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University of Bologna DEPARTMENT OF ELECTRICAL ENGINEERING FACULTY OF ENGINEERING Ph. D. in Electrical Engineering XXI year POWER ELECTRONICS, MACHINES AND DRIVES (ING-IND/32) Design optimization and control strategies for PM Multiphase Tubular Linear Actuators Final Dissertation in 2009 Ph. D. Thesis: Tutor: Filippo Milanesi Prof. Giovanni Serra Ph. D. Coordinator: Prof. Francesco Negrini

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  • University of Bologna

    DEPARTMENT OF ELECTRICAL ENGINEERING FACULTY OF ENGINEERING

    Ph. D. in Electrical Engineering XXI year

    POWER ELECTRONICS, MACHINES AND DRIVES (ING-IND/32)

    Design optimization and control strategies for PM

    Multiphase Tubular Linear Actuators

    Final Dissertation in 2009

    Ph. D. Thesis: Tutor:

    Filippo Milanesi Prof. Giovanni Serra

    Ph. D. Coordinator:

    Prof. Francesco Negrini

  • TomyparentsandtoSilvia

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    TABLEOFCONTENTS

    TABLEOFCONTENTS Page9

    CHAPTER1

    Basic of Tubular Linear Machines

    1.1. Introduction....................................................................................................................1.2. Structureoflinearactuators.................................................................................

    A. Geometries...................................................................................................................B. Forcesandfundamentalprincipleofoperations........................................C. Applications................................................................................................................

    1.3. Briefcomparisonamongdifferentlinearactuators................................A. Linearinductionactuators....................................................................................B. Linearpermanentmagnetsynchronousactuators....................................C. Linearreluctancesynchronousactuators......................................................D. Linearswitchedreluctanceandstepperactuators....................................

    1.4. Technicalcharacteristicsoftubularlineardrives...................................A. Lineartubularactuator..........................................................................................B. Lineartubularactuatorcompetitors................................................................

    1.5. IroncorePMtubularlinearactuator...............................................................A. Permanentmagnets.................................................................................................B. Magneticcircuit.........................................................................................................C. Windings.......................................................................................................................

    1.6. Conclusions......................................................................................................................

    Page11Page11Page11Page12Page13Page15Page15Page15Page16Page17Page17Page17Page20Page20Page21Page22Page25Page26

    CHAPTER2

    Basic of Multiphase Drives

    2.1. Introduction....................................................................................................................2.2. Multiphasespacevectorrepresentation........................................................2.3. MathematicalmodelofmultiphasePMsynchronousmachines.......2.4. Mainfeatureofmultiphasedrives.....................................................................2.5. Multiphasedrivesfortorquecapabilityenhancement.........................2.6. Faulttolerantdrives..................................................................................................2.7. MultiMotordrives......................................................................................................

    A. MultiMotordriveconcept....................................................................................B. Machineconnectivity...............................................................................................

    Page27Page27Page29Page30Page31Page33Page34Page37Page39

  • Tableofcontents

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    C. Control...........................................................................................................................2.8. Conclusion........................................................................................................................

    Page41Page42

    CHAPTER3

    Basic of Multiphase Drives

    3.1. Introduction....................................................................................................................3.2. Reviewofthedesignoptimizationmethodologies..................................3.3. Electromagneticmodel.............................................................................................

    A. Preliminarydesign...................................................................................................B. Thrustforcecontributions....................................................................................C. Fundamentalhypothesis.......................................................................................D. Designofthemagnets.............................................................................................E. Sliderandstatordesign.........................................................................................F. Thrustforcecalculations.......................................................................................

    3.4. Thermalmodel..............................................................................................................3.5. Mechanicalmodeloftheslider............................................................................

    A. Staticequilibriumanalysis....................................................................................B. Maximumsliderdisplacement............................................................................C. Reactionforcesonthebearings.........................................................................

    3.6. Optimizeddesignalgorithm..................................................................................A. Inputparameteranddesignconstraints........................................................B. Preliminarydesign:solutionprocedureoftheintegratedmodel

    equations......................................................................................................................C. Thermalmodelsolution.........................................................................................D. Finiteelementanalysis...........................................................................................E. Numberofwiresperslotoptimization...........................................................

    3.7. Conclusion........................................................................................................................

    Page43Page43Page44Page44Page44Page45Page46Page47Page48Page50Page52Page52Page55Page56Page57Page57

    Page59Page59Page60Page61Page63

    CHAPTER4

    Optimized design of a PM tubular linear actuator andexperimental results

    4.1. Introduction....................................................................................................................4.2. Definitionofthedesignconstraints..................................................................4.3. Analysisoftheinfluenceoftheairgaplengthonthemotor

    performance...................................................................................................................4.4. Optimizeddesignofatubularlinearactuator............................................

    A. Preliminarydesign...................................................................................................B. FEMrefining................................................................................................................C. Optimizationofthenumberofwiresperslot..............................................

    4.5. Experimentalresults.................................................................................................

    Page65Page65

    Page67Page69Page69Page70Page72Page74

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    A. Thrustforce.................................................................................................................B. Thermalbehaviour...................................................................................................

    4.6. Conclusion........................................................................................................................

    Page74Page77Page77

    CHAPTER5

    Fivephase dualmotor seriesconnecteddrive: simulations andexperimental results

    5.1. Introduction....................................................................................................................5.2. Descriptionofthefivephasedualmotorseiresconnecteddrive..

    A. Motorconnections....................................................................................................B. Invertervoltagelimit...............................................................................................C. Magneticdesignofthefivephasetubularmotor.......................................D. Controlscheme...........................................................................................................E. Tuningoftheregulators.........................................................................................

    5.3. Experimentalresults..................................................................................................A. Startuptest.................................................................................................................B. Positionreferencevariation.................................................................................C. Velocityvariation......................................................................................................C. Independentcontrotest.........................................................................................

    5.4. Conclusion........................................................................................................................

    Page79Page79Page79Page81Page84Page88Page89Page90Page91Page92Page92Page93Page93

    Conclusions Page95

    Bibliography Page97

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    CHAPTER

    1BasicsofTubularLinearMachines1.1 Introduction

    Thedemandforlinearelectricmachine,speciallyforcontrolledmotion,hasregisteredacontinuousgrowth inrecentyears,since its integration in industrialapplications leadstoimportant advantages [1]. Furthermore, the tubular structure seems to be attractive forindustrial purposes due to both its closed form and the inherently absence of attractiveforcebetweenthestatorandthemover.

    Inthischapterthefundamentalcharacteristicsofthelinearmachineswillbeillustratedinordertohighlinetheirmainadvantagesanddrawbacks.Particularattentionwillbepaidtothe technologic developments in soft and hard magnetic materials, which enable theindustrialrealizationofmotorwithhighforcedensity,improveddynamicbehaviourwithinarelativelylowcost.

    Finallytheattentionwillbefocusedonpermanentmagnet(PM)tubularlinearmotors.

    1.2 StructureoflinearactuatorsA. GeometriesLinear electric actuators can drive payloads in a straight line providing thrust force

    directly,withoutany intermediatemechanical transmissionsuchasgears,screwsorcrankshaft[2],[3].

    Therangeoflinearmachinetypesavailabletodaycorrespondsalmostexactlywiththerangeofrotatingcounterpart[4],[5],[6],infactItisknownthatcuttingarotatingmachinewitharadialemiplaneandunrollingit,asdepictedinFigure1,aflatsinglesidedtopologyisobtained.IftherotatingmotoriscutbyaradialplaneandsquashedasshowninFigure2,a flatdoublesided topologyoccur.Finally, the tubular linear topology isderived from thelinearflatsinglesidedconfigurationrollingitupalongthedirectionofmotion,asdepictedinFigure3.Thesetransformationsturnthestatoroftherotatingmachine intotheprimaryofthe linearmachine,which isfedbyavariableorfixedfrequencysupply,andtherotor intothesecondary,whichisequippedwithpermanentmagnet,withamultiphasewindingoritis

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    characterizedbyavariablereluctanceferromagneticstructuredependingonthekindofthemachine.

    It is evident that, due to the finite length of the machine, linear or reciprocatingmotionsareachievableonlybyanextensionofthe linearstatororofthe linearrotor.Theresult, shown inFigure4, is the socalled long statoror short statormachine. Inpracticalapplicationsbothofthemareuseddependingontheappliancerequirements,forexample,short stator solutions are often used in packaging andmanufacturing sectors,while longstatortypesareadoptedfortransportationsystems.Themovingpartofthelinearmachineisoftencalled forcer,moveror sliderandcancoincidebothwith theprimaryorwith thesecondaryofthemachine.

    B. ForcesandfundamentalprincipleofoperationsIn classical rotarymachines there are two forces acting on the rotor surface [7]: a

    tangentialforcewhichproducesthetorqueandaradialforce,actingperpendicularlytotherotorsurface.Thislatterforcesumstozeroaroundthecircumferenceduetothecylindrical

    Figure1Imaginaryprocessofsplittingandunrollingarotarymachinetoproduceasinglesidelinearmotor.

    Figure 2Imaginary process of cutting and squashing a rotary machine toproduceadoublesidelinearmotor.

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    symmetryofthemachine.Whenthemachine istransformedtocreatea linearmachine,thetangentialforcessumeachothertoconstitutethedrivingforce.Thenormalforcesinflatconfigurationsarenotnowbalancedanda largenetforceactsbetweentheunrolledrotorand stator.The ratiobetween the latterand the thrust force isworthyof considerations,typical value vary in the range810.Thepropulsion forceon themovingpartofa linearactuatorisgeneratedindifferentways,accordingtothemachinesoperationprinciple:

    linearinductionmotorsproducethrustforcebytheelectromagneticinteractionbetween the primary translating magnetic field, generated by an alternatecurrentmultiphasewinding,andtheinducedcurrentonthemultiphasewindingoronthebarofthesecondary;

    linearsteppingorswitchedreluctancemotorscreatethrustforcebytheactionofamagneticfield,producedbyelectronicallyswitcheddirectcurrentwindings,andanarrayofferromagneticpolesoravariablereluctanceferromagneticrail;

    linearsynchronousmotorsproducethrustbytheactionoftravellingmagneticfield,producedbyanalternatingcurrentmultiphasewinding,andanarrayofmagneticpolesoravariablereluctanceferromagneticrail;

    The aforementioned list suggests structural differences among the range of linear

    machine types; in thisway the presence/absence ofwindings in the primary and in thesecondary,theuseofpermanentmagnets,theutilizationofinnovativematerialsdeterminethespecificlinearmotorcharacteristics.

    C. ApplicationsTheuseof linearactuatorshasreceivedagrowing interest inseveralapplications[8],

    [9],sincealargenumberoflinearorreciprocatingmotionsisrequired.

    Figure 3Conceptual transformations to obtain a tubular motor starting to arotarymachine,passingthroughaflatlinearactuator.

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    Linear machines show performances comparable with the ones of the rotatingcounterpart; developments in materials, design methodologies and in manufacturetechnologieshaveincreasedforcedensity,dynamicbehaviour,robustnessandreliability.Insomeapplicationsthedesignconstraintlimitsorreduceperformancesoflinearmotors,viz.in transport sectorwhere large airgap are obliged, but these restrictions aremore thancompensatedbymany characteristicsof linearmotors that cannotbe achievedby rotaryequivalents.

    Historically the first linearmotor topology that tookplace in industrialand transportapplicationswas the induction one, but in the last 25 years the improvements in powerelectronics and permanentmagnetmaterials has driven the linear synchronousmachinetowardincreasingnumberofutilizations.

    Nowadaysflatandtubular linearmotorsareappliedforcontrolledmotionsandservoactuation in highspeed packaging and pickandplacemachines, in automated structuresuch as industrial robots [10], fastmanipulators and automaticmachine tool feeder [11].Other applications are reciprocating compressors and refrigerators [12]. Theaforementioned applications share relatively low power requirement and they constitutethe main commercial demand of linear drives, furthermore they are often customizedsolutions.

    Highmediumpowerapplicationsarelessfrequent,howeverresearcharebeingcarriedout inthisparticularfield.Lineardriveshavebeenproposed inenergyproductionsystems,freepiston energy converterwas studied in [13]while tidal andwaveenergy conversionsystemswereanalyzedin[14],[15].

    An important application field concernswith transportation systems for public andfactories [16]; linear peoplemovers are currently used in operations at severalinternationalairports,whilemagnetically levitatedorwheels supported trainsareused inJapan[1]andEurope.

    Linearmotor drives are strong candidates for both lowspeed and highspeedmasstransitapplications.Systemsusinglinearinductionandsynchronousmachinesareprevalentinindustry.

    (a)

    (b)

    Figure4Linearmotorwiththesocalledshortstator(a)orlongstator(b).

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    The futureprospectsofapplicationof lineardrivesareexpected to involvenotonlytraditionalsectors,suchastransport,materialprocessingandautomationequipments,butalsonewone,likeinaerospaceandautomotivesectors.

    1.3 BriefcomparisonamongdifferentlinearactuatorsResearchesandstudieshavebeencarriedoutforalltypeof linearmachines,butonly

    someofthemhavebeenfocusedforrealapplications.Thefeasibility,thecosteffectiveness,the robustness and the reliability are key aspects thatmust be owned by an industrialproduct.Forsomekindofmotorsmostofthesepropertiesare inherentlysatisfiedbut forother motor types these requirements have been satisfied with the developments ofmaterialsandofdigitalcontrollers.

    Thelatterscoveredanimportantrolefortheindustrialdiffusionoflinearactuators.Infact, control strategies for direct drive systems and servo actuations have becomemorecomplexandprecise,capabletogeneratesmoothmotioneveninharddynamicoperations.

    Theoperationalprinciplestronglyaffectsthemotorstructure,thusitdeterminestheirattitudetobeusedforspecificapplications.

    A. LinearinductionactuatorsLinearinductionactuators(LIAs)havebeenbuiltwithana.c.multiphasewindinginthe

    primary, whilst the secondary houses a cage or a shortcircuited three phase winding.However, conventional secondary configurations have a conducting plate on a solid ironplateormayhaveonlytheformer[4],[17].Theplatesolutiongivesaneasymanufacturedmachine,witharobustandrelativelycheapsecondary,suitedforhostileenvironmentorforlowmaintenanceapplications.ThecoreofprimaryandsecondaryofflatLIAs,exceptfortheplate secondary solution, is laminated in order to reduce core losses and magnetizingcurrent,furthermoreopenslotorsemiclosedslotgeometryfacilitatetheirconstruction.Fortubular structure, open slot geometry has been used since recent developments of softmagneticmaterials,viz.softmagneticcomposites. Infact, laminationsoftubularcorewithstandardtechnologiesmanifestproblemsforfluxpathsorforconstructionfeasibility.

    TheLIAsconfigurationdependonthetravellength:forshorttravelapplications,viz.0.5m,thetubularstructureisfavouritewhilstfortravellengthupto23mthemovingprimaryflatconfiguration ispreferred,andaflexiblecablesuppliesthewindings.Forgreatertravellength longprimaryflatconfigurationsareused;theprimarysectionsareplacedalongthetrackandthesecondaryactsasthemover,theformerareturnedononlywhenthelatterisintheirproximity.

    B. LinearpermanentmagnetsynchronousactuatorsLinear permanentmagnet synchronous actuators (LPMSAs) has penetrated into the

    market and established the practical application, since developments in hard and soft

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    magnetic material occurred [16]. Progress of highenergy rareearth magnets, whoseremanencecanbeupto1.2Tat20Candwhoseoperatingtemperaturecanbeupto150C,haspermittedtoreachproperperformancesforseveraldirectdriveapplications.Thereisnodoubt that permanent magnet machines are more expensive than induction machines.However,theyofferimprovedperformanceespeciallyintermsofforcedensityanddynamicresponse.Thisisparticularlytrueinsmallsizesactuators,wherethemagnetisingcurrentforinduction machines becomes important to the disadvantage of the thrust force.Unfortunately,thepresenceofmagnetscanbeaproblem inpollutedenvironmentduetostraymagnetfields,particularlyinpresenceofironparticles,sincetheycanbeattractedandcauseamachinefault.

    PrimarycorestructureforflatandtubularLPMSAsissimilartotheprimaryofLIAs.TheLPMSAssecondary ismadeupofpermanentmagnets,used forexcitation fluxproduction,and a laminated core. Magnets can be buried in the core structure or can be surfacemounted.

    MovingprimaryflatLPMSAsareusedfortraveluptoabout3m,whilstforlongertravelmovingmagnets flat configurations are used. For travel length of less than 1m tubularconfigurationsmay be preferred, since tubular structuresmake better use ofmaterials,resultingincompactactuators.

    C. LinearreluctancesynchronousactuatorsLinearreluctancesynchronousactuators(LRSAs)havethesameprimarycharacteristic

    ofLIAsandLPMSAs,but thesecondarydoesnothousewindingsorPMs.The thrust forceproduction is entrusted to the interaction between the secondary magnetic circuitanisotropy and the primary travelling magnetic field. These actuators are specificallydesigned to emphasize the magnetic anisotropy between qd axes, in fact the forceproductionisproportionaltothesecondarysaliency.Recentresearchinrotatingreluctancemachine has led to development also in the linear counterpart, resulting in a relativelyinexpensivemotorwithgoodperformanceindices.

    Good energy conversion is achieved by thin airgap, but this feature implies largenormal forces, viz. large attractive force between the stator and the mover. Thus themechanicalpartofflatconfigurationmeritsspecialtreatment,inordertoobtainasufficientreliabilityandrobustness.

    LSRAscanconstituteanalternativetoLPMSAsandLIAs.Comparedwith the formers,they have lower costs and can operate at very high speed because of the easier fieldweakeningcapabilityandtheruggedrotor.Itdoesn'thavesecondarywindingthattravelsatsynchronousspeed,sothecontroller issimplerthanotherofACmachines. IfcomparedtoLIAs, ithastheoreticallynosecondary lossesandacomparableforcedensitydependingonthedesignofthemotors.

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    Inconclusion,LRSAarea rarity in industrialdrives,although theyarenotcompletelyunknown[18],[19].

    D. LinearswitchedreluctanceandstepperactuatorsLinear switched reluctancemotor (LSRM) isanelectricmachinewhichposses simple

    structurewithawindingonlyonthestator.Thecostsarelowerthanothermotorsduetoitssimpleconstruction[20].

    Further,thewindingsareconcentratedratherthandistributed,makingthem idealforlowcostmanufacturingandmaintenance.Concentratedwindingsenableanaturallyfailsafesystem that can operate even with a phase shorted or open [21]. The availability ofnumerous lowcostconvertertopologiestodrivethemmakestheLSRAadvantageousoverlinearinductionandpermanentmagnetsynchronousmotors,whichincurahighercostandhavethermaldrawbacks[22],[23].AlthoughLSRAshaveplentyofadvantages,thefactthatthe force rippledeteriorates theperformance restricts itsapplication toprecisiondevices.Muchresearchhasbeenreportedtoreduceforceripple,andimprovedperformanceshavebeenenhanced [24], [25].Tubularconstructionof thiskindofmotorpermits toovercometheattractionforcebetweenexcitationandmover,sincetheaxialsymmetryofthemachineallowsainherentneutralizationofthisforce[26].

    In linearstepperactuators (LSAs) themotion iscausedby the tangential forcewhichtendstoalignpolesofthemovingpartwiththeexcitationpolesofthestaticpart[27],[28].Incontrasttorotarycounterpart,wherethenormalforceofattractionbetweenstatorandrotorpolefacesisneutralizedbythesameforceactingbetweenthediametricallyoppositepairofstatorandrotorpoles,thenormalforceinthelinearcounterpartishighandcanbeaseriousmechanicalproblem.Theproblemmay,tosomeextent,beovercomebytheuseofasymmetricaldoublysidedversionoftherectilinearcounterpart.

    1.4 TechnicalcharacteristicsoftubularlineardrivesTubular lineardriveoffersseveraladvantages incomparisonwithboth the flat linear

    andtherotarycounterpart.Somebenefitsareinherentlyintroducedbythetubularstructureofthemotor,hencetheybelongonlytothiscategoryofmotor,whereasotheradvantagesarecommonlypresentintheotherlinearmachines.

    A. LineartubularactuatorTubular linear actuator isobtained through the transformationdepicted in Figure 3,

    whichawardsspecialfeaturetothemotorifcomparedtotherotatingcounterpart.Thefirstoneconcerns themotorwindings, indeed theydonothave theendwindings, thusall theconductorsandthephasecurrentactivelyparticipatetothemagneticfieldproduction[6].The endwinding Joule losses disappear, hence an increased coil efficiency and a betterpowertoweight ratio compared to both the flat and the rotating topology results. The

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    windingshapelookslikearingofconductors,soitiseasytoimplementanditissuitedtobeindustriallyproduced.Furthermore,thetubulargeometryariseswithinherentfeatureofselfneutralization of normal forces acting between the stator and themover, since they aresymmetricallydistributedoverthemovercircumference.Thepistonlikeappearanceofthetubular linearmachinegives thedesignera firm foothold toconnect themechanical load,andtheclosedframemakesitcapabletooperateinpollutedandhostileenvironments[29].Ontheotherhand,thetubulartopologyimposeslimitsinthemaximummoverstroke,sinceexcessivemoverblendingmaycausevibrationsandmechanicaldamageofthemotor.

    Among the various linear actuators, tubular topologies with permanent magnetexcitationareparticularlyattractive,sincetheydonotshowthetypicalassemblyproblemsof linearopenmachines,having indeedacompactstructureandahigh forcedensity [30][31], [32].Unfortunately, thepermanentmagnets interactwith stator coredeterminingadisturbance force,which cancausenegativeeffects suchasvibrations,noises,positioningerrorsandnonuniformmovementatlowspeed.Suchaforceinlinearactuatorsisthesumoftwocomponents:thecoggingforceandtheextremityforce[33].Theformerisgeneratedbythetendencyofthemagnetstoalignwiththestatoratpositionswherethepermeanceofthemagnetic circuit ismaximized and has a spatial wavelength depending on the polenumber,theslotnumberandthepolepitch,whereasthelatterdependsontheinteractionsbetween the finite lengthof thearmaturecoreand themovingmagnet,andhasaspatialwavelengths of one polepitch. The effects of disturbance force can beminimized at themotordesignstage[34],[35],employingmethodssuchasskewingandoptimallydisposingthemagnets,andoptimizingthelengthofthearmaturecore[36].Furtherimprovementstogetsmoothmotioncanbeobtainedbyemployingacurrentshapingcontrolstrategy[37].

    The production of thrust force makes all linear drives suited to simplifying themechanics inautomatedmachinethroughdirectdriveactuators, jointlytotheadoptionofadvancedcontrolmethodologies.

    Toachieveprecisemotioncontrol,mostofthesemachinesuserotaryelectricalmotorsas their prime motion actuators, and couple their output shafts to mechanical motiontransmissionsas reductiongear,belt,ball screw,pinionrack,etc.Though this is themostwidely used method, mechanical transmissions suffer of several drawbacks, which areemphasizedthroughthehighdynamicperformancesofmodernPMdrivesandcontrols.

    Theuseofmechanicaltransmissions increasethewholedimensionsofmachinesandtheir costs. Figure 5 shows some applications where the linear drives can replace therotatingcounterparts.

    Themostevidentdisadvantagesofindirectdrivesare:

    deadpointsinmechanicalchainthatcancauseproblemforpositioncontrol. backlashes, which decrease the machine precision particularly during the

    inversionofthemotion.

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    stiffness torsional elasticity and bending of the shafts, imputable of resonancewhich

    represents inmany cases an insuperable limit both for the accuracy of thepositioningcontrolandforthemaximumspeed.

    additionalfrictionswhichcausepowerlosses,hencereducedefficiency. additionalmechanicalpartsreducethesystemreliability.

    All these aspects engrave negatively on the performance of the system in terms of

    accuracy, resolutionand repeatability.Applicationswith short strokeenlarge these limits,since the reduction of the positioning times increase the accelerations, viz. the dynamicaspects.

    The directdrive philosophy,which provide thrust force directly to a payload, offernumerousadvantagesover their rotarytolinear counterparts.All thepreviousdrawbacksareovercame,sincethemechanicaltransmission isnoweliminated.Linearmotorsbecomean integral part of the automatic machines, which are capable of higher dynamic

    Figure5Exampleofsomeapplicationswherethelineardrivescanreplacetherotatingcounterpart.

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    performance, improved reliability and higher control bandwidth than the traditionalmachines.

    B. LineartubularactuatorcompetitorsPneumatic and hydraulic pistons have the same appearance of tubular electric

    actuators,thustheyconstituteanalternativechoicetotheelectriccounterpart.Industrial applications often require controlled motion and, in some cases, also

    controlled force.Traditionalhydraulic andpneumatic thrustersoftenuseopenloop forcecontrol.Theforceiscreatedbyapplyingpressuretoaworkinggasorfluidinahydraulicorpneumaticcylinder.Apistonthenconvertsthepressuretoaforceappliedtotheload.

    Thissolutionshowsseveraldisadvantagesifcomparedwithtubularlinearelectricone,firstofall,anelectrictubulardrivehaslargeraccuracy,readinessandcontrolreliability,itisfreemaintenanceandithaseasierenergysupplyavailability,andsecondlyanelectricdrivedoesnotpresentanyfluiddripping.

    1.5 IroncorePMtubularlinearactuatorAmong the various linear actuators, tubular topologies with permanent magnet

    excitationareparticularlyattractive,sincetheydonotshowthetypicalassemblyproblemsoflinearopenmachines,havingindeedacompactstructureandahighforcedensity.Figure6depictsasimplifiedthreedimensionalsightofaPMtubularmotorwithaxiallymagnetizedmagnets.

    Uptofewyearsagothetubularmotorwasnotusedinindustrybecauseofconstructiveandperformancesreasons.Ironlessactuatorsexhibitedlowspecificforcesincehighenergyrareheartmagnetshadbeendeveloped,whileironcoremotorshadtherealizationproblemofamagneticcircuitsuitedforthefluxlinepath.

    Usually, inthetraditionalrotatingmotorstheplanar laminationofthemagneticcore

    Figure6ThreedimensionalsightofaPMtubularmotorwithaxiallymagnetizedmagnets.

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    interrupts the circulation of eddy current induced by timevariablemagnetic fields. In atubular motor eddy currents flow around the actuator circumference, hence theirinterruption shouldbe realizedwithpieslice lamination [38],as shown inFigure7.Thelatter ispracticallyunfeasible, thusother solutionwere adopted.Core lamination able toshortenthepathsofthe inducedcurrentcanbe longitudinalortransversewithrespectofthe direction ofmotion [5], [6]. Both of them show drawbacks: the former exhibits thepracticaldifficultytopreciselyalignthelaminationsofeverycoreallaroundthemotionaxle,as shown in Figure 8, while the latter determines an increase of the magnetic circuitreluctanceduetothesubstantialtotalthicknessoftheinsulationsamongthelaminationsonthemotorbackironwhichhas tobecrossedby the flux lines.Theseproblemshavebeenovercamethankstothescientificand industrialdevelopment,whichhasproducedtheSoftMagneticComposites(SMC).ThisclassofmaterialsisbasicallythatofironpowderparticlescoatedwithanelectricallyinsulatedlayerasshownschematicallyinFigure9.Thesepowderscanbecompactedincomplexformand,afterathermaltreatment,arereadytouseshowingisotropicmagneticproperties.

    Finally,windings of this kind ofmotor have not frontal connections, resulting in acompactstructureandagreatdensityofforce.

    TheprincipalpartsthatformaPMtubular linearactuatorwillbeshortlydescribed inthefollowingsections.

    A. PermanentmagnetsPermanentmagnettubularactuatorscanbeconstructedwithsurfacemountedradially

    magnetizedmagnets,depicted inFigure10,orwithaxiallymagnetizedmagnets,shown inFigure 11.An interesting comparison between the two solutionswas carried out in [39],where it isconcluded that theaxiallymagnetizedmachinehasahigher forcedensity,butmorepermanentmagnetmaterial isrequired. Ifthesamevolumeofpermanentmagnet isused, the two topologies lead to the same force density. However, axially magnetizedmachineshouldbepreferredbecauseaxiallyanisotropicrareearthmagnetsareusuallyless

    Figure7conceptualpieslicelaminationformagneticcoreoftubularactuator.

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    expensiveandwidelyavailable.Magnet typology for this actuators have to be selected considering the motor

    performancetargetfixedindesignstage.However,themostdiffusedmagnetsareinFerriteor inNdFeB.The firstones represent themosteconomicchoice,having low remanence,lowenergyproductandastrongtemperaturederating.

    The NdFeB magnets have at present the most elevated specific energy, similarconsideration concerns the remanence value and the intrinsic coercive field, even withmagnettemperatureof100150C.TypicalvalueofmainparameterforFerriteandNdFeBmagnetaresummarizedinTableI.

    Alloftheseelementshavetobeconsideredduringthemotordesigntomakesurethatthe magnet doesn't suffer permanent damages that jeopardize the reliability and theperformancesofthemotor.

    Figure8Practicalarrangementofstatorcoreusinglongitudinallaminateswithrespectofthedirectionofmotion.

    Figure9SchematicpictureofSMCmaterial.

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    TableIComparisonamongmainparameterofferriteandNdFeBmagnets.

    Parameter Ferrite NdFeB

    Br[T] 0.20.5 1.11.3

    Hc[kA/m] 150295 7001000

    Relativerecoilpermeability

    1.1 1.08

    Temperaturecoefficientof

    Br[%C]0.11/0.12 0.54/0.60

    TemperaturecoefficientofHc[%C]

    0.2 +0.3

    Figure10Axialsectionofatubularmovercomposedofradiallymagnetizedmagnets.

    Figure11Axialsectionofatubularmovercomposedofaxiallymagnetizedmagnets.

    B. MagneticcircuitTheuseofSMCmaterialallowstoovercometheproblemofapropermagneticcircuit

    for tubular actuators, since flux paths have both radial and axial direction. Magnetic

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    propertiesofsuchamaterialhavetobeanalyzedinordertofullyunderstanditspotentiality[40].

    Ingeneral,SMCshavelowerinitialpermeability,lowersaturationinductionandhigheriron lossesat5060Hz than the laminated steel.These limitationscanbecancelled if theisotropic 3D behaviour of SMCwill be exploited [41], [42]. Further, the aforementionedcharacteristics are strongly dependant on the kind of powder and on the transformationprocess used to obtain the finished component. In particular, the electromagnetic andmechanicalpropertieswilldependon theadded lubricant/binderandon theprocess, forexample,coldorwarmcompaction.

    A low amountof additive results in apowdermix that canbe compacted tohigherdensities and heat treatedwith higher temperature,which is beneficial for themagneticproperties,asshown inFigure12.Duringcompactionstress is introduced in theparticles,whichdeterioratesthesoftmagneticproperties.Thehighertheheattreatmenttemperatureisset,thehigherthedegreeofstressrelief.However,itiscrucialthatnosinteringtakeplace

    Figure12HeattreatmenteffectinhysteresisloopcharacteristicofSomaloy500.

    Figure13EffectofmaterialdensityonBHcurvesofSomaloy500.

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    betweentheparticles,asthedynamiclosseswouldincreaserapidlywithfrequency.AspresentedinFigure12,themagneticinductiondependsheavilyonthecompaction

    pressure of thematerial, because it determines thematerial density. A highermagneticinductionforacertainappliedfieldmeans improvedperformanceofthemotor.Comparedto laminations,SMChasnormallyaslightly lower induction level,thoughtthedifference issmallathigher fields.Hysteresis lossesareworse ifcompared to thatoneof laminations,insteaddynamicbehaviourisbetterduetoreducededdycurrentlosses.Thismeansthatatacertaincriticalfrequency,typicallyaroundacoupleof100Hz,the losseswillbe lower inthe SMCmaterial than in laminations. Themaximum relative permeability is significantlylowerinSMCscomparedtotheinplanepropertyofsteellaminations.

    OneofthemainadvantagewithSMCisits3Dproperties;thematerialisisotropic,thusmagneticfluxcanflowequallywellinalldirections.Anotheradvantageistheopportunitytoobtain3Dshapesbyacompactionprocess.Thecombinationofthesetwofeaturesappliedtoparticularmachines, such as thePM tubularone, can compensate the lowermagneticproperties than laminations. Tubular PM machines have high natural reluctance due topresenceofmagnets,thusthepermeabilityisoflessimportanceandgoodperformancecanbestillachieved.

    TheinfluenceofthechoiceofSMCsontheperformanceoftubularpermanentmagnetmachineswas carried out in [43],where it is shown that SMCmachines have a slightlyinferiorperformance ifcomparedwith laminatedone,buttheyareeasiertomanufacture,and,therefore,potentiallybelowercost.

    C. WindingsTheprimarywindingoftubularmotorshassimplediskshape,likethatonedepictedin

    Figure14.Thetubularmotor,foritsownnature,hasaverysimplestatoricstructure,thusitsconstructionconsistsonlyinanalternativelystackofSMCdisksandwindings.

    Thewinding can be preassembled and precompacted in order to obtain good slotfillingcoefficient.

    Thebenefitsarosebythisoperationaremanifold:theelectricexploitationoftheslotisincreased, and finally the thermal exchange ofwinding is improved, bringing to amoreuniformdistributionofthewindingtemperature.

    Figure14Diskshapedstatorwindingsoftubularlinearactuator.

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    1.6 ConclusionAn overall comparison among different topologies of linearmachines has beenwell

    developed, showing merits and demerits of each type of motor. Hence, it results that,among the various linear actuators, tubular topologies with permanent magnets areparticularlyattractive.Infacttheydonotshowthetypicalassemblyproblemsoflinearopenmachines, then theyhaveacompactstructure,moreoversoftandhardmagneticmaterialdevelopment bring them to higher force density. These machines have excellentcharacteristicsasservomotor,whichmakethemsuitedtotheuse inautomated industrialmachines.

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    CHAPTER

    2BasicsofMultiphaseDrives2.1 Introduction

    The conventional structure forvariablespeeddrives consistsofa threephasemotorsuppliedbya threephasevoltagesource inverter (VSI).Theconvertercanbeviewedasadecoupling interfacebetween the threephasemainsand themotor.Thus, theneed foraspecific number of phases, such as three, disappears. Furthermore, the development ofpower electronicsmakes it possible to consider the number of phases as an additionaldesignvariable.

    Nowadays, there is an increasing attention towardsmultiphase drives. Indeed theyoffer a greater number of degrees of freedom comparedwith threephasemotor drives,which can be utilized to improve the drive performance [44], [45]. In this chapter thefundamentalaspectsofmultiphasedriveswillbepresented,focusingtheattentiontothesocalledmultimotormultiphasedrive.

    2.2 MultiplespacevectorrepresentationAs known, a multiphase motor drive cannot be analyzed using the space vector

    representation inasingledqplane. It isknownthat,tocompletelydescribeamultiphaseelectromagnetic system, the space vector representation inmultiple dq planes (multiplespace vectors) must be adopted [46]. Some modern approach to the modulation ofmultiphase invertershaveovercome the inherentdifficultyof synthesizingmore thanoneindependentspacevectorsimultaneouslyindifferentdqplanes,inordertofullyexploitthepotentialmultiphasemotor drives [47]. They have shown that themultiple space vectornotationiswellsuitedtorepresentandtoanalyzemultiphasesystems.

    In a nphase system, for a given set of n real variables x1, x2, x3,...xn, a new set ofvariablescanbeobtainedbymeansofthefollowingsymmetricallineartransformations[48]:

    ( )=

    =n

    k

    khkh xn

    x1

    11 ,(h=0,1,2,...,n1) (2.1)

    where ( )nj /2exp = .

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    Withtheexception for 0=h (and 2/nh = forevennumberofphases),thequantityhx isacomplexnumberand it iscalled spacevectorcomponentof sequenceh,or space

    vectoronthe(dq)hplane.Itcanbeshownthatthespacevectorcomponentofsequence h isrelatedtothespace

    vector component of sequence hn through a complex conjugate relationship,furthermorethespacevectorcomponentofsequence hn coincidewiththespacevectorcomponent of sequence h . These properties can be synthesized by the followingequation:

    *hh xx = (2.2)

    Therealvariable 00 xx = isobtainedfor 0=h andiscalledzerosequencecomponent,whereastheadditionalrealspacevectorcomponentofsequence 2/n appearsforanevennumberofphasesonly.

    Using the aforementioned classifications and applying (2.2), the space vectortransformationscanberewrittenasfollows:

    ( )=

    =n

    k

    khkh xn

    x1

    12 (h=1,2,...,r) (2.3)

    =

    =n

    kkxn

    x1

    01

    (2.4)

    =

    =n

    k

    kkn xn

    x1

    12/ )1(

    1 (2.5)

    where 12/ = nr for an even phase number and 2/)1( = nr for an odd phase

    number.Thecorrespondinginversetransformationsare:

    ( ) ( )=

    ++=r

    h

    khh

    knk xxxx

    1

    112/0 )1( (k=1,2,...,n) (2.6)

    wherethesymbolrepresentsthedotproduct.

    Intheparticularcaseofbalancedandsinusoidaloperatingconditions(sequence1),the

    spacevector 1x assumesaspecialrelevancebeingtheonlyspacevectordifferentfromzero.However,itisopportunetoemphasizethat,inthegeneralcase,alltherspacevectors

    aswellasthezerosequencecomponentarenecessarytocompletelydescribethenphasesystem.

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    2.3 MathematicalmodelofmultiphasePMsynchronousmachines

    ThebehaviouroftherotatingnphasesynchronousmachinewithsurfacemountedPMcanbedescribedintermsofmultiplespacevectorsbythefollowingequationswritteninthestatorreferenceframe:

    dtdiRv shshssh+= (2.7)

    jhehseshshsh eiMiL ,+= (2.8)

    [ ] jejeeje edtdeiRev += (2.9)

    [ ]=

    +=r

    hshhse

    jee

    je ieM

    neiLe1

    ,2 (2.10)

    Fromthemagneticcoenergy,thetorqueequationcanbeobtainedasfollow:

    ( )=

    =r

    h

    jheshhse ejiiMhp

    nT1

    ,2 (2.11)

    wherepisthepolepairsnumber,histhemultiplespacevectorindex,andistheelectricalpositionoftherotorreferenceframe.

    Multiphase drives enable the independent control of a series ofmultiphasemotorssuppliedbyonemultiphase inverter,provided that themachinesaredesigned toproducesinusoidalfield distribution in their airgap. The latter feature simplifies the torqueexpression(2.11),infactinthiscaseonlythefirstcurrentspacevectorinfluencesthetorqueproduction.Moreover, considering a PM brushless tubular linear actuator, the equations(2.7)(2.10) remain unchanged, whereas equation (2.11) becomes the expression of thethrustforce(2.12)undertheassumptionofnegligibleendeffects.

    ( ) jestsep ejiiMnF = 11,2 (2.12)

    where p is the pole pitch, etse iM 1, is the voltage constant expressed in )//( smV and therelationbetweenelectricposition andlinearposition x is:

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    xp = (2.13)

    2.4 MainfeatureofmultiphasedrivesMultiphasevariablespeeddrivehasreceivedgrowinginterestsincethesecondhalfof

    the1990s.Theareaofelectricshippropulsionandtheirdevelopmentsactedinthatperiodaspropeller for the research [49], [50].Nevertheless,oldest records concerning this topicdatesbacktothesecondhalfofthe1960s.Historicaltechnicalreasonsthaturgedtoadoptthemultiphasedrivesolutioninsteadofthreephaseonearelistedbelow[51]:

    1) Multiphase variable speed drive reduces the stator current per phase, for a

    givenmotoroutputpower.2) Theuseofmorethanthreephasesofferanimprovedreliabilityduetothefault

    tolerancefeaturesofmultiphasedrives3) Multiphase machines present reduced pulsating toques produced by time

    harmoniccomponentsintheexcitationwaveform.As stated at point n1, the use ofmultiphase drive instead of the threephase one

    helped toovercome theproblems related tohighpowerapplicationswithcurrent limiteddevices.Foragivenmotorpower,an increase inphasenumberdeterminesareduction inpowerperphase,enablingtheuseofsmallerpowerelectronicdevicesineachinverterleg,withoutincreasingthevoltageperphase[52].Thisisstillasolutionadoptedforhighpowerapplications,suchastransportandshippropulsiondrives.Infact,largemultiphasemachinesforshippropulsionhavealreadybeenprototypedindustrially,andarecurrentlyundergoingcommercialevaluation.

    The improvedreliabilityfeaturesofmultiphasedrives, listedatpointn2,enabletheiruse also in faulty conditions, in fact ifonephaseof amultiphasemachinebecomeopencircuit,themachineisabletoselfstartandtorunwithonlyaderating,whichdependsonpostfaultcontrolstrategyandinthenumberofthephases.Inthethreephasecase,thelossof one phase determine an important derating of the machine while it is running,furthermore themachine isnot selfstartingand, for thispurpose, it requiresanexternalmean.Furtherdetailswillbegiveninparagraph2.5and2.6.

    Finally,theadvantagesderivedfromstatementatpointn3wasveryimportantinthe60s,whenthreephaseinverterfedacdrivesoperatewithsixstepmode.Timeharmonicofvoltages and currents introducedby thisoperationmodeproduced low frequency torqueripple, leading to difficulties on speed control and noise production. Since in a nphasemachine torquepulsationsarecausedbysupply timeharmonicsof theorder2n1,whichresult intorquerippleharmonic2ntimeshigherthanthesupply frequency,an increase in

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    the number of phases seem the best solution to the problem. This aspect ofmultiphasedriveshas lost importance since thediscoverofPWMofVSI,whichallows the controlofinverteroutputvoltageharmoniccontent.

    Another featureofmultiphasemachines concerns theirefficiency [53], indeed statorwindings create a magnetic field in the machine airgap with a lower spaceharmoniccontent. It results, for two identicalmachineswith the same airgap field and the samefundamental component of stator current, a reduction of stator Joule losses, hence anincreased efficiency. The stator Joule losses savingobtainedby increasing thenumberofphasesfromthree isrelatively low.Unlikely, itapproachesanasymptote,thustheuseofafifteenphasemachine insteadof twelvephaseoneproducesa stator Joule losses saveofonly0.2%[1].

    Someoftheaforementionedtechnicalcharacteristicsarenowadaysstillasrelevantastheywere intheearlydays,suchasthatoneatpointn1andatpointn2ofthepreviouslist.Overtheyears,multiphasedrivecharacteristicshavebeenexploredmoreindetailsandotherbeneficialfeatureshavebeenrecognized,suchasthepossibilitytoenhancethedrivetorque capability [54], [55]and theopportunity to realizeamultiphasemultimotordrivesystemwithsingleinvertersupply[56].

    The two latter aspects, togetherwith the fault tolerant feature, appear as naturalsolutions for some industrialproblems, thus theywillbedescribedmore indetails in thefollowingparagraphs.

    Allthefeaturesabovepresentedaresuitedbothformultiphaserotarymachineandformultiphaselinearones.

    2.5 MultiphasedrivesfortorquecapabilityenhancementApossibleway touse the largenumberofdegreesof freedomofmultiphasedrives

    consistsintheutilizationofthespatialthirdharmoniccomponentofthefluxdensityintheair gap for increasing the torque production capability. This feature could be particularlyinterestinginapplicationswherevolumetricorweightconstraintsareimportant,indeedforagiven torquevalue, thevolumeand theweighof themultiphasemachine controlledasstatedabovearesmallerthantheonesofthethreephasemotor.

    The relation between time and spaceharmonic fields produced by a multiphasewindingpresentedin[53]explainsclearlytheincreasingtorquecapabilityfeature.Underthehypothesis of stator current with fundamental component of frequency f together withtimeharmonic components at integermultiplesof f, theexcited P2 polenphase statorwinding can bemodelled by blocks of current and then resolved into rotating harmonicsurfacecurrentdistributions,thusitresultsoftheform:

    =k

    PtkjkSS eJtJ

    )(2Re),( (2.14)

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    32|P a g e

    where:

    f 2= (2.15) In(2.14),kisapositive integerwhichrangesoveralltimeharmonicsproducedbythe

    inverter, and is the azimuthal coordinate. The rms phasor of stator surface currentdensityisrepresentedby kSJ ,ithas P2 polesandrotatesataspeedof Pk = radianspersecondwithrespecttothestator.

    Theexpressionfor kSJ is:

    pdkk

    S KKDZIn

    J = (2.16)

    where Z is the number of seriesconnected conductors per phase, kI is the kth timeharmonic component of phase current, and D is themean airgap diameter. Pitch anddistributionfactorforthe thharmonicarerespectively dK and pK .

    Itisshownin[53]that kSJ isnonzeroonlyforvaluesof thatarerelatedtokandtothephasenumbernbytheexpression:

    ink 2= ,...3,2,1,0 =i (2.17) Equation (2.17) contains the basic information about the time and space harmonic

    fieldsproducedbyannphaseexcitation. Inafivephasemachine(n=5),thefundamentalfrequency excitation ( 1=k ) produces spaceharmonic fields of order = 1, 9, +11, ...whereasthethirdtimeharmonic( 3=k )producesspaceharmonicfieldsoforder = 3,7,+13, ecc. It is shown that the fundamental timeharmonic in excitation controls thefundamental spaceharmonic in the airgap of the machine, moreover the third timeharmonicinexcitationcontrolsthethirdspaceharmonic.

    The independent control of the first and third spatial harmonic component of themagneticfieldintheairgapofamachinewithconcentratedwindings(oneslotperpoleperphase, i.e.q=1)canbeusedtosynchronizethethirdspatialharmonicwiththefirstone inorder to obtain an airgap flux density nearly trapezoidal, leading to backEMF voltageshavingnonsinusoidalwaveforms,sothatthetorqueperamperecanbe increasedby10%[53],[55],[57].Ahighperformance inductionmotordrivewithhightorquedensitycanbeobtained using an Extended Field Oriented Control [58], [59] or an Extended DirectTorqueControl[60].

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    Furthermore,multiphase PM synchronousmotorswith surfacemountedmagnets intherotorhaveanearlyrectangularairgapfluxdensitydistribution,whereasthearmatureisrealized with fractional slot windings in order to minimize the cogging torque effect.Nevertheless, the use of concentratedwindingwith a fullpitch is able tomaximize thetorque per ampere. These motors have the advantages of both BLDC (rectangularfedbrushless dcmotors)motors and PMSMs (sinusoidal fed PM synchronousmotors). Thismeans that ithas thecontrollabilityofaPMSMwhilehaving thehigh torquedensityofaBLDCmotor[61].

    2.6 FaulttolerantdrivesThe increased number of phases in multiphase drives offers considerable benefits

    becauseofthecapabilitytocontinueoperationwhenasingleormultiplephaselossoccurs[62].

    Threephasedrive issensitivetodifferentkindsof faults,both inmotorphaseand ininverterleg.Whenoneofthesefaultsdoesoccurinonephase,thedriveoperationhastobestopped for a nonprogrammedmaintenance schedule. Themotor in faulty conditions isabletorunbutitisnotstillselfstarting.Thecostofthisschedulecanbehigh,thusjustifyingthedevelopmentoffaulttolerantmotordrivesystems.

    On the contrary, multiphase machines in postfault condition can continue to beoperatedwith an asymmetricalwinding structure andunbalancedexcitation,producing ahigher fraction of their rated torqueswith little pulsationswhen compared to the threephasemachines[63],[64],[65].

    Such a feature is very useful in safety critical applications,where thewhole systemreliabilityhas tobehigh,evenwhena faultoccurs.Applicationswhichuse the bywiretechnology, naval propulsion systems, moreelectric aircraft and electric and hybridvehiclesaretypicalexamples.

    Fault tolerant feature can be easily obtained inmultiphasemachine designed as nphasemachinecomposedofbgroupsorwindingsofaphasesystems [66], [67], [68][69].Thewholemotorhasn=abphases,butexistsb isolatedneutralpoints andeachaphasesystem is fed by b aphase drives. The postfault strategy for this kind ofmotor is verysimple, theaphasewinding inwhich the faulthasoccurred isdisconnected,whereas theremaining aphase windings can continue to operate without any control algorithmmodification.Thetorqueandpowerderatingcoefficientforeveryaphasesystemdetachedis(b1)/b.

    Amultiphasemachinewith singleneutralpointallows to fullyexploit thedegreesoffreedomofmultiphasedrives, thus itoffersbetter characteristics inpostfaultoperationsthanthemultineutralpointsolution[70].Theexclusionofonlythefaultyphasewithinthepossibilitytocontinuetooperatethehealthyphases, leavingoutproblemsrelatedtofaultisolation,permitstominimizethetorqueandpowerderating. Indeed,foragivennphase

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    motor, the single neutral point solution permits the exclusion of the only faulty phase,whereas inmultineutral point solution is necessary to take out of service the aphasesystemwhichcontainsthefaultyphase,thusaphasesaredetached.

    Thecontrolstrategy forpostfaultoperation inasingleneutralpointmachinehas todetermine the relationsbetween thecurrents in the remaininghealthyphases inorder topursue a specific aim. The effect of the strategies used, for a given multiphase drive,depends on the load operating point and on the load torque characteristic. Keepingmagnitudeandphaseof thecurrents in theremainingunfaultedphasesat theirprefaultvalues, it results ina stator joule losses reductionbya factor (n1)/nandaconsequentialreduction in developed torque, but the slip increase alongwith the rotor loss. This easycontrol strategy, does not exploit the whole number of degrees of freedom of themultiphasedrive.

    Analternativesolutioncanallowan increase inthemagnitudeofthecurrent ineachunfaultedphase.Hence,thewholestator joule lossremainsat itsprefaultvalueand lowtorquepulsationsresult.Thissolutiongivesbothasmallerdropinspeedandasmallerrotorjoulelossesthanthepreviousstrategy.

    A thirdstrategycouldpursue theaim tomaintain the torqueand thepowerat theirprefaultvalue.Anincreaseofthemagnitudeofthecurrentineachunfaultedphase,higherthan the previous strategy, is necessary; consequently it determines also an increase instatorjoulelosses.Thusthissolutionisnotsuitedforoperationssustainedforalongperiodoftime,sinceoverheatingproblemsreveal.

    Thepractical implementationofanyof these strategiesmightproveotherproblems,suchastotheratingofthepowerelectronicswitchesandthelimitationimposedbytheDClinkvoltage.

    2.7 MultiMotordrivesAnotherpossibilityofferedbymultiphasedrivesisrelatedtothesocalledmultimotor

    drives. A welldefined number of multiphase machines, having series connected statorwindings with an opportune permutation of the phases, as shown in Figure 1, can beindependentlycontrolledwithasinglemultiphaseinverter[71][72].

    Avectorcontrolalgorithmcanbeappliedtoeachmachineseparatelytogeneratetheinverter legcontrolsignals,andtosupplythestatorwindingsofthemultimachinesystemfromasinglecurrentcontrolledbyavoltagesourceinverter(VSI).Invertercurrentcontrolisperformed inthestationaryreferenceframe,using inverterphasecurrents.Thenumberofmultiphasemotorsconnected inseries,orrather thenumberofdrivingaxles,dependsonthe number of the phases of themultiphase inverter. Themaximum number ofmotorconnectable inseries, forasystemwithnphases, is 2/)2( = nM , incaseofn isaevennumber,whileitis 2/)1( = nM ifnisanoddnumber.

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    Thenecessity to control severalelectricalmachineswitha coordinatedmotion is anusual requirement inautomatic industrialapplications.Someexamplesare theXYplacer,highspeed packaging machine, pickandplace machine used for electronic boards andsiliconmonocrystalproductionequipments.

    Thetraditionalsolutionsusethreephasedrives,orratheronethreephasemotorfedbyone inverter foreachdrivenaxleshown inFigure2.Thecontrolarchitectureofsuchasystem canbe centralizedordecentralized. In the first case the controlofeachmotor iselaboratedby a central controlunitorbyone controlunitbelonging to amore complexhierarchicalcontrolsystem,andeachinverterbehavesinmanycasesasapoweramplifierofthecontrolsignalselaboratedbythecentralizedcontrol. Inthesecondcasethecontrolof

    Figure1Conceptualwindingconnectionsof 2/)1( n machines inmultimotordrivessuppliedbyanphaseinverter.

    Figure2Multimotorsconfigurationwiththreephasedrives.

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    36|P a g e

    eachmotor is directly implemented on its own inverter, and themotion synchronizationamongthewholedrivingaxlesismanagedbyonemasterdrivethroughanelectroniccam.

    Taking intoaccount thepowerconnectionpointofview,each invertercanbe fedbythe threephasegridorcanbeconnected toaDC linkasshownschematically inFigure3,enablingacompleteenergyrecoverysystem.

    Multiphasemultimotorsdrivesprovidedifferentsolutionsratherthanthethreephaseone, both for power feature and for control structure. The former results in a hardwaresaving, indeed takingasexamplea twomotordrives, itresults that theminimum inverterleg number required for multiphase solution is five, whereas the classical threephasesolutionsusessixinverterlegs.Thehardwaresavingconcerns,besidesdiscretecomponentsoftheinverterleg,alltheauxiliarycircuitandcomponentsusedtocontrolthecomponentsof the leg, such as drivers, protection circuits, electronic power supplies etc. The savingexistsforaphasenumbergreaterthanfive,andingeneral,higheristhenumberofphases,greateristhenumberofsavedlegsis.

    Another excellent featureof themultiphasemultimotordrives consists in a controlcomponents saving, in fact the implementationof thevector controlalgorithm forall themotorsconnectedinseriescanbemanagedwithinasingleDSP.Thecontrolofeachmotorcan be executed in parallel to give the proper control references that, after appropriatesummation, results in thephase current control references. If the threephase solution isconsidered,oneDSPwillbeused foreachdrivebecauseeach inverter iscontrolledwithaDSP.

    Theonlydrawbackofmultiphasemultimotordrivesisanincreaseofthewholestatorwindinglossesduetotheflowofthephasecurrentsthroughthestatorwindingsofallseriesconnectedmotors.Consideringthesituationwhereonlyonemotor isrunning, itresults in

    Figure3MultimotorsconfigurationwiththreephasedrivesconnectedwithcommonDCbus.

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    statorJuolelossesinthemotoritselfandintheothermotorsconnectedinthemultimotordrive.Theefficiencyofthemultimotordrivedecreaseswhencompared to theequivalentthreephase case, but the advantages above discussed seem to overcome the efficiencydrawback.

    A. MultiMotordriveconceptVectorcontrolofmultiphasemachinerequiresatminimumonlytwocurrent, i.e.only

    one space vector current, if only the fundamental field is utilized, as shown by equation(2.12).The additionaldegreesof freedomofferedbymultiphasedrives canbeutilized tocontrolindependentlyothermachineswithinamultiphasemultimotordrivesystem.Therearenotanyconstraintsconcerningwiththemotortypologiesconnectableinseries,thenthemultimotor drive canbe composed of induction and/or permanentmagnet synchronousand/orsynchronousreluctancemotors.

    If amachine is designed to produce sinusoidalfield distribution [51], then standardmodellingassumptionappliesandonlythefirstharmonicof inductancetermsexists inthephasevariablemodel.TheapplicationsofClarkes transformation,given in theexpression(2.18) for an odd phase number, to the phase voltage equations produces a set of nequations in the stationary frame reference.The firstpairofequations is identical to thecorrespondingoneforathreephasemachine,andtheyarethetwocomponentsofthefirstspacevector.Thelastrowin(2.18),orthelasttworowsforevenphasenumbers,representsthe zerosequence equation. The other couple of rows in expression (2.18), which are(n3)/2foroddphasenumberand(n4)/2forevenphasenumber,correspondtothehthspacevectors,wherethespacevectorindexhhasthesamemeaningexpressedin(2.3).

    Asdiscussedabove,thefirstspacevectorofthecurrentistheonlyoneabletoproducetorque/force inthemotor.Theothercurrentspacevectorsdonotaffectthetorque/forceandtheirvaluesareinfluencedbytheirrelativevoltagespacevectorsandthestatorwindingleakageimpedances.Thesecurrentspacevectorsrepresentdegreesoffreedomthatcanbeusedtocontrolothermachinesconnected inserieswiththefirstmachine.Ifthecontrolofthemachineshastobedecoupledonefromtheother,torque/forceproducingcurrentspacevector of onemachinemust not produce torque/force in all the othermachines in thegroup.Thiscanbeachievedwithaproperseriesconnectionofthemotors,abletotransformeach current space vector into the first current space vector for its relativemotorof thegroup.Inotherwords,thestatorwindingconnectionsofmultiphasemachinesmustbesuchthatwhatonemachineseesasthefirstcurrentspacevector,theothermachinesseeashthcurrentspacevector,andviceversa.

    In order to do so, it is necessary to connect statorwindings of all themultiphasemachines in serieswithanappropriatephase transposition,and finallycloseallphasesofthelastmotorinstarconnection,asshowninFigure1.ThenecessaryphasetranspositionistakendirectlyfromtheClarke'stransformationmatrix(2.18),readingitpercolumnsinorder

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    tounderstand the spatial angle among the rows,which represents thephase step. Thus,phases1ofallthemachineswillbeconnected inserieswithouttransposition, indeedthephasestepamongtherowsincolumn1iszero.Phase2ofthefirstmachine,viz.column2,hasaphasestepequalto;thismeansthatitwillbeconnectedtophase3ofthesecondmachine,whichwillbefurtherconnectedtophase4ofthethirdmachineandsoon.Inthesame way, phase 3 of the first machine is connected to the phase 5 of the secondmachine,whichfurthergetsconnectedtophase7ofthethirdmachine,andsoon,becauseof the phase step is 2. This explanation provides all the information to construct theconnectiontableorconnectivitymatrixproposed in[56],[75],andreportedhere inTableIinthegeneralcaseofanphasesystem.Thefirstcolumnin

    TableIshowsthemachinenumberconnectedinseries,thefirstrowshowsthephasenumber.

    Then it becomes possible to completely independently control themachines whilesupplyingthedrivesystemfromasinglecurrentcontrolledvoltagesourceinverter.

    Afterthedescriptionofthemaincharacteristicsofmultiphasedrives,itisworthnotingthat themultiplespacevectorsassumedifferentmeaningcasebycase.Asanexample, infivephasemachineswithconcentratedwindings,thefirstspacevectorofthestatorcurrentsisresponsibleofthe firstspatialharmoniccomponentofthemagnetic field intheairgap,whereasthethirdspacevector isrelatedtothethirdharmoniccomponent[57],[58],[74].On theotherhand, inmultiphasemultimotordrives, the first spacevectorof theoutputcurrents is responsible for the first spatial harmonic component of the first machine,whereas the third space vector generates the first spatial harmonic component of thesecondmachine[71].

    ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

    =

    21

    21...

    21

    21

    21

    21

    21sin2

    21sin...3

    21sin2

    21sin

    21sin0

    21cos2

    21cos...3

    21cos2

    21cos

    21cos1

    .....................)3(sin)6(sin...)9(sin)6(sin)3(sin0)3cos()6cos(...)9cos()6cos()3cos(1)2(sin)4(sin...)6(sin)4(sin)2(sin0)2cos()4cos(...)6cos()4cos()2cos(1)(sin)2(sin...)3(sin)2(sin)(sin0)cos()2cos(...)3cos()2cos()cos(1

    2

    nnnnn

    nnnnnn

    C

    (2.18)

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    TableIConnectivityMatrixforgeneralnphasecase.

    A B C D E F G H I L M

    M1 a b c d e f g h i l ...

    M2 a b+1 c+2 d+3 e+4 f+5 g+6 h+7 i+8 l+9 ...

    M3 a b+2 c+4 d+6 e+8 f+10 g+12 h+14 i+16 l+18 ...

    M4 a b+3 c+6 d+9 e+12 f+15 g+18 h+21 i+24 l+27 ...

    ... ... ... ... ... ... ... ... ... ... ... ...

    B. MachineconnectivityThe phase number n influences the number of connectable machines and their

    individualphasenumbers.Thisissueiswelldescribedin[75],wherethemultimotordriveisdividedinthreecategoriesdetailedasfollows.

    1) Thenumberofphasesnisaprimenumber,thenthenumberofmachinesconnected

    inserieswithphasetranspositionis:

    2)1( = nM (2.19)

    It corresponds to the number of independent space vectors obtained withtransformation(2.3).Everymachineofthegrouphasanumberofphasesequalton.Exampleofmultiphasesystemsthatbelongtothiscategoryaren=3,5,7,11,13,17,19,etc.

    2) The number of phases does not belong to the previous category, but it can be

    calculatedas:

    mln = (m=2,3,4,...) (2.20) The number of machines in series is still calculated with (2.19), but not all Mmachineshavenphases. For example,with aninephase system, themultimotordriveiscomposedofM=4machines:oneofthemisathreephasemachine,whereas

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    the remaining threemachines have nine phases. The connection diagram for thiscase is shown inFigure4.For thegeneralcaseofm>1, themachinesof the serieshaveanumberofphaseswhichbelongtothefollowingsuccession(2.21):

    12 ,....,,, mln

    ln

    lnn (2.21)

    Multiphasesystemsbelongingtothiscategoryhaveanumberofphasesn=9,25,27,49,81,etc.

    3) Thenumberofphasesdoesnotbelong to thecategory listedatpointn1andn2.

    However,n isdivisibleby twoormoreprimenumbers,denotedasn1,n2,n3,etc.Thenn=n1n2n3nj.Thus,themaximumnumberofmachinesconnectableinseriesiscalculatedby(2.22):

    2)1( < nM (2.22)

    Orderingrulesrequirethatallthenphasemachinesareatfirstconnectedinseriestothe source,withphase transposition. Then all theothermachines follow,orderedwithadecreasingphasenumber.Thus,ifthesequenceofthelargestprimenumberwhichdividesnisn1,n2,n3,...,thenthesequenceofthemotorsconnectedinseriesisordered as the prime number sequence. The observation of this rule permits theseriesconnectionofthewholegroupofmotors,otherwiseitshouldbeimpossibleto

    Figure4Connectiondiagramwithphasetranspositionfortheninephasecase.

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    connectamotorwithhighernumberofphasesafteralowerphasenumbermachine.Thus the latest nphase machine has to be connected with the first n1phasemachine, the latestn1phasemachinehas tobe connectedwith the firstn2phasemachine,andsoon.Theconnectionofmachineswithdifferentprimenumbers,suchasn1andn2,seemstobeimpossible,sincetheration1/n2isnotaintegernumber.Anattempt to connect these kind ofmachines leads to the shortcircuiting of someterminals.Thegroupofmachineswillbecomposedofmotorswithdifferentphasenumber,ingeneralthemachinephasesareequaltonandoneoftheprimenumbern1,n2,n3,etc.Thatis,thenumberofphasesofmachinesaren,n1orn,n2orn,n3orn,...orn,nj.This class of odd phase numbers encloses the situationwhere, besides the sameprimenumber,thereisatleastoneotherprimenumberinthesequence,suchasn=n1n2n3njl

    m.Thephasenumbersbelongingtothiscategoryaren=15,21,33,35,39,45,51,55,57,63,65,69,75,77,etc.

