1Grenoble Electrical Engineering Laboratory, University of Grenoble/Université Joseph Fourier/CNRS UMR 5269,Grenoble 38402, France
2GRUCAD/EEL/CTC/UFSC, Universidade Federal de Santa Cantarina, Santa Cantarina 88040-900, Brazil
A volume integral formulation to compute eddy currents in nonmagnetic conductive media is presented. The current distributionis approximated with facet finite elements. The formulation is general and leads to an equivalent lumped elements circuit. To ensurethe solenoidality of the current distribution, an algorithm detecting the independent loops is then used for the resolution. Theformulation is tested on TEAM workshop Problem 7. Even with coarse meshes, its accuracy is demonstrated.
Index Terms— Eddy currents, facet element, generalized partial element equivalent circuit (PEEC) method, integral equationmethod.
I. INTRODUCTION
THE partial element equivalent circuit (PEEC) methodis a well-known integral equation technique leading to
an equivalent circuit representation of an electromagneticdevice. It is mainly used for the modeling of complex inter-connect problems and can be favorably applied to deviceswhere the air region is dominant [1]. However, the classi-cal PEEC method does not enable easily the modeling of3-D conductive media. Moreover, it requires the generationof a structured mesh associated with uniform current densityon each element. Thus, its use is still limited for generalgeometries.
Previously, eddy-current integral formulations based ongeneral Whitney elements have been proposed [2], [3]. Ourapproach has the particularity of being based on the useof facet elements [4]. The reliability of such elements hasalready been shown for finite-element magnetostatic formula-tions [5], integral magnetostatic formulation [6], and also forsteady conduction problems [7]. Different quantities have beenapproximated, such as the magnetization in [6] and the currentdistribution in [5] and [7].
This paper proposes a generalization of the classical induc-tive PEEC formulation using facet elements enabling the useof general meshes. A similar approach has been recentlyproposed in [8] but limited to the modeling of surface regions.In our approach, the current density is linearly interpolatedwith first-order facet elements. An equivalent electrical cir-cuit representation, whose branches are the facets and thenodes of these branches are the centroids of elements, isproposed (i.e., an equivalent electrical circuit build on thedual mesh). The integral inductive PEEC formulation is thenadapted to this configuration. The solenoidality of the currentdistribution is ensured because of an independent loop searchtechnique.
Manuscript received June 28, 2013; revised August 28, 2013; acceptedSeptember 16, 2013. Date of current version February 21, 2014.Corresponding author: T.-T. Nguyen (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2013.2282957
II. FORMULATION
A. PEEC Integral Equation
From Maxwell–Faradays equation, a magnetic vector A anda scalar electric potential U exist such as
E = − jωA − gradU (1)
where E is the electric field and ω is the pulsation.We consider the conductive and nonmagnetic region �C
representing a set on conductors. The electric field E in theconductor satisfies Ohm’s law
Jσ
= − jωA − gradU (2)
with σ is the material conductivity and J is the current density.Without any magnetic material, we can write the magnetic
vector potential A as
A = μ0
4π
∫
�C
Jr
d�C (3)
where μ0 is the vacuum permeability. By introducing thisexpression in (2), we obtain
Jσ
+ jωμ0
4π
∫
�C
Jr
d�C = −gradU. (4)
The current density has to satisfy the several conditions. Inthe conductive region �C
div J = 0 (5)
and on the boundary of �C
n · J = 0 (6)
where n is the external normal.
B. Facet Element Interpolation
The current density interpolates with first-order facet ele-ments such as
7013504 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
Fig. 1. Representation of shape functions of the faces {i, j, k} and {i, j, k, l}for the reference tetrahedral and reference hexahedra.
where w j is shape function [see Fig. 1.] and I j is flux acrossthe j th facet. Expressions for w j can be easily computedfor different kind of element [4]. The main property of suchelement family is that the normal component of w j is con-served through each facet, so also ensuring the conservationof current.
Let us remember some of the properties for these shapefunctions
w j · n = ± 1
s j(8)
divw j = ± 1
ve(9)
where s j is the surface of facet j and ve is the volume ofelement e, which contains the facet j . The sign (±) dependson the facet orientation.
C. System Assembly Using Facet Elements
Applying the Galerkin method to (4) using wi as projectionfunctions, a system of linear equations is obtained
[Zb] If = ( [R] + jω [L] ) If = Ub (10)
with
Ri j =∫
�C
wi .w j
σd�C (11)
Li j = μ0
4π
∫
�C
wi .
∫
�C
w j
rd�Cd�C (12)
Ubi = −∫
�C
wi .gradUd�C . (13)
Matrix [Zb] can be seen as the impedance matrix of theelectrical equivalent circuit generated, If is the vector ofcurrents through the facets, [R] is the matrix of the resistiveterms, and [L] is the matrix of the mutual inductances. Letus notice that the resistive matrix [R] is sparse and similarto a matrix obtained during a classical finite-element method(FEM) assembly. On the other hand, the inductive part [L] isfully dense and more representative of integral method-basedassembly. To avoid integral singularity inaccuracies whilethe computation of [L], different Gauss points repartitionare considered especially for the computation of the self-inductance (i.e., Lii ).
Fig. 2. Primal and dual meshes, the black points are the centroid ofelements, red ones are the centroid of the faces on the boundary. (a) BothUbi configurations are represented (internal and border facets) and (b) part ofequivalent electrical circuit, Zbj is the impedance, Ubj is the voltage of thej th branch and Ifj is the current across the j th branch.
Let us define Ue the average values of the voltage on eachfinite-volume element �e and U f its average value on eachface �f of the mesh. We have
Ue = 1
ve
∫
�e
Ud�e (14)
U f = 1
s f
∫
� f
Ud� f (15)
with ve is the volume of element e and s f is the surface offacet f. Let us apply divergence theorem on (12), we obtain
Ubi = −∫
�
(wi .n)Ud� +∫
�C
div(wi )U d�C (16)
where � is the boundary of region �C .Let us now consider the computation of Ubi for an internal
facet i [see Fig. 2(a)]. The first term in (16) is null becausethe value of wi vanishes on all others facets of both adjacentelements sharing facet i . Thus, from (9) and (14), we candeduce that Ubi is the difference between the averaged voltagesof both elements sharing i .
Let us now consider Ubi for a border facet (i.e., belongingto the boundary of region �C). From (8) and (15), the valueof the first term in (16) becomes equal to the averagedpotential on the facet. The second term becomes equal to thevolume averaged voltage on the only elements to whom facet ibelongs. Facet orientations have to be considered for properlycomputing the values of Ubi .
From the previous considerations, an equivalent electricalcircuit (10) can be generated [see Fig. 2(b)]. The branchesof this circuit are represented by the facets of initial mesh.Each element of the mesh can then be seen as a node of thiscircuit. This electrical circuit is an equivalent representationof the dual mesh.
D. Resolution of the Electrical Circuit
To ensure the solenoidality of J, we can use a similartechnique to which used for classical structured PEEC method.In our approach, we have decided to use a circuit solverresolution based on the determination of independent loops [9].The general principle consists in finding small topologicalloops (minimum number of branches), allowing a very quick
NGUYEN et al.: INTEGRAL FORMULATION FOR THE COMPUTATION OF 3-D EDDY CURRENT 7013504
research of independent loops. Furthermore, the choice tomaximize the number of small loops allows significantlyimproving the matrix condition number of the final systemof equations.
In addition, we can notice that the boundary condition (5)can be easily considered in the formulation, by simply sup-pressing the associated degree of freedom in the equivalentcircuit. If the meshed conductor is electrically coupled with anexternal circuit, the electric coupling can naturally be imposedin the formulation without any difficulty.
