O De La Barrière

C Ragusa

C Appino

F Fiorillo

M Lobue

F Mazaleyrat

A computationally effective dynamic hysteresis model taking into account skin effect in magnetic laminations

We propose a simplified dynamic hysteresis model for the prediction of magnetization behavior of electrical steel up to high frequencies, taking into account the skin effect. This model has the advantage of predicting the hysteresis loop and loss behavior versus frequency with the same accuracy provided by the Dynamic Preisach Model with a largely reduced computational burden. It is here compared to experimental results obtained in Fe-Si laminations under sinusoidal flux up to 2 kHz.

I. INTRODUCTION

The problem of high frequency behavior of magnetic laminations is of primary importance in modern electrical engineering problems [1], but difficult problems arise in the prediction and assessment of magnetization process and losses, because the skin effect compounds with the nonlinear hysteretic response of the material. In order to cope with the inhomogeneous profile of the flux density over the lamination thickness, the magnetic loss is generally calculated by numerically solving the diffusion equation over the sample cross-section, and using a dynamic hysteresis model for the material constitutive law [2]. The Dynamic Preisach Model (DPM) is the model of choice, because it is accurate and solidly established from the physical viewpoint [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF] [START_REF] Bottauscio | [END_REF]. Its application is, however, particularly time consuming, because calculations must be done for each finite element of the spatial mesh until convergence is reached. A faster approach, preserving the special virtues of DPM, would therefore be appropriate. We apply in this paper the DPM to the broadband behavior of nonoriented Fe-Si laminations through a simplified method, drastically reducing computing time and complexity of the full method, requiring the computation of the dynamics of each elementary hysteron distributed in the Preisach plane [5]. We start our discussion from the differential relation found by Bertotti [6] (formula (9), page 4609, of reference [6]) for the excess magnetic field due to the dynamic behavior of the magnetization process

( ) ( ) ( ) ( ) exc stat stat d 4 sign 3
H H t H t H t H k ɺ ɺ ≡ - = , (1)
where

H(t) = H a (t) -H cl (t)
is the difference between the applied field H a (t) and the counterfield H cl (t) generated by the macroscopic eddy currents (classical field). H stat (t) is the field that would provide under static conditions the same irreversible polarization J irr and k d [A -1 s -1 m] is the DPM constant. Eq.

(1) is derived under the simplifying assumptions of triangular H a (t) and uniform Preisach distribution function. The extent to which such a restriction can be circumvented and the full DPM approach for generic exciting conditions and shape of the Preisach density function can be approximated will be discussed in the following. We find first that a numerical analysis based on (1) (Model 1) does not lead to highly accurate dynamic loop shapes, especially at high frequencies. A slight modification of (1) is therefore proposed (Model 2) and implemented in a non-linear magneto-dynamical model for the computation of the field distribution inside Fe-Si laminations, taking into account skin effect. The numerical results are compared with the experiments performed under sinusoidal flux up to 2 kHz.

II. THE SIMPLIFIED MODEL

A. The simplified DPM-Model 2

The Preisach distribution function, including the reversible contribution, was determined in a 0.35 mm thick Fe-(3.2wt%)Si-(0.5wt%)Al lamination, and the dynamic constant k d =350 A -1 s -1 m was identified. The full and the simplified (Model 1) DPM are compared in terms of the excess field. In this model benchmark, a sinusoidal dynamic field

H(t) = H a (t) -H cl (t) (peak value H p = 100A•m -1 ,
frequency f = 200 Hz) has been applied. For the full DPM case, the excess field

H exc (t) = H(t) -H stat (t)
is derived applying the DPM to H(t) in order to get the irreversible polarization J irr (t), and then using the inverse static Preisach model to compute the static field H stat (t) from J irr (t). For the model 1, a numerical solution of (1) directly provides H stat (t) from the sinusoidal H(t). A comparison between the results of two approaches is illustrated in Fig. 1 (the J irr waveform obtained from the DPM has also been represented). Discrepancies are found between the two predicted H exc (t) waveforms, particularly around the reversal points of the irreversible magnetization J irr . With the simplified Model 1 the zero of H exc is, according to (1), coincident with the maximum of H(t), whereas from the same picture, it appears that it occurs when J irr is maximum. Consequently, (1) is formulated as

( ) ( ) ( ) ( ) stat exc stat stat d 4 sign 3 H H t H t H t H k ɺ ɺ ≡ - = , (2)
on account of the fact that the sign of stat H ɺ is the same as that of irr J ɺ . The simplified DPM based on

(2) (DPM-Model 2) appears now to provide (Fig. 1) an H exc (t) waveform in good agreement with the one provided by the full DPM.

Once the static field is known, J irr (H stat ) is computed by means of the Static Preisach Model, while the reversible component J rev (H) is calculated ignoring any dynamic effect linked to the reversible contribution [1]. This procedure, which permits to obtain the constitutive law of the material J(H), is summed up in Fig. 2. An example of hysteresis loop prediction (in this case with nested minor loop) is illustrated in Fig. 3, confirming the good agreement between the results provided by the full DPM and the simplified DPM-Model 2.

A. Numerical implementation of DPM-Model 2

This model requires solving the non-linear differential equation ( 2) for H stat knowing the field H.

This can be done putting (2) under an equivalent canonical form and adding the periodicity conditions (period T=1/f):

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 d stat stat stat stat stat 9 sign 16 0 k H H t H t H t H t H H T ɺ  = - ⋅ ⋅ -    =  . ( 3
)
and numerically solving it using a Runge-Kutta method. The application of this newly defined dynamical model to the computation of flux distribution inside the steel lamination can dramatically reduce the computational burden, as demonstrated in the next section.

III. MAGNETO DYNAMICAL MODELING AND EXPERIMENTAL

A. Magneto-dynamical model of the lamination

The solution of the diffusion equation over the lamination thickness with a dynamic hysteretic constitutive law requires a special numerical treatment of the non-linearity using the Fixed Point method [3][7]. The method proposed in [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF] has been here implemented. The diffusion problem on the lamination thickness is one-dimensional, with the spatial coordinate x ranging over the lamination cross-section [-e/2; e/2] (where e = 0.35 mm is the lamination thickness). To solve the diffusion equation, the non-linearity of the material constitutive law J(H) is contained in a spatio-temporal function R(x,t), called the residual [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF] and the relationship between H and J is written as

( ) ( ) ( ) FP , , , H x t B x t R x t υ = ⋅ + , ( 4
)
where FP υ is a properly chosen constant, ensuring convergence [1]. Using (4), the diffusion equation can be written, with imposed mean flux density B MEAN (t), in terms of vector potential A on a half lamination [0; e/2]:

( ) ( ) ( ) ( ) 2 FP 2 MEAN 0, 0 and / 2, / 2
, is -periodic

A A R t x x A t A e t B t e A x t T υ σ  ∂ ∂ ∂ ⋅ - =  ∂ ∂ ∂   = = - ⋅      . ( 5
)
The time derivative and the time periodic condition are dealt with using temporal Fourier series [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF].

More precisely, the residual function R(x,t) is decomposed into complex Fourier series for what concerns the time dependence. Equation ( 5) is then solved for each time harmonic, the time derivative becoming an algebraic multiplication by the corresponding harmonic pulsation. An inverse Fourier transformation permits to retrieve the function A(x,t). For each time harmonic, the spatial second order derivative and the boundary conditions are dealt with using a numerical finite difference scheme (the half lamination is subdivided into 50 intervals). At the beginning of the iterative procedure, the residual R is initialized to zero and at each iteration step (index number i), the following process is performed:

1. Knowing the residual at the previous stage R (i-1) (x,t), the differential equation ( 5) is solved to compute the vector potential A (i) (x,t). The magnetic field is obtained as

( ) ( ) ( ) ( ) ( ) 1 FP , , i i i A H x t R x t x υ - ∂ = - + ∂ .
from H (i) and the new induction value B (i) (x,t)=J (i) (x,t)+µ 0 •H (i) (x,t) is calculated. In [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF], the dynamic hysteresis model is the DPM, and is replaced in this paper by the simplified model proposed in Fig.

2.

3. The new residual is then computed by

( ) ( ) ( ) ( ) ( ) ( ) FP , , , i i i R x t H x t B x t υ = - ⋅
. The process is repeated until convergence of the residuals is obtained.

B. Results

The dynamic hysteretic constitutive law J(H) of the material, which was in [START_REF] Bottauscio | Studies in Applied Electromagnetics and Mechanics. Non-linear electromagnetic systems[END_REF] given by the full DPM, has been here replaced by the model proposed in Fig. 2 where the Model 2 of the local excess field is applied. The Preisach distribution function has been identified following the method proposed in [1], in the previous 0. T dB e W t t

σ   =     ∫ (6)
It is found that ignoring the skin effect brings about a large overestimation of the loss above about f = 400 Hz, for the considered B MEAN (t) peak value of 0.5 T considered in Fig. 5. An example of calculated induction profile B(x, t o ) across the lamination thickness at a given instant of time t o is shown in Fig. 6.

