Magnetics basis B

MAGNETICSAstecCustomPowerLectureOutline•MagneticQuantitiesandUnits•BasicConcepts•TransformersandTransformerCores•TheB-HCurve•HysteresisLoopsandVolt-SecondBalance•TransformerLosses•DesignConsiderationsMagneticVariablesandUnitsQuantityVarDefinitionSIUnit=CGSUnit=ImperialUnitFluxDensityB/A=HTesla104Gauss64.5Klines/in2FieldIntensityHmmf/lengthA-turn/m0.004Oersted(Gilbert/cm)0.0254A-turn/inPermeabilityor=B/HHenry/m107/4Gauss/Oersted0.0254Henry/inPerm.ofAiro(constant)4x10-7H/m1Gauss/Oersted1.02x10-7H/mFluxBA=mmf/Weber108Maxwell108linesMagnetomotiveForcemmfNI=A-turn0.4GilbertA-turnReluctancelength/AA-turn/Wb4(10-9)Gilbert/Maxwell10-8A-turn/linePermeance1/=ALWb/A-turn25(107)/Maxwell/Gilbert108line/A-turnInductanceLN2/=Vt/IHenry109abHenryHenryEnergyW½LI2=½BHVJoule107Erg0.7376ft-lbEffectiveMagneticAreaACm2104cm26.452Sq.in.WindowAreaAWm2104cm26.452Sq.in.AreaProductAPAWACm4cm441.62in4Ampere’sLaw•Acurrent-carryingwiregeneratesamagneticfieldperpendiculartothecurrent,withthemagneticfieldlinesfollowingtheRightHandRule.•Ampere’sLawrelatesmagneticfieldintensityandcurrentdensity,thus:lH•dl=AJ•dA=IT•Integratingthemagneticfieldintensityaroundaclosedmagneticpathgivesthetotalcurrentfollowingthroughthewire.Therefore,magnetomotiveforceisgivenasmmf=NI.ClosedpathClosedpathFaraday’sLaw•Faraday’sLawisthecomplementofAmpere’sLawandrelateselectricfieldintensityandmagneticfluxdensity,thus:lE•dl=-/tAB•dA•Thevoltageinducedinaconductorisproportionaltothechangeinmagneticfluxpassingthroughthesurfaceenclosedbytheconductor.v=Nd/dtv+-AlB(t)CurrentandMagneticFields•Currentflowingthroughaclosedpathgeneratesconcentric“fluxrings”perpendiculartotheconductor.•Theloopsofacurrent-carryingcoilaresaidtobe“linked”bythesemagnetic“fluxrings”.Thesefluxlinesconstituteamagneticfield,withaFLUXDENSITYof:B=/Aeff•FluxdensityismoreorlessproportionaltotheMAGNETICFIELDINTENSITYHbyafactorcalledPERMEABILITY:=B/HExample:BaroundalongwireQuestion:Whatisthefluxdensityaroundaverylongstraightconductor(surroundedbyair)carryingacurrentIT?ThecurrentwillgenerateconcentricfluxringsaroundtheconductorwithconstantHalongeachfluxring.FromAmpere’sLaw:lH•dl=ITSinceH=B/,and=oforair,wehave:ringH•dl=(B/o)(2r)=ITB=oIT/2r[tesla]rExample:Binasingle-turncoilQuestion:Givenasingle-turncoilofradiusrwithacurrentIcoilflowingthroughit,whatisthefluxdensityinsidethecoil?FromAmpere’sLaw:lH•dl=ITSinceone“fluxring”isapproximatelycircularforasingle-turncoil,integratingalongthefluxringwiththesameradiusasthecoil,wehave:ringH•dl=(B/)(2r)=IcoilB=oIcoil/2rForanN-turnflatcoil,B=oNIcoil/2r[tesla]rrB=0.63NIcoil/r[gauss]Example:BinalongcoilQuestion:GivenanN-turncoilwhoselengthislargecomparedtoitsradius(length30r),withacurrentIcoilflowingthroughit,whatisthefluxdensityinsidethecoil?Again,fromAmpere’sLaw:l(B/)•dl=ITNowIT=NIcoil,andthetotallengthofthefluxpathisapproximatelytwicethecoillength.Thisyields:(B/)(2length)NIcoilB½NIcoil/length[tesla]whereisthepermeabilityoftheinductorcore.B=0.63NIcoil/length[gauss]Example:BinatoroidalcoreQuestion:GivenanN-turncoilwoundaroundatoroidalcorewithinnerradiusrandouterradiusR,withacurrentIcoilflowingthroughit,whatisthefluxdensityofthecore?WecandeducefromAmpere’sLawthatthefluxdensityincreasesastheradiusdecreases.TogetBAVE,wecanuseaneffectiveradiusofrEFF=(R+r)/2.Thus,BAVEcanbecalculatedas:B=EFFNIcoil/(R+r)[tesla]whereEFFistheeffectivecorepermeability.RrB=1.26NIcoil/length[gauss]Transformers•Themagneticfluxpathisprovidedbyhigh-permeabilitymaterial(e.g.,iron),therebytherebyforcingthefluxtoflowthroughbothwindings.•Anychangeinvoltageintheprimarywillinduceaspecificamountoffluxtoflowthroughthecorewhich,inturn,inducesaproportionalchangeinvoltageinthesecondary.(Ampere’s&Faraday’sLaw)•Thetotalpowerhandlingcapabilityofthetransformerisrelatedtoitsphysicalsize.ThiscanbeseenbyapplyingAmpere’slawtothewindowareaAWandFaraday’sLawtothecorecrosssectionAC:AP(areaproduct)=AWAC=(NI/J)(Vt/NB)VI=PACAW•Devicesmadeupoftwoormorecoilscoupledtogetherthroughacommonmagneticfluxpath.TransformerCharacteristics•Outputvoltagedependsonturnsratio:VP/VS=nP/nS•Outputisisolatedfrominput.•Currentatthelowvoltagewindingisalimitingfactor:(ideally)PP=PSVHIGHILOW=VLOWIHIGHIHIGHdictateswiresizeandwindowsizeCoreSaturation•FromAmpere’sLaw,thecurrentinatransformerwindingisproportionaltothemagneticfieldintensityinthecore.HI•Increasingthemagneticfieldintensityhastheeffectofaligningthemoleculesinthemagneticmaterial,therebyincreasingthenumberoffluxlinesthroughthematerial.•ThereeventuallycomesapointwhenallthemoleculedipolesbecomealignedandnoamountofcurrentincreasewillcauseasignificantriseinmagneticfluxdensityB.•TheoperatingpointisthussaidtohaveenteredtheSATURATIONregion.IIlinearsat0IdealActualEffectofCoreSaturation•Duringsaturation,thechangeinfluxperchangeincurrentapproacheszero.•KnowingfromFaraday’sLawthatV=Nd/dt,ifd/dt=0thenitfollowsthatV=0.•Aneffectiveshortcircuitoccursatthesecondary.•Iftheinputvoltageiscyclical,severecyclicalpeakcurrentswillbeobservedattheprimary.I