modeling and parametric design of permanent-magnet ac machines using computationally efficient...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 6, JUNE 2012 2403

Modeling and Parametric Design ofPermanent-Magnet AC Machines Using

Computationally Efficient Finite-Element AnalysisGennadi Y. Sizov, Student Member, IEEE, Dan M. Ionel, Senior Member, IEEE, and

Nabeel A. O. Demerdash, Life Fellow, IEEE

Abstract—Computationally efficient finite-element analysis(FEA) (CE-FEA) fully exploits the symmetries of electric andmagnetic circuits of sine-wave current-regulated synchronous ma-chines and yields substantial savings of computational efforts.Motor performance is evaluated through Fourier analysis and aminimum number of magnetostatic solutions. The major steady-state performance indices (average torque, ripple and coggingtorque, back-electromotive-force waveforms, and core losses) aresatisfactorily estimated as compared with the results of detailedtime-stepping (transient) FEA. In this paper, the CE-FEA methodis presented and applied to a parametric design study for aninterior-permanent-magnet machine. Significant reduction of sim-ulation times is achieved (approximately two orders of magnitude),permitting a comprehensive search of large design spaces foroptimization purposes. In this case study, the influence of threedesign variables, namely, stator tooth width, pole arc, and slotopening, on three performance indices, namely, average torque,efficiency, and full-load torque ripple, is examined, and designtrends are derived. One hundred candidate designs are simulatedin less than 20 minutes on a state-of-the-art workstation.

Index Terms—Design optimization, electromagnetic analysis,electromagnetic modeling, finite-element (FE) methods, Fourierseries, permanent-magnet (PM) machines, PM motors, synchro-nous machines.

NOMENCLATURE

A Magnetic vector potential [(MVP); in we-bers per meter].

J , JPM Stator and equivalent permanent-magnet(PM) current densities (in amperes persquare meter).

μ Magnetic permeability (in henries permeter).

Manuscript received December 20, 2010; revised March 31, 2011, May 12,2011, and May 23, 2011; accepted May 23, 2011. Date of publicationAugust 15, 2011; date of current version February 10, 2012. This work wassupported in part by the National Science Foundation’s Grant Opportunitiesfor Academic Liaison with Industry under Grant 1028348 and in part byA. O. Smith Corporation.

G. Y. Sizov and N. A. O. Demerdash are with the Department of Electricaland Computer Engineering, Marquette University, Milwaukee, WI 53201-1881USA (e-mail: [email protected]; [email protected]).

D. M. Ionel is with Vestas Technology R&D Americas, Marlborough, MA01752 USA (e-mail: [email protected]), and also with the Department ofElectrical and Computer Engineering, Marquette University, Milwaukee, WI53201-1881 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2011.2163912

ν, νM , νW Harmonic order, maximum order of fluxlinkage harmonics and stored energy har-monics, respectively.

λa, λν Instantaneous value and amplitude of theνth time harmonic of the flux linkage (inweber turns).

θ, θm Angular positions in electrical and mechan-ical radians, respectively.

ω, ωm Synchronous frequencies in electricaland mechanical radians per second,respectively.

ea Instantaneous induced voltage (in volts).ia Instantaneous phase current (in amperes).s Number of static finite-element (FE)

solutions.β Torque angle (current and the d-axis).Wstored Instantaneous stored energy (in joules).Wν Amplitude of the νth space harmonic of the

stored energy (in joules).Ns, Nφ, P , Ncog Numbers of slots, phases, poles, and cog-

ging periods per revolution, respectively.Tem Instantaneous electromagnetic torque (in

newton meters).Br,t Radial and tangential stator core flux densi-

ties (in teslas).Bν Amplitude of the νth harmonic of the flux

density (in teslas).wh, we Hysteresis and eddy-current specific losses

(in watts per kilogram).khν , keν Harmonic core-loss coefficients.θs Slot pitch in electrical radians.

I. INTRODUCTION

THE latest generation of high-efficiency and high-specific-power-density electric motor drives is based on brushless

PM technology [1], [2]. PM alternating-current (PMAC) ma-chines, which are also referred to as PM synchronous machines,employ sine-wave current-regulated drives and yield superiorperformance, which justifies the continued interest in theirstudy. The design of PMAC machines is complicated by typicalheavy magnetic saturation, complex geometries, and relativelylarge number of design variables.

Recent developments in computer hardware and softwaretechnologies have made possible the extensive use of finite

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2404 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 6, JUNE 2012

Fig. 1. Block diagram showing model-based design optimization employingCE-FEA.

element analysis (FEA) at all stages of the design process.While traditionally lumped parameter (magnetic equivalent cir-cuit) and analytical closed-form models [3]–[7] have been usedfor fast initial sizing and analysis, the design framework pro-posed in Fig. 1 employs for these tasks a special computation-ally efficient FEA (CE-FEA), which will be described in thispaper. The focus of CE-FEA is on the electromagnetic perfor-mance, and for a comprehensive design approach, the methodcan be combined with thermal [8], [9] and mechanical analysis[10], [11].

Analytical modeling of machines with interior-PM (IPM)rotors is facing many challenges due to the presence of bothgeometric and magnetic-saturation-induced saliencies that af-fect motor performance. Furthermore, due to the very largevariety of IPM rotor topologies, the generalization of suchmodels is difficult. On the other hand, the choice of an FE-basedmodel, in its classical formulation, significantly constrains thedesign optimization search space due to the prohibitively longexecution times.

The need for fast and accurate FE-based models for PMACmachines has prompted the development of various simpli-fied techniques by various authors, e.g., [12]–[17]. Early on,Fouad et al. proposed and described in [12] a method forestimating the stator core flux density and core losses basedon a series of static solutions and on the slot-pitch symmetryof the magnetic circuit. Ionel et al. published a method forcomputing the average torque from a single magnetostaticsolution [13], and Barcaro et al. followed by studying thetorque ripple calculations from several static solutions [14].Recently, Ionel and Popescu have proposed FE techniques,which employ the minimum number of magnetostatic solutionsby exploiting the symmetries of the magnetic [15] and electric[16] circuits, in order to estimate the average torque, inducedvoltage (EMF), and torque ripple, as well as core losses underany load condition.

This paper introduces additional modeling contributions,which include a minimum-effort estimation of the coggingtorque component and torque ripple and the calculation ofboth the radial and tangential flux density waveforms at anylocation in the stator core. Furthermore, the overall results ofthe method are systematically studied for errors that could becaused by the aliasing effects associated with Fourier analysis.

Fig. 2. Cross section of the case-study nine-slot six-pole motor showing theopen-circuit flux distribution, design parameters, and points used for stator coreflux density estimation.

The new technique makes possible a comprehensive designanalysis, with the full benefits of FEA capabilities in termsof accurate magnetic circuit geometry and magnetic saturationrepresentation, while achieving a significant reduction, of up totwo orders of magnitude, of the execution time in comparisonto time-stepping FE (TSFE) methods.

The CE-FEA method, which is generally applicable to a wideclass of sine-wave current-regulated synchronous machines,is presented in this paper through an example three-phasenine-slot six-pole IPM design case with concentrated windings(Fig. 2). It is demonstrated that design optimization, which issolely based on FE for magnetic circuit analysis (Fig. 1), cannow be performed with present state-of-the-art hardware andsoftware technologies. The capabilities are illustrated by a de-tailed parametric study in which more than 100 candidate IPMmotor designs have been evaluated in less than 20 minutes on aPC-based computational setup typical for industrial engineeringenvironments.

In this paper, the work presented in [17] is extended topresent a comprehensive parametric study highlighting theconflicting and nonconflicting nature of various design objec-tives. Also, application of the CE-FEA technique to modelingand optimization of PM machines operating in the constant-torque region (maximum torque per ampere) and in the field-weakening mode is discussed.

A comprehensive verification of the results of the newCE-FEA technique with respect to the well-established/validated and more detailed TSFE method is also presented.This approach was preferred for systematic validation as thetwo computational methods share the same geometrical andmaterial models and because it avoids the inherent uncertain-ties that could be introduced by manufacturing tolerances andtesting errors.

II. CE-FEA MATHEMATICAL FORMULATION

The main assumption used throughout this work is that themachine is supplied by a purely sinusoidal set of armaturewinding currents. In other words, time harmonics present inthe current waveforms due to the electronically controlledpulsewidth-modulation inverter supply are neglected. This as-sumption is justifiable in motor-drive systems that employ well-tuned current regulators.

SIZOV et al.: MODELING AND PARAMETRIC DESIGN OF PERMANENT-MAGNET AC MACHINES USING CE-FEA 2405

Fig. 3. Rated-load (3600 r/min, 23 A, 120◦ torque angle) (a) flux linkage, λa, and (b) back EMF, ea, obtained using CE-FEA (with five magnetostatic FEsolutions) and TSFE. Shown in (a) are the 30 flux linkage sample points obtained from five magnetostatic solutions.

Fig. 4. Cogging torque (open circuit).

The proposed approach employs a 2-D magnetostaticFE formulation based on the following quasi-Poissonianequation [18]:

∂

∂x

(1μ

∂A

∂x

)+

∂

∂

(1μ

∂A

∂y

)= −J − JPM. (1)

In this formulation, the magnetic field in the machine is solvedat a successive number of rotor positions, in which the modelis excited by instantaneous values of a set of sinusoidallytime-varying currents. These instantaneous values correspondto a specific value of the load torque and rotor position. Theinstantaneous values of the rotor position θm and the phasecurrents ia, ib, and ic are inputs to the model, and the MVPsA are the outputs, which are used in the postprocessing stageto generate flux linkages, flux densities, back EMFs, andenergy/coenergy values.

A. Flux Linkages and EMFs

It has been shown that, for a three-phase PMAC machine un-der symmetrical steady-state operating conditions, based on theelectric and magnetic circuit symmetries, a single magnetostaticFE solution provides three equidistantly spaced (in time) sam-ples of the average MVPs in the coil sides and, consequently,of both phase flux linkage and back-EMF waveforms [16]. Fur-thermore, using the half-wave symmetry, the number of pointsobtained per solution is doubled to six. Additional samples maybe obtained by carrying out additional magnetostatic solutions

Fig. 5. Electromagnetic torque: (top) rated-load, (middle) half-load,(bottom) open-circuit.

at equidistant increments of rotor position with correspondinginstantaneous values of currents over the winding phase belt. AFourier series of both phase flux linkages and back EMFs canthen be constructed

λa(θ) =νM∑ν=1

λν cos (νθ + φν) (2)

ea(θ) = − dλa

dθ

dθ

dt= ω

νM∑ν=1

νλνsin (νθ + φν) . (3)

Similar expressions can be developed for the remainingphases b and c. It should be noted that this development,which is straightforward for concentrated windings (Fig. 2),can be extended to distributed windings by considering theactual layout of the coil sides in the slots. Here, the relationshipbetween the maximum harmonic order νM and the number ofmagnetostatic solutions s is

νM = 3s − 1. (4)

Shown in Fig. 3(a) and (b) are the phase flux linkage andinduced voltage waveforms at rated-load conditions obtainedusing the procedure previously outlined above with five magne-tostatic solutions over 60 electrical degrees and a correspond-ing comprehensive TSFE simulation, respectively. Using fivemagnetostatic solutions yields a total of 30 sample points offlux linkage over a complete electrical cycle, allowing one toaccount for the harmonics up to the 14th order.


Fig. 6. Cross section of the nine-slot six-pole IPM showing the locations ofsister elements separated by one slot pitch θs. First-order elements with onevalue of flux density per element are assumed.

B. Electromagnetic Torque

The electromagnetic torque Tem calculation used in CE-FEAfollows the virtual work principle [19], which is based on theenergy stored in the magnetic circuit of the machine Wstored,for every magnetostatic field solution. The Fourier series of thestored energy and its derivative with respect to the mechanicalposition angle are then constructed as follows:

Wstored(θm) =νW∑ν=1

Wν cos(νNcogθm + φν) (5)

dWstored

dθm= − Ncog

νW∑ν=1

νWν sin(νNcogθm + φν) (6)

where Ncog is determined as follows:

Ncog =NsP

gcd(Ns, P )(7)

where gcd stands for the greatest common divisor. In this case,the maximum harmonic order of the estimated stored energyνW is related to the number of solutions s, such that νW =s/2. Hence, for satisfactory modeling of the cogging torque, asufficient number of solutions are required in order to capturethe variations of energy. Using the expressions developed forthe back EMF in (3) and the derivative of the stored energyin (6), the electromagnetic torque developed by a three-phasemachine can be estimated using (8) (shown at the bottom ofthe page). Torque under various load conditions is calculatedusing (8) with five magnetostatic solutions. Simulations arealso performed using a substantially more computationally

Fig. 7. Estimation of stator core flux densities using elements e1, e2, and e3

(Fig. 8) for five solutions. Assuming half-wave symmetry, the number of pointsis doubled to 30.

intensive TSFE, which employs Maxwell-stress-tensor-basedcalculations. The results from Figs. 4 and 5 show very goodagreement between the two approaches under all investigatedload conditions.

The estimation of the derivative of the stored magneticenergy in (6) is highly dependent on the mesh density and FEorder around the air-gap region, and hence, the recommendedair-gap mesh should be fine enough in order to capture the vari-ation of the stored energy (Fig. 4) with rotor position and, con-sequently, of the cogging torque component. This is not a lim-itation of this approach since such mesh density requirementsare even more important for the Maxwell stress tensor torquecalculations. The average electromagnetic torque 〈Tem〉 canbe estimated from the fundamental components of the phaseinduced voltage and phase current as given in the following:

〈Tem〉 =〈Pem〉ωm

=Nφe1i12ωm

cos(θe1 −

(β − π

2

))(9)

where e1 is the peak value of the fundamental component ofthe phase induced back EMF, θe1 is the phase angle of thefundamental component of the induced back EMF, and i1 is thepeak value of the fundamental component of the phase current.

C. Stator Flux Density and Core Losses

Based on the magnetic circuit symmetry, the values of theradial and tangential flux densities at steady-state conditionscan be estimated as follows [12], [15]:

Br,t

(t +

kθs

ω, r, θ

)= Br,t(t, r, θ + kθs) (10)

Tem =P

2

(ia

dλa

dθ+ ib

dλb

dθ+ ic

dλc

dθ

)− dWstored

dθm

=P

2

(ia(θ)

νM∑ν=1

νλν sin [νθ + φν ] + ib(θ)νM∑ν=1

νλν sin [ν(θ − 120◦) + φ] + ic(θ)νM∑ν=1

νλν sin [ν(θ − 240◦) + φ]

)

+ Ncog

νW∑ν=1

νWν sin(νNcogθm + φν) (8)


Fig. 8. Radial and tangential flux densities at the locations identified in Fig. 2. (a) Yoke and tooth–yoke junction. (b) Midtooth and tooth tip. The radial fluxdensity component in the yoke and the tangential component in the midtooth are negligible and are not shown. The CE-FEA results virtually overlap the TSFEcalculations.

where k is a positive integer. For a machine with Ns statorslots and P poles, a single magnetostatic solution would yieldNs/(P/2) samples spaced in time by θs/ω. Assuming the lackof even-order harmonics (half-wave symmetry), the number ofsamples per electrical cycle is doubled to 2Ns/(P/2). For thecase-study machine having a 9-slot and 6-pole combination, asingle magnetostatic solution yields six samples of elementalflux densities. For additional points, static FE solutions have tobe carried out at equidistant increments of rotor positions overa complete slot pitch, hence increasing the number of solutionsin addition to those used for the calculations of the flux linkagesin (2). For some slot–pole combinations, the computationaleffort can be substantially reduced by utilizing the same staticsolutions that are employed in the calculation of flux linkages.This procedure can be visualized with reference to Figs. 6 and 7,in which additional samples and half-wave symmetry are usedto populate a complete 360◦e cycle. Applying the logic used forthe estimation of phase flux linkages to elemental flux densitiesresults in the following Fourier series:

Br,t(θ) =νM∑ν=1

Bν cos(νθ + φν). (11)

Shown in Fig. 8 are the profiles of the radial and tangential fluxdensities at four locations (shown in Fig. 2). These locationsare at the following: 1) the yoke; 2) the tooth–yoke junction;3) the midtooth; and 4) the tooth tip. Good agreement betweenthe values obtained using CE-FEA and detailed TSFE can beobserved in Fig. 8.

Using the radial and tangential components of the core fluxdensities obtained from (11), the specific core losses per unitmass can be calculated as follows [20]:

wh =νM∑ν=1

khν(νf1, Bν)(νf1)B2ν

we =νM∑ν=1

keν(νf1, Bν)(νf1)2B2ν (12)

where the core-loss coefficients khν and keν are dependent onthe harmonic frequency νf1 and the peak value of the harmonicflux density Bν . A comparison between the results obtained

TABLE ICORE LOSSES. MEASURED AND CALCULATED USING TSFE AND

CE-FEA WITH A BASIC FOUR-BLOCK SUBDIVISION OF THE

MAGNETIC CIRCUIT (FIG. 2)

with the numerical models (CE-FEA and TSFE) is provided inTable I. Table I also includes the results of experimental mea-surements that are provided solely for reference. The differencebetween the numerical models and the experimental tests can beattributed to manufacturing and measurement uncertainties thatare not accounted for in the numerical models. The preferredvalidation for CE-FEA is provided by a TSFE of a motormodel employing the same materials, geometry, and mesh.This approach ensures a fair basis of comparison as it avoidsuncertainties and imperfections associated with prototypingor the effects of measurement errors. Satisfactory agreementbetween the core-loss values obtained using the detailed TSFEand CE-FEA methods is observed (see Table I). In this case,a basic subdivision of the stator magnetic circuit into foursubdomains, namely, yoke, tooth–yoke junction, tooth stem,and tooth tip, was employed as shown in Fig. 2. The accuracy ofthe CE-FEA calculation can be increased further by consideringa larger number of region discretizations of the magnetic circuit.It should be noted that the CE-FEA-based core-loss calculationcan be implemented on a per-element basis with an additionalrequirement of a repetitive FE mesh on a slot-pitch basis.

D. Constant-Torque and Field-Weakening Operations

A PM machine with an IPM rotor develops two distinctaverage torque components, namely, an alignment torque dueto the interaction of the PM field and the armature current anda reluctance torque due to the interaction of the rotor saliencyand the armature field. In the constant-torque region, the motoris typically controlled at “maximum torque per ampere,” which


Fig. 9. Fast search algorithm for optimal IPM performance in terms ofmaximum-torque-per-ampere production.

corresponds to a torque angle βmax. Within a design procedure,the convergence of the search process for the optimal operatingconditions can be accelerated through a numerical bisectionalgorithm as shown in Fig. 9. This approach also benefits fromthe fact that, for most practical designs, the peak of the shafttorque curve is fairly flat. It should be noted that a similar searchalgorithm could be used for calculating the peak value of thecogging torque.

In a typical IPM machine, operating in the field-weakeningregion, the motor currents are controlled to minimize the result-ing phase flux linkage and, consequently, the induced voltageswhile providing maximum torque [21], [22]. This is accom-plished by advancing the torque angle, which, in turn, decreasesthe resultant induced voltage as shown in Table II. Provided inTable II are the values of the average electromagnetic torque,torque harmonics, and torque ripple for different torque angles.Also provided in Table II are the harmonics of the inducedvoltage and the total harmonic distortion. The torque–speedcharacteristic corresponding to the constant-torque operationunder maximum-torque-per-ampere condition and the field-weakening operation for the machine in Fig. 2 is shown inFig. 10. Good agreement between the results of the CE-FEAand a detailed TSFE model can be observed for a wide range ofoperating conditions. Both maximum-torque-per-ampere per-formance modeling and field-weakening operation modelingmay be problematic when conventional TSFE is employed,because it significantly increases the computational burden, re-quiring additional costly TSFE evaluations. This is not of signi-ficant concern with simplified approaches such as the CE-FEA,where several magnetostatic solutions may be used for theextraction of quantities such as the average torque and the fun-damental components of induced voltages. In the preferred pro-cedure, the abc-frame-of-reference analysis is carried out basedon a minimum number of magnetostatic FE solutions as previ-ously described. Simulations are performed for different torqueangles in order to determine the optimal performance. Thisapproach increases the savings of computational effort as com-pared with techniques based on a more detailed TSFE analysis.

E. Discussion on Possible Aliasing Effectsand Implementation

In the CE-FEA, Fourier analysis is used for the constructionof MVPs, flux linkages, back EMFs, and energy waveforms.Since the Fourier analysis is applied to a finite set of samples,

Fig. 10. Torque–speed characteristic showing the constant-torque and field-weakening operations.

aliasing of higher order harmonics is possible, with the errordepending on the magnitudes of the harmonics that are cutoff (truncated) from the Fourier series. Such errors in the fluxlinkages will be further amplified in the back EMF due to thedifferentiation process and will therefore also affect the torqueestimation. This places a constraint on the minimum numberof magnetostatic FE solutions s, which depends mainly on themagnitude and order of the highest harmonic present in thewaveform. In other words, if a small number of solutions areperformed, resulting in a number of samples insufficient forcapturing higher order harmonics (which may have significantmagnitudes), aliasing will occur, leading to errors. The magni-tude of the aliasing error is highly dependent on the machinedesign parameters and operating point conditions. For somefractional-slot concentrated-winding machines, such as the oneused here for demonstration, the error may be more significantdue to the fact that the on-load flux linkage and induced back-EMF waveforms (Fig. 3) have a rich harmonic content, whichis dependent on the current amplitude and the torque angle. Thefrequency spectrum of the phase induced back-EMF waveformunder rated-load condition (Fig. 11) shows that, in the examplemotor, the harmonics higher than the 13th order are negligible.Hence, by applying (4), one can choose the number of solutionss = 5 in order to avoid aliasing. The effect of aliasing on theestimation of the fundamental component of the induced backEMF and on the average torque calculation in (9) is illustratedin Fig. 12, showing that, in this case, a further increase (be-yond s = 5) in the number of static solutions will not provideadditional accuracy in the estimation of the average torque. Onthe other hand, it is expected that, in the cases of distributed-winding machines, particularly those with short-pitched wind-ings, and/or machines with well-shaped (sinusoidal) rotor MMFwaveforms, as few as one FE solution may be required in orderto obtain an accurate estimate of the average torque. This isdue to the fact that such machines have a low harmonic contentand, hence, an insignificant error due to aliasing. Here, it shouldbe noted that, when the ripple torque needs to be consideredwith high precision in the overall torque production, one couldincrease the number of solutions in order to provide sufficientharmonic resolution of the stored energy and, hence, of thecogging torque component. This is further verified with addi-tional results shown in Figs. 13–15. Shown in Fig. 13(a) and (b)are the flux density and flux distributions corresponding to themachine operating at 120◦ and 170◦ torque angles, respectively.


TABLE IIPERFORMANCE PARAMETERS FOR VARIOUS TORQUE ANGLES ESTIMATED WITH CE-FEA EMPLOYING FIVE MAGNETOSTATIC

FE SOLUTIONS AND DETAILED TSFE

Fig. 11. Fourier spectrum of the back EMF ea from Fig. 3(b). In this case,five static FE solutions are sufficient to avoid aliasing.

As expected, under the deep field-weakening operating condi-tion, corresponding to 170◦ torque angle [Fig. 13(b)], strongreduction of the resulting flux densities due to high-armature-MMF demagnetization is observed. This results in a significantreduction of the fundamental induced voltage component, ascan be observed from Fig. 14. Shown in Fig. 14 are the inducedvoltages for the machine operating under 120◦ and 170◦ torqueangles, respectively. Also shown in Fig. 15 are the developedtorque profiles, corresponding to the two operating conditions,estimated using CE-FEA and TSFE, respectively. Again, fromFigs. 14 and 15 and Table II, excellent agreement between theCE-FEA employing five magnetostatic solutions and detailedTSFE simulations for a wide range of operating conditions isobserved.

The effects of mesh density on the performance parametersand execution time are presented in Table III. The resultspresented in Table III are compared to a detailed TSFE modelemploying a fine mesh and second-order elements. As canbe seen from Table III, the best accuracy in the estimationof the average torque, torque ripple, and induced voltage isobtained with second-order elements. It should be noted thatan FEA based solely on first-order elements may be sufficientfor the estimation of the average torque. Hence, a CE-FEAbased on first-order elements may be used in the search foran optimum operating condition (maximum torque per ampere)

Fig. 12. Error of the average torque computed with CE-FEA with respect to adetailed TSFE simulation.

or field-weakening capability evaluation. On the other hand, amore detailed and computationally intensive CE-FEA based onsecond-order FEs can be used for detailed analysis that includesaccurate cogging torque and torque ripple calculation.

III. PARAMETRIC DESIGN STUDY

Utilizing the CE-FEA, described in the previous section, arapid search of very large design spaces is possible. In thissection, the CE-FEA is used to create response surfaces repre-senting the relationships between three main design parameters,namely, pole arc, slot opening, and tooth width (Fig. 2), anda set of performance indices, namely, average shaft torque,efficiency, and torque ripple, which includes the cogging torquecomponent. The current density through the stator winding isconsidered constant. The steady-state operation of 100 candi-date designs was simulated through CE-FEA, based on fivemagnetostatic FE solutions for each design. The magnetostaticFE simulations have been performed with the commerciallyavailable software MagNet [23], which was scripted to performCE-FEA computations.

For the following examples, a single pole pair of the6-pole IPM motor geometry shown in Fig. 2 was used. Thecomputational time required to evaluate a single design ata single load condition on an Intel-Xeon (E5520) 2.27-GHz


Fig. 13. Flux density distributions for (a) maximum-torque-per-ampere condition (120◦ torque angle) and (b) flux-weakening operation (170◦ torque angle).(a) Rated load, 23 A, 120◦ torque angle. (b) Field weakening, 23 A, 170◦ torque angle.

Fig. 14. Induced voltage profiles corresponding to maximum-torque-per-ampere condition (120◦ torque angle) and flux-weakening operation(170◦ torque angle).

4-GB-RAM workstation was dependent on operating condi-tions and magnetic saturation and had an average of approx-imately 10 seconds. The total time required to evaluate thewhole set of 100 designs was below 20 minutes. In comparison,the simulation time for the same set of 100 candidate designswith similar mesh density using a detailed TSFE model with360 samples per electrical cycle would be substantially longerat approximately 40 hours. Because the three motor designparameters, which were previously mentioned, have somewhatpredictable influence on motor performance, the results of theparametric study can serve as a proof of concept. Furthermore,the problem is of practical engineering interest as it correspondsto a potential real-life situation in which a motor design has tobe optimized for a given frame that imposes the stator outerdiameter and has to be produced with existing winding tooling,which would require a certain stator inner diameter. The resultsof the parametric study are summarized in Figs. 16–18.

In line with expectations, for any of the values consideredfor the slot opening, the shaft torque and the efficiency areproportional to the tooth width and inversely proportional tothe pole arc (Fig. 16). If, for example, maximum efficiency isthe sole objective of an optimization study, a value of 94.6%can be achieved for a shaft torque of 16.6 N · m as shown in

Fig. 15. Developed torque profiles corresponding to maximum-torque-per-ampere condition (120◦ torque angle) and flux-weakening operation(170◦ torque angle).

TABLE IIIEFFECTS OF MESH DENSITY AND FE ORDER ON THE CE-FEA

ESTIMATION ACCURACY AND COMPUTATIONAL TIME

Fig. 18(a) and for a combination of design parameters that canbe identified from Fig. 16.

The rated-load torque ripple, which includes the coggingtorque component, has a strong nonlinear behavior with distinctminimum values and is influenced by all three design parame-ters as shown in Fig. 17. Should minimum torque ripple bethe only design objective, a value of 10.8% can be achieved;however, in this case, the motor would be rated at only14.1 N · m, and the efficiency would be limited at 93.4%.


Fig. 16. Efficiency at rated-load conditions as a function of design parameters.

Fig. 17. Torque ripple at rated-load conditions as a function of design parameters.

Fig. 18. Plots of the motor performance illustrating the conflicting and/or nonconflicting trends between average shaft torque, efficiency, and torque ripple for anexample parametric study.

The collection of parametric results and the identified trendscan also be employed to achieve the best “real-life compro-mise” that takes into account all three performance indices,

namely, efficiency, specific output, and torque ripple, as wellas any constructive limitations for the design parameters. In theconsidered parameter range, the example IPM topology yields


a substantially linear correlation between the efficiency and theaverage shaft torque as illustrated by the results from Fig. 18(a).This means that design requirements/specifications or objectivefunctions for maximum torque output and maximum efficiencyare nonconflicting and that, for simplification, either of themcould be eliminated from a multiobjective study. On the otherhand, the correlation between the third performance index, i.e.,torque ripple, and the efficiency and output torque is highlynonlinear, as shown in Fig. 18(b) and (c), respectively. In thiscase, a mathematical technique may be employed in order toachieve a multiobjective optimization in which a cost func-tion incorporates application-dependent weighted contributionsof the conflicting performance indices. Another optimizationapproach would be to consider Pareto criteria [24] for allperformance indices, i.e., design objectives, yielding a family of“best compromise” designs. The use of CE-FEA in conjunctionwith evolutionary-type optimization techniques, such as geneticalgorithm, differential evolution, and/or particle swarm [25], isthe subject of ongoing research, which will be published in afuture paper.

IV. CONCLUSION

Reduced-computational-effort FEA techniques, such as theCE-FEA at hand, are good candidates for comprehensive large-scale model-based design optimization studies. The results ofthe CE-FEA method are in excellent agreement with thoseprovided by a detailed and significantly more computationallyintensive and well-verified TSFE model, as demonstrated onan example IPM motor operating over a wide range of condi-tions. The two computational methods provide a sound basisof comparison, which avoids possible variations introduced bymaterial characteristics, prototyping, and testing.

The CE-FEA introduced in this paper is a very fast method,which is able to estimate the main steady-state performanceof sine-wave current-regulated PMAC machines, and can suc-cessfully replace less accurate analytical/lumped-parameter cir-cuit models. The comprehensive set of results includes thewaveforms of ripple and cogging torque and of the inducedvoltages, as well as stator losses and electric machine effi-ciency. The method employs Fourier analysis based on theresults provided by a minimum number of magnetostatic FEAsolutions and achieves a good balance between speed andprecision, which makes it well suited for parametric and op-timization studies that involve thousands of candidate designevaluations.

