Transformer Efficiency⚠ unverified

Understanding the Parameters

• $P_{out}$, $P_{in}$ — real powers; their gap is the internal loss.
• $\eta$ — very close to 1. The two loss families are copper loss ($I^2R$ in the windings, varying with load²) and core loss (hysteresis + eddy currents, roughly constant with load).

Derivation (Approaching a Proof)

By energy conservation the input splits into output plus losses:

$$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{cu} + P_{core}}.$$

Because copper loss $P_{cu}\propto I^2$ rises with load while core loss $P_{core}$ is nearly fixed, differentiating shows efficiency is maximum when the two losses are equal, $P_{cu} = P_{core}$ — transformers are therefore designed so this crossover falls near their typical operating load. The absence of friction/windage (no rotation) is why transformer efficiency exceeds that of rotating machines (see Motor Efficiency).

History

Efficiency rose dramatically through core‑material advances: from early iron to grain‑oriented silicon steel (1930s+) and, more recently, amorphous‑metal cores that slash hysteresis loss. Because the world's installed transformers run continuously, even fractional efficiency gains save vast amounts of energy — now enforced by minimum‑efficiency standards (e.g. DOE, EU EcoDesign).

Related Concepts: Transformer Turns Ratio, Transformer Current Ratio, Motor Efficiency, Watt's Law

Notes: Registry calculator transformer-efficiency (unverified). Single‑point ratio; full characterisation gives efficiency versus load, peaking where copper loss = core loss.