AC generator. An AC generator converts mechanical energy into electrical energy using electromagnetic induction. A coil of $N$ turns and area $A$ is rotated with angular speed $\omega$ in a uniform field $B$. The flux through it is $\Phi = NBA\cos\omega t$, so by Faraday's law the induced EMF is $\varepsilon = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t$, where the peak EMF is $\varepsilon_0 = NBA\omega$. The output is sinusoidal AC, delivered through slip rings.
Transformer. A transformer changes the voltage of an AC supply using mutual induction. Two coils — the primary ($N_p$ turns) and secondary ($N_s$ turns) — are wound on a common laminated soft-iron core. An alternating current in the primary creates a changing flux that links the secondary and induces an EMF in it.
Turns ratio. For an ideal transformer, the same flux links each turn, so the voltages are in the ratio of the turns:
- $\frac{V_s}{V_p} = \frac{N_s}{N_p}$ — the transformation ratio.
- A step-up transformer has $N_s > N_p$, so $V_s > V_p$ (raises voltage, lowers current).
- A step-down transformer has $N_s < N_p$, so $V_s < V_p$ (lowers voltage, raises current).
- For an ideal (lossless) transformer, power is conserved: $V_pI_p = V_sI_s$, so $\frac{I_s}{I_p} = \frac{N_p}{N_s}$.
Why transformers matter. Power is transmitted over long distances at very high voltage and low current to minimise $I^2R$ heat losses in the cables; step-down transformers then reduce the voltage to a safe $220\,\text{V}$ before it reaches homes. A transformer works only on AC, since a steady DC produces no changing flux.
Transformer losses. Real transformers are not perfect:
- Copper loss: $I^2R$ heating in the windings — reduced by using thick low-resistance wire.
- Eddy-current loss: currents induced in the core — reduced by laminating the core.
- Hysteresis loss: energy spent repeatedly magnetising and demagnetising the core — reduced by using soft iron with a narrow hysteresis loop.
- Flux leakage: not all flux links both coils — reduced by good core design.
LC oscillations. When a charged capacitor is connected to an inductor, energy oscillates back and forth between the electric field of the capacitor and the magnetic field of the inductor — just like a mechanical oscillator exchanges potential and kinetic energy. The charge oscillates sinusoidally at the natural frequency $f = \frac{1}{2\pi\sqrt{LC}}$. In an ideal (resistance-free) LC circuit, the total energy $U = \frac{q^2}{2C} + \frac{1}{2}LI^2$ stays constant; in practice, resistance damps the oscillations.