Skin effect formulas with imaginary numbers problems

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.

Answer 1

[Fabio Freschi]

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

[Pawel Kolodziejski]

Should the value of J be lower than 0 for imaginary values? that seems suspicious.

[Fabio Freschi]

I simply have implemented your formulas: if anything is suspicious, can you spot the problem in my code? As I told you, I think you don't have a clear idea of phasors and complex numbers in AC analysis. If you want to see the "perfect" exponential decay, you must plot the magnitude of J

[Pawel Kolodziejski]

Secondly, although i really appreciate the formula you wrote can u leave a feedback regarding second formula for oval steel conductor?

[Fabio Freschi]

Assuming that and are the bessel functions of first kind

% params
f = 50;         % (Hz)
sigma = 35e5;   % (S/m)
mir = 1000;     % (-)
mi0 = 4e-7*pi;  % (H/m)
I = 785;        % (A)
rmax = 0.005;   % (m)
R = 0.02;       % (m)

% angular frequency
w = 2*pi*f;

% Gamma squared/Gamma
Gamma2 = -1j*w*sigma*mi0*mir;
Gamma = sqrt(Gamma2);

% r coordinate
r = linspace(0,rmax,100);

% current density (complex number)
J = I*Gamma*besselj(0,Gamma*r)./(2*pi*R*besselj(1,Gamma*R));
figure, hold on
plot(r,real(J))
plot(r,imag(J))
legend('real(J)','imag(J)')

[Pawel Kolodziejski]

@Fabio Freschi Thank you for providing quick and high quality answers these are are really helpful codes.

As a follow-up I would like to ask you if it's possible to combine the real and imaginary values of those graphs into one diagram using the square root of a sum of squared values of real and imaginary based on theoretical formula J = sqrt((J_real)^2+(J_imag)^2) ?

[Pawel Kolodziejski]

@Fabio Freschi Although the formula for the second code including the shape of the curve looks good the values are totally off. I am sending a photo of simulated graph that shows correct values. How come we see such a huge difference?

[Fabio Freschi]

If you want to plot the magnitude of J, simply use

plot(z,abs(J))

[Fabio Freschi]

Can you provide more insight regarding the formulas for oval conductor? The scan of the page or the complete reference would be of help. I guess there is a problem in the interpretation of the parameters.

In any case, using numbers close to the ones used in the graph you attached, results seems reasonable

clear variables, close all

% params
f = [50 150 450];         % (Hz)
sigma = 56e6;   % (S/m)
mir = 1;        % (-)
mi0 = 4e-7*pi;  % (H/m)
I = 785;        % (A)
rmax = 0.02;    % (m)
R = 0.02;       % (m)

% open figure
figure, hold on

for i = 1:length(f)
    % angular frequency
    w = 2*pi*f(i);

    % Gamma squared/Gamma
    Gamma2 = -1j*w*sigma*mi0*mir;
    Gamma = sqrt(Gamma2);
    % r coordinate
    r = linspace(0,rmax,100);
    % current density (complex number)
    J = I*Gamma*besselj(0,Gamma*r)./(2*pi*R*besselj(1,Gamma*R));
    plot(r,abs(J),'DisplayName',[num2str(f(i)),' Hz'])
end
legend

[Pawel Kolodziejski]

@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

http://www.wydawnictwopw.pl/index.php?s=karta&id=184

In English it would be:

Laboratory of the basics of electromagnetism

group work

J. Starzyński (ed.)

Publisher: OWPW

[Fabio Freschi]

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