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Static and Dynamic Hysteresis Modeling of Power Ferrites Up to Curie Temperatures for Power Electronic Applications

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Static and Dynamic Hysteresis Modeling of Power Ferrites Up to Curie Temperatures for Power Electronic Applications
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Nakmahachalasint, Paiboon ( Author, Primary )
Copyright Date:
2008

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Curie temperature ( jstor )
Ferrites ( jstor )
Flux density ( jstor )
High temperature ( jstor )
Hysteresis ( jstor )
Magnetism ( jstor )
Magnetization ( jstor )
Mathematical independent variables ( jstor )
Parametric models ( jstor )
Room temperature ( jstor )

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University of Florida
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University of Florida
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Copyright Paiboon Nakmahachalasint. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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5/1/2013
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78536812 ( OCLC )

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STATIC AND DYNAMIC HYSTERESIS MODELING OF POWER FERRITES UP TO CURIE TEMPERATURES FOR POWER ELECTRONIC APPLICATIONS By PAIBOON NAKMAHACHALASINT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003

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ii ACKNOWLEDGMENTS Iwouldliketoexpressmysincereappreciationtomyadvisor,Dr.KhaiD.T.Ngo, for his excellent academic guidance and continued nancial support; and my co-advisor, Dr. Loc Vu-Quoc, for his enthusiastic discussions. I would also like to thank Dr. Kenneth K. O, Dr. Vladimir A. Rakov, Dr. Dennis P. Carroll, and Dr. Oscar D. Crisalle for serving as my supervisory committee. I am very grateful to the National Science Foundation and the State of Florida Integrated Electronics Center for supporting this research; my employer, Thammasat University,forgrantingmyPh.D.scholarship;andtheRoyalThaiGovernmentforpaying my monthly salary while I am on leave. I wish to thank Dr. Vittorio Basso (of IEN Galileo Ferraris), Mr. Mark A. Swihart (ofMagnetics,Inc.),andMr.JanisM.Niedra(ofNASAGlennResearchCenter)fortheir valuable comments during the course of this work; Mr. George E. Schaller (the Executive VicePresidentofCeramicMagnetics,Inc.)andAdamsMagneticProductsCo.(distributor ofMagneticsandFerroxcubeferrites)forprovidingthesampletoroids.Thanksalsogoto my brother, Dr. Paisan Nakmahachalasint, for help with some proofreading. Mostofall,Iwouldliketoexpressmygreatestlovetomymotherwhodevotesher life to working and taking good care of her six children. She will always be my perfect role model of perseverance. Finally, this dissertation is dedicated to my wife, Oraphan, for her true love and understanding.

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iii TABLE OF CONTENTS page ACKNOWLEDGMENTS............................................................................................ii TABLE OF CONTENTS..............................................................................................iii LIST OF TABLES........................................................................................................v LIST OF FIGURES......................................................................................................vi KEY TO SYMBOLS OR ABBREVIATIONS............................................................viii ABSTRACT..................................................................................................................xi CHAPTER 1INTRODUCTION..................................................................................................1 2STATIC HYSTERESIS MODELING AT ROOM TEMPERATURE..................5 2.1Introduction..................................................................................................... 5 2.2Domain-Wall Motion Model for Static Hysteresis......................................... 6 2.3Synthesis of Domain-Wall Surface Function................................................. 14 2.4Parameter Extraction and Experimental Verification..................................... 17 2.5Conclusion...................................................................................................... 19 3DYNAMIC HYSTERESIS MODELING AT ROOM TEMPERATURE.............21 3.1Introduction..................................................................................................... 21 3.2Dynamic Hysteresis in the Small-Signal Region............................................ 23 3.3Dynamic Hysteresis in the Large-Signal Region............................................ 28 3.4Complete Dynamic Hysteresis Model............................................................ 38 3.5Parameter Extraction and Experimental Verification..................................... 41 3.6Conclusion...................................................................................................... 43 4DYNAMIC HYSTERESIS MODELING AT HIGH TEMPERATURE...............47 4.1Introduction..................................................................................................... 47 4.2Temperature Characteristics of Magnetic Properties/Model Parameters....... 49

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iv 4.3Conversion of Magnetic Properties/Model Parameters.................................. 55 4.4Parameter Extraction and Experimental Verification..................................... 58 4.5Conclusion...................................................................................................... 60 5HIGH-TEMPERATURE CHARACTERIZATION SYSTEM..............................66 5.1Introduction..................................................................................................... 66 5.2Instrumentation............................................................................................... 67 5.3Measurement Procedures and Algorithms...................................................... 72 5.4Error Analysis and Calibration....................................................................... 77 5.5Measured Data................................................................................................ 80 5.6Conclusion...................................................................................................... 81 6SUMMARY............................................................................................................83 APPENDIX ADERIVATIONS AND DEFINITIONS..................................................................85 A.1Mean Domain-Wall Position...........................................................................85 A.2Initial Susceptibility and Differential Susceptibility.......................................86 A.3Coercive Force and Remanent Magnetic Flux Density...................................87 A.4Initial Permeability and Maximum Permeability............................................89 A.5Reversible Contribution of the Mean Domain-Wall Position.........................90 A.6Saturation Magnetic Flux Density...................................................................91 BMATLAB CODES..................................................................................................92 B.1Static DWM Model..........................................................................................92 B.2Dynamic DWM model....................................................................................93 B.3Dynamic DWM model with Temperature.......................................................94 CSTATIC AND DYNAMIC HYSTERESIS LOOPS..............................................96 C.1MN8CX Ferrite................................................................................................97 C.2P Ferrite...........................................................................................................129 C.33F3 Ferrite.......................................................................................................170 DMEASUREMENT ALGORITHMS.......................................................................208 REFERENCE LIST......................................................................................................212 BIOGRAPHICAL SKETCH........................................................................................216

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v LIST OF TABLES page T able 2-1Squared 2-norm error (resnorm) for MN8CX ferrite.......................................... 18 4-1Extracted model parameters for MN8CX ferrite (n = 1).................................... 62 4-2Extracted model parameters for P ferrite (n = 1)................................................ 63 4-3Extracted model parameters for 3F3 ferrite (n = 2)............................................ 64 4-4Temperature Coefficients of the test ferrites....................................................... 65 5-1High-Temperature Materials............................................................................... 69 5-2Material, size, dimensions, and effective parameters of the test toroids............. 81

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vi LIST OF FIGURES page Figure 1-1Summary of the modeling approaches used in this study...................................4 2-1Elemental hysteresis loops..................................................................................7 2-2Irreversible Preisach distribution function..........................................................8 2-3 x H and B H loops...............................................................................................9 2-4Comparison of theoretical and measured R ( m ) for MN8CX ferrite...................11 2-5Normalized magnetization vs mean domain-wall position.................................12 2-6Measured and modeled major and minor loops using R ( m ) = 1 m2.................13 2-7Comparison of theoretical and measured d R ( m )/d m for MN8CX ferrite...........13 2-8Parameter extraction diagram for the static DWM model..................................20 2-9Measured and modeled major and minor loops for MN8CX ferrite...................20 3-1Block diagram of the dynamic domain-wall motion (DDWM) model...............23 3-2Effect of location to apply the filter....................................................................26 3-3Waveforms of v , i , B , H , xrev, xirr, x ....................................................................31 3-4 x (squares), xrev (circles), xirr (crosses), and kirrxirr, dc (solid lines) vs. H ...........32 3-5Extracted (crosses) and computed (solid line) kirr vs. frequency........................37 3-6Block diagram of the DDWM model with equation numbers............................40 3-7Parameter extraction diagram for the dynamic DWM model.............................42 3-8Measured (dashed) and modeled (solid) B-H loops for MN8CX ferrite............44

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vii 3-9Measured (dashed) and modeled (solid) B-H loops for Pferrite........................45 3-10Measured (dashed) and modeled (solid) B-H loops for 3F3 ferrite....................46 4-1Block diagram of the dynamic domain-wall motion (DDWM) model at high temperature..........................................................................................................49 4-2Extracted and fit model parameters vs. temperatures for MN8CX ferrite..........51 4-3Extracted and fit model parameters vs. temperatures for P ferrite......................52 4-4Extracted and fit model parameters vs. temperatures for 3F3 ferrite..................53 4-5Example of the permeability in Eq.4-3 or Eq.4-4 described by a linear combination of two gamma distribution functions.............................................55 4-6Parameter extraction diagram for the DDWM model with temperature.............60 5-1Automatic high-temperature hysteresis measurement system............................68 5-2Test fixture for measurements up to 250oC.......................................................69 5-3Experimental procedure and algorithms for the automatic acquisition of multiple B H loops at different temperatures and frequencies............................73 5-4Blockdiagramfortheautomaticacquisitionofasingle B H loopwithvariable flux, along with a sample Labview program.......................................................76 5-5Blockdiagramfortheautomaticacquisitionofasingle B H loopwithconstant peak magnetic flux density.................................................................................76 5-6Uncompensated and compensated primary current............................................79 5-7Static hysteresis loops of MN8CX ferrite measured at 100oC............................82 5-8Dynamic hysteresis loops of MN8CX ferrite measured at 100oC......................82 A-1Preisach Diagrams for the derivation of the mean domain-wall motion............85 C-1Measured and modeled hysteresis loops for MN8CX ferrite.............................97 C-2Measured and modeled hysteresis loops for P ferrite.........................................129 C-3Measured and modeled hysteresis loops for 3F3 ferrite.....................................170

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viii KEY TO SYMBOLS OR ABBREVIATIONS a Shape parameter used in Eq.4-8 . AeEffective cross-sectional area described by Eq.5-4 . AvVoltage gain used in Eq.5-5 . AWGAmerican wire gauge. B Magnetic ux density. b Scale parameter used in Eq.4-8 . B0Magnetic ux density pertaining to a turning point in Figure2-3 . b1-b25Temperature coefcients used in Eq.4-1 through Eq.4-7 . BmaxMaximum ux density. BpPeak ux density . BrRemanent ux density. BsSaturation ux density. c Weighting coefcient of the reversible process. c Maximum normalized differential susceptibility. CADComputer-aided design. CsCapacitor of the current-sense resistor model in Eq.5-7 . DDWMDynamic domain-wall motion. DWMDomain-wall motion. e Measured uncertainty used in Eq.5-12 and Eq.5-13 . f Frequency in Hz. feqEquivalent frequency described by Eq.3-18 . firrCut-off frequency of the irreversible process. opsNumber of oating-point operations.

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ix frevCut-off frequency of the reversible process. G Gamma probability density function described by Eq.4-8 . g Gamma function used in Eq.4-8 . h Height of a toroid in Eq.5-4 . H Magnetic eld intensity. H0Magnetic eld intensity pertaining to a turning point in Figure2-3 . HcCoercive force. hcHalf the width of the elemental hysteresis loop in Figure2-1 . Hc,irrCoercive force of the irreversible process. huBias eld of the elemental hysteresis loop in Figure2-1 . i1Excitation current in Eq.5-1 . kirrAttenuated gain described by Eq.3-17 . leEffective mean path length described by Eq.5-3 . LsInductor of the current-sense resistor model in Eq.5-7 . lsqnonlinLeast-square non-linear tting function in Matlab. M Magnetization; B = mo( H + M ). m = M / MsNormalized magnetization. m0Permeability of free space. m0Normalized magnetization pertaining to a turning point. miInitial permeability. mmaxMaximum differential permeability. MnZnManganese-zinc. MsSaturation magnetization. mtNormalized magnetization that denes the transition to the saturation region. n Positive integer of the irreversible process used in Eq.2-2 . NpNumber of excitation or primary turns in Eq.5-1 . NsNumber of sense or secondary turns in Eq.5-2 .

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x R ( m )Domain-wall surface function given in Eq.2-12 and Eq.2-15 . resnormSquared 2-norm of the residual from lsqnonlin. riInner radius of a toroid in Eq.5-3 . roOuter radius of a toroid in Eq.5-4 . RsResistor of the current-sense resistor model in Eq.5-7 . S Sensitivity as shown in Eq.5-9 through Eq.5-11 . T Temperature. t Time. t0Time pertaining to a turning point used in Eq.3-18 . TcCurie temperature. tendTime pertaining to the end point of each return branch used in Eq.3-18 . v1Measured voltage across the current-sense resistor in Eq.5-7 . v2Induced voltage in Eq.5-2 . VgGenerator peak output voltage described by Eq.5-5 . w Frequency in radians/second. wrevCut-off frequency of the reversible process in radians/second. wirrCut-off frequency of the irreversible process in radians/second.xDynamic mean domain-wall position. x Mean domain-wall position. x0Domain-wall position pertaining to a turning point in Figure2-3 . xdownMean domain-wall position of any downward branch in Figure2-3 . xinitialMean domain-wall position of the initial magnetization curve in Figure2-3 . xirrIrreversible contribution to the mean domain-wall position. xreturnMean domain-wall position of any return branch. xrevReversible contribution to the mean domain-wall position x . xupMean domain-wall position of any upward branch in Figure2-3 . Y Admittance function described by Eq.5-8 . x

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xiABSTRACTAbstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulllment of the Requirements for the Degree of Doctor of Philosophy STATIC AND DYNAMIC HYSTERESIS MODELING OF POWER FERRITES UP TO CURIE TEMPERATURES FOR POWER ELECTRONIC APPLICATIONS By Paiboon Nakmahachalasint May 2003 Chair: Dr. Khai D. T. Ngo Cochair: Dr. Loc Vu-Quoc Major Department: Electrical and Computer Engineering Asthehigh-temperaturesemiconductortechnologyprogresses,thedesignof magneticcomponentsforpowerconvertersthatoperateinharshtemperature environmentsisbecominganevenmorechallengingtask.Suchhigh-temperature magnetichysteresismodelsoughttobeintegratedintoacomputer-aideddesign(CAD) systemforpowerelectronicapplications.Thisdissertationpresentsstaticanddynamic hysteresismodeling,alongwithmeasurements,ofmanganese-zinc(MnZn)powerferrites uptotheirCurietemperatures,undersinusoidalexcitationsinthe10kHzto1MHz frequencyrange.MnZnpowerferritesaresoftmagneticmaterialsmostcommonlyusedin powermagneticcomponentsforpowerconverters.Themodelsareveriedwith experimentaldataofcommercialMnZnpowerferritesintoroidalcoreshapes.The procedurestoextractthecorrespondingmodelparametersandtousethemodelsfor simulation are also described.

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xii Basso-Bertotti’sdomain-wall-motion(DWM)modelofstatichysteresisisrst reviewedwithemphasisonthemathematics,numerics,andformulationofthemodeling equations.Experimentaldataarethenpresentedtojustifytheneedforimprovementof oneofthetwomodelfunctions,thedomain-wallsurfacefunction R ( m ),tobetter characterizetheverygradualsaturationbehaviorofacommercialMnZnpowerferrite. Alternate R ( m )functionisproposedbyintroducingamodelparameter mtthatdenesthe transition between low eld/minor loops and high eld/major loop. TheimprovedDWMmodelofstatichysteresisisgeneralizedtoincludethe frequencydependenceofhysteresis.Thegeneralizationisdevelopedbasedonthe dynamicsofDWMinsmall-signalandlarge-signalregions.Inthesmall-signalregion wherethereversibleDWMisdominant,thedynamicofDWMismodeledbyarst-order differentialequation.Experimentaldataarethenpresentedtoverifytheexistenceof irreversiblecontributiontothedynamicofDWMinthelarge-signalregion,andthe proceduretoseparateitisformulated.Asaresultofthisseparation,analgebraicequation isusedtomodeltheattenuationoftheirreversiblecontributionwithincreasingfrequency. Twomodelparameters frevand firrareintroducedthatdenethecut-offfrequenciesofthe reversible and irreversible processes. Usingthegeneralizedmodel,allmodelparametersareextractedfromthe hysteresisloopsofeachtestcore,inthe10kHzto1MHzfrequencyrange,ataconstant temperaturestartingfromroomtemperaturewith5oCincrementtoitsCuriepoint. Temperaturedependenceofhysteresisisthenincorporatedintothemodelbycurve-tting the extracted model parameters versus temperatures using mathematical functions.

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1 CHAPTERCHAPTER 1 INTRODUCTION Modern technological advances in high-temperature semiconductor devices make power converters able to operate in harsh temperature environments without cooling. Recently, a development of high-temperature power converters using silicon carbide devices has been reported that can function at elevated ambient temperatures up to 250oC [ 1 ]. The development of such high-temperature circuits requires designers to optimize theperformanceandreliabilityofmagneticcomponentsthroughouttheentiretemperature operating range. High-temperature characterization of magnetic cores is thus needed. Their high-temperature magnetic hysteresis models ought to be integrated into a computer-aided design (CAD) system for power electronic converters. In power electronic applications, manganese-zinc (MnZn) power ferrites are the most commonly used soft magnetic materials. Their Curie temperatures or high-temperaturelimitsatwhichtheywillloseallmagneticpropertiesvaryaround180oC to 250oC. This dissertation focuses on static and dynamic hysteresis modeling of MnZn power ferrites up to their Curie points, under sinusoidal excitations in the 10 kHz to 1MHz frequency range. Experimental data are measured with commercial MnZn power ferrites in toroidal shapes. This dissertation is outlined as follows. Chapter2 presents static hysteresis modeling at room temperature. The chapter rst reviews the domain-wall motion (DWM) model for static hysteresis proposed by

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2 BassoandBertotti[ 2 ].Itthenpresentsexperimentdatatojustifytheneedformodication of one of the two model functions, the domain-wall surface function R ( m ), to better characterizetheverygradualsaturationbehaviorofacommercialpowerMnZnferrite[ 3 ]. A model parameter mt is introduced that denes the transition between low eld/minor loops and high eld/major loop. An explicit expression for magnetization in term of domain-wall position is synthesized to make the hysteresis model numerically attractive. The procedure to extract model parameters and the algorithms to use the model for simulation are described. The improved Basso-Bertotti’s static DWM model is experimentally veried with the quasi-static (10 kHz) major loop and the minor loops of the tested MN8CX [ 4 ] (from Ceramic Magnetics, Inc.) ferrite at room temperature. Chapter3 presents dynamic hysteresis modeling at room temperature. The improved Basso-Bertotti’s static DWM model in Chapter2 is generalized to model the frequency dependence of hysteresis in MnZn ferrites. The generalization is based on the dynamics of DWM contributed by the reversible and irreversible processes. Two model parameters frev and firr are introduced that dene the cut-off frequencies of the reversible and irreversible processes. In the small-signal region where the reversible contribution is dominant, the generalized model requires only a rst-order differential equation to describe the dynamics of DWM. Experimental data are then presented to justify the need to model the frequency dependence of the irreversible contribution in the large-signal region.Theresultsindicatethattheirreversiblecontributionattenuateswiththeincreasing frequency. An algebraic equation is used to describe the frequency dependence of the irreversible contribution. The complete formulation of the generalized model is summarized. The procedure to extract model parameters and the algorithms to use the

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3 model for simulation are described. The generalized dynamic DWM model is experimentallyveriedwiththehysteresisloopsofthetestedMN8CX[ 4 ](fromCeramic Magnetics,Inc.),P[ 5 ](fromMagnetics,Inc.),and3F3[ 6 ](fromFerroxcube,Inc.)ferrite cores at room temperature, in the frequency range of 10 kHz to 1 MHz. Chapter4 presents dynamic hysteresis modeling at high temperature. In this chapter, the magnetic properties including saturation ux density, coercive force, initial permeability, and maximum permeability are related to the model parameters of the dynamic DWM model. Their temperature characteristics are examined and then t with temperatures using mathematical functions. The generalized dynamic DWM model with temperature is experimentally veried with the hysteresis loops of the tested MN8CX [ 4 ] (from Ceramic Magnetics, Inc.), P [ 5 ] (from Magnetics, Inc.), and 3F3 [ 6 ] (from Ferroxcube, Inc.) ferrites from room temperature to their Curie points, in the frequency range of 10 kHz to 1 MHz. Chapter5 presents an automated, low-cost magnetic test system that can measure hysteresisloopsofMnZnpowerferritesuptotheirCurietemperatures,inthe10kHzto1 MHz frequency range. Low-cost test xtures are constructed that can withstand elevated temperatures as high as 250oC. Also included are the experimental procedures and block diagrams for the automated acquisition of dynamic hysteresis loops with constant and with variable peak ux density. The usefulness of the test system is demonstrated by the experimental results from commercial MnZn power ferrites. The modeling approaches used in this study is summarized in Figure 1-1 .

