Question

The root locus of the system
$G\left(s\right)H\left(s\right)=\frac{K}{s(s+2)(s+3)}$
has the break-away point located at

• A. $(-0.5,0)$
• B. $(-2.548,0)$
• C. $(-4,0)$
• D. $(-0.784,0)$

Answer 1

The correct option is D $(-0.784,0)$

$1+G\left(s\right)H\left(s\right)=0$
$\Rightarrow 1+\frac{K}{s(s+2)(s+3)}=0$
$\Rightarrow 1+\frac{K}{s(s^2+5s+6)}=0$
$\Rightarrow K=-1.(s^3+5s^2+6s)$
$\Rightarrow \frac{dK}{ds}=-(3s^2+10s+6)$
For break away point
$\frac{dK}{ds}=0$
$\Rightarrow 3s^2+10s+6=0$
$s=\frac{-10\pm \sqrt{100-72}}{6}$
$=\frac{-10\pm 5.3}{6}=-0.784,-2.55$

$Centroid=\frac{-2-3}{3}=\frac{-5}{3}=-1.66$
$angleofasymptots=(2K+1)\frac{\pi }{P-Z}$
$=(2K+1)\frac{\pi }{3}$
$\frac{\pi }{3},\pi ,\frac{5\pi }{3}$
$\therefore$ Break away point is $(-0.784,0)$
$as-2.55$ does not lie on RL