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

  • Thread starter Thread starter Dustinsfl
  • Start date Start date
  • Tags Tags
    Lti System
AI Thread Summary
The transfer function \(H(s)\) for the causal LTI system is derived as \(H(s) = \frac{1}{s^2 + s + 1}\). The region of convergence (ROC) for this system is specified as \(\text{Re} \{s\} < -\frac{1}{2}\), ensuring stability. The inverse Laplace transform of \(H(s)\) results in a time-domain response of \(\frac{2}{\sqrt{3}}e^{-\frac{1}{2}t}\sin\Big(\frac{\sqrt{3}}{2}t\Big)\). A causal system is characterized by the condition \(\lim_{z\to\infty}H(z) < \infty\). This analysis confirms that the system is both causal and stable.
Dustinsfl
Messages
2,217
Reaction score
5
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
\[
\frac{2}{\sqrt{3}}e^{-\frac{1}{2}t}\sin\Big(\frac{\sqrt{3}}{2}t\Big).
\]
 

Attachments

  • Screenshot from 2014-03-11 22:17:34.png
    Screenshot from 2014-03-11 22:17:34.png
    1.8 KB · Views: 89
Mathematics news on Phys.org
So a causal system is when
\[
\lim_{z\to\infty}H(z) < \infty
\]
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

Similar threads

Back
Top