Analyzing the Magnetic Flux Linkage Characteristics of Alternating Current Rotating Machines by...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011 2283

Analyzing the Magnetic Flux Linkage Characteristics of AlternatingCurrent Rotating Machines by Experimental Method

Tine Marcic�, Gorazd Stumberger��, and Bojan Stumberger��

TECES, Research and Development Centre for Electric Machines, Maribor SI-2000, SloveniaFaculty of Electrical Engineering and Computer Science, University of Maribor, Maribor SI-2000, Slovenia

Faculty of Energy Technology, University of Maribor, Krsko SI-8270, Slovenia

The magnetically nonlinear behavior of electric alternating current rotating machines is characterized by current dependent magneticflux linkage characteristics. In this work, the impacts of iron core saturation, iron core saliency, permanent magnets, and squirrel cagein the rotor design on the machines’ behavior, and the respective flux linkage characteristics in forms of hysteresis loops is analyzed. Theanalysis is based on experimental data by utilizing a stator with two phase-windings that produce magnetomotive forces displaced byelectrical 90 , enabling the direct evaluation of flux linkages by measurement of the windings’ voltage and current on the machine termi-nals. Five rotors of different electromagnetic construction and manufactured from equal materials have been tested in combination withthe aforementioned stator. Thus, induction machines, synchronous reluctance machines, permanent-magnet synchronous machines, andtheir hybrid types are incorporated in the analysis. The work provides a deeper insight into the magnetically nonlinear behavior of theaforementioned machines, which is not available in the current literature.

Index Terms—AC machines, magnetic cores, permanent magnets, rotating machine measurements, squirrel cage.

I. INTRODUCTION

T HE magnetically nonlinear behavior of electromagneticdevices can be characterized by different methods. At the

device design stage, methods which rely on detailed magneticproperties of the employed materials and detailed geometricdata are applied [1]–[10]. But when the device is manufacturedand to be used in a specific application, other methods areapplied which describe the magnetic behavior of the deviceby means of measurable parameters on the device terminals[11]–[23], i.e., by employing current dependent magnetic fluxlinkage characteristics. There, the exact design details of thedevice are not needed because its current dependent charac-teristics provide an integral view into the machine’s magneticbehavior.

When referring to electric ac rotating machines, the re-specting magnetic flux linkage characteristics depend notonly on the employed electrical steel sheet material and thecurrent excitation, but also on the different design measureswhich define a specific machine topology and type. Apart fromthe stator winding and stator iron core design, this includesthe design of magnetic flux barriers (FBs) in synchronousreluctance machines (SRMs), design and energy product ofpermanent magnets (PMs) in PM synchronous machines, andsquirrel-cage (SC) design in SC induction machines (SCIMs),or any combination of the aforementioned in the rotor’s ironcore applied in hybrid machines.

Therefore, this paper presents a unified and comprehensivestudy which incrementally analyses and demonstrates the im-pacts of all the aforementioned design related effects on the ma-chines’ current dependent magnetic flux linkage characteristics

Manuscript received January 23, 2011; revised March 20, 2011; acceptedApril 14, 2011. Date of publication April 21, 2011; date of current version Au-gust 24, 2011. Corresponding author: T. Marcic (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMAG.2011.2146266

in forms of hysteresis loops, obtained by experimental method.The study basically covers all essential and most widely usedtypes of contemporary ac rotating machines, which includes in-duction machines (IMs) [8], [11], SRMs [1], [11], interior PMsynchronous machines (IPMSMs) [4]–[7], [24] and also theirhybrid versions, i.e., SCSRMs [19] and SCIPMSMs. The latterare also called line-start PM machines [8], [9], [19], [25].

As far as the authors of this paper are aware, a comparablestudy can not be found in the available literature. An illustra-tive representation of the conducted experiments is shown inFig. 1. For the purpose of this work a stator with two electri-cally independent phase-windings, producing magnetomotiveforces displaced by electrical 90 , and five rotors of differentelectromagnetic construction made from equal materials havebeen employed and tested. Section II describes the conductedexperiments in detail, Section III discusses the obtained resultswhich were directly determined from measurements on proto-type machines of IEC frame size 90, and Section IV concludesthe paper.

II. DESCRIPTION OF EXPERIMENTS

A. Machine Details

The unified analysis of current dependent magnetic fluxlinkage characteristics of ac machines was conducted by usinga special stator winding. This was constructed of two windings(phase a winding and phase b winding) which had the equalwire diameter and equal number of turns. Their magnetic axeswere displaced by mechanical 45 since the experimentalmachines had 2 pole pairs. They produce magnetomotiveforces displaced by electrical 90 . The stator had embeddedthermocouples for temperature control through all tests.

Five different rotors of different electromagnetic construc-tion have been tested in combination with the aforementionedstator. First, an IM rotor was used for the determination of ini-tial saturation levels, since it does not exhibit any inherent ironcore saliency. The minor and usually neglected saliency effect

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2284 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

Fig. 1. Schematic representation of the employed experiments on (a)squirrel-cage induction machine, (b) squirrel-cage synchronous reluctancemachine, (c) synchronous reluctance machine, (d) squirrel-cage interiorpermanent-magnet synchronous machine, and (e) interior permanent-magnetsynchronous machine.

of slotting, known from SCIMs [26] was mitigated by skewingthe slots for one rotor slot pitch in all analyzed machines. Then,additional FBs were manufactured in another rotor which wasinitially equal to the first one; yielding a SCSRM rotor whichwas used for identification of inherent iron core saliency effectsor FB effects. PMs of Nd-Fe-B type were inserted in FBs of athird rotor which was equal to the aforementioned second one;yielding a SCIPMSM rotor which was used for identification ofPM effects on the magnetic flux linkage characteristics. Anothertwo rotors equal to the last two but without the SC were alsoemployed, yielding a SRM rotor and an IPMSM rotor. Thus,all the rotors were constructed of equal materials and with theequal outer diameter. This enabled the direct and incrementalcomparison of the respective machine magnetic flux linkagecharacteristics.

B. Experimental Method

All other rotors but the SCIM rotor were magnetically salientand these were firstly aligned with the magnetic axis of the phasea winding by applying a constant value of current (dc current) inthe phase a winding. Then the rotors were mechanically lockedthrough all successive tests, respectively. The individual rotorconstructions and their positions in respect to the magnetic axesof the stator windings are depicted in the lower right corner ofFigs. 3–8.

The machines’ general stator terminal voltage equations havethe form of (1) and (2), respectively. Where the subscripts and

denote variables of the phase a winding and phase b winding,respectively; , denotes the voltages, the currents,

the magnetic flux linkages, and the stator phase-windingohmic resistance

(1)

(2)

If a SC is present in the rotor, then the stator magnetic fluxlinkages and depend on the (usually) not measurable cur-rents induced in the SC also [19]. The effects of SC can be rep-resented by two short-circuited damping windings with mag-netically orthogonal axes [27]. Their voltage equations, writtenin the stator reference frame defined with the magnetic axes ofstator windings a and b, are given by (3) and (4), respectively;where the subscripts and denote variables of the dampingwindings in the axes a and b, respectively

(3)

(4)

All the windings’ magnetic flux linkages , , , anddepend on the current excitation, the magnetic flux linkage dueto PM excitation and on the electrical rotor position . There-fore, the windings’ magnetic flux linkages can be written inthe form of (5), where represents the magnetic flux linkagedue to current excitation and the magnetic flux linkage dueto PM excitation, which is considered as current independent

(5)

The time derivatives of magnetic flux linkages in (1)–(4)therefore include terms which depend on the time derivative ofthe electrical rotor position (i.e., the electrical rotor speed). Butwhen the rotor of the machine is locked, then these terms equal

MARCIC et al.: ANALYZING THE MAGNETIC FLUX LINKAGE CHARACTERISTICS OF AC ROTATING MACHINES BY EXPERIMENTAL METHOD 2285

zero because the time derivative of the electrical rotor positionequals zero; thus providing relation (6)

(6)

Furthermore, if the frequency of the stepwise-changing voltageimposed on the stator terminals is low enough, so that the re-sponding winding current reaches steady-state value, then theeffects caused by the (not measurable) SC currents, inducedduring the transient, die out [19]. Thus, the transient behaviorof the SC currents does not affect the final values of the mag-netic flux linkage in steady-state, but it influences their time be-havior and obtained characteristics and , aspresented in the results and detailed in Section III.

Considering the given explanation, the magnetic flux linkagesand of all respective machines can be determined from

measurement of voltage and current directly at the machine ter-minals, by using (7) and (8), respectively

(7)

(8)

The descriptions of the applied method and solutions re-garding changes in the resistance are discussed in detail in[14], [19], and [23]. If only one of the phase windings is excitedand the other is left open, then the magnetic flux linkage ofthe excited winding depends only on its current excitation.But when both of the phase windings are excited, then thecorresponding magnetic flux linkages depend on both excitingcurrents because of cross-coupling saturation effects [17], [18],[24], [28].

Furthermore, if the machine incorporates PMs, then thewinding’s magnetic flux linkages additionally depend on thePM flux linkage which offsets the current excited fluxlinkage [19]. Since the previously presented method fordetermining magnetic flux linkages does not distinguish be-tween the offset due to PMs and the offset due to the ironcore remanent flux, which is considered as zero [19], [23],the PM flux linkage is to be determined by separate tests, i.e.,by measurement of the back-EMF [29]. And therefore, thePM flux linkage is also not explicitly shown in the presentedresults, but its impact on the machines’ magnetic flux linkagecharacteristics is exhibited by higher saturation as described inSection III in detail.

Similar methods had been applied previously. However, thiswork comprehensively presents current dependent magneticflux linkage characteristics of all the previously described ac

Fig. 2. Applied voltage, responding current, and magnetic flux linkage of thesquirrel-cage induction machine (SCIM), where the phase a winding was ex-cited, while the phase b winding was left open.

rotating machines made of equal materials, and where theircharacteristics are determined based on measurements directlyon actual windings, and not indirectly through model windingsin an arbitrary reference frame as in [11], [14], [18], [19], [24],[25], and [28].

C. Experimental System

The experimental setup consisted of the tested machine, i.e.,the combination of the stator with two phase-windings andone of five previously described rotors, a mechanical brakewhich was used to lock the machine’s rotor at the chosenposition, a data acquisition system, and two different voltagesources—one controlled voltage source inverter for generatingstepwise-changing voltage pulses and one controlled dc currentsource (see Fig. 1). The frequency of the stepwise-changingvoltage was set by the control system dSpace 1103 PPC,where also the data acquisition system has been set up throughutilization of the Matlab/Simulink Real Time Workshop. Theduty cycle of power electronics switching components was setto 1, in order that their switching behavior does not affect themeasurement of the applied voltage. Thus, the amplitude ofthe stepwise-changing voltage pulses was set by the voltagesource inverter dc-link voltage. The machine’s winding currentswere measured by LEM current sensors and the voltages weremeasured by Hameg differential probes.

III. RESULTS AND DISCUSSION

Fig. 2 presents the measured waveforms of voltage, current,and by the above described method determined magnetic fluxlinkage in time-dependence. If the magnetic flux linkage isplotted in dependence of current then we obtain a magneticflux linkage characteristic in form of a hysteresis loop. For


Fig. 3. Current dependent magnetic flux linkage characteristics of the squirrel-cage induction machine (SCIM), where the phase a winding was excited, whilethe phase b winding was left open.

the particular test results shown in Fig. 2, the correspondinghysteresis loop is the largest one shown in Fig. 3.

As it can be seen from Fig. 2 the applied voltage to the phasea winding was changing in the stepwise manner; while the otherphase b winding was left open. The voltage step period has beenset for the time, which was needed in order that the respondingcurrent reaches its steady-state value. Although the referencevoltage was set to be a perfect step waveform, the measuredvoltage does not exhibit perfect stepwise changing shape, whichis caused by limitations of the voltage source. The same test wasthen repeated for different values of applied voltage, thus ob-taining the full set of characteristics in form of hysteresis loopsshown in Fig. 3 for the SCIM. For the SCIM tests were per-formed by utilizing phase a winding only, since the SCIM doesnot exhibit any inherent iron core saliency. On all the other mag-netically salient machines the tests were performed by utilizingalso the phase b winding.

When a dc current is applied to the phase a winding, thenthe SCSRM and SRM rotors align their axis of lower reluc-tance with the magnetic axis of the phase a winding. Contrarily,the rotors of SCIPMSM and IPMSM align the axis of the PMflux linkage vector, i.e., direction in which the PMs are magne-tized, with the magnetic axis of the phase a winding. Note thealignment of individual magnetically salient rotors in respectto the magnetic axes of the stator windings in the lower rightcorner of Figs. 4–8. Therefore, the phase a winding magneticflux linkage characteristics of the SCSRM and SRM reach sub-stantially higher values than the ones from the phase b winding;and vice-versa for the SCIPMSM and IPMSM.

By comparing the magnetic flux linkage characteristics of theSCIM and SCSRM in Fig. 3 and Fig. 4(a) one can see that themagnetic flux linkage produced by the same excitation currentis smaller in the SCSRM. This is due to the fact that the SCSRMrotor incorporates FBs, which steer the magnetic flux in specificdirections (saliency) and also because it incorporates less iron

Fig. 4. (a) Current dependent magnetic flux linkage characteristics of thesquirrel-cage synchronous reluctance machine (SCSRM), where the phase awinding was excited, while the phase b winding was left open. (b) Current de-pendent magnetic flux linkage characteristics of the squirrel-cage synchronousreluctance machine (SCSRM), where the phase b winding was excited, whilethe phase a winding was left open.

material. Thus, the SCSRM’s iron core is locally more saturatedby the same value of current than the one from the SCIM. Fur-thermore, by comparing the magnetic flux linkage characteris-tics of the SCSRM and SCIPMSM in Fig. 4(a) and Fig. 6(b),respectively; or Fig. 4(b) and Fig. 6(a), respectively; one cansee that the magnetic flux linkage produced by the same valueof current is smaller in the SCIPMSM. This is due to the pres-ence of PMs, which additionally saturate the SCIPMSM ironcore. This impact of the PMs on the iron core saturation can bealso observed by comparing the magnetic flux linkage charac-teristics of the SRM and IPMSM in Fig. 5(a) and Fig. 7(b), re-spectively; or Fig. 5(b) and Fig. 7(a), respectively. As mentionedbefore, the PMs actually offset the current dependent magneticflux linkage characteristics but this offset is not explicitly shownin the results.


Fig. 5. (a) Current dependent magnetic flux linkage characteristics of the syn-chronous reluctance machine (SRM), where the phase a winding was excited,while the phase b winding was left open. (b) Current dependent magnetic fluxlinkage characteristics of the synchronous reluctance machine (SRM), wherethe phase b winding was excited, while the phase a winding was left open.

The presence of PMs has also an additional effect on themachines’ magnetic flux linkage characteristics. PMs aremagnetized in a specific direction; therefore, their magneticflux linkage may be either supported or opposed by an ex-ternal magnetic field. This effect is consequently exhibitedeither as an increase or a decrease of the machine’s iron coresaturation level. The magnetic flux linkage characteristics ofthe SCIPMSM shown in Fig. 8 were measured by applyingthe step-wise changing voltage in one winding and leavingthe other winding open or supplying it by a positive value ofdc current or negative value of dc current, respectively. FromFig. 8(b) one can see that the magnetic flux linkage character-istic reached higher values when the negative dc current valuehas been applied in the phase a winding; and vice-versa for thepositive dc current value. The magnetic flux linkage charac-teristic, when the phase a winding was left open, lies between

Fig. 6. (a) Current dependent magnetic flux linkage characteristics of thesquirrel-cage interior permanent magnet synchronous machine (SCIPMSM),where the phase a winding was excited, while the phase b winding was leftopen. (b) Current dependent magnetic flux linkage characteristics of thesquirrel-cage interior permanent magnet synchronous machine (SCIPMSM),where the phase b winding was excited, while the phase a winding was leftopen.

the two aforementioned ones. This is a direct consequence ofthe additional magnetic field produced by the dc current inthe phase a winding because the PM flux linkage vector wasaligned with the magnetic axis of the phase a winding. Thus, theadditional magnetic field produced by the positive dc currentin the phase a winding directly supports the PM flux linkage,which increases the iron core saturation. And the additionalmagnetic field produced by the negative dc current in thephase a winding directly opposes the PM flux linkage, whichdecreases the iron core saturation. Contrarily, this effect cannot be seen from Fig. 8(a) where the additional magnetic fieldwas produced by the dc current in the phase b winding becausethe PM flux linkage vector was aligned with the magnetic axisof the phase a winding. The additional magnetic field produced


Fig. 7. (a) Current dependent magnetic flux linkage characteristics of the in-terior permanent magnet synchronous machine (IPMSM), where the phase awinding was excited, while the phase b winding was left open. (b) Current de-pendent magnetic flux linkage characteristics of the interior permanent magnetsynchronous machine (IPMSM), where the phase b winding was excited, whilethe phase a winding was left open.

by either the positive value of dc current or the negative valueof dc current in the phase b winding only additionally saturatesthe SCIPMSM’s iron core. And similar observations havebeen made for the SCIM, SCSRM, and SRM, which do notincorporate PMs.

