How to Solve a Serie RLC AC Circuit with Complex Numbers

The schematic represents part of an alternating current circuit, with the potential difference (voltage) between nodes A and B given by an AC voltage sourcet.

We will assume that such a source supplies an instantaneous cosine voltage

There are three passive elements in the circuit, the inductor represented by L, the capacitor represented by by C and the resistor represented by R.

The inductor obeys the relationship between voltage and current according to the following law

The capacitor follows this law

And the resistor follows Ohm’s law,

The voltage law in electricity says that, in a circuit, the voltage in a closed loop is equal to the sum of all the voltages around the loop and is equal to zero. Applying this property to our circuit, we have:

or

We then write the derivative of this expression as a function of t.

Whereas

then,

than substituting into the derivative expression and rearranging

Calculating the derivative of AB voltage cosine, we have

which allows us to write the expression:

or

The objective now is to solve this equation, determining the algebraic expression of the electric current i(t). This is a second-order linear differential equation, under which we will make a transformation by means of the following expression for complex numbers

and so the differential equation is rewritten as

or

Initially solving it in the homogeneous form, that is,

whose auxiliary equation has a root

onde

This gives the possibility to write the complementary function of the solution

The particular solution will be of the form

where, A is a coefficient to be determined, and

Computing the derivatives of the proposed particular solution, we have

Substituting this into the complex differential equation

and simplifying in the following steps, we have:

solving for A:

where

Thus,

And the solution to the complex differential equation will be:

The RLC circuit has two operating states: transient and steady state. The transient state corresponding to the complementary solution and the steady state corresponding to the particular solution. So that if we consider the condition of t tending to ∞, we have

and,

thus,