-- High current → low resistance → efficient power transfer
-- Straighter path (less branching waste)
-- Thermal ionization more efficient than repeated photoionization
-- Energy accumulates in single hot channel
-
-## Comparison Table
-
-| Property | Streamers | Leaders |
-|----------|-----------|---------|
-| **Diameter** | 10-100 μm | 1-10 mm |
-| **Velocity** | ~10⁶ m/s | ~10³ m/s |
-| **Temperature** | 1000-3000 K | 5000-20,000 K |
-| **Current** | mA | 100 mA - A |
-| **Resistance** | MΩ | kΩ |
-| **Color** | Purple/blue | White/orange |
-| **Branching** | Highly branched | Straighter |
-| **Persistence** | <1 ms | Seconds |
-| **Mechanism** | Photoionization | Thermal ionization |
-| **ε (J/m)** | 50-150+ | 5-20 |
-| **Efficiency** | Poor | Good |
-
-## The 6-Step Transition Sequence
-
-Streamers can transition to leaders if sufficient current and time are provided:
-
-### Step 1: High E-Field Creates Initial Streamers
-
-```
-t = 0 μs
-- High voltage applied to topload
-- E_tip exceeds E_inception (~2-3 MV/m)
-- Photoionization avalanche begins
-- Multiple thin streamers form from topload
-- Characteristics: Fast, purple, branched
-- Temperature: ~2000 K
-- Current: mA per streamer
-```
-
-### Step 2: Sufficient Current Flows → Joule Heating
-
-```
-t = 10-100 μs
-- Circuit provides sustained current (not just brief discharge)
-- Current concentrates in one or few dominant streamers
-- Joule heating: P = I²R
-- Channel temperature begins rising
-- Temperature: 2000 → 3000 K
-- Resistance begins decreasing
-```
-
-### Step 3: Heated Channel → Thermal Ionization Begins
-
-```
-t = 100 μs - 1 ms
-- Temperature reaches ~5000 K (thermal ionization threshold)
-- Collisional ionization adds to photoionization
-- Ionization density increases dramatically
-- Resistance drops further → more current → more heating
-- Positive feedback loop: heat → ionization → conductivity → current → heat
-- Temperature: 3000 → 8000 K
-- Current increasing to 100+ mA
-```
-
-### Step 4: Leader Forms from Base
-
-```
-t = 1-3 ms
-- Hottest region (base, near topload) becomes fully ionized
-- True leader channel established at base
-- Leader characteristics appear: thick, white, hot
-- Temperature: 8000 → 15,000 K at base
-- Current: several 100 mA
-- Diameter expands to ~1-3 mm
-```
-
-**Critical insight:** Leader forms **from base** (topload) and grows **downward**, not from tip!
-
-### Step 5: Leader Tip Launches New Streamers
-
-```
-t = 3-10 ms
-- Hot leader base established
-- Leader tip (interface) still has high E-field
-- Tip launches new streamers ahead (photoionization)
-- Streamers probe forward, find path
-- Temperature gradient: 15,000 K (base) → 5000 K (tip) → 2000 K (streamers)
-```
-
-### Step 6: Fed Streamers Convert to Leader
-
-```
-t = 5-20 ms (continuous process)
-- Current flows through newly formed streamers
-- Streamers heat up → thermal ionization
-- Hot leader channel "catches up" to streamer paths
-- Leader extends forward
-- Process repeats: tip launches streamers → streamers heat → leader extends
-- Continuous growth cycle
-
-Final state:
-- Main channel: hot leader (white, thick, efficient)
-- Active tip: transition zone with streamers
-- Failed branches: cool streamers (purple, thin)
-```
-
-{image:streamer-to-leader-transition-sequence}
-
-## Why This Transition Matters
-
-### For QCW Coils (Designed for Leader Formation)
-
-```
-Timeline optimized for transition:
-t = 0-1 ms: Streamer inception
-t = 1-5 ms: Transition to leader
-t = 5-20 ms: Leader growth dominates
-Result: Low ε (5-15 J/m), long sparks
-```
-
-**QCW design requirements:**
-- Sustained current capability (not just brief pulse)
-- Moderate ramp time (5-20 ms allows transition)
-- Adequate voltage maintenance
-- Result: Efficient leader formation
-
-### For Burst Mode (Mostly Streamers)
-
-```
-Timeline too short for transition:
-t = 0-50 μs: Streamer inception
-t = 50-200 μs: Brief heating begins
-t = 200 μs: Pulse ends (typical)
-t = 200 μs - 5 ms: Cooling (no power)
-Result: High ε (30-100+ J/m), short bright sparks
-```
-
-**Burst mode characteristics:**
-- High peak power creates bright streamers
-- Pulse too short for full leader transition
-- Channel cools between pulses
-- Next pulse restarts from streamers
-- Result: Spectacular but inefficient
-
-### Hybrid Modes (Mixed Behavior)
-
-```
-Timeline allows partial transition:
-t = 0-0.5 ms: Streamers
-t = 0.5-2 ms: Partial leader formation at base
-t = 2-5 ms: Mixed streamer/leader growth
-Result: Moderate ε (20-40 J/m), balanced performance
-```
-
-## Physical Intuition: The "Thermal Runway"
-
-Think of the transition as climbing a thermal runway:
-
-**Altitude (Temperature) vs Time:**
-
-```
-0 K ▬▬▬▬▬ Ground (cold air, insulator)
-
-2000 K ━━━━━ Streamer plateau (photoionization)
- ▲
- │ Need sustained current to climb
- │
-5000 K ━━━━━ Leader threshold (thermal ionization begins)
- ▲
- │ Positive feedback: easier to climb
- │
-15000 K ━━━━━ Fully developed leader
-
- Time →
-```
-
-**Burst mode:** Brief rocket boost (high power) gets to 2000 K, but fuel runs out (pulse ends) before reaching 5000 K. Falls back to ground.
-
-**QCW mode:** Sustained climb (continuous power) reaches 5000 K and beyond. Once at leader plateau, stays there efficiently.
-
-## Practical Observations
-
-### High-Speed Photography Evidence
-
-Time-resolved imaging shows:
-
-**0-100 μs:**
-- Multiple thin purple streamers from topload
-- Branching, exploring paths
-- No thick core visible
-
-**1-3 ms:**
-- White glow appearing near topload
-- Base region brightening
-- Purple streamers still at extremities
-
-**5-20 ms:**
-- Thick white core from topload partway down
-- Purple streamers at tip only
-- Clear leader/streamer boundary
-
-**After power off:**
-- White leader core persists (seconds, rising)
-- Purple streamers disappear immediately
-
-{image:high-speed-photography-leader-formation}
-
-### Energy Measurements
-
-Direct calorimetry and electrical measurements confirm:
-
-```
-Same total energy (100 J):
-
-Burst mode: 100 J → 1.2 m spark
- ε ≈ 83 J/m
- Mostly streamers
-
-QCW mode: 100 J → 8 m spark
- ε ≈ 12.5 J/m
- Mostly leaders
-
-Ratio: 6.7× better length efficiency for leaders!
-```
-
----
-
-## WORKED EXAMPLE: Estimating Transition Time
-
-**Given:**
-- Initial streamer resistance: R₀ = 10 MΩ
-- Initial current: I₀ = 20 mA (from voltage source)
-- Power deposition: P = I²R = (0.02)² × 10×10⁶ = 4000 W
-- Channel mass per meter: m ≈ 0.001 kg/m (100 μm diameter, 1 m long)
-- Heat capacity of air: c_p ≈ 1000 J/(kg·K)
-- Target temperature for leader: T_leader = 5000 K (from T_amb = 300 K)
-
-**Find:** Estimated heating time to leader threshold (simplified model)
-
-### Solution
-
-```
-Energy required to heat channel:
-Q = m × c_p × ΔT
- = 0.001 kg/m × 1000 J/(kg·K) × (5000 - 300) K
- = 1 kg·J/(kg·K) × 4700 K
- = 4700 J per meter
-
-Time to deliver this energy:
-t = Q / P
- = 4700 J/m / 4000 W
- = 1.175 s per meter (!)
-```
-
-**Wait, this seems too long!** What's wrong?
-
-**Reality check - positive feedback:**
-1. As temperature rises, resistance drops
-2. Lower resistance → more current (V = I×R, fixed V)
-3. More current → more heating (P = I²R)
-4. Exponential growth, not linear!
-
-**Improved estimate with feedback:**
-
-```
-R(T) ≈ R₀ × (T₀/T)^2 (approximate scaling)
-
-At T = 5000 K:
-R ≈ 10 MΩ × (300/5000)² ≈ 36 kΩ (250× reduction!)
-
-Current increases dramatically:
-I ≈ 20 mA × √(10 MΩ / 36 kΩ) ≈ 330 mA
-
-Power increases:
-P ≈ (330 mA)² × 36 kΩ ≈ 3,920 W (similar, but delivered more efficiently)
-
-More realistic time (accounting for exponential feedback):
-t_transition ≈ 1-5 ms (observed in experiments)
-```
-
-**Key insight:** Positive feedback accelerates the transition once started. This is why leaders form "explosively" after threshold.
-
----
-
-## Key Takeaways
-
-- **Streamers**: Thin (10-100 μm), fast (~10⁶ m/s), cold (1000-3000 K), photoionization-driven, high ε (50-150 J/m)
-- **Leaders**: Thick (1-10 mm), slower (~10³ m/s), hot (5000-20000 K), thermal-ionization-driven, low ε (5-20 J/m)
-- **6-step transition**: High E-field → current flows → Joule heating → thermal ionization → leader forms from base → tip launches streamers → fed streamers convert
-- **Leader formation requires**: Sustained current (not brief pulse) + adequate time (ms range) + sufficient voltage maintenance
-- **QCW optimized**: 5-20 ms ramps allow full leader development, ε ≈ 5-15 J/m
-- **Burst mode limitation**: <500 μs pulses too short for leader transition, ε ≈ 30-100+ J/m
-- **Efficiency difference**: Leaders ~6-10× more efficient than streamers for length extension
-
-## Practice
-
-{exercise:phys-ex-06}
-
-**Problem 1:** Explain why streamers propagate faster than leaders despite being at lower temperature. What fundamental mechanisms are different?
-
-**Problem 2:** A coil produces 2 m sparks in burst mode (ε = 70 J/m). If converted to QCW with ε = 12 J/m and same total energy, estimate the new spark length. What physical transition enables this improvement?
-
-**Problem 3:** In the 6-step transition sequence, why does the leader form from the base (topload) first, rather than from the tip? Consider where current density and heating are highest.
-
-**Problem 4:** High-speed photography shows purple streamers at t = 0.1 ms, then white glow at base by t = 2 ms, then white core extending by t = 10 ms. Which step(s) of the transition correspond to each observation?
-
----
-**Next Lesson:** [Capacitive Divider Problem](07-capacitive-divider.md)
diff --git a/spark-lessons/lessons/03-spark-physics/07-capacitive-divider.md b/spark-lessons/lessons/03-spark-physics/07-capacitive-divider.md
deleted file mode 100644
index aec4a82..0000000
--- a/spark-lessons/lessons/03-spark-physics/07-capacitive-divider.md
+++ /dev/null
@@ -1,471 +0,0 @@
----
-id: phys-07
-title: "The Capacitive Divider Problem"
-section: "Spark Growth Physics"
-difficulty: "advanced"
-estimated_time: 45
-prerequisites: ["fund-04", "fund-05", "phys-01", "phys-02"]
-objectives:
- - Understand how voltage divides between C_mut and C_sh
- - Calculate V_tip as a function of spark length
- - Recognize why tip voltage drops as spark grows
- - Apply capacitive division to predict sub-linear scaling
-tags: ["capacitive-divider", "voltage-division", "C_mut", "C_sh", "V_tip", "sub-linear"]
----
-
-# The Capacitive Divider Problem
-
-A critical limitation affects all Tesla coils: as the spark grows longer, the voltage at the tip **decreases** even if topload voltage is maintained. This "capacitive divider effect" creates progressively harder conditions for continued growth.
-
-## Review: Spark Circuit Topology
-
-From Fundamentals, recall the spark circuit:
-
-```
- [C_mut]
-Topload ----||---- Node_spark (spark base)
- |
- [R]
- |
- [C_sh]
- |
- GND
-```
-
-**Components:**
-- **C_mut**: Mutual capacitance between topload and spark
-- **C_sh**: Shunt capacitance from spark to ground
-- **R**: Spark resistance (varies with ionization)
-
-**Key insight:** The spark sees a **voltage divider** between topload and ground!
-
-## Voltage Division Equation
-
-The general voltage divider with complex impedances:
-
-```
-V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
-
-where:
- Z_mut = (1/jωC_mut) || R (parallel combination of capacitance and resistance)
- Z_sh = 1/(jωC_sh) (capacitive reactance)
-```
-
-**In complex form:**
-
-```
-Y_mut = jωC_mut + 1/R (admittance of parallel combination)
-Z_mut = 1/Y_mut
-
-Y_sh = jωC_sh
-Z_sh = 1/Y_sh
-
-V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
-```
-
-This is complex-valued (magnitude and phase).
-
-## Open-Circuit Limit (No Current Flow)
-
-**Simplified case:** When R → ∞ (no conduction, purely capacitive):
-
-```
-V_tip = V_topload × C_mut / (C_mut + C_sh)
-```
-
-This is the **capacitive voltage divider** formula.
-
-**Physical interpretation:**
-- Charges distribute between two capacitors in series
-- Voltage splits proportionally to inverse capacitances
-- As C_sh increases, V_tip decreases
-
-### The Problem: C_sh Grows with Length
-
-**Empirical relationship:**
-
-```
-C_sh ≈ 2 pF/foot × L_feet
-
-Or in SI units:
-C_sh ≈ 6.6 pF/m × L_meters
-```
-
-**As spark grows:**
-- Length L increases
-- C_sh increases (proportional to length)
-- Denominator (C_mut + C_sh) increases
-- V_tip decreases!
-
-**This is self-limiting:** Longer sparks make it harder to grow even longer.
-
----
-
-## WORKED EXAMPLE: Open-Circuit Voltage Division
-
-**Given:**
-- V_topload = 400 kV (constant, maintained by primary)
-- C_mut = 8 pF (approximately constant)
-- Spark grows from 1 ft to 6 ft
-
-**Find:** V_tip at L = 1, 2, 3, 4, 5, 6 feet
-
-### Solution
-
-**At L = 1 ft:**
-
-```
-C_sh = 2 pF/ft × 1 ft = 2 pF
-
-V_tip = 400 kV × 8/(8+2)
- = 400 kV × 8/10
- = 320 kV (80% of V_topload)
-```
-
-**At L = 2 ft:**
-
-```
-C_sh = 4 pF
-
-V_tip = 400 × 8/12
- = 267 kV (67%)
-```
-
-**At L = 3 ft:**
-
-```
-C_sh = 6 pF
-
-V_tip = 400 × 8/14
- = 229 kV (57%)
-```
-
-**At L = 4 ft:**
-
-```
-C_sh = 8 pF
-
-V_tip = 400 × 8/16
- = 200 kV (50%)
-```
-
-**At L = 5 ft:**
-
-```
-C_sh = 10 pF
-
-V_tip = 400 × 8/18
- = 178 kV (44%)
-```
-
-**At L = 6 ft:**
-
-```
-C_sh = 12 pF
-
-V_tip = 400 × 8/20
- = 160 kV (40%)
-```
-
-### Summary Table
-
-| Length | C_sh | V_tip | % of V_top | E_avg (MV/m) |
-|--------|------|-------|------------|--------------|
-| 1 ft (0.3 m) | 2 pF | 320 kV | 80% | 1.07 |
-| 2 ft (0.6 m) | 4 pF | 267 kV | 67% | 0.89 |
-| 3 ft (0.9 m) | 6 pF | 229 kV | 57% | 0.76 |
-| 4 ft (1.2 m) | 8 pF | 200 kV | 50% | 0.67 |
-| 5 ft (1.5 m) | 10 pF | 178 kV | 44% | 0.59 |
-| 6 ft (1.8 m) | 12 pF | 160 kV | 40% | 0.53 |
-
-**Observations:**
-- V_tip drops to 40% of V_topload by 6 ft
-- E_avg = V_tip/L decreases even faster
-- Growth becomes progressively harder
-
-{image:voltage-division-vs-length-plot}
-
----
-
-## With Finite Resistance
-
-Real sparks have finite resistance R ≈ R_opt_power (from optimization):
-
-```
-R_opt_power ≈ 1/(ω(C_mut + C_sh))
-```
-
-**Effect of finite R:**
-
-```
-Z_mut = R || (1/jωC_mut)
-
-For R ≈ R_opt:
-Z_mut ≈ (1-j)/(2ωC_mut) (complex, 45° phase lag)
-
-V_tip magnitude is LOWER than open-circuit case
-V_tip has phase shift relative to V_topload
-```
-
-**Result:** Voltage division is **worse** than the open-circuit case!
-
-### Detailed Calculation (Advanced)
-
-For R = R_opt_power = 1/(ω(C_mut + C_sh)):
-
-```
-Y_mut = jωC_mut + 1/R
- = jωC_mut + ω(C_mut + C_sh)
- = ω(C_mut + C_sh) + jωC_mut
-
-Z_mut = 1/Y_mut
- = 1 / [ω(C_mut + C_sh)(1 + jC_mut/(C_mut + C_sh))]
-
-Z_sh = 1/(jωC_sh)
-
-Ratio:
-V_tip/V_top = Z_mut/(Z_mut + Z_sh)
-
-After algebra (details omitted):
-|V_tip/V_top| ≈ C_mut/(C_mut + C_sh) × (1/√2)
-
-Approximately 0.707× the open-circuit value!
-```
-
-**Practical conclusion:** With conduction current, voltage division is ~30% worse than capacitive-only case.
-
-## Impact on E_tip and Growth
-
-Recall the tip field:
-
-```
-E_tip = κ × V_tip / L
-```
-
-**As L increases:**
-
-**Numerator effect (voltage division):**
-```
-V_tip ∝ C_mut / (C_mut + C_sh)
- ≈ C_mut / (C_mut + αL) (where α = 6.6 pF/m)
- ≈ 1 / (1 + αL/C_mut)
-
-For large L: V_tip ∝ 1/L
-```
-
-**Denominator effect (geometry):**
-```
-Division by L
-```
-
-**Combined:**
-```
-E_tip ∝ V_tip / L
- ∝ (1/L) / L
- ∝ 1/L²
-
-E_tip decreases as L²!
-```
-
-**This is devastating for long spark growth.**
-
-## Sub-Linear Scaling Prediction
-
-From the capacitive divider effect, we can predict scaling:
-
-**Growth stops when:**
-```
-E_tip(L_max) = E_propagation
-
-κ × V_tip(L_max) / L_max = E_propagation
-```
-
-**Substituting voltage division:**
-```
-κ × [V_topload × C_mut/(C_mut + αL_max)] / L_max = E_propagation
-
-Rearranging:
-V_topload × C_mut / (C_mut + αL_max) = E_propagation × L_max / κ
-
-V_topload × C_mut = E_propagation × L_max × (C_mut + αL_max) / κ
-```
-
-**For large L (C_sh >> C_mut):**
-```
-V_topload × C_mut ≈ E_propagation × L_max × αL_max / κ
-
-V_topload × C_mut ≈ (E_propagation × α / κ) × L_max²
-
-Solving for L_max:
-L_max ∝ √(V_topload × C_mut)
- ∝ √(V_topload) (if C_mut approximately constant)
-```
-
-**Connection to energy:**
-
-If topload voltage is limited by breakdown, V_top ∝ √E (from capacitor energy):
-```
-E_cap = ½ C_top V_top²
-V_top ∝ √E
-
-Therefore:
-L_max ∝ √V_top ∝ √(√E) ∝ E^(1/4) to E^(1/2)
-
-Approximately: L ∝ √E
-```
-
-**This explains Freau's empirical observation:** For burst mode (voltage-limited), spark length scales as square root of energy!
-
----
-
-## WORKED EXAMPLE: Scaling Prediction
-
-**Given:**
-- Coil A: V_top = 300 kV, produces L = 1.2 m spark
-- Coil B: Same design, but V_top = 450 kV (1.5× voltage)
-
-**Find:** Predicted length for Coil B using:
-(a) Linear scaling (naive)
-(b) Sub-linear scaling (capacitive divider)
-
-### Solution
-
-**Part (a): Linear scaling (incorrect)**
-
-```
-If L ∝ V:
-L_B = L_A × (V_B/V_A)
- = 1.2 m × (450/300)
- = 1.2 m × 1.5
- = 1.8 m
-```
-
-**Part (b): Sub-linear scaling (more realistic)**
-
-```
-If L ∝ √V (from capacitive divider):
-L_B = L_A × √(V_B/V_A)
- = 1.2 m × √(450/300)
- = 1.2 m × √1.5
- = 1.2 m × 1.225
- = 1.47 m
-
-Only 1.47 m instead of 1.8 m!
-```
-
-**Actual measurements typically show:** L_B ≈ 1.4-1.5 m, confirming sub-linear scaling.
-
-**Percentage improvement:**
-- Linear prediction: 50% longer (wrong)
-- Sub-linear prediction: 23% longer (correct)
-- Capacitive divider limits gains from higher voltage
-
----
-
-## Mitigation Strategies
-
-How can we fight the capacitive divider effect?
-
-### 1. Increase C_mut
-
-**Larger topload:**
-```
-C_top increases → C_mut increases
-→ C_mut/(C_mut + C_sh) ratio improves
-→ Better V_tip retention
-```
-
-**Effect:**
-- Diminishes relative impact of C_sh
-- Requires larger topload (practical limits)
-
-### 2. Active Voltage Ramping (QCW)
-
-**Strategy:**
-```
-Ramp V_topload upward as spark grows
-Compensate for voltage division
-Maintain E_tip above threshold longer
-```
-
-**This is the QCW advantage:**
-- Not fighting capacitive divider directly
-- But actively increasing numerator (V_topload)
-- Allows longer sparks than fixed voltage
-
-### 3. Reduce C_sh (Limited Options)
-
-**Physical constraints:**
-- C_sh ∝ L (fundamental geometry)
-- Cannot eliminate
-- Thin spark slightly better (smaller cross-section)
-- But thermal/ionization requirements limit how thin
-
-### 4. Accept the Limitation
-
-**Reality:**
-- Capacitive divider is fundamental
-- Cannot be eliminated
-- Design around it (optimize topload, use QCW ramping)
-- Accept sub-linear scaling
-
----
-
-## Comparison: QCW vs Burst Mode
-
-### Burst Mode (Fixed Voltage)
-
-```
-V_topload = constant (capacitor discharge)
-
-As spark grows:
-- V_tip decreases (capacitive divider)
-- E_tip decreases rapidly
-- Growth stalls at voltage limit
-- L ∝ √E scaling dominates
-```
-
-### QCW Mode (Ramped Voltage)
-
-```
-V_topload(t) increases with time
-
-As spark grows:
-- V_tip still affected by divider
-- But V_topload increasing compensates partially
-- Can maintain E_tip > E_propagation longer
-- Better scaling: L ∝ E^0.6 to E^0.8
-```
-
-**QCW doesn't eliminate the divider, but actively fights it!**
-
----
-
-## Key Takeaways
-
-- **Voltage divider**: V_tip = V_topload × C_mut/(C_mut + C_sh)
-- **C_sh grows with length**: C_sh ≈ 6.6 pF/m × L, making growth self-limiting
-- **V_tip drops dramatically**: Can reach 40% of V_topload by 6 ft
-- **E_tip ∝ 1/L²**: Combined effect of voltage division and geometric scaling
-- **Sub-linear scaling**: L ∝ √E for voltage-limited burst mode (Freau's observation)
-- **Finite R worsens effect**: Conduction current creates additional voltage drop
-- **QCW mitigation**: Active voltage ramping compensates for divider effect
-- **Fundamental limit**: Cannot be eliminated, only managed through design
-
-## Practice
-
-{exercise:phys-ex-07}
-
-**Problem 1:** V_top = 350 kV, C_mut = 10 pF. Calculate V_tip for:
-(a) L = 1 ft (C_sh = 2 pF)
-(b) L = 5 ft (C_sh = 10 pF)
-What percentage of voltage is lost?
-
-**Problem 2:** A spark needs E_propagation = 0.6 MV/m and κ = 3 to grow. For a 2 m spark, calculate the required V_tip. Then, if C_mut = 8 pF and C_sh = 13 pF (for 2 m), what V_topload is needed?
-
-**Problem 3:** Explain why spark length scales as L ∝ √E for voltage-limited burst mode. Connect this to the capacitive divider effect and the E_tip ∝ 1/L² relationship.
-
-**Problem 4:** Two coils: Coil A has C_mut = 6 pF, Coil B has C_mut = 12 pF (larger topload). Both operate at V_top = 400 kV and grow 1.5 m sparks. Calculate V_tip for each. Which suffers less from voltage division?
-
----
-**Next Lesson:** [Freau's Empirical Relationship](08-freau-relationship.md)
diff --git a/spark-lessons/lessons/03-spark-physics/08-freau-relationship.md b/spark-lessons/lessons/03-spark-physics/08-freau-relationship.md
deleted file mode 100644
index 3cb85fe..0000000
--- a/spark-lessons/lessons/03-spark-physics/08-freau-relationship.md
+++ /dev/null
@@ -1,457 +0,0 @@
----
-id: phys-08
-title: "Freau's Empirical Relationship"
-section: "Spark Growth Physics"
-difficulty: "advanced"
-estimated_time: 35
-prerequisites: ["phys-03", "phys-04", "phys-07"]
-objectives:
- - Understand Freau's empirical L ∝ √E scaling for burst mode
- - Derive the physical explanation from capacitive divider effects
- - Recognize differences between burst mode and QCW scaling
- - Apply scaling laws to predict performance changes
-tags: ["freau", "scaling-laws", "sub-linear", "burst-mode", "QCW", "empirical"]
----
-
-# Freau's Empirical Relationship
-
-Tesla coil community observations have revealed consistent patterns in how spark length scales with energy. Understanding these **scaling laws** helps predict performance and set realistic expectations.
-
-## The Empirical Observations
-
-Daniel Freau and others in the Tesla coil community documented:
-
-### Single-Shot Burst Mode
-
-```
-L ∝ √E
-
-where:
- L = spark length [m]
- E = bang energy (capacitor energy per pulse) [J]
-```
-
-**Example measurements:**
-- 25 J → 0.8 m
-- 100 J → 1.6 m (4× energy → 2× length)
-- 400 J → 3.2 m (16× energy → 4× length)
-
-**Sub-linear scaling:** Doubling energy does NOT double length; only increases by √2 ≈ 1.41×.
-
-### Repetitive Burst Operation
-
-```
-L ∝ P_avg^n
-
-where:
- P_avg = average power [W]
- n ≈ 0.3 to 0.5 (empirical exponent)
-```
-
-**Example:**
-- 10 kW → 1.2 m
-- 40 kW → 2.0 m (4× power → 1.67× length, n ≈ 0.4)
-
-**Still sub-linear:** More power helps, but with diminishing returns.
-
-### QCW Mode
-
-```
-L ∝ E^m
-
-where:
- m ≈ 0.6 to 0.8 (closer to linear than burst)
-```
-
-**Example:**
-- 50 J → 3.5 m
-- 200 J → 9.0 m (4× energy → 2.6× length, m ≈ 0.7)
-
-**Less sub-linear:** QCW shows better scaling than burst mode.
-
-## Physical Explanation: Voltage-Limited Burst Mode
-
-The L ∝ √E relationship for burst mode comes from the interplay of capacitive divider effects and voltage limitations.
-
-### Derivation from First Principles
-
-**Step 1: Growth stops when E_tip = E_propagation**
-
-```
-E_tip = κ × V_tip / L
-
-At stall:
-κ × V_tip / L_max = E_propagation
-
-Solving for L_max:
-L_max = κ × V_tip / E_propagation
-```
-
-**Step 2: Voltage division affects V_tip**
-
-From capacitive divider (Lesson 07):
-
-```
-V_tip ≈ V_topload × C_mut / (C_mut + C_sh)
-
-For long sparks (C_sh >> C_mut):
-C_sh ≈ αL (α ≈ 6.6 pF/m)
-
-V_tip ≈ V_topload × C_mut / (αL)
- ∝ V_topload / L
-```
-
-**Step 3: Substitute into stall condition**
-
-```
-L_max = κ × V_tip / E_propagation
- = κ × (V_topload/L_max) / E_propagation
-
-Multiply both sides by L_max:
-L_max² = κ × V_topload / E_propagation
-
-Solving for L_max:
-L_max = √(κ × V_topload / E_propagation)
- ∝ √V_topload
-```
-
-**Step 4: Connect to energy**
-
-For a capacitor discharge (burst mode):
-
-```
-E_bang = ½ C_primary V_primary²
-
-If transformer ratio is fixed:
-V_topload ∝ V_primary ∝ √E_bang
-
-Therefore:
-L_max ∝ √V_topload ∝ √(√E_bang) ∝ E_bang^(1/4) to E_bang^(1/2)
-```
-
-**The exact exponent depends on:**
-- Whether topload voltage saturates (breakdown limit)
-- Impedance matching (affects voltage transfer)
-- Spark loading (changes transformer ratio during pulse)
-
-**Empirically observed:** The exponent clusters around **0.5**, giving **L ∝ √E**.
-
-### Simplified Intuition
-
-**The vicious cycle:**
-
-```
-Longer spark → Higher C_sh → Lower V_tip → Lower E_tip → Harder to grow
-
-E_tip ∝ V_tip/L ∝ (V_top/L)/L ∝ V_top/L²
-
-Growth requires: V_top/L² ≥ E_propagation/κ
- V_top ≥ (E_propagation/κ) × L²
-
-For fixed V_top:
-L_max² ≤ κ × V_top/E_propagation
-L_max ∝ √V_top ∝ √E
-```
-
-**Physical meaning:** The capacitive divider creates a **quadratic penalty** (E_tip ∝ 1/L²), resulting in square-root scaling with energy/voltage.
-
----
-
-## WORKED EXAMPLE: Burst Mode Scaling
-
-**Given:**
-- Coil operates in burst mode
-- Test 1: E_bang = 40 J → L = 1.1 m
-- Test 2: E_bang = 160 J → L = ?
-
-**Find:** Predicted length for Test 2 using L ∝ √E
-
-### Solution
-
-```
-L₂/L₁ = √(E₂/E₁)
-
-L₂ = L₁ × √(E₂/E₁)
- = 1.1 m × √(160/40)
- = 1.1 m × √4
- = 1.1 m × 2
- = 2.2 m
-
-Predicted: 2.2 m for 160 J
-```
-
-**Verification:**
-- 4× energy (40 J → 160 J)
-- 2× length (1.1 m → 2.2 m)
-- Consistent with √E scaling ✓
-
-**If scaling were linear (wrong):**
-```
-L₂ = 1.1 m × (160/40) = 4.4 m (incorrect!)
-```
-
-**Key insight:** Quadrupling energy only doubles length in voltage-limited burst mode.
-
----
-
-## Why QCW Shows Different Scaling
-
-QCW mode shows less sub-linear scaling (L ∝ E^0.6 to E^0.8) because of active mitigation:
-
-### QCW Advantages
-
-**1. Voltage ramping:**
-```
-V_topload(t) increases during ramp
-Actively compensates for capacitive divider
-Can maintain E_tip > E_propagation longer
-```
-
-**2. Leader formation:**
-```
-Lower ε (5-15 J/m vs 30-100 J/m for burst)
-Same energy produces longer spark
-Better inherent efficiency
-```
-
-**3. Thermal accumulation:**
-```
-Channel stays hot (no cooling between pulses)
-Effective ε decreases during ramp
-Later growth more efficient than early growth
-```
-
-### Modified Scaling
-
-**Effective relationship:**
-
-```
-L_max ∝ (V_top(t_final) / ε_effective)
-
-Both numerator and denominator improve during QCW ramp:
-- V_top(t) increases (ramping)
-- ε_effective decreases (thermal accumulation)
-
-Result: L ∝ E^m where m ≈ 0.6-0.8
-```
-
-**Still sub-linear, but better than burst mode:**
-- Burst: L ∝ E^0.5
-- QCW: L ∝ E^0.7 (typical)
-
-**Ratio improvement:**
-```
-For 4× energy increase:
-Burst: 4^0.5 = 2.0× longer
-QCW: 4^0.7 = 2.64× longer
-
-QCW gains 32% more length for same energy increase!
-```
-
----
-
-## WORKED EXAMPLE: Comparing Modes
-
-**Given:**
-- Burst mode coil: 100 J → 1.5 m (baseline)
-- QCW conversion: Same 100 J total energy
-- Burst scaling: L ∝ E^0.5
-- QCW scaling: L ∝ E^0.7
-
-**Find:**
-(a) Predicted QCW length at 100 J
-(b) Energy needed for 3 m in each mode
-(c) Which mode is more "scalable"?
-
-### Solution
-
-**Part (a): QCW length at 100 J**
-
-Need calibration point for QCW. Assume QCW has lower ε:
-
-```
-From ε perspective:
-Burst: ε_burst = 100 J / 1.5 m = 67 J/m
-QCW: ε_QCW ≈ 12 J/m (typical)
-
-Linear estimate:
-L_QCW = 100 J / 12 J/m = 8.3 m
-
-But voltage limit will reduce this.
-Realistic with same topload: ~4-5 m
-
-We'll use 4.5 m as calibration point.
-```
-
-**Part (b): Energy for 3 m in each mode**
-
-**Burst mode:**
-```
-L ∝ E^0.5
-L₁ = 1.5 m at E₁ = 100 J
-L₂ = 3 m at E₂ = ?
-
-(L₂/L₁)² = E₂/E₁
-(3/1.5)² = E₂/100
-4 = E₂/100
-E₂ = 400 J needed for 3 m
-```
-
-**QCW mode:**
-```
-L ∝ E^0.7
-L₁ = 4.5 m at E₁ = 100 J
-L₂ = 3 m at E₂ = ?
-
-(L₂/L₁)^(1/0.7) = E₂/E₁
-(3/4.5)^1.43 = E₂/100
-0.667^1.43 = E₂/100
-0.568 = E₂/100
-E₂ = 56.8 J needed for 3 m
-
-Actually, 3 m < 4.5 m, so less energy needed.
-Correct calculation:
-(3/4.5)^1.43 = E₂/100
-E₂ ≈ 56.8 J
-```
-
-Wait, let me recalculate for going DOWN in length:
-
-```
-If QCW produces 4.5 m at 100 J, then for 3 m:
-(E₂/E₁) = (L₂/L₁)^(1/0.7)
-E₂/100 = (3/4.5)^1.43
-E₂ = 100 × 0.568 ≈ 57 J
-
-QCW needs only 57 J for 3 m
-Burst needs 400 J for 3 m
-
-QCW is 7× more energy-efficient!
-```
-
-**Part (c): Which is more scalable?**
-
-```
-Scalability = how much length increases per energy increase
-
-Burst: L ∝ E^0.5
- Doubling energy: 2^0.5 = 1.41× length gain
-
-QCW: L ∝ E^0.7
- Doubling energy: 2^0.7 = 1.62× length gain
-
-QCW is more scalable: 15% better length gain per energy doubling
-```
-
-**Practical implication:** QCW benefits more from increased energy/power than burst mode.
-
----
-
-## Repetitive Operation Scaling
-
-For repetitive burst mode (many pulses per second):
-
-```
-L ∝ P_avg^n where n ≈ 0.3-0.5
-```
-
-**Physical explanation:**
-
-**Thermal memory between pulses:**
-- If repetition rate is fast enough (~100+ Hz)
-- Some ionization/thermal memory carries over
-- Effective ε decreases slightly
-- Better scaling than single-shot (n > 0.5)
-
-**Power vs energy:**
-```
-P_avg = E_bang × f (f = pulse rate)
-
-For fixed E_bang:
-L ∝ P^n ∝ (E × f)^n ∝ f^n
-
-More frequent pulses help, but sub-linearly
-```
-
-**Example:**
-```
-100 Hz, 40 J per pulse: P_avg = 4 kW → L₁
-200 Hz, 40 J per pulse: P_avg = 8 kW → L₂
-
-L₂/L₁ = (8/4)^0.4 = 2^0.4 = 1.32
-
-Only 32% longer despite doubling pulse rate
-```
-
----
-
-## Practical Implications
-
-### Design Decisions
-
-**For maximum length:**
-- Use QCW mode (better scaling, lower ε)
-- Large topload (fight capacitive divider)
-- Modest energy with long ramp (exploit thermal accumulation)
-
-**For visual spectacle:**
-- Use burst mode (bright, branched)
-- High peak power (dramatic but short sparks)
-- Accept poor energy efficiency
-
-### Performance Predictions
-
-**When upgrading primary capacitance:**
-
-```
-C_primary doubles → E_bang doubles (same V_primary)
-
-Burst mode: L increases by √2 = 1.41×
-QCW mode: L increases by 2^0.7 = 1.62×
-
-QCW benefits more from the upgrade
-```
-
-**When adding more power:**
-
-```
-QCW mode: More sensitive to power increases
- Can ramp voltage higher/faster
- Better return on investment
-
-Burst mode: Less sensitive
- Voltage-limited earlier
- Diminishing returns
-```
-
----
-
-## Key Takeaways
-
-- **Burst mode scaling**: L ∝ √E (square root of energy)
-- **Physical origin**: Capacitive divider creates E_tip ∝ 1/L² penalty
-- **QCW scaling**: L ∝ E^0.7 (less sub-linear, better than burst)
-- **QCW advantages**: Voltage ramping + lower ε + thermal accumulation
-- **Repetitive burst**: L ∝ P^0.3-0.5, slight improvement over single-shot
-- **Design implication**: QCW is more "scalable" - better returns on energy/power increases
-- **Realistic expectations**: Quadrupling energy only doubles burst-mode length
-
-## Practice
-
-{exercise:phys-ex-08}
-
-**Problem 1:** A burst coil produces 1.4 m sparks with 60 J per pulse. Using L ∝ √E, predict:
-(a) Length with 135 J per pulse
-(b) Energy needed for 2.1 m sparks
-
-**Problem 2:** Compare two upgrade paths for a QCW coil currently at 80 J, 3.2 m (assume L ∝ E^0.7):
-- Option A: Upgrade to 160 J
-- Option B: Upgrade to 240 J
-Calculate expected length for each option.
-
-**Problem 3:** Explain why QCW shows L ∝ E^0.7 instead of L ∝ √E. What three mechanisms contribute to better-than-square-root scaling?
-
-**Problem 4:** A repetitive burst coil runs at 150 Hz with 30 J/pulse (4.5 kW average) and produces 1.0 m sparks. If pulse rate increases to 300 Hz (9 kW, same energy/pulse) and L ∝ P^0.4, predict new length.
-
----
-**Next Lesson:** [Part 3 Review & Exercises](09-review-exercises.md)
diff --git a/spark-lessons/lessons/03-spark-physics/09-review-exercises.md b/spark-lessons/lessons/03-spark-physics/09-review-exercises.md
deleted file mode 100644
index 399a909..0000000
--- a/spark-lessons/lessons/03-spark-physics/09-review-exercises.md
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----
-id: phys-09
-title: "Part 3 Review: Spark Growth Physics"
-section: "Spark Growth Physics"
-difficulty: "intermediate"
-estimated_time: 60
-prerequisites: ["phys-01", "phys-02", "phys-03", "phys-04", "phys-05", "phys-06", "phys-07", "phys-08"]
-objectives:
- - Synthesize understanding of spark growth physics
- - Apply multiple concepts to realistic design problems
- - Troubleshoot common performance issues
- - Make informed design decisions based on physics principles
-tags: ["review", "synthesis", "design", "troubleshooting", "comprehensive"]
----
-
-# Part 3 Review: Spark Growth Physics
-
-This lesson synthesizes the spark growth physics concepts from Part 3 and provides comprehensive practice problems integrating multiple topics.
-
-## Concepts Summary
-
-### Electric Field Thresholds (Lesson phys-01)
-
-**Key equations:**
-```
-E_inception ≈ 2-3 MV/m (initial breakdown)
-E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
-E_tip = κ × E_average = κ × V/L
-Growth criterion: E_tip > E_propagation
-```
-
-**Key concepts:**
-- Tip enhancement factor κ ≈ 2-5
-- Altitude/humidity effects: ±20-30%
-- Voltage-limited when E_tip < E_propagation
-
-### Maximum Voltage-Limited Length (Lesson phys-02)
-
-**Key equations:**
-```
-L_max ≈ κ × V_top / E_propagation
-
-FEMM provides: E_tip(V_top, L, geometry)
-```
-
-**Key concepts:**
-- Both voltage AND power are necessary
-- FEMM computes realistic field distributions
-- Environmental effects reduce E_propagation at altitude
-
-### Energy Per Meter (Lesson phys-03)
-
-**Key equations:**
-```
-ΔE ≈ ε × ΔL
-dL/dt = P_stream / ε (when E_tip > E_propagation)
-T = ε × L / P_stream (time to grow)
-```
-
-**Key concepts:**
-- ε [J/m] is energy per meter of growth
-- Includes ionization, heating, radiation, branching
-- Theoretical minimum ε ≈ 0.3-0.5 J/m
-- Practical values 20-300× higher
-
-### Empirical ε Values (Lesson phys-04)
-
-**Typical ranges:**
-```
-QCW: ε ≈ 5-15 J/m (efficient leaders)
-Hybrid DRSSTC: ε ≈ 20-40 J/m (mixed)
-Burst mode: ε ≈ 30-100+ J/m (inefficient streamers)
-```
-
-**Key concepts:**
-- Calibration: ε = E_delivered / L_measured
-- Thermal accumulation: ε(t) = ε₀/(1 + α∫P dt)
-- Operating mode choice trades efficiency vs aesthetics
-
-### Thermal Memory (Lesson phys-05)
-
-**Key equations:**
-```
-τ_thermal = d² / (4α) where α ≈ 2×10⁻⁵ m²/s
-v_convection ≈ √(g × d × ΔT/T_amb)
-```
-
-**Typical times:**
-```
-Thin streamers (d ~ 100 μm): τ ~ 0.1-0.2 ms
-Thick leaders (d ~ 3 mm): τ ~ 50-300 ms
-Effective persistence: 1-5 ms (streamers), seconds (leaders)
-```
-
-**Key concepts:**
-- Convection extends persistence beyond pure diffusion
-- QCW ramp time << leader thermal time (stays hot)
-- Burst gap >> streamer thermal time (cools completely)
-
-### Streamers vs Leaders (Lesson phys-06)
-
-**Comparison:**
-```
- Streamers Leaders
-Diameter: 10-100 μm 1-10 mm
-Velocity: ~10⁶ m/s ~10³ m/s
-Temperature: 1000-3000 K 5000-20,000 K
-Mechanism: Photoionization Thermal ionization
-ε: 50-150+ J/m 5-20 J/m
-```
-
-**6-step transition:**
-1. High E-field creates streamers
-2. Current flows → Joule heating
-3. Thermal ionization begins
-4. Leader forms from base
-5. Leader tip launches streamers
-6. Fed streamers convert to leader
-
-### Capacitive Divider (Lesson phys-07)
-
-**Key equations:**
-```
-V_tip = V_topload × C_mut/(C_mut + C_sh)
-C_sh ≈ 6.6 pF/m × L
-E_tip ∝ V_tip/L ∝ 1/L² (combined effect)
-```
-
-**Key concepts:**
-- Voltage division worsens as spark grows
-- Self-limiting: longer sparks harder to extend
-- Causes sub-linear scaling
-- QCW mitigation: active voltage ramping
-
-### Freau's Scaling Laws (Lesson phys-08)
-
-**Empirical relationships:**
-```
-Burst mode: L ∝ √E (sub-linear)
-QCW mode: L ∝ E^0.7 (less sub-linear)
-Repetitive burst: L ∝ P^0.4 (moderate)
-```
-
-**Key concepts:**
-- Physical origin: capacitive divider + voltage limitation
-- QCW advantages: ramping + low ε + thermal accumulation
-- Realistic expectations: 4× energy → 2× length (burst)
-
----
-
-## Comprehensive Practice Problems
-
-### Problem 1: Integrated Design Analysis
-
-**Scenario:**
-You are designing a QCW Tesla coil with the following targets:
-- Target spark length: L = 2.5 m
-- Ramp time: T = 15 ms
-- Operating frequency: f = 150 kHz
-
-**Measurements from FEMM:**
-- At L = 2.5 m, V_top = 550 kV: E_tip = 0.65 MV/m
-- C_mut ≈ 9 pF
-- C_sh ≈ 16.5 pF (for 2.5 m spark)
-
-**Questions:**
-
-**(a)** If E_propagation = 0.6 MV/m at your altitude, can the spark reach 2.5 m with 550 kV? Calculate the margin.
-
-**(b)** Assuming ε = 11 J/m for your QCW mode, calculate:
-- Total energy required
-- Average power required
-
-**(c)** Calculate V_tip using the capacitive divider formula. Compare to the voltage needed if there were no division (C_sh = 0). What percentage is lost?
-
-**(d)** If thermal accumulation reduces ε by 20% during the ramp (ε_effective = 8.8 J/m), recalculate the required power. How much benefit does thermal accumulation provide?
-
----
-
-### Problem 2: Mode Comparison
-
-**Scenario:**
-You have a coil that can operate in either burst mode or QCW mode with the same primary energy E = 120 J.
-
-**Burst mode characteristics:**
-- ε_burst = 55 J/m
-- No thermal accumulation
-- Voltage-limited to L_max = 2.0 m
-
-**QCW mode characteristics:**
-- ε_QCW = 13 J/m (initial)
-- With thermal accumulation: ε_effective ≈ 10 J/m (average)
-- Can ramp voltage to overcome divider partially
-- Voltage-limited to L_max = 4.5 m
-
-**Questions:**
-
-**(a)** Calculate predicted spark length for each mode using the power-limited formula L = E/ε. Which limit (power or voltage) dominates in each case?
-
-**(b)** For burst mode at 200 Hz repetition (P_avg = 24 kW), estimate whether thermal memory between pulses affects performance. Use τ_thermal ≈ 0.15 ms for thin streamers.
-
-**(c)** If you want 3 m sparks, which mode should you use? If neither reaches 3 m, what design changes would help?
-
----
-
-### Problem 3: Thermal Physics Analysis
-
-**Scenario:**
-High-speed photography of your QCW coil shows:
-- t = 0-0.5 ms: Purple streamers, d ≈ 80 μm
-- t = 2-15 ms: White core at base, d ≈ 3 mm
-- t > 15 ms (after ramp): Glowing channel rises for ~2 seconds
-
-**Questions:**
-
-**(a)** Calculate thermal diffusion time for:
-- Thin streamers (d = 80 μm)
-- Thick leaders (d = 3 mm)
-
-**(b)** The observation of leader persistence suggests thermal time constants alone don't explain the 2-second glow. Calculate convection velocity for the 3 mm leader with ΔT = 12,000 K. How does this explain the extended visibility?
-
-**(c)** Your ramp time is 15 ms. Compare this to the leader thermal time constant. Does the leader cool significantly during the ramp? (Use exponential cooling: T(t) ≈ T₀ × exp(-t/τ))
-
-**(d)** Estimate at what time during the ramp the streamer-to-leader transition occurs, given that thermal ionization requires ~5000 K and Joule heating provides ~20 kW to a 1.5 m channel. Use:
-- Channel mass: m ≈ d² × L × ρ_air ≈ (3×10⁻³)² × 1.5 × 1.2 ≈ 1.6×10⁻⁵ kg
-- Heat capacity: c_p ≈ 1000 J/(kg·K)
-
----
-
-### Problem 4: Scaling and Optimization
-
-**Scenario:**
-You have experimental data from three runs:
-
-| Run | V_primary | E_bang | L_measured | Notes |
-|-----|-----------|--------|------------|-------|
-| 1 | 300 V | 45 J | 1.3 m | Burst mode |
-| 2 | 400 V | 80 J | 1.65 m | Burst mode |
-| 3 | 400 V | 80 J | 4.2 m | QCW mode, 12 ms ramp |
-
-**Questions:**
-
-**(a)** Calculate ε for each run. What do the values tell you about the operating modes?
-
-**(b)** Check if Runs 1 and 2 follow L ∝ √E scaling (burst mode). Calculate the predicted L for Run 2 based on Run 1 data.
-
-**(c)** The QCW mode (Run 3) uses the same energy but produces 4.2 m vs 1.65 m for burst. Calculate the efficiency ratio. Where does the "extra length" come from physically?
-
-**(d)** You want to reach 2.5 m in burst mode. Using the L ∝ √E relationship from Runs 1-2, estimate the required energy. Is this upgrade worth it compared to just using QCW mode?
-
----
-
-### Problem 5: Capacitive Divider Deep Dive
-
-**Scenario:**
-Your coil has C_mut = 8.5 pF and operates at V_topload = 480 kV. You want to analyze voltage division effects.
-
-**Questions:**
-
-**(a)** Create a table showing L, C_sh, V_tip, and E_tip (with κ = 3.2) for spark lengths: 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m, 3.0 m. Use C_sh ≈ 6.6 pF/m × L.
-
-**(b)** If E_propagation = 0.55 MV/m, at what length does growth stall (E_tip = E_propagation)? Use your table and interpolate if needed.
-
-**(c)** Calculate what V_topload would be required to reach 3.0 m if E_propagation = 0.55 MV/m and κ = 3.2. Compare to your current 480 kV capability.
-
-**(d)** Propose two design changes to improve maximum length without increasing V_topload. For each, explain the physical mechanism and estimate the improvement.
-
----
-
-### Problem 6: Troubleshooting Scenario
-
-**Scenario:**
-A coiler reports the following symptoms:
-- Coil produces bright, purple, highly-branched 0.8 m sparks
-- Primary energy: E_bang = 95 J
-- Topload voltage measured: V_top ≈ 420 kV (from FEMM calibration)
-- Expected much longer sparks based on energy
-
-**Your analysis:**
-- FEMM shows E_tip ≈ 1.1 MV/m at 0.8 m length with 420 kV
-- C_mut ≈ 7 pF, C_sh ≈ 5.3 pF (for 0.8 m)
-- Operating mode: Hard-pulsed burst, 150 μs pulse width, 200 Hz
-
-**Questions:**
-
-**(a)** Calculate ε from the observed performance. Compare to expected values for burst mode. What does this indicate?
-
-**(b)** The E_tip = 1.1 MV/m is well above typical E_propagation ≈ 0.6 MV/m. Is the coil voltage-limited? What other limit explains the short sparks?
-
-**(c)** The symptom "bright, purple, highly-branched" suggests what type of discharge mechanism? Explain using the streamer vs leader concepts.
-
-**(d)** Calculate thermal diffusion time for a 100 μm streamer. Compare to the 150 μs pulse width and 5 ms gap between pulses. Does thermal memory persist between pulses?
-
-**(e)** Recommend three specific changes to improve spark length. For each, explain the physical principle and estimate the potential improvement.
-
----
-
-## Conceptual Questions
-
-### Question 1: Synthesis
-Explain the complete chain of physics that causes burst mode to scale as L ∝ √E:
-- Start with capacitive divider effect
-- Connect to E_tip ∝ 1/L²
-- Relate to voltage-limited stall condition
-- Conclude with scaling relationship
-
-### Question 2: Design Trade-offs
-Compare QCW and burst mode for:
-- Energy efficiency (ε values)
-- Thermal memory utilization
-- Voltage division mitigation
-- Practical applications
-Conclude: when would you choose each mode?
-
-### Question 3: Physical Mechanisms
-The streamer-to-leader transition requires three things:
-1. Sufficient current
-2. Sufficient time
-3. Sufficient voltage maintenance
-
-Explain WHY each is necessary using the physics of:
-- Joule heating
-- Thermal ionization threshold
-- Positive feedback mechanisms
-
-### Question 4: Limitations
-A coiler claims: "I have 200 kW available, so I should easily get 10 m sparks!"
-
-Identify the flaws in this reasoning. Discuss:
-- Voltage vs power limitations
-- Energy per meter constraints
-- Capacitive divider effects
-- Realistic expectations
-
----
-
-## Part 3 Mastery Checklist
-
-Before proceeding to Part 4, ensure you can:
-
-### Electric Fields
-- [ ] Calculate E_average and E_tip from V and L
-- [ ] Apply tip enhancement factor κ
-- [ ] Determine growth criterion (E_tip vs E_propagation)
-- [ ] Account for altitude/environmental effects
-
-### Energy and Power
-- [ ] Calculate total energy from ε and L
-- [ ] Apply growth rate equation dL/dt = P/ε
-- [ ] Predict growth time for target length
-- [ ] Distinguish voltage-limited from power-limited
-
-### Operating Modes
-- [ ] Explain ε differences between QCW, hybrid, burst
-- [ ] Calculate expected length from energy and ε
-- [ ] Recognize mode from observed spark characteristics
-- [ ] Choose appropriate mode for design goals
-
-### Thermal Physics
-- [ ] Calculate thermal diffusion times for different diameters
-- [ ] Estimate convection velocity from temperature excess
-- [ ] Explain QCW advantage via thermal memory
-- [ ] Predict streamer vs leader formation based on timescales
-
-### Discharge Mechanisms
-- [ ] Distinguish streamers from leaders (6 key properties)
-- [ ] Describe the 6-step transition sequence
-- [ ] Explain photoionization vs thermal ionization
-- [ ] Predict which mechanism dominates in a given mode
-
-### Capacitive Divider
-- [ ] Calculate V_tip from C_mut, C_sh, V_topload
-- [ ] Explain how C_sh increases with length
-- [ ] Derive E_tip ∝ 1/L² relationship
-- [ ] Identify mitigation strategies
-
-### Scaling Laws
-- [ ] Apply L ∝ √E for burst mode predictions
-- [ ] Explain physical origin of sub-linear scaling
-- [ ] Recognize QCW shows better scaling (L ∝ E^0.7)
-- [ ] Set realistic expectations for energy/power increases
-
----
-
-## Advanced Challenge Problem
-
-**Scenario:** Design a QCW coil from scratch to achieve 3.5 m sparks.
-
-**Given constraints:**
-- Budget allows C_primary up to 1.0 μF
-- V_primary limited to 600 V (safety)
-- Topload options: 20 cm toroid (C_top ≈ 25 pF) or 35 cm toroid (C_top ≈ 45 pF)
-- Target ramp time: 10-15 ms
-- Sea level operation (E_propagation = 0.6 MV/m)
-
-**Your task:**
-
-1. **Energy calculation:**
- - Choose ε for QCW mode
- - Calculate total energy required for 3.5 m
- - Verify this is achievable with C_primary and V_primary
-
-2. **Voltage requirement:**
- - Estimate C_mut for each topload option (use C_mut ≈ 0.7 × C_top as approximation)
- - Calculate C_sh for 3.5 m spark
- - For each topload, calculate V_topload needed to achieve E_tip = 0.7 MV/m at 3.5 m (assume κ = 3.0)
- - Include capacitive division effects
-
-3. **Power analysis:**
- - For T_ramp = 12 ms, calculate required average power
- - Estimate peak power (assume 1.5× average for QCW)
- - Check if this is reasonable for DRSSTC primary
-
-4. **Thermal verification:**
- - Estimate leader diameter (2-4 mm typical)
- - Calculate thermal time constant
- - Verify ramp time << thermal time (QCW condition satisfied)
-
-5. **Final recommendation:**
- - Which topload should be used? Why?
- - Is the 3.5 m target achievable with given constraints?
- - If not, what would you change and why?
-
----
-
-**Next Section:** [Part 4: Advanced Modeling](../04-advanced-modeling/01-introduction.md)
-
----
-
-## Solutions Provided Separately
-
-{exercise:phys-ex-comprehensive}
-
-Detailed solutions to all practice problems are available in the solutions guide to allow self-assessment and learning.
diff --git a/spark-lessons/lessons/04-advanced-modeling/01-lumped-model.md b/spark-lessons/lessons/04-advanced-modeling/01-lumped-model.md
deleted file mode 100644
index 5e0b560..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/01-lumped-model.md
+++ /dev/null
@@ -1,440 +0,0 @@
----
-id: model-01
-title: "Lumped Spark Model Theory"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 35
-prerequisites: ["phys-09", "phys-10", "phys-11"]
-objectives:
- - Understand single-element lumped model structure and assumptions
- - Learn when lumped models are appropriate vs distributed models
- - Master the complete workflow for building lumped spark models
- - Integrate lumped spark models with full Tesla coil circuit analysis
-tags: ["modeling", "lumped-model", "circuit-theory", "SPICE"]
----
-
-# Lumped Spark Model Theory
-
-The **lumped spark model** treats the entire spark as a single equivalent circuit element. This is the simplest and most computationally efficient approach for Tesla coil spark modeling, suitable for most practical engineering applications.
-
-## What is a Lumped Model?
-
-### Circuit Structure
-
-The lumped spark model represents the spark channel as three components:
-
-```
-Topload (V_top)
- |
- +---[C_mut]---+---[R]---+---[C_sh]---+
- | |
- Node Node GND
-```
-
-**Components:**
-
-1. **C_mut (Mutual Capacitance):** Capacitance between topload and spark channel
- - Typical range: 5-15 pF
- - Extracted from FEMM electrostatic analysis
-
-2. **R (Plasma Resistance):** Effective resistance of the entire spark
- - Typical range: 10-500 kΩ at 200 kHz
- - Optimized for maximum power transfer
- - Variable, depends on plasma state
-
-3. **C_sh (Shunt Capacitance):** Capacitance from spark to ground
- - Typical rule: ~2 pF/foot of spark length
- - Also extracted from FEMM
- - Critical for capacitive divider effect
-
-### Physical Meaning
-
-**The lumped model assumes:**
-- Uniform current distribution along spark
-- Single averaged resistance value
-- Quasi-static voltage distribution
-- Spark can be treated as electrically short at operating frequency
-
-**This works when:**
-- λ >> L (wavelength much greater than spark length)
-- At 200 kHz: λ = 1500 m, sparks typically <3 m
-- Distributed effects are second-order corrections
-
-## When to Use Lumped Models
-
-### Appropriate Applications
-
-**Use lumped models for:**
-
-1. **Short to Medium Sparks (<1-2 m)**
- - Uniform properties dominate
- - Single R approximation valid
-
-2. **Impedance Matching Studies**
- - Quick evaluation of different topload sizes
- - Coil-level optimization
- - Matching network design
-
-3. **First-Order Power Estimates**
- - Energy transfer calculations
- - Efficiency predictions
- - Quick design iterations
-
-4. **Engineering Estimates**
- - Performance predictions
- - Component selection
- - Safety margins
-
-**Computational cost:** <1 second per simulation
-
-### When Lumped Models Fail
-
-**Switch to distributed models when:**
-
-1. **Long Sparks (>2-3 m)**
- - Base vs tip properties differ significantly
- - Leader/streamer transition critical
- - Current distribution non-uniform
-
-2. **Current Distribution Matters**
- - Measuring actual current along spark
- - Validating against detailed measurements
- - Research applications
-
-3. **Extreme Parameters**
- - Very low frequency (λ approaches L)
- - Very high voltage (breakdown physics critical)
- - Unusual geometries
-
-4. **Publication-Quality Results**
- - Peer review requires distributed model
- - Detailed physics validation
-
-**Trade-off:** Distributed models 1000-2000× slower
-
-## Complete Lumped Model Workflow
-
-### Step 1: FEMM Electrostatic Analysis
-
-**Setup requirements:**
-```
-Geometry:
-- Axisymmetric (r-z coordinates)
-- Topload: toroid or sphere
-- Spark: vertical cylinder
-- Ground plane below
-
-Problem type:
-- Electrostatic (frequency = 0)
-- Two conductors: topload (V=1V), spark (floating)
-- Ground boundary condition
-
-Solve:
-- Extract 2×2 capacitance matrix [C]
-```
-
-Detailed FEMM procedure covered in next lesson.
-
-### Step 2: Extract Circuit Elements
-
-**From FEMM capacitance matrix:**
-
-```
- [Topload] [Spark]
-[Top] [ C₁₁ C₁₂ ]
-[Spark][ C₂₁ C₂₂ ]
-
-Where:
-- C_ii > 0 (diagonal: self-capacitance)
-- C_ij < 0 (off-diagonal: mutual capacitance, negative)
-- C₁₂ = C₂₁ (symmetric)
-```
-
-**Extraction formulas:**
-
-**Mutual capacitance:**
-```
-C_mut = |C₁₂| = |C₂₁|
-```
-Take absolute value of off-diagonal element.
-
-**Shunt capacitance:**
-```
-C_sh = C₂₂ + C₂₁
- = C₂₂ - |C₁₂| (since C₂₁ < 0)
-```
-
-This is spark-to-ground capacitance with topload present.
-
-### Step 3: Calculate Optimal Resistance
-
-**Power-optimal resistance formula:**
-```
-R_opt_power = 1 / (ω × C_total)
-
-Where:
- ω = 2πf (angular frequency)
- C_total = C_mut + C_sh
-```
-
-**Physical basis:** Hungry streamer theory
-- Plasma adjusts to maximize power extraction
-- R = 1/(ωC) gives optimal power transfer for capacitive load
-- Valid for streamer-dominated discharge
-
-**Apply physical bounds:**
-```
-R_min = 5 kΩ (hot leader, best case)
-R_max = 500 kΩ (cool streamer, worst case)
-
-R_clipped = clip(R_opt_power, R_min, R_max)
-```
-
-Use R_clipped in final model.
-
-### Step 4: Build SPICE Netlist
-
-**Example SPICE implementation:**
-
-```spice
-* Lumped spark model - Tesla coil discharge
-.param freq=200k
-.param omega={2*pi*freq}
-
-* Operating frequency
-* Angular frequency
-
-* Test voltage source (or connect to coil model)
-V_topload topload 0 AC 1V
-
-* Spark circuit elements
-C_mut topload spark_node {C_mut_value}
-R_spark spark_node spark_r {R_value}
-C_sh spark_r 0 {C_sh_value}
-
-* AC analysis
-.ac lin 1 {freq} {freq}
-
-* Output admittance at topload
-.print ac v(topload) i(V_topload) vp(topload) ip(V_topload)
-
-.end
-```
-
-### Step 5: Run AC Analysis and Extract Results
-
-**Calculate admittance:**
-```
-Y = I / V (complex admittance)
-
-Re{Y} = real part (conductance)
-Im{Y} = imaginary part (susceptance)
-```
-
-**Convert to impedance if needed:**
-```
-Z = 1/Y
-
-|Z| = magnitude
-φ_Z = phase angle
-```
-
-**Calculate power (for actual operating voltage):**
-```
-P_spark = 0.5 × |V_actual|² × Re{Y}
-
-Example:
-If V_actual = 320 kV, Re{Y} = 1.5 μS
-P_spark = 0.5 × (320×10³)² × 1.5×10⁻⁶
- = 76.8 kW
-```
-
-### Step 6: Validation Checks
-
-**1. Phase angle check:**
-```
-Expected: φ_Z = -55° to -75°
-(Capacitive-resistive, more capacitive than resistive)
-
-If outside range:
-- Check C values (FEMM errors?)
-- Check R (unphysical value?)
-- Review frequency
-```
-
-**2. Resistance range check:**
-```
-At 200 kHz:
-- Short spark (0.5 m): R ≈ 50-150 kΩ
-- Medium spark (1.5 m): R ≈ 100-300 kΩ
-- Long spark (3 m): R ≈ 200-500 kΩ
-
-If much higher: likely streamer-dominated (OK but low power)
-If much lower: check calculations
-```
-
-**3. Capacitance validation:**
-```
-C_sh ≈ 2 pF/foot × L_spark
-
-Within factor of 2 is acceptable:
-- Higher: concentrated field near ground
-- Lower: elevated geometry, less ground coupling
-
-Exact match not expected (geometry dependent)
-```
-
-**4. Compare to measurements:**
-```
-If available:
-- Ringdown frequency shift → Y_spark
-- E-field probe + current probe → Z_spark
-
-Adjust R within bounds to match measurements
-```
-
-## Integration with Full Coil Model
-
-### Connection to Secondary Circuit
-
-The lumped spark model appears as a **load impedance** at the topload terminal:
-
-```
-[Primary] → [Coupled Transformer] → [Secondary L_sec, R_sec] → [C_topload] → [Z_spark]
- ↓
- GND
-```
-
-**Effects on coil performance:**
-
-1. **Loaded Q reduction:**
- ```
- Q_loaded < Q_unloaded
-
- More resistive spark → lower Q → faster ringdown
- ```
-
-2. **Resonant frequency shift:**
- ```
- f_loaded ≠ f₀
-
- Spark adds capacitance → lowers frequency
- Magnitude: Δf ≈ 1-5 kHz typical
- ```
-
-3. **Power extraction:**
- ```
- P_spark = fraction of total power
-
- Well-matched: 50-70% to spark
- Poorly matched: <30% to spark
- ```
-
-### Impedance Matching
-
-**For maximum power transfer:**
-```
-Want: Z_spark ≈ Z_secondary*
-
-Where Z_secondary* is complex conjugate of secondary impedance
-
-Practical approach:
-- Adjust C_topload to tune frequency
-- Spark length determines Z_spark
-- Iterate to find optimal balance
-```
-
-**Trade-offs:**
-- Larger topload: better coupling, heavier load
-- Smaller topload: higher voltage, weaker coupling
-- Spark impedance: fixed by physics (less control)
-
-## Worked Example: Complete Lumped Model
-
-**Given parameters:**
-- Frequency: f = 190 kHz
-- FEMM results: C_mut = 9.5 pF, C_sh = 7.2 pF
-- Physical bounds: R_min = 5 kΩ, R_max = 500 kΩ
-
-**Step 1: Calculate R_opt_power**
-```
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-C_total = C_mut + C_sh
- = 9.5 + 7.2
- = 16.7 pF
-
-R_opt = 1/(ω × C_total)
- = 1/(1.194×10⁶ × 16.7×10⁻¹²)
- = 1/(1.994×10⁻⁵)
- = 50.2 kΩ
-```
-
-**Step 2: Check bounds**
-```
-R_min = 5 kΩ
-R_opt = 50.2 kΩ ✓ Within bounds
-R_max = 500 kΩ
-
-Use R = 50.2 kΩ
-```
-
-**Step 3: Build SPICE model**
-```spice
-V_test topload 0 AC 1V
-C_mut topload n1 9.5p
-R_spark n1 n2 50.2k
-C_sh n2 0 7.2p
-
-.ac lin 1 190k 190k
-.end
-```
-
-**Step 4: Simulate** (example results)
-```
-Y = I/V = 5.23 μS ∠74.5°
-
-Re{Y} = 5.23 × cos(74.5°) = 1.39 μS
-Im{Y} = 5.23 × sin(74.5°) = 5.04 μS
-
-Convert to Z:
-|Z| = 1/5.23×10⁻⁶ = 191 kΩ
-φ_Z = -74.5°
-```
-
-**Step 5: Validate**
-```
-✓ φ_Z = -74.5° in expected range (-55° to -75°)
-✓ R_eq ≈ 51 kΩ close to R_opt = 50.2 kΩ
-✓ Physical: Between 5-500 kΩ
-
-C_sh check:
-L ≈ 7.2 pF / (2 pF/ft) = 3.6 ft ≈ 1.1 m
-✓ Reasonable for medium spark
-```
-
-**Step 6: Power calculation** (if V_topload = 320 kV actual)
-```
-P = 0.5 × |V|² × Re{Y}
- = 0.5 × (320×10³)² × 1.39×10⁻⁶
- = 71.2 kW
-```
-
-Model complete and ready for coil integration!
-
-## Key Takeaways
-
-- **Lumped model** treats spark as single R-C-C network: simple, fast, accurate for most cases
-- **Use for:** sparks <2 m, impedance matching, engineering estimates, quick iterations
-- **FEMM extraction:** C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂| from Maxwell matrix
-- **Optimal resistance:** R = 1/(ω × C_total) from hungry streamer theory, with physical bounds
-- **Validation checks:** phase angle, resistance range, C_sh ≈ 2 pF/ft, compare to measurements
-- **Integration:** appears as load impedance at topload, affects Q, frequency, power transfer
-- **When to upgrade:** long sparks (>2 m), current distribution needed, research applications
-
-## Practice
-
-{exercise:model-ex-01}
-
----
-**Next Lesson:** [FEMM Extraction for Lumped Models](02-femm-extraction-lumped.md)
diff --git a/spark-lessons/lessons/04-advanced-modeling/02-femm-extraction-lumped.md b/spark-lessons/lessons/04-advanced-modeling/02-femm-extraction-lumped.md
deleted file mode 100644
index b3bac34..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/02-femm-extraction-lumped.md
+++ /dev/null
@@ -1,703 +0,0 @@
----
-id: model-02
-title: "FEMM Extraction for Lumped Models"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 45
-prerequisites: ["model-01", "phys-08"]
-objectives:
- - Master FEMM setup for two-body electrostatic problems (topload + spark)
- - Extract and interpret Maxwell capacitance matrices
- - Apply correct sign conventions for mutual and shunt capacitances
- - Validate extracted capacitances against empirical rules
-tags: ["FEMM", "electrostatics", "capacitance-matrix", "extraction", "validation"]
----
-
-# FEMM Extraction for Lumped Models
-
-This lesson covers the detailed procedure for using FEMM (Finite Element Method Magnetics) to extract capacitances for lumped spark models. We'll focus on the two-body problem: topload and spark channel.
-
-## The Maxwell Capacitance Matrix
-
-### Mathematical Definition
-
-FEMM outputs the **Maxwell capacitance matrix** [C] which relates charges to voltages:
-
-```
-[Q] = [C] × [V]
-
-Where:
-Q_i = charge on conductor i (coulombs)
-V_i = potential of conductor i (volts)
-[C] = capacitance matrix (farads)
-```
-
-### Matrix Properties
-
-The Maxwell matrix has specific mathematical properties:
-
-**1. Symmetry:**
-```
-C_ij = C_ji
-
-Physical basis: Maxwell's equations are symmetric
-Numerical check: |C_ij - C_ji| / |C_ij| < 0.01
-```
-
-**2. Diagonal elements positive:**
-```
-C_ii > 0 (self-capacitance)
-
-Physical meaning: Charge required to raise conductor i to 1V
-```
-
-**3. Off-diagonal elements negative:**
-```
-C_ij < 0 for i ≠ j
-
-IMPORTANT: This is the Maxwell convention!
-
-Physical meaning: Negative charge induced on conductor j
-when conductor i is at +1V (field lines terminate)
-```
-
-**4. Row sum equals zero:**
-```
-Σ_j C_ij = 0 for each row i
-
-Conservation: Total charge to ground = 0 when far-field grounded
-```
-
-### Two-Body System
-
-For topload (conductor 1) and spark (conductor 2), with ground implicit:
-
-```
- [1] [2]
-[1] [ C₁₁ C₁₂ ]
-[2] [ C₂₁ C₂₂ ]
-
-Example values:
- [Top] [Spark]
-[Top] [ 30 -8 ] pF
-[Spark][ -8 14 ] pF
-```
-
-**Interpretation:**
-
-- **C₁₁ = 30 pF:** Topload self-capacitance (to infinity/ground at ∞)
-- **C₂₂ = 14 pF:** Spark self-capacitance (to infinity)
-- **C₁₂ = C₂₁ = -8 pF:** Mutual capacitance (negative per convention)
-
-**Note:** These are NOT the circuit elements we need directly. Extraction required!
-
-## FEMM Setup for Lumped Model
-
-### Problem Type and Geometry
-
-**Problem configuration:**
-```
-Type: Electrostatic, axisymmetric
-Coordinates: r-z (cylindrical)
-Frequency: 0 Hz (pure electrostatic)
-Precision: 1e-8 (default)
-```
-
-**Geometry components:**
-
-**1. Topload (Conductor 1):**
-```
-Typical: Toroid
-- Major diameter: 20-50 cm
-- Minor diameter: 5-15 cm
-- Or sphere: radius 10-25 cm
-
-Position: Origin at center
-Material: Perfect conductor (grouped as Conductor 1)
-```
-
-**2. Spark channel (Conductor 2):**
-```
-Shape: Vertical cylinder
-- Length: Target spark length (e.g., 1.5 m)
-- Diameter: 1-3 mm (typical plasma channel)
-- Position: Base at topload bottom, extending downward
-
-Material: Perfect conductor (grouped as Conductor 2)
-
-Note: Small gap (0.1 mm) between topload and spark base
- This is numerical convenience; results insensitive
-```
-
-**3. Ground plane:**
-```
-Position: Below spark tip
-Distance: 10-20 cm below tip (or room floor distance)
-Extent: Large radius (3-5× max dimension)
-
-Boundary: V = 0 (Dirichlet condition)
-```
-
-**4. Outer boundary:**
-```
-Shape: Large cylindrical volume
-Radius: 3-5× maximum geometry dimension
-Height: Extends above and below structure
-
-Boundary condition: V = 0 (or mixed, grounded at ∞)
-```
-
-**5. Medium:**
-```
-All regions: Air
-ε_r = 1 (vacuum permittivity)
-```
-
-### Step-by-Step FEMM Procedure
-
-**Step 1: Create geometry**
-
-```
-1. Draw toroid in r-z plane (right half only, axisymmetric)
- - Use arc and line segments
- - Close contour
-
-2. Draw spark cylinder
- - Rectangle in r-z coordinates
- - r: [0, radius], z: [z_base, z_tip]
-
-3. Draw ground plane
- - Horizontal line at z = z_ground
- - r: [0, r_max]
-
-4. Draw outer boundary box
- - Enclose all geometry
- - Large enough to avoid boundary effects
-```
-
-**Step 2: Define materials**
-```
-Create air material block:
-- Name: "Air"
-- Relative permittivity: ε_r = 1
-- Apply to all regions
-```
-
-**Step 3: Define conductors**
-
-```
-Property → Conductors → Add Conductors:
-
-Conductor 1 (Topload):
-- Select all topload surface nodes/segments
-- Group: "1"
-- Voltage: 1V (test voltage)
-
-Conductor 2 (Spark):
-- Select all spark cylinder surfaces
-- Group: "2"
-- Voltage: (floating potential)
-
-Ground plane:
-- Boundary condition: V = 0 (not a separate conductor)
-```
-
-**Step 4: Mesh generation**
-
-```
-Mesh → Create Mesh
-
-Automatic meshing with refinement near conductors:
-- Typical element size: 1-5 mm near spark
-- 10-50 mm in far field
-- Total elements: 5,000-20,000 for lumped model
-
-Check mesh quality visually (no overly elongated triangles)
-```
-
-**Step 5: Solve**
-
-```
-Analysis → Solve
-
-Solver runs (typically <10 seconds for lumped model)
-
-Check for convergence:
-- Should converge in <100 iterations
-- Final residual < 1e-8
-- No warnings about poor mesh quality
-```
-
-**Step 6: Extract capacitance matrix**
-
-```
-View → Circuit Props
-
-Output shows:
-- Conductor properties (V, Q for each)
-- Capacitance matrix [C]
-
-Copy matrix values to spreadsheet or script
-```
-
-### Example FEMM Output
-
-**Conductor properties:**
-```
-Conductor 1 (Topload):
- Voltage: 1.0000 V (fixed)
- Charge: 3.52e-11 C = 35.2 pC
-
-Conductor 2 (Spark):
- Voltage: 0.2982 V (computed, floating)
- Charge: 1.68e-11 C = 16.8 pC
-```
-
-**Capacitance matrix [C]:**
-```
- [1] [2]
-[1] [ 35.2 -10.5 ] pF
-[2] [-10.5 16.8 ] pF
-```
-
-**Verify properties:**
-```
-✓ Symmetric: C₁₂ = C₂₁ = -10.5 pF
-✓ Diagonal positive: C₁₁, C₂₂ > 0
-✓ Off-diagonal negative: C₁₂, C₂₁ < 0
-✓ Row sum: 35.2 + (-10.5) = 24.7 ≈ 0? NO - ground implicit!
-
-Row sum ≠ 0 is OK: ground is not in matrix (infinite conductor)
-```
-
-## Extracting Circuit Elements
-
-### Formula Derivation
-
-**Goal:** Extract C_mut and C_sh for this circuit:
-
-```
-Topload ---[C_mut]--- Spark ---[C_sh]--- Ground
-```
-
-**C_mut (Mutual Capacitance):**
-
-Mutual capacitance is the capacitance *between* topload and spark.
-
-```
-C_mut = |C₁₂| = |C₂₁|
-
-Take absolute value of off-diagonal element
-
-Why absolute?
-- Circuit element capacitances are positive
-- Maxwell convention uses negative for mutual coupling
-- |C₁₂| converts to standard circuit convention
-```
-
-**Example:**
-```
-C₁₂ = -10.5 pF
-C_mut = |-10.5| = 10.5 pF ✓
-```
-
-**C_sh (Shunt Capacitance to Ground):**
-
-Shunt capacitance is spark-to-ground with topload present.
-
-**Method 1: From row sum**
-
-The charge on spark (row 2) with V₁=V_topload, V₂=V_spark is:
-```
-Q₂ = C₂₁ × V₁ + C₂₂ × V₂
-
-Charge to ground = -(Q₂) assuming no other charges
-But this includes charge from topload coupling!
-
-Actual spark-to-ground capacitance:
-C_sh = C₂₂ + C₂₁
- = C₂₂ - |C₁₂| (since C₂₁ = C₁₂ < 0)
-```
-
-**Derivation:**
-```
-Consider: Topload grounded (V₁ = 0), spark at V₂ = 1V
-
-Charge on spark: Q₂ = C₂₁ × 0 + C₂₂ × 1 = C₂₂
-But part of this is coupled to topload!
-
-Spark-to-actual-ground capacitance:
-Total capacitance to ∞ = C₂₂
-Minus coupling through topload = -C₂₁ = |C₁₂|
-Net shunt: C_sh = C₂₂ - |C₁₂|
-```
-
-**Example:**
-```
-C₂₂ = 16.8 pF
-C₁₂ = -10.5 pF
-C_sh = 16.8 - 10.5 = 6.3 pF ✓
-```
-
-**Method 2: Direct measurement** (verification)
-
-Run second FEMM simulation:
-```
-- Topload: V = 0 (grounded)
-- Spark: V = 1V
-- Ground: V = 0
-
-Measure charge on spark → this is C_sh directly
-
-Should match Method 1 result
-```
-
-### Sign Convention Summary
-
-**CRITICAL: Understand the sign conventions!**
-
-```
-Maxwell Matrix:
- C_ij < 0 for i ≠ j (negative mutual elements)
-
-Circuit Elements:
- All capacitances > 0 (positive values)
-
-Conversion:
- C_mut = |C₁₂| (absolute value)
- C_sh = C₂₂ - |C₁₂| (subtract absolute value)
-```
-
-**Common error:** Using C₁₂ directly as C_mut without absolute value
-**Result:** Negative capacitance in SPICE → error or nonsensical results
-
-## Validation Checks
-
-### 1. Matrix Symmetry
-
-```
-Check: |C₁₂ - C₂₁| / |C₁₂| < 0.01
-
-If not symmetric:
-- Mesh too coarse → refine near conductors
-- Convergence issue → lower tolerance
-- Geometry problem → check closed contours
-```
-
-### 2. Physical Value Ranges
-
-**C_mut (Mutual):**
-```
-Expected: 5-20 pF for typical Tesla coil toploads
-
-Too high (>30 pF): Check geometry (topload too large?)
-Too low (<2 pF): Check geometry (spark too short/far?)
-```
-
-**C_sh (Shunt):**
-```
-Empirical rule: C_sh ≈ 2 pF/foot × L_spark
-
-Example: L = 1.8 m = 5.9 ft
-Expected: C_sh ≈ 2 × 5.9 = 11.8 pF
-
-Acceptable range: 0.5× to 2.5× empirical prediction
-```
-
-**Why deviations occur:**
-```
-Higher than expected:
-- Nearby ground objects (walls, floor close)
-- Wide spark base (cone shape)
-- Ground plane too close in simulation
-
-Lower than expected:
-- Elevated spark (no ground plane modeled)
-- Thin diameter (<1 mm)
-- Topload shielding effect strong
-- Empirical rule may include mutual capacitance
-```
-
-**Important note for distributed models:**
-When using distributed models (Part 4, Lesson 4), the total C_sh from summing all segments may differ from the 2 pF/foot rule by a larger factor. This is because:
-- Matrix extraction method sums individual contributions
-- Mutual couplings between segments affect total
-- Distributed geometry changes field distribution
-- Factor of 2-3 deviation is normal and acceptable
-- Use FEMM value (more accurate for specific geometry)
-
-### 3. Energy Conservation Check
-
-```
-Total energy stored should be conserved:
-
-W = 0.5 × V^T × C × V
-
-For V = [1, V₂]:
-W = 0.5 × (C₁₁ + 2×C₁₂×V₂ + C₂₂×V₂²)
-
-Check: Should be positive, finite
-```
-
-### 4. Ground Distance Sensitivity
-
-**Test:** Vary ground plane distance, check C_sh
-
-```
-Ground at z = -2.0 m: C_sh = 6.8 pF
-Ground at z = -3.0 m: C_sh = 6.2 pF
-Ground at z = -5.0 m: C_sh = 6.0 pF
-
-Expect: C_sh decreases as ground moves away
-Convergence: <5% change when distance > 2× spark length
-```
-
-If C_sh changes significantly (>20%) with ground distance:
-- Ground plane too close
-- Move ground further away
-- Or accept measured geometry (e.g., actual room)
-
-## Worked Example: Complete Extraction
-
-**Given:**
-- Spark length: 1.8 m = 5.9 feet
-- FEMM simulation output (see above)
-- Operating frequency: 200 kHz
-
-**FEMM capacitance matrix:**
-```
- [1] [2]
-[1] [ 35.2 -10.5 ] pF
-[2] [-10.5 16.8 ] pF
-```
-
-**Step 1: Extract C_mut**
-```
-C_mut = |C₁₂| = |-10.5| = 10.5 pF ✓
-```
-
-**Step 2: Extract C_sh**
-```
-C_sh = C₂₂ + C₂₁
- = C₂₂ - |C₁₂|
- = 16.8 - 10.5
- = 6.3 pF ✓
-```
-
-**Step 3: Validate C_sh**
-```
-Empirical prediction:
-C_sh_predicted = 2 pF/ft × 5.9 ft = 11.8 pF
-
-FEMM result:
-C_sh_FEMM = 6.3 pF
-
-Ratio: 6.3 / 11.8 = 0.53
-
-This is LOWER than expected by factor ~2
-```
-
-**Analysis of discrepancy:**
-
-**Possible explanations:**
-```
-1. Empirical rule assumes straight vertical spark
- - If spark is angled or curved: less capacitance
- - FEMM models idealized vertical cylinder
-
-2. Empirical rule from community measurements
- - May include some C_mut in "measured" value
- - Difficult to separate mutual from shunt experimentally
- - Pure C_sh might be lower
-
-3. Ground plane distance matters
- - FEMM: specific ground geometry (15 cm below tip)
- - Empirical rule: "typical" room (floor 1-2 m away)
- - Closer ground in measurements → higher C_sh
-
-4. Diameter assumption
- - Thinner diameter → lower C_sh (logarithmic dependence)
- - C ∝ 1/ln(h/d), so d = 1 mm vs 3 mm changes C by ~30%
-```
-
-**Decision: Use FEMM value**
-```
-For modeling: Use C_sh = 6.3 pF (FEMM result)
-Reason: More accurate for specific geometry
-Empirical rule: Rough check only
-
-Within factor of 2-3: Acceptable agreement
-```
-
-**Step 4: Calculate total capacitance**
-```
-C_total = C_mut + C_sh
- = 10.5 + 6.3
- = 16.8 pF
-```
-
-**Step 5: Calculate R_opt**
-```
-f = 200 kHz
-ω = 2π × 200×10³ = 1.257×10⁶ rad/s
-
-R_opt = 1/(ω × C_total)
- = 1/(1.257×10⁶ × 16.8×10⁻¹²)
- = 47.3 kΩ ✓
-
-Within physical bounds (5-500 kΩ)
-```
-
-**Step 6: Build circuit**
-```
-SPICE netlist:
-C_mut topload spark_n 10.5p
-R_spark spark_n spark_r 47.3k
-C_sh spark_r 0 6.3p
-```
-
-Ready for simulation!
-
-## Common FEMM Errors and Troubleshooting
-
-### Problem: Matrix not symmetric
-
-**Symptoms:**
-```
-|C₁₂ - C₂₁| / |C₁₂| > 0.05
-```
-
-**Causes and fixes:**
-```
-1. Mesh too coarse
- → Refine mesh near conductors
- → Increase total element count
-
-2. Poor convergence
- → Lower precision requirement (1e-9 or 1e-10)
- → Check mesh quality
-
-3. Geometry errors
- → Verify all contours closed
- → Check no overlapping regions
-```
-
-### Problem: Negative C_sh
-
-**Symptoms:**
-```
-C_sh = C₂₂ - |C₁₂| < 0
-```
-
-**Causes:**
-```
-This should NEVER happen physically!
-
-1. Wrong extraction formula used
- → Double-check: C_sh = C₂₂ - |C₁₂|, not C₂₂ + C₁₂
-
-2. FEMM simulation error
- → Check conductor assignments
- → Verify boundary conditions
- → Remake geometry from scratch
-
-3. Conductors not properly grouped
- → Each conductor must be single contiguous group
-```
-
-### Problem: C_sh >> empirical rule (factor >5)
-
-**Symptoms:**
-```
-C_sh = 50 pF for 1 m spark (expected: 6 pF)
-```
-
-**Causes:**
-```
-1. Ground plane too close
- → Move ground plane further away
- → Check z-coordinate
-
-2. Spark diameter too large
- → Should be 1-3 mm, not 1-3 cm!
- → Check units
-
-3. Multiple ground connections
- → Check only one ground boundary condition
-```
-
-### Problem: C_mut unreasonably large
-
-**Symptoms:**
-```
-C_mut > 50 pF for medium toroid
-```
-
-**Causes:**
-```
-1. Topload size too large
- → Check diameter in correct units
-
-2. Spark embedded in topload
- → Should have small gap (0.1-1 mm)
-
-3. Scale error
- → Check all dimensions (cm? m? mm?)
-```
-
-## Best Practices
-
-**1. Consistent units:**
-```
-Recommended: Centimeters throughout FEMM
-- Easy to work with Tesla coil scales
-- Avoid mixing mm/cm/m
-- Output still in standard SI units
-```
-
-**2. Mesh refinement:**
-```
-Start coarse → check matrix → refine if needed
-
-Adequate: Symmetry <1% error
-Overkill: >50,000 elements for lumped model (slow, no benefit)
-```
-
-**3. Parametric studies:**
-```
-Vary one parameter at a time:
-- Spark length: C_sh should scale linearly
-- Ground distance: C_sh should saturate at large distance
-- Diameter: C_sh logarithmic dependence (weak)
-
-Check trends make physical sense
-```
-
-**4. Documentation:**
-```
-Save for each simulation:
-- Geometry parameters (toroid size, spark length, ground position)
-- Mesh statistics (elements, convergence)
-- Raw matrix output
-- Extracted C_mut, C_sh
-- Validation checks
-
-Build database for future reference
-```
-
-## Key Takeaways
-
-- **Maxwell matrix** uses negative off-diagonals for mutual capacitance (standard convention)
-- **Extraction formulas:** C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂| (absolute value critical!)
-- **FEMM setup:** Axisymmetric, two conductors (topload at 1V, spark floating), ground boundary
-- **Validation:** Check symmetry, C_sh ≈ 2 pF/ft ± factor 2, physical value ranges
-- **Discrepancies:** FEMM more accurate than empirical rules for specific geometry
-- **Common errors:** Wrong sign conversion, mesh too coarse, units mismatch, ground too close
-- **Use FEMM values** in circuit model, not empirical estimates
-
-## Practice
-
-{exercise:model-ex-02}
-
----
-**Next Lesson:** [Distributed Model Theory](03-distributed-model.md)
diff --git a/spark-lessons/lessons/04-advanced-modeling/03-distributed-model.md b/spark-lessons/lessons/04-advanced-modeling/03-distributed-model.md
deleted file mode 100644
index c757054..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/03-distributed-model.md
+++ /dev/null
@@ -1,576 +0,0 @@
----
-id: model-03
-title: "Distributed Model Theory"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 40
-prerequisites: ["model-01", "model-02"]
-objectives:
- - Understand when and why distributed models are necessary
- - Master nth-order segmentation strategy and circuit topology
- - Learn the trade-offs between lumped and distributed approaches
- - Apply distributed models to long sparks and research applications
-tags: ["distributed-model", "segmentation", "nth-order", "circuit-topology"]
----
-
-# Distributed Model Theory
-
-The **distributed spark model** divides the spark into multiple segments, each with its own resistance and capacitance network. This captures spatial variations in current, voltage, and plasma properties along the spark length.
-
-## Why Distributed Models?
-
-### Limitations of Lumped Models
-
-Lumped models treat the entire spark as a single element, which **fails to capture:**
-
-**1. Current distribution along spark**
-```
-Base: Full current (directly coupled to topload)
-Middle: Reduced current (capacitive shunting)
-Tip: Much lower current (weak coupling, high shunt)
-
-Lumped model: Assumes uniform current everywhere (wrong!)
-```
-
-**2. Voltage distribution**
-```
-Actual: Non-linear voltage drop due to distributed capacitance
-Lumped: Assumes simple voltage divider (oversimplified)
-
-Capacitive divider effects occur at EACH point along spark
-```
-
-**3. Base vs tip physical differences**
-```
-Base properties:
-- Hot plasma (continuously heated)
-- Well-coupled to topload
-- Low resistance (leader regime)
-- High current density
-
-Tip properties:
-- Cool plasma (sporadic heating)
-- Weakly coupled
-- High resistance (streamer regime)
-- Low current density
-
-Lumped model: Single R averages this out (loses physics!)
-```
-
-**4. Leader/streamer transitions**
-```
-Long sparks: Base forms leader, tip remains streamer
-Different physics: Different R, different behavior
-Lumped R: Cannot represent this transition zone
-```
-
-**5. Very long sparks (>3 m)**
-```
-Distributed effects dominate
-Single lumped R is poor approximation
-Error: Can be factor of 2-5 in current distribution
-```
-
-### When to Use Distributed Models
-
-**Use distributed when:**
-
-1. **Spark length > 1-2 meters**
- - Spatial variations become significant
- - Base-to-tip differences critical
-
-2. **Current distribution matters**
- - Measuring actual current profile along spark
- - Validating against detailed experimental data
- - Understanding leader formation dynamics
-
-3. **Research applications**
- - Physics investigations
- - Leader/streamer transition studies
- - Publication-quality results
-
-4. **Extreme parameters**
- - Very low frequency (λ comparable to L)
- - Very high voltage (breakdown physics critical)
- - Unusual geometries (horizontal, branched)
-
-**Stick with lumped when:**
-
-1. **Quick design iterations**
- - Impedance matching studies
- - Component selection
- - Performance estimates
-
-2. **Short sparks (<1 m)**
- - Uniform properties adequate
- - Computational efficiency critical
-
-3. **Engineering estimates**
- - ±20% accuracy sufficient
- - Fast turnaround needed
-
-**Computational trade-off:**
-```
-Lumped model: <1 second
-Distributed (n=10): ~10-30 seconds
-Distributed (n=20): ~1-5 minutes
-
-Speedup factor: 600-18000×
-
-Use distributed only when benefits justify cost!
-```
-
-## Segmentation Strategy
-
-### Dividing the Spark
-
-**Equal-length segments:**
-```
-n = number of segments (typically 5-20)
-L_segment = L_total / n
-
-Segment numbering:
- i = 1: Base (connected to topload)
- i = 2, 3, ..., n-1: Middle sections
- i = n: Tip (furthest from topload)
-```
-
-**Example: 2.4 m spark, n=6 segments**
-```
-L_segment = 2.4 / 6 = 0.4 m each
-
-Segment 1 (base): z = 0 to -0.4 m
-Segment 2: z = -0.4 to -0.8 m
-Segment 3: z = -0.8 to -1.2 m
-Segment 4: z = -1.2 to -1.6 m
-Segment 5: z = -1.6 to -2.0 m
-Segment 6 (tip): z = -2.0 to -2.4 m
-```
-
-### Why Equal Lengths?
-
-**Advantages:**
-```
-1. Simple FEMM geometry
- - Uniform cylinder sections
- - Easy to script/automate
-
-2. Uniform discretization
- - No bias toward any region
- - Straightforward convergence analysis
-
-3. Easy implementation
- - Regular array indexing
- - Simple matrix structure
-
-4. Standard practice
- - Literature comparisons
- - Validated approach
-```
-
-**Non-uniform segmentation possible:**
-```
-Alternative: Finer near tip (where R changes rapidly)
-
-Example: Geometric progression
- L[i] = L_base × ratio^(i-1)
-
-Benefits: Better captures tip physics with fewer segments
-
-Drawbacks:
- - More complex FEMM setup
- - Harder to interpret results
- - Diminishing returns for extra complexity
-
-Recommendation: Use equal lengths unless specific research need
-```
-
-### Choosing n (Number of Segments)
-
-**Convergence vs computational cost:**
-
-```
-n = 1: Lumped model (fastest, least accurate for long sparks)
-n = 5: Coarse distributed (captures main trends)
-n = 10: Standard distributed (good balance)
-n = 20: Fine distributed (research quality)
-n = 50: Overkill (no improvement, much slower)
-```
-
-**Rule of thumb:**
-```
-L < 1 m: Use lumped (n=1)
-L = 1-2 m: n = 5-10
-L = 2-4 m: n = 10-15
-L > 4 m: n = 15-20
-
-Convergence test: Double n, check if results change <10%
-If yes: Original n sufficient
-If no: Use higher n
-```
-
-**Practical limitations:**
-```
-FEMM: (n+1)×(n+1) matrix, scales as O(n²)
-SPICE: Network complexity, scales as O(n²-n³)
-Optimization: R sweep, scales as O(n)
-
-Total time ≈ t_FEMM × n² + t_SPICE × n² + t_optimize × n
-
-Diminishing returns beyond n ≈ 20
-```
-
-## Circuit Topology
-
-### Per-Segment Components
-
-**Each segment i has:**
-
-**1. Resistance R[i]**
-```
-Physical meaning: Plasma resistance of that segment
-Units: Ohms (typically kΩ to MΩ)
-Variable: To be optimized
-Expectation: Monotonically increasing from base to tip
-```
-
-**2. Mutual capacitances C[i,j]**
-```
-Coupling to:
- - Topload (j=0)
- - All other segments (j=1 to n, j≠i)
-
-Extracted from FEMM (n+1)×(n+1) matrix
-
-Expectation:
- - Stronger coupling to nearby segments
- - Weaker coupling to distant segments
- - C[i,j] decreases with |i-j|
-```
-
-**3. Shunt capacitance to ground**
-```
-Included in capacitance matrix diagonal
-NOT a separate component in circuit
-
-C[i,i] (diagonal) represents self-capacitance
-Includes ground coupling implicitly
-```
-
-### Network Structure
-
-**Full distributed network:**
-
-```
-Topload (node 0, V_top)
- |
- +---[C[0,1]]---+
- | |
- +---[C[0,2]]---|---+
- | | |
- +---[C[0,3]]---|---|---+
- | | | |
- ... | | |
- | | |
- [R[1]] | |
- | | |
- Node 1 | |
- | | |
- [C[1,2]]| |
- [C[1,3]]|---|
- | | |
- [R[2]] | |
- | | |
- Node 2 | |
- | | |
- [C[2,3]]|---|
- | | |
- [R[3]] | |
- | | |
- Node 3 | |
- | | |
- | | |
- GND GND GND
- (implicit in C matrix)
-```
-
-**Matrix representation:**
-```
-For n=3 segments + topload (4×4 matrix):
-
- [0] [1] [2] [3]
-[0] [ C₀₀ C₀₁ C₀₂ C₀₃ ] Topload
-[1] [ C₁₀ C₁₁ C₁₂ C₁₃ ] Segment 1 (base)
-[2] [ C₂₀ C₂₁ C₂₂ C₂₃ ] Segment 2
-[3] [ C₃₀ C₃₁ C₃₂ C₃₃ ] Segment 3 (tip)
-
-Plus resistances:
-R[1], R[2], R[3] (one per segment)
-
-Total unknowns: 3 R values (n in general)
-```
-
-### Complexity Analysis
-
-**For n segments:**
-```
-Capacitance matrix: (n+1)×(n+1) = n² + 2n + 1 elements
-Due to symmetry: (n+1)(n+2)/2 unique values
-
-Resistances: n values
-
-Circuit nodes: n+1 (including topload)
-
-SPICE equations: O(n²) for capacitance network
- O(n) for resistances
-
-Total complexity: O(n²) dominated by capacitance couplings
-```
-
-## Physical Expectations
-
-### Resistance Distribution
-
-**Expected profile:**
-```
-R[1] < R[2] < R[3] < ... < R[n]
-
-Monotonically increasing from base to tip
-```
-
-**Typical values at 200 kHz:**
-```
-Base (segment 1):
- R[1] ≈ 5-20 kΩ
- Hot leader, well-coupled
- High current, low resistance
-
-Middle (segments 2 to n-1):
- R[i] ≈ 10-100 kΩ
- Transition region
- Moderate coupling
-
-Tip (segment n):
- R[n] ≈ 100 kΩ - 10 MΩ
- Cool streamer, weakly coupled
- Low current, high resistance
-```
-
-**Total resistance:**
-```
-R_total = Σ R[i]
-
-Expected: 50-500 kΩ at 200 kHz for 2-3 m spark
-
-Compare to lumped: Should be similar order of magnitude
-If factor >5 different: Check model carefully
-```
-
-### Capacitance Patterns
-
-**Mutual capacitance C[i,j] (i≠j):**
-```
-Nearby segments: Larger |C[i,j]|
- Example: |C[2,3]| > |C[2,5]|
-
-Distant segments: Smaller |C[i,j]|
- Example: |C[1,10]| << |C[1,2]|
-
-Topload coupling: Decreases with distance
- |C[0,1]| > |C[0,2]| > ... > |C[0,n]|
-```
-
-**Self-capacitance C[i,i] (diagonal):**
-```
-Positive (always)
-Includes shunt to ground
-Typically: 5-15 pF per segment
-
-Total shunt: Σᵢ (C[i,i] - |C[i,0]|) ≈ 2 pF/ft × L_total
-(Approximate, factor of 2-3 variation acceptable)
-```
-
-### Current Distribution
-
-**Expected behavior:**
-```
-|I[1]| > |I[2]| > ... > |I[n]|
-
-Current decreases from base to tip
-```
-
-**Physical reason:**
-```
-Capacitive shunting at each segment:
-- Some current diverts to ground through C_sh
-- Less current reaches next segment
-- Accumulates along spark length
-
-Weak coupling at tip:
-- High R, low current naturally
-- Capacitive shunting reduces current further
-- Tip current can be 10-50× lower than base
-```
-
-**Validation:**
-```
-After simulation, plot I[i] vs position
-Should be monotonically decreasing
-If not: Check R distribution, C matrix
-```
-
-### Voltage Distribution
-
-**Expected behavior:**
-```
-V[1] > V[2] > ... > V[n]
-
-Voltage decreases from base to tip
-```
-
-**But NOT linear!**
-```
-Simple resistor chain: ΔV = I × R (linear)
-
-Distributed spark: Capacitive divider at each point
- - Voltage "leaks" to ground through shunt capacitance
- - Non-linear profile
- - Steeper drop near base (high current)
- - Flatter near tip (low current)
-```
-
-## Lumped vs Distributed Comparison
-
-### Equivalent Impedance
-
-**Both models should give similar Z_spark at topload:**
-```
-Lumped: Z = R + 1/(jωC_total)
-Distributed: Z = [complex network impedance]
-
-At topload port, similar order of magnitude
-Difference: Typically 10-30% for well-designed models
-```
-
-**If very different (factor >2):**
-```
-Check:
-1. Total resistance: Σ R[i] vs R_lumped
-2. Total capacitance: C_total_distributed vs C_mut + C_sh
-3. Matrix extraction errors
-4. Convergence of n (try higher n)
-```
-
-### Power Dissipation
-
-**Lumped:**
-```
-P_total = 0.5 × I² × R
-
-Single power value
-```
-
-**Distributed:**
-```
-P[i] = 0.5 × I[i]² × R[i]
-P_total = Σ P[i]
-
-Can see where power is dissipated:
-- Base: High current, moderate R → high power
-- Middle: Moderate current and R → moderate power
-- Tip: Low current, high R → low power (often <10% of base)
-```
-
-**Insight from distributed model:**
-```
-Most power dissipated in base 1/3 of spark
-Tip contributes little to total power
-But tip electric field critical for growth!
-
-This explains why:
-- Short sparks easier (more efficient power coupling)
-- Long sparks harder (tip poorly coupled)
-- QCW benefits (maintains hot base channel)
-```
-
-## Worked Example: 3-Segment Model
-
-**Given:**
-- Total spark: 1.5 m
-- Divide into n = 3 equal segments
-- Each segment: 0.5 m
-
-**Segment locations:**
-```
-Segment 1 (base): z = 0 to -0.5 m
-Segment 2 (middle): z = -0.5 to -1.0 m
-Segment 3 (tip): z = -1.0 to -1.5 m
-```
-
-**Expected capacitance matrix (example values):**
-```
- [0] [1] [2] [3]
-[0] [ 30.0 -9.0 -3.5 -1.5 ] pF
-[1] [ -9.0 14.0 -3.0 -1.0 ]
-[2] [ -3.5 -3.0 10.5 -2.5 ]
-[3] [ -1.5 -1.0 -2.5 8.0 ]
-
-Properties:
-✓ Symmetric
-✓ Diagonal positive
-✓ Off-diagonal negative
-✓ Nearby segments more strongly coupled
-```
-
-**Expected resistance distribution:**
-```
-R[1] = 30 kΩ (base, hot)
-R[2] = 60 kΩ (middle, moderate)
-R[3] = 150 kΩ (tip, cool)
-
-Total: 240 kΩ
-
-Monotonically increasing ✓
-```
-
-**Circuit implementation:**
-```
-Convert capacitance matrix to SPICE (see next lesson)
-Add resistances R[1], R[2], R[3]
-Simulate to get currents and voltages
-```
-
-**Expected results (qualitative):**
-```
-If V_topload = 1 V (test):
-
-I[1] ≈ 15 μA (base current)
-I[2] ≈ 8 μA (middle current, ~50% of base)
-I[3] ≈ 3 μA (tip current, ~20% of base)
-
-V[1] ≈ 0.8 V (base voltage)
-V[2] ≈ 0.5 V (middle voltage)
-V[3] ≈ 0.2 V (tip voltage, non-linear drop!)
-
-P[1] ≈ 7 μW (base power, 50% of total)
-P[2] ≈ 4 μW (middle power, 30%)
-P[3] ≈ 3 μW (tip power, 20%)
-```
-
-## Key Takeaways
-
-- **Distributed models** divide spark into n segments, capturing spatial variations in current, voltage, and resistance
-- **Use when:** sparks >2 m, current distribution needed, research applications, extreme parameters
-- **Segmentation:** equal-length segments, n = 5-20 typical, convergence test by doubling n
-- **Circuit topology:** (n+1)×(n+1) capacitance matrix plus n resistances, O(n²) complexity
-- **Physical expectations:** R monotonically increasing, current decreasing, voltage non-linear, power concentrated at base
-- **Trade-off:** 1000-2000× slower than lumped, use only when benefits justify computational cost
-- **Validation:** Compare to lumped model (similar Z_spark), check physical trends (I, V, R distributions)
-- **Next steps:** FEMM extraction for n-segment geometry (Lesson 4), resistance optimization (Lesson 5)
-
-## Practice
-
-{exercise:model-ex-03}
-
----
-**Next Lesson:** [FEMM Extraction for Distributed Models](04-femm-extraction-distributed.md)
diff --git a/spark-lessons/lessons/04-advanced-modeling/04-femm-extraction-distributed.md b/spark-lessons/lessons/04-advanced-modeling/04-femm-extraction-distributed.md
deleted file mode 100644
index 657ffb5..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/04-femm-extraction-distributed.md
+++ /dev/null
@@ -1,681 +0,0 @@
----
-id: model-04
-title: "FEMM Extraction for Distributed Models"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 50
-prerequisites: ["model-02", "model-03"]
-objectives:
- - Set up multi-body FEMM geometries for n-segment spark models
- - Extract and validate (n+1)×(n+1) capacitance matrices
- - Implement capacitance matrices in SPICE with correct sign handling
- - Apply passivity checks and matrix validation procedures
-tags: ["FEMM", "distributed-model", "capacitance-matrix", "SPICE", "validation"]
----
-
-# FEMM Extraction for Distributed Models
-
-This lesson covers the complete procedure for extracting capacitance matrices from FEMM for distributed spark models and implementing them in SPICE circuit simulators.
-
-## Multi-Body Electrostatic Setup
-
-### Geometry Definition
-
-**For n segments + topload → (n+1) conductors:**
-
-```
-Example: n=5 segments
-
-Conductors:
- Body 0: Toroid topload
- Body 1: Cylinder segment 1 (base)
- Body 2: Cylinder segment 2
- Body 3: Cylinder segment 3
- Body 4: Cylinder segment 4
- Body 5: Cylinder segment 5 (tip)
- Ground: Boundary condition (not explicit conductor)
-```
-
-**Cylindrical segments:**
-```
-Each segment i:
- Length: L_segment = L_total / n
- Diameter: d (typically 1-3 mm, uniform)
- Position: Vertical stack from topload to ground
-
-Gap between segments: 0.1 mm (numerical convenience)
- - Prevents touching in FEMM
- - Results insensitive to small gap
- - Represents continuous channel physically
-```
-
-### FEMM Axisymmetric Coordinates
-
-**r-z coordinate system:**
-
-```
-Example: 2.0 m spark, n=5, each segment 0.4 m
-
-Topload (toroid):
- Major diameter: 30 cm → r_major = 15 cm
- Minor diameter: 10 cm → r_minor = 5 cm
- Center: z = 0
- Lowest point: z = -5 cm
-
-Segment 1 (base):
- r = 0.1 cm (diameter = 2 mm)
- z from -5.1 cm to -45.1 cm
- Length: 40 cm
- Gap: 0.1 cm below topload
-
-Segment 2:
- z from -45.2 cm to -85.2 cm
- Gap: 0.1 cm above segment 1
-
-Segment 3:
- z from -85.3 cm to -125.3 cm
-
-Segment 4:
- z from -125.4 cm to -165.4 cm
-
-Segment 5 (tip):
- z from -165.5 cm to -205.5 cm
-
-Ground plane:
- z = -220 cm (15 cm below tip)
- r = 0 to 300 cm (large extent)
- Boundary: V = 0
-
-Outer boundary:
- r = 300 cm
- z = -250 cm to +50 cm
- Boundary: V = 0 (far field)
-```
-
-**Critical: Consistent numbering!**
-```
-FEMM conductor numbers must match array indices:
- Conductor 0 = Topload = C[0,:]
- Conductor 1 = Segment 1 (base) = C[1,:]
- ...
- Conductor n = Segment n (tip) = C[n,:]
-```
-
-### Step-by-Step FEMM Procedure
-
-**Step 1: Problem setup**
-```
-File → New
-Problem Type: Electrostatic, Axisymmetric
-Frequency: 0 Hz
-Length units: Centimeters (recommended)
-Precision: 1e-8
-```
-
-**Step 2: Draw geometry**
-```
-1. Draw toroid (arcs + lines, right half only)
-2. Draw n rectangles for spark segments
- - Each: width = r_spark, height = L_segment
- - Stack vertically with small gaps
-3. Draw ground plane (horizontal line)
-4. Draw outer boundary (large rectangle)
-5. Close all contours (check with "Show Points")
-```
-
-**Step 3: Define materials**
-```
-Materials → Add Material:
- Name: "Air"
- Relative permittivity: εr = 1
-
-Apply "Air" to all regions (click inside each)
-```
-
-**Step 4: Define conductors**
-```
-Properties → Conductors → Add Property:
-
-For i = 0 to n:
- Name: "Conductor_i"
- Voltage:
- i = 0: V = 1V (topload excitation)
- i = 1 to n: (floating)
-
-Assign conductor properties:
- - Select all boundary nodes/segments for each body
- - Right-click → Set Conductor
- - Choose corresponding conductor number
-
-CRITICAL: Verify numbering matches geometry!
-```
-
-**Step 5: Boundary conditions**
-```
-Ground plane and outer boundary:
- Select boundary segments
- Properties → Boundary → Add Property:
- Name: "Ground"
- Type: Fixed Voltage V = 0
- Apply to ground plane and outer boundary
-```
-
-**Step 6: Meshing**
-```
-Mesh → Create Mesh
-
-Automatic mesh with adaptive refinement:
- Near conductors: ~0.5 mm triangle size
- Mid-field: ~5 mm
- Far field: ~50 mm
-
-Expected element count:
- n = 5: ~15,000-30,000 elements
- n = 10: ~30,000-60,000 elements
- n = 20: ~60,000-120,000 elements
-
-Visual check: No extremely elongated triangles
-```
-
-**Step 7: Solve**
-```
-Analysis → Solve
-
-Convergence:
- - Should complete in <1 minute for n≤10
- - Iterations: 50-200 typical
- - Final residual < 1e-8
-
-Check for warnings:
- - Mesh quality issues
- - Conductor connectivity problems
- - Non-convergence (increase iterations or refine mesh)
-```
-
-**Step 8: Extract capacitance matrix**
-```
-View Results → Circuit Props
-
-Conductor properties window shows:
- - Voltage on each conductor
- - Charge on each conductor
- - Capacitance matrix [C]
-
-Copy matrix to file:
- - Select all text
- - Copy to spreadsheet or script
- - Save for processing
-```
-
-## Capacitance Matrix Output
-
-### Matrix Structure
-
-**For n=5 segments (6×6 matrix):**
-
-```
- [0] [1] [2] [3] [4] [5]
-[0] [ C₀₀ C₀₁ C₀₂ C₀₃ C₀₄ C₀₅ ]
-[1] [ C₁₀ C₁₁ C₁₂ C₁₃ C₁₄ C₁₅ ]
-[2] [ C₂₀ C₂₁ C₂₂ C₂₃ C₂₄ C₂₅ ]
-[3] [ C₃₀ C₃₁ C₃₂ C₃₃ C₃₄ C₃₅ ]
-[4] [ C₄₀ C₄₁ C₄₂ C₄₃ C₄₄ C₄₅ ]
-[5] [ C₅₀ C₅₁ C₅₂ C₅₃ C₅₄ C₅₅ ]
-
-All values in pF (picofarads)
-```
-
-**Example numerical values:**
-```
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
-
-(Illustrative values for 2 m spark, n=5)
-```
-
-### Matrix Properties
-
-**1. Symmetry:**
-```
-C[i,j] = C[j,i]
-
-Check: For all i 0 for all i
-
-Self-capacitance (conductor to infinity)
-Always positive by definition
-
-Example:
-C[0,0] = 32.5 pF ✓
-C[1,1] = 14.8 pF ✓
-...all positive
-```
-
-**3. Off-diagonal negative:**
-```
-C[i,j] < 0 for all i ≠ j
-
-Maxwell convention: Mutual capacitances negative
-
-Example:
-C[0,1] = -9.2 pF ✓
-C[2,4] = -0.7 pF ✓
-...all negative
-```
-
-**4. Row sum ≈ 0:**
-```
-Σⱼ C[i,j] ≈ 0 (but not exact due to ground at infinity)
-
-Check: Sum should be small compared to diagonal
-
-Example row 2:
--3.1 + (-2.8) + 10.4 + (-2.1) + (-0.7) + (-0.3) = 1.4 pF
-Compared to C[2,2] = 10.4: ratio = 13%
-
-Acceptable if <20%
-```
-
-## Matrix Validation
-
-### Check 1: Symmetry
-
-**Procedure:**
-```python
-# Pseudocode
-for i in range(n+1):
- for j in range(i+1, n+1):
- error = abs(C[i,j] - C[j,i]) / abs(C[i,j])
- if error > 0.01:
- print(f"Asymmetry at [{i},{j}]: {error*100:.2f}%")
- # ACTION: Refine mesh, check convergence
-```
-
-**If not symmetric:**
-- Mesh too coarse → refine near conductors
-- Poor convergence → increase precision or iterations
-- Geometry error → check conductor assignments
-
-### Check 2: Positive Semi-Definite (Passivity)
-
-**Eigenvalue test:**
-```
-Calculate eigenvalues λ of matrix C
-
-Physically passive if:
- - All λ ≥ 0 (non-negative)
- - One λ = 0 (ground reference freedom)
- - Rest λ > 0 (strictly positive)
-
-If any λ < 0 (within numerical precision):
- - Matrix not physically realizable
- - Check FEMM setup (conductor assignments)
- - Refine mesh
- - Verify boundary conditions
-```
-
-**Why this matters:**
-```
-Negative eigenvalue → negative energy stored
-Physically impossible for passive capacitance network
-Indicates error in simulation or extraction
-```
-
-### Check 3: Physical Value Patterns
-
-**Nearby vs distant coupling:**
-```
-Expectation: |C[i,j]| decreases with |i-j|
-
-Example: Row 3 (segment 3)
-C[3,0] = -1.2 (distant from topload)
-C[3,2] = -2.1 (adjacent segment)
-C[3,4] = -1.8 (adjacent segment)
-C[3,5] = -0.5 (distant, tip)
-
-Check: |C[3,2]| = 2.1 > |C[3,5]| = 0.5 ✓
- |C[3,4]| = 1.8 > |C[3,0]| = 1.2 ✓
-
-Adjacent segments most strongly coupled ✓
-```
-
-**Topload coupling:**
-```
-Expectation: |C[0,i]| decreases with i (distance from topload)
-
-|C[0,1]| = 9.2 (base, closest)
-|C[0,2]| = 3.1
-|C[0,3]| = 1.2
-|C[0,4]| = 0.6
-|C[0,5]| = 0.3 (tip, farthest)
-
-Monotonically decreasing ✓
-```
-
-### Check 4: Total Shunt Capacitance
-
-**Approximate formula:**
-```
-C_sh_total ≈ Σᵢ₌₁ⁿ (C[i,i] - |C[i,0]|)
-
-This sums shunt capacitance of all segments
-
-Empirical check: C_sh_total ≈ 2 pF/foot × L_total
-```
-
-**Example calculation:**
-```
-Segment 1: C[1,1] - |C[1,0]| = 14.8 - 9.2 = 5.6 pF
-Segment 2: C[2,2] - |C[2,0]| = 10.4 - 3.1 = 7.3 pF
-Segment 3: C[3,3] - |C[3,0]| = 8.6 - 1.2 = 7.4 pF
-Segment 4: C[4,4] - |C[4,0]| = 7.4 - 0.6 = 6.8 pF
-Segment 5: C[5,5] - |C[5,0]| = 5.8 - 0.3 = 5.5 pF
-
-C_sh_total = 5.6 + 7.3 + 7.4 + 6.8 + 5.5 = 32.6 pF
-
-Expected: 2 pF/ft × 6.56 ft = 13.1 pF
-
-Ratio: 32.6 / 13.1 = 2.5
-
-Higher than expected, but within factor of 2-3 (acceptable)
-```
-
-**Why discrepancy?**
-```
-1. Matrix interpretation method
- - C[i,i] includes all field terminations
- - Simple sum may overcount mutual terms
- - Exact extraction more complex
-
-2. Distributed vs lumped geometry
- - Segmentation changes field distribution
- - Not directly comparable to continuous cylinder
-
-3. Empirical rule uncertainty
- - ±50% variation typical
- - Geometry and environment dependent
-
-Conclusion: Factor of 2-3 deviation is NORMAL for distributed models
-Use FEMM values (more accurate for specific geometry)
-```
-
-## Implementing in SPICE
-
-### The Challenge: Negative Off-Diagonals
-
-**Problem:**
-```
-SPICE capacitor syntax:
- C_name node1 node2 value
-
-Value must be positive!
- C1 n1 n2 10p ← OK
- C1 n1 n2 -10p ← ERROR! Unphysical
-
-But Maxwell matrix has C[i,j] < 0 for i≠j
-Cannot use directly in SPICE!
-```
-
-### Solution 1: Partial Capacitance Transformation
-
-**Convert Maxwell → Partial (all positive):**
-
-**Formula:**
-```
-For off-diagonal (between nodes):
- C_partial[i,j] = -C_Maxwell[i,j] (flip sign!)
-
-For diagonal (to ground):
- C_partial[i,ground] = C[i,i] - Σⱼ₌₀ⁿ C_partial[i,j]
- (j≠i)
-```
-
-**SPICE implementation:**
-```spice
-* Partial capacitance network
-* Between every node pair i,j where i 1 and both i,j ≥ 1
-```
-
-**When acceptable:**
-```
-Large n (≥10): Distant couplings small (<10% of adjacent)
-Quick estimates: Engineering accuracy sufficient
-Weak segment-to-segment coupling: Topload dominates
-
-Example: |C[2,8]| << |C[2,3]|
-Dropping C[2,8] has negligible effect
-```
-
-**Validation:**
-```
-Compare:
- Full matrix: Z_spark = Z_full
- Nearest-neighbor: Z_spark = Z_approx
-
-If |Z_full - Z_approx| / |Z_full| < 0.1:
- Approximation acceptable
-
-Typically valid for n ≥ 10
-```
-
-## Worked Example: n=5 Complete Extraction
-
-**Given FEMM output (from earlier):**
-```
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ] pF
-```
-
-**Validation:**
-```
-✓ Symmetric: C[i,j] = C[j,i] for all i,j
-✓ Diagonal positive: All C[i,i] > 0
-✓ Off-diagonal negative: All C[i,j] < 0 for i≠j
-✓ Adjacent > distant: |C[2,3]| = 2.1 > |C[2,5]| = 0.3
-✓ Total C_sh ≈ 32.6 pF vs expected 13.1 pF (factor 2.5, acceptable)
-```
-
-**Convert to partial capacitances (selected):**
-```
-Between nodes (flip signs):
-C_0_1 = 9.2 pF
-C_0_2 = 3.1 pF
-C_1_2 = 2.8 pF
-C_2_3 = 2.1 pF
-C_3_4 = 1.8 pF
-C_4_5 = 1.4 pF
-... (15 between-node caps total)
-
-To ground:
-C_0_gnd = 32.5 - (9.2+3.1+1.2+0.6+0.3) = 18.1 pF
-C_1_gnd = 14.8 - (9.2+2.8+0.9+0.4+0.2) = 1.3 pF
-... (calculate for all nodes)
-```
-
-**SPICE implementation (abbreviated):**
-```spice
-* 5-segment distributed spark model
-.param freq=190k
-
-V_test topload 0 AC 1V
-
-* Partial capacitances (between nodes)
-C_0_1 topload seg1 9.2p
-C_0_2 topload seg2 3.1p
-C_1_2 seg1 seg2 2.8p
-C_2_3 seg2 seg3 2.1p
-C_3_4 seg3 seg4 1.8p
-C_4_5 seg4 seg5 1.4p
-* ... (add all others)
-
-* To ground
-C_0_gnd topload 0 18.1p
-C_1_gnd seg1 0 1.3p
-* ... (add all segments)
-
-* Resistances (to be optimized, placeholder values)
-R1 seg1 seg1_r 50k
-R2 seg2 seg2_r 80k
-R3 seg3 seg3_r 120k
-R4 seg4 seg4_r 180k
-R5 seg5 seg5_r 300k
-
-.ac lin 1 190k 190k
-.print ac v(topload) i(V_test) v(seg1) v(seg2) v(seg3) v(seg4) v(seg5)
-.end
-```
-
-**Next step:** Optimize R values (Lesson 5)
-
-## Key Takeaways
-
-- **(n+1)×(n+1) matrix** for n segments + topload, extracted from FEMM multi-body electrostatic simulation
-- **FEMM setup:** Axisymmetric, equal-length cylinder segments, 0.1 mm gaps, conductor numbering consistent with indices
-- **Matrix validation:** Check symmetry (<1% error), positive semi-definite (passivity), physical patterns (adjacent > distant)
-- **Total C_sh check:** Σ(C[i,i] - |C[i,0]|) vs 2 pF/ft rule, factor 2-3 deviation normal for distributed models
-- **SPICE implementation:** Three methods - partial capacitance (flip signs), controlled sources (direct), nearest-neighbor (approximation)
-- **Partial capacitance:** C_partial[i,j] = -C_Maxwell[i,j], all positive values, standard for SPICE
-- **Passivity check:** All eigenvalues ≥ 0, ensures physical realizability, critical validation step
-- **Use FEMM values** over empirical rules for distributed models (more accurate for segmented geometry)
-
-## Practice
-
-{exercise:model-ex-04}
-
----
-**Next Lesson:** [Resistance Optimization Methods](05-resistance-optimization.md)
diff --git a/spark-lessons/lessons/04-advanced-modeling/05-resistance-optimization.md b/spark-lessons/lessons/04-advanced-modeling/05-resistance-optimization.md
deleted file mode 100644
index 0473942..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/05-resistance-optimization.md
+++ /dev/null
@@ -1,703 +0,0 @@
----
-id: model-05
-title: "Resistance Optimization Methods"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 45
-prerequisites: ["model-03", "model-04"]
-objectives:
- - Master iterative resistance optimization algorithm with damping
- - Apply position-dependent physical bounds to resistance values
- - Understand circuit-determined resistance as simplified alternative
- - Validate total resistance ranges and convergence behavior
-tags: ["optimization", "resistance", "iterative-algorithm", "convergence", "validation"]
----
-
-# Resistance Optimization Methods
-
-This lesson covers methods for determining the resistance values R[i] for each segment in a distributed spark model. We present two approaches: a rigorous iterative optimization and a simplified circuit-determined method.
-
-## The Optimization Problem
-
-### Goal and Challenges
-
-**Objective:**
-```
-Find R[i] for i = 1 to n that maximizes total power dissipation:
-
- P_total = Σᵢ P[i]
- where P[i] = 0.5 × |I[i]|² × R[i]
-
-Subject to physical constraints:
- R_min[i] ≤ R[i] ≤ R_max[i]
-```
-
-**Challenge: Coupled optimization**
-```
-Changing R[j] affects current in segment i:
- - Alters network impedance
- - Changes voltage distribution
- - Modifies all currents I[1], I[2], ..., I[n]
-
-Cannot optimize each R[i] independently!
-Must use iterative approach
-```
-
-**Computational complexity:**
-```
-For each R[i]:
- - Sweep through candidate values (20-50 points)
- - Run SPICE AC analysis for each
- - Calculate power P[i]
-
-Total simulations per iteration: n × n_sweep
- n = 10, n_sweep = 20: 200 simulations
- Iterations: 5-10 typical
- Total: 1000-2000 AC analyses
-
-Compare to lumped: 1 analysis
-Trade-off: Accuracy vs computational cost
-```
-
-## Iterative Optimization Algorithm
-
-### Initialization: Tapered Profile
-
-**Physical expectation:**
-```
-Base: Hot, well-coupled → low R
-Tip: Cool, weakly-coupled → high R
-
-Monotonically increasing R[i] from base to tip
-```
-
-**Initialize with gradient:**
-```python
-# Pseudocode
-R_base = 10e3 # 10 kΩ (hot leader)
-R_tip = 1e6 # 1 MΩ (cool streamer)
-
-for i in range(1, n+1):
- position = (i-1) / (n-1) # 0 at base (i=1), 1 at tip (i=n)
- R[i] = R_base + (R_tip - R_base) * position**2
-```
-
-**Why quadratic taper?**
-```
-Linear: R[i] = R_base + (R_tip - R_base) × position
- - Simple, but too gradual
- - Doesn't capture rapid rise near tip
-
-Quadratic: position**2
- - Gentle rise at base
- - Steeper rise near tip
- - Better matches physics
-
-Exponential: also valid, similar results
-```
-
-**Example: n=5, R_base=10k, R_tip=1M**
-```
-i=1: position=0.00 → R[1] = 10 + (1000-10)×0.00 = 10 kΩ
-i=2: position=0.25 → R[2] = 10 + 990×0.0625 = 72 kΩ
-i=3: position=0.50 → R[3] = 10 + 990×0.25 = 258 kΩ
-i=4: position=0.75 → R[4] = 10 + 990×0.5625 = 567 kΩ
-i=5: position=1.00 → R[5] = 10 + 990×1.0 = 1000 kΩ
-```
-
-### Position-Dependent Bounds
-
-**Physical limits vary with position:**
-
-**Minimum resistance R_min[i]:**
-```
-Base can achieve low R (hot, well-coupled)
-Tip unlikely to reach very low R (cool, weak coupling)
-
-Formula:
- position = (i-1) / (n-1)
- R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
- = 1 kΩ at base → 10 kΩ at tip
-```
-
-**Maximum resistance R_max[i]:**
-```
-Base unlikely to reach very high R (good power coupling)
-Tip can reach very high R (streamer regime)
-
-Formula:
- position = (i-1) / (n-1)
- R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position²
- = 100 kΩ at base → 100 MΩ at tip
-```
-
-**Example bounds: n=5**
-```
-Segment 1 (base, pos=0.00):
- R_min[1] = 1.0 kΩ
- R_max[1] = 100 kΩ
-
-Segment 2 (pos=0.25):
- R_min[2] = 3.25 kΩ
- R_max[2] = 6.3 MΩ
-
-Segment 3 (pos=0.50):
- R_min[3] = 5.5 kΩ
- R_max[3] = 25.1 MΩ
-
-Segment 4 (pos=0.75):
- R_min[4] = 7.75 kΩ
- R_max[4] = 56.3 MΩ
-
-Segment 5 (tip, pos=1.00):
- R_min[5] = 10.0 kΩ
- R_max[5] = 100 MΩ
-```
-
-**Rationale:**
-```
-1. Prevents unphysical solutions
- - Tip with R = 1 kΩ: impossible (not hot enough)
- - Base with R = 100 MΩ: impossible (too much power)
-
-2. Guides optimization
- - Narrows search space
- - Faster convergence
- - More stable
-
-3. Based on physics
- - Leader regime: R ∝ 1/T, T high at base
- - Streamer regime: R very high, weakly coupled
-```
-
-### Iterative Loop with Damping
-
-**Algorithm structure:**
-
-```python
-# Pseudocode: Iterative resistance optimization
-
-# Initialize
-R = initialize_tapered_profile(n, R_base, R_tip)
-alpha = 0.3 # Damping factor
-max_iterations = 20
-tolerance = 0.01 # 1% convergence threshold
-
-for iteration in range(max_iterations):
- R_old = R.copy()
-
- for i in range(1, n+1):
- # Sweep R[i] while keeping other R[j] (j≠i) fixed
- R_test = logspace(R_min[i], R_max[i], 20) # 20 test points
- P_test = []
-
- for R_candidate in R_test:
- R[i] = R_candidate
- # Run SPICE AC analysis
- results = run_spice_ac(R, C_matrix, freq)
- I_i = results.current[i]
- P_i = 0.5 * abs(I_i)**2 * R_candidate
- P_test.append(P_i)
-
- # Find R that maximizes power in segment i
- idx_max = argmax(P_test)
- R_optimal[i] = R_test[idx_max]
-
- # Apply damping for stability
- R_new[i] = alpha * R_optimal[i] + (1 - alpha) * R_old[i]
-
- # Clip to physical bounds
- R[i] = clip(R_new[i], R_min[i], R_max[i])
-
- # Check convergence
- max_change = max(abs(R[i] - R_old[i]) / R_old[i] for i in range(1,n+1))
- print(f"Iteration {iteration}: max change = {max_change*100:.2f}%")
-
- if max_change < tolerance:
- print("Converged!")
- break
-
-# Final result: optimized R[1], R[2], ..., R[n]
-```
-
-**Key components:**
-
-**1. Logarithmic sweep:**
-```
-R_test = logspace(log10(R_min), log10(R_max), 20)
-
-Why logarithmic?
- - R varies over orders of magnitude (1k to 100M)
- - Linear spacing: wastes points at low end
- - Log spacing: uniform coverage across decades
-```
-
-**2. Power calculation:**
-```
-P[i] = 0.5 × |I[i]|² × R[i]
-
-AC steady-state: Factor of 0.5 for sinusoidal
-RMS values: P = I_rms² × R (without 0.5)
-
-Maximize power in segment i, not total power
- - Each segment optimized to extract maximum power
- - Self-consistent with hungry streamer physics
-```
-
-**3. Damping factor α:**
-```
-R_new[i] = α × R_optimal[i] + (1-α) × R_old[i]
-
-α = 0.3 to 0.5 typical
-
-Lower α (e.g., 0.2):
- - More stable (smaller steps)
- - Slower convergence (more iterations)
- - Use if oscillations occur
-
-Higher α (e.g., 0.7):
- - Faster convergence (larger steps)
- - Risk of oscillation (overshooting)
- - Use if convergence slow
-
-Start with α = 0.3, adjust if needed
-```
-
-**4. Clipping:**
-```
-R[i] = clip(R_new[i], R_min[i], R_max[i])
-
-Ensures R stays within physical bounds
-Prevents optimizer from exploring unphysical regions
-```
-
-### Convergence Behavior
-
-**Well-coupled base segments:**
-```
-Power curve P[i](R[i]) has sharp peak
-
-Example: Segment 1
- R = 10k: P[1] = 5.2 kW
- R = 20k: P[1] = 8.1 kW
- R = 30k: P[1] = 9.4 kW ← maximum (sharp peak)
- R = 40k: P[1] = 8.9 kW
- R = 50k: P[1] = 7.8 kW
-
-Characteristics:
- - Clear optimal R
- - Fast convergence (2-3 iterations)
- - Stable solution
-```
-
-**Weakly-coupled tip segments:**
-```
-Power curve P[i](R[i]) is FLAT
-
-Example: Segment 5 (tip)
- R = 100k: P[5] = 0.82 kW
- R = 500k: P[5] = 0.85 kW
- R = 1M: P[5] = 0.83 kW
- R = 5M: P[5] = 0.81 kW
-
-All values give similar power!
-
-Characteristics:
- - Optimal R poorly defined
- - Slow/no convergence to unique value
- - May oscillate between similar R values
- - Physical: weak coupling, low power anyway
-```
-
-**Convergence criteria:**
-```
-Base segments: Converge quickly (<5 iterations)
-Middle segments: Moderate convergence (5-10 iterations)
-Tip segments: May not converge fully
-
-Solution:
- - Allow tip segments to remain at reasonable values
- - Check that change <5% for tip segments
- - Focus convergence on base/middle (where most power is)
-```
-
-**Expected final distribution:**
-```
-At 200 kHz, 2 m spark:
-
-R[1] ≈ 5-20 kΩ (base leader)
-R[2] ≈ 10-40 kΩ
-R[3] ≈ 20-80 kΩ
-...
-R[n-1] ≈ 50-200 kΩ
-R[n] ≈ 100 kΩ - 10 MΩ (tip streamer, wide range)
-
-Total: R_total = Σ R[i] should be 50-500 kΩ at 200 kHz
-```
-
-### Worked Example: n=3 Iterative Optimization
-
-**Given:**
-```
-3 segments, f = 200 kHz
-Capacitance matrix from FEMM (simplified example)
-Initial: R[1]=50k, R[2]=100k, R[3]=500k
-Damping: α = 0.4
-```
-
-**Iteration 1:**
-
-**Optimize R[1]** (keeping R[2]=100k, R[3]=500k fixed)
-```
-Sweep R[1] in [1k, 100k] (20 points, log scale)
-
-Results (example):
- R[1]=10k → P[1]=5.2 kW
- R[1]=20k → P[1]=8.1 kW
- R[1]=30k → P[1]=9.4 kW ← maximum
- R[1]=40k → P[1]=8.9 kW
- R[1]=50k → P[1]=7.8 kW (current value)
- ...
-
-R_optimal[1] = 30 kΩ
-```
-
-**Apply damping:**
-```
-R_new[1] = 0.4 × 30k + 0.6 × 50k
- = 12k + 30k
- = 42 kΩ
-
-Check bounds: 1k < 42k < 100k ✓
-Update: R[1] = 42 kΩ
-```
-
-**Optimize R[2]** (with R[1]=42k, R[3]=500k)
-```
-Sweep R[2], find maximum at R_optimal[2] = 60 kΩ
-
-Current: R[2] = 100 kΩ
-
-R_new[2] = 0.4 × 60k + 0.6 × 100k
- = 24k + 60k
- = 84 kΩ
-
-Update: R[2] = 84 kΩ
-```
-
-**Optimize R[3]** (with R[1]=42k, R[2]=84k)
-```
-Sweep R[3], power curve is FLAT:
- R[3]=200k → P[3]=0.80 kW
- R[3]=500k → P[3]=0.85 kW
- R[3]=1M → P[3]=0.83 kW
-
-Maximum at 500k, but very weak peak (±5%)
-Tip segment: poorly coupled
-
-R_optimal[3] = 500 kΩ (no change)
-R_new[3] = 0.4 × 500k + 0.6 × 500k = 500 kΩ
-
-Update: R[3] = 500 kΩ
-```
-
-**Convergence check:**
-```
-Changes:
- R[1]: 50k → 42k (change = -16%)
- R[2]: 100k → 84k (change = -16%)
- R[3]: 500k → 500k (change = 0%)
-
-Max change = 16% > 1% tolerance
-→ Not converged, continue
-```
-
-**Iteration 2:**
-
-Repeat with new R values...
-Typically base segments converge within 3-5 iterations
-
-**Final result (example):**
-```
-After 5 iterations:
-
-R[1] = 35 kΩ (converged, change <1%)
-R[2] = 75 kΩ (converged, change <1%)
-R[3] = 500 kΩ (tip, flat curve, acceptable)
-
-Total: 610 kΩ at 200 kHz
-✓ Within expected range (50-500 kΩ, high end due to tip)
-```
-
-## Simplified Method: Circuit-Determined Resistance
-
-### Key Insight
-
-**Hungry streamer physics:**
-```
-Plasma adjusts diameter to seek R_opt_power for maximum power
-R_opt = 1 / (ω × C_total)
-
-For each segment:
- - Segment sees total capacitance C_total[i]
- - Adjusts to R[i] = 1 / (ω × C_total[i])
- - Self-consistent with power optimization
-```
-
-**Capacitance weakly depends on diameter:**
-```
-C ∝ 1 / ln(h/d)
-
-Logarithmic dependence:
- - 2× diameter → ~10% capacitance change
- - R_opt also changes ~10%
- - Small error from assuming fixed C
-```
-
-### Formula
-
-**For each segment i:**
-```
-C_total[i] = Σⱼ₌₀ⁿ |C[i,j]|
-
-Sum of absolute values of all capacitances involving segment i
-
-Then:
- R[i] = 1 / (ω × C_total[i])
- R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Why this works:**
-```
-1. Hungry streamer: Seeks R_opt = 1/(ωC)
-2. Diameter self-adjusts: Matches R to C
-3. Logarithmic C(d): Error ~10-15% (small)
-4. Other uncertainties: FEMM ±5-10%, physics ±30-50%
-5. Diameter error is SMALL compared to total uncertainty
-```
-
-### When to Use
-
-**Simplified method good for:**
-```
-1. Standard cases
- - Typical geometries (vertical spark, toroid topload)
- - Typical frequencies (100-300 kHz)
- - Typical lengths (1-3 m)
-
-2. First-pass analysis
- - Initial design evaluation
- - Quick parameter studies
-
-3. Engineering estimates
- - ±20% accuracy sufficient
- - Fast turnaround needed
-
-4. Educational purposes
- - Understanding physics
- - Building intuition
-```
-
-**Iterative method when:**
-```
-1. Research/validation
- - Publication-quality results
- - Detailed physics studies
-
-2. Extreme parameters
- - Very long sparks (>5 m)
- - Very short sparks (<0.5 m)
- - Very low frequency (<50 kHz)
-
-3. Measurement comparison
- - Highest accuracy required
- - Factor of 1.5 differences matter
-
-4. Unusual geometries
- - Horizontal sparks
- - Branched discharge
- - Non-uniform diameter
-```
-
-**Computational savings:**
-```
-Iterative:
- 5-10 iterations × 20 points × n segments
- = 1000-2000 AC analyses
- Time: 100-500 seconds for n=10
-
-Simplified:
- 1 AC analysis (after R calculation)
- Time: <1 second
-
-Speedup: 1000-5000× faster!
-
-Use simplified unless specific need for iterative
-```
-
-### Worked Example: Simplified Calculation
-
-**Given (same matrix as before):**
-```
-f = 190 kHz
-ω = 2π × 190×10³ = 1.194×10⁶ rad/s
-
-Capacitance matrix (n=5):
- [0] [1] [2] [3] [4] [5]
-[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
-[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
-[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
-[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
-[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
-[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ] pF
-```
-
-**Calculate R[i] for each segment:**
-
-**Segment 1 (base):**
-```
-C_total[1] = |C[1,0]| + |C[1,2]| + |C[1,3]| + |C[1,4]| + |C[1,5]|
- = 9.2 + 2.8 + 0.9 + 0.4 + 0.2
- = 13.5 pF
-
-R[1] = 1 / (ω × C_total[1])
- = 1 / (1.194×10⁶ × 13.5×10⁻¹²)
- = 1 / (1.612×10⁻⁵)
- = 62.0 kΩ
-
-Check bounds: 1k < 62k < 100k ✓
-```
-
-**Segment 2:**
-```
-C_total[2] = 3.1 + 2.8 + 2.1 + 0.7 + 0.3 = 9.0 pF
-
-R[2] = 1 / (1.194×10⁶ × 9.0×10⁻¹²)
- = 93.0 kΩ ✓
-```
-
-**Segment 3:**
-```
-C_total[3] = 1.2 + 0.9 + 2.1 + 1.8 + 0.5 = 6.5 pF
-
-R[3] = 1 / (1.194×10⁶ × 6.5×10⁻¹²)
- = 129 kΩ ✓
-```
-
-**Segment 4:**
-```
-C_total[4] = 0.6 + 0.4 + 0.7 + 1.8 + 1.4 = 4.9 pF
-
-R[4] = 1 / (1.194×10⁶ × 4.9×10⁻¹²)
- = 171 kΩ ✓
-```
-
-**Segment 5 (tip):**
-```
-C_total[5] = 0.3 + 0.2 + 0.3 + 0.5 + 1.4 = 2.7 pF
-
-R[5] = 1 / (1.194×10⁶ × 2.7×10⁻¹²)
- = 310 kΩ ✓
-```
-
-**Summary:**
-```
-R[1] = 62 kΩ (base, lowest)
-R[2] = 93 kΩ
-R[3] = 129 kΩ
-R[4] = 171 kΩ
-R[5] = 310 kΩ (tip, highest)
-
-✓ Monotonically increasing
-✓ All within position-dependent bounds
-
-Total: R_total = 765 kΩ
-```
-
-### Validation
-
-**Total resistance check:**
-```
-Expected at 190 kHz for 2 m spark:
- Lower bound: ~50 kΩ (very hot, efficient)
- Typical: 100-300 kΩ
- Upper bound: ~500 kΩ (cool, streamer-dominated)
-
-Result: 765 kΩ
-
-Higher than typical, but reasonable because:
- 1. Long spark (2 m)
- 2. Distributed model (tip high R, 310 kΩ)
- 3. Tip weakly coupled (high R expected)
-
-Within factor of 2-3 of typical: Acceptable
-```
-
-**If result very different:**
-```
-R_total < 20 kΩ:
- - Check formula (missing units conversion?)
- - Check C values (pF vs F?)
- - Too low for plasma physics
-
-R_total > 2 MΩ:
- - Check frequency (Hz vs kHz?)
- - Tip resistance very high (check tip coupling)
- - May need iterative method to find lower solution
-```
-
-## Total Resistance Validation Ranges
-
-**Frequency dependence:**
-```
-At 100 kHz:
- Typical: 100-600 kΩ (higher R at lower f)
-
-At 200 kHz:
- Typical: 50-300 kΩ
-
-At 400 kHz:
- Typical: 25-150 kΩ (lower R at higher f)
-
-Rule: R_total ∝ 1/f (approximately)
-```
-
-**Length dependence:**
-```
-At 200 kHz:
- 0.5 m: 30-100 kΩ
- 1.0 m: 50-150 kΩ
- 2.0 m: 100-300 kΩ
- 3.0 m: 150-500 kΩ
-
-Rule: R_total ∝ L (approximately, distributed effects complicate)
-```
-
-**Operating mode:**
-```
-QCW (long ramp):
- - Lower R (hot channel)
- - Factor 0.5-1× above estimates
-
-Burst (short pulse):
- - Higher R (cooler channel)
- - Factor 1-2× above estimates
-```
-
-## Key Takeaways
-
-- **Iterative optimization** maximizes power per segment, uses damping (α ≈ 0.3-0.5) for stability, 5-10 iterations typical
-- **Position-dependent bounds:** R_min increases 1k→10k, R_max increases 100k→100M from base to tip (quadratic)
-- **Convergence:** Base segments converge fast (sharp power peak), tip segments slow (flat curve, weakly coupled)
-- **Simplified method:** R[i] = 1/(ω × C_total[i]) from circuit theory, 1000× faster, ±20% accuracy
-- **When simplified:** Standard cases, first-pass analysis, engineering estimates, educational use
-- **When iterative:** Research, extreme parameters, measurement comparison, publication quality
-- **Validation:** R_total should be 50-500 kΩ at 200 kHz for 1-3 m sparks, monotonic increase base→tip
-- **Total resistance:** Scales as R ∝ 1/f and R ∝ L approximately, QCW lower than burst mode
-
-## Practice
-
-{exercise:model-ex-05}
-
----
-**Next Lesson:** [Part 4 Review and Comprehensive Exercises](06-review-exercises.md)
diff --git a/spark-lessons/lessons/04-advanced-modeling/06-review-exercises.md b/spark-lessons/lessons/04-advanced-modeling/06-review-exercises.md
deleted file mode 100644
index 9dfa366..0000000
--- a/spark-lessons/lessons/04-advanced-modeling/06-review-exercises.md
+++ /dev/null
@@ -1,699 +0,0 @@
----
-id: model-06
-title: "Part 4 Review and Comprehensive Modeling Project"
-section: "Advanced Modeling"
-difficulty: "advanced"
-estimated_time: 90
-prerequisites: ["model-01", "model-02", "model-03", "model-04", "model-05"]
-objectives:
- - Synthesize all advanced modeling concepts from Part 4
- - Apply complete workflow from FEMM to validated spark model
- - Compare lumped vs distributed approaches systematically
- - Execute comprehensive modeling project integrating all skills
-tags: ["review", "integration", "project", "validation", "comprehensive"]
----
-
-# Part 4 Review and Comprehensive Modeling Project
-
-This lesson reviews all advanced modeling concepts from Part 4 and guides you through a comprehensive project that integrates FEMM extraction, circuit implementation, resistance optimization, and validation.
-
-## Part 4 Concepts Summary
-
-### Lesson 1: Lumped Model Theory
-
-**Key concepts:**
-```
-Structure: C_mut - R - C_sh network
- - C_mut: Topload to spark coupling
- - R: Effective plasma resistance
- - C_sh: Spark to ground shunt
-
-When to use:
- ✓ Sparks <1-2 m
- ✓ Impedance matching studies
- ✓ Quick design iterations
- ✓ Engineering estimates
-
-Workflow:
- 1. FEMM electrostatic (2-body)
- 2. Extract C_mut, C_sh from 2×2 matrix
- 3. Calculate R = 1/(ω × C_total)
- 4. Build SPICE, simulate
- 5. Validate: φ_Z, R range, C_sh ≈ 2 pF/ft
-```
-
-### Lesson 2: FEMM Extraction - Lumped
-
-**Key concepts:**
-```
-Maxwell matrix convention:
- - Diagonal: C_ii > 0 (self-capacitance)
- - Off-diagonal: C_ij < 0 (mutual, negative!)
- - Symmetric: C_ij = C_ji
- - Row sum ≈ 0 (ground at infinity)
-
-Extraction formulas:
- C_mut = |C₁₂| (absolute value!)
- C_sh = C₂₂ - |C₁₂| (subtract absolute)
-
-Sign convention critical:
- - Maxwell: negative off-diagonals
- - Circuit: positive capacitances
- - Conversion: Take absolute value
-
-Validation:
- ✓ Symmetry <1% error
- ✓ C_sh ≈ 2 pF/ft ± factor 2
- ✓ Physical value ranges
- ✓ Ground distance sensitivity test
-```
-
-### Lesson 3: Distributed Model Theory
-
-**Key concepts:**
-```
-Why distributed:
- - Long sparks (>2 m)
- - Current distribution matters
- - Leader/streamer transitions
- - Research applications
-
-Segmentation:
- - Equal-length segments
- - n = 5-20 typical
- - Convergence test: double n
-
-Circuit topology:
- - (n+1)×(n+1) capacitance matrix
- - n resistance values
- - O(n²) complexity
-
-Physical expectations:
- - R monotonically increasing
- - Current decreasing base→tip
- - Voltage non-linear drop
- - Power concentrated at base
-
-Trade-off: 1000-2000× slower than lumped
-```
-
-### Lesson 4: FEMM Extraction - Distributed
-
-**Key concepts:**
-```
-Multi-body setup:
- - n conductors + topload
- - 0.1 mm gaps between segments
- - Consistent numbering critical
-
-Matrix validation:
- ✓ Symmetry
- ✓ Positive semi-definite (passivity)
- ✓ Adjacent > distant coupling
- ✓ Total C_sh vs 2 pF/ft rule
-
-SPICE implementation:
- 1. Partial capacitance (flip signs)
- 2. Controlled sources (direct)
- 3. Nearest-neighbor (approximation)
-
-C_sh discrepancy:
- - Factor 2-3 normal for distributed
- - Matrix method vs empirical rule
- - Use FEMM values (more accurate)
-```
-
-### Lesson 5: Resistance Optimization
-
-**Key concepts:**
-```
-Iterative method:
- - Initialize: tapered profile
- - Optimize each R[i] sequentially
- - Apply damping (α ≈ 0.3-0.5)
- - Position-dependent bounds
- - Convergence: <1% change
-
-Position-dependent bounds:
- R_min: 1 kΩ → 10 kΩ (base to tip)
- R_max: 100 kΩ → 100 MΩ (quadratic)
-
-Simplified method:
- R[i] = 1/(ω × C_total[i])
- - 1000× faster
- - ±20% accuracy
- - Use for standard cases
-
-Validation:
- ✓ R_total: 50-500 kΩ at 200 kHz
- ✓ Monotonic increase
- ✓ Scales as R ∝ 1/f, R ∝ L
-```
-
-## Complete Modeling Workflow Checklist
-
-### Phase 1: Problem Definition
-
-```
-[ ] Define spark length L_total
-[ ] Specify operating frequency f
-[ ] Choose model type:
- [ ] Lumped (if L < 2 m)
- [ ] Distributed n=___ (if L ≥ 2 m)
-[ ] Gather topload geometry data
-[ ] Determine ground plane position
-```
-
-### Phase 2: FEMM Geometry and Solve
-
-```
-[ ] Create FEMM geometry:
- [ ] Axisymmetric (r-z)
- [ ] Topload (toroid/sphere)
- [ ] Spark segment(s)
- [ ] Ground plane
- [ ] Outer boundary
-[ ] Define materials (Air, ε_r=1)
-[ ] Assign conductors:
- [ ] Conductor 0: Topload, V=1V
- [ ] Conductors 1-n: Segments, floating
- [ ] Boundary: Ground, V=0
-[ ] Generate mesh (check quality)
-[ ] Solve electrostatic problem
-[ ] Extract capacitance matrix [C]
-```
-
-### Phase 3: Matrix Validation
-
-```
-[ ] Check symmetry: |C[i,j] - C[j,i]| / |C[i,j]| < 0.01
-[ ] Check diagonal positive: C[i,i] > 0 for all i
-[ ] Check off-diagonal negative: C[i,j] < 0 for i≠j
-[ ] Check passivity: Eigenvalues ≥ 0
-[ ] Check physical patterns:
- [ ] Adjacent > distant coupling
- [ ] Topload coupling decreases with distance
-[ ] Check total C_sh vs 2 pF/ft rule (factor 2-3 OK)
-```
-
-### Phase 4: Resistance Determination
-
-```
-[ ] Choose method:
- [ ] Iterative (research, extreme cases)
- [ ] Simplified (standard cases, engineering)
-
-If Iterative:
-[ ] Initialize tapered profile
-[ ] Define position-dependent bounds
-[ ] Set damping factor α
-[ ] Run optimization loop
-[ ] Check convergence (<1% or <5% for tip)
-[ ] Validate R distribution (monotonic, ranges)
-
-If Simplified:
-[ ] Calculate C_total[i] for each segment
-[ ] Compute R[i] = 1/(ω × C_total[i])
-[ ] Apply bounds: R[i] = clip(R[i], R_min[i], R_max[i])
-[ ] Validate total R_total (50-500 kΩ at 200 kHz)
-```
-
-### Phase 5: SPICE Implementation
-
-```
-[ ] Convert C matrix to SPICE format:
- [ ] Partial capacitances (most common)
- [ ] Or controlled sources (advanced)
- [ ] Or nearest-neighbor (approximation)
-[ ] Add resistance elements R[i]
-[ ] Define voltage source (test or from coil)
-[ ] Set up AC analysis at operating frequency
-[ ] Verify netlist syntax
-```
-
-### Phase 6: Simulation and Analysis
-
-```
-[ ] Run SPICE AC analysis
-[ ] Extract results:
- [ ] Voltages V[i] at each node
- [ ] Currents I[i] through each segment
- [ ] Admittance Y_spark at topload
- [ ] Impedance Z_spark = 1/Y_spark
-[ ] Calculate power distribution:
- [ ] P[i] = 0.5 × |I[i]|² × R[i]
- [ ] P_total = Σ P[i]
-[ ] Plot distributions:
- [ ] V vs position
- [ ] I vs position
- [ ] P vs position
-```
-
-### Phase 7: Validation
-
-```
-[ ] Phase angle: -55° < φ_Z < -75°
-[ ] Total resistance: 50-500 kΩ at 200 kHz
-[ ] Current distribution: Decreasing base→tip
-[ ] Voltage distribution: Non-linear, physical
-[ ] Power balance: Concentrated at base
-[ ] Compare to lumped model (if applicable)
-[ ] Compare to measurements (if available)
-```
-
-### Phase 8: Documentation
-
-```
-[ ] Save FEMM geometry and results
-[ ] Save capacitance matrix
-[ ] Save resistance values
-[ ] Save SPICE netlist
-[ ] Save simulation results
-[ ] Document validation checks
-[ ] Record any issues/assumptions
-```
-
-## Lumped vs Distributed Comparison
-
-### When Results Should Agree
-
-**Equivalent impedance at topload:**
-```
-Lumped: Z_spark = R + 1/(jωC_total)
-Distributed: Z_spark (from network)
-
-Expected: Within 20-30% for well-designed models
-
-Example:
- Lumped: |Z| = 180 kΩ ∠-70°
- Distributed: |Z| = 195 kΩ ∠-68°
- Difference: 8% ✓ Good agreement
-```
-
-**Total resistance:**
-```
-Lumped: Single R value
-Distributed: R_total = Σ R[i]
-
-Should be similar order of magnitude
-Factor <2 difference: Excellent
-Factor 2-3: Acceptable
-Factor >5: Investigate
-```
-
-**Total capacitance:**
-```
-Lumped: C_total = C_mut + C_sh
-Distributed: More complex (matrix network)
-
-At topload, should see similar capacitive reactance
-```
-
-### When Results May Differ
-
-**Current distribution:**
-```
-Lumped: Assumes uniform (no spatial info)
-Distributed: Non-uniform, physically realistic
-
-Cannot compare directly - distributed provides extra detail
-```
-
-**Power distribution:**
-```
-Lumped: Single power value (total)
-Distributed: Spatial distribution P[i]
-
-Lumped gives total only
-Distributed shows WHERE power dissipated
-```
-
-**Tip behavior:**
-```
-Lumped: Averaged properties
-Distributed: Can show tip streaming (low current, high R)
-
-Distributed more realistic for long sparks
-```
-
-**Short spark (e.g., 0.8 m):**
-```
-Lumped and distributed should agree closely
-Spatial variations small
-Use lumped (simpler, faster)
-```
-
-**Long spark (e.g., 3 m):**
-```
-Distributed shows significant spatial variation
-Lumped may over-predict tip current/power
-Use distributed for accuracy
-```
-
-## Comprehensive Modeling Project
-
-### Project Goal
-
-**Design and model a complete spark system:**
-```
-Objective: Predict performance of 2.5 m spark at 200 kHz
-Approach: Use distributed model (n=10)
-Output: Current, voltage, power distributions + validation
-```
-
-### Project Specifications
-
-```
-Tesla coil system:
- - Operating frequency: f = 200 kHz
- - Topload: Toroid, 40 cm major dia, 12 cm minor dia
- - Target spark length: 2.5 m = 8.2 feet
- - Ground plane: 20 cm below spark tip
- - Topload voltage: 350 kV (estimate)
-
-Model requirements:
- - Distributed model: n = 10 segments
- - Each segment: 0.25 m length
- - FEMM extraction: Full 11×11 matrix
- - Resistance: Simplified method
- - Validation: All checks
-```
-
-### Step 1: FEMM Setup
-
-**Geometry parameters:**
-```
-Topload (toroid):
- - Major radius: 20 cm
- - Minor radius: 6 cm
- - Center at z = 0
- - Lowest point: z = -6 cm
-
-10 spark segments:
- - Each length: 25 cm
- - Diameter: 2 mm (uniform)
- - Positions:
- Segment 1 (base): z = -6.1 to -31.1 cm
- Segment 2: z = -31.2 to -56.2 cm
- ...
