A Tesla coil is two resonant circuits that share one note. A capacitor bank dumps a bang of energy into a fat copper spiral; a tall, thin coil next to it catches that energy in about 17 microseconds and turns it into hundreds of kilovolts. On the bench above, drag the clip along the spiral to tune it, then bring the tube or the little lamp close and watch them light up with no wires at all.
Two circuits, one note
A Tesla coil is not one coil but two resonant circuits. The primary is a 25 nF capacitor bank wired to a flat spiral of copper tube; with the clip on 8¼ turns it has 20.9 µH and rings at 220 kHz. The secondary is 1000 turns of thin wire on a 110 mm pipe, 19.9 mH, capped by a toroid. Its 26 pF of capacitance comes from the toroid (17 pF) and the winding's own stray capacitance (9 pF).
Each circuit has its note, f = 1/(2π√LC). Moving the clip changes the primary's inductance, and so its note: from 455 kHz at 4 turns to 147 kHz at 12. The secondary's note is fixed; without the toroid it would be 378 kHz. Get within about a tenth of a turn and the two sing together.
f = 1 / (2π √(L C)) the note of each circuitThe spark gap is a switch
Between bangs a transformer charges the capacitors. At 400 W from the wall and 120 bangs a second, each charge is about 2.7 J at 14.6 kV. Then the gap fires: the air between the electrodes breaks down and becomes a conductor, which closes the primary circuit in a fraction of a microsecond. Capacitor and spiral now form an oscillator, and a current of about 500 A swings back and forth 220 000 times a second.
A spark gap is a switch that turns itself on. The hard part is turning it off. It should stop conducting at the instant the primary current passes through zero with almost no energy left. Fail, and energy flows back into the primary and burns in the arc. A rotary gap helps: its spinning electrodes pull the arc apart.
E = ½ C V² = η · P / 120 energy per bang: 2.7 J at 400 WEnergy crosses over
The primary's magnetic field threads the secondary. Here only 13 % of the coupling a perfect transformer would have (k = 0.13), which is typical. Two coupled circuits tuned to the same note do not ring at one frequency but two, 207 and 236 kHz, and their beat hands the energy from one side to the other. After about 17 µs, some four swings, the primary current is almost gone and the secondary holds most of the energy. If the gap goes out right then, the energy stays trapped up top.
Energy conservation sets the voltage: ½CpVp² = ½CsVs², so the gain is √(Cp/Cs) ≈ 31. The ideal answer is 450 kV. Our model, with losses, gives 379 kV. Detune the primary and only a fraction crosses before it swings back.
Vs = Vp · √(Cp / Cs) ideal gain ≈ 31Where lightning starts
A toroid is a smooth, fat electrode. At 379 kV its surface field is about 2.4 MV/m, still below the roughly 3 MV/m at which air breaks down. That is the point: it lets the voltage build instead of leaking away as corona. The breakout point does the opposite. It is a thin rod that concentrates the field at its tip, so the discharge starts there.
Once launched, a positive streamer needs far less: about 0.5 MV/m keeps it growing. And because the channels are still hot 8 ms later, each bang extends the last one, much as a lightning leader grows step by step. Hobbyists found an empirical rule, Freau's equation: reach in inches ≈ 1.7 × √watts. 400 W gives 86 cm; four times the power, twice the length.
L ≈ 1.7 √P inches Freau's empirical rule (well-tuned coil)The numbers
| Quantity | Note | Value |
|---|---|---|
| Secondary resonance | 19.9 mH with 26.3 pF (8.9 pF self + 17.4 pF toroid) | 220kHz |
| Primary at tune | 20.9 µH, 25 nF; first-notch window 8.11–8.33 turns (218–224 kHz) | 8¼turns |
| Coupling coefficient k | typical 0.05–0.2 | 0.13 |
| Energy per bang (400 W) | at 14.6 kV; 120 bangs/s, 80 % wall-to-capacitor | 2.7J |
| Primary peak current | V₀ / √(Lp/Cp), Z₀ = 29 Ω | ≈ 500A |
| Transfer time | ≈ 4 cycles; 17.3 µs lossless (half a beat of the 207/236 kHz modes) | ≈ 17µs |
| Top voltage | model; 450 kV ideal; real sparks load it lower | ≈ 380kV |
| Spark reach (400 W / 800 W) | Freau's empirical law | 86cm / 1.22m |
| Air breakdown / streamer growth | sea level; growth field for positive streamers (negative ones need about twice as much) | 3 / 0.47MV/m |
| Secondary wire | on a 110 mm × 550 mm winding | 346m |
What's simplified
- Lumped secondary. The secondary is a lumped L and C; its voltage profile comes from a simple transmission-line approximation (θ = 53.5°). A real secondary is a distributed helical resonator.
