Three Phase Power⚠ unverified
Electrical / Basic · Compute the real power delivered by a balanced three-phase system
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| line_voltage | linevoltage | V | 1.0 | Line-to-line voltage |
| line_current | linecurrent | A | 1.0 | Line current |
| pf | pf | — | 1.0 | Dimensionless power factor of the load |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | P | W | Real three-phase power, in watts (W) |
The science & history
Understanding the Parameters
-
Line voltage $V_L$ — measured between two of the three lines (what "$400\,\text{V}$ three‑phase" refers to), distinct from the phase (line‑to‑neutral) voltage.
-
Line current $I_L$ — the current in each supply conductor.
- Power factor $\mathrm{PF}$ — accounts for the phase shift of a reactive load; $P$ is the real power, not the apparent power $S = \sqrt{3}\,V_L I_L$.
Derivation (Approaching a Proof)
A balanced three‑phase load carries equal power in each of its three phases, so the total real power is
$$P = 3\,V_\phi I_\phi \cos\varphi,$$
where $V_\phi$, $I_\phi$ are the per‑phase voltage and current. Convert to line quantities:
-
Wye (star): $V_L = \sqrt{3}\,V_\phi$ and $I_L = I_\phi$, so $V_\phi = V_L/\sqrt3$ and $$P = 3\left(\tfrac{V_L}{\sqrt3}\right) I_L \cos\varphi = \sqrt{3}\,V_L I_L \cos\varphi.$$
-
Delta: $V_L = V_\phi$ and $I_L = \sqrt{3}\,I_\phi$, giving the same result $P = \sqrt{3}\,V_L I_L \cos\varphi$.
So regardless of connection, in line quantities
$$P = \sqrt{3}\,V_L I_L \,\mathrm{PF}.$$
The $\sqrt3$ is not a fudge factor — it is exactly the ratio between line and phase quantities forced by the $120^\circ$ separation of the three phases. A further elegance of balanced three‑phase: the sum of the three instantaneous phase powers is constant in time (no $2\omega$ ripple), which is why three‑phase motors run smoothly.
History
Polyphase power grew from the 1880s work of Galileo Ferraris and Nikola Tesla (rotating magnetic fields, induction motors) and Mikhail Dolivo‑Dobrovolsky, who built practical three‑phase generators, motors, and transmission (the celebrated 1891 Lauffen–Frankfurt demonstration). Three‑phase won out because it transmits power with less conductor material and drives self‑starting motors — and remains the backbone of electrical generation and distribution worldwide.
Related Concepts: Power Factor, Watt's Law, Apparent Power, Three-Phase Power
Notes: Registry calculator basic-three-phase-power (unverified). Assumes a balanced load;
unbalanced systems require per‑phase (or symmetrical‑component) analysis.