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A TIME-STEPPINGCOUPLEDFINITE ELEMENT-STATE SPACE MODEL FORUCTION MOTOR DP A R T l : M

N.A. Demerdash, Fellow J. F.Bangura A. A. Arkadan, Senior MemberDepartment ofElectricaland Computer EnsineeringMarquetteUniversity,Milwaukee,WiscOnSin 53201-1881,USA.

Abstract- A Time-stepping coupled Finite Element4tate4pacemodel for induction motor drives is developed.?he model utiIizesm iterative approach to include the effects of magneticnonlinearities, and space harmonics due to the machine magneticcircuits opdogy and discretewinding layouts. Model formulationand development, which include an improvement in the layout ofthe cage circuit representation, are given in this paper. Thisimprovement leads to an enhancement of the %ell-posedness,that is, reduction of illconditioning in the overall numericalconvergence of the model. Meanwhile, in a companion paperresults of induction motor performance simulation will becompared with no-load and load tests for sinusoidal and inverteroperating conditions. Particular attention is given to comparisonbetween sinusoidal and inverter operating osses obtained from thisgeneralizedmodel.INTRODUCTIONA more reliable and rigorous simulation model forpredicting the performance and operating characteristics ofinduction motors is presented. The simulation model takesinto account the full impact of inherent nonlinearities andspace harmonics due to the machine magnetic circuitconfiguration, discrete winding layouts and nonlinearmagnetic circuit m aterial properties. Moreover, the model iscapable of ncorporating different configurationsof rotor andstator fault conditions. At the heart of this model is theimplementation of magnetic field based motor windingparameter calculation techniques. Therefore, a Time-Stepping Finite Element (TSFE) based machineparameters(inductanc) ComputationaIalgorithm constitutesone part of the complete simulation model. Furthermore,when field based calculation techniques are utilized, as inthis case, t is much more straight-forward and easier to usethe actual(mturaI) machine winding flw linkages, currentsand voltages. Accordingly, a second portion of he completemodel is a State-Space-(SS) algorithm for timedomainsimulation of the steady-state pedonnance, that is based onthe circuit relationships between the motor windings fluxlinkages, and terminal voltages. Contrary to the presentmodel, conventional d q based transformation modeling

of >reference s used throughout, The key parameters in thisS S model are the nonsinusoidal periodic windinginductances, which are obtained from the TSFE model[1,2,3]. These winding inductances, that is, all the apparemself and mutual inductances, depend on the rotor positjon aswell as the stator and rotor winding currents at a giveninstant in the ac operating cycle. These two models arecoupled together to form the so-called Time-SteppingCoupled Finite Element-State Space (TSCFE-SS) Modeldepicted in th e flowchart representation in Figure (1).

1chcckfaconvagtncr;Figure1:Flow ChartRepresentationof the TSCFE-SS ModelMODELDEVELOPMENTIn three-phase induction motors, the three phase statorarmature shown in Figure(2), can be represented by threeelectrical windings which are magneticaIIy coupled IO eachother, and to the rotor circuits.

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Figure2 MagneticCirarh, layoutandAxis Labelings

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cage is represented by a set of adjacent loops. Thecouplingbetween he loops isboth magneticandconductive.This is because each of these loops is formed by twoadjacent bars linked electtically through the bars and theend-ring segm ents connecting them on both ends of the rotorcage. Thus, each loop shares a bar with adjacent loops onboth sides, see Figure(3). This appmch, wi l l be shown nthis paper and its com panion to have significant modelingadvantages over that detailed in [I], in that it is quitegeneral, allowing for any asymmetrical and symmetricalconfiguration s of breakages in the rotor cage. It alsoenhancesboth the well-posedness of the formulation,andthe numerical convergence characteristics of the overallmodel, as will be shown in the paper. In addition, the d qrepresentation of [I] inherently imposes restrictions on thexange of bar and connector failures(breakages) that can beeasily simulated using the present model. In the d qrepresentation, any asymmetrical configuration offailures(breakages) that destroys the symmetry of the rotorcage further introduces numerically undesirable ill-conditioning in the inductance matrix of the SS model.Further details on these aspects will be included in the fullpaper.The squirrelcage rotor of the 1.2hp case-study motor has34 bars, c o ~ e c t e dt both ends hrough end-ring connectors.Thus, he complete rotor cage can be represented by 34 rotorcircuits. The respective currents flowing in these circuits,shown in Figure (3), are labeled i,,though irM These 34rotor circuit currents are used to compute the rotor cageindividualbar curr ents which, for a symmetrical rotor, areofequal magnitude with regular phase. angle progressionaround the circumference of the rotor cage.Accordingly, the 34 rotor circuitsplus the 3 stator arm aturecircuits constitute a total of 37 circuits by which the case-study induction m otor can be represented in the SS portionof the model. Therefore, the case-study motor has 37 statevariables. These 37 degrees of freedom eliminate theconventionally aocepted simplifying assumptions ofsinusoidally distributed d,urrent sheets and othersimilarconcepts associatedwith d q rame based models andtransformations. The se assumptions obscure the true natureof the spatial distribution and time harmonics in the barcurren ts, and consequently, the space harmonics affecting allthe motor inductances. fn other words, d q transformationbased models hinder the natural inherent coupling betweenspace and time effects in machine winding flux linkages,inductances, currents, and associated performanceparametercalculations.

figure3:RotorCageCircuits

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FINITEELEMENTMODELThe Finite Element based portion of the model is used tocompute all of the machine parameters(inductancs) ofinterest These parameters nclude, but are not limited to, thewinding inductance promes(in space and time), midgapradial flux density profles and profiles of elemental fluxdensities usefid for core loss computations. The crucialmachine parameters linking the TSFE model to the SSmodel are the machine winding inductances. Theinductances are directly utilized in the SSmodel to computethe steady state periodic nonsinusoidal current waveforms ofall the 37 circuits. Th e computation of the inductancesrequires a Finite Element grid of the machine cross section,as shown in Figure(4), which is excited by the iterativelyupdated instantaneous winding current profiles in a time-stepping samplin g fashion covering the complete ac cycle. Itshould be pointed out that every time sampling instant isass6ciatedwith a specific rotor an gular position in the steadystate ac cycle. T he upda ting of the currentsoccursonce everyround of a converged SS solution of the motors time-domain steady-state flux linkages and currents is achieved.Each converged SS solution is followed by a TSFE set ofsolutions to obtain th e inductance profiles over a steady stateac cycle. This process is repeated as shown schematically i nFigure(1) till convergence, which is described below, is

Figun 4 EntireFinitc Elemart Gridnese inductance profiles are computed by utilizing anenergy/current perturbation-finite element field solutiontechnique detailed earlier in many references such as [4,5].For the sake of completeness, the expressions used in thecomputation of these apparent inductances are given below:

*Aik are the current perturbations around the quiescentpoint solution obtained for the j-th and k-th windingcurrents at the given instant of time(rotor position) underconsideration, and the W s re the energies computed fromperturbed field solutions. These inductances, under steadystate conditions, are periodic nonsinusoidal functions of

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time, t, (that is rotor angular position, 8).Therefore, eachof these inductances can be expressed in harmonic Fourierseries form to be detailed in the paper.THE STATE-SPACEMODELApplying Faraday's lawand KVL formulation, the terminalvoltages in terms of the flux linkages and winding currentsof the 37 circuits of the motor can be written in compactmatrix form as follows:

y =g*i+-( l l ) ( 5 )where, Q , V_ andI represent the instantaneous fluxlinkage, terminal voltage and current vectors, respectively.The resistance matrix,& , onsists of the armaturewindingresistances and the selfand mutual conductive couplings of,and between, the circuits of the rotor cage in Figure(3).Further details on the & matrix will be given in the paper.The w inding flux linkages of the 37 magnetically coupledcircuits of the case-study motor are related to the cu rrents bythe windings' apparent self and mutual inductancesaccording to the relationship[4]:

ddt

37

where the L~~ re the apparent selfand mutual inductancesof, and between, the 37 circuits representing the case-studymotor, and where k=1,2, .... 37 and j=1,2, .... 37.Accordingly, (6)can be written in compact matrix form asfollows:Upon solving for the current from (7), and substituting into(9,ne obtains he key motor SS quation as follows:

1\=L,.r (7)

dd l . --A=( -&&- ' )&- t -y (8)Equation (8) is numerically integrated until nonsinusoidal,yet periodic, steady state is achieved for all the currents. Itmust be pointed out that the inductance matrix and itsinverse must be updated from the TSFE/current-energyperturbation magnetic field computations at each rotorangular position in the steady state ac cycle. Again, eachrotor angular position ap esp ond s to one of the sampledinstances of rime in the ac cycle. The resuIting steady state

circuit current, inductance,and lu x linkage profiles, as wellas developed torque profile, consequently include thecomprehensive mpact of the magnetic circuit saturation andassociated space h o n i c s . T h e terative xwycling throughthe SS model to obtain current profiles followed by themagnetic field TSFEbased nductance computation model toobtain inductance profiles is continued until Convergence.Here, convergenceis considered to have been achieved whensuccessive iteration s yield stator phase current profiles thatdo not exhibit no more than 5% change in amplitude(or rms

values), as well as no significant change in the harmoniccontent.SINUSOIDAL MODEL SIMULATIONThe developed model can easily be used to simulatesinusoidal voltage excitation by using the closed formexpressions for the W i a r abcpositive sequence of 3-phasebalanced voltages of the stato r phases.INVERTER MODELSIMULATIONUnlike the sinusoidal voltage excitation case, the statorphase voltages in the inverter excitation case are notrepresented by closed form expressions or by rectangularvoltage impulses. In this case, he sw itching inverter and themotor networks, as shown in Figure(5), were integrated toform an equivalent network graph of the inverter-motorsystem. ?... .......................... . m.................

: *: *

" i

Figure 5: Invnier-Moim Network Graph SchematicDetailed formulation of the e q a d e n t network graph modelis given in [2]. The Combined inverter-machineSSmodel is

dof the form, -A = A-&+V_,nd its details will be givendt -in the paper.CONCLUSIONSA more general TSCFE-SS model for induction motordrives has been developed. The model will be verified bycomparing simulation and test results for a 3phase, 2pole,1.2hp, squirre lcage case-study induction motor. A sample ofthese results of the simulation and test is given in thesummary of the companion paper.REFERENCES[11 Dancrdash, N. A and Baldassari,P.. "A combined Finite Elemenl-SlateSpaceModeling Envirmcn~or Indudon Mot m n the ABC FrameorReference:Th e No-Load condition," IEEE Transactionson EnergyConvasion, Vol. 7,No. 4, pp. 698-709, Deeanber 1992.[Z]Bangura J. F., "ATime-SteppingCoupled Finite Element-State Modcling

ofSinuSoidal-and Inverta-Fed Induction Motor Drives," MS Thesis.Marqucttc University,May 1996.131 Dens F. and Demerdash,N . A . "A Coupled F i E l r r n t c n t StataspaceApproach for SynchronousGmeratorsPart 1: Model Dcvelcpncnt,"IEEETratlsadionsonAerospaa and Elec(ronic Systems, VoL 32.No. 2, pp.7-15-784, ~ p r i i996.[4] Nehl, T.W., Fouad,F.AsndIk"h& .A.."m"ofSaturatedValuesofRotatingMachinery InaanenlalandM F t a n c e s b y a n E M F g Y P a t u b a t i o a M ~ " EEE TratsactmaaPowa Apparatusarid Syslars. Vol. PAS-100, p. 4112-4122.1981.[SI W ang R and Demerdash,N. A. "ComputatiotlOfLaad Pedtmmce andother Parametas fExfn HighSpeedModifiedLundell Attanstors %xn3D-FEMagneticFieldSolutiom," IEEE T d m mEnergyConversion,VoL EC-7, No. 2, pp.342352.1992.

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