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