It refers to f = 1 kHz and a time t o corresponding to the average flux density B MEAN (t o ) = 0. Again, the full and simplified DPMs predict very close results.

IV. CONCLUSIONS

In this work we have discussed a novel dynamic model for energy losses and hysteresis loops in soft magnetic laminations based on simplified treatment of the Dynamic Preisach Model (DPM). It is applied on experiments performed up to the kHz range in Fe-Si sheets, in the presence of substantial skin effect, showing good agreement both with the experimental results and the prediction made using the full machinery of the DPM. A remarkable advantage in computing time, which is reduced by a factor around 60, is obtained substituting the full DPM with the present model.

Figure captions

  35 mm thick Fe-(3.2wt%)Si-(0.5wt%)Al lamination (conductivity σ = 1.773•10 6 S•m -1 , k d =350 A -1 s -1 m). The magnetic loss and hysteresis loops have been measured under sinusoidal flux at different frequencies and peak inductions up to 2 kHz. The magnetic flux distribution and loss have been computed by applying the dynamical model, using both the full and the simplified DPM (Model 2). Fig. 4 provides an example of computed and experimental loops (J p = 0.5 T, frequency f = 1 kHz). In Fig. 5 the comparison is made for energy loss versus frequency behavior. It shows the good predicting capability of the simplified dynamic Model 2, in spite of a computing time reduced by a factor around 60 with respect to the full DPM. Fig. 5 also shows the predicted loss behavior if the skin effect is ignored. In this case the induction is assumed uniform across the sample thickness and equal to the mean flux density B MEAN (t). The classical loss is thus calculated according to

Fig. 1 -

 1 Fig. 1-Excess fields provided by the full DPM, Model 1, and Model 2, for a sinusoidal dynamic field H(t) = H a (t) -H cl (t) (frequency f = 200 Hz, field peak value H p = 100A•m -1 ) and corresponding

Fig. 2 -

 2 Fig. 2 -Block diagram of the simplified constitutive law J(H) of the material, using the Model 2 as a

Fig. 3 -Fig. 4 -

 34 Fig. 3 -Example of the reconstructed J(H) hysteresis loop with nested minor loops (f = 200 Hz, H p = 150A•m -1 )

Fig. 5 -

 5 Fig.5-Same as Fig.4for the energy loss versus magnetizing frequency (DC -2 kHz).

Fig. 6 :

 6 Fig. 6: Instantaneous induction profile B(x,t o ) across the lamination thickness as obtained by the full and the simplified (Model 2) DPM (f = 1 kHz, -0.175 mm ≤ x ≤ 0.175 mm). The time t o considered in

+

  Fig.4

A dynamic hysteresis model, providing the dynamic constitutive law J(H), is used to evaluate J(i)


C Beatrice

C Appino

Olivier De

La Barrière
email: barriere@satie.ens-cachan.fr

F Fiorillo

C Ragusa

O De La Barrière

Broadband magnetic losses in Fe-Si and Fe-Co laminations

Keywords: Magnetic losses, Fe-Si and Fe-Co laminations, Dynamic Preisach Model, Skin effect

I. INTRODUCTION

HE PREDICTION of power losses in magnetic laminations up to high frequencies is an important issue in the design of high-speed high-torque density machines [START_REF] Cho | Rotor Natural Frequency in High-Speed Permanent-Magnet Synchronous Motor for Turbo-Compressor Application[END_REF]. It is, however, a challenging task, because skin effect cannot be neglected and an accurate and computationally efficient model is required for a satisfactory treatment of the losses at high frequencies. Some authors have tried empirical approaches, derived from the Steinmetz equation corrected by appropriate frequency dependent coefficients, to take into account the skin effect [START_REF] Boglietti | Predicting iron losses in soft magnetic materials with arbitrary voltage supply: an engineering approach[END_REF]. The dependence of these coefficients from frequency is obtained through a linear skin effect formulation, assuming constant material permeability. This approximation proves effective only at low induction levels, far from magnetic saturation. At technically relevant inductions the non-linear hysteretic material behavior must be considered, as put in evidence in [START_REF] Rasilo | Importance of Iron-Loss Modeling in Simulation of Wound-Field Synchronous Machines[END_REF].

To address the whole problem, significant work has been performed in the literature, aimed at a physically based model, taking into account both skin effect and the non-linear hysteretic constitutive law of the material. The solution of the diffusion equation using a finite element approach, and the modeling of the material constitutive law given by the Static Preisach Model (SPM) has been considered in [START_REF] Appino | Power losses in thick steel laminations with hysteresis[END_REF]. The hysteretic B(H) relationship is dealt with using the fixed point method and an iterative algorithm is applied in [START_REF] Hantila | Polarization method for static fields[END_REF]. Although computationally efficient, thanks to the Everett formalism [START_REF] Mayergoyz | Mathematical models of hysteresis[END_REF], the SPM has the drawback of not taking into account the role of the domain walls in the magnetization dynamics (i.e. the excess loss contribution). This has been then addressed by means of a dynamic hysteretic relationship, like the viscoustype model [START_REF] Zirka | A Viscous-Type Dynamic Hysteresis Model as a Tool for Loss Separation in Conducting Ferromagnetic Laminations[END_REF], or Bertotti's Dynamic Preisach Model (DPM) [START_REF] Bertotti | Dynamic Generalization of the Scalar Preisach Model of Hysteresis[END_REF]. Because of its solid physical basis and the fact that it only requires the experimental identification of a single material constant (i.e. the dynamic constant k d [START_REF] Rouve | Determination of the parameter k of the generalized dynamic Preisach model[END_REF]), the DPM has become very popular [START_REF] Dupre | Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations[END_REF]. The implementation of the DPM into a numerical scheme for solving the diffusion equation of the magnetic field can well take into account skin effect, as discussed in [START_REF] Bottauscio | A Fixed Point Iteration H Scheme in the Solution of Hysteretic Electromagnetic Field Problems[END_REF] [START_REF] Basso | Power losses in magnetic laminations with hysteresis: finite element modelling and experimental validation[END_REF], where the fixed point method is used to deal with the non-linearity. Although accurate, this procedure is computationally cumbersome, because the state of every hysteron must be evaluated at each time step [START_REF] Bertotti | Dynamic Generalization of the Scalar Preisach Model of Hysteresis[END_REF] on every finite element, until convergence of the iterative scheme is attained.

The purpose of this work is to derive a simplified dynamic hysteresis model based on the DPM and its physical framework. It requires greatly reduced computation time, (the computational burden being approximately that of the SPM). This model and the full DPM are compared with the experimental magnetic energy losses in 0.2 mm thick Fe-Si and Fe-Co laminations, measured in the frequency range DC-10 kHz. It is shown that the prediction by the simplified DPM closely agrees with the experiments and the full DPM simulations, at a drastically reduced computational cost.

II. MODELING

A. Computational scheme

The concept of static magnetic field H stat is first introduced. It is the field that would give, under quasi-static excitation, the same irreversible polarization J irr (t) obtained under dynamic conditions through the applied field H(t). Once the static field H stat is known, the irreversible polarization J irr can be efficiently computed by application of the SPM. The main question is how to derive the static field H stat from knowledge of the dynamic magnetic field H(t). In the following, it is shown that a simplified relationship between H and H stat can be derived in terms of a non-linear first order differential equation in H stat , whose solution can provide the static field H stat for any given field H. This model is summed up in Fig. 1, where the dynamic operator F takes into account the dynamic properties of the hysteretic model. Once J irr is known, it is added to the reversible component J rev to obtain the total polarization J(t). The irreversible and reversible density functions are identified using the same technique as in [START_REF] Dupre | Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations[END_REF].

B. Differential relation between H and H stat

The dynamic operator F is derived from the DPM under appropriate simplifying assumptions. In the following, the dynamic constant k d of the DPM is assumed to be known (it is identified according to [START_REF] Rouve | Determination of the parameter k of the generalized dynamic Preisach model[END_REF]). We consider the case of an increasing magnetic field. It has been shown in [START_REF] Bertotti | Generalized Preisach model for the description of hysteresis and eddy current effects in metallic ferromagnetic materials[END_REF] that the set of switching hysterons in the dynamic Preisach plane α-β (α and β being the up and down switching fields, respectively) can be represented as a strip of width ∆H (Fig. In the SPM the boundary between the positively and negatively saturated hysterons is instead a horizontal line of coordinate H stat , and the function ϕ is therefore a step like one (Fig. 2). Assuming H is increasing, the following function µ(α) is introduced [START_REF] Bertotti | Generalized Preisach model for the description of hysteresis and eddy current effects in metallic ferromagnetic materials[END_REF]:

( ) ( ) LIM s irr , J p d α β µ α α β β = ∫ , ( 1
)
where p irr (α,β) is the normalized irreversible part of the Preisach density function, and J s is the saturation polarization.

According to the definition of H stat , the following relation holds

( ) ( ) ( ) ( ) stat stat , d d d H H H H H H H H t µ α φ α α µ α α µ α α -∆ -∆ = - ∫ ∫ ∫ . ( 2
)
Equation ( 2) defines the static field at each instant of time, given the state function ϕ and the width ∆H. According to the DPM theory [START_REF] Bertotti | Dynamic Generalization of the Scalar Preisach Model of Hysteresis[END_REF], we express the flux derivative as

d ( ) k H φ α = ⋅ - ɺ
. From the time derivative of (2) we consequently obtain

stat stat ∆ ( ) ( ) d 2 (
)

H d H H k H H H α µ α α µ - = - ∫ ɺ , (3)
representing the dynamical part of the model. The solution of (3) requires some simplifying assumptions, as explained in the following.

Assumption of constant µ function on the strip. On the interval [H-∆H,H], the function µ is assumed to be constant, equal to its mean value (the validity of this hypothesis is discussed later). Equation (2) therefore becomes

( ) ( ) stat 1 , 2 H H H t d H H H φ α α -∆ + = ⋅ -+ ∆     ∫ (4)
and (3) provides

2 d stat 4 k H H = ∆ ɺ . ( 5
)
Assumption of local constant time derivative of the magnetic field. In this case, as shown in [START_REF] Bertotti | Generalized Preisach model for the description of hysteresis and eddy current effects in metallic ferromagnetic materials[END_REF], the ϕ function is given by the parabolic function

( ) ( ) 2 2 2 , 1 t H H φ α α = -+ - ∆ , (6)
where the strip width ∆H has been introduced by taking into account that ϕ(H-∆H, t) = 1. Introducing ( 6) in ( 4), the width ∆H is obtained:

( ) stat 3 2 H H H ∆ = - . (7)
Eq. ( 7) is further introduced in (5) providing the following differential equation

( ) ( ) ( ) ( ) 2 stat stat d stat 9 sign 16 H H t H t k H t H t = - -         ɺ , (8)
where sign = ± 1 according to the ascending or descending direction of the magnetic field. Equation ( 8) eventually defines the dynamic operator F of Fig. 1. This equation can be numerically solved by a Crank-Nicholson method, applying periodic conditions on the static field H stat . The involved computational burden of the model is around that required by SPM.

We have introduced this simplified DPM in a numerical scheme for the computation of the magnetic field and flux distribution in the lamination. The results are compared with those provided by the full DPM and the DC 10 kHz experiments.

III. EXPERIMENTAL

The measurements were performed on commercial nonoriented Fe-(3 wt%)Si and Fe 50 Co 50 sheets. There thickness d is equal to 0.194 mm and 0.201 mm, respectively. The measurements were performed up to 10 kHz using the 200turn Epstein frame, with the test strips cut half along and half transverse to the rolling direction, according to the measuring standard IEC 60404-10 [START_REF]Methods of measurement of magnetic properties of magnetic steel sheet and strip at medium frequencies[END_REF]. The DC normal magnetization curve was obtained up to magnetic saturation by means of the point-by-point method (calibrated fluxmeter Lake-Shore mod. 480). The hysteresis loops and losses were measured versus frequency for several defined peak polarization values J p by means of a hysteresisgraph-wattmeter with digital control of the flux waveform. The form factor of the secondary voltage was kept at all frequencies within the interval FF = 1.1107 ± 0.4 %. The primary winding was supplied, depending on J p , and frequency either by a 200 W NF HSA4101 or a 1 kW NF 4510 power amplifier, driven by an Agilent 33210A function generator.

For each material, the constant of the DPM k d has been obtained measuring hysteresis loops at 50 Hz, 100 Hz and 200 Hz, for peak inductions of 0.5 T, 1 T, and 1.5 T, and by fitting the related excess loss values. It is provided, together with the main physical parameters of the investigated laminations, in Table I. A. Loss results Application of simplified and full DPMs to the energy loss measurements in the Fe-Si laminations are shown in Fig. 3a. The results obtained in the Fe-Co sheets are presented in Fig. 3b. Fe-Co, with its high saturation polarization (J s = 2.35 T), permits one to perform broadband measurements up to pretty high inductions. Remarkably, the measured losses are excellently described by the abovementioned models, including the high-frequency results, which are considerably influenced by the skin effect.

B. Measured and computed cycles

Hysteresis cycles, experimentally obtained and computed by the DPM and the simplified model, are compared in Fig. 4.

-1000 0 1000 -

Fe-Co

Applied field (A/m)

f = 1 kHz f = 1 0 k H z Polarization (T)
Fe-Si The dynamic loops are accurately reproduced at the frequency f = 1 kHz in both materials. Minor discrepancies are found at f = 10 kHz, where the induction profile in the lamination becomes highly inhomogeneous. This is understood as due to the fact that at the highest frequencies, the dynamic strip of the DPM (Fig. 2) can be quite large, and the assumption of constant µ function becomes untenable. But the loop area is still predicted with acceptable accuracy.

J p = 1 T Applied field (A/m) f = 1 kHz f = 1 0 k H z

B. Influence of the skin effect

In the absence of appreciable skin effect, we assume that the mean polarization J(t) and the local one are the same, and the classical loss is simply obtained, for sinusoidal induction of peak value

B p 2 2 2 class p p ( , ) 6 W J f d B f π σ = ⋅ . ( 9
)
In this case, the hysteresis and excess loss contributions can be found by direct inversion of the hysteresis model presented in this paper. Energy loss calculations in Fe-Co using the model with and without skin effect are compared in Fig. 5 with the experimental values. While the need for considering the skin effect in loss modeling is apparent, we see that when this is ignored, different kinds of discrepancies arise at low and high inductions. Thus, for J p =0.5 T the conventional loss model, where uniform induction is assumed in the lamination crosssection, overestimates the actual loss, while the opposite occurs at high inductions ( e.g. J p = 1.7 T). It turns then out that no differences arise at intermediate inductions, as shown for J p = 1.5 T in Fig. 6. This evolution follows from the progressively increasing depth of the magnetically saturated layer on increasing the J p value. This imposes a corresponding evolution of the induction profile across the lamination thickness, as illustrated in Fig. 6 for J p = 0.5 T and J p = 1.7 T (f = 10 kHz). We see here that for J p = 1.7 T the saturated region penetrates a fraction of the sample cross-sectional area while for J p = 0.5 T, the profile has the freedom to adjust close to the predictable behavior of the local peak induction B p,LOC (x) (-d/2 ≤ x ≤ d/2) in a constant permeability material.

IV. CONCLUSIONS

We have discussed DC-10 kHz loss measurements carried out for several peak polarization values in 0.2 mm thick Fe-Si and Fe-Co laminations. They are assessed by means of a computationally efficient dynamic hysteresis model based on the Dynamic Preisach Modeling (DPM), here proposed and implemented. This model has been introduced in a fixed point iterative procedure taking into account magnetic field diffusion across the lamination depth. The calculated energy losses and dynamic hysteresis loops and their evolution with frequency are in excellent agreement with the experiments and with the full DPM prediction, while showing strongly reduced computation time.

V. ACKNOWLEDGMENT

Fig. 1 :

 1 Fig. 1: General scheme of the model

Fig. 2 :

 2 Fig. 2: Representation of the dynamic and static Preisach planes providing the same irreversible polarization contribution Jirr

Fig. 3 -

 3 Fig. 3 -Energy loss versus frequency at different polarization values Jp (full symbols: experiments, solid line: full DPM, open triangle: simplified DPM, a) case of Fe-Si laminations, and b) case of Fe-Co laminations.

Fig. 4 -

 4 Fig. 4 -Dynamic cycles in Fe-Si (Jp = 1.0 T) and Fe-Co (Jp = 1.7 T) sheets at 1 kHz and 10 kHz. The experimental loops (symbols) are compared with the prediction by the full (solid line) and simplified (line plus symbols) DPMs.

Fig. 5 -Fig. 6 -

 56 Fig. 5 -The prediction of energy losses by the standard model ignoring skin effect overestimates the actual figures at low inductions and underestimates them at high inductions.

  2), in the interval [H-∆H,H].This relates to the fact that under dynamic conditions hysteron switching takes time. On the contrary, in the Static Preisach Model, the state of each hysteron is assumed to instantaneously switch from -1 to 1. All hysterons in the S + area are positively saturated, whereas the ones in the S

-area are negatively saturated. A function ϕ, defined in

[START_REF] Bertotti | Dynamic Generalization of the Scalar Preisach Model of Hysteresis[END_REF]

, represents the state of each hysteron in the interval [-1, 1]. An example is reported in Fig.

2

where the ϕ function depends only on α, because we are on the ascending branch of the field H. Since the hysterons in the interval [H-∆H, H] are commuting and display an intermediate state, Φ makes a smooth transition between -1 and +1.

This work was partly supported by the MIUR-PRIN2009 Project MEA 2009LKCT3E "Advanced characterization and modeling of magnetic materials for the More Electric Aircraft" and the e-méca project of the French ANR ('Agence Nationale de la Recherche').


O De La Barrière

M Lobue

F Mazaleyrat

Semi-analytical and analytical formulae for the classical loss in granular materials with rectangular and elliptical grain shapes

Keywords: classical loss computation, granular materials, rectangular shape, elliptical shape

Granular soft magnetic materials, such as ferrites or Soft Magnetic Composites, are widely spread in modern electrical engineering applications. An important loss contribution is the classical one. It is known that in these materials, two kinds of classical loss must be distinguished: eddy currents flowing at the scale of the whole sample (therefore called macroscopic), and current lines inside the grains (called microscopic). For the macroscopic eddy currents computation, the sample cross sections are often square or rectangular. For eddy currents prediction inside the grains, two cases can be distinguished: the case of low density materials, for which circular or elliptical grain shapes are often realistic, and the case of high density materials. In this last class of granular materials, it is more difficult to identify a precise grain shape, because the deformations occurring during the compaction process often give to the grains a random shape. For carrying out the eddy current computation, rectangular shapes are often considered, because they allow a complete filling of the available space. To our knowledge, analytical eddy current formulae only exist for cylindrical or spherical regions. For other shapes such as squares, rectangles, or ellipses, the finite element method must be used to compute the eddy current distribution, which can be time consuming. This paper overcomes this difficulty by proposing respectively a semi-analytical formula for classical loss in rectangular geometries, and an analytical formula for elliptical cases.

I. INTRODUCTION

RANULAR MAGNETIC MATERIALS, such as ferrites [START_REF] Magni | Domain wall processes, rotations, and high-frequency losses in thin laminations[END_REF] or 2 Soft Magnetic Composites (SMC) [START_REF] Anhalt | Dynamic losses in FeSi filled polymer bonded soft magnetic composites[END_REF] are nowadays very 3 popular in modern electrical engineering applications, such as electrical machines [START_REF] Guo | Core Loss Calculation for Soft Magnetic Composite Electrical Machines[END_REF], or power electronics [START_REF] Salas | Equivalent Electrical Model of a Ferrite Core Inductor Excited by a Square Waveform Including Saturation and Power Losses for Circuit Simulation[END_REF]. Indeed, these magnetic materials, made of magnetic particles coated by a dielectric insulator, potentially allow a significant reduction of the classical loss component [START_REF] Appino | Computation of eddy current losses in Soft Magnetic Composites[END_REF][6], compared to laminated materials [START_REF] Jiancheng | Homopolar 2-Pole Radial Permanent-Magnet Biased Magnetic Bearing With Low Rotating Loss[END_REF], in applications for which a magnetic permeability decrease is acceptable [START_REF] Cyr | Methodology to study the influence of the microscopic structure of soft magnetic composites on their global magnetization curve[END_REF].

The starting point of the loss separation analysis [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF] is an accurate computation of the classical loss component [START_REF] Magni | Magnetization Process in Thin Laminations up to 1 GHz[END_REF]. Actually, in granular materials, classical loss computation is far to be trivial. Indeed, it was shown that in these materials (ferrites or SMC), two classical loss terms must be distinguished [START_REF] Appino | Computation of eddy current losses in Soft Magnetic Composites[END_REF][2] [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF]. The first one is linked to eddy currents flowing at the scale of the single particle (therefore called microscopic). The second term (called macroscopic) is due to eddy currents flowing at the scale of the whole sample, because of capacitive effects through the dielectric insulator, or random contacts appearing between grains during the compaction process. The two terms can be separately computed and summed up [START_REF] Appino | Classical eddy current losses in soft magnetic composites[END_REF].

-For the microscopic loss component, analytical classical loss formulations are only available for cylindrical or spherical grains [START_REF] Anhalt | Dynamic losses in FeSi filled polymer bonded soft magnetic composites[END_REF] [START_REF] Bozorth | Ferromagnetism[END_REF]. These shapes are realistic for low density materials [START_REF] Anhalt | Dynamic losses in FeSi filled polymer bonded soft magnetic composites[END_REF]. However, it should be suitable, concerning low density materials, to also have formulae for elliptical grain shapes which can be encountered when dealing with anisotropic media [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF]. For highly dense materials, such as SMC which undergo a high pressure compaction, it is more difficult to define a precise grain size. Indeed, the grains often take random shapes due to plastic deformations occurring during the compaction process [START_REF] De La Barrière | Loss separation in soft magnetic composites[END_REF]. For classical loss estimation, square or rectangular shapes seem quite realistic, because they allow a complete filling of the available space. Concerning ferrites, the rectangular shape is often taken for the grains when modeling the microscopic classical loss [START_REF] Beatrice | Magnetic loss analysis in Mn-Zn ferrite cores[END_REF]. -Concerning the macroscopic classical loss prediction, the toroidal samples in which typical experiments are carried out often exhibit square or rectangular cross sections [START_REF] Lobue | Analysis of volume distribution of power loss in ferrite cores[END_REF]. Therefore, the problem of eddy current computation in a square or rectangular region still arises (in this case, the resistivity to consider is the bulk resistivity of the sample, and the dimensions are the ones of the cross section). This clearly emphasizes the need to accurately and efficiently compute the eddy current loss in a square, rectangular, or elliptical region. As no analytical classical loss formula exists for squares, rectangles, or ellipses, a finite element computation is often used to solve the Poisson equation [START_REF] De La Barrière | Loss separation in soft magnetic composites[END_REF]. This procedure can be quite cumbersome, and does not offer explicit expressions as analytical formulae.

The purpose of this paper is to propose respectively a semianalytical formulation for the eddy current loss problem in a rectangular (or square) region, and an analytical solution in the elliptical one. This can be useful both for microscopic classical loss computation (in this case, the considered region is the grain, and the resistivity to consider is the one of the grain material), and for macroscopic eddy currents (then, the outer geometry of the region is the cross section, and the bulk resistivity of the material is used). In this paper, all computations are done under the hypothesis of skin depth much larger than the region edges. In the first part, the analytical method is detailed. In the second part, the results are G compared with finite elements, and approximation formulae are also worked out. Finally, as an example, these methods are applied to the microscopic eddy current computation for a SMC.

II. ANALYTICAL FORMULATIONS

A. Problem to solve and simplifying assumptions

The simplifying assumptions are the followings.

-The eddy current paths are considered as bidimensional, flowing in the plane perpendicular to the flux density. The flux density is assumed purely alternative, i.e. applied in a given direction.

-The particle is assumed to be small enough to neglect skin effect in the range of frequencies of interest. A Cartesian axis system xyz is introduced. The applied induction B a (t), which is also the local one since skin effect is ignored, is directed along the z axis, as well as the magnetic field H(x,y,t), that can therefore be considered as a scalar. The current density ( ) ( )

, , ,

x y J x y t J J = , on the contrary, is a vector located in the xy plane. Under this set of simplifying assumptions, it has been shown in [START_REF] Bertotti | Hysteresis in Magnetism[END_REF] that the equation giving the magnetic field is a Poisson equation:

2 2 2 2 a dB H H H dt x y σ ∂ ∂ ∆ = + = ∂ ∂ (1)
where σ the conductivity of the material. The boundary condition on the region edges is H=H a (t), H a (t) being the external applied field. The problem to solve has been represented in Fig. 1. The rectangular has a width L along the x axis, and a height l along the y axis. For the elliptical shape, we take the major axis A along the x and the minor axis B along the y axis. Once the magnetic field H(x,y,t) is known, the current density can be obtained by the following partial derivations:

x y H J y H J x ∂  =  ∂   ∂  = -  ∂  (2)
And the classical loss, in terms of energy lost per cycle of period T and per unit volume, is (S is the region cross section):

( ) ( )

2 2 class 0 1 1 T x y S W J J dxdydt S σ = + ∫ ∫∫ (3)
B. Analytical developments Expression (3) of the classical loss will be written, whatever the shape, under the following form (see the Appendix for a detailed explanation):

[ ] 2 class a T dB W K S dt dt σ   = ⋅ ⋅ ⋅     ∫ (4)
In the previous expression, K is a factor depending on the region shape only (and not on its absolute dimensions). It can be shown that the expression (4) is valid for any region shape and size (provided skin effect can be neglected). In the following, we will give an analytical, or when not possible a semi-analytical (i.e. under the form of a converging infinite sum) expression of the shape factor K, for rectangular and elliptical grain shapes.

1) Rectangular and square shapes In this case, a semi-analytical expression of the shape factor K is searched.

The solution of (1) is written:

( ) ( ) ( ) ( ) 1 2 , , , , , a H
The term H 1 (x,t) has been chosen to verify the Poisson equation, and only depends on the x coordinate. The following expression is proposed (this potential is chosen equal to zero on the segments MQ and NP of Fig. 1(a)):

( ) 2 2 1 1 , 2 4 a dB L H x t x dt σ   = -     (6)
The field H 2 (x,y,t) is harmonic (i.e. verifies ∆H 2 =0). It should be equal to zero on MQ and NP (Fig. 1(a)), and equal to the opposite of H 1 (x,t) on the segments MN and QP. H 2 (x,y,t) is made fictitiously periodic along the x direction, so the following mathematical expression can be proposed (the a k functions are a family of unknown functions):

0 x z y 0 x z y ( ) ( ) ( ) 2 1
, , , cos 2 1

k k H x y t a y t k x L π ∞ =   = -     ∑ (7)
Injecting [START_REF] Jiancheng | Homopolar 2-Pole Radial Permanent-Magnet Biased Magnetic Bearing With Low Rotating Loss[END_REF] in the Laplace equation ∆H 2 =0, a second order differential equation is obtained for each a k function:

( ) ( ) 2 2 2 1, 2 1 , 0 k k a k k x a y t L y π ∂   ∀ ≥ - - =   ∂   (8)
The solution of this equation is a linear combination of hyperbolic functions. The expression of H 2 (x,y,t) is therefore given by:

( ) ( ) ( ) ( ) ( ) 2 1
, , sinh 2 1 2

sinh 2 1 cos 2 1 2 k k L H x y t h t k y L L k y k x L L π π π ∞ =      = - -                  + - + ⋅ -              ∑ (9)
The set of functions h k (t) only depends on time. After an expansion of the field H 1 (x,t) into Fourier series of the x coordinate, the boundary condition H 2 (x,y=±l/2,t)=-H 1 (x,t) on the segments MN and QP leads to the following expression, for the h k coefficients:

( ) ( ) ( ) ( ) 2 1 3 3 4 1 2 1 sinh 2 1 k a k dB L h t dt k k l L σ π π - = - -   - -     (10)
Knowing the terms H 1 (x,t) and H 2 (x,y,t) of ( 5), the currents J x and J y can be obtained by the derivation [START_REF] Anhalt | Dynamic losses in FeSi filled polymer bonded soft magnetic composites[END_REF]. The surface integral (3) can then be carried out to obtain the classical loss W class . All computations made, and given the fact that the region cross section S is equal to L.l for rectangular shapes, it turns out that the K (RECT) coefficient, which only depends on the height to width ratio R=l/L, is given by the following infinite series:

( ) ( ) ( ) ( ) ( ) ( )
( )

4 4 1 2 5 8 1 2 1 1 exp 2 1 2 1 1 exp 2 2 1 2 1 RECT k K R k k R R k R k π π π π ∞ =   =  -      - - -    -    - - - -     ∑ (11)
Equation ( 11) is an infinite sum. However, its numerical computation requires a finite number of terms. Therefore, the maximal number of terms is chosen using the following method: this number is increased until the partial sum do not change anymore, which means convergence of the computation (a relative difference of 0.5% is accepted). More precisely, calling K N (RECT) the partial sum of the series (12) with N terms, K N (RECT) is said to be equal to K (RECT) for the first number N verifying the condition:

( ) ( ) ( ) 1 0.5% RECT RECT N N RECT N K K K + - ≤ (12)
Using this criterion, as the series (11) exponentially converges, a maximal number of terms inferior to 20 is sufficient to compute K (RECT) whatever the R value. This can be achieved very rapidly, the computation time being divided by more than 100 in comparison with the finite element method to solve (1), which requires a cumbersome matrix inversion. This coefficient is plotted in Fig. 2 in function of R=l/L. This figures illustrates the fact that for a given excitation induction B a (t) and a given cross section S of the rectangle, the loss is reduced if the region exhibits a "flat" shape, that is to say if its height to width ratio is small. The maximal value is obtained for a square shape. Due to the fact that square shapes are often adopted in the literature [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF], it might be interesting to give the numerical value of the coefficient for this particular case: K (SQUARE) =3.51•10 -2 . An extended table of the K (RECT) coefficients in function of R is given in the Appendix. The validity of the previous approach can be also shown if the expression [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF] asymptotically converges when R→0 towards the specific loss in a lamination. Indeed, a lamination is a rectangular whose width L is much larger than its height l. The smallest dimension l is kept constant, and the ratio R=l/L tends towards zero. It is well known [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF] that the specific loss in a lamination without skin effect is equal to:

( ) [ ] 2 2 LAM class 12 a T dB l W dt dt σ ⋅   =     ∫ (13)
Equation ( 4), in the case of a rectangular, can be written under the following form:

K (RECT) K (ELLIP) K(R) R ( ) ( ) ( ) [ ] 2 RECT 2 class RECT a T K R dB W l dt R dt σ   = ⋅ ⋅ ⋅     ∫ (14)
(13) and ( 14) are in correct agreement if the ratio K (RECT) /R converges towards 1/12 when R→0. An asymptotic expansion to the third order gives:

( ) ( ) ( ) ( ) ( ) ( ) 2 3 0 1 exp 2 1 2 1 1 2 1 2 24 1 exp 2 2 1 R k R k R k R k R π π π π →   - - - -    → - -       - - -   (15)
Injecting ( 15) in [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF] and dividing by R, it can be found:

( ) ( ) 0 2 2 1 2 1 3 2 1 RECT R k K R k π → ≥  → - ∑ (16)
A summation provides:

( ) 2 2 1 1 8 2 1 k k π ∞ = = - ∑ (17)
Finally, the following result can be established:

( ) 0 1 12 RECT R K R →  → (18)
(18) demonstrates the consistency between ( 11) and ( 13). This convergence can also be illustrated in Fig. 3. 2) Elliptical shapes For non isotropic materials, the case of elliptical particles for the computation of microscopic classical loss can be of interest [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF]. A is the major axis of the ellipse, and B is its minor axis. The ratio R=B/A represents the shape factor of the elliptical particle, analogue to the ratio R=l/L of the rectangular one. Calling ( ) ( )

2 2 2 2 A B a = - and ( ) argtanh r R =
, the analytical developments are carried out in the following ρθ coordinate system:

( ) ( ) ( ) ( ) cosh cos sinh sin x a y a ρ θ ρ θ = ⋅ ⋅    = ⋅ ⋅   (19)
The Poisson equation becomes:

( ) ( ) 2 2 2 2 2 cosh 2 cos 2 2 a dB H H a dt σ ρ θ ρ θ ∂ ∂ + = -     ∂ ∂ (20)
Separating the variables ρ and θ, and taking into account the boundary conditions, the following solution can be found:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , cosh 2 cosh 2 8 cosh 2 1 cos 2 cosh 2 a a dB a H H t r dt r ρ θ σ ρ ρ θ = + -      + -          (21)
The derivation of this expression gives the current density components of (2) that can be substituted in [START_REF] Guo | Core Loss Calculation for Soft Magnetic Composite Electrical Machines[END_REF]. The integral of (3), performed on the elliptical cross-section, takes the form:

( ) ( ) ( ) [ ] 2 2 ELLIP class 1 cosh 2 16 cosh 2 a T dB a W r dt r dt σ     = -           ∫ (22)
Therefore, the ellipse cross section being equal to S=π/4•A•B, the K (ELLIP) coefficient of formula (4), which is only dependant on the ratio R=B/A and not on the absolute lengths A and B, has the following expression:

( ) ( ) ( ) ( ) ( ) ELLIP 1 1 1 tanh cosh 2 16 tanh cosh 2 K r r r r π     = - -             (23)
The previous expression has been represented in Fig. 2 in function of the shape factor R=B/A for the ellipse (it is recalled that r=argtanh(R)). It is interesting to notice that for a given region cross section S, and for a given shape ratio R of the particle (equal to l/L for the rectangular and B/A for the ellipse), the loss is approximately 10% higher for the elliptical shape than for the rectangular shape.

It is recalled that for the circular case [START_REF] Bozorth | Ferromagnetism[END_REF] (radius A/2), the loss is equal to:

( ) [ ] 2 2 CIRCLE class 1 8 2 a T dB A W dt σ     =         (24)
This means that the K (CIRCLE) factor for the circle is equal to 1/(8π)≈3.98•10 -2 . As shown in Fig. 2, this is the limit of expression (23) when R=B/A→1.

A. Finite element verification

The Poisson equation ( 1) can also be solved by finite elements (FEM) 1 , and the obtained loss or K factor can be compared to the one provided by the analytical method. This verification is shown in Fig. 4 for what concerns the K (RECT) factor of the rectangular. Exact correspondence can be observed. However, the computation time is more than 100 times faster for the semi-analytical method. The current lines can also be computed by the analytical method, and the FEM method (they are the iso-level lines of the field H over the cross section). The results are shown in Fig. 5 for the rectangular case. The analytical and FEM results are also very similar. Similar results and conclusions can be found for elliptical regions.

B. Simplified formulations for the rectangular

Formula ( 11) provides an exact semi-analytical computation of the K (RECT) coefficient. However, the fact it is given under the form of an infinite series can be a limit for its practical use. In this section, we propose a very simple analytical formula for the loss in rectangular shapes, based on a simplification of [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF]. This simplified formula is compared with other existing simplified formulae for the loss in rectangular regions available in the literature. The simplified formula we propose is based on the fact that the ratio K (RECT) /R, represented in Fig. 3 as a function of R, looks like a linear curve with negative slope. A slope value equal to -1/20 has been empirically determined to fit the ratio K (RECT) /R with a satisfying precision on the whole interval [0;1] for R. Taking into account, as demonstrated before, that when R tends towards 0, the ratio K (RECT) /R tends towards 1/12 (the lamination case), the following formula is proposed:

( ) 1 1 12 20 RECT K R R = -⋅ (25)
Some authors [START_REF] Bunet | Courants de Foucault[END_REF] have tried to compute the eddy current distribution in rectangular regions using the approximation of rectangular eddy current paths. These rectangular paths have the same height to width ratio than the region and the same center (Fig. 6). Fig. 6: Derivation of the simplified loss formula in a rectangular region presented in [START_REF] Bunet | Courants de Foucault[END_REF], based on the approximation of rectangular eddy current lines The conductance of an elementary current path as the one represented in Fig. 6 has the following expression (the vertical portions of this elementary current path are between the coordinates x and x+dx):

2 4 1 R dx dG x R σ = + (26)
The RMS value of the electromotive force in the elementary current path is given by:

( ) [ ] 2 2 1 4 a T dB E x Rx dt T dt   =     ∫ (27)
The classical loss in the region is therefore given by integration:

( ) 2 BUNET 2 class 0 1 L W T E dG S = ∫ (28)
After calculation, it is obtained:

( ) [ ] 2 BUNET class 2 1 16 1 a T dB R W S dt dt R σ   = ⋅ ⋅ ⋅ ⋅   +   ∫ (29)
This implies a K (BUNET) factor equal to:

( ) BUNET 2 1 16 1 R K R = ⋅ + (30)
Although simple, it is clear that the formula (30), contrary to [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF], fails to reproduce the loss behaviour observed for a lamination, i.e. when R→0. Indeed, it has been shown in Section II.B.1) that the ratio K/R should tend towards 1/12 when R→0 to correctly reproduce the lamination behaviour, while from formula (30), K (BUNET) /R→1/16. This error comes from the fact that the assumption of concentric current lines with the same height to width ratio than the region is wrong for small ratios R=l/L.

The exact formula [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF], the approximate one (25), and the one of Bunet (30) are compared in Fig. 7 (in terms of the K/R ratio). Although conceptually and analytically simple, the formula of Bunet (30) is quite far from the exact analytical result. The linear approximation (25) appears to be accurate on the whole range R, which is a clear advantage. Fig. 7: K/R ratio in function of R obtained for the rectangular (limit case of the lamination when R→0, exact formula K (RECT) /R [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF], linear approximation (25), and K (BUNET) /R ratio obtained by Bunet (30))

C. Application: computation of the microscopic classical loss in a Soft Magnetic Composite using a statistical model of the grain sizes and shapes

In [START_REF] De La Barrière | Characterization and prediction of magnetic losses in Soft Magnetic Composites under distorted induction waveform[END_REF], a method has been proposed to compute the classical loss in soft magnetic composites. It is based on a statistical analysis of micrographs such as the one shown in Fig. 8. As done in [START_REF] De La Barrière | Characterization and prediction of magnetic losses in Soft Magnetic Composites under distorted induction waveform[END_REF], the sample size is assumed to be small enough to neglect the macroscopic classical loss contribution (i.e. the currents flowing from grain to grain at the scale of the sample). Only microscopic eddy currents flowing at the scale of the grain are taken into account. Therefore, the region to consider in the computations is the single grain. As the grains are made of pure iron, the conductivity to consider is the one of iron σ Fe =9.93•10 6 S•m -1 . Using an image processing software, each grain has been isolated, and made equivalent to a rectangular of same cross section S, and same height to width ratio R (Fig. 9). The reason for choosing a rectangular shape is the important material density due to high pressure compaction, which cannot be obtained by an array of cylinders. Dealing with several micrographs and using appropriate statistical tools [START_REF] De La Barrière | Characterization and prediction of magnetic losses in Soft Magnetic Composites under distorted induction waveform[END_REF], a bivariate lognormal distribution (parameters (S,R)) has been identified for the material:

( ) ( ) ( ) ( )( ) 2 , 2 , ln 1 1 , exp 2 1 2 ln ln ln S S S R S R S R S R R R S R S f S R S R S R R α ξ πξ ξ χ χ α α α ξ ξ ξ  -  =  ⋅ -    - - -  + -    (31)
The classical loss is computed using the following integral:

( ) ( ) [ ] ( ) ( ) ( ) class p 2 RECT 0 0 , , SMC a T W J f f S R K S dRdS dB dt R dt σ ∞ ∞ = ⋅   ⋅ ⋅     ∫ ∫ ∫ (32)
The double integral in (S,R) has to be computed numerically. This can be time consuming if the K (RECT) factor is computed using finite elements, because a quite important number of shape factor evaluations can be required for an accurate integral computation. This was the method followed in [START_REF] De La Barrière | Characterization and prediction of magnetic losses in Soft Magnetic Composites under distorted induction waveform[END_REF], and several minutes were required. On the contrary, using the simplified formula (25) for the K factor, the computation becomes instantaneous (less than one second), for a similar accuracy (Fig. 10). This case for which a statistical distribution of the grain sizes is used is a typical case for which a simplified analytical formula for the K (RECT) coefficient (like (25)) is valuable from the computational point of view. IV. CONCLUSION

In this paper, a semi-analytical formula for the classical loss in rectangular-shaped regions and an analytical formula in elliptical ones have been proposed. It has been shown that the formula for the rectangular tends towards the well-known loss expression in laminations if the height to width ratio of the rectangular tends towards zero. Finite element validations are also proposed. The loss expression in rectangular regions can be both used for the computation of the macroscopic classical loss component (i.e. currents flowing at the scale of the sample which exhibits a rectangular cross section), and for microscopic classical loss evaluation for highly compacted materials. Elliptical shapes are useful for microscopic loss prediction in low density materials with anisotropy. An important decrease of the computation time compared to conventional methods such as finite elements has been observed.

In the second part of the article, a very simplified linear formula has been proposed for the classical loss in rectangular regions. This formula has been applied to rapidly and accurately compute the microscopic classical loss in a SMC, which requires a statistical computation to take into account the various shapes and sizes of the grains. The statistical properties of the grain have been retrieved by micrograph analysis.

V. APPENDIX

A. Explanation on formula (4)

Rather than using the magnetic field H(x,y,t) in this section, it is more convenient to use the electric vector potential T(x,y,t) chosen as:

( ) ( ) ( ) , , , , a x y t x y t T H H t = - (33)
In this way, the Poisson equation (1) written in terms of electric vector potential T verifies homogeneous boundary conditions on the domain boundary Γ:

0 a dB T dt T σ Γ  ∆ =    =  (34)
The current density is given by the same spatial derivatives as with the magnetic field:

x y T J y T J x ∂  =  ∂   ∂  = -  ∂  (35)
Consider a region of arbitrary shape Ω (spatial coordinates (x,y)), and normalize it to a region Ω n (normalized coordinates (x n ,y n )), chosen in such a way that the domain cross-section S n in the normalized coordinate system is equal to 1. The homothetic factor between the normalized region Ω n and the real region Ω has the dimension of a length and is called k (Fig. 11). The cross-section of Ω is equal to S=k 2 . In the real region Ω, the electric potential T(x,y,t) is obtained solving (34) in the (xy) system. In the normalized coordinate system (x n ,y n ), one can find the normalized potential t(x n ,y n ) verifying the following Poisson equation: Carrying out the spatial and temporal means of (3), and remembering that S=k 2 , it is found for the specific classical loss per cycle:

( ) 2 2
( ) [ ] 2
σ       ∂ ∂     = ⋅ + ⋅ ⋅       ∂ ∂           ∫∫ ∫ (40)
Therefore, noticing that the coefficient K defined as in (41) does not depend on the absolute dimensions of the region Ω, formula (4) is found.

