In a series LCR circuit connected to an ac source of variable frequency and voltage v $v_m= \sin, $ draw a plot showing the variation of current (I) with angular frequency ($\omega$) for two different values of resistance R1 and R2(R1 > R2 ). 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.
CBSE DELHI - SET 1 2013
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At certain frequency $\omega$, the flow of current through the series combination
$\text{I}_{m} = \frac{\text{v}_{m}}{\sqrt{\text{R}^{2} + (\text{x}_{L} - \text{x}_{C}^{2}})}$
Where $X_L = \omega_{L}$
and
$\text{X}_{C} = \frac{1}{\omega\text{C}}.$
Condition of resonance – If systemis (LCR) of natural frequency $\omega_{0}$ is driven by an energy source at a frequency $\omega$, the amplitude of the current flow increases, however the amplitude of the current rises to its maximum value, if frequency of the energy source becomes exactly equal to the natural frequency.
For resistance $R_2 < R_1$, series LCR shows a sharp resonance.
Q-factor – The ratio of reactance (either inductive or capacitive) at natural frequency to the resistance of the current is called Q - factor
$\text{Q} = \frac{\text{X}_{L}}{\text{R}} = \frac{\omega_{0}\text{L}}{\text{R}}$
Significance:
- If resistance R is low or inductance L is large then Q – factor is large and the circuit is more selective.
- If resonance is less sharp, tunning of the circuit will not be good.
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