DESIGN OF ROTATING ELECTRICAL MACHINES pdf

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DESIGN OF ROTATING
ELECTRICAL
MACHINES
i
Design of Rotating Electrical Machines Juha Pyrh¨onen, Tapani Jokinen and Val´eria Hrabovcov´a
© 2008 John Wiley & Sons,
Ltd. ISBN: 978-0-470-69516-6
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DESIGN OF ROTATING
ELECTRICAL
MACHINES
Juha Pyrh
¨
onen
Department of Electrical Engineering, Lappeenranta University of Technology, Finland
Tapani Jokinen
Department of Electrical Engineering, Helsinki University of Technology, Finland
Val
´
eria Hrabovcov
´
a
Department of Power Electrical Systems, Faculty of Electrical Engineering,

University of
ˇ
Zilina, Slovak Republic
Translated by
Hanna Niemel
¨
a
Department of Electrical Engineering, Lappeenranta University of Technology, Finland
iii
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This edition first published 2008
C

2008 John Wiley & Sons, Ltd
Adapted from the original version in Finnish written by Juha Pyrh
¨
onen and published by Lappeenranta University
of Technology
C

Juha Pyrh
¨
onen, 2007
Registered office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
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Library of Congress Cataloging-in-Publication Data
Pyrh
¨
onen, Juha.
Design of rotating electrical machines / Juha Pyrh
¨
onen, Tapani Jokinen, Val
´
eria Hrabovcov
´
a ; translated by
Hanna Niemel
¨
a.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-69516-6 (cloth)

1. Electric machinery–Design and construction. 2. Electric generators–Design and construction. 3. Electric
motors–Design and construction. 4. Rotational motion. I. Jokinen, Tapani, 1937– II. Hrabovcov
´
a, Val
´
eria.
III. Title.
TK2331.P97 2009
621.31

042–dc22
2008042571
A catalogue record for this book is available from the British Library.
ISBN: 978-0-470-69516-6 (H/B)
Typeset in 10/12pt Times by Aptara Inc., New Delhi, India.
Printed in Great Britain by CPI Antony Rowe, Chippenham, Wiltshire
iv
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Contents
About the Authors xi
Preface xiii
Abbreviations and Symbols xv
1 Principal Laws and Methods in Electrical Machine Design 1
1.1 Electromagnetic Principles 1
1.2 Numerical Solution 9
1.3 The Most Common Principles Applied to Analytic Calculation 12
1.3.1 Flux Line Diagrams 17

1.3.2 Flux Diagrams for Current-Carrying Areas 22
1.4 Application of the Principle of Virtual Work in the Determination
of Force and Torque 25
1.5 Maxwell’s Stress Tensor; Radial and Tangential Stress 33
1.6 Self-Inductance and Mutual Inductance 36
1.7 Per Unit Values 40
1.8 Phasor Diagrams 43
Bibliography 45
2 Windings of Electrical Machines 47
2.1 Basic Principles 48
2.1.1 Salient-Pole Windings 48
2.1.2 Slot Windings 52
2.1.3 End Windings 53
2.2 Phase Windings 54
2.3 Three-Phase Integral Slot Stator Winding 56
2.4 Voltage Phasor Diagram and Winding Factor 63
2.5 Winding Analysis 71
2.6 Short Pitching 72
2.7 Current Linkage of a Slot Winding 81
2.8 Poly-Phase Fractional Slot Windings 92
2.9 Phase Systems and Zones of Windings 95
2.9.1 Phase Systems 95
2.9.2 Zones of Windings 98
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vi Contents
2.10 Symmetry Conditions 99

2.11 Base Windings 102
2.11.1 First-Grade Fractional Slot Base Windings 103
2.11.2 Second-Grade Fractional Slot Base Windings 104
2.11.3 Integral Slot Base Windings 104
2.12 Fractional Slot Windings 105
2.12.1 Single-Layer Fractional Slot Windings 105
2.12.2 Double-Layer Fractional Slot Windings 115
2.13 Single- and Two-Phase Windings 122
2.14 Windings Permitting a Varying Number of Poles 126
2.15 Commutator Windings 127
2.15.1 Lap Winding Principles 131
2.15.2 Wave Winding Principles 134
2.15.3 Commutator Winding Examples, Balancing Connectors 137
2.15.4 AC Commutator Windings 140
2.15.5 Current Linkage of the Commutator Winding and
Armature Reaction 142
2.16 Compensating Windings and Commutating Poles 145
2.17 Rotor Windings of Asynchronous Machines 147
2.18 Damper Windings 150
Bibliography 152
3 Design of Magnetic Circuits 153
3.1 Air Gap and its Magnetic Voltage 159
3.1.1 Air Gap and Carter Factor 159
3.1.2 Air Gaps of a Salient-Pole Machine 164
3.1.3 Air Gap of Nonsalient-Pole Machine 169
3.2 Equivalent Core Length 171
3.3 Magnetic Voltage of a Tooth and a Salient Pole 173
3.3.1 Magnetic Voltage of a Tooth 173
3.3.2 Magnetic Voltage of a Salient Pole 177
3.4 Magnetic Voltage of Stator and Rotor Yokes 177

3.5 No-Load Curve, Equivalent Air Gap and Magnetizing Current
of the Machine 180
3.6 Magnetic Materials of a Rotating Machine 183
3.6.1 Characteristics of Ferromagnetic Materials 187
3.6.2 Losses in Iron Circuits 193
3.7 Permanent Magnets in Rotating Machines 200
3.7.1 History and Characteristics of Permanent Magnets 200
3.7.2 Operating Point of a Permanent Magnet Circuit 205
3.7.3 Application of Permanent Magnets in Electrical Machines 213
3.8 Assembly of Iron Stacks 219
3.9 Magnetizing Inductance 221
Bibliography 224
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Contents vii
4 Flux Leakage 225
4.1 Division of Leakage Flux Components 227
4.1.1 Leakage Fluxes Not Crossing an Air Gap 227
4.1.2 Leakage Fluxes Crossing an Air Gap 228
4.2 Calculation of Flux Leakage 230
4.2.1 Air-Gap Leakage Inductance 230
4.2.2 Slot Leakage Inductance 234
4.2.3 Tooth Tip Leakage Inductance 245
4.2.4 End Winding Leakage Inductance 246
4.2.5 Skewing Factor and Skew Leakage Inductance 250
Bibliography 253
5 Resistances 255
5.1 DC Resistance 255

5.2 Influence of Skin Effect on Resistance 256
5.2.1 Analytical Calculation of Resistance Factor 256
5.2.2 Critical Conductor Height 265
5.2.3 Methods to Limit the Skin Effect 266
5.2.4 Inductance Factor 267
5.2.5 Calculation of Skin Effect Using Circuit Analysis 267
5.2.6 Double-Sided Skin Effect 274
Bibliography 280
6 Main Dimensions of a Rotating Machine 281
6.1 Mechanical Loadability 291
6.2 Electrical Loadability 293
6.3 Magnetic Loadability 294
6.4 Air Gap 297
Bibliography 300
7 Design Process and Properties of Rotating Electrical Machines 301
7.1 Asynchronous Motor 313
7.1.1 Current Linkage and Torque Production of an
Asynchronous Machine 315
7.1.2 Impedance and Current Linkage of a Cage Winding 320
7.1.3 Characteristics of an Induction Machine 327
7.1.4 Equivalent Circuit Taking Asynchronous Torques and Harmonics
into Account 332
7.1.5 Synchronous Torques 337
7.1.6 Selection of t he Slot Number of a Cage Winding 339
7.1.7 Construction of an Induction Motor 342
7.1.8 Cooling and Duty Types 343
7.1.9 Examples of the Parameters of Three-Phase Industrial
Induction Motors 348
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viii Contents
7.1.10 Asynchronous Generator 351
7.1.11 Asynchronous Motor Supplied with Single-Phase Current 353
7.2 Synchronous Machine 358
7.2.1 Inductances of a Synchronous Machine in Synchronous Operation
and in Transients 359
7.2.2 Loaded Synchronous Machine and Load Angle Equation 370
7.2.3 RMS Value Phasor Diagrams of a Synchronous Machine 376
7.2.4 No-Load Curve and Short-Circuit Test 383
7.2.5 Asynchronous Drive 386
7.2.6 Asymmetric-Load-Caused Damper Currents 391
7.2.7 Shift of Damper Bar Slotting from the Symmetry Axis of the Pole 392
7.2.8 V Curve of a Synchronous Machine 394
7.2.9 Excitation Methods of a Synchronous Machine 394
7.2.10 Permanent Magnet Synchronous Machines 395
7.2.11 Synchronous Reluctance Machines 400
7.3 DC Machines 404
7.3.1 Configuration of DC Machines 404
7.3.2 Operation and Voltage of a DC Machine 405
7.3.3 Armature Reaction of a DC Machine and Machine Design 409
7.3.4 Commutation 411
7.4 Doubly Salient Reluctance Machine 413
7.4.1 Operating Principle of a Doubly Salient Reluctance Machine 414
7.4.2 Torque of an SR Machine 415
7.4.3 Operation of an SR Machine 416
7.4.4 Basic Terminology, Phase Number and Dimensioning of
an SR Machine 419
7.4.5 Control Systems of an SR Motor 422

7.4.6 Future Scenarios for SR Machines 425
Bibliography 427
8 Insulation of Electrical Machines 429
8.1 Insulation of Rotating Electrical Machines 431
8.2 Impregnation Varnishes and Resins 436
8.3 Dimensioning of an Insulation 440
8.4 Electrical Reactions Ageing Insulation 443
8.5 Practical Insulation Constructions 444
8.5.1 Slot Insulations of Low-Voltage Machines 445
8.5.2 Coil End Insulations of Low-Voltage Machines 445
8.5.3 Pole Winding Insulations 446
8.5.4 Low-Voltage Machine Impregnation 447
8.5.5 Insulation of High-Voltage Machines 447
8.6 Condition Monitoring of Insulation 449
8.7 Insulation in Frequency Converter Drives 453
Bibliography 455
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Contents ix
9 Heat Transfer 457
9.1 Losses 458
9.1.1 Resistive Losses 458
9.1.2 Iron Losses 460
9.1.3 Additional Losses 460
9.1.4 Mechanical Losses 460
9.2 Heat Removal 462
9.2.1 Conduction 463
9.2.2 Radiation 466

9.2.3 Convection 470
9.3 Thermal Equivalent Circuit 476
9.3.1 Analogy between Electrical and Thermal Quantities 476
9.3.2 Average Thermal Conductivity of a Winding 477
9.3.3 Thermal Equivalent Circuit of an Electrical Machine 479
9.3.4 Modelling of Coolant Flow 488
9.3.5 Solution of Equivalent Circuit 493
9.3.6 Cooling Flow Rate 495
Bibliography 496
Appendix A 497
Appendix B 501
Index 503
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About the Authors
Juha Pyrh
¨
onen is a Professor in the Department of Electrical Engineering at Lappeenranta
University of Technology, Finland. He is engaged in the research and development of electric
motors and drives. He is especially active in the fields of permanent magnet synchronous ma-
chines and drives and solid-rotor high-speed induction machines and drives. He has worked on
many research and industrial development projects and has produced numerous publications
and patents in the field of electrical engineering.
Tapani Jokinen is a Professor Emeritus in the Department of Electrical Engineering at
Helsinki University of Technology, Finland. His principal research interests are in AC ma-
chines, creative problem solving and product development processes. He has worked as an
electrical machine design engineer with Oy Str
¨

omberg Ab Works. He has been a consul-
tant for several companies, a member of the Board of High Speed Tech Ltd and Neorem
Magnets Oy, and a member of the Supreme Administrative Court in cases on patents. His
research projects include, among others, the development of superconducting and large per-
manent magnet motors for ship propulsion, the development of high-speed electric motors
and active magnetic bearings, and the development of finite element analysis tools for solving
electrical machine problems.
Val
´
eria Hrabovcov
´
a is a Professor of Electrical Machines in the Department of Power
Electrical Systems, Faculty of Electrical Engineering, at the University of
ˇ
Zilina, Slovak
Republic. Her professional and research interests cover all kinds of electrical machines, elec-
tronically commutated electrical machines included. She has worked on many research and
development projects and has written numerous scientific publications in the field of electrical
engineering. Her work also includes various pedagogical activities, and she has participated
in many international educational projects.
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Preface
Electrical machines are almost entirely used in producing electricity, and there are very few
electricity-producing processes where rotating machines are not used. In such processes,
at least auxiliary motors are usually needed. In distributed energy systems, new machine
types play a considerable role: for instance, the era of permanent magnet machines has now

commenced.
About half of all electricity produced globally is used in electric motors, and the share of
accurately controlled motor drives applications is increasing. Electrical drives provide proba-
bly the best control properties for a wide variety of processes. The torque of an electric motor
may be controlled accurately, and the efficiencies of the power electronic and electromechan-
ical conversion processes are high. What is most important is that a controlled electric motor
drive may save considerable amounts of energy. In the future, electric drives will probably
play an important role also in the traction of cars and working machines. Because of the
large energy flows, electric drives have a significant impact on the environment. If drives
are poorly designed or used inefficiently, we burden our environment in vain. Environmen-
tal threats give electrical engineers a good reason for designing new and efficient electric
drives.
Finland has a strong tradition in electric motors and drives. Lappeenranta University of
Technology and Helsinki University of Technology have found it necessary to maintain and
expand the instruction given in electric machines. The objective of this book is to provide stu-
dents in electrical engineering with an adequate basic knowledge of rotating electric machines,
for an understanding of the operating principles of these machines as well as developing el-
ementary skills in machine design. However, due to the limitations of this material, it is not
possible to include all the information required in electric machine design in a single book,
yet this material may serve as a manual for a machine designer in the early stages of his or
her career. The bibliographies at the end of chapters are intended as sources of references
and recommended background reading. The Finnish tradition of electrical machine design is
emphasized in this textbook by the important co-authorship of Professor Tapani Jokinen, who
has spent decades in developing the Finnish machine design profession. An important view of
electrical machine design is provided by Professor Val
´
eria Hrabovcov
´
a from Slovak Republic,
which also has a strong industrial tradition.

We express our gratitude to the following persons, who have kindly provided material for
this book: Dr Jorma Haataja (LUT), Dr Tanja Hedberg (ITT Water and Wastewater AB),
Mr Jari J
¨
appinen (ABB), Ms Hanne Jussila (LUT), Dr Panu Kurronen (The Switch Oy),
Dr Janne Nerg (LUT), Dr Markku Niemel
¨
a (ABB), Dr Asko Parviainen (AXCO Motors Oy),
xiii
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xiv Preface
Mr Marko Rilla (LUT), Dr Pia Salminen (LUT), Mr Ville Sihvo and numerous other col-
leagues. Dr Hanna Niemel
¨
a’s contribution to this edition and the publication process of the
manuscript is highly acknowledged.
Juha Pyrh
¨
onen
Tapani Jokinen
Val
´
eria Hrabovcov
´
a
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Abbreviations and Symbols
A linear current density [A/m]
A magnetic vector potential [V s/m]
A temperature class 105
◦
C
AC alternating current
AM asynchronous machine
A1–A2 armature winding of a DC machine
a number of parallel paths in windings without commutator: per phase, in
windings with a commutator: per half armature, diffusivity
B magnetic flux density, vector [V s/m
2
], [T]
B
r
remanence flux density [T]
B
sat
saturation flux density [T]
B temperature class 130
◦
C
B1–B2 commutating pole winding of a DC machine
b width [m]
b
0c
conductor width [m]

b
c
conductor width [m]
b
d
tooth width [m]
b
dr
rotor tooth width [m]
b
ds
stator tooth width [m]
b
r
rotor slot width [m]
b
s
stator slot width [m]
b
v
width of ventilation duct [m]
b
0
slot opening [m]
C capacitance [F], machine constant, integration constant
C temperature class >180
◦
C
C1–C2 compensating winding of a DC machine
C

f
friction coefficient
c specific heat capacity [J/kg K], capacitance per unit of length, factor,
divider, constant
c
p
specific heat capacity of air at constant pressure
c
th
heat capacity
CTI Comparative Tracking Index
c
v
specific volumetric heat [kJ/K m
3
]
D electric flux density [C/m
2
], diameter [m]
DC direct current
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xvi Abbreviations and Symbols
D
r
outer diameter of the rotor [m]
D

ri
inner diameter of the rotor [m]
D
s
inner diameter of the stator [m]
D
se
outer diameter of the stator [m]
D1–D2 series magnetizing winding of a DC machine
d thickness [m]
d
t
thickness of the fringe of a pole shoe [m]
E electromotive force (emf) [V], RMS, electric field strength [V/m], scalar, elastic
modulus, Young’s modulus [Pa]
E
a
activation energy [J]
E electric field strength, vector [V/m]
E temperature class 120
◦
C
E irradiation
E1–E2 shunt winding of a DC machine
e electromotive force [V], instantaneous value e(t )
e Napier’s constant
F force [N], scalar
F force [N], vector
F temperature class 155
◦

C
FEA finite element analysis
F
g
geometrical factor
F
m
magnetomotive force

H · dl [A], (mmf)
F1–F2 separate magnetizing winding of a DC machine or a synchronous machine
f frequency [Hz], Moody friction factor
g coefficient, constant, thermal conductance per unit length
G electrical conductance
G
th
thermal conductance
H magnetic field strength [A/m]
H
c
, H
cB
coercivity related to flux density [A/m]
H
cJ
coercivity related to magnetization [A/m]
H temperature class 180
◦
C, hydrogen
h height [m]

h
0c
conductor height [m]
h
c
conductor height [m]
h
d
tooth height [m]
h
p
height of a subconductor [m]
h
p2
height of pole body [m]
h
s
stator slot height [m]
h
yr
height of rotor yoke [m]
h
ys
height of stator yoke [m]
I electric current [A], RMS, brush current, second moment of an area, moment
of inertia of an area [m
4
]
IM induction motor
I

ns
counter-rotating current (negative-sequence component) [A]
I
o
current of the upper bar [A]
I
s
conductor current
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Abbreviations and Symbols xvii
I
u
current of the lower bar, slot current, slot current amount [A]
IC classes of electrical machines
IEC International Electrotechnical Commission
Im imaginary part
i current [A], instantaneous value i(t)
J moment of inertia [kg m
2
], current density [A/m
2
], magnetic polarization
J Jacobian matrix
J
ext
moment of inertia of load [kg m
2

]
J
M
moment of inertia of the motor, [kgm
2
]
J
sat
saturation of polarization [V s/m
2
]
J
s
surface current, vector [A/m]
j difference of the numbers of slots per pole and phase in different layers
j imaginary unit
K transformation ratio, constant, number of commutator segments
K
L
inductance ratio
k connecting factor (coupling factor), correction coefficient, safety factor, ordinal
of layers
k
C
Carter factor
k
Cu
, k
Fe
space factor for copper, space factor for iron

k
d
distribution factor
k
E
machine-related constant
k
Fe
,
n
correction factor
k
k
short-circuit ratio
k
L
skin effect factor for the inductance
k
p
pitch factor
k
pw
pitch factor due to coil side shift
k
R
skin effect factor for the resistance
k
sat
saturation factor
k

sq
skewing factor
k
th
coefficient of heat transfer [W/m
2
K]
k
v
pitch factor of the coil side shift in a slot
k
w
winding factor
k
σ
safety factor in the yield
L self-inductance [H]
L characteristic length, characteristic surface, tube length [m]
LC inductor–capacitor
L
d
tooth tip leakage inductance [H]
L
k
short-circuit inductance [H]
L
m
magnetizing inductance [H]
L
md

magnetizing inductance of an m-phase synchronous machine, in d-axis [H]
L
mn
mutual inductance [H]
L
pd
main inductance of a single phase [H]
L
u
slot inductance [H]
L

transient inductance [H]
L

subtransient inductance [H]
L1, L2, L3, network phases
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xviii Abbreviations and Symbols
l length [m], closed line, distance, inductance per unit of length, relative
inductance, gap spacing between the electrodes
l unit vector collinear to the integration path
l

effective core length [m]
l
ew

average conductor length of winding overhang [m]
l
p
wetted perimeter of tube [m]
l
pu
inductance as a per unit value
l
w
length of coil ends [m]
M mutual inductance [H], magnetization [A/m]
M
sat
saturation magnetization [A/m]
m number of phases, mass [kg],
m
0
constant
N number of turns in a winding, number of turns in series
N
f1
number of coil turns in series in a single pole
Nu Nusselt number
N
u1
number of bars of a coil side in the slot
N
k
number of turns of compensating winding
N

p
number of turns of one pole pair
N
v
number of conductors in each side
N Nondrive end
N set of integers
N
even
set of even integers
N
odd
set of odd integers
n normal unit vector of the surface
n rotation speed (rotation frequency) [1/s], ordinal of the harmonic (sub),
ordinal of the critical rotation speed, integer, exponent
n
U
number of section of flux tube in sequence
n
v
number of ventilation ducts
n

number of flux tube
P power, losses [W]
P
in
input power [W]
PAM pole amplitude modulation

PMSM permanent magnet synchronous machine (or motor)
PWM pulse width modulation
P
1
, P
ad
, P additional loss [W]
Pr Prandtl number
P
ρ
friction loss [W]
p number of pole pairs, ordinal, losses per core length
p
Al
aluminium content
p
∗
number of pole pairs of a base winding
pd partial discharge
Q electric charge [C], number of slots, reactive power [VA],
Q
av
average number of slots of a coil group
Q
o
number of free slots
Q

number of radii in a voltage phasor graph
Q

∗
number of slots of a base winding
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Abbreviations and Symbols xix
Q
th
quantity of heat
q number of slots per pole and phase, instantaneous charge, q(t)[C]
q
k
number of slots in a single zone
q
m
mass flow [kg/s]
q
th
density of the heat flow [W/m
2
]
R resistance [], gas constant, 8.314 472 [J/K mol], thermal resistance,
reactive parts
R
bar
bar resistance []
RM reluctance machine
RMS root mean square
R

