Concept of Transfer Function Eng R L Nkumbwa
Concept of Transfer Function Eng. R. L. Nkumbwa Copperbelt University 2010
Personal 2 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Concept l 3 Consider a single input, single output linear system: 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Where, 4 l 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. l Taking the Laplace transform of the state and output equations, we get: 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
We get 5 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
l 6 Let x 0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s). 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
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Transfer Function G(s) is called the transfer function, and represents the input-output relation for a given system in the s-domain. l 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. l 8 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Transfer Function Definition l The transfer function is sometimes defined as: – l 9 The Laplace transform of the time impulse response with zero initial conditions. The development directly above is where this definition comes from. 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
In Time Domain 10 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
In Laplace Domain Convolution in the time domain = Product in the Laplace domain. 11 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Notion of Poles and Zeros l 12 In the above, the transfer function G(s) was found to be a fraction of two polynomials in s. 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
l 13 The denominator, D(s), comes from the determinant of (s. I-A), which appears from taking the inverse of (s. I-A). 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Values of “s” l l 14 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) = ¥. 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Transfer Function Analysis 15 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Alternatively put, l 16 The poles are the roots of D(s), and the zeroes are the roots of N(s). 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Realization condition l 17 The realization condition states that the order of the numerator is always less than or equal to the order of the denominator. 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
Wrap-Up 18 9/10/2020 Eng. R. L. Nkumbwa @ CBU 2010
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