COMPEL - The international journal for computation andmathematics in electrical and electronic engineeringFINITE-ELEMENT SOLUTION OF STEADY-STATE SKIN-EFFECT PROBLEMS IN STRAIGHTFLAT CONDUCTORSG. COSTACHE

Article information:To cite this document:G. COSTACHE, (1983),"FINITE-ELEMENT SOLUTION OF STEADY-STATE SKIN-EFFECTPROBLEMS IN STRAIGHT FLAT CONDUCTORS", COMPEL - The international journal forcomputation and mathematics in electrical and electronic engineering, Vol. 2 Iss 2 pp. 35 - 39Permanent link to this document:http://dx.doi.org/10.1108/eb009974

Downloaded on: 13 July 2015, At: 12:03 (PT)References: this document contains references to 0 other documents.To copy this document: permissions@emeraldinsight.comThe fulltext of this document has been downloaded 24 times since 2006*

Users who downloaded this article also downloaded:Mary B. Teagarden, Ellen A. Drost, Mary Ann Von Glinow, (2005),"The Life Cycle of AcademicInternational Research Teams: Just When You Thought Virtual Teams Were All the Rage HereCome the AIRTS!", Advances in International Management, Vol. 18 pp. 303-336

Access to this document was granted through an Emerald subscription provided by emerald-srm:478489 []

For AuthorsIf you would like to write for this, or any other Emerald publication, then please use our Emeraldfor Authors service information about how to choose which publication to write for and submissionguidelines are available for all. Please visit www.emeraldinsight.com/authors for more information.

About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The companymanages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as wellas providing an extensive range of online products and additional customer resources and services.

Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of theCommittee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative fordigital archive preservation.

*Related content and download information correct at time ofdownload.

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)

COMPELThe International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 2, No. 2. 35-39 1983 BOOLE PRESS LIMITED

FINITE-ELEMENT SOLUTION OF STEADY-STATE SKIN-EFFECT PROBLEMS IN STRAIGHT FLAT CONDUCTORS

G. COSTACHE University of Ottawa, Ontario, Canada

Abstract. Using an integrodifferential approach to steady-state skin effect problems, the current density distribution in straight flat conductors is solved by the finite-element method. The approach takes into account a combination of one-dimensional finite elements corresponding to the flat conductors and triangular finite elements for the remaining domain outside conductors. The results obtained for a flat conductor placed inside a ferromagnetic medium are compared with analytical solutions provided by finite Fourier transforms.

As a final output, besides current density distribution, one can calculate parameters useful to designers such a a.c. resistance and reactance of the straight flat conductors.

1. INTRODUCTION

The problems of skin effect in electrical machines and transformers are very important, especially now when new design concepts and complicated geometries are necessary for a better adjustment to the new technology. Many papers have been produced on these topics and an excellent list of references is given in [1].

In this paper, the finite-element solution of steady-state skin-effect problems in straight flat conductors inside ferromagnetic materials is presented. The approach uses an integrodifferential finite element formulation [1, 2] in terms of vector magnetic potential which for the straight conductors has only one component.

The application selected to illustrate the method is a fiat conductor placed in a ferromagnetic medium with a winding made of continuous sheets. The results obtained for current density by the finite-element method were compared with analytical results provided by finite Fourier transform [3].

The finite-element approach has the advantage that it can be applied to any arbitrary shape of the conductor, the method not being restricted by the geometry as usually happens when dealing with analytical techniques.

2. FORMULATION OF THE PROBLEM

The geometry selected to be studied represents the skin-effect problems for a transformer having primary windings made of sheets. A cross section of the transformer window is shown in Fig. 1.

For simplicity, let us consider one flat conductor representing the low voltage winding and carrying the a.c. current i(t) = sin t.

In this model, the high voltage winding of the transformer is considered an

35

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)

36 G. Coslache, Finite-element solution of steady-state skin-effect problems

equipotential line for the vector magnetic potential. The magnetic material of the transformer core is assumed to be of infinite magnetic permeability in order to simplify the boundary conditions of the vector magnetic potential.

As the straight flat conductor of width g is considered infinitely long, the magnetic vector potential A(, x, y) and the complex current density J(, y) have only components on the z direction.

The system of equations to be solved in terms of the vector magnetic potential is [2]

-2A = 0 (2) where Sc is the cross section of the straight conductor.