The effect of the presence of a SC can be seen by comparingmagnetic flux linkage characteristic of magnetically identicalmachines with and without a SC, i.e., by comparing Figs. 4 and5 or Figs. 6 and 7, respectively. If a SC is present, then a statorwinding’s magnetic flux linkage depends on the applied voltage,responding stator current and also the currents induced in theSC. These have a damping effect on the stator current wave-form and consequently on the magnetic flux linkage waveform.Thus, the voltage-step period has to be long enough so the SCcurrents die-out and the stator current reaches its steady-state

Fig. 8. (a) Current dependent magnetic flux linkage characteristics of thesquirrel-cage interior permanent magnet synchronous machine (SCIPMSM),where the phase a winding was excited by stepwise changing voltage and thephase b winding was excited by dc current values of�4 A, 0 A, and�4 A. (b)Current dependent magnetic flux linkage characteristics of the squirrel-cageinterior permanent magnet synchronous machine (SCIPMSM), where the phaseb winding was excited by stepwise changing voltage and the phase a windingwas excited by dc current values of �4 A, 0 A, and �4 A.

value. The steady-state values of magnetic flux linkages are notaffected by the SC effect; therefore the final values of magneticflux linkages at specific values of current have the same valuefor magnetically identical machines with or without a SC [19].Thus, as it can be seen from comparison of Figs. 4 and 5 orFigs. 6 and 7, the SC has an “inflating” effect on the current-de-pendent magnetic flux linkage characteristics in forms of hys-teresis loops. And the current dependent magnetic flux linkagecharacteristics of SRMs [14] and IPMSMs [24] in form of a hys-teresis loop can be therefore determined by utilizing just one testwith the maximum allowed steady-state current value; whereasseveral tests with different steady-state current values have tobe employed in determining current dependent magnetic fluxlinkage characteristics of machines with a SC.


TABLE IRESULTS SUMMARY

The alignment with dc current is a prerequisite for appropriate rotororientation before locking it.

Achieved value of magnetic flux linkage at equal current excitation, byconsidering that the PM flux linkage is not explicitly shown in the results.

Table I summarizes the identified effects of the direct andincremental comparison of the respective ac machine magneticflux linkage characteristics.

The aforementioned results show that the machines with aSC exhibit an “inflated” measured current dependent magneticflux linkage characteristic in form of a hysteresis loop. Thisis difficult to account for in dynamic models of the machines.Therefore, as far as the authors are informed, such character-istics are approximated by unique current dependent magneticflux linkage characteristics in form of a virgin magnetizationcurve [14], [23]. These can be then included in nonlinear dy-namic models, satisfying constraints imposed by lossless fields[29], [30]. The effects of hysteresis and eddy current losses arein such models normally accounted for by additional constantor variable resistances [31].

Furthermore, the machines without a SC exhibit rathernarrow current dependent magnetic flux linkage characteristicsin form of a hysteresis loop and therefore, the approximationin form of a virgin magnetization curve for these machines isa very good one.

Flux linkage characteristics determined by the describedexperimental method became indispensable in cases whenexact motor geometry or used magnetic materials for particularmachine are not known. However, approximated—unique cur-rent dependent flux linkage characteristics in a form of a virginmagnetization curve (obtained from the measured hysteresisloops) can be compared with the unique current dependentmagnetic flux linkage characteristics of different machines,obtained by the finite-element analysis as well. Fig. 9 presentsunique current dependent magnetic flux linkage characteristicsof the SCIPMSM obtained by finite-element analysis [24].These characteristics were used in [32] to obtain by the linearmodel needed stator inductance values. The agreement betweenthe experimentally obtained final values of magnetic flux link-ages at specific values of current presented in Fig. 6 and theones presented in Fig. 9 is good, thus validating the experimen-tally obtained results. The relation between the magnetic fluxlinkage characteristic in form of a virgin magnetization curveand in form of a hysteresis loop is described in [14] and [23].

Despite that the presence of a SC aggravates the determina-tion of current dependent magnetic flux linkage characteristics;

Fig. 9. (a) Unique current dependent magnetic flux linkage characteristicsof the phase a winding for the squirrel-cage interior permanent-magnetsynchronous machine (SCIPMSM) obtained by finite-element analysis. (b)Unique current dependent magnetic flux linkage characteristics of the phase bwinding for the squirrel-cage interior permanent magnet synchronous machine(SCIPMSM) obtained by finite-element analysis.

there are several applications where the employment of a SC isfavorable. The SC represents a damping winding [19], [32] andas such it damps small load disturbances in synchronous op-eration of a SCSRM or SCIPMSM. More importantly, the SCprovides the so called line-starting capability [32], and thus aSCIM, SCSRM, or SCIPMSM can be started from standstill bybeing fed directly from a constant frequency and constant ampli-tude voltage supply. Contrarily, the usage of a SRM or IPMSMdemands accurate information of its rotor position (via sensors,observers, or estimators) and the employment of power elec-tronic converters in open- or closed-loop controlled drives. TheSCSRM or SCIPMSM can be employed in such drives as well.The SCIM is generally applicable; however its rather low effi-ciency [8], [32] makes it rather inferior to the other topologies.


The SCIPMSM and IPMSM topologies are advantageous in theflux-weakening operating regime [4], [7].

IV. CONCLUSION

A unified analysis of current dependent magnetic flux linkagecharacteristics in forms of hysteresis loops of contemporary andmost widely used ac rotating machines was presented. The im-pacts of the presence of FBs, PMs, and SC in the rotor design onthe machines’ current dependent magnetic flux linkage charac-teristics have been pointed out. The analysis was based on exper-imental data by utilizing a two-phase stator winding which en-abled the direct determination of magnetic flux linkages by mea-surement of the windings’ voltage and current. The prototyperotors were made of equal materials, thus enabling the incre-mental comparative analysis, provisioning a deeper insight intothe magnetically nonlinear behavior of ac rotating machines,which is not available in the current literature.

However, in this study no attempt was made to employ the ex-perimentally obtained characteristics in practical applications.This is part of future work.

ACKNOWLEDGMENT

This work was supported in part by the Slovenian ResearchAgency, Projects No. L2-1180 and L2-2060.

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Tine Marcic (M’04) was born in Maribor, Slovenia, in 1978. He receivedthe B.Sc. and Ph.D. degrees in electrical engineering from the University ofMaribor, Maribor, Slovenia, in 2002 and 2008, respectively.

Since 2003, he has been with TECES, Research and Development Centrefor Electric Machines, where he currently is a Researcher at the Department of


Electrical Machines. His current research interests include the electromechan-ical design, modeling, analysis and testing of electric rotating machines.

Dr. Marcic had been awarded the Prof. Dr. Vratislav Bedjanic Award for hisdoctoral thesis and the Best Student Paper Award at the 13th IEEE CEFC, bothin 2008.

Gorazd Stumberger (M’92) was born in Ptuj, Slovenia, in 1964. He receivedthe B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Universityof Maribor, Maribor, Slovenia, in 1989, 1992, and 1996, respectively.

In 1989, he joined the Faculty of Electrical Engineering and Computer Sci-ence, University of Maribor, where he currently is a Professor of Electrical Engi-neering. He was a Visiting Researcher with University of Wisconsin, Madison,in 1997 and 2001, and with Katholieke Universiteit Leuven, Belgium, in 1998and 1999. His current research interests include power systems research and thedesign, modeling, analysis, and control of electrical machines and drives.

Dr. Stumberger is a member of the International Compumag Society andSlovenian Committee CIGRE.

Bojan Stumberger (M’99) received the B.Sc., M.Sc., and Ph.D. degrees in elec-trical engineering from the University of Maribor, Maribor, Slovenia, in 1993,1997, and 1999, respectively.

Since 1993, he has been with the Faculty of Electrical Engineering and Com-puter Science, University of Maribor, where, since 2001, he has been an As-sistant Professor of electrical engineering. His current research interests are theelectromechanical design and modeling of electrical machines, CAD of electro-magnetic devices, and magnetic material modeling.

Dr. Stumberger is a member of the International Compumag Society and Eu-ropean Physical Society.

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