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Transformer efficiency is output power divided by output plus losses. Losses are of two kinds: iron (core) loss Pfe, which stays essentially constant at any load (it depends on voltage and frequency, not current), and copper loss, which grows with the square of the load current: Pcu(x) = x²·Pcu(rated). Because one loss is flat and the other rises with the square of load, efficiency traces a hump-shaped curve that peaks where the two losses are equal, at load fraction x* = √(Pfe/Pcu).
| Quantity | Formula |
|---|---|
| Output power at load x | Pout = x × S × cosφ |
| Copper loss at load x | Pcu(x) = x² × Pcu(rated) |
| Efficiency | η = Pout / (Pout + Pfe + Pcu(x)) |
| Maximum-efficiency load | x* = √(Pfe / Pcu(rated)) |
| Voltage regulation | %Reg ≈ x×(%R·cosφ ± %X·sinφ) |
Voltage regulation is the drop in output voltage from no-load to full-load, caused by the transformer's internal resistance (%R) and reactance (%X). Lagging (inductive) loads worsen regulation (voltage sags more); leading (capacitive) loads can actually improve it or even raise the output voltage above no-load, which is why the sign flips for leading power factor.
It is the ratio of useful output power to input power (output plus losses): η = Pout/(Pout+Pfe+Pcu). Well-designed transformers reach 97–99% efficiency at their optimal load.
Iron (core) loss Pfe comes from magnetising the core and is essentially constant, depending only on voltage and frequency. Copper loss Pcu comes from winding resistance (I²R) and grows with the square of the load current.
Because copper loss grows with the square of load while iron loss stays flat, the total loss is minimised relative to output at the load where the two losses are equal — typically 50–70% of rated load, not 100%.
Use x* = √(Pfe/Pcu(rated)), where Pfe is the constant iron loss and Pcu(rated) is the copper loss at full rated current. At this load, copper loss equals iron loss.
Since efficiency peaks below full load, a transformer that mostly runs at its efficiency-peak load wastes less energy over its lifetime than one that is lightly or heavily loaded most of the time.
It is the percentage drop in output voltage from no-load to full-load: %Reg = (VNL−VFL)/VFL×100. It is approximated from the transformer's per-unit resistance (%R) and reactance (%X) and the load power factor.
Regulation depends on both resistive and reactive voltage drops, which combine differently depending on whether the load current lags or leads the voltage. Lagging (inductive) loads add both drops; leading (capacitive) loads can partially cancel them.
Yes. With a sufficiently leading (capacitive) power factor, the reactive drop can exceed the resistive drop in the opposite sense, causing the full-load voltage to be slightly higher than the no-load voltage — a negative regulation.
Small distribution transformers often have %R around 1–2% and %X around 3–6%, giving typical full-load regulation of 2–5% at lagging power factors. Larger transformers tend to have relatively lower %R and higher %X.
Copper loss scales with the square of the load fraction: Pcu(x) = x²×Pcu(rated). At half load, copper loss is only a quarter of its full-load value.
Yes. Output power is x×S×cosφ, so a lower power factor reduces the real output power for the same current and losses, which lowers the calculated efficiency at that load.
The %R·cosφ±%X·sinφ formula is accurate to a fraction of a percent for typical power transformers and is the standard method used in practice; an exact phasor calculation gives only a marginally different result.
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