EEE 431 Computational Methods in Electrodynamics

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

2. Charged Conducting Plate Moment Method Solution

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

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

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

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

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

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. Charged Conducting Plate/ MoM Solution • So,

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

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

15. Charged Conducting Plate/ MoM Solution • The capacitance (Cont.):

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

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

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

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. 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. Moment Method/ Review • Where,

23. Moment Method/ Review • Or,

24. Moment Method/ Review • Where,

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

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

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

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

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

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

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. Moment Method/ Review • Hence: