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[SOLVED] Causal LTI system

dwsmith

Well-known member
Feb 1, 2012
1,673
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*}
Screenshot from 2014-03-11 22:17:34.png

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

dwsmith

Well-known member
Feb 1, 2012
1,673
So a causal system is when
\[
\lim_{z\to\infty}H(z) < \infty
\]