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**

**Constraints on optimization:**
- Insufficient source current/voltage (primary limited)
- Inception field not achieved (spark doesn't form)
- Physical conductivity limits (R_min, R_max)
- Thermal time constants (can't adjust faster than ~ms)

When constraints prevent reaching R_opt_power, the spark operates sub-optimally or stalls.

---

## Part 3: Impedance Measurement at Topload Port

### 3.1 Why V_top/I_base is Wrong

Measuring "impedance" as V_top/I_base is incorrect because I_base includes **all** displacement currents returning to ground:
- Every secondary section's capacitance to ground
- Strike ring coupling
- Primary-to-secondary capacitance
- **AND** the spark current

This mixes the spark load with all parasitic return paths.

### 3.2 Correct Measurement Port

**The measurement port is topload-to-ground** where the spark physically connects. All impedance and power calculations reference this port.

### 3.3 Thévenin Equivalent Extraction (Recommended)

This method separates Tesla coil characterization from load analysis.

**Step 1: Measure Z_th (output impedance with drive off)**
- Set primary drive source to AC 0V (short voltage source)
- Keep all tank components (MMC, L_primary, damping resistors) in circuit
- Apply 1V AC test source at topload-to-ground
- Measure current: I_test
- Calculate: **Z_th = 1V / I_test = R_th + jX_th**

**Step 2: Measure V_th (open-circuit voltage with drive on)**
- Remove test source
- Turn primary drive source ON at operating frequency
- Remove spark load (open-circuit topload)
- Measure: **V_th = V(topload)** (complex magnitude and phase)

**Step 3: Calculate power to any load**
For candidate load impedance Z_load:
```
P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
```

**Theoretical maximum power (sanity check):**
If conjugate match were achievable (Z_load = Z_th*):
```
P_max = 0.5 × |V_th|² / (4×Re{Z_th})
```
Actual spark power will be less than this due to topological constraints.

**Advantages:**
- Characterize coil once, evaluate many loads instantly
- No re-simulation for different spark parameters
- Separates "coil behavior" (Z_th) from "drive conditions" (V_th)

**Enhancement:** Measure Z_th(ω) and V_th(ω) over a frequency band (±10% of operating frequency) to account for frequency tracking as spark loads the system.

### 3.4 Direct Power Measurement (Alternative)

Keep full coupled model with spark load present:
- Drive primary at operating frequency and amplitude
- Run AC analysis
- Measure power in spark: P = 0.5 × Re{V(top) × conj(I(spark))}
- Step R to find maximum
- **Critical:** For each R, retune to loaded pole frequency (resonance shifts with loading)

---

## Part 4: DRSSTC Operating Modes and Pole Frequencies

### 4.1 Coupled System Poles

A Tesla coil is a coupled resonant system. Even without a spark, coupling between primary and secondary creates two resonant modes (eigenfrequencies):
- **Lower pole:** Below the geometric mean
- **Upper pole:** Above the geometric mean

The spark modifies both pole **frequency and damping**, not just frequency.

### 4.2 Frequency Shift with Loading

As spark grows:
- C_sh increases (~2 pF/foot)
- Both poles shift and become more damped
- Comparing different R values at fixed frequency measures detuning, not inherent matching quality

**Best practice:** For each R value, sweep frequency to find loaded pole (max |V_top|), then measure power at that frequency. This gives true matched performance.

---

## Part 5: Spark Growth Physics and Energy Requirements

### 5.1 Voltage Limit: Field Threshold

A spark continues to grow while the electric field at its tip exceeds a threshold.

**Field requirements (at sea level, standard conditions):**
```
E_inception ≈ 2-3 MV/m (initial breakdown from smooth topload)
E_propagation ≈ 0.4-1.0 MV/m (sustained leader growth)
E_tip = κ × E_average (tip enhancement factor κ ≈ 2-5)
```

**Maximum voltage-limited length:**
Solve: E_tip(V_top_peak, L) = E_propagation

Use FEMM to compute E_tip for given V_top and length L. As spark grows, E_tip decreases due to:
- Increased distance from topload
- Geometric field dilution
- Capacitive voltage division (see below)

**Note:** E_propagation varies with altitude and humidity by ±20-30%.

### 5.2 Power Limit: Energy per Meter

Growth consumes approximately constant energy per unit length ε [J/m]:

**Growth rate equation:**
```
dL/dt = P_stream / ε  (when E_tip > E_propagation)
dL/dt ≈ 0  (when E_tip < E_propagation, stalled)
```

**Over time T to reach length L:**
```
E_total ≈ ε × L
P_avg ≈ ε × L / T
```

### 5.3 Empirical Energy per Meter Values

Requires calibration per coil. Starting values:

**QCW-style growth:**
- ε ≈ 5-15 J/m
- Long ramp times (5-20 ms)
- Leader-dominated channels
- Energy efficiently extends length

**High duty cycle DRSSTC:**
- ε ≈ 20-40 J/m
- Hybrid streamer/leader formation
- Some thermal accumulation
- Moderate efficiency

**Hard-pulsed DRSSTC (burst mode):**
- ε ≈ 30-100+ J/m (single-shot)
- Short pulses, mostly streamers
- Much energy → brightening/branching
- Poor length efficiency

**Advanced refinement:** ε decreases during heating due to thermal accumulation:
```
ε(t) = ε₀ / (1 + α∫P_stream dt)

where α has units [1/J] and ∫P_stream dt is accumulated energy
```

### 5.4 Thermal Memory and Operating Regimes

**Pure thermal diffusion time constant:**
```
τ_thermal = d² / (4α)

where α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air

For thin streamers (d ~ 100 μm): τ ~ 0.1-0.2 ms
For thick leaders (d ~ 5 mm): τ ~ 300-600 ms
```

**Observed channel persistence is longer than pure thermal diffusion** due to:
- Buoyancy and convection maintaining hot gas column
- Ionization memory (recombination slower than thermal diffusion)
- Broadened effective channel diameter

**Effective persistence times:**
- Thin streamers: ~1-5 ms (convection/ionization dominated)
- Thick leaders: seconds (buoyancy maintains hot column)

**QCW advantage:**
- Ramps of 5-20 ms exploit ionization/convection persistence
- Channel stays hot throughout growth
- Continuous energy injection maintains E_tip
- Transitions streamers → leaders efficiently

**Burst mode characteristics:**
- Widely spaced bursts: channel cools/deionizes between pulses
- Must re-ionize repeatedly
- High peak current → bright, thick but short
- Voltage collapse limits length before leader formation

### 5.5 Streamers vs Leaders

**Streamers:**
- Thin (10-100 μm), fast (~10⁶ m/s), low current (mA)
- Photoionization propagation
- High resistance, short-lived (μs thermal time)
- Purple/blue, highly branched
- High ε (inefficient)

**Leaders:**
- Thick (mm-cm), slower (~10³ m/s), high current (A)
- Thermally ionized (5000-20000 K)
- Low resistance, persistent (seconds with convection)
- White/orange, straighter
- Low ε (efficient)

**Transition sequence:**
1. High E-field creates streamers
2. Sufficient current → Joule heating
3. Heated channel → thermal ionization → leader
4. Leader grows from base
5. Leader tip launches new streamers
6. Fed streamers convert to leader

### 5.6 The Capacitive Divider Problem

As spark grows, voltage division limits tip voltage:

```
V_tip = V_topload × Z_mut/(Z_mut + Z_sh)

where Z_mut = (1/jωC_mut) || R  (complex)
      Z_sh = 1/jωC_sh
```

**Open-circuit limit (R → ∞):**
```
V_tip ≈ V_topload × C_mut/(C_mut + C_sh)
```

**With finite R ≈ R_opt_power:** V_tip is lower and complex. Since C_sh ∝ L:
- As spark grows, C_sh increases
- V_tip decreases even if V_topload maintained
- E_tip decreases
- Growth becomes harder

This creates sub-linear scaling of length with energy.

### 5.7 Freau's Empirical Relationship

Community observations suggest:
```
Single-shot burst: L ∝ √(bang energy)
Repetitive operation: L ∝ P_avg^(0.3 to 0.5)
```

**The single-shot √E relationship** applies when there's no thermal accumulation between events - each spark starts cold.

