plasma-expert/context/energy-and-growth.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
energy-and-growth	Energy Budget and Spark Growth Dynamics	established	spark-physics.txt: Part 5 Sections 5.2-5.3 (lines 236-279), Part 6 Section 6.3 (lines 428-438)	[field-thresholds thermal-physics streamers-and-leaders capacitive-divider empirical-scaling power-optimization qcw-operation branching-physics coupled-resonance lumped-model distributed-model equations-and-bounds open-questions]	[growth-rate energy-total power-average epsilon-thermal-refinement power-balance]	[epsilon dL_dt P_stream E_propagation E_tip QCW burst_mode volumetric_energy_density corona_to_spark_energy]	[energy-budget-breakdown.png epsilon-by-mode-comparison.png length-vs-energy-scaling.png]	[spark-growth-timeline.md]	[How does epsilon vary with current density and ambient humidity? Can epsilon be predicted from first principles without calibration? What is the functional form of alpha in the thermal accumulation model epsilon(t)? How does branching split power and affect effective epsilon? What determines the transition point between power-limited and voltage-limited regimes?]

Energy Budget and Spark Growth Dynamics

This topic covers the fundamental energy relationships governing Tesla coil spark growth: how much energy is consumed per unit length, how growth rate connects to delivered power, and how different operating modes produce dramatically different efficiencies. Together with field-thresholds and capacitive-divider, these energy relationships form the complete framework for predicting spark length.

The Central Concept: Energy per Meter (epsilon)

Growth of a Tesla coil spark consumes approximately constant energy per unit length, denoted epsilon [J/m]. This is the single most important parameter for predicting spark behavior. It encapsulates all the complex plasma physics -- ionization, heating, radiation, branching -- into one empirically measurable quantity.

The constancy of epsilon is an approximation. In reality, epsilon depends on channel type (streamer vs leader), thermal history, current density, and ambient conditions. However, for a given operating mode and coil, epsilon remains sufficiently constant to enable useful predictions.

Physical Origin of epsilon

Energy delivered to the spark channel is consumed by several processes:

• Ionization energy: Creating free electrons and ions in the gas (~15.6 eV per nitrogen molecule, ~12.1 eV per oxygen molecule)
• Thermal energy: Heating the channel gas from ambient to plasma temperatures (1000-20000 K)
• Radiation losses: UV, visible, and IR emission from the hot channel
• Mechanical work: Expanding the channel against atmospheric pressure (shock waves in early phases)
• Branching: Energy diverted into side branches that do not contribute to forward propagation

The sum of these per unit length of forward propagation gives epsilon. Because streamers and leaders have very different physical properties, epsilon varies strongly with channel type.

Volumetric Energy Density Threshold

The energy per meter (epsilon) can be connected to the volumetric energy density required for channel formation. From the gas discharge literature:

Minimum specific energy for spark channel formation: 0.6 - 1 J/cm^3

[Becker et al. 2005, Ch 2, p. 59]

This is the energy density that must be deposited locally in the channel volume to achieve the corona-to-spark transition (see streamers-and-leaders). The relationship between epsilon (J/m) and volumetric energy density (J/cm^3) depends on the channel cross-section:

epsilon = rho_energy * A_channel * (1 + f_overhead)

where rho_energy is the volumetric energy density (~1 J/cm^3), A_channel is the channel cross-sectional area, and f_overhead accounts for all losses (radiation, branching, heating overhead, expansion work).

For a leader channel (d = 3 mm): epsilon_min = 1 J/cm^3 * pi * (0.15 cm)^2 = 0.07 J/m For a streamer channel (d = 100 um): epsilon_min = 1 J/cm^3 * pi * (0.005 cm)^2 = 0.00008 J/m

The observed epsilon values (5-100 J/m) are 50-1000x higher than these bare minimums. The multiplier is explained by well-understood loss mechanisms:

• Initial gas heating from ambient to 5000+ K (thermal energy >> ionization energy)
• Radiation losses (UV, visible, IR)
• Branching energy diverted to side channels
• Shock wave and expansion work
• Re-ionization of decayed segments between pulses

This analysis confirms that the empirically observed epsilon values are physically reasonable, and establishes an independent lower bound. It also confirms that predicting epsilon from first principles requires modeling all the loss channels, not just the ionization energy -- which is why empirical calibration remains necessary (see open question in open-questions).

Energy Ceiling from Tip Capacitance

An independent upper bound on the energy available for channel formation comes from the electrostatic energy stored in the spark tip:

W_max = pi * epsilon_0 * U^2    [J/m]

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), pp. 703-704]

This arises because the tip (hemisphere) has capacitance C_tip = 2piepsilon_0 (independent of radius), while the channel body has capacitance per unit length C_1 = 2piepsilon_0 / ln(L/r). The tip stores ln(L/r) times more energy per unit length than the body — this concentrated energy is what heats the first segment of new channel.

Application to TC sparks:

Topload Voltage	W_max (J/m)	Heated channel radius to 5000 K
100 kV	2.8	~0.2 mm
300 kV	25	~0.6 mm
600 kV	100	~1.2 mm

At 300 kV, W_max ~ 25 J/m — remarkably close to observed epsilon for QCW leaders (5-15 J/m) and within the range for burst mode (30-100 J/m). This is not a coincidence: the tip capacitance energy sets the scale of what is energetically possible per meter of new channel.

Important caveat: W_max is the energy available from tip charge alone at the instant of new channel formation. The TC resonant circuit continuously supplies additional energy through the conducting channel during the burst. The total energy delivered over the full burst duration far exceeds W_max. However, W_max constrains the energy available for initiating each new leader step before the conducting core extends to deliver circuit current.

Independent Epsilon Check: Heating to 4000 K

An independent estimate of epsilon from thermal energy requirements:

Energy to heat 1 mm diameter air column to 4000 K: ~8 J/m

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 716; scaled from their 1 cm/800 J/m calculation]

This assumes heating the full cross-section uniformly to the minimum self-sustaining temperature (4000 K, where associative ionization N+O->NO++e provides field-free electrons). For a 3 mm leader channel, this scales to ~72 J/m.

Combined with the eta_T ~ 10% heating efficiency at ambient (see thermal-physics), the actual electrical energy needed is ~80 J/m (for 1 mm) to ~720 J/m (for 3 mm) in the initial cold phase. As the channel warms and eta_T approaches 1.0, the effective epsilon drops dramatically — exactly as observed in QCW mode where epsilon falls from ~15 J/m early to ~5-8 J/m later.

Growth Rate Equation

The fundamental growth rate equation is:

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

Where:

• dL/dt is the rate of spark length increase [m/s]
• P_stream is the real power delivered to the spark channel [W]
• epsilon is the energy per unit length [J/m]
• E_tip is the electric field at the spark tip [V/m]
• E_propagation is the threshold field for sustained growth [V/m]

This equation has two distinct regimes:

1. Active growth: When E_tip exceeds E_propagation (see field-thresholds), the spark extends at a rate proportional to delivered power and inversely proportional to epsilon. More power means faster growth. Lower epsilon means more efficient growth.

2. Stalled: When E_tip falls below E_propagation, growth stops regardless of available power. Extra power heats and brightens the existing channel but does not extend it. This is the voltage-limited regime governed by the capacitive-divider effect.

The growth rate equation is a statement of energy conservation: the energy arriving per unit time (P_stream) is consumed at epsilon joules per meter of new channel, yielding dL/dt meters per second of extension.

Integrated Energy and Power Relations

Over a total growth time T to reach final length L:

E_total = epsilon * L
P_avg = epsilon * L / T

Where:

• E_total is the total energy consumed by the spark for growth [J]
• L is the final spark length [m]
• T is the total growth time [s]
• P_avg is the time-averaged power required for growth [W]

These are lower bounds. The actual energy delivered to the coil system will be substantially larger due to secondary losses, corona, radiation, and excess heating during voltage-limited phases (see Power Balance below).

Numerical Examples

QCW coil aiming for 2 m spark:

epsilon = 10 J/m (efficient QCW)
L = 2.0 m
T = 12 ms (typical QCW ramp)

E_total = 10 * 2.0 = 20 J (minimum for growth)
P_avg = 20 / 0.012 = 1.67 kW (minimum average spark power)

Burst mode coil aiming for 1 m spark:

epsilon = 60 J/m (streamer-dominated burst)
L = 1.0 m
T = 0.2 ms (short burst)

E_total = 60 * 1.0 = 60 J (minimum for growth)
P_avg = 60 / 0.0002 = 300 kW (minimum average spark power)

The burst mode requires 3x the energy for half the length, and 180x the average power. This starkly illustrates why operating mode matters.

