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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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