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  716 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 2, FEBRUARY 2009 Comparison of Models for Estimating Magnetic Core Losses in ElectricalMachines Using the Finite-Element Method Emad Dlala Department of Electrical Engineering, Helsinki University of Technology, 3000 Helsinki, FI-02015 TKK, Finland This paper focuses on the modeling and prediction of core losses in nonoriented magnetic materials of electrical machines. The aimis to investigate the accuracy, efficiency, and stability of certain models, including the commonly used and the advanced ones, and todiscuss theiradvantagesand disadvantages whenthey are implementedin thefinite-elementmethod (FEM). It is shownin thepaper thatthe traditional technique based on the loss separation theory can efficiently produce reasonable results in specific operation conditionsbut can, on the other hand, over- or underestimate the core losses in other circumstances. The advanced model based on solving theone-dimensional(1-D) Maxwell equations cangiveaccurate resultsfor theprediction ofcore losses in alamination strip,but itsaccuracy,stability, and computational burden are put under scrutiny when it is applied to the prediction of core losses in an electrical machine. Athird technique, referred to as the hybrid model, which captures the advantages of the traditional and advanced techniques and mergesthem into one, has been found to be the best compromise. The principal aim of the hybrid model is to avoid the numerical procedure of the 1-D Maxwell equations while maintaining relatively accurate predictions with a reasonable computational burden. A comparativeinvestigation has been conducted for the three core-loss models that have been incorporated into the 2-D FEM analysis of a 37-kWinduction motor on which experiments were carried out for comparisons.  Index Terms— Core loss, dynamic hysteresis, eddy currents, excess loss, finite-element method (FEM), harmonics, rotating electricalmachines, soft magnetic materials, time-stepping. I. I NTRODUCTION T HE characteristics of magnetic materials are importantto the performance and efficiency of electrical devices.Nonoriented materials are utilized in electrical machine coresto direct and maximize the magnetic field that acts as a mediumin the energy conversion process. This utilization results in pro-ducing large torque or large machine output per unit machinevolume. The magnetic field variation inside the magnetic mate-rials causes energy loss dissipations in electrical machines, theso-called iron losses or the core losses. The core losses are con-ceptually separated into three loss components, known as thehysteresis, classical eddy-current, and excess losses [2].Themagneticfluxpatternsappearinginanelectricalmachineare complicated and, therefore, they hinder the development of adequate methods for the prediction of core losses. In the ma- jority of cases, the complexity has been grossly reduced to theuse of simplistic techniques that are based upon postprocessingthe magnetic field solution and separating the core losses ac-cordingly [1], [3]–[5]. These techniques are widely believed toprovide reasonable results, but their limitations and imperfec-tions in generally obtaining accurate results are commonly ac-knowledged [6]–[9].Achieving high accuracy while the core losses are incorpo-rated into the field solution requires the application of magne-todynamic models that can track the magnetization behavior inthe magnetic materials under sinusoidal or distorted, unidirec-tional, or rotating flux conditions [10]–[12]. Following this pathwill not only guarantee the accuracy of the modeled core losses Manuscript received August 26, 2008; revised October 14, 2008. Currentversion published February 11, 2009. Corresponding author: E. Dlala ( versions of one or more of the figures in this paper are available onlineat Object Identifier 10.1109/TMAG.2008.2009878 but also will ensure the accuracy of the modeled overall perfor-mance of the electrical device. The latter feature seems to beunimportant when dealing with an electrical steel sheet alonewhere one is mainly interested in the core loss. In electrical ma-chine applications, however, the shape of the loops determinethe shape of the current waveforms, and hence, they are impor-tant for the evaluation of the machine characteristics and otherelectromagnetic losses.Consequently, the accurate analysis of the core losses in elec-tricalmachinesrequiresin-depthtreatmentofthemagneticfieldand the ferromagnetic materials. The coupling of the magneticfieldsolutionobtainedbythefinite-elementmethod(FEM)withthe hysteresis models needs therefore to be performed rigor-ously [13]. The eddy currents in the laminations must be mod-eled either, for example, by generalizing the hysteresis modelsorbyreformulatingthemagneticfieldequations[14],theexcessloss being included through the dynamization of the hysteresismodels.Although