AC Generator (Alternator)
Converts mechanical energy to electrical energy using electromagnetic induction.
Key Result
For a coil of \(N\) turns and area \(A\) rotating with constant angular speed \( \omega \) in a uniform magnetic field \(B\):
\[ e(t) = e_0 \sin(\omega t), \qquad e_0 = N B A\, \omega \]
Frequency \( f=\dfrac{\omega}{2\pi} \) (e.g., \(50\) Hz in India); RMS voltage \( E_\text{rms} = \dfrac{e_0}{\sqrt{2}} \).
Principle
Based on Faraday’s law of electromagnetic induction: a change in magnetic flux through a coil induces an emf. Lenz’s law fixes the polarity so that the induced current opposes the change in flux.
Construction (Main Parts)
- Armature (coil): \(N\) turns, area \(A\).
- Magnetic field: provided by permanent magnets or an electromagnet (stator/rotor arrangement).
- Rotor shaft: mechanically driven (turbine, engine, etc.).
- Slip rings and carbon brushes: take the alternating voltage from the rotating coil to the external circuit.
Working with Full Derivation
- Flux through the coil at time \(t\): take the area vector \( \mathbf{A} \) making angle \( \theta=\omega t \) with \( \mathbf{B} \).
\[ \Phi_B(t) = N\, B\, A \cos(\omega t) \] - Induced emf (Faraday’s law):
\[
e(t) = -\frac{d\Phi_B}{dt}
= -\frac{d}{dt}\big(N B A \cos\omega t\big)
= N B A\, \omega \sin(\omega t)
\]
Hence \( e(t) = e_0 \sin(\omega t) \), with \( e_0 = N B A \omega \). - Direction / polarity changes every half turn, so the output is alternating. (With a commutator instead of slip rings, polarity would be rectified → DC generator.)
Diagrams
Salient Features & Formulas
- Instantaneous flux: \( \Phi_B(t) = N B A \cos(\omega t) \)
- Instantaneous emf: \( e(t) = N B A\, \omega \sin(\omega t) \)
- Max (peak) emf: \( e_0 = N B A\, \omega \)
- Frequency: \( f=\dfrac{\omega}{2\pi} \);\; Period: \( T=\dfrac{1}{f} \)
- RMS voltage: \( E_\text{rms} = \dfrac{e_0}{\sqrt{2}} \)
AC vs DC Generator (Quick Compare)
| Feature | AC Generator | DC Generator |
|---|---|---|
| Output | Alternating \( e(t)=e_0\sin\omega t \) | Unidirectional (rectified) |
| Collector | Slip rings + brushes | Split-ring commutator + brushes |
| Polarity | Reverses every half turn | Maintained (after commutation) |
Applications
- Power stations (hydro/thermal/nuclear) — alternators provide grid AC.
- Vehicle alternators, portable generators.
- Wind turbines: mechanical rotation → electrical AC.
Worked Example
A coil of \( N=100 \) turns and area \( A=0.05\,\text{m}^2 \) rotates at \( f=50\,\text{Hz} \) in a uniform field \( B=0.10\,\text{T} \). Find the peak emf.
\( \omega=2\pi f = 100\pi \,\text{rad s}^{-1} \). So \( e_0 = N B A \omega = 100 \times 0.10 \times 0.05 \times 100\pi = 50\pi \approx 157\,\text{V} \).