StableLevitationofaPassiveMagneticBearing
KevinD.Bachovchin,JamesF.Hoburg,Fellow,IEEE,andRichardF.Post
DepartmentofElectricalandComputerEngineering,CarnegieMellonUniversity,Pittsburgh,PA15213USA
LawrenceLivermoreNationalLaboratory,Livermore,CA94550USA
Adesignforapassivemagneticbearingsystemthatcanstablylevitatearotorinalldirectionsisdescribed.ThebearingsystemconsistsoflevitationmagnetscoupledwithaHalbacharraystabilizer,whichinducescurrentsinstabilizationcoils,inordertoovercometheinherentinstabilityofasystemcomposedonlyofpermanentmagnets.Thelevitationmagnetsystemconsistsoftwopairsofannularringmagnetswhichprovideanupwardmagneticlevitationforcetocounteractthedownwardgravitationalforceoftherotor.TheHalbacharraystabilizerconsistsoftwostabilizationcoilsshiftedinangularpositionwithrespecttooneanotherandcenteredintheverticaldirectionbetweentworotatingHalbacharrays.Magneticfieldsfrompermanentmagnetsarecalculatedusingsuperpositionoffieldsduetopatchesofmagnetizationchargeatsurfaceswherethemagnetizationisdiscontinuous.Inducedcurrentsinthestabilizationcoilsarecalculatedbycomputingthetimederivativeofthemagneticfluxthroughthosecoils.Magneticforcesontherotorarecomputedusingasuperpositionofforcesoneachpatchofmagnetizationcharge.Theentiremagneticbearingsystem,consistingofboththelevitationmagnetsandtheHalbacharraystabilizer,isstabletobothverticalandlateraldisplacements.ResultsarecomparedwithasimplerstraightenedapproximationoftheHalbacharraystabilizer.
IndexTerms—Halbacharraystabilizer,magneticbearings,magneticlevitation,magnetizationsurfacecharge.
I.INTRODUCTION
passivemagneticbearingsystemthatisstableinalldirec-tionscanbeformedbyusingpermanentmagnets,alongwithHalbacharraystabilizers.In[1],alevitationmagnetsystemusingpermanentmagnetswasdescribed.Theresultantmag-neticforcesandstiffnesswereanalyzedusingamagneticsur-facechargeformulation,andthesystemwasdeterminedtobeunstabletorotordisplacementsinonedirection.ThisinstabilityisconsistentwithEarnshaw’stheorem,whichstatesthatitisim-possibletolevitateanobjectstablyinalldirectionsusingonlymagneticfieldsderivedfromfixedcurrentsorpermanentmag-nets[2].ByaddingHalbacharraystabilizers,whichinducecur-rentsinstabilizationcoils,theinstabilitycanbeovercome.ThisideahasbeenpursuedbyLawrenceLivermoreNationalLab(LLNL)[3]andisanalyzedhere.Amoredetailedanalysisisdescribedin[4].
Passivemagneticbearingshaveseveraladvantagescomparedtoalternativeoptions.Comparedtomechanicalbearings,mag-neticbearingsrequirenolubricationandarenotsubjecttowearfrommechanicalfriction.Forapplicationsinvolvinghighrota-tionalspeed,therearesignificantfrictionallossesinmechan-icalbearings,whereaswell-designedmagneticbearingsexhibitnear-zerolosses.
Comparedtoactivemagneticbearings,passivemagneticbearingshaveafarlowercost.Activemagneticbearings,asdescribedin[5]and[6],usepositionsensorsandelectroniccircuitsthatcontrolelectromagnetstoachievestablelevitationoftherotatingelement.
AnalternativeapproachpursuedbyArgonneNationalLaboratoryistousesuperconductingbearingelements[7].
A
ManuscriptreceivedMarch13,2012;revisedMay25,2012;acceptedJuly11,2012.DateofpublicationJuly17,2012;dateofcurrentversionDecember19,2012.Correspondingauthor:K.D.Bachovchin(e-mail:kbachovc@andrew.cmu.edu).
Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp://ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TMAG.2012.2209123
Becausesuperconductorshavediamagneticproperties,theyevadeEarnshaw’sTheoremandcanstablylevitateanobjectinalldirections[8].Examplesofsuperconductingmagneticbearingsarepresentedin[9]foramaglevsystemandin[10]foraflywheelenergystoragesystem.However,sincesuper-conductorsrequirecryogenictemperatures(below150C),itisdoubtfulthatsuperconductingmagneticbearingscanbeappliedinmostrotatingmachinery.Ontheotherhand,thepassivemagneticbearingsbeingpursuedbyLLNLcanoperateatambienttemperature.
Thispaperdescribesadesignforanambienttemperaturepas-sivemagneticbearingsystemthatcanstablylevitatearotorinalldirectionsandanalyzesthemagneticforcesandstiffnessoftheentiresystem.Inordertolevitatetherotor,alowerlevita-tionmagnetpairisusedtoexertarepellingupwardforceontherotorandanupperlevitationmagnetpairisusedtoexertanattractingupwardforceontherotor.Thelevitationmagnetsystemisstabletolateraldisplacementsbutunstabletoverticaldisplacements.Thisdesigndiffersfromthedesigndescribedin[1],whichconsistedofonlyonemagnetpairandwasstabletoverticaldisplacementswhileunstabletolateraldisplacements.Annularringmagnetsareusedbecausetheirlevitationforceissufficienttocounteractthedownwardgravitationalforceoftherotor.Forapplicationswithaheavierrotor,Halbacharrayscouldbeemployedinsteadofannularrings.InaHalbacharray,themagnetizationsrotatefromonemagnettothenextinordertoaugmentthemagneticfieldononesideofthearraywhilenearlycancellingthefieldontheotherside.Atsmallgaps,Hal-bacharrayscanprovideanorderofmagnitudelargerforcethanannularringmagnetshavingthesamesizeandsamemagneticmaterial[1].
TheverticalinstabilityofthelevitationmagnetsystemisstabilizedbyemployingaHalbacharraystabilizer.Thestabi-lizerconsistsoftwostabilizationcoils,bothcenteredinthever-ticaldirectionbetweentworotatingHalbacharrays.Ifthesta-bilizationcoilsareexactlycenteredbetweenthetwoHalbacharrays,themagneticfluxcontributionsthroughthecoilsfromtheupperandlowerarrayscancelandnocurrentisinduced.
0018-9464/$31.00©2012IEEE
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Fig.1.Magneticbearinggeometry.ThelevitationmagnetsystemiscomposedofMagnets1–4.TheHalbacharraystabilizeriscomposedoftwostabilizationcoilscenteredintheverticaldirectionbetweenthetwoHalbacharrays.(Figurenotdrawntoscale)
Fig.2.Shapeofeachlevitationmagnet.Theannularringhasaheight,innerradius,andouterradius.
magnets.Whentheaxialdisplacementoftherotor,vertdisp,iszero,theupperattractingmagnetsare0.88cmclosertoeachotherthanthelowerrepellingmagnets,whichmakesthesystemstableinthelateraldirectionbutunstableintheverticaldirec-tion.If,alternatively,thelowerrepellingmagnetswereposi-tionedclosertoeachotherthantheupperattractingmagnets,thenthesystemwouldbestableintheverticaldirectionandun-stableinthelateraldirection.
B.MagneticFieldsFromPermanentMagnets
Themagneticfieldfromapermanentmagnetcanbecom-putedbymodelingthemagnetastwouniformchargedsurfacesusingthecoulombianapproach[1],[11]orequivalentlymod-elingthemagnetastwosheetsofuniformcurrentdensityusingtheamperiancurrentapproach[12],[13].Usingthecoulombianapproach,anannularringmagnetthatispolarizedintheposi-tiveverticaldirectionismodeledasapatchofpositivesurfacechargedensityonthetopsurfaceoftheringandapatchofneg-ativesurfacechargedensityonthebottomsurfaceofthering.Thetopandbottomsurfacesoftheringhavetheshapeofannuli.Themagnitudeofthesurfacechargedensityforeachannulusis
(1)
whereisthediscontinuityinthenormalcomponentofthemagnetizationacrossthemagnetboundary.
Expressionsforthefieldresultingfromeachannularpatcharegivenin[1].Thefieldfromanannularringmagnetiscomputedbysummingthecontributionsfromthetwoannularpatches.C.MagneticForcesBetweenPermanentMagnets
Asdescribedin[1],theforceexertedbyMagnet1onMagnet2iscalculatedbysummingtheindividualforcesoneachof
If,however,thestabilizationcoilsbecomenotexactlycentered,thetime-varyingfluxinducesacurrentineachcoil.Thiscur-renttheninteractswiththemagneticfieldoftheHalbachar-raystoprovideastabilizingforceontherotorintheverticaldirection.Halbacharraysareusedforthestabilizerinsteadofannularringsinordertomakethestabilizerverystifftover-ticaldisplacementsfromtheequilibriumposition.Thisservestokeepthestabilizationcoilsveryclosetothenullfluxplane,whichminimizeslossesfrominducedcurrentsinthestabiliza-tioncoils.
II.LEVITATIONMAGNETS
A.Overview
Amagneticbearingdesign,basedonanearlyLLNLdesign,isshowninFig.1.Magnets1–4formthelevitationmagnetsystem.Magnets1and4arestationarywhileMagnets2and3areattachedtothebottomandtopoftherotor,respectively.Neodymiumpermanentmagnets(NdFeB)witharemanentfieldof1.35Tareusedforallfourmagnets.EachmagnethastheshapeofanannularringasportrayedbyFig.2.Eachringhasaheightof1.27cm,aninnerradiusof14.48cm,andanouterradiusof19.56cm.
Whentheradialdisplacementoftheaxisoftherotor,latdisp,iszero,therotormagnetsarecoaxialwiththestationary
BACHOVCHINetal.:STABLELEVITATIONOFPASSIVEMAGNETICBEARINGFig.3.Verticalforcesasafunctionofverticaldisplacementforthelevitationmagnets.thetwomagnetizationsurfacechargepatchesonMagnet2.Theforceonanannulusofmagnetizationchargeis
(2)
where
isthemagneticsurfacechargedensityofthepatchinArray2andisthemagneticfieldfromArray1,whichiscalculatedusingtheexpressionsgivenin[1]foranannularpatchofcharge.
ToevaluatethedoubleintegralsinMATLAB,the“dblquad”functionisusedwithanerrortoleranceof1e-3.WhenMagnet2iscoaxialtoMagnet1,thereisonlyaverticalforce.HoweveriftheaxisofMagnet2isdisplacedofftheaxisofMagnet1,thenthereisalsoalateralforceinthedirectionofdisplacement.TheforceexertedbyMagnet4onMagnet3iscalculatedusingthesameprocedure.ThetotalforceontherotoristhesumoftheforcebyMagnet1onMagnet2andtheforcebyMagnet4onMagnet3.(Sincethedistancebetweentheupperpairandthelowerpairisverylarge,theforcebyMagnet1onMagnet3andtheforcebyMagnet4onMagnet2arenegligible.)Fig.3showstheverticalforcesasafunctionofatseveralvaluesofandFig.4showsthelateralforcesasafunctionofatseveralvaluesof.Attheequilibriumposition(,),thereisnolateralforceandtheupwardmagneticverticalforceexactlyequalsthedownwardgravitationalforceof2046N.Theequilibriumisstableinthelateraldirectionbutunstableintheverticaldirection.
Theverticalstiffnessiscalculatedbycomputingthenega-tiveoftheslopeoftheverticalforceversuscurvewhilethelateralstiffnessiscalculatedbycomputingthenega-tiveoftheslopeofthelateralforceversuscurve.Fig.5showstheverticalstiffnessasafunctionofatsev-eralvaluesofandFig.6showsthelateralstiffnessasafunctionofatseveralvaluesof.Aposi-tivestiffnessmeansthesystemisstableinthatdirectionwhileanegativestiffnessmeansthesystemisunstableinthatdirec-tion.Sinceonlyaxiallysymmetricpermanent-magnetelements
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Fig.4.Lateralforcesasafunctionoflateraldisplacementforthelevitationmagnets.