- Segment 10 (tip): z = -231.5 to -256.5 cm
-
-Ground plane:
- - z = -270 cm (20 cm below tip)
- - r = 0 to 400 cm
-
-Outer boundary:
- - r = 400 cm
- - z = -300 to +50 cm
- - V = 0 boundary condition
-```
-
-**Expected mesh:**
-```
-Elements: 40,000-70,000
-Refinement: 0.5 mm near spark, 50 mm at boundary
-Solve time: 30-60 seconds
-```
-
-### Step 2: Matrix Extraction (Example Results)
-
-**Hypothetical FEMM output (11×11 matrix):**
-
-```
- [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
-[0] [ 38.2 -10.5 -4.2 -2.1 -1.2 -0.8 -0.5 -0.4 -0.3 -0.2 -0.1 ]
-[1] [ -10.5 16.2 -3.5 -1.4 -0.7 -0.4 -0.3 -0.2 -0.2 -0.1 -0.1 ]
-[2] [ -4.2 -3.5 12.8 -3.2 -1.3 -0.6 -0.4 -0.3 -0.2 -0.1 -0.1 ]
-[3] [ -2.1 -1.4 -3.2 11.4 -2.9 -1.2 -0.5 -0.3 -0.2 -0.1 -0.1 ]
-[4] [ -1.2 -0.7 -1.3 -2.9 10.6 -2.7 -1.1 -0.5 -0.3 -0.2 -0.1 ]
-[5] [ -0.8 -0.4 -0.6 -1.2 -2.7 9.8 -2.5 -1.0 -0.4 -0.2 -0.1 ]
-[6] [ -0.5 -0.3 -0.4 -0.5 -1.1 -2.5 9.2 -2.3 -0.9 -0.4 -0.1 ]
-[7] [ -0.4 -0.2 -0.3 -0.3 -0.5 -1.0 -2.3 8.6 -2.1 -0.8 -0.2 ]
-[8] [ -0.3 -0.2 -0.2 -0.2 -0.3 -0.4 -0.9 -2.1 8.2 -1.9 -0.5 ]
-[9] [ -0.2 -0.1 -0.1 -0.1 -0.2 -0.2 -0.4 -0.8 -1.9 7.6 -1.6 ]
-[10] [ -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.2 -0.5 -1.6 6.8 ] pF
-```
-
-*Note: These are illustrative values for the exercise.*
-
-### Step 3: Matrix Validation
-
-**Check symmetry:**
-```
-Example: C[2,5] = -0.6 pF, C[5,2] = -0.6 pF
-Error: |(-0.6) - (-0.6)| / 0.6 = 0%
-✓ Symmetric (check all pairs)
-```
-
-**Check patterns:**
-```
-Topload coupling: |C[0,1]| = 10.5 > |C[0,5]| = 0.8 > |C[0,10]| = 0.1 ✓
-Adjacent coupling: |C[3,4]| = 2.9 > |C[3,7]| = 0.3 ✓
-Diagonal positive: All C[i,i] > 0 ✓
-Off-diagonal negative: All C[i,j] < 0 for i≠j ✓
-```
-
-**Total shunt capacitance:**
-```
-C_sh_total = Σᵢ₌₁¹⁰ (C[i,i] - |C[i,0]|)
- = (16.2-10.5) + (12.8-4.2) + ... + (6.8-0.1)
- = 5.7 + 8.6 + 9.3 + 9.4 + 9.8 + 9.7 + 9.6 + 9.4 + 9.3 + 6.7
- = 87.5 pF
-
-Expected: 2 pF/ft × 8.2 ft = 16.4 pF
-
-Ratio: 87.5 / 16.4 = 5.3
-
-Higher than lumped expectation, but within factor 2-6 for distributed
-Matrix method includes all couplings - acceptable ✓
-```
-
-### Step 4: Calculate Resistances (Simplified Method)
-
-**Frequency:**
-```
-f = 200 kHz
-ω = 2π × 200×10³ = 1.257×10⁶ rad/s
-```
-
-**Segment 1 (base):**
-```
-C_total[1] = |C[1,0]| + |C[1,2]| + ... + |C[1,10]|
- = 10.5 + 3.5 + 1.4 + 0.7 + 0.4 + 0.3 + 0.2 + 0.2 + 0.1 + 0.1
- = 17.4 pF
-
-R[1] = 1 / (ω × C_total[1])
- = 1 / (1.257×10⁶ × 17.4×10⁻¹²)
- = 45.7 kΩ
-
-Bounds: R_min[1] = 1 kΩ, R_max[1] = 100 kΩ
-Check: 1 < 45.7 < 100 ✓
-```
-
-**Calculate similarly for all segments:**
-
-```
-Results (example):
-R[1] = 45.7 kΩ (position 0.00)
-R[2] = 58.3 kΩ (position 0.11)
-R[3] = 71.2 kΩ (position 0.22)
-R[4] = 86.5 kΩ (position 0.33)
-R[5] = 105 kΩ (position 0.44)
-R[6] = 128 kΩ (position 0.56)
-R[7] = 157 kΩ (position 0.67)
-R[8] = 195 kΩ (position 0.78)
-R[9] = 248 kΩ (position 0.89)
-R[10] = 320 kΩ (position 1.00)
-
-Total: R_total = 1415 kΩ = 1.42 MΩ
-```
-
-**Validation:**
-```
-✓ Monotonically increasing
-✓ Each within position-dependent bounds
-✓ Total: Expected 50-500 kΩ, got 1.42 MΩ
-
-Higher than typical - long spark (2.5 m), tip-dominated
-Within factor 3-5 of estimates - acceptable for distributed model
-```
-
-### Step 5: Build SPICE Netlist
-
-**Partial capacitance conversion (selected):**
-```spice
-* 10-segment distributed spark model - 2.5 m at 200 kHz
-.param freq=200k
-
-* Test voltage source
-V_test topload 0 AC 1V
-
-* Partial capacitances - between nodes (sample)
-C_0_1 topload seg1 10.5p
-C_0_2 topload seg2 4.2p
-C_1_2 seg1 seg2 3.5p
-C_2_3 seg2 seg3 3.2p
-C_3_4 seg3 seg4 2.9p
-* ... (continue for all pairs) ...
-
-* Partial capacitances - to ground (sample)
-C_0_gnd topload 0 {38.2 - (10.5+4.2+2.1+1.2+0.8+0.5+0.4+0.3+0.2+0.1)}
-C_1_gnd seg1 0 {16.2 - (10.5+3.5+1.4+0.7+0.4+0.3+0.2+0.2+0.1+0.1)}
-* ... (continue for all nodes) ...
-
-* Resistances
-R1 seg1 seg1_r 45.7k
-R2 seg2 seg2_r 58.3k
-R3 seg3 seg3_r 71.2k
-R4 seg4 seg4_r 86.5k
-R5 seg5 seg5_r 105k
-R6 seg6 seg6_r 128k
-R7 seg7 seg7_r 157k
-R8 seg8 seg8_r 195k
-R9 seg9 seg9_r 248k
-R10 seg10 seg10_r 320k
-
-* AC analysis
-.ac lin 1 200k 200k
-
-* Output
-.print ac v(topload) v(seg1) v(seg2) v(seg3) v(seg4) v(seg5)
-+ v(seg6) v(seg7) v(seg8) v(seg9) v(seg10)
-.print ac i(V_test) i(R1) i(R2) i(R3) i(R4) i(R5)
-+ i(R6) i(R7) i(R8) i(R9) i(R10)
-
-.end
-```
-
-### Step 6: Simulation Results (Example)
-
-**Voltage distribution (normalized, V_topload = 1V test):**
-```
-V[topload] = 1.000 V
-V[seg1] = 0.842 V (16% drop from topload)
-V[seg2] = 0.714 V
-V[seg3] = 0.608 V
-V[seg4] = 0.518 V
-V[seg5] = 0.441 V (56% of topload)
-V[seg6] = 0.375 V
-V[seg7] = 0.318 V
-V[seg8] = 0.269 V
-V[seg9] = 0.227 V
-V[seg10] = 0.192 V (tip, 19% of topload)
-
-Non-linear drop ✓ Expected for distributed capacitance
-```
-
-**Current distribution:**
-```
-I[seg1] = 18.4 μA (base, highest)
-I[seg2] = 12.2 μA (66% of base)
-I[seg3] = 8.54 μA
-I[seg4] = 6.00 μA
-I[seg5] = 4.20 μA (23% of base)
-I[seg6] = 2.93 μA
-I[seg7] = 2.03 μA
-I[seg8] = 1.38 μA
-I[seg9] = 0.91 μA
-I[seg10] = 0.60 μA (tip, 3% of base)
-
-Monotonically decreasing ✓ Capacitive shunting effect
-```
-
-**Power distribution:**
-```
-P[1] = 0.5 × (18.4×10⁻⁶)² × 45.7×10³ = 7.74 μW
-P[2] = 0.5 × (12.2×10⁻⁶)² × 58.3×10³ = 4.34 μW
-P[3] = 0.5 × (8.54×10⁻⁶)² × 71.2×10³ = 2.60 μW
-...
-P[10] = 0.5 × (0.60×10⁻⁶)² × 320×10³ = 0.058 μW
-
-Total: P_total ≈ 21.5 μW (at 1V test)
-
-Base segments (1-3): 14.7 μW (68% of total)
-Middle (4-7): 5.8 μW (27%)
-Tip (8-10): 1.0 μW (5%)
-
-Power concentrated at base ✓ Physical expectation
-```
-
-**Impedance at topload:**
-```
-Y = I_test / V_test = 18.4 μA / 1V = 18.4 μS
-|Z| = 1/18.4×10⁻⁶ = 54.3 kΩ
-φ_Z ≈ -62° (calculated from Re{Y}, Im{Y})
-
-Check: -55° < -62° < -75° ✓ Expected range
-```
-
-### Step 7: Scale to Actual Voltage
-
-**Given: V_topload = 350 kV actual**
-
-**Power scaling:**
-```
-P_actual = P_test × (V_actual / V_test)²
- = 21.5 μW × (350×10³ / 1)²
- = 21.5×10⁻⁶ × 1.225×10¹¹
- = 2.63 MW
-
-Total power to spark: 2.63 MW
-```
-
-**Segment powers:**
-```
-P[1] = 7.74 μW × scale = 949 kW (36%)
-P[2] = 4.34 μW × scale = 532 kW (20%)
-P[3] = 2.60 μW × scale = 319 kW (12%)
-...
-
-Base heavily loaded, tip lightly loaded ✓
-```
-
-### Step 8: Final Validation
-
-```
-✓ Phase angle: φ_Z = -62° in range (-55° to -75°)
-✓ Total resistance: 1.42 MΩ (high end, but acceptable for 2.5 m)
-✓ Voltage distribution: Non-linear, physically reasonable
-✓ Current distribution: Decreasing base→tip monotonically
-✓ Power distribution: 68% in base 1/3, physical
-✓ Matrix validation: All checks passed
-✓ Resistance monotonic: Increasing base→tip
-
-Model complete and validated!
-```
-
-## Key Takeaways from Part 4
-
-- **Lumped models:** Fast (<1s), accurate for short sparks (<2 m), C_mut-R-C_sh structure
-- **FEMM extraction:** Maxwell matrix has negative off-diagonals, C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|
-- **Distributed models:** Necessary for long sparks (>2 m), captures spatial variations, 1000× slower
-- **Segmentation:** Equal lengths, n = 5-20, convergence test by doubling n
-- **Matrix validation:** Symmetry, passivity (eigenvalues ≥ 0), physical patterns critical
-- **SPICE implementation:** Partial capacitance method (flip signs), controlled sources, or nearest-neighbor
-- **Resistance optimization:** Iterative (rigorous, slow) or simplified R = 1/(ωC) (fast, ±20%)
-- **Position-dependent bounds:** R_min 1k→10k, R_max 100k→100M, prevents unphysical solutions
-- **Validation ranges:** R_total 50-500 kΩ at 200 kHz typical, factor 2-3 variation acceptable
-- **C_sh discrepancy:** Factor 2-3 from 2 pF/ft rule normal for distributed (use FEMM values)
-- **Current distribution:** Decreases base→tip due to capacitive shunting (can be 20:1 ratio)
-- **Power concentration:** 60-70% in base 1/3 of spark, tip contributes <10%
-
-## Practice
-
-{exercise:model-ex-06}
-
----
-**Congratulations!** You have completed Part 4: Advanced Modeling. You now have the skills to:
-- Build lumped spark models for quick analysis
-- Extract capacitance matrices from FEMM for single and multi-body problems
-- Construct distributed models for long sparks and research applications
-- Optimize resistance distributions using iterative or simplified methods
-- Validate models against physical expectations and measurements
-- Apply complete modeling workflow from geometry to validated predictions
-
-**Next Steps:**
-- Part 5: Integration and Calibration (coming soon)
-- Apply these techniques to your own Tesla coil designs
-- Validate against measurements and refine models
-- Contribute to the community knowledge base
diff --git a/spark-lessons/reference/equation-sheet.md b/spark-lessons/reference/equation-sheet.md
deleted file mode 100644
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--- a/spark-lessons/reference/equation-sheet.md
+++ /dev/null
@@ -1,414 +0,0 @@
-# Tesla Coil Spark Physics - Equation Sheet
-
-Quick reference for all key equations in spark modeling and circuit analysis.
-
-**Convention:** All phasor quantities use **peak values** (not RMS). Power formulas include the 0.5 factor: P = 0.5 × Re{V × I*}.
-
----
-
-## Circuit Analysis
-
-### Admittance Components
-
-**Input admittance at topload (looking into spark):**
-
-```
-Y = ((G + jB₁) · jB₂) / (G + j(B₁ + B₂))
-```
-
-Where:
-- G = 1/R (conductance)
-- B₁ = ωC_mut (mutual capacitance susceptance, positive)
-- B₂ = ωC_sh (shunt capacitance susceptance, positive)
-
-**Real part of admittance:**
-
-```
-Re{Y} = GB₂² / (G² + (B₁ + B₂)²)
-```
-
-**Imaginary part of admittance:**
-
-```
-Im{Y} = B₂[G² + B₁(B₁ + B₂)] / (G² + (B₁ + B₂)²)
-```
-
-### Phase Angles
-
-**Admittance phase angle:**
-
-```
-θ_Y = atan(Im{Y}/Re{Y})
-```
-
-**Impedance phase angle (what we typically measure):**
-
-```
-φ_Z = -θ_Y = atan(-Im{Y}/Re{Y})
-```
-
-**Minimum achievable impedance phase angle:**
-
-```
-φ_Z,min = -atan(2√(r(1 + r)))
-```
-
-Where:
-- r = C_mut/C_sh (capacitance ratio)
-
-*Note:* When r ≥ 0.207, achieving φ_Z = -45° becomes mathematically impossible regardless of R value.
-
----
-
-## Optimization
-
-### Critical Resistance Values
-
-**R_opt_power - Maximum power transfer:**
-
-```
-R_opt_power = 1 / (ω(C_mut + C_sh))
-```
-
-*Example:* At f = 200 kHz with C_mut + C_sh = 12 pF:
-```
-R_opt_power = 1/(2π × 200×10³ × 12×10⁻¹²) ≈ 66 kΩ
-```
-
-**R_opt_phase - Closest to resistive:**
-
-```
-R_opt_phase = 1 / (ω√(C_mut(C_mut + C_sh)))
-```
-
-*Note:* R_opt_power < R_opt_phase always
-
-### Segment-Level Optimization (nth-order model)
-
-**Simplified circuit-determined resistance:**
-
-```
-For each segment i:
- C_total[i] = C_shunt[i] + sum(C_mutual[i,:])
- R[i] = 1/(ω × C_total[i])
- R[i] = clip(R[i], R_min[i], R_max[i])
-```
-
-**Tapered initialization for iterative optimization:**
-
-```
-position = i/(n-1) # 0 at base, 1 at tip
-R[i] = R_base + (R_tip - R_base) × position²
-```
-
-Typical: R_base = 10 kΩ, R_tip = 1 MΩ
-
-**Damped iterative update:**
-
-```
-R_new[i] = α × R_optimal[i] + (1 - α) × R_old[i]
-```
-
-Where α ≈ 0.3-0.5 for stability
-
----
-
-## Thévenin Equivalent
-
-### Measurement Procedure
-
-**Output impedance (drive off, test source on):**
-
-```
-Z_th = 1V / I_test = R_th + jX_th
-```
-
-**Open-circuit voltage (drive on, no spark):**
-
-```
-V_th = V(topload) [complex magnitude and phase]
-```
-
-### Power Calculations
-
-**Power to any load:**
-
-```
-P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
-```
-
-**Theoretical maximum power (conjugate match):**
-
-```
-P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
-```
-
-*Note:* Actual spark power will be less due to topological constraints.
-
----
-
-## Spark Growth
-
-### Electric Field Thresholds
-
-**Field requirements (at sea level, standard conditions):**
-
-```
-E_inception ≈ 2-3 MV/m (initial breakdown from smooth topload)
-E_propagation ≈ 0.4-1.0 MV/m (sustained leader growth)
-E_tip = κ × E_average (tip enhancement factor κ ≈ 2-5)
-```
-
-*Note:* E_propagation varies with altitude and humidity by ±20-30%.
-
-### Growth Rate Equation
-
-**When field threshold is met:**
-
-```
-dL/dt = P_stream / ε (when E_tip > E_propagation)
-dL/dt ≈ 0 (when E_tip < E_propagation, stalled)
-```
-
-Where:
-- L = spark length [m]
-- P_stream = power delivered to spark [W]
-- ε = energy per meter [J/m]
-
-**Energy and power over time:**
-
-```
-E_total ≈ ε × L
-P_avg ≈ ε × L / T
-```
-
-### Energy per Meter (ε)
-
-**By operating mode:**
-
-```
-ε ≈ 5-15 J/m (QCW-style growth, leader-dominated)
-ε ≈ 20-40 J/m (High duty cycle DRSSTC, hybrid)
-ε ≈ 30-100+ J/m (Hard-pulsed burst mode, streamer-dominated)
-```
-
-**Advanced time-dependent model:**
-
-```
-ε(t) = ε₀ / (1 + α∫P_stream dt)
-```
-
-Where:
-- α has units [1/J]
-- ∫P_stream dt = accumulated energy
-
----
-
-## Thermal Physics
-
-### Thermal Time Constants
-
-**Pure thermal diffusion:**
-
-```
-τ_thermal = d² / (4α)
-```
-
-Where:
-- d = channel diameter [m]
-- α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air
-
-**Examples:**
-```
-d = 100 μm → τ ≈ 0.1-0.2 ms (thin streamers)
-d = 5 mm → τ ≈ 300-600 ms (thick leaders)
-```
-
-*Note:* Observed persistence is longer due to convection and ionization memory:
-- Thin streamers: ~1-5 ms (effective)
-- Thick leaders: seconds (effective)
-
----
-
-## Capacitive Divider
-
-### Voltage Division Effect
-
-**General formula:**
-
-```
-V_tip = V_topload × Z_mut/(Z_mut + Z_sh)
-```
-
-Where:
-- Z_mut = (1/jωC_mut) || R [complex]
-- Z_sh = 1/jωC_sh
-
-**Open-circuit limit (R → ∞):**
-
-```
-V_tip ≈ V_topload × C_mut/(C_mut + C_sh)
-```
-
-*Note:* Since C_sh ∝ L, as spark grows, V_tip decreases even if V_topload is maintained.
-
----
-
-## Ringdown Method
-
-### Quality Factor Relations
-
-**At loaded resonance ω_L:**
-
-```
-Q_L = ω_L C_eq R_p = R_p/(ω_L L)
-```
-
-### Equivalent Resistance
-
-**From Q and capacitance:**
-
-```
-R_p = Q_L/(ω_L C_eq)
-```
-
-**From Q and inductance:**
-
-```
-R_p = Q_L ω_L L
-```
-
-### Total Conductance
-
-**From Q and capacitance:**
-
-```
-G_total = ω_L C_eq/Q_L
-```
-
-**From Q and inductance:**
-
-```
-G_total = 1/(Q_L ω_L L)
-```
-
-### Capacitance Change
-
-**Equivalent capacitance after loading:**
-
-```
-C_eq = C₀(f₀/f_L)²
-ΔC = C_eq - C₀
-```
-
-### Spark Admittance Extraction
-
-**Step-by-step:**
-
-```
-1. Measure unloaded: f₀, Q₀, C₀
-2. Measure with spark: f_L, Q_L
-3. C_eq = C₀(f₀/f_L)²
-4. ΔC = C_eq - C₀
-5. G_total = ω_L C_eq/Q_L
-6. G_0 = ω₀ C₀/Q₀
-7. Y_spark ≈ (G_total - G_0) + jω_L ΔC
-```
-
----
-
-## FEMM Extraction
-
-### Maxwell Capacitance Matrix
-
-**For lumped model:**
-
-```
-C_mut = -C[topload, spark] = |C_12|
-C_sh = C[spark, spark] + C[spark, topload] = C_22 - |C_12|
-```
-
-*Note:* Maxwell matrix has C_ii > 0 (self-capacitance) and C_ij < 0 for i≠j (mutual capacitance, negative).
-
-**Validation check:**
-
-```
-C_sh ≈ 2 pF per foot (empirical rule)
-```
-
----
-
-## Empirical Scaling Laws
-
-### Freau's Relationships
-
-**Single-shot burst (no thermal accumulation):**
-
-```
-L ∝ √(E_bang)
-```
-
-**Repetitive operation (with thermal memory):**
-
-```
-L ∝ P_avg^(0.3 to 0.5)
-```
-
-**QCW with voltage ramping:**
-
-```
-L ∝ E^(0.6 to 0.8) (closer to linear)
-```
-
----
-
-## Self-Consistency Check
-
-### Diameter Back-Calculation
-
-**For validation:**
-
-```
-ρ_typical = 10 Ω·m (partially ionized plasma)
-L_segment = L_total/n_segments
-d_implied = sqrt(4 × ρ_typical × L_segment / (π × R_opt))
-```
-
-If d_implied ≈ d_nominal (within factor of 2), model is self-consistent.
-
----
-
-## Physical Bounds Formulas
-
-### Position-Dependent Resistance Bounds
-
-**For nth-order model:**
-
-```
-position = i/(n-1) # 0 at base, 1 at tip
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position
-```
-
----
-
-## Power Balance Validation
-
-**Total power equation:**
-
-```
-P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation
-```
-
-**Efficiency check:**
-
-```
-η = P_spark / P_primary_input
-```
-
-Expected η varies widely by design and operating mode.
-
----
-
-*Total equations: 45+ key formulas across all categories*
diff --git a/spark-lessons/reference/physical-bounds.md b/spark-lessons/reference/physical-bounds.md
deleted file mode 100644
index 2173944..0000000
--- a/spark-lessons/reference/physical-bounds.md
+++ /dev/null
@@ -1,499 +0,0 @@
-# Tesla Coil Spark Physics - Physical Bounds and Typical Ranges
-
-Reference for validation criteria, physical constraints, and empirical values.
-
----
-
-## Resistance Bounds
-
-### Lumped Model
-
-**Physical limits:**
-
-```
-R_min ≈ 1 kΩ (very hot, thick leader plasma)
-R_max ≈ 100 MΩ (cold, thin streamer plasma)
-
-R_actual = clip(R_opt_power, R_min, R_max)
-```
-
-### nth-Order Model (Position-Dependent)
-
-**Base segments (position = 0):**
-- R_min = 1 kΩ
-- R_max = 100 kΩ
-
-**Tip segments (position = 1):**
-- R_min = 10 kΩ
-- R_max = 100 MΩ
-
-**Interpolated formula:**
-
-```
-position = i/(n-1)
-
-R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
- = 1 kΩ + 9 kΩ × position
-
-R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position
- ≈ 100 kΩ + 99.9 MΩ × position
-```
-
-### Typical Total Resistance (by operating mode)
-
-**At 200 kHz for 1-3 meter sparks:**
-
-| Operating Mode | Total R | Characteristics |
-|---------------|---------|-----------------|
-| Burst/Streamer-dominated | 50-300 kΩ | Short pulses, thin channels |
-| QCW/Leader-dominated | 5-50 kΩ | Long ramps, hot thick channels |
-| Very low frequency (<100 kHz) or very long sparks | 1-10 kΩ | Thick leaders, high power |
-
-**Validation flag:** If R_total is significantly outside these ranges for your frequency and length, investigate potential issues.
-
----
-
-## Capacitance Values
-
-### Mutual Capacitance (C_mut)
-
-**Typical values:**
-
-| Spark Length | Typical C_mut | Notes |
-|-------------|---------------|-------|
-| 1 foot (0.3 m) | 3-5 pF | Small topload |
-| 2 feet (0.6 m) | 5-8 pF | Medium topload |
-| 3 feet (0.9 m) | 7-12 pF | Large topload |
-| 5 feet (1.5 m) | 10-15 pF | Very large topload |
-
-*Depends on topload size and geometry*
-
-### Shunt Capacitance (C_sh)
-
-**Empirical rule:**
-
-```
-C_sh ≈ 2 pF per foot of spark length
-```
-
-**Examples:**
-
-| Spark Length | Typical C_sh |
-|-------------|--------------|
-| 1 foot (0.3 m) | 2 pF |
-| 2 feet (0.6 m) | 4 pF |
-| 3 feet (0.9 m) | 6 pF |
-| 5 feet (1.5 m) | 10 pF |
-| 10 feet (3.0 m) | 20 pF |
-
-**Validation:** Use this rule to verify FEMM extraction accuracy.
-
-### Capacitance Ratio (r)
-
-```
-r = C_mut/C_sh
-```
-
-**Typical geometries:**
-
-| Configuration | r value | φ_Z,min |
-|--------------|---------|---------|
-| Large topload, short spark | 0.5 - 2.0 | -50° to -70° |
-| Medium topload, medium spark | 0.3 - 0.8 | -48° to -60° |
-| Small topload, long spark | 0.1 - 0.4 | -43° to -52° |
-
-**Critical threshold:** When r ≥ 0.207, achieving φ_Z = -45° becomes impossible.
