Question

The open loop transfer fucntion of a unity feedback system is
$G\left(s\right)=\frac{k(s+2)}{(s+1+j1)(s+1-j1)}.$
The root locus plot of the system has

• A. two breakaway points located at s = -0.59
• B. one breakaway point located at s =-0.59
• C. one breakaway point located at s =-3.41
• D. one breakaway point located at s =-1.41

Answer 1

The correct option is C one breakaway point located at s =-3.41


R.L $\to$ Root locus part
NRL $\to$ Not root locus part
$G\left(s\right)H\left(s\right)=\frac{K(s+2)}{s^2+2s+2}$
For the break-away point
$\frac{dG\left(s\right)H\left(s\right)}{ds}=0$
$\frac{K(s^2+2s+1)1-K(s+2)(2s+2)}{(s^2+2s+2{)}^2}=0$
Solving this we get
$s^2+4s+2=0$
$s=\frac{-4\pm 2\sqrt{2}}{2}=-2\pm \sqrt{2}$
$s=-3.414,-0.59$
From the root locus part it is obvious that the valid break away point = -3.424