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Gate 2016 EE - Set 2 -Question52

G(S)=\frac{kS}{(S-1)(S-4)}

Characteristic equation \Rightarrow 1+G(S)H(S)=0

\Rightarrow (S-1)(S-4)+kS=0

S^{2}-4S-S+4+kS=0

S^{2}-5S+4+kS=0

k=-\frac{(S^{2}-5S+4)}{S}

k=-(S-5+\frac{4}{S})

For break away point, \frac{\mathrm{d} k}{\mathrm{d} S}=0

\frac{\mathrm{d} k}{\mathrm{d} S}=-\left [ 1-0-\frac{4}{S^{2}} \right ]=0

-1+\frac{4}{S^{2}}=0

-S^{2}+4=0

S=\pm 2

Valid break away point is at S = 2

Therefore, gain is at S = 2

K = Product of distances from all poles to breakaway point (denominator) / Product of distance from all the zeros to break away point (numerator)

K = \frac{1\times 2}{2}=1

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