FACTA UNIVERSITATIS (NI Š)
S ER .: E LEC . E NERG . vol. 23, No. 2, August 2010, 207-215.
Skin Effect Analysis in a Free Space Conductor
Marian Greconici, Gheorghe Madescu, and Martian Mot
Abstract: The low-frequency skin effect in free space conductors is analyzed numerically. The electromagnetic field in conductors has been calculated numerically using a
program based on finite elements method (FEM). The results, presented in a graphical
form, are compared with the similar analytical results, assuming some approximations.
Keywords: Low-frequency skin effect, free space conductor, finite element method,
FEM.
1 Introduction
an alternating current generates an alternating flux in the conductor material,
a so called skin effect is occurred. This phenomenon leads to an uneven
distribution of current density in the cross section area of the conductor and it is
known as the skin effect, [1–3].
The skin effect increases the resistance of the conductors and thus also can
produce significant losses in the conductor, and is, therefore, of interest in electrical equipments and especially in electric machines. In most of cases, this is an
undesired phenomenon.
Present paper analyses this effect for a free space conductor and presents some
numerical computations for conductors of different shapes and different cross sections area. Both, the magnetic field and current density distributions, on the cross
section of the conductor are presented using the FEM.
A
S
Manuscript received on November 19, 2009. An earlier version of this paper was presented at
9th International Conference on Applied Electromagnetics August 31 - September 02, 2009, Niš,
Serbia.
M. Greconici is with Politehnica University of Timisoara, Physical Foundation of Enginering Department, V. Parvan 2, 300223-Timisoara, Romania (e-mail: marian.greconici
@et.upt.ro). G. Madescu and M. Mot are with Romanian Academy Timisoara Branch,
M. Viteazul Bl.
24, 300223-Timisoara, Romania (e-mails: [gmadescu, martian]
@d109lin.utt.ro).
207
M. Greconici, G. Madescu, and M. Mot:
208
2 Field Equations and Finite Element Formulation
If a conductor with a cross-section area large enough carries an alternating current,
according to the Faradays induction law, an electric field strength curl is induced in
the internal path of the conductor:
E =−
∂E
∂t
(1)
which in turn creates an eddy current density:
J = σE
(2)
In order to reach an analytical solution, the problem must be simplified considerably and thus approximate relations are obtained. Such relations introduce in
consequence some errors in the evaluation of the skin effect in most of the cases, but
such approximate relations are very useful in technical design area. For example,
to estimate the skin effect in a free space conductor of some electrical equipment,
there occur some errors in additional losses calculations derived from the classical
curves or approximate coefficients that are used during the classical design process.
Some examples of the low-frequency skin effect in conductors of finite size and
free space are developed in present paper, using the Opera 12 software (of Vector
Fields) based on the finite element method (FEM).
In the conductor domain, the magnetic potential vector satisfied the relation:
∂A
1
A) = J 0 − σ
∇ × ( ∇ ×A
µ
∂t
(3)
in which J 0 is the prescribed current density and σ (∂ A /∂ t) is the induced current
density in conductor.
Outside the conductor, a Laplace equation is satisfied:
∇2A = 0
(4)
The boundary of the analyzed models, of circular shape, was set as a field line
far enough of the conductor.
3 Numerical Examples
Two types of cross-sections, circular shaped and square shaped, have been considered during the analyzed models.
The used material of the conductor is copper with µ = µ0 and conductivity
σ = 50 × 106 S/m and the prescribed rms value of the current density that flows
Skin Effect Analysis in a Free Space Conductor
209
through the conductor is J0 = 3 × 106 A/m2 , and f = 50 Hz. The penetration deep
for the analyzed models is:
1
σ=√
π f µ0 σ
Using this data, the field lines and the current density distribution for the circular shaped conductor of r = 25 mm radius, generated by Opera, are presented in Fig.
1. In Fig. 1b is presented the current density distribution along a diameter of the
conductor. The lowest value of the current density is about Jmin = 2.25 × 106 A/m2
in the center of the conductor, while the largest value is about Jmax = 5.7×106 A/m2
on the periphery of the conductor.
(a)
(b)
Fig. 1. Field (a) and current (b) density distribution in circular shaped conductor.
The field lines (Fig. 2a), and the current density distribution (Fig. 2b) are
generated for a square shaped conductor with the side l = 40 mm.
4 Conductor AC-Resistance
A parameter of interest is the ac-conductor resistance. Do to the skin effect, the
resistance in alternating current, Rac, becomes larger as the resistance in direct
current, Rdc. The coefficient of ac-resistance increasing (KR) is the ratio of the
two resistances:
Pac
Rac
=
(5)
KR =
Rdc Pdc
where Pac is the ac dissipated power in unit length of conductor, and Pdc is the dc
dissipated power in unit length of conductor.
M. Greconici, G. Madescu, and M. Mot:
210
(a)
(b)
Fig. 2. Field (a) and current (b) density distribution in square shaped conductor.
