ELECTRIC
MACHINES
Steady State, Transients,
and Design with MATLAB®
ION BOLDEA
LUCIAN TUTELEA
Lop) CRC Press
^ ^
J
Taylor & Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an i n f o r m a business
Contents
Preface
Part I
xvii
Steady State
1
Introduction
1.1 Electric Energy and Electric Machines
1.2 Basic Types of Transformers and Electric Machines
1.3 Losses and Efficiency
1.4 Physical Limitations and Ratings
1.5 Nameplate Ratings
1.6 Methods of Analysis
1.7 State of the Art and Perspective
1.8 Summary
1.9 Proposed Problems
References
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Electric Transformers
2.1 AC Coil with Magnetic Core and Transformer
Principles
2.2 Magnetic Materials in EMs and Their Losses
2.2.1 Magnetization Curve and Hysteresis Cycle
2.2.2 Permanent Magnets
2.2.3 Losses in Soft Magnetic Materials
2.3 Electric Conductors and Their Skin Effects
2.4 Components of Single- and 3-Phase Transformers
2.4.1 Cores
2.4.2 Windings
2.5 Flux Linkages and Inductances of Single-Phase
Transformers
2.5.1 Leakage Inductances of Cylindrical Windings
2.5.2 Leakage Inductances of Alternate Windings
2.6 Circuit Equations of Single-Phase Transformers
with Core Losses
2.7 Steady State and Equivalent Circuit
2.8 No-Load Steady State (I2 = 0)/Lab 2.1
2.8.1 Magnetic Saturation under No Load
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2.9 Steady-State Short-Circuit Mode /Lab 2.2
2.10 Single-Phase Transformers: Steady-State
Operation on Load/Lab 2.3
2.11 Three-Phase Transformers: Phase Connections
2.12 Particulars of 3-Phase Transformers on No Load
2.12.1 No-Load Current Asymmetry
2.12.2 Y Primary Connection for the 3-Limb Core
2.13 General Equations of 3-Phase Transformers
2.13.1 Inductance Measurement/Lab 2.4
2.14 Unbalanced Load Steady State in 3-Phase
Transformers/Lab 2.5
2.15 Paralleling 3-Phase Transformers
2.16 Transients in Transformers
2.16.1 Electromagnetic (R,L) Transients
2.16.2 Inrush Current Transients/Lab 2.6
2.16.3 Sudden Short Circuit from No Load
(Vi, = 0 ) / L a b 2.7
2.16.4 Forces at Peak Short-Circuit Current
2.16.5 Electrostatic (C,R) Ultrafast Transients
2.16.6 Protection Measures of Anti-Overvoltage
Electrostatic Transients
2.17 Instrument Transformers
2.18 Autotransformers
2.19 Transformers and Inductances for Power Electronics
2.20 Preliminary Transformer Design (Sizing) by Example
2.20.1 Specifications
2.20.2 Deliverables
2.20.3 Magnetic Circuit Sizing
2.20.4 Windings Sizing
2.20.5 Losses and Efficiency
2.20.6 No-Load Current
2.20.7 Active Material Weight
2.20.8 Equivalent Circuit
2.21 Summary
2.22 Proposed Problems
References
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Energy Conversion and Types of Electric Machines
3.1 Energy Conversion in Electric Machines
3.2 Electromagnetic Torque
3.2.1 Cogging Torque (PM Torque at Zero Current)
3.3 Passive Rotor Electric Machines .
3.4 Active Rotor Electric Machines
3.4.1 DC Rotor and AC Stator Currents
3.4.2 AC Currents in the Rotor and the Stator
3.4.3 DC (PM) Stator and AC Rotor
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Contents
Fix Magnetic Field (Brush-Commutator) Electric
Machines
3.6 Traveling Field Electric Machines
3.7 Types of Linear Electric Machines
3.8 Summary
3.9 Proposed Problem
References
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3.5
Brush-Commutator Machines: Steady State
4.1 Introduction
4.1.1 Stator and Rotor Construction Elements
4.2 Brush-Commutator Armature Windings
4.2.1 Simple Lap Windings by Example:
Ns = 16, 2 P l = 4
4.2.2 Simple Wave Windings by Example:
Ns = 9,2p1=2
4.3 Brush-Commutator
4.4 Airgap Flux Density of Stator Excitation MMF
4.5 No-Load Magnetization Curve by Example
4.6 PM Airgap Flux Density and Armature Reaction by
Example
4.7 Commutation Process
4.7.1 AC Excitation Brush-Commutation Winding
4.8 EMF .
4.9 Equivalent Circuit and Excitation Connection
4.10 DC Brush Motor/Generator with Separate (or PM)
Excitation/Lab 4.1
4.11 DC Brush PM Motor Steady-State and Speed Control
Methods /Lab 4.2
4.11.1 Speed Control Methods
4.12 DC Brush Series Motor/Lab 4.3
4.12.1 Starting and Speed Control
4.13 AC Brush Series Universal Motor
4.14 Testing Brush-Commutator Machines/Lab 4.4
4.14.1 DC Brush PM Motor Losses, Efficiency, and
Cogging Torque
4.15 Preliminary Design of a DC Brush PM Automotive
Motor by Example
4.15.1 PM Stator Geometry
4.15.2 Rotor Slot and Winding Design
4.16 Summary
4.17 Proposed Problems
References
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Contents
Induction Machines: Steady State
5.1 Introduction: Applications and Topologies
5.2 Construction Elements
5.3 AC Distributed Windings
5.3.1 Traveling MMF of AC Distributed Windings
5.3.2 Primitive Single-Layer Distributed Windings
(q > 1, Integer)
5.3.3 Primitive Two-Layer 3-Phase Distributed
Windings (q = Integer)
5.3.4 MMF Space Harmonics for Integer q
(Slots/Pole/Phase)
5.3.5 Practical One-Layer AC 3-Phase Distributed
Windings
5.3.6 Pole Count Changing AC 3-Phase Distributed
Windings
5.3.7 Two-Phase AC Windings
5.3.8 Cage Rotor Windings
5.4 Induction Machine Inductances
5.4.1 Main Inductance
5.4.2 Leakage Inductance
5.5 Rotor Cage Reduction to the Stator
5.6 Wound Rotor Reduction to the Stator
5.7 Three-Phase Induction Machine Circuit Equations
5.8 Symmetric Steady State of 3-Phase IMs
5.9 Ideal No-Load Operation/Lab 5.1
5.10 Zero Speed Operation (S = 1)/Lab 5.2
5.11 No-Load Motor Operation (Free Shaft)/Lab 5.3
5.12 Motor Operation on Load (1 > S > 0)/Lab5.4
5.13 Generating at Power Grid (n >f1/pl,S
< 0)/Lab 5.5
5.14 Autonomous Generator Mode (S < 0)/Lab 5.6
5.15 Electromagnetic Torque and Motor Characteristics
5.16 Deep-Bar and Dual-Cage Rotors
5.17 Parasitic (Space Harmonics) Torques
5.18 Starting Methods
5.18.1 Direct Starting (Cage Rotor)
5.18.2 Reduced Stator Voltages
5.18.3 Additional Rotor Resistance Starting
5.19 Speed Control Methods
5.19.1 Wound Rotor IM Speed Control
5.20 Unbalanced Supply Voltages
5.21 One Stator Phase Open by Example
5.22 One Rotor Phase Open
5.23 Capacitor Split-Phase Induction Motors
5.24 Linear Induction Motors
5.24.1 End and Edge Effects in LIMs
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Contents
5.25 Regenerative and Virtual Load Testing of IMs/Lab 5.7
5.26 Preliminary Electromagnetic IM Design by Example
5.26.1 Magnetic Circuit
5.26.2 Electric Circuit
5.26.3 Parameters
5.26.4 Starting Current and Torque
5.26.5 Breakdown Slip and Torque
5.26.6 Magnetization Reactance, X m , and Core
Losses, piron
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5.26.7 No-Load and Rated Currents, 1Q and I„
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5.26.8 Efficiency and Power Factor
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5.26.9 Final Remarks
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5.27 Summary
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5.28 Proposed Problems
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References
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Synchronous Machines: Steady State
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6.1 Introduction: Applications and Topologies
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6.2 Stator (Armature) Windings for SMs
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6.2.1 Nonoverlapping (Concentrated) Coil SM Armature
Windings
307
6.3 SM Rotors: Airgap Flux Density Distribution
andEMF
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6.3.1 PM Rotor Airgap Flux Density
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6.4 Two-Reaction Principle via Generator Mode
318
6.5 Armature Reaction and Magnetization Reactances,
Xdm and Xqm
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6.6 Symmetric Steady-State Equations and Phasor Diagram . . . 323
6.7 Autonomous Synchronous Generators
325
6.7.1 No-Load Saturation Curve/Lab 6.1
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6.7.2 Short-Circuit Curve: (I sc (J F ))/Lab 6.2
326
6.7.3 Load Curve: Vs (I s )/Lab 6.3
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6.8 Synchronous Generators at Power Grid/Lab 6.4
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6.8.1 Active Power/Angle Curves: P e (5V)
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6.8.2 V-Shaped Curves
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6.8.3 Reactive Power Capability Curves
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6.9 Basic Static- and Dynamic-Stability Concepts
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6.10 Unbalanced Load Steady State of SGs/Lab 6.5
339
6.10.1 Measuring X d , Xq, Z_, a n d X 0 / L a b
340
6.11 Large Synchronous Motors
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6.11.1 Power Balance
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6.12 PM Synchronous Motors: Steady State
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6.13 Load Torque Pulsations Handling by Synchronous
Motors /Generators
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6.14 Asynchronous Starting of SMs and Their
Self-Synchronization to Power Grid
6.15 Single-Phase and Split-Phase Capacitor PM Synchronous
Motors
6.15.1 Steady State of Single-Phase
Cageless-Rotor PMSMs
6.16 Preliminary Design Methodology of a 3-Phase PMSM
by Example
6.17 Summary
6.18 Proposed Problems
References
Part II
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Transients
Advanced Models for Electric Machines
7.1 Introduction
7.2 Orthogonal (dq) Physical Model
7.3 Pulsational and Motion-Induced Voltages in
dq Models
7.4 dq Model of DC Brush PM Motor (cub = 0)
7.5 Basic dq Model of Synchronous Machines (cut, = ш г)
7.6 Basic dq Model of Induction Machines
(шь = 0,iur,a>i)
7.7 Magnetic Saturation in dq Models
7.8 Frequency (Skin) Effect Consideration in dq Models
7.9 Equivalence between dq Models and AC Machines
7.10 Space Phasor (Complex Variable) Model
7.11 High-Frequency Models for Electric Machines
7.12 Summary
7.13 Proposed Problems
References
Transients of Brush-Commutator DC Machines
8.1 Introduction
8.2 Orthogonal (dq) Model of DC Brush Machines with
Separate Excitation
8.3 Electromagnetic (Fast) Transients
8.4 Electromechanical Transients
8.4.1 Constant Excitation (PM) Flux, ¥ d r
8.4.2 Variable Flux Transients
8.4.3 DC Brush Series Motor Transients
8.5 Basic Closed-Loop Control of DC Brush PM Motor
8.6 DC-DC Converter-Fed DC Brush PM Motor
8.7 Parameters from Test Data/Lab 8.1
8.8 Summary
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8.9 Proposed Problems
References
9 Synchronous Machine Transients
9.1 Introduction
9.2 Phase Inductances of SMs
9.3 Phase Coordinate Model
9.4 dqO Model—Relationships of 3-Phase
SM Parameters
9.5 Structural Diagram of the SM dqO Model
9.6 pu dqO Model of SMs
9.7 Balanced Steady State via the dqO Model
9.8 Laplace Parameters for Electromagnetic Transients
9.9 Electromagnetic Transients at Constant Speed
9.10 Sudden 3-Phase Short Circuit from a Generator at
No Load/Lab 9.1
9.11 Asynchronous Running of SMs at a Given Speed
9.12 Reduced-Order dqO Models for Electromechanical
Transients
9.12.1 Neglecting Fast Stator Electrical Transients
9.12.2 Neglecting Stator and Rotor Cage Transients
9.12.3 Simplified (Third-Order) dq Model Adaptation for
SM Voltage Control
9.13 Small-Deviation Electromechanical Transients (in PU) . . . .
9.14 Large-Deviation Electromechanical Transients
9.14.1 Asynchronous Starting and Self-Synchronization
of DC-Excited SMs/Lab 9.2
9.14.2 Asynchronous Self-Starting of PMSMs to
Power Grid
9.14.3 Line-to-Line and Line-to-Neutral Faults
9.15 Transients for Controlled Flux and Sinusoidal
Current SMs
9.15.1 Constant d-Axis (i|^) Flux Transients
in Cageless SMs
9.15.2 Vector Control of PMSMs at Constant л1>ао (ко = const)
9.15.3 Constant Stator Flux Transients in Cageless SMs at
cosi[>i = 1
9.15.4 Vector Control of SMs with Constant Flux (i|)s) and
cos cps = 1
9.16 Transients for Controlled Flux and Rectangular
Current SMs
9.16.1 Model of Brushless DC Motor Transients
9.16.2 DC-Excited Cage Rotor SM Model for Rectangular
Current Control
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Contents
9.17 Switched Reluctance Machine Modeling
for Transients
9.18 Split-Phase Cage Rotor SMs
9.19 Standstill Testing for SM Parameters/Lab 9.3
9.19.1 Saturated Steady-State Parameters, L^m and Lqm,
from Current Decay Tests at Standstill
9.19.2 Single Frequency Test for Subtransient Inductances,
I/j and Ц
9.19.3 Standstill Frequency Response Tests
9.20 Linear Synchronous Motor Transients
9.21 Summary
9.22 Proposed Problems
References
10 Transients of Induction Machines
10.1 Three-Phase Variable Model
10.2 dq (Space Phasor) Model of IMs
10.3 Three-Phase IM-dq Model Relationships
10.4 Magnetic Saturation and Skin Effects in the dq Model . . . .
10.5 Space Phasor Model Steady State: Cage and Wound
Rotor IMs
10.6 Electromagnetic Transients
10.7 Three-Phase Sudden Short Circuit/Lab 10.1
10.7.1 Transient Current at Zero Speed
10.8 Small-Deviation Electromechanical Transients
10.9 Large-Deviation Electromechanical Transients/Lab 10.2 . .
10.10 Reduced-Order dq Model in Multimachine Transients . . . .
10.10.1 Other Severe Transients
10.11 m/Nr Actual Winding Modeling of IMs with
Cage Faults
10.12 Transients for Controlled Magnetic Flux and Variable
Frequency
10.12.1 Complex Eigenvalues of IM Space Phasor
Model
10.13 Cage Rotor Constant Stator Flux Transients and
Vector Control Basics
10.13.1 Cage-Rotor Constant Rotor Stator Flux Transients
and Vector Control Basics
10.13.2 Constant Rotor Flux Transients and Vector Control
Principles of Doubly Fed IMs
10.14 Doubly Fed IM as a Brushless Exciter for SMs
10.15 Parameter Estimation in Standstill Tests/Lab 10.3
10.15.1 Standstill Flux Decay for Magnetization Curve
Identification: W^ (I m )
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10.15.2 Identification of Resistances and Leakage
Inductances for Standstill Flux Decay Tests
10.15.3 Standstill Frequency Response Tests
10.16 Split-Phase Capacitor IM Transients/Lab 10.4
10.16.1 Phase Variable Model
10.16.2 dq Model
