Question

Consider a unity feedback system with forward path transfer function G(s) = $\frac{K(s+3)(s+5)}{(s-2)(s-4)}$
The range of K to ensure stability is

• A. $K>\frac{3}{4}$
• B. $K<-1\left(or\right)K>\frac{3}{4}$
• C. k < - 1
• D. $-1<K<\frac{3}{4}$

Answer 1

The correct option is B $K<-1\left(or\right)K>\frac{3}{4}$

Closed loop transfer function is,

$T\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)}=\frac{K(s+3)(s+5)}{(K+1)s^2+(8K-6)s+(15K+8)}$

For stability all coefficients should be positive or all negative
Case-I: All coefficient are positive,
$K+1>0\Rightarrow K>-1$
$8K-6>0\Rightarrow K>0.75$
$15K+8>0\Rightarrow K>\frac{-8}{15}$
$\therefore$ K > 0.75

Case-II : All coefficients are negative:

$K+1<0\Rightarrow K<-1$
$8K-6<0\Rightarrow K<0.75$
$15K+8<0\Rightarrow K<\frac{-8}{15}$
$\therefore$ K < - 1
So, for stability either K > 0.75 or K < - 1