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# Series LCR Circuit - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Series LCR circuit is considered one the most difficult concept.

• 88 Questions around this concept.

## Solve by difficulty

In a  LCR circuit capacitance is changed from C to 2C . For the resonant frequency to remain unchanged, the inductance should be changed from L to

In a series resonant LCR circuit, the voltage across R is 100 volts and R = 1 k $\Omega$ with C = 2 $\dpi{100} \mu F$. The resonant frequency $\dpi{100} \omega$ is 200 rad/s. At resonance the voltage across L is

In an LCR series a.c. circuit, the voltage across each of the components, L, C, and R is 50 V. The voltage across the LC combination will be

If $\mathrm{R,X_{L}\: and\: X_{C}}$  represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

## Concepts Covered - 1

Series LCR circuit

Series LCR circuit-

The Figure given above shows a circuit containing a capacitor ,resistor and inductor connected in series through an alternating / sinosoidal 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 VL,VC and VR the potential drop across L,C and R respectively then
$\begin{array}{l}{\mathrm{v}_{\mathrm{R}}=\mathrm{i} \mathrm{R} \rightarrow \text { Voltage is in phase with i }} \\ \\ {\mathrm{v}_{\mathrm{L}=\mathrm{i} \omega \mathrm{L}} \rightarrow \text { voltage is leading i by } 90^{\circ}} \\ \\ {\mathrm{v}_{\mathrm{c}}=\mathrm{i} / \mathrm{\omega} \mathrm{c} \rightarrow \text { voltage is lagging behind i by } 90^{\circ}}\end{array}\varepsilon$

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

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

\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 \\ \text { where, } & \\ Z &=\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} \end{aligned}

Here, Z is called Impedence 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,     $\dpi{100} \omega L \ > \ \frac{1}{\omega C}$

then, tanφ is positive i.e. φ is positive and voltage leads the current i.
(ii) When $\dpi{100} \omega L \ < \ \frac{1}{\omega C}$

then, tanφ is negative i.e. φ is negative and voltage lags behind the current i.
(iii) When  $\dpi{100} \omega L \ = \ \frac{1}{\omega C}$ ,

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

## Study it with Videos

Series LCR circuit

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