Question

What is meant by resonant circuit? Write down the required condition for the L-C-R series resonant circuit and expression for the frequency in resonant condition.

Hint:

In an L-C-R series circuit, resonance occurs when the reactances of the inductor and capacitor cancel each other out, resulting in minimum impedance. At this point, the circuit can oscillate freely at the resonant frequency.

Answer 1

Resonant Circuit:
A resonant circuit, also called a tuned circuit, is an electrical circuit that resonates at a particular frequency, called the resonant frequency. In such a circuit, the inductive reactance (\( X_L \)) and capacitive reactance (\( X_C \)) are equal and cancel each other out, resulting in a minimum impedance. Condition for Resonance in L-C-R Series Circuit:
In a series L-C-R circuit, resonance occurs when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase. The condition for resonance is: \[ X_L = X_C, \] where: - \( X_L = \omega L \) is the inductive reactance,
- \( X_C = \frac{1}{\omega C} \) is the capacitive reactance,
- \( \omega \) is the angular frequency (\( \omega = 2 \pi f \)),
- \( L \) is the inductance,
- \( C \) is the capacitance. At resonance, \( X_L = X_C \), so: \[ \omega L = \frac{1}{\omega C}. \] Solving for the angular frequency \( \omega \) at resonance: \[ \omega^2 = \frac{1}{LC}, \] \[ \omega = \frac{1}{\sqrt{LC}}. \] The resonant frequency \( f_0 \) is then: \[ f_0 = \frac{1}{2 \pi \sqrt{LC}}. \]