BASSO V.

BERTOTTI G.

BOTTAUSCIO O.

CHIAMPI M.

FIORILLO F.

PASQUALE M.

REPETTO M.

English

CHIAMPI, Mario

REPETTO, MAURIZIO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEPower Losses in Magnetic Laminations with Hysteresis: Finite Element Modeling and Experimental Validation / BASSOV.; BERTOTTI G.; BOTTAUSCIO O.; CHIAMPI M.; FIORILLO F.; PASQUALE M.; REPETTO M.. - In: JOURNAL OFAPPLIED PHYSICS. - ISSN 0021-8979. - 81(1997), pp. 5606-5608.OriginalPower Losses in Magnetic Laminations with Hysteresis: Finite Element Modeling and ExperimentalValidationPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1398036 since:AIPPower losses in magnetic laminations with hysteresis: Finite elementmodeling and experimental validationV. Basso, G. Bertotti, O. Bottauscio, F. Fiorillo, and M. PasqualeIEN Galileo Ferraris, INFM and GNSM, C. so Massimo d’Azeglio 42, 1025 Torino, ItalyM. Chiampi and M. RepettoPolitecnico di Torino, C. so Duca degli Abruzzi 24, 1029 Torino, ItalyDynamic hysteresis loop shapes and magnetic power losses are studied in nonoriented Fe-Silaminations exhibiting significant excess losses. Measurements are carried out under controlledsinusoidal induction in the frequency range from 1 Hz to 1.6 kHz, at various peak inductions from0.25 to 1.5 T. Excess losses are found to obey a f 3/2 law up to frequencies of 200–400 Hz,depending on peak induction. Beyond this limit, definite deviations are observed, due to eddycurrent shielding. Detailed information on the flux and field distribution in this high frequencyregime is obtained by finite element solutions of Maxwell equations employing the dynamicPreisach model to describe quasi-static hysteresis and dynamic wall processes. The agreementbetween theoretical predictions and measurements is discussed. © 1997 American Institute ofPhysics. @S0021-8979~97!39708-4#I. INTRODUCTIONA problem of increasing interest in applications is theaccurate prediction of losses in magnetic cores operating inwide frequency intervals, under highly distorted flux wave-forms. In such cases, the dynamic response of the core de-pends on the complex interplay of various factors ~i! thehysteresis properties of the material; ~ii! the incomplete fluxpenetration due to eddy current shielding and, ~iii! the mo-tion of domain walls, giving rise to excess losses.In Ref. 1 we studied, both theoretically and experimen-tally, dynamic hysteresis and power losses in thick ~1 mm!SiFe sheets, a case where excess losses play a negligible roleand the interplay of static hysteresis and eddy current shield-ing could be analyzed in detail. Following these results, inthis article we consider Fe-Si laminations whose thickness~0.33 mm! and grain size ~of the order of 100 mm! are suchthat a significant excess loss contribution is expected. Wehave found that, up to frequencies of 200–400 Hz, excesslosses follow the f 3/2 law predicted by the statistical losstheory.2 Beyond this limit, a more complex behavior sets in,as a consequence of substantial eddy current shielding. Inthis high frequency regime, a numerical study was carriedout, based, as recently proposed by several authors,3,4 on thedynamic Preisach model ~DPM!.5 DPM gives, in the form ofa rate-dependent material constitutive law, a unified descrip-tion of quasi-static hysteresis as well as dynamic domaineffects responsible for excess losses. By inserting this con-stitutive law in a Maxwell equation finite element solver, it ispossible to study in some detail the behavior of power lossesand dynamic loop shapes when eddy current shielding anddomain effects have comparable roles.II. EXPERIMENTAL RESULTSMeasurements were performed on commercial ~AST 27/35! nonoriented Fe–Si alloys ~2.9 wt. % Si, 0.4 wt. % Al!,with average grain size s'75 mm. The lamination thicknesswas 0.334 mm. Epstein strips were cut at 0°, 30°, 60°, 90°angles to the rolling direction and inserted in the four framelegs in the order 0°, 30°, 90°, 60°, to emulate an isotropic5606 J. Appl. Phys. 81 (8), 15 April 1997 0021-8979/97/Downloaded¬02¬May¬2006¬to¬130.192.14.64.¬Redistribution¬subjesituation. Dynamic hysteresis loops, measured under con-trolled sinusoidal induction waveform at frequencies be-tween 1 Hz and 1.6 kHz and peak inductions between 0.25and 1.5 T, were recorded and the corresponding power losseswere determined. A general overview of the measured en-ergy loss per cycle versus frequency is given in Fig. 1, whiledetails on the 1 and 1.5 T loss data are shown in Fig. 2.The loss data were analyzed by means of the statisticalloss theory,2 which predicts for the power loss per unit massin nonoriented steelsP5Ph1Pc1Pe5Ph1p26d sd2~Bp f !218dAsGSV0~Bp f !3/2, ~1!where Ph is the quasi-static hysteresis loss, Pc is the classi-cal loss, and Pe is the excess loss. As to the other symbols,d57.653103 kg m23 is the mass density, s52.043106V21 m21 is the electric conductivity, G50.1357, S51025m2 is the strip cross-sectional area, V0 is a characteristic fielddescribing the influence of the material microstructure ondomain wall processes, f is the magnetization frequency, andBp is the peak induction. The validity of this law can betested by plotting (P2Pc)/ f as a function of f 1/2. This plotpermits one to make a reliable estimate of the quantities Phand V0 , which, once inserted in Eq. ~1!, lead to the theoret-ical curves shown in Figs. 1 and 2. According to Ref. 2 whenthe correlation regions controlling the magnetization processcoincide with individual grains, the field V0 should be relatedto the grain size s and the static loss Ph through the relation-ship V0'2s2Hhyst^Bs&/SBp , where ^Bs&51.7 T is the satu-ration flux density of an assembly of randomly orientedgrains where rotation effects are negligible and Hhyst is anaverage coercivity value, Hhyst5dPh/4Bp f . At 1.5 T, wherePh538 mJ kg21, i.e., Hhyst548.5 A m21, we find V0'0.06A m21. This theoretical estimate is about one third of theexperimental value of 0.17 A m21, which indicates that theactual size of the correlation regions is slightly larger thanthe grain size.Figures 1 and 2 show that Eq. ~1! is able to reproduce81(8)/5606/3/$10.00 © 1997 American Institute of Physicsct¬to¬AIP¬license¬or¬copyright,¬see¬http://jap.aip.org/jap/copyright.jspwell the overall trend of measured losses. Yet, some signifi-cant discrepancies are observed. In general, the theoreticalprediction overestimates the loss at low inductions and un-derestimates it at high inductions. The crossover is at 1 T,FIG. 1. Loss per cycle in nonoriented SiFe lamination. Symbols measureddata. Dotted lines predictions of Eq. ~1!. Values of Ph ~in mJ kg21! and V0~in A m21! used in calculations are as follows: 0.25 T: ~1.45,0.07!; 0.5 T:~4.45,0.1!; 0.75 T: ~8.6,0.12!, 1 T: ~14,0.14!, 1.25 T: ~22,0.15!; 1.5 T:~38,0.17!.FIG. 2. Detail of loss per cycle at 1 and 1.5 T. Solid symbols and dottedlines: same as in Fig. 1. Open symbols: prediction of FEM calculation basedon DPM.J. Appl. Phys., Vol. 81, No. 8, 15 April 1997Downloaded¬02¬May¬2006¬to¬130.192.14.64.¬Redistribution¬subjewhere Eq. ~1! is in remarkable agreement with experimentsin the whole investigated frequency interval. At 1.5 T, thedeviation of measured data from the f 1/2 law is quite signifi-cant when f exceeds '400 Hz. This is the consequence ofdistorted flux penetration due to eddy current shielding, afact not taken into account by the theory leading to Eq. ~1!.In this regime, quasi-static hysteresis, classical eddy currentshielding, and domain effects all have comparable roles andit is difficult to make accurate loss estimates without study-ing the full dynamic problem.III. NUMERICAL STUDYThe dynamic flux and field distribution in the laminationcross-section is obtained by solving the equation¹3¹3H˜ ~x ,t !52s]B˜@H˜ ~x ,t !