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Question

For an RLC circuit driven with a voltage of amplitude vm and frequency ω0=1LC the current exhibits resonance. The quality factor, Q is given by


A

Rω0C

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B

CRω0

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C

ω0LR

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D

ω0RL

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Solution

The correct option is C

ω0LR


Step 1: Given,

Amplitude of voltage =vm

Frequency ω0=1LC

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=R2+(XL-XC)2

(Where R is the resistance, XL is the inductive resistance, and XC 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=RL

(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.

Q=ω0BW=ω0RL=ω0LR

Hence, option C is the correct option.


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