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id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
field-thresholds	Electric Field Thresholds and Spark Inception/Propagation	established	spark-physics.txt: Part 5 Section 5.1 (lines 213-234), Part 5 Section 5.6 (lines 338-360), Part 6 (lines 389-438), Part 11 (lines 763-777)	[energy-and-growth thermal-physics streamers-and-leaders capacitive-divider empirical-scaling power-optimization branching-physics qcw-operation lumped-model distributed-model femm-workflow equations-and-bounds open-questions]	[E_inception threshold E_propagation threshold E_tip with enhancement factor Voltage-limited maximum length Capacitive voltage division V_tip Altitude correction for E thresholds Growth rate dL/dt = P_stream / epsilon Townsend ionization coefficient Paschen minimum Streamer criterion (Meek) Paschen density scaling E_breakdown proportional to N]	[inception field propagation field tip enhancement factor field dilution capacitive voltage division stall point streamer leader altitude correction Paschen curve Townsend coefficient streamer criterion reduced field E/N Townsend (Td) electron attachment corona shielding voltage rate limit dynamic threshold coupled voltage-power limit]	[electric-field-enhancement.png voltage-division-vs-length-plot.png femm-field-plot-example.png]	[spark-growth-timeline.md]	[How does the tip enhancement factor kappa vary during the transition from streamer to leader? What is the correct E_propagation for branched vs. single-channel sparks? How does UV pre-ionization from the topload corona affect E_inception for secondary streamers? Can E_propagation be measured directly in a controlled Tesla coil experiment? How does the effective E_propagation change when the spark grows into regions of non-uniform background field? What role do runaway electrons play in Tesla coil spark inception at fields exceeding 3x the stationary breakdown threshold? What is the gas temperature 1-10 mm ahead of a QCW leader tip during active growth? Does an accelerating voltage ramp produce longer QCW sparks than a linear ramp of the same peak voltage and energy?]

Electric Field Thresholds and Spark Inception/Propagation

This document establishes the field-based criteria that govern whether a Tesla coil spark can form and continue to grow. Two distinct thresholds exist: the inception field (required to start a spark) and the propagation field (required to sustain growth). The interplay between topload voltage, geometric field dilution, tip enhancement, and capacitive voltage division determines the maximum voltage-limited spark length for any given operating condition.

1. Two Distinct Field Thresholds

1.1 Inception Field (E_inception)

The inception field is the electric field required to initiate electrical breakdown from the topload surface in ambient air.

E_inception ~ 2 - 3 MV/m     (at sea level, standard conditions)

Physical basis: Breakdown in air requires that an electron avalanche achieves sufficient multiplication to become self-sustaining (the Townsend criterion, or equivalently, the streamer criterion). For air at atmospheric pressure, this requires approximately 3 MV/m for a uniform field. The range 2-3 MV/m reflects:

Smooth, large-radius topload: E_inception closer to 3 MV/m. The field is relatively uniform near the surface, and breakdown requires the full Paschen-like threshold.
Sharp points, small radius of curvature: E_inception closer to 2 MV/m (or even lower). Field enhancement at sharp features means the local field can exceed the breakdown threshold even when the average field is below it.
Surface condition: Dust, moisture, surface roughness, and residual ionization from previous sparks can reduce E_inception by providing seed electrons and reducing the statistical lag time.

Practical note for Tesla coils: Most toploads have relatively smooth surfaces (toroidal or spherical), so E_inception is typically near the upper end of the range (2.5-3 MV/m). However, breakout points (deliberately placed sharp features) are designed to lower E_inception at a specific location to control where the spark initiates.

1.2 Propagation Field (E_propagation)

The propagation field is the electric field required at the spark tip to sustain continued growth after initial inception.

E_propagation ~ 0.4 - 1.0 MV/m     (at sea level, standard conditions)

Why E_propagation << E_inception: Once a spark channel exists, it extends the conductor from the topload. The spark tip concentrates the field (see Section 2 on tip enhancement), and the ionized channel behind the tip provides a low-impedance path for current. The spark effectively "sharpens" the electrode, reducing the field required for continued avalanche propagation.

Additionally, the air ahead of the advancing tip has been partially pre-conditioned:

UV photoionization from the existing channel provides seed electrons
Shock heating from the advancing wavefront raises the gas temperature slightly
Previous streamer branches may have left residual ionization

Modeling value: For simulation purposes, E_propagation = 0.6-0.7 MV/m is a good starting point for typical conditions. This should be calibrated against measurements (see Section 6).

Independent confirmation: Bazelyan & Raizer (2000) report the critical average field for positive streamer propagation in air as E_cr(+) ~ 4.5-5 kV/cm (0.45-0.5 MV/m), and for negative streamers E_cr(-) ~ 10-12 kV/cm (1.0-1.2 MV/m). The positive streamer value is at the lower end of our E_propagation range, consistent with the fact that TC sparks propagating from a positive topload benefit from the easier positive streamer criterion. The 2:1 ratio between negative and positive streamer thresholds also explains observed polarity effects in TC spark length. [Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 703]

1.3 Sharp vs. Smooth Electrode Inception

Electrode Type	Approximate E_inception	Physical Reason
Smooth sphere (R > 10 cm)	3 - 4 MV/m	Near-uniform field at surface
Smooth toroid	2.5 - 3.5 MV/m	Mild curvature enhancement
Breakout point (R ~ 1 mm)	1 - 2 MV/m	Strong geometric enhancement
Wire tip (R ~ 0.1 mm)	0.5 - 1 MV/m	Extreme enhancement

Note: These are approximate surface field values at inception. The voltage required depends on the electrode geometry and its distance to ground.

1.4 Breakdown Physics: Ionization and Attachment in Air

The inception and propagation field values above arise from the balance between electron impact ionization and electron attachment in air. This subsection summarizes the underlying physics from the gas discharge literature.

Reduced Electric Field (E/N)

Breakdown behavior in gases is governed by the reduced electric field E/N, measured in Townsend (Td), where 1 Td = 10^-21 V*m^2. At standard temperature and pressure (STP), E/N = 100 Td corresponds to approximately E ~ 25 kV/cm (2.5 MV/m). [Becker et al. 2005, Ch 2, p. 26]

Townsend Ionization Coefficient

The electron impact ionization coefficient in air follows:

alpha/N = A * exp(-B * N / E)

where:
  A = 1.4 * 10^-20 m^2
  B = 660 Td
  Valid range: 10-150 Td (roughly 2.5-37.5 kV/cm at 1 atm)

[Becker et al. 2005, Ch 2, p. 32; after Wagner 1971, Moruzzi & Price 1974]

More sophisticated analytical approximations for ionization and attachment coefficients covering wider E/N ranges can be found in Morrow & Lowke (1997) or Chen & Davidson (2003).

Ionization-Attachment Crossover

In air, ionization (alpha) exactly balances three-body electron attachment (eta) at:

E/p ~ 25 kV/cm/bar    (equivalently E/N ~ 100 Td at STP)

Below this field, attachment dominates and no self-sustaining discharge is possible. Above it, ionization dominates and avalanches grow. This crossover IS the physical basis for the ~2.5 MV/m breakdown field in uniform gaps. [Becker et al. 2005, Ch 2, p. 33]

The electron lifetime in cold air at STP (below the crossover) is approximately 16 ns, dominated by three-body attachment to O2. This extremely short lifetime means free electrons are essentially instantaneously lost in cold, unperturbed air -- sustaining a discharge requires continuous energy input at a rate that exceeds the attachment loss. [Becker et al. 2005, Ch 1, p. 7]

Streamer Criterion (Meek Criterion)

An electron avalanche transitions to a self-propagating streamer when the total avalanche multiplication reaches:

N_critical ~ 10^8 electrons    (alpha * d ~ 18-20)

At this point, the space charge field of the avalanche head becomes comparable to the applied field, and the avalanche becomes self-propagating via its own enhanced field. [Becker et al. 2005, Ch 2, p. 35]

This criterion connects the microscopic (ionization coefficient) to the macroscopic (gap breakdown): given alpha(E) from the Townsend formula above, the minimum field for streamer formation in a gap of width d is the field where alpha(E) * d reaches 18-20.

Mean Electron Energy at Breakdown

At the breakdown threshold (E/N ~ 100 Td), the mean electron energy is approximately 3 eV (~35,000 K electron temperature). This is far above the gas temperature (~300 K), confirming that breakdown in air is a fundamentally non-equilibrium process: the electrons are "hot" while the gas remains "cold." [Becker et al. 2005, Ch 2, p. 26]

Runaway Electron Threshold

At fields exceeding approximately 3x the stationary breakdown field, electrons can "run away" -- gaining energy faster than they lose it through collisions. This threshold may be relevant during the initial moments of streamer head formation in Tesla coil sparks, where tip enhancement can push local fields well above 3 * 2.5 MV/m = 7.5 MV/m. [Becker et al. 2005, Ch 2, p. 39]

1.5 Paschen Curve Quantitative Data

The Paschen curve for air (breakdown voltage vs. pressure-distance product) has a well-characterized minimum:

V_min = 230 - 370 V
(pd)_min ~ 0.6 torr*cm

[Becker et al. 2005, Ch 2, p. 33]

The range in V_min reflects different electrode materials and surface conditions. For clean electrodes in dry air, V_min ~ 327 V (the classic Paschen minimum for air).

For Tesla coil applications, the Paschen curve is most relevant for understanding breakdown in small gaps (e.g., breakout point spacing, spark gap switches) rather than for the long-gap streamer propagation that governs spark length. Long-gap breakdown is dominated by the streamer mechanism (Section 1.4) rather than the Townsend mechanism that underlies the Paschen curve.

2. Tip Enhancement Factor

2.1 Definition

The electric field at the spark tip is enhanced relative to the average field (V/distance) by a geometric factor kappa:

E_tip = kappa * E_average

where E_average is the nominal field computed as if the spark were absent (e.g., V_tip / distance_to_ground for a simple geometry).

2.2 Physical Origin

The spark tip is a small-radius conductor protruding into a region of lower field. Electric field lines concentrate at the tip, just as they concentrate at any sharp conducting feature. The enhancement depends on:

Tip radius r_tip: Smaller radius -> higher enhancement. For a hemispherical cap on a cylinder: kappa ~ L_channel / r_tip (for long channels).
Channel geometry: A straight, thin channel has higher enhancement than a thick, blunt one.
Nearby conductors: Ground planes, strike rails, or other sparks in the vicinity can increase or decrease the local field.

2.3 Typical Values

kappa ~ 2 - 5     (for cylindrical spark channels with typical aspect ratios)

kappa ~ 2-3: Thick leader channels (d ~ several mm), relatively blunt tip
kappa ~ 3-5: Thin streamer channels (d ~ 100 um), sharp tip
kappa > 5: Very thin, very long channels (unusual in Tesla coil sparks)

For modeling: kappa = 3 is a reasonable default. FEMM simulation of the specific geometry provides a more accurate value (see femm-workflow).

2.4 FEMM Determination

The most reliable way to determine kappa for a specific configuration is to run a FEMM electrostatic simulation:

Model the topload, spark channel (as a thin conductor), and ground plane.
Set the topload to a known voltage V_top.
Solve for the electric field.
Read E_tip at the spark tip.
Compute E_average = V_tip / L (where V_tip accounts for voltage division and L is distance to ground).
kappa = E_tip / E_average.

3. Voltage-Limited Maximum Length

3.1 The Growth Condition

A spark continues to grow while:

E_tip(V_top_peak, L) > E_propagation

where E_tip is a function of the topload peak voltage and the current spark length L.

Growth stalls when:

E_tip(V_top_peak, L_max) = E_propagation

This defines the voltage-limited maximum length L_max for a given V_top_peak.

3.2 Why E_tip Decreases with Length

As the spark grows longer, three effects reduce E_tip:

Effect 1: Increased distance from topload. The spark tip moves farther from the topload (and from ground objects behind the topload). The geometric field at the tip would decrease even for a fixed-potential tip conductor, simply because the source of the field (the topload at V_top) is farther away.

Effect 2: Geometric field dilution. The field from a finite-size charged conductor (the topload) falls off with distance. For a point charge, E ~ 1/r^2. For a distributed charge on a toroid, the falloff is slower at short range (near-field) but eventually follows the 1/r^2 trend. The spark tip, being farther from the topload, sees a weaker driving field.

Effect 3: Capacitive voltage division (the most important effect for long sparks). As derived in capacitive-divider, the voltage at the spark tip is NOT equal to V_topload. The spark circuit forms a voltage divider between C_mut (coupling to topload) and C_sh (coupling to ground):

V_tip = V_topload * Z_mut / (Z_mut + Z_sh)

Open-circuit limit (R -> infinity):

V_tip ~ V_topload * C_mut / (C_mut + C_sh)

Since C_sh ~ 6.6 pF/m * L (proportional to spark length), V_tip decreases as the spark grows, even if V_topload is maintained constant. For a 2-meter spark with C_mut = 8 pF and C_sh = 13 pF:

V_tip / V_topload ~ 8 / (8 + 13) ~ 0.38

The spark tip sees only 38% of the topload voltage. The field at the tip is correspondingly reduced, making further growth harder.

With finite R (R ~ R_opt_power): V_tip is even lower and has a complex (not purely real) value, but the magnitude is still reduced.

3.3 Solving for L_max

The voltage-limited length is found by solving:

kappa * E_average(V_top, L_max) = E_propagation

where E_average depends on the FEMM field solution at the tip position. This is generally not solvable in closed form; it requires:

Iterative FEMM simulation: For a series of spark lengths L, compute E_tip. Find the L where E_tip = E_propagation.
Approximate analytic model: Using the capacitive divider and an assumed field geometry:

E_tip ~ kappa * V_tip / (effective_gap)
      ~ kappa * V_topload * C_mut / ((C_mut + C_sh) * (d_ground - L))

Set equal to E_propagation and solve for L. This gives a transcendental equation that must be solved numerically.

3.4 Practical Example

Consider: V_topload_peak = 400 kV, C_mut = 8 pF, C_sh = 6.6 pF/m * L, kappa = 3, E_propagation = 0.7 MV/m, distance to ground = 5 m.

At L = 2 m:

C_sh = 6.6 * 2 = 13.2 pF
V_tip = 400 * 8 / (8 + 13.2) = 400 * 0.377 = 151 kV
E_average ~ 151 kV / (5 - 2) m = 50.3 kV/m = 0.050 MV/m
E_tip = 3 * 0.050 = 0.15 MV/m

This is well below E_propagation = 0.7 MV/m. The simple E_average estimate is too conservative because it uses the wrong field geometry. FEMM accounts for the actual field distribution, which gives higher fields near the tip.

This example illustrates why FEMM simulation is essential: naive field estimates significantly underestimate E_tip because they do not account for the field concentration geometry.

4. Environmental Corrections

4.1 Altitude

Air density decreases with altitude, reducing the breakdown field proportionally:

E(altitude) = E(sea_level) * (P / P_0)

P / P_0 ~ exp(-h / 8500 m)

where h is the altitude in meters and P_0 is sea-level pressure.

Altitude (m)	P/P_0	E_propagation (if 0.7 MV/m at sea level)
0 (sea level)	1.000	0.70 MV/m
500	0.943	0.66 MV/m
1000	0.889	0.62 MV/m
1500	0.838	0.59 MV/m
2000	0.790	0.55 MV/m

At 2000 m altitude (e.g., Denver, Colorado), the propagation threshold is ~21% lower than at sea level. This means longer sparks for the same voltage, which is consistent with observations from high-altitude Tesla coil operators.

4.2 Humidity

Water vapor affects breakdown through two mechanisms:

Electron attachment: H2O has a significant electron attachment cross-section, removing free electrons and increasing the effective breakdown field. This effect INCREASES E_inception and E_propagation.
Reduced density: Water vapor is lighter than N2/O2, slightly reducing air density and thus the breakdown field. This effect DECREASES the thresholds.

The net effect is small and variable:

Humidity correction: +/- 10-20%

High humidity generally increases E_inception slightly (harder to start sparks) but has a less clear effect on E_propagation (mixed reports in the literature).

Quantitative humidity data: The breakdown voltage in air at atmospheric pressure has a minimum at approximately 1% water vapor content. At low humidity, adding water vapor reduces breakdown voltage (the reduced density effect dominates). Above ~1%, the electron attachment effect of H2O begins to dominate and raises the breakdown voltage again. For typical indoor conditions (30-70% RH at 20-25 C, corresponding to roughly 0.5-2% water vapor by volume), the humidity effect on E_inception is modest and may go in either direction depending on the specific humidity level. [Becker et al. 2005, Ch 2, p. 30; Protasevich 2000]

4.3 Temperature

Air density decreases with temperature, reducing breakdown fields:

Temperature correction: +/- 5-10%

At 40 C vs. 20 C: air density drops by ~7%, reducing breakdown thresholds by a similar amount.

4.4 Frequency Dependence of Breakdown

The breakdown voltage in air shows a frequency dependence, with a minimum near ~1 MHz. At frequencies well below this minimum, breakdown is governed by quasi-static (DC) processes. Near and above 1 MHz, electrons can survive the field reversal between half-cycles (the electron lifetime at STP is only 16 ns, see Section 1.4), reducing the effective breakdown threshold. [Becker et al. 2005, Ch 2, p. 30; Kunhardt 2000]

Tesla coil relevance: Typical DRSSTC operating frequencies (50-400 kHz) are below this minimum, so frequency effects are relatively minor:

At 50 kHz: Essentially DC-like breakdown behavior
At 200-400 kHz: Possibly 5-10% reduction in effective inception field compared to DC predictions
At ~1 MHz (some small SSTCs): Approaching the minimum, with potentially significant (~20-30%) reduction

This frequency dependence is a rarely discussed factor that could contribute to observed discrepancies between DC Paschen predictions and Tesla coil inception behavior, and to performance differences between coils operating at very different frequencies.

4.5 Corona Shielding and Voltage Rate Limit

When voltage rises slowly on a rounded electrode, a stable corona (continuous low-level discharge) can form and persist indefinitely, shielding the electrode from streamer inception. This occurs because the space charge from slowly-drifting ions stabilizes the surface field at the inception level. The maximum voltage growth rate at which this shielding corona can be sustained is:

A_u_max ~ 2 * mu_i * E_s^2 ~ 3.6 kV/us

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 5, pp. 269-270]

where mu_i ~ 2 cm^2/(V*s) is ion mobility and E_s ~ 30 kV/cm is the corona stabilization field.

If the voltage rises faster than A_u_max, the ions cannot drift fast enough to maintain the shielding charge cloud. The surface field increases beyond the streamer criterion, and the corona undergoes an abrupt transition to a streamer flash, which can then initiate a leader.

