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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on Power and Energy Systems
- Sprache: Englisch
Comprehensive resource on the fundamentals of electric machinery and variable speed drives, and their many conventional and emerging applications
Electric Machinery and Drives: An Electromagnetics Perspective provides advanced concepts of electrical machinery with control/drives and emphasizes the necessity of integration of power electronics and control strategy when studying modern electrical machinery. The text incorporates the fundamentals of electric machinery, variable speed drives, and motor controls, with the scope of including both the introduction of detailed operating principles as well as the electromagnetic design and control details from scratch.
The authors start with the introduction of electric circuit notations and elementary concepts of electrical circuits, power electronics, magnetostatics, magnetic circuits, and fundamentals of electromechanical energy conversion. Later, the book elaborates on the operating principles of polyphase induction machines and synchronous machines, as well as the associated scale and vector controls of these machines.
To aid in reader comprehension, the text includes a solutions manual and accompanying video animations.
Electric Machinery and Drives also contains information on:
- Real and reactive power in single-phase and balanced three-phase circuits and devices using consumer system concepts and notations
- Forces and torques in simple magnetically linear and nonlinear, multi-excited electromechanical devices and systems
- Simplified T-equivalent circuit model and its use in performance calculations of induction machines and associated torque-slip (speed) characteristics
- Brush-commutator and brushless DC machines, and natural ABC frame and Park’s two-reaction DQO frame state-space modeling of synchronous and induction machines
- Special machines, including single-phase induction machines, switched reluctance machines, and others
Electric Machinery and Drives is an ideal learning resource in undergraduate or graduate-level courses for all universities with electrical engineering programs across the world. Additionally, the text may be used as a fundamental reference by researchers and engineers in electrical, mechanical, automotive, aerospace, and automation engineering.
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Seitenzahl: 524
Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
About the Companion Website
1 Electric Circuit Notations and Elementary Concepts
1.1 Frequency‐Domain RMS Phasor Representation of Time‐Domain AC Voltages and Currents
1.2 Time‐Domain and RMS Frequency‐Domain Power Concepts Using Consumer System Formulation and Notations
1.3 Elementary Concepts of Complex Real and Reactive Power in Balanced Three‐Phase Circuits and Devices Using Consumer System Notations
2 Power Electronics and Converters
2.1 Semiconductor Devices
2.2 DC–DC Converters
2.3 Voltage Source Inverter
2.4 Pulse Width Modulation (PWM)
Homework Problems
References
3 Review of Magnetostatics, Magnetic Circuits, and Fundamentals of Electromechanical Energy Conversion
3.1 Magnetostatic Fields Notations and Ampere's Law
3.2 Magnetic Circuits with Ferrous Magnetic Cores
3.3 Magnetic Flux Linkages, Inductance, and Electromotive Forces
3.4 Energy Storage, Motion, and Forces in Magnetic Circuits
3.5 Lorentz's Law‐Induced Voltage and Forces in Magnetic Fields
3.6 Electromagnet Polarity and Permanent Magnets
3.7 Elementary Electromechanical Torque Production and Magnetomotive Force (MMF) Distribution Concepts
3.8 Elements of Three‐Phase Stator and Rotor Windings
3.9 Elementary Three‐Phase Synchronous and Induction Machines
3.10 Elementary Brush‐Commutator DC Machines
3.11 Mechanical Torque Production in AC and DC Machines
3.12 Forces and Torques in Magnetically Linear Singly Excited Electromechanical Devices and Systems
Homework Problems
4 Electromechanical Concepts in Electric Machines
4.1 Introductory Discussion
4.2 Motor‐Mechanical Load Dynamics
4.3 Mechanical Load Torque–Speed Characteristics
4.4 Mass Polar Moment of Inertia
4.5 Effects of Belt and Gear Couplings on Mechanical Dynamic Formulations
4.6 Operating Modes of Electric Machines
4.7 On Time Constants of the Mechanical Dynamics of Motor‐Mechanical Load Systems
4.8 Modeling and Simulation of Motor Starting Transients
Homework Problems
Reference
5 Electric Machinery Windings and Associated Electromotive and Magnetomotive Forces
5.1 AC Winding Layouts
5.2 DC Winding Layouts
5.3 Induced Electromotive Forces in Single‐Phase and Poly‐Phase AC Windings
5.4 Induced Electromotive Forces in Brush‐Commutator‐Type DC Windings
5.5 MMF in Distributed and Concentrated AC and DC Machine Windings and Associated Flux Density Distributions/Waveforms in Airgaps
5.6 Relationships Among Magnetomotive Forces, Flux Density Distributions, Volume, Developed Torque, and Power
5.7 Relationship Among Electric Machinery Volume and Developed Torque and/or Volt–Ampere Capabilities
5.8 Worked Examples
Homework Problems
6 The Poly‐Phase Induction Machine
6.1 Main Constructional Features
6.2 The
T
‐Equivalent Circuit Model and Its Fundamental Formulation
6.3 Performance Calculations Using the
T
‐Equivalent Circuit Model
6.4 The Simplified
T
‐Equivalent Circuit Model and Its Use in Performance Calculations and Associated Torque–Slip (Speed) Characteristics
6.5 On Torque–Slip (Speed) Simplification and Its Formulation
6.6 The Motor Starting Transient Modeling Using the Simplified
T
‐Equivalent Circuit
6.7 On Legacy and Modern Constant Volts Per Hertz Control of Torque and Speed of Induction Motors
6.8 Field‐Oriented Control of Induction Motors
Homework Problems
References
7 The Poly‐phase Synchronous Machine
7.1 Main Constructional Features of the Salient‐Pole and Cylindrical Rotor Varieties
7.2 The Types of Synchronous Machine Exciter‐Conventional and Modern Brushless Excitation Systems
7.3 The Equivalent Circuit Model of the Idealized Cylindrical Rotor Synchronous Machine
7.4 Phasor‐Vector Diagrams of Synchronous Motors and Generators
7.5 Elementary Treatment of Effects of Rotor Saliency
7.6 Synchronous Motor Under Constant Volts Per Hertz Control
7.7 Vector Control of Synchronous Motors in ASDs
Homework Problems
References
8 Brush‐Commutator and Brushless DC Machines
8.1 Main Constructional Features of Brush‐Commutator and Brushless DC Machines
8.2 Armature Windings of Brush‐Commutator DC Machines
8.3 Separately Excited DC Motors, Modeling, and Formulations
8.4 Shunt‐Excited DC Motors Modeling and Formulations
8.5 Series‐Excited DC Motors Modeling and Formulations
8.6 Compound‐Excited DC Motors Modeling and Formulations
8.7 DC Motors Input–Output Powers, Losses, and Efficiency
8.8 Worked Examples on DC Motors
8.9 Operating Principles and Modeling of Brushless DC Motors
8.10 Brushless DC Motors Torque–Speed Control
Homework Problems
Problem for Graduate Students
References
9 State‐Space Modeling of Synchronous Machines Including Full Effects of Rotor Saliency
9.1 Discrete Representation of the Windings of a Synchronous Machine in the Natural Frame of Reference
9.2 State‐Space Modeling Derivations of a Synchronous Machine in the Natural ABC Frame of Reference of the Poly‐Phase Stator Windings
9.3 A Magnetic Field Flux‐Map Finite‐Element Computed Point of View of Salient‐Pole and Cylindrical Rotor Machines
9.4 Park's
DQ
0 Frame of Reference Transformation
9.5 Applying Park's
DQ
0 Transformation to the
ABC
State‐Space Model Voltages, Currents, and Flux Linkages
9.6 State‐Space Modeling Derivation in the
DQ
0 Reference Frame of a Synchronous Machine Including Saliency Effects – The Inductance and Flux Linkage Matrices and Vectors
9.7 Synchronous Machine
DQ
0 State‐Space Model in the
DQ
0 Flux Linkage Frame of Reference
9.8 Power and Torque Formulations for a Synchronous Machine in the
DQ
0 Frame of Reference and
ABC
Frame of Reference
9.9 Derivation of the Synchronous Machine Phasor and Space‐Vector Diagrams in the
DQ
0 Frame of Reference
9.10 Initial Conditions in Transient Synchronous Machinery State‐Space Time‐Domain Simulations
9.11 Synchronous Machine ABC Frame State‐Space Models with Permanent Magnet Excitations
9.12 Synchronous Machine
DQ
0 Frame State‐Space Models with Permanent Magnet Excitations
9.13 Vector Control of Wound‐Field Synchronous Machines
Homework Problems
References
10 State‐Space Modeling of Induction Machines
10.1 Discrete Representation of the Windings of a Wound‐Rotor Induction Machine in the Natural Frame of Reference
10.2 State‐Space Modeling Derivations of an Induction Machine in the Natural
ABC
–
abc
Frame of Reference for the Poly‐phase Stator and Rotor Windings
10.3 On the Relationship Between the Terms of an Induction Machine's Inductance Matrix and the Inductances and Reactances of Its Conventional
T
‐Equivalent Circuit
10.4 State‐Space Model of an Induction Machine Using the Winding Flux Linkages as State Variables
10.5 Transformation of the State‐Space Model of an Induction Machine from the
ABC
–
abc
Frame to the
DQ
0–
dq
0 Frame of Reference
10.6 Three Types of DQ0–dq0 Frame of Reference Transformation and the Resulting State‐Space Models
10.7 On Torque and Power Computations in an Induction Machine in the
DQ
0–
dq
0 and
ABC
–
abc
Reference Frames
10.8 On Inductance Parameters in Induction Machine State‐Space Models
10.9 On Design and Parameter Computation Aspects in Induction Machines
10.10 Direct Torque Control of Induction Motors
Homework Problems
References
11 Single‐Phase Induction Motors and Other Special Motors
11.1 Single‐Phase Induction Motors
11.2 Torque–Speed Characteristics of Single‐Phase Induction Motors
11.3 Field Analysis of Split‐Phase and Capacitor‐Start Induction Motors
11.4 Power and Torque for Split‐Phase Induction Motors
11.5 Power and Torque for Capacitor‐Start Induction Motors
11.6 Switched Reluctance Motors
Reference
12 Emerging Applications, Technical Trends, and Challenges
12.1 Emerging Applications
12.2 Future Technical Trends
12.3 Technical Challenges
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Switching patterns and output voltages of the 3-phase VSI.
Chapter 3
Table P6.1 Test results for motors (
A
), (
B
), and (
C
).
Chapter 6
Table P5.1
Table P5.2 Results of motor performance during a starting transient.
Table P5.3
Chapter 9
Table P3.1 Inverter output voltage functions and conduction periods.
Table P3.2 EMF parameters.
Table P3.3 15 hp brushless DC motor inductances and resistances.
Chapter 10
Table 10.1 Comparison between the DTC and FOC methods.
List of Illustrations
Chapter 1
Figure 1.1 A summary graphical and formulation representation of the consume...
Figure 1.2 A balanced three‐phase set of current phasors, ...
Figure 1.3 Another approach to negative
a
,
c
,
b
sequencing in balanced three...
Figure 1.4 A balanced
Y
‐connected impedance load and its associate line‐to‐n...
Figure 1.5 Balanced Δ‐connected impedance load and its...
Chapter 2
Figure 2.1 Symbol and output characteristic of a power diode.
Figure 2.2 Internal structure, symbol, and output characteristics of an SCR ...
Figure 2.3 Internal structure, symbol, and output characteristic of an IGBT ...
Figure 2.4 Half‐bridge SiC MOSFET modules with various packages: (a) GE 1.7 ...
Figure 2.5 Performance comparison among silicon, SiC, and GaN switches [1]/w...
Figure 2.6 Symbol and basic lateral structure of a GaN HEMT.
Figure 2.7 Cascode normally‐off GaN transistor based on a D‐mode GaN HEMT.
Figure 2.8 Basic circuit of a buck converter.
Figure 2.9 Current flow of the buck converter when (a) the switch is turned ...
Figure 2.10 Illustration of turn‐on time, a switching period, and duty cycle...
Figure 2.11 Basic circuit of a boost converter.
Figure 2.12 Current flow path of a boost converter under the condition (a) w...
Figure 2.13 Three‐phase PWM voltage source inverter.
Figure 2.14 Illustration of the sinusoidal PWM (SPWM).
Figure 2.15 Relationship between the line‐to‐line voltage amplitude and the ...
Figure P3.1 An IGBT half‐bridge circuit connected to an inductive load.
Figure P4.1 A DC–DC buck converter.
Figure P5.1 A DC–DC boost converter for an EV battery charging system.
Figure P6.1 A battery‐powered inverter‐motor traction drive system for EVs....
Chapter 3
Figure 3.1 Ampere's law in integral form.
Figure 3.2 Faraday's law in integral form.
Figure 3.3 A magnetic circuit with ferrous core and one winding.
Figure 3.4 The magnetic and electric circuits analogy.
Figure 3.5 A ferrous core with an airgap and an
N
‐turns excitation coil.
Figure 3.6 Flux linkage and electromotive force.
Figure 3.7
B
–
H
characteristic nonlinearity.
Figure 3.8 A magnetic circuit with an airgap and moving ferrous part.
Figure 3.9 The electrical equivalent circuit with ohmic, transformer, and mo...
Figure 3.10 Toroidal
N
‐turns‐coil ferrous core device.
Figure 3.11 The concept of energy and co‐energy density per unit volume.
Figure 3.12 A sinusoidally time‐varying excitation current,
i
(
t
)
.
Figure 3.13 The
B
–
H
hysteresis phenomena for soft and hard magnetic material...
Figure 3.14 DC magnetization characteristics for common permanent magnet (PM...
Figure 3.15 Laminated ferrous core excited by an AC current‐carrying coil.
Figure 3.16 Cross‐section of a lamination with varnish insulation.
Figure 3.17 Induced EMF in a conductor moving with a velocity, , in the pre...
Figure 3.18 The orientation of forces, flux densities, and current flow in s...
Figure 3.19 The cases of a “generator” and a “motor” for a straight current‐...
Figure 3.20 An electromagnet shown in elevation and plane including MMF spat...
Figure 3.21 Part (a) A cylindrical device with ferrous cores and permanent m...
Figure 3.22 A six‐slot stator and a six‐slot rotor including the three‐phase...
Figure 3.23 Some basic discrete winding terminology: one‐turn coil and three...
Figure 3.24 A six‐slot stator including the three‐phase
A
,
B
, and
C
phase co...
Figure 3.25 Single‐layer winding with dissimilar coils and ends (concentric‐...
Figure 3.26 Single‐layer winding with similar coils and ends (lattice‐coils ...
Figure 3.27 Single fractional slot winding with dissimilar coils and coil en...
Figure 3.28 Double‐layer winding short‐pitched lattice‐coils
τc ≠ τP
...
Figure 3.29 An elementary two‐pole, three‐phase salient‐pole synchronous mac...
Figure 3.30 An elementary 4‐pole, 12 stator slots, 3‐phase salient‐pole sync...
Figure 3.31 The magnetic circuit of a 4‐pole, salient‐pole, 24 stator slots ...
Figure 3.32 The magnetic circuit of a 4‐pole, salient‐pole, 24 stator slots ...
Figure 3.33 An elementary two‐pole, three‐phase squirrel‐cage induction mach...
Figure 3.34 Armature coils embedded in a stator‐slotted core of an AC synchr...
Figure 3.35 An elementary two‐pole, six‐armature slots DC machine and commut...
Figure 3.36 Lap‐type lattice DC armature coils layout.
Figure 3.37 Wave‐type lattice DC armature coils layout.
Figure 3.38 (a) Rotating field in the stator of a two‐pole, salient‐pole syn...
Figure 3.39 Rotating field in a wound‐rotor induction machine at
ωt = 0
...
Figure 3.40 Rotating field in a squirrel‐cage induction machine at
ωt = 0
...
Figure 3.41 A single‐excited electromechanical system.
Figure P1.1 Geometry of an inductor.
Figure P2.1 Cross‐section of an elementary 2‐pole DC machine.
Figure P3.1 Cross‐section of an elementary 2‐pole 3‐phase synchronous machin...
Figure P4.1 Cross‐section of an elementary 2‐pole 3‐phase wound rotor induct...
Figure P5.1 Cross‐section of an elementary 2‐pole 3‐phase PM machine.
Chapter 4
Figure 4.1 Electromechanical system. (a) The EM rotational machine: motor co...
Figure 4.2 Motor and mechanical load torque–speed characteristics and speed ...
Figure 4.3 Third‐order torque load example with least mean‐squared curve fit...
Figure 4.4 Constant power load. (a) Take‐up reel. (b) Torque–speed character...
Figure 4.5 Moment of inertia. (a)
dm
in a cylindrical coordinate system. (b)...
Figure 4.6 Belt coupling (a and b) and gear coupling (c). (a) End view. (b) ...
Figure 4.7 The four basic operating modes of an electric machine.
Figure 4.8 Induction machine phase sequencing.
Figure 4.9 Induction machine operating modes.
Figure 4.10 Motor and load torque–speed characteristics.
Chapter 5
Figure 5.1 A double‐layered 48 slots stator.
Figure 5.2 The start of coils of phases (
A
), (
B
), and (
C
).
Figure 5.3 The phase (
A
) slots and possible connections.
Figure 5.4 The phase (
B
) slots.
Figure 5.5 The phase (
C
) slots.
Figure 5.6 The entire phase (
A
), (
B
), and (
C
) coils in this four Pole 48 slo...
Figure 5.7 The top and bottom coil sides of phase (
A
) – one parallel path.
Figure 5.8 Rotor with field excitation (
f
) and (−
f
) and a one‐turn coil (
a
) ...
Figure 5.9 Effects of short‐pitched turns/coils.
Figure 5.10 Distribution factor – effects of distributed windings.
Figure 5.11 Effects of skewing winding turns/coils.
Figure 5.12 Flux density waveform with harmonics.
Figure 5.13 Waveform of mechanical rectified induced voltage in an armature ...
Figure 5.14 A single‐coil model of a single‐phase AC armature surrounding a ...
Figure 5.15 A 2‐pole, 3‐phase, and 24 slots stator.
Figure 5.16 The MMF waveform of phase (a) for the 24 slots, 2‐pole, and thre...
Figure 5.17 A two‐pole salient field winding and resulting MMF waveform.
Figure 5.18 Computation of sinusoidally distributed flux.
Figure 5.19 A two‐pole 12 slots armature DC commutator‐brush‐type machine.
Figure 5.20 Profiles of field and armature winding waveforms in DC and AC co...
Figure 5.21 Idealized armature MMF waveform in DC and AC commutator‐brush ma...
Figure 5.22 The fundamental harmonic component of the MMF, , of a single‐ph...
Figure 5.23 Standing pulsating MMF waveform of a single‐phase winding at
ωt
...
Figure 5.24 Graphical depiction of the forward and backward (reverse) travel...
Figure 5.25 The schematic layout of a two‐phase two‐pole stator for phases (
Figure 5.26 Phases (
a
) and (
b
) in a 24‐slots lamination of a 2‐pole balanced...
Figure 5.27 Locus of tip of space vectors representing the resultant MMFs of...
Figure 5.28 The case of an unbalanced two‐phase winding, phases (
a
) and (
b
),...
Figure 5.29 Another case of an unbalanced two‐phase winding, where
Na ≪ Nb
...
Figure 5.30 The elliptic field case.
Figure 5.31 A 24‐slot stator wound for a three‐phase and two‐pole armature o...
Figure 5.32 A schematic simplified representation of a three‐phase and two‐p...
Figure 5.33 The time‐domain and phasor representation of the three‐phase AC ...
Figure 5.34 The resultant fundamental MMF,
F
1
(
θ
,
t
)
, as a rotating/trav...
Figure 5.35 A qualitative graphic view of the rotating MMF/field produced by...
Figure 5.36 Torque production and quantification in relation to stator and r...
Figure 5.37 Relationship among resultant MMF space vector, , peak resultant...
Chapter 6
Figure 6.1 A stator lamination in a 24‐slot, 3‐phase, 2‐pole single‐layer wi...
Figure 6.2 A squirrel‐cage rotor lamination of an induction motor with 36‐ro...
Figure 6.3 A stator lamination in a large horsepower motor for a 48‐slot, 3‐...
Figure 6.4 A stator lamination in a large horsepower motor for a 48‐slot, 3‐...
Figure 6.5 A stator of a three‐phase induction motor.