m
reluctance [A/V s = 1/H]
R
th
thermal resistance [K/W]
Re real part
Re Reynolds number
Re
crit
critical Reynolds number
RR Resin-rich (impregnation method)
r radius [m], thermal resistance per unit length
r radius unit vector
S1–S8 duty types
S apparent power [VA], cross-sectional area
SM synchronous motor
SR switched reluctance
SyRM synchronous reluctance machine
S
c
cross-sectional area of conductor [m
2
]
S
p
pole surface area [m
2
]
S
r

rotor surface area facing the air gap [m
2
]
S Poynting’s vector [W/m
2
], unit vector of the surface
s slip, skewing measured as an arc length
T torque [N m], absolute temperature [K], period [s]
Ta Taylor number
Ta
m
modified Taylor number
T
b
pull-out torque, peak torque [N m]
t
c
commutation period [s]
TEFC totally enclosed fan-cooled
T
J
mechanical time constant [s]
T
mec
mechanical torque [N m]
T
u
pull-up torque [N m]
T
v

counter torque [N m]
T
l
locked rotor torque, [N m]
t time [s], number of phasors of a single radius, largest common divider,
lifetime of insulation
t tangential unit vector
t
c
commutation period [s]
t
r
rise time [s]
t* number of layers in a voltage vector graph for a base winding
U voltage [V], RMS
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xx Abbreviations and Symbols
U depiction of a phase
U
m
magnetic voltage [A]
U
sj
peak value of the impulse voltage [V]
U
v
coil voltage [V]

U1 terminal of the head of the U phase of a machine
U2 terminal of the end of the U phase of a machine
u voltage, instantaneous value u(t) [V], number of coil sides in a layer
u
b1
blocking voltage of the oxide layer [V]
u
c
commutation voltage [V]
u
m
mean fluid velocity in tube [m/s]
V volume [m
3
], electric potential
V depiction of a phase
V
m
scalar magnetic potential [A]
VPI vacuum pressure impregnation
V1 terminal of the head of the V phase of a machine
V2 terminal of the end of the V phase of a machine
v speed, velocity [m/s]
v vector
W energy [J], coil span (width) [m]
W depiction of a phase
W
d
energy returned through the diode to the voltage source in SR drives
W

fc
energy stored in the magnetic field in SR machines
W
md
energy converted to mechanical work while de-energizing the phase
in SR drives
W
mt
energy converted into mechanical work when the transistor is conducting
in SR drives
W
R
energy returning to the voltage source in SR drives
W

coenergy [J]
W1 terminal of the head of the W phase of a machine
W2 terminal of the end of the W phase of a machine
W
Φ
magnetic energy [J]
w length [m], energy per volume unit
X reactance []
x coordinate, length, ordinal number, coil span decrease [m]
x
m
relative value of reactance
Y admittance [S]
Y temperature class 90
◦

C
y coordinate, length, step of winding
y
m
winding step in an AC commutator winding
y
n
coil span in slot pitches
y
φ
coil span of full-pitch winding in slot pitches (pole pitch expressed in
number of slots per pole)
y
v
coil span decrease in slot pitches
y
1
step of span in slot pitches, back-end connector pitch
y
2
step of connection in slot pitches, front-end connector pitch
y
C
commutator pitch in number of commutator segments
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Abbreviations and Symbols xxi
Z impedance [], number of bars, number of positive and negative phasors of the

phase
Z
M
characteristic impedance of the motor []
Z
s
surface impedance []
Z
0
characteristic impedance []
z coordinate, length, integer, total number of conductors in the armature winding
z
a
number of adjacent conductors
z
b
number of brushes
z
c
number of coils
z
p
number of parallel-connected conductors
z
Q
number of conductors in a slot
z
t
number of conductors on top each other
α angle [rad], [

◦
], coefficient, temperature coefficient, relative pole width of the
pole shoe, convection heat transfer coefficient [W/K]
1/α depth of penetration
α
DC
relative pole width coefficient for DC machines
α
i
factor of the arithmetical average of the flux density
α
m
mass transfer coefficient [(mol/sm
2
)/(mol/m
3
)=m/s]
α
ph
angle between the phase winding
α
PM
relative permanent magnet width
α
r
heat transfer coefficient of radiation
α
SM
relative pole width coefficient for synchronous machines
α

str
angle between the phase winding
α
th
heat transfer coefficient [W/m
2
K]
α
u
slot angle [rad], [
◦
]
α
z
phasor angle, zone angle [rad], [
◦
]
α
ρ
angle of single phasor [rad], [
◦
]
β angle [rad], [
◦
], absorptivity
Γ energy ratio, integration route
Γ
c
interface between iron and air
γ angle [rad], [