taking the magnetodynamic effects into account isimportant for the analysis of electrical machines, incorporatingthenonlinear models, including hysteresis, excess,and classicaleddy-current models, into FEM is indeed very complicated andcan easily lead to divergence for the numerical analysis; partic-ularly, when complex structures such as electrical machines areapplied. Therefore, one may have to compromise the accuracyforthesakeofstabilitybecausethesegoalsmayconflict.Forex-ample, a highly accurate method for predicting core losses maynot be stable for certain problems. Tradeoffs between accuracy,robustness, and speed are central issues in numerical analysis,and here they receive careful consideration.In this paper, three different techniques for the prediction of core losses in electrical machines will be investigated. The prin-cipal purpose of the work is to evaluate the performances of the core-loss models when implemented in the FEM analysisof electrical machines and discuss the advantages and disadvan-tages of each technique, where they fail and where they prevail. 0018-9464/$25.00 © 2009 IEEE  DLALA: COMPARISON OF MODELS FOR ESTIMATING MAGNETIC CORE LOSSES 717 The developed methods are applied in a two-dimensional (2-D)in-house FEM code, specialized for the design and analysis of electrical machines. The FEM simulations and the analysis of core losses are conducted on a 37-kW induction motor, and thenumerical results are validated experimentally.II. T ECHNIQUES FOR  M ODELING  M AGNETIC  M ATERIALS AND C ORE  L OSSES The history of the methods employed for the prediction of core losses goes back to the famous formula introduced bySteinmetz in the early twentieth century [15]. Today, severalapproaches are followed and, in this section, the most importantand common ones are discussed and analyzed.  A. Traditional Technique Traditionally, thecore losses in electrical machines are calcu-lated by postprocessingthe magnetic fieldsolution usingempir-ical equations or statistical laws [2], [3], [16], which have beenreinforced on several occasions [17]–[19].Fiorillo and Novikovmade a generalization to the statistical loss theory of Bertotti[2] in order to account for arbitrary (nonsinusoidal) flux den-sity waveforms [20]. The total power loss, , per unit volumedissipated in a ferromagnetic strip lamination, having thicknessand conductivity and periodically magnetized with funda-mental frequency , can then be given as the sum of the hys-teresis, , classical eddy-current, , and excess, , losses(1)where is the amplitude of the th harmonic of the flux den-sitywaveformsobtainedbythetime-steppingFEMsolutionandis the total number of harmonics considered. If the field isassumed to be 2-D, independent of the coordinate parallel to theshaft of the machine ( -direction), then represents the am-plitude of the flux density as . The clas-sical eddy-current term is derived from Maxwell equationsassuming a uniform flux distribution in the -direction; hence,. The coefficients and can be identifiedexperimentally from Epstein frame, core ring, or single-sheettester measurements. The flux density is obtained by the 2-DFEM using a single-valued magnetization curve in which themagnetic flux density and the magnetic field strength areassumed to be collinear. Thus, the nonlinear relation is handledas follows.1) Compute the components of the flux density andat each time step from the 2-D FEM solution.2) Compute the magnitude of the flux density asand the magnitude of the mag-netic field strength from the single-valued relation as(2)3) Compute the components of the field strength as(3)The  traditional  approach (1), although widely consideredto be grossly simplistic, is the most popular technique appliednowadays in research and commercial software of FEM pack-ages. The impetus for the popularity of the postprocessingmethods is attributed mainly to the following reasons: 1) themagnetic field is not required to be solved using hysteresismodels, and hence, only a lossless single-valued magnetizationcurve, which permits the stability and efficiency of the iterativeprocedure involved, is sufficient; 2) the identification problemof the core-loss model is simple, leading to the determinationonly of three parameters; and 3) the accuracy of the calculatedtotal core losses integrated over the volume of the machineis quite satisfactory in specific regimes of frequencies andvoltages.However,thedrawbacksofusingthepostprocessingmethodsoutweigh the advantages and they are numerous too: 1) the sta-tistical loss theory [2] and its various extensions [19] are suit-able only for low-frequency applications, because their deriva-tion is based on neglecting the skin effect [1]; 2) the effects of minor loops, which are important especially in the teeth coresand rotor surfaces, are not modeled properly by the postpro-cessingmethods;and 3)theincorporation ofcorelosses intothefield solution is hindered by the use of a lossless single-valuedmagnetizationmodelsothattheeffectsofcorelossesonthema-chine characteristics cannot be examined.  B. Advanced Technique The aforementioned shortcomings of the postprocessingmethods have led researchers in the area to the use of advancedmethods that are able to model the core losses more adequately.For example, the study of certain phenomena such as hysteresistorque [13], [21] can be achieved only if appropriate mod-eling of the vector hysteresis relation (hysteresis loop shapesincluding minor loops under alternating or rotating flux) is per-formed. Furthermore, cooling system designers, who nowadaysrequire the distribution of the losses in the electrical machinefor thermal analysis, cannot rely heavily on oversimplifiedapproaches. More importantly, if one pursues a proper wayfor reducing core losses, even the very small details must beadequately taken into account, or otherwise, the desired resultswould not be achieved.However, the eddy-current loss generated in the conductingsteel creates a difficult problem to deal with [14]. Electrical ma-chine cores are usually made of laminated materials in order tominimize the eddy-current loss induced. The eddy-current lossin the lamination intrinsically creates a 3-D magnetodynamicproblem enforced by Maxwell equations(4)  718 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 2, FEBRUARY 2009 The 3-D analysis is not commonly considered because of itshigh computation time, especially if hysteresis models are used.Therefore, if the edge effects are neglected, then (4) can be re-duced to the solution of two 1-D coupled penetration equations[14](5)Although these may appear as separate equations, they arestrongly coupled through the vectorial hysteretic relationshipbetween and , and also through the boundary conditions[22].Whendealingwith2-DFEMproblems,suchaselectricalma-chines, the 1-D magnetodynamic model (5), which representsthe solution of two 1-D coupled penetration equations, has tobe coupled with the 2-D FEM formulations. Despite its reason-able accuracy, the resulting 1-D-2-D  advanced   model increasesthe computation time and is easily vulnerable to convergenceproblems since it involves two iterative solutions of two cou-pled nonlinear problems [23].The inclusion of the hysteretic behavior in the eddy-currentproblem is vital, not only for obtaining accurate results but alsofor validating the numerical model by experiments. The rota-tionalcore losscallsupona vectorialhysteretic relationshipthatmust,toacertainextent,satisfythematerialhystereticbehavior.Modeling vector hysteresis is not the main focus of this paperand thus will not be dealt with thoroughly. Nevertheless, theMayergoyz vectorhysteresismodel [24] willbe appliedtocom-plete the analysis of the core losses. The model idea is based onprojecting the input data (magnetic field strength) in all feasibledirections, say , while using a scalar model that keeps its his-tory in each direction. The vector Mayergoyz model has beenrather successful, except for its rotational loss property, whichwill not be discussed here [25]. On the other hand, when themodel is used to deal with eddy currents, the vector model com-plicates the iterative procedure and increases the computationtime remarkably [25]; the 1-D model of the lamination, whichinvolves an iterative time-stepping procedure, has to be appliedineverydirection,creating penetrationequationstobesolved......(6)These penetration equations are strongly coupled through thevector hysteresis model. C. Hybrid Technique A technique that has been recently proposed in [26] but hasnot yet been tested in the FEM analysis of electrical machineswill be discussed in more detail here. The technique, referredto here as the  hybrid   technique, captures the advantages of thetwo aforementioned techniques and merges them into one: 1)the simplicity and stability of the postprocessing formula (tra-ditional technique) and 2) the accuracy and generality of the1-D magnetodynamic model with the hysteresis and eddy-cur-rent effects taken into account (advanced technique.) The prin-cipalaimofthehybridtechniqueistoomitthenumericalproce-dure of the nonlinear penetration equations (1-D model) by de-termining, using simple concepts, three magnetic field strengthcomponents: a hysteresis component, , excess component,, and classical eddy-current component, , which are re-sponsible for generating and calculating the core losses.The Fiorillo-Novikov method [20] can be expressed in a rela-tively different way to calculate the power loss per unit volumeover a time period of the fundamental frequency as(7)where . Equation (7) can be also applied to cal-culate the total energy loss as(8)and since , (8) becomes(9)By comparing (9) with (8), the total applied field is determinedas(10)where the directional parameter iscontrolled by the magnetic flux density, whether it is increasingor decreasing.The equivalence of the loss separation (9) and the fieldstrength separation (10) has been