# [SOLVED]Causal LTI system

#### dwsmith

##### Well-known member
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*}

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