Let us consider the series RCL circuit shown in the figure below where the initial conditions
are I_{L}(0) = I_{0},
v_{C}(0)= V_{0}, and u_{0}(t) is the unit step function. We want to find an
expression for the curren i(t) for t > 0.

For the circuit above sum of voltage drops across each element is equal to the source

and by differentiation

To find the forced response, we must first specify the nature of the excitation v_{s}, that is, DC or AC.

If v_{s} is DC, then:

The characteristic equation is:

Simplifying the above we get

From the quadratic eqation:

we have

We will use the following notations:

where the subscript stands for series circuit.

Substitute into the solution above we get:

The three solution cases based on the difference under the squareroot:

Depending on the circuit constants R, L, and C, the total response of a series RLC circuit that is excited by a DC source, may be overdamped, critically damped, or underdamped. In this section we will derive the total response of series RLC circuits that are excited by DC sources

For the circuit of Figure 1.5, *i _{L}(0) = 5A, v_{C}(0) = 2.5 V*, and the 0.5 Ohm resistor represents the
resistance of the inductor. Compute and sketch

This circuit can be represented by the integrodifferential equation

Differentiating

Substituting R, C, L values

The roots of the characteristic equation are s_{1} = -200 and s_{2} = -300 .

The constants k_{1} and k_{2} can be evaluated from the initial conditions

Then:

Also, at t = 0^{+}

As a reminder: