Iron Loss Calculation in a Three.docx
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Iron Loss Calculation in a Three.docx
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IronLossCalculationinaThree
IronLossCalculationinaThree-Phase-Laminated-CoreVariableInductorBasedonReluctanceNetworkAnalysis
KenjiNakamura1,ShuheiHisada1,KenjiArimatsu2,TakashiOhinata2,KunioSakamoto2,andOsamuIchinokura1
GraduateSchoolofEngineering,TohokuUniversity,Sendai,Miyagi980-8579,Japan
TohokuElectricPowerCo.,Inc.,Sendai,Miyagi981-0952,Japan
Variableinductors,whichconsistofonlyamagneticcoreandprimarydcandsecondaryacwindings,cancontroleffectiveinductanceofthesecondaryacwindingbyprimarydcexcitation.Therefore,thevariableinductorscanbeappliedasavarcompensatorin
electricpowersystems.Inapreviouspaper,wehaveproposedanovelthree-phase-laminated-corevariableinductor.Ithasonlyonelaminatedcoreinwhichthethree-phasesecondaryacwindingsareinstalledtoreducesizeandweightincomparisonwithaconventional
single-phasevariableinductor.Goodcontrollabilityandsinusoidaloutputcurrentoftheproposedvariableinductorweredemonstrated.
Thispaperpresentsamethodforcalculatingironlossofthethree-phase-laminated-corevariableinductorbasedonreluctancenetworkanalysis(RNA).Thevalidityoftheproposedmethodisprovedbycomparingwithmeasuredvalues.IndexTerms—Ironloss,reluctancenetworkanalysis(RNA),three-phase-laminated-corevariableinductor,varcompensator.
I.INTRODUCTION
Inelectricpowersystems,linevoltagestabilizationisoneofthemajorsubjectsbecauserenewableanddistributedenergyresourcessuchaswindandphotovoltaicpowergenerationsarebeingintroducedrapidly.Toregulatethelinevoltage,astaticvarcompensator(SVC)andstaticvargenerator(SVG),whichareoneofthesemiconductorpowerconverters,havebeenintroduced[1],[2].Theseapparatuses,however,havesomeproblemsincludingharmoniccurrent,electromagneticinterference(EMI),andhighcost.
Avariableinductorconsistsofonlyamagneticcoreandwindings[3],butcancontrolitswindinginductancequicklyandcontinuouslybymagneticsaturationeffect.Thus,itcanbeusedasavarcompensatorbycombiningwithfixedpowercapacitors[4].Thevariableinductorhasasimpleandrobuststructure,highreliability,andlowcost.Thesefeaturesareadvantagesforapplicationsinelectricpowersystems.
Inpreviouspapers,wehaveintroducedsomekindsofvariableinductorsusedinelectricpowersystems:
anorthogonal-corevariableinductor[5],andanEIE-corevariableinductor[6].Amongthem,theEIE-corevariableinductorissuitableforalargercapacityincomparisonwiththeothervariableinduc-torsbecauseitismadeoflaminatedsiliconsteelsheetsandhasnogapsforharmoniccurrentreduction.Atrial6.6kV–300kVAvarcompensatormadeoftheEIE-coreshasbeenmanufactured.However,alltheconventionalvariableinductorsincludingtheEIE-corehaveasingle-phasestructure.Therefore,atleastthreevariableinductorsarerequiredforvarcompensationinelectricpowersystems.
Toovercometheaboveproblem,wehaveproposedathree-phase-laminated-corevariableinductor[7].Theproposedvariableinductorhasonlyonelaminated-coreinwhichthethree-phasesecondaryacwindingsareinstalledtogether.Itsweightislighterby30percent
Fig.1.Basicconfigurationofathree-phase-laminated-corevariableinductor.
thantheEIE-corevariableinductor.A4.0-kVAthree-phase-laminated-corevariableinductorwasdesignedandmanufacturedbythetrial,anditsgoodcontrollabilityandsinusoidaloutputcurrentwereprovedinexperiments.
Thispaperpresentsamethodforcalculatingironlossofthethree-phase-laminated-corevariableinductorbasedonreluctancenetworkanalysis(RNA).Theestablishmentofanironlosscalculationmethodisindispensablefortheoptimumdesignofthevarcompensator.Theusefulnessoftheproposedmethodisprovedbycomparingwithmeasuredvalues.
II.RNAMODELOFTHREE-PHASE-LAMINATED-COREVARIABLEINDUCTOR
A.BasicConfigurationandOperatingPrincipleoftheVariableInductor
Fig.1illustratesabasicconfigurationofathree-phase-laminated-corevariableinductor,whichconsistsofsixlegs,ringyokes,andprimarydcandsecondaryacwindings.Thesecondaryacwindingsarecoiledaroundeachcoupleoflegsonthesameline,respectively.Thesewindingsareconnectedindelta,andtothethree-phaseacsource.Ontheotherhand,theprimarydcwindingsarecoiledaroundtheoutsideringyokesothatthedcfluxcirculatesintheyoke,andconnectedtothedcsource.Therefore,thelevelofmagneticsaturationintheringyokecanbe
Fig.2.Specificationsofthetrialthree-phase-laminated-corevariableinductor
regulatedbytheprimarydccurrent,thatis,theeffectiveinductanceofthesecondaryacwindingscanbecontrolledbytheprimarydcexcitation[7].
