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Question
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.
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Solution
Inductance, L = 3.0 H
Capacitance, C = 27 μF = 27 × 10−6 F
Resistance, R = 7.4 Ω
At resonance, the angular frequency of the source for the given LCR series circuit is given as:
ωr = `1/sqrt"LC"`
= `1/sqrt(3 xx 27 xx 10^-6)`
= `10^3/9`
= 111.11 rad s−1
Q-factor of the series:
Q = `(ω_"r""L")/"R"`
= `(111.11 xx 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 changingωr, we need to reduce R to half i.e.,
Resistance = `"R"/2 = 7.4/2` = 3.7 Ω
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