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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, \({\rm{k}} = 12\)

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