Ncert Exercises & Solutions

7.21 Obtain the resonant frequency and Q-factor of a series LCR circuit with L = 3.0 H, C = 27µF, and R = 7.4 Ω. It is desired to improve the sharpness of the resonance of the circuit by reducing its ‘full width at half maximum’ by a factor of 2. Suggest a suitable way.

Given,

Inductance, 𝐿 = 3 H

Capacitance, 𝐶 = 27 μF

Resistance, 𝑅 = 7.4 Ω

Resonant frequency,

$\begin{array}{l} ω_{r} = \frac{1}{\sqrt{L C}} = \frac{1}{\sqrt{3 \times 27 \times 10^{- 6}}} = 111 r a d s^{- 1} \\ Q - factor of the circuit, \\ Q = \frac{ω_{r} L}{R} = \frac{111 \times 3}{7.4} = 45 \end{array}$

For improvement in sharpness of resonance by a factor of 2, 𝑄𝑄 should be doubled. To double 𝑄 without changing frequency 𝑅𝑅 should be reduced to half, i.e., to
3.7 Ω.

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