AC Voltage Applied to Series LCR Circuit

Bookmark added to your notes.
View Notes

RLC Circuit Problems with Solutions

By connecting a constant source of voltage or battery through a resistor, the current is developed. The developed current has a single direction in which it flows and constant magnitude as well. Generally, current flows from negative to the positive terminal of the battery. If the direction of the current varies alternately across the resistor then it refers to alternating current. Some of the RLC series circuit problems with solutions are discussed in this handout. 

What is an LCR Circuit?

LCR circuit is well-known as a tuned or resonant circuit. It refers to an electrical circuit that comprises an inductor (L), a capacitor (C), and a resistor (R). Here, resistor, inductor, and capacitor are connected in series due to which the same amount of current flow in the circuit. Generally, the RLC circuit differential equation is similar to that of a forced, damped oscillator. To derive the LCR circuit equations, some of the terms like impedance are used. Impedance refers to the resistance related to the LCR series circuit. It includes the resistance offered by the inductor, the resistor, and the capacitor. 

Impedance is denoted by Z and given by the equation:

Z = √(R2 + (XC – XL)2)

Its SI unit is denoted by Ω (ohm).

LCR Circuit Derivation

Consider an electrical circuit having an inductor, capacitor, and resistor joined in series.  Suppose an AC voltage is applied to the circuit. 

[Image will be Uploaded Soon]

The diagram shows that inductor (L), a capacitor (C), and a resistor (R) are connected with each other in series.

Considering the source voltage as:

V = Vm sin (ωt)


Vm  refers to the amplitude of the voltage applied 

w refers to the frequency of the voltage applies

Let q be the charge on the capacitor connected and 

I be the current flowing at any time t

Applying Kirchhoff’s loop rule to this circuit:

We get;  

L (dI/dt) + IR + q/C = v

In this equation, 

I is the current passing through the LCR circuit

R is the resistance of the resistor connected

C = Capacitance of the capacitor connected

Now, we have to determine the instantaneous phase or current of the above-mentioned relationship. It can be determined by following the analytical analysis of the circuit:


current = rate of change of charge/ time that is;

I = dq/dt

It can be written as:

dI/dt = d2q/dt2

Hence, the equation of voltage in series in terms of charge q can be given by the equation:

Net EMF = V (voltage from the source) + e (self-induced emf) = IR (voltage drop across the resistor) + (q/C) ( voltage drop across the circuit capacitor)

that is;   Vm sinωt  - L (dI/dt) = IR +(q/C)

that is; Vm sinωt  = IR +(q/C)  + L (dI/dt)

By putting;

I = (dI/dt) :-  Vm sinωt  = R (dq/dt) + (q/C) + L(d2q/dt2)

By rearranging;

L(d2q/dt2) + R (dq/dt) + (q/C) =  Vm sinωt

This equation can be considered equivalent to the equation of damped oscillator. By considering the RLC series circuit problems with solutions, the above problem can be solved by:

q = qm sin (wt + θ)


dq/ dt = qm w cos (wt + θ) and

d2q/ dt  = qm w sin2 (wt + θ)

By putting this values in the equation of voltage in series:

qm w [R cos (wt + θ) + (Xc – XL) sin (wt + θ)] = Vm sinωt

After substituting the values of Xc, XL in the above equation

that is; 

Xc = 1/wC and XL = wL also, put Z = √(R2 + (XC – XL)2)

So, the equation becomes:

(qm ω Z) =  [(R/Z) cos (ωt + q) + (1/Z)( XC - XL) sin (ωt + q) ]=Vm sinωt

LCR circuit analysis formulas

To derive the general AC circuit analysis formulas, suppose:

R/Z = cos ø

(Xc – XL)/ Z = sin ø

From these equations:

tan ø = (Xc – XL)/ R

that is; 

ø = tan-1 (Xc – XL)/ R

(qm ω Z) [cos (wt + θø)] = Vm sinωt

Now, comparing the two equations:

qm ω Z = Vm = Im Z

Also, the equation for the current in LCR series circuit is given by:

I = dq/dt = dq/ dt = qm w cos (wt + θ)

I = Im cos (wt + θ)

I = Im sin (wt + θ)

From the LCR equations, some points can be concluded that is:

Current and voltage in series are in or out of phase depends on the angle θ:

If θ = 0, then voltage and current in LCR circuit are in-phase 

If θ = 90=degree, then then voltage and current in LCR circuit are out-phase 

FAQ (Frequently Asked Questions)

1. Explain the Working of the LCR Circuit.

In the LRC circuit, it mainly consists of the resistor, inductor, and capacitor. All three are connected in the series to each other. Thus it is an oscillating circuit. Thus the voltage present in the capacitor will give rise to the current in the circuit. Thus it finally results in the oscillation in the circuit. And in this way an LCR circuit works. The frequency of resonance is all dependent upon the inductance and capacitance in the circuit. 

2. How will you Define Resonance Frequency in an LCR Circuit?

The resonance in the series of LCR circuit happens when the capacitance equals the inductance in magnitude in the circuit. However, they cancel each other as they make a 180-degree angle with each other in the opposite direction. However, there must be sharp minimum impedance in the circuit that must cause the object to vibrate. A resonant circuit responds to the selectively signals in the particular frequency.