Question

If the open loop transfer function of negative feedback control system is $G\left(s\right)H\left(s\right)=\frac{K}{(s+1{)}^3}$, then the gain K for a closed loop pole to be at
$(-\frac{1}{2}+j\frac{\sqrt{3}}{2})is$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

Open loop T.F,

$G\left(s\right)=\frac{K}{(s+1{)}^3}$

Closed loop T.F,

$\Rightarrow \frac{G\left(s\right)}{1+G\left(s\right)H\left(s\right)}=\frac{K}{(s+1{)}^3+K}$

$ats=(-\frac{1}{2}+j\frac{\sqrt{3}}{2})$

For poles, the denominator should be equated to zero.

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

${[-\frac{1}{2}+j\frac{\sqrt{3}}{2}+1]}^3+K=0$

${[\frac{1}{2}+j\frac{\sqrt{3}}{2}]}^3+K=0$

$\Rightarrow 1-j3\sqrt{3}+j3\sqrt{3}-9+8K=0$

$\Rightarrow -8+8k=0$

$\Rightarrow 8K=8$

$\therefore K=1$