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

2. Personal 7/3/2015 2 Eng. R. L. Nkumbwa @ CBU 2010

3. Concept Consider a single input, single output linear system: 7/3/2015 3 Eng. R. L. Nkumbwa @ CBU 2010

4. 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: 7/3/2015 4 Eng. R. L. Nkumbwa @ CBU 2010

5. We get 7/3/2015 5 Eng. R. L. Nkumbwa @ CBU 2010

6. Let x 0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s). 7/3/2015 6 Eng. R. L. Nkumbwa @ CBU 2010

7. 7/3/2015 7 Eng. R. L. Nkumbwa @ CBU 2010

8. 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. 7/3/2015 8 Eng. R. L. Nkumbwa @ CBU 2010

9. 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. 7/3/2015 9 Eng. R. L. Nkumbwa @ CBU 2010

10. In Time Domain 7/3/2015 10 Eng. R. L. Nkumbwa @ CBU 2010

11. In Laplace Domain Convolution in the time domain = Product in the Laplace domain. 7/3/2015 11 Eng. R. L. Nkumbwa @ CBU 2010

12. Notion of Poles and Zeros In the above, the transfer function G(s) was found to be a fraction of two polynomials in s. 7/3/2015 12 Eng. R. L. Nkumbwa @ CBU 2010

13. The denominator, D(s), comes from the determinant of (sI-A), which appears from taking the inverse of (sI-A). 7/3/2015 13 Eng. R. L. Nkumbwa @ CBU 2010

14. 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) = ¥. 7/3/2015 14 Eng. R. L. Nkumbwa @ CBU 2010

15. Transfer Function Analysis 7/3/2015 15 Eng. R. L. Nkumbwa @ CBU 2010

16. Alternatively put, The poles are the roots of D(s), and the zeroes are the roots of N(s). 7/3/2015 16 Eng. R. L. Nkumbwa @ CBU 2010

17. Realization condition The realization condition states that the order of the numerator is always less than or equal to the order of the denominator. 7/3/2015 17 Eng. R. L. Nkumbwa @ CBU 2010

18. Wrap-Up 7/3/2015 18 Eng. R. L. Nkumbwa @ CBU 2010