Figure 6.6 A skewed wound‐rotor of an induction motor, with slip rings and b...
Figure 6.7 Three examples of skewed squirrel‐cage rotors of induction motors...
Figure 6.8 Circuit schematic of a wound‐rotor induction machine and a half w...
Figure 6.9 Stator phase (
A
) and rotor phase (
a
) coil schematics and associat...
Figure 6.10 Equivalent rotor circuit per phase of an induction motor with a ...
Figure 6.11 Equivalent rotor circuit per phase of an induction motor with a ...
Figure 6.12 Equivalent rotor circuit per phase of an induction motor with a ...
Figure 6.13 Equivalent rotor circuit per phase of an induction motor referre...
Figure 6.14 Equivalent rotor circuit per phase of an induction motor referre...
Figure 6.15 The
T
‐equivalent circuit of a three‐phase induction motor per ph...
Figure 6.16 The
T
‐equivalent circuit‐all quantities referred to the stator s...
Figure 6.17 The simplified
T
‐equivalent circuit per phase referred to the st...
Figure 6.18 The Thevenin equivalent to the source, , and the stator‐side up...
Figure 6.19 Thevenin equivalent of the
T
‐equivalent circuit per phase referr...
Figure 6.20 The simplified
T
‐equivalent circuit of an induction motor per ph...
Figure 6.21 The input–output power (energy) flow in an induction motor using...
Figure 6.22 The torque–slip/speed characteristic of the induction motor.
Figure 6.23 The induction motor equivalent circuit. (a) All parameters refer...
Figure 6.24 Torque–speed characteristic of the example motor.
Figure 6.25 Motor‐fan load system.
Figure 6.26 Torque–speed characteristics of NEMA classes “A, B, C, D, and F....
Figure 6.27 Approximate shapes of bars of NEMA classes A, B, C, D, and F squ...
Figure 6.28 An ASD system consisting of a 3‐phase diode rectifier, a DC/DC c...
Figure 6.29 Effect of constant volts per Hertz (V/f) control on shaping an i...
Figure 6.30 (V/f) an induction motor adjustable speed drive (IM‐ASD).
Figure 6.31 Two approaches to ASD torque–speed control of three‐phase induct...
Figure 6.32 Effects of adjustable voltage and frequency under the constant (
Figure 6.33 The flux‐weakening‐extended speed range.
Figure 6.34 Stator current space vector and its components in the
αβ
...
Figure 6.35 Stator current space vector and its components in the
dq
two‐coo...
Figure 6.36 Block diagram of field‐oriented control of an induction motor.
Figure 6.37 Block diagram of a field‐oriented control system of an induction...
Figure 6.38 Torque–speed characteristics.
Chapter 7
Figure 7.1 A typical cylindrical rotor of a wound‐field synchronous machine....
Figure 7.2 A typical salient‐pole rotor of a wound‐field synchronous machine...
Figure 7.3 The stator of a typical turbogenerator. Source: Yang Zhao et.al.,...
Figure 7.4 The stator of a typical hydro‐turbine‐driven generator. Source: C...
Figure 7.5 The slotted cylindrical‐shell of a laminated stator core of a syn...
Figure 7.6 A stator core with the three‐phase winding embedded in the stator...
Figure 7.7 The laminated rotor structure of a salient‐pole synchronous machi...
Figure 7.8 The complete rotor structure of a salient‐pole synchronous machin...
Figure 7.9 The rotor slotted forging of a synchronous turbogenerator.
Figure 7.10 The complete assembly of a cylindrical rotor of a synchronous tu...
Figure 7.11 A conventional brush‐slipring excitation system for wound‐field ...
Figure 7.12 A modern brushless (rotating diode‐bridge) excitation system for...
Figure 7.13 Schematic of an elementary three‐phase synchronous machine inclu...
Figure 7.14 Effect of a field‐excitation winding on equivalent circuits of t...
Figure 7.15 The equivalent circuit of an idealized cylindrical rotor synchro...
Figure 7.16 The phasor‐space‐vector diagram of an over‐excited synchronous m...
Figure 7.17 The torque‐power (torque) angle, , characteristic for motoring ...
Figure 7.18 The phase‐space‐vector diagram of an under‐excited synchronous m...
Figure 7.19 The V‐curves of a synchronous motor modes of operation from the ...
Figure 7.20 The phasor‐space‐vector diagram of an over‐excited generator.
Figure 7.21 The phasor‐space‐vector diagram of an under‐excited generator.
Figure 7.22 The input–output powers and losses and efficiency in a synchrono...
Figure 7.23 A synchronous motor torque–speed characteristic in a constant vo...
Figure 7.24 Vector control of a three‐phase permanent magnet synchronous mot...
Figure 7.25 Variations of the stator voltage and
q
‐axis current during the c...
Chapter 8
Figure 8.1 Brush‐commutator and brushless DC machines. (a) Wound field brush...
Figure 8.2 Main constructional features of brush
‐
commutator DC machine...
Figure 8.3 A schematic representation of windings in a brush
‐
commutato...
Figure 8.4 Main constructional features of a two‐pole SPM brushless DC machi...
Figure 8.5 Main constructional features of two types of two‐pole interiorper...
Figure 8.6 Simple lap winding schematic for brush‐commutator DC machines,
m
...
Figure 8.7 Simple wave winding schematic for brush‐commutator DC machines,
m
Figure 8.8 A typical armature coil is shown at three successive instances in...
Figure 8.9 The MMFs of the armature and field windings, and , in a brush‐...
Figure 8.10 Schematic winding representation and equivalent circuit model of...
Figure 8.11 Equivalent circuit model of separately excited DC motors.
Figure 8.12 Torque–speed characteristic of a separately excited DC motor.
Figure 8.13 A one‐quadrant chopper circuit topology, PWM control and output ...
Figure 8.14 Field voltage control of separately excited DC motors by DC chop...
Figure 8.15 Armature voltage control of separately excited DC motors by DC c...
Figure 8.16 A shunt field control connected DC motor and its equivalent circ...
Figure 8.17 Torque–speed characteristic of a shunt‐field‐excited DC motor.
Figure 8.18 Field chopper control of torque–speed characteristics of shunt m...
Figure 8.19 Armature chopper control of torque–speed characteristics of shun...
Figure 8.20 Torque–speed characteristic under field chopper control.
Figure 8.21 Torque–speed characteristic under armature chopper control.
Figure 8.22 A series‐field connected DC motor and its equivalent circuit.
Figure 8.23 Torque–speed characteristic of a series‐field‐excited DC motor....
Figure 8.24 The torque–speed characteristic of a series‐field‐excited DC mot...
Figure 8.25 DC chopper line‐voltage control of torque–speed characteristics ...
Figure 8.26 Torque–speed control of a series‐field‐excited DC motor by a DC ...
Figure 8.27 A long‐shunt field‐connected‐compound DC motor and its equivalen...
Figure 8.28 A short‐shunt‐field connected‐compound DC motor and its equivale...
Figure 8.29 Torque–speed characteristic of a compound‐excited DC motor.
Figure 8.30 Input–output and losses in a compound‐excited DC motor and its e...
Figure 8.31 Six‐pole samarium–cobalt PM Brushless DC motor.
Figure 8.32 Six‐pole strontium–ferrite PM brushless DC motor.
Figure 8.33 Functional block diagram of the components of a PM brushless DC ...
Figure 8.34 Power electronic inverters for PM brushless DC motors. (a) Volta...
Figure 8.35 120° Conduction cycles of switches for...
Figure 8.36 Three‐phase voltage source feeding a 3‐phase load.
Figure 8.37 Current–source inverter with a two‐quadrant DC chopper with curr...
Figure 8.38 Duty cycles of switches,
Q
1
through
Q
6
, and the idealized phase ...
Figure 8.39 Status of phase currents and EMFs during commutation of phase cu...
Figure 8.40 The discrete hopping nature of the stator MMF,
F
S
.
Figure 8.41 Representation of equivalent magnetic effect for the magnet shap...
Figure 8.42 The process of torque production in brushless DC motors.
Figure 8.43 Torque variation during one of the six states (periods) of switc...
Figure 8.44 The MMFs and at the beginning of state #1.
Figure 8.45 The MMFs and at the middle of state #1.
Figure 8.46 The MMFs and at the end of state #1.
Figure 8.47 The MMFs and at the beginning of state #2.
Figure 8.48 Effect of commutation angle,
δ
c
, on range of the torque ang...
Figure 8.49 Inverter switching sequence during motoring,...
Figure 8.50 Inverter switching sequence during motoring,...
Figure 8.51 Effect of the commutation angle,
δ
c
, on the torque profile ...
Figure 8.52 Idealized characteristics of PM brushless DC motors.
Figure 8.53 Idealized torque–speed characteristics of PM brushless DC motors...
Chapter 9
Figure 9.1 Schematic of the windings in a wound‐field salient‐pole synchrono...
Figure 9.2 The discrete winding (coil) representation of the
a
,
b
,
c
phases,...
Figure 9.3 Schematic of armature and two‐field windings synchronous machine....
Figure 9.4 Phase (
a
) winding to field winding mutual inductance.
Figure 9.5 The self‐inductance of phase (
a
),
L
aa
(
σ
)
.
Figure 9.6 The profile of variation of
L
ab
with the position,
σ
.
Figure 9.7 Stator and rotor MMF waveforms and corresponding space vectors,
Figure 9.8 Stator and rotor MMF waveforms and corresponding space vectors,
Figure 9.9 A 2‐pole, 3‐phase, 30‐stator slots salient‐pole synchronous machi...
Figure 9.10 The FE grid for magnetic field computation of its distribution i...
Figure 9.11 FE‐computed flux plot in the 2‐pole synchronous generator at no‐...
Figure 9.12 The FE‐computed mid‐airgap flux density waveform of the two‐pole...
Figure 9.13 FE‐computed flux plot in the two‐pole synchronous generator at r...
Figure 9.14 The FE‐computed mid‐airgap flux density waveform of the two‐pole...
Figure 9.15 A 2‐pole cylindrical rotor generator with 36 stator‐slots, three...
Figure 9.16 A 2‐pole cylindrical rotor generator with 36 stator‐slots, three...
Figure 9.17 The FE‐grid of a 2‐pole cylindrical rotor generator with 36 stat...
Figure 9.18 One quadrant of the FE grid of the 2‐pole cylindrical rotor gene...
Figure 9.19 The rated voltage FE‐computed flux plot at no‐load, see Refs. [6...
Figure 9.20 Mid‐airgap radial flux density at rated voltage no‐load, see Ref...
Figure 9.21 The rated load and rated voltage FE‐computed flux plot, see Refs...
Figure 9.22 Mid‐airgap radial flux density waveform at 0.85 power factor ful...
Figure 9.23 2D‐FE‐computed self‐inductance
L
aa
(no‐load case), see Refs. [6,...
Figure 9.24 2D‐FE‐computed mutual inductance
L
ab
(no‐load case), see Refs. [...
Figure 9.25 2D‐FE‐computed self‐inductance
L
ff
, see Refs. [6, 7].
Figure 9.26 2D‐FE‐computed self‐inductance
L
aa
(full‐load case), see Refs. [...
Figure 9.27 2D‐FE‐computed mutual inductance
L
ab
(full‐load case), see Refs....
Figure 9.28 2D‐FE‐computed self‐inductance
L
ff
, see Refs. [6, 7].
Figure 9.29 The development of Park's transformation from the geometric natu...
Figure 9.30 The
D
–
Q
phasor diagram of an overexcited synchronous generator....
Figure 9.31 The
D
–
Q
phasor diagram of an overexcited synchronous generator i...
Figure 9.32 Radial and tangential (circumferential) surface‐mounted permanen...
Figure 9.33 Cross‐section of the samarium–cobalt machine, see Refs. [1–3, 13...
Figure 9.34 Cross‐section of the strontium–ferrite machine, see Refs. [1–3, ...
Figure 9.35 Finite‐element grid of samarium–cobalt machine at a given rotor ...
Figure 9.36 Finite‐element grid of strontium–ferrite machine at a given roto...
Figure 9.37 No‐load equal VP contours (flux plot) of the samarium–cobalt mac...
Figure 9.38 No‐load equal MVP contours (flux plot) of the strontium–ferrite ...
Figure 9.39 Mid‐airgap flux density waveform at no‐load in the samarium–coba...
Figure 9.40 Mid‐airgap flux density waveform at no‐load in the strontium–fer...
Figure 9.41 Permanent magnet in a magnetic circuit with an airgap.
Figure 9.42 The demagnetization characteristic of a permanent magnet materia...
Figure 9.43 The operating point of a permanent magnet.
Figure 9.44 Equivalent magnetic circuit with electromagnet replacing the per...
Figure 9.45 Schematic of armature and PM equivalent “field winding” (PM mach...
Figure 9.46 Vector control block diagram of wound‐field synchronous motors....
Figure 9.47 Current limit circle and voltage limit ellipses for WFSMs.
Figure 9.48 Unity power factor ellipses for WFSMs in
dq
plane.
Figure P3.1 Inverter switching sequence for...
Figure P3.2 Inverter output voltage for 180°
e
...
Figure P3.3 Inverter output voltage for...
Figure P3.4 EMF in phase (
A
) and ...
Figure P4.1 Voltage–source inverters schematic of switches.
Figure P4.2 Current–source inverters with a two‐quadrant chopper with curren...
Chapter 10
Figure 10.1 The layout of the three stator phase belts, (
A
, −
A
), (
B
, −
B
), (
C
Figure 10.2 Three‐phase equivalent coil schematic of the
A
,
B
, and
C
stator ...
