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Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou Laboratoire d ‘Electrotechnique et d’Electronique de Puissance de Lille, Universite´ des Sciences et Technologies de Lille, Villeneuve d’Ascq, France

Preisach and Jiles-Atherton models 825 Received September 2002 Revised January 2003 Accepted January 2003

Keywords Finite element analysis, Electromagnetism, Energy Abstract In this communication, the Preisach and Jiles-Atherton models are studied to take hysteresis phenomenon into account in finite element analysis. First, the models and their identification procedure are briefly developed. Then, their implementation in the finite element code is presented. Finally, their performances are compared with an electromagnetic system made of soft magnetic composite. Current and iron losses are calculated and compared with the experimental results.

1. Introduction Hysteresis phenomenon modelling is very useful in electrical engineering, especially for ferromagnetic material modelling. So, many hysteresis models have been developed and two of them are widely used in the static case: the Preisach model (Preisach, 1935) and the Jiles-Atherton ( J-A) model ( Jiles and Atherton, 1986). The first one is based on a mathematical description of the material behaviour and the second one on a physical description using energy balance. The implementation of the magnetic constitutive relationship in field computation allows us to have a more accurate description of an electromagnetic system. According to the used formulation, scalar (f) or vector (A) potential, the hysteresis model must be chosen, respectively, with the magnetic field H or the magnetic flux density B as entry. In this communication, we compare the accuracy of both the models cited earlier with B as entry in the case of A-formulation. First, we present both models and their identification procedures. Implementation of such models in finite element analysis is briefly described. Then, these models are applied to study a coil with a soft magnetic composite (SMC) core (Cros and Viarouge, 2002a). 2. The models The constitutive relationship of a magnetic material can be described by: B ¼ m0 ðH þ M Þ

ð1Þ

This is a revised and enhanced version of a paper which was originally presented as a conference contribution at the XVII Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC), held in Leuven, Belgium, on 1-3 July 2002. This is one of a small selection of papers from the Symposium to appear in the current and future issues of COMPEL.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 825-834 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410510794

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where B is the magnetic flux density, H the magnetic field, M the magnetisation and m0 the air permeability. In this section, the Preisach and the J-A models are presented. In their original form, they give the magnetisation M versus the magnetic field H. The constitutive relationship B (H) of the material is then calculated using equation (1). The use of the A-formulation requires a model with B as entry, which can be deduced from the original form of the Preisach (Park et al., 1993) and J-A (Sadowski et al., 2001) models. In this work, the use of both the models is restricted to the study of isotropic magnetic materials and in the case of a quasi-static behaviour. Moreover, we assume that the magnetic field does not rotate. In this context, B and H are collinear and a scalar model linking their modulus is sufficient to represent the material behaviour. 2.1 The Preisach model A ferromagnetic material can be described as a set of commutators ga, b, which have a rectangular shape and two switching fields a and b, respectively, the upper and the lower switching fields (Figure 1) (Park et al., 1993). In this model, a ferromagnetic material is characterised by a density distribution p(a, b) of these commutators. Applying some physical constraints, parameters a and b, we can define the Preisach triangle D shown in Figure 1. All couples (a, b) belong to the triangle D. A ferromagnetic material state is characterised by a given distribution of couples (a, b). The total magnetisation is given by: ZZ pða; bÞga;b da db ð2Þ M ¼ M sat D

Introducing, the Everett function (Everett, 1955): ZZ pða; bÞ da db Eðx; yÞ ¼ M sat

ð3Þ

Tðx; yÞ

with T(x, y) the right-angled triangle in the Preisach triangle with (x, y) the right angle coordinates, we can write: M ðBÞ ¼ M ðBm Þ ^ 2EðB; Bm Þ

ð4Þ

where Bm is the last return point (i.e. extremum) of the magnetic flux density. Then, from the Everett function, the magnetisation can be calculated without any numerical derivation or integration, provided that the magnetic flux density extrema are known.

Figure 1. (a) Elementary magnetic commutator; and (b) Preisach triangle, D

2.2 The J-A model The original J-A model presented by Jiles and Atherton (1986) gives the magnetisation M versus the external magnetic field H. The model is based on the magnetic material response without hysteresis losses. This is the anhysteretic behaviour which Man(H ) curve can be described with a modified Langevin equation:

 He a M an ðH Þ ¼ M sat coth ð5Þ 2 He a where He¼ H+aM is the effective field experienced by the domains, a is the mean field parameter representing inter-domain coupling. The constant a is linked to the temperature. By considering the losses induced by domain wall motions, the energy dissipated through pinning during a domain wall displacement is calculated ( Jiles and Atherton, 1986). The magnetisation energy is assumed to be the difference between the energy that would be obtained in the anhysteretic case minus the energy due to the losses induced by domain walls motions. Consequently, the differential susceptibility of the irreversible magnetisation Mirr can be written as: dM irr M an 2 M irr ¼ dBe m 0 kd

ð6Þ

where the constant k is linked to the average pinning sites energy. The parameter d takes the value +1 when dH =dt $ 0 and 2 1 when dH =dt , 0: However, during the magnetisation process, domain walls do not only jump from one pinning site to another: they are flexible and bend when being held on pinning sites. Domain wall bending is associated with reversible changes in the magnetisation process (Jiles and Atherton, 1986). Assuming that the total magnetisation is the sum of the reversible and irreversible components, we have the following expression: M ¼ M irr þ cðM an 2 M irr Þ

ð7Þ

where the reversibility parameter c[ [0,1]. As we are interested here in the M (B) model, the differentiation of equation (7) with respect to B and using Be¼ m0He lead to the differential equation (8) (Sadowski et al., 2001): dM an irr ð1 2 cÞ dM dM dBe þ c dBe ¼ dM an irr dB 1 þ m0 ð1 2 cÞð1 2 aÞ dM dBe þ m0 cð1 2 aÞ dBe

ð8Þ

Five parameters a, a, k, c and Msat have to be determined from the experimental results. 3. Identification procedures 3.1 The Everett function The Preisach model is fully determined by the Everett function. Determination of this function is achieved using the experimental measures. Having a set of experimental centred minor loops, we can calculate the corresponding Everett functions using an interpolation method. This one must respect the Everett function continuity on the whole studied domain and then for hysteresis curves. The proposed method detailed by Cle´net and Piriou (2000) satisfies these conditions and is accurate.

Preisach and Jiles-Atherton models 827

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However, the experimental measure procedure has to be made carefully, especially for high excitation fields where the hysteresis loops have to be rigorously centred otherwise the interpolation method fails. 3.2 The J-A model parameters Determination of parameters c, a, k and a requires an iterative procedure presented by Jiles et al. (1992) using a major hysteresis loop, the first magnetisation and the anhysteretic curves. This method is numerically sensitive and does not systematically converge. We use a slightly different procedure (Benabou, 2001). This first calculation step gives a good estimation of the parameter values but these can still be improved. Then, in a second step, an optimisation procedure to improve the five parameters is used. In fact, this enables us to have a set of parameters close to the best solution (in the sense of the chosen objective function) which makes easier the convergence of the optimisation procedure. This procedure is carried out using a set of centred hysteresis loops as presented by Cle´net et al. (2001). 3.3 Application to the SMC material Both identification procedures have been applied to the SMC material on a torus sample. For the Everett function identification, 20 centred minor loops were measured. The 3D Everett function is given in Figure 2. We can see that this function is quite smooth. This is linked to the fact that, at the coercive field of the SMC material hysteresis loop, there is no abrupt variation of the slope. For the J-A parameters, five centred hysteresis loops were used for the optimisation procedure. The obtained parameters are given in Table I. In Figure 3, a comparison between a measured loop and simulated ones with both the models is shown. They give good results for a wide range of the magnetic field magnitude (0 to 8,000 A/m). 4. Finite element analysis In magnetostatics, the equations to be solved on a domain D bounded by a surface S, are:

Figure 2. Three-dimensional Everett function for the SMC material

div B ¼ 0

curl H ¼ J

and n £ H ¼ 0

on

n·B ¼ 0

on

Sb

ð9Þ

Sh

with Sb and Sh two complementary parts of S, J the current density and n the outward normal vector of S. To consider the material behaviour, a constitutive relationship, denoted by H ¼ f ðBÞ; is added. To model ferromagnetic material, this relationship can be one of the two models presented earlier (Section 2). All these equations are generally solved using a potential formulation. In the 2D case, the vector potential formulation is generally preferred to the scalar potential one, then we have: curl½f ðcurl AÞ
¼ J

n£A¼0

on S b

and n £ H ¼ 0

on S h

Preisach and Jiles-Atherton models 829

ð10Þ

where A represents the magnetic vector potential (i.e. B ¼ curl A). To be solved, equation (10) can be discretised using the finite element method. In the case of hysteretic constitutive relationship, it leads to a non-linear system. The numerical solution of equation (10) including hysteresis cannot be done with the same method as the one used with univoc functions (Newton-Raphson scheme for example). We choose the fixed-point method already presented by Bottauscio et al. (1995). The hysteretic constitutive relationship is then rewritten under the form: Parameter

Value

a a (A/m) k (A/m) c Msat (A/m)

1.83 £ 102 3 1,642 1,865 0.79 1,122,626

Table I. J-A parameters for the SMC material

Figure 3. Hysteresis loops calculated from both the models and the corresponding measured loop

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H ¼ f ðBÞ ¼ nFP B 2 M FP ðBÞ

ð11Þ

The reluctivity n FP is a constant and must respect some conditions to have convergence (Ionita et al., 1996). Nevertheless, the studied hysteretic models are scalar models whereas we need a vectorial model in equation (11). Then, to evolve this latter, as it was written earlier, we assume that B and H are collinear (Section 2). The magnitude of H is calculated from the one of B, by means of the hysteresis model, and the direction of H is the same as the one of B (obtained by curl A). Consequently, the magnetisation MFP has the same direction as n FPB. Its magnitude is obtained by calculating M PF ¼ f ðBÞ 2 n FP B: Finally, the equation to be solved can be written as: curl n FP curl A ¼ J þ curl M FP

ð12Þ

The discretisation with nodal shape functions of equation (12) using the Galerkin method leads to the matrix system: ½S FP
½A
¼ ½ J
þ ½M FP


ð13Þ

where the vector [A ] represents the nodal values of the vector potential, [SFP] a square matrix, [MFP] and [J ] vectors which considers the magnetisation MPF and the current density J. One can note that the matrix [SFP] is constant because the permeability nFP is constant as well. The non-linearities introduced by the ferromagnetic materials are reported in the source term [MFP] which depend on B (i.e. A). To take the coupling with the external circuit into account, vector [J ] can be expressed as a function of the coil current i (we suppose to have only one coil): ½J
¼ ½D
i

ð14Þ

The coil is supposed to be formed by thin conductors, skin effects can be neglected. Under this condition, the distribution of the current density is given in the windings. To represent this, a vector of turns density N is defined (Piriou and Razek, 1992). Its modulus is given by the ratio of turns number to the coil section and null everywhere else. Under these conditions, the terminal voltage u is linked to the vector potential using the Faraday’s law: ZZ  d A·N dD ð15Þ u ¼ Ri þ dt D where R is the resistance of the coil winding. After the discretisation, introducing a vector [G ] the previous equation becomes: u ¼ Ri þ ½G


d ½A
dt

ð16Þ

Gathering equations (13) and (16), the final system is: "

S PF

2D

0

R

#"

A i

#

# " # " # " 0 M PF d A þ ¼ þ 0 G 0 dt i u "

0

0

#

ð17Þ

5. Experimental and simulation results 5.1 The studied system To compare the performances of both the models in finite element analysis, we study a coil with an SMC core (Cros et al., 2002b). The geometry of the system is shown in Figure 4. The coil is supplied by a 60 Hz-90 V RMS sinusoidal voltage. We aim at comparing in the steady-state current, iron losses and local evolution of B(H) curve. For this purpose, three different meshes of the electromagnetic system have been considered. The first mesh M1 has 1,284 elements, the second M2 2,987 elements and the last M3 5,448 elements. This will show the influence of the quality of the mesh.

Preisach and Jiles-Atherton models 831

5.2 Comparison of the results First of all, we can compare the B(H) curves obtained with both the models. For example, hysteresis loops for the point P1 (Figure 4) are shown in Figure 5.

Figure 4. Geometry of the studied system (mm)

Figure 5. Hysteresis loops calculated from both the models for the same location

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It appears clearly that both the models give similar results. Tests have been made elsewhere in the system and both the models gave still close results. The measured current is compared with the current obtained by field calculation. The current RMS values are reported in Table II for the three meshes. The mesh M3 gives results very close to those of mesh M2, then only current waveshapes for M1 and M2 are given in Figures 6 and 7, respectively. First, we notice that mesh M2 gives a slightly less important magnitude for the current than mesh M1. Influence of the hysteresis model is more pronounced in the case of mesh M1. In fact, for this mesh, the J-A model gives results with an error of 9 per cent and the Preisach model an error of 6.5 per cent in comparison with the experience. For the mesh M2, both the models give an error of 13 per cent. The gap between the experimental and simulation results increases with the mesh refinement. There are two error sources during the process to obtain a numerical model: . the modelling error which represents the difference between the actual device and the mathematical model equation (10), . the numerical errors which are the difference between the numerical model equation (17) and the mathematical one. In Figure 8, a representation of the evolutions of both errors versus the mesh quality (i.e. the number of elements) is shown. The modelling error is constant as it depends only on the mathematical equations, whereas the numerical error decreases with the mesh quality.

Measurements

Table II. Comparison between experience and calculation

Figure 6. Experimental and calculated currents for mesh M1

IRMS¼ 0.76 A IRMS M1 (A) IRMS M2 (A) IRMS M3 (A) ILexp¼4.3 W IL1 (W) IL2 (W)

J-A model

Preisach model

0.69 0.66 0.66

0.71 0.67 0.66

4.3 4.3

4.17 4.38

Preisach and Jiles-Atherton models 833 Figure 7. Experimental and calculated currents for mesh M2

Figure 8. Evolution of the error sources

Then, in the case of mesh M1, the numerical and the modelling errors make up for each other. Instead, with meshes M2 and M3, the numerical error is weak and almost only the modelling errors remain (see Figure 8). Iron losses are also given in Table II in the case of mesh M3. Iron losses denoted as IL1 are those obtained from magnetic quantities, i.e. from the sum of iron losses in each element of the mesh. Iron losses denoted as IL2 are those obtained from electrical quantities, i.e. IL2 ¼ kuðtÞiðtÞ 2 Ri 2 ðtÞl: Both methods IL1 and IL2 give close results. So, both can be used to estimate the iron losses. It must also be noticed that, for a given model, iron losses calculated from electrical quantities have the same values for the three meshes. Then, to estimate iron losses in our case, a fine mesh is not required. Computation times for both the models are presented in Table III. It shows clearly that the Preisach model is more time-consuming than the J-A model. This is more Mesh M1 M2 M3

No. of elements

J-A model

Preisach model

1,284 2,987 5,448

1 1 1

2.8 3.1 3.9

Table III. Computation time ratio (J-A model is the reference)

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important for the finest mesh M3. In this case, the Preisach model is almost four times more time-consuming than the J-A model. 6. Conclusion In this work, we have compared the J-A and Preisach models implemented in a finite element analysis with a SMC core coil. These models are well adapted for electrical devices with no rotating field. The studied system, a coil with revolution axis symmetry, satisfies this assumption. Both the models give similar results for the local behaviour (B(H) curve) and global quantities (current, iron losses). Moreover, the J-A model, that is easier to implement and faster, has shown results similar to those of the Preisach model for the studied system. References Benabou, A. (2001), “Identification et optimisation des parame`tres du mode`le de Jiles-Atherton pour la mode´lisation de l’hyste´re´sis magne´tique”, JCGE’01, 13-14 November, Nancy, France, pp. 229-34. Bottauscio, O., Chiarabaglio, D., Chiampi, M. and Repetto, M. (1995), “A hysteretic periodic magnetic field solution using Preisach and fixed point technique”, IEEE Trans. Magnetics, Vol. 31 No. 6, pp. 3548-50. Cle´net, S. and Piriou, F. (2000), “Identification de la fonction d’Everett pour le mode`le de Preisach”, MGE 2000, 13-14 December, Lille, France, pp. 71-4. Cle´net, S., Cros, J., Piriou, F., Viarouge, P. and Lefebvre, L.P. (2001), “Determination of losses local distribution for transformer optimal designing”, Compel, Vol. 20 No. 1, pp. 187-204. Cros, J. and Viarouge, P. (2002a), “Design of inductors and transformers with soft magnetic composites”, ICEM’02, CDROM, paper No 514, Bruges, Belgium. Cros, J., Perin, A.J. and Viarouge, P. (2002b), “Soft magnetic composites for electromagnetic components in lighting applications”, IAS 2002, 13-18 October, Pittsburgh, Pennsylvania, USA. Everett, D. (1955), “A general approach to hysteresis”, Trans. Faraday Soc., Vol. 51, pp. 1551-7. Ionita, V., Cranganu-Cretu, B. and Iona, B. (1996), “Quasi-stationary magnetic field computation in hysteretic media”, IEEE Trans. Magnetics, Vol. 32 No. 3, pp. 1128-31. Jiles, D.C. and Atherton, D.L. (1986), “Theory of ferromagnetic hysteresis”, Journal of Magnetism and Magnetic Materials, Vol. 61, pp. 48-60. Jiles, D.C., Thoelke, J.B. and Devine, M.K. (1992), “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. on Magnetics, Vol. 28, pp. 27-35. Park, G.S., Hahn, S.Y., Lee, K.S. and Jung, H.K. (1993), “Formulation of the Everett function using least square method”, IEEE Trans. Magnetics, Vol. 29 No. 2, pp. 1542-5. Piriou, F. and Razek, A. (1992), “Finite element analysis in electromagnetic systems accounting for electric circuit equations”, IEEE Trans. Magnetics, Vol. 28, pp. 1295-8. Preisach, F. (1935), “U¨ber die magnetische nachwirfung”, Zeitschrift fu¨r Physik, Vol. 94, pp. 277-302. Sadowski, N., Batistela, N.J., Bastos, J.P.A. and Lajoie-Mazenc, M. (2001), “An inverse Jiles-Atherton model to take into account hysteresis in time stepping finite element calculations”, Compumag 2001, 2-5 July, France, Evian, Vol. 4, pp. 246-7.