Chapter9：DesignofHigh-FrequencyInductorsandTransformers9-1Introduction9-2BasicsofMagneticDesign9-4Area-ProductMethod9-5DesignExampleofanInductor9-6DesignExampleofaTransformerforaForwardConverter9-7LossmechanismsinmagneticcircuitsFromReference2--?FirstCourseonPowerElectronics?19-1IntroductionHigh-frequencyinductorsandtransformsaregenerallynotavailableoff-the-shelf,andmustbedesignedbasedontheapplicationspecifications.Sofar,wehavemostlyconsideredanalysis.Designismorechallenging,andispartlyanart.Inthischapter,asimpleandacommonlyusedapproachcalledArea-Productmethodispresented,wherethethermalconsiderationsareignored.(AdetaileddesigndiscussionispresentedinChapter30oftheTextbook)29-2BASICSOFMAGNETICDESIGNIndesigninghighfrequencyinductorsandtransforms,adesignerisfacedwithcountlesschoices,including:Corematerials(permeability

μisdependentofmaterials)Coreshape(someofferbetterthermalconductionwhereasothersofferbettershieldingtostrayflux)Coolingmethods(naturalconvectionversusforcedcooling)Losses(lowerlossesofferhigherefficiencyattheexpenseofhighersizeandweight)31Designconsistsof:(1)Selectingappropriatecorematerial,geometry,andsize(2)Selectingappropriatecopperwindingparameters:wiretype,size,andnumberofturns.

Core(doubleE)WindingBobbinAssembledcoreand

winding42Overviewofcorematerial:(1)Iron-basedalloylaminatedcores(oftentermedmagneticsteels):comprisedofalloysprincipallyofironandsmallamountsofotherelements,including:VariouscompositionsFe-Si(fewpercentSi)Fe-Cr-MnImportantpropertieslowerResistivity=(10~100)ρCulargevaluesofsaturationfluxdensityBs=1~1.8TUsedinlow-frequencyapplications.5(2)Powderedironalloycores:consistofsmallironparticleselectricallyisolatedfromeachother.VariouscompositionsFe-Si(fewpercentSi)Fe-Cr-MnImportantpropertieshavelargereffectiveresistivitythanlaminatedcores6(3)FerritecoresVariouscompositionsIronoxidesFe-Ni-MnoxidesImportantproperties

Resistivity

ρverylarge(insulator)-noohmiclossesandhenceskineffectproblemsathighfrequencies.Bs=0.3T(T=tesla)7Corematerialscomparison83Magneticcoreshapes ?

FerritecoresavailableasU,E,andIshapesaswellaspotcoresandtoroids. ?

Laminated(conducting)materialsavailableinE,U,andIshapesaswellastapewoundtoroidsandC-shapes..insulatinglayermagneticsteellamination91011 ?

OpengeometriessuchasE-coremakeforeasierfabricationandbetterthermalconduction

butmorestrayfluxandhencepotentiallymoresevereEMIproblems.

?

ClosedgeometriessuchaspotcoresmakeformoredifficultfabricationandworsethermalconductionbutmuchlessstrayfluxandhenceEMIproblems.124Twobasicquantitiesneedbeingcalculatedindesign-optimizationproblems:Thepeakfluxdensity

Bmax

inthemagneticcore(wemostnotexceedtheallowedfluxdensityofthematerial.Bmax<Bsat)tolimitcorelosses,andThepeakcurrentdensity

Jmax

inthewindingconductors(winemustthickenoughtocarrycurrentwithoutoverheating)tolimitconductionlosses.Ingeneral,Jmax<500A/cm213inductortransformer9-4AREA-PRODUCTMETHOD☆141CoreWindowArea

Awindow

BasedonJmaxandBmax,choosecoresize.Theconductorcross-sectionalareaAcond

dependsonthemaximumrmscurrent.

SelectpeakcurrentdensityJmaxandpeakfluxdensityBmax.