B.RNAModelConsideringIronLossinLaminatedSiliconSteelSheets
Fig.2showsthespecificationsofthetrialthree-phaselaminated-corevariableinductor.Thecorematerialisnonorientedsiliconsteelwithathicknessof0.35mm.Theratedcapacityandvoltageare4.0kVAand200V,respectively.Theworkingfluxdensityinthelegsis1.2T.
Tocalculatetheironlossofthevariableinductor,a3-DRNAisemployedbecauseitsanalyticalmodelisverysimple,calculationspeedandaccuracyarehigh,andthecoupledanalysisisreadilyperformed[5]–[8].
First,thethree-phase-laminated-coreisdividedintomultipleelementsasshowninFig.3(a).Toallowconsiderationofleakageflux,thesurroundingspaceisalsodivided.Thedividedelementscanbeexpressedina3-DunitmagneticcircuitasshowninFig.3(b).Incasetheunitmagneticcircuitisinthecoreregion,thereluctancesexpressthemagneticnonlinearity,andthemagneticinductancesrepresenttheironlossinlaminatedsiliconsteelsheets.Theseelementscanbeobtainedasfollows.
Whenconsideringtheironloss,arelationshipbetweenthemagneticfieldandfluxdensitycanbegivenby.Thefirstandsecondtermsof
(1)denotethemagneticnonlinearity,andthethirdandfourthtermsexpresstheironloss[8].
Fig.3.Three-dimensionalRNAmodelofathree-phase-laminated-coreconsideringironlossinlaminatedsiliconsteelsheets.(a)Divisionofthecore;(b)3-Dunitmagneticcircuit.
Fig.4.-_curveandcorelosscurveofthecorematerial.(a)-_curve;(b)corelosscurve.
ThecoefficientsandaredeterminedbyacurveofcorematerialshowninFig.4(a),andthecoefficientsandareobtainedfromacorelosscurveshowninFig.4(b),whichincludeshysteresisandeddycurrentlossesinthecore.Thesecoefficientsarerespectively.
From
(1),arelationshipbetweenthemagnetomotiveforce(MMF)andthefluxisgivenbywherethemagneticpathlengthandcrosssectionofeachdividedelementareand,respectively.Byusing
(2),thereluctancesandmagneticinductancescanbedetermined.
Ontheotherhand,thereluctanceintheaxisdirectionshouldbedeterminedconsideringalaminatedstructureofthecore.Apparentpermeabilityinaxisisgivenbywherethevacuumandcorepermeabilityareand,andaspacefactorofsiliconsteelsheetsis,whichis0.975.Accordingly,isobtainedasalinearreluctancefrom(3).
Incasetheunitmagneticcircuitisinthesurroundingspace,thelinearreluctancesareobtainedfromdimensionsofeachdividedelementandthevacuumpermeability,andthemagneticinductancesare0.
Fig.5.Electric-andmagnetic-coupledmodelofthethree-phase-laminated-corevariableinductor
Finally,iftheRNAmodelobtainedinthewaydescribedaboveiscoupledwiththeprimarydcandsecondaryaccircuitsasshowninFig.5,thecharacteristicsofthevariableinductorcanbecalculatedbyageneralpurposecircuitsimulatorSPICE[5].Inthefigure,theMMFsarear-rangedineachlegandyokeofthethree-phase-laminated-core.
Fig.6.Measuredandcalculatedironlossesofthethree-phase-laminated-corevariableinductor
Fig.6showsthemeasuredandcalculatedironlossofthethree-phase-laminated-corevariableinductor.Bothvaluesareingoodagreementwhenaprimarydccurrentis0A.Contrarytothis,theerrorincreasesastheprimarydccurrentbecomeslarger.
Fig.7.Distributionofleakagefluxfromanuppersurfaceofthelaminatedsiliconsteelwhenaprimarydccurrentis30AandU-phasesecondaryaccurrentisatthemaximum.
Fig.7illustratesthedistributionofleakagefluxfromanuppersurfaceofthelaminatedsiliconsteelwhenaprimarydccurrentis30A.Suchdistributioncanbeobtainedfromfluxespassingthougheach-axisreluctanceintheRNAmodel.Thisfigureindicatesthattheleakagefluxfromtheuppersurfaceofthelaminatedsiliconsteelcannotbeneglectedsincethevariableinductorutilizesthemagneticsaturationeffect.Hence,eddycurrentsonthelaminatedsiliconsteelmustbeconsideredintheRNAmodel.
Fig.8.DistributionoftheeddycurrentsassumedintheRNAmodel.
III.RNAMODELCONSIDERINGEDDYCURRENTSONLAMINATEDSILICONSTEEL
Ingeneral,eddycurrentsonthelaminatedsiliconsteelaredistributedinacomplicatedmanner.TotakethisintoconsiderationintheRNAmodel,thefollowingassumptionisapplied:
ThedistributionoftheeddycurrentsisbasedonthedivisionoftheRNAmodelasshowninFig.8,andtheeddycurrentineachelementflowsuniformly.
Fig.9.3-Dunitmagneticcircuitcombinedwithaneddycurrentcircuit.
Fig.10.Eddycurrentcircuitmodel
IntheRNAmodel,a3-DunitmagneticcircuitandaneddycurrentcircuitarecombinedasshowninFig.9.Inthefigure,theleakagefluxgivestheinducedvoltage.Fromtheinducedvoltageandtheresistance,theeddycurrentcanbeobtained,whichgivestheMMFinthe3-Dunitmagneticcirc
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