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Abstract In this paper the unbalanced magnetic forces which act upon the rotor of a salient-pole synchronous generator due to eccentricmotionoftherotorshaftinthepresenceofmagnetic fieldinno-loadandloadedconditionarecalculatedusingthe finite-elementmethod.Thedisplacementoftherotorismodeled usingtheactualshaftorbitmeasuredona5MVAsalient-pole generatordrivenbyagasturbineinacogenerationplant.The influencesofthestatorwindingparallelpathsandtherotor damper winding on attenuation of unbalanced magnetic pull are analyzed using the finite-element method. It is shown that under linearconditionsthecorrelationsbetweenseparateharmonic componentsoftheshaftorbit,theresultingharmonic componentsoftheunbalancedmagneticpullandtheinduced statorwindingcurrentscanbeestablishedinboththeno-load and loaded condition. IndexTermsGenerators,vibrations,electromagneticforces, force, damping , finite element methods, rotors, stators, armature NOMENCLATURE UMagnitude of the vector of forward precession VMagnitude of the vector of backward precession UPhase shift of the vector of forward precession VPhase shift of the vector of backward precession Angular speed of precession cDistancebetweenshaftscenteranditscenterof gravity e Angular speed of rotation of the shaft around its center K Shaft orbit in the complex form ss complex vector of static displacement ks complex vector of kinetic displacement s(t)instantaneous displacement of the rotor ManuscriptreceivedJanuary05,2011.AcceptedforpublicationJune1, 2011. Copyright 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. Damir arko and Drago Ban are with the Faculty of Electrical Engineering andComputing,DepartmentofElectricMachines,DrivesandAutomation, Unska 3, 10000 Zagreb, Croatia (phone: +385 1 6129 706; fax: +385 1 6129 705; e-mail: damir.zarko@fer.hr; drago.ban@fer.hr).IvanVazdariswithKONARGeneratorsandMotorsInc.,Fallerovo etalite 22, 10000 Zagreb, Croatia (e-mail: ivazdar@koncar-gim.hr). VladimirJariiswithMARTINGd.o.o.,AlbertaFortisa171,10000Zagreb, Croatia (e-mail: vladimir.jaric1@zg.t-com.hr). an, en, nAmplitude,frequencyandphaseshiftofthenth harmonic component of the shaft orbit Fx, FyConstant average forces in x and y direction DrRotor outer diameter DStator inner diameter o0Air gap size along the centerline of the pole shoe kcCarter factor lAxial length of the stator core Bo1 Fundamental component of the air-gap flux density EphaseRms value of the induced no-load phase voltage fRated frequency wNumber of stator winding turns connected in series pNumber of pole pairs fd1 Statorwindingdistributionfactorforfundamental component fp1Stator winding pitch factor for fundamental component qNumber of slots per pole and phaseyCoil pitch of the stator winding tpPole pitch ts Slot pitch NsNumber of stator slots doWidth of the slot opening FradialTotal radial force on the rotor Fx0, Fx1, Fx2, xcomponentsoftheUMPoriginatingfromthe staticdeflection,thefirstandthesecond harmonic components of the shaft orbit Fy0, Fy1, Fy2 ycomponentsoftheUMPoriginatingfromthe staticdeflection,thefirstandthesecond harmonic components of the shaft orbit NsNumber of stator slots Fa, FfArmature and field winding MMFs Ff1Fundamental component of the field winding MMF Fad1, Fai1Magnitudesofthedirectandinverse components of the armature winding MMF ad0, ai0 Initialphaseshiftsofthedirectandinverse components of the armature winding MMF usAngular coordinate along the stator circumference ee Electrical angular frequency ATotal air-gap permeance AHarmonic component of the air-gap permeance without eccentricity Calculation of Unbalanced Magnetic Pull in a Salient-Pole Synchronous Generator Using Finite-element Method and Measured Shaft Orbit Damir arko, Member, IEEE, Drago Ban, Member, IEEE, Ivan Vazdar, Vladimir Jari > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2AseAir-gappermeancecomponentduetostatic eccentricity or, ouRadialandtangentialcomponentsoftheMaxwell stress tensor 0Permeability of vacuum I.INTRODUCTION HEmagneticfieldinasynchronousmachine interactingwiththearmaturewindingcurrentsorthe salientpolescreatestangentialelectromagneticforcesonthe rotorwhichproducestorque.Theradialforcesaregenerated at the same time, but in the case when the stator and rotor are perfectlyconcentrichavingasymmetricalairgaparoundthe perimeter,theradialforcesarecancelledout.Theseforces dependnotonlyonelectromagneticparametersofthe machine,butalsoonthegeometricshapeofthestatorand rotorandthepresenceofaneccentricmotionoftherotor. Whentherotoriseccentrictheunbalancedmagneticpull (UMP) occurs in which case the net radial force is developed andininteractionwiththemechanicalsystemmaycause unwanted,harmfulvibrations.Adetailedmodelofthis interactionispresentedin[1]and[2]forthecaseofatwo-pole turbogenerator whose rotor is modeled by means of finite beamelements,thebearingeffectsbystiffnessanddamping coefficientsandthefoundationbypedestals.TheUMPis addedtootherexcitingforcestocorrectlyreproducethe dynamic behavior of a real machine during no-load operation. Thegeneralcaseofeccentricmotionincludesdynamic eccentricitywhentherotorcenterfollowsanarbitrary trajectorywhosegeometriccenterdoesnotcoincidewiththe center of the stator. In addition, the stator and rotor shape may deviatefromanidealsymmetricshape.In[3]ithasbeen shown that shape deviations in a hydrogenerator may give rise todifferentwhirlingmotions,includingbothbackwardand forwardprecession.InbothcasestheUMPwillarisedueto asymmetry in the air gap. The two important parameters of the unbalanced forces are frequency and amplitude which implies their periodic nature due to rotating motion of the machine. Insalient-polesynchronousmachinesthestatorwinding very often has parallel paths. The eccentric motion of the rotor induces circulating currents within parallel paths which have a dampingeffectontheUMP.Thisinfluenceisnoticeablein induction[4]andpermanent-magnetmotors[5]aswell.In addition,thesalient-polesusuallycontainthesquirrel-cage typerotordamperwindingwhichfurtherattenuatesthe UMP [6]. Both influences have been studied in this paper. Thecalculationofunbalancedmagneticforcesisessential for the analysis of vibrations and evaluation of the mechanical stresswhicharisesinvariouspartsofthemachine.Thetwo commonapproachesareanalyticalmethodsandthefinite-elementmethod(FEM).Theanalyticalsolutionsareusually basedonwindingfunctions,airgappermeanceandlumped parametersorconformalmappingcanbeusedaswell.In[7] the permeance function has been used to calculate analytically thewaveformoftheair-gapfluxdensityforthepurposeof detectingeccentricityinasynchronousgenerator.Simondet al.[8]usedtheair-gappermeanceapproachtoderive measurablemodulationfunctionswhichcarryinformation aboutstaticanddynamiceccentricitiesandstatorandrotor bore deformations of a salient-pole synchronous generator. In [9] the air-gap MMF for a machine with eccentricity has been formulatedinananalyticalmodelinamannertodistinguish theeffectofthehomopolarfluxwhichissignificantfor two-polemachines.Forthepurposeofdetectingeccentricityin inductionmachinestheanalyticalmodelingbasedonair-gap permeanceandmodifiedwindingfunctions[10]-[12]orthe MMFfunction[13]hasbeusedtodistinguishthefrequency componentsinthestatorcurrentorinthecomplexapparent powermodulus[14]thatappearduetorotoreccentricity.In [15] the effect of a ball bearing fault has been represented by permeancevariationwhichisacomplexsumofaninfinite number of rotating eccentricities. Those permeance variations reflect on the harmonic content of the stator current which can beusedfordiagnosingthebearingfaults.Variousdiagnostic techniquesfordetectionoffaultsininductionmachines, which includes eccentricity as well, have been summarized in [16].Thispaperprovidesacomprehensivelistofuseful references in the field of induction motor diagnostics. In[17]conformalmappinghasbeenusedtocalculatethe air-gappermeancefunction,fluxdensityandUMPina permanent-magnet motor with rotor eccentricity. Thecommonproblemoftheanalyticalsolutionsistheir limitationtolinearproblems,buttheiradvantageisshort calculation time. The nonlinearity can be taken into account as in[6]wherelumpedparametersofthenonlinearanalytical model are found by means of optimization using the results of finite-element(FE)simulationsinwhichcasetheparameters arevalidonlyforasingleoperatingpoint.Theanalytical permeancenetworkmodelscaninherentlytakeintoaccount saturation and also model the eccentric motion of the rotor. In [18]thisapproachisusedfordynamicsimulationofair-gap eccentricity in an induction motor. TheFEMincludessaturationlocallybydefault,which makesitinherentlymoretimeconsumingandinthecaseof salient pole synchronous machines with high number of poles andfractionalslotwindingsmayresultinthenecessityto modelalargeportionofthemachinetotakeinto considerationthegeometricsymmetry.Inaddition,inthe presenceofeccentricitythereisnogeometricsymmetry,so thewholecrosssectionofthemachinemustbemodeled regardless of the number of slots and poles. This may result in an extremely large number of elements in the FE mesh. In the FE based studies of UMP Perers et al. [19] reports a mesh size of 85000 elements for a 14 pole generator, while Lin Wang et al. [20] reports a mesh size of 158760 elements for an 88 pole generator.TheeccentricmotionoftherotorinFEMforany typeofelectricmachineismosteffectivelymodeledusinga timesteppingmethod,whichalsoallowsfortheinclusionof the electric circuits into the model to account for the effects of thestatorwindingandtherotordamperwinding.Time-stepping is used in [21] to model eccentricity in a permanent-T > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3magnetsynchronousmotorforthepurposeoffaultdiagnosis andin[22]tocalculateadditionallossesduetorotor eccentricity in an induction motor with PWM-voltage supply. However,thecalculationofforcesintheFEMissensitiveto discretizationerrors.InthatregardtheMaxwellstresstensor approachismoresensitivethanthevirtualworkmethod.In [23]ithasbeenshownthatthevalueoftorquecalculatedby thevirtualworkmethodisveryconsistentirrespectiveofthe mesh refinement and changes marginally with a change in the numberoflayersintheair-gapmesh.Thiscanbeimportant forcalculationofUMPbecause rotordisplacementisusually expressedinmicrometers,whichcanbesignificantlysmaller than the size of the finite-element mesh in the air gap. In[24]thedynamiceccentricityinamechanicalrunprior toexcitationofthegeneratorismodeledbyobservingthe whirlingmotionatasinglefrequency,whichassumesthat shaftorbithasacircularshape.Inreality,theshaftorbit consistsofmultipleharmoniccomponentswhichalltogether affecttheUMPinthemachine.Inlinearmodelseach harmoniccomponentcanbetreatedseparatelyandthetotal UPMisequaltothesumofUMPsforindividualharmonic components.However,thisapproachisnotvalidunder nonlinear conditions in which case permeance networks or the FEM are the most suitable approaches. Thecontributionofthispaperisamodelofthemeasured shaftorbitofasalient-polesynchronousgeneratorandthe associatedvibrationdisplacementsintwodirections, perpendiculartooneanother,withalltheirsignificant harmoniccomponentswhichhavebeenusedtocalculatethe unbalanced magnetic pull by means of the FEM coupled with circuitequationsforthestatorwindingandtherotordamper winding. For calculation of forces the virtual work method has been applied. The no-load operation and the loaded condition have both been included in the study and the damping effects ofthestatorwindingparallelpathsandtherotordamper winding have been taken into account.II. THEORETICAL BACKGROUND Shaftvibrationsaredefinedasrapidmotionsoftheshaft whicharerelatedtoitsdeflectioninrotatingmotion.The deflection of the shaft is a deformation of its elastic line which connects all geometric centers of the shaft cross sections along theaxisofthebearings.Thisaxisisanimaginaryline which connects geometric centers of the bearing bores and is aligned with the z axis of the absolute reference frame. Since the shaft bendsundertheactionofstaticradialforces,itscentersof rotationformastaticdeflectionline.Inrotationthekinetic displacement sk is added to the static displacement ss. (Fig. 1) Themotionofashaftcrosssectioniscomposedofamotion of the shaft center C around the point S of the static deflection line and also of a rotation of the cross section around the shaft centerC.Thismotionoftheshaftiscalledprecession.The precessionisdefinedbythemotionoftheshaftscenterof gravity which can be given in the complex plane by equation ( ) ( ) j j je e eU Vt t tz U V c e + += + + (1) whereUandVarethemagnitudes,andUandVarethe phaseshiftsofthevectorsofforwardandbackward precession respectively, is the angular speed of precession, c is the distance between the shafts center and the shafts center ofgravityandeistheangularspeedofrotationoftheshaft around its center. Thepathoftheshaftcenterintheradialplaneiscalled shaft orbit (marked asKin Fig. 1) and is given by a general equation in the complex form ( ) ( ) j js k se eU Vt tK s s s U V + += + = + + (2) where ssis the complex vector of static displacement and ksis the complex vector of kinetic displacement. When>0andV = 0theprecessionisgivenbyavector Uejt,whichrepresentsthepureforwardprecession.Inthat case the shaft orbit has the same orientation as the rotation of the shaft. When REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 Fig. 1.Kinetic shaft orbit III.MEASURED ECCENTRICITY Thesalient-polesynchronousgeneratorforwhichthe detaileddesigndatasheetsandtheresultsofmeasuredrotor vibrations are available is used in a cogeneration plant with a gas turbine as a prime mover. The basic data of the generator are shown in TABLE I. Themeasurementandanalysisofbearingandshaft vibrationshavebeencarriedoutbymeansofBrel&Kjr VIBROTEST60vibrationanalyzerandDEWE-50-PCI-16 data acquisition system. The electrodynamic sensors (Brel & KjrVS080)formeasurementofbearingvibrationshave been fixed on bearings housings in two directions (x, y). The noncontactsensors(Brel&KjrIN085)formeasurement ofshaftvibrationshavebeenfixedthroughholesintheshaft casingneardriveend(DE)bearingintwodirections(x,y). Therefore,theshaftvibrationshavebeenmeasuredrelatively to the bearing casing which is mounted on the machine frame. Thetrackingsignalfordeterminationofthephaseangleof fundamentalcomponentofvibrationspeedvectorshasbeen obtainedfromtherotorshaftbymeansofaphotosensitive transducer.Themeasuredvariablesarevibration displacement,vibrationvelocityandvibrationacceleration. Themostusefulquantityforouranalysisisthevibration displacement which is given in the form of a shaft orbit graph. The principle measurement scheme is shown in Fig. 2. The shaft orbit has been initially recorded for the cold unit inamechanicalrunat1500rpm(Fig.3)withbothfieldand armaturewindingcurrentsequaltozero.Therotorshaft exhibitsdynamiceccentricitywithforwardsynchronous precession. The coordinates of time integrated mean values of staticshaftdisplacementare(-11.2,82.6) m.Inthiscase there is no electromagnetic field in the machine and hence no UMPthatmightcontributetoeccentricmotionoftheshaft. Theeccentricmotionoccursmainlyduetogravityandfinite stiffnessoftheshaftwhichcauseittodeflectandshiftits centerofgravityawayfromtheaxisofthebearings.Asthe shaftrotatesarotatingcentrifugalforceappearswhichacts from the bearing axis towards the center of gravity. This force causes the eccentric motion of the shaft. In addition, eccentric motion in a mechanical run without excitation can also appear due to wear of the bearings or errors during assembly. The recorded shaft orbit for the case of no-load operation at synchronousspeedof1500rpmwithanexcitationcurrentof 36.5 Aandinducedratedarmaturevoltageof10500 Vis showninFig.4a.Theorbitgraphfortheloadedcondition (P = 1.84MW,Q=0.18MVAr)withathermallystableunit at 1500 rpm is shown in Fig. 4b. This generator operates in a cogenerationplantofanaturalgasprocessingfacility.The powerdemandinthefacilityatthetimewhenthe measurements were carried out did not allow the rated load of thegeneratortobeachieved.Thecoordinatesoftime integrated mean values of static shaft displacement for no-load operationare(-12.57,78.26) m,andforloadedgenerator (9.49, 89.91) m. FromFig.3andFig.4aitisapparentthatthepresenceof magneticfieldinno-loadoperation,andconsequentlythe unbalancedmagneticpull,affectstheshaftorbittoasmall extent.However,whenthegeneratorisloaded(Fig.4b)the sizeoftheorbitincreasesbyapproximately100%andthe static displacement is altered as well. IV.FINITE-ELEMENT MODEL FOR CALCULATION OF UNBALANCED MAGNETIC PULL Theeccentricmotionoftherotorinthecasewhenthe generatorismagnetizedeitherinno-loadorloadedoperation willresultinunbalancedelectromagneticforceswhichact upontherotorandconsequentlyonthebearingsofthe machine. TABLE I. DATA OF THE SYNCHRONOUS GENERATOR Rated power5000 kVA Rated power factor0.8 Rated voltage10500 V Voltage regulation 10% Rated frequency50 Hz Rated speed1500 rpm Overspeed1800 rpm Moment of inertia304 kgm2 Stator inner diameter800 mm Air gap size at the centerline of the pole shoe5.5 mm Air gap size at the edge of the pole shoe8.8 mm Width of the slot opening14.2 mm Axial length820 mm Fig. 2.Principleschemeformeasurementofsynchronousgenerator vibrations > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5 Fig. 3.Recorded shaft orbit in mechanical run of the cold unit at 1500 rpm. Field and armature winding currents are equal to zero. The goal is to use the finite-element method to simulate the measuredeccentricmotionoftheshaftandcalculatethe resultingunbalancedmagneticpull.Sincetheshaftorbitfor everyrevolutionoftherotorisslightlydifferent(seeFig.4), forthepurposeofFEsimulationtheaverageorbitsare calculated. Theharmoniccontentoftheaverageorbitsisshownin TABLE II. Since there is no backward precession, i.e. V in (3) equalszero,theshapesoftheorbitscanbereconstructed usingdatafromTABLEIIif(3)iswritteninaslightly different form, i.e. ( ) ( )( ) ( )s2cos j sincos j sinx yxn ynx U y Ukxn a yn anK s U t U ta n t a n t == + + + + (+ + + + (4) where ss , Ux, Uy, Ux, Uy, axn, ayn, axn and ayn are given in TABLEII.NotethatUx,Uy,UxandUyrefertothe fundamentalcomponentoftheshaftorbit.Onlythefirst10 harmoniccomponentsoftheFourierseriesaregiven,i.e. k = 10.AccordingtoFig.5,thefirst10componentsare sufficient for good reconstruction of the shaft orbit. Theeccentricmotionforonerevolutionoftherotoris modeledusingInfolyticaMagNet7.1.1softwareby simultaneouslycombiningtherotationoftherotoraboutits axis and the motion of the rotor axis with respect to the stator centeraccordingtoFig.5afortheno-loadconditionand according to Fig. 5b for the loaded generator. The time needed fortherotoraxistogothroughonefulleccentricorbitisthe same as the time it takes the rotor to make one full revolution aboutitsaxisatthespeedof1500rpm.Thetime-stepping methodisusedtomodelthemotionoftherotor.Theair-gap isdividedintwolayersalongitscenterline.Thelayeralong the stator bore is stationary, while the other layer is attached to therotorandslidesalongthestationaryoneastherotor moves. In each time step the air-gap mesh along the boundary betweenthetwolayersismodifiedsothatnodesalongthe boundary are mutual for both layers. Simultaneous rotary and linearmotionsoftherotorarethusallowed.Thestator windingcoilsareobjectsintheFEmodelandtheirmutual connections are modeled as an electric circuit which is solved simultaneouslywiththerestofthemodel.Theinduced voltagesduetobothrotaryandlinearmotionsarecalculated byFEM,andexternalcurrentorvoltagesources,resistors, capacitors,inductorsandotherelementscanbeaddedtothat circuitaswell.Theend-windingleakageinductancehasalso beenaddedtothemodel.Ithasbeencalculatedanalytically usinga3-Dmethodbasedonclosedformsolutionof Neumann integrals [26]. (a) (b) Fig. 4.Recordedshaftorbitin(a)no-loadcondition(1500 rpm,10500V), (b) loaded condition (P = 1.84 MW, Q = 0.18 MVAr, 1500 rpm, 10500 V) > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6TABLE II. HARMONIC CONTENT OF THE AVERAGE SHAFT ORBITS NO-LOAD CONDITION 2t25 rad/s ss -12.57+j78.26 m Ux, mUy, mUx, radUy, rad 22.41524.242-0.763-2.059 naxn, mayn, maxn, radayn,rad 22.4831.6742.708-0.825 30.7071.1713.0871.420 40.9350.883-2.9000.302 50.2630.690-2.644-0.730 60.2930.243-0.012-2.109 70.2410.112-3.0420.288 80.1080.0480.286-2.275 90.2330.2760.878-0.332 100.1800.139-0.3980.899 LOADED CONDITION 2t25 rad/s ss 9.49+j89.91 m Ux, mUy, mUx, radUy, rad 52.02843.073-1.860-0.240 naxn, mayn, maxn, radayn,rad 22.7351.515-1.1891.621 31.6830.4930.408-1.102 40.7610.5050.7420.146 51.2321.813-0.032-1.792 60.4470.765-0.311-1.098 70.2200.2572.326-0.473 80.0790.145-2.394-0.504 90.2090.294-2.1520.029 100.1920.197-1.478-0.690 In these simulations it is assumed that the generator exhibits parallel precession, i.e. the rotor axis is always parallel to the stator axis. In reality the shaft is not infinitely rigid, so it will alwayshaveaslightlyarchedshapebecauseofwhichthe orbitsmeasuredatthebearingswilldifferfromtheorbitsin othercrosssectionsoftheshaft.Theactualshapesofthe orbits along the shaft affect the UMP and vice versa, so for the accurateanalysistheelectromagneticandthestructural analysisproblemsmustbecoupled.Suchcouplingwasnot possibleinourcaseduetolimitationsoftheFEsoftware designated for solving only electromagnetic problems. Thenonlinearstructuralanalysismodelsoftherotorshaft basedonfinitebeamelementswhichtakeintoaccount stiffness,damping,gyroscopiceffects,weightandexternal forces(UMPatno-load,inherentunbalanceandstiffness asymmetry)tomodelthestaticanddynamicdeflectionlines alongtheshaftofatwo-poleturbogeneratorwithcylindrical rotor have been reported in [1]-[2]. The results indicate that it is possible to reliably model the shaft motion when comparing thesimulatedandmeasuredorbitsatthebearings.Itisthen justifiable to assume that such a model also works well for the othernodesalongtheshaftwhereitwasnotpossibleto measure the shaft orbits. -50 -40 -30 -20 -10 0 10 205060708090100110120Shaft displacement in x direction, mShaft displacement in y direction, m Measured shaft orbitReconstructed shaft orbit usingfirst 10 harmonic components (a) -50-40-30-20-10 0 10 20 30 40 50 60 70405060708090100110120130140150160Shaft displacement in x direction, mShaft displacement in y direction, m Measured shaft orbitReconstructed shaft orbit usingfirst 10 harmonic components (b) Fig. 5.Measured (averaged) and reconstructed shaft orbit for (a) no-load and (b) loaded condition V. SIMULATION RESULTS TheFEsimulationsusedforcalculationoftheUMP resultingfromtheeccentricmotionoftherotoraccordingto therecordedshaftorbitsforno-loadandloadedconditions have been carried out for the following three cases: 1)Forcecalculationwithoutstatorwindingparallelpaths and without rotor damper winding, 2)Forcecalculationwithstatorwindingparallelpathsand without rotor damper winding, 3)Forcecalculationwithstatorwindingparallelpathsand with rotor damper winding. A.Case 1 This is the simplest case in which the damping effects of the statorwindingparallelpathsandtherotordamperwinding havenotbeenincluded.Therefore,theresultingUMPwill > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7originateentirelyfrominstantaneousdisplacementsofthe rotor shaft center with respect to the center of the stator bore. The results of simulations for both operating conditions are unbalanced electromagnetic forces given as a function of time, which are shown in Fig. 6. The maximum radial force for no-loadconditionoccursat t = 12.4msand equals 2359 N. This timeinstantcorrespondstothemaximumaverageshaft displacementof100.2m.Inaddition,duetostatic displacementoftheshaft,theconstantaverageforce F = 1750 N will appear. Similarly,fortheloadedgeneratorthemaximumradial forceoccursatt = 37.4msandequals2518N.Thistime instantcorrespondstothemaximumaverageshaft displacementof 133.8m.Theconstantaverageforceinthe case of the loaded generator equals F = 1612 N. Themaximuminstantaneousforcedependsonthe maximum instantaneous displacement. On the other hand, the sizeoftheorbitreflectsonthepeak-to-peakvariationofthe force.Thepeak-to-peakvalueofthetotalradialforceinno-load condition is Fp-p = 1294 N, while for the loaded generator Fp-p = 1742 N. 0 0.01 0.02 0.03 0.04100020003000400050006000Time, sForce, N FEM: Fe=FEM: with saturationAnalytical (a) 0 0.01 0.02 0.03 0.040100020003000400050006000700080009000Time, sForce, N FEM: Fe=FEM: with saturationAnalytical (b) Fig. 6.Waveformsofthetotalunbalancedmagneticpullin(a)no-loadand (b)loadedconditioncalculatedanalyticallyandnumericallyusingFEM (simulation WITHOUT parallel paths and WITHOUT damper winding) In order to establish a link between the harmonic content of thetotalunbalancedforceontherotorandtheshapeofthe shaft orbit, a simple analytical model for calculation of forces isemployed.Theinstantaneoussinglesidedmagneticpullat eccentric rotor position can be calculated according to [27] ( )( )2 r10 c 04s t D lF t Bkot o= (5) whereDr = D-2o0istherotorouterdiameter(Disthestator innerdiameter),o0istheairgapsizealongthecenterlineof the pole shoe, kc is the Carter factor, l is the axial length of the statorcore,s(t)istheinstantaneousdisplacementoftherotor (Fig. 5a) and Bo1 is the fundamental component of the air-gap fluxdensity.ThegeometricparametersD,lando0aregiven in TABLE I. The Carter factor is used to take into account the presenceofstatorslotsandobtaintheeffectiveairgapsize. The air gap flux density can be calculated from phase d1 p1 1122DlE fwk k Bpot = (6) whereEphaseisthermsvalueoftheinducedno-loadphase voltage,fistheratedfrequency,wisthenumberofstator windingturnsconnectedinseries,pisthenumberofpole pairs, kd1 and kp1 are the winding distribution and pitch factors for fundamental component. The no-load FE simulation has in this particular case been carried out using measured rated no-loadfieldcurrent.Thecalculatedrmsvalueoftheinduced phasevoltageinthatcaseequals5747V,whichis5.2% smallerthanratedvoltageof6062 Vmeasuredforthatsame field current. The distribution and pitch factors are given by ( )( )( )( )0d10sin 6 10 2sin 20.9561sin 2 6sin 10 2qkqoo= = = (7) ( ) ( ) ( ) ( )p1sin 2 sin 14 2 18 0.9397pk y t t t (= = =( (8) whereqisthenumberofslotsper pole and phase and y/tp is thecoilpitchtopolepitchratio.Theair-gapfluxdensityis then phase1d1 p125747 2 22 2 50 96 0.9561 0.9397 0.8 0.820.9144 TE pBfwf f Dlot t= = =(9) The Carter factor follows from the well known expression sc2o o 0 os0 o 01.15972atan ln 12 2kd d ddtott o o= = (| | ( + `| (\ . )(10) wherets = (Dt)/(Ns)istheslotpitch,Nsisthenumberof stator slots and do is the width of the slot opening. Sinceallparametersin(5),excepts(t),areconstant,it appearsthatthewaveformoftheunbalancedmagneticpull > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8willbedirectlyproportionaltotheshapeoftheshaftorbit, whichisdeterminedbys(t).Thecalculationofforcein(5) assumesthatironisinfinitelypermeable,socomparisonwith FEsimulationispossibleifthesameassumptionismadein theFEmodel.Thisisaccomplishedbysettingtherelative permeability of the stator and rotor core to 109 to simulate an infinitely permeable material. However, in order to obtain the samefundamentalcomponentoftheinducedphasevoltage and thus the same fundamental component of the air-gap flux density as in the case when the actual B-H curves of the stator androtorcorearetakeninto account, the field current had to bereduced.Otherwise,withthesamefieldcurrentasinthe nonlinearsimulation,duetoincreasedoverallpermeanceof themagneticcircuitinthelinearmodel,highervaluesofthe air-gapfluxdensityandinducedvoltagewouldhavebeen obtained. The waveforms of the total unbalanced force on the rotor in no-loadconditioncalculatedanalyticallyusing(5)and numericallyusingFEMwithsaturationandwithinfinitely permeableironareshowninFig.6a.Thesmalloscillations thataresuperimposedonthebasicwaveformsaredueto slottingeffectandduetodiscretizationerrorswhichoccur because the shifts of the rotor are small relative to the size of the FE mesh in the air gap.Thewaveformsindicatethatfundamentalfrequencyis 25 Hz. The fundamental frequency of the force in the case of whirling motion is equal to the frequency of precession [6]. Since in our case is equal to the angular speed of rotation of the shaft around its center, in a four pole machine at 1500 rpm this corresponds to the frequency of 25 Hz.Thereisaverygoodagreementbetweenanalyticallyand numericallycalculatedwaveformswithinfinitelypermeable iron.Thisresultclearlyindicatesthattheshapeoftheshaft orbit is the dominant factor which determines the waveform of the UMP. Thestronginfluenceofsaturationinthestatorandrotor coreonthemagnitudeofforceisalsoapparent.Thepeak value of force is reduced by a factor of 2.36 when saturation is takenintoaccount.Inasensebyadjustingthefieldcurrent andthevalueofinducedvoltagetobethesameinnonlinear andlinearmodels,thesaturationhasbeentakenintoaccount tosomeextent.However,theequalityofinducedvoltages doesnotnecessarilyleadtotheequalityofcalculatedforces with and without the presence of saturation in the FE model. The correlation with the shaft orbit is also visible in Fig. 7a which compares harmonic content of the shaft orbit and of the totalunbalancedforceontherotor(withouttheaverage component).Themagnitudesarenormalizedwithrespectto their fundamental components (F1 and s1). Similaranalysiscanbedonefortheloadedconditionas well. However, in this case the fundamental component of the air-gapfluxdensitydependsontheresultingactionsofthe fieldwindingandthearmaturewindingMMFs.IntheFE modelaniterativeprocedureisusedtodeterminethefield currentandthepositionofthearmaturecurrentvectorwhich yieldtherequiredactiveandreactivepoweroutput (P = 1.84 MW, Q = 0.18 MVAr) in the cases of both saturated machineandunsaturatedmachinewithinfinitelypermeable iron.Sincethepresenceofeccentricitydoesnotaffectthe averagepoweroutput,inordertosavetimeandreducethe geometryof the FE model to one pole pitch, the field current andthepositionofthearmaturecurrentvectorhavebeen determinedbymodelingcircularmotionwithouteccentricity. The field and armature currents thus calculated are introduced intomodelswitheccentricmotiondefinedaccordingtoFig. 5b in order to calculate the UMP. ThefluxdensityBo1usedin(5)iscalculatedfromthe fundamentalcomponentoftheinducedphasevoltage resultingfromtheFEsimulationoftheloadedmachinewith saturation.InthatcaseEphase = 5986V,whichyields Bo1 = 0.9525 T.The waveforms of the total unbalanced force on the rotor in theloadedconditioncalculatedanalyticallyusing(5)and numericallyusingFEMwithsaturationandwithinfinitely permeable iron are shown in Fig. 6b. In this case there is also averygoodagreementbetweenanalyticallyandnumerically calculatedwaveformswithinfinitelypermeableiron. According to Fig. 7b the correlation of the harmonic contents oftheshaftorbitandoftheUMPcalculatedanalyticallyand numerically is apparent in the loaded generator as well. B.Case 2 Whentheeccentricmotionoccursinthepresenceof parallelpathsinthestatorwinding,itinducescurrentswhich circulatewithinthebranchesconnectedinparallel.Forthe loaded condition, the circulating components can be extracted from the total phase currents by subtracting the currents in the parallelbranchesofeachphaseanddividingtheirdifference bytwo.Thesecirculatingcurrentshaveadampingeffecton the UMP because they try to cancel the variation of magnetic fieldthatcausedthem.TheattenuationoftheUMPisvisible in the waveforms of the total UMP shown in Fig. 8. C.Case 3 Theactualgeneratorhasarotordamperwindingwith12 bars on every pole shoe which are shorted by end rings. There arenoconnectionsbetweendamperwindingsofadjacent poles. The eccentric motion of the rotor causes the variation of thefluxlinkageinthedamperwindingandinducesthe currentswhichalsoproducethedampingeffectandfurther attenuate the UMP. This effect is visible in the waveforms of the total UMP shown in Fig. 8. ThewaveformsoftheUMPforallthreecasesinno-load and loaded condition are compared in Fig. 8 and the resulting valuesofthepeakradialforcearecomparedinTABLEIII. Similarly,theharmoniccontentsarecomparedinFig.9.The attenuatingeffectsofthestatorwindingparallelpathsand particularlyoftherotordamperwindingarealsoapparent fromthosefigures.Moreover,thedampingdoesnotonly affectthemagnitude,butalsotheharmoniccontentofthe UMP.NotethatinCase2theinducedcurrentinthestator windinggivesrisetoafairlyhigh100Hzcomponent.The > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 9originofthisparticularharmoniccomponentcanbe determinedbysimulatingtheeccentricmotionusingone particularharmoniccomponentoftheshaftorbitatatime. Unfortunately,thisapproachcanonlybeusedinalinear model,sinceitemploystheprincipleofsuperpositionwhich cannotbeusedinanonlinearmodel.Nevertheless,thelinear approachcanalsogiveavalidinsightintotheoriginofthe 100 Hzcomponent.Forthispurposealinearfinite-element modelhasbeenusedforthecasesofno-loadoperationwith andwithoutstatorwindingparallelpaths.Inbothcasesthe rotordamperwindinghasnotbeenincluded.Foreachcase threesimulationshavebeenperformed.Thestaticdeflection ss , the first and the second harmonic components of the shaft orbitdefinedinTABLEIIforno-loadoperationhavebeen simulatedrespectively.Fromeachsimulationthexandy componentsoftheUMPhavebeencalculated(Fig.10and Fig. 11). In addition, for the case when stator winding parallel paths are included, the induced stator currents which circulate withinparallelbranchesofthephasewindingshavebeen calculated(Fig.12).Thetotalradialforceresultingfromall three simulations is equal to ( ) ( )22radial 0 1 2 0 1 2 x x x y y yF F F F F F F = + + + + + (11) where Fx0, Fx1, Fx2, Fy0, Fy1 and Fy2 are the x and y components of the UMP originating from the static deflection, the first and thesecondharmoniccomponentsoftheshaftorbit respectively.Thesametotalradialforcecanbecalculated from a single FE simulation in which the static deflection and thefirsttwo harmonic components of the shaft orbit are used simultaneously to define the eccentric motion of the rotor. Of course,thesameprinciplecanbeexpandedtosimulate independentlyanarbitrarynumberofshaftorbitharmonic components.However,inthatcasethesimulationswould requireverydenseFEmeshandsmalltimestepsinorderto correctlyaccountforindividualhigherharmoniccomponents oftheshaftorbitduetotheirsmallmagnitudeandhigh frequency. Note that the total radial force in Fig. 10b is comparable to the force in Fig. 6a. It is basically the same waveform, except thatinFig.10bthestaticdeflectionandtwoharmonic components of the shaft orbit are used, while in Fig. 6a all 10 harmoniccomponentsareusedtosimulatetheeccentric motion of the rotor. Similarly, the total radial force in Fig. 11b can be compared with the force in Fig. 8a calculated for Case 2.Thedifferenceisinthemagnitudeoftheforcesincethe resultinFig.11bhasbeenobtainedusinglinearmodelwith staticdeflectionandtwoharmoniccomponentsoftheshaft orbit, while in Fig. 8a all 10 harmonic components are used in a nonlinear model. 25 50 75 100 125 150 175 200 225 25000.20.40.60.81Frequency, HzF/F1 and s/s1 Force: FEM with saturation (F1=512.7 N)Force: FEM with Fe= (F1=1230.5 N)Force: Analytical (F1=1230.2 N)Shaft orbit (s1=23.21 m) (a) 25 50 75 100 125 150 175 200 225 25000.20.40.60.81Frequency, HzF/F1 and s/s1 Force: FEM with saturation (F1=704.1 N)Force: FEM with Fe= (F1=2367.8 N)Force: Analytical (F1=2398.1 N)Shaft orbit (s1=41.64 m) (b) Fig. 7.Comparison of normalized harmonic content of the shaft orbit and of thetotalunbalancedmagneticpullin(a)no-loadand(b)loadedcondition calculatednumericallyandanalytically(simulationWITHOUTparallelpaths and WITHOUT damper winding) 0 0.01 0.02 0.03 0.04050010001500200025003000Time, sForce, N WITHOUT stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITH rotor damper winding (a) 0 0.01 0.02 0.03 0.04050010001500200025003000Time, sForce, N WITHOUT stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITH rotor damper winding (b) Fig. 8.Waveformsofthetotalunbalancedmagneticpullin(a)no-loadand (b) loaded condition for all three cases > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1025 50 75 100 125 150 175 200 225 2500100200300400500600Frequency, HzForce, N WITHOUT stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITH rotor damper winding (a) 25 50 75 100 125 150 175 200 225 2500100200300400500600700800900Frequency, HzForce, N WITHOUT stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITHOUT rotor damper windingWITH stator parallel paths and WITH rotor damper winding (b) Fig. 9.Comparison of harmonic contents of the UMP in (a) no-load and (b) loaded condition TABLE III.COMPARISON OF PEAK RADIAL FORCE FOR ALL THREE CASES No-load Loaded Average force Peak force (% attenuation) Average force Peak force (% attenuation) Case 11790 N2359 N (0 %)1736 N2518 N (0 %) Case 21220 N2022 N (14.3 %)1133 N2087 N (17.1 %) Case 3645 N1005 N (57.4 %)654 N1233 N (51.0 %) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2000-1000010002000300040005000Time, sForce, NFx0Fx1Fy2Fx2Fy1Fy0 (a) 0 0.01 0.02 0.03 0.042500300035004000450050005500Time, sTotal radial force, N (b) Fig. 10.Forcecomponentsinxandydirectioninno-loadoperation WITHOUTstatorparallelpathsandWITHOUTrotordamperwinding calculatedusinglinearFEMresultingfromthefollowingharmonic componentsoftheshaftorbit:(a)staticdeflection(constantterm),1st harmonic, and 2nd harmonic, (d) static, 1st and 2nd together 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2000-100001000200030004000Time, sForce, NFy0Fx1Fy1Fx0Fx2Fy2 (a) 0 0.01 0.02 0.03 0.04010002000300040005000Time, sTotal radial force, N (b) Fig. 11.Forcecomponentsinxandydirectioninno-loadoperationWITH statorparallelpathsandWITHOUTrotordamperwindingcalculatedusing linearFEMresultingfromthefollowingharmoniccomponentsoftheshaft orbit:(a)staticdeflection(constantterm),1stharmonicand2ndharmonic,(b) static, 1st and 2nd together > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 11TheresultsinFig.11indicatethatthe100Hzcomponentof the UMP emerges in the case when there is static deflection of the rotor and the circulating currents are induced in the stator windingparallelbranches.Ifoneconsiderstheprinciple cross-sectionofafour-polemachine,thenthewindingaxes andthereferencedirectionsofthephasecurrentsfornormal operationaregivenaccordingtoFig.13a.However,inthe case when there are only induced circulating currents flowing in the stator winding, the instantaneous currents in the parallel branchesofonephasehavethesamevalue,buttheyflowin theoppositedirections.Therefore,thereferentdirectionof currentinadjacentparallelbranchofeveryphasewinding changes resulting in the winding structure and the positions of thewindingaxesasshowninFig.13b.Therefore,the oppositedirectionsofcurrentsintheparallelbranchesofthe samephasehavetheeffectofturningthefour-polewinding into a two-pole winding with winding axes of phases U and W mutuallyshiftedby60mechanicaldegrees.Thiseffectis illustratedinFig.14whichshowsthefluxlinesofthe armaturewindingfieldduetoinducedcirculatingcurrents showninFig.12aatt = 10 mswithfieldwindingcurrentset tozero.Thephaseshiftsbetweenthewindingaxesandthe phase shifts of induced circulating currents indicate that this is not a symmetrical three-phase system, which gives rise to both direct and inverse components of the armature winding MMF whosefundamentalcomponentinafour-polegeneratoris given by ( ) ( )a ad1 s e ad0 ai1 s e ai0cos cos F F t F t u e u e = + + + + (12) whereusistheangularcoordinatealongthestator circumference,eeistheelectricalangularfrequency,Fad1, Fai1,ad0andai0arethemagnitudesandtheinitialphase shiftsofthedirectandinversecomponentsrespectively.The fundamental component of the field winding MMF equals ( )f f 1 s ecos F F p t u e = (13) whereFf1isthemagnitudeofthefundamentalcomponent MMFandpisthenumberofpolepairs.Inthepresenceof static deflection of the rotor, i.e. static eccentricity, the air-gap permeance can be approximated using a term [28], [29] ( )s se s se00,2 ,4cos cosep ptpvvev u u u= (| |A = A + A (| ( \ . (14) where Av is the harmonic component of the air-gap permeance withouteccentricity,Aseisanadditionalpermeance componentduetostaticeccentricityanduse0istheangular positionofthemaximumair-gapreduction.Theradial component of the air-gap flux density is then equal to ( )r a fB F F = + A(15) TheUMPcanbecalculatedfromtheradialorand tangentialoucomponentsoftheMaxwellstresstensorgiven by 0 0.01 0.02 0.03 0.04-1-0.75-0.5-0.2500.250.50.751Time, sStator current, A iU0iV0iW0 (a) 0 0.01 0.02 0.03 0.04-0.4-0.3-0.2-0.100.10.20.30.4Time, sStator current, A iU1iV1iW1 (b) 0 0.01 0.02 0.03 0.04-0.03-0.02-0.0100.010.020.030.04Time, sStator current, A iU2iV2iW2 (c) 0 0.01 0.02 0.03 0.04-1-0.75-0.5-0.2500.250.50.7511.251.5Time, sStator current, A iU0+iU1+iU2iV0+iV1+iV2iW0+iW1+iW2 (d) Fig. 12.Induced stator winding currents in no-load condition calculated using linearFEMwhichcirculatewithinthebranchesconnectedinparallel (WITHOUTrotordamperwindingtakenintoaccount)resultingfromthe followingharmoniccomponentsoftheshaftorbit:(a)staticdeflection (constantterm),(b)1stharmonic,(c)2ndharmonic,(d)static,1stand2nd together > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 12(a) (b) Fig. 13.Principlecross-sectionofthefour-polegeneratorwithdoublelayer shortpitchedstatorwindingarrangement,referentcurrentdirectionsand positions of the phase winding axes for the cases of (a) normal operation, and (b)operationwithopen-circuitedstatorwindingandinducedcirculating currents in the parallel branches Fig. 14.Fluxlinesofthearmaturewindingfieldduetoinducedcirculating currentsinthewindingparallelpathswithfieldwindingcurrentsettozero without rotor damper winding and with only static deflection of the shaft orbit taken into account ( ) ( )2 20 02 ,u u uo o = =r r rB B B B (16) where 0 is the permeability of vacuum. The usual assumption forsalient-polemachinesisthatBuisnegligiblecomparedto Br.Thedetailedanalysiswhichconfirmsthisassumptionis given in [30]. Hence, the x and y components of the UMP are given by 2 2s s0 02 2s s0 0cos2 2sin2 2rxryB DF l dB DF l dttu uu u==}}(17) Forsimplicity,ifonlyadditionalpermeanceduetostatic eccentricityisusedin(14)combinedwith(15)and(17),the resulting expressions for Fx and Fy are ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )1 se0 ad0 ad0e ad0 se0 2 se0 ai0e ai 0 e ai0 se01 se0 ad0 ad0e ad0 se0 2 se0 ai0e ai0 e ai0cos 2 2coscos 2 2 cos 22cos 2 cos 2 2sin 2 2sinsin 2 2 sin 22sin 2 sin 2xyF ft ft tF ft ft tu e u u e e uu e u u e e = + +( + + + ( + + + = + +( + + + + + ( )se02u ((18) where 2 21 se f1 ad1 2 se f 1 ai10 0,16 16f F F Dl f F F Dlt t = A = A .(19) From these solutions it is apparent that Fx and Fy consist of constanttermsandharmoniccomponentsat2ee,which correspond to the frequency of 100 Hz. The FEM results (Fig. 11)alsoindicatethepresenceofthisharmoniccomponent. Themorecompletesolutionisobtainedbyaddingalsothe dominantair-gappermeancetermsA0+A2pcos[2(pus-eet)]in (14)and(15),buttheyareomittedduetoverylengthy solutionof(17)thusobtained.Forinstance,theinclusionof onlyA0yieldsadditionalconstanttermsandharmonic components at 2ee, while A2pcos[2(pus-eet)] yields additional constant terms and harmonic components at 2ee and 4ee. VI.CONCLUSION Thecalculationofunbalancedmagneticpullina5 MVA, 4 polesynchronousgeneratorbasedonknowngeometryof the machine, finite-element method and measured shaft orbits in no-load operation at 10500 V, 50 Hz, 1500 rpm and loaded condition has been presented. TheUMPhasbeenanalyzedforthreedifferentcaseswith respect to damping effects of the stator winding parallel paths andtherotordamperwinding.Ithasbeenshownthatinthe casewhenthedampingeffectsarenottakenintoaccountthe harmoniccontentoftheshaftorbitiscloselyrelatedtothe harmoniccontentoftheUMPsincethentheUMPis proportionaltotheinstantaneousdisplacementoftherotor. Whendampingisincluded,boththemagnitudeandthe harmoniccontentoftheUMPareaffected.Themoreexact correlationsbetweenharmoniccomponentsoftheUMP, individualharmoniccomponentsoftheshaftorbitandthe dampingeffectsofthestatorwindingandtherotordamper windingcanbeestablishedbysimulatingtheseparate harmoniccomponentsoftheshaftorbitinbothno-loadand loadedcondition.However,thisapproachinnotdirectly applicabletononlinearfinite-elementsimulationsbecause superpositioncannotbeusedundernonlinearconditions.A way around this problem would be to simulate the no-load or loadedconditionwiththeactualshaftorbitandstorethe valuesofpermeabilityrelatedtothenodesoftheFEmesh. AfterthattheUMPcanbecalculatedseparatelyforeach harmonic component of the shaft orbit always using the same stored nodal values of permeability in order to establish all the > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 13correlationscorrectly.Thisprinciplecanbeappliedtoall three cases studied in the paper. Theresultsshowninthepaperindicatethatforthis generator the stator winding parallel paths alone attenuate the UMPby14.3 %inno-loadand17.1%inloadedcondition. Therotordamperwindingandthestatorwindingparallel paths together yield the total attenuation of 57.4 % in no-load and 51.0 % in loaded condition. 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CurrentlyheisanAssistantProfessoratthe DepartmentofElectricalMachines,Drivesand Automation,FacultyofElectricalEngineering andComputing,UniversityofZagreb,Croatia. Hisresearchactivitiesarerelatedtodesign, modeling and optimization of electrical machines and power transformers and testing of electrical machines. > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 14Drago Ban (M01) was born in 1939 in Vrpolje-Sibenik,Croatia.HereceivedtheB.Sc.,M.Sc. and Ph.D. degrees in electrical engineering from the Faculty of Electrical Engineering, University ofZagreb,Zagreb,Croatiain1965,1975and 1987 respectively.From1968till1989heworkedasa development and project engineer in KONAR-ElectricalIndustry,Zagreb,Croatia.Heis currentlyafullProfessorattheDepartmentof ElectricalMachinesDrivesandAutomation, FacultyofElectricalEngineeringand Computing,UniversityofZagreb,Croatia.His scientific and professional activities have been related to the field of electrical machinery, electrical drives and technical diagnostics. ProfessorBanhasalongexperienceinorganizationofinternational conferences,seminarsandlectures.Hewasthechairmanofseveral internationalconferencesonelectricaldrivesandpowerelectronics:EDPE 1996, EDPE 98 and EDPE 2000. In 2002 he was the general chairman of the 10thInternationalPowerElectronicsandMotionControlConference,EPE-PEMC2002,9-11.Sept.Dubrovnik,Croatia.HeisamemberofEditorial BoardoftheJournalofElectricalEngineering,severalinternational professional associations, and a member of Croatian Academy of Engineering.