Empirical epsilon Values by Mode

Epsilon must be calibrated per coil from measurements. The following starting values reflect community experience:

QCW-Style Growth (epsilon = 5-15 J/m)

• Long ramp times: 5-20 ms
• Leader-dominated channels: thick, hot, low-resistance
• Energy efficiently extends length because leaders have low epsilon
• Continuous energy injection maintains channel temperature above thermal ionization threshold
• Streamer-to-leader transition occurs early in ramp (see streamers-and-leaders)
• Most efficient mode for producing long sparks per joule of input energy

Measured QCW energy budget: Loneoceans' QCW v1.5 delivers 275 J of total input energy for a 1.78 m spark, giving an apparent epsilon of 155 J/m (total input / length). At an estimated 30-50% system efficiency, the spark epsilon is 45-75 J/m. However, this includes the early inefficient growth phase (first ~2-4 ms at high epsilon) — the leader-dominated late-stage epsilon is significantly lower. [Phase 6 QCW community survey]

Measured QCW growth rate: ~170 m/s. At this rate, a 10 ms ramp produces 1.7 m, and a 20 ms ramp produces 3.4 m — matching observed QCW spark lengths. The growth rate is limited by the driven leader step time (~60 us per step, set by the conductance heating time constant tau_g = 40 us). See streamers-and-leaders for the detailed derivation.

High Duty Cycle DRSSTC (epsilon = 20-40 J/m)

• Intermediate between QCW and burst
• Hybrid streamer/leader formation: base segments become leaders, tip remains streamer
• Some thermal accumulation between closely spaced pulses
• Moderate efficiency: better than single-shot but worse than continuous ramp
• Represents many practical DRSSTC operating points

Hard-Pulsed DRSSTC / Burst Mode (epsilon = 30-100+ J/m)

• Short pulses with long gaps: channel cools between events
• Mostly streamer-dominated: thin, high-resistance, inefficient
• Much energy goes to brightening and branching rather than forward propagation
• High peak current produces visually impressive but short sparks
• Poor length efficiency: voltage collapse from capacitive-divider limits extension before leader formation can occur
• Single-shot bang energy determines length via Freau scaling (see empirical-scaling)

Measured burst ceiling: Steve Ward's DRSSTC-0.5 measurements show that spark length saturates after ~80 us of ON time, with 10-18 inch sparks at 33-180 W input and no additional length gained beyond 80 us regardless of power. This is consistent with the thermal time constant for 100 um streamers (~125 us). See thermal-physics for analysis. [Phase 6 QCW community survey]

The image epsilon-by-mode-comparison.png provides a bar chart comparison of these ranges.

Advanced: Thermal Accumulation Refinement

During sustained operation, the channel accumulates thermal energy, making subsequent extension easier. This is modeled as:

epsilon(t) = epsilon_0 / (1 + alpha * integral(P_stream dt))

Where:

• epsilon_0 is the initial (cold-channel) energy per meter [J/m]
• alpha has units [1/J] and represents the thermal benefit rate
• integral(P_stream dt) is the accumulated energy delivered to the spark [J]

Physical Interpretation

As energy accumulates in the channel:

• Temperature rises, reducing the energy needed for further ionization
• Hot channel gas already partially ionized requires less incremental energy
• Thermal inertia maintains conductivity, reducing re-ionization overhead
• Net effect: epsilon decreases with time/energy, favoring longer growth

This refinement is most important for QCW mode where the ramp is long enough for significant thermal accumulation. For single-shot bursts, the integral is small and epsilon remains approximately epsilon_0.

Practical Considerations

The alpha parameter is difficult to measure directly. It can be inferred by:

1. Running QCW at different ramp durations to the same target length
2. Measuring actual spark power (from SPICE model calibrated to measurements)
3. Fitting the epsilon(t) model to observed growth trajectories
4. Typical alpha values: 0.01-0.1 per joule (highly dependent on conditions)

For most practical calculations, using the mode-appropriate constant epsilon is sufficient. The refinement matters primarily for detailed growth simulation and for understanding why QCW efficiency improves during the ramp.

Calibration Procedure

Epsilon and E_propagation must be calibrated per coil from actual measurements. This is fundamental -- the framework provides the structure for prediction, but the parameters come from experiment.

Step 1: Measure Energy per Meter (epsilon)

1. Run the coil with known drive parameters (voltage, frequency, pulse width)
2. Measure spark length L (photograph with scale reference, take multiple measurements)
3. Compute delivered energy from SPICE simulation calibrated to measured primary waveforms:
   • E_delivered = integral(P_spark dt) over the growth period
   • P_spark is the real power delivered to the spark load in the circuit model
4. Calculate epsilon: epsilon = E_delivered / L

Step 2: Measure Field Threshold (E_propagation)

1. Use FEMM to compute E_tip for the measured V_top and final stall length L
2. E_propagation is approximately E_tip at the stall point
3. Typical result: 0.4-1.0 MV/m at sea level, standard conditions
4. Verify: E_propagation should be consistent across different operating conditions for the same coil geometry

Measurement Tolerances

• FEMM capacitance extraction: +/-10%
• Resistance estimation: +/-30-50%
• Epsilon: +/-30-50% (largest uncertainty source)
• Expected prediction precision: length +/-20-40%, power +/-30-50%, phase +/-5-10 degrees

These tolerances are inherent to the empirical nature of epsilon. Plasma physics variability, environmental conditions, and measurement uncertainty all contribute. The framework is designed to be useful within these ranges, not to provide precision beyond them.

