NapkinCalc

Power Systems — Distribution

Three-phase power

ELI5: the grid uses three sine waves, 120° out of step. The payoff is huge: their combined power is constant (no flicker), the three conductors share return paths (far less copper), and motors start themselves. The bookkeeping rule is P = √3·V_LL·I·PF.

VLL:=480VV_{LL} := 480 V = 480 V line-to-line, industrial standard
Iline:=65AI_{line} := 65 A = 65 A
PF=0.88PF = 0.88
S3ph=3⋅VLL⋅IlineS_{3ph} = \sqrt{3} \cdot V_{LL} \cdot I_{line} = 54040 W apparent power (VA)
P3ph=S3ph⋅PFP_{3ph} = S_{3ph} \cdot PF = 47555 W real power doing work (W)

Real-world hook: three-phase feeds every factory, data center, and EV fast-charger, and it's why big motors don't need a separate starter — the rotating field is built in.

Try it yourself: the apparent power (in VA, just the number) of a three-phase load at 480 V drawing 50 A? (S = √3·V·I.)

Syou=3⋅480⋅50S_{you} = \sqrt{3} \cdot 480 \cdot 50 = 41569.219441569.2194 ✏️ Your turn: compute √3 × 480 × 50 (the apparent power in volt-amps).
✓ pass abs(Syou−3⋅480⋅50)<1\mathrm{abs}\left(S_{you} - \sqrt{3} \cdot 480 \cdot 50\right) < 1 green when your apparent power is correct