**The repetitive power scaling** applies when thermal/ionization memory carries over between pulses.

**Physical explanation for voltage-limited burst mode:**
```
E_field ≈ V_top/L
Need: V_top > E_propagation × L
Power to maintain voltage: P ∝ V_top²/Z_spark
If Z_spark ∝ L, then: L ∝ √P
```

**QCW shows different scaling** (closer to linear, maybe L ∝ E^0.6-0.8) because:
- Active voltage ramping compensates for divider
- Leader formation more energy-efficient
- Still fights capacitive divider but with mitigation

---

## Part 6: Practical Simulation Workflow

### 6.1 Calibration Procedure

**Required measurements (one-time per coil type):**

1. **Energy per meter (ε):**
   - Run coil with known drive, measure final spark length L
   - From SPICE, compute E_delivered = ∫P_spark dt
   - Calculate: ε = E_delivered/L

2. **Field threshold (E_propagation):**
   - Use FEMM to compute E_tip for measured V_top and final L
   - E_propagation ≈ E_tip at stall point
   - Typical: 0.4-1.0 MV/m

### 6.2 Prediction Workflow

**Step 1: Voltage capability check**
- Simulate to determine V_top(t)
- Use FEMM: E_tip(V_top, L_target) ≥ E_propagation?
- If not, target length is voltage-limited

**Step 2: Power/energy requirement**
- Choose growth time T (e.g., 10 ms for QCW)
- Required: P_avg ≈ ε × L_target/T
- Required: E_total ≈ ε × L_target

**Step 3: Verify in SPICE**
- Verify delivered P_stream meets requirement
- Check coil stays near loaded pole

**Step 4: Power balance validation**
```
P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation

Check: P_spark / P_primary_input = expected efficiency
```

### 6.3 Growth Simulation (Advanced)

For each time step dt:
```
1. Check: E_tip(V_top(t), L) ≥ E_propagation?
2. If yes: dL/dt = P_stream(t)/ε(L,t)
3. If no: dL/dt = 0 (stalled)
4. Update: L = L + (dL/dt)×dt
5. Update spark model parameters for new L
6. Optionally track frequency to follow loaded pole
```

---

## Part 7: Lumped Spark Model Theory

### 7.1 Model Structure

Single lumped element:
```
        C_mut
Topload ----||---- Node_spark
                      |
                     [R]
                      |
                   [C_sh]
                      |
                     GND
```

### 7.2 FEMM Extraction

**Electrostatic simulation:**
- Topload at potential V
- Spark as cylindrical conductor
- Ground plane/boundaries
- Solve for 2×2 capacitance matrix

**Extract values from Maxwell capacitance matrix:**

The Maxwell matrix has C_ii > 0 (self-capacitance) and C_ij < 0 for i≠j (mutual capacitance, negative).

```
C_mut = -C[topload, spark] = |C_12|  (take absolute value of negative off-diagonal)
C_sh = C[spark, spark] + C[spark, topload] = C_22 - |C_12|  (total to ground)
```

**Sign convention note:** We're using the Maxwell capacitance matrix convention. If using partial capacitances, the extraction differs.

**Typical validation:** C_sh ≈ 2 pF per foot confirms model accuracy.

### 7.3 Determining R

**Default (recommended):**
```
R = R_opt_power = 1/(ω(C_mut + C_sh))
```

**Physical bounds:**
```
R_min ≈ 1 kΩ (very hot, thick leader plasma)
R_max ≈ 100 MΩ (cold, thin streamer plasma)
R_actual = clip(R_opt_power, R_min, R_max)
```

If clipping occurs, check if source can provide required power/voltage for this impedance.

