CAD package for electromagnetic and thermal analysis using finite elements. The modeling of magnetic core made of laminations in Flux

Transcription

1 CAD package for electromagnetic and thermal analysis using finite elements The modeling of magnetic core made of laminations in Flux

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3 Table of contents Table of contents 1. Introduction Aim of magnetic cores made of laminations: limiting losses Overview Magnetic losses: general Limiting the magnetic losses: improving efficiency Model soft materials Overview Behavior laws B(H) and modeling Hysteresis a univocal relationship Saturation linear / nonlinear approximation Anisotropy isotropic / anisotropic model The sheet packages crystalline anisotropy and pseudo-anisotropy Modeling of stacked magnetic sheets Overview Dimension issues Simplifying by a homogenization technique Homogenization: presentation of a basic model Homogenization: Lamination, a more sophisticated model Modeling of static phenomena Overview Some explanations on the modeling context Case 1: low field / linear model 2D or 3D Case 2: «average/strong» field / non linear model 3D Lamination Modeling of dynamic phenomena Overview Some explanations on the modeling context Concretely, in Flux...38 The modeling of magnetic core made of laminations in Flux PAGE A

4 Table of contents Flux PAGE B The modeling of magnetic core made of laminations in Flux

5 Flux Introduction 1. Introduction About this document This document describes different calcilation modes to take into account the magnetic cores made of laminations in the 2D and 3D Flux models. To treat this topic, in this document the reader will find pieces of information on: the magnetic cores made of laminations interest and the various types of losses in the electro-mechanical devices (. 2 «Aim of magnetic cores made of laminations: limiting losses») the modeling of soft magnetic materials and the choice of a comportamental law permitting to take into account the hysteresis, saturation, crystalline anisotropy and of the pseudo-anisotropy associated with the laminations (. 3 «Model soft materials») the laminations model focusing on the dimensional problems and the possibilities of simplification associated to the homogenization techniques (. 4 «Modeling of stacked magnetic sheets») Starting from this information, the various possibilities offered by the Flux software are then treated in more detail. in a first approach, named «static», which does not consider the induced currents/eddy currents in the laminations, two cases are presented: the first for weak magnetic fields, with a linear magnetization model, and the second for medium or strong magnetic fields with a non-linear model (. 5 «Modeling of static phenomena») in a second approach, named «dynamic», a method is presented to take into account the induced currents in the laminations (. 6 «Modeling of dynamic phenomena») Note In this document, complex permeability is not approached. The modeling of magnetic core made of laminations in Flux PAGE 1

6 Introduction Flux PAGE 2 The modeling of magnetic core made of laminations in Flux

7 Flux Aim of magnetic cores made of laminations: limiting losses 2. Aim of magnetic cores made of laminations: limiting losses 2.1. Overview Introduction The magnetic cores of electrical machines and of transformers are in general made of laminated materials (stacked magnetic sheets) in view of limiting the eddy current losses. This section reminds some basic notions on the magnetic losses in electrical devices. Contents This section contains the following topics: Sujet Page Magnetic losses: general 4 Limiting the magnetic losses: improving efficiency 6 The modeling of magnetic core made of laminations in Flux PAGE 3

8 Aim of magnetic cores made of laminations: limiting losses Flux 2.2. Magnetic losses: general Losses: general aspects The power losses in the electromechanical devices are mainly of three types: the magnetic losses in the magnetic cores (also called iron losses ) the losses by Joule effect in the coils (also called copper losses ) the mechanical losses (mainly by friction and ventilation in the rotating machines) Magnetic losses The magnetic losses in the magnetic cores represent an important part of the total losses in a. c. electrical devices. The magnetic losses can be decomposed in the following manner : losses generated by the induced currents, also called eddy current losses losses associated with the hysteresis phenomena in a.c. magnetic fields Induced currents/ Eddy currents losses Losses generated by the eddy currents: This type of losses is associated with the electrical behavior of the material. The ferromagnetic materials have good electro-conductive properties. An a.c. magnetic flux across the magnetic core creates induced currents in it, also named eddy currents, which return over themselves. The eddy current losses are due to the Joule effect of these currents. A part of the a.c. electromagnetic energy is thus converted into heat in the magnetic cores. PAGE 4 The modeling of magnetic core made of laminations in Flux

9 Flux Aim of magnetic cores made of laminations: limiting losses Losses by hysteresis Losses by hysteresis: This type of losses is associated with the magnetic behavior of the magnetic cores. The hysteresis losses are due to the hysteresis phenomenon. The energy absorbed by the material during magnetization, when H increases, is only partially restored during the demagnetization, when H decreases. The B(H) hysteresis cycle is carried out with a loss of energy. It is demonstrated that the losses associated with the hysteresis phenomenon are proportional to the area of the hysteresis cycle. B (T) Dissipated volume energy (J/m 3 ) H (A/m) The modeling of magnetic core made of laminations in Flux PAGE 5

