Computation of three-dimensional electromagnetic field including ...

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999

Computation of Three-Dimensional Electromagnetic Field Including Moving Media by Indirect Boundary Integral Equation Method Dong-Hun Kim, Song-Yop Hahn, Il-Han Park, and Gueesoo Cha

Abstract— We present a general analysis method for threedimensional (3-D) eddy current problems with a moving conductor. The indirect boundary integral equation method (IBIEM) is employed for 3-D electromagnetic field problems including an arbitrarily shaped conductor with constant relative velocity. Since the 3-D motion effect is taken into account in the fundamental Green’s function for the governing equation of diffusion type, this approach gets rid of spurious oscillations which usually occur in solutions obtained by the Galerkin finite element method. That is, the proposed method uses an elaborate fundamental Green’s function for magnetic diffusion instead of artificial upwind techniques of the finite element method. In addition, a new accurate integration technique for very local functions related to the Green’s function for magnetic diffusion is adopted, a technique that plays an important role in accuracy and stability of numerical solutions. The electromagnetic field and eddy current at any point are calculated through the numerical integration of the equivalent magnetic surface sources obtained by the integral system equation. The proposed method is numerically tested and validated through the analysis model where a conducting slab under a fixed rectangular coil moves with a constant velocity. Index Terms—Eddy current, Green’s function, indirect integral formulation, three-dimensional motion effect.

I. INTRODUCTION

U

P until now, the electromagnetic phenomena in conductors with relative motion to current source have attracted much attention in some applications such as the magnetic levitation of repulsion type, the linear induction motor, etc., and various studies have been undertaken for the accurate analysis of these phenomena. These studies can be classified into two categories. The first one adopts analytic approaches [1], [2], whereas the second one employs numerical methods [3]–[8]. Both of them have some drawbacks in practical use. The analytic approaches have severe restrictions on the geometry of the analysis model. It is not easy to deal with analytic solutions because of their infinite polynomials. On the other hand, the finite element method (FEM) using upwind techniques has been used for twoManuscript received January 12, 1998; revised January 4, 1999. D.-H. Kim is with the Living System Research Laboratory, LG Electronics Inc., Seoul 153-023, Korea (e-mail: [email protected]). S.-Y. Hahn is with the Department of Electrical Engineering, Seoul National University, Seoul 151-742, Korea. I.-H. Park is with the School of Electrical and Computer Engineering, Sungkyunkwan University, Suwon, Kyungki 440-746, Korea. G. Cha is with the Department of Electrical Engineering, Soonchunhyang University, Asan, Choongman 336-745, Korea. Publisher Item Identifier S 0018-9464(99)02785-5.

dimensional (2-D) modeling of electromagnetic problems with moving conductors since the late 1970’s. This method has been successful in analyzing 2-D problems. However, the upwind FEM is also faced with some difficulties in treating threedimensional (3-D) problems. Even though a few researchers [6]–[8] tried to model the 3-D moving effect recently, they could not deal with the electromagnetic diffusion phenomena in 3-D appropriately. One of the reasons is that the 3-D upwind FEM that they adopted could not avoid degrading the accuracy of the numerical solutions since 3-D upwind modeling of the moving effect is nothing but space expansion of a one-dimensional upwind scheme. Another reason is that the finite element analysis of 3-D eddy currents in moving media requires massive resources in computer memory and computing time. In this paper, a powerful algorithm for the accurate analysis of 3-D electromagnetic fields in the presence of moving media is proposed. The indirect boundary integral equation method (IBIEM) is used for the electromagnetic problems which may concern any current sources and arbitrary conductors in relative motion. The IBIEM is formulated in terms of the equivalent magnetic surface charge density, the equivalent magnetic surface current density, and Green’s functions [9]–[10]. Three-dimensional moving effect is taken into account in the fundamental Green’s function which is an analytic solution of the magnetic diffusion equation. Due to the narrow local behavior of the Green’s function for magnetic diffusion as well as the minimum order of boundary integral equation, the size of the computer memory and the length of the computing time for 3-D electromagnetic fields are reduced considerably. The boundary integral equations are discretized with constant triangular elements. A new accurate integration technique for singular integrals with a very local behavior is used, which plays an important role toward improving the accuracy and stability of numerical solutions. Basically, the proposed method suppresses nonphysical oscillations in numerical solutions because it does not employ any artificial approach like the upwind technique but the elaborate fundamental Green’s function. The electromagnetic field and eddy current at any point are calculated by the numerical integration of the equivalent magnetic surface sources obtained by the integral system equation. In order to validate the proposed method, numerical results are compared with their analytic solutions in the case of a rectangular coil above the moving conducting

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KIM et al.: COMPUTATION OF 3-D ELECTROMAGNETIC FIELD

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The IBIEM is introduced to transform the above partial differential equations into the boundary integral equations. In the indirect formulation, the equivalent magnetic surface current density and the equivalent magnetic surface distributed over the interface are unknown charge density and , can be reprevariables. The magnetic fields, sented in terms of the equivalent surface sources and their fundamental Green’s functions (7) (8) where

Fig. 1. A general configuration.

(9)

slab.

(10)

II. INDIRECT FORMULATION OF BOUNDARY INTEGRAL EQUATION Fig. 1 shows a general configuration of electromagnetic problems with moving media. A homogeneous isotropic conhas conductivity and permeability ducting region The conductor is moving at a constant velocity in a fixed Cartesian coordinate system. The nonconducting region and the conducting region are divided by the interface The source region has a direct current density Taking the velocity along direction, this problem is expressed as partial differential equations with the boundary conditions from Maxwell’s equations in or

in on the interface

(1) or

(2) (3)

and the superscripts and denote the where and , respectively. region in the nonconducting The magnetic field intensity region can be decomposed into two terms (4) in the right-hand side of (4) comes from the The first term and the second term is due to the applied current in eddy current or the magnetization of the conducting region. is expressed with the magnetic scalar potential The field because it is irrotational. is obtained directly from Biot–Savart law (5)

When a unit source is located at the source point , the funand which damental Green’s functions satisfy the respective differential equations (1) and (2) in infinite domain, correspond to the magnetic field intensity at the observer’s point It is necessary to investigate the nature of the Green’s for magnetic diffusion. When a unit source at function the origin (0, 0, 0) is moving toward direction with two m/s and m/s, the change of velocities the contour lines of on plane is depicted in Fig. 2. The directional motion of the unit source not only causes the to the direction deformation of the contour lines of direction. From Fig. 2, we can see that but also to the the directional material motion produces the deformation of electromagnetic fields in all directions. As a result of this, the shows the local behavior which makes Green’s function become nearly zero except at the source point the value of or upstream direction. In the proposed method, 3-D moving effect is taken into account on numerical solutions by the of (9). If the singular integrals related to Green’s function are successfully performed, this method always provides stable and accurate solutions regardless of the Peclet number where is the length of the element in the direction of motion). Applying (6)–(8) to (3), the indirect boundary integral equations of minimum order are obtained as follows:

(11)

(6) In (6), the subscripts and indicate the where observer’s point and the source point in the reference frame, respectively.

(12)

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999

(a) Fig. 3. Local coordinates and two basis vectors.

element equations as follows:

(15) (b) Fig. 2. Contour lines of Green’s function G+ for two different velocities under  = 3:82 107 moh/m and  = 4 1007 H/m. (a) vx = 0 m/s. (b) vx = 50 m/s.

2

2

(16) III. THREE-COMPONENT ALOGRITHM The boundary integral equations are discretized to yield the boundary element equations. In this paper, we use constant triangular elements where the equivalent magnetic surface sources are distributed uniformly over each boundary element. Since the equivalent magnetic surface current density has no normal component on the surface, the current density vector on the th element is expressed in two basis vectors, and , which define the surface of the th element

(17) where

(13) The local coordinate system with the associated basis vectors is shown in Fig. 3. The relationship between the Cartesian unit in the global coordinate system and two local vectors is given by unit vectors

(14) boundary elements generated over the Applying (13) and interface to (11) and (12), we can get three scalar boundary

In the above equations, the subscripts and refer to the considered boundary elements. The resultant coefficient matrix tends to become sparse when the velocity because of the very local behavior of of moving media increases. IV. NUMERICAL INTEGRATION VERY LOCAL FUNCTION

OF THE

Generally, singular integrals occur in the boundary element equations. Especially in the analysis of 3-D electromagnetic fields in moving media, singular integrals play an important

KIM et al.: COMPUTATION OF 3-D ELECTROMAGNETIC FIELD

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Fig. 4. Three divided regions in singular element. (a)

role in accuracy and stability of numerical solutions. Because in motion has the very local behavior the Green’s function as shown in Fig. 2, several hundreds of integration points have been employed to provide sufficient accuracy for the solution in 2-D [12]. Thus, it is not suitable to apply the above method in 3-D problems because it requires a long computing time. For efficient treatment of the narrow local functions, a new integration technique is adopted. is not mentioned The integration of the singular function here because it is well-known through previously published papers or books [13]–[16]. In this paper, we deal with the alone. The singular integrals for the Green’s function and in (15) and (16) always become resultant values of and are parallel to the normal vector zero because of the th element. Three components of Let’s consider the gradient of are expressed in the Cartesian coordinate system as follows:

(b) Fig. 5. Subelements and polar coordinates. (a) Subdivided ith element. (b) Polar coordinates in k th subelement.

component of singular integrals in (18) is presented here because the others are similar to the procedure described below. For the surface integrals over the th subelement, it is convenient to use a coordinate transformation from global to polar coordinates coordinates

(18) All components in (18) involve the vewhere locity term and show dominant behaviors around the singular possesses the point. Particularly, the component of very local behavior near the singularity. For the accurate numerical integration of the above singular functions, the boundary element including singularity is divided into three regions as shown in Fig. 4. The region I is a hemispherical surface which has an infinitesimal radius within region II. The regions II and III are distinguished from the in (18). The borderline between two regions is behavior of within the region II should determined such that value of be one with an error that is smaller than 0.1%. The area of each region depends on the velocity of moving media. It is found that the higher the velocity, the smaller the each area of regions I and II. Performing the numerical integration over the singular element with three divided regions, the element subdivision technique is applied. In Fig. 5, the th singular element is subtriangles and the polar coordinate system divided into is introduced. The observation point is at the centroid of the th element and has global coordinates Only the

(19) where

(20) denotes the height of the th subelement. is composed of three singular The component of functions and each surface integral over the th subelement can be divided into three parts which belong to the regions I–III, respectively. In the sense of Cauchy’s principle value, the singular integral in the region I is analytically evaluated. within the borderline is approximately Because the value of one, the surface integral over the region II can be regularly where is a positive treated as the singular function in region III decreases abruptly to zero as the integer. But, distance between the source and observer’s points becomes larger. The surface integrals on region III are given in the polar coordinate system as shown below (21)

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999

Fig. 6. An analysis model.

TABLE I CALCULATION CONDITIONS

(22)

(23) where

(24) denotes the surface corresponding to the region III and represents the th subelement which and the subscript belongs to the th element. The above equations, (21)–(23), are integrated analytically with respect to coordinate from to the upper bound in the region the lower bound III as (25) (26) (27) is the Euler polynomials of the first order and After (25)–(27) are calculated numerically with respect to coordinate by standard Gauss quadrature, the surface integral over the region III of the th subelement can be obtained. When the Peclet number is greater than 10 , it is not necessary to integrate over the regions II and III because the resulting integration values are almost zero. The above procedure for singular integrals with the very local behavior is used successfully at large Peclet number. In (27),

V. EVALUATION OF FIELDS AND EDDY CURRENTS The electromagnetic field and eddy current at any point are calculated by the numerical integration of the equivalent magnetic surface sources that are obtained on each boundary and , are element. The magnetic fields, denoted by given from (4), (7), and (8) (28)

(29) on But caution must be taken in calculating the field the interface between the nonconducting and the conducting and tangential regions. Because normal component of are discontinuous on the equivalent surface component of source layer, it is difficult to get an accurate result of by using only one of (28) and (29). Therefore, is usually obtained from combination of (28) and (29) as follows: (30) The eddy currents induced within the moving conductor can be evaluated in terms of the magnetic vector potential in the region The relation between the magnetic vector potential and the equivalent magnetic surface current is given from (7) as (31) in (31) is continuous function at any It is known that Both points in the conducting region including the interface are also continuous normal and tangential component of on a single surface current. Therefore, the eddy currents in the conductor as well as on the boundary are represented as the following manner [1]:

(32) The above procedure for calculating the eddy currents reduced the numerical errors due to the singular pole of higher order.

KIM et al.: COMPUTATION OF 3-D ELECTROMAGNETIC FIELD

Fig. 7. Distribution of the magnetic field

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H 0e around the moving coil.

(a)

(a)

(b) Fig. 9. Eddy current distribution on the surfaces of the conducting slab: (a) on the upper surface and (b) on the lower surface.

(b) Fig. 8. Comparisons of 3-D magnetic fields on the test lines: (a) on the test line A and (b) on the test line B.

VI. CERTIFICATION OF THE PROPOSED METHOD In order to validate the proposed method, it is applied to the analysis model of Fig. 6 where a rectangular coil above the conducting slab moves with a constant velocity. The analysis model has an analytic solution for 3-D magnetic fields at all points [2]. Table I indicates calculation conditions for both analytic and numerical computation of 3-D electromagnetic fields. The numerical analysis by the proposed method is car-

ried out with coarse boundary elements whose Peclet number ranges from 27–340 under the condition of Table I. Fig. 7 shows the field around the coil with the velocity of 30 m/s when the exciting current of the rectangular coil flows counter-clockwise. As shown in Fig. 7, the magnetic field distribution due to the eddy currents in the conducting slab is asymmetric around the moving coil. are computed along two test lines Three components of in Fig. 6. The comparisons of the analytic and the numerical solutions of 3-D magnetic fields is shown in Fig. 8. From this result, it can be concluded that the IBIEM provides very reliable solutions for 3-D electromagnetic fields in the presence of moving media. Under the above situation, the eddy currents induced in the conducting slab are obtained from (32). Fig. 9 shows the eddy current distribution on both surfaces of the slab. The eddy currents on two surfaces flow in the reverse direction to the exciting current. The moving effect on the distribution of the magnetic fields and the eddy currents appears in all numerical results.