15ThewindowareaislinearlyproportionaltothenumberofturnsNy,wherekwisthewindowfill-factor.Winemustfitinthewindowarea.windowfill-factor

kw

tellusthefractionofactualwindowareathatcanbefilledwithcopper.(typically,Kw=0.5,butdependsonwindingmethod)162CoreCross-SectionalArea

AcoreThecorecross-sectionalareaAcore

dependsonthepeakflux.Thecorecross-sectionalareaAcore

isinverselyproportionaltothenumbersofturns.17Forforwardconverter,kconv=DForfull-bridgeconverter,kconv=D/2183CoreArea-Product

Ap

=AcoreAwindow

TheArea-ProductAp

representstheoverallsizeofthedeviceandindependentsofthenumberofturns.☆194DesignProcedureBasedonArea-ProductAp→selectingtheappropriatecore(selectingspecificAcoreandAwindow):(thereareinfinitecombinationsofAcoreandAwindow)CalculatingAp:20→calculatingthenumbersofturns:21

Inainductor,becauseofμmcanvarywithH,Temperature,time,etc,socanL.ForveryaccurateandstablevaluesofL,weneedtodoadditionwork.→onemethod:Addanairgapofanappropriatelengthlginthepathoffluxlines.229-5DESIGNEXAMPLEOFANINDUCTOR☆I(lǐng)nthisexample,wewilldiscussthedesignofaninductorthathasaninductanceL=100μH

.Theworst-casecurrentthroughtheinductorisshowninFig.9-3,wheretheaveragecurrentI=5.0A,andthepeak-peakrippleΔI=0.75A

attheswitchingfrequencyfs=100kHz.Wewillassumethefollowingmaximumvaluesforthefluxdensityandthecurrentdensity：Bmax

=0.25T,andJmax

=6.0A/mm2

(forlargercores，thisistypicallyinarangeof3to4A/mm2).Thewindowfillfactorisassumedtobekw=0.5.23Soluction:1CalculatingAp.24FromtheMagnetics,Inc.catalog[2],wewillselectaP-typematerial,whichhasthesaturationfluxdensityof0.5Tandisquitesuitableforuseattheswitchingfrequencyof100kHz.Soluction:2selectingcorematerialtoensureBselect>Bmax25Apotcore26×16,whichisshowninFig.9-4foralaboratoryexperiment,hasthecoreAreaAcore=93.1mm2

andthewindowAreaAwindow

=39mm2.Therefore,wewillselectthiscore,whichhasanArea-ProductAp=93.1×39=3631mm4>

3583mm4.Soluction:3selectingcoreshapetoensurepracticalAp>calculatedAp.26Soluction:4calculatingN.WindingwirecrosssectionalareaAcond

=Irms/Jmax=5.0/6.0=0.83mm2.WewillusefivestrandsofAmericanWireGaugeAWG25wires[3],eachwithacross-sectionalareaof0.16mm2,inparallel.Soluction:5selectingAcond.27Soluction:6calculatinglg.289-6DESIGNEXAMPLEOFATRANSFORMERFORAFORWARDCONVERTER☆TherequiredelectricalspecificationsforthetransformerinaForwardconverterareasfollows:fs

=100kHz

andV1

=V2

=V3

=30V

.Assumethermsvalueofthecurrentineachwindingtobe2.5A

.Wewillchoosethefollowingvaluesforthisdesign:Bmax=0.25T,Jmax=5A/mm2,kw=0.5,kconv=0.5.29Soluction:1CalculatingAp.

Soluction:2selectingcoreshapetoensurepracticalAp>calculatedApSelectpotcore22×13,Acore=63.9mm2,Awindow=29.2mm2,andthereforeAp=1866mm4>1800mm4.30Soluction:3calculatingN31UsethreestrandsofAWG25wires[3],eachwithacross-sectionalareaof0.16mm2,inparallelforeachwinding.Soluction:4selectingAcond329-7Lossmechanismsinmagneticcircuits☆Thesizeofamagneticcomponentisoftendeterminedbyloss.Generally,thelossescanbedividedinto2components:windingassociatedlossandcoreassociatedloss.331WindinglossAtlowfrequency(includingdc),windinglossisjustduetothedcresistanceinthewindingandiseasytocalculate.

Pdiss=(irms)2RwireAthigherfrequency,thereareadditionaleffectwemustconsiderof:skineffectandproximityeffect.34SkineffectSkineffectisthe“self-shielding”effectofconductors:Duetoeddycurrentsgeneratedbychangesinmagneticfieldofanaccurrent,thefieldsandcurrentsmaynotpenetrateinsideaconductorathighfrequency.I(t)H(t)I(t)J(t)J(t)0Eddycurrentsraa(a)(b)(c)Time-varyingcurrenti(t)↓MagneticfieldsH(t)↓Eddycurrents35AccordingtoLenz’slaw,magneticfieldswithinthecoreinducecurrents(“eddycurrents”)toflowwithinthecore.TheeddycurrentsflowsuchthattheytendtogenerateafluxwhichopposeschangesinthecorefluxΦ(t).Theeddycurrentstendtopreventfluxfrompenetratingthecore.36EddyCurrentsIncreaseWindingLosseseddycurrentscauseanonuniformcurrentdensityintheconductor.Effectiveresistanceofconductorincreasedoverdcvalue.Forsinusoidalcurrents:currentdensityisanexponentiallydecayingfunctionofdistanceintotheconductor,withcharacteristiclengthδknownasthepenetrationdepthorskindepth.37Numericalexampleusingcopperat100°C.Frequency50Hz5kHz20kHz500kHzSkinDepth10.6mm1.06mm0.53mm0.106mmForcopperatroomtemperature:38So,ifweneedtocarryhighfrequencycurrent,wineofradius>δisnotuseful,sincethecurrentwillbecarriedonlyonthesurfaceofthewine.

☆

Thesolutiontothisproblemistoparallelisolatedwireofthickness<δoruseLitzwirebundle.Eachconductorinbundlehasadiameterlessthanaskindepth.39ProximityEffect

FurtherIncreasesWindingLossesAccurrentinaconductorinduceseddycurrentsinadjacentconductorsbyaprocesscalledtheproximityeffect.

Amulti-layerwinding,withd>>δ.Eachlayercarriesnetcurrenti(t).40Proximityeffectcausessignificantpowerlossinthewindingsofhigh-frequencytransformersandacinductors,especiallyinmulti-layerwindings.Thesolutiontominimizeproximitylossesis:☆I(lǐng)ninductors,windingscanbewithsingle-layerconstruction.Intransforms,windingscanbeinterleavedandavoidedhigh’soflayer.41Example:atwo-windingtransformerPrimaryturnsarewoundinthreelayers,assumethateachlayerisoneturn.Thesecondaryisasimilarthree-layerwinding.Eachlayercarriesnetcurrenti(t).Portionsofthewindingsthatlieoutsideofthecorewindowarenotillustrated.Eachlayerhasthicknessh>δ.42(1)Distributionofcurrentsonsurfacesofconductors:SkineffectcausescurrentstoconcentrateonsurfacesofconductorsSurfacecurrentinducesequalandoppositecurrentonadjacentconductorNetconductorcurrentisequaltoi(t)foreachlayer,sincelayersareconnectedinseriesCirculatingcurrentswithinlayersincreasewiththenumbersoflayers43(2)Estimatingproximityloss:high-frequencylimitThecurrenti(t)havingrmsvalueIisconfinedtothicknessδonthesurfaceoflayer1.Hencetheeffective“ac”resistanceoflayer1is:Rac

=(h/δ)RdcThisinducescopperlossP1

inlayer1:P1=I2Rac44PowerlossP2

inlayer2is:P2=P1+4P1=5P1PowerlossP3

inlayer3is:P3=(22+32)P1=13P145Adduplossesineachlayer:Comparewithdccopperloss:IffoilthicknesswereH=δ,thenatdceachlayerwouldproducecopperlossP1.ThecopperlossofM-layerwindingwouldbePdc=I2MRdcSotheproximityeffectincreasesthecopperlossbyafactorof:46Thesolutiontominimizeproximitylossesis:☆I(lǐng)ninductors,windingscanbewithsingle-layerconstruction.Intransforms,windingscanbeinterleavedandavoidedhigh’soflayer.47(3)Two-windingtransformerMMFdiagramWindinglayoutMMFdiagram(a)withoutproximityeffect48(b)withproximityeffect49(c)Interleavingthewindings:MMFdiagramGreatlyreducesthepeakMMF,leakageflux,andproximitylosses502Coreloss=Eddycurrentloss+hysteresislossSkineffectdB/dtthroughcoregeneratesvoltagewhichdriveeddycurrentsaroundcore,theseeddycurrentsflowsuchthattheytendtogenerateafluxwhichopposechangecoreflue!xdx-xLdwBsin(wt)xyzEddycurrentflowpath51xdx-xLdwBsin(wt)xyzEddycurrentflowpath