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Three-Phase Real Power

Also known as 1.732 formula · three phase kW · √3 power formula

P=3 VLIL PFP = \sqrt{3} \, V_{L} I_{L} \, \text{PF}

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Constant used — built into this formula, no need to enter
3=1.7320508075688772\sqrt{3} = 1.7320508075688772Square Root of Three · exact

Learning zone

The √3 is the whole story of three-phase power. Each of the three phases carries Vphase × Iphase, but the meter and the nameplate quote line quantities, and the 120° spacing between phases turns the sum of three phase powers into √3 × VL × IL. Multiply by the power factor and you have the real, billable, heat-and-torque-producing watts. A 480 V feeder carrying 100 A at 0.85 PF delivers 1.732 × 480 × 100 × 0.85 ≈ 70.7 kW.

The trap is mixing line and phase values: the √3 belongs in the formula only when VL and IL are both line quantities. Drop the power factor and you get apparent power in VA, not watts — the two differ by 15% on that feeder, which is exactly why a 100 kVA transformer will not carry 100 kW of motor load. Nikola Tesla's polyphase patents of 1888 made this arithmetic the foundation of every industrial plant since.

Three-Phase Real Power
P=3 VLIL PFP = \sqrt{3} \, V_{L} I_{L} \, \text{PF}
Where
  • PP= Real power
  • VLV_{L}= Line-to-line voltage
  • ILI_{L}= Line current
  • PF\text{PF}= Power factor
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