The formulas
and the power triangle that ties them together:
Real power P (kW) does the work; apparent power S (kVA) is what the electrical infrastructure — cables, transformer, generator — must actually carry; reactive power Q (kVAR) shuttles energy in and out of motor magnetic fields without doing net work. A poor power factor means paying for infrastructure (and often utility penalties) to move power that does nothing.
Worked example
A motor draws 100 A at 415 V, power factor 0.85:
- S = 1.732 × 415 × 100 ÷ 1000 = 71.9 kVA
- P = 71.9 × 0.85 = 61.1 kW
- Q = √(71.9² − 61.1²) = 37.9 kVAR
Correcting the power factor to 0.95 would deliver the same 61.1 kW at only 89 A — an 11% current reduction, felt in every cable and breaker upstream.
Quick reference — full-load current per kW
| System | A per kW at PF 0.85 | A per kW at PF 1.0 |
|---|---|---|
| 415 V 3φ (India) | 1.64 | 1.39 |
| 400 V 3φ (EU) | 1.70 | 1.44 |
| 480 V 3φ (US) | 1.42 | 1.20 |
| 230 V 1φ | 5.12 | 4.35 |
Field notes
- Line-to-line, not line-to-neutral. The √3 already accounts for the phase geometry; feeding 240 V into the 3φ formula is the classic √3 error.
- Transformers and generators are rated in kVA precisely because they don't know your power factor — size them on S, not P.
- Unbalanced loads break the formula: the √3 expression assumes balanced phases. For seriously unbalanced systems, compute per-phase and sum.
- Motor nameplate kW is shaft output, not electrical input — divide by efficiency (~0.9–0.95) before estimating current.
Frequently asked questions
What is the three-phase power formula?
P (kW) = √3 × V_LL × I × cosφ ÷ 1000, using the line-to-line voltage and line current. Apparent power drops the power factor: S (kVA) = √3 × V_LL × I ÷ 1000.
How many amps is 100 kW at 415 V three-phase?
At power factor 0.85: I = 100,000 ÷ (1.732 × 415 × 0.85) ≈ 164 A. At unity power factor it would be 139 A.
What is the difference between kW, kVA and kVAR?
kW is real power doing work, kVA is apparent power (what the cables and transformer must carry), and kVAR is reactive power circulating in magnetic fields. They relate as kVA² = kW² + kVAR², with power factor = kW ÷ kVA.
Do I use line-to-line or line-to-neutral voltage?
The √3 formula uses LINE-TO-LINE voltage (e.g. 415 V in India, 400 V EU, 480 V US). Using line-to-neutral (240 V) with the √3 formula understates power by √3 — a classic error.
Provided for reference and education. Verify independently before use in safety-critical work. See our disclaimer.