plasma-expert/context/empirical-scaling.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
empirical-scaling	Empirical Scaling Laws for Spark Length	established	spark-physics.txt: Part 5 Section 5.7 (lines 362-386), Part 6 Section 6.1 (lines 389-401)	[energy-and-growth capacitive-divider field-thresholds thermal-physics streamers-and-leaders power-optimization lumped-model equations-and-bounds open-questions]	[freau-single-shot freau-repetitive qcw-scaling voltage-limited-derivation]	[Freau_scaling bang_energy epsilon QCW burst_mode capacitive_divider E_propagation]	[length-vs-energy-scaling.png epsilon-by-mode-comparison.png]	[spark-growth-timeline.md]	[What is the exact exponent for QCW scaling across different coil designs? How does repetition rate quantitatively affect the transition from single-shot to repetitive scaling? Can the scaling exponents be derived rigorously from the growth equation and divider model? How does the scaling change at very high power levels where thermal saturation occurs? What is the effect of topload geometry on the scaling exponents?]

Empirical Scaling Laws for Spark Length

Community observations and measurements have established empirical relationships between spark length and input energy or power. These scaling laws, notably Freau's relationships, provide practical tools for predicting spark performance and connect to the theoretical framework through the capacitive-divider and energy-and-growth models. Understanding when and why different scaling laws apply is essential for coil design and performance prediction.

Freau's Empirical Relationships

The Tesla coil community, through extensive measurement and documentation by Freau and others, has observed consistent scaling relationships:

Single-Shot Burst Scaling

L proportional to sqrt(E_bang)

Or equivalently:

L = k_burst * sqrt(E_bang)

Where:

• L is the spark length [m]
• E_bang is the total energy delivered in a single burst [J]
• k_burst is a coil-specific constant [m/J^0.5]

Conditions for validity:

• Single isolated pulse (no thermal memory between events)
• Each spark starts from a cold, deionized state
• Streamer-dominated (insufficient time for leader transition)
• Voltage-limited growth (E_tip drops to E_propagation)

Typical bang energies: 1-100 J for DRSSTC bursts

Repetitive Operation Scaling

L proportional to P_avg^(0.3 to 0.5)

Where:

• L is the spark length [m]
• P_avg is the time-averaged power [W]
• The exponent varies from 0.3 to 0.5 depending on repetition rate and mode

Conditions for validity:

• Repetitive pulses with partial thermal/ionization memory between events
• Effective persistence of channels across pulse gaps (see thermal-physics)
• Higher repetition rates push the exponent toward 0.5
• Lower repetition rates (more cooling between pulses) push toward 0.3

Physical interpretation: Thermal memory from previous pulses reduces the effective epsilon for subsequent pulses, improving efficiency compared to single-shot operation. The accumulated benefit produces a power-law relationship with an exponent less than the single-shot case.

Physical Derivation of Single-Shot Scaling

The sqrt(E) scaling for single-shot burst mode can be derived from the voltage-limited growth model:

Starting Assumptions

1. Spark growth is voltage-limited (E_tip = E_propagation at stall)
2. Topload voltage is roughly constant during a burst (not ramping)
3. Spark impedance is dominated by capacitive reactance: Z_spark ~ 1/(omega*C_sh)
4. Shunt capacitance scales linearly with length: C_sh = C_sh_per_meter * L

Derivation

The electric field at the spark tip (open-circuit approximation from capacitive-divider):

E_tip ~ kappa * V_topload * C_mut / [(C_mut + C_sh_per_meter * L) * L]

At the stall length L_max, E_tip = E_propagation. For sparks where C_sh >> C_mut (long enough that the growing shunt capacitance dominates):

E_propagation ~ kappa * V_topload * C_mut / (C_sh_per_meter * L_max^2)

Therefore:

L_max^2 ~ kappa * V_topload * C_mut / (E_propagation * C_sh_per_meter)
L_max ~ sqrt(V_topload)

Now connect voltage to energy. The power delivered to the spark:

P ~ V_topload^2 / |Z_spark|
  ~ V_topload^2 * omega * C_sh_per_meter * L

For the total bang energy in burst time T_burst:

E_bang ~ P * T_burst ~ V_topload^2 * omega * C_sh_per_meter * L * T_burst

Since L ~ sqrt(V_topload) and V_topload ~ L^2:

E_bang ~ L^4 * omega * C_sh_per_meter * L * T_burst ~ L^5 * (constants)

This gives L ~ E_bang^(1/5) = E_bang^0.2, which is weaker than the observed sqrt(E) scaling. The discrepancy arises because the simplified derivation ignores:

• The time-varying nature of the spark impedance during growth
• The fact that not all delivered energy contributes to growth (see energy-and-growth)
• The interaction between R_opt_power and the changing capacitances

A more careful analysis using the growth equation dL/dt = P/epsilon with the full impedance model yields exponents closer to 0.4-0.5, consistent with observations. The key insight is that the sub-linear scaling is a robust consequence of C_sh increasing with L, regardless of the exact exponent.

Alternative Direct Argument

A simpler argument that captures the essential physics:

For a voltage-limited spark:

Need: V_topload > E_propagation * L (approximately)
Therefore: L_max ~ V_topload / E_propagation

The impedance at stall:

Z_spark ~ L / (omega * epsilon_0 * ...) ~ proportional to L

Power:

P ~ V_topload^2 / Z_spark ~ V_topload^2 / L ~ V_topload^2 / V_topload = V_topload

Energy:

E ~ P * T ~ V_topload * T
L ~ V_topload ~ E/T

And since for a single-shot burst T is roughly fixed:

L ~ E_bang^(~0.5)

This simplified argument, while not rigorous, shows why the sqrt relationship emerges. The exact exponent depends on details of the impedance model, but the sub-linear character is robust.

QCW Scaling: Better But Still Sub-Linear

QCW mode shows improved scaling compared to burst mode:

L ~ E^(0.6 to 0.8)    (QCW)

This is closer to linear than burst mode (exponent 0.5) but still sub-linear. Three factors explain the improvement:

1. Active Voltage Ramping Compensates Divider

In QCW, V_topload increases throughout the ramp:

V_topload(t) = V_max * (t / T_ramp)

As C_sh grows and attenuates V_tip, the rising V_topload partially compensates. The net attenuation of V_tip is slower than for constant-voltage operation, allowing growth to continue longer before the field threshold is reached.

2. Leader Formation Reduces epsilon

The streamer-to-leader transition (see streamers-and-leaders) reduces epsilon from 30-100 J/m (streamer) to 5-15 J/m (leader). From the growth equation:

dL/dt = P / epsilon

Lower epsilon means more length per watt. The leader-dominated QCW spark converts energy to length more efficiently.

3. Thermal Accumulation Further Reduces epsilon

During the long QCW ramp, thermal energy accumulates in the channel (see thermal-physics):

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

As the accumulated energy grows, epsilon decreases further, improving efficiency throughout the ramp. This effect is negligible for short bursts but significant over 5-20 ms ramps.

Combined Effect

All three mechanisms push the scaling exponent above 0.5 toward 0.8. The capacitive-divider still prevents linear scaling (exponent = 1.0), but QCW operates closer to the theoretical limit than burst mode.

Calibration: From Empirical to Predictive

The scaling laws provide empirical relationships, but for quantitative prediction, the underlying parameters must be calibrated per coil.

Parameters to Calibrate

1. epsilon (energy per meter): The most important parameter. Determined by:

   • Running the coil at known conditions
   • Measuring spark length L
   • Computing delivered energy E from SPICE model
   • epsilon = E / L
   • See energy-and-growth for detailed procedure

2. E_propagation (field threshold): Determines the voltage limit. Determined by:

   • Using FEMM to compute E_tip at the measured stall length and voltage
   • E_propagation = E_tip(V_top, L_stall)
   • See field-thresholds for details

3. Scaling constant k: Once epsilon and E_propagation are known, the coil-specific proportionality constant in L = k * f(E) can be calculated and used to predict performance at different operating conditions.

Calibration Procedure

1. Measure at baseline condition: Run coil, measure L_1, compute E_1. Calculate epsilon_1 = E_1/L_1.
2. Measure at different condition: Change power or ramp time. Measure L_2, compute E_2. Calculate epsilon_2 = E_2/L_2.
3. Verify consistency: epsilon_1 and epsilon_2 should agree within measurement uncertainty (+/-30-50%) for the same operating mode.
4. Predict new conditions: Use calibrated epsilon and E_propagation with the growth equation and voltage limit to predict L for untested conditions.

Expected Prediction Precision

The framework is inherently approximate due to plasma physics variability:

Parameter	Precision
Spark length prediction	+/-20-40%
Power prediction	+/-30-50%
Impedance phase prediction	+/-5-10 degrees
FEMM capacitance	+/-10%
Resistance estimation	+/-30-50%
epsilon calibration	+/-30-50%

These uncertainties are not a failure of the model. They reflect the fundamental variability of plasma discharge physics. The framework provides the correct scaling relationships and physically meaningful parameters; calibration fills in the quantitative values.

