Question

Consider the points $s_1=-3+j4$ and $s_2=-3-j2$ in the s-plane. Then, for a system with the open-loop transfer function
$G\left(s\right)H\left(s\right)=\frac{K}{(s+1{)}^4}$

• A. $s_1$ is on the root locus, but not $s_2$
• B. $s_2$ is on the root locus, but not $s_1$
• C. both $s_1$ and $s_2$ are on the root locus
• D. neither $s_1$ nor $s_2$ is on the root locus

Answer 1

The correct option is B $s_2$ is on the root locus, but not $s_1$

$s_1=-3+4j$ $G\left(s\right)H\left(s\right)=\frac{K}{(-3+4j+1{)}^4}=\frac{K}{(-2+4j{)}^4}$ $\angle G\left(s\right)H\left(s\right)=-4[\pi -ta{n}^{-1}2]$
$=-{466}^{\circ }\ne \text{odd multiple of}\pi$
So, $s_1$ does not lie on root locus $s_2=-3-j2$
$G\left(s\right)H\left(s\right)=\frac{K}{(-3-j2+1{)}^4}=\frac{K}{(-2-2j{)}^4}$ $\angle G\left(s\right)H\left(s\right)=-4\left[ta{n}^{-1}1\right]=-4\times \frac{\pi }{4}=-\pi$
Odd multiple of $\pi$
$\therefore s_2$ lies on root locus