Fig.5.Verticalstiffnessasafunctionofverticaldisplacementforthelevitationmagnets.
Fig.6.Lateralstiffnessasafunctionoflateraldisplacementforthelevitationmagnets.
areused,theverticalstiffnesstoverticaldisplacementsfromtheequilibriumisalwaysnegativetwotimesthelateralstiffnesstolateraldisplacementsfromtheequilibrium[3].Forexample,whenand,theverticalstiff-nessis202N/cmwhilethelateralstiffnessis101N/cm.Theforceandstiffnesscalculationsinthissectionarestatic,andthegyroscopiceffectfromtherotationalvelocityofthe
612Fig.7.TopviewoftheupperandlowerHalbacharrays.Themagnetizationsrotatebetweenthez-directionandtheazimuthaldirectioninbotharrays.
rotorisnotspecificallyanalyzed.Weexpectthatthestaticlateralstabilityisonlyenhancedbygyroscopiceffectsastherotationalspeedincreases.
III.HALBACHARRAYSTABILIZER
A.Overview
TheHalbacharraystabilizershowninFig.1isdesignedtostabilizetheentiresystemintheverticaldirection.Thestabilizerconsistsoftwostationarystabilizationcoilscenteredinthever-ticaldirectionbetweentwoHalbacharraysattachedtotherotor.TopviewsoftheupperandlowerHalbacharraysareshowninFig.7.Ineacharray,thereare24wavelengthsand96magnets.Thepolarizationsoftheindividualmagnetsrotatebetweenthe-directionandtheazimuthaldirection.FortheupperHalbacharray,therotationofthepolarizationscausesthestrongsidefieldtobebelowthearraywhileforthelowerHalbacharray,therotationofthepolarizationscausesthestrongsidefieldtobeabovethearray.Forbotharrays,theinnerandouterradiiare14.48and19.56cm,respectively.
EachindividualmagnetinbothHalbacharrayshasthesamegeometry,atrapezoidalprismwithaheightof1.27cm.Eachtrapezoidextends5.08cmintheradialdirection.Thelongandshortedgesofeachtrapezoidare1.27and0.95cm,respectively.Againneodymiumpermanentmagnetswitharemanentfieldof1.35Tareused.
Fig.8showsatopviewofoneofthestabilizationcoils,whichconsistsof48straightsegmentsand48semicircularloops.Thereare24wavelengthsaroundthecoil,andthewave-lengthofthecoilequalsthewavelengthoftheHalbacharrays.Thesecondstabilizationcoilisidenticaltothefirstcoilandisplacedinthesameverticalplane,butthesecondcoilisrotatedonequarterofawavelengthinthecounter-clockwisedirectionwithrespecttothefirstcoil.
Ifthestabilizationcoilsareexactlycenteredbetweenthetwoarrays,thefluxthrougheachcoilfromtheupperarraycancelsthefluxfromthelowerarrayandthereiszeronetfluxthrougheachcoil.When,however,therotorisdisplacedfromtheequi-libriumpositionintheverticaldirection,thereisatime-varyingflux,whichinducesacurrentineachcoil.Thiscurrentthenin-teractswiththemagneticfieldoftheHalbacharraystoprovideastabilizingforceintheverticaldirection.
TheforceasafunctionoftimeexertedbyeachstabilizationcoilontheHalbacharraysconsistsofadoublefrequencysinu-
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Fig.8.Topviewofoneofthestabilizationcoils.
soidalcomponentandaconstantcomponent.Twostabilizationcoils,ratherthanjustone,areusedinordertomakethetotalforcenearlyconstantasafunctionoftime.B.MagneticFieldsFromHalbachArrays
AmagnetintheHalbacharraywithpolarizationintheazimuthaldirectionismodeledasapatchofpositivesurfacechargeatthesidewallwherethemagnetizationvectortermi-natesandapatchofnegativesurfacechargeatthesidewallwherethemagnetizationvectorinitiates.Bothpatchesarerectangles,andthemagnitudeofthesurfacechargedensityofeachrectangleiscalculatedusing(1).Themagneticfieldfromarectangleofmagnetizationchargeisgivenin[14].
