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Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations Application of the Newton-Raphson method

A Jiles-Atherton vector hysteresis model 685

J. Gyselinck and P. Dular Department of Electrical Engineering (ELAP), University of Lie`ge, Lie`ge, Belgium

N. Sadowski, J. Leite and J.P.A. Bastos Department of Electrical Engineering (GRUCAD), Federal University of Santa Catarina, Floripanopolis, Brazil Keywords Finite element analysis, Vector hysteresis, Magnetic fields, Newton-Raphson method Abstract This paper deals with the incorporation of a vector hysteresis model in 2D finite-element (FE) magnetic field calculations. A previously proposed vector extension of the well-known scalar Jiles-Atherton model is considered. The vectorised hysteresis model is shown to have the same advantages as the scalar one: a limited number of parameters (which have the same value in both models) and ease of implementation. The classical magnetic vector potential FE formulation is adopted. Particular attention is paid to the resolution of the nonlinear equations by means of the Newton-Raphson method. It is shown that the application of the latter method naturally leads to the use of the differential reluctivity tensor, i.e. the derivative of the magnetic field vector with respect to the magnetic induction vector. This second rank tensor can be straightforwardly calculated for the considered hysteresis model. By way of example, the vector Jiles-Atherton is applied to two simple 2D FE models exhibiting rotational flux. The excellent convergence of the Newton-Raphson method is demonstrated.

1. Introduction In the domain of numerical electromagnetism, an increasing number of publications is devoted to the inclusion of hysteresis models in finite-element (FE) field computations. Mostly the well-known Preisach (1935) model and Jiles-Atherton model (Jiles et al., 1992) are used. A comparison of these two scalar models is presented by Benabou et al. (2003). They are applicable to 1D, 2D and 3D FE models displaying unidirectional flux (Chiampi et al., 1995; Sadowski et al., 2002; Saitz, 1999). In applications having rotational flux in part of the computation domain, a vector hysteresis model should be used (Dupre´ et al., 1998; Gyselinck et al., 2000). To date the inclusion of a scalar or vector hysteresis model in FE computations remains challenging. One possible difficulty may reside in the fact that the basic variable of the FE formulation does not coincide with the (main) input variable of The research was carried out in the frame of the Inter-University Attraction Pole IAP P5/34 for fundamental research funded by the Belgian federal government. P. Dular is a Research Associate with the Belgian National Fund for Scientific Research (FNRS).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 685-693 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540601

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the hysteresis model. This problem may be solved by inverting the hysteresis model (Gyselinck et al., 2000; Sadowski et al., 2002). Further, a suitable iterative method to deal with the nonlinearity of the hysteretic media has to be found. For nonlinear problems with isotropic nonhysteretic materials, the NR method is the obvious choice as it offers a quadratic convergence rate near the solution and is easy to implement (Bastos and Sadowski, 2003). In the presence of hysteretic media, the implementation of the NR method is somewhat less self-evident (Gyselinck et al., 2000; Sadowski et al., 2002), and the fixed-point method is often adopted (Chiampi et al., 1995; Saitz, 1999). In Section 2, it is shown that a straightforward elaboration of the NR method naturally leads to the explicit use of the differential reluctivity tensor when considering, e.g. the 2D magnetic vector potential formulation. In the third section, an orginal method to calculate this tensor for a vector generalisation of the Jiles-Atherton model (Bergqvist, 1996) is presented. This vector model is then applied to a simple 2D FE model exhibiting rotational flux. 2. Newton-Raphson method applied to 2D FE model We consider a 2D magnetic field problem in a domain V in the xy-plane (Bastos and Sadowski, 2003). In a subdomain Vs, the current density j s ðx; y; tÞ ¼ js ðx; y; tÞ1 z ; along the z-axis, is given. The magnetic field vector h ðx; y; tÞ and the induction vector b ðx; y; tÞ both have a zero z-component and are related by a hysteretic model in (part of) V. For any continuous vector potential a ¼ aðx; y; tÞ1 z ; the induction b ¼ curl a satisfies div b ¼ 0: The FE discretisation of V leads to the definition of basis functions a l ¼ aðx; yÞ1 z for the potential a : aðx; y; tÞ ¼

n X l¼1

a l ðx; yÞal ðtÞ or aðx; y; tÞ ¼

n X

al ðx; yÞal ðtÞ:

ð1Þ

l¼1

Commonly triangular elements and piecewise linear nodal basis functions are adopted. The weak form of Ampe`re’s law curl h ¼ j s reads ðcurl h; a 0k ÞV ¼ ð j s ; a 0k ÞVs ) ð h; curl a 0k ÞV þ k · ; · l›V ¼ ð j s ; a 0k ÞVs ;

ð2Þ

where a 0k is a continuous test function and ð · ; · ÞV denotes the integral of the scalar product of the two vector arguments over the domain V. For the sake of brevity, the contour integral k · ; · l›V ; originating from the partial integration, is disregarded in the following. By using the n basis functions a k as test functions as well, we obtain a system of n nonlinear algebraic equation (2). Because of the hysteretic material behaviour in V, magnetostatic equation (2), with given excitation js ðx; y; tÞ; has to be solved in the time domain (time stepping). Starting from the known solution a 2 ðx; yÞ and known material state ðb 2 ¼ curl a2 ; h 2 ; . . .Þ at a time instant t 2 , the solution at the next instant t þ ¼ t 2 þ Dt can be obtained by means of an iterative NR scheme. The ith NR þ þ iteration, i ¼ 1; 2; . . .; produces the ith approximation aþ ðiÞ ¼ aði21Þ þ DaðiÞ ; where þ the increment DaðiÞ follows from linearisation of equation (2) around the ði 2 1Þth þ 2 solution aþ ði21Þ : The iterative scheme is initialised with að0Þ ¼ a : The linearisation of equation (2) requires the evaluation of its derivatives with respect to the nodal values al. Given that

›h ›h curl a l ; ¼ ›a l ›b

ð3Þ

the differential reluctivity tensor ›h=›b emerges quite naturally and linearised equation (2) can be written as  n
X ›h curl a l ; curl a k DalðiÞ ¼ ð j s ; a k ÞVs 2 ð h ði21Þ ; curl a k ÞV : ð4Þ ›b V l¼1 For nonhysteretic nonlinear isotropic materials this amounts to the well-known expression of the Jacobian matrix. Indeed, the differential reluctivity tensor can be written as

›h dn ¼ n1 þ 2 2 b b; ›b db

ð5Þ

where the scalar reluctivity n(b) is a single-valued function of the magnitude of b; b b is the dyadic square of b and 1 is the unit tensor. In the xy coordinate system, the matrix representation of reluctivity tensor (5) thus is 3 2 ›hx ›hx " # " # " # 6 7 bx bx bx by 1 0 ›h 6 ›bx ›by 7 d n 7 6 ð6Þ ›b ¼ 6 ›hy ›hy 7 ¼ n 0 1 þ 2 db 2 by bx by by : 5 4 ›bx ›by The integrand




 ›h curl a l · curl a k ›b

in the lefthand-side of equation (4) can then be written as dn ðgrad al · grad aÞðgrad ak · grad aÞ: db 2 For hysteretic material models, the differential reluctivity tensor also depends on the history of the material. For instance, for the vector Preisach model considered by Dupre´ et al. (1998) and Gyselinck et al. (2000), the history consists of extreme values of the magnetic field projected on a number of spatial directions. The vectorised Jiles-Atherton model (Bergqvist, 1996) is dealt with in Section 3.

ngrad al · grad ak þ 2

3. Scalar and vectorised Jiles-Atherton model In the scalar Jiles-Atherton model, the material is characterised by five (scalar) parameters. The determination of these parameters, commonly denoted by a; a; ms ; c and k, is discussed by Benabou et al. (2003) and Jiles et al. (1992). The equations relevant to its vectorisation and FE implementation are given hereafter. The scalar magnetisation m ¼ b=m0 2 h consists of a reversible part mr and an irreversible part mi: m ¼ mr þ mi with mi ¼ ðm 2 cman Þ=ð1 2 cÞ and mr ¼ cðman 2 mi Þ;

ð7Þ

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where the anhysteretic magnetisation man is a single-valued function of the effective field he:

 he a man ðhe Þ ¼ ms coth ð8Þ with he ¼ h þ am: 2 he a The irreversible character of the material is given by


 dmi 1 dh ¼ ðman 2 mi Þ with d ¼ sign : dt dhe dk

ð9Þ

An alternative definition, based on Bergqvist (1996), may be adopted in order to prevent dmi =dhe (and db=dh) from becoming negative: dmi jman 2 mi j dmi if dh · ðman 2 mi Þ . 0; else ¼ ¼ 0: k dhe dhe

ð10Þ

The differential susceptibility dm=dh and the differential permeability db=dh can then be calculated for given b, h and sign(dh):
 db dm dm ¼ m0 1 þ ðb; h; signðdhÞÞ ¼ and dh dh dh

dman dmi þ ð1 2 cÞ dhe dhe : ð11Þ dman dmi 1 2 ac 2 að1 2 cÞ dhe dhe c

For a given state ðh 2 ; b 2 Þ and a given b +, the corresponding h + can be obtained by numerically integrating dh=db : Z bþ dh ðb; h; signðh þ 2 h 2 ÞÞ db: ð12Þ h þ ¼ h þ ðh 2 ; b 2 ; b þ Þ ¼ h 2 þ b 2 db We now outline the vector extension as proposed by Bergqvist (1996), but limit the analysis to the isotropic case. In the vector generalisation of equations (7)-(12), the scalar fields are replaced by vector fields, e.g. b becomes b, while the scalar differential quantities are replaced by tensors, e.g. dh=db becomes ›h=›b: The division in equation (11) is replaced by the multiplication of the nominator by the inverse of the denominator. The scalar 1 is replaced by the unit tensor 1 where necessary. The vector extension of equations (8) and (10) needs special attention. m an and ›m an =›h e are single-valued functions of h e : ! he ›m an man h eh e dman h e h e and m an ¼ man ðjh e jÞ ¼ 12 2 : ð13Þ þ jh e j ›h e he dhe h2e he According to Bergqvist (1996), the vector extension of equation (10) consists in assuming that the increment dm i is parallel to m an 2 m i ; proportional to jm an 2 m i j=k and nonzero only if dh · ðm an 2 m i ). Considering a local coordinate system x 0 y 0 , with the x 0 -axis along the vector m an 2 m i (Figure 1), we thus have " # " # ›m i jm an 2 m i j 1 0 ›m i ¼ ¼ 0: ð14Þ if dh · ðm an 2 m i Þ . 0 else ›h e k ›h e 0 0 0 0 x y

The matrix representation of ›m i =›h e in a coordinate system xy is then " #     cos u sin u ›m i ›m i T : ¼R R with R ¼ ›h e xy ›h e x 0 y 0 2sin u cos u

ð15Þ

Using all the above equations (or their vector extension), ›b =›h can be calculated for given b and h; and given direction of dh: By inverting (the matrix representation of) ›b=›h; ›h=›b is obtained.

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4. Application examples By way of example, the vector Jiles-Atherton model presented in Section 3 will now be applied to two simple 2D FE models. The five parameter values of the hysteresis model are those of the electrical steel (FeSi) considered in Benabou et al. (2003): ms ¼ 1; 145; 500 A=m; a ¼ 59 A=m; k ¼ 99 A=m; c ¼ 0:55 and a ¼ 1:3 £ 1024 : 4.1 FE model with spatially uniform field In a calculation domain in the xy-plane, in which the same material model is adopted throughout, a spatially uniform magnetic field h ðtÞ can be obtained by imposing a current layer k ðx; y; tÞ1 z ¼ hðtÞ · ð1 n £ 1 z Þ on the outer boundary of the domain, where 1 n ðx; yÞ is the outward unit vector (perpendicular to 1 z ). Note that the magnetic field and induction are uniform regardless of the FE discretisation. The unknowns of the FE problem are all the nodal values of the magnetic vector potential, except for one, which is, e.g. set to zero. We consider a square domain comprising 68 first-order triangular elements (Figure 2). (A FE mesh having only two elements could do the job as well.) A magnetic field with sinusoidally varying x and y components: h ðtÞ ¼ hx ðtÞ1 x þ hy ðtÞ1 y ¼ h^ x cosð2pftÞ1 x þ h^ y cosð2pft þ fÞ1 y ;

ð16Þ

is imposed as described above. As a static hysteresis model is considered, the frequency f is arbitrary. We take, e.g. f ¼ 1 Hz: If the phase angle f is nonzero (and h^ y is nonzero as well), the excitation is rotational; if f is zero, the flux is unidirectional. Calculations have been carried out with a unidirectional magnetic field ðf ¼ 0Þ of maximum value rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 hmax ¼ h^ þ h^ x

y

equal to 200, 400 and 800 A/m. It has been verified that the FE results do not depend on the spatial direction of the field, as can be expected for an isotropic material.

Figure 1. Local coordinate system x 0 y 0 with x 0 -axis along m an 2 m i

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Figure 2. Model with uniform field and FE discretisation – magnetic field vectors in the case of rotating excitation at t ¼ 0.65 (left) and 0.755 s (right)

Figure 3. bh-loci at alternating excitation (left) and bxhx-loci at rotational excitation (right), with hmax¼ 200 (dashed line) and 400 A/m (full line)

Calculations with a purely rotational flux, f ¼ p=2 and hmax ¼ h^ x ¼ h^ y ; have also been done, considering the same values for hmax (see also Figure 2). Two periods, from t ¼ 0 to 2 s, are time stepped using the implicit Euler scheme. The time step Dt is taken to be either 1/200, 1/400 or 1/800 s. During the initial interval [0,trelax], with, e.g. trelax ¼ 0:25 s; the imposed current layer kðx; y; tÞ1 z (or magnetic field h ðtÞ) is multiplied by a smooth step function f relax ðtÞ ¼ ð1 2 cosðpt=t relax Þ=2; in order to step smoothly through the first magnetisation curve of the hysteretic material. It has been verified that in steady-state and in the case of purely rotational flux, the h and b loci are circular, as, again, can be expected for an isotropic material. Two bh-loci at alternating excitation and two bxhx-loci at rotational excitation are shown in Figure 3. The nonlinear FE equations are solved by means of the NR method, as explained above. A relative tolerance of 102 4 is adopted. The average number of NR iterations per time step is listed in Tables I and II for the different calculations. One observes an excellent convergence of the NR scheme, requiring only three or four iterations in the case of alternating and rotational flux, respectively. In the considered hmax range, the number of iterations depends little on hmax. The number of iterations decreases as the time step is decreased, as could be expected.

4.2 T-joint of three-phase transformer The same vector Jiles-Atherton model is next applied to a 2D FE model of a T-joint of a three-phase transformer, which is shown in Figure 4. The Dirichlet condition a ¼ 0 is imposed on the upper boundary. On the other boundaries, the Neumann condition (zero tangential magnetic field) is implicitly considered. In the lower left and right square corners (of side 1 m) of the model, a uniform current density corresponding with a magnetomotive force of 400 cos (2pft) and 400 cos (2pft+2p/3) A t, respectively, is imposed. As in the first example, two periods are time stepped (with f ¼ 1 Hz; t relax ¼ 0:25s). A FE mesh having 458 first-order triangular elements is used. Figure 4 shows the flux pattern obtained at four equidistant instants in the first half of the second period. The b-loci and h-loci obtained in the six points indicated in Figure 4(a) are shown in Figure 5.

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Dt (s) hmax (A/m)

1/200

1/400

1/800

200 400 800

3.31 3.26 3.26

3.14 3.02 2.91

2.96 2.84 2.61

Table I. Number of NR iterations per time step in the case of alternating flux

Dt (s) hmax (A/m)

1/200

1/400

1/800

200 400 800

4.01 4.09 4.21

3.78 4.06 4.10

3.58 3.72 4.00

Table II. Number of NR iterations per time step in case of rotating flux

Figure 4. 2D FE model of T-joint of transformer - location of six points in the model – flux lines at t ¼ 1, 1.125, 1.25 and 1.375 s (a to d)

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Figure 5. The b-loci (left) and h-loci (right) in the six points indicated in Figure 4a

A fast convergence of the NR scheme is again observed: the average number of iterations per time step is 4.6, 3.89 and 3.38 with Dt equal to 1/200, 1/400 and 1/800 s, respectively. 5. Conclusions The implementation of a vector Jiles-Atherton model in 2D FE magnetic field computations has been studied. When solving the nonlinear equations by means of the NR method, the differential reluctivity tensor naturally emerges. This tensor is also a natural output of the considered vector hysteresis model. The vector Jiles-Atherton model has been successfully applied to two simple 2D FE models with rotational flux. The Newton-Raphson scheme has been shown to converge very well. References Bastos, J.P.A. and Sadowski, N. (2003), “Electromagnetic modeling by finite element methods”, Electrical Engineering and Electronic Series, 117, Marcel Dekker, New York, NY. Benabou, A., Cle´net, S. and Piriou, F. (2003), “Comparison of Preisach and Jiles-Atherton models to take into account hysteresis phenomenon for finite element analysis”, Journal of Magnetism and Magnetic Materials, Vol. 261, pp. 305-10. Bergqvist, A. (1996), “A simple vector generalisation of the Jiles-Atherton model of hysteresis”, IEEE Trans. Magn., Vol. 32 No. 5, pp. 4213-5. Chiampi, M., Chiarabaglio, D. and Repetto, M. (1995), “A Jiles-Atherton and fixed-point combined technique for time periodic magnetic field problems with hysteresis”, IEEE Trans. Magn., Vol. 31 No. 6, pp. 4306-11. Dupre´, L., Gyselinck, J. and Melkebeek, J. (1998), “Complementary finite element methods in 2D magnetics taking into account a vector Preisach model”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3048-51. Gyselinck, J., Vandevelde, L. and Melkebeek, J. (2000), “Calculation of noload lsses in an induction motor using an inverse vector Preisach model and an eddy current loss model”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 856-60.

Jiles, D., Thoelke, B. and Devine, M. (1992), “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. Magn., Vol. 28, pp. 27-35. Preisach, F. (1935), “Uber die magnetische nachwirkung”, Zeitschrift ¨fr Physik, Vol. 94, pp. 277-302. Sadowski, N., Batistela, N.J., Bastos, J.P.A. and Lajoie-Mazenc, M. (2002), “An inverse Jiles–Atherton model to take into account hysteresis in time-stepping finite-element calculations”, IEEE Trans. Magn., Vol. 32 No. 2, pp. 797-800. Saitz, J. (1999), “Newton-Raphson method and fixed-point technique in finite element computation of magnetic field problems in media with hysteresis”, IEEE Trans. Magn., Vol. 35 No. 3, pp. 1398-401.

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