MHB Find \(H(s)\) for Causal LTI System: Region of Convergence

[Dustinsfl]

Determine \(H(s)\) and specify its region of convergence. Your answer should be consistent with the fact that the system is causal and stable.

In order to find \(H(s)\), we need to find \(X(s)\) and \(Y(s)\).
\begin{align*}
x(t) &= Ri + L\frac{di}{dt} + \frac{1}{C}\int i(t)dt\\
X(s) &= \mathcal{L}\bigg\{i + \frac{di}{dt} + \int i(t)dt\bigg\}\\
&= I(s) + sI(s) - I(0) + \frac{1}{s}I(s)\\
&= I(s)\bigg(1 + s + \frac{1}{s}\bigg)\\
y(t) &= \frac{1}{C}\int i(t)dt\\
&= \mathcal{L}\bigg\{\int i(t)dt\bigg\}\\
&= \frac{1}{s}I(s)\\
H(s) &= \frac{\frac{1}{s}}{1 + s + \frac{1}{s}}\\
&= \frac{1}{s^2 + s + 1}
\end{align*}
View attachment 2097

What is a causal system?
For convergece, \(\text{Re} \ \{s\} < -\frac{1}{2}\) since the inverse Laplace of H is
\[
\frac{2}{\sqrt{3}}e^{-\frac{1}{2}t}\sin\Big(\frac{\sqrt{3}}{2}t\Big).
\]

 

[Dustinsfl]

So a causal system is when
\[
\lim_{z\to\infty}H(z) < \infty
\]