Part (a): Series LCR Circuit Analysis
Step 1 (Governing Laws & Phasor Representation): [1 Mark]
In a series LCR circuit, the current $I = I_m \sin(\omega t)$ is common to all components. The voltage across the resistor ($V_R$) is in phase with $I$, the voltage across the inductor ($V_L$) leads $I$ by $\pi/2$, and the voltage across the capacitor ($V_C$) lags $I$ by $\pi/2$.
The net voltage $V$ is the phasor sum: $\vec{V} = \vec{V_R} + \vec{V_L} + \vec{V_C}$.
Step 2 (Mathematical Derivation): [2 Marks]
From the phasor diagram, the resultant voltage $V$ is given by:
$$V = \sqrt{V_R^2 + (V_L - V_C)^2}$$Substituting $V_R = IR$, $V_L = IX_L$, and $V_C = IX_C$:
$$V = \sqrt{(IR)^2 + (IX_L - IX_C)^2} = I \sqrt{R^2 + (X_L - X_C)^2}$$The impedance $Z$ is defined as $Z = V/I$:
$$Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}$$The phase difference $\phi$ between $V$ and $I$ is:
$$\tan \phi = \frac{V_L - V_C}{V_R} = \frac{X_L - X_C}{R}$$Step 3 (Final Concluded Answer): [1 Mark]
- (i) Impedance is minimum: When $X_L = X_C$ (Resonance condition), $Z = R$. This occurs at $\omega = \frac{1}{\sqrt{LC}}$.
- (ii) Wattless current: Flows when the average power consumed is zero. This happens in a purely inductive or purely capacitive circuit where $\phi = \pi/2$, so $\cos \phi = 0$.
💡 EL10 Concept Insight
Think of the LCR circuit like a tug-of-war: the inductor pulls the voltage "up" and the capacitor pulls it "down." When they pull with equal strength, they cancel each other out, leaving only the resistor to resist the flow, which makes the path of least resistance (minimum impedance)!
Alternative (OR): A.C. Generator
Step 1 (Principle & Construction): [1.5 Marks]
Principle: Based on Electromagnetic Induction (Faraday's Law). When a coil rotates in a uniform magnetic field, the magnetic flux linked with it changes, inducing an emf.
Construction: Consists of a rectangular coil (armature), strong permanent magnets (field magnets), slip rings, and carbon brushes.
Step 2 (Deduction of Induced EMF): [2 Marks]
Magnetic flux $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos \theta$. Since $\theta = \omega t$:
$$\Phi_B = BA \cos(\omega t)$$By Faraday's Law, $\varepsilon = -N \frac{d\Phi_