10 KiB
Tesla Coil Spark Physics - Physical Bounds and Typical Ranges
Reference for validation criteria, physical constraints, and empirical values.
Resistance Bounds
Lumped Model
Physical limits:
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)
nth-Order Model (Position-Dependent)
Base segments (position = 0):
- R_min = 1 kΩ
- R_max = 100 kΩ
Tip segments (position = 1):
- R_min = 10 kΩ
- R_max = 100 MΩ
Interpolated formula:
position = i/(n-1)
R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
= 1 kΩ + 9 kΩ × position
R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position
≈ 100 kΩ + 99.9 MΩ × position
Typical Total Resistance (by operating mode)
At 200 kHz for 1-3 meter sparks:
| Operating Mode | Total R | Characteristics |
|---|---|---|
| Burst/Streamer-dominated | 50-300 kΩ | Short pulses, thin channels |
| QCW/Leader-dominated | 5-50 kΩ | Long ramps, hot thick channels |
| Very low frequency (<100 kHz) or very long sparks | 1-10 kΩ | Thick leaders, high power |
Validation flag: If R_total is significantly outside these ranges for your frequency and length, investigate potential issues.
Capacitance Values
Mutual Capacitance (C_mut)
Typical values:
| Spark Length | Typical C_mut | Notes |
|---|---|---|
| 1 foot (0.3 m) | 3-5 pF | Small topload |
| 2 feet (0.6 m) | 5-8 pF | Medium topload |
| 3 feet (0.9 m) | 7-12 pF | Large topload |
| 5 feet (1.5 m) | 10-15 pF | Very large topload |
Depends on topload size and geometry
Shunt Capacitance (C_sh)
Empirical rule:
C_sh ≈ 2 pF per foot of spark length
Examples:
| Spark Length | Typical C_sh |
|---|---|
| 1 foot (0.3 m) | 2 pF |
| 2 feet (0.6 m) | 4 pF |
| 3 feet (0.9 m) | 6 pF |
| 5 feet (1.5 m) | 10 pF |
| 10 feet (3.0 m) | 20 pF |
Validation: Use this rule to verify FEMM extraction accuracy.
Capacitance Ratio (r)
r = C_mut/C_sh
Typical geometries:
| Configuration | r value | φ_Z,min |
|---|---|---|
| Large topload, short spark | 0.5 - 2.0 | -50° to -70° |
| Medium topload, medium spark | 0.3 - 0.8 | -48° to -60° |
| Small topload, long spark | 0.1 - 0.4 | -43° to -52° |
Critical threshold: When r ≥ 0.207, achieving φ_Z = -45° becomes impossible.
Diameter Dependence
Weak logarithmic scaling:
C ∝ 1/ln(h/d)
Where:
- h = height above ground
- d = channel diameter
Typical change: 2× diameter → ~10-15% change in C
Electric Field Thresholds
Inception Field
Smooth electrode breakdown:
E_inception ≈ 2-3 MV/m (sea level, standard conditions)
Variations:
- Sharp electrodes: 1-2 MV/m (lower threshold)
- Very smooth, large radius: 3-4 MV/m (higher threshold)
Propagation Field
Sustained leader growth:
E_propagation ≈ 0.4-1.0 MV/m (typical range)
Common values:
- Conservative estimate: 0.8-1.0 MV/m
- Optimistic/ideal conditions: 0.4-0.6 MV/m
- Typical use for modeling: 0.6-0.7 MV/m
Tip Enhancement Factor
E_tip = κ × E_average
Typical values:
- κ ≈ 2-5 for cylindrical channels
- Higher for sharper geometries
- Use FEMM to calculate actual enhancement
Altitude and Environmental Effects
Altitude correction (rough approximation):
E(altitude) = E(sea level) × (P/P_0)
where P/P_0 ≈ exp(-h/8500 m)
Examples:
| Altitude | Pressure Ratio | Field Scaling |
|---|---|---|
| Sea level | 1.0 | 1.0 |
| 1500 m (Denver) | ~0.83 | ~0.83 |
| 3000 m | ~0.69 | ~0.69 |
Humidity effects: ±10-20% variation (higher humidity → slightly lower threshold)
Temperature: ±5-10% variation over normal range
Total variability: E_propagation can vary ±20-30% with environmental conditions
Energy per Meter (ε)
By Operating Mode
QCW-style growth:
ε ≈ 5-15 J/m
Characteristics:
- Long ramp times (5-20 ms)
- Leader-dominated channels
- Energy efficiently extends length
- White/orange appearance
High duty cycle DRSSTC:
ε ≈ 20-40 J/m
Characteristics:
- Hybrid streamer/leader formation
- Some thermal accumulation
- Moderate efficiency
- Mixed appearance
Hard-pulsed DRSSTC (burst mode):
ε ≈ 30-100+ J/m (single-shot)
Characteristics:
- Short pulses, mostly streamers
- Much energy → brightening/branching
- Poor length efficiency
- Purple/blue, highly branched
Calibration Requirements
Essential: Calibrate ε for your specific coil from measurements.
