Powe system analysis by grainger
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POWER SYSTEM ANALYSIS
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POWER SYSTEM ANALYSIS
Professor,' Department
John J. Grainger
of EleClrica/ and Compl/{er Engineering
.Yonh larolina S{o{e Uniccrsi{y
WUliam
D. Stevenson, Jr.
Lale Professor uf E/ec/m:o/ Engineering
Nonh Camfit/(J .\{(J/c Ulllcersi{y
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POWER SYSTEM ANALYSIS
International Editions 1994
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'This book was set in Times Roman by Science Typographers Inc.
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the p roduc t i on supervisor was Elizabeth J. Strange.
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,
Library of Congress Cataloging-in-Publication Data
Grainger, John J
Power system analysis / John J. Grainger, William D. Stevenson.
p.
cm.
Based on: Elements of power system anal ys i s by William D. Stevenson.
.
,
0-07-061293-5
1. Electric power di stri b ut i on
Includes index.
ISBN
Stevenson, W illi am D.
power system analys i s
I.
TK3001.G73
621.319-dc20
.
1994
2. Electric power systems.
II. Stevenson, William D. Elements of
.
III. Title.
When ordering this title, use ISBN 0-07-113338-0
Printed i1' Singapore
93-39219
To THE MEMORY OF
William D. Stevenson, Jr.
1912-1988
True friend and colleague
ABOUT THE AUTHORS
1" '-
1
John J . Grainger is P rofes sor of E l e ct r i c a l and Computer Engineering a t North
Carolina State Unversity.He is a graduate of the ;'\ ati o n a l University of Ireland
and re c e i ve d his M.S.E.E. and Ph�D. 'degrees at - the' L:"nivcrsity of Wisconsin
G ra inger is the founding Director of the El e c t ric Power Research
Cente r (It North Carolin,-l State University. a joint uni\'er�ity/industry coopera
tive research center in electric power systems engineering, He l e ads the
Center's major research programs in transmission and distribution systems
planning, design, (}uto01ation, and c on t rol areas. as well as power system
Madison.
Dr.
dynamics_
Madison, The I llinois Institute of Technology, 't>larquette University, and North
Profes�or
Grainger has
aho
«llight
at
the
of
University
Wisconsin
ity Supply Board of Ireland; Commonwealth Edison Company, Chicago; \Vis
consin Electric Po\vcr Company, MilwaUKee; and Carolina Pmver & Light
Company, Raleigh. Dr. Grainger is an active consullant with t h e Pacific Gas and
El e ct ric Company, San Francisco; Southern California Edison Company, Rose
mead; a n d mimy other power industry organizJtions. His educational ane!
Carolina State University. His industrial experience has been with the Electric
Am eri c a n Society of Engineering Education, the American Power Conference,
technical
involvements
erRED, and CIGRE.
include
the
IEEE
Povv'cr
Engineering Society,
The
Dr. Grainger is the aut h or of numerous papers in the IEEE Power
Engineering So c i e ty ' s Transactiotls and was recognized by the IEEE Transmis
s ion and Distribution Committee for the 1985 Prize Paper Award.
In 1984, P ro fessor Grainger was chosen by the Edison Electric Institute
for the EEl Power Engineering Ed u ca tor Award.
William D. Stevenson, Jr. (deceased) was a professor and the As oci at e Head of
the Electrical Engineering De p a rt m e n t of N ort h Carolina State University. A
Fellow of the I n sti tu te of Electrical and Electronics E ng i n ee r s , he worked in
private industry and taught at both Clemson U ni v e rs i ty and P r inc e t on Univer
s
en?ineering for t h e McGraw-HilI Encyc!opedin of Science and Technology, He
was the recipient of several teaching and professional awards,
sity.
Dr.
Stevenson also served
as a consulting editor in electrical power
CONTEN'fS
Preface
1
Basic Concepts
1.1
1.2
1.3
1.4
1.6
1.5
1.7
1.9
1.8
1.10
1.11
I.J2
1.13
1.14
US
2
Introduction
Single-Subscript NO{dtioll
Do ub le -Sub scr i pt N o ta tio n
Power in Single-Phase AC Circuits
Complex Power
The Power Tr i a n gle
Direction of Pow er flow
Vol tag e anc1 Current in BalanccJ Three-Phase Circuits
Power in Balllnccu Three-Phase Circuits
Per-Unit Qualltities
C ha n ging the B,lse of Per-Unit OU(lntities
Node Equations
The S i ngle - L i n e or One-Line Diagram
Impedance and Reactance Diag r ams
Summary
Proble ms
Transformers
2.1 The Ideal Transformer
2.2 Magnetically Coupled Coils
2.3 The Equivalent Circuit of a Single-Phase Transformer
2.4 Per-Unit Impedances in Single-Phase Transformer Circuits
2.5 Three-Phase Transformers
2.6 Three-Phase Transformers: Phase Shift and Equivalent Circuits
2.7 The Autotransformer
2.8 Per-Unit Impedances of Three-Winding Transformers
2.9 Tap-Changing and Regulating Transformers
2.10 The Advantages of Per-Unit Computations
2.11 Summary
P ro bl ems
1
.-,
.)
4
1
5
0
10
14
1J
24
25
29
30
34
36
17
37
41
41
46
51
56
59
64
71
72
76
80
82
82
XI
xii
3
CONTENTS
The Synchronous Machine
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
4
5
Description of the Synchronous Machine
Three-Phase Generation
Synchronous Reactance and Equivalent Circuits
Real and Reactive Power Control
Loading Capabi lity Diagram
Thc Two-Axis Machinc Modcl
Voltagc Equations: Sal icnt-Polc Machine
Transient and Subtransicnt Effects
Short-Circuit Currents
Summary
Problcms
Series Impedance of Transmission Lines
4.1
Typ es of Conductors
4.2
Resistance
4.3
Tab u lated Resistance Values
4.4
Inductance of a Conductor Due to Internal Flux
4.5
Flux: Linkages between Two Points External t o an Isolated
Conductor
4.6
Inductance of a Single-Phase Two-Wire Line
Flux Linkages of One Cond u ctor in a Group
4.7
4.8
Inductance of Composite-Conductor Lines
4.9
The Use of Tables
4.10 Inductance of Three-Phase Lines with Equilateral Spacing
4.11 Inductance of Three-Phase Lines with Unsymmetrical Spacing
4.12 Inductance Calculations for Bundled Conductors
4.13 Summary
Problems
136
141
142
143
146
146
149
151
153
155
159
161
161
164
166
167
Capacitance of Transmission Lines
Electric Field of a Lo ng Straight Conductor
5.1
5.2
The Potential Differen ce between Two Points Due to a Charge
5.3
Capacitance of a Two-Wire Line
Capacitance of a Three-Phase Line with Equilateral Spacing
5.4
Capacitance of a Three-Phase Line with Unsymmetrical Spacing
5.5
Effect of Earth on the Capacitance of Three-Phase Transmission
5.6
Lines
170
171
172
5.7
186
,
5.8
5.9
6
87
88
91
100
105
110
117
123
127
132
136
Capacitance Calculations for Bundled Conductors
Parallel-Circuit Three-Phase Lines
Summary
Problems
173
1 77
180
183
188
190
191
Current and Voltage Relations on a Transmission
Line
6. 1
62
.
Representation of Lines
The Short Transmission Line
193
195
196
CONTENTS
200
6.4
6.3
The Medium-Length Line
6.5
The Long Transmission Line: lnterpretation of the Equations
6.6
6.7
6.8
6.9
Equ a t i on s
The Long Transmission Line: Solution of the D i ffere ntial
The Long Transmission Line: Hyperbolic Form of the Equations
207
The Equivalent Circuit of a Long Line
212
Power Flow through a Transmission Line
215
Reactive Compensation of Transmission Lines
218
221
222
6.12
Transient Analysis: Traveling Waves
Transient Analysis: Rct1ections
226
233
231
Direct-Current Transmission
Summary
233
Problems
The Admittance Model and Netw o r k
Node
7.1
Branch and
7.2
Mutually Coupled Branches in Y hu,
An Equivalent AdmittZlncc i'-Jc(wnrk
7.�
7.4
Admittances
Calculations
238
239
245
251
,-
-))
7.6
Modifie,l(ion o r VioL"
The Network Incidence Matri\ and Y QUI
The Method of Successive Elimination
274
7.0
Triangular F,lctoriz(ltioll
Summar),
280
7.5
7.7
7.8
7.10
Node Elimination
Sparsity
ancl
(Kron
Reduction)
Ncar-Optilll(li Or(lcring
Prohlcms
The Impedance Model and Network Calcu lations
8.2
8.1
8.4
8.3
The Bus Admittance and Impedance Matrices
Thcvcnin's Thcorcm and Zbus
Mpclification of (Ill Existing Zhu:>
8.5
Direct Determination of Zou:>
Calculation of Zous Elements from YbU5
8.7
Mutually Coupled
8.8
Summary
8.6
Power Invariant
Transformations
B r an ch e s
in Zbus
Power-Flow Solutions
9.1 The Power-Flow Problem
9.3
The Gauss-Seidel Method
9.5
Power-Flow
9.2
9.4
9.6
257
263
271
279
280
283
284
287
30 1
294
306
310
316
324
324
Problems
9
205
Transmission-Line Transients
6.13
6.14
8
202
6.1 0
6. 1 1
7
xiii
329
329
335
The Newton-Raphson Method
F
342
347
The Newton-Raphson Power - l o w Solution
356
Regulating Transformers
361
S tu d i e s
in System Design and Operation
xiv
CONTENTS
9.7
9.8
10
11
The Decoupled Power-Flow Method
S ummary
Problems
Symmetrical Faults
10.1
10.2
10.3
10.4
10.5
10.6
13
380
Transients in RL Series Circuits
I n ternal Voltages of Loaded Machines under Fault Conditions
Fault Calculations Using Zbus
Fault Calculations Using ZhU5 Equivalent Circuits
The Selection of Ci rcu it Breakers
Summary
Problems
Symmetrical Components and Sequence Networks
11.1 Synt hesis of U nsymmetrical P hasors from Thei r Symmetrical
Components
11.2 The Symmetrical Components o f U nsymmetrical P h asors
11.3 Symmetrical Y and!:::. Circuits
11.4 Power in Terms of Symmet rical Com ponents
11.5 Sequence Circuits of and tl I mpedances
11.6 Sequence Circui ts of a Symmetrical Transmission Lin e
11.7 Seq uence Circuits of the Synchronous M achine
1 1 8 Sequence Circuits of -tl Transformers
11.9 Unsymmetrical Series Impedances
11.10 Sequence Networks
11.11 Summary
Problems
Y
Y
.
12
368
374
376
Unsymmetrical Faults
The Transmission-Loss Equation
Interpre tation of Transformation C
13.5 Classical Economic Dispatch with Losses
13.6 Automatic Generation Con t rol
13.7 Unit Commitment
13.4
An
402
411
412
416
41 7
418
422
427
429
435
442
449
459
461
467
467
512
523
527
Economic Operation of Power Systems
13.3
]95
470
470
482
488
494
500
12.1 Unsymmetrical Faults on Power Systems
12.2 Single Line-to-Ground Faults
1 2.3 Line-to-Line Faults
12.4 Double Line-to-Ground Fau l ts
12.5 Demonstration Problems
12.6 Open-Conductor Faults
12.7 Summary
Problems
13.1 D istribution of Load between U n i ts within
13.2 Distribution of Load between P l an ts
381
383
390
a
Plant
531
532
540
543
552
555
562
572
13.8
13.9
14
14.2
578
586
Problems
587
591
14.1
Adding and Removing Multiple Lines
14.3
Analysis of Single Contingencies
611
Analysis of Multiple Contingencies
620
14.5
Contingency Analysis by dc Model
626
System Reduction for Contingency and Fault Studies
Summary
636
Problems
636
14.6
14.7
Piecewise Solution of Interconnected Systems
State Estimation of Power Systems
592
601
628
641
15.1
The Method of Least SqU;:HCS
642
1 5.3
Test for Bad Data
655
Power System State Estimation
664
Sumnnry
688
15.2
15.4
1 is
1 5.6
16
Commitment Problem
xv
Summary
Zbus Methods in Contingency Analysis
14.4
15
So lv ing the Unit
CONTENTS
Statistics, Errors. ,wd Estimates
Thc Structure «(Ill! r:orm;ltion of H \
Problems
Power System Stability
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
Rotor Dynamics ;lnd the Swing Equation
Further Considerations of the S wing Equation
Thc Power-Angle Equ(ltion
Synchronizing rowcl Cocflicicnls
EClual-Area Criterion of S t abili ty
Further Applications of the Equal-Area Criterion
Multimachine Stability Studies: Classical Representatio!1
Step-by-Slep Solution of the Swing CunT
16.10 Computer Programs for Transient Stability Studies
16.11 Factors Affecting Transient S t abi li ty
16.12 Summary
Problems
A.i
A.2
Quantities
Distributed Windings of the Synchronous Machine
P-Transformation of Stator
Appendix B
()77
687
69S
The Stability Problem
Appendix A
650
695
60S
702
707
714
717
724
727
734
741
743
746
745
748
754
763
766
B.l
Sparsity and Near-Optimal Ordering
766
B.2
Sparsity of the Jacobian
771
Index
777
PREFACE
Elemen ts
the long-standing McGraw-Hill textbook by Professor William D.
This book embodies the principles and objectives of
Analysis,
of Power System
Stevenson, Jr., who was for many years my friend and colleague emeritus at
North Carolina State University. Sadly, Professor Stevenson passed away on
great efforts to continue the student -oriented style and format of his own famous
May
1, 1988, shortly after planning this joint venture. In
students for a considerable number of years.
my writing I have made
textbook that has guided the education of numerous power system engineering
The aim here is to instill confidence and understanding of those concepts
of power system analysis that are likely to to
be
encountered in the study and
practice of electric power engineering. The presentation is tutorial with empha
sis on a thorough understanding of fundamentals and underlying principles. The
approach and level of treatment are directed toward the senior undergraduate
and universities. The coverage, however, is quite comprehensive and spans a
and first-year graduate student of electrical engineering at technical colleges
wide range of topics commonly encountered in electric power system engineer
ing practice. In this regard, electric utility and other industry-based engineers
will find this textbook of much benefit in their everyday work.
Modern power systems have grown larger and more geographically expan
sive with many interconnections between neighboring systems. Proper planni ng,
operation, and control of such large-scale systems require advanced computer
based techniques, many of which are explained in a tutorial manner by means of
numerical examples throughout this book. The senior undergraduate engineer
ing student about to embark on a career in the electric power industry will most
certainly benefit from the exposure to these techniques, which are presented
here in the detail appropriate to an i ntroductory level. Lik�wise, electri c utility
engineers, even those with a previous course in power system analysis, may find
that the explanations of these commonly used analytic techniques more ade
quately prepare them to move beyond routine work.
Power System Analysis
can serve as a basis for two semesters of undergrad
uate study or for first-semester graduate study. The wide range of topics
facilitates versatile selection of chapters and sections for completion in the
semester or quarter time frame. Familiarity with the basic principles of electric
XVII
xviii
PREFACE
circuits, p h asor a lgebra, a n d t h e rudiments of d ifferential equations is assumed.
The reader should a lso h ave some u n d e rstanding of matrix operations and
notation a s they a re used t hroughout the t ext. The coverage includes newer
. /
tOPlCS suc h a s state estima tion a n d unit commitment, as wel l as more detailed
presen tations and newer approaches to traditional s u bj e ct s such as transform
ers, synchronous machines, and n etwork fau lts. Where appropriate, summary
tables allow quick reference of i mporta n t i d e as. Basic concepts of computer
based a lgorithms are presented so that students can implement their own
compu ter p rograms.
Chapters 2 and 3 a re devo ted to the t r a n sfo rm e r and sync hrono us ma
chine, respectively, and should comple ment m a te r i al covered in other electric
circuits and machines courses. Transmiss ion-line p a r a m e t e rs and c alc u l at i on s
are studied in Chapters 4 t h ro u gh 6. N e tw o rk m ode l s based on the adm i t t ance
and impedance representations a re developed i n Ch ap t er s -; and 8, which also
intro duce gaussian elimination, Kron reduction, triangular factorization, a nd the
Zbus building a lgorith m . The power-flow p roblem, sym m e trical com p onents, and
unsymmetrical fau lts are presented in Chapters 9 thro u g h 12: \\h e reas Chap ter
13 p rovides a self-contained d evelopmen t of economic dispatch and the basics
of u n i t commitment. Con t i ngency analysis a n d external equivalents are the
subjects of Chapter 14. Power system state estima t ion is covered in Chapter 15,
whi l e power system stability is introduced i n Cha p t e r 16. Homework problems
and exercises a re provi ded a t the end of each c h apter.
I am most p leased to acknowledge the assistance given to me by a number
of people with whom I h ave been associated within the Department of Electri
cal a n d Computer Engineering at North Carolina State University. Dr. Stan
S. H. Lee , my colleague a n d friend for m any yea rs, has always willingly g iven his
time a n d effort when I needed help, advice, or suggestions at the various stages
of d evelopment of this textbook. A n u mber of the homework problems a n d
solutions were contributed b y h i m and b y D r. Gamini Wickramasekara, o n e o f
m y former gradu ate s t u d e n t s at No rt h Carolina S t a te University. Dr. Michael 1.
Gorm a n , another of my recent gradu;lle sLucJents, gave ullstintingly or himselr in
d eveloping t he com p u ter-based figures and solu tions fo r many of the n u me r ic a l
examples throughou t the various chapters of the text . Mr. W. Ad r i a n Buie, a
recen t graduate of the Department of E l ectrical and Computer Engineering,
-un dertook the chall enge of committing the e ntire textbook to the computer and
p roduced a truly professional m a n uscript; i n this regard , Mr. Barry W . Tyndall
was also most he l pfu l in t h e early stages of the w r it ing My loyal secretary, Mrs.
Paulette Cannady-Kea, has a lways enth usiastically assisted in t he overall pro
ject. I am greatly indebted a n d extremely grateful to each and a l l of these
individuals for their generous efforts.
Also within the Department of Electrical and Computer Engineering at
North Carol ina State Univers i ty, the successive le adership of Dr. Larry K.
Monteith (now Chancellor of the University), Dr. Nino A. Mas nari (now
Director of the Engineering R esearch Center for Advanced Electronic M ateri
als Process i n g), and Dr. Ralph K. Cavin III ( p re s entl y Head of the Depart�ent),
.
with my faculty colleagues, particularly Dr. AJ fred 1.
environment of s u p po r t that I am very ple a sed to record.
xix
PREFACE
provided
an
source of patient understanding and encouragement during the preparation
of
along
Goetze,
The members of my family, especially my wife, Barbara, have been a great
this book. I ask
sincere t hanks.
each
of
them,
and
my frie nd Anne Stevenson, to accept my
would like to thank the following reviewers for
h e l p fu l comments and suggestions: Vernon D. Albertson, U niversity of
Mi n nesota; David R. Brown, University of Texas at Austin; Mehdi
Etezadi-Amoli, Un i ve r s i ty of Nevada. Reno; W. Mack Grady, l)nivcrsity of
McGraw-Hill and
I
many
their
Texas at Austin; Clifford Grigg, Rose-Hulman Institute of Technology; William
H. Kersting, Ne"v Mexico State University;
of
Kenneth KsuempeI, Iowa State
G. Phadke, Virginia Polytechnic Institute and State Uni\'ersily; B. Don Russell,
Texas A & M University; Peter W. Sauer, Uni\'ersit) of Illinois, Urbana
Champaign;
University;
Mangalore
A.
Pai, Unin'rsiry
Illinois. Urbana-Champaign; Arlin
.
John 1. Grainger
CHAPTER
1
BASIC
CONCEPTS
Normal and abnormal conditions of op e rati o n of the s ys tem are the concern of
the power system engineer who mu s t be very fa mili ar with steady-state ac
circuits, part icu l arly three-phase circuits. The purpos e of th i s chapter is to
review a few of the fund3mental i de Cl s o f s u ch circuits; t o es t a bl i sh the notation
used throughout the book; and to introduce the expression o f va lues of vol t age,
current, imp e da nce a nd power i n per u n i t . Moder n power system ana lysis rel ies
a lm os t exclusively on n o da l n erwork repres e n tatio n wh ich is i n t ro d uce d in the
for m o f the b u s admittance a nd the b u s impedclnce matrices.
,
1.1
INTRODUCTION
The waveform of voltage at the b u s es of a power system ca n be assumed to be
purely sinusoidal and of constant frequency. In developing most of the theory in
this book, we are concerned with the phasor representations of sinusoidal
volt?ges and currents a n d use the capital letters V and I to i ndicate t h ese
phasors (with appropriate subscripts where necessary). Vertical bars e nclosin g V
and I, that is, I VI and III, designate the magnitudes of the phasors. Magnitude s
of complex numbers such as impedance Z and admittance Ya re a lso indic a te d
by vertical bars. Lowercase letters generally indicate in s t antane o u s v a lues.
Where a generated voltage [electromotive force (emf)] is specified, the lett e r E
rather than V is often used for voltage to emphasize the fact that an e m f r a t her
tha n a general p o te n ti a l difference between two points is being considered.
1
2
CHAPTER 1
BASIC CONCEPTS
If a voltage and a current are expressed as fu nctions of time, such as
v = 141 .4 cos ( w t + 30° )
i = 7 .07 cos w t
and
their m ax i m u m values are obviously Vmax = 1 41.4 V and I m = 7.07 A, respec
t ively. Vertical bars are not needed when t he subscript max with V and I is used
to indicate
value. The term magnitude refe rs to root-mean-squ a re (or
rms) values, which equal the maximum values divided by Ii. Thus, for the
above expressions for v and i
ax
maximum
I VI = 1 00 V
III
and
=
SA
These are the values read b y the ordinary types of voltmeters a n d ammeters.
Another n a me for the rms value is the effective value. The average p ower
expended i n a resistor by a current of magnitude III is 1 I12R.
To exp ress these quanti ties as p hasors, we e mploy Euler's i d e n ti ty SiG =
cos e + j sin e, which gives
cos e
=
Re{8JO} = Re{cos e + j sin e}
(1 . 1 )
where Re m e ans the real part of. We now write
If t h e c u r rent is
the referen ce
I
=
p hasor, we h ave
58 jO°
=
5LQ:
=
�
5 + j0 A
a n d t he voltage which leads the refe rence phasor by 30°
°
V = 100£J30
=
1 00
IS
= 86.6 + j50 V
Of course, we might not c hoose as the refe rence phasor either the vol tage
or the curre n t whose instantaneous expressions are v and i, respectively, i n
w hich case their p hasor expressions would i nvolve other angles.
In circuit d iagrams it is often most convenient to use polarity marks in the
form of p lus and minus signs to i ndicate the terminal assumed positive when
specifying voltage. An arrow on t he diagram specifies the di rection assumed
positive for t h e fl ow of current. In t h e single-phase equivalent of a three-p hase
circuit single-subscript notation is usually sufficient, but double-subscrip t ,nota
tion is u su a l ly simpler when deal i ng with all three p hases.