In Fig. 3 has been drawn the coefficient of ac-resistance increasing, KR , versus
conductor cross section area, S, for a circular shaped conductor. For this ideal and
unique case there is known the exact solution of the field problem.
Fig. 3. Dependence KR over cross section area S of the circular shaped conductor.
The curve 1 has been drawn using the analytical well known solution of the
problem, while the curve 2 has been drawn using the numerical simulation. The
comparison between the two curves shows very good agreements, pointing out the
accurate of model used by the numerical computation analyzes (FEM).
Fig. 4 depicts the dependence coefficient of ac-resistance increasing, KR , versus
conductor cross section area, S, for a square shaped conductor.
Skin Effect Analysis in a Free Space Conductor
211
Fig. 4. Dependence KR over cross section area S of the square shaped conductor.
The curve 1 has been drawn using an approximate analytical solution proposed
by Press, [4–7], while the curve 2 has been drawn using the numerical simulation.
The differences between the two curves are explained by the simplified hypothesis
that Press has considered to achieve an analytical solution (closed form solution).
In this work (1916), Press supposed that the flux density line is overlapped with
the periphery of the rectangular conductor: [4, Fig. 227], [6, Fig. 5.3.8]. This
hypothesis can not be accepted today. In this case, we estimate that the numerical
results (curve 2) are more accurate as the analytical one (curve 1) obtained by Press
using approximate formulas.
Fig. 5. Dependence KR over cross section area S of the rectangle shaped conductor with the ratio b/a = 2.
212
M. Greconici, G. Madescu, and M. Mot:
Fig. 5 depicts the dependence coefficient of ac-resistance increasing, KR , versus
conductor cross section area, S, for a rectangle shaped conductor with the side ratio
b/a = 2.
Fig. 6 depicts the dependence coefficient of ac-resistance increasing versus
conductor cross section area for a rectangle shaped conductor with the ratio b/a =
5.
Fig. 6. Dependence KR over cross section area S of the rectangle shaped conductor with the ratio b/a = 5.
The comparison between the curves proposed by Press for the rectangular
shaped conductors and the curves obtained using the numerical simulation (Fig.
4-Fig. 6), shows major differences, especially in Fig. 6. The differences between
the two types of curves are larger as the side ratio b/a of the rectangular conductor
is higher. It happens because the approximation made by Press, considering the
conductor surface as a magnetic field line.
5 Skin Effect in Some Subconductor Groups Placed in Free Space
Using the FEM, has been analyzed the skin effect of identical cross section subconductors that fill the same total area. The material of the conductors is cooper and
the prescribed rms value of the current density that flows through the conductors is
J0 = 3 × 106 A/m2 , and frequency f = 50 Hz.
In Fig. 7a is drawn the current density distribution for a single conductor. The
range of the current density distribution is between 2.7 A/mm2 around conductor
center and 4.3 A/mm2 in the corners of the conductor.
The same current density distribution is generated by symmetry of two subconductors, Fig. 8a and Fig. 8b, or four subconductors, Fig. 7b.
Skin Effect Analysis in a Free Space Conductor
(a)
(b)
Fig. 7. Equivalent current density distributions in the same total area: (a) a single conductor; (b) four subconductors
213
(a)
(b)
Fig. 8. Equivalent current density distributions in the case of two subconductors placed
in two different modes.
The equivalent current density distribution from Fig. 7 and Fig. 8 is according
with [5, 6] and is obtained only for two or four subconductors. If the number
of subconductors is greater than four, unequivalent current density distributions in
total area is obtained.
The Fig. 9, Fig. 10 and Fig. 11 point out an unsymmetrical distribution of the
current density using a different number of conductors. The current density range
is between 2.6 A/mm2 and 3.5 A/mm2 .
(a)
(b)
Fig. 9. Current density distribution in the case
of six subconductors placed in two different
modes.
(a)
(b)
Fig. 10. Current density distribution in the
same total area divided in: (a) eight subconductors; (b) nine subconductors.
M. Greconici, G. Madescu, and M. Mot:
214
(a)
(b)
Fig. 11. Current density distribution in the same total area divided in:
(a) twelve subconductors; (b) sixteen subconductors.
6 Conclusion
This paper presents the low-frequency skin effect in a free space copper conductor
of some electrical equipment, high power transformers or some end bares windings
of electrical machines.
The numerical results show that the ac-resistance of a free space conductor
estimated with FEM is higher than ac-resistance calculated with classical approximated relations recommended in practical design. In consequence the additional
copper losses are larger and the efficiency of the actual machine becomes lower
than the estimated one.
The aim of the present paper is to show that some coefficients and curves used
in electrical engineering design must be reconsidered, using modern techniques
like numerical computation of the electromagnetic field with finite element method
(FEM).
References
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Skin Effect Analysis in a Free Space Conductor
215
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