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Tesla Coil Spark Physics — Cheat Sheet

Everything you need to see the whole picture, in order.

1. The Spark Is a Circuit Element

A spark hanging off a topload is not magic plasma — it's a lossy capacitive load on a resonant circuit. It has exactly three electrical properties that matter:

C_mut (mutual capacitance): coupling between topload and spark channel (~3-15 pF)
C_sh (shunt capacitance): coupling between spark channel and ground (~2 pF/foot)
R (resistance): the lossy part where power is dissipated (1 kohm to 100 Mohm)

These three form a bridged-T network. That's the entire circuit model of a spark.

2. The Phase Constraint

Because C_mut and C_sh form a capacitive divider, the impedance phase at the topload is always more negative than -45 degrees for typical TC geometries. You can't achieve a conjugate match. This is a topological fact, not a design flaw.

Typical impedance phase at optimum: -55 to -75 degrees.

3. The Plasma Self-Optimizes

The spark resistance R isn't fixed — it adjusts itself through heating and ionization. The key result:

The plasma drifts toward R_opt_power = 1/(omega * C_total) because:

Too high R → less power → less heating → R rises further (unstable, spark dies or branches)
Too low R → less power (past optimum) → but stronger heating prevents R from dropping much below optimum

This is the hungry streamer principle: the spark "eats" as much power as the circuit can deliver, automatically finding the impedance that maximizes power transfer.

4. Two Ways to Grow

A spark extends its length when two conditions are met:

Condition 1 — Field threshold: E_tip > E_propagation The electric field at the spark tip must exceed the propagation threshold. If it doesn't, the spark stalls regardless of available power.

Condition 2 — Energy supply: dL/dt = P_stream / epsilon Growth rate equals available power divided by energy cost per meter.

Critical nuance: E_propagation is NOT a fixed constant. In cold air, E_propagation ~ 0.5 MV/m. But at a driven leader tip, four mechanisms — UV pre-ionization, thermal pre-conditioning, residual ionization, and gas expansion — converge to dynamically reduce it. This is why QCW achieves 2+ m sparks at only 40-70 kV: the leader creates its own favorable conditions. Voltage and power are coupled limits, not independent ones (Section 4A).

The spark is always limited by whichever constraint binds first: voltage-limited (can't push field high enough even with dynamic threshold) or power-limited (can extend field but not fast enough).

5. Epsilon: The Central Parameter

Epsilon (J/m) = energy required per meter of spark growth. It varies enormously:

Mode	Epsilon	Why
QCW (leader)	5-15 J/m	Hot, efficient single channel
Burst (streamer)	30-100+ J/m	Cold, branched, inefficient

The difference is almost entirely explained by channel type (Section 7) and branching (Section 10).

6. The Capacitive Divider Problem

As the spark grows, C_sh increases (more conductor length to ground). This divides down the tip voltage:

V_tip = V_topload * C_mut / (C_mut + C_sh)

Longer spark → more C_sh → lower V_tip → weaker E_tip → harder to keep growing.

This creates sub-linear scaling: doubling energy does NOT double spark length. Burst mode follows L ~ sqrt(E). QCW is somewhat better (L ~ E^0.6-0.8) because leader channels have lower C_sh per unit length than branched streamers.

6A. The Dynamic Threshold

The capacitive divider predicts QCW sparks should stall at well under 1 m with only 40-70 kV topload. Yet 2+ m sparks are routinely achieved. The resolution: E_propagation is not a fixed constant — at a driven leader tip, four mechanisms converge to reduce it:

UV photoionization — corona creates seed electrons ahead of the tip
Thermal pre-conditioning — heat reduces gas density (E_breakdown proportional to N proportional to 1/T)
Residual ionization — previous streamers leave persistent electron density (~50 us decay)
Gas expansion — lower N means lower absolute field threshold

These are mutually reinforcing: more leader current drives all four harder. The result is a coupled voltage-power limit — power modifies the conditions that set the voltage threshold. More power → lower effective E_propagation → spark extends further at the same voltage.

But there is a floor: E_propagation can't reach zero. The capacitive divider wins eventually. The "too long" QCW regime (>25 ms) is exactly the point where even maximal pre-conditioning can't keep E_tip above the reduced threshold.

7. Two Kinds of Channel

This is the fork in the road that explains almost everything:

	Streamer	Leader
Temperature	300-3000 K	5,000-20,000 K
Diameter	10-100 um	1-10 mm
Resistance	Very high	Low
Persistence	Microseconds	Seconds
Branching	Extensive	Minimal
Epsilon	High (30-100+)	Low (5-15)
Color	Purple/blue	White/yellow

Streamers are cold, thin, branched, and inefficient. Leaders are hot, thick, straight, and efficient. The entire game is getting from streamer to leader.

8. The Thermal Ratchet

The transition from streamer to leader requires heating the channel past ~5000 K (through intermediate thresholds at 2000 K and 4000 K). But thin channels cool fast:

tau_thermal = d^2 / (4 * alpha)      alpha ~ 2e-5 m^2/s for air

A 100 um streamer cools in ~125 us. You have to heat it faster than it cools.

The conductance relaxation is asymmetric:

Heating: tau_g = 40 us (fast — ionization responds quickly to current)
Cooling: tau_g = 200 us (slow — recombination and thermal diffusion take longer)

This 5:1 asymmetry creates a one-way thermal ratchet: each RF cycle heats a little more than the previous one cooled. Over many cycles, temperature accumulates monotonically upward through the critical zone.

