---
title: "Generation of sinusoidal emf"
subject: "Basic Electrical Engineering"
chapter: 5
key: "semester_1/basic_electrical_engineering:sinusoidal-emf-generation-angular-velocity"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/sinusoidal-emf-generation-angular-velocity
site: BCT Engineering
---

# Generation of sinusoidal emf

Basic Electrical Engineering · Chapter 5: Alternating Quantities

- Asked in: 2069 Chaitra, 6 marks
- Video: [EMF & flux equation (& graph) of AC generator | Electromagnetic induction | Physics | Khan Academy](https://www.youtube.com/watch?v=Sp1m6GYo5iw) — Khan Academy India - English
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/sinusoidal-emf-generation-angular-velocity
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-5

## Question

Explain generation of sinusoidal emf with diagram and define angular velocity.

## Short answer

When a coil rotates at constant angular velocity ω in a uniform magnetic field, the induced emf is e = Em sin ωt, with Em = NBAω. Angular velocity is the rate of angular displacement: ω = 2πf rad/s.

## Answer

**Principle:** When a coil rotates in a uniform magnetic field, its conductors cut the flux and an emf is induced (Faraday's law). The emf varies sinusoidally with the angle turned.

#### Generation

Take a rectangular coil of N turns, area A, rotating at constant angular velocity ω in a uniform field B between N and S poles.

```text
        N                S
      ┌───────────────────────┐
      │    ┌───────────┐      │
      │    │   coil    │ ──►ω │
      │    └───────────┘      │
      └───────────────────────┘
   Slip rings + brushes → load
```

At t = 0 the coil plane is perpendicular to the field (flux maximum, emf zero). After time t the coil has turned through θ = ωt.

- Flux linked: φ = B A cos ωt
- Induced emf: e = −N dφ/dt = N B A ω sin ωt

So **e = Em sin ωt**, with Em = N B A ω.

- At θ = 0° and 180°: coil sides move parallel to the field, e = 0.
- At 90° (emf +max) and 270° (−max): sides cut the field at right angles.

Plotting e against ωt gives a sine wave. One revolution of a two-pole machine gives one cycle. Slip rings carry the emf to the external circuit.

#### Angular velocity

Angular velocity is the rate of change of angular displacement of the rotating coil:

ω = θ/t = 2π f rad/s

where f is the frequency in Hz. For f = 50 Hz, ω = 314 rad/s.

## Check yourself

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

1. Principle?
   A. Ohm's law
   B. Faraday's law
   C. Coulomb
   D. Lenz only
   Answer: B. Flux cutting. (Key point: Principle)

2. Emf?
   A. B A
   B. N B A
   C. N B A cos
   D. N B A omega sin omega t
   Answer: D. e=-N dphi/dt. (Key point: Equation)

3. Zero emf positions?
   A. 45
   B. none
   C. 0 and 180
   D. 90 and 270
   Answer: C. Sides move parallel. (Key point: Zero emf)

4. Angular velocity at 50 Hz?
   A. 314 rad/s
   B. 157
   C. 50
   D. 100
   Answer: A. 2 pi f. (Key point: Omega)

## Related in Basic Electrical Engineering

- [RL coil on sinusoidal source](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Arl-coil-200sin314t-20ohm-0-1h) — Basic Electrical Engineering, chapter 6
- [Explain emf, potential difference and current](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Aemf-potential-difference-current) — Basic Electrical Engineering, chapter 1

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