1. 1 Green’s Functions ECE 6382 David R. Jackson Note: j is used in this set of notes instead of i.

2. Green’s Functions 2 Green's Mill in Sneinton (Nottingham), England, the mill owned by Green's father. The mill was renovated in 1986 and is now a science centre. George Green (1793-1841) The Green's function method is a powerful and systematic method for determining a solution to a problem with a known forcing function on the RHS. The Green’s function is the solution to a “point” or “impulse” forcing function. It is similar to the idea of an “impulse response” in circuit theory.

3. Green’s Functions 3 Consider the following SOLDE: or where ( f is a “forcing” function.)

4. Green’s Functions 4 Problem to be solved: or

5. Green’s Functions (cont.) 5  We can think of the forcing function as being broken into many small rectangular pieces.  Using superposition, we add up the solution from each small piece.  Each small piece can be represented as a delta function in the limit as the width approaches zero. Note: The  function goes to infinity, while the actual function f (x) does not. However, as we will see, it is only the area of the rectangle pulse region that is important.

6. Green’s Functions (cont.) 6 The solution to the original differential equation is then The Green’s function G(x,x) is defined as the solution with a delta-function RHS at x = x. x a b

7. Green’s Functions (cont.) 7 There are two general method for constructing Green’s functions. Method 1: Find the solution to the homogenous equation to the left and right of the delta function, and then force boundary conditions at the location of the delta function. x a b  The Green’s function is assumed to be continuous.  The derivative of the Green’s function is allowed to be discontinuous. The functions u 1 and u 2 are solutions of the homogenous equation.

8. Green’s Functions (cont.) 8 Method 2: Use the method of eigenfunction expansion. Eigenvalue problem: We then have: or

9. Green’s Functions (cont.) 9 Method 1 x a b Note:

10. Green’s Functions (cont.) 10 Method 1 (cont.) x a b Also, we have (from continuity):

11. Green’s Functions (cont.) 11 Method 1 (cont.) x a b We then have: where

12. Green’s Functions (cont.) 12 Method 1 (cont.) x a b We then have:

13. Green’s Functions (cont.) 13 Method 1 (cont.) x a b Observation: The Green’s function automatically obeys “reciprocity.” Note: The Green’s function remains the same when we replace

14. Green’s Functions (cont.) 14 Method 2 x a b n=1 n=2 n=3 n=4 The Green’s function is expanded as a series of eigenfunctions. where The eigenfunctions corresponding to distinct eigenvalues are orthogonal (from Sturm-Liouville theory).

15. Green’s Functions (cont.) 15 Method 2 (cont.) x a b n=1 n=2 n=3 n=4 Orthogonality Delta-function property

16. Green’s Functions (cont.) 16 Method 2 (cont.) x a b n=1 n=2 n=3 n=4 Therefore, we have Hence

17. Application: Transmission Line 17 A short-circuited transmission line with a distributed current source: The distributed current source is a surface current. + -

18. 18 An illustration of the Green’s function: Application: Transmission Line (cont.) The total voltage due to the distributed current source is + -

19. 19 Take the derivative of the first and substitute from the second: Telegrapher’s equations for a distributed current source: (Please see the Appendix.) Application: Transmission Line (cont.)

20. 20 Hence where Application: Transmission Line (cont.) or

21. 21 Application: Transmission Line (cont.) The general solution of the homogeneous equation is: Method 1 + -

22. 22 Application: Transmission Line (cont.) The Green’s function is: + -

23. 23 Application: Transmission Line (cont.) The final form of the Green’s function is: where + -

24. 24 Application: Transmission Line (cont.) The eigenvalue problem is: Method 2 + - or where

25. 25 Application: Transmission Line (cont.) This may be written as where + -

26. 26 Application: Transmission Line (cont.) We have so that

27. 27 Application: Transmission Line (cont.) For the eigenvalue value problem we have: The solution is:

28. 28 Application: Transmission Line (cont.) The final solution is then

29. 29 Appendix

30. Appendix 30 Allow for distributed sources v+v+ v -v - + - + - zz i+i+ i-i- Appendix: Derivation of telegrapher’s equations with distributed sources

31. 31 Appendix (cont.)

32. 32 Appendix (cont.)

33. 33 In the phasor domain: Appendix (cont.)