Concept of Transfer Function

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

2. Personal Eng. R. L. Nkumbwa @ CBU 2010

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

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

6. Let x0 = 0. We are interested in finding the input-output relation, which is the relation between Y(s) and U(s). Eng. R. L. Nkumbwa @ CBU 2010

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

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

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

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

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