Question

(i) Mention the factors on which the resonant frequency of a series LCR circuit depends. Plot a graph showing variation of impedance of a series LCR circuit with the frequency of the applied AC source.

Hint:

The resonant frequency \( f_r \) is inversely proportional to \( \sqrt{LC} \). At resonance, impedance is minimum, and current is maximum.

Answer 1

Factors Affecting Resonant Frequency
- The resonant frequency \( f_r \) of a series LCR circuit is given by: \[ f_r = \frac{1}{2\pi \sqrt{LC}} \] It depends on:
1. Inductance (L): Increasing \( L \) decreases the resonant frequency.
2. Capacitance (C): Increasing \( C \) decreases the resonant frequency. Graph: Variation of Impedance with Frequency - The impedance \( Z \) of a series LCR circuit is: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where:
- \( X_L = \omega L \) (Inductive reactance),
- \( X_C = \frac{1}{\omega C} \) (Capacitive reactance). \includegraphics[]{q33 i.PNG} At resonance (\( f = f_r \)):
- \( X_L = X_C \) → Minimum impedance (Only \( R \) remains).
- High current flows through the circuit.