Five-Phase Permanent Magnet Synchronous Machine Operating as Generator - Design, Parameter Determination by FEM and Performance

Luis Alberto Pereira Electrical Engineering Graduate Program

Catholic University - PUCRS Porto Alegre, Brazil lpereira@ee.pucrs.br

Vicente Mariano Canalli Electrical Engineering Graduate Program

Catholic University - PUCRS Porto Alegre, Brazil canalli@ee.pucrs.br

Abstract— The paper describes the design and parameter

calculation of a five-phase permanent magnet synchronous machine intended for operation as generator connected to the load over a rectifier bridge. The main parameters and machine dimensions are determined using the Finite Element Method in 2 dimensions. Calculated no-load induced voltage and inductances are compared with measured values obtained from a prototype machine. A circuit based mathematical model is also developed and simulated using MatLab/Simulink and the results are used to determine the machine performance as generator. The simulation results are checked versus measured values, and a good agreement between the simulation and measurements was observed. The implemented model gives results with sufficient accuracy, allowing to determine the machine behavior in early design stages. The method described here can be successfully applied for designing and performance calculation of high phase machine.

Keywords—permanent magnet machine, high-phase machine.

I. INTRODUCTION Permanent magnet machines have been used in the last

three decades. They appear in several different stator and rotor structures and can be used in almost all kinds of application where traditional machines are used. Nowadays their main features and advantages are well known: higher efficiency, high torque/volume relationship [13][14]. Unfortunately, some of these advantages refer only to machines built with high energy magnets, which are still expensive, despite the introduction of new materials and improved production techniques. This feature has restricted the use of permanent magnet machines built with rare earth magnets to applications where cost is of secondary concern. Conversely, many published papers have shown that higher power density can be obtained if the phase number can be made higher than three [5][6][8][10][11][12]. Three phase machines have been intensively researched and their mathematical model can be easily stated. The classical model of three phase machine includes in general only the fundamental wave of airgap induction, the effects of higher harmonics are in general neglected. On the contrary, high phase machines have up to now no general mathematical model. In addition, higher time and space harmonics play a more important role in this kind of machine. Furthermore, high phase machine open new and interesting possibilities in what relates their winding connection, design and control schemes, as has been shown in [16]. Finally, one of the most important features is the

improved reliability, being possible to operate the machine even under failure of one phase winding as shown recently in [15].

The first aim of the paper is to present the design and determination of the main parameters of five-phase permanent magnet machine operating as generator over a rectifier bridge. The design and optimization of the main dimensions was performed with the aid of a software package based on the Finite Element Method (FEM) in 2 dimensions. The second aim is the development of an accurate FEM based mathematical model and its implementation. An accurate mathematical model is necessary in the design stage for optimization purpose, performance prediction and for the choice of the best control scheme. Due to the five-phase winding in the stator, a higher number of electronic switches will be necessary for the operation with inverter or rectifier. This however does not mean an increase in the overall cost for the switches, given that each individual switch can be rated with a lower current or voltage capacity [11]. In addition, electronic switches show a decreasing cost while the machine raw materials show increasing costs considering the past three decades. Finally, the paper also aims to compare simulation results with corresponding test results. The machine was tested working over a five-phase rectifier bridge and feeding a resistive load. The winding is star connected without neutral wire connection, although other connections are also possible. Based on the FEM calculations, the mathematical model was stated and implemented using the Power System Toolbox from Matlab/Simulink. Calculated power, currents and voltages are compared versus the corresponding values from tests.

II. MACHINE DESIGN A cross sectional view of one pole of the prototype

machine is shown in Figure 1. The machine was built in the frame of a standard ¾ HP 4 pole TEFC induction machine with 40 slots in the stator. The winding was designed as full coil pitch with 2 slots/pole/phase. The rotor has surface mounted neodymium-iron-bore permanent magnets with four segments mounted side by side. The choice for this pole arrangement lead to a reduction in the volume of the magnets without reducing the stator flux. The magnets are magnetized in the radial direction and cover about 80% of the pole area. A FEM analysis has shown that the leakage flux between the poles begins to increase if the magnets cover more than 80% of the pole area, making the use of the magnets less effective. The machine ratings and main dimensions are listed in table I.

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0-7803-7420-7/02/$17.00 © 2002 IEEE

III. AIRGAP INDUCTION The induction in the airgap at no-load is of fundamental

importance for the machine design, modeling and operation, because it determines the induced voltage, the power and torque capability. The radial component of the airgap induction under one pole at no-load is shown in Fig. 2. It was determined based on the field solution obtained with FEM (see Fig. 1). The air space between two consecutive magnets of the same pole introduces additional space harmonics in the airgap induction, whose effects can be seen in Fig. 2. Their amplitude is about twice the amplitude of the harmonics introduced by the slotting. However, the effect on the induced voltage is smaller given that they are filtered by the winding. The calculation of the induced voltage will be based on a Fourier Series including ten harmonics terms of the radial airgap induction. As it has been shown in many papers [6][7][9], higher harmonics can also produce useful torque in five phase machines. In this way, a careful choice of the

harmonic content in the airgap induction is very important for the performance optimization of the machine.

IV. INDUCED VOLTAGES The induced voltage is represented by a Fourier Series

containing the first ten voltage harmonics. Each harmonic was calculated from the airgap induction and considering the winding factor. The conductors are supposed to lie in the airgap and under the action of the induction shown in Fig. 3. Because of the symmetry, only uneven harmonics appear in the airgap induction curve. By the same reason only sine-terms will appear in the Fourier Series. Given that the winding has full coil pitch and each phase has 2 coils in series, the winding factor for the n-th harmonic can be given as follows:

⋅α= n2

cosk snw . (1)

sα is the angle in electrical degrees between two consecutive slots. As the machine has 40 slots, 5 phases and 4 poles sα is 18 electrical degrees. Therefore, the amplitude of the phase to neutral induced voltage for the nth-harmonic is given by the following equation:

nwnn kvBN4E ⋅⋅⋅⋅⋅= l . (2)

N – number of conductors in series in a coil; nB - amplitude of nth-harmonic of the airgap induction; l - axial length (m); v – rotor peripheral speed (m/s) Thus, the phase-induced voltage, expressed as a Fourier

Series, will assume the following form:

( ) ( )[ ]∑

∞

=

⋅ω⋅⋅=1n

na tnsinEte

( ) ( )[ ]tnsinkvBN4te n

wn1n

a ⋅ω⋅⋅⋅⋅⋅⋅⋅=∑∞

=

l

(3)

TABLE II – HARMONIC PHASE VOLTAGES (%) order calculated measured error (%)

1 +105 +100 +5.03 +28.3 +27.1 +1.25 +8.5 +12.8 -4.37 +2.3 +7.5 -5.29 -2.1 -0.4 +1.711 +0.4 +0.7 -0.313 +2.4 +2.2 +0.215 +3.4 +2.9 +0.517 +9.0 +12.3 -3.319 +12 +0.2 +11.8

TABLE I – MACHINE RATING AND DIMENSIONS

voltage 90 volts

current 1 ampere

base speed 1500 rpm

poles 4

axial length 64 mm

stator outer radius 65.15 mm

stator inner radius 40 mm

airgap 0.7 mm

remanent induction 1.1 Tesla

Figure 1- cross view of one pole and flux distribution at no load

0 30 60 90 120 150 1800

200

400

600

800

1000

stator ang le (e lec . degrees)

rad

ial

ind

uc

tio

n (

mT

)

Figure 2 - radial component of airgap induction at no load

399

For the purpose of modeling only ten voltage harmonics will be taken into account. The calculated and the measured values of each individual harmonic are shown in table II. The values are normalized using the measured voltage of the first harmonic as the base value. For the first three harmonic terms, the calculated values are in good agreement with the measured values. It was observed that the difference between measured and calculated values increases for harmonics higher than the 7th. This is because the FEM calculations can not correctly predict fast variations of the normal induction in the airgap, introducing larger errors in the higher harmonics. In case a better accuracy is needed, a more precise method for the induced voltage should be used, as for example the calculation of the linked flux with the winding for different rotor positions as described in [13]. The phase induced voltage calculated under consideration of those harmonics in table II is shown in Fig. 3 and the corresponding measured voltage is shown in Fig. 4. Both curves agree very well, showing that the procedure can be successfully applied for this kind of machine. It must be pointed out that the curve shown in Fig. 4 will be used later for the theoretical calculation of the machine performance working as a generator on a diode bridge.

Besides the phase voltage, the induced voltage between two phases was calculated from the airgap induction and is shown in Fig. 5 to 8. Fig. 5 refers to the calculated voltage between phase 1 and 2 and Fig. 6 refers to the corresponding measured voltage. As in the proceeding case, the calculated curve shows a nice agreement with the measured one. The calculated voltage between phase 1 and 3 is illustrated in Fig. 7 and the corresponding measured voltage can be seen in Fig. 8. The difference between the theoretical and the measured curves is within an acceptable range.

From the induced phase to neutral and phase to phase

voltages can be concluded that the FEM provides an acceptable accuracy for the determination of the induced voltage, even if the conductors are supposed to lie in the airgap. The knowledge of these induced voltages during the design stage of the machine is of crucial importance, because they determine the machine behavior.

V. INDUCTANCES AND STATOR FIELD A complete mathematical model must consider the effects

of the field produced by the stator (armature reaction field).

0 30 60 90 12 0 15 0 180-80

-60

-40

-20

0

20

40

60

80

wt (deg rees)

ph

as

e v

olt

ag

e (

vo

lts

)

Figure 3- calculated induced voltage

Figure 4- measured induced voltage

0 3 0 60 9 0 12 0 1 50 1 80-15 0

-10 0

-5 0

0

5 0

10 0

15 0

wt (d eg ree s)

ph

as

e 1

to

ph

as

e 3

vo

lta

ge

(v

olt

s)

Figure 5- calculated induced voltage between phase 1 and 2

Fig. 6- measured voltage between phase 1 and 2 (50V/div)

0 30 60 90 120 15 0 1 80-15 0

-10 0

-5 0

0

5 0

10 0

15 0

wt (deg rees)

ph

as

e 1

to

ph

as

e 2

vo

lta

ge

(v

olt

s)

Figure 7- calculated induced voltage between phase 1 and 3

Fig. 8- measured voltage between phase 1 and 3 (50V/div)

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This field is accounted by means of the self and mutual inductances of the winding phases. Given that the relative permeability of the permanent magnets is very close to the unity, the machine has a very large effective airgap and low inductances as a consequence. The self and mutual inductances can be directly calculated from the stator field produced by one phase acting alone, as shown by the flux plot in Fig. 9 and the corresponding airgap induction in Fig. 10. The self-inductance is calculated from the flux linked with the carrying current phase in the leftmost and rightmost slots. The linked flux is calculated using the difference of the vector potential for points in the middle of the slot belonging to the same coil [11]. The values of the vector potential are obtained from the FEM solution. The part of leakage flux corresponding to the slot leakage was calculated considering that part of the flux linked with the phase but not crossing the airgap. The remaining part of leakage flux corresponds to the end-winding connection and can not be determined from two dimensional FEM solutions. This would requires a three dimensional FEM calculation. Therefore, this part was calculated from analytical expressions described in [11]. Accordingly, the calculated components of self inductance are as follows:

main self inductance (FEM): 7.16Lm = (mH) slot leakage inductance (FEM): 2.5L s =δ (mH) end winding leakage inductance : 2L e =δ (mH) total self inductance: 9.23L11 = (mH).

Considering the main and leakage flux, the calculated inductance is 23.9 mH, which agrees with the value measured

directly with a commercial bridge (25 mH). The mutual inductances between the phases were calculated using the same procedure and their values are summarized in table III. The errors in the mutual inductances are larger than the error for the self inductance. This is because the inductances were measured indirectly and with the magnets mounted in the rotor. However, the influence of these errors should not impact the results because their absolute values are small. Better results for the inductances could be obtained with the method described in [4]. However, this method would require more computational efforts and an extra time for data preparation and numerical modeling.

VI. STATOR RESISTANCE The winding resistance can be determined based on the

following equation:

NA

lR

c

w ⋅⋅ρ

= (4)

wl - mean length of a coil

cA - cross area of a conductor ρ - conductor specific resistance The calculated value of the resistance of a phase is 6.7

ohms, which compares well with the measured value of 7 ohms.

VII. MATHEMATICAL MODEL The accurate knowledge of the performance and the

operating limits of electrical machines are important during its design stage. This knowledge makes possible to determine if the machine complies with the requirements of the specific application. For this purpose a large number of software tools are now available and can be used, as for example Matlab, Mathcad, Vissim... However, all of them need a mathematical model describing the dynamic behavior of the machine and its electrical parameters. The correct determination of the parameters for the model is in general a difficult task during the design stage and rise difficulties. In the last decades the FEM has been successfully applied in the analysis of the field inside the machine and to determine its main parameters [4]. For the case of permanent magnet machines, no-load induced voltages and inductances have to be calculated in order to implement its mathematical model. Considering a three-phase machine, there are a large number of published works focusing its modeling and simulation. The most popular software packages have built-in models of all kinds of electrical machines, which can be used for their simulation

TABLE III – SELF AND MUTUAL INDUCTANCES (MH)

Inductance FEM Measured error (%)

11L 23.9 25 -4.4

12L 3.6 4.9 -26.5

13L -11.1 -9 +23.3

14L -11.1 -9 +23.3

15L 3.6 5 +26.5

Figure 9- field produced by one stator phase

0 30 60 90 120 150 180-50

-20

10

40

70

100

130

stato r angle (e lec. degrees)

No

rma

l In

du

cti

on

(m

T)

Figure 10- radial field in the airgap produced by one phase

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and design. Given that five phase machines has attracted the attention of researchers more recently, there are fewer published works on the design and modeling of this kind of machine. Higher time harmonics and space harmonics play a more important role than in the traditional three-phase machine, where they are in general neglected. Thus, higher harmonics must be considered not only in design but also for simulation purposes.

This part of the paper aims to implement the model of a five-phase prototype machine built with permanent magnet and working as generator. The induced voltages and inductances are those determined in the foregoing sections and based on FEM static field analysis. As the prototype machine is intended to operate as generator, it has been tested working over a five-phase rectifier bridge and feeding a resistive load. The winding is star connected without neutral connection, although other connections are also possible. The mathematical model was implemented using the Power System Toolbox from Matlab/Simulink. Calculated power, currents and voltages are compared versus the corresponding values from tests. The method presented shows a good agreement between theoretical and measured values and can be applied for the design and performance prediction of high phase machines.

In matrix form the mesh equations describing the machine are given as:

[ ]

+

⋅

+

⋅=

5

4

3

2

1

5

4

3

2

1

5545352515

4544342414

3534332313

2524232212

1514131211

5

4

3

2

1

5

4

3

2

1

eeeee

iiiii

LLLLLLLLLLLLLLLLLLLLLLLLL

iiiii

RR

vvvvv

L (5)

ke - no load induced voltage of phase k

kv - terminal voltage of phase k

ki - phase current of phase k kjL - mutual inductance between phase k and j

R - resistance of one phase The induced voltages is represented by a Fourier Series

containing ten harmonic terms and was calculated from the

normal component of the induction in the airgap obtained from FEM solutions. All the inductances are considered as constant values, considering that the machine has a large effective airgap and the slot effects on the inductance are very reduced.

VIII. IMPLEMENTATION The arrangement of the machine and the diode bridge is

shown in Fig. 11. The diodes are numbered from P1 to P5, N1 to N5 and were implemented with the built-in models of the Power System Toolbox from Matlab/Simulink. The mutual inductances had to be represented by voltage dependent voltage sources; the reference voltages being the voltages across the self-inductance of one phase. The built-in mutual inductances model could not be used, because their are tailored for three phase devices. The machine need 10 diodes for working on a full rectifier bridge, instead of 6 that would be necessary for a three-phase machine. However, the diodes can be rated with a lower current capacity compared to the three phase case.

IX. SIMULATION AND TEST RESULTS In order to prove the accuracy of the implemented model,

the machine was simulated under several load conditions. In what follows the simulation results are compared with test results. The main steady state characteristics for the operation

Figure 11 – machine working over a diode bridge

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 51 00

1 10

1 20

1 30

1 40

1 50

1 60

1 70

1 80

1 90

2 00L o a d V o l ta g e X L o a d C u r re n t

c urre nt [A ]

volta

ge

[A]

calculate dm e as ure d

Figure 12 – Mean value of the load voltage for operation over a diode bridge .

0 1 2 3 4 50

100

200

300

400

500

600Power X Current

current [A]

measured calculated

Figure 13 – Mean output power for operation over a diode bridge as a function of the current.

402

as generator were determined based on simulations and are presented and discussed below together with measured values. All the measurements and calculations were performed for a constant rotor speed of 1500 rpm (50 Hz).

A. Load Voltage versus Current One of the most important characteristics of the machine

operating as generator is the mean value of the rectified load voltage as a function of the load current for different loads. This curve was calculated using the model implemented with MatLab and compared with measured values. The calculated curve and the measured points are plotted in Figure 12. Each calculated point in the curve was obtained integrating the load voltage over one period. The test results show a lower rectified voltage when compared to the corresponding value from the simulation. These differences are introduced by the errors in the calculated induced voltage harmonics and also by the errors in determination of the parameters, specially the inductance. The errors are also influenced by variation of the load resistance and winding resistance with temperature. This influence has not been taken into account in the simulation. However, the errors are lower than 5% in the worst case and therefore well acceptable. Except for the first part of the curve, it can be observed that the voltage drops linear with the current.

B. Output Power The most important characteristic of generators is the

output power capability. For this case, the mean output power as a function of the load current was calculated based on the simulation results and plotted in Figure 13. The mean output power was also calculated integrating the product of instantaneous rectified current and voltage over one period. The errors in the mean power are lower than 1%. It is much lower than the error appearing in the load voltage curve (see

Fig. 12). In what respects the power output the implemented model give very accurate results, as the measured values confirm.

C. Rectified Voltage Waveform In addition to the mean value of the rectified voltage, it is

important to be able to determine the instantaneous rectified voltage, in order to determine the filter size and ripple content in the output. The calculated instantaneous voltage at the load is shown in Figure 14. This curve corresponds to a mean load current of 2.13 A. The corresponding measured rectified voltage is shown in Fig. 15. The measured curve shows a stronger influence of the higher harmonic than the measured one. This can be explained considering that the errors in the determination of higher harmonics are larger for lower order harmonics. The estimated value of the ripple is about 18 volts and the calculated ripple is 20 volts. Therefore, the method proves to be efficient in what relates the rectified voltage, given that it can predict with reasonable accuracy the instantaneous values of the output voltage.

D. Phase to Neutral Voltage Waveform The accuracy of the simulation can also be checked

comparing the calculated and the measured waveform of the phase to neutral and the phase to phase voltages. Fig. 16 refers to the simulated phase to neutral voltage and Fig. 17 to the measured phase to neutral voltage. Both curves were obtained for a load current equal to 2.13 A. As can be seen, there is a good agreement between both curves. The difference between the curves is due the error in harmonic voltage calculation.

E. Phase to Phase Voltage The conducting sequence of the diodes depends basically

on the voltage between two neighboring phases. This simulated phase to phase voltage is shown in Fig. 18 and the measured voltage can be seen in Fig. 19. As was the case for the phase to neutral voltage, there is a good agreement between theoretical and practical curves resulting from tests. The curves shown in Fig. 18 and 19 also refer to load current of 2.13 A.

F. Phase Current The knowledge of the phase current waveform is important

for the diode rating and for the calculation of the winding losses. Thus, the phase current was calculated for a load current of 2.13 A and is illustrated in Fig. 20. The corresponding measured current is shown in Figure 21. As can be seen, the waveforms of both currents have shown a nice agreement. The errors in the harmonic calculation are smaller in the current waveform, because the machine inductances and resistance damp the higher voltage harmonics.

X. CONCLUSION The paper presented a FEM modeling and simulation of a

five-phase permanent magnet machine operating as generator over a diode bridge and feeding a resistive load. The main parameters were determined from static FEM solutions and the working characteristic were obtained using Matlab/Simulink. Simulated results were then compared with test results. This comparison show that the simulated load voltage is less than 5% higher than the corresponding

20 21 22 23 24 25 26 27 28 29 3040

60

80

100

120

140

160

180Load Voltage X Time

time [ ms ] Figure 14- calculated rectified voltage (Volts)

Figure 15- measured rectified voltage (Volts, 10 V/division)

403

measured values and the output power agrees within 1% with the measured values. These errors came from the errors in the calculated harmonic waves, calculated parameters and

temperature effects. The later was not accounted in the model. The accuracy of the model was also checked through a comparison between simulated and measured waveforms of voltages and phase current, from which a close match could be observed. The test results proves that the method so far presented produce results with good accuracy and can be applied for the design and performance prediction of high phase machines, leading to a close match between theoretical and measured values.

XI. REFERENCES [1] E. E. Ward and H. Harer, “Preliminary investigation of an inverter-fed 5-

phase induction motor”, Proc. Inst. Elect. Eng., Nr. 6, pp. 980-984, 1969.

[2] P. W. Franklin, “Theory of the three phase salient pole type generator with bridge-rectified output – part I and II”, IEEE PES – Winter Meeting, 1972, pp. 1960 – 1975.

[3] P. W. Franklin, “Theory of the bridge-rectified synchronous generator with a large number of stator phases – part i and ii”, IEEE PES – Summer Meeting, 1972, pp. 137 – 148.

[4] N. A. Demerdash, F. A. Fouad and T. W. Nehl, “Determination of winding inductances in ferrite type permanent magnet machinery by finite elements”, IEEE Transactions on Magnetics, vol. MAG 18, Nr. 6, 1982, pp. 1052-1054.

[5] T. A. Lipo, F. X. Wang, “Design and performance of a converter optimized ac machine”, IEEE Transactions on Industry Applications, Vol. IA-20, Nr. 4, pp. 834-844, 1984.

[6] H. Weh and U. Schrode, “Static inverter concepts for multiphase machines with square-wave current field distribution”, 1st European Conf. Power Electronics and Applications, Oct. 16-18, pp. 3143-3147, 1985.

[7] K. N. Pavithran, R. Parimelalagan and M. R. Krishnamurthy, “Studies on inverter-fed five-phase induction motor drive”, IEEE Trans. on Power Electronics, vol. 3, Nr. 2, pp. 224-235, 1988.

10 15 20 25 30 35 40 45 50 55 60-120

-80

-40

0

40

80

120Phase to Neutral Voltage [ V ]

time [ ms ]

Figure 16- calculated phase to neutral voltage

Fig. 17- measured phase to neutral voltage (20V/division)

40 42 44 46 48 50 52 54 56 58 60-200

-150

-100

-50

0

50

100

150

200Phase to Phase Voltage X Time

time [ ms ] Figure18- calculated phase to phase voltage

Figure 19- measured phase to phase voltage

40 42 44 46 48 50 52 54 56 58 60-3

-2

-1

0

1

2

3Phase Current X Time

time [ ms ] Figure 20- calculated phase current (A)

Figure 21- measured phase current (A)

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[8] P. J. McCleer, J. M. Bailey, J. S. Lawler and B. Banerjee, “Five phase trapezoidal back emf pm synchronous machines and drives”, 4th European Conf. Power Electronics and Applications, Italy, 1991, pp. 4128-4133.

[9] H. A. Toliyat, T. A. Lipo, J. C. White, “Analysis of a concentrated winding induction for adjustable speed drive operation - part II (motor design and performance)”, IEEE Trans. Energy Conversion, vol. 6, Nr. 4, pp. 685-692, 1991.

[10] H. A. Toliyat, L. Y. Xue and T. A. Lipo, “A five phase reluctance motor with high specific torque”, IEEE Trans. Ind. Applications, vol. 28, Nr. 3, pp- 659-667, 1992.

[11] C. C. Chan, J. Z. Jiang, G. H. Chen, X. Y. Wang and K. T. Chau, “A novel polyphase multipole square-wave permanent magnet motor drive for electric vehicles”, Trans. on Industry Applications, vol. 30, Nr. 5, pp. 1258-1266, 1994.

[12] L. A. Pereira, “Comparison between the dc and the electronically commutated synchronous machine”, PhD. Thesis, University of Kaiserslautern, Germany, march 1997 (in german).

[13] J.R Hendershot, T.J.E. Miller, “Design of brushless permanent-magnet motors”, Clarendon Press, Oxford, 1994.

[14] E. S. Hamdi, “Design of small electrical machines”, John Wiley & Sons, 1998.

[15] H. A. Toliyat, “Analysis and simulation of five-phase variable-speed induction motor drives under asymmetrical connections”, IEEE Transactions on Power Electronics, vol. 13, Nr. 4, pp. 748-756, 1998.

[16] H. A. Toliyat, S. P. Waikar, T. A. Lipo, “A|nalysis and simulation of five-phase synchronous reluctance machines including third harmonic of airgap mmf”, IEEE Trans. on Industry Applications, vol. 34, Nr. 2, pp. 332-339, 1998.

[17] L. Xu, “Rotor structure selections of nonsine five-phase synchronous reluctance machines for improved torque capability”, IEEE Trans. on Industry Applications, vol. 36, Nr. 4, pp. 1111-1117, 2000.

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