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