Power in ac circuits and transformers

In an ac circuit, the average power is not simply VI, because the current and voltage are out of step:

P = V I cos φ, with rms V and I. cos φ = R / Z is the power factor.

• Pure resistor: φ = 0, cos φ = 1, P = VI.
• Pure L or C: φ = 90°, cos φ = 0, P = 0. The current flows but does no work: a wattless current.
• At resonance: cos φ = 1, and P = V²/R.

Example: a motor on 220 V draws 5 A with power factor 0.8: P = 220 × 5 × 0.8 = 880 W.

A low power factor is costly. To deliver the same power, a larger current must flow, and the losses I²R in the supply lines grow. Factories with many motors (inductive loads) add capacitors to cancel the lagging current and bring cos φ close to 1.

Transformer: two coils on one iron core. An ac current in the primary makes a changing flux, which induces an emf in the secondary (mutual induction). For an ideal transformer:

V_s / V_p = N_s / N_p = I_p / I_s

• Step-up (more secondary turns): raises voltage, lowers current.
• Step-down: lowers voltage, raises current.

Power stays the same (V_p I_p = V_s I_s), so a transformer cannot raise both.

Example: 220 V to 11 V needs a turns ratio of 20 : 1, say 2000 primary and 100 secondary turns.

Why high-voltage transmission: sending 10 kW at 200 V needs 50 A; at 20 000 V only 0.5 A. Since the line loss is I²R, a hundredth of the current gives a ten-thousandth of the loss. The voltage is stepped up at the power station and stepped down near our homes.

Real transformers lose some energy: in the copper windings (heat), in eddy currents in the core (reduced by laminating it), and in magnetising the core again and again.