---
id: equations-and-bounds
title: "Equations and Physical Bounds Reference"
status: established
source_sections: "spark-physics.txt: Part 11 (lines 736-803), Part 1-10 (all equations), Part 12 (validation)"
related_topics: [circuit-topology, power-optimization, thevenin-method, coupled-resonance, field-thresholds, energy-and-growth, thermal-physics, streamers-and-leaders, capacitive-divider, empirical-scaling, lumped-model, distributed-model, femm-workflow, open-questions]
key_equations:
  - "All equations listed in this document"
  - "Townsend ionization coefficient"
  - "Recombination rate coefficients"
  - "Plasma conductivity from n_e"
  - "Power to sustain plasma"
key_terms:
  - "validation"
  - "physical bounds"
  - "red flags"
  - "measurement tolerance"
  - "sanity check"
images:
  - position-dependent-bounds.png
  - validation-total-resistance.png
  - power-vs-resistance-curves.png
  - phase-constraint-graph.png
  - epsilon-by-mode-comparison.png
  - thermal-diffusion-vs-diameter.png
  - length-vs-energy-scaling.png
examples:
  - calculating-ropt.md
  - thevenin-extraction.md
  - spark-growth-timeline.md
  - femm-lumped-extraction.md
  - distributed-model-complete.md
open_questions:
  - "Should bounds be tightened as more empirical data becomes available, or kept conservative to avoid rejecting valid edge cases?"
  - "Can statistical distributions (rather than hard bounds) be assigned to each parameter for probabilistic modeling?"
---

# Equations and Physical Bounds Reference

This document is the combined quick-reference for all formulas, validation ranges, and sanity checks used throughout the Tesla coil spark modeling framework. It is organized by category and cross-references the topic documents where each equation is derived and explained. Every calculated result in the framework should be checked against the bounds listed here.

**Convention:** All phasor quantities use peak values (not RMS). Power formulas include the factor of 0.5: P = 0.5 * Re{V * I*}.

---

## 1. Circuit Analysis (from [[circuit-topology]])

### 1.1 Input Admittance at Topload

At angular frequency omega = 2*pi*f, with G = 1/R, B_1 = omega*C_mut (positive susceptance), B_2 = omega*C_sh (positive susceptance):

```
Y = ((G + jB_1) * jB_2) / (G + j(B_1 + B_2))
```

**Real part (conductance component):**
```
Re{Y} = G * B_2^2 / (G^2 + (B_1 + B_2)^2)
```

**Imaginary part (susceptance component):**
```
Im{Y} = B_2 * [G^2 + B_1*(B_1 + B_2)] / (G^2 + (B_1 + B_2)^2)
```

### 1.2 Phase Angles

**Admittance phase:**
```
theta_Y = atan(Im{Y} / Re{Y})
```

**Impedance phase (what is typically measured):**
```
phi_Z = -theta_Y = atan(-Im{Y} / Re{Y})
```

Sign convention: phi_Z is between -90 degrees (purely capacitive) and 0 degrees (purely resistive).

### 1.3 Fundamental Phase Constraint

```
phi_Z_min = -atan(2 * sqrt(r * (1 + r)))

where r = C_mut / C_sh
```

**Critical threshold:** When r >= 0.207, achieving phi_Z = -45 degrees is mathematically impossible regardless of R value. This is a topological constraint of the circuit.

**Typical values:**
| Configuration | r = C_mut/C_sh | phi_Z_min |
|---------------|---------------|-----------|
| Large topload, short spark | 1.0 - 2.0 | -55 to -70 deg |
| Medium topload, medium spark | 0.5 - 1.0 | -50 to -55 deg |
| Small topload, long spark | 0.2 - 0.5 | -45 to -50 deg |

---

## 2. Power Optimization (from [[power-optimization]])

### 2.1 Optimal Resistance for Maximum Power

```
R_opt_power = 1 / (omega * (C_mut + C_sh))
```

**Numeric example:** At f = 200 kHz with C_mut + C_sh = 12 pF:
```
R_opt_power = 1 / (2*pi * 200e3 * 12e-12) = 66.3 kilohm
```

### 2.2 Optimal Resistance for Minimum Phase

```
R_opt_phase = 1 / (omega * sqrt(C_mut * (C_mut + C_sh)))
```

**Key relationship:**
```
R_opt_power < R_opt_phase    (always)
```

R_opt_power typically produces phase angles of -55 to -75 degrees, not -45 degrees.

### 2.3 Distributed Model: Per-Segment Resistance

**Simplified (circuit-determined):**
```
C_total[i] = C_shunt[i] + sum_j(|C_mutual[i,j]|)
R[i] = 1 / (omega * C_total[i])
R[i] = clip(R[i], R_min[i], R_max[i])
```

**Tapered initialization:**
```
position = (i - 1) / (n - 1)          (0 at base, 1 at tip)
R[i] = R_base + (R_tip - R_base) * position^2
R_base = 10 kilohm, R_tip = 1 megohm
```

**Damped iterative update:**
```
R_new[i] = alpha * R_optimal[i] + (1 - alpha) * R_old[i]
alpha = 0.3 to 0.5
Clip to bounds after each update
Converge when max relative change < 1%
```

---

## 3. Thevenin Equivalent (from [[thevenin-method]])

### 3.1 Extraction

```
Z_th = 1V / I_test       (drive off, AC test source at topload-to-ground)
V_th = V(topload)         (drive on, spark load removed, open-circuit voltage)
```

### 3.2 Power to Any Load

```
P_load = 0.5 * |V_th|^2 * Re{Z_load} / |Z_th + Z_load|^2
```

### 3.3 Theoretical Maximum Power

If conjugate match were achievable (Z_load = Z_th*):
```
P_max = 0.5 * |V_th|^2 / (4 * Re{Z_th})
```

Actual spark power is always less than P_max due to the topological phase constraint.

---

## 4. Spark Growth (from [[energy-and-growth]], [[field-thresholds]])

### 4.1 Growth Rate

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

### 4.2 Total Energy and Average Power

```
E_total = epsilon * L            (total energy to reach length L)
P_avg = epsilon * L / T          (average power over growth time T)
```

### 4.3 Time-Dependent Epsilon

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

where [alpha] = 1/J (units of inverse joules)
integral(P_stream dt) is the accumulated energy deposited in the channel
```

### 4.4 Field Thresholds

```
E_inception = 2 - 3 MV/m         (initial breakdown from smooth topload)
E_propagation = 0.4 - 1.0 MV/m   (sustained leader growth in air at sea level)
E_tip = kappa * E_average         (tip enhancement factor kappa = 2 to 5)
```

---

## 5. Thermal Physics (from [[thermal-physics]])

### 5.1 Pure Thermal Diffusion Time Constant

```
tau_thermal = d^2 / (4 * alpha)

alpha = k / (rho_air * c_p) approximately 2e-5 m^2/s for air
```

**Computed values:**
| Channel diameter | tau_thermal (diffusion only) | Effective persistence (with convection/ionization) |
|-----------------|------------------------------|--------------------------------------------------|
| 100 micrometers (thin streamer) | 0.1 - 0.2 ms | 1 - 5 ms |
| 1 mm (thick streamer) | 12.5 ms | 10 - 50 ms |
| 5 mm (leader) | 300 - 600 ms | seconds |

Observed persistence is longer than pure diffusion due to buoyancy-driven convection maintaining the hot gas column and ionization memory (recombination slower than thermal diffusion).

---

## 6. Capacitive Divider (from [[capacitive-divider]])

### 6.1 Tip Voltage

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

where Z_mut = (1/(jomega*C_mut)) || R    (complex impedance)
      Z_sh = 1/(jomega*C_sh)
```

### 6.2 Open-Circuit Limit (R -> infinity)

```
V_tip = V_topload * C_mut / (C_mut + C_sh)
```

Since C_sh increases with spark length (C_sh proportional to L), V_tip decreases as the spark grows even if V_topload is maintained. This creates sub-linear scaling of length with energy.