1.2
1.2
SINGLE..SUBSCRIPT NOTATION
SINGLE-SUBSCRIPT NOTATION
3
Eg, and the voltage between nodes a and 0 is identified as v,. The current in
the circuit is lL and the voltage across ZL is VL. To specify these voltages as
phasors, however, the + and - markings, called polarity marks, on the diagram
Figure
1.1
shows an ac circuit with an emf represented by a circle. The emf is
and an arrow for current direction are necessary.
terminal marked - for half a cycle of voltage and is negative with respect to the
In an ac circuit the terminal marked + is positive with respect to the
other terminal during the next half cycle. We mark the terminals to enable us to
say that the voltage between the terminals is positive at any instant when the
is positive when the
terminal marked plus'is actually at a higher potential than the terminal marked
tenninal marked plus is actually at a higher porential th a n the terminal marked
minus. For instance, in Fig. 1.1 the instantaneous voltage
V
,
with a negative sign. During the next half cycle the posi t ive ly marked terminal is
is negative. Some authors use an arrow but must
specify whether the arrow points toward the te rm i na l which would be labeled
plus or toward the terminal which would be labeled m i n us In the convention
actua)ly negative, and
v,
The current arrow p e r form s
similar function. The sub sc r ipt, in this case
L, is not necessary llnles� other currents arc �rcsenl. Obviollsly, the actual
direction of current flow in an ac circuit reverses each half cycle . The arrow
points in the direction which is to be called positive for current. \Vhen the
current is actually flowing in the dircciion op posit e to that of the arrow, the
current is negative. The phasor ,'urrenl is
described above.
1
I.
a
vI -vf.
( 1.2)
= -z,.,j .-
( 1.3)
and
Since certain nodes ill t he circuit ha ve been assigned letters, the voltages
voltages are expressed with respect to a reference node.
In Fig. 1.1 the
may be designated by the single-letter subscripts identifying the node whose
instantaneous voltage
a
Va
and the phasor voltage
with respect to the reference node
0,
and
V
a
v"
express the voltage of node
is positive when
a
is at a higher
FIGURE 1.1
An ac circuit with emf E� and load impedance Z[.
4
CHA PTER 1
B AS I C CONCEPTS
potential than o. T hus,
1.3
DOUBLE-SUBSCRIPT NOTATION
T h e use of polarity marks fo r vol tages a n d d i rect ion a r rows for cu rre n ts can be
avoided by double-subscript notation. The u n dersta n d ing of t h ree-phase c i rcuits
is considerably cla ri fied by adop ting a syst e m of double subscripts. The conven
t ion to be fol lowed i s quite s i mp le.
In denoting a current the order of the s ubscripts assigned to the symbol
for current d efines the d irection of the flow of current when the curre n t is
considered to b e positive. In Fig. 1 . 1 the arrow pointing from a to b defi nes the
positive d i rection for the curren tIL associated with t he arrow. The i ns ta nta
neous c urre nt i L is positive when the current is actua ll y in the dire ction from a
to b, a n d in double-subscript n otation this current is iab' The cur rent iab is
equal to iba'
In double-subscript notation the letter subs cripts on a vol tage ind icate the
n odes of the circ uit between which the voltage e xists. We shall fol low t he
convention which says that the fi rst s ubsc ri pt d e n otes the voltage of t h a t n ode
with respect to the node identified by the secon d subscript. This m e a ns tha t the
instantaneous voltage Vab across Z A of the circu i t of Fig. 1 . 1 is the voltage of
node a with r espect to node b a n d t h a t vab is positive du ring that half cycle
when a is a t a higher potenti al than b. The cor responding phasor voltage is V,,/J'
which is rel ated to the cu rrent fal) flowi ng from node a to node b by
-
( 1 .4 )
and
where 2/1 is the complex impedance (a l so cal l e d 2(/h) and Y/1 = 1/2/1 is t he
c omplex admittance (also called Y:II).
Reversing the order of the subscripts of either a current or a voltage gives
a cur rent or a voltage 1 800 out of p hase with the original; that is,
vbu
=
Vab c j 1800
=
Vab
/ 1800
=
-
Vab
The relation of single- a n d doub le- subscript notat ion for the ci rcu i t of Fig.
1.1 is summ arized as follows:
