Question

A fascinating behaviour of the series RLC circuit is the phenomenon of resonance.
a) Explain Resonance in an LCR circuit.
b) Draw a graphical representation of variation of current amplitude $i_m$ with frequency $\omega$.
c) What do you mean by sharpness of resonance? Explain it.

Answer 1

a) In LCR circuit, the impedance is given by:
$Z=\sqrt{R^2+(X_L-X_C{)}^2}$
where $R$ is resistance, $X_L$ is inductive resistance and $X_C$ is capacitive resistance.
At Resonance in LCR circuit,
$X_L=X_C$
$\omega L=\frac{1}{\omega C}$
$\therefore \omega =\frac{1}{\sqrt{LC}}$ which is known as resonance frequency.
Further, impedance is minimum at resonance, $Z_{min}=R$ which means that the current becomes maximum, $I_{max}=\frac{V_{rms}}{R}$.
b) The figure shows variation of current amplitude $i_m$ with frequency $\omega$ in LCR circuit for two values of resistance $R_1$ and $R_2$ ($R_1>R_2$).
c) Quality factor or Q-factor of a resonant LCR circuit is defines 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. If resonance is less sharp, not only the maximum current is less, the circuit is closer to resonance for larger values of $△\omega$ of frequencies and tuning of the circuit will not be good. So, less sharp the resonance, less the selectivity of the circuit while higher the Q, sharper the resonance curve and lesser will be the loss of energy in the circuit.