(a) (i) AC Generator: Principle, Diagram, and Working
Principle: An AC generator works on the principle of Electromagnetic Induction. When a closed coil is rotated in a uniform magnetic field, the magnetic flux linked with the coil changes continuously, which induces an alternating emf in the coil.
Diagram Description: A rectangular coil $ABCD$ is placed between the pole pieces of a strong permanent magnet. The ends of the coil are connected to two slip rings $R_1$ and $R_2$, which rotate with the coil. Two carbon brushes $B_1$ and $B_2$ press against the rings to conduct current to the external load.
Working: As the coil rotates, the magnetic flux $\phi = BA \cos(\omega t)$ changes. According to Faraday’s law, the induced emf is $\varepsilon = - \frac{d\phi}{dt} = NBA\omega \sin(\omega t)$. The direction of the induced current is determined by Fleming’s Right-Hand Rule.
(a) (ii) Numerical Solution
Step 1 (Given Data & Governing Laws): [1 Mark]
- $R = 400\ \Omega$, $L = \frac{5}{\pi}\ \text{H}$, $C = \frac{50}{\pi} \times 10^{-6}\ \text{F}$
- Source: $v = 140 \sin(100\pi t) \implies V_0 = 140\ \text{V}$, $\omega = 100\pi\ \text{rad/s}$
- $V_{\text{rms}} = \frac{V_0}{\sqrt{2}} = \frac{140}{\sqrt{2}} = 70\sqrt{2} \approx 99\ \text{V}$
Step 2 (Mathematical Working): [2 Marks]
Calculate Reactances:
$$X_L = \omega L = 100\pi \times \frac{5}{\pi} = 500\ \Omega$$ $$X_C = \frac{1}{\omega C} = \frac{1}{100\pi \times \frac{50}{\pi} \times 10^{-6}} = \frac{10^6}{5000} = 200\ \Omega$$Calculate Impedance ($Z$):
$$Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{400^2 + (500 - 200)^2} = \sqrt{160000 + 90000} = 500\ \Omega$$Calculate Current ($I_{\text{rms}}$):
$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{99}{500} = 0.198\ \text{A}$$Calculate rms voltages across elements:
$$V_R = I_{\text{rms}}R = 0.198 \times 400 = 79.2\ \text{V}$$ $$V_L = I_{\text{rms}}X_L = 0.198 \times 500 = 99\ \text{V}$$ $$V_C = I_{\text{rms}}X_C = 0.198 \times 200 = 39.6\ \text{V}$$Step 3 (Final Conclusion & Explanation): [1 Mark]