Three-Phase Power Calculator

About the Three-Phase Power Calculator

Three-phase power in two modes. Balanced takes the line-to-line voltage, line current and power factor and gives real power (kW), reactive power (kvar) and apparent power (kVA), plus the voltage, current and impedance of each phase for both star (Y) and delta (Δ) connections.

Unbalanced is for real distribution boards, where each phase carries a different load: enter the current and power factor of L1, L2 and L3 and get the totals, the neutral current as a true phasor sum, the percentage unbalance and the symmetrical components — with a phasor diagram that shows why the neutral current is what it is.

Balanced three-phase formulas

With V_L the line-to-line voltage, I_L the line current and φ the angle between phase voltage and current:

• Real power P = √3 × V_L × I_L × cos φ
• Reactive power Q = √3 × V_L × I_L × sin φ
• Apparent power S = √3 × V_L × I_L, and S² = P² + Q²
• Star (Y): phase voltage V_ph = V_L ÷ √3, phase current = I_L
• Delta (Δ): phase voltage = V_L, phase current I_ph = I_L ÷ √3

The totals are the same for both connections; only how the voltage and current are shared inside the load differs. The impedance of each phase is V_ph ÷ I_ph, so a delta load has three times the impedance per phase of a star load that draws the same line current.

Unbalanced loads and the neutral current

For a four-wire star (230/400 V) board, each phase is a single-phase load: P = V_ph × I × cos φ and Q = V_ph × I × sin φ, and the totals are the sums. The neutral carries the phasor sum of the three line currents, because the currents are 120° apart. Three equal currents at the same power factor cancel completely; 20, 15 and 10 A in phase with their voltages leave √(20² + 15² + 10² − 20×15 − 15×10 − 10×20) = √75 = 8.66 A in the neutral.

With different currents in each phase there are two ways to total the apparent power, and the calculator shows both: the arithmetic sum S₁ + S₂ + S₃, and the vector value √(P² + Q²), which is never larger. IEEE Std 1459-2010 also defines an effective apparent power that accounts for the neutral current; it is not calculated here.

Unbalance and symmetrical components

The percentage unbalance uses the NEMA MG 1 (§14.36) definition: 100 × the largest deviation from the average ÷ the average. NEMA defines it for motor supply voltages; the calculator applies the same formula to the line currents.

The symmetrical components (Fortescue, 1918) split the currents into a positive-sequence set (I₁, the balanced part), a negative-sequence set (I₂) and a zero-sequence set (I₀). The neutral current is 3 × I₀. The ratio I₂ ÷ I₁ is the stricter measure of unbalance; EN 50160 and the IEC power-quality standards use this negative-to-positive-sequence ratio for supply-voltage unbalance.

Assumptions

• The supply voltages are balanced and sinusoidal: equal in size and 120° apart.
• The phase order is L1 → L2 → L3 (positive sequence): L1 at 0°, L2 at −120°, L3 at +120°. With different power factors per phase, the reverse order would give a different neutral current.
• Unbalanced mode models a four-wire star load. A three-wire delta load has no neutral.
• Lagging current (inductive loads: motors, transformers, fluorescent ballasts) gives positive kvar; leading current (capacitor banks, some electronic supplies on light load) gives negative kvar.

Sources

• IEEE Std 1459-2010, Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions
• IEC 60038:2009, IEC standard voltages — 230/400 V, 400/690 V, 120/208 V, 277/480 V and 347/600 V systems
• NEMA MG 1, Motors and Generators, §14.36 — definition of percentage unbalance
• C. L. Fortescue, “Method of symmetrical co-ordinates applied to the solution of polyphase networks”, Transactions of the AIEE 37 (1918)

Limitations

• Sinusoidal waveforms only. Harmonic currents from electronic loads — especially the third harmonic, which adds up in the neutral instead of cancelling — can make the real neutral current much larger than calculated here.
• Unbalanced mode assumes the supply voltages themselves are balanced; it does not take three separate phase voltages.
• Delta-connected unbalanced loads are not covered.
• The impedances in balanced mode describe the load as seen from the line terminals; they say nothing about the cable or supply impedance.

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