FACTA UNIVERSITATIS (NI S)SER.: ELEC. ENERG. 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 numeri-cally. The electromagnetic field in conductors has been calculated numerically using aprogram based on finite elements method (FEM). The results, presented in a graphicalform, are compared with the similar analytical results, assuming some approxima-tions.

Keywords: Low-frequency skin effect, free space conductor, finite element method,FEM.

1 Introduction

AS an alternating current generates an alternating flux in the conductor material,a so called skin effect is occurred. This phenomenon leads to an unevendistribution of current density in the cross section area of the conductor and it isknown as the skin effect, [13].

The skin effect increases the resistance of the conductors and thus also canproduce significant losses in the conductor, and is, therefore, of interest in elec-trical equipments and especially in electric machines. In most of cases, this is anundesired phenomenon.

Present paper analyses this effect for a free space conductor and presents somenumerical computations for conductors of different shapes and different cross sec-tions area. Both, the magnetic field and current density distributions, on the crosssection of the conductor are presented using the FEM.

Manuscript received on November 19, 2009. An earlier version of this paper was presented at9th International Conference on Applied Electromagnetics August 31 - September 02, 2009, Nis,Serbia.

M. Greconici is with Politehnica University of Timisoara, Physical Foundation of Enginer-ing 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

208 M. Greconici, G. Madescu, and M. Mot:

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 inthe internal path of the conductor:

EEE =EEE t (1)

which in turn creates an eddy current density:

JJJ = EEE (2)In order to reach an analytical solution, the problem must be simplified con-

siderably and thus approximate relations are obtained. Such relations introduce inconsequence some errors in the evaluation of the skin effect in most of the cases, butsuch 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 classicalcurves or approximate coefficients that are used during the classical design process.

Some examples of the low-frequency skin effect in conductors of finite size andfree space are developed in present paper, using the Opera 12 software (of VectorFields) based on the finite element method (FEM).

In the conductor domain, the magnetic potential vector satisfied the relation:

( 1 AAA) = JJJ0AAA t (3)

in which JJJ0 is the prescribed current density and (AAA/ t) is the induced currentdensity in conductor.

Outside the conductor, a Laplace equation is satisfied:

2AAA = 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 consid-ered during the analyzed models.

The used material of the conductor is copper with = 0 and conductivity = 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 = 3106 A/m2, and f = 50 Hz. The penetration deepfor the analyzed models is:

=1

pi f 0Using this data, the field lines and the current density distribution for the circu-

lar 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 theconductor. The lowest value of the current density is about Jmin = 2.25106 A/m2in the center of the conductor, while the largest value is about Jmax = 5.7106 A/m2on 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) aregenerated 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, theresistance in alternating current, Rac, becomes larger as the resistance in directcurrent, Rdc. The coefficient of ac-resistance increasing (KR) is the ratio of thetwo resistances:

KR =RacRdc

=PacPdc

(5)

where Pac is the ac dissipated power in unit length of conductor, and Pdc is the dcdissipated power in unit length of conductor.

210 M. Greconici, G. Madescu, and M. Mot:

(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, versusconductor cross section area, S, for a circular shaped conductor. For this ideal andunique 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 theproblem, while the curve 2 has been drawn using the numerical simulation. Thecomparison between the two curves shows very good agreements, pointing out theaccurate of model used by the numerical computation analyzes (FEM).

Fig. 4 depicts the dependence coefficient of ac-resistance increasing, KR, versusconductor 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 proposedby Press, [47], while the curve 2 has been drawn using the numerical simulation.The differences between the two curves are explained by the simplified hypothesisthat Press has considered to achieve an analytical solution (closed form solution).In this work (1916), Press supposed that the flux density line is overlapped withthe periphery of the rectangular conductor: [4, Fig. 227], [6, Fig. 5.3.8]. Thishypothesis can not be accepted today. In this case, we estimate that the numericalresults (curve 2) are more accurate as the analytical one (curve 1) obtained by Pressusing approximate formulas.

