XiNanC.R.Sullivan
FoundinIEEEPowerElectronicsSpecialistsConference,June2004,pp.873–879.
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2004 35th Annual IEEE Power Electronics Specialists ConferenceAachen, Germany, 2004
SimplifiedHigh-AccuracyCalculationofEddy-CurrentLossinRound-WireWindings
XiNan
ThayerSchoolofEngineeringDartmouthCollege,USAEmail:xi.nan@dartmouth.edu
CharlesR.Sullivan
ThayerSchoolofEngineeringDartmouthCollege,USA
Email:CharlesR.Sullivan@dartmouth.edu
Abstract—Ithasrecentlybeenshownthatthemostcommonlyusedmethodsforcalculatinghigh-frequencyeddy-currentlossinround-wirewindingscanhavesubstantialerror,exceeding60%.Previousworkincludesaformulabasedonaparametricsetoffinite-elementanalysis(FEA)simulationsthatgivesproximity-effectlossforalargerangeoffrequencies,usingtheparametersfromalookuptablebasedonwindinggeometry.Weimprovetheformulabydecreasingthenumberofparametersintheformulaandalso,moreimportantly,byusingsimplefunctionstogettheparametersfromwindinggeometrysuchthatalargelookuptableisnotneeded.Thefunctionwepresentisexactinthelowfrequencylimit(diametermuchsmallerthanskindepth)andhaserrorlessthan4%athigherfrequencies.
Wemakeournewmodelcompletebyexaminingthefieldexpressionneededtogetthetotalproximity-effectlossandbyincludingtheskin-effectloss.Wealsopresentexperimentalresultsconfirmingthevalidityofthemodelanditssuperioritytostandardmethods.
HhvOdd SymmetryEven SymmetryRegion for FEA simulationI.INTRODUCTION
Inthedesignandoptimizationofmagneticcomponentssuchasinductorsandtransformersusedinpowerelectronicsapplications,accuratepredictionofhigh-frequencywindinglossisveryimportant.Eddy-currentwindingloss,whichincludesskin-effectlossandproximity-effectloss,increasesrapidlywithfrequency.Duetothecomplexityofwindinggeometriesandinteractionsbetweenconductorsinwindings,itisdifficulttofindageneralanalyticalsolutionfortheeddy-currentlossesinwindings.Severalmethodshavebeenusedtopredicthigh-frequencywindinglossesinwindingsofroundconductorsasreviewedin[1].Onetypeofthesemethods[2],[3],[4],[5],[6],oftencalledtheDowellmethod,istousetheanalyticalsolutionforafoilconductorasanequivalenttoroundconductorsinthesamelayerwiththesametotalcross-sectionalarea.AnothertypeofmethodiscalledtheFerreiramethodortheBessel-functionmethod[7],[8],[9],[10],[11],whichistousetheanalyticalfieldsolutionofasingleisolatedroundconductorwhichissubjectedtoanexternaluniformfield.
BoththeDowellmethodandtheBessel-functionmethodcanhavelargeerror(upto60%to150%)athighfrequencies[12].Anotherkindofapproachusedtocalculateeddy-currentlossistoemploynumericalmethodssuchasfiniteelementanalysis(FEA)tofindthefieldsolutions.ThroughFEA,itispossibletofindthelossforanygivenconfigurationtoanydesireddegreeofaccuracy,thoughitmaybeverytime-consumingandonesolutioncanonlybeappliedtoonecertainconfiguration.Severalapproaches[13],[14]havebeenusedtoovercomethelimitationsofdirectnumericalmethodsasdiscussedin[12].
0-7803-8399-0/04/$20.00 ©2004 IEEE.
Fig.1.FEAsimulationconfigurationforarectangularwindingofroundconductors.Thepowerlossintheshadedarearepresentshalftheproximity-effectlossperunitlengthineachturnofawindingwithinterwiredistancevandinterlayerdistanceh.
Tofindthebehaviorofaroundwireinawinding,[12]usedFEAforasinglewirewithsymmetryboundaryconditionsasshowninFig.1.Lossinthesimulationregioncanrepresentproximity-effectlossinhalfofaroundconductorinawindingwithinterlayerdistancehandinterwiredistancev.Thesetupoftheboundaryconditionswasdiscussedindetailin[12].Reference[12]collecteddataofpowerlossandfieldsolutionsforarangeofwirespacingsintwodirectionsandforratiosofwirediametertoskindepthrangingfrom0.6to60,andshowedthattheproximity-effectlossfactor,whichisproximity-effectlossinaconductornormalizedbythesquareoftheexternalfield,notonlyincreaseswithfrequency,butalsodependsontheinterwiredistanceinalayerandtheinterlayerdistance.Basedontheseresults,[12]givesafunctionthatapproximatesthesimulationresultsmuchbetter(errorlessthan2%),andprovidesatableinwhichtheparametersofthefunctioncanbelookedupaccordingtothewirespacings.Inthispaper,weimproveonthemodelprovidedin[12].Asin[12],thenewmodelgivesamoreaccuratelosspredictionthantheDowellmethodortheBessel-functionmethod,andworksforalargerangeoffrequenciesandforanywiresizeandwindinggeometryusedinpractice.However,thefunctionshavebeensimplifiedandadjustedtogiveexactresultsmatch-inganalyticalsolutionsinthelow-frequencylimit,andthenewmodeldoesnotrequirealargetableofparametersasin[12],butinsteadusessimplefunctionstogettheparametersfromwindinggeometry.Inadditiontopresentingthisimprovedmodelforproximity-effectlossfactor,weexaminethefieldexpressionneededtogetthetotalproximity-effectloss,discuss
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theinclusionoftheskin-effectloss,andshowthatusingasimplemodelforskin-effectlossgivesonlysmallerrors.Thus,weareabletoprovideacompletemodelforcalculatingthewindinglossesforroundconductorsconfiguredasshowninFig.1.Wealsopresentexperimentalresultsconfirmingthevalidityofthemodelanditssuperioritytostandardmethods.SectionIIdiscussesindetailhowwedecomposeeddy-currentlossintoskin-effectlossandproximity-effectloss.InSectionIII,themodelforcalculatingproximity-effectlossfactorispresentedanditsaccuracyisanalyzed.SectionIVshowshowtocalculatethetotal‘proximityfield’inawinding,whichisanotherimportantfactorthatdeterminesthetotalproximity-effectlossbesidesproximity-effectlossfactor.InSectionV,wediscussthecalculationofskin-effectloss.ExperimentalresultsaregiveninSectionVIthatprovethevalidityofourmodel.
II.DEFINITIONSANDDECOMPOSITIONOFSKIN-EFFECT
LOSSANDPROXIMITY-EFFECTLOSSWindinglossathighfrequenciesiscausedbyeddy-currenteffects.Generally,eddy-currenteffectsaredividedintoskineffectandproximityeffect.Theclassicaldefinitionofskin-effectlossistheextraAClossinasingleisolatedconductorwhichiscarryingatime-varyingcurrent.Andthecorrespond-ingdefinitionofproximity-effectlossinawindingisdefinedasthetotaleddy-currentlossminustheclassicalskin-effectloss.
Theclassicaldefinitionsofskin-effectlossandproximity-effectlosscanhelpusbetterunderstandthebehaviorofaconductorathighfrequencies.However,theydon’thelpmuchinsolvingtheeddy-currentproblem.Tocalculatetheproximity-effectlossbasedontheclassicaldefinition,onehastoknowthefielddistributionandcurrentdistributionbeforehand,whichisalmostimpossible.Todecomposelossintotwopartsthatareeasytocalculateandavoidanalyzingthelocalproximityfieldineachconductor,weuseadifferentdefinitionofproximity-effectloss.Wedefineproximity-effectlossinawindingasthelossduetotheexternalfieldappliedonamatrixofwire(thewinding),andthecorrespondingdefinitionofskin-effectlossisthetotaleddy-currentlossminusthatproximity-effectloss,whichisequaltothelossinaconductorcarryingtime-varyingnetcurrentandsubjectedtospecifiedboundaryconditionsasshowninFig.2.
Fig.2showsindetailhowthetotalcurrent(includingeddycurrent)inawindingisdecomposedintoskin-effectcurrentinAandproximity-effectcurrentinBbyourdefinitionsandalsohowthefieldonthewindingisdecomposedcorrespondingtothecurrentdecomposition.Thefactthatthecurrentsincon-figurationAandBadduptocurrentinCdoesn’tnecessarilymeanthatthelossinAandthelossinBadduptolossinC.ThetotallossescanbeobtainedbyaddinglossesinAandBonlyifwecanprovethatorthogonalityexistsbetweenskineffectandproximityeffectinFig.2.
Reference[7]discussedconditionsunderwhichorthogo-nalityisvalid.Asufficient(butnotnecessary)conditionforthesumofthelossinAandthelossinBtobeidenticaltothetotallossisthattheconductorhasanaxisofsymmetryingeometryandthecurrentdistributioninAhasanodd
Symmetry axisCurrent: Even symmetrySymmetry axisCurrent: Odd symmetrySymmetry axisH1H2…..Layer analyzed-H0H0HH….….Total Loss in one layerBoundary conditionsFig.2.Decompositionoftotallossintoskin-effectlossandproximity-effectlossinawinding.
symmetryaboutthisaxiswhilethecurrentdistributioninBhasanevensymmetry,orviceversa.ItisstraightforwardthattheconfigurationandcurrentdistributionsinAandBinFig.2satisfythesetwoconditions.Thusthetotaleddy-currentlossinoneconductorinCwillbe:
Ptotal,ac=Pproximity−effect+Pskin−effect
Pproximity−effect
ˆ2GH=
σ
(1)(2)
ˆisthenormalizedunitlessproximity-effectlossfactorwhereG
asin[12]:theproximity-effectlossperunitlengthinaconductorinawinding,normalizedbytheexternalfieldwhichthewindingsubjectedtoandbytheconductivity,andHisthemagnitudeofexternalfieldorso-called‘proximityfield’thewindingissubjectedto.FromFig.2,wecanseethatthetotalfieldontheconductorcanbedecomposedintoauniform‘externalfield’componentHplusH0or−H0causedbynetcurrentintheconductor,whereHistheexternalfieldusedtocalculateproximity-effectlossandistheaverageofthefieldsoneachsideofthewindingH1andH2:
H1+H2
(3)
2
Similarfielddecompositionmethodandorthogonalitybetweenskineffectandproximityeffectalsoapplytofoil-conductorwindings.Intheappendix,weshowthattheanalysisofwindinglossbasedonorthogonalityleadstothesameresultastheanalyticalsolutionoffieldandlossinfoil-conductorwindings.
H=
III.MODELFORCALCULATINGPROXIMITY-EFFECT
LOSSESINAWINDINGIn[12],weperformed4000simulationsforvariouswindinggeometriesandvariousfrequenciesandthendidcurvefittingˆbasedonthesimulationdata.TheresultsweregiveninofG
theformofafunctionwhoseparametersaredeterminedby
…..CASkin-effect lossWire carrying currentI/l: net current per height.BProximity-effect lossWire carrying no net currentH=H1+H22H2−H1=I/l=2H0874
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thewindinggeometryandcanbelookedupinatable.Ouraimhereistocreateasimilarfunctionbasedonthesamedatafrom2-Dsimulationsbuttoavoidtheunwieldytableneededin[12]byprovidingafunctiontocalculateparametersdirectlyfromthegeometry.
Thenewfunctionalformofourmodeloftheproximity-ˆistheweightedaverageoftwodifferenteffectfactorG
functions.OneiscalledthemodifiedDowellfunction:
ˆ1(X)=3πk−3Xsinh(kX)−sin(kX)G
16cosh(kX)+cos(kX)
(4)
w,kandbversush/dandv/d.Bylookingatplotsofkand
b,wefoundthatthemodel
s1−s2
f(Y,s1,s2,q)=−1+s2(8)
Y+q−1wouldbeabletodescribeaccuratelyhowbandwchangewith
normalizedinterwiredistancev/dandnormalizedinterlayerdistanceh/d,whereYistheinput—wirespacingv/dorh/d—whereass1,s2andqareparameters.Itisasimplifiedformofthedual-slopefunctionwheres1ands2decidethetwoslopes,andqdefineswherethecurvetransitionsbetweenthesetwoslopes.
ˆshowsanSimulationdataofproximity-effectlossfactorG
overshootatthetransitions.ThisovershootisexhibitedbythemodifiedDowellfunctionanddoesnotexistinthedual-slopefunction.Thus,weightingwdeterminestheextentofovershootin(7).Amodelforthevariationofwforvariouscombinationsofv/dandh/dcanbeconstructedusingcurvesoftheform
w(Y)=c1±(u1−u0e−Y0)2.
Y
wherekisafunctionofwirespacingsv/dandh/dandXisdefinedas:
d
X==dπσµf(5)
δwheredisthediameteroftheconductor,σistheconductivity,µisthepermeabilityoftheconductormaterial,andfisthefrequency.
Theotherfunctioniscalledthedual-slopefunction:
Xˆ2(X)=πG
32(X−3+b3)
(6)
(9)
Thedual-slopefunctionwasdiscussedindetailin[12].
Thefactorsof3π/16in(4)andπ/32in(6)ensurethat
πd4
(δ)atverylowbothofthemgivetheexactsolution32frequenciesd/δ1,whichisderivedfromtheBessel-functionmethod’ssolution.Atlowfrequencies,theTaylor
sinh(kX)−sin(kX)1333
expansionofcosh(kX)+cos(kX)is6xk.Thekfactorin
ˆ1thisexpansioncancelsthek−3factorin(4),makingG
independentofkatlowfrequencies.Sincekistheonly
ˆ1isparameterin(4)whichisrelatedtowirespacings,G
independentoftheturnspacing,whichmakesphysicalsense,becauseatlowfrequencies,thefieldcausedbyeddycurrentinnearbyconductorsisverysmallandnegligible.Comparedtothefunctionformsin[12],(4)and(6)aresimplerandensuretheaccuracyofthemodelinthelow-frequencyrangewhend/δismuchsmallerthanone.
Asshownin[12],the(6)providesabetterfitforsomege-ometries,whereas(4)providesabetterfitforothergeometries.Toallowfittingdatawitheithershape,oranyintermediateshape,weusedaweightedaverageofthetwofunctions(4)and(6),withweightingw:
ˆ=(1−w)Gˆ1(X)+wGˆ2(X)G
(7)
Theshapeofthew(Y)curveislikeaparaboliccurvewitha
smallinputvalueYandturnsintoaconstantwithlargevaluesofY.Severalsuchcurvesarecombinedwithaweightingdeterminedbyh/d.
Second,wedidcurve-fittingfork,bandw.Forexample,tofindtheparametersforthemodelofk(v/d,h/d),wefirstfound10setsofs1,s2andqfor(8)fordifferentvaluesofh/d,andthenchosethecurve-fittingfunctionss1(h/d),s2(h/d)andTp(h/d).Aftertheinitialfunctionformsandparametervaluesarefoundfork,bandw,weadjustalltheparametersofk(v/d,h/d),b(v/d,h/d)andw(v/d,h/d)simultaneouslytomake(7)fitthesimulationdatabest.Finally,thefunctionsforb,kandwcanbegivenas:
b(v/d,h/d)=fv/d,f(h/d,s1b,1,s2b,1,qb,1),
f(h/d,s1b,2,s2b,2,qb,2),f(h/d,s1b,3,s2b,3,qb,3)(10)
k(v/d,h/d)=fh/d,f(v/d,s1k,1,s2k,1,qk,1),
f(v/d,s1k,2,s2k,2,qk,2),f(v/d,s1k,3,s2k,3,qk,3)(11)w(v/d,h/d)=(h/d)w1(v/d)+w2(v/d)w1(v/d)=c11−(u11−u01e−Y01)2w2(v/d)=c21+(u21−u02e
−Yv/d02v/d
Byfitting(7)tothe100setsofdata(ineachsetofdatad/δsweepsfrom0.6to60with40samplesevenlydistributedonalogscale),weobtained100setsofw,k,andbvalues,definingcurveswhichfitthedatafrom2-DFEAsimulationsmuchbetterthanoriginalDowellmodel.Eachsetofvaluescorrespondstoadifferentv/dandh/d.
Tousetheresultsofcurve-fittingandtoavoidalargetableasprovidedin[12],westudieddifferentpossiblecurve-fitfunctionsthatwouldgivevaluesofw,kandbbasedonthevaluesofv/dandh/d.
Inordertofindw(h/d,v/d),k(h/d,v/d)andb(h/d,v/d),first,wechosetheappropriatemodelsforthecurve-fittingof
)2
(12)
Theparametersin(10),(11)and(12)areinTable.I.
Theerrorofourmodel((4),(6),(7),(8)and(10)–(12))iswithin4%intherangeoffrequencyuptod/δ=60,v/dfrom0.02to1.40,andh/dfrom0.02to1.90,comparedtoFEAresultsbothfromtheoriginal4000simulationsonwhichourcurve-fittingisbased,andalsoanother800simulationswedidfordifferentinterpolationvaluesofh/d,v/dandfinthesamerange.
Fig.3comparesmaximumerrorforanygeometryatacertainfrequencygivenbyeachofthreemodels:ourmodel,theDowellmethod,andtheBessel-functionmethod.Errorsare
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TABLEI
PARAMETERSFORb,kANDwTOBEUSEDIN(10),(11)AND(12)
s1b,j−0.00371.81670.7053s1k,j1.02610.47320.0930u11=0.1558u21=0.1912s2b,j0.04320.00740.8378s2k,j0.81490.80230.2588u01=0.3477u02=0.2045qb,j−0.06610.219523.8755qk,j9.39181.2225−0.0334Y01=1.0673Y02=1.3839100b
Maximum error (%)j=1j=2j=3j=1j=2j=3=0.0596=0.001851k
c11c21
0.1w
Below the simulation error 0.1% Our modelDowell methodBessel−function method0.11d/δ101000.01relativetothesimulationresultswhichhavelessthan0.1%error.Wecanseethatatfrequencieswhered/δislargerthanone,boththeBesselfunctionmethodandtheDowellmethodcangiveverylargeerror,from60%to120%,whileourmodelonlygiveserrorlessthan4%.Atlowfrequencies,theDowellmethod’serrorgoesto4.7%,whileboththeBessel-functionmethodandourmodelgiveerrormuchsmallerthanthat.Atd/δ1,theproximity-effectfactorsgivenbyBessel-functionmethodandourmodelconverge,andtheirerrorisbelowthesimulationerrorof0.1%.
AcontourplotofmaximumerrorswithvariousgeometriesisshowninFig.4.FromFig.4,wecanseethatthelargesterrorofourmodel—exceeding3%—happensingeometrieswheretheinterlayerdistancehissmallandtheinterwiredistancevislarge,whichisararesituationinpracticaldesign—errortheredoesn’tmattermuch.Thelargesterroralsohappenswhenbothvandhareverylarge.Basedon2-Dsimulationresultsshownin[12],proximity-effectfactorincreaseswiththeincreaseofv/dandthedecreaseofh/d.Tominimizetheproximity-effectloss,generallywewouldliketohavesmallv/dandlargeh/d.Fig.4showsthatintheregionofsmallv/dandlargeh/d,ourmodelgiveserrorsmallerthan3%.
IV.FIELDCALCULATIONFORATRANSFORMER
AsdiscussedinSectionII,thefieldmagnitudeHusedforcalculatingproximity-effectlossistheaverageofthefieldsoneachsideoftheconductor.InthissectionwewilldiscusshowtocalculateHforasimpletransformerwinding.However,the
ˆisindependentoftheoverallproximity-effectlossfactorG
fieldandcanbeusedbroadlyinvariousfieldshapes.
Inthepthlayerofasimplelayer-woundtransformerwindingofmlayers,themagnitudeofexternalfieldis:(2p−1)NmI
(13)
2bw
wherebwisthebreathofwindingwindow,Nisthetotalnumberofturns,andIisthepeakcurrentcarriedbyeachturn.
Fortheconvenienceofusing(13)in(1),wecanobtaintheaverageofsquareofthefieldinmlayers:
H=H2=
1(NI)21
(1−)
3bw24m2(14)
Fig.3.Thisplotcomparesthemaximumerrorforanygeometryata
certainfrequencygivenbyeachofthethreemodels:ourmodel,theDowellmethod,andtheBessel-functionmethod.Atlowfrequencies,theDowellmethod’serrorgoesto4.7%,whileboththeBessel-functionmethodgiveerrormuchsmallerthanthat.Atd/δ1,theproximity-effectfactorsgivenbyBessel-functionmethodandourmodelconverge,andtheirerrorisbelowthesimulationerrorof0.1%.Athigherfrequencies,ourmodelalwaysgivesamoreaccuratepredictionthantheDowellmethod(upto60%error)andtheBessel-functionmethod(upto120%error).
3.53.232.22.52.82.81.23.82.8333.233.53.83.53.533.212.80.8v/d2.83.233.80.63.23.52.582.2.23.2.550.42.52.53.22.8330.2Fig.4.Maximumerror(%)ofourmodel((7)-(12))relativetosimulationresultsoverthefrequencyrangecorrespondingtod/δfrom0.6to60iswithin4%.Generallyourmodelgivesasmallererror(within3%)atsmallerinterwiredistancev,whichisthegeometrymostfrequentlyusedinpractice.
themostgenerallyusedaveragefieldforamultilayerwindingis[15]:
1(NI)22H=(15)
3b2wAlthough(14)isabetterformulatouseespeciallyforaone
ortwolayerwinding,whenthenumberoflayersislarge,theerrorof(15)comparedto(14)isverysmall.Forexample,theerrorislessthan3%atm=3.
V.DISCUSSIONOFSKIN-EFFECTLOSSESOFA
CONDUCTORINAWINDING
Thelossincreaseathighfrequenciescausedbyskineffectforasingleisolatedroundconductorcanbeexpressedinthe
3.80.22.83.232.82.52.20.40.60.81h/d
1.21.41.61.8Assumingthatthefieldvarieslinearlyoverthewindinglayerthicknessandthatthenumberoflayersisverylarge,
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formoftheratioofACresistanceandDCresistance[6]:
γberγbeiγ−beiγγberγRac,skin=−1Rdc2ber2γ+bei2γ
whereγis:
1d1
γ=√=√X
2δ2(16)
1002010Rac/RdcEquation(16)isforaconductorcarryingaspecifiedcurrent
whichisnotsubjectedtoanyexternalfield.Butaccordingtoourdefinitionofskin-effectloss,theskin-effectlossofconductorsinawindingwillbelargerthanthatin(16)becausethecurrentdistributionwillbeaffectedbyotherconductors,eveninalayerwhichisnotsubjectedtoanyexternal‘proximity’field.
Weperformed2-DFEAsimulationforaconfigurationasshowninFig.5tofindoutthescopeoferrorcausedbyusing(16)tocalculateskin-effectlossinourdefinition.Fig.5isanequivalentsetupforAinFig.2.Atd/δ=8,thedifferencebetweenthelosspredictionof(16)andthesimulationresultisabout20%oftheskin-effectlossand11%oftheskin-effectlossplusresistiveloss.However,thisdifferenceisonlyabout1%ofthetotallosswhichalsoincludestheproximity-effectlossinaone-layerwinding.Whenthenumberoflayersincreases,proximity-effectlossbecomesmoresignificantanddominatesthetotalloss,theerrorincalculatingskin-effectlossisnegligibleinthetotalloss.
Inaone-layerwinding,thepercentageoferrorcausedbyusingapproximateskin-effectlosscalculationisgiveninFig.6.Inamulti-layerwinding,thiserrorwillcontinuetodecreaseasthenumberoflayersincreases.Thuswecanuse(16)forcalculatingtheskin-effectlossinourmodel.Error (%)(17)
10.10.01Percentage (%) of total lossPercentage (%) of skin−effect loss% of skin−effect loss plus resistive loss0.00112d/δ10100Fig.6.Inaccuracyproducedbyusingapproximateformula(16)isbelow1%ofthetotalwindingloss.Thoughusing(16)tocalculatetheskin-effectloss(byourdefinition)canproduceerroraslargeas20%,thiserrorisnegligibleinthetotaleddy-currentloss,inwhichskin-effectlossisalwaysasmallpartoverthefrequencyrange.Theplotisforaone-layerwindingwithv/d=0.4286andh/d=0.4286.Inamultilayerwinding,inaccuracyincalculatingskin-effectlosswillbeevenmoreinsignificantcomparedtothetotalloss,becauseproximity-effectlossdominates.
