IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 10, OCTOBER 2018 8104708

Evaluation of the Torque in High-Speed PMSMs Witha Shielding Cylinder and BLDC Control

Bert Hannon , Peter Sergeant , and Luc Dupré

Electrical Energy Laboratory, Department of Electrical Energy, Metals, Mechanical Constructionsand Systems, Ghent University, 9000 Ghent, Belgium

EEDT, Flanders Make, the strategic research centre for the manufacturing industry, Belgium

High-speed electric machines with permanent magnets are sometimes equipped with a shielding cylinder; a conductive sleeve that iswrapped around the magnets in order to reduce the rotor losses and/or retain the magnets. Despite the every increasing popularityof such high-speed permanent-magnet synchronous machines (PMSMs), there are still a lot of question regarding the effect ofthe shielding cylinder. Earlier work in literature evaluates how the shielding cylinder’s design affects the machine’s performance.However, that work assumes a sinusoidal current, hence neglects the stator current’s higher time-harmonic orders. Therefore, thispaper aims at evaluating how different stator sources affect the way in which the shielding cylinder influences the torque productionin high-speed PMSMs. The study is performed using a 2-D analytical subdomain model that accounts for slotting and the eddycurrent reaction field. Both current and voltage sources are considered. The torque is divided in two components, the classicaltorque due to interaction between the magnets and the stator currents and the torque due to interaction with the eddy currents inthe shielding cylinder. This approach results in a better insight in the machine’s physics.

Index Terms— Fourier-based model, permanent-magnet synchronous machine (PMSM), shielding cylinder (SC).

I. INTRODUCTION

TO IMPROVE efficiency and compactness ofmachines, high-speed permanent-magnet synchronous

machines (PMSMs) have gained a lot of interest. Despitemany publications on high-speed electric machines [1]–[8],there are still a lot of questions considering their design.

One aspect of high-speed PMSMs that require particularattention is the shielding cylinder (SC) [8]–[10]: a conductivecylinder around the magnets. Its goal is to reduce the overallrotor losses, thereby limiting the risk of overheating themagnets. Indeed, asynchronous harmonics in the field willinduce eddy currents in the SC. These eddy currents counteracttheir origin, thereby shielding the magnets. By shifting theeddy currents from the poorly conductive magnets to thehighly conductive SC, the rotor’s eddy current losses can bereduced as shown in [11]. However, the SC has to be carefullydesigned because a bad design may increase the eddy currentlosses instead of decreasing them.

A. Goal and Method

Earlier, the authors presented a paper that discusses torqueproduction in high-speed PMSMs with an SC [12]. It wasshown that there are two different torque-producing mecha-nisms. A first component of the torque is due to the classicalinteraction between the magnets and the stator currents. It isreferred to as the torque produced in the magnets (T (PM)).This component of the torque is dominated by the synchronousharmonic combinations.

Manuscript received March 19, 2018; revised June 18, 2018; acceptedAugust 3, 2018. Date of publication August 27, 2018; date of currentversion September 18, 2018. Corresponding author: P. Sergeant (e-mail:peter.sergeant@ugent.be).

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

Digital Object Identifier 10.1109/TMAG.2018.2864179

A second component of the torque is the result of interactionbetween the magnetic field and the eddy currents in the SC.It is referred to as the torque produced in the SC (T (SC)).This component of the torque is dominated by asynchronousharmonic combinations. Synchronous combinations do notinduce eddy currents in the SC. Note that T (SC) can intuitivelybe explained by considering the SC as the squirrel cage ofan induction machine. The slip with respect to synchronouscombinations is then zero; only asynchronous harmonic com-binations can produce torque.

This division of the torque in its components was usedin [11] to study the effect of the SC on the torque productionin PMSMs with an SC. Although this paper provided a numberof interesting conclusions, its premise that the stator currentis sinusoidal, is not realistic. Therefore, this paper aims atcomplementing [11] by considering how the chosen controlstrategy, i.e., the stator source, affects the impact of the SC onthe torque production.

To ensure a fair and easy comparison with [11], the samemethodology will be used. This means that the effect ofthe SC’s conductivity (σSC) and its thickness (tSC) will beconsidered and that all of the studies will be performed usingthe Fourier-based models of [13] and [14] and the torquecalculations presented in [12].

