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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on Power and Energy Systems
- Sprache: Englisch
New edition of the popular reference on machine analysis, focusing on reference frame theory with techniques for derivation of equations
Analysis of Electric Machinery and Drive Systems covers the concepts needed to understand the evolution of electrical and magnetic variables for designing the power-electronic circuits that supply or extract electrical energy from a variety of machines, comprehensively addressing the varied needs of readers in the electric machinery, electric drives, and electric power industries.
This fourth edition has been extensively revised and updated to include nine new or updated chapters on symmetrical three-phase stators, symmetrical induction machines, brushless DC machines, synchronous machines, neglecting electric transients, eigenvalues and voltage-behind-reactive machine equations, direct current machine and drive, and torque control of permanent-magnet and synchronous reluctance machines.
Introductory concepts related to the subject have also been expanded upon, detailing stationary magnetically coupled circuits, energy balance relationships, energy in coupling field, and steady-state and dynamic performance of electromechanical systems. The fourth edition also includes illustrations of the free-acceleration characteristics of induction and brushless dc machines viewed from various reference frames and many other topics.
With problems at the end of each chapter to reinforce learning, the book explores additional topics including:
- Operational impedances and time constraints of synchronous machines, covering Park’s equations in operational form and parameters from short-circuit and frequency-response characteristics
- Fully controlled three-phase bridge converters, covering six-step, sine-triangle, space-vector, hysteresis, and delta modulations, along with open- and closed-loop voltage and current regulations
- Motor drives, covering volts-per-hertz, constant slip current, field-oriented, and direct torque control as well as slip energy recovery drives
- Brushless DC motor drives, covering average-value analysis, steady-state performance, and transient and dynamic performance of voltage-source inverter drives
Analysis of Electric Machinery and Drive Systems, Fourth Edition, is a perfect resource for electrical engineering students and an essential, up-to-date reference for electrical and mechanical engineers working with drives.
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Seitenzahl: 570
Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
About the Authors
Preface
Acknowledgments
About the Companion Website
1 Introductory Concepts
1.1 Introduction
1.2 Stationary Magnetically Coupled Circuits
1.3 Energy Balance Relationships
1.4 Energy in Coupling Field
1.5 Electromagnetic Forces
1.6 Steady‐State and Dynamic Performance of an Electromechanical System
References
Problems
2 Symmetrical Three‐Phase Stator
2.1 Introduction
2.2 Stator Winding Configuration and Air‐Gap mmf
2.3 Transformation Equations
2.4 Voltage Equations in Arbitrary Reference Frame
2.5 Transformation Between Reference Frames
2.6 P‐Pole Machines
2.7 Transformation of a Balanced Set
2.8 Instantaneous and Steady‐State Phasors
2.9 Variables Observed from Several Frames of Reference
References
Problems
3 Symmetrical Induction Machine
3.1 Introduction
3.2 Induction Machine
3.3 Transformation of Rotor Windings to the Arbitrary Reference Frame
3.4 Voltage, Flux‐Linkage Equations, and Equivalent Circuit
3.5 Torque Expressed in Arbitrary Reference Frame Variables
3.6 Computer Simulation in the Arbitrary Reference Frame
3.7 Per Unit System
3.8 Steady‐State Equivalent Circuit and Common Modes of Operation
3.9 Free‐Acceleration Torque Versus Speed Characteristics
3.10 Free‐Acceleration Characteristics Viewed from Various Reference Frames
3.11 Dynamic Performance During Sudden Changes in Load Torque
References
Problems
4 Brushless DC Machine
4.1 Introduction
4.2 Voltage Equations in Machine Variables
4.3 Voltage and Torque Equations in Rotor Reference Frame Variables
4.4 Instantaneous and Steady‐State Phasors
4.5 Field Orientation of a Brushless DC Drive
References
Problems
5 Synchronous Machines
5.1 Introduction
5.2 Windings of a Synchronous Machine
5.3 Voltage Equations in Rotor Reference Frame Variables
5.4 Torque Expressions Positive for Motor Action
5.5 Time‐Domain Block Diagram
5.6 Rotor Angle and Angle Between Rotors
5.7 Per Unit System
5.8 Analysis of Steady‐State Operation
5.9 Stator Currents Positive out of Machine—Synchronous Generator Operation
References
Problems
6 Neglecting Electric Transients
6.1 Introduction
6.2 Neglecting Stator Electric Transients
6.3 Induction Machine with Stator Transients Neglected
6.4 The Synchronous Machine with Stator Transients Neglected
References
Problems
7 Machine Equations in Operational Impedances and Time Constants
7.1 Introduction
7.2 Park's Equations in Operational form
7.3 Operational Impedances and
G
(
P
) for a Synchronous Machine with Four Rotor Windings
7.4 Standard Synchronous Machine Reactances
7.5 Standard Synchronous Machine Time Constants
7.6 Derived Synchronous Machine Time Constants
7.7 Parameters from Short‐Circuit Characteristics
7.8 Parameters from Frequency‐Response Characteristics
References
Problems
8 Eigenvalues and Voltage‐Behind‐Reactance Machine Equations
8.1 Introduction
8.2 Machine Equations to be Linearized
8.3 Linearization of Machine Equations
8.4 Small‐Displacement Stability—Eigenvalues
8.5 Eigenvalues of Typical Induction Machines
8.6 Eigenvalues of Typical Synchronous Machines
8.7 Detailed Voltage‐Behind‐Reactance Model
8.8 Reduced‐Order Voltage‐Behind‐Reactance Model
References
Problems
9 Semi‐Controlled Bridge Converters
9.1 Introduction
9.2 Single‐Phase Load Commutated Converter
9.3 Three‐Phase Load Commutated Converter
9.4 Conclusions and Extensions
References
Problems
10 Fully Controlled Three‐Phase Bridge Converters
10.1 Introduction
10.2 The Three‐Phase Bridge Converter
10.3 Six‐Step Operation
10.4 Six‐Step Modulation
10.5 Sine‐Triangle Modulation
10.6 Extended Sine‐Triangle Modulation
10.7 Space‐Vector Modulation
10.8 Hysteresis Modulation
10.9 Delta Modulation
10.10 Open‐Loop Voltage and Current Regulation
10.11 Closed‐Loop Voltage and Current Regulation
References
Problems
11 Direct‐Current Machine and Drive
11.1 Introduction
11.2 Commutation
11.3 Voltage and Torque Equations
11.4 Permanent‐Magnet dc Machine
11.5 dc Drive
Reference
Problems
12 Torque Control of Permanent‐Magnet and Synchronous Reluctance Machines
12.1 Introduction
12.2 Torque Control of a Permanent‐Magnet AC Machine
12.3 Simulation of a Permanent‐Magnet AC Machine with Torque Control
12.4 Torque Control of a Synchronous Reluctance Machine
References
Problems
13 Induction Motor Drives
13.1 Introduction
13.2 Volts‐Per‐Hertz Control
13.3 Constant Slip Current Control
13.4 Field‐Oriented Control
13.5 Direct Field‐Oriented Control
13.6 Robust Direct Field‐Oriented Control
13.7 Indirect Rotor Field‐Oriented Control
13.8 Direct Torque Control
13.9 Slip Energy Recovery Drives
13.10 Conclusions
References
Problems
14 Permanent‐Magnet AC Motor Drives
14.1 Introduction
14.2 Voltage‐Source Inverter Drives
14.3 Equivalence of Voltage‐Source Inverters to an Idealized Source
14.4 Average‐Value Analysis of Voltage‐Source Inverter Drives
14.5 Steady‐State Performance of Voltage‐Source Inverter Drives
14.6 Transient and Dynamic Performance of Voltage‐Source Inverter Drives
14.7 Case Study: Voltage‐Source Inverter‐Based Speed Control
14.8 Current‐Regulated Inverter Drives
14.9 Voltage Limitations of Current‐Regulated Inverter Drives
14.10 Current Command Synthesis
14.11 Average‐Value Modeling of Current‐Regulated Inverter Drives
14.12 Case Study: Current‐Regulated Inverter‐Based Speed Controller
References
Problems
Appendix A: Abbreviations, Constants, Conversions, and IdentitiesAbbreviations, Constants, Conversions, and Identities
Appendix B: Phasors and Phasor Diagrams
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.9-1 Induction Machine Parameters.
Chapter 5
Table 5.9-1 Hydro Turbine Generator.
Table 5.9-2 Steam Turbine Generator.
Chapter 7
Table 7.5-1 Standard Synchronous Machine Time Constants.
Table 7.6-1 Derived Synchronous Machine Time Constants.
Chapter 8
Table 8.5-1 Induction Machine Eigenvalues.
Figure 8.6-1 Synchronous Machine Eigenvalues for Rated Conditions.
Chapter 10
Table 10.2-1 Converter Voltages and Currents.
Table 10.7-1 Modulation Indices versus State.
Table 10.7-2 State Sequence.
Chapter 13
Table 13.8-1 Achievable Voltage Vectors and Corresponding Switching State.
Table 13.8-2 Switching Table for Direct Torque Control.
Chapter 14
Table 14.7-1 Drive System Parameters.
List of Illustrations
Chapter 1
Figure 1.2-1 Magnetically coupled circuits.
Figure 1.2-2 Equivalent
T
circuit with winding 1 selected as reference windi...
Figure 1.2-3 Typical
B–H
curve for silicon steel used in transformers....
Figure 1.2-4 Magnetization curve.
Figure 1.2-5
f
(
λ
m
)
versus
λ
m
from Figure 1.2-4.
Figure 1.2-6 Time‐domain block diagram of a two‐winding transformer with sat...
Figure 1.3-1 Block diagram of an elementary electromechanical system.
Figure 1.3-2 Energy balance.
Figure 1.3-3 Electromechanical system with magnetic field.
Figure 1B-1 Electromechanical system.
Figure 1.4-1 Stored energy and coenergy in a magnetic field of a singly exci...
Figure 1.6-1 Steady‐state operation of an electromechanical system in Fig. 1...
Figure 1.6-2 Dynamic performance of the electromechanical system shown in Fi...
Figure 1.6-3 Dynamic performance of the electromechanical system shown in Fi...
Figure 1.6-4 System response shown in Fig. 1.6-2.
Figure 1P-1 Two‐winding, iron‐core transformer.
Figure 1P-2
R‐L
circuit.
Figure 1P-3 Spring‐mass‐damper system.
Figure 1P-4 Core configuration.
Chapter 2
Figure 2.2-1 Stator windings of a multiphase machine.
Figure 2.2-2 (a) The
as
winding. (b) Ampere's Law (c)
mmf
as
.
Figure 2.3-1 Elementary two‐pole three‐phase sinusoidally distributed stator...
Figure 2.3-2 Fictitious
qs
and
ds
windings.
Figure 2.3-3 Tesla's balanced steady‐state rotating magnetic field viewed ...
Figure 2.4-1 Arbitrary reference frame equivalent circuits for stator substi...
Figure 2.5-1 Transformation between two reference frames portrayed by trigon...
Figure 2.6-1
P
‐pole stator for
θ
esi
(0) = 0
.
Figure 2.7-1 Direction of rotation of
mmf
s
and the phase relation between ...
Figure 2.8-1 The q and d complex plane.
Figure 2.9-1 Variables of three‐phase stator circuits in the stationary refe...
Figure 2.9-2 Variables of three‐phase stator circuits in the synchronously r...
Figure 2.9-3 Variables of three‐phase stator circuits. First with
ω = − ω
...
Figure 2P-1 Uniformly distributed winding.
Figure 2P-2 Three‐phase RL load.
Figure 2P-3 Two‐phase stator.
Chapter 3
Figure 3.2-1 Four‐pole three‐phase 6.5‐Hp 460 V severe‐duty, squirrel‐cage i...
Figure 3.2-2 A two‐pole three‐phase symmetrical induction machine.
Figure 3.3-1 Elementary two‐pole three‐phase sinusoidally distributed rotor ...
Figure 3.4-1 Arbitrary reference frame equivalent circuits for a three‐phase...
Figure 3.6-1 Time‐domain block diagram for induction machine in the arbitrar...
Figure 3.8-1 Equivalent single‐phase circuit for a three‐phase symmetrical i...
Figure 3.8-2 Steady‐state torque versus speed characteristics of a symmetric...
Figure 3.8-3 Steady‐state torque versus speed characteristics of a symmetric...
Figure 3.8-4 Steady‐state torque versus speed characteristics of a symmetric...
Figure 3.9-1 Torque versus speed characteristics during free acceleration—3‐...
Figure 3.9-2 Torque versus speed characteristics during free acceleration—50...
Figure 3.9-3 Torque versus speed characteristics during free acceleration—50...
Figure 3.9-4 Torque versus speed characteristics during free acceleration—22...
Figure 3.9-5 Machine variables during free acceleration of a 3‐hp induction ...
Figure 3.9-6 Machine variables during free acceleration of a 2250‐hp inducti...
Figure 3.9-7 Comparison of steady‐state and free‐acceleration torque versus ...
Figure 3.9-8 Comparison of steady‐state and free‐acceleration torque versus ...
Figure 3.10-1 Free‐acceleration characteristics of a 10‐hp induction motor i...
Figure 3.10-2 Free‐acceleration characteristics of a 10‐hp induction motor i...
Figure 3.10-3 Free‐acceleration characteristics of a 10‐hp induction motor i...
Figure 3.10-4 Free‐acceleration characteristics of a 10‐hp induction motor i...
Figure 3.11-1 Dynamic performance of a 3‐hp induction motor during step chan...
Figure 3.11-2 Dynamic performance of a 2250‐hp induction motor during step c...
Figure 3.11-3 Torque versus speed for 2250‐hp induction motor during load to...
Figure 3P-1 A two‐pole two‐phase induction machine.
Chapter 4
Figure 4.2-1 (a) Four‐pole three‐phase 28‐V permanent‐magnet ac machine....
Figure 4.2-2 Two‐pole three‐phase permanent‐magnet ac machine with sensors....
Figure 4.3-1 Time‐domain block diagram of brushless dc motor.
Figure 4.5-1 Free‐acceleration characteristics of a brushless dc drive with
Figure 4.5-2 Torque versus speed characteristics for the free acceleration s...
Figure 4.5-3 Phasor diagram for brushless dc drive operation at rad/s with...
Figure 4.5-4 Torque versus speed characteristics for free acceleration with ...
Figure 4.5-5 Phasor diagram for brushless dc drive operation at with
φv =
...
Figure 4.5-6 Torque versus speed characteristics for free acceleration with
Figure 4.5-7 Phasor diagram for brushless dc drive operation at
ωr = 50π rad
...
Chapter 5
Figure 5.2-1 Two‐pole, three‐phase, wye‐connected salient‐pole synchronous m...
Figure 5.3-1 Equivalent circuits of a three‐phase synchronous machine with t...
Figure 5.5-1 Time‐domain block diagram for synchronous machines in the rotor...
Figure 5.9-1 Two‐pole three‐phase wye‐connected salient‐pole synchronous mac...
Figure 5.9-2 Equivalent circuits of a three‐phase synchronous machine with t...
Figure 5.9-3 Phasor diagram for generator operation with currents defined po...
Figure 5.9-4 Dynamic performance of a hydro‐turbine generator following a st...
Figure 5.9-5 Torque versus rotor‐angle characteristics for the study shown i...
Figure 5.9-6 Dynamic performance of a steam‐turbine generator following a st...
Figure 5.9-7 Torque versus rotor‐angle characteristics for the study shown i...
Figure 5.9-8 Dynamic performance of a hydro‐turbine generator during and fol...
Figure 5.9-9 Torque versus rotor‐angle characteristics for the study shown i...
Figure 5.9-10 Dynamic performance of a steam‐turbine generator during and fo...
Figure 5.9-11 Torque versus rotor‐angle characteristics for the study shown ...
Figure 5P-1 Two‐pole, two‐phase synchronous machine.
Chapter 6
Figure 6.3-1 Torque versus speed characteristics during free acceleration pr...
Figure 6.3-2 Torque versus speed characteristics during free acceleration pr...
Figure 6.3-3 Machine variables during free acceleration of a 3‐hp induction ...
Figure 6.3-4 Machine variables during free acceleration of a 2250‐hp inducti...
Figure 6.4-1 Dynamic performance of a hydro‐turbine generator during a three...
Figure 6.4-2 Torque versus rotor‐angle characteristics for the study shown i...
Figure 6.4-3 Dynamic performance of a steam‐turbine generator during a three...
Figure 6.4-4 Torque versus rotor‐angle characteristics for the study shown i...
Chapter 7
Figure 7.3-1 Equivalent circuit with two damper windings in the quadrature a...
Figure 7.3-2 Calculation of
X
d
(
s
) and
G
(
s
) for two rotor windings in direct...
Figure 7.7-1 Plot of transient and subtransient components of the envelope ...
Figure 7.8-1 Plot of
X
q
(
s
) and
X
d
(
s
) versus frequency for a solid iron sync...
Figure 7.8-2 Two‐rotor winding approximation of
X
q
(
s
).
Figure 7.8-3 Three‐rotor winding approximation of
X
q
(
s
).
Figure 7.8-4 Two‐rotor winding direct‐axis circuit with unequal coupling.
Figure 7P-1 Short‐circuit stator currents.
Chapter 8
Figure 8.3-1 Interconnection of small‐displacement equations of a synchronou...
Figure 8.5-1 Plot of eigenvalues for a 3‐hp induction motor.
Figure 8.5-2 Plot of eigenvalues for a 2250‐hp induction motor.
Figure 8.7-1 Example canonical branch circuit element.
Figure 8.7-2 Machine/rectifier model using PVCC form.
Figure 8.7-4 Operational impedance of PVVBR model and PVVBR model with appro...
Figure 8.7-3 Machine/rectifier model using the PVVBR model.
Figure 8.7-5 Operational impedance of PVVBR model and PVVBR model with auxil...
Chapter 9
Figure 9.2-1 Single‐phase full‐bridge converter.
Figure 9.2-2 Single‐phase, full‐bridge converter operation for constant outp...
Figure 9.2-3 Single‐phase, full‐bridge converter operation for constant outp...
Figure 9.2-4 Single‐phase, full‐bridge converter operation for constant outp...
Figure 9.2-5 Single‐phase, full‐bridge converter operation for constant outp...
Figure 9.2-6 Average‐value equivalent circuit for a single‐phase full‐bridge...
Figure 9.2-7 Single‐phase, full‐bridge converter operation with RL load. (a)...
Figure 9.2-8 Single‐phase, full‐bridge converter operation with RL and an op...
Figure 9.2-9 Single‐phase, full‐bridge converter operation with RL and an ai...
Figure 9.3-1 Three‐phase full‐bridge converter.
Figure 9.3-2 Three‐phase, full‐bridge converter operation with RL load. (a)
Figure 9.3-3 Three‐phase, full‐bridge converter operation with RL and an opp...
Figure 9.3-4 Three‐phase, full‐bridge converter operation with RL and an aid...
Figure 9.3-5 Average‐value model of three‐phase full‐bridge converter.
Figure 9.3-6 Comparison of average‐value dynamic response with actual respon...
Chapter 10
Figure 10.2-1 The three‐phase bridge converter topology.
Figure 10.2-2 One phase leg.
Figure 10.2-3 Phase leg equivalent circuits. (a) Upper switch on;
i
xs
> 0. (...
Figure 10.3-1 Line‐to‐line voltages for six‐step operation.
Figure 10.3-2 Line‐to‐neutral voltage for six‐step operation.
Figure 10.3-3 Frequency spectrum of six‐step operation.
Figure 10.3-4 Voltage and current waveforms for a six‐stepped converter feed...
Figure 10.3-5 Comparison of
a‐
phase voltage to
q‐
and
d‐
ax...
Figure 10.4-1 Six‐step modulation control schematic (deadtime logic not show...
Figure 10.4-2 Six‐step modulation control signals.
Figure 10.4-3 Voltage and current waveforms for six‐step modulated converter...
Figure 10.5-1 Sine‐triangle modulation control schematic (deadtime logic not...
Figure 10.5-2 Operation of a sine‐triangle modulator.
Figure 10.5-3 Voltage and current waveforms using sine‐triangle modulation....
Figure 10.5-4 Overmodulation.
