Rotor Flux-Barrier Design for Torque Ripple Reduction ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 45, NO. 3, MAY/JUNE 2009

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Rotor Flux-Barrier Design for Torque Ripple Reduction in Synchronous Reluctance and PM-Assisted Synchronous Reluctance Motors Nicola Bianchi, Member, IEEE, Silverio Bolognani, Member, IEEE, Diego Bon, and Michele Dai Pré, Student Member, IEEE

Abstract—The torque produced by a synchronous reluctance machine (including the permanent-magnet-assisted machine) is studied analytically, with the aim of pointing out the effect of the position of the flux barriers on the torque ripple. It is verified that the position of the flux-barrier ends highly influences the torque waveform. With the aim of reducing torque harmonic contents, a new strategy is proposed based on the choice of couples of flux barriers of different shapes. The flux-barrier geometry is chosen so as to obtain a compensation between the torque harmonics produced by each couple. Experimental results on two prototypes confirm the analytical prediction. Index Terms—Low torque ripple, permanent-magnet (PM)-assisted synchronous reluctance machine, synchronous reluctance machine.

Fig. 1. Sketch of (a) a synchronous reluctance motor with three flux barriers per pole and (b) a synchronous PM-assisted reluctance motor with two flux barriers per pole.

I. I NTRODUCTION

T

HE SYNCHRONOUS reluctance machine with transversally laminated rotor, shown in Fig. 1(a), is a good competitor in applications requiring high dynamic, high torque density, and fault-tolerant capability. A permanent magnet (PM) can be inserted in each rotor flux barrier with the aim of saturating the iron bridges and increasing the power factor [1], which is generally low in this machine. The resulting configuration is called PM-assisted synchronous reluctance machine. Fig. 1(b) shows a sketch of such a machine with two flux barriers per pole. The added PM is minimum to maintain the intrinsic fault-tolerant capability of the reluctance machine. As a consequence, the back EMF and the shortcircuit current are low [2], as well as the corresponding braking torque [3]. A common drawback of synchronous reluctance machines is their high torque ripple [4]. The interaction between the spatial

Paper IPCSD-08-084, presented at the 2006 Industry Applications Society Annual Meeting, Tampa, FL, October 8–12, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review November 1, 2006 and released for publication November 7, 2008. Current version published May 20, 2009. This paper was supported in part by the Electric Drive Laboratory, Department of Electrical Engineering, University of Padova, and in part by the Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) under Research Project PRIN 2003. The authors are with the Department of Electrical Engineering, University of Padova, 35131 Padova, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2009.2018960

harmonics of electrical loading and the rotor anisotropy causes a torque ripple that is intolerable in most of applications. In [5], it has been shown that the rotor skewing (commonly adopted in PM machines [6], [7]) is not enough to smooth the torque. In any case, only step skewing is possible when PMs are used: the rotor is split into two or more parts, each of them is skewed with respect to the others. In [5], it has been shown that a reduction of the torque ripple can be achieved by means of a suitable choice of the number of flux barriers with respect to the number of stator slots. In this case, the flux-barrier ends are uniformly distributed along the air gap (similarly to the stator slot distribution). Alternatively, in [8] and then in [9], the flux barriers are shifted from their symmetrical position. In this way, a sort of compensation of the torque harmonics is achieved. This technique is similar to that proposed in [10] for cogging torque reduction in surface-mounted PM motors. This paper presents a novelty strategy to compensate the torque harmonics of the synchronous reluctance motor. It is based on the following two-step design procedure. 1) At first, a set of flux-barrier geometries is identified so as to cancel a torque harmonic of given order. 2) Then, couples of flux barriers belonging to this set are combined together so as the remaining torque harmonics of one flux-barrier geometry compensate those of the other geometry. The term “to cancel” is adopted when a harmonic of given order is zero according to the geometry configuration of the rotor flux barriers. Conversely, the term “to compensate” is

