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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
Vol.97(1) March 2006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 45
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
46 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.97(1) March 2006
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].
Vol.97(1) March 2006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 47
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-
48 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.97(1) March 2006
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
Vol.97(1) March 2006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 49
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%.
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