Figure 10.5-5 Voltage and current waveforms during overmodulated operation....
Figure 10.7-1 Space‐vector diagram.
Figure 10.8-1 State transition diagram.
Figure 10.8-2 Allowable current band.
Figure 10.8-3 Voltage and current waveforms using a hysteresis modulator.
Figure 10.10-1 Voltage regulation using a six‐step modulator.
Figure 10.10-2 Voltage regulation using an extended sine‐triangle modulator....
Figure 10.11-1 Synchronous regulator.
Figure 10.11-2 Voltage‐source modulator based current regulator.
Figure 10P-1 The
a‐
phase line‐to‐ground voltage of a three‐phase bridg...
Figure 10P-2 Circuit than can be used to avoid shoot‐through.
Chapter 11
Figure 11.1-1 Two‐pole 0.1‐hp 8 V 12,000 r/min permanent‐magnet dc motor (Co...
Figure 11.2-1 A dc machine with parallel armature windings.
Figure 11.2-2 Same as Fig. 11.2-1 with rotor advanced approximately 22.5° co...
Figure 11.3-1 Equivalent circuit of a dc machine.
Figure 11.4-1 Steady‐state torque versus speed characteristic of a permanent...
Figure 11.5-1 Two‐quadrant chopper drive.
Figure 11.5-2 Steady‐state operation of a two‐quadrant dc converter drive.
Figure 11.5-3 Average‐value model of two‐quadrant dc converter drive.
Figure 11.5-4 Starting characteristics of a permanent‐magnet dc machine with...
Figure 11.5-5 Torque control.
Figure 11.5-6 Armature current versus speed trajectory during switching.
Chapter 12
Figure 12.2-1 Two‐pole interior permanent‐magnet ac machine.
Figure 12.2-2 Contour plot for various values of torque,
T
e
. Points A, B, an...
Figure 12.2-3 Optimal currents versus torque for
Is ≤ Is, max
...
Figure 12.2-4 Maximum and minimum torque as a function of electrical rotor s...
Figure 12.2-5 Stator currents for maximum torque.
Figure 12.2-6 Stator voltages for maximum torque.
Figure 12.2-7 Maximum mechanical power versus rotor speed.
Figure 12.2-8 Efficiency for maximum
T
e
.
Figure 12.2-9 Phasor diagrams at maximum torque for
ωr = 0
...
Figure 12.2-10 Plot of .
Figure 12.2-11 Plot of .
Figure 12.3-1 Block diagram of torque‐controlled drive system.
Figure 12.3-2 Current control.
Figure 12.3-3 Time‐domain block diagram of permanent‐magnet ac machine with ...
Figure 12.3-4 System‐level simulation block diagram.
Figure 12.3-5 Torque command stepped from 6.57 to 400
N · m
Figure 12.3-6 Expanded view near 2 and 10 s. Electrical and control transien...
Figure 12.3-7 Torque versus speed trajectories for step changes in torque co...
Figure 12.3-8 Reduced‐order simulation block diagram.
Figure 12.3-9 Same as Fig. 12.3-5 with the electric machine and control tran...
Figure 12.4-1 Contour plots for various values of torque,
T
e
.
Figure 12.4-2 Optimal currents versus torque without voltage constraint.
Figure 12.4-3 Maximum and minimum torque as a function of electrical rotor s...
Figure 12.4-4 Optimum stator currents versus speed for maximum torque.
Figure 12.4-5 Voltages needed for maximum torque.
Figure 12.4-6 Maximum mechanical power versus rotor speed.
Figure 12.4-7 Efficiency for maximum
T
e
.
Figure 12.4-8 Plot of .
Figure 12.4-9 Plot of .
Figure 12.4-10 Torque command stepped from 6.57 to 400
N · m
...
Figure 12.4-11 Same as Fig. 12.4-10 with the electric machine and control tr...
Chapter 13
Figure 13.2-1 Elementary volts‐per‐hertz drive.
Figure 13.2-2 Performance of elementary volts‐per‐hertz drive.
Figure 13.2-3 Compensated volts‐per‐hertz drive.
Figure 13.2-4 Performance of compensated volts‐per‐hertz drive.
Figure 13.2-5 Start‐up performance of compensated volts‐per‐hertz drive.
Figure 13.3-1 Constant slip frequency drive.
Figure 13.3-2 Performance of constant slip frequency drive (maximum torque‐p...
Figure 13.3-3 Performance of constant slip frequency drive (maximum efficien...
Figure 13.3-4 Start‐up performance of constant slip controlled drive.
Figure 13.3-5 Speed control.
Figure 13.4-1 Torque on a current loop.
Figure 13.4-2 Torque production in an induction motor.
Figure 13.4-3 Generic rotor flux oriented control.
Figure 13.5-1 Rotor flux calculator.
Figure 13.5-2 Direct field‐oriented control.
Figure 13.6-1 Flux control loop.
Figure 13.6-2 Torque control loop.
Figure 13.6-3 Robust direct rotor field oriented control.
Figure 13.6-4 Start‐up performance of robust direct field oriented drive.
Figure 13.7-1 Indirect rotor field oriented control.
Figure 13.7-2 Start‐up performance of indirect field oriented drive.
Figure 13.7-3 Start‐up performance of indirect field oriented drive with err...
Figure 13.8-1 Direct torque control of an induction motor.
Figure 13.8-2 Stator flux and achievable voltage vectors.
Figure 13.8-3 Step response of an induction motor with DTC.
Figure 13.9-1 Circuit/block diagram of a slip energy recovery drive system....
Figure 13.9-2 Feedforward control for a slip energy recovery drive system.
Chapter 14
Figure 14.1-1 Permanent‐magnet ac motor drive.
Figure 14.2-1 Permanent‐magnet ac motor drive circuit.
Figure 14.3-1 Electrical diagram of a permanent‐magnet ac machine.
Figure 14.3-2 Semiconductor switching signals.
Figure 14.3-3 Steady‐state performance of a six‐stepped permanent‐magnet ac ...
Figure 14.3-4 Steady‐state performance of a six‐step‐modulated permanent‐mag...
Figure 14.3-5 Steady‐state performance of sine‐triangle‐modulated permanent‐...
Figure 14.5-1 Steady‐state voltage‐source inverter‐based permanent‐magnet ac...
Figure 14.6-1 Start‐up performance of a sine‐triangle‐modulated permanent‐ma...
Figure 14.6-2 Start‐up performance of a sine‐triangle‐modulated permanent‐ma...
Figure 14.6-3 Start‐up performance of a sine‐triangle‐modulated permanent‐ma...
Figure 14.6-4 Response of a sine‐triangle‐modulated permanent‐magnet ac moto...
Figure 14.6-5 Response of a sine‐triangle‐modulated permanent‐magnet ac moto...
Figure 14.7-1 Speed control system.
Figure 14.7-2 Frequency response of the open‐loop permanent‐magnet ac motor ...
Figure 14.7-3 Frequency response of the compensated permanent‐magnet ac moto...
Figure 14.7-4 Frequency response of the closed‐loop permanent‐magnet ac moto...
Figure 14.7-5 Start‐up response of the closed‐loop permanent‐magnet ac motor...
Figure 14.8-1 Hysteresis‐modulated current‐regulated drive control.
Figure 14.8-2 Steady‐state performance of a hysteresis‐modulated current‐reg...
Figure 14.8-3 A sine‐triangle‐modulator based current regulator.
Figure 14A-1 Step response of a feedforward sine‐triangle‐modulated current‐...
Figure 14.9-1 Response of hysteresis‐modulated current‐regulated permanent‐m...
Figure 14.10-1 Selection of
q‐
and
d‐
axis currents.
Figure 14.12-1 Current‐regulated‐inverter based speed control.
Figure 14.12-2 Start‐up response of current‐regulated‐inverter based speed c...
Appendix B
Figure B-1 Phasor diagram for (B-22).
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
About the Authors
Preface
Acknowledgments
About the Companion Website
Begin Reading
Appendix A Abbreviations, Constants, Conversions, and Identities
Appendix B Phasors and Phasor Diagrams
Index
End User License Agreement
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Analysis of Electric Machinery and Drive Systems
Fourth Edition
Paul C. Krause
PC Krause and Associates, IncWest Lafayette, Indiana, United States
Oleg Wasynczuk
Purdue UniversityWest Lafayette, Indiana, United States
Scott D. Sudhoff
Purdue UniversityWest Lafayette, Indiana, United States
Steven D. Pekarek
Purdue UniversityWest Lafayette, Indiana, United States
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About the Authors
Paul C. Krause is a former professor in the School of Electrical and Computer Engineering at Purdue University. He is the founder of PC Krause and Associates, Inc., and the recipient of the 2010 IEEE Nikola Tesla Award.
Oleg Wasynczuk is a professor emeritus of Electrical and Computer Engineering at Purdue University. He also served as Chief Technical Officer of PC Krause and Associates, Inc., and is the recipient of the 2008 IEEE PES Cyril Veinott Electromechanical Energy Conversion Award.
Scott D. Sudhoff is the Michael and Katherine Birck Distinguished Professor of Electrical and Computer Engineering at Purdue University. He is former Editor‐in‐Chief of the IEEE Transactions on Energy Conversion and the recipient of the 2025 IEEE Nikola Tesla Award.
Steven D. Pekarek is the Edmund O. Schweitzer, III, Professor of Electrical and Computer Engineering at Purdue University. He has served as an editor of the IEEE Transactions on Energy Conversion and the IEEE Transactions on Power Electronics, and is the recipient of the 2018 IEEE PES Cyril Veinott Electromechanical Energy Conversion Award.
Preface
This book is written for graduate students and engineers interested in machines and drives analysis. Chapter 1 covers some basic concepts that are common to books in this area. This fourth edition differs from previous editions in several ways. For example, the transformation for both the q and d variables is obtained from the expression of the rotating magnetomotive force or mmf. This is a very straightforward approach that provides an analytic origin of the transformation. Also, the analysis of each machine is focused on motor action to set the stage for electric drives, although generator action is considered in the case of the synchronous machine. Also, since for analysis purposes the stators of the AC machines considered in this text are the same, the stators are considered once in Chapter 2 rather than repeating the analysis for each machine. However, the rotors are different and are treated separately for each machine. This reduces the work considerably.
The induction machine is considered in Chapter 3. Most induction motors have squirrel‐cage rotors. However, if the stator has sinusoidally distributed windings, the rotor may also be considered as having sinusoidally distributed windings even though the rotor may consist of solid bars. The transformation of the rotor variables to the q and d axes differs only in that the rotor windings are rotating relative to the stator. The permanent‐magnet AC machine and the synchronous generator are considered in Chapters 4 and 5, respectively. In Chapter 4, we treat the brushless DC machine with Ld = Lq. Three different values of angle between and , or φv, are considered. These are: φv = 0, which is the most common operating mode, φv = φv,MT/V or maximum torque per volt, and φv = φv,MT/A or maximum torque per ampere. In this case, the permanent‐magnet rotor is considered to be magnetized sinusoidally.
The first part of Chapter 5 is devoted to motor action of a synchronous machine. The second part is devoted to generator action with positive current assumed out of the machine. This latter mode of operation was treated by Park in his classic paper written in 1929. The basic analysis of AC machines covered in this text ends with Chapter 5. Power systems engineers could continue with Chapters 6, 7, and 8. The drives engineer would not cover these chapters, but would skip to Chapters 10 through 14, and would likely omit some of the material in Chapter 5.
In Chapter 6, the concept of neglecting stator transients is treated. This chapter would be of most interest to the power systems engineer since it deals with the basis of transient stability programs used in stability studies for power systems. Both power systems and drives engineers could find Chapter 9 interesting. Drives engineers would want to study Chapter 10, as it describes the most commonly used modulation strategies. Chapter 11 deals with DC drives. This chapter is brief but relevant to electric drive engineering.
In Chapter 12, the torque control of permanent‐magnet AC and synchronous reluctance machines are considered. The analysis of the permanent‐magnet machine is similar to the material in Chapter 4. The difference is that Ld ≠ Lq and a reluctance torque exists. The parameters of the machine considered are representative of electric drive motors used in hybrid and electric vehicles. The synchronous reluctance machine is considered with the permanent magnets removed, whereby only a reluctance torque exists. Synchronous reluctance machines are also considered as viable candidates as electric drive motors in hybrid and electric vehicles. It is shown that with power‐electronic‐based current control, the electric transients are so fast that they may be neglected when considering the mechanical dynamics.
Induction motor control is considered in Chapter 13, including the volt‐per‐hertz, constant‐slip, and field‐oriented control methods. Each is considered in substantial detail. Finally, the control of permanent‐magnet AC machines is considered in Chapter 14.
Although this is a graduate text, the first six or seven chapters could be used at the senior‐level with the remaining chapters used as a graduate text.
February 2025
Paul C. Krause
Oleg Wasynczuk
Scott D. Sudhoff
Steven D. Pekarek
West Lafayette, Indiana
Acknowledgments
To Our Families
About the Companion Website
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www.wiley.com/go/krause_aem4e
The website includes Solution Manuals.
1Introductory Concepts
1.1 Introduction
This chapter is a review for most since the material is covered in undergraduate courses in the analysis of electromechanical devices [1]. The material is presented to start everyone with the same background. The chapter begins with coupled circuits (transformers) where the phasor equivalent circuit is established. Since phasors are not always taught the same, they are covered briefly in Appendix B to make sure everyone understands the concept of phasors as used in this text. Although we will give several approaches for the calculation of torque of electric machines; Section 1.1-3 sets forth a method of calculating force and torque that is generally taught at the undergraduate level.
Some instructors may choose to skip some material and/or select topics that were not covered in undergraduate courses at their school. As mentioned, the material will be a review for most and can be covered rather fast. On the other hand, Chapter 2 dives into machine analysis that contains new material and can be taught at a much slower pace.
1.2 Stationary Magnetically Coupled Circuits
Magnetically coupled electric circuits are central to the operation of transformers and electromechanical motion devices. In the case of transformers, stationary circuits are magnetically coupled for the purpose of changing the ac voltage and current levels. The two windings shown in Fig. 1.2-1 consist of turns N1 and N2, and they are wound on a common core, which is a ferromagnetic material with a permeability large relative to that of air. The magnetic core is illustrated in two dimensions.
Figure 1.2-1 Magnetically coupled circuits.
The flux produced by each winding can be separated into two components: a leakage component denoted by the subscript l and a magnetizing component denoted by the subscript m. Each of these components is depicted by a single streamline with the positive direction determined by applying the right‐hand rule to the directions of current flow in the winding. The leakage flux associated with a given winding links only that winding, whereas the magnetizing flux, whether it is due to current in winding 1 or winding 2, links both windings.
The flux linking of each winding may be expressed as
The leakage flux Φl1 is produced by current flowing in winding 1, and it links only the turns of winding 1. Likewise, the leakage flux Φl2 is produced by current flowing in winding 2, and it links only the turns of winding 2. The flux Φm1 is produced by current flowing in winding 1, and it links all turns of windings 1 and 2. Similarly, the magnetizing flux Φm2 is produced by current flowing in winding 2, and it also links all turns of windings 1 and 2. Both Φm1 and Φm2 are called magnetizing fluxes. With the selected positive directions of current flow and the manner in which the windings are wound, the magnetizing flux produced by positive current flowing in one winding can add to or subtract from the magnetizing flux produced by positive current flowing in the other winding. Thus, the mutual inductance can be positive or negative. In Fig. 1.2-1, it is positive.
It is appropriate to point out that this is an idealization of the actual magnetic system. It seems logical that all of the leakage flux will not link all the turns of the winding producing it; hence, Φl1 and Φl2 are “equivalent” leakage fluxes. Similarly, all of the magnetizing fluxes of one winding may not link all of the turns of the other winding.
The voltage equations may be expressed as
In matrix form,
The resistances r1 and r2 and the flux linkages λ1 and λ2 are related to windings 1 and 2, respectively. Since it is assumed that Φ1 links the equivalent turns of winding 1 (N1) and Φ2 links the equivalent turns of winding 2 (N2), the flux linkages may be written as
where Φ1 and Φ2 are given by (1.2-1) and (1.2-2), respectively.
If we assume that the magnetic system is magnetically linear (i.e., core losses and saturation are neglected), we may apply Ohm's law for magnetic circuits to express the fluxes. Thus, the fluxes may be written as
where k = 1 or 2 and and are the reluctances of the leakage paths, and is the reluctance of the path of magnetizing fluxes. Typically, the reluctances associated with leakage paths are much larger than the reluctance of the magnetizing path. The reluctance associated with an individual leakage path is difficult to determine exactly, and it is usually approximated from test data or by using the computer to solve the field equations numerically. On the other hand, the reluctance of the magnetizing path of the core shown in Fig. 1.2-1 may be computed with sufficient accuracy.
For the iron
where li is the length of the path in iron, μr is the relative permeability of iron, μ0 is the permeability of free space, and Ai is the cross‐sectional area of the flux in the iron. In electromechanical devices, we will find that the magnetizing flux must transverse air gaps and
Substituting (1.2-8) and (1.2-9) into (1.2-1) and (1.2-2) yields
Substituting (1.2-12) and (1.2-13) into (1.2-6) and (1.2-7) yields
When the magnetic system is linear, the flux linkages are generally expressed in terms of inductances and currents. We see that the coefficients of the first two terms on the right‐hand side of (1.2-14) depend on N1 and the reluctance of the magnetic system, independent of the existence of winding 2. An analogous statement may be made regarding (1.2-15) with the roles of winding 1 and winding 2 reversed. Hence, the self‐inductances are defined as
where Ll1 and Ll2 are the leakage inductances and Lm1 and Lm2 are the magnetizing inductances of windings 1 and 2, respectively. From (1.2-16) and (1.2-17), it follows that the magnetizing inductances may be related as
which is .
The mutual inductances are defined as the coefficient of the third term on the right‐hand side of (1.2-14) and (1.2-15). In particular,
We see that L12 = L21 and, with the assumed positive direction of current flow and the manner in which the windings are wound as shown in Fig. 1.2-1, the mutual inductances are positive. If, however, the assumed positive directions of the current or the direction of the windings were such that Φm1 opposed Φm2, then the mutual inductances would be negative.
The mutual inductances may be related to the magnetizing inductances. Comparing (1.2-16) and (1.2-17) with (1.2-19) and (1.2-20), we see that
The flux linkages may now be written as
where L11 and L22 are defined by (1.2-16) and (1.2-17), respectively, and L12 and L21 by (1.2-19) and (1.2-20), respectively. The self‐inductances L11 and L22 are always positive; however, the mutual inductances L12(L21) may be positive or negative, as previously mentioned.
Although the voltage equations given by (1.2-3) and (1.2-4) may be used for purposes of analysis, it is customary to perform a change of variables that yields the well‐known equivalent T circuit of two windings coupled by a linear magnetic circuit. To set the stage for this derivation, let us express the flux linkages from (1.2-22) and (1.2-23) as
With λ1 in terms of Lm1 and λ2 in terms of Lm2, we see two logical candidates for substitute variables, in particular, (N2/N1)i2 or (N1/N2)i1. If we let
then we are using the substitute variable , which, when flowing through winding 1, produces the same mmf as the actual i2 flowing through winding 2; . This is said to be referring the current in winding 2 to winding 1 or to a winding with N1 turns, whereupon winding 1 becomes the reference or primary winding and winding 2 is the secondary winding and is negative. On the other hand, if we let
then is the substitute variable that produces the same mmf when flowing through winding 2 as i1 does when flowing in winding 1; . This change of variables is said to refer to the current of winding 1 to winding 2 or to a winding with N2 turns, whereupon winding 2 becomes the reference or primary winding and winding 1 the secondary with .
We will demonstrate the derivation of the equivalent T circuit by referring the current of winding 2 to a winding with N1 turns; thus, is expressed by (1.2-26). We want the instantaneous power to be unchanged by this substitution of variables. Therefore,
Hence,
Flux linkages, which have the units of V · s, are related to the substitute flux linkages in the same way as voltages. In particular,
Now, replace (N2/N1)i2 with in the expression for λ1, given by (1.2-24). Next, solve (1.2-26) for i2 and substitute it into λ2 given by (1.2-25). Now, multiply this result by N1/N2 to obtain and then substitute (N2/N1)2Lm1 for Lm2 in . If we do all this, we will obtain
where
The flux linkage equations given by (1.2-31) and (1.2-32) may also be written as
where
and L22 is defined by (1.2-17).
