Theory
A Tesla coil transformer operates in a significantly different fashion from a conventional (i.e., iron core) transformer. In a conventional transformer, the windings are very tightly coupled and voltage gain is determined by the ratio of the numbers of turns in the windings. This works well at normal voltages but, at high voltages, the insulation between the two sets of windings is easily broken down and this prevents iron cored transformers from running at extremely high voltages without damage.
With Tesla coils, unlike a conventional transformer (which may couple 97%+ of the fields between windings) a Tesla coil's windings are "loosely" coupled, with a large air gap, and thus the primary and secondary typically share only 10–20% of their respective magnetic fields. Instead of a tight coupling, the coil transfers energy (via loose coupling) from one oscillating resonant circuit (the primary) to the other (the secondary) over a number of RF cycles.
As the primary energy transfers to the secondary, the secondary's output voltage increases until all of the available primary energy has been transferred to the secondary (less losses). Even with significant spark gap losses, a well designed Tesla coil can transfer over 85% of the energy initially stored in the primary capacitor to the secondary circuit. The voltage achievable from a Tesla coil can be significantly greater than a conventional transformer, because the secondary winding is a long single layer solenoid widely separated from the surroundings and therefore well insulated, Also, the voltage per turn in any coil is higher because the rate of change of magnetic flux is high at high frequencies.
With the loose coupling the voltage gain is instead proportional to the square root of the ratio of secondary and primary inductances. Because the secondary winding is wound to be resonant at the same frequency as the primary, this voltage gain is also proportional to the square root of the ratio of the primary capacitor to the stray capacitance of the secondary.
Read more about this topic: Tesla Coil
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