StrandedWireWithUninsulatedStrandsasaLow-CostAlternativetoLitzWireXuTa - PDF document

StrandedWireWithUninsulatedStrandsasaLow-CostAlternativetoLitzWireXuTangC.R.SullivanFoundinIEEEPowerElectronicsSpecialistsConference,June2003,pp.289{295.2003IEEE.Personaluseofthismaterialispermitted.However,permissiontoreprintorrepublishthismaterialforadvertisingorpromotionalpurposesorforcreatingnewcollectiveworksforresaleorredistributiontoserversorlists,ortoreuseanycopyrightedcomponentofthisworkinotherworksmustbeobtainedfromtheIEEE. StrandedWireWithUninsulatedStrandsasaLow-CostAlternativetoLitzWireXuTangandCharlesR.Sullivanxu.tang@dartmouth.educharles.r.sullivan@dartmouth.eduhttp://engineering.dartmouth.edu/inductor8000CummingsHall,DartmouthCollege,Hanover,NH03755,USATel.+1-603-646-2851Fax+1-603-646-3856—High-frequencylossintransformerwindingsusingstrandedwireisanalyzed.Acompletelosspredictionmethodispresented.Theinterstrandresistivity,whichisanimportantparametertodeterminethepowerloss,ismeasuredexperimen-tally.Theanalyticalmodelissolvedtogetanoptimalpitch,whichspeciesthedegreeoftwistingthatresultsinminimumloss.Atransformerusingastrandedwirewindingisbuiltandmeasured.Themodelpredictionisveriedtohavehighaccuracyinthefrequencyrangeupto100kHz.Comparedwiththesametransformerusingasolidwirewinding,about67percentlesspowerlossat100kHzisachievedusingstrandedwire.Usingtheloss-predictionmodelprovidedinthispaper,engineerswillbeabletocontroleddy-currentlossesinhigh-frequencytransformerandinductorwindingsusingstrandedwireatarelativelylowcostcomparedtousinglitzwire.I.INTRODUCTIONSpecialwindingconstruction,usinglitzwireorfoil,isoftennecessarytocontroleddy-currentlossesinhigh-frequencytransformerandinductorwindings.Thehighcostofthesetechniquesisamajorlimitationindevelopinghigh-power-density,high-efciencycomponentsforpowerelectronics.Amuchlower-costalternativeissimplestrandedcopperwirewithuninsulated,barestrands.Inadditiontothewirecostadvantagearisingfromavoidingtheinsulationprocess,thereisacomponent-manufacturingcostadvantagearisingfromtheeasierterminationofbarestrands.Thelosswithuninsulatedstrandswillcertainlybehigherthanintruelitzwirewithindi-viduallyinsulatedstrands.However,thehigh-frequencylosscanbesubstantiallylowerthaninsolidwire—theseparationintostrandsimpedeseddycurrents,evenifitdoesnotcompletelystopthem.Anecdotalevidencehassupportedtheideathatthiscanworkwellinsomeapplications.However,untilnow,therehasnotbeenamodelavailabletopredicttheeddy-currentlossesinstrandedwirewithuninsulatedstrands,whichwewillrefertosimplyasstrandedwirefortheremainderofthispaper.Thus,ithasnotbeenpossibleforadesignertoevaluatethislow-costalternativeanddeterminewhetheritisagoodchoiceforaparticularapplication,anditisalmostneverused.Inthispaper,wedevelopamodeltopredictlossesinstrandedwire,bycombiningstandardanalysisoflossinlitzwire[1],[2],[3],[4]withnewanalysisofthecurrentsthatcirculatebetweenuninsulatedstrands.Animportantparameterinthismodel,theresistancebetweenstrands,neededtobedeterminedexperimentally;thesemeasurementsaredescribedinSectionIII.ThemodelhasbeenveriedexperimentallyasdescribedinSectionV.ThediscrepancybetweenthepredictionandthemeasurementatveryhighfrequencywasexplainedinThisworkwassupportedinpartbytheUnitedStatesDepartmentofEnergyundergrantDE-FC36-01GO1106.SectionV-A.Theexperimentsalsoconrmperformancemuchbetterthanthatofsolidwire.Withaveriedmodelforthelossesinstrandedwire,itispossibletoquantitativelyevaluateitsperformanceinaparticularapplication.Insomecases,itsperformancecanbeverysimilartothatoftruelitzwire,whereasinothersituations,itslossesaresubstantiallyhigher.Thus,ourmodelprovidesanimportanttooltodesigners,allowingthemtotakeadvantageofthecostsavingsofuninsulatedstrandswheretheyworkwell,whileavoidingthemwheretheydonot.Becauseweprovideananalyticalmodelofloss,itispossibletouseitinanalyticaldesignoptimization.InSectionIV,wepresenttheoptimizationofoneimportantparameter:thedegreeoftwistinginabundle,asquantiedbypitch,thedistanceforonefulltwist.Tighttwisting(shorterpitch)resultsinsmallerloopslinkingacux,andthuslowercirculatingcurrentsandacloss.However,shortpitchalsoresultsinlongerlengthofeachstrand,duetothetwistedpathitmustfollow,andthisleadstoanincreaseinothertypesofeddy-currentlossandtolargerdcresistance.Ouroptimizationprovidesthepitchgivingminimumtotallossinstrandedwire.II.LOSSALCULATIONSkineffectandproximityeffectarethetwoeffectsthatmakethecurrentdensitydistributionsnon-uniformathighfrequency,andthusincreasethepowerloss.Skineffectisthetendencyforhigh-frequencycurrentstoowonthesurfaceofaconductor.Proximityeffectisthetendencyforcurrenttoowinotherundesirablepatterns—loopsorconcentrateddistributions—duetothepresenceofmagneticeldsgeneratedbynearbyconductors.Ordinarily,proximityeffectsdominateskineffectsinatransformerorinductorbecauseinamulti-layerwindingthetotalmagneticeldismuchlargerthantheeldgeneratedonlybyonestrandorturn.Inmulti-strandwindings,skinandproximityeffectsmaybefurtherdividedintostrand-levelandbundle-leveleffects,asillustratedinFig.1.Bundle-leveleffectsrelatetocurrentcirculatinginpathsinvolvingmultiplestrands,whereasstrand-leveleffectstakeplacewithinindividualstrands.Strand-levelproximityeffectmayoptionallybestillfurtherdividedintointernalproximityeffect(theeffectofothercurrentswithinthebundle)andexternalproximityeffect,butweinsteadconsiderthetotalproximityeffectasaresultofthetotaleldatanygivenstrand.Strand-leveleffectsarenotsignicantlyaffectedbythepresenceorabsenceofinsulationexceptasdiscussedinSec-tionV-A.Thus,standardlitz-wireanalysis[1],[2],[3],[4]canbeappliedtouninsulatedstrands.Predictingbundle-level ProximityEffect SkinEffectStrand-levelBundle-level intextext"Internal""External" Fig.1.Typesofeddy-currenteffectsinbundledwire.proximityeffectwithniteconductivitybetweenstrandsismorecomplex.Thepotentialbetweenanypairofstrandsinducedbychangingmagneticeldiscalculated,andthenthecurrentandlossaredeterminedbaseduponthemeasuredresistancebetweenBecausewendthatthebundle-levelproximityeffectlossesarereducedbyusingsmallerpitch,itisimportanttoincludetheeffectofpitchondcresistanceandonstrand-levelproximityeffectinouranalysis.Thecalculationsofeachtypeofloss,includingpitcheffects,areoutlinedbelow.A.DCresistanceThedistanceastrandtravelsislongerwhenitistwistedthanwhenitgoesstraight.TheeffectoftwistingonthelengthofstrandisillustratedinFig.2,whichshowsasinglecylindricalshelloflengthequaltothepitch,unwoundtoshowatonthepage.Withsimpletwisting,eachstrandwillstaywithinonesuchshellataradius,andthuswillbelongerthantheoverallbundlebyafactorof p=1 cos(µ)=p +(2¼r isthepitch,istheangleasshowninFig.2,andistheactuallengthofthestrand—thediagonalinFig.2.Thislengthincreasedirectlycorrespondstotheincreaseddcresistanceofagivenstrand.Theoveralldcresistanceofatwistedbundleistheparallelcombinationoftheresistances d 2 r Fig.2.Theeffectoftwistingonthelengthofstrand.Seetext.ofmanysuchstrands,eachatadifferentradius.Becauseofthedifferentresistanceofstandsatdifferentradii,thedccurrentwillnotbeexactlyequalforeachstrand.However,theexpressionfortotalresistanceisgreatlysimpliedbyassumingthatthedccurrentowingineachstrandisthesame.Byperformingboththesimpliedandexactcalculations,wefoundthattheapproximationofequaldcstrandcurrentsisgoodtobetterthan2%whenthepitchismorethansixtimesthediameterofthebundle.Thus,wechosetousethatsimpliedcalculationforfurtheranalysis.Becausethestrandlengthdependsonradius,theoveralldiameterorradiusofthebundlemustbeknowninordertocalculatethedcresistance.Theoverallbundlediameterdependsonthestrandpackingfactor,whichwedeneas istheoverallbundlearea(¼d)andisthesumofthecrosssectionalareasofallthestrands,witheachstrandareatakenperpendiculartothebundle,notperpendiculartothestrand.Thus,theareaofeachstrandistakenatadifferentangle,,tothestrandaxis,resultinginaellipticalarea,asshowninFig.3.Weassumethatpackingfactorconstantindependentofthepitch.However,thebundlediameterincreaseswithsmallerpitch,ascanbeseeninthisexpression,derivedinAppendixI, nd2 (1+ isthenumberofstrandsandisthediameterofeachstrand.Using(3),AppendixIderivesthedcresistance,basedonequalcurrentsineachstrand: ¼nd(1+ isthelengthofthebundleandistheresistivityofcopper.Thefactor ¼ndrepresentsresistancewithouttwisting,andtheexpressionintheparenthesesrepresentstheeffectof s,e s,c bundle axis strand axis Fig.3.Inthecrosssectionofthebundlethestrandcrosssectionbecomes ap Fig.4.Integrationloopusedtondvoltagethatinducescurrentowalongthemarkedpath.B.Strand-leveleddy-currentlossForasinglestrandoflengthplacedinasinusoidallyvaryingmagneticeld,theproximity-effectpowerlosscanbemodelledas[1],[3],[4],[5]¼! istheradianfrequencyandisthepeakuxdensity,assumedtobeconstantthroughoutthestrand.Thisassumptionisequivalenttoassumingthestrandtobesmallcomparedtoaskindepthatthefrequencyofinterest[6].Ingooddesigns,thiswillinfactbethecase.Forthecaseofatransformerwinding,varieswithposition.Thusinordertocalculatethepowerlossforthewholewinding,weusethespatialaverageof,whichwewriteas Thestrand-leveleddy-currentlossforthewholetransformerwinding,takingintoaccountoftheeffectofpitchasinthecalculationofdcresistance,canbewrittenaseddy,strand¼! ^B2d4n` (1+ Intypicaltransformerdesigns,astandardone-dimensionalmodeloftheeldissufcienttoobtaintheaveragevalueof ^B2=1 3¢Ã¹0N^I isthewidthofthewindingwindow,isthenumberofturnsandisthepeakcurrent.C.Bundle-leveleddy-currentlossInatwistedbundlewithsignicantresistancebetweenstrands,thepotentialbetweenapairofstrandscanbecalculatedasthederivativeoftheintegraloftheuxlinkedbythepathshowninFig.4.TheareaoftheloopinFig.4varieswiththebetweenthepositionswherepotentialisevaluated.Weassumethattheuxisuniformthroughoutthebundle;thattheeddycurrentsarenotlargeenoughtosignicantlyreducetheux.InSectionV-A,wediscussthesituationwhereeddycurrentislargeenoughtoreducetheux.Inagivencrosssectionthroughthebundle,differentstrandsareatdifferentpointsinthetwistcycle,correspondingtodifferentvaluesof.Thus,ifwedescribethebundlecrosssectioninpolarcoordinatesr;Á,thepotentialdifferencebetweenastrandandthestrandinthecorrespondingpositionontheoppositesideofthebundledependson(whichdetermines),andontheradius.Takingthecenterofthebundleaszeropotential,wederiveinAppendixIIanexpressionforthepotentialatanypointinagivencrosssection.r;Á)=rp !tThispotentialdrivesthecurrentsbetweenstrands.Insteadofcalculatinginterstrandcurrentsforparticularstrands,weapproximatethenetworkofdiscreteresistancesbe-tweenstrandsasacontinuousmediumdescribedbyaresistivityintheplaneperpendiculartotheaxis.Thus,wecancalculatecurrentandlossfromtheelectriceldwhichisfoundfromthegradientofpotential.Theresultingtime-averagebundle-levelproximity-effectlossiscalculatedinAppendixIItobeeddy,bundle ^B2nd2` ¼K(1+ Again,fortypicaltransformerdesigns, isgivenby(7),basedonstandardone-dimensionalanalysis[7].Wehavenowcalculatedalloftheimportantlossesinstrandedwire.Weseethatinterstrandresistivity,,andpitch,areimportantparametersdeterminingtheamountofbundle-levelproximityeffectloss.Thus,wewilladdressthesetwoparametersinmoredetailinSectionsIIIandIV.III.INTERSTRANDESISTIVITYEASUREMENTSTheresistivityofthebundleintheplaneperpendiculartotheaxis,,isanimportantparameterindeterminingthebundle-levelproximity-effectloss.Iftheresistancebetweenstrandsisknown,itispossibletocalculate.Becausewewereunabletocalculateallofthefactorsaffectingresistancebetweenstrands,weusedtheapparatusshowninFig.5todirectlymeasureresistivityofalargenumberofstrandspackedintoarectangularregion.Whenstrandsarebundled,thecontactareaincreaseswiththeincreasingpressurepackingthestrands,andsotheinterstrandresistivitydecreaseswithpressure.Toenablequantifyingthiseffect,theapparatusshowninFig.5allowsvaryingpressurethroughtheforceappliedtoaplunger.Inordertomeasureonlytheresistanceofcontactsbetweenstrands,withoutmeasuringthecontactresistancebetweentheapparatusandthestrands,interdigitatedelectrodesontopandbottomwereusedwithafour-wiremeasurementtechnique.Wendthatwhenstrandsarepackedtightly(i.e.,withhighpressure,around80kPa)theinterstrandresistivityceasestochangemuchwithpressure.Thus,theresistivityathighpressurecanbeusedfordesignwithouttheneedtodeterminetheactualpressureinagivenwindingdesign.TableIshowstheexperimentalresultsat80kPa.Thereisnoconsistentpatterntothedata;theresultsmayberelatedmoretothedegreeofoxidationorothercontaminationonthesurfaceofthewirethantothediameter.Giventheinconsistencyofthedata,itseemswisetodesignbasedontheworst-caselowestvalueof20m.However,even20misoveronethousandtimeshigherthantheresistivityofbulkcopper.Thus,substantial TABLEINTERSTRANDRESISTIVITYUNDER80APRESSUREWiresize(AWG)323844 Barecopperwire240m75m90Tinnedcopperwire50m20m110decreasesinlosscanbeexpectedevenwithnospecialeffortstoincreaseresistivity.Itisalsopossibletoestimateinterstrandresistivityfromseveraldifferenttypesofac-lossmeasurements,asdiscussedinSectionV.IV.OPTIMALITCHFrom(9),weseethatthebundle-leveleddy-currentlossdecreasesaspitchisreduced.ThisisadirectresultofthereducedsizeoftheloopinFig.4.However,theotherlosses,rmsisgivenby(4))andthestrand-leveleddy-currentlosseddy,strandgivenby(6),increaseaspitchisreduced.Thus,thetotalloss,eddy,strandeddy,bundlecanbeexpectedtohaveaminimumvalueatanintermediate,optimalpitch,.Thisvaluecanbefoundbysettingthederivativeofwithrespecttopitchequaltozeroandsolvingforpitch. ¼4½ssnd4 rms !2 Becauseofbundle-leveleddy-currentloss,theoptimalpitchistypicallysmallerthanthatfoundinstandardcommercialwire.However,thereisnotechnicalbarriertomanufacturingwirewithsmallerpitches,nowthatthereisanincentivetodoso.Withanexpressionfortheoptimalpitchin-hand,theremain-ingchoicesarethenumberanddiameterofstrands.ThisdesignproblemisthensimilartothedesignproblemforstandardlitzV.EXPERIMENTALERIFICATIONOFOSSREDICTIONToverifytheeddy-currentlosspredictedby(6)and(9),weconstructeda40-turnto40-turntransformeronanETD39 F 50 mm V 12.5 mm50 mm 12.5 mm Fig.5.Experimentalapparatusformeasuringtheinterstrandresistivity.Adjustableforcecanbeappliedsuchthattheeffectofpressureontheinterstrandresistivitycanbetakenintoaccount.Therightsideshowstheinterdigitatedelectrodes.Theseelectrodeswereplacedonthetopandbottomoftherectangularregionwherewireswerepacked.Withtheseelectrodes,contactresistancebetweenelectrodesandstrandswillnotbeincludedinthesizeferritecore.Bothwindingsusedwirewith66uninsulatedstrandsof40AWG(80mdiameter)tinnedcopper,twistedwithapitch=8mm.Theoverallbundleisinsulatedwithathermoplasticcoating.Toreducecapacitanceanddielectriclosses,andtofacilitatehigh-frequencymeasurements,thickpolypropylenetapewasusedbetweenwindinglayers.Mea-surementswereperformedwiththetwowindingsconnectedinseriesopposition,toensureperfectlybalancedleakageex-citation[8].Withthepolypropylenetape,theself-resonantfrequencyinthiscongurationwas15.8MHz,wellabovethefrequencyrangeof1kHzto500kHzthatwemeasured.Fig.6showsthemeasuredacresistancefactor,alongwithtwotheoreticalcurvesbasedonourmodelwithdifferentvaluesofinterstrandresistivity:theworst-casevalueof20andthevaluethatgivesthebestttothemeasureddatainthelow-frequencyregion,25m.Wechoseavaluetotthemeasurementsinthelow-frequencyregionbecausethedeviationinthehigh-frequencyregionislikelyduetoself-shieldingofthebundle.Wewilldiscusstheself-shieldingeffectinmoredetailinsectionV-A.AlsoinFig.6isthemeasuredacresistancefactorofasingle-strand22AWG(0.64mmdiameter)winding.Thediameterofthesinglestrandisslightlysmallerthantheoverallstrandedbundle(0.7mmdiameter),butitstillhasabout10%lowerdcresistance.Consideringdcresistanceandacresistancefactor,thenetacresistanceofthetwodesignsisequalat15kHz;abovethisfrequencytheperformanceofthestranded-wirewindingissuperior,typicallybyalargemargin.Notethatthisiswithoutusingtheoptimalpitch,ascalculatedinSectionIV.At100kHz,forexample,theoptimalpitchwouldbe4.6mminsteadoftheactual8mm,andwoulddecreasetheoveralllossby8%;at300kHz,theoptimalpitchis2.9mm,andiscalculatedtoreducetheoveralllossby43%. 10 100 500 1 2 3 4 5 6 7 8 9 10 Frequency (kHz)Fr Measured stranded wire Fig.6.Experimentalacresistancefactorforatransformerwoundwithstrandedwirewith66uninsulated40AWGstrands,orwith22AWGsolidwire.Comparedwithasolid-wirewinding,astranded-wirewindingprovidesagreatlossreduction.Ourmodelpredictsthelossinstrandedwirewithhighaccuracyupto100kHzandreducedaccuracyupto500kHz. V Stranded wireunder test goes here Fig.7.Experimentalapparatususedtomeasureproximityeffecteddy-currentlossinstrandedwire.Twowindingsareused,asinastandardcore-lossmeasurement,suchthatthemeasurement(performedwithanimpedanceanalyzer)doesnotincludelossinthesewindings.However,themeasurementdoesincludeeddy-currentlossinducedinawiresampleplacedinthegap.Thewindingshaveatotaloftenturnseach,veoneachcore-half(FerroxcubeU93/76/30sizeof3C85ferrite).A.Self-shieldingeffectThebundle-levelself-shieldingeffectoccurswhenthemag-neticeldgeneratedbythebundle-leveleddycurrentislargeenoughtoreducetheoriginalmagneticeldwhichinducestheeddycurrent.Thusthemagneticeldinthecenterofthebundleisreduced.AsdenedandcalculatedinAppendixIII,theonsetfrequencyoftheself-shieldingeffectis¼½ Tolookattheself-shieldingeffectinmoredetail,wesetupameasurementapparatusasshowninFig.7todirectlymeasureproximity-effectloss.Withthisapparatusweusefourterminalimpedancemeasurementtodetectlossesinthisgappedtransformer.Whenthereisnowireinthegap,itdetectsthecoreloss;whenweputwireintheairgap,eddycurrentlossisinducedinthewireandthislosswillsimplyaddtothemeasuredloss.Todeterminethelossinthewire,wesubtractthecorelossfromthetotalloss.Supposethereisasinusoidalcurrentwithrmsamplitudeowingintheprimarywinding.Withthesecondarywindingopen,theuxdensityintheairgapis 2I isthelengthofairgap.Thusthepowerlossinthewirewillbewire=(1+ ¼! 128½c+p2!2^B2nd2` ¼KSubstituting(12)into(13)anddividingbothsidesof(13)by=(1+ ¼! 128½c+p2!2nd2` ¼K isthemeasuredincreaseintherealpartofimpedancewhenthewireisinsertedintothegap.Thusinthemeasurement,wedonotneedtoknowthecurrentowingintheprimarywinding. 10 100 1000 0.0001 0.001 0.01 0.1 1 10 100 Frequency (kHz)Rpe (W) measured data Fig.8.Measurementofproximity-effectlossinwirewith65strandsofAWG30tinnedcopperwire,usingtheapparatusinFig.7.Theverticalaxis()isthemeasuredincreaseintherealpartofimpedancewhenthewireisinsertedintothegap.Thedecreasedslopeofthemeasuredlossathighfrequenciesisbelievedtobeduetotheself-shieldingeffectofbundle-leveleddycurrents.Thecalculatedonsetfrequencyofthiseffectis237kHz.Toestimatetheexperimentalonsetfrequency,thedottedlinewithslope0.5isdrawntangenttothemeasureddata.Itcrossesthelow-frequencyassymptoteatabout130kHz,indicatingaloweronsetfrequencythancalculated,butitisdifculttodeterminethisaccuratelybecausetheaccuracyofthemeasurementisdegradedathighfrequenciesbyaresonanaceintheapparatusaround800kHz.Parametersforthestrandedwiretestedare=32=25=100TwowiresamplesweremeasuredandthedataareshowninFig.8andFig.9.Inbothgures,thepredictedresistanceisplottedalongwithbundle-levelandstrand-levelresistances.Foreachwiresample,ainterstrandresistivity,,ischosentotthemeasureddata.Using(11),wecalculatetheonsetfrequenciesoftheself-shieldingeffecttobe237kHzand625kHzforthetwosamples.Wewishtocomparetheexperimentalresultwiththiscal-culation.However,itisdifculttoclearlyidentifytheonsetfrequencyintheexperimentaldata.Weexpectthat,asinsolidwire,theacresistancewillbeproportionaltothesquarerootoffrequencyinthehigh-frequencyregionwellabovetheonsetofself-shielding.InFigs.8and9thishigh-frequencyasymptoteisrepresentedbyadottedlinewithaslopeof0.5,chosentobetangenttothemeasuredcurve.Theintersectionwiththelow-frequencypredictedresistancelinethendenesthemea-suredonsetfrequency.Unfortunately,theresonantfrequencyoftheapparatusoccursneartheonsetfrequency,atabout800kHz.Thus,thedatadoesnotincludealargeenoughportionwithslopeof0.5toallowanaccuratedeterminationofthisasymptote,andourexperimentalmeasurmentsareonlyroughestimatesoftheonsetfrequency.ThisdifcultyisparticularlysevereinFig.9whichhasthehigheronsetfrequency.Despitetheselimitations,theexperimentisausefulconr-mationofthetrendsinonsetfrequency,anditconrmsthepredictedorderofmagnitudeofthefrequencyatwhichtheself-shieldingeffectbecomessignicant:Forthewirewith65strands,themeasuredonsetfrequency(aswehavedeneditgraphically)isabout130kHz,whileforthewirewith41strands,itisabout300kHz.Bothmeasurementsareaboutafactoroftwobelowthecalculatedonsetfrequencies. 