-
-### Diameter Dependence
-
-**Weak logarithmic scaling:**
-
-```
-C ∝ 1/ln(h/d)
-```
-
-Where:
-- h = height above ground
-- d = channel diameter
-
-**Typical change:** 2× diameter → ~10-15% change in C
-
----
-
-## Electric Field Thresholds
-
-### Inception Field
-
-**Smooth electrode breakdown:**
-
-```
-E_inception ≈ 2-3 MV/m (sea level, standard conditions)
-```
-
-**Variations:**
-- Sharp electrodes: 1-2 MV/m (lower threshold)
-- Very smooth, large radius: 3-4 MV/m (higher threshold)
-
-### Propagation Field
-
-**Sustained leader growth:**
-
-```
-E_propagation ≈ 0.4-1.0 MV/m (typical range)
-```
-
-**Common values:**
-- Conservative estimate: 0.8-1.0 MV/m
-- Optimistic/ideal conditions: 0.4-0.6 MV/m
-- Typical use for modeling: 0.6-0.7 MV/m
-
-### Tip Enhancement Factor
-
-```
-E_tip = κ × E_average
-```
-
-**Typical values:**
-- κ ≈ 2-5 for cylindrical channels
-- Higher for sharper geometries
-- Use FEMM to calculate actual enhancement
-
-### Altitude and Environmental Effects
-
-**Altitude correction (rough approximation):**
-
-```
-E(altitude) = E(sea level) × (P/P_0)
-
-where P/P_0 ≈ exp(-h/8500 m)
-```
-
-**Examples:**
-
-| Altitude | Pressure Ratio | Field Scaling |
-|----------|---------------|---------------|
-| Sea level | 1.0 | 1.0 |
-| 1500 m (Denver) | ~0.83 | ~0.83 |
-| 3000 m | ~0.69 | ~0.69 |
-
-**Humidity effects:** ±10-20% variation (higher humidity → slightly lower threshold)
-
-**Temperature:** ±5-10% variation over normal range
-
-**Total variability:** E_propagation can vary ±20-30% with environmental conditions
-
----
-
-## Energy per Meter (ε)
-
-### By Operating Mode
-
-**QCW-style growth:**
-
-```
-ε ≈ 5-15 J/m
-
-Characteristics:
-- Long ramp times (5-20 ms)
-- Leader-dominated channels
-- Energy efficiently extends length
-- White/orange appearance
-```
-
-**High duty cycle DRSSTC:**
-
-```
-ε ≈ 20-40 J/m
-
-Characteristics:
-- Hybrid streamer/leader formation
-- Some thermal accumulation
-- Moderate efficiency
-- Mixed appearance
-```
-
-**Hard-pulsed DRSSTC (burst mode):**
-
-```
-ε ≈ 30-100+ J/m (single-shot)
-
-Characteristics:
-- Short pulses, mostly streamers
-- Much energy → brightening/branching
-- Poor length efficiency
-- Purple/blue, highly branched
-```
-
-### Calibration Requirements
-
-**Essential:** Calibrate ε for your specific coil from measurements.
-
-**Procedure:**
-1. Run coil with known drive power and time
-2. Measure final spark length L
-3. From SPICE, compute E_delivered = ∫P_spark dt
-4. Calculate: ε = E_delivered/L
-
-**Expected precision:** ±30-50% due to variability in plasma conditions
-
----
-
-## Thermal Time Constants
-
-### Pure Thermal Diffusion
-
-**Formula:**
-
-```
-τ_thermal = d² / (4α)
-
-where α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air
-```
-
-**By diameter:**
-
-| Diameter | Type | τ_thermal | Observed Persistence |
-|----------|------|-----------|---------------------|
-| 100 μm | Thin streamer | 0.1-0.2 ms | ~1-5 ms |
-| 1 mm | Thick streamer | 12-25 ms | ~10-50 ms |
-| 5 mm | Leader | 300-600 ms | seconds |
-| 1 cm | Thick leader | 1-2 seconds | 10+ seconds |
-
-**Note:** Observed persistence is longer than pure thermal diffusion due to:
-- Buoyancy and convection maintaining hot gas column
-- Ionization memory (recombination slower than thermal diffusion)
-- Broadened effective channel diameter
-
-### Operating Regime Implications
-
-**QCW advantage:**
-- Ramp times 5-20 ms match streamer-to-leader persistence
-- Channel stays hot throughout growth
-- Efficient energy coupling
-
-**Burst mode:**
-- Pulse spacing > 5 ms → channel cools between pulses
-- Must re-ionize repeatedly
-- Less efficient for length
-
----
-
-## Phase Angles
-
-### Impedance Phase (φ_Z)
-
-**Typical ranges:**
-
-```
-R = R_opt_power typically gives: φ_Z ≈ -55° to -75°
-```
-
-**By capacitance ratio:**
-
-| r = C_mut/C_sh | φ_Z,min | Typical at R_opt |
-|----------------|---------|------------------|
-| 0.1 | -42° | -55° to -60° |
-| 0.3 | -50° | -60° to -65° |
-| 0.5 | -55° | -62° to -68° |
-| 1.0 | -65° | -68° to -73° |
-| 2.0 | -73° | -72° to -76° |
-
-**Important:** The commonly cited "-45°" is often unachievable due to circuit topology.
-
-### Admittance Phase (θ_Y)
-
-```
-θ_Y = -φ_Z
-```
-
-Typical ranges: +55° to +75° (positive, capacitive)
-
----
-
-## Frequency Ranges
-
-### Operating Frequencies
-
-**Typical Tesla coil operating frequencies:**
-
-| Coil Type | Frequency Range | Notes |
-|-----------|----------------|-------|
-| Small DRSSTC | 150-400 kHz | Higher frequency, smaller secondary |
-| Medium DRSSTC | 100-250 kHz | Most common range |
-| Large DRSSTC | 50-150 kHz | Lower frequency, larger secondary |
-| SSTC | 100-500 kHz | Wide range possible |
-| QCW | 50-200 kHz | Typically lower frequencies |
-
-### Loaded vs Unloaded
-
-**Frequency shift with spark:**
-- Typical shift: 5-20% lower when loaded
-- Larger sparks → larger shift
-- Track frequency to loaded pole for accurate measurements
-
----
-
-## Power Levels and Efficiencies
-
-### Typical Power Ranges
-
-**By coil size:**
-
-| Coil Class | Primary Power | Spark Power | Typical η |
-|-----------|---------------|-------------|-----------|
-| Small DRSSTC | 0.5-2 kW | 0.1-0.5 kW | 15-30% |
-| Medium DRSSTC | 2-5 kW | 0.5-1.5 kW | 20-35% |
-| Large DRSSTC | 5-15 kW | 1.5-5 kW | 25-40% |
-| QCW | 1-10 kW | 0.5-4 kW | 30-50% |
-
-**Efficiency components:**
-- Spark power delivery: 15-50%
-- Secondary losses (heating): 10-30%
-- Primary circuit losses: 20-40%
-- Corona and radiation: 5-15%
-
-### Power Density
-
-**Typical values in spark channel:**
-
-```
-P/L ≈ 50-500 W/m (power per unit length)
-```
-
-Higher for burst mode (bright but short), lower for QCW (efficient leaders).
-
----
-
-## Geometric Constraints
-
-### Minimum Capacitance Bounds
-
-**For stable operation:**
-
-```
-C_mut + C_sh ≥ 5 pF (typical minimum for 100+ kHz)
-```
-
-Below this, impedance becomes very high and matching becomes difficult.
-
-### Maximum Practical Length
-
-**Voltage-limited:**
-
-```
-L_max ≈ V_top_peak / E_propagation
-
-Typical: V_top = 300-600 kV → L_max = 3-6 feet at E_prop = 1 MV/m
-```
-
-**Power-limited:**
-
-```
-L_max ≈ P_available × T / ε
-
-where T is growth time available
-```
-
-**Practical limit:** Whichever is more restrictive.
-
----
-
-## Plasma Properties
-
-### Conductivity Range
-
-**Partially ionized air plasma:**
-
-```
-σ ≈ 0.01 - 10 S/m (wide range depending on temperature and ionization)
-```
-
-**Equivalent resistivity:**
-
-```
-ρ ≈ 0.1 - 100 Ω·m
-```
-
-**Typical for modeling:**
-- Hot leader: ρ ≈ 1-10 Ω·m
-- Warm streamer: ρ ≈ 10-100 Ω·m
-
-### Temperature Ranges
-
-**Streamer:**
-```
-T ≈ 1000-3000 K
-```
-
-**Leader:**
-```
-T ≈ 5000-20000 K
-```
-
-**Arc (strike):**
-```
-T > 10000 K
-```
-
----
-
-## Validation Criteria
-
-### Self-Consistency Checks
-
-**Capacitance:**
-- C_sh/L ≈ 2 pF/foot ± 30%
-
-**Total resistance:**
-- Within expected range for operating mode (see above)
-- R_base < R_tip in distributed model
-
-**Power balance:**
-- P_spark + losses = P_input (within 20%)
-
-**Phase angle:**
-- φ_Z,actual ≥ φ_Z,min (within numerical precision)
-
-**Diameter self-consistency:**
-- d_implied ≈ d_nominal (within factor of 2-3)
-
-### Warning Flags
-
-**Red flags indicating potential errors:**
-
-- C_sh/L < 1 pF/foot or > 4 pF/foot
-- R_total < 500 Ω or > 10 MΩ at typical frequencies
-- φ_Z > -30° or < -85°
-- Power efficiency > 70% (unrealistically high)
-- ε < 1 J/m or > 200 J/m
-- Growth rates > 100 m/s (unphysical for leaders)
-
----
-
-## Measurement Tolerances
-
-### Expected Precision
-
-**Capacitance extraction (FEMM):**
-- ±10% typical accuracy
-- ±5% with careful meshing
-
-**Resistance measurement:**
-- ±30-50% (plasma variability dominates)
-
-**Field measurements:**
-- E_propagation: ±20-30% (environmental variability)
-
-**Energy per meter:**
-- ±30-50% (high variability)
-
-**Overall model predictions:**
-- Length: ±20-40% typical
-- Power: ±30-50% typical
-- Phase: ±5-10° typical
-
-Use these tolerances when validating model against measurements.
-
----
-
-*This reference compiled from empirical data, community observations, and validated modeling across multiple Tesla coil systems.*
diff --git a/spark-lessons/requirements.txt b/spark-lessons/requirements.txt
deleted file mode 100644
index cf7301b..0000000
--- a/spark-lessons/requirements.txt
+++ /dev/null
@@ -1,20 +0,0 @@
-# Tesla Coil Spark Course - PyQt5 Application Dependencies
-
-# Core PyQt5
-PyQt5>=5.15.0
-PyQtWebEngine>=5.15.0
-
-# Markdown rendering
-markdown>=3.4.0
-pymdown-extensions>=10.3.0
-
-# Data formats
-PyYAML>=6.0.1
-
-# Plotting and images
-matplotlib>=3.8.0
-Pillow>=10.1.0
-numpy>=1.26.0
-
-# Optional but recommended
-python-dateutil>=2.8.0
diff --git a/spark-lessons/resources/database/schema.sql b/spark-lessons/resources/database/schema.sql
deleted file mode 100644
index 4fc14e6..0000000
--- a/spark-lessons/resources/database/schema.sql
+++ /dev/null
@@ -1,138 +0,0 @@
--- Tesla Coil Spark Course - Database Schema
--- SQLite database for progress tracking, exercises, and user data
-
--- User profiles and preferences
-CREATE TABLE IF NOT EXISTS users (
- user_id INTEGER PRIMARY KEY AUTOINCREMENT,
- username TEXT UNIQUE NOT NULL,
- email TEXT,
- created_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- last_login TIMESTAMP,
- current_learning_path TEXT DEFAULT 'intermediate',
- theme_preference TEXT DEFAULT 'light',
- font_size INTEGER DEFAULT 14,
- auto_save_enabled BOOLEAN DEFAULT 1,
- show_hints BOOLEAN DEFAULT 1
-);
-
--- Lesson progress tracking
-CREATE TABLE IF NOT EXISTS lesson_progress (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- lesson_id TEXT NOT NULL,
- status TEXT CHECK(status IN ('not_started', 'in_progress', 'completed')) DEFAULT 'not_started',
- first_opened TIMESTAMP,
- last_accessed TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- time_spent INTEGER DEFAULT 0, -- Total seconds spent
- scroll_position FLOAT DEFAULT 0.0, -- 0.0 to 1.0
- completion_percentage INTEGER DEFAULT 0, -- 0 to 100
- completed_at TIMESTAMP,
- notes TEXT,
- bookmarked BOOLEAN DEFAULT 0,
- FOREIGN KEY (user_id) REFERENCES users(user_id),
- UNIQUE(user_id, lesson_id)
-);
-
--- Exercise attempts (all attempts recorded)
-CREATE TABLE IF NOT EXISTS exercise_attempts (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- exercise_id TEXT NOT NULL,
- lesson_id TEXT,
- attempt_number INTEGER NOT NULL,
- user_answer TEXT,
- is_correct BOOLEAN NOT NULL,
- points_earned INTEGER DEFAULT 0,
- points_possible INTEGER NOT NULL,
- hints_used INTEGER DEFAULT 0,
- time_taken INTEGER, -- Seconds
- attempted_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- FOREIGN KEY (user_id) REFERENCES users(user_id)
-);
-
--- Exercise completion (best performance)
-CREATE TABLE IF NOT EXISTS exercise_completion (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- exercise_id TEXT NOT NULL,
- best_score INTEGER NOT NULL, -- Points earned
- max_possible INTEGER NOT NULL, -- Total points available
- total_attempts INTEGER DEFAULT 1,
- first_attempted TIMESTAMP,
- first_completed TIMESTAMP,
- last_attempted TIMESTAMP,
- average_time INTEGER, -- Average seconds per attempt
- FOREIGN KEY (user_id) REFERENCES users(user_id),
- UNIQUE(user_id, exercise_id)
-);
-
--- Study sessions for streak tracking
-CREATE TABLE IF NOT EXISTS study_sessions (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- session_date DATE NOT NULL, -- Just the date (YYYY-MM-DD)
- session_start TIMESTAMP NOT NULL,
- session_end TIMESTAMP,
- lessons_viewed INTEGER DEFAULT 0,
- lessons_completed INTEGER DEFAULT 0,
- exercises_attempted INTEGER DEFAULT 0,
- exercises_completed INTEGER DEFAULT 0,
- points_earned INTEGER DEFAULT 0,
- time_active INTEGER DEFAULT 0, -- Seconds
- FOREIGN KEY (user_id) REFERENCES users(user_id),
- UNIQUE(user_id, session_date)
-);
-
--- Achievements and badges
-CREATE TABLE IF NOT EXISTS achievements (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- achievement_id TEXT NOT NULL, -- e.g., 'quick_learner', 'streak_master'
- achievement_name TEXT NOT NULL,
- achievement_description TEXT,
- earned_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- details TEXT, -- JSON with achievement specifics
- FOREIGN KEY (user_id) REFERENCES users(user_id),
- UNIQUE(user_id, achievement_id)
-);
-
--- Bookmarks and notes
-CREATE TABLE IF NOT EXISTS bookmarks (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- resource_type TEXT CHECK(resource_type IN ('lesson', 'example', 'reference', 'exercise')) NOT NULL,
- resource_id TEXT NOT NULL,
- title TEXT,
- note TEXT,
- created_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- updated_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- FOREIGN KEY (user_id) REFERENCES users(user_id)
-);
-
--- Learning path progress
-CREATE TABLE IF NOT EXISTS learning_path_progress (
- id INTEGER PRIMARY KEY AUTOINCREMENT,
- user_id INTEGER NOT NULL,
- path_id TEXT NOT NULL, -- 'beginner', 'intermediate', etc.
- lessons_completed INTEGER DEFAULT 0,
- total_lessons INTEGER NOT NULL,
- last_lesson_id TEXT,
- started_at TIMESTAMP,
- updated_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
- FOREIGN KEY (user_id) REFERENCES users(user_id),
- UNIQUE(user_id, path_id)
-);
-
--- Create indexes for performance
-CREATE INDEX IF NOT EXISTS idx_lesson_progress_user ON lesson_progress(user_id);
-CREATE INDEX IF NOT EXISTS idx_lesson_progress_lesson ON lesson_progress(lesson_id);
-CREATE INDEX IF NOT EXISTS idx_lesson_progress_status ON lesson_progress(status);
-CREATE INDEX IF NOT EXISTS idx_exercise_attempts_user ON exercise_attempts(user_id);
-CREATE INDEX IF NOT EXISTS idx_exercise_attempts_exercise ON exercise_attempts(exercise_id);
-CREATE INDEX IF NOT EXISTS idx_study_sessions_user_date ON study_sessions(user_id, session_date);
-CREATE INDEX IF NOT EXISTS idx_achievements_user ON achievements(user_id);
-CREATE INDEX IF NOT EXISTS idx_bookmarks_user ON bookmarks(user_id);
-
--- Insert default user (for single-user desktop app)
-INSERT OR IGNORE INTO users (user_id, username, email)
-VALUES (1, 'Student', 'student@teslacourse.local');
diff --git a/spark-lessons/resources/symbols_definitions.json b/spark-lessons/resources/symbols_definitions.json
deleted file mode 100644
index 6914758..0000000
--- a/spark-lessons/resources/symbols_definitions.json
+++ /dev/null
@@ -1,335 +0,0 @@
-{
- "variables": {
- "ω": {
- "name": "omega",
- "definition": "Angular frequency",
- "formula": "ω = 2πf",
- "units": "rad/s",
- "category": "frequency"
- },
- "f": {
- "name": "f",
- "definition": "Frequency",
- "formula": "f = ω/(2π)",
- "units": "Hz (hertz)",
- "category": "frequency"
- },
- "π": {
- "name": "pi",
- "definition": "Mathematical constant pi",
- "formula": "π ≈ 3.14159",
- "units": "dimensionless",
- "category": "constant"
- },
- "j": {
- "name": "j",
- "definition": "Imaginary unit (engineers use j instead of i)",
- "formula": "j = √(-1), j² = -1",
- "units": "dimensionless",
- "category": "complex"
- },
- "φ": {
- "name": "phi",
- "definition": "Phase angle",
- "units": "degrees or radians",
- "category": "angle"
- },
- "φ_Z": {
- "name": "phi_Z",
- "definition": "Impedance phase angle",
- "formula": "φ_Z = atan(X/R)",
- "units": "degrees or radians",
- "category": "angle"
- },
- "φ_Y": {
- "name": "phi_Y",
- "definition": "Admittance phase angle",
- "formula": "φ_Y = -φ_Z",
- "units": "degrees or radians",
- "category": "angle"
- },
- "θ": {
- "name": "theta",
- "definition": "Angle or phase",
- "units": "degrees or radians",
- "category": "angle"
- },
- "κ": {
- "name": "kappa",
- "definition": "Tip enhancement factor for electric field",
- "formula": "E_tip = κ × E_average",
- "units": "dimensionless (typically 2-5)",
- "category": "field"
- },
- "ε": {
- "name": "epsilon",
- "definition": "Permittivity",
- "units": "F/m (farads per meter)",
- "category": "field"
- },
- "ε₀": {
- "name": "epsilon_0",
- "definition": "Permittivity of free space",
- "formula": "ε₀ = 8.854×10⁻¹² F/m",
- "units": "F/m",
- "category": "constant"
- },
- "R": {
- "name": "R",
- "definition": "Resistance - opposition to current that dissipates energy",
- "units": "Ω (ohms)",
- "category": "circuit"
- },
- "L": {
- "name": "L",
- "definition": "Inductance - energy storage in magnetic field",
- "units": "H (henries)",
- "category": "circuit"
- },
- "C": {
- "name": "C",
- "definition": "Capacitance - energy storage in electric field",
- "formula": "Q = CV",
- "units": "F (farads)",
- "category": "circuit"
- },
- "C_mut": {
- "name": "C_mut",
- "definition": "Mutual capacitance between topload and spark channel",
- "units": "F (typically pF - picofarads)",
- "category": "capacitance"
- },
- "C_sh": {
- "name": "C_sh",
- "definition": "Shunt capacitance from spark to ground",
- "formula": "≈ 2 pF per foot of spark",
- "units": "F (typically pF - picofarads)",
- "category": "capacitance"
- },
- "G": {
- "name": "G",
- "definition": "Conductance - inverse of resistance",
- "formula": "G = 1/R",
- "units": "S (siemens)",
- "category": "circuit"
- },
- "B": {
- "name": "B",
- "definition": "Susceptance - imaginary part of admittance",
- "units": "S (siemens)",
- "category": "circuit"
- },
- "B₁": {
- "name": "B_1",
- "definition": "Susceptance of C_mut",
- "formula": "B₁ = ωC_mut",
- "units": "S (siemens)",
- "category": "circuit"
- },
- "B₂": {
- "name": "B_2",
- "definition": "Susceptance of C_sh",
- "formula": "B₂ = ωC_sh",
- "units": "S (siemens)",
- "category": "circuit"
- },
- "Y": {
- "name": "Y",
- "definition": "Admittance - inverse of impedance",
- "formula": "Y = G + jB = 1/Z",
- "units": "S (siemens)",
- "category": "circuit"
- },
- "Z": {
- "name": "Z",
- "definition": "Impedance - total opposition to AC current",
- "formula": "Z = R + jX",
- "units": "Ω (ohms)",
- "category": "circuit"
- },
- "X": {
- "name": "X",
- "definition": "Reactance - imaginary part of impedance",
- "units": "Ω (ohms)",
- "category": "circuit"
- },
- "X_C": {
- "name": "X_C",
- "definition": "Capacitive reactance",
- "formula": "X_C = -1/(ωC)",
- "units": "Ω (ohms), negative for capacitors",
- "category": "circuit"
- },
- "X_L": {
- "name": "X_L",
- "definition": "Inductive reactance",
- "formula": "X_L = ωL",
- "units": "Ω (ohms), positive for inductors",
- "category": "circuit"
- },
- "V": {
- "name": "V",
- "definition": "Voltage (potential difference)",
- "units": "V (volts)",
- "category": "circuit"
- },
- "V_top": {
- "name": "V_top",
- "definition": "Voltage at topload terminal",
- "units": "V (volts), often kV for Tesla coils",
- "category": "voltage"
- },
- "V_th": {
- "name": "V_th",
- "definition": "Thévenin equivalent voltage",
- "units": "V (volts)",
- "category": "voltage"
- },
- "I": {
- "name": "I",
- "definition": "Current - flow of electric charge",
- "units": "A (amperes)",
- "category": "circuit"
- },
- "I_base": {
- "name": "I_base",
- "definition": "Current at base of secondary coil",
- "units": "A (amperes)",
- "category": "current"
- },
- "P": {
- "name": "P",
- "definition": "Real power (dissipated)",
- "formula": "P = 0.5 × Re{V × I*}",
- "units": "W (watts)",
- "category": "power"
- },
- "Q": {
- "name": "Q",
- "definition": "Reactive power or charge",
- "units": "VAR (volt-amperes reactive) or C (coulombs)",
- "category": "power"
- },
- "r": {
- "name": "r",
- "definition": "Capacitance ratio",
- "formula": "r = C_mut/C_sh",
- "units": "dimensionless",
- "category": "ratio"
- },
- "R_opt_phase": {
- "name": "R_opt_phase",
- "definition": "Resistance for minimum phase angle",
- "formula": "R_opt_phase = 1/[ω√(C_mut(C_mut + C_sh))]",
- "units": "Ω (ohms)",
- "category": "optimization"
- },
- "R_opt_power": {
- "name": "R_opt_power",
- "definition": "Resistance for maximum power transfer",
- "formula": "R_opt_power = 1/[ω(C_mut + C_sh)]",
- "units": "Ω (ohms)",
- "category": "optimization"
- },
- "Z_th": {
- "name": "Z_th",
- "definition": "Thévenin equivalent impedance of coil",
- "units": "Ω (ohms)",
- "category": "impedance"
- },
- "E": {
- "name": "E",
- "definition": "Electric field strength",
- "formula": "E = -dV/dx or E ≈ V/d",
- "units": "V/m (volts per meter) or MV/m",
- "category": "field"
- },
- "E_tip": {
- "name": "E_tip",
- "definition": "Electric field at spark tip",
- "formula": "E_tip = κ × E_average",
- "units": "V/m or MV/m",
- "category": "field"
- },
- "E_average": {
- "name": "E_average",
- "definition": "Average electric field along spark",
- "formula": "E_average = V/L",
- "units": "V/m or MV/m",
- "category": "field"
- },
- "E_inception": {
- "name": "E_inception",
- "definition": "Field required to initiate breakdown",
- "formula": "E_inception ≈ 2-3 MV/m at sea level",
- "units": "V/m or MV/m",
- "category": "field"
- },
- "E_propagation": {
- "name": "E_propagation",
- "definition": "Field required to sustain spark growth",
- "formula": "E_propagation ≈ 0.4-1.0 MV/m",
- "units": "V/m or MV/m",
- "category": "field"
- },
- "L": {
- "name": "L",
- "definition": "Length (of spark or conductor)",
- "units": "m (meters)",
- "category": "geometry"
- },
- "d": {
- "name": "d",
- "definition": "Distance or diameter",
- "units": "m (meters)",
- "category": "geometry"
- },
- "A": {
- "name": "A",
- "definition": "Area",
- "units": "m² (square meters)",
- "category": "geometry"
- },
- "h": {
- "name": "h",
- "definition": "Height above ground",
- "units": "m (meters)",
- "category": "geometry"
- },
- "Re": {
- "name": "Re",
- "definition": "Real part of complex number",
- "formula": "Re{a + jb} = a",
- "units": "depends on quantity",
- "category": "complex"
- },
- "Im": {
- "name": "Im",
- "definition": "Imaginary part of complex number",
- "formula": "Im{a + jb} = b",
- "units": "depends on quantity",
- "category": "complex"
- },
- "|Z|": {
- "name": "|Z|",
- "definition": "Magnitude of impedance",
- "formula": "|Z| = √(R² + X²)",
- "units": "Ω (ohms)",
- "category": "complex"
- },
- "|Y|": {
- "name": "|Y|",
- "definition": "Magnitude of admittance",
- "formula": "|Y| = 1/|Z|",
- "units": "S (siemens)",
- "category": "complex"
- },
- "∠": {
- "name": "angle",
- "definition": "Phase angle notation",
- "formula": "Z = |Z| ∠ φ",
- "units": "degrees or radians",
- "category": "complex"
- }
- }
-}
diff --git a/spark-physics.txt b/spark-physics.txt
index f49c8ad..194b9b5 100644
--- a/spark-physics.txt
+++ b/spark-physics.txt
@@ -115,6 +115,8 @@ R_opt_power typically gives phase angles of -55° to -75°
- Plasma conductivity adjusts toward new R_opt_power
- **Stable equilibrium achieved when R_actual ≈ R_opt_power**
+**Causality insight (Richie Burnett):** "It is not early quenching that produces good sparks, but rather good spark loading that leads to an early quench." The causality runs: spark efficiently absorbs energy → secondary voltage drops → gap quenches (SGTC) or primary current drops (DRSSTC). Maximum power transfer produces maximum damping. Attempts to optimize spark performance by adjusting quench timing attack the symptom, not the cause.
+
**Constraints on optimization:**
- Insufficient source current/voltage (primary limited)
- Inception field not achieved (spark doesn't form)
@@ -216,13 +218,23 @@ As spark grows:
A spark continues to grow while the electric field at its tip exceeds a threshold.
+**Physical basis:** In air at STP, ionization and electron attachment balance at a reduced field of E/N ≈ 100 Td (≈25 kV/cm or 2.5 MV/m). Below this field, free electrons attach to O₂ within ~16 ns and no discharge can sustain. Above it, electron avalanches grow exponentially until reaching the streamer criterion (N_cr ≈ 10⁸ electrons, α·d ≈ 18-20), at which point the space charge field becomes self-reinforcing. [Becker et al. 2005, Ch 2]
+
**Field requirements (at sea level, standard conditions):**
```
E_inception ≈ 2-3 MV/m (initial breakdown from smooth topload)
-E_propagation ≈ 0.4-1.0 MV/m (sustained leader growth)
+E_propagation ≈ 0.4-1.0 MV/m (sustained growth; recommended modeling value: 0.6-0.7 MV/m)
E_tip = κ × E_average (tip enhancement factor κ ≈ 2-5)
```
+**Why E_propagation << E_inception (factor of ~3-4x):**
+Once a spark channel exists, the conducting channel acts as a sharp electrode extension of the topload. Three pre-conditioning mechanisms lower the threshold for continued breakdown ahead of the tip:
+1. **Tip geometry**: The spark tip (d ~ 100 μm for streamers, mm for leaders) concentrates the field far more than the smooth topload (R ~ 10+ cm). This geometric sharpening is the dominant factor.
+2. **UV photoionization**: The existing channel emits UV that ionizes O₂ up to ~1 mm ahead, providing seed electrons and eliminating statistical lag for new avalanches.
+3. **Residual ionization**: Previous streamer branches may have left partial ionization in nearby air, reducing the avalanche distance needed.
+
+**Important distinction:** The dramatic thermal effects (5000-20000 K) occur *within* the existing channel behind the tip. The air *ahead* of the advancing tip is only modestly pre-heated by the shock wave from the streamer front (perhaps hundreds of K, not thousands). The Paschen/density-reduction argument applies to maintaining the existing hot channel, not to reducing the inception threshold ahead of the tip.
+
**Maximum voltage-limited length:**
Solve: E_tip(V_top_peak, L) = E_propagation
@@ -231,11 +243,11 @@ Use FEMM to compute E_tip for given V_top and length L. As spark grows, E_tip de
- Geometric field dilution
- Capacitive voltage division (see below)
-**Note:** E_propagation varies with altitude and humidity by ±20-30%.
+**Note:** E_propagation varies with altitude and humidity by ±20-30%. Humidity has a specific effect: breakdown voltage reaches a minimum at ~1% water vapor content [Becker et al. 2005, Ch 2, p. 30]. There is also a frequency dependence with a breakdown voltage minimum near ~1 MHz; typical DRSSTC frequencies (50-400 kHz) are below this minimum but approaching it [Kunhardt 2000].
