Dupré, Luc

GARSHELIS, I

Sergeant, Peter

TOLLENS, S

Vandenbossche, Lode

Undetermined

Ghent University Academic Bibliography

IEEE TRANSACTIONS ON MAGNETICS 1Numerical model for the determination of hysteresisloss through the drag force methodPeter Sergeant, Luc Dupre´, Lode Vandenbossche, Ivan Garshelis and Stijn TollensAbstract To determine the hysteresis loss in a sample, usuallythe enclosed area of the B − H loop is evaluated. In the dragforce method, based on a proper energy balance, the hysteresisloss is obtained by analyzing the drag force prole when slowlymoving the sample forward and backward through the strongeld of permanent magnets. A numerical time-stepping modelis presented that calculates the drag force prole. At everytime step, the sample is slightly moved. The model is basedon 2D-FE computations including magnetic hysteretic materialbehaviour using the Preisach model. In order to improve thenumerical stability, we reformulated the Maxwell equations insuch a way that the material behaviour is described throughdifferential permeabilities. Consequently, the basic unknown forthe FEM becomes the time derivative of the vector potential.The drag force is obtained by using the Maxwell stress tensor.Simulations were carried out in which a sample was moved backand forth through the eld of one or two permanent magnets.The numerical model is validated by measurements and bycomparing the drag force hysteresis losses with the hysteresislosses computed conventionally by the integral of H .dB.I. INTRODUCTIONIn the drag force method, the hysteresis loss is obtainedby measuring the drag force that arises from the forward andbackward movement of a sample relative to the strong fieldof permanent magnets [1]. The distance between magnet andsample is kept constant in time. In the calculation, the magnetis moved to the left over the fixed sample – see Fig. 1a. Toavoid eddy currents, the movement should be sufficiently slow.The drag force profile is calculated by a numerical time-stepping model that uses a finite element model (FEM) com-bined with a hysteresis model. We first focus on these twomodels and then explain the whole calculation scheme.MagnetMmSampleMagnets=Mm x1Sample(a) (b)xyxy∂Mm∂t =-Mm x1∂Mm∂t=Mm x1∆Movement directionFig. 1. (a) Geometry of the drag force method (not on scale) in case of onemagnet that moves over the sample to the left. (b) Implementation of themethod in the model using differential permeability. The moving magnet isreplaced by two thin magnet slices with the movement step ∆ as thickness.The first three authors are with the Department of Electrical Energy,Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41, 9000Gent, Belgium. I. Garshelis is with Magnova, Inc., Pittsfield, MA, USA. S.Tollens and I. Garshelis are with MagCanica, Inc., San Diego, CA, USA.This work was supported by the FWO projects G.0322.04 and G.0082.06.The first author is a postdoctoral fellow with the FWO.II. NUMERICAL MODELA. FEM with differential permeabilityAs there are no current sources in the problem, the classicalFEM with vector potential A – where B = ∇ ×A – solves∇×(1µ∇×A)= 0 or ∇× (∇×A) = µ0(∇×M). In aproblem with hysteresis, it is not possible to solve the vectorpotential FE problem using the classical permeability, becausethis permeability may become negative, zero or infinite. Toovercome this, the problem is rewritten in terms of the timederivative of A. In the magnet, the equation to solve is:∇×(∇×∂A∂t)= µ0(∇×∂Mm∂t)(1)with Mm the magnetization of the magnet (see Fig. 1a). Forthe sample, we introduce the differential permeability µ¯d thatrelates the time derivatives of the flux density and the magneticfield vectors B and H: ∂B∂t= µ¯d∂H∂t. In the sample we have:∇×(1µ¯d∇×∂A∂t)= 0 (2)The flux density at time instant tk +∆t is obtained by adding∂B∂t·∆t to the previous B-value at tk. The differential perme-ability is function of the position in the sample. Its distributionis determined by the hysteresis model (see next paragraph)based on the magnetic field history. As the distribution of µ¯donly depends on the previous field values and as eddy currentsare not taken into account, the FEM is static.B. Preisach hysteresis model in FEMThe differential permeability µ¯d is described by the classicalscalar Preisach model [3]. For the studied material, an Everettmap is generated by measuring a high number of hysteresisloops in a single sheet tester. For a sequence of scalar Hvalues, the corresponding differential permeabilities are cal-culated by the derivatives of the Everett map [4]. All relevantminima Hmin and maxima Hmax are stored in memory. Thishysteresis model is combined with the FEM. To handle the2D-behaviour of the magnetic field vector H, the followingapproximation is considered. As in the drag force method withconfiguration of Fig. 1a, the field is mainly along the x-axis,the scalar hysteresis model is applied only to the x-componentof H. For Hy, a nonlinear characteristic without hysteresis isused. The non-diagonal elements of tensor µ¯d are 0.C. Time-stepping procedureAfter the initialization procedure in Fig. 2, the magnet ismoved backward and forward. During the corresponding timestepping procedure, tk = tk−1 + ∆t, k = 1, 2, 3, . . ., equation2 IEEE TRANSACTIONS ON MAGNETICSInitializationMove magnet one step DSolve FEM forCalculate == +AB AB B B/ t/ t / t¶ ¶¶ ¶ ¶¶ ¶ × D¶ Ñ ´k/ t tk-1Calculate = /H B md¶ ¶¶ ¶ × DH H Hk k= +-1 / t tUpdate m md d H= ( )Determine forceEndFig. 