Consider the 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 currents of each element is equal to the source
and by differentiation
To find the forced response, we must first specify the nature of the excitation is, that is, DC or AC.
If is is DC, then the natural response is found from the homogeneous equation:
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 the Figure below, iL(0) = 2A, vC(0) = 5.0 V Compute and sketch i(t) for t > 0.
We start with the equation for the total responce:
When steady-state conditions have been reached we will have
Steady-state conditions is a result of the forced responce
Since excitation applied is a DC current steady state responce of an inductor will be 0V.
To find out whether the natural response is overdamped, critically damped, or oscillatory, we need to compute the values of α0 and ω0 as well as values of s1 and s2.
s1 = -4 and s2 = -16
Therefore, the natural response is overdamped
The constants k1 and k2 can be evaluated from the initial conditions. From the initial condition vC(0) = v(0) = 5V. And we get
The second equation that is needed for the computation of the values of k1 and k2 is found from other initial condition, that is, iL(0) = 2A.
evaluate it at t = 0+, and we equate it with this initial condition.Then,
As a reminder:
We get the total response as