Thesis_-_Aleksandar_Borisavljevic1.
Limits, Modeling and Design
of
High-Speed Permanent Magnet Machines
Aleksandar Borisavljević
Limits, Modeling and Design
of
High-Speed Permanent Magnet Machines
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 15 augustus 2011 om 15.00 uur
door
Aleksandar BORISAVLJEVIĆ
Diplomirani inženjer elektrotehnike, Univerzitet u Beogradu
geboren te Kragujevac, Servië
Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.-Eng. J.A. Ferreira
Copromotor:
Dr.ir. H. Polinder
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof.dr.-Eng. J.A. Ferreira, Technische Universiteit Delft, promotor
Dr.ir. H. Polinder, Technische Universiteit Delft, copromotor
Prof.dr. E. Lomonova, Technische Universiteit Eindhoven
Prof.Dr.-Ing.habil. A. Binder, Technische Universität Darmstadt
Prof.dr.ir. L. Dupré, Universiteit Ghent
Prof.dr.ir. J. van Eijk, Technische Universiteit Delft
Prof.dr.ir. P.P. Jonker, Technische Universiteit Delft
This research was supported by MicroNed.
Cover design by Jelena Popadić
Printed by Wörmann Print Service, Zutphen, The Netherlands
ISBN 978-90-8570-837-7
c 2011 by Aleksandar Borisavljević
Copyright All rights reserved. No part of the material protected by this copyright notice may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage or retrieval system without permission from the author.
to my lovely parents
Acknowledgments
In September 2006 I moved to The Netherlands and began my PhD project in the Electrical
Power Processing (EPP) group of TU Delft. Both working on a PhD project and living in
Netherlands have helped me gain valuable personal and professional experience. Today, I look
back with content on the past five years that enriched me as a person more than any other period
in my life. But foremost, I feel I owe gratitude to many wonderful people that I was fortunate
to become acquainted with, both personally and professionally.
Firstly, I would like to express my gratitude to my co-promoter and daily supervisor, Henk
Polinder, whose supervision, support and friendship were crucial for me to persevere and overcome the challenges of doctoral work. I feel privileged to have had such a wonderful person as
a supervisor and true friend.
To my promoter, Prof. Braham Ferreira, I am sincerely grateful for helping me manage my
PhD work. Without our monthly meetings and his brilliant feedback, I would have never been
able to find my way in engineering research.
I would like to thank Jelena and Mark Gerber for their support and, especially, for taking
care of me like a family when I first moved to Delft. Their encouragement and friendship were
such an important influence, allowing me to quickly adapt and enjoy life in my new environment.
Everyone who has ever done something in the EPP lab knows what Rob Schoevaars means
to the group and PhD students. His experience, patience and good spirit helped me overcome
all those stubborn practical problems one must always face when doing experiments.
I am most grateful to the secretary of the department, Suzy Sirks-Bong, who has always
been ready to go out of her way to help me and other PhD students with any kind of problem.
I would like to express my gratitude to all the people of the Microfactory project team for
their collaboration and, in particular, to Hans Langen and Maarten Kimman whose work and
ideas formed the basis of my PhD research. I would also like to thank Prof. Rob Munnig
Schmidt - discussions with him were always inspiring.
This thesis would certainly not be the same without Petros Tsigkourakos, who designed the
air-bearings setup and inverter which I used to test my designs. It was a very demanding assignment, but Petros worked relentlessly until we had a working setup. Thanks to his diligence, we
managed to overcome many problems and both learned a lot.
A university is a very vibrant intellectual environment, and the feedback and advice of many
people have had an influence on my work. I must distinguish the great help I received from my
colleague and office-mate, Deok-Je Bang, whose vast experience and brilliant ideas helped me
work out the mechanical aspects of my designs. His ideas are interwoven into many solutions
i
ii
presented in this thesis.
I would also like to thank Zhihui Yuan for his help in DSP programming, Navin Balini for
advice on control and Domenico Lahaye for his assistance with finite element modeling. Dr.
Frank Taubner from Rosseta GmbH gave very valuable input to the design of the rotor retaining
sleeve.
It was a great pleasure working in the EPE group, I am happy that we established such a
friendly atmosphere among the PhD students, enhanced by both mutual support and fun. I was
particularly lucky to share an office with such great guys: Deok-Je Bank, Frank van der Pijl and
Balazs Czech. Our office was always known for its homelike atmosphere.
My wife, Veronica Pišorn, improved my academic writing immensely and spent long hours
in editing my papers. I learned from her how the good and professional presentiation of ideas
can improve scientific work.
In Delft, I met a lot of amazing people and gained very close friends. It is privilege to be
surrounded by so many interesting and smart people. I am particularly proud of my friends
from ex-Yugoslavia who reflect the best of the culture and spirit of our turbulent region. I am
thankful for their love and support throughout all these years.
I am very happy that I volunteered in Filmhuis Lumen, one of the nicest places in Delft.
Hanging out with cozy and interesting people from Lumen helped me learn the language and
get to know with Dutch people, along with doing something that I genuinely like.
Finally, I would like to mention those people that have personally meant the most to me. To
them I owe my deepest gratitude.
To Jasmina and Darko for sharing their lives with me, for their love and support, for giving
me ease when I needed it the most.
To Nada, Zoran and Nemanja, my wonderful family, for their love and dedication, for being
so sincere, for showing how truth and honesty is invincible.
To Veronica, love of my life, for her support and optimism, for unfolding so much beauty,
taste and joy in every day.
And to my dear friends and family in Serbia, who I carry in my heart wherever I go.
Aleksandar Borisavljević
Eindhoven, July 2011
Contents
Acknowledgments
i
Contents
iii
List of symbols and abbreviations
vii
1
2
3
Introduction
1.1 Machinery for micromilling . . . . . . . . . . . . . . . . . . . .
1.2 High-speed spindle drive . . . . . . . . . . . . . . . . . . . . . .
1.3 Problem description . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Thesis objectives . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Test setup: A high-speed PM motor in aerostatic bearings
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
1
4
5
6
6
7
High-speed PM machines: applications, trends and limits
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 PM machines: overview . . . . . . . . . . . . . . . .
2.3 Defining high-speed . . . . . . . . . . . . . . . . . . .
2.4 Survey of high-speed machines . . . . . . . . . . . . .
2.5 Speed limits of PM machines . . . . . . . . . . . . . .
2.6 Limits and rated machine parameters . . . . . . . . . .
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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11
11
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15
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Electromagnetic modeling of slotless PM machines
3.1 Introduction . . . . . . . . . . . . . . . . . . .
3.2 Model geometry and properties . . . . . . . . .
3.3 Modeling of the magnetic field . . . . . . . . .
3.3.1 Field of the permanent magnet . . . . .
3.3.2 Armature field . . . . . . . . . . . . .
3.3.3 Combined field . . . . . . . . . . . . .
3.4 Derived quantities . . . . . . . . . . . . . . . .
3.4.1 No-load voltage . . . . . . . . . . . . .
3.4.2 Torque and power . . . . . . . . . . . .
3.4.3 Phase inductance . . . . . . . . . . . .
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23
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24
27
29
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iv
Contents
3.5
3.6
3.7
4
5
6
Unbalanced magnetic pull and machine stiffness
Losses in the machine . . . . . . . . . . . . . .
3.6.1 Stator core losses . . . . . . . . . . . .
3.6.2 Copper losses . . . . . . . . . . . . . .
3.6.3 Air-friction loss . . . . . . . . . . . . .
3.6.4 Rotor loss . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . .
Structural aspects of PM rotors
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 Stress in a rotating cylinder . . . . . . . . . . .
4.2.1 Isotropic modeling . . . . . . . . . . .
4.2.2 Orthotropic modeling . . . . . . . . . .
4.3 Mechanical stress in a PM rotor . . . . . . . .
4.3.1 Test rotor: Analytical models . . . . . .
4.3.2 Test rotor: 2D FE model . . . . . . . .
4.3.3 Test rotor: Results comparison . . . . .
4.4 Structural limits and optimization of PM rotors
4.5 Conclusions . . . . . . . . . . . . . . . . . . .
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Rotordynamical aspects of high-speed electrical machines
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
5.2 Vibration modes . . . . . . . . . . . . . . . . . . . . .
5.3 Threshold of instability . . . . . . . . . . . . . . . . .
5.3.1 Stability of the Jeffcott rotor with damping . .
5.3.2 Jeffcott rotor with non-synchronous damping .
5.4 Critical speeds calculation . . . . . . . . . . . . . . .
5.4.1 Hard-mounted shaft . . . . . . . . . . . . . .
5.4.2 General case: bearings with a finite stiffness . .
5.5 Rigid-rotor dynamics . . . . . . . . . . . . . . . . . .
5.5.1 Rigid critical speeds . . . . . . . . . . . . . .
5.5.2 Unbalance response . . . . . . . . . . . . . .
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
Bearings for high-speed machines
6.1 Introduction . . . . . . . . . .
6.2 Mechanical bearings . . . . .
6.3 Air (fluid) bearings . . . . . .
6.4 Active magnetic bearings . . .
6.5 Conclusions . . . . . . . . . .
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Contents
7
8
9
v
Design of the high-speed-spindle motor
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 New spindle concepts . . . . . . . . . . . . . . . . . . . . .
7.3 Conceptual design of the motor . . . . . . . . . . . . . . . .
7.3.1 Stator core . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Conductors . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Permanent magnet . . . . . . . . . . . . . . . . . .
7.3.4 Magnet retaining sleeve . . . . . . . . . . . . . . .
7.4 Motor optimization . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Rotor shaft design . . . . . . . . . . . . . . . . . .
7.4.2 Electromagnetic optimization of the motor geometry
7.4.3 Optimization of conductors . . . . . . . . . . . . . .
7.5 FEM design evaluation . . . . . . . . . . . . . . . . . . . .
7.5.1 2D FEM: motor parameters . . . . . . . . . . . . .
7.5.2 3D FEM: no-load voltage and phase inductance . . .
7.5.3 2D FEM: conductor eddy-current loss . . . . . . . .
7.6 Design of the rotor retaining sleeve . . . . . . . . . . . . . .
7.6.1 Material considerations: PPS-bonded NdFeB magnet
7.6.2 Material considerations: carbon-fiber composite . .
7.6.3 Sleeve optimization . . . . . . . . . . . . . . . . . .
7.6.4 Final design . . . . . . . . . . . . . . . . . . . . . .
7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
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111
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Control of the synchronous PM motor
8.1 Introduction . . . . . . . . . . . . . . . . . . . . .
8.2 Stability analysis . . . . . . . . . . . . . . . . . .
8.3 Stabilization control . . . . . . . . . . . . . . . . .
8.4 I/f control method . . . . . . . . . . . . . . . . . .
8.5 Controller implementation and experimental results
8.5.1 Description of the test setup . . . . . . . .
8.5.2 I/f controller implementation . . . . . . . .
8.5.3 Experimental results . . . . . . . . . . . .
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . .
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Experimental results
9.1 Introduction . . . . . . . . . . . . . .
9.2 Practical setup . . . . . . . . . . . . .
9.2.1 Stator assembly . . . . . . . .
9.2.2 Rotor assembly . . . . . . . .
9.2.3 Air-bearings test setup . . . .
9.2.4 High-frequency inverter . . .
9.3 Motor phase resistance and inductance
9.3.1 Phase resistance . . . . . . .
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vi
Contents
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161
164
168
173
174
10 Conclusions and recommendations
10.1 Models presented in the thesis . . . . . . .
10.2 Speed limits of permanent magnet machines
10.3 Design evaluation . . . . . . . . . . . . . .
10.4 Thesis contributions . . . . . . . . . . . . .
10.5 Recommendations . . . . . . . . . . . . . .
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177
177
180
181
182
184
9.4
9.5
9.6
9.7
9.3.2 Phase inductance . . . . .
Speed-decay tests . . . . . . . . .
Locked-rotor tests . . . . . . . . .
Motor operation and performance
Conclusions . . . . . . . . . . . .
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Appendices
186
A Structural relationships in a rotating cylinder
187
B One explanation of rotor instability
189
C Stator core properties
191
Bibliography
193
Summary
209
Samenvatting
213
List of publications
217
Biography
219
List of symbols and abbreviations
Latin letters
→
−
A, A
A
Ac
→
−
B, B
Bi j
b
b
bei
ber
C
C
Cf
Ci
c
d
Di
E, e
E
→
−
F, F
F()
F , G, H
Fd
f
G
Gi j
G()
g
→
−
H, H
h
I, i
Magnetic vector potential
Surface area
Current loading
Magnetic flux density
Compliance matrix (element)
Width
Compliance
Kelvin function (imaginary part)
Kelvin function (real part)
Capacitor, Capacitance
Iron-loss coefficient
Air friction coefficient
Boundary coefficient
Coefficient of viscous damping
Diameter
Boundary coefficient
No-load voltage
Young’s modulus
Force
Function
Positive two-dimensional stress functions
Force density
Frequency
Shear modulus
Gyroscopic matrix (element)
Function
Effective air gap
Magnetic field intensity
Height
Electrical current
vii
[Wb·m−1 ]
[m2 ]
[A·m−1 ]
[T]
[N·m−1 ]
[m]
[N·m−1 ]
[F]
[various]
[]
[T]
[N·s·m−1 ]
[m]
[T·m]
[V]
[Pa]
[N]
[Ps]
[N·m−2 ]
[Hz]
[Pa]
[kg·m2 ]
[m]
[A·m]
[m]
[A]
viii
I
J, j
J
Ki j
k
k
k
L
l
Mi j
m
N
n
n
P, p
p
p
q
R
Re
S
r
T
T
Ta
t
U, u
u
V
v
W
w
x
y
z
Contents
Surface moment of inertia
Electrical current density
Moment of inertia
Stiffness matrix (element)
Portion of frequency-dependent losses in total copper loss
Roughness coefficient
Stiffness
Inductance
Length
Inertia matrix (element)
Mass
Number of winding turns
Angular conductor density
Number of conductor strands
Real power
Power density
Static pressure
Displacement function
Resistance
Reynolds number
Surface area
Radius
Torque
Temperature
Taylor number
Time moment
Voltage
Displacement
Volume
Velocity
Weight
Width
x-coordinate
y-coordinate
z-coordinate
[m4 ]
[A·m−2 ]
[kg·m2 ]
[N·m−1 ]
[]
[]
[N·m−1 ]
[H]
[m]
[kg]
[kg]
[turns]
[turns·rad−1 ]
[]
[W]
[W·m−3 ]
[Pa]
[m]
[Ω]
[]
[m2 ]
[m]
[N·m]
[◦ C, K]
[]
[s]
[V]
[m]
[m3 ]
[m·s−1 ]
[kg]
[m]
Ratio between active and total rotor length
Coefficient of linear thermal expansion
Gyroscopic moment, non-dimensional
[]
[K−1 ]
[]
Greek letters
α
α
Γ
Contents
δ
δ
ε
ε
ε
θ
λ
µ
µ
ν
ρ
ρ
σ
σ
σM
τ
Φ
φ
ϕ
χ
χ
ψ
Ω
ω
ix
Skin depth
Clearance
Rotor eccentricity
Strain
Static unbalance
Angle
Rotor slenderness
Magnetic permeability
Dynamic viscosity
Poisson’s ratio
Mass density
Electrical resistivity
Electrical conductivity
Stress
Maxwell’s stress
Temperature increment
Magnetic flux
Ratio between conductor diameter and the skin-depth
Angular position
Shear parameter
Couple unbalance
Flux linkage
Angular velocity
Angular frequency
Latin subscripts
a, b, c
AC, ac
BH
c
cr, crit
Cu
DC, dc
diss
e
e
e, eddy
ef f
ei
eo
Phase
Alternated current - ac
BH curve
Current
Critical value
Copper
Direct current - dc
Power dissipation
Electrical
Sleeve/enclosure
Eddy currents
Effective value
Sleeve/enclosure - inner
Sleeve/enclosure - outer
[m]
[m]
[]
[]
[m]
[rad]
[]
[H·m−1 ]
[kg·m−1 · s−1 ]
[]
[kg·m−3 ]
[Ω·m]
[S·m−1 ]
[Pa]
[Pa]
[K]
[Wb]
[]
[rad]
[]
[rad]
[Wb]
[rad·s−1 ]
[rad·s−1 ]
x
Contents
f
Fe
g
h
lin
m
mag
max
mech
n
nom
opt
PM
p
prox
r
r
rec
re f
rem
res
s
s
skin
so
st
t
t
t
U
vM
w
wc
x, y, z
z
0
(Air) Friction
Iron
Air gap
Hysteresis
Linearized
Magnet
Magnetic
Maximum or limiting value
Mechanical
Non-rotating
Nominal (rated) value
Optimal
Permanent magnet
Polar
Proximity effect
Rotor, rotating
Radial coordinate/direction
Recoil
Reference
Remanent
Resonant
Stator
Slot
Skin effect
Stator outer
Strand
Tooth
Tangential coordinate/direction
Transversal
Ultimate or yield
Von Mises
Winding
Winding center
Dimensions in Cartesian coordinate system
Axial coordinate/direction
Air/vacuum
Greek subscripts
θ
ϕ
Tangential coordinate/direction
Tangential coordinate/direction
Contents
xi
Abbreviations
AMB
CHP
CTE
DN
EDM
EM
FE
FEM
MEMS
PM
RMS
1D
2D
3D
Active magnetic bearings
Combined heat and power
Coefficient of thermal expansion
DN number: product of bearing diameter and rotational speed
Electro-discharge machining
Electromagnetic
Finite element
Finite element model
Microelectromechanical systems
Permanent magnet
Root mean square
One-dimensional
Two-dimensional
Three-dimensional
Vectors
→
− →
− →
−
ir , iϕ , iz
Unity vectors of r, ϕ and z axes in Cylindrical coordinate system
Chapter 1
Introduction
1.1
Machinery for micromilling
Miniature, micro/mesoscale components with 3D features are in demand for various industries such as electro-optics, biomedical, electronics, aerospace, etc. [1]. Applications for which
micro-products are needed include optical molds and assembly, medical diagnostic devices,
medical implants, electronic devices and chemical micro-reactors [2]. The microsystem technology market has grown steadily [3, 4] and with it the demand for high accuracy, complex
shapes and a broader assortment of materials.
Figure 1.1: Application market of microproducts a) by industry; b) by discipline (taken from
[4])
1
2
Chapter 1. Introduction
Different fabrication processes have been used for the production of miniature components
in large batches: chemical etching, photo-lithography, laser, ultrasonic and ion-beam cutting
and electro discharge machining. A majority of these methods is limited to silicon-based materials and/or planar geometries [1, 5]. For obtaining essentially 3D components, with dimensions ranging from a few hundred microns to a few millimeters and with micron-sized features,
micro-mechanical machining has become the most viable solution. This kind of machining represents, in effect, the scaling down of traditional cutting technologies such as milling, drilling
and grinding. By accommodating conventional macro processes on a micro scale, we are enabled to produce even small batches of micro components at lower costs and in a wide range of
materials: metallic alloys, ceramics, glass etc [4]. In this way, micromachining bridges the gap
between silicon-based MEMS processes and conventional precision machining [6].
Atomic/Material Science
SCALE
Nano
Micro
MEMS
Micro Mechanical
Machining
Meso
Macro
Conventional Ultra Precision Machining
Precision
High Precision
Ultra Precision
ACCURACY
Figure 1.2: Dimensional size for the micro-mechanical machining (according to [1])
Very strict requirements - in terms of high structural stiffness and damping, low thermal
distortion, environmental isolation - for the mechanical structure of micro-scale machinery [4]
have caused commercial machines be extremely large and heavy [7]. Their high cost and limited
flexibility bring about high initial cost and limit them to a narrow range of applications that
require a large number of high-precision components with relatively simple geometries. As
stated in [2]: ”Despite all of the progress in mechanical machining (...), the fact that it often
takes a two-ton machine tool to fabricate microparts where cutting forces are in the milli- to
micro-Newton range is a clear indication that a complete machine tool redesign is required for
the fabrication of micromachines.”
”Small Equipment for Small Parts” [2] is a good paradigm for the global trend towards
the miniaturization of manufacturing equipment and systems for micro-scale products. Many
academic institutions, mainly in Europe and Asia, have been directing efforts toward the development and commercialization of smaller, even desktop-sized micro-cutting machines. These
efforts have resulted in a number of compact, three to five-axis, mainly micromilling machines [7–17].
Important advantages of small micro-cutting machines can be stated as follows [7, 8]:
• small footprint and weight;
• ease of localized environmental and safety control;
1.1. Machinery for micromilling
3
• energy efficiency;
• portability and reduced transportation costs;
• reduced initial investment costs and costs from energy consumption and maintenance;
• a full spectrum of micro-machining applications;
• higher speed and acceleration due to smaller spindle size and inertia;
• positing effect on monitoring of the cutting process [18];
• shortened ramp-up process in production;
• allowing machine modular design and reconfiguration;
Two major trends towards machine miniaturization could be distinguished. In Japan and,
recently, Korea the ”Microfactory” concept has been dominant according to which one or more
small machines are placed on a desktop to produce microparts in a fully automated process [8,
15]. However, microfactories did not offer a major contribution in the fabrication of microparts
or final accuracy and most related projects have remained in the research phase [7, 19].
Most European projects, on the other hand, have been focusing on machine and process [2].
This approach practically represents a trade-off between large conventional machines and microfactories: the machinery is downsized and flexibility in producing complex 3D parts is
reached, while stiffness and accuracy of the precision machines is maintained. A typical representative of this concept is Ultramill - a bench-top 5-axis micromilling machine developed in
Brunel University, UK [7].
The spindle is a key part of micro-cutting machinery since it has the most significant effect
on the quality of the machined components [4]. The spindle needs to have high motion accuracy
and be capable of operating at very high-speeds. The dynamic of the rotor-bearings system is
very important: when machining with micro-scale tools, small spindle deviations can result in
large run-out [19].
In micromilling the speed requirement becomes particularly severe. As sizes of the features
to be machined decrease, the diameters of the required tools decrease accordingly. In order to
achieve cutting speeds which are standard for macroscopic milling tools, extremely high rotational speeds of a spindle are needed [20]. For instance, for milling materials such as stainless
steel or brass with a 50 µm tool, required spindle speeds are beyond 380.000 and 950.000 rpm
respectively [21]. At the moment of writing, speeds of machining spindles do not exeed 200.000
rpm and advanced spindle technologies are expected to support higher speeds in future.
Most existing small and medium size high-speed spindles use aerostatic bearings [4]. Because of their very small air-gaps (in the order of tens of µm) air bearings require flawless
geometry with very low tolerances. Extremely high motion accuracy can therefore be achieved
with high precision and repeatability. Compared to their dynamic counterpart aerostatic bearings consume less energy and can efficiently operate in a much wider speed range [7].
Active magnetic bearings (AMB) represent another promising technology for micro-machining
spindles. AMB offer a possibility of creating practically arbitrary damping or stiffness that can
4
Chapter 1. Introduction
be tailored to the operating mode (speed) [22]. In spite of thia and a number of other advantages
- such as high load capacity, no inherent instabilities, modularity - AMB spindles are still in the
research stage with regard to micro-machining applications.
Motorized spindles are suitable for high-speed rotation. Such a design eliminates the need
for power-transmission devices [23] and offers higher torque and efficiency than in the case
of air-turbine-driven spindles [1, 7]. Incorporating a motor into the spindle reduces its size
and rotor vibrations [4, 23], at the same time complicating thermal and mechanical design and
modeling [23]. When defining important elements of the spindle - bearings, rotor, motor - a
holistic approach to design may be preferred [7].
Regarding types of built-in spindle motors, mostly permanent magnet (PM) machines have
been used [7]. PM machines are well-suited for very-high-speed applications due to high efficiency and high power density, particularly in low volumes [24, 25]. Conventionally, electrical machines are optimized according to output torque/power requirement and permissible
heat dissipation due to electromagnetically induced losses. In miniature machining spindles,
however, motor design and optimization becomes closely connected with the design of other
key-elements of the spindle.
1.2
High-speed spindle drive
Using the Japanese ”micro-factory” as an inspiration and paradigm, the Dutch Microfactory
project centers its research on process and assembly technology for the fabrication of 3D products on a micro scale. The focus is on downsized equipment for micro-structuring, -assembly,
-sensing and -testing. The Microfactory project [26] virtually combines the ongoing research
in micro-systems technologies with the ”micro-factory” concept: the researched machinery is
desktop-sized and different process techniques are even intended to be combined on the same
device (e.g. high-speed cutting and EDM).
Within the Microfactory project, the study in microstructuring is mostly concerned with
micromilling as the most promising technology for production of 3D micro-parts. In order
to make milling technology competitive for the micro-scale products, micromilling requires
spindles with speeds beyond 300.000 rpm and a positioning accuracy in the order of 1 µm. New
specialized spindles with small rotors and frictionless - magnetic or air - bearings are necessary
to reach such rotational speeds and accuracy.
Spindles with built-in motors facilitate precise control of rotor acceleration and deceleration. Furthermore, the inclusion of a motor allows for a compact machine, as explained in
the previous section. However, spindle drives that support extremely high rotational speed and
are compatible with frictionless, low-stiffness bearings are not readily available on the market.
Electrical drives specially designed for high-speed machining spindles motivated work on this
thesis.
Recently, very high-speed machines have been developed for applications such as microturbine generators, turbocompressors and aircraft secondary power systems. In these examples,
the demand for compact and efficient electromechanical systems has created the necessity of
high-speed machine use [27]. The elimination of gears through the use of direct-driven electri-
1.3. Problem description
5
cal machines has resulted in improved system reliability.
Permanent magnet machines have become prevalent in low volume applications [24]. This
could be ascribed to their magnetic excitation - the air-gap flux density of PM machines is
determined predominantly by the quality of utilized permanent magnets and does not depend on
the size of the machine. Several extremely high-speed PM machines have been reported, both on
the market and in academia, with speeds up to 1 million rpm [28–30]. These relatively simple
machines are proposed for direct-drive generators; the inclusion of a motor in a machining
spindle, however, brings about new challenges in the area of electrical machines.
The advance in speed of spindle motors is linked with overcoming or avoiding a number
of machine limitations. Various physical parameters (stress, temperature, resonant frequencies)
can limit the speed of an electrical machine. Aside from speed, these variables are also affected
by power, size and electrical and magnetic loading. They need to be carefully designed so that
the spindle temperature, structural integrity and rotordynamic stability are not jeopardized at
the operating speed [25].
Additional challenges lie in the control of high-speed machines. Due to limitations of
the processing powers of microcontrollers, very high-speed machines are usually controlled
in open-loop [29]. However, the open-loop speed response of PM machines becomes unstable after exceeding a certain speed [31], therefore, special control algorithms are required that
would ensure stability without a great computational burden.
1.3
Problem description
The project commenced in a research group within the Microfactory project. The group has
worked towards new desktop machinery for micromilling, focusing on the spindle as the machine’s key element. Active magnetic bearings (AMB) were recognized as a particularly attractive technology for micromilling spindles. Beside frictionless operation, AMB facilitate
supercritical operation and exploiting self-centering of the rotor thus, with a reduction in the
rotor size, desired high rotational speeds could be achieved. Yet, a new electrical drive for the
spindle was needed.
The project goal was to design and build a high-speed electrical drive which would be compatible with a soft-mounted spindle for speeds beyond 300.000 rpm. Hence, it was primarily expected that the spindle motor has low stiffness (unbalanced pull), generates minimum
frequency-dependent losses (heat) and that the motor design does not compromise the strength
and robustness of the rotor.
In the initial phase of the project, a permanent magnet machine was chosen as the best candidate for low-volume high-speed applications. To minimize losses in the rotor, a synchronous
type of PM machine with sinusoidal currents in the stator was chosen. That implied the necessity of choosing or designing a power inverter that would supply the motor with sinusoidal
currents and, preferably, facilitate development of an adequate motor controller.
However, it was apparent that the application requires more than a conventional electromagnetic design of an electrical machine and its drive. Structural and rotordynamical aspects
imposed equally great challenges to the design. The reduction in size of the rotor would also
6
Chapter 1. Introduction
demand a closer integration of the motor and bearings and, in turn, their higher mutual dependence. New solutions were sought among unconventional spindle concepts [32].
At the same time, the project gave incentive for scientific research in the field of high-speed
machines and, particularly, for the modeling and design of high-speed PM motors. The intention
was to identify phenomena, both mechanical and electromagnetic, that take precedence in highspeed machines. By doing that, it was possible to single out a suitable motor configuration
(in terms of number of poles, winding configuration, materials, etc.). Moreover, an analytical
representation of those phenomena would enable the design and optimization of very high-speed
machines including a spindle motor.
The demand for such extremely high speeds spurred an inquiry into the limitations of the
machines: what does limit the speed of current machinery? Could those limits be overcome
or avoided? This study implicitly formed an approach for designing machines for high-speed
applications: (i) recognize the speed limits of the machines, (ii) correlate the limits with machine parameters and (iii) overcome those limits with new solutions which are adequate for a
particular application.
Eventually, new concepts and models needed validation in a practical demonstrator/setup.
It was envisaged that the setup would be used for testing the spindle drive and its suitability for
the application as well as verifying or refuting developed models.
1.3.1 Thesis objectives
To summarize the foregoing problem description, this thesis aims to achieve the following:
• The identification and systematization of phenomena, both mechanical and electromagnetic, that take precedence in high-speed machines.
• Define the speed limits of permanent magnet machines and correlating those limits with
basic parameters of the machines.
• Electromagnetic, structural (elastic) and rotordynamical modeling of a high-speed permanent magnet machine.
• The design and realization of a low-stiffness high-speed spindle drive.
1.3.2 Test setup: A high-speed PM motor in aerostatic bearings
The work on the high-speed spindle drive resulted in a practical setup. Since it is used throughout this thesis for validation of the models and concepts, a very short preview of the setup is
given here. Details on the design of the setup are presented in Chapters 7.
The setup consists of a radial-flux slotless PM motor with a disc-shaped rotor which is
supported by aerostatic bearings. A drawing of the stator and the rotor is given in Figure 1.3.
A magnet is applied onto the rotor in a ring shape and magnetized diametrically. The stator is
wound toroidally and fitted into the bearings’ housing.
In the thesis the setup will be referred to as the test setup and the PM machine as the test
machine or the test motor.
1.4. Thesis outline
7
Figure 1.3: The stator and the rotor with the dimensions in mm
1.4
Thesis outline
Chapter 2 - High-speed PM machines: applications, trends and limits
In the chapter, an overview of current and prospective applications of permanent magnet machines is presented and speed limits of those machines are studied with particular focus on
small-size high-speed applications. Several prominent and promising applications of very highspeed machines are reported. The term high-speed is discussed and its meaning in this thesis is
specified. The results of an empirical survey on the correlation between rated powers and speeds
of existing high-speed machines are presented. Physical factors that shape the speed limits of
PM machines in general are defined. The last section of the chapter attempts to theoretically
correlate the maximum rated powers and speeds of PM machines with respect to their inherent
limits, as defined in the preceding sections.
Chapter 3 - Electromagnetic modeling of slotless PM machines
The chapter presents magnetostatic modeling of a slotless PM machine that will form the basis
of the electromagnetic optimization of the test machine. Based on the model of the magnetic
field, other quantities of the machine - no-load voltage, torque, inductance, unbalanced magnetic
force and losses - are derived. The main purpose of the chapter is to distinguish and model the
most important electromagnetic parameters of high-speed PM machines. The modeling aims
at suitable representation of dominant phenomena that can serve as a good basis for designing
high-speed (slotless) PM machines.
Chapter 4 - Structural aspects of PM rotors
Structural modeling and optimization of high-speed PM rotors are the focus of this chapter. The
aim of the chapter is to model the influence of rotational speed and mechanical fittings on stress
in a high-speed rotor, while also considering the operating temperature. Through analytical
modeling, structural limits for the rotor speed are determined and quantified. At the same time,
an analytical representation of the limiting parameters implies a relatively simple approach for
8
Chapter 1. Introduction
the optimization of the rotor structure. This optimization approach will form the basis for the
design of a carbon-fiber sleeve for the rotor of the test machine.
Chapter 5 - Rotordynamical aspects of high-speed electrical machines
The chapter gives qualitative insight into important dynamical aspects of high-speed rotors
through analytical modeling. The goal of the chapter is to define the dynamical limits for
the rotor speed and to correlate those limits with machine parameters. A theoretical study on
rotation stability gives an assessment of the threshold speed of unstable, self-excited vibrations.
Based on this, practical speed limits of electrical machines are defined with respect to the critical
speeds of the rotor-bearings system and bearing properties. Basic correlations between critical
speeds and parameters of the rotor-bearing system are modeled. Lastly, the unbalance response
of rigid rotors is analytically modeled and the suitability of different rotor geometries for highspeed rotation is analyzed. The conclusions drawn from this chapter strongly influenced the
final spindle concept.
Chapter 6 - Bearings for high-speed machines
The goal of the chapter is to study different types of bearings with respect to their suitability for
high-speed rotation. The chapter is primarily concerned with bearings that have been the most
promising for high-speed rotation: (hybrid) ball bearings, externally pressurized (or static) air
bearings and magnetic bearings. A general overview and comparison of these bearing types is
given.
Chapter 7 - Design of the high-speed-spindle motor
The chapter presents the design of the spindle (test) motor, from conceptual design to the electromagnetic and structural optimization of the motor. The analyses and models presented in
Chapters 2-6 determined the spindle-motor design: the definition of speed limits of PM machines greatly affected the conceptual design of the new spindle drive while the developed
models formed an analytical basis for the motor design and optimization. In the first part of the
chapter, the development of new spindle concepts within the Microfactory project group at TU
Delft is presented. Geometric and electromagnetic design of the spindle motor is explained and
then evaluated using FEM. Lastly, the optimization of the rotor retaining sleeve is presented and
the method for rotor production is defined.
Chapter 8 - Control of the synchronous PM motor
The chapter presents a new control method developed for control of the high-speed motor.
For stable, sensorless control of a high-speed PM synchronous motor an open-loop, I/f control
method is proposed in which stabilizing control from the standard V/f approach has been incorporated. The controller was successfully implemented in a standard control-purpose DSP and
tested in the practical setup.
1.4. Thesis outline
9
Chapter 9 - Experimental results
The chapter gives a description of the test setup and experiments which were carried out to
gather practical data which further allowed the assessment of the developed models and chosen
design approach. The main tests performed for verification of the models developed in the thesis
are the speed-decay and locked-rotor tests. The overall performance of the electric drive during
operation is also reported and discussed in the chapter.
Chapter 10 - Conclusions and recommendations
In the last thesis chapter, theoretical and practical work presented in the thesis is revisited and
evaluated. Important conclusions are drawn with respect to the validation of the developed
models, speed limits of PM machines and performance and applicability of the spindle-drive
design. The chapter lists the main contributions of the thesis and gives recommendations for
future work.
10
Chapter 1. Introduction
Chapter 2
High-speed PM machines: applications,
trends and limits
2.1
Introduction
High-speed permanent magnet machines are in focus of this thesis; this chapter offers an
overview of their current and prospective applications and theoretical study on their limits.
The following section explains the reasons behind increasing use of PM machines with
particular attention to small-size high-speed applications. Several prominent and promising
applications of very high-speed machines are reported. In Section 2.3 the term high-speed is
discussed and its meaning in this thesis is specified. Section 2.4 presents results of an empirical
survey of the correlation between rated powers and speeds of existing high-speed machines; in
particular, the section points out trends of speed increase of slotted and slotless PM machines.
The rest of the chapter is dedicated to speed limits of PM machines. Physical factors that
lie behind speed limits of PM machines in general are defined in Section 2.5. Finally, Section
2.6 attempts to theoretically correlate maximum rated powers and speeds of PM machines with
respect to their inherent limits defined in the preceding section.
2.2
PM machines: overview
Permanent magnet machines have become increasingly popular in the last two decades. They
have replaced induction machines in a great number of converter-fed electrical drives and motion control systems1 [33,34]. Besides, many new applications which require high-performance
electrical machines are invariably linked to PM machines. Two factors may be singled out as
crucial for such development.
1
A simple indicator of prevalence of PM machines can be achieved by examining increase of the relative
number of associated academic papers. For illustration, ratio between number of papers that is associated with
PM/PM synchronous/brushless DC and induction/asynchronous motors at IEEE internet base grew from 0.13 for
period before 1990 to 0.54 after 1990 and to 0.65 after year 2000.
11
12
Chapter 2. High-speed PM machines: applications, trends and limits
Cost competitiveness and availability of rare-earth magnets. After years of rather slow advance of SmCo magnets in market, in the eighties rapid commercialization of NdFeB magnets
took place [35]. In the beginning of nineties the production of rare-earth magnets started booming in China, relying on its large deposits of rare earths and cheap mining [35]. The price of
rare-earth magnets went down and that removed the main obstacle for using strong magnets in
wide range of applications; accordingly, PM machines became cost competitive [36, 37]. However, not only that the price of the magnet has been reduced in past 20 years but also the palette
of different high-energy magnets has become readily available: sintered and plastic-bonded
magnets with different energy products, Curie’s temperatures, corrosion resistance, shape flexibility, etc [38, 39]. That gave opportunities for use of PM machines by even highly demanding
customers such as aircraft industry or military.
Emergence of efficiency-driven applications. Importance of economical exploitation of resources (fuel in particular) has brought about need for compact and efficient electromechanical
systems. In practice it means that more-efficient electrical machines are to either replace traditionally pneumatic, hydraulic and combustion engines (e.g. in aircraft sub-systems, electric
cars) or become physically integrated and directly coupled with mechanical systems (e.g. machining spindles, flywheels, turbines).
Qualities of PM machines make them a preferred choice for such systems. Owing to strong
magnets, PM machines can have high power densities and also achieve high efficiencies due to
no excitation losses, no magnetizing currents and very low rotor losses.
Virtue of high power density becomes particularly important for applications where low
volume/weight and high-speeds are important. Undoubtedly, PM machines dominate the field
of small high-speed machines [24]. This can be ascribed to their magnetic excitation - airgap flux density of PM machines is determined mainly by the quality of utilized permanent
magnets and does not depend on the size of the machine. Current-excited machines, on the
other hand, lack space for conductors in small volumes and thus have comparatively smaller
power densities.
For the same power requirement, a high-frequency design reduces size and weight of an
electrical machine. High-frequency also often means elimination of power transmission elements. Thus, with downsizing and integration the resulting machinery becomes more efficient,
lighter and even portable [27]. In the rest of this section a few typical applications for highspeed PM machines are discussed.
Miniature gas turbines are an exemplary application for high-speed generators. Small gas
turbines are a promising means of converting fuel energy into electricity. Unlike large turbines
whose output power is transmitted to generators via gears, these small gas turbines are conceived to be directly coupled with high-speed generators [27]. Resulting device would be a
highly efficient power unit suitable as a portable power supply or a part of a distributed power
network. Reliable gas turbines that utilize PM generators are available on the market [40]. Still,
there is a great interest in academia for developing reliable high-speed generators that would
keep pace with newly developed turbines which are capable of rotating at speeds beyond 1
million rpm [41, 42]. A good example of such an effort in academia is work on high-speed
generators in ETH Zurich [43].
Waste heat from a gas turbine may be further used to heat water or space. As stated in
2.3. Defining high-speed
13
[44]: ”Because electricity is more readily transported than heat, generation of heat close to
the location of the heat load will usually make more sense than generation of heat close to the
electrical load.” Such an idea lies behind concept of combined heat and power (CHP) [27, 45]
systems which are, according to current predictions, be employed as highly efficient heating,
cooling and power system of buildings in coming years [46].
Machining spindles were already thoroughly discussed in Introduction: integration of a PM
motor with a spindle gives way to efficiency, compactness and high speeds for production of
complex 3D parts. Spindles for medical tools represent also an attractive application for small
high-speed machines. Today, dental drilling spindles are mostly driven by air turbines, however,
replacing the turbines with electrical drives would facilitate adjustable speed and torque of the
spindles and reduce number of hand-pieces needed by a dentist [30].
The new trend of replacing mixed secondary power systems in aircraft with electrical ones
has brought forth requirements for light-weight, fault-tolerant machines [47]. To achieve powerful engines in small volumes high-speed machines are necessary.
Integrated with flywheel a high-speed PM machine forms an electro-mechanical (EM) battery which is efficient and long-lasting device for energy storage [48, 49]. These EM batteries
have been used in many applications, such as hybrid cars, locomotives [24] and spacecraft [50].
2.3
Defining high-speed
It is wise here to identify the speed which will be taken as a decisive factor in this thesis for
naming a machine a high-speed machine. As long as technology permits, arbitrary high rotational speeds could be simply achieved by scaling down the machine. Therefore, it would not
be sensible to take rotational speed as a sole criterion for the high speed.
The tangential speed at the outer rotor radius is often taken as a criterion when defining
high-speed because it also takes into account the size of a machine. Such reasoning could make
sense since one of major limiting factors for the rotational speed, mechanical stress in the rotor,
is dependent on the tangential rotor speed. However, this criterion practically represents the
degree of machine’s mechanical utilization and it would favor very large generators that operate
at 50/60 Hz which are hardly perceived as high-speed machines [24]. For that reason, as Binder
and Schneider [24] point out, only inverter-fed or variable-speed machines can, in common
understanding, be called high-speed.
In [51] super high-speed machines are classified according to operating power and rotational
speed. A numerical limit proposed in [52] correlating power limit with the rotational speed of an
electrical machine was used as a criterion for super high-speed in [51]. This type of relationship
has commonly been used to evaluate operating speed range of machines [43, 53]. Binder and
Schneider [24] also empirically found a correlation between rated powers and speeds of superhigh-speed machines.
Consequently, this thesis will focus on variable-speed PM machines of small and medium
size that have high speed with respect to their power. This correlation will be discussed more
in the rest of this chapter. Rated power of the machines of interest is typically below 500 kW
- the power range where PM machines appear to be prevalent. Nevertheless, a good part of the
14
Chapter 2. High-speed PM machines: applications, trends and limits
analyses found in the thesis is applicable to a broad range of high-speed electrical machines.
2.4
Survey of high-speed machines
Based on collected data on commercially available high-speed machines and machines developed in academia [28, 30, 40, 43, 49, 50, 54–76], a diagram of rated powers and speeds of highspeed permanent magnet and induction machines was made.
6
Rated speed [rpm]
10
induction machines
PM machines
slotless and very large air−gap PM machines
5
10
3.6
P~1/f
4
10 1
10
2
10
3
10
4
10
5
10
6
10
7
10
Rated power [W]
Figure 2.1: Diagram of rated speeds and powers of existing high-speed machines, [28, 30, 40,
43, 49, 50, 54–76]
Relation between rated powers and speeds in the diagram is in a good agreement with
the correlation empirically found by Binder and Schneider [24]. Namely, from a study on
published data on high-speed AC machines, the authors obtained the relationship: log f =
4.27 − 0.275 log P (consequently, P ∼ 1/ f 3.6 ), which is presented with a dashed line in the
diagram.
The diagram (Fig. 2.1) shows prevalence of PM machines among small high-speed machines. To the author’s knowledge, no machine other than permanent magnet has been reported
to operate beyond speed of 100.000 rpm.
Slotless PM machines show a trend of the highest increases in speeds and extremely fast
examples have been reported both in market and in academia [28, 30]. Bianchi et al. [77]
illustrated advantages of using slotless rather than a slotted stator in high-speed PM machines.
The authors optimized, constructed and, finally, assessed performance of machines of both
types. Optimum flux density in very high-speed PM machines is usually low [77], affecting
designs to result in machines with large effective air-gaps. Instead of increasing mechanical
air-gap, it is sensible to replace stator teeth with conductors. In this manner, the increase of
conductors’ area enables rise of the rated current, and that, in turn, partly compensates loss of
power density due to reduced air-gap flux density.
2.5. Speed limits of PM machines
15
slotted stator
slotless stator
rotor
rotor
a)
b)
Figure 2.2: Cross-section of a a) slotted machine with a large air gap and b) slotless machine
2.5
Speed limits of PM machines
Boost in speeds of PM machines is linked with overcoming or avoiding a number of machine
limitations. Various physical parameters (stress, temperature, resonant frequencies) can limit
the speed of an electrical machine. Aside from speed, these variables are also affected by
power, size and machine electrical and magnetic loading.
Only a small number of academic papers discuss the limits of high-speed machines. In
the paper by G.R. Slemon [78] parameters of surface-mounted PM machines were correlated
with respect to physical and technological constraints. The paper derived general approximate
expressions for acceleration and torque limits.
Bianchi et al. [77] evaluated thermal and PM-demagnetization limits of different types of
high-speed PM machines. Those limits were taken into account in the proposed optimization
procedure through constraints imposed on the design variables. In a later paper by the same
authors [79] the demagnetization limit was disregarded in slotless PM machine design as it is
too high to be reached.
In extremely high-speed machines, however, mechanical factors such as stress and vibrations, rather than electromagnetically induced heating, are likely to cause failure of the machine.
A good modeling of elastic behavior and constraints of a rotor of an electrical machine can be
found in the thesis of Larsonneur [80]. Other authors also took mechanical constraints into account when designing a PM rotor, e.g. [81, 82]. Finally, stability of rotation has been analyzed
comprehensively in the field of rotordynamics [83–86], however some important conclusions
on rotordynamical stability have not been included in literature on electrical machines.
Speed limitations of PM machines in general are the topic of this section. Only physical
limits that are inherent to PM machines will be discussed; speed limitations associated with
bearing types, power electronics, control or technological difficulties will not be considered.
Defining and quantification of the machine speed limits in the rest of the thesis will have an
essential impact on the design of the test machine.
The limit that is common to all machines is the thermal limit. The thermal behavior of a
machine depends on power losses that are further dependent on current and magnetic loading,
as well as rotational speed. These parameters will be correlated with machine size and rated
16
Chapter 2. High-speed PM machines: applications, trends and limits
power in the next section in order to see how the speed is limited.
The strength or stress limit is inherently connected with the elastic properties of the materials
used in the rotor. Mechanically, a PM rotor can be modeled as a compound of two or three
cylinders: a rotor magnet that either leans on a rotor iron shaft or is a full cylinder, and an
enclosure of the magnet. Given the interference fits between the adjacent cylinders, maximum
tangential speed vt,max of the rotor can be determined (see Chapter 4) at which either maximum
permissible stress is reached in one of the cylinders or contact at boundary surface(s) is lost.
Third limit that will be considered is related to rotordynamical properties of the mechanical
system. For every rotor-bearing system two types of vibrations can occur that can (or will) limit
the rotational speed of the rotor: resonant and self-excited. Resonant vibrations occur when
the speed of the rotor coincides with one of the resonant frequencies. In literature are those
rotor speeds referred to as critical speeds, among which those connected with flexural modes in
the rotor are particularly dangerous. Self-excited vibrations make rotation impossible, that is,
unstable, and they commence after a certain threshold speed [87].
k
k
l
Ω
lFe
k
d
k
Figure 2.3: Rotor-bearings configuration
Tackling the problem of rotor vibrations is a very complicated task that includes the mechanical design of the rotor and design of the bearings. However, the speed range in which
the rotor will operate is usually decided on in the initial design stage: the working speed range
must be sufficiently removed from the resonant speeds and below the threshold of instability.
For a given rotor geometrical profile and bearings’ stiffness the maximum ratio between the
rotor length and diameter can be defined, Fig. 2.3:
λmax =
l
lFe
=
.
d α2rr
(2.1)
so as to insure stable rotation at the target maximum rotor speed.
From (2.1) maximum ratio between active rotor length (active machine length) and the rotor
radius is determined:
lFe
λFe,max =
= αλmax ,
(2.2)
2rr
where α = lFe /l is ratio between active and total rotor length.
Parameter λmax can be viewed as a figure of the rotor slenderness or, conversely, robustness,
with lower values for a more robust rotor.
2.6. Limits and rated machine parameters
2.6
17
Limits and rated machine parameters
The goal of this section is to theoretically correlate rated power, speed and size of PM machines and, as a result, account for speed-power relationship in the diagram in Section 2.4. The
correlation will be drawn considering the limiting factors mentioned in the previous section.
The maximum slenderness λmax and tangential speed vt,max will be assumed known, as defined
by mechanical properties of the system. Temperature in a machine depends on losses and on
the cooling capability thus the surface loss power density will be maintained constant. The
derivation is similar to the one in [24], with focus on loss influence in different types of PM
machines.
To analyze interdependence of power, size and speed the following approximate equation
for output power will be used:
Pnom = ΩT = 2π fnom · Fd · 2πr s lFe ·r s ≈ 2π2 Bg Ac r2s lFe fnom .
| {z } | {z }
Ω
| F{z
}
(2.3)
T
In the equation Pnom , T and fnom are, respectively, rated power, torque and frequency and
Fd ≈ Bg Ac /2 [88] denotes machine’s maximum force density, where Bg is maximum air-gap
flux density and:
(
s
kw k f ill Jh s bsb+b
, for a slotted machine
t
Ac ≈
(2.4)
kw k f ill Jlw , for a slotless machine
is current loading. In (2.4) J is amplitude of sinusoidal stator current density, h s , b s and bt are
slot height, slot width and tooth width of a slotted machine, lw is conductors’ area thickness of
a slotless machine and kw and k f ill are winding and fill factors, respectively. If the ratio of slot
to tooth width is kept constant and having h s , lw ∼ rr , the expressions (2.4) yield:
Ac ∼ Jrr .
(2.5)
Using (2.1) in equation (2.3) and neglecting the air-gap length (r s ≈ rr ) we obtain:
Pnom = 4π2 Bg Ac αλmax rr3 fnom .
(2.6)
Due to increased losses, to maintain a given cooling capability, magnetic and current loading
must be lowered if frequency is increased. Consequently, the ratio between maximum permissible power dissipation and cooling surface area is maintained constant:
Pdiss
= const,
S
(2.7)
while keeping in mind that S ∼ rr2 .
An analysis of the influence of the machine size and frequency on dominant, frequencydependent losses will follow.
At very high frequencies iron losses in the stator are estimated as:
2
PFe = C · mFe fe2 B2g ∼ rr3 fnom
B2g ,
(2.8)
18
Chapter 2. High-speed PM machines: applications, trends and limits
where fe :
N poles
fnom
(2.9)
2
is electrical frequency and mFe is mass of the iron core.
Copper losses can be divided into two parts: (i) the conduction loss part PCu,skin , that also
includes a rise in loss caused by the reduction of effective conductors cross-section due to skineffect; and (ii) the proximity loss part PCu,prox , that accounts for eddy-current loss in the conductors due to the pulsating magnetic fields from the rotor magnet and neighboring conductors:
fe =
PCu = PCu,skin + PCu,prox .
(2.10)
According to [89] we can estimate the skin-effect conduction loss as:
PCu,skin = F (φ) · PCu,DC = (F (φ) − 1) I 2 RDC + I 2 RDC ,
|
{z
} |{z}
DC
skin - effect
where φ is the ratio between conductor diameter and the skin-depth:
r
πσCu µ0 fe
dCu
φ=
,
√ = dCu
2
δ skin 2
(2.11)
(2.12)
and function F (φ) is defined as:
F (φ) =
φ (ber(φ)bei0 (φ) − bei(φ)ber0 (φ))
.
·
2
ber0 2 (φ) + bei0 2 (φ)
(2.13)
For reasonable conductor diameters and electrical frequencies of high-speed machines the skineffect part of eq. (2.11) comprises less than one percent of the total value of PCu,skin - see Figure
3.17 in Subsection 3.6.2 - and can thus be neglected. Therefore:
PCu,skin ≈ PDC = I 2 RDC = J 2 S w ρCu lCu ∼ J 2 rr2 lFe ∼ J 2 rr3 .
(2.14)
In equation (2.14) S w is the copper cross-section area of the windings and ρCu is the copper
resistivity.
Regarding the proximity-effect part of the copper losses, in the example of slotless PM
machines, it will primarily be influenced by the rotor permanent magnet field rather than by the
field of the neighboring conductors. This part of losses can be estimated as [90] (for detail see
Subsection 3.6.2):
2
B2g (2π fnom )2 dCu
2
PCu,prox,slotless =
VCu ∼ rr3 B2g fnom
.
(2.15)
32ρCu
where VCu is volume of the conductors (copper).
In a slotted machine the influence of neighboring conductors on proximity loss is dominant,
but dependence of this loss on frequency takes on a more complicated form. The loss in a single
conductor can be expressed in a following form (see [89]):
1
PCu,prox
=
G(φ)
2
· He,rms
· lFe
σCu
(2.16)
2.6. Limits and rated machine parameters
19
where He,rms is the rms value of magnetic field strength of the neighboring conductors and:
G (φ) = 2πφ ·
(ber2 (φ)ber0 (φ) − bei2 (φ)bei0 (φ))
.
ber2 (φ) + bei2 (φ)
(2.17)
For a constant conductor diameter dependence of the expression G (φ) on frequency is very
closely quadratic - see Figure 3.18 and eq. (3.105) in Subsection 3.6.2 - and the total loss caused
by proximity effect for the slotted machine case can be approximated as:
2
2
PCu,prox,slotted ∼ J 2 fnom
VCu ∼ J 2 fnom
rr3
(2.18)
To maintain the permissible surface loss density (2.7) in a slotless machine, from (2.8),
(2.15) and (2.14) restrictions for magnetic flux density and current density are obtained as:
and
√
Bg ∼ 1/( rr fnom )
(2.19)
√
J ∼ 1/ rr .
(2.20)
Finally from (2.5), (2.6), (2.19) and (2.20) we obtain correlation between the rated power
and size for slotless PM machines:
Pnom ∼ rr3 .
(2.21)
In the slotted machine case, expressions for DC- (2.14) and frequency-dependent copper
√
√
loss (2.18) yield different restrictions for current density: J ∼ 1/ rr and J ∼ 1/( rr fnom ). Correlation between rated parameters of slotted PM machines can be expressed in a more general
form as:
r3
(2.22)
Pnom ∼ kr , 0 < k < 1,
fnom
where k depends on portion of frequency-dependent losses in total copper loss.
The last two equations explain trend of higher increase in rated speeds of slotless PM machines with respect to their slotted counterparts, as presented in Section 2.4.
Further, from (2.21) and (2.22) it is evident that it is not possible to gain power density by
merely scaling down the machine and increasing its rated speed. Unfortunately, with increasing operating speed (to obtain the same machine power), losses in the machine will increase
rapidly. To maintain a given cooling capability, magnetic and current loading must be lowered
if frequency is increased.
However, the rationale behind using high-speed machines is compactness and efficiency of
a particular electro-mechanical system and not of the machine itself.
Air-friction loss has not been analyzed yet in this section and it represents a great portion of
overall losses in high-speed machines. Magnetic and current loading, however, do not influence
air-friction and it is dependent solely on mechanical quantities: size, speed and roughness of
the rotor and length of the air-gap. Power of the air-friction drag is given by:
Pa f = k1C f vt , lg ρair πΩ3 rr4 l.
(2.23)
20
Chapter 2. High-speed PM machines: applications, trends and limits
Empirical estimations of the friction coefficient C f for typical rotor geometries can be found
in the thesis of Saari [91]. If the air-friction power is divided with the rotor surface area 2πrr l a
surface density of air-friction loss power is obtained:
1
qa f = k1C f vt , lg ρair v3t .
2
(2.24)
Temperature in the air-gap is proportional to this power density. Whether this temperature
will jeopardize operation of a machine depends on the coefficient of thermal convection on the
rotor surface and that coefficient is rather difficult to assess. In any case, tangential rotor speed
is a potential limiting figure for machine with respect to air-friction loss.
If tangential speed vt = 2πrr fnom is used in (2.21) and (2.22) instead of the radius, the
following correlations are obtained:
( 3 3+k
vt / fnom , for a slotted machine
(2.25)
Pnom ∼
3
v3t / fnom
, for a slotless machine
Equations (2.25) practically establish connection between the nominal powers of PM machines, their rotor radii (sizes) and nominal rotational frequencies. Usually, tangential speed in
PM machines is limited by stress in the rotor and today does not exceed 250 m/s (the highest
reported rotor tangential speed is 245 m/s and it is associated with a very large turbine generator in Germany, operating at 50 Hz [24]). Hence, correlation between nominal powers of PM
machines with their highest permissible nominal speeds can be expressed in a general form as:
Pnom ∼
1
3+k
fnom
, 0 < k < 1,
(2.26)
which Binder and Schneider [24] found with an approximate value of k = 0.6 for the frequency
exponent (Fig. 2.1), however, by including also induction and homo-polar machines in the
study.
2.7
Conclusions
The chapter gives an introduction to high-speed permanent magnet machines: it analyzes prominent and prospective applications of these machines, defines the attribute ”high-speed” for the
purpose of this thesis and presents a general theoretical study of speed limits of PM machines.
By doing so, the chapter explains the reasoning behind choosing the slotless PM machine as the
type of the test-motor and, more importantly, it defines the scope and outlook for the modeling
and design presented in the thesis.
Two reasons are singled out for the prevalence of PM machines in the last two decades:
(i) rapid commercialization and availability of rare-earth magnets and (ii) the emergence of
efficiency-driven applications. Qualities of PM machines - high efficiency and power density have made them a preferred choice for high-performance applications, particularly at medium
and low powers.
2.7. Conclusions
21
High power density of PM machines can be ascribed to their magnetic excitation - air-gap
flux density of PM machines is determined mainly by the quality of utilized permanent magnets
and does not depend on the size of the machine. At the same time, relatively simple, cylindrical
rotor can be robust enough to sustain the stress caused by the centrifugal force at high speeds.
In the thesis, high-speed machines are defined as variable-speed machines of small and
medium size (typically, below 500 kW) that have high speed with respect to their power. Empirical study on such machines developed in industry and academia also showed the predominance
of PM machines. Obtained relation between rated powers and speeds is in a good agreement
with the correlation empirically found by Binder and Schneider [24].
Slotless PM machines show a trend of the highest increase in speeds and extremely fast
examples have been reported both on the market and in academia. Since the optimal air-gap
flux density in high-speed PM machines for highest efficiency is usually low it is sensible to
replace stator teeth with conductors and, in that way, partly compensates loss of power density
due to reduced air-gap flux density.
Finally, the chapter identifies inherent (physical) speed limits of PM machines - thermal,
structural (elastic) and rotordynamical. Taking into account their physical limits, the chapter
theoretically correlates speed, power and size of PM machines and accounts for the correlations
appearing from the empirical survey. It is concluded that, if the cooling method is maintained,
it is not possible to gain power density by merely scaling down the machine and increasing its
rated speed. This study can be viewed as the main contribution of this chapter.
22
Chapter 2. High-speed PM machines: applications, trends and limits
Chapter 3
Electromagnetic modeling of slotless PM
machines
3.1
Introduction
The adequate modeling of the electromagnetic (EM) field is the starting point for a design of an
electrical machine. The models derived in this chapter will form the basis for the geometrical
optimization of the test machine described in Chapter 7.
The applicability of slotless PM machines for very high-speed operation, which was illustrated in the previous chapter, motivated the choice of a slotless configuration for the test
machine. Therefore, the modeling in this chapter will take into account only permanent magnet
machines with slotless geometry.
An analytical approach to the machine modeling and design was adopted because only analytical expressions represent an efficient means for machine optimization. In addition, it is often
stated that analytical models offer greater insight into the modeled phenomena than numerical
methods. That is true, however, to a certain extent. Strictly applied, the modeling in this chapter
leads to rather long and complex analytical expressions which hardly offer any insight into the
influence of the underlying parameters. On the other hand, the reason for the seeming lack of
given insight of the finite element (FE) method usually lies in limited knowledge of its users
and not in (seemingly obscured) logic behind the used FE software.
Analytical EM modeling of slotless machines is not strictly confined to such machines and
can be applied, with certain corrections, to a broader selection of PM machines. Namely, slotted
machines are frequently modeled by transforming their geometry into an equivalent slotless
geometry in which the resulting air gap length is somewhat extended using the Carter’s factor
which accounts for the fringing of the magnetic field around slots. However, such a model,
naturally, cannot explain the effects of slotting on the magnetic field and correlated phenomena
such as cogging torque and slot-harmonic induced losses.
The available literature on analytical modeling of electrical machines, including PM machines, is immense and a literature overview on the subject would be too broad to be of much
interest or use to the reader. Furthermore, the model to be presented in this chapter is developed in a rather standard manner - starting from the Maxwell equations, the model reduces to
23
24
Chapter 3. Electromagnetic modeling of slotless PM machines
Poisson’s or Laplace’s equations expressed over magnetic (vector) potential - which is generally known and used by electrical machines experts. Still, as a good reference for the topic one
could point to the outstanding work of Z.Q. Zhu whose series of articles on analytical modeling of PM machines of different topologies [92–96] represents a comprehensive resource for
the accurate modeling of PM machines. For the modeling in this chapter, though, the author
primarily referred to the theses of H. Polinder [97] and S.R. Holm [98].
The EM model in this chapter is two-dimensional since a slotless machine with its large
effective air gap cannot be accurately modeled without accounting for the curvature of the field
in the air gap. Magnetic vector potential was chosen as the basic variable from which other field
quantities are derived. The model is magnetostatic meaning that it neglects the reaction field of
the eddy currents induced in the conductive parts of the machine. There are several reasons for
this approach. Firstly, the influence of the reaction field of the (rotor) eddy currents on the total
field and, consequently, other EM quantities, is normally very small (see Zhu’s article [96]).
Therefore, the magnetostatic model is quite sufficient for the machine design. Furthermore,
a practical reason for the magnetostatic modeling is that the magnet of the test motor is of a
plastic-bonded type which is chosen to maximally suppress eddy currents in the rotor. Finally,
the power of eddy-current losses, particularly critical for the heating of the rotor magnet, can
still be assessed based on the harmonics of the magnetostatic field [99, 100].
Although eddy currents are neglected when the magnetic field in the machine is modeled,
their effect on losses in conductive machine parts is accounted for in Section 3.6.
Based on the model of the magnetic field in the machine, other quantities of the machine
- no-load voltage, torque, inductance, unbalanced magnetic force and losses - will be derived
and used in the design optimization. It is important to stress that the purpose of this chapter is
not to offer major contribution in machine modeling, but rather to distinguish and model most
important EM parameters of high-speed PM machines. The modeling does not look for ultimate
accuracy but for adequate representation of dominant phenomena that can serve as a good basis
for designing high-speed PM machines and assessing their limits.
The chapter starts with presenting model geometry and properties that will be used for
modeling of slotless PM machine. Section 3.3 presents the analytical modeling of the magnetic
field that forms the basis for most of models in the rest of the chapter. Main EM parameters
of the machine - no-load voltage, torque, inductance - are presented in Section 3.4. Models for
unbalanced magnetic force in a PM machine are investigated in Section 3.5 and comprehensive
modeling of different frequency-dependent losses is given in Section 3.6.
3.2
Model geometry and properties
A slotless machine is adequately represented as a compound of several coaxial cylinders which
renders the machine suitable for modeling in a cylindrical system. The machine will be modeled
in 2D with all the variables dependent only on radial and circular coordinate (r and ϕ); all endeffects will be neglected. The cross-section of the modeled machine is represented twice in
Figure 3.1. The machine configuration is depicted on the left-hand side and variables which
define the model geometry are shown on the right.
3.2. Model geometry and properties
25
non-magnetic rotor enclosure
c
AT
ST
b’
OR
.
x
Rm
a’
x
R
permanent magnet
O
TO
rso . rs
rotor iron core
rw re
rm
Rm
RR
.
R
a
x
rFe
lm
RR
φ
x
z
RS
r
RS
lag
.l
w
μr = 1
μr→∞
μr=μrec
.
air gap
x
.
x
windings in the air gap
b
μr→∞
c’
stator iron core
Figure 3.1: Cross-section of the slotless PM machine (twice)
Material properties of all machine parts are assumed to be stationary, isotropic and linear;
the last condition facilitates use of superposition of the field vectors resulting from different
sources. Both iron parts in the model have infinite permeability. The magnet working point is
assumed to be in the second quadrant of the BH plane and recoil permeability of the magnet is
µrec . All other machine parts have relative permeability equal to 1.
The most inner rotor part is the iron shaft on which permanent magnet ring resides. The
magnet has two poles as it is usual with very high-speed machines. The PM ring is magnetized
diametrically1 yielding the following expression for remanent flux density:
−−−→
→
−
→
−
Brem = B̂rem sin (ϕr ) ir + B̂rem cos (ϕr ) iϕ ,
(3.1)
where B̂rem is the arithmetic value of the magnet remanent flux density.
In reality, the magnet is retained with a non-magnetic enclosure/sleeve; however, since the
sleeve has, practically, the same magnetostatic properties as air, that region will not be separately treated in the modeling. Thus, external rotor radius re and the sleeve thickness are irrelevant for the EM modeling and in the further text in this chapter the term air gap will actually
determine mechanical air gap and rotor sleeve combined.
The stator has a slotless iron core. The conductors are shown in Figure 3.1 to be toroidally
wound around the stator core, as it is the case of the test machine. However, the magnetic
field within the machine is the same for the case of, more common, air-gap windings since
only parts of the windings that are inside of the air gap matter for the magnetic field in this
model. Furthermore, magnetic field outside of the stator core is neglected and the stator surface
at r = r so is the surface with a flux-parallel boundary condition.
1
The magnetization is equivalent to the ideal Halbach array with two poles.
26
Chapter 3. Electromagnetic modeling of slotless PM machines
Distribution of the conductors of the phase a of the test machine is given in Figure 3.2 in
terms of angular conductor density (number of conductors per radian) over the stator circumference. The distribution can be decomposed in a Fourier series as follows:
π
∞
sin k
X
4
6 cos(kϕ ),
nk cos(kϕ s ) = nmax
na (ϕ s ) =
s
π
k
k=1,3,5...
k=1,3,5...
∞
X
where:
nmax =
(3.2)
3N
;
π
(3.3)
N is the number of turns per phase2 .
The conductor distributions of the phases b and c are phase shifted for ±120◦ :
!
2π
nb (ϕ s ) = na ϕ s −
3
!
2π
nc (ϕ s ) = na ϕ s +
3
(3.4)
(3.5)
na
nmax
5π
6
7π
6
π
6
11π
6
2π
φs
-nmax
Figure 3.2: Angular density of conductors of the phase a
Alternatively, the conductor distribution (3.2) can be represented in an equivalent form:
na (ϕ s ) =
∞
X
k=−∞
n6k+1 cos(kϕ s ) +
∞
X
n6k+3 cos(kϕ s )
(3.6)
k=0
Finally, it will be assumed that the rotor revolves in synchronism with the stator field, thus:
ωr = ωe = ω,
where ωr and ωe are mechanical (rotor) and electrical angular frequency, respectively.
2
In the example of toroidal windings N is physically a half number of turns per phase.
(3.7)
3.3. Modeling of the magnetic field
27
