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Question

The open loop transfer function of a unity feedback control system is given as G(s)=as+1s2. The value of 'a' to give a phase margin of 45 is equal to.

A
0.141
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B
0.441
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C
0.841
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D
1.141
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Solution

The correct option is C 0.841

G(s)=as+1s2
ωg=gain crossover frequency, at which open gain is loop .
Phasemargin=180+G(jω)
=45=180+G(jωg)
G(jωg)=135
G(jωg)=jaωg+1(jωg)2
=2×90+tan1aωg
G(jωg)=180+tan1aωg=135
tan1aωg=45
aωg=1
ωg=1a
Gain=|G(jωg)|=1
1+a2ω2gω2g=1
1+a2×1a21/a2=1
1a2=2
a=121/4=0.841

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