Measurement Tolerances and Error Propagation

Source Uncertainties

• Spark length: Measured from photographs with scale reference. Branching and 3D geometry introduce ~10-20% uncertainty.
• Input energy: Calculated from primary waveforms and SPICE model. Coupling uncertainty (~5%), component tolerances (~5%), and waveform measurement (~10%) combine to ~15-20%.
• FEMM capacitances: Mesh density, boundary conditions, and geometry simplifications contribute ~10% uncertainty.
• Plasma resistance: Most uncertain parameter. Physical bounds span orders of magnitude (1 kohm to 100 Mohm). Effective R depends on temperature, which depends on operating conditions.

Propagation to Predictions

When using epsilon = E/L with both E and L uncertain:

delta_epsilon / epsilon = sqrt((delta_E/E)^2 + (delta_L/L)^2)
                        ~ sqrt(0.20^2 + 0.15^2)
                        ~ 25%

When predicting length L = E / epsilon:

delta_L / L = sqrt((delta_E/E)^2 + (delta_epsilon/epsilon)^2)
            ~ sqrt(0.20^2 + 0.25^2)
            ~ 32%

This is consistent with the stated +/-20-40% length prediction precision.

Scaling Regimes Summary

Operating Mode	Scaling Law	Exponent	Physical Basis
Single-shot burst	L ~ sqrt(E_bang)	0.5	No thermal memory, voltage-limited, streamer-dominated
Repetitive burst	L ~ P_avg^(0.3-0.5)	0.3-0.5	Partial thermal memory, frequency-dependent
High duty DRSSTC	L ~ E^(0.5-0.7)	0.5-0.7	Partial leader transition, some thermal accumulation
QCW	L ~ E^(0.6-0.8)	0.6-0.8	Voltage ramping, leader-dominated, thermal accumulation
Ideal (no divider)	L = E / epsilon	1.0	Theoretical limit never achieved in practice

All real scaling exponents are less than 1.0 because the capacitive-divider always attenuates V_tip as the spark grows. The variation in exponents reflects how effectively different operating modes mitigate the divider's effect.

The image length-vs-energy-scaling.png plots these curves on a log-log scale, showing the different slopes for different modes.

Practical Application

Using Scaling Laws for Quick Estimates

Given a known coil performance at one condition:

Example: A burst-mode coil produces 0.8 m sparks at 25 J bang energy. What length is expected at 50 J?

Using L ~ sqrt(E):
L_2 / L_1 = sqrt(E_2 / E_1)
L_2 = 0.8 * sqrt(50 / 25)
L_2 = 0.8 * 1.414
L_2 = 1.13 m

Example: A QCW coil produces 1.5 m sparks at 30 J total energy. What length is expected at 60 J?

Using L ~ E^0.7 (mid-range QCW exponent):
L_2 / L_1 = (E_2 / E_1)^0.7
L_2 = 1.5 * (60 / 30)^0.7
L_2 = 1.5 * 1.625
L_2 = 2.44 m

When Scaling Laws Break Down

The empirical scaling laws fail when:

• Operating at the absolute voltage limit (no more voltage headroom)
• Transitioning between operating modes (e.g., burst length transitions overlap QCW length scales)
• Environmental conditions change significantly (altitude, humidity)
• Coil topology changes (different topload, secondary, etc.)
• Power supply limitations cap the achievable V_topload

In these cases, the full simulation workflow using energy-and-growth growth equations and femm-workflow field simulations is required.

Connection to Spark Growth Timeline

The worked example in spark-growth-timeline.md provides a concrete illustration of these scaling concepts. Key findings from that example:

• Target: 2.0 m QCW spark at 420 kV, 12 ms ramp
• Achieved: 1.0 m (voltage-limited at 50% of target)
• At stall: 200 kW power available but unused for growth
• Growth efficiency: Only 1.2% of delivered energy went to length extension
• Confirms: Sub-linear scaling and voltage limitation as dominant constraint

The example demonstrates that having sufficient power is necessary but not sufficient. The voltage limit imposed by the capacitive-divider is the binding constraint for spark length.

Key Relationships

• Derives from: capacitive-divider (sub-linear scaling is a direct consequence of voltage division)
• Derives from: energy-and-growth (growth equation dL/dt = P/epsilon provides the theoretical basis)
• Derives from: field-thresholds (voltage limit E_tip = E_propagation sets the stall condition)
• Differentiated by: streamers-and-leaders (channel type determines epsilon, affecting scaling exponent)
• Differentiated by: thermal-physics (thermal persistence determines whether inter-pulse memory improves scaling)
• Calibrated via: Experimental measurement of epsilon and E_propagation per coil
• Verified in: lumped-model and distributed-model (circuit simulations should reproduce observed scaling)
• Connected to: power-optimization (R_opt_power determines power delivery, which drives the energy term in scaling laws)