Single-phase and three-phase power
Homes in North America are fed with split-phase 120/240 V single-phase power. Commercial and industrial buildings usually get three-phase power, three alternating currents offset by 120°, which delivers power more smoothly and with less copper. Common US three-phase voltages are 208 V (with 120 V to neutral) and 480 V (with 277 V to neutral).
The formulas
Single-phase: S (kVA) = V × I ÷ 1,000
Three-phase: S (kVA) = √3 × V(line-to-line) × I ÷ 1,000
Then:
- Real power P (kW) = S × power factor
- Reactive power Q (kVAR) = S × √(1 − PF²)
- Current from power: I = P × 1,000 ÷ (V × √3 × PF) in three-phase
√3 ≈ 1.732 because the voltage between two phases 120° apart is √3 times the voltage from each phase to neutral.
Worked example
A motor control center runs at 480 V three-phase, drawing 50 A at a power factor of 0.85.
- Apparent power = 1.732 × 480 × 50 ÷ 1,000 ≈ 41.57 kVA
- Real power = 41.57 × 0.85 ≈ 35.33 kW (about 47 hp)
- Reactive power = 41.57 × √(1 − 0.85²) ≈ 21.90 kVAR
The same 50 A on a 240 V single-phase circuit at unity power factor delivers only 12 kW.
kW, kVA and power factor
- kW is the power that does useful work: turning a shaft, heating an element.
- kVA is what the transformer, generator and cables must carry.
- Power factor = kW ÷ kVA. Resistive loads such as heaters are close to 1; motors run at 0.8 to 0.9 at full load and lower at light load.
Generators and transformers are rated in kVA, because their heating depends on current, not on real power. A 50 kVA generator can supply only 40 kW at a 0.8 power factor.
Why power factor matters
A low power factor means more current for the same useful power, which means larger cables, more losses and, for many commercial customers, utility penalties below 0.9 or 0.95. Capacitor banks or drives with power factor correction bring it closer to 1.
Practical tips
- Use the line-to-line voltage in three-phase formulas, not the line-to-neutral voltage.
- For wire and breaker sizing, the NEC requires continuous loads to be calculated at 125% of their current.
- Motor nameplates give full-load amps; use them rather than computing from horsepower when sizing circuits.
- Balance single-phase loads across the three phases to avoid overloading one of them.