Transformer principle and maximum efficiency

A transformer is a static device that transfers electrical energy from one circuit to another at the same frequency by electromagnetic induction, usually changing the voltage level.

Principle (mutual induction). It has two windings, primary and secondary, wound on a laminated silicon-steel core. When an alternating voltage $V_1$ is applied to the primary, an alternating current flows and produces an alternating flux $\Phi$ in the core. This flux links the secondary winding and, by Faraday's law, induces an emf in it:

$$e=-N\frac{d\Phi}{dt}$$

The same flux links both windings, so the emf per turn is the same, giving

$$\frac{E_2}{E_1}=\frac{N_2}{N_1}=K$$

If $N_2>N_1$ it is a step-up transformer; if $N_2<N_1$ it is step-down. There is no electrical connection between windings, and ideally input power = output power ($V_1I_1=V_2I_2$).