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4 Figure 1-1.Summary of the modeling approaches used in this study Modify R ( m ) to characterize the very gradual saturation in MnZn power ferrites Generalize the improved model using rst-order differential and algebraic equations Fit temperature characteristics of Temperature-dependent 1) Implement high-temperatureautomatic hysteresis measuring systemMeasure the test cores 10 kHz B H loops at room temperature10 kHz to 1MHzB H loops at room temperature verify 2) Construct high-temperature test xtures with test cores 10 kHz to 1 MHzB H loops up to Curie temperatures Dynamic DWM model for power ferrites Parameters: n , Bs, Hc, c , c , mt, frev, firrDependent Variables: B , x,Independent Variables: H , t x Static DWM model for power ferrites Parameters: n , Bs, Hc, c , c , mtDependent Variables: B , xIndependent Variable: H Basso-Bertotti’s Static DWM model Parameters: n , Bs, Hc, c , cDependent Variables: B , xIndependent Variable: H the model parameters using mathematical functionsDynamic DWM model for power ferrites Parameters: n ,b1b25, TcDependent Variables: B , x,Independent Variables: H, t , T x verify verify

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5 CHAPTER 2 STATIC HYSTERESIS MODELING AT ROOM TEMPERATURE 2.1 Introduction Hysteresis models for power ferrites are an integral part of a computer-aided design (CAD) system for power electronic converters. The models need to capture the dependence of hysteresis on such physical phenomena as irreversible magnetization, reversible magnetization, dynamic effects, and temperature effects. This chapter deals primarily with static hysteresis at room temperature. The Stoner-Wolhfarth, Jiles-Atherton, Globus, and Preisach models are four well-known physics-based macroscopic models of static hysteresis. Their characteristics and applicability are discussed and compared by Liorzou et al. [7] . Among these models, the Preisach model has been studied extensively for soft magnetic materials in recent years. Basso and Bertotti explicitly include domain-wall motion (DWM) as the main magnetizationmechanism [2] .Thismodelwasexperimentallyveriedforamorphousand nanocrystalline alloys [8] . Its application to power ferrites, however, is less successful because the very gradual saturation behavior of power ferrites is governed by coherent rotation instead of DWM [3] . This chapter presents a domain-wall surface function that would enable the Basso-Bertotti DWM model to describe those static hysteresis phenomena that are not necessarily governed by DWM. The motivation for the chapter has been the need to characterize both the major loop and the minor loops of the power ferrites used in power

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6 electronic applications. A model parameter mt is introduced that denes the onset of the saturation region, thus characterizing the saturation gradualness. In other words, the approach to saturation becomes more gradual as mt decreases from one toward zero. Following this section, the DWM model for static hysteresis is reviewed in Section2.2 . A domain-wall surface function suitable for power ferrites is synthesized in Section2.3 . Emphasis is placed more on the mathematics, numerics, and formulation/implementation of the modeling equations/procedures, and less on the physics of static hysteresis, which can be found in the textbook by Bertotti [9] . Parameter extractionandexperimentalvericationarediscussedin Section2.4 .Themainresultsare summarized in Section2.5 . 2.2 Domain-Wall Motion Model for Static Hysteresis Similar to the classical Preisach model, Basso and Bertotti’s DWM model uses a collection of non-interacting, statistically distributed elemental loops like those shown in Figure2-1 to describe the macroscopic hysteresis phenomena [2] . Each elemental loop is associated with an “idealized” one-dimensional domain-wall that is allowed to move freely when an external magnetic eld intensity H is applied. DWM could be reversible (no loss) or irreversible (with loss). The reversible DWM is modeled by the zero-width elemental hysteresis loop shown in Figure2-1A and the irreversible DWM by the elemental hysteresis loop in Figure2-1B . The elemental loops are distributed statistically according to the Preisach distribution function p(hc,hu) . According to Basso [8] , p(hc,hu) is uniform with respect to hu and depends on hc according to (2-1) p hc() 1 c – () pirrcprev+ =

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7 (2-2) (2-3) The function p(hc) consists of two terms, pirr(hc) and prev(hc) , describing irreversible and reversible processes, respectively. Since p(hc) , prev(hc) , and pirr(hc) are dened such that ,,and,thecoefcient c isused toweighthecontributionofthereversibleprocess;thecoefcient(1c )thenrepresentsthe contribution of the irreversible process. The function pirr(hc) in Eq.2-2 is plotted in Figure2-2 for various values of n . To facilitate parameter extraction, Hc,irr should be replaced by Hc, the coercive force of the major loop. According to Basso [8] , (2-4) Substitution of Eq.2-1 through Eq.2-3 into the preceding equation yields (2-5) Figure 2-1.Elemental hysteresis loops.A) For reversible DWM. B) For irreversible DWM. hu H x + d x 0 d x hchu H x 0+ d x d xpirrhc() = 1 (n-1)! ------------n Hcirr ,-------------nhc n-1nhc– Hcirr ,-------------exp prevhc()d hc() = phc() hcd0 1 = pirrhc() hcd0 1 = prevhc() hcd0 1 = Hchcphc() hcd0 = Hc1 c – () hcpirrhc() hcd0 chcprevhc() hcd0 +1 c – () Hcirr ,== A)B)

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8 Thefunction prev(hc) istheDiracdeltafunctionin Eq.2-3 .AsexplainedbyBasso [8] andderivedin AppendixA (see SectionA.2 ), prev(hc) enablestheinitialsusceptibility and the differential susceptibility after a turning point (e.g., point ( H0,down, x0,down) or ( H0,up, x0,up) in Figure2-3 ) to be non-zero. In the classical Preisach model, the magnetization under a certain eld history is found by proper integration of the distribution function p(hc,hu) . In the Basso-Bertotti DWM model, however, the magnetization is calculated from the domain-wall position that, inturn, is found by proper integration of p(hc) as derived in AppendixA (see SectionA.1 ): (2-6) (2-7) where Figure 2-2.Irreversible Preisach distribution function pirr(hc) in Eq.2-2 for Hc,irr=50A/m. 0 0.5 1 1.5 2 2.5 3 0 0.005 0.01 0.015 0.02 0.025 0.03 hc/ Hc,irrpirr( hc)n =1 n =2 n =5 n =10 xinitialH () = c H () sgn1 c – () PirrH () cPrevH () + [] xH () x02 c HH0– () 1 c – () PirrHH 0 – 2 ------------------() cPrevHH 0 – 2 ------------------() + [] sgn + =

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9 Figure 2-3. x H and B H loopswith Hc,irr = 50 A/m, c = 0.01 m/A, c = 0, and Bs=0.45T. A) H vs x . B) B vs x . C) B vs H . -500 0 500 -0.5 0 0.5 -5 0 5 -0.5 0 0.5 B)Magnetic Flux Density B (T)Mean Domain-Wall Position x -500 0 500 -5 0 5 Magnetic Field Intensity H (A/m)Mean Domain-Wall Position xA) ( H0,up, x0,up) ( H0,down, x0,down) xup xdown xinitialMagnetic Flux Density B (T)C) ( B0,down, H0,down) Bup Bdown Binitial ( B0,up, H0,up) Magnetic Field Intensity H (A/m)

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10 (2-8) (2-9) Themovement/locationofthedomainwallisdescribedby Eq.2-6 and Eq.2-7 ,whichare special cases of the expression for the mean domain-wall position function described by Basso and Bertotti [2] . In simulation of static hysteresis, one is interested in reproducing the measured initial magnetization curve after ac demagnetization (the B H curve originatingfrom B = H =0)andanyreturnbranchofa B H looporiginatingfromaturning point ( B0, H0). For the initial magnetization curve, the mean domain-wall position xinitialcorresponding to an applied H is computed from p(hc) according to Eq.2-6 , assuming xinitial( H =0)=0.Aplotof xinitialversus H isexempliedin Figure2-3 ,whereitisnoted that xinitialcouldreachinnityas H approachesinnity.Foranyreturnbranchoriginating from ( B0, H0), which corresponds to the mean domain-wall position x0, the mean domain-wallposition x correspondingtoanapplied H iscomputedfrom p(hc) accordingto Eq.2-7 . The plots of x versus H are exemplied in Figure2-3 for a downward branch originating from a large positive value of H , and for an upward branch originating from a large negative value of H . In this one-dimensional DWM model, the domain-wall surface function R(m) is used to relate a differential change in domain-wall position d x to a differential change in the normalized magnetization m : PirrD H () pirrhc()D Hhc– () hcd0 D H= D HHcirr ,– () Hcirr ,n D H – Hcirr ,-------------exp k nnk – () ! ------------------------n D H Hcirr ,-------------nk – k 1 = n+ = PrevD H () prevhc()D Hhc– () hcd0 D HD H ==

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11 ; (2-10) or (2-11) whereGistheinversefunctionof x ( m )(i.e.,G( x ( m ))= m ).Onechoiceof R ( m )issuggested by Kadar on the basis of population dynamics [10] : (2-12) whichisplottedin Figure2-4 asthecurvelabeled1m2.Thecorrespondingmagnetization functiongivenby Eq.2-11 is m(x) =tanh( x )asshownin Figure2-5 .As x increases, m ( M ) approachesunity( Ms),and R ( m )approacheszero.Thus,themagnetizationis“clamped”to Ms by R ( m ) although x is allowed to be arbitrarily large. Although the R ( m ) given by Eq.2-12 has been found to be satisfactory for amorphous and nanocrystalline alloys [8] , good t between modeling and measurement hasbeenfoundtobedifcultfortheverygradualsaturationcharacteristicofpowerMnZn ferrites such as MN8CX [4] ferrite, as is evident in Figure2-6 . Two arguments are now giventoexplainthedifcultyinttingnearsaturation.First,thefunction R ( m )isshownto Figure 2-4.Comparison of theoretical and measured R ( m ) for MN8CX ferrite. dmRm () dx =1 – m 1 xm () m d Rm () ------------= mx () Gx () = Rm () 1 m2– = measured Eq.2-15 with mt = 0.825 1m2Domain-wall Surface R ( m )Normalized Magnetization m

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12 be proportional to the normalized differential susceptibility from the saturation loop [8] . Thus, R ( m ) ought to approach zero very slowly as m approaches unity for MnZn powerferrites,fortheofpowerferritesapproacheszeroveryslowlyinthesaturation region. This is not seen in Figure2-4 . In fact, the (magnitude of the) slope of R ( m ) increases monotonically as m increases from 0 to 1, or as R ( m ) decreases from 1 to 0. Secondly,thegradualsaturationofthestatichysteresisloopcanbequantiedtobe the rate of change of dm / dH with respect to m , which can be obtained from the upward branch of the major loop using Eq.A-12 : (2-13) Thus, the suitability of R ( m ) for a given magnetic material can be assessed by comparing d R ( m )/d m with the measured data. This is done in Figure2-7 to evaluate the Figure 2-5.Normalized magnetization vs mean domain-wall position, using Eq.2-17 . mt = 0.825 mt = 0 mt = 1 ( m(x) = tanh( x )) Mean Domain-wall Position x Normalized Magnetization mdm dH ------dm dH ------d dm ------dm dH -------H0 – c d dm ------Rm () =

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13 appropriatenessof R ( m )=1m2forthismaterial.Theneedtosynthesizeanalternateform for R ( m ) is evident. Figure 2-6.Measured and modeled major and minor loops using R ( m ) = 1 m2for the test ferrite at room temperature and 10 kHz. Figure 2-7.Comparison of theoretical and measured d R ( m )/d m for MN8CX ferrite. measured modeled Magnetic Field Intensity H (A/m) Magnetic Flux Density B (T) measured Eq.2-14 with mt= 0.825 -2 m Normalized Magnetization m d R ( m )/d m

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14 2.3 Synthesis of Domain-Wall Surface Function As introduced by Basso and Bertotti [2] , R ( m ) is a positive even function with so that the susceptibility vanishes at saturation. It could be selected from a physicsorfromamathematicalstandpoint.Aphysics-based R ( m )wouldattempttoreect the mechanism underlying the magnetization behavior. An example is the parabolic form R ( m ) = 1 m2 that is suitable when the magnetization is dominated by DWM [8] . Thus, this form provides an excellent t with measurements for nanocrystalline alloys [8] and fortheminorloopsofthepowerferritesshownin Figure2-6 .Appreciableerror,however, could result if one attempts to use R ( m ) = 1m2 to model the magnetization behavior not governedbyDWM.Thisisevidentin Figure2-6 ,whichexhibitsapoortbetweentheory and measurements for the major loop in the saturation region, where coherent rotation of domain magnetization dominates [3] . Recognizingthatan R ( m )encompassingmorethanonemagnetizationmechanism might be difcult to nd, a mathematical approach is now presented to identify an R ( m ) forpowerferrites.Inthisapproach,afunctionissynthesizedtotthemeasuredd R ( m )/d m and integrated to obtain R ( m ). For the measured d R ( m )/d m of MN8CX ferrite plotted (as dots) in Figure2-7 , the function is piecewise linear as shown (by the solid line) in Figure2-7 : (2-14) where if and if. The new model parameter denes the transition between the low eld region where the magnetization R 1 () 0 = dRm () dm ---------------2 m mt----- – formmt 2 sgnm () m – 1 m –t---------------------------- – formtm 1 < = m () sgn1 = mmt> m () sgn1 – = mm –t< 0 mt1

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15 processisdominatedbyDWMandthehigheldregionwherethemagnetizationprocess isnotdominatedbyDWM.Thisparameterneedstobeextractedtoachievegoodtforall minor and major loops. Note that so that. Integration of Eq.2-14 yields R ( m ), plotted in Figure2-4 for mt = 0.825: (2-15) From Eq.2-11 and Eq.2-15 , a closed-form expression for x in terms of m can be derived: (2-16) More importantly, Eq.2-16 can be inverted thanks to its simple form to yield an explicit expression for m in terms of x : (2-17) where (2-18) dRm t() dm 2 = R 0 () 1 = R m () 1 m2mt------ – formmt m () sgn m – ()21 m –t----------------------------------formtm 1 < = xm () m d Rm () ------------mtm mt--------atanh formmt m () sgn 1 m –t1 m – --------------mtmt() atanh1 – + formtm 1 < == mx () m t x m t ---------tanh forxxt x () sgn1 1 m – t 1 xtx + – ------------------------- – forxxt> = x t x mm t = m t m t () atanh ==

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16 Eq.2-17 is plotted in Figure2-5 for mt = 0, 0.825 and 1. Note that mt = 1 corresponds to R ( m )=1m2.Awiderangeofsaturationapproachiscapturedbythemodelvia mt.Thus, analternateinterpretationof mtisthat mt“quanties”theonsetofthesaturationregion.An mt closer to zero means the magnetization approaches saturation more gradually (i.e., gradualsaturation).Conversely,an mt=1meansthemagnetizationapproachessaturation abruptly at Ms (i.e., abrupt saturation). Yet another (looser) interpretation of mt is that mt“quanties” the transition between the minor loops () and the major loop (). Torecap,theimprovedDWMmodelforstatichysteresisofsoftferritesconsistsof Eq.2-6 , Eq.2-7 , Eq.2-16 , and Eq.2-17 . The pseudo-code for the implementation of the precedingequationsisoutlinedin Algorithm2-1 .Notethat Hc,irrin Eq.2-2 isreplacedby Hcinaccordancewith Eq.2-5 and Bsisdenedas m0Msin AppendixA (see SectionA.6 ). The Matlab code for Algorithm2-1 is listed in AppendixB (see SectionB.1 ). Algorithm 2-1. Generate any hysteresis branch from H : 1.Input: H and the last turning point ( H0, x0) 2.Data: model parameters ( c , c , Hc, Bs, mt, n ). 3.Calculate Hc,irr = Hc/(1c ). 4.If ( H0 = 0) and ( x0 = 0), compute x using Eq.2-6 with Eq.2-8 and Eq.2-9 . 5.Otherwise, compute x using Eq.2-7 with Eq.2-8 and Eq.2-9 . 6.Compute m using Eq.2-17 with Eq.2-18 . 7.Calculate B = m0H + mBs. 8.Output: B and x . mmt mmt>

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17 2.4 Parameter Extraction and Experimental Verification Themodelhasoneintegerparameter( n )andverealparameters( Bs, Hc, c , c ,and mt) to be extracted. The rst step in parameter extraction is the measurement of the quasi-static hysteresis loops using a 10 kHz sinusoidal excitation at room temperature. Hysteresisloopdatawereacquiredusingtheautomaticmagnetichysteresismeasurement systemandthetestproceduredescribedin Chapter5 .Thematerialsanddimensionsofthe toroids measured are the “standard” sizes listed in Table5-2 . The next step is the estimation of the initial guesses for the parameters to be extracted. From the upward branch of the major loop, R(m) using Eq.A-12 and, then, d R ( m )/d m using Eq.2-13 are computed. The normalized magnetization at which d R ( m )/d m is the lowest is recorded for later use in estimation of the initial guess for mt in the parameter extraction procedure. In addition, AppendixA suggests the following equations to estimate the initial values for Bs, c , and c : (2-19) (2-20) (2-21) (2-22) where Bmax is the maximum magnetic ux density. Using the preceding estimations with Br = 0.08 T and mi=2500 m0, the initial guesses of the ve real model parameters for Bsm0MsBmax = HcmaxHB 0 ={} = c dm dH -------H0m0, () 00 , () = HH0lim c --------------------------------------------------------mic Bs-------= c max dm dH -------H0 – max c Rm () {} == BrBsHc-----------

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18 MN8CXferritearefoundtobeT,A/m,,m/A,and . Matlab [11] was then used to extract the six model parameters. (Indirectory ~paiboon/thesis/matlab, execute a Matlab command “bfit(m2stat,25,10)”.) All the B-H data were interpolated to 200 points/loop and compiled into matrix inputs as shown in Figure2-8 .Since n isaninteger,itcouldnotbeextractedusingtheleast-squarenon-linear ttingfunction “ lsqnonlin”inastraightforwardmanner.Thus, n wassweptbetween1and 5. For each n , the ve real parameters Bs, Hc, c , c , and mt were extracted using lsqnonlin. The rst quadrants of the measured and modeled major and minor loops using R ( m ) = 1m2 are compared for MN8CX ferrite in Figure2-6 ; and those using the R ( m ) described by Eq.2-15 in Figure2-9 . The minimum squared 2-norm errors (resnorm in Matlab) based on 200 data points per loop are compared in Table2-1 . The numbers of oating-point operations (ops) of the improved model is about thesameastheopsoftheoriginalmodel.Ittakesapproximately3500opsor150mson aSunUltra5workstationtogenerate100pointsofeachreturnbranch.Thecorresponding resnorm of the major loop for the proposed R ( m ) is at least two times better (lower) than that using R ( m ) = 1m2, whereas the resnorm of the minor loops for the proposed R ( m ) is very close (within 5%) to that using R ( m ) = 1m2. The close agreement between theory Table 2-1. Squared 2-norm error (resnorm) for MN8CX ferrite R ( m ) Major loop resnorm Minor loops resnorm 1m20.0298950.015893 Eq.2-15 0.0128930.016168 Bs0.47 Hc14 c 0.56 c 0.012 mt0.8

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19 and experiment justies the use of the formulation for magnetic materials with very gradual saturation. 2.5Conclusion The DWM model described by Basso and Bertotti uses two functions, a Preisach distribution function p(hc) and a domain-wall surface function R ( m ), to shape a static hysteresis loop. To better characterize both the major loop and the minor loops of power ferrites, a piecewise-parabolic function is proposed for R ( m ) ( Eq.2-15 ) based on the piecewise linear shape of the measured d R ( m )/d m . The new R ( m ) contains a model parameter mt that denes the onset of the saturation region and characterizes saturation gradualness. The resulting explicit expression m ( x ) ( Eq.2-17 ) has resulted in good agreement between measured and t data for the major and minor loops of the tested MnZn ferrites.

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20 Figure 2-8.Parameter extraction diagram for the static DWM model. Figure 2-9.Measured and modeled major and minor loops for MN8CX ferriteat room temperature and 10 kHz, using R ( m ) in Eq.2-15 with Bs = 0.4891 T, Hc = 12.2276 A/m, c =0.5839, c = 0.01366 m/A, mt = 0.8250, and n = 1. . . . H1 H2HNH3. . .B1 B2BNB3Static DWM Model Integer parameter: N loops Bmeasured n S + BmodeledLeast-square non-lineartting Real parameters: Bs, Hc, c , c , mtErrors measured (dashed) modeled (solid) Magnetic Field Intensity H (A/m) Magnetic Flux Density B (T)

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21 CHAPTER 3 DYNAMIC HYSTERESIS MODELING AT ROOM TEMPERATURE 3.1 Introduction Dynamic hysteresis modeling is often developed via a generalization of the existingstatichysteresismodel.Forinstance,Bertotti [12] generalizedthePreisachmodel by equipping each elementary loop with dynamic behaviors so that it can synthesize the nite rate of switching between magnetization polarities. This model was experimentally veriedforsoftferromagneticmaterialsincludingamorphousalloys [13] .However,there remain concerns with its simulation speed and memory usage [14] . Another approach, perhaps more efcient computationally, is to regenerate a dynamic solution from the output of the static hysteresis model regarded as the equilibrium state. This can be achieved by either applying a second-order differential equation [15] or a second-order low-pass lter [16] to the output magnetization. Although this approach was veried especiallywithexperimentaldataofmanganesezinc(MnZn)powerferrites,itsmodeling accuracy over a wide range of magnetization has not yet been demonstrated [15, 16] . MnZn power ferrites are used extensively in high-frequency power converters becausetheirhighresistivitycanyieldloweddycurrentlossesuptoafewMHzrange.But they also have high permittivity that promote dielectric losses. The core loss study of MnZn ferrites by Saotome and Sakaki [17] , however, reveals that the eddy-current and dielectric losses of small test toroids account for only 0.1% and less than 10% of the total

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22 losses, respectively. Note that the small toroids in this chapter have core dimensions comparable with those tested by Saotome and Sakaki [17] . Thischapterpresentsadynamicgeneralizationofthedomain-wallmotion(DWM) model described in Chapter2 . The dynamic domain-wall motion (DDWM) model was experimentally veried against three commercial power MnZn ferrites at room temperature in the peak magnetic ux density range of 0.05 to 0.4 T and the frequency range of 10 kHz to 1MHz. The DDWM model shown in Figure3-1 adds two main equations to the DWM model described in Chapter2 : a differential equation to regenerate the dynamic (mean) domain-wallpositionfromitsequilibriumposition x ,thesumofitsreversible( xrev)and irreversible contributions ( xirr); and an algebraic equation to attenuate xirr as frequency increases. The equations contains new model parameters frev and firr that denes the cut-off frequencies of the reversible and irreversible processes, respectively. The suitability of the differential equation for bulk MnZn ferrites can be assessed in the small-signal region (i.e., in the reversible magnetization range), whereas that of the algebraic equation in the large-signal region (i.e., in the irreversible magnetization range). In the small-signal region, the DWM is mainly contributed by xrev. This corresponds to the non-zero initial permeability characterized by the Dirac delta distributionfunctionintheDWMmodeldescribedin Chapter2 .Theresultingformulafor the complex (initial) permeability using the DDWM in Figure3-1 is equivalent to those previously reported in the literature [17, 18] . In the large-signal region, a mathematical procedure is formulated to obtain xirrfrom the measured hysteresis data. The measured magnetic ux density B is rst inverted x

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23 tothento x andthemeasuredmagneticeldintensity H to xrev.The xirristhusfoundby subtracting xrevfrom x .Theresulting xirrsuggestsasimplealgebraicequationtodescribe its attenuated gain ( kirr) with increasing frequency. Because the algebraic equation was obtained with sinusoidal excitations under steady-state conditions, the equivalent frequency feq is used to improve the accuracy of the DDWM in Figure3-1 prior to steady state. Following this section, frequency modeling in the small-signal and large-signal regions are presented in Section3.2 and Section3.3 , respectively. The complete dynamic hysteresis model is reviewed in Section3.4 . Parameter extraction and experimental verication are discussed in Section3.5 . The main results are summarized in Section3.6 . 3.2 Dynamic Hysteresis in the Small-Signal Region 3.2.1 Domain-Wall Motion Dynamics The Doring theory suggests that the DWM dynamics can be described by a second-order differential equation similar to that in a spring-mass system [19, 20] . With this theory Jiles [15] generalized the Jiles-Atherton model by applying the second-order differentialequationtothedisplacementbetweentheoutputmagnetization M attime t and M as time approaches innity (i.e., the equilibrium state). Hsu and Ngo also introduced a Figure 3-1.Block diagram of the dynamic domain-wall motion (DDWM) model. H Irreversible S xrevxirr Differential x x Domain-Wall Surface B Algebraic Equation feq kirr + + Equation/ Process Reversible Process Function Filter x

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24 similar approach based on the Hammerstein conguration in which a non-linear static hysteresis model is immediately followed by a second-order low-pass lter [16] . The DDWM model as shown in Figure3-1 , however, exploits the Doring theory in a straightforward way. Thankstotheexplicitexpressionof x intheDWMmodeldescribedin Chapter2 ,a rst-order differential equation deemed sufcient for bulk MnZn ferrites can be applied directly to the displacement between the output domain-wall position and its equilibrium state x : (3-1) where wrev (in radians/second) is a model parameter that denes the cut-off frequency of the reversible process; wrev = 2 p frev where frev is the cut-off frequency in Hz. Rearrangement of Eq.3-1 yields (3-2) Thetransferfunctionbetween x andcanthenberealizedbyarst-orderlow-passlter, located after x as shown in Figure3-1 : (3-3) One major advantage of placing the lter after x as in Figure3-1 , instead of placingafter B ,isthatanyfrequencyharmonicsintroducedbyasaturationlimitersuchas the domain-wall surface function in Figure3-1 will not get ltered out. As can be observedin Figure3-2 ,themodeled B-H loopswiththelterlocatedafter x closelyagree x 1 wrev---------dx t () x – [] dt -------------------------x t () x – [] +0 = 1 wrev---------dx t () dt -----------x t () + x = x x s () xs () --------1 s wrev----------1 + ------------------=

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25 withthemeasured,whereasthosewiththelterlocatedafter B displaydroopingtipswhen approaching saturation. The effect of location to apply the lter, however, is least noticeable in the low-eld region. The close agreement between theory and experiment justies the use of the rst-order differential equation for characterizing the frequency dependence of hysteresis in bulk MnZn ferrites. 3.2.2 Complex Permeability According to the MMPA [21] , the complex (initial) permeability of a magnetic material should be measured using a small alternating eld with zero bias provided the peak magnetic ux density is less than 1 mT. Under such small-signal test conditions, the very low applied eld is just sufcient for the domain walls to move back and forth without dissipating energy. In other words, the magnetization process is totally reversible if the variation of the applied eld from a turning point is lower than the threshold over which the domain walls jump to the next metastable minimum. Thus, the irreversible process along with its frequency effects can be temporarily deactivated. In general the initial permeability mi is dened as the starting slope of the initial magnetization curve from the demagnetized state ( H0 = 0, B0 = 0): (3-4) With this denition, the Laplace transform of the initial permeability predicted by the DDWM model in Figure3-1 is (3-5) midB dH -------H0B0, () 00 , () = HH0lim B H ----H0B0, () 00 , () = HH0lim == mis ()m0c c Bss wrev----------1 + ------------------+ =

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26 Figure 3-2.Effect of location to apply the lter.A)after x .B)after B .Themeasuredand modeled B-H loops are for MN8CX ferrite at room temperature and 300kHz. -200 -150 -100 -50 0 50 100 150 200 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 H (A/m)B (T)300 kHz 25oC -200 -150 -100 -50 0 50 100 150 200 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 H (A/m)B (T)300 kHz 25oC A)B) measured modeled measured (dashed) modeled (solid)

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27 whereisthepermeabilityoffreespace, c theweightingcontributionofthe reversible process, c the maximum (normalized) susceptibility, and Bs the saturation magnetic ux density. Note that in static hysteresis as derived in AppendixA (see SectionA.4 ). Consequently, the complex (initial) permeability mi(j w ) is (3-6) or in series terms commonly found in manufacturers’ data sheets (3-7) where w isthefrequencyinradians/second; w =2 p f where f isthefrequencyinHz.Thus, the real part and the imaginary part of mi(j w ) predicted by the DDWM model in Figure3-1 are ; (3-8) The formulae of and in Eq.3-8 are equivalent to those previously reported by Saotome and Sakaki [17] , Johnson and Visser [18] . Note that frev in Eq.3-8 canberelatedtothedynamiclossparameter [17] andtheresonancefrequency [18] .Thus, the suitability of the rst-order differential equation for bulk MnZn ferrites is also conrmedbytheandformulaepublishedintheliterature [17,18] .Yet,the applicability of the DDWM model beyond the initial permeability range has not been evaluated. m04 p7 – 10 = mim0c c Bs+ = mij w ()m0c c Bs1 j f frev--------+ --------------------+ = mij w ()miw () j miw () – = miw ()miw () miw ()m0c c Bs1 f frev--------2+ --------------------------+ = miw () c c Bsffrev 1 f frev--------2+ ----------------------------= miw ()miw () miw ()miw ()

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28 3.3 Dynamic Hysteresis in the Large-Signal Region 3.3.1 Procedure to Obtain the Irreversible Contribution To study the applicability of the DDWM model in Figure3-1 for large-signal dynamic hysteresis, a mathematical procedure is formulated to obtain the amount of xirrfromthemeasured B and H waveformsateachtestfrequency.Themeasured B isinverted to then to x and the measured H converted to xrev. The xirr is thus found by subtracting xrev from x . Thenormalizedmagnetization m = M / Mswhere Msisthesaturationmagnetization and d m /d t are rst obtained from the measured B and H waveforms: (3-9) (3-10) With the improved domain wall surface function R ( m ) for MnZn ferrites described in Chapter2 as shown in Eq.3-11 , m can be inverted to by integrating: (3-11) (3-12) x m B m0H – Bs-------------------= dm dt ------1 Bs----dB m0H – () dt ---------------------------= x dmRm () dx = Rm () 1 m2mt------ – formmt m () sgn m – ()21 m –t----------------------------------formtm 1 < = x m () mtm mt--------atanh formmt m () sgn 1 m –t1 m – --------------mtmt() atanh1 – + formtm 1 < =

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29 where mt is the normalized magnetization that denes the transition between minor and major loops. Substitution of and in Eq.3-2 yields (3-13) With the reversible Preisach distribution function prev(hc) = d ( hc), xrev derived in AppendixA (see SectionA.5 ) is linearly proportional to H : (3-14) Thus, subtraction of xrev in Eq.3-14 from x in Eq.3-13 gives (3-15) To visualize how this procedure works, the waveforms of, x , xrev, xirr, B , H , v , and i for MN8CX ferrite at 10 kHz and 1 MHz are plotted in Figure3-3A and Figure3-3B ,respectively.Asisevident, B at1MHzin Figure3-3B hasasignicantphase delay referred to H (a contributing factor mainly responsible to the increase of hysteresis loop area at high frequency) whereas B at 10 kHz in Figure3-3A has zero phase delay. Thefollowingscanbeobservedonthewaveformsof Figure3-3 :1)iscomputedfrom B using a one-to-one mapping ( Eq.3-12 ); therefore, it retains the same phase as B . 2) x is then found from by removing its dynamic effects (e.g., time delay and amplitude reduction) introduced by the differential equation/lter ( Eq.3-13 ); therefore it is in phase with H . 3) xrev is dened such that it varies linearly with H ( Eq.3-14 ); therefore it retains thesamephaseas H .4)Thus, xirrobtainedbysubtracting xrevfrom x ( Eq.3-15 )isalsoin phase with H . As can be observed in Figure3-3B , xrev and x at 1 MHz are almost equal in magnitude (i.e., xirr gets close to zero) whereas those at the quasi-static frequency x dmRm () dx = xm () 1 wrevRm () ----------------------dm dt ------x m () + = xrevH () c c H = xirrxm () xrevH () – = x x x

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30 (10kHz) in Figure3-3A are apart noticeably. The results suggests that the amount of the irreversible contribution could have faded at high frequency. 3.3.2 Frequency Attenuation of the Irreversible Contribution Withtheaboveprocedure, x (markedwithsquares), xrev(circles),and xirr(crosses) plotted vs. H in Figure3-4 were obtained from the measured B and H data of MN8CX ferrite under sinusoidal steady-state conditions at ten different frequencies ranging from 10 kHz to 1 MHz, in the same 50 A/m amplitude range. In this low-eld range, the measured waveforms are nearly pure sinusoids having total harmonic distortion (THD) < 4%. As is evident in Figure3-4 , the xirrH loop area shrinks as the frequency increases. Assuming that the irreversible Preisach distribution ( pirr) scales down with increasing frequency, the steady-state xirr at any frequency is proportional to that at the quasi-static frequency ( xirr, dc): (3-16) whereisaproportionalcoefcientthatdenestheattenuatedgainof xirrwith theincreasingfrequency.Forstatichysteresis, kirr=1.Notethat10kHzisselectedasthe quasi-static frequency of MnZn ferrites. The kirr plotted as crosses in Figure3-5 was extracted by least-square tting the xirrH loop obtained from the measured B-H data at each frequency with that from Eq.3-16 .Thebestt xirrH loopsplottedwithsolidlinesarealsoshownforcomparisonin Figure3-4 . The extracted kirr in Figure3-5 is identied by an algebraic function of frequency in Eq.3-17 also plotted with a solid curve in Figure3-5 : (3-17) xirrf () kirrf () xirrdc ,= 0 kirr1 kirrf () 1 f firr-------21 + -----------------------------=

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31 Figure 3-3.Waveforms of v , i , B , H , xrev, xirr, x , for MN8CX ferrite at room temperature. A) 10 kHz. B) 1 MHz. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.2 0 0.2 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 -10 0 10 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 1 MHz 25oC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 0 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 10 kHz 25oC v i H B x xirr xrev x v i H B x xirr xrev x Time (0.1ms) Time ( m s)i (A) H (A/m) xv (V) B (T)i (A) H (A/m) xv (V) B (T)A) B) x

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32 Figure 3-4. x (squares), xrev (circles), xirr (crosses), and kirrxirr, dc (solid lines) vs. H for MN8CX ferrite at room temperature and different frequencies: A) 10 kHz. B)30kHz.C)50kHz.D)70kHz.E)100kHz.F)200kHz.G)300kHz.H) 500 kHz. I) 700 kHz. J)1MHz. -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x10 kHz 25oC -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x30 kHz 25oC A) B)

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33 Figure 3.4. Continued -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x50 kHz 25oC -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x70 kHz 25oC C) D)

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34 Figure 3.4. Continued -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x100 kHz 25oC -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x200 kHz 25oC E) F)

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35 Figure 3.4. Continued (g) (h) -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x300 kHz 25oC -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x500 kHz 25oC G) H)

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36 Figure 3.4. Continued (i) (j) -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x700 kHz 25oC -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H (A/m)x1 MHz 25oC I) J)

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37 where firr is the cut-off frequency at which the amount of the irreversible contribution decreases to times its static value. Because Eq.3-17 wasobtainedassumingsteadystate,itisnotvalidbeforesteady state is reached. Thus, an equivalent frequency feq is estimated to improve the accuracy prior to steady state according to (3-18) where tend is the end time of each monotonically increasing or decreasing hysteresis trajectoryand t0itsstartingtime.Intime-domainsimulation, feqin Eq.3-18 isusedinplace of f to compute kirr in Eq.3-17 . Figure 3-5.Extracted (crosses) and computed (solid line) kirr vs. frequencyforMN8CX ferrite at room temperature, using Eq.3-17 . 12 104 105 106 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Extracted Eq.3-17 Frequency (Hz)kirrfeqtendt0– 4 -----------------for the initial magnetization curve tendt0– 2 -----------------for any return branches =

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38 3.4 Complete Dynamic Hysteresis Model Thecompletedynamichysteresismodelcomprisesasetofequationsthatdescribe the initial magnetization curve and all minor and major B-H loops. Each point on a B-H trajectoryisassociatedwithadomainwallposition x, or xinitialifthetrajectoryhappensto be the initial magnetization curve. The rst point of each B-H trajectory is called the “turning point,” and is characterized by ( B0, H0, x0,, t0); the rst point of the initial magnetizationcurvehas( B0, H0, x0,, t0)=(0,0,0,0,0).Withthereversible( prev)and irreversible ( pirr) distributions described in Chapter2 , the integrations of prev ( Prev) and pirr ( Pirr) over the Preisach plane vary as H is varied from H0 by: (3-19) (3-20) where n is a positive integer which controls the shape of the irreversible Preisach distribution, c the weighting coefcient denoting the amount of contribution of the reversible process, c the maximum (normalized) susceptibility, and Hc,irr = Hc/(1c ) the coerciveforceassociatedwiththeirreversibleprocess.Thechangeindomainwallposition ( x x0) is the weighted average of Prev and Pirr: (3-21) (3-22) with (3-23) x 0x 0D HHH0– = PrevD H ()D H = PirrD H ()D HHcirr ,– () Hcirr ,n D H – Hcirr ,-------------exp k nnk – () ! ------------------------n D H Hcirr ,-------------nk – k 1 = n+ = xinitialHfeq, ()c H () sgn kirrfeq() 1 c – () PirrH () cPrevH () + [] = x D Hfeq, () x02 cD H () kirrfeq() 1 c – () PirrD H2 ------------() cPrevD H2 ------------() + [] sgn + = kirrfeq() 1 feqfirr-------21 + -----------------------------=

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39 where firr is a new model parameter denoting the cut-off frequency of the irreversible process and feq the equivalent frequency dened in Eq.3-18 . The dynamic domain wall position is described by ; (3-24) where frev is a new model parameter denoting the cut-off frequency of the reversible process.Thedomainwallsurfacefunctionsuggestedin Chapter2 isusedtodeterminethe normalized magnetization m from: (3-25) with (3-26) where mt is the normalized magnetization that denes the transition between minor and major loops. Finally, the magnetic ux density is computed using the constitutive law: (3-27) where Bs is the saturation ux density dened as m0Ms in AppendixA (see SectionA.6 ). The block diagram of the complete dynamic hysteresis model is illustrated in Figure3-6 ,andthepseudo-codeimplementationofthemodeloutlinedwiththepreceding equations in Algorithm3-1 and Algorithm3-2 . The Matlab code for both algorithms are listed in AppendixB (see AppendixB.2 ). x 1 2 p frev---------------dx Ht , () dt -------------------x Ht , () + xHfeq, () = x H0t0, () x 0= x m m t x m t ---------tanh forx x t x () sgn1 1 m – t 1 x tx + – ------------------------- – forx x t> = x t m t m t () atanh = B m0HmBs+ =

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40 Algorithm 3-1. Generate any hysteresis branch from H ( t ): 1.Input: H , t , and the last turning point ( H0, x0,, t0). 2.Data: model parameters ( c , c , Hc, Bs, mt, frev, firr, n ). 3.Compute feq using Eq.3-18 . 4.Compute kirr using Eq.3-23 . 5.Find x via Algorithm3-2 . 6.Compute using Eq.3-24 . 7.Compute m using Eq.3-25 with Eq.3-26 . 8.Compute B using Eq.3-27 . 9.Output: B ,, x . Algorithm 3-2. Compute x from H : 1.Input: H , kirr, the last turning point ( H0, x0). 2.Data: model parameters ( c , c , Hc, n ) 3.Calculate Hc,irr = Hc/(1c ). Figure 3-6.Block diagram of the DDWM model with equation numbers. DWM via PreisachIntegrations (Equations 3-19 through 3-22 ) First-order Differential Equation ( Eq.3-24 ) Domain-wall Surface Function (Equations 3-25 through 3-27 ) x t () xt () Bt () Algebraic Equation ( Eq.3-23 ) kirrfeq() c , c , Hc, n Bs, mt frev firr Ht () t t x 0x x

