Concept of Transfer Function

Concept of Transfer Function

Eng. R. L. NkumbwaCopperbelt University2010

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04/22/232 Eng. R. L. Nkumbwa @ CBU 2010

Concept

Consider a single input, single output linear system:


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:


We get


Let x0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s).


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.


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.


In Time Domain


In Laplace Domain

Convolution in the time domain = Product in the Laplace domain.


Notion of Poles and Zeros

In the above, the transfer function G(s) was found to be a fraction of two polynomials in s.


The denominator, D(s), comes from the determinant of (sI-A), which appears from taking the inverse of (sI-A).


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) = ¥.


Transfer Function Analysis


Alternatively put,

The poles are the roots of D(s), and the zeroes are the roots of N(s).


Realization condition

The realization condition states that the order of the numerator is always less than or equal to the order of the denominator.


Wrap-Up


Concept of Transfer Function

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