LCR Series Circuit - Differential Equation & Analytical Solution

Introduction

LCR Series Circuit has many applications. In electronics, components can be divided into two main classifications namely active and passive components. Resistors, capacitors, and inductors are some of the passive components. The combination of these components gives RC, RL, LR, LC, and LCR circuits. An RC circuit comprises a resistor and a capacitor. An RL circuit includes a resistor and an inductor. In an LR circuit, an inductor and resistor are connected together. Also, an LC circuit, an inductor, and a capacitor are joints. LCR is the abbreviation for L- inductance, C-capacitance, and R- resistance. An LCR circuit comprises an inductor, a capacitor, and a resistor. It can also be called an RLC circuit.

Role of R, L, and C in a circuit

• Resistor − In a circuit, the resistor is represented by R. Generally, energy has degenerated as heat. There will be a constant voltage drop for constant current flowing through it.

• Inductor − Inductors are represented by L. The energy is stored in the form of a magnetic field. It opposes the change in electric current. It consists of a coil wound wire.

• Capacitor − Capacitors are represented by C. It is similar to the inductor. It stores energy in the form of an electric field. It also opposes the change in current flow.

What is an LCR Circuit?

An LCR circuit is an electrical circuit composed by joining an inductor, a capacitor, and a resistor connected in series or parallel combination. In a series LCR circuit, resistor, inductor, and a capacitor are coupled in series with the input source. The input source V s phasor sum of all the three elements ie., resistor, capacitor, and inductor. Current is the same for all the three elements.

Impedance of a series LCR circuit

Consider voltage amplitude across resistor R, inductor L, and capacitor C is given as 𝑽_𝑹, 𝑽_𝑳, 𝒂𝒏𝒅 𝑽_𝑪.

since, 𝑉_𝑹 = 𝑖𝑅

where i is current, R is resistance

$$\mathrm{V_L=iX_L=i\omega L}$$

where X_L is inductive reactance, ω is angular frequency, L is inductance $\mathrm{V_C=iX_C=i(\frac{1}{\omega C})}$

where X_Cis capacitive reactance, C is capacitance

$$\mathrm{We\: have\: V^2=V_R^2+(V_L-V_C )^2}$$

On substitution we get −

$$\mathrm{V^2=(iR)^2+(iX_L-iX_C )^2}$$

$$\mathrm{V^2=i^2 (R^2+(X_L-X_C )^2 )}$$

Then current −

$$\mathrm{ i=\frac{v}{\sqrt{(R^2+(X_L-X_C )^2)}}=\frac{v}{Z}}$$

where Z is impedance

$$\mathrm{Z=\sqrt{(R^2+(X_L-X_C )^2)}}$$

On substituting the value of inductive and capacitive reactance −

The impedance (Z) of an LCR circuit −

$$\mathrm{Z=\sqrt{R^2+\lbrace \omega L-\frac{1}{\omega C}\rbrace^2}}$$

LCR Circuit Derivation Resonance in an LCR Circuit and analytical solution

Let us suppose that a resistor, an inductor, and a capacitor are coupled in series with the input source. AC input is given by −

$$\mathrm{v=v_m sin\omega t}$$

Where ω is angular frequency and v_m is amplitude of the input source, “t”is time

Using Kirchhoff’s rule −

$$\mathrm{L\frac{\text{d}i}{\text{d}t}+iR+\frac{q}{C}=v …. (1)}$$

Where L is inductance, i is current, R is resistance, q is charge, and C is capacitance Analysing the circuit −

$$\mathrm{i=\frac{\text{d}q}{\text{d}t}……………… … (2)}$$

On differentiation of equation (2), we have

$$\mathrm{\frac{\text{d}i}{\text{d}t}=\frac{\text{d}^2q}{\text{d}t^2} …. (3) }$$

Substitute equation (2) and (3) in equation (1) −

$$\mathrm{L \frac{\text{d}^2q}{\text{d}t^2}+R\frac{\text{d}q}{\text{d}t}+q/C=v_m sin\omega t …. (4)}$$