TC implications — corona shielding is always defeated:

A typical DRSSTC topload reaches V_top ~ 300 kV in ~1 us (one RF half-cycle at 200 kHz), giving a voltage rate of:

dV/dt ~ 300 kV / 1 us = 300 kV/us  >> 3.6 kV/us

This is ~80x faster than the corona shielding limit. The practical consequence is that Tesla coils cannot maintain a stable corona at the topload — every voltage cycle that exceeds inception field strength immediately produces streamers, bypassing the corona shielding phase entirely. This is consistent with the observation that TC sparks appear as bright streamer bursts from the very first cycle, not as a gradual corona-to-streamer evolution.

Comparison to lightning: In natural lightning, the field rise rate at a grounded object from an approaching leader is much slower (~kV/ms range), allowing ultracorona to persist until the leader approaches within ~200 m altitude, at which point the rate exceeds the shielding limit and a counterleader launches. TC toploads effectively start in the "counterleader launch" regime from the first RF cycle.

Design implication: Corona rings and smooth toploads on Tesla coils do not suppress sparks through corona shielding (the voltage rate is far too fast for that). They work by reducing the peak surface field through geometric smoothing, delaying the point during the voltage ramp when E_surface exceeds E_inception.

4.6 Combined Uncertainty

The total uncertainty in E_propagation from environmental factors is:

E_propagation (total uncertainty) ~ +/- 20-30%

This is comparable to the intrinsic variability due to spark geometry and channel conditions. For modeling purposes, calibrate E_propagation against actual spark lengths rather than relying on theoretical values (see Section 6).

4.7 Dynamic E_propagation at Driven Leader Tips

The cold-air E_propagation values in Sections 1-4.6 (0.4-1.0 MV/m) apply to streamer propagation into undisturbed ambient air. At the tip of an actively driven leader — the regime relevant to QCW Tesla coil operation — the local conditions are fundamentally different, and the effective propagation threshold is substantially lower. This section develops the physics of this dynamic threshold and argues that it resolves the apparent paradox of QCW spark lengths.

The QCW Voltage Puzzle

The most striking empirical fact about QCW Tesla coil operation [T2]:

QCW topload voltage: 40-70 kV     (davekni measurement, 6+ independent coils)
QCW spark length:    2+ meters     (multiple builders, see [[qcw-operation]])

Compare to burst mode [T2]:

Burst topload voltage: 200-600 kV
Burst spark length:    similar or shorter

The voltage ratio is ~10-15:1. If E_propagation were a fixed constant, a 15x lower voltage should produce dramatically shorter sparks. The naive capacitive divider calculation (Section 3.4) confirms this — 70 kV with typical TC parameters predicts stall at well under 1 meter using cold-air E_propagation.

Two factors resolve this paradox:

Field geometry: Naive E_avg = V_tip/distance vastly underestimates E_tip. FEMM-computed fields at the tip of a thin conductor are much higher than average-field estimates because the field concentrates at the sharp tip (see Section 2). This is a geometric effect, not a plasma physics effect, and it accounts for a significant portion of the discrepancy. [T0 — electrostatics]
Dynamic threshold reduction: The effective E_propagation at a driven leader tip is much lower than the cold-air value, because the gas ahead of the tip has been pre-conditioned by multiple converging physical mechanisms. [T3 — this section]

Both factors are needed. Proper field geometry alone cannot fully explain the observations, and dynamic threshold alone cannot either. The QCW spark exploits both: concentrated tip fields pushing into pre-conditioned gas with a reduced ionization threshold.

Four Mechanisms That Reduce E_propagation

A leader grows by launching streamer corona from its tip into the gas ahead (see streamers-and-leaders). In undisturbed air, these streamers require E_propagation ~ 0.5 MV/m to sustain. At a driven leader tip, four physical mechanisms converge to lower this requirement:

Mechanism 1: UV Photoionization [T1 — mechanism established; T3 — quantitative effect at TC leader tips]

The active leader tip continuously generates intense UV from the streamer corona zone. Photons with energy >12.1 eV ionize O2, creating seed electrons ahead of the advancing front.

Cold air contains essentially zero free electrons (attachment kills them in ~16 ns [T1, Becker et al. 2005])
A single electron must undergo ~18-20 doublings to reach the streamer criterion (10^8 electrons) [T1]
With UV-generated seed density of 10^7-10^8 cm^-3 [T1, simulation data], new avalanches start from a pre-existing electron cloud rather than from zero
This eliminates the statistical lag (waiting for a lucky first electron) and reduces the net multiplication needed for self-propagation
More leader current → more intense corona → more UV → denser seed cloud → lower effective field threshold [T3]

The effect is strongest within ~1-5 mm of the leader tip, limited by the UV absorption length in air at atmospheric pressure.

Mechanism 2: Thermal Pre-conditioning [T0 — Paschen scaling; T3 — application to QCW tip]

Heat conducts and convects forward from the hot leader trunk (5,000-20,000 K). The gas immediately ahead of the tip is warmer than ambient, reducing its density.

The fundamental relationship is Paschen scaling [T0]: breakdown field is proportional to gas number density N.