II. GEOMETRY

The proposed model is valid for PMSMs with surfacemagnets and SC. The model can deal with both the innerand outer rotors, any number of phases, poles, and slots. Thestudy of the torque is illustrated for an example machine withinner rotor, two pole pairs, and 12 slots. A cross section of thestudied machine is shown in Fig. 1, the two-pole-pair machineis equipped with a single layer distributed winding that hasone slot per pole and per phase. All of the slots are indicated

0018-9464 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

8104708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 10, OCTOBER 2018

Fig. 1. Geometry and subdomains of the studied machine.

TABLE I

PARAMETERS OF THE STUDIED MACHINE

with an index i . In the stator coordinate system, their angularposition is δi

δi = − δ

2+ 2π

Nsi with 0 ≤ i ≤ Ns − 1 (1)

where Ns is the number of slots. An overview of the machine’sparameters, and their values is listed in Table I.

Two different source types are applied to the machine; anidealized brushless dc (BLDC) current and a (more realistic)BLDC voltage. Fig. 2 shows the idealized BLDC current,which has an amplitude of 66.56 A and a rise angle of 10◦.Fig. 3 shows the BLDC voltage, i.e., voltage blocks of 120◦electrical. The peak value (52.65 V) and time shift of theseblocks are chosen so that the resulting current density, whichis also shown in Fig. 3, has a small phase shift compared tothe electromotive force, resulting in a non-symmetrical currentwaveform.

Fig. 2. Idealized BLDC current.

Fig. 3. BLDC voltage and resulting current. (a) BLDC voltage. (b) BLDCcurrent.

III. TORQUE CALCULATION

As already mentioned in Section I, an extensive discussionon how the torque is calculated can be found in [12]. Thissection will briefly introduce those computations and theFourier-based model on which they are based. As shown inthe following, the total torque (T ), the torque in the magnets(T (PM)), and the torque in the SC (T (SC)) all consist of a partthat is time independent, i.e., the net torque, and a part that istime dependent, i.e., the torque ripple. Both the parts will bestudied in this paper.

A. Assumptions

In order to analytically calculate the machine’s magneticfield, the following assumptions are adopted:

1) infinite permeability of the rotor and stator iron;2) relative permeability μr = 1 for SC, PM, and stator

slots;3) no eddy currents in the slots, the magnets or the rotor,

and stator iron;4) radial slot boundaries;5) no end effects.

HANNON et al.: EVALUATION OF THE TORQUE IN HIGH-SPEED PMSMs 8104708

B. Fourier-Based Model

In order to calculate the machine’s torque, first its magneticfield has to be known. This magnetic field is calculatedusing the Fourier-based model that was presented in [13] andextended to account for voltage sources in [14]. A validation ofthis model by the finite-element method (FEM) is done in [11].Such a Fourier-based model calculates the magnetic vectorpotential (A) by dividing the machine in a number of regions,called subdomains. The subdomains, shown in Fig. 1, arereferred to an index ν; ν = 1 indicates the magnet area, ν = 2indicates the SC, and ν = 3 indicates the air gap. Finally, everyslot is a separate subdomain, indicated by ν = 4i , where i isthe slot number.

The result is an analytical expression for the magnetic vectorpotential

A(ν)(r, ϕ, t) =∞∑

n=−∞

∞∑

k=−∞A(ν)

n,k(r)e j (kϕ+(k−n)ωmt+kϕ0) (2)

if ν = 1, 2, 3 and

A(ν)(r, ϕ, t)

=∞∑

n=−∞

∞∑

k=−∞A(ν)

n,k(r)ej(

kπδ (ϕ−δi )+

(kπδ −n

)ωmt+ kπ

δ ϕ0

)

(3)

if ν = 4i .Here, the time- and spatial-harmonic orders are denoted by

n and k, the machine’s mechanical rotational speed is ωm .In(̃2) and (3), all of the time- and spatial-harmonic combi-nations (n, k) are separately available. This will be of greatuse when analyzing the results of the studies in Section IV.Note that the formulation of A(ν)

n,k(r) is discussed in [13].Note as well that in (2) and (3), mechanical periodicitiesare considered. This implies that the fundamental harmonicorders in the two-pole-pair machine of Fig. 1 are n = ±2 andk = ±2.

Finally, it is important to note that the rotational speed ofthe field’s (n, k) harmonic component can be calculated as

ωn,k = k − n

kωm . (4)

As the magnetic flux density is the curl of the magnetic vectorpotential, the results can be used to visualize the flux density.Figs. 4–6 do so for every harmonic combination of the fluxdensity in the center of the air gap. For Fig. 4, a sinusoidalcurrent was applied; for Fig. 5, the idealized BLDC currentwas applied; and for Fig. 6, the BLDC voltage was applied.Note that the synchronous harmonic combinations are shownon a green background, the oversynchronous combinations areshown on a yellow background, and the undersynchronouscombinations have a white background.

It is interesting to see that the asynchronous harmonic con-tent of the flux density is considerably larger when applyingthe idealized BLDC current or the BLDC voltage than whenthe sinusoidal current is applied. Especially, in the case of theidealized BLDC current, combinations (10,−2) and (14, 2),indicated in red, are important.

Fig. 4. Harmonic map of |B| in the center of the air gap when a sinusoidalcurrent (nominal current) is applied.

Fig. 5. Harmonic map of |B| in the center of the air gap when an idealizedBLDC current (nominal current) is applied.

Fig. 6. Harmonic map of |B| in the center of the air gap when a BLDCvoltage is applied.

C. Torque Calculations

This section considers three torque waveforms: the torqueT (PM)(t) produced by the stator current and the permanentmagnet flux, the torque T (SC)(t) produced by the statorcurrent and the induced currents in the SC, and the total

8104708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 10, OCTOBER 2018

torque T (t). Using results from the above magnetic vectorpotential calculation, the torque acting on a given volume canbe computed by integrating Maxwell’s stress tensor over asurface that encloses that volume. The total torque can thenbe computed by choosing the outer boundary of the SC asintegration surface. This results in

T (t) = lsr23

μ0

∫ 2π

0B(3)

r (r3, ϕ, t)B(3)ϕ (r3, ϕ, t) dϕ

= 2πlsr2

3

μ0

∞∑

n=−∞

∞∑

s=−∞Tn,se− j (n+s)ωmt (5)

where

Tn,s =∞∑

k=−∞B(3)

r,n,k(r3)B(3)ϕ,s,−k(r3). (6)

where Br,n,k(r) and Bϕ,s,−k(r) are the r -dependent parts ofBr,n,k(r, ϕ, t) and Bϕ,s,−k(r, ϕ, t), which, in turn, are foundfrom the definition of the magnetic vector potential.

To compute T (PM), the inner boundary of the SC is used asan integration surface

T (PM)(t) = 2πlsr2

2

μ0

∞∑

n=−∞

∞∑

s=−∞T (PM)

n,s e− j (n+s)ωmt (7)

where

T (PM)n,s =

∞∑

k=−∞B(1)

r,n,k(r2)B(1)ϕ,s,−k(r2). (8)

Finally, T (SC) can be calculated by subtracting T (PM) from T

T (SC)(t) = T (t) − T (PM)(t). (9)

As already mentioned, both the net torque and the torqueripple will be studied. The net torques (Tnet, T (PM)

net , and T (SC)net )

are simply calculated as the averages of T (t), T (PM)(t), andT (SC)(t), respectively. The torque ripples are expressed as apercentage of Tnet

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

�T% = 100% · max(T ) − min(T )

Tnet

�T (PM)% = 100% · max(T (PM)) − min(T (PM))

Tnet

�T (SC)% = 100% · max(T (SC)) − min(T (SC))

Tnet.

(10)

Note that (5), (7), and (9) are exponential Fourier series of realvalues. This implies that the opposing harmonic combination,i.e., (n, s) and (−n,−s) are complex conjugate and shouldthus be considered together. Therefore, the following notationwill be used:

X |y,z| = X y,z + X−y,−z (11)

where X can refer to any quantity that is expressed as acomplex Fourier series.

Fig. 7. Torque as a function of σSC in the case of idealized BLDC current.

Fig. 8. Torque T (PM) caused by the magnets and stator current, as a functionof σSC, detail of Fig. 7.

