1. 1 EEE 431 Computational Methods in Electrodynamics Lecture 17 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. 2 Charged Conducting Plate Moment Method Solution

3. 3 Charged Conducting Plate/ MoM Solution Consider a square conducting plate 2a meters on a side lying on the z=0 plane with center at the origin.

4. 4 Charged Conducting Plate/ MoM Solution Let represent the surface charge density on the plate. Assume that the plate has zero thickness.

5. 5 Charged Conducting Plate/ MoM Solution Then, V(x,y,z): Where;

6. 6 Charged Conducting Plate/ MoM Solution Integral Equation: When This is the integral equation for

7. 7 Charged Conducting Plate/ MoM Solution Method of Moment Solution: Consider that the plate is divided into N square subsections. Define: And let:

8. 8 Charged Conducting Plate/ MoM Solution Substituting this into the integral equation and satisfying the resultant equation at the midpoint of each, we get:

9. 9 Charged Conducting Plate/ MoM Solution Where: is the potential at the center of due to a uniform charge density of unit amplitude over

10. 10 Charged Conducting Plate/ MoM Solution Let : denote the side length of each the potential at the center of due to the unit charge density over its own surface.

11. 11 Charged Conducting Plate/ MoM Solution So,

12. 12 Charged Conducting Plate/ MoM Solution The potential at the center of can simply be evaluated by treating the charge over as if it were a point charge, so,

13. 13 Charged Conducting Plate/ MoM Solution So, the matrix equation:

14. 14 Charged Conducting Plate/ MoM Solution The capacitance:

15. 15 Charged Conducting Plate/ MoM Solution The capacitance (Cont.): Number of sub areas C/2a Approx. C/2a Exact 131.5 937.336.8 1638.237.7 3639.238.5

16. 16 Charged Conducting Plate/ MoM Solution Harrington, Field Computation by Moment Methods The charge distribution along the width of the plate

17. 17 Moment Method/ Review Consider the operator equation: Linear Operator. Known function, source. Unknown function. The problem is to find g from f.

18. 18 Moment Method/ Review Let f be represented by a set of functions scalar to be determined (unknown expansion coefficients. expansion functions or basis functions.

19. 19 Moment Method/ Review Now, substitute (2) into (1): Since L is linear:

20. 20 Moment Method/ Review Now define a set of testing functions or weighting functions Define the inner product (usually an integral). Then take the inner product of (3) with each and use the linearity of the inner product:

21. 21 Moment Method/ Review It is common practice to select M=N, but this is not necessary. For M=N, (4) can be written as:

22. 22 Moment Method/ Review Where,

23. 23 Moment Method/ Review Or,

24. 24 Where, Moment Method/ Review

25. 25 Moment Method/ Review If is nonsingular, its inverse exists and. Let

26. 26 Moment Method/ Review The solution (6) may be either approximate or exact, depending upon on the choice of expansion and testing functions.

27. 27 Moment Method/ Review Summary: 1)Expand the unknown in a series of basis functions. 2) Determine a suitable inner product and define a set of weighting functions. 3) Take the inner products and form the matrix equation. 4)Solve the matrix equation for the unknown.

28. 28 Moment Method/ Review Inner Product: Where:

29. 29 Moment Method/ Review Inner product can be defined as:

30. 30 Moment Method/ Review If u and v are complex:

31. 31 Moment Method/ Review Here, a suitable inner product can be defined:

32. 32 Moment Method/ Review Example: Find the inner product of u(x)=1-x and v(x)=2x in the interval (0,1). Solution: In this case u and v are real functions.

33. 33 Moment Method/ Review Hence: