plasma-expert/spark-physics.txt

# 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**

**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)
- 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.

**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 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

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%. 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]. 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:**
```
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: 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)

**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 — 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)
- 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 (30-100+ J/m, inefficient)

**Leaders:**
- Thick (mm-cm), slower (~10³ m/s), high current (A)
- 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 (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

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.

### 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

### 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)
  Positive streamer critical field: E_cr(+) ≈ 4.5-5 kV/cm [Bazelyan & Raizer 2000]

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)

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
```
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°
```

### 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:
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 (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

**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

**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

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.