Equation (1) corresponds to the conductor region and eq. (2) to the rest of the transformer window.

The boundary conditions satisfied by the vector magnetic potential are [3]

on the ferromagnetic wals (3)

A = 0 on the conducting wall (4)

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)

G. Costache, Finite-element solution of steady-state skin-effect problems 37

3. FINITE-ELEMENT APPROACH

The general functional associated with eqs. (1) and (2) for the boundary conditions (3) and (4) is [2]

where

S being the total domain. By considering 2b g, the straight conductor can be assumed thin with a current

density depending on y direction only. With this assumption, the functional (5) becomes

In eq. (7), (A)2 is the integrand of two integrals: first integral extends over the region S (divided in triangular finite elements), while the second integral cor-responds to the flat conductor (divided in linear finite elements).

For the region outside the straight conductor, the vector magnetic potential can be approximated in each triangular finite element by a linear combination of inter-polation polynomials as follows [1]:

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)

38 G. Costache, Finite-element solution of steady-state skin-effect problems

For the elements located on the straight conductor only one-dimensional elements have been used. Their generic trial function is given by

A possible division into triangular and linear finite elements is shown in Fig. 2. The average value of the vector magnetic potential can be calculated as

where NS is the total number of linear finite elements, NC is the total number of vertices located on the linear elements, Le is the length of a generic linear element and i are coefficients obtained by integration along each element.

The trial functions (8) and (9) and the average value (10) are replaced into the functional and the derivatives with respect to node potentials are set to zero. As a result of this operation, the final system of equations is

where NTR is the total number of triangular finite elements and N is the total number of vertices where the vector magnetic potential is unknown. The above system of complex equations has the following general form:

[S1 + j 0 S 2 ]A = B (12)

which can be easily solved. Using the values obtained for the magnetic vector potential, one can obtain the

current density in the straight conductor [2]:

The solution obtained by the finite-element method has been compared with an analytical solution established by a finite Fourier transform [3] for the following geometrical dimensions: h = 0.52 m, b = 0.5 m, g = 0.001 m, L1 = 0.0109 m and L2 = 0.052 m.

The results of the normalized current density for a frequency of 60 Hz are given in Table 1.

Once the vector magnetic potential is calculated, other field quantities, such as magnetic field H and electric field E, can be evaluated. They can be used to calculate

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)

G. Costache, Finite-element solution of steady-slate skin-effect problems 39

Table 1 Normalized current densities obtained by finite element method and finite Fourier transform

y/h

0.000 0.192 0.384 0.576 0.768 0.869 0.961

Finite Element

0.9558 - j0.0421 1.0123 - j0.0432 1.0110 - j0.0431 1.0089 - j0.0362 1.0007 - j0.0242 0.9687 + j0.1134 0.9307 + j0.1590

Fourier transform

0.9613 - j0.0461 1.0117 - j0.0448 0.0101 - j0.0437 1.0091 - j0.0378 1.0008 + j0.0234 0.9691 + j0.1197 0.9315 + j0.1610

the design parameters a.c. resistance and a.c. reactance per unit length, defined by

In eqs. (14) and (15) the integral of the complex Poynting vector is taken through a cylindrical surface which contains the unit length of the straight conductor.

4. CONCLUSIONS

The integrodifferential finite-element method has been applied to the skin-effect problem in straight conductors situated inside ferromagnetic media. The results obtained by this method are in agreement with those obtained by other methods, such as finite Fourier transform.

A similar formulation can be used for the transient skin-effect problems in fiat conductors, in which situation the final solution is obtained by solving a differential system of equations, instead of an algebraic one.

REFERENCES

[1] A. K. Konrad, Integrodifferential finite element formulation of two-dimensional steady-state skin-effect problems, IEEE Trans. Magn. MAG-18(1) (1982) 284-292.

[2] G. Costache, Calculation of eddy-currents and skin-effect in nonmagnetic conductors by the finite element method, Rev. Roum. Sci. Techn. Electrotechn. & Energ. 21(3) (1976) 357-363.

[3] G. Costache, Skin-effect in straight flat conductors placed in a ferromagnetic medium, Rev. Roum. Sci. Techn. Electrotechn. & Energ. 21(2) (1976) 175-180.

Dow

nloa

ded

by U

ERJ A

t 12:

03 1

3 Ju

ly 2

015

(PT)