- Ideal spark gap. A switch with a fixed 0.6 Ω resistance that goes out at the first current zero after the current has dropped below 15 % of its peak. Real gaps are nonlinear; first-notch quenching is very hard to get in practice, and they usually go out at the 2nd or 3rd notch.
- Fixed coupling and supply. k = 0.13 whatever the tap position; the mains transformer and charging are idealized (80 % efficiency, exactly 120 bangs per second, schematic charging curve).
- Sparks don't load the circuit. In reality they add capacitance (real coils are tuned a little "long") and drain energy, so the displayed top voltage is the unloaded value.
- Spark length and shape. The length comes from Freau's empirical law, scaled by how much energy tuning delivers, not from a plasma simulation. The shapes are procedural but built the way real channels behave: crooked random walks that branch more toward their cold tips, hot channels re-used bang after bang and lifted by the warm air, a strike channel that wanders toward the ball, and a struck target that captures the streamers heading its way. Real sparks flicker 120 times a second; here at most one channel is replaced at a time, at most 20 times a second, with an afterglow (the eye and a camera blend several bangs), so the light never strobes. The violet light they cast on the toroid and the ball follows their light averaged over half a second, never a single bang.
- Component formulas. Inductances and capacitances use classic empirical formulas (Wheeler, Medhurst, toroid fit), good to about 5–10 %.
- Fields. Field lines, equipotentials and the breakdown zone come from simple ring-charge and current-loop models; the number of lines drawn is schematic.
- The tube and the lamp. The tube's ΔV comes from the ring-charge model; its glow is schematic: the gas strikes above a few hundred volts, then shines roughly with the voltage across it (as its discharge current does), white when strong, violet when weak. Its stand stops 0.95 m from the axis: closer, a real coil would arc to it, and if you drag it into the sparks' reach they strike its end cap. The pickup lamp's power comes from the same current-loop model (Faraday's law, a 4-turn loop into a 15 Ω bulb) and ignores its tiny pull on the primary; its filament glows steadily because it cools far more slowly than the 8 ms between bangs. The rings on the bench are the same models: equipotentials at the tube's height, and where the bulb's power halves.
- Slow motion. The Flow view replays one bang on its real clock: charging slowed ×1/300, the ring-down ×1/250 000, streamer growth ×1/10. The Exploded view's "Follow the current" is a guided tour: each of its nine stages shows its own slice of the same bang (a few milliseconds of charging, a few microseconds of ringing) slowed by the factor shown on screen, and the charge packet that runs along each wire is a narrative: in reality current flows through the whole charging path at once. The rotary gap stays in step: a quarter turn per bang, so an electrode faces each stator exactly when the gap fires; in the Whole view it spins at its real 1 800 rpm, a blur.
Myths
Sources
- Tesla coil — Wikipedia
- Operation of a Tesla coil — Richie Burnett
- Coupling, split frequencies and quenching — Richie Burnett
- Formulas for Tesla Coils — T. Paulin
- Medhurst self-capacitance table — Tesla Coil Mailing List
- Topload design — Kaizer Power Electronics
- Simulations of positive streamers in air: the stability field — arXiv
- Peek's law — Wikipedia
- Wireless Power Transfer via Strongly Coupled Magnetic Resonances — Kurs et al., Science (2007)
- Drude (1904) on Tesla transformers, English translation — arXiv
- Tesla coil construction — Kevin's Workbench
