Electronics/RCL frequency domain

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Figure 1: RCL circuit
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:

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:

Figure 2: Example 1 Underdamped Response
Figure 2: Underdamped Resonse