DESIGN OF ROTATINGELECTRICALMACHINES

DESIGN OF ROTATINGELECTRICALMACHINES

Second Edition

Juha PyrhonenLappeenranta University of Technology, Finland

Tapani JokinenAalto University, School of Electrical Engineering, Finland

Valeria HrabovcovaFaculty of Electrical Engineering, University of Zilina, Slovakia

This edition first published 2014C© 2014 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Pyrhonen, Juha.Design of rotating electrical machines / Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova. – Second edition.

pages cmIncludes bibliographical references and index.

ISBN 978-1-118-58157-5 (hardback)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. Hrabovcova, Valeria.III. Title.

TK2331.P97 2013621.31′042–dc23

2013021891

A catalogue record for this book is available from the British Library.

ISBN: 978-1-118-58157-5

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

1 2014

Contents

Preface xi

About the Authors xiii

Abbreviations and Symbols xv

1 Principal Laws and Methods in Electrical Machine Design 11.1 Electromagnetic Principles 11.2 Numerical Solution 81.3 The Most Common Principles Applied to Analytic Calculation 12

1.3.1 Flux Line Diagrams 161.3.2 Flux Diagrams for Current-Carrying Areas 22

1.4 Application of the Principle of Virtual Work in the Determination ofForce and Torque 25

1.5 Maxwell’s Stress Tensor; Radial and Tangential Stress 321.6 Self-Inductance and Mutual Inductance 361.7 Per Unit Values 421.8 Phasor Diagrams 45

Bibliography 47

2 Windings of Electrical Machines 482.1 Basic Principles 49

2.1.1 Salient-Pole Windings 492.1.2 Slot Windings 532.1.3 End Windings 54

2.2 Phase Windings 542.3 Three-Phase Integral Slot Stator Winding 572.4 Voltage Phasor Diagram and Winding Factor 642.5 Winding Analysis 722.6 Short Pitching 742.7 Current Linkage of a Slot Winding 812.8 Poly-Phase Fractional Slot Windings 942.9 Phase Systems and Zones of Windings 97

2.9.1 Phase Systems 972.9.2 Zones of Windings 99

vi Contents

2.10 Symmetry Conditions 1012.10.1 Symmetrical Fractional Slot Windings 101

2.11 Base Windings 1042.11.1 First-Grade Fractional Slot Base Windings 1042.11.2 Second-Grade Fractional Slot Base Windings 1052.11.3 Integral Slot Base Windings 106

2.12 Fractional Slot Windings 1082.12.1 Single-Layer Fractional Slot Windings 1082.12.2 Double-Layer Fractional Slot Windings 117

2.13 Single- and Double-Phase Windings 1242.14 Windings Permitting a Varying Number of Poles 1272.15 Commutator Windings 129

2.15.1 Lap Winding Principles 1332.15.2 Wave Winding Principles 1362.15.3 Commutator Winding Examples, Balancing Connectors 1392.15.4 AC Commutator Windings 1432.15.5 Current Linkage of the Commutator Winding and

Armature Reaction 1442.16 Compensating Windings and Commutating Poles 1462.17 Rotor Windings of Asynchronous Machines 1492.18 Damper Windings 152

Bibliography 153

3 Design of Magnetic Circuits 1553.1 Air Gap and its Magnetic Voltage 161

3.1.1 Air Gap and Carter Factor 1613.1.2 Air Gaps of a Salient-Pole Machine 1663.1.3 Air Gap of Nonsalient-Pole Machine 172

3.2 Equivalent Core Length 1733.3 Magnetic Voltage of a Tooth and a Salient Pole 176

3.3.1 Magnetic Voltage of a Tooth 1763.3.2 Magnetic Voltage of a Salient Pole 180

3.4 Magnetic Voltage of Stator and Rotor Yokes 1803.5 No-Load Curve, Equivalent Air Gap and Magnetizing Current of the Machine 1833.6 Magnetic Materials of a Rotating Machine 186

3.6.1 Characteristics of Ferromagnetic Materials 1893.6.2 Losses in Iron Circuits 194

3.7 Permanent Magnets in Rotating Machines 2033.7.1 History and Development of Permanent Magnets 2033.7.2 Characteristics of Permanent Magnet Materials 2053.7.3 Operating Point of a Permanent Magnet Circuit 2103.7.4 Demagnetization of Permanent Magnets 2173.7.5 Application of Permanent Magnets in Electrical Machines 219

