A Tesla coil works by putting spikes of HV into a pair of LC circuits tuned to match their respective resonances. unlike a traditional transformer which uses the close coupling of the windings to magnetically couple them together to step voltage up, the Tesla coil is very loosely coupled and uses the resonance to achieve the voltage amplification.
in this simulation the switch is a spark gap. close the switch to simulate a spark. current will flow through the spark, through the primary side of the coil and back into the primacy capacitor to start oscillation. this causes the secondary side to oscillate as well, and the closely matched resonance amplifies the voltage.
its also a lot of maths
http://www.richieburnett.co.uk/operation.html#operation