IV. PARAMETER STUDY FOR ENFORCED

BLDC CURRENT

Using the above calculations, this section provides variousparameter studies of the SC’s thickness and conductivity.In this first parameter study, the idealized BLDC current ofFig. 2 is applied to the machine. It should be noted that thoseparameter studies are theoretical: their goal is to identify trendsrather than to study an actual machine. Therefore, both theSC conductivity σSC and the SC thickness tSC may extend tounrealistic values.

A. Conductivity of the Shielding Cylinder

The net torques, as plotted in Fig. 7, show a very similarcourse as when the machine is excited with a sinusoidalcurrent [11], i.e., the behavior of T (SC)

net is similar to what canbe expected from the torque in an induction machine. As thedirect effect of σSC is limited to the asynchronous contentof the magnetic field, it was expected that T (PM)

net would notchange in this parameter study. However, Fig. 8 shows T (PM)

netseparately, demonstrating an initial increase before decreasing.As this is a non-evident observation, it is important to clearlyexplain why this happens: the insight in the behavior of

HANNON et al.: EVALUATION OF THE TORQUE IN HIGH-SPEED PMSMs 8104708

Fig. 9. Harmonic study of the net torque in the magnets in the case of idealized BLDC current; the component of B that changes more quickly is highlighted.(a) Evolution of T (PM)

|2,−2| and T (PM)|6,−6|, relative to σSC = 0. (b) Radial B-component B(PM)

r,|2,−2|. (c) Radial B-component B(PM)r,|6,−6|. (d) Radial B-component

B(PM)r,|14,−14|. (e) Tangential B-component B(PM)

φ,|2,−2|. (f) Tangential B-component B(PM)φ,|6,−6|. (g) Tangential B-component B(PM)

φ,|14,−14|.

the torque components is one of the goals of this paper.Analyzing the harmonic content of the torque shows that thevariation in T (PM)

net is mainly caused by variations in T (PM)|2,−2| and

T (PM)|14,−14|. Moreover, at very low conductivities (σSC < 6.105),

the variation of T (PM)|14,−14| dominates that of T (PM)

|2,−2|. As T (PM)|14,−14|

increases at these low conductivities, so does T (PM)net . This

is illustrated in Fig. 9. If the conductivity increases further,the torque T (PM) decreases, until the conductivity value σδ

where the penetration depth of the harmonic T (PM)|2,−2| is reached

(see also Fig. 9). Conductivities higher than σδ do not causesignificant changes any more in T (PM).

As shown in Fig. 9, the underlying cause for the changesin those torque components are changes in the synchro-nous content of the magnetic field. As already mentioned,it was not expected that the field’s synchronous contentwould be affected by the SC’s conductivity. The reason isthat variations in the asynchronous harmonic combinationspropagate to the synchronous combinations via the slottingeffect.

8104708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 10, OCTOBER 2018

Fig. 10. Torque ripple as a function of σSC.

Fig. 11. Torque ripple T (PM) caused by the magnets and stator current, as afunction of σSC, detail of Fig. 10

We now study the effect of σSC on the torque ripple�T (PM)

% . As can be seen in Fig. 10 and the zoomed-in view ofFig. 11, the evolution of �T (PM)

% , for example, shows an initialdecrease until σSC = 6.105 s/m. After this local minimum,the torque ripple briefly increases before it steadily decreasesagain. The reason for an initial decrease in �T (PM)

% is thatsome of the asynchronous combinations with higher time-harmonic order, e.g., (10,−2) and (14, 2), are already greatlyshielded when σSC = 6.105 s/m.

The brief increase that follows is mainly due to the evo-lution of B|6,−6|: in Fig. 5, this harmonic is the secondlargest in the “green zone,” which are the synchronousharmonics. Although this component of the flux densitygets shielded more effectively at higher σSC, initially itsdecrease is entirely due to the decrease in its radialcomponent. At low σSC, its angular component increases.Moreover, proportionally, this increase is larger than thedecrease in the radial component. The result is an increasing�T (PM)

% .Once the SC’s conductivity exceeds the point at which the

penetration depth of (6,−6) equals tSC, the angular componentstarts decreasing as well. Note that, as already mentioned,the behavior of this asynchronous combination propagates tothe synchronous combination with the same time-harmonicorder, i.e., (6, 6).

