--- title: "Transformer losses; maximum efficiency" subject: "Electrical Machine" chapter: 2 key: "semester_4/electrical_machine:transformer-losses-max-efficiency" source: https://www.bctengineer.com/q/semester_4/electrical_machine/transformer-losses-max-efficiency site: BCT Engineering --- # Transformer losses; maximum efficiency Electrical Machine · Chapter 2: Transformer - Asked in: 2068 Bhadra, 8 marks - Video: [Transformers 12 Maximum Efficiency [Electric Machinery]](https://www.youtube.com/watch?v=TVmbgRRy85E) — Ognjen Grozdanovic - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machine/transformer-losses-max-efficiency - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-2 ## Question Describe different types of losses on the transformer. Also derive the expression for the maximum efficiency of the transformer. ## Short answer Core loss Pi constant, copper loss ∝ I². Efficiency η = x S cos φ / (x S cos φ + Pi + x² Pcu). Maximum η occurs when iron loss equals copper loss: Pi = x² Pcu. Load fraction x = √(Pi/Pcu); η_max = x S cos φ / (x S cos φ + 2Pi). ## Answer #### Types of losses **1. Core (iron) losses** occur in the core and are practically constant from no-load to full load (since V1 and f are constant). - *Hysteresis loss:* Pₕ = ηₕ Bm^1.6 f V, reduced by using silicon steel / CRGO. - *Eddy current loss:* Pₑ = kₑ Bm² f² t² V, reduced by thin insulated laminations. **2. Copper (I²R) losses** occur in the primary and secondary windings and vary as the square of the load current: Pcu = I1² R1 + I2² R2 = I1² R01. **3. Stray and dielectric losses (small):** leakage-flux losses in the tank and structure, and insulation losses (insignificant at normal voltages). Since a transformer has no moving part, there are no friction or windage losses, so efficiency is very high (95–99 %). #### Condition and expression for maximum efficiency Let *x* be the fraction of full-load kVA, S the rating (VA), cos φ the load power factor, Pi the iron loss and Pcu the full-load copper loss. ```text η = x S cos φ / ( x S cos φ + Pi + x² Pcu ) ``` Efficiency is maximum when the denominator-to-numerator ratio is minimum. Divide numerator and denominator by x: ```text η = S cos φ / ( S cos φ + Pi/x + x Pcu ) ``` η is maximum when (Pi/x + x Pcu) is minimum: ```text d/dx (Pi/x + x Pcu) = −Pi/x² + Pcu = 0 => Pi = x² Pcu ``` **Condition for maximum efficiency: iron loss = copper loss at that load.** **Load at maximum efficiency:** x = √(Pi/Pcu), i.e. output kVA = S √(Pi/Pcu). **Maximum efficiency:** ```text η_max = x S cos φ / ( x S cos φ + 2 Pi ) ``` since the copper loss equals the iron loss at that point. **Example:** S = 50 kVA, Pi = 400 W, Pcu = 900 W, cos φ = 1: x = √(400/900) = 0.667; η_max = 33 333/(33 333 + 800) = 97.7 %. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the condition for maximum efficiency of a transformer? A. Iron loss = copper loss at that load B. Copper loss = twice the iron loss C. Full-load copper loss = iron loss at full load only D. Output power = input power Answer: A. Setting d/dx(Pi/x + x Pcu) = 0 gives Pi = x² Pcu, i.e. iron loss equals copper loss. (Key point: Condition for maximum efficiency) 2. In the derivation, the numerator and denominator of η are divided by x. What must then be minimised? A. Pi x + Pcu/x B. Pi/x + x Pcu C. Pi + Pcu D. Pi/x + Pcu/x Answer: B. η = S cos φ / (S cos φ + Pi/x + x Pcu), so the sum Pi/x + x Pcu must be minimum. (Key point: Derivation step) 3. At what fraction of full load does a transformer have maximum efficiency? A. x = Pi/Pcu B. x = Pcu/Pi C. x = sqrt(Pcu/Pi) D. x = sqrt(Pi/Pcu) Answer: D. From Pi = x² Pcu, x = sqrt(Pi/Pcu). (Key point: Load at maximum efficiency) 4. Which expression gives the maximum efficiency? A. x S cos φ / (x S cos φ + 2 Pi) B. x S cos φ / (x S cos φ + Pi) C. x S cos φ / (x S cos φ + 2 Pcu + Pi) D. S cos φ / (S cos φ + 2 Pi) Answer: A. At that point copper loss equals iron loss, so total loss is 2 Pi. (Key point: Maximum efficiency expression) ## Close variants set in other exams - [Transformer losses and efficiency expression](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Atransformer-losses-efficiency-derivation) — Electrical Machine, chapter 2, asked 2× ## Related in Electrical Machine - [Transformer losses and efficiency expression](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Atransformer-losses-efficiency-expression) — Electrical Machines, chapter 2 - [Transformer losses and maximum efficiency condition](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Atransformer-losses-max-efficiency) — Electrical Machines, chapter 2, asked 4× - [Transformer losses and condition for maximum efficiency](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Atransformer-losses-max-efficiency) — Electrical Machines I, chapter 2, asked 3× - [Transformer losses, efficiency and maximum efficiency condition](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Atransformer-losses-efficiency-max-condition) — Electrical Machines I, chapter 2 - [Iron loss and maximum efficiency of 20 kVA 220/2200 V transformer](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3A20kva-iron-loss-max-efficiency-current) — Electrical Machines I, chapter 2, asked 2× - [Iron loss and maximum efficiency of 20 kVA 250/2500 V transformer](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3A20kva-250-2500-iron-loss-max) — Electrical Machines I, chapter 2, asked 2× ## Related in other subjects - [Condition for maximum transformer efficiency](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Amax-efficiency-condition-transformer) — Basic Electrical and Electronics Engineering, chapter 2 - [Transformer principle and maximum efficiency](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Atransformer-principle-efficiency) — Basic Electrical and Electronics Engineering, chapter 2 - [Distribution transformer efficiency and number of tubes](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Adistribution-efficiency-cooling-tubes) — Electric Machine Design, chapter 3, asked 2× - [300 kVA transformer efficiency calculation](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Atransformer-300kva-efficiency) — Basic Electrical and Electronics Engineering, chapter 2