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adopted when any technique (as step-skewing or pole shifting) is used to decrement the effect of one or more harmonics. In this case, the torque harmonics of given order are not zero in each configuration, but their sum is zero. This second step can be achieved in the following two ways: either 1) by forming the rotor with laminations of two different kinds or 2) by adopting two different flux-barrier geometries in the same lamination. The first is called “Romeo and Juliet” (R and J) configuration, since the rotor is formed by two different and inseparable kinds of lamination. The second is called “Machaon” configuration, which is the name of a butterfly with two large and two small wings, since the resulting flux barriers will be large and small alternatively under the adjacent poles. An accurate analytical model of the torque production of the motor is first presented. After a validation by means of a finite element (FE) analysis, the model is adopted to determine the more profitable flux-barrier geometry. Promising results have been reached analytically and by simulations. Experimental tests on two rotor prototypes are presented in the following, confirming the predictions.

Fig. 2. Reference frames of a synchronous reluctance motor. If the PM is inset to assist the motor, it is placed with magnetization axis along the negative q-axis.

II. I NVESTIGATION OF THE T ORQUE H ARMONICS For a given winding arrangement, the electrical loading (defined as the current linear density along the stator inner diameter corresponding to the stator currents) can be expressed as the Fourier series expansion, as ˆ ν sin (νpϑs − pϑm − αe ) K Ks (ϑs ) = (1) i ν

ˆ ν is the peak of electric where ν is the harmonic order, K loading harmonic of νth order, p is the number of pole pairs, ϑs is the coordinate angle in stator reference frame (mechanical degrees), ϑm is the rotor position (mechanical degrees), and αie is the angle of current vector (electrical degrees) [11]. Superscript e is used to highlight when an angle is expressed in electrical degrees, so as pϑm = ϑem . It is worth noticing that, during steady-state operations, the rotor position is linked to the rotor speed by the relationship pϑm = ωt, where it is assumed that ϑm = 0 at t = 0. The symbol ν can be positive or negative (it has to be considered with sign). Adopting a standard three-phase winding, with an integer number of slot per pole per phase, only harmonics of odd order, not multiple of three, exist. Thus, ν can be expressed as ν = 6k + 1 with k = 0, ±1, ±2, . . . [12]. Only the main harmonic (ν = p) is synchronous with the rotor, while the other harmonics move asynchronously. The electrical loading in (1) is expressed in the stator reference frame. Since pϑs = p(ϑr + ϑm ), as shown in Fig. 2, the electrical loading can be expressed in the rotor reference frame as ˆ ν sin [νpϑr + (ν − 1)pϑm − αie ] . (2) K Ks (ϑr ) = ν

The stator magnetic potential Us (ϑr ) along the stator bore periphery (see Fig. 3), in the rotor reference frame, can be

Fig. 3.

Main magnetic quantities in stator and rotor.

computed as Us (ϑr ) =

Ks (ϑr )

D dϑr 2

(3)

that results in Us (ϑr ) =

ν

−

ˆν D K cos [νpϑr + (ν − 1)pϑm − αie ] . ν 2p

(4)

Because of the small air gap, let us suppose that the airgap flux density has only radial component. In the adopted coordinate reference system, positive direction is from rotor to stator. Thus, it is Bg (ϑr ) = μ0

−Us (ϑr ) + Ur (ϑr ) g

(5)

where g is the air-gap thickness, and Ur (ϑr ) is the magnetic potential of the rotor, as shown in Fig. 3. This is considered to be constant in each rotor magnetic “island” and zero elsewhere. Rotor island is that region of rotor iron magnetically insulated and bordered by flux barriers and air gap (iron bridges are not considered since they are saturated). The difference of magnetic potential between the two edges of each flux barrier

BIANCHI et al.: ROTOR FLUX-BARRIER DESIGN FOR TORQUE RIPPLE REDUCTION IN MOTORS

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is computed as the product of the magnetic flux φb crossing the flux barrier π 2p +ϑb

φb =

Bg (ϑr )Lstk π 2p −ϑb

D dϑr 2

(6)

by the magnetic reluctance of the flux barrier itself Rb =

tb μ0 Lstk lb

(7)

where D and Lstk are the bore diameter and the stack length, respectively, and tb and lb are the thickness and the length of the flux barrier, respectively (see Fig. 3). Therefore, the resulting Ur (ϑr ) is a function of the electrical loading and the fluxbarrier geometry. When the rotor is formed by several flux barriers per pole, the computation becomes recursive. For each jth flux barrier, the magnetic flux φbj and the magnetic potential Urj are computed as described earlier, according to the geometry (i.e., tb , lb , and ϑb ) of each flux barrier. The torque is obtained by integrating the Lorentz’s force density −Bg (ϑr )Ks (ϑr ) along the air-gap surface and multiplying the result by the radius D/2, yielding τm

D =− 2

2π Bg (ϑr )Ks (ϑr )

DLstk dϑr . 2

(8)

0

Since Us (ϑr ) is due to Ks (ϑr ), as expressed in (3), the interaction between these two quantities does not yield any overall torque. Thus, after some manipulations, the torque can be rewritten as τm

μ0 D2 Lstk = g 4

2π Ur (ϑr )Ks (ϑr ) dϑr .

(9)

Since Ur (ϑr ) is a function of Ks (ϑr ), as discussed above, it can be inferred that the motor torque exhibits an average term, due to the fundamental harmonic of the electrical loading, together with oscillating terms, due to the interaction between the harmonics of the electrical loading of different orders. This will be clarified in the next section. III. R OTOR W ITH T WO F LUX B ARRIERS PER P OLE Let us consider the case of a rotor with two flux barriers per pole, spanning angles 2ϑb1 and 2ϑb2 , as shown in Fig. 2. Letting νπ + (ν − 1)pϑm − αie (10) λν = 2 the values of Ur (ϑr ) of the two rotor magnetic “islands” become Ur1

Ur2 = − D

K ˆν ρν2 cos(λν ) (νp)2 ν

where ρν1 and ρν2 are dimensionless parameters depending on rotor geometry only, i.e., on the following: 1) the air-gap thickness; 2) the lengths tb and lb of both flux barriers; 3) the angles ϑb1 and ϑb2 of the flux-barrier ends. The motor torque expressed in (9) results in K ˆν cos λν τm = kτ ν2 ν ⎡ K ˆξ sin λξ sin(ξpϑb1 ) × ⎣ρν1 ξ ξ ⎤ K ˆξ sin λξ [sin(ξpϑb2 ) − sin(ξpϑb1 )] ⎦ + ρν2 × ξ ξ

0

K ˆν = −D ρν1 cos(λν ) (νp)2 ν

Fig. 4. Torque behavior of a four-pole reluctance motor with assigned fluxbarrier angles ϑb1 and ϑb2 : comparison between analytical and FE results.

(11) (12)

(13) where λν and λξ are given by (10) and kτ = μ0

D3 Lstk 1 . g p2

(14)

Observing from (10) that λν and λξ depend on the harmonic order (ν and ξ, respectively) and on the rotor position ϑm , it can be verified that the torque (13) is given by the sum of an average component (according to ν = ξ = 1) and other oscillating components (for any combination of ν and ξ different from the previous case). These latter components can be reduced by means of a suitable design of the flux-barrier ends (i.e., of the angles ϑb1 and ϑb2 ). A. FE Comparison With the aim of validating the analytical computation, a FE analysis is carried out, referring to the same geometry. The curve of the torque versus rotor position predicted analytically is compared with the same curve computed by means of FE method [13]. Two comparisons between the analytical and FE results are reported in Figs. 4 and 5.

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Fig. 6.

Design map of torque harmonic of 6th order.

Fig. 7.


Fig. 8.


Fig. 5. Torque behavior of a four-pole reluctance motor with assigned fluxbarrier angles ϑb1 and ϑb2 : comparison between analytical and FE results.

At first, the good agreement between analytical and FE results should be noted. This confirms the accuracy of the analytical model. On the basis of this result, the analytical method will be used in the following to predict the torque harmonic contents and to select the more suited flux-barrier geometries. B. Simulation of Torque Ripple Reduction Some considerations on the torque ripple could be done. In Fig. 4, the torque harmonic of 12th order is evident. Fig. 5 shows that an appropriate choice of the angles θb1 and θb2 allows one to cancel this torque harmonic of 12th order. Among the remaining torque harmonics, the higher harmonics are of 6th and 36th orders. IV. C ONTOUR M APS OF THE T ORQUE H ARMONICS From the analysis of the torque harmonics presented earlier, it is possible to draw the amplitude of the various torque harmonics according to the angles ϑb1 and ϑb2 of the fluxbarrier ends. Referring to a four-pole 24-slot stator with one-slot chorded winding, some maps are shown in Figs. 6–8. They display the maximum amplitude of the torque harmonics of orders 6, 12, and 24, respectively, in the (ϑb1 , ϑb2 ) plane, in mechanical degrees. The top left side of the maps is not to be taken into account, since the values of ϑb2 < ϑb1 + 6◦ have not been considered, being one flux barrier too close to the other. The contour maps offer valid help to the rotor design. They allow one to identify those combinations of flux-barrier angles suited to cancel (or to minimize) some torque harmonics. For instance, Fig. 6 shows that the torque harmonic of 6th order disappears when ϑb1 ≈ 17◦ and ϑb2 ≈ 36◦ are chosen. Similarly, Figs. 7 and 8 show which combinations of ϑb1 and ϑb2 allow the torque harmonics of 12th and 24th orders to be cancelled. The number of combinations increases when the harmonic order increases, as can be inferred from comparing Figs. 6–8.

Although the maps presented earlier are valid help to the design the rotor geometry, it is not possible to cancel torque harmonic of more than one order. Then, a further strategy has to be adopted.


Fig. 9.

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R- and J-type laminations of the “R and J” rotor.

V. R OTOR W ITH D IFFERENT L AMINATIONS : T HE “R AND J” R OTOR The first proposed strategy consists in assembling the rotor using different laminations. Two different laminations are enough for an adequate reduction of torque ripple. The geometries of the flux barriers of both laminations are chosen with the aim of canceling the torque harmonic of one order and to compensate those of other orders. In this way, a smooth total torque is obtained. This rotor is called “R and J” rotor. It is formed by two parts, where one adopts lamination of the first kind and the second part adopts lamination of the other kind. Of course, the two parts are inseparable. According with the adopted stator winding, the flux-barrier angles are as follows. 1) R-type module. ϑb1 = 14.8◦ , and ϑb2 = 27.4◦ . 2) J-type module. ϑb1 = 22.2◦ , and ϑb2 = 40.2◦ . The two rotor laminations exhibit the torque harmonics of 12th order out of phase of almost 180◦ . The simulated torque ripple at nominal load has been computed to be equal to 5.45% (peak-to-peak torque ripple). The selected laminations are shown in Fig. 9. They will be adopted for the first rotor prototype (see Section VII). VI. R OTOR W ITH D IFFERENT F LUX B ARRIERS PER P OLE : T HE “M ACHAON ” R OTOR An alternative strategy lies in designing a single lamination in which the flux barriers present a different geometry under various poles. The aim is, again, to cancel the torque harmonic of one order and to compensate those of other orders. Let us refer to a four-pole motor. Among the various combinations, a couple of poles is designed with small (or R-type) flux barriers, while the other couple of poles is designed with large (or J-type) flux barriers. This configuration can be looked upon as an evolution of the “R and J” configuration, as shown in Fig. 10: The new rotor lamination adopts the optimized flux barriers of the two-part rotor. A couple of poles is designed with R-type flux barriers, while the other couple of poles is designed with J-type flux barriers. In order to avoid an unbalanced rotor, the poles of equal shapes are symmetrical to the shaft axis. This latter configuration overcomes the drawback of the “R and J” configuration whose realization requires two different rotor laminations. Although the use of two laminations yields

