The gain at the breakaway point of the root locus of a unity feedback system with open loop transfer function $G (s) = \frac{K s}{(s - 1) (s - 4)}$ is

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Solution

The correct option is A 1
$O L T F \Rightarrow G (s) = \frac{K s}{(s - 1) (s - 4)}$

Now, characteristics equation,

$1 + G (s) H (s) = 0$

$\frac{K s}{(s - 1) (s - 4)} + 1 = 0$

$\Rightarrow K s + (s^{2} - 5 s + 4) = 0$

$K = - \frac{(s^{2} - 5 s + 4)}{s} = [s - 5 + \frac{4}{s}]$

For break away point :

$\frac{d K}{d s} = 0$

$\frac{d K}{d s} = - [1 + 0 - \frac{4}{s^{2}}] = 0$

$w e g e t, s = \pm 2$

Therefore valid break away point is s = 2.

Now gain at s = 2 is

$\Rightarrow K = \frac{P r o d u c t o f d i s t a n c e f r o m a l l p o l e s t o b r e a k a w a y p o i n t}{p r o d u c t o f d i s t a n c e f r o m a l l t h e z e r o s t o b r e a k a w a y p o i n t}$

$G a i n, K = \frac{1 \times 2}{2} = 1$

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