The three-phase power triangle
Three-phase AC systems have three related power quantities that beginners often confuse:
- Real power (P, kW) — power that actually does work: turns motors, heats resistances, drives lights.
- Reactive power (Q, kVAR) — power that flows back and forth to magnetize inductive components (motors, transformers). Does no useful work but must be supplied.
- Apparent power (S, kVA) — the vector sum of P and Q. This is what the supply, transformer, and cables must actually be sized to handle.
- Power factor (cos φ) — the ratio P/S. A perfect PF of 1.0 means all supplied power is real; typical industrial loads run at 0.7 to 0.9.
These form a right triangle: S² = P² + Q² and PF = cos φ = P / S. Given any two of {P, Q, S, PF}, the calculator finds the other two — plus voltage and line current when they’re relevant.
How to use this calculator
Pick which quantities you know from the “I know” dropdown at the top. Six scenarios are supported:
- Voltage, Current & Power Factor — the classic setup when you’ve measured line values with a meter.
- Real Power & Power Factor — for spec-sheet inputs. Add voltage to also get line current.
- Apparent Power & Voltage — sizing based on transformer or generator kVA rating.
- Real Power & Reactive Power — when you’ve measured both P and Q directly.
- Real Power, Voltage & Current — derive PF from actual measurements.
- Power Factor Correction — size a capacitor bank to improve PF from an existing value to a target.
Formulas
Standard three-phase convention: voltage is line-to-line, current is line current.
S (VA) = √3 × VLL × ILP (W) = √3 × VLL × IL × cos φQ (VAR) = √3 × VLL × IL × sin φS² = P² + Q²(Pythagorean relationship)PF = cos φ = P / S
Why the √3 factor?
In a balanced three-phase system, the three phase currents are 120° apart. When you sum the instantaneous power across all three phases using line-to-line voltage and line current, the geometry gives you a factor of √3 (≈ 1.732). If you instead used phase voltage and phase current, no √3 factor appears — but line voltage and line current are what you actually measure in the field, so the √3 form is what electricians use.
Power factor correction — a bonus mode
An inductive load (motors, transformers, welders) drops the power factor below 1.0, forcing the supply to deliver more apparent power (kVA) than the actual work (kW) requires. Utilities penalize industrial customers for low PF because it wastes the supply infrastructure. The fix: parallel capacitors that supply the reactive power locally.
Sizing formula:
Qcapacitor = P × (tan φold − tan φnew)
The calculator’s PFC mode returns kVAR needed, the new (reduced) line current, the new apparent power, and the per-phase capacitance in μF at both 50 Hz and 60 Hz (delta-connected). Round up to the next standard capacitor bank size — units commonly come in 2.5, 5, 10, 15, 25 kVAR steps.
Watch out for harmonics
Modern industrial loads often include VFDs, LED drivers, and switched-mode power supplies — non-linear loads that draw harmonic currents. Plain power-factor capacitors can resonate with these harmonics and burn out, or amplify voltage distortion. If your facility has significant non-linear load, use detuned capacitor banks (with series reactors tuned above the 7th or 5th harmonic) rather than plain caps. This calculator sizes the fundamental-frequency reactive power; harmonic mitigation is a separate analysis.
Important limitations
Assumptions built into this calculator:
- Balanced three-phase — all three phases carry equal current at equal PF. Unbalanced systems need per-phase analysis.
- Lagging (inductive) power factor — the default assumption for motors, transformers, and most industrial loads. Leading (capacitive) PF gives negative Q, but the calculator returns absolute values.
- Fundamental frequency only — no harmonic distortion accounted for.
- Line-to-line voltage — 400 V in Europe, 480 V in US industrial. If you have a phase (line-to-neutral) voltage, multiply by √3 before entering.
For power system analysis with unbalanced loads, harmonics, or detailed capacitor bank design, this tool gives you the starting point — final designs need proper power quality analysis and coordination with your utility.