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,

LCR series circuit

AC voltage applied across an LCR circuit

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:

LCR series circuit

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

LCR series circuit

, we can write

LCR series circuit

Hence, writing the voltage equation in terms of the charge q through the circuit, we can write,

LCR series circuit

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,

LCR series circuit

So,

LCR series circuit

And

LCR series circuit

Substituting these values in the voltage equation, we can write,

LCR series circuit

Here, we have substituted the value of Xc and XL by Xc = 1/ωC and XL = ω L.

As we know,

LCR series circuit

hence, substituting this value in the above equation, we get,

LCR series circuit

Now, let

LCR series circuit

So we can say,

LCR series circuit

Now, comparing the two sides of the equation, we can write,

LCR series circuit

And,

LCR series circuit

Hence, the equation for current in the circuit can be given as,

LCR series circuit

Also read:
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.

Test your knowledge on AC voltage across LCR

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