    Categoriesatpointn2andn3ofthelistcanbeconsideredasubcategoriesofamore

    general case, where n is not a prime number. The common feature among these twocategories is that theseriesconnection iscomposedwithmotorswith twoormorephasenumbers.

    C. ControlThe torque/force produced by a nphase PM synchronous machine is directly

    controllablethroughthecurrentspacevectors,asshown in(2.11).Amachinedesignedformultimotordrivehastoproducesinusoidalfielddistributionintheairgapinordertomakethemachine controllable though the only first current space vector,whereas the othercurrent space vectorsmust not influence torque/force, viz. themotion control. CurrentcontrolVSIisthusrequiredtosupplythemultimotorsystemwithdecoupledcontrolofeachmotor. Any of the available vector control algorithm is applied in conjunctionwith eachmachineinordertogenerateindividualmachinephasecurrentreferences,whicharefurthersummed in an appropriateway to obtain the inverter phase current references. Vectorcontrol of multimotor drives can be achieved by using either current control in thestationaryreferenceframeorintherotatingreferenceframe[73].Thelattershowsin[76][77]anincreasedparametersensitivity.Thisheppenssincethedecouplingvoltagesbecomefunctionsofparametersofbothmachines,duetotheneedforcompensatingvoltagedropsin onemachine caused by the flow of the torque/force producing currents of the othermachinethroughout itswindings.Thus,control inthestationaryreferenceframe isbettersuitedseriesconnectedmultiphasemultimotordrivesystems.

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    2.8 ConclusionMultiphasedriveshavelargenumberofdegreesoffreedomifcomparedtotheclassic

    threephasesystem.Though themultiphasedriveconcept isnotnew, ithasattracted theattention of the research community since the 90s. Thus, new features have beendeveloped, such as thedrivepower capabilityenhancement, faulttolerant characteristicsand, last but not least, the multimotor drive configuration. The latter represents aninterestingandpromisingsolutionforautomaticmachinewherecoordinatedmotioncontrolamongseveralmotorsisrequired.Thenitwillbeinvestigatedinthefollowingchapters.

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    CHAPTER

    3PMTubularMotormodellingandoptimizeddesignalgorithm3.1 Introduction

    AnoptimaldesignofPMtubular linearactuatorspermitsto fullyexploitthe limitsofsoftandhardmagneticmaterial,resultinginhighperformancemotors.Thisobjectivecanbereachedpayingattentionnotonlytotheelectromagneticissues,butalsotothethermalandmechanicalconstraints.

    Anintegratedelectromagnetic,thermalandmechanicaldesignisdevelopedinordertomaximize the thrust force or the dynamic performance of a series of PM tubular linearactuators.Furthermore, thedynamicbehaviourof themotor isvery importantdue to theinherentreciprocatingmotionofthemover,thusanoptimizednumberofwiresperslot isdetermined through numerical simulations which take into account also the inverterconstraints.

    3.2 ReviewofdesignoptimizationmethodologiesA variety of techniques has been employed to facilitate the design optimization,

    predictionofmagnetic fielddistributions throughanalytical field computationwas carriedout in [78], [79], [4],whereas themost common approach employs a lumpedequivalentcircuit[80],[81],[82],[83],finallyFiniteElementMethod(FEM)representsanotheranalysistool. The first one gives good calculation results of field distribution and thrust force,considering also cogging force effect, but it does not allow tomanage the relationshipbetweencriticaldesignparametersandmachineperformance,ashappenswiththesecondmethodology. Finally, FEM analysis provide an accurate mean to determine the fielddistribution,thattakesintoaccountthesaturation,butitremaintimeconsuming.

    Itdoesnotexistabettersolution,becauseeachtechniquehavemeritsanddemerits,dependingalsoonthemotordesignconstraintsandinthemotortopology.

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    3.3 ElectromagneticmodelThestructureofthePMtubularactuatorisshowninFigure1.Thesliderisconstituted

    by a tube containing axiallymagnetized NdFeBmagnets alternatedwith ferromagneticdisks. The stator is composed of a magnetic yoke, and several inner disks. The statorwindings,placedbetweenthedisks,havetheformofcylindricalcoils.

    A. PreliminaryconsiderationsInorder todefineanoptimizedalgorithm, themain technical issuesof tubular linear

    actuatorshavetobeconsidered.Magneticcircuit laminationcannotbeadoptedbecauseof the3D fluxpaths,but the

    use of SoftMagnetic Composites (SMCs) can solve the problem tomould the yoke, thestatorinnerringsandthesliderpoles.AlthoughSMCshavelowerstrength,higherlossesatlowfrequenciesandlowermaximumpermeabilitythanlaminations,insulatedironpowderspresent isotropicmagneticpropertiescombinedwith thepossibility tomouldcomplex3Dmagneticcircuits.

    A preliminarymachine design can be carried out on the basis of a simplified onedimensional fieldanalysis inordertodeterminethemainactuatordimensions,oncesomekeyparametersareprefixed.Thisallows,forgivenvolumespecifications,thedeterminationof themain actuator dimensions and performance, considering thermal andmechanicalissues.

    B. ThrustforcecontributionsIntheactuatortopologyshowninFigure1,theconstantcomponentofthethrustforce

    is produced by the permanentmagnet andwinding current interaction. There is also anadditionalconstantcomponentofforce,thereluctanceforce,duetothevaryingreluctanceofthemagneticcircuit.Inthistypeofactuators,therearefoursourcesoftheforceripple:

    1) thecoggingeffect, i.e.thetendencyofpermanentmagnetstoalignwiththestator

    teethatpositionswherethereluctanceofthemagneticcircuitisminimized,2) theendeffects,duetotheopenmagneticcircuitnature,3) thepresenceofharmonics intheairgap fluxdensitydistributionofthepermanent

    magnets,4) thepresenceofhighorderharmonicsinthereluctanceofthemagneticcircuits.

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    Thefirstcontributionproducesthecoggingforce,thesecondcontributionthesocalledendeffect force, and the third contribution is responsible of the field harmonicelectromagneticforce.ThecausesofforcepulsationscomingfromthePWMsupplyarenotconsideredinthispaper.

    Unfortunately,thetraditionalmethodsusedinrotaryPMbrushlessmachinestoreducethe cogging torque, i.e. use of skewed or asymmetrical distributions of the permanentmagnets[34],arenoteasilyapplicableintubularmachines.Asaconsequence,inthesetypesofmachines,thecoggingforcecanbereducedactingonlyonthegeometriesofstatorslots,magnetsandsliderpoles.

    Theendeffect force, instead,canbecompensatedadding ferromagneticdisks inoneor inbothextremityof the statorassembly: thisaspectwillbediscussedparagraphs3.6,sectionD.

    C. FundamentalhypothesisSome hypothesis has used to simplify the definition of themodel equation for the

    consideredactuatortopology.Inparticular,ithasbeenassumed:

    onedimensionalfieldanalysis; infiniteironpermeability; cylindricalsymmetry; negligibleendeffects; negligibleironsaturation.

    Despitesomeoftheaforelistedassumptionsseemtobeveryrestrictive,theyresults

    fromapproximationssuitableforthekindofmotoranalyzed.Thestatementinthefirstpointsimplifying the problemmodelling,whereas the simplification stated in the second pointconsider that PM tubular machines have low natural permeance due to presence ofmagnets, thus the iron reluctance is less important and canbeneglected. The cylindricalsymmetrylistedinthethirdpointisainherentpropertiesofthemotor.Theendeffectcanbe neglected as stated in the fourth point if proper strategies are used. Finally, ironsaturationdoesnotoccurwhenthemachine isdesignedsuchthatthemagnetic load is inthelinearpartofthefirstmagnetizationcurveofiron.

    Figure1Basicschemeofatubularpermanentmagnetmotorwithaxialmagnets.

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    D. DesignofthemagnetsThe main geometric parameters used to define the electromagnetic model of the

    motoraredepictedinFigure2.Themagnetization curve of theNdFeBmagnet can be expressed by the following

    relationship

    rmrm BHB += (3.1)

    where Bm is themagnet flux density in axial direction, Hm themagnetizing force, Br themagnetremanenceandrthemagnetpermeability.

    Themagnetdiametercanberelatedtothepolepitchbyequatingtheslidercorefluxtotheairgapfluxasfollows

    pgg2sib

    2sim td BddB = ))((2 4 (3.2)

    wheretp isthepolepitch,Bg istheairgapfluxdensitycalculated inthemeandiameterdg,thelatterdefinedas

    g 2hdd tsig ++= (3.3) Themagnetwidthcanbeexpressed in termsof theairgap fluxdensityandmagnet

    magnetizingforceasfollows:

    hs

    wp wm

    wt ws

    tp

    ts

    dse dex

    wep

    hep

    hb

    dsib

    ht

    g

    Figure2Drawingofthetubularactuatorshowingthegeometricalparameters.

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    C0

    g

    mm K

    BH2w 0= (3.4)

    where0isthemagneticairgapgivenby

    ght +=0 (3.5)

    and CK istheCarterscoefficientthattakesthestatorandslidersloteffectintoaccount.It

    canbeexpressedby

    ( )( ) ( )( ))/(5 /)/)((5 /)( 002

    0

    0

    02

    0

    M

    MP

    P

    eps

    epsS

    SC

    wwt

    t

    wtwt

    t

    tK

    ++

    = .(3.6)

    E. SliderandstatordesignTheslidercorefluxdensityBpcanberelatedtothemagnetfluxdensityby

    ))(( 2sib2si4

    mPsip ddB2 wd B = . (3.7)

    Thetoothfluxisequaltothepolefluxdividedbythenumberofteethinapolepitch,

    andcanbeexpressedas

    sppggt /N td B = (3.8)

    where

    PSsp NNN /= . (3.9) Theradialcrosssectionalareaofapolarshoefacingtheairgapandthatofthetooth

    canbecalculatedrespectivelyby

    epgep wgdA )( += (3.10)

    and

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    tepgt whgdA )2( ++= . (3.11)

    Thetoothfluxdensityis

    spepgt

    pgg

    t

    tt Nhgdw

    tdBAB

    )2( ++== (3.12)

    whereasthestatorpoleshoefluxdensityresults

    spgep

    pgg

    ep

    tep Ngdw

    tdBAB

    )( +== . (3.13)

    Theslotwidthcanbecalculatedasfollows:

    tSacts wNww = / (3.14)

    where actw isthetotalactuatorlength.

    Takingintoaccountthepolefluxtobackironfluxratio,assumingthetoothfluxdensityequaltotheslidercorefluxdensity,thecrosssectionalareaofthebackironisgivenby

    2/sptb NAA = (3.15) andthebackironheightbecomesasfollows:

    )( bsseb

    b h2h2gdA

    h +++= . (3.16) Finally,theslotheightcanbeexpressedby

    ( )bseexs 2h2gddh = 21 (3.17)

    F. ThrustforcecalculationThethrustforcecanbecalculatedasfollows:

    =

    =3

    1kk

    k

    s

    idt

    dv1F

    (3.18)

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    wherek isthemagnetflux linkedwiththekthphase, ik isthecorresponding linecurrentandvsthesliderspeed.

    Assumingsinusoidalwindingcurrents,athrustforceisconstantwhenalsotheinducedEMFs are sinusoidal.Applying Steinmetzs transformation to (3.18) leads to the followingexpressionofthethrustforce:

    =

    =3

    1k

    kks

    Ijv1F (3.19)

    Thesliderspeed in(3.20) isrelatedtotheangularfrequencyofthesupplyvoltageas

    follows:

    p

    s

    tv = . (3.20)

    Themaximumvalueofthecoilfluxlinkagecanbecalculatedas

    sp

    pggc N

    tdB2= (3.21)

    andthemaximumvalueofthephasefluxlinkageis

    cdgM KQnN = (3.22)

    where gN is thenumberof seriesconnectedgroupsof coils, Q is thenumberof coilsper

    group, n isthenumberofwirespercoil,and dK isthewindingfactor.

    ThecurrentdensitycanbeexpressedintermofJoulelossesandgeometricparametersby

    ( ) sssssiscu

    Khwh2dNP

    J ++= (3.23) wherePcuisthewindinglosses, sK istheslotfillfactorand isthecopperresistivity.ItisworthnotingthatthevalueofPcuin(3.23)shouldbechosentakingintoaccountthecoolingcapabilityofthetubularactuator.

    Thepeakvalueofthetotalslotcurrentvalueis

    JKhw2I sssc = . (3.24)

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    Finally, assuming that themachineoperates in fieldoriented control conditions, theelectromagneticthrustforcecanbeexpressedas

    nI

    t

    23F cM

    p

    = (3.25)

    3.4 ThermalmodelThe thermal analysis allows to verify the compatibility between the temperatures

    insidetheactuatorandtheselectedinsulationclass.Thethermalanalysiscanbeperformedby means of lumped equivalent thermal circuit or FEM analysis of the temperaturedistributions.

    The firstmethod is particularly suitable in the preliminary design to determine theaveragetemperatureofdifferentinnerpartsoftheactuator,buttheprecisionoftheresultsisinfluencedbytheaccuracyofthelumpedcircuitschematization.

    Forsimplicity,thetemperatureoftheframe,mouldedinaluminiumorcastiron,canbeconsidered uniform because of the high value of thermal conductivity coefficient.Considering the actuator fixed on an adiabatic surface,with the simplified frame shapedepicted inFigure3,andwith the slider inhorizontalposition, the frametoenvironment

    temperaturerise ef canbecalculatedbymeansofthefollowingequation:

    radconv

    eftot QR

    Q += (3.26) whereQtotisthetotalpowerlosses,Rconvthethermalframetoenvironmentresistancedueto natural convection, and Qrad the radiated thermal power. Rconv can be expressed asfollows:

    HhcVvcconv SS

    R +=1

    (3.27)

    SVvc

    SHhc

    Figure3Simplifiedframeshapeusedforthermalmodelling.

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    whereSVandSHare thevaluesof theverticalandhorizontal framesurfaces,whereasvcand hc are the natural convection coefficients for vertical and horizontal surfaces. Thevaluesofvcandhccanbecalculatedasfollows:

    4

    env

    efvc Th

    5.6a = (3.28)

    vchc 5.0= (3.29)

    wherehistheframeheightandTenvtheambientabsolutetemperature.Finally,theradiatedpowercanbecalculatedasfollows:

    ( ) ( )[ ]44 envefenvradradrad TTSQ += (3.30)

    whereradistheframeradiationcoefficientandSradistheradiationsurfacevalue,calculatedasthesumofSVandSH.

    Undertheassumptionofwindinglossesuniformlydistributedinthecoils,theaveragevalueofthewindingtoframetemperaturerisecanbeexpressedas

    += 42sw

    32

    s

    SNP s

    s

    cufw '

    (3.31)

    wheresisthethicknessoftheslotinsulation,Sisthelateralsurfaceofthewinding,isthethermalconductivityoftheslot insulation,and istheequivalentthermalconductivityofthewinding,dependingontheenamelling,the impregnationandthe fillfactoroftheslot.Themain slotparameters arehighlighted in Figure4,where a axial sightof themotor isshown(a)anda3Dviewofonewindingisdepicted(b).

    Finally,thewindingtoenvironmenttemperatureriseis

    effwew += (3.32)

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    3.5 MechanicalmodelofthesliderThesliderof tubularPMmotorsusuallypresentsaslighteccentricitywith respect to

    thesymmetryaxisofthestator,duetotheassemblingmisalignmentandclearanceofthebearings[84][85].Asaconsequence,theresultantradialforceactingontheslider,duetothemagnets, isnotzeroandthestaticequilibriumconfigurationoftheslideraxistendstobend.Thecalculationofthemaximumdisplacementoftheslideraxisisextremelyimportantinorder toverify that there isno interferencebetween sliderand stator. Inaddition, thereaction forceson thebearingshave tobedetermined,because they increase the frictionreducingthelifeoftheactuator.

    Theanalysisof the staticequilibriumconfigurationof theactuator is reportedundertheassumptionofelasticmaterialsandsmalldisplacements.

    A. StaticequilibriumanalysisToanalyzethestaticequilibriumoftheslider, it isconvenienttousetheschemeofa

    simplysupportedbeamloadedwithadistributedforceq,showninFigure5.Asknown,thestaticequilibriumequationofthedeflectedbeamisasfollows:

    0)()(44

    =+ zqdz

    zvdEI (3.33)

    S

    (b)

    S

    hs

    ws

    (a)

    s

    Figure 4Drawing of the tubular actuator showing the geometricalparametersofonehighlightedcoil(a),and3Dsightofthecoil(b).

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    ThebendingmomentM(z)isrelatedtothedisplacementvasfollows:

    2

    2 )()(dz

    zvdEIzM = (3.34) Equations (3.33)and (3.34)canbesolved imposingtheboundaryconditionsthatv(z)

    andM(z)vanishforz=0andz=Loncethedistributedloadqisknown.Inordertodetermineq,referenceismadetoFigure6,showingatransversesectionof

    thetubularactuator,wherethesliderislocatedinaneccentricpositionwithrespecttothestatorsymmetryaxis.Undertheassumptionofsmalldisplacements,themagneticgapcanbeexpressedasfollows:

    =0+ycos. (3.35)

    z

    v(z)

    L

    q(z)

    Figure 5Scheme of the simplysupported beam for thestudyofthesliderstaticequilibriumconfiguration.

    y

    0 Rsi

    df

    Figure6Transversesectionofthetubularmotorshowingtheeccentricityoftheslider.

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    TheelementaryforceactingontheelementarysurfacedSoftheslideris:

    dSBdf0

    2

    2)()(

    = . (3.36) Thedistributedforceq,resultantofallelementarycontribution,isdirecteddownwards

    andisgivenby

    dBdq si )cos(4

    )(

    0

    22

    0= . (3.37)

    Toexplicittheexpression(3.37),thefunction )(B hastobedetermined.Considering

    themotorsectiondepictedinFigure7andcalculatingtheMMFdropalongthedashedlineitfollows:

    )(21)( 0

    MM HWB = . (3.38) Substituting (3.35) in (3.38), the resultingequation canbeexpressed inMc Laurens

    series,asfollows:

    += ...cos1)(

    00

    yBB (3.39)

    whereB0(z)isthevalueofthefluxdensityintheoriginalconfiguration.Substituting(3.39)in(3.37)leadstothefollowingexpressionofq(z):

    N-S S-NS-N

    wM

    Figure7Longitudinalmotorsection.

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    yBR

    q si00

    20

    2 = . (3.40) Inordertotakethenonuniformityofthemagneticfieldintheactuatorintoaccount,it

    can be demonstrated that in (3.40) B0 can be approximated by BRMS. In addition, thedisplacementycanbeexpressedasthesumoftwoterms,namelytheeccentricityv0andthedisplacementvduetobending.Inthisway,thedistributedloadqbecomes:

    ( )000

    2

    2vv

    BRq RMSsi +=

    . (3.41)

    Substituting(3.41)in(3.33)andsolvingthedifferentialequationitispossibletoobtain

    theanalyticalexpressionofv(z).Addingthemaximumvalueofv(z)totheinitialeccentricityv0leadstothemaximumdisplacementymaxoftheslideraxis.

    B. MaximumsliderdisplacementAssumingthatv0istheeccentricityoftheslideraxiswithrespecttothesymmetryaxis

    ofthestator,themaximumdisplacementoftheslidercanbeexpressedas

    ymax=v0 (3.42)

    whereisacoefficientgreaterthan1definedas

    +

    =

    11

    2cos

    2cosh

    21 KLKL . (3.43) In(3.43)ListhedistancebetweenthebearingsandKisdefinedasfollows

    4

    00

    2

    2 IEBRK RMSsi

    = (3.44)

    whereBRMSistheRMSvalueinapolepitchofthespatialdistributionofthefluxdensity,EistheYoungsmodulusand I is themomentof inertiaof theslider tubewithrespect to thebendingaxis.

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    ThebehaviourofisshowninFigure8asafunctionoftheproductKL.Accordingto(3.42), the maximum displacement is proportional to the slider eccentricity. Thereforeattentionhastobepaidtothechoiceofbearingstypeandtheassemblingaccuracy.

    Finally,itisworthnotingthatifKLisamultipleof,then(3.42)hasaninfinitevalue,i.e.astaticequilibriumconfigurationisnotpossible.ThecaseKL=1givesthecriticaldistancebetweenthebearings,thatis

    42

    300

    RMSsicr BR

    IEK

    L == (3.45)

    Obviously,thestatorlengthshouldbefarenoughfromthecriticallengthLcrinorderto

    avoidinterferencebetweenstatorandslider.

    C. ReactionforcesonthebearingsTheresultantforceRduetotheradialmagneticforcescanbecalculatedbymeansof

    thefollowingequations:

    LvBdR rmsse 000

    2

    = (3.46)

    whereisaconstantgreaterthan1,depictedinFigure9,whoseexpressionisasfollows:

    +=2

    tan2

    tanh1 KLKLKL

    (3.47)

    0 0.5 1 1.5 2 2.5 31

    10

    KL

    Figure8Behavioroftheparameter.

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    57|P a g e

    ThereactionforcesonthebearingsareequaltoR/2.Theyproducefrictionforc