Gain Margin

Ask

The gain margin of system under loop unity negative feedback G(s)H(s) = ${100 \over s(s+10)^2}$

(a)0 dB

(b)20 dB

(c)26 dB

(d)46 dB

Answer is ( c ) 26 dB

Concept

Gain Margin under closed loop unity negative feedback is =

$G(s)H(s) = {100 \over s(s+10)^2} ... Eq(1)$

Phase $\phi = -90° - 2arctan(ω/10)$

Phase cross - over frequency = -180°

$-180° = -90° - 2arctan(ω/10)$

$90° = 2arctan(ω/10)$

$tan(45°) = ω/10 \therefore ω = 10$

In Eq(1) , put s = jω

$\therefore G(jω)H(jω) = 100/jω(jω+2)^{2}$

$|G(jω)H(jω)| = {100\over ω(ω2 + 100)}$...Eq(2)

Substituting for ω = 10,Eq (2) becomes, ${100\over 10(100+100)} = 1/20$

Gain Margin = $1 \over |G(jω)H(jω)|$ = 1/(1/20) = 20

In dB, G.M (in dB) = 20 log1020 = 26dB