In the nine-slot six-pole IPM motor example presented inthis paper, the influence of three main design parameters,namely, rotor pole arc, stator tooth width, and slot opening, wassystematically examined through 100 CE-FEA-based candidatedesigns, with each of them employing five magnetostatic so-lutions. Yet, all calculations have been completed in less than20 minutes on a typical PC workstation. If conventional time-stepping FEA would have been used for the same design study,the computations would have brought in only marginal im-provements in the precision of steady-state motor performanceprediction and would have taken approximately 40 hours tocomplete.

ACKNOWLEDGMENT

The authors would like to thank A. O. Smith Corporation forthe engineering support.

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Gennadi Y. Sizov (S’07) received the B.S. and M.S.degrees in electrical engineering from MarquetteUniversity, Milwaukee, WI, in 2005 and 2007, re-spectively, where he is currently working toward thePh.D. degree in electrical engineering.

From 2003 to 2005, he was an ElectricalEngineering Intern with Bucyrus International,Milwaukee. From 2005 to 2006, he was an ElectricalEngineering Intern with the Department of Researchand Development of Low-Voltage AC Drives, ABB,New Berlin, WI. His areas of interest include the

analysis, modeling, design, and condition monitoring of electric machines andadjustable-speed drives.

Dan M. Ionel (M’91–SM’01) received the M.Eng.and Ph.D. degrees in electrical engineering from the“Politehnica” University of Bucharest, Bucharest,Romania. His doctoral program included aLeverhulme Visiting Fellowship at the University ofBath, Bath Spa, U.K.

He was with the Research Institute for Electri-cal Machines (ICPE-ME), Bucharest, with InvensysBrook Crompton Company, Huddersfield, U.K., andwith A. O. Smith Corporation, Milwaukee, WI. Hewas a Postdoctoral Researcher with the SPEED Lab-

oratory, University of Glasgow, Glasgow, U.K. His industrial career includedappointments with the Research Institute for Electrical Machines (ICPE-ME),Bucharest, with Invensys Brook Crompton Company, Huddersfield, U.K., andwith A. O. Smith Corporation, Milwaukee, WI. He is currently Chief Scientistwith Vestas Technology R&D Americas, Marlborough, MA, and also an Ad-junct Professor with the Department of Electrical and Computer Engineering,Marquette University, Milwaukee, WI.

Dr. Ionel is the Vice Chair of the IEEE Power and Energy Society ElectricMotor Subcommittee and an Associate Editor of the IEEE TRANSACTIONS

ON INDUSTRY APPLICATIONS.

Nabeel A. O. Demerdash (M’65–SM’74–F’90–LF’09) received the B.Sc.E.E. degree (with first-class honors) from Cairo University, Giza, Egypt, in1964 and the M.S.E.E. and Ph.D. degrees from theUniversity of Pittsburgh, Pittsburgh, PA, in 1967 and1971, respectively.

From 1968 to 1972, he was a DevelopmentEngineer with the Large Rotating Apparatus De-velopment Engineering Department, WestinghouseElectric Corporation, East Pittsburgh, PA. From 1972to 1983, he was an Assistant Professor, an Associate

Professor, and a Professor with the Department of Electrical Engineering,Virginia Polytechnic Institute and State University, Blacksburg. From 1983to 1994, he was a Professor with the Department of Electrical and ComputerEngineering, Clarkson University, Potsdam, NY. Since 1994, he has been a Pro-fessor with the Department of Electrical and Computer Engineering, MarquetteUniversity, Milwaukee, WI, where he was the Department Chair from 1994 to1997. He is the author or coauthor of more than 100 papers published in variousIEEE transactions. His current research interests include power electronicapplications to electric machines and drives, electromechanical propulsion andactuation, computational electromagnetics in machines and drives, and faultdiagnostics and modeling of harmonic effects in machine-power systems.

Dr. Demerdash is a member of the Electric Machinery Committee of theIEEE Power Engineering Society (PES) and its various subcommittees, aswell as the Electric Machines Committee of the IEEE Industry ApplicationsSociety. He is listed in the Distinguished Lecturer Program of IEEE PES andthe Distinguished Speaker Program of the IEEE Industrial Electronics Society.He was the recipient of the 1999 IEEE Nikola Tesla Technical Field Award, two1994 Working Group Awards from both IEEE PES and its Electric MachineryCommittee, and two 1993 Prize Paper Awards from both IEEE PES and itsElectric Machinery Committee. He is a member of the American Society forEngineering Education, the Sigma Xi, and the Electromagnetics Academy.

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