Motor Efficiency⚠ unverified

Understanding the Parameters

• $P_{out}$ — the useful shaft power, $T\omega$ (see Motor Power).

• $P_{in}$ — the electrical power supplied, $VI$ for DC or $\sqrt3\,V_L I_L\,\mathrm{pf}$ for three‑phase (see Watt's Law, Three-Phase Power).

• $\eta$ — the fraction converted to useful work; $1-\eta$ is lost.

Derivation (Approaching a Proof)

Efficiency is a direct application of energy conservation: input power splits into useful output plus losses,

$$P_{in} = P_{out} + P_{loss} \quad\Rightarrow\quad \eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{loss}}.$$

The losses have distinct physical origins: copper loss ($I^2R$ in the windings, rising with load), iron loss (hysteresis + eddy currents in the core, roughly constant with load), and mechanical loss (friction and windage). Because copper loss grows with load while iron loss does not, efficiency peaks at a partial load where the two are comparable, then falls off — the shape captured empirically by an Efficiency Map.

History

Efficiency accounting dates to the 19th‑century establishment of energy conservation and the mechanical equivalent of heat (Joule). It became a regulated design target in the late 20th century, when minimum‑efficiency standards (e.g. IEC IE classes, NEMA Premium) were imposed on the vast installed base of industrial motors, which consume a large share of global electricity.

Related Concepts: Motor Power, Watt's Law, Efficiency Map, Temperature Rise

Notes: Registry calculator motor-efficiency (unverified). Instantaneous ratio at one operating point; full characterisation uses an efficiency map over the torque–speed plane.