IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 1463

Magnetic Forces for Axisymmetric Eddy CurrentProblems Using a Hybrid FE/BE Method

V. Maló Machado1 and J. F. Borges da Silva2

Centro de Electrotecnia Teórica e Medidas Eléctricas do IST, Department of Electrical and ComputerEngineering, Instituto Superior Técnico, Technical University of Lisbon, 1049-001-Lisbon, Portugal

Centro de Electrotecnia Teórica e Medidas Eléctricas do IST, Technical University of Lisbon, 1049-001-Lisbon, Portugal

In this paper, the magnetic force acting on the currents induced in a nonmagnetic ring of finite cross section, taking into accountthe presence of eddy currents, is determined by the use of Laplace’s force formula. The required magnetic flux and current densitiesare obtained from vector potential values evaluated by a hybrid finite-element/boundary-element method. The magnetic flux density isfound from the potentials using a linear operator capable of yielding accurate results notwithstanding the discrete nature of the problem.Equivalent results obtained by the virtual-work principle are presented for comparison.

Index Terms—Eddy currents, finite element method (FEM), magnetic forces, quasistatic fields.

I. INTRODUCTION

THIS PAPER is a contribution to the magnetic force evalu-ation in axisymmetric problems with open boundaries in-

volving eddy currents.The device to which the method is applied is described in the

TEAM problem 17 “The Jumping Ring” [3].The evaluation of forces remains an important research area

in electromagnetics [5]–[7]. Different methods have been used[4]–[10] based mainly on Maxwell’s stress tensor, the virtualwork principle [4]–[8] and the method of equivalent currentsfor the Laplace magnetic force [8]–[10].

Global electromagnetic forces acting on a body may be de-scribed as the flux of Maxwell’s stress tensor over a closedsurface, placed in free space surrounding the body. In the vir-tual work principle approach, the magnetic forces are foundeither from the derivatives of energy taken at constant flux orfrom the derivatives of co-energy taken at constant current andwith respect to corresponding configuration coordinates sup-posed to undergo infinitesimal virtual displacements. Numericalevaluation of those space derivatives requires a special treatmentto avoid gross numerical errors. For this purpose, perturbationmethods or sensitivity approaches have been used dealing di-rectly with the problem results in discrete form [4]–[6]. Bothmethods are equivalent for the global force [5], the one based onMaxwell’s stress tensor where a flux surface integral is used andthe other based on the virtual work principle where an energyvolume integral is used instead. In particular, in [4] the integra-tion is performed over a frame in free space having a moveableinner boundary surrounding the body yielding a result equiva-lent to the flux of Maxwell’s stress tensor through that boundary.Another approach, equivalent to the former, under continuousfield theory, is based on the volume integration of the local forcegiven by Laplace’s formula.

The method developed in the present paper, uses Laplace’sforce formula to evaluate the force acting on a conducting

Digital Object Identifier 10.1109/TMAG.2006.871382

Fig. 1. (a) Cross section on the meridian plane of the axisymmetric magneticfield problem consisting in a magnetic core on the axis of a main coil and aconducting ring. The problem dimensions are given in [1]. The FEM mesh forthe inner region is represented in (a). In (b), the region of the conducting ring isrepresented showing the ring position coordinate � (� = 0 is set for a distancebetween the main coil and the ring equal to d = 12:5mm). In (c), the subspaceof the conducting ring is represented, showing the hexagonal mesh, surroundedby the frame used to find the force by the virtual work principle.

nonmagnetic ring, Fig. 1, due to the interaction of the currentsin the ring, induced by a variable time-harmonic magneticfield, with the magnetic field itself, the nonuniform currentdistribution in the ring cross section due to eddy currents beingtaken into account.

To allow for the open boundary nature of the field problem,the magnetic vector potential is evaluated by employing the hy-brid finite-element/boundary-element (FE/BE) method devel-oped in [1]. To avoid excessive numerical errors inherent incomputing the magnetic flux density components, through thetypical FE interpolation approach, the method described in [2]

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1464 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

has been used instead, consisting in setting up a linear operatorto find the fields directly from the calculated node potentials.The application of the method from [2] for the evaluation offorce is an original contribution.

As a check, results calculated using the principle of virtual-work following [4] are also presented.

II. MAGNETIC FIELD FORMULATION

The device under analysis, represented in Fig. 1(a), has axialsymmetry relative to the vertical axis and consists in a fixedmain coil wound around a coaxial, finite length cylindrical coreof high permeability magnetic material and a movable, nonmag-netic conducting ring, also coaxial with the core, and placedabove the main coil.

The problem is formulated in terms of the magnetic vector po-tential, described by its single azimuthal component, . Space isdivided in two regions separated by a fictitious spherical surface,centered on the symmetry axis. The inner region, containing thefield sources and material inhomogeneties is covered with a suit-able mesh and treated by the finite element method FEM, whilethe outer homogeneous region is handled by the boundary ele-ment method (BEM) using an expansion in terms of sphericalharmonics (associated Legendre functions) for the solution inthe outer region [1].

Instead of , it is more convenient to use , the magnetic fluxfunction, related to the former by

where (1)

and being, respectively, the radial and zenithal spherical co-ordinates and the radial distance to the axis.

III. MAGNETIC FORCE EVALUATIION

The total magnetic force acting on the currents induced inthe ring is given by

(2)

where is the only component of the current density in the az-imuthal direction, taken over the cross section of the ring and

is the flux function corresponding to the part of the field orig-inated by external sources outside the ring. If magnetic linearityof the material media is assumed

(3)

where corresponds to the field that would have been pro-duced by the ring currents alone acting in empty space. Thisquantity may be found by integration using Green’s functionfor a ring shaped current filament of intensity

(4)

where are, respectively, the radial and axial coordinates ofthe observation point and the coordinates of the sourcefilament on the ring cross section, and

(5)

The above integral (4) is only needed to find on the nodesover the boundary surface of a volume enclosing the ring,Fig. 1(c), where it is subtracted from the previously obtainedsolution for the total field. The resultant values are then usedas Dirichlet conditions to find the values on the nodes insidethe ring cross section, as if it were an empty space. The lattervalues in turn enable the field required by (2) to be found.

Practical advantages result from adopting this strategy. Nu-merical cancellations are avoided and an efficient method forevaluating the flux function gradient [2] becomes available.

IV. EVALUATION OF THE GRADIENT

It has been found [2] that, by suitable differentiation of theFEM equation coefficients relative to the node coordinates,linear operators may be established which, applied to the nodepotentials, will yield accurate values for the components of thepotential gradient at the nodes. The method is based on the par-ticular properties of functions satisfying Laplace’s equation andthus applies to homogeneous regions devoid of field sources. Ithas been shown that particular advantages, from the stand pointof accuracy, result if regular or nearly regular triangular meshelements, Fig. 1(c), are used to set up the FEM equations whenthis method is adopted [2].

V. NUMERICAL RESULSTS

Numerical results were obtained for the device dimensionsdescribed in [1] and [3] which, according to Fig. 1, are: radius ofthe magnetic core: 25 mm; height of the magnetic core: 250 mm;inner radius of the main coil: 40 mm; outer radius of the maincoil: 77.5 mm; height of the main coil: 100 mm; inner radiusof the conducting ring: 38.5 mm; outer radius of the conductingring: 44.85 mm; height of the conducting ring: 25.4 mm;

mm; mm. The radius of the spherical fictitioussurface (Fig. 1) was set equal to 300 mm. The magnetic core wastaken with a relative magnetic permeability equal to 300. Themain coil has turns. The conducting ring is made froman aluminum alloy of conductivity Sm . TheFEM mesh depends on the ring position. For the cases treatedin the paper, mm, the number of the FEM nodesgoes from 627 to 663. Results are presented for the frequency of50 Hz corresponding to the penetration depth for the conductingring equal to mm.

The instantaneous magnetic flux density lines are representedin Fig. 2. These lines are obtained as the contour lines for theflux function given by (1), when equal flux increments withthe value of Wb are taken. The conducting ring isplaced at the position (Fig. 1). Lines for the instant whenthe current in the main coil has a maximum value are represented

MACHADO AND DA SILVA: MAGNETIC FORCES FOR AXISYMMETRIC EDDY CURRENT PROBLEMS 1465

Fig. 2. Instantaneous magnetic flux density lines for the problem describedin Fig. 1 and frequency of 50 Hz, the conducting ring being at position � =

0. In (a), the lines are for the instant occuring when the main coil current hasa maximum value, and in (b) when the current has a zero value, going frompositive to negative values.

in Fig. 2(a) while Fig. 2(b) is for the case when the current goesthrough zero varying from positive to negative values.

The same field lines, now for the instant when the currentin the main coil has a maximum value and for increments of

Wb, are represented in Fig. 3 for the region of theconducting ring [Fig. 1(c)]. The conducting ring is kept at posi-tion . Lines in Fig. 3(a) correspond to the field obtainedfrom the complete solution given by (1). Lines in Fig. 3(b)correspond to the field of the ring currents alone, being obtainedfrom the term of the flux (3). Finally, Fig. 3(c) correspondsto the field originated solely by the external sources outside theconducting ring, being obtained from the term of the flux (3).

Magnetic forces were evaluated by using three different ap-proaches. The first one is characterized by the use of Laplace’sforce formula, the force being obtained from (2), where the gra-dient is evaluated applying the method described in [2].The second approach is a consequence of the virtual work prin-ciple described in [4] where the energy integral is taken over theframe-like volume of empty space surrounding the conductingring and represented in Fig. 1(c). The third approach is obtained,like the first one, from Laplace’s force formula but now by usingin (2) the gradient of the complete solution , instead of

, The field being assumed uniform in each finite elementin conformity with the usual linear FEM interpolation methodused to solve the problem.