E_breakdown proportional to N proportional to P/(k_B * T)    (ideal gas at constant pressure)

If the gas ahead of the leader tip is heated from 300 K to T_local, the effective breakdown field drops by the ratio 300/T_local:

T_local (K)	T_local / T_ambient	E_prop reduction factor	Effective E_prop (from 0.5 MV/m)
300 (ambient)	1.0	1.0	0.50 MV/m
600	2.0	0.50	0.25 MV/m
1000	3.3	0.30	0.15 MV/m
1500	5.0	0.20	0.10 MV/m
2000	6.7	0.15	0.075 MV/m

[T0: the Paschen scaling math. T3: the actual temperature reached ahead of a QCW leader tip is unknown.]

How hot does the gas get ahead of the tip? Pure thermal diffusion over distance delta in time t:

delta ~ sqrt(alpha_thermal * t)    where alpha_thermal ~ 2*10^-5 m^2/s

Over 1 ms (the timescale for a leader step): delta ~ 0.14 mm [T0]. This is tiny — pure conduction barely reaches ahead of the tip.

But additional transport mechanisms push hot gas further forward [T3]:

Convective outflow from the expanding leader channel displaces hot gas forward
The shock wave from rapid channel heating creates a transient low-density zone ahead
Radiation from the hot channel core heats surrounding gas volumetrically

The net effect is that gas within ~1-10 mm of the leader tip is significantly above ambient temperature [T3]. Even modest heating to 600-1000 K halves or thirds the effective E_propagation.

Mechanism 3: Residual Ionization [T1 — recombination data; T3 — application to TC]

Previous streamer passages leave residual ionization in the zone ahead of the leader tip. This residual charge persists because recombination is slow relative to the propagation timescale:

tau_recomb ~ 1/(alpha_recomb * n_e) ~ 50 us    (at n_e ~ 10^13 cm^-3)

[T1, Becker et al. 2005, Ch 4]

In QCW operation, the leader tip corona is continuously refreshed. New streamers propagate into the fading remnants of previous ones, not into pristine air. The residual electron density means:

The effective seed electron density is orders of magnitude above zero [T3]
Avalanches start from a pre-ionized state, requiring less multiplication
The gas retains partial conductivity, reducing the field needed to drive current through it
Each successive streamer cycle starts from a higher baseline ionization [T3]

This mechanism is cumulative during the QCW ramp: the longer the leader has been active, the more thoroughly pre-ionized the zone ahead of its tip becomes [T3].

Mechanism 4: Gas Expansion and Density Reduction [T0 — gas dynamics; T3 — magnitude at TC tips]

When the leader channel heats, the gas expands at approximately constant pressure (the acoustic transit time across the channel, ~d/v_sound ~ 1 mm / 340 m/s ~ 3 us, is fast compared to the heating timescale). This expansion:

Reduces gas density within and near the channel [T0]
Creates an outward flow that pushes lower-density gas forward [T3]
Means the region immediately ahead of the tip is at lower N than ambient [T3]

Since E/N ~ 100 Td is the fundamental breakdown parameter [T1], lower N means breakdown occurs at lower absolute E. This is the same physics as the altitude correction (Section 4.1), but locally produced by the leader's own heating.

For a channel at 5000 K, the internal density is 300/5000 = 6% of ambient [T0]. The gas ahead of the tip won't reach 5000 K, but even partial heating produces significant density reduction (see Mechanism 2 table).

The Convergent Nature of the Effect

The four mechanisms are not independent — they reinforce each other [T3]:

Leader current → UV + heating + residual ionization + expansion
     │
     ├─→ UV creates seed electrons ahead of tip
     │
     ├─→ Heat reduces gas density ahead of tip
     │         │
     │         └─→ Lower density + seed electrons
     │               → lower field needed for avalanche
     │               → streamer propagates at lower E
     │               → leader extends further
     │
     ├─→ Residual ionization from previous streamers
     │     → next streamers start from pre-ionized gas
     │     → further reduces required field
     │
     └─→ Gas expansion reduces local N
           → E/N threshold reached at lower absolute E

Each mechanism makes the others more effective. More current produces more UV, more heating, and more residual ionization simultaneously. The net reduction in effective E_propagation is greater than any single mechanism alone would produce [T3].

The Coupled Voltage-Power Limit

This convergent dynamic has a profound consequence: voltage and power are not independent limits on spark length [T3].

In the traditional model, there are two separate constraints:

Voltage limit: E_tip must exceed E_propagation (fixed constant)
Power limit: must deliver enough energy per unit time at rate P/epsilon

The dynamic threshold framework couples these: power delivery modifies the conditions that determine the voltage threshold. Specifically:

More power through the leader → more heating, UV, ionization at the tip [T3]
This reduces the effective E_propagation [T3]
Which allows growth to continue at lower V_tip [T3]
Which means the spark can grow longer before the capacitive divider stalls it [T3]

The "voltage limit" is therefore not a fixed line that the capacitive divider marches toward. It is a moving target that retreats as power increases — but with diminishing returns.

