How to Calculate Power Factor

In an alternating circuit the voltage and current fall out of step, so only part of the product of the two does any work. That share is the power factor. From the real power, the voltage and the current, this finds the power factor, the apparent and reactive powers and the phase angle.

In an alternating circuit the voltage and current can fall out of step. When they do, only part of their product does any work, and that share is the power factor.

\cos\phi = \dfrac{P}{S}, \qquad S = V I

Here $P$ is the real power, $S$ the apparent power, $V$ and $I$ the root-mean-square voltage and current, and $\phi$ the phase angle.

How the three powers relate

Real, reactive and apparent power form a right-angled triangle.

S^2 = P^2 + Q^2

The quantity $Q$ is the reactive power. It does no work, merely shuttling back and forth between the supply and the inductance or capacitance of the load, but it still flows in the cables, which must be sized to carry it. That is why correcting the power factor lets the whole installation be smaller.

Worked example

With 800 W of real power at 100 V and 10 A, the power factor is 0.8.

The apparent power is 100 × 10 = 1000 VA, so the factor is 800 ÷ 1000 = 0.8. The reactive power is $\sqrt{1000^2 - 800^2} = 600$ var, and the phase angle is $\cos^{-1} 0.8 = 36.87$ degrees.

Points to watch

A power factor cannot exceed 1. If the real power entered is greater than the apparent power, one of the readings is wrong and the calculation stops rather than returning a nonsense figure.

Enter root-mean-square values. Where the waveform is distorted, the figure here is the displacement power factor, which differs from the true power factor once harmonics are counted.