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The "Tesla coil"

Tesla invented his "Tesla Coil" around 1891 while he was repeating and then expanding on Heinrich Hertz' experiments that had discovered electromagnetic radiation three years earlier. Tesla decided to power his setup with the high speed alternator he had been developing as part of an improved arc lighting system but found that the high frequency current overheated the iron core and melted the insulation between the primary and secondary windings in the Ruhmkorff coil originally used in Hertz setup. To fix this problem Tesla changed the design so that there was an air gap instead of insulating material between the primary and secondary windings and made it so that the iron core could be moved to different positions in or out of the coil. Tesla also found he needed to put the capacitor normally used in such setups between his alternator and the coil's primary winding to avoid burning out the coil. By adjusting the coil and the capacitor Tesla found he could take advantage of the resonance set up between the two to achieve even higher frequencies.
In Tesla's coil transformer the capacitor, upon break-down of a short spark gap, became connected to a coil of a few turns (the primary winding set), forming a resonant circuit with the frequency of oscillation, usually 20–100 kHz, determined by the capacitance of the capacitor and the inductance of the coil. The capacitor was charged to the voltage necessary to rupture the air of the gap during the input line cycle, about 10 kV by a line-powered transformer connected across the gap. The line transformer was designed to have higher than normal leakage inductance to tolerate the short circuit occurring while the gap remained ionized, or for the few milliseconds until the high frequency current had died away.
The spark gap is set up so that its breakdown occurs at a voltage somewhat less than the peak output voltage of the transformer in order to maximize the voltage across the capacitor. The sudden current through the spark gap causes the primary resonant circuit to ring at its resonant frequency. The ringing primary winding magnetically couples energy into the secondary over several RF cycles, until all of the energy that was originally in the primary has been transferred to the secondary. Ideally, the gap would then stop conducting (quench), trapping all of the energy into the oscillating secondary circuit. Usually the gap reignites, and energy in the secondary transfers back to the primary circuit over several more RF cycles. Cycling of energy may repeat for several times until the spark gap finally quenches. Once the gap stops conducting, the transformer begins recharging the capacitor. Depending on the breakdown voltage of the spark gap, it may fire many times during a mains AC cycle.
A more prominent secondary winding, with vastly more turns of thinner wire than the primary, was positioned to intercept some of the magnetic field of the primary. The secondary was designed to have the same frequency of resonance as the primary using only the stray capacitance of the winding itself to ground and that of any "top hat" terminal placed at the top of the secondary. The lower end of the long secondary coil must be grounded to the surroundings.
The later and higher-power coil design has a single-layer primary and secondary. These Tesla coils are often used by hobbyists and at venues such as science museums to produce long sparks. The American Electrician gives a description of an early Tesla coil wherein a glass battery jar, 15 × 20 cm (6 × 8 in) is wound with 60 to 80 turns of AWG No. 18 B & S magnet wire (0.823 mm²). Into this is slipped a primary consisting of eight to ten turns of AWG No. 6 B & S wire (13.3 mm2) and the whole combination is immersed in a vessel containing linseed or mineral oil. (Quelle/Text: https://en.wikipedia.org/wiki/Tesla_coil)

Resonant Transformer
The specialized transformer used in the Tesla coil, also called the resonance transformer, oscillation transformer or radio-frequency (RF) transformer, functions differently from an ordinary transformer used in AC power circuits. While an ordinary transformer is designed to transfer energy efficiently from primary to secondary winding, the resonant transformer is also designed to temporarily store electrical energy. Each winding has a capacitance across it and functions as an LC circuit (resonant circuit, tuned circuit), storing oscillating electrical energy, analogously to a tuning fork. The primary winding (L1) consisting of a relatively few turns of heavy copper wire or tubing, is connected to a capacitor (C1) through the spark gap (SG). The secondary winding (L2) consists of many turns (hundreds to thousands) of fine wire on a hollow cylindrical form inside the primary. The secondary is not connected to an actual capacitor, but it also functions as an LC circuit, the inductance (L2) resonates with (C2), the sum of the stray parasitic capacitance between the windings of the coil, and the capacitance of the toroidal metal electrode attached to the high voltage terminal. The primary and secondary circuits are tuned so they resonate at the same frequency, they have the same resonant frequency. This allows them to exchange energy, so the oscillating current alternates back and forth between the primary and secondary coils. (Quelle/Text: https://en.wikipedia.org/wiki/Tesla_coil)

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