Generation of sinusoidal emf

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.

        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.