GATE EC 2014 Set 1 — Question 30
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Control Systems → Stability Analysis → First & Second-Order Response
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Question
The forward path transfer function of a unity negative feedback system is given by . The value of which will place both the poles of the closed-loop system at the same location, is ______.
Correct answer
2.24 to 2.26
Solution
For a unity negative feedback system, the closed-loop transfer function is given by:
Substitute the given :
Expand the denominator:
So, the closed-loop transfer function is .
The poles of the closed-loop system are the roots of the characteristic equation:
We are given that both poles are at the same location. This means the quadratic equation must have repeated roots. For a quadratic equation to have repeated roots, its discriminant must be zero ().
In our case, , , and .
So,
Alternatively, if the poles are at the same location, say , the characteristic equation can be written as , which expands to .
Comparing this with :
Coefficient of : .
Constant term: .
Substitute into the constant term equation:
.
Substitute the given :
Expand the denominator:
So, the closed-loop transfer function is .
The poles of the closed-loop system are the roots of the characteristic equation:
We are given that both poles are at the same location. This means the quadratic equation must have repeated roots. For a quadratic equation to have repeated roots, its discriminant must be zero ().
In our case, , , and .
So,
Alternatively, if the poles are at the same location, say , the characteristic equation can be written as , which expands to .
Comparing this with :
Coefficient of : .
Constant term: .
Substitute into the constant term equation:
.
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