2. Scheme of the numerical model(1) must be solved at each time point k. At time step k,the magnet is slightly moved over a distance ∆. Here, themovement of the magnet results in two subregions with width∆ where ∂Mm∂t6= 0. These two subregions are defined by theposition of the magnet at time point k− 1 and time point k –see Fig. 1b. As dynamic effects are not taken into account, wemay take for simplicity ∆t = 1. Hence, for the 2 subregionsin Fig. 1b, ∂Mm∂tbecomes Mm 1x and −Mm 1x respectively.The distribution µ¯d in (2) in the sample at time tk is foundby the hysteresis model using Hk−1 and previous extrema.The FE problem is solved resulting in the solution ∂A∂tand∂B∂t= ∇ × ∂A∂t. The flux density Bk = Bk−1 + ∂B∂t · ∆tis updated. Now, Hk = Hk−1 + 1µ¯d∂B∂t·∆t and µ¯d(Hk) areupdated iteratively. Finally, the drag force at time k is obtainedby integrating the Maxwell stress tensor applied to Hk alonga path in the air surrounding the sample at a small distance.We define a trajectory of motion during which the magnetis moved backward and forward with constant speed andfor which the start and the end position of the magnet arethe same. The hysteresis loss can be obtained by the energybalance: ∮trajectFx · dx =∫V(∮loopH · dB)dV (3)The work done by the drag force Fx along the trajectoryof motion is equal to the conventional losses integrated overthe whole volume of the sample. Consequently, (3) allows toestimate the hysteresis losses in the sample by measuring thedrag force Fx.III. EXPERIMENTAL VALIDATION AND CONCLUSIONSimulations were carried out for the geometry of the ex-perimental setup [1] in which one NdFeB permanent magnetwas moved back and forth over a sample of 1.6 mm thickness.The magnet is positioned as in Fig. 1a. Fig. 3a shows the dragforce profile during the movement, and the arrows indicate themovement direction of the magnet. Fig. 3b shows the sameprofile as a function of the magnet position. The correspondingB − H behaviour in the center of the sample is shown inFig. 3c. Initially, the magnet is at the right of the sample inposition (1). The dotted line in Fig. 3a and b shows that Fxin N per meter length in z-direction becomes high (264 N/m)as the magnet is moved to the left, comes closer to the 15 cmlong sample and attracts it. Then, the force profile becomesmore or less flat up to position (2). Fig. 3c shows that theconsidered point (the center of the sample) is magnetized untilsaturation and back to a negative H in (2). Now, the movementdirection is changed and we follow the solid line along (3) toposition (4). The force is higher due to hysteresis. Finally, themovement direction is changed again. The sample is broughtback to position (5)=(2), see Fig. 3b. The area enclosed by thesolid line in Fig. 3b indicates the hysteresis loss. The methodis validated numerically by checking the equality of both sidesof eq. (3). The correspondence was good.Similar to the experiments in [2], also simulations weredone with two magnets, symmetrical to the y = 0 plane. Thecalculated drag force profile is similar to the experimental oneshown in [2], with the advantage that the huge normal forcecomponent Fy (547 N/m) appearing in the one-magnet setupis cancelled.The full paper will describe a four-magnet configuration.The two extra magnets magnetize in opposite direction, sothat the hysteresis loops range from −Bsat to +Bsat, whilethe one and two magnet configurations achieve +Bsat only.(a)(b)(c)0 500 1000 1500 2000 2500 3000−1001020304050Time  [× ∆t]F x  [N/m](1)(2) (3) (4) (5)←←→→ ← ←−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1001020304050Position of magnet relative to sample center [m]F x  [N/m] Sample←←←←↑↓→ → (1)(2) (3)(4)(5)−500 0 500 1000 1500 2000 2500 3000−0.500.511.5Hx [A/m]Hmax = 8863 A/mB x [T](1)(2)(3)→(4)(5)Fig. 3. (a) Calculated force profile with indication of the magnet movementdirection, (b) force profile as a function of the position of the sample relativeto the magnet and (c) corresponding hysteresis loops in the sample center.REFERENCES[1] I. J. Garshelis, S. P. L. Tollens, R. J. Kari, L. Vandenbossche, andL. Dupre´, “Drag force measurement: A means for determining hysteresisloss,” J. Appl. Phys., Vol. 99, Art. No. 08D910, Apr. 2006.[2] I. Garshelis, S. Tollens, L. Vandenbossche, et al., “Determination ofhysteresis loss by drag force measurements with cancelled normal fieldcomponents,” Intermag Conf., 8–12 May 2006, San Diego, CA, p. 593.[3] I. Mayergoyz, Mathematical Models of Hysteresis and their Applications,Academic Press, 2003.[4] D. Everett, “A general approach to hysteresis - part 4, An alternativeformulation of the domain model,” Transactions Faraday Society, Vol.51, pp. 1551–1557, 1955.

Numerical model for the determination of hysteresis loss through the drag force method

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Numerical model for the determination of hysteresis loss through the drag force method

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