# LCR Circuit: Analysis of a LCR Series Circuit

In our article about the types of circuit, we discussed that there two major types of circuit connection : Series and Parallel. From the article, we understood that a series circuit is one in which the current remains the same along with each element. With this context, let us discuss the LCR circuit and its analysis in detail.

An LCR circuit, also known as a resonant circuit, tuned circuit, or an RLC circuit, is an electrical circuit consisting of an inductor (L), capacitor (C) and resistor (R) connected in series or parallel. 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.

## 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. 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.