If we multiply (1.2-4) by N1/N2 to obtain , the voltage equations become
where
The previous voltage equations, (1.2-37), together with the flux linkage equations, (1.2-34) and (1.2-35), suggest the equivalent T circuit shown in Fig. 1.2-2. This method may be extended to include any number of windings wound on the same core.
Figure 1.2-2 Equivalent T circuit with winding 1 selected as reference winding.
Example 1A The equivalent T circuit.
It is instructive to illustrate the method of deriving an equivalent T circuit from open‐ and short‐circuit measurements. When winding 2 of the two‐winding transformer shown in Fig. 1.2-2 is open circuited and a 60 Hz voltage of 110 V (rms) is applied to winding 1, the average power supplied to winding 1 is 6.66 W. The measured current in winding 1 is 1.05 A (rms). Next, with winding 2 short‐circuited, the current flowing in winding 1 is 2 A (rms) when the applied voltage is 30 V at 60 Hz. The average input power is 44 W. If we assume , an approximate equivalent T circuit can be determined from these measurements with winding 1 selected as the reference winding.
With and then the average power supplied to winding 1 may be expressed as
where
Here, and are phasors with the positive direction of taken in the direction of the voltage drop, and θev(0) and θei(0) are the phase angles of and , respectively. Phasors are covered in Appendix B. Solving for φpf during the open‐circuit test, we have
Although φpf = − 86.7∘ is also a legitimate solution of (1A-3), the positive value is taken since leads in an inductive circuit. With winding 2 open‐circuited, the input impedance of winding 1 is
With as the reference phasor, , . Thus,
If we neglect core losses, then, from (1A‐5), r1 = 6 Ω. We also see from (1A‐5) that Xl1 + Xm1 = 104.6 Ω. For the short‐circuit test, we will assume that since transformers are designed so that at rated frequency . Hence, using (1A‐1) again,
In this case, the input impedance is . This may be determined as
Hence, and, since it is assumed that , both are 10.2/2 = 5.1 Ω. Therefore, Xm1 = 104.6 − 5.1 = 99.5 Ω. In summary, r1 = 6 Ω, Ll1 = 13.5 mH, , . Make sure we converted from X's to L's correctly.
1.2.1 Nonlinear Magnetic System
Although the analysis of transformers and electric machines is often performed assuming a magnetically linear system, economics and physics dictate that in the practical design of many of these devices, some saturation occurs and that heating of the magnetic material exists due to hysteresis loss [2]. The magnetization characteristics of transformer or machine materials are typically given in the form of the magnitude of flux density versus magnitude of field strength (B–H curve) as shown in Fig. 1.2-3.
Figure 1.2-3 Typical B–H curve for silicon steel used in transformers.
If it is assumed that the magnetic flux is uniform through most of the core, then B is proportional to Φ and H is proportional to magnetomotive force (mmf). Hence, a plot of flux versus current is of the same shape as the B–H curve. A transformer is generally designed so that some saturation occurs during normal operation. During transients, saturation may occur resulting in large currents during startup transients. Electric machines are also designed similarly in that a machine generally operates slightly in the saturated region during normal, rated operating conditions. Since saturation causes coefficients of the differential equations describing the behavior of an electromagnetic device to be functions of the coil currents, transient analysis is difficult without the aid of a computer. Our purpose here is not to set forth methods of analyzing nonlinear magnetic systems. A method of incorporating the effects of saturation into a computer representation is of interest.
Formulating the voltage equations of stationary coupled windings appropriate for computer simulation is straightforward and yet this technique is fundamental to the computer simulation of ac machines. Therefore, it is to our advantage to consider this method here. For this purpose, let us first write (1.2-31) and (1.2-32) as
where
Solving (1.2-39) and (1.2-40) for the currents yields
If (1.2-42) and (1.2-43) are substituted into (1.2-37), and if we solve the resulting equations for flux linkages, the following equations are obtained:
Substituting (1.2-42) and (1.2-43) into (1.2-41) yields
where
We now have the equations expressed with λ1 and as state variables. In the computer simulation, (1.2-44) and (1.2-45) are used to solve for λ1 and and (1.2-46) is used to solve for λm. The currents can then be obtained from (1.2-42) and (1.2-43).
If the magnetization characteristics (magnetization curve) of the coupled winding are known, the effects of saturation of the mutual flux path may be incorporated into the computer simulation. Generally, the magnetization curve can be adequately determined from a test wherein one of the windings is open‐circuited (winding 2, for example) and the input impedance of the other winding (winding 1) is determined from measurements as the applied voltage is increased in magnitude from zero to say 150% of the rated value. With information obtained from this type of test, we can plot λm versus as shown in Fig. 1.2-4 wherein the slope of the linear portion of the curve is Lm1. From Fig. 1.2-4, it is clear that in the region of saturation we have
Figure 1.2-4 Magnetization curve.
where f(λm) may be determined from the magnetization curve for each value of λm. In particular, f(λm) is a function of λm given by (1.2-48) and shown in Fig. 1.2-5. Therefore, the effects of saturation of the mutual flux path may be taken into account by replacing (1.2-41) with (1.2-48) for λm. Substituting (1.2-42) and (1.2-43) for i1 and , respectively, into (1.2-48) yields the following equation for λm
Figure 1.2-5f(λm) versus λm from Figure 1.2-4.
Hence, the computer simulation for including saturation involves replacing λm given by (1.2-46) with (1.2-49) where f(λm) is a generated function of λm determined from the plot shown in Fig. 1.2-5. The time‐domain block diagram of a two‐winding transformer with saturation is shown in Fig. 1.2-6.
Figure 1.2-6 Time‐domain block diagram of a two‐winding transformer with saturation.
1.3 Energy Balance Relationships
Electromechanical systems consist of an electric system, a mechanical system, and a means whereby the electric and mechanical systems can interact. Interactions can take place through any and all electromagnetic and electrostatic fields that are common to both systems, and energy is transferred from one system to the other as a result of this interaction [3]. We will focus on the electromagnetic system, and the electrostatic system is treated in [2]. An electromechanical system with one electric subsystem, one mechanical subsystem, and one coupling field is depicted in Fig. 1.3-1. Electromagnetic radiation is neglected, and it is assumed that the electric system operates at a frequency sufficiently low so that the electric system may be considered a lumped‐parameter system.
Figure 1.3-1 Block diagram of an elementary electromechanical system.
