Transformer

Mutual inductance is the operating principle behind every kind of transformer -- a device that changes an alternating voltage from one value to another (step-up or step-down), commonly seen mounted on roadside poles supplying local power distribution. A transformer consists of two electrically-insulated coils, the primary (input) and secondary (output), both wound on a common soft-iron core, which channels almost all of the primary's flux through the secondary as well (i.e. couples them as tightly as practically possible, $K\approx1$).

When an alternating voltage is applied to the primary, the resulting sinusoidally-varying primary current produces a correspondingly sinusoidal flux in the shared iron core; since this same changing flux links BOTH coils, an emf is induced in each. If $\Phi$ is the flux linked per turn common to both coils at instant t, and $N_p$, $N_s$ are the numbers of turns in the primary and secondary respectively, the flux linked with each coil is $\Phi_p=N_p\Phi$ and $\Phi_s=N_s\Phi$, so the induced emfs are $e_p=-N_p\frac{d\Phi}{dt}$ and $e_s=-N_s\frac{d\Phi}{dt}$; dividing, $\frac{e_s}{e_p}=\frac{N_s}{N_p}$ -- the transformer equation. The ratio $N_s/N_p$ is called the turns ratio (or transformer ratio).

For an IDEAL transformer (no losses), input power exactly equals output power: $e_pi_p = e_si_s$, i.e. $\frac{e_s}{e_p}=\frac{i_p}{i_s}$. Combining this with the transformer equation gives $\frac{e_s}{e_p}=\frac{N_s}{N_p}=\frac{i_p}{i_s}$ -- voltage and current transform in EXACTLY inverse ratio to each other, so power is conserved. Two cases follow directly: if $N_s > N_p$, then $e_s > e_p$ (a step-up transformer) but $i_p > i_s$ -- the secondary delivers a higher voltage at correspondingly LOWER current; if $N_s < N_p$, then $e_s < e_p$ (step-down) but $i_p < i_s$ -- the secondary delivers a lower voltage at correspondingly higher current. …