Question

A unity negative feedback system has the open-loop transfer function $G\left(s\right)=\frac{K}{s(s+1)(s+3)}$
The value of the gain $K(>0)$ at which the root locus crosses the imaginary axis is

• A. 12

Answer 1

The correct option is A 12

$G\left(s\right)=\frac{K}{s(s+1)(s+3)}$
The characteristic equation
$=1+G\left(s\right)H\left(s\right)=0$
$=s(s+1)(s+3)+K=0$
$=s^3+4s^2+3s+K=0$
Using Routh's tabular form
$\begin{array}{ccc}{s}^3& 1& 3\\ s^2& 4& K\\ s^1& \frac{12-K}{4}& 0\\ s^0& K& \end{array}$
In order to cross the imaginary axis, system should be marginally stable
$\therefore \frac{12-K}{4}=0$
or $K=12$