### 7.4 User Measurement Integration

**Ringdown method (improved):**

For a parallel RLC equivalent at the loaded resonance ω_L:
```
Q_L = ω_L C_eq R_p = R_p/(ω_L L)

Therefore: R_p = Q_L/(ω_L C_eq)  or equivalently  R_p = Q_L ω_L L

And: G_total = 1/R_p = ω_L C_eq/Q_L  or equivalently  G_total = 1/(Q_L ω_L L)
```

**Measurement procedure:**
1. Measure unloaded: f₀, Q₀, C₀ (from geometry or separate measurement)
2. Measure with spark: f_L, Q_L
3. Calculate equivalent capacitance: C_eq = C₀(f₀/f_L)²
4. Calculate capacitance change: ΔC = C_eq - C₀
5. Calculate total conductance: G_total = ω_L C_eq/Q_L  (using either form above)
6. Calculate unloaded conductance: G_0 = ω₀ C₀/Q₀
7. Spark admittance: Y_spark ≈ (G_total - G_0) + jω_L ΔC

**Note:** This method is sensitive to primary coupling effects. The Thévenin port method (Section 3.3) is more robust.

**Direct measurement:**
- Use E-field probe for V_top (isolated, calibrated)
- Use Rogowski/CT for I_spark return current (not I_base)
- Calculate: Y = I/V, extract R from circuit model
- Low-level option: VNA with capacitive pickup (no spark) to verify Z_th

### 7.5 Limitations

**Good for:**
- Impedance matching studies
- Fast simulation
- Coil design optimization

**Cannot capture:**
- Current distribution along spark
- Tip vs. base differences
- Streamer/leader transitions
- Very long sparks (>10 feet)

---

## Part 8: nth-Order Distributed Spark Model

### 8.1 Model Structure

Divide spark into n segments (typically n=10):
```
Topload
   |
[C_01][R_1][C_1,gnd]
   |
[C_12][R_2][C_2,gnd]
   |
  ...
   |
[C_n-1,n][R_n][C_n,gnd]
```

Each segment: mutual capacitances, shunt capacitance, resistance. Optional: inductances if magnetic effects significant.

### 8.2 FEMM Extraction

**Electrostatic:**
- n cylindrical segments + topload + environment
- Solve for (n+1)×(n+1) capacitance matrix
- Includes all segment-to-segment and segment-to-environment couplings

**SPICE implementation challenge:**
Maxwell C-matrix has negative off-diagonals (C_ij < 0 for i≠j). Direct implementation as literal capacitors problematic. Solutions:
1. **Partial-capacitance matrix:** Use capacitances to ground with all others grounded (positive definite)
2. **Controlled sources:** Implement via MNA: I_i = Σ_j C_ij dV_j/dt
3. **Nearest-neighbor approximation:** Approximate with local couplings, validate against full matrix

**Passivity check:** Ensure C-matrix is symmetric positive semi-definite (SPD). If numerical noise creates slight non-passivity, add small diagonal term (+0.1 pF) or small series R for numerical stability.

### 8.3 Resistance Optimization: Iterative Power Maximization

**Initialization (tapered, recommended):**
```
position = i/(n-1)  # 0 at base, 1 at tip
R[i] = R_base + (R_tip - R_base)×position²
R_base = 10 kΩ, R_tip = 1 MΩ
```

**Iterative algorithm with damping:**
```
Iterate until convergence:
  For each segment i:
    Sweep R[i] to find value maximizing P[i]
    Apply damping: R_new[i] = α×R_optimal[i] + (1-α)×R_old[i]
      where α ≈ 0.3-0.5 for stability
    Clip to bounds: R[i] = clip(R_new[i], R_min[i], R_max[i])
  Check convergence: max relative change < 1%
  
If poles shifted >5%, re-optimize at new frequency
```

**Physical bounds (position-dependent):**
```
R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ)×position
R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ)×position
```

**Convergence behavior:**
- Well-coupled base segments: sharp power peak, fast convergence to low R
- Poorly-coupled tip segments: flat power curve, may not converge to unique value, stays at high R
- This naturally produces leader (base) + streamer (tip) distribution

**Typical total resistance validation:**

At 200 kHz for 1-3 meter sparks:
- **Streamer-dominated (burst mode):** Total R ≈ 50-300 kΩ
- **Leader-dominated (QCW):** Total R ≈ 5-50 kΩ (hot, thick channels)
- **Very low frequency (<100 kHz) or very long sparks:** Can approach 1-10 kΩ

Calculate total: R_total = Σ R[i]

Flag if significantly outside these ranges for your frequency and length.

