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Control system miscellaneous

  1. The frequency response of G(s) =
    1
    plotted in the complex G(jω)
    [s(s + 1)(s + 2)]

    plane (for 0 < ω < ∞) is



Correct Option: A

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

G(jω) =
1
jω(jω + 1)(jω + 2)

M =
1
ω√ω² + 1ω² + 4

∠ φ = -90 - tan-1ω - tan-1
ω
2


For ω = 0, M = ∞ ; ∠ φ = – 90
For ω = ∞, M = 0; ∠ φ = – 90 – 90 – 90 = – 270
So Cutting Real Axis
Imaginary part of G(jω) = 0
i.e. Img {G(jω)} = 0
⇒ Im
1
= 0
jω(-ω2 + 3)ω + 2

⇒ Im
1
= 0
-jω3 - 3ω2 + 2jω

⇒ Im
1
= 0
- 3ω2 + j(2ω - ω3)

⇒ Im
-3ω2 - j(2ω - ω3)
= 0
(3ω2)2 + (2ω - ω3)2

∴ 2ω – ω3 = 0
⇒ ω2 = 2 . ω = 0
Neglecting ω = 0, we have
ω = √2
∴ M |at ω = √2 =
1
=
1
1
< 3
232 + 43664



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