--- title: "Transformer losses and maximum efficiency condition" subject: "Electrical Machines" chapter: 2 key: "semester_4/electrical_machines:transformer-losses-max-efficiency" source: https://www.bctengineer.com/q/semester_4/electrical_machines/transformer-losses-max-efficiency site: BCT Engineering --- # Transformer losses and maximum efficiency condition Electrical Machines · Chapter 2: Transformer - Asked in: 2077 Chaitra, 8 marks; 2075 Baisakh (old course), 5+3 marks; 2068 Bhadra, 8 marks; 2066 Magh (old course), 8 marks (asked 4 times) - 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-max-efficiency - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-2 ## Question What are the different types of losses in a transformer? Derive the condition for maximum efficiency of the transformer. ## Short answer Transformer losses are constant core (iron) loss Pi and variable copper loss Pcu = I2^2 R02. Efficiency η = (V2 I2 cosφ2) / (V2 I2 cosφ2 + Pi + I2^2 R02). Maximum occurs when d/dI2 denominator zero, giving Pi = I2^2 R02, i.e., copper loss equals iron loss. ## Answer #### Losses in a transformer A transformer has no moving parts, so there are only two kinds of loss. 1. **Core (iron) losses $P_i$:** present whenever the primary is energised at rated voltage and frequency. They are practically constant from no load to full load. - *Hysteresis loss:* $P_h = \eta B_{max}^{1.6} f V$, from the repeated reversal of magnetisation. It is reduced by using silicon steel. - *Eddy current loss:* $P_e \propto B_{max}^2 f^2 t^2$, from currents induced in the core. It is reduced by thin insulated laminations. 2. **Copper losses $P_{cu}$:** $I_1^2R_1 + I_2^2R_2 = I_2^2R_{02}$ in the windings. They vary as the square of the load current. Stray (leakage flux) and dielectric losses are small and usually ignored. #### Condition for maximum efficiency Let the load current be $I_2$, secondary voltage $V_2$ (assumed constant), power factor $\cos\phi_2$ and $R_{02}$ the total resistance referred to the secondary. $$ \eta = \frac{V_2I_2\cos\phi_2}{V_2I_2\cos\phi_2 + P_i + I_2^2R_{02}} $$ Divide numerator and denominator by $I_2$: $$ \eta = \frac{V_2\cos\phi_2}{V_2\cos\phi_2 + \dfrac{P_i}{I_2} + I_2R_{02}} $$ For constant $V_2$ and $\cos\phi_2$, $\eta$ is maximum when the denominator is minimum: $$ \frac{d}{dI_2}\left(\frac{P_i}{I_2} + I_2R_{02}\right) = 0 \Rightarrow -\frac{P_i}{I_2^2} + R_{02} = 0 $$ $$ P_i = I_2^2R_{02} $$ **Condition:** efficiency is maximum when the variable (copper) loss equals the constant (iron) loss. Load at maximum efficiency, with $P_{cu,fl}$ the full-load copper loss: $$ x = \frac{I_2}{I_{2,fl}} = \sqrt{\frac{P_i}{P_{cu,fl}}},\qquad \text{kVA at max } \eta = \text{rated kVA}\times\sqrt{\frac{P_i}{P_{cu,fl}}} $$ The second derivative is positive, so this is a minimum of the denominator and a maximum of $\eta$. The condition does not depend on the power factor, but the value of the maximum efficiency is higher at a higher power factor. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which are the two kinds of loss in a transformer? A. Core (iron) loss and copper loss B. Friction loss and copper loss C. Windage loss and core loss D. Core loss and mechanical loss Answer: A. A transformer has no moving parts, so there is no friction or windage; only iron and copper losses matter. (Key point: Types of loss) 2. How do the iron and copper losses vary with load? A. Iron loss varies as load squared; copper loss is constant B. Iron loss is practically constant; copper loss varies as the square of load current C. Both are constant D. Both vary directly with load current Answer: B. Iron loss depends on rated voltage and frequency (constant flux), while copper loss is I^2R. (Key point: Behaviour of losses) 3. What is the condition for maximum efficiency? A. Copper loss is twice iron loss B. Iron loss is zero C. Variable (copper) loss equals constant (iron) loss D. The load is at full load Answer: C. Setting d/dI2 (Pi/I2 + I2 R02) = 0 gives Pi = I2^2 R02. (Key point: Condition for max efficiency) 4. Load at maximum efficiency as a fraction of full load is: A. Pi / Pcu,fl B. sqrt(Pcu,fl / Pi) C. Pcu,fl / Pi D. sqrt(Pi / Pcu,fl) Answer: D. From Pi = x^2 Pcu,fl, x = sqrt(Pi/Pcu,fl). (Key point: Load at max efficiency) 5. How does power factor affect the maximum-efficiency condition and value? A. The condition is independent of pf, but the maximum efficiency is higher at higher pf B. The condition changes with pf and the maximum efficiency is independent of pf C. Both depend on pf D. Neither depends on pf Answer: A. Pi = I2^2 R02 contains no pf, but a higher pf gives more output for the same losses. 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