Jingde Gao
Linzheng Zhang
Xiangheng Wang
AC Machine Systems
Mathematical Model and Parameters, Analysis, and System
Performance
Jingde Gao
Linzheng Zhang
Xiangheng Wang
AC Machine Systems
Mathematical Model and Parameters, Analysis, and
System Performance
With 158 figures
Authors
Jingde Gao
Dept. of Electrical Engineering
Tsinghua University
100084, Beijing, China
Linzheng Zhang
Dept. of Electrical Engineering
Tsinghua University
100084, Beijing, China
E-mail: LZZhang@umac.mo
Xiangheng Wang
Tsinghua University
100084, Beijing, China
E-mail: wangxh@mail.tsinghua.edu.cn
ISBN 978-7-302-19342-5
Tsinghua University Press, Beijing
ISBN 978-3-642-01152-8
e-ISBN 978-3-642-01153-5
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: pending
© Tsinghua University Press, Beijing and Springer-Verlag Berlin Heidelberg 2009
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Abstract
This book is written on the basis of the authors’ achievements in teaching and
research over many years in the field of ac machines and systems at Tsinghua
University, and reflects the new developments in and applications of science and
technology in recent years. When researching on the systems related to ac
machines as main roles, analysis methods and operating performance are stressed
in this book.
This book is divided into six chapters. Chapter 1 makes salient-pole electric
machines as the main subject of study. Considering ac machine as a circuit model
composed of stator and rotor having relative motion, and starting from a single
coil, the mathematic models and parameter calculation methods are discussed.
Chapter 2 discusses electromagnetic relations and parameters of synchronous
machines and their several operating conditions at synchronous speed, such as
3-phase sudden short circuit, sudden increase of load, opening short circuit, etc.
Chapter 3 introduces some special operating modes of synchronous machine
systems at constant speed and variable speed, such as two phase sudden short
circuit, asynchronous operation of synchronous machines, asynchronous starting
of synchronous motors, synchronous machine in connection with capacitances,
cycloconverter-fed synchronous motor, ac-dc hybrid loads of synchronous
machines, etc. Chapter 4 analyses oscillation, stability and excitation regulation
of synchronous machine systems. Chapter 5 introduces electromagnetic relations
of induction machines, some of their operating modes, such as starting
characteristics, reswitching process, self-excitation during motor in series with
capacitances, and combination system for polyphase induction generator with
dual windings and rectified load. Chapter 6 provides description of inner asymmetric
problems in the stator winding and squirrel-cage rotor of ac machines.
This book is a textbook for the postgraduates in the field of electric machines
and control, and also a reference book for technicists as well as for the teachers
and students of relevant specialties in universities and colleges.
i
Preface
With the increase of single unit capacity and broader application scope of ac
machines, as well as the development and application of power electronics,
computer, micro-electronic technology and control theory, the operating conditions
of ac machines and systems become more complicated, their automation level is
higher, and their operation demands are also stricter. Therefore, new projects and
new research results are presented one by one.
With development of science and technology, more attention has been paid to
the performance analysis of the following systems: variable speed drive system
of ac machines with power electronics and control devices, power energy system
composed of ac machines and power transmission lines, electric machine system
consisting of ac machines and excitation control devices. Therefore, a monographic
book on ac machine systems is needed urgently to reflect the new demand and
new developments.
This monographic book is written mainly on the basis of the following 5
monographic books: Analysis of AC Machines and Their Systems (1st Edition) in
1993 and 2nd Edition in 2004, Fundamental Theory and Analysis Methods of
Transients in Electric Machines Volumeĉin 1982 and VolumeĊ in 1983, AC
Machine Transients and Operating Modes Analysis in 1963, in which the first
book was awarded the First Prize of National Excellent Science and Technology
Book in 1995 and the third and fourth books also won the First Prize of National
Excellent Science and Technology Book in 1983, together with the Excellent
Prize of National Teaching Material in 1988. The first, third, and fourth books
were finished under the leadership of Professor Gao Jingde, and the fifth book
was also written by Prof. Gao, a Member of Chinese Academy of Sciences,
the former President of Tsinghua University, Fellow of IEEE, and the founder of
transient theory of electric machines and power systems in China.
For international academic exchange, this monographic book has been written
in English. Professor Wang Xiangheng wrote Chapters 1 and 6 (except for
Sections 6.2 and 6.3), Dr. Wang Shanming wrote Section 3.7, Dr. Wu Xinzhen
iii
wrote Section 5.6, Dr. Sun Yuguang wrote Sections 6.2 and 6.3 and Professor
Zhang Linzheng wrote Preface, Introduction, Appendices and Chapters 2, 3, 4
and 5 (except Sections 3.7 and 5.6) together with modifying and unifying all
chapters in this book.
In the process of compiling this book, some books and papers were referred to,
and some books and papers written by our colleagues were also used. Main part
of them have been listed in the references in order to help the readers know their
origins and learn about some interesting problems in more depth. We are grateful
to the authors of those books and papers, especially to our colleagues.
It is very difficult to avoid defects and careless omissions in this book owing
to the finite acquirements of the authors although we have made our efforts.
Please send your comments and suggestions regarding this book to Department
of Electrical Engineering, Tsinghua University, Beijing, China.
Authors
iv
Introduction
AC machines are basic equipment of electric energy production and application.
Electric energy used in industry, agriculture and daily life is mostly produced by
alternators and the generated electric energy is likely utilized through ac motors.
The capacity of a single unit has been increased greatly for modern alternators,
the application domain is extended day by day, the automation grade of ac
machine systems is heightened increasingly, and the operating condition is more
complicated. Therefore, it is very important for electro-technical experts to
understand and master the operating performance and the analysis method of ac
machine system.
The ac machine system is a relatively mature but quickly developing product
of science and technology. Emergence and development of ac machines have a
history of more than one hundred years, in which ac machines of various types
have been built for various uses. During design, manufacture, and operation of ac
machines and through researches and improvement, people have comprehended
the structure principles, manufacture technique, and operating performance
deeply. However, due to restraints of technical development and research
conditions in the past, people paid attention to normally operating performance,
outer faults and economic parallel-operation in safety for ac machines, which is
natural because those problems often exist in practice.
During the past decades, the power electronic technique has been widely used,
the dc transmission had a great progress, ac-dc hybrid transmission systems
emerged, large capacity alternators were built to bring about some new requirements,
and modern control theory, micro-electronics and computer technology also
progressed greatly, thus making the theory and operating performance of ac
machine systems undergo a new development. For example, the analysis of inner
unsymmetrical windings, the study of control system consisting of ac machines,
power electronic devices, micro-electronics and computer technology, the analysis
of electromagnetic harmonics effect and restraint are all the new topics.
During the operation of the ac machine, due to manufacturing defect or
misoperation, some fault occurs in the machine winding that creates unnecessary
loss. For large hydraulic generator, its stator winding often has multi-branches. If
inter-turns or inter-phases short circuit occurs or the winding is broken partly, the
large fault current will damage the winding or the iron core, and the negative
v
sequence magnetic field during fault may burn the rotor too. When the
turbo-generator and the induction machine are working, some serious inner faults
may occur in their stator winding. As far as the rotor of an ac machine is
concerned, weld defects, and the breaking or short circuit of conductors may
occur, too. Therefore, it is necessary to study the various state quantities during
inner faults in order to understand the harm of faults, to improve the design and
manufacturing technology and to put forward the corresponding protection relays.
So far as the dc transmission system goes, due to the combination of alternator
and rectifiers, it is quite different from the ac transmission system. As for the
alternator, it is often on unsymmetrical operating mode and in the system exist
serious harmonics due to nonlinearity of rectifying elements, so the corresponding
special topics need studying. In addition, the ac excitation system is widely used
and sometimes the excitation bars are installed in the upper positions of stator
slots for the main alternator to provide excitation current for the excitation
system of main alternator through the rectifiers, which also brings about the
similar problems as mentioned above.
Since they connect computer, micro-electronics and power electronic technique,
the speed regulation and control of ac motors have been developed swiftly, and
some electromagnetic processes, which were hardly measured and controlled in
the past, can be regulated and controlled today through on-line measurement,
computer analysis and control.
With the development of computer technology and the application of numerical
method, some problems of the electromagnetic field of an electric machine, which
were hardly calculated in the past, can be solved today to enhance the capability
of design and operating performance analysis, and the accuracy of parameters
calculation greatly for ac machines. In addition, the achievements in some aspects
are also gotten, such as pole-changing speed-regulation of induction motors,
principles and methods of starting, the combination of capacitors and stator
winding, special machine provided with rectangular waveform current, storage
electric machine, etc.
For a deep analysis of the ac machine systems, it is necessary to put forward
new theory and methods.
As you know, as far as the electromagnetic relation of ac machine is concerned,
it can be considered as the combination of several coils coupled magnetically and
studied according to usual circuit rules. However, due to rotor rotation, there
exists relative motion between those coils. 3-phase symmetrical winding is adopted
for the stator winding of an ac machine to make the air-gap magnetic field be of
sinusoidal waveform and revolve synchronously when 3-phase symmetrical
current passes through the 3-phase winding. In so doing, various targets of ac
machines are all good. Therefore, people used to assume the air-gap flux to be
sinusoidal and to merge small harmonic flux into leakage flux during the analysis
of ac machines. Analyzing their operating performance, the parameters used are
based on the criterion that they can reflect the total action of the air-gap magnetic
vi
field, such as d- and q-axes armature reaction reactances xad and xaq for
synchronous machines, magnetization reactance xm for induction machines,
etc. Correspondingly, other parameters such as xd , xq , xdc , xqc , xdcc and xqcc for
synchronous machines, are all some values to reflect the total electromagnetic
effect. The mathematical patterns built by using the above parameters and the
corresponding analytical results can meet the practice requirements; so in the
text-books and references, the phase winding has been the basis and the total
effect of multiphase winding has been taken to analyze ac machines for many
years.
As stated before, with the development of science and technology, there came
to exist the inner unsymmetrical problem of electric machine winding, the new
combination system of electric machine and power electronic devices and some
electric machines of special structure, which make taking a single machine as the
analysis object and symmetrical phase winding as the basis of theory and
methods be unsuitable for the study of some topics. Thus, it is necessary to take
the electric machine and the associated outer devices as a system to consider the
influence of electromagnetic harmonics. Based on the aforementioned points, a
single coil is adopted as the basis in this book to describe the basic electromagnetic
relation of ac machines and to build the corresponding expressions. After getting
the relations, they can be used to study not only the problems that are analyzed
by using the existing methods but also some new topics as mentioned before.
Since the basic electromagnetic relations of ac machines are based on single coil,
in the light of the requirements for different research problems compose the
corresponding loops to get the associated electromagnetic relations, inclusive of
the electromagnetic relations obtained when taking the phase winding as the
basis. Because electromagnetic harmonics are counted in the basic relations, we
can study not only some topics which need to count harmonic influence but also
a certain problem that can neglect harmonic effect for convenience.
The ac machine parameters used in the new method are self inductances and
mutual inductances for various coils, so it is not difficult to find out the
parameters and the basic relations to reflect the total electromagnetic effect of ac
machine winding according to the self inductances and mutual inductances if
necessary.
The study starts from a single coil and combines the corresponding loop
equations according to the actual loops connection of ac machines to research
into needful topics, which is termed the Multi-Loop Theory and is the main basis
of this book.
This book is divided into six chapters, referring to Contents of this book in detail.
Firstly, based on a single coil and using the Multi-Loop Theory created by
some authors of this book, we get various inductances and parameters which are
the same as classical method, and then analyze several operating modes of
synchronous machine systems, such as transformations of reference axes systems
connecting the relevant mmf to form a clear concept, short circuit and
vii
asynchronous operation, combination system for synchronous machine and
capacitances, cycloconverter-fed synchronous motor system, synchronous
generator system with ac and dc hybrid loads, oscillation, stability and excitation
regulation. Also, we analyze some operating modes of induction machine
systems, such as why the transient starting characteristic is quite different from
steady-state starting one, why the reclosing current is larger than the starting
current, discovery of new self-excitation regions during induction motor in series
with capacitances, and special induction generator with dual windings and
rectified load. By using the Multi-Loop Theory, we study the stator winding
inner faults and squirrel cage faults, which are hardly found in other books and
references. This book provides a new way for analyzing ac machine systems.
viii
Contents
Abstract ................................................................................................................ i
Preface ................................................................................................................ iii
Introduction ........................................................................................................ v
1
Circuit Analysis of AC Machines—Multi-Loop
Model and Parameters ................................................................................. 1
1.1 Electromagnetic Relations of AC Machine Loops ................................ 2
1.2 Air-Gap Permeance Coefficient of Electric Machines .......................... 6
1.3 Air-Gap Magnetic Field Produced by Single Coil and the
Corresponding Inductance .................................................................... 8
1.4 Inductances of Stator Loops ................................................................ 19
1.5 Inductances of Rotor Loops ................................................................ 23
1.6 Mutual Inductances Between Stator Loops and Rotor Loops ............. 32
1.7 Influence of Saturation on Parameters ................................................ 34
1.8 Inductances Produced by Leakage Magnetic Field ............................. 37
1.9 Electromagnetic Torque and Rotor Motion Equation of
AC Machines ...................................................................................... 40
1.10 Analysis of Parameters and Performance of Single-Phase
Induction Motors............................................................................... 44
References.................................................................................................... 56
2
Electromagnetic Relations and Parameters of Synchronous
Machines and Analysis of Their Several Operating Conditions
at Synchronous Speed ................................................................................ 58
2.1 Basic Relations of Synchronous Machines ......................................... 59
2.2 Basic Relations of Synchronous Machines in d, q, 0 Axes ................. 78
2.3 Per-Unit Systems in Synchronous Machines ...................................... 89
2.4 Basic Equations in Per-Unit, Operational Reactances and
Electromagnetic Torque of Synchronous Machines.......................... 100
2.5 3-Phase Sudden Short Circuit and Transient Parameters of
Synchronous Machines ......................................................................111
2.6 Voltage Dip during Sudden Increase of Load for Synchronous
Machines ........................................................................................... 142
ix
2.7 Transient EMFs of Synchronous Machines ...................................... 147
2.8 Electromagnetic Torque after 3-Phase Sudden Short Circuit ............ 153
2.9 Suddenly Opening 3-Phase Short Circuit of Synchronous
Machines ........................................................................................... 159
References.................................................................................................. 165
3
Some Special Operation Modes of Synchronous Machine Systems at
Constant Speed and Variable Speed ....................................................... 166
3.1 Transformations of Reference Axes Systems and Their Formulas in
Synchronous Machines ..................................................................... 168
3.2 2-Phase Sudden Short Circuit of Synchronous Machines................. 180
3.3 Steady-State Asynchronous Operation of Synchronous Machines ... 203
3.4 Asynchronously Starting of Synchronous Motors ............................ 214
3.5 Analysis of Combination System for Synchronous Machine and
Capacitances ..................................................................................... 224
3.6 Mathematic Model and Performance Analysis of Cycloconverter-fed
Synchronous Motor Systems with Field-Oriented Control............... 235
3.7 Analysis of Synchronous Generators with AC and DC Stator
Connections....................................................................................... 249
References.................................................................................................. 266
4
Oscillation, Stability and Excitation Regulation of Synchronous
Machine Systems ...................................................................................... 267
4.1 Small Perturbation and Linearization of AC Machine Systems ........ 268
4.2 Steady-State Small Oscillation and Torque Coefficients of
Synchronous Machine Systems ........................................................ 270
4.3 Static Stability of Synchronous Machine Systems and Influence of
Excitation Regulation on Static Stability .......................................... 301
4.4 Dynamic Stability and Analysis Methods of Synchronous Machine
Systems ............................................................................................. 315
References.................................................................................................. 338
5
Electromagnetic Relations of Induction Machine Systems and
Analyses of Some Operating Modes........................................................ 339
5.1 Basic Relations and Parameters of Induction Machines ................... 340
5.2 Basic Relations and Parameters of Induction Machines
in d, q, 0 Axes ................................................................................... 354
5.3 Analysis of the Starting Process for Induction Motors...................... 363
5.4 Transients of Reswitching on Induction Motors ............................... 371
5.5 Self-Excitation when Connecting Induction Motor in Series
with Capacitance and Counting the Inertia Effect in ........................ 386
5.6 Dual-Stator-Winding Multi-Phase High-Speed Induction
Generator........................................................................................... 402
x
References...................................................................................................411
6
Internal Asymmetric Analysis of AC Machines ..................................... 412
6.1 Analysis of Rotor Winding Failures of Squirrel Cage
3-Phase Induction Machines ............................................................. 415
6.2 Internal Fault Analysis of Stator Winding of Salient Pole
Synchronous Machines ..................................................................... 431
6.3 Internal Fault Analysis of Stator Winding of Non-Salient
Pole Synchronous Machines ............................................................. 447
References.................................................................................................. 454
Appendix A Air-Gap Permeance Coefficients in Consideration of
Slot Effect................................................................................ 455
Appendix B
Expressions of Effective Air-Gap .......................................... 458
Appendix C
Inductances Due to Leakage Flux of Stator
Winding End........................................................................... 461
Appendix D
Heaviside’s Operational Calculus and Its
Application to Transients Analysis ....................................... 474
Appendix E
Basic Relations and Equivalent Circuits of
Transformers .......................................................................... 480
Index ................................................................................................................ 484
xi
1 Circuit Analysis of AC Machines—Multi-Loop
Model and Parameters
Abstract The Multi-Loop Model and parameters are introduced and
described in this chapter as the basis of this book, which is more convenient
than the classical method for the analysis of inner faults. According to the model,
an ac machine is considered as an electric circuit linked with ferromagnetic
circuit. The difference between ac machines and general static circuits is the
existence of relative movement between the stator and the rotor for the former.
Hence, some loop inductances of ac machines are relative with the rotor
position, that is, these inductances are time-variant coefficients. The salientpole synchronous machines are analyzed as structure models of ac machines,
because a non-salient-pole synchronous machine is a special case of salientpole type. The single coil is taken as the fundamental unit in order to study and
discuss the electro-magnetic relation of ac machines. This chapter discusses
the circuit model of ac machines, inclusive of loop voltage equations and fluxlinkage equations of stator and rotor, loop parameters—mainly inductances
(main inductances and leakage inductances), electro-magnetic torque and
rotor motion equations, etc. Finally, as an example of using the Multi-Loop
Model, the parameters and performance analysis of a single phase induction
motor are presented. Evidently, the difference between the calculated values
and experimental data is bigger when considering only the fundamental of
air-gap magnetic field by the use of ordinary designed program. However,
when counting the higher harmonics in by using the Multi-Loop Method, the
calculated values are close to the experimental data.
AC machines are main equipment which convert mechanical energy into electric
energy (generators) or vice versa (motors). According to varying demands,
synchronous machines and induction machines with different capacities and
various sorts are produced and applied. Though the structures, principles and
performance of these machines are different and numerous, the stator windings
of ac machines with 3 phases or single phase are connected with several coils or
components; the structures of the rotor are of salient pole or non-salient pole, the
rotor windings are composed of conducting bar and end-ring or connected by
several coils; therefore the electromagnetic relations of the salient-pole synchronous
machine with non-uniform air-gap are more complicated. From the point of view
AC Machine Systems
of the rotor structures, the synchronous machine with damper winding and excitation
winding is more representative. From the point of view of electromagnetic relations,
all ac machines are composed of several coils having movement with each other,
hence after finding one coil’s relationship with current, flux-linkage and voltage,
the corresponding loop equations are obtained based on the concrete structures
and operation status, then the performance can be calculated and analyzed. In this
chapter, the salient-pole synchronous machines are considered as structure
models of ac machines, the single coil as the fundamental unit in order to study
and discuss the electromagnetic relation of ac machines, including the inductance
of single coil and the inductances of the loop composed of these coils, etc. As the
basis of the Multi-Loop Theory and Method, thoroughly understanding and
mastering it is the key for reading the rest of this book[1– 6].
1.1 Electromagnetic Relations of AC Machine Loops
AC machines are circuit structures with iron, the difference between ac machines
and general static circuits is the existence of relative movement between the
stator and the rotor for ac machines, hence some loop inductances of ac machines
depend upon the rotor position, i.e. these inductances are time-variant coefficients.
When studying the basic electro-magnetic relations of ac machines a lot of
theorems and rules about static circuits may still be applied.
Taking the coupled circuit in Fig. 1.1.1 as an example, in terms of the positive
directions in Fig. 1.1.1 and the right hand screw rule of the current and flux-linkage,
the voltage and flux-linkage equations are as follows:
\ 1 L1i1 M 12 i2
\ 2 M 21i1 L2 i2
u1
p\ 1 r1i1
u2 p\ 2 r2 i2
These electro-magnetic relations and concepts of simple and basic circuits are
often used in discussing the electro-magnetic relations of ac machines.
Figure 1.1.1 Coupled circuit with mutual inductance
2
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
So far as electro-magnetic relations are concerned, the salient-pole synchronous
machines have more general meaning. The non-salient-pole synchronous machine
with uniform air gap may be treated as a special example of the salient-pole
synchronous machine with non-uniform air gap. If we need to consider the action
of the eddy current in the rotor iron of the non-salient-pole synchronous machine,
its action can be replaced by equivalent damper winding. Therefore, some principles
and concepts drawn from salient-pole synchronous machine may be used in
non-salient-pole synchronous machine under certain conditions. As for the
induction machine, there exists not only a uniform air gap but its rotor is also
symmetrical to arbitrary axis in electric and magnetic aspects, so it may be treated
as a special example of salient-pole synchronous machine under some instances.
In order to analyze the operation performance of salient-pole synchronous
machines, firstly it is needed to list the voltage and flux-linkage equations of the
stator and rotor loops. Generally, the stator phase winding is considered as a loop
under symmetric 3 phases. But, in the inner asymmetric condition of the stator
winding, it is difficult to consider the stator phase winding as a loop, so selecting
the loop according to concrete problem is required. For example, when an inner
fault occurs in the stator winding, the short circuit part needs to be treated as a
single loop. For a normal single phase machine, although its phase winding may
be treated as a loop, under some special structure, e.g. for the single phase
machine with concentrated winding (q 1) , its harmonic magnetic field has to be
calculated, or else more errors will be brought in. So, based on a single coil, it is
significant to discuss the electro-magnetic relation of each loop. In the following,
we analyze concretely the electro-magnetic relation of each loop of the salient-pole
synchronous machine from two sides of the stator and rotor.
1.1.1
Stator Loops
When analyzing the performance of ac machines, it is necessary to list the loop
equations. Many loops exist in the stator, in which some loop may be a phase
winding, a branch of stator winding, or the special circuit connected by a coil or
several coils. For example, for a loop K, the positive directions of current,
flux-linkage and voltage are drawn in Fig. 1.1.2, i.e. positive current produces
negative flux-linkage, positive current produces positive voltage drop along the
load direction. Such regulation is consistent with the positive direction of the
stator loop-K current and its axis-K direction in Fig. 1.1.3, namely, the positive
direction of loop-K flux-linkage.
Under the above regulation, the voltage equation of arbitrary loop K of the stator
is written as
uK
p\ K rK iK
(1.1.1)
3
AC Machine Systems
where uK ,\ K , rK , iK are the terminal voltage, flux-linkage, resistance and current
d
is differential operator.
of the loop K respectively; p
dt
Figure 1.1.2 Positive direction relation
of i, u ,\ of stator loop
1.1.2
Figure 1.1.3 Positive direction regulation
of stator and rotor of electric machine
Excitation Loop
Selecting the direction of direct axis d and quadrature axis q as in Fig. 1.1.3, the
direct axis is consistent with pole central line, the quadrature axis leads the direct
axis by S / 2 electrical radians along the rotation direction of the rotor. The fluxlinkage positive direction of the excitation loop j is consistent with the positive
direction of d-axis. The positive direction of the current of excitation loop j is as in
Fig. 1.1.3, i.e. the positive excitation current
produces positive excitation loop fluxlinkage. The positive direction of excitation
voltage is as in Fig. 1.1.4, i.e. the voltagedrop positive direction is consistent with
the positive direction of excitation winding Figure 1.1.4 Positive direction regulation
of the voltage and current of excitation
current.
Based on the above positive direction winding
regulation of each quantity, the voltage equation of excitation loop j is written as
u fd j
p\ fd j rfd j i fd j
(1.1.2)
where u fd j ,\ fd j , rfd j , i fd j are the terminal voltage, flux-linkage, resistance and current
of excitation loop j, respectively.
1.1.3
Damper Loops
The damper bars of the salient-pole synchronous machine are all short-circuited
or partly short-circuited, these bars constitute a mesh circuit in which the
4
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
selection of current loops is arbitrary. According to the requirement of the
analyzed problem, the damper loops may be treated as an equivalent d-axis loop
and q-axis loop on each pole, or they can be selected based on practical mesh
circuit. The positive direction regulation of every quantity of the damper loops is
similar to excitation loop.
In this book, the damper loops will be analyzed in terms of practical mesh circuit
as in Fig. 1.1.5.
Figure 1.1.5
Sketch map of damper loops of salient-pole synchronous machine
As an example, the voltage equation of damper loop i is
0
p\ i rii i rc (ii 1 ii 1 )
(1.1.3)
where \ i , ri , ii are the flux-linkage, resistance and current of damper loop i,
respectively; rc is the bar resistance; ii 1 and ii 1 are the current of damper loops
( i 1 ) and ( i 1 ), respectively.
The voltage Eqs (1.1.1), (1.1.2), and (1.1.3) all include the loop flux-linkage. On
the basis of the circuit theorem, the relation between the flux-linkage and current
(flux-linkage equation) is listed as follows:
ª \ s1 º ª Ls1
«
» « #
« # » «
« \ sm » « M sm s1
«
» «
«\ fd1 » « M fd1s1
« # » « #
«
» «
«\ fdn » « M fdn s1
« \ » « M
1s1
« 1 » «
« # » « #
« \ » « M
¬ l ¼ ¬«
ls1
or written as
"
M s1sm
M s1 fd1
#
"
#
Lsm
"
M s1 fdn
M s11
"
#
"
M sm fd1
" M sm fdn
#
M sm 1
" M fd1sm
L fd1
" M fd1 fdn
M fd11 "
#
" M fdn sm
#
M fdn fd1
"
#
L fdn
"
M 1 fdn
L1
"
"
#
M lfdn
#
M l1
"
"
M 1sm
M 1 fd1
"
#
M lsm
#
M lfd1
\
LI
#
M fdn 1 "
M s1l º ª is º
» 1
# »« # »
« »
M sml » « is »
»« m »
M fd1l » « i fd »
1
# »« # »
»« »
M fdn l » «i fd »
n
M 1l »» « i1 »
« »
# »« # »
»
Ll ¼» ¬« il ¼»
(1.1.4)
5
AC Machine Systems
where L is self inductance; M is mutual inductance; each subscript represents the
corresponding loop, where s1 , s2 ," , sm represent the sequence of stator loops,
1,2, " , l represent the sequence of damper loops, fd1 , fd 2 ," , fd n represent the
sequence of excitation loops.
In the above formula, the corresponding mutual inductances are reversible, i.e.,
M s1s2
1.2
M s2 s1 ,
M fdn sm
M sm fdn ,
M sml
M lsm ,
M 1l
M l1 , "
Air-Gap Permeance Coefficient of Electric Machines
The parameters of electric machines is the key for analyzing their performance,
and the self inductances and mutual inductances are the most basic parameters of
electric machines which represent the capability of certain loop current producing
certain loop flux-linkage. Generally, the flux-linkage of certain loop depends upon
the currents in the related loops, as in the formula (1.1.4), and the elements of
inductance matrix L include the stator loop inductances, the rotor loop inductances
and the inductances between stator and rotor loops.
The flux-linkages produced by loop currents in ac machines are divided into
two kinds, one is the leakage, and the corresponding inductances are considered
to be independent of the rotor position, i.e. to be constant; another flux-linkage is
the main part which links the stator and rotor through the air-gap. The inductances
corresponding with these flux-linkages will vary when the reluctance variation
occurs owing to rotor rotation. The permeance analysis method of magnetic
circuit and the analysis method of magnetic field may be applied to research
these inductances. Here we discuss mainly the permeance analysis method of
magnetic circuit. While researching the inductances by using the permeance
analysis method, the effect of iron saturation is not taken into account, considering
that the reluctance of magnetic circuit does not vary with the flux density, and the
effect of iron reluctance is considered by magnifying the air gap appropriately.
Suitabe parameter value is applied to consider the effect of iron saturation
according to the operation condition, if necessary. In addition, the effect of some
secondary factors such as the hysteresis and eddy-current also will be neglected in
the analysis.
In this chapter, the effect of slots is represented by the Cater’s coefficient; the
slot harmonic variation of the air-gap reluctance is neglected. Please see Appendix
A for the inductance including the slot harmonics.
The effect of iron saturation and slots may be calculated by the magnetic field
analysis in which the magnetic field method should be used to count the
inductances of ac machines, and obviously the time spent in the process will be
greater. As there are specialized books on this topic, this book does not intend to
discuss it further.
6
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
When the inductances are calculated by using the permeance analysis method,
at first, the air-gap mmf and air-gap permeance are analyzed in terms of the basic
concept of inductance coefficients[7]; then the air-gap magnetic field can be
calculated, the self and mutual flux-linkages are found, and finally the corresponding
inductances will be obtained.
As mentioned above, setting out from a single coil and then to arbitrary loop
composed of the coils, the inductances of electric machine loops are studied,
therefore the air-gap mmf is analyzed in terms of the air-gap mmf of the single
coil. Here, the air-gap mmf includes integral and fractional harmonics with the
1 2 3
orders , , ," , in which P is pole-pair number.
P P P
The relation between the mmf F, the permeance / and magnetic flux ) of
the magnetic circuit is
F/ )
S
, where S is the cross-section area
l
of the magnetic circuit, l is length of the magnetic circuit, and P is permeability.
For magnetic flux of unit area (namely flux density) there is
For uniform magnetic circuit, there is /
P
FO
B
For uniform magnetic circuit, O
P
(1.2.1)
1
represents the permeance of unit area that
l
is called the permeance coefficient.
Along the different positions of the rotor coordinates system, the air-gap
length of salient pole synchronous machine is different owing to non-uniform
air-gap, hence the air-gap permeance /G and the permeance of unit area OG are
varied, and they are relative to the mmf and rotor position. This issue is discussed
further here.
Figure 1.2.1
Rotor coordinates of salient pole synchronous machine
The permeance of unit area is represented by OG ( x) which is called air-gap
permeance coefficient, and here x is the abscissa on the rotor as in Fig. 1.2.1.
Owing to the rotor symmetry relative to the d-axis, the air-gap lengths in the
7
AC Machine Systems
positions x and x of the d-axis are equal, hence their air-gap permeance
coefficients are equal too, i.e. OG ( x) OG ( x ) . The distributing status of air-gap
permeance coefficient under each pole is homologous, thus OG ( x ) OG (S r x) .
Therefore, the air-gap permeance coefficient is the even function of the variable
x only with even harmonics, and its general expression is
OG ( x)
O0
2
¦ O2l cos 2lx
l 1, 2,"
(1.2.2)
l
If the equivalent air-gap G ( x) of salient pole synchronous machine with
non-uniform air-gap is known, then the air-gap permeance coefficient OG ( x) is
OG ( x)
P0
G ( x)
(1.2.3)
The following expression obtained according to Fourier series is
O0
4 S
4 S P
OG ( x)dx = ³ dx
³
S 0
S G ( x)
(1.2.4)
O2l
4 S
OG ( x) cos 2lxdx
S ³
(1.2.5)
4 S P 0
cos 2lxdx
S ³ 0 G ( x)
Based on the structure features of the salient pole synchronous machine, the
equivalent air-gap lengths G ( x) of the pole shoe space and interpolar space may
be dealt with respectively. In the pole shoe space, it is considered that all flux lines
are through the air-gap along the radial direction, and the equivalent air-gap
length may be given according to geometric size. But, on the basis of the results
of graphic method and analysis method, in the interpolar space, the distributing
status of the magnetic field is relative to the position of the corresponding mmf,
hence the expressions of equivalent air-gap length are given respectively
according to two statuses in which the zero value or the maximum value of air-gap
mmf is on the central line of the interpolar space. The expressions of equivalent
air-gap lengths G ( x) are given in Appendix B.
After getting the air-gap mmf and air-gap permeance coefficient, the air-gap
magnetic field can be obtained, then the loop flux-linkage and corresponding
inductance are obtained.
1.3 Air-Gap Magnetic Field Produced by Single Coil and
the Corresponding Inductance
As mentioned above, several coils constitute each loop of the stator. No matter
what connection the stator winding has or how asymmetrical the stator winding is,
8
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
the stator loops are composed of several corresponding coils. Therefore, researching
and realizing the air-gap magnetic field (main magnetic field) have general
meaning. Besides the air-gap magnetic field (main magnetic field), the leakage
magnetic fields exist in electric machines, and the points with regard to the
leakage magnetic field and the corresponding inductances will be discussed in
Section 1.8.
The air-gap magnetic field produced by a single coil and the related inductances
are discussed as follows.
1.3.1
Air-Gap MMF Produced by Single Coil’s Current
Assume that electric current passes through the stator coil AAc , and rectangular
mmf is formed as in Fig. 1.3.1. Harmonic analysis is made for the rectangular
mmf along the inner circumference of the stator, and then exists
F (T )
¦a
n
n
Figure 1.3.1
n
cos T
P
n 1, 2,"
(1.3.1)
Air-gap mmf produced by the current of stator coil AAc
The above formula has only the cosine component, because the rectangular mmf
is symmetrical to the axis of the coil AAc . Therefore, there is
an
1 PS
§n ·
F (T ) cos ¨ T ¸ dT
³
P
S
PS
©P ¹
2
nE S
iwK sin
nS
2P
(1.3.2)
where P is pole-pair number, E is the ratio of actual pitch to full pitch, wK is
turn number of the coil, T is the electric angle along circumferential direction of
the electric machine.
It is seen that in the rectangular mmf of the single coil, the fractional harmonics
and lower harmonics are very strong.
In formula (1.3.1), the fundamental wave of mmf is a1 cos
T
, its period is
P
2PS ; the P-th harmonic of mmf is aP cosT , its period is 2S ; the KP-th harmonic
of mmf is akP cos kT , its period is 2S / k .
9
AC Machine Systems
If the space wave of 2S electric radians is considered as fundamental wave
according to general custom, then the formula (1.3.1) may be rewritten as
¦F
F (T )
k
cos kT
1 2 3
, , ,"
P P P
k
k
(1.3.3)
in which
Fk
2
kES
iwK sin
kPS
2
iwK k yk
kP S
k ES
is pitch factor of the coil.
2
Owing to non-uniform air-gap of salient pole synchronous machine, the flux
density produced by the same air-gap mmf is different in different rotor positions.
Obviously, when the mmf is located on d-axis or q-axis, the air-gap is different,
hence the corresponding air-gap magnetic field is different too. For the sake of
convenience for analyzing the air-gap magnetic field, the stator mmf is divided
into two components, Fdk ( x ) and Fqk ( x) , so that their positions are not varied
relative to the rotor, and the corresponding air-gaps relative to the two mmf are
not varied too. Therefore, it is needed to transform the air-gap mmf formula
(1.3.1) written in the stator coordinates system to rotor coordinates system. Since
T x J (see Fig. 1.3.1), if the d-axis position is considered as the origin of the
abscissa x established on the rotor, J is the rotor position angle (it is electric
angle by which the rotor d-axis leads the stator coordinates axis along the rotation
direction), then there is
and k yk
sin
J
t
³ Z dt J
0
0
where J 0 is the electric angle between rotor d-axis and stator coordinates axis at
t 0 , i.e. the rotor initial position angle; Z is rotor angular velocity.
When the rotor rotates at uniform speed, there is J Z t J 0 , and ( x J ) is
substituted into T , then formula (1.3.3) is changed to
F (T )
¦F
k
cos k ( x J )
k
¦[F
dk
k ( x) Fqk ( x)]
(1.3.4)
k
where
Fdk ( x)
Fk cos kJ cos kx
Fdkm cos kx
(1.3.5)
Fqk ( x)
Fk sin kJ sin kx
Fqkm sin kx
(1.3.6)
It is seen that the amplitude of Fdk ( x) is
10
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
Fdkm
2
k ES
iwK sin
cos kJ
kPS
Fk cos kJ
That is, the amplitude is relative to the current and position angle J , and the
position of the amplitude is located at x 0 . When the current and rotor position
are changed, only the amplitude of component Fdk ( x ) is changed, not its position,
i.e. it is a pulsating mmf component which has no motion relative to rotor d-axis.
Therefore, the mmf is called the direct axis component of the k-th harmonic mmf.
Similarly, the amplitude of Fqk ( x) is
Fqkm
2
kE S
iwK sin
sin kJ
kPS
Fk sin kJ
which has the same features for the rotor q-axis, hence it is called the quadrature
axis component of the k-th harmonic mmf.
1.3.2
Air-Gap Magnetic Field Produced by Single Coil’s Current
Based on the air-gap mmf and air-gap permeance coefficient, the air-gap
magnetic field can be obtained. The air-gap magnetic field has both integral
harmonics and fractional harmonics, because the harmonic orders of the air-gap
1 2 3
mmf are k
, , ," , and besides the constant component O0 / 2 , the even
P P P
harmonics exist in the air-gap permeance coefficient OG ( x) .
As mentioned above, the air-gap magnetic fields produced by the same
amplitude mmf of d- and q-axes of the salient pole synchronous machine are
different, so the d- and q-axes flux densities produced by the d- and q-axes mmf
should be discussed, respectively. For example, the air-gap flux density produced
by the direct axis component Fdk ( x) of k -th harmonic mmf is
Bd ( x)
Fdk ( x)OG ( x)
§O
Fdkm cos kx ¨ 0 © 2 l
­O
Fdkm ® 0 cos kx l
¯2
·
f
¦
O2l cos 2lx ¸
¹
1,2,3,"
f
¦
1,2,3,"
½
[cos(2l k ) x cos(2l k ) x]¾
2
¿
O2l
i.e. the direct axis k -th harmonic mmf will produce many different harmonics of
air-gap flux density. Through Fourier analysis, it is seen that the amplitude Bdkjm
of j -th air-gap flux density produced by the direct axis component Fdk ( x ) of k -th
harmonic mmf is
11
AC Machine Systems
Bdkjm
2 PS
Bd ( x) cos jxdx
PS ³ 0
PS ­ O
O
2
½
Fdkm ³ ® 0 cos kx ¦ 2l [cos(2l k ) x cos(2l k ) x]¾ cos jxdx
0
2
PS
l
¯
¿
(1.3.7)
where l 1, 2," , f. When | 2l k |z j and k z j , the integral result of the above
formula is zero; and when | 2l r k | j (i.e., 2l | k r j | ) and k z j , the integral
result of the above formula is
Bdkjm
1
Fdkm u (O2l 2l |k j| O2l
2
1
Fdkm u (O|k j| O|k j| )
2
Fdkm Odkj j | 2l r k |
2l |k j|
)
in which
Odkj
1
(O|k j| O|k j| )
2
| k r j | 2l
(1.3.8)
which is called the harmonic permeance coefficient of j -th harmonic flux density
produced by the d-axis k - th harmonic mmf. Here,
O|k r j|
O2l
l 1, 2,3,"
When k j and 2l k j , i.e., l k (N.B. the term corresponding to 2l
k j 0 does not exist, since l does not include zero ( l z 0 )), the integral result
of formula (1.3.7) is
Bdkkm
1
Fdkm u (O0 O2 k )
2
Fdkm Odkk
where
Odkk
1
(O0 O2 k )
2
(1.3.9)
is harmonic permeance coefficient of k -th harmonic flux density produced by
the d-axis k -th harmonic mmf.
After comparing formula (1.3.8) with (1.3.9), it is known that formula (1.3.9)
can be obtained through substituting k j into formula (1.3.8). Hence, whether
k j or k z j , if only 2l | k r j | , l 0,1, 2," , the direct axis harmonic
permeance coefficient can be calculated by formula (1.3.8).
12
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
Similarly, the j -th harmonic flux density produced by the quadrature axis k - th
harmonic mmf and harmonic permeance coefficient Oqkj are obtained as follows:
Bqkjm
Oqkj
Fqkm Oqkj
1
(O|k j| O|k j| )
2
(1.3.10)
where
O|k r j|
O2l
l
0,1, 2, "
Based on the above results, we can observe the following facts:
(i) Simple relation exists between the amplitude of k -th harmonic mmf and
the amplitude of j -th harmonic flux density produced by it, i.e.,
Bdkjm
Fdkm Odkj ,
Bqkjm
Fqkm Oqkj
Furthermore, simple relation exists between the air-gap harmonic permeance
coefficient Odkj and Oqkj as well as the air-gap permeance coefficient O2l ( l
0,1, 2," ). Hence, Odkj and Oqkj are obtained easily according to O2l . The Odkj and
Oqkj are not changed with the rotor position angle since O2l is constant which is
independent of rotor position.
(ii) In salient pole synchronous machine, the order j of harmonic flux density
produced by k -th harmonic mmf and the order k of harmonic mmf have
some relations, i.e., j | k r 2l | (l 0,1, 2,") . For example, when k 1 , then
1
1 9 11 19
, then j
, , , ,"
j 1,3,5, 7," ; when k
5
5 5 5 5
Therefore, definite relation exists between the subscripts k and j which represent
the harmonic order in Odkj and Oqkj , i.e., | k r j | 2l (l 0,1, 2,") . If | k r j |z 2l ,
then both Odkj and Oqkj are zero.
(iii) Based on formulas (1.3.8) and (1.3.9), it is seen that the harmonic permeance
coefficients are reversible, i.e.,
Odkj
Oqkj
1.3.3
Odjk ½°
¾
Oqjk °¿
(1.3.11)
Self Inductance of Single Coil
As mentioned above, according to air-gap harmonic permeance coefficient and
13
AC Machine Systems
direct axis and quadrature axis k -th harmonic mmf produced by the current of
single coil AAc , the direct axis and quadrature axis j -th harmonic flux densities
produced by the corresponding mmfs can be obtained, i.e.,
Bdkjm
Fdkm Odkj
Bqkjm
Fqkm Oqkj
From the direct axis air-gap flux density, the self flux-linkage of coil AAc can
be obtained as follows:
\ dkj
s2
wK ³ Bdkjm cos jxds
s1
§ x · Wl
d ¨ Rl ¸
dx , R is the stator inner radius, l is stator iron length,
©P ¹ S
W is pole pitch. Hence,
where ds
wKW l J E2S
ES Bdkjm cos jxdx
S ³ J 2
\ dkj
4 wK2 W l
k yk k yj Odkj i cos kJ cos jJ
PS kj
(1.3.12)
Similarly, the self flux-linkage of coil AAc corresponding to the quadrature axis
air-gap flux density is
wKW l J E2S
ES Bqkjm sin jxdx
S ³ J 2
\ qkj
4 wK2 W l
k yk k yj Oqkj i sin kJ sin jJ
PS kj
(1.3.13)
Self inductances of the coil corresponding to the direct axis and quadrature axis
flux densities are
Ldkj
Lqkj
\ dkj
i
\ qkj
i
4 wK2 W l
k yk k yj Odkj cos kJ cos jJ
PS kj
(1.3.14)
4 wK2 W l
k yk k yj Oqkj sin kJ sin jJ
PS kj
(1.3.15)
Accordingly, based on the total of every harmonic flux density produced by
every harmonic mmf, the self inductance of coil AAc by total air-gap magnetic
field is
14
1
LG
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
¦¦ ( L
dkj
k
Lqkj )
j
k yk k yj
4 wK2 W l
(Odkj cos kJ cos jJ Oqkj sin kJ sin jJ )
¦¦
PS k j kj
where k is the harmonic order of mmf, k
of flux density, j | k r 2l |, l
(1.3.16)
1 2 3
, , ," ; j is the harmonic order
P P P
0,1, 2,"
From formula (1.3.16), it is seen that the self inductance of single coil is a
double series which represents the total effects of every harmonic, in which
fractional harmonics and lower harmonics are so strong as to compare with the
fundamental wave. If we calculate only the effect of the fundamental magnetic
field produced by fundamental mmf, then there may be large error that is not
permitted. For example, for a 550kW salient pole synchronous machine (its polepair number is P 6 ), the self inductance of single coil (constant part L0 ) is
0.106 u 102 H, but if we consider only the fundamental wave, the self inductance is
only 0.884 u 104 H. It is obvious that the harmonic mmf and harmonic flux density
produced by the single coil current are very strong and need to be calculated.
The formula (1.3.16) can be changed to
LG
k yk k yj
2 wK2 W l
[(Odkj Oqkj ) cos(k j )J (Odkj Oqkj ) cos( k j )J )]
¦¦
PS k j kj
(1.3.17)
Owing to | k r j | 2l , l
0,1, 2," , the above formula can be written as
LG (J )
L0G L2 cos 2J L4 cos 4J "
(1.3.18)
The above formula indicates that when the rotor position angle varies S electric
radians, the self inductance of single coil will repeat once.
Only the self inductance produced by the air-gap magnetic field is discussed
above. After calculating the self inductance L0l produced by slot leakage flux
and end leakage flux (for leakage inductances, please see Section 1.8), the self
inductance of the coil AAc is
L(J )
L0 L2 cos 2J L4 cos 4J "
(1.3.19)
where
L0
L0l L0G
2
º
2 wK2 W l ª§ k yk ·
«
»
(
O
O
)
L0l ¨
¸
¦
dkk
qkk
P S 2 k «© k ¹
»¼
¬
(1.3.20)
15
AC Machine Systems
L2
º
2wK2 W l ­° ª k yk k y (2 k )
(Odk (2 k ) Oqk (2 k ) ) »
2 ®¦ «
PS ¯° k ¬ k (2 k )
¼
ª k yk k y ( k 2)
º ½°
2¦ «
(Odk ( k 2) Oqk ( k 2) ) » ¾
k ¬ k ( k 2)
¼ ¿°
In the summation symbol
¦ of
(1.3.21)
formula(1.3.20) and the second summation
k
1 2 3
, , ," , and in the first summation
P P P
k
1 2
2P 1
symbol ¦ of formula(1.3.21) there exists k
, ," ,
.
P
P
P
k
From formula (1.3.8) and formula (1.3.10), we can get
symbol
¦ of formula (1.3.21), there is k
Odkk Oqkk
O0
(1.3.22)
Odk (2 k ) Oqk (2 k )
O2
(1.3.23)
Odk ( k 2) Oqk ( k 2)
O2
(1.3.24)
Hence, formula (1.3.20) and formula (1.3.21) are simplified further as
1.3.4
L0
2 w2 W l O
L0l K 2 0
PS
L2
2wK2 W l O2
PS 2
§k ·
¦k ¨ kyk ¸
©
¹
2
ª k yk k y ( k 2) º °½
°­ ª k yk k y (2 k )
2¦ «
®¦ «
»¾
k ¬ k ( k 2) ¼ ¿
°¯ k ¬ k (2 k )
°
(1.3.25)
(1.3.26)
Mutual Inductance Between Coils
Calculating the mutual inductance between coils of stator winding is similar to
the calculation of self inductance, and the only difference is that the angle D of
the two coils should be considered. If the axis of coil BBc lags behind the axis of
coil AAc by D electric angle, then in the process of calculating the mutual fluxlinkage of coil BBc caused by the current of coil AAc , the upper limit and lower
ES
ES
and J D , respectively, i.e.,
limit of the integral should be J D 2
2
16
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
wKW l J D ES
ES Bdkjm cos jxdx
S ³ J D \ dkjAB
4wK2 W l
k yk k yj Odkj i cos kJ cos j (J D )
PS2 kj
(1.3.27)
wKW l J D ES
ES Bqkjm sin jxdx
S ³ J D \ qkjAB
4 wK2 W l
k yk k yj Oqkj i sin kJ sin j (J D )
PS2 kj
(1.3.28)
The corresponding mutual inductances are
M dkjAB
4 wK2 W l
k yk k yj Odkj cos kJ cos j (J D )
PS2 kj
(1.3.29)
M qkjAB
4 wK2 W l
k yk k yj Oqkj sin kJ sin j (J D )
PS2 kj
(1.3.30)
Considering the total effects of every harmonic flux density produced by every
harmonic mmf, the mutual inductance between the single coil AAc and coil BBc
owing to air-gap magnetic field is
M ABG
¦¦ (M
k
dkjAB
M qkjAB )
j
k yk k yj
2 wK2 W l
^(Odkj Oqkj ) cos[(k j)J jD ]
¦¦
PS k j kj
(Odkj Oqkj ) cos[(k j )J jD ]`
(1.3.31)
1 2 3
, , ,"; j | k r 2l |; l 0,1, 2,"
P P P
After spreading formula (1.3.31), use the following relations:
where k
M dkjAB M qkjAB M djkAB M qjkAB
4wK2 W l k yk k yj ­
ª
( j k )D
D ·º
§
cos «(k j ) ¨ J ¸ »
®(Odkj Oqkj ) cos
PS
kj ¯
2
2 ¹¼
©
¬
( j k )D
D ·º ½
ª
§
cos «(k j ) ¨ J ¸ » ¾
(Odkj Oqkj ) cos
2
2 ¹¼ ¿
©
¬
(1.3.32)
17
AC Machine Systems
2
M dkkAB M qkkAB
2wK2 W l § k yk ·
¨
¸ ^(Odkk Oqkk ) cos kD
P S © k ¹
D ·½
§
(Odkk Oqkk ) cos 2k ¨ J ¸ ¾
2 ¹¿
©
(1.3.33)
where k and j are the certain values, the multiplier (k j ) or ( k j ) before the
D·
§
angle ¨ J ¸ is even only, and there is also
2¹
©
Odkk Oqkk
1
1
(O0 O2 k ) (O0 O2 k )
2
2
O2 k
Only when k is a positive integer, O2k has its value, otherwise O2 k 0 .
The formula (1.3.31) can be written as even harmonic cosine series of the
D·
§
angle ¨ J ¸ , i.e.,
2¹
©
M ABG
D·
D·
§
§
M AB 0G M AB 2 cos 2 ¨ J ¸ M AB 4 cos 4 ¨ J ¸ "
2¹
2¹
©
©
(1.3.34)
When the axis of the rotor pole is located at the center of the two coils, i.e.,
J
D
0 , the mutual inductance reaches its extremum.
2
After considering the mutual inductance M AB 0l caused by the leakage flux, the
mutual inductance between coils AAc and BBc is
M AB
D·
D·
§
§
M AB 0 M AB 2 cos 2 ¨ J ¸ M AB 4 cos 4 ¨ J ¸ "
2¹
2¹
©
©
(1.3.35)
where
M AB 0
M AB 2
M AB 0l
2
º
2 wK2 W l ª§ k yk ·
«
¸ (Odkk Oqkk ) cos(kD ) »
2 ¦ ¨
Pʌ k «© k ¹
»¼
¬
(1.3.36)
º
2 wK2 W l °­ ª k yk k y (2 k )
(Odk (2 k ) Oqk (2 k ) ) cos(1 k )D »
2 ®¦ «
PS °¯ k ¬ k (2 k )
¼
ª k yk k y ( k 2)
º °½
2¦ «
(Odk ( k 2) Oqk ( k 2) ) cos(1 k )D » ¾
k ¬ k ( k 2)
¼ °¿
(1.3.37)
Based on formulas (1.3.22), (1.3.23), and (1.3.24), the above formula can be
18
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
simplified further as
2 w2 W l O
K 2 0
PS
M AB 0
M AB 0l
M AB 2
2 wK2 W l O2
PS2
2¦
k
ª§ k yk ·2
º
«
»
k
cos
D
¨
¸
¦k « k
»¼
¹
©
(1.3.38)
­° ª k yk k y (2 k )
cos(1 k )D
®¦ «
¯° K ¬ k (2 k )
º ½°
cos(1 k )D » ¾
k (k 2)
¼ ¿°
k yk k y ( k 2)
(1.3.39)
It is needed to point out that when the axes of coils BBc and AAc coincide with
each other, i.e. D 0 , the formulas (1.3.38) and (1.3.39) become formulas (1.3.25)
and (1.3.26), which accords with the physical concept.
In addition, in formulas (1.3.19) and (1.3.35), the stator reference axis is the
axis of coil AAc . If the reference axis is another axis, and the axis of coil AAc
lags behind the new axis by D a electric angle, then J in the formulas should be
replaced by ( J D a ).
In the mutual inductance between two coils, the influence of fractional harmonics
and low order harmonics of the air-gap magnetic field is very strong. For example,
in the 550kW salient-pole synchronous machine mentioned above, its mutual
inductance between two adjacent coils (merely counting the constant value part
which does not vary with the rotor position) is 0.962 u 103 H ; but if only the
fundamental wave is considered, then this mutual inductance will be 0.830 u 10 4 H
only, the difference between the two values is very large, which is produced by
very strong fractional harmonics and low order harmonics of the air-gap magnetic
field.
Finally, it should be pointed out that, for calculating self inductance and
mutual inductance according to formulas (1.3.19) and (1.3.35), generally, taking
only two items (constant item and second harmonic item) will satisfy the precision
demand.
1.4
Inductances of Stator Loops
In practice, it is often needed to research some problems related to the electric
machine loops. But, the constitution of the electric machine loop is different from
the researched problems. Because the electric machine stator loops are composed
of related coils, it is very easy to calculate the inductances of every loop based on
the inductances of associated coils.
At first, let us study the simple loop composed of two coils a1 , a2 as in Fig. 1.4.1.
Suppose that La1 , La2 are self inductances of coils a1 and a2 , M a1a2 is mutual
19
AC Machine Systems
inductance between the two coils, then, in terms of the basic definition of the
inductance, it is known that the self inductance of the whole loop-a (branch) is
La
\a
\ a \ a
i
i
1
2
The flux-linkages of coils a1 and a2 are, respectively
\a
\ a a \ a a
\a
\ a a \ a a
1
2
1 1
1 2
2 1
2 2
where \ a1a1 ,\ a2 a2 are self flux-linkages, \ a1a2 or \ a2 a1 is the mutual flux-linkage of
the two coils. Hence,
\ a a \ a a \ a a \ a a
La
1 1
1 2
2 1
i
Figure 1.4.1 Self inductance of the
circuit composed of coils a1 and a2
2 2
La1 La2 2M a1a2
Figure 1.4.2 Mutual inductance of
two loops a and b
Similarly, if the stator loop a is composed of two coils a1 , a2 , stator loop b is
composed of coils b1 , b2 as shown in Fig. 1.4.2, then the mutual inductance of
loops a and b is formed by the mutual inductances of these coils, i.e.,
M ab
M a1b1 M a2b1 M a1b2 M a2b2
If M S ( i )Q ( j ) represents the mutual inductance between i -th coil of loop S and
j -th coil of loop Q of the stator, then in terms of formula (1.3.35), the mutual
inductance between loops S and Q will be
m
M SQ
n
¦¦ M
S (i )Q ( j )
i 1 j 1
M SQ 0 M SQ 2 cos 2(J D SQ 2 ) M SQ 4 cos 4(J D SQ 4 ) "
20
(1.4.1)
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
where m is the coil number of loop S, n is coil number of loop Q.
As is the same as formulas (1.3.19) and (1.3.35), for formula (1.4.1), generally,
taking only the first two items will satisfy the demand of engineering calculation.
If the constant item of mutual inductance between i -th coil of loop S and
j -th coil of loop Q is denoted by M S ( i ) Q ( j )0 , then the constant item of formula
(1.4.1) can be calculated by
m
M SQ 0
n
¦¦ M
(1.4.2)
S ( i ) Q ( j )0
i 1 j 1
As for the amplitude M SQ 2 and phase angle D SQ 2 of the second harmonic item
in formula (1.4.1), it can be calculated in terms of the following approach.
Suppose that the amplitude and phase angle of the second harmonic item of
mutual inductance between i-th coil of loop S and j -th coil of loop Q is
denoted by M S ( i ) Q ( j )2 and D S (i ) Q ( j )2 , respectively (the phase angle equals half of
the included angle between the axes of these two coils), then
m
M SQ 2 cos 2(J D SQ 2 )
n
¦¦ M
S ( i ) Q ( j )2
cos 2(J D S (i ) Q ( j )2 )
(1.4.3)
i 1 j 1
When 2J
0 , then
m
M SQ 2 cos 2D SQ 2
n
¦¦ M
S ( i ) Q ( j )2
cos 2D S ( i )Q ( j )2
(1.4.4)
sin 2D S ( i )Q ( j )2
(1.4.5)
i 1 j 1
When 2J
S
, then there is
2
m
M SQ 2 sin 2D SQ 2
n
¦¦ M
S ( i ) Q ( j )2
i 1 j 1
M SQ 2 and tan 2D SQ 2 can be calculated from formulas (1.4.4) and (1.4.5), then
D SQ 2 will be obtained.
The effect of the harmonics of air-gap magnetic field on the inductances of
stator branches is also very strong, especially for fractional harmonics and low
order harmonics. In the same example of the 550kW salient pole synchronous
machine, the self inductance of a branch and mutual inductance of adjacent
branches in the same phase (merely counting the constant part independent of the
rotor position) are 0.185 u 101 H and 0.134 u 103 H , respectively, but if only
the air-gap fundamental magnetic field is considered, then the two inductances
are 0.293 u 102 H . It is obvious that the effect of harmonics on the inductances of
stator branches is very strong.
21
AC Machine Systems
It can be seen from formulas (1.3.19), (1.3.35) and (1.4.1) that the self
inductance and mutual inductance of stator single coil and the inductances of
stator loops are relative to the rotor position angle J , i.e. they are time-variant
parameters. For non-salient pole electric machine and induction machine, there
is OG ( x)
O0
2
, i.e. the above inductances are constants independent of rotor
position.
Finally, it should be stated that in most instances the phase winding can be
considered as a researched unit. In anordinary designed electric machine, the
distributed and fractional-pitch winding is adopted generally in the phase winding,
hence the harmonics of the air-gap magnetic field are weakened greatly. Please
refer to details related to the problem in Section 2.1.
When the inductances are calculated concretely in terms of formulas (1.3.25),
(1.3.26), (1.3.38), and (1.3.39), it is needed to sum the infinite series, and it is
enough to calculate the finite terms by the ordinary method. But, for the
inductances of single coil, only calculating the several terms of the series is not
sufficient owing to strong harmonic effect of air-gap magnetic field. The relation
between the inductance M 0 ( j ) of stator single coil of a 550kW salient pole
electric machine and the calculated highest harmonic order k is listed in Table
1.4.1, in which M 0 ( j ) represents the constant item of the inductance between
coil j and the first coil owing to air-gap magnetic field. The stator coil pitch of
the electric machine is 8, and in the table the constant items of inductances
M 0 ( j ) are listed at k 7, 27,127 . It can be seen that taking 27 as the calculated
highest order harmonic is sufficient for calculating single coil inductances, of
course, much precise result can be obtained by taking higher harmonic order, but
the effect is not obvious already.
Table 1.4.1 Relation between M 0 ( j ) of a 550 kW salient-pole electric machine
and harmonic order k
j
M0 ( j) / H
1
2
3
4
5
6
7
7
1.059
0.962
0.794
0.657
0.505
0.353
0.214
27
1.089
0.950
0.802
0.653
0.505
0.357
0.208
127
1.096
0.950
0.802
0.653
0.505
0.356
0.208
12
30
50
k
j
M0 ( j) / H
8
k
22
9
10
11
7
0.052 4
0.068 7 0.094 2 0.086 3 0.089 2 0.088 5 0.082 8
27
0.059 6
0.083 7 0.089 0 0.088 7 0.088 6 0.088 5 0.088 0
127
0.059 7
0.087 6 0.088 8 0.088 7 0.088 6 0.088 6 0.088 0
1
1.5
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
Inductances of Rotor Loops
Besides excitation winding, normally the damper winding is located on the rotor
of the salient pole synchronous machine. Their inductances will be analysed in
the following section.
1.5.1
Inductance of Excitation Winding
In general, the excitation winding on each pole is concentrated winding, the
windings on each pole may be in series or in parallel connection. The inductance
of excitation winding is discussed here. When other connection is adopted for
excitation winding or when internal fault occurs in it, its inductance can be
obtained by a similar method, the details are not repeated here.
The self inductance of excitation winding consists of two parts, i.e.,
L fd
L fdG L fdl
(1.5.1)
where L fdl is the inductance corresponding to end region leakage flux, leakage
flux between rotor poles and leakage flux of pole surface; L f d G is the inductance
corresponding to air-gap flux density.
The mmf produced by the current of excitation winding is rectangular wave,
the excitation mmf F ( x) and the corresponding air-gap flux density B ( x) are
drawn in Fig. 1.5.1. Suppose that the turn number of excitation winding on each
pole is w fd , the parallel branch number of excitation winding is a fd , total
excitation current is i fd , then the branch current of excitation winding will be
i fd
a fd
. Each pole excitation mmf is Ff
w fd
i fd
a fd
, and it can be considered as
Figure 1.5.1 Excitation mmf and its air-gap flux density
23
AC Machine Systems
rectangular wave with a width of S radians approximately. Here, the air-gap flux
density is
B f ( x)
P 0 Ff
G ( x)
i fd
P 0 w fd
1
(
a fd G x)
Wl
i fd
(1.5.2)
Each pole flux is
)fd
Wl
S³
S
S
B f ( x)dx
w fd
S
a fd
³
S
S
P
dx
G ( x)
From formula (1.2.4), we can see
)fd
W lw fd i fd
2
a fd
O0
Each pole flux-linkage is
\ fd
w fd)fd
Wl
2
w2fd
i fd
a fd
O0
Therefore, the self inductance of excitation winding corresponding to air-gap
flux density is
L fdG
1.5.2
2 P \ fd
a 2fd i fd
a fd
W lP
a 2fd
w2fd O0
(1.5.3)
Inductance of Damper Winding
Normally, the damper winding of the salient pole synchronous machine is cage
type with damping bars and damping rings; the damping bars are short-circuited
by damping rings fully or partially, and it is usually symmetric to the direct axis
or quadrature axis. In such cage circuit, the selection of current loop may be
arbitrary. Figure 1.5.2 represents damper winding with 4 damping bars per pole;
in the figure the selected damping loops are symmetric to the direct axis or
quadrature axis, and loops 1d , 2d are symmetric to direct axis, loops 1q, 2q are
symmetric to quadrature axis. Actual currents in the damping bars are algebraic
sum of the corresponding direct and quadrature axis loop currents. When only the
space fundamental of the air-gap magnetic field is considered, obvious merits
exist for such selected damping loops as follows: since the space angle difference
S
electric radians, and in
between direct and quadrature axis current loops is
2
24
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
structure the rotor is symmetric to direct axis and quadrature axis, mutual
inductance and mutual resistance do not exist between the direct and quadrature
axis current loops, then the actual damping winding can be considered as two
imaginary damping winding groups shown in Fig. 1.5.3, namely direct axis
damping winding and quadrature axis damping winding.
Figure 1.5.2
Damping winding of salient pole synchronous machine
Figure 1.5.3 Direct axis and quadrature axis damping winding of salient pole
synchronous machine
In practical salient pole synchronous machine, there are many damping bars,
accordingly, there are many direct axis damping windings 1d , 2d ," , nd and many
quadrature axis damping windings 1q, 2q," , nq. Therefore, normally two methods
25
AC Machine Systems
are adopted for the analysis and calculation of damping winding; one is to calculate
the parameters of each damping winding in terms of practical distribution of
damping windings, the other is to replace all damping winding loops by a direct
axis and a quadrature axis equivalent damping windings. Respective merits and
shortcomings exist for these two methods: using the former the practical instance
of every damping bar can be obtained, but the workload for calculating it is greater;
using the latter the analysis can be simplified when the internal status of the
damping winding does not need to be researched, but its equivalent parameters
are difficult to be calculated and the error of calculation is difficult to be avoided.
For more general instance, when strong space harmonics of air-gap magnetic
field exist or internal fault of damping winding occurs, the above direct axis and
quadrature axis damping windings will be relative to each other. In such an
instance, it is more suitable to select damping loops based on Fig. 1.5.4. The
inductances of such damping loops are discussed in the following. As for the
parameter calculation relative to d-axis and q-axis damping windings or equivalent
damping loops, the details are available in various books and hence do not need
to be discussed here.
Figure 1.5.4 Selection of damping loops of salient-pole synchronous machine
Referring to Fig. 1.5.5, assume the damping loop 11c to lead ahead of d-axis
by D1 electric angle; when a current i flows into the damping loop a rectangular
wave mmf appears, using the Fourier analysis we can obtain
F ( x1 )
¦a
n
n
cos
n
x1
P
n 1, 2,"
(1.5.4)
Figure 1.5.5 Model for calculating inductances of salient pole synchronous machine
26
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
nE S
2
wr i sin 1 , wr is the turn number of the damping loop (normally
2P
nS
wr 1 ), E1 is short pitch ratio of the damping loop; x1 is the abscissa on the
rotor, its origin is located on the axis of damping loop 11c .
Formula (1.5.4) contains only cosine items owing to the symmetry of rectangular
wave mmf to the axis of damping loop.
For the convenience of representation, the wave with 2S electric radians is
considered as the fundamental wave, in this instance formula (1.5.4) should be
rewritten as
1 2 3
F ( x) ¦ Fk cos kx1 k
(1.5.5)
, , ,"
P
P P
k
where
2E S
2
Fk
wr i sin 1
kPS
where an
When the mmf formula is transformed to the abscissa x in which the origin is on
d-axis, then the corresponding mmf expression is
F c( x)
¦F
k
cos k ( x D1 )
k
k
1 2 3
, , ,"
P P P
(1.5.6)
The air-gap flux density produced by the mmf is
BG ( x)
F c( x)OG ( x)
¦F
k
k
ªO
º
cos k ( x D1 ) « 0 ¦ O2l cos 2lx »
l
¬2
¼
l 1, 2,"
(1.5.7)
in which the air-gap flux density caused by k - th harmonic mmf is
B( x)
ªO
º
Fk cos k ( x D1 ) « 0 ¦ O2l cos 2lx »
2
l
¬
¼
Fk
O0
2
O2l
cos k ( x D 1 ) ¦ Fk
2
l
cos[(k 2l ) x kD1 ]`
^cos[(k 2l ) x kD1 ]
l 1, 2,"
(1.5.8)
From the above formula, the k - th harmonic flux density in it is
Bkk ( x)
Fk
O0
2
cos k ( x D1 ) Fk
Fk cos kD1
O0 O2 k
O2 k
2
cos k ( x D1 )
cos kx Fk sin kD1
O0 O2 k
2
2
Fk cos kD1Odkk cos kx Fk sin kD1Oqkk sin kx
sin kx
(1.5.9)
27
AC Machine Systems
The j - th harmonic flux density is
Bkj ( x)
Fk
O|k j|
2
cos( jx kD1 ) Fk
Fk cos kD1
O|k j| O|k j|
O|k j|
2
cos( jx kD1 )
cos jx Fk sin kD1
O|k j| O|k j|
2
Fk cos kD1Odkj cos jx Fk sin kD1Oqkj sin jx
2
sin jx
(1.5.10)
In the above formula, | k j | and | k j | are obtained in such a way: In
formula (1-5-8), when j k r 2l , j k r2l , so there is 2l | k j | ; when
j k 2l (since k and 2l are positive, j cannot equal k 2l ), then
2l | k j | .
If according to formula (1.5.6), the k - th harmonic mmf is decomposed into
direct axis and quadrature axis mmf s firstly, then k - th and j - th harmonic flux
densities produced by the mmfs are obtained respectively, its result is the same
as formulas (1.5.9) and (1.5.10).
From formulas (1.5.9) and (1.5.10), it is known that j - th air-gap harmonic
flux density is
B j ( x)
Fj
O0
2
cos j ( x D1 ) ¦
¦
Fk
O2l
2l |k j |
Fk
2l |k j |
O2l
2
cos( jx kD1 )
2
cos( jx kD1 )
2l
2, 4,"
After arranging the above formula we get
B j ( x)
¦
2l |k j |
Fk
O2l
2
cos( jx kD1 ) ¦
2l |k j |
Fk
O2l
2
cos( jx kD1 )
(1.5.11)
1 2 3
, , ,";| k j | 0, 2, 4,";| k j | 2, 4,"
P P P
Suppose D1 0 in formula (1.5.11), the formula will be simplified as
where k
B j ( x)
ª O2l
O2l
º
« Fk 2 2l |k j| Fk 2 2l |k j| » cos jx
¬
¼
Fk Odkj cos jx | k j | 0, 2, 4,"; | k j | 2, 4,"
Here, only the d-axis mmf exists, i.e. the above formula is the expression
of j - th harmonic flux density produced by d-axis k - th harmonic mmf when
the axis of the damping loop11c and the rotor d-axis overlap each other.
28
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
After getting the expression of the air-gap flux density, the flux-linkage of each
loop is obtained, and the inductance of each loop will be gained.
Firstly, the self inductance of damping loop 11c is calculated. The self fluxlinkage caused by j -th harmonic flux density B j ( x) in damping loop 11c is
wrW l E1S
E S B j ( x )dx1
S ³ \j
wrW l E1S D1
B j ( x)dx
ES
S ³ D1
O
2 wr2W l ­
kE S
jE S
i ® ¦ 2l sin 1 sin 1 cos( j k )D1
PS ¯ 2l |k j| kj
¦
O2l
kj
2l |k j |
sin
k E1 S
jE S
½
sin 1 cos( j k )D1 ¾
¿
(1.5.12)
where | k j | 0, 2, 4,";| k j | 2, 4," . The corresponding self inductance is
O2l
2 wr2W l ­
kE S
jE S
sin 1 sin 1 cos( j k )D1
® ¦
PS ¯ 2l |k j| kj
Lj
¦
2l |k j |
O2l
kj
sin
½
k E1S
jE S
sin 1 cos( j k )D1 ¾
¿
(1.5.13)
where | k j | 0, 2, 4,";| k j | 2, 4," . After counting each harmonic of the
air-gap magnetic field, the total self inductance of damping loop 11c is
L
¦L
j
j
¦
2l |k j|
O2l
2 wr2W l ­
kE S
jE S
sin 1 sin 1 cos( j k )D1
¦® ¦
PS j ¯ 2l |k j| kj
O2l
kj
sin
½
k E1S
jE S
sin 1 cos( j k )D1 ¾
¿
(1.5.14)
1 2 3
, , ,";| k j | 0, 2, 4,";| k j | 2, 4,"; normally, it is sufficient
P P P
to take the definite order of k , j , 2l .
The mutual inductance between damping loops 11c and 22c is calculated in
the following (see Fig. 1.5.5). Suppose current i flows into the damping loop 11c ,
then j -th harmonic air-gap flux density produced by it is expressed as formula
(1.5.11), the flux-linkage of damping loop 22c is
where j
29
AC Machine Systems
wrW l E2 S
E S B j ( x )dx2
S ³ 2
\ 12 j
wrW l E2 S D 2
B j ( x)dx
E S
S ³ 2 D 2
O
kE S
jE S
2 wr2W l ­
i ® ¦ 2l sin 1 sin 2 cos( jD 2 kD1 )
PS ¯ 2l |k j| kj
O2l
¦
2l |k j |
kj
sin
½
k E1S
jE S
sin 2 cos( jD 2 kD1 ) ¾
¿
(1.5.15)
where | k j | 0, 2, 4,";| k j | 2, 4," ; x2 is the abscissa on the rotor and its
origin is on the axis of damping loop 22c ; E 2 is the short pitch ratio of damping
loop 22c . Corresponding mutual inductance is
M 12 j
O2l
2 wr2W l ­
kE S
jE S
sin 1 sin 2 cos( jD 2 kD1 )
® ¦
PS ¯ 2l |k j| kj
¦
O2l
2l |k j |
M 12
¦M
kj
sin
½
k E1S
jE S
sin 2 cos( jD 2 kD1 ) ¾
¿
(1.5.16)
12 j
j
O2l
2wr2W l ­
kE S
jE S
sin 1 sin 2 cos( jD 2 kD1 )
¦® ¦
PS j ¯ 2l |k j| kj
¦
2l |k j |
O2l
kj
sin
½
k E1S
jE S
sin 2 cos( jD 2 kD1 ) ¾
¿
(1.5.17)
1 2 3
, , ,";| k j | 0, 2, 4,";| k j | 2, 4,"
P P P
If D 2 D1 and E 2 E1 are substituted into formula (1.5.17), formula (1.5.14) is
obtained. That is, self inductance is a special example of mutual inductance.
where j
1.5.3
Mutual Inductance Between Excitation Winding and
Damping Loop
As mentioned above, suppose that current i fd flows into excitation winding, its
mmf is rectangular wave, the air-gap flux density B ( x ) produced by it (refer to
Fig. 1.5.1) is
30
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
B( x)
F ( x)OG ( x)
F ( x)
P0
G ( x)
(1.5.18)
From Fig. 1.5.1, we can see
B( x)
B( x),
B( x)
B (S r x)
So, after the Fourier analysis, only cosine items exist in B ( x ) , and only odd
order harmonics exist, its general expression is
B( x)
¦B
fkm
cos kx
k 1,3,5,"
(1.5.19)
k
where
B fkm
2 S
B( x) cos kxdx
S ³
P
2 S
F ( x) 0 cos kxdx
³
S
G ( x)
4 S Ff P0
cos kxdx
S ³ G ( x)
(1.5.20)
Corresponding permeance coefficient is
Odk
B fkm
Ff
4 S P 0
cos kxdx
S ³ G ( x)
k 1,3,5,"
(1.5.21)
After comparing formula (1.5.21) with formula (1.2.5), it is known that the
calculation formulas of air-gap permeance coefficient O2l of salient pole
synchronous machine and the permeance coefficient Odk corresponding to
harmonics flux density produced by rectangular wave excitation mmf are similar
in form. Their only difference is that 2l in formula of O2l is even, and k in
formula of Odk is odd.
The flux-linkage of k - th harmonic air-gap flux density in damping loop 11c
produced by the current i fd of excitation winding is
\ 1 fk
wrW l E1S D1
B fkm cos kxdx
ES
S ³ 1 D1
2 wr w fdW l i fd Odk
kE S
sin 1 cos kD1
S
a fd k
(1.5.22)
The mutual inductance corresponding to k -th harmonic air-gap flux density is
31
AC Machine Systems
2wr w fdW l 1 Odk
kE S
sin 1 cos kD1
S
a fd k
M 1 fk
(1.5.23)
Considering all the air-gap flux density harmonics, the total mutual inductance
between damping loop 11c and excitation winding is
M1 f
¦M
1 fk
k
2 wr w fdW l 1
S
a fd
¦
k
Odk
k
sin
k E1S
cos kD1
k 1,3,5,"
(1.5.24)
If we suppose that the current flows into damping loop 11c firstly, then when we
calculate the flux-linkage of excitation winding, the same result as formula (1.5.24)
will be obtained.
Finally, it should be pointed out that the inductances of rotor loops are
independent of the rotor position, and formulas (1.5.14), (1.5.17), and (1.5.24)
reflect this feature. This is in accord with the rule, because stator iron is round.
1.6 Mutual Inductances Between Stator Loops and Rotor
Loops
Inductances between stator loops and rotor loops are changed with the rotor
position owing to the rotor rotation, i.e. they are time-variant coefficients.
Firstly, the mutual inductance between stator single coil AAc and excitation
winding is calculated. Suppose that the excitation winding connection is as in
Section 1.5, the air-gap flux density produced by excitation current i fd is seen from
formula (1.5.19). The flux-linkage in stator coil AAc owing to k -th harmonic air
gap flux density is
\ fak
wKW l ES
ES B fkm cos kxdT
S ³ wKW l ES J
ES B fkm cos kxdx
S ³ J
where T x J .
The expression of B fkm in formula (1.5.20) is substituted into the above
formula, there is
\ fak
2 w fd wKW l i fd Odk
kES
sin
cos kJ
S
a fd k
Corresponding mutual inductance is
32
(1.6.1)
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
2 w fd wKW l 1 Odk
kES
sin
cos kJ
S
a fd k
M fak
(1.6.2)
When the total of all air-gap flux density harmonics is obtained, the total mutual
inductance between stator coil AAc and excitation winding is
M fa
¦M
2 w fd wKW l 1
S
a fd
fak
k
¦
k
Odk
k
sin
kES
cos kJ
k 1,3,"
(1.6.3)
It may be seen that the mutual inductance depends upon the rotor position.
Then, the mutual inductance between stator single coil AAc and damping loop
11c is analyzed. Referring to Fig. 1.5.5, suppose the current i flows into damping
loop 11c , then j -th harmonic air-gap flux density produced by it is as in formula
(1.5.11), and the flux-linkage relative to B j ( x) for coil AAc is
\ 1aj
wKW l ES
E S B j ( x)dT
S ³ wKW l ES J
E S B j ( x )dx
S ³ J
O
kE S
2 wK wrW l ­
jE S
i ® ¦ 2l sin 1 sin
cos( jJ kD1 )
PS
¯ 2l |k j| kj
¦
O2l
2l |k j|
kj
sin
½
k E1 S
jE S
sin
cos( jJ kD1 ) ¾
¿
(1.6.4)
where | k j | 0, 2, 4,";| k j | 2, 4,"
Considering all the air-gap flux harmonics, the total mutual inductance between
stator single coil AAc and damping loop 11c is obtained as
M 1a
O2l
2wK wrW l ­
kE S
jE S
sin 1 sin
cos( jJ kD1 )
® ¦
¦
PS
kj
j ¯ 2l |k j |
¦
2l |k j |
O2l
kj
sin
½
k E1S
jE S
sin
cos( jJ kD1 ) ¾
¿
(1.6.5)
where | k j | 0, 2, 4,"; | k j | 2, 4,"
1 2 3
, , ,"
In formulas (1.6.4) and (1.6.5), j
P P P
In formulas (1.6.4) and (1.6.5), the mutual inductance is calculated by the
current flowing into the rotor. If it is calculated by the current flowing into the
stator, the result is the same.
33
AC Machine Systems
Comparing formula (1.6.5) with (1.5.17), it is seen that the difference of both
is to make the following substitution:
wK o wr ,
E o E 2 , J o D 2
But, the difference of the two formulas in meaning is obvious: the result of formula
(1.5.17) is constant, i.e. the mutual inductances between damper loops are not
relative to rotor position; and the result of formula (1.6.5) is function of the
angle J , i.e. the mutual inductances between damper loop and stator coil is
relative to rotor position.
After obtaining the mutual inductances between stator single coil and excitation
winding or damping loop, the mutual inductance between stator loop and rotor
loop consisting of them will be obtained easily.
Finally, it should be pointed out that in the calculation of the self inductance of
excitation winding, mutual inductance between excitation winding and damping
loop, and mutual inductance between excitation winding and stator loop as in front,
all excitation winding is considered as series or parallel connection. If other
connections are adopted for excitation winding, or internal fault of excitation
winding occurs, the excitation winding will be divided into several loops; at this
time the inductances of excitation loops and mutual inductances between them
and damper loops or stator loops need to be calculated, respectively. It is not
discussed further here.
1.7
Influence of Saturation on Parameters
In discussing the inductances related to air-gap magnetic field in the above
sections, the concept of air-gap permeance and harmonic analysis method are
adopted, and it is supposed that mmfs are all spent in air-gap; as for iron
reluctance, it is considered by enlarging air-gap length suitably.
In the unsaturation state of electric machine, the iron section works in the
linear part of the B-H curve, and the relation of excitation current and the no-load
voltage of electric machine is also at linear part. Under this instance, compared
with air-gap reluctance the iron reluctance is less, so it may be ignored, or air-gap
is enlarged in terms of the ratio between iron reluctance and air-gap reluctance or
according to the linear part of the no-load characteristic from experiment.
Normally, an electric machine operates near saturation state, at this time the
effect of iron reluctance is boosted obviously, here the influence of saturation
may be considered as follows.
If the electric machine operates in no-load, the saturation coefficient K may be
confirmed according to the no-load characteristic of the electric machine, and the
34
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
air-gap should be enlarged to K times. As in Fig. 1.7.1, the excitation current
corresponding to rating voltage U N is i f 0 ; if saturation is not considered, the
excitation current is icf 0 according to air-gap line, so the saturation coefficient is
K
if 0
i cf 0
df
de
Figure 1.7.1 Confirmation of saturation coefficient based on no-load characteristic
Figure 1.7.2 Phasor diagram of electric machine operation on load
If the electric machine operates on load, the corresponding phasor diagram is
shown in Fig. 1.7.2. Here, air-gap emf corresponding to air-gap flux density is EG .
Based on the relation as in Fig. 1.7.2 and after getting EG , the saturation coefficient
will be obtained in terms of EG and no-load characteristic (refer to Fig. 1.7.1) as
follows:
Kc
d cf c
d cec
The effect of saturation is researched by using this method based on fundamental
wave magnetic circuit under normal operation. If the researched problem is related
35
AC Machine Systems
to the instance with strong space harmonics of air-gap magnetic field, using this
method will definitely lead to error. If needed, numerical calculation may be used
to analyze the distribution of the magnetic field in terms of the loops and the
magnetic field of the electric machine. When this kind of problem is researched,
normally the electromagnetic field of the whole electric machine needs to be
calculated. Since the air-gap of the salient pole synchronous machine is non-uniform,
analysis and calculation of many electromagnetic fields are also needed. Therefore,
in the instance of definite computer memory and calculation speed, sometimes it is
difficult to adopt very thin mesh to reach the expected goal.
Each loop parameter under saturation is calculated by the above two methods
respectively on a 30 kVA simulation electric machine (6-pole salient pole
synchronous machine). As an example, the mutual inductance of the first coil and
j -th coil on stator is
M ( j)
M 0 ( j ) M 2 ( j ) cos 2(J D ( j ))
where j 1, 2," , Z1 ( Z1 is stator slot number). When j 1 , the mutual inductance
becomes self inductance. Figure 1.7.3 shows M 0 ( j ) obtained by two methods
(including the inductance M 0G ( j ) related to air-gap magnetic field and leakage
inductance M 0l ( j ) ). When the electromagnetic field of the whole electric machine
is calculated, the cell number is taken as 3132, the node number is 1621. From
the figure, we can see that the results of the two methods are the same. The pitch
of the stator coil of the electric machine is 7. When j ı 8 , the overlapping part
no longer exists between the j -th coil and the first coil, hence using the permeance
analysis method M 0G ( j ) is constant; then adopting the numerical analysis method,
the larger the j is, the less the absolute value of M 0G ( j ) is. The latter result is
more rational.
Figure 1.7.3 Comparison of M 0 ( j ) by two methods
36
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
1.8 Inductances Produced by Leakage Magnetic Field
Inductances produced by air-gap magnetic field (main magnetic field) are discussed
in the above sections. Leakage magnetic field is not the main part of the whole
magnetic field, but the leakage magnetic field should not be ignored, and it
should be considered in calculating the inductances.
1.8.1
Inductance Produced by Rotor Leakage Magnetic Field
The problem has been discussed in a lot of literatures[8], here we only list the result.
(1) Leakage self inductance of excitation winding of salient pole synchronous
machine is
Lf V
2 PP0
w2 fd
a 2fd
Of l
(1.8.1)
where O f is the specific leakage permeance of excitation winding; l is rotor iron
length; and
§ dt
·
§a
·
0.25 ¸ 1.75 ¨ P 0.20 ¸
© cP
¹
© cP
¹
O f | 4.00 ¨
2
§h ·
§a
·
§b ·
1.27 ¨ P 0.50 ¸ 1.15 ¨ m ¸ 0.44 ¨ m ¸
© l ¹
© cP
¹
© cm ¹
in which
dt
cP
cm
bP2
4 D1
2Sdt
W bP 2P
ʌ
(hm 2hP 2G )
W bm 2P
hP G The size of each part of the magnetic pole of the salient pole synchronous machine
is shown in Fig. 1.8.1, here D1 is the inter diameter of the stator iron; G is air-gap
length; hP and bP are the height and width of the pole shoes respectively; hm
and bm are the height and width of the pole body, respectively.
(2) Leakage self inductance of damping winding of the salient pole synchronous
machine is
Lc
P 0lc (Oc Ot )
37
AC Machine Systems
where lc is rotor pole length; Oc is the specific slot leakage permeance; Ot is the
specific tooth top leakage permeance; and
Oc
0.30 0.64
1 bs / d c hs
1 bs / d c bs
(for circular bar)
2
Ot
§ 2G ·
1 ¨ ¸
1
2G
2G G
© bs ¹
ln
arctan
ʌ
2
ʌbs
bs bs
The size of each part is referred to in Fig. 1.8.2.
Figure 1.8.1 Size of each part of magnetic
pole of salient pole synchronous machine
Figure1.8.2 Size of damping bar hole
of salient pole synchronous machine
(3) Leakage inductance of each segment of the end ring of the salient pole
synchronous machine is
Le
·
t2 P 0 § 2le
0.25 ¸
¨ ln
2 ʌ © re
¹
where t2 is length of each segment of the end ring; le is the interval between the
end ring and rotor end surface; re is the equivalent radius of the cross-section of
the end ring.
1.8.2
Inductances Produced by Stator Leakage Magnetic Field
In analyzing the inductance of each stator loop produced by the air-gap magnetic
field, the harmonic magnetic field is considered already, so the effect of stator
differential leakage reactance is included in this inductance. The inductances
produced by slot leakage and end part leakage are discussed below:
(1) Inductance produced by slot leakage flux
With regard to inductance produced by slot leakage flux, detailed discussion
and well known calculation method already exist. If the leakage flux is not in
38
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
saturation state and the reluctance of iron part is ignored, then the mutual
inductance produced by slot leakage flux exists only between the coil bars in the
same slot, and the mutual inductance produced by slot leakage flux does not exist
between the coil bars in different slots. After knowing the connection of stator
winding, the inductance of each stator loop produced by slot leakage flux may be
calculated by adopting the method of searching coil bars in the same slots one by
one. It is very convenient to do this using computer.
(2) Inductance produced by end part leakage flux
It is very difficult to calculate exactly the inductance produced by end part
leakage flux; relatively rigorous method of calculating the end part leakage flux
is numeration of the end region magnetic field. Under the state of 3-phase symmetry,
the 3-phase currents are treated as equivalent current sheet. But, this method is not
suitable for the inductance of stator single coil, discrete mode may be used to deal
with it. Concrete method is as follows. When calculating the end region magnetic
field produced by single coil current, the coil end part is divided into several
current elements, and the magnetic field produced by the current elements is
calculated in terms of Biot-Savart law. The magnetic field on each point of end
region is the superposition of the magnetic field produced by all current elements.
After getting the magnetic field, the flux-linkage of the end part of each coil can be
calculated, and then the inductance produced by end region leakage flux will be
obtained. In the calculation, the effect of stator iron is replaced by image current,
the effect of air-gap is replaced by air-gap current; now the end region magnetic
field problem produced by a single coil current becomes the problem of the
magnetic field produced by coil end part current, air-gap current and their image
currents altogether in uniform medium. In the calculation process, both cartesian
coordinates system and cylindrical coordinates system may be adopted.
The process of calculating the inductance produced by stator end region
leakage flux can be found in Appendix C.
The calculation results of end part inductance of stator single coil of a 630kW
salient pole synchronous machine are listed in Table 1.8.1. From the table we can
Table 1.8.1 Comparison of calculated results of end part inductances in two
coordinates systems
unit: H
Slot number
between two coils
0
1
2
3
4
5
Calculated results in
Cartesian coordinates
0.914 u 104
0.311 u 104
0.206 u 104
0.550 u 105
0.395 u 105
0.466 u 105
Calculated results in
Cylindrical coordinates
0.895 u 104
0.298 u 104
0.198 u 104
0.513 u 105
0.380 u 105
0.485 u 105
39
AC Machine Systems
see that difference between the calculated results of the two coordinates systems is
not obvious. In the example, generally absolute error does not exceed 0.19u105 H ,
relative error is below 7%.
1.9
Electromagnetic Torque and Rotor Motion Equation
of AC Machines
Electromagnetic torque and rotor motion equation are important foundation for
analyzing the electric machine operation performance. The electromagnetic torque
of salient pole synchronous machine is discussed[9] here.
Suppose that the stator winding has m loops, the m-th loop in stator is
represented by sm ; the excitation winding has n loops, the n -th loop of
excitation winding is represented by fd n ; the damper winding has l loops, the
l -th loop of damper winding is represented by l ; then the total magnetic field
energy of the electric machine is
1
(\ s1 is1 \ s2 is2 " \ sm ism \ fd1 i fd1 \ fd2 i fd2 "
2
\ fdn i fdn \ 1i1 \ 2 i2 " \ l il )
Wm
(1.9.1)
where the flux-linkages of various loops are, respectively:
\s
Ls1 is1 " M s1sm ism M s1 fd1 i fd1 " M s1 fdn i fdn
1
M s11i1 " M s1l il
"
\ fd
1
(1.9.2)
M fd1s1 is1 " M fd1sm ism L fd1 i fd1 " M fd1 fdn i fdn
M fd11i1 " M fd1l il
"
\1
(1.9.3)
M 1s1 is1 " M 1sm ism M 1 fd1 i fd1 " M 1 fdn i fdn
L1i1 " M 1l il
"
(1.9.4)
In the above formulas, the minus before the related items of stator is caused by the
regulation that the positive current produces negative flux-linkage.
Total magnetic field energy Wm of electric machine is expressed by the
following matrix:
40
1
Wm
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
1
[is " ism i fd1 " i fdn
2 1
" M s1sm
M s1 fd1
ª Ls1
«
#
#
« #
«M
"
L
M
sm
sm fd1
« sm s1
« M fd1s1
M fd1sm
L fd1
«
u« #
#
#
«M
" M fdn sm M fdn fd1
« fdn s1
« M 1s " M 1s
M 1 fd1
1
m
«
#
#
« #
« M
M lfd1
¬ ls1 " M lsm
i1 " il ]
"
M s1 fdn
M s11
"
#
" M sm fdn
#
M sm 1
"
" M fd1 fdn
M fd11 "
"
#
L fdn
"
M 1 fdn
L1
#
#
M lfdn
M l1
"
#
M fdn 1 "
"
"
M s1l º ª is º
1
»
# » «« # »»
M sml » « is »
»« m »
M fd1l » « i fd »
»« 1 »
# »« # »
M fdn l » « i fd »
»« n »
M 1l » « i1 »
»«
»
# »« # »
Ll »¼ «¬ il »¼
1
i L iT
2
(1.9.5)
In the above formula, the minus appears before the stator current items of the
current row matrix, and the positive signs are taken for the whole elements of the
inductance matrix which is different from the sign in matrix L of formula (1.1.4).
Based on the virtual displacement principle when each loop current is kept as
constant, there is
f
wWm
wg
(1.9.6)
i constant
where f is broad sense force, g is broad sense displacement. If rotor mechanical
angle T is considered as broad sense displacement, then broad sense force should
be electromagnetic torque Te of electric machine, hence
Te
wWm
wT
i constant
§1
·
w ¨ i L iT ¸
2
©
¹
wT
i
1 wL T
i
i
2 wT
constant
(1.9.7)
In formula (1.9.7), Te is consistent with rotor rotation direction, that is Te is
driving torque.
41
AC Machine Systems
If P is the pole-pair number of the electric machine, J is the electric angle by
which the rotor d -axis leads ahead of the stator coordinates system axis (i.e. rotor
position angle), then there is
T
J /P
If according to generator convention, positive electro-magnetic torque is considered
as braking torque, then the electro-magnetic torque can be written as
Te
P wL T
i
i
2 wJ
(1.9.8)
When the rotor motion equation of the electric machine is set up, the generator
is considered as a standard, such as in Fig. 1.9.1. Here, Tm is the mechanical
torque of the prime mover which is consistent with rotor general rotation direction;
Te is the electro-magnetic breaking torque; surplus torque Tm Te is the accelerating
torque, and the rotor motion equation can be set up as
J
dZ
dt
Tm Te
where J is rotary inertia; Z is mechanical angular speed;
(1.9.9)
dZ
is angular
dt
acceleration.
Figure 1.9.1 Sketch map for setting rotor motion equation
If the mechanical angle between rotor d -axis and phase a axis is expressed
dJ
by J , then Z
, thus formula (1.9.9) becomes
dt
J
d 2J
dt 2
Tm Te
(1.9.10)
where the unit of J is mechanical radian; the unit of t is s ; the unit of Tm and
Te is N ˜ m; the unit of J is kg ˜ m 2 .
42
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
If the unit of flywheel torque GD 2 of rotation part is t ˜ m 2 , then
1
GD 2 u 103 (kg ˜ m 2 )
4
J
If J in formula (1.9.10) is expressed by electric radian, then the electric
machine rotor motion equation is
J
1 d 2J
P dt 2
Tm Te
(1.9.11)
where P is the pole pair number of the electric machine.
In practice, per-unit value is adopted mostly in the above formula. Generally,
the base value of torque is selected as
TG
S N u 103
(N ˜ m)
2Sn
60
where S N is the normal apparent power of the electric machine (kV ˜ A); n is
synchronous speed (rpm).
1
Generally, the base value of time is selected as tG
(s), it corresponds to
2Sf
the time when the rotor of the synchronous machine rotates by one electric radian
at synchronous speed. Two sides of formula (1.9.11) are divided by these base
values, and we get
ª 1 GD 2 3 2 § n · º d 2J
2S f ¨ ¸ »
«
2
© 60 ¹ ¼ § t ·
¬ P SN
d¨ ¸
© tG ¹
Tm Te
TG TG
(1.9.12)
i.e., the per-unit value form of motion equation is
H
d 2J
dt 2
H
dZ
dt
Tm Te
(1.9.13)
where
H
1 GD 2 3 2 § n ·
2S f ¨ ¸
P SN
© 60 ¹
2S3 f
GD 2 § n ·
¨ ¸
S N © 60 ¹
2
(1.9.14)
H is called as inertia constant, and simple relation exists between it and the
per-unit value of the time. From Eq. (1.9.13), we can get
d 2J
dt 2
dZ
dt
Tm Te
H
43
AC Machine Systems
Suppose that the surplus torque Tm Te 1 , W is the time when the electric machine
rotation speed is raised from zero to synchronous speed ( Z 1 ), then
W
³
W
0
dt
³
1
0
HdZ
H
(1.9.15)
It can be seen that while constant surplus torque with per-unit value 1 ( Tm Te 1 )
is exerted on original standstill electric machine rotor, the per-unit value of the
time equals H when the rotation speed is raised from zero to synchronous speed.
1.10
Analysis of Parameters and Performance of SinglePhase Induction Motors
In the above sections, the basic electromagnetic relations of the salient pole
synchronous machine are analyzed mainly, that is, related physical concept, basic
voltage equations and flux-linkage equations, calculation method of each loop
parameter of electric machine, electro-magnetic torque and rotor motion equation,
etc. are analyzed. After getting these relations, various operation performances of
the ac machine can be analyzed and calculated. In this section, as an example, the
single-phase induction machine is presented for explaining the calculation method
of the electric machine loop parameters, for writing voltage equations and fluxlinkage equations according to the actual loops, and for analyzing the performance
of the electric machine.
There are various types of induction machines, their stator windings have both
symmetric distribution and asymmetric distribution, and the winding connection
has various styles. In the operation of a single induction machine, normally its
air-gap magnetic field is not round; besides the positive sequence rotating magnetic
field, the negative sequence magnetic field exists in air-gap. The air-gap harmonic
magnetic field of a single phase induction machine is very strong, and its effect
on the electric machine’s performance is greater.
Normally, symmetric component method is used for researching the single
phase electric machine; in this method, the above factors would bring about many
inconveniences, and bigger error would be caused.
Adopting the Multi-Loop Analysis method for researching the single phase
electric machine, writing equations and calculating parameters can be realized
according to actual loops, and the effects of many factors can be considered more
comprehensively. This method is suitable particularly for the condition with
stronger space harmonics of air-gap magnetic field and special winding connection.
In the following, the Multi-Loop Method is used to research the single phase
electric machine.
Figure 1.10.1(a) is the stator connection sketch of a single phase induction
machine. Here, m and a represent main phase winding and auxiliary phase
44
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
winding, respectively, C is series capacitor. Figure 1.10.1(b) is the sketch of the
rotor loops of a single phase induction machine.
Figure 1.10.1 Sketch map of single phase induction machine circuit
(a) Stator connection; (b) Rotor loops
When parameter calculation and performance analysis of single phase induction
machine are carried out, the stator phase windings can be considered as stator
loops. At the same time, it is needed to calculate the inductances of stator single coil,
then by combining them the inductances of stator phase winding are obtained. In
the following, firstly the inductances related to air-gap main magnetic field are
researched, sum of the inductances caused by the main magnetic field and leakage
one are the inductances of stator phase winding.
1.10.1
Inductances of Single Phase Induction Machine
Induction machine is with uniform air-gap, its air-gap permeance is constant.
Hence, formula (1.2.3) can be simplified as
OG ( x)
O0
2
P0
G
(1.10.1)
in which the effect of slot and iron reluctance is considered in choosing G .
Suppose that the electric angle per stator slot is T , then the electric angle between
i -th coil and j -th coil of the stator is (i j )T , according to formula (1.3.38)
the mutual inductance between these two coils is
M ij
ª§ k yk ·2
º
2 wK2 W l
«
»
O
cos
(
)
T
k
i
j
¨
¸
¦
0
P S2
»¼
k «© k ¹
¬
(1.10.2)
1 2 3
, , ,"
P P P
The mutual inductance between the stator coils of the induction machine is
independent of rotor position.
Suppose that the stator phase winding consists of 2P series coils, each coil
where k
45
AC Machine Systems
group has q coils. When the slot number per pole per phase is an integer, the
serial number of i -th coil under m -th pole is (m 1)2q i , the serial number
of j -th coil of the same phase coil group under n -th pole is (n 1)2q j (the
serial number of the first coil under the first pole is considered as 1). Here, 2q is
the slot number per pole of the single phase induction machine. It is known that
the self inductance LI of stator phase winding is the sum of the inductances of
all coils (including 2 Pq u 2 Pq coil) in the phase winding, that is
2P 2P
q
q
¦¦¦¦ M
LI
minj
m 1n 1 i 1 j 1
2
§k ·
2 wK2 W l
O0 ¦ ¨ yk ¸ (1)m n
¦¦¦¦
2
m 1 n 1 i 1 j 1 PS
k © k ¹
u cos k[(m 1) u 2q i (n 1) u 2q j ]T
2P 2P
q
q
2
§k ·
2 wK2 W l
O0 ¦ ¨ yk ¸ (1)m n
¦¦¦¦
2
m 1 i 1 j 1 n 1 PS
k © k ¹
u cos k[(m 1)S (i j )T ]
q
2P
q
2P
(1.10.3)
1 2 3
, , ," ; M minj represents the inductance between i -th coil under
P P P
m -th pole and j -th coil under n -th pole. N.B. The winding direction of the coil
where k
group under the adjacent pole is opposite, so the multiple (1)m n appears in
formula (1.10.3).
Based on formula (1.10.3), it can be seen that if m, i and j are fixed, and n is
changed from 1 to 2P , then the following summation is zero while k is fraction
or even, i.e.,
2P
¦ (1)
mn
cos k[(m n)S (i j )T ] 0
n 1
that is, when k is fraction or even, the formula (1.10.3) is zero.
When k is odd
2P
¦ (1)
mn
cos k[(m n)S (i j )T ] 2 P cos k (i j )T
(1.10.4)
n 1
Substituting the above formula into formula (1.10.3), we get
q
q
§k ·
(2 P ) 2 u 2 wK2 W l
O0 ¦ ¨ yk ¸
2
PS
k © k ¹
46
2
§k ·
2 wK2 W l
O0 ¦ ¨ yk ¸ u 2 P cos k (i j )T
¦¦¦
2
m 1 i 1 j 1 PS
k © k ¹
2P
LI
2
q
q
¦¦ cos k (i j )T
i 1 j 1
(1.10.5)
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
where k 1,3,5"
Following the triangular formula
qT
2 cos § D q 1T ·
¨
¸
T
2 ¹
©
sin
2
sin
q
¦ cos(D iT )
i 1
(1.10.6)
we get
qkT
2 cos § kiT q 1 kT ·
cos k (i j )T ¦
¦¦
¨
¸
kT
2
©
¹
i 1 j 1
i 1
sin
2
2
qkT ·
§
sin
¨
2 ¸ cos § q 1 kT q 1 kT ·
¨
¸
¨
¸
2
2
©
¹
¨ sin kT ¸
2 ¹
©
q
q
q
sin
qkT
§
¨ sin 2
¨
¨ sin kT
2
©
·
¸
¸
¸
¹
2
(1.10.7)
Therefore,
qkT
§
§ k yk · ¨ sin 2
(2 P) 2 w W l
O0 ¦ ¨ ¸ ¨
kT
PS 2
k © k ¹ ¨
sin
©
2
2
LI
2
2
K
·
¸
¸
¸
¹
2
qkT
§
§ k yk · ¨ sin 2
(2 PqwK ) u 2W l
O0 ¦ ¨ ¸ ¨
kT
PS2
k © k ¹ ¨
q sin
©
2
2
2
2 w12W l
§k ·
O0 ¦ ¨ wk ¸
P S2
k © k ¹
·
¸
¸
¸
¹
2
2
(1.10.8)
where k 1,3,5"; w1 2 PqwK is the series turn number per phase of stator winding;
qkT
2
k yk
kT
q sin
2
sin
kwk
k yk k pk
is stator winding factor.
Based on the inductance of single coil, the mutual inductance between main
47
AC Machine Systems
phase m winding and auxiliary phase a winding can be obtained similarly. If
the turn number per coil of stator main phase winding is wKm , and the turn number
per coil of auxiliary phase winding is wKa , the pitches of the main phase and
auxiliary phase windings are the same, and the serial number of stator coil begins
from the first coil under the first pole of the main phase winding, then the serial
number of i -th coil under n -th pole of the main phase winding is (n 1)2q i ,
the serial number of j -th coil under nc -th pole of the auxiliary phase winding is
(nc 1)2q q j. Accordingly, the mutual inductance between main phase winding
and auxiliary phase winding is
2P 2P
q
q
¦¦¦¦ M
M ma
nmincaj
n 1 nc 1 i 1 j 1
2
§k ·
2 wKm wKaW l
O0 ¦ ¨ yk ¸ (1)n nc
¦¦¦¦
2
PS
n 1 nc 1 i 1 j 1
k © k ¹
u cos k[(n 1) u 2q i (nc 1) u 2q q j ]T
2P 2P
q
q
2
§k ·
2 wKm wKaW l
O0 ¦ ¨ yk ¸ (1)n nc
¦¦¦¦
2
PS
n 1 i 1 j 1 nc 1
k © k ¹
u cos k[(n nc)S (i j )T qT ]
2P
q
q
2P
(1.10.9)
1 2 3
, , ,"; M nmincaj represents the mutual inductance between i -th
P P P
coil under n -th pole of winding m and j -th coil under nc - th pole of winding a .
When k is odd
in which k
2P
¦ (1)
n nc
cos k[(n nc)S (i j )T qT ]
nc 1
2 P cos[k (i j )T kqT ]
When k is fraction or even, the above formula is zero. Hence, formula (1.10.9)
can be simplified further as
M ma
§k ·
(2 P)2 u 2 wKw wKaW l
O0 ¦ ¨ yk ¸
2
PS
k © k ¹
2
q
q
¦¦ cos[k (i j )T kqT ]
i 1 j 1
where k 1,3,"
Using formula (1.10.6), we get
q
q
¦¦ cos[k (i j )T kqT ]
i 1 j 1
q
¦
i 1
48
qkT
2 cos § kiT kqT q 1 kT ·
¨
¸
kT
2
©
¹
sin
2
sin
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
2
qkT
§
¨ sin 2
¨
¨ sin kT
2
©
·
¸
q 1
q 1 ·
§
kT kT ¸
¸ cos ¨ kqT 2
2
©
¹
¸
¹
qkT
§
¨ sin 2
¨
¨ sin kT
©
2
·
¸
¸ cos kqT
¸
¹
2
Therefore,
qkT
§
§ k yk · ¨ sin 2
(2 P)2 u 2 wKm wKaW l
O0 ¦ ¨ ¸ ¨
kT
PS2
k © k ¹ ¨
sin
©
2
2
M ma
qkT
§
§ k yk · ¨ sin 2
2 PqwKm u 2 PqwKaW l
O0 ¦ ¨ ¸ ¨
kT
PS 2
k © k ¹ ¨
q sin
©
2
2
2
·
¸
¸ cos kqT
¸
¹
2
·
¸
¸ cos kqT
¸
¹
2
2 wm waW l
§k ·
O0 ¦ ¨ wk ¸ cos kqT
2
PS
k © k ¹
(1.10.10)
in which k 1,3," ; wm and wa are the series turn number per phase of the main
phase winding and auxiliary phase winding, respectively; kqT is the electric angle
between k -th harmonic central lines of the main phase and auxiliary phase
windings.
Formulas (1.10.8) and (1.10.10) are deduced according to the phase winding
with single branch. If the phase winding is with multi-branches, then w1 of formula
(1.10.8) is changed into w1 / a1 , and wm and wa of formula (1.10.10) are changed
into wm / am and wa / aa , respectively, where a1 , am and aa are the parallel branch
numbers of the corresponding phase windings respectively.
As mentioned above, the self inductance and mutual inductance of stator phase
winding of the single phase induction machine are deduced according to single
coil inductance, as shown in formulas (1.10.8) and (1.10.10). It can be seen that
the inductances of stator phase winding of the single phase induction machine are
independent of rotor position owing to uniform air-gap.
It should be illustrated that formulas (1.10.8) and (1.10.10) can be calculated
directly too, based on main phase and auxiliary phase windings; that is, according
to the air-gap mmf produced by the main phase and auxiliary phase windings, the
corresponding air-gap flux density is calculated. If several different connection
types of the main phase and auxiliary phase windings exist, then respective
49
AC Machine Systems
calculations are needed based on actual connection. If the single coil inductance
is calculated firstly, in spite of coil connection mode it is very convenient to
calculate the inductances of windings, branches, or certain several coils connected,
if needed, in terms of the obtained single coil inductances. The reason why the
above method is adopted is to illustrate its concrete usage.
The rotor circuit of the single phase induction machine is shown in Fig. 1.10.1(b).
When the inductances of the rotor loops of the single phase induction machine
are researched, the formulas (1.5.14) and (1.5.17) of the rotor damper loop
inductances of the salient pole synchronous machine can be cited. But, the air-gap
permeance of the single phase induction machine has only the constant term
O0 / 2 (harmonic terms O2 O4 " 0 ) owing to uniform air-gap of the single
phase induction machine, i.e., only when 2l
k j
0 , there are values in the
two formulas, hence the self inductance of rotor i - th loop is
Li
2wr2W l
kE S
1
O0 ¦ 2 sin 2 r
P S2
k
2
k
(1.10.11)
The mutual inductance between rotor i - th loop and j - th loop is
M ij
2wr2W l
kE S
1
O0 ¦ 2 sin 2 r cos k ( j i )M
2
PS
2
k k
(1.10.12)
1 2 3
, , ,"; E r is the short pitch ratio of the rotor loop; wr is the turn
P P P
number of the rotor loop, generally wr 1 ; i, j are the serial number of rotor
loops; M is the electric angle between the central lines of rotor adjacent coils.
Actually, the self inductances of all rotor loops have the same value, the mutual
inductances between all rotor loops have another same value, and they are all
independent of rotor position. Formulas (1.10.11) and (1.10.12) reflect this fact.
In the following, still based on the inductance between stator single coil and
rotor loop, the inductance between stator winding and rotor loop is calculated.
Cite formula (1.6.5) of the mutual inductance between stator single coil and
damper loop of the salient pole synchronous machine, and pay attention to the fact
that only O0 has its value for the induction machine, then the mutual inductance
between stator i - th coil and rotor j -th loop of the single phase induction machine
can be obtained
where k
M sij
2 wK wrW l
1
k E S k Er S
O0 ¦ 2 sin
sin
cos k (J D )
2
PS
2
2
k k
(1.10.13)
where J is the electric angle by which the rotor d axis leads the central line of
the stator coil; D is the electric angle by which the central line of the rotor loop
50
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
leads the rotor d axis; (J D ) is the electric angle by which the central line of
the rotor loop leads the central line of the stator coil.
If the electric angle by which the central line of rotor 0 - th loop leads the
central line of stator 0 - th coil is taken as angle J c , then formula (1.10.13) can
be written as
2 wK wrW l
1
k E S k Er S
O0 ¦ 2 sin
sin
cos k (J c iT jM )
2
PS
2
2
k k
M sij
(1.10.14)
N.B. The serial number of stator and rotor loops is along reverse rotation direction.
If the stator phase winding consists of 2P series coil groups, there are q coils
in each coil group, and the first coil under the first pole is taken as the initial coil,
then the total serial number of n - th coil under m -th pole is (m 1)2q n . If
M smnj expresses the mutual inductance between stator n -th coil under m -th pole
and rotor j -th loop, then the mutual inductance between stator phase winding
and rotor j -th loop is
2P
q
¦¦ M
M ij
smnj
m 1n 1
2P q
2 wK wrW l
kE S
1
k ES
O0 ¦ 2 sin
sin r (1) m 1
2
2
2
PS
m 1n 1
k k
u cos k{J c [(m 1) u 2q n]T jM }
2P q
2 wK wrW l
kE S
1
k ES
O0 ¦ 2 sin
sin r (1) m 1
¦¦
2
2
2
PS
m 1n 1
k k
c
u cos k[J (m 1) S nT jM ]
¦¦
(1.10.15)
When k is odd
2P
¦ (1)
m 1
cos k[J c (m 1)S nT jM ] 2 P cos k (J c nT jM )
m 1
When k is fraction or even, the value of the above formula is zero. Hence, formula
(1.10.15) can be simplified further as
M sj
2 wK wrW l u 2 P
kE S
1
k ES
O0 ¦ 2 sin
sin r
PS2
k
2
2
k
q
u ¦ cos k (J c nT jM )
k 1, 3, 5, "
n 1
Using formula (1.10.6), finally we get
51
AC Machine Systems
M sj
2(2 PqwK ) wrW l
kE S
1
k ES
O0 ¦ 2 sin
sin r
2
PS
2
2
k k
qkT
sin
2 cos k § J c jM q 1T ·
u
¨
¸
kT
2 ¹
©
q sin
2
2w1 wrW l
kE S
1
O0 ¦ 2 kwk sin r
2
PS
2
k k
q 1 ·
§
u cos k ¨ J c jM T¸
2 ¹
©
k 1, 3, "
(1.10.16)
q 1 ·
§
where ¨ J c jM T ¸ is the electric angle between the central line of the
2 ¹
©
stator phase winding and the central line of rotor j -th loop. It can be seen that
the mutual inductance between stator phase winding and rotor loop of the single
phase induction machine depends on rotor position, that is, it is time-variant
parameter at rotor rotation.
Similarly, formula (1.10.16) is also deduced according to single-branch phase
winding; if the phase winding is with multi-branches, branch number is a1 , then
w1 of the formula should be changed into w1 / a1 .
Only the air-gap magnetic field is considered above, but the leakage flux
should be considered in calculating stator phase winding inductances and rotor
loop inductances, that is, corresponding leakage inductance terms should be
added to formulas (1.10.8), (1.10.10), (1.10.11), and (1.10.12), description of which
is not needed here.
1.10.2
Basic Relations of Single Phase Induction Machine
After knowing the inductance of each loop of the single phase induction machine,
the voltage equations and flux-linkage equations can be written and researched.
(1) Steady state expressions of stator and rotor currents
Suppose that the stator power supply of the single phase induction machine is
sine wave voltage, the corresponding stator current is also fundamental sine current
owing to uniform air-gap of the induction machine and symmetric multi-phase
winding in the rotor. The reason is that no matter what rotation speed the
harmonic magnetic field has, produced by the stator fundamental current, only
the corresponding harmonic magnetic field with the same rotation speed is
produced by the induced current in symmetric multi-phase winding in the rotor.
If rotor angular speed is Z , the rotation speeds of positive rotation and reverse
rotation magnetic fields of k -th harmonic produced by the fundamental phase
52
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
current of the stator single phase winding in air gap are Z 0 / k and Z 0 / k ,
respectively, where Z 0 2ʌf is fundamental angular frequency, k 1,3,5,"
(suppose the stator is integer slot winding), then the rotation speeds of these
positive rotation and reverse rotation magnetic field relative to the rotor are
Z
§ Z
·
r 0 Z , accordingly, the frequencies of rotor induced currents are ¨ r 0 Z ¸ k ,
k
© k
¹
that is, Z0 kZ and Z 0 kZ . If the stator current and rotor current are written
as sine and cosine functions, then the current of stator main phase winding is
im
I m cos Z0t I mc sin Z0 t
(1.10.17)
The current of stator auxiliary winding is
ia
I a cos Z0 t I ac sin Z0 t
(1.10.18)
The current of rotor first loop is
i1
¦ {I
1k
cos(Z0 kZ )t I 2 k sin(Z0 kZ )t I 3k cos(Z 0 kZ )t
k
I 4 k sin(Z0 kZ )t}
(1.10.19)
k 1, 3, 5, "
Considering the phase angle relation of induced currents in rotor loops produced
by positive rotation and reverse rotation magnetic fields, the current of rotor
n - th loop is
in
¦ {I
1k
cos[(Z0 kZ )t k (n 1)M ] I 2 k sin[(Z0 kZ )t
k
k (n 1)M ] I 3k cos[(Z0 kZ )t k (n 1)M ]
I 4 k sin[(Z 0 kZ )t k (n 1)M ]}
k 1, 3, 5, "
(1.10.20)
where M is the electric angle between the central lines of adjacent loops.
(2) Voltage equation and flux-linkage equation of each winding of single phase
induction machine
When voltage equation and flux-linkage equation are written, the following
positive direction regulation is adopted: the current flowing into the electric
machine is considered as positive, and the positive direction of voltage drop is
consistent with the current positive direction; positive current produces positive
flux-linkage for both stator and rotor. And then, the voltage equation of the main
phase winding is
um
p\ m rm im
(1.10.21)
The voltage equation of auxiliary phase winding is
53
AC Machine Systems
ua
p\ a ra ia
(1.10.22)
The voltage equation of rotor n -th loop is
un
p\ n rr in rc (in 1 in 1 )
n 1, 2," , Z 2
(1.10.23)
d
is differential operator; Z 2 is rotor slot number; rr is resistance of
dt
rotor loop; rc is resistance of rotor bar.
The flux-linkage equation of each loop is
where p
ª\ m º
«\ »
« a»
«\ 1 »
« »
« # »
«\ n »
« »
¬« # ¼»
ª Lm
«M
« am
« M 1m
«
« #
« M nm
«
¬« #
M ma
La
M 1a
#
M na
#
M m1 " M mn
M a1 " M an
L1 " M 1n
#
#
M n1 " Ln
#
#
"º ªim º
"»» «« ia »»
"» « i1 »
»« »
»« # »
"» « in »
»« »
¼» ¬« # ¼»
(1.10.24)
In the inductances of the above formula, the inductances of stator phase windings
(left-upper-angle square matrix) and the inductances of rotor loops (right-downangle square matrix) are all constant, the mutual inductances between stator
phase windings and rotor loops are time-variant parameters, that is, functions of
rotor position angle J c
t
³ Z dt J
0
0
Z t J 0 (at constant rotation speed), here
J 0 is rotor initial position angle. All the inductances can be obtained by formulas
(1.10.8), (1.10.10), (1.10.11), (1.10.12), and (1.10.16).
(3) Forming steady-state linear algebraic equation group
The expressions of stator and rotor currents (1.10.17), (1.10.18), (1.10.19), and
(1.10.20) are substituted into the flux-linkage equation (1.10.24), then the voltage
equations (1.10.21), (1.10.22), and (1.10.23) are brought in, a group of equations
will be obtained in which the amplitudes of the sine parts and cosine parts are
unknown quantities, including each harmonic sine function and cosine function
of Z t . Pay attention to the fact that the rotor loop of the induction machine is short
circuit (un 0) , the electric source of stator winding is fundamental voltage (refer to
Fig. 1.10.1(a)), then
um
ua
54
u U c cos Z0t U cc sin Z0t
(1.10.25)
1
( I a cos Z 0t I ac sin Z 0t )dt
C³
I
Ic
U c cos Z0t U cc sin Z0 t a sin Z 0t a cos Z 0t
(1.10.26)
Z0C
Z0C
u uC
U c cos Z0t U cc sin Z0t 1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
It is enough to analyze only according to a certain loop equation of rotor, because
the amplitude of each loop current of the rotor is the same, and the phase angle
difference of the adjacent loops is also the same. As for the current of another loop
of the rotor, it is not difficult to be calculated based on equation (1.10.20). In this
way, three equations of transcendental function with multi harmonics are obtained.
Each harmonic of the equations can form an equation itself, if the number of
the researched harmonics is k1 , then a total of 3k1 equations will be got. For
each harmonic equation, two special times are selected in order to make the sine
term or cosine term to be zero (for example, for fundamental order Z t 0 or
Z t ʌ / 2 ), then two corresponding linear algebraic equations are got, in this way
a total of 6k1 linear algebraic equations are obtained. In these equations, the
unknowns are the amplitudes of the sine term and cosine term of each current,
the coefficients of the equations are the inductance and resistance of each loop,
the right terms of the equations are either zero or the amplitudes of the sine term
or cosine term of the electric source voltage. Solving this linear algebraic equation
group, all loop currents of the machine can be obtained.
After getting the input current and voltage of the electric machine, the input
power of the electric machine can be obtained, and various losses of the electric
machine are subtracted from the input power, then the output power of the electric
machine will be obtained.
Electro-magnetic torque is produced under the interaction of flux and current.
The electro-magnetic torque of the single phase induction machine is (refer to
formula (1.9.7))
Te
1 wL T 1
i
i
>im ia i1
2 wT
2
ª Lm M ma M m1
«M
La
M a1
« am
L1
w « M 1m M 1a
u
«
#
#
wT « #
« M nm M na M n1
«
#
#
«¬ #
" in "@
" M mn
" M an
" M 1n
#
" Ln
#
"º ªim º
"»» ««ia »»
"» «i1 »
»« »
» «# »
"» «in »
»« »
»¼ «¬# »¼
(1.10.27)
where T is the rotor position angle (mechanical angle).
Considering only the inductances between stator windings and rotor loops to
be a function of the rotor position angle in inductance matrix, the formula
(1.10.27) can be written as
Z2
Te
im ¦ ii
i 1
Z2
wM mi
wM ai
ia ¦ ii
wT
wT
i 1
(1.10.28)
As an example, an electric machine for an electric fan with pole-pair number
55
AC Machine Systems
P 2 , stator winding full pitch and q 1 is calculated by using the Multi-Loop
Method. The winding factor of each harmonic of the electric machine is the same as
the fundamental, and the amplitude ratio of harmonic mmf to fundamental mmf is
Fk
F1
1
k
k 1,3,5,"
Therefore, the stator harmonic mmf of this single phase induction machine is very
strong, in the calculation it is necessary to consider the effect of different harmonic
air-gap magnetic field. The experimental data, the calculated values of the original
designed program, and the calculated values by using the Multi-Loop Method are
shown in Table 1.10.1. In the table, I m is the main phase winding current, I a is
the auxiliary phase winding current, I l is the total current, P1 is input power, k
is the calculated harmonic order of stator mmf. Operation conditions of the electric
machine are: U 220V, n 1250rpm, phase-splitting capacitor C 1.202PF .
Table 1.10.1 Comparison of operation performance of a single phase induction motor
I m /A
I a /A
I l /A
I 1 /A
Experimental data
Calculated values of original designed
program
k 1
Calculated values by using k 1, 3
0.205 6
0.176 0
0.309 0
62.56
0.230 0
0.146 9
0.357 7
74.33
0.246 2
0.149 0
0.328 3
66.21
the Multi-Loop Method
0.215 8
0.161 4
0.315 9
64.36
k
1, 3, 5
0.205 9
0.167 0
0.315 6
64.54
k
1, 3, 5, 7
0.205 0
0.169 3
0.315 2
64.41
It can be seen from the table that while only considering the fundamental of
the air-gap magnetic field, the calculated values by using the Multi-Loop Method
are close to the calculated values of the original designed program, but the difference
is bigger compared with the experimental data. The higher the considered harmonic
order of air-gap magnetic field is, the closer to experimental data the calculated
results are.
References
[1] Gao J D, Wang X H (1987). Multi-loop theory of AC machines. Journal of Tsinghua
University (Science and Technology), 1: 1 8 (in Chinese)
[2] Gao J D, Wang X H, Jin Q M (1987). Calculation of loop parameters of salient-pole
synchronous machine. Journal of Tsinghua University (Science and Technology), 1: 9 19
(in Chinese)
56
1
Circuit Analysis of AC Machines — Multi-Loop Model and Parameters
[3] Gao J D, Wang X H, Li F H (1993). Analysis of AC Machines and Their Systems, First
Edition. Beijing: Tsinghua University Press (in Chinese)
[4] Gao J D, Wang X H, Li F H (2005). Analysis of AC Machines and Their Systems, 2nd
Edition. Beijing: Tsinghua University Press (in Chinese)
[5] Wang X H, Sun Y G, Ouyan g B, Wang W J, Zhu Z Q, Howe D (2002) Transient behavior
of salient-pole synchronous machines with internal stator winding faults. IEE Proc.-Electr.
Power Appl. 2: 143 151
[6] Wang X H, Chen S L, Wang W J , Sun Y G, Xu L Y (2002). A study of armature winding
internal faults for turbogenerators. IEEE Trans. on IA, 3: 625 631
[7] Gao J D, Zhang L Z (1982). Fundamental Theory and Analysis Mothods of Transients in
Electric Machines. Beijing: Science Press (in Chinese)
[8] Danileviq Y B, Dombloski V V, Kazovski E Y (1965). Parameters of AC machines.
Beijing: Science Press (in Russian)
[9] Wang X H, Lo K L (1997). New development of torque analysis of synchronous machine.
Electric Machines and Power Systems, 8: 827 838
57
2
Electromagnetic Relations and Parameters of
Synchronous Machines and Analysis of Their
Several Operating Conditions at Synchronous Speed
Abstract In this chapter, firstly study electromagnetic relations and
parameters of conventional 3-phase synchronous machines by using multi-loop
method, whose results are the same as classical method. Secondly, the actual
a, b, c axes are converted into d, q, 0 axes according to both mmfs being
equal to form a clear concept. Furthermore, two types of per-unit systems
are introduced for synchronous machines, inclusive of distinct formula
deduction, and then basic equations in per-unit, operational reactances and
electromagnetic torque formula are also built. In order to help readers grasp
how to use those basic equations in per-unit and operational equivalent
circuits to solve practical problems, several symmetrical operating modes at
synchronous speed are analysed, which may occur in practical operation.
Among of them, 3-phase sudden short circuit will threaten synchronous
machines. Thus, comprehension of the transients may provide scientific basis
not only for reasonable design and reliable operation of synchronous
machines but also for a careful choice of relay protection. Voltage dip during
sudden increase of load for synchronous machines occurs in practice and is
also studied in this chapter. Transient emfs of synchronous machines are
introduced, which are proportional to the flux-linkages of rotor loops. The
flux-linkage of rotor winding, especially the flux-linkage of excitation
winding, under a certain condition, can be considered as a constant, which
will make analysis and calculation simplified. Electromagnetic torque after
3-phase sudden short circuit, inclusive of pulsating torque and average
torque, are also analysed in this chapter. Finally, suddenly opening 3-phase
short circuit of synchronous machines is also paid attention to.
In Chapter 1, electromagnetic relations and parameters of ac machines have been
studied according to salient-pole type, writing down voltage and flux-linkage
equations and giving out inductance computation of stator and rotor loops on the
basis of single coil. In this chapter, electromagnetic relations and parameters are
studied for conventional 3-phase synchronous machines.
During normal operation of 3-phase synchronous machine, its stator winding
is symmetrical. Even if exist unbalanced loads and asymmetrical faults, the stator
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
winding it self is still symmetrical, so it is convenient to study those problems on
the basis of one phase winding and corresponding magnetic field.
In electric machinery, fundamental flux in air-gap is mainly considered and
harmonic flux effect can be counted in by using the differential leakage reactance,
thus causing a certain deviation. In Chapter 1 we have studied the influence of airgap harmonic flux on parameters and performance of single phase induction motors.
In this chapter, firstly study the effect of harmonics on 3-phase synchronous
machines, and then research on their several operating conditions on the basis of
one phase winding.
In general, air-gap harmonics can be reduced a lot by means of special design
for structure and windings, such as good pole-arc form, fractional-pitch and
distributed stator winding, fractional-slot winding, etc. During small air-gap
harmonics, we use coordinates conversion, especially d, q, 0 axes to analyse
operating conditions of synchronous machines, which is convenient because the
basic equations in d, q, 0 axes are of constant coefficients referring to Refs.[1,3,5].
Therefore, d, q, 0 axes are paid more attention to in this chapter.
By using basic equations in d, q, 0 axes, we analyse some symmetrical operating
modes of synchronous machines at synchronous speed, such as symmetrical
steady-state operation, the transient process when rotor winding is short-circuited
and voltages are suddenly applied to the stator winding, the currents after 3-phases
are suddenly short-circuited, the voltage dip when the load of a synchronous
machine increases suddenly, the transient emfs of synchronous machines,
electromagnetic torque after 3-phases are suddenly short-circuited, opening
3-phase short-circuit, referring to Refs.[2,4,6].
2.1
Basic Relations of Synchronous Machines
In the stator of synchronous machines there are phase a, b and c windings, and in
the rotor there is excitation winding besides damping winding that can be
represented by equivalent d-axis damping winding 1d and q-axis damping
winding 1q. According to reference directions stated in Chapter 1, various
winding voltage equations in matrix form for 3-phase salient-pole synchronous
machines can be written as
p\ RI
U
(2.1.1)
in which
U
[ua
ub
I
[ia
ib
\
[\a
uc
ic
0 0]T
u fd
i fd
i1d
i1q ]T
\ b \ c \ fd \ 1d \ 1q ]T
59
AC Machine Systems
R
ª r
«
r
«
r
«
«
«
«
«
¬
R fd
R1d
º
»
»
»
»
»
»
R1q »¼
and subscripts a, b, c, fd , 1d , and 1q indicate stator phase a, b and c windings,
rotor excitation winding, and equivalent d-axis and q-axis damping windings,
respectively, and r is stator phase resistance.
In a similar way, the relations for multi-loop damping winding can be found
out if necessary.
Various winding flux-linkage equations in matrix form for 3-phase salient-pole
synchronous machines can be expressed as
\
(2.1.2)
LI
in which
L
ª Laa
« M
ba
«
« M ca
«
« M fda
« M 1da
«
«¬ M 1qa
M ab
Lbb
M cb
M fdb
M 1db
M 1qb
M ac
M bc
Lcc
M fdc
M 1dc
M 1qc
M afd
M bfd
M cfd
L ffd
M 1 fd
0
M a1d
M b1d
M c1d
M f 1d
L11d
0
M a1q º
M b1q »»
M c1q »
»
0 »
0 »
»
L11q »¼
(2.1.3)
L with corresponding subscript indicates self-inductance for the relevant winding,
and M with related subscript shows corresponding mutual inductance. The
corresponding mutual inductances are reversible, namely
M ab M ba ,
M afd M fda ,
M a1d M 1da ,
M a1q M 1qa ,
M f 1d M 1 fd
M bc M cb ,
M bfd M fdb ,
M b1d M 1db ,
M b1q M 1qb ,
M ac M ca
M cfd M fdc
M c1d M 1dc
M c1q M 1qc
In addition, mutual inductance between q-axis damping winding 1q and d-axis
damping winding 1d or and excitation winding fd is zero because winding 1q is
set S electric radians apart from winding 1d and winding fd .
Various self-inductances and mutual inductances in Eq. (2.1.3) are important for
study of synchronous machine operation. Computation of single coil inductance
60
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
has been discussed in Chapter 1, by which the inductance for phase winding
composed of several coils is analysed here. For convenience, ferro-magnetic
saturation, eddy current and hysteresis effect, and slot influence are neglected. If
it is necessary to consider slot influence, refer to Appendix A in this book.
2.1.1
Inductances of Stator Phase Windings
In Chapter 1, Eq. (1.3.31) indicates stator single coil inductance corresponding to
air-gap flux, in which D is electric angle between two coils. If two coils are
numbered as i c and j c , respectively, and each slot electric angle is T , then their
inductance corresponding to air-gap flux is
M icj cG
k yk k yj
2 wK2 W l
^(Odkj Oqkj )
¦¦
PS k j kj
u cos[(k j )(J i cT ) j ( j c i c)T ]
(Odkj Oqkj ) cos[(k j )(J i cT ) j ( j c ic)T ]`
(2.1.4)
in which
k
1 2 3
, , , "; j | k r 2l |; l
P P P
0,1, 2, "
(J i cT ) is the electric angle, at which rotor d-axis leads ahead of stator ic -th
coil when taking stator 0-th coil central line as stator reference axis.
Supposing that stator phase winding is composed of 2P coil-groups in series,
each coil-group has q coils during q integer, and the first coil of the phase
winding under the first pole is taken as No.1 coil, then the number of ic -th coil
of the phase winding under m -th pole is 3q (m 1) ic, the number of j c -th coil
of the same phase under n -th pole is 3q (n 1) j c, and the stator phase winding
self-inductance is
2P 2P
LI
q
q
¦¦¦¦ M
mi cnj c
m 1 n 1 ic 1 jc 1
k yk k yj
2 wK2 W l
(Odkj Oqkj )(1)m n
^
¦¦¦¦
2 ¦¦
S
P
kj
m 1 n 1 ic 1 jc 1
k
j
u cos[( k j )(J i cT ) j (3q (n 1)
j c 3q(m 1) i c)T ] (Odkj Oqkj )(1) m n
2P 2P
q
q
u cos[( k j )(J i cT ) j (3q(n 1)
j c 3q(m 1) i c)T ]`
61
AC Machine Systems
k yk k yj
2 wK2 W l
^(Odkj Oqkj )(1)mn
¦¦¦¦
2 ¦¦
P
kj
S
c
m 1n 1 i 1 n 1
k
j
u cos[(k j )(J i cT ) j (3q (n m)T ( j c i c)T )]
q
2P
q
2P
(Odkj Oqkj )(1) m n
u cos[(k j )(J i cT ) j u 3q (n m)T ( j c i c)T ]`
in which
3qT
S
If making m, i c and j c constant and letting n change from 1 to 2P in the
summation term above, then the summation term equals zero during j fraction
and also zero during j even number, namely,
2P
¦ (1)
mn
cos[(k B j ) (J i cT ) B j ((n m)S ( j c ic)T )] 0
n 1
During j odd number exists
2P
¦ (1)
mn
cos[(k B j ) (J i cT ) B j ((n m)S ( j c i c)T )]
n 1
2 P cos[(k B j ) (J i cT ) B j ( j c i c)T )]
Hence, stator phase winding self-inductance is also written as
k yk k yj
2 P u 2wK2 W l
{(Odkj Oqkj )
¦¦
2
Pʌ
kj
m 1 ic 1 j c 1
k
j
u cos[(k j )(J icT ) j ( j c ic)T ]
(Odkj Oqkj ) cos[(k j )(J icT ) j ( j c ic)T ]
2P
q
q
¦¦¦
LI
where k 1, 3, "; j | k r 2l | .
By using Eq. (1.10.6) exists
q
q
¦¦ cos[(k B j )(J icT ) B j ( j c ic)T ]
ic 1 jc 1
qjT
2 cos ª(k B j )(J i cT ) r jicT B q 1 jT º
¦
«
»
j
T
2
¬
¼
ic 1
sin
2
2
qjT ·
§
sin
¨
2 ¸ cos ª(k B j )J q 1 kT B q 1 jT º
¨
¸
«
»
2
2
¬
¼
¨ sin jT ¸
©
2 ¹
q
62
sin
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
qjT
§
¨ sin 2
¨
¨ sin jT
2
©
2
·
¸
q 1 ·º
ª
§
T¸
¸ cos «(k B j ) ¨ J 2 ¹ »¼
©
¬
¸
¹
so we can get
LI
­
qjT
§
¨ sin 2
k yk k yj °°
2(2 PqwK ) 2 W l
®(Odkj Oqkj ) ¨
¦¦
Pʌ 2
kj °
k
j
¨ q sin jT
©
2
¯°
q 1 ·
§
u cos(k j ) ¨ J T¸
2 ¹
©
(Odkj
qjT
§
¨ sin 2
Oqkj ) ¨
¨ q sin jT
2
©
·
¸
¸
¸
¹
2
2
½
·
¸
q 1 · °°
§
T ¸¾
¸ cos(k j ) ¨ r 2 ¹°
©
¸
¹
°¿
kwk kwj
2w2W l
2 ¦¦
Pʌ k j kj
(Odkj
q 1 ·
­
§
T¸
®(Odkj Oqkj ) cos(k j ) ¨ J 2 ¹
©
¯
q 1 ·½
§
Oqkj ) cos(k j ) ¨ J T ¸¾
2 ¹¿
©
(2.1.5)
where
k 1, 3, "; j | k r 2l |
Referring to Eq. (1.3.18), the formula above can also be written as
LI
q 1 ·
q 1 ·
§
§
T ¸ L4 cos 4 ¨ J T ¸ "
L0G L2 cos 2 ¨ J 2 ¹
2 ¹
©
©
L0G
2
º
2w2W l ª§ kwk ·
(Odkk Oqkk ) »
¸
2 ¦ «¨
Pʌ k ¬«© k ¹
¼»
2w2W l
§k ·
O0 ¦ ¨ wk ¸
2
Pʌ
k © k ¹
L2
(2.1.6)
2
k 1, 3, "
(2.1.7)
2w2W l ­ 2
®k w (Od 11 Oq11 )
Pʌ 2 ¯ 1
ª k wk k w( k 2)
º ½°
(Odk ( k 2) Oqk ( k 2) ) » ¾
2¦ «
k ¬ k ( k 2)
¼ ¿°
63
AC Machine Systems
ªk k
º ½°
2 w2W l ­° 2
O2 ®kw1 2¦ « wk w( k 2) » ¾
2
Pʌ
k ¬ k ( k 2) ¼ ¿
°
¯°
k 1, 3, "
(2.1.8)
in which w 2 Pqwk is total number of turns for stator phase winding.
The self-inductance above corresponds to air-gap flux only, so the phase winding
self-inductance formula (2.1.6), after considering leakage self-inductance L0l
relevant to slot and end leakage flux, is reformed as
LI
q 1 ·
§
T ¸ "
L0 L2 cos 2 ¨ J 2 ¹
©
q 1 ·
§
T ¸ "
( L0l L0G ) L2 cos 2 ¨ J 2 ¹
©
(2.1.9)
where L0G and L2 are indicated, respectively, by formulas (2.1.7) and (2.1.8).
q 1
T is the included electric angle between phase
In the formulas above,
2
winding central axis and 0-th coil central line, namely reference axis, so rotor d-axis
q 1 ·
§
leads ahead of stator phase winding central axis by electric angle ¨ J T ¸.
2 ¹
©
The phase winding self-inductance formula above is obtained in accordance
with single branch. If there are as branches in parallel, several branch currents for
the phase winding are the same during symmetrical operation or external
asymmetrical operation, so there is no need for distinction between single branch
and several branches. However, the branch current, namely coil current, for as
branches in parallel per phase, is 1/as of single branch current for the same phase
current, so will the air-gap flux density. In addition, number of each branch
coil-groups is 2P/as for several branches in parallel, so the flux-linkage for each
branch winding is 1/a2s of single branch condition. Hence, phase winding
self-inductance during several branches in parallel is 1/a2s of single branch
condition, which is the only difference between them.
Evidently, referring to formulas (2.1.7) and (2.1.8), the phase self-inductance
components during as branches in parallel are:
L0G
L2
64
2w2W l O0
Pʌ 2 as2
§k ·
¦k ¨© kwk ¸¹
2
k 1, 3, "
ª k wk k w( k 2) º ½°
2w2W l O2 ­° 2
k 2¦ «
»¾
2
2 ® w1
Pʌ as ¯°
k ¬ k ( k 2) ¼ ¿
°
(2.1.10)
k 1, 3, "
(2.1.11)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Mutual inductance between phase a and phase b windings will be discussed as
follows. Numbering stator coils and starting from the first coil of phase a winding
under the first pole, then number of ic -th coil for phase a under m-th pole is
[3q (m 1) i c] , and number of j c -th coil for phase b under n-th pole is
[3q (n 1) 2q j c] . During single-branch and integer-slot winding, mutual
inductance between phase a and phase b corresponding to air-gap flux can be
written as
2P 2P
M abG
q
q
¦¦¦¦ M
mai cnbj c
m 1 n 1 ic 1 jc 1
2P 2P
q
q
k yk k yj
2wK2 W l
{(Odkj Oqkj )
2 ¦¦
kj
1 Pʌ
k
j
¦¦¦¦
m 1 n 1 ic 1 jc
u (1)m n cos[(k j )(J i cT ) j (3q (n 1) 2q j c
3q(m 1) ic)T ] (Odkj Oqkj )(1) m n
u cos[( k j )(J icT ) j (3q(n 1) 2q j c
3q(m 1) i c)T ]}
q 2P
2P q
k yk k yj
2wK2 W l
{(Odkj Oqkj )(1) m n
¦¦¦¦
2 ¦¦
P
kj
ʌ
m 1 ic 1 jc 1 n 1
k
j
u cos[(k j )(J i cT ) j (3q(n m)T 2qT ( j c i c)T ]
(Odkj Oqkj )(1) m n cos[(k j )(J i cT )
j (3q (n m)T 2qT ( j c i c)T ]}
in which
k
1 2 3
, , , "; j | k r 2l |; 3qT
P P P
ʌ
M maicnbj c is the mutual inductance between ic -th coil for phase a under m-th
pole and j c -th coil for phase b under n-th pole.
In a silimar way as before, during j is odd number exists
2P
2ʌ
ª
º
cos «(k B j )(J icT ) B j ((n m)ʌ ( j c ic)T ) »
3
¬
¼
n 1
ª
§ 2ʌ
·º
2 P cos «(k B j )(J icT ) B j ¨ ( j c ic)T ¸ »
3
©
¹¼
¬
¦ (1)
mn
and the formula above equals zero during j
there is
fraction or even number. Hence,
65
AC Machine Systems
k yk k yj
2 P u 2wK2 W l
¦¦¦
¦¦
2
Pʌ
kj
m 1 ic 1 jc 1
k
j
2P
q
q
­
®(Odkj Oqkj )
¯
2ʌ
ª
º
u cos «(k j )(J icT ) j
j ( j c i c)T »
3
¬
¼
2ʌ
ª
º½
(Odkj Oqkj ) cos «(k j )(J i cT ) j
j ( j c i c)T » ¾
3
¬
¼¿
M abG
where
k 1, 3, "; j | k r 2l |
By use of formula (1.10.6), there is
q
q
ª
¦¦ cos «¬(k B j )(J icT ) r j
ic 1 jc 1
2ʌ
º
B j ( j c i c)T »
3
¼
qiT
2 cos ª(k B j )(J i cT ) r j 2ʌ r ji cT B q 1 jT º
¦
«
»
T
j
3
2
¬
¼
ic 1
sin
2
2
qjT ·
§
sin
¨
2 ¸ cos ª(k B j )J r j 2ʌ q 1 kT B q 1 jT º
¨
¸
«
»
3
2
2
¬
¼
¨ sin jT ¸
©
2 ¹
q
sin
qjT
§
¨ sin 2
¨
¨ sin jT
©
2
2
·
¸
ª
q 1 ·
2ʌ º
§
T¸r j »
¸ cos «(k B j ) ¨ J 2 ¹
3¼
©
¬
¸
¹
so we can get
­
qjT
§
¨ sin 2
k yk k yj °°
2(2 PqwK ) W l
®(Odkj Oqkj ) ¨
¦¦
Pʌ 2
kj °
k
j
¨ q sin jT
2
©
°̄
2
M abG
2ʌ º
q 1 ·
ª
§
T ¸ j » (Odkj
u cos «(k j ) ¨ J 2 ¹
3¼
©
¬
½
2ʌ º °°
q 1 ·
ª
§
T¸ j » ¾
u cos «(k j ) ¨ J 2 ¹
3¼ °
©
¬
°¿
66
·
¸
¸
¸
¹
2
qjT
§
¨ sin 2
Oqkj ) ¨
¨ q sin jT
2
©
·
¸
¸
¸
¹
2
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
k wk k wj
2 w2W l
2 ¦¦
Pʌ k j kj
(Odkj
­
ª
2ʌ º
q 1 ·
§
T¸ j »
®(Odkj Oqkj ) cos «(k j ) ¨ J 2
3¼
©
¹
¬
¯
ª
2ʌ º ½
q 1 ·
§
Oqkj ) cos «(k j ) ¨ J T ¸ j »¾
2 ¹
3 ¼¿
©
¬
where
k 1, 3, "; j | k r 2l |
Like the relation in formula (1.3.34), the mutual inductance between phase a and
phase b windings can be written as
M abG
q 1
ʌ·
§
M ab 0G M ab 2 cos 2 ¨ J T ¸
2
3¹
©
q 1
ʌ·
§
T ¸ "
M ab 4 cos 4 ¨ J 2
3¹
©
(2.1.12)
in which
2
M ab 0G
2 w2W l § k wk ·
2kʌ
(Odkk Oqkk ) cos
¦
3
Pʌ 2 k ¨© k ¸¹
2
2w2W l
2kʌ
§k ·
O0 ¦ ¨ wk ¸ cos
3
Pʌ 2
k
¹
k ©
M ab 2
k 1, 3, "
(2.1.13)
2 w2W l ­ 2
®k w1 (Od 11 Oq11 )
Pʌ 2 ¯
ª k wk k w( k 2)
2ʌ º °½
(Odk ( k 2) Oqk ( k 2) ) cos(1 k ) » ¾
2¦«
3 ¼ ¿°
k ¬ k ( k 2)
k k
2 w2W l ­ 2
2ʌ ½
O2 ®kw1 2¦ wk w( k 2) cos(1 k ) ¾ k 1, 3, "
2
Pʌ
k
(
k
2)
3¿
k
¯
(2.1.14)
q 1
ʌº
ª
T » is the included electric angle between rotor
In formula (2.1.12), «J 2
3¼
¬
d-axis and the central line that is between phase a and phase b, and M abG reaches
extreme value when rotor d-axis coincides with the central line between phase a
q 1
ʌº
ª
and phase b, namely, «J T » 0.
2
3¼
¬
In formula (2.1.13), only consider the mutual inductance component caused by
air-gap flux, and leakage mutual inductance component M abl should be added in
after considering slot leakage flux and end leakage flux. Obviously, M abl is a
constant and independent of rotor position, which has negative value because phase
67
AC Machine Systems
a and phase b windings have a difference of 120ein space. Let M abl
total mutual inductance between phase a and phase b windings is
M ab
M ml , then
q 1
ʌ·
§
M ab 0 M ab 2 cos 2 ¨ J T ¸
2
3¹
©
q 1
ʌ·
§
T ¸ "
M ab 4 cos 4 ¨ J 2
3¹
©
(2.1.15)
where
M ab 0G M ml
M ab 0
In a similar way, the mutual inductance components for as branches in parallel are:
M ab 0G
M ab 2
ª§ k wk ·2
2kʌ º
¦k «¨© k ¸¹ cos 3 »
«¬
»¼
2 w2W l O0
Pʌ 2 as2
(2.1.16)
k wk kw( k 2)
2 w2W l O2 ­ 2
2ʌ ½
cos(1 k ) ¾
k 2¦
2
2 ® w1
3¿
Pʌ as ¯
k (k 2)
k
(2.1.17)
where
k 1, 3, "
Clearly, the stator phase winding inductances of salient-pole synchronous
machines depend upon rotor position, namely, variable parameters with time.
It should be noted that the discussion above is appropriate to number of stator
slots per pole per phase q integer. If q fraction, then distribution factor and
harmonic orders in self-inductance and mutual inductance formulas are different
to make some accommodation.
2.1.2
Inductances Between Stator and Rotor Windings
The mutual inductance between stator single coil and rotor excitation winding
has been deduced for salient-pole synchronous machines in formula (1.6.3). If
taking the central line of stator 0-th coil as the stator reference axis, then the
mutual inductance between stator i-th coil and rotor excitation winding is
M ifd
2w fd wKW l 1
ʌ
a fd
¦
k
Odk
k
sin
kE ʌ
cos k (J iT )
2
k 1, 3, "
Supposing that the stator phase winding is composed of 2P coil-groups in series
and each coil-group has q coils, then the mutual inductance between corresponding
stator phase winding and rotor excitation winding for q integer is
68
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
2P
q
¦¦
M I fd
m 1 i 1
u (1)
Because 3qT
2 w fd wKW l 1
ʌ
a fd
m 1
¦
Odk
k
k
sin
kE ʌ
2
cos k{J [(m 1)3q i ]T }
k 1, 3, "
(2.1.18)
ʌ , there is
2P
¦ (1)
m 1
2P
¦ (1)
m 1
cos k{J [(m 1)3q i ]T }
m 1
cos k{J (m 1)ʌ iT }
m 1
2 P cos k (J iT )
In the light of formula (1.10.6) exists
qkT
2 cos k § J q 1T ·
¨
¸
kT
2 ¹
©
sin
2
sin
q
¦ cos k (J iT )
i 1
Hence, formula (2.1.18) can be reformed as
M I fd
qkT
sin
kE ʌ
2 cos k § J q 1T ·
¦k k sin 2
¨
¸
kT
2 ¹
©
q sin
2
2 ww fdW l 1
Odk
q 1 ·
§
¦ kwk cos k ¨© J 2 T ¸¹ k 1, 3, "
a fd k k
ʌ
2 PqwK 2w fdW l 1
a fd
ʌ
Odk
(2.1.19)
q 1 º
ª
where «J T is the included electric angle between rotor d-axis and the
2 »¼
¬
central line of stator phase a winding.
If taking the phase a central line as stator reference axis, then the mutual
inductance between phase a winding and excitation winding can be written as
M afd
2 ww fdW l 1
ʌ
a fd
¦
k
Odk
k
kwk cos kJ
k 1, 3, "
(2.1.20)
2ʌ ·
2ʌ ·
§
§
If substituting ¨ J ¸ and ¨ J ¸ for J , respectively, then the mutual
3 ¹
3 ¹
©
©
inductances between phase b or phase c and excitation winding can be got as:
M bfd
2 ww fdW l 1
ʌ
a fd
¦
k
Odk
k
2ʌ ·
§
kwk cos k ¨ J ¸
3 ¹
©
k 1, 3, "
(2.1.21)
69
AC Machine Systems
2ww fdW l 1
ʌ
a fd
M cfd
¦
Odk
k
k
2ʌ ·
§
kwk cos k ¨ J ¸
3 ¹
©
k 1, 3, "
(2.1.22)
Evidently, the mutual inductances between stator phase windings and excitation
winding depend upon rotor position.
The formulas above are appropriate to single branch condition. If having as
branches in parallel, then the right sides of corresponding formulas above should
1
be multiplied by
. For example, the mutual inductance between phase a winding
as
and excitation winding, referring to formula (2.1.20), should be changed to
2 ww fdW l
M afd
ʌas a fd
¦
k
Odk
k
k wk cos kJ
k 1, 3, "
(2.1.23)
The mutual inductances between stator phase windings and damping loops
will be discussed next.
The mutual inductance between stator single coil and damping loop has been
shown in formula (1.6.5). If taking stator 0-th coil axis as stator reference axis,
then the mutual inductance between stator i-th coil and damping loop 11c is
M i1
O2l
2wK wrW l ­
kE ʌ
jE ʌ
sin 1 sin
cos[ j (J iT ) kD1 ]
® ¦
¦
2
Pʌ
kj
2
2
j ¯ 2l |k j |
¦
2l |k j |
O2l
kj
sin
½
k E1 ʌ
jE ʌ
sin
cos[ j (J iT ) kD1 ]¾
2
2
¿
(2.1.24)
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j
1 2 3
, , ,"
P P P
For single branch and integer number of slots, study the mutual inductance
between stator phase winding and damping loop as follows. Taking the first coil
of the phase winding under the first pole as No.1 coil, then the mutual inductance
between stator phase winding and damping loop 11c is
q
M I1
O2l
2 wK wrW l ­
kE ʌ
jE ʌ
sin 1 sin
(1) m 1
® ¦
¦
2
P
ʌ
kj
2
2
j ¯ 2l |k j |
1
2P
¦¦
i 1 m
u cos{ j[J (( m 1)3q i )T ] kD1}
O
kE ʌ
jE ʌ
¦ 2l sin 1 sin
(1) m 1
2
2
2 l |k j | kj
½
u cos{ j[J ((m 1)3q i )T ] kD1}¾
¿
70
(2.1.25)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; 3qT
ʌ
Analysing the summation term of formula (2.1.25), making m change from 1 to
2P, and supposing i to be constant and j to be odd number, then there is
2P
¦ (1)
m 1
2P
¦ (1)
m 1
cos{ j[J ((m 1)3q i )T ] r kD1}
m 1
cos{ j[J (m 1)ʌ iT ] r kD1}
m 1
2 P cos{ j (J iT ) r kD1}
which equals zero during j fraction or even number. Thus, formula (2.1.25) can
be written as
M I1
­
O2l
kE ʌ
jE ʌ
sin 1 sin
® ¦
2
2
j 1,3," ¯ 2 l | k j | kj
q
O
kE ʌ
jE ʌ
u ¦ cos[ j (J iT ) kD1 ] ¦ 2l sin 1 sin
2
2
i 1
2 l |k j | kj
2wK wrW l u 2 P
Pʌ 2
¦
q
½
u ¦ cos[ j (J iT ) kD1 ]¾
i 1
¿
where
| k j | 0, 2, 4, "; | k j | 2, 4, "
Referring to formula (1.10.6) exists
¦ cos[ j (J iT ) r kD ]
1
i 1
qjT
2 cos ª j § J q 1T · r kD º
1»
¸
« ¨
jT
2 ¹
¬ ©
¼
sin
2
sin
q
so the mutual inductance between stator phase winding and damping loop 11c is
M I1
qjT
­
sin
O2l
2 PqwK 2 wrW l °
k E1 ʌ
jE ʌ
2
¦j ®2l¦|k j| kj sin 2 sin 2
jT
Pʌ 2
°
q sin
2
¯
O2l
k E1 ʌ
ª §
º
q 1 ·
sin
u cos « j ¨ J T ¸ kD1 » ¦
2 ¹
2
¬ ©
¼ 2l |k j| kj
71
AC Machine Systems
qjT
½
sin
jE ʌ
q 1 ·
ª §
º°
2
T ¸ kD1 » ¾
cos « j ¨ J u sin
2 q sin jT
2 ¹
¬ ©
¼°
2
¿
O2l
kE ʌ
2 wwrW l ­
q 1 ·
ª §
º
sin 1 kwj cos « j ¨ J T ¸ kD1 »
® ¦
¦
2
Pʌ
2
2 ¹
j ¯ 2 l |k j | kj
¬ ©
¼
½
O
kE ʌ
q 1 ·
ª §
º
T ¸ kD1 » ¾
¦ 2l sin 1 k wj cos « j ¨ J kj
2
2
¹
2 l |k j |
¬ ©
¼¿
(2.1.26)
in which
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
q 1 º
ª
«J 2 T » is the electric angle, by which rotor d-axis leads ahead of stator
¬
¼
phase winding axis.
If taking the central line of phase a winding as stator reference axis, then the
mutual inductance between phase a winding and damping loop 11c is
O2l
2wwrW l ­
kE ʌ
sin 1 k wj cos( jJ kD1 )
® ¦
¦
2
Pʌ
2
j ¯ 2 l |k j | kj
M a1
¦
O2l
2l |k j |
kj
sin
½
k E1 ʌ
k wj cos( jJ kD1 ) ¾
2
¿
(2.1.27)
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
2ʌ ·
2ʌ ·
§
§
Substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in formula (2.1.27),
3 ¹
3 ¹
©
©
the mutual inductances between phase b or phase c and damping loop 11c can be
obtained.
If stator phase winding has as branches in parallel, like the mutual inductance
between stator phase winding and excitation winding, the mutual inductance between
stator phase winding and damping loop should be changed from formula (2.1.26) to
M I1
O2l
2wwrW l ­
kE ʌ
ª §
º
q 1 ·
sin 1 k wj cos « j ¨ J T ¸ kD1 »
® ¦
¦
2
Pʌ as j ¯ 2l |k j| kj
2
2 ¹
¬ ©
¼
¦
2l |k j |
72
O2l
kj
sin
k E1 ʌ
ª §
º½
q 1 ·
kwj cos « j ¨ J T ¸ kD1 » ¾
2
2 ¹
¬ ©
¼¿
(2.1.28)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
where
j 1, 3, "; | k j | 0, 2, 4, "; | k j | 2, 4, "
Clearly, the mutual inductance between stator phase winding and damping
loop depends upon rotor position.
As mentioned before, an equivalent d-axis damping winding and an equivalent
q-axis one are used to represent damping winding, so several parameters have to
be replaced in associated mutual inductance formulas. For example, during
calculation of mutual inductance between stator phase winding and equivalent
d-axis damping winding, the number of turns w1d for equivalent d-axis damping
winding will replace wr , the fractional pitch ratio E1d for equivalent d-axis one
will replace E1 , and let D1
M I 1d
0 . Hence, there is
2 ww1dW l ­
O
kE ʌ
ª §
q 1 ·º
¦ ® ¦ 2l sin 21d kwj cos «¬ j ¨© J 2 T ¸¹»¼
Pʌ 2 as j ¯ 2l |k j| kj
¦
O2l
2l |k j|
kj
sin
k E1d ʌ
ª §
q 1 ·º ½
T¸ ¾
kwj cos « j ¨ J 2
2 ¹ »¼ ¿
¬ ©
(2.1.29)
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
In addition, during the calculation of mutual inductance between stator phase
winding and equivalent q-axis damping winding, the number of turns w1q for
equivalent q-axis damping winding will supersede wr , the fractional pitch ratio
E1q for equivalent q-axis one will supersede E1 , and let D
ʌ/2 since rotor
q-axis leads ahead of rotor d-axis by ʌ/2 electric radians. Thus exists
M I 1q
2 ww1qW l
­
O2l
k E1q ʌ
ª §
q 1 · kʌ º
k wj cos « j ¨ J T¸
Pʌ as j ¯ 2l |k j| kj
2
2 ¹ 2 »¼
¬ ©
k E1q ʌ
O
ª §
q 1 · kʌ º ½
¦ 2l sin
kwj cos « j ¨ J T¸
(2.1.30)
¾
2
2 ¹ 2 »¼ ¿
2 l | k j | kj
¬ ©
2
¦® ¦
sin
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
The discussions above are targeted on stator 3-phase winding inductances and
mutual inductances between stator phase windings and rotor windings. As for
rotor winding inductances, the formulas (1.5.3), (1.5.14), (1.5.17), and (1.5.24) can
still be used. If using an equivalent d-axis damping winding and an equivalent
q-axis one, then the self-inductance of d-axis damping winding is
73
AC Machine Systems
2 w12dW l ­
O2l
kE ʌ
jE ʌ
sin 1d sin 1d
2 ¦® ¦
Pʌ
2
2
j ¯ 2 l |k j | kj
L11d
¦
2l |k j|
O2l
kj
sin
k E1d ʌ
jE ʌ ½
sin 1d ¾
2
2 ¿
(2.1.31)
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
The self-inductance of equivalent q-axis damping winding is
L11d
2 w12qW l
Pʌ
­
¦
O2l
¦® ¦
2
j
O2l
kj
2 l |k j |
¯ 2l
sin
|k j|
kj
k E1q ʌ
2
sin
sin
k E1q ʌ
2
j E1q ʌ
2
sin
cos
j E1q ʌ
2
cos
( j k )ʌ
2
( j k )ʌ ½
¾
2 ¿
(2.1.32)
where
| k j | 0, 2, 4, "; | k j | 2, 4, "; j 1, 3, "
The mutual inductance between excitation winding and equivalent d-axis
damping winding is
M f 1d
¦
k
2.1.3
2 w1d w fdW l 1 Odk
kE ʌ
sin 1d
ʌ
a fd k
2
k 1, 3, "
(2.1.33)
3-Phase Synchronous Machine Inductances in Accordance
with Fundamental Air-Gap Flux
In comparison with single coil inductances, the stator phase winding inductances
for integer number of slots and 60ephase belt only comprise fundamental and
odd harmonics effect except fractional harmonics and even harmonics as in
single coil. In addition, the fractional pitch and distributed winding is used to
make air-gap flux approximate to sinusoidal curve, so the calculation error of
inductances, during only consideration of fundamental only, is usually within
10%.
Considering fundamental air-gap flux only, then phase a winding self-inductance
is
Laa
L0 L2 cos 2J
(2.1.34)
in which J is the electric angle the rotor d-axis leads ahead of stator phase a
74
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
winding axis by, and
2 w2W l O0 2
kw1
Pʌ 2 as2
L0
L0l L2
2w2W l O2 2
k w1
Pʌ 2 as2
(2.1.36)
When rotor d-axis coincides with phase a winding axis, namely J
to formula (2.1.34) there are
Laa
L0 L2
Laad
L0G L2
(2.1.35)
0 , referring
L0l Laad
2 w2W l
(O0 O2 )k w21
Pʌ 2 as2
2
4W lP § wkw1 ·
Od 11
as2 ʌ ¨© P ¸¹
(2.1.37)
where Laad is phase a winding inductance caused by air-gap flux when rotor d-axis
coincides with phase a winding axis.
When rotor q-axis coincides with phase a axis, namely J ʌ / 2 , there exist
Laa
L0 L2
Laaq
L0G L2
L0l Laaq
2 w2W l
(O0 O2 )k w21
Pʌ 2 as2
2
4W lP § wk w1 ·
Oq11
as2 ʌ ¨© P ¸¹
(2.1.38)
where Laaq is phase a winding inductance caused by air-gap flux when rotor
q-axis coincides with phase a winding axis.
In the light of formulas (2.1.37) and (2.1.38) there are
L0G
L2
1
( Laad Laaq )
2
1
( Laad Laaq )
2
Hence, phase a winding self-inductance corresponding to fundamental air-gap
flux can be written as
75
AC Machine Systems
Laa
1
1
L0l ( Laad Laaq ) ( Laad Laaq ) cos 2J
2
2
L0l Laad cos 2 J Laaq sin 2 J
(2.1.39)
2ʌ ·
2ʌ ·
§
§
Substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in formula (2.1.34), then
3
3 ¹
©
¹
©
phase b and phase c winding inductances Lbb and Lcc can be obtained.
The mutual inductances between phase a and phase b windings are:
ʌ·
§
M ab 0 M ab 2 cos 2 ¨ J ¸
3¹
©
M ab
w2W l
O0 kw21 M ml
Pʌ 2 as2
M ab 0
M ab 2
2w2W l
O k2
2 2 2 w1
Pʌ as
(2.1.40)
(2.1.41)
(2.1.42)
Indicating mutual inductance between phase a and phase b by using Laad and Laaq ,
there is
M ab
1
1
ʌ·
§
M ml ( Laad Laaq ) ( Laad Laaq ) cos 2 ¨ J ¸
4
2
3¹
©
2ʌ ·
2ʌ ·
§
§
M ml Laad cos J cos ¨ J ¸ Laaq sin J sin ¨ J ¸
3
3 ¹
©
¹
©
(2.1.43)
2ʌ ·
2ʌ ·
§
§
Substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in Eq. (2.1.40), then the
3
3 ¹
©
¹
©
mutual inductance between phase b and phase c and that between phase c and
phase a can be found out.
The mutual inductance between phase a winding and excitation winding is
M afd
2 ww fdW l 1
Od 1kw1 cos J
ʌas a fd
(2.1.44)
Let
M afd 0
2ww fdW l
ʌas a fd
Od 1kw1
(2.1.45)
then
M afd
M afd 0 cos J
(2.1.46)
2ʌ ·
2ʌ ·
§
§
Substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in Eq. (2.1.46), then the
3 ¹
3 ¹
©
©
76
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
mutual inductances between phase b winding or phase c winding and excitation
winding can be got.
When considering fundamental flux produced by fundamental mmf only,
namely taking j 1 and k 1 in Eq. (2.1.29), we can get the mutual inductance
between phase a winding and equivalent d-axis damping winding. If taking phase
a winding axis as reference axis, namely ordinate, then J is substituted for
q 1 ·
§
T ¸ , so there is
¨J 2 ¹
©
2 ww1dW l
E ʌ
sin 1d k w1 (O0 O2 ) cos J
2
Pʌ a s
2
E ʌ§ w
8W l
·
w1d sin 1d ¨
kw1 ¸ Od 11 cos J
2
ʌ as
2 © 2P ¹
M a1d
(2.1.47)
Let
E ʌ§ w
8W l
·
w1d sin 1d ¨
kw1 ¸ Od 11
2
ʌ as
2 © 2P ¹
M a1d 0
(2.1.48)
then
M a1d
M a1d 0 cos J
(2.1.49)
2ʌ ·
2ʌ ·
§
§
Substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in Eq. (2.1.49), the mutual
3 ¹
3 ¹
©
©
inductances between phase b or phase c windings and d-axis damping winding
can be obtained.
When considering fundamental flux produced by fundamental mmf only,
namely taking j 1 and k 1 in Eq. (2.1.30), we can get the mutual inductance
between phase a winding and equivalent q-axis damping winding after substituting
q 1 ·
§
J for ¨ J T ¸ as follows:
2 ¹
©
M a1q
2 ww1qW l
2
Pʌ as
sin
E1q ʌ
2
kw1 (O0 O2 )( sin J )
E1q ʌ § w
8W l
·
w1q sin
kw1 ¸ Oq11 sin J
¨
2
ʌ as
2 © 2P ¹
(2.1.50)
E1q ʌ § w
8W l
·
w1q sin
kw1 ¸ Oq11
¨
2
ʌ as
2 © 2P
¹
(2.1.51)
Let
M a1q 0
77
AC Machine Systems
then
M a1q
M a1q 0 sin J
(2.1.52)
2ʌ ·
2ʌ ·
§
§
Similarly, substituting ¨ J ¸ and ¨ J ¸ for J , respectively, in Eq. (2.1.52),
3
3 ¹
©
¹
©
then the mutual inductances between phase b or phase c windings and q-axis
damping winding can be got.
It is noted that Eq. (2.1.45) differs from Eqs. (2.1.48) and (2.1.51) because
excitation winding fundamental flux is assumed to be produced by rectangular
mmf and d-axis or q-axis damping winding fundamental flux is assumed to be
produced by fundamental mmf.
2.2
Basic Relations of Synchronous Machines in d, q, 0
Axes
In the section above, votage and flux-linkage equations together with several
inductances have been found out for 3-phase salient-pole synchronous machines,
which are a set of differential equations with variable coefficients to solve them
in a difficulty.
As stated above, the ideal machine pattern can be used to analyse synchronous
machine operation and the caused error is limited and allowed, which is discussed
next.
2.2.1
Flux-Linkages and Parameters of Stator Phase Windings
As mentioned before, the flux-linkage of phase a produced by stator currents
ia , ib , and ic is
\ as
Laa ia M ab ib M ac ic
(2.2.1)
Referring to Eq. (2.1.39) and (2.1.43), the self- and mutual-inductances are:
Laa
M ab
L0l Laad cos 2 J Laaq sin 2 J
2ʌ ·
§
M ml Laad cos J cos ¨ J ¸
3 ¹
©
2ʌ ·
§
Laaq sin J sin ¨ J ¸
3 ¹
©
Substituting them into Eq. (2.2.1), we can get
78
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
\ as
ª
2 ·
2 ·º
§
§
Laad «ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ » cos J
3 ¹
3 ¹¼
©
©
¬
ª
2 ·
2 ·º
§
§
Laaq «ia sin J ib sin ¨ J ʌ ¸ ic sin ¨ J ʌ ¸ » sin J
3 ¹
3 ¹¼
©
©
¬
[ L0l ia M ml (ib ic )]
(2.2.2)
Various terms in the formula above have the following meanings.
After stator phase windings are passed with currents, they will constitue a mmf
in air-gap each. As stated before, the mmf amplitude of the fundarnental produced
by each phase current should be proportional to the phase current, so although
ia , ib , and ic are current quantity, they also represent the mmf amplitudes
produced by various phase currents. The mmf amplitudes lie on their own phase
winding axes, so after ia , ib and ic are multiplied by cos J ,
2 ·
2 ·
2 ·
§
§
§
cos ¨ J ʌ ¸ , and cos ¨ J ʌ ¸ , respectively, i.e. ia cos J , ib cos ¨ J ʌ ¸ ,
3 ¹
3 ¹
3 ¹
©
©
©
2 ·
§
and ic cos ¨ J ʌ ¸ , they can represent the mmfs produced on rotor d-axis
3 ¹
©
ª
2 ·
§
by stator phase currents and the sum of them «ia cos J ib cos ¨ J ʌ ¸ 3 ¹
©
¬
2 ·º
§
ic cos ¨ J ʌ ¸» can represent the resultant mmf produced on rotor d-axis by all
3 ¹¼
©
stator phase currents.
Obviously, the resultant d-axis mmf acts on the permeance Od 11 , so the d-axis
flux produced by all stator phase currents should be proportional to the product
ª
2 ·
2 ·º
§
§
of the current «ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ » and the permeance Od 11 .
3
3 ¹¼
©
¹
©
¬
Multiplying the result above by corresponding winding data, we can get d-axis
flux-linkage produced by all stator phase currents. Therefore, in Eq. (2.2.2), the item
ª
2 ·
2 ·º
§
§
Laad «ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ »
3
3 ¹¼
©
¹
©
¬
2
ª
4W lP § wk w1 ·
2 ·
2 ·º
§
§
Od 11 «ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ »
¸
2 2 ¨
as ʌ © P ¹
3 ¹
3 ¹¼
©
©
¬
is the d-axis air-gap flux-linkage produced by all stator phase currents.
In order to get a clear conception, now we may suppose that on the stator
d-axis there is a fictitious winding which has the same turns number and
form as the practical stator phase winding, and then we can see that when
79
AC Machine Systems
ª
2 ·
2 ·º
§
§
«ia cos J ib cos ¨ J 3 ʌ ¸ ic cos ¨ J 3 ʌ ¸ » is multiplied by Od 11 and corres©
¹
©
¹¼
¬
ponding winding data, i.e. by Laad , the d-axis air-gap flux-linkage of the fictitious
d-axis winding may be written as
ª
2 ·
2 ·º
§
§
Laad «ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ »
3 ¹
3 ¹¼
©
©
¬
\ dsG
(2.2.3)
If a fictitious current i( d ) circulates through the fictitious d-axis winding, then
we can know that the air-gap fulx-linkage produced in the d-axis winding by the
fictitious current is Laad i( d ) . Comparing the result with Eq. (2.2.3), we can
understand that when both flux-linkages are equal the fictitious current in the
fictitious d-axis winding is
i( d )
2 ·
2 ·
§
§
ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸
3 ¹
3 ¹
©
©
(2.2.4)
That is to say, so far as the air-gap flux density is concerned, the d-axis air-gap
flux density produced by stator currents may be calculated according either to
stator 3-phase currents or to fictitious current i( d ) , both results being equal.
Because the air-gap flux density is distributed sinusoidally in space, the
flux-linkage of phase a corresponding to d-axis air-gap flux density is \ dsG cos J .
As for q-axis we have the similar explanation, i.e.,
ª
2 ·
2 ·º
§
§
«ia sin J ib sin ¨ J ʌ ¸ ic sin ¨ J ʌ ¸ »
3 ¹
3 ¹¼
©
©
¬
can represent the resultant mmf produced on q-axis by all stator phase currents.
When multiplying it by Laaq , we can get the q-axis air-gap flux-linkage of the
fictitious q-axis winding which has the same turns number and form as the practical
stator phase winding. Let its value be \ qsG , then
\ qsG
ª
2 ·
2 ·º
§
§
«ia sin J ib sin ¨ J ʌ ¸ ic sin ¨ J ʌ ¸ » ( Laaq )
3 ¹
3 ¹¼
©
©
¬
(2.2.5)
ª
2 ·
2 ·º
§
§
«ia sin J ib sin ¨ J ʌ ¸ ic sin ¨ J ʌ ¸ »
3 ¹
3 ¹¼
©
©
¬
(2.2.6)
Similarly, let
i( q )
be the fictitious q-axis winding current, then the q-axis air-gap flux-linkage
produced in the q-axis winding by the fictitious q-axis current is also \ qsG .
80
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Multiplying the flux-linkage by sin J and taking negative sign, we can obtain the
flux-linkage of phase a winding corresponding to the q-axis air-gap flux density,
namely, \ qsG sin J .
L0l and Lml in Eq. (2.2.2) are stator leakage self-inductance and leakage mutual
inductance, respectively. Therefore, [ L0l ia M ml (ib ic )] is the leakage fluxlinkage not across air-gap produced in phase a winding by stator 3-phase currents.
The resultant flux-linkage of phase a winding is equal to the sum of various
flux-linkages mentioned above, and Eq. (2.2.2) can be rewritten as
\ as \ dsG cos J \ qsG sin J [ L0l ia M ml (ib ic )]
( Laad i( d ) ) cos J ( Laaq i( q ) )sin J [ L0l ia M ml (ib ic )]
(2.2.7)
In practice, we shall use d-axis and q-axis armature reaction inductances Lad
and Laq to calculate some problems instead of inductances Laad and Laaq . Lad is
the stator equivalent inductance when rotor d-axis coincides with the axis of
resultant magnetic field produced by stator 3-phase currents, and Laq is the
corresponding equivalent inductance when rotor q-axis coincides with the axis of
the resultant magnetic field. There is a simple relation between Lad and Laad or
between Laq and Laaq , and the simple relation can be easily found as follows.
Let 3-phase balanced currents be
ªia º
«i »
« b»
«¬ ic »¼
sin Z t
ª
º
«
I m «sin(Z t 120e
) »»
«¬sin(Z t 120e
) »¼
Referring to Eq. (2.2.2), the flux-linkage corresponding to air-gap flux-density
produced in phase a winding by stator 3-phase currents is
\ asG
ª
2 ·
2 ·
§
§
Laad « I m sin Z t cos J I m sin ¨ Z t ʌ ¸ cos ¨ J ʌ ¸
3
3 ¹
©
¹
©
¬
2 ·
2 ·º
§
§
I m sin ¨ Z t ʌ ¸ cos ¨ J ʌ ¸» cos J
3
3 ¹¼
©
¹
©
2 · §
2 ·
ª
§
Laaq « I m sin Z t sin J I m sin ¨ Z t ʌ ¸ sin ¨ J ʌ ¸
3 ¹ ©
3 ¹
©
¬
2 · §
2 ·º
§
I m sin ¨ Z t ʌ ¸ sin ¨ J ʌ ¸» sin J
3 ¹ ©
3 ¹¼
©
3
3
Laad I m sin(Z t J ) cos J Laaq I m cos(Z t J )sin J
2
2
81
AC Machine Systems
When J
Z t , there is
\ asG
3
Laaq I m sin Z t
2
3
Laaq ia
2
and the corresponding equivalent inductance is
\ asG
ia
3
Laaq
2
i.e., when J Z t , rotor speed is equal to the speed of revolving field produced
by stator 3-phase currents and rotor q-axis coincides with the revolving field axis.
Hence the corresponding equivalent inductance is Laq , namely,
Laq
When J
3
Laaq
2
2
3 4W lP § wkw1 ·
Oq11
2 as2 ʌ 2 ¨© P ¸¹
(2.2.8)
Z t ʌ / 2 exists
\ asG
3
Laad I m sin Z t
2
3
Laad ia
2
and the corresponding equivalent inductance is
\ asG
ia
3
Laad
2
i.e., when J
Z t ʌ / 2 , rotor d-axis coincides with the axis of revolving field.
Thus, the corresponding equivalent inductance is Lad , namely,
Lad
3
Laad
2
2
3 4W lP § wkw1 ·
Od 11
2 as2 ʌ 2 ¨© P ¸¹
(2.2.9)
Using inductances Lad and Laq to rewrite Eq. (2.2.7), there is
\ as
2 º
2 º
ª
ª
« Lad 3 i( d ) » cos J « Laq 3 i( q ) » sin J
¬
¼
¬
¼
[ L0l ia M ml (ib ic )]
(2.2.10)
Supposing
82
id
2
i( d )
3
iq
2
i( q )
3
2ª
2 ·
2 ·º ½
§
§
ia cos J ib cos ¨ J ʌ ¸ ic cos ¨ J ʌ ¸ » °
«
3¬
3 ¹
3 ¹¼ °
©
©
¾
2ª
2 ·
2 ·º
§
§
«ia sin J ib sin ¨ J ʌ ¸ ic sin ¨ J ʌ ¸ » °
3¬
3 ¹
3 ¹ ¼ °¿
©
©
(2.2.11)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
and calling them to be the stator current d-axis and q-axis components, respectively,
then exists
\ as
> Lad id @ cos J ª¬ Laq iq º¼ sin J
[ L0l ia M ml (ib ic )]
(2.2.12)
i.e., when keeping flux-linkage the same, for convenience, we have substituted
3
inductances Lad and Laq which are amplified by a factor of
for inductances
2
Laad and Laaq , and have also substituted currents id and iq which are reduced by
2
for currents i( d ) and i( q ) .
3
Out of Eq. (2.2.11) we can see that from the view-point of mathematics currents
id and iq are two new variables constituted by original variables ia , ib and ic .
However, if there are three original variables, then usually exist three new
variables. For example, when a 3-phase machine operates asymmetrically, three
original variables cannot be replaced by two new variables, so we must choose
the third new variable as follows:
a factor of
i0
1
(ia ib ic )
3
(2.2.13)
The relation between i0 and ia , ib , ic has the same form as zero sequence current
in ordinary Symmetrical Components Method, but currents i0 , ia , ib and ic here
are all instantaneous values and not complex phasors that represent currents
changing sinusoidally as in Symmetrical Components Method. The current i0 is
called zero-axis component of stator current. From the winding connection we can
see that the zero-axis component is practically one third of neutral-line current. If
having no neutral line, the zero-axis component of stator current will be zero.
When the same single-phase current circulates through 3-phase-windings,
i.e. ia ib ic I m sin Z t , obviously there is i0 I m sin Z t and the pulsating
fundamental mmfs produced by currents ia , ib and ic have the same instantaneous
2
value, but their space positions differ from one another by ʌ. Therefore, the
3
resultant fundamental mmf in air-gap is equal to zero thus having no air-gap flux,
but each phase winding can produce its own leakage flux.
Thus, currents id , iq and i0 can be written as
83
AC Machine Systems
ªid º
«i »
« q»
«¬ i0 »¼
ª2
« 3 cos J
«
« 2
« sin J
« 3
« 1
«
¬ 3
2
2 ·
2
2 ·º
§
§
cos ¨ J ʌ ¸
cos ¨ J ʌ ¸ »
3
3
3
3
©
¹
©
¹»
ªia º
2 §
2 ·
2 §
2 ·» « »
sin ¨ J ʌ ¸ sin ¨ J ʌ ¸ » « ib »
3 ©
3 ¹
3 ©
3 ¹»
«i »
»¬ c¼
1
1
»
3
3
¼
Transforming the equation above inversely, we can describe currents ia , ib and ic as
ªia º
« »
«ib »
«¬ ic »¼
cos J
sin J
1º ªid º
ª
« cos(J 120e
) sin(J 120e
) 1»» «« iq »»
«
«¬cos(J 120e
) sin(J 120e
) 1»¼ «¬ i0 »¼
(2.2.14)
Substituting (2.2.14) into (2.2.12) and arranging it in order, there is
\ as
( Lad Ll )id cos J ( Laq Ll )iq sin J L0 i0
(2.2.15)
where
Ll
L0l M ml
(2.2.16)
is called stator leakage inductance that consists of leakage self-inductance and
leakage mutual inductance for stator winding. In the discussion before, harmonic
mmfs have been neglected, but during analysis of transients those harmonics can
be considered by a differential leakage reactance, i.e. during calculation of
leakage inductance Ll count a differential leakage inductance in.
In (2.2.15), the expression
L0
L0l 2 M ml
(2.2.17)
is called stator zero-axis inductance, which also consists of leakage self-inductance
and leakage mutual inductance of stator winding but has a different value from
the stator leakage inductance Ll . It is easily shown that L0 is stator equivalent
inductance when the steady-state zero-sequence current circulates through 3-phasewindings.
Supposing
Ld
Lq
Lad Ll ½
Laq Ll ¾¿
(2.2.18)
then exists
\ as
84
Ld id cos J ( Lq iq )sin J L0 i0
(2.2.19)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
or
\ as \ ds cos J \ qs sin J \ 0
(2.2.20)
in which
\ ds Ld id
\ qs Lq iq
\ 0 L0i0
where Ld is called stator d-axis inductance and Lq the stator q-axis inductance.
Ld and Lq have the following physical conceptions:
If steady-state 3-phase currents circulate through stator phase windings and
rotor d-axis coincides with the axis of revolving field produced by 3-phase
currents, then referring to Eqs. (2.2.14) and (2.2.19) there are
i0
0, iq
0, id
ia
id cos J , Ld
Im
\ as
ia
Thus, Ld is stator equivalent inductance on the condition above. Compared with
Lad , the Ld includes leakage-flux effect.
Similarly, when rotor q-axis coincides with the revolving field axis mentioned
above, referring to Eqs. (2.2.14) and (2.2.19) exist
i0
0,
id
ia
iq sin J ,
0,
iq
Lq
Im
\ as
(ia )
Hence, Lq is stator equivalent inductance when rotor q-axis coincides with the
revolving field axis. Lq also includes the leakage-flux effect.
From Eq. (2.2.20) we can see that phase a flux-linkage produced by all stator
phase currents can be indicated by three flux-linkages \ ds , \ qs and \ 0 . \ ds and
\ qs are called d-axis and q-axis components of stator flux-linkage produced by
all stator phase currents, respectively. Compared with \ dsG
Lad id and \ qsG
Laq iq , the d-axis component \ ds and q-axis component \ qs all include the
stator leakage flux-linkage. \ 0 is called zero-axis component of stator flux-linkage
that has no magnetic connection with rotor winding.
When studying phase b and phase c flux-linkages in a similar way or substituting
85
AC Machine Systems
2 ·
§
directly ¨ J B ʌ ¸ for J in Eq. (2.2.20), we can write the flux-linkages of phase
3 ¹
©
b and phase c produced by all stator phase currents as follows:
ª\ bs º
« »
¬\ cs ¼
ª §
2 ·
2 · º
§
«cos ¨ J 3 ʌ ¸ sin ¨ J 3 ʌ ¸ 1» ª\ ds º
¹
©
¹ »« »
« ©
\ qs
« §
2 ·
2
§
· »« »
«cos ¨ J ʌ ¸ sin ¨ J ʌ ¸ 1» «¬\ 0 »¼
3 ¹
3 ¹ ¼
©
¬ ©
(2.2.21)
The stator phase a flux-linkage produced by rotor winding currents is
\ ar
M afd i fd M a1d i1d M a1q i1q
M afd 0 i fd cos J M a1d 0 i1d cos J M a1q 0 i1q sin J
( M afd 0 i fd M a1d 0 i1d ) cos J M a1q 0 i1q sin J
\ dr cos J \ qr sin J
(2.2.22)
in which
\ dr
\ qr
M afd 0 i fd M a1d 0 i1d
M a1q 0 i1q
Comparing Eq. (2.2.22) with Eq. (2.2.20), we can see that \ dr and \ qr are
d-axis and q-axis components of stator flux-linkage produced by rotor currents.
In addition, excitation winding current and d-axis damping winding current
produce only d-axis component of stator flux-linkage, q-axis damping winding
current produces only q-axis component of stator flux-linkage, and both cannot
produce zero-axis component of stator flux-linkage. That is not difficult to
understand, because currents i fd and i1d produce only d-axis flux and current i1q
produces only q-axis flux.
According to the results analysed above, we can describe the resultant
flux-linkage produced in phase a of stator winding by stator and rotor currents as
\ a \ as \ ar \ d cos J \ q sin J \ 0
(2.2.23)
in which
\d
\q
\0
Similarly, there are
86
Ld id M afd 0 i fd M a1d 0 i1d ½
°
Lq iq M a1q 0 i1q
¾
°
L0 i0
¿
(2.2.24)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
ª\ b º
«\ »
¬ c¼
ª §
2 ·
2 · º
§
«cos ¨ J 3 ʌ ¸ sin ¨ J 3 ʌ ¸ 1» ª\ d º
¹
©
¹ »« »
« ©
\q
« §
2 ·
2 · »« »
§
«cos ¨ J ʌ ¸ sin ¨ J ʌ ¸ 1» «¬\ 0 »¼
3 ¹
3 ¹ ¼
©
¬ ©
(2.2.25)
Inversely, from Eqs. (2.2.23) and (2.2.25) exist
ª\ d º
«\ »
« q»
«¬\ 0 »¼
2.2.2
ª2
« 3 cos J
«
« 2
« sin J
« 3
« 1
«
¬ 3
2
2 ·º
§
cos ¨ J ʌ ¸ »
3
3 ¹»
©
ª\ a º
2 ·
2 §
2 ·»
ʌ ¸ sin ¨ J ʌ ¸ » ««\ b »»
3 ¹
3 ©
3 ¹»
«\ »
»¬ c¼
1
»
3
¼
2
2 ·
§
cos ¨ J ʌ ¸
3
3 ¹
©
2 §
sin ¨ J
3 ©
1
3
(2.2.26)
Flux-linkages and Parameters of Rotor Windings
Referring to Eq. (2.1.2), the excitation winding flux-linkage is
\ fd
M fad ia M fbd ib M fcd ic L ffd i fd M f 1d i1d
According to Eq. (2.2.14) and mutual inductance relations in Eq. (2.1.46), or
according to the concept that the stator d-axis ficticious winding couples
electromagnetically the excitation winding, we can obtain
\ fd
M afd 0 i( d ) L ffd i fd M f 1d i1d
\ fd
3
M afd 0 id L ffd i fd M f 1d i1d
2
or
(2.2.27)
The formula above shows that in excitation winding flux-linkage there are not
only flux-linkage components L ffd i fd and M f 1d i1d produced by excitation winding
current and d-axis damping winding current but also flux-linkage component
§ 3
·
¨ M afd 0 id ¸ produced by all stator phase currents. That is not difficult to
© 2
¹
understand because air-gap flux-linkage produced on d-axis by stator currents
depends only upon d-axis mmf produced by stator currents or i( d ) . In Eq. (2.2.27)
exists id
2
i( d ) , so the corresponding mutual inductance is multiplied by 3/2.
3
87
AC Machine Systems
In a similar way, the flux-linkage equations of d-axis and q-axis damping
windings can be described as
\ 1d
\ 1q
2.2.3
3
½
M a1d 0 id M 1 fd i fd L11d i1d °
°
2
¾
3
°
M a1q 0 iq L11q iq
°¿
2
(2.2.28)
Voltage Equations of Synchronous Machines
We can see that phase a voltage equation is
p\ a ria
p(\ d cos J \ q sin J \ 0 )
ua
r (id cos J iq sin J i0 )
cos J p\ d \ d sin J pJ sin J p\ q \ q cos J pJ
p\ 0 rid cos J riq sin J ri0
( p\ d \ q pJ rid ) cos J
( p\ q \ d pJ riq )sin J ( p\ 0 ri0 )
where pJ Z is rotor instantaneous speed.
If letting
ud
uq
u0
p\ d \ qZ rid ½
°
p\ q \ d Z riq ¾
°
p\ 0 ri0
¿
(2.2.29)
ud cos J uq sin J u0
(2.2.30a)
then exists
ua
i.e., ua has a similar expression as ia and \ a referring to Eqs. (2.2.14) and (2.2.23).
2ʌ ·
§
Substituting ¨ J B
¸ for J in (2.2.30a), ub and uc can be written as
3 ¹
©
ub
uc
88
2 ·
2 ·
½
§
§
ud cos ¨ J ʌ ¸ uq sin ¨ J ʌ ¸ u0 °
3 ¹
3 ¹
°
©
©
¾
2 ·
2 ·
§
§
ud cos ¨ J ʌ ¸ uq sin ¨ J ʌ ¸ u0 °
°¿
3 ¹
3 ¹
©
©
(2.2.30b)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Making inverse transformation in Eqs. (2.2.30a) and (2.2.30b), the voltage
conversion formulas, just as Eqs. (2.2.11), (2.2.13), and (2.2.26), can be described as
ª ud º
«u »
« q»
¬« u0 ¼»
ª2
« 3 cos J
«
« 2
« sin J
« 3
« 1
«
¬ 3
2
2 ·º
§
cos ¨ J ʌ ¸ »
3
3 ¹»
©
ª ua º
2 ·
2 §
2 ·» « »
ʌ ¸ sin ¨ J ʌ ¸ » «ub »
3 ¹
3 ©
3 ¹»
«u »
»¬ c¼
1
»
3
¼
2
2 ·
§
cos ¨ J ʌ ¸
3
3 ¹
©
2 §
sin ¨ J
3 ©
1
3
(2.2.31)
Just as stator flux-linkages and currents, ud , uq and u0 are called the stator voltage
d-axis, q-axis and zero-axis components, respectively.