3.8 Assembly of Iron Stacks 226Bibliography 227

Contents vii

4 Inductances 2294.1 Magnetizing Inductance 2304.2 Leakage Inductances 233

4.2.1 Division of Leakage Flux Components 2354.3 Calculation of Flux Leakage 238

4.3.1 Skewing Factor and Skew Leakage Inductance 2394.3.2 Air-Gap Leakage Inductance 2434.3.3 Slot Leakage Inductance 2484.3.4 Tooth Tip Leakage Inductance 2594.3.5 End Winding Leakage Inductance 260Bibliography 264

5 Resistances 2655.1 DC Resistance 2655.2 Influence of Skin Effect on Resistance 266

5.2.1 Analytical Calculation of Resistance Factor 2665.2.2 Critical Conductor Height in Slot 2765.2.3 Methods to Limit the Skin Effect 2775.2.4 Inductance Factor 2785.2.5 Calculation of Skin Effect in Slots Using Circuit Analysis 2795.2.6 Double-Sided Skin Effect 287Bibliography 292

6 Design Process of Rotating Electrical Machines 2936.1 Eco-Design Principles of Rotating Electrical Machines 2936.2 Design Process of a Rotating Electrical Machine 294

6.2.1 Starting Values 2946.2.2 Main Dimensions 2976.2.3 Air Gap 3056.2.4 Winding Selection 3096.2.5 Air-Gap Flux Density 3106.2.6 The No-Load Flux of an Electrical Machine and the Number of

Winding Turns 3116.2.7 New Air-Gap Flux Density 3166.2.8 Determination of Tooth Width 3176.2.9 Determination of Slot Dimensions 3186.2.10 Determination of the Magnetic Voltages of the Air Gap,

and the Stator and Rotor Teeth 3236.2.11 Determination of New Saturation Factor 3266.2.12 Determination of Stator and Rotor Yoke Heights and

Magnetic Voltages 3266.2.13 Magnetizing Winding 3276.2.14 Determination of Stator Outer and Rotor Inner Diameter 3296.2.15 Calculation of Machine Characteristics 329Bibliography 330

viii Contents

7 Properties of Rotating Electrical Machines 3317.1 Machine Size, Speed, Different Loadings and Efficiency 331

7.1.1 Machine Size and Speed 3317.1.2 Mechanical Loadability 3337.1.3 Electrical Loadability 3377.1.4 Magnetic Loadability 3387.1.5 Efficiency 340

7.2 Asynchronous Motor 3427.2.1 Current Linkage and Torque Production of an

Asynchronous Machine 3427.2.2 Impedance and Current Linkage of a Cage Winding 3497.2.3 Characteristics of an Induction Machine 3567.2.4 Equivalent Circuit Taking Asynchronous Torques and Harmonics

into Account 3617.2.5 Synchronous Torques 3677.2.6 Selection of the Slot Number of a Cage Winding 3697.2.7 Construction of an Induction Motor 3717.2.8 Cooling and Duty Types 3737.2.9 Examples of the Parameters of Three-Phase Industrial

Induction Motors 3787.2.10 Asynchronous Generator 3807.2.11 Wound Rotor Induction Machine 3827.2.12 Asynchronous Motor Supplied with Single-Phase Current 383

7.3 Synchronous Machines 3887.3.1 Inductances of a Synchronous Machine in Synchronous Operation

and in Transients 3907.3.2 Loaded Synchronous Machine and Load Angle Equation 4007.3.3 RMS Value Phasor Diagrams of a Synchronous Machine 4077.3.4 No-Load Curve and Short-Circuit Test 4177.3.5 Asynchronous Drive 4197.3.6 Asymmetric-Load-Caused Damper Currents 4237.3.7 Shift of Damper Bar Slotting from the Symmetry Axis of the Pole 4247.3.8 V Curve of a Synchronous Machine 4267.3.9 Excitation Methods of a Synchronous Machine 4267.3.10 Permanent Magnet Synchronous Machines 4277.3.11 Synchronous Reluctance Machines 456

7.4 DC Machines 4687.4.1 Configuration of DC Machines 4687.4.2 Operation and Voltage of a DC Machine 4707.4.3 Armature Reaction of a DC machine and Machine Design 4747.4.4 Commutation 475

7.5 Doubly Salient Reluctance Machine 4797.5.1 Operating Principle of a Doubly Salient Reluctance Machine 4797.5.2 Torque of an SR Machine 4807.5.3 Operation of an SR Machine 481

Contents ix

7.5.4 Basic Terminology, Phase Number and Dimensioning ofan SR Machine 485

7.5.5 Control Systems of an SR Motor 4897.5.6 Future Scenarios for SR Machines 491Bibliography 492

8 Insulation of Electrical Machines 4958.1 Insulation of Rotating Electrical Machines 4978.2 Impregnation Varnishes and Resins 5038.3 Dimensioning of an Insulation 5068.4 Electrical Reactions Ageing Insulation 5098.5 Practical Insulation Constructions 510

8.5.1 Slot Insulations of Low-Voltage Machines 5118.5.2 Coil End Insulations of Low-Voltage Machines 5128.5.3 Pole Winding Insulations 5128.5.4 Low-Voltage Machine Impregnation 5138.5.5 Insulation of High-Voltage Machines 513

8.6 Condition Monitoring of Insulation 5158.7 Insulation in Frequency Converter Drives 518

Bibliography 521

9 Losses and Heat Transfer 5239.1 Losses 524

9.1.1 Resistive Losses 5249.1.2 Iron Losses 5269.1.3 Additional Losses 5269.1.4 Mechanical Losses 5279.1.5 Decreasing Losses 5299.1.6 Economics of Energy Savings 533

9.2 Heat Removal 5349.2.1 Conduction 5349.2.2 Radiation 5389.2.3 Convection 541

9.3 Thermal Equivalent Circuit 5489.3.1 Analogy between Electrical and Thermal Quantities 5489.3.2 Average Thermal Conductivity of a Winding 5499.3.3 Thermal Equivalent Circuit of an Electrical Machine 5509.3.4 Modeling of Coolant Flow 5609.3.5 Solution of Equivalent Circuit 5659.3.6 Cooling Flow Rate 568Bibliography 568

Appendix A 570

Appendix B 572

Index 575

Preface

Electrical machines are almost entirely used in producing electricity, and there are very fewelectricity-producing processes where rotating machines are not used. In such processes, atleast auxiliary motors are usually needed. In distributed energy systems, new machine typesplay a considerable role: for instance, the era of permanent magnet machines has commenced.

About half of all electricity produced globally is used in electric motors, and the share ofaccurately controlled motor drives applications is increasing. Electrical drives provide probablythe best control properties for a wide variety of processes. The torque of an electric motor maybe controlled accurately, and the efficiencies of the power electronic and electromechanicalconversion processes are high. What is most important is that a controlled electric motor drivemay save considerable amounts of energy. In the future, electric drives will probably playan important role also in the traction of cars and working machines. Because of the largeenergy flows, electric drives have a significant impact on the environment. If drives are poorlydesigned or used inefficiently, we burden our environment in vain. Environmental threats giveelectrical engineers a good reason for designing new and efficient electric drives.

Finland has a strong tradition in electric motors and drives. Lappeenranta University ofTechnology and Aalto University have found it necessary to maintain and expand the instruc-tion given in electric machines. The objective of this book is to provide students in electricalengineering with an adequate basic knowledge of rotating electric machines, for an under-standing of the operating principles of these machines as well as developing elementary skillsin machine design. Although, due to the limitations of this material, it is not possible to includeall the information required in electric machine design in a single book, this material will serveas a manual for a machine designer in the early stages of his or her career. The bibliographiesat the end of chapters are intended as sources of references and recommended backgroundreading. The Finnish tradition of electrical machine design is emphasized in this monographthrough the important contributions of Professor Tapani Jokinen, who has spent decades indeveloping the Finnish machine design profession. Equally important is the view of electricalmachine design provided by Professor Valeria Hrabovcova from Slovak Republic, which alsohas a strong industrial tradition.

In the second edition, some parts of the first edition have been rewritten to make the textproceed more logically and many printing errors have been corrected. Especially, permanentmagnet machine and synchronous reluctance machine chapters are now much more com-prehensive including new research results. Also the Eco-design principles and economicalconsiderations in machine design are shortly introduced.

xii Preface

The authors are thankful for Dr. Hanna Niemela for translating the original Finnish materialfor the first edition.

We express our gratitude to the following persons, who have kindly provided material forthis book: Professor Antero Arkkio (Aalto University), Dr Jorma Haataja, Dr Tanja Hedberg(ITT Water and Wastewater AB), Mr Jari Jappinen (ABB), Dr Hanne Jussila (LUT), Dr PanuKurronen (The Switch Oy), Dr Janne Nerg (LUT), Dr Markku Niemela (ABB), Dr AskoParviainen (AXCO Motors), Dr Sami Ruoho (Teollisuuden Voima), Dr Marko Rilla (Visedo),Dr Pia Salminen (LUT), Dr Ville Sihvo (MAN Turbo), Mr Pavel Ponomarev, Mr Juho Monto-nen, Ms Julia Alexandrova, Dr. Henry Hamalainen and numerous other colleagues. Dr HannaNiemela’s contribution to the first edition and the publication process of the original manuscriptis particularly acknowledged.

Juha PyrhonenTapani Jokinen

Valeria Hrabovcova

About the Authors

Juha Pyrhonen is a Professor in the Department of Elec-trical Engineering at Lappeenranta University of Technol-ogy, Finland. He is engaged in the research and develop-ment of electric motors and drives. He is especially activein the fields of permanent magnet synchronous machinesand drives and solid-rotor high-speed induction machinesand drives. He has worked on many research and industrialdevelopment projects and has produced numerous publi-cations and patents in the field of electrical engineering.

Tapani Jokinen is a Professor Emeritus in the School ofElectrical Engineering at Aalto University, Finland. Hisprincipal research interests are in AC machines, creativeproblem solving and product development processes. Hehas worked as an electrical machine design engineer withOy Stromberg Ab Works. He has been a consultant forseveral companies, a member of the Board of High SpeedTech Ltd and Neorem Magnets Oy, and a member of theSupreme Administrative Court in cases on patents. Hisresearch projects include, among others, the development

of superconducting and large permanent magnet motors for ship propulsion, the developmentof high-speed electric motors and active magnetic bearings, and the development of finiteelement analysis tools for solving electrical machine problems.

xiv About the Authors

Valeria Hrabovcova is a Professor of ElectricalMachines in the Department of Power Electrical Sys-tems, Faculty of Electrical Engineering, at the Universityof Zilina, Slovak Republic. Her professional and researchinterests cover all kinds of electrical machines, electron-ically commutated electrical machines included. She hasworked on many research and development projects andhas written numerous scientific publications in the fieldof electrical engineering. Her work also includes variouspedagogical activities, and she has participated in manyinternational educational projects.

Abbreviations and Symbols

A linear current density, [A/m]A magnetic vector potential, [Vs/m]A magnetic vector potential scalar value, [Vs/m]A temperature class 105 ◦CAC alternating currentAM asynchronous machineA1-A2 armature winding of a DC machineA1n, A2n, A3n factors for defining permanent magnet flux densitya number of parallel paths in windings without commutator: per phase, in

windings with a commutator: per half armature, diffusivityB magnetic flux density, vector [Vs/m2], [T]B magnetic flux density scalar value, [Vs/m2]Br remanent flux density, [T]Bsat saturation flux density, [T]B temperature class 130 ◦CB1-B2 commutating pole winding of a DC machineb width, [m]b0c conductor width [m]bc conductor width [m]bd tooth width, [m]bdr rotor tooth width, [m]bds stator tooth width, [m]br rotor slot width, [m]bs stator slot width, [m]b0 slot opening, [m]bv width of ventilation duct, [m]C capacitance, [F], machine constant, integration constant, fabrication cost, [€]C temperature class >180 ◦CC1-C2 compensating winding of a DC machineCf friction coefficientCM torque coefficientCs saving cost per year, [€/a]

xvi Abbreviations and Symbols

c specific heat capacity, [J/kgK], capacitance per unit of length, factor, divider,constant

Cdiff increase of the purchase cost, [€]ce energy cost, [€/kWh]cp specific heat capacity of air in constant pressureCpw cost per one kilowatt of loss over the life of motor, [€/kW]cth heat capacityCTI Comparative Tracking Indexcv specific volumetric heat, [kJ/Km3]D electric flux density [C/m2], diameter [m]DC direct currentDOL direct- on-lineDs inner diameter of the stator, [m]Dse outer diameter of the stator, [m]Dr outer diameter of the rotor, [m]Dri inner diameter of the rotor, [m]D1-D2 series magnetizing winding of a DC machined thickness, [m]dt 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], bearing loadEa activation energy, [J]E electric field strength, vector, [V/m]E electric field strength scalar value, [V/m]E temperature class 120 ◦CE irradiation intensity [W/m2]E1-E2 shunt winding of a DC machinee electromotive force [V], instantaneous value e(t)e Napier’s constantemf electromotive force, [V]F force, [N], scalarF force, [N], vectorF temperature class 155 ◦CFEA Finite Element AnalysisFg geometrical factorFm magnetomotive force

∮H · dl , [A], (mmf)

F1-F2 separate magnetizing winding of a DC machine or a synchronous machinef frequency, [Hz], Moodyfriction factorfBr factor for defining permanent magnet radial flux densityfBθ factor for defining permanent magnet tangential flux densityg coefficient, constant,thermal conductance per unit lengthG electrical conductanceGth thermal conductanceH magnetic field strength, [A/m]H magnetic field strength scalar value, [A/m]Hc, HcB coercivity related to flux density, [A/m]

Abbreviations and Symbols xvii

HcJ coercivity related to magnetization, [A/m]H temperature class 180 ◦CHn number of partial dischargesh height, [m]h0c conductor height [m]hc conductor height [m]hd tooth height [m]hp height of a subconductor, [m]hp2 height of pole body, [m]hys height of stator yoke, [m]hyr height of rotor yoke, [m]hs stator slot height, [m]I electric current, [A], RMS, brush current, second moment of inertia of an

area, [m4]IM induction motorIns counter-rotating current (negative sequence component), [A]Io current of the upper bar, [A]Iu current of the lower bar, slot current, slot current amount, [A]Is conductor currentIC classes of electrical machinesIEC International Electrotechnical CommissionIm imaginary parti current, [A], instantaneous value i(t), per unit value of current, [pu], annual

rate of interestJ moment of inertia, [kgm2], current density [A/m2], magnetic polarizationJ0PM current density on the PM surface, [A/m2]JPM eddy current density, [A/m2]J Jacobian matrixJext moment of inertia of load, [kgm2]JM moment of inertia of the motor, [kgm2]Jsat saturation of polarization, [Vs/m2]Js surface current, vector, [A/m]Js surface current vector scalar value, [A/m]j difference of the numbers of slots per pole and phase in different layersj imaginary unitK transformation ratio, constant, number of commutator segmentsKBr factor for defining permanent magnet radial flux densityKBθ factor for defining permanent magnet tangential flux densityKL inductance ratiok connecting factor (coupling factor), correction coefficient, safety factor,

ordinal of layers, roughness coefficientkE machine-related constantkC Carter factorkCu, kFe space factor for copper, space factor for ironkd distribution factor, correction factor, saliency factor in d- axiskq saliency factor in q- axis

xviii Abbreviations and Symbols

kdsat saliency factor taking into account saturation in d- axiskqpar saliency factor taking into account parallel magnetic lines in q- axiskFe,n correction factorkk short circuit ratiokL skin effect factor for the inductancekp pitch factorkpw pitch factor due to coil side shift, present worth factor of an equal payment

serieskR skin effect factor for the resistanceksat saturation factorksq skewing factorkth coefficient of heat transfer, [W/m2K]kv pitch factor of the coil side shift in a slotkw winding factorkσ safety factor in the yieldL self inductance [H]L characteristic length, characteristic surface length, tube length [m]LC inductor-capacitorLd tooth tip leakage inductance, synchronous inductance in d- axis [H]Lq synchronous inductance in q- axis [H]Ld/ Lq inductance ratioLk short-circuit inductance, [H]Lm magnetizing inductance, [H]Lmd magnetizing inductance of an m-phase synchronous machine, in d-axis,[H]Lmq magnetizing inductance of an m-phase synchronous machine, in q-axis, [H]Lmn mutual inductance, [H]Lmp magnetizing inductance of single-phase winding, [H]Lpd main inductance of a single phase, [H]Lsq skew leakage inductance, [H]Lu slot inductance, [H]Lw end winding leakage inductance, [H]Lδ air-gap leakage inductance, [H]Lmδ magnetizing inductance of synchronous machines with non-salient

poles, [H]L ′ transient inductance, [H]L ′′ subtransient inductance, [H]L1, L2, L3 network phasesl length [m], closed line, distance, inductance per unit of length, relative

inductance (inductance per unit value), gap spacing between the electrodesl unit vector collinear to the integration pathl’ effective core length, [m]lew average conductor length of winding overhang, [m]lp wetted perimeter of tube, [m]lpu inductance as a per unit valuelw length of coil ends, [m]lsub length of one sub- stack, [m]

