torque ripple reduction in reluctance synchronous machines using magnetic wedges
TRANSCRIPT
Torque Ripple Reduction in Reluctance
Synchronous Machines using Magnetic
Submitted to the Department of Electrical Engineering in partial fulfilment of the
requirements for the Baccalaureus Technologiae in Electrical
SUPERVISOR: E. VOSS
Torque Ripple Reduction in Reluctance
Synchronous Machines using Magnetic
Wedges
By
Robert Albert Smith
Submitted to the Department of Electrical Engineering in partial fulfilment of the
requirements for the Baccalaureus Technologiae in Electrical Engineering at the
SUPERVISOR: E. VOSS
NOVEMBER 2010
Torque Ripple Reduction in Reluctance
Synchronous Machines using Magnetic
Submitted to the Department of Electrical Engineering in partial fulfilment of the
Engineering at the
ii
DECLARATION
I declare that Torque ripple reduction in reluctance synchronous machines using magnetic
wedges is my own, unaided work and that all the sources I have used or quoted have been
indicated and acknowledged as complete references. This thesis has not been submitted before
for any degree or examination at any other University.
Robert Albert Smith
_______________________________________________
(Signature of candidate)
Signed in Cape Town the _________ day of ________________ 2010
iii
ACKNOWLEDGEMENTS
I would like to thank those who I believe have assisted me getting to this point in my academic
career and therefore with my thesis.
My Heavenly Father
Jesus Christ, without You none of this would have been possible and I thank You for Your
guidance, support and endless love You have given me throughout these years.
Mr. Egon Voss
Thank you for your help and supervision throughout the whole research process. I appreciate all
the time and effort you put aside for me and all the assistance you have given me. You are an
inspiration to many at this institution.
My university friends
Thank you, Deon Heyns, Albertu Prins and all of my fellow BTech engineering students for your
assistance, guidance and support throughout this year.
My family
Thank you for giving me the opportunity to be educated in a tertiary institution doing what I love
most. Your support, love and belief in me made all of this possible.
iv
ABSTRACT
This paper compares different types of magnetic wedges within a reluctance synchronous
machine’s (RSM) stator to determine the effect they have on the torque ripple, as well as the
average torque produced. The torque ripple in any motor is a major disadvantage and it is,
therefore, an important task to find methods of reducing this ripple without considerably
affecting the average torque. The role of magnetic wedges in the stator slots are discussed in
detail. Various case studies were performed and their torques compared. This was completed in
linear and non-linear materials of different permeabilities. The different magnetic materials were
also combined, without changing the wedge geometry, to form double wedges in the stator. This
determined the effect that linear- and non-linear materials have on the torque ripple. From these
case studies, an optimum linear permeability was found, while from the range of non-linear
permeabilities, the lowest torque ripple was produced for a mild steel wedge. After determining
the optimum wedge materials for the specified application, the effect which these different
magnetic materials have on the motor’s torque was also determined.
Index Terms—magnetic wedges, permeability, reluctance synchronous machine, torque ripple.
v
TABLE OF CONTENTS
Declaration.................................................................................................................. ii
Acknowledgements..................................................................................................... iii
Abstract....................................................................................................................... iv
List of figures.............................................................................................................. vii
List of tables................................................................................................................ viii
List of symbols............................................................................................................ ix
List of abbreviations.................................................................................................... x
1. Introduction...................................................................................................... 1
1.1 The history of the RSM....................................................................... 1
1.2 General RSM operation....................................................................... 1
1.3 The effect of magnetic wedges............................................................ 3
2. Research statement.......................................................................................... 4
3. Methodology.................................................................................................... 5
3.1 Simulation specifications..................................................................... 5
3.2 Description of case studies................................................................... 7
3.3 Linear wedge material.......................................................................... 8
3.4 Non-linear wedge material................................................................... 9