10 100 1000 0.0001 0.001 0.01 0.1 1 10 Frequency (kHz)Rpe (W) measured data Fig.9.Measurementofproximity-effectlossinwire,similartoFig.8,butwith41strandsofAWG30tinnedcopperwire.Again,thedecreasedslopeofthemeasuredlossathighfrequenciesisbelievedtobeduetotheself-shieldingeffectofbundle-leveleddycurrents.Thecalculatedonsetfrequencyofthiseffectis625kHz,andtheestimatedexperimentalonsetfrequencyisabout300kHz.However,itisverydifculttodeterminethispointaccuratelybecauseofitsproximitytothe800kHzresonanceoftheapparatus.Parametersforthestrandedwiretestedare=20=25=100VI.CONCLUSIONStrandedwirecanbeausefullow-costalternativetohigh-costlitzwire.Boththecostofinsulatingindividualstrands,andthecostofterminatingthelitzwirecanbeavoidedordecreased.Inordertomaketheuseofstrandedwireaviablealternative,itisnecessarytobeabletopredicthigh-frequencylossesinit.Theselossesincludethesamestrand-leveleddy-currentandresistivelossesasinlitzwire,plusbundle-leveleddy-currenteffectsarisingfromtheniteinterstrandresistivity.Wehavepresentedananalysisofthisloss,plusimprovementstomodelsofstrand-leveleddy-currentandresistivelossestopreciselyaccountfortheeffectofpitch.Importantparametersindeterminingtheperformanceofstrandedwirearetheresistancebetweenstrandsandthepitch.Interstrandresistivityhasbeencharacterized,andaworst-casevalueproposedforuseindesign.Anoptimalpitchhasbeenfoundforminimumtotalloss.Experimentalmeasurementsverifythemodelandshowdra-maticadvantagesovertheperformanceofasolid-wirewindingwithoutthehighcostoflitzwire.Discrepanciesbetweenpredictionandmeasurementatveryhighfrequencyduetothebundleself-shieldingeffecthavebeendiscussed.Abovetheself-shieldingonsetfrequencywherethelosscalculationisnotaccurate,itoverestimatesactuallossessuchthatdesignsbasedonourmodelwillbeconservative.Withthemodelsanddataweprovide,designerscannowtakeadvantageofthisopportunityforlargesavingsincostandloss.PPENDIXALCULATIONOFDCRESISTANCEThediameterofthebundlewillincreasewithtwisting.Asdenedin(2),isthepackingfactorandisassumedtobeconstantindependentofpitch.Nowweconsiderthesituationwhenabundleofstrandsistwisted.Inthebundlecrosssection,eachstrandareaiselliptical,atanangle,,tothestrand,showninFig.3.Notethatatdifferentradii,hasdifferentvalues.Fromtheellipticalarea,wecancalculatecrosssectionalareaperpendiculartothestrand.isthecrosssectionalareaofthestrandperpendiculartothestrandandisthecrosssectionalareaofthestrandperpendiculartothebundle.Asisindependentofpitch, istheoverallbundleareawhenthereisnotwisting.Thetotalcrosssectionalareaofcopperperpendiculartoeachstrandis¼d isthebundlediameterwithouttwisting.Inatwistedbundle,canbecalculatedasAndthiscanbeapproximatedas ¼rdrCombining(17)and(19),wegetthebundlediameterwith (1+ Substitutingin nd2 resultsintheequationforbundlediameter(3).TheDCpowerlossofasinglestrandis 1 ¼disthermscurrentineachstrand.Inthecrosssectionofatwistedbundle,DCpowerlossperunitareais As;e Ka=16KaI2s½c` WeintegrateoverthebundletogetthetotalDCpowerloss ¼rdr(1+ isDClossofthebundlewithouttwisting.NowitiseasytocalculatetheDCresistanceofthetwistedbundle ¼nd(1+ 