Tesla's Coil
To investigate high-frequency and high-voltage, Tesla invented an apparatus that pushed the limits of electrical understanding. None of the circuit's typical components were unknown at the time, but its design and operation together achieved unique results,not the least because of Tesla's masterful refinements in construction of key elements, most particularly of a special transformer, or coil, which is at the heart of the circuit's performance (a picture of the original schematic is shown below).
Such a device first appeared in Tesla's US patent No. 454,622 in 1891 for use in new, more efficient lighting systems. In its basic form, the circuit calls for a power supply, a large capacitor, the coil (an air core transformer) itself, and adjustable spark-gap electrodes. Why these components, and what do they accomplish?
Oscillators and Tuned Circuits
Capacitors (or condensers) and inductors (or coils) are, electrically speaking, somewhat opposite in operation. Whereas current builds quickly in a capacitor as it charges up, voltage lags. In an inductor, voltage is felt immediately, while current is retarded as it works against the magnetic field its own passage builds in the coil. If a coil and condenser are sized and selected to act with exactly opposite timing—with voltage peaking in the coil just as it reaches a minimum in the capacitor—then the circuit may never reach an electrically quiet, stable state. A bit like the sloshing of water back and forth in a tub or pushing a child on a swing, current and voltage can be made to chase each other back and forth, from end to end of the circuit. (An oscillator of this kind is often called a tank circuit.)
The Spark Gap
To set the oscillator "ringing," Tesla employed sudden discharges, sparks, across an adjustable gap between two electrodes. Voltage on a capacitor builds until it reaches a level at which air in the gap breaks down and ceases to be an insulator; at that point the gap fires and completes the circuit.
The initial impulse is very powerful, all the energy stored over several microseconds is released in a rush, and that impulse is itself transformed to a somewhat higher voltage in passing from the primary coil windings to those of its secondary. This completes but a single cycle in the circuit's operation. The air gap restores itself as an insulator, and the capacitor begins to charge until it reaches a breakdown value once again. The whole process can repeat itself many thousands times per second.
The transformer's secondary reacts quickly to a sudden energy spike and, most importantly, to concentrate voltage at one end as a standing wave. Its length is calculated so that wave crests, as they reach the end and are reflected back, meet and exactly reinforce the waves behind them. The net effect is a wave, a voltage peak, which appears to stand still (check out the link to Ritchie's site on my home page, he has some great animations that show this).
So what can you use it for?
If, as happened in practice, Tesla made an antenna of the high-voltage end of his secondary, it became a powerful radio transmitter. In fact, in the early decades of radio, most practicable radios utilized Tesla coils in their transmission antennas. In 1943 Tesla was actually credited with the invention of radio (rather than Marconi). Even though I am old, 52 LOL, I went to school well after 1943 and the text books still introduced "Marconi" as the "Father of Radio" (The case was resolved in the U.S. Supreme Court by overturning most of Marconi's patents. Tesla is just now getting the full credit he deserves for his many inventions). Tesla himself used larger or smaller versions of his invention to investigate fluorescence, x-rays, radio, wireless power, biological effects, and even the electromagnetic nature of the earth and its atmosphere.
Today, high-voltage labs often operate such devices, and amateur enthusiasts (like us) around the world build smaller ones to create arcing, streaming electrical displays—it is not difficult to reach a quarter million volts. The coil has become a commonplace in electronics, used to supply high voltage to the front of television picture tubes, in a form known as the flyback transformer and is used in almost all automobiles ignition circuits. The PBS web site has additional great information on Tesla History and his other inventions, I highly recommend it.