- you forgot the empy space permeability
- you are misinterpretating the concept of complex number in AC analysis
- minor issues

# Skin effect formulas with imaginary numbers problems

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Pawel Kolodziejski
on 3 Oct 2023

Edited: Fabio Freschi
on 6 Nov 2023

Dear MathWorks Community,

As the summmary points out I am dealing with "Skin effect formulas with imaginary numbers problems". I am trying to draw graphical representation of skin effect current density J, using two formulas:

- Photos: "current density - copper conductor formula", "copper conductor formula 2", "teta2 to copper conductor formula"

J - current density [A/m^2] (imaginary value)

I - current [A] (imaginary value)

w = 2*pi*f, where pi = 3,1415.... , f = 50 Hz or 150 Hz - frequency

y - variable of the distance from one egde of the conductor paralell to y-axis = 0 to 0.06 [m]

a,b - dimensions of the copper conductor = 60 x 5 [mm] = 0.06 x 0.005 [m]

teta^2 (Picture "teta2 t ocopper conductor formula") (imaginary value)

2. Photo "skin effect formula for oval conductor"

J - current density [A/m^2] (imaginary value)

I - current [A] (imaginary value)

, where: r ∈ < 0, R >

R - the length of the radius of a circular conductor [m],

J¬0(x) - Bessel function of the first kind of zero order,

J¬1(x) - Bessel function of the second kind of zero order,

ω = 2πf - circular frequency [rad/s], where f is the frequency expressed in [Hz],

γ - electrical conductivity [S/m],

μ - electrical permeability [F/m].

The above formulas come from a script of electromagnetism laboratory. The excercise is called "Skin effect", which purpose is to show this phenomenon for two conductors with diffrent shapes and made of different materials tested in different environments while applying 50 Hz or 150 Hz frequency. (1) copper conductor with rectangular cross section in air or ferromagnetic environment and (2) steel conductor with oval cross section only in air environement. Their current density can be measured with a theoretical formulas included in attachment.

For measurements I have gathered voltage values across the whole cross section of each conductor with different surrounding environment. I also have the current values for each frequency, conductor and surrounding environment combination. All the other data are constant values.

My question is how can i successfully apply this formula knowing that the data is supported with imaginery numbers. I have attemted to apply the formulas above to solve my task, but i am doing something wrong, I am attaching the code in the attachment. If anyone can help me out I would really appreciate it.

##### 0 Comments

### Accepted Answer

Fabio Freschi
on 4 Oct 2023

There are several problems in your script

I tried to rewrite your code for the first case, the second one should be simple. Note that I've used sigma for the electrical conductivity, to avoid misunderstanding with the function Γ

% params

f = 50; % (Hz)

sigma = 58e6; % (S/m)

mir = 0.99999; % (-)

mi0 = 4e-7*pi; % (H/m)

I = 785; % (A)

zmax = 0.06; % (m)

a = 0.06; % (m)

b = 0.005; % (m)

% angular frequency

w = 2*pi*f;

% Gamma squared/Gamma

Gamma2 = 1j*w*sigma*mi0*mir;

Gamma = sqrt(Gamma2);

% z coordinate

z = linspace(0,zmax,100);

% current density (complex number)

J = I*Gamma*cosh(Gamma*z)./(b*sinh(Gamma*a));

figure, hold on

plot(z,real(J))

plot(z,imag(J))

legend('real(J)','imag(J)')

##### 10 Comments

Fabio Freschi
on 6 Nov 2023

Edited: Fabio Freschi
on 6 Nov 2023

If you use "reasonable" values for the params, values look reasonable as well.

I slightly changed the notation to use formulas more familiar to me

% params

f = [50 150 450]; % (Hz)

Sigma = 56e6; % (S/m)

MuR = 1; % (-)

Mu0 = 4e-7*pi; % (H/m)

I = 785; % (A)

r0 = 0.02; % (m)

% r coordinate

r = linspace(0,r0,100);

figure, hold on

legend

for i = 1:length(f)

% angular frequency

w = 2*pi*f(i);

% penetration depth

delta = sqrt(2/(w*MuR*Mu0*Sigma));

% k

k = (1-1j)/delta;

% current density (complex number)

J = k*I/(2*pi*r0)*besselj(0,k*r)./besselj(1,k*r0);

plot(r,abs(J),'DisplayName',['abs(J) @',num2str(f(i)), 'Hz'])

end

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