Fig. 10. Evolution from “R and J” rotor to “Machaon” rotor.

a higher number of degrees of freedom in the torque ripple reduction, the additional manufacturing costs could represent an impediment to its realization. Since the final rotor presents two poles with large flux barriers and two poles with short flux barriers, this second solution is called “Machaon” configuration, which is the name of a butterfly with two large wings and two small wings. The final lamination is shown in the lower part of Fig. 10. Whereas the flux-barrier angles have been not modified, the thickness of the flux barriers has been slightly optimized in order to improve the iron paths. The simulated peak-to-peak torque ripple at nominal load has been computed to be lower than 5%. This lamination shown in Fig. 10 has been adopted for the second rotor prototype (see Section VII). VII. E XPERIMENTAL R ESULTS Three motor prototypes have been manufactured. The same three-phase stator has been used for the three motors. It is a standard four-pole 24-slot induction motor stator, with a double-layer one-slot chorded winding. A. “R and J” Rotor The “R and J” rotor is formed by the laminations of two different types, each of them has two flux barriers per pole. The rotor laminations have been chemically eroded. They are shown in Fig. 11. In the center of the flux barrier, small PMs (the assisting PMs) have been added so as to saturate the iron bridge and to increase the power factor. Then, both laminations show a hole of the same size to hold the PM. The PM flux is quite low, to limit the short-circuit current and braking torque in case of fault. The rotor has been step skewed. At first, two stacks formed by both laminations are packed. Then, one stack is skewed with respect to the other.

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Fig. 11. Photos of the “R and J” laminations. (a) R-type. (b) J-type. Fig. 14. rotor.

Insertion of the assisting PMs in the flux barriers of the “Machaon”

Fig. 14 shows a step of the rotor assembling: The PM is going to be inserted into the rotor flux barrier. Finally, the rotor has been split into two parts that have been step skewed. C. IPM Rotor

Fig. 12. Photos of the “Machaon” lamination. (a) Actual waveform. (b) Average and fundamental waveforms.

In order to compare the performance of the “R and J” and “Machaon” motors with the classical interior PM (IPM) motor, a further rotor has been designed and built. For the sake of comparison, the IPM rotor has two flux barriers per pole as well. It is labeled IPM2B and is shown in Fig. 1(b). The fluxbarrier ends are uniformly distributed along the air gap, i.e., they are the same under each pole. Their position was obtained by means of an optimization procedure so as to minimize the torque ripple. In addition, a two-step skewing has been adopted to cancel a higher order torque harmonic. D. Motor Tests

Fig. 13. Air-gap flux density distribution.

B. “Machaon” Rotor The “Machaon” rotor is formed by laminations that are chemically eroded, shown in Fig. 12. Small PMs have been again added in the center of the flux barrier. The PM width and thickness are the same of those used in the “R and J” rotor. Fig. 13 shows the air-gap flux density distribution and the average value, together with the fundamental waveform.

Fig. 15 shows the measured torque behaviors of the IPM2B, “R and J,” and “Machaon” motors, referred to the nominal current Iˆ = 2.64 A (peak value). The same scale of 0.1 N · m/div is used, so that one can easily observe the effective reduction of the torque ripple achieved by the proposed solutions. The torque ripple of the “R and J” and “Machaon” motors is about one third of the torque ripple of the classical IPM2B motor. Fig. 16 shows the measured torque behaviors referred to a current of Iˆ = 5.30 A (peak value). As mentioned earlier, both innovative motor configurations exhibit a lower torque ripple. In this case, the torque ripple becomes almost a half of that of the IPM2B motor. Table I reports a numerical comparison between average torque (Tavg ) and torque ripple (ΔT ), at various currents. It is worth noticing that the proposed solutions yield always an appreciable reduction of the torque ripple. A reduction of the average torque of about 8% is not negligible in the “R and J” motor. On the contrary, the average torque appears slightly higher in the “Machaon” motor.