Fundamental circuit equations are expressed in the presenceof external source voltages
[M] (Ub + Us) = 0 (17)
with [M] is the branch-fundamental independent loop transi-tion matrix where the value of each element can be −1, 0,or 1 and Us the vector of external voltage sources (mostpart of time equal to zero). We can write a new system oflinear equations where unknowns Im are currents flowing inindependent loops
[M] [Zb] [M] t Im = − [M] US. (18)
Once linear system has been solved, we obtain the currentsflowing on each branch by applying the following equation:
Im = [M]t Im . (19)
E. Addition External Source Field
In presence of an external source like (a coil for instance)in which the current density is imposed, (4) has to be adapted
Jσ
+ jωμ0
4π
∫
�C
Jr
d�C + jωA0 = −gradU (20)
where A0 is the source magnetic vector potential created bythe external source. Equation (10) becomes
[Zb] If = Ub + U0 (21)
where
U0i = − jω∫
�C
wi .A0d�C . (22)
Independent loop search technique and resolution are thenapplied to (20) as previously.
III. NUMERICAL EXAMPLE
To validate the proposed formulation and show its per-formances, we consider Problem 7 of the TEAM workshopbenchmark problems [10]. The results will be compared withthose obtained with FLUX software, a commercial FEMprogram.
The geometry of Problem 7 is shown in Fig. 3. The problemconsists in an asymmetrical conductor with a hole and anexciting coil. The coil is placed 30 mm above the plate andexcited with 2472 A turns at 50 and 200 Hz. The drivingcurrent reaches the maximum at a phase of 0°.
Fig. 3. Geometry of TEAM problem 7: coils and conducting plate with hole(all dimension are in mm).
Fig. 4. Eddy-current distribution in the plate at (a) 50 and (b) 200 Hz.
In this problem, we have to add an additional sourcemagnetic vector potential representing the contribution of thecoils in which the current distribution is known. We have
A0 = μ0
4π
∫
�0
J0
rd�0 (23)
where J0 is the current density vector of the coil and �0 is thecoil region. A0 is evaluated numerically due to a tetrahedraldiscretization of the coil and a Gauss point quadrature.
We focus on the computed eddy-current distribution andlosses in the plate [Fig. 4(a) and (b) and Table I]. The resultsshow that the relative error of the formulation in comparison
7013504 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
TABLE I
RELATIVE ERROR OF OUR FORMULATIONS COMPARED
WITH THE FEM SOLUTION
Fig. 5. Convergence of Joules losses value in the plate at (a) 50 and(b) 200 Hz.
with the 3-D FEM associated to a very high mesh densityis good (<5%). Fig. 5(a) and (b) show the convergence ofthe Joules losses value of two methods follows the number ofelements in the plate at 50 and 200 Hz. Results provided byour formulation are very encouraging, the convergence beingreached with a very few number of elements in comparisonwith the FEM one.
Another advantage of the new formulation is its capacity totreat any multiply connected problems because of the efficientdetermination of independent loops of the equivalent electricalcircuit. This point is really valuable for the modeling ofindustrial problem, where the techniques of automatic cutssearch can be poorly efficient [11].
IV. CONCLUSION
In this paper, we have presented an original eddy-currentvolume integral formulation using facet elements dedicated tothe modeling of nonmagnetic conductors. The formulation hasa small relative error in comparison with a 3-D reference FEMwith a high mesh density. Moreover, the convergence of theresults is quickly reached and results with very coarse meshesremain acceptable.
This new formulation seems very attractive because itenables an easy treatment of the multiply connected problemswithout cuts technique and meshing the air. Moreover, thecoupling with the classical PEEC formulation is natural, bothformulation having same theoretical bases. It opens inter-esting perspective for the modeling for multiply connectedvolume/surface regions with direct electrical connections acomplex wiring system.
The obtained matrix is fully dense but this problem canbe overcome by the use of a compression technique likefast multipole method [12] enabling the save of memoryand reducing the computation time. It remains to study therobustness of this approach in this context where convergencescan sometimes be difficult to reach.
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