AC Voltage Applied to Series LCR Circuit

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 the 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 flows 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 + (X_C – X_L)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. 

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

Considering the source voltage as:

V = V_m sin (ωt)

Here, 

Vm  refers to the amplitude of the voltage applied 

w refers to the frequency of the voltage that applies

Let q be the charge on the capacitor connected and 

I will 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:

As;

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;   V_m sinωt  - L (dI/dt) = IR +(q/C)

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

By putting;

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

By rearranging;

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

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

q = qm sin (wt + θ)

Now,

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:

q_m w [R cos (wt + θ) + (X_C – X_L) sin (wt + θ)] = V_m 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:

(q_m ω Z) =  [(R/Z) cos (ωt + q) + (1/Z)( X_C - X_L) sin (ωt + q) ]=V_m sinωt

LCR circuit analysis formulas

To derive the general AC circuit analysis formulas, suppose:

R/Z = cos ø

(X_c – X_L)/ Z = sin ø

From these equations:

tan ø = (X_c – X_L)/ R

that is; 

ø = tan^-1 (X_c – X_L)/ R

(q_m ω Z) [cos (wt + θ – ø)] = V_m sinωt

Now, comparing the two equations:

q_m ω Z = V_m = I_m 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 = I_m cos (wt + θ)

I = I_m 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 depending on the angle θ:

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

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

Resonance in an LCR Circuit

The ability to resonate at a specific frequency called the resonance frequency, f0 is a unique feature of electric circuits such as the LCR circuit. All frequencies are measured in units of hertz. For calculation purposes, angular frequency, ω0, is used because it is more mathematically convenient. One must remember that angular frequency is not measured in hertz, but in radians per second. Angular and resonant frequency are related to each other by the equation:

\[\omega = 2\pi f_{0}\]

The phenomenon of resonance occurs because energy is stored in two different ways: in an electric field when the capacitor is charged and in a magnetic field as electric current flows through the inductor device. These two stored energies are transferable from one device to the other within the circuit and this can also be oscillatory in nature. A mechanical analogy can be the example of a weight suspended on a spring that will oscillate up and down when released. Note that this is not an accidental analogy as an oscillation of weight on a spring is explained in the same second-order differential equation as the LCR circuit and for all the properties of one system, there will be found an analogous property of the other. The mechanical property that is analogous to the resistor in the circuit is friction in the spring–weight system. This frictional force will slowly bring down any oscillation to cessation if there is no external force acting upon it to keep it oscillating. Similarly, the resistance in the LCR circuit will "damp"  the oscillation (gradually come to halt), fading it with passing time if there is no driving AC power source in the circuit.

This resonance frequency that we discussed above, can be explained as the frequency at which the impedance (opposition to the flowing current) of the circuit is at a minimum. In other words, it can also be defined as the frequency at which the impedance is purely resistive. The effect arises due to the fact that the impedances of the inductor and capacitor are equal but of opposite phase and therefore cancel out during resonance.