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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∘+tan−1aωg ∠G(jωg)=−180∘+tan−1aωg=−135∘ tan−1aωg=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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