Design of a Permanent-Magnet
Synchronous Machine with NonOverlapping Concentrated Windings
for the Shell Eco Marathon Urban Prototype
DANIEL MARTÍNEZ
Degree project in
Electrical Engineering
Master of Science
Stockholm, Sweden 2012
XR-EE-E2C 2012:020
Design of a Permanent-Magnet Synchronous Machine
with Non-Overlapping Concentrated Windings for the
Shell Eco Marathon Urban Prototype
DANIEL MARTÍNEZ
Royal Institute of Technology
School of Electrical Engineering
Electrical Energy Conversion
Stockholm 2012.
XR-EE-E2C 2012:020
Design of a Permanent-Magnet Synchronous Machine with Non-Overlapping
Concentrated Windings for the Shell Eco Marathon Urban Prototype
DANIEL MARTÍNEZ
c DANIEL MARTÍNEZ, 2012.
School of Electrical Engineering
Department of Electrical Energy Conversion (E2C)
Kungliga Tekniska Högskolan
SE–100 44 Stockholm
Sweden
Abstract
This thesis deals with the design of a permanent-magnet synchronous inner rotor motor
for an in-wheel application for the Shell Eco Marathon Urban concept vehicle.
First of all, concepts related to permanent magnet motors are studied. Likewise, different features of permanent magnet motors are qualitatively evaluated in order to choose
the most suitable. A radial flux motor is selected based on its solid, economic and acceptable characteristics.
Next, a detailed study of concentrated windings is carried out. Through this investigation, undesirable configurations of pole and slot numbers due to unbalanced magnetic
pull or a low fundamental winding factor will be avoided and how to determine the different winding layouts for different pole and slots configuration will be explained. As well,
based on this study, and the magnetic and electric behavior of the machine, an analytical
model is created. This model calculates the optimum size and characteristics of a machine
in order to obtain lightweight design.
After that, the design of a program based on a finite element method that simulates
different situations for the machine is accomplished, dealing with the difficulties that entails the concentrated windings.
Finally, through the use of this program, the machine calculated by the analytical
model is analyzed, specially regarding that it does not surpass some important margin in
order not to be demagnetized or not to surpass the maximum phase voltage supplied by
the batteries.
Keywords: Finite element analysis, in-wheel motor, concentrated windings,
permanent-magnet synchronous machine, fundamental winding factor.
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Sammanfattning
Detta examensarbete fokuserar på designen av en permanentmagnetiserad synkronmotor
av ytterrotortyp. Motorn är avsedd att användas som en hjulmotor i ett konceptfordon
(Shell Eco Marathon Urban Concept). En motor av radialflödestyp studeras på grund av
konceptets enkelhet och acceptabla prestanda. Sedan presenteras en noggrann studie kring
koncentrerade lindningar. Pol- och spårtalskombinationer som ger upphov till obalanserade magnetiska krafter eller en låg fundamental lindningsfaktor härleds och en metod
för att bestämma lindningsutbredningen beskrivs. En analytisk modell tas sedan fram av
motorn vilken används för att minimera motorns aktiva vikt. Efter detta tas ett designprogram baserat på finita-elementmetoden fram i vilken den resulterande motordesignen
härrörande från den analytiska modelleringen utvärderas. Speciellt studeras risken för demagnetisering av permanentmagneterna samt att fasspänningen inte överstiger den gräns
som stipuleras av givna batterispänningen.
Nyckelord: Finita elementmetoden, hjulmotorer, koncentrerade lindningar,
permanentmagnetiserade synkronmotorer, fundamental lindningsfaktor.
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Acknowledgements
First of all, I would like to sincerely thank Oskar Wallmark for all his help and
guidance provided throughout these months of work. He has taught me a lot of things and
transmitted his knowledge and experience to me, making me a better engineer.
I am also thankful to Mats Leksell for his advices during the thesis and, above
all, for his willingness to help me getting this Master Thesis in a moment that was quite
difficult and critical for me.
I am also very grateful to Rosa and Mercedes, for their help and advice along these
months, and for taking care of me in difficult times. I also appreciate them for creating
such a good work atmosphere in the office, which has contributed to perform this thesis
easier.
Within my group of friends, I would like to specially mention Guillem and Elena
for all the help they have provided me with throughout the year, which has been priceless.
I would also like to thank Manuel, Miguel, Alberto, Alejandro, Navarrete, Pere, Jesus,
Lorena, Pastor, Sarah... their help and support throughout the year, especially in the moments that have been critical to me, and for providing me with so many unforgettable
moments. Finally, the most important ones... I would like to thank my mother, my father
and my sister for all the support, advice, help and patience I have received along this year
and throughout my life, which have made possible who I am right now. Thank you so
much!
Daniel Martı́nez Jiménez
Stockholm, Sweden
November 2012
vii
viii
Contents
Abstract
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Sammanfattning
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Acknowledgements
vii
Contents
ix
1 Introduction
1.1 Motivation and Objectives
1.2 Outline of Thesis . . . . .
1.3 Methodology and Material
1.4 Shell Eco Marathon 2013 .
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2 Low-Speed Direct-Drive Applications. In-Wheel PMSM
2.1 In-Wheel Motors . . . . . . . . . . . . . . . . . . . . .
2.1.1 Low-Speed and High-Speed Machines . . . . . .
2.2 PM Synchronous Machines . . . . . . . . . . . . . . . .
2.2.1 Permanent Magnets . . . . . . . . . . . . . . . .
2.2.2 Advantages and Drawbacks . . . . . . . . . . .
2.2.3 Radial Flux PM Synchronous Machine (RFPM) .
2.2.4 Axial Flux PM Synchronous Machines, (AFPM)
2.3 Summary and Conclusions . . . . . . . . . . . . . . . .
3 Shell Eco Marathon Circuit Analysis. Requirements
3.1 Track and Car Requirements . . . . . . . . . . .
3.1.1 Driving Cycle . . . . . . . . . . . . . . .
3.1.2 Cycle Optimization . . . . . . . . . . . .
3.2 Analysis of the Energy Consumption . . . . . . .
3.3 Conclusion . . . . . . . . . . . . . . . . . . . .
4 Non-Overlapping Concentrated Windings Analysis
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25
ix
Contents
4.1
4.2
4.3
4.4
4.5
4.6
Concentrated or Distributed winding . . . . . . . . . . . .
Double or Single Layer . . . . . . . . . . . . . . . . . . .
Concentrated Winding Layout . . . . . . . . . . . . . . .
Fundamental Winding Factor . . . . . . . . . . . . . . . .
4.4.1 Winding Factor Based on Cros’ Method. . . . . .
4.4.2 Winding Factor Based on Pole-slot Combinations.
Desirable Configurations of Poles and Slots . . . . . . . .
4.5.1 Unbalanced Magnetic Pull . . . . . . . . . . . . .
4.5.2 Cogging Torque and Torque Ripple . . . . . . . .
Summary and Conclusion . . . . . . . . . . . . . . . . . .
5 Analytical Design
5.1 Geometric Design . . . . . . . . . . . . . . . .
5.1.1 Magnet Sizing . . . . . . . . . . . . .
5.1.2 Iron Sizing . . . . . . . . . . . . . . .
5.2 Electric Model . . . . . . . . . . . . . . . . .
5.3 Loss Model . . . . . . . . . . . . . . . . . . .
5.3.1 Copper Losses . . . . . . . . . . . . .
5.3.2 Core Losses . . . . . . . . . . . . . . .
5.3.3 Magnet Losses . . . . . . . . . . . . .
5.4 Analytical Program . . . . . . . . . . . . . . .
5.4.1 Objective . . . . . . . . . . . . . . . .
5.4.2 Additional Equations . . . . . . . . . .
5.4.3 Design Variables . . . . . . . . . . . .
5.4.4 Constants . . . . . . . . . . . . . . . .
5.4.5 Fixed Values and Constraints of Design
5.4.6 Matlab Program . . . . . . . . . . . .
5.4.7 Results . . . . . . . . . . . . . . . . .
6 Finite Element Analysis
6.1 FEMM Model . . . . . . . .
6.1.1 Input Parameters . .
6.1.2 Machine Geometry .
6.1.3 Material Settings . .
6.1.4 Accuracy Parameters
6.1.5 Data Obtaining . . .
6.1.6 Analysis . . . . . .
6.1.7 Program Flow . . . .
6.2 Machine Analysis . . . . . .
6.2.1 Target . . . . . . . .
6.2.2 No Load . . . . . .
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Contents
6.2.3
6.2.4
6.2.5
Rated Point Conditions . . . . . . . . . . . . . . . . . . . . . . .
Overload Condition . . . . . . . . . . . . . . . . . . . . . . . . .
Different Working Points . . . . . . . . . . . . . . . . . . . . . .
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7 Conclusions and Further Work
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . .
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A Dynamic simulation
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B Analytical Motor Data
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C Fundamental Winding factor
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D Iron Loss Constants
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E List of Symbols, Subscripts and Abbreviations
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References
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xi
Contents
xii
Chapter 1
Introduction
1.1 Motivation and Objectives
It is widely recognized that permanent magnet synchronous machines (PMSMs) have
several advantages, such as a high efficiency, that make them suitable for vehicle propulsion. In fact, many automakers are implementing these systems in hybrid electric vehicles
(HEVs) and electric vehicles (EVs). In-wheel (IW) motor technology is a novel solution
for EVs. Through the use of this type of devices, gearboxes are removed from the drive
train increasing the efficiency of the propelling system. The multi-pole PMSM designed
in this thesis uses concentrated double-layer windings. This type of coil arrangements exhibits better properties compared to the traditional distributed windings, being their best
characteristic the reduction of copper inside the machine. This entails also a great reduction of the losses of the machine as well as its cost. The aim of this thesis is to design an
electric propelling system to drive the ”Shell Eco Marathon Urban Concept”. Since the
main objective of this contest is the reduction of energy consumption, the design will be
focused on the machine and the architecture exhibited above. An analytical model able to
calculate the parameters of this motor is developed, and in order to test it, a tool based on
Finite Element Method Magnetics (FEMMR ) is programmed to be able to verify different properties.
1.2 Outline of Thesis
The thesis is structured as follows:
• Chapter 2 reports a study of the possible solutions which can propel the vehicle,
showing their advantages and drawbacks. Permanent Magnets (PM) are studied
showing their relevant properties.
1
Chapter 1. Introduction
• Chapter 3 explains how the requirements of the machine are obtained, based on the
driving cycle.
• Chapter 4 focuses on the design of a PMSM with non-concentrated windings. Methods to calculate the fundamental winding factor and winding layout are exposed.
Undesirable effects on this windings are also studied.
• Chapter 5 shows the magnetic and the electric behavior that the electrical machine
experience, as well as the losses. Based on concentrated windings theory, an analytical model is developed to calculate the parameters of the machine which is the aim
of this thesis. The lightest design is selected and its election is motivated in order to
achieve a lightweight design,
• Chapter 6 introduces a program based on the FEMMR package that is developed
to analyze the machine selected in the previous chapter in order not to surpass
certain constraints of design, such as the minimum flux density in the magnets or
the maximum phase-neutral voltage.
• Chapter 7 discusses the implementation of this machine in Shell Eco Marathon
vehicle, and proposes further improvements for some elements of the program and
the machine designed.
1.3 Methodology and Material
The guideline followed during this project is quite similar to its structure. During the first
stages, information of the different type of motors and IW concepts were gathered. Next,
requirements were calculated based on the information gathered between the different
parts of the Shell Eco 2013 team through a simple simulation based on Newton dynamics. After that, an analytical model of a PMSM was carried out, choosing concentrated
windings as its coil arrangement. Fundamental winding factor based on different theories
and winding layout were the main tasks of this part of the study. Next, the design of a
program to analyze the machine designed by the analytical model was carried out. This
program was implemented using the FEMMR software. This task entailed several challenges, such as the determination of the starting angle of the rotor, for the electrical period.
Since FEMMR calculates properties on static simulations, the program developed had to
analyze the different characteristics of the machine as well as in a whole electric period,
in all operating points of the machine. In order to validate the results obtained by this
Finite Element tool, they were checked with E METORR . To conclude, the machine was
analyzed based on the analytical model data calculated and different information gathered
in the literature review.
2
1.4. Shell Eco Marathon 2013
1.4 Shell Eco Marathon 2013
Shell Eco Marathon is a student approached competition which challenge them to design
and build a vehicle which aims to be as energy-efficient as possible. This contest takes
place in three annual events in different continents (America, Asia and Europe). In such
events, young engineers have the opportunity of showing their technical skills to push the
boundaries of fuel efficiency as well as other interesting issues as developing the competition vehicle. For instance, some of the vehicles are able to drive a similar distance as from
Paris to Moscow (2485 km) consuming only a single liter of fuel [24]. The origin of this
Fig. 1.1 Rotterdam Ahoy Urban Circuit [24].
competition was a friendly bet between employees of Shell Oil Company in 1939. The
wager consisted of who could travel furthest using the same quantity of fuel. Since then,
it has been in constant evolution, expanding the contest to other propelling technologies
such as bio-fuel, electric vehicles, hybrids, solar vehicles, etc [24].
The Shell Eco Marathon 2013 circuit will take place in Rotterdam, which is illustrated in Fig. 1.1. The Ahoy urban track will challenge the different teams with five 90o
corners, so driving skills such as braking, accelerating and cornering will be critical factors in order to succeed in the competition. Thus, it is essential that all members in the
vehicle development will be involved when creating the vehicle [24].
The aim of ”Shell Eco Marathon 2013” is the design of an EV. This thesis is focused on the propulsion system of the mentioned EV. In order to save as much energy as
possible, a novel approach is taken. The designed motor will be located inside the wheel
3
Chapter 1. Introduction
(motor hub), enabling interesting advantages, such as the removal of the gearbox in the
drive train.
4
Chapter 2
Low-Speed Direct-Drive Applications.
In-Wheel PMSM
The following chapter presents the most suitable alternatives to propel the Shell Eco
Marathon vehicle. Several PMSM topologies are put forward, setting out their advantages and drawbacks. Finally, based on the literature exposed and concerning designing
constraints, a configuration is selected.
2.1 In-Wheel Motors
Nowadays, vehicles that use electric drive trains can locate the motor in different places
inside the car. A novel and original trend is to locate it inside the wheel, where the transmission path between the electric machine and the wheel can be reduced and even eliminated, avoiding a big source of losses. The fact that determines whether the transmission
is reduced or eliminated is the speed of the machine attached to the rim [6]. Furthermore,
the fact of bedding electric machines inside the hub gives the chance of obtaining better control over the driving wheels. Since IW vehicles usually have a motor hub in each
wheel, they can be controlled independently, performing the necessary amount of torque
required at real-time. This is essential for vehicles that face traction problems, such as
slippery surfaces or damps.
Nonetheless, setting an IW motor entails some problems. The unsprung mass, which
refers to all components not supported by the car’s suspension, is increased with these
types of motors, and this leads to problems with the suspension and steering, since it is
not shielded from bumps and potholes. Though there are some articles which deny this
unsprung mass problem and they conclude its analysis asserting that the loss of comfort,
safety and drive steering is imperceptible for an average driver [2].
5
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
Wheel Bearing
Rotor
Rim
Suspension
Housing
Stator
Permanent
Magnets
Fig. 2.1 Protean Hub motor exploded view [5].
2.1.1 Low-Speed and High-Speed Machines
Electrical machines are designed to perform a certain quantity of mechanical power. Nevertheless, the dimensions of the machine are not straight determined through this parameter. Mechanical power is defined as a product of the torque performed by the motor at
some rotational speed. Hence, in order to perform the required power, electrical machines
can be designed to develop a high torque at low speed or low torque at high speed. The fact
that mainly defines the size of these machines is the torque, as it is displayed in equation
2.1
T = KD 2 Lact = KVrot
(2.1)
where K is a constant which depends on the parameters of the machine, D is the diameter
of the rotor, L is the active length of the machine and Vrot is the volume of the rotor [26]
[8] [11]. Therefore, electrical machine design can be approached in two different ways
taking into account the torque performed, and consequently the working speed:
• High-speed machines. (Fig 2.2) Since their working speed is high, a fixed speed
gear becomes necessary to achieve the needed velocity in the wheel. A planetary
gear is usually used in this kind of IW motor and as all gearboxes, it is a source of
losses [6]. During the last few years, Michelin designed and presented in 2008 its
”active wheel” concept, as example of this type of technology [1].
• Low-speed machines. (Fig. 2.3) The torque that runs these machines is high, thus
they are bigger and heavier than high-speed machines. Within IW devices, the rotor
is directly coupled to the wheel rim, therefore no gearbox is required [6]. As an
example of this approach of hub motor, Siemens designed the ”eCorner Module”
[25].
6
2.2. PM Synchronous Machines
Electrical suspension
motor
Brake disk
Electrical machine.
30 kW
Suspension spring
In-wheel
active suspension
Brake Caliper
a)
b)
Fig. 2.2 a) Michelin Active Wheel [1]; b) Cross-section of high-speed IW [6].
Another key point is that most of the commercial applications require a low rotating speed
and a high torque in the load, i.e. wind mills, elevators, EVs. Thus, high speed machines
are usually coupled to a gearbox, wasting energy that could be useful. The use of direct
drives, which are low-speed electric machines, entails several advantages mainly related
to the removal of the gearbox [16] such as:
• Higher efficiency. This is the fact that justifies the implementation of this devices.
The elimination of the gearbox take out the losses due to friction.
• Reduce maintenance. Gearboxes are one of the main causes of failures of mechanic systems. Moreover, they do not need to be lubricated as regularly as the
ones with gears.
• Higher reliability.
• Reduction of noise. Gearboxes are noisy since the teeth of the gears collide at high
speed. Again, this problem is reduced with IW motors.
• Simpler design. The number of pieces on the power train is reduced.
2.2 PM Synchronous Machines
2.2.1 Permanent Magnets
There are two types of magnetic materials, soft materials and hard materials. On the one
hand, soft materials, also called ferromagnetic materials, can be easily magnetized and
7
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
PMSM
Active suspension
Active steering
Brake
Fig. 2.3 Siemens eCorner [1] and low-speed hub motor scheme [6].
demagnetized and they are used to facilitate the magnetizing guidance. Laminated steel
is an example. On the other hand, hard materials are magnetized and demagnetized with
difficulty. They are characterized by a wide hysteresis loop [7] [19]. Regarding the microscopic structure of hard materials, they are characterized by the anisotropy of the material
and the shape of the crystals they are composed of, which impedes the orientation of its
Weiss domains, once they are oriented. This is the reason, from a microscopic point of
view, for a high coercivity Hci and a high magnetic hardness [11]. Hence, they can be
magnetized and produce magnetic field in the air gap without the use of excitation winding. Permanent magnets (PMs) are magnetized in quadrant I (or III) of the B-H hysteresis
loop (Fig. 2.4) and they operate in quadrant II (or IV). Data sheets that manufacturing
companies normally provide only depict the II quadrant. Within Fig. 2.4, the main parameters which characterize a PM are described [7] [11].
• Saturation magnetic flux density, Bsat and Hsat . All the magnetic moments of
the PM domains are aligned in the direction of the external magnetic field applied.
• Remanent magnetic flux density or remanence, Br . It is the magnetic flux density
when zero magnetic field intensity is applied, H = 0. The higher remanence, the
higher magnetic flux density in the air gap can support the magnet.
• Coercive field strength or coercivity, Hc . This is the value of the magnetic field
applied to bring the magnetic flux density to zero. The higher this value, the thinner
magnets can be implemented to withstand the same demagnetizing field.
• Intrinsic demagnetization curve, Bi (Hi ). Bi = B − µoHi
• Intrinsic coercivity, i Hc . It is the value needed to permanently demagnetize the
material.
8
2.2. PM Synchronous Machines
B
Operation
(II)
oil
Br
rec
e
lin
Magnetizing
(I)
Bknee
Hci Hc Hknee
H
Intrinsic demagnetization
Bi -Hi curve
Magnetizing
(III)
Normal Bm –Hm
curve
Operation
(IV)
Fig. 2.4 B-H Hysteresis loop and characteristic parameters for a permanent magnet [19].
• Knee magnetic flux density, Bknee . Once this threshold is exceeded, the material
enter in a region where it is demagnetized.
• Recoil magnetic permeability, µrec B = Br + µrec µo H
• Maximum magnetic energy per unit of volume, (BH)max . This value shows a
relative measure of the strength of the PM.
• Temperature coefficients of Br and HcJ . Magnets are temperature sensitive, and
their properties vary from Tamb = 20◦ C to the operating temperature.
• Resistivity. PM are exposed to magnetic field variations in electric machines, therefore, they are a source of eddy currents. It is important that this value is high.
• Chemical characteristics. There are some hard materials which cannot be used
in corrosion atmosphere. NeFeB magnets have low corrosion resistance, thus they
react quicker in high temperature or humidity conditions. Furthermore, adhesives
with acid content must be avoided, since they lead to fast decomposition of the PM.
SmCo Magnets are more tolerant with corrosion issues [7].
PMs are designed to operate in the recoil line (Fig. 2.4), the linear part of the demagnetization curve between Br and Hc . Hence, the operating point is located along the
mentioned line. The linearity of this recoil line finishes when the magnetic field density reaches Bknee . If B is higher than this value, magnetizing and demagnetizing cycles
are reversible. If B is further reduced, the magnet is partially demagnetized and it falls
9
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
down in a new recoil line and thus, in an irreversible cycle. It is important to highlight
that Bknee depends on the temperature: the higher the temperature, the higher Bknee and
therefore, the easier demagnetization as is illustrated in Fig. 2.5 [19]. In general, regard-
B
o
iC
1.2
0.8
ΔBrem
20ºC
70ºC
-800
ΔT
120ºC
ΔHc
-400
-600
-200
field strength(kA/m)
0.6
flux density(T)
1
N
Al
1.2
1
0.8
B
Fe
Ne
0.6
Co
Sm
0.4
te
rri
Fe
0.2
H
1.4
-1000
-800
-600
-400
-200
field strength(kA/m)
flux density(T)
B
0.4
0.2
H
Fig. 2.5 a) Effect of the temperature on PM demagnetization curve. b) Demagnetization curves of
common permanent magnets at 20o C. Adaptation from [19].
ing the properties shown above, depending the application which the PMSM is built for,
a good PM has to have a good trade-off wide hysteresis loop, high Br , high (BH)max
and elevate working temperature since the operating point has a deep impact on the PM
characteristics. In Table 2.1 general properties of the most common PMs are displayed.
Table 2.1: General properties of permanent magnets [19]
Br
Hc
TCurie Tmax density (BH)max ∆Br / ∆T
[T] [kA/m] [◦ C] [◦ C] [kg/m3 ] [kJ/m3 ]
[%/◦ C]
Ferrite
0.38
250
450
300
4800
30
-0.20
AlNiCo 1.20
50
860
540
7300
45
-0.02
SmCo
0.85
570
775
250
8300
140
-0.04
NdFeB 1.15
880
310
180
7450
260
-0.12
2.2.2 Advantages and Drawbacks
The principal difference of PMSM relies on the rotor, since the stator used is the same as
the normal AC machines stator. PMs are used as excitation system in the rotor substituting
the electromagnetic excitation. This can lead to an improvement of several characteristics
on synchronous motors [20] [19].
• Higher torque density and/or power density. The space occupied by the PMs in
the rotor is smaller than the electromagnetic excitation. Hence, the machine can be
more compact performing the same power and torque.
10
2.2. PM Synchronous Machines
• Higher efficiency. Copper losses caused by the electromagnetic excitation are removed. Nevertheless, eddy currents are induced in the PM (it is a conductive material) due to the variation of the magnetic field on them. However, these losses can
be minimized by cutting axially the magnets [11].
• Simplification of construction and maintenance. The absence of conductors in
the rotor makes the construction easier. Moreover, the rotor is more solid and the
maintenance is easier as well.
• Better dynamic performance due to higher magnetic flux density in the air gap.
Nevertheless, the use of this materials as excitation system involves also some disadvantages [20]:
• The high price of PM. The price of NeFeB has increased during the last years,
reaching a peak of 800% of its value in 2007. At the present, it has been slightly
reduced to values between 300-200% compare to the one in 2007 as it is depicted
in Fig. 2.6.
Fig. 2.6 Neodymium price evolution [18].
• Magnets are temperature sensitive. As depicted in Fig. 2.5 the properties of the
PMs are deteriorated when the temperature rises. This can vary the properties of the
machine during the operating function. It is important to avoid magnets trespassing
certain temperature since they can be permanently demagnetized, as it has been
mentioned above.
• A sensor position or rotor angle estimation method is required.
11
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
• Excitation cannot be controlled directly.
Depending where the PM is placed in the rotor, it leads to a variation on the magnetization
direction (radially or axially). Therefore, there are several topologies of PMSM. The most
suitable for the PMSM which is designed in this thesis are the RFPM and the AFPM.
2.2.3 Radial Flux PM Synchronous Machine (RFPM)
RFPMs are the most common PMSM due to their rotor simplicity and their similarity
to the synchronous AC motors. The stator of this machines is the same one as in the
synchronous AC motors. As has been mentioned above the difference lies in the rotor.
Since the electromagnetic excitation is removed, the direct control of the excitation from
a DC disappears. Instead, the flux is controlled with the stator currents [16]. Fig. 2.7
shows the path followed by the flux through the airgap. It crosses radially the airgap. The
currents of the stator flow in the same direction of the rotating axis of the machine.
flux direction
Iron
current direction
rotation
PM
shaft
Fig. 2.7 Flux, current direction and rotation in a Radial flux PMSM [16].
Rotor Configurations
Notwithstanding, despite the stator is the same as in the synchronous AC machines, the
rotor configurations can assume different topologies depending where the magnets are
located. They can be divided in three main groups [19]:
• Surface-mounted PM machine (SPM). (Fig. 2.8a) The main advantage of these
machines is the simplicity of construction and therefore, a consequently reduction
of cost compared to other magnets topologies. Nevertheless, this configuration exposes the PM to demagnetization fields. Besides Ld ≃ Lq , since the relative permeability of the magnet is similar to air (µr = 1 − 1.05), the rotor is isotropic. Hence,
this configuration is not able to perform reluctance torque [16].
12
2.2. PM Synchronous Machines
a)
b)
c)
Fig. 2.8 Different RFPM synchronous motors. a) SPM motor; b) Inset PM motor; c) IPM motor.
.
• Inset-PM machine. (Fig. 2.8b) The rotor is similar to the rotor of SPM motors.
The difference is that there is iron between two adjacent PMs. Since the relative
permeability of the iron is higher compared with the magnet permeability, the rotor
is anisotropic. Thus, Ld 6= Lq and the torque is composed by two sources: the torque
component that the SPM motors exhibit (PM torque) and the reluctance torque.
• Interior-PM machine (IPM). (Fig. 2.8c) The rotor is completely different to the
others shown above. PMs are located inside the rotor, and they can be adapted
to different shapes depending on the direction of magnetization inside the rotor,
which can be tangential or radial. All IPM motors are characterized by their ability
to perform reluctance torque due to the different permeance paths that they exhibit.
Moreover, they can achieve a high open-circuit air gap flux density, specially the
ones magnetized tangentially, and they are protected against demagnetization and
mechanical stress. However, they are not as simple as the other machines, and thus
leads to more expensive solutions.
Despite IPM and Inset-PM motors seem to exhibit better properties for the purpose of this
thesis, SPM synchronous topology is the final choice due to its simplicity, its solid and its
cost.
Inner vs Outer rotor
The rotor of RFPM machines can be placed as the outside or the interior part of the PMSM
as depicted in Fig. 2.9. Both of them have different properties, which are important to take
into account [16].
• Suitability for mechanical drives. Outer rotor design is usually a better solution
for applications such as IW motors and wind turbines. The rotor can be directly
attached to the wind turbine or the rim. This makes the system really compact.
13
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
• Thermal design. In some applications it is a critical aspect. As it has been mentioned above, magnets are temperature sensitive and winding isolation cannot surpass certain temperature (Isolation H can handle a maximum temperature of 180◦ C).
Otherwise, both of them can be damaged. The differential factor between them lies
in where the main source of losses is located, the stator windings. Inner rotor configurations are easier to refrigerate, as the cooling surface which is directly in contact
with the copper is larger and directly attached to the aluminium housing. Outer rotor
commercial designs are usually refrigerated by inner coolant.
airgap
diameter
• Magnetic design. Outer rotor PMSM usually has larger D for the same external
diameter, as it is illustrated in Fig. 2.9. Since the torque is directly proportional to
the square value of D and the active Lact (equation (2.1)) the axial length of outer
rotor designs is smaller. Hence, a reduction of active weight is achieved. Outer rotor
PMSM motors are usually 15% lighter than inner rotor designs [16].
Fig. 2.9 Comparison of Dgap in Inner and Outer design. Adapted from [16].
• Winding wound. It is slightly easier to wind outer rotor than inner rotor designs
since the teeth point outwards.
• PM detachment. PMs withstand centrifugal force due to the rotation of the machine. This force is proportional to the distance they are located from the rotation
axis and to the square of the speed. Despite the air gap diameter being usually higher
in outer rotor design, magnets push against the rotor yoke. On the other hand, inner
rotor magnets are only glued to the rotor surface, and there is not iron part opposing
force. That is for this reason that they are usually bandaged or use some protection
to avoid the detachment.
2.2.4 Axial Flux PM Synchronous Machines, (AFPM)
Unlike RFPM machines, the flux inside this machine flows axially through the air gap
while the currents flow in the radial direction [16]. This type of motor is commonly used
in solar car competitions, due to its high efficiency and low weight, as the Aurora Car. It
performs efficiencies close to 98% and its torque density is the highest among all types of
architecture [21]. AFPMs have several advantages over the counterparts RFPMs [10] [14]:
14
2.3. Summary and Conclusions
Windings
Stator Iron
Rotor Iron
PM
Shaft
Direction of magnetization
Current direction
Rotation
Fig. 2.10 Flux, current direction and rotation in a Axial flux PMSM. Adapted from [16].
• Higher power density. RFPMs do not exploit the rotor yoke, which is hardly used
as magnetic circuit. AFPMs take advantage of the machine’s iron in a better way
than RFPMs, thus it is possible to design more compact machines.
• Planar and adjustable air gap. AFPMs have the capability of adjust the air gap.
• Lower vibrations and noise levels.
• Better heat removal. Since the inner diameter of AFPMs is greater than the RFPMs,
their capability of ventilation and cooling is expected to be better.
• Larger diameter to length ratio relation. AFPMs usually have larger diameters than
RFPM but an smaller axial length.
• Higher number of poles. Since the outer diameter of the core is larger, a higher
number of poles can be placed, making this choice really interesting for low-speed
applications.
Its main drawback lies in the manufacturing procedure. The manufacturing process of
the stator is difficult, due to the need of avoiding eddy currents, and thus, it is extremely
expensive.
2.3 Summary and Conclusions
Throughout this chapter, the different possibilities for an IW motor are raised, both motor
architecture and different types of PM motors. Regarding the motor architecture, in order
to remove losses and increase the efficiency of the system, the most suitable option is an
15
Chapter 2. Low-Speed Direct-Drive Applications. In-Wheel PMSM
IW low-speed machine directly attached to the rim. This removes any gearbox from the
motor to the rim, removing unnecessary losses.
Related to the type of machine selected, it is important to take into account that the
machine is air-cooled. It is an essential fact to be aware of during the design procedure,
since the magnets and the winding insulations cannot reach certain threshold temperature
of (150◦ C), as well as the difficulty of attaching the rotor to the carbon fiber rim which is
currently constructed for the ”Shell Eco 2012”. Thus, RFPM with an outer rotor design is
discarded since the risk of thermal heating is higher than in the two proposed alternatives.
Between the remaining designs, AFPMs provides more advantages than the RFPM of
inner rotor design. AFPMs are lighter since their power density ratio and their cooling
system is better. Vehicles involved in solar car contests, as the ”Aurora”, usually use
them. However, since the manufacturing cost of the stator is extremely high, it is also
discarded. Therefore, the machine selected is an RFPM of inner design. This is the most
conservative and solid solution for accomplishing the goal of a light-weight and efficient
design. Despite RFPM is not the best solution in terms of light-weight and efficiency, it
has a good trade-off power and torque density as well as a good efficiency ratio as well as
it is the harder machine.
16
Chapter 3
Shell Eco Marathon Circuit Analysis.
Requirements
In this chapter, the main requirements of the machine are calculated. 1) Peak torque
Tpeak , 2) Steady state torque Tsteady and 3) Maximum rotational speed ωmax . Based on an
ideal simulation of the vehicle and its maximum requirements, these are estimated. This
estimation of the parameters is done based on a trade-off reduction of energy per cycle
and feasibility of the machine
3.1 Track and Car Requirements
Before calculating the motor requirements, which have to be defined as torque and speed
values, the contest requirements and the circuit characteristics are depicted to further meet
the machine requirements. In Table 3.1 the principal characteristics of the ”Ahoy Urban
Circuit 2013” are summarized. Two possible vehicle concepts are available in this comTable 3.1: ”Ahoy Urban Circuit 2013” characteristics [24].
Circuit
Distance
1630 [m]
Slope
0 [%]
90◦ corner 5
petition and both have to fulfill different requirements. This thesis focuses on the ”Urban
Concept”. Regarding ”Shell Eco Marathon” rules, the machine that will be designed has
to fulfill several conditions, taking into account some relevant constructive aspects of the
race, listed below:
• 10 laps of 1630 meters at 25 km/h. That means 234 seconds per lap.
• Reduction of energy consumption as much as possible.
17
Chapter 3. Shell Eco Marathon Circuit Analysis. Requirements
Table 3.2: Different vehicle concepts [24].
Prototype Concept Urban concept
Wheels
3-4
4
Min Driver Weight
50 [kg]
70 [kg]
Max Vehicle Weight
140 [kg]
205 [kg]
• Be able to drive through corners. It would be desired not to use the breaks during
cornering.
• The ratio between peak torque and steady state torque cannot be extremely high.
The machine is designed taking the steady state torque as a reference. If the peak
torque is so high it is possible that the machine cannot perform this due to the iron
saturation.
In order to further determine the car requirements, it is assumed that some characteristics
of this vehicle are the same as the last ”Shell Eco Marathon Urban Concept 2012”. Likewise, some parameters are expected and estimated by the Shell Eco Marathon Team, such
as the mass of the vehicle. These parameters are detailed in Table 3.3.
Table 3.3: Parameters of the Shell Eco Marathon 2013.
Vehicle Characteristics
Mass
110 [kg]
Drag coefficient, cd
0.3
Rolling resistance coefficient, µrr
0.014
Frontal Area
0.8 [m2 ]
Number of driven wheels
1
Radius of the wheel
275 [mm]
3.1.1 Driving Cycle
First of all, the speed cycle has to be determined. The principal aim of this cycle is to
perform a good trade-off between two of the issues described above: the reduction of
energy consumption and the torque ratio. Moreover, since the maximum speed is not
expected to be higher than 30 km/h, the car should not need to use the brake in any of
the corners of circuit. Thus, the shape of the cycle proposed is illustrated in Fig. 3.1.
It has to be pointed out that this simulation cycle is an ideal cycle which does not take
into account some important matters as the efficiency map of the motor, points which the
converters works at highest efficiency, and the importance of the braking energy recovery.
This simple cycle is the most suitable for the competition since there is only one period
18
3.1. Track and Car Requirements
V(t)
vav=25km/h
Δt
t
Δt
T=234 s
Fig. 3.1 Driving cycle.
of acceleration and a deceleration one. In terms of energy consumption, this is a good
choice because a lot of energy is used during acceleration and not all is recovered during
the deceleration period, since the energy from the wheels to the battery have to pass
through some elements which generate losses, as the motor or the converters. Regarding
the peak torque issue, it can be reduced by tempering the speed profile at the final part of
the acceleration slope. This can be decreased if the final part of the acceleration slope is
tempered as in Fig. 3.2.
V(t)
t
Fig. 3.2 Tempered driving cycle.
3.1.2 Cycle Optimization
Once the main shape of the driving cycle is selected, it has to be optimized to accomplish
with the requirements exposed above. In order to optimize this cycle and calculate the motor requirements, an ideal simulation of the vehicle is carried out using S IMULINKR . The
energy consumption during the complete cycle, the energy consumption of each contribution of power (dynamics, drag resistance and rolling resistance) and the torque profile are
estimated. Furthermore, the maximum speed (or the steady-state speed in this case) that
the vehicle reaches is calculated. The equations which this simulation model is based on
are attached in Appendix A. The main difference between the diverse driving cycles is the
19
Chapter 3. Shell Eco Marathon Circuit Analysis. Requirements
acceleration and deceleration time, ∆t (both periods are considered equal, because it is an
ideal simulation). Since the average speed is fixed if ∆t varies, consumption, Tpeak , Tsteady ,
energy consumption distribution and other magnitudes change. Thus the study is carried
out throughout ∆t = 1, 2...234/2. Facing the energy consumed and the Tpeak in every cy-
v(km/h)
60
40
20
0
0
50
100
150
200
time (s)
Fig. 3.3 Different driving cycles. Average speed of both cycle is the same.
cle, one can realize that there is a wide area where the energy consumption is minimum,
below 120 Wh as it is illustrated in Fig. 3.4. Regarding Fig. 3.2 and Fig. 3.3 this minimum
region is extended form 20 Nm to 70 Nm. This interval of Tpeak corresponds to a Tsteady
of 6-7 Nm, as is illustrated in Fig. 3.5. Since the machine cannot be overloaded more than
3-4 times its nominal torque value, and the Tsteady is around 7 Nm, the feasible area is
the one belonging to 20-30 Nm of Tpeak . Moreover, the maximum speed that is reached
in this area is lower than the one reached in speed cycles with smaller peak torque, as is
Fig. 3.5 shows. That means that the losses related to frequency issues would be reduced
(Iron and magnet losses).
d)
E(Wh)
150
140
130
120
110
0
10
20
30
40
50
60
70
Tpeak
Fig. 3.4 Relation between energy consumption and peak torque.
20
3.2. Analysis of the Energy Consumption
3.2 Analysis of the Energy Consumption
The energy supplied by a battery is not only used to move the car, but also to overcome
the drag and the rolling resistance. In the whole process of the energy transformation,
there are some losses in the electrical system (iron, copper and semiconductors losses)
as well as mechanical losses (bearings and gears). This section shows how the energy
is ideally used (without losses), and its distribution in the different cycles. Regarding
App. A, the power performed by the machine is divided in three components, the drag,
the rolling resistance and the dynamic power, and consequently, the torque. Looking at
Fig. 3.6, the energy used to overcome the rolling friction is constant since it only depends
on the gravity and the mass of the car. However, the energy consumption due to the drag
increases almost the double since the drag power depends on speed. App. A illustrates
the relation between the energy consumption and the interval of acceleration for a same
average speed. Likewise, in Fig. 3.6, one can realize that the energy consumption of the
acceleration period increases when the interval of acceleration is higher. The vehicle will
not be able to recover the whole amount of dynamic energy since there will be losses in
the motor, the power electronics, despite the fact that in this ideal simulation the dynamic
energy consumption is zero as illustrates Fig. 3.6. Furthermore, Fig. 3.4 illustrates the
Tpeak tendency to decrease quite quickly, maintaining the energy consumption in similar
values, then, it reaches a point where Tpeak is stabilized and the energy consumption rises.
As a result of this, some points with the same Tpeak consume different energy. This can
be explained by looking at Fig. 3.7. For small acceleration periods, the dynamic component of the torque leads the torque composition used to move the car. However, as long
as this acceleration interval increases, dynamic features decrease since the acceleration
needed is smaller and the drag component continuously increase. For big time intervals
of acceleration, where the maximum speed of the cycle is almost double than the smaller
periods, that the drag component is the one that leads the torque distribution, and there
is an acceleration interval which is a minimum (Fig. 3.7). Hence, there are points with
different energy consumption and the same Tpeak . The dynamic component of the torque
decreases and the drag component increases quicker, as well as the energy consumed by
the drag resistance increases raises as well.
Mass and Area Sensitivity
An important issue to analyze is how penalizes, in terms of energy consumption, the
increase of weight or area of the vehicle, or on the other hand, what benefits their reduction can provide. The energy consumption of every source of energy per lap that
the motor has to face is calculated in Appendix A. Mass and area are linearly dependant to energy consumption. Therefore, the sensitivity of increasing the mass, ∆mv , is
αmass = µrr gvav T nlap = 0.6695 Wh/kg and the sensitivity of increasing the frontal area
∆Af is αarea = 54.422 Wh/m2. Both sensitivities have been calculated using ∆t = 15s.
21
Chapter 3. Shell Eco Marathon Circuit Analysis. Requirements
Thus, if this value changes, so does the sensitivity.
3.3 Conclusion
The selection of the driving cycle for the ”Shell Eco Marathon 2013” has been proved to
be an important part of the design of the motor. Depending on the cycle selected, different
requirements for the machine are needed and varied energy consumptions are achieved.
The cycle selected is the one with an acceleration period of ∆t = 15. The corresponding
design parameters are illustrated in Table 3.4
Table 3.4: Requirements for the PMSM.
Requirements
Tpeak
22 [Nm]
Tr
7 [Nm]
ωr
275 [rpm]
22
3.3. Conclusion
Tsteady (Nm)
a)
ωmax (rpm)
b)
15
10
5
0
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
20
40
60
80
100
120
600
400
200
0
c)
Tpeak (Nm)
75
50
25
0
E(Wh)
d)
150
125
100
∆t(s)
Fig. 3.5 a) Variation of Tsteady with the interval of acceleration; b) Variation of wmax with the interval of acceleration; c) Variation of Tpeak with the interval of acceleration; d) Variation
of the energy consumption with the interval of acceleration.
23
Chapter 3. Shell Eco Marathon Circuit Analysis. Requirements
Energy (Wh)
80
60
40
20
0
0
20
40
60
80
100
120
∆t (s)
Fig. 3.6 Different energy components. a) Drag: GREEN; b) Rolling: RED; c) Dynamic: BLUE;
d) Dynamic peak: MAGENTA.
20
T(Nm)
16
12
8
4
0
0
20
40
60
80
100
120
∆t(s)
Fig. 3.7 Evolution of the different sources of resistance torque, at overload conditions. BLUE:
Peak torque; GREEN: Drag resistance torque; RED: Rolling resistance torque; MAGENTA: Acceleration torque.
24
Chapter 4
Non-Overlapping Concentrated
Windings Analysis
Along this chapter concentrated windings are described. Different methods to calculate
their winding layout are explained as well as the fundamental winding factor. Further
some undesirable effects due to the PM and the winding layout are depicted and it is
explained how to avoid them. This chapter mainly shows how to choose a good configuration of poles and slots.
4.1 Concentrated or Distributed winding
The type of stator winding used in an electrical machine is determined by the number
of slots Qs , the number of poles p, and the number of phases m [12]. These parameters
of the machine determine q, the number of slots per pole and phase (4.1). Depending on
whether the value of q is an integer or a decimal, the winding type is different.
q=
Qs
mp
(4.1)
• Distributed winding. (q = integer). It is the conventional winding. The higher the
value of q, the more sinusoidal MMF-wave is produced by the windings [23]. The
fundamental winding factor, kw1 , in this type of winding is 1 when q = 1.
• Concentrated windings. (q 6= integer). The machines that use this type of winding
can have each coil wound around one tooth. Depending the type of winding, they
can be single-layered or double-layered as Fig. 4.1.
Concentrated winding presents better properties for low-drive applications.
• Short end-windings. Since copper losses are much higher compared with iron
losses for frequencies lower than 100 Hz (low-speed applications) [12], it is important to reduce them as much as possible in order to increase the efficiency of
25
Chapter 4. Non-Overlapping Concentrated Windings Analysis
A
A
A
A
A
B
C
B
C
A
B A
C
B
A
A
C
B
C
B
C
A
A
A
Fig. 4.1 End winding distribution of different winding topologies. a) Distributed winding with
Qs = 36, p = 12 and q = 1; b) one-layer concentrated winding with Qs = 12, p = 8
and q = 0.5; c) double-layer concentrated winding with Qs = 12, p = 8 and q = 0.5 Slot
divided vertically. While one-layer concentrated winding wound alternate teeth, doublelayer concentrated winding wound all the teeth in the machine.
the motor. Concentrated windings contain smaller end-windings compared to distributed windings, since they are wound around one tooth. Thus, there is less copper
in the machine and the losses decrease [16] [12], as they are directly related to the
volume of copper, as it is exhibited in equation (5.41).
• Low cogging torque. This oscillatory torque which can be seen in absence of currents, is owing to the tendency of the magnets to line up with the stator trying to
maximize the permeance of the magnetic circuit from the standpoint of the permanent magnets [16] [12]. According to [20], it is possible to realize that the ripple
is quite small in concentrated windings compared to distributed windings, which
usually need skewing on the rotor.
• Good fault-tolerant capability. Distributed windings are overlapped, hence there
is a risk of failure if the insulation fails. On the other hand, since the coils in concentrated windings are wound around one tooth, the contact between conductors of
different phases is drastically reduced, even eliminated in single-layer winding.
• High constant power speed can be achieved [20].
• Insulation and manufacturing systems are easier in concentrated windings [16]
[12].
• Reduction of mutual coupling among the phases [9].
26
4.2. Double or Single Layer
4.2 Double or Single Layer
In Fig. 4.2 cross-sectional area of of the stator is shown. It can be appreciated the different wound between single-layer and double-layer windings concentrated windings.
Depending on the final application of the machine, it is preferred one of these type of
a)
b)
Fig. 4.2 Teeth in both type of windings. a) single-layer. b) double layer [16].
winding or the other. They exhibit different characteristics which can adapt better for a
particular application. Characteristics shown in Table 4.1 are the most important ones
concerning this thesis. As it has been mentioned above, light-weight and efficiency are
Table 4.1: Comparison of properties between single-layer and double-layer windings [16]
Fundamental winding factor
End-windings
. Eddy currents
Overload Torque capability
Harmonic content of EMF
Torque ripple
Single-layer
higher
longer
higher
higher
higher
higher
Double-layer
lower
shorter
lower
lower
lower
lower
critical in the design of this motor. In order to accomplish both objectives, double-layer
winding seems to be more appropriated. Since the end-windings are shorter and the eddy
currents computed in the double-layer winding are lower, the losses that this machine performs during its operation are significantly lower than its counterpart single-layer winding. Moreover, the value of the torque ripple is lower in double-layer windings and there
are more possible combinations of slots and poles [16]. However, it is important to remark
that the overload torque capability is lower in double-layered windings. This point is important in this design because, as it will be further described in Chapter 5, the machine is
designed for the steady-state, and it has to be able to reach a peak torque between three or
four times the steady-state torque, as mentioned in Chapter 3.
4.3 Concentrated Winding Layout
Unlike distributed windings, concentrated windings perform a winding factor below the
unit. This means that due to the different distribution of slots Qs and poles p, the electro27
Chapter 4. Non-Overlapping Concentrated Windings Analysis
motive force (EMF) induced in each phase is not the addition of the absolut value of the
EMF induced in each conductor, since they are phase shifted. According to the Faraday’s
Induction law [11]
e=
dψ
dφ
= ns kw1
dt
dt
(4.2)
where e is the electromotive force induced. This entails an effect on the torque performance, since the fundamental winding factor is directly related to the torque. Therefore,
the stator needs to be fed with higher currents in order to deliver the same torque compared with those with larger winding factor. The fundamental winding factor kw1(Qs , p),
depends on the number of poles and the number of slots, as well as the winding layout.
But there are other properties of the machine that also depend on this configuration, such
as the torque ripple and the unbalanced magnetic pull. Thus, the selection of the number
of poles and slots the PMSM is designed for, is a critical issue to perform a good trade-off
of all properties described above [16] [30].
First of all, two methods are proposed to place the conductors of each winding
phase. Both methods dispose the conductors following some key rules taking into account
each configuration of Qs and p:
• Obtain the maximum amplitude of the main EMF harmonic waveform. This means
the highest winding factor possible.
• The waveform has to be equal in each phase.
• The phases of the winding have to be electrically displaced by 2π/m radians,
among them.
Winding Layout Using Cros’ Method
This method is based on the decomposition of q in its most simplified fraction. It is
based on the method used in large synchronous machines with a fractional value of q.
The method is described in the following steps and it is illustrated in Fig. 4.3 [16] [13].
The parameters used in this example are Qs = 18 and p = 14, as well as a double-layer
winding.
1. Simplification of q. The number of slots per pole per phase, q, is canceled down to
its lowest terms. q = bc ; d = c − b.
2. Partial one-zeros sequence. In this step, a sequence of b ones (”1”) and c − b zeros
(”0”) is distributed as regularly as possible.
3. Complete ”ones-zeros sequence”. The sequence obtained in the last part, is repeated
3p/d = Qs /b times, and is compared to the layout of the distributed winding, q = 1.
28
4.3. Concentrated Winding Layout
4. Comparison to distributed winding layout. As it is illustrated in Fig. 4.3 conductors
from the distributed winding that corresponds to ”1” in the ”one-zero sequence” are
kept and they form one side of the double-layer layout. The other side is obtained
by writing the returning coil in the other side of the tooth.
~ that describes the layout of one phase of the machine is created.
5. S vector. A vector S
The final layout obtained is numbered by slots, from 1 to Qs . For each slot which
contains a conductor of the A phase, the number of the slot with the sign of the phase
~ Thus, S
~ has 2Qs /3 elements. This last step is essential to calculate
is written in S.
the fundamental winding factor kw1 , as it can be appreciated in equation (4.8).
3 x "1"
a) q=3/7=b/d
4 x "0"
b) 1010100
c)
1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0...
A C' B A' C B' A C' B A' C B' A C' B A' C B' A C' B... '
A
B
C
C'
A'
B'
B
C
A
d) A' A A' B B' C C' C' C A' A B' B B B' C C' A
1
slot
number
e)
2
3
4
5
6
7
8
9
S = [-1 1 -2 -5 6 9 -10 -10 11 14 -15 -18 ]
Fig. 4.3 Illustration of winding layout Cros’ method determination for Qs = 18 and p = 14
This method is not suitable to be used in single-layer winding, since there are some layouts
which are difficult to be found [16]
Winding layout from the Star of Slot
This method, as the one explained above, is based on the design of large synchronous machines with a high number of poles. Its principal aim is to maximize the main harmonics
of the EMF induced in the windings [4]. It illustrates the phasor representation of the main
EMF harmonic induced in each coil in each slot. Some parameters have to be calculated
before implementing this method [22].
• Machine periodicity.
t = GCD{Qs , p/2}
(4.3)
29
Chapter 4. Non-Overlapping Concentrated Windings Analysis
• Number of spokes.
Nspk =
• Angle between two adjacent slots.
(4.5)
p/2
α
Qs
(4.6)
2π
αe
=
t
Qs /t
p/2
(4.7)
αe =
αph =
(4.4)
2π
Qs
α=
• Angle between two spokes.
Qs
t
Moreover, it has to be taken into account that the winding is feasible only when there are
the same number of spokes per phase. Procedure [22]:
• Phase Region. Phasors’ space is divided into 2m equal sectors, each sector covering π/m radians. Opposite sectors belong to the same phase and each sector and
its opposite have different polarity (Fig. 4.4). To determine the other phases, both
sectors have to be rotated by an angle of 2kπ/m radians, k = 1, 2, ..(m − 1)
C+
B-
A-
A+
B+
C-
Fig. 4.4 Phase Region.
• Star of slots. There are Nspk spokes that have to be equally divided over the phasor
space. The angle between spokes is αph . One of the spokes have to be placed at 0o ,
coinciding with the common ”x axis”. After that, the phasors have to be numbered.
The first one has to be positioned in 0o . The angle between two adjacent slots is αe ,
so starting from phasor ”1”, the following have to be shifted αe non-clockwise Qs
times.
30
4.4. Fundamental Winding Factor
7
11
5
4
9
αse
15
αph= αse
2
6
17
2
8
3
12
14
αph
10
1
4
7
1
10
6
5
8
18
12
11
13
a)
3
8
16
b)
Fig. 4.5 Star of Slots. a) Qs = 12, p = 8. With this parameters the results are: t = 4, αse =
αph = π/3. Thus, there are 4 spokes with 4 phasors each; b) Qs = 18, p = 14. With this
parameters the results are: t = 1, αse = 7αph = 7π/9. Thus, there are 18 spokes with 1
phasors each
• Overlaying both draws (Fig. 4.6), the phase and the sign of all coils are determined.
The slots that are inside the different sectors of the phase regions, belong only to this
phase and sign. With this method, only one coil side is determined, the return side
of each coil is in the adjacent slot, with the opposite polarity. This method is widely
explained in [4], and it is also extended to a single-layer winding. It is important to
notice that there are some configurations where some spokes are in the boundaries
of the phase region, thus the phase and sign of these spokes are not determined. A
solution to this problem is to rotate the phase region in non-clockwise for αph /2.
This happens when αph = k1 π6 ; k = 1, 2..., N (for a three phase winding).
