Transformer Efficiency and Regulation Formulas
Transformer efficiency formulas: efficiency from iron and copper loss, load at maximum efficiency, voltage regulation, a worked example and a table by load.
The formulas
| To find | Formula |
|---|---|
| Output power at load x | Pout = x × S × pf |
| Copper loss at load x | Pcu(x) = x² × Pcu(full load) |
| Efficiency | η = Pout / (Pout + Piron + x² × Pcu) × 100 |
| Load at maximum efficiency | x = √(Piron / Pcu) |
| Resistance in percent | %R = Pcu / (S × 1000) × 100 (S in kVA, Pcu in kW) |
| Voltage regulation, lagging | %reg ≈ %R × pf + %X × sin φ, with %X = √(%Z² − %R²) |
S is the rated kVA, x the load as a fraction of rating, Piron the no-load (iron) loss and Pcu the load (copper) loss at full load, both in kW. The regulation formula is the usual approximation for load near rating. For a leading power factor, subtract the %X term.
Worked example: 1,000 kVA at power factor 0.8
Iron loss 1.7 kW, copper loss 10.5 kW at full load, impedance 5%.
- At full load Pout = 1 × 1000 × 0.8 = 800 kW and the losses are 12.2 kW
- η = 800 / (800 + 12.2) × 100 = 98.5%
- Maximum efficiency is at x = √(1.7 / 10.5) = 0.4024 of full load, where η = 98.95%
Worked example: regulation
- %R = 10.5 / 1000 × 100 = 1.05% and %X = √(5² − 1.05²) = 4.889%
- %reg = 1.05 × 0.8 + 4.889 × 0.6 = 3.773%
Efficiency by load at power factor 0.8
| Load | Output | Losses | Efficiency |
|---|---|---|---|
| 25% | 200 kW | 2.356 kW | 98.84% |
| 50% | 400 kW | 4.325 kW | 98.93% |
| 75% | 600 kW | 7.606 kW | 98.75% |
| 100% | 800 kW | 12.2 kW | 98.5% |
Frequently asked questions
Why is the best efficiency below full load?
Iron loss is fixed and copper loss rises with the square of the load. They are equal at x = √(Piron / Pcu), which is the point of best efficiency.
Which losses are on the test report?
The no-load test gives the iron loss and the short-circuit test gives the copper loss at rated current. Use the figures from your transformer’s test certificate.
Does a lower power factor lower the efficiency?
Yes. The output power in kW falls while the losses stay the same, so the efficiency falls.
Is the regulation formula exact?
It is an approximation. It is close for loads up to rating and for impedances typical of distribution transformers.
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