---

## 7. Ringdown Measurement (from [[lumped-model]])

### 7.1 Parallel RLC Equivalence

At the loaded resonant frequency omega_L:
```
Q_L = omega_L * C_eq * R_p = R_p / (omega_L * L)

R_p = Q_L / (omega_L * C_eq)
R_p = Q_L * omega_L * L

G_total = 1/R_p = omega_L * C_eq / Q_L
G_total = 1 / (Q_L * omega_L * L)
```

### 7.2 Measurement Extraction

```
C_eq = C_0 * (f_0 / f_L)^2           (frequency shift gives capacitance change)
delta_C = C_eq - C_0                   (added capacitance from spark)
G_total = omega_L * C_eq / Q_L        (total conductance at loaded frequency)
G_0 = omega_0 * C_0 / Q_0             (unloaded conductance)
Y_spark = (G_total - G_0) + j*omega_L * delta_C
```

---

## 8. FEMM Extraction (from [[femm-workflow]])

### 8.1 Lumped Model

```
C_mut = |C_12|                         (absolute value of negative off-diagonal)
C_sh = C_22 - |C_12|                   (diagonal minus absolute off-diagonal)
C_total = C_mut + C_sh = C_22          (identity for 2-conductor system)
```

### 8.2 Distributed Model

For each segment i in an (n+1) x (n+1) Maxwell matrix:
```
C_total[i] = sum_{j != i} |C[i,j]|    (sum of absolute off-diagonals in row i)
```

Partial capacitance transformation:
```
C_branch[i,j] = |C_maxwell[i,j]|      (positive capacitor between nodes i and j)
C_ground[i] = C_maxwell[i,i] - sum_{j != i} |C_maxwell[i,j]|   (to ground)
```

---

## 9. Empirical Scaling Laws (from [[empirical-scaling]])

### 9.1 Freau's Relationships

```
Single-shot burst:      L proportional to sqrt(E_bang)
Repetitive operation:   L proportional to P_avg^(0.3 to 0.5)
QCW ramp:              L proportional to E^(0.6 to 0.8)
```

### 9.2 Physical Explanation

- Single-shot sqrt(E) arises from voltage-limited growth: E_field approximately V_top/L, P proportional to V^2/Z, Z proportional to L, so L proportional to sqrt(P).
- QCW shows closer-to-linear scaling because active voltage ramping compensates for the capacitive divider and leader formation is more energy-efficient.
- Repetitive power scaling (exponent 0.3-0.5) applies when thermal/ionization memory carries over between pulses.

---

## 10. Physical Bounds

### 10.1 Resistance Bounds

**Lumped model:**
```
R_min = 1 kilohm        (very hot, thick leader plasma)
R_max = 100 megohm      (cold, thin streamer plasma)
```

**Distributed model (position-dependent):**
```
position = (i - 1) / (n - 1)       (0 at base, 1 at tip)

R_min[i] = 1 kilohm + 9 kilohm * position
R_max[i] = 100 kilohm + 99.9 megohm * position^2
```

**Total resistance by operating mode (at 200 kHz, 1-3 m sparks):**
| Mode | Total R range |
|------|--------------|
| Burst/streamer-dominated | 50 - 300 kilohm |
| QCW/leader-dominated | 5 - 50 kilohm |
| Very low frequency (<100 kHz) or very long sparks | 1 - 10 kilohm |

### 10.2 Capacitance Bounds

**Shunt capacitance:**
```
C_sh approximately 2 pF/foot (+/- factor of 2-3)
C_sh approximately 6.6 pF/meter (same rule, converted)
```

Factor of 2-3 variation is normal due to topload shielding, ground distance, and channel geometry.

**Mutual capacitance:**
```
C_mut = 3 - 15 pF    (for 1-5 foot sparks with typical toroidal toploads)
```

C_mut depends primarily on topload geometry and is relatively insensitive to spark length.

### 10.3 Field Thresholds

```
E_inception = 2 - 3 MV/m                (initial breakdown in air at sea level)
E_propagation = 0.4 - 1.0 MV/m          (sustained growth, sea level)
Tip enhancement factor kappa = 2 - 5     (depends on tip geometry)
```

E_propagation varies +/-20-30% with altitude and humidity.

### 10.4 Energy per Meter (epsilon)

| Operating mode | epsilon range (J/m) | Channel type |
|----------------|-------------------|--------------|
| QCW-style growth | 5 - 15 | Leader-dominated, long ramps (5-20 ms) |
| High duty cycle DRSSTC | 20 - 40 | Hybrid streamer/leader |
| Hard-pulsed burst mode | 30 - 100+ | Streamer-dominated, single-shot |

### 10.5 Impedance Phase

```
phi_Z at R_opt_power: typically -55 to -75 degrees
phi_Z_min (absolute limit): depends on r = C_mut/C_sh
phi_Z = -45 degrees: impossible when r >= 0.207
```

### 10.6 Frequency Parameters

```
Operating frequency range: 50 - 400 kHz (typical Tesla coil range)
Frequency shift with spark loading: 5 - 20% (both poles shift and damp)
```

### 10.7 Power Parameters

```
Primary input power: 0.5 - 15 kW (typical amateur DRSSTC range)
Spark power efficiency: 15 - 50% of primary input
```

Power balance: P_primary = P_spark + P_secondary_losses + P_corona + P_radiation

### 10.8 Thermal Parameters

```
Thermal diffusivity of air: alpha approximately 2e-5 m^2/s
Streamer temperature: 300 - 1000 K (weakly ionized)
Leader temperature: 5,000 - 20,000 K (thermally ionized)
Streamer diameter: 10 - 100 micrometers
Leader diameter: 1 mm - 1 cm
Streamer lifetime (thermal): microseconds
Leader persistence (effective): seconds (with convection)
```

### 10.9 Plasma Parameters

```
Plasma conductivity: sigma = 0.01 - 10 S/m
Plasma resistivity: rho = 0.1 - 100 ohm*m
```

These span the range from cold, weakly-ionized streamer plasma to hot, fully-ionized leader plasma.

