Series LCR Circuit - Practice Questions & MCQ

Series LCR circuit-

The Figure given above shows a circuit containing a capacitor, resistor and inductor connected in series through an alternating/sinusoidal voltage source.

As they are in series so the same amount of current will flow in all the three circuit components and for the voltage, vector sum of potential drop across each component would be equal to the applied voltage.

Let 'i' be the amount of current in the circuit at any time and $\mathrm{V}_{\mathrm{L}}, \mathrm{V}_{\mathrm{C}}$ and $\mathrm{V}_{\mathrm{R}}$ the potential drop across $\mathrm{L}, \mathrm{C}$ and R respectively then

$
\begin{aligned}
& \mathrm{v}_{\mathrm{R}}=\mathrm{iR} \rightarrow \text { Voltage is in phase with } \mathrm{i} \\
& \mathrm{v}_{\mathrm{L}=\mathrm{i} \omega \mathrm{~L}} \rightarrow \text { voltage is leading } \mathrm{i} \text { by } 90^{\circ} \\
& \mathrm{v}_{\mathrm{c}}=\mathrm{i} / \omega \mathrm{c} \rightarrow \text { voltage is lagging behind } \mathrm{i} \text { by } 90^{\circ}
\end{aligned}
$
 

By all these we can draw phasor diagram as shown below -

One thing should be noticed that we have assumed that V_L is greater than V_C which makes i lags behind V. If V_C > V_L then i lead V. So as per our assumption, their resultant will be (V_L -V_C). So, from the above phasor diagram V will represent resultant of vectors V_R and (V_L -V_C). So the equation becomes - 

$
\begin{aligned}
V & =\sqrt{V_R^2+\left(V_L-V_C\right)^2} \\
& =i \sqrt{R^2+\left(X_L-X_C\right)^2} \\
& =i \sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2} \\
& =i Z
\end{aligned}
$

where,

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

Here, Z is called Impedance of this circuit.
Now come to the phase angle. The phase angle for this case is given as -

$
\tan \varphi=\frac{V_L-V_C}{V_R}=\frac{X_L-X_C}{R}=\frac{\omega L-\frac{1}{\omega C}}{R}
$

Now from the equation of the phase angle three cases will arise. These three cases are -
(i) When,

$
\omega L>\frac{1}{\omega C}
$

then, $\tan \varphi$ is positive i.e. $\varphi$ is positive and voltage leads the current i .
(ii) When $\omega L<\frac{1}{\omega C}$
then, $\tan \varphi$ is negative i.e. $\varphi$ is negative and voltage lags behind the current $i$.
(iii) When $\omega L=\frac{1}{\omega C}$,

      then tanφ is zero i.e. φ is zero and voltage and current are in phase. This is called electrical resonance.