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Chapter1MagneticCircuitsandMagneticMaterials7/10/2024MagneticCircuitsandMagneticMaterials1IntroductionThemagneto-quasi-staticformofMaxwell’sequationsareusedatlowfrequenciestypicalofenergy-conversiondevicesAmpere’slawandGauss’slaw:7/10/2024MagneticCircuitsandMagneticMaterials2CoilorwindingofNturnsonamagneticcoreofhighpermeabilitym>>moFluxismostlyconfinedtothecoreSimplifyAmpere’slaw:coilMMF=Ni7/10/2024MagneticCircuitsandMagneticMaterials3Gauss’slaw:fluxlinesareclosed,sonetfluxisfluxdensityBctimecross-sectionalareaAc Fc=BcAcSinceweareusingaveragefluxdensityBcthenthecorrespondingaveragemagneticfieldintensityHcwillbeapproximately7/10/2024MagneticCircuitsandMagneticMaterials4lcistheaverage(mean)corelengthwhichisperpendiculartothecross-sectionalareaAcThematerialpropertiesgivetherelationbetweenBandH,suchasB=mHforalinearmaterial7/10/2024MagneticCircuitsandMagneticMaterials5Magneticcircuitwithairgap7/10/2024MagneticCircuitsandMagneticMaterials6AnalogybetweenDCelectriccircuit(a)andmagneticcircuit(b):VoltageVisanalogoustoMMFFCurrentIisanalogoustofluxfResistanceRisanalogoustoreluctanceRFluxlinkage,inductanceandenergyFluxlinkageofacoilisl=NfwherethefluxfisassumedtolinkallNturns,thenAmpere’slawgivesthecoilinducedvoltagee=dl/dtCoilonprecedingslidehasNturnsandinductance
L=l/i=N2/Rtot
whereRtot=Rc+RgPowertothecoilp=ei=idl/dtEnergystoredinmagneticfieldW=pdt7/10/2024MagneticCircuitsandMagneticMaterials7PropertiesofmagneticmaterialsFerromagneticmaterialssuchasironhaveveryhighpermeabilitym,
idealforfocusingmagneticfluxinmachineryalsohighlynonlinearifdrivenintosaturation7/10/2024MagneticCircuitsandMagneticMaterials8HysteresisinBversusHforferromagneticmaterialACoperationMagneticcoreunderACexcitation,havingsinusoidalflux,willhaveanon-sinusoidalexcitationcurrentif7/10/2024MagneticCircuitsandMagneticMaterials9Hysteresisandeddy-currentlossesTheenergyperunitvolumelostpercycleistheareaenclosedbythehysteresisloopThereisadditionalenergylostduetoeddycurrentsinducedintheironbythetime-varyingfluxdensityEddy-currentlossisminimizedbylaminatedcores,builtupfromsheetselectricalsheetsteelFiguresinthetextillustratepropertiesofM-5grainorientedsheetsteel(propertiesareobtainedbytherollingprocessesusedinthemill)7/10/2024MagneticCircuitsandMagneticMaterials10PermanentmagnetsPermanentmagnetresidualmagnetismBrandcoercivityHcareshowninthefigureBrgivesthefluxdensitywithnodemagnetizingmmfHcgivesthedemagnetizingfieldneededtoreduceBtozeroTheloadlineshownillustratestheeffectofcuttinganairgapinthemagneticcore(seeExample1.9)SecondquadranthysteresisloopforAlnico57/10/2024MagneticCircuitsandMagneticMaterials117/10/2024MagneticCircuitsandMagneticMaterials12ThenumericalsolutionforExample1.9(seetheloadlineshownonthepreviousslide):Bg=0.30TNoticethattheair-gapfluxdensityisabout¼ofBrApplicationofpermanentmagnetsOperatingapermanent-magnetdevicewillsubjectittodemagnetizingforcesThefigureshowstheeffectsofthisasaminorhysteresiscurve,approximatedasarecoilline7/10/2024MagneticCircuitsandMagneticMaterials13SummaryThetheoryofmagneticcircuitsisdevelopedandusedforanalysisofwindingsonmagneticcoresThenonlinearmagneticpropertiesofironcorescauseshysteresisandeddycurrentlossunderACexcitationPermanent-magnetpropertieswereintroduced7/10/2024MagneticCircuitsandMagneticMaterials14Chapter2Transformers7/10/2024Transformers152.1IntroductionThetransformerconsistsoftwoormorecoils(orwindings)coupledbymutualmagneticfluxIftheprimarywindingisconnectedtoanalternatingvoltagesource,analternatingfluxwilllinkthesecondarywinding,inducinganalternatingsecondaryvoltageMosttransformersconsideredherehavehigh-permeabilityironcorestoincreasethedegreeofcouplingbetweenthewindings7/10/2024Transformers162.2No-loadconditionsWithsecondaryopen,thereisnoloadonthetransformerAlloftheprimarycurrentisexcitingcurrent,thatis,itgoestomagnetizethecoreandtoovercomecorelossesCurrentsandvoltagesareequivalentsinusoidalvalues7/10/2024Transformers17TransformerwithopensecondaryNo-loadphasordiagramNo-loadexcitationcurrentIe=Im+IcMagnetizingcurrentImlagsvoltageE1by90