Fig. 12. Torque as a function of tSC.

Fig. 13. Torque ripple as a function of tSC.

Note that, in contrast with the net torque, the evolution ofthe torque ripple when applying an idealized BLDC currentshows a very significant difference with respect to the evo-lution that is observed when applying a sinusoidal current.This was indeed expected as the effect of the asynchronousharmonic combinations on the torque ripple is much larger.

B. Thickness of Shielding Cylinder

As a function of the thickness of the SC, the evolutionof the net torque is shown in Fig. 12. The conductivity ofthe SC is given in Table I. Using Carter’s coefficient and areference value for the torque, T (PM)

net can again accuratelybe predicted, while T (SC)

net accounts for the deviation betweenCarter’s prediction and Tnet. The Carter coefficient can befound in [15]. The torque ripples as a function of tSC areshown in Fig. 13 and a zoomed-in view in Fig. 14. Theseripples are subject to the same mechanisms as in Section IV-A.The shielding of harmonic combinations (10,−2) and (14, 2)

at tSC = 0.01 mm explains the initial decrease in �T (PM)%

and the very steep increase in �T (SC)% . �T (PM)

% then increasesagain until (6,−6) is effectively shielded.

V. PARAMETER STUDY FOR ENFORCED

BLDC VOLTAGE

In this final set of parameter studies, a voltage instead ofa current is applied to the machine. As will be shown in the

HANNON et al.: EVALUATION OF THE TORQUE IN HIGH-SPEED PMSMs 8104708

Fig. 14. Torque ripple in the magnets as a function of tSC.

Fig. 15. Torque as a function of σSC.

Fig. 16. Torque in the magnets as a function of σSC, detail of Fig. 15.

following, fixing the stator voltage implies that the currentmay change. Indeed, changing the SC affects the machine’sinductance. For this reason, only the SC’s conductivity isconsidered in this section. The effect of changing the SC’sthickness on the machine’s inductance is too big to drawmeaningful conclusion as to tSC’s direct effect on the torque.

One can see from Fig. 15 that the net torque in the magnetsbehaves differently than in Section IV. As shown in more detailin Fig. 16, T (PM)

net now changes more drastically and increasesinstead of decreasing.

Fig. 17. rms value of the current density in phase A as a function of σSC.

Fig. 18. Torque ripple as a function of σSC.

The reason is that, in addition to the indirect effect ofthe SC on the synchronous harmonic combinations, the SC’sconductivity also affects the current through the windings.Indeed, it has an effect on the machine’s inductance. As thevoltage is now fixed, varying σSC implies changing thecurrent. Evidently, this affects the torque in the magnetsas well. The evolution of the rms value of the currentdensity in phase A is shown in Fig. 17. Fig. 18 showsthe evolution of the torque ripples as a function of theSC’s conductivity. It is clear that the evolution is verysimilar to that of the parameter study in which the ideal-ized BLDC current is imposed. However, �T (PM)

% reachesits minimum at a different conductivity, i.e., 16 · 105 s/minstead of 6 · 105 s/m. This is, among others, due to thechanging of the current waveform at different conductivities,which in turn affects the relative importance of differentharmonic combinations.

VI. CONCLUSION

The effect of the SC’s conductivity and thickness on thetorque has been studied for various source terms. The noveltyof this paper is the division of the torque in a componentrelated to the magnets and a component related to the SC.

8104708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 10, OCTOBER 2018

This gives a lot of insight in how torque is produced, andleads to the following most important findings.

1) The total torque cannot be predicted using Carter’sfactor; only the torque component T (PM)

net is predictedby Carter’s factor.

2) The torque component T (SC)net in the SC has the same

characteristics as the torque in an induction machine,because it is caused by interaction with asynchronousharmonic combinations in the field.

3) The sum of the torque ripples in the SC and the PMdoes not equal the ripple of the total torque. The reasonis that the ripples of the different torque components areshifted in time with respect to each other.

4) Unlike expected, the torque in the magnets T (PM)net

depends on the SC’s conductivity, because variationsin the asynchronous content of the machine propagateto the synchronous harmonic combinations through theslotting effect.