The field quantities in the model will be represented in one of two reference frames: the
stationary frame, fixed to the stator, and the rotating frame, fixed to the rotor. Correlation
between these frames is depicted in Figure 3.3. Angular positions in two coordinate systems
are correlated in time as:
ϕ s = ϕr + θ,
(3.8)
where θ = θ(t) determines the rotor angular position in stationary reference frame:
θ (t) = ωt + θ0 .
(3.9)
In (3.9) θ0 is the rotor initial position.
ωr
r
P
φr
φs
θ=ωrt+θ0
z
r
Figure 3.3: A position vector in the stationary and rotating reference frame
3.3
Modeling of the magnetic field
→
−
The field in the machine will be solved using magnetic vector potential A. The vector potential
is implicitly defined by its curl and divergence:
→
−
→
−
B = ∇ × A,
→
−
∇ · A = 0;
(3.10)
(3.11)
the latter expression is known as the Coulomb gauge.
As mentioned in the beginning of the chapter, the modeling will conform to the magnetostatic assumption and the effect of induced eddy currents on the field will be neglected. Therefore, Ampere’s law simply yields:
→
− →
−
∇ × H = Js,
(3.12)
where J s is current density of the stator (air-gap) conductors.
Combining (3.12) and constitutive equation for magnetic field in a linear medium:
→
−
→
− −−−→
B = µ H + Brem
(3.13)
28
Chapter 3. Electromagnetic modeling of slotless PM machines
and by incorporating (3.10) and (3.11), one can obtain the governing equation for the magnetic
vector potential of a magnetostatic field in the slotless PM machine:
→
−
→
−
−−−→
∆ A = −µ J s − ∇ × Brem .
(3.14)
The symbol ∆ = ∇2 is the Laplace operator and µ = µ0 µr .
The equation (3.14) is a Poisson’s differential equation and it will be used for obtaining
expressions of the magnetic field quantities. The equation will be solved using the principle of
superposition: two terms on the right-hand side of (3.14) can be considered as sources of the
field and the resulting vector potentials from each source can be separately determined and then
→
−
summed. Finally, after solving (3.14) for the vector potential, magnetic flux density B and field
→
−
intensity H can be simply obtained from (3.10) and (3.13).
The governing equation (3.14) is the general one and it is valid for the whole machine
from Figure 3.1. However, the equation takes on different, simpler forms in different machine
regions. The model geometry defined in the previous section can be subdivided into 5 regions
with different magnetic properties and/or governing equations. The regions are enumerated for
easier representation in the equations and represented in Figure 3.4. The governing equations
of different regions are listed in Table 3.1.
armature
{
5
4
3
2
stator iron core
air gap windings
air gap
permanent magnet
(excitation)
r2
Rm
r1
r3 r4
r5
RR
1
rotor iron core
2
3
4
5
r1=rFe
r2=rm
r3=rw
r4=rs
r5=rso
Figure 3.4: Model geometry
Finally, the governing equation can be also written in a scalar form. It is evident (Fig.
3.1) that current density vector has only a z-component and the same is true for a curl of the
remanence vector. Therefore, magnetic vector potential has also only z-component:
→
−
→
−
A = Az (r, ϕ) · iz
(3.15)
3.3. Modeling of the magnetic field
29
Table 3.1: Properties and field equations of the modeled regions
i
region
range for r
µr
1
rotor iron
2
magnet
0 ≤ r < rFe
∞
3
air gap
4
windings
5
stator iron
rFe ≤ r < rm
µrec
rw ≤ r < r s
1
rm ≤ r < rw
1
r s ≤ r < r so
∞
governing equation
→
−
∆A = 0
→
−
−−−→
∆ A = −∇ × Brem
→
−
∆A = 0
→
−
→
−
∆ A = −µ0 J s
→
−
∆A = 0
constitutive equation
→
−
H=0
→
−
→
− −−−→
H = ( B − Brem )/µ0 µrec
→
− →
−
H = B/µ0
→
− →
−
H = B/µ0
→
−
H=0
and the equation (3.14) is equivalent to:
Brem,ϕ ∂Brem,ϕ 1 ∂Brem,r
∂2 Az 1 ∂2 Az 1 ∂Az
+
+
=
−µJ
−
−
+
.
s,z
∂r2
r2 ∂ϕ2
r ∂r
r
∂r
r ∂ϕ
(3.16)
−−−→
The radial and angular components Brem,r and Brem,ϕ of the remanent flux density vector Brem
were defined in (3.1).
Boundary conditions for solving equation proceed from Ampere’s and flux-conservation
law and may be defined as follows:
− −−→
→
−n · →
Bi − Bi+1 = 0,
(3.17)
→ −−−→
→
−
→
−n × −
Hi − Hi+1 = Ki ,
(3.18)
−n is a unit vector normal on the boundary surface and directed from the region i to the
where →
→
−
region i + 1 while Ki is the surface current density at the boundary surface.
For the example of the modeled machine these boundary conditions could be rewritten in
simpler forms:
Bi,r (r = ri ) = Bi+1,r (r = ri ) , i = 1..5,
Hi,ϕ (r = ri ) = Hi+1,ϕ (r = ri ) , i = 1..5.
(3.19)
(3.20)
−
→
There is no magnetic field outside the modeled machine regions given in Fig. 3.4, thus B6 = 0
−→
and H6 = 0.
3.3.1 Field of the permanent magnet
The field from the permanent magnet, or the excitation field, will be modeled first. The governing equation for this case reduces to:
→
−
−−−→
∆ A = −∇ × Brem .
(3.21)
It can be easily shown that the curl of the remanent field vector defined as (3.1) is zero:
−−−→
∇ × Brem = 0,
(3.22)
30
Chapter 3. Electromagnetic modeling of slotless PM machines
and that further reduces the governing equation (3.21) to the Laplace’s equation:
→
−
∆ A = 0.
(3.23)
With respect to the type of the magnetization, the solution for the magnetic potential can be
found in the following form:
−
→
→
−
A1 = −C1 r cos (ϕr ) · iz ,
Di →
−
→
−
cos (ϕr ) · iz , i = 2..5
Ai = − C i r +
r
(3.24)
Magnetic flux density can then be calculated from (3.10) as:
−
→
→
−
→
−
B1 = C1 sin (ϕr ) · ir + C1 cos (ϕr ) · iϕ ,
(3.25)
Di Di →
−
→
−
→
−
Bi = Ci + 2 sin (ϕr ) · ir + Ci − 2 cos (ϕr ) · iϕ , i = 2..5
r
r
→
−
and magnetic field intensity H is calculated in each region from the corresponding constitutive
equation.
Boundary coefficients Ci , Di were calculated symbolically in Matlab from the system of
boundary conditions (3.19) and (3.20). The final analytical expressions for the field vectors are
rather lengthy and will not be presented here.
Table 3.2: Parameters of the test machine
parameter
symbol value
rotor shaft radius
rfe
10.5 mm
magnet outer radius
rm
14.5 mm
winding region inner radius
rw
17.5 mm
stator inner radius
rs
18.5 mm
stator outer radius
r so
27.3 mm
magnet remanent flux density
0.504 T
B̂rem
magnet recoil permeability
µrec
1.15
Results of the modeling of the PM field are presented for the test motor (the parameters are
shown in Table 3.2). Distribution of radial and tangential flux density at various radii in the
machine are given in Figures 3.5 and 3.6 and flux lines are shown in Figure 3.7. As expected,
diametrical magnetization in the rotor causes a perfectly sinusoidal spatial distribution of the
magnetic field throughout the whole machine. Analytical model calculations match results of
2D FEM modeling.
3.3.2 Armature field
The field of the stator currents - armature field - will be modeled in this subsection. The governing equation for this case yields:
→
−
→
−
∆ A = −µ J s ,
(3.26)
3.3. Modeling of the magnetic field
31
0.3
middle of the magnet
winding surface
stator inner surface
middle of the stator
B r(ϕ) [T]
0.2
0.1
0
−0.1
−0.2
−0.3
0
pi/2
pi
3pi/2
2pi
ϕ [rad]
Figure 3.5: PM field: spatial distribution of the radial flux density; the dashed lines represent
results of the corresponding 2D FEM model
magnet middle
winding surface
stator inn. surf.
stator middle
0.6
B ϕ(ϕ) [T]
0.4
0.2
0
−0.2
−0.4
−0.6
0
pi/2
pi
3pi/2
2pi
ϕ [rad]
Figure 3.6: PM field: spatial distribution of the tangential flux density
→
−
where J s denotes the vector of total current density of the stator conductors which reside in the
air-gap winding region. This vector has only the z-component:
→
−
→
−
J s = J s · iz .
(3.27)
32
Chapter 3. Electromagnetic modeling of slotless PM machines
permanent
magnet
φ
r
Figure 3.7: Flux lines of the field of the permanent magnet (contour plot)
The stator current density J s is calculated by summing the current density of each phase in
the following way:
na (ϕ s ) ia (t) + nb (ϕ s ) ib (t) + nc (ϕ s ) ic (t)
J s = J s (ϕ s , t) =
,
(3.28)
rwc lw
where ia , ib and ic are phase currents, lw = r s − rw is the winding thickness and rwc is the radius
at the center of the windings :
rw + r s
.
(3.29)
rwc =
2
The phase currents can be represented as sums of their time harmonics. Since currents of a
balanced three-phase system do not contain even and triplen harmonics [97], the phase currents
can be represented as:
∞
P
î6m+1 cos [(6m + 1) ωt]
!#
"
2π
ib =
î6m+1 cos (6m + 1) ωt −
3
m=−∞
"
!#
∞
X
2π
ic =
î6m+1 cos (6m + 1) ωt +
3
m=−∞
ia =
m=−∞
∞
X
(3.30)
Using trigonometric identities and equations (3.28), (3.30) and (3.6) it can be shown that
the current density can be represented in the following form:
Js =
∞
∞
X
X
k=−∞ m=−∞
J s,6k+1,6m+1
∞
∞
3 X X n6k+1 î6m+1
cos (6k + 1) ϕ s − (6m + 1) ωt
=
2 k=−∞ m=−∞ rwc lw
(3.31)
3.3. Modeling of the magnetic field
33
The governing equation takes on the full form (3.26) only in the winding region:
−
→
∆A4 = −µJ s ,
(3.32)
while it reduces to the Laplace’s equation in other regions:
→
−
∆Ai = 0, i = 1, 2, 3, 5
(3.33)
Magnetic vector potential is found in the following form:
∞
∞
X
X
→
−
→
−
Ai =
Ai,6k+1,6m+1 · iz ,
(3.34)
k=−∞ m=−∞
|6k+1|
A1,6k+1,6m+1 = −C
J s,6k+1,6m+1 ,
1,6k+1 r
|6k+1|
Ai,6k+1,6m+1 = − Ci,6k+1 r
+ Di,6k+1 r−|6k+1| J s,6k+1,6m+1 , i = 2, 3, 5
!
µ0 r 2
−|6k+1|
|6k+1|
A4,6k+1,6m+1 = − C4,6k+1 r
+ D4,6k+1 r
+
J s,6k+1,6m+1 .
(6k + 1)2 − 4
(3.35)
0.02
middle of the magnet
winding surface
stator inner surface
middle of the stator
0.015
0.005
r
B (ϕ) [ T ]
0.01
0
−0.005
−0.01
−0.015
−0.02
0
pi/2
pi
3pi/2
2pi
ϕ [rad]
Figure 3.8: Armature field: spatial distribution of the radial flux density; the dashed lines represent results of the corresponding 2D FEM model
Magnetic flux density is then obtained using (3.10) and boundary coefficients are again
solved using the Matlab symbolic solver. Spatial distribution of radial and tangential flux density at different radii of the test
√ machine are plotted in Figures 3.8 and 3.8 for the maximum
amplitude of the current - 2 2 A - at the moment t = 0; results were confirmed by 2D FEM
modeling. Time harmonics of the current do not influence the field distribution and are not
considered.
34
Chapter 3. Electromagnetic modeling of slotless PM machines
0.04
magnet middle
winding surface
stator inn. surf.
stator middle
0.03
B ϕ(ϕ)[T ]
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
0
pi/2
pi
3pi/2
2pi
ϕ [rad]
Figure 3.9: Armature field: spatial distribution of the tangential flux density
For the given value of the current, the magnitude of the armature field is much smaller than
the magnitude of the field of the magnet even though a magnet with rather small remanence is
used. The reason for this is a very large effective air gap which comprises the magnet thickness,
sleeve thickness, mechanical air gap and the winding thickness. Additionally, it is apparent that
higher spatial harmonics of the flux density attenuate when approaching the rotor shaft. This
property is beneficial for alleviation of the rotor induced losses.
The attenuation of spatial harmonics is also demonstrated in Figure 3.10 where field lines
of the first, fifth and seventh harmonics (k equals to 0, -1 and 1, respectively) of the armature
field are plotted.
φ
φ
r
φ
r
r
air-gap windings
Figure 3.10: Flux lines of the armature field: 1st , 5th and 7th spatial harmonic (contour plot)
3.3. Modeling of the magnetic field
35
3.3.3 Combined field
Using the principle of superposition, field vectors of the combined field can be obtained by
summing the field vectors resulting from the field of the permanent magnet and stator currents
separately. Therefore, combined magnetic vector potential can be represented in the stator
reference frame as:
−−−→
→
− −−→
A = A pm (r, ϕ s − θ (t)) + Acurr (r, ϕ s )
(3.36)
and, similarly, magnetic flux density of the combined field becomes:
−−−→
→
− −−→
B = B pm (r, ϕ s − θ (t)) + Bcurr (r, ϕ s ) .
(3.37)
Magnetic vector potentials from the separate sources were derived in the previous two subsections and given by equations (3.24) and (3.35). Correlation between angular coordinates of
the two reference frames are given by (3.8) and (3.9).
Figure 3.11 shows distribution of the combined radial flux density for the case when stator
currents field leads the magnet field for π/2. The distribution is plotted for the maximum amplitude of the currents at the moment ωt = π/2. It is evident that the influence of the armature
field on the total field in machine is practically negligible.
0.3
middle of the magnet
winding surface
stator inner surface
middle of the stator
0.2
B r(ϕ) [ T ]
0.1
0
−0.1
−0.2
−0.3
0
pi/2
pi
3pi/2
2pi
ϕ [rad]
Figure 3.11: Combined field: spatial distribution of the radial flux density, the armature field
leading the permanent magnet for π/2. Dashed lines represent the flux density of the magnet
only.
36
Chapter 3. Electromagnetic modeling of slotless PM machines
3.4
Derived quantities
3.4.1 No-load voltage
Flux linkage of a coil can be found as a contour integral of magnetic vector potential along the
coil, thus:
I
−
→
− →
ψ=
A ·d l.
(3.38)
C
Therefore, PM-flux linkage of the phase a can be calculated by:
ψa = 2l s
Zπ/2
na (ϕ s ) Azpm (r = rwc , ϕr ) dϕ s ,
(3.39)
−π/2
or, after taking into account (3.6) and (3.8):
ψa = 2l s
∞
X
n6k+1 Azpm
k=−∞
(rwc )
Zπ/2
−π/2
cos (6k + 1) ϕ s cos (ϕ s − θ) dϕ s .
(3.40)
In the equations above l s represents stator axial length and Azpm (rwc ) denotes the part of the
vector potential which is not dependent on ϕr (see equation (3.24)):
Azpm (r = rwc , ϕr ) = Azpm (rwc ) · cos (ϕr ) .
(3.41)
The integral in (3.40) has a non-zero value only when k = 0. Hence, the flux linkage
becomes:
ψa = πl s n1 Azpm (rwc ) cos θ,
(3.42)
which, after substituting expressions for n1 and θ (equations (3.2), (3.3) and (3.9)), yields:
ψa =
6
Nl s Azpm (rwc ) cos (ωt + θ0 ) .
π
(3.43)
Finally, induced voltage in the phase a, no-load voltage, is obtained as the time derivative
of the flux linkage:
d
(3.44)
ea = ψa = −ê sin (ωt + θ0 ) ,
dt
6
pm
ê = ωNl s Az (rwc ) .
(3.45)
π
No-load voltage of the other two phases is, naturally, phase-shifted for ±2π/3 with respect
to the phase a:
!
2π
,
eb = −ê sin ωt + θ0 −
3!
(3.46)
2π
.
ec = −ê sin ωt + θ0 +
3
3.4. Derived quantities
37
The expression (3.45) represents the amplitude of the machine no-load voltage. In general,
exact analytical expression for ê is lengthy, however, it takes on a shorter form for µrec = 1:
êµrec =1 =
2
rm2 − r2f e r2s + rwc
3
ωNl s 2
B̂rem .
π
r s − r2f e rwc
(3.47)
The last expression can be written in another form:
êµrec =1
!
rm + r f e 1 r s
lm
rwc
= 2ωkw Nl s r s B̂rem ·
·
+
,
g
r s + r f e 2 rwc
rs
(3.48)
where kw = 3/π is the machine winding factor, lm = rm − r f e is the magnet thickness and
g = r s − r f e is the effective air gap, or:
!
rm + r f e 1 r s
rwc
·
+
,
(3.49)
êµrec =1 = ê1D,µrec =1 ·
r s + r f e 2 rwc
rs
where ê1D,µrec =1 is the no-load voltage would result from a one-dimensional model which does
not consider curvatures of the field in the air gap.
The last term in the product on the right side of the equation (3.49) is very closely equal to
1, therefore, that equation can be quite accurately expressed as:
êµrec =1 ≈ ê1D,µrec =1
rm + r f e
.
rs + r f e
(3.50)
With the last equation one can assess the accuracy of an one-dimensional model of a slotless
machine. For the example of the test machine, the 1D model would overestimate the machine
voltage for, approximately, 16% with respect to the voltage which would result from more
accurate 2D modeling.
3.4.2 Torque and power
Since the developed model does not account for induced losses in the machine, the easiest
way to calculate machine torque and power is from the equilibrium of input/electrical and output/mechanical power:
Pe = Pm ,
(3.51)
or, in another form:
ea ia + eb ib + ec ic = T ω,
(3.52)
where T is the machine torque.
Using equations (3.44), (3.46) and (3.30) it is not difficult to show that expression for the
machine power is:
∞
3 X
î6m+1 sin (6mωt − θ0 ).
(3.53)
P = Pe = ê
2 m=−∞
38
Chapter 3. Electromagnetic modeling of slotless PM machines
It can be concluded that oscillations of power and torque of the machine is, practically,
influenced solely by non-fundamental harmonics of the current. This conclusion is generally
valid for slotless machines regardless of the rotor magnetization since the time harmonics of the
field have predominant influence on torque oscillations and rotor losses (see [98]).
The average power is given by:
3
Pavg = − êî sin (θ0 )
2
(3.54)
and its maximum is reached when the rotor position lags behind the maximum field of the
currents for π/2:
π 3
Pavg,max = θ0 = − = êî.
(3.55)
2
2
This form could have also been anticipated from the sinusoidal forms of the phase currents and
no-load voltages. In the equations î = î1 is the amplitude of the phase currents.
The maximum torque is then:
3 êî
T avg,max =
.
(3.56)
2ω
3.4.3 Phase inductance
Similarly to Subsection 3.4.1, linkage of the phase a with the field of the currents can be calculated by:
Zπ/2
(r = rwc , ϕ s ) dϕ s
ψa = 2l s
na (ϕ s ) Acurr
(3.57)
z
−π/2
After taking into account expressions (3.6), (3.31) and (3.35), the last equation can be expressed as:
∞
∞
∞
X
X
X
3 n6k1 +1 n6k2 +1 curr
Az,6k2 +1 (rwc )
î6m+1 · ...
ψa = 2l s
2 lw rwc
m=−∞
k =−∞ k =−∞
1
... ·
Zπ/2
−π/2
2
cos (6k1 + 1) ϕ s cos (6k2 + 1) ϕ s − (6m + 1) ωt dϕ s ,
where:
Acurr
z,6k+1 (rwc ) =
Acurr
z,6k+1,6m+1 (rwc , ϕ s )
J s,6k+1,6m+1 (ϕ s )
.
(3.58)
(3.59)
The integral in the last equation has a non-zero value only for k1 = k2 = k, thus the last
expression can be reduced to:
∞
∞
X
π X
3 n26k+1 curr
(rwc )
A
î6m+1 cos [(6m + 1) ωt],
ψa = 2l s
2 lw rwc z,6k+1
2 m=−∞
k=−∞
(3.60)
3.5. Unbalanced magnetic pull and machine stiffness
39
and, with respect to (3.30), further to:
∞
X
n26k+1 curr
3
(rwc ) · ia .
A
ψa = πl s
2 k=−∞ lw rwc z,6k+1
(3.61)
The total (synchronous) phase inductance is than obtained as:
∞
X
n26k+1 curr
ψa 3
(rwc ).
= πl s
A
L=
ia
2 k=−∞ lw rwc z,6k+1
(3.62)
The expression for inductance (3.62) comprises the self-inductance of a phase, mutual inductance and the air-gap leakage inductance since all the correlated effects have been included
in the developed 2D model of the machine field. On the other hand, the expression does not account for change of the inductance with frequency since the frequency-dependent effects (eddy
currents) have not been modeled.
The exact form of equation (3.62) is rather long and will not be presented here. The expression for inductance resulting from a 1D model, on the other hand, is simple and is given
by:
(kw N)2
6
L1D = µ0 l s r s
.
(3.63)
π
lm /µrec + lag + lw
However, this expression overestimates inductance of the test machine for about 20%.
3.5
Unbalanced magnetic pull and machine stiffness
Asymmetric displacement of the electromagnetic components of an electrical machine brings
about unbalanced magnetic pull (UMP) between the rotor and stator. Reasons for the asymmetry
of PM machines can be put down to two main factors: i) central asymmetry of the armature field
due to inherently asymmetric pole/slot configuration and ii) rotor eccentricity.
The former reason is fairly irrelevant for slotless machines since their armature is rarely
asymmetric; besides, the armature field is rather weak with respect to the field of permanent
magnets.
On the other hand, due to irregularities in the bearings, some eccentricity of the rotor is
always present. As a result, during machine’s operation a revolving attraction force between
rotor and stator exists which, in turn, might cause severe rotor vibrations and bearings wear.
This phenomenon is particularly unwanted in machining and hard-disc-drive spindles whose
applications are exceptionally susceptible to spindle runout [101, 102].
The attraction between stator and rotor counteracts the bearing force and appears as a machine negative stiffness. In the example of soft-mounted high-speed machines this effect must
be considered early in the design phase of the system. Machine stiffness, thus, must be considerably lower than the stiffness of the bearings; furthermore, maximum tolerated unbalanced
force must be below force capacity of the bearings.
For electromagnetic design, in most of cases, a reasonable estimation of the unbalanced pull
is sufficient. Space harmonics of UMP might also be interesting for rotordynamical analysis
40
Chapter 3. Electromagnetic modeling of slotless PM machines
[103], however, this will not be addressed in the thesis since it is of little relevance for the test
machine.
Therefore, this section focuses on finding a sufficiently accurate model for unbalanced magnetic force in slotless PM machines. The estimation of the unbalanced force will be utilized in
Chapter 9 when the requirements for the stiffness and force capacity of the bearings are set.
y
xc
os
φ
s
q
rm
+Δ
l
φs
Os Δx Or
d
rm
φr’
θ
x
rs
Figure 3.12: Simplified model geometry of a slotless machine with an eccentric rotor
The geometry which will be used for modeling in this section is depicted in Figure 3.12.
The stationary reference frame xy has its origin in the center of the stator. Center of the rotor
is shifted from the stator center for the displacement ∆x. The rotating reference frame dq is
connected to the center of the rotor with the d-axis coinciding with the rotor magnetization
vector. Angular position of the rotor is defined by the angle θ which is given by equation (3.9).
For very small displacements it is reasonable to approximate the angular position ϕ0r in the
rotating frame with:
∆x
sin ϕ s ≈ ϕ s − θ.
(3.64)
ϕ0r ≈ ϕ s − θ +
rm
For such an irregular geometry, Maxwell’s stress method is arguably the easiest way to calculate the unbalanced force. The linearity assumption will be made - the effect of iron saturation
can be easily neglected due to machine’s very large effective air gap. In the stationary reference
frame the force per unit length that acts on the rotor can be determined using the following
equations [102, 104]:
I h
i
1
0
Fx = −
B2ε,ϕ − B2ε,r cos ϕ s + 2Bε,r Bε,ϕ sin ϕ s r (ϕ s )dϕ s ,
(3.65)
2µ0 l
3.5. Unbalanced magnetic pull and machine stiffness
Fy0
1
=−
2µ0
I h
l
i
B2ε,ϕ − B2ε,r sin ϕ s − 2Bε,r Bε,ϕ cos ϕ s r (ϕ s )dϕ s ,
41
(3.66)
where l is a closed surface in the air gap and Bε,r and Bε,ϕ are radial and tangential components
of the magnetic flux density in the machine with eccentricity:
−
→
→
−
→
−
Bε = Bε,r · ir + Bε,ϕ · iϕ .
(3.67)
In order to find the unbalanced force, it is essential to adequately represent magnetic field
in the machine with rotor eccentricity. One way to analytically calculate the field in the machine with eccentricity is to directly model the field in the eccentric-rotor machine. This was
done, for instance, in [105], by applying perturbation method when the governing equation and
boundary conditions were formulated. However, the method leads to rather complex derivation
and lengthy expressions despite the truncation of the higher-order components in the solution
of the governing equation.
Another, more common way, is to correlate the air-gap flux density in the machine with and
without rotor eccentricity using different, indirect approaches and useful simplifications. That
correlation is usually given as a function that represents normalized flux density in the eccentricrotor machine with respect to the flux density of the perfectly concentric machine. Since it also
indicates the perturbation of the air-gap permeance as a function of the rotor displacement, the
function is commonly referred to as relative permeance function f (r, ϕ s ). The resulting flux
density yields:
−
→
→
−
→
−
Bε ≈ fr (r, ϕ s ) Br ir + fϕ (r, ϕ s ) Bϕ iϕ ,
(3.68)
where Br and Bϕ denote components of the flux density in the machine without eccentricity.
To find relative permeance function conformal mapping is often used [106–108]. In [108]
magnetic field which exists between two eccentric bodies representing rotor and stator of an
electrical machine is determined. Each of these abstract bodies is assumed to have constant
magnetic potential. Magnetic flux density of the homopolar field between the bodies is found
after transforming the given geometry to a similar, but concentric configuration using conformal
mapping. Finally, the 2D relative permeance function is found in an analytical form as a ratio
between the determined flux density and the flux density in the same abstract system without
eccentricity.
The relative permeance function obtained in [108] was used to perturbate flux density of
an actual, concentric PM machine to obtain the flux density of the same machine with rotor
eccentricity. Similar approach was adopted also in [107] and findings of these studies complied
with the outcome of FE methods.
The function that modulates air-gap flux density according to the rotor displacement can be
also estimated using quite simple approximations. Surprisingly, effectiveness of such an approach has rarely been explored in recent literature, with an exception of [109]. It is a straightforward guess, though, to assume that the radial flux density in the eccentric-rotor machine is
inversely proportional to the effective air gap at a given radius. This assumption, in turn, leads
to the following correlation:
ge f f
Bε,r
=
,
(3.69)
Br
gε,e f f
42
Chapter 3. Electromagnetic modeling of slotless PM machines
where the effective air gap of the machine without and with eccentric rotor are given by:
ge f f =
lm
+ lag ,
µrec
gε,e f f = ge f f − ∆x cos ϕ s .
(3.70)
(3.71)
Using similar approximation in his paper as early as in 1963, S.A. Swann [106] obtained
flux density distribution in an eccentric-rotor machine which lay quite closely to the result of a
rigorous method which included conformal transformation.
For small rotor displacements (∆x ge f f ) equation (3.69) can be further expressed in the
following way:
Bε,r (ϕ s ) = Br (ϕ s )
1
≈ Br (ϕ s ) · (1 + ε cos ϕ s ) ,
∆x
cos ϕ s
1−
ge f f
(3.72)
where ε = ∆x/ge f f represents rotor eccentricity.
Evidently, from (3.72), a very simple perturbation- or relative permeance function is obtained:
f = f (ϕ s ) = 1 + ε cos ϕ s .
(3.73)
The function f obtained using this, very simple approach is plotted in Figure 3.13 against
radial component of the 2D relative permeance function obtained using the method outlined in
[108]. It is noticable that the shape of the simple function from (3.73) is very similar to the shape
of the function obtained using conformal mapping. The difference lies in the magnitude: the
simple function overestimates perturbation of the flux density. Effectiveness of both permeance
functions for representing the deformed flux density and force calculation will be examined in
the rest of this section.
The most appropriate surface for integration of the Maxwell’s stress according to equations
(3.65) and (3.66) is the inner stator surface because at that surface the magnetic flux density in
the air is strictly radial. Therefore, the Maxwell’s stress at that surface reduces to:
σ M (r = r s , ϕ s ) =
B2ε,r (r = r s , ϕ s )
,
2µ0
(3.74)
and, from (3.65) and (3.66), magnetic force on the rotor can be calculated by:
F x0 = r s
Z2π
σ M (r = r s , ϕ s ) cos ϕ s dϕ s ,
(3.75)
Z2π
σ M (r = r s , ϕ s ) sin ϕ s dϕ s .
(3.76)
0
Fy0
= rs
0
3.5. Unbalanced magnetic pull and machine stiffness
43
conformal mapping
approximation
1.06
1.04
1.02
f (ϕ )
s
1
0.98
0.96
0.94
0
pi/2
pi
ϕ
3pi/2
2pi
s
Figure 3.13: Relative permeance function f obtained using the conformal mapping method
from [108] and using approximation - equation (3.73) - for 10% eccentricity
Radial flux density in the air gap at the inner stator surface of the concentric machine can
be expressed as (eq. (3.25)):
!
D4
Br (r = r s , ϕ s ) = B4,r (r = r s , ϕ s ) = C4 + 2 cos (ϕ s − θ)
(3.77)
rs
Radial flux density at the inner stator surface of the eccentric-rotor machine will be separately obtained using the permeance functions from Figure 3.13. Using the simple form (3.73),
the flux density in the eccentric rotor machine can be estimated as:
!
D4
(3.78)
Bε,r = C4 + 2 (1 + ε cos ϕ s ) cos (ϕ s − θ) ,
rs
or using the permeance function fcm obtained from conformal mapping [108]:
!
D4
Bε,r,cm = C4 + 2 fcm cos (ϕ s − θ) .
rs
(3.79)
Adequate representation of the flux density along the stator inner surface is, thus, necessary
for accurate estimation of the force. Expressions (3.78) and (3.79) are calculated for the example
of the test machine when the rotor magnetization coincides with the x-axis (θ = 0) and the
results are compared with results from 2D FEM in Figure 3.14.
Both permeance functions are good in representing the deformed radial flux density and the
estimations are quite comparable. The distribution obtained using conformal-mapping function
44
Chapter 3. Electromagnetic modeling of slotless PM machines
0.25
0.25
2D FEM
approximation
conformal mapping
0.2
0.2
0.15
0.15
10% eccentricity
s
0.05
r
s
s
0.05
r
30% eccentricity
0.1
s
B (r=r ,ϕ ) [ T ]
0.1
B (r=r ,ϕ ) [ T ]
2D FEM
approximation
conformal mapping
0
0
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
0
pi/2
pi
−0.2
0
pi/2
ϕ [rad]
s
pi
ϕ [rad]
s
Figure 3.14: Radial flux density at the inner stator surface of the test machine calculated by
(3.78), (3.79) and 2D FEM, for 10% and 30% eccentricity
400
simple f
f from conformal mapping
2D FEM
350
250
200
x
F ’ [N/m]
300
150
100
50
0
0
0.05
0.1
0.15
0.2
0.25
ε = ∆x / g eff
Figure 3.15: Unbalanced force on the rotor of the test machine calculated using i) equation
(3.81), ii) conformal mapping and iii) 2D FEM
3.6. Losses in the machine
45
is slightly more accurate than the distribution when the simple correlation is used and that
difference is more evident when the eccentricity is increased.
The force per unit length that acts on the rotor in the x direction can be calculated from
(3.75). Using the flux density expression (3.78) and the expression for the Maxwell’s stress
(3.74), it can be shown, after integration, that the force is closely equal to:
!2
#
"
rs
D4
1
π
0
(3.80)
C4 + 2
1 + cos (2θ) ∆x.
Fx ≈
2µ0
r s ge f f
2
As anticipated, it can be shown that the force in the y direction is equal to zero.
For this study it is important to determine the maximum unbalanced force for a given rotor
displacement. From the last equation it is evident that the maximum force occurs when θ = kπ
i.e. when the rotor magnetization is parallel to the x axis and it yields:
!2
3π
D4
rs
0
0
F x,max = F x (θ = kπ) ≈
C4 + 2
∆x.
(3.81)
4µ0
r s ge f f
From (3.81) machine stiffness per unit length can be expressed as:
!2
0
∆F
3π
rs
D4
x,max
0
≈−
C4 + 2
kPM = −
∆x ∆x=0
4µ0
r s ge f f
(3.82)
and it is negative because the magnetic force tends to remove the rotor from the center.
Finally, the force can be obtained in the same way just using the flux density with the relative
permeance function obtained using the conformal-mapping method from [108], eq. (3.79).
This approach is fully two-dimensional since it utilizes 2D models of both relative permeance
function and the magnetic field of a non-eccentric machine.
Results of both analytical models are compared with results from 2D FEM modeling for the
test machine example. Both methods seem to overestimate the total force on the rotor, however,
the force estimation obtained using conformal mapping is considerably more accurate. On the
other hand, estimating the field using the simple permeance function is much easier. It also
results in a short analytical form which offer much more insight into the unbalanced force
than the complex analytical form of the function obtained from conformal mapping. From
the machine design point of view 50÷60 percent of overestimation resulting from the simple
method is quite tolerable and can be accounted as a safety margin in the system design.
3.6
Losses in the machine
3.6.1 Stator core losses
Modeling of induced losses in the stator feromagnetic core is a rather difficult task for a number
of reasons. Losses in the core stem from a combination of linear (such as eddy-currents) and
non-linear phenomena (such as hysteresis) whose combined effect cannot be accurately predicted by superposition. Besides, hysteresis, as an inherently non-linear phenomenon, does not
46
Chapter 3. Electromagnetic modeling of slotless PM machines
easily lend itself to an analytical modeling. Finally, classical representation of core losses as a
combination of hysteresis and eddy-current losses is not sufficient to account for all the losses at
high frequencies. In literature, additional, rather obscured loss named anomalous, stray or excess loss is often introduced to account for the loss increase without much attempting to explain
its physical nature [33, 97].
At the same time, it is extremely difficult to verify models for a particular loss mechanism
since it is practically impossible to separate different loss phenomena. In order to get figures on
losses that can be utilized in modeling, designers usually rely on empirical correlations rather
than on classical EM modeling. Empirical formula that is regularly used to calculate the core
losses in an iron core is the Steinmetz equation which will be represented here in the following
form:
pFe = C B̂αm f β ,
(3.83)
where p is a mass (or volume) density of the total loss power, B̂m and f are flux density amplitude and frequency of the magnetic field in the core, respectively, and C, α and β are empirically
obtained parameters.
However, when the core is subjected to a field whose frequency varies throughout a broad
range of values, it is difficult to compose a single equation which can represent the losses accurately enough for all possible frequencies and flux densities. Furthermore, the whole loss modeling depends on (availability of) the manufacturer’s data which are often incomplete and/or
inadequate for the desired operating mode.
In this section both classical and empirical approaches will be taken in calculating the core
losses and shown for the test machine case. The purpose of the section is to find an adequate
analytical expression for power of the losses that can be used for the machine optimization.
→
−
Hysteresis loss represents the energy that is dissipated in the material while flux density B
→
−
and field intensity H oscillate in cycles over the hysteresis loop. Volume density of the energy
dissipated in each cycle is proportional to the area encircled by the hysteresis loop in the BH
characteristic of the material. Therefore, the power density of the loss is equal to:
pFe,h = ABH f,
(3.84)
where ABH is the area of the hysteresis loop in J/m3 .
Dependence of the hysteresis-loop area on the flux density is rather complex and is property
of the material; it may be expressed, however, using the approximate correlation:
ABH ≈ Ch Bαmh ,
(3.85)
where the parameters Ch and αh can be adjusted according to magnetic properties of the material.
Therefore, the equation for power density of hysteresis loss becomes:
pFe,h ≈ Ch B̂αmh f.
(3.86)
For the silicon-steel core used in the test machine the parameters of the hysteresis loss were
obtained from a plot of hysteresis loops given by the manufacturer - Appendix C. The hysteresis
3.6. Losses in the machine
47
loss density is roughly estimated as:
3
pFe,h ≈ 70 B̂1.8
m f [W/m ]
(3.87)
where Bm and f are expressed in SI units.
Eddy current loss in a laminated core can be estimated from a simplified model of eddy
currents in thin laminations induced by a one-dimensional field. The equation for the eddy
current loss density is given by [33, 98]:
pFe,e
π2 f 2 d2 B̂2m
,
=
6ρFe
(3.88)
where d is the lamination thickness and ρFe is resistivity of the core material.
Total power of losses can be calculated by integrating (3.86) and (3.88) over the stator core
volume. For the modeled machine (Figure 3.1), however, it is appropriate to express the flux
density amplitude Bm as an equivalent flux density in the stator yoke By [98]:
B̂m = B̂y =
Φmax
,
(r so − r s ) l s
(3.89)
where Φmax is the maximum flux in the stator yoke and l s is the stator axial length.
Maximum flux can be calculated as:
Φmax = −l s
Zrso
B5,ϕ (r, ϕ = 0) dr
(3.90)
rs
and, using (3.25), it can be shown that:
√
D5
B̂y = B5,ϕ r = r s r so = −C5 +
.
r s r so
(3.91)
After substituting (3.89) into (3.86) and (3.88) the total loss in the core calculated in the
classical way can be simply estimated as:
PFe = PFe,h + PFe,e = pFe,h + pFe,e V s ,
(3.92)
where V s is the stator core volume:
V s = r2so − r2s πl s .
(3.93)
Another approach to model the losses is to adjust the parameters of the Steinmetz equation
(3.83) so that its results fit the manufacturer’s data. Loss curves for the used core material of
the test machine are obtained from the manufacturer - Appendix C.
The parameters of the equation were first fitted using all the available data from the material
loss curves for the frequency range [50÷10k] Hz and flux density range [0.1÷1] T:
PFe ≈ 7.0 · 10−4 B̂1.75 f 1.50 · V s [W].
(3.94)
48
Chapter 3. Electromagnetic modeling of slotless PM machines
3
P h (classical)
P (classical)
e
2.5
P h+P e (classical)
Ploss [W]
manufacturer’s data (overall estimate)
manufacturer’s data (2−5 kHz estimate)
2
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
frequency [Hz ]
Figure 3.16: Power of the core losses in the stator of the test machine, B̂y = 0.43 T , resulting
from the different models
Since, for a high-speed machine, it is most important to adequately represent losses in the
high-frequency operating region, another estimate was made using loss curves for 2 and 5 kHz
only (see Figure C.2). Around those frequencies the loss approximation yields:
PFe,h f ≈ 4.7 · 10−4 B̂1.86 f 1.53 · V s [W]
(3.95)
Results of the models presented in this section are plotted in Figure 3.16 for the stator core
of the test machine. The figure shows estimation of the loss power vs. frequency for the actual
machine whose equivalent flux density in yoke was calculated by (3.91).
The discrepancy between loss approximation based on classical modeling and manufacturer’s data is huge; classical modeling evidently fails to model core losses for the high-frequency
operation of the machine. Not only that the non-modeled (excess) losses become significant at
high frequencies, but also manufacturing processes which are used to assemble machine cores,
such as cutting and punching, have a great influence on the final core properties [110].
Additionally, the plot shows considerable discrepancy between results of the Steinmetz
equation fitted to all the available loss curves and the equation fitted to the specific frequency
region. The Steinmetz equation, in its original form, can hardly account for iron losses at a
great range of frequencies and magnetic fields. Therefore, for the machine design it is the most
effective if the losses are represented only in the frequency range of the most interest. For
the example of the test machine equation (3.95) will be used for the machine optimization in
Chapter 7.
3.6. Losses in the machine
49
3.6.2 Copper losses
Pulsating magnetic field of both rotor magnets and conductors current of a high-speed machine
will influence the distribution of the current within the conductors and will cause formation of
eddy currents in the copper. The higher the frequency of the magnetic field is the stronger its
influence on losses in the conductors will be. As a result, at very high electric frequencies the
total losses in conductors can significantly differ from the standard I 2 R conduction loss. In this
section, losses in the copper of a slotless PM machine will be assessed with particular focus on
their frequency-dependent part.
Skin- and proximity effect are two main sources of frequency-dependent copper losses in
power transformers and they were studied in detail in literature. In works of J.A. Ferreira
[89, 111, 112] it was shown that skin- and proximity effect can be independently treated, due to
their orthogonality. Additionally, useful expressions for the correlated losses were derived and
verified and those will be the starting point for the study in this section.
Copper losses of a slotless PM machine will be divided here into three parts: (i) the conduction loss part PCu,skin , which also includes a rise in loss caused by the reduction of effective
conductors cross-section due to skin-effect; (ii) the proximity loss part PCu,prox , which accounts
for eddy-current loss in the conductors due to the pulsating magnetic field of the neighboring
conductors; and (iii) PCu,eddy , which accounts for the eddy-current loss due to the pulsating
magnetic field of the rotor magnet:
PCu = PCu,skin + PCu,prox + PCu,eddy .
(3.96)
The total conduction loss can be expressed in the following way [89, 112]:
PCu,skin = F (φ) · PCu,DC = (F (φ) − 1) I 2 RDC + I 2 RDC .
|
{z
} |{z}
DC
skin - effect
(3.97)
In equation (3.97) the increase of conduction losses as the result of skin effect is distinguished from the regular DC conduction loss. Function F (φ) was derived in [89, 112] as:
F (φ) =
φ (ber(φ)bei0 (φ) − bei(φ)ber0 (φ))
·
,
2
ber0 2 (φ) + bei0 2 (φ)
(3.98)
where parameter φ is proportional to the ratio between conductor diameter and the skin-depth:
r
πσCu µ0 f
dCu
φ=
.
(3.99)
√ = dCu
2
δ skin 2
The second and the third term in equation (3.96) take on the following forms for the example
of a slotless machine [89, 112]:
Z
G (φ)
PCu,prox =
· Ĥc2 dl,
(3.100)
σ
Cu
lCu
Z
G (φ)
· Ĥm2 dl,
(3.101)
PCu,eddy =
lCu,ag σCu
50
Chapter 3. Electromagnetic modeling of slotless PM machines
where Ĥc and Ĥm are amplitudes of magnetic field intensity in the conductor due to neighboring
conductors and rotor permanent magnet, respectively; lCu is the total conductor length and lCu,ag
is the total conductor length in the air gap.
Function G (φ) is also derived in [89, 112] as:
G (φ) = 2πφ ·
(ber2 (φ)ber0 (φ) − bei2 (φ)bei0 (φ))
.
ber2 (φ) + bei2 (φ)
(3.102)
In a slotless PM machine the field of the conductors is very small compared to the field of
the permanent magnet, therefore, proximity-effect loss can be neglected. Eddy-current loss in
conductors, when end effects are neglected, can be simply calculated as:
PCu,eddy =
G (φ)
· Ĥm2 lCu,ag .
σCu
(3.103)
In order to assess the influence of frequency-dependent losses on the total copper loss in the
machine, it is necessary to analyze the actual values of the functions F (φ) and G (φ).
It is not reasonable to expect conductor diameters of high-speed machines larger than 1 mm
and (fundamental) electrical frequencies higher than 10 kHz. For those two particular values,
at room temperature, parameter φ can be calculated from (3.99) to be 1.05. Therefore, it is
sufficient to analyze the functions F and G for parameter φ in the range [0, 1].
−3
6
x 10
5
F(φ)−1
4
3
2
1
0
0
0.2
0.4
φ
0.6
0.8
1
Figure 3.17: Plot of the function (F (φ) − 1)
Function (F (φ) − 1) is plotted in Figure 3.17 for the given range of φ. It is evident from
the values of the plotted function that the skin-effect part of equation (3.97) certainly comprises
less than one percent of the total value of PCu,skin and can thus be neglected. Hence:
PCu,skin ≈ PCu,DC = I 2 RDC .
(3.104)
3.6. Losses in the machine
51
0.4
G(φ)
0.35
4
π/8⋅φ
0.3
G(φ)
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
φ
0.6
0.8
1
Figure 3.18: Plot of the function G (φ) and of its approximation
In Figure 3.18 function G (φ) is presented. Although it has a complex analytical expression
(eq. (3.102)), the function can be very well approximated with the following function:
G (φ) ≈
2
4
µ20 f 2 dCu
π 4 π3 σCu
φ =
,
8
4
(3.105)
which was also shown in Fig. 3.18.
After inserting expression (3.105) into (3.103) and knowing that B̂m = µ0 Ĥm , one can obtain
the approximate expression for the eddy-current loss in the air-gap conductors:
PCu,eddy =
2
σCu
B̂2m (2π f )2 dCu
VCu,ag ,
32
(3.106)
which is sometimes used in literature on machines (e.g. [90]). In (3.106) VCu,ag is the total
volume of conductors in the air gap:
VCu,ag = n
2
dCu
πlCu,ag ,
4
(3.107)
where n is a number of parallel conductors in a phase.
According to (3.25) amplitude of flux density in the air-gap windings can be calculated as:
B̂m ≈ B4,r (r = rwc ) = C4 +
D4
.
2
rwc
(3.108)
Finally, the total copper loss in a slotless PM machine can be calculated as:
PCu ≈ I 2 RDC + PCu,eddy ,
(3.109)
52
Chapter 3. Electromagnetic modeling of slotless PM machines
where the total DC resistance is given by:
RDC =
4ρCu
lCu .
2
ndCu
π
(3.110)
Although DC resistance does not depend on frequency it is strongly dependent on temperature since copper resistivity changes significantly with temperature.
For a toroidally wound machine it holds:
lCu = [2l s + (r so − r s ) π] 6N,
(3.111)
lCu,ag = l s 6N,
(3.112)
where 2N is the number of turns per phase.
The optimization of stator conductors is presented in Subsection 7.4.3 of Chapter 7.
3.6.3 Air-friction loss
Both prediction and measurement of shear stress, loss and temperature increase in rotor as
result of air friction on the rotor surface are important, but very difficult and thankless tasks.
The importance of estimation of air-friction loss in a high-speed machine lies in the strong
dependence of the loss on rotational frequency. Although virtually immaterial at low speeds,
air-friction drag has the highest rate of increase with rotational speed in comparison with other
loss factors and it will inevitably take a predominant portion of overall losses as the machine
speed rises.
Therefore, in design of very-high-speed machines good assessment of air-friction loss is
essential. Not only does air friction greatly influence the drag torque at high speeds, it also
affects temperature in the rotor. Excessive rise of the rotor temperature is unacceptable because
the rotor is, usually, barely cooled.
Flow of air (fluid) in the machine gap and interaction between the rotor and the air is very
complex for modeling and those phenomena remain elusive for majority of machine designers.
Although laminar air flow can be analytically modeled, that type of flow is dominant only at
very low speeds at which air friction is fairly irrelevant. Air friction becomes important when
turbulences and vortices act in the air gap and modeling of those requires, at least, comprehensive knowledge of fluid dynamics. For practical needs of engineers, however, empirical
correlations are satisfactory and they are greatly utilized for estimating air-friction loss and
thermal convection in the air gap. These correlations rely on empirically-obtained expressions
for coefficients such as friction coefficients or convection coefficient.
It is the empirical coefficients, however, and their usage in academic papers that makes modeling of air friction rather dubious. Either there is lack of recent studies on empirical coefficients
for friction-correlated parameters or today’s researchers of electrical machines are not aware of
them. Coefficients, which are obtained several decades ago and whose forms are too unfamiliar to have any analytical value, are repeatedly used in literature on machines, copied from one
paper to another, disregarding the experimental context in which the coefficients are actually obtained. When authors, after introducing several empirical coefficients, without explanation and
3.6. Losses in the machine
53
often missing to refer to their original source, obtain results which perfectly match analytical
predictions, the results of such studies are, at most, suspicious.
Based on a few studies that give some reasonable practical evaluation of the used empirical
coefficients for high-speed rotors, this section will analyze power of air-friction loss in the
test machine. Since the author was not able to seriously examine validity of different models,
this section will only offer a rough prediction of the total air-friction loss rather than rigorous
modeling.
The power that is required from machine to overcome air-friction drag at the cylindrical
surface of the rotor is given by (e.g. [91]):
Pa f = kr C f ρπω3 r4 l,
(3.113)
where ρ is the air mass density, ω = 2π f is rotor angular frequency, r and l are external radius
and axial length of the cylinder, respectively. The coefficient C f is called friction coefficient and
is obtained empirically3 . Parameter k f represents roughness coefficient and is equal to 1.0 for
perfectly smooth surfaces.
For air-friction loss at faces of a rotating disc the following equation is used:
1
Pa f = kr C f ρω3 r5 − ra5 ,
2
(3.114)
where ra is the inner radius of the disc.
Expressions for friction coefficients are obtained from experimental work. Although they
take on analytical forms in which geometrical, kinetic and material parameters of the studied
bodies and the gap fluids are correlated, those expressions offer very little, if any, insight into
the air friction phenomenon. Besides, different experimental studies give different expressions
for the coefficients. It is, thus, of crucial importance for trustful modeling that the used friction
coefficient is applied to the same or similar conditions under which the coefficient was originally
obtained.
An abundance of different friction coefficients can be found in the engineering literature;
however, the amount of information on origin of those coefficients is sparse. In recent years,
thesis by J. Saari [91] has become an indispensable, if not only, resource for air-friction modeling for machine designers. Saari gave some qualitative analysis of several experimental studies
on friction coefficients including the conditions under which the coefficients were obtained.
For friction at cylindrical surfaces Saari presented, among others, friction coefficients reported by Bilgen and Boulos [113]:
δ 0.3
C f = 0.515 r 0.5 ,
Reδ
δ 0.3
C f = 0.0325 r 0.2 ,
Reδ
3
500 < Reδ < 104
(3.115)
104 < Reδ
Equation (3.113) is actually derived from the definition of the friction coefficient. The coefficient is defined as
the ratio between shear stress and dynamic pressure at the cylinder surface.
54
Chapter 3. Electromagnetic modeling of slotless PM machines
and Yamada [114]:
Cf =
0.0152
,
Re0.24
δ
800 < Reδ < 6 · 104 .
(3.116)
In (3.115) and (3.116) Reδ is the Reynolds number for cylinders in enclosure (stator):
Reδ =
ρωrδ
,
µ
(3.117)
where δ is the air-gap length in the radial direction and µ is dynamic viscosity of air. The
Reynolds number is generally used to determined the nature of the particular gas flow.
The aforementioned coefficients were originally obtained for liquid fluids (water and oil);
however, with use of coefficients from [113], Saari’s predicted correlation between the loss
power at the cylindrical rotor surfaces and frequency had a very similar shape as that obtain from
experiments. Relatively small discrepancy between the results was put down to inappropriate
roughness coefficient.
In his articles, Aglen [76, 115, 116] used coefficients from Yamada [114] to predict the air
friction loss. He conducted extensive measurements on a 70.000 rpm PM generator, including direct measurements of friction losses and calorimetric tests, and the results matched the
predictions pretty well.
In their paper on a very high-speed PM motor authors of [117] used the following friction
coefficients, taken from the paper of Awad and Martin [118]:
C f = 0.0095 · T a−0.2 ,
where T a is the Taylor number:
(3.118)
δ
.
(3.119)
r
Zwyssig et. al [119] used expression very similar to (3.118) for friction coefficients for
turbulent flow (T a > 400) and verified the air-friction loss estimation, however, up to relatively
moderate speeds.
For the example of an enclosed rotating disc, expressions for the friction coefficients obtained by Daily and Nece [120] are regularly used:
T a = Reδ
2π
Cf = s
, Regime I
Rer
r 0.1
s
3.7
r , Regime II
Cf =
Re0.5
r
0.08
, Regime III
C f = 0.167
s
0.25
Rer
r
s 0.1
0.0102
r , Regime IV
Cf =
0.2
Rer
(3.120)
3.6. Losses in the machine
55
where s is the distance between the disc surfaces and the enclosure and Rer is the tip Reynolds
number:
ρωr2
.
(3.121)
Rer =
µ
s/r
0.1
II
IV
0.05
I
III
4
10
10
8
6
10
Rer
Figure 3.19: Fluid flow regimes according to Daily and Nece [120]
The different regimes of the gas flow appearing in equations (3.120) are shown in Figure
3.19. Saari also obtained good approximations for the loss using the coefficients (3.120) for end
rings and discs of his test rotor.
ra
δa
la
bearings + housing
s
rotor
stator
ld
s
rd
δd
Figure 3.20: Simplified sketch of a half of the test-setup cross-section
The friction coefficients represented in this section will be applied to equations (3.113) and
(3.114) for example of the test machine in order to compare the different models and check their
consistency. A simplified geometry of the test rotor and its setting is shown in Figure 3.20 and
the parameters are given in Table 3.3.
56
Chapter 3. Electromagnetic modeling of slotless PM machines
Table 3.3: Parameters of the test rotor geometry (Figure 3.20)
parameter
symbol value
shaft radius
ra
4 mm
shaft clearance
δa
14 µmm
bearing length
la
5 mm
rotor disc radius
rd
16.5 mm
machine air gap
δd
≈1 mm
rotor disc length
ld
8 mm
disc-housing clearance
s
mm
Since the diameter of the rotor disc is much larger than the diameter of the shaft, all models
suggest that the air-friction loss at the shaft is negligible in comparison with the losses at the
surfaces of the disc. Therefore, only losses at those surfaces will be presented here. The loss
at the external cylindrical surface of the disc is separately calculated using equations (3.115),
(3.116) and (3.118), while the loss at the disc faces is calculated using (3.120). Total, combined
losses resulting from these models are shown in Figure 3.21 with respect to the rotor frequency.
90
total loss: Yamada / Daily and Nece
total loss: Bilgen and Boulos / Daily and Nece
total loss: Awad and Martin / Daily and Nece
cylindrical surface only: Yamada
cylindrical surface only: Bilgen and Boulos
cylindrical surface only: Awad and Martin
80
P loss [W]
70
60
50
40
30
20
10
0
0
50000
100000
150000
200000
rpm
Figure 3.21: Air-friction loss of the test machine vs. rotational speed; predictions formed using
friction coefficients (3.115), (3.116), (3.118) and (3.120)
From the loss plot from Figure 3.21 it is evident that the air friction will certainly be the main
source of drag and loss in the test machine. Different loss models show significant difference
in prediction at very high speeds, but the rate of the loss increase with frequency is consistent.
Therefore, it is necessary to take this loss into account when the power and torque of the test
machine are sized.
3.6. Losses in the machine
57
On the other hand, it is hardly possible that the air flows at the side and faces of the disc
can be independently treated and their resulting effects simply summed. Furthermore, the test
rotor is suspended in air bearings: compressed air from the bearings contributes significantly
to the total air flow, modeling of which goes far beyond intentions of this thesis. However,
the analytical expressions represented in this section can still be used for rough estimation of
air-friction loss that is so important for the machine design.
3.6.4 Rotor loss
In Subsection 3.3.2 magnetic vector potential of the armature field in a slotless PM machine
was expressed as:
∞
∞
X
X
→
−
→
−
Ai,6k+1,6m+1 (r, ϕ s ) · iz ,
(3.122)
Ai =
k=−∞ m=−∞
where algebraic values of the vector-potential harmonics were defined by equations (3.31) and
(3.35).
Following the derivation from Subsection 3.3.2, harmonic components of the vector potential can also be formulated in the following way:
Ai,6k+1,6m+1 (r, ϕ s ) = Ai,6k+1,6m+1 (r) cos (6k + 1) ϕ s − (6m + 1) ωt , i = 1..5,
(3.123)
where Ai,6k+1,6m+1 (r) depends on the radial coordinate only.
If the angular coordinate ϕ s in the stationary reference frame is expressed via the angular
position in the rotor reference frame (equations (3.8) and (3.9)):
ϕ s = ϕr + ωt + θ0
(3.124)
and when this is applied to (3.123), harmonic components of the vector potential in the rotor
reference frame are obtained:
Ai,6k+1,6m+1 (r, ϕr ) = Ai,6k+1,6m+1 (r) cos (6k + 1) (ϕr + θ0 ) + 6 (k − m) ωt , i = 1..5. (3.125)
Total z component of the magnetic vector potential in the rotor reference frame is then given
by:
Ai,z (r, ϕr )
∞
∞
X
X
Ai,6k+1,6m+1 (r, ϕr ), i = 1..5.
(3.126)
k=−∞ m=−∞
Vector components of corresponding magnetic flux density in the rotating frame are obtained using (3.10):
Ai,6k+1,6m+1 (r)
sin (6k + 1) (ϕr + θ0 ) + 6 (k − m) ωt ,
Bi,6k+1,6m+1,r = − (6k + 1)
r
∂Ai,6k+1,6m+1 (r)
Bi,6k+1,6m+1,ϕ = −
cos (6k + 1) (ϕr + θ0 ) + 6 (k − m) ωt , i = 1..5.
∂r
(3.127)
From the last equations it is evident that the rotor also observes traveling waves of the
flux density as a result of combination of armature spatial- and time harmonics with unequal
58
Chapter 3. Electromagnetic modeling of slotless PM machines
harmonic numbers: k , m. Therefore, the rotor conductive regions (in particular, permanent
magnet and rotor iron, i = 1, 2) are prone to induction of eddy-current losses due to the existence
of the armature traveling harmonics.
Based on the magnetostatic modeling of magnetic field presented in this section, it is not
possible to rigorously model rotor eddy currents; hence, the corresponding loss can only be
estimated using approximate expressions. In general, rotor induced loss does not comprise a
great portion of the overall losses, however, it may still be a cause of rotor failure since rotors
are seldom cooled.
In his thesis H. Polinder [97] showed different approaches for calculating rotor losses from
a magnetostatic field model. For the machine defined in Section 3.2 it would be of particular
interest to assess losses in the magnet since it is particularly exposed to the armature harmonics
and rather sensitive to heat at the same time.
For a cylindrical rotor magnets the eddy-current loss can be found from the Faraday’s law:
→
−
∂B
→
−
,
∇× E =−
∂t
(3.128)
∂ →
−
→
−
ρm ∇ × J = −
∇× A ,
∂t
(3.129)
1 ∂Az (r, ϕr )
,
ρm
∂t
(3.130)
or, alternatively:
→
−
→
−
→
−
where E = ρm J is electric field, ρm is resistivity of the magnet and J is current density vector
of induced eddy currents.
In analogy with the vector potential, it can be assumed the current density vector only has the
z component. Therefore, eddy-current density in the z direction can be calculated by (see [97]):
Jz (r, ϕr ) = −
having, naturally, magnetic vector potential and current density expressed in the rotor coordinates.
Finally, the loss in the magnet can be calculated by integrating power density of the induced
currents pm,eddy = ρm Jz2 throughout the magnet volume:
Pm,eddy =
$
Vm
ρm Jz2
ls
(r, ϕr ) dVm =
ρm
Zrm Z2π
rfe
0
∂Az (r, ϕr )
∂t
!2
dϕr dr.
(3.131)
However, the magnet of the test machine does not lend itself to such calculation of the
magnet eddy-current loss. Namely, an injection-molded plastic-bonded magnet is used in the
test rotor in which small magnet particles are blended with a plastic binder. Although magnet
has a cylindrical form, its conductive component (NdFeB magnet) is powdered and spread
around the whole magnet volume so it does not form a homogeneous cylinder as assumed in
derivation of the last equation.
3.7. Conclusions
59
For such a magnet structure Polinder’s approach for calculating loss in segmented magnet
seems more appropriate [97, 99]. For such a case volume density of the magnet eddy-current
loss is approximately given by:
!
b2m dBr
,
(3.132)
pm,eddy,seg ≈
12ρm dt
where bm is the width of the magnet segment and Br is the radial flux density.
Unfortunately, details of the physical structure of the test-rotor magnet, in particular, size
and distribution of permanent magnet particles within the magnet composite, are unknown to
the author and no approximation of the magnet eddy-current loss will be made. Nevertheless,
since the actual size of the magnet particles (bm ) is very small and the armature field in the
magnet is rather weak it is expected that the induced loss in the magnet is negligible.
3.7
Conclusions
The chapter presents analytical electromagnetic modeling of a (high-speed) slotless PM machine. The goal of the chapter is to distinguish dominant EM phenomena in the machine and to
provide adequate representation of those phenomena that can serve as a good basis for designing
high-speed PM machines and assessing their limits.
Governing differential equations that represent field in machines are expressed over magnetic vector potential in a cylindrical 2D system. Geometry and properties of the model are
adapted to the test machine which is a two-pole toroidally-wound slotless PM motor; the modeling can also account for other types of slotless machines. The model is magnetostatic because
of the small influence of the reaction field of the rotor eddy-currents on the total field in the
machine. Results of the analytical field models are confirmed by 2D FEM which maintains,
however, the same model geometry and assumptions as the analytical model. Based on the
field equations, machine parameters - no-load voltage, phase inductance, torque and power are derived and compared to results of 1D models.
The chapter presents an original study on calculation of unbalanced magnetic pull (force)
and stiffness in a PM machine with rotor eccentricity. An approximate analytical expression
for distorted magnetic field in an eccentric-rotor machine is used to determine the unbalanced
magnetic force and stiffness of slotless PM machines. The effectiveness of such a model in
representing the field in the air gap and unbalanced force is compared to the results of a model
based on conformal mapping [108] (the method frequently reported to provide accurate results)
and 2D FEM. It is shown that the simplified model gives predictions of magnetic flux density in
the air gap similar to the predictions of the conformal-mapping method and 2D FEM. Furthermore, the prediction of the force is, despite noticeable overestimation, useful and effective for
machine-design purposes.
Modeling of machine losses with particular attention to their frequency dependence is given
in the chapter. It is again demonstrated in the example of the test-machine laminations that
classical representation of iron losses as a combination of eddy-current and hysteresis losses
fails to account for the losses in actual laminated iron cores. Therefore, the modeling of iron
losses relies on the manufacturer’s data.
60
Chapter 3. Electromagnetic modeling of slotless PM machines
Using analytical models derived by Ferreira [89, 112], the dominant causes of copper losses
in slotless PM machines are distinguished. It is shown that classical I2 R loss and eddy-current
losses in the air-gap conductors have a dominant influence on the overall copper loss in a slotless
machine while the skin- and proximity-effect influence can be neglected. Additionally, the
section on copper losses adapts a Ferreira’s equation to calculate eddy-current losses in the
air-gap conductors of a slotless machine and correlates that expression to a simplified formula
reported in literature (e.g. [90]). This study is one of the thesis’ contributions.
Three different models of power of air-friction loss at the rotor cylindrical surface are combined with another model which represents the friction loss of a rotating disc. Since the author
was not able to seriously examine validity of different models, these estimations represent a
rough prediction of the total air-friction loss rather than a rigorous model.
An injection-molded plastic-bonded magnet is used in the test rotor; analytical models of
eddy-current losses in rotor magnets can hardly be applied to such a magnet material. Nevertheless, given the material structure of plastic-bonded magnets (small magnetic particles in a
plastic binder), it is expected that induced losses in the magnet are negligible.
Chapter 4
Structural aspects of PM rotors
4.1
Introduction
Structural design of a rotor of a high-speed electrical machine represents a challenging task.
At high rotational frequencies centrifugal forces and, accordingly, stress in the rotor material
become very high. Temperature increase in the rotor due to induced eddy currents and friction
will produce additional, thermal stress between the rotor parts with different thermal properties.
High-speed rotors must be capable of withstanding those stresses and also transfer of electromagnetic torque must be ensured over the whole rotor.
In a PM rotor, the magnet represents the most mechanically vulnerable part. While the
compressive strength of permanent magnets is good, their flexural and tensile strengths are very
low [121]. Magnets cannot sustain tension caused by centrifugal forces during high-speed rotation. Besides, magnets are usually very brittle and cannot be pressed onto the shaft. Therefore,
a permanent magnet in a high-speed rotor must be contained in a non-magnetic enclosure or
sleeve which would limit tension in the magnet and guarantee the transfer of torque from the
magnet to the shaft at elevated speeds.
As a rule, the magnet in a high-speed rotor is either in the form of a full cylinder or consists
of separated blocks that are glued on the steel shaft. Such a rotor structure is generally preferred
for high-speed applications to a rotor with interior (buried) PMs since the latter is prone to much
higher stress concentrations at high speeds [75]. The retaining sleeve is, most often, pressed on
the magnet, although a more subtle technique is applying the sleeve on a cold-shrunken rotor.
Little has been published on mechanical design of high-speed PM rotors and optimization of
the rotor structure. Still, wide literature on structural mechanics, including textbooks, provides
sufficient information to analyze the subject. The thesis by Larsonneur [80] gives a general
stress calculation for axisymmetrical rotors. He distinguished two main structural limitations
for rotational speed: reaching yielding stress in one of the rotor parts or loss of contact between adjacent rotor parts. Larsonneur also observed the existence of an optimal interference fit
between two rotating press- or shrink-fitted rings for maximum permissible speed using numerical solution for the stress in the rings. Binder et al. [75] showed advantages of using surfacemounted magnets for high speed and also the validity of analytical mechanical modeling for
the case of magnets without inter-pole gaps. In choosing materials for high-speed generator,
61
62
Chapter 4. Structural aspects of PM rotors
Zwyssig et al. [43] considered not only strength of rotor materials, but also compatibility of
their thermal properties.
Aim of this chapter is to model the influence of rotational speed and mechanical fittings
on stress in a high-speed rotor, while also considering the operating temperature. Through
analytical modeling, structural limits for the rotor speed are determined and quantified. At the
same time, a relatively simple approach of optimizing the rotor structure is achieved. This
optimization approach will be revisited in Chapter 7 where design of a carbon fiber sleeve for
the rotor of the test machine is presented.
Basic correlations between the structural quantities will not be elaborated in this chapter and
the reader is referred to numerous textbooks on structural mechanics and elasticity.
The modeling of the stress first considers a single rotating cylinder and then a compound of
cylinders, which is regarded as a good representative of high-speed PM rotors. The analytical
models are then tested against finite element modeling for the example of the test-machine rotor.
Finally, structural limits and the approach for the retaining sleeve design are presented.
4.2
Stress in a rotating cylinder
A model of a rotating hollow cylinder, whose cross-section is presented in Fig. 4.1, will be
used in this section so as to form a basis for the structural analysis of PM rotors in the ensuing
sections. The cylinder rotates with a rotational speed Ω and is subjected to an internal static
pressure pi and an external static pressure po . Radial distribution of temperature increment
τ = τ(r) is known:
τ = T − T0,
(4.1)
where T 0 is the initial (room) temperature.
po
τ = τ(r)
pi
Ω
ri
ro
Figure 4.1: Cross-section of a hollow rotating cylinder
4.2. Stress in a rotating cylinder
63
To facilitate a relatively simple analytical solution of the stress within the cylinder, an approach suitable for plane elastic problems will be taken. Figure 4.2 represents two types of
cylinders whose stress or strain can be modeled planar. If the axial length of a cylinder is
much smaller than its radius the cylinder can be modeled using the plane stress condition under which the stresses occur only in the plane perpendicular to the z-axis. Conversely, if the
length of the cylinder is very large, plane strain condition under which, in general, axial strain
is constant [122]. The plane strain condition is, thus, suitable for slender cylinders, thus for
long rotors, while the plane stress condition is valid for disks, thus for short, disk-shaped rotors
and laminated long rotors.
z
ro
L>>ro,
σz=const
z
ro
L
l
l<<ro, σz=0
a)
b)
Figure 4.2: Cylinders with plane a) stress and b) strain elastic conditions
4.2.1 Isotropic modeling
In this subsection the material of the cylinder is assumed to be isotropic and linear. Hooke’s law
for the case of the axisymmetrical isotropic cylinder will be expressed in a generalized form to
comprise both plane stress and strain conditions [122]:
"
# " 1
" ∗ #
# " ∗ #
#"
ν∗
εr
σ
α
c
−
r
∗
∗
E
= Eν∗
+
τ− ∗
εθ
α∗
c
σθ
− E ∗ E1∗
|{z} |
|{z}
{z
} | {z } |{z}
ε
where:
∗
E =
(
M
E
,
1−ν2
σ
(*)
(**)
( E,
(1 + ν) α, (*)
α∗ =
α,
(**)
(*) - under the plane strain condition
(**) - under the plane stress condition
α
c
(
ν
,
1−ν
(*)
(**)
( ν,
νε
,
(*)
0
c∗ =
0,
(**)
∗
ν =
(4.2)
(4.3)
64
Chapter 4. Structural aspects of PM rotors
In (4.2) and (4.3) ε and σ denote 2D strain and stress, respectively, E is Young’s modulus,
ν is Poisson’s ratio and α is the coefficient of linear thermal expansion. The term ε0 designates
residual axial strain under the plane strain condition, i.e. εz = ε0 .
From (4.2) the vector of stress components is:
(4.4)
σ = M −1 ε + c − M −1 α · τ
To obtain the expression for stress distribution in the cylinder, the displacement technique
will be employed. The correlations between radial displacement and strain are:
εr =
du
u
, εθ =
dr
r
(4.5)
and the force-equilibrium equation is:
r
dσr
+ σr − σθ + ρr2 Ω2 = 0.
dr
(4.6)
Equations (4.5) and (4.6) are explained in more details in Appendix A.
By combining equations (4.4),(4.5) and (4.6) we obtain the governing differential equation
for radial displacement:
r2
3 2
du
d2 u
∗ 2 dτ
∗2 ρr Ω
∗
)
(1
α
r
+
r
−
u
=
−
1
−
υ
+
+
υ
,
du2
dr
E∗
dr
(4.7)
and its solution is given by:
ρr3 Ω2
B ∗1
∗
)
(1
α
u = Ar + − 1 − υ∗2
+
+
υ
r
8E ∗
r
Z
τrdr.
(4.8)
r
After substituting (4.8) into (4.5) and then into (4.4), expressions for the stress components
yield:
Z
E∗
E ∗ B 3 + ν ∗ 2 2 α∗ E ∗
σr =
A−
−
ρr Ω − 2
τrdr
1 − ν∗
1 + ν∗ r 2
8
r
Z
r
(4.9)
E ∗ B 1 + 3ν∗ 2 2 α∗ E ∗
E∗
∗ ∗
A
+
−
ρr
Ω
+
τrdr
−
α
E
τ
σθ =
1 − ν∗
1 + ν∗ r 2
8
r2
r
In the model of the rotating cylinder from Fig. 4.1 the boundary coefficients A and B are
obtained by equating the radial stresses at the external cylinder surfaces with the static pressures:
σr (ri ) = −pi , σr (ro ) = −po
(4.10)
After applying values for the boundary coefficients, stress components can be subdivided
into parts based on rotation, static pressure and temperature increment:
σr = σΩr + σrp + στr
σθ = σΩθ + σθp + στθ
(4.11)
4.2. Stress in a rotating cylinder
65
2
t
*
Non−dimensional stress components: σ =σ/(ρ v )
1
1
1
r /r = 0.5
r /r = 0.2
r /r = 0.8
i o
i o
i o
*
θ
σ
*
θ
σ
*
σθ
0.5
0.5
0.5
*
r
σ
σ*
r
*
r
σ
0
0
ri
ro
0
ri
ro
ri
ro
Figure 4.3: Stress components resulting from the cylinder rotation
The rotation-influenced parts have the following values:
(3 + ν∗ ) ρΩ2 2
=
ro + ri2 −
8
(3 + ν∗ ) ρΩ2 2
ro + ri2 +
σΩθ =
8
σΩr
!
ro2 ri2
2
−r
r2
!
ro2 ri2 1 + 3ν∗ 2
−
r
r2
3 + ν∗
(4.12)
The stress components are scaled down with the factor 1/(ρv2t ), where vt = Ωro is the tip
tangential speed, and they are plotted in Figure 4.3 for ν∗ = 0.3. Evidently, the tangential
stress is dominant throughout the whole cylinder’s volume and its maximum occurs at the inner
surface of the cylinder.
In order to avoid cracking, the value of stress in every point of an elastic body needs to be
kept lower than a certain value σU , which is either yield or ultimate stress, depending on type of
the material. Theories of material failure (Tresca’s, von Mises’: see e.g. [123]), whose findings
are confirmed with experimental results [124], define a reference stress as a criterion for elastic
failure. For the plane stress condition Tresca’s reference stress is defined as [80]:
(r) = max (|σr (r) − σθ (r)| , |σr (r)| , |σθ (r)|) ,
σTreresca
f
(4.13)
and von Mises’ reference stress as:
σvM
re f
q
(r) = σ2r + σ2θ − σr σθ .
(4.14)
Values of the reference stresses resulting from these theories are quite close, with Tresca’s
criterion being slightly more conservative (see Fig. 4.4).
66
Chapter 4. Structural aspects of PM rotors
Figure 4.4: Tresca’s and von Mises’ yielding criteria in σ plain
Reference stress must be lower than the yielding/ultimate stress of the material in all points
of the cylinder throughout the whole speed range:
σre f (r) < σU , ri ≤ r ≤ ro , Ω ≤ Ωmax .
(4.15)
When considering the rotating cylinders it can be shown ( [80], see also Fig. 4.3) that the
Tresca’s reference stress is equal to the tangential stress, thus, with a maximum value at the
inner cylinder surface:
σre f,max = σθ (ri ) = σΩθ (ri ) + σθp,τ (ri ) .
(4.16)
Only the rotation-dependent part of Eq. (4.16) will be analyzed:
σΩre f,max = σΩθ (ri ) =
or, in a different form:
σΩre f,max
i
ρΩ2 h
(3 + ν∗ ) ro2 + (1 − ν∗ ) ri2 ,
4
ρv2
ρv2 ri
= (3 + ν ) t + (1 − ν∗ ) t ·
4
4
ro
∗
!2
.
(4.17)
(4.18)
Apparently, the maximum reference stress in the cylinder is proportional to the square of the
tip tangential speed, which can be used as an adequate limitation figure of speed of a rotating
axisymmetrical body. Practically, from (4.18), since ν∗ < 1, the value of ρv2t can be approximately taken as the maximum stress in a hollow rotating cylinder and that value must be below
the material ultimate stress:
σΩre f,max ≈ ρv2t < σU
(4.19)
4.2. Stress in a rotating cylinder
67
In a lesser extent, the reference stress is also dependent on the ratio of the cylinder radii:
the stress is lower for the smaller inner radius. However, the equation (4.18) cannot be used
for calculation of the maximum stress in the example of a full cylinder (ri = 0) because the
equations (4.9) and (4.12) are not defined for r = 0. In that case the boundary equations (4.10)
change to:
u (0) , ∞, σr (ro ) = −po .
(4.20)
After calculating values of the boundary coefficients using (4.8), (4.9) and (4.20), the rotationinfluenced parts of the rotor stress components yield:
(3 + ν∗ ) ρΩ2 2
ro − r 2
8
!
(3 + ν∗ ) ρΩ2 2 1 + 3ν∗ 2
=
r
ro −
8
3 + ν∗
f ull
σΩ,
=
r
f ull
σΩ,
θ
(4.21)
Maximum reference stress of a full cylinder occurs in the center of the cylinder:
f ull
Ω, f ull
(0) = (3 + ν∗ )
σΩ,
re f,max = σθ
ρv2t
.
8
(4.22)
The maximum stress value in this case can be approximated as:
f ull
σΩ,
re f,max ≈
ρv2t
< σU
2
(4.23)
Remarkably, the maximum reference stress of a full cylinder is equal to a half of the reference stress of a hollow cylinder of the same external radius with an infinitely small inner
hole:
1
f ull
σΩ,
lim+ σΩre f,max
(4.24)
re f,max =
r
2 i →0
√
Consequently, the maximum achievable tangential speed of a full cylinder is, at least, 2
times higher than that of a hollow cylinder with the same external diameter.
4.2.2 Orthotropic modeling
An assumption about isotropy of the cylinder’s material was introduced in the beginning of
this section. However, rotors of electrical machines sometimes include materials, such as fiber
composites, with strong orthotropic nature, having, thus, quite different material properties in
different directions. Analytical solution for stress in an orthotropic rotating cylinder is rather
lengthy; therefore, only important correlations will be presented here which can, however, lead
to desired expressions for the stress.
For the sake of simplicity, the temperature increment within the cylinder will be assumed
uniform, thus:
τ = ∆T = const.
(4.25)
68
Chapter 4. Structural aspects of PM rotors
Generalized Hooke’s law for plane stress/strain conditions in a hollow orthotropic cylinder
has the following form:
"
#  1
# " ∗ #
" ∗ #
ν∗  "
 Er∗ − Erθr∗  σr
εr
αr
cr

=  νθr∗
+
(4.26)
,

∗ τ−
1
εθ
σθ
αθ
c∗θ
− E∗ E∗
|{z} | θ{z θ } | {z } |{z}
| {z }
ε
where:
(
M
σ
Er
,
1−νrz νzr
α
(
c
Eθ
,
1−νθz νzθ
(*)
=
,
(**)
( Eνrθr+ν
rz νzr
, (*)
∗
1−νrz νzr
νrθ
=
νrθ ,
(**)
(*)
,
(**)
( Eνθrθ+ν
θz νzθ
, (*)
∗
1−νθz νzθ
νθr
=
νθr ,
(**)
α∗r
α∗θ
Er∗
(
αr + νrz EErz αz , (*)
=
(**)
( αr ,Ez
νrz Er ε0 , (*)
c∗r =
0,
(**)
(*) - under the plane strain condition
(**) - under the plane stress condition
Eθ∗
=
(
αθ + νθz EEθz αz , (*)
(**)
( αθ ,Ez
νθz Eθ ε0 , (*)
c∗θ =
0,
(*)
=
(4.27)
(The reader may refer, for instance, to literature on mechanics of composite materials, e.g.
[125].)
For an orthotropic material the following symmetry relation also holds:
νrθ νθr
=
.
(4.28)
Er
Eθ
Similarly as in the beginning of this section, the Hooke’s law in the stiffness form is obtained using (4.4). After combining the resulting expressions with (4.5) and (4.6), a differential
equation over radial displacement yields the following form:
d2 u 1 du
u P
+
− h 2 + ∆T + QΩ2 r = 0,
(4.29)
2
dr
r dr
r
r
where h, P and Q are constants dependent on material properties:
Eθ
(4.30)
h= ,
Er
P = (1 − νrθ ) αθ h − (1 − νθr ) αr ,
(4.31)
(1 − νrθ νθr ) ρ
.
(4.32)
Q=
Er
Solution of the differential equation (4.29) gives the expression for displacement:
√
√
P
Q
u = Ar h + Br− h +
∆T r +
Ω2 r 3 ,
(4.33)
h−1
h−9
and further derivation of the stress components can be performed in the same way as for the
isotropic example.
It is noticeable that the analytical solving of the stress of an orthotropic body is rather burdensome. However, the orthotropic solution will provide a means for testing applicability of the
isotropic assumption for structural modeling of a PM rotor.
4.3. Mechanical stress in a PM rotor
4.3
69
Mechanical stress in a PM rotor
A PM rotor will be modeled as a compound of three adjacent cylinders which represent an (iron)
shaft, permanent magnet and the magnet retaining sleeve (Fig. 4.5). Rotors of small high-speed
PM generators consist sometimes, though, only of a solid magnet enclosed with a sleeve [29,81]
(i.e. rFe = 0): the configuration that renders lower stress in the magnet, as shown in the previous
section. Still, conclusions from this and the subsequent chapter will remain valid for that rotor
configuration too.
δ
re
rotor enclosure
rm
rei
permanent
magnet
rFe
shaft (iron)
δ = rm - rei
Figure 4.5: Cross-section of the model of a PM rotor
In the rotor compound contact pressure between the adjacent cylinders must be maintained
throughout the whole speed range. In other words, radial stress on the boundary surfaces must
be smaller than zero. In Section 4.2.1 (Eq. (4.12)) it was seen that radial stresses based on
rotation at the boundaries is zero. Therefore a static pressure needs to be applied in order to
maintain contact between the magnet and other rotor parts. The contact pressure is achieved by
press- or shrink-fitting a non-magnetic retaining sleeve over the permanent magnet. The amount
of shrinkage, or the interference fit, is defined as:
δ = rm − rei .
(4.34)
With help of the previous section, stress in the rotor will be correlated with the rotational
speed, interference fit and operating temperature. The assumption (4.25) about a constant temperature increment in the rotor will be maintained for simplicity; having different temperatures
(∆T 1,2,3 ) of different rotor parts would add, though, a little to complexity of the solution.
With the aforementioned assumption, regarding the parts of a fully isotropic PM rotor whose
cross-section is given in Fig. 4.5, expressions (4.8) and (4.9) for displacement and stress components in rotating cylinders take on the following form:
ρr3 Ω2
Bi − 1 − υ∗2
, i = 1, 2, 3
(4.35)
ui = Ai r +
r
8E ∗
70
Chapter 4. Structural aspects of PM rotors
and
E ∗ Bi 3 + ν∗ 2 2
E∗
A
−
−
ρr Ω
i
1 − ∗ν∗
1 + ∗ν∗ r2
8 ∗
i = 1, 2, 3
(4.36)
E Bi 1 + 3ν 2 2
E
∗ ∗
A
+
−
ρr
Ω
−
α
E
∆T
σθi =
i
1 − ν∗
1 + ν∗ r 2
8
where 1, 2 and 3 denote cylindrical rotor components: the shaft, the magnet and the sleeve,
respectively.
Boundary coefficients Ai and Bi in each region are obtained from the boundary equations
which correlate radial stresses and displacements at the boundary regions:
σri =
σr1 (0) , ∞
σr2 (rFe ) − σr1 (rFe ) = 0
ur2 (rFe ) − ur1 (rFe ) = 0
σr3 (rm ) − σr2 (rm ) = 0
ur3 (rm ) − ur2 (rm ) = δ
σr3 (re ) = 0
(4.37)
Solution of the system of equations (4.37) is achieved using a symbolic solver. Finally,
after substituting obtained coefficients Ai and Bi into (4.36) analytical expressions for radial
and tangential stress in the rotor are obtained. The expressions are rather lengthy; however,
influences of static pressure (fitting), centrifugal force (rotation) and temperature increment on
the stress can be clearly distinguished. Thus, both radial and tangential stresses can be expressed
in the following way:
σr/θ (r) = Fr/θ · Ω2 + Gr/θ · δ + Hr/θ · ∆T
(4.38)
where F , G and H are functions of radius r, external cylinders’ radii (rFe , rm and re ) and
material properties (E ∗ , ν∗ , α∗ ).
Similarly, for those regions (cylinders) with orthotropic properties, equations (4.35) and
(4.36) for displacement and stress components can be replaced by corresponding expressions
valid for the orthotropic cylinders. Again, solving system of the equations (4.37) will result in
analytical expressions for stress components that retain the form given in (4.38).
In the remainder of this section validity of these models will be tested against 2D finite
element (FE) modeling using the rotor of the test machine. The rotor will be referred to as the
test rotor.
4.3.1 Test rotor: Analytical models
The test rotor has higher polar than transversal inertia having a disc which dominates its volume.
A plastic-bonded magnet is applied onto the stainless steel disc in a ring shape. Finally, a
carbon-fiber ring is pressed over the magnet with δ = 95 µm of the press fit. The rotor was
designed for operation at rotational speeds up to 200.000 rpm and temperatures up to 85◦ C.
Dimensions and properties of the parts of the test-rotor disc are given in Table 4.1.
In the table ro denotes outer radius of the corresponding cylinder (rFe , rm or re ). The elastic
moduli represent tensile moduli of the materials; it was assumed, however, that the compressive
4.3. Mechanical stress in a PM rotor
71
Table 4.1: Properties of the test-rotor parts
ro ρ
Er Eθ νθr
νrz
mm g/cm3
GPa
shaft: stainless steel
10.5 7.8
200
0.3
magnet: PPS-bonded NdFeB 14.5 4.8
31.7
0.3
sleeve: carbon fibers
16.5 1.6 9.5 186 0.3 0.59
αr αθ
µm/m/◦ C
12
4.7
-1
5
and tensile moduli of each material are equal. Details on acquiring mechanical properties of the
rotor materials are given in Chapter 7.
Calculations of mechanical stress in the rotor were performed based on two analytical models: one that assumes isotropic behavior of the carbon-fiber ring and other that takes into account
full orthotropic data of the carbon fiber composite using the orthotropic cylinder modeling presented in Subsection 4.2.2. In the former model the carbon-fiber properties in the direction of
fibers were assumed to be valid in all directions, thus: E = Eθ , α = αθ and ν = νθr . Since the
test rotor has a disc shape, the plane stress condition was appropriate.
Both models resulted in components of stress in the form (4.38). For instance, after substitution of values of the mechanical properties from Table 4.1, the stress components in the
magnet and retaining sleeve (carbon fibers) modeled with the isotropic assumption yield:









iso
σθ (r) = 
















iso
σr (r) = 







0.4 −
1.2e−5
r2
0.3 −
3.4e−5
r2
0.4 +
1.2e−5
r2
0.3 −
3.4e−5
r2
− 1.1e3r2 · Ω2 + −1.0e12 +
· δ + −8.3e4 + 22
· ∆T
r2
(rFe < r ≤ rm )
1.2e9
2
2
− 370r · Ω + 4.3e12 + r2 · δ + 6.4e5 + 170
· ∆T
r2
(rm < r ≤ re )
4.8e7
r2
(4.39)
(4.40)
· δ + −8.3e4 − 22
· ∆T
r2
(rFe < r ≤ rm )
1.2e9
2
2
− 640r · Ω + 4.3e12 − r2 · δ + 6.4e5 − 170
· ∆T
r2
(rm < r ≤ re )
− 2.0e3r2 · Ω2 + −1.0e12 −
4.8e7
r2
In equations (4.39) and (4.40) values of the variables are in SI units.
Similar expressions are obtained by having orthotropic assumption for the carbon-fibers
region.
4.3.2 Test rotor: 2D FE model
Finite element modeling was performed using structural analysis of the Ansys FEM software.
The model used PLANE82 elements for 2D structural modeling while the contact area between
the magnet and enclosure was modeled using contact elements: CONTA172 and TARGE169.
Generated mesh is presented in Figure 4.6.
72
Chapter 4. Structural aspects of PM rotors
Figure 4.6: Cross-section of the PM rotor - FE model
4.3.3 Test rotor: Results comparison
Outcome of the three 2D models are presented in Table 4.2. The table represents stresses at two
critical regions: inner surfaces of the magnet and sleeve, including three examples: the rotor at
standstill and the rotor rotating at the target speed of 200.000 rpm at both room and maximum
operating temperature. The index vM denotes von Mises reference stress.
Table 4.2: Stress components at critical regions based on different analytical 2D models
Mechanical stress
[MPa]
sleeve, σr (rm )
sleeve, σθ (rm )
magnet, σr (rFe )
magnet, σθ (rFe )
magnet, σvM (rFe )
20◦ C, 0 rpm
iso
ortho FEM
-118.3 -109.8 -109
921
940.6
935
-137.8
-128
-127
-56
-52
-51.6
120
111.3
110
20◦ C, 200.000 rpm
iso
ortho FEM
-117 -108.7 -108
1088
1108
1110
-26.8 -17.1 -13.1
-0.44
3.52
5.18
26.6
19.1
16.3
85◦ C, 200.000 rpm
iso
ortho FEM
-128 -122.8 -117
1176
1188
1160
-43.7 -37.4 -25.9
6.54
9.13
8.75
47.3
42.7
31.2
It can be seen that the isotropic model is, generally, in a good agreement with other two
modes. An exception is stress in the magnet at the maximum speed: when the compression
weakens, discrepancies between models become larger.
Nevertheless, as calculated by the FE model, compression in the magnet at stand-still and
tension in the fibres at the maximum speed is slightly smaller than predicted by the isotropic
model. Contact pressure between magnet and shaft is maintained throughout the whole speed
range. Temperature rise causes additional stress in the rotor but also increases the contact
pressure between the cylinders.
4.4. Structural limits and optimization of PM rotors
73
Although orthotropic analytical model predict the stress more accurately, the accuracy improvement does not justify great increase in complexity with respect to the isotropic model.
In conclusion, the fully isotropic analytical model of mechanical stress is sufficiently accurate for modeling of PM rotors even when they include composite enclosures with strong
orthotropic nature. The isotropic model will be used in this thesis for structural optimization of
the test machine rotor. Design will be finally evaluated using 3D FE modeling in Chapter 7.
4.4
Structural limits and optimization of PM rotors
In the preceding sections the influence of the geometry, operating speed, fitting and temperature on mechanical stress in a PM rotor was determined through analytical modeling. In this
section, using the developed models, structural limits for the rotor speed will be determined and
quantified. At the same time, a relatively simple approach of optimizing the rotor structure will
be presented.
In the example of the rotor consisting of cylindrical parts the structural design narrows
down to defining thicknesses (or radii) of the rotor parts and the interference fits between them.
However, in the case of a PM rotor, dimensions of the iron shaft and the magnet will strongly
influence rotordynamic and electromagnetic performance of the machine. The structural design
of the rotor can hardly be removed from the electromagnetic design of the whole machine. Here,
it will be assumed that the dimensions of the shaft and magnet were defined beforehand during
the electromagnetic design. Therefore, the optimization of the rotor structure will focus on
determining thickness and fitting of the magnet retaining sleeve so as to maintain the structural
integrity of the PM rotor throughout the whole ranges of operating speed and temperature.
The thickness of the sleeve (re − rm ) and the interference fit (δ) must be adequately chosen
so that the contact pressure between the adjacent cylinders is always preserved, thus:
σr (rFe , rm ) < 0
(0 ≤ Ω ≤ Ωnom , 0 ≤ ∆T ≤ ∆T max ) .
(4.41)
At the same time, the reference stress in each rotor part must be considerably below the
yielding/ultimate stress of the corresponding material:
σref (r) < σiU
(0 ≤ r ≤ re , 0 ≤ Ω ≤ Ωnom , 0 ≤ ∆T ≤ ∆T max )
(4.42)
The purpose of the sleeve is to prevent high tension in the magnet and ensure the transfer
of torque from the magnet to the shaft. The highest tension in the whole rotor occurs at the
inner surface of the sleeve. Suitable materials for the magnet retainment have, thus, high tensile
strength and low weight (glass and carbon composites, titanium).
At the same time, after shrink- or press-fitting the sleeve, the magnet is subjected to compression. Compressive strength of sintered magnets (which are regularly used in high-performance
applications) is relatively high and, usually, compressive stress in the magnet does not represent
a limitation for the fitting of the sleeve.
Hence, in this type of high-speed rotors (Figure 4.5), the most critical stresses are radial
(contact) stress at the magnet-iron boundary and tangential stress (tension) at the sleeve inner
74
Chapter 4. Structural aspects of PM rotors
surface. As a result of modeling from the previous sections and after having the dimensions of
the shaft and magnet specified, analytical expressions for these stresses can be presented as:
σmr,crit = σr (rFe ) = F1 (re ) · Ω2 − G1 (re ) · δ − H1 (re ) · ∆T
(4.43)
σeθ,crit = σeθ (re ) = F2 (re ) · Ω2 + G2 (re ) · δ + H2 (re ) · ∆T
(4.44)
where F1,2 and G1,2 are positive functions of the sleeve outer radius re , re > rm . Sign of the
functions H1,2 depends on the difference between the coefficients of thermal expansion of the
magnet and the sleeve. For the sake of the analysis the functions H1,2 will be also assumed
positive (equivalent to αe < αm )
If the interference fit is very high, contact pressure between magnet and iron will be maintained (σmr,crit < 0), but the maximum stress σeU in the enclosure will be reached at a certain
speed. Conversely, if the fit is very low, loss-of-contact limit (σmr,crit = 0) will be met with increasing speed. It can be thus inferred from (4.43) and (4.44) that for an expected operating
temperature there is an optimal value of the interference for which both limits defined by (4.41)
and (4.42) are reached at a same rotational speed Ωmax (see Figure 4.7 and [80]). This speed can
be adjusted by the enclosure radius re so that the theoretical maximum rotational speed Ωmax is
a considerable margin higher than the operating speed Ωnom .
σ
δ>δ
opt
δ
opt
yield s trength
σU
δ<δ
opt
σ
θ
δ<δ
opt
los s of contact
0
δopt
δ>δ
opt
σr
0
Ωmax
Ω
Figure 4.7: Finding the optimal interference fit, similarly as in [80]
Hence, the optimal fit δopt and theoretical maximum speed Ωmax are obtained as functions
4.5. Conclusions
of the sleeve radius re from the following system of equations:
σmr,crit Ω = Ωmax , δ = δopt , ∆T = 0 = 0
σeθ,crit Ω = Ωmax , δ = δopt , ∆T = ∆T max = σeU
75
(4.45)
In reality, the higher the sleeve thickness is (re − rm ) the higher the maximum permissible
rotational speed will be. On the other hand, that thickness is restricted by the available space in
the machine air-gap, which is, as assumed, also decided on in the electromagnetic design.
However, structural and electromagnetic optimization can be carried out simultaneously.
Namely, for defined dimensions of the permanent magnet there is an optimal value of the interference fit and a necessary sleeve thickness for reaching requested rotational speed. If this
correlation is considered during design of the machine, requested electromagnetic performance
(torque, power, losses) can be achieved with an optimal structural design of the rotor for the
desired operating speed.
4.5
Conclusions
The permanent magnet represents the most mechanically vulnerable part of a PM rotor. In order
to prevent high tension in the magnet and ensure the transfer of torque from the magnet to the
shaft, high-speed PM rotors are usually enclosed with strong non-magnetic retaining sleeves.
Good design of the retaining sleeve is crucial for the rotor structural integrity.
In the chapter, a high-speed rotor is represented as a compound of three concentric cylinders
which represent the (iron) shaft, permanent magnet and magnet retaining sleeve. The chapter
models the influence of rotational speed and mechanical fittings on stress in such a rotor, while
also considering the operating temperature. The modeling of stress in a rotating PM rotor is
based on equations which can be found in textbooks on structural mechanics. The suitability of
2D analytical models to represent a PM rotor without magnet-pole spacers is demonstrated by
Binder in [75].
This thesis shows that a model which assumes isotropic behavior of the carbon-fiber retaining sleeve by assigning the properties in the direction of fibers to all directions makes almost
equally good predictions as a fully orthotropic model of the rotor. Results of these two analytical
models were compared to results of 2D FEM and agreement of the models is quite satisfactory.
The critical, thus limiting stresses in the rotor are radial (contact) stress at the magnet-iron
boundary and tangential stress (tension) at the sleeve inner surface. It is shown in that for an
expected operating temperature there is an optimal value of the interference fit between the
sleeve and magnet for which both tension and contact limits are reached at an equal rotational
speed. This speed can be adjusted by the enclosure thickness so that the theoretical maximum
rotational speed is a considerable margin higher than the operating speed.
In the chapter, structural limits for speed of PM rotors are identified and the limiting parameters (stresses at the rotor material boundaries) are represented in a simple analytical form
that clearly indicates optimal geometry of the rotor retaining sleeve. In this way, a relatively
simple approach of optimizing the retaining sleeve is achieved; the approach takes into account
76
Chapter 4. Structural aspects of PM rotors
the influence of rotational speed, mechanical fittings and operating temperature on stress in a
high-speed rotor. This represents the main contribution of this chapter.
The presented structural design of the rotor follows the electromagnetic design of the machine. However, the analytical approach for the sleeve optimization lends itself to inclusion
into a simultaneous, structural and electromagnetic optimization process. Namely, for defined
dimensions of the permanent magnet there is an optimal value of the interference fit and a necessary sleeve thickness for reaching requested rotational speed. If this correlation is considered
during the machine design, electromagnetic performance requirements (torque, power, losses)
can be achieved with an optimal structural design of the rotor for the desired operating speed.
Chapter 5
Rotordynamical aspects of high-speed
electrical machines
5.1
Introduction
For very high-speed machines study of rotordynamics becomes very important. Rotors of today’s high-speed machines regularly operate in supercritical regime, therefore, issues such as
unbalance response and stability of rotation need to be addressed early in the design phase to
prevent machine’s failure.
Naturally, the field of rotordynamics concerns with a great number of rather complex problems, study of which goes well beyond scope of this thesis. Still, machine geometry, construction, even magnetic field have great influence on dynamical behavior of the rotor and that
influence is particularly important when designing high-speed machines. In this chapter, a
qualitative insight into important dynamical aspects of high-speed rotors will be given through
analytical modeling. The goal of the chapter is to define the dynamical limits for the rotor speed
and to correlate those limits with machine parameters.
The phenomenon which is of practical concern is rotor vibrations, their cause and influence
on the system. Two types of vibrations can be distinguished: resonant and self-excited [86].
Resonant vibrations are excited by an oscillating force whose frequency coincides with
one of the natural frequencies of the rotor-bearings system. Vibrations that are caused by an
(external) oscillating force are generally referred to as forced vibrations.
Typically, the oscillating force comes from the rotor mass unbalance. Influence of the rotor
unbalance (which is inevitably present in realistic rotors) can be modeled as a force which rotates around a perfectly balanced rotor and has the same rotational frequency as the actual rotor
frequency. Therefore, the rotor unbalance will excite the resonant vibrations when rotational
speed is equal to a natural frequency of the rotor-bearings system. Those rotational speeds are
referred to as critical speeds.
Evidently, resonant vibrations occur at certain frequencies and are influenced by the amount
of unbalance. They can be damped by external damping and may be passed if sufficient energy
(dissipated in bearings’ dampers) is invested.
However, not only mass unbalance of the rotor can cause resonant vibrations. Circulating
77
78
Chapter 5. Rotordynamical aspects of high-speed electrical machines
fluids in the bearings or harmonics of unbalanced magnetic pull of the electrical machine whose
forces may not be synchronous with the rotor can also excite resonant vibrations. Nevertheless,
these phenomena will not be analyzed in detail in this chapter and the reader will be referred to
literature.
Self-excited vibrations, on the other hand, require no external force for inception1 [86, 126].
They arise within range(s) of rotational speed, usually after a certain (threshold) speed which
is correlated with intrinsic properties of the system. The external (non-rotating) damping can
increase threshold speed value but has virtually no influence when the vibrations occur. Unbalance has no influence on these vibrations - they would also occur in a perfectly balanced
rotor.
Self-excited vibrations are unstable and hazardous; the rotor should not operate in the speed
range where these vibrations occur. Hence, the threshold speed is the definite speed limit of
the rotor. A rotor operating at speed higher than the threshold speed can be viewed as is in the
state of unstable equilibrium - any perturbation will bring about vibrations whose amplitude
will (theoretically) grow to infinity.
This chapter starts by defining different vibrational modes. A theoretical study on stability
of rotation follows that will assess threshold speed of self-excited vibrations and, consequently,
define practical speed limits of electrical machines with respect to critical speeds and properties
of the bearings. The study lastly gives basic correlations between critical speeds and the rotor
and bearings parameters. In the rest of the chapter only rigid rotors are considered. Unbalance
response of rigid rotors is modeled analytically and suitability of different rotor geometries for
high-speed rotation is analyzed.
The analysis in this chapter had a defining influence on the new high-speed-spindle concept
presented in Chapter 7. The new spindle design has its foundation in conclusions drawn in the
current chapter: the design is developed so that the major rotordynamical stability limits are
simply avoided.
The list of literature that is relevant for the subject of this chapter is quite long; this study
was mainly influenced by the works of G. Genta [86, 127, 128], D. Childs [126, 129] and A.
Muszynska [130, 131].
5.2
Vibration modes
In literature on electrical machines rigid and flexural (bending) vibration modes are often distinguished based on whether the rotor is deformed while vibrating. However, this division is
rather conditional. Truly rigid modes can occur only in the case of free rotors. Otherwise, the
modes are mixed. In practice, if the rotor stiffness is much higher than the stiffness of bearings
(soft-mounted rotor), term rigid-body modes refers to the modes in which rotor deformation
is negligible when compared to the bearing deformation. If the condition for stiffness is not
fulfilled, it is not possible to speak about rigid modes.
1
The term self-excited can be disputed since a rotor that resonates due to its own unbalance is, physically, selfexcited. However, by self-excitation is emphasized that no disturbing force is required for the vibration. On the
other hand, the force coming from the rotor unbalance is considered external to the ideal rotor.
5.3. Threshold of instability
79
Rigid-body vibrations occur at relatively low speeds. Since shape of the rotor is maintained
internal rotor damping has no influence on the dynamics of the system.
Regarding the relative motion of the rotor with respect to geometrical axis cylindrical and
conical modes are recognized.
cylindrical
conical
Figure 5.1: Rigid-body vibration modes
Flexural (or bending) modes are higher order vibrations in which both rotor and bearings
are deformed, therefore internal damping of rotor also plays role in the rotor dynamics. Flexural
vibrations occur at higher speeds with respect to rigid vibrations.
Figure 5.2: The first flexural vibration mode
Finally, torsional and axial vibrations can also occur, however, if flexural and torsional/axial
modes are not coupled radial forces (e.g. unbalance) will not excite these vibrations.
5.3
Threshold of instability
Purpose of this section is to describe mechanisms that lie behind rotational instability of highspeed machines. Through simplified analytical models of the rotor-bearing systems basic expressions for limiting rotational speed will be given and, more importantly, implications for
high-speed electrical machines will be drawn.
To find the threshold of instability the Jeffcott rotor model will be used. It is rather simplified
model in which a rotor is represented as a massless (generally compliant) shaft with the mass
concentrated at one point of the shaft - Fig. 5.3. However, this model gives a qualitative
insight into rotors’ dynamical behavior and will be used here to infer about the stability limit in
general [86, 129].
80
Chapter 5. Rotordynamical aspects of high-speed electrical machines
5.3.1 Stability of the Jeffcott rotor with damping
For calculating the instability threshold the example of a perfectly balanced Jeffcott rotor is
considered, thus, center of rotor inertia coincides with the geometrical center of the rotor C.
The center is always located in the xy plane.
z
Ω
y
C
m
x
Figure 5.3: Balanced Jeffcott rotor
The equations of motion of the Jeffcott rotor in the stationary reference frame are:
!
d2 xC
dxC
dxC
m 2 + cn
+ cr
+ ΩyC + kxC = F x
dt
dt
dt
!
dyC
dyC
d2 yC
+ cr
− ΩxC + kyC = Fy
m 2 + cn
dt
dt
dt
(5.1)
or in the matrix form:




m 0
0 m




ẍC
ÿC
 
 
 
+
cn + c r
0
0
cn + cr




ẋC
ẏC
 
 
 
+
k 0
0 k



+


Ω
0 cr
−cr 0




xC
yC
 
 
 
=
Fx
Fy




(5.2)
In (5.2) Ω is rotational speed, m the rotor mass, k stiffness of the bearings, cn is coefficient of
viscous-type non-rotational (external) damping of the rotor-bearings system and cr is coefficient
of viscous-type rotational (usually rotor internal) damping.
Position of the point C can be also expressed as a complex vector:
rC = xC + jyC = rCo e st .
(5.3)
Exponential form of rC from (5.3) can be applied to (5.2) and the free whirling case (F x =
Fy = 0) yields characteristic equation of the system:
ms2 + (cr + cn ) s + k − jΩcr = 0.
(5.4)
5.3. Threshold of instability
81
Solutions of Eq. (5.4) yield eigenvalues of the system:
s
(cr + cn )2 − 4m (k − jΩcr )
cr + c n
±
.
s = σ + jω = −
2m
4m2
(5.5)
Evidently, there are two complex eigenvalues:
s1,2 = σ1,2 + jω1,2 .
(5.6)
Imaginary part of the solutions represents frequency of the rotor free whirl and from (5.5) it
yields the following form:
rq
ω∗1,2 = ±
Π∗2 + Ω∗2 ζr2 + Π∗
(5.7)
where:
h
i
Π∗ = 1 − (ζn + ζr )2 /2
(5.8)
and ω∗ , Ω∗ , ζn and ζr are non-dimensional values of frequencies and dampings:
ω
Ω
ω∗ = √
Ω∗ = √
k/m
k/m
cr
cn
ζr = √
ζn = √
2 km
2 km
(5.9)
The solution of the characteristic equation (5.4) which has positive natural frequency (ω1 )
represents the forward rotor whirl, which is in the direction of rotation, and the other solution
with (ω2 ) represents the backward whirl. In the case of an undamped rotor (ζn = ζr = 0)
frequencies of the resonant whirls yield:
r
k
ω1,2 = ±
= ±Ωcr
(5.10)
m
√
where Ωcr = k/m denotes critical speed of the undamped Jeffcott rotor.
Real parts of the eigenvalues (5.5) are given by:
rq
σ∗1,2 = − (ζn + ζr ) ±
where:
Π∗2 + Ω∗2 ζr2 − Π∗ ,
(5.11)
σ
(5.12)
k/m
In order for the rotation to be stable it is necessary that both complex solutions (5.6) of (5.4)
have negative real parts, thus σ1,2 < 0. It is apparent from (5.11) that the backward whirl is
always stable (σ2 < 0) while it is easy to show that the condition for stability of the forward
whirl (σ1 < 0) is equivalent to:
r
!
cn
k
1+
,
(5.13)
Ω<
m
cr
σ∗ = √
82
Chapter 5. Rotordynamical aspects of high-speed electrical machines
1.5
1
*
ω*
ζr=0.3
0.2
ζr=0
critical speed
0.2
unstable
stable
0
*
ω =Ω
0.5
forward
backward
0.4
ζr=0.3
0.1
ζr=0
−0.2
σ*
forward
backward
0
−0.4
0.1
−0.6
−0.5
0.2
−0.8
ζ =0
r
−1
−1
ζr=0.3
−1.5
0
1
2
ζr=0.3
−1.2
3
0
*
1
2
3
*
Ω
Ω
Figure 5.4: Non-dimensional parts of complex eigenvalues of the Jeffcot rotor for different
values of rotating damping and ζn = 0.1
or, in another form:
Ω < Ωcr
!
cn
.
1+
cr
(5.14)
Vibrations (whirls) defined by (5.3) and (5.6) are essentially self-excited since the rotor is
assumed perfectly balanced and no external force is acting. If the condition (5.14) is fulfilled
these vibrations are hardly noticeable - the motions damp out very quickly. However, if the
rotational speed is higher than the limit defined by (5.14) the amplitude of the vibrations will
grow and will be limited only by the system non-linearities [86]. The rotor-bearings system
from Fig. 5.3 is unstable in that case regardless of the presence of exciting forces such as mass
unbalance or unbalanced magnetic force.
From (5.14) it is evident that the presence of internal or rotating damping can negatively
influence the stability of rotation in the supercritical regime. If no rotating damping is present
the system is always stable; when rotating damping is increased the instability threshold lowers
(Fig. 5.4). On the other hand, non-rotating damping has always stabilizing effect [86, 129].
An insightful demonstration of the destabilizing effect of the rotating damping in the supercritical regime is presented in [87]. Similarly to an example from the paper, an intuitive
explanation of the rotor instability is given in Appendix B.
The aforementioned analysis was performed for a simplified rotor model and the dampings
are assumed to be of viscous type which is usually applicable only to the rotating damping.
Still, important conclusions resulting from expression (5.14) also hold true when more complex
models are used. More rigorous mathematical modeling was presented for instance in [84] using
the Timoshenko beam model with incorporated viscous and hysteretic damping. The modeling
5.3. Threshold of instability
83
showed that the the internal viscous damping has stabilizing effect on the rotor stability in the
subcritical regime but destabilizing effect in the supercritical regime. It was pointed out, however, that the hysteretic damping is also destabilizing in the subcritical regime. The influence of
internal and external damping on the presence of self-excited vibrations was practically shown,
for instance, in [85].
In the light of the forgoing modeling some important conclusions on rotordynamical stability of electrical machines will be given, as follows.
Rotation can become unstable in the supercritical regime of a certain vibration mode if
rotating damping is present in that mode. The threshold speed can be increased with additional
external damping.
During rigid-body vibrations, no deformation of rotor occurs and the internal rotor damping
plays no role. Rotors are usually stable in supercritical range pertaining to rigid modes and
today’s high-speed rotors operate regularly in that speed range. An exception is rotors which are
supported in fluid journal bearings - as result of non-synchronously circulating fluid, rotating
damping influences also rigid-body modes and a rotor can become unstable beyond the rigid
critical speed(s) [130–133] (see the next subsection).
Rotors which possess some internal damping, can easily become unstable in supercritical
range pertaining to flexural modes. Sources of rotating damping [83,126] in electrical machines
in general are eddy-currents [87], press fits, material damping, interaction with fluids [134] etc.
Moreover, flexural modes are often poorly damped externally - bearings are located at the ends
of the rotor and/or have insufficient bandwidth to cope with those vibrations.
Rotors of electrical machines are receptive to eddy-currents, always comprised of fitted
elements, and often contain materials, such as composites, with significant material damping.
Therefore, these rotors are prone to be unstable in flexural supercritical regime. The first flexural
critical speed practically represents rotordynamical speed limit of an electrical machine.
5.3.2 Jeffcott rotor with non-synchronous damping
It is of practical interest to study the case of the rotor-bearings system in which an energy
dissipating element (damper) exists which rotates with a speed other than the speed of the rotor.
A typical example for such systems are rotors in lubricated journal bearings or hydrodynamic
bearings [135]. The average tangential speed of the bearing fluid differs, as a rule, from the
tangential speed of the rotor giving a way to non-synchronous damping forces.
The equation of motion of the balanced Jeffcott rotor in presence of non-synchronous damping is as follows [127] (see [136] for gyroscopic rotors):
mr̈C + cṙC + (k − iΩcr − iΩd cd ) rC = 0,
(5.15)
where Ωd is rotational speed of the non-synchronously rotating damper, cd is equivalent viscous
damping correlated with that damper and c = cn + cr + cd .
After seeking a solution in the form (5.3), the characteristic equation becomes:
ms2 + cs + k − iΩcr − iΩd cd = 0.
(5.16)
84
Chapter 5. Rotordynamical aspects of high-speed electrical machines
Solving the characteristic equation results in complex eigenvalues s1,2 = σ1,2 + iω1,2 whose
real parts in the non-dimensional form (5.12) are given by [127]:
q
√
∗
(5.17)
σ = −ζ ± −Π∗ + Π∗2 + Ξ∗2 ,
where:
Ξ∗ = Ω∗ ζr + Ω∗d ζd
(5.18)
and:
1 − ζ2
c
, ζ= √ .
2
2 km
It can be shown that the condition for rotation stability is equivalent to:
Π∗ =
Ω∗ <
ζd
ζ
− Ω∗d
for Ξ∗ > 0,
ζr
ζr
(5.19)
(5.20)
ζd
ζ
− Ω∗d
for Ξ∗ < 0.
(5.21)
ζr
ζr
If the spinning speed of the rotating damping is proportional to the rotor speed, it is reasonable to introduce the ratio between those two speeds α = Ωd /Ω; the stability condition can be
then expressed as:
ζ
Ω∗ <
for Ξ∗ > 0,
(5.22)
ζr + αζd
ζ
Ω∗ < −
for Ξ∗ < 0.2
(5.23)
ζr + αζd
It is interesting to analyze the example of a rigid rotor. In that case, rotating damping has
no influence on the system and, if the damper rotates in the same direction as the rotor, the
condition of stability becomes:
!
Ωcr
cn
Ω<
1+
(5.24)
α
cd
Ω∗ < −
where Ωcr is the critical speed of the undamped Jeffcott rotor.
Finally, if the rotating damper has the dominant influence on the rotor damping, then cd >>
cn and the condition for stability is approximately:
Ω<
Ωcr
.
α
(5.25)
In fluid journal bearings value of the ratio α is near 0.5 [127, 130, 131, 133–135] and the
rotation becomes unstable at the speed which is approximately twice of the first critical speed.
Subsynchronous rotor vibrations that exist due to subsynchronous spinning of the damper become unstable when their frequency reaches critical frequency. The phenomenon is known as
oil whirl and at the threshold rotational speed defined by (5.25) it is actually replaced by the
2
From (5.22) and (5.23) it is evident that for α = −ζr /ζd (counter-rotating damping) the system is always stable,
which can be useful in design of active dampers.
5.4. Critical speeds calculation
85
oil whip, a particularly hazardous form of instability in journal bearings [130, 131, 133] (see the
next chapter for further discussion on this issue).
Nature of these phenomena is very complex [130], explanation of which exceeds the outreach of this study. However, it is important to notice that rigid rotors can also become unstable
in the presence of non-synchronous damping. In the study in the previous subsection it was
shown that rotors supported by purely stationary bearings cannot become unstable unless rotor
deforms and rotating damping becomes active. Here, an additional mechanism of rotational
instability is described which is connected solely to the rotor-bearings interaction.
5.4
Critical speeds calculation
The goal of this section is to correlate values of critical speeds, in particular of those connected
with bending modes, to machine parameters. In order to calculate critical speeds of a typical
rotor-bearings system, the model of an axisymmetrical Timoshenko’s beam is going to be used.
The system with a rotating beam is depicted in Figure 5.5.
k, cn
x
k, cn
y
Ω
l
z
compliant shaft (E,I)
d
ux(z)
k, cn
k, cn
Figure 5.5: A compliant rotating beam with supports
The Timoshenko’s beam is a comprehensive model of a continuous shaft that takes into
account shear deformation and gyroscopic coupling between dynamic behavior of the shaft in
different directions.
Displacement of the shaft u = u x + juy can be represented as:
u (z, t) = q (z) e jωt ,
(5.26)
where q(z) - mode shapes - must comply with:
d2 q (z)
d4 q (z)
Eχ 2
EIy
+ ρIy ω 1 +
+
− 2ωΩ
2
dz4 "
G
# dz
ρIy χ
ρIy χ 3
+ρ ω4
−2
ω Ω − Aω q (z) = 0
G
G
(5.27)
(see e.g. [86] for details).
In (5.27) Iy = I x = d4 π/64 is the surface moment of inertia of the shaft cross-section, and
χ is the shear parameter that has the value of 10/9 for circular cross-sections. Besides, ρ is the
86
Chapter 5. Rotordynamical aspects of high-speed electrical machines
mass density, E is Young’s modulus and G is the shear modulus and it holds:
E
= 2 (1 + ν) ,
G
(5.28)
where ν is Poisson’s ratio.
From the boundary conditions determined by how the rotor is borne - free, supported,
clamped, etc. - characteristic equations are obtained from which critical frequencies can be
calculated.
5.4.1 Hard-mounted shaft
For a hard-mounted rotor (k → ∞) supported at the rotor ends boundary conditions yield:
q (0) = q (l) = 0
d2 q d2 q =
=0
dz2 z=0 dz2 z=l
(5.29)
and it can be shown that, as result of (5.29) q (z) has the following form:
q (z) = qi0 sin (iπz/l) .
(5.30)
After substituting (5.30) into (5.27) the characteristic equation of the motion is obtained:
!
2 2
4λ2
i2 λ2
i4 λ4
∗
∗4
∗ ∗3
∗2 i λ
(5.31)
ω − 2Ω ω − 4ω 2 ∗ 1 + χ + 2 2 + 8Ω∗ ω∗ 2 ∗ + 16 4 ∗ = 0.
πχ
iπ
πχ
πχ
In (5.31) λ = l/d is the rotor slenderness while ω∗ , Ω∗ and χ∗ are non-dimensional natural
frequency, rotational frequency and shear parameter of the rotor, respectively:
ω∗ =
π2
l2
ω
q
EIy
ρA
, Ω∗ =
π2
l2
Ω
q
EIy
ρA
, χ∗ =
χE
G
(5.32)
Equation (5.31) implicitly defines correlation between natural frequencies of the rotorbearings system and rotational speed. When plotted, lines ω∗ (Ω∗ ) are referred to as a Campbell
diagram of the system and critical speeds represent intersections between those lines and line
ω∗ = Ω∗ (Figure 5.6). Hence, from (5.31) and ω∗ = Ω∗ = Ω∗cr the characteristic equation yields:
!
2 2
4λ2
i4 λ4
∗
∗4
∗2 i λ
(5.33)
Ωcr + 4Ωcr 2 ∗ −1 + χ + 2 2 − 16 4 ∗ = 0.
πχ
iπ
πχ
For i = 0 no real solutions for Ω∗cr exist thus no rigid modes are present (refer to eq. (5.30)).
For i > 0 solutions of (5.33) are given by:
v
u


u
u
u
r


u
∗
u
2
u −1 + χ∗ + 4λ −1 + 1 + 4χ 
t

i2 π2 

2 2
−1+χ∗ + 4λ
2 π2
2iλ
i
∗
.
(5.34)
Ωcr =
π
2χ∗
5.4. Critical speeds calculation
87
15
15
i=3
λ=5
i=2
10
λ = 15
10
ω*= Ω*
ω*= Ω*
i=3
i=1
i=3
5
5
i=2
i=2
*
ω
*
ω
i=1
0
i=1
0
i=1
i=1
i=2
i=2
i=3
−5
−5
i=1
i=3
i=2
i=3
−10
−15
0
2
−10
4
6
8
10
*
−15
0
2
4
6
8
10
*
Ω
Ω
Figure 5.6: Campbell diagram of a hard-mounted shaft based on Eq. 5.31; plotted for first three
orders of critical speeds and two values of slenderness
Figure 5.7 shows non-dimensional values of first four critical speeds for different slendernesses with χ = 10/9 and ν = 0.3. For very slender hard-mounted beams non-dimensional
critical speeds approach to the square of the order i, namely:
Ω∗ = i2 , i ≥ 1,
(5.35)
cr λ→∞
the result following also from the Euler-Bernoulli beam model [86] which neglects influences
of gyroscopic effect and shear deformation.
All the critical speeds in this case are connected with flexural vibrational modes. From
(5.32) and (5.35) value of the first flexural critical speed in the example of a hard-mounted shaft
can be approximated as:
s
s
π2 EIy π2 d E
= 2
.
(5.36)
Ωcr ≈ 2
l
ρA
4l
ρ
5.4.2 General case: bearings with a finite stiffness
High-speed rotors are predominantly soft-mounted, meaning that the stiffness of their bearings
is by far lower than the rotor internal stiffness. For a soft-mounted rotor finite stiffness k of the
bearings must be taken into account when calculating values of the critical speeds.
Unbalanced magnetic force in a machine can also have noticeable impact on values of critical speeds. Resultant attraction force between the stator and the rotor appears as a negative
88
Chapter 5. Rotordynamical aspects of high-speed electrical machines
Ω
16
cr4
14
12
Ω*cr
10
Ω
cr3
8
6
Ω
cr2
4
Ω
2
0
cr1
0
5
10
15
λ
20
25
30
Figure 5.7: First four critical speeds of a hard-mounted beam with respect to slenderness
stiffness of the machine, causing critical speeds to decrease. However, the influence of the machine stiffness will be neglected in this section and the reader is referred to literature on that
subject (e.g. [103]).
To model the bearings stiffness in general and also to infer about vibration modes at calculated critical speeds, non-dimensional stiffness will be used:
k∗ =
kl3
EIy
(5.37)
Non-dimensional stiffness k∗ practically represents ratio between the bearings stiffness and
flexural stiffness of the beam (e.g. flexural stiffness of a simply supported beam is 48EIy /l3 ).
If k∗ is very small (k∗ < 1), first two critical speeds (or only first in the example of a disc, see
Section 5.5) may be considered rigid. Otherwise, it cannot be spoken about rigid nor flexural
critical speeds - the modes are mixed. Finally, when k∗ >> 1 the modeling from the previous
subsection is applicable: practically, only flexural vibrations exist.
General solution of the differential equation (5.27) has the following form:
q (z) = C1 eaz/l + C2 e−az/l + C3 e jbz/l + C4 e− jbz/l ,
(5.38)
where ±a/l and ± jb/l are roots of the polynomial corresponding to Eq. (5.27).
Characteristic equation is obtained from boundary conditions. For a shaft supported at the
ends shear force at the coordinates of the bearings must be equal to the bearings reaction:
k∗
d3 q (z) = 3 q (z)
.
(5.39)
3
dz z=0,l
l
z=0,l
5.4. Critical speeds calculation
89
Condition of free rotation yields second boundary condition - bending moment vanishes at
the bearings coordinates:
d2 q (z) = 0.
(5.40)
dz2 z=0,l
After substituting (5.38) into the boundary conditions (5.39) and (5.40) and setting ω∗ =
Ω∗ = Ω∗cr a four equations system is obtained which can be presented in a matrix form:
[A] · [C] = 0,
(5.41)
[A] = F Ω∗cr ,
(5.42)
where:
and:
[C] =
h
C1 C2 C3 C4
iT
.
(5.43)
The system (5.41) has a non-trivial solution [C] , 0 if and only if:
det [A] = 0.
(5.44)
From equation (5.44) values of critical speeds can be calculated. However, those values
cannot be obtained in a closed analytical form and have to be calculated numerically. The
characteristic equation is solved numerically for a range of the non-dimensional stiffness and
three values of slenderness and values of first four critical speeds are plotted in Fig. 5.8. Again,
circular rotor cross-section (χ = 10/9) and the Poisson’s ratio of 0.3 were assumed.
λ=5
λ = 10
λ = 15
16
14
soft−mounted
hard−mounted
12
10
Ω*cr4
*
cr
Ω
8
6
Ω*cr3
4
*
Ωcr2
2
0 −1
10
Ω*cr=1
Ω*cr1
0
10
1
10
k
*
2
10
3
10
4
10
Figure 5.8: First four critical speeds of a shaft vs bearing stiffness for different values of slenderness
90
Chapter 5. Rotordynamical aspects of high-speed electrical machines
When the bearings’ stiffness is small compared to the rotor stiffness (the soft-mounted case)
the stiffness of the bearings has influence only on values of the low-order, rigid critical speeds.
Values of the higher order critical speeds, which certainly include flexural modes, are primarily
influenced by the shaft slenderness.
In the context of hard-mounted rotors, the concept of ’rigid critical speeds’ becomes irrelevant. Flexural vibrations occur together with other, translational and conical movements of the
rotor at all critical speeds. When k∗ → inf, value of the first critical speed of the Timoshenko
beam can be determined in a closed form according to equation (5.36).
The rotor slenderness remains the most critical factor for stable rotation of the rotor at the
maximum desired speed. Hence, for a given maximum working speed of a machine, maximum
slenderness λmax of the rotor can be evaluated so that the rotor operates at speeds below the first
flexural critical speed (as pointed out in Section 2.5).
In today’s practice, critical speeds of rotor-bearings systems are usually determined using
finite element modeling (FEM). However, in the case of relatively slender rotors, results from
this analytical modeling are in good agreement with FEM and measurements, as can be seen
in [137, 138].
Finally, the gyroscopic moment resulting from additional mass on the rotor (such as a rotor disc or press fitted elements) also influences values of critical speeds. In the example of
asymmetrical rotors, whose center of gravity is not collocated with the center of the shaft, the
gyroscopic moment causes an increase of critical speeds, the effect known as gyroscopic stiffening (see the next section). However, this effect has importance only with short and thick,
gyroscopic rotors for which the stability threshold at the first flexural critical speed is too high
to be reached and it ceases to be the limiting factor for the rotational speed.
5.5
Rigid-rotor dynamics
In the previous sections dynamical stability limit and critical speeds of, generally compliant,
rotors was studied. Although, mathematically, no apparent distinction between rigid and flexural vibration modes was made, based on qualitative analysis in Section 5.3 practical dynamical limit of electrical machines was connected with the first flexural critical speed. Since for
soft-mounted rotors this critical speed is influenced solely by the rotor slenderness, it can be
correlated with the flexural critical speeds and viewed as the limiting machine parameter when
it comes to rotordynamical stability.
In this section, dynamic behavior of rigid, thus, infinitely stiff rotors will be analyzed with
particular attention to two important phenomena: (rigid) critical speeds and unbalance response.
The Jeffcott rotor model will be extended in this section to include rotor moments of inertia
and, in turn, gyroscopic effect. Additionally, in order to analyze unbalance response, static and
couple unbalance will be introduced into the rotor model.
In Figure 5.9 a rigid rotor in compliant bearings is presented. Total radial stiffness of the
bearings is k. The rotor spins with an angular speed in axial direction: Ωz = Ω. The rotor is
characterized with its mass m, polar and transversal moment of inertia J p and Jt , respectively,
and its length and position of the center of mass. It is assumed that geometrical axis of the rotor
5.5. Rigid-rotor dynamics
91
χ
Ω
k/4
inertia axis
try a
geome
k/4
xis
Jp, Jt
z
y
O
k/4
C ε
P
x
k/4
Figure 5.9: Rigid rotor in compliant bearings
does not coincide with its inertia axis. The distance between center of geometry C and center
of mass P represents rotor static unbalance while the angle χ represents the couple unbalance inclination of the inertia axis with respect to geometrical axis.
Figure 5.10 shows arrangement of relevant reference frames and variables which will be
used in the modeling. The frame xyz is the stationary reference frame and ξηζ is the frame
connected to the rotor with its geometry axis coinciding with the ζ-axis. The static unbalance is
assumed to lead the couple unbalance for the angle α (the angle χ resides in a plane parallel to
the ξζ plane). Movements of the rotor will be represented with respect to the stationary frame,
namely, using displacements of the rotor’s center C: xC = x, yC = y, zC = z, and inclinations of
the geometry axis: ϕ x , ϕy .
In order to study dynamical behavior of the rigid rotor, the static and couple unbalances
will be represented as external forces which spin with the same angular speed as the rotor and
which act on an ideal, thus, perfectly-balanced rotor. Equations of movements of an undamped,
92
Chapter 5. Rotordynamical aspects of high-speed electrical machines
ζ’
χ
ζ
z’
φx
φy
y’
ε α
ξ’
P
ξ
C
axis
Ωt
x’
ine r
tia
geometr y
y
axis
z
O
η
x
Figure 5.10: Reference frames used in the modeling
spinning rotor in the stationary reference frame are expressed as3 [86]:
m ẍ + k11 x + k12 ϕy = mεΩ2 cos (Ωt + α) ,
mÿ + k11 y − k12 ϕ x = mεΩ2 sin (Ωt + α)
,
2
Jt ϕ̈ x + J p Ωϕ̇y − k12 y + k22 ϕ x = −χΩ Jt − J p sin (Ωt) ,
Jt ϕ̈y − J p Ωϕ̇ x + k12 x + k22 ϕy = χΩ2 Jt − J p cos (Ωt) .
(5.45)
In equations (5.45) the terms which include elements of the stiffness matrix k11 , k12 = k21
and k22 represent forces that are linked to elastic reaction of the bearings. For the example of the
rigid rotor from Figure 5.9 these stiffness terms can be calculated by inverting the compliance
matrix B:
"
#
"
#−1
k11 k12
b11 b12
−1
K=
=B =
.
(5.46)
k12 k22
b12 b22
Elements of the compliance matrix can be found using adequate theoretical (or practical)
experiments: b11 is the displacement of the center of mass P caused by a unit force applied
at the same point, b12 is the rotation caused by the same force (or, alternatively, displacement
cause by a unit moment) and b22 is the rotation of the rotor in the center of mass caused by a
3
Equations of this form can also represent rotors with a massless compliant shaft supported by (generally)
compliant bearings.
5.5. Rigid-rotor dynamics
93
unit moment at the same point. For the example from Figure 5.9 the compliance matrix is as
follows:


a 2 ! 2 a
a 
 2
1−2 +2
1 − 2 

l
l
kl
l 
(5.47)
B =  k

a
4
2


1−2
kl
l
kl2
and the stiffness matrix is calculated accordingly. In (5.47) l represents axial distance between
the bearings and a defines the position of the center of mass (refer to Figure 5.11).
l
a
P
Figure 5.11: Position of the center of mass in a cross-section of the rotor
After introducing complex rotor displacement and inclination:
r = x + iy,
ϕ = ϕy − iϕ x ,
(5.48)
the equations 5.45 can be presented in a matrix form:
#" #
#" # "
"
#" #
"
#
iα
mεe
m 0
0 0
k11 k12
r̈
ṙ
r
2
eiΩt
− iΩ
+
=Ω
0 Jt
0 Jp
k12 k22
ϕ̈
ϕ̇
ϕ
χ Jt − J p
| {z }
| {z }
| {z } |{z}
|
{z
}
"
M
G
K
q
(5.49)
f
The damping can also be included into the matrix equation (5.49) in the following way
[86, 139]:
(5.50)
M q̈ + Cn + Cr − iΩG q̇ + K − iΩCr q = Ω2 f eiΩt
where matrices of non-rotating and rotating damping Cn and Cr , respectively, have a form similar to the form of the stiffness matrix [86]. Introducing the damping into the problem, however,
would bring about a great complexity into this study [128, 139]. Moreover, damping barely
influences positions of the critical speeds (as seen also in Section 5.3) while qualitative insight
into the unbalance response can be well drawn using the adopted model of the undamped rotor.
94
Chapter 5. Rotordynamical aspects of high-speed electrical machines
5.5.1 Rigid critical speeds
In order to find critical speeds of the undamped system from Figure 5.9, the case of rotor
free whirling will be studied. One should, therefore, look for homogeneous solution of the
differential equation (5.49), thus:
"
#" # "
#" #
"
#" #
0 0
k11 k12
r̈
ṙ
r
m 0
− iΩ
+
=0
(5.51)
0 Jp
ϕ̈
ϕ̇
k12 k22
ϕ
0 Jt
| {z }
| {z } |{z}
| {z }
G
M
K
q
After introducing a solution of type:
st
q = q0 e =
"
#
r0
e st
ϕ0
the following system of equations is obtained:
ms2 + k11 r0 + k12 ϕ0 = 0,
k12 r0 + Jt s2 − isJ p Ω + k22 ϕ0 = 0.
The system 5.53 has non-trivial solutions if and only if:
"
#
ms2 + k11
k12
det
= 0.
k12
Jt s2 − isJ p Ω + k22
(5.52)
(5.53)
(5.54)
Since there is no damping in the system, the solutions of the equation (5.54) are purely
imaginary. After substitution of s = iω, the equation (5.54) becomes:
2
ω4 mJt − ω3 ΩmJ p − ω2 (k11 Jt + mk22 ) + ωΩk11 J p + k11 k22 − k12
= 0,
(5.55)
and it gives an implicit correlation between the natural frequencies of the system and the rotational speed based on which a Campbell diagram can be plotted.
To find values of critical speeds, ω = Ω = Ωcr is substituted into (5.55) and that leads to the
following biquadratic equation over the critical speed:
h i
2
m J p − Jt Ω4cr − k11 J p − Jt − mk22 Ω2cr − k11 k22 − k12
=0
(5.56)
Positive solutions of Eq. (5.56) are given by:
v
u
qh u
u
i2
u
t 2
k11 J p − Jt − mk22 + 4 k11 k22 − k12
m J p − Jt
k11 J p − Jt − mk22 ±
(5.57)
Ωcr =
2m J p − Jt
It can be noticed that two real solutions for Ωcr exist only if transversal moment of inertia
is higher than the polar moment (Jt > J p ); in the opposite case (J p > Jt ) only one real solution
5.5. Rigid-rotor dynamics
95
exist. Rotors whose polar moment of inertia is higher than the transversal are referred to as
short or disc-shaped rotors and they have one critical speed less than long (slender) rotors.
Using (5.46) and (5.47) it can be shown that the stiffness matrix of the rigid rotor from Fig.
5.9 can be expressed in the following form:
#
" 0
0
k11 lk12
,
(5.58)
K=k
0
0
lk12
l2 k22
0
0
0
where terms k11
, k12
and k22
depend solely on the ratio a/l (Fig. 5.11).
After replacing the stiffness terms of (5.57) with the adequate elements of the stiffness matrix (5.58) values of critical speeds can be expressed in the following non-dimensional form:
v
t
q
2
0
0
0 0
02
0
0
Γ∗ k11
− k22
+ 4 k11
k22 − k12
Γ∗ k11
− k22
±
Γ∗
∗
,
(5.59)
Ωcr =
2Γ∗
where Ω∗cr is ratio between the critical rotational speed and the critical speed of the undamped
Jeffcott rotor:
Ωcr
Ω∗cr = √
,
(5.60)
k/m
and Γ∗ is a non-dimensional equivalent of the gyroscopic moment [86]:
J p − Jt
Γ∗ =
.
ml2
(5.61)
The critical speeds calculated from Eq. (5.59) are plotted in Figure 5.12. Again, it is evident
that short rotors are associated with only one critical speed. Cylindrical and conical movement
of rotors are generally coupled (unless a/l = 0.5) and so are the corresponding vibrations. Still,
vibrations of the rotor are usually dominantly cylindrical or conical and the lower critical speed
is usually considered cylindrical while the higher one (if existent) is considered conical.
In the special case of a symmetrical rotor whose center of mass is in the middle of the shaft
(a/l = 0.5) critical speeds take on the following values:
r
r
k11
k
Ωcr,cyl =
=
,
(5.62)
m
m
s
s
l
k22
k
=
.
(5.63)
Ωcr,con = 2
Jt − J p
Jt − J p
Naturally, conical critical speed is imaginary in the case of short rotors.
Gyroscopic effect influences rise of critical speeds; the phenomenon which is sometimes
referred to as ”gyroscopic stiffening” although no real stiffening is taking place [129, 140]. It
should be pointed out that the non-dimensional gyroscopic moment of most of realistic rigid
rotors lie withing a small range around zero. Namely, if the gyroscopic moment is very small
96
Chapter 5. Rotordynamical aspects of high-speed electrical machines
2
a/l = 0.1
a/l = 0.2
a/l = 0.3
a/l = 0.4
a/l = 0.5
1.8
1.6
1.4
Ω*cr
1.2
1
0.8
0.6
long rotors
0.4
−1
short rotors
−0.5
0
0.5
1
*
Γ
Figure 5.12: Critical speeds of rigid rotors with respect to the gyroscopic moment
(Jt >> J p ), the rotor is too slender to be considered rigid; conversely, if J p >> Jt the rotor is
practically a thin disc and again cannot be regarded as rigid [86].
Finally, a special case that should be considered is the case of rotors with Jt ≈ J p . This rotor
type is highly unsuitable for high speed rotation. Namely, if a Campbell diagram is plotted
based on Eq. (5.55) it can be shown [86] that, at high speeds, the resonant frequency associated
with the forward conical mode asymptotically approaches the line ω = ΩJ p /Jt . If J p = Jt the
rotor would always be in the vicinity of the critical speed (although never at the critical speed)
and the vibrations would steadily grow as the rotor speed increases. The rotor would not have
a preferred position and would not be able to benefit from self-centering at high speeds (see the
next subsection).
5.5.2 Unbalance response
Dynamic response of the rotor to the unbalance is found as the particular solution of the differential equation (5.49). The equation is presented here again in a compact form:
M q̈ − iΩGq̇ + Kq = Ω
2
"
#
iα
mεe eiΩt
χ Jt − J p
(5.64)
The solution can be found in the following form:
q = q0 e
iΩt
=
"
#
r0
eiΩt ,
ϕ0
(5.65)
5.5. Rigid-rotor dynamics
97
which, after substitution into (5.64), leads to the expressions for amplitudes of rotor’s cylindrical
and conical whirl:
"
#
i−1 " mεeiα #
h
r0
2
2
.
(5.66)
= Ω −Ω M − G + K
q0 =
ϕ0
χ Jt − J p
It can be inferred from (5.66) that the amplitudes r0 and ϕ0 are, in general, complex numbers since the static and couple unbalance do not lie in the same plane. Yet, in other to make
important conclusions about the unbalance response, it is sufficient to study the case in which
phase angle of the static unbalance vanishes (α = 0). In that case, the last equations yields the
following expression:


χ Jt − J p k12 
Ω2  mε k22 − Jt − Jp Ω2 −
(5.67)
q0 =
 ,

∆ 
−mεk12 + χ Jt − J p k11 − mΩ2
where:
2
∆ = m Jt − J p Ω4 − k11 Jt − J p + mk22 Ω2 + k11 k22 − k12
.
(5.68)
If the elements of the stiffness matrix are represented according to (5.58), considering (5.61),
the amplitudes of the unbalance response can be expressed using non-dimensional quantities:
 r 
r0  0 
" 0
#
∗ 0
∗2
 ε 
k22 + Γ∗ Ω∗2
Γ k12
,

χ=0 χl ε=0  = Ω
(5.69)
0
0
ϕ0  ϕ0 
−k12
−Γ∗ k11
− Ω∗2
∆∗


ε/l
χ
χ=0
ε=0
where the rotational speed is rated by the critical speed of the Jeffcott rotor:
Ω∗ = √
Ω
,
k/m
(5.70)
and non-dimensional determinant is defined as:
∆∗ =
∆
2
0 0
0
0 ∗2
Ω + k11
k22 − k12
.
= −Γ∗ Ω∗4 + Γ∗ k11
− k22
2
2
kl
(5.71)
For high-speed rotors it is particularly important to analyze the rotor behavior in the supercritical regime. As the speed increases, the influence of cross-coupling terms of Eq. (5.69)
becomes immaterial. Here, a special case of the symmetrical rotor will be examined whose
center of mass is in the middle of the shaft or, equivalently, whose cylindrical and conical
movements are decoupled (k12 = 0). The rotor of the test machine belongs to this category.
0
0
From (5.46), (5.47) and (5.58) for the rigid symmetrical rotor it holds k11
= 1 and k22
= 1/4.
Hence, expressions for amplitude response to static and couple unbalance reduce to:
Ω∗2
r0
=
,
ε
1 − Ω∗2
(5.72)
98
Chapter 5. Rotordynamical aspects of high-speed electrical machines
ϕ0
=−
χ
Ω∗2
1
+ ∗
4Γ
Ω∗2
.
(5.73)
Amplitudes of the unbalance response are plotted against rotational speed in Figures 5.13
and 5.14 according to expressions (5.72) and (5.73). In supercritical regions amplitudes of the
rotor’s translational and conical movements tend to −ε and −χ respectively. Physically, it means
that the rotor changes its axis of rotation from geometrical to inertia axis as the speed advances
in the supercritical regime. This phenomenon is referred to as self-centering and it is a result of
the fact that in the supercritical region inertia forces overcome bearing forces causing the rotor
to spin around its principal axis of inertia.
5
4
3
2
1
r0/ε
0
−1
−2
−3
−4
−5
0
0.5
1
1.5
2
2.5
3
*
Ω
Figure 5.13: Amplitude of response to static unbalance of symmetrical rotors
In the example of the response to static unbalance and also to couple unbalance in long
rotors the change of preferred axis of rotation happens very fast as the rotor crosses the critical
speed: the whirl of the geometric center in the stationary reference frame inverts its phase with
respect to the excitation. In the example of short rotors, on the other hand, the alignment of the
rotor to couple unbalance proceeds gradually.
Self-centering is a greatly exploited phenomenon in design of supercritical rotating machinery [86, 138, 140]. Instead of making a geometrically perfect rotor in a fixed support, it is often
more reasonable to arrange a flexible support for the rotor and let the rotor inertia offset the
influence of the unbalance at high speeds.
Distinguishing case is, however, rotors with Γ∗ = 0 (J p = Jt , see Figure 5.14). Such a rotor
type cannot benefit from self-alignment and is generally avoided when it comes to supercritical
rotors.
5.6. Conclusions
99
5
Γ = −0.2
Γ = −0.1
Γ=0
Γ = 0.1
Γ = 0.2
4
3
2
1
φ0/χ
0
Ω*=Ω*
−3
−4
−5
0
0.5
1
cr,con2
−2
Ω*=Ω*
cr,con1
−1
1.5
2
2.5
3
*
Ω
Figure 5.14: Amplitude of response to couple unbalance of symmetrical rotors
It is important to mention that the model of an undamped rotor is not adequate to describe
rotor behavior in the vicinity of a critical speed. Unbalance response at a critical speed highly
depends on damping which attenuates the vibrations and the response is also influenced by the
rate of the speed increase through the critical speed [129].
5.6
Conclusions
This chapter studies vibrations of a rotor of a high-speed electrical machine, their cause and
influence on the rotor-bearings system. Two types of vibrations are distinguished: resonant
(forced) and self-excited; the latter vibrations draw particular concern since they are unstable
and hazardous. The speed at which self-excited vibration occur is referred to as instability
threshold speed and it represents rotordynamical limit for the rotor rotational speed.
Rotors of high-speed machines are usually soft-mounted meaning that their stiffness is much
higher than the stiffness of their bearings. For such rotors it is possible to recognize two modes
of vibrations based on whether the rotor is deformed while vibrating: rigid-body and flexural
(bending) vibration modes. Using a simple, Jeffcott rotor model, the chapter shows that rotation
can become unstable in the supercritical regime of a certain vibration mode if rotating damping
of the rotor-bearing system affects that mode.
Rotors are usually stable in a supercritical speed range which corresponds to rigid-body
vibrational modes and today’s high-speed rotors regularly operate in that speed range. An
exception are rotors supported by bearings with a non-synchronous damping such as lubricated
journal bearings and hydrodynamic bearings.
100
Chapter 5. Rotordynamical aspects of high-speed electrical machines
Rotors which possess some internal damping can easily become unstable in a supercritical
range corresponding to flexural modes. Rotors of electrical machines are receptive to eddycurrents, always comprised of fitted elements and often contain materials, such as composites,
with significant material damping. Therefore, these rotors are prone to be unstable in flexural
supercritical regimes. The first flexural critical speed practically represents the rotordynamical
speed limit of an electrical machine.
Critical speeds of a high-speed rotor (represented as a cylindrical Timoshenko beam) are
correlated with the rotor slenderness and bearing stiffness. It was shown that flexural critical
speeds of a rotor-bearings system depend solely on the rotor slenderness. The results of the
analytical modeling comply with FEM calculations available in literature [137]. Flexural critical
speeds are far above the operating speed range of the test motor and the calculations of these
speeds are irrelevant for the test machine.
The rotor slenderness is the most critical factor for stable rotation of a rotor in a high-speed
electrical machine. Hence, a maximum slenderness of the rotor can be evaluated so that the
rotor operates at speeds below the first flexural critical speed.
Last section of the chapter analyzes behavior of rigid rotors i.e. rotors which operate well
below the speeds in which flexural vibrations occur. For the analysis purpose, the Jeffcott rotor
model is extended to include gyroscopic effect.
Rotors whose polar moment of inertia is higher than the transversal are referred to as short or
disc-shaped rotors and they have one critical speed less than long (slender) rotors. Gyroscopic
effect influences rise of critical speeds; the phenomenon which is sometimes referred to as
gyroscopic stiffening.
At supercritical regime, rigid rotors change their axis of rotation from geometrical to inertia
axis as the speed advances. This phenomenon is referred to as self-centering and it can be exploited in design of supercritical rotating machinery: instead of making a geometrically perfect
rotor in a fixed support, it is often more reasonable to arrange a flexible support for the rotor
and let the rotor inertia offset the influence of the unbalance at high speeds.
The dynamic of a rotor-bearings system and its important aspects - stability of rotation and
critical speeds - presented in this chapter are fairly known and well-researched in the field of
rotordynamics; most models can be found in textbooks. The importance and main contribution
of this chapter is that it highlights the effects of those phenomena relevant to rotational stability
of electrical machines.
Chapter 6
Bearings for high-speed machines
6.1
Introduction
Advance in electrical machines is characterized, among other, by pursuit of ever higher rotational speeds, particularly regarding turbomachinery and machining spindles [27]. Demands
for increasing tangential speeds and, often, positioning accuracy could not be achieved by using
standard journal or ball bearings. Therefore, the requirement for speed of the machinery brought
about improvements in bearing technology. Enhancements have been either sought within the
standard (mechanical) bearing technology that would become suitable for required high speeds
or alternatives have been looked for in the form of contactless bearings.
Speed capability of rotational bearings is usually represented in terms of the DN number
which represents product of the bearing inner diameter in mm and rotational speed of the rotor
in rpm. Standard ball bearings have the DN number below 500.000 [24]. On the other hand, DN
values that are currently needed in certain high-tech applications such as turbines for aircraft
or high-speed spindles amount to a few millions [141]. Under such conditions mechanical
bearings are subjected to great centrifugal loading and high temperatures which both reduce
bearings load capacity and life-time [141].
Bearing type and stiffness have strong influence on rotational accuracy which is of particular
importance in machining spindles. The accuracy can be improved, though, by performing rotor
balancing, the process that may be time-consuming and costly, or by exploiting self-centering
of high-speed rotors if low-stiffness bearings are used (see the previous chapter).
Capability of bearings to support required speed and accuracy of the machine along with
their cost and durability play the most important role in choice of bearings for high-speed machines.
Goal of this chapter is to study different types of bearings with respect to their applicability
for high-speed rotation. The chapter is primarily concerned with the bearings that have been
the most promising for high speed: (hybrid) ball bearings, externally pressurized (or static) air
bearings and magnetic bearings. A general overview and comparisons will be given in the end
of the chapter.
101
102
6.2
Chapter 6. Bearings for high-speed machines
Mechanical bearings
Conventional, mechanical bearings are still predominantly used among commercial machines.
The greatest advantages of mechanical bearings are robustness and low cost. However, they
have limited operational temperature making it the main restriction for the rotational speed.
Furthermore, great increase in speed tremendously intensifies wear due to friction which, in
turn, shortens life-time of the bearings.
Particular problem with high-speed rotation is centrifugal loading which rises with square of
the speed and lowers load capacity of the bearings [141]. In order to mitigate this problem, rotor
balancing is often required. For rigid rotors, though, balancing in two planes to compensate for
static and couple unbalance is usually sufficient [142].
Although a lot of effort has been invested, particularly in academia, in developing and promoting contactless (or frictionless) bearings, mechanical bearings are still widely used for highspeed applications due to their simplicity and low cost. For majority of commercial applications
conventional bearings will suffice making fluid (air) and magnetic bearings reserved for special applications with stringent requirements [143]. At the same time, considerable research is
aimed at improving quality of mechanical bearings [144].
For high-speed applications ball-bearings are mostly used; lubricated journal bearings are
generally avoided because of their issues with instability [130, 133]. Today, specially-designed
ball bearings can be found for rotational speeds up to 100.000 rpm and values of the DN number
as high as 1.5 million [141].
A distinctive example of a very high-speed commercial machine running on a ball bearings
is Dyson’s 104.000 rpm DC brushless motor (DDM V2 [145]) which has a simple and small,
balanced permanent magnet rotor without retainment.
Ball bearings are commonly found in dental spindles where they can support speeds up to
500.000 rpm [30]. When it comes to machining spindles, air and magnetic bearings replace
mechanical bearings in precision machinery to an ever greater extent.
Zwyssig et al. from ETH Zurich have published successful designs of several high-speed
motors that run on ball bearings up to speed of 1 million rpm [30]. The DN value of the
used bearings is around 1.6 million [146], however, no data on durability of those bearings is
available.
In high-speed turbomachinery a trend of transition from mechanical to air bearings is noticeable [27]. Ball bearings limited the speed of a gas turbine designed at University of Leuven
to 160.000 rpm [147]. The authors suggested that special ball bearings would allow speeds over
200.000 rpm; however, air bearings would be needed for the desired speed of 420.000 rpm.
Improvements of ball bearings consist in reduction of size of the balls and transition to
ceramic materials to accommodate high speeds [144]. These changes have been followed by
novel designs of geometry, curvature and seals [144]. Oil lubrication is replaced with a small
amount of grease and inclusion of the seals allow that bearings are sealed and greased for
life [144,148]. Combination of ceramic balls and grease lubrication allow very long operational
time of the bearings with the DN index well above 1.5 million [149]. Such bearings can also be
used at cryogenic temperatures at which standard lubricants would solidify [117].
Exceptionally good result have been obtained with silicon-nitride hybrid bearings which
6.3. Air (fluid) bearings
103
Figure 6.1: SKF ceramic hybrid bearings
represent bearings consisting of ceramic, silicon-nitride balls and steel rings. Very small wear
in these bearings is result of the fact that ”silicon nitride rubbing on M-50 steel offers friction
and wear characteristics as good as those of silicon nitride rubbing on itself” [150], which results
in a very long bearing life. Due to the comparably low density of ceramics, silicon-nitride balls
reduce centrifugal loading on the outer, steel, raceway of the bearings [141]. Ceramics, in
general, are also less sensitive to lubricant type and lubricant contamination [141].
Authors of [141] report hybrid silicon nitride bearings with DN values up to 4 millions. According to [151] machining spindles can be supported with high stiffness and rotational accuracy
by using ceramic hybrid bearings. ”In general, compared to all-steel bearings, rolling contact
ceramic bearings can more easily meet requirements of higher efficiency, higher speed, higher
reliability, higher accuracy, greater stiffness, longer life, marginal lubrication, lower friction,
corrosion resistance and non-conductivity and with less maintenance action.” [141]
6.3
Air (fluid) bearings
Air bearings use a thin fluid film or pressurized air to support the rotor. Fluid (air) forms a layer
between the bearing housing and the shaft (”gas lubrication”) transferring, at the same time, the
force which supports the shaft. This principle of operation has been established for more than
50 years; nowadays, technology of air bearings is quite mature making them frequently applied
in a large number of applications.
Because of very small air-gaps (in order of tens of µm) air bearings require flawless geometry with very low tolerances. Extremely high motion accuracy can be, therefore, achieved
with high precision and repeatability [4] which makes air bearings very attractive for precision
machinery.
According to how the pressure in fluid is generated, two types of air bearings are recognized:
static and dynamic. Dynamic bearings use relative motion between the moving body (shaft)
104
Chapter 6. Bearings for high-speed machines
and bearing housing to generate hydrodynamic pressure in the fluid. Liquid (oil) is used as the
fluid rather then gas (air) because of higher viscosity; these bearings are usually referred to as
hydrodynamic. Hydrodynamic bearings are essentially frictionless only at high speeds - at zero
and low speeds the shaft is in contact with the housing (or rather, with the foil journal lining in
foil bearings).
Static air bearings use externally pressurized air to levitate the body (rotor) at all possible
speeds including zero speed. From the pressure supply the air is directed through either small
holes in the bearing (orifices) or porous material.
oil inlet
w
journal
(rotor)
bearing
w
w
w
At rest
oil flow
minimum film
thickness
Slow rotation
w
resultant oil
film force
Fast rotation
Figure 6.2: Hydrodynamic bearing
Figure 6.3: Aerostatic bearing
For high speed machinery static air bearings are often used. Although having higher load
capacity, hydrodynamic bearings can operate efficiently only in a narrow speed range [7] and
have larger thermal deformation than aerostatic bearings [4]. Dynamic bearings are preferably
used in heavy-load precision machines while static air bearings are used in small and medium,
high-speed and/or precision machines [4].
Very high speeds have been achieved in laboratory environments using air bearings in the
example of gas turbines. A micromachined air turbine supported with air bearings was designed
at MIT and operated at 1.3 million rpm with tangential speed of 300 m/s [152].
High-speed machining spindles with aerostatic bearings have µm motion accuracy at speeds
up to 200.000 rpm, such as the micromilling spindle designed at Brunel University [7].
6.3. Air (fluid) bearings
105
Air bearing can have very high static and dynamic stiffness, comparable with ball bearings [153]. On the other hand, static air bearings generally have low load capacity. Another disadvantage of air bearings lies in the need for preloading for certain geometries and directions.
While journal bearings are usually self-preloaded, trust bearings need preloading to increase the
stiffness and maintain a fixed air-gap.
The most salient problem with air bearings, though, is stability of rotation. Hydrodynamic
journal bearings have a self-excited instability, which is commonly called whirl instability, and
it is correlated with behavior of the fluid film in the bearings. The fluid in the journal bearings
moves at an average speed which is close to half of tangential speed of the rotor [86, 133, 135].
This movement is reflected in subsynchronous whirling of the rotor with frequency ω = Ω/2
which is superimposed to other whirling motions. At speed near twice of the first critical speed
the whirl frequency reaches the value of the first resonant frequency of the system. The whirl is
then replaced by oil whip - a particularly destructive ”lateral forward precessional subharmonic
vibration of the rotor” [130]. Independently of the further increase of speed oil whip maintains
the constant frequency (see the cascade plot of the rotor vibrations in Figure 6.4).
f res
Figure 6.4: Cascade plot of rotor vibrations measured in an oil-lubricated bearing, taken
from [131]: cascade lines represent amplitudes of the rotor whirl at given rotational speeds
and frequencies. The plot shows transitions of self-excited vibrations from whirl to whip and
again from whip to a second-mode whirl.
Partially, phenomenon of whirl instability was explained in Section 5.3.2 using the Jeffcott
rotor model: the fluid in the bearing provides a subsynchronously rotating damping which
rotates with a speed close to Ω/2. However, the behavior is far more complex; that model cannot
account for the phenomenon of oil whip. As Muszynska points out [130]: ”Researchers and
106
Chapter 6. Bearings for high-speed machines
engineers do not always agree upon the physical description of the shaft/bearing or shaft/seal
solid/fluid interaction dynamic phenomena. The complexity of these phenomena and the long
list of factors affecting them make the picture tremendously obscure.”
A heuristic explanation for the whirl instability is given by Crandall in [133] and analytical
models and explanations can be found in papers by Muszynska [130, 131].
Aerostatic bearings have, in general, higher dynamic stiffness and lower viscous drag than
hydrodynamic bearings, both properties having positive influence on their stability [154]. Nevertheless, this type of bearings is not immune to the whirl instability [154] since aerodynamic
forces completely overwhelm aerostatic forces at very high speeds [7].
In order to avoid instability the system is designed with constraint of having a maximum
operating speed of the rotor well below the frequency of whirl instability, i.e. double value of
the first critical speed [155]. At the same time, bearings can be optimized to maximally increase
the threshold of instability [156, 157] and/or innovative bearing designs can be used [158].
Unfortunately, air bearings can suffer from other forms of, mainly pneumatic, instabilities,
prevention of which must be considered in the design (see [159, 160]. Emergence of pneumatic
instabilities depends on mechanical design of the bearings which is beyond the scope of this
chapter.
6.4
Active magnetic bearings
Active magnetic bearings (AMB) use force of an electromagnet for levitation of a body (rotor).
An active magnetic bearing consists of an electromagnet and a power amplifier that supplies
the electromagnet with currents (Fig. 6.5). The bearing needs continuous current input and a
closed-loop controller since active magnetic bearings are unstable in open loop. A magnetically
levitated body may have certain degrees of freedom (DOF) passively controlled, thus, by means
of permanent magnets only; however, for stable levitation at least one DOF must be actively
controlled, according to the theorem of Earnshaw [161].
As a rule, rotors are supported by pairs of AMB in differential mode that enables linear
control of the magnetic force and, in turn, rotor position. Quite often, a permanent magnet is
utilized to create the bias flux in the bearings and current of the coils is then used to control
the rotor position. The most suitable for support of high speed rotors is the homopolar bearing
structure in which the rotor (ideally) does not experience changes of the bias flux while rotating
(Fig. 6.6).
For a long time magnetic bearings were considered too complex and expensive to be commercially appealing [163]. However, in course of the last two decades AMB have proven their
effectiveness and reliability and shown great potential for an increasing number of applications [162, 163]. AMB have several important advantages with respect to other types of bearings. They provide purely contact- and frictionless operation with no contamination and no
need for lubricants.
Beside frictionless operation, AMB offer possibility of creating practically arbitrary damping or stiffness, the property which can be greatly utilized to adjust the dynamical properties of
the system as the rotational speed changes.
6.4. Active magnetic bearings
107
Figure 6.5: Basic setup of an active magnetic bearing carrying a rotor [162]
ro
cont
l flu x
N
b
lu x
i as f
S
N
S
Figure 6.6: A homopolar magnetic bearing designed by Kimman [140]
All these properties make AMB particularly attractive for high speed applications. They
do not suffer from instabilities connected with air or lubricant flow as fluid bearings do. Their
stiffness is generally lower than the stiffness of air bearings; however, having larger clearances,
AMB rotors can exploit self-centering and operate stably at supercritical speeds.
Still, magnetic bearings are rather complex electromechanical systems which include sensors, power electronics/amplifiers, advanced digital controllers and electromagnets that, in the
end, affect the price and reliability of the bearings. Their stiffness declines with the operating
frequency [162] and they have limited control bandwidth [164] due to presence of coils and sensors and limited processing power of controllers. Gyroscopic effect and/or non-linearities can
destabilize the rotor-bearings system at high speeds [165] and more complex control methods
are needed such as cross-feedback [165] or non-linear feedback control [161].
The benefit of adaptable damping and stiffness has virtually no influence on high-frequency
108
Chapter 6. Bearings for high-speed machines
whirl and flexural vibrations. AMB are usually located at points where flexural vibrations
are very small. Furthermore, the frequency range in which AMB can counteract forces that
cause displacement of the rotor is limited by the control loop bandwidth [166]. Frequencies of
flexural vibrations of high-speed rotors are usually far above the bandwidth of the bearings so
the vibrations cannot be suppressed.
Nevertheless, AMB technology is advancing and the bearings appear in many precision and
high speed applications [167]. Improvements have been made in design of specialized power
amplifiers and their integration with AMB [163], control algorithms [168] and design of AMB
for high speed [164].
There have been an increasing number of AMB in turbomachinery for which low maintainance and long life-time of AMB under sever conditions is a great advantage [169–171]. In
general, turbomachinery is currently the main commercial application of AMB [167] among
rotating machines although improving reliability of AMB is a key requirement for making them
widely embraced by this industry.
AMB are considered less mature technology for high-speed production machinery in comparison with air bearings; however, high-speed machining spindles are a promising application
of magnetic bearings. Considerable research on AMB for machining applications has been
conducted at University of Virginia [172] and at TU Delft [164]. Hybrid solutions have also
emerged in high-speed machining spindles where AMB have been combined with other types
of bearings and actuators to take advantage of certain virtues of AMB such as accurate positioning during rotation [15].
Today, AMB are still mainly used in rather special applications where bearings either need
to work under special conditions (vacuum, harsh environment [167]) or must conform to very
strict limitations (no lubrication, no contamination [143]). An example of such a special application is 60.000 rpm, 4.1 kW magnetically levitated flywheel which NASA intends to use as a
replacement for batteries on the space stations [173].
Finally, AMB find use in many important additional tasks such as force monitoring [18],
system identification [138], machine state diagnosis [174], monitoring and suppressing vibration levels [166], etc.
6.5
Conclusions
In this chapter different bearing types have been studied with respect to their applicability in
high-speed machinery.
Ball bearings are still a predominant bearing type among commercial high-speed machines
due to their robustness and low costs. Different types of ball bearings are regularly present in
machines with rotational speeds up to 100.000 rpm. Hybrid ball bearings with silicon-nitride
balls represent the most promising type of mechanical bearings in terms of not only speed
performance but also reliability, stiffness, life-time and low contamination.
Nevertheless, ball bearings have apparent limitations for extremely high rotational and tangential speeds and alternatives are sought in different types of contactless bearings.
Air bearings are well-known and widely utilized technology for frictionless support of ro-
6.5. Conclusions
109
Table 6.1: Advantages and disadvantages of different bearing types
ball bearings
+ low cost
+ robust
air bearings
+ no friction/wear
+ ultra precision
+ high stiffness
+ low maintenance
- temperature limited
- wear
- need lubrication
- instabilities
- low load capacity
- require flawless geometry
- need preloading
- susceptible to dirt,
temperature
- need maintenance
active magnetic bearings
+ no friction/wear
+ zero contamination
+ no maintenance
+ adjustable force,
damping
+ positioning during rotation
+ modular design
+ operate in harsh settings
+ facilitate monitoring
- complex
- expensive
- low reliability
- require control
- require constant power
supply, sensors, electronics
tors. Air bearings offer high stiffness and great accuracy and repeatability of rotation which
makes them ideal for precision machinery. Aerostatic bearings are particularly suitable for lowand middle volume high-speed machines. However, a salient problem of air bearings are instabilities which limit rotational speed and complicate design of the bearings.
Active magnetic bearings seem to have the greatest potential for high-speed applications:
perfect conditions for rotation (no friction, wear, lubricants), modularity in design and possibility of regulating stiffness and damping of the bearings due to their active nature. On the
other hand, they are not seldom considered too complex, costly and unreliable. Nowadays,
they are mostly used in special applications with harsh working conditions or very stringent requirements for maintenance. However, the advance of active magnetic bearings in the last two
decades has made them an increasingly mature and auspicious technology that can be applied
to a wide range of high-speed rotating machines.
An overview of advantages and drawbacks of studied bearing types is given in Table 6.1.
110
Chapter 6. Bearings for high-speed machines
Chapter 7
Design of the high-speed-spindle motor
7.1
Introduction
As explained in Sections 1.2 and 1.3 of the thesis Introduction, the PhD project started within
the Dutch Microfactory framework with the goal of development of a built-in electrical spindle
drive which would facilitate high rotational speed of the spindle and accurate micro-milling.
This chapter will present the design of the spindle motor, from a conceptual design to electromagnetic and structural optimization of the motor.
In the time when the project started, some experience with machining spindles had already
been gained within the Microfactory project and important limitations of high-speed spindles
had been foreseen. This greatly influenced the design of the spindle drive. Namely, not only
that the design aimed at improvements in electromagnetic actuation of the existing high-speed
spindles, but it also looked for radically new spindle concepts which would overcome speed
limits of existing spindles. As a result, a concept of a frictionless short-rotor spindle was born.
Influence of the analyses and models presented in Chapters 2-6 on the spindle-motor design
was twofold. Definition of speed limits of permanent magnet machines greatly affected initial,
conceptual design of the new spindle drive. At the same time, the developed models formed an
analytical basis for the motor design and optimization.
The Chapter will start with presenting development of new spindle concepts in the Microfactory project group - Section 7.2. Thereafter, geometric and electromagnetic design of the
spindle motor will be explained in Section 7.3 and 7.4, then evaluated using FEM in Section 7.5
and, lastly, optimization of the rotor retaining sleeve is presented in Section 7.6.
7.2
New spindle concepts
In the Microfactory project a small spindle in active magnetic bearings (AMB) was realized by
M. Kimman [164, 175]. A relatively slender rotor is supported by two radial bearings and an
axial bearing which exerts force over a small rotor disc (Figure 7.1). Rotation of the spindle is
controlled by a commercially available PM motor. The maximum attained speed of the spindle
is 150.000 rpm. It was shown that miniaturization of a spindle has positive effects on actuation
111
112
Chapter 7. Design of the high-speed-spindle motor
and cutting force monitoring [18, 164].
Figure 7.1: Section view of the AMB spindle, taken from [164]
The first flexural critical speed (approximately 180000 rpm) of the spindle has represented
an obstacle of utilizing the motor up to its maximum speed constrained by the structural limit of
the PM rotor (250000 rpm). Additionally, relatively high negative stiffness of the motor caused
runout higher than it was expected [164].
It was apparent that, for reaching higher rotational speeds, the spindle length would need to
be significantly decreased. However, that was hardly achievable with the same motor-bearings
configuration. Hence, Kimman et al. [140] proposed a whole new approach for high-speed
spindles: to use a short (disc-shaped) rotor suspended in AMB. The inspiration was found in
an idea of 3DOF combined axial and radial magnetic bearings envisaged in [176]. Kimman
et al. [140] proposed using such bearings for supporting 5DOF of a disc thus benefiting from
reducing rotor tilting and higher resonance frequencies (Figure 7.2). In essence, it would mean
that all the bearings from the original setup - Figure 7.1 - would be grouped around the axialbearing disc and that would, in turn, drastically reduce the spindle volume.
Advantages of using short rotor follow also from analyses of Chapter 5. A rigid short rotor
has one critical speed less than its long/slender counterpart and it may also benefit from increase
of critical speeds as a result of gyroscopic stiffening (see Figure 5.12 in Section 5.5). Still,
the greatest advantage of such a rotor clearly comes from the increase of flexural resonance
frequencies as a result of the great reduction of the rotor slenderness. In that way, stability
threshold at the first flexural critical speed is too high to be reached and it ceases to be the
limiting factor for the rotational speed.
7.2. New spindle concepts
113
N S
Fe
S N
Fe
r
roto
Fe
S N
b
ng
eari
N S
coils
Figure 7.2: 5DOF magnetic bearings, as proposed by M. Kimman [32]
The next step in the development of a short-rotor spindle was to integrate an electrical motor into the 5DOF bearings from Figure 7.2. It was apparent that close spatial integration of
AMB and motor was needed. Several concepts for spindle motor were considered including
bearingless motors [177, 178] and axial-flux machines. It was a standard, radial-flux PM motor,
however, that offered possibility of motor-bearings integration without merging their function
and without changing the original AMB concept. The conceptual design of the new spindle is
depicted in Figures 7.3 and 7.4.
control flux
axial bearing
control flux
radial bearing
SN
NS
NS
S
NS
N
NS
S
SN
stator
conductors
bias flux from
magnets
motor
stator core
NS
rotor
magnet ring,
mounted on the rotor
Figure 7.3: Short-rotor spindle - concept
In proposed magnetic bearings (Fig. 7.3) control actions in axial and radial directions are
decoupled: permanent magnets are utilized both to create bias flux and to separate control fluxes
114
Chapter 7. Design of the high-speed-spindle motor
stator core
housing of the bearings
magnet ring
toroidally-wound conductors
short rotor
Figure 7.4: Short-rotor spindle - a cross-section
in different directions. Armature flux of the PM motor passes through the rotor disc, without
interfering with the bearings’ control fluxes. In that way, performance of the motor and the
AMB are not simultaneously compromised in the design stage and their control is virtually
independent during operation. A slotless stator of the motor is fitted into the axial bearings to
facilitate good thermal contact with the environment. Toroidal windings represent a sensible
solution for a motor with such a small stack length in comparison with the radius.
Finally, it was realized, after work on the spindle design started, that it would be convenient
to have a setup with a passive bearing type for testing the electrical drive separately, without
any coupling with magnetic bearings. For this purpose, aerostatic bearings were chosen for a
number of reasons. They are relatively simple, frictionless and have high motion accuracy and
stiffness. Simple orifices were chosen as air restrictors since they have not been associated with
any pneumatic instability [179]. The concept of the air-bearings setup is presented in Figures
7.5 and 7.6.
The aerostatic bearings for supporting the short rotor were developed by P. Tsigourakos
[180] under supervision of M. Kimman and the thesis’ author. Two journal air bearings support
the rotor in radial directions. The axial/thrust bearing is preloaded by a permanent magnet to
provide high bearing stiffness. The PM motor shares again the same housing with the bearings
- it was planned that, for the motor, magnetic- and air bearings from Figures 7.3 and 7.5 are
interchangeable.
The air-bearings setup was used as a test setup for the motor in this thesis. More details on
the setup are given in Section 9.2.
7.3
Conceptual design of the motor
At the time of the motor design many of the application requirements were still missing.
Namely, the project group at TU Delft lacked practical experience with micromilling and information on torque requirements for high-speed micromilling was rather scarce. Additionally, the
7.3. Conceptual design of the motor
115
journal bearing 1
air supply
rotor
air supply
journal bearing 2
preload magnet
(axially magnetized)
air supply
thrust bearing
Figure 7.5: Aerostatic bearings for the short rotor - concept [180]
work on the motor design started long before models presented in this thesis were developed.
Therefore, most of the actual requirements were imposed by the designers themselves and not
directly by the prospective application. However, the goal of the design was to offer a good
proof of concept rather than a definite solution for micromilling spindles. Taking into consideration available data and technological limits in an academic environment, the designers looked
for sensible and adequate requirements for the new spindle and the spindle motor in particular.
They are presented as follows:
1. The motor needs to fit both proposed bearings and its stator should have a good thermal
contact with the (bearings’) housing.
2. It should be possible for both setups to be manufactured using technology of a standard/university workshop and of-the-shelf components. In other words, miniaturization
of the rotor is limited by technological capabilities in the university environment. This
is particularly important in the example of AMB whose extreme miniaturization would
require utilization of advanced processing techniques and/or components (miniature sensors, coils, etc.).
3. In light of the previous requirement, it was concluded that a rotational speed of 200.000
rpm of a disc-shaped rotor in proposed bearings would be technically achievable (see also
the next section). Therefore, that speed was set as the speed requirement for the motor.
4. No data on required torque for micromilling were available. However, available estimations of cutting forces for milling with sub-millimeter tools suggested rather small
load forces, in order of a few newtons and lower [164, 181]. It was, thus, expected that
the load torque would be significantly smaller then the drag resulting from windage and
116
Chapter 7. Design of the high-speed-spindle motor
Figure 7.6: Setup with the aerostatic bearings - exploded view [180]
eddy-currents. Taking into account worst predictions of air-friction loss (Figure 3.21 in
Subsection 3.6.3) it was concluded that 200 W of power would be certainly sufficient for
operation at the required maximum speed.
5. For a high-frequency-operating machine which would be completely enclosed in magnetic bearings, minimization of frequency-dependant losses was extremely important.
Besides, due to inability to reliably model air-friction loss and losses in the permanent
magnet (Subsections 3.6.3 and 3.6.4), thermal model was not developed, thus, mitigation
of losses was inevitable for a safe design. Therefore, minimum loss (= maximum efficiency) was taken as a decisive criterion for both component choice and electromagnetic
design.
6. As pointed out in Section 3.5, negative motor stiffness, which results from the unbalanced
magnetic force, must be, at least, an order of magnitude lower than the stiffness of the
radial bearings and the unbalanced force must be lower than the bearing force capacity.
The stiffness limit was critical for the case of AMB (estimated in order of 105 N/m) while
the force limit was critical for the aerostatic bearings.
7. Structural robustness of the rotor was an equally important requirement for the design.
Proper retaining of the magnet was crucial, particularly for a high-speed rotor with a high
7.3. Conceptual design of the motor
117
diameter to length ratio.
All these requirements affected the design of the motor whose conceptual design is depicted
in drawings in Figures 7.7 The motor concept is explained in the rest of this section.
b)
a)
c)
Figure 7.7: Spindle motor drawing: a) top view, b) lateral view and c) a lateral cross-section
A laminated, slotless stator core has protrusions corresponding to the axial direction for
good thermal contact with the housing. Advantages of slotless machines for very-high-speed
operation were discussed in Chapter 2. Exclusion of stator teeth removes slotting-effect harmonics from the PM field while, at the same time, reduces impact of armature-field harmonics
in the PM rotor. As a whole, a slotless motor is prone to be more efficient and less susceptible
to rotor overheating than its slotted counterpart.
Conductors are wound toroidally over the core, thus, dispensing with, for this case, unavoidably long end windings. The windings are non-overlapping, i.e. each phase winding is
uniformly wound over two 60◦ -sections of the stator circumference.
A plastic-bonded magnet of the injection molded type is applied onto the incised part of
the rotor disc (see Figure 7.10). An incision is previously made in the shaft for a better fit
of the magnet. The injection molded magnet contains very small magnet particles that are
blended with a plastic binder - PPS. After applying this mixture onto the shaft at a very high
temperature, the magnet will apply a stress on the shaft during the cooling in the mould. Such
a magnet is very resistive to eddy-current losses. At the same time, low remanent flux density
of the magnet, as a result of the plastic binder overtaking a great portion of the magnet volume,
118
Chapter 7. Design of the high-speed-spindle motor
is quite adequate for a very-high-speed machine (see Section 2.4). The magnet is diametrically
magnetized providing a perfectly sinusoidal back emf.
Finally, in order to sustain very strong centrifugal force at high speeds and ensure transfer of
torque in the rotor throughout the whole range of speeds, the magnet needs to be contained in a
non-magnetic enclosure/sleeve. A non-conductive sleeve has been conceived as a combination
of glass and carbon fiber: details on the sleeve design are presented in Section 7.6.
glass fiber
magnet ring
carbon fiber
Figure 7.8: Rotor design - concept
In the rest of the section the materials used for the motor parts - stator core, conductors,
permanent magnet and sleeve - are discussed.
7.3.1 Stator core
Resistivity to induced losses was a decisive factor in choosing the stator core material. Amorphous iron has excellent figures of losses and very favorable magnetic properties [146, 182] in
comparison with other, more common core materials. However, amorphous iron is brittle and
available only in form of ring tape cores and could not be processed to fit into the bearings.
Different types of silicon steel were considered. It is well known [183] that alloys with high,
approximately 6.5%, content of Si show excellent properties in terms of minimum induced loss
and maximum permeability. However, steels with such a high Si content used to suffer from
hardness and brittleness.
Workability of 6.5% Si-steel has lately been improved [183] and such steel has been used in
this project. The loss vs. frequency characteristics of this steel are compared with amorphous
iron and with high-frequency 0.12 mm Si-steel with a lower percentage of Si. The characteristics
are presented in a graph (Figure 7.9). Loss figures for Si-steel with 6.5% Si are comparable to
amorphous iron and significantly better than high-quality standard Si-steel.
7.3. Conceptual design of the motor
119
B= 1T
400
2
P [ W/kg ]
300
200
1
3c
100
3b
3a
0
0
1000
2000
3000
4000
5000
f [ Hz ]
Figure 7.9: Loss power density of different lamination materials vs. frequency at 1 T flux
density: 1 - 6.5% Si-steel [184]; 2 - 0.12 mm Si-steel laminations [185]; 3 - different samples
of amorphous iron [182]
7.3.2 Conductors
A drawback of the slotless type of PM machines is deteriorated cooling of the conductors in the
air-gap. Namely, the absence of stator teeth makes transfer of heat from the air-gap conductors
towards the stator core difficult [79]. In order to alleviate this effect, a special type of selfbonded wires is used. The wire [186] contains an adhesive surface varnish that interconnects
wires after curing, enhancing the thermal conduction.
7.3.3 Permanent magnet
As already mentioned, an injection-molded plastic-bonded magnet has been used in the rotor.
This type of magnet was chosen for the following reasons:
• High resistivity: the rotor is barely cooled and preventing magnets from overheating is
essential;
• Shape flexibility: injection-molded magnet could be directly applied onto the shaft in a
ring form, keeping the high-speed rotor relatively well-balanced;
• A relatively small remanent field of the magnet (0.5 T) is adequate for the application.
120
Chapter 7. Design of the high-speed-spindle motor
However, very low yield stress, both compressive and tensile, is a great drawback of this
magnet type and a lot of attention was given to structural designing of the rotor. High temperature polymer PPS, favorable for injection molding, has been chosen as the plastic binder.
This binder type provides comparably the best mechanical characteristics among plastic-bonded
magnet types [187, 188].
7.3.4 Magnet retaining sleeve
Carbon fibres were chosen as the enclosure material for their light weight and exceptional
strength. The main drawback of using carbon fibres is their negligible thermal expansion in
contrast with that of permanent magnets; therefore, additional stress on the enclosure is expected at elevated rotor temperatures. Furthermore, while able to withstand extreme tensions,
carbon fibres are rather sensitive to bending [75] and they need to be protected from being cut
at edges of neighbouring materials.
7.4
Motor optimization
7.4.1 Rotor shaft design
Design of the rotor iron shaft was an initial step of the motor design since the rotor dimensions
were also decisive for the bearing design. Length and diameter of the narrow shaft parts were
determined first as a part of the conceptual design of the magnetic bearings. The length of 5
mm was estimated as a minimum required by the bearings to be available at both sides of the
disc so that the bearings can be manufactured in the university workshop and can provide the
desired stiffness. Consequently, the length of the narrow shaft part was set at 5.5 mm.
Diameter and length of the rotor disc were determined so that the polar inertia of the whole
rotor is considerably higher than the transversal inertia so that the motor can benefit from rotor
self-aligning. Another considered factor was eventual force density of the rotor - with the
resulting area of the incised part of the disc a prospective required force density of the motor
for the given power demand (200 W at 200.000 rpm) should not be too high. Force density
estimation gives:
P
Fd <
(7.1)
2πωr2 l s
where r is the disc radius.
With the dimensions given in Figure 7.10 the ratio between polar and transversal moments
of inertia of the final rotor was going to be higher than 1.3. Additionally, with the given rotor
dimensions, force density of the motor at the maximum power was going to be lower than 1.7
kN/m2 , which is more than acceptable when compared with corresponding values for commercially available PM machines.
7.4. Motor optimization
121
8
5.5
23
8
Figure 7.10: Rotor shaft with dimensions in mm
7.4.2 Electromagnetic optimization of the motor geometry
The motor optimization was carried out in two steps. In the first step, which will be explained
here, the machine dimensions and number of stator-conductor turns were determined for a minimum total loss in the stator. In the second optimization step, reported in the next subsection,
frequency-dependent copper losses are considered and the conductors are optimized.
As it was pointed out in Section 7.3, loss minimization was taken as the ultimate criterion for
the motor design. However, it was not overall motor efficiency that was sought in the machine
design, since that energy efficiency of the spindle drive was never a goal per se. The main
intention of the design was, actually, to mitigate the overheating of the motor that would result
from frequency-dependent losses. This criterion was partly taken as a safe option for the motor
design since a good thermal model of the motor was not available.
In the rotor, the permanent magnet is the most susceptible to heat that would result from the
losses. However, protection of the magnet from overheating was greatly taken care of in the
conceptual design: plastic-bonded magnet material is highly resistive to eddy-current loss and
slotless machine design additionally alleviates the influence of armature-field harmonics on the
rotor.
It was also assumed that air friction does not have a great influence on the temperature inside
the high-speed rotor for two reasons: turbulent air flow at high speeds greatly improves removal
of the resulting heat at the rotor surface and, also, carbon fibers serve as a good thermal insulator
for the rotor due to their very low thermal conductivity.
On the other hand, air friction greatly influences temperature in the air gap and, consequently, at the inner stator surface and air-gap conductors. However, at the time of the design,
the author had neither a good representation of the air-friction loss nor a model of convection
between the disc and stator. Although air friction has a huge impact on the overall drag torque
122
Chapter 7. Design of the high-speed-spindle motor
(Subsection 3.6.3), how the air friction at the disc surface correlates with temperatures in the
rotor and stator has remained undetermined.
Eventually, it was concluded that a relatively large friction loss is inevitable for the discshaped rotor so the air-friction loss was removed from the design focus. It was clear that without a thermal model of the machine the optimization goal of minimizing machine operating
temperature was hardly going to be reached. A level of arbitrariness in the design was present
and accepted from the beginning.
Minimization of the total electromagnetic loss in the stator was taken as the optimization
goal. The objective function to be minimized was the following:
Ploss = PFe + PCu,DC ,
(7.2)
where the stator core loss PFe is estimated using equation (3.95) from Subsection 3.6.1.
The copper loss was simply estimated as DC conduction loss for the given conductor dimensions, thus, excluding losses due to proximity- and skin-effect:
PCu,DC = I 2
lCu
,
σCu ACu
(7.3)
Aw
k f ill kCu ,
(7.4)
6N
where Aw is the cross-section of the winding region, k f ill is the estimated fill factor of rounded
conductors and kCu is roughly estimated copper cross-section within a single conductor. Total
conductor length lCu is given by equation (3.111).
Three motor parameters were taken as the design variables: thickness of the magnet ring
lm , the number of turns per phase 2N and the thickness of the stator core ly . The optimization
is performed using MATLAB Optimization Toolbox. Optimization constraints were defined as
follows:
ACu =
1. In order to obtain sinusoidal currents for the motor, it was planned to design a PWM
inverter drive for the motor. To support the desired fundamental frequency of 3.33 kHz
(200.000 rpm), very fast switching, in order of 100 kHz, was expected. The machine
voltage is limited by the capability of a PWM inverter to perform hard-switching at such
a high frequency. It was therefore decided to limit the voltage of the machine to a value for
which fast-switching MOSFETs are readily available. Amplitude of the no-load voltage
at the maximum speed was set at 50 V:
ê ( f = 3.33 kHz) = 50 V
(7.5)
For the optimization purpose the expression (3.47) for the no-load voltage was used.1
1
In the actual design, a mistake was made when the expression (3.47) was applied. Instead of setting 50 V
as the no-load voltage amplitude, the desired rms value ( √502 ≈ 36 V) is set for the voltage amplitude. Therefore,
the actual winding flux linkage and power output of the test motor is lower than expected. It is important to
notice, though, that the power requirement for the motor was set somewhat arbitrarily, based only on the projected
frequency-dependent losses, and it is not crucial for the thesis.
7.4. Motor optimization
123
With the given power reference (200 W) and the set value for the no-load voltage, the
value of the stator current was practically determined:
I=
P
3Ek
(7.6)
where k is a coefficient that accounts for the excess voltage over the phase impedance,
arbitrarily set at 0.9.
2. The electromagnetic design was carried out before the modeling and design of the rotor
carbon-fiber enclosure. The speed of sound in air was taken as the absolute limit for the
rotor tip tangential speed which led to the maximum value for the outer radius of the rotor
disc:
re,max ≤ 16.5 mm.
(7.7)
According to analysis presented in Section 4.2 on the stress in a rotating cylinder, the
maximum rotation-influenced tangential stress in the carbon fiber sleeve with this outer
radius would be smaller than:
σc f,re f,max < ρc f v2t = 190 MPa,
(7.8)
which is still around 10% of the maximum tensile stress in the fibers. This margin of
stress consequently allows for additional stress in the fibers as a result of their fitting over
the magnet.
3. In order to limit leakage of the magnet flux in the axial direction, the magnet thickness is
set to comprise, at least, a half of the effective air gap:
le + g + lw ≤ lm
(7.9)
where le is the sleeve thickness, g is the mechanical air-gap and lw is the thickness of the
winding area.
4. The mechanical air gap is set at 0.5 mm
5. Eventually, an additional constraint of equalizing copper and iron losses is added, yielding
somewhat balanced distribution of loss in the stator:
PFe = PCu,DC .
(7.10)
6. Naturally, saturation limit for the stator core is imposed. This limit hardly affected the
optimization since saturation flux density in the core is, for the given magnet material,
too high to be reached.
Optimization results are given in Table 7.1 and presented in drawings in Figure 7.11.
124
Chapter 7. Design of the high-speed-spindle motor
Table 7.1: Optimization results: parameters of the motor geometry
parameter
symbol value
stator stack length
ls
6 mm
magnet inner radius
rm
10.5 mm
magnet thickness
lm
4 mm
maximum sleeve thickness
le
2 mm
winding area thickness
lw
1.5 mm
stator yoke thickness
ly
8.8 mm
number of phase turns
2N
88 (80∗ )
phase resistance
R
0.35 Ω∗∗
phase inductance
L
43 µH∗∗
flux linkage amplitude
ψmax
1.72 mWb∗∗
rotor moment of inertia
J
3.6 mg·m2
∗
value actually implemented in the motor
model estimations based on the actually implemented number of turns
∗∗
7.4.3 Optimization of conductors
Optimization in the previous subsection determined, practically, magnetic field in the machine,
the available space for conductors and the number of conductor turns. In the second optimization step optimal conductors that fit the available winding space were selected so as to minimize
copper loss for the estimated phase currents and the maximum current (electrical) frequency
(3.33 kHz). The optimization procedure is quite similar to the one presented in [146].
In order to alleviate skin- and proximity effect losses in a transformer or high-speed motor,
parallel/stranded conductors are used. For very-high-frequency applications litz-wires are used
which represent bundles of individually enameled strands. The bundle of strands is usually
finally coated with cotton or silk. In this optimization step number and diameter of the strands
within a phase conductor of the designed machine was defined.
In Subsection 3.6.2 losses in copper of a high-speed slotless machine were modeled. It was
shown that skin effect has a very little influence on the losses in copper for reasonable speeds
and conductor diameters of electrical machines. Therefore, only the DC conduction loss and
eddy-current loss in the air-gap conductors were considered for the conductor optimization. Analytical expressions (3.104) and (3.106) for these two loss components can be, after considering
equations (3.107) and (3.110), rewritten here in the following form:
PCu,DC =
k1
,
2
ndCu
4
PCu,eddy = k2 ndCu
,
(7.11)
(7.12)
where n and dCu are number and (copper) diameter of parallel conductors or strands within
a single phase conductor and k1 and k2 are coefficients that depend on parameters which are
determined in the previous subsection.
7.4. Motor optimization
125
Figure 7.11: Stator and rotor geometry with dimensions in mm
Hence, the total copper loss can be represented as:
PCu = PCu,DC + PCu,eddy =
k1
4
+ k2 ndCu
.
2
ndCu
(7.13)
For a given number of strands, an optimal strand diameter can be found so that the copper
loss power is minimal (see Figure 7.12):
∂PCu k1
3
dCu,opt = −2 3
+ 4k2 ndCu,opt
= 0.
∂dCu
ndCu,opt
Thus, from (7.14) the optimal strand diameter yields:
r
1 6 k1
dCu,opt = √3
,
n 2k2
(7.14)
(7.15)
where coefficients k1 and k2 stem directly from expressions for copper losses: (3.104), (3.110)
and (3.106), (3.107), respectively:
k1 =
4I 2
(2l s + (r so − r s ) π) 6N,
πσCu
(7.16)
π B̂2m ω2 σCu
l s 6N.
(7.17)
128
The expression (7.15) shows that the optimal diameter of copper within a strand decreases
proportionally to the third root of the number of strands. After substituting expression (7.15)
into equation (7.13), it can be seen that the same proportionality is valid also for the total copper
loss:
1
(7.18)
PCu ∼ √3 .
n
k2 =
126
Chapter 7. Design of the high-speed-spindle motor
5
2 strands
4 strands
8 strands
4.5
4
P Cu [W]
3.5
3
2.5
2
1.5
1
0.5
1
2
3
4
d [mm]
x 10
st
−4
Figure 7.12: Total copper loss vs. strand diameter for different numbers of strands; dashed lines
indicate optimal strand diameter
Total copper loss
Optimal strand diameter
0.45
2.6
0.35
d
st,opt
P Cu[W]
[mm]
2.2
1.8
0.25
1.4
0.15
2
4
6
n
st
8
10
2
4
6
8
10
n
st
Figure 7.13: Optimal conductor-strand diameter and corresponding total copper loss vs. strand
number
The trends of decrease of the optimal strand diameter and loss power for the machine optimized in the previous subsection is shown in Figure 7.13.
7.4. Motor optimization
127
With increasing number of strands and accordingly adjusted copper diameter, the minimum
copper loss decreases. However, for an assigned number of turns this increase in number of
strands is limited by available conductor area in the air gap. It is, thus, necessary to estimate the
area comprised by the optimized conductors.
For a calculated copper diameter within a strand the total strand diameter including insulation build is found first. For wires in the range of 30÷60 AWG2 (0.254÷0.009 mm) the total
strand diameter can be estimated as [189]:
!
dCu,opt β
d st,opt = dr α
.
(7.19)
dr
Coefficients in eq. (7.19) were adjusted to fit manufacturer’s data [190]: α = 1.12, β = 0.97
and dr = 0.079 mm.
For thicker wires, the following approximation is used [189]:
d st,opt = dCu,opt + 10
X−AWG
44.6
· Y,
(7.20)
where X = 0.518 and Y = 0.0254 mm.
If several strands are used within a conductor, they are usually twisted together to form a
bundle. It was assumed that the conductors are bundled if at least 3 strands are used. Diameter
of a bundle was calculated in the following way:
√
(7.21)
dbundle = p f d st,opt n,
where the packing factor p f has the following values [190]:

1,
n = 1, 2






1.25,
3
≤ n ≤ 12



1.26, 13 ≤ n ≤ 18
pf = 




1.27, 19 ≤ n ≤ 25



 1.28, n > 25
(7.22)
The thickness of coating adds between 0.03 and 0.04 mm to the diameter of the bundle [189]:
dbundle,tot = dbundle + 0.035 mm.
(7.23)
Finally the area required for the optimized conductors can be calculated as:
Aw,opt
2
1 dbundle,tot
=
π · 6N,
k f ill
4
(7.24)
where k f ill = 0.4 is the fill factor of rounded conductors.
It is evident from equations (7.15) and (7.21) that the increase of the bundle diameter with
respect to number of strands is faster than corresponding decrease of diameter of individual
strands. Therefore, the total area Aw,opt will also increase with the number of strands. The
limitation on the number of strands is eventually imposed by the available conductor area:
h
i
Aw,opt < Aw = π r2s − (r s − lw )2
(7.25)
128
Chapter 7. Design of the high-speed-spindle motor
350
2
A w,opt [mm ]
250
Aw
150
50
1
2
3
4
5
6
7
8
9
10
n
st
Figure 7.14: Required area for optimized windings; available area is marked in red
Total required cross-sectional area of optimized conductors for the designed machine is
plotted with respect to number of strands and compared with available conductor area in Figure
7.14. For the designed motor only two parallel optimized conductors could fit the available
area. Due to rather week magnet in the rotor, eddy-current losses in the air-gap conductors are
not as pronounced as they would be in a slotless machine with high-energy magnets and that
was decisive for such a low number of optimal parallel conductors.
Optimal diameter of the conductors was found to be 0.335 mm and the closest available
diameter of Thermibond wires - 0.314 mm - was used. Since only two parallel wires were
needed, they were simply wound in parallel without additional twisting and coating.
7.5
FEM design evaluation
The motor electromagnetic design presented in the previous sections and some of the underlying
models will be evaluated using finite-element simulations in this section. The FE models were
built using Cedrat Flux2D/3DTM software.
7.5.1 2D FEM: motor parameters
The developed analytical models of the motor field have already been checked and confirmed
by 2D FEM in Subsections 3.3.1 and 3.3.2. However, that FE model adopted the same assump2
American Wire Gauge - a standard set of wire conductor sizes in the US; the AWG number can be converted
36−AWG
into mm using the approximate equation: d [mm] = 0.127 × 92 39
7.5. FEM design evaluation
129
stator iron
air gap
sleeve
magnet
rotor iron
winding areas
(coils)
Figure 7.15: Geometry and mesh of the 2D FE model of the designed motor
Figure 7.16: Field lines and magnitude of the armature field modeled by 2D FEM
tions as the 2D analytical model and, consequently, emulated the same physical model of the
machine. The model does not take into account actual, toroidal distribution of the conductors
130
Chapter 7. Design of the high-speed-spindle motor
since it assumes zero magnetic field outside the stator iron. External leakage of the armature
field is, thus, neglected and the model geometry is equivalent to the geometry of a standard
slotless machine with inserted air-gap conductors (see Figure 3.4 in Chapter 3).
Here, results of a more-adequate 2D FEM of the test machine are presented in which the field
is allowed to cross outside the motor external boundary. Calculations of those electromagnetic
parameters of the motor that are directly derived from the magnetic field are compared to the
results of the analytical model. The FEM geometry and mesh is given in Figure 7.15 and results
are shown in Table 7.2.
Table 7.2: 2D FEM vs. analytical model results
parameter
analytical
amplitude of flux density in the windings [mT]
209.5
PM-flux linkage [mWb]
1.7285
phase inductance [µH]
43.427
rotor loss [mW]
/
2D FEM
209.249
1.7264
64.02
<1
While the analytical representation of the permanent-magnet field is maintained - estimations of the PM flux density and the winding linkage of the PM flux of both models match very
well - the total phase inductance of the motor seems to be significantly higher than analytically
predicted. According to the FE model the leakage of the armature field outside the stator iron is
significant (see Figure 7.16) which is not surprising taking into account large effective air gap
of the machine. This result shows inadequacy of the developed analytical model to represent
the armature field in a toroidally-wound machine.
Rotor iron loss was modeled in a transient FE simulation for maximum expected current (2
A) and rotational frequency. The result partly confirms suitability of the magnetostatic approach
in motor modeling since the losses in the rotor iron are extremely small. However, it was not
possible to model eddy-current loss in the plastic-bonded magnet because no suitable means to
represent the magnet electrical conductivity has been found. Still, the physical nature of such
a magnet - permanent-magnet powder in a plastic binder - suggests that those losses are quite
small.
7.5.2 3D FEM: no-load voltage and phase inductance
To make a more realistic estimation of the motor parameters, a 3D FE model of the motor was
created. Due to a rather short rotor and a very large gap between the magnet surface and stator
inner surface, it was anticipated that leakage of the flux of the permanent magnet in the axial
direction can have a significant influence on the motor performance. Furthermore, the effect
of the external leakage of the armature field on the phase inductance was expected to be much
higher than the 2D FE model suggested due to, beside else, a very short stator stack length.
From the simulation PM-flux linkage and, accordingly, no-load voltage appear to be noticeably lower (10 percent less) than predicted by the 2D analytical model which is a result of
the axial flux leakage. The leakage is primarily enhanced by 1 mm extrusions of the rotor iron
7.5. FEM design evaluation
131
axial flux
leakage
magnet
rotor iron
Figure 7.17: Zoom-view of the 3D FEM geometry: axial leakage of the PM flux
Figure 7.18: One-eight of the motor 3D FEM and coil conductors of a single phase
which support axial sides of the magnet - their presence partly short-circuits the field of the
magnet (see Figure 7.17).
132
Chapter 7. Design of the high-speed-spindle motor
Motor phase inductance is simulated in 3D FEM using non-meshed rectangular coils (it was
not possible to create toroidal coils in the used software) - Figure 7.18. The analytical model
(and, also, 2D FEM) greatly underestimates the phase inductance: the analytical prediction is
almost an order of magnitude lower than the prediction of the 3D FEM.
Table 7.3: PM-flux linkage and phase inductance according to analytical and FE models
analytical 2D FEM 3D FEM
PM-flux linkage [mWb]
1.7285
1.7264
1.5528
phase inductance [µH]
43.4
64.0
350.0
7.5.3 2D FEM: conductor eddy-current loss
It would be extremely difficult to directly simulate losses in the conductors in FEM because of
their disproportionally small size with respect to dimensions of the motor parts. Eddy-current
loss in the conductor strands is, therefore, indirectly simulated. Firstly, the field in the center
of the windings is calculated and simulated using 2D FEM. As already mentioned, 2D FEM
upholds analytical field calculations and also affirms very low influence of the field of conductors on the total field in the winding area. It is also reasonable to assume that field of the
eddy-currents do not have a noticeable influence on the PM field.
Figure 7.19: A round conductor in a solenoid - finite element model
Based on the calculation of the field in the winding, an abstract FE model of a thin solenoid
is developed (Figure 7.19) which creates the same flux density in the solenoid center as it is
in the middle of the motor windings. Subsequently, a round conductor (strand) is added in the
middle of the solenoid and, in a transient analysis, the losses in the conductor are calculated for
different field frequencies and conductor diameters. Finally, the total eddy-current loss in the
7.6. Design of the rotor retaining sleeve
Eddy−current loss for d
Cu
133
=d
Eddy−current loss for d
Cu,opt
2
Cu
=2d
Cu,opt
35
analytical
2D FEM
analytical
2D FEM
1.8
30
1.6
25
20
Cu,eddy
[W]
1.2
1
P
P
Cu,eddy
[W]
1.4
15
0.8
0.6
10
0.4
5
0.2
0
1000
2000
3000
4000
Frequency [Hz]
5000
6000
0
1000
2000
3000
4000
5000
6000
Frequency [Hz]
Figure 7.20: Eddy-current loss in the air-gap conductors: analytical and FE model predictions
stator conductors is calculated by scaling the simulated loss with the total number of conductors
(strands).
The eddy-current loss in the conductors calculated in this way perfectly matches analytical
model predictions (Figure 7.20). The FE simulations also confirm great importance of proper
sizing of the air-gap conductors because eddy-current losses in conductors increase proportionally to the fourth power of the strand diameter (see equation (7.12)).
7.6
Design of the rotor retaining sleeve
Motor design represented in Section 7.4 determined electromagnetic properties of the motor.
It is the magnet retaining sleeve that is supposed to ensure rotor structural integrity. Proper
enclosing of the magnet became crucial for success of the design, particularly for a high-speed
rotor with a high diameter to length ratio.
Theoretical framework for structural modeling and design of a PM rotor is presented in
Chapter 4 of this thesis. Particularly, in Section 4.4 an approach for the retaining sleeve optimization is developed; the approach was followed in the design of the enclosure of the test rotor
and will be presented in this section.
7.6.1 Material considerations: PPS-bonded NdFeB magnet
Injection-moulded bonded NdFeB magnets offer important advantages such as high resistivity
to magnetically induced losses and shape flexibility, while their remanent field is sufficient for
134
Chapter 7. Design of the high-speed-spindle motor
the application. Still, the possibility of plastic deformation and creep of the polymer material
has remained a great concern.
High temperature polymer polyphenylene-sulfide (PPS), favourable for injection molding
[187], was chosen as the plastic binder. Although PPS allows somewhat smaller volume fraction
of permanent magnet material than some other polymer binders (e.g. nylon) and, accordingly,
poorer magnetic properties [188], PPS-based bonded magnets have superior mechanical properties with respect to other bonded magnet types in terms of strength and processability [187,188].
PPS also exhibits, for a polymer, high melting temperature (280◦ C) and PPS-based magnets can
maintain structural integrity even at the temperature of 180◦ C [191]. Yet, at about 85◦ C glass
transition occurs in PPS [187] causing softening of the amorphous phase and a great reduction
in strength [192]. Therefore, that temperature is taken as the rotor temperature limit.
Table 7.4: Properties of the bonded magnet and its constituents
material property
PPS
NdFeB
bond∗
density [g/cm3 ]
1.35-1.7 7.35-7.6
4.81
compressive modulus [GPa] 2.8-3
140-170
31.72
compressive strength [MPa] 125-185 800-1100 120
Poisson’s ratio
0.36-0.4 0.24
0.33
CTE [µm/m/◦ C]
40-55
4-8
4.73
∗
1
values used in modeling
measured; 2 taken from [191]; 3 calculated using the rules of mixtures
Scarce of data on mechanical properties of bonded magnets hampered the work on modeling of stress in the rotor. A valuable study on this issue was performed by M.G. Garell et
al. [191] who offered data on tensile modulus and strength. However, the magnet endures compression from the enclosure and the properties must have been assessed for that condition. In
the modeling it was assumed that the compressive modulus is equal to the tensile modulus. Further, compression was assumed to be limited by the compressive strength of PPS and the value
of 120 MPa was imposed on the maximum stress in the magnet. Finally, Poisson’s ratio and
coefficient of thermal expansion (CTE) were estimated from the rules of mixtures [125]:
νbond = (1 − p) · ν pps + p · νNdFeB ,
αbond =
(1 − p) · α pps E pps + p · αNdFeB E NdFeB
,
(1 − p) · E pps + p · E NdFeB
(7.26)
(7.27)
where p (= 0.6) is the volume fraction of permanent magnet material in the bond and E pps and
E NdFeB are tensile module of PPS and NdFeB magnet respectively.
7.6.2 Material considerations: carbon-fiber composite
Carbon fibers exhibit strong orthotropic nature since the fibers’ mechanical properties differ
considerably in different directions and strongly depend on the cross-sectional type [193]. This
makes precise analytical modelling burdensome. Analyses in Chapter 4, however, showed that
7.6. Design of the rotor retaining sleeve
135
mechanical stress in the rotor can be quite adequately represented by assuming the composite
properties in the direction of fibers (Table 7.5: || fibers) be valid in all directions.
Table 7.5: Properties of the carbon fibres
material property
k fibers ⊥ fibers
density [g/cm3 ]
1.55
elastic modulus [GPa]
186.2
9.5
∗
maximum stress [MPa]
1400
100
Poisson’s ratio
ν12 = 0.3
CTE [µm/m/◦ C]
-1
54
∗
the values assumed a great safety margin
7.6.3 Sleeve optimization
Goal of the optimization of the rotor enclosure/sleeve was maximizing the rotational speed that
the rotor could withstand. Maximum permissible thickness of the sleeve that could be fit in the
air-gap is 2 mm. Temperature of the PPS glass-transition - 85◦ C, ∆T max ≈ 60◦ C - is set as the
rotor temperature limit.
As pointed out is Chapter 4, the thickness of the enclosure and interference fit must be
adequately chosen so that the contact pressure between the rotor parts is maintained and that
the equivalent stress in each rotor part must be below the material ultimate stress (equations
(4.41) and (4.42)) throughout the whole speed range at the operating temperature.
Critical stresses in the rotor - are radial (contact) stress at the magnet-iron boundary and
tangential stress (tension) at the enclosure inner surface - take on the following forms (equations
(4.43) and (4.44)):
σmr,crit = F1 (re ) · Ω2 − G1 (re ) · δ − H1 (re ) · ∆T,
(7.28)
σeθ,crit = F2 (re ) · Ω2 + G2 (re ) · δ + H2 (re ) · ∆T,
(7.29)
where F1,2 , G1,2 and H1,2 are positive functions of the sleeve outer radius re .
The optimal fit and theoretical maximum speed are obtained as functions of the sleeve radius
re from the following system of equations:
σmr,crit Ω = Ωmax , δ = δopt , ∆T = 0 = 0
.
(7.30)
σeθ,crit Ω = Ωmax , δ = δopt , ∆T = 60◦C = σemax
For the optimal interference fit, both critical stresses are reached at a same maximum rotational speed which is, in turn, determined by the enclosure radius.
As it was expected, the highest speed could be reached with the highest allowable enclosure
thickness of 2 mm. The optimal interference fit is obtained from (7.30) to be 110 µm.
Finally, in this design, additional limit is forced on the fitting by the magnet compression
strength. Compliance of the calculated interference fit to that limit needs to be ensured. Namely,
because of the polymer used in the magnet, compression strength of the bonded magnet is much
136
Chapter 7. Design of the high-speed-spindle motor
lower than that of sintered magnets (Table 7.4), maximum amount of a press-fit that can be
applied on the magnet is expected to be rather low.
To account for this, maximum permissible fit δmax is calculated from the equation:
σmeq,crit (Ω = 0, ∆T = 0) = σmmax ,
(7.31)
where σmeq,crit is von Mises stress at the magnet inner surface:
σmeq,crit =
q
m2
m
m
σm2
r,crit + σθ,crit − σr,crit σθ,crit
(7.32)
and σmmax is assumed compression strength of the magnet.
From (7.31) maximum interference fit is calculated to be 95 µm and, since it is smaller than
the previously calculated optimal fit, it is set as the definite value of the interference fit between
the magnet and enclosure.
7.6.4 Final design
Although it is a commonly applied technique in high-speed rotors, the test application rotor
could not be simply enclosed by pressing a fiber ring over a bare magnet without causing its
damage. Polymer magnet would also buckle if it was simply pressed by the carbon fibers.
Therefore, the rotor structure was designed so as to ensure structural integrity of both magnet
and fibers. A quarter of a cross-section of the final rotor is given in Figure 7.21.
iron
PPS-NdFeB
glass fibres
carbon fibres
Figure 7.21: Final rotor structure
In the final design glass fibres were used to enable safe pressing of the carbon fibre ring and
to protect carbon fibres from bending at corners of the magnet.
The procedure for retaining the magnet was as follows: First the glass fibre rings are pressed
on the shaft over top and bottom faces of the magnet. Outer surfaces of the magnet and the glass
fibre rings are then polished before pressing of the enclosure. At the same time, carbon fibres
7.6. Design of the rotor retaining sleeve
137
Figure 7.22: Stress in the magnet as result of the sleeve fitting
Figure 7.23: Stress in the magnet at 180.000 rpm, room temperature
are wound around a very thin glass fibre ring whose inner radius is for 95 µm smaller then the
outer radius of the magnet. Finally, the carbon/glass fibre ring is pressed onto the rotor.
The rotor final structure was modeled in 3D using Ansys Workbench software. Compression
at the magnet inner surface resulting only from the press-fit is much smaller than calculated from
the 2D modelling. However, what concerns is a large stress concentration at the line close to
boundary between the iron shaft and glass fibre ring (pointed by arrows at Figure 7.22).
Further, according to this model, if the rotor remained at room temperature, the contact
between the magnet and iron would be lost beyond 180.000 rpm (Figure 7.23). Maximum
possible speed is increased, though, if the operating temperature rises.
Maximum equivalent stress in the magnet at the maximum speed and temperature is at the
138
Chapter 7. Design of the high-speed-spindle motor
Figure 7.24: Stress in the magnet at 200.000 rpm, 85◦ C
outer magnet surface and amounts to 110 MPa (Figure 7.24) which is still below the compression limit.
Maximum tensile stress in carbon fibres - 1147 MPa - is in a very good agreement with
results from 2D modelling.
7.7
Conclusions
The chapter presents the design of the high-speed spindle motor, from a conceptual design to
electromagnetic and structural optimization of the motor.
Two new spindle concepts3 are presented; in both concepts a slotless toroidally-wound PM
motor with a short rotor is combined with 5DOF frictionless - active magnetic or aerostatic
- bearings. The motor is spatially integrated with bearings without merging their functions
even in the example of active magnetic bearings. The spindle with a short rotor benefits from
rotordynamical advantages of such rotors; most importantly, stability threshold becomes too
high to be reached and the dynamical stability problems are simply avoided.
Loss minimization is taken as the ultimate criterion for the motor electromagnetic design.
More precisely, the main intention of the design was to mitigate the overheating of the motor
that would result from frequency-dependent losses. Chosen materials, conceptual design and
electromagnetic optimization resulted in a relatively simple machine with very low electromagnetic losses.
Design of the rotor shaft was the initial step of the motor design since the rotor dimensions
were also decisive for the bearing design. Diameter and length of the rotor disc were determined
3
developed together with M. Kimman [175]
7.7. Conclusions
139
so that the polar inertia of the whole rotor is considerably higher than the transversal inertia so
that the motor can benefit from rotor self-aligning.
The electromagnetic optimization was carried out in two steps. In the first step the machine
geometry and number of stator-conductor turns were determined for a minimum total loss in the
stator. In the second step the stator conductors are optimized for the given number of turns and
machine loading defined in the preceding step. Exact analytical formulas for the optimization
of air-gap conductors in slotless PM machines are derived which is an originality of this thesis.
The electromagnetic design of the motor is evaluated by FEM. The model of the PM field is
confirmed by magnetostatic 2D FEM simulations and the field reduction influenced by the axial
flux leakage is observed in 3D FE simulations. This leakage is enhanced due to the presence of
the small extrusions of the rotor iron shaft which protect the magnet.
The armature field model developed in the thesis is not fully suitable to represent a motor
with toroidal windings. The model does not take into account the actual, toroidal distribution of
the conductors since it neglects the magnetic field outside the stator iron. However, as seen in
the FE simulations and the external leakage of the armature field is immense, causing the motor
phase inductance to be almost an order of magnitude higher than analytically predicted.
Eddy-current losses in motor air-gap conductors are indirectly simulated using an abstract
FE model of a thin solenoid which creates the same flux density in the solenoid center as it
is in the middle of the motor windings. The eddy-current loss in the conductors calculated in
this way perfectly matches analytical model prediction and confirms great importance of proper
sizing of the air-gap conductors.
Finally, transient FE models showed a negligible influence of eddy-current losses in the
rotor iron on machine performance.
The rotor retaining sleeve is designed using the optimization approach described in Chapter
4. An innovative final design of a retaining sleeve for a short PM rotor is presented; the design
consists of a combination of glass- and carbon-fiber retaining rings.
140
Chapter 7. Design of the high-speed-spindle motor
Chapter 8
Control of the synchronous PM motor
8.1
Introduction
Permanent magnet machines have become prevalent among very high-speed machines and, to
the author’s knowledge, no machine other than permanent magnet has been reported to operate beyond the speed of 130.000 rpm. So-called PM synchronous machines (PMSM) with
sinusoidal phase currents are generally preferred when low loss and smooth torque are important [194]. On the other hand, using high-frequency sinusoidal currents in the stator makes the
design of a power converter more difficult and also complicates the implementation of control
algorithm given a short switching period. A control method for PMSM must ensure stable,
synchronous operation of the machine at high speeds having, at the same time, computational
complexity tolerable by the given microcontroller. Since small high-speed machines, as a rule,
lack space for a position/speed sensor, sensorless operation is usually required.
Unlike induction machines, PM machines become unstable or marginally stable beyond a
certain frequency when driven in open-loop [31]. Namely, due to the virtual absence of a component of the machine’s torque which is proportional to the rotor speed, PMSM has a pair of
poorly or negatively damped rotor poles [31, 195] and the machine is prone to lose synchronism when subjected to disturbance. Large synchronous machines have damper windings that
suppress oscillations of speed during any transition by producing stabilizing slip torque like
in an induction machine. However, having damper windings on a high-speed rotor is beyond
consideration and necessary damping has to be introduced in a feedback control.
For achieving high performance, PM synchronous machines without damper windings are
controlled using vector/field-oriented control. In the sensorless variant of the vector control,
the position of the rotor is estimated by processing measured phase currents [65]. Nevertheless,
sensorless vector control methods require a lot of computation while the processing power of
standard microcontrollers and DSPs constrain their complexity when sample periods become
very low. Therefore, very powerful or even dual DSPs are needed for realization of sensorless,
close-loop control [29] and their price and complexity outweigh benefit that they bring [196].
Fortunately, applications where high-speed PMSM are used usually require no high performance of the speed control so that open-loop controllers can be used [29, 195, 197]. The
open-loop control method that is mostly used for high-speed PMSM is V/f control with fre141
142
Chapter 8. Control of the synchronous PM motor
quency modulation. In its basis, the method resembles V/f control of an induction motor: applied voltage is increased proportionally to frequency. In order to achieve stability, however,
the reference frequency is modulated using perturbations of the active power and this frequency
modulation, in effect, introduces necessary damping to the speed response. The method has
been used regularly in open-loop control of high-speed PM machines [195, 197–199]; moreover, Itoh et al. [198] showed that, at high speeds, performance of this method is as good as or
even better than performance of the sensorless vector control.
Voltage reference in the V/f control can be a priori set with respect to frequency, knowing
machine parameters and the load torque. However, voltage is then applied irrespective of the
actual current and load torque which may lead to overcurrents or, conversely, stalling and loss
of synchronism. Dynamics of speed regulation in such a controller is very slow and only limited
speed ramps can be supported [200].
Reference voltage is often actively calculated using measurements of the currents [195,199].
Setting the voltage in this manner still has disadvantages: the voltage reference is strongly
dependent on the machine’s parameters and the calculation of the voltage involves a lot of
demanding computations. Furthermore, such methods have been reported for rather low-speed
machines [195].
Another possibility for adjusting the reference voltage to an optimum value is to use efficiency control [198] in which an a priori set value of reference voltage is modified during
operation so as to achieve the d-axis current practically equal to zero.
Authors of [200] also used measurements of the reactive power but for perturbation of both
reference voltage and frequency. Their solution offers faster dynamics of the speed control
with respect to controllers in which only reference frequency is perturbated. Additionally, the
reliance of the voltage-reference setting on machine parameters is not serious - it primarily
depends on estimation of the phase inductance [200].
In the I/f control method a reference current is set instead of the voltage and the command
voltages are generated through a current regulator [201, 202]. In papers in which the method
has been reported the command current was set somewhat empirically and the method was used
only for the start-up and low-speed range to prevent high current deviations in those operating
regions; the control would be subsequently handed over to the standard V/f controller [201,202].
In the thesis, the I/f method is used as a basis for stable, sensorless control of a high-speed
PMSM. The method was developed for control of the spindle drive whose design is presented in
the previous chapter. Stabilizing control (frequency modulation) from the standard V/f approach
has been incorporated into the I/f method. A computationally simple controller with an inherent
control of the current (torque) has been developed. The controller was successfully tested in the
practical setup.
This chapter starts with state-space modeling of an open-loop controlled PM machine and
its stability is analyzed, Section 8.2. The V/f control method with modulation of the reference
frequency is then presented in Section 8.3. The I/f control method, which is ultimately used
for control of the spindle motor, is explained in Section 8.4. Finally, the control method is
implemented in DSP and tested in the practical setup; results of the experiments are presented
in Section 8.5.
8.2. Stability analysis
8.2
143
Stability analysis
Two rotating reference frames will be used to represent the electromagnetic quantities of the
test motor: dq reference frame with the d-axis coinciding with the direction of the permanent
magnet flux ψm and γδ as controller’s reference frame. In order to analyze open-loop stability of
the rotor, δ-axis will be connected to the voltage phasor of the motor (as in the V/f control, Fig.
8.1). The reference frames rotate with electrical and mechanical angular frequency ωe and ωr ,
respectively, and the power angle δ also represents angular displacement between the frames.
q
δ
E
V
δ
φ
I
ωe
γ
ωr
Ψm
d
Figure 8.1: Rotating reference frames, voltage control
State-space electromechanical model of the motor in γδ reference frame is given in the
following equations:
d
R
ψm
1
iγ = − iγ + ωe iδ −
ωr sin δ + vγ
dt
L
L
L
d
R
ψm
1
iδ = − iδ − ωe iγ −
ωr cos δ + vδ
dt
L
L
L
3 ψm
3 ψm
T load (ωr )
d
ωr = −
iγ sin δ +
iδ cos δ −
dt
2 J
2 J
J
d
δ = ωe − ωr
dt
(8.1)
In (8.1) R and L denote phase resistance and inductance respectively, while i and v represent
instantaneous current and voltage in the corresponding reference frame. After linearization of
equations (8.1), the small-signal model of the motor is obtained:






 ∆i̇γ 
 ∆iγ 
 ∆vγ 
 ∆i̇δ 
 ∆iδ 






∆vδ  ,
(8.2)
 ∆ω̇r  = A ·  ∆ωr  + B · 





∆ωe
∆δ̇
∆δ
144
Chapter 8. Control of the synchronous PM motor
where:




A = 


− RL
−ω0
3 ψm
sin δ0
2 J
0
ω0
− ψLm sin δ0
− ψLm cos δ0
− RL
3 ψm
cos δ0
− kTJ1
2 J
0
−1
3 ψm
2 J
 1
 L 0 Iδ0
 0 1 −I
γ0
L
B = 
0
0
0

0 0 1
− ψLm ω0 cos δ0
ψm
L ω0 sin δ0
Iγ0 cos δ0 − Iδ0 sin δ0
0



 .