9. Frequency Matters

The ratchet only works if the RF period is much shorter than tau_thermal:

At 400 kHz (T_half = 1.25 us): streamer experiences ~100 RF cycles per tau_thermal. Heating is effectively continuous. Ratchet works. → Swords.
At 100 kHz (T_half = 5 us): thin streamers cool significantly between cycles. Ratchet is intermittent. → Branchy, noisy sparks.

The community-observed threshold: 300-600 kHz for sword sparks. This is not about breakdown physics — it's about whether the thermal ratchet can outrun cooling.

10. Branching Is a Competition

Discharges branch because of Laplacian instability at the propagating tip (same physics as viscous fingering). Streamers branch every ~10-20 diameters.

But branches compete for current. The channel resistance follows a nonlinear power law:

R = A / I^b      b = 1.84 for TC currents (1-10 A)

Because b > 1, the V-I curve has negative slope. A branch that gets slightly more current heats up, becomes more conductive, steals more current from its neighbors. This is positive feedback — one branch wins, the rest die.

Competition timescale: ~120-200 us (a few tau_g).

Burst mode (70-150 us pulses): too short for competition to resolve → many branches survive → bushy
QCW mode (10-20 ms ramp): competition resolves in <1 ms → single dominant channel → sword
Pulse-skip: intermediate — competition operates but with jitter → "sword-like but still branches"

11. QCW vs Burst: The Complete Picture

	QCW	Burst
Voltage	40-70 kV (!!)	200-600 kV
Duration	10-20 ms	70-150 us
Frequency	300-600 kHz	50-200 kHz
Channel type	Leader	Streamer
Branching	Suppressed by competition	Extensive
Epsilon	5-15 J/m	30-100+ J/m
Spark:secondary ratio	7-16x	2.5-3.6x
Morphology	Straight sword	Bushy tree
Mechanism	Thermal ratchet over many ms	Brute-force high voltage

The 15:1 voltage ratio (measured by davekni) is the single most striking number. QCW achieves leader formation at 40-70 kV because it has time — the ratchet accumulates thermal energy over 10-20 ms. Burst needs 200-600 kV because it must reach leader temperature in a single ~100 us pulse.

12. The Three Ramp Regimes

QCW ramp duration selects three distinct outcomes:

Too short (<5 ms): Insufficient time for streamer-to-leader transition. Segmented, gnarly sparks.
Optimal (10-20 ms): Leader forms within 1-2 ms, grows as single channel for remainder. Straight swords.
Too long (>25 ms): Leader reaches voltage-limited max length (capacitive divider). Excess energy drives lateral breakouts. "Hot, fat, bushy."

13. Putting It All Together

The complete causal chain:

RF drive at frequency f
    │
    ├─→ Resonant voltage gain → V_topload
    │
    ├─→ E_tip = kappa * V_tip / L  → inception when E_tip > E_inception
    │
    ├─→ Streamer channels form (cold, branched, high R)
    │
    ├─→ Hungry streamer: R drifts toward R_opt_power
    │       │
    │       ├─→ Power delivered: P = f(V_th, Z_th, R)
    │       │
    │       └─→ Growth: dL/dt = P / epsilon
    │
    ├─→ Thermal evolution (depends on mode):
    │       │
    │       ├─→ QCW: sustained ramp → thermal ratchet → leader formation
    │       │     → branch competition selects single channel
    │       │     → low epsilon → efficient growth → sword
    │       │
    │       └─→ Burst: short pulse → no time for leader transition
    │             → branches coexist → high epsilon → bushy
    │
    ├─→ Dynamic threshold (QCW only):
    │       Leader current → UV + heat + residual ionization + expansion
    │       → E_propagation_effective drops well below cold-air value
    │       → spark extends further at lower voltage
    │       → coupled V-P limit, not independent constraints
    │
    ├─→ Capacitive divider: C_sh grows with L
    │       → V_tip decreases → E_tip drops
    │       → eventually E_tip < E_propagation_effective → stalls
    │       → sub-linear scaling: L ~ sqrt(E) for burst
    │
    └─→ Final length set by:
            min(dynamic voltage limit, energy limit, ramp duration)

14. The Numbers That Matter

Quantity	Value	Why it matters
C_sh per foot	~2 pF	Sets voltage division rate
R_opt_power	10-100 kohm	Where plasma naturally sits
E_propagation (cold air)	0.4-1.0 MV/m	Field floor for cold streamer growth
E_propagation (leader tip)	Much lower (T3)	Dynamically reduced by UV/heat/ionization
tau_thermal (100 um)	~125 us	Streamer cooling timescale
tau_g (heating)	40 us	Conductance response speed
tau_g (cooling)	200 us	5:1 asymmetry drives ratchet
Competition time	~120-200 us	Branch winner decided
Burst ceiling	~80 us	ON time saturation (Steve Ward)
QCW optimal ramp	10-20 ms	Sweet spot for leader growth
Frequency threshold	300-600 kHz	Below this, no swords
QCW voltage	40-70 kV	15:1 less than burst
da Silva b exponent	1.84	b > 1 → current hogging
Fractal dimension	~2.2	Streamer tree space-filling

15. What We Don't Know

Exact branching power division — no validated current-sharing rule
Epsilon from first principles — still requires calibration
Time-resolved impedance during QCW ramp — never measured
Spectroscopic temperature of QCW sparks — 5000 K inferred, not measured
Arc current in any QCW spark — secondary current unmeasured
How C_sh scales with branching — qualitative only
Branching fraction of epsilon — how much energy goes to side branches vs other overhead
Dynamic threshold magnitude — how much is E_propagation reduced at a QCW leader tip?
Gas temperature ahead of leader tip — spectroscopic measurement needed

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