---

## 11. Validation Red Flags

These are conditions that should trigger a review of calculations, model setup, or input data. A red flag does not necessarily indicate an error, but it means the result is outside the expected range and deserves investigation.

### 11.1 Capacitance Red Flags

| Check | Expected range | Red flag |
|-------|---------------|----------|
| C_sh / L_spark | 1 - 3 pF/foot | Outside 0.5 - 6 pF/foot |
| C_mut | 3 - 15 pF (1-5 ft sparks) | Below 1 pF or above 30 pF |
| C_mut / C_sh ratio | 0.2 - 2.0 | Below 0.05 or above 5.0 |
| C_total | C_22 from matrix | C_total != C_22 (for lumped 2x2) |
| Matrix symmetry | C_ij = C_ji | Difference > 0.1% |
| Off-diagonal signs | All negative | Any positive off-diagonal |

### 11.2 Resistance Red Flags

| Check | Expected range | Red flag |
|-------|---------------|----------|
| R_opt_power (lumped) | 10 - 500 kilohm | Below 1 kilohm or above 10 megohm |
| R_total (distributed, burst) | 50 - 300 kilohm | Below 10 kilohm or above 1 megohm |
| R_total (distributed, QCW) | 5 - 50 kilohm | Below 1 kilohm or above 200 kilohm |
| R distribution monotonicity | Increasing base to tip | Decreasing or non-monotonic |
| Adjacent segment R ratio | < 3:1 | Ratio exceeding 5:1 |

### 11.3 Phase Red Flags

| Check | Expected | Red flag |
|-------|----------|----------|
| phi_Z at R_opt_power | -55 to -75 deg | More negative than -85 deg or less negative than -40 deg |
| phi_Z compared to phi_Z_min | phi_Z >= phi_Z_min | phi_Z < phi_Z_min (physically impossible) |

### 11.4 Efficiency and Power Red Flags

| Check | Expected | Red flag |
|-------|----------|----------|
| Spark efficiency (P_spark/P_primary) | 15 - 50% | Below 5% or above 80% |
| Power balance | Sum of losses = P_primary | Imbalance > 20% |

### 11.5 Growth and Energy Red Flags

| Check | Expected | Red flag |
|-------|----------|----------|
| epsilon (QCW) | 5 - 15 J/m | Below 2 J/m or above 30 J/m |
| epsilon (burst) | 30 - 100 J/m | Below 10 J/m or above 200 J/m |
| Growth rate (QCW) | 50 - 500 m/s | Below 10 m/s or above 2000 m/s |
| E_tip at stall | 0.4 - 1.0 MV/m | Below 0.1 MV/m or above 3 MV/m |

### 11.6 Distributed Model Red Flags

| Check | Expected | Red flag |
|-------|----------|----------|
| Tip/base current ratio | 0.3 - 0.5 (for 10 segments) | Below 0.1 or above 0.8 |
| Peak power location | Segments 1-4 (base/middle) | Peak at tip segment |
| Convergence iterations | 2 - 5 | More than 15 |
| Simplified vs. iterative R agreement | Within 5% | Difference > 20% |

---

## 12. Measurement Tolerances

These tolerances reflect the combined uncertainty of the measurement method and the physical variability of the quantity being measured. They are important for setting realistic expectations when comparing model predictions to experimental observations.

| Quantity | Method | Typical tolerance |
|----------|--------|------------------|
| C_mut, C_sh | FEMM extraction | +/- 10% (well-meshed), +/- 5% (carefully refined) |
| R (spark resistance) | Ringdown or probe | +/- 30 - 50% (high variability in plasma conditions) |
| E_propagation | FEMM + stall observation | +/- 20 - 30% (environmental dependence) |
| epsilon (energy per meter) | Energy/length from multiple shots | +/- 30 - 50% (mode-dependent variability) |
| Spark length | Visual measurement | +/- 20 - 40% (branching, curvature, definition ambiguity) |
| V_top | Calibrated probe | +/- 10 - 20% (probe coupling uncertainty) |
| f_0, f_L | Frequency counter | +/- 0.1% (excellent precision) |
| Q factor | Ringdown decay | +/- 10 - 20% (depends on signal quality) |

**Guidance on combining uncertainties:** When multiple uncertain quantities multiply (e.g., R = 1/(omega*C)), add fractional uncertainties in quadrature. For example, if C has +/-10% uncertainty and omega has +/-0.1% uncertainty, R has approximately +/-10% uncertainty.

When comparing model to experiment, agreement within the combined measurement tolerance should be considered excellent. Agreement within twice the tolerance is acceptable. Disagreement beyond three times the tolerance indicates a systematic error or modeling deficiency.

---

## 13. Connection to Other Topics

### Key Relationships

This document serves as the combined reference for all other topics in the knowledge graph:

- **[[circuit-topology]]**: Sections 1 (admittance, phase constraint)
- **[[power-optimization]]**: Section 2 (R_opt_power, R_opt_phase, distributed R)
- **[[thevenin-method]]**: Section 3 (Z_th, V_th, power formulas)
- **[[coupled-resonance]]**: Section 10.6 (frequency parameters)
- **[[field-thresholds]]**: Sections 4.4, 10.3 (E_inception, E_propagation, kappa)
- **[[energy-and-growth]]**: Sections 4, 9 (epsilon, growth rate, scaling laws)
- **[[thermal-physics]]**: Sections 5, 10.8 (tau, temperatures, diameters)
- **[[streamers-and-leaders]]**: Sections 10.8, 10.9 (plasma parameters)
- **[[capacitive-divider]]**: Section 6 (V_tip, voltage division)
- **[[empirical-scaling]]**: Section 9 (Freau's relationships)
- **[[lumped-model]]**: Sections 7, 8.1 (ringdown, FEMM extraction)
- **[[distributed-model]]**: Sections 2.3, 8.2, 10.1 (per-segment R, bounds)
- **[[femm-workflow]]**: Section 8 (matrix extraction formulas)
- **[[open-questions]]**: Section 12 (measurement tolerances define what is and is not resolvable)
- **[Becker et al. 2005]**: Section 14 (plasma physics constants, breakdown, recombination, conductivity)
- **[Liu 2017]**: Sections 14.9, 14.10 (leader inception, Gallimberti critique, ionization threshold)
- **[Yang et al. 2022]**: Section 14.8 (Mayr/Cassie arc model parameters)
- **[da Silva et al. 2019]**: Sections 14.11-14.13 (nonlinear resistance power law, heating efficiency, channel expansion)
- **[Bazelyan & Raizer 2000]**: Sections 14.14-14.23 (V-I characteristic, leader velocity, energy ceiling, temperature thresholds, physical constants, conductance relaxation, transmission line parameters, equilibrium composition, streamer parameters, breakdown voltage)
- **[Phase 6 QCW Survey 2026]**: Section 14.24 (QCW operating parameters, coupling data, growth rate, frequency threshold, timing comparisons)

---

## 14. Plasma Physics Constants

Fundamental plasma physics parameters relevant to Tesla coil spark modeling, sourced from K.H. Becker, U. Kogelschatz, K.H. Schoenbach, R.J. Barker, "Non-Equilibrium Air Plasmas at Atmospheric Pressure," IOP Publishing, 2005.

