Question

For a unity feedback control system, if its open-loop transfer function is given by \(\frac{10}{(s+5)^3}\), then its gain margin will be
1. 20 dB
2. 40 dB
3. 60 dB
4. 80 dB

Answer 1

Correct Answer - Option 2 : 40 dB

Concept:

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

The phase Crossover frequency (ω_pc) 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:

\(G(s)=\frac{10}{(s+5)^3}\)

∠G(jω)H(jω) = -180°

\(-3\times tan^{-1}({\frac{ω_{pc} }{5}})= - 180^\circ \)

\(ω_{pc}=5\sqrt{3}\) rad/sec

\(\left| {G\left( {j{ω _{pc}}} \right)H\left( {j{ω _{pc}}} \right)} \right| = 0.01\)

\(G.M.\; = 20\log \frac{1}{a} = 20\log \frac{1}{{0.01}} = 40\;dB\)