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## Capacitance method of moments

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**Capacitance**method of moments**Q**a Capacitance of hollow sphere**Capacitance of a circular disk**r R Voltage at origin = 0 a 2 - **Capacitance of a circular disk**r a R 2 - Charge density is uniform s0**Capacitance of a circular disk**r a R 2 - Charge density decreases s0(1-r/a)**This technique was developed**in a Ph.D. thesis by Ken Mei in 1962. His adviser was Jean van Bladel.**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**V(a)**V(b) Q2 Q1 2 equations 2 unknowns**V(a)**V(b) Q2 Q1**>> 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**complications???**r1 r2 • The potential @ each charge sphere is specified as V1 or V2 • Locations are with respect to an origin.**solution to singularity**Assume that the potential is uniform within the sphere and it is equal to the value at its edge r = a.**1**1 1 1 1 1**Q =**• 4.69 2.56 4.69 -4.69 -2.56 -4.69 • >>**a**b -a -a a -b -b b Capacitor C = Q / V rj ri**b**-b -b b Capacitor C = Q / V Harrington – p. 27 Dwight 200.01 & 731.2**b**-b -b b Capacitor C = Q / V**MATLAB**• clear; clf; • N=9; • az=37.5; • el=5;**%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**%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**%set voltages on plates**• for h=1:N*N • V(h) = +1/2; • V(h+N*N) = -1/2; • end**%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);**%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)**QT=0;**for j=1:N*N QT=QT+Q(j); end QT**What have you done?**You have learned a technique, to accurately calculate the capacitance of a parallel plate capacitor! Big deal!**Capacitance of a circular disk**r a R 2 - Charge density is uniform s0**Capacitance of a circular disk**r a R 2 - Charge density is noniform**C = 3.9420**C = 3.2367 C = 2.7026 C = 3.6332**capacitance of a unit cube**Hwang & Mascogni – “Electrical capacitance of a unit cube” – Journal of Applied Physics 3798-3802 (2004).**2a**C a**note the charge**density at the corner**a**h Sava V. Savov February 1, 2004**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]**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**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**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**MATLAB**• [V] = [matrix] [Q] V V r r x x**Calculate vector “V”**• clear • clf • m=input('What is the size of the matrix? ...'); • for i=1:m • v(i)=NaN; • end • v**Calculate matrix “a”**• for i=1:m • for j=1:m • a(i,j)=NaN; • end • end**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