
Figure 1: RCL circuit
Define the pole frequency
and the dampening factor
as:
To analyze the circuit first calculate the transfer function in the s-domain H(s). For the RCL circuit in figure 1 this gives:
When the switch is closed, this applies a step waveform to the RCL circuit. The step is given by
. Where V is the voltage of the step and u(t) the unit step function. The response of the circuit is given by the convolution of the impulse response h(t) and the step function
. Therefore the output is given by multiplication in the s-domain H(s)U(s), where
is given by the Laplace Transform available in the appendix.
The convolution of u(t) and h(t) is given by:
Depending on the values of
and
the system can be characterized as:
3. If
the system is said to be underdamped The solution for h(t)*u(t) is given by:
Given the following values what is the response of the system when the switch is closed?
R |
L |
C |
V
|
0.5H |
1kΩ |
100nF |
1V
|
First calculate the values of
and
:
From these values note that
. The system is therefore underdamped. The equation for the voltage across the capacitor is then:

Figure 2: Underdamped Resonse