Three-Phase Power Formulas
Three-phase power formulas for kW, kVA, kVAR and line current, star and delta relations, two worked examples and a table of currents at 400 V.
The formulas
| Quantity | Three-phase (line values) | Single-phase |
|---|---|---|
| Apparent power S (kVA) | √3 × VL × IL / 1000 | V × I / 1000 |
| Real power P (kW) | √3 × VL × IL × pf / 1000 | V × I × pf / 1000 |
| Reactive power Q (kVAR) | √3 × VL × IL × sin φ / 1000 | V × I × sin φ / 1000 |
| Line current from kW | I = P × 1000 / (√3 × VL × pf) | I = P × 1000 / (V × pf) |
| Line current from kVA | I = S × 1000 / (√3 × VL) | I = S × 1000 / V |
VL is the line-to-line voltage in volts, IL the line current in amperes, pf the power factor and φ = arccos(pf). The loads are balanced and the waveforms are sinusoidal.
Star and delta
| Connection | Phase voltage | Phase current |
|---|---|---|
| Star (wye) | VL / √3 | IL |
| Delta | VL | IL / √3 |
Worked example: power from voltage and current
A balanced load draws 50 A at 400 V line-to-line with a power factor of 0.85.
- S = √3 × 400 × 50 / 1000 = 34.64 kVA
- P = S × pf = 34.64 × 0.85 = 29.44 kW
- φ = arccos(0.85) = 31.79°, so Q = S × sin φ = 18.25 kVAR
Worked example: current from power
A 75 kW load on 415 V with a power factor of 0.9.
- I = 75 × 1000 / (√3 × 415 × 0.9) = 115.9 A
Line current at 400 V and power factor 0.85
| Power | Line current |
|---|---|
| 1 kW | 1.698 A |
| 5 kW | 8.49 A |
| 10 kW | 16.98 A |
| 25 kW | 42.45 A |
| 50 kW | 84.9 A |
| 100 kW | 169.8 A |
Frequently asked questions
Why is there a √3 in three-phase power?
The line voltage is √3 times the phase voltage in a star connection, and the three phases are 120° apart, so the total of the three phase powers works out to √3 × VL × IL × pf.
Is the voltage in the formula line-to-line or line-to-neutral?
The formulas above use the line-to-line voltage. With the line-to-neutral voltage VLN the power is 3 × VLN × IL × pf.
Does kVA include the power factor?
No. kVA is apparent power and does not depend on the power factor. kW is kVA multiplied by the power factor.
Do these formulas work for an unbalanced load?
Not directly. They assume equal currents in the three lines. For an unbalanced load, add the power of each phase.
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