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Transformers take advantage of electromagnetic induction and the properties of wire coils to change the voltage and current of a circuit. They are made up of three parts: the primary coil, which is a coil of copper wire, the secondary coil, a second copper coil with a different number of windings than the primary, and the core, which is a ring of iron on which the primary and secondary are wound.
When a current is sent through the primary coil, it creates a magnetic field that induces a current in the core, which in turn induces a current in the secondary. While the core is not necessary, it is more energy-efficient than simply inducting the electricity directly from one coil to the other. In accordance with Faraday's Law, this only occurs when the magnetic field of the primary is changing, which means the current in the primary must change direction. Thus, only alternating current will allow a transformer to function.
The key part of a transformer's function lies in the ratio of turns of wire from the primary to the secondary coils. If there are more turns in the secondary than the primary, then the output voltage will be higher. If the opposite is true, then the voltage will be lower. For example, if the ratio of turns from the primary to the secondary is 1:10, or one turn for every ten turns, then the output voltage will be ten times greater.
Choosing the Transformer is the beginning of any Tesla coil build. Its output voltage and current determines how big the coil can be, and how long the resulting streamers will be. My coil uses a 15,000 volt, 60 milliamp neon sign transformer, the most common kind of transformer used in Tesla coils. It is an old, heavy one without any sophisticated electronics so the Tesla coil cannot hurt it with random spikes in current. More modern NSTs may have sensitive circuitry or even automatic shutoff circuits built in, which make them ill-suited to powering a Tesla coil.