AC voltage applied to a series LCR circuit: Analytical Solution

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.

AC voltage applied across an 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

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

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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