Power factor

With v = vm sin ωt and i = im sin(ωt + φ), the average power over a cycle is P = (vm im/2) cos φ = VI cos φ. It can also be written P = I²Z cos φ.

cos φ is the power factor. The power depends on the phase angle as well as on V and I.

Purely resistive circuit: φ = 0, cos φ = 1, and the power is the largest possible for that V and I.

Pure L or pure C: φ = π/2, cos φ = 0, and no power is used even though current flows. Such a current is called wattless current.

In any LCR circuit, power is used up only in the resistor. At resonance φ = 0, cos φ = 1 and P = I²R, the most power for the circuit.

Example 7.7: for a given power at a given voltage, a low power factor means a bigger current and so bigger I²R loss in the lines. A lagging wattless current can be cancelled by a capacitor connected in parallel, which raises the power factor towards 1.

Example 7.8: peak 283 V at 50 Hz on R = 3 Ω, L = 25.48 mH, C = 796 μF. XL = 8 Ω, XC = 4 Ω, Z = 5 Ω, φ = −53.1° (current lags), I = 40 A, P = 4800 W and power factor 0.6.

Example 7.9: the same circuit resonates at ω₀ = 1/√(LC), about 222 rad/s or 35.3 Hz. There Z = R = 3 Ω, the rms current is 66.7 A and the power is 13.35 kW, more than at 50 Hz.