In Section 2.1, we have got excitation winding and damping winding voltage
equations
u fd
0
0
p\ fd R fd i fd ½
°
p\ 1d R1d i1d ¾
p\ 1q R1q i1q °¿
(2.2.32)
Formulas (2.2.29) and (2.2.32) compose voltage equations in d, q, 0 axes for
synchronous machines.
2.2.4
Park’s Formulas of Synchronous Machines
Equations (2.2.29), (2.2.32), (2.2.24), (2.2.27), and (2.2.28) constitute together
the basic relations of synchronous machines in d, q, 0 axes, which are named
Park’s formulas and the corresponding quantities are shown in actual values.
Advantages of actual values are clear concept and distinct unit dimension, but
practical calculation is not convenient. Hence, per-unit systems are often adopted
to analyse several problems and discussed in the following section.
2.3
Per-Unit Systems in Synchronous Machines
In sections above we have discussed electromagnetic relations of Salient-pole
synchronous machines on the basis of actual values. Advantages of actual values
are clear physical concept and explicit unit dimension. However, it is not convenient
to calculate practical problems. Therefore, per-unit systems are often used to research
into many problems of electric machines. By using per-unit systems, the bases
have to be selected at first. Generally speaking, stator rated amplitudes are chosen
89
AC Machine Systems
as the stator bases of synchronous machines. However, so far as the choice of rotor
bases is concerned, there are different methods that are discussed as follows.
2.3.1
Per-Unit System of Reversible Mutual Inductances
If I mG , U mG , and \ G are stator current base, voltage base and flux-linkage base that
are all corresponding rated amplitudes, I fdG , U fdG , and \ fdG are excitation winding
current base, voltage base and flux-linkage base, I1dG , U1d G , and \ 1dG are d-axis
damping winding current base, voltage base and flux-linkage base, tG is time
base and let
kifd
I mG
I fd G
kufd
U mG
U fdG
\ G / tG
\ fdG / tG
\G
\ fdG
I mG
I1dG
U mG
U1dG
\ G / tG
\ 1dG / tG
\G
\ 1dG
ki1d
ku1d
½
°
°
°
°
°
¾
°
°
°
°
°
¿
(2.3.1)
we can see that the coefficients kifd , kufd , ki1d , and ku1d represent ratios of stator
current base (or flux-linkage base) to rotor corresponding bases, respectively.
Compared with transformers, those coefficients are similar to transformation
ratio of stator winding to rotor winding. In order to keep the per-unit equations
having the same form as original equations, when choosing various winding
bases we must maintain the following expressions,
voltage base current base u impedance base
power base voltage base u current base
reactance base angular frequency base u inductance base
flux-linkage base inductance base u current base
voltage base flux-linkage base/time base
time base 1/angular frequency base
Two sides of the flux-linkage equations
ª\ d º
«\ »
« fd »
«¬\ 1d »¼
90
ª
« Ld
« 3
« M afd 0
« 2
« 3
«¬ 2 M a1d 0
M afd 0
L ffd
M 1 fd
º
M a1d 0 »
» ª id º
M f 1d » «i fd »
»« »
» «¬ i1d »¼
L11d »
¼
(2.3.2)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
are divided by \ G , \ fdG , and \ 1dG , respectively, and by use of Eq. (2.3.1) and
\G
LG I mG , after arrangement we can obtain
ª\ d º
« »
«\ fd »
« »
«¬\ 1d »¼
ª Ld
«
« M fad
« M 1ad
¬
M afd
L ffd
M 1 fd
M a1d º ª i d º
»
M f 1d » ««i fd »»
L11d »¼ «¬ i1d »¼
(2.3.3)
The quantities with bottom-bars above represent corresponding per-unit values,
in which
½
Ld
°
Ld
°
LG
°
M afd 0 1
°
M afd
LG kifd °
°
3
°
M afd 0
°
2
M fad
kufd °
LG
°
M a1d 0 1 °
M a1d
LG ki1d °
°
3
°
M a1d 0
¾
(2.3.4)
2
M 1ad
ku1d °
LG
°
°
L ffd kufd
L ffd
°
LG kifd
°
°
L11d ku1d
L11d
°
LG ki1d
°
M 1 fd ku1d °
M 1 fd
°
LG kifd °
M 1 fd kufd °
M f 1d
°
LG ki1d °
¿
We can see that Eqs. (2.3.2) and (2.3.3) have the same form, and various mutual
inductances in Eq. (2.3.3) can be reversible or not, i.e. M afd and M fad , M a1d
and M 1ad , or M f 1d and M 1 fd may be equal, respectively or not, which depends
upon the choice of transformation coefficients kifd , kufd , ki1d , and ku1d at all.
If those mutual inductances are made reversible, the following relations have
to be maintained, namely,
91
AC Machine Systems
3
kufd
2
3
ku1d
2
kufd
ki1d
1 ½
°
kifd °
1 °°
¾
ki1d °
ku1d °
°
kifd °¿
(2.3.5)
2½
3 °°
¾
2°
3 °¿
(2.3.6a)
or
kufd kifd
ku1d ki1d
From Eq. (2.3.6a) it is clear that the products kufd kifd , and ku1d ki1d each must equal
2/3 in order to get reversible mutual inductances after stator bases are taken.
It is well known that in transformers the condition making their mutual
inductances reversible is ku ˜ ki 1. In synchronous machines there is an
electromagnetic connection among stator d-axis winding, excitation winding and
d-axis damping winding as in three-winding transformers, but here the condition
making their mutual inductances reversible becomes ku ˜ ki 2 / 3 , because
current id is taken as 2/3 times the fictitious d-axis winding current i( d ) that
produces equivalent d-axis air-gap flux density.
It is clear that Eq. (2.3.6a) is easily sufficed and coefficients kifd and ki1d (or kufd
and ku1d ) can be chosen arbitrarily. The result shows that the methods of choosing
rotor d-axis bases are still a lot although stator bases are already chosen and the
mutual inductances are made reversible. However, once coefficients kifd and ki1d
(or kufd and ku1d ) are chosen, kufd and ku1d (or kifd and ki1d ) are also determined,
i.e. only one between kifd and kufd (or ki1d and ku1d ) can be chosen arbitrarily.
The rotor q-axis bases can be selected in the same way and we can reach the
conclusion, i.e., when mutual inductances are needed to be reversible, only one
between coefficients ki1q I mG / I1qG and ku1q \ G /\ 1qG can be selected arbitrarily,
in which I1qG and \ 1qG are q-axis damping winding current base and flux-linkage
base, respectively, and there is
ki1q ku1q
92
2
3
(2.3.6b)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
In practice, when we need to choose rotor bases of a synchronous machine,
kifd , ki1d and ki1q are selected at first, i.e. rotor current bases are chosen firstly.
Although coefficients kifd , ki1d and ki1q are selected arbitrarily, per-unit systems
used in practice have only several types. We shall discuss a per-unit system used
in this book as below.
In the per-unit system, current base of excitation winding is so chosen that the
excitation current base produces stator no-load voltage of actual amplitude xad
I mG that is evaluated according to air-gap line when a synchronous machine
rotates at synchronous speed and its stator winding is open-circuited. Here xad is
the synchronous reactance of d-axis armature reaction, xad ZG Lad , and ZG is
angular frequency base, namely synchronous angular frequency Z s 2ʌf . On
the basis of the conditions mentioned above, there is
ZG M afd 0 I fdG
ZG Lad I mG
xad I mG
or
kifd
I mG
I fdG
M afd 0
(2.3.7)
Lad
Substituting Eq. (2.3.7) into Eq. (2.3.4) exists
M afd
M afd 0 Lad
LG M afd 0
L ad
(2.3.8)
i.e., when excitation current base is chosen according to the way above, M afd in
per-unit is equal to Lad in per-unit. Substituting Eqs. (2.1.45) and Eq. (2.2.9) into
Eq. (2.3.7) we can get the following relation:
3 4 § wkw1 · § I mG
¨
2 ʌ ¨© 2 P ¸¹ © as
·
¸ Od 11
¹
w fd
I fd G
a fd
Od 1
which shows that fundamental air-gap flux produced by stator 3-phase base currents
is equal to that produced by excitation winding current base.
Similarly, if letting
ki1d
I mG
I1dG
M a1d 0
,
Lad
ki1q
I mG
I1qG
M a1q 0
Laq
and referring to Eq. (2.3.4), then there are
M a1d
L ad ,
M a1q
L aq
93
AC Machine Systems
Out of the relations above, it is clear that the per-unit system makes not only
mutual inductances reversible but also some mutual inductances equal that are
originally not equal to one another, so referring to Eqs. (2.3.4) and (2.3.6) we can
get
M afd
M fad
M a1d
M 1ad
M a1q
M 1aq
L aq
M f 1d
M 1 fd
2 M f 1d
3 LG
2
L ffd
2 L ffd
3 LG
§ I fdG ·
¨
¸
© I mG ¹
L11d
2 L11d
3 LG
§ I1dG ·
¨
¸
© I mG ¹
L11q
2 L11q
3 LG
§ I1qG ·
¨
¸
© I mG ¹
2
2
L ad ½
°
°
§ I fdG I1dG · °
¨
¸°
2
© I mG ¹ °
°
°
¾
°
°
°
°
°
°
°
¿
(2.3.9)
Substituting those results into Eq. (2.3.3) and considering inductance in per-unit
to be equal to reactance in per-unit corresponding to rated angular frequency, we
can write the following equation:
ª\ º
« d»
«\ »
« fd »
«¬\ 1d »¼
ª xd
«
« x ad
«
¬ x ad
x ad
x ffd
x1 fd
x ad º ª i d º
»
x f 1d » ««i fd »»
x11d ¼» «¬ i1d »¼
(2.3.10)
Similarly, there are
ª\ q º
« »
«¬\ 1q »¼
ª xq
« x
¬ aq
x aq º ª i q º
x11q »¼ «¬i1q »¼
(2.3.11)
Because
kufd kifd
§ U mG
¨¨
© U fdG
·§ I mG
¸¨
¸¨ I
¹© fd G
·
¸¸
¹
2
3
the excitation voltage base can be determined as
U fdG
94
3
U mG
2
§ I mG
¨¨
© I fdG
·
¸¸
¹
(2.3.12)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
The resistance (or impedance) base of excitation winding should be so chosen
that there is no remainder coefficient in equations expressed in per-unit. When
current and voltage bases are already chose, resistance (or impedance) base
should be taken as
Z fdG
U fdG
§ I mG
¨¨
© I fdG
3
U mG
2
I fdG
3 U mG
2 I mG
§ I mG
¨¨
© I fdG
·
¸¸
¹
· 1
¸¸
¹ I fd G
2
3
ZG
2
§ I mG
¨¨
© I fdG
·
¸¸
¹
2
(2.3.13)
where U mG / I mG ZG xG LG ZG is impedance base of stator winding.
Similarly, impedance bases of d-axis and q-axis damping windings are
2
§ I mG · ½
¨
¸ °
© I1d G ¹ °°
2 ¾
3 § I mG · °
ZG ¨
¸
2 ¨© I1qG ¸¹ °°¿
3
ZG
2
Z1dG
Z1qG
(2.3.14)
When voltage equations of stator and rotor windings are described in per-unit,
the equation forms are similar to those expressed in actual values. For example,
ua p\ a ria for phase a, two sides of which are divided by U mG , and then exists
ua
U mG
d\ a 1
ri
a
dt U mG U mG
d(\ a /\ G ) r ia
d(t / tG )
ZG I mG
i.e.
ua
d\ a
dt
r ia
(2.3.15)
In addition, u fd p\ fd rfd i fd for excitation winding, two sides of which are
divided by U fd G , and then there is
u fd
U fdG
d\ fd
1
dt U fdG
rfd i fd
d(\ fd /\ fd G )
U fdG
d(t / tG )
rfd
i fd
Z fdG I fdG
i.e.
u fd
p\
fd
r fd i fd
(2.3.16)
95
AC Machine Systems
2.3.2
Park’s Per-Unit System of Irreversible Mutual Inductances
When rotor has only an excitation winding and no damping winding, it is also
convenient to use the per-unit system of irreversible mutual inductances, by
which Park formulated basic equations of synchronous machines.
Without damping windings, the relations corresponding to Eq. (2.3.4) are
Ld
M afd
M fad
L ffd
Ld
LG
½
°
°
M afd 0 1 °
°
LG kifd °
°
3
¾
M afd 0
°
2
kufd °
LG
°
°
L ffd kufd
°
LG kifd
°¿
(2.3.17)
Evidently, if mutual inductances M afd and M fad are not asked to be reversible
then transformation coefficients kufd and kifd are all chosen arbitrarily. Park chose
the bases according to the following way. At first, excitation current base can
produce stator rated no-load voltage that is evaluated according to air-gap line
when a synchronous machine rotates at synchronous speed. Secondly, excitation
voltage base can produce excitation current base and thirdly, excitation current
base may produce flux-linkage base of excitation winding. In the light of the
conditions above, there is
ZG M afd 0 I fdG
U mG
ZG\ G
ZG LG I mG
i.e. we have selcted
kifd
In addition, exists \ fdG
kufd
I mG / I fd G
M afd 0 / LG
(2.3.18)
L ffd I fdG , i.e., we have chosen
\ G /\ fdG
LG I mG / L ffd I fdG
M afd 0 / L ffd
(2.3.19)
According to the results above, it is not difficult to reduce Eq. (2.3.17) to the
following formulas:
96
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
M afd 0
M afd
LG
LG
M afd 0
1
3
M afd 0 M
afd 0
2
LG
L ffd
M fad
L ffd M afd 0
L ffd
LG
L ffd M afd 0
LG
½
°
°
3 2 °
M afd 0 °
°
2
¾
LG L ffd °
°
°
1
°
°¿
(2.3.20)
Substituting the results above into flux-linkage equation of a synchronous
machine without damping winding referring to Eq. (2.3.3), the corresponding
equations indicated by Park’s per-unit system are
ª \ dp º
«
»
«¬\ fdp »¼
ª x dp 1º ª i dp º
«M
1»¼ «¬i fdp »¼
fadp
¬
(2.3.21)
It is noted that the corresponding quantities of Eq. (2.3.21) are all added with
subscript P to distinguish Park’s per-unit system from the per-unit system of
reversible mutual inductances.
As stated before, Park chose the resistance voltage drop produced by excitation
current base as the voltage base of excitation winding, namely
U fdG
in which Td 0
L ffd
rfd
rfd I fdG
rfd
I fdG L ffd
rfd
L ffd
L ffd
\ fdG
1
\ fdG
Td 0
(2.3.22)
is time constant of excitation winding when stator winding is
opened.
It should be noted that here U fdG z \ fd G / tG since tG
1
z Td 0 .
2ʌf
Because
p\ fd rfd i fd
u fd
we can get
u fd
U fdG
i.e.
Td 0
tG
u fdP
§\ ·
d ¨ fd ¸
¨\ ¸ r i
© fdG ¹ fd fd
rfd I fd G
§ t ·
d¨ ¸
© tG ¹
T d 0 p\
fdP
i fdP
(2.3.23)
97
AC Machine Systems
Comparing the equation expressed by Park’s per-unit system with the original
equation indicated by actual values, we can see that there is an obvious
difference between them as for their forms. That is because during choice of
2
excitation winding bases we do not comply with the rules of kufd kifd
and
3
\G
U
flux linkage base
z mG .
, to bring about kufd
voltage base
time base
\ fd G U fdG
If describing the corresponding equations of synchronous machine without
damping windings by the per-unit system of reversible mutual inductances
mentioned before, referring to Eqs. (2.3.10) and (2.3.16) we can obtain
x d i d x ad i fd ½
°°
x ad i d x ffd i fd ¾
°
p\ fd r fd i fd °
¿
\d
\ fd
u fd
(2.3.24)
Comparing the first equation in Eq. (2.3.24) with the corresponding equation in
Park’s per-unit system, we can get
\d
\ dP
½
°
i d i dP
°
¾
x d x dP
°
x ad i fd i fdP °¿
(2.3.25)
The first three expressions in Eq. (2.3.25) indicate the same choice of stator bases
for two per-unit systems. The last expression in Eq. (2.3.25) shows a difference
of x ad times between excitation current per-unit values for two per-unit systems,
because their excitation current bases are different from each other.
Rewriting the second equation in Eq. (2.3.24) as the following form:
x ad \
x ffd
2
fd
x ad
i d x ad i fd
x ffd
2
x ad
i d i fdP
x ffd
and comparing it with the corresponding equation in Park’s per-unit system
referring to Eq. (2.3.21), we can get
2
M fadP
\ fdP
98
½
°
°
¾
x ad
\ °
x ffd fd °¿
x ad
x ffd
(2.3.26)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
i.e., there is a difference of x ad / x ffd times between excitation winding flux-linkage
values for two per-unit systems.
2
x
Because ad x d xcd , \ fdP usually has the following form:
x ffd
\ fdP
( x d xcd )i dP i fdP
(2.3.27)
in which xcd is called d-axis transient reactance of synchronous machines.
Rewriting the third equation in Eq. (2.3.24) as the following form:
x ffd
x ad
u fd
r fd
r fd
x ffd
r fd
§x
p ¨ ad \
¨x
© ffd
p\
fdP
·
¸ x ad i fd
fd ¸
¹
i fdP
and comparing it with the corresponding Eq. (2.3.23) in Park’s per-unit system,
in which x ffd / r fd L ffd / r fd T d 0 , we can easily know
u fdP
x ad
u fd
r fd
(2.3.28)
i.e., there is a difference of x ad / r fd times between excitation voltages expressed
by two per-unit systems.
From now on, we shall use the per-unit system of reversible mutual inductances
stated before. In the per-unit system, excitation current in per-unit, during no-load
rated voltage on air-gap line, is equal to 1/ x ad , and corresponding excitation
voltage in per-unit is r fd i fd
r fd / x ad . In many cases, machine parameters in
per-unit are known, and excitation voltage in actual value u fdx corresponding to
no-load rated stator voltage and relevant excitation current in actual value i fdx are
also known. If saturation effect is neglected, then according to excitation current
1
in actual value i fdx and its per-unit
we can write its base as
x ad
I fdG
i fdx
1/ x ad
x ad i fdx
(2.3.29)
Similarly, on the basis of excitation voltage in actual value u fdx and its per-unit
r fd / x ad we can describe its base as
99
AC Machine Systems
u fdx
U fdG
r fd / x ad
x ad
u fdx
r fd
(2.3.30)
2.4 Basic Equations in Per-Unit, Operational Reactances
and Electromagnetic Torque of Synchronous Machines
2.4.1
Basic Equations in Per-Unit of Synchronous Machines1
In Section 2.2, flux-linkage equations and voltage equations in d, q, 0 axes have
been found out for synchronous machines, which are named Park’s formulas. In
Section 2.3, per-unit systems are also discussed and corresponding voltage
equations in per-unit are:
ud
uq
u0
u fd
0
0
p\ d \ qZ rid ½
°
p\ q \ d Z riq ¾
°
p\ 0 ri0
¿
(2.4.1)
p\ fd R fd i fd ½
°
p\ 1d R1d i1d ¾
p\ 1q R1q i1q °¿
(2.4.2)
The flux-linkage equations in per-unit are:
\d
\q
\0
\ fd
\ 1d
\ 1q
xd id xad i fd xad i1d ½
°
xq iq xaq i1q
¾
°
x0 i0
¿
(2.4.3)
xad id X ffd i fd X f 1d i1d ½
°
xad id X 1 fd i fd X 11d i1d ¾
°
xaq iq X 11q i1q
¿
(2.4.4)
From the equations above we can see that original quantities in a, b, c axes
disappear and there are the quantities corresponding to d, q, 0 axes. Now exist
three circuits that are fictitious circuits obtained by reference axis conversion.
The first circuit is called stator d-axis circuit whose axis coincides with rotor
d-axis and rotates synchronously with rotor d-axis. Its flux-linkage is \ d , current
1
The signs with bottom-bars indicate per-unit values, but bottom-bars are omitted from now on.
100
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
is id , self-inductance is xd and resistance is r. It has a magnetic connection with
rotor d-axis windings and the mutual inductance is xad . The second circuit is
called stator q-axis circuit whose axis coincides with rotor q-axis and also rotates
synchronously with rotor q-axis. Its flux-linkage is \ q , current is iq ,
self-inductance is xq and resistance is r. It has a magnetic connection with rotor
q-axis winding and the mutual inductance is xaq . The third circuit is called stator
0-axis circuit that has no electromagnetic connection with other windings. Its
flux-linkage is \ 0 , current is i0 , self-inductance is x0 and resistance is r.
From Eq. (2.4.1) it is clear that stator d-axis voltage consists of three parts. The
first part p\ d is transformer emf caused by change of d-axis flux-linkage. The
second part \ qZ is speed emf produced in stator d-axis winding by revolving
q-axis flux-linkage and the third part rid is the resistance voltage-drop of stator
d-axis winding. Their algebraic sum should be equal to terminal voltage of stator
d-axis winding and according to the reference direction rules described in Chapter
1 the sum is ( p\ d \ qZ rid ). Comparing this result with Eq. (2.2.29), we can
see that ud in Eq. (2.2.29) is just the value. Therefore, d-axis component of stator
voltage converted by Eq. (2.2.31) gets clear physical concept now, i.e., it is just
the terminal voltage the fictitious stator d-axis winding should have.
Similarly, stator q-axis voltage consists of three parts too, in which p\ q is
transformer emf caused by change of q-axis flux-linkage, \ d Z is speed emf
produced in stator q-axis winding by revolving d-axis flux-linkage and riq is the
resistance voltage drop of stator q-axis winding. Their algebraic sum ( p\ q \ d Z riq ) should be equal to terminal voltage of stator q-axis winding. Just as
stator d-axis winding, the terminal voltage is also uq , the q-axis component of
stator voltage converted by formula (2.2.31).
So far as stator 0-axis winding is concerned, because it has no electromagnetic
connection with other windings, its terminal voltage is only the algebraic sum of
transformer emf p\ 0 caused by change of its own flux-linkage and resistance
voltage-drop ri0 , namely ( p\ 0 ri0 ) . That is also u0 the 0-axis component of
stator voltage converted by formula (2.2.31).
When fundamental flux is considered only, 0-axis flux-linkage has no
magnetic connection with rotor winding so that it doesn’t produce any torque.
During calculation, 0-axis component can be processed alone.
Comparing basic Eqs. (2.4.1) (2.4.4) of synchronous machines with those of
dc machines except 0-axis component, there is some similarity between them.
The fact shows that 3-phase ac machine after conversion may be replaced by a dc
machine. The physical model of dc machine is shown in Fig. 2.4.1. It is a dc
machine with two pairs of brushes that are set on d-axis or q-axis respectively.
The armature circuit determined by d-axis brushes corresponds to stator d-axis
winding of synchronous machines and the armature circuit determined by q-axis
brushes corresponds to stator q-axis winding of synchronous machines. Excitation
101
AC Machine Systems
winding of synchronous machines corresponds to excitation winding set on d-axis
pole of dc machine, and d-axis and q-axis damping windings of synchronous
machines correspond to two short-circuited coils set on d-axis and q-axis poles of
dc machine, respectively.
Figure 2.4.1 Physical pattern of 3-phase ac machine after axis conversion
Obviously, 0-axis component can be represented by a static circuit as shown in
Fig. 2.4.2.
Figure 2.4.2 Static circuit corresponding to 0-axis component
2.4.2 Equivalent Circuits and Operational Reactances of
Synchronous Machines
After obtaining the basic equations of synchronous machines with reversible and
constant mutual inductances in per-unit, we can draw the corresponding equivalent
circuits. Using Heaviside’s or Laplace’s operational calculus to analyse many
102
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
operating modes of synchronous machines, those equivalent circuits are significant.
Now, according to different types of machines, two conditions will be discussed.
(1) Having Excitation Winding and No Damping Winding on Rotor
Under that condition, the stator d-axis flux-linkage equation and excitation
winding voltage equation can be changed into
\d
xd id xad i fd
u fd
§
xad id ¨ X ffd
©
p
½
°
R fd · ¾
i
¸ fd
p ¹ °¿
(2.4.5)
According to the expressions above, the equivalent circuit is plotted in Fig. 2.4.3.
Figure 2.4.3 d-axis operational equivalent circuit when having excitation winding
only on rotor
That circuit shows the relation between flux-linkages and currents. Comparing
it with usual static circuits, the flux-linkage in Fig. 2.4.3 corresponds to a voltage,
reactance corresponds to a resistance and the resistance corresponds to a capacitance.
In Fig. 2.4.3, ( xl xd xad ) is stator leakage reactance and ( X fdl X ffd xad )
is excitation winding leakage reactance, which can be proved as follows.
Referring to (2.3.9) and (2.3.7) there is
X ffd
L ffd
2 L ffd § I fdG ·
¨
¸
3 LG © I mG ¹
2
2 L ffd
3 LG
§ Lad ·
¨¨
¸¸
© M afd 0 ¹
2
and on the basis of Eqs. (1.5.3), (2.1.45) and (2.2.9) exist
L ffd
M afd 0
W lP
a 2fd
w2fd O0 L fdl
2ww fdW l
ʌas a fd
Od 1k w1
2
Lad
3 4W lP § wk w1 ·
Od 11
2 as2 ʌ 2 ¨© P ¸¹
so we can obtain
103
AC Machine Systems
X ffd
ª 3 1 wk w1
º
« S a P Od 11 »
ª
º
2 1 W lP 2
s
»
« 2 w fd O0 L fdl » «
1
»
3 Lį «¬ a fd
»¼ «
« a w fd Od 1 »
fd
¬
¼
L ad
2
O0 Od 11 2 L fdl
Od21
3 LG
§O O
xad xad ¨ 0 2d 11
© Od 1
2
§ I fdG ·
¨
¸
© I mG ¹
·
1¸ X cfdl
¹
or
X fdl
X ffd xad
§O O
·
X cfdl xad ¨ 0 2d 11 1¸
© Od 1
¹
in which X cfdl is excitation winding leakage reactance component corresponding
to end leakage flux, pole-to-pole leakage flux and leakage flux between two
§O O
·
adjacent pole-surfaces, xad ¨ 0 2d 11 1¸ is excitation winding differential leakage
© Od 1
¹
reactance, and X fdl is excitation winding total leakage reactance.
In many cases only stator quantities need studying, so it is preferable to delete
rotor quantities from basic equations to obtain simpler equations. Using
Thevenin’s Law and the equivalent circuit in Fig. 2.4.3, we can get the following
simple equation:
\d
G ( p )u fd xd ( p )id
(2.4.6)
in which G ( p )u fd is the voltage when terminals a, b in Fig. 2.4.3 are opened, i.e.,
G ( p )u fd
u fd
p
u fd
xad
xad X fdl R fd
p
xad
p( xad X fdl ) R fd
xad
u fd
pX ffd R fd
Hence, there is
G( p)
104
xad
pX ffd R fd
(2.4.7)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
xd ( p ) is short circuit impedance observed from terminals a, b when the supply
u fd / p is short-circuited, i.e.,
xd ( p)
xl 1
1
xad
xd xad xd xd xd 1
X fdl R fd
p
pxad X fdl xad R fd
pX ffd R fd
pxad X ffd pxad X fdl xad R fd xad R fd
pX ffd R fd
pxad ( X ffd X fdl )
pX ffd R fd
2
pxad
pX ffd R fd
(2.4.8)
G ( p ) is called operational conductance and xd ( p ) is called d-axis operational
reactance. They are all operational expressions that are independent of machine
speed. Therefore, they can be used to study any operating mode of synchronous
machine at any speed. Supposing u fd 0 in Eq. (2.4.6), we can get \ d xd ( p)id .
As stated above, if the source u fd / p is short-circuited, the impedance observed
from terminals a, b will be operational reactance xd ( p ). Thus, this type of circuit
is called equivalent circuit of stator d-axis operational reactance for synchronous
machines.
Without damping windings, the stator q-axis flux-linkage equation is
\q
xq iq
(2.4.9)
The corresponding equivalent circuit is shown in Fig. 2.4.4. Accordingly, this
type of circuit is called equivalent circuit of stator q-axis operational reactance
for synchronous machines.
Figure 2.4.4 q-axis operational equivalent circuit without damping windings
105
AC Machine Systems
(2) Having a Set of d-axis and q-axis Damping Windings besides Excitation
Winding on Rotor
Under that condition, the stator d-axis flux-linkage equation, excitation
winding and d-axis damping winding voltage equations can be described as
ª\ d º
«u »
« fd »
« p »
« »
¬ 0 ¼
ª x
« d
«
«
« xad
«
«
« xad
¬
xad
X ffd º
»
»ªi º
»« d »
X f 1d » «i fd »
» «i »
¬ 1d ¼
R1d »
X 11d »
p ¼
xad
R fd
p
X 1 fd
(2.4.10)
The equivalent circuit of d-axis operational reactance corresponding to the
matrix equation above is shown in Fig. 2.4.5.
Figure 2.4.5 d-axis operational equivalent circuit when having a d-axis damping
winding besides excitation winding on rotor
Similarly, after deleting rotor currents from stator flux-linkage equation exists
\d
G ( p )u fd xd ( p )id
(2.4.11)
in which
G( p)
p ( X 11d xad X f 1d xad ) xad R1d
A( p)
xd ( p )
xd B( p)
A( p )
A( p )
p 2 ( X 11d X ffd X 2f 1d ) p( X 11d R fd X ffd R1d ) R1d R fd
B( p)
2
2
2
2
2
p 2 ( X 11d xad
2 X f 1d xad
X ffd xad
) p( xad
R1d xad
R fd )
(2.4.12)
(2.4.13a)
If neglecting the mutual leakage reactance between excitation winding and
d-axis damping winding, i.e., considering that there is no mutual flux that links
106
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
only the two windings and does not link stator winding, then we can write the
mutual inductance between excitation winding and d-axis damping winding as
M f 1d
M f 1dG M f 1dl
M f 1dG
2w1d w fdW l 1
E S
Od 1 sin 1d
2
S
a fd
Referring to Eqs. (2.3.9) and (2.3.7), its per-unit can be described as
M f 1d
2 M f 1d § I fd G I1dG ·
¨
¸
3 LG © I m2G ¹
2 M f 1d § Lad · § Lad ·
¨
¸¨
¸
3 LG ¨© M afd 0 ¸¹ © M a1d 0 ¹
After substituting the relations of Lad , M afd 0 , M f 1d and M a1d 0 into the equation
above referring to Eqs. (2.2.9), (2.1.45), and (2.1.48) and arranging it in order,
we can get
M f 1d
L ad
X f 1d
xad
namely,
Accordingly, there is
xd ( p)
xd 2
3
2
2
2
( X 11d xad
2 xad
X ffd xad
) p 2 ( xad
R1d xad
R fd ) p
2
( X 11d X ffd xad
) p 2 ( X 11d R fd X ffd R1d ) p R1d R fd
(2.4.13b)
and the equivalent circuit in Fig. 2.4.5 can be reduced to that in Fig. 2.4.6.
Figure 2.4.6 The reduced d-axis operational equivalent circuit
In the circuit in Fig. 2.4.6, X 1dl X 11d xad is the leakage reactance of d-axis
damping winding, which is proved in the same way as X fdl . Considering
harmonic effect, we have to add differential leakage reactance of d-axis damping
winding to leakage reactance X 1dl .
107
AC Machine Systems
The stator q-axis flux-linkage equation and q-axis damping winding voltage
equation are
ª\ q º
«0»
¬ ¼
ª xq
«
« x
«¬ aq
xaq
º
ª iq º
»
R1q » « »
i1q
p »¼ ¬ ¼
X 11d
(2.4.14)
The equivalent circuit of q-axis operational reactance corresponding to the
equations above is shown in Fig. 2.4.7.
Figure 2.4.7 q-axis operational equivalent circuit with one q-axis damping winding
In the circuit in Fig. 2.4.7, ( X 1ql X 11q xaq ) is the leakage reactance of q-axis
damping winding. Considering harmonic effect, we have to add differential
leakage reactance of q-axis damping winding to leakage reactance X 1ql , too.
After deleting rotor current, there is
\q
xq ( p )iq
(2.4.15)
in which
xq ( p )
xl 1
1
xaq
xq 1
X 1ql R1q
2
pxaq
pX 11q R1q
(2.4.16)
p
xq ( p) is called the operational reactance of stator q-axis.
So far as the non-salient pole synchronous machine is concerned, its d-axis
and q-axis equivalent circuits are similar to Figs. 2.4.5 2.4.7, but xad xaq only.
As for an induction machine, there is xad xaq xm owing to uniform air-gap.
Since induction machine rotor is electromagnetically symmetrical with respect to
any axis, its d-axis and q-axis operational reactances are equal to each other, i.e.,
xd ( p)
xq ( p)
x( p )
in which xs is stator leakage reactance,
108
xs 1
1
xm
(2.4.17)
1
Xr Rr
p
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
X r and Rr are rotor equivalent leakage reactance and resistance, respectively
referring to Chapter 5.
The equivalent circuit of operational reactance is shown in Fig. 2.4.8.
Figure 2.4.8 Equivalent circuit of operational reactance x( p) for induction machines
Because the rotor winding of induction machine is often short-circuited, stator
d-axis and q-axis flux-linkages can be written as
\d
\q
x( p )id
x( p)iq
Therefore, the corresponding equivalent circuits of d-axis and q-axis operational
reactances are shown in Figs. 2.4.9(a) and 2.4.9(b).
Figure 2.4.9 d-axis and q-axis operational equivalent circuits of induction machines
2.4.3
Output Power and Electromagnetic Torque in Synchronous
Machines
The instantaneous output power of 3-phase synchronous machine, when expressed
in actual values, is
p
ua ia ub ib uc ic
(2.4.18)
The power base is
PG
3UI
§ U ·§ I ·
3 ¨ mG ¸¨ mG ¸
© 2 ¹© 2 ¹
109
AC Machine Systems
in which U and I are rms values of stator rated phase voltage and current, U mG
and I mG are instantaneous bases of phase voltage and current. Thus, instantaneous
output power in per-unit can be described as
2
(ua ia ub ib uc ic )
3
p
(2.4.19)
Substituting the variables in d, q, 0 axes for variables in a, b, c axes as stated
before, we can write the instantaneous output power in per-unit according to d, q, 0
axes as
p
ud id uq iq 2u0 i0
(2.4.20)
Substituting Eq. (2.4.1) into Eq. (2.4.20) exists
p
id ( p\ d \ qZ rid ) iq ( p\ q \ d Z riq ) 2i0 ( p\ 0 ri0 )
(id p\ d iq p\ q 2i0 p\ 0 ) (iq\ d id\ q )Z r (id2 iq2 2i02 )
(2.4.21)
Observing various terms in the equation above, we can see that the first term
(id p\ d iq p\ q 2i0 p\ 0 ) is total products of currents and rates of flux-linkage
change. Assuming negative flux-linkage to be produced by positive current as
mentioned before, the first term will represent decrease rate of stator magnetic
field energy. The third term r (id2 iq2 2i02 ) indicates stator resistance loss. On
the basis of power balance, the second term (iq\ d id\ q )Z should be the power
delivered to stator across air-gap.
Because
electromagnetic torque
the power delivered to stator across air-gap
rotor speed
the electromagnetic torque of a synchronous machine is
Te
iq\ d id\ q
(2.4.22)
It should be pointed out that the basic equations of synchronous machines,
after converting a, b, c axes to d, q, 0 axes, are changed from differential
equations with variable coefficients to those with constant coefficients, but
u , i, \ , Z are still variables with time. The equations as stated before include
xd ( p ), xq ( p ) and p together with time-variables u (t ), i (t ), \ (t ) and Z (t ), which
are called Heaviside’s operational calculus equations and different from pure
operational calculus equations with u ( p ), i ( p ) and z ( p). If necessary, refer to
Reference [2] or Appendix D in this book.
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2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
2.5
3-Phase Sudden Short Circuit and Transient
Parameters of Synchronous Machines
As mentioned before, basic equations of synchronous machines in d, q, 0 axes
are differential equations with constant coefficients, by which we can study several
operating modes of synchronous machines simply, especially those when stator
phase windings are connected symmetrically.
Among of them, 3-phase sudden short circuit will threaten synchronous
machines. Thus, comprehension of the transients may provide scientific basis not
only for reasonable design and reliable operation of synchronous machines but
also for a careful choice of relay protection. In addition, the analysis methods and
parameters derived for study of the problems above are of importance to analyse
other transients, too. Therefore, this section will lay a foundation for researching
further into other transients of synchronous machines.
Studying 3-phase sudden short circuit, we can consider the short circuit not to
have happened but the stator terminals to be suddenly applied on by 3-phase
voltages that are equal but opposite to the phase voltages before short circuit. In
thinking so, the problem of 3-phase sudden short circuit of synchronous
machines is turned into the synthetic problem of the following two operation
conditions:
(i) Steady-state operation condition that is the same as the condition before
short circuit.
(ii) Excitation winding is short-circuited and stator terminals are suddenly
applied on by 3-phase voltages that are equal but opposite to the phase voltages
before short circuit.
Because basic equations are linear under that condition, we can use superposition
theorem to treat the problem above.
2.5.1
Original Steady-State Operation Condition
In steady-state operation, stator emfs can be described as
ea
eb
ec
E sin J
½
°
E sin(J 120e
)¾
E sin(J 120e
) °¿
(2.5.1)
When a synchronous machine is loaded, there is an angle between its terminal
voltage and no-load emf due to load current. If terminal voltage lags behind
no-load emf by a phase angle G, then instantaneous terminal voltages can be
written as
111
AC Machine Systems
U sin(J G )
ua
½
°
U sin(J G 120e
)¾
U sin(J G 120e
) °¿
ub
uc
(2.5.2)
in which U is terminal voltage amplitude. By use of d, q, 0 axes, the equations
above can be converted into
ª ud º
« »
« uq »
«¬ u0 »¼
ª cos J
«
2«
« sin J
3«
1
«
¬ 2
cos(J 120e
)
cos(J 120e
)º
» ª ua º
»
) sin(J 120e
) » ««ub »»
sin(J 120e
» «u »
1
1
»¬ c¼
2
2
¼
ªU sin G º
«U cos G »
«
»
¬« 0 ¼»
(2.5.3)
i.e., ud and uq are constant. In the same way it can be proved that stator currents
id and iq are also constant. Therefore, according to Eq. (2.4.3) the flux-linkages
can be written as
\d
xad i fd xd id
\q
xq iq
E xd id
Constant
Constant ½°
¾
°¿
(2.5.4)
Substituting the expressions above into Park’s equations, we can obtain
ud
\ q rid
uq
\ d riq
xq iq rid
½°
¾
E xd id riq °¿
(2.5.5)
Solving the equation above, we get
id
iq
rud xq ( E uq )
2
r xd xq
xd ud r ( E uq )
r 2 xd xq
rU sin G xq ( E U cos G ) ½
°
r 2 xd xq
°
¾
xd U sin G r ( E U cos G ) °
°
r 2 xd xq
¿
(2.5.6)
In practice, the stator resistance r is small and may be neglected, so the expressions
above can also be simplified as
id
iq
E uq
E U cos G ½
°
xd
xd
°
¾
ud U sin G
°
°
xq
xq
¿
(2.5.7)
After having current and voltage relations, we can further analyse the output
power, reactive power and electromagnetic torque.
112
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
(1) Output power
Substituting the results mentioned above into Eq. (2.4.20), we can describe
output power as
P
ud id uq iq
ª rU sin G xq ( E U cos G ) º
U sin G «
» U cos G
r 2 xd xq
¬«
¼»
1
r xd xq
2
ª xd U sin G r ( E U cos G ) º
«
»
r 2 xd xq
¬«
¼»
xd xq 2
ª
º
2
U sin 2G ( xq sin G r cos G )UE »
« rU 2
¬
¼
(2.5.8)
If stator resistance r is neglected, we can obtain
P
§ 1
EU
1 ·U 2
sin G ¨ ¸
sin 2G
¨x
¸
xd
© q xd ¹ 2
(2.5.9)
Because d-axis and q-axis synchronous reactances are equal to each other in
non-salient-pole synchronous machines, output power under the condition r 0 is
P
EU
sin G
xd
(2.5.10)
The curve of output power P changing with angle G is termed power-angle
characteristic of a synchronous machine. The stator resistance neglected,
power-angle characteristic of a non-salient pole synchronous machine is shown
EU
and corresponding angle
in Fig. 2.5.1 where the maximum output power is
xd
G 90e. The power-angle characteristic of a salient-pole machine is shown in
Fig. 2.5.2 after neglecting stator resistance, and the maximum output power will
appear in the range of 45e 90ebecause of the second-order harmonic.
Figure 2.5.1 Power-angle characteristic of
non-salient-pole synchronous machines
Figure 2.5.2 Power-angle characteristic
of salient-pole synchronous machines
113
AC Machine Systems
(2) Reactive power
If there is no 0-axis component, on the basis of Eq. (2.5.3) we can obtain
ud2 uq2 ½°
¾
id2 iq2 °¿
U
I
(2.5.11)
in which U and I are amplitudes of stator voltage and current, respectively.
So the reactive power is
Q
S 2 P2
(ud2 uq2 )(id2 iq2 ) (ud id uq iq )2
(2.5.12)
uq id ud iq
in which S UI is the apparent power of a synchronous machine. Substituting
Eqs. (2.5.3) and (2.5.6) into the formula above, we can obtain
Q
ª
º
1
U2
U2
cos 2G »
( xd xq )
«( xq cos G r sin G )UE ( xd xq )
r xd xq ¬
2
2
¼
2
(2.5.13)
If stator resistance is neglected, we can get
Q
§1
EU
1 ·U 2 § 1
1 ·U 2
¨ ¸
cos G ¨ ¸
cos 2G
¨x
¸
¨
¸
xd
© q xd ¹ 2 © xq xd ¹ 2
(2.5.14)
In non-salient-pole synchronous machine, there is
Q
EU
U2
cos G xd
xd
(2.5.15)
(3) Electromagnetic torque
Substituting expressions (2.5.4) and (2.5.6) into formula (2.4.22) and arranging
it in order, we can get
Te
iq\ d id\ q
r (r 2 xq2 )
2
(r xd xq )
2
E2 xd xq
ª
( xd sin 2 G xq cos 2 G )r
(r xd xq ) 2 «¬
2
1
EU
º
( xd xq r 2 )sin 2G » U 2 2
[(rxd sin G xq2 cos G )2r
2
(r xd xq )2
¼
( xd xq r 2 )(r cos G xq sin G )]
114
(2.5.16)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
or
Te
Tk Tr Ts
(2.5.17)
in which the first term
r (r 2 xq2 )
Tk
(r 2 xd xq )2
E2
(2.5.18)
is the short circuit torque of a synchronous machine. It depends upon no-load emf
E, is independent of terminal voltage U, and corresponds to the torque produced
when excitation current is constant and stator winding is short-circuited.
When excitation current is constant and stator winding is short-circuited, referring
to equation (2.5.6) the corresponding stator current can be written as
xq E
½
°
r xd xq °
¾
rE °
r 2 xd xq °¿
id
2
iq
(2.5.19)
i.e. the amplitude of stator short-circuit current is
id2 iq2
ik
r 2 xq2
r 2 xd xq
E
Therefore the stator loss caused by short circuit current is
ik2 r
r (r 2 xq2 )
2
(r xd xq )
2
E2
which is equal to the short circuit torque, i.e.
ik2 r
Tk
(2.5.20)
Obviously, when r 0, i.e. there is no resistance loss in stator, the torque will
be equal to zero.
The torque Tk is an asynchronous torque produced through the interaction
between the rotating field caused by excitation current and that caused by stator
short-circuit current induced by excitation field. Compared with an induction
machine, its slip is equal to 1, so Tk ik2 r can be obtained.
The second term
115
AC Machine Systems
Tr
xd xq
1
ª
º
( xd sin 2 G xq cos 2 G )r ( xd xq r 2 )sin 2G » U 2
(r xd xq ) 2 «¬
2
¼
2
(2.5.21)
is the salient pole torque, which depends upon terminal voltage and is independent
of no-load emf. That is the torque caused by difference between d-axis and q-axis
synchronous reactances, i.e. by difference between d-axis and q-axis air-gap
permeances.
If air-gap is uniform, for example in a non-salient pole synchronous machine,
then there are
xd
xq ,
Tr
0
If stator resistance is neglected, then
Tr
1§ 1
1 · 2
¨¨ ¸¸U sin 2G
2 © xq xd ¹
(2.5.22)
The third term
Ts
UE
[(rxd sin G xq2 cos G )2r ( xd xq r 2 )(r cos G xq sin G )]
2
(r xd xq )
2
(2.5.23)
is the synchronous torque, which depends upon not only terminal voltage but also
no-load emf.
If stator resistance is neglected, then exists
Ts
UE
sin G
xd
(2.5.24)
From the analysis above we can see that when a synchronous machine operates
symmetrically in steady-state, various components of the electro-magnetic torque
are all constants, i.e. there is only average torque and no pulsating torque. That is
natural, because under this condition there are only synchronously rotating fields
and no others.
When stator resistance is neglected, the resultant electromagnetic torque is
Te
UE
U2 § 1
1·
sin G ¨¨ ¸¸ sin 2G
xd
2 © xq xd ¹
(2.5.25)
Comparing the formula above with equation (2.5.9), we can see that the output
power and electromagnetic torque of a synchronous machine are equal to each
other in values when r 0. That is natural, because under this condition the
116
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
power across air-gap to stator corresponding to the electromagnetic torque is
delivered to network without any loss.
2.5.2
Current Components Caused by 3-phase Sudden Short Circuit
Before short circuit, referring to (2.5.3) there are
ud
U sin G
ud 0
uq
U cos G
uq 0
so after short circuit, it is equivalent to the situation that ud 0 and uq 0 are applied
suddenly to stator terminals during closed excitation winding as mentioned
before, i.e. during synchronous speed, namely Z 1 exist
ud
ud 0
U sin G
uq
uq 0
U cos G
p\ d \ q rid ½°
¾
p\ q \ d riq °¿
Because excitation winding is short-circuited, i.e. u fd
can be reduced to
\d
\q
(2.5.26)
0, flux-linkage equations
xd ( p)id ½°
¾
xq ( p )iq °¿
(2.5.27)
Substituting (2.5.27) into (2.5.26) and solving them, we can obtain
id
iq
[ pxq ( p ) r ]ud 0 xq ( p )uq 0
½
°
[ pxd ( p ) r ][ pxq ( p ) r ] xd ( p ) xq ( p) °
¾
[ pxd ( p ) r ]uq 0 xd ( p )ud 0
°
[ pxd ( p ) r ][ pxq ( p ) r ] xd ( p ) xq ( p) °¿
(2.5.28)
According to machines with or without damping windings, change of those
currents will be discussed respectively as follows.
(1) Current change of a machine without damping windings
A machine having no damping winding, its operational reactances are
xd ( p)
xd xq ( p)
xq
2
½
pxad
°
R fd pX ffd ¾
°
¿
(2.5.29)
The corresponding current expressions are
117
AC Machine Systems
id
iq
1
½
°
°
°°
¾
1
2
2
°
x
X
x
u
p
x
R
rX
u
{(
)
[(
)
d
ffd
ad
q0
d fd
ffd
q0
°
ap 3 bp 2 cp d
°
2
( xd X ffd xad )ud 0 ] p (rR fd uq 0 xd R fd ud 0 )}
°¿
{xq X ffd ud 0 p 2 [(rX ffd R fd xq )ud 0
ap bp 2 cp d
xq X ffd uq 0 ] p (rR fd ud 0 xq R fd uq 0 )}
3
(2.5.30)
in which
a
b
c
d
2
xd xq X ffd xad
xq
½
°
2
r ( xd xq ) X ffd rxad
xd xq R fd
°°
¾
2
2
xd xq X ffd xad xq ( xd xq )rR fd r X ffd °
°
xd xq R fd r 2 R fd
°¿
(2.5.31)
Its characteristic equation is
ap 3 bp 2 cp d
0
(2.5.32)
The general solution of the equation above is
p1
p2
p3
a
3
½
°
°
1
3
a°
(u1 v1 ) ¾
(u1 v1 ) j
2
2
3°
1
3
a°
(u1 v1 ) °
(u1 v1 ) j
2
2
3¿
u1 v1 (2.5.33)
in which
u1
3
q
q 2 m2
2
4 27
v1
3
q
q 2 m2
2
4 27
q
m
2a 3 ab
c
27
3
a2
b
3
The calculation is tedious and it is not easy to predict the effect of machine
118
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
parameters on currents. Therefore, we shall find approximate solutions according
to actual condition of machine parameters.
Because stator and rotor resistances are much smaller than reactances, they
may be neglected during approximate calculation. Now we find the approximate
solutions according to two conditions as follows.
(a) Neglecting excitation winding resistance
Referring to equation (2.4.8), the operational reactance under this condition
will be a constant, i.e.
xd ( p) |R fd
0
xd 2
xad
X ffd
xl 1
1
1
xad X fdl
xdc
(2.5.34)
which is termed d-axis transient reactance of a synchronous machine.
When the excitation voltage and resistance of a synchronous machine without
damping windings are all equal to zero, its equivalent circuit of operational
reactance changes from Fig. 2.4.3 to Fig. 2.5.3.
Figure 2.5.3 Equivalent circuit of d-axis transient reactance
From Eq. (2.5.28) we can see that the characteristic equation under this condition
is reduced to
( pxdc r )( pxq r ) xdc xq
(2.5.35)
0
whose solution is
p1,2
r§ 1
1·
r2 § 1
1·
¨ ¸ r j 1 ¨ ¸
¨
¸
¨
2 © xdc xq ¹
4 © xdc xq ¸¹
2
(2.5.36)
2
In practice,
1
r2 § 1
1·
¨¨ ¸¸ | 1, so
4 © xdc xq ¹
p1,2 | 1
r xdc xq
rj rj
c
2 xd xq
Ta
(2.5.37)
119
AC Machine Systems
in which
1 2 xdc xq
r xdc xq
Ta
x2
r
(2.5.38)
and
x2
2 xdc xq
(2.5.39)
xdc xq
is the negative sequence reactance.
(b) Neglecting stator winding resistance
From equation (2.5.28) we can know that the characteristic equation under this
condition is
p 2 xd ( p ) xq xd ( p) xq
0
i.e.
2
§
pxad
( p 2 1) xq ¨ xd ¨
R fd pX ffd
©
·
¸¸ 0
¹
(2.5.40)
Its roots are
p1,2
p3
rj
xd R fd
2
ad
xd X ffd x
xd 1
xdc Td 0
xd R fd
xdc X ffd
1
Tdc
in which
Tdc
xdc
Td 0
xd
ª
1 «
1
« X fl 1
1
R fd «
«¬
xad xl
º
»
»
»
»¼
(2.5.41)
is the time constant of excitation winding when stator winding is short-circuited
whose resistance r is equal to zero referring to Fig. 2.4.3.
Obviously, the roots of equation (2.5.40) p1,2 r j correspond to the roots
p1,2
120
1
r j obtained above referring to equation (2.5.37) because during r
Ta
0
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
both are equal.
According to analysis of the two approximate conditions, the roots of
characteristic equation (2.5.32) have the following approximate values:
p1,2
p3
1
½
r j°
Ta
°
¾
1
°
°¿
Tdc
(2.5.42)
Therefore, the characteristic equation can be written as
§
·§
·
1 ·§
1
1
¨ p ¸¨ p j ¸¨ p j ¸
Tdc ¹©
Ta
Ta
©
¹©
¹
0
(2.5.43)
Substituting the result above into equation (2.5.30), we can get
{xq X ffd ud 0 p 2 [(rX ffd R fd xq )ud 0 xq X ffd uq 0 ] p (rR fd ud 0 xq R fd uq 0 )}
id
·§
· °½
1 ·§
1
1
°­ §
®a ¨ p ¸¨ p j ¸¨ p j ¸ ¾
Tdc ¹©
Ta
Ta
°¯ ©
¹©
¹ °¿
iq
2
2
{( xd X ffd xad
)uq 0 p 2 [( xd R fd rX ffd )uq 0 ( xd X ffd xad
)ud 0 ] p
(rR fd uq 0 xd R fd uq 0 )}
­° §
·§
· ½°
1 ·§
1
1
®a ¨ p ¸¨ p j ¸¨ p j ¸ ¾
Tdc ¹©
Ta
Ta
¹©
¹ ¿°
¯° ©
Because
a
2
xd xq X ffd xad
xq
xdc xq X ffd
we can get
id
iq
­°
ª§ r
º
§ r ud 0 uq 0 · ½°
1 ·
2
¸¸ ud 0 uq 0 » p ¨¨
¸¸ ¾
®ud 0 p «¨¨ °¯
© xq Td 0 Td 0 ¹ ¿°
¬«© xq Td 0 ¹
¼»
ª §
1 ·§ 2 2
1 ·º
« xdc ¨ p ¸¨ p p 1 2 ¸ »
Tdc ¹©
Ta
Ta ¹ »¼
«¬ ©
ª§ 1
º
§ r
u · °½
r ·
°­
2
¸ uq 0 u d 0 » p ¨
uq 0 d 0 ¸ ¾
®uq 0 p «¨
Tdc ¹ ¿°
© xdc Td 0
¬© Tdc xdc ¹
¼
¯°
ª §
1 ·§ 2 2
1 ·º
« xq ¨ p ¸¨ p p 1 2 ¸ »
Tdc ¹©
Ta
Ta ¹ »¼
«¬ ©
½
°
°
°
°
°
°
¾
°
°
°
°
°
°
¿
(2.5.44)
121
AC Machine Systems
Using Decomposition Theorem to solve the equation above, we can obtain
id
A Be
t
Tdc
2Ce
t
Ta
cos(t T )
in which
§ r
·
Tdc
Tc
1
uq 0
¨¨ ud 0 uq 0 ¸¸ | d uq 0
xdc Td 0
xd
§
1 · © xq
¹
xdc Td 0 ¨1 2 ¸
© Ta ¹
­°§ 1 · 2
º § r u
uq 0 · ½°
1 ª§ r
1 ·
d0
®¨ ¸ ud 0 «¨ ud 0 uq 0 » ¨
¸
¸¸ ¾
¨
Tc
Tdc ¬«¨© xq Td 0 ¸¹
¼» © xq Td 0 Td 0 ¹ ¿°
¯°© d ¹
A
B
­° 1 ª§ 1 · 2 2 1
1 º ½°
c
«
»¾
1
x
® d
¨ ¸
Ta2 » °
°¯ Tdc «¬© Tdc ¹ Ta Tdc
¼¿
º ª1
1 ª 1
1
1º
|
u u
u
1 ¬« Tdc q 0 Td 0 q 0 ¼» ¬« xdc xd »¼ q 0
xdc
Tdc
2
­° ª§ 1
· § r
· r 1 º
ª 1
1 ·§ 1
«
» ud 0 « j
j
¸¸ ¨ j ¸ ® ¨
¸ ¨¨ T
x
T
T
x
T
d0 ¹©
a
q d0 »
¹ © q
¹
¬ Ta
°¯ «¬© a
¼
Ce jT
|
1 º °½
» uq 0 ¾
Td 0 ¼ ¿°
·§ 1
1·
°­ § 1
°½
® xdc ¨ j ¸¨ j ¸ (2 j) ¾
Tdc ¹
¹© Ta
¯° © Ta
¿°
( j) 2 ud 0 juq 0
2 jxd ( j)
1
(uq 0 jud 0 )
2 xdc
2
U jG
e
2 xdc
i.e.
C
1
2 xdc
ud2 0 uq20
T
arctan
ud 0
uq 0
sin T
cosT
Therefore,
122
G
ud 0
2
d0
u uq20
uq 0
ud2 0 uq20
U
2 xdc
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
t
t
½
ª§ 1
1 · T c 1 º
1 T
«¨ ¸ e d » uq 0 e a ª¬ud 0 sin t uq 0 cos t º°
¼
xd »¼
xdc
¬«© xdc xd ¹
°°
¾
t
t
ª§ 1
°
1 · Tdc 1 º
U Ta
«¨ ¸ e » U cos G e cos(t G )
°
xd ¼»
xdc
«¬© xdc xd ¹
°¿
id
(2.5.45)
Similarly, we can obtain
t
iq
1
1 ud 0 e Ta [ud 0 cos t uq 0 sin t ]
xq
xq
t
U
U sin G e Ta sin(t G )
xq
xq
Transforming them to a, b, c axes, we can get
ia
t
ª§ 1
u
1 · T c 1 º
«¨ ¸ e d » uq 0 cos(t J 0 ) d 0 sin(t J 0 )
xd »¼
xq
¬«© xdc xd ¹
t
1§ 1
1 · Ta
¨ ¸ e [uq 0 cos J 0 ud 0 sin J 0 ]
2 ¨© xdc xq ¸¹
t
1§ 1
1 · Ta
¨ ¸ e [uq 0 cos(2t J 0 ) ud 0 sin(2t J 0 )]
2 ¨© xdc xq ¸¹
(2.5.46)
) for J 0 in the formula above, we can obtain the
Substituting (J 0 B 120e
expressions for ib and ic .
As mentioned above, after voltages ud 0 and uq 0 are suddenly applied to
stator winding, there is not only fundamental current but also aperiodic current
and second harmonic current in ti. Its physical concept is mentioned as follows.
Wher 3-phase symmetrical voltages are suddenly applied to stator winding, in it
the corresponding 3-phase fundamental currents will be produced which can
cause the flux-linkage values in stator and rotor windings changing suddenly.
Therefore, the aperiodic currents in stator and rotor windings will be caused to
keep flux-linkage constant. Because the rotor revolves at a synchronous speed,
aperiodic current in stator will cause an ac current of basic-frequency in rotor
winding, which will produce a pulsating magnetic field owing to asymmetrical
rotor winding. The pulsating magnetic field can be divided into two opposite
revolving fields, one of which will cause the second harmonic current in stator
winding.
In addition, the stator aperiodic current and second harmonic current will
123
AC Machine Systems
decay to zero at a time-constant Ta . The stator fundamental current will decay to
its steady-state value at a time-constant Tdc. The time-constant Ta is referred to as
the decay time-constant of stator aperiodic current, which is approximately equal
to the ratio of negative sequence reactance to stator resistance. The timeconstant Tdc is called the decay time-constant of d-axis transient current, which is
approximately equal to the ratio of excitation winding reactance when the stator
winding having no resistance is short-circuited to excitation winding resistance.
(2) Current change of a machine with damping windings
The machine having damping windings, its operational reactance can be
expressed as equation (2.4.13). Those expressions substituted into equation
(2.5.28), the corresponding characteristic equation will become an equation of
fifth degree. Solving directly the equation not only brings about complicated
result but also makes it not easy to predict the effect of machine parameters on
currents. Therefore, we shall use an approximate method to solve the current, i.e.
at first to find the initial value and steady-state value of the current and then to
search for the corresponding decay time-constants. At last, we can write the
expression for the current to change with time according to those results. Under
general condition, the results obtained by this approximate method are sufficiently
accurate.
(a) The initial value of stator current
According to Initial Value Theorem in operational calculus we can see that the
value of the operational expression when p f is the corresponding initial
value. However, supposing directly p f in equation (2.5.28), we can only find
the initial values id 0 and iq 0, and can not obtain the initial values expressed
as a function of time mentioned above. Therefore, we shall use another method to
find the initial values.
Each inductive loop can keep its flux-linkage constant at the beginning of
transients, and the superconductor loop will keep its flux-linkage constant for
ever after transients. Thus, we can use the concept of superconductor loop to find
the initial value of stator current.
If rotor circuits are considered as superconductor loops, i.e. the rotor circuit
resistances are all equal to zero, referring to equation (2.4.13b) d-axis operational
reactance has the following constant,
xd ( p) |R fd
R1d 0
xd xl 124
2
3
2
X 11d xad
2 xad
X ffd xad
2
X 11d X ffd xad
1
1
1
1
xad X fl X 1dl
xdcc
xd ( p ) | p
f
(2.5.47)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
xdcc is known as d-axis subtransient reactance, and the corresponding equivalent
circuit is shown in Fig. 2.5.4. It is not difficult to understand that the equivalent
circuit is similar to that of the operational reactance when U fd 0 and
R fd R1d 0 as shown in Fig. 2.4.6.
Figure 2.5.4 Equivalent circuit of d-axis subtransient reactance
From equation (2.5.47) we can realize that d-axis operational reactance when
rotor resistances are zero has the same result as the expression of d-axis
operational reactance when p f. That is not difficult to understand. According
to Initial Value Theorem we can realize that the value during p f corresponds
to that during t 0, and during t 0 the passive loop has the property that its
flux-linkage can not change abruptly, i.e. the property the superconductor loop
has. Similarly, rotor resistances being equal to zero, q-axis operational reactance
has the following constant,
xq ( p ) |R1q
0
xq ( p) | p
xl f
xq 1
1
1
xaq X 1ql
2
xaq
X 11q
xqcc
(2.5.48)
xqcc is termed q-axis subtransient reactance, and the corresponding equivalent circuit
is shown in Fig. 2.5.5. It is not difficult to understand that the equivalent circuit is
similar to that of the operational reactance when R1q 0 as shown in Fig. 2.4.7.
Figure 2.5.5 Equivalent circuit of q-axis subtransient reactance
125
AC Machine Systems
If the machine has no damping winding, the corresponding results are
xd ( p ) |R fd
xq ( p)
0
xd ( p) | p
xd 2
xad
X ffd
xqc
xq
f
xl xdc
1
1
1
xad x f
(2.5.49)
which are the results of equations (2.5.34) and (2.5.29) as mentioned before.
From formulas of xdcc and xqcc we can recognize that their values depend
mainly upon the leakage reactances of stator and rotor windings because xad
and xaq are much larger than xl , X 1dl , X fl and X 1ql and their equivalent circuits
are similar to that of a transformer whose secondary is short-circuited. However,
the stator current is alternating and rotor current is aperiodic, which is different
from the static transformer.
According to equation (2.5.28), operational formulas of currents, after stator
and rotor resistances are neglected, can be reduced to the following forms:
pud 0 uq 0 ½
°
xdcc ( p 2 1) °
¾
puq 0 ud 0 °
xqcc( p 2 1) °¿
(2.5.50)
uq 0
½
ud 0
sin t (1 cos t ) °
xdcc
xdcc
°
¾
uq 0
ud 0
sin t (1 cos t ) °
°
xqcc
xqcc
¿
(2.5.51)
id
iq
Solving them, we can obtain
id
iq
or
id
U
[cos G cos(t G )]
xdcc
iq
U
[sin G sin(t G )]
xqcc
When t 0, these currents will be equal to zero. That is because the machine has
no current before voltages are suddenly applied to it.
Transforming the results above to a, b, c axes, we can get
126
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
ia
ª uq 0
º 1§ 1
u
1·
cos(t J 0 ) d 0 sin(t J 0 ) » ¨ ¸ [ud 0 sin(2t r0 )
«
xqcc
«¬ xdcc
»¼ 2 ©¨ xdcc xqcc ¹¸
1§ 1
1·
uq 0 cos(2t r0 )] ¨ ¸ (uq 0 cos r0 ud 0 sin r0 )
¨
2 © xdcc xqcc ¸¹
(2.5.52)
or
ia
U
2
§ 1
1·
U§ 1
1·
¨¨ ¸¸ cos(t J 0 G ) ¨¨ ¸¸ cos(t J 0 G )
2 © xdcc xqcc ¹
© xdcc xqcc ¹
U
2
§ 1
1·
U§ 1
1·
¨¨ ¸¸ cos(2t J 0 G ) ¨¨ ¸¸ cos(J 0 G )
2 © xdcc xqcc ¹
© xdcc xqcc ¹
Substituting (J 0 B 120e
) for J 0 in the formula above, we can obtain
ib
ic
½
°
°
°
1§ 1
1·
°
) uq 0 cos(2t J 0 120e
)]°
¨ ¸ [ud 0 sin(2t J 0 120e
¨
¸
2 © xdcc xqcc ¹
°
°
1§ 1
1·
°
) ud 0 sin(J 0 120e
)]
¨ ¸ [uq 0 cos(J 0 120e
°
2 ¨© xdcc xqcc ¸¹
°
¾
ª uq 0
º
ud 0
°
sin(t J 0 120e
)»
cos(t J 0 120e
)
«
°
xqcc
¬« xdcc
¼»
°
°
1§ 1
1·
) uq 0 cos(2t J 0 120e
)]°
¨ ¸ [ud 0 sin(2t J 0 120e
°
2 ©¨ xdcc xqcc ¹¸
°
°
§
·
1 1
1
°
) ud 0 sin(J 0 120e
)]
¨ ¸ [uq 0 cos(J 0 120e
2 ©¨ xdcc xqcc ¹¸
¿°
ª uq 0
º
u
cos(t J 0 120e
) d 0 sin(t J 0 120e
)»
«
xqcc
¬« xdcc
¼»
(2.5.53)
(b) The steady-state value of stator current
According to Final Value Theorem of operational calculus it is not difficult to
find the current steady-state value. Because
xd ( p) | p
0
xq ( p) | p
0
xd ½°
¾
xq °¿
(2.5.54)
taking p 0 in equation (2.5.28) we can get the steady-state value of stator
current as follows:
127
AC Machine Systems
id
iq
rud 0 xq uq 0 ½
°
r 2 xd xq °
¾
ruq 0 xd ud 0 °
r 2 xd xq °¿
(2.5.55)
or
ia
1
[(rud 0 xq uq 0 ) cos(t J 0 ) ( xq ud 0 ruq 0 )sin(t J 0 )]
r xd xq
2
(2.5.56)
The stator resistance being neglected, the steady-state value of stator current is
id
iq
uq 0 ½
°
xd °
¾
u
d0 °
xq °¿
(2.5.57)
or
ia
uq 0
xd
cos(t J 0 ) ud 0
sin(t J 0 )
xq
(2.5.58)
(c) Time constants
As mentioned before, the initial and steady-state values of stator current have
been found when resistances are neglected. Those results are sufficiently accurate
because resistances of practical machines are small. From those results we can
see that in steady-state the aperiodic and second harmonic components of stator
current will decay from their initial values to zero, and the fundamental component
can decay from its initial value to steady-state value. Time constants at which
those current components will decay are discussed as follows.
(i) The time constant Ta
Because the aperiodic and second harmonic components in stator current
correspond to the fundamental component in rotor current, those components
will decay at the same time constant. Obviously, the time constant depends upon
parameters of stator and rotor windings. However, the rotor reactance of basic
frequency is much larger than its resistance, so the rotor resistance may be
considered as zero when time constant Ta is determined. Under this condition,
we have
xd ( p )
xq ( p )
128
xdcc ½°
¾
xqcc °¿
(2.5.59)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
so equation (2.5.28) can be reduced to
id
iq
( pxqcc r )ud 0 xqccuq 0
½
°
( pxqcc r )( pxqcc r ) xdcc xqcc °
¾
p ( xdcc r )uq 0 xdccud 0 °
( pxdcc r )( pxqcc r ) xdcc xqcc °¿
(2.5.60)
The corresponding characteristic equation is
( pxdcc r )( pxqcc r ) xdcc xqcc
(2.5.61)
0
and its roots are
p1,2
r§ 1
1·
r2 § 1
1·
¨ ¸ r j 1 ¨ ¸
2 ¨© xdcc xqcc ¸¹
4 ¨© xdcc xqcc ¸¹
2
(2.5.62)
The corresponding time constant is
2 xdcc xqcc
r xdcc xqcc
Ta
x2
r
(2.5.63)
in which
x2
2 xdcc xqcc
xdcc xqcc
is negative sequence reactance.
Observing equation (2.5.62), we can know that when rotor resistance is
neglected but stator resistance is taken into account, in currents id and iq there is
no basic-frequency component but an approximate basic-frequency component
whose frequency is
2
r2 § 1
1·
1 ¨ ¸ |1
¨
4 © xdcc xqcc ¸¹
In the corresponding stator phase currents ia , ib and ic there are no aperiodic and
second harmonic components but a low frequency component and an ac component
whose frequency is approximately equal to double basic-frequency. However,
during calculation of these currents the difference is not taken into account.
(ii) Time constants Tdcc, Tqcc and Tdc
The fundamental component in stator current corresponds to the aperiodic
component in rotor current, so the two components will decay at the same time
constant. Of course, the time constants depend also upon the parameters of stator and
129
AC Machine Systems
rotor windings. However, the stator reactance of basic frequency is much larger
than its resistance, so stator resistance may be considered as zero when time
constant is determined. Under this condition, equation (2.5.28) can be reduced to
pud 0 uq 0 ½
°
xd ( p)( p 2 1) °
¾
puq 0 ud 0 °
xq ( p )( p 2 1) °¿
id
iq
(2.5.64)
The corresponding characteristic equations are
xd ( p )( p 2 1)
xq ( p )( p 2 1)
0 ½°
¾
0 °¿
(2.5.65)
in which the roots p1,2 r j of equation ( p 2 1) 0 correspond to the fundamental
component in currents id and iq , i.e. the aperiodic and second harmonic components
in stator current mentioned before, so the roots of equations xd ( p) 0 and
xq ( p) 0 correspond to the aperiodic component in currents id and iq , i.e. the
fundamental component in stator current. These time constants can be determined
as follows. First of all, we find out the root of equation xq ( p ) 0. Because
xq ( p)
2
( X 11q xq xaq
) p R1q xq
pX 11q R1q
(2.5.66)
the corresponding time constant is
Tqcc
2
X 11q xq xaq
X 11q xqcc
xq R1q
R1q xq
Tq 0
in which Tq 0
X 11q
R1q
xqcc
xq
1 ª
1
º
X 1ql «
»
1
1
R1q
»
«
xaq xl »¼
¬«
(2.5.67)
is the time constant of q-axis damping winding when stator
winding is open-circuited and Tqcc is the time constant of q-axis damping winding
when stator winding is short-circuited whose resistance is equal to zero. Thus Tqcc
is called the decay time constant of q-axis subtransient current.
Now we search for the roots of equation xd ( p) 0. Because
130
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
xd ( p)
xd 2
3
2
2
2
[ p 2 ( X 11d xad
2 xad
X ffd xad
) p ( xad
R1d xad
R fd )]
2
[ p 2 ( X 11d X ffd xad
) p( X 11d R fd X ffd R1d ) R1d R fd ]
(2.5.68)
xd ( p) 0 is an equation of second degree which is also tedious. In practice, an
approximate method is also used. Because the damper resistance is much larger
than excitation winding resistance in actual machines the aperiodic current of
damping winding decays much faster than that of excitation winding. Therefore,
we can approximately consider that during the initial period after voltages are
suddenly applied to the stator winding, the aperiodic current of excitation winding
does not decay and only that of the damping winding decays.
The aperiodic current of excitation winding will not decay when its resistance
is equal to zero, so we can find the time constant by R fd 0 in equation (2.5.65).
The corresponding characteristic equation is
xd 2
3
2
2
p ( X 11d xad
2 xad
X ffd xad
) xad
R1d
2
p ( X 11d X ffd xad
) X ffd R1d
0
(2.5.69)
and the corresponding time constant is
Tdcc
2
2
3
2
2 xad
X ffd xad
xd ( X 11d X ffd xad
) X 11d xad
2
R1d ( xd X ffd xad
)
Tdcc0
xdcc
xdc
1 §
1
·
X 1dl ¨
1
1
1¸
R1d
¸
¨
¨
xad X fl xl ¸¹
©
(2.5.70)
in which
X 11d Tdcc0
R1d
2
xad
X ffd
1 §
1
·
X 1dl ¨
1
1 ¸
R1d
¨
¸
¨
xad X fl ¸¹
©
(2.5.71)
is the time constant of d-axis damping winding when stator winding is opencircuited and excitation winding is short-circuited referring to Fig. 2.4.6.
Tdcc is the time constant of d-axis damping winding when stator and excitation
windings are all short-circuited in which r 0 and R fd 0, and it is called the
decay time constant of d-axis subtransient current.
After a certain time, the aperiodic current of damping winding has basically
decayed out. It can be approximately considered that decay of the aperiodic
current in excitation winding will determine decay of the fundamental current in
131
AC Machine Systems
stator winding and damping winding is open-circuited after that time, so
xd xd ( p )
2
pxad
pX ffd R fd
and the time constant corresponding to equation xd ( p)
2
xd X ffd xad
Tdc
Td 0
xd R fd
xdc
xd
(2.5.72)
0 is
(2.5.73)
If damping winding resistance Rld is not very different from excitation
winding resistance R fd , during estimation of time constants Tdc and Tdcc it can not
be considered that R fd 0 in the initial period and Rld f after that period.
Both resistances must be considered simultaneously. The corresponding time
constants Tdc and Tdcc can be obtained according to the following way. Let
c
xad
2
xd xad xad
xd
1
1
1
xad xl
(2.5.74)
be the equivalent mutual reactance between d-axis damping winding and excitation
winding when stator winding is short-circuited whose resistance is zero.
c
T ffd
X cffd
2
X ffd xd xad
R fd
R fd xd
X fl 1
1
1
xad xl
R fd
c
X fl xad
(2.5.75)
R fd
is the time constant of excitation winding when stator winding is short-circuited
and damping winding is open-circuited.
T11c d
X 11c d
R1d
X 1dl 2
X 11d xd xad
R1d xd
1
1
1
xad xl
R1d
c
X 1dl xad
R1d
(2.5.76)
is the time constant of d-axis damping winding when stator winding is shortcircuited and excitation winding is open-circuited.
132
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
V cf 1d
1
c2
xad
X cffd X 11c d
(2.5.77)
is the leakage-flux factor between d-axis damping winding and excitation winding
when stator winding is short-circuited.
The factors above substituted into equation (2.5.68), the equation of second
degree xd ( p) 0 can be reduced to
c T11c d p 2 (T ffd
c T11c d ) p 1 0
V cf 1d T ffd
(2.5.78)
Solving it, we can get
p1,2
c )(r q)
(T11c d T ffd
c T11c d
2V cf 1d T ffd
(2.5.79)
in which
q
1
c V cf 1d
4T11c d T ffd
c )2
(T11c d T ffd
|1
(2.5.80)
The corresponding time constants are
c T11c d
2V cf 1d T ffd
Tdcc 1
p1
c )
(1 q )(T11c d T ffd
Tdc
1
p2
c T11c d
2V cf 1d T ffd
c )
(1 q)(T11c d T ffd
|
c T11c d
V cf 1d T ffd
c T11c d
T ffd
1
c ) | T11c d T ffd
c
(1 q )(T11c d T ffd
2
(2.5.81)
(2.5.82)
(d) The general formulas of stator current
Having the initial values, steady-state values and time constants obtained
above, we can easily write the general expressions of stator current as
id
t
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º
1 T
«¨ ¸ e d ¨ ¸ e d » uq 0 e a [ud 0 sin t uq 0 cos t ]
xd »¼
xdcc
© xdc xd ¹
¬«© xdcc xdc ¹
(2.5.83)
iq
t
ª§ 1 1 · Tt cc 1 º
1 «¨ ¸ e q » ud 0 e Ta [ud 0 cos t uq 0 sin t ]
xq »
xqcc
«¬¨© xqcc xq ¸¹
¼
(2.5.84)
133
AC Machine Systems
or
id
iq
t
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º
U T
«¨ ¸ e d ¨ ¸ e d » U cos G e a cos(t G )
xd »¼
xdcc
© xdc xd ¹
¬«© xdcc xdc ¹
(2.5.85)
t
ª§ 1 1 · Tt cc 1 º
U «¨ ¸ e q » U sin G e Ta sin(t G )
xq »
xqcc
«¬©¨ xqcc xq ¹¸
¼
(2.5.86)
2.5.3 Currents after 3-phase Sudden Short Circuit of Synchronous
Machines
3-phase sudden short circuit is a severe accident produced possibly in synchronous
machine operation and it is also a basis for analysis of other sudden short circuit
phenomena. As stated before, during analysis of 3-phase sudden short circuit, we
can consider the short circuit not to have happened but the stator terminals to be
suddenly applied on by 3-phase voltages which are equal but opposite to the
voltages before short circuit. In thinking so, the problem of sudden short circuit
is turned into the synthetic problem of the following two operation conditions:
(i) Original steady-state operation condition before short circuit, referring to
2.5.1.
(ii) Stator terminals are suddenly applied on by 3-phase voltages which are
equal but opposite to original voltages before short circuit, referring to 2.5.2.
Because the basic equations are linear under this condition, we can use
Superposition Theorem to treat this problem.
(1) Stator currents after 3-phases are suddenly short-circuited
If the amplitudes of terminal voltage and excitation emf before short circuit are
U and E respectively and the terminal voltage lags behind excitation emf by an
angle G, then referring to equation (2.5.3) we can write stator terminal voltage as
ud 0
uq 0
U sin G ½°
¾
U cos G °¿
(2.5.87)
According to Superposition Theorem, it is not difficult to understand that the
stator current after 3-phases are suddenly short-circuited consists of the original
steady-state current and transient current caused by applying the voltages
(ud 0 ) and (uq 0 ) to stator winding.
(a) The original steady-state current
If stator resistance is neglected, according to equation (2.5.7) the currents can
134
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
be written as
id
iq
E U cos G ½
°
xd
°
¾
U sin G
°
°¿
xq
(2.5.88)
(b) The transient current caused by applying the voltages (ud 0 ) and (uq 0 ) to
stator winding
Referring to equations (2.5.85) and (2.5.86), we can write transient currents as
t
t
t
½
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º
U Ta
e
e
U
cos
e
cos(t G ) °
G
Ǭ
»
¸
¨
¸
xd »¼
xdcc
«¬© xdcc xdc ¹
© xdc xd ¹
°°
¾
t
t
ª§ 1
°
1 · Tqcc 1 º
U Ta
iq «¨ ¸ e » U sin G e sin(t G )
°
¨ xcc x ¸
xq »
xqcc
q ¹
°¿
¬«© q
¼
(2.5.89)
(c) Stator actual currents after 3-phase sudden short circuit
Adding equation (2.5.88) to equation (2.5.89), we can write stator actual
current after short circuit as
id
id
t
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc º
E U Ta
e
e
U
cos
e
cos(t G )
G
Ǭ
»
¸
¨
¸
xd xdcc
«¬© xdcc xdc ¹
»¼
© xdc xd ¹
(2.5.90)
t
iq
t
§ 1 1 · T cc
U ¨ ¸ e q U sin G e Ta sin(t G )
¨ xcc x ¸
xqcc
q ¹
© q
(2.5.91)
Having id and iq , we can transform them into ia , ib , ic . For convenience, we
suppose the machine to be no-load before short circuit, and then there are
G 0
U E
ia
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º
«¨ ¸ e d ¨ ¸ e d » E cos(t J 0 )
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
t
º
§ 1
E ª§ 1
1·
1·
e Ta «¨ ¸ cos J 0 ¨ ¸ cos(2t J 0 ) »
¨ xcc xcc ¸
2
«¬¨© xdcc xqcc ¸¹
»¼
q ¹
© d
(2.5.92)
(J 0 120e
) and (J 0 120e
) substituted respectively for J 0 in the formula above,
we can obtain the expressions for ib and ic .
135
AC Machine Systems
In nonsalient-pole machines such as turbogenerators which have strong damping
windings, xdcc | xqcc, so the formula above can be reduced to
ia
t
ª§ 1
1 · Tdcc § 1
1
¨ e
Ǭ
¸
cc
c
c
«¬© xd xd ¹
© xd xd
t
t
· Tdc 1 º
E Ta
˜
e
E
cos(
t
)
e
cos J 0
J
»
¸
0
xd »¼
xdcc
¹
(2.5.93)
If the sudden short-circuit current does not decay, according to formula (2.5.92)
we can express phase a current as
ia
§ 1
§ 1
E
1 ·E
1 ·E
cos(t J 0 ) ¨ ¸ cos J 0 ¨ ¸ cos(2t J 0 )
¨
¸
¨
¸
xdcc
© xdcc xqcc ¹ 2
© xdcc xqcc ¹ 2
(2.5.94)
From here we can know that the amplitude of stator fundamental current
is E / xdcc ; in comparison with the corresponding short-circuit current E / xd in
steady-state the difference between them is to change xd into xdcc. The value of xdcc
depends mainly upon stator and rotor leakage reactances, whose physical meaning
can be stated as follows.
If there were no leakage flux in rotor winding, the rotor flux-linkage would
induce the same emf in stator winding as before short-circuit, so the stator
fundamental current amplitude would be E / xl , i.e. the subtransient reactance xdcc
would equal the stator leakage reactance xl . In fact, leakage flux exists in rotor
winding and the stator armature reaction increases during sudden short-circuit.
Therefore, rotor current will rise so as to keep the rotor flux-linkage unchanged,
and the rotor leakage flux will increase too. Thus the mutual flux between rotor
and stator will decrease, i.e. the stator emf will be less than the value E which is
the emf before short-circuit. Provided that the stator emf during sudden
short-circuit is ( E 'E ), the stator fundamental current amplitude will become
( E 'E ) / xl . However, it is not convenient to calculate 'E directly since
'E depends upon the increase of rotor leakage flux or the decrease of mutual
flux. In practical estimation, we can suppose the stator emf after short-circuit is
the same as before, but xl changes into ( xl 'x); that is to say, the
corresponding current ( E 'E ) / xl can be considered to equal E /( xl 'x ), so
there is xdcc xl 'x. Therefore, it is evident that xdcc depends basically upon the
stator and rotor leakage reactances.
From equation (2.5.92) we are clear that the stator short-circuit current is
dependent on the rotor position angle J 0 at the short-circuit moment; taking
non-salient pole synchronous machines as example we shall discuss the following
two cases.
During J 0 90e, expression (2.5.93) will be turned into
136
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
ia
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º
«¨ ¸ e ¨ ¸ e » E sin t
xd »¼
«¬© xdcc xdc ¹
© xdc xd ¹
(2.5.95)
From here we can recognize that no aperiodic component exists in stator
current ia in the case. That is natural because \ a is just equal to zero at the
short-circuit moment, so there is no need to cause an aperiodic current to keep
the original flux-linkage. The curve of stator current ia is shown in Fig. 2.5.6. It
should be noted that the aperiodic component still exists in current ib or ic .
Figure 2.5.6
ia -curve during 3-phase sudden short circuit at J 0
90e
Under that condition, the stator fundamental current consists of the following
three parts:
E
(i) Steady-state short-circuit current
sin t , xd
t
§ 1
1· (ii) Subtransient current ¨ ¸ e Tdcc E sin t ,
© xdcc xdc ¹
t
§ 1
1· (iii) Transient current ¨ ¸ e Tdc E sin t.
© xdc xd ¹
During J 0
ia
0, formula (2.5.93) will change into
t
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º
E Ta
e
e
E
cos
t
e
Ǭ
»
¸
¨
¸
xd »¼
xdcc
«¬© xdcc xdc ¹
© xdc xd ¹
(2.5.96)
whose curve is shown in Fig. 2.5.7. On this condition, phase a fundamental
current has the same three components as before except a phase-angle difference
of 90e. In addition, an aperiodic component appears in stator current ia to keep
the original flux-linkage of phase a unchanged suddenly. The aperiodic component
will make the curve of ia asymmetrical about the abscissa. Current ia will reach
its maximum in about half a cycle after short circuit, whose value is around twice
as large as the maximum value during J 0 90e. Incidentally, the initial value of
137
AC Machine Systems
aperiodic current is equal but opposite to that of the fundamental current. That is
natural because the current ia at t 0 must be zero.
Figure 2.5.7
ia -curve during 3-phase sudden short circuit at J 0
0
Because the stator fundamental current decays slowly, its rms value or amplitude
change is significant. Obviously, the rms value or amplitude at any instant, with
reference to formulas (2.5.95) and (2.5.96), can be written as
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º
«¨ ¸ e d ¨ ¸ e d » E
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
(2.5.97)
whose curve is shown in Fig. 2.5.8.
Figure 2.5.8 Maximum-value envelope of stator fundamental current during 3-phase
sudden short circuit
(2) Rotor currents after 3-phase sudden short circuit
The rotor currents after short circuit can be calculated in the same way as the
stator currents. It may be estimated according to the following two parts:
(i) The rotor currents before short circuit,
(ii) The rotor current caused by applying voltages (ud 0 ) and (uq 0 ) to stator
winding.
138
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
If the machine operates symmetrically in steady-state before short circuit, the
first part can be easily calculated, i.e.
I fd 0
I1d 0
U fd
R fd
I1q 0
½
°
¾
0 °¿
(2.5.98)
The solution of the second part is also tedious, so it can be calculated by the
same approximate method as the stator current, i.e. we shall estimate only the
values at two or three instants and then write its expression according to the time
constants obtained before. The equivalent circuits shown in Figs. 2.4.3 to 2.4.7
have given the relation between currents id , iq and rotor currents when U fd 0.
Because currents id and iq have been found, rotor currents can be obtained
according to those equivalent circuits.
(a) The initial values of rotor currents caused by applying voltages (ud 0 ) and
(uq 0 ) to stator winding
During calculation of the initial values, all rotor resistances can be considered
as zero, so it is convenient to use the equivalent circuits shown in Figs. 2.5.3 and
2.5.4 to solve the problem because there is no operator p in those equivalent
circuits.
When there is no damping winding on rotor, according to Fig. 2.5.3 we can get
i fd
xad
id
X fl xad
xad
id
X ffd
xad U
[cos G cos(t G )]
X ffd xdc
(2.5.99)
When there is a damping winding on rotor, according to Fig. 2.5.4 we can
obtain
i fd
2
X 11d xad xad
i
2 d
X 11d X ffd xad
2
X 11d xad xad
U
[cos G cos(t G )]
2
X 11d X ffd xad xdcc
i1d
2
X ffd xad xad
2
X 11d X ffd xad
(2.5.100)
id
2
X ffd xad xad
U
[cos G cos(t G )]
2
X 11d X ffd xad xdcc
(2.5.101)
139
AC Machine Systems
From the formulas above we can see that during the same value of id the value of
I fd when there is a damping winding is less than that when there is no damping
winding. That is natural, because the damping winding shares part of the induced
current. From the formulas above we can also know
2
X 11d xad xad
2
X ffd xad xad
i fd
i1d
X 1dl
X fl
(2.5.102)
In addition, according to Fig. 2.5.5 we can see
i1q
xaq
X 11q
iq
xaq U
[sin G sin(t G )]
X 11q xqcc
(2.5.103)
(b) The steady-state values of rotor currents caused by applying voltages
(ud 0 ) and (uq 0 ) to stator winding
Obviously, the current values are equal to zero because there is p 0 in the
equivalent circuit of operational reactance shown in Fig. 2.4.6.
(c) General expressions of rotor currents caused by applying voltages (ud 0 )
and (uq 0 ) to stator winding
Because there are resistances in the machine, the rotor current will decay
gradually from the initial value to zero. The decay time constants are similar to
those of the corresponding components of stator current. Therefore, when there is
a damping winding on rotor, general expressions of rotor currents caused by
applying voltages (ud 0 ) and (uq 0 ) to stator winding are
i fd
­° ª§ X x x 2
11d ad
ad
® «¨¨
2
°¯ «¬© X 11d X ffd xad
2
§ X x xad
¨ 11d ad
2
¨
© X 11d X ffd xad
i1d
2
§ X ffd xad xad
¨¨
2
© X 11d X ffd xad
t
· 1
xad 1 º Tdcc § xad
¨
e
»
¸¸
¨
cc
c
x
X
x
»
d
ffd
d
¹
© X ffd
¼
· 1 Ttc ½°
¸¸ e d ¾U cos G
°¿
¹ xdc
· U Tt
¸¸ e a cos(t G )
¹ xdcc
2
· U Ttcc
§ X ffd xad xad
¸¸ e d cos G ¨¨
2
¹ xdcc
© X 11d X ffd xad
(2.5.104)
· U Tt
¸¸ e a cos(t G )
¹ xdcc
(2.5.105)
t
i1q
140
xaq U Tqcc
xaq U Tta
e sin G e sin(t G )
X 11q xqcc
X 11q xqcc
(2.5.106)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
When there is no damping winding on rotor, we can get
t
t
xad U Tdc
x U Ta
e cos G ad
e cos(t G )
X ffd xdc
X ffd xdc
i fd
(2.5.107)
The rotor actual current after 3-phase sudden short circuit can be obtained by
sum of the two parts as mentioned before.
When there is no damping winding on rotor, we have
i fd
U fd
R fd
t
t
xad U Tdc
x U Ta
e cos G ad
e cos(t G )
X ffd xdc
X ffd xdc
(2.5.108)
whose curve is shown in Fig. 2.5.9.
Figure 2.5.9 Excitation winding current after 3-phase sudden short circuit and
without damping winding
When there is a damping winding on rotor, the actual current in excitation
winding is
i fd
U fd
R fd
t
t
ª§ X x x 2 1
xad 1 · Tdcc xad 1 Tdc º
ad
e
e
Ǭ 11d ad
» U cos G
¸
2
¨
cc
c¸
X ffd xdc
¬«© X 11d X ffd xad xd X ffd xd ¹
¼»
t
2
X 11d xad xad
U Ta
e cos(t G )
2
X 11d X ffd xad xdcc
(2.5.109)
whose curve is shown in Fig. 2.5.10. It is noted that the component contained in
i fd and decaying at time constant Tdcc is generally negative value. Therefore, the
curve of aperiodic component of excitation winding current i fd is bent down in
its initial part.
Because damping winding current is zero before short circuit, the damping winding
actual current after short circuit is the same as expressions (2.5.105) and (2.5.106).
141
AC Machine Systems
Figure 2.5.10 Excitation winding current after 3-phase sudden short circuit and
with damping winding
2.6
Voltage Dip during Sudden Increase of Load for
Synchronous Machines
If 3-phase symmetrical constant loads are suddenly connected to a synchronous
generator, so far as calculation of its currents is concerned, we can consider that
condition as 3-phase sudden short circuit after the load resistance and reactance
per phase are added to stator phase resistance and leakage reactance respectively.
Provided that the load is a starting induction motor, it can also be treated as a
constant impedance because it may be represented approximately by a short
circuit impedance during part of the starting process.
If the load reactance and resistance are X L and RL , the corresponding
operational reactance and stator circuit resistance are
xdG ( p ) X L ½
°
xq ( p ) xqG ( p ) X L ¾
°
R rG RL
¿
xd ( p )
(2.6.1)
in which the subscript G represents the values corresponding to the synchronous
generator.
According to the concept of 3-phase short circuit and referring to equations
(2.5.28) and (2.5.3), we can find the basic relations after a 3-phase symmetrical
load is suddenly connected to a synchronous generator as follows:
142
ud 0
U sin G
uq 0
U cos G
0 ½°
¾
E °¿
(2.6.2)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
xq ( p )uq 0
id
iq
½
°
[ pxd ( p ) R][ pxq ( p ) R ] xd ( p ) xq ( p ) °
¾
[ pxd ( p ) R]uq 0
°
[ pxd ( p ) R][ pxq ( p ) R ] xd ( p ) xq ( p ) °¿
(2.6.3)
It should be noted that deriving those relations we have supposed the speed to
be constant, because the speed change is much slower than the voltage dip.
It is very complicated to solve equation (2.6.3) directly, and we still use an
approximate method to solve it. Because the time constant of damping winding is
usually very small and the corresponding subtransient component decays very
fast, studying the voltage dip we don’t take the effect of damping winding into
account, i.e. we can consider the machine having no damping winding. This
problem is now studied as follows.
