Engineering · Electrical Engineering · Power Systems
Three-Phase Power Calculator
Calculates three-phase real, reactive, and apparent power from line voltage, line current, and power factor.
Calculator
Formula
P is real (active) power in watts (W); Q is reactive power in volt-amperes reactive (VAR); S is apparent power in volt-amperes (VA); V_L is line-to-line voltage in volts (V); I_L is line current in amperes (A); cos φ is the power factor (dimensionless, 0–1); sin φ is derived from the power factor angle φ = arccos(PF).
Source: IEC 60038 standard voltages and IEEE Std 141 (Red Book) — Recommended Practice for Electric Power Distribution for Industrial Plants.
How it works
Three-phase AC power systems deliver electricity through three conductors, each carrying a sinusoidal current displaced by 120° from the others. This arrangement enables constant power delivery, reduced conductor material compared to equivalent single-phase systems, and the ability to drive three-phase motors directly. In any balanced three-phase system, the total power can be characterised by three interrelated quantities: real power, reactive power, and apparent power.
Real power (P), measured in watts (W) or kilowatts (kW), is the actual power consumed and converted to useful work — heat, mechanical motion, or light. It is calculated as P = √3 × V_L × I_L × cos φ, where V_L is the line-to-line voltage, I_L is the line current, and cos φ is the power factor. Reactive power (Q), measured in VAR or kVAR, represents energy oscillating between the source and inductive or capacitive loads without doing useful work. Apparent power (S), measured in VA or kVA, is the vector magnitude of P and Q and equals √3 × V_L × I_L — it represents the total current-carrying capacity the system must support. The power factor angle φ = arccos(PF) links all three quantities through the power triangle: S² = P² + Q².
These calculations are applied in a wide range of practical scenarios: sizing generators and transformers for industrial plants, selecting cable cross-sections and circuit breakers, assessing power quality and penalty charges from utilities for poor power factor, and specifying power factor correction capacitor banks. Renewable energy grid connections, variable-speed motor drives, and data centre power infrastructure all depend on accurate three-phase power calculations for safe and compliant design.
Worked example
Consider a three-phase industrial motor load connected to a 415 V (line-to-line) supply, drawing a line current of 100 A at a power factor of 0.85 (lagging).
Step 1 — Apparent Power:
S = √3 × 415 × 100 = 1.732 × 415 × 100 = 71,881 VA ≈ 71.88 kVA
Step 2 — Real Power:
P = S × cos φ = 71,881 × 0.85 = 61,099 W ≈ 61.10 kW
Step 3 — Power Factor Angle:
φ = arccos(0.85) ≈ 31.79°
Step 4 — Reactive Power:
Q = S × sin φ = 71,881 × sin(31.79°) = 71,881 × 0.5268 = 37,862 VAR ≈ 37.86 kVAR
Verification: P² + Q² = 61,099² + 37,862² = 3,733 × 10⁶ + 1,433 × 10⁶ = 5,166 × 10⁶; √(5,166 × 10⁶) ≈ 71,875 VA ✓ (matches S within rounding).
This motor would require a transformer or generator rated for at least 72 kVA, cable rated for at least 100 A, and if the site penalty threshold is PF < 0.90, a capacitor bank supplying approximately 20 kVAR would be needed to bring the power factor to unity-adjacent levels.
Limitations & notes
This calculator assumes a balanced three-phase system — equal voltages across all three phases and equal currents in each line. In unbalanced systems (common with mixed single-phase and three-phase loads), each phase must be calculated individually and results summed. The formula also assumes a purely sinusoidal waveform; in systems with significant harmonic distortion from variable-frequency drives, UPS systems, or switch-mode power supplies, the true power factor (displacement PF × distortion PF) differs from the displacement power factor entered here, and total harmonic distortion (THD) analysis is required. The power factor input must remain between 0 and 1 (inclusive); values outside this range are physically meaningless. This calculator uses line-to-line voltage — if only phase voltage (V_Ph) is known, multiply by √3 first (V_L = √3 × V_Ph). Results are valid for the steady-state condition only; motor starting currents, inrush effects, and transient events require separate analysis.
Frequently asked questions
What is the difference between real power, reactive power, and apparent power in a three-phase system?
Real power (P, in kW) is the energy per unit time actually converted to useful work such as heat or mechanical motion. Reactive power (Q, in kVAR) is power that oscillates between the source and inductive or capacitive loads — it does no net work but must be supplied by the source and carried by conductors. Apparent power (S, in kVA) is the total power the source must deliver and equals the vector sum of P and Q. The ratio P/S is the power factor.
Why does the three-phase power formula use √3?
The √3 factor (≈ 1.732) arises from the 120° phase displacement between the three voltage waveforms. When summing the contributions of all three phases expressed in terms of line-to-line voltage and line current, the geometric relationship between line and phase quantities introduces this factor. In a star (wye) connection, V_L = √3 × V_Ph; in a delta connection, I_L = √3 × I_Ph — both lead to the same √3 × V_L × I_L expression for total three-phase power.
How do I convert kVA to kW for a three-phase load?
Multiply the apparent power in kVA by the power factor (cos φ) to obtain real power in kW: P (kW) = S (kVA) × PF. For example, a 50 kVA load at 0.90 power factor draws 50 × 0.90 = 45 kW of real power. The remaining capacity (50 × sin(arccos(0.90)) ≈ 21.8 kVAR) is reactive power.
What power factor should I use if I only know the load type?
Typical power factors by load type are: induction motors at full load 0.85–0.92, induction motors at partial load 0.70–0.85, resistive heaters and incandescent lighting 1.00, fluorescent lighting with magnetic ballasts 0.50–0.70, modern LED drivers and switch-mode power supplies 0.90–0.99, and large synchronous motors 0.80–1.00 (can be leading). When in doubt, 0.85 is a common conservative assumption for mixed industrial loads.
Can this calculator be used for single-phase systems?
No — the √3 multiplier is specific to three-phase systems. For single-phase systems use P = V × I × cos φ, Q = V × I × sin φ, and S = V × I, where V is the supply voltage and I is the current. Running these numbers through the three-phase calculator would overestimate power by a factor of √3 ≈ 1.732.
Last updated: 2025-01-15 · Formula verified against primary sources.