Power in an AC Circuit: Average Power and Power Factor

Instantaneous power. The instantaneous power delivered to any AC circuit is simply the product of the instantaneous voltage and instantaneous current, $p=vi$. For a general series circuit with $v=v_0\sin\omega t$ and current lagging (or leading) by phase angle $\phi$, $i=i_0\sin(\omega t-\phi)$, so

Averaging over a full cycle. Using the trigonometric identity $\sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]$, this becomes

The first term, $\tfrac12 v_0 i_0\cos\phi$, is a CONSTANT (independent of time). The second term oscillates at TWICE the original frequency and, being a pure cosine, averages to exactly zero over any whole number of cycles. So the average power over a complete cycle is simply the constant first term:

Substituting the rms values $V_{rms}=v_0/\sqrt2$ and $I_{rms}=i_0/\sqrt2$ from Section 7.2,

Power factor. The quantity $\cos\phi$ appearing here is called the power factor of the circuit -- it is a pure number, between $0$ and $1$, that says what FRACTION of the product $V_{rms}I_{rms}$ (sometimes called the apparent power) is actually converted into genuine average power. When $\phi=0$ (voltage and current in phase, as in a pure resistor), $\cos\phi=1$ and every bit of $V_{rms}I_{rms}$ is real power; when $\phi=90^\circ$ (as in a pure inductor or pure capacitor), $\cos\phi=0$ and NO average power is delivered at all, however large the rms voltage and current individually are. …