For the damped oscillator equation, the above equation is analogous.

$$\mathrm{q=q_m sin(\omega t+ θ)}$$

$$\mathrm{\frac{\text{d}q}{\text{d}t}=q_m \omega cos(\omega t+ θ)}$$

$$\mathrm{\frac{\text{d}^2q}{\text{d}t2}=-q_m \omega ^2 sin(\omega t+ θ)}$$

Substitute the above equation in voltage equation −

$$\mathrm{q_m \omega [Rcos(\omega t+ θ)+(X_C-X_L )sin(\omega t+ θ)]=v_m sin\omega t}$$

We know capacitive reactance $\mathrm{X_C=\frac{1}{\omega C}}$ and inductive reactance $\mathrm{X_L=\omega L}$

Also, impedance (Z) of an LCR circuit −

$$\mathrm{ Z=\sqrt{(R^2+(X_L-X_C )^2 )}}$$

On substitution we get the values −

$$\mathrm{\frac{q_m \omega}{Z} [\frac{R}{Z} cos(\omega t+ θ)+(\frac{(X_C-X_L)}{Z})sin(\omega t+ θ)]}$$

On solving the equation for θ and ϕ, equation for current in the LCR circuit is obtained as −

$$\mathrm{i=\frac{\text{d}q}{\text{d}t}=q_m \omega cos(\omega t+ θ)=i_m \omega cos(\omega t+ θ)}$$

$$\mathrm{i=i_m sin(\omega t+ ϕ) }$$

Electrical Resonance −

The process of resonance is quite common among different systems that have a tendency to oscillate at a specific frequency. This frequency is termed as the natural frequency of oscillation of the system. In LCR system, this frequency is determined by the values of an inductor, capacitor and resistor.

Conclusion

In electronics, components can be divided into two main classifications namely active and passive components. Resistors, capacitors, and inductors are some of the passive components. Combination of these passive components gives RC, RL, LR, LC, and LCR circuits. An LCR circuit is an electrical loop comprised by joining an inductor, capacitor, and resistor in series or parallel combination. The impedance (Z) of an LCR circuit

$$\mathrm{Z=\sqrt{R^2+\lbrace \omega L-\frac{1}{\omega C}\rbrace^2}}$$

On finding the analytical solution, equation for current in the LCR circuit is obtained as −

$$\mathrm{i=\frac{\text{d}q}{\text{d}t}=q_m \omega cos(\omega t+ θ)=i_m \omega cos(\omega t+ θ)=i_m sin(\omega t+ ϕ)}$$

FAQs

1.Describe resonance frequency.

When the oscillation of a system has a frequency equal to its natural frequency is called resonance frequency.

2.What is common in an inductor, resistor, and capacitor?

These devices are two-terminal linear devices. These devices pass the current through the circuits linearly with increased applied voltage.

3. How important is the inductor in an LCR circuit?

An inductor can reserve the magnetic energy only if the electric current is applied. Inductor keeps the dynamic energy of accelerating electrons in the form of a magnetic.

4.What is the importance of an LCR circuit?

An LCR circuit is helpful due to following reasons −

• LCR circuits help to reduce the consumption of power by controlling the current flow through a component, which leads to overheating.

• It also helps to reduce the voltage fluctuations which lead to damaging the electronic devices.

• LCR circuits help in storing energy and it releases in a controlled manner which will help in preventing the flow of current through the resistor.

5.How series resonance LCR circuit is used in TV?

The application can be shown on radio and TV receiver sets. The antenna of the radio/TV intercept signals from many broadcasting stations. If we want to receive a specific radio station/TV Channel we tune the receiver set by making a change in the capacitance of a capacitor in the tuning circuit in such a way the resonance frequency of the circuit becomes equal to the frequency of that specific station.