(1) The initial value of stator current
Since the resistance of excitation winding is very small, evaluating the initial
value of stator current we can take the resistance as zero, i.e.
xd ( p ) |R fd
0 or p f
xdc
c XL
xdG
(2.6.4)
xq ( p ) |R fd
0 or p f
xq
xqG X L
(2.6.5)
The values above substituted into equation (2.6.3), it is not difficult to find the
initial value of stator d-axis current as follows,
xq E
id
[ pxdc R][ pxq R] xdc xq
̯
xq E
R 2 xdc xq
xq E
Z xdc xq ( R 2 xdc xq )
e
t
Ta
sin(Z t M )
(2.6.6)
in which
Z
M
Ta
2 ½
R2 § 1
1· °
1
¨ ¸
4 ¨© xdc xq ¸¹ °
°°
arctan ZTa
¾
°
c
2
x
x
1
d q
°
°
R xdc xq
°¿
(2.6.7)
Ta is the decay time constant of stator aperiodic current. The second term in
equation (2.6.6) is the aperiodic and second harmonic components of stator
143
AC Machine Systems
current, which are neglected when the voltage dip is calculated because those
components have a little effect on the voltage dip when R is small and they will
decay quickly when R is large. For example, when an induction motor starts, its
impedance is X L 0.3 and RL 0.1, and the decay time constant, if the generator
2 u 0.6 u 1.1
c
7.8
parameters are xdG
0.3, xqG 0.8 and r 0.005, is Ta
0.1005(0.6 1.1)
7.8
second, i.e. only about one cycle.
radians or
2S u The aperiodic current and second harmonic component neglected, the initial
value of stator d-axis current will be
xq E
id
2
R xdc xq
(2.6.8)
Similarly, the initial value of q-axis current is
RE
R 2 xdc xq
iq
(2.6.9)
Because the stator aperiodic current and second harmonic component are
caused by the stator transformer emf, those componnts will disappear when the
two terms p\ d and p\ q are neglected in Park’s equation. That is to say, we use
the voltage equations
\ q Rid ½°
¾
\ d Riq °¿
ud
uq
(2.6.10)
to analyse the problem, i.e. we shall neglect the stator aperiodic current and second
harmonic component.
For convenience, when analysing some problems, such as the voltage dip and
machine stability, we often use this kind of equation instead of original Park’s
equation.
Equation (2.6.10) used to analyse the voltage dip, equation (2.6.3) can be
reduced to
id
iq
If xd ( p)
144
xdc and xq ( p)
xq ( p ) E
½
°
R xd ( p ) xq ( p ) °
¾
RE
°
R 2 xd ( p ) xq ( p ) °¿
2
xq , the initial values of stator current are
(2.6.11)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
id
iq
xq E
½
°
R xdc xq °
¾
RE °
R 2 xdc xq °¿
2
(2.6.12)
that are equations (2.6.8) and (2.6.9) obtained by another method. Obviously, the
two methods have the same result.
(2) The steady-state value of stator current
Supposing p 0 in equation (2.6.3) or (2.6.11), we can obtain the steady-state
value of stator current
id
iq
xq E
½
°
R xd xq °
¾
RE °
R 2 xd xq °¿
2
(2.6.13)
(3) Time constant
The time constant of stator fundamental current depends upon the effective
reactance and resistance corresponding to the aperiodic current in excitation
winding. The flux-linkage of excitation winding in steady-state is
\ fd
X ffd I fd xad id
X ffd I fd xad xq E
R 2 xd xq
2
§
xad
xq ·
X
¨¨ ffd
¸¸ I fd
2
R
x
x
d
q
©
¹
Therefore, the corresponding time constant is
Tdc
1
R fd
§ \ fd
¨¨
© I fd
·
¸¸
¹
R 2 xdc xq
R 2 xd xq
Td 0
(2.6.14)
(4) The general formulas of current and voltage
After finding the initial value, steady-state value and time constant, we can
write the general formulas of stator current as
id
t
§
·
xq E
1
1
Tdc
2
¨¨ 2
¸¸ xq Ee 2
R xd xq
© R xdc xq R xd xq ¹
iq
t
§
·
RE
1
1
Tdc
2
¨¨ 2
¸¸ REe 2
R xd xq
© R xdc xq R xd xq ¹
R
id
xq
(2.6.15)
145
AC Machine Systems
According to the formulas above, we can obtain the amplitudes of stator
current and terminal voltage
| ia |
2
d
2
q
i i
§R
1 ¨
¨x
© q
2
·
¸¸ id
¹
ª§
º
· Ttc
1
1
1
2
«¨¨ 2
» R 2 xq2 E
¸¸ e d 2
c
R
x
x
R
x
x
R
x
x
d q
d q ¹
d q »
¬«©
¼
|u|
(2.6.16)
X L2 RL2 | ia |
ª§ R 2 xd xq
º ( RL2 X L2 )( R 2 xq2 )
· t
1¸ e Tdc 1»
E
«¨¨ 2
¸
R 2 xd xq
«¬© R xdc xq
»¼
¹
(2.6.17)
Supposing t 0 in the formula above, we can get the initial amplitude of terminal
voltage
| u |t
( RL2 X L2 )( R 2 xq2 )
0
R 2 xdc xq
E
(2.6.18)
From that formula we can see:
(a) If power factor of the load is very low, i.e. R may be neglected, then the
initial amplitude of terminal voltage is
| u |t
0
XL
E
xdc
XL
E
c
xdG X L
c id
E xdG
(2.6.19)
i.e. under this condition, the initial dip of terminal voltage depends mainly upon
d-axis transient reactance, and the voltage dip is severe.
(b) If power factor of the load is high, i.e. the load reactance X L can be
neglected, then the initial dip of terminal voltage is not large, and sometimes the
terminal voltage will rise.
For example, under this condition, xqG 1.0, RL 1.0, X L | rG | 0, we can get
| u |t
0
1.414
E
c
1 xdG
(2.6.20)
c 0.414, terminal voltage during initial period will not decrease but
i.e. when xdG
increase. Because q-axis synchronous reactance xq of a turbogenerator is larger
c is smaller than a hydraulic generator, the
and transient reactance xdG
turbogenerator can undertake a larger load switched on than the hydraulic
generator so far as the voltage dip is concerned.
146
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
2.7
Transient EMFs of Synchronous Machines
The flux-linkage of rotor winding, especially the flux-linkage of excitation
winding, under a certain condition, can be considered as a constant, which will
make analysis and calculation simplified. However, it is not convenient directly
to use the rotor flux-linkage, so we shall use other quantities which can represent
the property of rotor flux-linkage. Those new quantities are called d-axis
transient emf, d-axis or q-axis subtransient emf, and the transient emf behind
transient reactance, etc.
Because those emfs can represent the property of rotor flux-linkage, they are
important quantities when the main process of basic-frequency current is studied.
We shall discuss those emfs and their applications as follows.
(1) d-axis transient emf
If there is only an excitation winding and no damping winding on the rotor, the
flux-linkage equations can be written as
\ fd
\d
xad id X ffd I fd ½°
¾
xd id xad I fd °¿
(2.7.1)
Deleting I fd , we can get
\d
§
x2 ·
x
¨ xd ad ¸ id ad \ fd
¨
X ffd ¸¹
X ffd
©
x
xdc id ad \ fd
X ffd
(2.7.2)
or
Eqc \ d xdc id
in which Eqc
(2.7.3)
xad
\ fd is a quantity that is proportional to the flux-linkage of
X ffd
excitation winding and represents the property of excitation winding flux-linkage.
Eqc is called d-axis transient emf.
(2) d-axis subtransient emf
If there is not only an excitation winding but also a damping winding on rotor,
the flux-linkage equations can be written as
\ fd
\ 1d
\d
xad id X ffd I fd xad I1d ½
°
xad id xad I fd X 11d I1d ¾
°
xd id xad I fd xad I1d ¿
(2.7.4)
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AC Machine Systems
Deleting I fd and I1d , we can get
xdccid \d
2
2
X ffd xad xad
X 11d xad xad
\
\ 1d
fd
2
2
X 11d X ffd xad
X 11d X ffd xad
(2.7.5)
or
Eqcc \ d xdccid
(2.7.6)
in which
Eqcc
2
2
X ffd xad xad
X 11d xad xad
\
\ 1d
fd
2
2
X 11d X ffd xad
X 11d X ffd xad
(2.7.7)
is a quantity that is proportional to the flux-linkages of excitation winding and
damping winding and represents the property of rotor d-axis flux-linkages. Eqcc is
called d-axis subtransient emf.
(3) q-axis subtransient emf
On q-axis of a machine with damping winding, the flux-linkage equations can
be written as
\ 1q
\q
xaq iq X 11q I1q ½°
¾
xq iq xaq I1q °¿
(2.7.8)
Deleting I1q , we can obtain
\q
2
§
·
xaq
xaq
¨ xq \ 1q
iq ¸
¨
X 11q ¸¹
X 11q
©
xaq
xqcciq \ 1q
X 11q
(2.7.9)
or
Edcc \ q xqcciq
in which Edcc
xaq
X 11q
(2.7.10)
\ 1q is a quantity that is proportional to the flux-linkage of
q-axis damping winding and represents the property of q-axis damper
flux-linkage. Edcc is called q-axis subtransient emf.
(4) The transient emf behind transient reactance
When a synchronous machine without damping windings operates symmetrically
in steady-state, referring to equation (2.7.3) we can obtain the following relations:
148
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
\d
E xd id
\q
xq iq
ud
\ q rid
uq
\ d riq
Eqc xdc id ½
°
°
¾
°
°
¿
(2.7.11)
If taking d-axis as real axis, q-axis as imaginary axis and supposing
uˆ ud juq , iˆ id jiq , we can plot a vector diagram on complex-number plane
as shown in Fig. 2.7.1.
Figure 2.7.1 Vector diagram for transient emfs
In that figure, E c uˆ (r jxdc )iˆ ( xq xdc )iq jEqc is called the transient emf
behind d-axis transient reactance or simply the transient emf behind transient
reactance, and Eq E ( xd xq )id is called the emf behind q-axis synchronous
reactance.
Neglecting stator resistance, we have
E c uˆ jxdc iˆ
In practice, because the flux-linkage of q-axis damper \ 1q decays rapidly, it
can be neglected, i.e. Edcc 0. Therefore, there is only d-axis subtransient emf
Eqcc and transient emf Eqc , and we can call them the subtransient emf and transient
emf respectively.
If xdc xq , i.e. there is no transient saliency-effect, E c will be equal to Eqc . In
actual machines, the difference between transient emf Eqc and the transient emf
behind transient reactance E c is very small referring to Fig. 2.7.1, and the latter
can be found according to the current, terminal voltage and transient reactance.
Thus, we often use E c | Eqc to analyse some problems. Similarly, we can
obtain E cc uˆ (r jxdcc )iˆ | uˆ jxdcciˆ, which is called the subtransient emf behind
subtransient reactance. In practice, we can consider E cc | Eqcc. Because the
flux-linkage of damping winding changes much faster than that of excitation
149
AC Machine Systems
winding, E cc or Eqcc will be changed rapidly into E c or Eqc . Therefore, we often
neglect E cc or Eqcc but use E c or Eqc to study many problems, i.e. Only the effect
of excitation winding is taken into account.
(5) Application examples of the transient emf
We have mentioned definition of the transient emf as before. Now some
application examples are introduced as follows.
(a) Application of the transient emf to calculating the initial value of threephase short circuit current
When a synchronous machine that is operating symmetrically in steady-state is
suddenly 3-phase short-circuited, the initial values of aperiodic components in
currents id and iq , i.e. the initial values of stator basic-frequency current, are
(id 0 uq 0 / xdcc ) and (iq 0 ud 0 / xqcc) respectively referring to equations (2.5.90) and
(2.5.91). After arrangement, those values are equal to
1
(iq 0 xqcc ud 0 )
xqcc
1
(id 0 xdcc uq 0 )
xdcc
Eqcc
xdcc
and
Edcc
respectively. That is natural, because Eqcc and Edcc represent
xqcc
the emfs which are proportional to rotor d-axis and q-axis flux-linkages during
short circuit; Those flux-linkages can not change abruptly at the moment of short
circuit, so the corresponding emfs Edcc and Eqcc can not change abruptly, either. In
addition, the reactances corresponding to 3-phase sudden short circuit are
just xdcc and xqcc, so the corresponding currents are Edcc / xqcc and Eqcc / xdcc. If there is
no damping winding, then the initial value of aperiodic component in
current id is Eqc / xdc and that in current iq is equal to zero because there is no loop
on rotor q-axis and its flux-linkage can not be kept constant.
The above calculation is based on the stipulation that a synchronous machine
operates symmetrically in steady-state before short circuit. However, the formulas
can also be used under any operation mode before short circuit if Edcc , Eqcc and
Eqc have correct values. That is because those emfs are proportional to the rotor
flux-linkages and are some quantities which can not change abruptly. For example,
studying sudden asymmetrical short circuit,we can use the transient emf,
transient reactance and Symmetrical Components Method to solve it.
(b) Application of the transient emf to estimating the voltage dip when the load
of a synchronous generator increases suddenly
For convenience, we have studied the voltage dip when a synchronous generator
that has no load is suddenly loaded in Section 2.6. If the machine has undertaken
a certain load before it is suddenly loaded, the corresponding result is different.
Because the transient emf, when operation condition changes, does not change
abruptly, we can use it while studying the initial voltage dip after the load of a
synchronous generator increases suddenly.
150
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Letting RL and X L represent the parallel resistance and reactance of the
original load with the new load with reference to equation (2.5.6), the initial and
steady-state currents, when the aperiodic component of stator curent is neglected,
can be written as
idc
iqc
idy
iqy
xq Eqc( i ) ½
°
R 2 xdc xq °
¾
REqc( i ) °
R 2 xdc xq °¿
(2.7.12)
xq E
½
°
R xd xq °
¾
RE °
R 2 xd xq °¿
(2.7.13)
E ( xd xdc )id 0
(2.7.14)
2
in which
Eqc (i )
The subscript i indicates that the machine has an original load before loaded.
Because the expression of decay time-constant is the same as equation (2.6.14),
the formulas of current and voltage dip are similar to the corresponding expressions
when a synchronous generator that has no load is suddenly loaded. However,
there is a difference between them. If the machine has no load before loaded, the
emf in those formulas is excitation emf E and the impedance RL , X L is a
new-loaded impedance. If the machine has undertaken an original load before
loaded, the emf is the transient emf Eqc (i ) before a new load is switched on, and
the impedance RL , X L is the parallel impedance of original load with the new
load. The corresponding formulas are
id (i )
§ xq Eqc( i )
xq E · Ttdc
xq E
2
¨¨ 2
¸¸ e 2
R xd xq
© R xdc xq R xd xq ¹
iq ( i )
§ REqc( i )
· Ttc
RE
RE
2
¨¨ 2
¸¸ e d 2
c
R
x
x
R
x
x
R
xd xq
d q
d q ¹
©
| ia |( i )
R 2 xq2
xq
id ( i )
(2.7.15)
R
id (i )
xq
§ R2 x2
R 2 xq2 · Ttc
q
¨
c
E
E ¸e d
¨ R 2 xdc xq q (i ) R 2 xd xq ¸
©
¹
½
°
°°
¾
R 2 xq2 °
E
2
R xd xq °°
¿
(2.7.16)
151
AC Machine Systems
RL2 X L2 | ia |
| u |( i )
ª§ R 2 xd xq Eqc( i )
º ( R 2 xq2 )( RL2 X L2 )
· Ttc
d
1
e
1
E
Ǭ 2
»
¸¸
2
R
x
x
«¬©¨ R xdcc xq E
»
d
q
¹
¼
(2.7.17)
| u |( i )( t
( R 2 xq2 )( RL2 X L2 )
0)
R 2 xdc xq
Eqc( i )
(2.7.18)
Because the transient emf Eqc when the machine has no load is equal to the
excitation emf E, those formulas (2.7.15), (2.7.16), (2.7.17) and (2.7.18) can also
be used when a synchronous generator that has no load is suddenly loaded.
(c) Output power expressed by transient emf
The output power and electromagnetic torque are important quantities when
the rotor motion is studied. The machine has a disturbance and those quantities
will change accordingly. Because the flux-linkage of excitation winding changes
slowly in transient process and can be considered approximately as a constant in
the initial period, it is convenient that the output power or electromagnetic torque
is expressed by the transient emf Eqc which represents the property of excitation
flux-linkage.
Neglecting stator resistance and referring to equations (2.5.9) and (2.4.22), we
can write the output power and torque in steady-state as follows:
P Te
EU
1§ 1
1 ·
sin G ¨ ¸ U 2 sin 2G
¨
xd
2 © xq xd ¸¹
(2.7.19)
In addition, there are the following relations:
\d
E xd id
\q
xq iq
ud
\ q
uq
\d
Eqc xdc id ½°
¾
°¿
U sin G
xq iq
Eqc xdc id
½°
¾
U cos G °¿
(2.7.20)
(2.7.21)
Therefore, we can get the output power or torque expressed by the transient emf
Eqc as follows:
P
ud id uq iq
EqcU
Eqc uq
xdc
ud u d uq
xq
1§ 1
1·
sin G ¨ ¸ U 2 sin 2G
¨
xdc
2 © xq xdc ¸¹
152
(2.7.22)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Te
\ d iq \ q id
uq iq ud id
P
EqcU
1§ 1
1 ·
sin G ¨ ¸ U 2 sin 2G
¨
2 © xq xdc ¸¹
xdc
(2.7.23)
According to equations (2.7.19) and (2.7.22), we can plot two curves A and B
shown in Fig. 2.7.2, supposing U fd to be constant.
Figure 2.7.2
P - G curves for steady-state and transients
From those curves we can see that when power angle G changes very slowly,
i.e. the changing process may be considered as steady-state, the output power P
will change from the steady-state operation point ( P0 , G 0 ) along curve A shown
in Fig. 2.7.2. However, when power angle G changes quickly, the output power P,
if the excitation winding flux-linkage is constant, will change from the
steady-state operation point ( P0 , G 0 ) along curve B shown in Fig. 2.7.2. Curve A
is called the steady-state P - G curve, and curve B the transient P - G curve.
From curves A and B we can see that the maximum value of curve A is situated
at the region G 90eand that of curve B at the region G ! 90e. In addition, the
maximum value of the latter is much larger than that of the first. The reason for
producing the angle difference is that the salient-pole torques during steady-state
and transients have different signs, because usually xd ! xq and xdc xq . From
curves A and B we can also see that the ability for a machine to bear sudden
disturbance is much stronger than that for a machine to bear slow disturbance.
2.8
Electromagnetic Torque after 3-Phase Sudden
Short Circuit
Synchronous machines, after 3-phases are suddenly short-circuited, can produce
average and pulsating electromagnetic torque. Those torques are significant for
calculation of stresses in machine-parts and frames and analysis of operationstability. During analysis of those torques, it is tedious to solve strictly the basic
equations. If the approximate method like in analysis of 3-phase short circuit is
153
AC Machine Systems
used, i.e. the currents and flux-linkages are found when resistances are neglected
at first, then the corresponding time-constants are considered, and at last the
currents and flux-linkages are substituted into the electro-magnetic torque
formula, then the average torque is lost at all. However, the pulsating
electromagnetic torques obtained in this approximate method are sufficiently
accurate. In this section, we shall use the approximate method to analyse pulsating
electromagnetic torque components; as for the average electromagnetic torque,
another method is used to analyse it separately. In addition, we shall discuss only
the torque after 3-phase short circuit during no-load for convenience.
(1) Pulsating torque after 3-phase sudden short circuit
After 3-phases of a synchronous machine are suddenly short-circuited during
no-load, the currents id and iq , according to G 0, U E and equations (2.5.90)
and (2.5.91), can be written as
id
iq
t
t
t
½
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º
E Ta
«¨ ¸ e ¨ ¸ e » E e cos t °
xd »¼
xdcc
°
«¬© xdcc xdc ¹
© xdc xd ¹
¾
t
°
E Ta
e sin t
°
xqcc
¿
(2.8.1)
Because the machine before short circuit is not loaded, the initial flux-linkages
in phases a, b, c are
\ a0
\ b0
\ c0
E cos J 0
½
°
E cos(J 0 120e
)¾
E cos(J 0 120e
) °¿
(2.8.2)
The stator resistance neglected, from Park’s equations we can see that under steadystate condition of 3-phase short circuit the flux-linkages in phases a, b, c are
\ ay \ by \ cy
0
(2.8.3)
After 3-phase sudden short circuit, the stator flux-linkages will decay from
their initial values to zero. The decay time constant is Ta as mentioned before, and
the general expressions for flux-linkages in phases a, b, c are
154
\a
Ee
\b
Ee
\c
Ee
t
Ta
t
Ta
t
Ta
½
°
°
°
cos(J 0 120e
)¾
°
cos(J 0 120e
)°
°
¿
cos J 0
(2.8.4)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Transforming them into d, q components, we can get
\d
\q
t
Ta
½
cos t °
¾
t
°
Ta
Ee sin t ¿
Ee
(2.8.5)
Thus,
Te
iq\ d id\ q
2t
2t
E 2 Ta
E 2 Ta
e sin t cos t e sin t cos t
xqcc
xdcc
t
t
t
ª§ 1
§ 1
1 · 1 · 1º
«¨ ¸ e Tdcc ¨ ¸ e Tdc » E 2 e Ta sin t
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
t
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º 2 T
«¨ ¸ e d ¨ ¸ e d » E e a sin t
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
2t
1§ 1
1 · 2 Ta
¨ ¸ E e sin 2t
2 ¨© xdcc xqcc ¸¹
(2.8.6)
From the formula above we can see that when the stator and rotor resistances are
neglected, the electromagnetic torque after short circuit contains only the pulsating
torque components and has no average torque.
In addition, the pulsating torque components are the fundamental and second
harmonic components. The first is very large, whose initial maximum value is
inversely proportional to xdcc and decays slowly. The latter is small, especially in
turbogenerators, whose maximum value depends upon the difference between
xdcc and xqcc and is equal to zero when xdcc xqcc.
It is not difficult to understand that there are those torque components in a
synchronous machine after 3-phase sudden short circuit. The field produced by
stator fundamental current has a synchronous speed, the field caused by stator
aperiodic current is at a standstill and that produced by the stator second
harmonic current has double synchronous speed. The field produced by rotor
aperiodic current has synchronous speed and that caused by the rotor
fundamental current has two components whose speeds are double synchronous
speed and zero respectively. It is not difficult to know that the relative speeds
between those fields are zero, Z s and 2Z s and the fundamental and second
harmonic pulsating torque components can be produced. The interaction between
stator and rotor fields whose relative speed is zero will produce an average
torque, but the average torque is equal to zero because the stator and rotor
resistances have been neglected and their field-axes coincide with each other.
155
AC Machine Systems
The electromagnetic torque after 3-phase sudden short circuit is shown in
Fig. 2.8.1.
Figure 2.8.1 Pulsating torque after 3-phase sudden short circuit
(2) Average torque after 3-phase sudden short circuit
After 3-phases of a synchronous machine are suddenly short-circuited, in
stator winding there are the fundamental, second harmonic and aperiodic currents
and in rotor winding there are the fundamental and aperiodic currents, which is
the synthetic result when the aperiodic currents circulate through the stator and
rotor windings simultaneously. Because the fields produced by the two groups of
currents have different speeds, the average torque components corresponding to
them can be studied separately.
(a) The average torque component corresponding to the rotor aperiodic current
and stator fundamental current
The revolving fields produced by those currents have the synchronous speed
and constitute average torque. Because stator winding is short-circuited, the
average torque component is equal to the short circuit torque Tk referring to
equation (2.5.18), i.e. the copper loss caused by the fundamental current in stator
winding. That is natural, because under the condition of short circuit the energy
needed according to the copper loss can be obtained only through air-gap and the
value is equal to corresponding electromagnetic torque during synchronous speed.
The copper loss produced by the dc current in excitation winding is supplied by
an excitation source, so the loss can not produce average torque.
After the 3-phases of a synchronous machine are suddenly short-circuited, the
fundamental current in stator winding referring to equation (2.5.92) can be
written as
ia (50 )
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º
«¨ ¸ e ¨ ¸ e » E cos(t J 0 )
xd »¼
«¬© xdcc xdc ¹
© xdc xd ¹
(2.8.7)
According to equations (2.5.109), (2.5.108), (2.5.106) and (2.5.105), the aperiodic
current in rotor winding can be written out easily. If the decay process can be
considered as a series of steady-state conditions, according to the average torque
156
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
formula during 3-phase short circuit in steady-state, the average torque
component corresponding to the rotor aperiodic current and stator fundamental
current can be expressed as
2
Tecp (50 )
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º 2
«¨ ¸ e d ¨ ¸ e d » E r
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
(2.8.8)
It should be noted that after 3-phase sudden short circuit, in rotor winding
there is not only the dc current supplied by an excitation source but also the
aperiodic current changing with time which is produced in order to keep the rotor
flux-linkage constant. The copper losses caused by those currents in rotor
winding are provided with the excitation source and magnetic energy reserved in
rotor winding. Therefore, those copper losses can not produce average torque.
(b) The average torque components corresponding to the stator aperiodic and
second harmonic currents and rotor fundamental current
After a dc current circulates through stator winding, the fundamental current
will be produced in rotor winding. Because the rotor is electrically and magnetically
asymmetrical, the latter current will cause the second harmonic current in stator
winding. The revolving fields corresponding to those currents have double
synchronous speed and zero speed respectively, and a certain average torque will
be developed.
(i) So far as the stator second harmonic current is concerned, like the stator
fundamental current as mentioned before, it also causes the copper loss in stator
winding and produces the average torque component which is equal to the
resistance loss. That is because both are the short circuit torque, but the first is
caused by the revolving field corresponding to rotor dc current and the latter is
caused by the revolving field corresponding to rotor fundamental current.
After 3-phase sudden short circuit, the stator second harmonic current,
according to equation (2.5.92), can be written as
ia (100 )
t
E§ 1
1 · Ta
¨ ¸ e cos(2t J 0 )
2 ¨© xdcc xqcc ¸¹
(2.8.9)
If the decay process can be considered as a series of steady-state conditions, the
corresponding average electromagnetic torque component is
2
Tecp (100 )
2t
E2 § 1
1 · Ta
¨ ¸ e r
4 ¨© xdcc xqcc ¸¹
(2.8.10)
(ii) So far as the rotor fundamental current is concerned, it also causes the
copper loss in rotor winding and produces corresponding average torque. It can
be proved that the copper loss in rotor winding caused by the rotor fundamental
157
AC Machine Systems
current, when expressed in terms of the stator current components id and iq ,
1
[| id (50 ) |2 Rd 1 | iq (50 ) |2 Rq1 ], in which id (50 ) and iq (50 ) are the
2
fundamental components of currents id and iq , and Rd 1 and Rq1 are d-axis
is equal to
and q-axis equivalent resistances, i.e. the corresponding values in the following
expressions:
xd ( j )
xd ( p) | p
j
xq ( j )
xq ( p ) | p
j
X d 1 jRd 1 ½°
¾
X q1 jRq1 °¿
(2.8.11)
After 3-phase sudden short circuit, the fundamental components of currents
id and iq , according to equation (2.5.90), can be written as
id (50 )
iq (50 )
t
½
E Ta
e cos t °
xdcc
°
¾
t
E Ta
°
e sin t °
xqcc
¿
(2.8.12)
If the decay process can be considered as a series of steady-state conditions, the
corresponding average electromagnetic torque component is
Tecp (0 )
1
[| id (50 ) |2 Rd 1 | iq (50 ) |2 Rq1 ]
2
2
2
2t
2t
1 § E · Ta
1 § E · Ta
¨ ¸ e Rd 1 ¨¨ ¸¸ e Rq1
2 © xdcc ¹
2 © xqcc ¹
(2.8.13)
(iii) The stator aperiodic current also causes the copper loss in stator winding,
but the loss is provided with the magnetic energy reserved in stator winding and
can not produce average torque.
(c) The average electromagnetic torque after 3-phase sudden short circuit
According to the results obtained above, we can write the average torque after
short circuit as
Tecp
Tecp (50 ) Tecp (100 ) Tecp (0 )
2
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º 2
«¨ ¸ e ¨ ¸ e » E r
xd »¼
«¬© xdcc xdc ¹
© xdc xd ¹
2
2t
ª§ 1
2 Rd 1 2 Rq1 º E 2 Ta
1·
«
»
¨ ¸ r 2 2
e
xdcc
xqcc » 4
«¨© xdcc xqcc ¸¹
¬
¼
158
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
2
t
t
ª§ 1
1 · Tdcc § 1
1 · Tdc 1 º 2
e
e
» E r
Ǭ
¸
¨
¸
xd »¼
«¬© xdcc xdc ¹
© xdc xd ¹
2
2t
1§ 1
1·
1
1
ª
º ¨ ¸ E 2 «b 2 r (1 b)2 Rd 1 (1 b)2 Rq1 » e Ta
4 ©¨ xdcc xqcc ¹¸
2
2
¬
¼
(2.8.14)
in which
b
xqcc xdcc
xdcc xqcc
In practice, the copper loss and torque Tecp (100 ) caused by the stator second
harmonic current are very small and can be neglected. Therefore, the average
electromagnetic torque after short circuit can also be evaluated according to the
following formula:
Tecp
t
t
ª§ 1
1 · T cc § 1
1 · T c 1 º 2
«¨ ¸ e d ¨ ¸ e d » E r
xd »¼
© xdc xd ¹
¬«© xdcc xdc ¹
2t
1 2 § Rd 1 Rq1 · Ta
E ¨ 2 2 ¸e
2 ¨© xdcc
xqcc ¸¹
(2.8.15)
The analyses above show that the average torque Tecp after short circuit is equal
to the copper losses produced by the stator and rotor ac currents, and the copper
losses caused by the stator and rotor aperiodic currents are supplied by the
excitation source and magnetic energy reserved in the machine. The important
conclusion can be used not only to symmetrical short circuit but also to
asymmetrical short circuit.
At last, it should be noted that the instantaneous values of the currents and
flux-linkages, according to the state-variable equation, can be estimated by
digital computers, and the instantaneous values of the torque may also be
evaluated according to the torque formula Te iq\ d id\ q . By the numerical
method we can obtain the instantaneous values of the torque, but can not predict
the harmonic components of the torque.
2.9
Suddenly Opening 3-phase Short Circuit of
Synchronous Machines
It often appears that 3-phase short circuit of a synchronous machine is suddenly
opened. Analysing that problem, we can consider the 3-phase short circuit not to
be opened but a current to be suddenly injected into the stator which is equal but
159
AC Machine Systems
opposite to the original short-circuit current. If the machine has 3-phase short
circuit in steady-state before opened, the short circuit current, when stator
E
and iq 0 0. Therefore, the problem, that 3-phase
resistance neglected, is id 0
xd
short circuit is suddenly opened, can be considered as that the current
E
id 0 is suddenly injected into the stator. By Superposition Theorem the
xd
problem will be discussed as follows.
(1) Each quantity caused by the current (id 0 )
Because the flux-linkage equation of a synchronous machine is
\d
G ( p)U fd xd ( p )id
(2.9.1)
and that expression can be rewritten as
id
1
pG ( p ) § U fd ·
\d ¨
¸
xd ( p )
xd ( p ) © p ¹
(2.9.2)
from the equivalent circuit of d-axis operational reactance of a synchronous
machine shown in Fig. 2.4.6 we can see that the xd ( p ) in the formula above is
x ( p)
operational self-impedance and d
is operational mutual impedance. In
pG ( p )
addition, according to the Reciprocity Theorem we can see that when U fd 0,
i.e.
U fd
p
0 we have
I fd
pG ( p)
\d
xd ( p)
(2.9.3)
During this time there is
\d
xd ( p )id
I fd
pG ( p)id
so
(2.9.4)
As mentioned above, the problem that 3-phase short-circuit is suddenly
opened can be considered as that the current (id 0 ) is suddenly injected into the
stator, so we have
I fd
\d
160
pG ( p )id 0 ½°
¾
xd ( p)id 0 °¿
(2.9.5)
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
Having those expressions, we can solve the excitation current I fd and d-axis
flux-linkage \ d . According to two conditions the problem will be discussed as
follows.
(i) If there is no damping winding, we can get
2
ª
pxad
« xd R fd pX ffd
«¬
\d
º
» id 0
»¼
t
x2
̯ xd id 0 ad id 0 e Td 0
X ffd
id 0 [ xd ( xd xdc )e
t
Td 0
(2.9.6)
]
t
I fd
pxad
x
id 0 ̯ ad id 0 e Td 0
pX ffd R fd
X ffd
(2.9.7)
(ii) If there is a damping winding, we have
\d
2
3
2
2
2
ª
2 xad
X ffd xad
R fd xad
p 2 ( X 11d xad
) p( R1d xad
) º
x
« d
» id 0
2
2
p ( X 11d X ffd xad ) p( X 11d R fd X ffd R1d ) R1d R fd ¼»
¬«
(2.9.8)
For convenience, we shall use the same approximate method as in Section 2.5.
(a) Instantaneous values of \ d
Supposing R fd R1d 0 or p f in equation (2.9.8), we have
Supposing R fd
Supposing p
xdccid 0
½
°
0 and R1d f, we can get °
°°
\ d xdc id 0
¾
°
0, we can obtain
°
°
\ d xd id 0
°¿
\d
(2.9.9)
(b) Decay time constants of \ d
According to equation (2.9.8), its characteristic equation is
2
p 2 ( X 11d X ffd xad
) p ( X 11d R fd X ffd R1d ) R1d R fd
For convenience, taking approximately R fd
0
(2.9.10)
0, we have
2
p ( X 11d X ffd xad
) X ffd R1d
0
and the corresponding time constant is
161
AC Machine Systems
X 11d Tdcc0
2
xad
X ffd
(2.9.11)
R1d
f, we have
Supposing R1d
pX ffd R fd
0
and the corresponding time constant is
X ffd
Td 0
(2.9.12)
R fd
(c) The general expression of \ d caused by the current (id 0 ) is
\d
id 0 [ xd ( xd xdc )e
t
Td 0
( xdc xdcc )e
t
Tdcc0
]
(2.9.13)
(d) The excitation-winding current caused by the current (id 0 ) :
According to equation (2.9.5), we have
I fd
2
p 2 ( X 11d xad xad
) pxad R1d
id 0
2
2
p ( X 11d X ffd xad ) p( X 11d R fd X ffd R1d ) R1d R fd
Supposing R1d
R fd
0 or p
I fd
Supposing R1d
f and R fd
(2.9.14)
f in equation (2.9.14), we can obtain
2
X 11d xad xad
i
2 d0
X 11d X ffd xad
(2.9.15)
0 in equation (2.9.14), we can get
I fd
xad
id 0
X ffd
(2.9.16)
If p 0 in equation (2.9.14), then exists I fd 0.
The time constant of I fd is the same as that of \ d because their characteristic
equations are similar, so we can obtain
I fd
t
2
ªx
§ x
X x xad
id 0 « ad e Td 0 ¨ ad 11d ad
2
¨X
«¬ X ffd
© ffd X 11d X ffd xad
· Ttcc º
¸¸ e d 0 »
»¼
¹
(2.9.17)
(e) The d-axis damper current I1d caused by the current (id 0 ) :
The quantity can be obtained by using the same method as in Section 2.5. Its
expression is omitted here.
162
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
(2) The original short-circuit values
Neglecting stator resistance r, we have
id 0
id
iq
0
\d
uq
\q
u d
I fd
I fd 0
E
xd
0
0
U fd
R fd
xd
id 0
xad
(2.9.18)
(3) Various actual values when 3-phase short circuit is suddenly opened
The actual values of stator current are
in addition
id
id 0 id 0
iq
0
\q
0
0½
°
¾
°
¿
(2.9.19)
(a) When there is only excitation winding on rotor, we have
\d
t
ª
º
id 0 « xd ( xd xdc )e Td 0 »
«¬
»¼
I fd
I fd 0 t
xad
id 0 e Td 0
X ffd
(2.9.20)
ª x xdc Tt º
e d0 »
I fd 0 «1 d
x
d
¬«
¼»
t
ud
p\
uq
\d
d
1
( xd xdc )e Td 0
id 0
Td 0
E
Td 0
ª xdc Tt º
«1 e d 0 »
«¬ xd
»¼
t
ª
º
id 0 « xd ( xd xdc )e Td 0 »
«¬
»¼
t
ª §
xc · º
E «1 ¨1 d ¸ e Td 0 »
xd ¹
«¬ ©
»¼
(2.9.21)
(2.9.22)
(2.9.23)
in which ud is the transformer emf and uq is the speed emf.
Because ud uq , the amplitude of phase emf is
163
AC Machine Systems
|u|
2
d
2
q
u u | uq
t
ª §
xdc · Td 0 º
E «1 ¨1 ¸ e »
xd ¹
«¬ ©
»¼
(2.9.24)
According to equations (2.9.20) and (2.9.23), the curves of | u | and I fd are
shown in Fig. 2.9.1.
Figure 2.9.1 I fd - t curve and | u |- t curve after opening short circuit of
synchronous machines
From the results above we can see that after 3-phase short circuit is suddenly
opened, the stator current and armature reaction are decreased abruptly to zero.
Therefore, \ fd and the stator voltage are to increase suddenly, but \ fd can not
increase suddenly, so the stator voltage can not be increased abruptly to its
steady-state value. In the meantime, there is a negative aperiodic current in
excitation winding to keep its flux-linkage not changing abruptly, so
xc
the excitation current I fd is decreased suddenly to d I fd 0 . However, the decrease
xd
of I fd results in the decrease of leakage flux in excitation winding, so the mutual
flux-linkage still increases suddenly and the corresponding stator voltage will
xc
rise suddenly from zero to d E. In addition, because the negative aperiodic
xd
current in excitation winding will decay to zero according to the time constant
Td 0 , the corresponding stator voltage and \ d will rise to their steady-state values
according to that time constant.
(b) When there is not only an excitation winding but also a damping winding
on the rotor, we can get
| u || \ d
I fd
164
t
t
ª
º
Td 0
Tdcc0
c
c
cc
( xd xd )e »
id 0 « xd ( xd xd )e
«¬
»¼
t
2
ªx
§ x
x
X x xad
id 0 « d ad e Td 0 ¨ ad 11d ad
2
¨X
© ffd X 11d X ffd xad
¬« xad X ffd
· Ttcc º
¸¸ e d 0 »
¹
¼»
2 Analysis of Several Operating Conditions of Synchronous Machines at Synchronous Speed
It should be noted that when t
0 we have | u || \ d
id 0 xdcc
xdcc
E. Compared
xd
with the machine without damping windings, its value decreases by a factor
§ xcc
xc
xdcc ·
of ¨ d E d E
¸ , i.e. after 3-phase short circuit is suddenly opened, the
xd
xdc ¹
© xd
terminal voltage with damping windings is lower than without damping windings,
which has a benefit to quenching electric arc of the switch-gear.
References
[1] Gao J D (1963) AC machine transients and operating modes analysis (Chapter 2, 3, 4, in
Chinese). Science Press, Beijing
[2] Gao J D, Zhang L Z (1982) Fundamental theory and analysis mothods of transients in
electric machines, Vol 1 (Chapter 1, 4, 5, in Chinese). Science Press, Beijing
[3] Gao J D, Wang X H, Li F H (2004) Analysis of ac machines and their systems, 2nd Ed
(Chapter 2, in Chinese). Tsinghua University Press, Beijing
[4] Jackson W B, Winchester R L (1969) Direct- and quadrature-axis equivalent circuits for
solid-rotor turbine generators. J IEEE Trans. PAS 88(7): 1121 1136
[5] Randin A W (1945) Per unit impedances of salient-pole synchronous machines. J AIEE
Trans. Aug. Part I: 569 573, Dec. Part Ċ: 839 841
[6] Randin A W (1945) The direct- and quadrature-axis equivalent circuits of the synchronous
machine. J AIEE Trans. Aug. Part I: 569 573, Dec. Part Ċ: 867 868
165
3
Some Special Operation Modes of Synchronous
Machine Systems at Constant Speed and Variable
Speed
Abstract As is stated in Chapter 2, by use of d, q, 0 axes can get the
differential equations with constant coefficients for symmetrical faults of
synchronous machines at synchronous speed, but can’t for unsymmetrical
faults or variable speed. Therefore, in this chapter it is necessary to introduce
other reference axes systems first, such as D , E , 0 axes; 1, 2, 0 axes;
d c , qc , 0 axes; F, B, 0 axes; Fc , Bc , 0 axes. Then use D , E , 0 axes to analyse
two phases sudden short circuit of synchronous machines, inclusive of stator
and rotor loop parameters and currents, overvoltage, pulsating and average
torques. Furthermore, use 1, 2, 0 axes to study asynchronous operation of
synchronous machines to get corresponding equivalent circuits, currents,
torques and negative sequence reactance that depends upon the operating
modes. In a similar way, analyse asynchronously starting of synchronous
motors to get starting torque and pull-in one and suggest the optimal time
for applying a dc voltage to excitation winding. Analysis of combination
system for synchronous machine and capacitances is also an interesting
topic, in which by use of Herwitz’s, Routh’s and Mihainov’s criterions
determine the salient-pole synchronous self-excitation region, asynchronous
and repulsion synchronous self-excitation regions, explain the mechanism to
cause those self-excitation regions, and illustrate why the sub-synchronous
resonance occurs. Cycloconverters can change the source frequency to lower
frequency that is under 1/3 1/2 of source frequency, and cycloconverter-fed
synchronous motor systems with field-oriented control are widely used to
drive slow speed and large capacity production machines because of their
some advantages over the ac-dc-ac frequency converter. Therefore, their
mathematic patterns and performance analysis belong to another important
topic. Taking a steel-mill driven by a 4 000kW-cycloconverter-fed synchronous
motor as example, analyse the steady-state and transient performance of the
cycloconverter-fed synchronous motor system, simulative and experimental
results being in accord with each other and thus giving a suggestion that is
how to dispose damping bars. Some special power supply systems, such as
those used on ships, tractions, and aircraft, are often required to provide
high quality ac and dc electricity simultaneously, so analysis of synchronous
3 Some Special Operation Modes of Synchronous Machine Systems
generators with ac and dc stator connections is also distinct. The ac-dc
generator has five suits of independent star-connected 3-phase windings in
its stator, one is called ac winding to supply ac power and the others belong
to dc windings, which are connected to four 3-phase bridge rectifiers
respectively, and then the dc output sides of the four 3-phase bridge rectifiers
are in parallel. By use of multi-loop model analyse the ac-dc generator to
compared with experimental data, both results being in accord with each other.
In Chapter 2, we have studied some symmetrical operation modes of synchronous
machines by use of d, q, 0 axes. It is evident that analysis of several problems in
d, q, 0 axes is convenient especially when the stator connection of a synchronous
machine is symmetrical. However, the reference axes system is not more
convenient than other ones when asymmetrical operation modes of synchronous
machines are researched on. Therefore, with progress in science and technology
some new methods are developed for study of transients of synchronous
machines. With regard to axes transformation referring to [2,3], there are the
following reference axes systems to be used practically:
(i) The stationary reference axes systems that lie on the stator, namely a, b, c
axes, D , E , 0 axes and 1, 2, 0 axes.
(ii) The revolving reference axes systems that are put on the rotor and rotate
with it, namely d, q, 0 axes and F, B, 0 axes.
(iii) The synchronous-speed reference axes systems that rotate at synchronous
speed, namely d c , qc , 0 axes and Fc , Bc , 0 axes.
Those reference axes systems have their advantages and shortcomings. The
choice of either this reference axes system or that one depends upon the
following conditions:
(i) The needed accuracy and convenience of calculation.
(ii) The conditions of studied problems, such as steady-state or transient,
symmetrical or asymmetrical, speed-changing or constant, accelerative or oscillatory,
and so forth.
(iii) The used tools, such as mathematical analysis or experiment simulation,
digital computers, analogue computers or physical simulation, etc.
The same problem can often be solved by several methods, some of which are
used conveniently or by which more accurate results can be obtained and the
others are more tedious or not precise enough. Similarly, the same method is
used easily under some condition, but it is complicated on other conditions.
Therefore, the researchers should grasp various axes transformation methods and
use them very well according to the concrete condition. Generally speaking, if
167
AC Machine Systems
the stator or rotor is symmetrical and the other is asymmetrical, it is more
suitable to put the reference axes on the asymmetrical part. For example, when
the rotor of a synchronous machine is asymmetrical and the stator is symmetrical,
it is more convenient to make use of d, q, 0 axes or F, B, 0 axes. Moreover, the
rotor of an induction machine is normally symmetrical and it is more suitable to
take D , E , 0 axes or 1, 2, 0 axes when the operation problem about the
asymmetrical stator is studied. If both the stator and rotor are asymmetrical, the
basic equations of the machine expressed by any reference axes system are all
equations with variable coefficients even though the machine speed is constant.
Owing to the causes above, we shall discuss other reference axes systems in
addition to d, q, 0 axes and the transformation relations between them in this
chapter. In the meantime, we shall research into some special operation problems
of synchronous machines further by using some of the reference axes systems
mentioned above.
3.1
Transformations of Reference Axes Systems and Their
Formulas in Synchronous Machines
The transformation of reference axes is that of variables, i.e. a set of variables is
replaced by another set. Letting the original set of variables be
x1 , x2 , x3 ," , xn
and supposing
ª y1 º
«y »
« 2»
« y3 »
« »
«#»
«¬ yn »¼
ªM11
«M
« 21
«M31
«
« #
«¬M n1
M12 M13 " M1n º ª x1 º
M 22 M 23 " M 2 n »» «« x2 »»
M32 M33 " M3n » « x3 »
#
#
M n 2 M n3
»« »
# »« # »
" M nn »¼ «¬ xn »¼
(3.1.1)
or
Y
CX
we can get a new group of variables y1 , y2 , y3 " , yn , in which the element M in
the transformation matrix C is called the transformation coefficient which can be
taken as a constant (real or complex number) or a function of time. In the linear
transformation, M is independent of variables x and y.
From Linear Algebra we can see that in order to fulfil the transformation and
168
3 Some Special Operation Modes of Synchronous Machine Systems
have a single-value relation between the new and original variables, the
determinant corresponding to the transformation matrix C in equation (3.1.1)
must suffice the following condition,
M11 M12 M13 " M1n
M 21 M 22 M 23 " M 2 n
| C | M31 M 32 M33 " M 3n z 0
#
#
#
(3.1.2)
#
M n1 M n 2 M n3 " M nn
There are many methods to choose the transformation coefficients to suffice
equation (3.1.2). For example, the variables in a, b, c axes of a 3-phase
synchronous machine taken as original variables, the new variables after
transformation have various types. d, q, 0 axes mentioned before are an example,
whose transformation matrix is
C
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
¬«M31 M 32 M 33 »¼
2
2
ª2
º
cos(J 120e
)
cos(J 120e
)»
« 3 cos J
3
3
«
»
2
2
2
« sin J sin(J 120e
) sin(J 120e
)»
« 3
»
3
3
«
»
1
1
« 1
»
3
3
¬« 3
¼»
Obviously, | C |z 0.
It depends upon the actual need and mathematical convenience how to choose
the transformation matrix and corresponding reference axes. In practice, there are
already several transformation models. In the following we shall introduce other
reference axes except d, q, 0 axes. It should be pointed out that in various
reference axes the transformation coefficients M 31 , M 32 , M 33 are equal to
1
M31 M32 M33
, i. e. the 0-axis component has the same form in various
3
reference axes. It is also noted that in study of transient processes the three
variables — current, voltage and flux-linkage must all be transformed but they
have the similar transformation formula. Therefore, in the following we shall
take only the current as an example.
(1) D , E , 0 axes
In the reference axes, the corresponding transformation matrix is
169
AC Machine Systems
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
«¬M31 M 32 M 33 »¼
C
ª2
«3
«
«0
«
«
«1
«¬ 3
1
3
1
3
1
3
2
2
ª2
º
)
cos(0e 120e
)»
« 3 cos 0e 3 cos(0e 120e
3
«
»
2
« 2 sin 0e 2 sin(0e 120e
) sin(0e 120e
)»
« 3
»
3
3
«
»
1
1
1
«
»
«¬
»¼
3
3
3
º
»
»
»
3»
»
1 »
3 »¼
1
3
1
(3.1.3)
Taking the current as an example and substituting equation (3.1.3) into (3.1.1),
we can get
iD
iE
i0
i i ·½
2§
ia b c ¸ °
¨
3©
2 ¹°
°
1
(ib ic ) ¾
3
°
°
1
(ia ib ic ) °
3
¿
(3.1.4)
½
°
°
1
3
°
iE i0 ¾
iD 2
2
°
°
1
3
iE i0 °
iD 2
2
¿
(3.1.5)
The inverse-transformation is
ia
ib
ic
iD i0
The relation between D , E components and d, q components is
iD
iE
id cos J iq sin J ½°
¾
id sin J iq cos J °¿
(3.1.6)
iD cos J iE sin J ½°
¾
iD sin J iE cos J °¿
(3.1.7)
or
id
iq
170
3 Some Special Operation Modes of Synchronous Machine Systems
in which J is the included angle between d-axis and D-axis or a-axis.
In D , E , 0 axes, the formulas of the flux-linkage, voltage, out-put power and
electromagnetic torque are
\D
\E
[cos J xd ( p ) cos J sin J xq ( p )sin J ]iD
½
°
[cos J xd ( p)sin J sin J xq ( p ) cos J ]iE cos J G ( p )u fd °
¾
[sin J xd ( p) cos J cos J xq ( p )sin J ]iD
°
[sin J xd ( p )sin J cos J xq ( p ) cos J ]iE sin J G ( p)u fd °¿
(3.1.8)
p\ D riD ½°
¾
p\ E riE °¿
(3.1.9)
uD iD uE iE 2u0 i0
(3.1.10)
uD
uE
P
Te \ D iE \ E iD
(3.1.11)
It is noted that the flux-linkage equation (3.1.8) and votage equation (3.1.9) are
the expressions described by Heaviside’s Operation, so the arrangement order of
various physical quantities in the expressions can not be exchanged arbitrarily, i.e.
the operational reactances xd ( p ), xq ( p ) and operational conductance G ( p ) don’t
act on cos J or sin J before them but act on the functions of time written after
them referring to Appendix D in this book. It is not difficult to understand that
the flux-linkages \ D , \ E are some complicated functions of currents iD and iE .
However, the rotor resistance neglected, the equations above can be much
reduced and used easily.
Studying d, q, 0 axes, we have mentioned that the currents id and iq represent
the stator d-axis and q-axis mmfs which are observed from the reference axes
that rotate with the rotor. Discussing D , E , 0 axes now, from equations (3.1.3)
and (3.1.4) we can know that the currents iD and iE represent the stator D-axis
and E -axis mmfs respectively, in which D-axis coincides with the stator a-axis
S
and E -axis leads ahead of D-axis by an angle of
referring to Fig. 3.1.1, so those
axes are static ones relative to the stator. In addition, id , iq , \ d , \ q and ud , uq in
d, q, 0 axes represent the currents, flux-linkages and voltages in the fictitious
circuits of the stator d-axis and q-axis respectively. Similarly, iD , iE , \ D , \ E and
uD , u E in D , E , 0 axes represent the currents, flux-linkages and voltages in the
fictitious circuits of the stator D-axis and E -axis referring to Fig. 3.1.1. Thus, we
171
AC Machine Systems
have another physical model of synchronous machines which is an equivalent
2-phase synchronous machine.
Figure 3.1.1 D , E , 0 pattern of synchronous machines
From equation (3.1.9) we can see that there is on speed emf in D , E , 0 axes.
That is not difficult to understand, because D , E , 0 axes are static ones and
\ D , \ E don’t rotate relatively to the stator.
(2) 1, 2, 0 axes
The transformation matrix in the reference axes is
C
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
«¬M 31 M 32 M 33 »¼
ª1
«3
«
«1
«3
«
«1
«¬ 3
1
a
3
1 2
a
3
1
3
1 2º
a
3 »
»
1 »
a
3 »
»
1 »
3 »¼
(3.1.12)
1
3
e j120e j
is the complex operator.
2
2
Taking the current as an example and substituting equation (3.1.12) into (3.1.1),
we can obtain
in which a
i1
i2
i0
1
½
(ia aib a 2 ic ) °
3
°
1
°
(ia a 2 ib aic ) ¾
3
°
°
1
(ia ib ic ) °
3
¿
(3.1.13)
It is not difficult to understand that after transformation the original actual
instantaneous currents ia , ib and ic become the instantaneous currents i1 and i2
172
3 Some Special Operation Modes of Synchronous Machine Systems
expressed by complex-number, so the reference axes are sometimes called the
complex-number axes, which are motionless relatively to the stator.
The inverse-transformation is
ia
ib
ic
i0 i1 i2
½
°
i0 a i1 ai2 ¾
°
i0 ai1 a 2 i2 ¿
2
(3.1.14)
From the formulas above we can see that the basic formulas of the complexnumber axes look like those of the ordinary Symmetrical Components Method,
but both are essentially different. In complex-number axes the corresponding
variables are complex-number instantaneous values, but in ordinary Symmetrical
Components the corresponding variables are complex-number vectors representing
the quantities changing sinusoidally with time. In ordinary Symmetrical Components
the operator a e j120e means that the time-vector rotates by an angle of 120e, but
in complex-number axes the operator a e j120e sometimes means that the spacevector rotates by an angle of 120e. Besides, 1, 2 components in complexnumber axes are always conjugate to each other, but in Symmetrical Components
Method the positive and negative components have not the same relation as
above.
The relations among 1,2 components, D , E components and d, q components are
i1
i2
1
½
(iD jiE ) °
°
2
¾
1
(iD jiE ) °
°¿
2
(3.1.15)
or
iD
iE
i1
i2
i1 i2
½°
¾
j(i1 i2 ) °¿
(3.1.16)
1
½
(id jiq )e jJ °
°
2
¾
1
jJ °
(id jiq )e
°¿
2
(3.1.17)
or
id
iq
i1e jJ i2 e jJ
j(i1e
jJ
½°
¾
i2 e ) °¿
jJ
(3.1.18)
173
AC Machine Systems
In 1, 2, 0 axes, the formulas of flux-linkage, voltage, output power and
electromagnetic torque are
\1
\2
1 jJ
½
e G ( p )u fd e jJ xs ( p )e jJ i1 e jJ xD ( p )e jJ i2 °
°
2
¾
1 jJ
e G ( p )u fd e jJ xs ( p )e jJ i2 e jJ xD ( p )e jJ i1 °
°¿
2
(3.1.19)
in which
1
[ xd ( p ) xq ( p )]
2
xs ( p )
is the average operational reactance and
1
[ xd ( p ) xq ( p)]
2
xD ( p )
is the half-difference operational reactance.
p\ 1 ri1 ½
¾
p\ 2 ri2 ¿
(3.1.20)
2(u2 i1 u1i2 u0 i0 )
(3.1.21)
j2(\ 1i2 \ 2 i1 )
(3.1.22)
u1
u2
P
Te
It should be noted that the flux-linkage formula (3.1.19) is still expressed by
Heaviside’s Operation, so the arrangement order of various quantities in the
expressions can not be exchanged arbitrarily.
When the speed of a machine is constant, i. e. the angular velocity Z is
constant, according to Heaviside’s Shift Theorem the equation (3.1.19) can be
rewritten as the following simple form:
\1
\2
1 jJ
½
e G ( p )u fd xs ( p jZ )i1 e j2J xD ( p jZ )i2 °
°
2
¾
1 jJ
e G ( p )u fd xs ( p jZ )i2 e j2J xD ( p jZ )i1 °
°¿
2
(3.1.23)
Besides simplicity mentioned above, there is a property that 1, 2 components
are always conjugate to each other. we can use the property to check whether
the calculation results are correct. When xd ( p ) xq ( p ), 1, 2 components are
independent of each other, so it is sufficient to calculate any one only. The
induction machine having a symmetrical rotor falls into that condition. In
practice, it is convenient to use 1, 2 components simultaneously.
174
3 Some Special Operation Modes of Synchronous Machine Systems
(3) d c , qc , 0 axes
The transformation matrix in the reference axes is
C
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
¬«M31 M 32 M 33 »¼
2
ª2
)
« 3 cos(t J 0c ) 3 cos(t J 0c 120e
«
« 2 sin(t J c ) 2 sin(t J c 120e
)
0
0
«3
3
«
1
1
«
3
3
¬«
2
º
cos(t J 0c 120e
)»
3
»
2
sin(t J 0c 120e
)»
»
3
»
1
»
»¼
3
(3.1.24)
Taking the current as an example and substituting equation (3.1.24) into (3.1.1),
we can get
idc
iqc
i0
2
½
[ia cos(t J 0c ) ib cos(t J 0c 120e
) ic cos(t J 0c 120e
)] °
3
°
2
°
) ic sin(t J 0c 120e
)]¾ (3.1.25)
[ia sin(t J 0c ) ib sin(t J 0c 120e
3
°
1
°
[ia ib ic ]
°
3
¿
The inverse-transformation is
ia
ib
ic
idc cos(t J 0c ) iqc sin(t J 0c ) i0
½
°
idc cos(t J 0c 120e
) iqc sin(t J 0c 120e
) i0 ¾
°
idc cos(t J 0c 120e
) iqc sin(t J 0c 120e
) i0 ¿
(3.1.26)
The relations between d c , qc and d , q components are
idc
iqc
id cos 'J iq sin 'J ½°
¾
id sin 'J iq cos 'J °¿
(3.1.27)
idc cos 'J iqc sin 'J ½°
¾
idc sin 'J iqc cos 'J °¿
(3.1.28)
or
id
iq
in which
'J
t
³ 'Z dt J
0
0
J 0c
J (t J 0c )
175
AC Machine Systems
and
'Z
Z 1
is the difference between the actual speed and synchronous one.
When Z 1, then 'Z 0 and 'J J 0 J 0c , so idc , iqc are the same as id , iq
except that there is an angle of (J 0 J 0c ) between them. That is natural, because
d c , qc , 0 axes are d, q, 0 axes during synchronous speed.
In d c , qc , 0 axes, the formulas of flux-linkage, voltage, output power and
electromagnetic torque are
\ dc
\ qc
[cos 'J xd ( p ) cos 'J sin 'J xq ( p )sin 'J ]idc ½
°
[cos 'J xd ( p )sin 'J sin 'J xq ( p) cos 'J ]iqc °
°
cos 'J G ( p )u fd
°
¾
[sin 'J xd ( p) cos 'J cos 'J xq ( p)sin 'J ]idc °
[sin 'J xd ( p )sin 'J cos 'J xq ( p) cos 'J ]iqc °
°
°¿
sin 'J G ( p )u fd
(3.1.29)
p\ dc \ qc ridc ½°
¾
p\ qc \ dc riqc °¿
(3.1.30)
P
udc idc uqc iqc 2u0 i0
(3.1.31)
Te
\ dc iqc \ qc idc
(3.1.32)
udc
uqc
(4) F, B, 0 axes
The transformation matrix in the reference axes is
C
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
«¬M31 M 32 M 33 »¼
ª
«
«
«
«
«
«
«
«¬
176
2 jJ
e
3
2 jJ
e
3
1
3
2 jJ
ae
3
2 2 jJ
ae
3
1
3
2 2 jJ º
ae »
3
»
2 jJ »
ae »
3
»
»
1
»
3
»¼
(3.1.33)
3 Some Special Operation Modes of Synchronous Machine Systems
Taking the current as an example and substituting equation (3.1.33) into (3.1.1),
we can get
iF
2
(ia aib a 2 ic )e jJ
3
iB
2
(ia a 2 ib aic )e jJ
3
i0
1
(ia ib ic )
3
½
2i1e jJ °
°
°°
2i2 e jJ ¾
°
°
°
°¿
(3.1.34)
The inverse-transformation is
ia
ib
ic
1
2
1
2
1
2
½
°
°
°
2i0 ) ¾
°
°
2i0 ) °
¿
(iF e jJ iB e jJ 2i0 )
(a 2 iF e jJ aiB e jJ
(aiF e jJ a 2 iB e jJ
(3.1.35)
The relations between F, B and d, q components are
iF
iB
id
iq
1
½
(id jiq ) °
°
2
¾
1
(id jiq ) °
2
¿°
1
½
°°
¾
1
j
(iF iB ) °
°¿
2
2
(3.1.36)
(iF iB )
(3.1.37)
It is not difficult to understand that the original instantaneous currents ia , ib
and ic have been changed into the instantaneous currents expressed by complexnumbers iF and iB . Therefore, the components are also complex-number
components. However, their axes are stationary not relatively to the stator but to
rotor. Thus, discriminating them from 1, 2 components, we sometimes call them
the forward and backward components.
In F, B, 0 axes, the formulas of flux-linkage, voltage, output power and
electromagnetic torque can be written as
177
AC Machine Systems
\F
\B
1
½
G ( p)u fd xs ( p )iF xD ( p )iB °
°
2
¾
1
G ( p)u fd xD ( p )iF xs ( p )iB °
°¿
2
(3.1.38)
p\ F jZ\ F riF ½
¾
p\ B jZ\ B riB ¿
(3.1.39)
u F iB uB iF 2u0 i0
(3.1.40)
j(iB\ F iF\ B )
(3.1.41)
uF
uB
P
Te
(5) Fc , Bc , 0 axes
The transformation matrix in the reference axes is
ªM11 M12 M13 º
«M
»
« 21 M 22 M 23 »
¬«M 31 M 32 M 33 »¼
C
ª
«
«
«
«
«
«
«
«¬
2 j(t J 0 )
e
3
2 j(t J 0 )
e
3
1
3
2 j(t J 0 )
ae
3
2 2 j(t J 0 )
ae
3
1
3
2 2 j(t J 0 ) º
ae
»
3
»
2 j(t J 0 ) »
ae
»
3
»
»
1
»
3
»¼
(3.1.42)
Taking the current as an example and substituting equation (3.1.42) into (3.3.1),
we can get
iFc
2
(ia aib a 2 ic )e j( t J 0 )
3
iBc
2
(ia a 2 ib aic )e j( t J 0 )
3
i0
1
(ia ib ic )
3
½
2i1e j(t J 0 ) °
°
°
j( t J 0 ) °
2i2 e
¾
°
°
°
°¿
(3.1.43)
They are also complex-number components, but their axes rotate at a synchronous
speed.
178
3 Some Special Operation Modes of Synchronous Machine Systems
The inverse-transformation is
ia
ib
ic
½
°
2
°
1 2
°
j( t J 0 )
j( t J 0 )
aiBc e
2i0 ) ¾
(a iFc e
2
°
°
1
(aiFc e j( t J 0 ) a 2 iBc e j( t J 0 ) 2i0 ) °
2
¿
1
(iFc e j( t J 0 ) iBc e j(t J 0 ) 2i0 )
(3.1.44)
The relations between Fc , Bc and d c , qc components are
iFc
iBc
1
½
(idc jiqc ) °
°
2
¾
1
(idc jiqc ) °
°¿
2
(3.1.45)
or
idc
iqc
1
½
(iFc iBc ) °
°
2
¾
1
j
(iFc iBc ) °
°¿
2
(3.1.46)
In Fc , Bc , 0 axes, the formulas of the flux-linkage, voltage, output power and
electromagnetic torque are
\ Fc
\ Bc
1 j'J
e G ( p )u fd e j'J xs ( p )e j'J iFc
2
e j'J xD ( p )e j'J iBc
½
°
°
°°
¾
1 j'J
e G ( p )u fd e j'J xs ( p )e j'J iBc °
°
2
°
j'J
j'J
e xD ( p )e iFc
°¿
(3.1.47)
( p j)\ Fc riFc ½
¾
( p j)\ Bc riBc ¿
(3.1.48)
u Fc iBc uBc iFc 2u0 i0
(3.1.49)
uFc
uBc
P
Te
j(iBc\ Fc iFc\ Bc )
(3.1.50)
179
AC Machine Systems
3.2
2-Phases Sudden Short Circuit of Synchronous
Machines
When a 3-phase synchronous machine operates symmetrically in steady-state, its
many targets such as voltage, efficiency and torque are all good. Therefore, many
synchronous machines are designed and used according to the operation
condition. However, asymmetrical operation always arises, for example, unbalanced
loads — electric trains, two phases short circuit and one phase open circuit, etc. So
it is significant to analyse asymmetrical operation modes of synchronous machines.
In this section, we shall discuss the transient process after two phases of a
synchronous machine are suddenly short-circuited, in order to grasp the analysis
method of asymmetrical operation modes of synchronous machines.
As mentioned before, when either the stator or rotor is asymmetrical, it is
convenient to put the reference axes on the asymmetrical part. For example,
when studying 3-phase sudden short circuit of synchronous machines, we shall
use d, q, 0 axes, because the stator is symmetrical and rotor is asymmetrical. The
analysis result shows that the basic equations of synchronous machines have
constant coefficients when rotor speed is constant. However, when both stator
and rotor are asymmetrical, the constant coefficient equations can not be obtained
whether we put the reference axes on the stator or rotor. Under this condition, the
equations can be solved only approximately. But when analysing two phases sudden
short circuit of synchronous machines, we can use D , E , 0 axes conveniently if
neglecting the stator and rotor resistances refferring to [2,3,5,6,9]. Now we use
the reference axes to analyse two phases sudden short circuit of synchronous
machines under no-load condition if rotor speed remains synchronous.
After two phases are short-circuited, referring to Fig. 3.2.1 there is
ub
uc
ia
0
ib ic
½
°
¾
0 °¿
(3.2.1)
When transforming them into D , E , 0 components, we obtain
iD
i0
uE
0½
°
0 ¾
0 °¿
(3.2.2)
After substituting them into equations (3.1.9) and (3.1.8), we have
uD
0
180
p\ D
½°
¾
p\ E riE °¿
(3.2.3)
3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.2.1 Two phases short circuit of synchronous machines
\D
\E
[cos J xd ( p )sin J sin J xq ( p ) cos J ]iE cos J G ( p)u fd ½°
¾
[sin J xd ( p )sin J cos J xq ( p ) cos J ]iE sin J G ( p )u fd °¿
(3.2.4)
where J t J 0 ; J 0 is an angle by which the rotor direct axis leads ahead of the
stator phase a axis when t 0.
It is complicated to solve these equations directly. In practice, they can be
solved by the following approximate method and the calculation results are
sufficiently accurate.
(1) Steady-state values of currents and voltages
When two phases of a synchronous machine are suddenly short-circuited,
corresponding aperiodic currents are induced in the stator and rotor loops in
order that the flux-linkages of those loops don’t change abruptly. The stator
aperiodic currents produce a standstill magnetic field. In the rotor winding a
basic-frequency current is induced by that magnetic field because the rotor
revolves at a synchronous speed. Since the rotor is asymmetrical, the rotor
basic-frequency current produces a pulsating magnetic field relative to the rotor.
If the pulsating field is divided into two revolving fields — the positive revolving
field and negative revolving field, the negative revolving field is stationary
relatively to stator and the positive one has twice the synchronous speed.
Therefore, the voltage and current of twice the basic frequency are induced in
stator winding. As a result of the asymmetrical stator, the pulsating field of twice
the basic frequency produced by the stator current is also divided into the
positive and negative revolving fields at twice the synchronous speed. The
positive revolving field has a synchronous speed relatively to rotor and the
basic-frequency voltage is still induced in rotor by the field; besides, the negative
revolving field has triple the synchronous speed with respect to rotor and the
current of triple the basic frequency is induced in the asymmetrical rotor winding.
In doing so over and over again, at last, there are a series of even harmonics
currents in stator winding and odd harmonics currents in rotor winding referring
to Fig. 3.2.2(a).
In addition, the rotor aperiodic current, inclusive of the excitation current, has
the same condition; i. e. it causes the basic-frequency current in stator and the stator
basic-frequency current induces a rotor current of twice the basic-frequency due
181
AC Machine Systems
to asymmetrical stator. The rotor current of twice the basic-frequency also causes
the stator current of triple the basic-frequency as a result of asymmetrical rotor.
In doing so repeatedly, at last, there are a series of odd harmonics currents in
stator winding and even harmonics currents in rotor winding referring to
Fig. 3.2.2(b).
Figure 3.2.2 Stator and rotor currents of various frequencies during two phases
sudden short circuit
When two phases short circuit reaches steady-state, the stator aperiodic current
equals zero and the rotor aperiodic current becomes the constant excitation current
u fd
. Therefore, a series of harmonic currents corresponding to the stator
I fdo
R fd
aperiodic current become zero during steady-state and a series of harmonic currents
corresponding to the rotor aperiodic current reach their steady-state values. Thus,
in steady-state there are only odd harmonic currents in the stator and they can be
written as follows:
f
ib
¦
ic
f
¦
An cos nt n 1,3,5,"
Anc sin nt
(3.2.5)
n 1,3,5,"
From formula (3.2.5) we get
iE
1
3
(ib ic )
2 ª
«
3 ¬n
There is J
2
3
ib
f
¦
1,3,5,"
t when we choose J 0
n
¦
(3.2.6)
0. Because the steady-state excitation
voltage equals a constant u fd , we make p
182
º
Anc sin nt »
1,3,5,"
¼
f
An cos nt 0 in formula (2.4.12) and obtain
3 Some Special Operation Modes of Synchronous Machine Systems
xad u fd
G ( p )u fd
I fdo xad
R fd
E
in which E is no-load voltage of a synchronous machine.
Substituting the results above into equation (3.2.4), we can get
\E
ª¬sin txd ( p)sin t cos txq ( p) cos t º¼ iE E sin t
ª
sin txd ( p)sin t «
3
¬n
2
ª
u«
¬n
f
¦
º 2
cos txq ( p) cos t
Anc sin nt » 3
1,3,5,"
¼
f
An cos nt 1,3,5,"
n
¦
º
Anc sin nt » E sin t
1,3,5,"
¼
f
An cos nt 1,3,5,"
n
­
sin txd ( p) ®
3
¯n
1
f
¦
¦
f
¦
An [sin(n 1)t sin(n 1)t ]
1,3,5,"
½
Anc [cos(n 1)t cos(n 1)t ]¾
1,3,5,"
¿
f
n
¦
­
cos txq ( p) ®
3
¯n
1
f
¦
An [cos(n 1)t cos( n 1)t]
1,3,5,"
½
Anc [sin(n 1)t sin(n 1)t]¾ E sin t
1,3,5,"
¿
f
n
¦
Because
cos(n r 1)t
1 j( n r1)t
e j( n r1)t ]
[e
2
in steady-state exists
xd ( p) cos(n r 1)t
1
xd ( p )[e j( n r1) t e j( n r1)t ]
2
1
{xd [ j(n r 1)]e j( n r1)t xd [ j(n r 1)]e j( n r1) t }
2
If neglecting the rotor resistances, we have xd [r j(n r 1)]
and referring to formula (2.5.47) or Fig. 2.5.4; thus,
xd ( p ) cos(n r 1)t
xdcc during n r 1 ! 1
1
xdcc[e j( n r1)t e j( n r1) t ]
2
xdcc cos(n r 1)t
sin txd ( p) Anc cos(n r 1)t sin txdcc Anc cos(n r 1)t
183
AC Machine Systems
When n 1 0, we can get xd [r j(n 1)]
(2.4.13b). Therefore, there are
xd ( p ) cos(n 1)t xd
sin txd ( p ) Anc cos(n 1)t
xd (0)
xd referring to equation
xd A1c sin t
In doing so, when n r 1 ! 1 , we can obtain
sin txd ( p) An sin(n r 1)t
sin txdcc An sin(n r 1)t
cos txq ( p ) An cos(n r 1)t
cos txqcc An cos(n r 1)t
and
When n 1 0, there is
cos txq ( p ) An cos(n 1)t
cos txq A1
When n r 1 ! 1, we have
cos txq ( p ) Anc sin(n r 1)t
cos txqcc Anc sin(n r 1)t
Thus, the steady-state flux-linkage equation above can be rewritten as follows:
\E
1 ª
« A1cxd sin t xdcc sin t
3¬
n
f
¦
( An An 2 )sin(n 1)t
1,3,5,"
º 1
ª A1 xq cos t
( Anc 2 Anc ) cos(n 1)t » 3¬
1,3,5,"
¼
f
xdcc sin t
n
¦
f
xqcc cos t
¦
º
( Anc Anc 2 )sin(n 1)t »
1,3,5,"
¼
f
( An An 2 ) cos(n 1)t xqcc cos t
n 1,3,5,"
n
¦
E sin t
1 ­
® A1cxd sin t 3¯
n
½
1
xdcc ( Anc 2 Anc ) >sin(n 2)t sin nt @¾
1,3,5," 2
¿
f
1
xdcc ( An An 2 ) > cos nt cos(n 2)t @
1,3,5," 2
¦
f
n
¦
1 ­
® A1 xq cos t 3¯
n
½
1
xqcc( Anc Anc 2 ) >sin(n 2)t sin nt @¾ E sin t
1,3,5," 2
¿
f
1
xqcc( An An 2 ) > cos(n 2)t cos nt @
1,3,5," 2
¦
f
n
¦
(3.2.7)
184
3 Some Special Operation Modes of Synchronous Machine Systems
Neglecting stator resistance, from formula (3.2.3) we can obtain
p\E
0
Taking a derivative of equation (3.2.7), we get
p\ E
1 ­
® A1cxd cos t 3¯
n
f
1
xdcc ( An An 2 )[(n 2)sin(n 2)t n sin nt ]
1,3,5," 2
¦
½
1
xdcc ( Anc 2 Anc )[(n 2) cos(n 2)t n cos nt ]¾
1,3,5," 2
¿
f
n
¦
1 ­
® A1 xq sin t 3¯
n
½
1
xqcc( Anc Anc 2 )[(n 2) cos(n 2)t n cos nt ]¾ E cos t
1,3,5," 2
¿
f
1
xqcc( An An 2 )[(n 2)sin(n 2)t n sin nt ]
1,3,5," 2
¦
f
n
¦
0
(3.2.8)
or
1 ª
1
1
º
1 ª
1
A1 xq xdcc ( A1 A3 ) xqcc( A1 A3 ) » sin t A1cxd xdcc ( A3c «
«
2
2
2
3¬
3¬
¼
f
1
1
º
A1c) xqcc( A1c A3c ) 3E » cos t ¦ > xdcc ( An An 2 )
2
3 n 1,3,5,"
¼
xqcc( An An 2 ) xdcc ( An 2 An 4 ) xqcc( An 2 An 4 ) º¼
n2
sin(n 2)t
2
f
1
3n
¦
1,3,5,"
ª¬ xdcc ( Anc 2 Anc ) xqcc( Anc Anc 2 ) xdcc ( Anc 4 Anc 2 )
xqcc( Anc 2 Anc 4 )
n2
cos(n 2)t
2
0
(3.2.9)
From equation (3.2.9) we can see that in order to keep this formula always
being zero, the sine amplitudes at every frequency must be zero and the cosine
amplitudes must also be zero. That is,
1
1
xdcc ( A1 A3 ) xqcc( A1 A3 ) 0
2
2
xdcc ( An An 2 ) xqcc( An An 2 ) xdcc ( An 2 An 4 ) xqcc( An 2 An 4 )
A1 xq 0
1
1
xdcc ( A3c A1c) xqcc( A1c A3c ) 3E 0
2
2
xdcc ( Anc 2 Anc ) xqcc( Anc Anc 2 ) xdcc ( Anc 4 Anc 2 ) xqcc( Anc 2 Anc 4 ) 0
A1cxq in which n 1,3,5,"
185
AC Machine Systems
or
( xq xscc) A1 xDcc A3
0
½
°
xDcc An 2 xsccAn 2 xDcc An 4 0 °
¾
( xd xscc) A1c xDcc A3c
3E °
xDcc Anc 2 xsccAnc 2 xDcc Anc 4 0 °¿
(3.2.10)
where
1
( xdcc xqcc)
2
1
( xdcc xqcc)
2
xscc
xDcc
The first two equations in formula (3.2.10) constitute the following algebraic
equations:
( xq xscc) A1 xDcc A3
0½
°
xDcc A1 2 xsccA3 xDcc A5 0 °
¾
xDcc A3 2 xsccA5 xDcc A7 0 °
°
"
¿
(3.2.11)
They have a group of effective solutions,
A1
A3
A5
" 0
The latter two equations in formula (3.2.10) constitute the following algebraic
equations:
( xd xscc) A1c xDcc A3c
3E ½
°
xDcc A1c 2 xsccA3c xDcc A5c 0 °
¾
xDcc A3c 2 xsccA5c xDcc A7c 0 °
xDcc A5c 2 xsccA7c xDcc A9c 0 °¿
(3.2.12)
If
A3c
k1 A1c, A5c
k3 A3c , A7c
k5 A5c , " , Anc 2
kn Anc ,
(3.2.13)
the third equation in formula (3.2.12) may be rewritten as the following form:
k1 xDcc A1c 2k3 xsccA3c k5 xDcc A5c
0
Comparing this equation with the second equation in formula (3.2.12), we can
see that k1 , k3 and k5 must all be equal. In a similar way, according to the
186
3 Some Special Operation Modes of Synchronous Machine Systems
remainder equations in formula (3.2.12), it is not difficult to prove k5
k7 " kn ; that is to say, the ratios between any two neighbouring Anc and Anc 2
are all the same. Letting the ratio be –b, we obtain
k1
k3
k5
" kn
b
(3.2.14)
Substituting those values into the second equation in formula (3.2.12), we get
( xDcc 2 xsccb xDcc b 2 ) A1c 0
so
xDcc 2 xsccb xDcc b 2
0
Its solutions are
xqcc xdcc B 2 xdcc xqcc
2 xscc r 4 xscc2 4 xDcc2
2 xDcc
b1,2
xqcc xdcc
( xqcc B xdcc )2
( xqcc xdcc )( xqcc xdcc )
Obviously, the solution
( xqcc xdcc ) 2
xqcc xdcc
( xqcc xdcc )( xqcc xdcc )
xqcc xdcc
is not correct, because the harmonic current amplitudes can not increase with their
orders. Therefore, the suitable solution is
b
Substituting A3c
( xqcc xdcc ) 2
xqcc xdcc
( xqcc xdcc )( xqcc xdcc )
xqcc xdcc
(3.2.15)
bA1c into the first equation in formula (3.2.12), we obtain
A1c
3E
xd xscc xDcc b
3E
xd xdcc xqcc
(3.2.16)
After the An and Anc values are obtained as above, according to formula (3.2.5)
the steady-state currents of two phases short circuit can be written as follows:
ib
ic
3E
xd xdcc xqcc
[sin t b sin 3t b 2 sin 5t "]
(3.2.17)
187
AC Machine Systems
On the other hand, if the two phases short circuit in steady-state is analysed by
Symmetrical Components Method, its currents after neglecting resistances are
Ib
3E
x1 x2
Ic
3E
. Comparing
x1 x2
xd , we get
That is to say, the amplitude of the basic-frequency current is
this expression with formula (3.2.17) and taking x1
x2
xdcc xqcc
(3.2.18)
It should be pointed out that during different asymmetrical conditions the
negative sequence reactance x2 has various values and the value above is suitable
for two phases short circuit.
After the currents ib and ic are expressed, the expressions of flux-linkage \ D
and voltages ua , ub and uc may be obtained. On the basis of formula (3.2.4), the
relation of steady-state flux-linkage \ D is
\D
[cos txd ( p)sin t sin txq ( p) cos t ]
2
3
ib E cos t
Substituting current ib into the equation above and arranging it in order
referring to formula (3.2.7), we obtain
2 xdcc xqcc
\D
Because ia
xd xdcc xqcc
0 and i0
E (cos t b cos3t b 2 cos5t ")
0, we have
\ 0 x0i0 0
\ a \D \ 0 \D
ua p\ a ria p\ a
Since u0
ub
188
ua ub uc
uc
1
ua
2
(3.2.19)
2 xdcc xqcc
xd xdcc xqcc
0 and ub
p\ D
E (sin t 3b sin 3t 5b2 sin 5t ")
(3.2.20)
uc , we get
xdcc xqcc
xd xdcc xqcc
E (sin t 3b sin 3t 5b2 sin 5t ")
(3.2.21)
3 Some Special Operation Modes of Synchronous Machine Systems
Evaluating the steady-state currents before, we have taken J 0
In fact, J 0 may be an arbitrary value and J
the formulas above. Thus, we have
ib
ic
ub
xd xdcc xqcc
2 xdcc xqcc
\D
ua
3E
xd xdcc xqcc
uc
t J 0 must be substituted for t in
(sin J b sin 3J b 2 sin 5J ")
(3.2.22)
E (cos J b cos3J b2 cos5J ")
(3.2.23)
E (sin J 3b sin 3J 5b 2 sin 5J ")
(3.2.24)
xd xdcc xqcc
2 xdcc xqcc
0 for convenience.
xdcc xqcc
xd xdcc xqcc
E (sin J 3b sin 3J 5b 2 sin 5J ")
(3.2.25)
(2) Initial values of currents and voltages
Evaluating initial values, we may consider the flux-linkages of all the closed
loops to be kept constant, i. e. the resistances of all the closed loops to be zero.
On this condition, we have xd ( p) xdcc , xq ( p) xqcc and
\E
1
3
(\ b \ c )
1
3
(\ b 0 \ c 0 ) \ E 0
(3.2.26)
where \ E 0 , \ b 0 , \ c 0 are the corresponding flux-linkages before short circuit.
Because the synchronous machine has no load before short circuit, we may
obtain \ q 0 0, \ d 0 uq 0 E and
\ E 0 \ d 0 sin J 0
E sin J 0
Substituting the results above into formula (3.2.4), in the initial period after short
circuit we have
\E
\E0
[ xdcc sin 2 J xqcc cos 2 J ]iE sin J G ( p)u fd
(3.2.27)
If the excitation voltage u fd before and after short circuit is constant, we may
take p
0 in equation (2.4.12) and make G ( p )u fd
E; thus, the expression
above can be rewritten as follows:
189
AC Machine Systems
\E
\E0
[ xdcc sin 2 J xqcc cos 2 J ]iE E sin J
1
[ xdcc xqcc ( xdcc xqcc) cos 2J ]iE E sin J
2
(3.2.28)
in which E is no-load voltage induced by air-gap flux produced from the original
excitation current and E sin J is the flux-linkage of fictitious winding E
constituted by this air-gap flux, and
1
[ xdcc xqcc ( xdcc xqcc) cos 2J ]iE
2
is the flux-linkage of winding E produced by the transient current iE . Their sum
is \ E 0 . The result is obvious, because the sudden short circuit may be considered
as superposition of the two conditions, i. e. the steady-state condition before
short circuit and the transient condition when rotor winding is short-circuited and
stator winding is applied on by a voltage equal but opposite to the original voltage
before short circuit. We can conclude that the equal but opposite voltage is the
cause producing the stator current iE after short circuit, because the machine has
no load before short circuit and the stator current is equal to zero. Thus, in
flux-linkage equation (3.2.28), the flux-linkage corresponding to current iE should
be the flux-linkage caused by the equal but opposite voltage, and the flux-linkage
corresponding to E is the flux-linkage caused by the excitation voltage.
Solving equation (3.2.28), we can obtain
iE
ib
2( E sin J \ E 0 )
( xdcc xqcc) ( xdcc xqcc) cos 2J
ic
xdcc xdcc xqcc
3E (sin J sin J 0 )
( xdcc xqcc) ( xdcc xqcc) cos 2J
3
iE
2
3E
(3.2.29)
(sin J b sin 3J b 2 sin 5J ")
3\ E 0 § 1
·
2
3
¨ b cos 2J b cos 4J b cos 6J "¸
2
xdcc xqcc ©
¹
(3.2.30)
in which
b
xqcc xdcc
xqcc xdcc
is the ratio between two neighbouring odd-harmonic amplitudes or two neighbouring
even-harmonic amplitudes.
190
3 Some Special Operation Modes of Synchronous Machine Systems
Therefore, we can see that after two phases sudden short circuit the stator
current has a series of odd and even harmonics including the aperiodic component.
The two kinds of harmonics can be known to be caused by the stator and rotor
aperiodic currents. The stator odd-harmonic currents are produced by rotor
aperiodic current and the stator even-harmonic currents are produced by stator
aperiodic current referring to Fig. 3.2.2.
When two phases are suddenly short-circuited at J 0 0, i.e. the resultant fluxlinkage of phases b and c is equal to zero (\ E 0 0), the stator even harmonics
will disappear. It is not difficult to understand that condition, because at the
moment of short circuit the flux-linkage of the closed loop constituted by phases
b and c is just equal to zero and it is not necessary to have an aperiodic current in
stator to cancel the flux-linkage increased abruptly. When two phases are suddenly
short-circuited at J 0 90e, i.e. the resultant flux-linkage \ E 0 of phases b and c
is maximum, the stator even harmonics will have their maximum amplitudes.
That is because there is a maximum aperiodic current in the stator to cancel the
flux-linkage increased abruptly which is caused by the ac current.
After finding the current iE (or ib and ic ), we can obtain the remainder quantities.
Because
i0 0
\0
so
\ a \D \ 0 \D
x0 i0
0
( xdcc xqcc)sin J cos J iE E cos J
1
E cos J ( xdcc xqcc)sin 2J iE
2
( E sin J \ E 0 )( xdcc xqcc)sin 2J
E cos J ( xdcc xqcc) ( xdcc xqcc) cos 2J
ua
p\ a
E sin J ( xdcc xqcc)
b 2 sin 5J ") 2 xdcc xqcc
xdcc xdcc xqcc
(3.2.32)
d ª E sin 2J
«
(sin J b sin 3J
dt « xdcc xdcc xqcc
¬
·º
2
3
¨ b cos 2J b cos 4J b cos 6J "¸ »
xdcc xqcc © 2
¹»¼
\ E 0 sin 2J § 1
E (sin J 3b sin 3J 5b2 sin 5J ")
4b\ E 0 (cos 2J 2b cos 4J 3b 2 cos 6J ")
Because uE
(3.2.31)
0 and u0
p\ 0 ri0
(3.2.33)
0
191
AC Machine Systems
so
ub
uc
1
ua
2
xdcc xqcc
xdcc xdcc xqcc
E (sin J 3b sin 3J 5b 2 sin 5J ")
2b\E 0 (cos 2J 2b cos 4J 3b 2 cos 6J ")
(3.2.34)
From the formula above, we can see that during xdcc xqcc (i.e. b 0) the phase a
voltage after two phases (b and c) are suddenly short-circuited is the same as
before, i.e. ua E sin J . It can be considered that the change of the voltage
ua after phases b and c are suddenly short-circuited is caused by the asymmetry
of rotor. In fact, if the machine has a symmetrical rotor, the current iE (or ib and
ic ) has only the aperiodic component and fundamental current and the rotor
current has only the fundamental, aperiodic component and the second harmonic.
The magnetic field produced by the aperiodic component of current iE is static
relatively to stator and no voltage can be induced in phase a. The pulsating field
produced by the fundamental of current iE is perpendicular to the axis of phase a
and there is no mutual flux-linkage between them, thus not any voltage can be
induced in phase a. The revolving field at synchronous speed, which is produced
by the rotor extra-aperiodic current due to short circuit, is just equal but opposite
to that revolving field of a synchronous speed, which is produced by the rotor
second harmonic current; therefore, in phase a the voltages induced by the two
fields cancel each other. The rotor fundamental harmonic current produces a
revolving field of a synchronous speed relative to rotor, but the field is static
relatively to stator so that no voltage can be induced in phase a. Therefore, the
voltage of phase a depends still upon the rotor excitation current before short
circuit.
From the formula of ua , we can see that ua has its maximum amplitude only
when \E 0 reaches its maximum value (\E 0 E ) , i.e. the two phases are suddenly
short-circuited at J 0 90e. That is because the aperiodic currents of phases b
and c have their maximum value on that condition.
Under that condition, we get
ua
2 xdcc xqcc
xdcc xdcc xqcc
E (sin J 3b sin 3J 5b 2 sin 5J ")
4bE (cos 2J 2b cos 4J 3b 2 cos 6J ")
From the formula above we know that during J 270e
, i.e. t
maximum instantaneous value for the first time, whose value is
192
(3.2.35)
S , ua has its
3 Some Special Operation Modes of Synchronous Machine Systems
2 xdcc xqcc
xdcc xdcc xqcc
E (1 3b 5b 2 ") 4bE (1 2b 3b 2 ")
2 xdcc xqcc
ª
1
2 b º
1
4bE
E«
2 »
2
b
b) 2
(1
)
(1
xdcc xdcc xqcc ¬ (1 b )
¼
ª
1
2 b º
1
4bE
(1 b) E «
2 »
2
(1 b) ¼
(1 b)2
¬ (1 b )
E
E
(1 b)(1 b ) 2 2 b (1 b )2
(1 b )2 (1 b ) 2
1 6b b 2
(1 b)2
§ xqcc ·
E ¨ 2 1¸
© xdcc ¹
(3.2.36)
Deriving the formula above, we have used the following equation:
(1 x) 2
1 2 x 3x 2 4 x3 "
If two phases are suddenly short-circuited at J 0
0, then \ E 0
0, the maximum
xqcc
S·
§
90e¨ i.e. t
¸ and equals cc E. So we
xd
¹
©
can see that the damping winding may decrease the inner-overvoltage or operating
overvoltage, because the damper makes xqcc be equal to xdcc approximately.
value of voltage ua will appear at J
If the neutral point of a machine is free from earth and the fault-point is
grounded referring to Fig. 3.2.3, the voltage of phase a to earth is
ua 0
ua ub
3
ua
2
therefore, the maximum amplitude of voltage ua 0 during J 0
§ xqcc 1 ·
3 § xqcc ·
E ¨ 2 1¸ 3 E ¨ ¸
2 © xdcc ¹
© xdcc 2 ¹
(3.2.37)
90ewill be
(3.2.38)
Figure 3.2.3 Two phases sudden short circuit when the fault-point is earthed
193
AC Machine Systems
The initial values of the stator current and voltage have been obtained as above.
In order to obtain the initial value of rotor current, we must get currents id and iq
first according to the following formulas:
id
iq
iE sin J
iE cos J
2
½
ib sin J °
°
¾
ib cos J °
°¿
3
3
2
(3.2.39)
then use the equivalent circuit of operational reactance ( p f, u fd 0) and can
find the rotor current. The calculation method is the same as in 3-phase sudden
short-circuit; such as a machine without damping winding, we have
id
2 E sin 2 J 2\ E 0 sin J
( xdc xq ) ( xdc xq ) cos 2J
2 E sin J 2\ E 0
xdc xdc xq
E
xdc xdc xq
[1 (1 b) cos 2J b(1 b) cos 4J b 2 (1 b) cos 6J "]
2\ E 0
xdc xdc xq
E
xdc xdc xq
\E0
xdc xq
(sin J b sin 3J b 2 sin 5J ")
[sin J b sin 3J b 2 sin 5J "]
[1 (1 b) cos 2J b(1 b) cos 4J "]
(1 b)[sin J b sin 3J b 2 sin 5J "]
(3.2.40)
in which
b
xq xdc
xq xdc
Therefore, the current of excitation winding is
x
I fd I fd 0 ad id
X ffd
I fd 0 194
xad E
[1 (1 b) cos 2J b(1 b) cos 4J "]
X ffd [ xdc xdc xq ]
xad\ E 0
X ffd xdc xq
(1 b)[sin J b sin 3J b 2 sin 5J "]
(3.2.41)
3 Some Special Operation Modes of Synchronous Machine Systems
(3) Time constants
The even harmonics of the stator current and voltage and the odd harmonics of
the rotor current are all caused by the stator aperiodic current. Thus, those
harmonics will decay to zero with the stator aperiodic current, i.e. the steadystate values of those harmonics are all zero.
The odd harmonics of the stator current and voltage and the even harmonics of
the rotor current are all caused by the rotor aperiodic current including the
damper aperiodic current. So, these harmonics will decay with the rotor aperiodic
current. However, the rotor aperiodic current cannot finally decay to zero except
the damper aperiodic current but decays to the steady-state value determined by
the excitation voltage. Therefore, the corresponding harmonics finally decay to
some steady-state values, too.
The change process of the current and voltage from their initial values to
steady-state values is complicated. In practice, we can use the same approximate
method as in 3-phase sudden short circuit and various time constants are
discussed as follows.
(a) The decay time constant Ta 2 of the stator even harmonic currents and rotor
odd harmonic currents
This time constant depends upon the equivalent reactance and resistance
corresponding to the aperiodic current of phases b and c. Because phases b and c
in 3-phase machine are equivalent to phase Ein two phases D and E machine,
thus the time constant is also equal to the ratio of the equivalent reactance to
resistance corresponding to the aperiodic current of phase E. The equivalent
reactance can be found as follows.
From equation (3.2.30), we can see that if the synchronous machine rotates at
a synchronous speed, its excitation winding is closed and the aperiodic current of
phase E is I 0 / 2, then the resultant current caused by the aperiodic current is
iE
§1
·
I 0 ¨ b cos 2J b 2 cos 4J b3 cos 6J " ¸
©2
¹
f
1
I 0 I 0 ¦ (b)n cos 2nJ
2
n 1,2,3,"
J
in which
b
(3.2.42)
t J0
xqcc xdcc
xdcc xqcc
Referring to formula (3.2.4) and taking u fd
0 in it, we can describe the flux-
linkage of phase E as
195
AC Machine Systems
\E
f
ª1
º
[sin J xd ( p)sin J cos J xq ( p) cos J ] « I 0 I 0 ¦ (b) n cos 2nJ »
2
n 1,2,3,"
¬
¼
1
1
­1
® I 0 sin J xd ( p)sin J I 0 cos J xq ( p ) cos J I 0 sin J xd ( p )
2
2
¯2
f
u
¦
(b)n [sin(2n 1)J sin(2n 1)J ]
n 1,2,3,"
1
I 0 cos J xq ( p )
2
n
½
(b) n [cos(2n 1)J cos(2n 1)J ]¾
1,2,3,"
¿
f
¦
Neglecting rotor resistance and referring to equation (3.2.7), we have
xd [r j(2n r 1)]
xq [r j(2n r 1)]
xdcc
xqcc
where n 1, 2,3,"
\E
­1
1
1
® I 0 xdcc sin 2 J I 0 xqcc cos 2 J I 0 xdcc sin J
2
2
2
n
¯
sin(2n 1)J ] 1
I 0 xqcc cos J
2
n
1
xqcc
2 n
(b) n [sin(2n 1)J
1,2,3,"
½
(b) n [cos(2n 1)J cos(2n 1)J ]¾
1,2,3,"
¿
f
¦
1 ­1
1
1
I 0 ® ( xdcc xqcc) ( xdcc xqcc ) cos 2J xdcc
2 ¯2
2
2 n
cos(2n 2)J cos(2n 2)J cos 2nJ ]
f
¦
f
¦
(b)n [cos 2nJ
1,2,3,"
½
(b) n [cos 2nJ cos(2n 2)J cos(2n 2)J cos 2nJ ]¾
1,2,3,"
¿
(3.2.43)
f
¦
Thus, the average flux-linkage, i.e. the aperiodic flux-linkage produced by the
1
aperiodic current I 0 in phase E can be written as
2
\ E cp
196
1 ª1
1
1
º
I 0 « ( xdcc xqcc) bxdcc bxqcc »
2 ¬2
2
2
¼
xqcc xdcc º
1 ª
»
I 0 « xdcc xqcc ( xdcc xqcc)
4 «
xqcc xdcc »¼
¬
1
I 0 xdcc xqcc
2
(3.2.44)
3 Some Special Operation Modes of Synchronous Machine Systems
We can see that so far as the aperiodic current is concerned, the equivalent
reactance of phase E is xdcc xqcc x2 .
Because the resistance of phase E is r, the time constant can be written as
xdcc xqcc
Ta 2
r
x2
r
(3.2.45)
(b) The decay time constants Tdcc2 , Tdc2 of the stator odd harmonic currents and
rotor even harmonic currents
These time constants Tdcc2 and Tdc2 depend upon the equivalent reactance and
resistance corresponding to the aperiodic currents of rotor. If the machine has no
damping winding, then the average flux-linkage of excitation winding during
steady-state two phases short circuit can be written as
\fdcp
Xffd I fdcp xad idcp
X ffd I fd 0 X ffd I fd 0 xad E
xd xdc xq
2
xad
I fd 0
xd xdc xq
§ xdc xdc xq
¨
¨ xd xdc xq
©
·
¸ I fd 0 X ffd
¸
¹
Thus, the equivalent reactance of excitation winding is
xdc xdc xq
xd xdc xq
X ffd
The corresponding time constant is
Tdc2
X ffd xdc xdc xq
R fd xd xdc xq
Td 0
xdc x2
xd x2
(3.2.46)
The result is similar to that of 3-phase sudden short circuit, only the numerator
and denominator are added to the x2 respectively. That is well known from the
conception of Symmetrical Components.
When the rotor has a damping winding besides excitation winding, the two
phases short circuit can also be analysed in the same way as 3-phase sudden short
circuit, i.e. considering the stator odd harmonic currents in the initial stage to
decay approximately according to Tdcc2 then to Tdc2 . The first stage depends mainly
on the damping winding and the latter stage on the excitation winding. On the
basis of the same conception, the time constants can be written as
197
AC Machine Systems
Tdcc2
xdcc xdcc xqcc
Tdcc0
Tdc2
xdc xdcc xqcc
xdc xdcc xqcc
Td 0
xd xdcc xqcc
xdcc x2 ½
°
xdc x2 °°
¾
xdc x2 °
Td 0
°
xd x2 °
¿
Tdcc0
(3.2.47)
(4) General expressions of currents and votages
Having the initial values, steady-state values and time constants mentioned
above, we can write the general formulas of currents and voltages in the same
way as 3-phase sudden short circuit as follows:
ic
ib
ua
3F (sin J b sin 3J b 2 sin 5J ")
3\ E 0
xdcc xqcc
e
§1
·
2
¨ b cos 2J b cos 4J " ¸
2
©
¹
(3.2.48)
2 xdcc xqcc F (sin J 3b sin 3J 5b 2 sin 5J ")
4b\ E 0 e
ub
t
Ta 2
uc
t
Ta 2
(cos 2J 2b cos 4J 3b 2 cos 6J ")
(3.2.49)
xdcc xqcc F (sin J 3b sin 3J 5b 2 sin 5J ")
2b\ E 0 e
t
Ta 2
(cos 2J 2b cos 4J 3b 2 cos 6J ")
(3.2.50)
in which
t
t
§ 1
1 · Tdcc2 § 1
1 · Tdc 2
E
¨
¨
¸ Ee
¸ Ee
cc
c
c
xd x2
© xd x2 xd x2 ¹
© xd x2 xd x2 ¹
F
(3.2.51)
During two phases sudden short circuit, the excitation current of a machine
without damping winding is
I fd
I fd 0
t
ª§
xad
x
1
E
Tdc 2
e
ad (1 b) E Ǭ
X ffd xdc xdc xq
X ffd
Ǭ xdc xdc xq
©
º
» (cos 2J b cos 4J b 2 cos 6J ")
»
¼
· tc
1
¸ e Td 2 ¸
c
xd xd xq ¹
xd xdc xq
xad (1 b)
\ E 0 e Ta 2 (sin J b sin 3J b 2 sin 5J ")
X ffd xdc xq
1
t
198
(3.2.52)
3 Some Special Operation Modes of Synchronous Machine Systems
(5) Electromagnetic torque after two phases sudden short circuit
When two phases of a no-load synchronous machine are suddenly shortcircuited, referring to formula (3.2.33) the initial flux-linkage of phase D can be
written as follows:
1
E cos J ( xdcc xqcc)sin 2J iE
2
( E sin J \ E 0 )( xdcc xqcc)sin 2J
E cos J ( xdcc xqcc) ( xdcc xqcc) cos 2J
\D
2 xqccE cos J \ E 0 ( xdcc xqcc)sin 2J
( xdcc xqcc) ( xdcc xqcc) cos 2J
2 xdcc xqcc
xdcc xdcc xqcc
E (cos J b cos 3J b 2 cos 5J ")
2b\ E 0 (sin 2J b sin 4J b 2 sin 6J ")
(3.2.53)
Its steady-state value referring to equation (3.2.23) is
2 xdcc xqcc
\D
xd xdcc xqcc
E (cos J b cos 3J b 2 cos 5J ")
According to the initial value, steady-state value and time constants, the general
expression of flux-linkage \ D may be described as
2 F xdcc xqcc (cos J b cos 3J b 2 cos 5J ")
\D
2b\ E 0 e
t
Ta 2
(sin 2J b sin 4J b 2 sin 6J ")
2 Fxqcc( xdcc xdcc xqcc ) cos J
( xdcc xqcc) ( xdcc xqcc) cos 2J
( xqcc xdcc )\ E 0 sin 2J
( xdcc xqcc) ( xdcc xqcc ) cos 2J
e
t
Ta 2
(3.2.54)
in which F is evaluated with formula (3.2.51).
On the other hand, from equation (3.2.48) we can obtain
iE
2
3
ib
2 F (sin J b sin 3J b 2 sin 5J ")
2\ E 0
xdcc xqcc
e
t
Ta 2
§1
·
2
¨ b cos 2J b cos 4J "¸
©2
¹
2( xdcc xdcc xqcc ) F sin J
xdcc xqcc ( xdcc xqcc) cos 2J
2\ E 0
xdcc xqcc ( xdcc xqcc) cos 2J
e
t
Ta 2
(3.2.55)
199
AC Machine Systems
Substituting the results above into the torque equation, we can get
Te
\ D iE \ E iD
\ D iE
1
[2 F 2 xqcc( xdcc xdcc xqcc ) 2 sin 2J
[( xdcc xqcc) ( xdcc xqcc) cos 2J ]2
4 Fxqcc( xdcc xdcc xqcc )\ E 0 e
t
Ta 2
cos J
2( xdcc xqcc)( xdcc xdcc xqcc ) F\ E 0 e
2( xdcc xqcc)\ E2 0 e
2t
Ta 2
t
Ta 2
sin 2J sin J
(3.2.56)
sin 2J ]
Referring to Appendix 2 of Reference [3] exist
1
( xdcc xqcc) ( xdcc xqcc) cos 2J
sin J
cc
cc
( xd xq ) ( xdcc xqcc) cos 2J
d
dJ
1 §1
·
2
¨ b cos 2J b cos 4J " ¸
xdcc xqcc © 2
¹
1
xdcc xdcc xqcc
(sin J b sin 3J b 2 sin 5J ")
ª
º
2( xdcc xqcc )sin 2J
1
«
» [( xdcc xqcc) ( xdcc xqcc) cos 2J ]2
¬« ( xdcc xqcc) ( xdcc xqcc) cos 2J »¼
2b
(sin 2J 2b sin 4J 3b 2 sin 6J ")
xdcc xqcc
ª
º 2 xqcc cos J ( xdcc xqcc)sin 2J sin J
sin J
«
»
2
«¬ ( xdcc xqcc) ( xdcc xqcc) cos 2J »¼ [( xdcc xqcc) ( xdcc xqcc) cos 2J ]
1
(cos J 3b cos 3J 5b 2 cos 5J ")
xdcc xdcc xqcc
d
dJ
therefore there is
Te
2 F 2 xdcc xqcc (sin 2J 2b sin 4J 3b 2 sin 6J ")
2 F\ E 0 e
2b\ E2 0
xdcc xqcc
t
Ta 2
e
2t
Ta 2
(cos J 3b cos 3J 5b 2 cos 5J ")
(sin 2J 2b sin 4J 3b 2 sin 6J ")
(3.2.57)
It should be noted that the formula above is obtained after neglecting the stator
and rotor resistances but considering the effect of resistances on time constants.
From the torque expression we can see that the electromagnetic torque after two
200
3 Some Special Operation Modes of Synchronous Machine Systems
phases sudden short circuit is only the pulsating torque under this condition. The
torque contains a series of odd and even harmonics. These harmonic torques are
caused by a series of harmonic currents under the short circuit condition.
Because both the stator and rotor are asymmetrical, the stator aperiodic current
produces not only stator even harmonic currents but also rotor odd harmonic
currents. Similarly the rotor aperoidic current causes stator odd harmonic
currents and rotor even harmonic currents. The magnetic fields produced by
stator odd harmonic currents act with the magnetic fields produced by rotor even
harmonic currents to produce the even harmonic torques, and the field produced
by the stator fundamental current acting with the field produced by rotor aperiodic
current does not cause an average torque because we have supposed the
resistances to be zero. Those even harmonic torques depend obviously upon the
rotor aperiodic current and decay finally to the steady-state value, which are the
first group of harmonics in the torque formula (3.2.57). The fields produced by
stator odd harmonic currents act with the fields produced by rotor odd harmonic
currents to cause the odd harmonic torques, and the fields produced by stator
even harmonic currents act with the fields produced by rotor even harmonic
currents to cause the odd harmonic torques, too. Sum of the two kinds of torques
is the second group of harmonics in the torque formula (3.2.57). Obviously, the
group of harmonic torques is determined by the stator and rotor aperiodic
currents and decays finally to zero. The fields produced by stator even harmonic
currents act with the fields produced by rotor odd harmonic currents to cause the
even harmonic torques that are the third group of harmonics in torque formula
(3.2.57). This group of harmonic torques depends obviously upon stator aperiodic
current and decays finally to zero. In addition, during xdcc xqcc, i.e. b 0, the third
group of harmonic torques will disappear, because under this condition the rotor
is symmetrical, there is only the aperiodic current in stator even harmonic currents
and only the basic-frequency current exists in rotor odd harmonic currents. The
rotor basic-frequency current only produces the static field relative to stator because
the rotor is symmetrical, so the static field does not produce the harmonic torques
by acting with the field produced by stator aperiodic current. Of course, there is
no average torque because we have supposed resistances to be zero.
When the effect of stator and rotor resistances is considered, in the
electromagnetic torques after two phases sudden short circuit there are not only
the pulsating torques but also the average torque. Generally speaking, although
the average torque is not large, sometimes it must be considered when we study
some problems, such as the power system stability. However, the pulsating
torques calculated by neglecting resistances are accurate enough. The average
torque can be analysed as below.
As mentioned above, the operation condition of two phases sudden short
circuit is approximately considered as the condition of the aperiodic currents
circulating simultaneously through the stator and rotor windings. When evaluating
the average electromagnetic torque, we can study the two groups of currents
separately, because the fields produced by them have different speeds.
201
AC Machine Systems
(i) The rotor aperiodic current causes the odd harmonic currents in stator and
the even harmonic currents in rotor. It may be proved that the average
electromagnetic torque caused by this group of currents is equal to the effective
resistance losses produced by these harmonic currents; but the resistance loss of
the rotor aperiodic current is provided with the excitation supply and the rotor
storage magnetic-energy, so it does not produce the electromagnetic torque.
Namely, their average electromagnetic torque is
2
3m
c
Tecp
f
¦
| im |2 r 1,3,5,"
1
2s
f
¦
[| ids |2 sRds | iqs |2 sRqs ]
(3.2.58)
2,4,6,"
in which
| im | is the stator m-th harmonic current amplitude,
| ids |,| iqs | are the rotor s-th harmonic current amplitudes of the d and q axes,
s m 1,
Rds , Rqs are the corresponding values of the following expressions,
xd ( js )
X ds jRds
xq ( js )
X qs jRqs
The first term of the formula (3.2.58) represents the stator effective resistance
losses produced by all the stator odd harmonic currents and the second term
expresses the rotor effective resistance losses produced by all the rotor even
harmonic currents.
(ii) The stator aperiodic current causes stator even harmonic currents and rotor
odd harmonic currents. It can be similarly proved that the average electromagnetic torque caused by this group of currents equals the effective resistance
losses produced by these harmonic currents, but the resistance loss of the stator
aperiodic current is provided with the stator storage magnetic-energy, so it does
not produce the electromagnetic torque, i.e. their average electromagnetic torque is
cc
Tecp
2
3m
f
¦
| im |2 r 2,4,6,"
1
2s
f
¦
[| ids |2 sRds | iqs |2 sRqs ]
(3.2.59)
1,3,5,"
Therefore, the resultant average torque after two phases short circuit is
Tecp
c Tecp
cc
Tecp
2
3m
f
¦
1,2,3,"
| im |2 r 1
2s
f
¦
[| ids |2 sRds | iqs |2 sRqs ]
1,2,3,"
[the stator effective resistance losses produced by all the
stator harmonic currents] [the rotor effective resistance
losses produced by all the rotor harmonic currents]
202
(3.2.60)
3 Some Special Operation Modes of Synchronous Machine Systems
3.3
Steady-state Asynchronous Operation of Synchronous
Machines
In practice, the asynchronous operation of a synchronous machine often arises.
For example, starting of a synchronous motor, pulling a synchronous generator
into step, falling out of step and periodical oscillation of a synchronous machine
are all connected with the asynchronous operation. During asymmetrical operation
of a synchronous machine, its stator current has not only the fundamental
component but also a series of harmonics. Although the machine rotor rotates at a
synchronous speed at this time, there is asynchronous operation so far as a series
of harmonic currents are concerned.
When a synchronous machine operates in steady-state at an asynchronous
speed, its stator or rotor current is the sum of different harmonic currents of
various frequencies if the terminal voltages of its stator have symmetrical and
sinusoidal waves. If the supply frequency is f, the synchronous and actual speeds
are Z s and Z respectively which are all constant, and the machine slip is
Zs Z
, then the supply voltage will produce 3-phase symmetrical currents
s
Zs
of frequency f in the stator. However, the synchronous revolving-field produced
by the symmetrical currents has a relative speed sZ s with respect to the rotor, so
a current of frequency sf can be induced in the rotor. Owing to asymmetrical
rotor, that current will produce the revolving fields at speeds (Z r sZ s ), which
induce the symmetrical currents of frequencies f and (1 2 s ) f in stator
respectively. In addition, the rotor excitation current also induces the symmetrical
currents of frequency (1 s ) f in stator. Therefore, there are harmonic currents of
frequencies f, (1 s ) f and (1 2 s ) f in stator and direct current and harmonic
current of frequency sf in rotor referring to [2,3]. On the basis of the deduction,
the stator current may be written as follows,
ia
A1 cos t +A2 cos(1 2 s )t +A3 cos(1 s )t
A1c sin t +A2c sin(1 2s )t +A3c sin(1 s )t
ib
A1 cos(t 120e
)+A2 cos[(1 2 s )t 120e
]
A3 cos[(1 s )t 120e
] A1csin(t 120e
A2csin[(1 2s )t 120e
]+A3c sin[(1 s )t 120e
]
ic
A1 cos(t 120e
]
)+A2 cos[(1 2 s )t 120e
c
A3 cos[(1 s )t 120e
] A1sin(t 120e
A2csin[(1 2s )t 120e
]+A3c sin[(1 s )t 120e
]
½
°
°
°
°
°
¾
°
°
°
°
°
¿
(3.3.1)
203
AC Machine Systems
If taking
I1
I2
I3
1
( A1 jA1c),
2
1
( A2 jA2c ),
2
1
( A3 jA3c ),
2
I1*
I*2
I*3
1
½
( A1 jA1c) °
2
°
1
°
c
( A2 jA2 ) ¾
2
°
1
°
( A3 jA3c ) °
2
¿
(3.3.2)
We obtain
ia
I1e jt I2 e j(1 2s)t I3e j(1s)t I1*e jt
I* e j(1 2s)t I* e j(1s)t
2
ib
3
j (1 2s)t 120e
j (1 s)t 120e
@
@
I1e j(t 120e) I2 e >
I3 e >
j (1 2s)t 120e
j (1 s)t 120e
@
@
I1*e j(t 120e) I*2 e >
I*3e >
ic
j (1 2s)t 120e
j (1 s)t 120e
@
@
I3 e >
I1e j(t 120e) I2 e >
j (1 2s)t 120e
j (1 s)t 120e
@
@
I1*e j(t 120e) I*2 e >
I*3e >
½
°
°
°
°
¾
°
°
°
°
¿
(3.3.3)
It is noted that a current changing sinusoidally with time can usually be
represented by a rotating time-vector in ac Circuitry Theory, but here it is
represented by two conjugate rotating vectors which have half the amplitude of
that rotating time-vector.
Transforming the currents ia , ib , ic above into 1, 2, 0 components according to
formula (3.1.13), we can get
i1
i2
i0
½
I1e jt I2 e j(1 2 s ) t I3e j(1 s ) t
°
*
j
*
j(1
2
)
*
j(1
)
t
s
t
s
t
I1e I2 e
I3 e
¾
°
0
¿
(3.3.4)
If the 3-phase symmetrical terminal voltages are
ua
ub
uc
B cos t Bc sin t
½
°
B cos(t 120e
) Bc sin(t 120e
)¾
B cos(t 120e
) Bc sin(t 120e
) °¿
(3.3.5)
1
½
( B jBc) °
°
2
¾
1
( B jBc) °
°¿
2
(3.3.6)
and we take
U
U *
204
3 Some Special Operation Modes of Synchronous Machine Systems
then there are
u1 Ue jt ½
°
u2 U *e jt ¾
°
u0 0
¿
(3.3.7)
Provided that we choose J 0 at the moment t 0 , i.e. J 0 0, then during
steady-state asynchronous operation, according to formulas (3.1.23) and (3.3.4)
and taking Z 1 s together with J (1 s )t , we can obtain the following basic
relations,
\1
\2
½
°
°
°
s
D
1
2
°
[ xs ( js ) I2 xD ( js ) I1* ]e j(1 2 s )t °
°
\ 3e j(1 s )t \1e jt \ 2 e j(1 2 s )t
°
¾
1
ª
* x (0) I º e j(1 s )t °
E
x
I
(0)
s
D
3
3»
«2
°
¬
¼
°
t
*
j
[ xs ( js ) I1 xD ( js ) I2 ]e
°
°
*
j(1 2 s ) t
[ xs ( js ) I 2 xD ( js ) I1 ]e
°
\ *3e j(1 s )t \ 1*e jt \ *2 e j(1 2 s )t °¿
ª1
* º j(1 s )t
« 2 E xs (0) I 3 xD (0) I 3 » e
¬
¼
[ x ( js ) I x ( js ) I* ]e jt
(3.3.8)
in which
\1
\
*
1
\ 2
\ 2*
\ 3
\ 3*
xs ( js ) I1 xD ( js ) I2*
x ( js ) I* x ( js ) I
½
°
°
s
D
1
2
* °
xs ( js ) I 2 xD ( js ) I1 °
°
xs ( js ) I2* xD ( js ) I1
¾
°
1
E xs (0) I3 xD (0) I*3 °
2
°
°
1
*
E xs (0) I 3 xD (0) I 3 °
¿
2
(3.3.9)
From equations (3.1.20), (3.3.4) and (3.3.8) we can get
205
AC Machine Systems
u1 Ue jt
u2
½
p\ 1 ri1
°
[ j(1 s )\ 3 rI3 ]e j(1 s )t ( j\1 rI1 )e jt °
°
[ j(1 2 s )\ 2 rI2 ]e j(1-2s )t
°
¾
* jt
U e
p\ 2 ri2
°
°
*
*
*
j(1 s ) t
( j\1
[ j(1 s )\ 3 rI 3 ]e
°
rI1* )e jt [ j(1 2 s )\ 2* rI2* ]e j(1 2 s ) t °¿
(3.3.10)
Comparing various terms having the same exponential function in the equations
above and referring to formula (3.3.9), we obtain
r
j
jU
\1 I1
U *
j
\1* jrI1*
r
xs ( js ) I1 xD ( js ) I2* I1
j
½
°
°
¾
xs ( js ) I1* xD ( js ) I2 jrI1* °
°¿
r
I2
j(1 2s )
½
xs ( js ) I2 °
°
r
°
*
I2 °
xD ( js ) I1 s
j(1 2 )
°
¾
2
I*2 xs ( js) I*2 °
0 \ *2 j
°
(1 2 s )
°
r
*
°
xD ( js ) I1 j
I2
°¿
(1 2s )
0 \ 2 (3.3.12)
r
I3
j(1 s )
1
½
E xs (0) I3 °
2
°
r
°
*
I3
xD (0) I 3 °
j(1 s )
°
¾
r * 1
0 \ *3 j
I3
E xs (0) I*3 °
°
(1 s )
2
°
r *
°
xD (0) I3 j
I3
°¿
(1 s )
0 \ 3 (3.3.11)
(3.3.13)
On the basis of those equations we can draw these equivalent circuits of
steady-state asynchronous operation of a synchronous machine as shown in
Figs. 3.3.1 and 3.3.2. In Fig. 3.3.1(a) there is an equivalent circuit of calculating
or measuring the stator current component caused by the direct current in the
excitation winding. Figure 3.3.1(b) shows an equivalent circuit of calculating or
measuring the stator current components produced by the stator terminal voltages.
Figure 3.3.2 indicates the resultant equivalent circuit.
206
3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.3.1 Equivalent circuit
(a) Equivalent circuit for the stator current component caused by excitation emf;
(b) Equivalent circuit for the stator current components caused by stator terminal voltage
Figure 3.3.2 Resultant equivalent circuit for asynckronous operation of synchronous
machines
207
AC Machine Systems
The output power and electromagnetic torque of the machine can be measured
in the resultant equivalent circuit or estimated by the following formulas,
2(u2 i1 u1i2 u0 i0 )
2[ I U * I U *e j2 st I U *e jst
P
1
2
3
I1*U I2*Ue j2 st I3*Ue jst ]
Te
(3.3.14)
j2(i2\ 1 i1\ 2 )
j2[ I1\1* I1*\1 I2\ 2* I*2\ 2 I*3\ 3 I3\ 3* ]
( I*\ I \ * I*\ I \ * )e jst
2
3
3
2
3
1
1
3
( I1*\ 3 I3\1* I*3\ 2 I2\ 3* )e jst
( I1\ 2* I*2\1 )e j2 st ( I2\1* I1*\ 2 )e j2 st ]
(3.3.15)
The average electromagnetic torque of the machine is
Tecp
j2( I1\1* I1*\1 I21\ 2* I*2\ 2 I*3\ 3 I3\ 3* )
(3.3.16)
Therefore, we can see that various components of the average electromagnetic
torque are all constituted by the currents and flux-linkages whose frequencies are
the same. That is natural, because the revolving fields caused only by them have
the same speed.
It should be pointed out that the equivalent circuit for the direct current of the
excitation winding as shown in Fig. 3.3.1(a) is independent of the equivalent
circuit for applying supply voltage to stator as shown in Fig. 3.3.1(b). That is
clear according to the Superposition Theorem, because the machine has been
supposed to be connected to an infinite network, so the action of the direct
current of excitation winding is similar to the action of 3-phase short circuit.
It is also noted that there are harmonic torques and power of frequencies 2sf
and sf besides the average electromagnetic torque and power in the torque Te
and output power P. That is not difficult to understand, because in this operation
mode there are three kinds of revolving fields. The first is the revolving field of
speed Z s corresponding to stator basic-frequency current, the second is the
revolving field of speed (1–2s) Z s corresponding to the stator current of
frequency (1– 2s) f and the third is the revolving field of speed (1 –s) Z s
corresponding to the direct current of excitation winding or the stator current of
frequency (1–s)f. The relative speeds between these fields are 2sZ s and sZ s .
Therefore, the harmonic torques or power values of frequencies 2sf and sf are
produced.
The excitation winding current and damper current can be estimated on the
basis of the equivalent circuit of operational reactance. However, first we must
calculate currents id and iq by using the following formulas,
208
3 Some Special Operation Modes of Synchronous Machine Systems
id
iq
i1e jJ i2 e jJ
( I3 I3* ) ( I1 I2* )e jst ( I2 I1* )e jst
½
°
°
¾
j(i1e jJ i2 e jJ )
°
*
*
jst
*
jst °
j( I 3 I 3 ) j( I1 I 2 )e j( I 2 I1 )e ¿
(3.3.17)
As for study of the rotor motion behavior, the average torque of synchronous
machine under asynchronous operation condition is significant. Now we derive
the average-torque formula by use of the equivalent circuit during asynchronous
operation obtained above referring to Figs 3.3.1 and 3.3.2.
The complex power may be described as
*
P UI
P jQ
6I i2 Ri j6I i2 X i
in which P 6I i2 Ri is the resultant active power through a certain point of a
circuit whose voltage is U and whose current is I ; I i2 Ri is the consumed active
power of the branch i; Q 6I i2 X i is the resultant reactive power through the
same point of a circuit; I i2 X i is the absorbed reactive power of the branch
i .However, it should be noted that X i is positive value for an inductive branch
and negative value for a capacitive branch.
On the basis of the concept above, by use of the equivalent circuit as shown in
Figs 3.3.1(a) and 3.3.1(b) and equation (3.3.16) we can obtain the following
expressions:
(i) j2( I3*\ 3 I3\ *3 )
r 2
r 2º
ª
j2 « j
I3 j
I3 »
1 s
¬ 1 s
¼
2
r
2 I3
1 s
(3.3.18)
(ii) j2[ I1 \ 1* I1*\ 1 I2\ 2* I2*\ 2 ]
2
2
2
4 I1 Rqs 2 I1 I2* ( Rds Rqs ) 4 I2 Rqs 8 I2
2
2
2 I1 I 2* Rds 2 I1 I2* Rqs 8 I2
2
2
r
1 2s
r
1 2s
(3.3.19)
in which Rds and Rqs are included in the following expressions:
xd ( js )
X ds jRds
xq ( js )
X qs jRqs
209
AC Machine Systems
Impedances xd ( js ) and xq ( js ) may be measured or calculated by the
equivalent circuit of operational reactance during p js. In the expressions
above, both Rds and Rqs have a coefficient ( j), because the equivalent circuit of
operational reactance is usually made up of the equivalent resistance and
capacitance.
From formula (3.3.17) we can see
| id ( nep ) |2 Rds | iq ( nep ) |2 Rqs
4 | I1 I2* |2 Rds 4 | I1 I2* |2 Rqs
(3.3.20)
where | id ( nep ) | and | iq ( nep ) | are the amplitudes of ac components of currents id
and iq .
According to formulas (3.3.18), (3.3.19) and (3.3.20) we can get
Tecp | 2 I3 |2
r
2r
| 2 I2 |2
1 s
1 2s
1
[| id ( nep ) |2 Rds | iq ( nep ) |2 Rqs ]
2
(3.3.21)
Various terms in that average electromagnetic torque expression have the
following physical meanings:
r
(i) The first term | 2 I3 |2
is the short circuit torque (or asynchronous
1 s
torque), because at this time the excitation field has a relative speed (1 s ) Z s
with respect to the stator, so the stator winding resistance r should be divided by
(1 s). Obviously, this torque will be zero when there is no excitation source.
(ii) The latter two terms are the salient-pole torque (or reluctance torque) and
asynchronous torque caused by the applied voltage. When s 0.5, the current
I2 will be zero, so the second term of the average torque expression will become
zero.
If the excitation voltage is supposed to be zero in the analysis above, that is
steady-state asynchronous operation of a synchronous machine without
excitation source. Because under this condition E 0 and I3 0, the average
electromagnetic torque is
Tecp | 2 I2 |2
2r
1
[| id ( nep ) |2 Rds | iq ( nep ) |2 Rqs ]
1 2s 2
(3.3.22)
So far as the induction machine without excitation source is concerned, there
R
is the relation xd ( js ) xq ( js ) X j because its rotor is symmetrical, thus in
s
this asynchronous operation mode we obtain I2 0 and I3 0. In addition,
210
3 Some Special Operation Modes of Synchronous Machine Systems
according to formulas (3.3.20) and (3.3.22) we can get
| id ( nep ) | | iq ( nep ) | | 2 I1 |
Tecp
R
| 2 I1 |
s
2
| 2U |2
R·
§
¨r ¸
s¹
©
2
½
°
R °
°
¾
s
°
X2°
°¿
(3.3.23)
That is the usual torque formula of induction machines. The minus sign appears
before the formula because the basic equations are written according to generator
convention.
The equivalent circuit of steady-state asynchronous operation of the
synchronous machine without excitation source is shown in Fig. 3.3.1(b). When
r
becomes infinite and one of the corresponding equivalent
s 0.5, the value
1 2s
circuits is shown in Fig. 3.3.3.
Figure 3.3.3 One of the equivalent circuits for Fig. 3.3.1(b) at s 0.5
It is not difficult to see that under this condition there is only the basic-frequency
alternating current in stator winding and the corresponding impedance is
U
I1
Z
R jX
1
1
r ( Rds Rqs ) j ( X ds X qs )
2
2
(3.3.24)
Parameters xd ( js ) and xq ( js ) or X ds , Rds and X qs , Rqs are all dependent on
slip s, whose changes can be evaluated by the equivalent circuit of operational
reactance referring to Section 2.4. From the equivalent circuit, we can see that
when the value of slip s increases from zero to infinite, the values of reactances
X ds , X qs will decrease from the synchronous reactances xd , xq to subtransient
reactances xdcc , xqcc. However, because the rotor winding of a synchronous
machine has a small resistance, these reactances X ds , X qs are nearly equal to the
subtransient reactances xdcc , xqcc when slip s is more than a not large value. These
reactances X ds , X qs change strongly with slip s when the slip s becomes small
211
AC Machine Systems
values. This property is not difficult to be proved so long as we evaluate the
equivalent circuit of operational reactance of a typical machine. Of course, it can
also be proved through measurement of a specified machine. The test results of a
synchronous machine with an excitation winding short-circuited are shown in
Fig. 3.3.4. From those curves we can see that when slip s is more than 0.2, the
reactances X ds , X qs change little with the slip s and approximate to the subtransient
reactances xdcc , xqcc. When 0 s 0.2, the reactances X ds , X qs change strongly
with slip s.
Figure 3.3.4 Test results of a synchronous machine with an excitation winding
short-circuited
Therefore, when slip s has a large value, for example s ! 0.5 , we can neglect
the stator and rotor resistance for calculating currents and approximately consider
xd ( js )
xq ( js )
xdcc
xqcc
Under this approximate condition, one of the equivalent circuits shown in
Fig. 3.3.1(b) can be reduced to Fig. 3.3.5.
Figure 3.3.5 One of the equivalent circuits for Fig. 3.3.1(b) during xd ( js )
and xq ( js ) xccq
212
xdcc
3 Some Special Operation Modes of Synchronous Machine Systems
When s 2 , which is called the asynchronous braking of synchronous
machines, this condition corresponds to the operation that the stator terminals are
applied on by negative-sequence voltages. On this condition, the reactance
measured from the stator terminals should be the negative-sequence reactance
x2 . From Fig. 3.3.5 we can see that the reactance during asynchronous braking of
a synchronous machine, i.e. the negative-sequence reactance, has the following
value,
x2
xqcc 1
1
1
xqcc 1 ( xcc xcc)
d
q
2
2 xdcc xqcc
(3.3.25)
xdcc xqcc
That is the negative-sequence reactance we have used in Chapter 2 to calculate
time constants.
This result is obtained under the condition of non-sinusoidal currents and
sinusoidal voltages. If voltages are non-sinusoidal and currents sinusoidal, after
neglecting the stator and rotor resistances the negative-sequence reactance
corresponding to the asynchronous braking is
x2
xdcc xqcc
(3.3.26)
2
Studying two phases sudden short circuit, we have obtained the negativesequence reactance which is x2
xdcc xqcc . Now under asynchronous braking
condition there are other expressions referring to formulas (3.3.25) and (3.3.26).
This fact shows that the negative-sequence reactance is not a constant. Its value
changes with the operation modes of synchronous machines. After neglecting the
stator and rotor resistances, the negative-sequence reactance values in some
important operation modes are given in Table 3.3.1. From this table we can see
that the negative-sequence reactance x2 is a derivative parameter of xdcc , xqcc and x0 .
Table 3.3.1 Negative sequence reactance x2
x2
Operating modes
1. Two-phase short circuit at machine
terminals
2. Two-phase short circuit at lines
3. Asynchronous braking during sinusoidal
voltage and non-sinusoidal current
(2)
2
x
x2(~)
x2( a )
xccd xqcc
xdcc xqcc
2
2 xdcc xqcc
xdcc xqcc
213
AC Machine Systems
Continued x2
Operating modes
4. Asynchronous braking during sinusoidal
current and non-sinusoidal voltage
5. Single phase-to-neutral short circuit
x2(~)
x2(1)
6. Two phases-to-neutral short circuit
x2(2 0)
xccd xccq
2
§ cc 1 ·§ cc 1 · 1
¨ xq x0 ¸¨ xd x0 ¸ x0
2 ¹©
2 ¹ 2
©
2 x0 xccd xccq
(2 x0 xdcc )(2 x0 xqcc ) xccd xccq
xdcc xccq xdcc xqcc (2 x0 xccd )(2 x0 xqcc )
2 x0 xdcc xqcc
Notes: (1) If x0
(2) The value of
(1)
2
(2)
2
(2 0)
2
(a)
2
(a)
2 is
0, then x
x and x
x ;
x2(~) is maximum and the value of x
minimum.
When these parameters are known, the value of negative-sequence reactance can
be calculated indirectly.
In non-salient pole machines, we may consider xdcc | xqcc, so
x2 | xdcc | xqcc | constant
3.4
Asynchronously Starting of Synchronous Motors
As 3-phase induction motors used in the electric drive system without speed
regulation, the 3-phase synchronous motors are also widely used, for example, in
compressors, pumps, ventilators, grinders, crushers, rolling mills, clipping
machines, saw machines and the motor-generator sets. Compared with induction
motors, synchronous motors have several advantages, such as high over-load
ability, a good power factor, the invariable speed, the larger air-gap and the
torque affected less by the supply voltage. On some conditions, especially during
large power and at low speed, a good effect can be obtained by using
synchronous motors.
The basic equations of synchronous motors in various reference axes systems
are the same as those of synchronous generators, so the theory and study methods
of synchronous machines mentioned before can all be used. The torque-slip
characteristic during asynchronous starting of synchronous motors will be
discussed in this section.
It is well known that there is the auxiliary prime-mover method, the
frequency-changing method and the asynchronous starting method so far as the
starting method of synchronous motors is concerned. Of all those methods the
asynchronous starting method is most convenient, so the method is most
214
3 Some Special Operation Modes of Synchronous Machine Systems
noticeable and widely used.
There is a damping winding on the rotor of a synchronous motor for the
asynchronous starting. However, the starting winding of the synchronous motor
has higher resistance which is made of brass, aluminum bronze or other suitable
alloys. It is developed to use massive pole-shoes for the asynchronous starting in
recent years.
During starting a synchronous motor there are two processes, i.e. the
asynchronous operation process without excitation and the process of pulling into
step with excitation.
The asynchronous operation during starting of a synchronous motor is not a
steady-state asynchronous operation process. However, because of the large
inertia of the rotor, the rotor acceleration can be neglected so far as the
electromagnetic transient process is concerned. If the electromagnetic transient
process is not considered, the whole asynchronous starting process can be
regarded as a series of steady-state asynchronous operation conditions with
different slips.
During asynchronous starting, the excitation winding of a synchronous motor
is often closed through a discharge resistance whose value is about ten times of
the excitation winding resistance. The discharge resistance is usually necessary
because an over-voltage will exist at the terminals of excitation winding which is
open-circuited during starting. Moreover, when the excitation winding is opened,
the torque of pulling into step is lower, which is also adverse to the starting
process. On the contrary, if the excitation winding is short-circuited during
starting, the starting characteristic is bad because it has a sink at about half a
synchronous speed due to the single axis phenomenon, although the over-voltage
will not exist and there is higher pull-in torque.
When a synchronous motor speeds up close to its synchronous speed, the
synchronous motor will come into the stage of pulling into step if a dc voltage is
impressed on the excitation winding. Under certain conditions, the machine will
be brought into step and run normally. On some adverse conditions, the machine
can not be pulled in and will operate asynchronously, which is undesirable. The
process and conditions of pulling into step are complicated which fall into a
nonlinear problem of electromechanical transient process, so it can only be
studied approximately by analytical method in this section and will be analysed
by numerical method later referring to [2,3].
(1) The torque of a synchronous motor during asynchronous starting
Because the asynchronous starting process can be considered as a series of
steady-state asynchronous operation conditions with various slips, some
conclusions reached in Section 3.3 can also be used here. The stator resistance of
synchronous motor is usually small and can be neglected. Incidentally, the 3-phase
voltages can be regarded as symmetrical. Under that condition, according to
equations (3.3.11) and (3.3.12) we can obtain
215
AC Machine Systems
jU
xs ( js ) I1 xD ( js ) I2* °½
¾
xD ( js ) I1 xs ( js ) I2* °¿
0
(3.4.1)
Solving the simultaneous equations above, we can get
jxs ( js )U ½
°
x ( js) xD2 ( js ) °
¾
jxD ( js)U °
xs2 ( js) xD2 ( js ) °¿
I1
2
s
I
*
2
(3.4.2)
so with reference to formula (3.3.17), Id and Iq , namely the time vectors of
d-axis current id and q-axis current iq , are
Id
Iq
½
xs ( js ) xD ( js ) U
°
2
2
x s ( js ) x D ( js )
°
°°
jU
jU
¾
xs ( js ) xD ( js ) xd ( js )
°
°
j
U
U
°
I1 I*2 j
xs ( js ) xD ( js ) xq ( js ) °¿
I1 I*2
j
(3.4.3)
or
| id ( nep ) |
j2U
xd ( js )
| iq ( nep ) |
j2U
xq ( js )
½
°
X R °°
¾
U
°
2
2 °
X qs Rqs °¿
U
2
ds
2
ds
(3.4.4)
in which U 2 | U | is the terminal voltage of the synchronous motor.
The results above substituted into equation (3.3.22) and stator resistance r
neglected, the average torque during asynchronous starting can be written as
Tecp
ª Rds
Rqs º
2
« 2
2
2 »
«¬ X ds Rds X qs Rqs »¼
U2
2
ª 1
U2
1 º
Im «
»
2
«¬ xd ( js) xq ( js ) »¼
(3.4.5)
It is noted that we still adopt the same reference directions of current and torque
as the synchronous generator when studying synchronous motors, so here exists
216
3 Some Special Operation Modes of Synchronous Machine Systems
the negative sign.
The values of xd ( js ) and xq ( js ) in the formula above with reference to
Chapter 2, can be estimated as follows.
Because
X lld Tdcc0
2
xad
X ffd
xd
Td 0
2
§
xad
1
¨
xd X lld
©
Tldc
Tdcc
X lld
Rld
Tld 0
Rld
xd Tdc
2
xad
X ffd
·
¸ Tld 0
¹
Td 0
X ffd
R fd
2
2
3
xd ( X lld X ffd xad
) xad
( X lld X ffd ) 2 xad
2
Rld ( xd X ffd xad
)
we can write out
2
p 2 ( X lld X ffd xad
) p ( X lld R fd X ffd Rld ) Rld R fd
p 2 Rld X ffd Tdcc0 pRld R fd (Td 0 Tld 0 ) Rld R fd
Rld R fd ª¬Tdcc0Td 0 p 2 (Td 0 Tld 0 ) p 1º¼ ,
2
3
2
2
2
p 2 [ X lld xad
] p ( xad
Rld xad
R fd )
2 xad
X ffd xad
2
2
p 2 [ xd ( X lld X ffd xad
) ( xd X ffd xad
) Rld Tdcc]
§T
T ·
2
Rld R fd ¨ d 0 ld 0 ¸
pxad
¨X
¸
© ffd X lld ¹
p 2 xd Rld R fd [Td 0Tdcc0 TdccTdc ]
ª x2
º
x2
pRld R fd « ad Td 0 ad Tld 0 » ,
X lld
»¼
¬« X ffd
xd ( p)
xd 2
3
2
2
2 xad
X ffd xad
p 2 ( X lld xad
) pxad
( Rld R fd )
2
p 2 ( X lld X ffd xad
) p ( X lld R fd X ffd Rld ) Rld R fd
ª x2
º
x2
p 2 xd (Td 0Tdcc0 TdccTdc ) p « ad Td 0 ad Tld 0 »
X lld
¬« X ffd
¼»
xd 2
cc
Td 0Td 0 p (Td 0 Tld 0 ) p 1
217
AC Machine Systems
ª
º
x2
x2
TdccTdc p 2 p «Td 0 Tld 0 ad Td 0 ad Tld 0 » 1
xd X ffd
X lld xd
«¬
»¼
xd u
2
Tdcc0Td 0 p (Td 0 Tld 0 ) p 1
xd
TdccTdc p 2 (Tdc Tldc ) p 1
Tdcc0Td 0 p 2 (Td 0 Tld 0 ) p 1
(3.4.6)
Similarly, there is
xq ( p )
xq xq
2
pxaq
xq pxqccTq 0
pX llq Rlq
1 pTq 0
pTqcc 1
(3.4.7)
pTq 0 1
in which
Tq 0
X llq
Rlq
Tqcc Tq 0
xqcc
xq
According to xd ( p) and xq ( p ), we can get the expressions for
1
and
xd ( js )
1
as follows:
xq ( js )
1
xd ( js )
1
xq ( js )
1 (1 s 2Td 0Tdcc0 ) js (Td 0 Tld 0 ) ½
°
xd (1 s 2TdcTdcc) js (Tdc Tldc ) °
¾
1 1 jsTq 0
°
°
xq 1 jsTqcc
¿
(3.4.8)
It is complicated to use formula (3.4.8) for calculation and the damper
parameters in that formula are not easy to be evaluated accurately. In practice,
the following approximate formulas can often be adopted for estimation.
Generally speaking, the time constant of excitation winding is much larger
than that of the damping winding, i.e. Td 0 Tld 0 and Tdc Tldc , so equation (3.4.6)
can be rewritten approximately as
xd ( p )
218
TdccTdc p 2 (Tdc Tldc ) p 1
Tdcc0Td 0 p 2 (Td 0 Tld 0 ) p 1
( pTdcc 1)( pTdc 1)
| xd
( pTdcc0 1)( pTd 0 1)
xd
3 Some Special Operation Modes of Synchronous Machine Systems
Therefore,
1
1 § 1
1 · Tcp
|
¨ ¸ d
xd ( p ) xd © xdc xd ¹ 1 Tdc p
§ 1
1 · T ccp
¨ ¸ d
cc
© xd xdc ¹ 1 Tdccp
(3.4.9)
In addition, from equation (3.4.7) we can obtain
1
xq ( p )
1 § 1 1 · Tqccp
¨ ¸
xq ¨© xqcc xq ¸¹ 1 Tqccp
The results obtained from formulas (3.4.9) and (3.4.10) during p
(3.4.10)
js are
substituted into equation (3.4.5), and the average electromagnetic torque can be
written approximately as
Tecp
U2
2
ª§ 1
§ 1
1 · sTdc
1 · sTdcc
¨ ¸
«¨ ¸
2
2
«¬© xdc xd ¹ 1 ( sTdc ) © xdcc xdc ¹ 1 ( sTdcc)
§1
1 · sTqcc º
¨ ¸
¨ xcc x ¸ 1 ( sT cc) 2 »»
q ¹
q
© q
¼
(3.4.11)
According to that formula it is clear that the average electromagnetic torque
during asynchronous starting consists of three parts; the first reflects the main
effect of excitation winding, the second and third parts represent the main
influence of starting winding. Formula (3.4.11) can give an accurate result when
the slip value is larger, but there is an obvious error during small slips because
the stator and rotor resistances will play a more important role at this time, and
the reactance value rises rapidly from subtransient or transient reactance to
synchronous reactance with the slip decreasing. However, the approximate
formula can still be used since the initial starting torque Tn (during s 1), the
pull-in torque Tb (during s
0.05) and the maximum torque Tm are most important
so far as the torque-slip characteristic for asynchronous starting of a synchronous
motor is concerned, in which the corresponding slip values are all not very small
except the slip value corresponding to the pull-in torque.
The torque-slip characteristic of a synchronous motor is shown in Fig. 3.4.1, in
which the effect of the stator resistance is neglected for curve a and the resistance
effect is considered for curve b.
Because of the asymmetrical rotor of a synchronous motor, there is the
pulsating torque with a frequency of 2sf besides the average torque during
219
AC Machine Systems
starting process. Referring to the equivalent circuit in Fig. 3.3.1, we can see that
during r 0 there is
\ 2 \ 2*
\1
jU
\1*
jU *
0
The results above substituted into equation (3.3.15), the amplitude of pulsating
torque with a frequency of 2sf during starting process of a synchronous motor
can be described as
Tm (2 s )
j2 u 2(U ) 2 xD ( js )
xs2 ( js ) xD2 ( js )
U2 1
1
2 xq ( js ) xd ( js )
U2
2
( X ds X qs )2 ( Rds Rqs )2
( X ds2 Rds2 )( X qs2 Rqs2 )
(3.4.12)
The relation between the amplitude of pulsating torque with a frequency of 2sf
and the slip during starting process of a synchronous motor is shown in Fig. 3.4.2.
Because the rotor inertia of synchronous motors is ordinarily larger, the effect of
the pulsating torque on starting process is not important and can be neglected in
practical calculation.
From Fig. 3.4.2 we are also clear that the amplitude of pulsating torque will
rise to a large value with the slip decreasing and the corresponding pulsating
Figure 3.4.1 Torque-slip characteristic of a
synchronous motor
220
Figure 3.4.2 Pulsating torque amplitude
with frequency of 2sf-slip curve during the
starting process of a synchronous motor
3 Some Special Operation Modes of Synchronous Machine Systems
frequency (2sf )is small which is several Hz, so the speed oscillation may exist
near the synchronous speed, too.
According to equation (3.4.12) we can know that the pulsating torque with a
frequency of 2sf depends upon the asymmetrical condition of synchronous
machine rotor. When the rotor is symmetrical and there is xd ( p) xq ( p ) such as
in an induction motor, the pulsating torque will be zero. During s 0, i.e.
§1
1
1 ·
synchronous operation, the amplitude of pulsating torque will be U 2 ¨ ¸ ,
¨
2 © xq xd ¸¹
i.e. the torque amplitude at the synchronous speed will become the maximum
value of the salient synchronous torque. In addition, from equation (3.4.5) we are
also clear that the average electromagnetic torque Tecp (i.e. the asynchronous
torque) during s 0 will be equal to zero. That is natural, because during s 0 ,
i.e. the steady-state synchronous operation, the supply voltage will produce the
salient synchronous torque besides causing the synchronous torque with the
excitation current, but no other torque exists referring to Section 2.5.
(2) The calculation method of asynchronous starting torque of synchronous
motors according to the d-axis and q-axis equivalent circuits
Since a synchronous motor usually has a strong starting winding, the value of
xd ( js ) is close to that of xq ( js ) , i.e. the value of xD ( js ) is small. Accordingly, the
value of I2 is also not large referring to Fig. 3.3.1 and equation (3.3.11), so exists
jU
xs ( js ) I1 xD ( js ) I2* jrI1 | xs ( js ) I1 jrI1
i.e.
2U | {[r jxd ( js )] [r jxq ( js )]}( I1 )
(3.4.13)
which corresponds to the equivalent circuit shown in Fig. 3.4.3. If the difference
between xd ( js ) and xq ( js ) is small, that equivalent circuit can also be divided
into two equivalent circuits as shown in Fig. 3.4.4.
Figure 3.4.3 Equivalent circuit corresponding to formula (3.4.13)
221
AC Machine Systems
Figure 3.4.4 Corresponding equivalent circuits during small difference between
xd ( js ) and xq ( js )
Because xd ( js ) is ordinarily smaller than xq ( js ), Id is a little more than I1
and Iq is a little less than I1 .
2r
caused by the stator resistance
Therefore, when the small torque | 2 I2 |2
1 2s
is neglected referring to equation (3.3.21), the average torque during asynchronous
starting is
1
[| id ( nep ) |2 Rds | iq ( nep ) |2 Rqs ]
2
2[| I1 I2* |2 Rds | I1 I2* |2 Rqs ]
| 2[| Id |2 Rds | Iq |2 Rqs ]
1
[| 2 Id |2 Rds | 2 Iq |2 Rqs ]
2
(3.4.14)
The equivalent circuits can be reproduced as those in Fig. 3.4.5 and then the
average electromagnetic torque during asynchronous starting according to
equation (3.4.14) will correspond to the average value of two starting torques of
the following two induction motors: one has a symmetrical rotor with d-axis
characteristic and another has a symmetrical rotor with q-axis characteristic.
Figure 3.4.5 Equivalent circuits reproduced referring to Fig. 3.4.4
Some factories have adopted the method to evaluate the asynchronous starting
torque of a synchronous motor before the dc voltage is applied to excitaion
winding. The error of the calculation method will increase with the values
1
of xD ( js )
[ xd ( js ) xq ( js )] and stator resistance r rising. However, the error
2
222
3 Some Special Operation Modes of Synchronous Machine Systems
of this approximate method is usually not large except near the slip of 0.5. The
starting characteristic of a typical synchronous motor is shown in Fig. 3.4.6 for
reference, in which Ted is the torque of induction motor with d-axis characteristic,
Teq is the torque with q-axis characteristic and Tecp is the resultant starting torque.
Figure 3.4.6 Starting characteristic of a typical synchronous motor
(3) The starting torque after a dc voltage is applied to the excitation winding of
a synchronous motor
In order to pull a synchronous motor into step, a dc source must be provided
for the excitation winding when the rotor speed rises near to synchronous speed.
If the transient process is neglected, the average asynchronous torque produced
by excitation current, with reference to equation (3.3.21) and Fig. 3.3.1(a), is
Tecp ( f ) | 2 I3 |2
r
1 s
[r 2 (1 s )2 xq2 ](1 s )r
[r 2 (1 s )2 xd xq ]2
E2
(3.4.15)
Because stator resistance is small, we can take
r2
r3
r4
0
Those values substituted into formula (3.4.15), we have
Tecp ( f )
r §E·
|
¨ ¸
1 s © xd ¹
2
(3.4.16)
whose value is not large and not more than twenty percent of the rated torque and
whose sign is opposite to that of the average torque produced by the ac supply,
223
AC Machine Systems
i.e. the torque is a braking one. Therefore, it is adverse to provide an excitation
current unduly early.
Besides the average torque, the pulsating torque with a frequency of sf is
caused by the excitation current referring to equation (3.3.15). Neglecting stator
resistance and referring to equation (3.3.13) and Fig. 3.3.1, we have
I3
E
, \ 3
2 xd
0, \1
jU , \ 2
0
The values above substituted into formula (3.3.15), the amplitude of pulsating
torque with a frequency of sf will be
Tm ( s )
2 | I2*\ 3 I3\ 2* I3*\1 I1\ 3* |
E
2 jU
2 xd
EU
xd
(3.4.17)
That is the well known amplitude of the synchronous torque whose value is
large, so the excitation current will give rise to the serious oscillation of the power,
current, voltage and speed during a larger slip. From that view point, it is also
adverse to apply a dc voltage to excitation winding unduly early. However, that
torque will play an important role in the pull-in process since the torque will
become the synchronous one ultimately.
3.5
Analysis of Combination System for Synchronous
Machine and Capacitances
In order to raise the technical and economic targets of the electrical power
transmission, the capacity, voltage and distance of electrical power systems are
increasing day by day, but the capacitances between long transmission lines and of
lines to ground are large enough. Moreover, for the sake of raising the capacity of
power transmission, the series connection of capacitors is used to compensate the
inductances of long transmission lines. Therefore, the problem about synchronous
machines in connection with capacitances is obviously more important.
When there are capacitances in electrical power systems, some special
problems may be brought about such as the self-excitation and inner over-voltage
in synchronous machines referring to [2,3,10]. In the early days of 1970’s there
were two severe accidents in which the main shafts of two large turbo-generators
were damaged by torsion because series capacitors result in self-excitation
currents of low-frequencies. Thus, it is significant to analyse these special
problems when synchronous machines are in connection with capacitances and to
take corresponding measures.
(1) Basic equations during a synchronous machine in series with symmetrical
capacitances
224
3 Some Special Operation Modes of Synchronous Machine Systems
According to Fig. 3.5.1 we can write the following equations:
ua
ub
uc
x · ½
§
p\ a ¨ r c ¸ ia °
p¹ °
©
x · °°
§
p\ b ¨ r c ¸ ib ¾
p¹ °
©
x · °
§
p\ c ¨ r c ¸ ic °
p ¹ ¿°
©
(3.5.1)
or
pua
pub
puc
p 2\ a ( pr xc )ia ½
°
p 2\ b ( pr xc )ib ¾
°
p 2\ c ( pr xc )ic ¿
(3.5.2)
Figure 3.5.1 The system of synchronous machine in series with capacitances
Because
ud
uq
2
½
[ua cos J ub cos(J 120e
) uc cos(J 120e] °
°
3
¾
2
) uc sin(J 120e] °
[ua sin J ub sin(J 120e
°¿
3
(3.5.3)
we have
pud
puq
2
½
[cos J pua cos(J 120e
) pub cos(J 120e
) puc ] °
3
°
2
) uc sin(J 120e
)]Z °
[ua sin J ub sin(J 120e
°
3
¾
2
) pub sin(J 120e
) puc ]°
[sin J pua sin(J 120e
°
3
°
2
) uc cos(J 120e
)]Z °
[ua cos J ub cos(J 120e
3
¿
(3.5.4)
225
AC Machine Systems
By equations (3.5.2) and (3.5.3), the formulas above can be transformed into
the following forms,
pud uqZ
puq ud Z
2
½
[cos J p 2\ a cos(J 120e
) p 2\ b cos(J 120e
) p 2\ c ] °
3
°
2
) pib cos(J 120e
) pic ] °
r[cos J pia cos(J 120e
°
3
°
2
°
)ib cos(J 120e
)ic ]
xc [cos J ia cos(J 120e
°
3
¾ (3.5.5)
2
) p 2\ b sin(J 120e
) p 2\ c ]°
[sin J p 2\ a sin(J 120e
°
3
°
2
°
r[sin J pia sin(J 120e
) pib sin(J 120e
) pic ]
3
°
°
2
xc [sin J ia sin(J 120e
)ib sin(J 120e
)ic ]
°
3
¿
Because
p\ d
p\ q
p 2\ d
2
p \q
2
[cos J p\ a cos(J 120e
) p\ b cos(J 120e
) p\ c ] \ qZ
3
2
[sin J p\ a sin(J 120e
) p\ b sin(J 120e
) p\ c ] \ d Z
3
2
[cos J p 2\ a cos(J 120e
) p 2\ b cos(J 120e
) p 2\ c ]
3
2
) p\ b sin(J 120e
) p\ c ]Z
[sin J p\ a sin(J 120e
3
Z p\ q \ q pZ
½
°
°
°
°
°
°
¾
2
) p 2\ b sin(J 120e
) p 2\ c ] °
[sin J p 2\ a sin(J 120e
°
3
°
2
) p\ b cos(J 120e
) p\ c ]Z °
[cos J p\ a cos(J 120e
°
3
°
Z p\ d \ d pZ
¿
½
°°
¾ (3.5.6)
°
°¿
(3.5.7)
we can obtain
2
½
[cos J p 2\ a cos(J 120e
) p 2\ b cos(J 120e
) p 2\ c ] °
3
°
p 2\ d 2Z p\ q \ d Z 2 \ q pZ ;
°°
¾
2
) p 2\ b sin(J 120e
) p 2\ c ]°
[sin J p 2\ a sin(J 120e
°
3
°
2
2
p \ q 2Z p\ d \ qZ \ d pZ
°¿
226
(3.5.8)
3 Some Special Operation Modes of Synchronous Machine Systems
In addition, there is
pid
piq
2
½
[cos J pia cos(J 120e
) pib cos(J 120e
) pic ] iqZ °
°
3
¾
2
[sin J pia sin(J 120e
) pib sin(J 120e
) pic ] id Z °
°¿
3
(3.5.9)
Substituting equations (3.5.9) and (3.5.8) into (3.5.5) and simplifying them, we
can get
p (ud p\ d rid \ qZ ) xc id
Z (uq p\ q riq \ d Z )
(3.5.10)
p (uq p\ q riq \ d Z ) xc iq
Z (ud p\ d rid \ qZ )
(3.5.11)
According to equation (3.5.1) and the transformation formula for the zero-axis
component, it is not difficult to obtain
u0
§
x ·
p\ 0 ¨ r c ¸ i0
p¹
©
(3.5.12)
Equations (3.5.10), (3.5.11) and (3.5.12) are the basic equations when a
synchronous machine is in series with symmetrical capacitances. On the basis of
those equations, some problems about the system of synchronous machine in
series with capacitances can be analysed.
(2) Self-excitation of synchronous machines
If the parameters of a synchronous machine are not a suitable match for those
of electrical power system including capacitances, self-excitation may occur in
the synchronous machine system at synchronous or asynchronous speed. Under
this condition, regulation of the excitation current can not control the current and
voltage of the synchronous machine, and those quantities depend mainly upon
the parameters of the synchronous machine and power system. During
self-excitation, the mechanical energy of the synchronous machine will be
exchanged with the electromagnetic energy of the machine and power system,
which falls into the problem of electromechanical parameter-resonance. Because
the inertia constant of synchronous machine is ordinarily large, for convenience
it can be taken as infinite, i.e. the speed of synchronous machine can be
considered as constant during analysis of this problem. Thus, self-excitation of
synchronous machines will be reduced to a problem of electro magnetic
parameter-resonance. During self-excitation of synchronous machines, there is
not only the current of supply-frequency but also the self-excitation current
caused by parameter-resonance in the machine and power system, which will
227
AC Machine Systems
bring about the current increasing constantly or make the machine operate
abnormally. Now we analyse the causes and conditions producing self-excitation
so as to protect the system against it.
Because the excitation voltage of synchronous machine can not be controlled
during self-excitation, for convenience the excitation voltage may be supposed to
be zero, i.e. U fd 0. Thus,
\d
\q
xd ( p)id ½°
¾
xq ( p )iq °¿
(3.5.13)
Furthermore, supposing the speed of a synchronous machine to be asynchronous
constant one and letting it be Z 1 s, we can reduce the basic equations
(3.5.10) and (3.5.11) to the following forms:
{[(1 s ) 2 p 2 ]xd ( p )
½
°
pr xc }id (1 s )[2 pxq ( p ) r ]iq °
¾
(1 s )[2 pxd ( p ) r ]id
°
2
2
{[(1 s ) p ]xq ( p) pr xc }iq °¿
(3.5.14)
½
°
{[ p 2 (1 s) 2 ](1 s) xq ( p) (1 s) xc }uq º¼ A( p ) °°
¾
ª¬{[ p 2 (1 s) 2 ](1 s) xd ( p ) (1 s ) xc }ud
°
°
{[ p 2 (1 s )2 ]Z d ( p) pxc }uq ¼º A( p )
°¿
(3.5.15)
pud (1 s )uq
puq (1 s )ud
Solving them, we can obtain
id
iq
ª¬{[ p 2 (1 s ) 2 ]Z q ( p) pxc }ud
in which
A( p ) [ p 2 (1 s )2 ][ Z d ( p) Z q ( p) (1 s ) 2 xd ( p) xq ( p)]
xc [ pZ d ( p ) pZ q ( p ) (1 s ) 2 xd ( p) (1 s )2 xq ( p ) xc ]
Z d ( p)
pxd ( p ) r
Z q ( p)
pxq ( p) r
From the formulas above we can see that when the roots of the characteristic
equation [ A( p) 0] are positive values or complex-numbers with positive real
parts, currents id and iq will increase constantly, i.e. self-excitation occurs.
Therefore, the problem whether there is self-excitation has become the problem
whether the equation A( p) 0 has a positive real root or complex-number root
with a positive real part.
228
3 Some Special Operation Modes of Synchronous Machine Systems
There are many methods to identify if the equation has positive real roots or
complex-number roots with positive real parts. The usual methods are Herwitz’s,
Routh’s and Mihainov’s criterions.
The synchronous machine having a damping winding, the order of the equation
A( p) 0 is higher and the approximate digital solution can be obtained by a
computer. However, in order to make the problem clear, we shall analyse only
the self-excitation of synchronous machines without damping winding in the
following. Under this condition, we have
xd ( p)
xd xq ( p )
xq
s
2
pxad
pX ffd R fd
xdc Td 0 p xd
Td 0 p 1
0
so
A( p )
xdc xqTd 0 p 5 [ xd xq rTd 0 ( xdc xq )] p 4
[2 xdc xqTd 0 r ( xd xq ) Td 0 xc ( xdc xq ) Td 0 r 2 ] p 3
[2 xd xq rTd 0 ( xdc xq ) xc ( xd xq ) r 2 2rxcTd 0 ] p 2
[ xdc xqTd 0 r ( xd xq ) Td 0 xc ( xdc xq ) 2rxc Td 0 ( xc2 r 2 )] p
[ xd xq xc ( xd xq ) xc2 r 2 ]
(3.5.16)
That is the fifth degree equation and it is tedious to write the corresponding
discriminant. However, the practical values given, calculation is convenient. For
example, a machine has the following parameters:
xd
1.0,
xdc
0.30,
xq
0.6,
Td 0
1 000
According to those values, the parameter range causing self-excitation can be
determined as shown by full line in Fig. 3.5.2.
Figure 3.5.2 Self-excitation region of synchronous machine
229
AC Machine Systems
The self-excitation under several special conditions is discussed as follows.
a. Neglecting stator resistance (r 0)
Under this condition, equation (3.5.16) is turned into the following form:
A( p )
xdc xqTd 0 p 5 xd xq p 4 Td 0 [2 xdc xq xc ( xdc xq )] p3
[2 xd xq xc ( xd xq )] p 2 Td 0 [ xdc xq xc ( xdc xq ) xc2 ] p
[ xd xq xc ( xd xq ) xc2 ]
(3.5.17)
From Routh’s and Herwitz’s criterions we can see that the self-excitation can
be produced only under the following condition:
xc xq and xq xc xd or xc xd
(3.5.18)
Therefore, when stator resistance is neglected and there is xc xd , the
self-excitation will be caused.
In practice, because of the stator resistance the self-excitation region is smaller
than above; ordinarily the region in xdc xc xd . That conclusion can be known
from Fig. 3.5.2.
b. The excitation winding open-circuited (Td 0 0)
Under this condition, equation (3.5.16) will be simplified. The self-excitation
can be discriminated to be caused in the following region:
xd xq ( xd xq ) 2 4r 2
2
xc xd xq ( xd xq )2 4r 2
2
(3.5.19)
Obviously, the shade-line area in Fig. 3.5.2 corresponds to the self-excitation
region indicated by formula (3.5.19).
c. Neglecting excitation winding resistance (Td 0 f)
Under this condition, the self-excitation will be caused in the following region:
xdc xq ( xdc xq ) 2 4r 2
2
xc xdc xq ( xdc xq )2 4r 2
2
(3.5.20)
According to formulas (3.5.19) and (3.5.20) we can see that there is no
xd xq
xq xdc
or r !
. That conclusion is also known
self-excitation when r !
2
2
from the typical example in Fig. 3.5.2, in which the self-excitation region
indicated by formula (3.5.20) is shown by a dotted line. That self-excitation
region, i.e. the region when Td 0 f, is very close to the self-excitation region
between xq and xdc in a practical machine with larger Td 0 . Because the typical
generator without damping winding has large Td 0 , its self-excitation range can be
discriminated by formulas (3.5.19) and (3.5.20).
230
3 Some Special Operation Modes of Synchronous Machine Systems
On the basis of formula (3.5.19) it is not difficult to understand that there is no
self-excitation when xd xq and the excitation winding is open-circuited, i.e. a
non-salient-pole synchronous machine, when its excitation winding is
open-circuited, has no self-excitation. However, when the excitation winding is
short-circuited and xdc z xq , this type of machine may be self-excited referring to
formula (3.5.20). That is because the salient-pole machine, when its excitation
ª1
º
§ 1
1 ·
winding is open-circuited, can develop the active power « U 2 ¨ ¸ sin 2G »
«¬ 2 ¨© xq xd ¸¹
»¼
due to its asymmetrical magnetic circuit to balance the stator resistance loss, but
the non-salient-pole machine has no possibility. However, when the excitation
winding of a non-salient-pole machine is short-circuited, it can produce an active
ª1
º
§ 1
1·
power « U 2 ¨ ¸ sin 2G » due to the asymmetrical electric circuit of its
«¬ 2 ¨© xdc xq ¸¹
»¼
rotor to compensate the stator resistance loss. The first self-excitation
phenomenon is called the salient-pole synchronous self-excitation, and the latter
the repulsion synchronous one. The salient-pole synchronous self-excitation is
still maintained when the stator flux-linkage is constant, so this self-excitation
can finally reach a steady-state value and the growing rate of its current and
voltage is generally slow. However, the repulsion synchronous self-excitation
will grow constantly at a higher rate, or else the corresponding transient current
can’t be induced in the rotor circuit and there is no parameter xdc .
During the two synchronous self-excitation conditions above, stator selfexcitation current has supply-frequency and the machine has a synchronous
speed. The current during this self-excitation is shown in Fig. 3.5.3.
Figure 3.5.3 Stator self-excitation current during synchronous self-excitation
During self-excitation, the oscillation period of the stator self-excitation current
depends upon the inductance and capacitance in the stator circuit. Therefore, in
the stator there can also be another self-excitation current whose frequency is
different form the supply-frequency when the machine has a synchronous speed.
This self-excitation phenomenon is called the asynchronous self-excitation one.
Under the condition of asynchronous self-excitation, the speed of air-gap field
produced by the self-excitation current is different from the rotor speed. Letting
231
AC Machine Systems
Z f be the speed of air-gap field, Z s the synchronous speed, i.e. the practical rotor
speed and s
Z f Zs
the slip, there is a current of frequency sf in the rotor
Zs
circuit. If the rotor circuit is asymmetrical, the stator current will contain two
components of the frequencies (1 r s) f as shown in Fig. 3.5.4.
Figure 3.5.4 Stator self-excitation current during asynchronous self-excitation
When the rotor is symmetrical as in an induction machine, the stator current
contains only one component of frequency (1 s ) f and the slip s must be
negative. Under this condition, the induction generator will develop an active
power to compensate the stator resistance loss.
As discussed above, the machine is supposed to be a synchronous generator
with a synchronous speed. If the machine is a synchronous motor with a
synchronous speed, the conclusions above can also be utilized. When the
machine is a starting synchronous motor, i.e. its speed is below the synchronous
speed, the salient-pole synchronous self-excitation or asynchronous self-excitation
can also occur. However, the stator current frequency during the salient-pole
synchronous self-excitation must correspond to the rotor speed. For example,
when that self-excitation takes place at a synchronous speed, the self-excitation
current has a frequency of 50Hz; when the self-excitation appears at half a
synchronous speed, the self-excitation frequency is 25Hz. If there is an
asynchronous self-excitation in the starting process, the stator self-excitation
frequency will be lower, because only in this way the speed of the revolving field
produced by the asynchronous self-excitation current is below the rotor speed to
make the machine as an induction generator to maintain the asynchronous
self-excitation. During the repulsion synchronous self-excitation, the stator
self-excitation frequency also corresponds to the rotor speed. As mentioned
before, this type of self-excitation will increase constantly at a higher rate,
through which the self-excitation can exist. There is the self-excitation
phenomenon when the d-axis and q-axis transient parameters are not equal to
each other. If a pair of complex-number roots in the characteristic equation of the
asynchronous self-excitation change to a pair of equal roots, the asynchronous
self-excitation is turned into the repulsion synchronous self-excitation, so it is
difficult to differ from them.
232
3 Some Special Operation Modes of Synchronous Machine Systems
Because the salient-pole synchronous self-excitation can appear only under the
larger capacitance condition, the asynchronous self-excitation is a phenomenon
which often occurs in practice and must be prevented first. During the asynchronous
self-excitation, the speed of the revolving filed produced by the self-excitation
current is below the synchronous speed and there is an interaction between that
revolving field and the synchronous rotating-field produced by the supplyfrequency current to produce the pulsating torque and cause oscillation and noise.
If the frequency of the pulsating torque is equal to or near the natural frequency
of the main-shaft torsion-oscillation, the mechanical resonance may occur to
make the main-shaft broken or damaged, which is termed sub-synchronous
resonance. As mentioned before, the main shafts of two large turbogenerators
were damaged due to that cause.
When a rated voltage is applied to a generator without excitation the
corresponding stator current is named the charge current of a generator.
Obviously, the current is I Z U N / xd 1/ xd and the corresponding apparent
power is called the generator charge-capacity, which is PZ
U N2
xd
1
.
xd
Similarly, the apparent power of a capacitor-load at rated voltage is called the
U N2
1
. Therefore,
charge capacity of a capacitor-load which is equal to PZN
xc
xc
in order to protect the machine against any self-excitation phenomenon, it is
essential to keep xc ! xd or PZ ! PZN , i.e. the generator charge-capacity must be
larger than the charge capacity of a capacitor-load.
(3) The transient process when a synchronous machine is suddenly connected
to a capacitor-load
This problem is significant in practice. For example, the synchronous machine
with no-load is abruptly connected to a long-distance transmission line or the
transmission line in series with capacitors is suddenly short-circuited, which all
fall into that scope.
Now let us discuss the transient process when the synchronous machine
rotating at a synchronous speed without load is suddenly connected to a 3-phase
symmetrical capacitor-load.
Under this condition, we can equally consider that the generator in series with
capacitors is suddenly short-circuited. By the Superposition Theorem, it may be
thought that the generator is not short-circuited but suddenly applied on by a
voltage which is equal but opposite to original no-load voltage. Substituting the
following relations:
s
0, ud
0,
uq
E
into equation (3.5.15), we can obtain
233
AC Machine Systems
id
iq
( p 2 1) xq ( p ) xc
½
E °
A( p)
°
¾
2
( p 1) Z d ( p ) pxc °
E°
A( p )
¿
(3.5.21)
The equation can be solved by Operational Calculus, but the corresponding
characteristic equation, due to capacitors, has a higher order. Therefore, it is more
difficult to solve the equation. In practice, we only discuss its maximum short
circuit current and maximum voltage drop in the capacitor, because those
quantities are significant for use of the machine and choice of protection
equipment. Solving those quantities we can also neglect the stator and rotor
resistances much as in 3-phase short circuit.
After obtaining id and iq , we can get the maximum value of current ia by
virtue of the axes-transformation formulas and calculate the voltage drop in a
capacitor according to the following formula:
u xc
1
ia dt
C³
xc ³ ia dt
(3.5.22)
The possible maximum value of voltage u xc is generally not more than triple
the rated phase-voltage. However, the capacitor-load being asymmetrical, especially
the machine having no damping winding, the operating over-voltage can be
severe, whose causes are: 1) even if there is no capacitor, under the condition of
asymmetrical short circuit, there are a series of harmonics in the stator current
due to the asymmetrical rotor, which can result in higher instantaneous value of
the voltage; 2) when there is capacitor, the capacitance and inductance of the
machine and transmission line will produce resonance at a certain frequency,
so the instantaneous value of the voltage will further increase. Thus, when
xc n 2 xdc xq , the over-voltage is specially severe. n is positive and generally
n 3 and n 5 are most important. When a machine has damping winding, the
over-voltage will obviously decrease because the damping winding makes the
ratio xqcc xdcc close to 1. If there is only the q-axis damping winding, the effect is
also obvious. As mentioned before, when xqcc xdcc 1, the harmonics are not easy
to appear.
The conclusions above are obtained on the basis of the machine operating
separately. If the machine is parallel to other machines with damping windings or
it has an enough resistance or induction-motor-load, the hazard of over-voltage
can decrease.
234
3 Some Special Operation Modes of Synchronous Machine Systems
3.6
Mathematic Model and Performance Analysis of
Cycloconverter-fed Synchronous Motor Systems
with Field-Oriented Control
Cycloconverters can change the source frequency to lower frequency that is
under 1/ 3 1/ 2 of source frequency. Cycloconverter-fed synchronous motor
systems with field-oriented control are widely used to drive slow speed and large
capacity production machines such as steel-rolling machine, grinder, minehoister, etc. In comparison with ac-dc-ac frequency converter, the cycloconverter
has the following features:
(i) The intermediate dc filter can be saved and there is only one stage powerconverting circuit, so its efficiency is high.
(ii) The source can convert current directly, so there is no need for auxiliary
current-converting circuit.
(iii) A cycloconverter generally consists of 36 thyristors, so its structure is
complicated.
In general, large cycloconverters are of voltage type and the cycloconverter-fed
synchronous motor system is shown in Fig. 3.6.1, which is composed of a 3-phase
rectiformer with 3 separated secondary windings for each phase, cycloconverter,
synchronous motor, position and speed sensors, and field-oriented vector control
system, referring to [4,7].
Figure 3.6.1 A cycloconverter-fed synchronous motor system
The output voltage of cycloconverter is pieced together by the segments of its
input voltage, so there are plentiful harmonics and the harmonic frequency of its
output voltage is
f0h
KPfi r (2n 1) f 0
(3.6.1)
The harmonic frequency of its input current is
f ih
( KP r 1) fi r 6nf 0
(3.6.2)
235
AC Machine Systems
in which fi and f 0 are the fundamental frequency of input and output respectively,
k 1, 2,3," , n 0,1, 2," , and P is number of pulses for the cycloconverter.
Firstly the mathematical pattern is built up for the cycloconverter-fed
synchronous motor speed regulation system, then the steady-state and transient
performance can be analysed for the system, and finally some useful conclusions
will be drawn for design of damping winding for the cycloconverter-fed
synchronous moter.
3.6.1
Mathematical Patterns of the Cycloconverter-fed
Synchronous Motor System
The cycloconverter-fed synchronous motor system with field-oriented control
consists of synchronous motor, cycloconverter and field-oriented control system,
whose patterns will be discussed respectively as follows.
(1) Mathematical pattern of synchronous machine
The multi-loop pattern is adopted for the synchronous machine to consider the
influence of spacial harmonics in air-gap field fully. In addition, damping winding
has an important effect on the performance of cycloconverter-fed synchronous
machine, especially the dynamic performance. The multi-loop pattern is to write
down the corresponding equations according to actual damping winding, so the
actual currents of various damping loops can be got, which is essential for design
of damping winding about this type of synchronous machine.
The voltage equations can be written down respectively for 3-phase stator
windings and excitation winding of synchronous machine, and then take d-axis
and q-axis as central lines respectively to write loop voltage equations. For
example, there are 6 damping bars per pole in Fig. 3.6.2, so for d-axis exist 3
damping loops and for q-axis also exist 3 damping loops.
Figure 3.6.2 Damping loops selected
The voltage equation set of synchronous machine is
U
236
L
d
I (R Z G) I
dt
(3.6.3)
3 Some Special Operation Modes of Synchronous Machine Systems
in which
U
[ua ub uc u fd 0 0 0 0 0 0]T
is voltage matrix and
I
[ia ib ic i fd i1d i2 d i3d i1q i2 q i3q ]T
dL
and T is rotor
dT
position angle; L and R are inductance and resistance matrices respectively and
there is
is current matrix; Z is electric angular velocity of rotor; G =
ª Lss
«M
¬ rs
L
M sr º
Lrr »¼
where
ª Laa
«
« M ab
«¬ M ac
Lrr
ª Lrrd
«M
¬ rqd
M rdq º
, L
Lrrq »¼ ss
M sr
ª M afd
«
« M bfd
« M cfd
¬
M a1d
M b1d
M c1d
M rs
M srT
Lrrd
ª L ffd
«M
« f 1d
«M f 2d
«
¬« M f 3d
M f 1d
L11d
M 12 d
M 13d
M f 2d
M 12 d
L22 d
M 23d
Lrrq
ª L11q
«
« M 12 q
« M 13q
¬
M 12 q
L22 q
M 23q
M 13q º
»
M 23q »
L33q »¼
M a 2d
M b 2d
M c2d
M a 3d
M b 3d
M c3d
M ab
Lbb
M bc
M a1q
M b1q
M c1q
M ac º
M bc »»
Lcc »¼
M a 2q
M b2q
M c 2q
M a 3q º
»
M b 3q »
M c 3q »¼
M f 3d º
M 13d »»
M 23d »
»
L33d ¼»
When considering the fundamental and odd harmonic air-gap fields only, M rqd
and M rdq are all zero-matrices.
The resistance matrix R is
R
ª Rs
«
¬
º
Rr »¼
237
AC Machine Systems
where
Rs
Rr
ª r1
º
«
»
r1
«
»
«¬
r1 »¼
ª rfd
«
r1d r12 d
«
«
r12 d r2 d
«
r13d r23d
«
«
«
«
«
¬
r13d
r23d
r3d
r1q
r12 q
r13q
r12 q
r2 q
r23q
º
»
»
»
»
»
r13q »
»
r23q »
r3q »¼
In the formulas mentioned above, the subscripts a, b, c represent stator 3-phases
respectively; the subscripts fd , 1d , 2d , 3d , 1q, 2q, 3q indicate excitation winding
and various loops on d, q axes; r1 is stator phase resistance.
The electromagnetic torque formula is
Tem
I T GI
(3.6.4)
dZ
dt
TL Tem
(3.6.5)
and the rotor motion equation is
H
Calculation of various inductances in above formulas has been discussed in
Chapter 1.
(2) Mathematical pattern of cycloconverter
The main circuit of 3-phase bridge-type cycloconverter with 6 pulses is shown
in Fig. 3.6.3, in which the input terminals for 3 sets of current converters are
supplied by a 3-phase rectiformer with 3 separated secondary windings for each
phase, and each phase is composed of 12 thyristors, i.e. two 3-phase rectifierbridges controlled fully are connected in parallel inversely.
For cycloconverters the cosine-intersection method is usually adopted to
control thyristors conducting. The main circuit of i-th phase (i a, b, c) for
3-phase bridge-type cycloconverter with 6 pulses is shown in Fig. 3.6.4, in which
various thyristors are triggered into conduction according to the sequence of
1 2 3 4 5 6 1, and the triggering time and conducting length of time can
be determined for various thyristors by use of cosine-intersection method.
When considering the secondary leakage inductances of 3-phase rectiformer,
without circulating current the 3-phase bridge-type cycloconverter with 6 pulses
has three conducting modes, namely normal two thyristors conducting mode, two
238
3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.6.3 The main circuit of 3-phase bridge-type cycloconverter with 6 pulses
Figure 3.6.4 The main circuit of i-th phase (i
a, b, c) for 3-phase bridge-type
cycloconverter with 6 pulses
thyristors converting mode, and positive and negative sets interchanging mode,
which are illustrated as follows.
(a) Normal two thyristors conducting mode
Supposing that thyristors 1 and 2 of i-th phase are conducted as shown in
Fig. 3.6.5(a), then the effective voltage impressed on the cycloconverter is
U iNc
U as U cs
2L f
d
ii U iN
dt
(3.6.6)
in which U iN is the voltage between terminal i and converter neutral point N, ii is
the i-th phase load current, and L f is the phase leakage inductance of rectiformer
secondary windings.
239
AC Machine Systems
Figure 3.6.5 Conducting modes of cycloconverter with 6 pulses
(a) Normal two thyristors conducting mode;
(b) Two thyristors converting mode.
(b) Two thyristors converting mode
For rectiformer secondary windings exists leakage inductance L f , so switching-on
and switching-off of thyristors can’t be accomplished in an instant, and the load
current is transited from the switching-off thyristor to switching-on thyristor
gradually that is two thyristors converting mode.
If thyristors 1 and 2 of i-th phase are conducting, then thyristor 3 is triggered
to cause current converting from thyristor 1 to thyristor 3 and in the meantime
thyristors 1, 2, 3 are conducting simultaneously as shown in Fig. 3.6.5(b).
The loop voltage equation for ii is
U as U cs
2Lf
d
d
ii L f ihi U iN
dt
dt
where ihi is the converting loop current. The converting loop voltage equation is
U bs U as
2L f
d
d
ihi L f ii
dt
dt
During ihi ii the current conversion is finished. Deleting ihi from the two formulas
above, we can get the cycloconverter’s voltage equation under this mode as
follows:
U iNc
U as U bs
U cs
2
1.5 L f
d
ii U iN
dt
(3.6.7)
which is the effective voltage impressed on the cycloconverter under this mode.
Synthesizing formulas (3.6.6) and (3.6.7), the generalized expression of
cycloconverter’s voltage equation can be got under two thyristors conducting
mode and two thyristors converting mode of i-th phase as follows:
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3 Some Special Operation Modes of Synchronous Machine Systems
U iNc
L fi
d
ii U iN
dt
(3.6.8)
where
L fi
­° 2 L f
®
°̄1.5 L f
U iN
for two thyristors conducting mode
for two thyristors converting mode
U iO U NO
and subscripts N,O indicate the neutral points of cycloconverter and synchronous
motor stator winding respectively.
(c) Positive and negative sets interchanging mode
When a phase load-current changing its direction for the cyclconverter with 6
pulses and without circulating current, the cycloconverter needs to go into the
positive and negative sets interchanging stage, and in the meantime all thyristors
of positive and negative stes for this phase are blockaded to achieve cycloconverter’s
switching-over without circulating current, which is termed the dead-region stage
and its persistence time is about 2 8ms.
If i-th phase is under positive and negative sets interchanging mode, then exists
ii
0
(3.6.9)
Solving the state equation, we can decrease one dimension for state variables.
In the meantime, the thyristors of this phase are blocked and the output voltage
of cycloconverter is determined not ty cycloconverter itself but by the load
voltage impressed on this phase, so there is
U iNc
U iO U NO
(3.6.10)
(3) Mathematical pattern of air-gap flux-linkage oriented control system
In the mathematical pattern of air-gap flux-linkage oriented control system for
cycloconverter-fed synchronous motor, the speed regulator, flux regulator and
current regulator all use proportional-integral regulation, whose transfer function is
W ( p)
Kp
1 TN p
TN p
where K p is amplification and TN is integral time constant. Various feedback
quantities in the control system all contain filtering links and the amplitude
limiters are provided for various regulators. By use of the mathematical pattern
above make simulative calculation and experiment for a 4.5kW-reverse-mounted
cycloconverter-fed synchronous motor with field-oriented control, and the
simulative results are approximate to the experimental data, which also proves
the mathematical pattern above to be correct.
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AC Machine Systems
3.6.2
Simulation and Performance Analysis of Cycloconverter-fed
Synchronous Motor Systems
The cycloconverter-fed synchronous motor system with field-oriented control
and speed regulation is a nonlinear, multivariable and multiloop system, whose
mathematical pattern is a differential equation set with variable coefficients and
can be solved by use of numerical methods.
In comparison with usual synchronous motor, the cycloconverter-fed synchronous
motor system has some topics to need studying, such as the harmonics in
steady-state operation and the performance in transient operation. In addition, the
simulative calculation in steady-state operation is also different from that in
transient operation.
During simulation in steady-state the rotor speed is constant, and the action of
control system is to provide a standard voltage for the cycloconverter, whose
frequency is equal to the operating frequency of synchronous motor and its phase
angle depends upon the machine power factor and rotor initial position angle.
Therefore, so long as the frequency and power factor is predetermined for
steady-state operation, the action of control system can be superseded by setting
a standard voltage during simulative calculation in steady-state.
During simulation in transient operation, the synchronous motor has a
transient process from original steady-state to another steady-state, so the control
system has to join in simulative calculation. In order to shorten the simulative
process, parallel calculation is used for synchronous motor and series calculation
is adopted for control system.
Taking a steel-mill driven by a 4 000kW-cycloconverter-fed synchronous motor
as example, analyse the steady-state and transient performance of the cycloconverterfed synchronous motor system. The parameters of 4 000kW-cycloconverter-fed
synchronous motor are:
rated power: 4 000kW
rated voltage: 1 650V
rated current: 1 512A
number of phases: 3
rated power factor: 0.978
rated frequency: 4Hz
rated speed: 40rpm
number of pole-pairs: 6
excitation voltage: 147V
excitation current: 544.4A
(1) Steady-state operating performance of cycloconverter-fed synchronous
motor systems
In steady-state, the output voltage of cycloconverter contains plentiful harmonics,
and in the meantime the harmonic voltage, harmonic current, harmonic loss,
harmonic torque, vibration and noise must be paid attention to.
During rated operation of 4 000kW-cycloconverter-fed synchronous motor, the
wave-forms of U aN , U aO , ia , Tem , i f are shown in Fig. 3.6.6 and those of 6
damping bars currents for each pole are indicated in Fig. 3.6.7.
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3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.6.6 Simulative waveforms of U aO , ia , U aN , Tem , i f during rated operation
of 4 000kW-cycloconverter-fed synchronous motor
From Fig. 3.6.6 we can see, the cycloconverter’s output voltage U aN is not
equal to the synchronous motor voltage U aO because the neutral point of
synchronous motor is isolated from the neutral point of cycloconverter to bring
about the voltage difference between them, whose value is U NO . From Fig. 3.6.6
it is also evident that the waveforms of synchronous motor phase current and
electromagnetic torque are rather smooth although the waveform of its phase
voltage contains ripples or burrs, because the harmonic impedance of synchronous
motor is relatively large to filter current harmonics. At the same time, the excitation
current of synchronous motor is not a smooth straight-line any longer and contains
some harmonics. During steady-state operation, 6 damping bars currents also
contain stronger harmonics as shown in Fig. 3.6.7.
By use of FFT analysing program, we can evaluate the harmonic spectra of
various quantities mentioned above and then analyse the relations between them
and various factors.
(2) Dynamic performance of cycloconverter-fed synchronous motor systems
For a cycloconverter-fed synchronous motor system with field-oriented control
used to drive steel-mill and during rolling period, the synchronous motor is
loaded, unloaded or rotating forward or backward frequently and its load torque
is also variable, so all the system is in a transient process. Thus it is significant to
study dynamic performance of cycloconverter-fed synchronous motor.
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AC Machine Systems
Figure 3.6.7 Simulative waveforms of 6 damping bars currents for each pole
during rated operation
In order to improve the dynamic performance of cycloconverter-fed synchronous
motor, the damping winding is installed usually for this kind of synchronous
motor. There are two types of damping windings, i.e. half-damping type without
full end-ring and having no connection between adjacent poles and full-damping
type with full end-ring. The transient processes for this kind of synchronous
motor are studied respectively for sudden load, sudden reversal-rotation and
speed-up with weak field as follows.
(a) Sudden load
When a 4 000kW-cycloconverter-fed synchronous motor with flux-linkageoriented control operates at rated speed 40rpm and load torque is increased
suddenly from 0.1 per-unit to 1.0 per-unit, the changing processes of rotor speed
n, power angle G , stator current d-axis and q-axis components id and iq , stator
current’s torque-component iT , electromagnetic torque Tem and excitation
current i f are shown respectively in Fig. 3.6.8(a) (f) for full-damping type of
synchronous motor. According to the same situation also analyse the transient
processes of half-damping and no-damping synchronous motors, and the two
types of synchronous motors can’t operate stably due to oscillation.
From the above example it is clear that q-axis damping winding plays an
important role to keep air-gap flux-linkage constant and to raise the system
stability during the transient process of sudden load.
(b) Sudden reversal-rotation
Taking the 4 000kW-cycloconverter-fed synchronous motor as example,
assuming the rotor speed and load torque to be 1.0 per-unit and during sudden
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3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.6.8 Simulative waveforms of n, G , id , iq , iT , Tem , i f during transient process
of sudden load
change of rotation from backward to forward direction, the dynamic curves of
rotor speed n, stator current ia , stator current d-axis and q-axis components id
and iq , stator current’s torque-component iT , electromagnetic torque Tem and
power angle G are shown respectively in Fig. 3.6.9(a) (f) for full-damping type
of synchronous motor. In addition, also analyse the sudden reversal-rotation
process of half-damping and no-damping synchronous motors, and the calculation
results indicate that the two types of synchronous motors can’t operate stably.
When sudden reversal-rotation is compared with sudden load, both can cause
the sudden change of stator current’s torque-component iT and stator current
q-axis component iq and then the rather large rotor q-axis damping winding
245
AC Machine Systems
current is produced to compensate the change of air-gap flux-linkage and to keep
air-gap flux-linkage constant, but sudden reversal-rotation is more serious than
sudden load. In order to improve the system dynamic performance and to raise
the system stability, the cycloconverter-fed synchronous motor must have a
full-damping rotor to provide a passage for rotor q-axis damping current,which is
more important for the cycloconverter-fed synchronous motor used in frequent
change of rotating direction. In China, some half-damping synchronous motors
have burning marks on their rotor-pole dovetails during frequent change of
rotating direction, which has proved the above conclusion.
Figure 3.6.9 Dynamic simulative wave-forms of n, ia , id , iq , iT , Tem , G during
sudden change of rotating direction for synchronous motor
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3 Some Special Operation Modes of Synchronous Machine Systems
(c) Speed-up with weak field
Taking the 4 000kW-cycloconverter-fed synchronous motor as example,
keeping the load torque of synchronous motor as 0.5 per-unit, raising the rotor
speed from 1.0 per-unit to 1.5 per-unit and diminishing the air-gap flux-linkage
from 1.0 per-unit to 0.67 per-unit, calculate the change processes of synchronous
motor speed n, stator current ia , stator current d-axis and q-axis components id
and iq , stator current’s torque-component iT , electromagnetic torque Tme and
power angle Gas shown in Fig. 3.6.10(a) (f). Evidently, during a certain load
torque and speed-up with weak field, the stator current q-axis component changes
Figure 3.6.10 Dynamic simulative wave-forms of n, ia , id , iq , iT , Tem , G during a
certain load torque of synchronous motor and speed-up with weak field
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AC Machine Systems
a little so the rotor q-axis damping winding current caused by it is also small.
However, the stator current d-axis component is double or more so the rotor
d-axis damping winding current caused by it is also large.
Calculation results indicate that during speed-up with weak field and even if
small load torque exists serious oscillation for half-damping and no-damping
synchronous motors, thus the system being unstable. Therefore, the full-damping
winding is also essential even if speed-up with weak field
3.6.3
Design of Damping Winding for Cycloconverter-fed
Synchronous Motors with Field-oriented Control
From the above analysis we can see, the full-damping winding has to be installed
for cycloconverter-fed synchronous motor to get good dynamic performance.
Design of damping winding must meet three requirements: (i) Improve the system
dynamic performance (ii) Make the damping winding loss as small as possible
(iii) Let the damping bars currents and losses be distributed uniformly for each
pole. Therefore, it is necessary to make an overall calculation, especially
dynamic-performance calculation, and then the damping winding can be disposed
rationally.
