Nonlinear Modeling of Magnetization Loss in Permanent ... - IEEE Xplore

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Valeo Equipements Electriques Moteur, 94046 Créteil Cedex, France. Permanent magnets are commonly used in electromechanical energy conversion devices ...
IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 11, NOVEMBER 2012

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Nonlinear Modeling of Magnetization Loss in Permanent Magnets Radu Fratila

, Abdelkader Benabou , Abdelmounaim Tounzi , and Jean Claude Mipo

L2EP/Université Lille1, Bâtiment P2, Cité Scientifique, 59655 Villeneuve d’Ascq, France Valeo Equipements Electriques Moteur, 94046 Créteil Cedex, France Permanent magnets are commonly used in electromechanical energy conversion devices, especially in permanent magnet synchronous machines (PMSM). When designing such devices, it is necessary to take account for the operating conditions of the permanent magnets. Ideally, the operating point of the magnet lies on a linear reversible demagnetization characteristic. Nevertheless, the operating conditions (temperature, demagnetization field, etc.) may have great influence on the magnetic characteristic that becomes strongly nonlinear and may lead to magnetization loss in the magnet. Consequently, the impact is also observed on the electrical machine performances. This paper presents a nonlinear model accounting for magnetization loss of permanent magnets which magnetic characteristic presents knee points in the demagnetization curve. The presented model is applied to analyze the local magnetization loss in magnets of a PMSM. Index Terms—Magnetization loss, nonlinear demagnetization curve, permanent magnet.

I. INTRODUCTION

P

ERMANENT magnets (PM) are widely used in various types of electrical machines and devices. However, in some circumstances, they can be subjected to magnetization loss such as cases of high dynamic in variable speed conditions or after a short circuit. Furthermore, the temperature rise, due to eddy current losses within the PM, also leads to irreversible loss of magnetization. Therefore, it is important to use an accurate description of the magnetic behavior law of the PM in order to predict any demagnetization that will impact their performances. Nowadays, the Finite Element Method (FEM) is widely used to model electromagnetic devices in order to describe with accuracy their magnetic operational conditions. In the case of the PM, these conditions are directly linked to the electrical machine structure and supply characteristics [1]. In fact, when designing high performance PMSM, based on rare earth magnets, the demagnetizing field and the temperature parameters are critical in the design process. In fact, if high temperature conditions are combined with a large demagnetizing field, full demagnetization can occur. Some works have been focused only on the investigation of eddy current losses in magnets, by analytical or FE methods [2], [3]. Although it was shown that these losses, caused by the spatial and time harmonics of the magnetic field, induce important increase of temperature, no magnetization loss was considered. For predicting magnetization loss, Kral et al. [1] present a linear model of PM that takes into account the temperature and the associated demagnetization effects. Ruoho et al. [4] propose a nonlinear PM exponent function model to highlight the effects of temperature on the PM magnetization loss. Other works have been interested in implementing hysteresis models, as proposed by Xie et al. and Rosu et al. [5], [6] to describe the hysteretic

Manuscript received March 02, 2012; accepted April 02, 2012. Date of current version October 19, 2012. Corresponding author: A. Benabou (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2012.2193878

behavior of permanent magnets in electrical machines. Such approach is of interest if the PM behavior law presents minor loops in the recoil curves that lead to additional losses. Nevertheless, in high variable speed PMSM, the magnets are mostly subjected to eddy current losses [7]. Moreover, the nonlinear numerical scheme used to solve the FE problem must be robust when hysteresis is considered [8]. In this paper, a nonlinear PM model is presented using the nonlinear Marrocco equation. The demagnetization loss is considered when high demagnetizing field occurs. In that case, the proposed nonlinear model is extended by varying the parameters with respect to the operating point of the PM. Firstly, the PM model is presented. Secondly, the FE numerical approach is developed. Finally, as application, a virtual PMSM model, with ferrite magnets, under demagnetization conditions is studied. II. PERMANENT MAGNET MODEL A. Physical Model The demagnetization curve of a permanent magnet with knee point is shown in Fig. 1. The initial magnetic state of the magnet is given by the load line and operating point . Depending on the magnetic operating conditions, the demagnetization process can be separated into two distinct contributions: — a reversible part (above the knee point ) where, if a deis applied, the operating point of magnetization field the magnet is determined by the intersection of the load and the demagnetization curve 1. If the demagneline tization field is reduced to zero, the operating point of the magnet will return to the operating point . — an irreversible part (below the knee point ) where, if a is applied, the operating point demagnetization field determined by the of the magnet will fall at the point and the curve 1. If the intersection of the load line demagnetization field is decreased to zero, the operating point of the magnet is . This point is determined by the intersection of the original load curve L and the recoil line 2 of the permanent magnet. In this model, the partial undesirable demagnetization of the magnet occurs when the operating point goes below the knee curve. It is also assumed that the recoil curve has of the

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 11, NOVEMBER 2012

where is the magnetic vector potential, the current density and the magnetic permeability. The permanent magnet is modeled using the following behavior law: (5) where is associated to the nonlinear demagnetization curve and is the remanent magnetic flux density. Expression (4) is solved using the weak formulation such that

(6) Fig. 1. Physical model of the demagnetization process in a permanent magnet.

the same slope as the reversible region of the permanent magnet [1], [4], [9]. B. Demagnetization Behavior Law Model In this work, the permanent magnets are considered at fixed temperature and the demagnetization loss occurs only under demagnetizing field as described in Section II-A. The permanent magnet behavior model is based on the Marrocco equation (1) that is primarily used for the anhysteretic nonlinear approximation of soft magnetic material behavior. In the same way, the Marrocco equation is used to fit the demagnetization curve of hard magnetic materials after being translated in the first quadrant by its coercive field : (1) and are, respectively, the magnetic In this expression, is the vacuum magfield strength and magnetic flux density, , are parameters that have to netic permeability and be determined by a fitting procedure with the experimental behavior law. It has been identified that parameters and present a polycorrenomial evolution versus the magnetic flux density sponding to the knee point (2) (3) Therefore, when a permanent magnet is subjected to a demagnetizing field, the calculation procedure consists in checking if the operating point is below the knee. If so, a recoil line is generated, according to the Marrocco parameter identification, to replace the original demagnetization curve. III. NUMERICAL MODEL A. Mathematical Model To investigate the demagnetization loss of permanent magnets in a synchronous electrical machine, an FE model is used. The magnetostatic vector potential formulation is used (4)

where { is the test function and the studied domain including the sub-domain of the permanent magnets. The nonlinear problem is then solved using a substitution scheme until convergence of the vector potential . Moreover, to take account for the magnetization loss in the PM, the demagnetization curve is updated during the FE calculation procedure at each time step and in each element of the PM. This is done according to the physical model presented in Section II-A along with the modification of the Marrocco parameters using (2) and (3). IV. APPLICATIONS In order to illustrate the application of the proposed approach, a virtual permanent magnet synchronous machine is studied. First, the model of the ferrite magnets used is presented as well as the fitted Marrocco curves and the variation of the parameters and . Then, we introduce briefly the model of the PMSM. The last part focuses on the presentation and analysis of the results. A. Identification of the Permanent Magnet Model The curve that is used to illustrate the proposed permanent magnet model is extracted from the datasheet of Eneflux-Armtek Magnetics [10] for a hard ferrite magnet at 20 (black line in Fig. 2). Its characteristics are such that the remanent magnetic flux density and the magnetic coercive field . The recoil curves (gray lines in Fig. 2) are constructed following the principle of demagnetization described in the previous section. In Fig. 3, the fitted Marrocco equation is shown with the associated recoil curves. The parameters of the Marrocco expression are given in Table I. For and , the polynomial laws, as described in Section II-B, are detailed. B. Studied PMSM The electrical machine is a three-phase permanent magnet synchronous machine with 4 poles and 18 stator slots. The permanent magnets are nonconductive hard ferrite placed on the rotor. Fig. 4 shows a cross section of the machine with the magnets placed on the rotor surface, the armature windings and a 2-D view of the mesh used for the numerical model. The main dimensions of the machine are: — the stator outer diameter: 42 mm — the rotor outer diameter: 20 mm

FRATILA et al.: NONLINEAR MODELING OF MAGNETIZATION LOSS IN PERMANENT MAGNETS

Fig. 2. Demagnetization and recoil curves of a hard ferrite magnet at 20

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Fig. 5. Magnet after magnetization loss for different angles.

Fig. 3. Fitting of the Marrocco equation on the measured data.

TABLE I MARROCCO PARAMETERS

Fig. 4. FE model of the 4-poles surface magnet machine.

— air-gap: 1 mm — core length: 90 mm Using the symmetries of the system, only half of the motor is modeled. The mesh is constituted of 12 276 prismatic elements

and 9381 nodes. Initially, the magnets are considered with the nonlinear demagnetization curve given by the datasheet. C. Results and Discussions In order to investigate the magnetization loss in different operating conditions, simulations of the PMSM are carried out with the same three-phase armature currents but for different angles between the armature currents and the e.m.f. Initially, the electrical machine is operating at maximum . Additional torque which corresponds to an angle , 45 and 60 , angles are also considered such that which correspond to lower torques developed by the machine when the armature currents are unchanged. These cases can be encountered in variable speed or torque conditions. In Fig. 5, the saturation of one magnet of the machine is shown for the different angles. It can be noticed that the location of the magnetization loss strongly depends on the angle between the permanent magnet and the stator fields. The magnetization of the magnet remains unchanged in the inner part but is significantly impacted in the outer part where the demagnetizing field penetrates the magnet. The loss of magnetization is up to 30% which is correlating with results found in other researches [1], [9]. A closer insight into the permanent magnet for two elements and is illustrated in Figs. 6 and 7 (see Fig. 5) located at and 30 , respectively. for remain In the first case we notice that operating points of on the linear part of the magnetization curve, whereas the operare already in the irreversible magnetization ating points of area, i.e., below the knee of the behavior law.

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V. CONCLUSION The presented model for the magnetization loss in permanent magnets, based on a nonlinear behavior law with variable parameters to account for the knee point of the curve, is simple and easy to implement in finite elements. The approach has been illustrated by the study of a virtual permanent magnet synchronous machine with ferrite magnets. Different operating conditions showed different local magnetization loss. Another further improvement of the model will be to deal with the temperature. This can be investigated in the case of rare earth magnets that are most sensitive to the temperature.

REFERENCES Fig. 6. Operating points for elements E1 and E2 in the PM for a

Fig. 7. Operating points for elements E1 and E2 in the PM for a

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For the second case, when , we observe that the operating points for the element show a slight loss in magnehave tization whereas the operating points of the element reached the magnetization loss area. More precisely, the element shows two successive magnetization losses: one that . occurred at point and the second one at point

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