Electronics/RCL frequency domain

Figure 1: RCL circuitFigure 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: Example 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: Example 1 Underdamped ResponseFigure 2: Underdamped Resonse