---
title: "Transformer losses and efficiency expression"
subject: "Electrical Machines"
chapter: 2
key: "semester_4/electrical_machines:transformer-losses-efficiency-expression"
source: https://www.bctengineer.com/q/semester_4/electrical_machines/transformer-losses-efficiency-expression
site: BCT Engineering
---

# Transformer losses and efficiency expression

Electrical Machines · Chapter 2: Transformer

- Asked in: 2071 Bhadra, 8 marks
- Video: [13 Losses and Efficiency of Transformer](https://www.youtube.com/watch?v=jShXjmkMyB8) — SP Engg. ACADEMY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machines/transformer-losses-efficiency-expression
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-2

## Question

What are different types of losses in transformer? Derive the expression of efficiency of transformer.

## Short answer

Losses: core loss (hysteresis and eddy current) constant Pi, copper loss I1^2 R1 + I2^2 R2 varies as square of load. Efficiency for load fraction x is η = (x S cosφ) / (x S cosφ + Pi + x^2 Pcu,fl) where S is rated VA.

## Answer

#### Losses in a transformer

| Loss | Cause | Nature |
|---|---|---|
| **Hysteresis** $P_h = \eta B_{max}^{1.6} f V$ | Repeated reversal of core magnetisation | Constant (fixed $V$, $f$) |
| **Eddy current** $P_e \propto B_{max}^2f^2t^2$ | Currents induced in the core | Constant |
| **Copper** $I_1^2R_1 + I_2^2R_2$ | Resistance of the windings | Varies as (load current)$^2$ |

The first two together are the **iron (core) loss $P_i$**. Stray load loss and dielectric loss are small and neglected. Because there is no rotating part, there is no friction or windage loss.

#### Expression for efficiency

$$
\eta = \frac{\text{output power}}{\text{input power}} = \frac{\text{output}}{\text{output} + \text{losses}}
$$

For a load that takes a fraction $x$ of the full-load current, at power factor $\cos\phi$ and rated secondary voltage $V_2$:

$$
\begin{aligned}
\text{Output} &= x\,S\cos\phi\quad (S = V_2I_{2,fl}\ \text{rated VA}) \\
P_{cu} &= x^2P_{cu,fl} \\
\text{Losses} &= P_i + x^2P_{cu,fl}
\end{aligned}
$$

$$
\boxed{\eta = \frac{xS\cos\phi}{xS\cos\phi + P_i + x^2P_{cu,fl}}\times100\%}
$$

**Remarks:**
- Efficiency is zero at no load, rises with load, reaches a maximum when $P_i = x^2P_{cu,fl}$, and then falls.
- For a good power transformer $\eta$ is 95 to 99%.
- The efficiency is higher at a higher power factor, since the output power is larger for the same losses.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement about the losses in the table is correct?
   A. Hysteresis loss varies as load squared
   B. Copper loss is constant
   C. Hysteresis and eddy-current loss together form constant iron loss
   D. Eddy loss includes friction
   Answer: C. Both depend on flux and frequency, which are fixed, so they are constant. (Key point: Hysteresis loss)

2. At fraction x of full-load current the copper loss is:
   A. x Pcu,fl
   B. Pcu,fl / x^2
   C. Pcu,fl
   D. x^2 Pcu,fl
   Answer: D. Copper loss goes as I^2. (Key point: Copper loss variation)

3. Efficiency at fraction x of load and pf cosphi is:
   A. x S cosphi / (x S cosphi + Pi + x^2 Pcu,fl)
   B. x S cosphi / (x S cosphi + x Pi + Pcu,fl)
   C. S cosphi / (S cosphi + Pi + Pcu,fl)
   D. x S / (x S + Pi + x^2 Pcu,fl)
   Answer: A. Output = xS cosphi; losses = Pi + x^2 Pcu,fl. (Key point: Efficiency expression)

4. How does efficiency vary with load?
   A. Highest at no load
   B. Zero at no load, rises to a maximum when Pi = x^2 Pcu,fl, then falls
   C. Rises continuously to full load
   D. Constant at all loads
   Answer: B. At light load iron loss dominates; at heavy load copper loss grows as x^2. (Key point: Efficiency trend)

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