4.4 Fundamental Winding Factor
This parameter is one of the most important factors of this type of windings, since it
determines how efficiently the conductors have been arranged throughout the stator. It
explains how much fundamental flux of each harmonic the winding extracts from the flux
wave [30]. Several methods have been found to calculate the fundamental winding factor
~ calculated
and their harmonics. One is based on the Cros’ method and it uses the vector S
in the last step of this method [16] [13]. Other analytical methods, which are based on
the proportion of slots and poles, are described in [30] [22], and they explain the different
factors that are involved in this fundamental winding factor.
31
Chapter 4. Non-Overlapping Concentrated Windings Analysis
C+ 7
11
C+
B-
8
5
3
1
4
7
10
A+
6
8
12
17
4
2
2
A-
12
A-
9
15
14
10
1
5
6
18
C-
B+
B-
A+
11
13
B+
8
3
16
C-
b)
a)
Fig. 4.6 a) Layout of one coil-side: ABC|ABC|ABC|ABC; b) Layout of one coil-side:
ACC − B − A− ACBB − A− C − CBAA− C − B − B
4.4.1 Winding Factor Based on Cros’ Method.
This method uses the EMF phasors to calculate the winding factor. Looking up to the
~ was calculated. This vector, which contains the
Cros’ method, in the last step a vector S
number of slot that belongs to one phase of the machine, is used to obtain the electromo~ i , for the main harmonic order
tive force (EMF) phasor of conductor i from the phase A, E
of p/2 or the fundamental.
~
jπp|S(i)/Q
s|
~
~ i = sign(S(i))e
E
Then, the fundamental winding factor kw1 is calculated as
2Qs /3 X ~ i
E
i=1
kw1 =
nl Qs /3
(4.8)
(4.9)
~ and nl the number of layers, since the analysis is carried out in
where i is an element of S
a double-layer winding (nl = 2). Continuing with the example started above, the sum of
the EMF phasors is calculated in equation 4.10 and illustrated in Fig. 4.8.
12
X
i=1
p
p
p
p
p
p
p
p
p
~ i = ejπ Qs 2 − ejπ Qs − e4jπ Qs + e5jπ Qs − 2e9jπ Qs + e10jπ Qs + e13jπ Qs − e14jπ Qs − e17jπ Qs
E
(4.10)
The result for the of the fundamental winding factor is kw1 = 0.902
32
4.4. Fundamental Winding Factor
C+
9
B-
2
B-
C+
6
A-
12
B+
11
A+
7
A+
A-
4
C-
C-
B+
5
a)
1
8
3
10
b)
Fig. 4.7 a) Overlapping of spokes and boundaries of the phase region. b) Phase region shifted
π/12. Qs = 12,p = 10. With these parameters the results are: t = 1, αse = 5αph =
5π/6. Thus, there are 12 spokes with 1 phasors each. The layout of one coil-side is the
following one: ACC − B − BAA− C − CBB −A−
1
1
2
5
6
9
10
10
11
14
15
18
number of
slots
Fig. 4.8 Sum of EMF phasors for one phase of a 14 poles 18 slots SPM synchronous machine
4.4.2 Winding Factor Based on Pole-slot Combinations.
As is explained in [22] [30], the winding factor can be calculated as the product of the
pitch factor kp , the distribution factor kd , and the skew factor ks .
kw = kp kd ks
(4.11)
This method is detailed in [30] where it can be implemented to PMSM with m-phases and
single-layer and double-layer winding independently. Factors kp and kd are expressed in
different formulas between [30] and [22], but they are related by trigonometric relations;
thus they are the same equation. The calculation of the different parts is detailed
• Distribution factor. Distribution factor is given by the ratio of the MMF performed
in the concentrated windings compared to the distributed windings. Coils in the
stator are displaced from each other by a certain electrical angle, each coil produce a
33
Chapter 4. Non-Overlapping Concentrated Windings Analysis
sinusoidal MMF with a shift angle γ. For distributed windings, this angle is γ = 2π
and for concentrated windings γ 6= 2kπ (k = 1, 2..). Therefore, the sum of the
MMF of the distributed windings is the multiplication of the value of MMF of one
coil and, on the other hand, it is a sum of vectorsNv , since they are displaced. kd is
derived after some calculations [30].
kd =
sin[(Nv γ)/2]
Nv sin(γ/2)
(4.12)
Nv and γ have to be linked with the values of the machine. In order to validate 4.11
for all the harmonic order n, it is not as simple as multiplying γ per the harmonic
n
. Therefore, after some calculations
multiple number n′ , where n′ = GCD(Q
s ,p/2)
th
cited in [30], the distribution factor for each n harmonic order n is derived from
kdn =
sin(nπ/2m)
z sin(nπ/2mz)
(4.13)
where m is the number of phases and z:
z=
Qs
GCD(Qs , mp)
(4.14)
kdn is the distribution factor for the nth harmonic.
• Pitch factor or coil-span factor. The pitch factor is defined as the ratio between the
flux linked by a short pitch coil compared to a full-pitch coil, as illustrated in
Fig. 4.9 [30]
kpn =
Bnshort
= sin(α/2)
Bnf ull
(4.15)
where Bnshort is the magnetic flux density produced by a short pitch coil and Bnf ull
Fundamental flux distribution, B1
The 3-rd order harmonic, B3
slot pitch: α12
Fig. 4.9 Flux density distribution to short-pitch coils
34
4.4. Fundamental Winding Factor
is the magnetic flux density produced by a full pitch coil. In order to calculate the
pitch factor, the slot pitch angle, α12 is defined as
α12 =
2π p
π
=
Qs 2
mq
(4.16)
The coil-span angle ε is illustrated in Fig. 4.10
ε = π − α12
(4.17)
To conclude, the pitch factor can be calculated for any nth harmonic order from [22]
π
π
ε/2
ε/2
ε/2
ε/2
α12
α12
π-ε
π+ε
b)
a)
Fig. 4.10 a) Coil span is shorter than pole pitch of π; b) Coil span is larger than pole pitch of π
ε
kp = cos(n )
2
(4.18)
• Skew factor. A PMSM usually suffers torque ripple, and it cannot be removed
through the winding distribution, due to several issues further explained in section 4.5.2. In order to remove this ripple, the magnets or the stator windings are
sometimes displaced a certain angle. The fundamental skew factor ks is defined as
the ratio between the MMF produced by the machine skewed and the MMF produced by the machine unskewed [30].
ks =
Bnskew
sin(σ/2)
=
Bnnonskew
σ/2
(4.19)
where Bnskew is the magnetic flux density produced by the skewed machine, Bnnonskew
is the magnetic flux density produced by the non skewed machine and σ is the skew
angle.
35
Chapter 4. Non-Overlapping Concentrated Windings Analysis
Based on both ways of calculating the fundamental winding factor, results are deployed
in App. C. If these results are faced against the number of slots per pole per phase q
(Fig. 4.11), it can be realized that there are a lot of the combinations that are not suitable
for the design of concentrated winding because of their low fundamental winding factor.
To achieve a fundamental winding factor higher than kw1 ≥ 0.866, q has to belong to a
range of 0.25 ≤ q ≤ 0.5. Thus, all the combination which are not within this range are
not considered further [22].
1
kw1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
q
0.6
0.7
0.8
0.9
Fig. 4.11 Fundamental winding factor as function of the number of slots per pole and phase,
kw1 (q)
Winding Space Harmonics
Depending on the pole-slot configuration, as well as performing different fundamental
winding factors, different winding factors for all harmonic spectrum are achieved. According to the definition of the EMF, this can be used , somehow, as a filter to avoid
different undesirable frequencies in the back EMF. Fig. 4.12 illustrates the winding factors for all the frequencies in a machine with 18 slots and 14 poles [22].
4.5 Desirable Configurations of Poles and Slots
The selection of the number of poles and the number of slots is a critical issue when using
concentrated windings. As mentioned above, not only the fundamental winding factor is
involved in this selection, but there are some properties that are seriously dependent on
this configuration such as the torque ripple and the unbalanced magnetic pull. Being careful about this selection can avoid undesired forces which can create noise and vibrations
as well as the reduction of some sources of the torque ripple (cogging torque).
36
4.5. Desirable Configurations of Poles and Slots
kwn
1
0.5
0
0
10
20
30
40
50
nth
Fig. 4.12 Winding factor spectrum for a double-layer concentrated winding with Qs = 18 and
p = 14
4.5.1 Unbalanced Magnetic Pull
This is an important aspect in machines with concentrated windings, since they are a fractional winding type. When q is not an integer there are rotating attractive forces in RFPM
machines. As it has been mentioned above, noise and vibration depend on the selection
of the number of poles and slots. These are mostly due to the unbalanced magnetic forces
in the airgap. These forces are computed using the Maxwell stress tensor, using the flux
density distribution in a line along the airgap [16] [22] [9].
Z
Z
1
1
2
2
F~ =
[Bn (θ, t) − Bt (θ, t)]ds~n +
Bn (θ, t)Bt (θ, t)ds~t
(4.20)
2µo
µo
S
S
where equation (4.20) is the integral through the airgap, θ is the mechanical angle of the
stator, t is the time. This force in the airgap has two components:
• Radial component. It is the one which mainly causes the noise and the vibrations.
Z
1
~
Fn =
[Bn2 (θ, t) − Bt2 (θ, t)]ds~n
(4.21)
2µo
S
• Tangential component. It is the component involved in the torque.
Z
1
F~t =
Bn (θ, t)Bt (θ, t)ds~t
µo
(4.22)
S
The importance of the pole and slot combination resides in the flux density B along the
airgap. As it has been mentioned above, the radial force depends on both components
of the flux density in the airgap. Since the flux density waveform varies from one pole
slot combination to another, radial magnetic forces can be not regularly distributed along
the airgap, resulting in unbalanced pulling forces that generates noise and vibration in
the machine, since the rotor eccentricity is increased [16] [9]. Moreover, is important
37
Chapter 4. Non-Overlapping Concentrated Windings Analysis
to take into account that these forces do not excite any resonant frequency of the machine [22]. This unbalanced force is called unbalanced magnetic pull. Machines which
experiment this unbalanced magnetic pull are characterized by their lack of symmetry in
their winding layout, without taking into account the conductors orientation. According
to equation 4.21, magnetic forces are directly related to the square of the flux density,
therefore the conductor orientation has no influence on these forces [16]. The periodicity
of the winding layout and consequently, the structural periodicity on the airgap is given
by the greatest common divisor between the number of poles and the number of slots,
X ′ = GCD(p, Qs ). When the value of X ′ is 1, there is a high resulting unbalanced force,
since there is no symmetry on the winding layout. The combinations with Q = p ± 1
and some with odd Qs are the ones which accomplish this value. Thus, they have to be
avoided in order not to perform unbalanced magnetic pull in the machine, as illustrated
in Fig. 4.13 [16]. The higher X ′ the machine configuration has, the better the balance
concerning radial forces. Since X ′ describes the symmetry of the radial forces, it can
Fig. 4.13 Magnetic forces in the stator of a PMSM with different pole-slot configuration. a) and b)
68 poles and 69 slots; c)60 poles and 72 slots; d)10 poles and 12 slots. Figure extracted
from [16]
be noticed that there is a periodic force waveform. The fundamental wave length of the
attractive force can be computed as [22]:
τF,1 =
38
2π
|Qs − p|
(4.23)
4.5. Desirable Configurations of Poles and Slots
4.5.2 Cogging Torque and Torque Ripple
The torque performed by the machines is not perfectly constant, it oscillates around a
mean value, varying its value with the rotor position. It is caused by different sources [16]:
• The cogging torque.
• Variation of the permanence due to the magnetic saturation.
• Time harmonics. The inverter produces harmonics in the current stator windings.
Therefore, a pulsating torque is caused when this interacts whit the rotor field.
• Space harmonics. The field produced by the PM contains some harmonics that
interact with the magnetic field produced by the windings.
• Imperfections in the machine, such as not regular magnetization of the magnets
and rotor eccentricity.
The reduction of this torque ripple is important, since it is a source of noise and vibration,
specially in low-speed drives. This vibrations can cause problems in the control system,
since the estimation of the position and the speed of the rotor can be distorted [20]. In
Fig. 4.14 Definition of Cogging and Ripple torque [16].
order to reduce this ripple, several methods can be implemented. These methods try to
reduce the source of torque ripple shown above [20] [19].
• Reduction of cogging torque. Several strategies can be adopted to reduce it.
39
Chapter 4. Non-Overlapping Concentrated Windings Analysis
– Correct selection of the number of poles and slots [16] [20] [19] [9].
– Skew the rotor magnets or the stator windings.
– PM Pole arc with different width.
– Notches in the stator teeth.
– Shifting of the PMs.
• Reduction of the space harmonics, changing the shape of the magnets, to perform
a more sinusoidal flux in the airgap.
• Feed the windings with pulsating currents which create a pulsating torque opposite
to the ripple [20].
Since the machine designed is based on concentrated windings, its ripple should be small,
as [20] [9] point out. This thesis is only focused on the reduction of the cogging torque by
the selection of the correct number of poles and slots.
Cogging Torque
Cogging is the ripple torque which occurs in absence of currents in PMSMs, when the
rotor tries to align with the stator in order to maximize the permanence of the magnetic
circuit regarding the PMs [9] [19].
Tcog =
∂Wm
∂θm
(4.24)
where Wm is the magnetic energy in the airgap. This variation is not homogenous along
PM middle
point
PM
θm
Wm
θm
τ
edge
θm
Fig. 4.15 Elementary torque τedge illustration due to the variation of the magnetic energy Wm
along the relative position of the rotor θm
the airgap, Tcog has to be considered as the interaction of each edge of the rotor with the
40
4.6. Summary and Conclusion
stator slot openings. Since each PM edge is located in a different position regarding the
slot opening, they have to be considered independently and then, sum their contribution.
Fig. 4.15 shows the variation of the Wm when the PM edge reaches the slot opening. This
magnetic energy is formed by the contribution of the PM and the air, and as Fig. 4.15
shows they depend on the angular position of the rotor θm . The variation of Wm is low
when the PM is in the middle of the slot or far away from it. Thus, normally there is a peak
of variation in the edge of the PM τedge , although it is not a rule of thumb. So adding all
the τedge that occurs along the airgap of this interaction between PM and teeth the cogging
torque is obtained [19]. Therefore, all the applications mentioned above that reduce the
cogging torque are mainly focused in the counteraction of the different variations of the
Wm or avoiding the superimposing of them. One of the sorts of decreasing this source of
torque ripple is selecting an appropriate combination of the number of poles and slots, np ,
which is given by
np =
p
LCM(Qs , p)
=
GCD(Qs , p)
Qs
(4.25)
where LCM(Qs , p) is the number of Tcog periods per revolution. The lower np , the more
τedge will be superimposed (negatives and positives), yielding a high Tcog . On the other
hand, if np is high the elementary torques τedge are distributed and not superimposed along
the slot pitch, performing a low Tcog . Therefore, combinations of low GCD(Qs , p) or high
LCM(Qs , p) are sought in order to perform low Tcog [9] [19] [12]. Another rules to select
the correct number of poles and slots.
• The number of poles have to be an even number.
• The number of pole pairs cannot be multiple of the phase number, since this lead
to unbalanced windings.
• The number of poles cannot be equal to the number of slots.
4.6 Summary and Conclusion
Properties of concentrated winding show that they are more suitable than distributed winding in order to be built in the machine. However, the design of the winding layout is not
as intuitive as the one for distributed windings. When a design process of a PMSM based
on concentrated windings is carried out, the selection of the number of poles, p and the
number of slots Qs is critical to succeed. The selection of this configuration can lead to
machines with good performance characteristics as a high fundamental winding factor kw1
and low cogging torque, as well as avoid some undesired effects as the unbalanced magnetic pull. In Table 4.16 the parameters that have to be taken into account in the selection
process of the winding layout are summarized.
41
Chapter 4. Non-Overlapping Concentrated Windings Analysis
High winding factor
Maximize torque
per Ampere
Symetries in winding layout
Avoid unbalanced
magnetic pull
Low cogging torque
Minimize cogging
torque
Fig. 4.16 Selection process of concentrated windings depending Qs and p, adapted from [16].
42
Chapter 5
Analytical Design
This chapter explains the magnetic and electric properties that determine the dimensions
and the choice of some important parameters of the machine. An analytical model that
selects the dimensions of the machine is built, based on the theory framework exhibited
along the previous, and the current chapter.
5.1 Geometric Design
Before developing the analytical model which selects the appropriate dimensions for the
PMSM, in order to model the machine it is important to describe the relations among the
different parameters of the machine, such as the number of slots Qs , the number of poles
p, dimensions of the different ferromagnetic portions inside the machine, as well as the
volume of the magnets and the size of the conductors. Fig. 5.1 displays the more relevant
constructive parameters of a surface-mounted RFPM machine.
5.1.1 Magnet Sizing
An important parameter involved in the design of a PMSM is the fundamental flux density
expected in the air gap, B̂δ . This parameter determines the dimension of the PM in a
PMSM, both thickness hm , and coverage angle, 2α. Assuming that the shape of the flux
density above the magnets Bm is rectangular, as illustrated in Fig. 5.2, and the absence of
flux leakage, B̂δ is given by the following expression [23]
B̂δ =
4
Bm sin α
π
(5.1)
where α is the half angle of the magnet of the magnet, defined in electrical degrees. This
pole angle 2α is usually designed to 120◦ since the highest flux density in the airgap
per kilogram of magnet is obtained. I.e if the pole angle is increased to 180◦ the magnet
volume increases by 50% and the airgap flux density only is increased in 14% [23]. The
43
Chapter 5. Analytical Design
Dext
δ
b ss2
2α
w toot
b ss1
h
h sy
h ss
h
τs
hm
ry
Di
D
Fig. 5.1 Cross-section of a SMPM synchronous machine of 4 poles.
dependence between hm on B̂δ is studied through the analysis of the magnetic flux density
above the magnets Bm , which can be expressed as
Bm = Br,m
1
1 + µr hδme
(5.2)
where Br,m is the remanence flux of the magnets at operating temperature which typically
is 15◦ − 20◦ below the winding temperature [23], µr is the relative permeability of the
magnet and δe is the equivalent airgap length (including the effect of the stator slotting).
