Dodd & Deeds Equation - symbolic integration

Hi

I'm trying to write some code to solve the Dodd and Deeds equation for an electromagnetic coil above a material, to produce an impedance plane diagram (see attachments).

I'm having some difficulty using symbolic integration, and producing something that I can plot. The code that I have so far is below:

% Set up values
w = 100e6; % frequency
%w = 0:200:10000;
u0 = 1.257e-6; % magnetic permeability 
u = 1.26e-4; % permeability 
ur = u/u0; % relative permeability 
N = 200; % no. of coil turns 
r1 = 0.5; % coil inner radius
r2 = 1; % coil outer radius
l = 0.5e-3; % coil height
l0 = 1e-3;  % lift off
sigma = 6.21e6; % conductivity

syms a x 

xmin = a*r1;
xmax = a*r2;
f = @(x) x*besselj(1,x);
J = int(f,x,xmin,xmax);

%a1 = {};
Z5 = [];
for i = 1:length(w)
    Z5_(i) = (w(i)*pi*u0*N^2)/(l^2*(r2-r1)^2);
    %a1_ = @(a) sqrt(a^2 + 1j*w(i)*ur*u0*sigma);
    Z5 = [Z5;Z5_(i)];
    %a1 = {a1;a1_};
end 


a1 = @(a) sqrt(a^2 + 1j*w(i)*ur*u0*sigma);

R = @(a) (a*ur - a1(a))/(a*ur + a1(a));

Integrand = @(a) ((J^2)/a^6)*(2*(a*l + exp(-a*l) - 1) + ((exp(-a*(l+l0))...
    - exp(-a*l0)).^2)*(R(a)));


% limits
x1 = 0; 
x2 = Inf;
I = int(Integrand,a,x1,x2);

Z = Z5*I;

Answer 1

[benjamin ducharne]

Dear Oliver, I'll tried to run your code. Instead of Inf for the X2 values I used real values that I increased gradually to check on the influence of this parameter and it seems that beyond 10000 the results remain the same.

I used vpa to convert the symbolic values into numerical ones:

result = vpa (Z_final);

In the images you uploaded we have no information about the coil number of turns and by the way can you give us some information about the origin of the images (book ?)