# Electronics/RCL frequency domain

*4 November 2016*. There are template/file changes awaiting review.

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:

## Example:Edit

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: