Table Of ContentElectric Machines
Electric Machines
Transients, Control Principles, Finite Element
Analysis, and Optimal Design with
MATLAB®
Second Edition
Ion Boldea
and
Lucian N. Tutelea
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Library of Congress Cataloging-in-Publication Data
Names: Boldea, I., author. | Tutelea, Lucian, author.
Title: Electric machines : transients, control principles, finite element
analysis and optimal design with MATLAB / Ion Boldea, Lucian N. Tutelea.
Description: Second edition. | Boca Raton, FL : CRC Press, 2021. | Includes
bibliographical references and index.
Identifiers: LCCN 2021035679 (print) | LCCN 2021035680 (ebook) | ISBN
9780367375652 (hbk) | ISBN 9781032105727 (pbk) | ISBN 9781003216018 (ebk)
Subjects: LCSH: Electric machinery--Design and construction--Data
processing. | MATLAB.
Classification: LCC TK2331 .B584 2021 (print) | LCC TK2331 (ebook) | DDC
621.31/042--dc23
LC record available at https://lccn.loc.gov/2021035679
LC ebook record available at https://lccn.loc.gov/2021035680
ISBN: 978-0-367-37565-2 (hbk)
ISBN: 978-1-032-10572-7 (pbk)
ISBN: 978-1-003-21601-8 (ebk)
DOI: 10.1201/9781003216018
Typeset in Times
by SPi Technologies India Pvt Ltd (Straive)
Contents
Preface .......................................................................................................................xi
Authors ....................................................................................................................xiii
Chapter 1 Electric Machine Circuit Models for Transients and Control ..............1
1.1 Introduction ...............................................................................1
1.2 Orthogonal (dq) Physical Model ...............................................2
1.3 Pulsational and Motion- Induced Voltages in dq Models ...........4
1.4 dq Model of DC Brush PM Motor (ωb = 0) ...............................5
1.5 Basic dq Model of Synchronous Machines (ωb = ωr) ................6
1.6 Basic dq Model of Induction Machines (ωb = 0, ωr, ω1) ............8
1.7 Magnetic Saturation in dq Models ............................................9
1.8 Frequency (Skin) Effect Consideration in dq Models .............11
1.9 Equivalence between dq Models and AC Machines ................12
1.10 Space Phasor (Complex Variable) Model ................................15
1.11 High- Frequency Models for Electric Machines ......................17
1.12 Orthogonal Models of Multiphase A.C. Machines ..................19
1.13 Summary ..................................................................................22
1.14 Proposed Problems ..................................................................24
References ..........................................................................................25
Chapter 2 Transients and Control Principles of Brush–Commutator DC
Machines ............................................................................................27
2.1 Introduction .............................................................................27
2.2 Orthogonal (dq) Model of DC Brush Machines with
Separate Excitation ..................................................................27
2.3 Electromagnetic (Fast) Transients ...........................................30
2.4 Electromechanical Transients ..................................................32
2.4.1 Constant Excitation (PM) Flux, Ψdr ...........................32
2.4.2 Variable Flux Transients .............................................35
2.4.3 DC Brush Series Motor Transients .............................37
2.5 Basic Closed- Loop Control of DC Brush PM Motor ..............38
2.6 D.C.–D.C. Converter- Fed D.C. Brush PM Motor ...................39
2.7 Parameters from Test Data/Lab 2.1 .........................................41
2.8 Summary ..................................................................................42
2.9 Proposed Problems ..................................................................43
References ..........................................................................................44
Chapter 3 Synchronous Machine Transients and Control Principles ..................45
3.1 Introduction .............................................................................45
3.2 Phase Inductances of SMs .......................................................46
v
vi Contents
3.3 Phase Coordinate Model ..........................................................48
3.4 dq0 Model to Three- Phase SM Parameters Relationships ......49
3.5 Structural Diagram of the SM dq0 Model ...............................52
3.6 pu dq0 Model of SMs ..............................................................54
3.7 Balanced Steady State via the dq0 Model ...............................57
3.8 Laplace Parameters for Electromagnetic Transients ...............60
3.9 Electromagnetic Transients at Constant Speed ........................62
3.10 Sudden Three- Phase Short Circuit from a Generator at
No Load/Lab ............................................................................64
3.11 Asynchronous Running of SMs at a Given Speed...................66
3.12 Reduced- Order dq0 Models for Electromechanical
Transients .................................................................................71
3.12.1 Neglecting Fast Stator Electrical Transients...............71
3.12.2 Neglecting Stator and Rotor Cage Transients ............72
3.12.3 Simplified (Third- Order) dq Model Adaptation
for SM Voltage Control ..............................................72
3.13 Small- Deviation Electromechanical Transients (in PU) ..........75
