Let us consider the series RCL circuit shown in the figure below where the initial conditions are IL(0) = I0, vC(0)= V0, and u0(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 vs, that is, DC or AC.
If vs is DC, then:
The characteristic equation is:
Simplifying the above we get
From the quadratic eqation:
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, iL(0) = 5A, vC(0) = 2.5 V, and the 0.5 Ohm resistor represents the resistance of the inductor. Compute and sketch i(t) for t > 0.
This circuit can be represented by the integrodifferential equation
Substituting R, C, L values
The roots of the characteristic equation are s1 = -200 and s2 = -300 .
The constants k1 and k2 can be evaluated from the initial conditions
Also, at t = 0+
As a reminder: