Question

Complex conjugate roots of characteristic equation for lies on
$G\left(s\right)H\left(s\right)=\frac{1}{2s(s+\frac{1}{2})}$

• A. straight line
• B. parabola
• C. circle
• D. hyperbola

Answer 1

The correct option is C circle

Root locus plot of the system is drawn as K is varied from 0 to $\infty$.

• Poles (P) s = -1, -1
• Zero (Z) s = -3
• There will be one asymptotes at P - Z = 1
• Centroid, $\sigma =\frac{(-1-1)-(-3)}{1}=\frac{-2+3}{1}=1$
• Angle of asymptates = $\pi$
• Break away point, $\frac{dK}{ds}$ = 0

$1+\frac{K(s+3)}{{(s+1)}^2}=0$

$K=\frac{-{(s+1)}^2}{s+3}$

$\frac{dK}{ds}=\frac{(s+3)\{-2(s+1\left)\right\}+{(s+1)}^2}{{(s+3)}^2}$

$s^2+6s+5=0$
Breakaway point, s = -1, -5
Root locus can be drawn


It is clear from root locus that complex roots lie on circle.