4Kap2)(25)where4½c` ¼ndrepresentsresistancewithouttwisting. PPENDIXIIALCULATIONOFUNDLELEVELDDYCURRENTOSSFig.10showstheintegrationloop(markedwitharrows)usedtondvoltagethatinducescurrentowbetweenstrands,asinFig.4,butwithcoordinates.TheareaoftheloopinFig.10varieswithdistancebetweenthepositionswherepotentialisevaluated. zB x a x,z Fig.10.Integrationloopusedtondvoltagethatinducescurrentowalongthemarkedpath.Thecoordinatesofthemarkedpointontheloop()canbeexpressedas Thentheareaoftheloopprojectedonthexzplaneisloop=4 rp ¼a Thus,ifwedescribethebundlecrosssectioninpolarcoordi-nates(),thepotentialdifferencebetweenastrandandthestrandinthecorrespondingpositionontheoppositesideofthebundledependson(whichdetermines),andontheradius.Thetimevaryinguxdensityis!tAccordingtoFaraday'sLaw,thepotentialcanbefoundfromthederivativeofuxwithrespecttotime.Takingthecenterofthebundleaszeropotential,thepotentialatthepoint()isr;Á)= BA rp !tTheelectriceldcanbefoundfromthegradientofpotential.r;Á)= !tt^rsinÁ+^ÁcosÁ](30)Insteadofcalculatinginterstrandcurrentsforparticularstrands,weapproximatethenetworkofdiscreteresistancesbetweenstrandsasacontinuousmediumdescribedbyaresistivityintheplaneperpendiculartotheaxis.Thuswecalculatethelossfromtheelectriceld)= ½ssdÀ=p2!2^B2nd2` ¼K(1+ )sin!tTakingthetime-averageof(31)andusing(7),resultsinanexpressionforbundle-levelproximity-effectloss(9).PPENDIXIIIALCULATIONOFELF-SHIELDINGFFECTNSETREQUENCYTheself-shieldingeffectoccurswhenthemagneticeldgeneratedbytheeddycurrentislargeenoughtoreducetheoriginalmagneticeldwhichinducestheeddycurrent.Thefullcalculationoftheeldasacombinationofappliedeldandeddyeldiscomplicated.Tomakethecalculationsimpler,weassumethattheeddycurrentisonlyduetotheappliedeldandisnotaffectedbyeddycurrent.Thuswedenetheonsetfrequencyasthefrequencyatwhichtheeldduetoeddycurrentinthecenterofthebundlematchestheappliedeld.Havingtheelectriceldinagivenbundlecrosssection(30),wecancalculatethevolumecurrentdensity.r;Á)=r;Á Andthesurfacecurrentdensityataspecicangleis)= Onlytheaxialcomponentofthesurfacecurrentwillcontributetoeddyeldtoreducetheappliedeld.)=)cos(Nowwecanintegrateoverthecircletogettheuxdensityduetoeddycurrentinthecenterofthebundle. p!r ¼½!tSettingthemagnitudeofthisuxdensityequaltothemagnitudeofapplieduxdensity,wegettheonsetfrequencyoftheself-shieldingeffect.¼½ EFERENCES[1]P.N.Murgatroyd,“Calculationofproximitylossesinmultistrandedconductorbunches”,IEEProceedings,PartA,vol.36,no.3,pp.115–120,1989.[2]J.A.Ferreira,“Analyticalcomputationofacresistanceofroundandrectan-gularlitzwirewindings”,IEEProceedings-BElectricPowerApplicationsvol.139,no.1,pp.21–25,Jan.1992.[3]B.B.Austin,“Theeffectiveresistanceofinductancecoilsatradiofrequency”,TheWirelessEngineer,vol.11,pp.12–16,Jan.1934,SummaryofworkbyS.Butterworth.[4]C.R.Sullivan,“Optimalchoicefornumberofstrandsinalitz-wiretransformerwinding”,IEEETransactionsonPowerElectronics,vol.14,no.2,pp.283–291,1999.[5]E.C.Snelling,SoftFerrites,PropertiesandApplications,Butterworths,secondedition,1988.[6]XiNanandC.R.Sullivan,“Animprovedcalculationofproximity-effectlossinhigh-frequencywindingsofroundconductors”,34thAnnualIEEEPowerElectronicsSpecialistsConference,2003.[7]R.W.EricksonandD.Maksimovic,FundamentalsofPowerElectronicsKluwerAcademicPublishers,secondedition,2001.[8]J.G.Hayes,N.O'Donovan,M.G.Egan,andT.O'Donnell,“Inductancecharacterizationofhigh-leakagetransformers”,EighteenthAnnualIEEEvol.2,pp.1150–1156,2003.