Amagnetthatispolarizedinthepositiveverticaldirectionismodeledasapatchofpositivesurfacechargedensityonthetopsurfaceofthetrapezoidalprismandapatchofnegativesur-facechargedensityonthebottomsurfaceoftheprism.Bothpatchesaretrapezoids,andthemagnitudeofthesurfacechargedensityofeachtrapezoidisagaincalculatedusing(1).Sinceatrapezoidisthesumofarectangleandtwotriangles,thetotalmagneticfieldfromatrapezoidalpatchofchargeisfoundbysummingthecontributionsfromtherectangularpatchandthetwotriangularpatches.Themagneticfieldresultingfromatri-angularpatchisgivenin[4].ThetotalfieldfrombothHalbacharraysiscomputedbysummingthecontributionsfromeachofthe192rectangularpatchesandthe192trapezoidalpatches.C.FluxThroughStabilizationCoils
Themagneticfluxthroughoneofthestabilizationcoilsis
(3)
where
istheinnersurfaceenclosedbythestabilizationcoilandisthetotalmagneticfluxdensityfrombothHalbacharrays,whichiscalculatedusingtheexpressionsgivenin[14]forarectangularpatchofchargeandtheexpressionsgivenin[4]foratriangularpatchofcharge.
Tonumericallycomputethesurfaceintegrals,the“quad2d”functioninMATLABisusedwithanerrortoleranceof1e-7.Ifthestabilizationcoilisexactlycenteredbetweenthetwoarrays,thefluxthroughthecoilfromtheupperarraycancelsthefluxfromthelowerarrayandthereiszeronetfluxthroughthecoil.Whenisnonzero,how-
BACHOVCHINetal.:STABLELEVITATIONOFPASSIVEMAGNETICBEARINGFig.9.Fluxasafunctionoftimethrougheachstabilizationcoilwith
and.
ever,thereisatime-varyingfluxthroughthecoil.Foranominalrotationalspeedof1000r/min,thefluxisperiodicwithperiod
,theamountoftimeittakesthearraystorotateone
wavelength.
Sincethesecondstabilizationcoilisrotatedonequarterofawavelengthinthecounter-clockwisedirectionwithrespecttothefirstcoil,whenthelateraldisplacementiszero,thefluxthroughthesecondcoillagsthefluxthroughthefirstcoilbyex-actly90.Whenthelateraldisplacementisnonzero,thisisnolongertrue,butisstillagoodapproximationforlateraldisplace-mentsthataresmallcomparedtothecoilradii.Asanexample,Fig.9showsthefluxthrougheachofthestabilizationcoilsasafunctionoftimeforoneperiodwhenthearraysarerotatingat1000r/min,,and.
D.InducedCurrentsinStabilizationCoils
Thetime-varyingmagneticfluxfromtheHalbacharraysin-ducesacurrentineachofthestabilizationcoils,asdescribedby
(4)(5)
whereistheresistanceofeachcoil(20.1),istheself-in-ductanceofeachcoil(2.28),andisthemutualinduc-tancebetweenthetwocoils(0.554).Calculationsoftheseparametersaredescribedin[4].Thesystemofdifferentialequa-tionsissolvedusingtheMATLABdifferentialequationsolver“ode45.”Ifthemutualinductancewereneglected,theinducedcurrentinthe2ndcoilwouldmerelylagtheinducedcurrentinthe1stcoilby90.Howeverthemutualinductancecausestheamplitudeoftheinducedcurrentinthesecondstabilizationcoiltohaveaslightlygreateramplitudethanthecurrentinthefirstcoil.Fig.10showstheinducedcurrentineachofthestabiliza-tioncoilsforoneperiodusingthesameparametersasinFig.9.
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Fig.10.Inducedcurrentasafunctionoftimeineachstabilizationcoilwith
and
E.MagneticForcesonHalbachArrays
UsingtheLorentzForceLaw,theforceontheHalbacharrays
exertedbythestabilizationcoilis
(6)
whereisthecurrentinthestabilizationcoil,isthemagneticfluxdensityfrombotharrays,andisthecontourofthestabilizationcoil.Tonumericallyevaluatetheintegral,the“quad”functioninMATLABisusedwithanerrortoleranceof1e-6.Whenthelateraldisplacementofthearraysiszero,thereisonlyaverticalforce.Ifthearraysaredisplacedinthelateraldirection,thenthereisalsoalateralforceinthedirectionofdisplacement.
Fig.11andFig.12showtheverticalandlateralforcesexertedbyeachstabilizationcoilontheHalbacharraysforoneperiodusingthesameparametersasinFig.9.Ifthemutualinductancewereneglected,thetotalforceexertedbybothcoilswouldbeaconstant.However,theunequalcurrentamplitudesduetothemutualinductancecausethetotalforcetostillhaveadoublefre-quencysinusoidalcomponent,withanamplitudemuchsmallerthantheconstantpart.
Sincethesinusoidalcomponentofthetotalforceexertedbybothcoilsissmallandthefrequencyishigh,time-averageforcesareusedtoanalyzethestabilityoftheHalbacharraystabilizertosmallperturbationsintheverticalorlateraldirections.Fig.13showsthetime-averageverticalforceasafunctionofatseveralvaluesofandFig.14showsthetime-av-eragelateralforceasafunctionofatseveralvaluesof
.
TheHalbacharraystabilizerprovidesastabilizingforcetoverticaldisplacementsfromtheequilibriumpositionofthemag-neticbearingsystem(,),butprovidesadestabilizingforcetolateraldisplacements.Sincethelevitationmagnetsystemisstableinthelateraldirection,
614Fig.11.VerticalforceexertedbyeachstabilizationcoilonHalbacharraysas
afunctionoftimewith
and.Fig.12.LateralforceexertedbyeachstabilizationcoilonHalbacharraysasa
functionoftimewith
and.Fig.13.VerticalforcesasafunctionofverticaldisplacementfortheHalbach
arraystabilizer.
thetotalmagneticbearingsystemconsistingofboththelevita-tionmagnetsandtheHalbacharraystabilizerisstabletolateraldisplacements.
Fig.15showstheverticalstiffnessofthestabilizerasafunctionofatseveralvaluesofandFig.16
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Fig.14.LateralforcesasafunctionoflateraldisplacementfortheHalbach
arraystabilizer.
Fig.15.VerticalstiffnessasafunctionofverticaldisplacementfortheHalbach
arraystabilizer.
showsthelateralstiffnessasafunctionofatseveralvaluesof.Theminimumverticalstiffnessoccurswhen,andastheabsolutevalueof
increases,thestiffnessincreasesbecauseoneoftheHalbacharraysgetsclosertothestabilizationcoils.Themaximumabso-lutevalueofthelateralstiffnessoccurswhen,andasincreases,theabsolutevalueofthestiffnessdecreasesbecausetheHalbacharraysmovefurtherawayfromthestabilizationcoils.
F.EntireMagneticBearingSystem
ThetotalforceontherotoristhesumoftheforcesfromthelevitationmagnetsandfromtheHalbacharraystabilizer.Fig.17showsthetime-averageverticalforceasafunctionof
atseveralvaluesofwhileFig.18showsthe
time-averagelateralforceasafunctionofatseveralvaluesof.
Attheequilibriumposition(,
),thereisnolateralforceandtheupwardmagneticlevi-tationforceexactlyequalsthedownwardgravitationalforceof2046N.Theequilibriumisstableintheboththelateralandver-ticaldirections.Fig.19showstheverticalstiffnessasafunctionofatseveralvaluesofandFig.20showsthe
BACHOVCHINetal.:STABLELEVITATIONOFPASSIVEMAGNETICBEARINGFig.16.LateralstiffnessasafunctionoflateraldisplacementfortheHalbacharraystabilizer.