We still take the 4 000kW-cycloconverter-fed synchronous motor as example
to discuss the damping winding current distribution as follows. Assuming the
number of damping bars for each pole to be 5 and 6 and increasing load torque
suddenly from 0.1 per-unit to 1.0 per-unit at rated speed, the distribution curves
of various damping bars currents in rms value are shown respectively in
Figs. 3.6.11(a) and (b).
Figure 3.6.11 Distribution of various damping bars currents during sudden load
From Fig. 3.6.11 it is evident that during sudden load the damping bars
currents on the pole-fore are larger than those on the pole-aft for all the two
situations of 5 and 6 damping bars per pole. During 5 damping bars per pole, the
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3 Some Special Operation Modes of Synchronous Machine Systems
damping bar in the middle of pole has the largest current of all damping bars.
When the number of damping bars per pole is even, the damping bars currents
and losses are distributed more uniformly than odd number of damping bars per
pole, because during sudden load the stator current’s torque-component occurs
mainly on q-axis of the synchronous motor with field-oriented control.
Assuming the rotor speed and load torque to be 1.0 in per-unit, supposing the
number of damping bars per pole to be 5 and 6 respectively and during sudden
reversal-rotation, the distribution curves of various damping bars currents per
pole are shown in Figs. 3.6.12(a) and (b) for the 4000kW-cycloconverter-fed
synchronous motor, in which the damping bars currents during even number of
damping bars per pole are distributed more uniformly than odd number of
damping bars per pole.
Figure 3.6.12 Distribution of various damping bars currents during sudden
reversal-rotation
During speed-up with weak field, the load is relatively small, so the rotor
damping bars currents are also small. Therefore, it is not necessary for design of
damping winding to consider the dynamic process of speed-up with weak field.
To sum up, for cycloconverter-fed synchronous motors with field-oriented
control used in sudden load or sudden reversal-rotation, it is better to take the
number of damping bars per pole as even number.
3.7
Analysis of Synchronous Generators with AC and DC
Stator Connections
Some special power supply systems, such as those used on ships, tractions, and
aircraft, are often required to provide high quality ac and dc electricity
simultaneously. The synchronous generator with ac and dc stator connections
(ac-dc generator) is an effective way to satisfy this requirement. There are five
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AC Machine Systems
suits of independent star-connected 3-phase windings in the ac-dc generator
stator, one is called ac winding to supply ac power and the others belong to dc
windings1, which are connected to four 3-phase bridge rectifiers respectively, and
then the dc output sides of the four 3-phase bridge rectifiers are in parallel to
supply electricity to dc load, as shown in Fig. 3.7.1.Owing to the construction
with the ac and dc windings arranged in the same stator, the ac-dc generator has
many obvious advantages. Firstly, this generator has small volume, light weight
and low cost. In addition, there is no electrical connection between the ac
winding and dc winding, but only magnetic couple between them. So the voltage
harmonic distortion of ac winding caused by the commutation process of dc
winding becomes small, then the influence of dc load on the voltage of ac
winding decreases greatly and the electromagnetic compatibility performance is
improved.
Figure 3.7.1 The schematic diagram of an ac-dc generator
The mathematic model of ac-dc generators is complicated and number of
equations is high, so the simulation is used to analyze the performance, such as
the voltage harmonic distortion of ac voltage of ac winding, ripple of dc voltage,
power quality and electromagnetic compatibility. Moreover, the established
model and simulation method can be used to analyze the low frequency power
oscillation while the counter emf load is connected to dc side, and to find out the
effective way to restrain the oscillation and guarantee the reliable operation. Also
it can be applied to simulate the process of sudden short circuit at ac side or dc
1
In order to distinguish the ac winding with 3-phase bridge rectifier from the oridinary ac winding, the former is
defined as dc winding.
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3 Some Special Operation Modes of Synchronous Machine Systems
side, and provide the reliable protection.
In this section the multi-loop model is applied to analyze this system, which
implies that the a, b, c coordinates system is based. So the coordinates transformation
is not needed for simulation, and the state variables are the actual values, which
make the simulation results directly perceived and the debugging of program easy.
On the other hand, because some elements of inductance matrix are time-variant,
the inverse of inductance matrix at each simulation step increases the calculation.
Another advantage of using the multi-loop model is that the effects of the spatial
harmonic components of air-gap magnetic field are included naturally, which
improves the accuracy. On the stator of ac-dc generators there are five 3-phase
windings, the slot number per pole per phase of which can not be too much
because of the limited stator slot number. While the slot number per pole per
phase of dc winding is equal to 1, the spatial harmonic components of air-gap
magnetic field are rather strong. At last, the damper winding would influence the
commutation process of rectifier, so the multi-loop model helps to simulate the
commutation process with better accuracy, where the damper winding is dealt
with according to its actual structure.
Because of the frequent switching of rectifiers, the generator always works in
an unsymmetrical state, which makes the coordinates transformation have no
obvious advantage and even increase the complexity of programming. Although
the application of d, q, 0 coordinates system can obtain the constant coefficient
inductance matrix and save the calculation of inversing matrix, the transformation
of d, q, 0 to a, b, c coordinates system must be needed to get the voltage and
current in a, b, c system to judge the commutation, which makes the debugging
of program more complicated.
In brief, although the calculation of multi-loop model is a vast amount, the
multi-loop model is suitable to simulate the performance of ac-dc generators,
because the calculation of parameters is accurate, and the coordinates
transformation is not necessary for the judgment of commutation. If only the
fundamental component of air-gap magnetic filed is considered, the d, q, 0
coordinates system leads to a simple system equation, which is easy to make
theoretical analysis and understand the system rules, referring to [1,8,11].
3.7.1
Mathematical Model of AC-DC Generators and Loads
(1) Multi-loop model of ac-dc generators
Under normal condition, every stator phase winding is considered as a unit,
while all of the rotor field winding is regarded as a unit. The multi-loop method
enables one to deal with damper winding according to its actual structure. While
only considering the odd order harmonic field, the damper winding can be
divided into two groups, d-axis and q-axis windings, whose axes are orthogonal
251
AC Machine Systems
to each other. There is no electrical and magnetic link between d and q axis loops,
but mutual resistance and inductance within each group. Under fault condition,
the stator single coil can be dealt as a unit.
If the reference direction is set using generator convention for stator circuit
and motor convention for rotor circuit, positive current excites positive flux linkage,
and the positive directions of d-axis and q-axis flux linkage are consistent with
d-axis and q-axis positive direction, the multi-loop equation of generators can be
expresses as a matrix form:
UF
LF pI F ( pLF ) I F RF I F
d
, the current vector is
dt
where the differential operator p
iF
(3.7.1)
[i fd i1d " imd iA iB iC ia1 ib1 ic1 ia 2 ib 2 ic 2 ia 3 ib 3 ic 3 ia 4 ib 4 ic 4 ]T
where the subscript fd represents the field winding, 1d to md represent the
damper winding, A, B, C represent 3-phase ac winding respectively, a1, b1, c1,
a2, b2, c2, a3, b3, c3, a4, b4, c4 represent four suits of 3-phase dc windings
respectively.
The voltage vector in equation (3.7.1)
UF
[u fd 0 " 0 u A u B uC ua1 ub1 uc1 ua 2 ub 2 uc 2 ua 3 ub 3 uc 3 ua 4 ub 4 uc 4 ]T
where the voltages of damper loops are zero because the damper loops are short
circuited without external supplies.
The resistance matrix in equation (3.7.1) is
RF
diag(rfd , Rd , Rac , Rdc )
where the subscription fd, d, ac, dc represent the field winding, damper winding,
ac winding and dc winding respectively. The resistance matrix of damper winding
is
Rd
ª r11d " r1md º
«
»
# »
« #
¬« rm1d " rmmd ¼»
The self resistance matrix of ac winding is
Rac
diag(rac , rac ,rac )
The self resistance matrix of dc winding is
Rdc
252
diag(rdc ,rdc ," ,rdc )
3 Some Special Operation Modes of Synchronous Machine Systems
where rac is the phase resistance of ac winding, rdc is the phase resistance of dc
winding.
The mutual inductance matrix in equation (3.7.1) is
ª L fdfd
«L
« fdd
« L fdac
«
¬« L fddc
LF
L fdd
Ldd
Ldac
Lddc
L fdac
Ldac
Lacac
Lacdc
L fddc º
Lddc »»
Lacdc »
»
Ldcdc ¼»
The calculation method of the inductance parameters is presented in detail in
Chapter 1.
(2) Voltage equation of ac load for ac-dc generators
The voltage equation of ac load is
U ac
Lac
dǿ ac
Rac I ac
dt
(3.7.2)
where U ac [ua1 , ub1 , uc1 ]T is the voltage vector, I ac [ia1 , ib1 , ic1 ]T is the current
vector. For 3-phase symmetrical star-connected load, the inductance matrix is
Lac diag( Ll , Ll , Ll ) and the resistance matrix is Rac diag( Rl , Rl , Rl ). Ll and
Rl are the inductance and resistance of ac load respectively. The delta-connected
load can be equivalent to star-connected load. For the 3-phase asymmetrical load,
the inductance or resistance of 3-phases can be set as different values.
(3) Voltage equation of dc load for ac-dc generators
The voltage equation of dc load is
udc Edc
Ldc
didc
Rdc idc
dt
(3.7.3)
where Ldc , Rdc , Edc are the inductance, resistance and counter emf of dc load
respectively.
3.7.2
System Equation of ac-dc generator
Combining the ac-dc generator equation (3.7.1) with the load equations (3.7.2)
and (3.7.3), a system circuit equation is gotten as follows:
U
LpI ( pL) I RI
(3.7.4)
where
U
[U FT
U acT
U dc ],
I
[ I FT
I acT
I dc ]T
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AC Machine Systems
ª LF
L «« 0
«¬ 0
ª RF
R «« 0
«¬ 0
0
Lac
0
0 º
0 »»
Ldc »¼
0
Rac
0
0 º
0 »»
Rdc »¼
It must be noted that equation (3.7.4) does not represent the actual loops and
can not be solved. Under normal condition, the generator and loads must be
connected with each other to form the actual loops which alter with the operating
load condition of the ac and dc windings and the state of rectifiers. The
connection can be represented by a connection transformation matrix T in which
each row corresponds to an actual loop and each column to a component. If the
component and loop are related to each other and their reference directions are
the same, the element in the matrix is 1. If the component and loop are related
and their reference directions are opposite, the element is –1. If they are not
related, the element is 0.
Figure 3.7.2 The connection of ac winding and star-connected load
The connection of the ac winding and load is easy to understand. Figure 3.7.2
shows the connection between 3-phase star-connected ac winding and the
star-connected load. There are two independent loops formed between them,
which are represented as the 9th and 10th rows of the connection transformation
matrix in Fig. 3.7.3. Every row of matrix is corresponding to one actual loop, and
the total row number is the number of independent loops of system. In Fig. 3.7.3
the zero’s are not filled. The 9th row is corresponding to actual loop 1 composed
of phase A and B ac windings and phase A and B loads, where the reference
direction of B phase current is opposite with that of loop current. Similarly, the
10th row is corresponding to actual loop 2. The connection transformation matrix
can be established similarly, if the ac winding is abnormally connected.
The former 8 rows of the connection transformation matrix in Fig. 3.7.3, represent
one field loop and 7 damper loops respectively.
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3 Some Special Operation Modes of Synchronous Machine Systems
Figure 3.7.3 Example of connection transformation matrix
The connection between the dc winding and dc load is more complicated
because it is determined by the working state of rectifiers. While the state of
rectifiers changes, the connection transformation matrix must be modified.
Figure 3.7.4 shows the connection of dc windings, rectifier bridge and dc load at
a certain instant, where the shaded diode is in conduction state and the diode
without shade is switched off.
Figure 3.7.4 Connection of dc winding and dc load
As shown in Fig. 3.7.4, for the first 3-phase bridge rectifier there are three
diodes in conduction state and the rectifier is in commutation state, so two
independent loops are formed between dc windings, rectifier bridge and load.
The loop 1 is composed of winding a1, diode 3 of the first rectifier bridge, dc
load (E, L and R), diode 2 of the first rectifier bridge and winding c1, which is
represented by the 11th row of connection transformation matrix in Fig. 3.7.3.
The column corresponding to wingding c1 is –1 because the reference direction
of loop is opposite to that of wingding c1. The loop 2 is composed of winding b1,
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AC Machine Systems
diode 1 of the first rectifier bridge, dc load (E, L and R), diode 2 of the first
rectifier bridge and winding c1, which is represented by the 12th row of connection
transformation matrix in Fig. 3.7.3.
There are two diodes in conduction state for the 2nd, 3rd , 4th rectifier bridge,
and these bridges are in conduction state, so only one independent loop is formed
between dc windings, rectifier bridge and load. For the 2nd rectifier bridge, the
loop is composed of winding a2, diode 3 of the 2nd rectifier bridge, dc load (E, L
and R), diode 2 of the 2nd rectifier bridge and winding c2, which is represented
by the 13th row of connection transformation matrix in Fig. 3.7.3.
The switching of diode is determined by the voltage and current of the diode.
The simulation program can establish the connection transformation matrix
automatically in accordance with the states of diodes.
After founding the connection transformation matrix according to the relation
between ac and dc windings and their loads, the voltages and currents can be
transformed as
U c TU
I TTIc
Hence, the system equation is
Uc
LcpI c ( pLc) I c RcI c
where
Lc TLT T , Rc TRT T
Therefore, the standard state equation can be expressed as
pI c
AI c B
(3.7.5)
where
A Lc 1( pLc Rc), B
3.7.3
Lc 1U c
Model of Rectifier Bridge
Generally, the diode of rectifier bridge is considered as ideal component and the
conduction voltage drop is neglected in simulation. But it would lead to the
inconsistence between simulation and experiment under certain load condition,
even the divergence would happen. For example, the simulation results of 3phase rectifier bridge under short circuit state at dc side as shown in Fig. 3.7.5
show that the conduction voltage drop of diode must be considered to obtain
correct result. If the short circuit occurs at dc side, after considering the conduction
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3 Some Special Operation Modes of Synchronous Machine Systems
voltage drop of diode, because the sum of voltage across the two diodes in the
same arm is zero, the two diodes on the same arm of bridge would not conduct at
the same time, which is different from the equivalent 3-phase short circuit when
neglecting the conduction voltage drop. Figure 3.7.6 gives the measured
waveform of voltages across a diode of 3-phase rectifier bridge short-circuited at
dc side. It can be seen that each diode is not in conducting state for all time, and
when the voltage between two terminals of a diode is negative, the diode is shut
off. For the sake of this, the voltage drop is counted in and set to +0.7V.
Figure 3.7.5 3-phase rectifier bridge shorted at dc side
Figure 3.7.6 Voltage across each diode in the same arm of rectifier bridge during
short circuit of dc side
The correct calculation of the commutation of diodes is the key to the simulation
of all the ac-dc generator system. In the ac-dc generator, each one of the four
3-phase dc windings is connected to one 3-phase rectifier bridge independently
and then they are in parallel to supply dc power at the load side. The commutation
is much more complicated than general 3-phase rectifier bridge, which contains not
only commutation among the diodes of single 3-phase bridge, but also that
among various bridges. Once the working state of diodes alters, the topology of
circuit varies accordingly, and the state variables vary too.
Previous works suggested dividing all the possible state of rectifier into several
groups, each of which can be expressed as an equation, called mode classifying
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AC Machine Systems
method. Now in this system the number of diodes reaches 24, while the number
of modes reaches 192, which is much more complicated than that of general
3-phase bridge. Using mode classifying method, the program must be rewritten
when the number of rectifier bridges increases or decreases.
In this part, the connection transformation is applied to represent the topology,
which results in that only the elements of the connection transformation matrix
need to be changed with topology.
Because the four neutral points of 3-phase dc windings are independent of
each other, the switch on of diode is determined by the line voltage and dc
voltage. The diode is conducted while the line voltage is larger than or equal to
the dc load voltage and voltage drop of rectifier, otherwise it is in shut-off. The
time of shut-off is determined by the current of the diode, that is, the diode shutsoff when the current changes from positive to negative. In the simulation program,
an array is used to record the state of diodes. Firstly, the maximum line voltage is
found among the 24 line voltage of four rectifier bridges and the order number of
two diode are recorded too where the maximum voltage is applied. When the
maximum line voltage is larger than the sum of dc voltage and twice the voltage
drop of diode, then the two diodes are set to conduction. Any one diode would be
switched off once its current is less than zero.
3.7.4
Model of Automatic Voltage Regulator (AVR)
To establish the precise model of AVR is complicated. But for the analysis of ac
voltage waveform under steady state, only the steady state excitation current
must be calculated for specified load condition. Furthermore, only the steady-state
regulation precision is concerned, while the dynamic process is not considered.
So a first order PI model is applied to represent the AVR as follows:
ǻu U ~ U *
ǻu fd
½°
¾
K P ǻu K I ³ ǻudt °¿
(3.7.6)
where ǻu fd is the variation of excitation voltage, K P , K I are the proportional
and integral coefficients of PI regulator respectively, U * , U ~ and 'u are the
reference value, feedback value, and their error respectively. The feedback rms
value is defined as
U ~ (t0 )
1 t0 2
u (t )dt
T ³ t0 T
1
¦ u j 2 (t )'t
T j
(3.7.7)
where T , 't are the period of main frequency and the sampling interval of
voltage respectively, u (t ) is the instantaneous value of line voltage.
258
3 Some Special Operation Modes of Synchronous Machine Systems
In order to calculate the rms value of line voltage, various instantaneous values
of line voltage during the whole period ahead of the calculation instant t0 are
needed. Because the variable-step Runge-Kutta integration method is used, the
number of calculation intervals within one period of main frequency T varies,
which leads to the complexity of saving the instantaneous values of line voltage
and calculating the rms value. In simulation program the step is adjusted to
calculate the line voltage in fixed step, and an array is used to record the line
voltage, which would be updated during the simulation.
The simulation results of excitation regulating process show that the proper
coefficient of PI regulator can achieve high steady-state precision for this simple
PI regulator, i.e. the steady-state error between the reference and feedback values
would be less than 5‰ under rated load condition. The model of AVR is used to
calculate the field current for any specified ac voltage and any load condition,
which is important to the analysis of steady-state performance.
3.7.5
Motion Equation of Rotor and Speed Regulators
When studying some transient process of ac-dc generators, the variation of
rotation speed must be considered, therefore the motion equation of rotor should
be solved.
The electromagnetic torque of generator is
Te
P wL T
i
i
2 wJ
(3.7.8)
where i is the current vector, and L is the inductance matrix.
The motion equation of rotor is
d 2J
dt 2
dZ
dt
Tm Te
H
(3.7.9)
where J is the rotor position angle, Z is the rotation speed, Tm is the
mechanical torque of prime mover, H is the inertia constant.
The mathematic model of speed PI regulator is
'Z
'Tm
Z Z *
½°
¾
K P 'Z K I ³ 'Z dt °¿
(3.7.10)
where 'Tm is the variation of mechanical torque of prime mover, K P , K I are the
proportional and integral regulation coefficients of PI speed regulator respectively,
Z *, Z and 'Z are the reference value, feedback value of speed and their error
respectively.
259
AC Machine Systems
3.7.6
Simulation Procedure of AC-DC Generator System
In order that the simulation of a long time process must be finished without letup,
the record file is used to record the state of the last step. In the record file, all the
key information must be logged, including the loop currents, rotor position,
rotation speed, the working state of all the diodes, the integral values of excitation
regulator and speed regulator, and the samples of instantaneous line voltage of ac
side during the latest period, which are used to calculate the rms value of line
voltage of ac side. The flow chart of simulation program is shown in Fig. 3.7.7.
Figure 3.7.7 Flow chart of simulation of ac-dc generator systems
At the beginning of every simulation, the latest state must be read from the
record file, while at the end of existing simulation, the record file must be updated
with the new state. So the simulation can be continued from the latest step and
need not simulate from the original state, by which the simulation time can be
spared greatly, especially when the steady state must be reached. When the
simulation is executed for the first time, the no load condition can be used to
write the record file, where all of the loop currents are set as zero, except that the
field current is set as the ratio of field voltage to the resistance of field winding.
The Runge-Kutta integration method with variable step is applied to solve the
state equation. In order to find the exact commutation instant, the step must be
adjusted. Once the state of diode varies in next step, the existing step length is
shortened to a half until that the step reaches the minimal one or no commutation
occurs.
Because of the variable step simulation, in order to plot the calculated results
easily and use FFT to analyze the waveform, the simulation step must be
automatically adjusted to make the output of results in fixed step.
260
3 Some Special Operation Modes of Synchronous Machine Systems
3.7.7
Verification of Simulation Program
The calculated results are compared in detail with those of experiments on a
model ac-dc generator as follows, in which the dimensions and data of the model
ac-dc generator are shown in Table 3.7.1.
Table 3.7.1 Dimensions and data model ac-dc generator
Diameter of iron core
Rated frequency
Core length
Pole arc factor
Damper bar number per pole
Rated power
Rated voltage
Rated current
AC winding
Power factor
Winding pitch
Series turns per phase
250mm
50Hz
152mm
0.72
7
15kW
390V
27.8A
0.8
10
72
Pole-pair number
Stator slot number
Maximum airgap length
Minimum airgap length
Skew of stator core
Rated power
Rated voltage
Rated current
DC winding
Winding pitch
Series turns per phase
2
48
1.5mm
1mm
one slot pitch
3.3kW
510V
6.5A
10
64
Owing to a limited space, only one set of comparison between calculated and
measured results is given below. According to the Shannon’s sampling theorem,
in order to sample correctly the sinusoidal waveform whose frequency is equal to
nf 0 , the sampling frequency must be greater than 2nf 0 . The lowest frequency of
the high frequency components in the output dc voltage is 24 times the main
frequency 50Hz, so the sampling frequency must be greater than 48 times the
main frequency 50Hz, which means that the sampling period, also the maximum
simulation step length, must be less than 4.17 u 104 s .
The maximum simulation step length is 100ȝs , and the minimum is 0.1ȝs ,
which can achieve a compromise between the simulation speed and accuracy.
The minimum simulation step determines the accuracy of searching the commutation
instant of diodes, so it would not be too long.
If the pure resistant or nearly pure resistant load is connected to the ac winding
or dc winding, the variation rate of loop current would be great; while if the pure
inductive or nearly pure inductive load is connected, the variation rate of voltage
would be great. The step of Runge-Kutta integration method with variable step is
rather sensitive to these two kinds of loads. So for these two kinds of loads, the
maximum simulation step length must be decreased to improve the accuracy,
which is generally less than 10ȝs .
The comparisons of line voltages between the calculation and experiment
results for ac winding and dc winding, while the field current I fd is 1.3A, the
load current of ac winding I ac is 7.8A, power factor is 0.8, the line voltage of ac
261
AC Machine Systems
winding U ac is 180V, and the dc winding is no load, are shown in Figs. 3.7.8 and
3.7.9 respectively.
Figure 3.7.8 Calculated and measured waveforms of line voltage of ac winding
(a) Calculated; (b) Measured
Figure 3.7.9 Calculated and measured waveforms of line voltage of dc winding
(a) Calculated; (b) Measured
The colculated and measured harmonic components and the voltage harmonic
distortion in percentage of line voltage of ac winding are compared with each
other in Table 3.7.2, while those of line voltage of dc winding are shown in Table
3.7.3. The percentage of harmonic component is the ratio of amplitude of harmonic
component to fundamental component. It can be seen that the results of calculation
and experiment are consistent with each other.
262
3 Some Special Operation Modes of Synchronous Machine Systems
Table 3.7.2 Comparison between the calculated and measured harmonic components
of line voltage of ac winding
Harmonic order
Calculated harmonic
components /ă
5
6.18
7
4.43
11
6.10
13
4.56
17
1.45
19
0.49
23
3.45
25
0.31
VHD
11.4
Measured harmonic
components /ă
4.2
4.2
3.7
1.9
0.6
0.8
5.9
1.2
11.0
Table 3.7.3 Comparison between the calculated and measured harmonic components
of line voltage of dc winding
Harmonic order
Calculated harmonic
components /%
5
10.6
7
10.8
11
3.43
13
2.66
17
1.85
19
1.58
23
0.28
25
0.70
VHD
16.1
Measured harmonic
components /%
9.36
11.7
2.87
2.39
2.06
1.15
0.66
0.10
15.6
More simulation results of various conditions show that when the load current
is small (below half of the rated load), the good consistence is achieved between
the calculated and measured results. But with the increase of load currents, the
errors between calculated and measured results also increase. The errors are caused
by the assumption that the multi-loop model is linear model, and the saturation of
magnetic circuit is dealt with by use of the equivalent enlarged air-gap. The
enlargement of air-gap is according to the fundamental magnetic field, so when
the magnetic circuit is saturated, the harmonic magnetic field can not be analyzed
accurately, which leads to the larger error. From the comparison, it can be seen
that the calculated harmonic components are mostly greater than those of
experiments, which means the calculation results are more rigorous and make the
design more reliable. The accurate analysis of the effect of saturation on the
harmonic field must be simulated by combination of the multi-loop model and
the finite element method.
From the comparison between the simulation and experiment, it can be seen
that the calculation of voltage harmonics under steady state is accurate enough.
Furthermore, the design optimization of ac-dc generators for different dimensions
and load conditions can be conducted.
3.7.8
Simulation Research of Voltage Harmonics of AC-DC
Generators
In applications of the military, aerospace, and communication, the high quality of
power supply is required in order to make the high sensitivity equipment avoid
the interference. The relations between the VHD of ac winding and various
factors, especially the structure dimensions, are analyzed as below.
263
AC Machine Systems
(1) Influence of load current on the harmonic component and VHD
The load current has great effect on the VHD of line voltage of ac winding. So
the influence of currents at ac side and dc side are calculated, and six working
conditions are chosen as follows:
Condition 1: no load at ac side, and no load at dc side.
Condition 2: no load at ac side, rated load at dc side (6.5A, pure resistance).
Condition 3: half-load at ac side (14A, power factor 0.8), no load at dc side.
Condition 4: half-load at ac side (14A, power factor 0.8), rated load at dc side
(6.5A, pure resistance).
Condition 5: rated load at ac side (28A, power factor 0.8), no load at dc side.
Condition 6: rated load at ac side (28A, power factor 0.8), rated load at dc side
(6.5A, pure resistance).
The calculated results of harmonic components and VHD of line voltage of ac
winding are shown in Table 3.7.4 and Fig. 3.7.10.
Table 3.7.4 Relation of the harmonic components and VHD of line voltage of ac
winding with the working conditions
unit: ă
Harmonic
order
5
7
11
13
17
19
23
25
VHD
1
1.76
2.52
1.39h10–5
2.02h10–5
8.29h10–7
8.63h10–6
5.83h10–6
8.64h10–6
3.07
2
3.04
1.85
1.04
0.487
1.07
0.204
0.332
2.29
4.73
Working condition
3
4
2.65
2.97
3.16
3.18
1.66h10–2
1.86
6.70h10–3
0.382
1.33h10–4
0.954
–4
1.45h10
0.609
1.19h10–4
13.8
9.94h10–5
10.6
4.12
18.3
5
3.70
3.91
3.34h10–2
1.27h10–2
2.33h10–5
3.11h10–5
2.26h10–5
1.88h10–5
5.39
6
2.33
6.94
1.94
1.44
2.69
0.795
19.6
15.6
26.7
Figure 3.7.10 Relation chart between VHD of line voltage of ac winding and its
working conditions
264
3 Some Special Operation Modes of Synchronous Machine Systems
It can be seen that the VHD of ac windings increases with the increase of load
currents at ac side and dc side. For the working condition 1, 3 and 5, the dc side
is no load, and the VHD of ac winding increases slightly with the load current of
ac winding. For the working condition 2, 4 and 6, the dc side is rated load, and the
VHD of ac winding increases more rapidly with the load current of ac winding
than the working condition 1, 3 and 5. Comparing condition 2 with 1, condition 4
with 3, and condition 6 with 5, it can be seen that when the dc load current
increases, the VHD of ac winding increases greatly. It is clear that the load current
of dc winding would cause greater distortion in ac voltage of ac winding because
of the frequent commutation.
(2) Influence of dc winding pitch on the harmonic components and VHD
In the ac-dc generator, the commutation of rectifier bridge makes the ac voltage
distort through the mutual inductance coupling. Although there is no restriction
of VHD of line voltage of dc winding, the reasonable design of dc winding would
improve the VHD of ac winding. So here the influence of dc winding pitch on
the VHD of ac winding is studied. The harmonic components of ac winding are
calculated and the results are shown in Table 3.7.5, while the pitch of dc winding
varies from 2/3 1.
Table 3.7.5 Relation of the harmonic components and VHD of line voltage of ac
winding with the pitch of dc winding
unit: ă
Harmonic order
5
7
11
13
17
19
23
25
VHD
2/3
2.33
6.94
1.94
1.44
2.69
0.795
19.6
15.6
26.7
3/4
2.71
6.21
1.36
0.993
1.81
0.236
13.2
15.1
21.6
Short pitch ratio
5/6
4.42
2.62
1.82
0.510
0.868
1.01
8.00
9.38
13.7
11/12
2.07
5.33
1.06
0.508
1.74
1.11
7.79
4.96
11.4
1
2.71
3.93
1.34
0.521
1.90
1.26
12.5
9.64
16.9
The relation between the VHD and the pitch of dc winding is drawn as Fig. 3.7.11.
The VHD of ac winding reaches the minimum while the pitch is 11. It can be
concluded that a suitable pitch of dc winding can effectively make the VHD of ac
winding decrease.
The influence of the relative position of ac and dc windings in stator slot, the
relative spatial position of ac and dc windings, the number of damper bars per pole
and the open width of damper slot, on the harmonic components and VHD are
also studied. But the results show that their influence is not too much.
The model and simulation method established in this section can be used not
only to analyze the steady state performance like the harmonic distortion of ac
265
AC Machine Systems
Figure 3.7.11 Relation curve of VHD of line voltage for ac winding with the pitch
of dc winding
voltage, but also simulate the transient process, such as the low frequency power
oscillation and the sudden short circuits occurring at ac side or dc side.
References
[1] Franklin P W (1972) Theory of the 3-phase salient pole type generator with bridge rectifier
output. J IEEE Trans. PAS-91(5): Part I, 1960 1967, Part Ċ, 1968 1975
[2] Gao J D (1963) AC machine transients and operating modes analysis (Chapter 2, 3, 4, 7, in
Chinese). Science Press, Beijing
[3] Gao J D, Zhang L Z (1982) Fundamental theory and analysis mothods of transients in electric
machines, Vol 1 (Chapter 6 and Appendix 2, in Chinese). Science Press, Beijing
[4] Gao J D, Wang X H, Li F H (2004) Analysis of ac machines and their systems,2nd Ed
(Chapter 3 and Section 4.7, in Chinese). Tsinghua University Press, Beijing
[5] Goodman E D (1984) Troque, voltage and current harmonics of shychronous machines. J
IEEE Trans. IA-20(1): 209 215
[6] Kirschbaum H S(1945) Transient electrical torque of turbine generators during short circuits
and synchronizing. J AIEE Trans. Feb. 64: 65 70
[7] Liu T, Wang X H (2000) Simulation of dynamic performance for cycloconverter-fed
synchronous motor. Proc IPEMC’2000, Beijing, pp 676 679
[8] Wang S M, Wang X H, Li Y X, Su P S (2002) Steady-state performance of synchronous
generators with ac and dc connections considering saturation. IEEE Trans. on Energy
Conversion 17: 176 182
[9] Wong M C, Zhang L Z (1999) Analysis of asymmetric systems with harmonics and their
compensation. Proc CICEM’99, International Academic Publishers, Xi’an, pp 946 949
[10] Zhang L Z, Chen L F (1991) Analysis of subsynchronous resonance in a turbogenerator
and its shstem using compensation capacitors. Proc CICEM, Huazhong University of
Science and Technology Press, Wuhan, pp 104 108
266
4 Oscillation, Stability and Excitation Regulation of
Synchronous Machine Systems
Abstract The definition of static stability for synchronous machines is not
unified in some references. Static stability is to discuss a problem whether
the synchronous machine can restore its original steady-state mode or not
when undergoing as small perturbation and then abolishing it, but the rate and
fashion of small perturbation are not prescribed. If the small perturbation is
caused by small and slow increase of loads, the excitation current has not
any change and only the rotor position angle reaches a new balance position,
which is the first possible case. The second possible case is that the increase
rate of small loads is relatively fast to bring about change of excitation
current even if having no excitation regulator. Therefore, the static stability
can be divided into steady-state static stability and dynamic-state static
stability in some references. By static stability refer to dynamic-state static
stability in this chapter even if having no excitation regulator. Due to small
perturbation, the corresponding basic equations can be linearized and small
oscillation of synchronous machine systems is an example of linearization,
in which according to the linearized basic equations get synchronization
torque coefficient and damping torque coefficient to find out natural
oscillation frequency, decay time-constant, forced oscillation produced by
mechanical torque harmonics, resonant curves and influence of stator
resistance on the oscillation, and then analyse the oscillation causes from
physical concept in detail. Moreover, use Herwitz’s criteria to get the
prerequisites of static stability for synchronous machine systems, utilize
Routh’s criterion to find out the corresponding stable regions for generator
operation and motor one, and then analyse the influence of excitation
regulation on static stability. The dynamic stability problem of synchronous
machine systems is to study whether the synchronous machine can operate
synchronously after undergoing a certain or large disturbance, inclusive of
basic equations corresponding to dynamic stability analysis, the influence of
ferromagnetic saturation on parameters, corresponding mathematical patterns
and calculation example. Through calculation it is clear to cut off the fault
line in time to increase dynamic stability, but the area rule and graphic
method provide only the cut-off angle G OTK , so the dividing intervals
calculation should be used together with the area rule to search for the
cut-off time limit tOTK.
AC Machine Systems
This chapter includes small perturbation and linearization of ac machine systems,
steady-state small oscillation and torque coefficients of synchronous machine
systems, static stability of synchronous machine systems and influence of excitation
regulation on static stability, and dynamic stability and analysis methods of
synchronous machine systems.
4.1
Small perturbation and Linearization of AC Machine
Systems
In Chapter 2 and Chapter 3 we have discussed some problems of ac machines at
synchronous speed or constant speed, in which the basic equations are linear
ones with constant or variable coefficients, so it is not necessary to consider rotor
motion. However, in practical terms the rotor motion must be counted in for
some operating conditions of ac machines, where the electromagnetic transients
and mechanical process have to be considered simultaneously and both influence
each other. For example, the prime mover of synchronous generator is Dieselengine without uniform torque or a synchronous motor drives air-compressor, for
which the electromagnetic transients and mechanical process have to be
evaluated simultaneously to get correct analysis results.
During a synchronous machine in parallel operation with network, the machine
has to operate at synchronous speed strictly according to network frequency.
However, due to some causes the small oscillation of synchronous machine rotor
speed may occur, which will be studied in Section 4.2. In addition, the synchronous
machine and its connected network dispatch loads or encounter some accidents
to bring about a question whether they can still keep synchronous operation or
not, namely the stability problem of synchronous machine system. The stability
problem is a significant and interesting problem, because the main generator in a
power system is out of step to make stop electricity of large area and to cause
confusion. In order to answer that question above, the effect of speed change has
to be taken into account for synchronous machines. For convenience, the stability
problem of synchronous machines can be divided into two types, namely static
stability and dynamic stability, referring to [1,2,4,6].
Static stability is to discuss a problem whether the synchronous machine can
keep synchronous operation under a certain steady-state operating mode. If the
machine can’t restore its original state after undergoing a small perturbation
under that steady-state operating mode, it can be considered to be unstable even
though the machine can still operate. On the contrary, if the machine can restore
its original steady-state operating mode after sustaining a small perturbation, it
can be considered to be stable under that steady-state operating mode. Under the
definition of static stability mentioned above, nonlinear basic equations of
268
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
synchronous machines can be linearized to simplify the problem whether the
machine can be stable or not.
The dynamic stability problem is to discuss whether a synchronous machine
can keep synchronous operation or not when undergoing a large disturbance such
as change of parameters, dispatching loads or accidents. During study of the
problem above, the basic equations of synchronous machines can’t be linearized
any longer, so the problem will be studied only by numerical method. Generally
speaking, dynamic stability limit is lower than static stability one, but a synchronous
machine can’t operate normally if exceeding the static stability limit while it may
operate normally if exceeding the dynamic stability limit because the accidents
can’t often occur and are not so severe as calculation of dynamic stability limit
even though the accidents happen. Of course, the stability limits of two types are
all not permitted to be exceeded during design of power systems, but here
illustrate the difference between them only.
It should be pointed out, that the definition of static stability for synchronous
machines is not unified in some references and bibliography. As mentioned
before, static stability is to discuss a problem whether the synchronous machine
can restore its original steady-state operating mode or not when undergoing a
small perturbation and then abolishing it, but the rate and fashion of small
perturbation are not prescribed. If the small perturbation is caused by small
increase of loads and the increase rate of machine loads is very slow, the
excitation current has not any change and only the rotor position angle, namely
power angle, reaches a new balance position at a very low speed, which is the
first possible case. The second possible case is that the increase rate of small
loads is relatively fast to bring about change of excitation current due to change
of rotor position angle even if the synchronous machine has no excitation
regulation device. Of course, with excitation regulation device exists more
change of excitation current. In the second possible case, the static stability limit
is higher than the first possible case, namely constant excitation current.
Therefore, in accordance with the cases above, the static stability can be divided
into steady-state static stability and dynamic-state static stability in some
references. There are excitation regulation devices in modern generators, so
analysis of static stability has to consider influence of excitation current change
on static stability. Therefore, by static stability refer to dynamic-state static
stability in this book even if there is no excitation regulation device.
For combination system of synchronous machine and capacitances as discussed
in Chapter 3, when the rotor inertia is infinite the influence of speed change on
self-excitation can be neglected, i.e. the self-excitation is considered as an
electromagnetic parameter resonance. If the rotor inertia is relatively small, the
effect of rotor speed change must be taken into account, i.e. the self-excitation is
an electromechanical parameter resonance. Therefore, during analysis of the
269
AC Machine Systems
conditions to cause self-excitation, the influence of rotor motion should be
counted in and the small perturbation and Linearization theory can be used to
analyse the self-excitation phenomenon.
In brief, no matter what problem such as small oscillation, static stability or
self excitation it is, the basic equations of synchronous machines have been
nonlinear. However, those problems have a common feature, namely study of the
phenomena for synchronous machines after undergoing a small perturbation, so
the behaviors after sustaining a small perturbation depend mainly upon the initial
state and the corresponding basic equations can be linearized. By linearization
refer to the following way: when a synchronous machine undergoes a small
perturbation, the variable x can be written as ( x0 'x), in which x0 is the initial
state before perturbation and 'x is an increment caused by perturbation.
Substituting those variables into nonlinear basic equations and neglecting the
second-and higher-order increments, the linearized equations with variable 'x
can be got. Obviously, it is convenient to use the linearized equations to research
into several problems such as small oscillation, static stability, self-excitation
phenomenon, etc.
4.2
Steady-State Small Oscillation and Torque Coefficients
of Synchronous Machine Systems
In practical operation of synchronous machines there may be an oscillation due
to some adverse causes. During oscillation, the speed, current, voltage, power
and torque are all not constants, and possibly reach the hazardous values.
Therefore, the oscillation has an adverse effect on the machine itself and power
system connected with it, which sometimes brings about users not to be capable
of work, e.g. the electric lights blink and the motor speeds are unstable, which
possibly makes the machine out of synchronism and causes the electric service to
be interrupted, or even makes the machine and apparatus damaged. Thus, it is
significant to understand the oscillation phenomenon of synchronous machine
systems and to take some measures, for example fitting a damping winding,
increasing the machine-set inertia and installing protection equipment.
There are many causes to stimulate oscillations, e.g. the design and regulation
of controllers are not suitable, the prime-mover or load has a pulsating torque and
the outer load is unstable or asymmetrical, etc.
On the basis of the different causes producing oscillations, they can be divided
into the free oscillation and forced oscillation.
During an oscillation as mentioned before, the basic equations of a synchronous
machine are nonlinear, because its speed is not constant. However, here we only
study the small oscillation in steady-state of synchronous machine systems, so the
270
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
basic equations can be linearized according to the small perturbation theory
referring to [3,4,6]. Under this condition, the angle G between the rotor axis and
synchronous axis changes periodically around a certain value G 0 . Let us suppose
this function to be
G
G0 D
G 0 D m sin O t
(4.2.1)
in which
D D m sin O t
D m is the amplitude of the small oscillation and O is the angular velocity of the
machine oscillation whose value lies in the range from zero to 0.05, and 0.02
corresponds to an oscillation period per second.
In practical operation of a synchronous machine, its oscillation is more
complicated than mentioned above, but the oscillation, in virtue of its periodic
nature, can always be analysed into corresponding harmonics, so the supposed
function is still suitable as far as a certain harmonic is concerned.
Because
J
t G J 0c
t J 0c G 0 D m sin O t
(4.2.2)
we can obtain
Z
dJ
dt
1
dD
dt
1 OD m cos O t 1 s
(4.2.3)
dD
OD m cos O t is the instantaneous slip of a machine.
dt
If there is no oscillation in a machine, the corresponding relations, by virtue of
Park’s equations, can be gotten as follows:
in which s
ud 0
uq 0
\d0
\ q0
Te 0
U sin G 0
\ q 0 rid 0 ½
°
U cos G 0 \ d 0 riq 0 °
°
E xd id 0
¾
°
xq iq 0
°
°
\ d 0iq 0 \ q 0 id 0
¿
(4.2.4)
The steady-state values above, due to the small oscillation, will change a little
with time. Representing these small changes by 'ud , 'uq , '\ d , '\ q , 'id , 'iq
and 'Te , we can write the corresponding values ud , uq , \ d , \ q , id , iq and Te as
271
AC Machine Systems
ud
uq
\d
\q
id
iq
Te
ud 0 'ud ½
°
u q 0 'u q °
\ d 0 '\ d °
°°
\ q 0 '\ q ¾
°
id 0 'id
°
°
iq 0 'iq
°
Te 0 'Te °¿
(4.2.5)
The machine having no excitation-regulation, we have
U fd
constant,
G ( p )U fd
E
Substituting the values above into Park’s formulas (2.4.1), (2.4.11) and (2.4.15)
and torque equation (2.4.22), we can obtain
p (\ d 0 '\ d ) (\ q 0 '\ q )(1 s ) r (id 0 'id ) ½
°
p'\ d \ q 0 s\ q 0 '\ q s'\ q rid 0 r 'id °
uq 0 'uq p'\ q \ d 0 s\ d 0 '\ d s'\ d riq 0 r 'iq °
°
°°
\ d 0 '\ d E xd ( p)(id 0 'id ) E xd id 0 xd ( p)'id
¾
\ q 0 '\ q xq iq 0 xq ( p)'iq
°
Te 0 'Te (\ d 0 '\ d )(iq 0 'iq ) (\ q 0 '\ q )(id 0 'id ) °
°
°
\ d 0 iq 0 \ d 0 'iq '\ d iq 0 '\ d 'iq \ q 0id 0
°
\ q 0 'id '\ q id 0 '\ q 'id
°¿
u d 0 'u d
(4.2.6)
Subtracting equation (4.2.4) from the above equations, we have
'ud
'uq
p'\ d s\ q 0 '\ q s'\ q r 'id ½°
¾
p' \ q s\ d 0 '\ d s'\ d r 'iq °¿
'\ d
'\ q
xd ( p )'id ½°
¾
xq ( p )'iq °¿
'Te \ d 0 'iq '\ d iq 0 '\ d 'iq \ q 0 'id '\ q id 0 '\ q 'id
The machine connected to an infinite source, we can get
272
(4.2.7)
(4.2.8)
(4.2.9)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
½
°
U sin G U sin(G 0 D ) °
°
U sin G 0 cos D U cos G 0 sin D
¾
U cos G 0 'uq U cos G U cos(G 0 D ) °
°
U cos G 0 cos D U sin G 0 sin D
¿°
U constant
ud0 'ud U sin G 0 'ud
uq 0 'uq
Because D is small, we can approximately take cosD
the above equation and formula (4.2.4), we can obtain
'ud
U D cos G 0
'uq
U D sin G 0
1, sin D
D uq 0
½°
¾
D ud 0 °¿
(4.2.10)
D . Thus, from
(4.2.11)
Since the product of two small increments can be neglected as compared with
a small increment (s is also a small value), equations (4.2.7) and (4.2.9) can be
reduced to
'ud
D uq 0
'uq
D ud 0
'Te
p'\ d s\ q 0 '\ q r 'id ½°
¾
p'\ q s\ d 0 '\ d r 'iq °¿
\ d 0 'iq '\ d iq 0 \ q 0 'id '\ q id 0
(4.2.12)
(4.2.13)
Referring to equations (4.2.1) and (4.2.3) and substituting the time functions of s
and D and the equation (4.2.8) into formula (4.2.12), we have
D m uq 0 sin O t O\ q 0D m cos O t
½
°
pxd ( p )'id xq ( p )'iq r 'id °
¾
D m ud 0 sin O t O\ d 0D m cos O t
°
pxq ( p)'iq xd ( p)'id r 'iq °¿
(4.2.14)
which is a group of operational formulas with constant coefficients, in which
'id and 'iq seem to be produced by some emf that changes sinusoidally at an
angular velocity O. Therefore, its steady-state value is also a sinusoidal value
with an angular velocity O and we can use the Complex-number Vector Method
to solve it.
Let the complex-number time-vectors of 'id and 'iq be Id D m and IqD m
respectively. According to equation (4.2.14) we have
uq 0 jO\ q 0
ud 0 jO\ d 0
jO xd ( jO ) Id xq ( jO ) Iq rId ½°
¾
jO xq ( jO ) Iq xd ( jO ) Id rIq °¿
(4.2.15)
273
AC Machine Systems
After solving it, we can get
Id
Iq
[r jO xq ( jO )](uq 0 jO\ q 0 ) xq ( jO )(ud 0 jO\ d 0 ) ½
°
r 2 jO r[ xd ( jO ) xq ( jO )] (l O 2 ) xd ( jO ) xq ( jO ) °
¾
[r jO xd ( jO )](ud 0 jO\ d 0 ) xd ( jO )(uq 0 jO\ q 0 ) °
r 2 jO r[ xd ( jO ) xq ( jO )] (l O 2 ) xd ( jO ) xq ( jO ) °¿
(4.2.16)
Having Id and Iq , we can find \ d D m and \ qD m the complex-number time-vectors
of '\ d and '\ q according to equation (4.2.8) as follows:
xd ( jO ) Id ½°
¾
xq ( jO ) Iq °¿
\ d
\ q
(4.2.17)
Supposing
Id
I
q
I d jI dc
I q jI qc
\ d \ d j\ dc
\ q \ q j\ qc
we can obtain
'id
'iq
'\ d
'\ q
D m I d sin O t D m I dc cos O t
D m I q sin O t D m I qc cos O t
½
°
°
¾
D m\ d sin O t D m\ dc cos O t °
D m\ q sin O t D m\ qc cos O t °¿
(4.2.18)
Substituting the above equation into equation (4.2.13), we have
'Te
in which
Ts
TD
(\ d iq 0 \ d 0 I q \ q id 0 \ q 0 I d )D m sin O t
½
°
(\ dc iq 0 \ d 0 I qc \ qc id 0 \ q 0 I dc )D m cos O t °
°
TsD TD s
°
¾
°
\ d iq 0 \ d 0 I q \ q id 0 \ q 0 I d
°
°
1
°
(\ dc iq 0 \ d 0 I qc \ qc id 0 \ q 0 I dc )
O
¿
(4.2.19)
From the formulas above we can see, 'Te the additional electromagnetic
torque caused by the rotor oscillation consists of two components M sD and TD s.
274
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
The latter is proportional to slip and the first to the angle D which the rotor leaves
its balanced position by. Therefore, the corresponding coefficients Ts and TD are
called the synchronization torque coefficient and damping torque one
respectively.
Since 'Te also changes sinusoidally, it can be expressed in a complex-number
vector. Letting the complex-number vector of 'Te be TD m and according to
'Te TsD TD s TsD TD pD , we have
TD m
TsD m jOTDD m
or
T
Ts jOTD
\ d 0 Iq \ d iq 0 \ q 0 Id \ q id 0
[iq 0 xd ( jO ) \ q 0 ]Id [\ d 0 id 0 xq ( jO )]Iq
(4.2.20)
so Ts and TD can be found according to Id and Iq . This calculation method is
convenient and we shall use it later.
(1) The synchronization torque and damping torque coefficients when stator
resistance r can be neglected
We have stated the concept and calculation method of those coefficients as
before. However, those coefficients have rather complicated forms. In order to
illustrate their concepts and analyse them conveniently, we shall at first discuss
the synchronization torque and damping torque coefficients when stator resistance
r can be neglected.
The stator resistance r neglected, equation (4.2.16) can be simplified to
Id
Iq
Because r
0, we have ud 0
ud 0 O 2\ q 0 jO (\ d 0 uq 0 ) ½
°
(1 O 2 ) xd ( jO )
°
¾
uq 0 O 2\ d 0 jO (\ q 0 ud 0 ) °
°
(O 2 ) xq ( jO )
¿
(4.2.21)
\ q 0 and uq 0 \ d 0 , and according to
xd ( jO )
X d O jRd O
xq ( jO )
X qO jRqO
we can obtain
Id
1
ud 0
xd ( jO )
Iq
1
uq 0
xq ( jO )
X d O jRd O
½
ud 0 °
2
2
X d O Rd O
°
¾
X qO jRqO
uq 0 °
°
X q2O Rq2O
¿
(4.2.22)
275
AC Machine Systems
Substituting the above equation into formula (4.2.20), we can get
T
Ts jOTD
[iq 0 xd ( jO ) ud 0 ]
X qO jRqO
X d O jRd O
ud 0 [uq 0 id 0 xq ( jO )] 2
uq 0
2
2
X d O Rd O
X qO Rq2O
X qO jRqO 2
X d O jRd O 2
ud 0 2
uq 0
2
2
X d O Rd O
X qO Rq2O
(4.2.23)
º
RqO
U 2 ª Rd O
sin 2 G 0 2
cos 2 G 0 »
« 2
2
2
O ¬« X d O Rd O
X qO RqO
¼»
(4.2.24)
uq 0 id 0 ud 0 iq 0 Thus,
TD
uq 0 id 0 ud 0 iq 0 Ts
Q0 X qO U 2
X d OU 2
2
sin
G
cos 2 G 0
0
X d2O Rd2O
X q2O Rq2O
X qO U 2
X d OU 2
2
sin
G
cos 2 G 0
0
X d2O Rd2O
X q2O Rq2O
(4.2.25)
in which
Q0
uq 0 id 0 ud 0 iq 0
U (id 0 cos G 0 iq 0 sin G 0 )
U [ I 0 sin(G 0 M 0 ) cos G 0 I 0 cos(G 0 M 0 )sin G 0 ]
UI 0 sin M 0
is the output reactive power before the oscillation occurs, in which I 0
id20 iq20
and M 0 are the corresponding steady-state current and its phase-angle.
Because
Q0
UI 0 sin M 0
EU
U2
U2
cos G 0 cos 2 G 0 sin 2 G 0
xd
xd
xq
EU
U2
1§ 1
1 ·
cos G 0 ¨ ¸ (1 cos 2G 0 ) xd
xd
2 ¨© xq xd ¸¹
the formula (4.2.25) can be written as another form:
Ts
X qO U 2
X U2
EU
U2
U2
cos G 0 cos 2 G 0 sin 2 G 0 2 d O 2 sin 2 G 0 2
cos 2 G 0
xd
xd
xq
X d O Rd O
X qO Rq2O
§ 1
ª X
EU
1·
1º
cos G 0 ¨ ¸ U 2 (cos 2 G 0 sin 2 G 0 ) « 2 d O 2 » U 2 sin G 0
¨
¸
xd
¬ X d O Rd O xd ¼
© xq xd ¹
276
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
ª X qO
1º
« 2
» U 2 cos 2 G 0
2
«¬ X qO RqO xq »¼
xd xq 2
ª X
1
EU
cos G 0 U cos 2G 0 « 2 d O 2 xd
xd xq
¬ X d O Rd O xd
º 2 2
» U sin G 0
¼
ª X qO
1º
« 2
» U 2 cos 2 G 0
2
«¬ X qO RqO xq »¼
(4.2.25a)
From statements above we can see, the synchronization torque coefficient
EU
cos G 0 is the fundamental component
consists of four parts, in which the first
xd
that depends upon the corresponding steady-state excitation emf and terminal
xd xq 2
U cos 2G 0 is the salient-pole component. Those two
voltage; the second
xd xq
components are independent of the rotor-circuit condition, which are the
synchronization torque coefficient when the rotor oscillates at a very slow speed.
Letting Ts 0 represent those two components, we have
Ts 0
Ts |O
0
xd xq 2
EU
cos G 0 U cos 2G 0
xd
xd xq
The latter two components in formula (4.2.25a) depend upon the oscillation
currents in the rotor d-axis and q-axis windings respectively. Letting TsO represent
them, we have
TsO
ª X dO
1
« 2
2
¬ X d O Rd O xd
ª X qO
º 2 2
1º
» U 2 cos 2 G 0
» U sin G 0 « 2
2
«¬ X qO RqO xq »¼
¼
Likewise, the damping torque coefficient consists of two components, which
depend upon the oscillation currents in the rotor d-axis and q-axis windings
respectively.
Ts 0 and TsO varying with G 0 during O 0.03 in a typical machine are shown
in Fig. 4.2.1. The damping torque coefficient M D varying with G 0 in the same
machine is shown in Fig. 4.2.2. On the basis of those curves it is not difficult to
understand that the q-axis damping winding has a significant effect.
The TD and Ts expressions can also be rewritten as the following forms to use
them conveniently.
277
AC Machine Systems
Figure 4.2.1 Synchronization torque coefficient components Ts 0 -G 0 and TsO - G 0
curves during O
0.03
Figure 4.2.2
Damping torque coefficient TD - G 0 curve
(a) Without damping winding
1
xd ( jO )
X dO
R
j 2 dO 2
2
X Rd O
X d O Rd O
2
dO
1 jOTd 0
xd jO xdc Td 0
xd O 2 xdc Td20 jOTd 0 ( xd xdc )
xd2 O 2 xdc2Td20
278
(4.2.25b)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
i.e.
xd xdc O 2Td20
xd2 O 2 xdc2Td20
OTd 0 ( xd xdc )
xd2 O 2 xdc2Td20
X dO
2
X d O Rd2O
Rd O
X Rd2O
2
dO
In addition,
X qO
1
xq ( jO )
X
2
qO
R
X
2
qO
X
2
qO
j
2
qO
RqO
X
2
qO
Rq2O
1
xq
i.e.
X qO
Rq2O
RqO
Rq2O
1
xq
0
Substituting those values into equations (4.2.24) and (4.2.25a), we can get the
damping torque and synchronization torque coefficients of a synchronous
machine without damping winding and supposing the stator resistance r 0 as
follows:
( xd xdc )TdcU 2 sin 2 G 0
xd xdc (1 O 2Tdc2 )
TD
( xd xdc )Td 0 2 2
U sin G 0
xd2 O 2Td20 xdc2
Ts
xd xq 2
( x xc ) xc O 2T 2
EU
U cos 2G 0 d 2 d 2d 2 d20 U 2 sin 2 G 0
cos G 0 xd
xd xq
xd ( xd O Td 0 xdc )
(b) With a d-axis and q-axis windings,
(4.2.25b), and
1
has the same form as equation
xd ( jO )
1
can be obtained by a similar method as follows:
xq ( jO )
X qO
xq xqccO 2 Tq20
X q2O Rq2O
xq2 O 2 xqcc2 Tq20
Rq O
OTq 0 ( xq xqcc)
xq2 O 2 xqcc2 Tq20
X
2
qO
2
qO
R
Therefore, in accordance with equations (4.2.24) and (4.2.25a), the synchronization
279
AC Machine Systems
torque and damping torque coefficients of a synchronous machine with a d-axis
and q-axis windings, supposing the stator resistance r 0, can be written as
TD
( xq xqcc)Tq 0 2
( xd xdc )Td 0 2 2
U sin G 0 2
U cos 2 G 0
2
2 2
2
xd O Td 0 xdc
xq O 2Tq20 xqcc2
Ts
xd xq 2
( x xc ) xc O 2T 2
EU
cos G 0 U cos 2G 0 d 2 d 2d 2 d20 U 2 sin 2 G 0
xd
xd xq
xd ( xd O Td 0 xdc )
( xq xqcc) xqccO 2Tq20
xq ( xq2 O 2Tq20 xqcc2 )
U 2 cos 2 G 0
(c) If on the rotor there is not only an excitation winding but also a d-axis and
q-axis damping windings, and the excitation-winding resistance can be neglected,
then the d-axis operational reactance xd ( p ), according to equations (2.5.69),
(2.5.71), (2.5.47) and (2.5.49), can be reduced to
xd ( p)
xd 2
( xd xdcc ) X ffd Tdcc0 p xad
X ffd (1 pTdcc0 )
xdc xdccTdcc0 p
1 pTdcc0
According to the above formula and referring to equation (4.2.25b), we can get
X dO
X d2O Rd2O
Rd O
X Rd2O
2
dO
xdc xdccO 2Tdcc02
xdc2 O 2 xdcc2 Tdcc02
OTd 0 ( xdc xdcc )
xdc2 O 2 xdcc2Tdcc02
Therefore, according to equations (4.2.24) and (4.2.25a) and supposing the
stator and excitation winding resistances r R fd 0, the synchronization torque
and damping torque coefficients, when there are both excitation winding and
damping winding on the rotor, can be obtained as follows:
TD
( xq xqcc)Tq 0 2
( xdc xdcc )Tdcc0
U 2 sin 2 G 0 2
U cos 2 G 0
2
2
2
2
2 2
2
xdc O Tdcc0 xdcc
xq O Tq 0 xqcc
Ts
xd xq 2
( x xdcc ) xdccO 2Tdcc02 ( xd xdc ) xdc 2 2
EU
cos G 0 U cos 2G 0 d
U sin G 0
xd
xd xq
xd ( xdc2 O 2Tdcc02 xdcc2 )
( xq xqcc) xqccO 2Tq20
xq ( xq2 O 2Tq20 xqcc2 )
U 2 cos 2 G 0
(d) If there are excitation winding and d-axis and q-axis damping windings on
the rotor, but the excitation-winding resistance can’t be neglected, then xd ( jO )
has a more complicated form referring to equation (2.5.68). However, we can
280
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
reduce it as follows.
Because
X 11d Tdcc0
2
xad
X ffd
R11d
xd Tdc
2
xad
X ffd
xd
X ffd
Td 0
R fd
,
, T1d 0
Td 0 ,
Tdcc
X 11d
R11d
2
§
·
xad
¨1 ¸ T1d 0
xd X 11d ¹
©
2
2
3
) xad
( X 11d X ffd ) 2 xad
xd ( X 11d X ffd xad
T1cd
2
)
R11d ( xd X ffd xad
,
we can get
2
p 2 ( X 11d X ffd xad
) p ( X 11d R fd X ffd R11d ) R11d R fd
p 2 R11d X ffd Tdcc0 pR11d R fd (Td 0 T1d 0 ) R11d R fd
R11d R fd [Tdcc0Td 0 p 2 (Td 0 T1d 0 ) p 1],
2
3
2
2
2
p 2 [ X 11d xad
] p ( xad
R11d xad
R fd )
2 xad
X ffd xad
2
2
p 2 [ xd ( X 11d X ffd xad
) ( xd X ffd xad
) R11d Tdcc]
§T
T
2
R11d R fd ¨ d 0 1d 0
pxad
¨X
© ffd X 11d
·
¸¸
¹
ª x2
º
x2
p 2 xd R11d R fd [Td 0Tdcc0 TdccTdc ] pR11d R fd « ad Td 0 ad T1d 0 » ,
X 11d
«¬ X ffd
»¼
xd ( p)
xd 2
3
2
2
2 xad
X ffd xad
p 2 ( X 11d xad
) pxad
( R11d R fd )
2
p 2 ( X 11d X ffd xad
) p ( X 11d R fd X ffd R11d ) R11d R fd
ª x2
º
x2
p 2 xd (Td 0Tdcc0 TdccTdc ) p « ad Td 0 ad T1d 0 »
X 11d
¬« X ffd
¼»
xd 2
Tdcc0Td 0 p (Td 0 T1d 0 ) p 1
ª
º
x2
x2
TdccTdcp 2 p «Td 0 T1d 0 ad Td 0 ad T1d 0 » 1
xd X ffd
X 11d xd
»¼
¬«
xd
2
Tdcc0Td 0 p (Td 0 T1d 0 ) p 1
xd
TdccTdcp 2 (Tdc T1cd ) p 1
Tdcc0Td 0 p 2 (Td 0 T1d 0 ) p 1
281
AC Machine Systems
According to this expression of xd ( p ), the values corresponding to xd ( jO ) are
X dO
2
X d O Rd2O
(1 O 2Td 0Tdcc0 )(1 O 2TdcTdcc ) O 2 (Td 0 T1d 0 )(Tdc T1cd )
xd (1 O 2TdcTdcc ) 2 O 2 (Tdc T1cd ) 2
Rd O
2
X d O Rd2O
O (1 O 2Td 0Tdcc0 )(Tdc T1cd ) (1 O 2TdcTdcc )(Td 0 T1d 0 )
xd
(1 O 2TdcTdcc ) 2 O 2 (Tdc T1cd )2
According to those results and supposing the stator resistance r 0, the
synchronization torque and damping torque coefficients, when there are excitation
winding and d-axis and q-axis damping windings on the rotor, can be obtained as
follows:
TD
Ts
U 2 (1 O 2TdcTdcc )(Td 0 T1d 0 ) (1 O 2Td 0T1ccd 0 )(Tdc T1cd ) 2
sin G 0
xd
(1 O 2TdcTdcc )2 O 2 (Tdc T1cd )2
( xq xqcc)Tq 0 2
2
U cos 2 G 0
xq O 2Tq20 xqcc2
xd xq 2
EU
O2
cos G 0 U cos 2G 0 {(1 O 2TdcTdcc )(TdcTdcc Td 0Tdcc0 )
xd
xd xq
xd
(Tdc T1cd )(Td 0 T1d 0 Tdc T1cd )}/{(1 O 2TdcTdcc ) 2 O 2 (Tdc T1cd ) 2 }
˜ U 2 sin 2 G 0 O 2 ( xq xqcc) xqccTq20 2
U cos 2 G 0
xq ( xq2 O 2 xqcc2Tq20 )
We have found the synchronization torque and damping torque coefficients
during oscillation as above, and with them further discuss the conditions causing
the oscillation in a synchronous machine.
(2) The steady-state small-oscillation conditions of synchronous machines when
the stator resistance r neglected
As mentioned before, the electromagnetic torque during an oscillation consists
of the following parts:
Te
Te 0 'Te
Te 0 TD s TsD
(4.2.26)
Therefore, the rotor-motion equation during the steady-state small-oscillation can
be written as
H
d 2 (G 0 D )
TD s TsD Te 0
dt 2
or
2
H
282
dD
TD s TsD
dt 2
Tm Te 0
½
Tm °
°°
¾
°
°
°¿
(4.2.27)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
in which Tm is the mechanical torque of a prime-mover or a load that may be
constant or variable. In the latter, Tm varies periodically with time and can be
written as
Tm
Tm 0 ¦ TmO cos(O t M O )
(4.2.28)
Because the synchronous machine is in a steady-state small-oscillation state,
i.e. the average value of D is equal to zero, we have Te 0 Tm 0 , or else D will have
an average value. Thus, the rotor-motion equation can also be written as
H
d 2D
dD
TD
TsD
2
dt
dt
¦ T O cos(Ot MO )
m
(4.2.29)
That is a linear differential equation, so the harmonics of the mechanical torque
can be studied separately. So far as each of the harmonics is concerned, the
corresponding equation is equal to that of the R-L-C circuit shown in Fig. 4.2.3,
because the equation of the R-L-C circuit has the following form:
Um
Z
sin(Z t M )
or
U m cos(Z t M )
½
di 1
idt
dt C ³ °°
°
¾
°
2
di
di 1
L 2 R i°
dt
dt C °¿
Ri L
(4.2.30)
Figure 4.2.3 Equivalent R-L-C circuit
Equation (4.2.29) compared with equation (4.2.30), there is the following
relation between them,
L
R
p
H
p
TD
1
i Um
C
p p p
Ts D TmO
Z
p
O
and we can obtain the following results.
(a) When TD 0 (corresponding to R
0 in the electric circuit), even if the
mechanical torque is constant, i.e. TmO
0 (supposing Ts ! 0) , the oscillation
283
AC Machine Systems
1
in the electric
ZC
circuit). That phenomenon is called the no-decaying natural oscillation.
Accordingly, there are the following relations: the frequency of natural
1 Ts
oscillation f 0
, the angular velocity of natural oscillation O0 2Sf 0
2S H
may occur and does not decay (corresponding to Z L
Ts
, the period of natural oscillation T0
H
1
f0
2S
H
.
Ts
Thus, the conditions producing the no-decaying natural oscillation are
TD
0 (corresponding to R
0)
and
O2H
Ts
§
¨ corresponding to Z L
©
1 ·
Z C ¸¹
(b) When TD ! 0, the oscillation caused by some forcing-factor can not be
maintained for long after the forcing-factor disappeared. The decaying rate
depends upon the electromechanical parameters of the machine.
It is more complicated strictly to solve the decaying process, because the machine
is not in the steady-state small-oscillation state now and the corresponding TD
and Ts are already not constants. However, if the decaying rate and oscillation
frequency are all very low, TD and Ts can be taken approximately as constants,
i.e. the whole decaying process consists of a series of steady-state small natural
oscillations. Therefore, the rotor-motion equation (4.2.29) can still be used, i.e.
( Hp 2 TD p Ts )D
0
(4.2.31)
Solving it, we can obtain
p1,2
TD r TD2 4 HTs
2H
(4.2.32)
Observing equations (4.2.24) and (4.2.25) we can see, TD is positive and Ts
may be positive or negative if the stator resistance r is neglected. Thus, we can
know:
(i) When TD2 ! 4 HTs and Ts ! 0, the rotor will come back to its original position
aperiodically.
(ii) When TD2 4 HTs and Ts ! 0, the machine can maintain its oscillation at a
certain time, but its amplitude will gradually decay to zero and the oscillation
stops. Accordingly, there are the following relations,
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4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
4 HTs TD2
The oscillation angular velocity O
The oscillation frequency
f
The oscillation period
T
The decay time-constant
Td
2H
4 HTs TD2
4SH
4SH
4 HTs TD2
2H
TD
Therefore, if D m is the amplitude when the oscillation starts to decay (i.e. t
Dme
T
D t
2H
0),
is the amplitude when time is equal to t, and the next neighbouring
amplitude is D m e
TD
( t T / 2)
2H
. Thus, the ratio between two neighbouring amplitudes is
' e
TD
t
2H
e
TD
( t T / 2)
2H
TDT
e 4H
(4.2.33)
or
ln '
TD
T
t D (t T / 2)
2H
2H
TDT
4H
(4.2.33a)
in which ' is called the ratio of the oscillation decay.
(iii) when Ts 0, equation (4.2.31) has a positive real root. Therefore, the
machine can not stop its oscillation but will aperiodically increase its
rotor-position angle G to bring about out of step. This loss of synchronism is
called the crawling phenomenon, because during this process the rotor-position
angle G grows slowly. since Ts Ts 0 TsO and TsO is always positive referring to
Fig. 4.2.1, Ts is always more than Ts 0 . If O | 0, then TsO | 0 and Ts | Ts 0 ; thus,
Ts 0 0 is the condition causing the crawling phenomenon and falling out of step.
(c) Now we shall further discuss the forced oscillation produced by the
mechanical torque harmonic TmO cos(O t M O ).
(i) According to Fig. 4.2.3, the amplitude D mO can be written as
D mO
TmO / O
T ·
§
TD2 ¨ O H s ¸
O¹
©
TmO
2
2
2
D
O T (O 2 H Ts ) 2
(4.2.34)
When TD and Ts are all equal to zero, i.e. the stator and rotor windings are all
open-circuited, the oscillation amplitude D mc O and instantaneous value D Oc have
285
AC Machine Systems
D mc O
TmO
, D Oc
O2H
D mc O cos(O t M O )
Thus,
[O
D mO
D mc O
O2H
O 2TD2 (O 2 H Ts )2
(4.2.35)
is called the oscillation amplification or resonance coefficient, which represents
the amplitude amplification caused by the electromagnetic action. If O O0 , then
O02 H Ts and [O has its maximum value, which is
[ mO
0
O0 H
TD
(4.2.36)
D mO changing with O is shown in Fig. 4.2.4. These curves are called the
resonant curves.
Figure 4.2.4
Resonant curves
In order to prevent the oscillation amplitude from too big, the oscillation
angular velocity of the mechanical torque applied must be far away from the
angular velocity of the natural oscillation. So far as the machine itself is
concerned, the following measures during design can be used.
(a) The inertia constant H changes, i.e. the resonant curve moves horizontally;
(b) The damping winding is installed or strengthened to enhance the damping
torque coefficient, i.e. to move the resonant curve vertically.
When the damping winding designed, the leakage reactance has a little effect
on the damping torque coefficient and it is a good method to change its
resistances R11d and R11q . However, the cross-section of the damping bar
286
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
increases sometimes to bring about a difficulty of manufacture. Therefore, the
choice of the damping bar material and change of the ratio between d-axis and
q-axis damping resistances are also important measures. For example, we can
decrease the q-axis resistance R1q with the help of increasing the d-axis resistance
R1d , because the machine ordinarily operates at the angle G 0 20 ~ 30eand the
q-axis damping winding has a strong influence referring to Fig. 4.2.2.
(ii) The non-uniformity of the applied mechanical torque can usually be indicated
by v the non-uniformity of the angular velocity caused in non-uniform mechanical
torque when the stator and rotor windings are all open-circuited, in which
v
Z H G Z HM
Zs
(4.2.37)
in which Z H G , Z HM and Z s are the maximum, minimum and average angular
velocities respectively when all windings are opened. Ordinarily, v is smaller
1
1
.
than
250 150
So far as a certain harmonic angular velocity Ois concerned, we have
vO
(Z s 'Z OG ) (Z s 'Z OG )
Zs
2'Z OG
Zs
Because Z s
amplitude
2
dD Oc
dt
HG
Zs
2D mc O O
Zs
(4.2.38)
1, from equation (4.2.35) and the above formula we can get the
D mO
[ OD mc O
vO [ O
2O
(4.2.39)
As mentioned before, under the condition of the forced oscillation, the
synchronization torque coefficient Ts is always more than Ts 0 , and during
natural oscillation and Ts 0 0 the machine will crawl and lose synchronism.
Therefore, there is a problem whether we can, during Ts 0 0, use a mechanical
method to maintain a certain forced oscillation to reach Ts ! 0 and to prevent the
machine from out of step. The answer to that problem is “no”, i.e. during
Ts 0 0, even though the result Ts ! 0 can be obtained by a forced oscillation,
the crawling phenomenon may still occur. That is because the forced oscillation
only applies a new oscillation torque to the rotor. At a viewpoint of mathematics,
the forced oscillation torque can not change the root nature of the rotor-motion
characteristic equation but adds a mechanical oscillation of a new frequency
referring to Fig. 4.2.3.
287
AC Machine Systems
(3) The effect of stator resistance r
For convenience, as mentioned before, we have considered the stator resistance r
to be neglected, in which TD is always positive. Therefore, whether the machine
can operate stably or not depends upon the sign of Ts 0 . When the forced
oscillation occurs owing to some cause, if Ts 0 is negative the machine will still
crawl and lose synchronism even though the cause producing an oscillation
disappears. On the contrary, if Ts 0 is positive, in the machine there is only the
natural oscillation without decay or that with decaying to zero.
In practical operation, e.g. a synchronous machine connected to the infinite
network through the transmission line, sometimes the machine loses synchronism
when Ts is positive. Furthermore, the out-of-step modes include not only increasing
aperiodically the rotor-position angle G but also enhancing periodically the
oscillation amplitude whose frequency is ordinarily low and about 1 2Hz. The
latter is called the self-oscillation out of step phenomenon. There are many
causes to make that phenomenon, in which the large stator-resistance is a main
cause, because the damping torque coefficient TD decreases actually when the
stator resistance increases. In addition, when the stator resistance is more than a
certain value, TD will become negative. According to equation (4.2.32) we can
see, when TD is negative, equation (4.2.31) will have positive real roots or
complex-number roots with a positive real part. Under the first condition, the
machine will increase aperiodically its rotor-position angle and lose synchronism.
In the latter, the machine is out of step through self-oscillation. Thus, it is
significant to study the effect of stator resistance on the damping torque
coefficient and synchronization torque coefficient.
Substituting ud 0 and uq 0 in equation (4.2.4) into equation (4.2.16), we can get
Id
Iq
{[r 2 jO rxq ( jO )]iq 0 rxq ( jO )id 0 [(1 O 2 ) xq ( jO ) jO r ]\ q 0 ½
°
r\ d 0 }/{r 2 jO r[ xd ( jO ) xq ( jO )] (1 O 2 ) xd ( jO ) xq ( jO )} °°
¾
{[r 2 jO rxd ( jO )]id 0 rxd ( jO )iq 0 [(1 O 2 ) xd ( jO ) jO r ]\ d 0 °
°
r\ q 0 }/{r 2 jO r[ xd ( jO ) xq ( jO )] (1 O 2 ) xd ( jO ) xq ( jO )} °¿
(4.2.40)
Substituting the formulas above into equation (4.2.20) and arranging it, we can obtain
T
Ts jOTD
Q0 1
{(1 O 2 )[\ d20 xd ( jO )
r 2 jO r[ xd ( jO ) xq ( jO )] (1 O 2 ) xd ( jO ) xq ( jO )
\ q20 xq ( jO )] jO r[\ d20 \ q20 2\ d 0 id 0 xd ( jO ) 2\ q 0 iq 0 xq ( jO )
id20 xd ( jO ) xq ( jO ) iq20 xd ( jO ) xq ( jO )] r 2 [2\ q 0 iq 0 2\ d 0 id 0
id20 xq ( jO ) iq20 xd ( jO )]}
(4.2.41)
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4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
That expression is more complicated. For convenience, we shall take some
approximate conditions. At first, although the stator resistance r can not be neglected,
its value and O are all not very large. Therefore, we can neglect r 2 and higher
power terms and take (1 O 2 ) | 1, so the above formula can be reduced to
T
ª§ \
·
\ q20
\ d20
jO r «¨ d 0 id 0 ¸
Ts jOTD | Q0 ¨
¸
xq ( jO ) xd ( jO )
«© xq ( jO )
¹
2
¬
§ \ q0
·
¨
iq 0 ¸
O
(
j
)
x
© d
¹
2
º
»
»¼
(4.2.42)
Without damping winding, we have
1
xd ( jO )
1 jOTd 0
x xdc
1
|
j d
OTd 0 xdc2
xd xdc jOTd 0 xdc
1
xq ( jO )
1
xq
Substituting those values into equation (4.2.42) and supposing the imaginary part
1
of
to be much less than the real part, we can obtain
xd ( jO )
Ts | Q0 \ d20
xq
\ q20
(4.2.43)
xdc
2
2
2
ª§ \
· § \ q0
§ \ q0 ·
· º
1
d
0
«
»
TD | 2 ( xd xdc ) ¨
r
i
i
¨
¸
¸
¨
d0 ¸
q0 ¸
O Td 0
xdc
«©¨ xq
»
© xdc ¹
©
¹
¹
¬
¼
(4.2.44)
As mentioned before, when TD 0 there is a self-oscillation out of step and
when Ts 0 0 there is a crawling and loss of synchronism. Therefore, if TD or Ts 0
is negative, then the machine will lose synchronism, i.e. TD ! 0 and Ts 0 ! 0 are
the conditions under which the machine can operate in stability. Because there is a
little effect of r on Ts 0 , the condition determining TD ! 0 is also that determining
the critical value of r. According to equation (4.2.44) we can see, when
r!
§\ ·
1
( xd xdc ) ¨ q 0 ¸
2
O Td 0
© xdc ¹
2
2
§\ d0
· §\
·
id 0 ¸ ¨ q 0 iq 0 ¸
¨¨
¸
c
x
x
¹
© q
¹ © d
2
rk
(4.2.45)
289
AC Machine Systems
the machine can not operate in stability, in which rk is called the critical value of
the stator resistance.
Obviously, TD will decrease when r increases and TD will become negative
when r is more than rk . However, TD also depends upon a load, so rk will
change with the load. When a certain synchronous machine has no damping
winding, its critical resistance rk changes with a load, as shown in Figures 4.2.5.
and 4.2.6. According to those curves we can see, the lighter the load (i.e. the
smaller the value of I 0 or G 0 ) , the smaller the critical resistance rk , i.e. the more
easily does the oscillation occur. During no-load, i.e. G 0 | 0, we have \ q 0 | 0
and rk | 0, so the synchronous machine without damping winding can’t be used
as a synchronous condenser.
Figure 4.2.5 Critical resistance rk - I 0 curve
Figure 4.2.6 Critical resistance rk - G 0 curve
Because the value of rk is obtained according to the condition TD
Ts
. Since the value of Ts does not change greatly
H
accordingly we have O
O0
with r, there is O 2
Ts
in equation (4.2.45), so
H
O02
rk
0,
§\ ·
H
( xd xdc ) ¨ q 0 ¸
TsTd 0
© xdc ¹
2
2
§\ d0
· §\
·
id 0 ¸ ¨ q 0 iq 0 ¸
¨¨
¸
c
¹
© xq
¹ © xd
2
(4.2.45a)
It is clear that with a certain load the larger the inertia constant, the larger the
critical resistance rk . Thus, the machine stability can be raised through increasing
the inertia constant.
By calculating in detail we can see, the damping winding, especially the q-axis
290
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
damping winding, can make TD and rk increase. The stronger the damping
winding, the larger the value of rk . Therefore, the damping winding has an
important effect on the machine stability, especially when Ois close to O0 , the
damping winding will play an excellent role in limiting the oscillation amplitude.
Of course, when O is much different from O0 , its effect is not so significant,
which can be known from equation (4.2.34). In addition, installation of a
damping winding can decrease the need of raising the inertia constant, i.e. the
machine with damping winding may have a lighter rotor.
(4) Some physical concepts during machine oscillation
We have analysed some problems about small oscillation in detail as before. In
order to understand the change of the rotor-position angle G during an oscillation,
we shall further discuss it according to the following approximate assumptions:
(i) A non-salient pole synchronous machine without damping winding,
(ii) Only considering the transient process in excitation winding and not regarding
that in stator winding.
(iii) The speed of synchronous machine being synchronous speed, P | Te .
That is to say, we study this problem according to the following approximate
equations,
ud
\ q rid
uq
\ d riq
p\ fd R fd I fd
U fd
\d
xd id xad I fd
\q
xq iq
\ fd
xd id E
xd iq
xad id X ffd I fd
P | Te
E\ q
xd
At first, neglecting the stator resistance r, we discuss the influence of the excitationwinding self-inductance, because its value is large which has an important effect
on the excitation-winding flux-linkage or Eqc during an oscillation.
Because
U fd
Eqc
p\ fd R fd I fd
xad
\ fd
X ffd
we have
U fd
X ffd
xad
pEqc R fd I fd
(4.2.46a)
291
AC Machine Systems
If the machine has no oscillation, there is U fd R fd I fd 0 .
When the machine oscillates and U fd is kept constant, we have
U fd
X ffd
xad
p ( Eqc0 'E c) R fd ( I fd 0 'I fd )
(4.2.47)
Subtracting equation (4.2.46a) from equation (4.2.47), we can get
X ffd p'E c xad R fd 'I fd
0
(4.2.48)
or
0 Td 0 p'E c 'E
in which
'E
xad 'I fd
Because
\d
xd id xad I fd
Eqc xdc id
without an oscillation we have
Eqc 0 xdc id 0
xd id 0 xad I fd 0
(4.2.49)
and with an oscillation there is
Eqc 0 'E c xdc (id 0 'id )
xd (id 0 'id ) xad ( I fd 0 'I fd )
(4.2.50)
Subtracting equation (4.2.49) from the above equation, we have
'E c ( xd xdc )'id xad 'I fd
(4.2.51)
According to the above equation and formula (4.2.48) and after deleting 'I fd , we
can obtain
( pX ffd R fd )'E c R fd ( xd xdc )'id
or
(Td 0 p 1)'E c ( xdc xd )'id
(4.2.52)
If the machine operating in steady-state and the stator resistance r neglected, we have
292
E0
xad I fd 0
id 0
1
( E U cos G 0 )
xd
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
With an oscillation, there is
'E c ( xd xdc )'id
'E
xad 'I fd
'id
1
['E (U sin G 0 )D ]
xd
(4.2.53)
1
['E c ( xd xdc )'id (U sin G 0 )D ]
xd
(4.2.54)
or
'id
1
['E c (U sin G 0 )D ]
xdc
(4.2.55)
When oscillating exist
G
G0 D
E0 'E
Eqc Eqc 0 'E c
E
Substituting equation (4.2.55) into equation (4.2.52), we can get
§
xd ·
¨ Td 0 p ¸ 'E c
xdc ¹
©
( xdc xd )U sin G 0
D
xdc
(4.2.56)
If D D m sin O t , according to the above equation we can find the corresponding
'E c as follows:
'E c
E
( xd xdc )U sin G 0
2
§x ·
xdc ¨ d ¸ O 2Td20
© xdc ¹
O xc
arctan d Td 0
xd
½
D m sin(O t E ) °
°
°
¾
°
°
°¿
(4.2.57)
or
§
xc ·
'E c ¨1 d ¸ (U sin G 0 cos E )D m sin(O t E )
xd ¹
©
If X ffd
0, then Td 0
0 and E
'E c (4.2.57a)
0, so the above formula can be changed into
( xd xdc )U sin G 0
D
xd
293
AC Machine Systems
The above formula means that when the excitation winding reactance is neglected,
'E c has a linear relation to D and a reverse sign against DThat result coincides
with the physical process at all, because when G increases, id will be raised, so
the excitation winding flux-linkage \ fd xad id X ffd I fd will decrease, i.e. Eqc
will diminish, whose meaning is reflected by the negative sign in the formula
above. However, the excitation winding reactance considered, from equation
(4.2.57) we can see, the change of the flux-linkage, i.e. the change of Eqc , will
lag behind the change of G, i.e. when Gincreases, Eqc can’t decrease to its steadystate value at once, and when G decreases, Eqc can’t increase to its steady-state
value accordingly.
Because
D m sin(O t E ) D m sin O t cos E D m cos O t sin E
D cos E D m2 D 2 sin E
(4.2.58)
supposing
§
xc ·
A ¨1 d ¸U sin G 0 cos E
xd ¹
©
(4.2.59)
equation (4.2.57a) can be written as
'E c A[D cos E D m2 D 2 sin E ]
(4.2.57b)
Arranging it ,we have
'E c2 2 A cos E ('E c)D A2D 2
A2D m2 sin 2 E
or
2
§ 'E c ·
§ 'E c ·§ D · § D ·
¨
¸ 2cos ¨
¸¨
¸¨
¸
© AD m ¹
© AD m ¹© D m ¹ © D m ¹
2
sin 2 E
(4.2.58a)
which is an elliptical equation. Therefore, the relation between 'E c and D , due
to the excitation-winding self-inductance, become an elliptical equation instead
of a linear relation as X ffd 0.
As mentioned before, 'E Td 0 p'E c, so 'E jOTd 0 'E c, i.e. there is a
phase-angle difference of 90ebetween 'E and 'E c, whose amplitudes differ
from each other by OTd 0 times. Therefore, the relation between 'E and D is
similar to that between 'E c and D , i.e. the relation between 'E and D is also
an elliptical equation.
294
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
Now we further discuss the output power change of a non-salient pole machine
without stator resistance during an oscillation.
In steady-state, the output power or torque of a non-salient pole machine
without damping winding is
Te 0
P0
E0U
sin G 0
xd
(4.2.59a)
which is shown by a steady-state P - G curve in Fig. 4.2.7. The corresponding
transient emf is
Eqc 0
E0 ( xd xdc )id 0
E0 xd xdc
( E0 U cos G 0 )
xd
xdc
x xdc
E0 d
U cos G 0
xd
xd
From the two expressions above, the output power or torque can be indicated by
another formula as follows:
Te 0
Eqc 0U
1§ 1
1 ·
sin G 0 ¨ ¸U 2 sin 2G 0
xdc
2 © xdc xd ¹
P0
(4.2.59b)
When the machine has an oscillation and U is kept constant, neglecting the
second-power small increment we can obtain
Te
P
E\ q
xd
P0 'P
EU
sin G
xd
Te 0 'Te
( E0 'E )U
sin(G 0 D )
xd
E0U
EU
'EU
sin G 0 0 cos G 0D sin G 0
xd
xd
xd
'Te
'P
E0U
'EU
cos G 0D sin G 0
xd
xd
(4.2.60a)
or
Te
'Te
P
EqcU
1§ 1
1·
sin G ¨ ¸ U 2 sin 2G
2 © xdc xd ¹
xdc
Eqc0U
§ 1
1 ·
'E cU
cos G 0D sin G 0 ¨ ¸U 2 cos 2G 0D
'P
xdc
xdc
© xdc xd ¹
(4.2.61a)
295
AC Machine Systems
If D changes slowly, i.e. the transient process in the excitation winding can be
neglected, we have
'E
0
'Te
'P
E0U
cos G 0D
xd
(4.2.60b)
or
'E c 0
'Te
'P
ª Eqc0U
º
§ 1
1 ·
cos G 0 ¨ ¸U 2 cos 2G 0 » D
«
© xdc xd ¹
¬« xdc
¼»
(4.2.61b)
Obviously, when the machine has a small oscillation and G changes very slowly
and after the transient process in the excitation winding is neglected referring to
equation (4.2.60b) or (4.2.61b), we can approximately consider that the output
power or torque will change according to the tangent through point G 0 on the
steady-state P -G curve as shown in Fig. 4.2.7, in other words, according to the
steady-state P -G curve. If the excitation winding reactance is taken into account
referring to equations (4.2.60a) or (4.2.61a), the output power or electromagnetic
torque will change around angle G 0 with an elliptical curve shown in Fig. 4.2.7,
whose direction is indicated by arrows in that figure. That is because Eqc or E
can’t at once change to its steady-state value due to the effect of the excitation
winding reactance when angle G changing by virtue of an oscillation, so there is
an elliptical relation between 'E c or 'E and D as mentioned before.
Figure 4.2.7 Steady-state P -G curve
Figure 4.2.8
The ellipse moved from Fig. 4.2.7
In order to illustrate the change of the output power or torque during an
oscillation, the ellipse in Fig. 4.2.7 is moved to Fig. 4.2.8. From that figure we
can see, when G G 0 D m , i.e. operating at point a in Fig. 4.2.8, the machine has
a surplus torque ae so an acceleration to make angle G increase. As mentioned
296
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
before, owing to the effect of the excitation winding reactance and D D m at
this moment, 'E c become positive and the corresponding Eqc is more
than Eqc0 that is the steady-state value during G G 0 D m because of Eqc
Eqc 0 'E c, so the instantaneous output power is more than the corresponding
steady-state value, which makes point a above the steady-state P -G curve. On
the basis of the same cause, the instantaneous output power at other points is also
different from the value on the steady-state P -G curve and forms an elliptical
curve.
When point b in Fig. 4.2.8 is reached, the torques are balanced so there is no
acceleration, but this time the speed has been more than synchronous speed so the
surplus kinetic energy makes angle G increase continually. After passing through
point b, the speed will diminish due to the negative surplus torque. Reaching point
c, the machine speed is equal to synchronous speed again, but by virtue of
negative surplus torque cf the machine has a deceleration so angle G will decrease
gradually. When angle G reaches point d, the torques are balanced again, but now
the speed has been less than synchronous speed so angle G will go on to diminish.
During a small steady-state oscillation, P and G will change so repeatedly, in
which the change of the surplus kinetic energy is indicated in Table 4.2.1.
Table 4.2.1 Change of suplus kinetic energy
G
The area proportional to the
surplus kinetic energy
Notes
From a to b
aeba
Preservation
From b to c
bcfdb
Consumption
From c to d
cfdc
Consumption
From d to a
daebd
Preservation
From that table we can see, during a small steady-state oscillation, the preservation
of the surplus kinetic energy is equal to the consumption of that, so the average
energy from the prime-mover is constant. However the preserved energy aeba
from a to b is less than the consumed energy bcfdb from b to c, so the difference
between them must be provided with the pulsating torque of the prime-mover at
the positive half-cycle, whose value is proportional to area bcdb. On the contrary,
the surplus energy from c to d and from d to a has to be absorbed by the pulsating
torque of the prime-mover at the negative half-cycle, whose value is proportional
to area dabd. If the pulsating torque causing a small oscillation disappears,
because of preserved energy aeba being less than consumed energy bcfdb or
consumed energy cfdc being less than preserved energy daebd, angle G can’t
change from a to c or from c back to a, so the oscillation will gradually decay to
297
AC Machine Systems
zero. From the above condition we can see, when angle Gchanges clockwise
along the elliptical locus, the machine is stable. During r 0 and G 90e
, the
machine can suffice that condition so it is stable.
Now we further discuss the condition when r z 0. Under that condition,
during a small oscillation and referring to equation (4.2.46) we have
id
rU sin G xd ( E U cos G )
r 2 xd2
(4.2.62)
xd U sin G r ( E U cos G )
r 2 xd2
(4.2.63)
iq
P Te
E\ q
EU ( xd sin G r cos G )
rE 2
2
2
r xd
r 2 xd2
Eiq
xd
(4.2.64)
Supposing
T
arctan
r
xd
and considering the stator resistance loss to be small, we have
id
xd E
U
cos(G T )
2
2
r xd
r xd2
(4.2.62a)
2
rE 2
UE
sin(G T )
r 2 xd2
r 2 xd2
UE
|
sin(G T )
r 2 xd2
P Te
(4.2.64a)
If the machine oscillates around angle G 0 and voltage U is maintained constant,
corresponding to the steady-state operating point we can get
P0
id 0
298
Te 0
UE0
r 2 xd2
sin(G 0 T )
E0U
r 2 xd2
xd E0
U
cos(G 0 T )
2
2
r xd
r 2 xd2
xd E0
U
cos G 0c
2
2
2
r xd
r xd2
sin G 0c
(4.2.65)
(4.2.66)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
in which
G 0c G 0 T
Substituting E E0 'E and G G 0 D into equations (4.2.62a) and (4.2.64a)
and neglecting the second-power small increment, we can obtain
xd 'E
U
sin G 0cD
2
r 2 xd2
r xd2
'id
'P
UE0
'Te
r 2 xd2
cos G 0cD U 'E
r 2 xd2
(4.2.67)
sin G 0c
(4.2.68)
Substituting equation (4.2.53) into formula (4.2.67), we can get
'id
xd
U
['E c ( xd xdc )'id ] sin G 0cD
2
r 2 xd2
r xd2
or
'id
1
[ xd 'E c r 2 xd2 U sin G 0cD ]
r xd xdc
2
(4.2.69)
Substituting the above formula into equation (4.2.52), we have
(Td 0 p 1)'E c
xdc xd
[ xd 'E c r 2 xd2 U sin G 0cD ]
r 2 xd xdc
or
§
r 2 xd2
T
p
¨ d0
r 2 xd xdc
©
Supposing D
·
¸ 'E c
¹
( xd xdc ) r 2 xd2
U sin G 0cD
r 2 xd xdc
D m sin O t , we can obtain
'E c
( xd xdc ) r 2 xd2 U sin G 0c
2
§ r 2 xd2 ·
2 2
(r 2 xd xdc ) ¨ 2
¸ O Td 0
c
r
x
x
d d ¹
©
O (r 2 xd xdc )
E c arctg
Td 0
r 2 xd2
D m sin(O t E c)
(4.2.70)
This formula is similar to equation (4.2.57), so when the stator resistance r is
considered there is also an elliptical relation between 'E c and D . Similarly,
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AC Machine Systems
because of 0 Td 0 p'E c 'E referring to equation (4.2.48), there is a phase-angle
difference of 90ebetween 'E and 'E c, whose amplitudes differ from each other
by OTd 0 times, so there is also an elliptical relation between 'E and D .
Equation (4.2.68) compared with equation (4.2.60a) we can see, the relation
between output power or torque increment and power angle increment D
considering the stator resistance r is similar to that without the stator resistance,
but their coefficients differ a little from each other.
