# 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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## Answer (Detailed Solution Below) 12

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## Detailed Solution

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

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