### 8.4 Circuit-Determined Resistance (Simplified Alternative)

If plasma always adjusts to R_opt_power and C depends weakly on diameter (logarithmically):

```
For each segment:
  C_total[i] = C_shunt[i] + sum(C_mutual[i,:])
  R[i] = 1/(ω × C_total[i])
  R[i] = clip(R[i], R_min[i], R_max[i])
```

**Justification:**
- C ∝ 1/ln(h/d): weak diameter dependence
- R_opt ∝ 1/C: also weak diameter dependence
- 2× diameter → ~10-15% change in C, R
- Error acceptable given other uncertainties (FEMM ~10%, plasma variability ~50%)

**When to use:** Standard cases within typical parameter ranges.
**When to iterate:** Edge cases, validation studies, highest accuracy needs.

### 8.5 Diameter Considerations

**Circuit-first view (recommended):**
1. Use nominal diameter in FEMM (e.g., 1 mm for burst, 3 mm for QCW)
2. Calculate C matrices
3. Calculate R_opt from C
4. Plasma adjusts properties to match R_opt
5. Diameter is dependent variable

**Self-consistency check (optional):**
```
d_nominal = 1e-3 m  # 1 mm starting guess
C_mut, C_sh = FEMM(d_nominal)
R_opt = 1/(ω(C_mut + C_sh))

# Back-calculate implied diameter (typical partially ionized plasma):
ρ_typical = 10 Ω·m
L_segment = L_total/n_segments
d_implied = sqrt(4×ρ_typical×L_segment / (π×R_opt))

# If d_implied ≈ d_nominal (within factor of 2), self-consistent
# If not, iterate once with d = (d_nominal + d_implied)/2
```

Because dependence is logarithmic, typically converges in 1-2 iterations if needed.

---

## Part 9: Impedance Matching for Target Spark Length

### 9.1 QCW Matching Strategy

During QCW, spark grows from 0 to target length. Impedance changes dramatically.

**Recommendation: Match at 50-70% of target length**

**Reasoning:**
- Decent power transfer throughout ramp
- Spark grows fastest in middle phase
- Frequency tracking compensates for mismatch

**Rule of thumb: Match at 60% for first design iteration**

### 9.2 Optimization Approach

Minimize total energy over growth:
```
E_total = ∫₀ᵀ [ε × L(t)/η(t)] dt
η(t) = power transfer efficiency
```

**Procedure:**
1. Simulate growth with match points at 0%, 30%, 50%, 70%, 100%
2. Calculate E_total to reach target for each
3. Choose match point minimizing E_total

### 9.3 Burst Mode Matching

For non-ramping burst:
- Match to final spark length (100%)
- Coil rings up quickly
- Steady-state matching more important

---

## Part 10: Implementation Summary

### 10.1 Lumped Model Workflow

1. FEMM electrostatic: topload + single spark cylinder
2. Extract C_mut = |C_12|, C_sh = C_22 - |C_12| from Maxwell matrix
3. Calculate R = 1/(ω(C_mut + C_sh)), clip to bounds
4. Build SPICE: (C_mut||R) in series with C_sh at topload port
5. AC analysis: Thévenin equivalent or direct power measurement
6. Use for matching optimization and performance prediction

### 10.2 nth-Order Workflow

1. FEMM: n segments + environment → full C-matrix
2. Optional: magnetic analysis → L-matrix
3. Initialize R with tapered profile
4. Choose approach:
   - Full iterative optimization with damping (highest accuracy)
   - Simplified R = 1/(ωC_total) (good for typical cases)
5. Export to SPICE with proper C-matrix handling (partial capacitances or controlled sources)
6. AC analysis or transient simulation
7. Validate: power balance, total R in expected range, R distribution physical

### 10.3 Validation Strategy

**Tests:**
- Lumped vs. 1-segment nth-order (should match exactly)
- Convergence: n=5 vs. n=10 vs. n=20 (diminishing changes)
- Measurements: compare impedance, power, length to real coil
- Self-consistency: R distribution shows base < tip, total R reasonable

