Three-phase real power is √3 × line voltage × line current × power factor. At 400 V, 32 A and a 0.9 power factor that is 19.95 kW real from 22.17 kVA apparent. The √3 — 1.732 — comes from the 120-degree phase relationship between the lines, not from any convention: adding three sinusoids that peak at different moments does not give three times one of them.
The same factor is why a 400 V three-phase supply and a 230 V single-phase one come out of the same transformer. 230 × √3 is 398, which rounds to the 400 V quoted between any two lines.
What is line voltage versus phase voltage?
Phase voltage is measured from one line to neutral; line voltage is measured between two lines. In a star-connected system the line voltage is √3 times the phase voltage, which is exactly why 230 V and 400 V appear together.
| Measured between | Voltage | What uses it |
|---|---|---|
| One line and neutral | 230 V | Sockets, lighting, single-phase loads |
| Two lines | 400 V | Motors, ovens, three-phase equipment |
The formula uses line voltage and line current, which is what a meter clamped around one incoming conductor actually reads. That is deliberate: the quantities in the formula are the ones you can measure without opening anything up.
Why does three phase move more power down the same copper?
Because the return currents cancel. In a balanced system the three phases sum to zero at the neutral, so there is little or no neutral current and the conductors carry only what their own phase needs.
A single-phase circuit needs a live and a neutral both carrying the full current. A balanced three-phase circuit moves three times as much power over four conductors rather than six, and the fourth one is barely working. That is why industrial motors above a few kilowatts are almost always three phase, and why distribution networks are.
What does it mean for the current?
It falls by √3 for the same power. A 7.4 kW load is 32 A at 230 V single phase and only 10.7 A per line at 400 V three phase — which is the difference between a dedicated heavy circuit and an ordinary one.
This is the practical argument in domestic electrics too. A house with a three-phase supply can add a car charger, a heat pump and an induction hob without the arithmetic falling over; a house with 80 A on a single phase runs out much sooner, which is why load management devices exist.
Where does single phase stop?
Lower than most people assume, and the ceilings are set by the protective device rather than by the supply.
| Circuit | Ceiling |
|---|---|
| 120 V, 15 A | 1,800 W |
| 230 V, 13 A plug fuse | 2,990 W |
| 230 V, 16 A | 3,680 W |
| 230 V, 32 A | 7,360 W |
The 13 A plug fuse is why 3 kW is the practical ceiling for anything that plugs in. Electric showers, hobs and car chargers exceed it, which is why they get their own circuits — and why North American homes run high-power appliances on 240 V rather than the 120 V used for sockets.
What is the difference between kW and kVA?
Watts is the power doing work; volt-amps is the product of the voltage and the current the supply has to deliver. For a resistive load they are equal. For a motor the current lags the voltage and the two diverge by the power factor, so a 10 kVA supply feeding a 0.8 power factor load delivers 8 kW of useful power.
Generators, transformers and UPS units are rated in kVA for exactly that reason: they are limited by the current they can pass and the voltage they can hold, not by how much of it ends up doing work. Sizing a generator in kW and connecting a motor load to it is a standard way to buy something too small.
Does an unbalanced load break the formula?
It makes the simple version inaccurate rather than wrong. The √3 formula assumes the three phases carry equal current; when they do not, neutral current appears and the total power is the sum of the three phases calculated individually.
For a large motor the balance is inherent and the formula holds. For a distribution board full of single-phase circuits spread across three phases, balance is something the designer arranges, and a badly balanced board wastes capacity in the lightly loaded phases while the heaviest one limits the whole supply.
Questions people ask
Why 400 V and 230 V from the same supply? Because 400 is 230 × √3. Line-to-neutral gives 230, line-to-line gives 400, and both are available at the same board.
Is line current the same as phase current? In a star connection, yes. In a delta connection the phase current is the line current divided by √3, which is why the connection type has to be known before the formula is applied.
Why √3 rather than 3? Because the three voltages peak at different instants. Adding them as vectors 120 degrees apart gives √3 times one of them, not three times.
Does a low power factor cost me money? For a domestic customer usually not, since billing is on real power. For an industrial customer with a demand tariff it very much does, which is why power factor correction equipment exists.
The factor is geometry, and it is why the two voltages on a three-phase board are 230 and 400. The three phase power calculator applies it in both directions, and the single phase power calculator covers the supply most homes actually have.