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A transformer works because of two principles:
(a) magentic fields are created by the movement of charged particles - this includes electicity (electrons) moving through a regular cable.
(b) a conductor will induce a charge if it is moved through a magnetic field.

In a transformer the primary coil is powered which results in a magnetic field being produced, as per (a). Because the magnetic field generated by a single cable is very small a transformer uses a coil of several hundred loops to increase the magnetic field produced.

A transformer needs to be powered by AC (alternating current). In AC the direction of the current flow changes (e.g. mains power typically changes direction 50 to 60 times per second). As the AC power changes polarity so does the magnetic field and the North-South poles swap with the AC power.

This changing of magnetic poles is equivalent to moving the magnetic field around. Even though the secondary coil doesn't move, because the magnetic field is moving a current is induced in the secondary coil as per (b).

The number of loops in the primary coil compared to the number of coils in the secondary loop is used to control the strength of the magnetic field, the secondary coil induction effect and thus the output voltage. This is a simple ratio, so if the secondary coil has twice as many loops as the primary then the output voltage will be twice the input voltage.

There are two types of Transformers, "step up" where the voltage is increased and "step down" where the voltage is reduced.

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