concept of transfer function
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Concept of Transfer Function
Eng. R. L. NkumbwaCopperbelt University2010
Personal
04/22/232 Eng. R. L. Nkumbwa @ CBU 2010
Concept
Consider a single input, single output linear system:
04/22/233 Eng. R. L. Nkumbwa @ CBU 2010
Where,
A is an n-by-n matrix, b is a n-by-one vector, c is a one-by-n vector, and d is a scalar.
Taking the Laplace transform of the state and output equations, we get:
04/22/234 Eng. R. L. Nkumbwa @ CBU 2010
We get
04/22/235 Eng. R. L. Nkumbwa @ CBU 2010
Let x0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s).
04/22/236 Eng. R. L. Nkumbwa @ CBU 2010
04/22/237 Eng. R. L. Nkumbwa @ CBU 2010
Transfer Function
G(s) is called the transfer function, and represents the input-output relation for a given system in the s-domain.
The above equation is an important formula, but note that it may not necessarily be the easiest way to obtain the transfer function from the state and output equations.
04/22/238 Eng. R. L. Nkumbwa @ CBU 2010
Transfer Function Definition
The transfer function is sometimes defined as:– The Laplace transform of the time impulse
response with zero initial conditions.
The development directly above is where this definition comes from.
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In Time Domain
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In Laplace Domain
Convolution in the time domain = Product in the Laplace domain.
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Notion of Poles and Zeros
In the above, the transfer function G(s) was found to be a fraction of two polynomials in s.
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The denominator, D(s), comes from the determinant of (sI-A), which appears from taking the inverse of (sI-A).
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Values of “s”
These values of s have the same importance in the present discussion.
Values of s that make the numerator, N(s), go to zero are called zeros since they make G(s) = 0. Values of s that make the denominator, D(s), go to zero are called poles; they make G(s) = ¥.
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Transfer Function Analysis
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Alternatively put,
The poles are the roots of D(s), and the zeroes are the roots of N(s).
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Realization condition
The realization condition states that the order of the numerator is always less than or equal to the order of the denominator.
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Wrap-Up
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