3676IEEETRANSACTIONSONMAGNETICS,VOL.46,NO.9,SEPTEMBER2010

StatorSlottingEffectontheMagneticFieldDistributionofSalientPole

SynchronousPermanent-MagnetMachines

GurakuqDajaku1andDieterGerling2

FEAAMGmbH,D-85577Neubiberg,Germany

InstituteforElectricalDrives,UniversityofFederalDefenseMunich,D-85577Neubiberg,Germany

Thispaperdealswithanewanalyticalmethodfordeterminingtheair-gapfluxdensityofsalientpolesynchronouspermanent-magnet(PM)machinesincludingthestatorslottingeffect.Theapproachtoanalyzetheair-gapfluxdensitybasedonthenewmodelistodeter-minetheair-gapmagnetomotiveforce(MMF)andtheflux-pathpermeancefunction.Wedeterminetheflux-pathpermeancefunction,whichincludesthestatorslottingeffect,byusingasimplemagneticreluctancenetworkthattakesintoaccountthegeometryofthemachineandthematerialcharacteristics.Usingthenewmodelwithstatorslotting,wecalculatetheair-gapfluxdensitydistributionofdifferentelectricmachinesusingfinite-elementmethodstoprovetheaccuracyofthenewmodel.

IndexTerms—Air-gapfluxdensity,analyticalsolution,finite-elementmethod(FEM),flux-pathpermeance,permanent-magnet(PM)machine,statorslottingeffect.

I.INTRODUCTION

PERMANENT-MAGNETsynchronousmachines

(PMSMs)gainmoreandmoreimportanceforspe-

cialdriveapplications.Uptorecentyears,PMSMswereknown

forsmalldrives,e.g.,forservoapplications.Inthelastyears,

PMSMshavebeenincreasinglyappliedinseveralareassuch

astraction,automobiles,etc.Thistypeofelectricalmachines

offermanyadvantages,includingahighpower-to-weightratio,

constantpoweroverawidespeedrange,andhighefficiency.

MagneticfieldanalysisinPMmachinesisanimportantpre-

requisiteforthepredictionofmachineparameters,electromag-

netictorque,radialforces,andsoon.Inparticular,accurate

knowledgeofthefluxdensitydistributionintheair-gapises-

sentialfortheaccuratepredictionofthemotorperformances.

Magnetconfiguration,magnetizationdirection,air-gaplength,

andthenumberofthepoleandslotcombinationhavesignificant

effectsonthefluxdensitydistributionintheair-gap.Further,

thepresenceofstatororrotorslotshavealargeinfluenceonthe

air-gapmagneticfielddistributionandthereforeontheelectro-

magnetictorque.Theconsequencescanbecoggingtorqueand

torqueripplescausingvibration,noise,andspeedfluctuation.

Therefore,thedetailedknowledgeofthefielddistributionsin

theair-gapisofmainimportanceforpredictingandoptimizing

theperformanceofPMmachines.

Ofcourseaswellknown,slottinginfluencesthemagnetic

fieldintwoways.First,itreducesthetotalfluxperpole,an

effectwhichisusuallyaccountedbyintroducingtheCarterco-efficientintothecalculation.Second,itaffectsthedistribu-

tionofthefluxdensityintheair-gapoftheelectricmachines.

Therefore,theslottingeffectisimportanttobetakenintoac-

countduringdeterminationoftheair-gapfluxdensity.

ManuscriptreceivedFebruary26,2010;revisedApril09,2010;acceptedApril15,2010.DateofpublicationMay03,2010;dateofcurrentversionAu-gust20,2010.Correspondingauthor:G.Dajaku(e-mail:Gurakuq.Dajaku@unibw.de).Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp://ieeexplore.ieee.org.DigitalObjectIdentifier10.1109/TMAG.2010.2049269Theair-gapmagneticfieldwithslottingeffectscanbeeval-

uatedbyavarietyoftechniquesincludinganalyticalorsemi-

analyticalmethodsaswellasnumericaltechniqueslikefinite

elementsorboundaryintegralmethods.Finiteelementsgive

accurateresultsconsideringgeometricdetailsandnonlinearity

ofmagneticmaterials.However,thismethodiscomputertime

consumingandpoorlyflexibleforthefirststepofdesignstage

ofelectricalmachines.Analyticalmethodsareusefultoolsfor

firstevaluationofelectricalmotorsperformancesandforde-

signoptimizationsincecontinuousderivativesissuedfromthe

analyticalsolutionareofgreatimportanceinmostoptimization

methods.Asignificantnumberofpublicationsontheanalytical

solutionoftheair-gapmagneticfieldinslottedmachinescanbe

foundintheliterature,e.g.,[1]–[17].

Twoanalyticalmethodsaremainlydevelopedformodeling

thestatorslottingeffect.Thefirstoneconcernstheuseofcon-

formalmappingtoconsiderslottingeffects[1]–[7],andtheac-

tualair-gapfluxdensityiscalculatedbymultiplyingtherela-

tivepermeancefunctionwiththeradialfluxdensityofaslotless

motor.Thismethodhasanimportantdrawbacksinceitisas-

sumedthattheslotwidthismuchsmallerthanitsheight.Fur-

thermore,eventhoughthismethodgivesinformationaboutthe

air-gapfluxdensityfluctuationduetostatorslotting,itdoesn’t

takecorrectlyintoconsiderationtheslottingeffect.Thereason

isthatthemagneticfieldandtheslottingeffectaredetermined

separatelyandfordifferentconditions.Thesecondmethodcon-

sistsofanalyticalsolutionofLaplaceorPoissonequationsin

thedifferentsub-domains(magnet,air-gapandslots)byap-

plyingtheboundaryconditionsontheinterfacebetweensub-do-

mains[8]–[13].However,thesemodelsarenotabletopro-

videanaccurateevaluationofair-gapfluxdensityiftheboth

statorandrotorslottinghavetobeconsidered.Alsoaswillbe

shownlater,therotorsaliencyeffectonthestatorslottingisn’t

consideredintheseworks.Ontheotherhand,differentfrom

[1]–[13],in[14]–[17]anothermodelingwayforthestatorslot-

tingisused;theslottingeffectistakenintoconsiderationduring

derivationofthetotalpermeancefunctionforthemagneticflux

path.However,eventhismodelingproceduredescribesaccu-

ratelytheair-gappermeancetakingintoaccountthestatorand

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