Fig.17.Verticalforcesasafunctionofverticaldisplacementfortheentiremagnetsystem.
Fig.18.Lateralforcesasafunctionoflateraldisplacementfortheentiremagnetsystem.
lateralstiffnessasafunctionof
atseveralvaluesof
.
IV.APPROXIMATESTRAIGHTENEDML
Usinganapproximatestraightenedmodel,closedformex-pressionscanbeobtainedforthetime-averageverticalforceex-ertedbytheHalbacharraystabilizer.Theapproximatemodel
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Fig.19.Verticalstiffnessasafunctionofverticaldisplacementfortheentiremagnetsystem.
Fig.20.Lateralstiffnessasafunctionoflateraldisplacementfortheentiremagnetsystem.
neglectsthecurvatureofthegeometry,thehigherorderhar-monicsofthemagneticfieldsfromtheHalbacharrays,andthesemicircularsegmentsofthestabilizationcoil.Anadvantageoftheapproximatemodelisthatitismuchmorecomputationallyefficientsinceitdoesnotrequirenumericalintegrations.Adis-advantage,inadditiontothefactthatitislessaccurate,isthatitcanonlybeappliedwhenthelateraldisplacementoftherotor,
,iszero.
Fig.21portraysthestraightenedmodelfortheHalbacharraystabilizer.Forsmalldistancesfromthenullfluxplane,com-paredtothewavelengthoftheboththecoilandthearrays,themagneticfieldcomponentsattheobservationpoint()canbeapproximatedas
(7)(8)
Thewavenumberisandtheradianfrequencyis
,whereisthenumberofwavelengthsofthe
arraysandistherotationalvelocityofthearrays.
616Fig.21.ApproximatestraightenedrepresentationoftheHalbacharraystabi-lizer.Thefirststabilizationcoilisshowningreenandthesecondstabilizationcoilisshowninblue.Thereare24wavelengths,butonlythreeareshown.
Sincethesefieldcomponentsvarysinusoidallywithtimeatfrequency,theyareexpressedinphasornotationas
(9)(10)
Thefluxthrougheachstabilizationcoiliscomputedas
(11)
(12)
Thesefluxesalsovarysinusoidallyatfrequencyandareexpressedinphasornotationas
(13)(14)
Differential(4)and(5),whichgoverntheinducedcurrentsinthecoils,becomealgebraicequationsinthephasordomain.Theseequationsaresolvedforand
(15)(16)
Thetime-averageverticalforceexertedbyeachcoilonthearraysiscalculatedbyfindingthetime-averageoftheproductoftwosinusoids
(17)(18)
Finallythetotaltime-averageverticalforceonthearraysis
(19)
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TABLEI
PARAMETERSFORSIMPLEMODEL
Fig.22.Time-averageverticalforceasafunctionofverticaldisplacementwhenusingtheapproximatelinearmodelversuswhenusingtheexactsurfacechargemethod.