3.5 Flux barrier materials............. ............................................................. 10
4. Results............................................................................................................. 10
4.1 Linear materials for single wedge…………….……………………… 10
4.2 Non-linear materials for single wedge……………………………….. 11
vi
4.3 Combined linear- and non-linear materials for double wedge………. 13
4.4 Flux barrier permeabilities.................................................................. 14
5. Analysis of results…………………………………………………………… 15
5.1 Linear materials for single wedge…………….……………………….… 15
5.2 Non-linear materials for single wedge………………….…………..…… 16
5.3 Combined linear- and non-linear materials for double wedge………….. 16
5.4 Flux barrier permeabilities........................................................................ 16
6 Conclusion…………………………………………………………..……… 17
7 References……………………………………………………………..…….. 18
Appendix A: Geometry of RSM…………………………………………….….. 20
Appendix B: Torque results for linear materials……………………………….. 21
Appendix C: Torque results for non-linear materials………………………...… 22
vii
LIST OF FIGURES
Figure 1.2-1 A quarter of the RSM representing its d- and q-axes……………..... 2
Figure 1.2-2 Example of torque ripple without wedges………………………….. 3
Figure 1.3-1 The effect on the flux density of a stator with (a) and without (b)
magnetic wedges………………………………………….………… 4
Figure 3.1-1 Calculation path inside the air-gap along contour Γ……….……….. 6
Figure 3.1-2 Procedure for acquiring RSM torque values………………………… 6
Figure 3.2-1 RSM with a single wedge…………………………………………… 7
Figure 3.2-2 RSM with a double wedge…………………..………………………. 8
Figure 3.4-1 Maximum relative permeability……………………………..……… 9
Figure 4.1-1 The torque comparison for a range of linear magnetic wedges…….. 10
Figure 4.1-2 The torque graph for a linear magnetic wedge compared to the
absence of a magnetic wedge……………………………………….. 11
Figure 4.2-1 The torque comparison for a range of non-linear magnetic
wedges……………………………………………………………..... 12
Figure 4.2-2 The torque graph for a non-linear magnetic wedge compared
to the absence of a magnetic wedge………………………………… 12
Figure 4.4-1 Average torque as a function of the relative permeability of the
flux barriers.………………………………………………………… 14
viii
LIST OF TABLES
Table 4.2-1 List of non-linear materials to be simulated…………………….…… 11
Table 4.3-1 Combined wedge torque values……………………………………… 13
Table 5-1 Summary of findings for lowest torque ripple………………………. 14
ix
LIST OF SYMBOLS
Γ – Gamma (contour) [m]
�max – Maximum relative permeability
�0 – Permeability of vacuum [Vs/Am]
�r – Relative permeability
π – Pi
Bt – Magnetic flux density (tangential component) [Vs/m2]
Br – Magnetic flux density (radial component) [Vs/m2]
B – Magnetic flux density [Vs/m2]
F – Force [N]
H – Magnetic field strength [A/m]
I – Current [A]
ℓs – Stack length [m]
S – Surface contour [m2]
TAVG – Average torque [Nm]
TMAX – Maximum torque [Nm]
TMIN – Minimum torque [Nm]
Tm – Torque (Maxwell stress tensor) [Nm]
%TRIPPLE – Percentage torque ripple [%]
V – Voltage [V]
x
LIST OF ABBREVIATIONS
d – Direct
FEA – Finite element analysis
RSM – Reluctance synchronous machine
Rms – Root mean square
q – Quadrature
1
1 INTRODUCTION
1.1 The history of the RSM
Although the theory of magnetic reluctance has been known for more than 150 years, research on
the RSM has been conducted only since the early days of the 20th
century, with the researchers’
attention mainly focused on the geometry and structure of the rotor. At first, these findings were
only theoretical, but as technology advanced, more computer-based analysis methods were made
possible, giving the engineer fewer restrictions to experiment with the RSM’s parameters. A
large percentage of the research performed on the RSM covered the effect of changes in the rotor
geometry which included the analysis of inductances, magnetic fields, high anisotropy, high
torque-per-volume values, and permeance harmonics. Research on torque harmonics, however,
has not yet received adequate attention and this is the reason for this paper. [1, 2]
1.2 General RSM operation
As the name implies, the RSM operates on the principle of magnetic reluctance. This reluctance,
which is the reciprocal of permeability, is similar to electric resistance in the sense the magnetic
flux always tends to flow through the path of least reluctance [3]. The RSM rotor consists of
magnetic steel laminations, with no conducting windings, and is designed in such a manner that
the inductance varies as it rotates [4]. This is caused by its internal flux barriers and external cut-
outs and is what defines the rotor’s d- and q-axes. These axes (Figure 1.2-1) guide the magnetic
flux, produced by the current in the stator windings, and cause the rotor to move in the direction
of the stator phase sequence which finally reaches the speed synchronous to that of the stator
frequency. Consequently, this magnetic flux, that in effect “pulls” the rotor along the stator,
produces the RSM’s torque. Figure 1.2-1 also illustrates how the flux lines tend to follow the
path of least reluctance which is in effect through the d-axis. The key below the RSM shows the
range of the flux density throughout the stator and rotor. From this it can be seen that the flux
density is the largest around phase B’s winding slots and is distributed along the RSM, around
the flux barriers, and through the stators d-axis.
