Question

For an RLC circuit driven with a voltage of amplitude $v_m$ and frequency ${\omega }_0=\frac{1}{\sqrt{LC}}$ the current exhibits resonance. The quality factor, Q is given by

• A. $\frac{R}{{\omega }_{0}C}$
• B. $\frac{CR}{{\omega }_0}$
• C. $\frac{{\omega }_{0}L}{R}$
• D. $\frac{{\omega }_{0}R}{L}$

Answer 1

The correct option is C

$\frac{{\omega }_{0}L}{R}$

Step 1: Given,

Amplitude of voltage $=v_m$

Frequency ${\omega }_0=\frac{1}{\sqrt{LC}}$

Step 2: Find the bandwidth,

Resonance: In an LCR circuit, resonance occurs when the inductive and capacitive reactances are equal in magnitude but are $180°$ apart in phase.

Condition for resonance: $Z=\sqrt{R^2+{(X_L-X_C)}^2}$

(Where $R$ is the resistance, $X_L$ is the inductive resistance, and $X_C$ is capacitive resistance)

Bandwidth: Bandwidth is the range of frequencies for which the total power is more than half of the maximum power in a system.

The value of bandwidth is given by

$BW=\frac{R}{L}$

(BW is the bandwidth, R is the resistance and L is the inductance)

Step 3: Find the quality factor,

Quality factor: It is the sharpness of a resonant LCR circuit which is determined by the ratio of resonant frequency to the bandwidth.

$$\begin{gathered} Q=\frac{{\omega }_0}{BW} \\ =\frac{{\omega }_0}{{\displaystyle \frac{R}{L}}} \\ =\frac{{\omega }_{0}L}{R} \end{gathered}$$

Hence, option C is the correct option.