effects-of-eddy-currents-in-transformer-windings

EffectsofeddycurrentsintransformerwindingsP.L.Dowell,B.Sc.(Eng.),GraduateI.E.E.SynopsisTheeffectsofeddycurrentsintransformerwindingsareconsidered,andamethodisderivedforcalculatingthevariationofwindingresistanceandleakageinductancewithfrequencyfortransformerswithsingle-layer,multilayerandsectionalisedwindings.Themethodconsistsindividingthewindingintoportions,calculatingthed.c.resistancesandd.c.leakageinductancesofeachoftheseportions,andthenmultiplyingthed.c.valuesbyappropriatefactorstoobtainthecorrespondinga.c.values.Thesea.c.valuesarethenreferredto,say,theprimarywindingandsummedtogivethetotalwindingresistanceandleakageinductanceofthetransformer.Formulasarederivedandquotedforcalculatingthed.c.resistancesandleakageinductancesofthewindingportions.Theoreticalexpressionsarederivedforthevariationwithfrequencyetc.ofthefactorsbywhichthed.c.valuesmustbemultipliedtoobtainthecorrespondinga.c.values.Theseexpressionsarepresentedintheformofgraphs,permittingthefactorstobereadasrequired.ListofsymbolsMKSunitsareusedthroughoutthepaper.a=breadthofaconductor(Fig.7)B=magneticfluxdensityb=windingbreadth(Fig.7)D=2ahtanh%~FR=ratioofa.c.valueofRwtod.c.valueFL=ratioofa.c.valueofLwtod.c.valueFLo0=valueofFLforaportionwithaninfinitenumberoflayersg=heightofintersectiongap(Fig.7)H=magneticfieldstrengthh=heightofaconductor(Fig.7)/=totalcurrentinaconductor/=currentelement/=currentdensityinaconductorJh=currentdensityatthetopofaconductorLg=leakageinductanceduetoanintersectiongapL/=leakageinductanceLup—leakageinductanceduetopthinterlayergapLv=leakageinductanceduetoallinterlayergapsinaportionLw—leakageinductanceduetofluxcuttingtheconductorslT—meanturnlengthM=txhcothahMl/2=valueofMwithhreplacedbyhj2m=numberofwholelayersinawindingportionN=numberofturnsNt=numberofturnsperlayerNp=numberofturnsinawindingportion(i.e.mNtor(m+Wi)n=N2IN{P,Q—constantsofintegrationp=positiveintegermRi=leakageresistanceRw=resistanceofawindingportions=magneticpathlengthU=totalheightofallinterlayergapsinawindingportionu=heightofaninterlayergapVh=voltageacrossalayeratthetopsoftheconductorsVj=inducedvoltageacrossalayeratthetopsoftheconductorsPaper5058J,firstreceived20thJanuaryandinrevisedform12thApril1966Mr.DowelliswithTelephonFabrikAutomatic,Copenhagen,Den-mark,andwasformerlywithMullardResearchLaboratories,Redhill,Surrey,EnglandPROC.IEE,Vol.113,No.8,AUGUST1966Vjp=V,.=Wup=Wg=x=inducedvoltageineachofthelayersbetweenthepthlayerandthepositionofzerom.m.f.duetothefluxcrossingpthlayertotalvoltageacrosstheendsofalayerresistivevoltagedropacrossalayeratthetopsoftheconductorsmagneticenergystoredinpthinterlayergapmagneticenergystoredinanintersectiongaplengthmeasuredindirectionofwindingheight(Fig.7)Rw+JO)Lyy5=insulationthicknessN,a^T6=temperaturep=resistivity(f=magneticflux(fa,fb,(fc=partsofthefluxcuttingthewindingcf)p=fluxcrossingpthlayerjt—totalfluxcuttingthewindingOJ=angularfrequencySubscriptsetc.0=d.c.valueof—'—realpartof—=imaginarypartof—1IntroductionTheperformanceofatransformercanbecalculatedfromitsequivalentcircuit,andmethodsareavailableforcalculatingthevaluesoftheequivalent-circuitelementsfromaknowledgeofthegeometryofthetransformerandofthematerialsusedinitsconstruction.Thesemethodsusuallygivethed.c.valuesoftheelements,butthevariationwithfrequencyoftheshuntinductanceandshuntlosscanbeobtainedfromaknowledgeofthecomplexpermeabilityofthecore.Thefrequencyvariationofthewindingcapacitances(andtheirloss)isoftennegligible,butcanbeobtainedfromaknowledgeofthecomplexpermittivitiesofthedielectricmaterialsinvolved.Theotherelementsintheequivalentcircuitarethewindingresistanceandleakageinductance.Theseelementswillvarywithfrequencybecauseoftheeffectsoftheeddycurrentsinducedinthewindings,anditisthepurposeofthepapertoproduceamethodforcalculatingthevaluesoftheseelementsatanyfrequency.Theeffectsofeddycurrentsontheeffectiveresistanceofcoilshasbeenconsidered,byseveralauthorsinthepast.123Itis,inprinciple,possibletoapplytheseresultstotransformerwindings,althoughmostofthepapersdonotconsider1387transformerwindingsspecifically.Henceitcanbequitedifficulttoapplytheresultsto,forexample,asectionalisedtransformerwinding.Noneofthesepapersconsiderstheeffectsofeddycurrentsontheleakageinductanceoftransformers.Itisobviousthattheproblemofeddycurrentsintrans-formerwindingsisphysicallyrelatedtotheproblemofeddycurrentsinslot-wound-armatureconductors.Thislatterproblemhasalsobeentreatedbyauthorsinthepast,4-5andprobablytheresultsgivenforarmatureconductorshavebeenappliedtotransformerwindings.However,itdoesnotseemtobeeasytoapplytheseresultsinsuchawayastoobtainaccurateanswersfortransformerwindings.Theuseful-nessoftheresultsderivedforarmatureconductorsisfurtherreducedbythefactthattheyareintendedforuseatpowerfrequencies,andthedesignerofaradio-frequencytransformerwilloftenfindthatausefulfunctionisnotplottedtoasufficientlyhighvalueofthefrequency-dependentvariable.Itwasdecidednottotrytomodifytheresultsforslot-wound-armatureconductors,butinsteadtoderiveresultsfortrans-formersdirectly,usingthesametypeofmathematicalapproachas