--- title: "Transformer operating principle and emf equation" subject: "Electrical Machines I" chapter: 2 key: "semester_4/electrical_machines_i:transformer-principle-emf-equation" source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/transformer-principle-emf-equation site: BCT Engineering --- # Transformer operating principle and emf equation Electrical Machines I · Chapter 2: Transformer - Asked in: 2078 Chaitra, 8 marks; 2068 Bhadra, 8 marks (asked 2 times) - Video: [Transformers - Principle, Working, EMF equation, Construction and applications](https://www.youtube.com/watch?v=Vesx0eEmzX0) — BeTECH SIVA - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines_i/transformer-principle-emf-equation - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-2 ## Question Explain the operating principle of a single phase transformer and derive the emf equation. ## Short answer A transformer transfers AC power by mutual induction between primary and secondary windings on a common core. An alternating flux φ induces emfs E₁ = 4.44 f N₁ φₘ and E₂ = 4.44 f N₂ φₘ. The voltage ratio K = E₂/E₁ = N₂/N₁. ## Answer A **transformer** is a static device that transfers electrical energy from one ac circuit to another at the same frequency, usually changing the voltage level, by **mutual induction** between two windings on a common magnetic core. #### Construction (in brief) Two windings — primary ($N_1$ turns) connected to the supply and secondary ($N_2$ turns) connected to the load — are placed on a laminated silicon-steel core. They are electrically isolated but magnetically coupled. ```text phi ----> +---------------------+ o---|] [|---o V1 |] N1 N2 [| V2 --> load o---|] I1 I2 [|---o +---------------------+ ``` #### Operating principle 1. An alternating voltage $V_1$ applied to the primary drives an alternating current, which sets up an alternating flux φ in the core. 2. Almost all of this flux is confined to the core and links the secondary winding. 3. By Faraday's law the changing flux induces an emf $E_2$ in the secondary (mutually induced emf) and an emf $E_1$ in the primary (self-induced back emf, which nearly balances $V_1$). 4. If a load is connected, current $I_2$ flows and power is delivered. The secondary mmf tends to reduce φ, so the primary draws extra current to restore it; in this way power passes from primary to secondary through the magnetic field. 5. Emf per turn is the same in both windings, so the voltages are in the ratio of turns: step-up if $N_2 > N_1$, step-down if $N_2 < N_1$. A dc supply gives no changing flux, so a transformer does not work on dc. #### Derivation of emf equation Let the core flux be $\phi = \phi_m \sin\omega t$, where $\omega = 2\pi f$. Instantaneous emf in the primary: $$ \begin{aligned} e_1 &= -N_1\frac{d\phi}{dt} = -N_1 \phi_m \omega \cos\omega t \\ &= 2\pi f N_1\phi_m \sin(\omega t - 90^\circ) \end{aligned} $$ Maximum value: $E_{1,max} = 2\pi f N_1 \phi_m$. RMS value: $$ \begin{aligned} E_1 &= \frac{2\pi f N_1\phi_m}{\sqrt2} = 4.44\, f N_1 \phi_m \ \text{V} \end{aligned} $$ Similarly $E_2 = 4.44\, f N_2 \phi_m$ V. Since $\phi_m = B_m A$, $E = 4.44\, f N B_m A$. #### Voltage transformation ratio $$ K = \frac{E_2}{E_1} = \frac{N_2}{N_1} $$ The emf lags the flux by 90°. **Example:** a 230/115 V, 50 Hz transformer with 400 primary turns has $\phi_m = 230/(4.44\times50\times400) = 2.59$ mWb, and needs $400\times115/230 = 200$ secondary turns. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a transformer? A. A rotating machine converting ac to dc B. A device that changes frequency between windings C. A device that works equally on dc D. A static device transferring energy between ac circuits at the same frequency by mutual induction Answer: D. Two coupled windings on a common core, with no moving parts, change voltage but not frequency. (Key point: Static device) 2. Why does a transformer not work on dc? A. dc has too high a frequency B. dc gives constant flux, so no emf is induced C. dc makes the core lose permeability at once D. dc gives the windings infinite resistance Answer: B. Induction requires a changing flux. (Key point: DC supply) 3. What is the rms emf in the primary? A. E1 = 2.22 f N1 φm B. E1 = 6.28 f N1 φm C. E1 = 4.44 f N1 φm D. E1 = 4.44 N1 φm / f Answer: C. Maximum emf 2π f N φm divided by √2. (Key point: Emf equation) 4. For a 230/115 V, 50 Hz transformer with 400 primary turns, what is φm? A. About 25.9 mWb B. About 0.26 mWb C. About 5.2 mWb D. About 2.59 mWb Answer: D. φm = 230/(4.44×50×400) = 2.59×10^-3 Wb. (Key point: Example) ## Close variants set in other exams - [Ideal transformer principle and emf equation](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Aideal-transformer-principle-emf) — Electrical Machines I, chapter 2, asked 6× ## Related in Electrical Machines I - [Operating principle of 3-phase induction motor](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ainduction-motor-operating-principle) — Electrical Machines I, chapter 5 - [EMF equation of DC generator](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Adc-generator-emf-equation) — Electrical Machines I, chapter 3 - [DC motor working principle and torque equation](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Adc-motor-principle-torque-equation) — Electrical Machines I, chapter 4, asked 4× ## Related in other subjects - [Ideal transformer principle and emf equation](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Aideal-transformer-emf-equation) — Electrical Machine, chapter 2 - [Ideal transformer principle and emf equation](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aideal-transformer-principle-emf) — Electrical Machines, chapter 2 - [Transformer definition, principle, emf equation](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Atransformer-emf-equation) — Basic Electrical and Electronics Engineering, chapter 2 - [Output (kVA) equation of single phase transformer](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aoutput-equation-single-phase-transformer) — Electric Machine Design, chapter 3, asked 3×