---

## Part 11: Key Equations Reference

### Circuit Analysis
```
R_opt_power = 1/(ω(C_mut + C_sh))
Example: f=200 kHz, C_total=12 pF → R_opt ≈ 66 kΩ

R_opt_phase = 1/(ω√(C_mut(C_mut + C_sh)))

φ_Z,min = -atan(2√(r(1+r))), r = C_mut/C_sh

Y = ((G+jB₁)·jB₂)/(G+j(B₁+B₂))
where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh (positive susceptances)

φ_Z = -atan(Im{Y}/Re{Y})  (impedance phase)
```

### Thévenin Equivalent
```
Z_th = 1V/I_test (drive off, test source on)
V_th = V(topload) (drive on, no spark)
P_load = 0.5×|V_th|²×Re{Z_load}/|Z_th+Z_load|²

Theoretical maximum (conjugate match):
P_max = 0.5×|V_th|²/(4×Re{Z_th})
```

### Spark Growth
```
E_inception ≈ 2-3 MV/m (initial breakdown)
E_propagation ≈ 0.4-1.0 MV/m (sustained growth)

dL/dt = P_stream/ε (when E_tip > E_propagation)

ε ≈ 5-15 J/m (QCW), 20-40 J/m (hybrid), 30-100 J/m (burst)
ε(t) = ε₀/(1 + α∫P dt), where [α] = 1/J

V_tip ≈ V_topload×C_mut/(C_mut+C_sh) (open-circuit limit)

τ_thermal = d²/(4α), α ≈ 2×10⁻⁵ m²/s for air
d=100 μm → τ~0.1 ms; d=5 mm → τ~300 ms
(Observed persistence longer due to convection/ionization)
```

### Physical Bounds
```
R_min ≈ 1-10 kΩ (hot leader plasma, position-dependent)
R_max ≈ 100 kΩ - 100 MΩ (cold streamer, position-dependent)

Typical total spark resistance at 200 kHz for 1-3 m:
- Burst/streamer: 50-300 kΩ
- QCW/leader: 5-50 kΩ
- Low frequency/very long: can approach 1-10 kΩ

Typical impedance phase: -55° to -75°
```

### Ringdown Method
```
At loaded resonance ω_L:
Q_L = ω_L C_eq R_p = R_p/(ω_L L)

R_p = Q_L/(ω_L C_eq) = Q_L ω_L L
G_total = ω_L C_eq/Q_L = 1/(Q_L ω_L L)

C_eq = C₀(f₀/f_L)²
Y_spark ≈ (G_total - G_0) + jω_L(C_eq - C_0)
```

---

## Part 12: Open Questions and Future Work

### 12.1 Remaining Uncertainties

- ε variability with current density, frequency, ambient conditions
- E_propagation dependence on geometry, humidity, altitude
- Full thermal evolution including convection and radiation
- Branching: power division among multiple channels

### 12.2 Future Enhancements

**Advanced physics:**
- Dynamic capacitance: d_eff(E) = d₀×(1 + β×ln(E/E_threshold))
- Radial temperature profiles: hot core, cool edges
- Time-dependent ε with thermal memory
- Branching models: I_branch ∝ d_branch^1.5

**Simulation improvements:**
- Full transient with L(t) evolution
- 3D FEA for complex geometries
- Monte Carlo for stochastic breakout/branching
- Strike detection: R → few ohms when contact occurs

**Validation needs:**
- Systematic measurements across coil types, frequencies, power levels
- High-speed photography for growth rate validation
- RF current distribution measurements at multiple points
- Database correlating spark parameters to operating conditions

---

## Conclusion

This framework provides practical, implementable Tesla coil spark modeling:

**Core principles:**
1. Circuit topology imposes fundamental phase constraints
2. Plasma self-optimizes within constraints (hungry streamer)
3. R_opt_power maximizes power transfer
4. Capacitances depend weakly (logarithmically) on diameter
5. Circuit determines R; plasma adjusts to match
6. Growth requires E_tip > E_propagation AND sufficient energy (ε×L)

**For basic use:** Lumped model with R = R_opt_power

**For advanced use:** nth-order distributed model with iterative (highest accuracy) or simplified (good for typical cases) R optimization

**Critical:** Calibrate ε and E_propagation from measurements, then predict new operating conditions with validated power balance.

The framework balances theoretical rigor with practical implementation, acknowledging where empirical calibration fills gaps in complex plasma physics while maintaining solid circuit-theoretical foundations.