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41 4.If ( H0 = 0) and ( x0 = 0), compute x using Eq.3-21 with Eq.3-19 and Eq.3-20 . 5.Otherwise, compute x using Eq.3-22 with Eq.3-19 and Eq.3-20 . 6.Output: x . 3.5 Parameter Extraction and Experimental Verification The model has one integer parameter ( n ) and seven real parameters ( c , c , Hc, Bs, mt, frev, and firr) to be extracted. The rst step in parameter extraction is the measurement of hysteresis loops at room temperature, and frequencies ranging from 10 kHz to 1 MHz. Hysteresisloopdatawereacquiredusingtheautomaticmagnetichysteresismeasurement system and the test procedure described in Chapter5 . Hysteresis loop data were acquired using the automatic magnetic hysteresis measurement system and the test procedure described in Chapter5 . The materials and dimensions of the toroids measured are the “small” sizes listed in Table5-2 . Matlab [11] was then used to extract the eight model parameters at room temperature. The six static parameters ( n,Bs, Hc, c , c , and mt) were rst extracted with only the measured quasi-static hysteresis loops at 10 kHz using the parameter extraction procedure described in Chapter2 (see Figure2-8 ). (Indirectory ~paiboon/thesis/matlab, execute Matlab commands “bfit(m2stat,25,10),” “bfit(p2stat,25,10),” and “bfit(f2stat,25,10)”for MN8CX, P, and 3F3 ferrites, respectively.) As shown in Figure3-7 , the static parameters were then used as xed values when the remaining two dynamic parameters ( frev and firr) were extracted with all the dynamic hysteresis loops at room temperature ranging from 30kHz to 1MHz. (Indirectory ~paiboon/thesis/matlab, execute Matlab commands “bfit(m2t,25),” “bfit(p2t,25),” and “bfit(f2t,25)”for MN8CX,

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42 P, and 3F3 ferrites, respectively.) All the B , H , and t data were interpolated to 200 points/cycle and compiled into matrix inputs as shown in Figure3-7 . The rst and second quadrants of the measured and modeled major and minor loops using the DDWM model in Figure3-1 are compared for MN8CX, P, and 3F3 ferritesin Figure3-8 through Figure3-10 .Theminimumsquared2-normerrors(resnorm in Matlab) based on 100 data points per branch are 0.3001, 0.6945, and 0.8626 for MN8CX, P, and 3F3 ferrites, respectively. The close agreement between theory and experiment justies the use of the formulation for power ferrites. Figure 3-7.Parameter extraction diagram for the dynamic DWM model. . . . H1 H2HNH3. . .B1 B2BNB3Dynamic DWM Model Static parameters N loops Bmeasured n,Bs, Hc, c , c , mt S + BmodeledLeast-square non-lineartting Dynamic frev, firr, Errors. . .t1 t2tNt3 parameters: constant temperature

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43 3.6Conclusion The static DWM model described in Chapter2 is generalized by adding two new equations, a rst-order differential equation and an algebraic equation, to characterize frequencydependenceofhysteresisinMnZnferrites.Theuseoftherst-orderdifferential equation for MnZn ferrites is conrmed with the complex permeability formulae published in the literature [17, 18] . The mathematical form of the algebraic function is identied in Eq.3-17 based on the attenuated amount of the irreversible DWM contribution under steady-state sinusoidal conditions. The two newly added equations contain two new model parameters, frev and firr, that denes the cut-off frequencies of the reversible and irreversible processes, respectively. The generalized DDWM model has resulted in good agreement between measured and modeled dynamic hysteresis loops of the tested MnZn power ferrites over a wide range of magnetization.

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44 Figure 3-8.Measured (dashed) and modeled (solid) B-H loops for MN8CX ferriteat room temperature with c = 0.0130 m/A, c = 0.6047, Hc = 14.4192 A/m, Bs=0.4725T, mt=0.6981, frev=0.5045MHz, firr=0.1621MHz,and n =1. -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H (A/m)

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45 Figure 3-9.Measured (dashed) and modeled (solid) B-H loops for Pferriteatroom temperature with c = 0.0175 m/A, c = 0.4034, Hc = 19.0317 A/m, Bs = 0.4608 T, mt = 0.5259, frev = 0.4626 MHz, firr = 0.1550 MHz, and n = 1. -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H (A/m)

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46 Figure 3-10.Measured (dashed) and modeled (solid) B-H loops for 3F3 ferriteatroom temperature with c = 0.0281 m/A, c = 0.2722, Hc =23.6830 A/m, Bs = 0.4316 T, mt = 0.6854, frev = 0.4403 MHz, firr = 0.2168 MHz, and n = 2. -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H (A/m)

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47 CHAPTER 4 DYNAMIC HYSTERESIS MODELING AT HIGH TEMPERATURE 4.1 Introduction Power Ferrites are quite sensitive to temperature variations. The saturation ux density, for instance, can drop below 70% of that at room temperature when the temperature exceeds 100oC. The decrease in saturation ux density level at high temperatures pushes the ferrite cores to operate closer to the saturation region, causing greatercorelosses,self-heating,andariskofthermalrunawayfailure.Accuratemodeling of power ferrite characteristics at high temperature is thus needed to optimize the design with performance and reliability over the entire operating temperature range. This chapter presents dynamic hysteresis modeling at high temperature using the dynamic domain-wall motion (DDWM) model described in Chapter3 . Temperature dependencyisconsideredthroughinclusionoftemperatureeffectsonmagneticproperties that also are the model parameters of the DDWM model. The resulting temperature-dependent DDWM model was experimentally veried against three commercial power MnZn ferrites from room temperature to their Curie temperatures in thepeakmagneticuxdensityrangeof0.05to0.4Tandthefrequencyrangeof10kHzto 1MHz. Note that this test range far exceeds the normal operation of MnZn ferrites in most power electronic applications. Normally, power ferrite cores operate in temperature range of 20 to 100oC, and peak ux density level less than 0.2 T [22] .

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48 The DDWM model at high temperature shown in Figure4-1 uses seven mathematical functions with twenty ve temperature coefcients b1b25 and the Curie temperature Tc to describe temperature characteristics of the seven real magnetic properties/model parameters: 1) Saturation ux density Bs. 2) Coercive force of the irreversible process Hc,irr. 3) Initial permeability mi. 4) Maximum (differential) permeability mmax. 5) Normalized magnetization at the transition to the saturation region mt. 6) Cut-off frequency of the reversible process frev. 7) Cut-off frequency of the irreversible process firr,. In Figure4-1 , Hc,irr, mi,and mmaxinsteadof Hc(coerciveforce), c (weightingcontribution of the reversible process), c (maximum normalized susceptibility) used by the DDWM modelin Chapter3 areselectedasthetparameters.Theremainingtparameters( Bs, mt, frevand firr)arethesameasthoseusedbytheDDWMmodelin Chapter3 .Thet Hc,irr, mi, and mmax are then converted to Hc, c , and c , respectively. The nal seven real model parametersenteringtheDDWMmodelare Bs, Hc, c , c ,mt, frev,and firr.Theintegermodel parameter n also shown in Figure4-1 is assumed to be independent of temperature. Following this section, temperature characteristics of magnetic properties/model parametersareidentiedusingmathematicalrepresentationsin Section4.2 .Theequations for converting some t parameters to the model parameters used by the DDWM model in Chapter3 are formulated in Section4.3 . Parameter extraction and experimental verication are discussed in Section4.4 . The main results are summarized in Section4.5

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49 4.2 Temperature Characteristics of Magnetic Properties/Model Parameters Themathematicalfunctionsfordescribingtemperaturecharacteristicsoftheseven real magnetic properties/model parameters ( Bs, Hc,irr, mi, mmax, mt, frev and firr)are identied for: (4-1) (4-2) (4-3) (4-4) (4-5) (4-6) Figure 4-1.Blockdiagramofthedynamicdomain-wallmotion(DDWM)modelathigh temperature. Mathematical Functions DDWMmodel b1b25Tc T Temperature Coefcients Curie Temperature H t Hc,irr, mi, mmax, for Bs, mt, frev, firrConversionof Hc,irr, mi, mmaxto Hc, c , c for DynamicHysteresis B Hc, c , c , Bs, mt, frev, firr n TTC BsT () b11 T TC------ – b2= Hcirr ,T () b31 T TC------ – b4= miT ()m0b +5b6G TCT – b7b8,, () 1 b6– ()G TCT – b9b10,, () + [] = mmaxT ()m0b +11b12G TCT – b13b14,, () 1 b12– ()G TCT – b15b16,, () + [] = mtT () b17b181 T b19------- – 21 + b201 T TC------ – 21 + ----------------------------------------------------------------------------------------------= frT () b211 T TC------ – b22=

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50 (4-7) where b1 b25 are the unknown coefcients to be t with the corresponding magnetic properties/modelparameters; TcistheCurietemperatureabovewhichmagneticmaterials lose their magnetic properties; G is the gamma probability density function dened by (4-8) where is the gamma function; a is a shape parameter and b is a scale parameter. Eq.4-1 through Eq.4-7 plotted with lines are compared with the corresponding extracted values plotted as circles in subparts A through G of Figure4-2 through Figure4-4 for MN8CX, P, and 3F3 ferrites, respectively. The solid lines in Figure4-2 through Figure4-4 indicate that the t parameters are determined within the actual test range whereas the dashed lines represent those extrapolated by the corresponding mathematical functions. InthesubpartAof Figure4-2 through Figure4-4 , Bsvs.Tcurveisdescribedbya power-law function in Eq.4-1 . Hisatake [23] also used a similar power-law function to characterize the temperature dependence of saturation magnetization for bulk MnZn ferrites near Tc. However, its application over an extremely wide temperature range (i.e., from room temperature to the temperature point near Tc) has not yet been found or reported [23] . According to the MMPA test methods for soft ferrites, Tc is dened based on small-signal measurements as the temperature point at which the magnetic core inductance drops below 10% of its room temperature value [21] . In this chapter, Tc is, fiT () b23b241 T b25------- – 21 + ----------------------------------------------= GD T ab ,, () 1 baga () -----------------D T ()a 1 –eD T b ------- –= ga ()ta 1 –et –t d0 =

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51 Figure 4-2.Extracted and t model parameters vs. temperatures for MN8CX ferrite. A) Bs. B) Hc,irr. C) mi. D) mmax. E) mt. F) frev. G) firr. H) Hc. I) c . J) c . The t model parameters plotted with lines are computed using Eq.4-1 through Eq.4-11 with coefcients b1 to b25 and Tc listed in Table4-4 . 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 Bs (T) 0 50 100 150 200 250 0 10 20 30 40 50 Hc,irr(A/m) 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 mi/ m0 0 50 100 150 200 250 0 2000 4000 6000 8000 10000 mmax/ m0 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 mt 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 frev (MHz) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 firr (MHz) 0 50 100 150 200 250 0 5 10 15 20 25 Hc(A/m) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 cT (oC) 0 50 100 150 200 250 0 0.02 0.04 0.06 0.08 0.1 0.12 c (m/A)T (oC) A)B) C)D) E)F) G)H) I)J)

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52 Figure 4-3.Extracted and t model parameters vs. temperatures for P ferrite. A) Bs. B) Hc,irr. C) mi. D) mmax. E) mt. F) frev. G) firr. H) Hc. I) c . J) c . The t model parameters plotted with lines are computed using Eq.4-1 through Eq.4-11 with coefcients b1 to b25 and Tc listed in Table4-4 . 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 Bs (T) 0 50 100 150 200 250 0 10 20 30 40 50 Hc,irr(A/m) 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 mi/ m0 0 50 100 150 200 250 0 2000 4000 6000 8000 10000 mmax/ m0 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 mt 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 frev (MHz) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 firr (MHz) 0 50 100 150 200 250 0 5 10 15 20 25 Hc(A/m) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 cT (oC) 0 50 100 150 200 250 0 0.02 0.04 0.06 0.08 0.1 0.12 c (m/A)T (oC) A)B) C)D) E)F) G)H) I)J)

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53 Figure 4-4.Extracted and t model parameters vs. temperatures for 3F3 ferrite. A) Bs. B) Hc,irr. C) mi. D) mmax. E) mt. F) frev. G) firr. H) Hc. I) c . J) c . The t model parameters plotted with lines are computed using Eq.4-1 through Eq.4-11 with coefcients b1 to b25 and Tc listed in Table4-4 . 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 Bs (T) 0 50 100 150 200 250 0 10 20 30 40 50 Hc,irr(A/m) 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 mi/ m0 0 50 100 150 200 250 0 2000 4000 6000 8000 10000 mmax/ m0 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 mt 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 frev (MHz) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 firr (MHz) 0 50 100 150 200 250 0 5 10 15 20 25 Hc(A/m) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 cT (oC) 0 50 100 150 200 250 0 0.02 0.04 0.06 0.08 0.1 0.12 c (m/A)T (oC) A)B) C)D) E)F) G)H) I)J)

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54 however,denedbasedonlarge-signalmeasurementsasthetemperaturepointatwhich Bsapproacheszero.Thismethodofdetermining Tcissuitableforthehysteresismeasurement setup in Chapter5 . Thus, Tc is found by extrapolating the extracted Bs vs. T curve using Eq.4-1 . The extrapolated Tc is then used as a xed value for the remaining temperature-dependentmagneticproperties/modelparametersin Eq.4-2 through Eq.4-6 . In the subpart B and F of Figure4-2 through Figure4-4 , Hc,irr and frevdecreasing monotonically with increasing temperature are described by power-law functions in Eq.4-2 and Eq.4-6 identical to that used for Bs. To minimize the number of temperature coefcients, Hc,irrhasbeenselectedasthetparameterinsteadof HcusedbytheDDWM model in Chapter3 . As is evident in the subpart H of Figure4-2 through Figure4-4 , Hcpossesses a local minimum just above room temperature, which requires more complex polynomial tting. InthesubpartCandDof Figure4-2 through Figure4-4 , miand mmaxaredescribed bylinearcombinationsoftwogammaprobabilitydensityfunctionsin Eq.4-3 and Eq.4-4 , respectively.Because miand mmaxarewidelyrecognizedasthemainspecicationsforsoft magnetic materials, they have been selected as the t parameters instead of c and c used by the DDWM model in Chapter3 to make this model attractive for users. As exemplied in Figure4-5 , the relative permeability plotted with a solid line is the sum of the two weighted gamma probability density functions ( G1 and G2) computed using Eq.4-8 . Each weighted functions plotted with a dashed line is marked with coefcientvalues( D T , a , b )onitslabelin Figure4-5 .Thisfunctionalformissuitablefor describingthetemperaturecharacteristicofMnZnferritepermeabilitywithtwodistinctive peaks. Concerning the initial permeability, the rst peak of mi vs. T curve appears just

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55 belowtheCurietemperatureandthesecond,alsoknownasthetemperatureofanisotropy compensation, usually above room temperature. MnZn ferrites are typically compensated bychemicalmeanstoachievethehighestinitialpermeabilityaroundthedesignednormal operating temperatures of the magnetic cores [22] . In the subpart E and G of Figure4-2 through Figure4-4 , mt with a double-hump structure is described by a rational function with two pairs of complex conjugate poles in Eq.4-5 whereas firr with a single-hump structure is described by a rational function with one pair of complex conjugate poles in Eq.4-7 . 4.3 Conversion of Magnetic Properties/Model Parameters The t parameters Hc,irr, mi, and mmax can be converted to the model parameters Hc, c , and c used by the DDWM model in Chapter3 according to Figure 4-5.Exampleofthepermeabilityin Eq.4-3 or Eq.4-4 describedbyalinear combination of two gamma distribution functions. -100 -50 0 50 100 150 200 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.5 G10.5 G2+ m0--------------------------------Temperature T (oC)Relative Permeability m / m0 0.5 G1200 T –1.260 ,, () m0--------------------------------------------------------0.5 G2200 T 30 ,, () m0----------------------------------------------------

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56 (4-9) (4-10) (4-11) Eq.4-9 through Eq.4-11 are plotted with lines in the subparts H, I, and J for MN8CX, P, and 3F3 ferrites in Figure4-2 through Figure4-4 , respectively. Hc,irr can be related to Hc using Eq.4-15 as derived in Chapter2 : (4-12) Substitution of Eq.4-10 in Eq.4-12 yields Eq.4-9 . Inthischapter, miisdenedasthestartingslopeoftheinitialmagnetizationcurve; and mmax as the maximum differential slope of the upward branch of the major B H loop: (4-13) (4-14) where ( H0, B0) is the starting point of the hysteresis branch after reversal; and ( H0, B0) = (0,0) the demagnetized state. By the denitions in Eq.4-13 and Eq.4-14 , mi and mmax can be related to the model parameters used by the DDWM model in Chapter3 as derived in AppendixA (see SectionA.4 ): (4-15) Hcmmaxmi– () Hcirr ,mmaxm0– () ----------------------------------------= cT () miT ()m0– mmaxT ()m0– -------------------------------= c T () mmaxT ()m0– BsT () ----------------------------= Hcirr ,Hc1 c – () ---------------= midB dH -------H0B0, () 00 , () = HH0lim = mmaxmax dB dH -------H0 – = mim0c c Bs+ =

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57 (4-16) where Wb/A/m is the permeability of free space and Bs the saturation magnetic ux density dened by m0Ms as discussed in AppendixA (see SectionA.6 ), where Ms is the saturation magnetization. Arrangement of Eq.4-15 and Eq.4-16 yields Eq.4-10 and Eq.4-11 . The pseudo-code for the implementation of the preceding equations is outlined in Algorithm4-1 and Algorithm4-2 and the Matlab code is listed in AppendixB (see SectionB.3 ). Algorithm 4-1. Generate any hysteresis branch from H ( t ) at temperature T : 1.Input: H , t , T , and the last turning point ( H0, x0,, t0). 2.Data: integer model parameter ( n ), temperature coefcients ( b1to b25, Tc). 3.Find real model parameters ( Bs, Hc, c , c , mt, frev, firr) at T via Algorithm4-2 . 4.Find B via Algorithm 3. 5.Output: B at T . Algorithm 4-2. Find real model parameters ( Bs, Hc, c , c , mt, frev, firr) at T . 1.Input: T . 2.Data: temperature coefcients ( b1to b25, Tc). 3.Compute Bs, Hc,irr, mi, mmax, mt, frev and firrusing Eq.4-1 through Eq.4-7 . 4.Compute Hc, c , and c using Eq.4-9 through Eq.4-11 . 5.Output: Bs, Hc, c , c , mt, frev, firrmmaxm0c Bs+ = m04 p7 – 10 = x 0

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58 4.4 Parameter Extraction and Experimental Verification The model has the Curie temperature ( Tc) and twenty ve unknown coefcients ( b1 to b25) to be extracted. The rst step in parameter extraction is the measurement of hysteresis loops at a constant temperature starting from room temperature to the Curie temperature of the test material with 5oC increment. At each constant temperature, hysteresis loops were measuredusingsinusoidalexcitationsattendifferentfrequenciesrangingfrom10kHzto 1 MHz. Hysteresis loop data were acquired using the automatic magnetic hysteresis measurement system and the test procedure described in Chapter5 . The materials and dimensions of the toroids measured are the “small” sizes listed in Table5-2 . Matlab [11] wasthenusedtoextractthesevenrealmodelparameters( c , c , Hc, Bs, mt, frev, and firr) of the dynamic DWM model described in Chapter3 at each temperature as shown in Figure4-6 . The ve real static parameters ( Bs, Hc, c , c , and mt) were rst extracted using only the measured quasi-static hysteresis loops at 10 kHz using the parameter extraction procedure described in Chapter2 (see Figure2-8 ). (Indirectory ~paiboon/thesis/matlab, execute Matlab commands “bfit(m2stat),” “bfit(p2stat),” and “bfit(f2stat)”for MN8CX, P, and 3F3 ferrites, respectively.) The integer parameter n assumed to be independent of temperature was predetermined for each power ferrite at room temperature using the parameter extraction procedure described in Chapter2 . The extracted static parameters were then xed when extracting the remaining two dynamic parameters ( frev and firr) with all the dynamic hysteresis loops at that temperature ranging from 30 kHz to 1MHz using the parameter extraction procedure described in Chapter3

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59 (see Figure3-7 ). (Indirectory ~paiboon/thesis/matlab, execute Matlab commands “bfit(m2t),” “bfit(p2t),” and “bfit(f2t)”for MN8CX, P, and 3F3 ferrites, respectively.) As shown in Figure4-6 , each set of model parameters were extracted at a xed temperature,startingfrom25oCwith5oCincrementuptothecurietemperatureofthetest ferrite(approximatedas190oCforMN8CXferrite,230oCforPferrite,and215oCfor3F3 ferrite). The measured and modeled B H loops were plotted with 5oC increment for MN8CX, P, and 3F3 ferrites in FigureC-1 , FigureC-2 , and FigureC-3 , respectively (see AppendixC ). The extracted parameters are listed at each temperature, along with the minimumsquared2-normerrors(resnorminMatlab)basedon200datapointsperloop,in Table4-1 , Table4-2 , Table4-3 for MN8CX, P, and 3F3 ferrites, respectively. As shown in Figure4-6 , the extracted model parameters Hc, c , and c in Table4-1 through Table4-3 were converted to Hc,irr, mi, and mmax using Eq.4-12 through Eq.4-14 , respectively. The extracted parameters Bs, Hc,irr, mi, mmax, mt, frev and firrwere then curve-ttedwiththecorrespondingmathematicalfunctionsin Eq.4-1 through Eq.4-7 .(In directory~paiboon/thesis/matlab, execute Matlab commands “bfit(m2tcf),” “bfit(p2tcf),” and “bfit(f2tcf)”for MN8CX, P, and 3F3 ferrites, respectively.) The extrapolated value of Tcandthebesttvaluesof b1to b25usedin Eq.4-1 through Eq.4-7 arelistedin Table4-4 for MN8CX, P, and 3F3 ferrites. With the coefcients in Table4-4 , Eq.4-1 through Eq.4-7 , and Eq.4-9 through Eq.4-11 are plotted with lines for MN8CX, P, and 3F3 ferritesin Figure4-2 through Figure4-4 ,respectively.Thecorrespondingextractedvalues areplottedascirclesforcomparison.Thecloseagreementbetweentheoryandexperiment justies the use of the formulation for power ferrites.

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60 4.5 Conclusion Temperature dependence of hysteresis is incorporated into the dynamic hysteresis model described in Chapter3 by tting the temperature characteristics of its seven model parameters with the corresponding mathematical functions. The complete dynamic hysteresis model with temperature contains 25 temperature coefcient b1 to b25 and the Figure 4-6.Parameter extraction diagram for the DDWM model with temperature. t Bs( T ), Hc,irr( T ), mi( T ), mmax( T ), mt( T ), frev( T ),and firr( T ) Extractstatic parameters Bs, Hc, c , c , mtBs25 () Bs30 () Bs35 () BsTc() Hcirr ,25 () Hcirr ,30 () Hcirr ,35 () Hcirr ,Tc() mi25 ()mi30 ()mi35 ()miTc() mmax25 ()mmax30 ()mmax35 ()mmaxTc() mt25 () mt30 () mt35 () mtTc() frev25 () frev30 () frev35 () frevTc() firr25 () firr30 () firr35 () firrTc() every 5oC 25oC up to Tc Extractdynamic parameters frev, firr Convert Hc, c , c to Hc,irr, mi, mmax Extractstatic parameters Bs, Hc, c , c , mt Extractdynamic parameters frev, firr Convert Hc, c , c to Hc,irr, mi, mmax Mathematicalfunctions S+ Least-square non-lineartting TemperatureCoefcients: Errors T Eq.4-1 through Eq.4-7 b1to b25, Tc

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61 Curietemperature Tc.Theresultsshowagoodagreementbetweenmeasuredandmodeled hysteresis loops of the tested MnZn ferrites up to their Curie temperatures.

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62 Table 4-1. Extracted model parameters for MN8CX ferrite (n = 1). T (oC) c (m/A) c Hc(A/m) Bs(T) mtfrev(MHz) firr(MHz) resnorm 25 0.01300.604714.41920.47250.69810.50450.16210.3001 300.01340.620015.58630.45760.73350.52330.20340.3074 350.01380.621814.30720.43660.79020.52190.20420.3108 400.01380.613014.18850.43170.79590.52060.19970.3602 450.01410.640414.24910.41420.85970.50410.19220.3666 500.01440.614713.52680.40330.87070.50400.21560.4259 550.01460.594613.37520.39720.86040.50040.24530.4542 600.01520.610013.72300.38310.88590.48410.24340.4655 650.01550.587313.87090.37610.86920.48050.29930.4453 700.01620.568414.00210.36650.87540.47580.28390.4961 750.01600.562512.91130.36760.83790.46880.31640.4781 800.01730.527913.72950.35070.85170.47090.30980.4909 850.01780.540913.33120.34230.84790.46060.30560.5270 900.01860.535613.65900.33270.84200.45730.30790.5291 950.01950.524313.08140.32210.83600.45340.29740.5310 1000.02000.511112.37840.31830.79390.45090.29050.5639 1050.02080.516711.89710.30910.79300.44880.36440.3115 1100.02170.520712.16570.30030.78720.44520.34690.2996 1150.02230.517411.05760.29080.79120.44070.35320.3351 1200.02380.497311.02560.27970.77940.44350.32410.3701 1250.02530.533110.78110.26390.81900.43240.28520.3718 1300.02630.555110.45720.25820.78240.41820.27980.3612 1350.02880.532410.16000.24020.83650.41980.25440.4218 1400.03100.561310.12860.22720.85160.40200.17920.4743 1450.03260.54928.85990.21590.84320.39580.16990.5451 1500.03610.54808.21380.20040.85480.40600.25080.2252 1550.03870.58018.51160.18860.85260.38610.20180.2515 1600.04410.59717.83090.17070.87340.35490.11100.2872 1650.05000.58476.86890.15600.83010.32410.06880.3670 1700.06160.55346.40200.13350.86070.32820.07230.1967 1750.07110.53094.88320.11340.80570.29000.04610.2455 1800.08210.62593.47780.08230.77400.27410.06020.0576

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63 Table 4-2. Extracted model parameters for P ferrite (n = 1). T (oC) c (m/A) c Hc(A/m) Bs(T) mtfrev(MHz) firr(MHz) resnorm 250.01750.403419.03170.46080.52590.46260.15500.6945 300.01700.410317.41700.46090.49580.47530.26150.8581 350.01630.445417.17740.45880.51700.47160.21430.8087 400.01560.452116.58930.46300.49920.47210.22820.8095 450.01510.458216.57680.45170.55080.47970.24390.8038 500.01470.477115.79120.44330.57460.46970.24140.7981 550.01430.483816.30560.43960.59180.46590.24540.8080 600.01380.485915.76480.43410.60750.45930.27790.8633 650.01360.470316.73930.43420.57570.45630.30900.8937 700.01340.461516.05460.42510.59330.44770.37020.8968 750.01330.442916.16720.42370.57550.44420.42010.9365 800.01310.431916.14500.42050.56610.43650.48990.9292 850.01360.400016.07340.40540.58260.43240.54560.9357 900.01380.392816.44140.39830.57480.42870.58460.9190 950.01410.382216.96360.39060.57360.42600.59800.9132 1000.01420.362116.19810.38570.55820.41870.66480.9149 1050.01460.358516.69010.37630.57050.41830.64980.9796 1100.01550.343216.67080.36330.56730.41520.64210.9915 1150.01570.352816.64270.35630.56380.41210.67040.9664 1200.01620.293815.72750.34840.55290.41400.70100.9610 1250.01660.338415.90900.34200.55180.39630.71680.6481 1300.01700.330615.92010.34230.51370.39660.67420.6815 1350.01770.322815.44510.33050.51890.39480.68860.6640 1400.01860.301514.66940.31950.51550.39330.68310.6930 1450.01950.310714.12970.30620.54820.38860.69090.6916 1500.02020.315013.41430.29560.56670.38700.69100.6970 1550.02170.338813.58840.28260.58080.38630.61090.7546 1600.02290.365513.97920.27450.59540.37510.57550.5566 1650.02450.320412.62300.26210.60350.37870.54650.5799 1700.02610.341812.12180.25250.59540.37320.50540.6009 1750.02710.340911.74450.24170.62570.37800.50740.5742 1800.02970.327010.86080.22690.65450.37800.45780.6121 1850.03120.406110.26900.21300.70830.36580.40020.6016 1900.03450.37009.50920.20380.68110.36040.38220.3657 1950.03750.38369.09760.19030.69060.35670.36440.3566 2000.04200.39838.10980.17580.70820.34590.28790.3742 2050.04810.38807.65950.15850.73680.34390.24560.3825 2100.05510.46968.09910.14180.79190.32140.12930.3672 2150.06230.43026.53640.12750.72970.31390.11140.3250 2200.06840.46435.40260.11050.66880.29300.17300.1027 2250.07790.48193.98690.07700.80610.29140.11310.0761

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64 Table 4-3. Extracted model parameters for 3F3 ferrite (n = 2). T (oC) c (m/A) c Hc(A/m) Bs(T) mtfrev(MHz) firr(MHz) resnorm 25 0.02810.272223.68300.43160.68540.44030.21680.8626 300.02710.291222.40300.44220.62580.40590.24930.7065 350.02630.320221.87700.43850.63560.40820.24720.7135 400.02680.330421.44650.43160.63810.40450.22640.7500 450.02610.334219.73790.42380.65690.40720.24350.6486 500.02600.368019.16430.41970.64830.40210.21560.6354 550.02580.386818.51980.41400.65490.40170.19740.6030 600.02490.396117.04140.41040.65660.40480.20750.5811 650.02480.424416.93130.40790.64680.40020.17930.5414 700.02450.429816.34660.40100.66920.40760.17180.5562 750.02410.454616.40200.40070.65730.40220.16050.5508 800.02380.466815.74610.39070.68920.40660.15830.5524 850.02410.479415.15100.38360.68180.40390.14090.5028 900.02340.491515.03330.38190.69140.40750.13500.5123 950.02310.516813.99920.37280.69460.40040.14170.4545 1000.02330.535814.72190.36120.75800.39870.12940.5166 1050.02320.517312.75550.35660.72340.41110.14880.3840 1100.02310.526412.80320.35390.70570.41020.14440.3946 1150.02420.518113.05220.33840.74270.41210.13900.3981 1200.02460.530312.72920.32920.76100.40590.12520.4374 1250.02500.538012.33220.32040.75900.39690.14430.3813 1300.02570.523111.99100.31040.77240.39620.14260.4173 1350.02640.516011.73200.29980.80040.39520.14500.4451 1400.02770.516711.53130.28900.79510.39140.13880.4171 1450.02930.523411.34030.27770.80430.38080.12160.4182 1500.03130.500411.34830.26320.83470.38250.12430.4218 1550.03210.514711.08040.25860.79080.36740.12150.4026 1600.03460.46569.47190.24680.77230.37610.15330.3521 1650.03800.472210.15030.23340.78470.36470.13140.3469 1700.04140.45949.66850.22010.79410.35750.12290.3654 1750.04500.46769.14900.20650.80840.34240.10970.3767 1800.04800.44838.40710.19610.76660.33050.15110.3034 1850.05350.45667.90030.17990.80790.31060.11950.3790 1900.06070.51477.80320.16390.82300.27760.08250.2997 1950.07420.46267.25130.14740.76090.26640.06190.2936 2000.08860.46066.07690.13070.65380.23680.04610.2869 2050.10000.41584.23860.10760.65880.23300.07420.1344 2100.10000.46562.97060.06730.69590.22370.03770.0519

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65 Table 4-4. Temperature Coefficients of the test ferrites CoefcientsMN8CXP3F3 Tc190230.3213.1 b10.48990.50760.4803 b20.57760.51230.4708 b3 40.6733.4634.9 b40.50120.42470.3881 b5 1.3041.4871.056 b60.36990.20010.3333 b7 1.3441.2651.47 b8 59.1852.8135.09 b9 5.6235.6784.909 b10 44.9157.3535.24 b11 2.7848.694.781 b120.22830.17270.1955 b13 1.3851.1631.366 b14 45.52137.147.24 b15 2.8377.6064.536 b1611161.4766.03 b17 5.8446.5973.27 b180.30040.27740.03802 b19 32.6935.219.3 b20 9.66716.16.019 b210.53530.46980.4412 b220.21860.15560.164 b230.34140.72120.2247 b24 2.5354.4740.003117 b25 93.821278.152

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66 CHAPTER 5 HIGH-TEMPERATURE CHARACTERIZATION SYSTEM 5.1 Introduction The ferrite cores in power electronic converters normally operate at temperatures other than room temperature owing to self-heating or elevated ambient temperature. Tosupport the modeling of ferrites under these operating conditions, automatic measurement systems have been reported for the normal temperature range, typically up to 120oC [24] . Methods and apparatus for the characterization of ferrites up to Curie temperatureandintothemulti-megahertzfrequencyrangehavenotbeenwidelydescribed in the literature [24, 25] . Thischapterpresentstheimplementationofanautomaticmeasurementsystemfor ferrite cores up to 250oC, which exceeds the Curie temperatures of several commercial power ferrites. The system air-heats the magnetic cores in a computer-controlled temperature chamber. Hysteresis loops of bulk ferrites are measured at temperatures rangingfromroomtemperaturetotheCuriepoints,andfrequenciesrangingfrom10kHz to 1 MHz. Following this section, the instrumentation issues are discussed in Section5.2 . Section5.3 describes the measurement procedures and the corresponding software algorithms. Sampled hysteresis loops acquired are reported in Section5.5 . The main results are summarized in Section5.6 .

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67 5.2 Instrumentation 5.2.1 Automatic Hysteresis Loop Measurement System The automatic hysteresis measurement system is presented in Figure5-1 . A personal computer (PC) equipped with Labview automation software [26] supervises the operationsofthePM5192functiongenerator [27] ,theTEK2440digitaloscilloscope [28] , and the DELTA9039 temperature chamber (the “oven”) [29] via IEEE-488 (NI-488) interface [30] . The digital oscilloscope has 100 MHz bandwidth with 500 MHz sampling rateand2%accuracy.Theovencanheatasampleupto300oC.TheENI2100LRFpower amplier [31] , driven by the function generator, supplies (through an isolation transformer) a sinusoidal voltage excitation to the air-heated test toroid securely placed inside the temperature chamber. The power amplier has 10 kHz to 12 MHz frequency range, 100 W output power, and 50 dB gain. Besides providing electrical isolation, the isolation transformer prevents the small dc offset in the power amplier output voltage from entering to the test toroid. The transformer has 3:1 turns ratio, selected based on impedance matching in this measurement setup. As can be seen from Figure5-2 , bilar windings using the high-temperature wire specied in Table5-1 are wound on the test core. One winding is the excitation winding (i.e., it receives the excitation current). The other winding is the sense winding (i.e., it is left open-circuited so that its induced voltage can be measured and integrated to nd the ux in the core). The excitation current and the induced voltage are measured by the digital oscilloscope. The excitation current is measured via the voltage drop across a non-inductive current-sense resistor (50 m W, 1% tolerance [32] ) connected in series with the primary winding. The resistor is placed outside the temperature chamber to avoid the

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68 Figure 5-1.Automatic high-temperature hysteresis measurement system.A) Block diagram. B) Actual hardware. TEK 2440Isolation Transformer ENI 2100LPM 5192 Rsv1v2RF PowerCaddockTest ToroidAmplier50 m W , 1%DELTATemperature Chamber IEEE 488 (GPIB)PCSoftware: LabVIEW PMaG 250oC MP925 1.0 MHz Sin Function Generator 9039Cards: NI AT-GPIB & NI-DAQ/AT-MIO-E Thermocouple Type K to NI-DAQ Termination breadboard SC-2070 A) B)

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69 Figure 5-2.Test xture for measurements up to 250oC. Table 5-1. High-Temperature Materials Material Maximum temperature/voltage Manufacturer/ part number MIL-W-16878 Wire AWG #30 (5-mil PTFEa Teon insulation, NPCb conductors) 260oC/250 V Quirk Wire/ 301-XE-NPC [38] MIL-C-17 Coaxial Cable (PTFE Teon insulation, SPCc Braids) 250oC/1200 V Dearborn/ 6188A [39] PTFE Teon Washers (screw # 6) 260oC McMaster-Carr/ 95630A237 [40] NPC Ring Terminals (screw # 6) 480oC McMaster-Carr/ 69405K5 PTFE Teon Tape (2-mil Thick) 260oC McMaster-Carr/ 76475A11 a. PTFE = Polytetrauoroethylene b. NPC = Nickel Plated Copper c. SPC = Silvered Plated Copper

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70 directheatfromtheoven.Themeasureddataaretransferredfromthedigitaloscilloscope to the PC for post-processing in Labview, such as converting voltage and current measurements to B-H loops. The magnetic eld intensity H is computed from the excitationcurrent i1,thenumberofexcitationturns Np,andtheeffectivemeanpathlength le according to Ampere’s law: (5-1) Themagneticuxdensity B iscomputedfromtheinducedvoltage v2,thenumberofsense turns Ns, and the effective cross-sectional area Ae according to Faraday’s law: (5-2) TheIECstandards [33] suggestthefollowingformulaefor leand Aeoftoroidalcoreswith sharp corners: (5-3) (5-4) where ri and ro are the inner radius and the outer radius of a toroid. The IEC formulae approximatedforamagneticcorehavingnonuniformelddistributionarecommonlyused by most soft ferrites producers [34] . The le and Ae formulae for toroids with rounded corners and other core shapes can also be found the IEC standards [33] and the MMPA standards [34] . H t () Npi1t () le-----------------= B t () 1 NsAe-----------v2t () t d= le2 p rori---ln 1 ri--1 ro---- – ---------------------= Aeh rori---ln21 ri--1 ro---- – --------------------=

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71 The test core temperature is constantly monitored by a glass-braided insulated, exposed-junction type K (Nickel-Chromium/Nickel-Aluminium) thermocouple [35] . The thermocouple has the maximum service temperature up to 900oC, 0.8 second response time (time constant) in forced circulated air, and 2.2oC standard limits of error. The thermocouple leads are connected to the termination breadboard of the DAQ (data acquisition) card; this breadboard also contains a room temperature sensor IC for cold junction compensation [36, 37] . To keep the cost of the high-temperature feature low, all instruments, sensors, and probes to measure voltages and currents are mounted outside (and on the door of) the thermal chamber as shown in Figure5-1B . Thus, only the test xture that carries the core undertesthastobesubjectedtoextremetemperatures,andneedsspecialdesignattention. Thehigh-temperaturecoaxialcablesbetweenthemountedboardoutsidetheovenandthe testxtureinsidetheovenwerekeptshortandequalinlengthtominimizetimedelayand possible phase error. 5.2.2 High-Temperature Test Fixtures The words “test xture” refer to everything in Figure5-2 except the core and the bilar windings under test. Since the test xture is inside the oven, it must be assembled using the high-temperature materials listed in Table5-1[38, 39, 40] . PTFE Teflon is the primary high-temperature insulation for virtually all components on the test fixture. AlthoughthemeltingpointofPTFETeflonis327oC,itsmaximumoperatingtemperature is rated at 260oC for continuous use. For ultra high temperatures, mica glass tape (MGT) insulationsratedat450oCcouldbeselectedinstead.The5-milinsulationthicknessofthe wire is a compromise between voltage rating and ll factor.

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72 FortestinguptotheCurietemperature,thetestedferritecoreshouldnothaveany coating unless the coating can withstand such high temperature. Typically, the temperature ratings of coating materials range from 130oC to 200oC, whereas the Curie temperaturesofferritesvaryfrom100oCto250oC.SincetheCurietemperatureofferrites is far below the sintering temperature (700oC or above), testing of ferrites up to the Curie point should not permanently damage the material if extra precaution is taken. Thus, the rate of temperature change is kept below 5oC to 10oC per minute to avoid a sudden fracture of the test core [41] . 5.3 Measurement Procedures and Algorithms Figure5-3 illustratesblockdiagramsoftheexperimentalproceduresforautomatic B-H loopsmeasurementatdifferenttemperaturesanddifferentfrequencies.Theprocedure comprises nested temperature, frequency, and amplitude loops in AlgorithmD-1 that call AlgorithmD-2 through AlgorithmD-5 . The details of the algorithms can be found in Appendix D . In AlgorithmD-1 , the test equipment is initialized, and the test core demagnetized by slowly reducing the magnitude of the ac excitation voltage from the saturation value to zero [42, 43] . Generally, the slower the rate of magnitude decay is, the better demagnetization results can be obtained. In this experiment, a 10 kHz excitation voltage having a linearly decaying amplitude with 0.2 V/s was used. Note that the frequency of demagnetization should be the static or quasi-static frequency of the test magnetics materials to avoid demagnetization results being affected by dynamic effects such as eddy currents. Hysteresis loops are measured in 5oC steps between 25oC and the Curie temperature, which can be as high as 250oC.

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73 Theexperimentalprocedureforcontrollingtheoventemperatureissummarizedin AlgorithmD-2 . Securely placed inside the PC-controlled temperature chamber, the test coreisheatedtoeachtemperaturesetpointanditstemperaturemonitoredbytheattached thermocoupleeverytwoseconds.ThegeneratedthermocouplevoltageisreadbytheDAQ board and converted to the actual temperature reading using the standard polynomial Figure 5-3.Experimental procedure and algorithms for the automatic acquisition of multiple B-H loops at different temperatures and frequencies. Initialize Demagnetize Frequency Loop Amplitude Loop Equipment Test Core 1 i = i+1 30 40 250 50 i TEMP(i) Temperature Loop Heat Oven to Wait for Set Point T Stabilized T 2 j = j+1 j FREQ(j) 1 T kHzoC 350 50 300 Set Generator Set OscilloFrequency k = k+1 2 400 100 1000 200 10 f scope time/div AlgorithmD-3 AlgorithmD-2 B, H T AMPL(k) Controlled Bp? AlgorithmD-5 Plot & Save Measure a B-H Loop with Constant Flux Vg(mV) No Yes Bp(mT) AlgorithmD-4 Measure a B-H Loop with Variable Flux f

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74 approach for thermocouple type K. After reaching each temperature set point, the oven is allowed to rest for at least a few minutes until the core temperature is stabilized. This stabilization time is to allow elevated environmental heat to dissipate throughout the test core. The test core is deemed thermally stabilized when the corresponding points on two consecutively measured B-H loops differ by less than 1%. This was obtained in advance and used as a preset value in the automatic measuring process. The frequency of the excitation source is then swept to presetable values between 10 kHz and 1 MHz. The 10 kHz lower limit is set by the ENI 2100L RF Power Amplier.The 1 MHz upper limit is set by the driving voltage and current capability of the excitation source, rather than the capability of the measuring equipment. The experimental procedure for frequency control is summarized in AlgorithmD-3 .Ateachpresetfrequency,the B-H loopsaremeasuredfromthelargest(or major or saturation) loop to the smallest minor loop. Starting the measurement from the saturation loop helps erasing previous magnetic histories, thus resulting in better minor loops, as is evident in Figure5-7 . The minor loop tips can be connected to construct the initial magnetization curve. To prevent self-heating, the core is excited briey, allowing just sufcient time for the waveforms to reach steady state before each set of data is measured.Sufcientidletimeisthenallowedsothatthecorehasachancetocooldownto thesetambient(oven)temperaturebeforethenextmeasurementisinitiated.Theoretically, thesteadystatecanbedeterminedwhenthepresentandtheprevious B-H loopscoincide. However,toomuchunwantedself-heatingcanoccurifthetestcoreisoverlyexcited,thus affecting the hysteresis measurements. In this experiment, the excitation time is deemed sufcientwhenthepresent B-H loopstayswithin5%ofthepreviousone.Theidletimeis

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75 deemedsufcientwhenthesurfacetemperatureofthetestcoredropsbacktowithin0.1oC of the set point. Both times was determined beforehand and used as preset values in the automaticmeasuringprocess.Forexample,at10kHzthetestcoreisexcitedfor3seconds and idled for 6 seconds. At 1 MHz, for the same amount of excitation time, the core temperaturecanriseupto3oCbecauseofself-heating.Withthehelpoffan-circulatedair in the oven, the idle time of 20 seconds was experimentally found sufcient for keeping thecoretemperatureatthesetambient(oven)temperature.Theamountsoftheexcitation and the idle times are dependent on the frequency and temperature characteristics of the test magnetic material. As implied by AlgorithmD-3 and AlgorithmD-3 in Figure5-3 , the system can measure the hysteresis loops in two modes. In the “Variable-Flux” mode, the frequency and temperature are maintained constant, and hysteresis loops with varying peak ux density ( Bp) are acquired as the amplitude of the voltage exciting the core is varied. This modeisusefulforgeneratingthemajorandminorloops.Themeasurementprocedurefor this mode is captured in AlgorithmD-4 , outlined in Figure5-4 along with a sample Labview program. “Constant Flux” is the other mode of hysteresis-loop measurement offered by the system.Inthismode Bp,aswellastemperature,iskeptconstantasthefrequencyisvaried. This mode is useful for generating, e.g., a family of core loss density versus frequency, parametricwithrespecttopeakuxdensity.Themeasurementprocedureforthismodeis captured in AlgorithmD-5 , outlined in Figure5-5 . For sinusoidal excitation at a constant frequency f , the theoretical peak output voltage Vg that needs to be applied to the power amplier to achieve a given Bp can be derived from Eq.5-2 :

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76 Figure 5-4.Block diagram for the automatic acquisition of a single B-H loop with variable ux, along with a sample Labview program. Figure 5-5.Block diagram for the automatic acquisition of a single B-H loop with constant peak magnetic ux density. 1 3 2 Adjust Display Remove Offset Convert toB-H Start Timer Read V-I Signal OFF Wait Excitation Time Store Display Signal ON Wait Start Timer Plot & Save Idle Time 1 2 3 Start End LabVIEW programAlgorithmD-4 Vg* Bp/ Bp-Bp Initial Guess AlgorithmD-5 STARTVgB p B p ? AlgorithmD-4Measure aB-H Loop with Variable Flux Find BpEnd Yes No B-H Plot & Save B-H

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77 (5-5) where AvistheproductofthevoltagegainoftheRFpoweramplierandthevoltageratio oftheisolationtransformer.Inreality,however, Vgcannotsimplybecomputedfrom Bpvia theprecedingequationbecauseboththeexcitationandtheuxwaveformsaredistortedby coresaturation.Thus,thefollowingiterativeequationisemployedtocalculatethenext Vgfrom the present Vg ( Vg -) and the present Bp ( Bp -): (5-6) where the initial guess of Vg is calculated using Eq.5-5 . In the linear region (e.g., below 0.2 T at room temperature) the measured hysteresis loops converged to the desired Bpwithinafewattemptsusing5%tolerance,suggestingthatthelinearscalinginthepreceding equation is adequate. In the saturation region, ve or six attempts were needed to achieve convergencewith5%tolerance.Notethat Vgneedstobelimitedto1Vasspeciedbythe manufacturer of the ENI2100L power amplier. The algorithms summarized above are delineated in the Appendix D . All algorithms are written for the specic equipment illustrated in Figure5-1 . 5.4 Error Analysis and Calibration 5.4.1 Temperature Chamber AnimpedanceanalyzerHP4194A [44] wasusedtoevaluatethepossibilitythatthe temperature chamber used in this experiment may affect the measurements of magnetic properties. The impedance of the small test core of MN8CX ferrite [4] in Table5-2 measured inside the operating oven is compared with that measured outside the oven at room temperature for the entire test frequency range. No signicant differences are V g2 p fNsAeBpAv----------------------------= VgVg -BpBp -------=

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78 detected, other than random variation in measurements. The maximum magnitude and phase deviation are 0.22% and 0.0536o, respectively. 5.4.2 Current-sense Resistor The current-sense resistor is modeled as a resistor Rs in series with the parallel combinationofaninductor Ls,andacapacitor Cs.TheimpedanceanalyzerHP4194A [44] was used to measure the impedance of the current-sense resistor between 10 kHz and 1MHz. The “Equivalent Circuit” feature of the HP4194A then yielded Rs = 54.7 m W , Ls=4.22nH, and Cs = 1.21 m F. The compensated excitation current i1 can then be obtained from the measured voltage across the current-sense resistor v1 according to the second-order differential equation in Eq.5-7 or its transfer function in Eq.5-8 : (5-7) ; (5-8) In Figure5-6 , the compensated and uncompensated current are compared at room temperature and 1 MHz. The uncompensated current was calculated using only the resistance value. Since i1iscompensatedusing Eq.5-7 ,themeasurementuncertaintyof i1relieson the accuracy of Rs, Ls, and Cs elements measured by the HP4194A. To perform error analysis [45] , the relative sensitivity S of each elements are computed according to (5-9) d2i1t () dt2---------------1 RsCs----------di1t () dt -------------1 LsCs----------++ 1 Ls---dv1t () dt --------------1 RsLsCs----------------v1t () + = I1s () Ys () V1s () = Ys () sRsCs1 + s2RsLsCssLsRs++ ------------------------------------------------= SL Ys ()Ys () L ------------L Ys () ---------sLssRsCs1 + () – s2RsLsCssLsRs++ ------------------------------------------------==

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79 (5-10) (5-11) At 1 MHz, = -0.085-0.556j, = 0.332+0.243j, and = -0.583+0.798j, using Rs, Ls, and Cs values mentioned earlier. Since the resistor is placed outside the temperature chamber, temperature effects on the resistor are negligible. The worst-case magnitude and phase uncertainty of the admittance can then be calculated using (5-12) (5-13) Figure 5-6.Uncompensated and compensated primary currentobtained from the test MN8CX toroid at room temperature and 1 MHz. SC Ys ()Ys () L ------------L Ys () ---------sRs 2Css2RsLsCssLsRs++ () sRsCs1 + () ----------------------------------------------------------------------------------== SR Ys ()Ys () L ------------L Ys () ---------Rs– s2RsLsCssLsRs++ () sRsCs1 + () ----------------------------------------------------------------------------------== compensated uncompensated Time t ( m s) Primary Current i1 (A)SL Yj w ()SC Yj w ()SR Yj w () d Yj w () Yj w () -------------------e ReSx Yj w (){}xLsCsRs,, == d Yj w () –e ImSx Yj w (){}xLsCsRs,, ==

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80 where e isthemeasureduncertaintyoftheimpedanceanalyzerforgain-phasemeasurement whichisequalto1% [44] .Attheupperendofthefrequencyrange(1MHz),themagnitude and phase uncertainty of the admittance are 1% and 0.91o, respectively. The primary and secondary voltages v1 and v2 were measured using the oscilloscope with relative uncertainty of 2% [28] . Thus, the total relative uncertainty of current i1 based on normal distribution is estimated as 2.23% using the “root-sum-of-squares” method [46] . Since the coaxial cables for measuring both voltages are kept as short as possible (e.g. a few inches) and have the same length, no phase difference is detected between the two channels of the oscilloscope. Thus, the total phase uncertainty is estimated as 0.91o. 5.5Measured Data Static hysteresis loops were measured for the “standard” toroid of MN8CX [4] ferrite listed in Table5-2 , using a 10 kHz sinusoidal excitation. The cores were heated fromroomtemperaturetotheirCurietemperatures,185oCforMN8CXferrite.Aprimary windingandasecondarywindingwerewoundbilaronthecores,bothhaving25turnsof the high-temperature wire gauge AWG #24 in Table5-1 . The nished test core for the standard toroid of MN8CX ferrite is shown in Figure5-2 . The measured static minor loops of MN8CX ferrite at 100oC are shown in Figures 5-7 . It takes about 10 seconds to measure each loop. Dynamic hysteresis loops were measured for the “small” toroids of MN8CX, P, 3F3 ferrites listed in Table5.5 , using sinusoidal excitations. A primary winding and a secondary winding were wound bilar on the cores, both having 4 turns of the high-temperature wire gauge AWG #30 in Table5-1 . The temperature was swept from

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81 room temperature to the Curie point by the experimental procedure described in Section 5.2 . At each temperature, a set of dynamic B-H loops with the same Bp was measured at presetable frequencies from 10 kHz to 1 MHz. It takes about 30 seconds to measure each loop. The measured data for MN8CX ferrite at 100oC are shown in Figure5-8 . 5.6Conclusion Theexperimentalsetupandproceduresdescribedhavebeenfoundeffectiveforthe automaticacquisitionofhigh-temperatureandhigh-frequencydataforpowerferrites.The high-temperature capability requires an inexpensive test xture; no high-temperature voltage or current probes or measuring equipment are needed. It is hoped that sufcient information has been provided for the system to be reproduced elsewhere. Table 5-2. Material, size, dimensions, and effective parameters of the test toroids MaterialSize Outer radius ro, mm Inner radius ri, mm Height h , mm Effective length le, mm Effective area Ae, mm2MN8CXStandard11.647.477.7058.1231.63 MN8CXSmall6.353.172.5429.358.06 PSmall6.354.003.2032.237.52 3F3Small6.353.955.1032.0612.24

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82 Figure 5-7.Static hysteresis loops of MN8CX ferrite measured at 100oCand 10 kHz. Figure 5-8.Dynamic hysteresis loops of MN8CX ferrite measured at 100oC; 10 kHz, 100kHz, 200 kHz, 300 kHz, 500 kHz, and 1 MHz; 100 mT peak magnetic ux density. Magnetic Flux Density B (T)Magnetic Field Intensity H (A/m) 500 kHz 300 kHz 1 MHz 100 kHz 10 kHz 700 kHz 200 kHz Magnetic Field Intensity H , (A/m) Magnetic Flux Density B (T)

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83 CHAPTER 6 SUMMARY This dissertation has developed a high temperature, high-frequency magnetic hysteresismodelforpowerferritesbasedonBasso-Bertotti’sphysics-basedDWMmodel in such a way that it can be integrated into a CAD system for power electronic applications. The complete model has resulted in good agreement between measured and modeleddataforthemajorandminorloopsofthethreecommercialMnZnpowerferrites up to their Curie temperatures, under sinusoidal excitations in the 10 kHz to 1 MHz frequency range. As reviewed in Chapter2 , the DWM model described by Basso and Bertotti uses two functions, a Preisach distribution function p(hc) and a domain-wall surface function R ( m ), to shape a static hysteresis loop. To better characterize both the major loop and the minor loops of power ferrites, a piecewise-parabolic function is proposed for R ( m ) based onthepiecewiselinearshapeofthemeasuredd R ( m )/d m .Thenew R ( m )containsamodel parameter mt that denes the transition to the saturation region and characterizes saturation gradualness of power ferrites. The static DWM model for power ferrites described in Chapter2 is generalized in Chapter3 byaddingtwonewequations:arst-orderdifferentialequationandanalgebraic equation to characterize frequency dependence of hysteresis in MnZn ferrites. The use of the rst-order differential equation for MnZn ferrites is conrmed with the complex permeability formulae previously published. The mathematical form of the algebraic

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84 function is identied based on the attenuated amount of the irreversible DWM contribution under steady-state sinusoidal conditions. The two newly added equations contain two new model parameters, frev and firr, that denes the cut-off frequencies of the reversible and irreversible processes. In Chapter4 , temperature dependence of hysteresis is incorporated into the dynamichysteresismodeldescribedin Chapter3 byttingthetemperaturecharacteristics of its seven model parameters with the corresponding mathematical functions. The complete dynamic hysteresis model with temperature contains 25 temperature coefcient b1 to b25 and the Curie temperature Tc. The experimental setup and procedures described in Chapter5 have been found effective for the automatic acquisition of high-temperature and high-frequency data for power ferrites. The high-temperature capability requires an inexpensive test xture; no high-temperature voltage or current probes or measuring equipment are needed. This research work has focused entirely on toroidal cores, sinusoidal excitations without dc bias, and parameter extraction using large-signal measurements. Therefore, future development could focus on other core shapes, sinusoidal excitations with dc bias, arbitrary excitation waveforms with or without dc bias, and parameter extraction using only small-signal measurements. The complete model should be implemented in a commercial circuit simulator, offering power electronic designers another choice for magnetic core models.

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85 APPENDIXAPPENDIX A DERIVATIONS AND DEFINITIONS A.1 Mean Domain-Wall Position WiththePreisachdistributionfunction p ( hc)describedin Chapter2 andthe Preisachdiagramin FigureA-1 ,themeandomain-wallposition x fortheinitial magnetizationcurve,thedownwardbranch,andtheupwardbrancharederived, respectively,asshownin Eq.A-1 through Eq.A-3 .Thegeneralized Eq.A-4 foranyreturn branches can then be obtained by combining Eq.A-2 and Eq.A-3 . Figure A-1.Preisach Diagrams for the derivation of the mean domain-wall motion.A) Fortheinitialmagnetizationcurve.B)Forthedownwardhysteresisbranch. C) For the upward hysteresis branch. hu 0 H H H-hc downup huH0H huH0H (H0-H)/2 (H-H0)/2H0-hc H+hcH0+hc H-hc A)B)C) H hchchchchuswitch to + d x if H > hu+ hcor hu< H hcswitch to d x if H < huhcor hu> H + hcx + d x d x 0

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86 (A-1) (A-2) (A-3) (A-4) A.2 Initial Susceptibility and Differential Susceptibility By chain rule, the differential (normalized) susceptibility is (A-5) With x = xinitial as shown in Eq.A-1 , (A-6) Substitution of Eq.A-6 in Eq.A-5 yields (A-7) Since p ( hc)isdenedsuchthat,theinitial(normalized)susceptibility after a demagnetized state is (A-8) With x = xup as shown in Eq.A-3 , xinitialH ()c phc() huhcd d0 Hhc–0 Hc phc() Hhc– () hcd0 H== xdownx0c phc() hudhcdHhc+ H0hc–0 H0H – 2 -----------------– x02 c phc()H 0 H – 2 ------------------hc– hcd0 H0H – 2 -----------------– == xupx0c phc() huhcd dH0hc+ Hhc–0 HH0– 2 -----------------+ x02 c phc()HH 0 – 2 ------------------hc– hcd0 HH0– 2 -----------------+ == xH () x02 cHH 0 –() sgn phc()HH 0 – 2 ------------------hc– hcd0 HH0– 2 -----------------+ = dm dH ------dm dx ------dx dH ------Rm () dx dH ------== dx dH ------c d dH ------Hphc() hcd0 H d dH ------hcphc() hcd0 H – = c phc() hcd0 HHpH () + HpH () – = c phc() h c d0 H= dm dH ------c Rm () phc() hcd0 H= phc() hcd0 0+1 = dm dH ------H 0 lim m H ----H 0 lim c c ==

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87 (A-9) Substitution of Eq.A-9 in Eq.A-5 yields (A-10) Since p ( hc)isdenedsuchthat,thedifferentialsusceptibilityforthe upward branch after a turning point is (A-11) Since p ( hc)isdenedsuchthat,thedifferentialsusceptibilityforthe upward branch of the major loop is (A-12) A.3 Coercive Force and Remanent Magnetic Flux Density By denition, ,,(A-13) ,(A-14) dx dH -------2 c d dH -------HH 0 – 2 ------------------phc() hcd0HH 0 – 2 -----------------d dH ------hcphc() hcd0HH 0 – 2 -----------------– = 2 c 1 2 -phc() hcd0HH 0 – 2 ------------------HH 0 – 2 ------------------pHH 0 – 2 -----------------1 2 -+ HH 0 – 2 ------------------pHH 0 – 2 -----------------1 2 -- – = c phc() hcd0HH 0 – 2 ------------------= dm dH ------c Rm () phc() hcd0HH 0 – 2 ------------------= phc() hcd0 0+1 = dm dH -------HH0lim c c Rm () = phc() hcd0 1 = dm dH -------H0 – c Rm () = pirrhc() hcd0 1 = prevhc() hcd0 1 = hcpirrhc() hcd0 Hcirr ,= hcprevhc() hcd0 0 =

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88 With x = xinitialasshownin Eq.A-1 , x afterthetestcoreisdrivenupfromthedemagnetized state to the maximum magnetic ux intensity Hmax is (A-15) With x = xdownasshownin Eq.A-2 , x afterthetestcoreisdrivendownfrom HmaxtoHcis (A-16) Since p ( hc)isdenedsuchthat( Eq.2-1 ),substitutionof Eq.A-15 in Eq.A-16 yields (A-17) With the denitions in Eq.A-13 and Eq.A-14 , (A-18) Thus, (A-19) With x = xdownasshownin Eq.A-2 , x afterthetestcoreisdrivendownfrom Hmaxto0(i.e., B = Br = m0Mr) is (A-20) Since p ( hc)isdenedsuchthat( Eq.2-1 ),substitutionof Eq.A-15 in Eq.A-20 yields xHmaxc phc() Hhc– () hcd0 Hmax= xHC–xHmax2 – c phc() HmaxH c + 2 -------------------------hc– hcd0 HmaxHc+ 2 ------------------------0 == p hc() 1 c – () pirrcprev+ = xHC–c 1 c – () pirrhc() cprevhc() + [] Hhc– () hcd0 Hmax= 2 – c 1 c – () pirrhc() cprevhc() + [] HmaxH c + 2 -------------------------hc– hcd0 HmaxHc+ 2 ------------------------0 = xHC– Hmax lim c Hc1 c – () Hcirr ,– () 0 = = Hc1 c – () Hcirr ,= xBrxHmax2 – c phc() Hs2 -----hc– hcd0 Hmax2 -----------= p hc() 1 c – () pirrcprev+ =

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89 (A-21) Using the denitions in Eq.A-13 and Eq.A-14 , (A-22) Thus, (A-23) Using Eq.2-17 and the denition Bs = m0Ms in SectionA.6 , (A-24) Thus, if(A-25) A.4 Initial Permeability and Maximum Permeability Theinitialpermeability miisdenedasthestartingslopeoftheinitial magnetizationcurve;andthemaximumpermeability mmaxasthemaximumslopeofthe upward branch of the major hysteresis loop: (A-26) (A-27) xBrc 1 c – () pirrhc() cprevhc() + [] Hhc– () hcd0 Hmax= 2 – c 1 c – () pirrhc() cprevhc() + [] Hmax2 -----------hc– hcd0 Hmax2 -----------xBrHmax lim c Hs1 c – () Hcirr ,– [] 2 – c Hs2 ------1 c – () Hcirr ,– c 1 c – () Hcirr ,== xBrHmax lim c Hc= BrBs----m0Mrm0Ms------------m t c Hcm t ---------tanh == BrBs----c Hc c Hcm t ----------1 midB dH -------H0B0, () 00 , () = HH0lim = mmaxmax dB dH -------H0 – =

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90 Since B = m0( H + M ) and Bs = m0Ms as dened in SectionA.6 , (A-28) Substitution Eq.A-28 in Eq.A-26 yields (A-29) Using Eq.A-8 , (A-30) Substitution Eq.A-28 in Eq.A-27 yields (A-31) Using Eq.A-12 and R ( m ) dened in Eq.2-15 (i.e. maximum of R ( m ) = 1), (A-32) A.5 Reversible Contribution of the Mean Domain-Wall Position By denition, the reversible contribution of x is (A-33) WiththereversiblePreisachdistributionfunction prev( hc)= d ( hc)describedin Chapter2 , p ( hc)= cprev( hc),and Eq.A-1 through Eq.A-3 , x fortheinitialmagnetizationcurve,the downwardbranch,andtheupwardbranchofa x H looparederived,respectively,asshown in Eq.A-34 through Eq.A-36 .Thegeneralized Eq.A-37 for xrevdenedin Eq.A-33 can then be obtained by solving Eq.A-34 through Eq.A-36 . (A-34) (A-35) dB dH ------m0m0dM dH -------+ m0Bsdm dH ------+ == mim0Bs+ dm dH -------H0B0, () 00 , () = HH0lim = mim0c c Bs+ = mmaxm0Bs+ max dm dH -------H0 – = mmaxm0c Bs+ = xrevxphc() cprevhc() == xinitialH () c c H = xdownH () xrevinitial ,H0() c c H0H – () – =

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91 (A-36) (A-37) A.6 Saturation Magnetic Flux Density Thesaturationmagneticuxdensity Bscanbedenedinvariousways.Oftenitis denedinmaterialdatasheetsasthemaximumvalueofmagneticuxdensity Bmaxata specied level of applied eld strength Hmax: (A-38) where m0isthepermeabilityoffreespace,and M themagnetization.Forasoftmagnetic material,themagneticuxdensityoffreespaceisnegligiblecomparedtothatcontributed by the material. The saturation magnetic ux density can thus be approximated by (A-39) where Msisthesaturationmagnetization.Thisapproximationyieldsapproximately1% error at Hmax = 4000 A/m for a magnetic material having m0Ms = 0.5 T. Tomakeavailableamorecommonlyfoundmagneticpropertylike Bsasamodel parameterratherthan Ms, Bsisthusdenedas m0Ms,insteadof Eq.A-38 ,throughoutthis thesis.Notethatthisdenitiondoesnotyielderrorsprovideditisalwaysinterpreted exactly as dened. xupH () xrevdown ,H0() c c HH0– () + = xrevH () c c H = Bmaxm0Hmaxm0MHmax() + = Bmaxm0MHmax()m0Ms

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92 APPENDIX B MATLAB CODES B.1 Static DWM Model TheMatlabcodefor Algorithm2-1 in Chapter2 islistedbelowalongwith comments, signied by the% sign.function B = findB(H,H0,B0,chi,c,Hc,Bs,mt,n) % Data: model parameters {chi, c, Hc, Bs, mt, n} Hcirr = Hc/(1-c); xt = sqrt(mt)*atanh(sqrt(mt)); % x(mt) % Input: H and {H0, B0} % Goal: compute x, m, B if (H0 == 0) & (B0 == 0), % initial magnetization curve delH = abs(H); Pirr = fPirr(delH,Hcirr,n); Prev = delH; x = chi*sign(H-H0)*((1-c)*Pirr+c*Prev); else % any return branch m0 = B0/Bs; if abs(m0) <= mt, x0 = sqrt(mt)*atanh(m0/sqrt(mt)); else x0 = sign(m0)*((1-mt)/(1-abs(m0))+xt-1); end delH = abs(H-H0)/2; Pirr = fPirr(delH,Hcirr,n); Prev = delH; x = x0+2*chi*sign(H-H0)*((1-c)*Pirr+c*Prev); end if abs(x) <= xt, m = sqrt(mt)*tanh(x/sqrt(mt)); else m = sign(x)*(1-(1-mt)/(1-xt+abs(x))); end B = mu0*H+Bs*m; % subfunction Pirr(delH) function Pirr=fPirr(delH,Hcirr,n) sum = 0; for k = 1:n, sum = sum+k/n/prod(1:n-k)*(n*delH/Hcirr)^(n-k); end Pirr = (delH-Hcirr)+Hcirr*exp(-n*delH/Hcirr)*sum;

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93 B.2 Dynamic DWM model TheMatlabcodesfor Algorithm3-1 and Algorithm3-2 in Chapter3 arelisted below along with comments, signied by the% sign.function [B,xtilde,x] = H2B(H,t,x0,xtilde0,par,n) if nargin < 6, n = 1; end chi = par(1); c = par(2); Hc = par(3); Bs = par(4); mt = par(5); fr = par(6)*1e6; fi = par(7)*1e6; mu0 = 4*pi*1e-7; kirr = ones(size(H)); x = H2x(H,x0,kirr,chi,c,Hc,n); xtilde = x2xtilde(x,t,xtilde0,fr); m = xtilde2m(xtilde,mt); B = mu0*H+Bs*m; MaxIter = 10; MaxTol = 0.0001; iter = 0; tol = Inf; while (iterMaxTol), iter = iter+1; feq = B2feq(B,t); kirr = feq2kirr(feq,fi); x = H2x(H,x0,kirr,chi,c,Hc,n); xtilde = x2xtilde(x,t,xtilde0,fr); m = xtilde2m(xtilde,mt); Bprev = B; B = mu0*H+Bs*m; tol = max(abs(B-Bprev)); end %----------------------------------------------------------------------function kirr = feq2kirr(feq,fi) kirr = sqrt(1./((feq/fi).^2+1)); %----------------------------------------------------------------------function feq = B2feq(B,t) dBdt = gradient(B,t); for i = 1:length(B), feq(i) = 1/2/pi*sqrt(trapz(abs(dBdt(i)^2-dBdt(1:i).^2))./(trapz(abs(B(i)^2-B(1:i) .^2))+eps)); end %----------------------------------------------------------------------function m = xtilde2m(xtilde,mt) mt = min(max(mt,0),1); xt = sqrt(mt)*atanh(sqrt(mt)); for i = 1:length(xtilde), if abs(xtilde(i)) <= xt, m(i) = sqrt(mt)*tanh(xtilde(i)/sqrt(mt)); else m(i) = sign(xtilde(i))*(1-(1-mt)/(1-xt+abs(xtilde(i)))); end end %----------------------------------------------------------------------function xtilde = x2xtilde(x,t,xtilde0,fr)

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94 if isfinite(fr), wr = 2*pi*fr; sys = ss(-wr,wr,1,0); t0 = t(1); xtilde = lsim(sys,x,t-t0,xtilde0)’; else return end %----------------------------------------------------------------------function x = H2x(H,x0,kirr,chi,c,Hc,n) c = min(max(c,0),1); if c ~= 1, Hcirr = Hc/(1-c); else Hcirr = 0; end H0 = H(1); delH = H-H0; if H0 == 0 & x0 == 0, x = chi*sign(delH).*(... (1-c)*kirr.*Pirr(abs(delH),Hcirr,n)+c*Prev(abs(delH)) ); else x = x0 + 2*chi*sign(delH).*(... (1-c)*kirr.*Pirr(abs(delH/2),Hcirr,n)+c*Prev(abs(delH/2)) ); end %----------------------------------------------------------------------function P = Pirr(delH,Hcirr,n) if ~isfinite(Hcirr), P = zeros(size(delH)); return, end Hcirr = max(Hcirr,eps); for i = 1:length(delH), sum = 0; for k = 1:n, sum = sum+k/n/prod(1:n-k)*(n*delH(i)/Hcirr)^(n-k); end P(i) = (delH(i)-Hcirr)+Hcirr*exp(-n*delH(i)/Hcirr)*sum; end %----------------------------------------------------------------------function P = Prev(delH) P = delH; %-----------------------------------------------------------------------B.3 Dynamic DWM model with Temperature TheMatlabcodesforAlgorithms Algorithm4-1 and Algorithm4-2 in Chapter4 are listed below along with comments, signied by the% sign.function mn8cx(T,f) if nargin < 1, T = 100; end if nargin < 2, f = 100e3; end figure(1), clf wt = linspace(0,pi/2); for i = 1:4,

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95 wt = [wt; linspace((i-1)*pi+pi/2,i*pi+pi/2)]; end t = wt/(2*pi*f); Hmax = 200; H = Hmax*sin(wt); % MN8CX ferrite Tc = 190; b = [0.4899 0.5776 40.6700 0.5012 1.3040,... 0.3699 1.3440 59.1800 5.6240 44.9000,... 2.7820 0.2287 1.3840 45.5700 2.8420,... 110.8000 5.8380 0.2987 32.6400 9.6550,... 0.5353 0.2205 0.3372 2.3820 93.6700]; T = min(max(T,25),Tc-.1); par = T2par(T,Tc,b); n = 1; x0 = 0; xtilde0 = 0; color = ’b’; for i = 1:size(wt,1), [B(i,:),xtilde(i,:),x(i,:)] = H2B(H(i,:),t(i,:),x0,xtilde0,par,n); x0 = x(i,end); xtilde0 = xtilde(i,end); if i > size(wt,1)-2, color = ’r’; end figure(1), hold on, plot(H(i,:),B(i,:),color) end xlabel(’H (A/m)’) ylabel(’B (T)’) s = sprintf(’ T = %0.0f deg. C, F = %g Hz’,T,f); title(s) axis([-250 250 -.5 .5]) %----------------------------------------------------------------------function par = T2par(T,Tc,b) Bs = b(1)*(1-T/Tc).^b(2); Hcirr = b(3)*(1-T/Tc).^b(4); mui_mu0 = b(5)*( b(6)*gampdf(Tc-T,b(7),b(8)) + (1-b(6))*gampdf(Tc-T,b(9),b(10)) ); mumax_mu0 = b(11)*( b(12)*gampdf(Tc-T,b(13),b(14)) + (1-b(12))*gampdf(Tc-T,b(15),b(16)) ); mt = b(17)./(b(18)*(1-T/b(19)).^2+1)./(b(20)*(1-T/Tc).^2+1); fr = b(21)*(1-T/Tc).^b(22); fi = b(23)./(b(24)*(1-T/b(25)).^2+1); chi = mumax_mu0./Bs; c = mui_mu0./mumax_mu0; Hc = Hcirr.*(1-c); par = [chi’ c’ Hc’ Bs’ mt’ fr’ fi’]; %-----------------------------------------------------------------------

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APPENDIX C STATIC AND DYNAMIC HYSTERESIS LOOPS

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97 C.1 MN8CX Ferrite Figure C-1Measured and modeled hysteresis loops for MN8CX ferriteat10kHz,30kHz, 50 kHz, 70 kHz, 100 kHz, 200 kHz, 300kHz, 500 kHz, 700 kHz, and 1 MHz; constant temperature ranging from room temperature to the Curie temperature with 5oC increment. The modeled loops plotted with solid lines are simulated usingtheDDWMmodeldescribedin Chapter3 withthecorrespondingmodel parameters listed in Table4-1 . -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H (A/m)

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98 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 30oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 30oC H (A/m)

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99 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 35oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 35oC H (A/m)

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100 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 40oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 40oC H (A/m)

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101 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 45oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 45oC H (A/m)

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102 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 50oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 50oC H (A/m)

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103 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 55oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 55oC H (A/m)

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104 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 60oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 60oC H (A/m)

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105 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 65oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 65oC H (A/m)

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106 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 70oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 70oC H (A/m)

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107 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 75oC H (A/m)B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 75oC H (A/m)

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108 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 80oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 80oC H ( A/m )

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109 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 85oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 85oC H ( A/m )

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110 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 90oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 90oC H ( A/m )

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111 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 95oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 95oC H ( A/m )

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112 Figure C-1Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 100oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 100oC H ( A/m )

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113 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 105oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 105oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 105oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 105oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 105oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 105oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 105oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 105oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 105oC H (A/m)B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 105oC H (A/m)

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114 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 110oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 110oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 110oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 110oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 110oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 110oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 110oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 110oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 110oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 110oC H ( A/m )

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115 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 115oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 115oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 115oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 115oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 115oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 115oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 115oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 115oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 115oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 115oC H ( A/m )

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116 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 120oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 120oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 120oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 120oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 120oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 120oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 120oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 120oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 120oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 120oC H ( A/m )

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117 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 125oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 125oC H ( A/m )

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118 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 130oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 130oC H ( A/m )

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119 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 135oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 135oC H ( A/m )

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120 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 140oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 140oC H ( A/m )

PAGE 133

121 Figure C-1Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 145oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 145oC H ( A/m )

PAGE 134

122 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 150oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 150oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 150oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 150oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 150oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 150oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 150oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 150oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 150oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 150oC H ( A/m )

PAGE 135

123 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 155oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 155oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 155oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 155oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 155oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 155oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 155oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 155oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 155oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 155oC H ( A/m )

PAGE 136

124 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 160oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 160oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 160oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 160oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 160oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 160oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 160oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 160oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 160oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 160oC H ( A/m )

PAGE 137

125 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 165oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 165oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 165oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 165oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 165oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 165oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 165oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 165oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 165oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 165oC H ( A/m )

PAGE 138

126 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 170oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 170oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 170oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 170oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 170oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 170oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 170oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 170oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 170oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 170oC H ( A/m )

PAGE 139

127 Figure C-1Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 175oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 175oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 175oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 175oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 175oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 175oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 175oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 175oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 175oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 175oC H ( A/m )

PAGE 140

128 Figure C-1Continued -50 -25 0 25 50 0 0.1 10 kHz 180oC B (T) -50 -25 0 25 50 0 0.1 30 kHz 180oC -50 -25 0 25 50 0 0.1 50 kHz 180oC B (T) -50 -25 0 25 50 0 0.1 70 kHz 180oC -50 -25 0 25 50 0 0.1 100 kHz 180oC B (T) -50 -25 0 25 50 0 0.1 200 kHz 180oC -50 -25 0 25 50 0 0.1 300 kHz 180oC B (T) -50 -25 0 25 50 0 0.1 500 kHz 180oC -50 -25 0 25 50 0 0.1 700 kHz 180oC H ( A/m ) B (T) -50 -25 0 25 50 0 0.1 1 MHz 180oC H ( A/m )

PAGE 141

129 C.2 P Ferrite Figure C-2Measured and modeled hysteresis loops for P ferriteat 10 kHz, 30 kHz, 50kHz, 70 kHz, 100 kHz, 200 kHz, 300kHz, 500 kHz, 700 kHz, and 1 MHz; constant temperature ranging from room temperature to the Curie temperature with 5oC increment. The modeled loops plotted with solid lines are simulated usingtheDDWMmodeldescribedin Chapter3 withthecorrespondingmodel parameters listed in Table4-2 . -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H ( A/m )

PAGE 142

130 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 30oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 30oC H ( A/m )

PAGE 143

131 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 35oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 35oC H ( A/m )

PAGE 144

132 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 40oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 40oC H ( A/m )

PAGE 145

133 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 45oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 45oC H ( A/m )

PAGE 146

134 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 50oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 50oC H ( A/m )

PAGE 147

135 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 55oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 55oC H ( A/m )

PAGE 148

136 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 60oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 60oC H ( A/m )

PAGE 149

137 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 65oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 65oC H ( A/m )

PAGE 150

138 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 70oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 70oC H ( A/m )

PAGE 151

139 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 75oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 75oC H ( A/m )

PAGE 152

140 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 80oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 80oC H ( A/m )

PAGE 153

141 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 85oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 85oC H ( A/m )

PAGE 154

142 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 90oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 90oC H ( A/m )

PAGE 155

143 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 95oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 95oC H ( A/m )

PAGE 156

144 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 100oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 100oC H ( A/m )

PAGE 157

145 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 105oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 105oC H ( A/m )

PAGE 158

146 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 110oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 110oC H ( A/m )

PAGE 159

147 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 115oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 115oC H ( A/m )

PAGE 160

148 Figure C-2Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 120oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 120oC H ( A/m )

PAGE 161

149 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 125oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 125oC H ( A/m )

PAGE 162

150 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 130oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 130oC H ( A/m )

PAGE 163

151 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 135oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 135oC H ( A/m )

PAGE 164

152 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 140oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 140oC H ( A/m )

PAGE 165

153 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 145oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 145oC H ( A/m )

PAGE 166

154 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 150oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 150oC H ( A/m )

PAGE 167

155 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 155oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 155oC H ( A/m )

PAGE 168

156 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 160oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 160oC H ( A/m )

PAGE 169

157 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 165oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 165oC H ( A/m )

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158 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 170oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 170oC H ( A/m )

PAGE 171

159 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 175oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 175oC H ( A/m )

PAGE 172

160 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 180oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 180oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 180oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 180oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 180oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 180oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 180oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 180oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 180oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 180oC H ( A/m )

PAGE 173

161 Figure C-2Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 185oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 185oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 185oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 185oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 185oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 185oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 185oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 185oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 185oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 185oC H ( A/m )

PAGE 174

162 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 190oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 190oC H ( A/m )

PAGE 175

163 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 195oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 195oC H ( A/m )

PAGE 176

164 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 200oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 200oC H ( A/m )

PAGE 177

165 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 205oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 205oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 205oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 205oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 205oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 205oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 205oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 205oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 205oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 205oC H ( A/m )

PAGE 178

166 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 210oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 210oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 210oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 210oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 210oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 210oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 210oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 210oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 210oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 210oC H ( A/m )

PAGE 179

167 Figure C-2Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 215oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 215oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 215oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 215oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 215oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 215oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 215oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 215oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 215oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 215oC H ( A/m )