These values provide physical grounding for the empirical parameters (epsilon, E_propagation, R bounds) used in the circuit framework and enable first-principles plausibility checks.

### 14.1 Breakdown Parameters

| Quantity | Value | Reference |
|----------|-------|-----------|
| Breakdown E/N threshold | 100 Td (~25 kV/cm at 1 atm) | Ch 2, p. 26 |
| Mean electron energy at breakdown | ~3 eV (~35,000 K) | Ch 2, p. 26 |
| Ionization-attachment crossover | E/p ~ 25 kV/cm/bar (E/N ~ 100 Td) | Ch 2, p. 33 |
| Electron lifetime in cold air (STP) | 16 ns (three-body attachment to O2) | Ch 1, p. 7 |
| Paschen minimum for air | V_min = 230-370 V, (pd)_min ~ 0.6 torr*cm | Ch 2, p. 33 |
| Streamer criterion (Meek) | N_cr ~ 10^8 electrons (alpha*d ~ 18-20) | Ch 2, p. 35 |
| Runaway electron threshold | ~3x stationary breakdown field | Ch 2, p. 39 |

### 14.2 Ionization Coefficients

Townsend ionization coefficient for dry air:

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

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

[Becker et al. 2005, Ch 2, p. 32; after Morrow & Lowke 1997]

More sophisticated analytical approximations covering wider E/N ranges: see Morrow & Lowke (1997) or Chen & Davidson (2003).

### 14.3 Electron Density by Discharge Type

| Discharge Type | n_e (cm^-3) | Reference |
|----------------|-------------|-----------|
| Streamer outer boundary | ~10^11 | Ch 2, p. 37 |
| Streamer body (inner core) | >10^13 | Ch 2, p. 37 |
| Microdischarge filament | 10^14 - 10^15 | Ch 6, Table 6.2.1 |
| Pulsed breakdown channel | ~10^16 | Ch 2, p. 38 |
| Equilibrium air at 2900 K | 4 * 10^10 | Ch 5, p. 229 |
| Non-equilibrium DC discharge (700-2000 K) | >10^12 | Ch 2, p. 23 |

### 14.4 Recombination Rate Coefficients

| Reaction | Rate Coefficient (cm^3/s) | Reference |
|----------|--------------------------|-----------|
| O2+ + e- | 1.9 * 10^-7 * (300/T_e)^0.5 | Ch 4, p. 170 |
| N2+ + e- | 1.8 * 10^-7 * (300/T_e)^0.39 | Ch 4, p. 174 |
| NO+ + e- | 4.3 * 10^-7 * (300/T_e)^0.37 | Ch 4, p. 172 |
| H3O+ + e- | 6.3 * 10^-7 * (300/T_e)^0.5 (T_e < 1000 K) | Ch 4, p. 175 |
| All major atmospheric ions at 300 K | ~2 * 10^-7 | Ch 4, p. 174 |
| High-pressure three-body limit | up to 10^-4 | Ch 4, p. 175 |

### 14.5 Plasma Sustainability

| Quantity | Value | Reference |
|----------|-------|-----------|
| Average ionization energy in air | ~14 eV | Ch 7, p. 440 |
| Power to sustain n_e = 10^13 in 2000 K air | 14 kW/cm^3 | Ch 5, p. 230 |
| Power to sustain n_e = 10^13 in cold air | 1.4 kW/cm^3 (attachment-limited) | Ch 7, p. 440 |
| Electron-air collision cross section | 1.5 * 10^-15 cm^2 | Ch 5, p. 229 |
| N2 vibrational relaxation time (1 atm) | >100 us | Ch 5, p. 231 |

### 14.6 Conductivity from Electron Density

```
sigma = n_e * e^2 / (m_e * nu_e-air)

where:
  nu_e-air = N_air * sigma_collision * v_e    (electron-neutral collision frequency)
  sigma_collision = 1.5 * 10^-15 cm^2         (electron-air collision cross section)
  N_air ~ 2.5 * 10^19 cm^-3 at STP
  v_e ~ 10^6 m/s (mean electron speed at ~1 eV)
```

[Becker et al. 2005, Ch 5, p. 229]

Example: For n_e = 10^13 cm^-3 in air at STP, this yields sigma ~ 0.075 S/m, consistent with the "cold streamer" conductivity range (0.01-0.1 S/m) in Section 10.8.

### 14.7 Streamer and Spark Kinetics

| Quantity | Value | Reference |
|----------|-------|-----------|
| Primary streamer velocity | ~10^8 cm/s (10^6 m/s) | Ch 2, p. 59 |
| Secondary streamer/leader velocity | 10^5 - 10^6 cm/s (10^3 - 10^4 m/s) | Ch 2, p. 59 |
| Ionization front thickness | ~0.015 cm (~150 um) | Ch 2, p. 37 |
| Min. specific energy for spark channel | 0.6 - 1 J/cm^3 | Ch 2, p. 59 |
| Spark current rise rate (dI/dt) | ~10^7 A/s | Ch 2, p. 60 |
| Ion mobility in air (STP) | ~2 * 10^-4 m^2/(V*s) | Ch 2, p. 60 |
| Humidity: breakdown voltage minimum | at ~1% water vapor | Ch 2, p. 30 |
| Frequency: breakdown voltage minimum | ~1 MHz | Ch 2, p. 30 |

### 14.8 Arc Model Parameters (Mayr/Cassie)

The Mayr and Cassie arc models describe the time evolution of arc conductance. For Tesla coil sparks (low current, non-equilibrium), the Mayr model is appropriate.

**Mayr equation:**

```
dG/dt = (1/tau_m) * (P/P_0 - 1) * G

where:
  G = arc conductance [S]
  tau_m = thermal time constant [s]
  P = I^2/G = instantaneous power dissipated [W]
  P_0 = cooling power (power to sustain ionization) [W]
```

**Cassie equation** (for high-current arcs, NOT applicable to TC sparks):

```
dG/dt = (1/tau_c) * (u^2/U_c^2 - 1) * G

where:
  u = arc voltage
  U_c = characteristic voltage (= arc voltage at equilibrium)
```

| Parameter | TC Streamer | TC Leader | High-current arc | Reference |
|-----------|------------|-----------|-----------------|-----------|
| tau_m | 0.1-0.5 ms | 10-500 ms | 0.1-10 ms | Yang et al. 2022 |
| P_0 | ~1 W/m * L_seg | ~1 kW/m * L_seg | 1-100 kW | Yang et al. 2022 |
| tau_c | N/A | N/A | 0.1-1 ms | Yang et al. 2022 |

[Yang et al. 2022, "Arc Modeling Approaches: A Comprehensive Review," Frontiers in Physics]

**Key points for TC application:**

- TC sparks are firmly in the Mayr regime (low current, thin channels)
- Cassie model is irrelevant for TC sparks (applies to high-current industrial arcs)
- Sensitivity analysis shows tau_m and P_0 are the critical parameters; small changes produce large conductance variations
- LTE (Local Thermodynamic Equilibrium) assumption in both models breaks down at the low currents typical of TC streamers

**Hybrid transition function** for combined Mayr-Cassie (informational only):

```
sigma(i) = exp(-i^2/I_0^2)   (weighting function)
G = sigma * G_Mayr + (1-sigma) * G_Cassie
```

For TC sparks with i << I_0, sigma -> 1 and the combined model reduces to pure Mayr.

