Three-Phase Formulas

Power, current, voltage, power factor, and efficiency relationships.

Use the √3 relationships for balanced three-phase systems, keep line-to-line and line-to-neutral quantities distinct, and state whether the answer is apparent power, real input power, or estimated mechanical output.

Apparent power: kVA = √3 × VLL × I ÷ 1,000
Real input power: kW = √3 × VLL × I × PF ÷ 1,000
Estimated output: kWout = kWin × efficiency

1. Name every quantity

VLL is RMS line-to-line voltage; VLN is RMS line-to-neutral voltage; I is line current for the balanced three-phase relationship.
kVA is apparent power, kW is real power, and kvar is reactive power. Do not label one as another.
Power factor is a unitless ratio between real and apparent power; efficiency is a separate ratio between output and input power.
State whether the values are nameplate, calculated, estimated, or measured and whether the load is sufficiently balanced for the shortcut being used.

2. Balanced three-phase relationships

Apparent power: kVA = √3 × VLL × I ÷ 1,000.
Real input power: kW = √3 × VLL × I × PF ÷ 1,000.
Current from known real input power: I = kW × 1,000 ÷ (√3 × VLL × PF).
Estimated motor output: kWout = √3 × VLL × I × PF × efficiency ÷ 1,000. Use decimal values for power factor and efficiency.

3. Keep wye and delta relationships separate

The √3 power equation using line-to-line voltage and line current works for a balanced three-phase load without first converting branch quantities.
In a balanced wye connection, line voltage is √3 times phase voltage and line current equals phase current.
In a balanced delta connection, line voltage equals phase voltage and line current is √3 times phase current.
Do not mix a measured line current with a branch or winding voltage unless the connection and conversion are explicit.

4. Use measured data carefully

Measure all phase currents and applicable voltages with instruments rated for the environment and under an approved work procedure.
If phases are materially unbalanced, calculate or measure per-phase power rather than forcing averaged values into a balanced-load shortcut.
A clamp meter reading alone does not provide power factor; use verified nameplate/design data for an estimate or a suitable power analyzer for actual real power.
For VFD outputs, non-sinusoidal waveforms and instrument bandwidth can make ordinary meter readings unsuitable; follow the drive and instrument instructions.

5. Apply formulas to equipment planning

Use calculated current only as one input to a complete conductor, protection, voltage-drop, load, and equipment analysis.
For motors, distinguish electrical input from mechanical output and include efficiency only in the correct direction.
For transformers and generators, respect kVA ratings and load characteristics rather than treating every load as unity power factor.
For energy, multiply real power by time using consistent units; kW and kWh are not interchangeable.

Common mistakes

Using line-to-neutral voltage in the √3 equation while calling it line-to-line voltage.
Entering 85 for 85% instead of 0.85.
Applying both power factor and efficiency when converting kVA to input kW.
Using a balanced formula despite large voltage or current imbalance.
Treating a calculated full-load current as a universal conductor or breaker size.
Reporting excessive precision from approximate power factor, efficiency, or averaged measurements.

Worked field example

Worked example: estimate input and output power

Given a balanced 480 V line-to-line load at 32 A and power factor 0.86, calculate apparent power: √3 × 480 × 32 ÷ 1,000 ≈ 26.6 kVA.
Calculate real input power: 26.6 × 0.86 ≈ 22.9 kW.
If verified efficiency at that operating point is 91%, estimate mechanical output: 22.9 × 0.91 ≈ 20.8 kW.
Compare the result with nameplate and measured phase values, and state that power factor and efficiency are assumptions unless measured or manufacturer-provided.

What this proves: The same voltage and current can describe 26.6 kVA apparent power, 22.9 kW real input, and about 20.8 kW estimated output. Labeling the quantity and assumptions is as important as the arithmetic.

Verify before use

Official sources

Source set reviewed September 2026. Recheck the current adopted requirements before design, purchase, permit, or installation.

NEMA MG 1 — Motors and Generators ↗
Official NEMA reference for motor definitions, ratings, application, and performance context.
NFPA 70 — National Electrical Code ↗
Use the adopted edition for conductor, protection, motor, transformer, and load-calculation requirements; formulas alone do not establish compliance.