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Obtain the resonant frequency and varrho-factor of a series LCR circuit.

Q7.21 Obtain the resonant frequency and \varrho-factor of a series LCR  circuit  with L=0.3H, C=27\mu F, and R=7.4\Omega . 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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The inductance of the inductor L=0.3H

The capacitance of the capacitor C=27\mu F

The resistance of the resistor R=7.4\Omega

Now,

Resonant frequency

w_r=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{0.3*27*10^{-6}}}=111.11rad/sec

Q-Factor of the circuit

Q=\frac{w_rL}{R}=\frac{111.11*0.3}{7.4}=45.0446

Now, to improve the sharpness of resonance by reducing its full width at half maximum, by a factor of 2 without changingw_r,

we have to change the resistance of the resistor to half of its value, that is

R_{new}=\frac{R}{2}=\frac{7.4}{2}=3.7\Omega

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