10.17 Linear Induction Motor Transients
10.18 Summary
10.19 Proposed Problems
References
Part III
xiii
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FEM Analysis and Optimal Design
11 Essentials of Finite Element Method in Electromagnetics
11.1 Vectorial Fields
11.1.1 Coordinate Systems
11.1.2 Operations with Vectors
11.1.3 Line and Surface (Flux) Integrals of a
Vectorial Field
11.1.4 Differential Operations
11.1.5 Integral Identities
11.1.6 Differential Identities
11.2 Electromagnetic Fields
11.2.1 Electrostatic Fields
11.2.2 Fields of Current Densities
11.2.3 Magnetic Fields
11.2.4 Electromagnetic Fields: Maxwell Equations
11.3 Visualization of Fields
11.4 Boundary Conditions
11.4.1 Dirichlet's Boundary Conditions
11.4.2 Neumann's Boundary Conditions
11.4.3 Mixed Robin's Boundary Conditions
11.4.4 Periodic Boundary Conditions
11.4.5 Open Boundaries
11.4.5.1 Problem Truncation
11.4.5.2 Asymptotical Boundary Conditions
11.4.5.3 Kelvin Transform
11.5 Finite Element Method
11.5.1 Residuum (Galerkin's) Method
11.5.2 Variational (Rayleigh-Ritz) Method
11.5.3 Stages in Finite Element Method Application
11.5.3.1 Domain Discretization
11.5.3.2 Choosing Interpolation Functions
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Contents
11.5.3.3 Formulation of Algebraic System
Equations
11.5.3.4 Solving Algebraic Equations
11.6 2D FEM
11.7 Analysis with FEM
11.7.1 Electromagnetic Forces
11.7.1.1 Integration of Lorenz Force
11.7.1.2 Maxwell Tensor Method
11.7.1.3 Virtual Work Method
11.7.2 Loss Computation
11.7.2.1 Iron Losses
References
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12 FEM in Electric Machines: Electromagnetic Analysis
12.1 Single-Phase Linear PM Motors
12.1.1 Preprocessor Stage
12.1.2 Postprocessor Stage
12.1.3 Summary
12.2 Rotary PMSMs (6/4)
12.2.1 BLDC: Preprocessor Stage
12.2.2 BLDC Motor Analysis: Postprocessor Stage
12.2.3 Summary
12.3 The 3-Phase Induction Machines
12.3.1 Induction Machines: Ideal No Load
12.3.2 Rotor Bar Skin Effect
12.3.3 Summary
References
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13 Optimal Design of Electric Machines: The Basics
13.1 Electric Machine Design Problem
13.2 Optimization Methods
13.3 Optimum Current Control
13.4 Modified Hooke-Jeeves Optimization Algorithm
13.5 Electric Machine Design Using Genetic Algorithms
References
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14 Optimization Design of Surface PMSMs
14.1 Design Theme
14.2 Electric and Magnetic Loadings
14.3 Choosing a Few Dimensioning Factors
14.4 A Few Technological Constraints
14.5 Choosing Magnetic Materials
14.6 Dimensioning Methodology
14.6.1 Rotor Sizing
14.6.2 PM Flux Computation
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14.6.3 Weights of Active Materials
14.6.4 Losses
14.6.5 Thermal Verification
14.6.6 Machine Characteristics
14.7 Optimal Design with Genetic Algorithms
14.7.1 Objective (Fitting) Function
14.7.2 PMSM Optimization Design Using Genetic
Algorithms: A Case Study
14.8 Optimal Design of PMSMs Using Hooke-Jeeves Method . . .
14.9 Conclusion
References
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15 Optimization Design of Induction Machines
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15.1 Realistic Analytical Model for Induction Machine Design . . . 717
15.1.1 Design Theme
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15.1.2 Design Variables
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15.1.3 Induction Machine Dimensioning
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15.1.3.1 Rotor Design
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15.1.3.2 Stator Slot Dimensions
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15.1.3.3 Winding End-Connection Length
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15.1.4 Induction Machine Parameters
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15.2 Induction Motor Optimal Design Using Genetic
Algorithms
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15.3 Induction Motor Optimal Design Using Hooke-Jeeves
Algorithm
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15.4 Machine Performance
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15.5 Conclusion
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References
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Index
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