Magnets are temperature sensitive, so that the remanence flux Br,m variation is reflected
in
Br,m = Br,0 [1 − Tk (Tr − T0 )]
(5.3)
where Br,0 is the flux remanence at ambient temperature, Tk is the temperature coefficient
for the flux remanence density [1/◦ C], Tr is the temperature of the magnets at the operating
44
5.1. Geometric Design
B(θ)
Bδ
Bm
θ
2α
Fig. 5.2 Fundamental flux density above the magnets in an electric period.
point and T0 is the ambient temperature. Indeed, manufacturers of PMs usually give the
data of the magnet according to these parameters exhibited above, taking as reference
T0 = 20◦ C, as it is illustrated in [28]. Therefore, considering the equations (5.1) and (5.2)
the thickness of the magnet which is necessary to produce certain B̂δ in the airgap is
hm =
µr δe
Br,m 4 sin α
B̂δ π
−1
(5.4)
With regard to the airgap length, it can be chosen more freely compared to the asynchronous machines since the power factor cos φ of the PMSM is not penalized. The physical airgap δ which is suitable for PMSM machines belong to the range of 1–3 mm. In
order to choose within this range, some properties which depends on the airgap length are
exhibited [23]:
• Magnet height. The larger the airgap length, the more magnet material is necessary
in order to create the required flux density in the airgap. Thus, the magnet height
has to be higher and the magnet volume increases, which lead to an increase of the
machine cost.
• Magnetic inductance. When the airgap is larger, the magnetic inductance of the
machine decreases. Field-weakening is related to the magnetic inductance and a
drop of its value shortens the field-weakening operation range.
• Eddy current losses. The larger airgap, the more sinusoidal airgap flux density and
thus, less harmonics in this wave. As it is explained later, eddy currents depend on
the variation of the magnetic field. Therefore, the less content of harmonics, the
lower losses are performed.
Considering the information related to the airgap properties, the most suitable election for
the machine of this thesis is an small airgap, since the quantity of PM required is lower,
and consequently the weight of the machine.
45
Chapter 5. Analytical Design
5.1.2 Iron Sizing
This section describes the guidance of the magnetic flux through the ferromagnetic material in the motor. It is assumed the absence flux leakage among the different iron parts and
there is not saturation in the ferromagnetic material. Concerning the design of electrical
machines, the magnetic flux density in the different parts of the machine is usually a constraint of the design. In a PMSM, the values of the flux density in the different parts of the
machine, Bry , Bsy and Btooth varies between 1.2–1.5 T [11]. As depicted in Fig. 5.3, all
the flux produced by the magnets φm passes through the stator yoke and the rotor yoke
φm = Bm wm Lact = Bm
2α
(D − 2δ)Lact
p
(5.5)
where Bm is the magnetic flux density on the magnets, wm the width of the magnets, 2α
the magnet coverage and Lact the active length of the machine. The flux which passes
through the stator yoke and the rotor yoke is given by
φsy = Bsy hsy kj Lact
(5.6)
φry = Bry hry kj Lact
(5.7)
where kj is the stacking factor of the iron lamination and hsy and hry are the height of
Φsy
Φsy
Φtooth
Φm
Φry
Φry
Fig. 5.3 Flux guidance through a ferromagnetic portion of the machine.
the stator and the rotor. As Fig. 5.3 illustrates, the flux created passes through the teeth,
φtooth , and it splits in two equal fluxes in the stator yoke, φsy . These two equal parts, in
absence of leakage return to the magnet through the rotor yoke, φry
φry = φsy = φm /2
46
(5.8)
5.1. Geometric Design
According to this relation and (5.6) and (5.7), hry and hsy can be determined
hry =
αBm (D − 2δ)
αBm D
≈
pkj Bry
pBry
(5.9)
hsy =
αBm (D − 2δ)
αBm D
≈
pkj Bsy
pBsy
(5.10)
It can be appreciated in equations (5.10) and (5.9) that hsy and hry are quite similar.
They are usually equal since the magnetic flux which they are designed for is the same,
in order to exploit all the ferromagnetic material. To calculate the width of the tooth the
alternative direction of the magnets is ignored. The total flux that passes from the magnets
to the stator through the airgap is
φtotal = Bg Ag = pφm
(5.11)
where Bg , is the flux density in the airgap and Ag , is the area of the airgap. The whole
flux created by the magnets is divided among the total number of teeth in the stator, Qs ,
independently whether the flux is incoming or outcoming.
φtotal = Qs φtooth = pφm
(5.12)
φtooth = Btooth wtooth kj Lst
(5.13)
Linking (5.12) and (5.13) wtooth can be obtained as
wtooth =
Bm 2α(D − 2δ)
Bm 2αD
≈
Qs Btooth kj
Qs Btooth
(5.14)
Equations (5.14), (5.10) and (5.9) play a fundamental role in determining the cross-section
area available for the windings. These parameters hry , hsy and wtooth show a linear dependence on the diameter of the gap D and the magnetization of the magnets. However, it is
undoubtedly that there is a relationship between tooth width, stator yoke and rotor yoke if
some assumptions are taken into account. Taking care of exploiting as much as possible
the ferromagnetic material, the design for all the parts of the machine has to be done for
the same flux density, hence Btooth = Bry = Bsy and absence of flux leakage is assumed.
Eq 5.16 shows this relationship.
φtotal = pφm = p2φsy = Qs φtooth
(5.15)
φtooth
2p
wtooth
p
=
=
⇒ wtooth = 2 hry
φry
Qs
hry
Qs
(5.16)
47
Chapter 5. Analytical Design
The main difference of both is that hry and hsy exhibit an inversely dependence on the
number of poles whereas wtooth on the number of slots Qs . Thus, the number of poles
and slots for each diameter of machine is restricted, since hry , hsy and wtooth can be small
enough to be involved in mechanical troubles.
Once the cross-sections which guide the flux are calculated, the other dimensions
of the machine can be fixed or derived (Di , Dext and hss ). They have to be controlled in
the analytical process in order to not exceeding the design constraints.
Dext = D + 2hss + 2hsy
(5.17)
Di = D − 2(δ + hm + hry )
(5.18)
Once the dimensions of the teeth are calculated, the width of the slots can be determined
taking into account that the width of the tooth is constant.
τs =
πD
Qs
(5.19)
Where τs is the slot pitch. The width of the slot in the closes part of the tooth bss1 is given
by
bss1 = τs − wtooth
(5.20)
The width of the slot which is closest to the stator yoke is given by
bss2 = π
D + 2hss
− wtooth
Qs
(5.21)
To conclude the definition of the dimensions of the machine, the area of the slots is expressed as
bss1 + bss2
hss
(5.22)
2
The relation of how much bare copper is inside the slot is given by the fill in factor,
ff (Fig. 5.4). This factor takes into account the air between conductors as well as the
isolating layer which the conductors are sealed with. The maximum ideal fill in factor ff
reaches 0.79 [8].
Asl =
Acu = ff Aslot
(5.23)
where Acu is the bare copper area in the slot. Despite this ideal maximum, concentratedwinding machines reach ff values between 0.5 and 0.6 (round wire). This fill in factor
can vary depending the manufacturing technique used to implement the wire in the stator.
The conductor area is expressed as
Acond =
48
Acu
ns
(5.24)
5.2. Electric Model
b)
a)
Fig. 5.4 a) Slot filled with conductors; b) Maximum ideal fill factor with round wire.
where ns is the number of coil turns. The height of the conductor hss can be chosen
following different guidelines. Some ideas point out to reach the maximization of the
torque per ampere [26], whereas other try to reduce the copper resistance as much as
possible, since RAC depends on the height of the slot [11]. An important constraint once
the cross-sectional area of the machine is fixed is the product ns I. This describes the
MMF produced by each coil with ns turns carrying I Amps [8].
ns I = JAcu
(5.25)
J is the current density. Since this parameter is fixed, neither ns nor I can be changed
independently to maximize the torque, or reduce the copper losses as it can be appreciated
further, in equation 5.41. This constant is defined once the cross-sectional area of the
machine is defined and J is established as a design constraint. The current density is
usually an important design constraint since it determines the cooling system which the
machine is designed for. This constant varies depending the type of machine built [11].
Table 5.1: Permitted rms values for current densities for PMSM machines depending the
type of cooling system [11]
Irms [A]
Air cooling
2-3.5
Force cooling 7-10
5.2 Electric Model
Here are deployed the equations which model the electric behavior of the machine. Machine behavior is normally explained through the use of a dq frame reference. If harmonics
are neglected, steady state d- and q- voltage can be presented as
ud = Rs id − ωψq = Rs id − ωLq iq
(5.26)
uq = Rs iq + ωψd = Rs iq + ω(ψm + Ld id )
(5.27)
49
Chapter 5. Analytical Design
The peak value of the fundamental phase voltage Ê (back EMF) is obtained from the
derivation of the magnetic flux linkage at not no-load
dφm
dψm
=N
(5.28)
Ê = max
dt
dt
where N set the number of turns per phase and φm is the fundamental magnetic flux.
These magnitudes can be calculated as
p
N = qns kw1
2
(5.29)
π(D − δ) 2
sin(ωel t)
φm = φˆm sin(ωel t) = B̂δ L
p
π
(5.30)
where ωel is the electrical angular speed of the machine. Gathering equations 5.28, 5.29
and 5.30 in equation 5.31 the result is
Ê = qns kw1 ωel B̂δ L(D − δ)
(5.31)
As it has been shown in Chapter 2, SPM machines d- and q- inductance is quite similar
since the permeability of the magnets µr is close to one. Thus, Ld ≃ Lq = Ls . The
synchronous direct inductance is made up by the magnetizing inductance Lm and the
leakage inductance Lσ . The magnetizing inductance is calculated in [23] as
Lm =
3
µo
(qns kw1)2
L(D − δ)
π
δe + hµmr
(5.32)
The stator leakage inductance Lσ can be computed from the following equation which is
detailed in [17]
Lσ = pqn2s µo Lλ1
(5.33)
where the ratio λ1 is the specific permanence coefficient of the slot opening. Due to simplicity and the following 2D analysis end winding inductance is neglected. The external
voltage that the motor can perform is limited by the voltage range that the motor can perform. Since the DC supply is linked through a PWM converter, the maximum voltage in
the converter terminals is
q
UDC,min q 2
(5.34)
Umax = √
= ud + u2q = (E + Rs Imax )2 + (ωel Ls Imax )2
3
where id has been neglected, since field weakening is out of scope in this thesis. It is
important to take into account that the calculation of the synchronous inductance is an
approximation since its value is reduced when the machine is saturated [17].
50
5.3. Loss Model
5.3 Loss Model
One of the most important parameters of the electrical machines is how much power they
waste to produce power on the output shaft. In low-speed applications copper losses are
mainly the main source of wasting energy. Furthermore, magnets in SMPM synchronous
machines are exposed to harmonics, and they produce quantifiable losses. This section try
to show some models from this three different sources and suggest some ways of reducing
them. Stray and bearing losses are not modeled in this section.
Ploss = Pcu + Piron + Pmag
(5.35)
5.3.1 Copper Losses
Model a copper resistance of an electrical machine is important not only because of the
losses but also because its electric properties. The resistance is a propriety of all materials
which show the opposition of the material to let the current flow through it
ρl
(5.36)
A
where ρ is the resistivity [Ωm], l the length of the section and A the section of the conductor. The resistivity depends on the temperature which the material stands. Depending
the type of material, this dependency can be linear or exponential. For materials as the
copper [8]
R=
ρ(T ) = ρo (To )[1 + αt (T − To )]
(5.37)
where ρo is the resistivity at ambient temperature To . In the design process it is really
important to be aware about the warming-up temperature of the machine , since it determines the value of the resistance, the copper losses and thus the cooling system that the
machine needs. For copper wire, ρo,20◦ = 1.68 · 10−8 Ωm αt = 3.7 · 10−3◦ C−1 Taking into
account equation (5.36), and Fig. 5.5 the slot resistance for a concentrated winding can
be calculated as
ρntooth Lcoil
ρntooth Lcoil
Lcoil 1
Rslot =
= ρn2s
= Acu
(5.38)
Acoil
2 Acu
/ns
2
where ntooth is the number of turns per tooth, ns is the number of turns per slot, Lcoil is
the length of the coil, and the phase resistance is
Rs = Rslot
Lcoil 1
Lcoil 1 Qs
Qs
= ρn2s
= ρn2s pq
3
2 Acu 3
2 Acu
(5.39)
Eq. (5.39) shows that the resistance is dependance with the numbers of the turns squared.
Thus, the copper losses are defined as
Pcu = 3I 2 Rs = ρQs
Lcoil 1
(ns I)2
2 Aslot
(5.40)
51
Chapter 5. Analytical Design
where I is the rms value of the current in the machine. Introducing the constant (5.25)
in equation (5.40) confirms that the copper losses only depend on the volume of copper
used for the windings, the number of slots and the squared value of the current density
Pcu = 3I 2 Rs = ρQs Vcu J 2
(5.41)
where Vcu is the volume of the copper wires. Equations (5.40)– (5.41) shows that copper
losses for a cross sectional area fixed cannot be improved varying the number of turns, and
it only depends on the squared value of the constant ns I. The length of coil is calculated
as follows [3]
Average coil
2
Qs
Lact
1
Acu
ns
τs
wtooth
τs
b)
a)
Fig. 5.5 a) Cross-section of a stator with concentrated winding; b) Cross-section of a tooth with
concentrated windings. The striped area is the tooth.
Lcoil = 2Lact + 2Lend,av
(5.42)
where Lact is the active length of the machine and Lend,av is the average length of the
end-winding. Glancing Fig. 5.5 Lend,av is given by
τs − wtooth
1h
π
τs i
1
w
+
w
+
π
=
w
(2
−
(5.43)
)
+
π
Lend,av =
tooth
tooth
tooth
2
2
2
2
2
Introducing wtooth from equation (5.14) in (5.43) it can be concluded that Lend,av =
ktooth QDs . Thus, the value of Lcoil is given by
Lcoil = 2Lact + ktooth
D
Qs
(5.44)
and copper losses
Pcu = 3I 2 Rs = ρ(Qs Lact + ktooth D)
52
1
(ns I)2
Acu
(5.45)
5.3. Loss Model
It is important to emphasize that copper losses do not depend on the speed of the machine,
if the increase of the machine temperature is neglected during field weakening operation
[17]. In order to decrease this source of losses it is important to fill the slot as much
as possible (highest ff ), increasing the contact between wires. This makes better the heat
transfer between wires, therefore the temperature of the windings get lower, the resistance
decreases and also the copper losses [17].
5.3.2 Core Losses
The presence of an alternate magnetic field in the iron causes losses from different origin.
Hysteresis losses (Physt ∼ f ) are caused by the continue variation of the operating point
in the iron along the hysteresis loop. Eddy currents losses (Peddy ∼ f 2 ) are caused by
the EMF induced in the iron by the alternating field due to Faraday’s law. Modelling this
losses is not as straightforward as the copper losses [8]. Losses data given by manufacturers is often only available at 50–60 Hz only for sinusoidal current and there are parts
of the machine which transport flux with a high level of harmonics. But these estimations
are good enough to determine strategies on the machine in order to reduce these losses. If
it is assumed that the magnetic field density in the machine is sinusoidal, iron losses can
be computed as
piron = physt + peddy = khyst B βst f + keddy B 2 f 2
(5.46)
where physt and peddy are the hysteresis and the eddy current loss density, khyst and keddy
are hysteresis and eddy current constants, βst is the Steintmetz constant and f is the electrical frequency of the machine. The value of these constants depends on the ferromagnetic material used as well as the thickness of the lamination. Typical values for iron
laminations varies from khyst = 40 − 55, keddy = 0.04 − 0.07 and βst = 1.8 − 2 [17].
These parameters are usually calculated by fitting them to the data given from the manufacturers for sinusoidal conditions [8]. Considering βst = 2, khyst and keddy show a linear
dependence in (5.58) and these coefficients can be determined using linear least squares.
Assuming that there are not minor hysteresis loops, the hysteresis loss density in the whole
ferromagnetic material parts of the machine can be calculated as
physt = khyst B βst f
(5.47)
However, eddy currents are caused by other phenomena where harmonics play a fundamental role. According to Faraday’s law the current induced in a conductor is due to the
variation of the flux density dB
and the losses are directly related to the main square value
dt
,
similar
to
the
copper losses. The expression shown above (5.58) has to
of the rms of dB
dt
be changed, since this is only valid for sinusoidal flux waves and should be valid for all
the parts of the machine
B(t) = B sin(2πf t)
(5.48)
53
Chapter 5. Analytical Design
This is the sinusoidal flux density. Its derivative expression is
dB(t)
= 2πf B cos(2πf t)
dt
(5.49)
and the square value of the rms value is
dB(t)
dt
2
= 2π 2 f 2 B 2
(5.50)
Using (5.50) in the eddy current loss part in (5.58) leads to a more general expression of
eddy currents
peddy
keddy
=
2π 2
dB(t)
dt
2
(5.51)
Since eddy currents losses increase with the variations of the magnetic field, the teeth are
the ferromagnetic part of the machine that much suffer this phenomena. The flux density in the teeth rises quickly when the edge of the magnet is approaching and decreases
quickly when the lagging edge of the magnet passes long the teeth, leading the flux density from maximum to zero. This extreme variation leads to calculate teeth eddy current
losses density in the following way [23]
peddy,teeth =
4D
2
keddy f 2 Bts
πpwtooth
(5.52)
Some strategies can be adopted on the design process in order to decrease this source of
losses as much as possible [17] [8]
• Reducing lamination thickness. Eddy current losses are directly proportional to the
square of the thickness.
