Concept of Transfer Function
Eng. R. L. Nkumbwa
Copperbelt University
2010

Personal
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Eng. R. L. Nkumbwa @ CBU
2010

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Concept
 Consider a single input, single output linear system:
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Eng. R. L. Nkumbwa @ CBU
2010

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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:
Eng. R. L. Nkumbwa @ CBU
2010 4/11/2020

We get
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Eng. R. L. Nkumbwa @ CBU
2010

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 Let x
0
= 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s).
4/11/2020
Eng. R. L. Nkumbwa @ CBU
2010

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Eng. R. L. Nkumbwa @ CBU
2010

Transfer Function
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 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.
4/11/2020
Eng. R. L. Nkumbwa @ CBU
2010

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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.
Eng. R. L. Nkumbwa @ CBU
2010 4/11/2020

In Time Domain
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Eng. R. L. Nkumbwa @ CBU
2010

In Laplace Domain
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Convolution in the time domain = Product in the Laplace domain .
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Eng. R. L. Nkumbwa @ CBU
2010

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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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Eng. R. L. Nkumbwa @ CBU
2010

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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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Eng. R. L. Nkumbwa @ CBU
2010

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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) = ¥.
Eng. R. L. Nkumbwa @ CBU
2010 4/11/2020

Transfer Function Analysis
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Eng. R. L. Nkumbwa @ CBU
2010

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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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Eng. R. L. Nkumbwa @ CBU
2010

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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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Eng. R. L. Nkumbwa @ CBU
2010

Wrap-Up
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Eng. R. L. Nkumbwa @ CBU
2010