AC voltage applied to a series LCR circuit

In this article, we will discuss an AC circuit that contains a resistor, inductor, and capacitor in series, so let’s get started…

What is LCR Circuit?

An $LCR$ electronic circuit contains an $R$-ohm resistor, a $C$-farad capacitor, and an L-Henry inductor, all connected in series. Since the three elements of the $LCR$ circuit are connected in series, the current through them all is the same and equals the total current $I$ through the circuit. A circuit that contains $L$, $R$, and $C$ components at certain frequencies can cause $L$ and $C$ (or some of their electrical effects) to vanish entirely.

Explanation of LCR Circuit

The LCR circuit can function individually as just a capacitor, just a resistor, or just an inductor. The LCR circuit is also used for voltage boosting to increase the voltage flowing through the individual components of the circuit.

This voltage can be much higher than the external voltage applied to the circuit. LCR circuits are also useful for changing the circuit’s impedance to increase or decrease the resistance to current at different frequencies. All of these effects can be used separately or together to achieve desired results on electronic devices.

Components of an LCR circuit

The three Components of an LCR circuit work together to Produce different electrical Effects

Resistor

Resistance limits current flow. It helps control the power or voltage applied to the LCR circuit. Resistance is a component of an electronic device that limits the flow of electrical current. The resistor helps control the amount of current or voltage that is applied to the LCR circuit. This is important as it prevents too much current from flowing through the other components in the circuit.

Capacitor

A capacitor stores energy and releases it in a controlled manner. It helps control the voltage or power that is applied to the LCR circuit. The capacitor stores energy and releases it in a controlled manner, preventing too much current from flowing. via the resistor L.

Inductor

An inductor resists changes in current flow: it helps control fluctuations in current flow. The inductor resists changes in current flow, which helps stabilize the LCR circuit. LCR circuitry is used as part of electronic devices such as cell phones, televisions and computers to regulate the intensity of light emitted by these devices.

AC circuit containing $LCR$ in series

In the below figure, let’s consider a resistor $R$, inductor $L$, and capacitor $C$ are connected in series to a source of alternating EMF $\mathcal{E}$ that is given by $$\mathcal{E}=\mathcal{E_0}\sin\omega t$$

AC voltage applied to a series LCR circuit
AC circuit containing $LCR$ in series

Let $I$ be the current through the series circuit at any instant, then

1). Voltage $\overrightarrow{V_R}=\overrightarrow{I}R$ across the resistance $R$ will be in phase with the current $\overrightarrow{I}$. So phasors $\overrightarrow{V_R}$ and $\overrightarrow{I}$ are in the same direction, as shown in the below figure. The amplitude of $\overrightarrow{V_R}$ is $${V_0}^R=I_0 R$$

2). 2). Voltage $\vec{V}_L=X_L \vec{I}$ across the inductance $L$ is ahead of current $\vec{I}$ in phase by $\pi / 2$ rad. So phasor $\vec{V}_{\mathrm{L}}$ lies $\pi / 2 \mathrm{rad}$ anticlockwise w.r.t. the phasor $\vec{I}$, It amplitude is
$$V_0{ }^L=I_0 X_L $$

3). Voltage $\vec{V}_C=X_C \vec{I}$ across the capacitance $C$ lags behind the current $\vec{I}$ in phase by $\pi / 2$ rad. So phasor $\vec{V}_{\mathrm{C}}$ lies $\pi / 2 \mathrm{rad}$ clockwise w.r.t. the phasor $\vec{I}$, It amplitude is
$$V_0{ }^C=I_0 X_C $$

As $\vec{V}_L$ and $\vec{V}_C$ are in opposite direction, their resultant is $(\vec{V}_L – \vec{V}_C)$. By the parallelogram law, the resultant of $\vec{V_R}$ and $(\vec{V}_L – \vec{V}_C)$ must be equal to the applied EMF $\mathcal{E}$, given by the diagonal of the parallelogram.

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Using the Pythagoras theorem, we get

\begin{aligned}
\mathcal{E_0}^2 &=\left(V_0^R\right)^2+\left(V_0^L-V_0^C\right)^2=\left(I_0 R\right)^2+\left(I_0 X_L – I_0 X_C\right)^2 \\
&=I_0^2\left[ R^2+\left(X_L – X_C\right)^2\right] \\
I_0 &=\frac{\mathcal{E_0}}{\sqrt{\left[R^2+\left(X_L – X_C\right)^2\right]}}
\end{aligned}

Clearly, $\sqrt{\left[R^2+\left(X_L – X_C\right)^2\right]}$ is the effective resistance of the series LCR-circuit which opposes or impedes the flow of a.c. through it. It is called its impedance and is denoted by $Z$, and its unit is ohm $(\Omega)$. Thus

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

What is impedance triangle?

The relationship between the resistance $R$, inductive reactance $X_L$, capacitive reactance $X_C$, and the impedance $Z$ is shown below in the form of a right angled triangle called impedance triangle.

impedance triangle
impedance triangle
Phasor diagram for series LCR circuit
Phasor diagram for series LCR circuit

Special Cases

1. When $X_L>X_C$ or $V_L>V_C$, we can see from above phasor diagram that emf is ahead of current by phase angle $\phi$ which is given by $$ \tan \phi=\frac{X_L-X_C}{R} \text { or } \cos \phi=\frac{R}{Z} $$ The instantaneous current in the circuit will be $$ l=I_0 \sin (\omega t-\phi) $$ The series $L C R$-circuit is said to be inductive.

2. When $X_L<X_C$ or $V_L<V_C$, current is ahead of emf by phase angle $\phi$ which is given by $$ \tan \phi=\frac{X_C-X_L}{R} \text { or } \cos \phi=\frac{R}{Z} $$ The instantaneous current in circuit will be $$ I=I_0 \sin (\omega t+\phi) $$ The series $L C R$-circuit is said to be capacitive.

3. When $X_L=X_C$ or $V_L=V_C, \phi=0$, the emf and current will be in the same phase. The series LCR-circuit said to be purely resistive. It may also be noted that $$ I_0=\frac{E_0}{Z} \text { or } \frac{I_0}{\sqrt{2}}=\frac{E_0}{\sqrt{2} Z} \text { or } I_{r m s}=\frac{E_{r m s}}{Z} $$

What is Susceptance?

Susceptance definition: The reciprocal of the reactance of an AC circuit is called susceptance. Its SI unit is ohm-1 or mho.

What is Admittance?

Admittance definition: The reciprocal of the impedance of an AC circuit is called its admittance. Its SI unit is ohm-1 or mho.

Stay tuned with Laws Of Nature for more useful interesting content.

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