    ∂ ∂ = +     ∂ ∂     ∫∫ (41)
For example, for what concerns the lamination with very small height to width ratio R=l/L, the classical loss is given by ( 14). An obvious algebraic manipulation gives, since the cross section is equal to S=L•l: The coefficient K in this important case of thin laminations is equal to K=1/12•R. (RECT) coefficients in function of R An extended table of the K (RECT) coefficients in function of R, computed from expression [START_REF] Bottauscio | Electromagnetic phenomena in heterogeneous media: Effective properties and local behavior[END_REF], is given in Table I:

B. Table of the K

Fig. 1 :

 1 Fig. 1: Bi-dimensional Poisson equation to solve, (a): rectangular region, and (b): elliptical region.

Fig. 2 :

 2 Fig. 2: Representation of the K coefficient in function of the ratio R (case of the rectangular for which R=l/L, and case of the ellipse for which R=B/A)

Fig. 3 :

 3 Fig. 3: Convergence of the ratio K (RECT) /R towards 1/12 if R→0

Fig. 4 :

 4 Fig. 4: Computation of the K (RECT) coefficient in function of the ratio R=l/L, by the semi-analytical formula (11), and by FEM

(a): Analytical computation 1 Fig. 5 :

 15 Fig. 5: Current lines comparison between the analytical and FEM methods (rectangular of normalized absolute dimensions, normalized excitation dBa/dt, and shape ratio R=l/L equal to 1/2)

Fig. 8 :

 8 Fig. 8: Micrograph of a SMC (provided by the Höganäs company)

Fig. 9 :

 9 Fig. 9: Equivalence of each grain to a rectangular

Fig. 10 :

 10 Fig. 10: Classical loss in the SMC of Fig. 8 in function of the frequency, for a peak polarization Jp=1 T (use of the finite element method to compute K [19], and use of the simplified formula (25))

Fig. 11 :

 11 Fig. 11: Definition of a normalized region Ωn chosen in such a way that the normalized cross-section Sn is equal to 1

  using the linearity of the Poisson equation, it can be obtained:

TABLE I K

 I (RECT) COEFFICIENTS IN FUNCTION OF R (EXPRESSION (11))

	Homothetic factor k	y=k.y n
	y n
		W
	W n
	0	x n	0	x=k.x n
	G n		G
				R	K (RECT)
				0	0
				0.1	7.81•10 -3
				0.2	1.46•10 -2
				0.3	2.03•10 -2
				0.4	2.49•10 -2
				0.5	2.86•10 -2
				0.6	3.13•10 -2
				0.7	3.32•10 -2
				0.8	3.43•10 -2
				0.9	3.50•10 -2
				1	3.51•10 -2


O De La Barrière

C Appino

C Ragusa

F Fiorillo

F Mazaleyrat

M Lobue

High-frequency rotational losses in different Soft Magnetic Composites (SMC)

The isotropic properties of Soft Magnetic Composites (SMC) favor the design of new machine topologies and their granular structure can induce a potential decrease of the dynamic loss component at high frequencies. This paper is devoted to the characterization of the broadband magnetic losses of different SMC types under alternating and circular induction. The investigated materials differ by their grain size, heat treatment, compaction rate and binder type. It is shown that, up to peak polarization J p = 1.25 T, the ratios between the rotational and the alternating loss components (classical, hysteresis, and excess) are quite independent of the material structural details, quite analogous to the known behavior of nonoriented steel laminations. At higher inductions, however, this is no more the case. It is observed, in particular, that the J p value at which the rotational hysteresis loss attains its maximum, related to the progressive disappearance of the domain walls under increasing rotational fields, decreases with the material susceptibility.

I. INTRODUCTION

Embedded applications, such as hybrid vehicles or aircraft systems, are a challenge for machine designers. Indeed, the mass of these systems, and then the actuator torque density, are crucial parameters. Possible solution call either for the increase of the actuator working frequency, in order to improve the torque density [1], or the design of non-conventional machine topologies such as claw pole machines [2]. For both alternatives, the technological limitations of conventional laminated magnetic materials may be reached. Actually, the classical eddy current losses can become too important at high frequencies, while laminated materials only allow flux circulation in the lamination plane, and therefore are not suitable for claw pole machines with 3D flux paths [2]. Soft Magnetic Composites can be a satisfying solution, because their granular structure brings about a significant reduction of the classical losses [3], while exhibiting good isotropic properties [4]. Consequently, there is a clear necessity to provide machine designers loss results and models for such materials under multidimensional induction excitations.

The relatively low susceptibility of SMC, and the necessity to reach the range of frequencies for which these materials are designed, can make the experiments particularly difficult. For laminations, most of the 2D induction loss results are provided for the industrial frequency of 50 Hz [5], or up to 200 Hz [6]. On the contrary, SMC require testing in the kilohertz range [4]. A few papers present such results. In [4], the peak induction of 0.54 T has been reached for a maximum frequency of 1000 Hz under 3D induction loci, using a special winding of the measurement setup (the number of turns is decreased while the frequency increases). In [7] loss results under alternating and circular inductions in a SMC disk sample up to 1.25 T at 1000 Hz are proposed, using a specially designed three phase magnetizer. However, contrary to the alternating induction case [8], no comparative study of the loss under multi-dimensional induction waveforms in different SMC samples (with different grain sizes, different preparation methods or binder…) has been performed so far. This makes difficult the choice of the best SMC for a given application.

In this paper, three different SMC samples have been experimentally characterized under alternating and circular induction loci up to the kilohertz range, using the system presented in [7]. In the following we discuss first experiments made up to peak polarization J p = 1.25 T. It is shown that, in spite of different loss values and behaviors in the different materials, the ratio of the circular to the alternating hysteresis loss components appears to be quite independent of the sample, the same being also true for the excess loss. We consider then the material behavior at higher peak polarizations, experimentally showing that the J p value at which the hysteresis loss under circular polarization reaches its maximum is correlated with the material susceptibility.

II. EXPERIMENTAL

A. Measurement setup

The measurement setup for SMC testing under 2D induction is similar to the one used in [7]. Diskshaped samples have been pressed directly to the final size (diameter 80 mm, thickness 3 mm), i.e. no cutting has been made. For mechanical reasons, samples thinner than 3 mm could not be manufactured.

The disk samples have been centered in the bore of a three phases/two poles magnetizer endowed, for maximum field homogeneity, with three slots per pole and per phase and a toroidal winding, minimizing the copper losses and avoiding overhangs crossing. The geometry of the magnetizer has been optimized by 3D finite elements, in order to reach the best induction-frequency combination under power supply by a CROWN AUDIO 5000VZ linear amplifier.

The loss measurements were performed out by the fieldmetric method (double B-coil and H-coil [7]). The measurement zone is a square of 20 mm at the center of the sample, offering a good compromise between signal strength and induction homogeneity, the latter being verified by a finite element computation, as made in [9]). The H-coils have been tested by an impedance analyzer to check that the capacitive effects between turns were negligible in the measurement range of frequencies.

Induction loci are controlled thanks to a feedback algorithm [10]. For the case of alternating sinusoidal induction, the loss measurements have been carried out along two perpendicular directions and the results have been averaged. For measurements under circular polarization, averaging is made of the loss figures obtained for clockwise and counterclockwise rotation.

B. Investigated SMC materials

In this paper, three kinds of commercial SMC are characterized and compared. These materials are made of pure iron grains with electrical conductivity σ Fe =9.93•10 6 S/m (detailed information is given in Table I). The first two materials have been obtained using inorganic binders and are heat-treated at much higher temperature than Material 3, where an organic binder is used. The compacting pressure is also higher for the first two materials. The micrographic inspection shows that Material 1 has much larger average grain size than Materials 2 and 3 (167 µm versus 46 µm and 57 µm, respectively).

Material 3 shows quite larger dispersion of the grain size than Materials 1 and 2. The higher magnetic susceptibility of Material 1 makes it suitable for applications in electrical machines. Materials 2 and 3, with smaller grain size and higher resistivity (because of their lower density, the grain-to-grain contacts are reduced), have lower classical eddy current losses. They are consequently more suitable for highfrequency applications, such as power electronics inductors or very high speed machines [11].

III. ROTATIONAL AND ALTERNATING LOSSES UP TO J P = 1.25 T

In this Section, energy losses measured under circular and alternating sinusoidal induction are provided up to a few kHz for low and medium induction levels and maximum peak polarization J p = 1.25 T. Using a classical loss model, it is possible to separate the loss components and analyze for each material the ratios between hysteresis loss under circular and alternating conditions R hyst (J p )=W hyst (CIRC) (J p )/W hyst (ALT) (J p ), and the excess loss ratios R exc (J p )=W exc (CIRC) (J p ,f)/W exc (ALT) (J p ,f).

For each material, loss measurements at the polarization levels J p = 0.2 T, 0.5 T, 1 T, and 1.25 T have been carried out under circular and alternating sinusoidal induction loci. The results are shown in Fig. 1 for J p = 0.5 T, 1 T, and 1.25 T. Material 1 has the highest dynamic loss (both under circular and alternating polarization). This is chiefly due to its large grain size, which is responsible for a significant mesoscopic classical loss [8]. The small grain-sized Materials 2 and 3, present in fact quite reduced dynamic loss. This has, however, a counterpart in a higher hysteresis (quasi-static) loss. It is remarked that these behaviors, typically observed under alternating field, equally hold for circular polarization. In the following Section we will more precisely show that the ratios of rotational to alternating loss components (hysteresis, excess, and classical loss) are to a good extend independent of the material.

