Question

Loop transfer function of a feedback system is $G\left(s\right)H\left(s\right)=\frac{s+3}{s^2(s-3)}.$ Take the Nyquist contour in the clockwise direction. Then, the Nyquist plot of G(s)H(s) encircles -1+j0

• A. once in clockwise direction
• B. twice in colckwise direction
• C. once in anticlockwise direction
• D. twice in anticlockwise direction

Answer 1

The correct option is A once in clockwise direction

Nyquist plot of $G\left(s\right)H\left(s\right)=\frac{s+3}{s^2(s-3)}$ is as shown below:



From the Nyquist plot G(s) H(s) encircle -1+j0 once in clockwise direction.
Atlernate solution:
Characteristic equation,
$1+G\left(s\right)H\left(s\right)=0$
$s^2(s-3)+(s+3)=0$
$s^3-3s^2+s+3=0$
using Routh's array,
$\begin{array}{c}{s}^3\\ s^2\\ s^1\\ s^0\end{array}|\begin{array}{cc}1& 1\\ -3& 3\\ 2& 0\\ 3& \end{array}$
There are two sign changes, hence two poles in right half of s-plane exist.
$Z=2,P=1$
$N=P-Z=-1$
One encirclement in clockwise direction.