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999

VII. CONCLUSIONS The proposed computation method of 3-D electromagnetic fields in the presence of moving media has many advantages when compared to other methods: 1) the proposed method has no restriction on the geometry of the system; 2) because the fundamental Green’s function, which is the analytic solution of the magnetic diffusion equation, takes into account 3-D moving effect directly, acceptable solutions are obtained with coarse boundary elements. Thus, the other techniques are not needed for the moving effect; 3) since this formulation has the unknown variables in the minimum order of degree of freedom and the Green’s function for magnetic diffusion possesses the very local behavior, the computer memory and the computational time are reduced considerably. The numerical results show that the proposed method provides very stable and accurate solutions for 3-D electromagnetic fields when the Peclet number is much greater than two. Therefore, the proposed method based on the IBIEM can be widely used for the analysis of general applications with moving conducting media. REFERENCES [1] J.-L. Boulnois and J.-L. Giovachini, “The fundamental solution in the theory of eddy currents and forces for conductor in steady motion,” J. Appl. Phys., vol. 49, pp. 2241–2249, Apr. 1987. [2] J. R. Reitz and L. C. Davis, “Force on a rectangular coil moving above a conducting slab,” J. Appl. Phys., vol. 43, no. 4, pp. 1547–1553, 1972.

[3] S.-Y. Hahn et al., “An “upwind” finite element method for electromagnetic field problems in moving media,” Int. J. Numer. Methods Eng., vol. 24, pp. 2071–2086, 1987. [4] T. Furukwa, K. Komiya, and I. Muta, “An upwind Galerkin finite element analysis of linear induction motors,” IEEE Trans. Magn., vol. 26, pp. 662–665, Mar. 1990. [5] E. K. C. Chan and S. Williamson, “Factors influencing the need for upwinding in two-dimensional field calculation,” IEEE. Trans. Magn., vol. 28, pp. 1611–1614, Mar. 1992. [6] D. Rodger, T. Karaguler, and P. J. Leonard, “A formulation for 3-D moving conductor eddy current problems,” IEEE Trans. Magn., vol. 25, pp. 4147–4149, Sept. 1989. [7] H. Song and N. Ida, “Modeling of velocity terms in 3-D eddy current problems,” IEEE Trans. Magn., vol. 28, pp. 1178–1181, Mar. 1992. [8] H. T. Yu et al., “Upwind-linear edge elements for 3-D moving conductor eddy current problems,” IEEE Trans. Magn., vol. 32, pp. 760–763, May 1996. [9] I. D. Mayergoyz, “Boundary integral equations of minimum order for the calculation of three-dimensional eddy current problems,” IEEE Trans. Magn., vol. 18, pp. 536–539, Mar. 1982. [10] J. Yuan and A. Kost, “A three-component boundary element algorithm for three-dimensional eddy current calculation,” IEEE Trans. Magn., vol. 30, pp. 3028–3031, Sept. 1994. [11] M. T. Ahmed, J. D. Lavers, and P. E. Burke, “An evaluation of the direct boundary element method and the method of fundamental solutions,” IEEE Trans. Magn., vol. 25, pp. 3001–3006, July 1989. [12] M. Enokizono and S. Nagata, “Convection-diffusion analysis at high Peclet number by the boundary element method,” IEEE Trans. Magn., vol. 28, pp. 1651–1654, Mar. 1992. [13] E. Schlemmer et al., “Accuracy improvement using a modified GaussQuadrature for integral methods in electromagnetics,” IEEE Trans. Magn., vol. 28, pp. 1755–1758, Mar. 1992. [14] K. Hayami, “High precision numerical integration methods for 3-D boundary element analysis,” IEEE Trans. Magn., vol. 26, pp. 603–606, Mar. 1990. [15] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1961. [16] P. K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science. New York: McGraw-Hill, 1981. [17] S. Hasebe and Y. Kano, “New vector green theorem using dyadic operator,” IEEE Trans. Magn., vol. 24, pp. 2916–2918, Nov. 1988.