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’1:H/(A/m)NI(A)l(m)le2HB=0HJB=0(HM)BJM0410-7/(H/m)BJHMA/m1T=104GsAeVHzNBTAem23BH:H=NIlH=(A/m)2INle1Oe=79.58A/m11034:B=0.225VfNAeBsBrHcHsBsBrHcHs41)2)rr3)()H1mTB0.1mT4):gleg5)HHB6)7)1)~6)BH=0(1+)=absoluteMHBH=0r10BH(H0)=ii1+gi/le=e=a10BH10BHHDC==s-js=_sspp5XL=jL,Xc=RZsZp6tgdm:tgm=tgh+tge+tgr7'11p1jp1jCZs=Rs+jLsZp11/(jLp)+1/Rptgm==RsLsss'tgm==LpRp'pptgmitgi8(tg)gapei1:eiQHiHi=He-Hd=He-NMNHiHe(He=)M0~4N=0N9Q(Q)Q==RtotRtotQ10:Pm=Ph+Pe+PrPhPePr11i=i1,i2T1T2:i,ir12i1,i2t1,t2DA,DFiDF(tg)gape-1=tgi-1(tg)gape=tgitg.ei(tg)gap=NIle1tgLRtoti2-i1i11T2-T1ir==ii1i2-i1(i1)21T2-T1DA=i1-i2i11lg(t2/t1)DF==DAi1i1-i2i11lg(t2/t1)213ALNLAL=ALAL=.109LN100NALAL(nH):i=ALC1mm-1ALAL14--HDC:B/2():LN2nHN21N2Ts=L(nH)AL(nH/N2)C10.4nHN2H2H2H=HDC+sint=10BHHDC80%201.MagneticfieldCurrentinducesmagneticfield.Inspiralcoils,themagneticfield(H)inducedbycurrentcanbeexpressedas:WhereallparametersareinSIunitsystemandNisturnnumber,I(A)iscurrent,l(m)isthelengthofthespiralcoils.Inmagneticcore,thefieldstrengthHinducedbyalternatecurrentcanbecalculatedintermoftheeffectivelengthleofthespiralcoils:2.Magneticfluxdensity,magneticpolarizability,magnetization.Inmagneticmaterial,themagneticfluxdensityvariesasappliedfieldH.Itbehaviorsas:B=0H+JorB=0(H+M)WhereBismagneticfluxdensityalsocalledmagneticinduction,Jmagneticpolarization,Mmagnetization,and0vacuumpermeabilitywiththevalueof410–7H/m.TheunitsofBandJareTesla(T)andthoseofHandMareA/m.1Tesla=104GaussInmagneticcores,themagneticfluxdensitycanbecalculatedusingeffectiveareaAe:WhereViselectricpotentialinVolt,ffrequencyinHz,Nturnnumber,BinmTandAeinm2.3.Saturationmagnetization,remanentmagnetization,andcoercivity.BesidesthelinearrelationbetweenBandHinvacuum,BbehaviorsanonlinearrelationasHinmagneticmaterialsdisplayingthehysteresisshowninthefigure.H=NIlH=(A/m)2INle1Oe=79.58A/m11034B=Forsinewave0.225VfNAeMainconceptsanddefinitionsInthefigure,BSissaturationinduction,Brresidualinduction,HCcoercivity,andHSsaturationfield.Differentmagneticmaterialsdisplayvarioushysteresis,leadingtodifferentBS,Br,HC,andHS.4.Permeability1)calledabsolutepermeabilitywithdimension.2)rwhereriscalledrelativepermeability,whichisapurenumberwithoutdimension.Usuallyweusetherelativepermeability,neglectingthefootnoter.3)iscalledinitialpermeability.Itdependsontemperatureandfrequency.Themeasurementofµishouldbemadeinaclosedmagneticcircuitatcertaintemperatureandfrequencyinaveryweakappliedfield.Inmeasurement,itrequiresthatthechangeofmagneticfluxdensity(B)inducedbyHshouldbelessthan1mT,generallyB=0.1mT.4)Forunclosedmagneticcircuitwithagap,measuredpermeabilityiscalledeffectivepermeabilityexpressedas:wheregisthelengthofthegap,andletheeffectivelengthofthemagneticcircuit.Itnotesthatthisequationonlyanapproximationofµeforthesmallgap.Forlargegap,theeffectivepermeabilitywilllargerthanthatcalculatedusingaboveequation.5)WhenanappliedfieldHislargerwithoutaDCbiasfield,itinducesthemagneticfluxdensityB,inwhich6)InanalternatefieldwithaDCbiasfield,thepermeability.iscalledincrementalpermeability.FortheelectricinductancemeasuredintheACfieldsuperposedwithabiasDCfield,thepermeabilityisprobablyalsotheincrementalpermeability.7)Thepermeabilityinabove1)—6)areallobtainedinthelowfrequencyorneartoDCsituation.Whenthefrequencyishigh,thepermeabilityiscomplex.BH=0(1+)=absoluteMHBH=0r10BH(H0)=ii1+gi/le=e,iscalledamplitiudepermeability.10BH10BHHDCa==Inserialcircuit,=s-jsInparallelcircuit,=-s,s,p,p,areallthefunctionsoffrequency.5.ImpedanceInductiveimpedanceinanelectricinductanceisXL=jL,andcondenserimpedanceinacondenserisXC=,Thesetwoaregenerallycalledelectricalimpedance.AddingpureresistanceR,theyareinallcalledimpedance.Inmagneticdevices,weonlyconsiderinductiveimpedanceandpureresistancefortheissueofrelativelowfrequency,neglectingcondenserimpedance.Thereisthedifferencebetweenserialandparallelcircuit.ZsandZpdependonfrequency,andtheircharacteristicsarecalledimpedancefrequencycharacteristicsandrelatedtothefrequencycharacteristicsofmagneticmaterials,andtheyareconnectedwithwindingparameters.Incomplexpermeability,itsfrequencycharacteristicsisdeterminedbythefrequencycharacteristicsofbothand.Actually,theimpedancefrequencycharacteristicisthecharacteristicofthemagneticdevicebutthecharacteristicofmaterial.6.LossfactorLossfactorindicatesthelosspropertyofmaterialinsmallsignal.Itinducesphaseshiftofsignalduetomagneticcoreloss,whichcanbeexpressedas:wheretgmiscalledlossfactorindicatingtheratiooflosspowerandinputpower.Becausemagneticcorelossinduceshysteresisloss,eddyloss,andresidualloss,thelossfactorcanbeexpressedas:tgm=tgh+tge+tgr,Wheretgh,tge,andtgriscalledhysteresislossfactor,eddylossfactor,andresiduallossfactorrespectively(seethefollowingFigure).7.SpecificLossfactororiscalledspecificlossfactor,whichisindependentofgeometricalsizeofmaterial,indicatingsmallsignallosscharacteristicofthematerial.'''11p1jp1jCSeriesrepresentationParallelreparesentationSeriesrepresentationZs=Rs+jLsParallelreparesentationZp11/(jLp)+1/Rptgm==RsLsss''tgm==LpRp'pportgmitgi8.TheinfluenceofgapWhenthemagneticcircuitisunclosedwithagap,thelossfa