 ,

(8.3)
(8.4)
In (8.3) and (8.4) index 0 denotes steady-state value of the given variable and ωr0 = ωe0 = ω0
is the synchronous frequency. It is assumed that load torque T load linearly depends on the
rotational speed, i.e.:
T load (ωr ) = kT 1 · ωr + kT 0 .
(8.5)
Correlations between steady-state quantities are also derived in linearization of (8.1). Equilibrium between the electromagnetic and the load torque on the motor shaft yields the following
equation:
3
ψm Iq0 − kT 1 ω0 − kT 0 = 0
(8.6)
2
In further analysis it will be assumed that the steady-state d current is zero. Hence, the
expressions for components of the steady-state currents in γδ reference frame are:
Iγ0 = Iq0 sin δ0
Iδ0 = Iq0 cos δ0
(8.7)
Bearing in mind that Vγ0 = 0, equations that correlate operating voltage and currents in the
controller’s reference frame with operating speed are given as:
− RL Iγ0 + ω0 Iδ0 −
− RL Iδ0 − ω0 Iγ0 −
ψm
ω0
L
ψm
ω0
L
sin δ0 = 0
cos δ0 + L1 Vδ0 = 0
(8.8)
For different values of the operating speed ω0 steady-state quantities are calculated using
(8.6), (8.7) and system of equations (8.8). Finally, eigenvalues of the open-loop system represented by the state-space model (8.2) are plotted in Figure 8.2 for the example of the test
motor.
In the left-hand plot in Fig. 8.2 are presented system poles for different operating frequencies (100-1500 Hz). Two groups of the system poles can be distinguished. In the first group
are very well damped electrical poles which represent fast dynamics of the motor’s electrical
subsystem. The right-hand plot represents a zoom-in on the other group of poles. These mechanical poles, which are associated with dynamics of the electromechanical subsystem, are
poorly or negatively damped and have the dominant impact on motor stability.
8.3. Stabilization control
145
4
Pole−Zero Map
Pole−Zero Map
x 10
1
1500 Hz
80
0.8
60
500 Hz
0.6
40
Imaginary Axis
Imaginary Axis
0.4
0.2
0
−0.2
20
100 Hz
0
−20
−0.4
−40
−0.6
−60
−0.8
−1
−4000
−80
−2000
0
Real Axis
2000
−1.5
−1
−0.5
Real Axis
0
0.5
Figure 8.2: Plot of the system poles as a function of the applied frequency
8.3
Stabilization control
In order to add the necessary damping into the system, the reference frequency is modulated
using perturbations of the active power. The method has been repeatedly analyzed in literature
[195, 197–202]; it will be shortly recounted here.
After analyzing a reduced model of PM machine [195, 199] it can be observed that the
damping can be achieved by modulating applied frequency proportionally to the time derivative
of rotor frequency perturbations [199]:
d∆ωt
(8.9)
dt
Since measurements of the speed are not available, the desired modulating term can be
acquired from perturbations of the active power pm , thus:
!
d∆ωr
∆ωe = −k p ∆pm = −k p Jω0
+ 2kT 1 ω0 ∆ωr + kT 0 ∆ωr
(8.10)
dt
∆ωe = −K
Finally, perturbations of the output power will be shown in the input, electrical power pe ,
which is measurable:
3
3 ∆pm ≈ ∆pe = ∆ vδ iδ + vγ iγ = Vδ0 ∆iδ
(8.11)
2
2
With the steady-state values calculated as in the previous section and the value of the feedback gain set as:
const
,
(8.12)
kp =
ω0
146
Chapter 8. Control of the synchronous PM motor
vγ* = 0
ωe*
ωe
+
-
γδ
vδ*
Voltage
calculation
va*
*
b
v
*
c
abc v
1/s
A
PWM B
C
PM motor
θe
kp·Δpe
HPF
kp·pe
3/2·kp
×
iγ abc
ia
iδ
ib A/D
γδ
Figure 8.3: Block diagram: modified V/f control
eigenvalues of the new system are plotted in Figure 8.4. The electrical poles do not practically
change their value while mechanical poles are well-damped, having damping of about 0.7 at
high frequencies.
4
Pole−Zero Map
Pole−Zero Map
x 10
1
60
1500 Hz
0.8
500 Hz
40
0.6
0.4
Imaginary Axis
Imaginary Axis
20
0.2
0
100 Hz
0
−0.2
−20
−0.4
−0.6
−40
−0.8
−1
−4000
−2000
0
Real Axis
2000
−60
−60
−40
−20
Real Axis
0
Figure 8.4: Poles of the voltage-controlled motor including frequency modulation, k p =
4500/ω0
8.4. I/f control method
8.4
147
I/f control method
For the given application, implementing the V/f control method was troublesome, mainly due to
difficulty of setting adequate voltage references throughout the large speed range of the motor.
Calculation of the voltage from required torque/current invariably depended on the values of stator resistance, which changes with temperature, and inductance, which could not be accurately
measured. Additional voltage was needed for fast acceleration. During operation, overvoltage
would activate overcurrent protection, undervoltage would result in motor stalling. Eventually,
a lot of tuning of the reference voltage was needed to match different operating regimes and
motor acceleration rate was extremely limited.
Therefore, the I/f control method has been used. Since required current depends solely on
load torque and permanent magnet flux (eq. (8.6)), which both can be accurately measured in
the test setup, it was far simpler to set current as the reference instead of voltage.
A block diagram of the control system is given in Figure 8.5. The frequency modulation is
retained and only current regulators were added with respect to the V/f method. Coefficients of
PI regulators are adjusted so that the dynamic of the current controller loop is much faster than
the dynamics of the mechanical subsystem. Value of the δ-axis reference current is set with
respect to operating speed and, as a result, in steady state γδ frame is connected to the current
phasor (Fig. 8.6).
iγ* = 0
+
Current
calculation
ωe*
iδ*
+
-
PI
iγ
PI
iδ
+
-
vγ*
γδ
vδ*
ωe
va*
A
vb*
PWM B
C
*
c
abc v
1/s
PM motor
θe
kp·Δpe
HPF
kp·pe
3/2·kp
×
iγ abc
ia
iδ
ib A/D
γδ
Figure 8.5: Block diagram: I/f control with frequency modulation
8.5
Controller implementation and experimental results
8.5.1 Description of the test setup
Here, a short overview of the test setup will be given with focus on properties which are important for the controller implementation. Details on the test setup and experimental results are
presented in Chapter 9.
The setup consists of three parts: a 2-pole slotless PM motor which shares the same housing
with static air bearings, a high-frequency inverter which drives the motor and a DSP-based
controller board. The motor is thoroughly described in Chapter 7 and more details on the
bearings and inverter can be found in Chapter 9.
148
Chapter 8. Control of the synchronous PM motor
q
δ
E
V
δ
φ
I
ωr
α
d
Ψm
ωe
γ
Figure 8.6: Rotating reference frames, current control
The motor was designed to support high-speed micro-milling with a maximum speed of
200.000 rpm; properties of the motor are given in Table 7.1 in Section 7.4. As pointed out in
Section 7.3, it is expected that the main part of the load during milling at the maximum speed
comes from the dragging torque caused by air-friction and eddy currents in the stator core. The
load torque from the micro-milling itself is very small and can be regarded as a high-frequency
(synchronous) disturbance.
The gate-driving circuitry of the high-frequency inverter introduces a considerable signal
delay of about 800 ns. An LC filter with the resonant frequency of 13 kHz is placed at the
output of each inverter leg.
The microcontroller used is the Texas Instruments 100MHz TMS320F2808 fixed-point DSP
LC filters
Current measuring
to phases
N
S
Isolation + gate drivers
Signal conditioning
inverter board
command
ADC: current signals
QEP encoder pulses
RTDX: data
(real-time communication)
DSP board
Figure 8.7: Scheme of the test setup
PM motor
8.5. Controller implementation and experimental results
149
and its processing power has represented the main limitation for complexity of the control algorithm. Namely, at the desired switching frequency of 100 kHz this microcontroller would allow
only 1000 micro-instructions; for instance, only one fixed-point sinus operation requires at least
40 micro-instructions. Therefore, stringent code implementation of the control algorithm was
necessary to accommodate processing of signals within one switching period.
Although speed measurements have not been used to control the motor, a reflective optical
sensor was used for test-measurements of speed and its output was fed into the DSP. By using
RTDX (Real-Time Data eXchange) communication, real-time transfer of data between the DSP
and a computer has been enabled.
15
phase1
phase2
phase3
E=ψm⋅(2πf)
Voltage [V]
10
5
0
0
500
1000
1500
Frequency [Hz]
Figure 8.8: No-load voltage - measurements
8.5.2 I/f controller implementation
Stabilization control (modulating reference frequency) was implemented as described in Section
8.3 with the feedback gain k p set according to equation (8.12). Coefficients of PI regulators were
set using the IMC strategy [203] with the goal of ensuring fast dynamic of the current controller.
For measuring permanent magnet flux and load torque decay-speed test was used. The
current reference was set to a fixed value (1.5 A), the motor was driven to a very high speed
and then, after switching off the inverter, frequency decay vs. time and no-load voltage vs.
frequency was being registered. Measurements of the phase no-load voltage is given in Figure
8.8 and decay of the rotational frequency over time in Figure 8.9.
In order to estimate the load torque the frequency was first correlated with time using a
150
Chapter 8. Control of the synchronous PM motor
1400
measured frequency
kf3t3+kf2t2+kf1t+kf0
1200
Frequency [Hz]
1000
800
600
400
200
0
0
5
10
15
20
Time [s]
25
30
35
Figure 8.9: Frequency decay - measurement
2.5
based on measurements
kT1⋅(2πf)+kT0
Torque[mN⋅m]
2
1.5
1
0.5
0
0
200
400
600
Frequency [Hz]
800
1000
1200
Figure 8.10: Load torque vs. frequency
polynomial curve-fitting of the measured data. The load torque was then calculated as:
df
.
(8.13)
dt
Finally, by fitting the correlation between the load torque and frequency with a linear funcT load = −2πJ
8.5. Controller implementation and experimental results
151
tion - equation (8.6) - coefficients kT 1 and kT 0 are obtained (Fig. 8.10).
The correlation between the rotational frequency and current reference is obtained from the
torque equilibrium equation:
Iδ0 =
2
(kT 1 ω0 + kT 0 ) kerr kadd
3ψm
(8.14)
In (8.14) kerr = 1.05 represents a correction of the estimated torque for the maximum error
of the fitting (5%) and kadd is 5÷10% added torque that is needed for acceleration and to balance
some additional load torque resulting from armature-induced eddy currents.
Figure 8.11: A phase current at 200 Hz (12000 rpm) without and with frequency modulation
Imax = 2A
f = 1667 Hz
Figure 8.12: Current waveform at 100000 rpm
152
Chapter 8. Control of the synchronous PM motor
Figure 8.13: A phase current at 100000rpm
8.5.3 Experimental results
The proposed method proved to be very convenient for control of the high-speed motor. Only
a few sets of measurements of no-load voltage and speed decay were sufficient to estimate an
adequate current reference.
A relatively large current amplitude (1 A) was used for the rotor initial positioning and
speeding up. The initial aligning of the rotor with the stator flux was done by applying the
current phasor in two directions with 90◦ span consecutively. The current was reduced after
speed-up and further acceleration was performed with current set as in equation (8.14).
Since the current regulator takes care of adjusting the reference current to the load torque,
no compensation of the phase voltage drop was needed. At the same time, the load torque
and no-load voltage, according to which the current reference was set, hardly changed during
operation. Faster acceleration could easily be achieved by adequately increasing coefficient kadd
during speed-up.
Without frequency modulation high oscillations in the torque were already present at rather
low speeds (Fig. 8.11). After the oscillations would grow, the synchronism would be lost at
about 300 Hz (18.000 rpm). However, with the frequency modulation applied and the feedback
gain set as in (8.12), the torque and speed remained stable throughout the whole speed range.
The rotor speed also remained stable when it was disturbed when touching metal objects.
The motor was successfully drive up to highest speed achieve with this setup - approximately
2600 Hz (156.000 rpm). There was no need for change or tuning of initially obtained currentreference parameters and change of the stator temperature did not affect controller performance.
This setup was inadequate to offer comprehensive tests of the controller performance such
as bandwidth of the speed controller and response to changes in the load torque.
8.6. Conclusions
8.6
153
Conclusions
For very high-speed PM synchronous machines open-loop control methods are generally used
so as to facilitate controller implementations in standard microcontrollers. The V/f control
method with modulation of the reference frequency has proven its merits and has been regularly used for high-speed PM machines. Still, the method offers no means for controlling
current (torque) while setting a suitable voltage reference with respect to the applied frequency
is parameter-dependent and/or computationally difficult.
This chapter presents a new realization of the I/f method in which the voltage reference is
generated through a current regulator while retaining the frequency modulation as in the V/f
method. With a fast dynamic of the current control the stability of the system is preserved with
the apparent advantage of having supervision of the currents incorporated. The main benefit,
however, is that the required reference current can be quite simply and adequately determined
using knowledge or measurements of the machine PM flux and load torque.
The method was applied in control of a small high-speed PM motor up to the rotational
speed of almost 160.000 rpm. The control algorithm was easily implemented in a non-expensive,
standard digital signal controller. Based on measurements of no-load voltage and speed decay
the adequate current reference was set. The control was stable and good in disturbance rejection
throughout the whole speed range.
154
Chapter 8. Control of the synchronous PM motor
Chapter 9
Experimental results
9.1
Introduction
Experiments and accurate measurements of high-speed electrical machines often meet many
practical difficulties mainly as a consequence of constrained geometries and the fast rotation
itself. To test many of the phenomena discussed in this thesis, dedicated test setups would be
necessary although that would exceed the scope of the project. The setup that was developed for
the purpose of the thesis project was primarily designed to demonstrate the new spindle-drive
concept and test the overall motor performance. With the experiments reported in this chapter,
the test capabilities of the designed setup were explored to gather practical data on the motor
and bearings which would allow an assessment of the developed models and design approach.
The chapter starts with description of the practical setup including a short description of
stator and rotor assembly and important information about the air-bearing setup and highfrequency inverter. In Section 9.3 measurements of the motor phase impedance, i.e. phase
resistance and inductance, are reported.
The main tests performed for verification of the models developed in this thesis are the
speed-decay and locked-rotor tests as reported in Sections 9.4 and 9.5. The speed-decay tests
were performed to measure the no-load voltage and losses of the motor. The locked-rotor tests
were used to measure apparent phase impedance at high frequencies and to assess motor losses
under load.
Finally, the overall performance of the electric drive during rotation is reported and discussed.
9.2
Practical setup
9.2.1 Stator assembly
The stator cores are made of 0.1 mm laminations of non-oriented Si-steel 10JNEX900 [204].
The lamination sheets were cut to the design dimensions (see Figure 7.11) and glued together.
R
The stator cores are wound toroidally using two parallel self-bonding wires Thermibond
155
156
Chapter 9. Experimental results
158 [186]. A smaller number of turns per coil than initially designed (40 instead of 44) was
applied due to a restriction on the stator’s total axial length imposed by the bearing housing.
Windings were finally pressed and cured by means of a DC-current surge which was adjusted to heat the wires to 230◦ C for approximately 3 minutes. The high temperature causes
softening of the adhesive varnish before curing and, in turn, eliminates the need for impregnation of the windings.
Figure 9.1: A photo of the stator and rotor; the rotor is without the sleeve
Figure 9.2: A photo of the final rotor
9.2. Practical setup
157
9.2.2 Rotor assembly
The iron shaft of the rotor was first produced according to the dimensions from Figure 7.10.
The PPS-bonded NdFeB magnet was applied onto the incised part of the rotor disc using the
injection-molding technique.
The realization of the rotor bandage was the most demanding part of the rotor production.
Section 7.6 includes an explanation of the procedure which was eventually used to ensure the
structural integrity of both the plastic-bonded magnet and carbon fibers; the procedure is depicted in Figure 9.3. On the other hand, the pressing of the sleeve parts during assembly was
expected to cause a large mass unbalance in the rotor.
magnet
glass-fiber rings
iron shaft
1 - initial rotor
2 - glass-fiber rings pressed
carbon-fiber
sleeve
glass-fiber
layer
3 - outer surface polished
4 - carbon-fiber sleeve pressed
Figure 9.3: Steps in the assembly of the rotor retaining sleeve
For test purposes one rotor was produced in which the magnet was substituted with epoxy
(dummy rotor) and this rotor was used for measuring air-friction losses and assessing losses in
the magnet.
9.2.3 Air-bearings test setup
A setup with air bearings for testing the motor was developed as a master project assignment
[180]. The concept of the setup was presented in Section 7.2.
158
Chapter 9. Experimental results
The air-bearing housing was designed to accommodate the stator and facilitate good thermal
contact with environment. A transparent image of the air-bearing housing with the motor in it
is given in Figure 9.4 and a photo of the setup is shown in Figure 9.5.
rotor
stator
Figure 9.4: The motor inside the air-bearing housing (taken from 7.2)
Figure 9.5: A photo of the air-bearing setup
Two important physical requirements were set for the air bearings: the bearings were supposed to provide a sufficient load capacity to support the PM rotor and to ensure stability of
rotation for a maximum possible range of rotational speeds.
While the stiffness of air bearings is generally high, their load capacity is rather low and
this holds particularly true for air bearings with simple orifice restrictors, the type employed
9.2. Practical setup
159
in this test setup. To define maximum radial load force on the rotor, the expression (3.81)
for unbalanced magnetic force in the motor was used. With ∆xmax = 0.5 mm of maximum
anticipated rotor eccentricity (1 mm is the physical maximum) the highest possible value of the
unbalanced force was estimated (Table 9.1). Finally, the air bearings were designed so their
load capacity is, at least, an order of magnitude higher than the estimated value.
The main concern with the bearings was their influence on the stability of rotation. Instability problems with journal fluid bearings were discussed in Section 6.3 and a simplified theoretical explanation for the phenomena was given in Subsection 5.3.2: because of non-synchronous
aerodynamic forces in the bearings, rotors in aerodynamic- and lubricated journal bearings tend
to become unstable at speeds very close to twice their critical speed. On the other hand, static air
bearings with orifice restrictors have not yet been associated with any type of instability [179].
Nevertheless, a few reported high-speed spindles supported by static air bearings were intentionally designed to maintain their rotation below twice the first critical speed [7, 155]: the
authors expected that aerodynamic forces would overpower static bearing forces at very high
speeds.
The critical frequency of the test rotor is estimated by:
r
r
1
2k1
k
1
fcr =
=
,
(9.1)
2π
m
2π m
where m is the rotor mass, k1 is the stiffness of a single radial bearing and k = 2k1 is the total
radial stiffness of the air bearings.
Based on the models developed in [180] the first critical frequency is estimated to be somewhat below 900 Hz (54.000 rpm; Table 9.1) and potential stability problems were expected in
the speed range of 110.000-120.000 rpm.
Table 9.1: Estimations of dynamic parameters of the motor and the bearings
parameter
analytical FEM
maximum unbalanced force [N]
0.5780
0.3638
∗
radial load capacity (total) [N]
8.2
8.4
∗
radial stiffness (total) [kN/m]
1170
1200
critical frequency [Hz]
878.5
889.7
∗
values obtained from [Petros]
9.2.4 High-frequency inverter
At the time of the motor design there was no power converter on the market that would be
suitable for driving such a high-speed synchronous PM motor. Therefore, a high-frequency
PWM inverter was developed within the project as a master project assignment [180]. The
inverter was also supposed to be capable of driving the high-speed motor from the AMB setup
described in [164]. Moreover, an in-house built inverter provided the possibility to develop a
dedicated controller for the spindle motor.
160
Chapter 9. Experimental results
A functional scheme of the inverter is shown in Figure 8.7. The core of the design is a
MOSFET bridge capable of 200 kHz switching and tested for DC voltages up to 300 V. The
output phase currents of the inverter are filtered by LC filters and current measurements are
provided by the developed high-bandwidth circuitry.
Power converter
3-phase PWM
voltage
DC voltage
supply
Braking
chopper
MOSFET
bridge
LC filters
PM motor
Gate drivers
Isolation
circuits
Isolation
circuits
Current
measurement
switching
signals
Speed
command
DSP board
Figure 9.6: A functional scheme of the developed inverter [180]
9.3
Motor phase resistance and inductance
9.3.1 Phase resistance
Measurements of the motor phase resistance were done using a simple setup shown in Figure
9.7. The current in the series connection of two phases was adjusted using a voltage source
while voltage and current were also measured with an oscilloscope and current probe, respectively. Higher currents resulted in higher winding temperature which was monitored using a
thermocouple. The measurements were repeated for all three pairs of the phases.
Motor resistance can also be estimated using the equations presented in Subsection 3.6.2 on
motor copper losses. Resistance of a single phase is given by (see equation (3.110))1 :
R=
4
lCu ρCu ,
2
π
ndCu
(9.2)
where the length of a single phase conductor can be estimated as:
lCu = [2l s + (r so − r s ) π] 2N.
1
Notation from previous chapters is maintained.
(9.3)
9.3. Motor phase resistance and inductance
161
ot
or
3
m
R 3,L 3
R
R
L1
1,
2 ,L
2
1
2
current
probe
V
-
+
Figure 9.7: Phase-resistance measurement setup
Copper resistivity changes linearly with temperature:
ρCu = ρ0 (1 + αCu ∆T )
(9.4)
and for a base temperature of 20◦ C (∆T = T − 20◦ C) the thermal coefficient of the resistivity is
αCu = 4.03 · 10−3 Ωm/◦ C.
Predicted and measured values of the series resistance of two motor phases with respect to
temperature are presented in Figure 9.8. The resistance values stemming from the model fall
well within the range of the measured values. Apparently, equation (9.3) is quite adequate for
the estimation of conductor length.
From the measurements shown in Figure 9.8 there is a noticeable difference in the resistances of different phases. This can certainly be ascribed to the manual winding process; apparently there is a slight variation in the number of turns of toroidal coils in the stator.
The measured values were fitted with a linear curve to represent the actual phase resistance
of the motor. The measurement-fitted resistance yielded 0.3486 Ω at 20 ◦ C and the predicted
value for the same temperature is 0.3436 Ω.
9.3.2 Phase inductance
An accurate measurement of a motor DC inductance is difficult to achieve since it would require
measuring the winding flux linkage. Assessment of the inductance was attempted using the
following three measuring methods.
Firstly, the inductance between two motor phases was measured using an RLC meter. This
measurement seemed rather inaccurate: the meter showed largely different results in different
162
Chapter 9. Experimental results
0.79
0.78
0.77
R12
R23
R31
model prediction
fitted by measurements
R [ Ω]
0.76
0.75
0.74
0.73
0.72
0.71
0.7
25
30
35
40
45
50
55
°
Temperature [ C]
Figure 9.8: Measured and predicted values of phase-to-phase motor resistance
measurement modes (series or parallel inductance) therefore this measurements were abandoned.
This measurement indicated, however, that the setup suffers from very large eddy-current
losses in the housing. Because of the toroidally-wound stator fitted into a highly conductive,
aluminum housing, armature leakage flux causes rather strong eddy currents in the housing. The
difference in RLC-meter inductance readings on the machine with and without the top housing
was tremendous.
Another method to measure inductance was to include the motor windings in a resonant
circuit; the measurement setup is shown in Figure 9.9. The frequency of the AC voltage source
was increased until a maximum amplitude of the current (minimum impedance) was registered.
Given the resonant frequency of the circuit fres , the inductance of a single motor phase was
estimated by:
1
L=
.
(9.5)
2 (2π fres )2 C
Phase inductance estimations for two values of the capacitor are given in Table 9.2. Large
deviations of the estimated values from the model prediction are evident.
Table 9.2: Phase-inductance values measured with the resonant circuit and the predicted value
capacitor resonant frequency inductance - measured inductance - predicted
30 [µF]
1660 Hz
153 µH
43 µH
60 [µF]
1240 Hz
137 µH
9.3. Motor phase resistance and inductance
163
ot
or
3
m
R3,L3
R
L1
2 ,L
2
,
R1
1
2
Hall
sensor
R
V,f
C
Figure 9.9: Phase-inductance measurement setup: resonant circuit
Finally, inductance was estimated using a motor short-circuit test. Namely, the windings of
the machine were short-circuited, the rotor was levitated in the air bearings and rotated using
an air-blow gun. The frequency and amplitude of the current were registered and the phase
inductance was calculated as:
L=
1
ω
s
ê
î
!2
s
1 ωψmax 2
2
−R =
− R2 .
ω
î
(9.6)
The measurement of the phase resistance R were presented in the previous subsection; the fluxlinkage amplitude ψmax was also measured; the results are shown in the following section.
This measurement was rather inaccurate since it was very difficult to maintain rotor speed
using the air gun. The maximum achievable rotational speed was quite low (≈38 Hz) which,
in turn, resulted in a very small phase reactance and additionally decreased the measurement
accuracy. Estimations of the phase inductance based on this test varied greatly, between 150
and 350 µH.
In conclusion, the first and last method appeared to be rather inaccurate; the results obtained
by the resonant-circuit method will be maintained by the locked-rotor measurements in Section 9.5, however, those results are valid only at the frequencies corresponding to the resonant
frequencies of the established circuits.
164
9.4
Chapter 9. Experimental results
Speed-decay tests
Speed-decay tests were performed to measure the no-load voltage and losses of the motor. These
tests were also used to define the current reference for the controller as described in Chapter 8.
Hence, in a speed-decay test the rotor is driven to a certain speed, the drive (inverter) is then
switched off and the rotational-frequency decay vs. time and no-load voltage vs. frequency is
registered.
Measurements of the no-load phase voltage are shown in Figure 9.10. The measured voltage
matches the prediction of the 3D FE simulations; analytical and 2D FE model overestimated
the motor voltage due to axial flux leakage (see Section 7.5). Such a good match between the
measured and simulated motor voltage can be ascribed to very accurate data on magnet BH
curve which was measured after the magnet application on the rotor.
15
phase1 measured
phase2 measured
phase3 measured
3DFEM
analytical/2DFEM
Voltage [V]
10
5
0
0
500
1000
1500
Frequency [Hz]
Figure 9.10: No-load phase voltage: measurements and modeling predictions
For measurements of no-load losses, the decay of rotational frequency in time was registered. The frequency was correlated with time using a polynomial curve-fitting of the measured
data (see Figure 8.9 in Chapter 8):
f (t) = k f 3 t3 + k f 2 t2 + k f 1 t + k f 0 .
(9.7)
The drag torque was then calculated by:
T drag = −2πJ
df
dt
(9.8)
and the power of the drag (= no-load losses) was found as:
Pdrag = 2π f T drag .
(9.9)
9.4. Speed-decay tests
165
Firstly, the rotor without the magnet (dummy rotor) was tested to assess air-friction loss.
It was spun by the air gun up to maximally ∼280 Hz (∼16.800 rpm) and the speed-decay was
registered. The loss power, which consists of air-friction loss only, was calculated and compared
with the analytical predictions of models presented in Subsection 3.6.3, Figure 9.11. In the
figure no-load losses are also shown for the complete rotor in the same speed range (0÷500
Hz).
1
dummy rotor
balanced rotor
non−balanced rotor
a.f. loss: [115]−[121]
a.f. loss: [114]−[121]
core loss: manufacturer
0.9
0.8
0.6
0.5
P
loss
[W]
0.7
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
350
400
450
500
frequency [Hz]
Figure 9.11: No-load losses of the motor at low speeds based on the speed-decay tests; full lines
represent losses based on measurements and dashed lines represent predictions
It is evident from Figure 9.10 that models of air-friction losses based on empirically developed formulas (Subsection 3.6.3) cannot account for the air-friction losses in the test motor.
However, those models were developed for rotating conditions that significantly differ from
the conditions in the test machine: the test rotor is supported by air bearings in all directions,
influencing greatly the air flow.
From the measured no-load losses it is evident that electromagnetic losses dominate the low
frequency operation, as expected. The losses in the machine with a non-balanced rotor were
slightly higher than with a rotor which is mechanically balanced: some energy is lost in the
bearing dampers during rotor synchronous whirling [86].
In Figure 9.12 no-load losses are plotted for frequencies up to 1200 Hz (72.000 rpm). In this
figure, measured air-friction losses, using the dummy rotor, are extrapolated to account for the
loss increase at higher frequencies. It is hardly possible to obtain an accurate prediction using
such an extrapolation: at lower speeds, where these losses are measured, the air flow in the air
gap is predominantly laminar and the loss increase is closely quadratic to the speed; this would
surely not be the case when strong turbulences are present. Still, it is quite certain, judging from
166
Chapter 9. Experimental results
18
dummy rotor
dummy rotor: extrapolation
full rotor: measured
a.f. loss: [114]−[121]
core loss: manufacturer
16
14
P
loss
[W]
12
10
8
6
4
2
0
0
200
400
600
800
1000
1200
frequency [Hz]
Figure 9.12: No-load losses of the motor based on the speed-decay tests up to 1200 Hz (72.000
rpm); full lines represent losses based on measurements, the dashed-dotted line represents
measurement-based extrapolation and dashed lines represent predictions
both Figures 9.11 and 9.12, that overall no-load losses in the setup cannot be explained merely
as a combination of air-friction- and stator-core losses.
Eddy-current losses in the conductors could not be directly measured, but those losses hardly
represent a large portion of the overall losses: both analytical and FEM models (see Section
7.5) suggest extremely low eddy-current losses in the optimized conductors (below 0.1 W for
the frequencies shown in Figure 9.12); these predictions are, for that reason, not shown in the
figures.
To account for unexpectedly high no-load losses, transient 3D FE simulations were performed, this time taking into account also the presence of the housing. Namely, top and bottom
aluminum housings of the air bearings are placed only 0.5 mm above and below the axial sides
of the rotor disc, see Figure 9.13. Since the axial leakage of the permanent-magnet flux is,
apparently, rather pronounced in this motor, it is reasonable to expect significant losses in the
housing in regions close to the magnet.
Losses in the housing were calculated using a transient 3D FEM and the results for the low
frequency range (0÷600 Hz) are plotted in Figure 9.14 along with measurements and estimations of other dominant losses. It is difficult to judge the combined contribution of the loss
factors since the information about the various losses is obtained in different ways: measurements, analytical and FE modeling and manufacturer’s data. Still, it can be inferred from the last
plot that air friction and eddy currents in the housing are predominant sources of no-load losses.
Unfortunately, both of those loss factors could not have been analytically modeled: housing
losses are essentially a 3D phenomenon and no adequate model is available for the air friction
9.4. Speed-decay tests
167
aluminum
rotor
aluminum
Figure 9.13: Cross-section of the test setup; field lines of the permanent-magnet field are
sketched at regions where housing losses are expected
6
air−friction loss − measured
air−friction loss − extrapolated
housing loss − 3D FEM
core loss − manuf. data
no−load losses combined − estimation
no−load losses − measured
5
P loss [W]
4
3
2
1
0
0
100
200
300
400
500
600
frequency [Hz]
Figure 9.14: No-load losses of the motor at low speeds including 3D FEM estimation of losses
in the housing; full lines represent losses based on measurements, the dashed-dotted line represents measurement-based extrapolation and dashed lines represent predictions
in the test motor.
Causes for discrepancy between the measured no-load losses obtained by the speed-decay
tests and the estimation of the combined losses (obtained from the measurements and extrapolation of the air-friction loss and the estimations of housing and stator-core losses) can be
sought in various possible factors. It is quite conceivable that the manufacturer’s data underestimate losses in the core in a certain extent: the stator manufacturing process has an influence
168
Chapter 9. Experimental results
on the core properties; rotation of the field in the core additionally increases the losses (the
manufacturer’s data are, namely, given for an unidirectional magnetic field). Extrapolation of
the air-friction measurements is not a reliable means for the loss assessment as it was already
emphasized: the air-friction loss rises more rapidly at high frequencies than the low-frequency
trend of the increase suggests. Finally, the transient 3D FEM for the housing-loss calculation
used rather simplified geometry - an intricate model would, however, require too much time and
computation resources.
9.5
Locked-rotor tests
To assess losses in the motor under load, locked-rotor tests were performed. A scheme of the
test setup is shown in Figure 9.15. The inverter described in Subsection 9.2.4 was used to set
currents of different frequencies in the motor windings. Active power P and reactive power Q
flowing into the motor were measured using a power analyzer. The temperature in the windings
was also monitored using a thermocouple. From these measurements apparent (AC) phase
resistance and inductance were determined as:
ot
or
3
m
R 3,L 3
,
R1
1
L1
R
2 ,L
2
A
V
2
V
A
V
A
PWM
inverter
(I and f
controlled)
power analyzer
Figure 9.15: Locked-rotor test: measurement setup
P
,
3I 2
Q
=
.
3 (2π f ) I 2
RAC =
LAC
(9.10)
(9.11)
The AC resistance represents load-dependent losses in the test setup at the given current
and electrical frequency and the AC inductance tells about the influence of eddy-currents on
the armature field of the motor since the field of the eddy-currents counteracts the stator-current
field and, in effect, reduces apparent phase inductance.
The test results for frequencies up to 3500 Hz and rms currents up to 2.5 A are shown in
Figures 9.16 and 9.17. The AC resistance is normalized over the DC resistance which has been
9.5. Locked-rotor tests
169
corrected for the temperature impact using measurements of the winding temperature:
rAC =
RAC
RAC
=
,
RDC R0 (1 + αCu ∆T )
(9.12)
where the DC phase resistance at room temperature R0 was measured (Section 9.3) and the AC
resistance RAC was calculated from the measurements according to equation (9.10).
The tests were initially performed on the original setup - the motor fitted into the air-bearing
housing - and they were repeated for two different rotors: the normal, PM rotor and the rotor in
which the magnet has been replaced by epoxy (dummy rotor).
Normalized AC phase resistance vs. frequency
Normalized AC phase resistance vs. current
4
4
full motor
motor with the dummy rotor
full motor
motor with the dummy rotor
f = 3500 Hz
3.5
3.5
I = 2.5 A
3
dc
/R
2.5
ac
2.5
R
R
ac
/R
dc
3
I = 0.25 A
2
2
1.5
1.5
f = 100 Hz
1
0
500
1000
1500
2000
frequency [Hz]
2500
3000
3500
1
0
0.5
1
1.5
2
2.5
rms current [A]
Figure 9.16: Normalized AC resistance of the motor measured in the locked-rotor test
Several conclusions can be made based on the measurements. Firstly, it is quite evident that
the presence of the magnet in the rotor does not influence the results. Hence, neglecting losses in
the plastic-bonded magnet done in the modeling is thoroughly supported by the measurements.
Moreover, taking into account the very low conductivity of carbon-fiber composites and the FE
calculations from Section 7.5 (showing rotor-iron losses be negligible), it is quite suitable to
disregard rotor losses in the designed machine.
On the other hand, the measurements show that eddy-currents are present in the setup to
a great extent. The influence of eddy-currents is clearly visible in the measurements of the
phase inductance which decreases greatly with frequency. At the targeted frequency of 3300
Hz (corresponding to 200.000 rpm) the overall frequency-dependent losses amounted to almost
3.5 times the value of the classical conduction loss (I 2 RDC ) in the machine.
Both the measured apparent resistance and inductance changed very little with current (except the inductance values at very low frequencies). The small deviations were influenced not
170
Chapter 9. Experimental results
AC phase inductance vs. frequency
AC phase inductance vs. current
500
500
full motor
motor with the dummy rotor
full motor
motor with the dummy rotor
400
400
L ac [µH]
L ac [µH]
f = 100 Hz
300
300
I = 0.25 A
200
200
I = 2.5 A
100
0
500
1000
100
f = 3500 Hz
1500
2000
frequency [Hz]
2500
3000
3500
0
0.5
1
1.5
2
2.5
rms current [A]
Figure 9.17: AC inductance of the motor measured in the locked-rotor test
only by the armature field but also by the temperature in the core and the housing during different measuring sessions.
Already during measurements of the phase inductance (Section 9.3) it was noticed that the
presence of the housing has an impact on the armature field. Therefore, the locked-rotor test
was repeated for the motor outside the housing. The averaged measured values at different
frequencies from two test setups are shown in Figures 9.18 and 9.19.
By comparing the two test cases it is evident that the bulk of eddy-current losses is generated
in the housing: without the housing the change of the AC phase inductance with frequency is
insignificant. The frequency-dependent portion of losses under load at the maximum motor
frequency (speed) would add less than 70% to the regular DC conduction loss for the motor
outside the housing; with the housing, however, this figure amounts to 250%.
Armature field is generally considered to have a small impact on the performance of slotless machines; this seems, however, not quite correct for toroidally-wound machines - at least
for those placed in an electrically-conductive housing. A magnetostatic model is adequate to
describe performance of the test motor on itself, but cannot account for losses that are induced
outside the machine, losses which appear to be very high in this case.
Measurements of the phase inductance on the motor without housing agree with the 3D
FEM predictions (Table 7.3, Section 7.5). The total DC inductance was estimated from the
measurements to be around 325 µH and is just several percent lower than the 3D FEM value 350 µH.
The measured increase of the apparent phase resistance in the motor without housing still
cannot be explained using analytical and FE models so far developed in the thesis. All the
9.5. Locked-rotor tests
171
Normalized AC phase resistance vs. frequency
3.5
motor in the housing
motor without the housing
R ac /R dc
3
2.5
2
1.5
1
0
500
1000
1500
2000
2500
3000
3500
4000
frequency [Hz]
Figure 9.18: Normalized AC resistance of the motor measured in the locked-rotor test: with and
without housing
AC phase inductance vs. frequency
500
motor in the housing
motor without the housing
L ac [µH]
400
300
200
100
0
500
1000
1500
2000
2500
3000
3500
4000
frequency [Hz]
Figure 9.19: AC inductance of the motor measured in the locked-rotor test: with and without
housing
models agree that the armature field in the stator iron is very small, the field at which losses in
the iron should be negligible. On the other hand, it is not reasonable to guess that the proximity
172
Chapter 9. Experimental results
effect has such a significant influence on the AC copper losses given the small influence of the
armature field in the air gap.
It was observed in Section 9.3 that measurements of the resistances of different phases
indicate some unbalance between the phases: the toroidal coils appear to be unevenly wound
as a result of the manual production. In order to assess the influence of unbalanced coils in the
motor, the 2D FE model from Subsection 7.5.1 was adjusted to account for small variations of
numbers of turns of the coils.
Figure 9.20: Flux-density plots and field lines of the motor with balanced coils and with a single
coil having an additional turn - 2D FEM
In Figure 9.20 a flux-density plot and field lines are shown for examples of an evenly wound
machine and of a machine with only one extra turn in one of the two toroidal coils of the phase
a (precisely, 41 turns instead of 40). While adding an extra turn in the left-hand coil of the phase
does not increase by much the phase inductance (76 µH to 64 µH, as calculated by the 2D FEM),
flux density in the stator iron increases tremendously: 262 mT of the maximum flux-density in
the unevenly wound motor against only 20 mT in the balanced machine (simulated for ia = 1
A, ib = ic = −0.5 A). Finally, the losses in the core will quadratically follow the increase in
amplitude of the flux density which can, therefore, explain much higher iron loss than expected.
The effect is also depicted in Figure 9.21: instead of exiting the stator and crossing the air
gap, the field of the added turn links entirely through the stator iron whose extremely small
reluctance makes this field very high. At the same time, this additonal field penetrates two
serially-connected coils in opposite directions with respect to their orientations causing the two
flux linkages of the coils to cancel each other to a great extent and, accordingly, greatly reduce
the influence of the field increase on the phase inductance.
This phenomenon implies a potential disadvantage of toroidal windings. Namely, for an
efficient machine, the toroidal coils in the stator must have an equal number of turns and the
winding production process should be highly repetitive.
9.6. Motor operation and performance
173
stator
iron
rn
nal tu
o
i
it
d
ad
a’ .
flux line of
balanced coils
x
rotor
iron
.
x a
flux line of
the additional turn
Figure 9.21: Motor geometry cross-section: depiction of the effect of an additional turn in one
of the phase coils on the field in the stator core
9.6
Motor operation and performance
Originally, the manufactured rotor was not capable of reaching very high speeds because of rotor
unbalance - the maximum rotational speed attained with such a rotor was, approximately, 330
Hz (20.000 rpm). The rotor manufacturing, which included three press-fittings of the retainment
rings, resulted in a rather large unbalance: the measured static unbalance of the rotor was 1.7
g·mm which corresponds to a 45 µm shift of the inertia axis with respect to the rotor geometrical
center. For the air bearings with only 14 µm clearance this was far beyond permissible. After
the balancing process, however, the static unbalance was reduced to only 32 mg·mm.
The motor controller presented in Chapter 8 which used the developed realization of the I/f
control method performed very well. Once set coefficients for the controller current reference
were maintained in all the tests throughout the speed range. Figure 9.23 shows the setting of the
current versus rotational speed. Anticipated required current at the maximum rotational speed
- 2.38 A - was somewhat higher than the originally predicted value of 2 A, mostly due to the
10% reduction of the motor no-load voltage as a result of axial flux leakage (see Table 7.3).
The output electromagnetic power of the rotor required to sustain the rotation at 200.000 rpm
is, based on the current reference and no-load voltage, estimated at 216 W.
Initially, the critical speed was passed unnoticed since the rotor was very well-balanced.
However, after rotational frequency was gradually increased beyond 2000 Hz (120.000 rpm),
the rotor became noticeably unbalanced again: apparently, the rotor parts rested in a new position. The critical speed was observed in the vicinity of 850 Hz (51.000 rpm), slightly lower
than predicted (see Table 9.1).
Eventually, the newly established rotor unbalance became so large that it was more and
174
Chapter 9. Experimental results
Figure 9.22: Measured rotor static unbalance before and after mechanical balancing
more difficult to cross the critical speed. The rotor was touching the bearings when operating
near the critical speed and a higher motor torque was becoming necessary to overcome this.
The maximum speed attained with this setup was 2600 Hz (156.000 rpm). At that speed the
rotor tangential speed was 270 m/s and, to the author’s knowledge, that is more than previously
reported tangential speeds of electrical machines. The machine operated stably up to this speed:
rotor remained unheated, maximum registered temperature in the stator was 44◦ C. The rotor
structure seemed undamaged and the no-load voltage of the machine did not change. The
designed rotor retainment fulfilled the task of enabling very high-speed rotation; on the other
hand, it caused problems with the unbalance which appeared to be too high for the air bearings.
In one of the subsequent tests a major accident happened: with the rotor operating in the
vicinity of critical speed, the rotor crashed into the bottom air bearing. This damaged the bearing, disturbed the alignment between the top and bottom bearings and, consequently, prohibited
further testing. The last measurement data from the setup were recorded at the speed of 100.000
rpm (shown in Chapter 8).
One of the important results of this study is that the rotor remained stable at a much higher
speed than twice the critical speed. It shows, thus, that static air bearings represent a good
solution for stable high-speed operation and that the limits of rotational stability of such bearings
are much higher than those of aerodynamic and lubricated bearings. At the same time, very tight
tolerances of air bearings impose high precision standards on the rotor manufacturing.
9.7
Conclusions
The chapter reports results of practical evaluation of the test motor and the setup. Practical data
from the measurements are used for assessment of the developed models and design approach.
The chapter includes a short description of stator and rotor assembly and important information about the air-bearing setup and high frequency inverter.
Measurements of the DC phase resistance confirmed the model prediction thereby verifying
9.7. Conclusions
175
2.5
2
1.5
I [A]
1
0.5
0
0
50
100
150
200
krpm
Figure 9.23: Phase current reference (rms) with respect to the rotational speed setting
the model of the DC-conduction loss in the windings. Relatively large deviations of measured
resistances of different phases are noticed as a result of an uneven number of turns due to errors
in the manual winding process.
Direct measurements of the phase inductance were performed in different ways; however,
only the method of including motor phases in a resonant circuit gave reasonably accurate values.
The main tests carried out for verification of the models developed in this thesis are the
speed-decay and locked-rotor tests. Speed-decay tests were performed to measure the no-load
voltage and losses of the motor.
The measured values of no-load voltage of 3D FEM matched the predicted values of no-load
voltage very closely: this can also be ascribed to very accurate data on the magnet remanent
field and recoil permeability.
According to the results on the drag torque obtained from the speed-decay tests with a
rotor without permanent magnet, it is evident that the models generally used for more conventional machines with slender rotors fail to account for the air-friction loss in the test machine.
However, those models were developed for rotating conditions that significantly differ from the
conditions in the test machine.
Speed decay tests showed a great discrepancy between calculated and predicted no-load
losses. With help of 3D FEM, the high no-load losses can be explained by, apparently, very
pronounced losses in the housing caused by the significant axial flux leakage of the PM field.
Air friction and eddy currents in the housing are predominant sources of no-load losses; unfortunately, both of those loss factors could not have been analytically modeled.
Locked-rotor tests were performed to assess the losses in the motor under load. It is easy
to infer from the measurements that the presence of the magnet in the rotor does not influence
176
Chapter 9. Experimental results
either field or losses in the motor. The plastic-bonded magnet proved to be extremely resistive
to eddy-current losses, which is a great advantage for high-speed applications.
Nevertheless, the measurements show that eddy-currents are present in the setup to a great
extent. By comparing the losses generated in the setup with and without housing it is evident
that the bulk of eddy-current losses is generated in the housing. The losses in the aluminum
housing are, actually, the weakest point of the design; along with air friction, eddy currents
in the housing represent the most dominant loss factor in the setup, greatly influencing drive
performance and temperature.
These measurements also showed that the magnetostatic model cannot account for all important phenomena in a toroidally-wound machine. Namely, such a model cannot adequately represent the armature field in the machine, particularly if the machine is placed in an electricallyconductive housing.
Measurements of the phase inductance on the motor without housing agree with the 3D
FEM predictions.
Unevenly wound coils in motor phases caused an unnecessary increase of the armature field
and losses in the stator core. This phenomenon implies that an automatic and highly repetitive
winding process is strongly recommended for toroidally-wound machines.
After rotor balancing the motor was capable of reaching very high speeds. However, after
rotational frequency was gradually increased beyond 2000 Hz, the rotor became noticeably
unbalanced again. The problem of the recurring unbalance eventually resulted in the bearing
crash during rotation in vicinity of the critical speed.
The maximum rotational speed obtained with the setup was 156.000 rpm. At that speed,
the rotor developed the tip tangential speed of 270 m/s which is the highest tangential speed
of an electrical machine that has yet been reported in academic literature. Very high attained
tangential speed confirms validity of the approach for retaining sleeve optimization.
The motor I/f controller presented in Chapter 8 performed very well. Once set coefficients
for the controller current reference were maintained in all the tests throughout the speed range.
The test setup demonstrates the suitability of aerostatic bearings for very-high-speed operation: a rotor in aerostatic bearings can operate well above twice the critical speed. This renders
limits of their rotational stability higher than the limits of aerodynamic and lubricated journal
bearings.
Chapter 10
Conclusions and recommendations
10.1
Models presented in the thesis
Various analytical models are presented in the thesis. In this section the models will be reviewed; particular attention is given to model verification.
Electromagnetic models
• In principle, magnetostatic modeling is very well-suited for representing field within a
slotless PM machine, which is also confirmed in the thesis. The influence of the eddy currents induced in the rotor magnet and back iron as a result of armature-field fluctuations
is negligible, particularly for a rotor with a plastic-bonded magnet. This is demonstrated
by FE calculations in Section 7.5 and measurements in Section 9.5. On the other hand, it
is shown in Chapter 9 that eddy-current losses in the housing have a great influence on the
motor performance - both on the armature field and overall losses. This effect comes into
play in toroidally-wound machines placed in an electrically-conductive housing. Since
good thermal conductors used for cooling jackets and housings are also, as a rule, good
electrical conductors, the phenomenon of losses in the housing can be expected in most
toroidally-wound machines. Hence, the magnetostatic model cannot account for all important phenomena in a toroidally-wound machine and some improvement is needed for
a fully adequate model.
• 2D modeling in a plane perpendicular to the axis of rotation is usually sufficient for representation of rotating machines. However, axial flux leakage of the field of the permanent
magnet is very pronounced in the designed motor, mostly due to the presence of the small
extrusions of the rotor iron shaft made to protect the magnet on its axial sides. The leakage
effect causes a 10% reduction of the motor no-load voltage with respect to the 2D-model
(analytical and FEM) predictions.
• The model of the PM field is confirmed by magnetostatic 2D FEM simulations and the
field reduction influenced by the axial flux leakage is observed in 3D FE simulations.
The predictions of 3D FEM match the measured values of no-load voltage very closely:
177
178
Chapter 10. Conclusions and recommendations
this can also be ascribed to very accurate data on the magnet remanent field and recoil
permeability.
• The armature field model developed in the thesis is not fully suitable to represent a motor
with toroidal windings; as both FEM and measurements suggested, the model failed to
represent certain phenomena that have a strong impact on the test-motor performance.
Firstly, the model does not take into account the actual, toroidal distribution of the conductors since it neglects the magnetic field outside the stator iron. However, as seen in
the FE simulations and confirmed by the measurements, external leakage of the armature
field is immense, causing the motor phase inductance to be an order of magnitude higher
than analytically predicted. Secondly, as already mentioned, a magnetostatic model is not
suitable to represent a toroidally-wound machine in an electrically-conductive housing.
• The analytical model of the distorted flux density in a slotless PM machine with an eccentric rotor, based on the simple approximation of the relative permeance function (Section
3.5), gives similar predictions of the air-gap field in the machine as the models based
on conformal mapping and 2D FEM. The analytical expression for the unbalanced force
which proceeds from the simplified flux-density model significantly overestimates the
force (by about 50% to 60% with respect to 2D FEM; practical verification with the test
setup was not possible). However, this expression is still satisfactory for motor design
purposes - such a model is quite simple and, at the same time, accurate force predictions
are rarely needed.
• Manufacturer’s data were used to estimate losses in the stator iron core. These losses
could not be distinguished from the measurements of overall no-load losses because of
the strong impact of the eddy-current losses in the housing. However, the measurement
results, in combination with 3D FE simulations, indicated that the losses in the core are
somewhat higher than predicted by the manufacturer’s data.
• The prediction of the phase DC resistance agrees with the measured values, thereby verifying the model of the DC-conduction loss in the windings. Relatively large deviations
of measured resistances of different phases are noticed as a result of an uneven number of
turns due to errors in the manual winding process.
• The skin-effect influence on copper losses in the machine is neglected. Indeed, in conductors whose diameter is almost four times smaller than the copper skin depth at the
maximum electrical frequency the skin effect hardly has any impact.
• The eddy-current losses in the air-gap conductors of a slotless PM machine can become
extremely high unless the conductors are optimized. Analytical models show that the
eddy-current losses rise by the fourth power of the conductor-strand diameter while 2D
FEM confirms the analytical model predictions. The predictions are, however, neither
verified nor refuted by the measurements since these losses could not be separated from
accompanying, certainly far more dominant loss factors during the speed-decay tests.
10.1. Models presented in the thesis
179
• The proximity-effect loss in the conductors was also neglected as the amplitude of the
field of the neighboring conductors in a single conductor strand is expected to be much
smaller than the field of the permanent magnet. The measurements did not register any
significant influence of the proximity effect.
• No suitable model for representing air-friction loss in the designed machine was found.
Rather intricate study would be needed to develop a model for air flow and friction loss
in such an unconventional rotor supported by air bearings. According to the speed-decay
tests in Section 9.4 the models generally used for more conventional machines with slender rotors fail to account for the air-friction loss in the test machine.
• Rotor losses in the motor are neglected due to its large effective air gap and highlyresistive permanent magnet. This assumption is proven to be correct using both 2D FE
simulations and locked-rotor tests.
Structural models
• Modeling of stress in a rotating PM rotor is based on equations which can be found in
textbooks on structural mechanics. The suitability of 2D analytical models to represent a
PM rotor without magnet-pole spacers as a compound of concentric cylindrical regions is
demonstrated by Binder in [75]. This thesis shows that a model which assumes isotropic
behavior of the carbon-fiber retaining sleeve by assigning the properties in the direction
of fibers to all directions makes almost equally good predictions as a fully orthotropic
model of the rotor. Results of these two analytical models were compared to results of
2D FEM and agreement of the models is quite satisfactory.
• Results of 3D FE simulations of the actual rotor showed a greater discrepancy between the
predictions of different models, particularly in stresses at corners of different rotor parts;
naturally, 2D models cannot account for this. Still, the 3D simulations affirm that the 2D
isotropic analytical model is a good basis for the stress calculation and, more importantly,
for structural optimization even of such a complex rotor. It is reasonable to expect that
such an optimization approach would be even more appropriate for slender rotors with
an indeed concentric cylindrical structure. Finally, the very high tangential speed (≈270
m/s) attained with the test rotor confirms validity of the optimization approach.
Rotordynamic models
• Critical speeds of a high-speed rotor (represented as a cylindrical Timoshenko beam) are
correlated with the rotor slenderness and bearing stiffness in Section 5.4. This analytical
modeling has not been validated in the thesis; however, the results comply with FEM
calculations available in literature [137]. The modeling was carried out primarily to show
the dependence of flexural critical speeds on machine parameters. These critical speeds
are far above the operating speed range of the test motor and validation of these models
180
Chapter 10. Conclusions and recommendations
was not possible. The prediction of the rigid critical speed is confirmed by the tests on
the setup (Section 9.6).
10.2
Speed limits of permanent magnet machines
The identification and parameterization of the speed limits of PM machines was set as one of
the thesis’ main objectives. Here, a short recount of important limiting factors is given.
• The thermal limit is common to all machines. The thermal behavior of a machine depends
on power losses that are further dependent on current and magnetic loading, as well as
rotational speed. These parameters were theoretically correlated with machine size and
rated power in Section 2.6. Based on this study it is concluded that, if the cooling method
is maintained, it is not possible to gain power density by merely scaling down the machine
and increasing its rated speed. Slotless PM machines show the highest capability of speed
increase through downsizing; this trend is also confirmed by the empirical study reported
in Section 2.4.
• In order to prevent high tension in the magnet and ensure the transfer of torque from the
magnet to the shaft, high-speed PM rotors are usually enclosed with strong non-magnetic
retaining sleeves. In this type of high-speed rotors, the critical, thus limiting stresses are
radial (contact) stress at the magnet-iron boundary and tangential stress (tension) at the
sleeve inner surface. It was shown in Section 4.4 that for an expected operating temperature there is an optimal value of the interference fit between the sleeve and magnet for
which both tension and contact limits are reached at an equal rotational speed. This speed
can be adjusted by the enclosure thickness so that the theoretical maximum rotational
speed is a considerable margin higher than the operating speed.
• Rotation can become unstable in the supercritical regime of a certain vibration mode if
rotating damping of the rotor-bearing system affects that mode. Rotors are usually stable
in a supercritical speed range which corresponds to rigid-body vibrational modes (with
the exception of rotors in fluid journal bearings) and today’s high-speed rotors operate
regularly in that speed range. On the other hand, rotors which possess some internal
damping can easily become unstable in a supercritical range corresponding to flexural
modes. Rotors of electrical machines are receptive to eddy-currents, always comprised of
fitted elements and often contain materials, such as composites, with significant material
damping. Therefore, these rotors are prone to be unstable in flexural supercritical regimes.
The first flexural critical speed practically represents the rotordynamical speed limit of an
electrical machine.
10.3. Design evaluation
10.3
181
Design evaluation
High-speed PM motor
• Loss minimization was taken as the ultimate criterion for the motor electromagnetic design. More precisely, the main intention of the design was to mitigate the overheating
of the motor that would result from frequency-dependent losses. That goal is fulfilled
when considering electromagnetic losses within the motor: chosen materials, conceptual
design and electromagnetic optimization all resulted in a relatively simple machine with
very low electromagnetic losses. The slotless design and use of a plastic-bonded magnet
and carbon fibers in the rotor resulted in negligible eddy-current losses in the rotor.
• Air-friction loss is rather high given the rotor volume, as expected given the very large
diameter of the rotor disc. However, very strong turbulences, influenced partly by the air
bearings, very effectively removed the ensuing heat from the rotor and the temperature
in the air gap remained quite moderate even after spinning at very high tangential speeds
(up to 270 m/s).
• From the efficiency perspective, losses in the aluminum housing are the weakest point
of the design; along with air friction, eddy currents in the housing represent the most
dominant loss factor in the setup, greatly influencing drive performance and temperature. Although practically inevitable, these losses could have been mitigated with more
adequate modeling and dedicated optimization of the stator and housing geometry.
• Unevenly wound coils in motor phases caused an unnecessary increase of the armature
field and losses in the stator core. This phenomenon implies that an automatic and highly
repetitive winding process is strongly recommended for toroidally-wound machines.
• The rotor structural design facilitated the extremely high tangential speed of the rotor. On
the other hand, the design consisting of a few press-fitted elements caused the problem of
the recurring unbalance which, eventually, resulted in the bearing crash.
• The plastic-bonded magnet proved to be extremely resistive to eddy-current losses, which
is a great advantage for high-speed applications. The relatively small remanent field of
this magnet type is suitable for most high-speed machines since their optimum air-gap
flux density is usually low. However, the possibility of plastic deformation and creep of
the polymer material after long hours of operation has remained a great concern.
Air-bearings setup
• Apparently, static air bearings represent a good solution for stable high-speed operation:
thresholds of their rotational instability are higher than those of aerodynamic and lubricated bearings. At the same time, very tight tolerances of air bearings impose high
precision standards on rotor manufacturing.
182
Chapter 10. Conclusions and recommendations
• The experience with the motor tests indicates that an assembled rotor prone to unbalance
and poorly-damped bearings with very small clearance are not an auspicious match. Such
a rotor would require either large damping in the bearings to suppress the oscillations at
the critical speed or bearings with large air gaps, such as active magnetic bearings, so that
the rotor could benefit from self-aligning. On the other hand, a robust rotor with much
tighter tolerances would be appropriate for rotation in air bearings.
10.4
Thesis contributions
Whole thesis
• In this thesis, phenomena, both mechanical and electromagnetic, that take precedence in
high-speed permanent magnet machines are identified and systematized. The attribute
high-speed is, for the thesis purposes, defined in Section 2.3 and refers to variable-speed
PM machines of small and medium power (typically below 500 kW) that have high speed
with respect to their power. A majority of the analyses found in the thesis is applicable
to a broader range of (non-PM) high-speed electrical machines.
• The thesis identifies inherent (physical) speed limits of permanent magnet machines and
correlates those limits with the basic parameters of the machines. The analytical expression of the limiting quantities does not only impose solid constraints on the machine
design, but often also paves the way for design optimization, leading to the maximum
mechanical and/or electromagnetic utilization of the machine. Simply put, the analytical
expressions indicate where the optimum lies. The most evident example of this can be
seen in the rotor structural optimization in Chapter 4.
• The electromagnetic, structural (elastic) and rotordynamical modeling of a (slotless) highspeed permanent magnet machine is presented in the thesis. By juxtaposing the models
of different, yet equally important physical aspects of high-speed PM machines, emphasizing relevant machine parameters and presenting those parameters in a logical way, the
thesis represents a comprehensive resource for the design of high-speed permanent magnet machines.
• A low-stiffness high-speed spindle drive - a toroidally-wound slotless PM motor with a
short rotor - was designed and realized in a practical setup. The design and its practical evaluation offer insight into the merits and drawbacks of using short (gyroscopic)
rotor, plastic-bonded magnets and toroidal windings for high-speed machines. Important
conclusions are drawn from this design experience and they are listed in Section 10.3.
Chapter 2
• Section 2.6 theoretically correlates rated speed, power and size of PM machines, taking
into account their physical limits. This theoretical study is supported by an empirical
10.4. Thesis contributions
183
survey of the correlation between rated power and speeds of existing high-speed machines
(Section 2.4).
Chapter 3
• In Section 3.5 an approximate analytical expression for distorted magnetic field in an
eccentric-rotor machine is used to determine the unbalanced magnetic force and stiffness
of slotless PM machines. The effectiveness of such a model in representing the field in the
air gap and unbalanced force is compared to the results of a model based on conformal
mapping [108] (the method frequently reported to provide accurate results) and 2D FEM.
It is shown in the section that the simplified model gives predictions of magnetic flux
density in the air gap similar to the predictions of the conformal-mapping method and 2D
FEM and that the prediction of the force is, despite noticeable overestimation, useful and
effective for machine-design purposes.
• Using analytical models derived by Ferreira [89, 112], Subsection 3.6.2 distinguishes the
dominant causes of copper losses in slotless PM machines. Additionally, the section
adapts Ferreira’s equation to calculate eddy-current losses in the air-gap conductors of a
slotless machine and correlates that expression to a simplified formula reported in literature (e.g. [90]).
Chapter 4
• Structural limits for speed of PM rotors are identified and the limiting parameters (stresses
at the rotor material boundaries) are represented in a simple analytical form that clearly
indicates optimal geometry of the rotor retaining sleeve. In this way, a relatively simple
approach of optimizing the retaining sleeve is achieved; the approach takes into account
the influence of rotational speed, mechanical fittings and operating temperature on stress
in a high-speed rotor (Section 4.4).
Chapter 5
• The dynamic of a rotor-bearings system and its important aspects - stability of rotation
and critical speeds - presented in this chapter are fairly known and well-researched in
the field of rotordynamics; most models can be found in textbooks. The importance
of this chapter is that it highlights the effects of those phenomena relevant to rotational
stability of electrical machines. In Section 5.3 rotordynamical stability limits of speed
of electrical machines are identified. Section 5.4 analytically correlates critical speeds of
a rotor-bearings system using the Timoshenko beam model; the section shows that the
value of the first flexural critical speed (= the limiting speed of an electrical machine)
depends solely on rotor slenderness. The correlation is supported by FE models reported
in literature [137].
184
Chapter 10. Conclusions and recommendations
Chapter 7
• Section 7.2 reports a new spindle-drive concept: a PM motor with a short rotor supported
by frictionless (AMB/aerostatic) bearings (together with M. Kimman [175]).
• In Subsection 7.4.3 exact analytical formulas for the optimization of air-gap conductors
