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

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
    end
    
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')

Answer 1

[Ravi]

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.

1. Define relative permittivity "epsilon_r".
2. 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".
3. Define four variables, "diel_x_start", "diel_x_end", "diel_y_start", and "diel_y_end", denoting a square region for the dielectric.
4. 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