#]t~2!for the electromagnetic H field as a function of the depth xacross a slab of thickness d and infinite width. The constitu-tive law BH describes the hysteresis behavior of the mag-FIG. 3. Dynamic hysteresis loops at 1 T, for 0, 400, 1600 Hz. Continuouslines measurements. Dotted lines FEM prediction based on dynamic PM.Broken line. FEM prediction based on conventional PM.FIG. 4. Dynamic hysteresis loops at 1.5 T, for 0, 400, 1300 Hz. Continuouslines measurements. Dotted lines. FEM prediction based on dynamic PM.5607Basso et al.ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jap.aip.org/jap/copyright.jspnetic material. It is assumed that B and H are always colinearand pointing along the longitudinal axis of the slab, whichreduces BH to a scalar relationship B(H) between the fieldintensities. In this work, B(H) is described by the DPM,5which summarizes in a rate-dependent constitutive lawquasi-static hysteresis and domain wall dynamic processes.Equation ~2! is supplemented by the integral constraint* B[H(x ,t)]dx5F(t), where F(t) is proportional to theimposed sinusoidal magnetic flux.The problem is linearized following the fixed point tech-nique, which splits B(H) into a linear term and a residualpart S[B(H)5mTH1S], where S has to be iteratively esti-mated, starting from a trial value. Under periodic supply con-ditions, time dependencies are expanded in harmonic com-ponents, giving rise to a set of equations of the form¹3¹3Hk~n !52 jsvnmTHk~n !2 jsvnSk21~n ! ~3!with the constraint F (n)5*[mTHk(n)1Sk21(n) ]dx . Here v is thefundamental angular frequency, n is the harmonic order, k isthe iteration index, and the field quantities are complex vari-ables. At each iteration step, the finite element method~FEM!, is employed to solve Eqs. ~3!, by dividing the slabthickness into 40 elements. This calculation gives the har-monic content of Hk in each element. The time waveformHk(t) is then determined by inverse fast Fourier transform~FFT!. In order to compute the residual partSk(t)5Bk(t)2mTHk(t), the flux density Bk(t) is evaluatedthrough DPM. To this end, Hk(t) is applied as an input toDPM, starting from the demagnetized state, and, after theinitial transients have died out ~usually after 3–4 periods!,the asymptotic periodic flux density output Bk(t) is obtained.Once Sk(t) is computed in each element, its harmonic con-tent is calculated by FFT, the Sk(n) terms in the right handside of Eq. ~3! are updated and the whole procedure is iter-ated until convergence is reached.In the implementation of DPM, it is necessary to keeptrack of the time evolution of each point of the Preisachplane. The plane must be discretized in a number of elemen-tary hysteresis operators sufficiently large to accurately re-produce the global behavior of the system, but on the otherhand small enough to keep the computation time within rea-sonable limits. To this purpose, a new computationally fastalgorithm was developed, where a single elementary hyster-esis operator is associated with a finite region of the Preisachplane, and appropriate modified evolution equations for thewhole region are introduced.6The Preisach distribution p(a ,b) was reconstructedfrom the recoil curve measured at 1.5 T.1,7,8 The DPM re-quires an hypothesis on the value of the parameter k , definedin Ref. 5 by the relation ]w(a ,b;t)/]t5k(H2a) @or k(H2b)#, where w(a ,b;t) ~21<w<1! describes the state of thepoint ~a,b! of the Preisach plane and k measures the strengthof the eddy current damping acting on moving domain walls.In Ref. 9 it is shown that, when the grain size controls thesize of the correlation regions driving the magnetization pro-cess, k is expected to follow the relation k'1/sG^Bs&s2,which gives k'380 mA21 s21. In order to test this estimate,we carried out a set of simulations at 400 Hz and 1 T, even-5608 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997Downloaded¬02¬May¬2006¬to¬130.192.14.64.