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TABLE I TORQUE COMPARISON AT DIFFERENT CURRENTS AMONG IPM2B, R AND J, AND “MACHAON” MOTORS

TABLE II TORQUE HARMONIC COMPARISON

E. Torque Harmonics Fig. 15. Measured torque comparison (current amplitude Iˆ = 2.64 A). (a) IPM2B motor. (b) “R and J” motor. (c) “Machaon” motor.

Table II shows the torque harmonics computed from the torque behaviors of Fig. 15 (Iˆ = 2.64 A). In the same table, the measured torque harmonics are compared with the torque harmonics predicted by means of simulations. The good agreement between the measures and the predictions highlights again the effectiveness of the models used in the study. From Table II, it is interesting to observe that higher torque harmonics are achieved, corresponding to the slot harmonics (i.e., of 12th and 24th orders). In addition, Table II shows that the main reduction of the torque ripple is achieved in the torque harmonic of 12th order: It decreases from 5.36% in IPM2B motor to 1.35% in “R and J” motor and to 1.61% in “Machaon” motor. Other significative reductions are of the torque harmonics of 24th and 36th orders. VIII. C ONCLUSION

Fig. 16. Measured torque comparison (current amplitude Iˆ = 5.30 A). (a) IPM2B motor. (b) “R and J” motor. (c) “Machaon” motor.

The reason of the average torque decrease can be imputed to the higher saturation of the central rotor path of the J-type lamination (see Fig. 9) and to the leakage flux among the rotor islands of two different laminations.

An analytical model has been developed to study the dependence of the torque harmonics on the rotor geometry. It is the key tool in selecting the combinations of flux-barrier shapes, yielding a significative torque harmonic reduction. An effective torque ripple reduction is achieved by designing flux barriers of different geometries, both adopting different laminations forming the same rotor and a single lamination with flux barriers of different geometries. The results can be applied to synchronous reluctance machines, as well as to PM-assisted reluctance machines (as realized by the authors). Two motor prototypes have been manufactured and tested. The experimental results come up to the theoretical expectations: adopting the proposed solutions yields a sensible reduction of the torque ripple. At nominal current, the torque ripple becomes about one-third of that exhibited by the classical rotor design.

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ACKNOWLEDGMENT The authors would like to thank Dr. G. Terruzzi of Saimag S.p.a., Pogliano Milanese, Italy, for supplying the PMs; Eng. G. Perletti of Volonterio and C. S.p.A., Milan, Italy, for supplying the stator laminations; M. Trova and M. Bellomi of Magnetic S.p.a., Montebello Vicentino, Italy, for assembling the motor prototype; and M. Castiello for the help during the motor tests.

Nicola Bianchi (M’98) received the Laurea and Ph.D. degrees in electrical engineering from the University of Padova, Padova, Italy, in 1991 and 1995, respectively. Since 1998, he has been with the Department of Electrical Engineering, University of Padova, as a Senior Research Assistant in the Electric Drives Laboratory. His interest is in the field of the electromechanical design of brushless, synchronous, and induction motors with particular interest in drive applications. He is the author or coauthor of several technical papers and two books.

R EFERENCES [1] A. Fratta, A. Vagati, and F. Villata, “Permanent magnet assisted synchronous reluctance drive for constant-power application: Drive power limit,” in Proc. Intell. Motion Eur. Conf. (PCIM), Nürnberg, Germany, Apr. 1992, pp. 196–203. [2] N. Bianchi, M. Dai Pré, and S. Bolognani, “Design of a fault-tolerant IPM motor for electric power steering,” IEEE Trans. Veh. Technol., vol. 55, no. 4, pp. 1102–1111, Jul. 2006. [3] B. A. Welchko, T. M. Jahns, W. L. Soong, and J. M. Nagashima, “IPM synchronous machine drive response to symmetrical and asymmetrical short circuit faults,” IEEE Trans. Energy Convers., vol. 18, no. 2, pp. 291– 298, Jun. 2003. [4] A. Fratta, G. P. Troglia, A. Vagati, and F. Villata, “Evaluation of torque ripple in high performance synchronous reluctance machines,” in Conf. Rec. IEEE IAS Annu. Meeting, Toronto, ON, Canada, Oct. 1993, vol. I, pp. 163–170. [5] A. Vagati, M. Pastorelli, G. Franceschini, and S. C. Petrache, “Design of low-torque-ripple synchronous reluctance motors,” IEEE Trans. Ind. Appl., vol. 34, no. 4, pp. 758–765, Jul./Aug. 1998. [6] T. M. Jahns and W. L. Soong, “Pulsating torque minimization techniques for permanent magnet ac motor drives—A review,” IEEE Trans. Ind. Electron., vol. 43, no. 2, pp. 321–330, Apr. 1996. [7] S. Y. Jung and H. K. Jung, “Reduction of force ripples in permanent magnet linear synchronous motor,” in Proc. ICEM, Brugges, Belgium, Aug. 2002. CD-ROM. [8] N. Bianchi and S. Bolognani, “Reducing torque ripple in PM synchronous motors by pole-shifting,” in Proc. ICEM, Helsinki, Finland, Aug. 2000, pp. 1222–1226. [9] M. Sanada, K. Hiramoto, S. Morimoto, and Y. Takeda, “Torque ripple improvement for synchronous reluctance motor using asymmetric flux barrier arrangement,” in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, Oct. 12–16, 2003, pp. 250–255. [10] T. Li and G. Slemon, “Reduction of cogging torque in permanent magnet motors,” IEEE Trans. Magn., vol. 24, no. 6, pp. 2901–2903, Nov. 1988. [11] N. Bianchi, “Analysis of the IPM motor—Part I. Analytical approach in design, analysis, and control of interior PM synchronous machines,” in Conf. Rec. IEEE IAS Annu. Meeting, N. Bianchi and T. M. Jahns, Eds. Seattle, WA, Oct. 3, 2005, pp. 3.1–3.33. IEEE IAS Tutorial Course Notes, CLEUP ([email protected]). [12] M. Liwschitz-Garik and C. C. Whipple, Electric Machinery, vol. II. New York: Van Nostrand, 1960. [13] N. Bianchi, Electrical Machine Analysis Using Finite Elements, ser. Power Electronics and Applications Series. Boca Raton, FL: CRC Press, 2005.

Silverio Bolognani (M’98) received the Laurea degree in electrical engineering from the University of Padova, Padova, Italy, in 1976. In 1976, he joined the Department of Electrical Engineering, University of Padova, where he is currently a Full Professor of electrical drives and is engaged in research on advanced control techniques for motor drives and on the design of ac electrical motors for variable-speed applications. He is the author of more than 100 papers on electrical machines and drives. Prof. Bolognani is the President of the IEEE Industry Applications Society–Industrial Electronics Society–Power Electronics Society North Italy Joint Chapter.

Diego Bon was born in Vicenza, Italy, in 1977. He received the Laurea degree in electrical engineering from the Department of Electrical Engineering, University of Padova, Padova, Italy, in 2002, where he is currently working toward the Ph.D. degree. He is involved in the design of low-cost electric drives for the rural community.

Michele Dai Pré (S’06) was born in Verona, Italy, in 1979. He received the Laurea degree in electrical engineering from the University of Padova, Padova, Italy, in 2004, where he is currently working toward the Ph.D. degree in the Department of Electrical Engineering and is involved in the design of innovative electrical motors for automotive and aerospace applications, with special attention to fault-tolerant configurations. In 2004, he received a bursary scheme at the Laboratory of Electric Drives, Department of Electrical Engineering, University of Padova.