• Reduce the level of flux density in the ferromagnetic parts of the machine.
• Reduce the number of magnets poles. Have a great number of poles in the machine
entails lots of advantage i.e. reduction of the machine size, but it entails disadvantages too, as increasing the electric frequency and consequently, the losses. Thus, a
good trade-off of these proprieties have to be achieved with the number of poles.
• Increase the resistivity of the laminations in order to reduce eddy currents.
• Annealing laminations after they have been cut or stamped, since it eliminates the
influence of mechanical stress on iron losses.
54
5.4. Analytical Program
5.3.3 Magnet Losses
Rare earth permanent magnets are conductive materials, 5–10 times more conductive than
electric steel (110 − 170 · 10−8 [Ωm]) and they are exposed to an alternate magnetic field,
that is the reason why eddy current are induced in them. Indeed, the location of the permanent magnets in SMPM synchronous machines contributes larger in these losses since
they are exposed to slotting effect and great amount of harmonics. Moreover, harmonic
content in concentrated windings is higher than distributed windings. Nonetheless, it is
difficult to give an accurate analytical model of this losses. However, some techniques
can be applied to reduce the losses in the magnet [20] [11]
• Magnets can be segmented axially or circumferentially in order to limit eddy current
effects.
• The use of plastic bonded magnets in stead of sintered magnets can eliminate eddy
current. However, the magnetic proprieties of them are worse and the amount of
magnetic material needed is longer.
5.4 Analytical Program
Once the requirements are calculated and the theory of concentrated windings, magnetic
and electric behavior has been deployed, the constructive parameters of the machine can
be calculated. The design procedure is carried out by an own numerical tool in M ATL ABR , mainly based on the theory mentioned above. The machine is designed in order to
reach Tsteady . Thus, it has to be checked that they are able to reach the Tpeak through the
use of FEMMR .
5.4.1 Objective
With regard to the objective of this thesis, different selection criteria can be set up to design the motor. This design can be focused on one of the principal aims of this machine,
lightweight or efficiency. Motor efficiency seems to be more important than lightweight
considering the consumption of energy point of view. Nevertheless, since the motor is
located inside the wheel (unsprung mass problem) and the set of losses is not accurate
enough (PM losses), this program is concentrated on achieving the lightest design possible. Furthermore, a light design entails the advantage of less material, and as a result a
cheaper design.
55
Chapter 5. Analytical Design
5.4.2 Additional Equations
Torque
The obtention of the torque equation derived from [23] is explained in
D2
T = π kw1 B̂δ ŜLact sin β
4
(5.53)
where Ŝ is the current linear density of the motor and sin β is the saliency of the machine.
In the design process, the torque is fixed to its rated value. The other parameters shown
in equation 5.53) reveals somehow the relation between the constructive parameters from
the machine. S is directly related to the slot area. For a fixed external diameter, the area
of the slot influences the rotor diameter. The torque is not designed for field weakening
operation, id = 0, and this means that sin β = 1. The active length of the machine Lact
has a great influence on the machine weight, hence, it should be limited. If the constant
ns I is considered as Ŝτs = ns Iˆ = Acu Jˆ [23], the torque can be expressed as
T =
D
ˆ s Lact
kw1B̂δ Acu JQ
4
(5.54)
where the torque performed by the machine is derived from their constructive parameters.
Slot Height
As has been mentioned in Section 5.1.2, some parameters have to be established to calculate the whole geometry of the machine. In [26], it is explained that the torque performed
Dgap
by a PMSM per ampere can be maximized based on the slot ratio, d = Dgap
. The
+hss
shear stress in the airgap is considered as [26] [11]
σ = B̂δ Ŝ
(5.55)
This shear stress can be considered as the responsible of developing the torque on the
rotor
T = σAg
D
π
= D 2 Lact σ = 2Vrot σ
2
2
(5.56)
where Vrot is the volume of the rotor and Ag is the area of the airgap. As it has been
mentioned in Chapter 2, the torque is proportional to the volume of rotor, where the shear
stress is the product of magnetic and electric loading. This value should not overpass
48000 Pa [11]. Hence, it is easy to realize that torque performance is obtained tradingoff the volume of the rotor and the available area for the stator conductors. In fact, for a
fixed external diameter Dext fixed, the larger the diameter of the rotor D, the lower space
for the conductors will be. Therefore, it has to exist an optimum value of the rotor outer
diameter that maximize the torque performance. As mentioned above, stator teeth have
56
5.4. Analytical Program
parallel sides to exploit all the ferromagnetic material. The electric current linear density
S is roughly proportional to the area of the slot, thus A ∼ (1 − d2 ). So the expression of
the torque is given by
T ∼ σVr ∼ (1 − d2 )d2
(5.57)
This equation 5.57 has maximum value in d = √12 . Thus, the value of this proportion of
the slot in order to maximize the torque should be 0.71, but regarding machines designed
for a great number of poles it is convenient to increase this number between 0.8-0.85 so
as to reduce the quantity of magnetic material and consequently the iron losses [26].
5.4.3 Design Variables
These are the input variables that calculates all the geometric parameters of the machine in
each loop, using some fixed values and keeping the results within some constraint values.
They are shown in Table 5.2.
Table 5.2: Design variables.
Variable
Symbol
Range
Number of poles
p
4 ≤ p ≤ 60, p is even
Slots per pole and phase q
0.25 ≤ q ≤ 0.5
Diameter of the Rotor
D
100 ≤ D ≤ 300 mm
5.4.4 Constants
These values are considered constant in the analytical program. They are properties from
the materials used in the machine.
5.4.5 Fixed Values and Constraints of Design
There are some values fixed for the design calculations. The parameters calculated have
to be inside some margins of design.
5.4.6 Matlab Program
In Fig. 5.6 it is depicted how the analytical program based on the information shown above
works. After calculations shown in the big square of Fig. 5.6, the matrix is tidy regarding
the weight of the designs. After that, machines which are till 0.3 kg heavier are analyzed.
These machines have to be checked that they reach the peak torque Tpeak under overload
capacity. As well, since the models for inductance of the machine are not very accurate,
57
Chapter 5. Analytical Design
Table 5.3: Constants.
Constant
Symbol
Remanent flux density
Br
Relative permeability
µr
Magnet Temperature sensitivity Tk
Magnet density
ρm
Magnet temperature
Tmag
Steel density
ρiron
Eddy loss constant
keddy
Iron
Hysteresis loss constant khyst
Steinmetz constant
β
Copper density
ρcu
Copper resistivity
rcu
Winding
Copper fill factor
ff
Copper Temperature
Tcu
Value
1.2 T
1.05
0.001 T/K
7700 kg/m3
120◦ C
7700 kg/m3
4.26·10−5
0.0269
2
8930 kg/m3
2.392·10−8 Ω·m
0.6
150
Table 5.4: Fixed values and constraints.
Peak fund. flux density in stator yoke Bsy
1.6 T
Peak fund. flux density in teeth
Bst
1.6 T
Peak fund. flux density in rotor yoke Bry
1.6 T
Peak fund. flux density in airgap
B̂δ
0.9 T
ˆ
Max. peak fund. current density
J
3·106 A/m2
Air gap length
δ
1 mm
Electrical magnet angle
α
120◦
Rotor back thickness
hry
hry ≥ 5 mm
Active length
L
L ≤ 20 mm
Magnet Temperature
Tmag 120◦ C
the number of turns ns it has to be also checked if the phase voltage of the machine does
not reach the minimum voltage range of the battery times 0.9, in order to allow voltage
for the control. Otherwise, they have to be modified to fulfill this requirement.
UDC,min 0.9
√
Uˆph ≤
3
(5.58)
5.4.7 Results
From the program shown above, different motor models are obtained. The lightest model
is analyzed further through an own-developed program over FEMMR package. In App. B
are detailed the measures and the parameters of the lightest design.
58
5.4. Analytical Program
INPUT
Fixed values
Tsteady
Setting design variables variation
variable data each loop
Design Constraints
Concentrated windings
maximize
torque
CALCULATION MACHINE
GEOMETRIC PARAMETERS
WEIGHT & LOSSES
OUT
fulfill constraints?
OK
Keep in result Matrix
Selection
Criteria
Short Matrix
Lightest
Results
-0.3 kg
Cogging Torque
HIGH
CHECK
Overload Capacity
FEMM
Fig. 5.6 Flow program design
59
Chapter 5. Analytical Design
60
Chapter 6
Finite Element Analysis
Throughout this chapter, the machine calculated in the analytical model is evaluated.
This evaluation is carried out through the use of an own-developed program based on the
Finite Element Method Magnetics package (FEMMR ). In addition, the tooth shape of the
machine is selected through the analysis of the cogging torque. To conclude, the designed
machine is analyzed an evaluated at different load requirements.
6.1 FEMM Model
The Finite Element Method environment used in this thesis is FEMMR . It is a finite
element software developed by David Mecker which can process magnetics, electrostatic
and heat transfer problems. However, unlike other finite element magnetic softwares as
FLUX2DR and JMAGR , FEMMR does not provide any model of the wide variety
of electrical machines. For this reason, a SPM synchronous machine has to be modeled
in FEMMR environment. M AT L ABR language is used to design this model following
the command guidelines Octave FEMMR [15]. The purpose of the designed program
is to obtain different results such as the back EMF, the phase voltage of the machine,
the torque, the inductances, etc. for a given input, which gather, on the one hand the
constructive parameters of the machine and on the other hand the power input represented
in a dq frame reference. In the following steps it is described how the program is designed
and how it works.
6.1.1 Input Parameters
Constructive and Geometric Parameters
First of all, the machine parameters calculated in the analytical model have to be acquired by the program. This is computed as an input vector which contains the geometric
parameters, the number of poles, the number of slots and the number of turns. It is im61
Chapter 6. Finite Element Analysis
portant to point out that the number of turns ns calculated in the analytical model are
number of turns per slot. When the machine is drawn in FEMMR , the next step is labeling each block with its material properties. Since the machine designed uses double layer
concentrated-windings, each slot is divided in two blocks. Therefore, it is suggestible to
consider programming the number of turns per tooth to make easier the definition of the
materials.
Power Input Parameters
Once the machine is completely modeled (shape and materials determined), dynamics has
to be simulated. Hence, the machine input has to be related to the torque and the speed
performed by the machine. First, the mechanical speed, ωmech is a direct input. On the
other hand, the torque performed leads to a current input, q- current
3
T = pψm iq
4
(6.1)
In order to extract iq from this equation, first ψm is calculated as the flux linkage fundamental harmonic at no load, and used as input parameter. For this reason, a dq currents
frame reference is implemented as input. Nevertheless, since the machine is designed
with concentrated windings, some assumptions to implement a dq model are not accomplished [16]
• Windings should be sine distributed.
• Magnetic circuit should not be saturated.
Despite this fact, in [16], states that the dq model can be applied to concentrated windings.
In order to transform these d and q input currents to phase currents, the Park transformation has to be applied. This is necessary because current blocks in FEMMR have to be
labeled with phase currents. The Park transformation can change three-phase components
fa , fb , and fc , which can be either currents, voltage or fluxes into 2 variables, fd and fq ,
which are in other frame reference, and viceversa [29]. This is described in 6.2
  
 
fa
cos(θ)
− sin(θ)
1 fd
 fb  = cos(θ − 2π ) − sin(θ − 2π ) 1 fq 
(6.2)
3
3
2π
2π
fc
cos(θ + 3 ) − sin(θ + 3 ) 1
fo
| {z } |
{z
} | {z }
fph
Tdq,ph
fdq
where θ is the angle between the d axis of the rotor which is the direction of the north pole
of the rotor, and the starting position of the electric period in electric degrees. The starting
position is defined where is found the maximum magnetic flux linkage of the phase A,
as illustrated in 6.2. In a distributed winding, this starting position is straightforward to
be calculated, as described in Fig. 6.1a. However, as illustrated in Fig. 6.1b and Fig. 6.2,
62
6.1. FEMM Model
d
θi
θ
θ
Initial position
of the electric
period
d
A-
A
b)
a)
Fig. 6.1 a) Distributed winding rotor angle; b) Concentrated winding rotor angle and initial electric angle period, θi
.
it is difficult to calculate this starting position for concentrated windings, θi . One way
to determine θi is calculating ψa for the whole electric period and establish where is
the maximum. This is because concentrated winding layouts follow different patterns, as
illustrated in 6.2 Through the use of this own-developed program, different configurations
Initial position
of the rotor
B' B C' C A' A A A'B B' C C'
ψA
No load
analysis
Δθi,el
θel
θm
Fig. 6.2 Initial position of the rotor in a SPM synchronous machine and the initial phase difference
between the d− axis of the rotor and the flux linkage of A phase.
of slot and poles have been tested in order to obtain the angle where the flux linkage is
larger.
6.1.2 Machine Geometry
A sector that covers one slot pitch of the machine, τs has to be plotted in FEMMR using
the input vector above. After that, this sector is copied Qs times for the stator and p times
for the rotor. It is important to delimit properly the different areas, in this case specially
the conductors, since each slot is going to allocate two different types of wire phase.
63
Chapter 6. Finite Element Analysis
6.1.3 Material Settings
For each region created in the step above, material characteristics have to be set. Three
main issues have to be taken into account for a PMSM of concentrated winding. First of
all the direction of the magnets has to be established, taking into account that it is a part
of the machine that rotates. Moreover, the iron is not linear, so the hysteresis curve of the
steel has to be set as well. Finally, the windings have to be set according to the obtention
of the winding layout shown in Chapter 3 through the Cros’ method. Here the materials
used during FEMMR processing are detailed.
Magnets
The selection of the magnet is conditioned by the temperature risk. In order to deal with
this problem, VACODYM 890 TP has been the choice of this thesis since this type of NeFeB magnets can withstand higher temperatures. In Fig. 6.3 the demagnetization curves
Fig. 6.3 Demagnetization curves B-H for VACODYM 890 TP at different temperatures [28].
for different temperatures are deployed. As well, it will be analyzed if the magnet overpasses this threshold. It is important that the magnets do not reach this point of demagnetization since they can be irreversible demagnetized and consequently the motor would
behave different from how it was designed.
Electrical Steel
The ferromagnetic material chosen for guiding the magnetic flux density is laminated
steel. It is important to choose the laminations as thin as possible, in order to decrease the
64
6.1. FEMM Model
iron losses. The selected iron is the M235-35A (.35 mm) provided by Surahammars Bruk
AB [27]. BH curve is provided in Fig. 6.4 and loss density curves at different frequencies
2
B(T)
1.5
1
0.5
0
0
2000
4000
6000
8000
10000
12000
H(A/m)
Fig. 6.4 BH Hysteresis curve at 50 Hz for M235-35 steel [27].
Loss density (W/kg)
are depicted in Fig. 6.5.
50
40
30
frequency
increase
20
10
0
0
0.5
1
1.5
B(T)
Fig. 6.5 Iron loss density of the M235-35A magnetic steel at different frequencies 50, 100, 200,
400, 1000 and 2500 Hz [27].
6.1.4 Accuracy Parameters
Mesh Size
Mesh size is a very sensitive parameter. On the one hand the size of the mesh defines how
accurate the calculation will be. If the mesh size is big enough it can contribute to poor
calculations. On the other hand, this high accuracy with extremely small mesh size can
lead to long calculation times. Thus, it is important to achieve a good trade-off between the
accuracy of the mesh and the time consumption. Before defining the mesh in the different
blocks of the machine it is important to take into account that there are some parts of
the machine which contribution to the final computation result is larger. Therefore, the
airgap and the magnets have to use a smaller mesh size in order to achieve good results.
65
Chapter 6. Finite Element Analysis
This program uses a general mesh of 3 mm for all the elements in the machine excluding
the airgap and the magnets, for which a mesh size of 0.3 mm is used, as it illustrated in
Fig. 6.6
Fig. 6.6 Mesh sizing.
Iterations
This is the number of steps in which the rotor completes 2π electrical degrees. In each
iteration data is obtained for further processing. In the manner of the mesh size, it has to
be a good trade-off between the accuracy of the final wave obtained and the number of
iterations. For this machine, each electric period entails 360 iterations. When currents are
varied, 20 iterations per period is good enough.
6.1.5 Data Obtaining
FEMMR is able to get different data in each iteration. Flux linkage, current and voltage
drop of each phase can be obtained from the circuit. From the remaining parts of the
machine (stator back, rotor back, magnets, airgap and stator teeth) magnetic data from
relevant points is acquired. The torque can be computed in different ways, either using
the Maxwell stress tensor applied on the rotor or using a line integral of the force inside
the airgap [15]. As well, it can be obtained based on the electric power consumed by each
phase but it entails error since the back EMF is not directly taken from the FEMMR .
6.1.6 Analysis
Data obtained in each iteration is kept in different vectors for its later analysis. It is important to realize that the results obtained are not continuous, but discrete. Thus the accuracy
of the post processing results depends on the number of iterations. In order to derive the
66
6.1. FEMM Model
flux linkage, the discrete derivative is used, as described in equations 6.3 and 6.4
dψa
dψa dθ
dψa
=
=ω
dt
dθ dt
dθ
(6.3)
where the discrete derivative from the flux linkage is
dψa
dθ
=
i
ψa,i − ψa,i−1
∆θ
(6.4)
6.1.7 Program Flow
Fig. 6.7 shows the different features that the program calculates, keeps as data and plots
in M AT L ABR .
INPUT
POSITION
- Angle [0.360]
- q current
- d current
-Flux density curves
of certain regions
(Bδ ,Bmagnet)
-Flux density analysis
scaled.
-Airgap strength
analysis
Linear or No Linear?
