Three-Phase Power and Why the Square Root of Three Appears

Finds three-phase power as P = √3 × line voltage × line current × power factor. The √3 appears because the line voltage is √3 times the phase voltage. The apparent and reactive powers are given as well.

This finds the power in a three-phase alternating supply, the form in which nearly all industrial and large-building machinery is fed.

P = \sqrt{3} \, V I \cos\phi

Here $V$ is the line voltage, $I$ the line current, $\cos\phi$ the power factor and $P$ the real power.

Where the square root of three comes from

A three-phase supply carries three voltages 120 degrees apart. The line voltage, measured between any two lines, is $\sqrt{3}$ times the phase voltage of a single leg. Because the two voltages being subtracted are 120 degrees out of step, the difference is not simply double but $\sqrt{3}$ times one of them.

The power in three legs is $3 V_p I \cos\phi$ , and substituting $V_p = V / \sqrt{3}$ leaves $3 / \sqrt{3} = \sqrt{3}$ , which is the formula above.

Worked example

A line voltage of 200 V, a line current of 10 A and a power factor of 0.8 give about 2771 W of real power.

The apparent power is $\sqrt{3} \times 200 \times 10 = 3464$ VA, and multiplying by the power factor of 0.8 leaves 2771 W. The reactive power is $3464 \times 0.6 = 2078$ var, and the phase voltage is 200 ÷ 1.732 = 115.5 V.

Points to watch

The voltage and current here are the line values. Entering phase values instead puts the answer out by a factor of $\sqrt{3}$ .

The formula assumes a balanced load, with the three phases loaded equally. An unbalanced load has to be worked out phase by phase.