The gain margin in dBs of a unity feedback control system whose open-loop transfer function, \(\frac{1}{s(s+1)}\) is

Concept:

The gain margin is always calculated at the phase Crossover frequency (ωpc)

The phase Crossover frequency (ωpc) is calculated as:

\(\angle G\left( {j\omega } \right)H\left( {j\omega } \right){\left. \right|_{\omega = {\omega _{pc}}}} = \pm 180^\circ \)

Gain margin \( = \frac{1}{{{{\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right|}_{\omega = {\omega _{pc}}}}}}\)

Calculation:

Given:

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

\(G\left( {j\omega } \right)H\left( {j\omega } \right) = \frac{1}{{j\omega \left( {j\omega + 1} \right)}}\)

The phase Crossover frequency (ωpc) is calculated as:

\(\angle G\left( {j\omega } \right)H\left( {j\omega } \right){\left. \right|_{\omega = {\omega _{pc}}}} = \pm 180^\circ \)

\(- 90^\circ - {\tan ^{ - 1}}{{{\omega _{pc}}}} = 180^\circ \)

\({\tan ^{ - 1}}{{{\omega _{pc}}}} = 90^\circ \)

\({\omega _{pc}} = \infty\)

Now,

\(\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right| = \frac{1}{{\omega \sqrt {4 + {\omega ^2}} }}\)

\({\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right|_{\omega = {\omega _{pc}}}} = 0\)

Thus,

Gain margin \(= \frac{1}{{{{\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right|}_{\omega = {\omega _{pc}}}}}}\)