PAGE 180

168 Figure C-2Continued -50 -25 0 25 50 0 0.1 10 kHz 220oC B (T) -50 -25 0 25 50 0 0.1 30 kHz 220oC -50 -25 0 25 50 0 0.1 50 kHz 220oC B (T) -50 -25 0 25 50 0 0.1 70 kHz 220oC -50 -25 0 25 50 0 0.1 100 kHz 220oC B (T) -50 -25 0 25 50 0 0.1 200 kHz 220oC -50 -25 0 25 50 0 0.1 300 kHz 220oC B (T) -50 -25 0 25 50 0 0.1 500 kHz 220oC -50 -25 0 25 50 0 0.1 700 kHz 220oC H ( A/m ) B (T) -50 -25 0 25 50 0 0.1 1 MHz 220oC H ( A/m )

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169 Figure C-2Continued -50 -25 0 25 50 0 0.1 10 kHz 225oC B (T) -50 -25 0 25 50 0 0.1 30 kHz 225oC -50 -25 0 25 50 0 0.1 50 kHz 225oC B (T) -50 -25 0 25 50 0 0.1 70 kHz 225oC -50 -25 0 25 50 0 0.1 100 kHz 225oC B (T) -50 -25 0 25 50 0 0.1 200 kHz 225oC -50 -25 0 25 50 0 0.1 300 kHz 225oC B (T) -50 -25 0 25 50 0 0.1 500 kHz 225oC -50 -25 0 25 50 0 0.1 700 kHz 225oC H ( A/m ) B (T) -50 -25 0 25 50 0 0.1 1 MHz 225oC H ( A/m )

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170 C.3 3F3 Ferrite Figure C-3Measured and modeled hysteresis loops for 3F3 ferriteat 10 kHz, 30 kHz, 50kHz, 70 kHz, 100 kHz, 200 kHz, 300kHz, 500 kHz, 700 kHz, and 1 MHz; constant temperature ranging from room temperature to the Curie temperature with 5oC increment. The modeled loops plotted with solid lines are simulated usingtheDDWMmodeldescribedin Chapter3 withthecorrespondingmodel parameters listed in Table4-3 . -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 25oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 25oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 25oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 25oC H ( A/m )

PAGE 183

171 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 30oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 30oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 30oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 30oC H ( A/m )

PAGE 184

172 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 35oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 35oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 35oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 35oC H ( A/m )

PAGE 185

173 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 40oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 40oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 40oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 40oC H ( A/m )

PAGE 186

174 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 45oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 45oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 45oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 45oC H ( A/m )

PAGE 187

175 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 50oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 50oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 50oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 50oC H ( A/m )

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176 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 55oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 55oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 55oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 55oC H ( A/m )

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177 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 60oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 60oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 60oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 60oC H ( A/m )

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178 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 65oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 65oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 65oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 65oC H ( A/m )

PAGE 191

179 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 70oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 70oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 70oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 70oC H ( A/m )

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180 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 75oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 75oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 75oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 75oC H ( A/m )

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181 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 80oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 80oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 80oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 80oC H ( A/m )

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182 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 85oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 85oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 85oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 85oC H ( A/m )

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183 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 90oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 90oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 90oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 90oC H ( A/m )

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184 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 95oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 95oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 95oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 95oC H ( A/m )

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185 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 100oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 100oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 100oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 100oC H ( A/m )

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186 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 105oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 105oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 105oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 105oC H ( A/m )

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187 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 110oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 110oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 110oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 110oC H ( A/m )

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188 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 115oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 115oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 115oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 115oC H ( A/m )

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189 Figure C-3Continued -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 10 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 30 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 50 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 70 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 100 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 200 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 300 kHz 120oC B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 500 kHz 120oC -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 700 kHz 120oC H ( A/m ) B (T) -200 -100 0 100 200 0 0.1 0.2 0.3 0.4 1 MHz 120oC H ( A/m )

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190 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 125oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 125oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 125oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 125oC H ( A/m )

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191 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 130oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 130oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 130oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 130oC H ( A/m )

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192 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 135oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 135oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 135oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 135oC H ( A/m )

PAGE 205

193 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 140oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 140oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 140oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 140oC H ( A/m )

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194 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 145oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 145oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 145oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 145oC H ( A/m )

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195 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 150oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 150oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 150oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 150oC H ( A/m )

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196 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 155oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 155oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 155oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 155oC H ( A/m )

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197 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 160oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 160oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 160oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 160oC H ( A/m )

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198 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 165oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 165oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 165oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 165oC H ( A/m )

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199 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 170oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 170oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 170oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 170oC H ( A/m )

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200 Figure C-3Continued -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 10 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 30 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 50 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 70 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 100 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 200 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 300 kHz 175oC B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 500 kHz 175oC -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 700 kHz 175oC H ( A/m ) B (T) -150 -100 -50 0 50 100 150 0 0.1 0.2 0.3 1 MHz 175oC H ( A/m )

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201 Figure C-3Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 180oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 180oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 180oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 180oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 180oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 180oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 180oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 180oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 180oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 180oC H ( A/m )

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202 Figure C-3Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 185oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 185oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 185oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 185oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 185oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 185oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 185oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 185oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 185oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 185oC H ( A/m )

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203 Figure C-3Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 190oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 190oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 190oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 190oC H ( A/m )

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204 Figure C-3Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 195oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 195oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 195oC H ( A/m ) B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 195oC H ( A/m )

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205 Figure C-3Continued -100 -50 0 50 100 0 0.1 0.2 10 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 30 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 50 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 70 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 100 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 200 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 300 kHz 200oC B (T) -100 -50 0 50 100 0 0.1 0.2 500 kHz 200oC -100 -50 0 50 100 0 0.1 0.2 700 kHz 200oC H (A/m)B (T) -100 -50 0 50 100 0 0.1 0.2 1 MHz 200oC H (A/m)

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206 Figure C-3Continued -50 -25 0 25 50 0 0.1 10 kHz 205oC B (T) -50 -25 0 25 50 0 0.1 30 kHz 205oC -50 -25 0 25 50 0 0.1 50 kHz 205oC B (T) -50 -25 0 25 50 0 0.1 70 kHz 205oC -50 -25 0 25 50 0 0.1 100 kHz 205oC B (T) -50 -25 0 25 50 0 0.1 200 kHz 205oC -50 -25 0 25 50 0 0.1 300 kHz 205oC B (T) -50 -25 0 25 50 0 0.1 500 kHz 205oC -50 -25 0 25 50 0 0.1 700 kHz 205oC H ( A/m ) B (T) -50 -25 0 25 50 0 0.1 1 MHz 205oC H ( A/m )

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207 Figure C-3Continued -50 -25 0 25 50 0 0.1 10 kHz 210oC B (T) -50 -25 0 25 50 0 0.1 30 kHz 210oC -50 -25 0 25 50 0 0.1 50 kHz 210oC B (T) -50 -25 0 25 50 0 0.1 70 kHz 210oC -50 -25 0 25 50 0 0.1 100 kHz 210oC B (T) -50 -25 0 25 50 0 0.1 200 kHz 210oC -50 -25 0 25 50 0 0.1 300 kHz 210oC B (T) -50 -25 0 25 50 0 0.1 500 kHz 210oC -50 -25 0 25 50 0 0.1 700 kHz 210oC H ( A/m ) B (T) -50 -25 0 25 50 0 0.1 1 MHz 210oC H ( A/m )

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208 APPENDIX D MEASUREMENT ALGORITHMS Algorithm D-1 Multiple B-H loop measurements: 1.Input: temperature ( TEMP ), frequency ( FREQ ), amplitude ( AMPL ) arrays. 2.Data: Material ( matl ), Sample ( smpl ). 3.Initialize the function generator, oscilloscope, and temperature chamber. 4.Demagnetize the test core at 10 kHz. 5.Sort TEMP , FREQ , and AMPL . 6.Reset i = j = k = 0. 7.While i < the length of TEMP array, 8.i = i+1; 9.Call AlgorithmD-2 {Input: TEMP (i)} to heat the oven to TEMP (i); 10.While j < the length of FREQ array, 11.j = j+1; 12.Call AlgorithmD-3 to set the excitation frequency to FREQ (j); 13.While k < the length of AMPL array, 14.k = k+1; 15.Concatenate header = [ matl , smpl , “T”, TEMP (i), “f”, FREQ (j)]; 16.If the constant peak ux density mode is selected, 17.Call AlgorithmD-5 ; 18.Concatenate le = [ header , “B”, AMPL (k)];

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209 19.Otherwise, 20.Call AlgorithmD-4 ; 21.Concatenate le = [ header , “V”, AMPL (k)]; 22.Plot B versus H and save B-H data to le . Algorithm D-2 Temperature Control: 1.Input: temperature ( T ). 2.GPIB-write the oven temperature set point = T (oC). 3.GPIB-write to turn ON the heater. 4.While | T Toven| > 0.1oC, 5.Wait for the temperature scanning time, e.g., every 2 seconds; 6.GPIB-read and display the oven temperature Toven; 7.DAQ-read the voltage from the room-temperature IC sensor; 8.DAQ-read the voltage from the test thermocouple; 9.Compute and display the test core temperature. 10.Wait for the set-point temperature to stabilize. Algorithm D-3 Frequency Control. 1.Input: frequency ( f ). 2.Data: oscilloscope TIME/DIV array in 5-2-1 sequence. 3.GPIB-write f to the function generator. 4.Compute the period of one cycle cycl = 1/f . 5.Calculate cycl/div = cycl /20 to store the data in twenty divisions. 6.Find and GPIB-write to the oscilloscope the lowest TIME/DIV that is greater than or equal to cycl/div to store the data as close to one cycle as possible.

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210 Algorithm D-4 A Single B-H Loop Measurement: 1.Input: peak output voltage ( Vg). 2.Data: le, Ae, Np, Ns, and oscilloscope VOLT/DIV array in 5-2-1 sequence. 3.GPIB-write to the oscilloscope to set both channels to the highest VOLT/DIV . 4.GPIB-write Vg to the function generator. 5.Start the excitation timer ( texcit). 6.GPIB-write to turn ON the function generator. 7.GPIB-read from the oscilloscope the voltages from both channels ( V1 & V2) 8.Find the maximum values of V1 & V2 ( V1,max & V2,max) 9.Calculate V1/DIV = V1,max/4and V2/DIV = V2,max/4(Fourdivisionsinahalfof the oscilloscope screen is selected to maximize resolution). 10.Find and GPIB-write to the scope the lowest VOLT/DIV that is greater than or equal to V1/DIV & V2/DIV. 11.Wait for the preset value of texcit, e.g., 5 seconds. 12.GPIB-write to stop the oscilloscope to ensure the data from both channels are stored simultaneously. 13.GPIB-write to turn OFF the signal of the function generator. 14.Start the idle timer ( tidle). 15.GPIB-read from the oscilloscope V1, V2, and the sampling time ( D t). 16.Calculate the excitation current I1 = V1/ Rsense ( V2 is the induced voltage). 17.Calculate the number of data points in one cycle PTS = 1/( f * D t). 18.Find the index of the rst maximum value of I1 ( IDX ).

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211 19.Let i1=onecycledataof I1= I1( IDX , IDX +1, IDX +2,..., IDX + PTS -1),starting from the maximum to the minimum then returning for one complete cycle. 20.Let v2 = one cycle data of V2 = V2( IDX , IDX +1, IDX +2,..., IDX + PTS -1), corresponding point by point to I1. 21.Remove DC offset from i1 and v2. 22.Compute the magnetic eld strength H = Npi1/ le. 23.Compute the magnetic ux F by integrating v2/ Ns. 24.Remove dc component from F . 25.Compute the magnetic ux density B = F / Ae. 26.Wait for self-heating of the test core to cool off before restarting the next loop. 27.Output: B and H . Algorithm D-5 Constant Peak Flux Density Control: 1.Input: peak ux density ( Bp). 2.Data:frequency ( f ) 3.Compute an initial guess Vg = 2p fNsBpAe/Av. 4.Set tolerance (tol) = 5% of Bp. 5.Set the maximum Vg ( Vg, max) = 1 V (for ENI2100L). 6.While (| BpBmax| > tol) and ( Vp < Vp, max), 7.Call AlgorithmD-4 ; 8.Display the maximum ux density Bp-; 9. Vg = Vg * Bp/ Bp -. 10.Output: B and H .

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212 REFERENCE LIST 1.Garuda VR, Kazimierczuk MK, Ramalingam ML, Tunstall C, Tolkkinen L. High-temperature performance characterization of buck converter using SiC and Si devices.PESC98.Proceedingsofthe29thAnnualIEEEPowerElectronicsSpecialists Conference; 1998 May 17-22; Fukuoka, Japan. p. 1561-1567. 2.Basso V, Bertotti G. Hysteresis models for the description of domain wall motion. IEEE Trans Magn 1996; 32: 4210-4212. 3.Nakmahachalasint P, Ngo KDT. Generalized formulation for the description of hysteresis in soft magnetic materials. IEEE Trans Magn 2002; 38: 200-204. 4.Ceramic Magnetics. 1999. Mn-Zn ferrites. Ceramic Magnetics, Inc. Available from URL: http://www .cmi-ferrite.com/data/materialsstdummary .pdf .SitelastvisitedApril 2003. 5.Magnetics. 2001. Ferrite cores design manual. Magnetics, Inc. Available from URL: http://www .mag-inc.com/ferrites/fc601.asp . Site last visited April 2003. 6.Ferroxcube. 2002. Soft ferrites and accessories. Ferroxcube, Inc. Available from URL: http://www .ferroxcube.com/appl/info/HB2002.htm .SitelastvisitedApril2003. 7.LiorzouF,PhelpsB,AthertonDL.Macroscopicmodelsofmagnetization.IEEETrans Magn 2000; 36: 418-428. 8.Basso V. Hysteresis models for magnetization by domain wall motion. IEEE Trans Magn 1998; 34: 2207-2212. 9.Bertotti G. Hysteresis in magnetism. San Diego (CA): Academic Press; 1998. 10.KadarG.Thebilinearproductmodelofhysteresisphenomena.PhysicaScripta1989; T25: 161-164. 11.The MathWorks 1999. MATLAB: optimization toolbox (version 2). [Computer program]. Available from The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098. 12.Bertotti G. Dynamic generalization of the scalar Preisach model of hysteresis. IEEE Trans Magn 1992; 28: 2599-2601.

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213 13.Bertotti G, Fiorillo F, Pasquale M. Measurement and prediction of dynamic loop shapesandpowerlossesinsoftmagneticmaterials.IEEETransMagn1993;29:13-16. 14.Fuzi J, Ivanyi A. Features of two rate-dependent hysteresis models. Physica B 2001; 306: 137-142. 15.Jiles DC. Frequency dependence of hysteresis curves in non-conducting magnetic materials. IEEE Trans Magn 1993; 29: 3490-3492. 16.HsuJT,NgoKDT.AHammerstein-baseddynamicmodelforhysteresisphenomenon. IEEE Trans Power Electron 1997; 2: 406-413. 17.Saotome H, Sakaki Y. Iron loss analysis of Mn-Zn ferrite cores. IEEE Trans Magn 1997; 33: 728 -734. 18.Johnson MT, Visser EG. A coherent model for the complex permeability in polycrystalline ferrites. IEEE Trans Magn 1990; 26: 1987-1989. 19.DoringW.Uberdietragheitderwandezwischenweibschenbezirken.ZeitschriftFur Naturforschung 1948; 3a: 373-379. 20.Chikazumi S. Physics of magnetism. New York: John Wiley; 1994. p. 321-355. 21.Magnetic Materials Producers Association (MMPA). MMPA SFG-98: soft ferrites a user’s guides. Chicago (IL): MMPA; 1998. p. 34-36. 22.Goldman A, Magnetic components for power electronics. Norwell (MA): Kluwer Academic Publishers; 2002. 23.Hisatake K, Anomalous thermal change of permeability near Curie point in MnZn ferrite. Electronics and Communications in Japan 1975; 58-C (5): 114-121. 24.Batista AJ, Fagundes JCS, Viarouge P. An automated measurement system for core loss characterization. IEEE Trans Instrum Meas 1999; 48: 663-667. 25.Thottuvelil VJ, Wilson TG, Owen HA. High-frequency measurement techniques for magnetic cores. IEEE Trans Power Electron 1990; 5: 41-53. 26.National Instruments 1999. LabVIEW (version 5.1). [Computer program]. Available from National Instruments Corporation, 11500 N Mopac Expwy, Austin, TX 78759-3504. 27.Philips. PM5192: operating manual. Hamburg (Germany): Philips GmbH; 1988. 28.Tektronix.TEK2440:programmersreferenceguide.Beaverton(OR):Tektronix,Inc.; 1991.

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214 29.Delta Design. DELTA9039: instruction manual. San Diego (CA): Delta Design, Inc.; 1987. 30.Automated Instrumentation Technical Committee of the IEEE Instrumentation and MeasurementSociety.ANSI/IEEEStd488.1-1987:digitalinterfaceforprogrammable instrumentation. New York: IEEE; 1988. 31.ENI Technology. ENI2100L: operation manual. Rochester (NY): ENI Technology, Inc.; 1991. 32.CaddockElectronics.1999.MP900andMP9000seriesKool-Pakpowerlmresistors. Caddock Electronics, Inc. Available from URL: http://www .caddock.com/Online_catalog/current_sense/current_sense.html . Site last visited April 2003. 33.International Electrotechnical Commission (IEC). IEC 60205: calculation of the effective parameters of magnetic piece parts. 2nd ed. Geneva (Switzerland): IEC; 2001. 34.Magnetic Materials Producers Association (MMPA). MMPA SFG-98: soft ferrites a user’s guides. Chicago (IL): MMPA; 1998. p. 21-23. 35.Omega Engineering. 2000. The temperature handbook. Omega Engineering, Inc. Available from URL: http://www .ome g a.com/literature . Site last visited April 2003. 36.National Instruments. ATMIO/AI E Series user manual. Austin (TX): National Instruments Corporation; 1996. 37.NationalInstruments.SC207Xseriesusermanual.Austin(TX):NationalInstruments Corporation; 1995. 38.Wirecraft. 1997. MIL-W-16878 PTFE Teon wire. Quirk Wire Co., Inc. Available from URL: http://www .quirkwire.com/MIL-W -16878.html . Site last visited April 2003. 39.Dearborn.2001.Militarytypecoaxialcables.DearbornWireandCable,Inc.Available from URL: http://www .dearborn-cdt.com/catalog/CO AX-1.html . Site last visited April 2003. 40.McMaster-Carr. 2000. Find products. McMaster-Carr Supply Company. Available from URL: http://www .mcmaster .com . Site last visited April 2003. 41.Magnetics.2001.Designingwithmagneticcoresathightemperatures.Magnetics,Inc. AvailablefromURL: http://www .mag-inc.com/library .asp .SitelastvisitedApril2003.

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215 42.IEEE Std 393-1991 IEEE Standard for Test Procedures for Magnetic Cores, section 6.6.1.5, p. 48. 43.Bulte D, Langman R. Comparison of the normal and initial induction curve. IEEE Trans Magn 2001; 37: 3892 -3899. 44.Hewlett-Packard.HP4194A:impedance/gain-phaseanalyzeroperationmanual.Tokyo (Japan): Yokogawa-Hewlett-Packard; 1987. 45.Huelsman LP. Introduction to the theory and design of active lters. New York: McGraw-Hill; 1980. 46.TaylorBN,KuyattCE.1994.Guidelinesforevaluatingandexpressingtheuncertainty ofNISTmeasurementresults.NationalInstituteofStandardsandTechnology.United States Department of commerce. Available from URL: http://ph ysics.nist.go v/Document/tn1297.pdf . Site last visited April 2003.

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216 BIOGRAPHICAL SKETCH PaiboonNakmahachalasintreceivedtheB.Eng.degreeinindustrialinstrumentationfromKingMongkut’sInstituteofTechnologyLadkrabang,Thailand,in1991,andthe M.S.degreeinelectricalengineeringfromtheUniversityofFlorida,in1994.Since1992, hehasbeenwithThammasatUniversity,Thailand,whereheiscurrentlyanassistantprofessor. His current research interests include magnetic materials and components.