What will be the gain margin in dB of a system having the following open loop transfer function?

G(s)H(s)=2s(s+1)

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Answer (Detailed Solution Below)

Option 4 : ∞
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Gain margin (GM): The gain margin of the system defines by how much the system gain can be increased so that the system moves on the edge of stability.

It is determined from the gain at the phase cross-over frequency.

GM=1|G(jω)H(jω)|ω=ωpc

Phase crossover frequency (ωpc): It is the frequency at which the phase angle of G(s) H(s) is -180°.

G(jω)H(jω)|ω=ωpc=180

Phase margin (PM): The phase margin of the system defines by how much the phase of the system can increase to make the system unstable.

PM=180+G(jω)H(jω)|ω=ωgc=180

It is determined from the phase at the gain cross over frequency.

Gain crossover frequency (ωgc): It is the frequency at which the magnitude of G(s) H(s) is unity.

|G(jω)H(jω)|ω=ωgc=1

Calculation:

G(s)H(s)=2s(s+1)

G(jω)H(jω)=2jω(jω+1)

∠G(jω)H(jω) = - 90° - tan-1 (ω)

At phase crossover frequency, - 90° - tan-1pc) = -180°

⇒ ωpc = ∞

Now,

|G(jω)H(jω)|=2ω1+ω2

|G(jω)H(jω)|ω=ωpc=0

Thus,

Gain margin =1|G(jω)H(jω)|ω=ωpc

Gainmargin=10=

Important Points:

  • If both GM and PM are positive, the system is stable (ωgc < ωpc)
  • If both GM and PM are negative, the system is unstable (ωgc > ωpc)
  • If both GM and PM are zero, the system is just stable (ωgc = ωpc)
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