in slotless PM machines are derived.
• An innovative design of a retaining sleeve for a short PM rotor is presented in Section
7.6; the design consists of a combination of glass- and carbon-fiber retaining rings.
Chapter 8
• The chapter presents a new, open-loop control method for very-high-speed PM motors:
an I/f motor controller with reference-frequency modulation. The method is successfully
implemented in the test setup.
Chapter 9
• The measurements and FEM presented in the chapter observe certain weaknesses of
toroidally-wound machines that have not been emphasized in literature, such as susceptibility to losses in the housing and the necessity of having a flawless winding process in
order to avoid excessive core losses.
• The developed motor reached the highest tangential speed of an electrical machine that
has yet been reported in academic literature (Section 9.6).
• The test setup demonstrates the suitability of aerostatic bearings for very-high-speed operation: a rotor in aerostatic bearings can operate well above twice the critical speed.
This renders limits of their rotational stability higher than the limits of aerodynamic and
lubricated journal bearings. (Section 9.6).
10.5
Recommendations
• The representation of the machine in the thesis clearly lacked a thermal model; the designing was, therefore, too cautious and the author refrained from investigating possible
thermal limits of the motor in the given setup. The main question which remains unanswered is how to treat the complex fluid/thermal flow in the air gap in a pragmatic way
which would be suitable for machine design that would lead to maximum utilization of a
high-speed machine. A possible solution that could be explored is to simply bypass the
air gap in the thermal modeling - apparently, turbulent air flow at very high speeds is a
very efficient means of heat transfer and the effective thermal resistance of the air gap is
expected to be very small.
10.5. Recommendations
185
• The modeling of the armature field of a toroidally-wound machine should be improved
with respect to the model developed in this thesis. Phenomena that need to be addressed
are eddy-current losses in electrically-conductive housing and external leakage of the
armature field.
• The presented structural design of the rotor followed the electromagnetic design of the
machine. However, the analytical approach for the sleeve optimization lends itself to inclusion into a simultaneous, structural and electromagnetic optimization process. Namely,
for defined dimensions of the permanent magnet there is an optimal value of the interference fit and a necessary sleeve thickness for reaching requested rotational speed. If this
correlation is considered during the machine design, electromagnetic performance requirements (torque, power, losses) can be achieved with an optimal structural design of
the rotor for the desired operating speed.
• The thesis has shown suitability of using plastic-bonded magnets for high-speed machines
from the electromagnetic perspective. More research is, however, needed to investigate
their suitability with respect to their mechanical properties. In particular, it is important
to research whether plastic-bonded magnets can maintain their structural integrity after
long hours of operation.
• Practical realization of the proposed short-rotor spindle concept with 5DOF active magnetic bearings would be very interesting. In such bearings, the rotor could benefit from
self-aligning at high speeds and, provided that active position control of a gyroscopic rotor is achievable, the concept would result in an extremely compact and powerful spindle
drive.
186
Chapter 10. Conclusions and recommendations
Appendix A
Structural relationships in a rotating
cylinder
Governing differential equation for radial displacement in a rotating cylinder (4.7) is derived
from equations (4.2), (4.5) and (4.6) in Chapter 4. The first equation is a constitutive structural
relation for linear materials - Hooke’s law - expressed for the plane stress/strain conditions. The
second and third equation represent kinematic and force-equilibrium equation and they will be
shortly explained here.
u ( r + dr )
Ω
dFc = ρ r Ω 2 dV
σ r ( r + dr ) ⋅ ( r + dr ) dθ dz
dθ/2
u (r )
σ θ ( r ) ⋅ drdz
C
dz
dθ/2
dr
σ r ( r ) ⋅ rdθ dz
σ θ ( r ) ⋅ drdz
dθ
r
Figure A.1: A small element of a rotating cylinder
Figure A.1 presents a small element of a axisymmetrical rotating cylinder at a radius r,
along with directions and intensities of forces acting on that element. For symmetry reasons, all
the structural variables - stress, strain and displacement - are functions of the radius r only, thus
187
188
Appendix A. Structural relationships in a rotating cylinder
independent of the angle θ. Additionally, all shear stress components are zero due to the same
symmetry.
As a result of the centrifugal force, all the points in the cylinder undergo a displacement,
mainly in the radial direction (ur = u(r)) and certainly not in the tangential direction (uθ = 0).
Strain represents the relative displacement of the cylinder particles; since no shear components
are present, the strain components can be simply calculated as:
u (r + dr) − u (r) ∂u (r) du (r)
=
=
,
dr→0
dr
∂r
dr
εr = lim
(A.1)
(r + u (r)) dθ − rdθ u (r)
=
.
(A.2)
dθ→0
rdθ
r
By referring to Figure A.1, one can formulate the equilibrium of forces acting on the given
element in radial direction:
εθ = lim
σr (r + dr) · (r + dr) dθdz − σr (r) · rdθdz + 2 · σθ (r) · drdz ·
dθ
+ ρr2 Ω2 · drdθdz = 0. (A.3)
2
After higher order differential terms in (A.3), the force-equilibrium equation is obtained:
r
dσr
+ σr − σθ + ρr2 Ω2 = 0.
dr
(A.4)
Appendix B
One explanation of rotor instability
Figure B.1 displays a cross-section of a Jeffcott rotor which rotates at rotational frequency Ω
and, at the same time, whirls around the center of the bearings at frequency equal to the critical
frequency Ωw = Ωcr . The rotor is subjected to rotating and non-rotating viscous damping. The
force from non-rotating damping is proportional to the translational speed of the rotor in the
stationary reference frame xy:
−−−→
F = −c −
v→,
(B.1)
nrd
n xy
while the force from rotating damping is proportional to the rotor speed in the vu frame whose
origin coincides with the bearing center and which rotates at the rotor frequency Ω, thus:
−−→
Frd = −cr −
v→
uv .
(B.2)
(Mind that in a real case rotor displacement rC is far smaller than the rotor radius.)
Correlation between translational speeds in two reference frames yields:
− →
−
−→ →
−
v→
uv = v xy − Ω × rC
y
y
y≡v
u
u
v
v
Ω
rC
Ωt
x
=
Ωcr
rC
Ωt
x
Ωw = Ωcr
a)
Ω
Frd
Ωw
rC
(B.3)
+
x≡u
Ωw = 0
b)
Ω=0
c)
Figure B.1: Viscous forces acting on a whirling rotor
189
Fnrd
190
Appendix B. One explanation of rotor instability
Forces in the example of rotating and whirling rotor (Fig. B.1 a)) can be obtained as a
superposition of forces in two examples: b) the rotating rotor with a fixed displacement rC and
c) the non-rotating rotor whirling at the critical rotational frequency.
In the first example non-rotating example has no effect since the rotor is not moving with
respect to the stationary frame. Force resulting from the rotating damping is then (according to
reference direction in Fig. B.1 b)):
Frd = cr ΩrC .
(B.4)
In the example of only-whirling rotor (Fig. B.1 c)) the reference frames coincide and both
dampings are active. Force on the rotor in this case is:
Fnrd = (cr + cn ) Ωcr rC .
(B.5)
Apparently, the force in the first example is in the direction of whirling while the force in the
second example resists the whirling thus is restoring. If the restoring force dominates, the whirl
loses the energy and the motion is stable; conversely, if the total force supports the vibration,
the whirl will gain energy and increase the amplitude. Therefore, the condition of the rotation
stability is:
Frd < Fnrd ,
(B.6)
which is equivalent to:
Ω < Ωcr
!
cn
.
1+
cr
(B.7)
The expression (B.7) is identical to the stability limit analytically obtained in Section 5.3.
This intuitive example also demonstrates stability of the backward whirl. Namely, if the
whirling has the direction opposite to the direction assumed in Fig. B.1 the non-rotating viscous
force will also oppose the whirling and the movement is always stable.
Appendix C
Stator core properties
Figure C.1: Hysteresis loops of 10JNEX900 Si-steel [204]
191
192
Appendix C. Stator core properties
Figure C.2: Hysteresis loops and loss curves of 10JNEX900 Si-steel [204]
Bibliography
[1] J. Chae, S. Park, and T. Freiheit, “Investigation of micro-cutting operations,” International Journal of Machine Tools and Manufacture, vol. 46, no. 3-4, pp. 313 – 332, 2006.
[2] D. Bourell, K. F. Ehmann, M. L. Culpepper, T. J. Hodgson, T. R. Kurfess, M. Madou,
K. Rajurkar, and R. E. DeVor, International Assessment of Research and Development in
Micromanufacturing, World Technology Evaluation Center, Baltimore, Maryland, 2005.
[3] H. Wicht and J. Bouchaud, NEXUS Market Analysis for MEMS and Microsystems III
2005-2009, Setting the Pace for Micro Assembly Solutions, 2005.
[4] X. Luo, K. Cheng, D. Webb, and F. Wardle, “Design of ultraprecision machine tools with
applications to manufacture of miniature and micro components,” Journal of Materials
Processing Technology, vol. 167, no. 2-3, pp. 515–528, 2005.
[5] T. Masuzawa, “State of the art of micromachining,” CIRP Annals-Manufacturing Technology, vol. 49, no. 2, pp. 473–488, 2000.
[6] G. L. Benavides, D. P. Adams, and P. Yang, Meso-machining capabilities, Sandia Report,
2001.
[7] D. Huo, K. Cheng, and F. Wardle, “Design of a five-axis ultra-precision micro-milling
machine - UltraMill. Part 1: holistic design approach, design considerations and specifications,” The International Journal of Advanced Manufacturing Technology, vol. 47, no.
9-12, pp. 867–877, 2009.
[8] Y. Okazaki, N. Mishima, and K. Ashida, “Microfactory-concept, history, and developments,” Journal of manufacturing science and engineering, vol. 126, p. 837, 2004.
[9] Y. Bang, K. Lee, and S. Oh, “5-axis micro milling machine for machining micro parts,”
The International Journal of Advanced Manufacturing Technology, vol. 25, no. 9, pp.
888–894, 2005.
[10] S. K. R. D. K. E. M.P. Vogler, X. Liu, “Development of meso-scale machine tool (mMT)
systems,” Transactions of NAMRI/SME, vol. 30, pp. 653–661, 2002.
[11] E. Kussul, T. Baidyk, L. Ruiz-Huerta, A. Caballero-Ruiz, G. Velasco, and L. Kasatkina,
“Development of micromachine tool prototypes for microfactories,” Journal of Micromechanics and Microengineering, vol. 12, p. 795, 2002.
193
194
Bibliography
[12] S. W. Lee, R. Mayor, and J. Ni, “Dynamic analysis of a mesoscale machine tool,” Journal
of manufacturing science and engineering, vol. 128, p. 194, 2006.
[13] H. Li, X. Lai, C. Li, Z. Lin, J. Miao, and J. Ni, “Development of meso-scale milling machine tool and its performance analysis,” Frontiers of Mechanical Engineering in China,
vol. 3, no. 1, pp. 59–65, 2008.
[14] Y. Takeuchi, Y. Sakaida, K. Sawada, and T. Sata, “Development of a 5-axis control ultraprecision milling machine for micromachining based on non-friction servomechanisms,”
CIRP Annals-Manufacturing Technology, vol. 49, no. 1, pp. 295–298, 2000.
[15] J.-K. Park, S.-K. Ro, B.-S. Kim, J.-H. Kyung, W.-C. Shin, and J.-S. Choi, “A precision
meso scale machine tools with air bearings for microfactory,” 5th Int. Workshop on Microfactories, Besancon, France, 2006.
[16] Nanowave, Nano Corporation, 2006. [Online]. Available: http://www.nanowave.co.jp/
index e.html
[17] Robonano α-0iA brochure, Fanuc Ltd. [Online]. Available: http://www.fanuc.co.jp/en/
product/robonano/index.htm
[18] R. Blom, M. Kimman, H. Langen, P. v. d. Hof, and R. M. Schmidt, “Effect of miniaturization of magnetic bearing spindles for micro-milling on actuation and sensing bandwidths,” Proceedings of the Euspen International Conference, EUSPEN 2008, Zurich,
Switzerland, 2008.
[19] D. Dornfeld, S. Min, and Y. Takeuchi, “Recent advances in mechanical micromachining,”
CIRP Annals - Manufacturing Technology, vol. 55, no. 2, pp. 745 – 768, 2006.
[20] A. G. Phillip, S. G. Kapoor, and R. E. DeVor, “A new acceleration-based methodology
for micro/meso-scale machine tool performance evaluation,” International Journal of
Machine Tools and Manufacture, vol. 46, no. 12-13, pp. 1435–1444, 2006.
[21] T. Schaller, L. Bohn, J. Mayer, and K. Schubert, “Microstructure grooves with a width
of less than 50 µm cut with ground hard metal micro end mills,” Precision Engineering,
vol. 23, pp. 229–235, 1999.
[22] C. R. Knospe, “Active magnetic bearings for machining applications,” Control Engineering Practice, vol. 15, no. 3, pp. 307–313, 2007.
[23] C. W. Lin, J. F. Tu, and J. Kamman, “An integrated thermo-mechanical-dynamic model
to characterize motorized machine tool spindles during very high speed rotation,” International Journal of Machine Tools and Manufacture, vol. 43, no. 10, pp. 1035–1050,
2003.
[24] A. Binder and T. Schneider, “High-speed inverter-fed ac drives,” Electrical Machines
and Power Electronics, 2007. ACEMP ’07. International Aegean Conference on, pp. 9
–16, 10-12 2007.
Bibliography
195
[25] A. Borisavljevic, H. Polinder, and J. Ferreira, “On the speed limits of permanent-magnet
machines,” Industrial Electronics, IEEE Transactions on, vol. 57, no. 1, pp. 220 –227,
jan. 2010.
[26] H. Langen, “Microfactory research topics in the netherlands,” The 5th Int. Workshop on
Microfactories, Besancon, France, Oct 2006.
[27] J. F. Gieras, “High speed machines,” J. F. Gieras, Ed. Springer, 2008.
[28] High Speed, Calnetix Inc. [Online]. Available: http://www.calnetix.com/highspeed.cfm
[29] L. Zhao, C. Ham, L. Zheng, T. Wu, K. Sundaram, J. Kapat, L. Chow, and C. Siemens,
“A highly efficient 200 000 rpm permanent magnet motor system,” IEEE Transactions
on Magnetics, vol. 43, no. 6, pp. 2528–2530, 2007.
[30] C. Zwyssig, J. W. Kolar, and S. D. Round, “Megaspeed drive systems: Pushing beyond
1 million r/min,” IEEE/ASME Trans. on Mechatronics, vol. 14, no. 5, 2009.
[31] P. Mellor, M. Al-Taee, and K. Binns, “Open loop stability characteristics of synchronous
drive incorporating high field permanent magnet motor,” Electric Power Applications,
IEE Proceedings B, vol. 138, no. 4, pp. 175 –184, jul 1991.
[32] M. Kimman, H. Langen, J. van Eijk, and R. Schmidt, “Design and realization of a miniature spindle test setup with active magnetic bearings,” Advanced intelligent mechatronics,
2007 IEEE/ASME international conference on, pp. 1 –6, 4-7 2007.
[33] Z. Zhu, K. Ng, and D. Howe, “Design and analysis of high-speed brushless permanent
magnet motors,” Electrical Machines and Drives, 1997 Eighth International Conference
on (Conf. Publ. No. 444), pp. 381–385, 1-3 1997.
[34] K. Binns and D. Shimmin, “Relationship between rated torque and size of permanent
magnet machines,” Electric Power Applications, IEE Proceedings -, vol. 143, no. 6, pp.
417 –422, nov 1996.
[35] S. Trout, “Rare earth magnet industry in the USA: Current status and future trends,” 17th
Int. Workshop on Rare Earth Magnets and Their Applications, University of Delaware,
2002.
[36] T. Shimoda, “A prospective observation of bonded rare-earth magnets,” IEEE Translation
Journal on Magnetics in Japan, vol. 8, no. 10, pp. 701–710, 1993.
[37] P. Campbell, “Magnet price performance,” Magnetic Business and Technology, 2007.
[38] W. Rodewald and M. Katter, “Properties and applications of high performance magnets,”
18th international workshop on high performance magnets and their applications, pp.
52–63, 2004.
196
Bibliography
[39] W. Pan, W. Li, L. Y. Cui, X. M. Li, and Z. H. Guo, “Rare earth magnets resisting eddy
currents,” IEEE Transactions on Magnetics, vol. 35, no. 5 Part 2, pp. 3343–3345, 1999.
[40] E. A. Setiawan, “Dynamics behavior of a 30 kw capstone microturbine,” Institut fuer
Solare Energieversorgungstechnik eV (ISET), 2007, Kassel, Germany.
[41] K. Isomura, M. Murayama, H. Yamaguchi, N. Ijichi, N. Saji, O. Shiga, K. Takahashi,
S. Tanaka, T. Genda, and M. Esashi, “Development of micro-turbo charger and microcombustor as feasibility studies of three-dimensional gas turbine at micro-scale,” ASME
Conference Proceedings, vol. 2003, pp. 685–690, 2003.
[42] J. Peirs, D. Reynaerts, and F. Verplaetsen, “Development of an axial microturbine for
a portable gas turbine generator,” Journal of Micromechanics and Microengineering,
vol. 13, pp. S190–S195, 2003.
[43] C. Zwyssig, S. D. Round, and J. W. Kolar, “An ultra-high-speed, low power electrical
drive system,” IEEE Trans. on Industrial Electronics, vol. 55, no. 2, pp. 577–585, 2008.
[44] R. Lasseter and P. Paigi, “Microgrid: a conceptual solution,” Power Electronics Specialists Conference, 2004. PESC 04. 2004 IEEE 35th Annual, vol. 6, pp. 4285 – 4290 Vol.6,
20-25 2004.
[45] P. A. Pilavachi, “Mini-and micro-gas turbines for combined heat and power,” Applied
Thermal Engineering, vol. 22, no. 18, pp. 2003–2014, 2002.
[46] MTT recuperated micro gas turbine for micro CHP systems, Micro Turbine Technology
BV. [Online]. Available: http://www.mtt-eu.com/
[47] G. J. Atkinson, “High power fault tolerant motors for aerospace applications,” Ph.D.
dissertation, University of Newcastle upon Tyne, 2007.
[48] S. M. Jang, S. S. Jeong, D. W. Ryu, and S. K. Choi, “Comparison of three types of pm
brushless machines for an electro-mechanical battery,” IEEE Transactions on Magnetics,
vol. 36, no. 5, pp. 3540–3543, 2000.
[49] A. Nagorny, N. Dravid, R. Jansen, and B. Kenny, “Design aspects of a high speed permanent magnet synchronous motor / generator for flywheel applications,” Electric Machines
and Drives, 2005 IEEE International Conference on, pp. 635 –641, 15-15 2005.
[50] A. S. Nagorny, R. H. Jansen, and D. M. Kankam, “Experimental performance evaluation
of a high-speed permanent magnet synchronous motor and drive for a flywheel application at different frequencies,” Proceedings of 17th International Conference on Electrical
Machines - ICEM, 2006.
[51] M. Rahman, A. Chiba, and T. Fukao, “Super high speed electrical machines - summary,”
Power Engineering Society General Meeting, 2004. IEEE, pp. 1272 –1275 Vol.2, 10-10
2004.
Bibliography
197
[52] A. Maeda, H. Tomita, and O. Miyashita, “Power and speed limitations in high speed
electrical machines,” Proceedings of IPEC, Yokohama, Japan, pp. 1321–1326, 1995.
[53] J. Oliver, M. Samotyj, and R. Ferrier, “Application of high-speed, high horsepower, asd
controlled induction motors to gas pipelines,” 5th European Conference on Power Electronics and Applications, EPE ’93, pp. 430–434, 1993.
[54] Danfoss Turbocor Compressors, Danfoss Turbocor. [Online]. Available:
//www.turbocor.com/
http:
[55] Ultra-High Speed Motors & Generators, SatCon Applied Technology. [Online].
Available: http://www.satcon.com/apptech/mm/uhs.php
[56] Permanent magnet rotor with CFRP rotor sleeve, e+a Elektromaschinen und Antriebe
AG. [Online]. Available: http://www.eunda.ch/
[57] M. Caprio, V. Lelos, J. Herbst, and J. Upshaw, “Advanced induction motor endring design features for high speed applications,” Electric Machines and Drives, 2005 IEEE
International Conference on, pp. 993 –998, 15-15 2005.
[58] P. Beer, J. E. Tessaro, B. Eckels, and P. Gaberson, “High-speed motor design for gas
compressor applications,” Proceeding of 35th Turbomachinery Symposium, pp. 103–112,
2006.
[59] M. M. Harris, A. C. Jones, and E. J. Alexander, “Miniature turbojet development at
hamilton sundstrand the TJ-50, TJ-120 and TJ-30 turbojets,” 2nd AIAA ”Unmanned Unlimited” Systems, Technologies, and Operations Aerospace, Land, and Sea Conference
and Workshop & Exhibit, pp. 15–18, 2003.
[60] The smallest drive system in the world, Faulhaber Group, 2004. [Online]. Available:
http://www.faulhaber.com/n223666/n.html
[61] ATE Micro Drives, ATE Systems. [Online]. Available: http://www.ate-system.de/en/
products/ate-micro-drives.html
[62] U. Schroder, “Development and application of high speed synchronous machines on active magnetic bearings,” Proceedings of MAG’97, Industrial Conference and Exhibition
on Magnetic Bearings, Alexandria, Virginia, p. 79, August, 1997.
[63] M. Ahrens, U. Bikle, R. Gottkehaskamp, and H. Prenner, “Electrical design of highspeed induction motors of up to 15 mw and 20000 rpm,” Power Electronics, Machines
and Drives, 2002. International Conference on (Conf. Publ. No. 487), pp. 381 – 386, 4-7
2002.
[64] M. Larsson, M. Johansson, L. Naslund, and J. Hylander, “Design and evaluation of highspeed induction machine,” Electric Machines and Drives Conference, 2003. IEMDC’03.
IEEE International, vol. 1, pp. 77 – 82 vol.1, 1-4 2003.
198
Bibliography
[65] B.-H. Bae, S.-K. Sul, J.-H. Kwon, and J.-S. Shin, “Implementation of sensorless vector
control for super-high speed pmsm of turbo-compressor,” Industry Applications Conference, 2001. Thirty-Sixth IAS Annual Meeting. Conference Record of the 2001 IEEE,
vol. 2, pp. 1203 –1209, 30 2001.
[66] M. Aoulkadi, A. Binder, and G. Joksimovic, “Additional losses in high-speed induction
machine - removed rotor test,” Power Electronics and Applications, 2005 European Conference on, pp. 10 pp. –P.10, 0-0 2005.
[67] H.-W. Cho, S.-M. Jang, and S.-K. Choi, “A design approach to reduce rotor losses in
high-speed permanent magnet machine for turbo-compressor,” Magnetics, IEEE Transactions on, vol. 42, no. 10, pp. 3521 –3523, oct. 2006.
[68] I. Takahashi, T. Koganezawa, G. Su, and K. Ohyama, “A super high speed pm motor drive
system by a quasi-current sourceinverter,” IEEE Transactions on Industry Applications,
vol. 30, no. 3, pp. 683–690, 1994.
[69] PCB Spindles, Westwind Air Bearings.
westwind-airbearings.com/pcb/index.html
[Online].
Available:
http://www.
[70] J. Oyama, T. Higuchi, T. Abe, K. Shigematsu, and R. Moriguchi, “The development of
small size ultra-high speed drive system,” Power Conversion Conference - Nagoya, 2007.
PCC ’07, pp. 1571 –1576, 2-5 2007.
[71] Oil-free, motorized compressor systems, Mohawk Innovative Technology. [Online].
Available: http://www.miti.cc/
[72] J. Bumby, E. Spooner, J. Carter, H. Tennant, G. Mego, G. Dellora, W. Gstrein, H. Sutter, and J. Wagner, “Electrical machines for use in electrically assisted turbochargers,”
Power Electronics, Machines and Drives, 2004. (PEMD 2004). Second International
Conference on (Conf. Publ. No. 498), vol. 1, pp. 344 – 349 Vol.1, 31 2004.
[73] BorgWarner Turbo & Emission Systems. [Online]. Available:
bwauto.com
http://www.turbos.
[74] C. Zwyssig, M. Duerr, D. Hassler, and J. W. Kolar, “An Ultra-High-Speed, 500000 rpm,
1 kW electrical drive system,” Proceedings of Power Conversion Conference - PCC’07,
pp. 1577–1583, 2007.
[75] A. Binder, T. Schneider, and M. Klohr, “Fixation of buried and surface-mounted magnets
in high-speed permanent-magnet synchronous machines,” Industry Applications, IEEE
Transactions on, vol. 42, no. 4, pp. 1031 –1037, july-aug. 2006.
[76] O. Aglen and A. Andersson, “Thermal analysis of a high-speed generator,” Industry Applications Conference, 2003. 38th IAS Annual Meeting. Conference Record of the, vol. 1,
pp. 547 – 554 vol.1, 12-16 2003.
Bibliography
199
[77] N. Bianchi, S. Bolognani, and F. Luise, “Potentials and limits of high-speed pm motors,”
Industry Applications, IEEE Transactions on, vol. 40, no. 6, pp. 1570 – 1578, nov.-dec.
2004.
[78] G. Slemon, “On the design of high-performance surface-mounted pm motors,” Industry
Applications, IEEE Transactions on, vol. 30, no. 1, pp. 134 –140, 1994.
[79] N. Bianchi, S. Bolognani, and F. Luise, “High speed drive using a slotless pm motor,”
Power Electronics, IEEE Transactions on, vol. 21, no. 4, pp. 1083 – 1090, july 2006.
[80] R. Larsonneur, “Design and control ot active magnetic bearing systems tor high speed
rotation,” Ph.D. dissertation, Swiss Federal Institute of Technology Zurich, 1990.
[81] C. Zwyssig and J. W. Kolar, “Design considerations and experimental results of a 100
W, 500 000 rpm electrical generator,” Journal of Micromechanics and Microengineering,
vol. 16, no. 9, pp. 297–307, 2006.
[82] T. Wang, F. Wang, H. Bai, and J. Xing, “Optimization design of rotor structure for high
speed permanent magnet machines,” Electrical Machines and Systems, 2007. ICEMS.
International Conference on, pp. 1438 –1442, 8-11 2007.
[83] T. Iwatsubo, “Stability problems of rotor systems,” Shock and Vibration Information
Center The Shock and Vibration Inform. Digest, vol. 12, no. 7, 1980.
[84] J. Melanson and J. W. Zu, “Free vibration and stability analysis of internally damped
rotating shafts with general boundary conditions,” Journal of Vibration and Acoustics,
vol. 120, p. 776, 1998.
[85] W. Kim, A. Argento, and R. A. Scott, “Forced vibration and dynamic stability of a rotating tapered composite timoshenko shaft: Bending motions in end-milling operations,”
Journal of Sound and Vibration, vol. 246, no. 4, pp. 583–600, 2001.
[86] G. Genta, Dynamics of rotating systems. Springer Verlag, 2005.
[87] V. Kluyskens, B. Dehez, and H. B. Ahmed, “Dynamical electromechanical model for
magnetic bearings,” IEEE Transactions on Magnetics, vol. 43, no. 7, pp. 3287–3292,
2007.
[88] A. Grauers and P. Kasinathan, “Force density limits in low-speed pm machines due to
temperature and reactance,” Energy Conversion, IEEE Transactions on, vol. 19, no. 3,
pp. 518 – 525, sept. 2004.
[89] J. Ferreira, “Improved analytical modeling of conductive losses in magnetic components,” Power Electronics, IEEE Transactions on, vol. 9, no. 1, pp. 127 –131, jan 1994.
[90] E. Spooner and B. Chalmers, “’TORUS’: a slotless, toroidal-stator, permanent-magnet
generator,” Electric Power Applications, IEE Proceedings B, vol. 139, no. 6, pp. 497
–506, nov 1992.
200
Bibliography
[91] J. Saari, “Thermal analysis of high-speed induction machines,” Ph.D. dissertation, Acta
Polytechnica Scandinavica, 1998.
[92] Z. Zhu, D. Howe, E. Bolte, and B. Ackermann, “Instantaneous magnetic field distribution in brushless permanent magnet dc motors. i. open-circuit field,” Magnetics, IEEE
Transactions on, vol. 29, no. 1, pp. 124 –135, jan. 1993.
[93] Z. Zhu and D. Howe, “Instantaneous magnetic field distribution in brushless permanent
magnet dc motors. ii. armature-reaction field,” Magnetics, IEEE Transactions on, vol. 29,
no. 1, pp. 136 –142, jan. 1993.
[94] ——, “Instantaneous magnetic field distribution in brushless permanent magnet dc motors. iii. effect of stator slotting,” Magnetics, IEEE Transactions on, vol. 29, no. 1, pp.
143 –151, jan. 1993.
[95] ——, “Instantaneous magnetic field distribution in permanent magnet brushless dc motors. iv. magnetic field on load,” Magnetics, IEEE Transactions on, vol. 29, no. 1, pp. 152
–158, jan. 1993.
[96] Z. Zhu, K. Ng, N. Schofield, and D. Howe, “Improved analytical modelling of rotor eddy
current loss in brushless machines equipped with surface-mounted permanent magnets,”
Electric Power Applications, IEE Proceedings -, vol. 151, no. 6, pp. 641 – 650, nov.
2004.
[97] H. Polinder, “On the losses in a high-speed permanent-magnet generator with rectifier,”
Ph.D. dissertation, Delft University of Technology, 1998.
[98] S. Holm, “Modelling and optimization of a permanent magnet machine in a flywheel,”
Ph.D. dissertation, Delft University of Technology, 2003.
[99] H. Polinder and M. Hoeijmakers, “Eddy-current losses in the segmented surfacemounted magnets of a pm machine,” Electric Power Applications, IEE Proceedings -,
vol. 146, no. 3, pp. 261 –266, may. 1999.
[100] H. Polinder, M. Hoeijmakers, and M. Scuotto, “Eddy-current losses in the solid back-iron
of pm machines for different concentrated fractional pitch windings,” Electric Machines
Drives Conference, 2007. IEMDC ’07. IEEE International, vol. 1, pp. 652 –657, may.
2007.
[101] C. Bi, Z. Liu, and T. Low, “Effects of unbalanced magnetic pull in spindle motors,”
Magnetics, IEEE Transactions on, vol. 33, no. 5, pp. 4080 –4082, sep. 1997.
[102] Z. Liu, C. Bi, Q. Zhang, M. Jabbar, and T. Low, “Electromagnetic design for hard disk
drive spindle motors with fluid film lubricated bearings,” Magnetics, IEEE Transactions
on, vol. 32, no. 5, pp. 3893 –3895, sep. 1996.
Bibliography
201
[103] D. Guo, F. Chu, and D. Chen, “The unbalanced magnetic pull and its effects on vibration
in a three-phase generator with eccentric rotor,” Journal of Sound and Vibration, vol.
254, no. 2, pp. 297–312, 2002.
[104] Z. Zhu, D. Ishak, D. Howe, and C. Jintao, “Unbalanced magnetic forces in permanentmagnet brushless machines with diametrically asymmetric phase windings,” Industry
Applications, IEEE Transactions on, vol. 43, no. 6, pp. 1544 –1553, nov. 2007.
[105] U. Kim and D. Lieu, “Magnetic field calculation in permanent magnet motors with rotor
eccentricity: without slotting effect,” Magnetics, IEEE Transactions on, vol. 34, no. 4,
pp. 2243 –2252, jul. 1998.
[106] S. Swann, “Effect of rotor eccentricity on the magnetic field in the air-gap of a nonsalient-pole machine,” Electrical Engineers, Proceedings of the Institution of, vol. 110,
no. 11, pp. 903 –915, nov. 1963.
[107] J. Li, Z. Liu, and L. Nay, “Effect of radial magnetic forces in permanent magnet motors
with rotor eccentricity,” Magnetics, IEEE Transactions on, vol. 43, no. 6, pp. 2525 –2527,
jun. 2007.
[108] J.-P. Wang and D. Lieu, “Magnetic lumped parameter modeling of rotor eccentricity in
brushless permanent-magnet motors,” Magnetics, IEEE Transactions on, vol. 35, no. 5,
pp. 4226 –4231, sep. 1999.
[109] F. Wang and L. Xu, “Calculation and measurement of radial and axial forces for a bearingless pmdc motor,” Industry Applications Conference, 2000. Conference Record of the
2000 IEEE, vol. 1, pp. 249 –252 vol.1, 2000.
[110] A. Krings and J. Soulard, “Overview and comparison of iron loss models for electrical
machines,” Journal of Electrical Engineering, vol. 10, no. 3, pp. 162–169, 2010.
[111] J. Ferreira, “Analytical computation of ac resistance of round and rectangular litz wire
windings,” Electric Power Applications, IEE Proceedings B, vol. 139, no. 1, pp. 21 –25,
jan. 1992.
[112] ——, Electromagnetic modeling of power electronic converters.
sachusetts: Kluwer Academic Publishers, 1989, ch. 6.
Norwell, Mas-
[113] E. Bilgen and R. Boulos, “Functional dependence of torque coefficient of coaxial
cylinders on gap width and reynolds numbers,” Journal of Fluids Engineering, vol. 95,
no. 1, pp. 122–126, 1973. [Online]. Available: http://link.aip.org/link/?JFG/95/122/1
[114] Y. Yamada, “Torque resistance of a flow between rotating co-axial cylinders having axial
flow,” Bulleting of JSME, vol. 5, no. 20, pp. 634–642, 1962.
[115] O. Aglen, “Loss calculation and thermal analysis of a high-speed generator,” Electric
Machines and Drives Conference, 2003. IEMDC’03. IEEE International, vol. 2, pp. 1117
– 1123 vol.2, jun. 2003.
202
Bibliography
[116] ——, “Back-to-back tests of a high-speed generator,” Electric Machines and Drives Conference, 2003. IEMDC’03. IEEE International, vol. 2, pp. 1084 – 1090 vol.2, jun. 2003.
[117] L. Zheng, T. Wu, D. Acharya, K. Sundaram, J. Vaidya, L. Zhao, L. Zhou, K. Murty,
C. Ham, N. Arakere, J. Kapat, and L. Chow, “Design of a super-high speed permanent
magnet synchronous motor for cryogenic applications,” Electric Machines and Drives,
2005 IEEE International Conference on, pp. 874 –881, 15-15 2005.
[118] M. Awad and W. Martin, “Windage loss reduction study for tftr pulse generator,” Fusion
Engineering, 1997. 17th IEEE/NPSS Symposium, vol. 2, pp. 1125 –1128 vol.2, oct. 1997.
[119] C. Zwyssig, S. Round, and J. Kolar, “Analytical and experimental investigation of a low
torque, ultra-high speed drive system,” Industry Applications Conference, 2006. 41st IAS
Annual Meeting. Conference Record of the 2006 IEEE, vol. 3, pp. 1507 –1513, oct. 2006.
[120] J. Daily and R. Nece, “Chamber dimenstion effects on induced flow and frictional resistance of enclosed rotating disks,” ASME Journal of Basic Engineering, vol. 82, no. 1, pp.
217–232, 1960.
[121] Y. M. Rabinovich, V. V. Sergeev, A. D. Maystrenko, V. Kulakovsky, S. Szymura, and
H. Bala, “Physical and mechanical properties of sintered Nd—Fe—B type permanent
magnets,” Intermetallics, vol. 4, no. 8, pp. 641 – 645, 1996.
[122] N. Naotake, B. Hetnarski Richard, and T. Yoshinobu, Thermal Stresses.
Taylor&Francis, 2003, ch. 6.
New York:
[123] A. C. Fischer-Cripps, Introduction to contact mechanics. Springer Verlag, 2000.
[124] R. LeMaster, Steady Load Failure Theories - Comparison with Experimental Data,
Lectures on Machine Design, University of Tennessee at Martin. [Online]. Available:
www.utm.edu/departments/engin/lemaster/machine design.htm
[125] L. Kollar and G. Springer, Mechanics of Composite Structures.
Press, 2003.
Cambridge University
[126] F. Ehrich and D. W. Childs, “Self-excited vibrations in high performance turbomachinery,” Mechanical Engineering, vol. 106, no. 5, pp. 66–79, 1984.
[127] G. Genta and E. Brusa, “On the role of nonsynchronous rotating damping in rotordynamics,” International Journal of Rotating Machinery, vol. 6, no. 6, pp. 467–475, 2000.
[128] G. Genta and F. De Bona, “Unbalance response of rotors: a modal approach with some
extensions to damped natural systems,” Journal of Sound and Vibration, vol. 140, no. 1,
pp. 129–153, 1990.
[129] D. Childs and D. W. Childs, Turbomachinery rotordynamics: phenomena, modeling, and
analysis. Wiley-Interscience, 1993.
Bibliography
203
[130] A. Muszynska, “Whirl and whip–rotor/bearing stability problems,” Journal of Sound and
Vibration, vol. 110, no. 3, pp. 443–462, 1986.
[131] ——, “Stability of whirl and whip in rotor/bearing systems,” Journal of Sound and Vibration, vol. 127, no. 1, pp. 49–64, 1988.
[132] T. Kenull, W. R. Canders, and G. Kosyna, “Formation of self-excited vibrations in wet
rotor motors and their influence on the motor current,” European Transactions on Electrical Power, vol. 13, no. 2, pp. 119–125, 2007.
[133] S. H. Crandall, “A heuristic explanation of journal bearing instability,” Proceedings of the
Workshop on Rotordynamic Instability Problems in High-Performance Turbomachinery,
Texas A&M University, College Station, Texas, pp. 274–283, 1982.
[134] B. Murphy, S. Manifold, and J. Kitzmiller, “Compulsator rotordynamics and suspension
design,” Magnetics, IEEE Transactions on, vol. 33, no. 1, pp. 474 –479, jan 1997.
[135] B. H. Rho and K. W. Kim, “A study of the dynamic characteristics of synchronously
controlled hydrodynamic journal bearings,” Tribology International, vol. 35, no. 5, pp.
339–345, 2002.
[136] E. Brusa and G. Zolfini, “Non-synchronous rotating damping effects in gyroscopic rotating systems,” Journal of Sound and Vibration, vol. 281, no. 3-5, pp. 815 – 834, 2005.
[137] J. Ede, Z. Zhu, and D. Howe, “Rotor resonances of high-speed permanent-magnet brushless machines,” Industry Applications, IEEE Transactions on, vol. 38, no. 6, pp. 1542 –
1548, nov/dec 2002.
[138] T. Wang, F. Wang, H. Bai, and H. Cui, “Stiffness and critical speed calculation of magnetic bearing-rotor system based on fea,” Electrical Machines and Systems, 2008. ICEMS
2008. International Conference on, pp. 575 –578, 17-20 2008.
[139] J. T. Sawicki and G. Genta, “Modal uncoupling of damped gyroscopic systems,” Journal
of Sound and Vibration, vol. 244, no. 3, pp. 431–451, 2001.
[140] M. Kimman, H. Langen, J. van Eijk, and H. Polinder, “Design of a novel miniature
spindle concept with active magnetic bearings using the gyroscopic stiffening effect,”
Proceedings of the 10th Int. Symp. on Magnetic Bearing, Martigny, Switzerland, 2006.
[141] L. Wang, R. W. Snidle, and L. Gu, “Rolling contact silicon nitride bearing technology: a
review of recent research,” Wear, vol. 246, no. 1-2, pp. 159 – 173, 2000.
[142] E. Owen, “Flexible shaft versus rigid shaft electric machines for petroleum and chemical plants,” Petroleum and Chemical Industry Conference, 1989, Record of Conference
Papers.. Industrial Applications Society, 36th Annual, pp. 157 –165, 11-13 1989.
[143] J. Donaldson, “High speed fans,” High Speed Bearings for Electrical Machines (Digest
No: 1997/164), IEE Colloquium on, pp. 2/1 –210, 25 1997.
204
Bibliography
[144] A. Dowers, “The pursuit of higher rotational speeds; developments in bearing design and
materials,” High Speed Bearings for Electrical Machines (Digest No: 1997/164), IEE
Colloquium on, pp. 5/1 –5/5, 25 1997.
[145] Dyson Digital Motors, Dyson Ltd, 2009. [Online]. Available: http://www.dyson.com/
technology/ddmtabbed.asp
[146] C. Zwyssig, J. Kolar, W. Thaler, and M. Vohrer, “Design of a 100 w, 500000 rpm
permanent-magnet generator for mesoscale gas turbines,” Industry Applications Conference, 2005. Fourtieth IAS Annual Meeting. Conference Record of the 2005, vol. 1, pp.
253 – 260 Vol. 1, 2-6 2005.
[147] J. Peirs, D. Reynaerts, and F. Verplaetsen, “Development of an axial microturbine for
a portable gas turbine generator,” Journal of Micromechanics and Microengineering,
vol. 13, no. 4, p. S190, 2003.
[148] Radical design improvements with hybrid bearings in electric drives for core drill and
stone saw systems, SKF, 2001. [Online]. Available: http://www.skf.com/files/001278.pdf
[149] M. Weck and A. Koch, “Spindle bearing systems for high-speed applications in machine
tools,” CIRP Annals - Manufacturing Technology, vol. 42, no. 1, pp. 445 – 448, 1993.
[150] L. Burgmeier and M. Poursaba, “Ceramic hybrid bearings in air-cycle machines,” Journal of Engineering for Gas Turbines and Power, vol. 118, no. 1, pp. 184–190, 1996.
[151] H. Aramaki, Y. Shoda, Y. Morishita, and T. Sawamoto, “The performance of ball bearings
with silicon nitride ceramic balls in high speed spindles for machine tools,” Journal of
Tribology, vol. 110, no. 4, pp. 693–698, 1988.
[152] L. G. Frechette, S. A. Jacobson, K. S. Breuer, F. F. Ehrich, R. Ghodssi, R. Khanna, C. W.
Wong, X. Zhang, M. A. Schmidt, and A. H. Epstein, “High-speed microfabricated silicon
turbomachinery and fluid film bearings,” Journal of Microelectromechanical Systems,
vol. 14, no. 1, pp. 141–152, 2005.
[153] Air Bearing Application and Design Guide, New Way Precision, 2003. [Online].
Available: www.newwayprecision.com
[154] B. Majumdar, “Externally pressurized gas bearings: A review,” Wear, vol. 62, no. 2, pp.
299 – 314, 1980.
[155] I. Pickup, D. Tipping, D. Hesmondhalgh, and B. Al Zahawi, “A 250 000 rpm drilling
spindle using a permanent magnet motor,” Proceedings of Int. Conf. on Electrical Machines - ICEM’96, pp. 337–342, 1996.
[156] T. Osamu, T. Akiyoshi, and O. N. O. Kyosuke, “Experimental study of whirl instability
for externally pressurized air journal bearings,” Bulletin of the Japan Society of Mechanical Engineers (JSME), vol. 11(43), pp. 172–179, 1968.
Bibliography
205
[157] C.-H. Chen, T.-H. Tsai, D.-W. Yang, Y. Kang, and J.-H. Chen, “The comparison in stability of rotor-aerostatic bearing system compensated by orifices and inherences,” Tribology
International, vol. 43, no. 8, pp. 1360 – 1373, 2010.
[158] K. Czolczynski and K. Marynowski, “Stability of symmetrical rotor supported in flexibly
mounted, self-acting gas journal bearings,” Wear, vol. 194, no. 1-2, pp. 190 – 197, 1996.
[159] K. Stout and F. Sweeney, “Design of aerostatic flat pad bearings using pocketed orifice
restrictors,” Tribology International, vol. 17, no. 4, pp. 191 – 198, 1984.
[160] R. Bassani, E. Ciulli, and P. Forte, “Pneumatic stability of the integral aerostatic bearing:
comparison with other types of bearing,” Tribology International, vol. 22, no. 6, pp.
363–374, 1989.
[161] A. Mohamed and F. Emad, “Nonlinear oscillations in magnetic bearing systems,” Automatic Control, IEEE Transactions on, vol. 38, no. 8, pp. 1242 –1245, aug 1993.
[162] G. Schweitzer, “Active magnetic bearings - chances and limitations,” 6th International
Conference on Rotor Dynamics, pp. 1 –14, 2002.
[163] R. Larsonneur, P. Bhler, and P. Richard, “Active magnetic bearings and motor drive towards integration,” Proceedings 8th International Symposium Magnetic Bearing, Mito,
Japan, pp. 187 –192, 2002.
[164] M. Kimman, H. Langen, and R. M. Schmidt, “A miniature milling spindle with active
magnetic bearings,” Mechatronics, vol. 20, no. 2, pp. 224 – 235, 2010.
[165] M. Ahrens, L. Kucera, and R. Larsonneur, “Performance of a magnetically suspended
flywheel energy storagedevice,” IEEE Transactions on control systems technology, vol. 4,
no. 5, pp. 494–502, 1996.
[166] H. Fujiwara, K. Ebina, N. Takahashi, and O. Matsushita, “Control of flexible rotors supported by active magnetic bearings,” Proc. of the 8th International Symposium on Magnetic Bearings, Mito, Japan, 2002.
[167] G. Schweitzer and E. H. Maslen, Eds., Magnetic bearings: theory, design and application to rotating machinery. Springer, 2009.
[168] H. M. N. K. Balini, H. Koroglu, and C. W. Scherer, “Lpv control for synchronous disturbance attenuation in active magnetic bearings,” ASME 2008 Dynamic Systems and
Control Conference, vol. 2008, no. 43352, pp. 1091–1098, 2008.
[169] W. R. Canders, N. Ueffing, U. Schrader-Hausman, and R. Larsonneur, “MTG400: A
magnetically levitated 400 kW turbo generator system for natural gas expansion,” Proceedings of the 4th International Symposium on Magnetic Bearings, pp. 435–440, 1994.
[170] J. Schmied, “Experience with magnetic bearings supporting a pipeline compressor,” Proceedings of the 2nd International Symposium on Magnetic Bearings, pp. 47–56, 1990.
206
Bibliography
[171] Y. Suyuan, Y. Guojun, S. Lei, and X. Yang, “Application and research of the active
magnetic bearing in the nuclear power plant of high temperature reactor,” Proceedings of
the 10th International Symposium on Magnetic Bearings, 2006.
[172] C. R. Knospe, “Active magnetic bearings for machining applications,” Control Engineering Practice, vol. 15, no. 3, pp. 307–313, 2007.
[173] R. Hebner, J. Beno, and A. Walls, “Flywheel batteries come around again,” Spectrum,
IEEE, vol. 39, no. 4, pp. 46 –51, apr 2002.
[174] R. Humphris, P. Allaire, D. Lewis, and L. Barrett, “Diagnostic and control features with
magnetic bearings,” Energy Conversion Engineering Conference, IECEC-89, Proceedings of the 24th Intersociety, pp. 1491 –1498 vol.3, 6-11 1989.
[175] M. H. Kimman, “Design of a micro milling setup with an active magnetic bearing spindle,” Ph.D. dissertation, Delft University of Technology, 2010.
[176] J. H. W.-R. Canders, H. May, “Contactless magnetic bearings for flywheel energy storage
systems,” Proceedings of the 8th Int. Symposium on Magnetic Suspension Technology,
2005.
[177] A. Chiba, T. Deido, T. Fukao, and M. Rahman, “An analysis of bearingless ac motors,”
Energy Conversion, IEEE Transactions on, vol. 9, no. 1, pp. 61 –68, Mar. 1994.
[178] M. Ooshima, A. Chiba, T. Fukao, and M. Rahman, “Design and analysis of permanent
magnet-type bearingless motors,” Industrial Electronics, IEEE Transactions on, vol. 43,
no. 2, pp. 292 –299, Apr. 1996.
[179] A. van Beek, Machine lifetime performance and reliability.
nology, 2004.
Delft University of Tech-
[180] P. Tsigourakos, “Design of a miniature high-speed spindle test setup including power
electronic converter,” Master’s thesis, Delft University of Technology, 2008.
[181] R. Blom and P. van den Hof, “Estimating cutting forces in micromilling by input estimation from closed-loop data,” Proceedings of the 17th World Congress The International
Federation of Automatic Control, Seoul, Korea, 2008.
[182] W. Yuan, F. Liu, S. Pang, Y. Song, and T. Zhang, “Core loss characteristics of fe-based
amorphous alloys,” Intermetallics, vol. 17, no. 4, pp. 278 – 280, 2009.
[183] T. Yamaji, M. Abe, Y. Takada, K. Okada, and T. Hiratani, “Magnetic properties and
workability of 6.5% silicon steel sheet manufactured in continuous cvd siliconizing line,”
Journal of Magnetism and Magnetic Materials, vol. 133, no. 1-3, pp. 187 – 189, 1994.
[184] JNEX-Core 10JNEX900, JFE Steel Corporation, 2003. [Online]. Available: http:
//www.jfe-steel.co.jp/en/products/list.html
Bibliography
207
[185] NO12-Core, Cogent Power Ltd., 2010. [Online]. Available: http://www.sura.se/Sura/
hp main.nsf/startupFrameset?ReadForm
R 158, Von Roll Isola, 2004. [Online]. Available:
[186] Enamelled wires: Thermibond
http://products.vonroll.com/web/download.cfm?prd id=1485\&are id=2\&lng id=EN
[187] W. Sattich and J. Geibel, Reinforced poly(phenylene sulfide).
panies Inc., 2006, pp. 385–418.
The McGraw-Hill Com-
[188] Z. X.-L, M.-Y. Zhu, Y. Li, Q.-P. Yang, H.-M. Jin, J. Jiang, Y. Tian, and Y. Luo, “Study
on fabrication process of anisotropic injection bonded nd-fe-b magnets,” Journal of Iron
and Steel Research, International, vol. 13, no. Supplement 1, pp. 286 – 288, 2006.
[189] C. Sullivan, “Optimal choice for number of strands in a litz-wire transformer winding,”
Power Electronics, IEEE Transactions on, vol. 14, no. 2, pp. 283 –291, Mar. 1999.
[190] Enamelled Wires, Elektrisola. [Online]. Available:
enamelled-wire.html
http://www.elektrisola.com/
[191] M. G. Garrell, B.-M. Ma, A. J. Shih, E. Lara-Curzio, and R. O. Scattergood, “Mechanical
properties of polyphenylene-sulfide (pps) bonded nd-fe-b permanent magnets,” Materials
Science and Engineering A, vol. 359, no. 1-2, pp. 375 – 383, 2003.
[192] Ryton PPS Data Sheets, Chevron Phillips Chemical Company, 2000. [Online].
Available: http://www.cpchem.com/bl/rytonpps/en-us/Pages/RytonPPSDataSheets.aspx
[193] H.-B. Shim, M.-K. Seo, and S.-J. Park, “Thermal conductivity and mechanical properties
of various cross-section types carbon fiber-reinforced composites,” Journal of Materials
Science, vol. 37, pp. 1881–1885, 2002.
[194] P. Pillay and R. Krishnan, “Application characteristics of permanent magnet synchronous
and brushless dc motors for servo drives,” Industry Applications, IEEE Transactions on,
vol. 27, no. 5, pp. 986 –996, 1991.
[195] P. Chandana Perera, F. Blaabjerg, J. Pedersen, and P. Thogersen, “A sensorless, stable
V/f control method for permanent-magnet synchronous motor drives,” Applied Power
Electronics Conference and Exposition, 2002. APEC 2002. Seventeenth Annual IEEE,
2002.
[196] S. Berto, A. Paccagnella, M. Ceschia, S. Bolognani, and M. Zigliotto, “Potentials and
pitfalls of fpga application in inverter drives - a case study,” Industrial Technology, 2003
IEEE International Conference on, vol. 1, pp. 500 – 505 Vol.1, 2003.
[197] L. Zhao, C. Ham, Q. Han, T. Wu, L. Zheng, K. Sundaram, J. Kapat, and L. Chow, “Design
of optimal digital controller for stable super-high-speed permanent-magnet synchronous
motor,” Electric Power Applications, IEE Proceedings -, vol. 153, no. 2, pp. 213 – 218,
2006.
208
Bibliography
[198] J.-I. Itoh, N. Nomura, and H. Ohsawa, “A comparison between V/f control and positionsensorless vector control for the permanent magnet synchronous motor,” Power Conversion Conference, 2002. PCC Osaka 2002. Proceedings of the, 2002.
[199] R. Colby and D. Novotny, “An efficiency-optimizing permanent-magnet synchronous
motor drive,” Industry Applications, IEEE Transactions on, vol. 24, no. 3, pp. 462 –469,
1988.
[200] R. Ancuti, I. Boldea, and G.-D. Andreescu, “Sensorless V/f control of high-speed surface
permanent magnet synchronous motor drives with two novel stabilising loops for fast
dynamics and robustness,” Electric Power Applications, IET, vol. 4, no. 3, pp. 149 –157,
2010.
[201] L. Xu and C. Wang, “Implementation and experimental investigation of sensorless control schemes for pmsm in super-high variable speed operation,” Industry Applications
Conference, 1998. Thirty-Third IAS Annual Meeting. The 1998 IEEE, vol. 1, pp. 483
–489 vol.1, Oct. 1998.
[202] T. Halkosaari, “Optimal U/f-control of high speed permanent magnet motors,” Industrial
Electronics, 2006 IEEE International Symposium on, vol. 3, pp. 2303 –2308, 2006.
[203] C. E. Garcia and M. Morari, “Internal model control. a unifying review and some new results,” Industrial and Engineering Chemistry Process Design and Development, vol. 21,
no. 2, pp. 308–323, 1982.
[204] JFE Super Cores Magnetic Property Curves, JFE Steel Corporation, 2003. [Online].
Available: http://www.jfe-steel.co.jp/en/products/list.html#Electrical-Steels
Summary
Limits, Modeling and Design
of
High-Speed Permanent Magnet Machines
PhD thesis
by Aleksandar Borisavljević
Background
The incentive for the work on this thesis comes from a project that has centered its research on
process and assembly technology for the fabrication of 3D products on a micro scale (Dutch Microfactory project). These efforts have been directed towards the development of small, desktopsized micro-cutting machines with a focus on micromilling as the most promising technology
for producing 3D micro-parts in a wide variety of materials.
In order to make the milling technology competitive for micro-scale products, micromilling
requires spindles with speeds beyond 300.000 rpm and a positioning accuracy in the order of
1 µm. New, specialized spindles with small rotors, built-in drives and frictionless - magnetic
or air - bearings are necessary to reach such rotational speeds and accuracy. However, spindle
drives that support extremely high rotational speed and are compatible with frictionless, lowstiffness bearings are not readily available on the market. The need for electrical drives specially
designed for high-speed machining spindles which would meet these requirements motivated
work on this thesis.
Project goal
The goal of the project was to design and build a high-speed electrical drive which would be
compatible with a soft-mounted spindle for speeds beyond 300.000 rpm. Hence, it was primarily expected that the spindle motor has low stiffness (unbalanced pull), generates minimum
frequency-dependent losses (heat) and that the motor design does not compromise the strength
and robustness of the rotor. Indeed, this thesis presents the modeling, design and practical evaluation of a small, toroidally-wound slotless permanent magnet (PM) machine with a short rotor
supported by static air bearings. The merits and drawbacks of the designed spindle motor are
assessed.
209
210
Summary
Research goals
The project included research in the field of high-speed machines and, particularly, the modeling
and design of high-speed PM motors while using the designed spindle motor as a test case. A
slotless PM machine has been chosen as the motor technology because this type of machine
exhibits high power density at low volumes, low losses and a relatively simple rotor. The
suitability of slotless PM machines for very-high-speed applications is demonstrated in the
thesis through both theoretical and empirical studies.
The demand for extremely high speeds spurred also an inquiry into the limitations of the
machines: What does limit the speed of current machinery? Could those limits be overcome or
avoided? Various physical parameters (stress, temperature, resonant frequencies) can limit the
speed of an electrical machine. Aside from speed, these variables are also affected by power,
size and electrical and magnetic loading. They need to be carefully designed so that the spindle
temperature, structural integrity and rotordynamic stability are not jeopardized at the operating
speed.
In the thesis, a study of machine limits implicitly outlines an approach for designing machines for high-speed applications: (i) recognize the speed limits of the machines, (ii) correlate
the limits with machine parameters and (iii) overcome those limits with new solutions which are
adequate for a particular application. At the same time, the analytical expressions of the limiting quantities indicate the method of design optimization, leading to the maximum mechanical
and/or electromagnetic utilization of the machine.
Two new high-speed spindle concepts are presented in this thesis. In one, a slotless PM
motor is spatially integrated with 5DOF active magnetic bearings without merging their functions. In the other, the same motor is combined with 5DOF aerostatic bearings. Both concepts
use a short rotor to ensure the rotordynamical stability of the system. The concept with the air
bearings is realized in a practical setup and evaluated.
Modeling
Analytical modeling in the thesis is divided into three main sections: electromagnetic, structural
and rotordynamical. The models are developed to suit the test-machine; however, a substantial
number of the models found in the thesis is applicable to a broader range of (non-PM) highspeed electrical machines.
The electromagnetic behavior of a slotless PM machine is represented using a 2D magnetostatic model. Based on the model of the magnetic field in the machine, other quantities of the
machine - no-load voltage, torque, inductance, unbalanced magnetic force and losses - are derived. The purpose of the electromagnetic modeling is to adequately model most important EM
parameters of high-speed PM machines and distinguish dominant loss factors. While the models show good results in predicted machine quantities correlated with the field of the permanent
magnet, 2D magnetostatic model appears to be unsuitable for representing a toroidally-wound
machine fitted in an electrically-conductive housing.
Structural aspects of high-speed PM rotors are analyzed using 2D modeling of stress in a
compound of three concentric cylinders in rotation; the cylinders represent the shaft, permanent
magnet and retaining sleeve of the rotor. It is demonstrated that a fully isotropic representation
Bibliography
211
of the rotor materials is adequate even when the retaining sleeve exhibits strong orthotropic
nature, as it is the case with the test-machine rotor.
The rotordynamical stability of a rotor-bearing is analyzed using a simple, Jeffcott rotor
model; it is shown that the rotating damping has a negative influence on the rotational stability
in supercritical the regime and that the first flexural critical speed, practically, limits the speed
of an electrical machine. Using the Timoshenko-beam model the critical speeds of the rotorbearings system are correlated with the rotor geometry and bearing stiffness; it is demonstrated
that the value of flexural critical speeds depend almost solely on the rotor slenderness. Finally,
it is shown that short rotors with polar moment of inertia higher than the transversal moment
has important rotordynamical advantages with respect to their slender (long) counterpart.
While electromagnetic and structural models directly formed a basis for the analytical optimization of the test motor, the rotordynamical study had a defining influence on the new highspeed-spindle concept which is developed so that the major rotordynamical stability limits are
avoided.
Design
Loss minimization is taken as the ultimate criterion for the motor electromagnetic design. More
precisely, the main intention of the design was to mitigate the overheating of the motor that
would result from frequency-dependent losses. Chosen materials, conceptual design and electromagnetic optimization all resulted in a relatively simple machine with very low electromagnetic losses. The electromagnetic optimization was carried out in two steps: in the first step
the machine geometry is defined, in the second step the stator conductors are optimized for the
given number of turns and loading defined in the preceding step.
Structural optimization of the rotor retaining sleeve stems directly from the modeling of
the rotor critical (limiting) stresses. For an expected operating temperature there is an optimal
value of the interference fit between the sleeve and magnet for which both tension and contact
limits are reached at an equal rotational speed. The final rotor consists of a disc-shaped iron
shaft with an injection-molded plastic-bonded magnet and press-fitted glass- and carbon-fiber
retaining rings.
Control
An additional challenge for a high-speed application lies in the control of the high-speed machine. Due to limitations of the processing powers of microcontrollers, very high-speed machines are usually controlled in open-loop. A special open-loop control algorithm is developed
in the thesis that ensures stability of the speed response of a high-speed PM motor without
creating a great computational burden.
Practical design evaluation
The main tests performed for verification of the models developed in this thesis are the speeddecay and locked-rotor tests. The tests showed that air friction and eddy currents in the aluminum housing are the main loss factors in the test setup. The models used to represent air-
212
Summary
friction loss in standard rotating machines are not suitable to represent the friction loss of the
short rotor in the air bearings. Losses in the housing were not modeled and those represent the
weakest point of both the electromagnetic modeling and design. Other losses are effectively
minimized in the optimization procedure.
The developed motor controller performed well and the speed was stable throughout the
attained speed range. The maximum speed reached with the motor is 156.000 rpm which corresponds to a tip tangential speed of 270 m/s. The designed rotor retainment fulfilled the task of
enabling very high-speed rotation. On the other hand, press-fitted rotor elements caused problems with recurring unbalance, which appeared to be too high for the air bearings - in one of
the tests the rotor crashed into the bottom air bearing when operating in the vicinity of critical
speed.
The thesis demonstrates that static air bearings represent a good solution for stable highspeed operation and that the limits of rotational stability of such bearings are much higher than
those of aerodynamic and lubricated bearings. At the same time, very tight tolerances of air
bearings impose high precision standards on the rotor manufacturing.
In conclusion
In this thesis, phenomena, both mechanical and electromagnetic, that take precedence in highspeed permanent magnet machines are identified and systematized. By juxtaposing the models
of different, yet equally important physical aspects of high-speed PM machines, emphasizing
relevant machine parameters and presenting those parameters in a logical way, the thesis represents a comprehensive resource for the design of high-speed permanent magnet machines.
Samenvatting
Beperkingen, modelvorming en ontwerp
van
sneldraaiende permanente-magneet machines1
Proefschrift
door Aleksandar Borisavljević
Achtergrond
Het initiatief voor het onderzoek waarvan dit proefschrift het resultaat is, komt voort uit een
project dat is toegespitst op proces- en assemblagetechnologie voor de vervaardiging van driedimensionale producten op miniatuurschaal: het Dutch Microfactory project. Hierin staat de
ontwikkeling van kleine, tafelmodel microsnijmachines centraal. Er wordt in het bijzonder
gekeken naar microfrezen, wat als de meest veelbelovende technologie wordt gezien voor de
vervaardiging van 3D miniatuuronderdelen uit een breed spectrum aan materialen.
Wil freestechnologie de concurrentie aan kunnen gaan met bestaande productieprocessen
voor miniatuurproducten, dan zijn er voor deze fabricagemethode freesspindels nodig die op
snelheden hoger dan 300.000 rpm kunnen draaien en een positioneringsnauwkeurigheid hebben
die in de orde van 1 µm ligt. Om aan deze eisen te kunnen voldoen zijn er nieuwe, speciale spindels nodig die een kleine rotor en een geı̈ntegreerde aandrijving hebben, en gebruik maken van
wrijvingsloze (magnetische of lucht-) lagers. Helaas zijn er geen spindelsystemen verkrijgbaar
die bij een dergelijk hoge snelheid te gebruiken zijn en bovendien uit te rusten zijn met lage
stijfheids-, wrijvingsloze lagers. Het onderzoek voor dit proefschrift is ontstaan als een direct
gevolg van de behoefte aan sneldraaiende freesspindels die aan de genoemde eisen voldoen.
Doel van het project
Het doel van het project was het ontwerp en het construeren van een complete elektrische aandrijving voor hoge snelheden die bruikbaar is in combinatie met een zacht-opgegangen spindel
spindel voor snelheden hoger dan 300.000 rpm. Er werd aangenomen dat de spindelmotor een
lage stijfheid heeft (ongebalanceerde trekkracht), minimale frequentieafhankelijke verliezen
(warmte) genereert en dat het motorontwerp geen afbreuk doet aan de sterkte en robuustheid
1
Translated to Dutch by Maurice Roes
213
214
Summary
van de rotor. Dit proefschrift beschrijft daarom de modelvorming, het ontwerp en de praktische
evaluatie van een kleine toroı̈daal gewikkelde tandloze permanente magneet (PM) machine met
een korte rotor die ondersteund wordt door statische luchtlagers. Voorts worden de voor- en
nadelen van de ontworpen spindelmotor uiteengezet.
Onderzoeksdoelstellingen
In het project is er onderzoek gedaan naar sneldraaiende machines, in het bijzonder de modelvorming en het ontwerp van sneldraaiende PM motors, waarvoor de ontworpen spindelmotor
als testcase is gebruikt. Er is gekozen voor een tandloze PM machine als motortechnologie
omwille van een aantal redenen: dit type machine heeft een hoge vermogensdichtheid bij lage
volumes, lage verliezen en een relatief eenvoudige rotorconstructie. Door middel van theoretische en empirische studies is in het proefschrift is aangetoond dat tandloze PM machines
geschikt zijn voor applicaties waar zeer hoge omwentelingssnelheden vereist zijn.
De vraag naar extreem hoge snelheden leidde tot een onderzoek naar de beperkingen van
de machines: Wat limiteert de snelheid van huidige machines? Kunnen deze beperkingen overwonnen of vermeden worden? De snelheid van een elektrische machine is begrensd door een
aantal fysische grootheden: mechanische spanning, temperatuur, resonantiefrequenties. Behalve door de snelheid worden deze variabelen ook beı̈nvloed door het vermogen, de afmetingen
en de elektrische en magnetische belasting. Deze moeten zorgvuldig gekozen worden, zodat bij
de bedrijfssnelheid de temperatuur van de spindel niet te hoog wordt, er geen structurele schade
optreedt en de rotordynamische stabiliteit niet in het gedrang komt.
In het proefschrift wordt het ontwerpproces van machines voor hogesnelheidsapplicaties
impliciet afgetekend door een onderzoek naar de beperkingen van machines: (i) het herkennen
van de snelheidsbeperkingen van de machines, (ii) het correleren van de beperkingen met de
machineparameters, en (iii) het overwinnen van deze beperkingen door middel van nieuwe
oplossingen voor specifieke applicaties. Tegelijkertijd geven de analytische uitdrukkingen voor
de beperkende grootheden een handvat voor ontwerpoptimalisatie, waardoor het mogelijk wordt
om het volledige mechanisch en/of elektromagnetisch potentieel van de machine te bereiken.
In dit proefschrift worden twee nieuwe concepten voor sneldraaiende spindels gepresenteerd. Het ene ontwerp bestaat uit een tandloze PM motor die ruimtelijk geı̈ntegreerd is met
een 5DOF actief magnetisch lager, zonder dat hun functionaliteit wordt samengevoegd. In het
andere concept wordt dezelfde motor gecombineerd met een 5DOF aerostatisch lager. In beide
concepten wordt een korte rotor gebruikt om de rotordynamische stabiliteit van het systeem te
kunnen garanderen. Het concept waarin de luchtlagers worden toegepast is als een testopstelling
gerealiseerd en gebruikt voor evaluatiedoeleinden.
Modelvorming
De analytische modelvorming in het proefschrift is onderverdeeld in drie hoofddelen: elektromagnetisch, structureel en rotordynamisch. Alhoewel de modellen specifiek voor de testmachine zijn ontwikkeld, geldt voor een aanzienlijk deel van de modellen in het proefschrift dat ze
van toepassing zijn op een breder gamma aan (niet-PM) sneldraaiende elektrische machines.
Bibliography
215
Om het elektromagnetische gedrag van een tandloze PM machine te beschrijven wordt er
gebruik gemaakt van een tweedimensionaal magnetostatisch model. Andere grootheden kunnen
aan de hand van het magnetisch veld in de machine worden afgeleid: de open-klemspanning,
het koppel, de inductie, de ongebalanceerde magnetische krachten en de verliezen. Het doel van
het elektromagnetische model is het adequaat modelleren van de belangrijkste EM parameters
van sneldraaiende PM machines en het onderscheiden van de dominante verliescomponenten.
Hoewel de modellen goede resultaten geven bij het voorspellen van machinegrootheden die in
verband staan met het veld van de permanente magneet, lijkt het 2D magnetostatische model
onbruikbaar te zijn voor een toroı̈daal gewikkelde machine met een elektrisch geleidende behuizing.
De structurele eigenschappen van sneldraaiende PM rotoren zijn geanalyseerd door middel
van tweedimensionale modelvorming van mechanische spanning in een samenstelling van drie
concentrische roterende cilinders. De cilinders stellen de as, de permanente magneten en de
bandage van de rotor. Er is aangetoond dat een volledig isotrope beschrijving van de rotormaterialen voldoende is, zelfs als de magneetbandage een sterk orthotrope aard heeft, zoals het geval
is bij de rotor van de testmachine.
De rotordynamische stabiliteit van een rotorlager is geanalyseerd met behulp van een eenvoudig Jeffcot rotor model. Er is aangetoond dat de roterende demping een negatieve invloed
heeft op de rotatiestabiliteit in het superkritische regime, en dat de eerste buigingskritische
snelheid in de praktijk een beperking oplegt aan de maximaal haalbare snelheid van een machine. Aan de hand van het Timoshenko balkmodel kunnen de kritische snelheden van het
rotor-lagersysteem gecorreleerd worden met de rotorgeometrie en de lagerstijfheid. Er is aangetoond dat de waarden van de buigingskritische snelheden bijna volledig bepaald worden door
de slankheid van de rotor. Als laatste is er bewezen dat een korte rotor waarvan het polair
traagheidsmoment hoger is dan het transversale draaimoment belangrijke rotordynamische voordelen heeft in verhouding tot zijn slanke (lange) tegenhanger.
Hoewel de elektromagnetische en structurele modellen een directe basis vormden voor de
analytische optimalisatie van de testmotor, had de rotordynamische studie een grote invloed op
het nieuwe sneldraaiende-spindel concept, dat ontwikkeld is om de belangrijkste rotordynamische stabiliteitsbeperkingen uit de weg te gaan.
Ontwerp
De minimalisatie van verliezen is als ultiem streven voor het elektromagnetisch ontwerp van de
motor genomen. Om preciezer te zijn, het hoofddoel van het ontwerp was het tegengaan van
motoroververhitting door frequentieafhankelijke verliezen. De gekozen materialen, het conceptuele ontwerp en elektromagnetische optimalisatie hebben geleid tot een relatief eenvoudige
machine met zeer lage elektromagnetische verliezen. De elektromagnetische optimalisatie is
uitgevoerd in twee stappen: tijdens de eerste stap is de geometrie van de machine gedefinieerd,
en in de tweede stap zijn de statorwikkelingen geoptimaliseerd voor het tijdens de voorgaande
stap gedefinieerde aantal windingen en de belasting.
Het modelleren van de (beperkende) kritische spanning van de rotor leidde direct tot een
structurele optimalisatie van de bandage van de rotor. Er is een optimale waarde van de pers-
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Summary
passing tussen de bandage en de magneet bij een bepaalde bedrijfstemperatuur, waarbij zowel
de spannings- als de contactlimiet bereikt worden bij een gelijkwaardige omwentelingssnelheid.
Het uiteindelijke rotorontwerp bestaat uit een schijfvormige ijzeren as met een gespuitgiete kunstofgebonden magneet en geperspaste glas- en koolstofvezel borgringen.
Regeling
Een bijkomende uitdaging voor hogesnelheidstoepassingen ligt in de regeling van sneldraaiende
machines. Vanwege de beperkte rekenkracht van microcontrollers worden machines voor zeer
hoge snelheden meestal in een open loop configuratie aangestuurd. Een speciaal open loop
regelalgoritme is ontwikkeld in het proefschrift om ervoor te zorgen dat de snelheidsrespons
van de sneldraaiende PM motor stabiel is, zonder dat dit zeer veel rekenwerk oplevert.
Evaluatie van het praktijkontwerp
Om de opgestelde modellen te verifiren zijn hoofdzakelijk de snelheidsverval- en geblokkeerde
rotortest uitgevoerd. Deze experimenten toonden aan dat luchtwrijving en wervelstromen in de
aluminium behuizing de grootste bronnen van verliezen zijn in de testopstelling. De modellen
van luchtwrijving die voor normale roterende machines gebruikt worden zijn niet bruikbaar voor
het wrijvingsverlies van de korte, luchtgelagerde rotor. De verliezen in de behuizing zijn niet
gemodelleerd en deze vormen het zwakste punt in zowel de elektromagnetische modelvorming
als het ontwerp. De overige verliezen zijn doeltreffend geminimaliseerd in de optimalisatieprocedure.
De ontworpen motorregelaar presteerde zeer goed; de snelheid was stabiel in het gehele
snelheidsbereik. De maximaal door de motor behaalde snelheid is 156.000 rpm, overeenkomend
met een tangentile snelheid van het rotoroppervlak van 270 m/s. De ontworpen koolstofvezelbandage voor de rotor werkte goed en maakte hoge snelheden mogelijk. Daarentegen veroorzaakte de geperspaste rotorelementen problemen met betrekking tot een terugkerende onbalans,
die te hoog bleek te zijn voor de luchtlagers - tijdens een van de testen is de rotor tegen het
onderste luchtlager aangeslagen terwijl de motor in de buurt van zijn kritische snelheid draaide.
Het proefschrift toont aan dat statische luchtlagers een stabiel hogesnelheidsbedrijf mogelijk
maken, en dat de limiet van de rotationele stabiliteit van de dergelijke lagers veel hoger ligt dan
die van aerodynamische en gesmeerde lagers.
Ter conclusie
In dit proefschrift zijn de fenomenen, zowel mechanisch als elektromagnetisch van aard, die
optreden in sneldraaiende permanente-magneet machines geı̈dentificeerd en gesystematiseerd.
Door de modellen van verschillende, doch even belangrijke, fysische aspecten van sneldraaiende
PM machines naast elkaar te plaatsen, de relevante machineparameters te benadrukken en deze
op een logische wijze te presenteren, beschrijft dit proefschrift een uitgebreid hulpmiddel voor
het ontwerp van sneldraaiende permanente-magneet machines.
List of publications
Journals
1. A. Borisavljevic, H. Polinder, J.A. Ferreira: “On the Speed Limits of Permanent Magnet
Machines”, IEEE Transactions on Industrial Electronics, January 2010, Vol. 57, No. 1,
pp. 220-227.
2. A. Borisavljevic, H. Polinder, J.A. Ferreira: “Conductor Optimization for Slotless PM Machines”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2011, Submitted
Conference proceedings
1. A. Borisavljevic, H. Polinder, J.A. Ferreira: “Calculation of Unbalanced Magnetic Force in
Slotless PM Machines”, Electrimacs, 2011
2. A. Borisavljevic, H. Polinder, J.A. Ferreira: “Enclosure Design for a High-Speed Permanent
Magnet Rotor”, Power Electronics, Machines and Drives (PEMD), April 2010, Vol. 2,
pp. 817-822.
3. A. Borisavljevic, H. Polinder, J.A. Ferreira: “Realization of the I/f Control Method for
a High-Speed Permanent Magnet Motor”, International Conference on Electrical Machines (ICEM), September 2010, Vol. 4, pp. 2895-2900.
4. A. Borisavljevic, M.H. Kimman, P. Tsigkourakos, H.H. Langen, H. Polinder and J.A. Ferreira: “Integration of a Motor Drive within Active Magnetic Bearings for High-Speed
Micro-Milling”, Euspen International Conference (Euspen) 2009, Vol. 1, pp. 320-323.
5. A. Borisavljevic, M.H. Kimman, P. Tsigkourakos, H. Polinder, H.H. Langen, R. Munnig
Schmidt, J.A. Ferreira: “Drive for a Novel High-Speed Micro-Milling Spindle”, International Conference on Advanced Intelligent Mechatronics (AIM) 2009, pp. 1492-1497.
6. A. Borisavljevic, H. Polinder, J.A. Ferreira: “Overcoming Limits of High-Speed Machines”,
International Conference on Electrical Machines (ICEM) 2008
7. A. Borisavljevic, H. Polinder, M.H. Kimman, H.H. Langen, J.A. Ferreira: “Very High Speed
Motor Design for Magnetically Levitated Spindles”, Power Conversion Intelligent Motion
(PCIM) 2008
217
Biography
Aleksandar Borisavljević was born in Kragujevac, Serbia, in 1978. He graduated from Faculty of Electrical Engineering, University of Belgrade, in 2004. After graduation he worked
as a researcher at the Institute ”Mihajlo Pupin” in Belgrade. In September 2006 he joined the
Electrical Power Processing group at Delft University of Technology where worked toward a
PhD degree. His area of expertise are high-speed electric drives.
Since November 2011 he has been a post-doc researcher at Eindhoven University of Technology and a development engineer at Micro Turbine Technologies BV, Eindhoven. His work
is focused on electromechanical aspects of Combined Heat and Power systems.
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