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

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This question was previously asked in

ESE Electronics 2012 Paper 2: Official Paper

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Answer 1

Concept:

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

The phase Crossover frequency (ω_pc) is calculated as:

$∠ G (j ω) H (j ω) |_{ω = ω_{p c}} = \pm 180^{\circ}$

Gain margin $= \frac{1}{{| G (j ω) H (j ω) |}_{ω = ω_{p c}}}$

Calculation:

Given:

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

∠G(jω)H(jω) = -180°

$- 3 \times t a n^{- 1} (\frac{ω_{p c}}{5}) = - 180^{\circ}$

$ω_{p c} = 5 \sqrt{3}$ rad/sec

$| G (j ω_{p c}) H (j ω_{p c}) | = 0.01$

$G . M . = 20 \log \frac{1}{a} = 20 \log \frac{1}{0.01} = 40 d B$