In an AC circuit, voltage and current can be out of phase, so instantaneous power

$$p(t)=v(t)i(t)$$

can vary during the cycle and can even become negative temporarily as stored energy returns to the source.

Average real power

For sinusoidal voltage and current with phase difference $\phi$,

$$P=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi.$$

This is the average real power transferred irreversibly to loads such as resistors or mechanical output.

Reactive power

Energy that oscillates back and forth through ideal inductors and capacitors is described by reactive power:

$$Q=V_{\mathrm{rms}}I_{\mathrm{rms}}\sin\phi.$$

Complex and apparent power

Complex power is

$$S=P+iQ,$$

with magnitude

$$|S|=V_{\mathrm{rms}}I_{\mathrm{rms}}.$$

The magnitude is called apparent power.

Power factor

The power factor is

$$\mathrm{pf}=\cos\phi=\frac{P}{|S|}.$$

For the same real power and voltage, a low power factor requires greater RMS current, increasing conductor losses and equipment ratings. Power-factor correction aims to reduce unnecessary reactive current.