10090807060504030Bessel function methodDowell methodExperimental DataOur model0.4 mm0.4 mmOdd symmetry2010Even symmetry00246d/δ81012Diameter of the conductor is 0.28 mmFig.7.ComparisonofexperimentaldataforwindingtypeAtotheresultsgivenbyournewmodel
Fig.5.ConfigurationofFEAsimulationforskin-effectlossVI.EXPERIMENTALRESULTS
Tovalidatetheaccuracyofournewmodel,wedidlossmeasurementsonthreedifferenttypesofwindings.Allofthethreewindingsareonpotcores(42mm×29mm)ofMnZnferrite(Philips3F3).Allofthethreewindingscontainthreelayersofprimarywindingandthreelayersofsecondarywinding.Thetwowindingsarewoundinoppositedirectionsandareconnectedinseriesoppositiontoachievesmallinductanceinthewindingsothataccuratemeasurementofsmallwindingresistanceiseasier.Weusedanimpedance
analyzertomeasuretheACresistanceoverthefrequencyrangeof1kHzto2.5MHz.Also,theerrorcausedbytheparasiticcapacitancewascompensatedusingthecircuitmodelin[16].SpecificationsofthethreewindingtypesareinTableII.
InFig.7,Fig.8andFig.9,wecanseethatourmodelfitstheexperimentaldatabetterthaneithertheDowellmodelandtheBessel-functionmodel.
WindingtypeAandwindingtypeChavethesameinterwiredistancevandsamewirediameter,whilewindingChasalargerinterlayerdistancehthanwindingtypeA.WindingAwillhavelargerwindinglossaccordingtoourmodel’sprediction.ThoughthedifferenceinlossbetweenexperimentaldatainFig.7and9isnotobvious,acloserexaminationoftheexperimentaldatarevealedthatatd/δaround3the
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TABLEII
SPECIFICATIONSOFTHEWINDINGSMEASURED
WindingtypeABCWire22AWGmagnetwire22AWGsolidwire22AWGmagnetwireInsulationonwiresinglebuild0.25mmteflonsinglebuildInsulationbetweenlayersonelayeroftape—5layersofpolypropylenetape
Actualaveragev/d0.281.430.29Actualaverageh/d0.291.431.5040353025Rac/Rdc201510500Bessel function methodDowell methodExperimental DataOur modellargeinterlayerdistance,willbethebest.
VII.CONCLUSION
Inthispaper,basedon2-DFEAsimulationdata,wepresentacompletemodelforcalculatingtheeddy-currentwindinglossesinawindingofroundconductors.Comparedtoothermethods,ourmethodofcalculatingwindinglosshasthefollowingadvantages:First,ourmethod’sresultsarebasedonFEAsimulationsona2-Dmodelofthewinding,whichareinherentlymoreaccuratethanthepreviouslyusedapproximatemodelssuchasDowell’s1-DmodelandtheBessel-functionmodel,whichneglectstheinteractionsbetweenconductors.Second,ourlossmodelisabletodescribethebehaviorofawindingwithvariousgeometryparametersoverawidefre-quencyrange,andourmethodseparatesoverallfieldanalysisfromtheanalysisoflocaleddycurrentsthustheresultcanbeextendedtoanyfieldshape.Third,ourmodelispresentedasaclosed-formfunction,suchthatdesignerscanuseitdirectlytocalculatelossandthusavoidthepainofFEAsimulations.
ACKNOWLEDGMENT
ThisworkwassupportedinpartbytheUnitedStatesDepartmentofEnergyundergrantDE-FC36-01GO1106.
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[8]MassimoBartoli,NicolaNoferi,AlbertoReatti,andMarianK.Kaz-imierczuk,“Modelinglitz-wirewindinglossesinhigh-frequencypowerinductors”,in27thAnnualIEEEPowerElectronicsSpecialistsConfer-ence,June1996,vol.2,pp.1690–1696.
[9]WilliamR.Smythe,StaticandDynamicElectricity,McGraw-Hill,1968,
page411.
[10]J.A.Ferreira,“Analyticalcomputationofacresistanceofroundand
rectangularlitzwirewindings”,IEEProceedings-BElectricPowerApplications,vol.139,no.1,pp.21–25,Jan.1992.
[11]J.A.Ferreira,ElectromagneticModellingofPowerElectronicConvert-ers,KluwerAcademicPublishers,1989.
