# A unity negative feedback system has the open – loop transfer function ${\rm{G}}\left( {\rm{s}} \right) = \frac{{\rm{k}}}{{{\rm{s}}\left( {{\rm{s}} + 0 votes 119 views closed A unity negative feedback system has the open – loop transfer function \({\rm{G}}\left( {\rm{s}} \right) = \frac{{\rm{k}}}{{{\rm{s}}\left( {{\rm{s}} + 1} \right)\left( {{\rm{s}} + 3} \right)}}$. The value of the gain ${\rm{K}}( > 0)$ at which the root locus crosses the imaginary axis is _________.

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The characteristic equation is $1 + {\rm{G}}\left( {\rm{s}} \right){\rm{H}}\left( {\rm{s}} \right) = 0$

$\begin{array}{l} \Rightarrow 1 + \frac{{\rm{k}}}{{{\rm{s}}\left( {{\rm{s}} + 1} \right)\left( {{\rm{s}} + 3} \right)}} = 0\\ \Rightarrow {{\rm{s}}^3} + 4{{\rm{s}}^2} + 3{\rm{s}} + {\rm{k}} = 0 \end{array}$

Solving the Routh’s table,

 ${{\rm{s}}^3}$ 1 3 ${{\rm{s}}^2}$ 4 k ${{\rm{s}}^1}$ $\frac{{12 - {\rm{k}}}}{4}$ ${{\rm{s}}^0}$ k

For finding imaginary axis poles, we set $\frac{{12 - {\rm{k}}}}{4} = 0$

Thus, ${\rm{k}} = 12$