Transformer definition, principle, emf equation

Definition. A transformer is a static electrical device that transfers electric power from one circuit to another at the same frequency through electromagnetic induction, normally raising or lowering the voltage.

Operating principle. It works on mutual induction. An alternating voltage applied to the primary winding drives an alternating flux in the laminated iron core. This flux links the secondary winding and induces an emf in it (Faraday's law). The emf in each winding is proportional to its number of turns, so $\frac{E_2}{E_1}=\frac{N_2}{N_1}$.

emf equation. Let $N_1$ = primary turns, $N_2$ = secondary turns, $\Phi_m$ = maximum flux in the core (Wb), $f$ = frequency (Hz). The flux varies sinusoidally:

$$\Phi=\Phi_m\sin\omega t$$

The average rate of change of flux in a quarter cycle is

$$\frac{d\Phi}{dt}\Big|_{av}=\frac{\Phi_m}{T/4}=4f\Phi_m$$

So the average emf per turn is $4f\Phi_m$ volts. For a sine wave, the form factor is 1.11, so rms emf per turn $=1.11\times4f\Phi_m=4.44f\Phi_m$. Hence:

$$E_1=4.44\,f\,N_1\,\Phi_m \qquad E_2=4.44\,f\,N_2\,\Phi_m$$

Dividing, the transformation ratio is

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