### 14.9 Ionization Threshold Discrepancy

Two sources give slightly different values for the ionization-attachment crossover:

| Source | E/N threshold | Equivalent at STP |
|--------|--------------|-------------------|
| Becker et al. 2005, Ch 2, p. 26 | 100 Td | ~25 kV/cm |
| Stanley (2000), via Liu 2017 | 120 Td | ~30 kV/cm |

The 20% discrepancy is within measurement uncertainty for the ionization and attachment coefficients. For the TC framework, the difference is not operationally significant because: (1) E_inception (~2-3 MV/m) is determined by geometry and pressure, not by the bare crossover field; and (2) E_propagation (~0.4-1.0 MV/m) is an empirical parameter calibrated to specific coils regardless of the fundamental crossover value.

### 14.10 Leader Inception Parameters

| Quantity | Value | Reference |
|----------|-------|-----------|
| Minimum gas temperature for stable leader | Significantly >2000 K | Liu 2017, Ch 3 |
| Reason for overshoot | Convection losses during gas expansion | Liu 2017, Ch 3 |
| Dark period duration | ~1-5 ms between streamer bursts | Liu 2017; Les Renardieres 1977, 1981 |
| Ion mobility (drift recovery) | ~2 * 10^-4 m^2/(V*s) | Becker et al. 2005, Ch 2, p. 60 |
| Kinetic model complexity | 45 species, 192 reactions | Liu 2017, Ch 3 |
| Gallimberti model accuracy | Qualitative only; quantitative predictions unreliable | Liu 2017, Ch 3 |

### 14.11 Nonlinear Resistance Power Law

The equilibrium resistance per unit length of a spark/leader channel follows a power law in current:

```
R = A / I^b    (ohm/m)
```

[da Silva et al. 2019, "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance," JGR Atmospheres]

**Fitted parameters by current regime (at 10 ms timescale):**

| Regime | Current Range | A (ohm * A^b / m) | b | Dominant Cooling | Fit Error |
|--------|-------------|-------------------|------|-----------------|-----------|
| Region I | 1-10 A | 1.24 * 10^4 | 1.84 | Heat conduction | 9% |
| Region II | 10-1,000 A | 2.82 * 10^3 | 1.16 | Mixed conduction + radiation | 4% |
| Region III | 1,000-10,000 A | 0.18 * 10^3 | 0.75 | Radiation | 1% |
| King (1961) expt | 1-10,000 A | 2.87 * 10^3 | 1.16 | Experimental steady-state | 25% |

**Application to Tesla coil sparks:**

- **Region I (1-10 A)** is TC streamer/early leader territory. At I = 1 A: R ~ 12,400 ohm/m. At I = 10 A: R ~ 179 ohm/m.
- **Region II (10-1,000 A)** is DRSSTC burst mode territory. At I = 100 A: R ~ 13.5 ohm/m.
- Typical TC sparks at 1-3 A peak current: R ~ 3,000-12,000 ohm/m, giving total R of 3-36 kohm for a 1-3 m spark. This is consistent with the R bounds in Section 10 (5-300 kohm).

**Relationship to Mayr equation:** The R = A/I^b power law describes the *equilibrium* resistance the channel tends toward. The Mayr equation (Section 14.8) describes *how fast* the channel approaches that equilibrium. Together they form a complete dynamic resistance model:
- Mayr: dG/dt = (1/tau) * (P/P_0 - 1) * G  (dynamics)
- da Silva: R_eq = A/I^b  (target equilibrium)

**Key physical insight:** The channel "forgets" its initial conditions at the ~10 ms timescale. Regardless of starting electron density, radius, or temperature, the resistance converges to the R = A/I^b curve. This supports the hungry streamer self-optimization principle: the plasma state is determined by the current the circuit delivers, not by the plasma's initial conditions.

### 14.12 Air Heating Efficiency

Not all electrical power dissipated in the channel heats the neutral gas. At low temperatures, most energy goes into vibrational excitation of N2, which relaxes slowly:

```
eta_T = 0.1 + 0.9 * [tanh(T/T_amb - 4) + 1] / 2
```

[da Silva et al. 2019, after Flitti & Pancheshnyi 2009]

| Gas Temperature | eta_T | Meaning |
|----------------|-------|---------|
| 300 K (ambient) | ~0.10 | Only 10% of Joule heating goes to gas temperature |
| 600 K | ~0.10 | Still mostly vibrational excitation |
| 1200 K | ~0.55 | Transition zone: V-T relaxation accelerating |
| 2000 K | ~1.0 | Full thermalization: all power heats gas |

**Critical implication for TC sparks:** This explains why the streamer-to-leader transition takes milliseconds despite MW/m power densities in thin streamer channels — 90% of the electrical energy goes into N2 vibrational modes, not gas heating. Only after the gas reaches ~1000-2000 K does thermalization become efficient, triggering the thermal runaway to leader temperatures (>5000 K).

### 14.13 Channel Radius Expansion

The current-carrying radius and thermal radius expand by ambipolar diffusion and thermal conduction respectively:

```
rc ~ sqrt(4 * D_a * t)    (current-carrying radius)
rg ~ sqrt(4 * kappa_T / (rho_m * c_p) * t)    (thermal radius)
```

[da Silva et al. 2019]

Where D_a is the ambipolar diffusion coefficient. Both expand as sqrt(t), consistent with the thermal diffusion model in [[thermal-physics]].

**Streamer-to-leader channel evolution at 10 A:**

| Parameter | Initial (streamer) | After transition (~4 us at 10 A) |
|-----------|-------------------|----------------------------------|
| rc | 0.5 mm | 1 mm |
| rg | 5 mm | 10 mm |
| n_e | 10^14 cm^-3 | 9 * 10^11 cm^-3 |
| T | 300 K | 5000 K |

### 14.14 Bazelyan V-I Characteristic

A simple relationship between arc/leader current and internal electric field, valid at atmospheric pressure for moderate currents:

```
i * E = b

where:
  b = 300 V*A/cm  (empirical constant for air at 1 atm)
  i = channel current [A]
  E = internal electric field [V/cm]

Equivalently: R_per_meter = b / i^2 = 30,000 / i^2  [ohm/m]
```

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 706]

**Comparison with da Silva power law (Section 14.11):**

| Current | Bazelyan (b=300) | da Silva Region I | Ratio |
|---------|-----------------|-------------------|-------|
| 1 A | 30,000 ohm/m | 12,400 ohm/m | 2.4x |
| 3 A | 3,333 ohm/m | 1,575 ohm/m | 2.1x |
| 10 A | 300 ohm/m | 179 ohm/m | 1.7x |
| 100 A | 3 ohm/m | 13.5 ohm/m (Region II) | 0.2x |

The two formulas agree within a factor of ~2 for 1-10 A (TC-relevant range), diverging at higher currents where the simple i*E=b formula breaks down. The Bazelyan formula is a quick approximation; da Silva's three-regime power law is more accurate.

