Question

In the circuit shown in the figure, the ratio of the quality factor and the band width is__s

Answer 1

Correct Answer: 10

To find the ratio of the quality factor (Q) to the bandwidth (BW) of the given RLC circuit, we first determine the relevant parameters. The quality factor is given by the formula:
\( Q = \frac{\omega_0 L}{R} \)
The resonant angular frequency \( \omega_0 \) is given by:
\( \omega_0 = \frac{1}{\sqrt{LC}} \)
where \( L = 3\, \text{H} \), \( C = 27\, \mu\text{F} = 27 \times 10^{-6}\, \text{F} \), and \( R = 10\, \Omega \).
Calculating \( \omega_0 \):
\( \omega_0 = \frac{1}{\sqrt{3 \times 27 \times 10^{-6}}} = \frac{1}{\sqrt{81 \times 10^{-6}}} = \frac{1}{9 \times 10^{-3}} = \frac{1000}{9} \approx 111.11 \, \text{rad/s} \)
Then, substitute \( \omega_0 \) in the Q formula:
\( Q = \frac{111.11 \times 3}{10} = \frac{333.33}{10} = 33.33 \)
The bandwidth BW is given by:
\( \text{BW} = \frac{\omega_0}{Q} = \frac{111.11}{33.33} = 3.33\, \text{rad/s} \)
Finally, the ratio of the quality factor and the bandwidth is:
\( \frac{Q}{\text{BW}} = \frac{33.33}{3.33} = 10 \)
This computed value 10 falls within the provided range of (10, 10), confirming its accuracy.