◦
], coefficient
γ
c
commutation angle [rad], [
◦
]
γ
D
switch conducting angle [rad], [
◦
]
δ air gap (length), penetration depth [m], dissipation angle [rad], [
◦
], load angle
[rad], [
◦
]
δ
c
the thickness of concentration boundary layer [m]
δ
e
equivalent air gap (slotting taken into account) [m]
δ
ef
effective air gap(influence of iron taken into account)
δ
v
velocity boundary layer [m]

δ
T
temperature boundary layer [m]
δ

load angle [rad], [
◦
], corrected air gap [m]
δ
0
minimum air gap [m]
ε permittivity [F/m], position angle of the brushes [rad], [
◦
], stroke angle [rad],
[
◦
], amount of short pitching
ε
th
emissitivity
ε
0
permittivity of vacuum 8.854 × 10
−12
[F/m]
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xxii Abbreviations and Symbols

ζ phase angle [rad], [
◦
], harmonic factor
η efficiency, empirical constant, experimental pre-exponential constant,
reflectivity
Θ current linkage [A], temperature rise [K]
Θ
k
compensating current linkage [A]
Θ

total current linkage [A]
θ angle [rad], [
◦
]
ϑ angle [rad], [
◦
]
κ angle [rad], [
◦
], factor for reduction of slot opening, transmissivity
Λ permeance, [Vs/A], [H]
λ thermal conductivity [W/m K], permeance factor, proportionality factor,
inductance factor, inductance ratio
µ permeability [V s/A m, H/m], number of pole pairs operating simultaneously per
phase, dynamic viscosity [Pa s, kg/s m]
µ
r
relative permeability
µ

0
permeability of vacuum, 4π ×10
−7
[V s/A m, H/m]
ν ordinal of harmonic, Poisson’s ratio, reluctivity [A m/V s, m/H], pulse velocity
ξ reduced conductor height
ρ resistivity [ m], electric charge density [C/m
2
], density [kg/m
3
], reflection
factor, ordinal number of a single phasor
ρ
A
absolute overlap ratio
ρ
E
effective overlap ratio
ρ
ν
transformation ratio for IM impedance, resistance, inductance
σ specific conductivity, electric conductivity [S/m], leakage factor, ratio of the
leakage flux to the main flux
σ
F
tension [Pa]
σ
Fn
normal tension [Pa]
σ

Ftan
tangential tension [Pa]
σ
mec
mechanical stress [Pa]
σ
SB
Stefan–Boltzmann constant, 5.670 400 × 10
−8
W/m
2
/K
4
τ relative time
τ
p
pole pitch [m]
τ
q2
pole pitch on the pole surface [m]
τ
r
rotor slot pitch [m]
τ
s
stator slot pitch [m]
τ
u
slot pitch [m]
τ

v
zone distribution
τ

d
direct-axis transient short-circuit time constant [s]
τ

d0
direct-axis transient open-circuit time constant [s]
τ

d0
direct-axis subtransient open-circuit time constant [s]
τ

q
quadrature-axis subtransient short-circuit time constant [s]
τ

q0
quadrature-axis subtransient open-circuit time constant [s]
υ factor, kinematic viscosity, µ/ρ, [Pa s/(kg/m
3
)]
Φ magnetic flux [V s, Wb]
Φ
th
thermal power flow, heat flow rate [W]
Φ

δ
air gap flux [V s], [Wb]
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Abbreviations and Symbols xxiii
φ magnetic flux, instantaneous value φ(t) [V s], electric potential [V]
ϕ phase shift angle [rad], [
◦
]
ϕ

function for skin effect calculation
 magnetic flux linkage [V s]
ψ electric flux [C],
ψ

function for skin effect calculation
χ length/diameter ratio, shift of a single pole pair
Ω mechanical angular speed [rad/s]
ω electric angular velocity [rad/s], angular frequency [rad/s]
T temperature rise [K,
◦
C]
∇T temperature gradient [K/m,
◦
C/m]
p pressure drop [Pa]
Subscripts

0 section
1 primary, fundamental component, beginning of a phase, locked rotor torque,
2 secondary, end of a phase
Al aluminium
a armature, shaft
ad additional (loss)
av average
B brush
b base value, peak value of torque, blocking
bar bar
bearing bearing (losses)
C capacitor
Cu copper
c conductor, commutation
contact brush contact
conv convection
cp commutating poles
cr, crit critical
D direct, damper
DC direct current
d tooth, direct, tooth tip leakage flux
EC eddy current
e equivalent
ef effective
el electric
em electromagnetic
ew end winding
ext external
F force
Fe iron

f field
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xxiv Abbreviations and Symbols
Hy hysteresis
i internal, insulation
k compensating, short circuit, ordinal
M motor
m mutual, main
mag magnetizing, magnetic
max maximum
mec mechanical
min minimum
mut mutual
N rated
n nominal, normal
ns negative-sequence component
o starting, upper
opt optimal
PM permanent magnet
p pole, primary, subconductor, pole leakage flux
p1 pole shoe
p2 pole body
ph phasor, phase
ps positive-sequence component
pu per unit
q quadrature, zone
r rotor, remanence, relative

res resultant
S surface
s stator
sat saturation
sj impulse wave
sq skew
str phase section
syn synchronous
tan tangential
test test
th thermal
tot total
u slot, lower, slot leakage flux, pull-up torque
v zone, coil side shift in a slot, coil
w end winding leakage flux
xx-direction
yy-direction, yoke
ya armature yoke
yr rotor yoke
ys stator yoke
zz-direction, phasor of voltage phasor graph
δ air gap
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Abbreviations and Symbols xxv
γ ordinal of a subconductor
ν harmonic
ρ ordinal number of single phasor

ρ friction loss
ρw windage (loss)
σ flux leakage
Φ flux
Subscripts
ˆ peak/maximum value, amplitude

imaginary, apparent, reduced, virtual
* base winding, complex conjugate
Boldface symbols are used for vectors with components parallel to the unit
vectors i, j and k
A vector potential, A = i A
x
+ j A
k
+ kA
z
B flux density, B = i B
x
+ j B
k
+ kB
z
I complex phasor of the current
I bar above the symbol denotes average value
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1
Principal Laws and Methods in
Electrical Machine Design

1.1 Electromagnetic Principles
A comprehensive command of electromagnetic phenomena relies fundamentally on
Maxwell’s equations. The description of electromagnetic phenomena is relatively easy when
compared with various other fields of physical sciences and technology, since all the field
equations can be written as a single group of equations. The basic quantities involved in the
phenomena are the following five vector quantities and one scalar quantity:
Electric field strength E [V/m]
Magnetic field strength H [A/m]
Electric flux density D [C/m
2
]
Magnetic flux density B [V s/m
2
], [T]
Current density J [A/m
2
]
Electric charge density, dQ/dV ρ [C/m
3
]
The presence of an electric and magnetic field can be analysed from the force exerted by
the field on a charged object or a current-carrying conductor. This force can be calculated by
the Lorentz force (Figure 1.1), a force experienced by an infinitesimal charge dQ moving at a
speed v. The force is given by the vector equation
dF = dQ
(
E + v × B
)
= dQ E +
dQ

dt
dl × B = dQE +idl × B. (1.1)
In principle, this vector equation is the basic equation in the computation of the torque for
various electrical machines. The latter part of the expression in particular, formulated with a
current-carrying element of a conductor of the length dl, is fundamental in the torque produc-
tion of electrical machines.
Design of Rotating Electrical Machines Juha Pyrh¨onen, Tapani Jokinen and Val´eria Hrabovcov´a
© 2008 John Wiley & Sons,
Ltd. ISBN: 978-0-470-69516-6
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2 Design of Rotating Electrical Machines
i
dF
dl
B
β
Figure 1.1 Lorentz force dF acting on a differential length dl of a conductor carrying an electric cur-
rent i in the magnetic field B. The angle β is measured between the conductor and the flux density vector
B. The vector product i dl × B may now be written in the form i dl × B = idlB sin β
Example 1.1: Calculate the force exerted on a conductor 0.1 m long carrying a current of
10 A at an angle of 80
◦
with respect to a field density of 1 T.
Solution: Using (1.1) we get directly for the magnitude of the force
F =
|
il × B
|
= 10 A ·0.1m· sin 80

◦
· 1Vs/m
2
= 0.98VAs/m = 0.98 N.
In electrical engineering theory, the other laws, which were initially discovered empirically
and then later introduced in writing, can be derived from the following fundamental laws
presented in complete form by Maxwell. To be independent of the shape or position of the
area under observation, these laws are presented as differential equations.
A current flowing from an observation point reduces the charge of the point. This law of
conservation of charge can be given as a divergence equation
∇·J =−
∂ρ
∂t
, (1.2)
which is known as the continuity equation of the electric current.
Maxwell’s actual equations are written in differential form as
∇×E =−
∂B
∂t
, (1.3)
∇×H = J +
∂D
∂t
, (1.4)
∇·D = ρ, (1.5)
∇·B = 0. (1.6)