 # AC Voltage Applied to Series LCR Circuit: Analytical Solution

An LCR circuit is also known as a resonant circuit or tuned circuit. It consists of an inductor-L, capacitor-C, and resistor-R which are connected in either in series or parallel.

In this article, we will learn about an electrical circuit where an AC voltage is applied across an inductor, a capacitor, and a resistor connected in series.

Table of Content

## Derivation of AC Voltage Applied to Series LCR Circuit

Consider the circuit shown above. Here, we have an inductor, a resistor, and a capacitor connected through a series connection across an AC voltage source given by V. Here, the voltage is sinusoidal in nature and is given by the equation,  Here, vm is the amplitude of the voltage and ω is the frequency.

If q is the charge on the capacitor and i the current at time t, we have, from Kirchhoff’s loop rule: Here, q is the charge held by the capacitor, I is the current passing through the circuit, R is the resistance of the resistor and C is the capacitance of the capacitor. In order to determine the instantaneous current or the phase of the relationship, we will follow the analytical analysis of the circuit.

Analytical solution

As i , we can write Hence, writing the voltage equation in terms of the charge q through the circuit, we can write, The above equation can be considered analogous to the equation of a forced, damped oscillator. In order to solve the equation, we assume a solution given by, So, And Substituting these values in the voltage equation, we can write, Here, we have substituted the value of Xc and XL by Xc = 1/ωC and XL = ω L.

As we know, hence, substituting this value in the above equation, we get, Now, let So we can say, Now, comparing the two sides of the equation, we can write, And, Hence, the equation for current in the circuit can be given as, Impedance of an LCR Circuit

## Frequently Asked Questions – FAQs

### Why do we use LCR circuits?

The LCR circuit is used for the detection of the frequencies of radio waves in which the broad spectrum of the radio waves are on the narrow range.

### What is Q-factor in an LCR circuit?

The ratio of resonance frequency to the difference between the neighbouring frequencies with respect to current is 1/√2 times of the peak value is called the Q-factor in an LCR circuit.

### For an LCR circuit, when will the Q-factor be large?

There are two conditions for Q-factor to be large for an LCR circuit:

• When the L/C is large, Q-factor will be large.
• The value of R should be less for Q-factor to be large.

### What is the value of power factor of LCR circuit which is in the state of resonance?

The value of power factor of LCR circuit which is in the state of resonance is unity.

### What happens to the circuit with very high Q value?

If a circuit has a very high Q value, then the circuit will have a very sharp resonance. This is because the frequencies will be of a very narrow range.

To learn more about the analytical solution for AC voltage and current through a circuit with AC voltage applied across a combination of resistor, inductor and the capacitor and other related topics, download BYJU’S – The Learning App.

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