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

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Presentation transcript:

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

Personal 7/3/ Eng. R. L. CBU 2010

Concept Consider a single input, single output linear system: 7/3/ Eng. R. L. 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: 7/3/ Eng. R. L. CBU 2010

We get 7/3/ Eng. R. L. CBU 2010

Let x 0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s). 7/3/ Eng. R. L. CBU 2010

7/3/ Eng. R. L. 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. 7/3/ Eng. R. L. 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. 7/3/ Eng. R. L. CBU 2010

In Time Domain 7/3/ Eng. R. L. CBU 2010

In Laplace Domain Convolution in the time domain = Product in the Laplace domain. 7/3/ Eng. R. L. CBU 2010

Notion of Poles and Zeros In the above, the transfer function G(s) was found to be a fraction of two polynomials in s. 7/3/ Eng. R. L. CBU 2010

The denominator, D(s), comes from the determinant of (sI-A), which appears from taking the inverse of (sI-A). 7/3/ Eng. R. L. CBU 2010

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) = ¥. 7/3/ Eng. R. L. CBU 2010

Transfer Function Analysis 7/3/ Eng. R. L. CBU 2010

Alternatively put, The poles are the roots of D(s), and the zeroes are the roots of N(s). 7/3/ Eng. R. L. CBU 2010

Realization condition The realization condition states that the order of the numerator is always less than or equal to the order of the denominator. 7/3/ Eng. R. L. CBU 2010

Wrap-Up 7/3/ Eng. R. L. CBU 2010