Dupré, Luc

Olyslager, Femke

VAN DE WIELE, B

Undetermined

Van de Wiele, Ben

Ghent University Academic Bibliography

1Fast numerical 3D-Scheme for the Simulation ofHysteresis in ferromagnetic MaterialsBen Van de Wiele, Luc Dupre´, Member, IEEE, and Femke Olyslager, Fellow, IEEEAbstract— The paper deals with a two-level model for theevaluation of the electromagnetic hysteresis behavior of ferro-magnetic materials when applying a magnetic field. The modelcan be used to determine which mechanisms minimize materiallosses in electrical steels. The material is seen in a high-levelpart as an ensemble of interacting monocrystals, where eachmonocrystal is described in a low-level 3D micromagnetic model.In this paper the 3D micromagnetic model is presented in orderto find which mechanisms influence the magnetization loopsof monocrystals. The micromagnetic model is based on severalinteractions between the magnetization vectors M of elementarydipoles in the crystal and the applied field. These interactionscontribute to the effective field, composed of the applied field, theexchange field, the anisotropy field, the magnetostatic field andthe magneto-elastic field. The time variation of the magnetizationvector M in each space point obeys the Landau-Lifshitz equation.Macroscopic magnetization loops are reconstructed consideringthe average magnetization in the direction of the applied field.The influence of crystal defects on the magnetization is includedby means of the stress they induce in the lattice.Index Terms— crystal defects, hysteresis, Landau-Lifshitz,magnetization loop, micromagnetism, monocrystalI. INTRODUCTIONTO improve the performance of electromagnetic devices,like electrical machines and transformers, the overalllosses in general and the iron losses in particular must beminimized. Classically, electrical steel is characterized andqualified by means of standards which are based on simple,unidirectional sinusoidal flux excitations using e.g. Epstein,ring core or single sheet measurement equipment. Generally,the measured iron losses in electromagnetic devices exceed thevalue based on these standard measurements (unidirectionalsinusoidal excitation). This is caused by distorted flux distribu-tions, local rotational excitations, short-circuit between neigh-boring laminations, changes in the magnetic characteristics dueto the mechanical treatment, etc. Therefore, a profound studyof the material behavior of electrical steel under applicationconditions is very important. This study is important for theproducers of electrical steel as for the machine constructors,both aiming to maximize machine performances. Understand-ing the phenomena inside the machine core is an indispensablestep in the process of developing electromagnetic devices witha higher efficiency, thus lower core losses, resulting in a lowerThis work was supported by the Flemish government through the IWT PhDscholarship SB/51032.B. Van de Wiele and L. dupre´ are with the Department of Electrical Energy,Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium.F. Olyslager is with the Department of Information Technology, GhentUniversity, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium.energy consumption. Hence producers of electrical steels arenot only interested in the material behavior under prescribed,strongly simplified conditions, but also under the realisticconditions in electromagnetic devices. Here, the relation be-tween the electromagnetic behavior and the microstructure ofthe electrical steel is crucial as the microstructure is directlyrelated to the material production and treatment techniques.Also machine constructors are interested in material behaviorunder application conditions with regard to the material choicewhen optimizing the design of the device. A good descriptionof the magnetic behavior of ferromagnetic materials may startfrom microstructural features like the presence of lattice ef-fects, grains, textures, stresses, crystal defects, etc. A physicalmaterial model becomes valuable when these microstructuralfeatures are translated into quantitative predictions about themagnetization dynamics. The establishment of the correlationbetween the macroscopic magnetic properties, such as theelectromagnetic losses, and the microstructural features plays acrucial role in the adaption of material production technologiesto improve the material qualityII. MACROSCOPIC MAGNETIC MATERIAL MODELA ferromagnetic material can be magnetized when it isexposed to an external time varying magnetic field. However,there is no unique relation between the applied magneticfield and the magnetization of the crystal: no single valuedmagnetization curve is described, but instead a magnetizationloop or hysteresis loop.The macroscopic hysteresis loop characterizes the materiallosses when cyclic magnetic fields are applied. A two-levelmodel helps us to understand which microstructural featurescan decrease the electromagnetic losses. In this model themacroscopic material is seen as an ensemble of N monocrys-tals, each described by a 3D-micromagnetic model (the lowestlevel). A macroscopic model (the highest level) describes theensemble of monocrystals. Each monocrystal interacts with themagnetic field applied on the macroscopic material (Happlied)on the one hand and the mean magnetizations < Mi > ofall other monocrystals i = 1..N on the other hand, togetherresulting in a space varying effective applied field Ha,eff (r).The link between the two models is the relation between theeffective applied field felt by the monocrystal i, Ha,eff (ri)and the mean magnetization of the monocrystal <Mi >.Ha,eff (ri)↔<Mi > (1)In this paper the 3D-micromagnetic model describing themonocrystal is presented in order to find which mechanisms2influence the relation (1). The monocrystal is seen as an entitysubjected to an effective applied field Ha,eff (ri), for the sakeof clarity simply called the applied field Ha.The presented 3D-micromagnetic model is based on theLandau-Lifshitz-formalism [1], that describes the electromag-netic action at the level of magnetic dipoles, resulting inthe damped precession movement of the local magnetizationvector with constant amplitude. By means of the local (micro-scopic) effective field, the formalism allows us to account forboth the magnetic short distance effects (e.g. exchange field,anisotropy field and magnetoelastic field) and the magneticlong distance effects (magnetostatic field, imposed magneticfield). The magnetoelastic field takes lattice defects into ac-count via the stress they induce in the lattice.Starting from a predefined microstructural state, the mi-cromagnetic model must permit us to describe the meanmagnetization < M > of the monocrystal in the form ofmagnetization loops, caused by a time varying magnetic field.III. MICROMAGNETIC THEORYThe theory of ferromagnetic materials is based on theassumption, following Landau and Lifshitz [1], that the mag-netization of magnetic dipoles M varies with the position, butthat it has a fixed temperature dependent magnitude |M| =Ms (below Curie-temperature). In the theory, the Gibbs freeenergy (φG) represents the total energy density in the crystal.This energy density φG depends on the configuration of themagnetic dipoles M =Msm and is minimized in order to findequilibrium conditions for the magnetic dipoles [2], leading tothe definition of the effective magnetic field (Heff ) in eachpoint of the crystal. The static micromagnetic equilibrium isexpressed asm(r)×Heff (r) = 0 (2)The effective field contains all influences of the total crystalon the magnetic dipole M at that place.A. The effective field HeffThe effective field is composed of the applied field (Ha),the exchange field (Hexch), the anisotropy field (Hani), themagnetostatic field (Hms) and the magnetoelastic field (Hme).Heff = Ha +Hexch +Hani +Hms +Hme (3)Here the exchange and anisotropy fields are (using theEinstein summation convention: when two equal indices areused, summation is made)Hexch =2Aµ0Ms∇2γiei (4)Hani = − 1µ0Ms∂φani∂γiei (5)with A the exchange stiffness and the anisotropy energyφani = K1(α21α22 + α22α23 + α23α21) +K2α21α22α23 (6)where αi is the direction cosine of the m-vector with respectto the ith crystallographic axis of the based centered cubic ironlattice, while γi refer to the direction cosines of m with respectto the (x,y,z)-coordinate system. K1 and K2 are anisotropyconstants.The magnetostatic field follows from ∇ ·Hms = −∇ ·Mand ∇×Hms = 0 using Greens functions:Hms =Ms4pi∫V(m|r− r′|3 −(m · (r− r′)) (r− r′)|r− r′|5)dV ′(7)Here is V the volume of the considered crystal.The magnetoelastic field is given byHme = − 1µ0Msσij∂∂γk(²Qkj(α))ei (8)where σ is the local stress tensor and ²Q is the free magne-tostrain tensor with elements²Qij =32λ100(αiαj − 13)δij +32λ111αiαj(1− δij) (9)with λ100 andλ111 magnetostriction constants while δij is thekronecker symbol.B. Crystal defectsAs mentioned, the presence of crystal defects is takeninto account by there characteristic stress distribution. In themodel, edge dislocations, screw dislocations and point defectscontribute to the total stress tensor σ.The non-zero components of the stress tensor σedge char-acteristic for an edge dislocation running through the origin,with dislocation axis parallel to the z-axis and Burgers vectorb parallel with the x-axis are given by [3]σedge11 = −τ0by(3x2 + y2)(x2 + y2)2(10)σedge22 = τ0by(x2 − y2)(x2 + y2)2(11)σedge33 = 2τ0νyx2 + y2(12)σedge12 = σedge21 = τ0bx(x2 − y2)(x2 + y2)2(13)withτ0 =G2pi(1− ν) (14)Here is b the length of the Burgers vector of the edgedislocation and G and ν the modulus of rigidity and Poisson’sratio respectively. In the monocrystal the Burgers vector isparallel with a < 1 1 1 > direction, while the axis of thedislocation is parallel with a < 1 1 0 > direction [4]. Sothe stress tensor σedge has to be transformed in terms of aproper rotation matrix Redge to a stress tensor σ′edge thatincorporates the monocrystal byσ′edgeij = Redgeik Redgejl σedgekl (15)The non-zero components of the the stress tensor σscrcharacteristic for a screw dislocation running through the3origin and with dislocation axis parallel to the z-axis are givenby [3]σscr13 = σscr31 = −Gb2piyx2 + y2(16)σscr23 = σscr32 = −Gb2pixx2 + y2(17)For a screw dislocation the Burgers vector b is parallel withthe axis of the dislocation and parallel with a < 1 1 1 >-direction. So the stress tensor σscr should also be transformedto σ′scr to match the crystal lattice by a transformation similarto (15).The components of the strain tensor ²pd of a point defectare given by [5]²pdij = −32a3((1 + β)1/3 − 1))r3rirjr2(18)with ri=1,2,3 = (x, y, z) respectively, r =√x2 + y2 + z2and a the lattice constant of iron. The mismatch parameterβ is defined as the relative difference in volume between thedistorted volume (V ) and undistorted volume (V0)V = V0(1 + β) (19)An interstitial or foreign atom bigger than a lattice atomis described by a characteristic volume expansion, β > 1.A vacancy or foreign atom smaller than a lattice atom isdescribed by a characteristic volume reduction , β < 1.The stress components σpdij are calculated by the constitutiverelationσpdij = Cijkl²pdkl (20)with C a material dependent positive-definite symmetricfourth-order tensor. The expressions (18) and (20) areisotropic, thus independent of lattice directions and can beused unchanged.C. DynamicsThe evolution of the magnetic dipoles m is expressed interms of the effective field by the Landau-Lifshitz-equation(LL-equation)∂m∂t=γG1 + α2m×Heff − αγG1 + α2m× (m×Heff ) (21)with α and γG the damping constant and the gyromagneticconstant respectively. The static micromagnetic equilibriumcondition, usually formulated as m × Heff = 0, followsdirectly from the LL-equation (21).IV. 3D MICROMAGNETIC MODELIn Fig 1 the micromagnetic model of the monocrystal isshown as a cuboid subdivided in Nx × Ny × Nz cubeswith sides ∆. The crystallographic axes x′, y′, z′ are definedwith respect to the xyz-coordinate system in terms of Eulerangles (φ, θ, ψ) that define the rotation matrix R(φ, θ, ψ)which rotates the xyz-coordinate system towards the x′y′z′-coordinate system [7]. Each cell in the space discretizationcontains one single magnetization vector m. The space deriva-tives appearing in (4) are approximated by a classical finiteN cellsyyxzN cellsxN cellszy’x’z’DFig. 1. Scheme of the micromagnetic model of a cuboid monocrystaldifference method. The magnetostatic field (7) together withthe exchange field (4) is calculated by Fast Fourier Transformsin order to save computational time.The initialization of the total stress is done by adding thestress contributions of the different defect structures, eachaccording to there own orientation in the crystal as outlinedabove. Hereby, the intrinsic stress caused by magnetostrictionis neglected so (total) strain compatibility∇× (∇× ²)T = 0 (22)can not be guaranteed (² is the total strain tensor). Thisapproximation is called the relaxed approach and is provento be reasonable for thin materials with low magnetostriction[6].For the time discretization the quasi-static applied field Hais approximated with a piecewise constant time function. Itis assumed that at the moment the applied field Ha jumpsfrom a constant value to the next one, the material is in staticmicromagnetic equilibrium. Using the LL-equation (21), themagnetization dynamics in each basis cell is computed throughtime stepping until a new static micromagnetic equilibrium isobtained corresponding with the new constant value for theapplied field Ha. The magnetization dynamics is evaluatedanalytically at [ti, ti + 1] by introducing in each basis cell alocal (u,v,w)-system with the u-axis parallel with Heff (ti).The LL-equation (21) can be solved analytically in this frame-work when Heff (t) is kept constant during the time stepHeff (t) = Heff (ti)eu ti ≤ t ≤ ti+1 (23)During the time step m evolves from m(ti) = uieu+ viev +wiew at t = ti to m(ti+1) = ui+1eu + vi+1ev + wi+1ew att = ti +∆t withui+1 =eqα∆t(1 + ui)− e−qα∆t(1− ui)eqα∆t(1 + ui) + e−qα∆t(1− ui) (24)vi+1 = 2vi cos(q∆t)− wi sin(q∆t)eqα∆t(1 + ui) + e−qα∆t(1− ui) (25)wi+1 = 2vi sin(q∆t) + wi cos(q∆t)eqα∆t(1 + ui) + e−qα∆t(1− ui) (26)with q = γGHeff (ti)/(1 + α2). In order to improve the rateof convergence, α is chosen to be 1.4TABLE IMICROSCOPIC MATERIAL CONSTANTS FOR PURE IRONquantity value unitµ0Ms 2.16 [T ]A 1.5 10−11 [Jm−3]K1 0.48 105 [Jm−3]K2 −0.50 105 [Jm−3]λ100 22 10−6 []λ111 −21 10−6 []a 0.3 10−10 [m]C11 2.41 1011 [Pa]C44 1.12 1011 [Pa]C12 1.46 1011 [Pa]G 82 109 [Pa]ν 0.33 []γG −2.21 105 [mA−1s−1]V. INFLUENCE OF MICROSTRUCTURE ON THE HYSTERESISLOOPTo determine what mechanisms influence the shape of thehysteresis loop several numerical experiments were performed.Table I shows the material constants for pure iron, used in theexperiments with Cijkl in Voigt’s notation [2]. A first seriesof experiments is performed on a cuboid monocrystal withdimensions 1.28µm × 2.56µm × 1.28µm divided in cubeswith sides 10nm. Two cases are distinguished.In the first case the crystal has no defects while in the secondcase the crystal has five dislocations (edge and screw dislo-cations). For each case three hysteresis loops are simulated.A first one with the applied field Ha parallel with the y-axisand lattice axes parallel with the xyz-axes i.e. parallel withthe edges of the cuboid (see Fig 1), a second one with theapplied field Ha parallel to the space diagonal of the cuboidand the lattice axes parallel to the xyz-axes and a third onewith the applied field Ha parallel to the space diagonal andthe lattice axes parallel with the directions determined by theEuler angles (φ = pi/5, θ = pi/5, ψ = pi/5). The results areshown in Fig 2. Fig 3 shows the microscopic configuration ofthe magnetization vectors in three edge plains of the cuboidfor the remanent magnetization (Ha = 0) of the hysteresisloop in solid line of Fig 2(a).A second experiment is performed on a spherical monocrys-tal with a radius of 0.64µm divided in cubes