Although it remains difficult to extrapolate the above-mentioned findings quantitatively to other machines,the results do provide a better understanding of high-speedPMSMs.

ACKNOWLEDGMENT

This work was supported through the framework of theNWO-Vlaio Project “High Efficiency, High Speed ElectricalDrives.”

REFERENCES

[1] N. Bianchi, S. Bolognani, and F. Luise, “Potentials and limits of highspeed PM motors,” in Proc. Conf. Rec. 38th IAS Annu. Meeting. Ind.Appl. Conf., vol. 3, Oct. 2003, pp. 1918–1925.

[2] P.-D. Pfister and Y. Perriard, “Very-high-speed slotless permanent-magnet motors: Analytical modeling, optimization, design, and torquemeasurement methods,” IEEE Trans. Ind. Electron., vol. 57, no. 1,pp. 296–303, Jan. 2010.

[3] J. R. Bumby et al., “Electrical machines for use in electrically assistedturbochargers,” in Proc. 2nd Int. Conf. Power Electron., Machines Drives(PEMD), vol. 1, Mar. 2004, pp. 344–349.

[4] C. Zwyssig, M. Duerr, D. Hassler, and J. W. Kolar, “An ultra-high-speed,500000 rpm, 1 kw electrical drive system,” in Proc. Power Convers.Conf.-Nagoya, Apr. 2007, pp. 1577–1583.

[5] A. Binder, T. Schneider, and M. Klohr, “Fixation of buried andsurface-mounted magnets in high-speed permanent-magnet synchronousmachines,” IEEE Trans. Ind. Appl., vol. 42, no. 4, pp. 1031–1037,Jul. 2006.

[6] O. Aglen and A. Andersson, “Thermal analysis of a high-speed gen-erator,” in Proc. Conf. Rec. 38th IAS Annu. Meeting Ind. Appl. Conf.,vol. 1, Oct. 2003, pp. 547–554.

[7] J. Barta, C. Ondrusek, P. Losak, and R. Vlach, “Design of high-speed induction machine for the 6 kW, 120 000 rpm helium turbo-circulator,” in Proc. 22nd Int. Conf. Elect. Machines (ICEM), Sep. 2016,pp. 1552–1558.

[8] M. R. Shah and S.-B. Lee, “Rapid analytical optimization of eddy-current shield thickness for associated loss minimization in electricalmachines,” IEEE Trans. Ind. Appl., vol. 42, no. 3, pp. 642–649,May 2006.

[9] F. Zhou, J. Shen, W. Fei, and R. Lin, “Study of retaining sleeve andconductive shield and their influence on rotor loss in high-speed PMBLDC motors,” IEEE Trans. Magn., vol. 42, no. 10, pp. 3398–3400,Oct. 2006.

[10] S. R. Holm, H. Polinder, and J. A. Ferreira, “Analytical modeling ofa permanent-magnet synchronous machine in a flywheel,” IEEE Trans.Magn., vol. 43, no. 5, pp. 1955–1967, May 2007.

[11] B. Hannon, P. Sergeant, and L. Dupré, “Study of the effectof a shielding cylinder on the torque in a permanent-magnet synchronous machine considering two torque-producingmechanisms,” IEEE Trans. Magn., vol. 53, no. 10, Oct. 2017,Art. no. 8108208.

[12] B. Hannon, P. Sergeant, and L. Dupré, “Torque and torque com-ponents in high-speed permanent-magnet synchronous machines witha shielding cylinder,” Math. Comput. Simul., vol. 130, pp. 70–80,Dec. 2016.

[13] B. Hannon, P. Sergeant, and L. Dupré, “2-D analytical subdomain modelof a slotted PMSM with shielding cylinder,” IEEE Trans. Magn., vol. 50,no. 7, pp. 1–10, Jul. 2014.

[14] B. Hannon, P. Sergeant, and L. Dupré, “Voltage sources in 2d Fourier-based analytical models of electric machines,” Math. Problems Eng.,vol. 2015, Oct. 2015, Art. no. 195410.

[15] J. Pyrhönen, J. Tapani, and V. Hrabovcová, Design of Rotating ElectricalMachines. Chichester, U.K.: Wiley, 2008.