How can I add a square dielectric object into parallel/coplanar capacitance sensor?

Question

Cobus Labuschagne 2022-6-18

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1743185-how-can-i-add-a-square-dielectric-object-into-parallel-coplanar-capacitance-sensor

回答： Ravi 2024-1-30

My program is below and I am trying to add in a dielctric object into different places outside the capacitor.

I would appreciate any help on adding same.

I understand that Neumann boundaries can be added but I am not sure how to add the relative permittivity of the dielectric.

Thank you very much.

% 2D Poisson Equation for standard parallel plate capacitor using finite differences method

% Poisson eq : d²V(x,y)/dx²+d²V(x,y)/dy²= - ρ/ε

%-------------------------------------------------------------------------%

% PARAMETERS USED IN THIS PROGRAM

%-------------------------------------------------------------------------%

% E = Total electric field matrix using Poisson's equation

% V = Potential matrix

% ρ = volume density of free charges

% ε = permittivity of dielectric medium

%-------------------------------------------------------------------------%

% INITIAL PARAMETERS USED

%-------------------------------------------------------------------------%

Nx = 201; % Number of X-grids

Ny = 201; % Number of Y-grids

mpx = ceil(Nx/2); % Mid-point of x

mpy = ceil(Ny/2); % Mid point of y

Ni = 750; % Number of iterations for the Poisson solver

V = zeros(Nx,Ny); % Potential (Voltage) matrix

rho = zeros(Nx,Ny) % Linear charge density in coulomb per meter

rho = 201×201

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

epsilon_0 = 8.854E-12 % Permittivty of free space (F/m)

epsilon_0 = 8.8540e-12

T = 0; % Top-wall potential

B = 0; % Bottom-wall potential

L = 0; % Left-wall potential

R = 0; % Right-wall potential

%-------------------------------------------------------------------------%

% Initial Edge potentials

%-------------------------------------------------------------------------%

V(1,:) = L;

V(Nx,:) = R;

V(:,1) = B;

V(:,Ny) = T;

%-------------------------------------------------------------------------%

% Initial Corner potentials

%-------------------------------------------------------------------------%

V(1,1) = 0.5*(V(1,2)+V(2,1));

V(Nx,1) = 0.5*(V(Nx-1,1)+V(Nx,2));

V(1,Ny) = 0.5*(V(1,Ny-1)+V(2,Ny));

V(Nx,Ny) = 0.5*(V(Nx,Ny-1)+V(Nx-1,Ny));

%-------------------------------------------------------------------------%

length_plate = 70; % Length of plate in terms of number of grids

lp = floor(length_plate/2);

position_plate = 15; % Position of plate on x axis

pp1 = mpx+position_plate;

pp2 = mpx-position_plate;

for E = 1:Ni % Number of iterations

for i=2:Nx-1

for j=2:Ny-1

% The next two lines are meant to force the matrix to hold the

% potential values for all iterations

V(pp1,mpy-lp:mpy+lp) = 100;

V(pp2,mpy-lp:mpy+lp) = -100;

V(i,j)=0.25*(V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1) + rho(i,j)./epsilon_0);

end

% Take transpose for proper x-y orientation

%V = V';

[Ex,Ey]=gradient(V);

Ex = -Ex;

Ey = -Ey;

% Electric field Magnitude

E = sqrt(Ex.^2+Ey.^2);

x = (1:Nx)-mpx;

y = (1:Ny)-mpy;

% Charge

Q = Ex.*x.*sqrt(x.^2 + (lp).^2)/9E9

Q = 201×201

1.0e-05 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0002 -0.0002

% Contour Display for electric potential

figure(1)

contour_range_V = -101:0.5:101;

contour(x,y,V,contour_range_V,'linewidth',0.5);

axis([min(x) max(x) min(y) max(y)]);

colorbar('location','eastoutside','fontsize',14);

xlabel('x-axis in meters','fontsize',14);

ylabel('y-axis in meters','fontsize',14);

title('Electric Potential distribution, V(x,y) in volts','fontsize',14);

h1=gca;

set(h1,'fontsize',14);

fh1 = figure(1);

set(fh1, 'color', 'white')

% Quiver Display for electric field Lines

figure(2)

contour(x,y,E,'linewidth',0.5);

hold on, quiver(x,y,Ex,Ey,2)

title('Electric field Lines, E (x,y) in V/m','fontsize',14);

axis([min(x) max(x) min(y) max(y)]);

colorbar('location','eastoutside','fontsize',14);

xlabel('x-axis in meters','fontsize',14);

ylabel('y-axis in meters','fontsize',14);

h3=gca;

set(h3,'fontsize',14);

fh3 = figure(3);

set(fh3, 'color', 'white')

hold off

% Graph for capacitance

figure(3)

surf(x,y,Q)

axis([min(x) max(x) min(y) max(y)]);

colorbar('location','eastoutside','fontsize',14);

xlabel('x-axis in meters','fontsize',14);

ylabel('y-axis in meters','fontsize',14);

title('Capacitance distribution, C in C/m','fontsize',14);

h1=gca;

set(h1,'fontsize',14);

fh1 = figure(1);

set(fh1, 'color', 'white')

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Answer 1

Ravi 2024-1-30

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1743185-how-can-i-add-a-square-dielectric-object-into-parallel-coplanar-capacitance-sensor#answer_1399676

在 MATLAB Online 中打开

Hi Cobus Labuschagne,

When a dielectric is added, the permittivity of the medium changes. If the relative permittivity of the medium is

and permittivity of dielectric is

, then the effective permittivity is

.

This is how you can add a dielectric. Follow the steps below to add a square dielectric.

Define relative permittivity "epsilon_r".
Define "epsilon_matrix", a matrix that contains the permittivity values at every grid point. Initialize the matrix to contain all entries equal to "epsilon_0".
Define four variables, "diel_x_start", "diel_x_end", "diel_y_start", and "diel_y_end", denoting a square region for the dielectric.
Update the entries of the "epsilon_matrix" in the dielectric region.

epsilon_matrix(diel_x_start:diel_x_end, diel_y_start:diel_y_end) = epsilon_r * epsilon_0;

5. Modify the potential calculation equation to incorporate the epsilon_matrix.

V(i,j) = 0.25 * (V(i+1,j) + V(i-1,j) + V(i,j+1) + V(i,j-1) + rho(i,j) / epsilon_matrix(i,j));

In a nutshell, define the relative permittivity, specify the region of dielectric, and update the potential calculation to account for varying permittivity.

Hope this answer helps.

Regards,

Ravi