Transformer

A transformer is two coils wound around a common ferromagnetic core, coupled by mutual inductance M. Drive the primary coil (N₁ turns) with an AC voltage V₁, and the changing flux in the core induces an AC voltage V₂ in the secondary coil (N₂ turns) in the ratio V₂/V₁ = N₂/N₁. The currents go inversely: I₂/I₁ = N₁/N₂. Power is conserved at the ideal limit: V₁I₁ = V₂I₂ (real transformers are 95–99% efficient, losing the remainder to core hysteresis, eddy currents, and copper resistance).

The mathematics follows from Faraday's law. Assume a tightly-coupled core so every turn sees the same flux Φ. Then V₁ = N₁ dΦ/dt (primary back-EMF) and V₂ = N₂ dΦ/dt (secondary induced EMF). Dividing gives V₂/V₁ = N₂/N₁ directly. Conservation of energy plus ideal coupling gives the inverse ratio on the currents. The same core flux is therefore threading both windings; what's actually being transferred through the core is not electrical energy but magnetic flux, and Poynting's theorem applied to the core and the gap shows energy flowing through the space between the windings via the Poynting vector, not through the core material itself. Physics of the device: it is the classic example of AC electromagnetic energy transfer.

Transformers are the reason the electrical grid exists in its present form. Generating stations produce AC at about 20 kV. Step-up transformers raise this to 110–750 kV for long-distance transmission (at higher voltage, the same power carries less current, and I²R losses along the transmission line fall as the square of the current). At each substation a step-down transformer drops to 11 kV for local distribution, and at each street-level pole a further step-down transformer brings it to the 120/240 V of household outlets. Without transformers, grids larger than a few miles would lose more power in the transmission lines than they deliver. Inside equipment, transformers isolate circuits galvanically, match impedances between stages, and provide safety isolation for medical equipment, test instruments, and power supplies.