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Skin effect formulas with imaginary numbers problems

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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:
  1. 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.

Accepted Answer

Fabio Freschi
Fabio Freschi on 4 Oct 2023
There are several problems in your script
  1. you forgot the empy space permeability
  2. you are misinterpretating the concept of complex number in AC analysis
  3. minor issues
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
Pawel Kolodziejski
Pawel Kolodziejski on 30 Oct 2023
@Fabio Freschi I totally can provide a scan of that paper but its in polish i can translate it for you hope that helps:
4.3 A conductor of circular cross section
Let us consider the phenomenon of displacement of current in a sufficiently long (theoretically infinitely long) conductor of circular cross section placed in air. The radius of the wire is R. Let us assume that a sinusoidally alternating current of complex amplitude L flows through it. In this case, the vector of current density has only one component in accordance with the direction of the wire. The equation describing the distribution in the conductor of the complex amplitude of the current density takes the form
(d^2 J)/dr^2 + 1/r * dJ * dJ/dr - j*omega*gamma*mu*J = 0,
My comment: J - is described with an underscore, because it is a complex number
where r <0, R >.
In the case under consideration, the solution of the differential equation (4.7) is the expression
(4.8) "skin effect formula for oval conductor" picture
where: J0(x) - Bessel function of the first kind of zero order, J1(x) - Bessel function of the second kind of zero order. The values of the Bessel function are complex numbers, and for small values of the arguments we can read them in mathematical tables or easily determine them numerically. The distribution of the values of the amplitude modulus of the current density vector as a function of the radius of the conductor for pulsation values (pulsations in the range of single multiples of pulsations and (for different lnaterials is shown in Figure 4.5.
"Fig. 4.5"
4.5 Theoretical distribution of the modulus of the amplitude of the current density in a circular cable (copper and steel) placed in air
In the case of high frequency of current flowing in the conductor or when the conductor is made of ferromagnetic, we speak of a strong displacement (skinning) phenomenon. Such a case is modeled by the penetration of a plane wave into an unconfined environment (in the absence of a reflected wave). We can omit the reflected wave from the calculation if the equivalent penetration depth, defined by the relation
delta = sqrt(2/omega*gamma*mu (4.9)
omega = 2*pi*f,
gamma - electrical conductivity [S/m],
mu - electrical permeability [F/m].
Is significantly more than the prornien of the conductor.
In terms of the full paper it is (polish version in attachment - scan of the skin effect laboratory excercise):
"Laboratorium podstaw elektromagnetyzmu" - praca zbiorowa J. Starzyński wydawnictwo OWPW
In English it would be:
Laboratory of the basics of electromagnetism
Publisher: OWPW
Fabio Freschi
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
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'])

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