Question

Resonant frequency of series LCR circuit is $200Hz$ . Quality factor of the circuit is $5$ , then bandwidth is
A. $200Hz$
B. $100Hz$
C. $40Hz$
D. $50Hz$

Answer 1

Hint: In a series LCR circuit, at resonant frequency maximum current flows through the circuit. The quality factor in LCR circuits is the ratio of the resonant frequency to the bandwidth. Quality factor is dimensionless thus, units of bandwidth will be the same as frequency.

Complete step by step answer:
 In an LCR circuit, the impedance during resonance is only due to the resistor. Quality factor relates to the peak energy stored in the circuit to the energy dissipated during each cycle of oscillation. This implies that quality factor is the ratio of resonant frequency to the bandwidth.
This is denoted as:
$Q={\displaystyle \frac{f_r}{BW}}$
Where $Q$ is the quality factor;
$f_r$ is the frequency during resonance;
And $BW$ denotes the bandwidth.

Using above relation, we can calculate the bandwidth as follows:
$Q={\displaystyle \frac{f_r}{BW}}$
$\Rightarrow BW={\displaystyle \frac{f_r}{Q}}$
It is given that the resonant frequency is of $200Hz$. And the quality factor of the circuit is $5$. Substituting these values in the above equation, we have:
$\Rightarrow BW={\displaystyle \frac{200}{5}}$
$\therefore BW=40Hz$

Therefore, the bandwidth at resonant frequency is $40Hz$ .Option C is the correct answer.

Additional details:
Quality factor is a measure of performance of the coil, capacitor and inductor in terms of its losses and resonator bandwidth. If the quality factor increases then the bandwidth decreases. Higher quality factor indicates a lower rate of energy loss and the oscillations die out more slowly. Bandwidth is always measured between the 0.707 current amplitude points only. This value corresponds to half of the power points.

Note: The units of frequency and bandwidth are the same. Bandwidth of a resonant circuit is the total number of cycles below and above the resonant frequency. The frequency and bandwidth are directly proportional to each other. Quality factor is dimensionless quantity, it has no SI unit.