246d/δ81012Fig.8.ComparisonofexperimentaldataforwindingtypeBtotheresultsgivenbyournewmodel
10090807060Rac/Rdc504030201000246d/δ81012Bessel function methodDowell methodExperimental DataOur modelFig.9.ComparisonofexperimentaldataforwindingtypeCtotheresultsgivenbyournewmodel
lossintypeCis1%largerthanthelossintypeA,whichmatchesthepredictionofourmodel.Athigherfrequencies,ourexperimentaldataonwindingtypeAandCdoesn’tshowthesamelossdifference.Thismaybeduetothelimitationofourcompensationofparasiticcapacitances.
LossinwindingtypeBissmallerthanthatofwindingtypeAorthatofwindingtypeC.ThisisbecausetherearefewerturnsperlayerinwindingtypeBandsmallerH.Tominimizetheproximity-effectlossforthesamenumberofturns,windingtypeC,withminimuminterwiredistanceand
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2004 35th Annual IEEE Power Electronics Specialists ConferenceAachen, Germany, 2004
[12]XiNanandCharlesR.Sullivan,“Animprovedcalculationofproximity-effectlossinhighfrequencywindingsofroundconductors”,inPESC03,2003,vol.2,pp.853–860.
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[14]AlexanderD.Podoltsev,“Analysisofeffectiveresistanceandeddy-currentlossesinmultiturnwindingofhigh-frequencymagneticcompo-nents”,IEEETransactionsonMagnetics,vol.39,no.1,pp.539–548,Jan.2003.
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ThisshowsthatthedecompositionFig.2worksandthattheaverageofthevaluesofHoneachsideofawindinglayeristhecorrectvaluetousein(2)tocalculateloss.Forthepthlayerinamultilayerfoilwinding,
(p−1)I
bpIH2=
b(2p−1)IH=
2b
bρ(2p−1)2I2
Pproximity=G(h/δ)
δ4b2
H1=
(26)(27)(28)(29)
APPENDIX
Thedecompositionoforthogonalskineffectandproximity
effectinFig.2worksforanyconductororgroupofconductorswhichhasasymmetricshape.WecanusethefoilconductorasanexampletoexplainhowtheorthogonalityworksandshowthattheaverageofthefieldsoneachsideofthewindingistherightHtousefor(2).Thelossperunitlengthinafoilconductorisgivenin[17]:
1bρ
[(H1−H2)2F(h/δ)+2H1H2G(h/δ)](18)2δ
wherehisthethicknessoftheconductor,bisthewidthoftheconductor,ρistheresistivityoftheconductor,H1andH2arethefieldmagnitudesoneachsideofthefoil,andFandGarefunctionsoftheskindepthδ:
P=
F(x)=
sinh(2x)+sin(2x)cosh(2x)−cos(2x)sinh(x)−sin(x)cosh(x)+cos(x)
(19)(20)
Pproximity+Pskinbρ(2p−1)2I21ρI2=(F(h/δ)−G(h/δ))G(h/δ)+δ4b22δb2ρI2=[F(h/δ)+2p(p−1)G(h/δ)]2δb1bρ
(30)=[(H1−H2)2F(h/δ)+2H1H2G(h/δ)]
2δ
TheabovecalculationsagainprovethevalidityofthedecompositioninFig.2.
G(x)=
ThelossperunitlengthinafoilconductorwhichissubjectedtoanexternalfieldofpeakvalueHis:
bρ2
HG(h/δ)(21)δ
Theskin-effectlossinafoilconductorinawindingwindowwillbe:
IρI21
H1=−H2=;Pskin=(F(h/δ)−G(h/δ))
2b2δb2
(22)
Thenifweconsideraone-layerwinding,weknowthatforthatlayer:
Pproximity=
I
(23)
b
Fortheproximity-effectloss,ifweusefieldHasaverageofH1andH2:
ρI2
G(h/δ)(24)Pproximity=
δ4b
Equation(22)givestheexpressionforskin-effectloss.Eitherbyplugging(23)into(18)orbyadd(22)and(24),wegetanidenticaltotallossexpressionforaone-layerwinding:
H1=0;
H2=
ρI2
F(h/δ)P=
2δb
(25)
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