Question

The forward-path transfer function and the feedback-path transfer function of a single loop negative feedback control system, are given as $G\left(s\right)=\frac{K(s+2)}{s^2+2s+2}andH\left(s\right)=1$, respectively. If the variable parameter $K$ is real positive, then the location of the breakaway point on the root locus diagram of the system is

• A. -3.41

Answer 1

The correct option is A -3.41

$G\left(s\right)=\frac{K(s+2)}{s^2+2s+2};H\left(s\right)=1$
$1+G\left(s\right)H\left(s\right)=0$
$1+\frac{K}{\left(s^2\right)+2s+2}(s+2)=0$
$K=-\frac{(s^2+2s+2)}{(s+2)}$

$\frac{dK}{ds}=-\left(\frac{(2s+2)(s+2)-(s^2+2s+2)}{(s+2{)}^2}\right)$
For break away points, $\frac{dK}{ds}=0$
$(2s+2)(s+2)-(s^2+2s+2)=0$
$2s^2+6s+4-s^2-2s-2=0$
$s^2+4s+2=0$
$s=\frac{-4\pm \sqrt{16-8}}{2}$
$=\frac{-4\pm 2\sqrt{2}}{2}=-2\pm \sqrt{2}$
$s=-0.58;s=-3.41$
But, $s=-3.41$ lies on root locus
Hence $s=-3.41$