Fig. 5. Dependence KR over cross section area S of the rectangle shaped conduc-tor 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, versusconductor cross section area, S, for a rectangle shaped conductor with the side ratiob/a = 2.

Fig. 6 depicts the dependence coefficient of ac-resistance increasing versusconductor 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 conduc-tor with the ratio b/a = 5.

The comparison between the curves proposed by Press for the rectangularshaped conductors and the curves obtained using the numerical simulation (Fig.4-Fig. 6), shows major differences, especially in Fig. 6. The differences betweenthe two types of curves are larger as the side ratio b/a of the rectangular conductoris higher. It happens because the approximation made by Press, considering theconductor 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 subcon-ductors that fill the same total area. The material of the conductors is cooper andthe prescribed rms value of the current density that flows through the conductors isJ0 = 3106 A/m2, and frequency f = 50 Hz.

In Fig. 7a is drawn the current density distribution for a single conductor. Therange of the current density distribution is between 2.7 A/mm2 around conductorcenter and 4.3 A/mm2 in the corners of the conductor.

The same current density distribution is generated by symmetry of two subcon-ductors, Fig. 8a and Fig. 8b, or four subconductors, Fig. 7b.

Skin Effect Analysis in a Free Space Conductor 213

(a) (b)Fig. 7. Equivalent current density distribu-tions in the same total area: (a) a single con-ductor; (b) four subconductors

(a) (b)Fig. 8. Equivalent current density distribu-tions in the case of two subconductors placedin two different modes.

The equivalent current density distribution from Fig. 7 and Fig. 8 is accordingwith [5, 6] and is obtained only for two or four subconductors. If the numberof subconductors is greater than four, unequivalent current density distributions intotal area is obtained.

The Fig. 9, Fig. 10 and Fig. 11 point out an unsymmetrical distribution of thecurrent density using a different number of conductors. The current density rangeis between 2.6 A/mm2 and 3.5 A/mm2.

(a) (b)Fig. 9. Current density distribution in the caseof six subconductors placed in two differentmodes.

(a) (b)Fig. 10. Current density distribution in thesame total area divided in: (a) eight subcon-ductors; (b) nine subconductors.

214 M. Greconici, G. Madescu, and M. Mot:

(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 conductorof some electrical equipment, high power transformers or some end bares windingsof electrical machines.

The numerical results show that the ac-resistance of a free space conductorestimated with FEM is higher than ac-resistance calculated with classical approx-imated relations recommended in practical design. In consequence the additionalcopper losses are larger and the efficiency of the actual machine becomes lowerthan the estimated one.

The aim of the present paper is to show that some coefficients and curves usedin electrical engineering design must be reconsidered, using modern techniqueslike numerical computation of the electromagnetic field with finite element method(FEM).

References[1] M. Chari and Z. Csendes, Finite element analysis of the skin effect in current carrying

conductors, I.E.E.E. Trans. on Magnetics, vol. MAG-13, no. 5, pp. 11251127, 1977.[2] A. Konrad, Integrodifferential finite element formulation of two-dimensional steady-

state skin effect problems, I.E.E.E. Trans. on Magnetics, vol. MAG-18, no. 1, pp.284292, 1982.

[3] K. Preis, O. Biro, H. Reisinger, K. Papp, and I. Ticar, Eddy current losses in large aircoils with layered stranded conductors, I.E.E.E. Trans. on Magnetics, vol. 44, no. 6,pp. 13181321, 2008.

[4] R. Richter, Elektrische Maschinen. Basel: Verlag Birkhauser, 1951, Erster Band.[5] K. Vogt, Elektrische Maschinen Berechnung rotierender elektrischer Maschinen.

Berlin: Veb Verlag Technik, 1974.

Skin Effect Analysis in a Free Space Conductor 215

[6] G.Muller, K. Vogt, and B. Ponik, Berechnung Elektrischer Maschinen. Wiley-VCHVerlag, 2008.

[7] I. D. Sabata, Fundamentals of Electrical Engineering. Timisoara, Romania: IPTVPublishing House, 1974, vol. 2, (in Romanian).

[8] Opera 12, 2D Reference Manual, 2008.