Saturation and the Ultimate Limit

The dynamic threshold cannot reduce E_propagation to zero [T0 — ionization requires nonzero field]. Several effects create a floor:

Minimum E/N for net ionization: Even in pre-heated, pre-ionized gas, some minimum E/N is needed to drive ionization faster than attachment/recombination. In hot air (>2000 K), attachment is suppressed (see thermal-physics), but ionization still requires field-driven avalanches. [T1]
Diminishing returns on each mechanism [T3]:
- UV seed density saturates (finite photon production rate, absorption limits range)
- Thermal pre-conditioning is limited by how far ahead heat can propagate (~mm scale)
- Residual ionization decays between leader steps (tau_recomb ~ 50 us)
- Gas expansion is bounded by the pressure ratio (can't go below zero density)
The capacitive divider always wins eventually [T0]: V_tip = V_topload * C_mut/(C_mut + C_sh) decreases monotonically with spark length. Even with a very low E_propagation floor, there exists a length where E_tip drops below it.

The ultimate stall length for a QCW spark is therefore set by the intersection of two curves [T3]:

The decreasing E_tip curve (capacitive divider + field geometry, computed by FEMM)
The decreasing E_propagation_effective curve (dynamic threshold, set by delivered power)

Both curves decrease with spark length, but E_tip decreases faster (because the capacitive divider is relentless and C_sh grows linearly). Eventually E_tip drops below E_propagation_effective, and the spark stalls.

Connection to QCW Ramp Regimes

The dynamic threshold framework provides a unified explanation for the three QCW ramp regimes documented in qcw-operation [T3]:

Too short (<5 ms): Insufficient time for the thermal mechanisms to develop. The leader is young, the gas ahead is barely pre-conditioned, and E_propagation_effective is still close to the cold-air value. Growth is voltage-limited at a short length. Sparks are segmented, gnarly, high-epsilon.

Optimal (10-20 ms): The leader has time to fully develop. By 2-5 ms, the thermal ratchet has established a hot leader trunk, UV production is intense, and residual ionization ahead of the tip is high. E_propagation_effective is well below the cold-air value. The spark grows efficiently as a single channel (see branching-physics) at low epsilon. Growth continues until the capacitive divider finally overwhelms the dynamic threshold.

Too long (>25 ms): The spark has already reached the ultimate stall length — where E_tip equals E_propagation_effective even with maximal pre-conditioning. Additional power cannot reduce E_propagation further at the tip (saturation). The energy must go somewhere: it heats and thickens the leader trunk, eventually triggering lateral breakouts (see branching-physics Section 4.3). The spark gets "hot and fat but bushy" rather than longer.

Why This Doesn't Help Burst Mode

Burst pulses (70-150 us) are too short for the dynamic threshold to develop significantly [T3]:

UV is present but transient — dies with each pulse
Thermal pre-conditioning requires sustained heating (~ms) that a single 100 us pulse doesn't provide
Residual ionization from one pulse persists (~50 us tau_recomb) but decays during the inter-pulse gap (5-10 ms)
Gas expansion is localized and transient

Each burst pulse propagates streamers into approximately cold, un-conditioned air. The effective E_propagation is close to the cold-air value. This is why burst mode needs 200-600 kV to achieve similar spark lengths — it cannot exploit the dynamic threshold reduction, so it must rely on brute-force voltage [T3].

This provides another perspective on the 10-15:1 voltage ratio between burst and QCW [T2]: it is a rough measure of how much the dynamic threshold effects reduce the effective E_propagation during sustained QCW operation [T3].

Testable Predictions

The dynamic threshold framework makes specific predictions that could be tested experimentally [T4]:

Effective E_propagation at stall: At the moment a QCW spark stops growing, E_tip (measurable via FEMM + V_topload) equals E_propagation_effective. This should be much lower than 0.5 MV/m. No such measurement exists yet.
Power dependence: E_propagation_effective should decrease with increasing leader power. Two QCW sparks of the same length but different power levels should stall at different times (the higher-power one stalls later).
Frequency dependence: Higher RF frequency → more heating cycles per unit time → faster development of pre-conditioning → lower E_propagation_effective at a given time into the ramp. This is consistent with the observed 300-600 kHz threshold for QCW swords [T2], but the connection to the dynamic threshold specifically (as opposed to the thermal ratchet generally) is untested.
Temperature measurement: Spectroscopic measurement of gas temperature 1-10 mm ahead of a QCW leader tip should show significantly elevated temperature (>600 K, possibly >1000 K). No such measurement exists for TC sparks.
Ramp shape sensitivity: If the dynamic threshold is real, an accelerating voltage ramp (faster increase late in the ramp) should produce longer sparks than a linear ramp of the same peak voltage and energy, because it delivers more power at the end when E_propagation_effective is already low. This is a specific, testable prediction that distinguishes the dynamic threshold model from a fixed-threshold model.