10 Aim of magnetic cores made of laminations: limiting losses Flux 2.3. Limiting the magnetic losses: improving efficiency Introduction To improve the efficiency of the a.c. electric devices with magnetic cores, it is therefore necessary to reduce the magnetic losses. Various strategies can be adopted. Reducing eddy current losses Magnetic core made of laminations To reduce losses generated by the induced currents in the magnetic cores, the resistance associated with the passage of the currents must be increased. To do so, there are two solutions: the use of magnetic core made of laminations; this solution is used for low and medium frequencies, from 10 Hz to 100 khz the use of nonmetallic magnetic materials with very high resistivity; it is the case of high frequencies, of over 100 khz Magnetic core made of laminations: The massive magnetic core is replaced by a stack of magnetic sheets insulated from one another. The purpose of the insulation is to prevent the circulation of the currents from a sheet to the other. Massive magnetic circuit Laminated magnetic circuit Reducing the hysteresis losses To reduce the hysteresis losses, a soft magnetic material must be chosen, namely a material with a sufficiently narrow B(H) hysteresis cycle. PAGE 6 The modeling of magnetic core made of laminations in Flux

11 Flux Model soft materials 3. Model soft materials 3.1. Overview Introduction The soft materials used generally have an anisotropic, nonlinear, hysteretic behavior, and the corresponding material properties can also depend more or less strongly on other physical quantities, for example on temperature or frequency. The behavior law B(H) can therefore be very complex if we wish to get a very accurate representation of the materials. This section deals with the simplifications necessary to model and presents the choices made in Flux. Contents This section contains the following topics: Topic Page Behavior laws B(H) and modeling 8 Hysteresis a univocal relationship 9 Saturation linear / nonlinear approximation 11 Anisotropy isotropic / anisotropic model 12 The sheet packages crystalline anisotropy and pseudo-anisotropy 14 The modeling of magnetic core made of laminations in Flux PAGE 7

12 Model soft materials Flux 3.2. Behavior laws B(H) and modeling Consequence Generally, it is not possible to express the complexity of a behavior law into one single model, which would simultaneously take into consideration the various aspects previously mentioned. Indeed, in order to carry out the numerical simulation of a device with the given models of materials, one must take into consideration a certain number of requirements, such as: the possibility of evaluating all the model parameters within a reasonable time interval the possibility of carrying out the field computations utilizing these models, by the available numerical tools and within a reasonable time interval. The model concept Therefore, the same material can be modeled in several ways, the choice depending on the studied phenomena and the operation conditions. Definition The macroscopic dependence linking the local values of the magnetization M (or magnetic polarization J) and the magnetic field strength H in any point of a material is called the law of material magnetic behavior. This law renders the fact that the magnetization of matter is modified under the effect of a magnetic field. This law is written in the form: M m H with the magnetization vector M J H with the polarization vector J 0 m where m is the scalar or the tensor of magnetic susceptibility * Attention: the quantity m depends on H; H in all the relationships above. should be written in the form m m Notation The magnetization law is written in the form: H J B 1 H B H B H B 0 where is the scalar or the tensor of permeability 0 m 0 r * Attention: the quantities, m r H in all the relationships above., should be written in the form H, H m, r PAGE 8 The modeling of magnetic core made of laminations in Flux

13 Flux Model soft materials 3.3. Hysteresis a univocal relationship Hysteresis and hysteresis cycle Hysteresis is a complex phenomenon related to physically irreversible processes. It consists of the fact that, at a given moment, the value of a material property depends not only on the intrinsic properties of the material, but also on its history. Magnetic materials are generally characterized by a hysteresis cycle. It is represented by a closed surface in the (H, B) coordinates where all points are accessible. Therefore, an infinite number of relationships between B and H exist. The figure below shows a typical configuration of a hysteresis cycle. (1) law of electric behavior(1) law of electric behavior asymptote line of slope 0 first magnetization curve asymptote line of slope 0 Modeling hysteresis Modeling hysteresis is a difficult problem, which is not currently done in Flux. It is difficult to model hysteresis correctly since there is an infinite number of possible B(H) curves, as shown in the figure above. in Flux In Flux, the B(H) dependence is a univocal relationship: one value of B corresponds to one value of H and vice versa. With this approximation, the soft magnetic materials are modeled by their curve of first magnetization, or their anhysteretic curve, which is justified by the very low value of the coercive magnetic field strength. B H Continued on next page The modeling of magnetic core made of laminations in Flux PAGE 9

14 Model soft materials Flux Consequence for the magnetic losses computation Neglecting the hysteresis cycle is needed for simplifying the numerical simulation, but it is also accepted on the hypothesis that hysteresis does not essentially modify the distribution of magnetic flux within the electrotechnical device. It is precisely this hypothesis that certain authors utilize in order to calculate the losses in electrical machines, even if the electric motor sheets present nonnegligible hysteretic characteristics. The finite element computation of magnetic flux density repartition is usually carried out in magnetostatics, and then the magnetic losses are evaluated by means of theoretical or experimental formulas starting from the distribution of the magnetic flux density. PAGE 10 The modeling of magnetic core made of laminations in Flux

15 Flux Model soft materials 3.4. Saturation linear / nonlinear approximation Introduction The B(H) dependence is generally a nonlinear law, but in some cases this dependence can be approximated by a straight line. This is then the case of a linear approximation. Zones with linear approximation The zone where a linear approximation can be considered depends on the modeled material, as presented in the figure. B H The modeling of magnetic core made of laminations in Flux PAGE 11

16 Model soft materials Flux 3.5. Anisotropy isotropic / anisotropic model Introduction The studied materials can be isotropic or anisotropic. In other words, the magnetic law B(H) can be: independent of the direction of the applied field (isotropic material) dependent on the direction of the applied field (anisotropic material) These two cases are presented in the blocks below. Isotropic materials Isotropic materials are characterized by a magnetization law, which is independent of the direction of the applied field. The B and H vectors are always collinear. The dependence between B and H is a scalar relationship, written: B. H Anisotropic materials Anisotropic materials are characterized by a magnetization law which is dependent on the direction of the applied field. The B and H vectors are not collinear. The dependence between B and H is a vector relationship, written: B H. xx xy xz where is the permeability tensor : yx yy yz zx zy zz To define an analytical model describing the magnetic behavior of an anisotropic structure, it is essential to know the magnetic permeability tensor of this structure. To determine a tensor of «full» three-dimensional relative permeability that would take into consideration the global anisotropy of the material is not simple. PAGE 12 The modeling of magnetic core made of laminations in Flux

17 Flux Model soft materials Difficulties in modeling magnetic anisotropy For an anisotropic material, permeability appears as tensor quantity with each of the components depending on the applied field. The vector relationship B(H) is therefore written in the form of 3 families of curves: Bx (Hx, Hy, Hz) By (Hx, Hy, Hz) Bz (Hx, Hy, Hz) The description of this type of curve at the experimental level is quite delicate, as one must be able to measure simultaneously both the magnitude and direction of magnetic flux density as a function of the field H. Generally, we limit ourselves to measurements in the directions where the magnetic field strength H and the magnetic flux density B are parallel. Difficulties in modeling magnetic anisotropy (continued) Supposing that these characteristics exist, one must then be able to input them into the software, and the software must be able to carry out the necessary interpolations during the numerical calculations. For all these various reasons, at the present time: there are simplified models, but whose validity domains are limited the nonlinear anisotropic models require more research Simplified models Generally, simplified models are based on a separation of the phenomena along the main axes of the material. The methodology consists of expressing the vector relationship B(H) starting from three main directions.. H can therefore be expressed in the form of 3 curves: B x (H x ), B y (H y ), B z (H z ). The vector dependence between B and H which is written as: B H The permeability tensor is written: x 0 0 y z This purely mathematical formalism is very simple and convenient at the level of resolution. It renders the fact that at the microscopic level the magnetic flux density and the magnetic field strength are collinear on three main directions: privileged direction, transversal direction and a third direction. The simplified model provided in Flux incorporates the separation of axes with linear interpolation. The modeling of magnetic core made of laminations in Flux PAGE 13

18 Model soft materials Flux 3.6. The sheet packages crystalline anisotropy and pseudo-anisotropy Introduction Accurate modeling of the magnetic sheet anisotropy remains difficult. It is indeed possible to distinguish two forms of anisotropy: crystalline anisotropy due to the crystalline structure of the material the anisotropy associated with the heterogeneity of the structure (stacking up of magnetic sheets, composite material...) ; therefore, we speak about «pseudo-anisotropy». In function of objectives, the treatment of these two types of anisotropy can be quite different. Crystalline anisotropy (in the sheet plan) The models that permit the taking into consideration of the crystalline anisotropy are those models focused on the 2D anisotropy (i.e. only in the sheet plan). These models mainly observe the measured magnetic characteristics of the sheets in their main directions, the direction of lamination DL and the transversal direction DT. The following models can be cited from the Nahil HIHAT thesis: separation of the axes model [NIYS75] model of the two axes [HLN84] elliptical model and elliptical model with axes rotation [DNP83] empirical anisotropic models Preferred directions of a sheet: DL: direction of lamination DT: transversal direction DN: normal direction Crystalline anisotropy (in the normal direction) Most studies are focused on the experimental characterization in 2D with a view to analyzing the phenomena seen as principal: the distribution of the magnetic flux density in the sheet plan, in the DL and DT directions the currents induced by the variation of the magnetic flux in the transversal section of the sheet. The characterization of the sheets in the DN direction and the impact of taking it into consideration in finite element modeling, be it theoretical or experimental, are not much presented in the literature. Continued on next page PAGE 14 The modeling of magnetic core made of laminations in Flux

19 Flux Model soft materials Pseudoanisotropy As regards the second form of anisotropy, the pseudo-anisotropy, it can be treated by homogenization methods meant to simplify the laminated structures and thus to compute the fields whilst ensuring reasonable computation time. This part is dealt with in «Simplifying by a homogenization technique». Bibliography Complementary information on the modeling of anisotropic magnetic materials is available in the following documents: Quasi 3D Models for the Analysis of Structures Presenting a 3D Anisotropy - thesis of Nabil HIHAT Université Lille Nord de France Uartois Anisotropic and nonlinear laws of magnetization: modeling and experimental validation - thesis of Thierry PERA INPG () Contribution to the bi-dimensional and three-dimensional modeling of anisotropy phenomena in 3-phase transformers thesis of Jean Marc DEDULLE INPG () References regarding semi-analytical 2D models (plan of the sheet) : [NIYS75] T. Nakata, Y. Ishihara, K. Yamada et A. Sasano : Non-linear analysis of rotating flux in the t-joint of a three-phase, three-limbed transformer core. In proceedings of Soft Magnetic Materials 2 Conference, pages [HLN84] D. Huttenloher, H.W. Lorenzen et D. Nusheler : Investigation of the importance of the anisotropy of cold rolled electrical steel sheet. IEEE Transactions on Magnetics, 20(5): , [DNP83] A. Di Napoli et R. Paggi : A model of anisotropic grain-oriented steel. IEEE Transactions on Magnetics, 19(4): , The modeling of magnetic core made of laminations in Flux PAGE 15

20 Model soft materials Flux PAGE 16 The modeling of magnetic core made of laminations in Flux

21 Flux Modeling of stacked magnetic sheets 4. Modeling of stacked magnetic sheets 4.1. Overview Introduction The modeling of stacked magnetic sheets raises problems, as there is a strong dimensional disproportion between the length of the magnetic core and the thickness of the magnetic sheets and of the insulation between sheets. This section deals with the dimension issues and the possibilities of simplification associated with the homogenization techniques. Contents This section contains the following topics: Topic Page Dimension issues 18 Simplifying by a homogenization technique 20 Homogenization: presentation of a basic model 22 Homogenization: Lamination, a more sophisticated model 23 The modeling of magnetic core made of laminations in Flux PAGE 17

22 Modeling of stacked magnetic sheets Flux 4.2. Dimension issues The issue The use of the finite elements method requires dividing the modeled geometry into elementary meshes, while observing the borders of the various regions. In the case of laminated magnetic cores, the shape of the magnetic sheets and of the inter-sheets insulation raises problems, as there is a strong dimensional disproportion between the length of the magnetic core and the thickness of the «leaves», as presented in the sections below. Dimensions of the magnetic cores Dimensions of the magnetic cores: The dimensions of the magnetic core can be quite varied, from several mm in small transformers, to one meter for certain large devices. Magnetic cores of transformer Magnetic sheets thickness Sheet thickness: There is a great variety of sheets for the construction of magnetic circuits. Several thicknesses of magnetic sheets are given in the table below. Sheet type Standardized thickness, the most frequent ones in [mm] FeSi (NO) Conventional : 0.35 / 0.50 / 0.65 Low thickness : < 0.1 FeSi (GO) Conventional : 0.23 / 0.28 / 0.35 High permeability : 0.23 / 0.30 Low thickness : < 0.1 FeNi Conventional : de 0.1 to 0.3 Low thickness : < 0.1 PAGE 18 The modeling of magnetic core made of laminations in Flux

23 Flux Modeling of stacked magnetic sheets Thickness of inter-sheets insulation Insulation thickness: The magnetic sheets are covered by a fine insulating layer on each face. The type of chosen insulation and its thickness depend on the required properties. Obviously, the level of the electrical resistance of the inter-sheets insulation is very important, but other points are also to be taken into account, such as the capacity of being cut up, of being welded, the resistance to corrosion and the capacity to work at high temperature. Order of magnitude: Thickness of the insulating layer from m to mm 1 st approach: modeling the leaves entirely To accurately model the magnetic cores made of laminations, the 1 st approach, consisting in describing each of the leaves, proves to be complicated. The modeling difficulties are presented in the table below. Construction Geometric description Mesh generation Solving process Difficulties Slow work : the geometry of each leave must be described: magnetic sheet + insulation Difficult mesh : the low thickness of insulation requires a dense mesh over the assembly of the magnetic core Considerable computation time : the dense mesh demands a considerable computation time An example is shown in the figure below : 10-sheet package (10 mm x10 mm) 0.35 mm sheet thickness / filling factor : 0.9 (insulation thickness = 0.05 mm) Total number of nodes --> (2 nd order elements) Need for simplification It is therefore necessary to simplify the studied structure, as the taking into consideration of the full geometry of a structure is very expensive, almost impossible in some cases. The modeling of magnetic core made of laminations in Flux PAGE 19

24 Modeling of stacked magnetic sheets Flux 4.3. Simplifying by a homogenization technique Simplification A first solution for simplification consists in building up an equivalent macroscopic model by means of a homogenization technique. Homogenization is a technique which permits to simplify the behavior law of a structure or of a material. Starting from a heterogeneous structure consisting of several materials or defects, a homogeneous equivalent structure can be determined. Schematic diagram The block of insulated laminations is replaced by a homogeneous block. The physical properties of the homogeneous block take into consideration the made of laminations property. The magnetic sheet stacking and its equivalent are shown in the figure below. Continued on next page PAGE 20 The modeling of magnetic core made of laminations in Flux

25 Flux Modeling of stacked magnetic sheets Bibliography / references Homogenization has lead to the development of numerous approaches based on mathematic formulations. Among them, one can cite the method of energy, the asymptotic methods and those of averaging. Within the framework of the magnetic cores made of laminations, the most usual homogenization models are the following: basic homogenization of the sheet stack; work of M. L. Barton [Bar80] homogenization of an isotropic nonlinear problem; work of H.Waki [WIH06] homogenization of an isotropic nonlinear problem; work of J. M. Dedulle [WIH06] homogenization of magnetic stackings in magnetodynamic regime ; work of J. Gyselinck and P. Dular [DGG03, GD04, KDZB04, GSD06]. These models are presented in the thesis of Nabil HIHAT. References: Quasi 3D Models for the Analysis of Structures Presenting a 3D Anisotropy - thesis of Nabil HIHAT Université Lille Nord de France Uartois The modeling of magnetic core made of laminations in Flux PAGE 21

26 Modeling of stacked magnetic sheets Flux 4.4. Homogenization: presentation of a basic model Objective The objective is to replace the magnetic sheet stacking by a homogeneous block, as shown in the figure below, and to determine the tensor of the relative permeability equivalent to that stacking structure. Principle The simplest approach consists in the analogy with an electrical circuit in order to calculate the reluctance of the stacking structure. The tensor of the magnetic permeability of the homogeneous block equivalent to the stacking structure in function of the permeability of the magnetic sheet and of the filling factor is given by the relation : hom o. 1 x, fer 1 1 y, fer z, fer 1 1 Since the relative permeability of the magnetic sheets is much higher than 1, it gets : hom o. x, 0 0 fer 0 y, fer PAGE 22 The modeling of magnetic core made of laminations in Flux

27 Flux Modeling of stacked magnetic sheets 4.5. Homogenization: Lamination, a more sophisticated model A Flux model: Lamination A more sophisticated homogenization model is proposed in Flux 3D. This model permits the modeling of magnetic sheet packages by taking into consideration : the pseudo-anisotropy (leaves anisotropy) the saturation (nonlinear model) The user version including this model is named Lamination. Geometric description The user version Lamination permits the description of flat or cylindrical magnetic cores made of laminations as the ones presented in the figure below. Flat magnetic sheets package Cylindrical sheet package Equivalent homogeneous block = Parallelepipedic block Filling factor Equivalent homogeneous block = Cylindrical block Filling factor Physical description The user version Lamination permits the modeling of the ferromagnetic material by means of the Flux model: Analytical anisotropic saturation + control of bend (arctg, 3 coef.) The magnetic properties are described by user sub routine. The modeling of magnetic core made of laminations in Flux PAGE 23

28 Modeling of stacked magnetic sheets Flux PAGE 24 The modeling of magnetic core made of laminations in Flux

29 Flux Modeling of static phenomena 5. Modeling of static phenomena 5.1. Overview Introduction This section deals with the various possible modeling types with Flux. At a first stage, we are interested only in the study of static phenomena. The influence of induced currents is not taken into account. After several reminders of the working hypotheses and the general conditions for modeling, different working situations are presented. These are: on the one side, the modeling of magnetic sheet packages in low strength magnetic field (linear model) on the other side, the modeling of the magnetic sheet packages in «average/strong» magnetic field (nonlinear model). Contents This section contains the following topics: Topic Page Some explanations on the modeling context 26 Case 1: low field / linear model 2D or 3D 26 Case 2: «average/strong» field / non linear model 3D Lamination 27 The modeling of magnetic core made of laminations in Flux PAGE 25

30 Modeling of static phenomena Flux 5.2. Some explanations on the modeling context Introduction This paragraph reminds the working hypotheses and it describes the modeling conditions. Working hypotheses The working hypotheses are the following : the hysteresis phenomenon is ignored the eddy currents are ignored It is presumed, on one hand, that hysteresis does not fundamentally modify the repartition of the magnetic flux, and, on the other, that lamination fully plays its role of reducing eddy currents. The computation of the magnetic field distribution is carried out in static*, and losses are calculated after that by means of the Bertotti formulas or the LS model, starting from the distribution of the magnetic flux density. * that is, without taking into consideration the eddy currents Modeling conditions The modeling conditions in Flux are presented in the table below. Physical applications Geometric description Physical description Magnetic applications: Magneto Static / Transient Magnetic / Steady State AC Magnetic The magnetic sheet package is described as a homogeneous block Physical region: the physical region is of the type Non conductive magnetic region Material: the model of the material is explicit in the different cases presented below What can be done The main possibilities of modeling are listed in the table below and presented in detail in the following paragraphs. With Flux standard version 2D / 3D (MS / MT / MH) user version Lamination 3D (MS / MT) and in the conditions of low field / linear model «average/ strong» field / non linear model modeling of crystalline anisotropy pseudo-anisotropy (via formula) pseudo-anisotropy PAGE 26 The modeling of magnetic core made of laminations in Flux

31 Flux Modeling of static phenomena 5.3. Case 1: low field / linear model 2D or 3D Conditions of operation The conditions of operation are the following : The applied field is low, i.e. the magnetic core is not saturated the material is described by means of a linear anisotropic model Conditions of modeling The general modeling conditions are those described in the 1 st of this section Some explanations on the modeling context The Flux version is the «standard version». The study domain is a 2D plane domain or a 3D domain. Modeling of the anisotropy The material used to model the magnetic sheet package is described by means of the Flux model: Linear anisotropic This model permits the modeling of the anisotropy via the relative values of the magnetic permeability along the main axes: rx, ry in 2D rx, ry, rz in 3D There can be: crystalline anisotropy pseudo-anisotropy (lamination): formulas issued by homogenization techniques or a combination of the two aspects Examples Modeling of the crystalline anisotropy in 2D: transformers in 2D with FeSi GO (Oriented Grains) magnetic sheets Modeling of the pseudo-anisotropy (lamination) in 3D motors with FeSi NO (Non Oriented) magnetic sheets Result postprocessing The postprocessing results are as follows: Repartition / distribution of magnetic field (B and H) A posteriori computation of losses: - Bertotti (applications MT / MH) - Modèle LS (application MT) Continued on next page The modeling of magnetic core made of laminations in Flux PAGE 27

32 Modeling of static phenomena Flux Operating mode The construction of the Flux project is carried out in a «standard manner». The specific operations, relative to the use of an anisotropic material, are described in the table below. Step Action 0 Create a specific coordinate system for the SHEET_PACKAGE region (this coordinate system will be used to orient the anisotropic material) 1 Computation of the tensor of equivalent relative permeability (according to formulas below) 2 Create the material Enter the name of the material: MAT_LIN_ANI In the B(H) tab, click on Magnetic property choose the Linear anisotropic model enter the relative permeability values: rx, ry, rz 3 Assign the material MAT_LIN_ANI to the region SHEET_PACKAGE 4 Orient the material MAT_LIN_ANI to make the axes of anisotropy of the material coincide with the coordinate system axes Computation of the tensor of the equivalent relative permeabilitie: r x, équi = r x, tôle r y, équi = r y, tôle r z, équi = (1 - ) -1 with: r x, équi, r y, équi, et r z, équi : equivalent relative permeabilities along the axes x, y and z r x,tôle, r y, tôle : relative permeabilities of the magnetic sheets along the axes x and y : filling factor = sheet thickness / (sheet thickness + insulation thickness) ( between 0 and 1, usually between 0.95 to 0.97). PAGE 28 The modeling of magnetic core made of laminations in Flux

33 Flux Modeling of static phenomena 5.4. Case 2: «average/strong» field / non linear model 3D Lamination Conditions of operation The conditions of operation are the following: The applied field is «average/strong»; the magnetic core can work in the bend region or saturation region of the B(H) curve the material is described by means of a non linear model Conditions of modeling The general modeling conditions are those described in the 1 st of this section Some explanations on the modeling context The Flux version is the user version Lamination. The study domain is 3D domain. The physical applications are Magneto Static and Transient Magnetic applications. Flat sheet package Cylindrical sheet package Modeling of the anisotropy The material used to model the sheet package is described by means of the Flux model: User magnetic property This model is described by means of different user sub-routine that permit the taking into consideration of the : pseudo-anisotropy (homogenization technique) saturation (non linear model) The crystalline anisotropy cannot be taken into consideration with this model. Example Modeling of the FeSi NO (Non Oriented) magnetic sheets for motors Result postprocessing The postprocessing results are as follows: Repartition / distribution of magnetic field (B and H) A posteriori computation of losses: - Bertotti (applications MT / MH) - Modèle LS (application MT) Continued on next page The modeling of magnetic core made of laminations in Flux PAGE 29

34 Modeling of static phenomena Flux Operating mode (1) To select the user version Lamination Step Action 1 At supervisor level : in the menu bar, select the menu Versions and point on Lamination_104 The name of the user version Lamination_104 is displayed in the title bar Continued on next page PAGE 30 The modeling of magnetic core made of laminations in Flux

35 Flux Modeling of static phenomena Operating mode (2): parallelepiped block The construction of the Flux project is carried out in a «standard manner». The specific operations, concerning the creation of the equivalent material are described in the table below in the case of a parallelepiped magnetic sheet package. Step Action 1 Create the equivalent material 1a Enter the name of the material: LAMINATION_PLAN_XXX The name of the material must begin by lamination_plan 1b In the B(H) tab: click on Magnetic property choose the user magnetic property model 1c In the window User coefficients: choose the model Non linear property enter the values of the following 7 coefficients: Anisotropic analytical model (arctg, 3 coef.) Coef. 1 Permeability relative to the origin Coef. 2 Magnetization to saturation (in Tesla) Coef. 3 Bending coefficient (between 0 and 0.5) Direction of sheets / definition of vector V (see image below) Coef. 4 Vx : component X of vector V Coef. 5 Vy : component Y of vector V Coef. 6 Vz : component Z of vector V Filling factor of sheets Coef. 7 Factor between 0 and 1 (usually close to 1) 2 Assign the material LAMINATION_PLAN_XXX to the volume region SHEET_PACKAGE Definition within global coordinate system: of vector V perpendicular to the sheet plan V V x V y V z X Y Z If the vector V is parallel to the Y axis Vx = 0, Vy = 1, Vz = 0 Continued on next page The modeling of magnetic core made of laminations in Flux PAGE 31

36 Modeling of static phenomena Flux Operating mode (2): cylindrical block The construction of the Flux project is carried out in a «standard manner». The specific operations, concerning the creation of the equivalent material are described in the table below in the case of a cylindrical magnetic sheet package. Step Action 1 Create the equivalent material 1a Enter the name of the material: LAMINATION_CYL _XXX The name of the material must begin by lamination_cyl 1b In the B(H) tab: Click on Magnetic property Choose the user magnetic property model 1c In the window User coefficients: Choose the model Non linear property Enter the values of the following 7 coefficients: Anisotropic analytical model (arctg, 3 coef.) Coef. 1 Permeability relative to the origin Coef. 2 Magnetization to saturation (in Tesla) Coef. 3 Bending coefficient (between 0 and 0.5) Direction of sheets / definition of vector V (see image below) Coef. 4 Vx : component X of vector V Coef. 5 Vy : component Y of vector V Coef. 6 Vz : component Z of vector V Direction of sheets / definition of point M (see image below) Coef. 7 X 0 : X coordinate of the point M Coef. 8 Y 0 : Y coordinate of the point M Coef. 9 Z 0 : Z coordinate of the point M Filling factor of sheets Coef. 10 Factor between 0 and 1 (usually close to 1) 2 Assign the material LAMINATION_CYL_XXX to the volume region SHEET_PACKAGE Continued on next page PAGE 32 The modeling of magnetic core made of laminations in Flux

37 Flux Modeling of static phenomena Definition within global coordinate system: of vector V: direction of the axis of the cylinder of point M : point on the axis of the cylinder V V x V y V z X Y Z If the vector V is parallel to the Z axis Vx = 0, Vy = 0, Vz = 1 The modeling of magnetic core made of laminations in Flux PAGE 33

38 Modeling of static phenomena Flux PAGE 34 The modeling of magnetic core made of laminations in Flux

39 Flux Modeling of dynamic phenomena 6. Modeling of dynamic phenomena 6.1. Overview Introduction This section deals with the different possible modelings with Flux. The interest goes, at a second stage, towards the taking into consideration of the dynamic phenomena, and especially to the effect of the eddy currents. After several reminders on the working hypotheses and the general modeling conditions, the two working situations previously presented within cases 1 and 2 are dealt with by taking into consideration the eddy currents. Contents This section contains the following topics: Topic Page Some explanations on the modeling context 36 Concretely, in Flux 38 The modeling of magnetic core made of laminations in Flux PAGE 35

40 Modeling of dynamic phenomena Flux 6.2. Some explanations on the modeling context Introduction This paragraph reminds the working hypotheses and it describes the modeling conditions. Working hypotheses The working hypotheses are the following : the hysteresis phenomenon is ignored the eddy currents in the plan of the magnetic sheet are taken into consideration It is presumed, on one side, that the hysteresis phenomenon does not fundamentally modify the magnetic flux and eddy currents distribution and, on the other, that lamination plays its role of reducing the eddy currents in the transversal section of the sheet package. However, it is assumed at the same time that eddy currents may develop in the sheet plane generated by the «parasite magnetic fields» non parallel with the sheet plane. These eddy currents can thus perturb the «initial distribution of the magnetic field». About eddy currents Usually the magnetic field generated by the field sources is essentially parallel with the sheet plane and the eddy currents in the magnetic cores are reduced by the lamination. Residual eddy currents can develop only in the transversal section of each sheet of the magnetic core. In reality, there is a component of the magnetic field non parallel with the sheet plan, namely a transversal magnetic field that generates eddy currents in the sheet plan. Those are the eddy currents that we try to introduce in the 3D model. These eddy currents cannot be modeled in 2D. Examples of magnetic fields normal to the sheet plan : field generated by the winding part outside the slots at the lateral extremities of the rotating machines, leakage of magnetic flux of windings with respect to the external sheets of the transformers magnetic cores Continued on next page PAGE 36 The modeling of magnetic core made of laminations in Flux

41 Flux Modeling of dynamic phenomena Modeling conditions The modeling conditions in Flux are presented in the table below. Physical applications Geometric description Physical description Magnetic applications: Transient Magnetic / Steady State AC Magnetic The magnetic sheet package is described as a homogeneous block Region: the region is of solid conductor type Material: - B(H) model: as previously - J(E) model: isotropic resistivity Volume constraint: specific volume constraint, which requires the current to develop in the sheet plan* *according to the following paragraph What can be done The main possibilities of modeling are listed in the table below and presented in detail in the following paragraphs (same as previous section / 3D). With Flux 3D standard version (MT / MH) user version Lamination 3D (MT) and in the conditions of low field / linear model «average/ strong» field / non linear model modeling of crystalline anisotropy pseudo-anisotropy (via formula) pseudo-anisotropy and taking into consideration the effect of eddy currents in the sheet plan* * Attention: the eddy currents are minimized, as no return in the thickness is permitted. The modeling of magnetic core made of laminations in Flux PAGE 37

42 Modeling of dynamic phenomena Flux 6.3. Concretely, in Flux Choice of a basic case We are in one of the previously described cases: Case 1: «low» magnetic field (linear model) Case 2: «average/strong» magnetic field (non linear model) / 3D Lamination Conditions of modeling The general modeling conditions are those described in the 1 st of this section Some explanations on the modeling context. The Flux version is the «standard version» in the case 1 and the the user version Lamination in the case 2. The study domain is 3D domain. About volume constraints Some explanations on the volume constraints: The lamination of a magnetic circuit is made in order to avoid the development of Foucault currents orthogonal to the plane of the sheets. The loops in the xoz and yoz planes cannot exist. On the contrary, the loops in the xoy plane would appear. z Possible xy loops y x Forbidden yzloops Forbidden xz loops The current in the solid conductor region is deduced from the electric vector potential T by the relation : J= rot T By setting the two components Tx and Ty of the vector to zero, by means of a volume constraint, you ensure that the z component of the Foucault currents is zero, as shown by the expression J = curl T detailed below: J x = T z /y J y = -T z /x J z = 0 The volume constraints are not directly accessible Flux (via the menus). They can be manipulated by means of the PyFlux commands. Continued on next page PAGE 38 The modeling of magnetic core made of laminations in Flux

43 Flux Modeling of dynamic phenomena Result postprocessing The postprocessing results are as follows: Repartition / distribution of magnetic field (B and H) Computation of Joule losses (applications MT / MH) Operating mode The starting point of this operating mode is the Flux project corresponding to case 1 or case 2 previously described (in MT or in MH). The specific operations relative to the taking into consideration of the eddy currents are described in the table below. Step Action 1 Complete the description of the material 1a In the J(E) tab: click on Electrical property choose the isotropic resistivity model enter the resistivity value in [m] 2 Modify the volume region SHEET_PACKAGE 2a In the box Edit Volume region [SHEET_PACKAGE ] select the type: Region of solid conductor type 2b Click on the button Orient to make the material anisotropy axes coincide with the axes of the coordinate system 3 Modify the physical application 3a In the menu Application: click on Edit current application 3b In the Model Formulation tab, in the zone Type of approximation functions, select Nodal finite elements 4 Charge the macro 4a In the menu Extension, point on Macro and click on Load 4b Select the macro to load: C:\Cedrat\Extensions\Macros\Macros_Flux3D_Physics\ VolumeConstraint\ SolidConductorCurrentInPlaneCreate.PFM 5 Execute the macro 5a In the menu Extension, point on Macro and Click on Execute 5b In the box VolumeConstraintCreation: select the volumes corresponding to the region SHEET_PACKAGE choose the number relative to the sheet plane: 1 = plane XOY / 2 = planeyoz / 3 = plane ZOX The modeling of magnetic core made of laminations in Flux PAGE 39

44 Modeling of dynamic phenomena Flux Note Attention : New volume constraints are not visible in the data tree. In order to manage the volume constraints, it is necessary to load the following macros: SolidConductorCurrentInPlaneCreate.PFM: to create a volume constraint SolidConductorCurrentInPlaneEdit.PFM: to edit a volume constraint SolidConductorCurrentInPlaneDelete.PFM: to delete a volume constraint PAGE 40 The modeling of magnetic core made of laminations in Flux