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The open-loop transfer function of a unity-feedback control system is

\({\rm{G}}\left( {\rm{S}} \right) = \frac{{\rm{K}}}{{{{\rm{s}}^2} + 5{\rm{s}} + 5}}\)

The value of K at the breakaway point of the feedback control system’s root-locus plot is _______.

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Given the open-loop transfer function:

\({\rm{G}}\left( {\rm{s}} \right) = \frac{{\rm{K}}}{{\left( {{{\rm{s}}^2} + 55 + 5} \right)}}\)

\( 1 + {\rm{G}}\left( {\rm{s}} \right){\rm{H}}\left( {\rm{s}} \right)\)

\({\rm{\;}}1 + \frac{{\rm{K}}}{{\left( {{{\rm{s}}^2} + 5{\rm{s}} + 5} \right)}} = 0\)

\(\\ {\rm{K\;}} = {\rm{\;}} - \left( {{{\rm{s}}^2} + 5{\rm{s}} + 5} \right)\)

For Breakpoint calculation we need:

\(\frac{{{\rm{dK}}}}{{{\rm{ds}}}}{\rm{\;}} = {\rm{\;}}0{\rm{\;}} \)

\({\rm{\;}}2{\rm{s}} + 5 = 0{\rm{\;}}\)

s = - 2.5

At s = -2.5, G(s) will be:

\({\rm{\;G}}\left( {\rm{s}} \right) = \frac{{\rm{K}}}{{6.25 + 5\left( { - 2.5} \right) + 5}}\)

Form magnitude Criterion:

\(\left| {{\rm{G}}\left( {\rm{s}} \right)} \right|{\rm{\;}} = 1{\rm{\;at\;s\;}} = {\rm{\;}}2.5\)

\( \left| {\frac{{\rm{K}}}{{11.25 - 12.5}}} \right| = 1\)

\(\\ \left| {\frac{{\rm{K}}}{{ - 1.25}}} \right| = 1 \)

K = 1.25

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