Electronics/RCL frequency domain

Figure 1: RCL circuit

Define the pole frequency $\omega _{n}$ and the dampening factor $\alpha$ as:

${\frac {R}{L}}=2\alpha$

${\frac {1}{LC}}=\omega _{n}^{2}$

To analyze the circuit first calculate the transfer function in the s-domain H(s). For the RCL circuit in figure 1 this gives:

$H(s)={\frac {s{\big (}s+2\alpha {\big )}}{s^{2}+2\alpha s+\omega _{n}^{2}}}$

$H(s)={\frac {s{\big (}s+2\alpha {\big )}}{{\big (}s+\alpha +j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}{\big (}s+\alpha -j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}}}$

When the switch is closed, this applies a step waveform to the RCL circuit. The step is given by $Vu(t)$. 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 $Vu(t)$. Therefore the output is given by multiplication in the s-domain H(s)U(s), where $U(s)=V{\frac {1}{s}}$ is given by the Laplace Transform available in the appendix.

The convolution of u(t) and h(t) is given by:

$H(s)U(s)={\frac {V{\big (}s+2\alpha {\big )}}{{\big (}s+\alpha +j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}{\big (}s+\alpha -j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}}}$

Depending on the values of $\alpha$ and $\omega _{n}$ the system can be characterized as:

3. If $\alpha <\omega _{n}$ the system is said to be underdamped The solution for h(t)*u(t) is given by:

$h(t)*u(t)=Ve^{-\alpha t}{\big (}\cos({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t)+{\frac {\alpha }{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}}\sin({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t){\big )}$

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 $\alpha$ and $\omega _{n}$:

$\alpha ={\frac {R}{2L}}=1000$

$\omega _{n}={\frac {1}{\sqrt {LC}}}\approx 4472$

From these values note that $\alpha <\omega _{n}$. The system is therefore underdamped. The equation for the voltage across the capacitor is then:

$h(t)*u(t)=e^{-1000t}{\big (}\cos(4359t)+0.229\sin(4359t){\big )}$

Figure 2: Underdamped Resonse