Heat loss will occur in the mechanical system due to friction, and the electric system will dissipate heat due to the resistance of the current‐carrying conductors. Eddy current and hysteresis losses occur in the ferromagnetic materials. If WE is the total energy supplied by the electric source and WM the total energy supplied by the mechanical source, then the energy distribution could be expressed as
In (1.3-1), WeS is the energy stored in the magnetic fields, which are not coupled with the mechanical system. The energy WeL is the heat loss associated with the electric system excluding the coupling field losses. This loss occurs due to the resistance of the current‐carrying conductors as well as the energy dissipated in the form of heat owing to hysteresis and eddy current losses external to the coupling field. We is the energy transferred to the coupling field by the electric system. The energies common to the mechanical system may be defined in a similar manner. In (1.3-2), WmS is the energy stored in the moving member and compliances of the mechanical system, WmL is the energy loss in the mechanical system in the form of heat, and Wm is the energy transferred to the coupling field. It is important to note that, with the convention adopted, the energy transferred to the coupling field by either source is considered positive. Also, WE(WM) is negative when energy is supplied to the electric source (mechanical source).
If WF is defined as the total energy transferred to the coupling field, then
where Wf is energy stored in the coupling field and WfL is the energy dissipated in the form of heat due to losses within the coupling field (eddy current or hysteresis losses). In order to comply with convention, we will use Wf to denote the energy stored in the coupling field rather than WfS. The electromechanical system must obey the law of conservation of energy, thus,
which may be written as
This energy balance is shown schematically in Fig. 1.3-2.
Figure 1.3-2 Energy balance.
The actual process of converting electric energy into mechanical energy (or vice versa) is independent of (1) the loss of energy in either the electric or the mechanical systems (WeL and WmL), (2) the energies stored in the electric or magnetic fields that are not common to both systems (WeS), or (3) the energies stored in the mechanical system (WmS). If the losses of the coupling field are neglected, then the field is conservative and (1.3-5) becomes
An example of an elementary electromechanical system is shown in Fig. 1.3-3. It has a magnetic coupling field with the space between the movable and stationary members exaggerated for clarity. The voltage of the electric source is denoted v, and f is an externally applied mechanical force. The electromagnetic force is denoted fe. The resistance of the current‐carrying conductor is denoted by r, with l denoting the inductance of a linear (conservative) electromagnetic system that does not couple with the mechanical system. In the mechanical system, M is the mass of the movable member, and the linear compliance and damper are represented by a spring constant K and a damping coefficient D. The displacement x0 is the zero force or equilibrium position of the mechanical system, which is the steady‐state position of the mass with fe and f equal to zero.
Figure 1.3-3 Electromechanical system with magnetic field.
The voltage equation that describes the electric systems shown in Fig. 1.3-3 may be written as
where ef is the voltage drop due to the coupling field. The dynamic behavior of the translational mechanical systems may be expressed by employing Newton's law of motion. Thus,
Since power is the time rate of energy transfer, the total energy supplied by the electric source is
The total energy supplied by the mechanical source is
which may also be expressed as
Substituting (1.3-7) into (1.3-9) yields
The first term on the right‐hand side of (1.3-12) represents the energy loss due to the resistance of the conductors (WeL). The second term represents the energy stored in the linear electromagnetic field external to the coupling field (WeS). Therefore, the total energy transferred to the coupling field from the electric system is
Similarly, for the mechanical system
Here, the first and third terms on the right‐hand side of (1.3-14) represent the kinetic energy stored in the mass and the potential energy stored in the spring, respectively. The sum of these two stored energies is WmS. You should take a moment to look at the first term on the right‐hand side of (1.3‐14) and recognize that it can be written as . The second term is the heat loss due to friction (WmL). Thus, the total energy transferred to the coupling field from the mechanical system is
It is important to note from Fig. 1.3-3 that a positive force fe is assumed to be in the same direction as a positive displacement dx. Substituting (1.3-13) and (1.3-15) into the energy balance relation, (1.3-6), yields
The equations set forth may be readily extended to include an electromechanical system with any number of electric and mechanical inputs. Whereupon the field may be expressed as
wherein J electric and K mechanical inputs exist. The total energy supplied to the coupling field from the electric inputs is
The total energy supplied to the coupling field from the mechanical inputs is
In our analysis of electromechanical systems, we will consider devices with only one mechanical input, for example, the shaft of the electric machine or the moving arm of a magnetic solenoid. On the other hand, since machines may have more than one electric terminal, it is necessary to consider systems with multiple electric inputs. In all cases, however, the multiple electric inputs have a common coupling field. Therefore, we need not become too ambitious in the following derivations. More specifically, hereafter we will restrict our analysis to electromechanical devices with only one mechanical input. Thus, the k subscript will be dropped from fe, x, and Wm. This reduces our work considerably without restricting the practical application of our results. With one mechanical input, the energy balance equation becomes
In differential form, which will be the form we will use extensively,
Example 1B Calculation of inductance.
We will consider the electromechanical system shown in Fig. 1B-1. The system is at x0 when v and f are both zero. The value of x0 is 3 mm, which is very much exaggerated in Fig. 1B-1. The distance from c to d is 20 cm and 10 cm from a to b. The cross‐sectional area of the iron and air gap is Ai = Ag = 0.01 m2. The relative permeability of the iron is 4000, and the permeability of air is 4π × 10−7 H · m.
Figure 1B-1 Electromechanical system.
In Section 1.6, we let
Determine N (turns) for k = 6.283 × 10−5 H · m.
The reluctance to the magnetizing flux with x = x0
Clearly, the reluctance is dominated by the reluctance of the two air gaps.
Now, if we neglect the reluctance of the iron, then
Comparing (1B-1) and (1B-3) we see that
Solving for N yields
1.4 Energy in Coupling Field
Before using (1.3-21) to obtain an expression for the electromagnetic force fe, it is necessary to derive an expression for the energy stored in the coupling field. Once we have an expression for Wf, we can take the total derivative to obtain dWf, which can then be substituted into (1.3‐21). When expressing the energy in the coupling field, it is convenient to neglect all losses associated with the magnetic coupling field, whereupon the field is assumed to be conservative and the energy stored therein is a function of the state of the electrical and mechanical variables. Although the effects of the core losses of the coupling field may be functionally accounted for by appropriately introducing resistance in the electric circuit, this refinement is generally not necessary since the ferromagnetic material is selected and arranged in laminations so as to minimize the hysteresis and eddy current losses. Moreover, most of the energy stored in the coupling field is stored in the air gap of the electromechanical device. Since air is a conservative medium, all of the energy stored therein can be returned to the electric or mechanical systems. Therefore, the assumption of a lossless coupling field is not as restrictive as it might first appear.
The energy stored in a conservative field is a function of the state of the system variables and not the manner in which the variables reached that state. It is convenient to take advantage of this feature when developing a mathematical expression for the field energy. In particular, it is convenient to fix mathematically the position of the mechanical system associated with the coupling field and then excite the electric system with the displacement of the mechanical system held fixed. During the excitation of the electric inputs, dx = 0, hence, Wm
