Tesla coils are interesting devices. Many people think that they are nothing more than transformers with high turns ratios, but they aren't really much like conventional transformers. A conventional transformer is an inductive device; capacitive effects are usually minimal. Conventional transformers have very tight coupling between the primary and secondary windings; this is accomplished by using magnetic cores which have high permeability. Tesla coils have air cores. this is for several interesting reasons: standard magnetic materials typically have electrical conductivity, requiring massive insulation to prevent flashovers in a high-voltage transformer. The second reason is that many magnetic materials are very lossy at high frequencies, which is where Tesla coils operate.
A Tesla coil is more like a resonance device; it consists of two LC tank circuits that are weakly coupled. An LC tank circuit is like a mass on a spring; it has its own natural frequency of oscillation. What oscillates is the electric charge in the tank circuit. The electric current stop flowing periodically, just like a mass on a spring stops moving during part of its oscillation,. This occurs when the capacitor is fully charged and the electric current in the circuit has dropped to zero. If the system is allowed to resonate, the capacitor will discharge into the inductor and cause current to flow again.
The inductive part of the circuit; the L, shares electrical energy with the capacitance part of the circuit, the C. The current flowing through the inductance can be quite large. The energy of the inductor is equal to one half L. I. squared. that current charges the capacitor in the circuit. The energy of the capacitor is 1/2CV^2. Because C is typically a very, very small number (~100 pF for my big coil) the quantity V^2 is very, very high. This is how the Tesla coil creates high-voltage: an inductor force-feeds lots of charge into a small capacitor. The voltage across the capacitor is equal to the charge Q divided by the capacitance C. So when C is very small the voltage V across it will be very high when a fixed charge Q is forced into it. some numbers: if we force 0.1 millicoulombs of charge into a 100 pF capacitor, the voltage across it will be 10-4/10-10 or 1,000,000 volts! As the current in an inductor starts to decrease, the magnetic field transfers energy to the electrons in the conductor and forces current to keep flowing. This is how the capacitor can get charged to high voltages.
The LC tank circuit in a Tesla coil is typically very lossy; this is because electrical streamers or arcs discharge from the toroid. Thus the sinewave of current decays in amplitude over time. The following image shows the waveforms, corresponding to the current flow in both the primary and secondary windings of my Tesla coil. You can see that the envelope of the two sine waves decay in about 56 µs. The top curve is the current waveform of the primary circuit in my Tesla coil; the bottom is the current waveform of the secondary circuit.
The oscilloscope voltage scale is 5 V per division on the top curve. I used a 10X attenuator between the input of the scope and the current transformer. The current transformer that I used to measure the current puts out 1 V for every 40 amps of current. Therefore the peak-to-peak amplitude of this waveform is about three divisions or 150 V, corresponding to a current of 6000 amps peak-to-peak! Lots of current flows in Tesla coil primary circuits. The secondary waveform was captured with the voltage division of 200 mV. The peak-to-peak amplitude is about 600 mV or 2.4 amps. The time scale of the oscilloscope is 8 µs per division. The period of oscillation of this Tesla coil is about 7 µs, which corresponds to a resonant frequency of about 150 kHz.