2
Figure 1.2-1: A quarter of the RSM representing its d- and q-axes
Although the RSM has the advantage not to produce any copper loss in the rotor, it also has the
disadvantage of producing an inherently high torque ripple [4]. This torque ripple (Figure 1.2-2)
is defined as the “deviation of the maximum and minimum torque from the average value” [5]
and is given by the equation [2]:
%���� ��������������
� � 100% (1)
For equation (1) TMAX, TMIN, and TAVG is the maximum-, minimum-, and average torque
respectively. This torque ripple is caused by the combination of the rotor flux barriers and slots
in the stator which produce a “non-sinusoidal air-gap permeance variation”. Because this ripple
in the torque means an uneven pull, vibrations, and acoustic noise it is desirable to reduce it to a
minimum without reducing the average torque considerably [2], [4], [5].
Phase B
3
Figure 1.2-2: Example of torque ripple without wedges
There have been numerous contributions towards the reduction of the torque ripple, for instance
skewing of the rotor, or changing the flux barrier geometry within the rotor [1], [2], [4-8].
Although electronic drives have been developed, which are of tremendous help, the electrical
engineer still tries to do research on the roots of the problem.
1.3 The effect of magnetic wedges
Similar to the stator rotor, the wedges are made of a magnetic material which makes the air-gap
magnetically smooth. The absence of these magnetic wedges causes the radial force between the
rotor and stator to constantly change as the rotor turns, which in effect produces a ripple in the
torque [1]. A wedge, however, would reduce the average torque, but also produces a smoother
output torque. The mutual flux through the air-gap is now evenly distributed and hence reduces
the torque ripple. The challenge here is to find the optimum material which would distribute
enough magnetic flux to the rotor to maintain the average torque, whilst reducing the variation in
the d- and q-axes inductances. Figure 1.3-1 (a) and (b) illustrate the difference a magnetic wedge
makes to the flux distribution in the slotted region. It can also be seen that the flux density
between the tooth region and the rotor becomes less for the magnetic wedge’s presence. The low
average torque that these wedges create is a result of this reduced flux density.
40
42
44
46
48
50
52
54
56
58
0 30 60 90 120 150 180
Torq
ue (N
m)
Angle (elec°)
Ripple Torque
Avg Torque
Figure 1.3-1: The flux density
For Figure 1.3-1 (a) it can be seen that a certain region in the air
highlighted region covers five flux lines which are transferred from the stator teeth to
The same size region is highlighted in Figure 1.3
eight flux lines is found. This example clearly indicates how the magnetic wedges reduce the
RSM’s average torque.
2 RESEARCH STATEMENT
There have been numerous papers [1
within the stator slots have on the torque ripple. In
method for the reduction of the torque ripple by inserting magnetic wedges within the stator slots
whilst keeping the stator and rotor geometr
“magnetically smooth” which should theoretically produce sinusoidal flux linkages between the
stator and the rotor [5].
Experiments with several different types of linear and non
simulated to cover a variety of permeabilities. These materials will be simulated for single
wedges as well as combined double wedges with different permeabilities.
a
Wedge
4
The flux density in the air-gap with (a) and without (b) magnetic wedges
1 (a) it can be seen that a certain region in the air-gap is highlighted. This
highlighted region covers five flux lines which are transferred from the stator teeth to
The same size region is highlighted in Figure 1.3-1 (b), but in this case, an increased value of
found. This example clearly indicates how the magnetic wedges reduce the
TATEMENT
There have been numerous papers [1], [4], [5] dealing with the effect which magnetic wedges
within the stator slots have on the torque ripple. In this paper, however, the author
method for the reduction of the torque ripple by inserting magnetic wedges within the stator slots
whilst keeping the stator and rotor geometries in its initial form. These wedges make the air
“magnetically smooth” which should theoretically produce sinusoidal flux linkages between the
xperiments with several different types of linear and non-linear magnetic materi
of permeabilities. These materials will be simulated for single
wedges as well as combined double wedges with different permeabilities. The effect of a change
b
Air-gap with more flux lines
Wedge
Air-gap with
less flux lines
No wedge
with (a) and without (b) magnetic wedges
gap is highlighted. This
highlighted region covers five flux lines which are transferred from the stator teeth to the rotor.
n this case, an increased value of
found. This example clearly indicates how the magnetic wedges reduce the
magnetic wedges
this paper, however, the author proposes a
method for the reduction of the torque ripple by inserting magnetic wedges within the stator slots
ies in its initial form. These wedges make the air-gap
“magnetically smooth” which should theoretically produce sinusoidal flux linkages between the
linear magnetic materials will be
of permeabilities. These materials will be simulated for single
The effect of a change
gap with ore flux lines
5
in the flux barrier permeability will also be researched to determine the effect it has on the
average torque.