Procedure:
- Run coil with known drive power and time
- Measure final spark length L
- From SPICE, compute E_delivered = ∫P_spark dt
- Calculate: ε = E_delivered/L
Expected precision: ±30-50% due to variability in plasma conditions
Thermal Time Constants
Pure Thermal Diffusion
Formula:
τ_thermal = d² / (4α)
where α = k/(ρ_air × c_p) ≈ 2×10⁻⁵ m²/s for air
By diameter:
| Diameter | Type | τ_thermal | Observed Persistence |
|---|---|---|---|
| 100 μm | Thin streamer | 0.1-0.2 ms | ~1-5 ms |
| 1 mm | Thick streamer | 12-25 ms | ~10-50 ms |
| 5 mm | Leader | 300-600 ms | seconds |
| 1 cm | Thick leader | 1-2 seconds | 10+ seconds |
Note: Observed 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
Operating Regime Implications
QCW advantage:
- Ramp times 5-20 ms match streamer-to-leader persistence
- Channel stays hot throughout growth
- Efficient energy coupling
Burst mode:
- Pulse spacing > 5 ms → channel cools between pulses
- Must re-ionize repeatedly
- Less efficient for length
Phase Angles
Impedance Phase (φ_Z)
Typical ranges:
R = R_opt_power typically gives: φ_Z ≈ -55° to -75°
By capacitance ratio:
| r = C_mut/C_sh | φ_Z,min | Typical at R_opt |
|---|---|---|
| 0.1 | -42° | -55° to -60° |
| 0.3 | -50° | -60° to -65° |
| 0.5 | -55° | -62° to -68° |
| 1.0 | -65° | -68° to -73° |
| 2.0 | -73° | -72° to -76° |
Important: The commonly cited "-45°" is often unachievable due to circuit topology.
Admittance Phase (θ_Y)
θ_Y = -φ_Z
Typical ranges: +55° to +75° (positive, capacitive)
Frequency Ranges
Operating Frequencies
Typical Tesla coil operating frequencies:
| Coil Type | Frequency Range | Notes |
|---|---|---|
| Small DRSSTC | 150-400 kHz | Higher frequency, smaller secondary |
| Medium DRSSTC | 100-250 kHz | Most common range |
| Large DRSSTC | 50-150 kHz | Lower frequency, larger secondary |
| SSTC | 100-500 kHz | Wide range possible |
| QCW | 50-200 kHz | Typically lower frequencies |
Loaded vs Unloaded
Frequency shift with spark:
- Typical shift: 5-20% lower when loaded
- Larger sparks → larger shift
- Track frequency to loaded pole for accurate measurements
Power Levels and Efficiencies
Typical Power Ranges
By coil size:
| Coil Class | Primary Power | Spark Power | Typical η |
|---|---|---|---|
| Small DRSSTC | 0.5-2 kW | 0.1-0.5 kW | 15-30% |
| Medium DRSSTC | 2-5 kW | 0.5-1.5 kW | 20-35% |
| Large DRSSTC | 5-15 kW | 1.5-5 kW | 25-40% |
| QCW | 1-10 kW | 0.5-4 kW | 30-50% |
Efficiency components:
- Spark power delivery: 15-50%
- Secondary losses (heating): 10-30%
- Primary circuit losses: 20-40%
- Corona and radiation: 5-15%
Power Density
Typical values in spark channel:
P/L ≈ 50-500 W/m (power per unit length)
Higher for burst mode (bright but short), lower for QCW (efficient leaders).
Geometric Constraints
Minimum Capacitance Bounds
For stable operation:
C_mut + C_sh ≥ 5 pF (typical minimum for 100+ kHz)
Below this, impedance becomes very high and matching becomes difficult.
Maximum Practical Length
Voltage-limited:
L_max ≈ V_top_peak / E_propagation
Typical: V_top = 300-600 kV → L_max = 3-6 feet at E_prop = 1 MV/m
Power-limited:
L_max ≈ P_available × T / ε
where T is growth time available
Practical limit: Whichever is more restrictive.
Plasma Properties
Conductivity Range
Partially ionized air plasma:
σ ≈ 0.01 - 10 S/m (wide range depending on temperature and ionization)
Equivalent resistivity:
ρ ≈ 0.1 - 100 Ω·m
Typical for modeling:
- Hot leader: ρ ≈ 1-10 Ω·m
- Warm streamer: ρ ≈ 10-100 Ω·m
Temperature Ranges
Streamer:
T ≈ 1000-3000 K
Leader:
T ≈ 5000-20000 K
Arc (strike):
T > 10000 K
Validation Criteria
Self-Consistency Checks
Capacitance:
- C_sh/L ≈ 2 pF/foot ± 30%
Total resistance:
- Within expected range for operating mode (see above)
- R_base < R_tip in distributed model
Power balance:
- P_spark + losses = P_input (within 20%)
Phase angle:
- φ_Z,actual ≥ φ_Z,min (within numerical precision)
Diameter self-consistency:
- d_implied ≈ d_nominal (within factor of 2-3)
Warning Flags
Red flags indicating potential errors:
- C_sh/L < 1 pF/foot or > 4 pF/foot
- R_total < 500 Ω or > 10 MΩ at typical frequencies
- φ_Z > -30° or < -85°
- Power efficiency > 70% (unrealistically high)
- ε < 1 J/m or > 200 J/m
- Growth rates > 100 m/s (unphysical for leaders)
Measurement Tolerances
Expected Precision
Capacitance extraction (FEMM):
- ±10% typical accuracy
- ±5% with careful meshing
Resistance measurement:
- ±30-50% (plasma variability dominates)
Field measurements:
- E_propagation: ±20-30% (environmental variability)
Energy per meter:
- ±30-50% (high variability)
Overall model predictions:
- Length: ±20-40% typical
- Power: ±30-50% typical
- Phase: ±5-10° typical
Use these tolerances when validating model against measurements.
This reference compiled from empirical data, community observations, and validated modeling across multiple Tesla coil systems.