During G 0c ! 0, i.e. G 0 ! T , the coefficients of both equations have the same
sign, so the output power or torque considering the stator resistance still changes
around angle G 0 along the elliptical curve shown in Fig. 4.2.7 or 4.2.8, in which
the direction of change is the same as that without the stator resistance. That is to
say, during G 0 ! T the small oscillation, even if the effect of the stator resistance
is considered, will tend to the stable condition after a perturbation disappears.
During G 0c 0, i.e. G 0 T , the sign of sin G 0c in equation (4.2.68) is opposite
to that of sin G 0 in equation (4.2.60a). Therefore, although the output power or
torque considering the stator resistance still changes around angle G 0 along the
elliptical curve, the direction of change is counter-clockwise as shown in
Fig. 4.2.9. From Fig. 4.2.9 we can see, the area adbea from a to d, which
represents the preserved kinetic energy, is more than the area dfcd from d to c
which represents the consumed energy, so the oscillation, if any force is not
applied to, can not stop at point c, and its amplitude increases more and more to
step out ultimately, i.e. during G 0 T the machine may lose synchronism. The
larger the r, the biger the T . Therefore, the lighter the load, i.e. the smaller the
G , more easily does the oscillation occur by virtue of the stator resistance.
Figure 4.2.9 Steady-state P - G curve and corresponding ellipse during G 0 T
r
, the machine can operate stably only when r xd tanG .
xd
The approximate stable condition considering the effect of the stator resistance is
obtained according to a non-salient pole machine, but it can also be used
approximately for a salient-pole machine. However, in a salient-pole machine
Because T
300
arctan
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
there is xd ! xq , so r xq tanG is often used for a stable condition because it is
safer to use xq .
Figure 4.2.10 Critical resistance rk - G curve
The critical resistance rk changing with angle Gis shown in Fig. 4.2.10 which
is determined according to TD 0 or rk xq tanG . It is not difficult to understand
the approximate grade of using r xq tanG .
4.3
Static Stability of Synchronous Machine Systems and
Influence of Excitation Regulation on Static Stability
During a synchronous machine in parallel with network, the machine has to operate
at synchronous speed strictly according to network frequency. Therefore, the
synchronous machine and its connected network dispatch loads or encounter some
accidents to bring about a question whether they can still keep synchronous
operation or not, namely the stability problem of synchronous machine system.
The stability problem is a significant and interesting problem, because the
main generator in a network is out of step to make stop electricity of large area
and cause confusion. In order to answer the above question, the effect of speed
change has to be taken into account for synchronous machines, i.e. not only
electromagnetic transient process but also mechanical transients have to be
considered simultaneously. For convenience, the stability problem of synchronous
machines can be divided into two types, namely static stability and dynamic
stability (or transient stability), in which dynamic (or transient) stability will be
stated in Section 4.4 and static stability is discussed in the following.
Static stability is to discuss a problem whether the synchronous machine can
keep synchronous operation under a certain steady-state operating mode. If the
301
AC Machine Systems
machine can’t restore its original state after undergoing a small perturbation
under that steady-state operating mode, it can be considered to be unstable even
though the machine can still operate. On the contrary, if the machine can restore
its original steady-state operating mode after sustaining a small perturbation, it
can be considered to be stable under that steady-state operating mode. Under the
definition of static stability mentioned above, nonlinear basic equations of
synchronous machines can be linearized to simplify the problem whether the
machine can be stable or not.
It should be pointed out, that the definition of static stability for synchronous
machines is not unified in some references and bibliography. As mentioned
before, static stability is to discuss a problem whether the synchronous machine
can restore its original steady-state operating mode or not when undergoing a
small perturbation and then abolishing it, but the rate and fashion of small
perturbation are not prescribed. If the small perturbation is caused by small
increase of loads and the increase rate of machine loads is very slow, the
excitation current has not any change and only the rotor position angle, namely
power angle, reaches a new balance position at a very low speed, which is the
first possible case. The second possible case is that the increase rate of small
loads is relatively fast to bring about change of excitation current due to change
of rotor position angle even if the synchronous machine has no excitation
regulation device. Of course, with excitation regulation device exists more
change of excitation current. In the second possible case, the static stability limit
is higher than the first possible case, namely constant excitation current. Thus, in
accordance with the above cases, the static stability can be divided into steadystate static stability and dynamic-state static stability in some references. There
are excitation regulation devices in modern generators, so analysis of static stability
has to consider influence of excitation current change on static stability, referring
to [1,2,4,5]. Therefore, by static stability refer to dynamic-state static stability in
this book even if there is no excitation regulation device.
4.3.1
Basic Relations of Synchronous Machine Systems for
Analyzing Static Stability
When a synchronous machine operating in steady-state, according to Park’s
formulas the relations with various quantities are
ud 0 \ q 0 rid 0
½
°
uq 0 \ d 0 riq 0
°
\ d 0 E xd id 0
¾
°
\ q 0 xq iq 0
°
Tm 0 Te 0 \ d 0 iq 0 \ q 0 id 0 ¿
302
(4.3.1)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
If the machine undergoing a small perturbation such as a mechanical torque
increment 'Tm , then among various quantities exist the following relations:
p (\ d 0 '\ d ) (\ q 0 '\ q )(1 p'G ) r (id 0 'id ) ½
p'\ d \ q 0 \ q 0 p'G '\ q '\ q p'G rid 0 r 'id °°
uq 0 'uq p'\ q \ d 0 \ d 0 p'G '\ d '\ d p'G riq 0 r 'iq °
°
\ d 0 '\ d G ( p )(u fd 0 'u fd ) xd ( p )(id 0 'id )
°
E xd id 0 G ( p)'u fd xd ( p)'id
¾
°
\ q 0 '\ q xq iq 0 xq ( p)'iq
°
Te 0 'Te \ d 0 iq 0 \ q 0 id 0 '\ d iq 0 '\ q id 0
°
°
\ d 0 'iq \ q 0 'id '\ d 'iq '\ q 'id
°
2
Hp 'G Tm 0 'Tm Te 0 'Te
¿
(4.3.2)
u d 0 'u d
where the subscript “0” indicates original steady-state values.
Subtracting various equations in (4.3.1) from the corresponding equations in
(4.3.2) and neglecting multiplication of two small increments, there are
'ud p'\ d \ q 0 p'G '\ q r 'id
½
°
'uq p'\ q \ d 0 p'G '\ d r 'iq
°
'\ d G ( p)'u fd xd ( p)'id
°
¾
'\ q xq ( p)'iq
°
'Te '\ d iq 0 \ d 0 'iq '\ q id 0 \ q 0 'id °
°
Hp 2 'G 'Te 'Tm
¿
(4.3.3)
The above equations can be reduced further to
'ud pG ( p )'u fd
½
°
Z d ( p )'id xq ( p )'iq \ q 0 p'G
°
'uq G ( p)'u fd
°
¾
xd ( p )'id Z q ( p )'iq \ d 0 p'G
°
'Tm iq 0 G ( p )'u fd
°
[\ q 0 iq 0 xd ( p )]'id [\ d 0 id 0 xq ( p )]'iq Hp 2 'G °¿
(4.3.4)
where
Z d ( p)
Z q ( p)
pxd ( p ) r
pxq ( p) r
That is to say, through the procedures above, the nonlinear differential
equation set (4.3.2) has been changed to linear differential equation set (4.3.3) or
(4.3.4), by use of which we can study the static stability of synchronous machine
under a certain steady-state operating mode.
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AC Machine Systems
4.3.2
Approximate Analysis of Static Stability for Synchronous
Machine Systems
In order to illustrate the analytical steps of static stability, we take a synchronous
machine connected to infinite source as example referring to Fig. 4.3.1.
Figure 4.3.1 A synchronous machine connected to infinite source
Under that condition exist
U sin G 0 ½
U cos G 0 ¾¿
(4.3.5)
U cos G 0 'G uq 0 'G ½
U sin G 0 'G ud 0 'G ¾¿
(4.3.6)
ud 0
uq 0
'ud
'uq
Supposing the excitation winding voltage to be constant, namely 'u fd
equation (4.3.4) can be simplified as
½
0 Z d ( p )'id xq ( p)'iq (uq 0 \ q 0 p )'G
°
0 xd ( p )'id Z q ( p )'iq (ud 0 \ d 0 p )'G
¾
'Tm [\ q 0 iq 0 xq ( p )]'id [\ d 0 id 0 xq ( p )]'iq Hp 2 'G °¿
0, so
(4.3.7)
Deriving the equation set, we have assumed the factor of causing various quantity
change for synchronous machines to be 'Tm . Therefore, if a synchronous
machine can return to original steady-state operating mode when abolishing 'Tm ,
then the synchronous machine is stable; or it is unstable if incapable of returning
to original steady-state operating mode. The above problem is not difficult to
respond when knowing the root properties of characteristic equation corresponding
to the differential equation set above. Supposing that there is no positive real-root
or complex-root with positive real part for characteristic roots, the synchronous
machine is stable; or it is unstable when there is a positive real-root or complexroot with positive real part. Thus, judgement of static stability for synchronous
machines is converted into discriminating whether the real root or the real part of
a complex root is negative for the characteristic equation.
It should be noted that not only 'Tm but parameter and load changes also
belong to the perturbation types, but the basic equation set is similar after
304
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
cancelling the perturbation.
If taking 'Tm 0 and solving equation set (4.3.7), it is clear that the decay
regularity of 'id , 'iq and 'G will change with different roots of the
characteristic equation. Especially the variation of 'G expresses the varying
situation of rotor position for synchronous machines, so the rotor returns to its
original steady-state position directly without any oscillation if its characteristic
roots are all negative real roots; or the rotor becomes out-of-step without any
oscillation when its characteristic roots are all real roots in which exists one
positive real root at least, i.e. the rotor will step out through the crawl
phenomenon. If there is complex root but on complex root with positive real part
or no positive real root, then the rotor will return to its original steady-state position
through a certain oscillation. Supposing that in characteristic equation there are
complex root with negative real part together with positive real root, then the rotor
will leave its balance position to bring about out of step although its oscillation
amplitude decays step by step. If the real part of complex root is positive, then its
oscillation amplitude will increase step by step to lose synchronism finally, i.e.
the rotor will step out through self-oscillation. However, it should be noted that
the above conclusions are correct only under small change condition, so when
the rotor position is relatively far away from its balance position, i.e. 'G is not a
small quantity, the small perturbation, and linearization mentioned before can’t
illustrate the rotor motion condition any longer.
The characteristic equation for (4.3.7) is
D( p)
Z d ( p)
xd ( p )
xq ( p )
Z q ( p)
[\ q 0 iq 0 xd ( p )] [\ d 0 id 0 xq ( p )]
(uq 0 \ q 0 p)
(ud 0 \ d 0 p )
Hp 2
Hp 2 {Z d ( p ) Z q ( p ) xd ( p) xq ( p)}
[\ d 0 id 0 xq ( p)]{Z d ( p )[\ d 0 p ud 0 ] xd ( p )[\ q 0 p uq 0 ]}
[\ q 0 iq 0 xd ( p )]{Z q ( p)[\ q 0 p uq 0 ] xq ( p)[\ d 0 p ud 0 ]}
0
(4.3.8)
which is a high degree equation and even if there is no damping winding
D( p ) 0 is still fifth-order equation thus using numerical method.
In order to realize the main effect on static stability and its physical conception,
it is necessary to reduce the equations further. If neglecting the stator
flux-linkage change p\ d and p\ q , the equation D( p ) 0 will be degraded.
In Park’s voltage equations, \ qZ and \ d Z represent the speed emfs in stator
circuit, p\ d and p\ q express the transformer emfs in stator circuit, and rid and
riq indicate the voltage drops in stator resistance. Therefore, Park’s voltage
equations can be rewritten as
305
AC Machine Systems
u d \ qZ
uq \ d Z
p\ d rid ½
p\ q riq ¾¿
(4.3.9a)
Neglecting p\ d and p\ q , it corresponds to a pure-resistance circuit, so only
the speed emfs and terminal voltages cause the corresponding currents, and there
is no current produced by transformer emfs. Taking 3-phase sudden short circuit
as example, the former causes stator fundamental current and the latter gives rise
to the stator aperiodic current and second harmonic current. Because the torques
produced by the latter currents have a small influence on the rotor motion, during
discussion of some static stability problems the latter can be neglected, i.e. the
static stability can be studied according to the following equation,
ud
uq
\ qZ rid ½
\ d Z riq ¾¿
(4.3.9b)
If neglecting the stator resistance further, then Park’s equations can be reduced to
ud
uq
\d
\q
\ qZ
½
°°
G ( p )u fd xd ( p )id ¾
°
xq ( p)iq
°¿
\ dZ
(4.3.9c)
Now let’s use the above equations to study the static stability problems during
a synchronous machine without damping winding in parallel with infinite source.
Supposing that the excitation voltage is still constant and the machine undergoes
a small perturbation, then there are the following relations:
ud 0 'ud \ q 0 '\ q \ q 0 p'G '\ q p'G
uq 0 'uq \ d 0 '\ d \ d 0 p'G '\ d p'G
\ d 0 '\ d E id 0 xd xd ( p)'id
\ q 0 '\ q iq 0 xq xq 'iq
Hp 'G Tm 0 'Tm \ d 0 iq 0 \ q 0id 0 '\ d iq 0
'\ q id 0 \ d 0 'iq \ q 0 'id '\ d 'iq '\ q 'id
Substituting (4.3.1) into the above relations and neglecting multiplication of
two small increments and \ q 0 p'G together with \ d 0 p'G , we can get
'ud
'uq
'Tm
306
uq 0 'G '\ q xq 'iq
ud 0 'G '\ d xd ( p )'id
[\ q 0 iq 0 xd ( p )]'id (\ d 0 id 0 xq )'iq Hp 2 'G
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
It should be pointed out that \ d 0 p'G and \ q 0 p'G represent the speed emfs
caused when the rotor leaving synchronous speed, which will produce a certain
current and asynchronous torque but are neglected only to bring about a small
influence on static stability.
The characteristic equation for the above equation set is
xd ( p )[ xq Hp 2 (\ d 0 id 0 xq )uq 0 ud 0\ q 0 ] xq ud 0\ q 0
D( p)
0
into which substituting the following relations:
xd xdc Td 0 p
,
1 Td 0 p
xd ( p )
\d0
uq 0
U cos G 0 ,
\ q0
id 0
u d 0
U sin G 0
E U cos G 0
,
xd
iq 0
U sin G 0
xq
the characteristic equation can be rewritten as
a0 p 3 a1 p 2 a2 p a3
0
where
a0
a2
xdc xqTd 0 H , a1 xd xq H
xdc Td 0 (\ d 0 uq 0 \ q 0 ud 0 xq id 0 uq 0 ) xqTd 0 ud 0\ q 0
a3
ª EU
º
§1
§ 1
1·
1 ·
xdc xqTd 0 «
cos G 0 ¨ ¸ U 2 cos 2G 0 ¨ ¸ U 2 sin 2 G 0 »
¨x
¸
«¬ xd
»¼
© xdc xd ¹
© q xd ¹
xd (\ d 0 uq 0 \ q 0 ud 0 xq id 0 uq 0 ) xq ud 0\ q 0
ª EU
º
§1
1 ·
xd xq «
cos G 0 ¨ ¸ U 2 cos 2G 0 »
¨
¸
© xq xd ¹
¬« xd
¼»
According to Herwitz’s Criteria, the prerequisites of stable operation are
(i) a0 ! 0, '1
a1
(ii) ' 2
a3
a1
(iii) ' 3 a3
0
a1
a0
a2
a0
a2
0
! 0;
a1a2 a0 a3 ! 0;
0
a1
a3
a3
a1
a3
a0
!0
a2
(namely a3 ! 0)
In accordance with the coefficients of characteristic equation above, the
prerequisites have the following situations:
(i) The first criterion, namely a0 ! 0 and a1 ! 0, can be satisfied on any operating
condition.
307
AC Machine Systems
(ii) The second criterion
' 2 a1a2 a3 a0
§ 1
1 ·
xdc xd xq2Td 0 H ¨ ¸U 2 sin 2 G 0 ! 0
© xdc xd ¹
can also be met under any operating mode.
(iii) The third criterion ' 3 ! 0, namely a3 ! 0, can’t be satisfied for some
operating modes, so it is the only criterion to discriminate the range of static
stability. According to the criterion a3 ! 0, the main influence on static stability
is those quantities in coefficient a3 , so the following equation
§ 1
EU
1·
cos G m ¨ ¸ U 2 cos 2G m
¨x
¸
xd
© q xd ¹
0
can be used to find out the limiting angle G m of static stability.
During xd z xq and referring to equation (2.5.9), the output power is
P
EU
U2 § 1
1 ·
sin G ¨¨ ¸¸ sin 2G
xd
2 © xq xd ¹
(4.3.10)
Therefore, the criterion for static stability limit is the same as the approximate
dP
0 in Electric Machinery, and here provide a theoretical basis for
criterion
dG
dP
the approximate stable criterion
! 0.
dG
If needing more accurate static stability range, it is necessary to solve equation
(4.3.8), but writing out its general expression is difficult. Analysing practical
problems, the numerical solutions can be found out in the light of actual values
of machine parameters and by use of digital computer. Now take a synchronous
machine without damping winding in parallel with infinite source as example to
illustrate the problem. Let the machine parameters be U 1.0, xd 1.0, xq 0.6,
xdc 0.3, Td 0 1 000, H 1 000, E 1.0 or 1.5, according to which and equation
(4.3.8) together with Routh’s Criterion we can get the curves as shown in
Fig. 4.3.2, where the ordinate represents the stator resistance r, the abscissa
indicates the power angle G, the full line corresponds to the results for E 1.0,
and the dashed line corresponds to the results for E 1.5.
From Fig. 4.3.2 it is clear that the synchronous generator without damping
winding in parallel with infinite source can’t operate stably during larger stator
resistance r and small power angle G such as G 30e. Using it as synchronous
motor, its stable range is narrow and maybe it can’t operate as motor when the
stator resistance exceeding a certain value such as r ! 0.6.
From Fig. 4.3.2 it is also evident that the larger the stator resistance, the larger
308
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
the power angle G corresponding to stable operation. During zero stator resistance,
the machine may be unstable even if G 90ebecause there is xd z xq here.
Considering the rotor to be unsymmetrical on magnetic circuit, namely xd z xq ,
the output power or torque ( P Te during r 0) contains the second harmonic,
so the maximum power or torque can’t occur at G 90ebut G 90e.
Figure 4.3.2 Discriminating operation stability by use of Routh’s Criterion
dP
dG
angle G m corresponding to the maximum power Pm can be got as
Differentiating with respect to formula (4.3.10) and letting
cos G m
2
ª
º 1
xq E
r «
» 4( xd xq )
2
¬« 4( xd xq )U ¼»
xq E
0, the power
(4.3.11)
N.B. there are several solutions for G m , in which the solution less than 90e
should be taken. On the basis of expression (4.3.11), for the demonstrated
example exist G m | 65eduring E 1.0 and G m | 70eduring E 1.5.
Because the synchronous generator is usually connected to outer impedance
and operates at higher excitation emf E, i.e. under over-excitation condition, thus
ª1 § 1
º
1 ·
in formula (4.3.10) the salient-pole power « ¨ ¸U 2 sin 2G » can be
«¬ 2 ¨© xq xd ¸¹
»¼
neglected. That is to say, during calculation of static stability, the salient-pole
synchronous generator has a synchronous reactance equal to xd and can be
calculated according to the corresponding formulas of non-salient pole synchronous
generators.
When the stator resistance r neglected, output power in steday-state is equal to
309
AC Machine Systems
electromagnetic torque. Therefore, when the machine undergoing a small
perturbation to bring about very slow change, the output power P is also
approximate to electromagnetic torque Te in per-unit, so there is
dTe
dG
dP
dG
or
'Te
§ dP ·
¨
¸ 'G
© dG ¹
(4.3.12)
In addition, when the synchronous machine undergoing a small perturbation
and its power angle G changing slowly, the process can also be considered as
dG
steady-state small oscillation but its oscillating angular velocity is O
| 0. In
dt
Section 4.2, the additional torque during steady-state small oscillation is
'Te
Ts 'G TD
dG
| Ts 0 'G
dt
(4.3.13)
That is to say, when the machine undergoing a small perturbation and its power
dP 'Te
| Ts 0 . Hence the stable prerequisite
angle G changing slowly exists
dG 'G
dP
! 0 is also the synchronization torque coefficient Ts 0 ! 0 referring to
dG
Section 4.2, which can also be illustrated in the following way.
On the basis of equation (4.2.25b), during O | 0 exist X d O | xd , Rd O | 0,
X qO | xq and RqO | 0. Substituting the above relations into equations (4.2.24) and
(4.2.25a) and having supposed xd
xq , we can get
TD | 0
Ts
Ts 0
Seeing that P
EU
U2
U2
U2
U2
cos G 0 sin 2 G 0 cos 2 G 0 cos 2 G 0 sin 2 G 0
xd
xd
xq
xd
xq
EU
cos G 0
xd
EU
sin G , there is
xd
dP
dG G
G0
EU
cos G 0
xd
which demonstrates the stable prerequisite
synchronization torque coefficient Ts 0 ! 0.
310
Ts 0
(4.3.14)
dP
! 0 being in accord with the
dG
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
dP
! 0 doesn’t count in the
dG
influence of stator resistance and speed change, in which the damping torque
coefficient is always positive when neglecting stator resistance, so when
dP
! 0 there is no crawl out-of-step phenomenon
satisfying the stable criterion
dG
but maybe exists a self-oscillation out-of-step phenomenon of low frequency
such as 1 2Hz. The damping winding has no effect on the synchronization
torque coefficient Ts 0 , so it also has no influence on the crawl phenomenon, but
there is main effect on depression of self-oscillation out-of-step phenomenon. In
practical terms, the stator resistance is small and there is stronger damping
winding, so the possibility of self-oscillation out-of-step phenomenon is very low
for the synchronous generator without excitation regulation device. Therefore,
dP
! 0 is still an important and useful criterion.
dG
If considering the aperiodic component of stator current, i.e. counting in
p '\ d and p'\ q , then the electromagnetic torque also contains pulsating torque
It should be noted that the stable criterion
whose frequency is about the basic frequency such as 50Hz, so there is also high
frequency oscillation of approximate basic frequency besides crawl and low
frequency self-oscillation. Taking the synchronous machine without damping
winding as example, and counting in stator resistance and aperiodic component
of stator current, then its characteristic equation will be of fifth-order. Therefore,
besides one real root may exist two pairs of complex roots corresponding to high
frequency oscillation and low frequency self-oscillation. However, aperiodic
component of stator current corresponding to high frequency oscillation decays
very quickly and the rotor inertia is also large, so the amplitude of high frequency
oscillation is small and decays very quickly, which can’t be perceived in practical
operation.
In general, generators can’t operate on limiting power condition and the safe
margin is necessary. So far as turbogenerators are concerned, their rated power
angle G N is about 30e, and as for hydraulic generators the power angle G N is
some 25e.
4.3.3
Influence of Excitation Regulation on Static Stability
The analysis of static stability mentioned above is based on the supposition that
the excitation voltage keeps constant, i.e. the influence of excitation regulation
on static stability is neglected. In practical terms, modern generators all have
excitation regulation devices, so when the generator undergoing a perturbation
the excitation regulation is put into action to increase the stable limit of
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AC Machine Systems
synchronous machine systems and to decrease the requirements of machine
parameters.
As stated before, assuming the synchronous machine to be of non-salient pole
and neglecting its stator resistance, its output power is
P
EU
sin G
xG
(4.3.15)
so the output power can be increased if the excitation regulation device is put
into action to regulate the excitation emf E in time. Now take a synchronous
generator in parallel with infinite source as example to illustrate the problem.
Suppose the power system to be shown in Fig. 4.3.3, its equivalent circuit is
indicated in Fig. 4.3.4 and the phasor diagram in steady-state is shown by full
lines in Fig. 4.3.5.
Figure 4.3.3 The power system comprising a synchronous generator in parallel
with infinite source.
Figure 4.3.4 Equivalent circuit corresponding to Fig. 4.3.3
For the studied power system exists
P
where
xc
EU
sin G
xG xc
xT 1 (4.3.16)
1
xL xT 2
2
Evidently, if excitation current or excitation voltage is invariable, i.e. E is
constant, then after the power system changing its operating condition the
corresponding steady-state phasor diagram is indicated by dashed lines as shown
in Fig. 4.3.5. From Fig. 4.3.5 it is clear that during increase of power angle G the
phase angle of generator terminal voltage U G will change and the voltage
magnitude decrease. On the contrary, when an excitation regulation device is
installed at point U G of the generator, the change of terminal voltage due to
variation of generator operation will cause the change of excitation current or
312
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
excitation voltage to restore the terminal voltage to its original value or to
approximate to original terminal voltage. When U G being constant, the excitation
emf E of generator will increase with power angle G as shown in Fig. 4.3.6.
Figure 4.3.5 Steady-state phasor diagram during a synchronous machine in parallel
with infinite source
Figure 4.3.6 Steady-state phasor diagram during U G
constant
If plotting P-G curves for various values of E, we can get a family of
sinusoidal curves as shown in Fig. 4.3.7, which are termed inner power
characteristics or inner P-G curves. When an excitation regulation device is
installed at point U G of the generator, due to the change of excitation emf E its
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AC Machine Systems
P-G curve will shift from a sinusoidal curve to another sinusoidal one, and a
certain point on original curve also shifts to another corresponding curve. Thus
the points mentioned above will constitute a curve with an arrow as shown in
Fig. 4.3.7, which is called outer power characteristic or outer P-G curve. Under
this condition it can be seen that its output power P still increases with power
angle G in a certain range even if G ! 90ebecause the increase of excitation emf
E is more than the decrease of sin G when G is slightly more than 90ereferring to
formula (4.3.16).
Figure 4.3.7 Inner power characteristics and outer power characteristic
Of course, the above fact can also be explained according to the following
conception. Seeing that U G and U are all constant under this condition, output
power P can also be calculated in the light of U G and U , namely
P
U GU
sin G c
xc
(4.3.17)
where G c is the angle included between phasors U G and U .
U GU
Evidently, P Pm
during G c 90e
, so the maximum value of outer
xc
, not at G 90e. There is G ! G c due to
power characteristic exists at G c 90e
generator reactance xG referring to Fig. 4.3.5, so the maximum value of outer
power characteristic appears during G ! 90e.
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4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
As mentioned before, excitation regulation has an important influence on static
stability, which has been researched into for many years to get distinguished
results. However, the influence of excitation regulation is in many ways. For
example, it can regulate terminal voltage automatically and raise stable limit, but
it can also cause self-oscillation or self-oscillation out-of-step phenomenon to
affect the stable limit of power transmission and also to influence power system
operating quality. Studying the problems above, it is necessary to write down the
equations corresponding to generator, excitation regulation device and power
system, to analyse them further and then to take corresponding technique measures.
4.4
Dynamic Stability and Analysis Methods of
Synchronous Machine Systems
The dynamic stability problem of synchronous machine systems is to study whether
the synchronous machine can operate synchronously after undergoing a certain
or large disturbance. There are many disturbances, but they can be summarized
into the following several types: (i) Sudden change of load (ii) Cut-off of transmission line or transformer (iii) Disconnection of generators (iv) Short circuit fault,
in which the short circuit fault is the most dangerous so it has to be paid attention
to firstly. For a power system with neutral point earthed may occur the following
short circuit faults: (i) Single phase earthed (ii) Two phases short circuit (iii) Two
phases short circuit earthed (iv) 3-phases short circuit. Statistical data about
accidents indicate that the number of times for single phase earthed owns about
70% of the total short circuit faults for high voltage networks and the number of
times for 3-phases short circuit only owns 6% 7% of the total short circuit
faults. So far as the influence of short circuit faults on stability is concerned,
3-phases short circuit will cause the most difficulty because during 3-phases
short circuit the voltage will dip quickly to weaken the connection between
power stations or to lose the system stability finally. For single phase earthed or
two phases short circuit the voltage dip is relatively small, especially the former,
so it is of benefit to synchronous generators. For important transmission systems to
maintain stability is necessary not only for unsymmetrical short circuit but also
for 3-phases short circuit, because short circuit fault is the most dangerous from
the viewpoint of dynamic stability. Therefore, for networks with neutral point
earthed the stability calculation should be made respectively according to 3-phases
short circuit, two phases earthed and single phase earthed. As for 2-phases short
circuit people examine it only a few.
When a synchronous generator undergoing the above disturbances, its output
power will change suddenly but the prime mover’s torque can’t change abruptly
due to regulator inertia. Therefore, the output power of generator can’t balance
the prime mover power, thus making the generator speed variable and effecting
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AC Machine Systems
the electromechanical transient process of whole power system. Under some
adverse condition, the generator will lose synchronism. In order to discriminate
whether the synchronous generator will step out during the above situation, it is
necessary to study the whole electromechanical transient process. As stated
before, it is a complicated nonlinear problem. In practical terms, according to the
problems studied and considering the corresponding computation methods, get
the reduced mathematical patterns to analyse them referring to [1,3,4,7].
4.4.1
The Equations Corresponding to Analysis of Dynamic
Stability for Synchronous Machine Systems
Before analysing dynamic stability for synchronous machine systems, firstly
discuss the corresponding equations. On the rotor of synchronous machine exist
d-axis damping winding and q-axis one besides the excitation winding, whose
basic equations have been discussed in detail in Section 2.4. and can be used to
study dynamic stability of synchronous machine systems. Of course, for convenience
of analysis it is necessary to reform and reduce those basic equations.
Substituting formulas (2.4.3) and (2.4.4) into formulas (2.4.1) and (2.4.2) and
rearranging them, the voltage equation of synchronous machines can be written as
ª ud º
«u »
« q »
«U fd »
« 0 »
«
»
¬ 0 ¼
Z xq
pxad
pxad
Z xaq
ª (r pxd )
º
« Z x
»
(r pxq )
Z xad
Z xad
pxaq
d
«
»
0
( R fd pX ffd )
0
pX f 1d
« pxad
»˜
« px
»
0
( R1d pX 11d )
0
pX f 1d
ad
«
»
«¬
0
0
0
( R1q pX 11q ) »¼
pxaq
ª id º
«i »
« q »
« I fd »
«I »
« 1d »
¬« I1q ¼»
(4.4.1)
where Z is rotor speed in per-unit.
In addition, there is
J
t
t
³ Z dt J ³ (Z 'Z )dt J
Z t ³ 'Z dt J
³ Z dt J c Z t J c
0
0
s
0
t
s
Jc
316
0
0
t
0
s
0
s
0
0
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
Letting G be the angle included between d, q axes and d c , qc axes, then exist
G
J
t
J J c ³ 'Z dt (J 0 J 0c )
0
Zst G J 0 G 0
t
³ 'Z dt G
0
0
in which
G0
J 0 J 0c is the angle included between d, q axes and d c , qc axes at t 0;
'Z
Z Z s is the difference between rotor speed and synchronous speed.
Hence, the rotor speed in per-unit can also be written as
Z
dJ
dt
1 pG
(4.4.2)
Tm Te
(4.4.3)
In addition, the rotor motion equation is
Hp 2G
The flux-linkage equations are
\d
xd id xad I fd xad I1d
(4.4.4)
\ fd
xad id X ffd I fd X f 1d I1d
(4.4.5)
\ 1d
xad id X f 1d I fd X 11d I1d
(4.4.6)
\q
xq iq xaq I1q
(4.4.7)
\ 1q
xaq iq X 11q I1q
(4.4.8)
The electromagnetic torque equation is
Te
\ d iq \ q id
[id iq I fd I1d
Taking X f 1d
ª 0
« x
« d
I1q ] « 0
« 0
«
¬ 0
xq
0
0
0
0
0
xad
0
0
0
0
xad
0
0
0
xaq º ª id º
« »
0 » « iq »
»
0 » « I fd »
0 » « I1d »
»« »
0 ¼ ¬« I1q ¼»
(4.4.9)
xad , i.e. neglecting the mutual leakage reactance between
excitation winding and damping winding, the formulas of stator flux-linkages
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AC Machine Systems
\ d and \ q can be reduced and the corresponding relations, referring to Section
2.4, can be written as
\d
G ( p)U fd xd ( p )id
(4.4.10)
\q
xq ( p )iq
(4.4.11)
in which
1 p(Td 4 Td 5 ) p 2Td 4Td 6
1 p (Td 1 Td 2 ) p 2Td 1Td 3
xd ( p)
xd
xq ( p )
xq
G( p)
1 pTd 7
xad
2
1 p(Td 1 Td 2 ) p Td 1Td 3 R fd
Td 1
Td 2
Td 3
Td 4
Td 5
Td 6
Td 7
Tq1
Tq 2
1 pTq1
(4.4.12)
(4.4.13)
1 pTq 2
(4.4.14)
X ffd
Td 0
R fd
X 11d
T1d 0
R1d
1 §
¨ X 11d R1d ¨©
1 §
¨ X ffd R fd ©
2
xad
X ffd
·
¸¸ Tdcc0
¹
2
·
xad
¸ Tdc
xd ¹
x2 ·
ad ¸
xd ¹
§
¨ X 11d
©
2
xad
(2 xad xd X ffd ) ½°
1 ­°
X
® 11d
¾ Tdcc
2
R1d ¯°
xd X ffd xad
¿°
X 11d xad
R1d
2
·
xaq
1 §
¨¨ X 11q ¸ Tqcc
xq ¸¹
R1q ©
X 11q
Tq 0
R1q
1
R1d
Since Td 2 and Td 3 are much less than Td 1 , and Td 5 and Td 6 are much less than
Td 4 , thus the operational reactances xd ( p ), xq ( p ) and operational conductance
318
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
G ( p ) mentioned above can also approximate to
xd ( p)
xd
(1 pTdc )(1 pTdcc )
(1 pTd 0 )(1 pTdcc0 )
(4.4.15)
G( p)
1 pTd 7
xad
(1 pTd 0 )(1 pTdcc0 ) R fd
(4.4.16)
xq ( p )
xq
1 pTqcc
1 pTq 0
(4.4.17)
The error due to simplification of time constants above doesn’t exceed 5%.
In the light of formulas (4.4.15), (4.4.16) and (4.4.17), the stator flux-linkage
equations (4.4.10) and (4.4.11) can be rewritten as
\d
\q
A
B
id id xdccid
1 pTd 0
1 pTdcc0
Gc
G cc
U fd U fd
1 pTd 0
1 pTdcc0
xq xqcc
1 pTq 0
iq xqcciq
A | xd xdc
½
°
B | xdc xdcc
°
Tdc
°
xdc xd
°
Td 0
°
T cc
°
xdcc xdc d
Tdcc0
°
°
Tqcc
¾
cc
xq xq
°
Tq 0
°
Td 0 Td 7 xad °
˜
Gc
Td 0 Tdcc0 R fd °
°
Tdcc0 Td 7 xad °
G cc
˜
°
Td 0 Tdcc0 R fd °
¿
xd xdc
xc xdcc
id d
id xdccid
1 pTd 0
1 pTdcc0
Gc
G cc
U fd U fd
1 pTd 0
1 pTdcc0
(4.4.18)
(4.4.19)
(4.4.20)
\d | (4.4.21)
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AC Machine Systems
According to formulas (4.4.10) (4.4.14) and opening damping winding, i.e.
letting R1d f and R1q f, then there is
Td 2
Td 3
Td 5
Td 6
Td 7
Tq1
0
Therefore, if neglecting damping winding, then exist
1 pTdc
1 pTd 0
xad
1
G( p)
1 pTd 0 R fd
1 pTdc
xad
1
U fd xd
id
\d
1 pTd 0 R fd
1 pTd 0
xd ( p )
xd
Since
Gc
\d
there is
Td 0 Td 7 xad
x
| ad
Td 0 Tdcc0 R fd R fd
1 pTdc
Gc
U fd xd id
1 pTd 0
1 pTd 0
Referring to Section 2.7, the d-axis transient emf of synchronous machine
without damping winding is Eqc \ d xdc id , so another expression for Eqc is
Eqc \ d xdc id
x xdc
Gc
U fd d
id
1 pTd 0
1 pTd 0
(4.4.22)
In addition, the d-axis and q-axis transient emfs of synchronous machine with
damping winding are respectively
Eqcc \ d xdccid
(4.4.23)
Edcc \ q xqcciq
(4.4.24)
Substituting expressions (4.4.23) and (4.4.24) into formulas (4.4.19), (4.4.21)
and (4.4.22), and rearranging them, we can get the following relations:
320
pEqc
1
{G cU fd ( xd xdc )id Eqc }
Td 0
(4.4.25)
pEqcc
1
{G ccU fd ( xdc xdcc )id Eqcc Eqc } pEqc
Tdcc0
(4.4.26)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
1
{( xq xqcc)iq Edcc}
Tq 0
pEdcc
(4.4.27)
Substituting expressions (4.4.23) and (4.4.24) into formula (4.4.9), another
expression for electromagnetic torque is
Te
Edccid Eqcciq ( xqcc xdcc )id iq
(4.4.28)
As you know, the stator voltage equations of synchronous machine are
ud
p\ d Z\ q rid
(4.4.29)
uq
p\ q Z\ d riq
(4.4.30)
in which p\ d and p\ q represent d-axis and q-axis transformer emfs respectively,
Z\ d and Z\ q indicate d-axis and q-axis speed emfs respectively, and the
former will influence the aperiodic component of stator current mainly and the
latter the basic-frequency component of stator current. Therefore, for
convenience of analysis it is necessary to use some approximation. For example,
during analysis of some transients the rotor flux-linkage can be considered as
constant at initial stage of disturbance, and the aperiodic component of stator
current is also neglected, i.e. p\ d and p\ q are omitted referring to Sections 2.7
and 2.6. In other cases such as a generator with excitation regulation device, its
rotor flux-linkage after disturbance will change a lot to bring about main effect.
In order to count the influence in, sometimes use the following approximation:
p\ d
p ( Eqcc xdccid ) | pEqcc
(4.4.31)
p\ q
p ( Edcc xqcciq ) | pEdcc
(4.4.32)
Z\ d
Z ( Eqcc xdccid ) | Z Eqcc xdccid
(4.4.33)
Z\ q
Z ( Edcc xqcciq ) | Z Edcc xqcciq
(4.4.34)
Thus, equations (4.4.29) and (4.4.30) can be reduced to
ud
pEqcc Z Edcc xqcciq rid
(4.4.35)
uq
pEdcc Z Eqcc xdccid riq
(4.4.36)
Even if using the approximate formulas above, sometimes still make a further
approximation, such as:
(i) In formulas (4.4.35) and (4.4.36) take Z 1;
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AC Machine Systems
(ii) In formulas (4.4.35) and (4.4.36) neglect pEqcc and pEdcc;
(iii) In formula (4.4.35) take xqcc xdcc.
In practical terms, for more convenience sometimes neglect the damping
winding effect, by which the results can usually meet the practical requirement,
but in some cases such as relatively large change of rotor speed omit the damping
winding effect to bring about large error. In order to remedy the drawback of this
approximation, a damping torque is added in electromagnetic torque to consider
the damping winding effect referring to Section 4.2, i.e. the rotor motion equation
(4.4.3) can be replaced by
Hp 2G
Tm Te TD pG
in which the damping torque coefficient TD is taken approximately as a constant.
Generally TD 1 3 and for very strong damping winding TD 25.
Neglecting damping winding effect or without damping winding, the
corresponding relations are
xd xdc
xad
1
id xdc id U fd
1 pTd 0
1 pTd 0 R fd
\d
\q
xq iq
(4.4.38)
Eqc
xad
x xdc
1
U fd d
id
1 pTd 0 R fd
1 pTd 0
(4.4.39)
(4.4.37)
In addition exists
\d
Eqc xdc id
(4.4.40)
According to the relations above and using the approximation like formulas
(4.4.31) (4.4.34), the corresponding calculation results can be got. If using the
following approximation,
p\ d
p ( Eqc xdc id ) | 0
(4.4.41)
p\ q
p ( xq iq ) | 0
(4.4.42)
Z\ d
Z ( Eqc xdc id ) | Eqc xdc id
(4.4.43)
Z\ q
Z ( xq iq ) | xq iq
(4.4.44)
then equations (4.4.29) and (4.4.30) can be reduced to
ud
322
xq iq rid
(4.4.45)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
xdc id Eqc riq
uq
(4.4.46)
Rearranging equation (4.4.39) exists
pEqc
1
Td 0
°­ xad
°½
U fd ( xd xdc )id Eqc ¾
®
¯° R fd
¿°
(4.4.47)
and the corresponding electromagnetic torque formula is
Te
{Eqc ( xq xdc )id }iq
(4.4.48)
In analysis above, the influence of ferromagnetic saturation is not considered
to cause a certain error. There are several methods to count in ferromagnetic
saturation, in which it is the most convenient to use invariable saturatedparameters. However, this method still causes obvious error when the operating
condition of synchronous machine changes in a wide range. In the following we
shall introduce a method to adjust the saturated-parameters according to
operating condition change.
This method is based on the supposition that the leakage flux way is
unsaturated and the leakage flux also has no effect on magnetic saturation, and
the no-load characteristic of synchronous machine is simulated to consider the
saturation effect. It is well-known that the mutual flux-linkages \ md and \ mq on
d-axis and q-axis during linearity are
\ md
xad (id I fd I1d )
\ mq
xaq (iq I1q )
xad idt
(4.4.49)
xaq iqt
(4.4.50)
id I fd I1d ½
¾
iq I1q
¿
(4.4.51)
in which
idt
iqt
represent the total currents respectively on d-axis and q-axis and are proportional
to total mmfs respectively on d-axis and q-axis.
Neglecting saturation effect, the flux-linkages \ md and \ mq are proportional to
the currents idt and iqt i.e. xad and xaq are invariable. Considering saturation effect,
xad and xaq will change with saturation level. For convenience, assume the ratio
of xaq to xad to be invariable, namely xaq / xad
\ md
\ mq
xad idt
xaq iqt
K dq
constant, so exist
xm idt
K dq xm iqt
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AC Machine Systems
The saturation level depends upon the total flux produced by the total mmf
which is composed of d-axis and q-axis mmfs in perpendicular in space, so the
total mmf will be proportional to an equivalent current it and can be written as
idt2 K dq2 iqt2
it
(4.4.52)
Correspondingly, the saturated values of xad and xaq are
xad
xaq
xm f (it )
½
K dq xm K dq f (it ) ¾¿
(4.4.53)
In the following, we discuss how to use no-load characteristic to get the value
of xm f (it ).
At no-load exist idt
0, and the corresponding flux-linkage \ md
I fd and iqt
and no-load emf uq are respectively
\ md
uq
xm I fd
f ( I fd ) I fd
\ md
(4.4.54)
(4.4.55)
If the no-load characteristic is indicated by
uq
g ( I fd )
(4.4.56)
then there is
\ md
g ( I fd )
f ( I fd ) I fd
namely
f ( I fd )
g ( I fd )
I fd
(4.4.57)
When the synchronous machine is loaded, the equivalent excitation current is
it , so exists
xm
f (it )
g (it )
it
(4.4.58)
After finding out xm , we can use the saturated reactances xm and K dq xm to
supersede xad and xaq in the above formulas during neglect of saturation effect,
thus obtaining accurate calculation results.
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4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
4.4.2
Mathematical Patterns Corresponding to Analysis of Dynamic
Stability for Synchronous Machine Systems
During analysis of dynamic stability for synchronous machine systems, different
mathematical patterns can be adopted in terms of concrete research topics and
computation methods. Several mathematical patterns in the following are
suggested for reference only.
(1) Pattern 1
This is the simplest mathematical pattern of synchronous machines. Based on
the pattern to study dynamic stability, use the following approximation:
(i) Neglect transformer emfs in the stator voltage equations.
(ii) Except the rotor motion equation, assume rotor speed to be constant,
namely Z 1, in other equations.
(iii) Neglect the damping winding effect.
(iv) Neglect the ferromagnetic saturation effect.
(v) d-axis transient emf Eqc is constant.
Obviously, the above equations (4.4.45), (4.4.46) and (4.4.48) belong to the
relations of this mathematical pattern.
For analysis of some dynamic transients, sometimes neglect the salient-pole
effect further, i.e. take xq xdc approximately. Taking this approximation exists
Eqc E c, so the mathematical pattern can be reduced further to the pattern with
E c constant, in which E c is the transient emf behind the transient reactance
referring to Section 2.7.
(2) Pattern 2
In comparison with Pattern 1, this pattern will cancel the supposition that
d-axis transient emf Eqc is kept constant, so besides the above relations (4.4.45),
(4.4.46) and (4.4.48) the Pattern 2 also includes relation (4.4.47) to count the
change of Eqc in.
(3) Pattern 3
This is the simplest mathematical pattern of synchronous machines with
damping windings. Based on the pattern to study dynamic stability, suppose the
following conditions:
(i) Take Z 1 in stator voltage equations (4.4.35) and (4.4.36).
(ii) Let pEqcc pEdcc 0 in stator voltage equations (4.4.35) and (4.4.36).
xad
(iii) Take G c
in equation (4.4.25).
R fd
(iv) Take G cc 0 in equation (4.4.26).
The relations of pattern 3 are (4.4.25) (4.4.28), (4.4.35) and (4.4.36).
(4) Pattern 4
In comparison with Pattern 3, for this pattern G c and G cc have to be calculated
325
AC Machine Systems
according to formula (4.4.20), so the basic relations of Pattern 4 are (4.4.20),
(4.4.25) (4.4.28), (4.4.35) and (4.4.36).
(5) Pattern 5
This pattern can consider not only damping winding effect but also
ferromagnetic saturation influence, so the basic relations of Pattern 5 are (4.4.1),
(4.4.9), (4.4.51) (4.4.53) and (4.4.58). In comparison with other patterns, this
pattern has considered more factors, so the calculation results are more accurate
but the computation is complicated.
4.4.3
Calculation Example of Dynamic Stability for Synchronous
Machine Systems
By use of mathematical Pattern 1 analyze the dynamic stability of a synchronous
machine system. Therefore, firstly discuss the relation between electromagnetic
torque Te and power angle G. In the light of formula (4.4.48), the electromagnetic
torque expression of Pattern 1 is
{Eqc ( xq xdc )id }iq
Te
During steady-state symmetrical operation, referring to equation set (2.7.21)
there are
id
iq
Eqc U cos G
xdc
U sin G
xq
(4.4.59)
(4.4.60)
N.B. the angle G can be realized as the angle between terminal voltage and
no-load emf referring to equation (2.5.2), and is also considered as the angle
included between d, q axes and d c , qc axes referring to formula (4.4.2).
Substituting formulas (4.4.59) and (4.4.60) into (4.4.48), the torque formula of
steady-state symmetrical operation can be written as
Te
EqcU
1§ 1
1 ·
sin G ¨ ¸ U 2 sin 2G
2 ¨© xq xdc ¸¹
xdc
(4.4.61)
If using an equivalent machine without transient salient-pole effect to
represent actual machine, then formula (4.4.61) referring to Section 2.7 can be
reformed as
Te
326
E cU
sin G eq
xdc
(4.4.62)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
where G eq is the angle included between the transient emf E c behind transient
reactance and terminal voltage U for the equivalent machine. For convenience,
G eq can be written as G here, so exists
Te
E cU
sin G
xdc
(4.4.63)
Now let a synchronous generator be connected with infinite source to illustrate
the calculation of dynamic stability as shown in Fig. 4.4.1, in which the generator
passes through double transmission lines to connect infinite source.
Figure 4.4.1 A generator passes through double transmission lines to connect
infinite source
As you know, if neglecting stator resistance the output power of generator
during normal operation will equal the electro magnetic torque. The output power
before short circuit can be written as
P
E cU
sin G
xc
(4.4.64)
where
xc
xdc xT 1 1
xL xT 2
2
is the effective reactance between generator and infinite source; xT 1 and xT 2 are
short circuit reactances of transformer 1 and transformer 2 respectively; xL is the
reactance of each transmission line.
When the power system in Fig. 4.4.1 is in normal situation, its transient
P - G curve is indicated by curve I as shown in Fig. 4.4.2. Now supposing that the
unsymmetrical short circuit occurs at point K on the left of transmission line 2
referring to Fig. 4.4.1, then during initial stage of short circuit the equivalent circuit
327
AC Machine Systems
of power system is shown in Fig. 4.4.3, in which xK represents the equivalent
reactance caused by short circuit fault at point K. By use of Symmetrical
Components Method, the values of xK can be indicated in Table 4.4.1, in which
x2 and x0 are corresponding negative sequence and zero sequence reactances
viewed from the fault point K of the power system. It is noted that here only
consider the influence of positive component on rotor motion. In some cases,
especially during unsymmetrical fault near generator terminal, the negative
sequence current may produce obvious braking torque, which can be calculated
according to negative sequence network to get stator negative sequence current
and negative sequence resistance if necessary.
Figure 4.4.2
P - G curves before short circuit and after fault at point K.
Figure 4.4.3 Equivalent circuit during unsymmetrical short circuit at point
K of the power system in Fig. 4.4.1
Table 4.4.1 Values of xK for different faults
Types of unsymmetrical short circuit
Two phases short circuit
Values of x K
x2
Single phase earthed
x2 x0
Two phases short circuit earthed
x2 x0
x2 x0
In Fig. 4.4.3, U is the constant voltage of infinite source, E c is the transient
emf of generator and xdc is its transient reactance. Reaching the steady-state
finally, E c and xdc will change to E and xd respectively. Although E c can’t
change suddenly during short circuit, it has to decay with time. Therefore, the
328
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
actual output power is in accord with the transient power characteristic during
E c constant only at initial stage of short circuit. With time delayed, there is a
deviation of actual output power from the transient power characteristic during
E c constant. Figure 4.4.4(a) indicates the unstable condition of synchronous
machine system. Figure 4.4.4(b) shows the stable situation, in which the actual
power is approximate to the corresponding value on the transient power
characteristic during E c constant for the first oscillation period, then the
oscillation decays gradually, and finally the generator operates at point C of
power-angle curve during excitation emf E constant to reach a new stable
operation condition. Figure 4.4.4(c) belongs to another condition, in which the
actual output power is approximate to the corresponding value on the transient
power characteristic during E c constant for the first oscillation period, it seems
to be stable, but the synchronous generator can’t reach a new stable operating
condition because the P - G curve during E constant can’t intersect the line for
prime mover power P0 . Under this condition, the stability will be lost after
several oscillation periods. In practical terms, if the generator after fault doesn’t
lose synchronism during the first oscillation, then there is a few possibility to
step out after the first oscillation; at the same time, the subtransient component of
power system decays very quickly to neglect it, and the transient component
decays slowly, so during discrimination of dynamic stability, i.e. within the first
oscillation period, the transient component can be considered as constant, which
also gives the reason why mathematical Pattern 1 can be used to calculate
dynamic stability approximately. In the following we shall use mathematical
Pattern 1 to analyse dynamic stability of the power system shown in Fig. 4.4.1.
Figure 4.4.4
P - G curves for transients and in steady-state
Figure 4.4.3 indicates a T-form electric circuit, so it can be converted into a Sform electric circuit as shown in Fig. 4.4.5. The reactances xE and xu in
Fig. 4.4.5 are connected in parallel with E c and U respectively, so they can be
neglected for calculation of generator output power. Under this condition, the
output active power of generator is
P
E cU
sin G
xcc
(4.4.65)
329
AC Machine Systems
where xcc is the equivalent reactance of power system corresponding to the Sform electric circuit.
xcc
§x
·
( xdc xT 1 ) ¨ L xT 2 ¸
© 2
¹
xc xK
is substituted into formula (4.4.65), and the corresponding transient P - G curve
is shown by curve Ċin Fig. 4.4.2.
Figure 4.4.5
S-form electric circuit obtained from a T-form circuit in Fig. 4.4.3
From here we can see, due to sudden change of network parameters during
short circuit, the output power of generator changes from point a on P - G curve
ĉ in Fig. 4.4.2 to point b on P - G curve Ċ . In the meantime, because of
regulator lagging for prime mover, the input mechanical power of generator still
maintains constant and is equal to P0 . Due to the surplus power or surplus torque
effect, the generator is accelerated to make the power angle G grow. With the
power angle G growing, the generator output power also increases thus reducing
the surplus power or surplus torque. At point c in Fig. 4.4.2 the surplus power
will be equal to zero, but the rotor speed relative to d c , qc axes is maximum. Due
to rotor inertia the power angle G will continue growing to exceed point c in
Fig. 4.4.2. At this time, the output power of generator has been more than the
mechanical power provided by prime mover, so the surplus torque becomes
negative to make the rotor decelerate. For the deceleration period, the surplus
power of generator is supplied by the storage kinetic-energy during acceleration.
When reaching point d in Fig. 4.4.2, the kinetic energy stored in rotor has been
used up and the rotor will stop relative motion to d c , qc axes, i.e. the rotor speed
is restored to synchronous speed. However, the surplus torque at point d is of
braking effect, so the rotor motion relative to d c , qc axes begins in the opposite
direction. During opposite motion and due to rotor inertia, the rotor passes
through point c to reach the minimum value of power angle G and then stops
relative motion. Later on, the acceleration starts again. In so doing, the rotor
operates at point c through several oscillations with decaying amplitude, and the
point c is the stable point on new power-angle curve ĊIf the rotor exceeds the
angle G kp corresponding to P0 on new P - G curve Ċduring the first oscillation
referring to Fig. 4.4.2, then the surplus torque will become positive again to
make rotor accelerated. Later on, with the power angle G growing, the
330
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
acceleration torque increases constantly to bring about the stepout of synchronous
generator.
P - G curves in Fig. 4.4.2 provide a short-cut method to determine the
maximum position angle G max and to discriminate the stable possibility of power
system. It is noted that during oscillation Z | Z s so there is 'Te 'P. For the
acceleration period the rotor moves from G 0 to G CT , i.e. from point a to point c
in Fig. 4.4.2, and the work done by surplus torque 'Te is
Ayck
G CT
³G
'Te dG
0
G CT
³G
'PdG
f abc
(4.4.66)
0
where f abc represents the area of shade-line part abc. Therefore, for the
acceleration period, the kinetic energy stored in rotor is just equal to the area
f abc that is named the acceleration area.
When the rotor position exceeding point c on new P ~ G curve Ċthe surplus
torque will become negative to make rotor decelerated. During deceleration the
rotor moves from angle G CT to angle G max , and the kinetic energy stored in rotor
is negative, whose value can be expressed as
ATOP
G max
³G
CT
'Te dG
G max
³G
'PdG
f cde
(4.4.67)
CT
The area f cde is termed the deceleration area. During deceleration period, the
kinetic energy stored in rotor is returned to power system in order to compensate
the deficiency of prime mover power. When the whole kinetic energy stored in
rotor during acceleration period has been used up at the deceleration stage, i.e.
the work ATOP done during deceleration is balanced against the work Ayck done
during acceleration, then the rotor relative speed will be equal to zero to stop
relative motion and the rotor operates at the maximum position angle G max .
Therefore, according to the following formula,
or
Ayck ATOP
f abc f cde
0 ½°
¾
0 °¿
(4.4.68)
we can get the value of angle G max . As mentioned above, the acceleration area
must be equal to the deceleration area at the maximum position angle G max , so the
topic is converted into how to determine point d (see Fig.4.4.2) to suffice the
above condition, which can be solved by use of graphic method.
In Fig. 4.4.2 the maximum possible deceleration area is equal to f cdcc obviously.
If the area f cdcc being less than the acceleration area f abc , then the generator will
lose synchronism. The ratio of possible deceleration area to acceleration area
f cdcc / f abc is the storage mark of power system dynamic-stability, which is termed
the storage coefficient of dynamic stability.
331
AC Machine Systems
When possible deceleration area is less than acceleration area, if cutting off the
fault line quickly it is still possible to get stable operation because the transmitted
power by use of single transmission line is more than the power transmitted by
double lines during short circuit referring to Fig. 4.4.1. After cutting off the fault
line, the transient power equation is
P
E cU
sin G
xccc
(4.4.69)
xccc xdc xT 1 xL xT 2
where
Curve ċin Fig. 4.4.6 is the transient power characteristic corresponding to
equation (4.4.69). In Fig. 4.4.6, curves ĉand Ċ indicate the transient power
characteristics for normal situation and short circuit fault respectively.
Figure 4.4.6 Transient power characteristics for normal situation, short circuit fault
and cut-off of fault line
As stated before, during short circuit the generator output power will decrease
so the rotor begins to accelerate. If cutting off the fault line at point d on curve Ċ
then the operation point after cut-off will change to point e on curve ċ(see
Fig. 4.4.6), thus increasing the generator output power to make the maximum
possible deceleration area f cdefccc be more than the sustained short circuit
situation; furthermore, the more quickly cut off the fault line, i.e. the less the
angle G OTK is, the more the deceleration area grows. Therefore, the cut-off of
fault line can enhance dynamic stability of power system, which is simple and
effective so is widely used in practice to become an important measure for raise
of dynamic stability.
In Figs. 4.4.2 and 4.4.6, the method to use area to discriminate dynamic
stability is called the area rule. According to Fig. 4.4.6 we can use graphic
method to find out the limit value of angle G OTK , at which cut off the fault line to
guarantee stable operation, i.e. in the light of the rule that the acceleration area
equals the maximum possible deceleration area we can get the limit value of
cut-off angle G OTK . Similarly, by use of analytical method also obtain the limit
332
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
value of angle G OTK , i.e. let sum of the acceleration area and the maximum
possible deceleration area be zero, then exists
f abcdeccc
G OTK
³G
( P0 PmĊ sin G )dG
0
³
G KP
G OTK
(P0 Pmċ sin G )dG
(4.4.70)
0
where PmĊ and Pmċ are maximum values on the transient power characteristics
during short circuit and after cut-off of fault line respectively. After integral on
the basis of formula (4.4.70) there is
P0 (G OTK G 0 ) PmĊ (cos G OTK cos G 0 )
P0 (G KP G OTK ) Pmċ (cos G KP cos G OTK )
0
(4.4.71)
whose solution is
cos G OTK
P0 (G kp G 0 ) Pmċ cos G kp PmĊ cos G 0
Pmċ PmĊ
where the angles are expressed in radians.
Substituting the values of G 0 and G KP S arc sin( P0 / Pmċ ) into the above
formula, then the values of cos G OTK and G OTK can be found out. However, so far
as the practice is concerned, the above calculation is not enough, because it is not
necessary to know the angle G OTK when demanding interruption speed from the
breaker and relay, but need the time at which the generator rotor reaches the
angle G OTK , i.e. the permitted time limit of cutting off short circuit, shortly
cut-off time limit. The cut-off time limit can’t be obtained according to the area
rule above, so we have to adopt another method for analysis of dynamic stability,
namely dividing intervals calculation.
The dividing intervals calculation is also a routine method for analysis of
dynamic stability, whose merits are obviously to watch the short circuit process
at different instants and also to consider the influence of various factors on the
cut-off time limit. For example, the dividing intervals calculation can count in
effects of voltage regulator and armature reaction to determine the permitted time
limit of cutting off short circuit. In fact, the dividing intervals calculation is to solve
the rotor motion equation by use of difference method so it is an approximate
method.
The rotor motion equation in expression (4.4.3) can be reformed as
H d 2G
360 f dt 2
P0 P
(4.4.72)
where H is expressed in seconds and t also in seconds, the unit of G is electric
degree, P0 and P are all expressed in per-unit.
333
AC Machine Systems
In equation (4.4.72), the maximum power Pm for P Pm sin G will change
with different operating conditions such as normal operation, short circuit and
cut-off of fault line referring to formulas (4.4.64), (4.4.65) and (4.4.69). In the
light of equation (4.4.72), solve it for G f (t ) and then get a complete picture
of G changing with time to discriminate whether the generator can keep in
synchronism or not. However, equation (4.4.72) is nonlinear so there is no
analytical solution generally, and it is necessary to solve it by use of numerical
methods. As you know, there are various numerical methods, but here only
introduce difference method.
By use of difference method, the whole oscillation process can be divided into
a series of time interval 't , and then the angle increment 'G will be evaluated
step by step for each time interval. In general, the length of time interval 't is
taken as 0.05 second.
At the beginning of short circuit, the surplus power is assumed to be 'P(0) P0 Pm sin G 0 and will not change within the short time interval 't , so according to
the acceleration motion formula and after the first time interval 't we can get the
rotor angular speed increment 'Z(1) and power angle increment 'G (1) as follows:
d 2G
dt 2
'Z (1)
'G (1)
d 2G
dt 2
˜ 't D (0) 't
(4.4.73)
G G0
˜
G G0
't 2
2
D (0)
't 2
2
(4.4.74)
where D represents the angular acceleration. At the beginning of short circuit, the
rotor relative angular speed to d c , qc axes equals zero so after the first time
interval 't the rotor relative angular speed Z (1) equals the corresponding angular
speed increment 'Z (1) , namely Z (1)
'Z(1) . In addition. On the basis of equation
(4.4.72) exists
D (0)
360 f
'P(0)
H
(4.4.75)
so we can get
Z (1)
'Z(1)
D (0) 't
360 f 't
'P(0)
H
(4.4.76)
and
'G (1)
334
D (0) 't 2
2
360 f 't 2 'P(0)
2
H
k
'P(0)
2
(4.4.77)
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
where
360 f 't 2
H
k
After obtaining the angle increment 'G (1) for the first time interval, then the power
angle at the beginning of the second time interval is
G (1)
G 0 'G (1)
(4.4.78)
After having the value of G (1) , the surplus power 'P(1) for the second time interval
can be determined as
P0 Pm sin G (1)
'P(1)
During calculation of the angle increment 'G (2) after the second time interval,
besides the effect of acceleration D (1) exists the rotor relative angular speed Z (1)
at the beginning of the second time interval, i.e.
'G (2)
Z (1) 't D (1) 't 2
Z (1) 't k
2
'P(1)
2
N.B. it is not very accurate to get the rotor relative angular speed Z (1)
(4.4.79)
D (0) 't
by use of formula (4.4.76), because the surplus power 'P(0) and acceleration
D (0) are not constant for the first time interval. If taking the average acceleration
for the first time interval to calculate Z (1) , i.e. taking
D (0) D (1)
D (0) cp
2
then the result is more accurate. In so doing, the rotor relative angular speed Z (1)
will become
Z (1) D (0) cp 't
D (0) D (1)
2
't
Substituting the above formula into equation (4.4.79) and referring to formula
(4.4.77), there is
'G (2)
D (0) D (1)
2
D (0) 't 2
2
't 2 D (1) 't 2
D (1) 't 2
2
'G (1) k 'P(1)
(4.4.80)
335
AC Machine Systems
After getting 'G (2) we can evaluate G (2) , 'P(2) and 'G (3) as follows,
G (2)
G (1) 'G (2)
'P(2)
P0 Pm sin G (2)
'G (3)
'G (2) k 'P(2)
If cutting off the fault line at the k-th time interval referring to Fig. 4.4.7, then
the surplus power will change suddenly from 'P(cK 1) to 'P(ccK 1) . When evaluating
the power angle increment 'G ( K ) for the first time interval after cut-off, the surplus
power can be taken as the average value of 'P(cK 1) and 'P(ccK 1) , namely
'G ( K )
'G ( K 1) k
'P(cK 1) 'P(ccK 1)
2
(4.4.81)
For the later time intervals, we can still use formula (4.4.80) to estimate the
increment of power angle G. If the power angle G doesn’t begin decreasing or
can’t be predicted to be infinite, the calculation must go on.
The dividing intervals calculation can be used together with the area rule.
Firstly, the area rule is used to find out the power angle limit G OTK of cutting off
the fault line, and then the time tOTK is the cut-off time limit, referring to
Fig. 4.4.8 that is obtained by use of the dividing intervals calculation.
Figure 4.4.7
P -G curves if cutting off the fault line at the k-th time interval
Figure 4.4.8
G - t curve obtained by use of the dividing intervals calculation
and the cut-off time limit
336
4 Oscillation, Stability and Excitation Regulation of Synchronous Machine Systems
In order to illustrate the conception and computation of generator dynamic
stability, the simplest mathematical pattern 1 is adopted in the above example. Of
course, other mathematical patterns can also used if needing higher accuracy.
Finally, as should be pointed out, in this section we only discussed the basic
relations and mathematical patterns of synchronous machine systems, and then
discriminated the dynamic stability of power system during the transient emf
E c constant and according to the first oscillation period of the generator rotor
for convenience. In many cases, the dynamic process of synchronous machine
systems develops quickly, generally in 1 2 seconds after undergoing a disturbance
we can discriminate whether the power system is stable or not. In addition, the
output power of prime-mover can’t change obviously in a short time due to large
inertia of speed regulator for the prime-mover, so the effect of speed regulator
may be neglected temporarily to consider the output power of prime-mover to be
constant. Furthermore, due to large inductance of excitation winding its
flux-linkage also can’t change clearly in a short time, so the influence of
excitation regulation may be neglected to consider the transient emf E c to be
constant. Therefore, the method in this section, even if neglecting the effect of
speed regulator and excitation regulation, still can give satisfactory results.
However, it is different for long-distance transmission or weakly linking power
system, which may lose dynamic stability through many times of oscillations for
a long time such as several seconds or even tens seconds. Evidently, under this
condition it is not suitable to discriminate the dynamic stability according to the
first period of oscillation as mentioned before. As is well-known, the influence of
excitation regulation and prime-mover speed regulation on dynamic stability
depends upon not only the regulation speed but also cut-off time of the fault line.
For modern power systems, automation is high to make the cut-off time very
short, so the influence of those regulation systems on the first period of
oscillation is finite. However, the effect of regulation systems will be raised with
time to bring about an important effect on the generator rotor oscillation.
Therefore, when studying the dynamic stability for this kind of power system, the
effects of speed-regulation and excitation-regulation systems must be paid
attention to. In so doing, it is necessary not only to write down accurate
mathematical patterns of synchronous machine as mentioned before but also to
express the corresponding relations of speed-regulation and excitation-regulation
systems. Obviously, it is complicated to solve the non-linear simultaneous
differential equations. In addition, when the power system undergoing a
disturbance or fault, whether the synchronous machine loses synchronism is no
doubt an important problem, but whether the generator oscillation can be
depressed quickly is also an important problem since it influences the dynamic
quality of power system. Therefore, although the calculation as stated above is
complicated, it is still necessary in practice because so far as the regulation
systems are concerned, increase of dynamic stability limit is not in accord with
the requirement of dynamic quality improvement.
337
AC Machine Systems
References
[1] Arthur R Bergen (1986) Power systems analysis. Prentice-Hall, London
[2] Demello F P, Concordia C (1969) Concepts of synchronous machine stability as affected
by excitation control. J IEEE Trans. PAS-88(4): 316 329
[3] Gao J D (1963) AC machine transients and operating modes analysis (Chapter 4, 5, in
Chinese). Science Press, Beijing
[4] Gao J D, Zhang L Z (1982) Fundamental theory and analysis mothods of transients in
electric machines (Chapter 7, 8, in Chinese). Science Press, Beijing
[5] Gao J D, Wang X H, Li F H (2004) Analysis of ac machines and their systems, 2nd Ed
(Chapter 3, in Chinese). Tsinghua University Press, Beijing
[6] Zhang L Z, Chen L F (1991) Analysis of subsynchronous resonance in a turbogenerator
and its shstem using compensation capacitors. Proc CICEM, Huazhong University of
Science and Technology Press, Wuhan, pp 104 108
[7] Zhang L Z, Yao L (1996) Effect of transfoemer connection banks on analysis of a
synchronous generator and its system during unsymmetrical faults. Proc ICEMA’96,
International Academic Publishers, Harbin, pp 369 373
338
5 Electromagnetic Relations of Induction Machine
Systems and Analyses of Some Operating Modes
Abstract Induction machine systems are widely used in practice, so their
operating behavior, especially transient performance, must be paid attention
to. Firstly, we introduce the basic relations and parameters of induction
machines in d, q, 0 axes, and then analyze the starting process of induction
motors, in which dc, qc, 0 axes with synchronous speed are preferable to
d, q, 0 axes due to variable rotor speed. The relations in dc, qc, 0 axes can be
found out according to basic equations in d, q, 0 axes together with the
conversion formulas in Section 3.1. By using the 4th order Runge-Kutta
Method, we calculate the starting characteristics for different fly-wheel
moments and loads, compare the transient starting characteristics with the
steady-state ones, which are quite different, especially during no-load and small
rotating inertia, and then illustrate why the transient starting characteristics
are quite different from the steady-state ones. Furthermore, some fault is often
temporary, so the production loss caused by failure in power supply can be
decreased with the reswitching of the induction motors as quickly as possible
after supply recovery. Transients of reswitching of the induction motors
include two stages, one is the disconnection process due to source fault, and
the other is the reclosing process after supply recovery, whose main problems
are surge current and large electromagnetic torque; therefore we analyze
them using D , E , 0 axes. As you know, the reclosing surge current is more
serious than the direct starting current, which can be diminished in two ways:
monitor the stator voltage phase-angle at the reclosing instant and use the
extinguishing-flux method to make rotor current decay quick at the
disconnecting instant. Moreover, induction motors in series with capacitance
can keep the supply voltage constant, but sometimes self-excitation may
occur, which is analyzed by using the D-domain partition method to get two
self-excitation regions during lower compensation degree that is a new
discovery, because some scientists pointed out only one self-excitation
region before. In addition, a special generator is introduced as an important
part of ship’s integrated power system, in which the stator has two suits of
windings, one is a 12-phase power winding connected to a rectifier load and
another is a 3-phase control winding connected to a static excitation regulator,
and there is a squirrel-cage solid rotor. There are also the capacitors for
AC Machine Systems
self-excitation and inter-phase reactors in the system for performance
improvement. It can be divided into two sub-systems: one consisting of
12-phase power winding with a rectifier load calculated by the circuit
method, and another consisting of dual-stator-winding and squirrel-cage
solid rotor analyzed by the electromagnetic field finite element method, the
simulative results being approximate to experimental data.
As far as the structures and uses are concerned, induction machines are different
from synchronous machines, but their electromagnetic relations are the same, i.e.
some coils are coupled by fluxes. Therefore, the analysis methods and concepts
of synchronous machines can also be used for induction machines. Furthermore,
the basic equations of induction machines due to symmetrical rotor are simpler than
those of synchronous machines. In this chapter, we mainly discuss the conventional
3-phase induction machines; the other types can be studied in a similar way.
The voltage equations, flux-linkage equations and parameters of stator and
rotor windings are the basis for studying the performance of induction machines.
Learning the equations in steady-state and equivalent circuit, we can find out the
relation between multi-loop method and conventional method. The basic equations
and parameters in d, q, 0 axes are useful in the analysis of induction machine
performance. Those voltage and flux-linkage equations can easily be converted
to other axes, if necessary.
Starting characteristic is an interesting topic. Moreover, the reclosing transient
process of induction motors and self-excitation during motors in series with
capacitors are also important, which are discussed in this chapter, refer to
[1,2,3,5,10 14]. In addition, a dual-stator-winding multi-phase high-speed
induction generator is also studied here, refer to [6 9].
5.1
Basic Relations and Parameters of Induction
Machines
In chapter 1, the electromagnetic relations and parameters of ac machines have
been found out by using a single coil. In this section, conventional 3-phase
induction machines will be discussed according to the above mentioned results
and reference directions in Section 1.10, i.e. the current entering the machines is
positive, the reference directions of voltage drop and current are the same, and
the positive currents will produce positive flux-linkages for stator and rotor.
On the basis of reference directions, the voltage equations of induction machines
can be written as
340
5 Analyses of Some Operating Modes of Induction Machine Systems
ua
ub
uc
p\ a ria ½
°
p\ b rib ¾
p\ c ric °¿
(5.1.1)
where
ua , ub and uc are stator phase voltages, respectively,
ia , ib and ic are stator phase currents, respectively,
\ a , \ b and \ c are stator phase flux-linkages, respectively,
d
is differential operator.
dt
The rotor winding of induction machines may be wound-rotor type or squirrelcage one, whose number of phases may or may not be equal to stator phases.
r is stator phase resistance and p
5.1.1
Wound-Rotor Induction Machines
Phase voltage equations for wound-rotor are
uac
ubc
ucc
p\ ac r ciac ½
°
p\ bc r cibc ¾
p\ cc r cicc °¿
(5.1.2)
The corresponding flux-linkage equations can be written as
\a
\b
\c
\ ac
\ bc
\ cc
Laa ia M ab ib M ac ic M aac iac M abc ibc M acc icc ½
°
M ba ia Lbb ib M bc ic M bac iac M bbc ibc M bccicc °
M ca ia M cb ib Lcc ic M caciac M cbcibc M cccicc °°
¾
M aca ia M acb ib M acc ic Lacaciac M acbcibc M acccicc °
M bca ia M bcb ib M bcc ic M bcaciac Lbcbcibc M bcccicc °
°
M cca ia M ccb ib M ccc ic M ccaciac M ccbcibc Lccccicc °¿
(5.1.3)
in which the signs with prime belong to rotor quantities.
Just as 3-phase synchronous machines, the air-gap flux of 3-phase induction
machines approximates to sinusoidal wave by means of distribution, fractional
pitch and skew-slot windings; so they can be considered as ideal machines. If
necessary, we can count higher harmonics in by using the method stated in
Chapter 1. The hypotheses of ideal machines are as follows:
(i) The influences of ferromagnetic saturation, hysteresis and eddy current can
be neglected, and the skin effect of iron-cores and conductors is also omitted.
(ii) The rotor is cylindrical, air-gap is uniform, and the surfaces of stator and
rotor are all smooth without slot effect.
341
AC Machine Systems
(iii) 3-phase coils of stator are symmetrical, and each phase winding will produce
mmf and flux density distributed sinusoidally in space.
(iv) The rotor winding is of multi-phase and symmetrical.
Under the supposition of ideal induction machine and referring to (1.10.8), the
self-inductance of stator phase winding is
Laa
Lbb
2w2W l
O0 k w2 L0l
P S
Lcc
constant
(5.1.4)
Referring to (1.10.10), the mutual inductance between stator phase windings is
M ab
M bc
2 w2W l
O0 kw2 cos120e M ml
PS
M ca
w2W l
O0 kw2 M ml
P S
(5.1.5)
constant
Similarly, the self-inductance of rotor phase winding is
Lacac
Lbcbc
Lcccc
2( wc) 2 W l
O0 (kwc )2 L0cl
P S
constant
(5.1.6)
The mutual inductance between rotor phase windings is
M acbc
M bccc
M ccac
( wc) 2W l
O0 (kwc ) 2 M mlc
PS
constant
(5.1.7)
In which:
x L0l is the leakage self-inductance of stator phase winding.
x M ml is the leakage mutual inductance between stator phase windings.
x L0cl is the leakage self-inductance of rotor phase winding.
c is the leakage mutual inductance between rotor phase windings.
x M ml
x W and W c are number of turns in series per phase for stator and rotor,
respectively.
x kw and kwc are stator and rotor winding-factors for the fundamental wave.
respectively.
Referring to (1.10.16), the mutual inductance between the stator and rotor
windings is
M aac
M bbc
M ccc
2wwcW l
O0 kw kwc cos J
PS
(5.1.8)
where J is the included electric angle between stator phase a and rotor phase a'
winding axes, J
velocity of rotor.
342
t
³ Zdt J
0
0
, J 0 is the rotor initial-angle, Z is the electric angular
5 Analyses of Some Operating Modes of Induction Machine Systems
M abc
M acb
M bcc
M bcc
M cac
M cca
2 wwcW l
½
O0 kw kwc cos(J 120e
)°
°
PS ¾
2 wwcW l
°
c
e
O
k
k
cos(
J
120
)
0 w w
°¿
PS (5.1.9)
The above parameters are substituted into flux-linkage and voltage equations
of stator phase a, and there are
\a
ua
§ 2 w2W l
·
§ w2W l
·
O0 kw2 L0l ¸ ia ¨ O0 kw2 M ml ¸ (ib ic )
¨
© PS
¹
© PS
¹
2wwcW l
O0 kw kwc [cos J iac cos(J 120e
)ibc cos(J 120e
)icc ]
PS
p\ a ria
§ 2 w2W l
·
§ w2W l
·
2
O
O0 kw2 M ml ¸ p (ib ic )
k
L
pi
¨
¨
0 w
0l ¸
a
© PS
¹
© PS
¹
2wwcW l
O0 kw k wc [cos J piac iac sin J pJ cos(J 120e
) pibc
PS
ibc sin(J 120e
) pJ cos(J 120e
) picc
icc sin(J 120e
) pJ ] ria
(5.1.10)
Similarly, the flux-linkage and voltage equations of rotor phase a' are
\ ac
uac
2wwcW l
O0 kw kwc [cos J ia cos(J 120e
)ib
PS
§ 2 wc2W l
·
)ic ] ¨
O0 kwc2 L0cl ¸ iac
cos(J 120e
© PS
¹
§ wc2W l
·
O0 kwc2 M mlc ¸ (ibc icc )
¨
© PS
¹
p\ ac r ciac
2wwcW l
O0 kw kwc [cos J pia ia sin J pJ
P S
) pib ib sin(J 120e
) pJ
cos(J 120e
cos(J 120e
) pic ic sin(J 120e
) pJ ]
§ 2wc2W l
·
O0 kwc2 L0cl ¸ piac
¨
© PS
¹
2
§ wc W l
·
¨
O0 kwc2 M mlc ¸ p(ibc icc ) r ciac
© PS
¹
(5.1.11)
343
AC Machine Systems
In a similar way, we can get the flux-linkage and voltage equations of stator
phases b, c and rotor phases bc, cc. According to the above relations and parameters
together with electromagnetic torque and motion equations, we can study various
operational modes of wound-rotor induction machines.
Firstly, we analyze the 3-phase symmetrical operation in steady-state.
Suppose that the stator currents in steady-state are
ia
2 I sin(Z t M )
ib
2 I sin(Z t M 120e
)
ic
2 I sin(Z t M 120e
)
and the rotor currents are
iac
2 I c sin( sZ t M c)
ibc
2 I c sin( sZ t M c 120e
)
icc
2 I c sin( sZ t M c 120e
)
where M and M c are the initial phase angles of the corresponding currents, s is
rotor slip, and J (1 s )Z t J 0 .
The current expressions above are substituted into (5.1.10), and there exists
ua
§ 3w2W l
·
O0 kw2 L0l M ml ¸ Z u 2 I cos(Z t M )
¨
© PS
¹
3wwcW l
O0 kw kwc Z u 2 I c cos(Z t M c J 0 )
PS
r u 2 I sin(Z t M )
( x1 xm ) u 2 I cos(Z t M ) x12 u 2 I c
u cos(Z t M c J 0 ) r u 2 I sin(Z t M )
(5.1.12)
in which
x1 Z ( L0l M ml ) is the stator leakage reactance,
3w2W l
O0 kw2 is the magnetization reactance of induction machines,
P S
3wwcW l
O0 kw kwc is the mutual reactance between stator and rotor.
x12 Z
PS The same way is true of voltage equations for stator phases b and c.
The voltage equation of rotor phase a' is
xm
344
Z
5 Analyses of Some Operating Modes of Induction Machine Systems
uac
3wwcW l
O0 kw kwc sZ u 2 I cos( sZ t M J 0 )
P S
§ 3wc2W l
·
¨
O0 kwc2 L0cl M mlc ¸ sZ u 2 I c cos( sZ t M c)
S
P
©
¹
r c u 2 I c sin( sZ t M c)
sx12 u 2 I cos( sZ t M J 0 ) s( x2 xm 2 )
u 2 I c cos( sZ t M c) r c u 2 I c sin( sZ t M c)
(5.1.13)
Z ( L0cl M ml c ) is rotor leakage reactance,
Z 3wc2W l
O0 (kwc ) 2 is the magnetization reactance referred to rotor.
2
where x2
xm 2
PS
The same way is true of voltage equations for rotor phases b' and c'.
wkw
k wkw 1
is the
, in which k
Two sides of (5.1.13) are multiplied by
wckwc
s wck wc s
ratio of stator phase-winding effective turns number to rotor effective turns
number. For a wound-rotor induction machine, its rotor winding is usually
short-circuited, i.e., uac ubc ucc 0, hence
0
3w2W l
O0 kw2Z u 2 I cos( sZ t M J 0 )
PS
ª 3wwcW l
º
wk
«
O0 kw kwc w ( L0cl M mlc ) »
wckwc
¬ PS
¼
u Z u 2 I c cos( sZ t M c) wkw r c
u 2 I c sin( sZ t M c)
wckwc s
xm u 2 I cos( sZ t M J 0 )
( x12 kx2 ) u 2 I c cos( sZ t M c)
rc
k u 2 I c sin( sZ t M c)
s
(5.1.14)
N. B. The above formula is the voltage equation for angular frequency sZ .
If J 0 0, then (5.1.12) and (5.1.14) can be changed respectively into
ua
( x1 xm ) u 2 I cos(Z t M ) x12 u 2 I c cos(Z t M c)
r u 2 I sin(Z t M )
0
xm u 2 I cos( sZ t M ) ( x12 kx2 ) u 2 I c cos( sZ t M c)
rc
k u 2 I c sin( sZ t M c)
s
(5.1.15)
(5.1.16)
345
AC Machine Systems
Let
I 2c
wckwc
Ic
wkw
Ic
k
then (5.1.15) and (5.1.16) are changed respectively into
ua
( x1 xm ) u 2 I cos(Z t M ) xm u 2 I 2c cos(Z t M c)
r u 2 I sin(Z t M )
0
where x2c
r2c
xm u 2 I cos( sZ t M ) ( xm x2c ) u 2 I 2c cos( sZ t M c)
rc
2 u 2 I 2c sin( sZ t M c)
s
(5.1.17)
(5.1.18)
k 2 x2 is the rotor leakage reactance referred to stator,
k 2 r c is the rotor resistance referred to stator.
Formulas (5.1.17) and (5.1.18) are voltage equations expressed as instantaneous
values, which can be written as complex operator equations as follows:
2U sin(Z t M u ), then (5.1.17) and (5.1.18) can be converted into
If ua
Im(Ue jMu e jZ t )
Im[ j( x1 xm ) Ie jM e jZ t ] Im[ j( xm I 2c e jM c e jZ t )
Im(rIe jM e jZ t )
0
(5.1.19)
Im( jxm Ie jM e jsZ t ) Im[ j( xm x2c ) I 2ce jM c e jsZ t ]
§ rc
·
Im ¨ 2 I 2ce jM c e jsZ t ¸
©s
¹
(5.1.20)
Taking away the sign of imaginary part, there are
Ue jMu e jZ t
j( x1 xm ) Ie jM e jZ t jxm I 2c e jM c e jZ t
rIe jM e jZ t
0
jxm Ie jM e jsZ t j( xm x2c ) I 2c e jM c e jsZ t
rc
2 I 2ce jM c e jsZ t
s
(5.1.21)
(5.1.22)
Formulas (5.1.21) and (5.1.22) are not simultaneous equations due to different
frequencies. Multiplying (5.1.22) by e j(1 s )Z t , there exists
0
346
jxm Ie jM e jZ t j( xm x2c ) I 2ce jM c e jZ t
rc
2 I 2ce jM c e jZ t
s
(5.1.23)
5 Analyses of Some Operating Modes of Induction Machine Systems
Now (5.1.21) and (5.1.23) are simultaneous equations having the same frequency,
which can be reduced to
U
0
j( x1 xm ) I jxm I2c rI ½
°
r2c ¾
jxm I j( xm x2c ) I 2c I 2c °
s ¿
(5.1.24)
or
U
0
(r jx1 ) I jxm ( I I2c )
½
°
§ r2c
· ¾
jxm ( I I 2c ) ¨ jx2c ¸ I 2c °
©s
¹ ¿
(5.1.25)
According to (5.1.25), there exists the equivalent circuit of induction machines
as shown in Fig. 5.1.1, which is familiar to us in Electric Machinery.
Figure 5.1.1 Equivalent circuit of induction machines
After getting relations and equivalent circuit of induction machines in steadystate, the corresponding phasor diagram can be drawn and various characteristics
found out.
Just as Electric Machinery, the referring calculation has two steps, i.e. refer
different numbers of turns to the same, and then refer different frequencies to the
same too, for stator and rotor. After referring calculation, the electromagnetic
relations for stator and rotor can be drawn in the common circuit. Obviously, the
air-gap flux and its speed produced by actual rotor current are equal to those
caused by the rotor current referred to stator.
5.1.2
Squirrel-cage Rotor Induction Machines
For squirrel-cage rotor, suppose that the number of bars is Z 2 , and the number of
bars per pole-pair is Z 2 / P, i.e. the number of rotor phases is K Z 2 / P, in
which each phase is composed of P branches in parallel. Hence, voltage and
flux-linkage equations for stator and rotor can be written as
347
AC Machine Systems
ua
ub
uc
un
\a
\b
\c
\n
½
°
°
p\ b rib
°
p\ c ric
°
°
r
r
°
p\ n R in c (in 1 in 1 )
P
P
°
Laa ia M ab ib M ac ic M a1i1 " M aK iK ¾
°
M ba ia Lbb ib M bc ic M b1i1 " M bK iK °
°
M ca ia M cb ib Lcc ic M c1i1 " M cK iK °
M na ia M nb ib M nc ic M n1i1 " M nK iK °
°
n 1, 2," , K °¿
p\ a ria
(5.1.26)
where rR is the resistance of one loop for the squirrel-cage rotor,
rC is the bar resistance,
\ n is rotor flux-linkage of the n-th phase and the subscript n indicates the
order number of rotor-phases.
The stator inductances for squirrel-cage type are the same as those for woundrotor type, refer to (5.1.4) and (5.1.5).
Referring to (1.10.16), the mutual inductance between stator phase a and rotor
n-th phase is
M an
kE S
2 wW l
1
O0 ¦ 2 kwk sin r cos k[J (n 1)M 0 ]
2
PS
k k
(5.1.27)
where k 1, 3, "
J is the electric included angle between winding axes of rotor first phase and
stator phase a,
360e
.
M 0 is the electric angle between two adjacent rotor-phases, M 0
K
During the 3-phase symmetrical operation in steady-state, the air-gap flux is
approximate to sinusoidal, so (5.1.27) is changed to
M an
ES
2 wW l
O0 kw sin r cos[J (n 1)M 0 ]
PS
(5.1.28)
According to (1.5.14), the self-inductance of rotor phase winding is
L11
L22
" LKK
2W l
P 2 S
O0
¦j
j
2
sin 2
jEr S
L0cl
2
(5.1.29)
On the basis of (1.5.17), the mutual inductance between two rotor-phase
windings is
348
5 Analyses of Some Operating Modes of Induction Machine Systems
2W l
P 2 S
M mn
O0
¦j
j
2
sin 2
jEr S
(m n)360e
c
cos j
M mnl
K
(5.1.30)
N. B. the rotor phase winding is composed of P branches in parallel, so the
inductance expressions (5.1.29) and (5.1.30) have a sign P more in their
c is the rotor
denominators, in which L0lc is the rotor leakage self-inductance, M mnl
leakage mutual inductance between the m-th phase and n-th phase, (m n)360e/ K
is the electric included angle between the winding axes of rotor m-th phase and
n-th phase and j 1,3,"
In rotor inductance expressions, there are relative strong higher harmonics for
squirrel-cage type. However, during the 3-phase symmetrical operation in steadystate, the air-gap flux caused by rotor winding is still approximate to sinusoidal,
which can be considered as ideal electric machine, too. Therefore, the selfinductance and mutual inductance of rotor phase-winding can be expressed as
L11
M mn
ES
2W l
½
O sin 2 r L0cl constant °
2 0
PS
°
E
S
e
2W l
(
m
n
)360
°
2
r
c
O0 sin
M mnl
cos
¾
P 2 S
K
°
ES
°
2W l
c
O sin 2 r cos(m n)M 0 M mnl
constant °
2 0
PS
¿
L22
" LKK
(5.1.31)
The above inductances are substituted into the corresponding flux-linkage
equations, and the flux-linkage equations are substituted into voltage equations,
and then there are
\a
ua
§ 2 w2W l
·
§ w2W l
·
2
O
k
L
i
O0 kw2 M ml ¸ (ib ic )
¨
¨
0 w
0l ¸ a
© PS
¹
© PS
¹
E
S
2wW l
O0 kw sin r [cos J i1 cos(J M 0 )i2 "
PS
cos(J ( K 1)M 0 )iK ]
p\ a ria
§ 2 w2W l
·
§ w2W l
·
2
O
k
L
pi
O0 kw2 M ml ¸ p(ib ic )
¨
¸
¨
0 w
0l
a
© PS
¹
© PS
¹
Er S
2wW l
O0 kw sin
p[cos J i1 cos(J M 0 )i2 "
PS
cos(J ( K 1)M 0 )iK ] ria
(5.1.32)
349
AC Machine Systems
Similarly, the flux-linkage and voltage equations of rotor n-th phase winding are
\n
2 w1W l
ES
O0 kw sin r [cos(J (n 1)M 0 )ia
P S
cos(J (n 1)M 0 120e
)ib
cos(J (n 1)M 0 120e
)ic ]
ES
2W l
O sin 2 r [cos(n 1)M 0 i1 "
2 0
P S
c iK )
cos(n K )M 0 iK ] ( M 1cnl i1 " M Knl
un
r
rR
in c (in 1 in 1 )
P
P
2 w1W l
Er S
O0 kw sin
p[cos(J (n 1)M 0 )ia
P S
cos(J (n 1)M 0 120e
)ib cos(J (n 1)M 0
0
p\ n ES
2W l
O sin 2 r p[cos(n 1)M 0i1 "
2 0
P S
c iK )
cos(n K )M 0 iK ] p ( M 1cnl i1 " M Knl
120e
)ic ] r
rR
in c (in 1 in 1 )
P
P
(5.1.33)
The same way is true of the flux-linkage and voltage equations for stator phases
b, c and rotor other phases. According to those basic relations and parameters,
together with electromagnetic torque and rotor motion equations, we can study
various operation modes for squirrel-cage rotor induction machines.
Now we discuss the 3-phase symmetrical operation in steady-state.
If
ia
2 I sin(Z t M )
ib
2 I sin(Z t M 120e
)
ic
2 I sin(Z t M 120e
)
If
i1
2 I c sin( sZ t M c)
then
i2
2 I c sin( sZ t M c M 0 )
iK
2 I c sin[ sZ t M c ( K 1)M 0 ]
then
and
Supposing J 0
J
(1 s )Z t J 0
0, there is
J
(1 s )Z t
Substituting the above formulas into (5.1.32), there exists
350
5 Analyses of Some Operating Modes of Induction Machine Systems
ua
§ 3w2W l
·
O0 kw2 L0l M ml ¸ u 2 Ip sin(Z t M )
¨
P
S
©
¹
ES
wW l
O0 kw sin r u 2 I cKp sin(Z t M c)
PS r u 2 I sin(Z t M )
§ 3w2W l
·
O0 kw2 L0l M ml ¸ Z u 2 I cos(Z t M )
¨
P
S
©
¹
ES
wW lK
O0 kw sin r Z u 2 I c cos(Z t M c)
PS
r u 2 I sin(Z t M )
( x1 xm ) u 2 I cos(Z t M ) x12 u 2 I c cos(Z t M c)
r u 2 I sin(Z t M )
(5.1.34)
in which
ES
wW lk
O0 kw sin r is the mutual reactance between stator and rotor for
PS squirrel-cage type.
The same way is true of voltage equations for phases b and c.
The voltage equation of rotor n-th phase is changed to
x12
Z
0
ES
3wW l
O0 kw sin r sZ u 2 I cos[ sZ t M (n 1)M 0 ]
P S
ES
W lK
2 O0 sin 2 r sZ u 2 I c cos[ sZ t M c (n 1)M 0 ]
P S
§ Lr 2 Lc
·
cos M 0 ¸ sZ u 2 I c cos[ sZ t M c (n 1)M 0 ]
¨ P
©P
¹
§ r 2r
·
¨ r c cos M 0 ¸ u 2 I c sin[ sZ t M c (n 1)M 0 ]
©P P
¹
(5.1.35)
Lc
M (cn 1) nl M (cn 1) nl is the leakage mutual inductance between two
P
adjacent rotor-phases, which is zero for nonadjacent rotor-phases of squirrelcage type,
Lr
L0cl ,
P
Lc is the leakage inductance of one rotor-bar,
where Lr is the leakage inductance of one rotor-loop.
