Obtain the resonant frequency and Q-factor of a series LCR circuitwith L = 3.0 H, C = 27 μF, and R = 7.4 W. It is desired to improve thesharpness of the resonance of the circuit by reducing its ‘full widthat half maximum’ by a factor of 2. Suggest a suitable way.
Given: The inductance of inductor is 3 H , the capacitance of the capacitor is 27 μF and the resistance of resistor is 7.4 Ω .
The angular frequency of the source for the given RLC series circuit is given as,
ω r = 1 L C
Where, the inductance is L and the capacitance is C .
By substituting the given values in the above equation, we get
ω r = 1 3 × 27 × 10 − 6 = 10 3 9 = 111.11 rad / s
The quality factor of the series circuit is given as,
Q = ω r L R
Where, the resistance is R .
By substituting the given values in the above equation, we get
Q = 111.11 × 3 7.4 = 45.0446
Since, to improve the sharpness of the resonance of the circuit by reducing its full width at half maximum by a factor of 2 .
The new resistance of the circuit is given as,
R 1 = R 2
By substituting the given value in the above equation, we get
R 1 = 7.4 2 = 3.7 Ω
Thus, to improve the sharpness of the resonance of the circuit the value of resistance should be 3.7 Ω .
Given: The inductance of inductor is
The angular frequency of the source for the given RLC series circuit is given as,
Where, the inductance is
By substituting the given values in the above equation, we get
The quality factor of the series circuit is given as,
Where, the resistance is
By substituting the given values in the above equation, we get
Since, to improve the sharpness of the resonance of the circuit by reducing its full width at half maximum by a factor of
The new resistance of the circuit is given as,
By substituting the given value in the above equation, we get
Thus, to improve the sharpness of the resonance of the circuit the value of resistance should be
