torque performance of optimally...

Vol.97(1) March 2006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 43

TORQUE PERFORMANCE OF OPTIMALLY DESIGNED THREE- AND

FIVE-PHASE RELUCTANCE SYNCHRONOUS MACHINES WITH TWO

ROTOR STRUCTURES.

E.T. Rakgati and M.J. Kamper

Electrical Machines and Drives Laboratory, Department of Electrical and Electronic Engineering, University of

Stellenbosch, Private Bag X1, Matieland (Stellenbosch) 7602, South Africa

Abstract: In this paper the torque performance of optimally designed three- and five-phase reluctance synchronous

machines with different normal laminated rotor structures are studied. Both the round rotor with internal flux

barriers and salient-pole rotor with no internal flux barriers but only cut-outs are investigated. The effect on the

torque performance by adding third harmonic current component to the phase currents in a five-phase reluctance

synchronous machine is also studied. The magnetostatic finite-element field solution with skew taken into account is

used directly by an optimisation algorithm to optimise in multi-dimensions the design of the machines under same

copper losses and volume. It is found that the torque increase due to third harmonic current injection is only 4% in

the case of the five-phase machine with salient-pole rotor; the three-phase machine with round, internal-flux-barrier

rotor is shown to outperform this machine in terms of torque by 28%. The measured torque results of the three-phase

machine with round, internal-flux-barrier rotor are presented and compared with calculated results.

Key words: Reluctance synchronous machine, rotor structures, design optimisation, finite element method, five-phase

1. INTRODUCTION

Reluctance synchronous machines (RSMs) have attracted

the efforts of various researches especially on the rotor

design of the machine. The various research work done

focussed on the following types of rotor [1 - 8]: (i) salient-

pole rotor with cutouts and with/without internal flux

barriers; (ii) normal (transverse) laminated round rotor with

internal flux barriers; (iii) axially laminated rotors which

might be round or with salient poles. Examples of two-pole

RSMs with salient-pole and internal-flux-barrier rotors are

shown in Figure 1.

(a) (b)

Figure 1: Two-pole RSMs with (a) salient-pole and (b) internal-

flux-barrier rotors

In axially laminated rotors the laminations are bent in

accordance to the poles of the machine and then stacked

(axially) on the rotor shaft. From literature [4 - 6] the

normal laminated rotor is preferred over the axially

laminated rotor due to the following reasons: the rotor is

easier to manufacture, it can easily be skewed to reduce

torque ripple and it has no iron loss problem associated

with axially laminated rotors [8].

The outcomes of the various research work done on the

torque performance of 5-phase RSMs with injection of

third harmonic current are as follows: (i) 10% improvement

in torque for the RSMs with salient pole rotor [9, 11];

(ii) 24% torque boost and 17% torque decline for the

RSMs with salient pole and round axially laminated rotor

respectively at 2 per unit load [10, 12]. It is, however, not

clear how the torque performances of 3- and 5-phase RSMs

with different rotor structures and with/without the use of

third harmonic currents compare. A true comparison in

this regard is therefore necessary of all these RSMs. This

paper attempts to do a first comparison study by focussing

on the torque per copper-loss per volume performance of

different RSMs; the effect of iron losses is not investigated

in this first study. Unlike in previous studies, the rotors

investigated are all normal laminated rotors.

The application for the salient-pole RSMs (Figure 1a) is

high power (MW), high-speed drives for the petrochemical

industry where any arcing within the machine is not

allowed. The high number of phases is to handle the high

power by means of solid-state converters.

Two types of reluctance rotors are investigated namely

the rotor with only cut-outs (Figure 1a), abbreviated as

CR rotor, and the round rotor with internal flux barriers

(Figure 1b) abbreviated as FBR rotor. The four-pole

CR and FBR rotors investigated are shown in Figures 2

and 3 respectively. The different dimensions (variables)

defining the rotor structures, which have to be optimised

Copyright (c) 2004 IEEE. This paper was first published in AFRICON ‘04,

15-17 September 2004, Gabarone, Botswana

44 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.97(1) March 2006

in the design optimisation, are also shown in Figures 2 and

3. Note that the shaft diameter, dsh, is constant in the design

optimisation. The rotor outer diameter, dr, is varied with the

stator inner diameter, di, as the airgap is kept constant.

The stator of the RSM under investigation consists of a

conventional cylindrical structure. A cross section of a quarter

of the stator of the distributed 3-phase, four-pole RSM is

shown in Figure 4. The important dimensions of the stator

to be optimised shown in this figure are the tooth width,

tw, stator yoke height, s

yh, and stator inner diameter, d

i. The

investigation is extended to 5-phase RSMs with two stator

winding configurations namely, a distributed winding as

shown in Figure 5 and a concentrated winding as shown in

Figure 6. All the different RSMs are optimally designed in

multi-dimensions by using an optimisation algorithm together

with magnetostatic finite-element (FE) field solutions.

Figure 2: Structure (quarter section) of the 4-pole CR-rotor

Figure 3: Structure (quarter section) of the 4-pole FBR-rotor

Figure 4: Stator configuration (quarter section) of the 3-phase RSM

Figure 5: Stator configuration (quarter section) of the 5-phase

RSM with distributed winding

Figure 6: Stator configuration (quarter section) of the 5-phase

RSM with concentrated winding


In the sections that follow, the torque equations, finite

element analysis and design optimisation are briefly

described. This is then followed by a presentation of the

optimum dimensions and performance results of the

different RSMs. Finally, the measured and calculated

torque results of the 3-phase RSM with the FBR-rotor of

Figure 2 are compared.

2. TORQUE EQUATIONS

From literature [10, 11, 13] the electromagnetic torque

equations of the different RSMs are expressed as follows:

(i) 3-phase RSM:

(1)

(ii) 5-phase RSM:

(2)

(iii) 5-phase plus third harmonic current RSM:

(3)

In these equations Ldqm

are the magnetising inductances of

the d- and q-axis respectively. Subscripts 1 and 3 indicate

fundamental and third harmonic components respectively.

Comparing equations (2) and (3) shows that equation (3)

contains three components of torque: a torque component

due to fundamental current components that is similar to

equations (1) and (2), a torque component produced by

the coupling (Lm13

) between the fundamental and the third

harmonics and finally a torque component due to third

harmonic currents only. In this paper an important aspect

is the evaluation of the torque performance of the 5-phase

RSM using third harmonic current injection.

3. FINITE ELEMENT ANALYSIS

The FE package used in the study is a 2D non-commercial

package originally developed by the University of

Cambridge (UK). Since a current regulated PWM VSI is

assumed to provide the current waveforms, the stator phase

currents are known and can be used in the FE program to

calculate the torque of the RSM. The stator phase currents

of the 3-phase, 5-phase and 5-phase plus third harmonic

current RSMs are given by equations (4) – (6) respectively

as follows:

In these equations is the rotor position in electrical

degrees measured from the phase a magnetic axis to the

rotor’s q-axis, and is the current angle defined in Figure

7 in terms of dq currents. Note that the third harmonic

current component is 33% of the total current for optimum

performance [9 – 12].

Figure 7: dq-current phasor diagram


To be able to compare the machines under the same copper

losses, Pcu

, the maximum (peak) current Im in equations (4)

- (6) must be expressed as a function of copper losses for

each case. Equations (7) – (9) give these functions for the

different RSMs as follows:

s

cuphase)m(

R

PI

3

23 =

(7)

s

cuphase)m(

R

PI

5

25 =

(8)

s

curd)phasem(

R

P.I

5

8135 =+

(9)

In these equations Rs is the per phase resistance of the stator

winding that is calculated from the given slot dimensions of

the stator. With the rotor position, , and the copper losses,

Pcu

, known as constants in the design optimisation, and the

current angle, , known as a variable to be optimised, the

phase currents can be calculated according equations (4) –

(6). Knowing the phase currents and the dimensions of the

stator and rotor given in Figures 2 – 4, an FE field solution

can be obtained for the machine. From this field solution

the torque of the machine, amongst other things, can be

calculated. The FE analysis takes into account the effects

of saturation, cross magnetisation and skew as explained

by [6, 7].

4. DESIGN OPTIMISATION

The optimisation procedure for optimising the design of

the RSMs is explained by the flow diagram of Figure 8

[7]. Using the FE solution the optimisation algorithm

(Powell’s method [14]), assigns values to the machine

variables, X, that maximises the performance, Y, of the

machine. Note that with each iteration r, the optimisation

algorithm determines directions of search in a multi

dimensional space along which Y (torque) is maximised.

The optimisation algorithm is linked with the FE program

as such, each time the optimisation algorithm requires the

output Y for a given input X, it calls the FE program. The

FE program then generates a new mesh according to the

changed inputs (it must be mentioned that the meshes were

checked for ill-shaped triangles). The program then does

the pre-processing and the nonlinear solution to determine

the magnetic vector potentials. From this the FE program

calculates the flux linkage and torque. The FE program

can be called more than once by the algorithm. At the end

of each iteration, a test is carried out to determine if the

maximum is reached; if not a next iteration is executed.

The objective function is to maximise, without any

constraints, the torque per copper losses of the machine;

thus an unconstrained optimisation is used. Note that the

effect of iron losses was not taken into account in the

optimisation as the accurate calculation of these losses needs

further in-depth study, especially for machines with third

harmonic currents. Note that the average (fundamental)

torque is maximised in the design optimisation. Re-

running the optimisation with different initial values and

checking if it still converge to the same optimum values,

verified the design optimisation. There was no problem

experienced with local maxima in the optimisation process

as the objective function versus the optimisation variables

gives smooth curves with clear maximas. Furthermore, the

design optimisations of the three phase machines were done

by using the Powell and the quasi-Newton optimisation

methods. The results were found to be almost the same.

The machines are optimised at 2.5 per unit load, which

corresponds to a copper loss of 13.5 kW for the machines

investigated. The reason for the latter is that machines that

are optimised at 2.5 p.u. load were found to perform better

overall (from no-load to 2.5 p.u. load) than machines that

are optimised at 1.0 p.u. load.

Figure 8: Optimisation procedure using FE method.

4.1 Skew

The design optimisation is done for skewed machines

only. The effect of skew on the torque of the machine

is accounted for in the 2D FE analysis by dividing the

machine axially into five sub-machines as shown in Figure

9. The sub-machines are displaced by the angle and the

skewing is done over a slot-pitch angle . The torque of

the skewed machine is calculated by taking the average

of the torques of the sub-machines; this requires five FE

field solutions [7].


Figure 9: Representation of a skewed machine.

4.2 Variables to be optimised

The machine parameters that are kept constant in the design

optimisation are given in Table 1.

Table 1: Machine constants

Peak copper losses (kW) 13.5

Rated copper losses (kW) 2

Rated speed (r/min) 1500

Number of poles 4

Stack length (mm) 175

Stator outer diameter (mm) 340

For the RSMs with the CR-rotor, the following parameters

are optimised (see Figures 2, 4 and 5): tooth width, tw,

stator yoke height, syh

, stator inner diameter, di, rotor cut-

out depth, rc, rotor cut-out angle, , and the current angle,

. For the RSMs with the FBR-rotor nine variables are

optimised as follows (see Figures 3 - 5): four rotor barrier

widths, b1, b

2, b

3, b

4, rotor cut-out depth, r

c and the same

stator variables as for the RSMs with the CR-rotor namely

tw, s

yh, d

i and .

4.3 Optimisation results

The results of the optimum dimensions of the different

RSMs are given in Table 2. Note from Table 2 that for the

3-phase RSMs only the optimum dimensions are given of

the machines with distributed stator windings. For the 5-

phase RSMs the optimum dimensions are given for both

the distributed and concentrated stator winding machines.

Noticeable from Table 2 is the difference in di, and s

yh

between the FBR- and CR-rotor RSMs. For the CR-rotor

the cross coupling effect in the machine is severe, which

explains the smaller current angle and the narrow pole

arc . The narrow pole arc implies less flux in the machine,

which explains the small yoke height, syh

, as compared to

the yoke height of the FBR-rotor RSM. By maximising the

torque in the design optimisation the latter drop in flux is

compensated for by increasing the current of the machine;

the current is increased by increasing the slot copper area

by reducing the stator inner diameter di – this explains the

lower value for di of the CR-rotor RSM.

Table 2: Optimisation results (all in mm and deg.)

Optimised 3-phase RSM (distributed)

FBR-rotor CR-rotor

b1

5.8 16.3°

b2

6.5 rc

24.8

b3

4.8 tw

6.7

b4

4.1 syh

33.5

rc

2.9 di

192.0

tw

7.6 54.9°

syh

36.2

di

199.0

68.0°

Optimised 5-phase RSM (distributed | concentrated)

FBR-rotor CR-rotor

b1

4.4 | 4.9 15.9° | 15.7°

b2

7.0 | 6.2 rc

25.1 | 25.3

b3

6.1 | 5.7 tw

8.1 | 17.1

b4

4.1 | 5.0 syh

33.5 | 32.8

rc

2.2 | 1.7 di

193.5 | 194.5

tw

9.4 | 18.8 55.3° | 54.8°

syh

36.4 | 36.7

di

199.9 | 200.1

67.5° | 61.2°

Optimised 5-phase RSM + 3rd harmonic current

FBR-rotor CR-rotor

b1

5.3 | 5.1 15.6° | 15.7°

b2

6.1 | 5.1 rc

25.1 | 25.3

b3

5.0 | 6.0 tw

8.2 | 17.1

b4

5.5 | 5.9 syh

32.9 | 32.8

rc

4.2 | 3.0 di

194.3 | 194.5

tw

9.3 | 18.9 56.7° | 54.8°

syh

36.1 | 36.3

di

199.6 | 200.4

69.0° | 61.9°

The FE calculated torque performances of the different

RSMs are shown in Figures 10 and 11. Note that the torque

profiles are those of skewed machines. A summary of the

average torque performances of the machines is given in

Tables 3 and 4. Table 3 gives the rated torques (at 2 kW

copper losses) of the different optimum designed RSMs

and Table 4 gives the torque of the machines at 2.5 p.u.

load. Comparing the 3- and 5-phase RSMs with sinusoidal

current and with the same rotor structure it can be seen

that the 5-phase machines develop slightly higher torques

(3% in the case of the FBR-RSMs and 9% in the case of

the CR-RSMs). Adding a 3rd harmonic current in the case

of the 5-phase RSMs causes a further 4% improvement

in torque (in total thus 13%) in the case of the CR-RSM,

however, a drop of more than 3% in the case of the FBR-


RSM. The most important result from Table 3, however,

is the performance of the 3-phase FBR-RSM that still

outperforms the 5-phase-plus-3rd-harmonic-current CR-

RSM by almost 28%.

5. SOME MEASURED RESULTS

The optimised 3-phase RSM with the FBR-rotor of Figure 3

has been built with the rotor unskewed. The machine is fed

by a 3-phase PWM VSI. The current and speed control of the

drive system are done by means of a fixed-point digital signal

processor (DSP). Tests were conducted to determine the torque

of the machine at full load as a function of current angle. The

measured as well as FE-calculated results are shown in Figure

12. Hence, there is fairly good agreement between measured

and calculated results; the measured optimum current angle is

slightly higher than calculated due to the magnetostatic field

solutions that do not take into account the damping effect of

the eddy currents. It can also be observed that the measured

optimum current angle corresponds to the calculated optimum

current angle and also the optimum current angle through

design optimisation (Table 2).

Table 3: Torque in Nm of optimum designed RSMs at 1.0 p.u.

load (2 kW copper losses).

FBR-rotor CR-rotor

Winding Distr. Conc. Distr. Conc.

3-Phase RSM 303 - 209 -

5-phase RSM 313 294 228 221

5-Phase RSM

+ 3rd harmonic302 276 237 237

Table 4: Torque in Nm of optimum designed RSMs at 2.5 p.u.

load (13.5 kW copper losses).

FBR-rotor CR-rotor

Winding Distr. Conc. Distr. Conc.

3-Phase RSM 770 - 530 -

5-phase RSM 787 731 544 544

5-Phase RSM

+ 3rd harmonic757 691 568 574

Figure 12: Calculated and measured results of torque versus

current angle of the 3-phase RSM with the FBR-rotor.

6. CONCLUSIONS

From the finite element calculated results the following

conclusions are drawn on the torque performances of the

different RSMs for the same copper losses and the same

stack volume:

(i) With the same rotor structure the 5-phase RSMs develop

slightly more torque than the 3-phase RSMs. The 5-phase

RSMs also generate a much higher torque per rms phase

current.

(ii) Under sinusoidal 3- and 5-phase conditions the RSMs

with the internal flux barrier reluctance rotor develop more

than 33% higher torque as the RSMs with the cut-out

reluctance rotor.

(iii) The five-phase RSM with the injection of third

harmonic currents and using the cut-out reluctance rotor

results in only 4% increase (not 10% as found by previous

researchers [9, 11]) in torque as compared to the same

machine with sinusoidal currents. However, with the

internal flux barrier rotor the torque decreases by 4%. Figure 11: Torque profile of 5-phase RSMs w/o 3rd harmonic

current with different rotor structures

Figure 10: Torque profile of 3-phase RSMs with different rotor

structures


The increase in torque with 3rd harmonic current injection

depends, thus, on the type of rotor structure used. The latter

is also found by [10].

(iv) Even though there is an increase in torque due to the

injection of the 3rd harmonic current in the case of the 5-

phase RSM with the cut-out reluctance rotor, the 3-phase

RSM with the internal flux barrier rotor still outperforms

this machine by almost 28%.

7. REFERENCES

[1] Kostko J.K.:”Polyphase reaction synchronous motors”,

journal AIEE, vol. 42, pp 1162-1168, 1923.

[2] Brinkman J: “Theoretishe und experimentelle

untersuchungen an einem motor mit verbesserter

ausnuzung des reaktionsprinzips”, Dissertation,

Fakultät für Machinenwesen der Technischen

Hochschule Carolo-Wilhelmina, Braunschweig, Jan.

1965.

[3] Honsinger V.B.: “The inductances Ld and L

q of

reluctance machines”, IEEE Trans. PAS, vol. 90, no.

1, Jan. 1971.

[4] El-Antably A. and Hudson T.L.: “The design and

steady-state performance of a high efficiency

reluctance motor”, IEEE – IAS Annual meeting, pp.

770 – 776, 1985.

[5] Matsuo T. and Lipo T.A. : “Rotor design optimisation

of synchronous reluctance machine”, IEEE Trans. on

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1994.

[6] Kamper M.J. and Volschenk A.F.: “Effect of rotor

dimensions and cross magnetisation on Ld and L

q

inductances of reluctance synchronous machine with

cageless flux barrier rotor”, IEE Proc. – Electr. Power

Appl., vol. 141, no. 4, pp. 213-220, July 1994.

[7] Kamper M.J, Van der Merwe and Williamson S,

“Direct Finite Element design optimisation of the

cageless Reluctance Synchronous machine”, IEEE

Trans on Energy conversion Vol. 11 No. 3 Sept. 1996.

[8] Vagati A, Canova A, Chiamphi M, Pastrorelli M

and Repetto M.: “Design refinement of synchronous

reluctance motors through finite element analysis”,

IEEE Trans. on Industry Applications, vol. 36, no. 4,

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[9] Toliyat H.A, Xu L and Lipo T.A, “A five-phase

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May/June 1992.

[10] Xu L, “Rotor structure selections of non-sine five-

phase Synchronous Reluctance Machines for improved

Torque Capability”, IEEE Transactions on IAS, vol.

36, no.4, pp. 1111 – 1117, July/August 1992.

[11] Toliyat H.A, Waikar P.S. and Lipo T.A, “Analysis

and simulation of five phase Synchronous Reluctance

machines including third harmonic of airgap MMF”,

IEEE Transactions on IAS, vol. 34, no.2, pp. 332 –

339, March/April 1998.

[12] Xu L and Fu W.N, “Evaluation of 3rd Harmonic

component effects in 5-phase Synchronous Reluctance

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Method” IEEE IAS Conf. record – 2000, pp. 1 – 8.

[13] Toliyat H.A, Shi R and Xu H, “A DSP-based

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[14] Powell M.J.D.: “A fast algorithm for

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