6.9 KiB
Tesla Coil Spark Physics - Equation Sheet
Quick reference for all key equations in spark modeling and circuit analysis.
Convention: All phasor quantities use peak values (not RMS). Power formulas include the 0.5 factor: P = 0.5 × Re{V × I*}.
Circuit Analysis
Admittance Components
Input admittance at topload (looking into spark):
Y = ((G + jB₁) · jB₂) / (G + j(B₁ + B₂))
Where:
- G = 1/R (conductance)
- B₁ = ωC_mut (mutual capacitance susceptance, positive)
- B₂ = ωC_sh (shunt capacitance susceptance, positive)
Real part of admittance:
Re{Y} = GB₂² / (G² + (B₁ + B₂)²)
Imaginary part of admittance:
Im{Y} = B₂[G² + B₁(B₁ + B₂)] / (G² + (B₁ + B₂)²)
Phase Angles
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})
Minimum achievable impedance phase angle:
φ_Z,min = -atan(2√(r(1 + r)))
Where:
- r = C_mut/C_sh (capacitance ratio)
Note: When r ≥ 0.207, achieving φ_Z = -45° becomes mathematically impossible regardless of R value.
Optimization
Critical Resistance Values
R_opt_power - Maximum power transfer:
R_opt_power = 1 / (ω(C_mut + C_sh))
Example: At f = 200 kHz with C_mut + C_sh = 12 pF:
R_opt_power = 1/(2π × 200×10³ × 12×10⁻¹²) ≈ 66 kΩ
R_opt_phase - Closest to resistive:
R_opt_phase = 1 / (ω√(C_mut(C_mut + C_sh)))
Note: R_opt_power < R_opt_phase always
Segment-Level Optimization (nth-order model)
Simplified circuit-determined resistance:
For each segment i:
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])
Tapered initialization for iterative optimization:
position = i/(n-1) # 0 at base, 1 at tip
R[i] = R_base + (R_tip - R_base) × position²
Typical: R_base = 10 kΩ, R_tip = 1 MΩ
Damped iterative update:
R_new[i] = α × R_optimal[i] + (1 - α) × R_old[i]
Where α ≈ 0.3-0.5 for stability
Thévenin Equivalent
Measurement Procedure
Output impedance (drive off, test source on):
Z_th = 1V / I_test = R_th + jX_th
Open-circuit voltage (drive on, no spark):
V_th = V(topload) [complex magnitude and phase]
Power Calculations
Power to any load:
P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
Theoretical maximum power (conjugate match):
P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
Note: Actual spark power will be less due to topological constraints.
Spark Growth
Electric Field Thresholds
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)
Note: E_propagation varies with altitude and humidity by ±20-30%.
Growth Rate Equation
When field threshold is met:
dL/dt = P_stream / ε (when E_tip > E_propagation)
dL/dt ≈ 0 (when E_tip < E_propagation, stalled)
Where:
- L = spark length [m]
- P_stream = power delivered to spark [W]
- ε = energy per meter [J/m]
Energy and power over time:
E_total ≈ ε × L
P_avg ≈ ε × L / T
Energy per Meter (ε)
By operating mode:
ε ≈ 5-15 J/m (QCW-style growth, leader-dominated)
ε ≈ 20-40 J/m (High duty cycle DRSSTC, hybrid)
ε ≈ 30-100+ J/m (Hard-pulsed burst mode, streamer-dominated)
Advanced time-dependent model:
ε(t) = ε₀ / (1 + α∫P_stream dt)
Where:
- α has units [1/J]
- ∫P_stream dt = accumulated energy
Thermal Physics
Thermal Time Constants
Pure thermal diffusion:
τ_thermal = d² / (4α)
Where:
- d = channel diameter [m]
- α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air
Examples:
d = 100 μm → τ ≈ 0.1-0.2 ms (thin streamers)
d = 5 mm → τ ≈ 300-600 ms (thick leaders)
Note: Observed persistence is longer due to convection and ionization memory:
- Thin streamers: ~1-5 ms (effective)
- Thick leaders: seconds (effective)
Capacitive Divider
Voltage Division Effect
General formula:
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)
Note: Since C_sh ∝ L, as spark grows, V_tip decreases even if V_topload is maintained.
Ringdown Method
Quality Factor Relations
At loaded resonance ω_L:
Q_L = ω_L C_eq R_p = R_p/(ω_L L)
Equivalent Resistance
From Q and capacitance:
R_p = Q_L/(ω_L C_eq)
From Q and inductance:
R_p = Q_L ω_L L
Total Conductance
From Q and capacitance:
G_total = ω_L C_eq/Q_L
From Q and inductance:
G_total = 1/(Q_L ω_L L)
Capacitance Change
Equivalent capacitance after loading:
C_eq = C₀(f₀/f_L)²
ΔC = C_eq - C₀
Spark Admittance Extraction
Step-by-step:
1. Measure unloaded: f₀, Q₀, C₀
2. Measure with spark: f_L, Q_L
3. C_eq = C₀(f₀/f_L)²
4. ΔC = C_eq - C₀
5. G_total = ω_L C_eq/Q_L
6. G_0 = ω₀ C₀/Q₀
7. Y_spark ≈ (G_total - G_0) + jω_L ΔC
FEMM Extraction
Maxwell Capacitance Matrix
For lumped model:
C_mut = -C[topload, spark] = |C_12|
C_sh = C[spark, spark] + C[spark, topload] = C_22 - |C_12|
Note: Maxwell matrix has C_ii > 0 (self-capacitance) and C_ij < 0 for i≠j (mutual capacitance, negative).
Validation check:
C_sh ≈ 2 pF per foot (empirical rule)
Empirical Scaling Laws
Freau's Relationships
Single-shot burst (no thermal accumulation):
L ∝ √(E_bang)
Repetitive operation (with thermal memory):
L ∝ P_avg^(0.3 to 0.5)
QCW with voltage ramping:
L ∝ E^(0.6 to 0.8) (closer to linear)
Self-Consistency Check
Diameter Back-Calculation
For validation:
ρ_typical = 10 Ω·m (partially ionized plasma)
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), model is self-consistent.
Physical Bounds Formulas
Position-Dependent Resistance Bounds
For nth-order model:
position = i/(n-1) # 0 at base, 1 at tip
R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position
Power Balance Validation
Total power equation:
P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation
Efficiency check:
η = P_spark / P_primary_input
Expected η varies widely by design and operating mode.
Total equations: 45+ key formulas across all categories