Work and power of alternating current

The instantaneous power of alternating current is p = u·i; for u = Uₘcosωt and i = Iₘcos(ωt + φ) it varies in time and is negative during part of the period (energy returns to the source). The mean power over a period, in terms of the effective values of current and voltage, is P = U·I·cosφ (U and I effective), and the work of the current is A = U·I·t·cosφ. The factor cosφ is the power factor; for an R–L–C circuit cosφ = R/Z. In a circuit with only active resistance φ = 0, cosφ = 1 and P = U·I; in a circuit with only a coil or a capacitor φ = ±π/2, cosφ = 0 and P = 0: even though a current flows, the mean power is zero. If cosφ is small, a larger current is needed for the same power (I = P/(U·cosφ)), so the heat losses (I²r) in the wires grow. Electric motors have a large inductive resistance, so their cosφ is low; to raise it, compensating capacitors are connected to the factory's network and motors are not run idle.