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: