alternating-currents Question 31
Question: Q. 8. (i) A series $L C R$ circuit is connected to an ac source of variable frequency. Draw a suitáble phasor diagram to deduce the expressions for the amplitude of the current and phase angle.
(ii) Obtain the condition at resonance. Draw a plot showing the variation of current with the frequency of a.c. source for two resistances $R_{1}$ and $R_{2}\left(R_{1}>R_{2}\right)$. Hence define the quality factor, $Q$ and write its role in the tuning of the circuit.
U] [Delhi Comptt. I, II, III 2014]
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Solution:
Ans. (i) Try yourself similar Q. 5 Long Answer Type Question.
(ii) At resonance, $I_{m}$ is maximum.
$$ \Rightarrow \quad X_{L}=X_{\mathrm{C}^{\prime}} $$
[Alternatively : $\omega_{0}=\frac{1}{\sqrt{L C}}$ ]
(ii) plot is for $R_{1}$ (i) plot is for $\mathrm{R}{2}$ $\left(R{2}>R_{1}\right)$
Quality factor of $L C R$ circuit is defined as
$$ \begin{equation*} \frac{\omega_{0}}{2 \Delta \omega}=\frac{\omega_{0} L}{R} \tag{1} \end{equation*} $$
A larger value of quality factor corresponds to a sharper resonance.
[CBSE Marking Scheme, 2014]