3.14 Large- Deviation Electromechanical Transients .......................78
3.14.1 Asynchronous Starting and Self—
Synchronization of DC—Excited SMs/Lab 3.2 .........79
3.14.2 Asynchronous Self- Starting of PMSMs to
Power Grid .................................................................81
3.14.3 Line- to- Line and Line- to- Neutral Faults ....................81
3.15 Transients for Controlled Flux and Sinusoidal
Current PMSMs .......................................................................82
3.15.1 Constant d- Axis (ψd) Flux Transients in
Cageless SMs .............................................................83
3.15.2 Vector Control of PMSMs at Constant ψd0
(i = const) .................................................................86
d0
3.15.3 Constant Stator Flux Transients in Cageless
SMs at cos φ1 = 1 .......................................................87
3.15.4 Vector Control of SMs with Constant Flux (ψs)
and cos φs = 1 .............................................................91
3.16 Transients for Controlled Flux and Rectangular
Current SMs .............................................................................92
3.16.1 Model of Brushless DC- PM Motor Transients ..........93
3.16.2 DC- Excited Cage Rotor SM Model for
Rectangular Current Control ......................................95
3.17 Switched Reluctance Machine Modeling for Transients .........96
3.18 Split- Phase Cage Rotor SM Transients .................................102
3.19 Standstill Testing for SM Parameters/Lab 3.3 .......................104
3.19.1 Saturated Steady- State Parameters, L and
dm
L , from Current Decay Tests at Standstill .............104
qm
3.19.2 Single Frequency Test for Subtransient
Inductances, L′′ and L′′ ..............................................108
d q
Contents vii
3.19.3 Standstill Frequency Response Tests ........................108
3.20 Linear Synchronous Motor Transients ..................................109
3.21 Summary ................................................................................113
3.22 Proposed Problems ................................................................115
References ........................................................................................118
Chapter 4 Induction Machines Transients and Control Principles ....................119
4.1 Three- Phase Variable Model ..................................................119
4.2 dq (Space Phasor) Model of IMs ...........................................121
4.3 Three- Phase IM–dq Model Relationships .............................123
4.4 Magnetic Saturation and Skin Effects in the dq Model .........124
4.5 Space Phasor Model Steady State: Cage and
Wound Rotor IMs ..................................................................125
4.6 Electromagnetic Transients ...................................................132
4.7 Three- Phase Sudden Short Circuit/Lab 4.1 ...........................133
4.7.1 Transient Current at Zero Speed...............................136
4.8 Small- Deviation Electromechanical Transients .....................137
4.9 Large- Deviation Electromechanical Transients/Lab 4.2 .......139
4.10 Reduced- Order dq Model in Multimachine Transients .........141
4.10.1 Other Severe Transients ............................................143
4.11 m/N Actual Winding Modeling of IMs with Cage Faults .....143
r
4.12 Transients for Controlled Magnetic Flux and Variable
Frequency ..............................................................................147
4.12.1 Complex Eigenvalues of IM Space Phasor Model ...147
4.13 Cage Rotor IM Constant Rotor Flux Transients and
Vector Control Basics ............................................................149
4.13.1 Cage- Rotor IM Constant Stator Flux Transients
and Vector Control Basics ........................................154
4.13.2 Constant Rotor Flux Transients and Vector
Control Principles of Doubly Fed IMs .....................156
4.14 Doubly Fed IM as a Brushless Exciter for SMs ....................157
4.15 Parameter Estimation in Standstill Tests/Lab 4.3 ..................162
4.15.1 Standstill Flux Decay for Magnetization Curve
Identification: Ψ∗ (I ) .............................................162
m m
4.15.2 Identification of Resistances and Leakage
Inductances from Standstill Flux Decay Tests .........165
4.15.3 Standstill Frequency Response (SSFR) Tests ...........165
4.16 Split- Phase Capacitor IM Transients/Lab 4.4 ........................167
4.16.1 Phase Variable Model ...............................................168
4.16.2 dq Model ..................................................................169
4.17 Linear Induction Motor Transients ........................................170
4.18 Line-Start Self- Synchronizing Premium Efficiency IMs ......173
4.18.1 Line Start One Phase—Source Split-Phase
Capacitor Self- synchronizing Induction Motor
with PMs in the Rotor’s Flux Barriers .....................175
viii Contents
4.19 Summary ................................................................................179
4.20 Proposed Problems ................................................................183
References ........................................................................................185
Chapter 5 Essentials of Finite Element Analysis (FEA) in Electromagnetics ........187
5.1 Vectorial Fields ......................................................................187
5.1.1 Coordinate Systems ..................................................187
5.1.2 Operations with Vectors ...........................................189
5.1.3 Line and Surface (Flux) Integrals of a
Vectorial Field ..........................................................190
5.1.4 Differential Operations .............................................191
5.1.5 Integral Identities ......................................................193
5.1.6 Differential Identities ...............................................195
5.2 Electromagnetic Fields ..........................................................196
5.2.1 Electrostatic Fields ...................................................196
5.2.2 Fields of Current Densities .......................................197
5.2.3 Magnetic Fields ........................................................198
5.2.4 Electromagnetic Fields: Maxwell Equations ............199
5.3 Visualization of Fields ...........................................................199
5.4 Boundary Conditions .............................................................202
5.4.1 Dirichlet’s Boundary Conditions ..............................202
5.4.2 Neumann’s Boundary Conditions ............................203
5.4.3 Mixed Robin’s Boundary Conditions .......................203
5.4.4 Periodic Boundary Conditions .................................204
5.4.5 Open Boundaries ......................................................204
5.4.5.1 Problem Truncation ...................................204
5.4.5.2 Asymptotical Boundary Conditions ..........204
5.4.5.3 Kelvin Transform .......................................206
5.5 Finite Element Method ..........................................................208
5.5.1 Residuum (Galerkin’s) Method ................................208
5.5.2 Variational (Rayleigh–Ritz) Method ........................209
5.5.3 Stages in Finite Element Method Application ..........210
5.5.3.1 Domain Discretization ...............................210
5.5.3.2 Choosing Interpolation Functions .............210
5.5.3.3 Formulation of Algebraic System
Equations ...................................................210
5.5.3.4 Solving Algebraic Equations .....................210
5.6 2D FEM .................................................................................211
5.7 Analysis with FEM ................................................................213
5.7.1 Electromagnetic Forces ............................................215
5.7.1.1 Integration of Lorenz Force .......................215
5.7.1.2 Maxwell Tensor Method ............................216
5.7.1.3 Virtual Work Method .................................217
Contents ix
5.7.2 Loss Computation ....................................................218
5.7.2.1 Iron Losses .................................................218
References ........................................................................................219
Chapter 6 FEA of Electric Machines Electromagnetics ...................................221
6.1 Single- Phase Linear PM Motors ...........................................221
6.1.1 Preprocessor Stage ...................................................222
6.1.2 Postprocessor Stage ..................................................228
6.1.3 Summary ..................................................................235
6.2 Rotary PMSMs (6/4) .............................................................237
6.2.1 BLDC Motor: Preprocessor Stage ............................238
6.2.2 BLDC Motor Analysis: Postprocessor Stage ...........243
6.2.3 Summary ..................................................................261
6.3 The Three- Phase Induction Machines ...................................261
6.3.1 Induction Machines: Ideal No Load .........................266
6.3.2 Rotor Bar Skin Effect ...............................................271
6.3.3 Summary ..................................................................278
References ........................................................................................279
Chapter 7 Thermal FEA of Electric Machines..................................................281
7.1 Thermal Models .....................................................................281
7.1.1 The Single-body Thermal Model .............................281
7.1.2 The Two-body Thermal Model .................................283
7.1.3 Equivalent Thermal Circuit (Thermal Network) ......285
7.2 Thermal Analysis of Electric Machines by Finite Element ...285
7.2.1 Equivalent Coil Conductivity––Simplified Thermal
Model of the Slot ......................................................285
7.2.2 Boundary Conditions ................................................286
7.2.3 The Input Data ..........................................................287
7.3 Steady-state Simulation Results ............................................289
7.4 Thermal Transient Analysis of Electric Machines by
Finite Element .......................................................................292
7.5 Summary ................................................................................295
References ........................................................................................298
Chapter 8 Optimal Electromagnetic Design of Electric Machines: The Basics .....299
8.1 Electric Machine Design Problem .........................................299
8.2 Optimization Methods ...........................................................301
8.3 Optimum Current Control .....................................................307
8.4 Modified Hooke–Jeeves Optimization Algorithm .................311
8.5 Electric Machine Design Using Genetic Algorithms ............317
References ........................................................................................321
