Three-Phase Power⚠ unverified

Understanding the Parameters

• Line voltage $V_L$ is measured between two lines (e.g. "400 V three‑phase"), distinct from the phase (line‑to‑neutral) voltage $V_\phi$.

• Line current $I_L$ is the current in each conductor.

• Power factor $\mathrm{pf}$ converts apparent power $S=\sqrt3\,V_L I_L$ to real power $P$; the reactive part is $Q = \sqrt3\,V_L I_L \sin\varphi$ (see Reactive Power).

Derivation (Approaching a Proof)

A balanced load dissipates equal power in each of its three phases, so the total real power is $P = 3\,V_\phi I_\phi \cos\varphi$. Convert per‑phase to line quantities:

• Wye (star): $V_L = \sqrt3\,V_\phi$, $I_L = I_\phi$, so $V_\phi = V_L/\sqrt3$ and $$P = 3\left(\frac{V_L}{\sqrt3}\right)I_L\cos\varphi = \sqrt{3}\,V_L I_L \cos\varphi.$$

• Delta: $V_L = V_\phi$, $I_L = \sqrt3\,I_\phi$, which gives the same result.

Either way, in line quantities $\;P = \sqrt{3}\,V_L I_L\,\mathrm{pf}$. The $\sqrt3$ is exactly the line‑to‑phase ratio forced by the $120^\circ$ separation of the phases. A further virtue of balanced three‑phase is that the sum of the three instantaneous phase powers is constant in time — no $2\omega$ pulsation — so three‑phase motors deliver smooth torque.

History

Polyphase AC arose in the 1880s from Galileo Ferraris and Nikola Tesla (rotating fields, induction motors) and Mikhail Dolivo‑Dobrovolsky, whose three‑phase generator, motor, and transmission powered the landmark 1891 Lauffen–Frankfurt demonstration. Three‑phase uses less conductor per watt and self‑starts motors, and remains the global standard for generation and distribution.

Related Concepts: Three Phase Power, Apparent Power, Power Factor, Reactive Power, Watt's Law

Notes: Registry calculator three-phase-power (unverified). Assumes a balanced load; unbalanced systems need per‑phase or symmetrical‑component analysis.