PERIOD
- ωmech
- q current
- d current
- Iterations
- ψd(θel)
- ψq(θel)
- T(θel)
- Power (θel)
- Iron losses
- Copper losses
- Bsy(θel), Bry(θel),
Btooth(θel)
- Bδ(θel), Bm(θel),
- ψA(θel), IA(θel) ,UA(θel)
- dψ/dt(θel)
NOMINAL POINT
- Max current
- ωmech
- Iterations/period
- Iterations/Imax
- ψd(id): Ld
- ψq(iq): Lq
- T(iq)
- Peak2Peak(iq)
- Tripple(iq)
- Bsy, Btooth
Fig. 6.7 Interface menu and different features offered by the program.
67
Chapter 6. Finite Element Analysis
6.2 Machine Analysis
6.2.1 Target
The importance of this analysis lies in the fact of not surpassing certain values in determinate parts of the machine. First of all, two design conditions have to be differentiated.
At rated point conditions, magnetic flux parameters set in the analytical model have to be
check. On the other hand, overload capacity also has to be tested, which entails several
tasks
0.9U
dc,min
√
• Phase voltage must not exceed
, since this is the lowest value that the bat3
tery can operate, giving also some margin for the control system.
• Magnetic flux on the magnets must not surpass Bknee defined for 150◦ C, in order
to avoid the demagnetization effect.
• Saturation on the core entails bigger current to achieve the designed Tpeak .
Moreover, different important properties of the machine are analyzed through this chapter,
in order to show all the properties performed by the designed machine.
6.2.2 No Load
Tooth Shoe
Although the analytical model calculates most of the parameters of the machine, the tooth
shape of the machine is not considered. As it has been mentioned during chapter three,
cogging torque depends on the variation of the energy in the airgap. This variation was
higher when the reluctance changed, in other words, at the end of the tooth tip. These variations of reluctance are produced in all the teeth, and they can be cancelled by the torque
produced in other teeth, or increased by the same reason. This is the primary purpose
for tooth shoes [8]. Thus, if the proportion between the tooth opening and the slot width,
kopen is varied from 0 to 1, there must exist a configuration where the cogging torque is
smaller. Different slots openings have been tested in the machine designed, concluding
that the most suitable kopen is 0.75, as shows Fig. 6.8. So the final shape of the machine
analyzed is displayed in Fig. 6.9
Cogging
Some authors consider the cogging torque as the variation that occurs in the magnetic
energy in the airgap, as mentioned in Section 4.5.2. Concurrently, other authors consider
the cogging torque as a result of the variation of the reluctance along the gap [8]
1 dR
Tcog = − φ2
2 dθ
68
(6.5)
∆T(Nm)
6.2. Machine Analysis
0.32
0.24
0.16
0.08
0
0.2
0.4
0.6
0.8
Kopen
Fig. 6.8 Influence of the open width of the teeth with the cogging torque.
C+ A-
A+ A+
A-
B+
CB+
BC+
BB-
CC-
B+
A-
C+
A-
A+
C-
A+
B-
C+
C+
B+
B+
C-
C+
B+
B-
A+
A- A-
B-
A+ C-
Fig. 6.9 SPM Synchronous Machine designed. Cross-section.
Considering both statements, the variation undergone is related to the number of slots
that contains the machine. Their slot openings is the source of this undesirable torque.
Therefore, the frequency of this torque can be considered as
p
fcog = Qs f
(6.6)
2
where f is the frequency performed by the machine. Fig. 6.10 shows the cogging torque
performed by the machine in an electric period. It can be appreciated through its Fast
Fourier Transformation (FFT) analysis, the main harmonic is the 18th , as the number of
slots. This was the cogging frequency expected.
Airgap Flux
Slotting effect creates a great harmonic content in the sinusoidal flux density wave. It
is revealed through the analysis of the magnetic field density in the airgap, exhibited in
Fig. 6.11. Fig. 6.11 shows different peaks and wave-shapes far from a sine-wave, which
69
T(Nm)
Chapter 6. Finite Element Analysis
0.01
0
−0.01
−0.02
−0.03
0
60
120
180
240
300
T(Nm)
θe (rad)
0.015
0.01
0.005
0
0
5
10
15
20
25
30
35
40
nth harmonic
Fig. 6.10 No load torque.
indicates a great harmonic content. These harmonics are faced by the shoe of the teeth
and by the magnets. This great harmonic content as well as the lack of sine-distributed
windings, make concentrated winding machines to have a high quantity of losses in tooth
tips and magnets.
Back EMF
As it is illustrated in Fig. 6.12, it is a sine-wave harmonics free. According to equation 5.28, EMF is related to the winding factor of all the harmonics and the flux produced
by the magnets in the airgap. Regarding Fig. 4.12, which shows the whole spectrum of
winding factors, shows that the 5th harmonic is 0, and their neighbors are relatively low.
In Fig. 6.11, it can be realized that the 5th harmonic is the bigger one. Both sources of
are cancelled. In Table 6.1 some important parameters shown in the no load analysis on
the FEMMR are exhibited. All the values are lower than it was expected in the analytical
model. This can be due to the change of tooth tip. Increase the tooth tip involves that the
flux leakage between two neighboring tips also rises. This contributes to the drop of these
values. Likewise, slotting effect can also participate on the decline of the magnetic flux
above the magnets, Bm .
Table 6.1: No-load characteristics values
Analytical values
FEMM values
Êo (V)
11.77
10.97 (7.29%)
Bδ (T)
0.9
0.82 (9.75%)
Bm (T)
0.8162
0.7302 (11.77%)
ψm (Vs)
0.0526
70
6.2. Machine Analysis
B(T)
0.5
0
−0.5
0
60
120
180
240
300
B(T)
θe
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
nth harmonic
Fig. 6.11 Magnetic flux density in the airgap.
E0 (V)
10
0
−10
0
60
120
180
240
300
360
θe (rad)
Fig. 6.12 Back EMF.
6.2.3 Rated Point Conditions
Regarding Fig. 6.13, it can be noticed that the magnetic flux density in the machine is
symmetrical, GCD(Qs , p) = 2. This can exhibit that the machine does not perform an
unbalanced torque as explained in Section 4.5.1. Forces performed inside the machine are
compensated for opposite sizes. Moreover, concerning the magnetic flux density values
performed by the machine in an electric period, Table 6.2, their values are lower than it
was expected in the analytical design. This probably is attributable to the lack of linearity
in the iron, since values of magnetic flux density around 1.5 T indicates a little saturation.
The torque performed in Fig. 6.14 is close to 7 Nm, considering that it exists a little saturation level on the machine. The ripple performed by the machine at this designed point
is quite low, 4.28%, which is acceptable. As is depicted in Fig. 6.15, the harmonic flux
content in the magnet is considerable. The main harmonic of the magnet does not generate
any eddy currents, due to it magnet itself, but the harmonic does. Through the FFT anal71
Chapter 6. Finite Element Analysis
Fig. 6.13 Cross-section of the machine at nominal conditions, at maximum flux linkage in phase
A.
Table 6.2: Magnetic flux density at rated point
Analytical values FEMM values
Btooth (T)
1.6
1.505
Bsy (T)
1.6
1.521
Bry (T)
1.6
1.506
ysis of this figure, it can be noticed the great amount of harmonics. Since flux harmonics
are a critical aspect for the eddy currents generation, magnets should be segmented in
order to reduce this source of losses as much as possible. This was expected, since SPM
synchronous machines with concentrated windings perform a great quantity of harmonics in the airgap flux, as well as the magnets are not shielded from the slotting effect. In
Table 6.3 the values performed by the machine at rated point are described. Despite the
core losses are quite short since the electric frequency of the machine is low, the copper
losses are big enough to make the efficiency be under 90% of efficiency. Considering that
the value of the output power of this machine is close to 200 W, mainly because it is a
low-speed drive, it is easy that a low quantity of copper losses can lead to low efficiencies.
T (Nm)
7
72
ˆ
I(A)
13.58
Table 6.3: FEMMR values at rated point
Ua (V) Jˆ(A/mm2) dψ
(V) Pcu (W) Piron (W)
dt
12.9
2.74
11.24
34.32
1.53
η(%)
82.21
felec (Hz)
32.1
T(Nm)
6.2. Machine Analysis
7.1
7
6.9
6.8
0
60
120
180
240
300
360
Tripple (Nm)
θe (rad)
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
30
35
40
nth harmonic
Fig. 6.14 Torque at load condition.
6.2.4 Overload Condition
As it is depicted in Fig. 6.16, the machine is widely saturated. Iron losses will arise as
well as higher current will be need to achieve the peak torque estimated. This will lead
to a low efficiency. As well, since the losses are critically increased at this point, the
inside temperature of the machine will rise and some parameters calculated may vary,
as the resistance of the copper. This temperature will affect to the magnets properties,
and it is essential to evaluate where the value of Bknee is, in order to determine whether
the magnets can be demagnetized or not. In Table 6.4, maximum values achieved by
the magnetic flux density waveform are illustrated. Fig. 6.17shows the different sources
Table 6.4: Magnetic flux density at rated point
FEMM values
Btooth (T)
1.72
Bsy (T)
1.705
Bry (T)
1.71
of voltage in the machine. It can be appreciated that the phase difference between the
different sources of voltage is the one expected, compared to Fig. 6.18. As mentioned
above, magnets cannot surpass certain magnetic flux limit, Bknee , in order to not being
demagnetized. Through the study of Fig. 6.19, it can be appreciated that the lowest value
of magnetic flux density in the magnet is 0.4 T. Thus, regarding Fig. 6.3, magnets can
reach 240o C, which is far from the estimated designed temperature. In this figure, it can
be appreciated that not all the points in the magnet experience the same flux density. In
73
Chapter 6. Finite Element Analysis
B(T)
0.9
0.8
0.7
0.6
0
60
120
180
240
300
360
B(T)
θe
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
nth harmonic
Fig. 6.15 Magnetic flux density in the middle point of the magnet at rated conditions and its FFT
analysis
Table 6.5 are summarized the most important points of the machine at overload capacity.
The current has been increased more than 3 times, since the machine is saturated. As
well, the value of the phase voltage of the machine is lower than 18.7 V, which is the
maximum voltage that the machine can reach. Copper losses an iron losses have been
widely affected, since this is not the point they are designed for.
T (Nm)
22
Table 6.5: FEMMR values at overload point
ˆ
I(A)
Ua (V) dψ
(V) Pcu (W) Piron (W) η(%)
dt
43.45 18.51 13.92 368.89
1.75
41.497
felec (Hz)
32.1
6.2.5 Different Working Points
Throughout the following figures, different operating points of the machine are described,
showing their characteristics. First of all, in Fig. 6.20 it can be appreciated how the machine torque does not evolute linearly with the current, due to the saturation, as it was
expected. The more iq current on the machine, the more saturated it becomes. This can be
appreciated in Fig. 6.21. This figure describes the evolution of the magnetic flux density
in the stator back (green line) and in the tooth (blue line). At no load operating point, the
flux density in the stator back is higher than the one in the tooth. While iq increases, so
does both magnet densities, but since the saturation on the tooth is lower at the beginning,
the rise of this magnetic flux is higher. When the q current is close to 22 A, both magnetic
flux densities are equalized. When iq is close to the overload point (30 A), both flux densities grow equally. Furthermore, looking at the evolution of ψq , it is also noticeable the
saturation effect. Fig. 6.22 shows the drop of the inductance when iq reaches 25 A, which
74
6.2. Machine Analysis
Fig. 6.16 Cross-section of the machine at overload conditions, at maximum flux linkage in phase
A.
is almost double the designed current. Fig. 6.23 shows how the ripple grows according to
the definition of Tripple shown in Fig. 4.14.
75
Chapter 6. Finite Element Analysis
a
, Rs Ia (V)
Ua , dψ
dt
20
10
0
−10
−20
0
60
120
180
240
300
θe
20
Ua (V)
15
10
5
0
0
5
10
15
20
25
30
35
40
45
nth harmonic
Fig. 6.17 Voltage sources of the machine at overload conditions. RED: Rs Ia ; GREEN:
BLUE: Ua .
dψa
dt ;
q
-wLq iq
Rsiq
Rsiq
Ua
dψa/dt
Eo
d
Fig. 6.18 Phasor diagram of the voltage in the machine in a dq frame reference, over iq > 0 and
id = 0.
76
B(T)
6.2. Machine Analysis
0.9
0.8
0.7
0.6
0.5
0
60
120
180
240
300
360
B(T)
θe
0.2
0.1
0
0
10
20
30
40
50
nth harmonic
Fig. 6.19 Magnetic flux density in the weakest point of the magnet, in three electric periods.
25
T(Nm)
20
15
10
5
0
5
10
15
20
25
30
35
40
ıq (A)
Fig. 6.20 Torque evolution with q-current and its linear behavior.
B(T)
1.8
1.7
1.6
1.5
1.4
0
5
10
15
20
25
30
35
40
ıq (A)
Fig. 6.21 Evolution of the peak value of the magnetic flux density in the tooth (blue line) and in
the stator back (green line).
77
Chapter 6. Finite Element Analysis
ψq (Vs)
0.04
0.02
0
5
10
15
20
25
30
35
40
25
30
35
40
ıq (A)
Lq (mH)
1
0.95
0.9
0
5
10
15
20
ıq (A)
Tripple (%)
Fig. 6.22 Evolution of the value of the q-flux linkage ψq and the q-inductance of the machine Lq .
6
4
2
0
0
5
10
15
20
iq
25
30
Fig. 6.23 Evolution of torque ripple.
78
35
40
Chapter 7
Conclusions and Further Work
This chapter summarizes the conclusions obtained in this thesis, as well as criticize it.
Recommendations for further work are deployed here
7.1 Conclusions
Throughout this thesis, an electric motor prototype for the Shell Eco Marathon has been
designed. After creating an analytical model which calculates the parameters of the machine, it is analyzed in FEMMR . First of all, the topology of the machine was selected.
In order to improve the drive train as much as possible, gearboxes should be removed.
This leads to an low-drive direct IW motor design. Several topologies of PM machine are
qualitative studied, radial (SPM, Inset, IPM) and axial, showing their advantages and their
drawbacks concerning the design. Finally, based on this qualitative study and some design
constraints, such as the air-cooling system or economic reasons (axial design), a radial
SPM synchronous machine of inner rotor was chosen for the design. After that, the requirements of the machine are calculated. According to the requirements of the Shell Eco
Marathon 2013 urban concept prototype and based on dynamic Newton’s law, the maximum speed of the machine and the rated torque and the overload torque are calculated.
Regarding the winding arrangements, concentrated windings have been demonstrated the
best solution in efficiency terms, since they have the smallest end-windings. However, the
selection of the number of the poles and slots, since they can perform unbalance forces
or required with a high level of torque ripple. In order to avoid this configurations, lots
of configurations cannot be implemented. Following, an analytical model is developed to
calculate the dimensions of the machine and some parameters of the machine, based on
its magnetic and electric behavior, and some design constraints. The machine resulting is
analyzed through an own developed program created in FEMMR . This program for concentrated windings, entailed several difficulties as the design of the winding layout and
to determine the initial position of the rotor to apply the Park transformation. Finally, the
machine calculated in the analytical program, is implemented in the FEMMR program,
79
Chapter 7. Conclusions and Further Work
and adjusted some parameters such as the width of the tooth and the number of turns to
fulfill the design constraints. The efficiency obtained at the rated point is not as high as
expected, mainly because the required power of the vehicle is small. Furthermore, the
permanent magnets have been analyzed at overload conditions, showing that the temperature that the magnets have to reach in order to be demagnetized has to be higher than
200 o C. The magnets should not achieve this temperature because according to [23], the
copper wires operate on higher temperatures, and the highest limit temperature for the H
insulation is 180◦ C. Moreover, the machine performs the requirements calculated without
surpassing the maximum phase voltage provided by the minimum voltage of the battery.
The overload torque is reached over saturation conditions in the iron. To conclude, there is
not an analytical model implemented on the iron losses on the machine (almost neglected
since it is a low-speed machine) and on the magnets. Permanent magnets exhibits a great
quantity of harmonics, which can lead to a big source of losses. Thus, they should be segmented axially. Regarding the results, iron losses are negligible if they are compared to
copper losses, because the IW is a low-speed application and the electric frequency performed is quite low (32.1 Hz). The efficiency is lower than it was expected. The reason
of this is that the design of the motor is focused on a lightweight design, instead of a high
efficiency design. Thus, it would be recommendable to decrease the current density and
find a better trade-off between a lightweight and a efficiency design.
7.2 Recommendations for Future Work
• The analytical model used could be improved. Teeth shoes have not been taken into
account, and they can vary the weight of the machine and the amount of PM needed,
due to their relation to the karter factor. The inductance model can also be improved
also. Besides, an analytical model for the torque ripple could be implemented.
• FEMMR does not provide a result for iron losses in absence of frequency since the
design of the FEMMR has been done setting the frequency parameter to 0 due to
specifications of the program for PM. Thus, the iron model should be tested in other
program as F LUXR , or JMAGR .
• A thermal model should be implemented to check the most vulnerable parts of the
machine. This could be accomplished through the creation of a lumped circuit, and
test it with M OTOR -CADR , or on the other hand, in order to improve the program
designed in this thesis, create an thermal program based on FEMMR .
• PM losses model could be implemented in FEMMR and tested it in a reliable
software.
• In the program developed, end-winding has been introduced in the drop voltage
of the windings. However, leakage winding inductance has not been taken into ac80
7.2. Recommendations for Future Work
count. An analytical model could be designed to be implemented in FEMMR and
checked the end winding effect on FEMMR with a 3D FEM software.
• Other alternatives discussed on Chapter 2 could be analyzed, since some of them
might seem better to achieve the targets of ”Shell Eco Marathon 2013”. SPM inner
design improves the properties of this model, but its study should be coupled with
a careful study of the thermal transient of the machine, since it is air cooled.