Abbreviations and Symbols xix

M mutual inductance [H], magnetization [A/m]Msat saturation magnetization, [A/m]m number of phases, mass, [kg]mc mutual coupling factorm0 constantmmf magnetomotive force, [A]N number of turns in a winding, number of turns in seriesNf1 number of coil turns in series in a single poleNu Nusselt numberNu1 number of bars of a coil side in the slotNp number of turns of one pole pairNk number of turns of compensating windingNv number of conductors in each sideN Non-drive endN set of integersNeven set of even integersNodd set of odd integersn normal unit vector of the surfacen rotation speed (rotation frequency), [1/s], ordinal of the harmonic (sub),

ordinal of the critical rotation speed, integer, exponent, years of saving (motorlife time)

nv number of ventilation ductsnU number of section of flux tube in sequencenΦ number of flux tubeP power, losses [W]Pin input power, [W]PAM Pole-Amplitude-ModulationPM permanent magnetPMSM permanent magnet synchronous motor (or machine)PWM Pulse Width ModulationP1, Pad, PLL additional loss, [W]Pew end winding losses, [W]Pr Prandtl numberPρ friction loss, [W]Pdiff reduction of the purchase cost, [€]PPM eddy current loss in permanent magnet, [W]p number of pole pairs, ordinal, losses per core length, resistive losses per core

length, [W/m], pressure, [Pa]pAl aluminium contentp∗ number of pole pairs of a base windingpd partial dischargeQ electric charge, [C], number of slots, reactive power, [VA]Qav average number of slots of a coil groupQp number of slots per poleQo number of free slotsQ’ number of radii in a voltage phasor graph

xx Abbreviations and Symbols

Q∗ number of slots of a base windingQth quantity of heatq number of slots per pole and phase, instantaneous charge, q(t), [C]qk number of slots in a single zoneqm mass flow, [kg/s]qth density of the heat flow, [W/m2]R resistance, [�], gas constant, 8.314472 [J/K·mol], thermal resistance, reactive

partsRbar bar resistance, [�]RM reluctance machineRMS root mean squareRm reluctance, [A/Vs = 1/H]Rth thermal resistance, [K/W]Re real partRe Reynolds numberRecrit critical Reynolds numberRR Resin Rich impregnation methodr radius, [m], thermal resistance per unit length, per unit resistance [pu],

coefficient of radiationr radius unit vectorS1-S8 duty typesS apparent power, [VA], cross-sectional areaSM synchronous motorSR switched reluctanceSyRM synchronous reluctance machineSc cross-sectional area of conductor, [m2]Sp pole surface area, [m2]Sr rotor surface area facing the air gap, [m2]S Poynting’s vector, [W/m2], unit vector of the surfaces slip, skewing measured as an arc lengthsb slip at maximum torquessp skewing expressed as a number of slot pitchesT torque, [Nm], absolute temperature, [K], period, operating time of the motor

per year, [h/a]Ta Taylor numberTam modified Taylor numberTb pull-out torque, peak (maximum) torque [Nm]tc commutation period, [s]TEFC totally enclosed fan-cooledTJ mechanical time constant, [s]Tmec mechanical torque, [Nm]Tpb payback timeTs temperature of the planeTu pull-up torque, [Nm]Tv counter torque, [Nm]Tl locked rotor torque, [Nm]

Abbreviations and Symbols xxi

TC tooth coilt time, [s], number of phasors of a single radius, largest common divider,

lifetime of insulationt tangential unit vectortc commutation period, [s]tr rise time, [s]t∗ number of layers in a voltage vector graph for a base windingU voltage, [V], RMSU depiction of a phaseUcontact contact voltage drop, [V]Um magnetic voltage, [A]Ur resistive voltage, [V]Usj peak value of the impulse voltage, [V]Uv coil voltage, [V]U1 terminal of the head of the U-phase of the machineU2 terminal of the end of the U-phase of the machineu voltage, instantaneous value u(t), [V], number of coil sides in a layer, per unit

value of voltage, [pu]ub1 blocking voltage of the oxide layer, [V]uc commutation voltage, [V]um mean fluid velocity in tube, [m/s]V volume, [m3], electric potentialV depiction of a phaseVm scalar magnetic potential, [A]VPI Vacuum Pressure ImpregnationV1 terminal of the head of the V-phase of the machineV2 terminal of the end of the V-phase of the machinev speed, velocity, [m/s]v vectorW energy, [J], coil span (width), average coil span [m]W depiction of a phaseWfc energy stored in the magnetic field in SR machinesWd energy returned through the diode to the voltage source in SR drivesWmt energy converted into mechanical work when the transistor is conducting in

SR drivesWmd energy converted to mechanical work while de-energizing the phase in SR

drivesWR energy returning to the voltage source in SR drivesW’ co-energy, [J]W1 terminal of the head of the W-phase of the machineW2 terminal of the end of the W-phase of the machineWΦ magnetic energy, [J]w length, [m], energy per volume unitwPM permanent magnet width, [m]X reactance, [�]x coordinate, length, ordinal number, coil span decrease [m]

xxii Abbreviations and Symbols

xm relative value of reactanceY admittance, [S]Y temperature class 90◦Cy coordinate, length, step of windingym winding step in an AC commutator windingyn coil span in slot pitchesyQ coil span of full-pitch winding in slot pitches, pole pitch expressed in number

of slots per poleyv coil span decrease in slot pitchesy1 step of span in slot pitches, back end connector pitchy2 step of connection in slot pitches, front end connector pitchyC commutator pitch in number of commutator segmentsZ impedance, [�], number of bars, number of positive and negative phasors of

the phaseZM characteristic impedance of the motor, [�]Zs surface impedance, [�]Z0 characteristic impedance, [�]z coordinate, length, integer, total number of conductors in the armature

windingza number of adjacent conductorszb number of brusheszc number of coilszcs number of conductors in half slotzp number of parallel-connected conductorszQ number of conductors in a slotzt number of conductors on top each otherα angle, [rad], [◦], coefficient, temperature coefficient, relative pole width

of the pole shoe, convection heat transfer coefficient, [W/K], skew angle,[rad], [◦]

1/α depth of penetrationαDC relative pole width coefficient for DC machinesαi ratio of the arithmetical average of the flux density to its peak valueα m mass transfer coefficient, [(mol/sm2)/ (mol/m3) = m/s]α PM relative permanent magnet widthαSM relative pole width coefficient for synchronous machinesαr heat transfer coefficient of radiationαstr angle between the phase windingαth heat transfer coefficient [W/(m2K)]αph angle between the phase windingαu slot angle, [rad], [◦]αz phasor angle, zone angle, [rad], [◦]αρ angle of single phasor, [rad], [◦]β angle [rad], [◦]β absorptivityΓ energy ratio, integration routeΓ c interface between iron and air

Abbreviations and Symbols xxiii

γ angle, [rad], [◦], coefficientγ c commutation angle, [rad], [◦]γ D switch conducting angle, [rad], [◦]δ air gap (length), penetration depth [m], dielectric loss angle, [rad], [◦],

dissipation angle, [rad], [◦], load angle, [rad], [◦]δc the thickness of concentration boundary layer, [m]δd(x) air gap profile function in d- axis, [m]δq(x) air gap profile function in q- axis, [m]δe equivalent air gap (slotting taken into account), [m]δef effective air gap (influence of iron and slotting taken into account)δPM depth of penetration in PM, [m]δv velocity boundary layer, [m]δT temperature (thermal) boundary layer, [m]δ ’ load angle, [rad], [◦], corrected air gap, [m]δ0 minimum air gap, [m]δ0e air gap in the middle of the pole corrected with Carter factor, [m]δde equivalent d- axis air gap, [m]δqe equivalent q- axis air gap, [m]Δ2 damping factorε permittivity [F/m], position angle of the brushes, [rad], [◦], stroke angle, [rad],

[◦], amount of short pitchingεsp amount of short pitching expressed as slot pitchesεth emissitivityεthr relative emissitivityε0 permittivity of vacuum 8.854·10−12 [F/m]ζ phase angle, [rad], [◦], harmonic factor, saliency ratio, phase angle of the rotor

impedanceζ d harmonic factor in d-axisζ q harmonic factor in q-axisη efficiency, empirical constant, experimental pre-exponential constantη reflectivity, thermal conductivityΘ current linkage, [A], temperature rise (difference) [K]Θk compensating current linkage, [A]Θ� total current linkage [A]θ angle, position, [rad], [◦]ϑ angle, [rad], [◦]κ angle, [rad], [◦], factor for reduction of slot openingκ transmissivityΛ permeance, [Vs/A], [H]Λ′ specific permeance, [Vs/A/m2]Λ′

0 average of specific permeance, [Vs/A/m2]thermal conductivity [W/m·K], permeance factor, proportionality factor,inductance factor, inductance ratio

μ permeability [Vs/Am, H/m], number of pole pairs operating simultaneouslyper phase, friction coefficient

