# LCR Circuit : Analysis Of A LCR Series Circuit

We already know that there are broadly two types of connection: series and parallel. Series circuit is one in which current remains the same along each element. Now we will talk about LCR circuit. The LCR circuit analysis can be understood better in terms of phasors. A phasor is a rotating quantity.

Current Vs Voltage Graph

For an inductor (L), if we consider I to be our reference axis, then voltage leads by 90° and for the capacitor the voltage lags by 90°. But the resistance, current and voltage phasors are always in phase.

Following is the table explaining other related concepts of circuit:

## Analysis of an LCR circuit – series circuit

Let’s consider the following lcr circuit using the current across the circuit to be our reference phasor because it remains the same for all the components in a series lcr circuit.

RLC Series circuit

As described above the overall phasor will look like below:

Phasor diagram of current Vs voltage for resistor, inductor and capacitor for LCR series circuit

From the above phasor diagram we know that,

$V^2$ =$(V_R)^2 ~+~ (V_L~ –~ V_c)^2$   —– (1)

Now Current will be equal in all the three as it is a series LCR circuit. Therefore,

$V_R$ = $IR$—– (2)

$V_L$ = $IX_L$ —– (3)

$V_c$ = $IX_c$ —– (4)

Using (1), (2), (3) and (4)

$I$ = $\frac{V}{√{R^2~ +~(X_L~ -~ X_C)^2}}$

Also the angle between $V$ and $I$ is known phase constant,

$tan~ ∅$ = $\frac{V_L~-~V_C}{V_R}$

It can also be represented in terms of impedance,

$tan~ ∅$ = ${X_L~-~ X_C}{R}$

Depending upon the values of $X_L$ and $X_C$

we have three possible conditions,

1. If $X_L > X_c$, then $tan ∅ > 0$ and the voltage leads the current and the circuit is said to be inductive
2. If $X_L < X_c$ , then $tan ∅ < 0$ and the voltage lags the current and the circuit is said to be capacitive
3. If$X_L$ =$X_c$ , then $tan ~∅$ = $0$ and the voltage is in phase with the current and is known as resonant circuit.

This was just an introduction to LCR circuit. Know more about LCR circuit, types of circuit, inductance, capacitance, resistance and for more physics articles download BYJU’S The Learning App and fall in love with learning.

#### Practise This Question

A telephone wire of length 200 km has a capacitance of 0.014μF per km. If it carries an ac of frequency 2.5 kHz, what should be the value of an inductor required to be connected in series so that the impedance of the circuit is minimum