Capacitance Calculations using Method of Moments

1. Capacitance method of moments

2. Q a Capacitance of hollow sphere

3. Capacitance of parallel plates

4. Capacitance of a circular disk r R   Voltage at origin   = 0 a 2 - 

5. Capacitance of a circular disk r  a R  2 -  Charge density is uniform s0

6. Capacitance of a circular disk r  a R  2 -  Charge density decreases s0(1-r/a)

7. This technique was developed in a Ph.D. thesis by Ken Mei in 1962. His adviser was Jean van Bladel.

8. Method of moments • a known charge causes a potential • what is the potential? • a measured or known potential is caused by an unknown charge • what was that charge? • used in em to find current distributions on antennas, scattering objects such as cubes, sleds, planes, and missles • PhD theses, faculty publications

9. V(a) V(b) Q2 Q1 2 equations 2 unknowns

10. V(a) V(b) Q2 Q1

11. MATLAB!!

12. >> A = [1/4 1/5 ; 1/5 1/4] ; >> V = [1 ; 1] ; >> Q = V’ / A % ‘transpose Q = 2.2222 2.2222 >> QQ = A \ V; QQ = 2.2222 2.2222

13. complications??? r1 r2 • The potential @ each charge sphere is specified as V1 or V2 • Locations are with respect to an origin.

14. solution to singularity Assume that the potential is uniform within the sphere and it is equal to the value at its edge r = a.

15. a = 1

16. 1 1 1 1 1 1

17. Q = • 4.69 2.56 4.69 -4.69 -2.56 -4.69 • >>

18. equal charge

19. equal charge

20. a b -a -a a -b -b b Capacitor C = Q / V rj ri

21. b -b -b b Capacitor C = Q / V Harrington – p. 27 Dwight 200.01 & 731.2

22. b -b -b b Capacitor C = Q / V

23. MATLAB • clear; clf; • N=9; • az=37.5; • el=5;

24. %identify subareas • for m = 1:2 • for h = 1:N • for k = 1:N • a(m - 1)*N*N + (h - 1)*N + k, :)=[m, h, k]; • end • end • end

25. %calculate matrix elements • for h=1:2*N*N • for k=1:2*N*N • aa=norm(a(h, :) - a(k, :)); • if aa==0 • b(h, k) = 2 * sqrt(pi); • else • b(h, k) =1 / aa; • end • end • end

26. %set voltages on plates • for h=1:N*N • V(h) = +1/2; • V(h+N*N) = -1/2; • end

27. %calculate charges • Q = V*inv(b); • %top plate • QA(1:N*N) = Q(1:N*N); • %bottom plate • QB(1:N*N)=Q(N*N+1:2*N*N);

28. %plot • [x,y]=meshgrid(1:N); • for i=1:N • za(1:N,i)=QA((i-1)*N+1:(i-1)*N+N)'; • zb(1:N,i)=QB((i-1)*N+1:(i-1)*N+N)'; • end • mesh(x,y,za) • hold on • mesh(x,y,zb)

29. QT=0; for j=1:N*N QT=QT+Q(j); end QT

30. What have you done? You have learned a technique, to accurately calculate the capacitance of a parallel plate capacitor! Big deal!

33. Capacitance of a circular disk r  a R  2 -  Charge density is uniform s0

34. Capacitance of a circular disk r  a R  2 -  Charge density is noniform

35. C = 3.9420 C = 3.2367 C = 2.7026 C = 3.6332

36. capacitance of a unit cube

37. capacitance of a unit cube

38. capacitance of a unit cube Hwang & Mascogni – “Electrical capacitance of a unit cube” – Journal of Applied Physics 3798-3802 (2004).

39. 2a C a

41. note the charge density at the corner

42. a h Sava V. Savov February 1, 2004

43. 1 2 3 x. • E = s /2 • V = x s /2 • V1 = 2 [0/2 s1 + 1 s2 + 2/2 s3 ] • V2 = 2 [1/2 s1 + 0 s2 + 1/2 s3] • V3 = 2 [2/2 s1 + 1 s2 + 0/2 s3]

44. 1 2 3 • V1 0 1 2/2 s1 • V2 = 2 1/2 0 1/2 s2 • V3 2/2 1  0 s3 • pn junction • double layer • VLSI

45. pn junction • linear quadratic • V[-2d] - 0.36 - 2 • V[-d] - 0.18 - 3/2 • V[0] = 0 or 0 • V[d] + 0.18 + 3/2 • V[2d] + 0.36 + 2

46. matrix • 0 1 2 3 4/2 • 1/2 0 1 2 3/2 • 2/2 1 0 1 2/2 • 3/2 2 1 0 1/2 • 42 3 2 1 0

47. MATLAB • [V] = [matrix] [Q] V V r r x x

48. Calculate vector “V” • clear • clf • m=input('What is the size of the matrix? ...'); • for i=1:m • v(i)=NaN; • end • v

49. Calculate matrix “a” • for i=1:m • for j=1:m • a(i,j)=NaN; • end • end

50. for i=1:m • for j=i:m • if i==j • a(i,j)=0; • b(i)=i; • elseif i<j & j<m a(i,j)=(j-b(i)); elseif i<j & j==m • a(i,j)=(j-b(i))/2; • end • end • end