**More precise measured CVC (from full textbook):**

```
E = 32 + 52/i    [V/cm, i in Amperes]
```

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, p. 90, Eq. 2.48]

This measured current-voltage characteristic is more accurate than the simple i*E=b formula. The 32 V/cm floor represents irreducible radiation and convection losses; the 52/i term dominates at TC-relevant currents.

| Current | Simple (i*E=300) | Measured CVC | da Silva Region I | TC Context |
|---------|-----------------|--------------|-------------------|-----------|
| 0.5 A | 600 V/cm | 136 V/cm | — | TC streamer |
| 1 A | 300 V/cm | 84 V/cm | 124 V/cm | TC early leader |
| 3 A | 100 V/cm | 49 V/cm | 47 V/cm | TC leader |
| 10 A | 30 V/cm | 37 V/cm | 18 V/cm | DRSSTC burst |

The measured CVC agrees better with da Silva at TC currents (1-10 A) than the simple formula does.

### 14.15 Leader Velocity Formula

Empirical formula for leader propagation velocity as a function of tip potential:

```
v_L = a * sqrt(|Delta_U_t|)

where:
  a = 1500 cm/s / V^(1/2)  (= 0.474 m/s per sqrt(V))
  Delta_U_t = tip potential minus external potential at tip location [V]
```

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 709, Eq. 5]

Valid for both positive (continuous) and negative (stepped) leaders. Derived from extensive laboratory spark and lightning data.

**Application to Tesla coil sparks:**

| Tip Voltage | v_L | Propagation time for 2 m |
|-------------|-----|--------------------------|
| 100 kV | 4.7 km/s | 0.42 ms |
| 300 kV | 8.2 km/s | 0.24 ms |
| 600 kV | 11.6 km/s | 0.17 ms |

These velocities are consistent with the intermediate regime between laboratory sparks (~10 km/s) and lightning leaders (~100 km/s), and with the observation that TC spark growth takes ~1-10 ms during a QCW ramp (accounting for the fact that the leader must also wait for thermal transition at each step).

**Physical basis:** Leader velocity is set by two processes in series: (1) streamer propagation from the leader tip at v_s ~ 10^7 cm/s over a conducting length l ~ v_s/nu_a ~ 1 cm, and (2) thermal contraction instability build-up in tau_ins ~ 1 us. The net leader advance rate is v_L ~ l/tau_ins ~ 10^6 cm/s (10 km/s), modulated by the square root of tip voltage which controls streamer vigor.

### 14.16 Channel Capacitance and Energy Ceiling

The linear capacitance of a spark channel and the maximum energy available per unit length from tip charge alone:

```
Channel capacitance per unit length:
  C_1 = 2*pi*epsilon_0 / ln(L/r)  [F/m]
      = 0.0555 / ln(L/r)  [pF/m]

Maximum energy per unit length (from tip capacitance):
  W_max = pi * epsilon_0 * U^2  [J/m]
```

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), pp. 703-704, Eq. 1]

The tip (hemisphere) has capacitance C_1t = 2*pi*epsilon_0 (independent of radius), which is ln(L/r) times larger than the per-unit-length body capacitance. This means the tip stores disproportionately more energy for heating the next channel segment.

**Application to Tesla coil sparks:**

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

Note: W_max is the energy available from the tip charge alone. The TC resonant circuit continuously supplies additional energy through the conducting channel during the burst, so the total energy available for channel formation far exceeds W_max. However, W_max constrains the energy available for *new channel initiation* ahead of the leader tip before the leader has extended its conducting core.

**Validation:** The C_1 formula gives C_1 = 0.0555/ln(200/0.5) ~ 9.3 pF/m for a 2-m spark of 1-cm diameter. This is comparable to but distinct from the empirical C_sh ~ 6.6 pF/m (2 pF/foot), which includes the full ground-referenced capacitance extracted from FEMM (geometry-dependent). The difference arises because C_sh is extracted from the complete Maxwell matrix including ground plane effects, while C_1 is the free-space formula for an isolated conductor.

### 14.17 Temperature Thresholds for Self-Sustaining Plasma

Three distinct temperature thresholds govern leader channel viability:

| Threshold | Temperature | Physical Mechanism | Reference |
|-----------|------------|-------------------|-----------|
| Onset (Liu) | >2000 K (must overshoot) | Thermal ionization begins; gas expansion can abort if marginal | Liu 2017, Ch 3 |
| Associative ionization | >4000 K | N + O -> NO+ + e: field-free ionization kicks in; n_e ~ 7*10^12 cm^-3 at equilibrium | Bazelyan & Raizer 2000, pp. 715-716 |
| Full self-sustaining | >5000 K | Electron attachment virtually nonexistent; channel survives without external field | Bazelyan & Raizer 2000, p. 703 |

**Key insight:** These three thresholds are not contradictory — they describe different stages of the same transition:
- At 2000 K, thermal ionization *starts* but the channel is fragile (can be killed by expansion/convection)
- At 4000 K, associative ionization provides a field-independent electron source; the channel is robust
- At 5000 K, the plasma is fully self-sustaining; the channel cannot be extinguished by anything short of removing the gas

For TC sparks, the practical threshold is **4000-5000 K**: the channel must reach this range to survive as a leader. The 2000 K onset temperature (Liu) represents the minimum to *begin* the transition, while 5000 K (Bazelyan) represents the endpoint where the channel is truly persistent.

### 14.18 Additional Physical Constants

| Quantity | Value | Reference |
|----------|-------|-----------|
| Electron mobility in air (STP) | mu_e = 600 cm^2/(V*s) | Bazelyan & Raizer 2000, p. 714 |
| Ion mobility in air (STP) | mu_i = 1.5 cm^2/(V*s) | Bazelyan & Raizer 2000, p. 711 |
| Electron attachment time (cool air) | ~100 ns (10^-7 s) | Bazelyan & Raizer 2000, p. 703 |
| Thermal instability contraction time | ~1 us (10^-6 s) | Bazelyan & Raizer 2000, p. 704 |
| Leader channel wave impedance | ~500 ohm | Bazelyan & Raizer 2000, p. 709 |
| Specific enthalpy to heat air to 5000 K | 12 kJ/g | Bazelyan & Raizer 2000, p. 703 |
| Recombination coefficient (cold air) | beta ~ 10^-7 cm^3/s | Bazelyan & Raizer 2000, p. 714 |
| n_e at 4000 K equilibrium (1 atm) | 7 * 10^12 cm^-3 | Bazelyan & Raizer 2000, p. 716 |

### 14.19 Conductance Relaxation Model

An alternative to the Mayr equation (Section 14.8) for modeling time-dependent channel conductance, derived from return stroke physics but applicable to any spark channel with time-varying current:

```
dG/dt = [G_st(i) - G(t)] / tau_g

where:
  G = channel conductance per unit length [S/m]
  G_st(i) = i / E_L = stationary (equilibrium) conductance at current i [S/m]
  E_L = leader/channel field at steady state (~10 V/cm for leaders)
  tau_g = conductance relaxation time [s]

  tau_g = 40 us    (when current is rising — channel heating)
  tau_g = 200 us   (when current is decreasing — channel cooling)
```

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 4, pp. 194-195, Section 4.4.3]

**Key difference from Mayr:** The Mayr equation (dG/dt = (1/tau_m)*(P/P_0 - 1)*G) describes conductance as driven by power relative to a cooling power threshold. The Bazelyan relaxation model describes conductance as relaxing toward a current-dependent equilibrium value with an asymmetric time constant (faster heating than cooling). Both models converge for small perturbations but diverge for large transients.

**Physical basis for asymmetric tau_g:** Channel heating (ionization, thermal expansion) is driven by Joule energy deposition (fast, positive feedback). Channel cooling involves thermal conduction through hot gas, recombination against residual ionization, and radiative losses (slower, especially at intermediate temperatures where the gas is opaque to its own radiation).

**Application to Tesla coil sparks:**

- At TC frequencies (50-400 kHz, half-period 1.25-10 us), the conductance cannot reach equilibrium within a single half-cycle (tau_g = 40 us >> half-period). This means the channel conductance is effectively time-averaged over many RF cycles.
- The 200 us cooling time constant is comparable to thin streamer persistence (~100-200 us), confirming that conductance and thermal state decay on similar timescales.
- The 5:1 heating/cooling asymmetry (40 us vs 200 us) means the channel "remembers" high-current states longer than it takes to reach them — a hysteresis effect that favors leader maintenance once established.

**Connection to Mayr parameters:** For small perturbations around G_st, the Bazelyan model reduces to exponential relaxation with tau_g, while the Mayr model reduces to exponential relaxation with tau_m. The correspondence is tau_m ~ tau_g for steady-state conditions. The Bazelyan model is more physical for large transients (e.g., return strokes, burst mode transitions) because it explicitly models the target equilibrium state.

### 14.20 Channel Transmission Line Parameters

A leader/spark channel can be modeled as a lossy transmission line with distributed inductance, capacitance, and resistance:

```
Linear inductance:
  L_1 = (mu_0 / 2*pi) * ln(H/r_c)  [H/m]
      ~ 0.2 * ln(H/r_c)  [uH/m]

Linear capacitance:
  C_1 = 2*pi*epsilon_0 / ln(L/r)  [F/m]
      ~ 10 pF/m  (with corona envelope at R ~ 16 m, lightning scale)
      ~ 2-5 pF/m (TC scale, topload to tip)

Wave impedance (lossless limit):
  Z = sqrt(L_1 / C_1)  ~ 500 ohm

Wave velocity (lossless limit):
  v = 1 / sqrt(L_1 * C_1)  ~ 0.6-0.7 * c
```

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 4, pp. 184-185, Eq. 4.25-4.26]

**Numerical values:**

| Parameter | Lightning Leader | TC Spark (est.) | Units |
|-----------|-----------------|-----------------|-------|
| L_1 | 2.5-2.7 | 2-3 | uH/m |
| C_1 | 10 | 2-5 | pF/m |
| Z | 500 | 700-1200 | ohm |
| v | 0.6-0.7c | 0.3-0.5c | m/s |

**TC spark values** are estimated by scaling: TC sparks are shorter (1-3 m vs 1-4 km), thinner (1-5 mm vs 1-5 cm), and lack the corona envelope of lightning leaders, giving higher L_1/C_1 ratio and hence higher Z.

**Relevance to TC modeling:** The wave impedance Z ~ 500-1200 ohm sets the characteristic impedance seen by transient waves propagating along the spark channel. This is relevant for:
- Strike events (return stroke analogy): when a TC spark contacts ground, the impedance mismatch between Z and R_ground drives a reflection/recharging wave
- Distributed model validation: Z provides an independent check on the L_1*C_1 product
- The C_1 value (2-5 pF/m) is consistent with the empirical C_sh ~ 6.6 pF/m, with the difference accounted for by the ground plane capacitance contribution in the FEMM extraction

### 14.21 Equilibrium Air Plasma Composition

Thermodynamic equilibrium plasma parameters in air at atmospheric pressure:

| T (K) | N (10^18 cm^-3) | n_e (10^13 cm^-3) | N_O (10^16 cm^-3) | N_NO (10^16 cm^-3) | n_e/N (10^-5) |
|-------|-----------------|--------------------|--------------------|--------------------|--------------------|
| 4000 | 1.79 | 0.63 | 0.25 | 1.62 | 0.35 |
| 4500 | 1.60 | 1.70 | 1.15 | 4.54 | 1.06 |
| 5000 | 1.48 | 4.90 | 3.61 | 2.73 | 3.31 |
| 5500 | 1.35 | 11.2 | 9.92 | 1.67 | 8.30 |
| 6000 | 1.27 | 21.4 | 20.6 | 1.03 | 16.8 |

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, Table 2.2, pp. 85-86]

**Key observations:**
- n_e increases 34x from 4000 K to 6000 K (exponential Saha dependence)
- NO concentration peaks at ~4500 K then decreases (thermal dissociation of NO above 5000 K)
- At 5000 K: n_e = 4.9 * 10^13 cm^-3, ionization degree ~3.3 * 10^-5 (very weakly ionized)
- Effective ionization potential from fit: I_eff = 8.1 eV (close to NO ionization at 9.3 eV)
- Conductivity increases more steeply than n_e because the exponential sigma(T) relation magnifies temperature differences

**Associative ionization rate constant** (dominant at 4000-6000 K):

```
N + O + 2.8 eV -> NO+ + e

k_ass = 2.59 * 10^-17 * T^1.43 * exp(-31140/T)  [cm^3/s]
```

[Bazelyan & Raizer 2000, Ch 2, p. 85, Eq. 2.44-2.45]

Equilibrium establishment time: 20-50 us at T = 4000-6000 K. This is fast enough that a TC leader channel at these temperatures is in near-equilibrium.

### 14.22 Streamer Parameters

Fundamental streamer ionization wave parameters in air at atmospheric pressure:

| Quantity | Value | Reference |
|----------|-------|-----------|
| Maximum tip field E_m | 150-170 kV/cm | Ch 2, p. 42 |
| Ionization frequency at E_m | v_im = 1.1 * 10^10 s^-1 | Ch 2, p. 42 |
| Electron mobility at E_m | mu_e = 270 cm^2/(V*s) | Ch 2, p. 42 |
| Drift velocity at E_m | V_em = 4 * 10^5 m/s | Ch 2, p. 44 |
| Initial plasma density behind front | n_c = 9 * 10^13 cm^-3 (voltage-independent!) | Ch 2, pp. 42-43 |
| Minimum streamer velocity | (1.5-2) * 10^5 m/s at U_t = 5-8 kV | Ch 2, p. 44 |
| Channel field behind tip (1 m) | E_c = 4.2 kV/cm | Ch 2, p. 48 |
| Energy density per passage | 0.026 J/cm^3 -> Delta_T < 3 K | Ch 2, pp. 49-50 |
| Conductivity 100x drop time | ~300 ns behind tip | Ch 2, p. 53 |
| Streamer zone internal resistance | ~0.5 Megaohm (acts as current source) | Ch 2, p. 69 |

**Streamer velocity formula:**

```
V_s = v_im * r_m / [(2k-1) * ln(n_c/n_0)]

V_s is proportional to U_t  (because r_m = U_t / (2*E_m) at constant E_m)
```

### 14.23 Breakdown Voltage Formulas

Empirical minimum spark breakdown voltage for sharply non-uniform fields (rod-plane) in air:

```
U_50%_min = 3400 / (1 + 8/d)    [kV, d in meters]    (d < 15 m)
U_50%_min = 1440 + 55*d          [kV]                  (15 < d < 30 m)
```

[Bazelyan & Raizer 2000, Ch 2, p. 94, Eq. 2.52]

| Gap length d | U_50%_min | E_average | TC context |
|-------------|-----------|-----------|-----------|
| 0.5 m | 200 kV | 400 kV/m | Small DRSSTC |
| 1 m | 378 kV | 378 kV/m | Mid-range DRSSTC |
| 2 m | 680 kV | 340 kV/m | Large DRSSTC |
| 3 m | 927 kV | 309 kV/m | Competition coil |
| 5 m | 1308 kV | 262 kV/m | Very large coil |

**Optimal voltage risetime for breakdown:**

```
t_f_opt ~ 50 * d  [microseconds, d in meters]
```

For a 1 m gap, optimal risetime is ~50 us. QCW ramps are typically 5-20 ms (100-400x slower), operating far above optimal. Burst mode pulses at 50-500 us are closer to optimal for their gap lengths.

**TC relevance:** These formulas apply to the total topload-to-ground breakdown, not the incremental streamer propagation governed by E_propagation. They are useful for estimating the minimum secondary voltage needed for a target spark length, and for sanity-checking SPICE-predicted secondary voltages against known breakdown thresholds.

### 14.24 QCW Operating Parameters (Community Survey)

Consensus operating parameters from a comprehensive survey of QCW Tesla coil builders. [Phase 6 QCW community survey, 2026-02-10; sources: Loneoceans (4 builds), Steve Ward, davekni, flyglas, Lucasww, Rafft, Mathieu thm, Fat Coil, Dr. Kilovolt, LabCoatz, Kaizer DRSSTC IV]

**QCW vs Burst Parameter Comparison:**

| Parameter | QCW Range | Burst DRSSTC Range | Ratio |
|-----------|-----------|-------------------|-------|
| Coupling (k) | 0.3-0.55 | 0.05-0.2 | 2-5x higher |
| Operating frequency | 300-600 kHz | 50-110 kHz | 3-10x higher |
| Ramp/pulse duration | 10-22 ms | 70-150 us | 100-200x longer |
| Peak primary current | 50-200 A | 200-1000+ A | 3-10x lower |
| Secondary voltage | 40-70 kV | 200-600 kV | 5-15x lower |
| Spark:secondary ratio | 7-16x | 2-4x | 3-5x higher |
| Tank capacitance | 5-15 nF | 50-300 nF | 5-20x smaller |

**QCW Growth Parameters:**

| Quantity | Value | Source |
|----------|-------|--------|
| Growth rate | ~170 m/s | HVF visual estimate |
| Driven leader step time | ~60 us | Derived (0.01 m / 170 m/s) |
| Frequency threshold for swords | 300-600 kHz | 6+ independent observers |
| Burst ceiling (ON time) | ~80 us | Steve Ward DRSSTC-0.5 |
| Optimal ramp duration | 10-20 ms | Loneoceans QCW v1.5 |
| Energy per pulse (1.78 m) | 275 J | Loneoceans QCW v1.5 |
| Apparent epsilon (input/length) | 155 J/m | Derived |
| Estimated spark epsilon | 45-75 J/m | At 30-50% system efficiency |

**Critical Time Comparisons:**

| Timescale | Value | Significance |
|-----------|-------|-------------|
| RF half-period at 400 kHz | 1.25 us | << tau_thermal: effectively continuous heating |
| RF half-period at 100 kHz | 5 us | Marginal for continuous heating of thin streamers |
| Streamer tau_thermal (100 um) | ~125 us | 100x longer than RF period at 400 kHz |
| Conductance tau_g (heating) | 40 us | Time per driven leader step |
| Conductance tau_g (cooling) | 200 us | 5x longer than heating (hysteresis) |
| Burst pulse duration | 70-150 us | Comparable to streamer tau |
| QCW ramp duration | 10-22 ms | 100x longer than tau_g |
| Leader transition time | 0.5-2 ms | Within QCW ramp, exceeds burst pulse |

**Coupling Coefficient Data (all documented QCW builds):**

| Builder | k | Spark:secondary | Max spark | Notes |
|---------|---|----------------|-----------|-------|
| Loneoceans v1.0 | 0.32-0.35 | 7.3:1 | 40" | Initial |
| Loneoceans v1.5 | 0.38 | 13:1 | 70+" | Breakthrough came at 0.38, not 0.306 |
| Loneoceans QCW2 | 0.365 | 10:1 | 24" | Miniature |
| flyglas | 0.391 | ~12:1 | 170 cm | |
| Lucasww | 0.44 | 10:1 | 51" | |
| Dr. Kilovolt | 0.55 | — | 2-2.5 m | SiC PSFB |
| davekni | 0.71 | — | 2-2.5 m | Ferrite-assisted, highest documented |

**Loneoceans frequency tracking data (simulated streamer loading):**

| Condition | Frequency | Shift |
|-----------|-----------|-------|
| Unloaded secondary | 406-409 kHz | baseline |
| With 50 cm wire | 349 kHz | -14% |
| With 1 m wire | 310 kHz | -24% |
| During QCW spark growth | 413 → 377 kHz | -8.7% |

The 8.7% shift during actual QCW operation (with 1.78 m spark) is less than the 24% shift from a 1 m solid wire, confirming that real sparks have lower effective capacitance than solid conductors — consistent with the branched, partially conducting nature of real plasma channels.