The values of the time averaged axial component of magneticforce acting on the ring are presented in Fig. 4 as a functionof the ring position coordinate , as defined in Fig. 1(b), fordifferent values of the penetration depth in the conducting ring,. In Fig. 4, one unit of force has the value

(6)

Fig. 3. Lines of instantaneous magnetic flux density inside the conducting ring[Fig. 1(c)] for the instant when the main coil current has a maximum value.Frequency is 50 Hz, the conducting ring being at position � = 0. In (a), the linescorrespond to the complete solution, in (b) to the ring currents alone, and in (c)lines are for the field originated by the external sources outside the conductingring.

Fig. 4. Normalized (6) axial component of the magnetic force as a functionof the ring position � [Fig. 1(b)]. Thick curves are for forces computed byLaplace’s force formula, using the method of [2] for the potential gradientevaluation. Thin curves are for force computation based on the virtual workprinciple. The penetration depth in the conducting ring is (a) � = 20 mm, (b)� = 15:6 mm (ring of aluminum alloy given in the text, for the frequency of50 Hz), and (c) � = 5:66 mm.

in Newtons, where is the number of turns, and is the rootmean square (rms) value of the current intensity in the main coil.

For A, the unit force (6) just balances the ringweight, which has the approximate value of 85 g. Under theseconditions, the rate of energy dissipation in the ring is less than75 W.

In Fig. 4, the results obtained from the first approach, based onLaplace’s force formula using the method of [2] for the potentialgradient evaluation, are represented by the thick line curves, thethin lines representing the comparison results obtained by thevirtual work principle.

When compared with results obtained from the first approach,an error less than 1% is obtained by using the third approachbased on Laplace’s force formula for the action of the completefield on the currents in the conducting ring. The good agreementis a consequence of the refined discretization adopted for thefinite elements inside the conducting ring and confirms the well

1466 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

TABLE INUMERICAL RESULTS

known fact that the effect of the field originated by the currentsin the body itself cancels out in the global force evaluation.

Finally in Table I numerical results are presented as a func-tion of the conducting ring position [Fig. 1(b)], forcm. The first two rows show results for the values of the time av-eraged axial component of the magnetic force acting on the ringusing the first, , and the second, , approaches, nowwith absolute values, instead of normalized values as in Fig. 4.In the third and fourth rows of Table I, results for the phasor, ,of the ring current, , are presented showing the rms value andthe phase lag relative to the main coil current

(7)

where the surface integral is evaluated over the cross section ofthe conducting ring, . The fifth and sixth rows of Table I showthe values for the mutual inductance coupling between the ringand the main coil, , and the values for the self-inductance ofthe conducting ring, . These values were obtained from thesolutions of two field problems. One is the complete problemdescribed in the paper where the source current in the main coiland induced currents in the conducting ring are simultaneouslypresent. The other is the same problem in the absence of ringcurrents. This last solution is equivalent to the problem with thering replaced by empty space, since the ring is nonmagnetic. Theinductance coefficients are evaluated in both problems from thevalues of the stored magnetic energy, as indicated in [1]. In thisway, the self-inductance of the main coil may be obtained from

the second problem, found to be mH. The last rowgives the value of the ring resistance obtained from the ringlosses that correspond to the rate of Joule’s energy dissipationas was also indicated in [1].

VI. CONCLUSION

A numerical method for the evaluation of the magnetic forceacting on a conductor carrying induced currents that takes intoaccount the current nonuniformity due to the presence of eddycurrents was developed for axis symmetric time-harmonicfields. The method is based on the Laplace force formula, aFEM/BEM approach being used for the determination of thefield flux function and a linear operator method for extractingthe flux density from the flux function values. The method wasapplied to a “jumping ring” device, the force on the ring beingfound as a function of position along the device axis. Resultswere compared with those obtained using the virtual-workprinciple.

ACKNOWLEDGMENT

This work was supported in part by the Fundação paraa Ciência e a Tecnologia (FCT) of the Portuguese gov-ernment and the POCTI program of the European UnionFEDER funding, through the FCT pluriennial program and thePOCTI/ESE/46 998/2002 Project.

REFERENCES

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[3] E. Freeman and D. Lowther. Team Workshop Problem 17. The JumpingRing. [Online]. Available: http://ics.ec-lyon.fr/team.html

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[8] Z. Ren, “Comparison of different force calculation methods in 3D finiteelement modeling,” IEEE Trans. Magn., vol. 30, no. 5, pp. 3471–3474,Sep. 1994.

[9] S. Bobbio, F. Delfino, P. Girdinio, and P. Molfino, “Equivalent sourcesmethods for the numerical evaluation of magnetic force with extensionto nonlinear materials,” IEEE Trans. Magn., vol. 36, no. 4, pp. 663–666,Jul. 2000.

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Manuscript received June 28, 2005 (e-mail: pcvmalo@mail.ist.utl.pt).