Question

The open loop transfer function of a unity gain feedback system is given by:
$G\left(s\right)=\frac{k(s+3)}{(s+1)(s+2)}$
The range of positive values of k for which the closed loop system will remain stable is

• A. $1<k<3$
• B. $0<k<10$
• C. $5<k<\infty$
• D. $0<k<\infty$

Answer 1

The correct option is D $0<k<\infty$

The characteristic equation is
$1+\frac{k(s+3)}{(s+1)(s+2)}=0$
$(s+1)(s+2)+k(s+3)=0$
$s^2+3s+2+ks+3K=0$
$s^2+(3+k)s+(2+3K)=0$
According to R-H array:

For stability
$3+k>\Rightarrow k>-3$
and $2+3k>0\Rightarrow k>-\frac{2}{3}$
$\Rightarrow$ $k>-\frac{2}{3}$
From the options the appropriate of range k is $o<k<\infty$