3 METHODOLOGY
3.1. Simulation specifications
A 3-phase, 4-pole, 380V RSM with single layer windings will be simulated at a rms current of
16A and modelled using a Finite Element Analysis (FEA) software. The RSM is also simulated
as a two-dimensional model with the assumption that any fringing effects as well as end leakages
are neglected.
For optimum torque ripple results it was proposed [1] that the RSM rotor has an angle of 26°
between adjacent flux barriers and that the current angle be simulated at 50°. Appendix A gives a
description of the RSM’s specifications and geometries.
FEA produces 180 instantaneous torque values over a range of 180°el, which is equivalent to
90°mech (Figure 1.2-1). FEA calculates these individual torques using Maxwell’s stress tensor
which is derived from the resultant stress tensor σs, for a closed surface, S [1].
� !" ·" $% (3)
This equation is then simplified to a line integral along the closed contour Γ in the centre of the
air-gap (4), because only a two-dimensional slice of the RSM is simulated and then multiplied
with the effective stack length of the rotor [1].
�& '()
*+ ,-,.$Γ / ℓ" (4)
Where μ0 is the permeability of vacuum, Γ is the contour length of the defined path at which each
individual torque is calculated, r is the radius from the centre of the shaft to the contour Γ, Br and
Bt are the radial and tangential components respectively and ℓs the effective stack length of the
rotor [1]. Figure 3.1-1 illustrates the refined mesh description along the air-gap along with the
defined path. This path runs through the centre of the air-gap.
Figure 3.1-1: Calculation path inside the air
These instantaneous torque values are used to calculate the average torque and also to find the
maximum and minimum torque values.
procedure illustrated in Figure 3.1
Figure 3.1-2:
6
: Calculation path inside the air-gap along contour Γ
These instantaneous torque values are used to calculate the average torque and also to find the
maximum and minimum torque values. The torque ripple values are acquired using the
3.1-2.
: Procedure for acquiring RSM torque values
Refined air-gap mesh
Contour (Γ)
Γ
These instantaneous torque values are used to calculate the average torque and also to find the
acquired using the
7
The average torque, TAVG, can be calculated from the equation:
�123 ∑ �567)°58)°
9 (2)
Where Tn is the value (Nm) for each individual torque and n is the number of torque values
acquired ranging from 0°el to 180°el.
3.2. Description of case studies
The magnetic wedges are simulated with both linear- and non-linear permeabilities. Both of
these wedge materials will be simulated with single- as well as double wedges in the slots. The
double wedges will consist of a combination of both wedge materials. These combinations will
be determined by the best performing wedge, with low torque ripple as the main performance
parameter. It is important to note that the wedge height remains at 1.22 mm throughout the
various case studies. For the final case study, however, a RSM without any magnetic wedges will
be simulated with various flux barrier materials. Practically, this method can be considered as the
insertion of magnetic materials inside the flux barrier region. Figure 3.2-1 and Figure 3.2-2,
single- and double wedge respectively, show how the wedges fit into the stator slots.
8
Figure 3.2-1: RSM with a single wedge
Figure 3.2-2: RSM with a double wedge
3.3. Linear wedge material
The different linear magnetic wedges will be simulated for relative permeabilities ranging from
0.1 to 1 in steps of 0.2, followed by a range of 2 to 10 in steps of 2, after which the steps increase
from 10 to 100 in steps of 10, and finally increasing from 100 to 1000 000 by multiplication
factors of 10. This is to get an approximation of how the torque reacts to such a wide range of
linear permeabilities. The aim of this range is to find an optimum value for the linear wedges
used. The relative permeability for the wedges used in the stator can be calculated using equation
[9]
�. :(); (3)
Where B is the magnetic flux density, μ0 is the permeability of air, and H is the magnetic field
strength. The relative permeability is defined as “the ratio of the permeability of a specific
medium to the permeability of free space given by the magnetic constant μ0 = 4π*10-7
” Vs/Am
[10].
1
2
9
3.4. Non-linear wedge material
Equation (3) causes a non-linear material to have a vast range of relative permeabilities as the
values of B or H change along the B-H curve. According to McLyman [11], theses various
relative permeabilities can be categorized in more than one way. One of these methods
determines the B-H curve’s maximum relative permeability (µmax) which is defined as “the slope
of a straight line drawn from the origin tangent to the curve at its knee.” This means that the
values of both B and H are obtained at the point where the straight line is equal to the curve at
the knee-point. These values are then used to calculate the maximum relative permeability using
equation (3). Figure 3.4-1 is an example of a B-H curve and illustrates how the maximum
relative permeability of a non-linear material is obtained.
Figure 3.4-1: Maximum relative permeability [11]
The various non-linear materials were limited to the list of B-H curves supplied by FEA. From
these materials, a careful selection was used that ranged from a maximum relative permeability
of 100 to 260000.
10
3.5. Flux barrier materials
For this case study it is not necessary to find the material that produces the lowest torque ripple.
This case study is to purely determine how the average torque is affected. The insertion of the
magnetic materials inside the flux barriers will therefore only consist of linear magnetic
materials to provide a wide range of materials.
The linear materials are initially increased by relative permeable steps of 10 up to a value of 50
to illustrate the graph’s initial gradient. From 50 it is increased in larger steps to fit the average
torque values over a larger range of permeabilities.
4 RESULTS
4.1 Linear materials for single wedge
This specific range of linear materials is selected to provide a wide scope of permeabilities. The
graph below (Figure 4.1-1) shows the results for the whole range of linear materials that were
simulated. It is interesting to note how the percentage torque ripple decreases to a specific value
and then rises to a value where it saturates at a constant torque ripple.
Figure 4.1-1: The torque comparison for a range of linear magnetic wedges
4041424344454647484950
0%5%
10%15%20%25%30%35%40%45%50%
0.2
0.4
0.6
0.8 1 2 4 6 8
10
20
25
30
40
50
60
70
80
90
10
010
00
100
00
1000
00
10
00…
Aver
age Torq
ue (N
m)
Torq
ue Rip
ple (%
)
Relative Permeabilty
Linear Material Results
Average Torque
%Torque Ripple
11
Throughout the vast range of linear relative permeabilities it was found that the lowest torque
ripple occurred at a relative permeability of μr = 25. Using equation (2) it can be calculated that
this material produces a torque ripple percentage of 10.67% with an average torque of 44.38 Nm.
This is a remarkable result, as a RSM typically operates between relative permeabilities of 3 500
and 5 000. It is therefore interesting to note that the lowest torque ripple is produced for a
magnetic wedge with such a low relative permeability. For comparative reasons Figure 4.1-2
illustrates the torque ripple and average torque that this material produces as well as the torque
produced with no magnetic wedges. The average torque and torque ripple for the linear wedge is
denoted as “Avg torque 25” and “torque ripple 25” respectively. The same notation applies for
air.
Figure 4.1-2: The torque graph for a linear magnetic wedge compared to the absence of a
magnetic wedge
4.2. Non-linear materials for single wedge
Table 4.2-1 shows the different non-linear materials used as wedges for the RSM. These
materials are acquired from FEA software’s database. The method of obtaining the materials’
maximum relative permeabilities previously described is used. These values obtained are
rounded off to give an estimation of the material’s B-H curve and saturation region.
Table 4.2-1: List of non-linear materials to be simulated
40424446485052545658
0 30 60 90 120 150 180
Torq
ue (N
m)
Angle (elec°)
Torque comparison graph - Linear
Torque Ripple Air
Avg Torque Air
Torque Ripple 25
Avg Torque 25
12
Wedge Material
Maximum Relative
Permeabilty µmax
Wedge Material
Maximum
Relative
Permeabilty µmax
Unisil 27M4 100 A36 Steel 4000
Unisil 30M2H 400 M19FP 26GA 4500
Unisil Grain Orient 800 Newcor 32 4600
Unisil 23M3 1500 Losil 1000 5500
HCR Alloy 2000 EN9 Steel 6000
Cast Iron 2200 Grade 450 Steel 6200
Mild steel 2800 Alnico YCM-9B 34000
Unisil 27MOH 3500 Ferixite 260000
Rolled Steel 3800
Figure 4.2-1 shows the lowest percentage torque ripple of 10.20% was acquired for mild steel
which effectively produced an average torque of 43.41 Nm.
Figure 4.2-1: The torque comparison for a range of non-linear magnetic wedges
This low torque ripple mild steel produces is shown in figure 4.2-2. Again, the blue waveform is
the torque produced with no magnetic wedge inserted. This gives an idea of the effect the
magnetic wedges have.
40
41
42
43
44
45
46
47
48
49
50
0%
5%
10%
15%
20%
25%
10
0
40
0
80
0
15
00
20
00
22
00
28
00
35
00
38
00
40
00
45
00
46
00
55
00
60
00
62
00
34
00
0
26
00
00
Aver
age Torq
ue (N
m)
Torq
ue Rip
ple (%
)
Maximum Relative Permeabilty
Non-linear Material Results
Average Torque
%Torque Ripple
13
Figure 4.2-2: The torque graph for a non-linear magnetic wedge compared to the absence of a
magnetic wedge
4.3. Combined linear- and non-linear materials for double wedge
For the final two case studies the optimum wedges for the two previous case studies, with the
torque ripple as the main performance parameter, will be combined. It should be noted that the
total physical wedge geometry will remain constant and that each wedge material will only be
divided along the breadth of the wedge as illustrated in Figure 3.2-2. The order in which this set
of wedges is placed is also taken into account.
Table 4.3-1 shows the results for the two separate wedge combinations with wedge 1 being the
wedge furthest from the air-gap and wedge 2 closest to the air-gap as displayed in Figure 3.2-2.
Table 4.3-1: Combined wedge torque values
Wedge 1 Non-linear Linear
Wedge 2 Linear Non-linear
%TRIPPLE 10.92% 10.64%
TAVG (Nm) 43.92 43.95
40424446485052545658
0 30 60 90 120 150 180
Torq
ue (N
m)
Angle (elec°)
Torque comparison graph - Non-linear
Torque Ripple Air
Avg Torque Air
Torque Ripple Mild Steel
Avg Torque Mild Steel
14
4.4. Flux barrier permeabilities
According to the theory of magnetism, a force is created when a magnetic flux runs through two
materials having different permeabilities. In the case of the RSM, the rotor, having a relative
permeability of 4500, produces a force onto the flux barrier region, having a relative
permeability of 1. The force created in this specific case contributes to the torque produced by
the RSM.
By increasing the permeability of the flux barrier (which, in this case, would be inserting
different linear magnetic materials) the force produced between the rotor and flux barrier
decrease, which consequently decreases the overall torque. At the point where the flux barrier’s
permeability is equal to that of the rotor, the average torque would be expected to drop to 0 Nm.
As this permeability is increased even further, a force is created in the opposite direction. The
question in this case would be how this increased force between these two materials affects the
average torque produced. Figure 4.4-1 illustrates the decreasing average torque for an increase in
the flux barrier’s permeability.
Figure 4.4-1: Average torque as a function of the relative permeability of the flux barriers
-2
0
2
4
6
8
10
12
10
20
30
40
50
1 0
00
2 0
00
3 0
00
4 0
00
5 0
00
10
00
0
15
00
0
20
00
0
25
00
0
30
00
0
35
00
0
40
00
0
45
00
0
50
00
0
10
0 0
00
20
0 0
00
30
0 0
00
40
0 0
00
50
0 0
00
Avg T
orq
ue (N
m)
Relative Permeability
15
5 ANALYSIS OF RESULTS
Table 5-1 summarises the findings for the paper’s first four case studies. The materials that
produced the lowest torque ripples are documented with their respective average torque values.
For the bottom two rows, the combined wedges are categorised in the order in which each wedge
is inserted i.e. the linear & non-linear row represent the findings for the linear wedge closest to
the air-gap, while the non-linear & linear row represents the non-linear wedge closest to the air-
gap.
Table 5-1: Summary of findings for lowest torque ripple
Material type µr/µmax %TRIPPLE TAVG (Nm)
Linear 25 10.67% 44.38
Non-linear 2800 10.20% 43.41
Linear & non-linear 2800/25 10.92% 43.92
Non-linear & linear 25/2800 10.64% 43.95
5.1 Linear materials for single wedge
With the lowest torque ripple at a value of 10.67%, it was also noted that the average torque
reduced to 44.38 Nm when compared to a RSM with no magnetic wedges. This implies that
while the torque ripple was reduced by 67.1%, the average torque also reduced by 6.2%. It is
interesting to note how the torque ripple decreases up to a certain permeability (μr = 25) and then
rises to a point where the percentage ripple remains constant. It can also be seen that the average
torque is at its maximum where the relative permeability is equal to one, which is equivalent to
the absence of a magnetic wedge. As discussed in section 1.3, the reason for this is the increased
distribution of flux lines within the stator teeth.
16
5.2. Non-linear materials for single wedge
From all of the performed case studies, the non-linear wedge, Mild steel, produced the lowest
torque ripple (10.20%) and, unfortunately, the lowest average torque (43.41 Nm) as well. When
comparing these values to a RSM without any magnetic wedges, a reduction of 68.6% and 8.3%
was calculated for the torque ripple and average torque respectively.
