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

Inductance, L = 3.0 H,
Capacitance, C = 27 μF = 27 x 10^-6 F,
Resistance, R = 7.4 Ω
At resonance, angular frequency of the source for the given LCR series circuit is given as

ω_r = 1/√(LC) = 1/√(3 x 27 x 10^-6) = 111.11 rad/s

Q-factor of the series:

Q = ω_rL/R  = 111.11 x 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 ω_r, we need to reduce R to half i.e.,

Resistance = R/2 = 7.4/2 = 3.7 Ω