Three-phase power is the sum of the power in the three phases. For a balanced load there is a short formula. For a real site, a meter measures each phase and adds the results. This guide gives the formulas, explains line and phase values, and shows how to check a three-phase measurement.
This guide is the start of the power fundamentals series. The other guides explain kW and kWh, power factor and kVA, true RMS measurement and negative power readings.
Three-phase power formula
For a balanced three-phase load, with the same current and power factor on each phase:
| Quantity | With line-to-line voltage | With line-to-neutral voltage |
|---|---|---|
| Active power, P (W) | √3 × VLL × IL × PF | 3 × VLN × IL × PF |
| Apparent power, S (VA) | √3 × VLL × IL | 3 × VLN × IL |
| Reactive power, Q (var) | √(S² − P²) for a sinusoidal supply |
Divide by 1,000 for kW, kVA and kvar. PF is the power factor, the ratio of active power to apparent power.
A 400 V circuit with 100 A per line at a power factor of 0.9 uses about 62.4 kW:
With a power factor of 1, the same current would give 69.3 kW. The power factor changes the result by 10%, so use a measured value, not a guess. The kW, kVA, kvar and power factor calculator moves between kW, kVA, kvar and power factor, and kW to amps works the formula backwards to find the current.
Line voltage, phase voltage and √3
Line-to-line voltage (VLL) is measured between two phase conductors. Line-to-neutral voltage (VLN) is measured between one phase conductor and the neutral. In a star (wye) system:
- VLL = √3 × VLN, so 400 V line to line is 231 V line to neutral;
- the line current is the same as the phase current.
In a delta winding there is no neutral, the phase voltage is the line-to-line voltage, and the line current is √3 times the current in each winding. The formula in terms of VLL and IL is the same for both, because it uses only values you can measure at the terminals.
IEC 60038 gives 230/400 V as the standard low-voltage system for 50 Hz networks. Common three-phase systems:
| Region | Line to neutral | Line to line |
|---|---|---|
| Europe, UK, Ireland, Australia | 230 V | 400 V |
| North America, commercial | 277 V | 480 V |
| North America, small buildings | 120 V | 208 V |
Do not put 230 V into the √3 formula, and do not multiply a three-phase total by three again. Those are the two most common calculation errors.
Balanced and unbalanced loads
The formula assumes that each phase carries the same current at the same power factor. A motor on a good supply is close to balanced. A building is not: single-phase lighting, sockets and IT loads put different current on each phase, and the power factor differs too.
For an unbalanced load, calculate each phase separately and add the results:
P = V1 × I1 × PF1 + V2 × I2 × PF2 + V3 × I3 × PF3
where each V is a line-to-neutral voltage. Unequal line currents also mean current in the neutral, and unequal voltages stress motors. The voltage unbalance calculator gives the NEMA and IEC figures for three measured voltages.
How a meter measures three-phase power
A meter does not use the balanced formula. It samples the voltage and the current of each phase many times per cycle, multiplies each voltage sample by the current sample of the same phase, and averages the product. That average is the active power of the phase. The meter adds the three phase powers to get the total. This method is correct for any load, balanced or not, and it includes harmonics.
Blondel's theorem sets the number of measuring elements: a circuit with N conductors needs N − 1 elements.
| Circuit | Conductors | Elements | Voltage references |
|---|---|---|---|
| Three-phase four-wire (star with neutral) | 4 | 3 | L1-N, L2-N, L3-N |
| Three-phase three-wire (delta, or star without neutral) | 3 | 2 | L1-L2, L3-L2 |
The two-element (two-wattmeter) method gives the correct total for any three-wire load. Its two element readings are not phase powers, and one reading can be negative at a low power factor. Set the meter's wiring mode to match the real circuit: a four-wire mode on a three-wire circuit, or the other way round, gives wrong phase values.
Check the phase pairing
A plausible total can hide a connection error. At commissioning, compare the values of each phase under a normal load:
- Negative power on one phase on a site that does not export: the current sensor on that phase is reversed, or its two leads are swapped at the meter.
- Power factor near zero on one phase, with normal current: the current sensor is paired with the wrong voltage. A sensor moved to the wrong phase shifts the angle by 120°. With a true power factor of 0.9, that reads as about −0.83 or −0.07.
- Phase currents that do not match a clamp meter on the same conductors: wrong sensor ratio, wrong sensor type or a sensor on the wrong conductor.
Record the per-phase voltage, current, power factor and active power with a reference instrument at the same time. The negative readings guide gives a full check sequence.
From power to energy
Power is a rate. Energy is power over time: 62.4 kW for 8 hours is 499 kWh. A meter counts energy continuously, so its kWh register includes every change in load between readings. The kW vs kWh guide explains the difference and the kW to kWh calculator does the conversion with a tariff.
Three-phase measurement with a ZEM
The ZEM electricity monitor measures three-phase three-wire and four-wire circuits from 110 to 480 V line to line. Its current sensors ship connected and calibrated to the meter, so Class 0.5S (IEC 62053-22) applies to the meter and sensors together, from 0.1 A to 3 kA per phase. It has up to 30 data feeds, including RMS voltage, RMS current, power factor, kWh and kVAh, and sends them over the Zigbee mesh to a Gateway, where Edge stores and charts them.
For a large supply, check the transformer first: the transformer current calculator gives the full-load current on each side, which sets the current sensor range.
Common questions
What is the formula for three-phase power?
For a balanced load, active power in watts is √3 × V × I × PF, where V is the line-to-line voltage, I is the line current and PF is the power factor. Divide by 1,000 for kW. Apparent power in VA is √3 × V × I.
Why do you multiply by 1.732 in three-phase calculations?
1.732 is √3. In a star system the line-to-line voltage is √3 times the line-to-neutral voltage, so 3 × V(L-N) × I is the same as √3 × V(L-L) × I. Do not use √3 with a line-to-neutral voltage.
Can I work out three-phase kW from current readings alone?
Only as an estimate. kW needs the voltage and the power factor of each phase, and both change with the load. Use a meter that measures voltage and current on each phase when the result has to be right.
How many CTs do I need for a three-phase circuit?
Three for a four-wire circuit (three lines and a neutral). A three-wire circuit can be measured with two elements by Blondel's theorem, but many meters use three CTs on three-wire circuits too, so that each line current is visible.