μr relative permeability

xxiv Abbreviations and Symbols

μ dynamic viscosity, [Pa·s, kg/(s·m)]μ0 permeability of vacuum, 4·�·10−7 [Vs/Am, H/m]ν ordinal of harmonic, Poisson’s ratio, reluctivity, [Am/Vs, m/H], pole pair

number of harmonics, kinematic viscosity of the coolantν pulse velocityξ reduced conductor heightρ resistivity, [�m], electric charge density, [C/m2], density, [kg/m3], reflection

factor, ordinal number of a single phasorρA absolute overlap ratioρE effective overlap ratioρν transformation ratio for IM impedance, resistance, inductanceσ specific conductivity, electric conductivity [S/m], leakage factor, ratio of the

leakage flux to the main fluxσ δ air gap harmonic leakage factorσ F tension, [Pa]σ Fn normal tension, [Pa]σ Ftan tangential tension, [Pa]σmec mechanical stress, [Pa]σ SB Stefan-Boltzmann constant, 5.670400×10−8 W·m−2·K−4

τ relative time, span of the lamination thickness on one pole pitchτ 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 distributionτ ′

d direct transient short-circuit time constant, [s]τ ′

d0 direct transient open-circuit time constant, [s]τ ′′

d direct subtransient short-circuit time constant, [s]τ ′′

d0 direct subtransient open-circuit time constant, [s]τ ′′

q quadrature subtransient short-circuit time constant, [s]τ ′′

q0 quadrature subtransient open-circuit time constant, [s]υ factor, kinematic viscosity, μ/ρ, [Pa·s/(kg/m3)]Φ magnetic flux, [Vs], [Wb]Φ th thermal power flow, heat flow rate [W]Φδ air gap flux, [Vs], [Wb]φ magnetic flux, instantaneous value φ(t), [Vs], electric potential [V]ϕ phase shift angle, [rad], [◦]ϕ’ function for skin effect calculationΨ magnetic flux linkage, [Vs]ψ electric flux, [C]ψe electric flux, [C]ψm air gap flux linkage [Vs]ψmp magnetic flux linkage of phase winding [Vs]ψ function for skin effect calculation

Abbreviations and Symbols xxv

χ length/diameter ratio, shift of a single pole pairΩ mechanical angular speed [rad/s]ω electric angular velocity [rad/s], angular frequency [rad/s] difference, drop T temperature rise (difference) [K], [◦C]∇T temperature gradient [K/m], [◦C/m] p pressure drop [Pa]

Subscripts

0 section1 primary, fundamental component, beginning of a phase, locked rotor torque2 secondary, end of a phaseAl aluminuma armature, shaftad additional (loss)av averageB brushb base value, peak value of torque, blocking, damper barbar barbearing bearing (losses)C capacitorCu copperCuw End winding conductorconv convectionc conductor, commutationcf centrifugalcp commutating polescontact brush contactcr, crit criticalDC direct currentD direct, damperd tooth, direct, tooth tip leakage fluxdiff differenceE emfe equivalentef effectiveel electricem electromagneticew end windingext externalF forceFe ironf fieldFt eddy current

xxvi Abbreviations and Symbols

Hy hysteresisi internal, insulation, ordinalin inputk compensating, short circuit, ordinallam laminationsLL additional load lossesM motormax maximumm mutual, main, magnetizingmag magnetizing, magneticmec mechanicalmin minimummut mutualmp single-phase magnetizingN ratedn nominal, normalns negative-sequence componento starting, upper, overopt optimalout outputPM permanent magnetp pole, primary, subconductor, pole leakage flux, operational harmonicp1 pole shoep2 pole bodyph phasor, phaseps positive-sequence componentpu per unitq quadrature, zoner rotor, remanence, relative, damper ring short circuitres resultantS surfaces statorsj impulse wavesat saturationstr phase sectionsq skewsyn synchronoustan tangentialtest testth thermaltot totalu slot, lower, under, bottom, slot leakage flux, pull-up torquev zone, coil side shift in a slot, coilx x-directiony, y-direction, yokeya armature yoke

Abbreviations and Symbols xxvii

yr rotor yokeys stator yokew end windingz z-direction, phasor of voltage phasor graphρ ordinal number of single phasorρ friction lossρw windage (loss)δ air gapΦ fluxν harmonicσ flux leakageγ ordinal of a subconductorμ harmonic ordinal

Superscripts

ˆ peak/maximum value, amplitude‘ imaginary, apparent, reduced, virtual, referred to the stator∗ 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 Ax + j Ak + kAz

B flux density, B = i Bx + j Bk + kBz

I complex phasor of the currentI bar above the symbol denotes average value