with sides 10nm,the applied field Ha is chosen parallel with a lattice axis. Theresult is shown in Fig 4. For both experiments, the appliedmagnetic field Ha is approximated by a piecewise constantfunction, considering 400 equidistant time intervals. The meanmagnetization is calculated in the direction of the applied field.Comparison between the hysteresis loop with solid lineof Fig 2(a) and the hysteresis of Fig 4 proves that theshape of the ferromagnet influences the mean magnetizationdrastically. Comparison between the hysteresis loops for thecuboid monocrystal of Fig 2 learns that the direction of theapplied field Ha with respect to the lattice axes and the edgesof the cuboid influences the shape of the hysteresis loop.Comparison between the hysteresis loops of Fig 2(a) andFig 2(b) learns that dislocations also have a large influenceon the macroscopic magnetization. The configuration of themagnetization dipoles in Fig 3 shows some magnetization−8 −6 −4 −2 0 2 4 6 8x 105−1−0.8−0.6−0.4−0.200.20.40.60.81H [A/m]mean M [A/m](a) zero dislocations−8 −6 −4 −2 0 2 4 6 8x 105−1−0.8−0.6−0.4−0.200.20.40.60.81H [A/m]mean M [A/m](b) five dislocationsFig. 2. cuboid monocrystal with 0 and 5 dislocations, the solid line is thehysteresis loop for Ha parallel to the y-axis and lattice axes parallel to thexyz-axes, the dashed line for Ha parallel to a space diagonal and lattice axesparallel to the xyz-axes, the dash-dot line for Ha parallel to a space diagonaland lattice axes not parallel to the xyz-axes.mechanics known in the literature [8]. In the y = 0 plain90 degree domain walls come together in a vortex, the otherplains are separated in two magnetic domains by a 180 degreedomain wall.VI. CONCLUSIONThe presented micromagnetic proves that different mech-anisms influence the macroscopic hysteresis loop of amonocrystal. First, the shape of the monocrystal has a largeinfluence on the magnetization. Second, the directions of theapplied field with respect to the lattice axes and geometricalaxes are important. Third, lattice defects influence the shapeof the hysteresis loop.REFERENCES[1] L. D. Landau and E. M. Lifshitz, Electrodynamics of continiuous media,Pergamom Press X: Oxford-London-New York-Paris, 1960.5050100050100150200250020406080100120x [10−8 m]y [10−8 m]z [10−8  m]Fig. 3. Microscopic magnetization configuration for the hysteresis loop withsolid line of Fig 2(a) with Ha = 0 (remanent magnetization). The arrowsshow the magnetization in the three edge plains x = 0, y = 0 and z = 0−8 −6 −4 −2 0 2 4 6 8x 105−1−0.8−0.6−0.4−0.200.20.40.60.81H [A/m]mean M [A/m]Fig. 4. Hysteresis loop for spherical monocrystal with zero dislocations forHa parallel to a lattice axis[2] H. Kronmu¨ller and M. Fa¨nhle, Micromagnetism and the microstructureof ferromagnetic solids, Cambridge University Press: Max-Planck-Institutfu¨r Metallfoschung, Stuttgart, Germany, 1994.[3] F. R. N. Nabarro, ”The mathematical Theory of Stationary Dislocations”,Advances in Physics, Vol. 1(3), July 1952.[4] G. E. Dieter, Mechanical Metallurgy, McGraw-Hill Book Company,London, 1988.[5] D. T. Britton and M. Ha¨rting, ”The Influence of Strain on Point DefectDynamics”, Advanced Engineering Materials, Vol. 4(8), 2002.[6] Y. C. Shu, M. P. Lin and K. C. Wu, ”Micromagnetic Modeling of mag-netostrictive Materials under intrunsic Stress”, Mechanics of Materials,Vol. 36, pp. 975-997, 2004.[7] E. W. Weisstein, Euler Angles, From Mathworld–A Wolfram Web Re-source. http://mathworld.wolfram.com/EulerAngles.html[8] G. Bertotti, Hysteresis in Magnetism, Academic Press, 1994.

Fast numerical 3D-Scheme for the Simulation of Hysteresis in ferromagnetic Materials

Fast numerical 3D-scheme for the simulation of hysteresis in ferromagnetic materials

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Fast numerical 3D-scheme for the simulation of hysteresis in ferromagnetic materials

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Ghent University Academic Bibliography

Ghent University Academic Bibliography