103,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on Power Engineering
- Sprache: Englisch
- Veröffentlichungsjahr: 2021
Discover the analytical foundations of electric machine, power electronics, electric drives, and electric power systems In Introduction to the Analysis of Electromechanical Systems, an accomplished team of engineers delivers an accessible and robust analysis of fundamental topics in electrical systems and electrical machine modeling oriented to their control with power converters. The book begins with an introduction to the electromagnetic variables in rotatory and stationary reference frames before moving onto descriptions of electric machines. The authors discuss direct current, round-rotor permanent-magnet alternating current, and induction machines, as well as brushless direct current and induction motor drives. Synchronous generators and various other aspects of electric power system engineering are covered as well, showing readers how to describe the behavior of electromagnetic variables and how to approach their control with modern power converters. Introduction to the Analysis of Electromechanical Systems presents analysis techniques at an introductory level and at sufficient detail to be useful as a prerequisite for higher level courses. It also offers supplementary materials in the form of online animations and videos to illustrate the concepts contained within. Readers will also enjoy: * A thorough introduction to basic system analysis, including phasor analysis, power calculations, elementary magnetic circuits, stationary coupled circuits, and two- and three-phase systems * Comprehensive explorations of the basics of electric machine analysis and power electronics, including switching-circuit fundamentals, conversion, and electromagnetic force and torque * Practical discussions of power systems, including three-phase transformer connections, synchronous generators, reactive power and power factor correction, and discussions of transient stability Perfect for researchers and industry professionals in the area of power and electric drives, Introduction to the Analysis of Electromechanical Systems will also earn its place in the libraries of senior undergraduate and graduate students and professors in these fields.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 315
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Preface
About the Authors
1 Basic System Analysis
1.1 Introduction
1.2 Phasor Analysis and Power Calculations
1.3 Elementary Magnetic Circuits
1.4 Stationary Coupled Circuits – The Transformer
1.5 Two- and Three-phase Systems
1.6 Problems
2 Fundamentals of Electric Machine Analysis
2.1 Introduction
2.2 Coupled Circuits in Relative Motion
2.3 Electromagnetic Force and Torque
2.4 Winding Configurations
2.5 Rotating Air-gap mmf – Tesla's Rotating Magnetic Field
2.6 Change of Variables
2.7 Stator Voltage Equations in Arbitrary Reference Frame
2.8 Instantaneous and Steady-state Phasors
2.9
P
-pole Machines
2.10 Problems
References
3 Electric Machines
3.1 Introduction
3.2 Direct-current Machine
3.3 Permanent-magnet ac Machine
3.4 Symmetrical Induction Machines
3.5 Problems
References
4 Power Electronics
4.1 Introduction
4.2 Switching-circuit Fundamentals
4.3 dc–dc Conversion
4.4 ac–dc Conversion
4.5 dc–ac Conversion
4.6 Problems
References
Note
5 Electric Drives
5.1 Introduction
5.2 dc Drive
5.3 Brushless dc Drive
5.4 Induction Motor Drive
5.5 Problems
References
6 Power Systems
6.1 Introduction
6.2 Three-phase Transformer Connections
6.3 Synchronous Generator
6.4 Reactive Power and Power-Factor Correction
6.5 Per Unit System
6.6 Discussion of Transient Stability
6.7 Problems
References
Appendix A: Abbreviations, Constants, Conversions, and Identities
A. Constants and Conversion Factors
B. Trigonometric Identities
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Summary of diode trial method for half-wave rectifier with resisti...
Table 4.2 Summary of defining states in the diode trial method for a full-wa...
List of Illustrations
Chapter 1
Figure 1A.1 Phasor diagram.
Figure 1.1 Elementary magnetic circuit with one air gap.
Figure 1.2 Magnetic equivalent circuit for magnetic system shown in Figure 1...
Figure 1.3
λi
characteristic of a magnetically linear system.
Figure 1.4
λ
versus
i
for a magnetically nonlinear magnetic system.
Figure 1.5 Magnetically coupled circuits.
Figure 1.6 Transformer equivalent
T
circuit with winding 1 selected as refer...
Figure 1.7 Elementary two-pole two-phase concentrated stator windings.
Figure 1.8 Elementary two-pole three-phase concentrated stator windings.
Figure 1D.1 Three-phase source connected to symmetrical stator windings.
Figure 1.9 Series-parallel circuit.
Figure 1.10 Waveforms of the source voltages of Figure 1D.1.
Chapter 2
Figure 2.1 Elementary rotational electromechanical device. (a) End view; (b)...
Figure 2A.1 Windings in relative motion (stator housing omitted).
Figure 2.2 Block diagram of possible energy interchange in an elementary ele...
Figure 2.3 Energy balance.
Figure 2.4 Elementary rotational electromechanical device.
Figure 2.5 Concentrated winding. (a) Cutaway view. (b) Cross-sectional view....
Figure 2.6 Air-gap mmf of concentrated stator winding. (a) “Measuring” the a...
Figure 2.7 Applying Ampere's law to Figure 2.6a. (a) Ampere's law. (b) Devel...
Figure 2.8 Elementary sinusoidally distributed winding. (a) Winding connecti...
Figure 2C.1 Elementary two-pole single-phase stator winding over
90°
. (...
Figure 2.9 Elementary two-pole two-phase sinusoidally distributed stator win...
Figure 2.10 Elementary two-pole two-phase sinusoidally distributed stator wi...
Figure 2.11 Tesla's rotating magnetic field
viewed from
−
π
to
π
...
Figure 2.12 Elementary two-pole three-phase sinusoidally distributed stator ...
Figure 2.13 Elementary two-pole two-phase stator with
q
and
d
axes.
Figure 2.14 The
q
and
d
complex plane.
Figure 2.15 Stator winding arrangement of a four-pole, two-phase symmetrical...
Figure 2.16 Mutual coupling between four-pole stator and rotor windings.
Figure 2.17 Two stator windings and two rotor windings.
Figure 2.18 The stator windings and one rotor winding.
Chapter 3
Figure 3.1 A dc machine with parallel armature windings.
Figure 3.2 Same as Figure 3.1 with rotor advanced approximately 22.5° counte...
Figure 3.3 Two-pole 0.1-hp 6-V 12 000-r/min permanent-magnet dc motor.
Figure 3.4 Equivalent circuit of dc machine.
Figure 3.5 Steady-state torque-speed characteristic of a permanent-magnet dc...
Figure 3.6 Two-pole two-phase permanent-magnet ac machine.
Figure 3.7 Four-pole three-phase 28-V
hp permanent-magnet ac machine. (a) D...
Figure 3.8 (a) The
- and
-equivalent permanent-magnet ac machine, (b) quad...
Figure 3C.1 Phasor diagram for steady-state operation of two-phase permanent...
Figure 3C.2 Observing the stator and rotor rotating magnetic fields while ru...
Figure 3.9 Four-pole three-phase 6.5-Hp 460-V severe-duty, squirrel-cage ind...
Figure 3.10 Four-pole two-phase 1/10-Hp 115-V induction motor with reduction...
Figure 3.11 A two-pole, two-phase, symmetrical machine.
Figure 3.12 Two-phase rotating, identical, sinusoidally distributed symmetri...
Figure 3.13 The fictitious
qr
and
dr
windings.
Figure 3.14 Arbitrary reference frame equivalent circuits for a two-phase, s...
Figure 3.15 Free-acceleration characteristics of a two-pole two-phase 5-hp i...
Figure 3.16 Torque versus speed during free acceleration shown in Figure 3.1...
Figure 3.17 Same as Figure 3.15 – stationary reference variables.
Figure 3.18 Same as Figure 3.15 – synchronously rotating reference frame var...
Figure 3.19 Equivalent single-phase circuit for a two-phase symmetrical indu...
Figure 3.20 Steady-state torque-speed characteristics of a symmetrical induc...
Figure 3F.1 Equivalent circuit for steady-state operation of a single-fed in...
Figure 3F.2 Phasor diagram; motor action.
Figure 3F.3 Phasor diagram; generator action.
Chapter 4
Figure 4.1 A generic power-electronic circuit.
Figure 4.2 Ideal switch in (a) open/off and (b) closed/on configurations.
Figure 4.3 General periodic switching function
.
Figure 4.4 Ideal diode circuit representation.
Figure 4A.1 Switching function.
Figure 4.5 Switched (a) inductor and (b) capacitor circuits.
Figure 4.6 Ideal dc–dc conversion circuit.
Figure 4.7 Filtered square wave voltage dc–dc voltage conversion strategy.
Figure 4.8 Buck converter circuit.
Figure 4.9 Buck converter in (a) Configuration A, with
, and (b) Configurat...
Figure 4.10 Buck converter with “large” output capacitance approximated as a...
Figure 4.11 Buck converter inductor current waveform over one-and-a-half swi...
Figure 4.12 Buck converter output stage used to calculate
.
Figure 4.13 Buck converter approximate capacitor current.
Figure 4.14 Circuit waveforms from the 25 W buck converter designed in Examp...
Figure 4.15 Boost converter circuit.
Figure 4.16 Boost converter conversion ratio
for ideal (lossless) and loss...
Figure 4.17 Circuit waveforms from ideal 25 W boost converter with
= 50
μH
...
Figure 4.18 Half-wave rectifier circuit.
Figure 4.19 Voltage waveforms for a half-wave rectifier with resistive load;...
Figure 4.20 Load current and power waveforms for the half-wave rectifier wit...
Figure 4.21 Steady-state voltage waveforms from a half-wave rectifier with p...
Figure 4.22 Source current and power, and load (resistor) current, and power...
Figure 4.23 Steady-state voltage waveforms for a half-wave rectifier with se...
Figure 4.24 Source current and power, and load (resistor) current and power ...
Figure 4.25 Full-wave rectifier circuit.
Figure 4.26 Voltage waveforms from a full-wave rectifier with resistive load...
Figure 4.27 Load current and power waveforms for the full-wave rectifier wit...
Figure 4.28 Voltage waveforms from a full-wave rectifier with parallel
loa...
Figure 4.29 Source current and power, and load (resistor) current and power ...
Figure 4.30 Steady-state voltage waveforms from a full-wave rectifier with s...
Figure 4.31 Source current and power, and load (resistor) current and power ...
Figure 4.32 Full-wave rectifier with
output filter and resistive load.
Figure 4.33
‖
G
filt
‖
and normalized Fourier series harmonic coeff...
Figure 4.34 Steady-state voltage waveforms from a full-wave rectifier with
Figure 4.35 Single-phase dc–ac inverter circuit.
Figure 4.36 Single-phase inverter voltage.
Figure 4.37 Steady-state voltage, current, and power waveforms from a single...
Figure 4.38 Steady-state voltage, current, and power waveforms for the singl...
Figure 4.39 Resistive voltage divider circuit for Problem 1.
Figure 4.40 Symmetric square wave switching function for Problem 2.
Figure 4.41 Triangle waveform for Problem 3.
Figure 4.42 Boost converter circuit with resistive losses in the inductor; f...
Figure 4.43 Diode circuit for Problem 8.
Figure 4.44 Input voltage for diode circuit; for Problem 8.
Chapter 5
Figure 5.1 Two-quadrant chopper drive.
Figure 5.2 Steady-state operation of a two-quadrant dc converter drive.
Figure 5.3 Average-value model of two-quadrant dc converter drive.
Figure 5.4 Starting characteristics of a permanent-magnet dc machine with a ...
Figure 5.5 Torque control.
Figure 5.6 Drive operation during
switching.
Figure 5.7 Inverter-machine drive. (a) Inverter configuration, (b) transisto...
Figure 5.8 Two-pole three-phase permanent-magnet ac machine with sensors.
Figure 5.9 Plots of
as
and
i
as
and the components of
i
as
.
Figure 5.10 Free-acceleration characteristics of a three-phase brushless dc ...
Figure 5.11 Torque-speed characteristics for free acceleration shown in Figu...
Figure 5.12 Dynamic performance during step changes in load torque of a thre...
Figure 5.13 Free-acceleration characteristics of a brushless dc drive with h...
Figure 5.14 Controlling
T
e
with
= 0.
Figure 5.15 Block diagram of torque control of a three-phase permanent-magne...
Figure 5.16 Block diagram depicting field-oriented control principles. Note:...
Figure 5.17 Operation of induction motor drive with field orientation for st...
Figure 5.18 Phasor diagram for Operating Point 1.
Chapter 6
Figure 6.1 A Y–Y three-phase transformer connection.
Figure 6.2 A
Δ – Δ
three-phase transformer conne...
Figure 6.3 A
Y – Δ
three-phase transformer connection...
Figure 6.4 Three-phase ideal transformers for
abc
sequence: (a)
Y – Y
...
Figure 6A.1 Delta- and wye-connected impedances.
Figure 6.5 Three-phase round-rotor synchronous machine.
Figure 6.6 The
and
equivalent circuit for a three-phase round rotor synchr...
Figure 6B.1 Phasor diagram with
r
s
neglected for generator action and with t...
Figure 6.7 (a)
(b)
(c)
.
Figure 6.8 One-line diagram of an elementary power system with a series
R
s
a...
Figure 6.9 One-line diagram of an elementary power system with equivalent pa...
Figure 6.10 Phasor diagram for Figure 6.9.
Figure 6.11 One-line diagram of an elementary power system with power-factor...
Figure 6.12 Phasor diagram for Figure 6.11 with
X
C
=
X
p
.
Figure 6D.1 One phase of an elementary power system for part (a). Voltage le...
Figure 6.13 One-line diagram of system configuration for three-phase fault....
Figure 6.14 Dynamic performance of the steam turbine generator during a thre...
Figure 6.15 Torque versus angle characteristics for the study shown in Figur...
Figure 6.16 Per unit of Figure 6D.1.
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
About the Authors
Begin Reading
Appendix A Abbreviations, Constants, Conversions, and Identities
Index
End User License Agreement
Pages
ii
iii
iv
ix
x
xi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardEkram Hossain, Editor in Chief
Jón Atli Benediktsson
Anjan Bose
David Alan Grier
Elya B. Joffe
Xiaoou Li
Lian Yong
Andreas Molisch
Saeid Nahavandi
Jeffrey Reed
Diomidis Spinellis
Sarah Spurgeon
Ahmet Murat Tekalp
Introduction to the Analysis of Electromechanical Systems
Paul C. Krause
Oleg Wasynczuk
Timothy O'Connell
Copyright © 2022 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data is applied for
Hardback: 9781119829997
Cover Design: Wiley
Preface
In the past, a course in electric machines has generally been the first course in the study of electromechanical systems; however, this area now includes electric machines, power electronics, and electric drives, in addition to electric power systems. This has caused some concern as to the most appropriate first course. Although the analysis of electric machinery is fundamental to electromechanical systems, it may not be the most effective way to introduce this area to the student who is trying to decide on a career path. Some schools have instituted a survey course; however, this often lacks the depth to provide the student an introduction to the analysis common to electromechanical systems or to provide an analytical foundation on which to build in follow-on courses. An introductory course that establishes a useful analytical foundation seems appropriate. This book is an attempt to fulfill this need.
Tesla's rotating magnetic field is the basis of reference frame theory used in the analysis of electromechanical systems, since all known mathematical transformations that are used in this area are contained in the expression of Tesla's rotating magnet field knowledge of this is very important to the engineer working in machines, drives, power and power systems stability. This approach provides an analytical means of positioning the stator and rotor poles on the phasor diagram, thereby providing a straightforward and instructive illustration of the operation of the electromechanical system. Tesla not only invented the ac machine; his work is instrumental in analyzing and visualizing its operation.
Although reference frame theory in Chapter 2 should be covered, the reference frame derivations in later chapters may be deemphasized. Most of the material in Chapter 1 is common to all disciplines. The subsequent chapters contain material fundamental to the areas of electric machines, power electronics, electric drives, and electric power systems. Material that is somewhat more advanced is included in most cases. This allows the instructor to choose between a brief or a more in-depth coverage.
Electric machines are covered inChapters 2 and 3, focusing on the analysis of the dc, round-rotor permanent-magnet ac, and induction machines. Power electronics is covered in Chapter 4. The dc, the brushless dc, and the induction motor drives are each covered in Chapter 5. The synchronous generator and several other aspects of electric power system engineering are covered in Chapter 6. Hopefully, this book will help the student in deciding if he or she would want to pursue advanced study in electromechanical systems.
This book has several uses, it can be used as a one- to three-hour course in schools that offer only one course in the power and drives area. The material can be taught at the sophomore level or first-semester junior level. For schools that have a power program, the course could be used as the introductory course.
Obviously, there are other aspects that would be appropriate for an introductory text; however, the choices we have made seem to be representative at this time. Nevertheless, as the area continues to evolve through the twenty-first century, so must an introductory text.
We would like to acknowledge Dr Brett Robbins of PCKA for developing the drawings and formatting the computer traces and Chris Ramsey who typed the draft of the text.
West Lafayette, Indiana, USAApril 2021
Paul C. Krause, Oleg Wasynczuk, and Timothy O'Connell
About the Authors
PAUL C. KRAUSE, PhD, retired after 39 years as a professor at Purdue University School of Electrical and Computer Engineering. He founded P.C. Krause & Associates in 1983. He is a Life Fellow of IEEE and has authored or co-authored over 100 technical papers and five textbooks on electric machines. He was the 2010 recipient of the IEEE Nikola Tesla Award.
OLEG WASYNCZUK, PhD, is a Professor of Electrical and Computer Engineering at Purdue University. He has authored or co-authored over 100 technical papers and four textbooks on electric machines. He is a Fellow of IEEE and was the 2008 recipient of the IEEE Cyril Veinott Award. He also serves as Chief Technical Officer of P.C. Krause & Associates.
TIMOTHY O'CONNELL, PhD, is a Senior Lead Engineer at P.C. Krause & Associates and an Adjunct Professor of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. He is a Senior Member of IEEE, and an Associate Editor of the IEEE Transactions on Aerospace and Electronic Systems. He has co-authored over 20 technical papers and three textbooks on electric machines and has co-edited a book on electrified aircraft propulsion.
1Basic System Analysis
1.1 Introduction
The twentieth century began with the electric power industry in its infancy; Thomas Edison and Nikola Tesla were locked in battle with Edison advocating direct current (dc) and Tesla alternating current (ac). The century ended with the electric power industry expanding rapidly from the traditional power generation, transmission, and utilization into propulsion of air, ground, and sea transportation. The advent of the computer and the silicon-controlled rectifier in the mid-1900s brought about an expansion of the power area to include the smart-grid, microgrids, efficient and robust electric drives, more electric aircraft, ships, and land vehicles. A growth which is likely to continue into the foreseeable future.
Before the advent of the computer, engineers were essentially limited to steady-state analysis and therefore unable to conveniently deal with the analytical challenges of the expanding power industry. This chapter sets forth some of the basic concepts and analysis tools that are part of the present-day power and electric drives area. Although not inclusive, the material covered in this chapter is representative and common to most disciplines of the power area.
1.2 Phasor Analysis and Power Calculations
Since the early twentieth century, we have lived in an ac world. Thanks to George Westinghouse and Nikola Tesla, power systems are predominately ac; power is generated by large ac generators, transmitted by high-voltage transmission lines, and transformed to a low voltage and distributed to homes and factories. The evolution of the ac power system brought about many engineering challenges and, as we look back, it is difficult to comprehend how these problems were solved without a computer. Even steady-state ac-circuit analysis posed a problem until the early 1900s when Charles Stienmetz, who was a less flamboyant colleague of Edison and Tesla, came up with the concept of what is now known as phasors. Some may consider the phasor a casualty of the computer age along with the slide rule. It is, however, still a very useful means for understanding and portraying the steady-state performance of electric machines, power systems, and electric drives. Moreover, the phasor concept provides a means of visualizing sinusoidal variations from different frames of reference, and in Chapter 2 we will find that the voltage and current phasors combined with Tesla's rotating magnetic field provides a straightforward means of analyzing and portraying the steady-state operation of ac machines.
The phasor can be established by expressing a steady-state sinusoidal variable as
where the a subscript is used here to denote sinusoidal quantities. The sinusoidal variations are expressed as cosines, capital letters are used to denote steady-state quantities, and Fp is the peak value of the sinusoidal variation. Generally, F or f represents voltage (V or ) or current (I or i) in circuit analysis, but it could be any sinusoidal variable. For steady-state conditions, θef may be written as
where ωe is the electrical angular velocity in rad/s (2π times the frequency) and θef(0) is the time-zero position of the electrical variable. Substituting (1.2) into (1.1) yields
Now, Euler's identity is
and since we are expressing the sinusoidal variation as a cosine, (1.3) may be written as
where Re is shorthand for the “real part of.” Equations (1.3) and (1.5) are equivalent. Let us rewrite (1.5) as
We need to take a moment to define what is referred to as the root-mean-square (rms) of a sinusoidal variation. In particular, the rms value is defined as
where F is the rms value of Fa(t)and T is the period of the sinusoidal variation. It is left to the reader to show that the rms value of (1.3) is . Therefore, we can express (1.6) as
By definition, the phasor representing Fa(t), which is denoted with a raised tilde, is
which is a complex number. The reason for using the rms value as the magnitude of the phasor will be addressed later in this section. Equation (1.6) may now be written as
A shorthand notation for (1.9) is
Equation (1.11) is commonly referred to as the polar form of the phasor. The Cartesian form is
When using phasors to calculate steady-state voltages and currents, we think of the phasors as being stationary at t = 0; however, we know from (1.10) that a phasor is related to the instantaneous value of the sinusoidal quantity it represents. In other words, the real projection of the phasor rotating counterclockwise at ωe is the instantaneous value of . Thus, with θef(0) = 0 in (1.3)
the phasor representing (1.13) is
For
the phasor is
We will use degrees and radians interchangeably when expressing phasors. Although there are several ways to arrive at (1.16) from (1.15), it is helpful to ask yourself where must the rotating phasor be positioned at time zero so that, when it rotates counterclockwise at ωe, its real projection is ? It follows that a phasor of amplitude F positioned at 90° represents .
In other words, we are viewing a sinusoidal variation as the real projection in the real-imaginary plane of a rotating line equal in magnitude to the positive peak value of the variation and rotating at the electrical angular velocity of the sinusoidal variation. Since we are dealing with a steady-state variation, we can stop the rotation at any time and view it as a fixed line, but knowing full well that it, in fact, represents a sinusoidal variations and to represent the sinusoidal variation we must rotate it counterclockwise at ωe and take the real projection. Please understand that if we ran at ωe in unison with the rotating line it would appear as a constant to us. Therefore, this is no different than stopping the phasor at some arbitrary time zero; but realizing that it actually represents a sinusoidal variation. We'll talk more about this important aspect as we go along. See Example 1A.
To show the facility of the phasor in the analysis of steady-state performance of ac circuits and devices, it is useful to consider a series circuit consisting of a resistance, an inductance L and a capacitance C. Thus, using uppercase letters to indicate steady-state variables
Throughout the text, we will use either R or r to represent resistance. For steady-state operation, let
where we have dropped the functional notation, and the subscript a helps to distinguish the instantaneous value from the rms value of the steady-state variables. The steady-state voltage equation may be obtained by substituting (1.18) and (1.19) into (1.17), whereupon we can write
The second term on the right-hand side of (1.20), which is , can be written
Since , from (1.21), we can write
Since , (1.22) may be written
If we follow a similar procedure, we can show that
Differentiation of a steady-state sinusoidal variable rotates the phasor counterclockwise by or j; integration rotates the phasor clockwise by or −j.
The steady-state voltage equation given by (1.20) can now be written in phasor form as
We can express (1.25) compactly as
where the impedance, Z, is a complex number; it is not a phasor. It may be expressed as
where XL = ωeL is the inductive reactance and is the capacitive reactance. We should be careful here. Some prefer to write (1.27) as R + jX where X is XL + XC and let XC be negative. This is essentially a matter of choice and does not change the end result. We will deal primarily with XL and not XC, therefore, this will have little impact on our work; nevertheless, since some authors will use a negative XC we should make the reader aware of this difference.
It is appropriate to discuss the notation that will be used throughout the text. When an equation is written with the variables in lowercase letters it is valid for transient and steady state. If the variables are written with uppercase letters as in (1.17), the equation is a function of time and valid for instantaneous steady-state conditions. Equations (1.26) and (1.27) are phasor equations representing steady-state sinusoidal variables and are written in uppercase letters with an over tilde.
1.2.1 Power and Reactive Power
The instantaneous steady-state power is
where V and I are rms values. After some manipulation, we can write (1.28) as
The instantaneous steady-state power given by (1.29) varies about an average value at a frequency of 2ωe. That is, the second term of (1.29) has a zero average value and the average power Pave may be written
where and are V and I, respectively, which are the magnitudes of the phasors (rms value), is referred to as the power factor angleϕpf, and cos[] is the power factor. Power is in watts. If current is assumed positive in the direction of voltage drop, then (1.30) is positive if power is consumed and negative if power is generated. It is interesting to point out that in going from (1.28) to (1.29), the coefficient of the two right-hand terms is or one-half the product of the peak values of the sinusoidal variables. Therefore, it was considered more convenient to use the rms values for the phasors, whereupon average steady-state power could be calculated by the product of the magnitude of the voltage and current phasors as given by (1.30).
The reactive power is defined as
The units of Q are var (volt–ampere reactive). An inductance is said to absorb reactive power where the current lags the voltage by 90° and Q is positive. In the case of a capacitor, where the current leads the voltage by 90°, Q is supplied and is negative. Actually, Q is a measure of the interchange of energy stored in the electric (capacitor) and magnetic (inductor) fields. However, unlike instantaneous real power, the average value of instantaneous reactive power is zero. We'll talk more about reactive power later.
Example 1APhasor analysis
The parameters of a series RLC circuit are R = 6 Ω,L = 20 mH,C = 1 × 103 μF. The 60-Hz applied voltage is Va = 155.6 cos ωet. Calculate , Pave, Q and draw the phasor diagram and the sinusoidal variations as viewed running at counterclockwise with the phasor representing maximum . From the expression of Va
Now, ωe = 2πf = 2π × 60 = 377 rad/s and
where
The phasor diagram is shown in Figure 1A.1.
SP1.2.1
Express the instantaneous steady-state power for
Example 1A
. [Substitute into
(1.29)
]
SP1.2.2
Redraw the phasor diagram shown in
Figure 1A.1
showing
and
as individual voltages. [Show
and then from the terminus of
show
]
Figure 1A.1 Phasor diagram.
SP1.2.3
We know that
does
? [Yes]
SP1.2.4
If
and
in the direction of the voltage drop, calculate
Z
and
P
ave
. Is power generated or consumed? [
(−1 +
j
0) Ω
, 1 W, generated]
SP1.2.5
Express the instantaneous power for 60 Hz voltage,
, applied to a resistive circuit,
. [
1 + cos 754
t
]
SP1.2.6
Repeat
SP1.2.5
for (a) an inductance,
and (b) a capacitance,
. [
(a)
I
L
cos(754
t
− 90°), (b)
I
C
cos(754
t
+ 90°)
]
1.3 Elementary Magnetic Circuits
Electric machines and transformers, which are the backbone of the power industry, are electromagnetic systems. Therefore, magnetic circuits and magnetic coupling play a major role in power and drives systems, and it is necessary to establish the principles of magnetic systems sufficiently to convey the basic operation of the electromagnetic devices considered in later chapters. We will attempt to do this without becoming too involved.
An elementary magnetic circuit is shown in Figure 1.1. It consists of a ferromagnetic member (core) with a coil of wire of N turns wound on it and an air gap of length x. The ferromagnetic member could be iron, nickel, cobalt, or steel, for example. The voltage equation of the electric circuit may be written
where r is the total resistance of the circuit, is the source voltage, e is the voltage induced in the coil according to Faraday's law, and i is the current flowing in the circuit. The current flowing through the coil causes a magnetomotive force (mmf), which produces flux in the magnetic circuit denoted as Φm and Φl in Figure 1.1, much as an electromotive force (emf) or source voltage produces current in an electric circuit.
Figure 1.1 Elementary magnetic circuit with one air gap.
There are arrows associated with the dashed lines representing the flux paths in Figure 1.1. These arrows indicate the assumed positive direction of flux which is determined from the assumed positive direction of current by the so-called “right-hand” rule. If you grasp the coil with your right hand with your fingers in the assumed direction of positive current flow around the coil, your thumb will point in the direction of positive flux. Or, imagine grasping a turn of the coil with thumb in the assumed direction of positive current. If you ungrasp your fingers, they will point in the direction of positive flux.
The total flux, Φ, that travels through (links) all of the turns N is
where Φl is the equivalent flux that links all the turns of the coil but does not traverse the ferromagnetic member and