The effect of the heat treatment is apparent on comparing Materials 2 and 3. Although Material 2 has smaller grain size and is manufactured with a higher compaction pressure (Table 1), it has lower hysteresis loss under identical excitation conditions. Material 2 is prepared with inorganic binder and is heat-treated beyond 500°C, while Material 3, with its organic binder, is cured at low temperatures (200°C), to avoid binder deterioration. Higher hysteresis loss are inevitably associated with Material 3, because, as explained in [12], the heat treatment helps relieving the residual stresses arising during the compaction process.

B. Loss separation under alternating and circular polarization

The starting point of the loss separation is the classical loss computation. The main assumption is, like in [8], that the classical loss remains confined at the scale of the single grain (we speak of microscopic classical loss), and that eddy currents circulating on a larger scale because of random grain-to-grain contacts are negligible. This is a reasonable assumption when considering, as in the present case a few millimeter thick samples [8]. The classical loss computation under alternating field conditions W class (ALT) (J p ,f) is carried out using the microscopic eddy current loss computation proposed in [13], taking into account the grain size and shape dispersion observed on micrographs. Following [6], the classical loss under circular field conditions is obtained doubling the classical alternating loss, so that the ratio

R class = W class (CIRC) (J p , f) / W class (ALT) (J p ,f) = 2.
The ratios of the circular to the alternating hysteresis loss R hyst (J p ) (this loss component being obtained by extrapolating the total loss to zero frequency [7]) are given versus J p in Fig. 2 for each material. R hyst (J p ) decreases with J p [7], with the quite interesting property of being little dependent on the material type. This quite in agreement with previous results obtained in Fe-Si laminations having different grain size [START_REF] Ragusa | Comprehensive investigation of alternating and rotational losses in non-oriented steel sheets[END_REF].

A similar comparison for the excess loss ratio R exc (J p ) has been carried out (Fig. 3). It has been experimentally shown in [7] that this ratio is almost frequency independent. A somewhat similar conclusion can be drawn for Materials 1 and 3 (Fig. 3), although the results obtained in Material 3 are very noisy because of the small amount of dynamic loss. Material 2 has so small dynamic loss that an accurate evaluation of the dynamic loss component is not possible.

IV. HYSTERESIS LOSSES UP TO 1.6 T

In this section, higher peak polarization values, up to J p =1.6 T, are considered, both under circular and alternating flux, for the three materials. The actual measuring frequency f = 50 Hz can be considered as quasi-static in these materials, given the correspondingly negligible dynamic loss contribution. Thus, we can assume W tot (J p ,50 Hz) = W hyst (J p ).

Quite remarkably, it is observed that the rotational hysteresis loss W hyst (CIRC) (J p ) attains its maximum at different J p values in the different materials. In particular, J p,MAX drifts from 1.5 T to 1.25 T on going from Material 1 to Material 3. A similar trend cannot be observed in the conventional nonoriented Fe-Si laminations, where W hyst (CIRC) (J p ) invariably attains a maximum value close to J p = 1.5 T [START_REF] Ragusa | Comprehensive investigation of alternating and rotational losses in non-oriented steel sheets[END_REF]. The fact that W hyst (CIRC) passes through a maximum value, to eventually drop to zero value on the approach to saturation, points to the contribution of the coherent magnetization rotations, which are reversible in nature, becoming significant with respect to the domain wall processes. It is apparent that the evolution of W hyst (CIRC) (J p ) shown in Fig. 4 is a peculiar consequence of the granular structure of the SMC materials and the role of the internal demagnetizing fields. We actually see in Table I that the drift of the maximum of W hyst (CIRC) (J p ) correlates with a similar drift of the material susceptibility, in turn strictly depending on the material density. It is clear, looking also at the shape of the hysteresis loops, that the susceptibility is chiefly determined by the internal demagnetizing fields, which are pretty strong, in view of the non-negligible thickness of the non-magnetic grain boundary layer. Lower susceptibility SMC are associated with thicker layers, higher average internal demagnetizing effects, and broader distribution of the local demagnetizing coefficients. Very broad distribution of the internal demagnetizing fields implies that higher applied fields are required for achieving any given J p . It happens then that the domain wall processes are quite not exhausted, contrary to magnetic laminations, when the applied field is high enough to engender coherent rotations in the softer (i.e. surrounded by thinner layers) grains. The coherent rotations tend eventually to occur at lower inductions in the lower susceptibility SMC materials, bringing with them a similar trend for the turning point of W hyst (CIRC) (J p ).

V. CONCLUSION AND PERSPECTIVES

In this paper, measurements under alternating and circular inductions in different SMC types (grain size, binder, preparation) have been proposed, up to 3 kHz. The loss components have been separated, and it has been shown that the ratios of these components, measured under circular and alternating polarization, do not significantly depend on the material type.

The alternating and rotational hysteresis losses have also been measured up to high inductions (J p = 1.6 T). It has been shown that the rotational hysteresis loss W hyst (CIRC) (J p ) passes through its maximum value at lower J p values in lower susceptibility materials. This effect, never observed in the conventional soft magnetic laminations, can be qualitatively justified by taking into account that the susceptibility in SMC is chiefly determined by the thickness of the non-magnetic grain boundaries, that is, the strength of the internal demagnetizing fields. The higher these fields, as occurring in lower density materials, the broader their distribution and the stronger the field to be applied to reach a given J p value. This inevitably leads to earlier appearance of coherent magnetization rotations with respect to denser materials and anticipated fall of W hyst (CIRC) (J p ) versus J p .

Table captions

Table 1: Main physical properties of the SMC materials used in this work.

Figures captions

Fig. 1: Total energy loss per cycle under circular and alternating sinusoidal induction loci as a function of the frequency f, for the induction peak values J p =0.5 T, 1 T, and 1.25 T. Fig. 2: Hysteresis loss ratio R hyst (J p )=W hyst (circ) (J p )/W hyst (alt) (J p ) versus the peak polarization J p , for each material.

Fig. 3: Excess loss ratio R exc (J p )=W exc (circ) (J p ,f)/W exc (alt) (J p ,f) versus the peak polarization J p , for each material, at three different frequencies (f=100 Hz, 500 Hz, and 1000 Hz).

Fig. 4 :

 4 Fig. 4: Measured loss W tot (J p ,f=50Hz) under circular and alternating sinusoidal induction loci, in

b Measured relative magnetic susceptibility, defined for an alternating field as χ r =J p /(µ 0 H p ), J p is the peak polarization chosen equal to 1 T, H p is the corresponding peak magnetic field under static conditions. c The abbreviation rsd means "relative standard deviation", which is, in statistics, the ratio between the standard deviation and the mean of a set of data.


Pascal Audigane
email: p.audigane@brgm.fr

Alireza Ebrahimiyekta

M Pichavant

Evaluation of Hydrogen migration and geochemical reactivity into underground

Keywords: Underground storage, Gas migration, Porous rock, Fluid-rock chemical interactions

The use of hydrogen as an alternative for energy storage has emerged rather recently though it was originally more identified as a secondary energy carrier and storage medium. By way of electrolysis it becomes one of the major actors in the possible conversion of excess wind or solar energy with the favorable arguments of being environment-friendliness and having high storage densities [START_REF] Crotogino | Large-scale hydrogen underground storage for securing future energy supplies[END_REF]. The concept of "Power to Gas" comes from the conversion of electricity into gas via electrolysis process. Hydrogen and Oxygen produced under various forms can be reused for several purposes and applications. Research project related to investigate the feasibility of storing hydrogen in porous geological formations are under development like the H2STORE project in Germany [START_REF] Ganzer | The H2STORE Project -Experimental and Numerical Simulation Approach to Investigate Processes in Underground Hydrogen Reservoir Storage Society of Petroleum Engineers[END_REF], the European project HyUnder which proposes to evaluate thoroughly from a technical, economic and societal standpoint if hydrogen underground storage can become a potentially attractive solution [START_REF] Correas | Storing renewable energy: is hydrogen a viable solution?[END_REF]. In Austria, the Undergound SUN.STORAGE project, led by the Rohöl-Aufsuchungs Aktiengesellschaft (RAG), conduct research and analysis of the impact of hydrogen on underground gas (methane) storage systems [START_REF] Sun | [END_REF]. In Pentagonia, Argentina, Hychico S.A. is conducting operations to convert electricity produced from the Diadema wind farm into hydrogen to be injected into a depleted oil field converted into a methane gas storage reservoir [START_REF] Hychico | [END_REF]. In this paper we assess the transport properties of hydrogen gas injected into a sandstone formation like. Using a core sample of the rock, laboratory experiment is conducted to evaluate the permeability relative to hydrogen gas phase flow into a water saturated sample of sandstone rock representative of the Alsace Triassic formation. Geochemical reactivity of dissolved hydrogen with clay minerals of the rock is also investigated using autoclave experiments, meaning that crushed powder of rock is remained in contact with hydrogen and water for several weeks to evaluate the possible change of the chemical composition of the water in contact with Hydrogen and rock mineralogy. These measurements should serve at providing reservoir storage criteria for estimation with numerical modeling tools of storage capacity of underground sedimentary formation for large scale storage of hydrogen.