Growth Simulation Algorithm

For detailed time-domain simulation, the growth is computed step-by-step:

For each time step dt:
  1. Check: E_tip(V_top(t), L) >= E_propagation?
  2. If yes: dL/dt = P_stream(t) / epsilon(L, t)
  3. If no:  dL/dt = 0 (stalled)
  4. Update: L = L + (dL/dt) * dt
  5. Update spark model parameters for new L:
     - C_sh(L) = C_sh_per_meter * L
     - R_opt(L) = 1 / (omega * (C_mut + C_sh(L)))
     - Recompute Z_spark, I_spark, P_spark
  6. Optionally track frequency to follow loaded pole

Implementation Notes

• Time step selection: dt should be small enough that dL changes by less than ~10% per step. For QCW ramps, dt = 0.1-0.5 ms is typical. For burst mode, dt = 1-10 microseconds.

• Frequency tracking (Step 6): As the spark grows, the loaded pole frequency shifts (see coupled-resonance). If the drive does not track this shift, the coil detunes and power delivery drops dramatically. Proper simulation should either assume ideal tracking (drive always at loaded pole) or model the actual frequency control loop.

• Parameter coupling: Steps 4-5 create a coupled system. As L increases, C_sh increases, which changes R_opt, which changes Z_spark, which changes P_spark, which changes dL/dt. This coupling is what makes growth nonlinear and why simple linear extrapolation fails.

• The worked example in spark-growth-timeline.md demonstrates this algorithm for a QCW coil and reveals a critical finding: the coil reaches only 1.0 m of a 2.0 m target because the voltage limit (not the power limit) constrains growth.

Prediction Workflow

The complete prediction workflow uses calibrated epsilon and E_propagation:

Step 1: Voltage Capability Check

• Simulate to determine V_top(t) (from SPICE or analytical model)
• Use FEMM: Is E_tip(V_top, L_target) >= E_propagation?
• If not, the target length is voltage-limited, not power-limited
• Reduce L_target or increase V_top

Step 2: Power/Energy Requirement

• Choose growth time T (e.g., 12 ms for QCW, 0.2 ms for burst)
• Required average spark power: P_avg = epsilon * L_target / T
• Required total energy: E_total = epsilon * L_target
• Is the coil capable of delivering this power?

Step 3: SPICE Verification

• Build circuit model with spark load at target length
• Run AC or transient analysis
• Verify P_spark meets requirement from Step 2
• Check coil stays near loaded pole frequency

Step 4: Power Balance Validation

P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation

• P_spark / P_primary_input should equal expected efficiency (15-50% typical)
• If efficiency is unreasonably low or high, review model parameters
• Secondary losses: 10-30% of input power
• Corona and radiation: 5-15% of input power

The image energy-budget-breakdown.png shows a typical energy distribution pie chart.

Scaling Behavior

The growth equation dL/dt = P/epsilon, combined with the capacitive-divider effect on V_tip and E_tip, produces characteristic scaling relationships. These are explored in detail in empirical-scaling, but the key insight is:

• Power-limited regime: When E_tip is well above threshold, growth rate is proportional to P/epsilon. More power = proportionally more length per unit time.

• Voltage-limited regime: When E_tip approaches threshold, additional power cannot extend the spark. Length saturates regardless of energy input. The excess power goes into heating and brightening.

• Transition: Most real sparks operate in a crossover regime, initially power-limited (rapid growth) then transitioning to voltage-limited (stalling). The length vs energy curve is sub-linear: L proportional to E^0.5 for burst mode, L proportional to E^0.6-0.8 for QCW.

The image length-vs-energy-scaling.png shows these curves on a log-log plot for different operating modes.

Key Relationships

• Derives from: circuit-topology (provides the impedance framework that determines P_stream)
• Derives from: power-optimization (R_opt_power determines the maximum extractable power)
• Interacts with: field-thresholds (E_tip threshold determines the growth/stall boundary)
• Interacts with: capacitive-divider (voltage division limits E_tip as spark grows, creating sub-linear scaling)
• Interacts with: thermal-physics (thermal memory affects epsilon through the accumulation term)
• Interacts with: streamers-and-leaders (channel type determines epsilon magnitude)
• Enables: empirical-scaling (growth equation is the foundation for Freau's scaling laws)
• Enables: lumped-model (growth simulation uses lumped model for impedance at each time step)
• Enables: distributed-model (advanced growth simulation uses distributed model for accuracy)
• Calibrated via: femm-workflow (FEMM provides capacitances needed for power calculation)