Power factor and three-phase

Why it matters

Utilities bill large customers for poor power factor, motors run hotter when it is low, and conductors, breakers and transformers must be sized for the apparent power, not just the useful power. Factories, hospitals and data centres are supplied with three-phase power, and every electrician and electrical engineer needs to calculate three-phase loads. This lesson connects the phase shift from the last lesson to real, reactive and apparent power.

Three kinds of power

When current is out of phase with voltage, not all of the current does useful work. Engineers split power into three parts:

• Real (active) power, P, in watts (W): the power actually converted into heat, light or motion.
• Reactive power, Q, in volt-amperes reactive (VAR): power that flows back and forth to build the magnetic fields of motors and transformers (or the electric fields of capacitors). It does no net work but still flows in the wires.
• Apparent power, S, in volt-amperes (VA): simply Vrms × Irms. This is what the wires, transformer and breaker "feel".

They form a right triangle, the power triangle:

S² = P² + Q²
P = S × cos θ
Q = S × sin θ

where θ is the phase angle between voltage and current.

Power factor

power factor (pf) = P / S = cos θ

• pf = 1 (θ = 0°): purely resistive, like a heater. All current does work.
• pf below 1: some current is reactive. Induction motors at light load can have a power factor of 0.7 or less.
• Lagging pf: inductive loads (motors, transformers), current lags voltage.
• Leading pf: capacitive loads.

Worked example 1. A single-phase motor draws 10 A at 240 V with a power factor of 0.8 lagging.

S = 240 × 10 = 2,400 VA
P = 2,400 × 0.8 = 1,920 W
Q = √(2,400² − 1,920²) = 1,440 VAR

The wires carry 10 A, but only 1,920 W is doing useful work. If the power factor were 1.0, the same 1,920 W would need only 8 A.

Power-factor correction

Because capacitors draw leading reactive power and motors draw lagging reactive power, a capacitor bank placed near motors can cancel some of the motor's Q. The total current drops, losses fall, and the utility may stop charging a power-factor penalty. Sizing correction capacitors is engineering work; install and maintenance is done by qualified electricians under the site's safety program.

Three-phase power

A three-phase supply has three AC voltages of the same size, each shifted 120° from the others. Benefits:

• Power delivery is smooth (the total never drops to zero), so motors run with less vibration.
• Three-phase motors start and run without extra start windings.
• For the same power, less copper is needed than with single-phase.

Two voltages matter:

• Line voltage (VL): between any two phase conductors, for example 208 V, 480 V or 400 V.
• Phase voltage (Vph): between one phase conductor and neutral in a wye (star) system. VL = √3 × Vph. That is why 120/208 V and 277/480 V systems come in those pairs: 120 × 1.732 = 208, and 277 × 1.732 = 480.

For a balanced three-phase load:

S = √3 × VL × IL
P = √3 × VL × IL × pf

Worked example 2. A three-phase motor draws 20 A per line at 480 V with a power factor of 0.85.

S = 1.732 × 480 × 20 = 16,628 VA ≈ 16.6 kVA
P = 16,628 × 0.85 = 14,134 W ≈ 14.1 kW

Worked example 3. What line current does a balanced 30 kW, 0.9 pf load draw at 208 V?

IL = P / (√3 × VL × pf) = 30,000 / (1.732 × 208 × 0.9) = 92.5 A

Common mistakes

• Using P = V × I on a motor. That gives VA, not watts, unless you multiply by the power factor.
• Forgetting √3 in three-phase. Multiplying by 3 instead of √3 when using line voltage overstates power by 73%.
• Mixing line and phase voltage. Check whether a value is phase-to-phase or phase-to-neutral.
• Sizing equipment on watts. Transformers and generators are rated in kVA because they must carry the full current.

Safety and real-world requirements

Three-phase industrial systems carry high fault currents and serious arc-flash hazards. These calculations are paper exercises here. Working on such equipment requires employer training, an energized-work policy based on NFPA 70E (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e), lockout/tagout under OSHA rules (https://www.osha.gov/control-hazardous-energy) and, for installation work, a licensed electrician. Checked on 2026-10-23.

Recap

• Real power P (W) does work; reactive Q (VAR) sustains fields; apparent S (VA) = V × I.
• pf = P / S = cos θ; motors are usually lagging.
• Balanced three-phase: S = √3 × VL × IL, P = S × pf; VL = √3 × Vph in a wye system.

Try it

Open the Do tab and write the real-power formula in a single- and three-phase power calculator.