# 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}} + 0 votes 36 views closed 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