### 5.2 Power Limit: Energy per Meter
-Growth consumes approximately constant energy per unit length ε [J/m]:
+Growth consumes approximately constant energy per unit length ε [J/m]. The minimum volumetric energy density for spark channel formation is 0.6-1 J/cm³ [Becker et al. 2005, Ch 2, p. 59], which for a 3 mm leader gives ε_min ≈ 0.07 J/m. Observed ε values (5-100 J/m) are 50-1000× higher because most energy goes into gas heating (~14 eV per electron-ion pair), radiation, branching, and expansion work rather than just ionization.
**Growth rate equation:**
```
@@ -292,48 +304,96 @@ For thick leaders (d ~ 5 mm): τ ~ 300-600 ms
**Observed channel persistence is longer than pure thermal diffusion** due to:
- Buoyancy and convection maintaining hot gas column
-- Ionization memory (recombination slower than thermal diffusion)
+- Ionization memory: electron-ion recombination rate ≈ 2×10⁻⁷ cm³/s at 300 K [Becker et al. 2005, Ch 4], giving τ_recomb ≈ 50 μs at n_e = 10¹³ cm⁻³ — comparable to thin streamer thermal diffusion
+- N₂ vibrational relaxation time >100 μs at 1 atm [Becker et al. 2005, Ch 5] — acts as energy reservoir sustaining partial ionization after direct heating ceases
- Broadened effective channel diameter
**Effective persistence times:**
- Thin streamers: ~1-5 ms (convection/ionization dominated)
- Thick leaders: seconds (buoyancy maintains hot column)
-**QCW advantage:**
-- Ramps of 5-20 ms exploit ionization/convection persistence
-- Channel stays hot throughout growth
-- Continuous energy injection maintains E_tip
-- Transitions streamers → leaders efficiently
+**Conductance relaxation (Bazelyan & Raizer 2000, Ch 4, pp. 194-195):**
+```
+dG/dt = [G_st(i) - G(t)] / τ_g
+
+τ_g = 40 μs (current rising, channel heating)
+τ_g = 200 μs (current falling, channel cooling)
+```
+The 5:1 asymmetry between heating and cooling time constants creates a thermal ratcheting effect over many RF cycles: during high-current half-cycles, conductance rises quickly (τ_g = 40 μs); during low-current half-cycles, it falls slowly (τ_g = 200 μs). The net effect is that conductance ratchets upward over ~10-50 RF cycles. This is the microsecond-timescale mechanism underlying the millisecond-timescale streamer-to-leader transition. At 400 kHz, τ_g spans ~16 RF cycles, ensuring smooth conductance buildup.
+
+**QCW operating mode** (community survey data from 30+ forum threads, 6 builder sites, 2026):
+
+QCW uses voltage ramps of 10-22 ms at 300-600 kHz to grow thermally persistent leader channels. Key measured parameters:
+
+| Parameter | QCW | Burst DRSSTC |
+|-----------|-----|-------------|
+| Coupling (k) | 0.3-0.55+ | 0.05-0.2 |
+| Operating frequency | 300-600 kHz | 50-110 kHz |
+| Secondary voltage | 40-70 kV | 200-600 kV |
+| Spark:secondary ratio | 7-16× | 2-4× |
+
+**QCW secondary voltage is LOW.** Multiple builders measured only 40-70 kV topload voltage during QCW operation despite producing meter-length sparks. The critical comparison (davekni): ~600 kV for 2-3 m burst sparks vs ~40 kV for equivalent QCW sparks at 450 kHz — a 15:1 voltage ratio. This proves QCW growth is driven by sustained energy injection through a persistent leader, not high instantaneous voltage.
+
+**Leader formation voltage threshold:** A minimum ~300-400 kV is required for single-shot leader inception in air [Bazelyan & Raizer 2000, Ch 5, p. 271]. QCW bypasses this threshold because the conductance relaxation ratcheting mechanism (τ_g asymmetry above) accumulates energy from thousands of RF cycles, crossing critical temperature thresholds (2000→4000→5000 K) without requiring high instantaneous voltage.
+
+**Frequency threshold for sword sparks: 300-600 kHz.** Six or more independent builders observe straight "sword" sparks only above ~300 kHz. Below 100 kHz, QCW produces swirling/branchy sparks. Physical basis: at 400 kHz, the RF half-period (1.25 μs) is 100× shorter than τ_thermal for a 100 μm streamer (~125 μs). The channel sees effectively continuous heating. At 50-100 kHz (half-period 5-10 μs), thinner streamers experience significant cooling between cycles — the preferred path diffuses and branches.
+
+**Driven leader growth rate: ~170 m/s** (approximately half the speed of sound; community estimate, not directly measured with high-speed camera). Self-consistency: at 170 m/s over 10 ms → 1.7 m, matching observed QCW lengths. The step time (0.01 m / 170 m/s ≈ 60 μs) is close to τ_g = 40 μs, suggesting the advance rate is limited by how fast each new streamer segment can be heated to leader temperature. Note: this is the net growth rate averaged over many steps, NOT the Bazelyan instantaneous leader step velocity (~km/s).
+
+**Burst ceiling: ~80 μs** (Steve Ward, DRSSTC-0.5). Spark length saturates after ~80 μs ON time regardless of power. Consistent with τ_thermal ≈ 125 μs for 100 μm streamers — after one thermal time constant, channels cool as fast as they heat. This is the fundamental wall QCW overcomes with sustained drive.
+
+**Three ramp regimes** (Loneoceans QCW v1.5):
+- Too short (<5 ms): segmented sparks — insufficient time for leader transition
+- Optimal (10-20 ms): straight sword sparks — leader forms and grows continuously
+- Too long (>25 ms): hot/fat/bushy without extra length — leader hits voltage-limited L_max from capacitive divider; excess energy drives branching
+
+**Power envelope quality matters.** True QCW uses a linear voltage ramp → quadratic power (P ~ V²), the natural profile for growing against increasing capacitive loading. Pulse-skip modulation (H-bridge freewheeling at OCD threshold) delivers a sawtooth current envelope with per-cycle jitter. This is a continuum: coarse pulse-skip → Bresenham PDM linear ramp (more sword-like but still branches) → true analog QCW (full swords). Spark straightness improves progressively with envelope smoothness.
**Burst mode characteristics:**
- Widely spaced bursts: channel cools/deionizes between pulses
-- Must re-ionize repeatedly
+- Must re-ionize repeatedly — high ε overhead
- High peak current → bright, thick but short
- Voltage collapse limits length before leader formation
+- Growth saturates at ~80 μs ON time (burst ceiling)
+- Short bursts of high peak power outperform long bursts of low peak power at the same total energy (Steve Conner)
### 5.5 Streamers vs Leaders
**Streamers:**
- Thin (10-100 μm), fast (~10⁶ m/s), low current (mA)
-- Photoionization propagation
-- High resistance, short-lived (μs thermal time)
+- Electron density: 10¹¹-10¹³ cm⁻³ (non-equilibrium: T_e ≈ 35,000 K while T_gas ≈ 300-1000 K)
+- Ionization front at tip: ~150 μm thick [Becker et al. 2005, Ch 2]
+- Propagation via photoionization: UV from excited N₂ ionizes O₂ up to ~1 mm ahead of tip
+- High resistance (σ ≈ 0.01-0.1 S/m), short-lived (μs thermal time, 1-5 ms effective with ionization memory)
- Purple/blue, highly branched
-- High ε (inefficient)
+- High ε (30-100+ J/m, inefficient)
**Leaders:**
- Thick (mm-cm), slower (~10³ m/s), high current (A)
-- Thermally ionized (5000-20000 K)
-- Low resistance, persistent (seconds with convection)
+- Electron density: ~10¹⁵-10¹⁶ cm⁻³ (approaching thermal equilibrium at 5000-20000 K)
+- Thermally ionized: temperature sustains ionization without external field
+- Low resistance (σ ≈ 10-100 S/m), persistent (seconds with convection)
- White/orange, straighter
-- Low ε (efficient)
-
-**Transition sequence:**
-1. High E-field creates streamers
-2. Sufficient current → Joule heating
-3. Heated channel → thermal ionization → leader
-4. Leader grows from base
-5. Leader tip launches new streamers
-6. Fed streamers convert to leader
+- Low ε (5-15 J/m, efficient)
+
+**Two-stage spark formation** (observed in high-speed photography):
+1. **Primary streamer**: fast propagation at ~10⁶ m/s via photoionization
+2. **Secondary streamer/leader**: slower propagation at 10³-10⁴ m/s along same trajectory, driven by energy deposition into the existing channel (gas heating, vibrational excitation) rather than direct ionization [Becker et al. 2005, Ch 2, pp. 59-60]
+
+**Corona-to-spark energy threshold:** Minimum 0.6-1 J/cm³ deposited in channel volume [Becker et al. 2005, Ch 2, p. 59]. This is easily met in terms of total energy; the constraint is **power density** (current density >10⁶ A/m² sustained for 0.5-2 ms).
+
+**QCW transition sequence:**
+1. High E-field creates primary streamers (μs timescale)
+2. Space charge from first burst shields electrode → dark period (~1-5 ms)
+3. Ion drift restores field → subsequent streamer bursts (thermal ratcheting)
+4. Multiple aborted leaders may precede stable inception [Liu 2017; Les Renardieres 1977, 1981]
+5. Critical: gas temperature must **significantly exceed 2000 K** — convection losses during gas expansion can abort leaders at marginal temperatures [Liu 2017, Ch 3]
+6. Continuous current → Joule heating in base channels (0.5-2 ms cumulative)
+7. Heated channel → thermal ionization → leader (T > 5000 K, n_e → 10¹⁵+)
+8. Leader grows from base, resistance drops toward R_opt_power
+9. Leader tip launches new streamers into virgin air
+10. Fed streamers convert to leader as current heats them
+Note: Multiple stems share current simultaneously (Schlieren photography confirms); the stem receiving the most cumulative energy transitions first [Liu 2017, Ch 2]
### 5.6 The Capacitive Divider Problem
@@ -661,6 +721,91 @@ d_implied = sqrt(4×ρ_typical×L_segment / (π×R_opt))
Because dependence is logarithmic, typically converges in 1-2 iterations if needed.
+### 8.6 Time-Domain Plasma Evolution: Mayr Equation for Segment Conductance
+
+Sections 8.3-8.4 determine steady-state resistance distributions. For time-domain simulation (QCW ramps, burst transients), each segment's conductance must evolve dynamically. The Mayr arc equation provides this:
+
+```
+dG_i/dt = (1/τ_i) × (P_i/P_0i - 1) × G_i
+
+where:
+ G_i = conductance of segment i [S] (G = 1/R)
+ P_i = power dissipated in segment i = I_i² / G_i [W]
+ P_0i = equilibrium cooling power for segment i [W]
+ τ_i = plasma thermal time constant for segment i [s]
+```
+
+**Physical interpretation:** When power input P_i exceeds the cooling power P_0i, conductance increases (channel heats, ionization rises, R drops). When P_i < P_0i, conductance decreases (channel cools, recombination dominates, R rises). This IS the hungry streamer feedback loop (Section 2.3) expressed as a differential equation.
+
+**Connection to hungry streamer:** The Mayr equation naturally drives each segment toward its power-maximizing resistance. At equilibrium (dG/dt = 0), either G = 0 (extinguished) or P = P_0 (thermal balance). The segment's R self-adjusts toward R_opt_power because that maximizes P_i — and maximum P_i is the most stable equilibrium with P_i > P_0i.
+
+**Parameter estimation by channel type:**
+
+| Parameter | Streamer segment | Leader segment | Units |
+|-----------|-----------------|----------------|-------|
+| τ | 0.1-0.5 ms | 10-500 ms | s |
+| P_0 | ~1 W/m × L_seg | ~1 kW/m × L_seg | W |
+| G_initial | 10⁻⁸ - 10⁻⁵ | 10⁻⁴ - 10⁻² | S |
+
+τ values come from thermal diffusion: τ = d²/(4α_thermal), with d ~ 100 μm (streamer) to 5 mm (leader). P_0 values come from the power density required to sustain plasma ionization against attachment/recombination losses (see context/thermal-physics.md for derivation from first principles: 1.4 kW/cm³ for cold air, 14 kW/cm³ for hot air, at n_e = 10¹³ cm⁻³).
+
+**Combined growth algorithm with Mayr evolution:**
+```
+For each time step dt:
+ 1. For each segment i:
+ a. Compute I_i from circuit solution (SPICE AC or transient)
+ b. P_i = I_i² / G_i
+ c. dG_i/dt = (1/τ_i) × (P_i/P_0i - 1) × G_i
+ d. G_i_new = G_i + dG_i × dt
+ e. Clip: G_i = clip(G_i_new, G_min[i], G_max[i])
+ 2. At tip segment (last active):
+ a. Compute E_tip (from FEMM or approximate model)
+ b. If E_tip > E_propagation (~0.6-0.7 MV/m): activate next segment
+ c. E_propagation is the PROPAGATION threshold (NOT inception)
+ - Do NOT use 30 kV/cm (3 MV/m) — that is E_inception for cold air
+ - The air ahead of the advancing tip is pre-conditioned (see Section 5.1)
+ 3. Update spark length: L = n_active × L_segment
+ 4. Update capacitances for new length (C_sh grows linearly)
+ 5. Retune drive to loaded pole frequency (critical — see Part 4.2)
+```
+
+**Key advantage of Mayr approach:** It naturally produces the streamer-to-leader transition. Base segments receiving high current (P >> P_0) see G rise rapidly — resistance drops through the streamer range (MΩ) into the leader range (kΩ). Tip segments receiving low current (P ~ P_0) hover at streamer conductance. The composite leader-trunk / streamer-crown structure emerges from the physics without being imposed.
+
+**Mayr parameter ranges from literature:**
+
+Independent confirmation from arc modeling reviews [Yang et al. 2022, "Arc Modeling Approaches," Frontiers in Physics]:
+- TC sparks are firmly in the **Mayr regime** (low current, non-equilibrium)
+- Cassie model irrelevant for TC (applies to high-current industrial arcs only)
+- tau_m sensitivity: small changes produce large conductance variations → careful calibration needed
+- LTE assumption breaks down at low TC currents → Mayr equation is approximate, not exact
+- Hybrid Mayr-Cassie transition: sigma(i) = exp(-i²/I₀²); for TC sparks i << I₀, reducing to pure Mayr
+
+**Gallimberti model limitations** [Liu 2017, Ch 3]:
+- The widely-used Gallimberti (1972) streamer-to-leader model assumes constant stem field, simplified V-T relaxation, and single stem — all shown to be quantitatively unreliable by detailed 45-species kinetic modeling
+- Humidity effect on V-T relaxation is weak ("several orders of magnitude smaller" than other energy sources)
+- Use Gallimberti as conceptual framework only, not for quantitative predictions
+
+**Equilibrium resistance power law** [da Silva et al. 2019, JGR Atmospheres]:
+The Mayr equation drives each segment toward an equilibrium. That equilibrium R follows a power law in current:
+```
+R_eq = A / I^b (Ω/m)
+
+Region I (1-10 A): A = 12,400 b = 1.84 ← TC streamer/early leader
+Region II (10-1000 A): A = 2,820 b = 1.16 ← DRSSTC burst mode
+Region III (>1000 A): A = 180 b = 0.75 ← Lightning/high-current arcs
+```
+At 1 A: R ~ 12.4 kΩ/m. At 10 A: R ~ 179 Ω/m. At 100 A: R ~ 13.5 Ω/m.
+The steep b=1.84 in Region I is the quantitative expression of the positive feedback driving streamer-to-leader transition: doubling current cuts resistance by ~3.6×.
+
+**Air heating efficiency** [da Silva et al. 2019, after Flitti & Pancheshnyi 2009]:
+```
+η_T = 0.1 + 0.9 × [tanh(T/T_amb - 4) + 1] / 2
+```
+At ambient: only 10% of Joule heating goes to gas temperature (90% → N₂ vibrational modes).
+Above 2000 K: η_T → 1.0 (full thermalization). This explains why streamer-to-leader transition takes milliseconds despite MW/m power densities — the thermal runaway only accelerates after T > ~1000 K.
+
+**Simplification for steady-state:** If not modeling transients, set dG/dt = 0 for all segments. Each segment is either extinguished (G = 0, no current path) or at thermal equilibrium (P = P_0). This recovers the circuit-determined R of Section 8.4 as a limiting case.
+
---
## Part 9: Impedance Matching for Target Spark Length
@@ -764,6 +909,7 @@ P_max = 0.5×|V_th|²/(4×Re{Z_th})
```
E_inception ≈ 2-3 MV/m (initial breakdown)
E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
+ Positive streamer critical field: E_cr(+) ≈ 4.5-5 kV/cm [Bazelyan & Raizer 2000]
dL/dt = P_stream/ε (when E_tip > E_propagation)
@@ -775,6 +921,32 @@ V_tip ≈ V_topload×C_mut/(C_mut+C_sh) (open-circuit limit)
τ_thermal = d²/(4α), α ≈ 2×10⁻⁵ m²/s for air
d=100 μm → τ~0.1 ms; d=5 mm → τ~300 ms
(Observed persistence longer due to convection/ionization)
+
+Leader step velocity (instantaneous): v_L = 1500×√(ΔU) [cm/s, U in V]
+ At 300 kV: ~8.2 km/s [Bazelyan & Raizer 2000]
+ This is the speed of thermal instability contraction within a single step
+
+QCW net growth rate: ~170 m/s (community estimate)
+ Limited by τ_g: each step takes ~60 μs to thermalize
+ Leader advances in fast micro-steps at ~km/s, spends most time heating
+
+Conductance relaxation: dG/dt = [G_st(i) - G] / τ_g
+ τ_g = 40 μs (heating), 200 μs (cooling) [Bazelyan & Raizer 2000]
+ 5:1 asymmetry drives thermal ratcheting
+
+Burst ceiling: ~80 μs ON time (Steve Ward, DRSSTC-0.5)
+ Consistent with τ_thermal ≈ 125 μs for 100 μm streamers
+
+Energy ceiling from tip capacitance: W_max = π×ε₀×U² [J/m]
+ At 300 kV: ~25 J/m
+
+Temperature thresholds for self-sustaining channel:
+ >2000 K: thermal ionization onset (fragile) [Liu 2017]
+ >4000 K: associative ionization N+O→NO⁺+e (robust) [Bazelyan & Raizer 2000]
+ >5000 K: electron attachment negligible (persistent)
+
+Bazelyan V-I characteristic: i×E = 300 V·A/cm
+ (agrees with da Silva R=A/I^b within factor ~2 for 1-10 A)
```
### Physical Bounds
@@ -790,6 +962,27 @@ Typical total spark resistance at 200 kHz for 1-3 m:
Typical impedance phase: -55° to -75°
```
+### Mayr Equation (Segment Conductance Evolution)
+```
+dG_i/dt = (1/τ_i) × (P_i/P_0i - 1) × G_i
+
+P_i = I_i²/G_i (power dissipated in segment)
+P_0i ~ 1 W/m (streamer) to 1 kW/m (leader) × L_segment
+τ_i ~ 0.1-0.5 ms (streamer) to 10-500 ms (leader)
+
+Equilibrium: P = P_0 (thermal balance) or G = 0 (extinguished)
+
+Equilibrium resistance power law:
+R_eq = A / I^b (Ω/m)
+ Region I (1-10 A): A=12,400 b=1.84 (TC streamers)
+ Region II (10-1000 A): A=2,820 b=1.16 (DRSSTC burst)
+
+Air heating efficiency:
+η_T = 0.1 + 0.9×[tanh(T/T_amb - 4) + 1]/2
+ At 300 K: η_T ~ 0.1 (90% → vibrational modes)
+ At 2000 K: η_T ~ 1.0 (full thermalization)
+```
+
### Ringdown Method
```
At loaded resonance ω_L:
@@ -808,9 +1001,9 @@ Y_spark ≈ (G_total - G_0) + jω_L(C_eq - C_0)
### 12.1 Remaining Uncertainties
-- ε variability with current density, frequency, ambient conditions
-- E_propagation dependence on geometry, humidity, altitude
-- Full thermal evolution including convection and radiation
+- ε variability with current density, frequency, ambient conditions (lower bound established: 0.6-1 J/cm³ volumetric → ε_min ~ 0.07 J/m for leaders; observed 50-1000× higher due to overhead losses)
+- E_propagation dependence on geometry, humidity (minimum at ~1% H₂O), altitude
+- Full thermal evolution: recombination (~50 μs), vibrational relaxation (>100 μs), convection, and radiation now partially quantified from [Becker et al. 2005]
- Branching: power division among multiple channels
### 12.2 Future Enhancements
@@ -833,6 +1026,19 @@ Y_spark ≈ (G_total - G_0) + jω_L(C_eq - C_0)
- RF current distribution measurements at multiple points
- Database correlating spark parameters to operating conditions
+**Key references for plasma physics grounding:**
+- Becker, Kogelschatz, Schoenbach & Barker, "Non-Equilibrium Air Plasmas at Atmospheric Pressure" (IOP, 2005) — source for recombination rates, ionization coefficients, conductivity data, and energy thresholds used in this framework
+- Liu, "Electrical Discharges: Streamer-to-Leader Transition and Positive Leader Inception" (KTH Doctoral Thesis, 2017) — detailed kinetic modeling (45 species, 192 reactions) of streamer-to-leader transition; leader inception requires T >> 2000 K; Gallimberti model limitations; dark period physics
+- Yang, Meng, Niu et al., "Arc Modeling Approaches: A Comprehensive Review" (Frontiers in Physics, 2022) — Mayr/Cassie parameter ranges; TC sparks in Mayr regime; sensitivity analysis
+- Les Renardieres Group (1977, 1981) — comprehensive experimental studies of long spark formation; Schlieren photography; primary data for Liu's kinetic validation
+- Raether (1964), Meek & Craggs (1978) — classical textbooks on streamer/spark physics
+- Morrow & Lowke (1997) — ionization/attachment coefficients for air discharge modeling
+- da Silva et al., "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance" (JGR Atmospheres, 2019) — R = A/I^b equilibrium resistance power law; air heating efficiency η_T; channel expansion dynamics; rate coefficients available on Zenodo
+- Gallimberti (1972) — early streamer propagation simulation methodology (conceptually useful but quantitatively limited per Liu 2017)
+- Bazelyan & Raizer, "The mechanism of lightning attraction and the problem of lightning initiation by lasers" (Physics-Uspekhi 43(7), 2000) — leader velocity formula, V-I characteristic (i×E=300), temperature thresholds (4000-5000 K), energy ceiling from tip capacitance, electron mobility/attachment data
+- Bazelyan & Raizer, "Lightning Physics and Lightning Protection" (IOP, 2000) — full textbook; conductance relaxation model (τ_g = 40/200 μs), leader energy balance, maximum heatable channel radius, stepped vs continuous leaders, complex ion recombination rates, streamer velocity/density formulas
+- Phase 6 QCW Community Research Survey (2026) — 30+ forum threads, 6 builder sites; key findings: 40-70 kV QCW voltage (15:1 ratio vs burst), 300-600 kHz frequency threshold for sword sparks, ~170 m/s growth rate, 80 μs burst ceiling, three ramp regimes, pulse-skip envelope quality continuum. See phases/phase-6-qcw-community-research.md
+
---
## Conclusion
diff --git a/tools/extract_bazelyan_book.py b/tools/extract_bazelyan_book.py
new file mode 100644
index 0000000..6eabb3f
--- /dev/null
+++ b/tools/extract_bazelyan_book.py
@@ -0,0 +1,13 @@
+import fitz
+
+doc = fitz.open(r'C:\git\spark-lesson\reference\sources\cenrs-book.pdf')
+output = []
+for i in range(len(doc)):
+ text = doc[i].get_text()
+ if text.strip():
+ output.append(f'=== PAGE {i+1} ===')
+ output.append(text)
+full = '\n'.join(output)
+with open(r'C:\git\spark-lesson\reference\sources\bazelyan-raizer-lightning-physics-2000.txt', 'w', encoding='utf-8') as f:
+ f.write(full)
+print(f'Extracted {len(full)} chars from {len(doc)} pages')
diff --git a/tools/extract_cenrs.py b/tools/extract_cenrs.py
new file mode 100644
index 0000000..d3e865c
--- /dev/null
+++ b/tools/extract_cenrs.py
@@ -0,0 +1,11 @@
+import fitz
+import sys
+
+doc = fitz.open(r'C:\git\spark-lesson\reference\sources\cenrs-book.pdf')
+print(f'Pages: {len(doc)}')
+for i in range(min(len(doc), 10)):
+ text = doc[i].get_text()
+ if text.strip():
+ print(f'=== PAGE {i+1} ===')
+ print(text[:3000])
+ print('...')
diff --git a/tools/extract_liu.py b/tools/extract_liu.py
new file mode 100644
index 0000000..e315bbd
--- /dev/null
+++ b/tools/extract_liu.py
@@ -0,0 +1,12 @@
+import fitz
+import sys
+
+doc = fitz.open(r'C:\git\spark-lesson\reference\sources\liu-discharge-transitions-thesis.pdf')
+print(f'Pages: {len(doc)}')
+
+# Extract first 10 pages to see TOC
+for i in range(min(10, len(doc))):
+ text = doc[i].get_text()
+ if text.strip():
+ print(f'--- Page {i+1} ---')
+ print(text[:2000])
diff --git a/tools/extract_liu_full.py b/tools/extract_liu_full.py
new file mode 100644
index 0000000..8ccb7e2
--- /dev/null
+++ b/tools/extract_liu_full.py
@@ -0,0 +1,20 @@
+import fitz
+
+doc = fitz.open(r'C:\git\spark-lesson\reference\sources\liu-discharge-transitions-thesis.pdf')
+print(f'Total pages: {len(doc)}')
+
+output = []
+for i in range(len(doc)):
+ text = doc[i].get_text()
+ if text.strip():
+ output.append(f'=== PAGE {i+1} ===')
+ output.append(text)
+
+full_text = '\n'.join(output)
+
+with open(r'C:\git\spark-lesson\reference\sources\liu-discharge-transitions-thesis.txt', 'w', encoding='utf-8') as f:
+ f.write(full_text)
+
+print(f'Extracted {len(output)//2} pages')
+print(f'Total characters: {len(full_text)}')
+print(f'Total lines: {full_text.count(chr(10))}')
diff --git a/tools/extract_pdf.py b/tools/extract_pdf.py
new file mode 100644
index 0000000..0790005
--- /dev/null
+++ b/tools/extract_pdf.py
@@ -0,0 +1,27 @@
+"""Extract text content from PDF to plain text file."""
+import sys
+import fitz
+
+src = r'C:\git\spark-lesson\reference\sources\non-equilibrium-air-plasmas-becker-kogelschatz.pdf'
+dst = r'C:\git\spark-lesson\reference\sources\non-equilibrium-air-plasmas-becker-kogelschatz.txt'
+
+doc = fitz.open(src)
+print(f'Pages: {len(doc)}')
+print(f'Title: {doc.metadata.get("title", "N/A")}')
+print(f'Author: {doc.metadata.get("author", "N/A")}')
+
+text = []
+for i, page in enumerate(doc):
+ t = page.get_text()
+ if t.strip():
+ text.append(f'--- Page {i+1} ---\n{t}')
+
+full = '\n'.join(text)
+print(f'Total chars: {len(full):,}')
+print(f'Estimated size: {len(full.encode("utf-8"))/1024/1024:.1f} MB')
+
+with open(dst, 'w', encoding='utf-8') as f:
+ f.write(full)
+
+print(f'Written to {dst}')
+doc.close()
diff --git a/tools/extract_pdfs.py b/tools/extract_pdfs.py
new file mode 100644
index 0000000..0bd1758
--- /dev/null
+++ b/tools/extract_pdfs.py
@@ -0,0 +1,31 @@
+import fitz
+import sys
+
+files = [
+ (r'C:\git\spark-lesson\reference\sources\bazelyan-noaa-preprint.pdf',
+ r'C:\git\spark-lesson\reference\sources\bazelyan-noaa-preprint.txt'),
+ (r'C:\git\spark-lesson\reference\sources\plasma-nature-lightning-channels.pdf',
+ r'C:\git\spark-lesson\reference\sources\plasma-nature-lightning-channels.txt'),
+]
+
+for pdf_path, txt_path in files:
+ try:
+ doc = fitz.open(pdf_path)
+ print(f'\n=== {pdf_path} ===')
+ print(f'Pages: {len(doc)}')
+
+ output = []
+ for i in range(len(doc)):
+ text = doc[i].get_text()
+ if text.strip():
+ output.append(f'=== PAGE {i+1} ===')
+ output.append(text)
+
+ full_text = '\n'.join(output)
+ with open(txt_path, 'w', encoding='utf-8') as f:
+ f.write(full_text)
+
+ print(f'Extracted {len(output)//2} pages')
+ print(f'Total characters: {len(full_text)}')
+ except Exception as e:
+ print(f'Error with {pdf_path}: {e}')
diff --git a/tools/extract_ufn.py b/tools/extract_ufn.py
new file mode 100644
index 0000000..7b359e5
--- /dev/null
+++ b/tools/extract_ufn.py
@@ -0,0 +1,14 @@
+import fitz
+import sys
+
+doc = fitz.open(r'C:\git\spark-lesson\reference\sources\ufn-2000-paper.pdf')
+output = []
+for i in range(len(doc)):
+ text = doc[i].get_text()
+ if text.strip():
+ output.append(f'=== PAGE {i+1} ===')
+ output.append(text)
+full = '\n'.join(output)
+with open(r'C:\git\spark-lesson\reference\sources\ufn-2000-paper.txt', 'w', encoding='utf-8') as f:
+ f.write(full)
+print(f'Extracted {len(full)} chars from {len(doc)} pages')
diff --git a/spark-lessons/generate_images.py b/tools/generate_images.py
similarity index 100%
rename from spark-lessons/generate_images.py
rename to tools/generate_images.py
diff --git a/spark-lessons/generate_placeholders.py b/tools/generate_placeholders.py
similarity index 100%
rename from spark-lessons/generate_placeholders.py
rename to tools/generate_placeholders.py