Obtain the resonant frequency and Q-factor of a series LCR circuit with L = 3.0 H, C = 27 μF, 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.
Inductance, L = 3.0 H
Capacitance, C = 27 μF = 27 × 10−6 F
Resistance, R = 7.4 Ω
At resonance, angular frequency of the source for the given LCR series circuit is given as:
$\omega_{r}=\frac{1}{\sqrt{L C}}$
$=\frac{1}{\sqrt{3 \times 27 \times 10^{-6}}}=\frac{10^{3}}{9}=111.11 \mathrm{rad} \mathrm{s}^{-1}$
Q-factor of the series:
$Q=\frac{\omega_{r} L}{R}$
$=\frac{111.11 \times 3}{7.4}=45.0446$
To improve the sharpness of the resonance by reducing its 'full width at half maximum' by a factor of 2 without changing $\omega_{r}$, we need to reduce $R$ to half i.e.,
Resistance $=\frac{R}{2}=\frac{7.4}{2}=3.7 \Omega$