5. Spark Growth Dynamics

5.1 The Growth Equation

Spark growth rate is determined by the power available and the energy cost per meter (see energy-and-growth for detailed treatment):

dL/dt = P_stream / epsilon     (when E_tip > E_propagation)
dL/dt ~ 0                      (when E_tip < E_propagation, stalled)

The field threshold acts as a gate: growth can only occur when sufficient field exists at the tip. The rate of growth, when it occurs, is governed by the power-to-energy ratio.

5.2 Time-Stepped Growth Simulation

For each time step dt in a growth simulation:

1. Compute V_topload(t) from the drive model (or Thevenin equivalent + loaded frequency)
2. Compute V_tip from the capacitive divider (current C_mut, C_sh, R)
3. Compute E_tip from FEMM (or approximate formula) at current length L
4. Check: E_tip >= E_propagation?
   - If yes: dL/dt = P_stream(t) / epsilon(L, t)
   - If no:  dL/dt = 0 (stalled; spark cannot advance)
5. Update: L = L + dL/dt * dt
6. Update spark model parameters (C_sh, R_opt) for new L
7. Optionally: retune to loaded pole frequency (see [[coupled-resonance]])
8. Repeat

5.3 Stall and Recovery

When E_tip drops below E_propagation, the spark stalls but does not necessarily extinguish immediately:

The channel remains hot for a thermal time constant (see thermal-physics)
If V_topload increases (e.g., during QCW ramp), E_tip may recover above threshold
The spark resumes growth from its current length, not from zero (thermal memory preserves the channel)

This stall-recovery dynamic is common in QCW operation, where the voltage ramp may briefly lag behind the increasing field threshold as the spark lengthens.

6. Calibration Procedure

6.1 Determining E_propagation from Measurements

E_propagation is best determined empirically for each coil:

Measure spark length L for a known operating condition (drive voltage, pulse width, frequency).
Run FEMM simulation with topload at V_top_peak and a spark conductor of length L.
Read E_tip from the FEMM solution at the spark tip position.
At the stall point (spark at maximum length): E_tip ≈ E_propagation.

This gives E_propagation for the specific coil, environment, and operating mode. Typical results: 0.4-1.0 MV/m, with 0.6-0.7 MV/m being common for medium DRSSTCs at sea level.

6.2 Determining kappa from FEMM

Run the FEMM simulation described in Section 2.4 for several spark lengths to establish how kappa varies with geometry. For a self-consistent model, use the same kappa profile when predicting growth.

6.3 Validation

After calibrating E_propagation and kappa:

Predict spark length for a different operating condition (different drive voltage, different pulse width)
Compare to measurement
If prediction is consistently off, adjust E_propagation

A well-calibrated model should predict spark lengths to within +/-20% across a range of operating conditions.

7. Connection to Other Topics

Key Relationships

Derives from: Gas discharge physics (Townsend/streamer breakdown theory, Paschen's law)
Interacts with: capacitive-divider (voltage division directly determines V_tip and hence E_tip)
Enables: energy-and-growth (field threshold is one of two conditions for spark growth; the other is energy/power)
Interacts with: streamers-and-leaders (streamer vs. leader propagation has different effective E_propagation)
Interacts with: thermal-physics (temperature affects local gas density and thus local breakdown field)
Measured via: femm-workflow (FEMM provides E_tip for given V_top and spark geometry)
Constrains: lumped-model and distributed-model (the field condition determines whether each segment can grow)
Explains: empirical-scaling (the sub-linear L vs. E relationship arises from capacitive voltage division reducing E_tip)

Summary of Key Results

Two thresholds: E_inception ~ 2-3 MV/m (to start), E_propagation ~ 0.4-1.0 MV/m (to sustain growth).
Tip enhancement: E_tip = kappa * E_average, with kappa ~ 2-5 for typical spark channels.
Three mechanisms reduce E_tip with length: distance, geometric dilution, capacitive voltage division.
Capacitive voltage division (V_tip/V_topload = C_mut/(C_mut + C_sh)) is the dominant effect for long sparks.
Altitude correction: E(alt) = E(sea level) * exp(-h/8500). Humidity: +/-10-20%. Temperature: +/-5-10%.
Total environmental uncertainty in E_propagation: +/-20-30%. Calibration against measurements is essential.
The growth condition (E_tip > E_propagation) acts as a gate; growth rate is set by power/energy balance.
FEMM simulation is essential for accurate E_tip determination; naive field estimates are unreliable.
[T3] E_propagation is not fixed — at a driven leader tip, UV, heat, residual ionization, and gas expansion dynamically reduce it. Voltage and power are coupled limits, not independent.
[T3] The dynamic threshold explains QCW's 10-15:1 voltage advantage over burst mode, the three QCW ramp regimes, and why burst mode can't exploit the same physics.

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