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Question

The gain at the breakaway point of the root locus of a unity feedback system with open loop transfer function G(s)=Ks(sāˆ’1)(sāˆ’4) is

A
1
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B
2
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C
5
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D
9
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Solution

The correct option is A 1
OLTFG(s)=Ks(s1)(s4)

Now, characteristics equation,

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

Ks(s1)(s4)+1=0

Ks+(s25s+4)=0

K=(s25s+4)s=[s5+4s]

For break away point :

dKds=0

dKds=[1+04s2]=0

we get, s=±2

Therefore valid break away point is s = 2.

Now gain at s = 2 is

K=Product of distance from all poles to break away pointproduct of distance from all the zeros to break away point

Gain,K=1×22=1

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