also shown in [1] and [27].In this manner, the core losses are calculated by integratingthe loop areas that are created by the contributions of the mag-netic field strength of each loss component. This systematicprocedure has transformed the postprocessing method (7) intoanother method (10), which has retained all the main advan-tages of the postprocessing method and added several othersto them. In addition to its ability to incorporate the losses intothe field solution, (10) can be verified experimentally by usingthe instantaneous values of the and waveforms (loopshapes) and the integral of the core loss (loop area), unlike thepostprocessing method (7), which only uses the integral of thecore loss.The first term of (10) is calculated by applying any suitablestatic hysteresis model; the second term represents the excess  DLALA: COMPARISON OF MODELS FOR ESTIMATING MAGNETIC CORE LOSSES 719 field through the time delay of the magnetic flux density be-hind the magnetic field strength; the third term is the classicaleddy-current field, which is analytically derived from Maxwellequations assuming a uniform flux distribution over the sheetwhere no dependency on the magnetization law is con-sidered [1]. This particular assumption, which was srcinallymade in order to avoid the numerical solution of the Maxwellequations, directly implies that (10) can produce accurate re-sults only if the skin effect is negligible. The flux density av-erage is assumed to be independent of and, as a result, theclassical eddy-current loss becomes independent of thematerialrelation . To improve the predictions of (10), the classicaleddy-current field is forced to be implicitly dependent on themagnetization law through the use of a scaling function(11)Experiments [26] show that the scaling function can be ap-proximated by a second-order polynomial as(12)where is a predefined saturation value of the magnetic fluxdensity. The scaling function can be further revised in order toimprove the accuracy of minor loops by letting the coefficientbe linearly dependent on the reversals of the flux densities(13)The reversals of the flux density are tracked in time andwhenever changes sign, a new reversal occurs. The co-efficients , , and can be estimated by fitting thecalculated dynamic loops to the experimental ones [26]. Thevalue of can be assigned to be the same as . An optimalchoice of the coefficients in must ensure that the eddy-currentfield is nonlinearly dependent on , analogous to the real-istic case endured by (5).Since the hybrid technique will be used for the predictionof core losses in rotating electrical machines, a vector hys-teresis model is developed to account for the rotational losses,including the hysteresis, classical eddy-current, and excessrotational losses. The Mayergoyz model of vector hysteresiscan be applied in its inverted version to calculate the magneticfield strength [13]. In this way, the magnetic flux densityis projected over several directions specified by an angleto calculate the magnetic field strength using a scalar dynamicmodel such as (11). The magnetic field strength is the vectorialsum of each contribution(14)where is the number of directions along and is a pa-rameter needed for the identification. The projections of arecalculated as(15)where and .Thepolar angle specifies the direction of and the scalar functionis served by (11).The total power losses per unit volume are computed usingthe Poynting vector theorem as(16)Furthermore,separatingthealternatingpowerlossfromthetotalpower loss can be achieved as [29](17)where is the angle lag between and . The rotational losscomponent can be then computed as(18)which goes to zero under purely unidirectional (alternating)field excitations.III. F INITE -E LEMENT  A NALYSIS OF  E LECTRICAL  M ACHINES The magnetic material models developed in Section II are es-sentially useless if they cannot be applied to the prediction of the behavior of magnetic materials in electrical devices. The ac-curate solution of the magnetic field in a complicated geometrysuchasarotatingelectricalmachinerequiresrigorousnumericaltreatment. The spatial discretization of the geometry is neededand here it is accomplished by the use of the 2-D FEM cou-pled with the circuit equations of the supply circuit and the endwindings[30].InSectionII,itwasassumedthatthefluxdensitycomponents, and ,wereknown;theyarenotinrealityandhence they need to be calculated by FEM at each time step .Then, the magnetic field strength components, and , aredetermined by the material model and entered back to the 2-DFEM.The coupling between the FEM and the material model is in-tricate and receives considerable attention in the article. How-ever, the 2-D FEM will not be elaborated on as a whole. Moreemphasis will be placed on the part concerning the magneticlaminated materials. Nevertheless, the complete overall systemof equations will be briefly presented.
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