Ferixite, the material with the highest maximum relative permeability, produced the highest
average torque of all the other materials. The interesting fact about this discovery is the
material’s ability to produce almost the exact same average torque as a RSM without any
wedges. The average torque only decreased by 0.4% while the torque ripple reduced by 29.2%.
Depending on the application, this material still produces a high percentage torque ripple
(22.97%), but is a considerable decrease for applications where high average torque is an
important design factor. These findings agreed with that of Voss’ [5] in saying that the non-linear
materials hardly affect the average torque but produces significant reductions in the torque
ripple.
It is interesting to note that these materials do not follow a specified pattern as their respective
permeabilities increase. This is because the shape of their B-H curves has different
characteristics regarding its saturation region and knee-point.
5.3. Combined linear- and non-linear materials for double wedge
The results for the combined wedges show the torque ripple to be higher and lower when
compared to a single material wedge. It is also noted that a slight difference occurs for the order
in which the wedges are placed. A linear wedge close to the air-gap produces a larger percentage
torque ripple as well as a smaller average torque than with the case of a non-linear wedge closest
to the air-gap. This implies that a non-linear wedge being placed closest to the air-gap produces
better results in terms of its torque ripple and average torque.
5.4. Flux barrier permeabilities
As expected, the average torque is almost completely reduced where the flux barrier and rotor
have equal permeabilities. However, when the flux barrier’s permeability is increased even
further it can be seen that the average torque remains at a low value of around 0 Nm. From these
17
results it is clear that, for the permeabilities higher than that of the rotor’s permeability, the force
produced between the two materials do not contribute to the torque. When a further analysis is
conducted, it can be seen that the higher permeable values in the flux barrier create a magnetic
short circuit which, in effect, draw more magnetic flux. The magnetic flux between the rotor and
stator consequently reduces which reduces the torque.
6 CONCLUSIONS
The objective of this paper was to determine the effects magnetic wedges have on the torque
ripple. Along with previous papers, these results confirmed a definite reduction in the torque
ripple. It was also clear that the non-linear magnetic material provided to be more effective in
terms of reducing the torque ripple. From these results different conclusions can be drawn
depending on the RSM designer’s application. A RSM which is dependent on its torque but
needs to reduce the ripple torque would ideally need to employ Fexirite wedges which proved to
keep the average torque constant. A RSM which needs to operate on a more constant torque for
applications that require precision would possibly be employed with a Mild steel wedge. Finally
it can be concluded that the insertion of magnetic wedges seems essential and should be
considered in most RSM designs.
Although certain alterations were made on the materials of the flux barriers, the main objective
of this paper was to determine the effects magnetic wedges have on the torque ripple. Finally, it
can be concluded that a torque is only produced when the permeability of the flux barrier is less
than the rotor’s permeability.
18
7 REFERENCES
[1] A.N. Hanekom, “A torque ripple analysis on reluctance synchronous machines” master’s
thesis, Dept. of Electrical Engineering, Cape Peninsula University of Technology, Cape Town,
South Africa, September 2006
[2] A. Vagati, M. Pastorelli, G. Franceschini, and S. C. Petrache, “Design of Low-Torque-
Ripple Synchronous Reluctance Motors,” in IEEE Transactions on Inustry Applications, 1998,
pp. 758-764.
[3] T.R. Kuphaldt, “Reluctance motor,” 2003,
http://www.allaboutcircuits.com/vol_2/chpt_13/4.html
[4] E. Chiricozzi, G. Conti, F. Parasiliti, M. Villani, “Design solutions to optimize torque
ripple in synchronous reluctance motors” in Proceedings in International Conference on
Electrical Machines, Vigo, Spain, 1996, pp 148-153.
[5] E. Voss and M. Kamper, “On the use of magnetic wedges in synchronous reluctance
machines” in Southern African Universities Power Engineering Conference, University of
Stellenbosch, January 2004
[6] T. Ishikawa and G. R. Slemon, “A Method of Reducing Ripple Torque in Permanent
Magnet Motors without Skewing,” in IEEE Transactions on Magnetics, 1993, pp. 2028-2031.
[7] N. Bianchi, S. Bolognani, D. Bon and M. Dai Pré, "Rotor Flux-Barrier Desing for Torque
Ripple Reduction in Synchronous Reluctance and PM-Assisted Synhronous Motors," in IEEE
Transactions on Inustry Applications, 2009, pp. 921-928.
[8] J. W. Lee, H. S. Kim, B. I. Kwon and T. Kim, “New Rotor Shape Design for Minimum
Torque Ripple of SRM using FEM,” in IEEE Transactions on Magnetics, 2004, pp. 754-757.
[9] E. Voss, The Voss lectures on electrical machine design, 2nd
ed., Cape Peninsula
University of Technology, 2010
19
[10] Anon, “Relative Permeability,” July 2010;
http://en.wikipedia.org/wiki/Relative_permeability
[11] T. McLyman, Transformer and inductor design handbook, 2nd
ed., California Institute of
Technology, New York: Marcel Dekker, Inc., 1988
20
Appendix A: Geometry of RSM
aw = air-gap width (0.34mm)
ba = barrier angle (54.34°)
bh = barrier height (38.65mm)
bp = barrier pitch (11.9°)
ca = cut-out angle (51.26°)
ch = cut-out height (49.15mm)
cp = cut-out pitch (23.5°)
db = diagonal barrier length
(21.0mm)
dw = diagonal barrier width
(5.76mm)
hb = horizontal barrier lenght
(25.38mm)
hw = horizontal barrier width
(7.49mm)
rr = radius of rotor (63.15mm)
rs = radius of stator (101.6mm)
sh = slot height (19.913mm)
sp = slot pitch (10°)
sw = slot width (5.46°)
tw = tooth widht (4.54°)
wh = wedge height (1.22mm)
yh = yoke height (18.197mm)
21
Appendix B: Torque results for linear materials
Relative Permeability Tmax (Nm) Tmin (Nm) Tripple (%) Tavg (Nm)
0.1 57.21 35.83 46.04% 46.44
0.2 57.22 36.65 44.01% 46.75
0.3 57.07 37.82 41.01% 46.93
0.4 57.06 38.61 39.21% 47.04
0.5 56.84 39.35 37.11% 47.14
0.6 56.75 39.69 36.13% 47.21
0.7 56.61 40.07 35.00% 47.26
0.8 56.49 40.35 34.12% 47.28
0.9 56.33 40.60 33.24% 47.31
1 56.17 40.81 32.45% 47.33
2 54.91 42.03 27.25% 47.28
3 53.99 42.49 24.46% 47.02
4 53.08 42.84 21.88% 46.84
5 52.28 43.02 19.84% 46.65
6 51.58 43.13 18.17% 46.48
7 50.98 43.19 16.81% 46.31
8 50.45 43.17 15.78% 46.15
9 49.99 43.12 14.93% 46.00
10 49.58 43.07 14.19% 45.86
20 47.08 42.10 11.13% 44.76
30 46.34 41.37 11.29% 44.06
40 46.17 40.57 12.86% 43.55
50 46.09 39.97 14.19% 43.17
60 46.05 39.34 15.66% 42.88
70 46.05 38.72 17.18% 42.64
80 46.05 38.26 18.35% 42.45
90 46.07 37.93 19.24% 42.30
100 46.09 37.65 20.01% 42.17
1000 46.74 34.24 30.43% 41.07
10000 46.86 33.77 31.96% 40.95
100000 46.88 33.73 32.13% 40.93
1000000 46.88 33.72 32.14% 40.93
22
Appendix C: Torque results for non-linear materials
Wedge Material Tmax (Nm) Tmin (Nm) Tripple (%) Tavg (Nm)
Maximum Relative
Permeabilty
Unisil 27M4 48.84 41.49 16.69% 44.05 100
Unisil 30M2H 48.9 41.94 15.68% 44.37 400
Unisil Grain Orient 48.31 41.73 14.97% 43.95 800
Unisil 23M3 48.56 41.46 16.15% 43.96 1500
HCR Alloy 48.77 41.73 15.94% 44.14 2000
Cast Iron 52.25 43.49 18.97% 46.21 2200
Mildsteel 46.44 42.01 10.20% 43.41 2800
Unisil 27MOH 46.28 41.58 10.89% 43.16 3500
Rolled Steel 48.39 41.82 14.91% 44.08 3800
A36 Steel 48.35 41.86 14.71% 44.09 4000
M19FP 26GA 48.79 41.94 15.47% 44.30 4500
Newcor 32 48.76 41.54 16.37% 44.13 4600
Losil 1000 48.3 41.72 14.95% 44.02 5500
EN9 Steel 49.24 41.98 16.31% 44.47 6000
Grade 450 Steel 48.96 41.88 15.95% 44.34 6200
Alnico YCM-9B 52.63 43.36 19.92% 46.58 34000
Ferixite 53.77 42.94 22.97% 47.15 260000