Circuits & Electrical Power · Three-phase power
Where the √3 comes from
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Where the √3 comes from

Industrial power arrives on three conductors carrying three voltages 120° apart. P=3VLILPFP = \sqrt{3} \, V_{L} I_{L} \, \text{PF} — read aloud P equals root three V-L I-L times P F. PP is the real power in watts; VLV_{L} is the line-to-line voltage in volts, measured between any two of the three conductors, and the number printed on every panel; ILI_{L} is the line current in amperes, in one conductor; PF\text{PF} is the same bare power factor as before. The subscript L means LINE throughout — outside the machine — as opposed to phase, which is inside it.

Drop the power factor and you have the apparent power, S=3VLILS = \sqrt{3} \, V_{L} I_{L}, in volt-amperes. Read backwards, that same line is how a kVA nameplate becomes full-load amps, which is the calculation that sizes a transformer's secondary breaker.

The 3\sqrt{3} is not there because there are three phases. It is there because the voltage you measure spans two windings set 120° apart, while the current you measure belongs to one — and the geometry of that mismatch is 31.732\sqrt{3} \approx 1.732. Writing 3 instead is the single commonest arithmetic error in the trade: it overstates a load by 73%, and it looks completely reasonable on the page.