Question

In a series LCR circuit connected to an ac source of variable frequency and voltage $v=v_{m}sin\omega t$, draw a plot showing the variation of current (I) with angular frequency $\left(\omega \right)$ for two different values of resistance $R_{1}and{R}_2(R_1>R_2)$, Write the condition under which the phenomenon of resonance occurs. For which value of the resistance out of the two curves a sharper resonance is produced? Define Q-factor of the circuit and give its significance

Answer 1

Figure shows the variation of $i_m$ with $\omega$ in a LCR series circuits for two values of Resistance $R{F}_{1}and{R}_2(R_1>R_2)$,
The condition for resonance in the LCR circuit is,
${\omega }_0=\frac{1}{\sqrt{LC}}$
We can observe that the current amplitude is maximum at the resonant frequency $\omega$. Since $i_m=V_{m}lR$ at resonance, the current amplitude fro case $R_2$ is sharper to that for case $R_1$ Quality factor or simply the Q-factor of a resonance LCR circuit is defined as the ratio of voltage drop across the capacitor (or inductor ) to that of applied voltage
It is given by $Q=\frac{1}{R}\sqrt{\frac{L}{C}}$
The Q factor determines the sharpness of the resonance curve and if the resonance is less sharp, the maximum current decrease and also the circuit is close to the resonance for a larger range $\Delta \omega$ of frequencies and the regulation of the circuit will not be good. So, less sharp the resonance, less is the selectivity of the circuit while higher is the Q, sharper is the resonance curve and lesser will be the loss in energy of the circuit