4.5.2 The a.c. Generator
What is an a.c. generator?
An a.c. generator (alternator) is a device that turns mechanical energy into an alternating electric current. Think of it as a spinning magnet inside a coil of wire that creates a changing magnetic field, which in turn induces an electromotive force (e.m.f.) in the coil. The e.m.f. varies sinusoidally with time, just like a wave on a string.
Sketching the e.m.f. vs. time graph
The e.m.f. produced by a simple a.c. generator follows a sine wave:
\$E(t) = E_{\text{max}} \sin(\omega t)\$
where:
- \$E_{\text{max}}\$ = maximum e.m.f. (peak value)
- \$\omega\$ = angular frequency (rad s⁻¹)
- \$t\$ = time (s)
The graph looks like a smooth wave that rises to a peak, falls to a trough, and crosses zero twice every cycle. Use the emoji ⚡️ to mark peaks, 📉 for troughs, and ⏱️ for zero crossings.
⚡️ Peak → ⏱️ Zero → 📉 Trough → ⏱️ Zero → ⚡️ Peak
Interpreting the graph
Key points:
- The e.m.f. is zero when the magnetic field through the coil is not changing.
- Peaks occur when the magnetic flux changes most rapidly.
- The sign of the e.m.f. (positive or negative) tells you the direction of the induced current.
- One full cycle takes \$T = \frac{2\pi}{\omega}\$ seconds.
Coil position vs. e.m.f.
The position of the coil (measured as the angle \$\theta\$ from the reference position) determines the instantaneous e.m.f. The relationship is:
\$E(\theta) = E_{\text{max}} \sin(\theta)\$
where \$\theta = \omega t\$.
| Coil Position (°) | e.m.f. (Relative) | Interpretation |
|---|
| 0° | 0 | Zero crossing – magnetic flux not changing. |
| 90° | +\$E_{\text{max}}\$ | Peak – flux changing fastest. |
| 180° | 0 | Zero crossing – flux not changing. |
| 270° | –\$E_{\text{max}}\$ | Trough – flux changing fastest in the opposite direction. |
| 360° | 0 | Back to zero – cycle repeats. |
Example: Simple generator with one coil
Imagine a single coil spinning in a uniform magnetic field. As the coil rotates:
- When the coil is aligned with the field (0°), no flux change → e.m.f. = 0.
- At 90°, the coil is perpendicular to the field → maximum flux change → e.m.f. peaks.
- At 180°, the coil aligns again but opposite → e.m.f. returns to zero.
- At 270°, the coil is perpendicular again but in the opposite sense → e.m.f. trough.
- At 360°, the coil completes a full rotation → e.m.f. zero again.
This simple cycle repeats, producing a sinusoidal a.c. output that can be used to power devices.
Exam Tips
- Remember that the e.m.f. is proportional to the rate of change of magnetic flux: \$E = -\frac{d\Phi}{dt}\$.
- Use the sine wave shape to identify peaks, troughs, and zero crossings.
- When asked to sketch, label the axis: time (s) on the horizontal, e.m.f. (V) on the vertical.
- For coil position questions, convert angles to radians if needed: \$\theta{\text{rad}} = \theta{\text{deg}}\times\frac{\pi}{180}\$.
- Practice converting between frequency (Hz) and angular frequency: \$\omega = 2\pi f\$.
- Use the analogy of a rotating wheel: the faster it spins, the higher the frequency and the steeper the slope of the sine wave.