UsingtheparametersshowninTableI,Fig.22showsthetime-averageverticalforceasafunctionofverticaldisplace-mentofthearrays.(TheparameterinTableIwasobtainedbycalculatingthehorizontalfieldatthenullfluxplaneusingthesurfacechargeformulationwiththestraightenedgeometryshowninFig.21.Alternativelycouldbefoundusingtheapproximateformulagivenin[14],butthisapproximationin-troducesgreaterinaccuracyduetothefinitedepthofthearraysintheradialdirection.)Theresultsobtainedusingtheapproxi-matestraightenedmodelshowrelativelygoodagreementwiththeexactsurfacechargemethodresults,whichincludetheef-fectsofcurvature.Theforcesobtainedfromthestraightenedmodelaresomewhatsmallerbecausethisapproximatemethodneglectsthefluxthroughthesemicircularregionsofthecoils.Itisalsoofinteresttoexploretheeffectthattherotationalspeedhasontheverticalrestoringforce.Fig.23showsthetime-averageverticalforceasafunctionofr/minonasemilogplotusingtheparametersinTableIandusingaverticaldisplace-ment.Againtheforcesobtainedusingtheapproximatestraightenedmodelaresomewhatsmallerthantheexactsurfacechargemethodresults.Theverticalforceisalmostzeroatspeedslessthan100r/min.Therefore,atlowrotationalspeeds,theHalbacharraystabilizercannotstabilizetheverticalinstabilityofthelevitationmagnetsystem.Theverticalforceap-proachesitsmaximumabsolutevalueataround100000r/min
BACHOVCHINetal.:STABLELEVITATIONOFPASSIVEMAGNETICBEARINGFig.23.Time-averageverticalforceasafunctionofr/minwhenusingtheap-proximatelinearmodelversuswhenusingtheexactsurfacechargemethod.
andthenremainsalmostconstantasthevelocityincreases.Athighrotationalspeeds,(19)simplifiesto
(20)
Inthislimit,thestabilizationcoilsareinductancedominated,withtheinducedcurrentscompletelypreventingfluxpenetra-tionthroughthecoilareas.
V.CONCLUSION
TheHalbacharraystabilizerdescribedinthispaperisshowntobeeffectiveatstabilizingalevitationmagnetsystemthatisstabletolateraldisplacementsbutunstabletoverticaldisplace-ments.Adesignofthisbearingsystemiscurrentlybeingcon-structedandtestedatLLNL.Asmallscaleexperimentalsystemproducedexcellentagreementwiththecalculatedverticalforceasafunctionofrotationalvelocity.ExperimentalverificationforafullscalesystemisintendedaspartofongoingresearchatLLNL.
Alternativestabilizergeometries,presentedin[3]and[8],canstabilizealevitationmagnetsystemthatisstabletoverticaldis-placementsbutunstabletolateraldisplacements.Therearesit-uations,however,wheretherotorisalsounstableagainsttiltdisplacements,whichrequirechangesinthedesignofthesta-bilizer.Thesimplestchangeindesignforaverticalstabilizeristosubdividethesinglesnake-likewindingdescribedinthispaperintofourindependentquadrantwindings.Thisconfigura-tionwillthenprovidestabilizingforcesforbothverticalandtiltdisplacements.Aslightdisadvantageisthatthisconfiguration,comparedtothewindingdescribedinthispaper,hasaweakeraveragingeffectagainstslightgeometricormagnetizationde-viationsfromtheidealmodelconsideredinthispaper.
Amoreradical,butinsomecasesveryadvantageous,designchangeistoemployadifferentroutetotheflux-nullingeffect.ThoughalsoemployingdualHalbacharrays,theazimuthalphasingofthetwoarraysischangedsothatthetransversefieldcomponentsareadditive,whiletheazimuthalcomponents
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cancelatthemid-planebetweenthearrays.Atthesametimethewindingconfigurationischangedfromaplanarsnake-likeconfigurationtoanassemblyofwindow-frame-likeshortedloopsthattransverselyspanaportionofthegapbetweenthetwoarrays.Nowthedesiredflux-nullingeffectarisesfromthefactthattheazimuthalfieldcomponentsoftheupperandlowerarraysareoppositelydirected,sothefluxcancellationarisesfromthealgebraicsumofthetwoazimuthallydirectedfluxesthatpassthroughthewindow-framewindings,ratherthanfromthegeometriclocationofaplanarwindingatthemid-planebetweenthearrays.Futureworkinthisareawillinvolveassessmentsofinstabilitymodesandofalternativestabilizerdesignsofthekinddescribedabove.
ACKNOWLEDGMENT
LawrenceLivermoreNationalLaboratoryisoperatedbyLawrenceLivermoreNationalSecurity,LLC,fortheU.S.De-partmentofEnergy,NationalNuclearSecurityAdministrationunderContractDE-AC52-07NA27344.
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