81
Chapter 7. Conclusions and Further Work
82
Appendix A
Dynamic simulation
The simulation created in Simulink which is attached further in this appendix, is an ideal
simulation only based on Newton’s second law which does not take into account important
parameters as the efficiency map of the motor and the converters. The power that the motor
has to perform is described in (A.1).
Pmot = Pdyn + Pdrag + Prr
(A.1)
where Fmot is the driving force performed by the motor, Fdyn is the dynamic force, Fdrag is
the wind resistance and Frr is the rolling resistance. The dynamic force which is the one
involved in the acceleration of the vehicle is expressed as
Pdyn = mv
dv
v
dt
where mv is the mass of the vehicle, included the driver, and
vehicle. The rolling resistance force can be expressed as
(A.2)
dv
dt
is the acceleration of the
Prr = µrr mv gv
(A.3)
where µrr is the rolling resistance coefficient and g is the gravity acceleration. The wind
or drag resistance is given by
1
Pdrag = cd Af ρair v 3
2
(A.4)
where cd is the drag coefficient, Af is the vehicle’s front area, ρair is the density of the
air and v is the speed of the vehicle. Since there is only one wheel hub and there is no
gearbox, the torque required by the motor is.
Tm =
R
dv
1
[mv v + µrr mv gv + cd Af ρair v 3 ].
v
dt
2
(A.5)
where R is the radius of the wheel.
83
Appendix A. Dynamic simulation
Drag Consumption
One of the purposes of the cycle, is finishing the cycle with a fixed average speed vav =
27km/h. Depending on the interval of acceleration ∆t illustrated in Fig. 3.1 the maximum
speed varies its value
vmax = vav
T
T − ∆t
(A.6)
and as well the acceleration varies with different intervals
a=
vmax
∆t
(A.7)
To calculate the energy dissipated by the wind resistance
ZT
Edrag =
Pdrag =
0
ZT
Cv(t)3 dt
(A.8)
0
where C = 12 cd Af ρair . Integrating this expression throughout the speed cycle, the result is
the following
3 T − 23 ∆t
3
T
3
3
T − ∆t = C2
(A.9)
Edrag = Cvmax (T − ∆t) = C vav
2
T − ∆t
2
(T − ∆t)3
Rolling Resistance Consumption
This section gives an expression of the energy consumed by the rolling resistance.
Err =
ZT
Prr =
0
ZT
µrr mv gv(t)dt
(A.10)
0
Err = µrr mv gvav T
(A.11)
Acceleration Consumption
This section computes the energy used during the acceleration of the vehicle.
Edyn =
ZT
Pdyn =
0
Edyn
84
1
= mv vav
2
Z∆T
mv av(t)dt
(A.12)
0
T
T − ∆t
2
(A.13)
Appendix B
Analytical Motor Data
Motor 1
Motor Characteristics
Rated torque
Overload torque
Rated speed
Max. fund. phase rated current (rms)
Max. fund. phase overload current (rms)
Max. fund. current density (rms)
Min. dc-link voltage
Geometrical spec.
Number of slots
Number of poles
Physical air gap length
Shaft diameter
Outer Stator diameter
Core diameter
Total slot width
Total slot depth
Tooth width
Thickness of stator back
Thickness of rotor back
Active length
Magnet thickness
Active weight
Electrical data
Nm
Nm
rpm
A
A
A/mm2
V
7
22
275
13.48
43.45
√
3/ 2
32.4
mm
mm
mm
mm
mm
mm
mm
mm
mm
mm
mm
kg
18
14
1
124.878
192.826
135.829
15.17
18
11.01
5.48
5.48
20.3
3.84
2.89
85
Appendix B. Analytical Motor Data
Phase resistance
mΩ
Permanent magnet flux linkage
Vs
d−axis air gap inductance
mH
q−axis air gap inductance
mH
d−axis current at rated operation
A
q−axis current at rated operation
A
Loss data
Copper losses at rated operation
W
Iron losses at rated operation
W
Efficiency at rated operation
%
Winding spec.
Number of poles
Number of slots
Number of turns per slot
Fundamental winding factor
Copper fill factor
Flux density data
Air gap peak fund. density
Peak fund. flux density in stator back
T
Peak fund. flux density in stator teeth
T
Peak fund. flux density in rotor back
T
Magnet data
Remanent flux density at operating temp.
T
Relative permeability
Magnet density
kg/m3
Steel data
Eddy current loss density constant
Ws2 /kgT2 rad2
Steinmetz constant
Hysteresis loss density constant
Ws/kgTβ rad
Steel density
kg/m
Other spec.
Copper resistivity at operating temp.
nΩm
Copper density
kg/m3
86
130.2
0.0526
.95
.95
0
13.48
34.32
1.53
82.21
14
18
58
0.902
0.577
0.81
1.505
1.521
1.506
1.18
1.05
7700
4.26·10−5
2
0.0269
7700
23.9
8390
Appendix C
Fundamental Winding factor
In this appendix characteristics of different pole-slots combinations are deployed. Fig. C.3
describes the values of the fundamental winding factor kw1 for all the pole-slots combination, Fig. C.2 shows the configurations which are not desirable due to the unbalance
magnetic pull and Fig. C.1 illustrates the variation of the LCM(Qs , p), which is related with the cogging torque. The contents of the following tables have been retrieved
from [16] [13].
Fig. C.1 Lowest common multiple of different pole-slots combinations, LCM (Qs , p). The higher
this value, the lower is the cogging torque.
87
Appendix C. Fundamental Winding factor
Fig. C.2 Great common divisor of different pole-slots combinations, GCD(Qs , p). Configurations that GCD(Qs , p) = 1, perform unbalance magnetic pull.
88
Appendix C. Fundamental Winding factor
Fig. C.3 Fundamental winding factor of different pole-slots combinations for concentrated windings. Different colors illustrates some winding layout which is repeated. kw1 (q)
89
Appendix C. Fundamental Winding factor
90
Appendix D
Iron Loss Constants
As mentioned in section 5.3.2, eddy current constant, ke , and hysteresis constant, kh ,
depend on the material used. In this appendix both constants are calculated by the least
square method, considering the Steintmetz constant, βst = 2. Table D.1 shows the loss
density data given by Surahammars Bruk AB. Changing equation 5.58 to the conditions
Table D.1: Iron loss density data from SURAR M235-35A [27]
B(T) W/kg
W/kg
W/kg
W/kg
W/kg
W/kg
(50Hz) (100Hz) (200Hz) (400Hz) (1000Hz) (2000Hz)
0,1
0,02
0,04
0,08
0,19
0,93
3,89
0,2
0,06
0,14
0,32
0,87
3,55
14,3
0,3
0,11
0,3
0,73
1,88
7,45
29,6
0,4
0,2
0,49
1,21
3,17
12,3
50,2
0,5
0,29
0,71
1,78
4,73
18,5
76,7
0,6
0,38
0,97
2,44
6,56
25,8
110
0,7
0,5
1,25
3,19
8,67
34,6
153
0,8
0,62
1,57
4,03
11
45
205
0,9
0,77
1,92
4,97
13,8
57,2
270
1
0,92
2,31
6,01
16,9
71,5
349
1,1
1,1
2,75
7,19
20,3
88,3
1,2
1,31
3,26
8,54
24,3
1,3
1,56
3,88
10,1
28,9
1,4
1,92
4,67
12,2
34,8
1,5
2,25
5,54
14,4
41,2
1,6
2,53
1,7
2,75
1,8
2,94
exhibited above
piron = khyst B 2 f + keddy B 2 f 2
(D.1)
91
Appendix D. Iron Loss Constants
This equation can be adapted to calculate ke and kh . Considering pf the iron loss density
for each frequency
 2
0.1
khyst
 ..  2
(D.2)
pf =  .  f f
k
eddy
1.82
|
{z
}
[A f ]


0.12


where f is the frequency, and  ...  is the flux density for every loss density in the table.
1.82
Since [Af ] is not an square matrix, in order to obtain ke and kh , there must to be multiplied
by its transpose matrix to be able to calculate its inverse
khyst
T
T
[Af ] pf = [Af ] [Af ]
(D.3)
keddy
h
i−1
khyst
= [Af ]T [Af ]
[Af ]T pf
keddy
92
(D.4)
Appendix E
List of Symbols, Subscripts and
Abbreviations
Symbols
a
bss1
bss2
cd
cf
cp
d
e
felec
fcog
ff
g
hm
hry
hss
hsy
i
kw1
kj
kopen
mv
ns
nl
ntooth
p
acceleration
slot top width
slot base width
drag coefficient
Carter factor
specific heat at constant pressure
ratio between rotor diameter and slot height diameter
induced electromotive force
electric frequency
cogging frequency
copper fill factor
gravity acceleration
magnet height
rotor yoke height
stator slot height
height of the stator yoke
peak fundamental phase current, integer
fundamental winding factor
stacking factor
slot opening
vehicle weight
number of turns per slot
number of winding layers
number of coils per tooth
number of poles
93
Appendix E. List of Symbols, Subscripts and Abbreviations
q
r
t
u
v
wtooth
w
Af
Ag
Acond
Acu
B
Bm
Br
Br,m
Br,0
Bt
Bknee
Bry
Bsy
Btooth
Bδ
D
Dext
Di
Ê
Fd
Frr
Fw
H
Hc
iHc
J
Lact
Lcoil
Lend
Lm
Ls
Lσ
N
94
number of slots per pole and phase
resistivity
time, gearbox ratio, machine periodicity
peak fundamental voltage
velocity
tooth width
magnet segment width
vehicle front area
airgap area
stator slot area
bare copper area in stator slots
magnetic flux density
magnetic flux density above the magnets
remanent magnetic flux density, radial flux density
remanent magnetic flux density at rated temperature
remanent magnetic flux density at ambient temperature
tangential flux density
demagnetization magnetic flux density in the magnets
magnetic flux density in the rotor yoke
magnetic flux density in the stator yoke
magnetic flux density in the stator teeth
peak fundamental air gap flux density
rotor diameter
external diameter
shaft diameter
peak fundamental phase voltage
driving force
rolling resistance force
wind resistance force
magnetic field intensity
coercive field strength
intrinsic coercivity
peak fundamental current density
motor active length
coil length
end winding length
magnetic inductance
synchronous inductance
leakage inductance
number of turns per phase
Appendix E. List of Symbols, Subscripts and Abbreviations
Nspk
P
Qs
R
Rs
Rslot
Ŝ
~
S
T
Te
T0
Tpeak
Tcog
Tr
Tripple
Tsteady
Vrot
Wm
α
αe
αelec
αph
β
δ
δe
∆t
θ
θm
θel
γ
λ1
µ
µ0
µr
µrr
ρ
ρcu
ρm
ρiron
σ
number of spokes
power loss
number of slots
wheel radius
stator resistance
stator slot resistance
peak fundamental current loading
Cros’ method layout vector
temperature
electro-mechanical torque
ambient temperature
overload torque
cogging torque
rated temperature
ripple torque
steady-state torque
volume of the rotor
magnetic energy in the airgap
electric angle covered by one magnet, coefficient, angle between two adjacent slots
electric angle between two adjacent slots
electrical angle covered by one magnet
angle between two spokes
Steinmetz constant
airgap length
effective airgap length
acceleration interval
rotor angle in a dq reference
mechanical angle
electrical rotor angle (position)
shift electric angle of stator coils
specific coefficient of the slot opening
magnetic permeability
magnetic permeability of vacuum
relative magnetic permeability
rolling resistance coefficient
electrical resistivity, density
copper density
magnet density
steel density
shear stress
95
Appendix E. List of Symbols, Subscripts and Abbreviations
τs
τedge
φ
φm
φsy
φry
φtooth
ψ
ψm
ω
slot pitch
torque contribution of each slot in cogging torque
magnetic flux
magnetic flux in the magnets
magnetic flux in the stator yoke
magnetic flux in the rotor yoke
magnetic flux in the stator teeth
flux linkage
permanent magnet flux linkage
electrical rotor speed, angular frequency of the current
Subscripts
a
d
0
q
r
av
cu
eddy
mag
hyst
max
min
phase a component
direct-axis component
ambient temperature
quadrature-axis component
rated value
average
copper
eddy currents
magnet
hysteresis
maximum
minimum
Abbreviations
dc
rms
CW
EMF
EV
FEM
FFT
GCD
HEV
IPM
IW
LCM
MMF
PM
96
direct current
root mean square
concentrated windings
electromotive force
electric vehicle
finite element method
fast Fourier transformation
greatest common divisor
hybrid vehicle
interior mounted permanent magnet
in-wheel motor
lowest common multiple
magnetomotive force
permanent magnet
Appendix E. List of Symbols, Subscripts and Abbreviations
PMSM
SPM
permanent magnet synchronous machine
surface mounted permanent magnet
97
Appendix E. List of Symbols, Subscripts and Abbreviations
98
References
[1] “Michelin active wheel,” Paris Motor Show, Oct. 2008.
[2] M. Anderson and D. Harty, “Unsprung mass with in-wheel motors - myths and realities.”
[3] D. W. N. Ayman M. El-Refaie, Thomas M. Jahns, “Analysis of surface permanent
magnet machines with fractional-slot concentrated windings,” 2006.
[4] N. Bianchi and M. D. Pré, “Use of the star of slots in desingning fractional-slot
single-layer synchronous motors,” IEE Proc.-Electr. Power Appl., Vol. 153, No. 3,
May 2006.
[5] L. Brooke, “Protean electric tackles the unsprung-mass ’myth’ of in-wheel motors,”
Vehicle Electrification, Mar. 2011.
[6] C. C. Chan and K. T. Chau, Mordern electric vehicle technology. New York, USA:
Oxford Science Publications, 2001.
[7] J. F. Gieras and M. Wing, Permanent magnet motor technology.
Marcel Dekker, Inc., 2002.
New York, USA:
[8] D. D. Hanselman, Brushless Permanent Magnet Motor Design. Orono, USA: The
Writers’ Collective., 2003.
[9] P. F. J. A. Güemes, A. M. Iraolagoitia and M. P. Donsión, “Comparative study of
pmsm with integer-slot and fractional-slot windings,” ICEM, XIX International Conference on Electrical Machines.
[10] R.-J. W. Jacek F. Gieras and M. J.Kamper, Axial Flux Permanent Magnet Brushless
Machines. Springer Science, 2008.
[11] T. J. Juha Pyrhönen and V. Hrabovcová, Design of Rotating Electrical Machines.
Lappeenranta University of Technology, USA: John Wiley and Sons, 2008.
99
References
[12] H. Jussila, P. Salminen, M. Niemela, and J. Pyrhonen, “Guidelines for designing concentrated winding fractional slot permanent magnet machines,” POWERENG2007 International Conference on Power Engineering - Energy and Electrical Drives Proceedings, pp. 191–194, Apr. 2007.
[13] F. Libert and J. Soulard, “Investigation on pole-slot combinations for permanentmagnet machines with concentrated windings,” 16th International Conference on
Electrical Machines. Conference Proceedings, Sept. 2004.
[14] S. H. M. Aydin and T. A. Lipo, “Axial flux permnent magnet disc machines: A
review,” 2004.
[15] D. Meeker, “Finite element method magnetics: Octavefemm, version 1.2, user’s
manual,” 2006.
[16] F. Meier, “Permanent-magnet synchronous machines with non-overlapping concentrated windings for low-speed direct-drive applications,” Doctoral Thesis, Royal
Inst. Technol., Stockholm, Sweden, 2008.
[17] S. Meier, “Theorical design of surface-mounted permanet magnet motors with fieldweaking capability,” Master Thesis, Royal Inst. Technol., Stockholm, Sweden, 2002.
[18] (2012-08-28) Why is the price for neodymium magnets subject to such high
fluctuations? [Online]. Available: http://www.supermagnete.ch/eng/fag/price
[19] M. B. Nicola Bianchi and S. Bolognani, “A mordern approach to the analysis of pm
motors.”
[20] S. R. Perez, “Analysis of a light permanent magnet in-wheel motor for an electric vehicle with autonomous corner modules,” Master Thesis, Royal Inst. Technol.,
Stockholm, Sweden, 2011.
[21] H. P. R. Al Zaher, S. de Groot and P. Wieringa, “Comparison of an axial flux and a
radial flux permanent magnet motor for solar race cars,” XIX International Conference on Electrical Machines, Sept. 2010.
[22] R. N. S. E. Skaar, Ø. Krøvel, “Distribution, coil-span and winding factors for pm
machines with concetrated windings.”
[23] C. Sadarangani, Electrical Machines. Design and Analysis of Induction and Permanent Magnet Motors. Stockholm, Sweden: Royal Inst. Technol., 2006.
[24] (2012-07-16)
Shell
eco
marathon.
http://www.shell.com/home/content/ecomarathon/
100
[Online].
Available:
References
[25] (2012-08-24)
Siemens
evdo.
[Online].
Available:
http://www.siemens.com/innovation/en/publikationen/publications pof/pof spring 2007/technol
[26] W. L. Soong, “Sizing of electrical machines,” ICEM, XIX International Conference
on Electrical Machines, Sept. 2008.
[27] (2012) Surahammars bruck non oriented electrical steel brochure. [Online].
Available: http://www.sura.se/Sura/hp main.nsf/startupFrameset?ReadForm
[28] (2010)
Vacuumschmeltze
brochure.
[Online].
http://www.vacuumschmelze.de/index.php?id=242&L=2
Available:
[29] O. Wallmark, “Control of a permanent magnet synchronous motor with nonsinusoidal flux density distribution,” Master Thesis, Chalmers University of Technology, Göteborg, Sweden, 2001.
[30] E. K. Yang Tang, Johannes J. H. Paulides and E. A. Lomonova, “Investigation of
winding topologies for permanent magnet in-wheel motors,” COMPEL: The international Journal for Computation and Mathematics in Electrical and Electronic
Engineering, Vol. 31 Iss: 1 pp.88-107.
101