Question

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 Influencing Resonant Frequency
The resonant frequency \( f_r \) of a series LCR circuit is calculated using the formula: \[ f_r = \frac{1}{2\pi \sqrt{LC}} \] Its value is determined by:
1. Inductance (L): An increase in \( L \) leads to a decrease in the resonant frequency.
2. Capacitance (C): An increase in \( C \) leads to a decrease in the resonant frequency. Graph: Impedance vs. Frequency in a Series LCR Circuit - The impedance \( Z \) is defined as: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where:
- \( X_L = \omega L \) represents the inductive reactance,
- \( X_C = \frac{1}{\omega C} \) represents the capacitive reactance. 

 At the resonant frequency (\( f = f_r \)):
- Inductive reactance equals capacitive reactance (\( X_L = X_C \)), resulting in minimum impedance (only resistance \( R \) is present).
- A significant current flows through the circuit.