Figure 10.3 The choice of the locations of the
d
‐axis along the axis of phas...
Figure 10.4 Per phase
T
‐equivalent circuit of a three‐phase induction machin...
Figure 10.5 The
d
–
q
(or
D
–
Q
) frame of reference rotating at the ...
Figure 10.6 The stationary
DQ
0 stator reference frame with a rotor speed i...
Figure 10.7 The synchronously rotating
DQ
0 frame with a rotor speed in e.r...
Figure 10.8 The torque profile of the starting transient of the 100 hp motor...
Figure 10.9 The starting transient of the phase (
A
) current of the 100 hp mo...
Figure 10.10 Effect of slotting on airgap flux (from Reference [13]).
Figure 10.11 Effect of radial ventilation ducts on airgap flux (from Referen...
Figure 10.12 Carter coefficients for airgaps (from Reference [13]).
Figure 10.13 Leakage flux patterns (from Reference [13]). (a) Total flux. (b...
Figure 10.14 Various stator slot geometries and the related permeance calcul...
Figure 10.15 The layout for a two‐pole, ...
Figure 10.16 Slot leakage flux pattern.
Figure 10.17 Coil and end‐leads/turns geometry.
Figure 10.18 Magnetic circuit, winding layout, and axis labeling.
Figure 10.19 Stator slot dimensions in inches.
Figure 10.20 Rotor slot dimension in inches.
Figure 10.21 Rated load case FE‐computed flux plot at..
Figure 10.22 Functional block diagram of direct torque control of an inducti...
Chapter 11
Figure 11.1 (a) A split‐phase induction motor. (b) The phasor diagram of the...
Figure 11.2 A capacitor‐start induction motor and the phasor diagram of volt...
Figure 11.3 Comparison of motor cross‐sectional topologies between a three‐p...
Figure 11.4 The equivalent circuit of a split‐phase induction motor operatin...
Figure 11.5 Torque–speed characteristics of a split‐phase induction motor.
Figure 11.6 The elliptic field represents the most general case of magnetic ...
Figure 11.7 Equivalent circuit of a single‐phase induction motor at various ...
Figure 11.8 Equivalent circuit of a capacitor‐start induction motor. (a) Onl...
Figure 11.9 Topology of a four‐phase‐switched reluctance motor with eight po...
Chapter 12
Figure 12.1 Power topologies for various transportation propulsion structure...
Figure 12.2 Generators and power converters in the drivetrain of renewable p...
Figure 12.3 An integrated six‐phase motor‐drive concept.
Figure 12.4 Reflected voltage in long‐cable‐fed motor‐drive systems.
Figure 12.5 The voltage spike versus cable length and semiconductor rise tim...
Figure 12.6 Differential‐mode and common‐mode EMI in a motor‐drive system.
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Dedication
About the Authors
Preface
About the Companion Website
Begin Reading
Index
End User License Agreement
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Electric Machinery and Drives
An Electromagnetics Perspective
Nabeel A. O. Demerdash
Marquette UniversityMilwaukee, WI, USA
JiangBiao He
University of TennesseeKnoxville, TN, USA
Hao Chen
Zhejiang UniversityHangzhou, China
Copyright © 2025 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
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Library of Congress Cataloging‐in‐Publication Data
Names: Nabeel A.O. Demerdash, JiangBiao He, and Hao Chen.
Title: Electric machinery and drives : an electromagnetics perspective / Nabeel A. O. Demerdash, Marquette University, Milwaukee, WI, USA; JiangBiao He, University of Tennessee, Knoxville, TN, USA; Hao Chen, Zhejiang University, Hangzhou, China.
Description: First edition. | Hoboken, New Jersey : Wiley, [2025] | Includes index.
Identifiers: LCCN 2024008446 (print) | LCCN 2024008447 (ebook) | ISBN 9781119985723 (hardback) | ISBN 9781119985532 | ISBN 9781119985549 (epub)
Subjects: LCSH: Electric machinery.
Classification: LCC TK2000 .C486 2025 (print) | LCC TK2000 (ebook) | DDC 621.31/042–dc23/eng/20240319
LC record available at https://lccn.loc.gov/2024008446
LC ebook record available at https://lccn.loc.gov/2024008447
Cover Design: Wiley
Cover Images: © natatravel/Shutterstock, © Engineer studio/Shutterstock, © Phonlamai Photo/Shutterstock, © Fasttailwind/Shutterstock, © TebNad/Adobe Stock Photos
In Memory
of
My Dear Parents Mr. Aly O. Demerdash and Mrs. Aziza M. Demerdash
and
In Gratitude
to
The Support of My Dear Spouse Mrs. Esther A. Demerdash and My Dear Children Dr. Yvonne A. N. A. Demerdash‐Stubbs, Dr. Omar N. A. Demerdash, Dr. Nancy N. A. Demerdash, and All My Former Students.
About the Authors
Nabeel A. O. Demerdash received the BSc EE degree (distinction with first‐class Hons.) in electrical engineering from Cairo University, Giza, Egypt, in 1964, and the MS and PhD degrees in electrical engineering from the University of Pittsburgh, Pittsburgh, PA, USA, in 1967 and 1971, respectively.
From 1968 to 1972, he was a development engineer in the Large Rotating Apparatus Development Engineering Department, Westinghouse Electric Corporation, East Pittsburgh. From 1972 to 1983, he was an assistant professor, an associate professor, and then a professor in the Department of Electrical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA. From 1983 to 1994, he was a professor in the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY, USA. Since 1994, he has been a professor with the Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI, USA, where he was the department chair from 1994 to 1997. He was the director of the SEMPEED Consortium at Marquette University. He is the author or co‐author of over 150 papers published in various IEEE TRANSACTIONS and magazines. His current research interests include electric machines and drives, electromechanical propulsion and actuation, computational electromagnetics in machines and drives, and fault diagnostics and modeling of motor‐drive systems.
Prof. Demerdash is an IEEE Life Fellow and the recipient of the 1999 IEEE Nikola Tesla Technical Field Award for pioneering contributions to electric machine and drive system design using coupled finite‐element and electrical network models. He has also received two 1994 Working Group Awards and two 1993 Prize Paper Awards from the IEEE Power and Energy Society (PES) and its Electric Machinery Committee, and a 2012 Prize Paper Award from the PES, as well as a 2012 Prize Paper Award from the IEEE Industry Applications Society Electric Machines Committee and the 2015 Marquette University Lawrence G. Haggerty Faculty Award for Research Excellence.
JiangBiao He is an associate professor in electrical engineering at the University of Tennessee, Knoxville, TN, USA. He was an assistant professor and then associate professor in electrical engineering at the University of Kentucky between 2019 and 2024. He previously worked in industry, most recently as a lead engineer at GE Global Research, Niskayuna, New York. He received the PhD degree in electrical engineering from Marquette University, USA. His research interests focus on motor‐drive systems and power electronic converters for various emerging applications. He has authored and co‐authored over 150 technical articles and 10 patents in this area. Dr. He is an IEEE Senior Member and has served as an editor or associate editor for multiple prestigious IEEE journals. He also served in the organizing committees for numerous IEEE international conferences and has been an active member of multiple IEEE technical committees. He is a recipient of several achievement awards, including the 2018 GE Whitney Technical Excellence Award, 2019 AWS Outstanding Young Member Achievement Award with IEEE Industry Applications Society, and the 2023 Faculty Excellence in Research Award at the University of Kentucky.
Hao Chen received the BSc degree in electrical engineering from the School of Electrical Engineering, Beijing Jiaotong University, China, in 2012, and the PhD degree in control science and engineering from the School of Automation, Beijing Institute of Technology, Beijing, China, in 2019. From 2019 to 2021, he was a Postdoctoral Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From 2022 to 2023, he was a Researcher with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden. He is currently an Associate Professor with the College of Electrical Engineering, Zhejiang University, Hangzhou, China. His research interests include the design and optimization of electric machines, power electronic drives, and motor control.
Preface
There is a widespread and almost universally accepted realization that climate change and associated global warming, caused by carbon emissions attributed to energy generation from petroleum, coal, natural gas, and other forms of hydrocarbons, are leading to major efforts toward electrification of all modes of transportation on land, sea, and air. This includes automobiles, buses, trucks, trains, and other forms of transportation. This is besides the widespread desire and urge to move toward renewable energy generation from wind turbine generators and solar energy systems of various sorts. This is also in addition to robotic systems that are at the heart of automation of manufacturing processes. All these forms of new energy conversion and generation systems rely heavily on knowledge and deep understanding of the principles of modeling, performance analysis, and design of electric machinery and adjustable‐speed drives (ASDs). That is, motors and generators, as well as their associated control systems consisting of power electronic components. These electric motors, generators, and their control systems/drives are the subjects of the various chapters of this textbook, which is intended for a wide range of electrical/mechanical/automotive engineering senior/junior, first‐ and/or second‐year graduate MS and PhD‐level students, and engineers working in the related domains. Accordingly, this textbook includes topics and subjects on rotating electric machinery and drives, as well as associated electromechanical energy conversion concepts and devices that are hereby delineated below.
We start with Chapter 1 on electric circuit notations and elementary circuit concepts including AC phasor notations. This is in addition to linear algebra matrix and vector notations, as well as time‐domain instantaneous power and AC complex‐power, real and reactive, notations. In Chapter 2, we cover fundamentals of power electronic conditioning and processing. This includes the conventional silicon and the emerging wide bandgap semiconductor devices, DC choppers of the buck and boost DC–DC type, as well as three‐phase inverters.
Meanwhile, in Chapter 3 we review magnetostatic and magnetic circuit concepts and notations. This is in addition to the review of the fundamentals of electromechanical energy conversion, including magnetomotive forces (MMFs) and consequent concepts of production of electromechanical torque in both AC and DC machines. Accordingly, in Chapter 4 we address the mechanical dynamics of motor‐load and generator‐prime mover systems. This includes the modeling and approaches to simulation of motor‐load starting transients.
In Chapter 5, we review AC and DC winding layouts, induced electromotive forces (EMFs), MMFs, in such windings, and analytical relationships between MMFs and flux density waveforms (distributions) in airgaps of such machines. This is in addition to analytical relationships between such MMFs, flux density waveforms, and resulting electromechanical torques. Meanwhile, we present in Chapter 6 the poly‐phase AC induction machine, all the way till covering its scalar torque–speed control and its vector torque‐speed control in induction motor drives. Accordingly, we present in Chapter 7 the poly‐phase AC synchronous machine, both as a mainline generator of AC power and as a motor in special, more limited applications, all the way till covering its scalar torque–speed control, and its vector torque–speed control in synchronous motor‐ASD systems. Furthermore, in Chapter 8, we present the conventional/legacy‐type brush‐commutator DC machine and the more modern permanent magnet (PM) brushless DC machine. This includes modern power electronic based torque–speed control of brush‐commutator legacy DC motors and PM brushless DC motors.
In this textbook, Chapter 9 is dedicated to the more advanced graduate‐level state‐space modeling of three‐phase synchronous machines, including the all‐important full effects of rotor saliency, in the natural ABC frame‐of‐reference, and in the two‐reaction (Park's transformation) DQ0 frame of reference. This material includes the full details of the algebra and calculus of the DQ0 transformation models and the resulting phasor‐space vector diagrams. This also includes full details on the modeling and computation of initial conditions for a given load condition, including no‐load. Meanwhile, in Chapter 10 we give graduate‐level students and engineers full details of the derivation of three‐phase induction machines in the natural ABC–abc frame of reference, as well as three forms of DQ0–dq0 frame‐of‐reference transformations and corresponding/resulting three state‐space models. This chapter includes further details on the stator and rotor leakage inductances as key parameters in the modeling and performance analysis of induction motors. Also, in this chapter, we revisit modern torque–speed vector control in light of the advanced induction machine models developed here in Chapter 10. In this textbook, Chapter 11 is the subject of single‐phase induction motors and other special motors, such as split‐phase AC motors, and capacitor start motors, as examples of such special machines. Meanwhile, in the final chapter of this textbook, Chapter 12, we review the major emerging applications in electric machinery and drives, such as electric transportations, robotics, drones, wind turbine powertrains, and other applications. Additionally, prospective innovative technical trends in electric machinery and drives are discussed, such as ultra‐compact integrated motor‐drive systems, axial‐flux machines, superconducting machines, as well as additive manufacturing of electric machinery and drives.
Finally, throughout the chapters of this textbook, we give ample numerical examples with solutions, as well as an adequate amount of problems for use by instructors/professors as homework and/or projects for students, for purposes of assignments throughout a semester or two, whatever the class level is, junior, senior, or graduate level.
Nabeel A. O. Demerdash
Milwaukee, WI, USA
JiangBiao He
Knoxville, TN, USA
Hao Chen
Hangzhou, China
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/demerdash/electricmachineryanddrives
The website includes:
Animations
1Electric Circuit Notations and Elementary Concepts
In this introductory chapter, we start by defining the electric circuit notations and elementary circuit concepts, including symbolisms used in representation of time‐domain and AC phasor variables, such as voltage and currents, used throughout Chapters 1–12 of this textbook. These include multivariable arrays, the so‐called vectors in matrix linear‐algebra parlance, as well as concepts of time‐domain instantaneous power and complex power in AC phasor formulation of electric circuits.
A basic and fundamental complex‐algebra concept to all that will follow is the well‐known Euler formulation that yields the following identities:
and
With these identities, Eqs. (1.1)–(1.3), we are ready to discuss frequency‐domain root mean square (RMS) phasor representation of time‐domain AC voltages and currents.
1.1 Frequency‐Domain RMS Phasor Representation of Time‐Domain AC Voltages and Currents
Here, lower‐case variables (symbols) such as v, i, e, and p stand for time‐domain voltages, currents, electromotive forces (emfs), and real powers, respectively. Also here, upper‐case variables (symbols) such as, V, I, E, and P stand for RMS magnitudes (absolute values) of voltage phasors, current phasors, emf phasors, and average real power in phasor complex power computations, respectively. Meanwhile, variables (symbols) such as , , , and stand for RMS phasor complex form voltages, currents, emfs, and complex power (real and reactive) in phasor complex power computations, respectively.
Accordingly, in steady‐state AC circuit analysis, one can write the following:
where V is the RMS AC voltage magnitude (absolute value), ω = (2πf) is the angular frequency in electrical radians per second, f is the AC frequency in Hertz, t is the time in seconds, and φ is the phase angle of the voltage signal.
Hence, based on the Euler identities given earlier, Eqs. (1.1)–(1.3), one can rewrite v(t) as follows:
or
In AC phasor form, the term (ejωt) is common to all voltage, current, and other signals. Hence, all the information that is needed is in terms (V) and (ejφ). Hence, in RMS phasor notation, the voltage, , can be written as follows in exponential phasor form:
or in Cartesian coordinate polar and rectangular complex forms as follows:
Similarly, for an instantaneous steady‐state time‐domain, current, i(t), given by the following equation:
The corresponding Euler formulation gives
and for the corresponding RMS phasor notation, the current, , can be written as follows:
or, in Cartesian coordinate polar and rectangular complex forms, the current, , can be written as follows:
In this text, is the conjugate of the phasor and is the conjugate of the phasor . Therefore,
and
1.2 Time‐Domain and RMS Frequency‐Domain Power Concepts Using Consumer System Formulation and Notations
Consider the two‐terminal “system or device” shown in Figure 1.1. The current i (or ) is taken to be flowing in a positive orientation when flowing into the terminal designated with a positive voltage polarity, v (or ). In such a case, the instantaneous input power, p(t), is given by the following:
where a positive p(t) means that real power (watts) is being consumed and a negative p(t) means that real power (watts) is being generated.
Figure 1.1 A summary graphical and formulation representation of the consumer system in the time‐domain and frequency‐domain phasor power computation.
Meanwhile, in phasor frequency‐domain, the complex power, , is given by the following:
where P is the real power in watts and Q is the reactive power in vars, and once again, P is positive means that watts is being consumed (as in a motoring mode) and P is negative means that watts is being generated (as in a generating mode), and Q is the reactive power in vars, in which Q is positive means that vars is being consumed (as in inductive loads) and Q is negative means that vars is being generated (as in capacitive loads). The reader is urged to examine the complex powers, , , , and , associated with the current phasors, , , , and , respectively, relative to the terminal voltage phasor, , associated with the two‐terminal system (or device), shown in Figure 1.1.
In the Consumer System Notation:
Positive
p
(
t
)
means watts is consumed (load/motoring).
Negative
p
(
t
)
means watts is generated (generator/source).
Now, for a multiterminal (or port), n, system (or device), the voltages can be written in matrix/array (vector) form in the time‐domain, , or phasor frequency‐domain, , respectively, as follows:
and
Similarly, the currents can be written in matrix/array (vector) form in the time‐domain, , or phasor frequency‐domain, , respectively, as follows:
and
Therefore, the instantaneous time‐domain power, p(t), can be written as follows in an n‐polyphase device:
where is the transpose of .
Meanwhile, the frequency‐domain phasor computation of the complex power, , can be written as follows:
That is,
where is the transpose of .
1.3 Elementary Concepts of Complex Real and Reactive Power in Balanced Three‐Phase Circuits and Devices Using Consumer System Notations
A balanced three‐phase set of current phasors, , , and , is shown in Figure 1.2. We will always assume counterclockwise rotation for such phasor diagrams throughout this textbook, unless it is explicitly stated otherwise. Accordingly, for an “observer” located at the star‐point in this diagram, the observer will see an a, b, c, a, b, c, … sequence, that is, a positive (+) sequence as designated in this figure. Meanwhile, if the rotation of this phasor diagram is reversed to a clockwise orientation, the sequence would become an a, c, b, a, c, b, … one, that is, a negative (−) sequence.
Figure 1.2 A balanced three‐phase set of current phasors, , , and , and the concepts of positive a, b, c and negative a, c, b sequencing.
If one would interchange the locations of the and phasors, as shown in Figure 1.3, and still preserve the counterclockwise rotation and the location of the star‐point “observer,” the result will be an a, c, b, a, c, b, … sequence. That is, one will be seeing a negative (−) sequence as designated in Figure 1.3.
Figure 1.3 Another approach to negative a, c, b sequencing in balanced three‐phase set of currents, , , and .
Meanwhile, we discuss complex power, , in the context of a balanced three‐phase Y‐connected impedance load with an isolated neutral, n. See Figure 1.4 in which the impedance per‐phase is , and the phase‐currents, , , and , which are also equal to the a, b, c line‐currents as depicted in this figure. Also, given in Figure 1.4 are the formulations for the phase (line‐to‐neutral) voltages, , , and , as well as phase‐currents. The phasor diagram of the voltages depicts the graphical/schematic phasor representation of the line‐to‐line voltages, , , and . Here, one can write the complex power, , as follows:
where P is the real power in watts and Q is the reactive power in vars. Again, here, one can write the following:
and in the case of a balanced three‐phase load, the power factor, PF, is
where the angle, θ, is the power factor angle given by
which is the angle of the impedance per‐phase, , where ; see Figure 1.4. Furthermore, in this balanced three‐phase Y‐connected impedance load, one can write the following for :
Figure 1.4 A balanced Y‐connected impedance load and its associate line‐to‐neutral and line‐to‐line voltages and line currents, , , and .
The schematic of a balanced Δ‐connected impedance load is shown in Figure 1.5, in which an impedance, , interconnects lines a to b, lines b to c, and lines c to a, respectively. The formulations for the line‐to‐line voltages, , , and , are depicted in Figure 1.5, including the delta (Δ) branch currents, , , and . Also, depicted in Figure 1.5 are the line currents, , , and .
Figure 1.5 Balanced Δ‐connected impedance load and its line‐to‐line voltages and line currents, , , and .
Comparing the line currents, , , and , for the Y‐connected balanced load of Figure 1.4 and the Δ‐connected balanced load of Figure 1.5, one can conclude that for the two loads to draw the same line, a, b, and c, currents, one must be able to write the following for each line current, , , and :
and
Hence, for the two Y‐connected and Δ‐connected loads to be equivalent at their a, b, and c terminals, we must have
In summary, Eqs. (1.32) and (1.33) constitute the Δ‐to‐Y impedance transformation and the Y‐to‐Δ impedance transformation for balanced three‐phase impedance loads, respectively. The complex power, , drawn from the source is identical for both impedance loads in Eqs. (1.32) and (1.33).
