Active, Reactive & Complex Power in AC Circuits

For sinusoidal voltage and current at one frequency, RMS phasors separate average energy transfer from the oscillatory exchange associated with stored fields. The single expression

$$\underline S=\underline V\underline I^*=P+jQ$$

contains both.

The power triangle #

Let $\phi=\angle\underline V-\angle\underline I$. Then

$$P=VI\cos\phi,\qquad Q=VI\sin\phi,\qquad |S|=VI=\sqrt{P^2+Q^2}.$$

Under the passive sign convention, an inductive load has $Q>0$ and current lags voltage; a capacitive load has $Q<0$ and current leads voltage.

For impedance $Z=R+jX$:

$$\underline S=|\underline I|^2Z=\frac{|\underline V|^2}{Z^*}.$$

This makes resistor loss and reactance sign explicit.

Instantaneous power underneath #

For $v=\sqrt2V\cos\omega t$ and $i=\sqrt2I\cos(\omega t-\phi)$:

$$p(t)=VI\cos\phi+VI\cos(2\omega t-\phi).$$

The first term is $P$; the second averages to zero over a cycle. Reactive power does not mean no instantaneous energy moves. It quantifies an exchange that returns over the cycle in ideal storage elements.

Complex power balance #

With consistent branch references:

$$\sum_k \underline S_k=0.$$

Both real and imaginary parts must balance. Calculate source complex power with the same passive convention rather than forcing it into a separate "generated" formula.

For distorted waveforms, total power factor remains $P/(V_{rms}I_{rms})$, while $\cos\phi$ describes only the fundamental displacement factor. Harmonic current can reduce total power factor even when the fundamental voltage and current are in phase.