Introduction of Power in AC Circuits

The power of a circuit is the rate at which it consumes electric energy, given at any instant by the product of the instantaneous voltage and current. With $v=V_m\sin\omega t$ and, allowing for a general phase angle $\phi$ between voltage and current (as in a series RLC circuit), $i=I_m\sin(\omega t+\phi)$, the instantaneous power is

Expanding $\sin(\omega t+\phi)=\sin\omega t\cos\phi+\cos\omega t\sin\phi$ and multiplying out gives

Averaging this over a complete cycle, and using the standard time-averages $\langle\sin^2\omega t\rangle=\tfrac12$ and $\langle\sin\omega t\cos\omega t\rangle=0$, gives the average power

or, in terms of RMS values,

$P_{av} = V_{RMS}I_{RMS}\cos\phi \qquad (4.55)$ …