Core-losscurrentIcisinphasewithvoltageE17/10/2024Transformers18No-loadphasordiagram2.3Effectofsecondarycurrent:idealtransformerIdealtransformermodel:Neglectlossesandmagnetizingcurrent,andassumeallthefluxinthemagneticcorelinksbothwindingsFaraday’sLawstatesthatinducedvoltageequalstimerateofchangeoffluxlinkingacoilAmpere’sLawstatesthatprimarycoilMMFisequaltothesecondarycoilMMF7/10/2024Transformers19IdealtransformerandloadFaraday’slaw:v1=e1=N1df/dtv2=e2=N2df/dtv1/v2=N1/N2Ampere’slaw:N1i1=N2i2i1/i2=N2/N1Idealtransformer:v1/v2=N1/N2andi1/i2=N2/N1Losslesssincepowerin=poweroutv1i1=v2i27/10/2024Transformers20IdealtransformerReflectedimpedanceAnimpedanceonthesecondaryofanidealtransformermaybereflectedorreferredtotheprimarybythesquareoftheturnsratio7/10/2024Transformers212.4TransformerreactancesandequivalentcircuitsWeuseacircuitthatisequivalentatitsterminaltoincludethemainnon-idealeffectsofatransformer:LeakagereactancesX1andX2torepresenteffectsofleakagefluxWindingresistancesR1andR2torepresentconductorlossMagnetizingreactanceXmtorepresenteffectsofMMFtomagnetizethecoreCore-lossresistanceRctorepresentcorelosses7/10/2024Transformers227/10/2024Transformers23StepsindevelopmentoftransformerequivalentcircuitPrimaryresistanceandleakagereactance.Magnetizingreactanceandcorelossresistance.Addingidealtransformerandsecondaryimpedance.Reflectingsecondaryimpedancetoprimaryofidealtransformer.ApproximateequivalentcircuitsExcitationbranchimpedancesarelargecomparedtootherbranchesApproximateequivalentcircuitscalledcantilevercircuitsUsetheformthatismostconvenient7/10/2024Transformers24ApproximateequivalentcircuitsAtnormalload,theexcitingcurrentmaybeneglectedForlargepowertransformers,thewindingresistancesaresmall7/10/2024Transformers25Example2.5Example2.5:Atransformersuppliedbyafeeder.NeglectexcitingcurrentReferallparameterstothehigh-voltagesideofthetransformer7/10/2024Transformers26AutotransformersAutotransformersareveryefficientforsmallturnsratios(fromabout1:2to2:1)Efficiencyisgainedbymetallicandmagneticcouplingbetweenprimaryandsecondary7/10/2024Transformers27Three-phasetransformersThreesingle-phasetransformerscanbeconnectedasathree-phasetransformerbankinfourwaysshowninFig.2.19(nextslide)windingsattheleftaretheprimariesthoseattherightarethesecondariesprimarywindinginonetransformercorrespondstothesecondarywindingdrawnparalleltoitSeeExamples2.8and2.9fortypicalcalculations7/10/2024Transformers28Three-phasetransformerconnections7/10/2024Transformers29InstrumenttransformersInstrumenttransformersincludevoltagetransformersandcurrenttransformersVoltagetransformersorpotentialtransformers(PT’s)areusedasalternatingvoltagetransducers,usuallysteppingdownhighvoltagetolowfeedingahighimpedance(e.g.,avoltmeter)Currenttransformers(CT’s)areusedasalternatingcurrenttransducers,usuallysteppingdownhighcurrenttolowfeedingalowimpedance(e.g.,anammeter)7/10/2024Transformers30Per-unitsystemQuantitiessuchasvoltageV,currentI,powerP,reactivepowerQ,apparentpowerS,andimpedanceZ,
canbeexpressedinper-unitformasfollows:7/10/2024Transformers31BaseapparentpowerSisusedasbaseforPandQ:VAbase=Vbase×Ibase
Baseimpedance:Zbase=Vbase/Ibase
Rulesforusingper-unitsystemSelectacommonVAbaseforthesystemSelectabasevoltageatapointinthesystemChooseallotherbasevoltagesinthesameratioastheturnsratioofanytransformerencounteredConvertallquantitiesto
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