¬Redistribution¬subjetually choosing for k the value at which the calculated andthe measured loops had the same area. We found k5320mA21 s21, which, when compared to the theoretical estimatek'380 mA21 s21, points again to correlation regions of sizeslightly larger than the grain size. The value k5320mA21 s21 was kept constant in all subsequent simulations.Calculations were performed at 10, 50, 200, 400, 1000,1600 ~1300 at 1.5 T! Hz, at 1 and 1.5 T peak inductions. Inaddition to the DPM approach used here, calculations basedon the conventional PM method were also carried out, inorder to quantify the importance of domain effects. Figures 3and 4 show some of the measured hysteresis loops ~continu-ous lines! together with the prediction of DPM calculations~dotted lines!. The hysteresis loop calculated at 400 Hz, 1 T,by the conventional PM approach is also shown for compari-son. The calculations based on the conventional PM gaveunsatisfactory predictions for the loop shape in all the casesstudied here. Domain effects should therefore be included ina correct procedure for the calculation of power losses byFEM in this type of laminations. The use of DPM definitelyimproves the agreement with measurements, even though, athigh frequencies, the experimental loop shape appears morecomplicated than that which is predicted by the model. Inany case, the loss prediction is basically correct, as illustratedby the results shown in Fig. 2 ~open symbols!, where, by theuse of DPM, one can resolve discrepancies between the re-sults provided by Eq. ~1! ~dotted lines! and measured losses~solid symbols!.As a final comment about the possible origin of someobserved differences between simulations and measurementswe might stress that the method used to determine the Prei-sach distribution was probably oversimplified. In particular,it does not take into account the field and induction depen-dence of the reversible permeability and the possible pres-ence of mean-field effects.9 On the other hand, a more gen-eral problem resides in the fact that, with increasingfrequency and increasing flux shielding by eddy currents, themagnetization process tends to become highly inhomoge-neous in the lamination cross section. With such a rapidspace variation of the flux distribution, it is not clear if onecan still describe domain effects by a DPM-like local consti-tutive law, since the use of DPM necessarily implies someaveraging of dynamic wall processes over regions containingmany domains.1C. Appino, G. Bertotti, O. Bottauscio, M. Chiampi, F. Fiorillo, M.Repetto, and P. Tiberto, J. Appl. Phys. 79, 4575 ~1996!.2G. Bertotti, IEEE Trans. Magn. 24, 621 ~1988!; F. Fiorillo and A. No-vikov ibid. 26, 2904 ~1990!.3D. A. Philips, L. R. Dupre, and J. A. A. Melkebeek, IEEE Trans. Magn.30, 4377 ~1994!.4L. L. Rouve, F. Ossart, T. Waeckerle´, and A. Kedous-Lebouc, IEEETrans. Magn. 32, 4219 ~1996!.5G. Bertotti, IEEE Trans. Magn. 28, 2599 ~1992!.6O. Bottauscio, M. Chiampi, and M. Repetto, Proc. 7th Int. IGTE Sympo-sium, Graz, Austria, 1996, p. 508.7L. L. Rouve, T. Waeckerle´, and A. Kedous-Lebouc, IEEE Trans. Magn.31, 3557 ~1995!; V. Basso, M. Lo Bue, and G. Bertotti, J. Appl. Phys. 75,6577 ~1994!.8G. Kadar, J. Appl. Phys. 61, 4013 ~1987!.9G. Bertotti and M. Pasquale, IEEE Trans. Magn. 28, 2787 ~1992!.Basso et al.ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jap.aip.org/jap/copyright.jsp

Power Losses in Magnetic Laminations with Hysteresis: Finite Element Modeling and Experimental Validation

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Power Losses in Magnetic Laminations with Hysteresis: Finite Element Modeling and Experimental Validation

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