Question

A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit. Plot a graph to show the variation of current with frequency of the source, explaining the nature of its variation.

Answer 1

Phasor Diagram for a series LCR circuit is shown in the attached figure.

From definition of impedance,

For inductor, $V_L=I{X}_L=I\omega L$

For capacitance, $V_C=I{X}_C=\frac{I}{\omega C}$

For resistance, $V_R=IR$

From the phasor diagram,

$V_s^2=V_R^2+(V_L-V_C{)}^2$

$(IZ{)}^2=(IR{)}^2+(I\omega L-\frac{I}{\omega C}{)}^2$

$Z=\sqrt{R^2+(\omega L-\frac{1}{\omega C}{)}^2}$

Variation of current in the circuit on the variation of applied frequency is shown in the attached figure.

Current is given as:

$I=\frac{V_s}{Z}$

At resonance, impedance is minimum and hence, current is maximum as shown in the plot.

For frequencies smaller than the resonant frequencies, $V_C>V_L$ and hence circuit is capacitive.

For frequencies greater than the resonant frequencies, $V_L>V_C$ and hence circuit is inductive.