351
AC Machine Systems
Two sides of (5.1.35) are multiplied by
k
s
wk w
s sin
Er S
, and there is
2
0
3w W l
O0 kw2Z u 2 I cos[ sZ t M (n 1)M 0 ]
PS
ES
wW lK
O0 kw sin r Z u 2 I c cos[ sZ t M c (n 1)M 0 ]
PS
wkw
( L 2 Lc cos M 0 )Z u 2 I c cos[ sZ t M c (n 1)M 0 ]
ES r
P sin r
wkw rr 2rc cos M 0
u 2 I c sin[ sZ t M c (n 1)M 0 ]
Er S
s
P sin
xm u 2 I cos[ sZ t M (n 1)M 0 ]
( x12 kx2 ) u 2 I c cos[ sZ t M c (n 1)M 0 ]
rc
k u 2 I c sin[ sZ t M c (n 1)M 0 ]
s
(5.1.36)
If we write the voltage equation for rotor first phase, namely n 1, then (5.1.36)
is changed to
0
xm u 2 I cos( sZ t M ) ( x12 kx2 ) u 2 I c cos( sZ t M c)
rc
k u 2 I c sin( sZ t M c)
s
(5.1.37)
where
x2 Z ( Lr 2 Lc cos M 0 ) / P is rotor leakage reactance,
r c (rr 2rc cos M 0 ) / P is rotor resistance.
ES
K sin r
I c KI c
Let I 2c
3wk w
3k
and then (5.1.34) and (5.1.37) are changed respectively to
ua
( x1 xm ) u 2 I cos(Z t M ) xm u 2 I 2c cos(Z t M c)
r u 2 I sin(Z t M )
0
352
xm u 2 I cos( sZ t M ) ( xm x2c ) u 2 I 2c cos( sZ t M c)
rc
2 u 2 I 2c sin( sZ t M c)
s
(5.1.38)
(5.1.39)
5 Analyses of Some Operating Modes of Induction Machine Systems
in which
3 2
x2c
k x2 is the rotor leakage reactance referred to stator,
K
3 2
r2c
k r c is the rotor resistance referred to stator.
K
Equations (5.1.38) and (5.1.39) are identical with (5.1.17) and (5.1.18). Hence,
by the use of complex operators we can write the voltage equations for
squirrel-cage type as
U
(r jx1 ) I jxm ( I I2c )
½
°
§ rc
· ¾
jxm ( I I2c ) ¨ 2 jx2c ¸ I2c °
©s
¹ ¿
0
(5.1.40)
Similarly, the corresponding equivalent circuit can be plotted just as in Fig. 5.1.1.
In order to compare the equivalent circuit with that in Electric Machinery, take
rotor phase current as PI c , i.e. the rotor phase winding is composed of P bars in
parallel, which belong to the same phase under P pole-pairs. Thus, there exist
PI c
Ic
2sin
KI c
3k
I 2c
PI c
M0
2sin
2
KPI c
6k sin
3 2
k x2
K
x2c
Er S
Er S
K
PI c
6wk w
PI c
ki
3 2
k Z ( LR 2 Lc cos M 0 ) / P
K
Because LR 2( Lc Lk ), in which Lk is one ring-segment inductance for squirrelcage, there are
x2c
3
K
w2 k w2
sin
2
Er S
Z u 2[ LK Lc (1 cos M 0 )]/ P
§
12 Pw2 k w2 ¨
LK
Z ¨ Lc M
Z2
¨¨
2sin 2 0
©
2
§
·
¨
¸
LK
ke kiZ ¨ Lc ¸ P
M
2sin 2 0 ¸¸
¨¨
©
2 ¹
·
¸
¸
¸¸
¹
P
353
AC Machine Systems
r2c
§
¨
rK
3 2
k r c ke ki ¨ rc M
K
¨¨
2sin 2 0
2
©
·
¸
¸
¸¸
¹
P
where
ki
6 wk w
K
ke
2 wk w
k e ki
6 Pwk w
Z2
12 Pw2 k w2
Z2
The current transformation ratio ki , emf ratio ke and impedance ratio ke ki are
familiar to us in Electric Machinery.
In this section, considering only the fundamental flux and using the Multi-loop
method stated in Chapter 1, we get the same results as in Electric Machinery for
steady-state. However, during strong higher harmonic flux, special winding or
inner-asymmetrical fault, it is better to use the Multi-loop method, because of
accurate consideration to higher harmonics and precise results, which is clear
from the example of single phase induction machine in Section 1.10.
5.2
Basic Relations and Parameters of Induction Machines
in d, q, 0 Axes
In Section 5.1, we obtained the voltage and flux-linkage equations with
corresponding parameters according to phase-coordinates, analyzed the steadystate operation and obtained the complex operator equations and corresponding
equivalent circuit. However, for analyzing some special operation modes and
transients of induction machines, it is necessary to convert coordinates just as
synchronous machines. In this section, we discuss the basic relations and parameters
in d, q, 0 axes for induction machines. On the basis of conversion formulas stated
in Section 3.1, it is not difficult to find out induction machine relations in other
axes, if necessary. Again we take the ideal-machine hypothesis and the reference
directions of currents, voltages and flux-linkages in Section 5.1.
The stator winding of induction machines is of 3-phases, wound-rotor winding
is also of 3-phases, and the number of phases for squirrel-cage is K Z 2 / P. In
general, if the number of rotor phases is K, just as in (5.1.26), we can write the
stator and rotor voltage equations and flux-linkage equations in per-unit values as
follows1:
1
the signs with bottom-bars indicate per-unit values, but bottom-bars are omitted from now on.
354
5 Analyses of Some Operating Modes of Induction Machine Systems
p\ a ria
ua
½
°
°
°
p\ c ric
°
p\ n rr in
°
¾
xaa ia xab ib xac ic xa1i1 " xaK iK °
xba ia xbb ib xbc ic xb1i1 " xbK iK °
°
xca ia xcb ib xcc ic xc1i1 " xcK iK °
xna ia xnb ib xnc ic xn1i1 " xnK iK °¿
p\ b rib
ub
uc
un
\a
\b
\c
\n
(5.2.1)
in which rr is rotor phase resistance, and the mutual resistance between the rotor
phases is neglected and might be counted in if necessary. In (5.2.1), several
reactances, namely inductances in per-unit, can be obtained from (5.1.4), (5.1.5),
(5.1.27) and (5.1.31).
Referring rotor parameters to stator and taking per-unit values, there exist
xaa
xbb
xcc
xssc
xm xls
xab
xbc
xca
xms
x11
x22
"
xKK
constant
xrrc xm xlr
xnK
xm cos(n K )M 0 xnKl
xan
xm cos[J (n 1)\ 0 ]
constant
½
°
°
°
constant ¾
°
°
°¿
(5.2.2)
in which
xm is the mutual inductance between a stator phase and rotor phase, whose
axes coincide with each other,
xls and xlr are leakage inductances for stator and rotor phases, respectively,
xnKl is the mutual inductance between rotor K-th phase and n-th phase caused
by the flux that does not pass through the air-gap.
As given above, the basic equations of induction machines are also a set of
equations with variable coefficients. Although those coefficients vary more simply
than those of the synchronous machine, it is difficult to solve the equations directly.
For convenience, we make use of coordinates conversion or variables conversion
just as a study of synchronous machines.
Here, various rotor and stator quantities are converted into d, q, 0 axes
quantities, d, q axes revolving with the rotor. Since d, q axes are of two phases, it
is necessary to transform rotor currents, voltages and flux-linkages from
poly-phase to 2-phase. The conversion formulas are
Ydr
Yqr
2 K
½
¦ Yn cos(n 1)M0 °°
Kn1
¾
2 K
¦ Yn sin(n 1)M0 °°¿
Kn1
(5.2.3)
355
AC Machine Systems
in which Yn represents rotor current, flux-linkage or voltage.
Generally, induction machines have no neutral line, so the zero-axis component is
neglected here. If necessary, we can count it in according to the following formula,
Y0
1
K
K
¦Y
n
(5.2.4)
n 1
On the basis of (5.2.1), (5.2.2) and those conversion formulas, the rotor basic
equations can be written as
udr
uqr
u0 r
\ dr
2
K
p\ dr rr idrc ½
°
p\ qr rr iqrc ¾
°
p\ 0 r rr i0cr ¿
K
¦\
n
cos(n 1)M 0
n 1
2 K
2 K
cos
x
i
n
M
¦ ( n1) n n
¦ x(n 2) nin cos(n 1)M0
0
Kn1
Kn1
2 K
" ¦ x( n K 1) n in cos(n K 2)M 0
Kn1
K
2
2 K
¦ xna ia cos(n 1)M 0 ¦ xnb ib cos(n 1)M 0
Kn1
K n1
K
2
¦ xnc ic cos(n 1)M 0
Kn1
2 K
xrrc idrc ¦ [ x12l xm cos M 0 ]in cos nM 0
Kn1
K
2
¦ [ x13l xm cos 2M 0 ]in cos(n 1)M 0 "
Kn1
2 K
¦ [ x1Kl xm cos( K 1)M 0 ]in cos(n K 2)M 0
Kn1
2 K
¦ xm cos(n 1)M 0 {ia cos[J (n 1)M 0 ]
Kn1
ib cos[J 120e (n 1)M 0 ] ic cos[J 120e (n 1)M 0 ]}
xrrc idrc 2 K
¦ [ x12l xm cos M0 ]in [cos(n 1)M0 cosM0
Kn1
2 K
sin(n 1)M 0 sin M 0 ] ¦ [ x13l xm cos 2M 0 ]in
Kn1
u [cos(n 1)M 0 cos 2M 0 sin(n 1)M 0 sin 2M 0 ] "
xrrc idrc 356
(5.2.5)
5 Analyses of Some Operating Modes of Induction Machine Systems
2 K
¦ [ x1Kl xm cos( K 1)M0 ]in [cos(n 1)M0 cos( K 1)M0
Kn1
sin(n 1)M 0 sin( K 1)M 0
2 K
¦ xm cos(n 1)M0 [ia cos J cos(n 1)M0 ia sin J sin(n 1)M0
Kn1
) cos(n 1)M 0 ib sin(J 120e
)sin( n 1)M 0
ib cos(J 120e
) cos(n 1)M 0 ic sin(J 120e
)sin(n 1)M 0 ]
ic cos(J 120e
1
xm iqrc sin 2M 0 xm idr cos 2 2M 0
2
1
1
xm iqrc sin(2 u 2M 0 ) " xm idrc cos 2 ( K 1)M 0 xm iqrc
2
2
xrrc idrc xm idrc cos 2 M 0 K
K
n 2
n 2
u sin 2( K 1)M 0 ¦ idrc x1nl cos(n 1)M 0 ¦ iqrc x1nl sin(n 1)M 0
2 K
)
¦ xm cos 2 (n 1)M 0 [ia cos J ib cos(J 120e
Kn1
2 K
)] ¦ xm cos(n 1)M 0 sin(n 1)M 0
ic cos(J 120e
K n1
) ic sin(J 120e
)]
u [ia sin J ib sin(J 120e
xlr idrc xm idrc [1 cos 2 M 0 cos 2 2M 0 " cos 2 ( K 1)M 0 ]
1
xm iqrc [sin 2M 0 sin(2 u 2M 0 ) " sin 2( K 1)M 0 ]
2
3 K
3 K
¦ xm id cos 2 (n 1)M 0 ¦ xm iq cos(n 1)M 0 sin(n 1)M 0
K n1
Kn1
K
K
n 2
n 2
¦ idrc x1nl cos(n 1)M 0 ¦ iqrc x1nl sin( n 1)M 0
(5.2.6)
where udr , uqr , u0 r , \ dr , \ qr , \ 0 r , idrc , iqrc , i0cr are rotor voltages, flux-linkages and
currents in d, q, 0 axes, respectively.
Since
K
2
sin 0e+ sin2M 0 sin(2 u 2M 0 ) " sin 2( K 1)M 0
1 cos 2 M 0 cos 2 2M 0 " cos 2 ( K 1)M 0
id
iq
0
2
[ia cos J ib cos(J 120e
) ic cos(J 120e
)]
3
2
[ia sin J ib sin(J 120e
) ic sin(J 120e
)]
3
357
AC Machine Systems
K
¦ ic x
qr 1nl
sin(n 1)M 0
0
n 2
so
\ dr
xlr idrc K
K
3
xm idrc xm id idrc ¦ x1nl cos(n 1)M 0
2
2
n 2
K
K
3
ª
º
« xlr 2 xm ¦ x1nl cos(n 1)M 0 » idrc 2 xm id
n 2
¬
¼
3
xrr idrc xm id
2
(5.2.7)
where
xrr
K
K
xm ¦ x1nl cos(n 1)M 0
2
n 2
K
K 2
xrrc xm ¦ x1nl cos(n 1)M 0
2
n 2
xlr is the rotor self-inductance equivalent to two phases.
In a similar way, there exists
\ qr
xrr iqrc If rotor winding is of 3-phase, namely K
xrr
in which x12l
xmr
(5.2.8)
3, then there is
xm
x12l cos120e x13l cos 240e
2
x
xrrc m x12l xrrc xmr
2
xrrc (5.2.9)
x13l ,
x12l xm cos120eis the rotor mutual inductance between various phases.
\ 0r
358
3
xm iq
2
1 K
1 K
1 K
\ n xrrc i0cr ¦ x( n 1) n in ¦ x( n 2) n in
¦
Kn1
Kn1
K n1
K
K
1
1
1 K
" ¦ x( n K 1) n in ¦ ia xna ¦ ib xnb
Kn1
Kn1
Kn1
K
1
¦ ic xnc
Kn1
1 K
xrrc i0cr ¦ [ x12l xm cos M 0 ]in
K n1
5 Analyses of Some Operating Modes of Induction Machine Systems
1 K
¦ [ x13l xm cos 2M0 ]in "
K n1
1 K
¦ [ x1Kl xm cos( K 1)M0 ]in
K n1
1 K
ia ¦ xm cos[J (n 1)M 0 ]
K n1
1 K
ib ¦ xm cos[J 120e (n 1)M 0 ]
K n1
1 K
ic ¦ xm cos[J 120e (n 1)M 0 ]
K n1
K
K 1
§
·
¨ xrrc ¦ x1nl ¸ i0cr xm i0cr ¦ cos nM 0
n 2
n 1
©
¹
K
K
§
·
¨ xlr ¦ x1nl ¸ i0cr xm i0cr ¦ cos(n 1)M 0
n 1
n 2
©
¹
K
§
·
¨ xlr ¦ x1nl ¸ i0cr x0 r i0cr
n 2
©
¹
(5.2.10)
K
where x0 r
xlr ¦ x1nl .
n 2
If the rotor winding is of 3-phase, then there exists
x0 r
xlr x12l x13l
xrrc 2 xmr
xlr 2 x12l
xrrc xm 2 x12l
(5.2.11)
Transforming the stator quantities to d, q, 0 axes, there exist
ud
p\ d \ qZ r rs id
uq
p\ q \ d Z r rs iq
u0
p\ 0 rs i0
\d
2 K
¦ xanin cos J
3n1
2 K
2 K
) ¦ xcn in cos(J 120e
)
¦ xbn in cos(J 120e
3n1
3n1
2 K
( xssc xms )id ¦ in xm cos[J (n 1)M 0 ]cos J
3n1
K
2
)
¦ in xm cos[J 120e (n 1)M 0 ]cos(J 120e
3n1
xssc id xms id 359
AC Machine Systems
2 K
¦ in xm cos[J 120e (n 1)M0 ]cos(J 120e)
3n1
2 K
3
( xssc xms )id ¦ in xm cos(n 1)M 0
3n1
2
K
xss id xm idrc
2
in which xss xssc xms is the stator synchronous reactance.
In a similar way, we can get
\q
\0
x0
K
xm iqrc
2
( xssc 2 xms )i0 x0 i0
xssc 2 xms
xss iq The basic equations deduced above can be summarized as
ud
uq
u0
\d
\q
\0
udr
uqr
u0 r
\ dr
\ qr
\ 0r
360
p\ d \ qZ r rs id ½
°
p\ q \ d Z r rs iq ¾
°
p\ 0 rs i0
¿
(5.2.12)
K
½
xm idrc °
2
°
K
c
xss iq xm iqr ¾°
2
°
x0 i0
¿
(5.2.13)
xss id p\ dr rr idrc ½
°
p\ qr rr iqrc ¾
°
p\ 0 r rr i0cr ¿
3
½
xm id °
2
°
3
xrr iqrc xm iq ¾°
2
°
x0 r i0cr
¿
xrr idrc 5 Analyses of Some Operating Modes of Induction Machine Systems
3
K
xm in the formula above is not equal to
xm in
2
2
(5.2.13). For convenience, they can be equal after changing the current and
3
K
impedance base-values of the rotor or stator, i.e. let X m
xm , idr
idrc ,
2
3
K
3
3
3
K
i0cr , Rr
rr , X rr
xrr , X 0 r
x0 r .
iqr
iqrc , i0 r
3
K
K
K
3
After changing the base values, the basic equations in d, q, 0 axes for
induction machines can be written as
The mutual inductance
ud
uq
u0
\d
\q
\0
udr
uqr
u0 r
\ dr
\ qr
\ 0r
p\ d \ qZ r rs id ½
°
p\ q \ d Z r rs iq ¾
°
p\ 0 rs i0
¿
(5.2.14)
xss id X m idr ½
°
xss iq X m iqr ¾
°
x0 i0
¿
(5.2.15)
p\ dr Rr idr ½
°
p\ qr Rr iqr ¾
°
p\ 0 r Rr i0 r ¿
(5.2.16)
X rr idr X m id ½
°
X rr iqr X m iq ¾
°
X 0 r iq 0
¿
(5.2.17)
Substituting (5.2.15) into (5.2.14), there exist
ud
uq
u0
( pxss rs )id pX m idr Z r xss iq Z r X m iqr ½
°
Z r xss id Z r X m idr ( pxss rs )iq pX m iqr ¾
°
( p\ 0 rs )i0
¿
(5.2.18)
Substituting (5.2.17) into (5.2.16), there exist
udr
uqr
u0 r
pX m id ( pX rr Rr )idr ½
°
pX m iq ( pX rr Rr )iqr ¾
°
( pX 0 r Rr )i0 r
¿
(5.2.19)
361
AC Machine Systems
Equations (5.2.18) and (5.2.19) can be expressed in a matrix form as follows:
ª ud º
« »
« uq »
«udr »
« »
«uqr »
« u0 »
« »
¬«u0 r ¼»
ª pxss rs
«
« Z r xss
« pX m
«
« 0
« 0
«
¬« 0
Z r xss
pxss rs
0
pX m
0
0
pX m
Zr X m
pX rr Rr
0
0
0
Z r X m
pX m
0
pX rr Rr
0
0
0
0
0
0
px0 rs
0
0
º ª id º
»« »
0
» « iq »
» « idr »
0
»« »
0
» « iqr »
» « i0 »
0
»« »
pX 0 r Rr ¼» ¬«i0 r ¼»
(5.2.20)
For induction machines, their rotor windings are usually short-circuited, so
udr uqr 0.
For induction machines, the input power and electromagnetic torque are
P
ud id uq iq 2u0 i0
(5.2.21)
Te
\ d iq \ q id
(5.2.22)
X m (idr iq iqr id )
The rotor motion equation for induction machines is
H
dZ r
dt
Te Tm
(5.2.23)
in which Tm is the counter-torque of production machines, the inertia constant H
and time are all expressed in electric radians, namely per-unit values.
In practice, substituting slip s for speed Z r , there exist
Zr
1 s
dZ r
dt
H
ds
dt
ds
dt
Tm Te
(5.2.24)
(5.2.25)
(5.2.26)
If we delete the rotor current in (5.2.15), then there exist
\d
\q
362
§
Xm
pX m2 · ½
udr ¨ xss ¸ id °
pX rr Rr
pX rr Rr ¹ °
©
°
¾
G ( p )udr x( p)id
°
°
G ( p )uqr x( p)iq
°
¿
(5.2.27)
5 Analyses of Some Operating Modes of Induction Machine Systems
where G ( p )
Xm
is the operational conductance for induction machines,
pX rr Rr
pX m2
is the corresponding operational reactance.
pX rr Rr
In general, the rotor current base-value is so chosen that the reactances in
per-unit have the following relations:
x( p )
xss xss
X rr
xs X m
Xr Xm
in which xs and X r represent equivalent leakage reactances per phase, respectively,
for stator and rotor.
According to those formulas from (5.2.15) to (5.2.17) together with the relations
above, the operational equivalent circuits can be shown in Fig. 5.2.1 for induction
machines.
Figure 5.2.1 Operational equivalent circuits for induction machines
After getting the relations in d, q, 0 axes above, it is not difficult to research
into different problems to select appropriate axes by using the conversion
formulas in Section 3.1.
5.3
Analysis of the Starting Process of Induction Motors
The starting characteristics of induction motors, inclusive of starting current and
torque varying with rotor speed, are considered especially for large induction
motors. On the basis of the starting characteristics, we evaluate the effect of the
starting process on line voltage, estimate the minimum voltage needed for the
starting process, determine the starting current and time avoided for protection
relays of induction motors and service power, etc. In the past, the starting process
was often analyzed by using the equivalent circuit in Electric Machinery, which is
in steady-state. However, several phenomena occur during the tests of starting
the motor, which cannot be explained according to the steady-state theory. For
example, the maximum speed in the starting process may exceed the synchronous
speed, torque and speed may oscillate near the steady-state operation point, the
363
AC Machine Systems
maximum torque in the starting process is lower than that provided by
manufacturers, sometimes a 40% decrease, etc. Now, we will discuss those
differences and then explain why they occur.
5.3.1
Basic Equations for the Analysis of Starting Process
In the starting process, the induction motor speed is variable, so it is convenient to
select d c , qc , 0 axes with synchronous speed. On the basis of the basic equations
in d, q, 0 axes together with those conversion formulas in Section 3.1, the relations
in d c , qc , 0 axes can be found out. Due to symmetrical and short-circuited rotor
winding, take udr uqr 0 and neglect zero-axis component, then we can get
ªudcs º
«u »
« qcs »
« 0 »
« »
¬ 0 ¼
ª\ dcs º ª rsc
«\ » «
x
qcs »
p«
« ss
«\ dcr » « 0
«
» «
¬«\ qcr ¼» ¬ sX m
xss
rs
sX m
0
0
Xm
Rr
sX rr
X m º ªidcs º
« »
0 »» «iqcs »
sX rr » «idcr »
»« »
Rr ¼ ¬«iqcr ¼»
(5.3.1)
where
ª\ dcs º
«\ »
« qcs »
«\ dcr »
«
»
«¬\ qcr »¼
ª xss
« 0
«
«Xm
«
¬ 0
0
xss
0
Xm
0 º ªidcs º
« »
X m »» «iqcs »
0 » «idcr »
»« »
X rr ¼ «¬iqcr »¼
Xm
0
X rr
0
in which subscript s indicates stator quantity,
subscript r indicates rotor quantity,
subscripts d c and qc express the quantities in d c , qc axes,
xss is stator self-inductance,
X rr is rotor self-inductance,
X m is mutual inductance between stator and rotor,
rs and Rr are stator and rotor resistances respectively,
s is rotor slip, s 1 Z , Z is rotor angular velocity.
Rewriting (5.3.2) and (5.3.1) as the state equation, there is
ªidcs º
«i »
qcs
p« »
«idcr »
« »
«¬iqcr »¼
364
ª xss
« 0
«
«Xm
«
¬ 0
0
xss
0
Xm
Xm
0
X rr
0
0 º
X m »»
0 »
»
X rr ¼
1
­ ªudcs º
°« »
° «uqcs »
®
°« 0 »
° «¬ 0 »¼
¯
(5.3.2)
5 Analyses of Some Operating Modes of Induction Machine Systems
ª rs
« x
« ss
« 0
«
¬ X m
xss
rs
Xm
ª 0
« 0
Z «
« 0
«
¬Xm
0
0
Xm
0
0
0
Xm
Rr
X rr
0
0
0
X rr
X m º ªidcs º
« »
0 »» «iqcs »
X rr » «idcr »
»« »
Rr ¼ ¬«iqcr ¼»
0 º ªidcs º ½
« »°
0 »» «iqcs » °
¾
X rr » «idcr » °
»« »
0 ¼ «¬iqcr »¼ °¿
(5.3.3)
The rotor motion equation is
H
dZ
dt
Te Tm
(5.3.4)
The electromagnetic torque Te is
Te
X m (iqcs idcr idcs iqcr )
(5.3.5)
By using (5.3.3), (5.3.4) and (5.3.5), the torque and current can be calculated
for the starting process of induction motors.
5.3.2
Numerical Calculation of Starting Characteristics
In the starting process, the rotor speed is variable, so (5.3.3), (5.3.4) and (5.3.5)
are non-linear differential equations, which can be solved by numerical method
because it is difficult to use the analytical method.
The rotor speed changes more slowly than the electromagnetic quantities, so
(5.3.4) can be solved by using the first order Euler Method to make numerical
integral. Start at Z 0, make integral for a step length, take the integral result as
the initial value for the next step and so forth, until the calculation is finished.
The integral formula of the m-th step is
Z (mh) Z[(m 1)h] (Te Tm )h / H
in which h is integral step length, electromagnetic torque Te is evaluated by (5.3.5),
where the d c , qc components for stator and rotor currents are given by (5.3.3), so
the integral calculation needs simultaneous equations (5.3.3), (5.3.4) and (5.3.5).
Making numerical calculation according to (5.3.3), we use the 4th order
Runge-Kutta Method to get satisfactory results.
For each discrete step, the speed Z is considered as constant, so non-linear
equation (5.3.3) can be linearized for each integral step length.
365
AC Machine Systems
In general, the initial conditions for the starting process are all zero, i.e.,
idcs (t0 )
0, iqcs (t0 )
0
idcr (t0 )
0, iqcr (t0 )
0, Z (t0 )
0
where t0 is the starting instant.
For the calculation results to accord with practical condition, we have to
consider the following.
(i) Studying transients, we usually select rated maximum phase-voltage as
voltage base-value and rated maximum phase-current as current base-value.
However, in the design of electric machines, we often take active current as current
base-value, which can be estimated by machine rated output power. Because of
different current base-values, the impedance, power and torque base-values are
also different. Evaluating the starting process, it is necessary to convert those
parameters in per-unit used in electric machine design.
(ii) For deep-slot rotor or double-cage induction motors, their rotor parameters
vary obviously with rotor slip. In addition, ferromagnetic saturation will influence
the reactances, too. If necessary, according to the starting and operating parameters
provided by manufacturers, estimate the rotor resistance for a slip-interval as
follows:
r2c
r2c
( s se )(rstc 2 rec2 ) rec2
rec2
s ı se ½
¾
s se ¿
(5.3.6)
in which
r2c is rotor phase resistance at any slip s,
se is rated slip,
rstc 2 is rotor phase resistance at start,
rec2 is rotor phase resistance at rated operation.
The rotor leakage reactance x2c in a slip-interval can be evaluated as
x2c
x2c
s se ( xstc 2 xec2 ) xec2
xec2
s ı se ½
¾
s se ¿
(5.3.7)
where xstc 2 and xec2 indicate the rotor leakage reactance at the start and rated
operation, respectively.
5.3.3 Comparison and Analysis of Calculation Results in Two Ways
The no-load starting process is evaluated for a 2000 kW-induction motor, whose
results are shown in Fig. 5.3.1, in which continuous lines indicate T-s curve and
366
5 Analyses of Some Operating Modes of Induction Machine Systems
I-s curve in steady-state, and dash lines express those in the starting process.
Slip s 1 Z , where Z is the rotor speed in per-unit.
Figure 5.3.1 Comparison of T-s curves and I-s curves for steady-state and starting
process of induction motor
At middle speed, namely 0.7>s>0.2, two T-s curves or two I-s curves for
steady-state and transients are approximate to each other. At low speed, namely
s>0.7, the T-s curve and I-s curve in transients all contain strong oscillating
components, but their average values are approximate to those in steady-state. In
Fig. 5.3.1, the line-segments for s>0.7 cannot be drawn due to strong oscillating
components. Near the critical slip sK , the torque T in the starting process is less
than that in steady-state, especially for maximum Torque. In the following, we
will find out the main reasons that cause this difference. Due to small voltage
drop of stator resistance, rs can be neglected in (5.3.1), so the stator current in
steady-state can be estimated by the following formula:
ªidcs º
«i »
« qcs »
«idcr »
« »
«¬iqcr »¼
ª 0
« x
« ss
« 0
«
¬ sX m
ª B11
«B
¬ 21
xss
0
sX m
0
0
Xm
Rr
sX rr
ª udcs º
B12 º «« uqcs »»
B22 »¼ « 0 »
«
»
¬ 0 ¼
Xm º
0 »»
sX rr »
»
Rr ¼
1
ª udcs º
« u »
« qcs »
« 0 »
«
»
¬ 0 ¼
(5.3.8)
367
AC Machine Systems
in which exists p\
0 , and the expression
1
R ( sW X rr ) 2
B22
W
2
r
ª Rr
« sW X
¬
rr
sW X rr º
Rr »¼
xss X rr X m2
xss X rr
can be obtained by inversing the partition matrix. Other sub-matrices have no
effect on the following analysis, so they are omitted here.
Similarly, neglecting stator resistance, the stator and rotor currents in transients
can be obtained by
ªidcs º
«i »
« qcs »
«idcr »
« »
«¬iqcr »¼
ª udcs p\ dcs º
«
»
B12 º « uqcs p\ qcs »
»
p\ dcr
B22 »¼ «
«
»
p\ qcr
«¬
»¼
ª B11
«B
¬ 21
(5.3.9)
Near the critical slip, p\ dcs and p\ qcs can be neglected, so (5.3.9) is changed to
ªidcs º
«i »
« qcs »
«idcr »
« »
«¬iqcr »¼
ª udcs º
«
»
B12 º « uqcs »
B22 »¼ « p\ dcr »
«
»
«¬ p\ qcr »¼
ª B11
«B
¬ 21
(5.3.10)
In order to judge the difference between steady-state and transients near the
ª p\ dcr º
critical slip, pay attention to B22 «
» , which influences the rotor currents
¬ p\ qcr ¼
idcr , iqcr and then the stator currents idcs , iqcs .
s
Take an induction motor to illustrate this problem, whose parameters at
0.03 are rs 0.004 22, Rr 0.004 686, X m 3.577 4, xss 3.650 63, X rr
3.690 73. The applied voltages are udcs
0.544 021, uqcs
0.839 071 5. At speed
Z 0.97, the estimation results are
p\ dcs
0.000 2,
p\ qcs
0.000 7
p\ dcr
0.006 33,
p\ qcr
0.003 64
At the same speed Z 0.97, the stator and rotor currents together with electromagnetic torque are given in Table 5.3.1 for different operation conditions.
368
5 Analyses of Some Operating Modes of Induction Machine Systems
Table 5.3.1 Currents and torques evaluated for different operation conditions
Calculation
formula
idcs
iqcs
idcr
Estimation in transients
(5.3.3)
4.301
0.556
4.1539
Estimation in steady-state
(5.3.8)
4.115
0.402 9
3.965 2 0.558 35 2.93
Operation condition
In addition, according to the motor parameters at s
B22
iqcr
0.420 3
Te
2.089
0.03 , there exists
ª 88.764 104.695º
«104.695 88.764 »
¬
¼
which is multiplied by small rates of change for rotor flux-linkage, and then the
effect of transients on rotor currents is clear. The estimation result is
ª p\ dcr º
B22 «
»
¬ p\ qcr ¼
ª 0.180 79 º
« 0.985 82 »
¬
¼
(5.3.11)
whose values 0.180 79 and 0.98582 indicate the increments of idcr and iqcr
caused by small rates of change for rotor flux-linkage. The increments cannot be
neglected to compare the following steady-state rotor current
ªidcr º
«i »
¬ qcr ¼
ª 3.965 2 º
« 0.558 35»
¬
¼
(5.3.12)
because the addition of the corresponding values in (5.3.11) and (5.3.12) represents
transient rotor currents idcr and iqcr at s 0.03, as shown in Table 5.3.1.
Evidently, at critical slip, the stator and rotor currents in steady-state are quite
different from those in transients, which is caused mainly by the rates of change
for rotor flux-linkages \ dcr , \ qcr and motor parameters. For example, the smaller
the rotor resistance, i.e. the smaller the critical slip, the larger the various element
values of sub-matrix B22 are, so small change rates of rotor flux-linkages influence
rotor currents and then stator currents, obviously. However, increasing rotor slip,
several element values in B22 are decreased distinctly, so the change rates of rotor
flux-linkages have only marginal influence. Therefore, at middle speed Z 0.6,
namely s 0.4, the torque and current in steady-state are close to those in
transients, as shown in Fig. 5.3.1.
The calculation results for a 2 000kW-induction motor are shown in Fig. 5.3.2.
Clearly, starting the rotor near the steady-state operation point, the torque and rotor
speed swing around the equilibrium point several times. In addition, the maximum
speed in the starting process can exceed the synchronous speed at s 0.024,
which cannot appear for steady-state characteristics. This phenomenon can be
explained as follows. Although there is no excitation source for the rotor winding
369
AC Machine Systems
of induction motors, the starting process is accompanied with electromagnetic
transient process, and the rotor current at synchronous speed cannot decay suddenly
to zero, which is a temporary excitation source. Therefore, the electromechanical
oscillation exists just as drawing rotor in step for a synchronous motor.
Of course, the maximum speed in the starting process depends upon several
factors. Increasing rotor inertia and load, the maximum speed may decrease, and
the swing process will be completed quickly.
Figure 5.3.2 Speed and torque oscillation near the steady-state operation point in
starting process
The starting characteristics for different rotating inertias and loads are shown
in Fig. 5.3.3. During Z 0.7, namely s ! 0.3, rotating inertias and loads have a
little effect on the starting characteristics, so the line-segments during Z 0.7,
namely s ! 0.3, are omitted in Fig. 5.3.3. For evaluation, suppose the load to be a
ventilator, which can be indicated by the formula T 0.15 0.75Z 2 and whose
load-factor is 0.9. The total fly-wheel moment of motor and ventilator is GD 2
5 370kg ˜ m 2 , and the fly-wheel moment of motor itself is GD 2 870kg ˜ m 2 .
Curve ċ in Fig. 5.3.3 corresponds to a fictitious condition, i.e. the load is the
same but total fly-wheel moment is GD 2 870kg ˜ m 2 .
Figure 5.3.3 Effect of load and rotating inertia on starting characteristics
ĉ— steady-state operation, Ċ— GD 2
ċ— GD 2
370
870kg ˜ m 2 , no-load starting,
870kg ˜ m 2 , load starting, Č— GD 2 5 370kg ˜ m 2 , load starting
5 Analyses of Some Operating Modes of Induction Machine Systems
Clearly, in Fig. 5.3.3, the maximum torque can increase with rotating inertia
and load. If rotating inertia is large enough, T-s curve in transients is approximate
to that in steady-state, which can be explained as follows. The larger the rotating
inertia, the longer the starting process, thus approaching the steady-state process.
From calculation and analysis above, we can get the following conclusions:
(i) Using the steady-state theory to analyze the starting process brings about
some errors for a certain speed region, which must be paid attention to. In the
starting process, the rotor speed may exceed the synchronous speed in short time
to cause oscillation, but it cannot damage motors.
(ii) T-s curve and I-s curve in the starting process are different from those in
steady-state, which needs to be noted, too. The maximum torque provided by the
manufacturers is obtained in steady-state, so the maximum torque measured in
the starting process is less than that given by the manufacturers.
(iii) Load and rotating inertia have obvious effect on starting characteristics.
The maximum torque increases with load and rotating inertia. When the rotating
inertia is large enough, T-s curve in transients will approach that in steady-state.
Therefore, the two starting characteristics can be used respectively according to
different rotating inertias and loads. For example, T-s curve in steady-state is
used to determine the over-load capability for steady-state operation, and T-s
curve in transients is used to evaluate the starting time.
5.4
Transients of Reswitching on Induction Motors
The transient process of induction motors at variable speed is a practical topic
which include the starting process, plugging operation, disconnecting process
due to source fault, reclosing process due to supply recovery, etc. In this section,
we analyze the transient process of reswitching on the induction motors, by which
we can understand not only some problems about reswitching on but also other
characteristics of induction motors at variable speed.
Sometimes, temporary fault occurs for a supply system to bring about source
voltage dip and then tripping away for induction motors, such as short circuit of
power system, mis-action of protection relays, and switching over large motors.
Source fault is often temporary, so production loss caused by failure in power
supply can be decreased if we reswitch on the induction motors as quickly as
possible after supply recovery, especially for some continuous production process.
Transients of reswitching on the induction motors include two parts, one is the
disconnection process due to source fault, and the other is the reclosing process
after supply recovery. The main problems of reswitching on are surge current and
electromagnetic torque, which will be discussed as follows.
371
AC Machine Systems
5.4.1
Transients of Induction Motors after Disconnection
When interrupter’s contacts are separated, 3-phase currents should be changed to
zero at the same time, but it sometimes cannot be changed to zero simultaneously
due to electric arc or other reasons. For example, when phase a current passes
through zero and interrupter’s contacts are away from each other in the meantime,
phase a current can be considered as zero, but phase b and phase c currents still
exist until phase b or phase c current passes through zero to make the two phase
currents change to zero simultaneously. If 3-phase currents cannot be changed to
zero at the same time, the disconnecting process can be divided into two stages,
the first stage is from one phase cut off to the instant of all phases cut off, i.e.
from ia 0 to ia ib ic 0, and the second stage is from 3-phase zero currents
to reclosing instant, in which the first stage is much shorter than the second stage,
so the results in this way are approximate to those when assuming 3-phase
currents to be changed to zero at the same time. Of course, the latter is much
simple and used in practice. In order to understand those problems comprehensively,
the two methods will be discussed as follows.
(1) Analysis of 3-phase currents not to be zero simultaneously, i.e. the first
method
Evidently, before 3-phase currents are all changed to zero, the stator circuits are
unsymmetrical, so it is convenient to put coordinates axes on the unsymmetrical
stator such as D , E , 0 axes. Therefore, the relations of induction machines are
converted into those in D , E , 0 axes.
On the basis of (5.2.15), there are
\ d cos J
\ q sin J
xss id cos J X m idr cos J
xss iq sin J X m iqr sin J
Thus, referring to (3.1.6), there exists
\D
xss iD X m iD r
(5.4.1)
According to (5.2.17) and (3.1.6), we can also get
\E
\Dr
\ Er
xss iE X m iE r ½
°
X rr iD r X m iD ¾
X rr iE r X m iE °¿
Since
p (\ cos J )
cos J ( p\ ) \ sin J ( pJ )
p(\ sin J ) sin J ( p\ ) \ cos J ( pJ )
372
(5.4.2)
5 Analyses of Some Operating Modes of Induction Machine Systems
on the basis of (5.2.14) and (5.2.16), there exist
p\ D rs iD ½°
¾
p\ E rs iE °¿
(5.4.3)
p\ D r Z r\ E r Rr iD r ½°
¾
p\ E r Z r\ D r Rr iE r °¿
(5.4.4)
uD
uE
uD r
uE r
Substituting (5.4.1) and (5.4.2) into (5.4.3) and (5.4.4), there are
xss piD X m piD r rs iD ½°
¾
xss piE X m piE r rs iE °¿
(5.4.5)
X rr piD r X m piD Z r X rr iE r Z r X m iE Rr iD r ½°
¾
X rr piE r X m piE Z r X rr iD r Z r X m iD Rr iE r °¿
(5.4.6)
uD
uE
uD r
uE r
which can be written as matrix form,
U
XI Z r GI RI
U
ª uD º
«u »
« E », I
« uD r »
« »
«¬u E r »¼
ª iD º
«i »
« E»
« iD r »
« »
«¬iE r »¼
X
ª xss
« 0
«
«Xm
«
¬ 0
Xm
0
X rr
0
0
xss
0
Xm
R
ª rs
«
«
«
«
¬
G
ª 0
« 0
«
« 0
«
¬ X m
rs
Rr
0
0
Xm
0
º
»
»
»
»
Rr ¼
0
0
0
X rr
(5.4.7)
0 º
X m »»
0 »
»
X rr ¼
0 º
0 »»
X rr »
»
0 ¼
373
AC Machine Systems
Having the relations above, together with electromagnetic torque formula and
rotor motion equation, it is not difficult to analyze the situation for 3-phase currents
not to be zero simultaneously.
(a) The first stage
As mentioned above, the first stage is from one phase cut off to the instant of
all phases cut off. In the stage exists ia 0, but ib and ic are not equal to zero.
Relevantly, assume the steady-state stator currents before disconnection to be
ian
ibn
icn
½
°
°
§ 2S · °
I m sin ¨ t ¸¾
¹°
©
§ 2S · °
I m sin ¨ t ¸°
¹¿
©
I m sin t
(5.4.8)
which can be converted into D , E components
iD n
iE n
2­
1ª
§ 2S ·
§ 2S · º ½ ½
® I m sin t « I m sin ¨ t ¸ I m sin ¨ t ¸ ¾°
¹
¹ »¼ ¿°
3¯
2¬
©
©
¾
1 ª
§ 2S ·
§ 2S · º
°
« I m sin ¨ t ¸ I m sin ¨ t ¸ » I m cos t °
3¬
©
¹
©
¹¼
¿
(5.4.9)
Substituting the formulas above into (5.4.5) and (5.4.6), and supposing rotor
winding to be short-circuited, the stator and rotor relations before disconnection
can be written as
uE n
xss I m cos t X m pID rn rs I m sin t ½°
¾
xss I m sin t X m piE rn rs I m cos t °¿
(5.4.10)
uD rn
0
X rr piD rn X m I m cos t Z r X rr iE rn ½
°
Z r X m I m cos t Rr iD rn
°
¾
0 X rr piE rn X m I m sin t Z r X rr iD rn °
°
Z r X m I m sin t Rr iE rn
¿
(5.4.11)
uD n
uE rn
or
in which Z r
374
( X rr p Rr )iD rn Z r X rr iE rn
sX m I m cos t
Z r X rr iD rn ( X rr p Rr )iE rn
sX m I m sin t
1 s.
5 Analyses of Some Operating Modes of Induction Machine Systems
Evidently, the steady-state rotor currents iD rn and iE rn before disconnection are
sinusoidal form with operating frequency, which can be indicated as complex
operator form
( jX rr Rr ) ID rn Z r X rr IE rn
Z r X rr ID rn ( jX rr Rr ) IE rn
jsX m I m ½°
¾
sX m I m °¿
(5.4.12)
whose solutions are
s 2 X rr X m I m jsRr X m I m
Rr2 ( sX rr ) 2
jsX m I m
Rr jsX rr
ID rn
jIE rn
IE rn
js 2 X rr X m I m sRr X m I m
Rr2 ( sX rr )2
namely
iD rn
iE rn
½
s 2 X rr X m I m
sR X I
sin t 2 r m m 2 cos t °
2
2
Rr ( sX rr )
Rr ( sX rr )
°
¾
2
s X rr X m I m
sRr X m I m
°
t
t
cos
sin
°
Rr2 ( sX rr ) 2
Rr2 ( sX rr )2
¿
(5.4.13)
Substituting the results above into (5.4.10), the steady-state stator voltages
uD n and uE n are
uD n
uE n
½
ª
ª
s 2 X rr X m2 º
sR X 2 º
I cos t « rs 2 r m 2 » I m sin t °
« xss 2
2 » m
Rr ( sX rr ) ¼
Rr ( sX rr ) ¼
¬
¬
°
¾
2
2
2
ª
ª
s X rr X m º
sRr X m º
°
I sin t « rs 2
I cos t °
« xss 2
2 » m
2 » m
(
)
(
)
R
sX
R
sX
r
rr
r
rr
¬
¼
¬
¼
¿
(5.4.14)
As stated before, after interrupter’s contacts are separated, there is
ia
0
(5.4.15)
ib
ic
(5.4.16)
Without neutral line, there exists
Because phase b and phase c are still connected to the supply, the line voltage
ubc after the disconnection of phase a is the same as before, namely
ubc
ub uc
ubcn
(5.4.17)
375
AC Machine Systems
The boundary conditions above can be expressed in D , E components, namely
i i ·
2§
ia b c ¸
¨
3©
2 ¹
iD
1
uE
3
(ub uc )
0
1
3
(5.4.18)
ubc
1
3
ubcn
uE n
(5.4.19)
which are substituted into (5.4.7), thus getting
ª uD º
«u »
« En »
« 0 »
« »
¬ 0 ¼
ª xss
« 0
«
«Xm
«
¬ 0
0
xss
0
Xm
Xm
0
X rr
0
ª 0
« 0
Zr «
« 0
«
¬ X m
0
0
Xm
0
0 º ª 0 º ª rs
« »
X m »» « iE » ««
rs
0 » « iD r » «
»« » «
X rr ¼ ¬«iE r ¼» ¬
0
0 ºª 0 º
« »
0
0 »» « iE »
X rr » «iD r »
0
»« »
X rr
0 ¼ ¬«iE r ¼»
Rr
ºª 0 º
»«i »
»« E »
» « iD r »
»« »
Rr ¼ ¬«iE r ¼»
(5.4.20)
in which
uD
Xm
diD r
dt
(5.4.21)
so (5.4.20) can be reduced to
ªu E n º
« 0 »
« »
«¬ 0 »¼
ª xss
« 0
«
«¬ X m
0
X rr
0
ª 0
Z r «« X m
«¬ 0
X m º ª iE º ª rs
« »
0 »» « iD r » ««
X rr »¼ «¬iE r »¼ «¬
0
0
X rr
Rr
º ª iE º
» «i »
» « Dr »
Rr »¼ «¬iE r »¼
0 º ª iE º
« »
X rr »» « iD r »
0 »¼ «¬iE r »¼
which can be written as the standard state equation as follows:
ª iE º
d« »
iD r
dt « »
«iE r »
¬ ¼
376
ª xss
«« 0
«¬ X m
0
X rr
0
Xm º
0 »»
X rr »¼
1
rs
ª
«
u «(1 s ) X m
«¬
0
0
Rr
(1 s ) X rr
º ª iE º
« »
»
(1 s ) X rr » « iD r »
»¼ «¬iE r »¼
Rr
0
5 Analyses of Some Operating Modes of Induction Machine Systems
ª xss
«« 0
«¬ X m
Xm º
0 »»
X rr »¼
0
X rr
0
1
ªu E n º
« 0 »
« »
«¬ 0 ¼»
(5.4.22)
Referring to (5.4.9) and (5.4.13), the initial values of iE , iD r and iE r are indicated as
Im
iE 0
iD r 0
iE r 0
½
°
sR X I
2 r m m 2 °°
Rr ( sX rr ) ¾
s 2 X rr X m I m °°
Rr2 ( sX rr ) 2 °¿
(5.4.23)
Referring to (5.4.14), the voltage uE n is
uE n
ª
s 2 X rr X m2 º
x
« ss
» I m sin t
Rr2 ( sX rr )2 ¼
¬
ª
sR X 2 º
« rs 2 r m 2 » I m cos t
Rr ( sX rr ) ¼
¬
After interrupter’s contacts are separated, electromagnetic torque and rotor
speed will change, and the corresponding formulas, referring to (5.2.22) and
(5.2.26), can be written as
Te
H
X m (idr iq iqr id )
ds
dt
Tm Te
X m (iE iD r iD iE r )
(5.4.24)
(5.4.25)
Having the basic relations from (5.4.22) to (5.4.25), we can analyze the
transients of the first stage, i.e. from phase a cut off to the instant of all phases
cut off. The basic relations are non-linear differential equations, which can be
solved in the following steps:
(i) According to (5.4.23), we get several current initial values iE 0 , iD r 0 , iE r 0.
(ii) Since phase a is broken, and phase b and phase c are still connected to the
source, the line voltage ubc is the same as before, so uE will not change and can
be expressed by uE n before disconnection, referring to (5.4.14), into which t 0
is substituted to get uE n (0).
(iii) The change of rotor speed is much slower than that of electromagnetic
quantities, so slip s is considered as constant exactly for a short time interval 'T .
377
AC Machine Systems
Thus, (5.4.22) is approximately state equations with constant coefficient s. On
the basis of several initial values iE 0 , iD r 0 , iE r 0 , uE n (0), s0, and (5.4.22), by means
of the Runge-Kutta Method or other numerical methods, evaluate iE ('T ), iD r ('T ),
and iE r ('T ) for t 'T .
(iv) Substituting t 'T into (5.4.14), we get uE n ('T ).
(v) The instantaneous values of stator currents at t 'T are
3
iE ('T )
2
ia ('T ) iD ('T ) 0, ib ('T )
ic ('T )
3
iE ('T )
2
(vi) Referring to (5.4.24), the electromagnetic torque at t
Te ('T )
'T is
X m iE ('T )iD r ('T )
(vii) According to (5.4.25) and initial values s0, Tm (0), Te (0) X m iE 0 iD r 0 , by the
Runge-Kutta Method or other methods, calculate rotor slip s( 'T ) at t 'T .
Using the calculation procedures above, the evaluation steps for t 2'T are:
(i) According to iE ('T ), iD r ('T ), iE r ('T ), s ('T ), uE n ('T ), and state equation
(5.4.22), by means of the same numerical method, calculate iE (2'T ), iD r (2'T )
and iE r (2'T ) at t 2'T .
(ii) Substituting t 2'T into (5.4.14), we get uE n (2'T ).
(iii) The instantaneous values of stator currents at t 2'T are
ia (2'T ) iD (2'T )
ib (2'T )
ic (2'T )
(iv) Electromagnetic torque at t
Te (2'T )
0
3
iE (2'T )
2
3
iE (2'T )
2
2'T is
X m iE (2'T )iD r (2'T )
(v) On the basis of (5.4.25) and s ('T ), Tm ('T ) and Te ('T ), by using the same
method as above, estimate rotor slip s (2'T ) at t 2'T .
In so doing, the iterating calculation continues until iE (k 'T ) 0, i.e.
ib ( k 'T ) 0, thus completing the estimation for the first stage.
378
5 Analyses of Some Operating Modes of Induction Machine Systems
(b) The second stage
As mentioned above, at t k 'T , there exist iE (k 'T ) 0 and ib (k 'T ) 0,
which means that phase b is cut off and phase c is also open, so the second stage
begins, in which there is no stator current and rotor currents decay. The initial
values in the second stage are rotor currents iD r (k 'T ), iE r (k 'T ) and rotor slip
s (k 'T ) , respectively.
On the basis of (5.4.7), the basic relations in the second stage are
ª uD º
«u »
« E»
«0»
« »
¬0¼
0 Xm
0 ºª 0 º
ª xss
« »
« 0
0
xss
X m »» « 0 »
«
« X m 0 X rr
0 » « iD r »
«
»« »
0 X rr ¼ «¬iE r »¼
¬ 0 Xm
ª 0
ª rs
ºª 0 º
«
«
»« 0 »
r
s
» « » (1 s ) « 0
«
«
» « iD r »
« 0
Rr
«
»« »
«
Rr ¼ ¬«iE r ¼»
¬
¬ X m
0
0
Xm
0
0
0
0
X rr
0 ºª 0 º
« »
0 »» « 0 »
X rr » « iD r »
»« »
0 ¼ ¬«iE r ¼»
(5.4.26)
or
uD
uE
ª0º
«0»
¬ ¼
ª X rr
« 0
¬
diD r ½
dt °°
diE r ¾°
Xm
dt ¿°
Xm
0 º ª iD r º ª Rr
« »
X rr »¼ ¬iE r ¼ «¬ 0
0
ª
«
¬ (1 s ) X rr
(5.4.27)
0 º ª iD r º
« »
Rr »¼ ¬iE r ¼
(1 s ) X rr º ª iD r º
» «i »
0
¼ ¬ Er ¼
(5.4.28)
Writing (5.4.28) as the standard state equation, there is
d ªiD r º
« »
dt ¬iE r ¼
ª Rr
º
1 s»
« X
ªi º
rr
» « Dr »
«
«
Rr » ¬iE r ¼
« (1 s )
»
X rr ¼
¬
(5.4.29)
Having the basic equations above, it is not difficult to analyze the transients of
the second stage, i.e. from 3-phase zero currents to reclosing instant. The
following are the corresponding steps:
379
AC Machine Systems
(i) On the basis of iD r (k 'T ), iE r (k 'T ), s (k 'T ) and state equation (5.4.29), by
using the same numerical method, evaluate rotor currents iD r [(k 1)'T ] and
iE r [(k 1)'T ] at t
(k 1)'T .
(ii) According to (5.4.24), electromagnetic torque is always zero in the second
stage, i.e.,
Te [(k j )'T ] 0
j
0,1, 2,3,"
(iii) On the basis of (5.4.25) and s (k 'T ), Tm (k 'T ) and Te (k 'T ) 0, by using
the same method, estimate rotor slip s[(k 1) 'T ] at t (k 1)'T .
Following the calculation procedures above, the evaluation steps for
t (k 2)'t are:
(i) According to iD r [( k 1) 'T ], iE r [( k 1)'T ], s[(k 1) 'T ] and state equation
(5.4.29), by using the same method, evaluate rotor current components
iD r [(k 2) 'T ] and iE r [(k 2)'T ] at t (k 2)'T .
(ii) On the basis of (5.4.25) and s[(k 1)'T ], Tm [( k 1)'T ] and Te [( k 1) 'T ] 0,
in the same way, reckon rotor slip s[(k 2)'T ] at t (k 2)'T .
In so doing, the iterating calculation continues until the reclosing instant
t (k n)'T , thus completing the estimation for the second stage.
(2) Analysis for 3-phase currents to be zero simultaneously, i.e., the second
method
As stated above, 3-phase stator currents all change to zero when interrupter’s
contacts are separated for the second method, so it is simpler than the first
method. For comparison with the first method, assume the disconnection instant
to be t 0, and the stator currents before disconnection are expressed in (5.4.8).
Using D , E , 0 axes, substitute iD iE 0 and uD r u E r 0 into (5.4.7) and refer
to (5.4.26), there are
uD
uE
d ªiD r º
« »
dt ¬iE r ¼
380
diD r ½
dt °°
diE r ¾°
Xm
dt °¿
Xm
ª Rr
º
1 s»
« X
ªi º
rr
» « Dr »
«
«
Rr » ¬iE r ¼
« (1 s )
»
X rr ¼
¬
(5.4.30)
(5.4.31)
5 Analyses of Some Operating Modes of Induction Machine Systems
The stator currents iD 0 and iE 0 before disconnection can be evaluated by (5.4.9),
and the corresponding rotor currents iD r 0 and iE r 0 are still estimated by (5.4.23),
so due to the conservation of rotor flux-linkages after stator disconnection, the
rotor current initial values should be written as
iD r 0
iE r 0
X rr iD r 0 X m iD 0 ½
°
X rr
°
X rr iE r 0 X m iE 0 ¾°
°¿
X rr
(5.4.32)
The calculation procedures are:
(i) On the basis of rotor current initial values iD r 0, iE r 0 in (5.4.32), initial slip s0
and state equation (5.4.31), by using a certain numerical method, count rotor
currents iD r ('T ) and iE r ('T ) at t 'T .
(ii) According to (5.4.24), electromagnetic torque is always zero, namely
Te (k 'T )
k
0
0,1, 2,3,"
(iii) By (5.4.25) and initial values s0 , Tm (0) together with Te (0) 0, using the
same method as above, get rotor slip s ('T ) at t 'T .
Using the calculation steps above, the evaluation procedures for t 2'T are:
(i) On the basis of iD r ('T ), iE r ('T ), s ('T ) and state equation (5.4.31), by using
the same method, estimate rotor currents iD r (2'T ) and iE r (2'T ) at t 2'T .
(ii) According to (5.4.25) and s( 'T ), Tm ( 'T ), and Te ('T ) 0, in the same
way, obtain rotor slip s (2'T ) at t 2'T .
In so doing, the iterating calculation continues until the reclosing instant
t (k n)'T , in which we take t (k n)'T for comparison with the first
method.
5.4.2
Transients of Induction Motors after Reclosing
We still use D , E , 0 axes to analyze the reclosing transients. The basic equation
(5.4.7) can be written as standard state equation
ª iD º
« »
d « iE »
dt «iD r »
« »
«¬iE r »¼
ª xss
« 0
«
«Xm
«
¬ 0
0
xss
0
Xm
Xm
0
X rr
0
0 º
X m »»
0 »
»
X rr ¼
1
381
AC Machine Systems
rs
ª
«
0
u«
«
0
«
¬ (1 s ) X m
ª xss
« 0
«
«Xm
«
¬ 0
0
rs
(1 s ) X m
0
Xm
0
X rr
0
0
xss
0
Xm
0 º
X m »»
0 »
»
X rr ¼
0
0
Rr
(1 s ) X rr
1
0
º ª iD º
»«i »
0
»« E »
(1 s ) X rr » « iD r »
»« »
Rr
¼ ¬«iE r ¼»
ª uD n º
«u »
« En »
« 0 »
« »
¬ 0 ¼
(5.4.33)
where uD n and uE n are D , E components of stator voltage after reclosing, which
can be found out as follows.
Assume the reclosing instant to be the beginning time for calculation and let
3-phase stator voltages be
uan
ubn
ucn
U m sin(t T )
½
°
§ 2S
·
U m sin ¨ t T ¸ °°
©
¹¾
2
S
§
·°
T ¸°
U m sin ¨ t ©
¹°
¿
(5.4.34)
where T is the switching phase angle.
Converting 3-phase stator voltages into D , E components, there are
uD n
uE n
u ucn · ½
2§
uan bn
¸°
3 ¨©
2
¹°
uan U m sin(t T ) °
¾
1
°
(ubn ucn )
°
3
°
U m cos(t T )
¿
(5.4.35)
Having the relations above, the reclosing transients of induction motors can be
estimated as follows:
(i) Taking the reclosing instant as the beginning time for calculation and according
to one of the two methods above, evaluate iD r [(k n)'T ], iE r [(k n)'T ],
s[(k n)'T ], which are the corresponding initial values when reclosing, namely
iD r 0
s0
382
iD r [(k n)'T ],
s[(k n)'T ]
iE r 0
iE r [(k n)'T ]
5 Analyses of Some Operating Modes of Induction Machine Systems
In addition, there are
iD 0
(ii) Substituting t
0,
iE 0
0
0 into (5.4.35), get uD n (0) and uE n (0).
(iii) According to iD 0 , iE 0 , iD r 0 , iE r 0 , s0, uD n (0), u E n (0) and state equation (5.4.33),
by using a certain numerical method, estimate several current components iD ('T ),
iE ('T ), iD r ('T ) and iE r ('T ) at t
'T .
'T , the electromagnetic torque is
(iv) By (5.4.24) and t
Te ('T )
X m [iE ('T )iD r ('T ) iD ('T )iE r ('T )]
(v) Referring to (3.1.5), the stator phase currents at t
'T are
ia ('T ) iD ('T )
ib ('T )
ic ('T )
1
iD ('T ) 2
1
iD ('T ) 2
3
iE ('T )
2
3
iE ('T )
2
(vi) According to s0 , Tm (0), Te (0) 0 and (5.4.25), by using a certain numerical
method, evaluate rotor slip s ( 'T ) at t
'T .
Following the calculation procedures above, the evaluation steps for t
2'T
are:
(i) Substituting t
'T into (5.4.35), get uD n ('T ) and uE n ('T ).
(ii) According to iD ('T ), iE ('T ), iD r ('T ), iE r ('T ), s('T ), uD n ('T ), uE n ('T )
and state equation (5.4.33), in the same way, evaluate several current components
iD (2'T ), iE (2'T ), iD r (2'T ) and iE r (2'T ).
(iii) Electromagnetic torque at t
Te (2'T )
2'T is
X m [iE (2'T )iD r (2'T ) iD (2'T )iE r (2'T )]
(iv) The stator phase currents at t
2'T are
ia (2'T ) iD (2'T )
ib (2'T )
ic (2'T )
1
iD (2'T ) 2
1
iD (2'T ) 2
3
iE (2'T )
2
3
iE (2'T )
2
383
AC Machine Systems
(v) According to equation (5.4.25), s ('T ), Tm ('T ) and Te ('T ), in a similar
way, get s (2'T ) at t 2'T .
In so doing, the iterating calculation continues until the induction motor reaches
steady-state at t k 'T . In this way, we can obtain stator currents ia (t ), ib (t ),
ic (t ) curves, electromagnetic torque Te (t ) and slip s (t ) curves. In addition, we
can get the effect of different switching phase angle T on the reclosing surge
current and torque.
As an additional note, let us notice that the calculation methods above are
suitable not only for the analysis of electromechanical transients of reswitching
on the induction motors but also for the analysis of starting electromechanical
transients, whose difference is due to different initial conditions and the latter
is iD 0 iE 0 iD r 0 iE r 0 0, and s0 1.
Obviously, the rotor current at the reclosing instant has an important effect on
impact degree for reswitching on the induction motors, which can be illustrated
as follows. When the stator is disconnected from source, the rotor current plays a
role of excitation current to produce air-gap flux and then to induce 3-phase
voltages in stator winding. At the reclosing instant, if those induced voltages are
opposite to source voltages, the surge current is more serious than the starting
situation, which can be diminished by the following methods:
(i) Monitor the stator voltage phase-angles at the reclosing instant.
(ii) Use the extinguishing-flux method to make rotor current decay quick at the
disconnecting instant. Of course, the 3-phase flux-extinguishing resistors are
connected only in stator circuit for squirrel-cage type.
In the following, we analyze the transient process after the extinguishing resistors
are connected in stator circuit.
5.4.3
Transients of Induction Motors after Flux-extinguishing
Resistors are Connected in Stator Circuit
When motor stator is disconnected from source and then 3-phase resistors RL
are connected in stator circuit to extinguish flux, the stator resistance r is changed
to (r RL ) and there are uD 0 and uE 0. According to (5.4.7), the
corresponding matrix equation is
ª0º
«0»
« »
«0»
« »
¬0¼
384
ª xss 0 X m
« 0 x
0
ss
«
« X m 0 X rr
«
¬ 0 Xm 0
0 º ª iD º
0
0
ª 0
« »
»
«
X m » « iE »
0
0
0
Zr «
« 0
Xm
0 » « iD r »
0
» « »
«
X rr ¼ «¬iE r »¼
¬ X m 0 X rr
0 º ª iD º
« »
0 »» « iE »
X rr » « iD r »
»« »
0 ¼ «¬iE r »¼
5 Analyses of Some Operating Modes of Induction Machine Systems
ª(rs
«
«
«
«
¬
RL )
0
(rs
0
0
0
0 0 º ª iD º
« »
RL ) 0 0 »» « iE »
0
Rr 0 » « iD r »
»« »
0
0 Rr ¼ ¬«iE r ¼»
(5.4.36)
or
ª iD º
« »
d « iE »
dt «iD r »
« »
«¬iE r »¼
0
ª xss
« 0
xss
«
«Xm 0
«
¬ 0 Xm
ª rs RL
«
0
u«
«
0
«
¬ (1 s ) X m
Xm
0
1
0 º
X m »»
0 »
X rr
»
0 X rr ¼
0
rs RL
(1 s ) X m
0
0
0
Rr
(1 s ) X rr
0
º ª iD º
»«i »
0
» « E » (5.4.37)
(1 s ) X rr » « iD r »
»« »
Rr
¼ ¬«iE r ¼»
Initial slip s0 is known, stator current initial values are iD 0 0 and iE 0 0, and
rotor current initial values iD r 0 and iE r 0 are still calculated by (5.4.32).
Having the relations above, transients after connecting resistors in stator circuit
can be evaluated as follows:
(i) According to iD 0 0, iE 0 0, and iD r 0 , iE r 0 , s0 and state equation (5.4.37),
by using a certain numerical method, estimate stator and rotor current components
iD ('T ), iE ('T ), iD r ('T ) and iE r ('T ) at t 'T .
(ii) On the basis of (5.4.24), electromagnetic torque at t 'T is
Te ('T )
X m [iE ('T )iD r ('T ) iD ('T )iE r ('T )]
(iii) According to (5.4.25) and s0 , Tm (0), and Te (0) 0, by a certain numerical
method, evaluate rotor slip s ( 'T ) at t 'T .
Following the calculation procedures above, the evaluation steps for t 2'T are:
(i) According to iD ('T ), iE ('T ), iD r ('T ), iE r ('T ), s('T ), and state equation
(5.4.37), in the same way, evaluate stator and rotor current components iD (2 'T ),
iE (2'T ), iD r (2'T ) and iE r (2'T ) at t 2'T .
(ii) On the basis of (5.4.24), electromagnetic torque at t 2'T is
Te (2'T )
X m [iE (2'T )iD r (2'T ) iD (2'T )iE r (2'T )]
(iii) According to (5.4.25) and s ('T ), Tm ('T ) and Te ('T ), by using the same
method, get rotor slip s (2'T ) at t 2'T .
385
AC Machine Systems
In so doing, iterating calculation continues until the reclosing instant t k 'T ,
in which rotor currents iD r ( k 'T ), iE r (k 'T ), and rotor slip s (k 'T ) are the initial
values for the reclosing calculation, i.e., iD r 0 iD r (k 'T ), iE r 0 iE r (k 'T ), s0 s(k 'T ),
iD 0 0 and iE 0 0 . On the basis of those initial values and state equation (5.4.33),
by using the numerical method above, calculate transients after reclosing.
Evidently, the calculation of flux-extinguishing process has to link with that of
the reclosing process to identify what kind of resistors should be selected and
how the flux-extinguishing resistors influence surge currents after reclosing.
5.5 Self-Excitation when Connecting Induction Motor in
Series with Capacitance and Counting the Inertia
Effect in
In Section 3.4, we assumed speed to be constant to analyze the self-excitation of
synchronous machine in series with capacitance, i.e. machine’s inertia constant
to be infinite. Under that assumption, analysis and calculation are all simple. In
addition, the results are accurate enough for large inertia constant and stable
source voltage. However, while analyzing the self-excitation of induction motors,
it is necessary to consider variable speed, because induction motors often have
small inertia constant and sometimes there exist self-excitation region near rated
speed, which cannot be reflected by constant speed. In addition, we cannot
process the effect of source voltage and load on self-excitation region by constant
speed. Considering variable speed, we can get accurate results and understand
the influence factors and degree.
Considering the inertia effect, self-excitation of induction motors is a
resonance of electromechanical parameters. When motor parameters, such as
resistance, reactance, inertia constant, etc, are not a good match for network
parameters and series capacitance, a resonance of electromechanical parameters
may occur to bring about self-excitation oscillation of induction motors.
While counting the inertia effect in, the basic equations will be complicated.
For convenience, we use synchronously rotating d c , qc , 0 axes in Section 3.1
and the corresponding small perturbation equation to analyze that problem.
5.5.1
Basic Equations of Induction Machines when Counting
the Inertia Effect in
An induction machine is in series with capacitance and then connected to electric
source, whose scheme can be reduced as in Fig. 5.5.1.
386
5 Analyses of Some Operating Modes of Induction Machine Systems
Figure 5.5.1 One-line diagram for motor-capacitance-network
For comparison with the analysis of synchronous machine in series with
capacitance stated in Section 3.4, we still use generator convention to write the
following voltage relations:
ua
ub
uc
xC ½
ia
p °°
x °
ubc C ib ¾
p °
xC °
ucc ic °
p ¿
uac (5.5.1)
in which
ua , ub , uc are several phase voltages of electric source, respectively,
uac , ubc , ucc , are phase terminal voltages of machine, respectively,
ia , ib , ic are several phase currents of stator, respectively,
xC is series capacitance.
Converting them into d c , qc components, there are
udc
uqc
½
§i ·
2 ª§ ia ·
)°
xC «¨ ¸ cos(t J 0 ) ¨ b ¸ cos(t J 0 120e
3 ¬© p ¹
© p¹
°
°
º
§i ·
°
)»
¨ c ¸ cos(t J 0 120e
© p¹
°°
¼
¾
§ ib ·
2 ª§ ia ·
c xC «¨ ¸ sin(t J 0 ) ¨ ¸ sin(t J 0 120e
) °°
uqc
3 ¬© p ¹
p
© ¹
°
°
º
§ ic ·
)»
¨ ¸ sin(t J 0 120e
°
© p¹
°¿
¼
c udc
(5.5.2)
Because
387
AC Machine Systems
§ ia ·
¨ ¸ cos(t J 0 )
© p¹
§i ·
1 j( t J 0 )
e j(t J 0 ) ] ¨ a ¸
[e
2
© p¹
½
1­ 1
1
[ia e j( t J 0 ) ] [ia e j(t J 0 ) ]¾
®
2¯p j
p j
¿
1
{( p j)[ia e j(t J 0 ) ]
2( p 2 1)
( p j)[ia e j(t J 0 ) ]}
§i ·
sin(t J 0 ) ¨ a ¸
© p¹
§i ·
1 j( t J 0 )
[e
e j( t J 0 ) ] ¨ a ¸
j2
© p¹
½
1­ 1
1
[ia e j( t J 0 ) ] [ia e j(t J 0 ) ]¾
®
j2 ¯ p j
p j
¿
1
{( jp 1)[ia e j(t J 0 ) ]
2( p 2 1)
( jp 1)[ia e j(t J 0 ) ]}
(5.5.3)
there are
pudc uqc
c uqc
c pudc
2 xC
1
{( p 2 1)
3 2( p 2 1)
u [ia e j(t J 0 ) ib e j(t J 0 120e) ic e j( t J 0 120e) ]
( p 2 1)[ia e j( t J 0 ) ib e j( t J 0 120e)
ic e j( t J 0 120e) ]}
x
c uqc
c C [2ia cos(t J 0 )
pudc
3
2ib cos(t J 0 120e
)
2ic cos(t J 0 120e
)]
c uqc
c xC idc
pudc
puqc udc
c udc
c puqc
(5.5.4a)
2 xC
1
{ j(p 2 1)[ia e j( t J 0 )
3 2( p 2 1)
ib e j(t J 0 120e) ic e j( t J 0 120e) ]
j( p 2 1)[ia e j(t J 0 ) ib e j(t J 0 120e) ic e j( t J 0 120e) ]}
x
c udc
c C [2ia sin(t J 0 ) 2ib sin(t J 0 120e
puqc
)
3
2ic sin(t J 0 120e
)]
c udc
c xC iqc
puqc
(5.5.4b)
388
5 Analyses of Some Operating Modes of Induction Machine Systems
Referring to Section 3.1, there are
c
udc
c
uqc
p\ dc \ qc ridc ½°
¾
p\ qc \ dc riqc °¿
(5.5.5)
so
pudc uqc
puqc udc
( p 2 1)\ dc 2 p\ qc (rp xC )idc riqc ½°
¾
( p 2 1)\ qc 2 p\ dc (rp xC )iqc ridc °¿
(5.5.6)
Stator flux-linkage relations are
\ dc
\ qc
[cos 'J x( p ) cos 'J sin 'J x( p )sin 'J ]idc ½
[cos 'J x( p)sin 'J sin 'J x( p) cos 'J ]iqc °°
[sin 'J x( p ) cos 'J cos 'J x( p )sin 'J ]idc ¾°
[sin 'J x( p)sin 'J cos 'J x( p) cos 'J ]iqc °¿
(5.5.7)
in which 'J is the included angle between rotor axis and synchronous axis,
'J
t
³ 'Z d t ,
0
where 'Z
Z 1 is the difference between rotor speed and
synchronous speed.
x( p ) is the operational reactance of induction machines, whose equivalent
circuit is shown in Fig. 5.5.2.
Figure 5.5.2 Equivalent circuit of operational reactance for induction machines
Referring to Section 5.2, there is
x( p )
xss xcT0 p
1 T0 p
in which
xss is stator reactance when rotor circuit is open,
xss xs X m , where xs is stator leakage reactance per phase and X m is
magnetization reactance of induction machine,
X rr
T0 is time constant of rotor winding, T0
,
Rr
389
AC Machine Systems
xc is transient reactance of induction machines, whose relation to x( p ) is
xc [ x( p)] p
f
and there is
xc
§
X m2 ·
xss ¨1 ¸ W xss
xss X rr ¹
©
in which
X rr is rotor reactance when stator circuit is open,
X rr
X r X m , and X r is rotor leakage reactance,
W is leakage flux coefficient between stator and rotor windings, and there is
W 1
X m2
xss X rr
While counting the inertia effect in, we have to use the rotor motion equation
and the torque formula,
iqc\ dc idc\ qc ½
°
¾
d 2J
H 2 Tm Te °
dt
¿
Te
(5.5.8)
Equations (5.5.6), (5.5.7) and (5.5.8) constitute the basic equations for the
induction machine in series with capacitance to be connected to electric source,
in order to analyze self-excitation.
5.5.2
Basic Relations of Induction Machines in Steady-state
Assume source voltages to be
ua
ub
uc
U sin(t M )
½
°
U sin(t M 120e
)¾
U sin(t M 120e
) °¿
(5.5.9)
Converting them into udc and uqc components, there are
udc
uqc
in which T 0
390
M J 0.
U sin(M J 0 ) U sin T 0 ½
U cos(M J 0 ) U cosT 0 ¾¿
(5.5.10)
5 Analyses of Some Operating Modes of Induction Machine Systems
Referring to Section 3.1 and converting (5.5.7) into complex forward component
Fc with synchronous constant speed, the relation between flux-linkage and
current is
e j'J x( p )e j'J iFc
\ Fc
(5.5.11)
in which
1
\ Fc
2
1
iFc
2
(\ dc j\ qc )
(idc jiqc )
When steady-state operation is at constant speed Z r
t
t
'J
³ (Z
\ Fc
e jst x( p )e jst iFc
0
r
1)dt
³ sdt
0
1 s, there is
st
(5.5.12)
x( p js )iFc
(5.5.13)
so
or
\ dc j\ qc
x( p js )(idc jiqc )
[ X RP ( p ) j X IP ( p)](idc jiqc )
(5.5.14)
in which
X RP ( p )
X IP ( p )
1
½
[ x( p js ) x( p js )] is the real part of x(p + js )
°°
2
¾
1
[ x( p js ) x( p js )] is the imaginary part of x ( p j s ) °
j2
°¿
(5.5.15)
Because
x ( p js )
xss xcT0 ( p js)
1 T0 ( p js )
xss xc( sT0 )2 ( xss xc)T0 p xcT02 p 2
(1 T0 p) 2 ( sT0 ) 2
( xss xc) sT0
j
(1 T0 p ) 2 ( sT0 )2
391
AC Machine Systems
there are
X RP ( p )
X IP ( p )
xss xc( sT0 )2 ( xss xc)T0 p xcT02 p 2 ½
°
(1 T0 p) 2 ( sT0 ) 2
°
¾
( xss xc) sT0
°
°¿
(1 T0 p) 2 ( sT0 )2
(5.5.16)
According to (5.5.14), there exist
\ dc
\ qc
X RP ( p)idc X IP ( p)iqc ½°
¾
X IP ( p )idc X RP ( p )iqc °¿
(5.5.17)
Solving (5.5.6), (5.5.10) and (5.5.17) simultaneously, there are
[( p 2 1) X RP ( p ) 2 pX IP ( p ) ½
°
xC rp ]idc [( p 2 1) X IP ( p ) 2 pX RP ( p ) r ]iqc °
¾
pU cosT 0 U sin T 0 [( p 2 1) X IP ( p ) 2 pX RP ( p) °
r ]idc [( p 2 1) X RP ( p ) 2 pX IP ( p ) xC rp ]iqc °¿
pU sin T 0 U cosT 0
During asynchronous operation in steady-state, namely p
above can be reduced to
U cosT 0
U sin T 0
[ X ( s ) xC ]idc [r R( s ) ]iqc ½°
¾
[r R( s ) ]idc [ X ( s ) xC ]iqc °¿
(5.5.18a)
0, the equations
(5.5.18b)
where
X (s)
R( s )
xss xc( sT0 )2 ½
°
1 ( sT0 )2
°
¾
( xss xc)( sT0 ) °
X IP (0)
1 ( sT0 ) 2 °¿
X RP (0)
(5.5.19)
Solving simultaneous equations (5.5.18b), there exist
idc
iqc
[r R( s ) ]U sin T 0 [ X ( s ) xC ]U cosT 0 ½
°
[r R( s ) ]2 [ X ( s ) xC ]2
°
¾
[r R( s ) ]U cosT 0 [ X ( s ) xC ]U sin T 0 °
°
[r R( s ) ]2 [ X ( s ) xC ]2
¿
(5.5.20)
According to (5.5.17), the stator flux-linkages at asynchronous operation in
steady-state are
392
5 Analyses of Some Operating Modes of Induction Machine Systems
\ dc
\ qc
X ( s ) idc R( s ) iqc ½°
¾
R( s ) idc X ( s ) iqc °¿
(5.5.21)
From the results above, it is clear that the currents and flux-linkages at
steady-state asynchronous operation are constants if using synchronously rotating
d c , qc axes, thus making the study of self-excitation convenient.
5.5.3
Small Perturbation Equations of Induction Machines
When an induction machine is operating in steady-state and then is perturbed,
there is a small increment for each quantity. After small perturbation, there is
J
t
³ Z dt J
0
t
³ (Z
0
0
r
'Z r )dt J 0
t
Z r t J 0 ³ 'Z r dt
0
(1 s )t J 0 D
(5.5.22)
where
Zr
D
1 s
½°
¾
³ 0'Zr dt °¿
(5.5.23)
t
Z r is rotor speed before perturbation, and D is the increment of rotor positionangle, so
'J
t
t
t
0
0
0
³ 'Z dt ³ (Z 1)dt ³ (Z
st ³ 'Z dt st D
r
'Z r 1)dt
t
0
r
(5.5.24)
According to (5.5.11), the relation after small perturbation is
\ Fc 0 '\ Fc
e j( st D ) x( p)e j( st D ) (iFc 0 'iFc )
e jD x( p js )e jD (iFc 0 'iFc )
(5.5.25)
in which
\ Fc 0 and iFc 0 are steady-state flux-linkages and current before perturbation,
respectively,
'\ Fc and 'iFc are flux-linkage and current increments after perturbation,
respectively.
Angle D is a small increment, so there exists
e r jD
cos D r jsin D | 1 r jD
393
AC Machine Systems
According to (5.5.25), we can get
\ Fc 0 '\ Fc
(1 jD ) x( p js )(1 jD )(iFc 0 'iFc )
(1 jD ) x( p js )(iFc 0 jD iFc 0 'iFc jD'iFc )
x( p js )(iFc 0 jD iFc 0 'iFc jD'iFc )
jD x( p js )(iFc 0 jD iFc 0 'iFc jD'iFc )
(5.5.26)
In the light of (5.5.11), the relation before perturbation is
\ Fc 0
x( p js )iFc 0
x( js )iFc 0
(5.5.27)
Neglecting the second-order small increment, (5.5.26) can be rewritten as
'\ Fc
x( p js )'iFc jx( p js )D iFc 0 jD x( p js )iFc 0
x( p js )'iFc jiFc 0 x( p js )D jD x( js )iFc 0
(5.5.28)
namely
'\ dc j'\ qc
[ X RP ( p ) jX IP ( p)]('idc j'iqc )
j(idc 0 jiqc 0 )[ X RP ( p ) jX IP ( p)]D
j(\ dc 0 j\ qc 0 )D
(5.5.29)
Separating the real part from the imaginary part, there are
'\ dc
X RP ( p )'idc X IP ( p)'iqc
½
°
[\ qc 0 idc 0 X IP ( p) iqc 0 X RP ( p)]D °
¾
X IP ( p)'idc X RP ( p )'iqc
°
[\ dc 0 idc 0 X RP ( p) iqc 0 X IP ( p)]D °¿
'\ qc
(5.5.30)
In accordance with (5.5.6), the voltage equations after small perturbation are
p (udc 0 'udc ) (uqc 0 'uqc )
½
°
( p 1)(\ dc 0 '\ dc ) 2 p (\ qc 0 '\ qc ) °
(rp xC )(idc 0 'idc ) r (iqc 0 'iqc ) °°
¾
p (uqc 0 'uqc ) (udc 0 'udc )
°
( p 2 1)(\ qc 0 '\ qc ) 2 p(\ dc 0 '\ dc ) °
°
(rp xC )(iqc 0 'iqc ) r (idc 0 'idc )
°¿
2
and the relations before perturbation are
pudc 0 uqc 0
puqc 0 udc 0
394
( p 2 1)\ dc 0 2 p\ qc 0 (rp xC )idc 0 riqc 0 ½°
¾
( p 2 1)\ qc 0 2 p\ dc 0 (rp xC )iqc 0 ridc 0 °¿
5 Analyses of Some Operating Modes of Induction Machine Systems
Subtracting the second set of formulas from the first set above, the increment
relations are
p'udc 'uqc
p'uqc 'udc
( p 2 1)'\ dc 2 p'\ qc (rp xC )'idc r 'iqc ½°
¾
( p 2 1)'\ qc 2 p'\ dc (rp xC )'iqc r 'idc °¿
(5.5.31)
According to (5.5.8), in a similar way, we can obtain
'Te
2
\ dc 0 'iqc iqc 0 '\ dc \ qc 0 'idc idc 0 '\ qc ½°
Hp D Ha0
'Tm 'Te
¾
°¿
(5.5.32)
where a0 is angular acceleration before perturbation.
Formulas (5.5.30), (5.5.31) and (5.5.32) constitute the relations of several
increments after small perturbation, which can be used to study the change
condition of increments.
5.5.4
Determination of Self-excitation Region for Induction
Machines when Counting the Inertia Effect in
For an induction machine in series with capacitance and not in good match with
the electromechanical parameters, self-excitation may occur. If we connect the
series capacitance before starting the motor, self-excitation may occur in the starting
process, thus causing static self-excitation around a certain low speed and not
reaching the normal speed. If we connect the series capacitance at rated speed,
self-excitation may also cause the motor return to a certain lower speed to bring
about static self-excitation. In some special cases, self-excitation may occur in the
starting process, but the motor still passes through the self-excitation region and
finally reaches the normal speed. When the load, connected to the network behind
capacitance, increases to a large value, self-excitation may occur at high speed.
In the starting process or in the sudden connection of series capacitance or
during a certain large load behind capacitance, the self-excitation is always formed
gradually. Therefore, whether self-excitation happens for a certain speed can be
examined by applying a small perturbation to the state before self-excitation. In
the starting process or in the sudden connection of series capacitance, there exists
a transient component that decays quickly. Especially in the starting process,
self-excitation is formed when the rotor speed increases to a certain value. Similarly,
when the load behind capacitance increases to a certain value, self-excitation
happens under the original static operation condition and at a perturbation. Thus,
it is accurate enough to consider the motor at steady-state operation before
self-excitation, that is to say, the small perturbation equations deduced from the
original static asynchronous operation, as stated before, are all suitable for those
395
AC Machine Systems
conditions above. Obviously, if several increments 'idc , 'iqc and D approximate
to zero after a small perturbation, self-excitation does not occur; otherwise it may
occur. Therefore, according to the roots of the characteristic equation for the small
perturbation relations as before, we can ascertain whether self-excitation occurs
or not. For convenience, the beginning time of small perturbation is selected so
that idc 0 0. In accordance with the choice above and (5.5.20) and (5.5.21), the
corresponding values iqc 0 , \ dc 0 and \ qc 0 are
U
iqc 0
2
[r R( s ) ] [ X ( s )
\ dc 0
\ qc 0
When taking idc 0
R( s ) iqc 0
X ( s ) iqc 0
½
°
xC ] °
°
¾
°
°
°
¿
2
(5.5.33)
0 as the beginning instant, there is
ia 0
idc 0 cos(t J 0 ) iqc 0 sin(t J 0 ) i0
iqc 0 sin(t J 0 )
If electric source is infinite, after small perturbation, there exist
'udc
'uqc
0½
0 ¾¿
(5.5.34)
On the basis of (5.5.30), (5.5.31), (5.5.32), (5.5.33) and (5.5.34), there are
0
Z1 ( p )'idc Z 2 ( p )'iqc Z A ( p)iqc 0D
0
Z 2 ( p )'idc Z1 ( p )'iqc Z B ( p )iqc 0D
'Tm Ha0
½
°
°
Z 3 ( p)iqc 0 'idc Z C ( p )iqc 0 'iqc ¾°
°
[ Hp 2 iqc2 0 Z 3 ( p)]D
¿
(5.5.35)
where
Z1 ( p)
Z 2 ( p)
Z3 ( p)
Z 4 ( p)
Z A ( p)
Z B ( p)
ZC ( p)
396
(1 p 2 ) X RP ( p ) 2 pX IP ( p ) ( xC rp) ½
°
(1 p 2 ) X IP ( p) 2 pX RP ( p) r
°
°
X ( s ) X RP ( p)
°°
R( s ) X IP ( p)
¾
°
(1 p 2 ) Z 3 ( p) 2 pZ 4 ( p )
°
2
°
(1 p ) Z 4 ( p ) 2 pZ 3 ( p )
°
R( s ) X IP ( p )
°¿
(5.5.36)
5 Analyses of Some Operating Modes of Induction Machine Systems
The characteristic equation of (5.5.35) is
A( p )
Z1 ( p )
Z 2 ( p)
Z A ( p )iqc 0
Z 2 ( p)
Z1 ( p )
Z B ( p )iqc 0
Z C ( p )iqc 0 [ Hp 2 iqc2 0 Z 3 ( p)]
Z 3 ( p )iqc 0
[ Hp 2 iqc2 0 Z 3 ( p )][ Z12 ( p ) Z 22 ( p )]
[ Z 3 ( p ) Z A ( p) Z B ( p ) Z C ( p )]iqc2 0 Z1 ( p )
[ Z 3 ( p ) Z B ( p ) Z A ( p ) Z C ( p )]iqc2 0 Z 2 ( p )
0
Dividing the two sides of the equation above by iqc2 0 , the characteristic equation
can be changed to another form
A( p)
ªH 2
º 2
2
« 2 p Z 3 ( p) » [ Z1 ( p ) Z 2 ( p )]
«¬ iqc 0
»¼
[ Z 3 ( p) Z A ( p ) Z B ( p) Z C ( p )]Z1 ( p)
[ Z 3 ( p) Z B ( p ) Z A ( p ) Z C ( p)]Z 2 ( p )
0
(5.5.37)
In the light of (5.5.37), whether self-excitation occurs for a rotor speed Z r can
be judged. However, it is the twelfth order equation, whose solutions are tedious.
In practice, according to concrete topics, select appropriate criterion. In general,
it is convenient to use the Mihainov’s method or the D-domain partition method.
5.5.5
Effect of Inertia Constant on Self-excitation Region
Motor inertia constant H has obvious effect on the self-excitation region under
some conditions. Especially for small inertia constant, if assuming rotor speed to
be constant without mechanical oscillation to study self-excitation, the results are
sometimes quite different from experimental data.
Considering the inertia effect and analyzing the self-excitation region on the
basis of (5.5.37), it is convenient to use the D-domain partition on single
complex parameter plane. Thus, (5.5.37) can be reformed to
HR( p) Q( p )
A( p )
H
namely
in which
R( p)
1
2
qc 0
i
0
Q( p)
R( p)
[ Z12 ( p ) Z 22 ( p )] p 2
(5.5.38)
(5.5.39)
(5.5.40)
397
AC Machine Systems
Q( p) [ Z12 ( p ) Z 22 ( p )]Z 3 ( p )
[ Z 3 ( p) Z A ( p ) Z B ( p ) Z C ( p )]Z1 ( p )
[ Z 3 ( p) Z B ( p ) Z A ( p ) Z C ( p )]Z 2 ( p )
Let p
(5.5.41)
jk , there is
H ( jk )
Q ( jk )
R jk
U ( k ) jV ( k )
(5.5.42)
According to (5.5.42), use the D-domain partition on single complex parameter
plane to determine inertia constants at the self-excitation region boundary points
for a certain speed Z r . During the calculation, take k f ~ f to get a series
of U (k ) and V (k ) values and then to plot the projection curve of the
characteristic root plane imaginary axis on to U - V plane, namely H complexplane, which is the D-domain partition boundary on H-plane. A 1.6kW-motor
has the following parameters: r 0.167 1, xs 0.138 7, X r 0.078 3, X m 1.720 ,
Rr 0.056 and series capacitance xc 0.1086. At rotor speed Z r 2 880rpm,
estimate the D-domain partition boundary curve as shown in Fig. 5.5.3. Since the
Figure 5.5.3 D-domain partition boundary curve for an induction motor in series
with capacitance
398
5 Analyses of Some Operating Modes of Induction Machine Systems
D-domain partition boundary curve is symmetrical to the real axis, namely the
U-axis, it is sufficient to make the calculation for k 0 ~ f only. According to
these evaluation results, draw a half of boundary curve and then plot the remainder
curve on the basis of symmetrical relation. In the light of the self-excitation features
of induction machines, it is enough to take k 0.1 ~ 1.0 for practical estimation.
After having the D-domain partition boundary curve, move along the curve
from k f to k f and plot shade lines on the left side of the curve.
Obviously, if we pass through the D-domain partition boundary curve on the
H-plane from the shade-line side to the no shade-line side, then a characteristic
root passes through the imaginary axis from left half plane to right half plane on
the characteristic equation root plane. The next step is to partition and mark
regions in the following way. When passing through the D-domain partition
boundary curve once from the no shade-line side to the shade-line side, region
grade will be raised once. According to the principle, region ċin Fig. 5.5.3 has
the highest grade and is certainly a stable region.
It should be pointed out that the highest grade region is a possible region
without self-excitation but sometimes it does not exist, so it is necessary to check
whether the highest grade region is a stable region. However, H value for stable
operation can be estimated practically to save this check. For example, during
H f, there is no self-excitation region at high speed, and for small inertia
constant H, there is no self-excitation region at low speed.
During V (k ) 0 in (5.5.42), there is H ( jk ) U (k ) real number. Thus, when
the inertia constant H is equal to U (k ) corresponding to V (k ) 0, there exists
Q ( jk )
HR( jk ), namely
HR( jk ) Q( jk )
A( jk )
0
(5.5.43)
which indicates that the characteristic equation A( p ) 0 has a pair of imaginary
roots r jk for the inertia constant stated above that corresponds to the boundary
of the self-excitation region. According to Fig. 5.5.3, U (k ) corresponding to
H
124
V (k ) 0 is 124 per-unit, namely
0.395s. Therefore, during inertia
314 314
H
! 0.395s and at speed Z r 2880 rpm or s 0.04, self-excitation
constant
314
does not occur because region ċhas the highest grade, which coincides with
the estimation result that during infinite inertia constant there is no self-excitation
region at high speed.
On the basis of motor parameters and series capacitance xc 0.1086, making
similar calculation at other speeds, the self-excitation region can be obtained as
shown in Fig. 5.5.4, in which sign “ ” indicates the experimental datum-points
H
0.354s or 1.07s at Z r 2880 rpm, thus calculation results
for
314
approximating experimental data.
ƻ
399
AC Machine Systems
Figure 5.5.4 Self-excitation region for an induction motor in series with capacitance
From (5.5.43) it is evident that k-value corresponding to self-excitation boundary
point is stator self-excitation current frequency in d c , qc axes at that speed. If m
indicates stator self-excitation current frequency in a, b, c axes, there is m 1 k .
Calculation results and test data all show that m has a range of 0.1 1.0, by
which we get the corresponding k to make estimation. As for rotor self-excitation
current frequency n, there are
n
(1 k ) Z r
sk
(5.5.44)
and
k 1 m
(5.5.45)
Thus, k-value represents rotor mechanical oscillation frequency or pulsating
torque frequency.
H
When
0.395s in Fig. 5.5.3, the corresponding k-value is 0.276, so there
314
exist stator self-excitation current frequency (1 0.276) u 50 36.2Hz and rotor
mechanical oscillation frequency 0.276 u 50 13.8Hz during static self-excitation
H
0.395s.
at Z r 2880rpm and
314
Considering the inertia influence, the calculation of the self-excitation region for
induction machines is complicated, so it is appropriate to use computers. For
accuracy of evaluation, interval 'k is taken as a small value, such as 'k 0.001
or 0.0001. In the calculation results of H ( jk ) U (k ) jV (k ), only those data
corresponding to U (k ) positive and V (k ) 0 are useful, and the other data
need not be printed. While studying the self-excitation regions of induction motors,
several speeds in the starting process have to be examined, in which rotor slip s
has a range of 0 1.0 and interval 's is taken as 0.05 or 0.01 to suffice for demands.
Figures 5.5.5 to 5.5.7 show the calculation results of the self-excitation regions
400
5 Analyses of Some Operating Modes of Induction Machine Systems
for a 1.6kW-induction motor in series with capacitances xc 0.04, 0.07 and 0.108 6,
respectively. From those curves we can see the important properties, i.e. there are
two self-excitation regions during lower compensation degree, the two regions
are broadened with the increase of the compensation degree, and the two regions
are to be connected to form a self-excitation region when the compensation
degree exceeds a certain value. If assuming rotor speed to be constant to analyze
the self-excitation region, only one self-excitation region exists no matter what
compensation degree there is, thus bringing about serious error sometimes.
Figure 5.5.5 Self-excitation region for an induction motor in series with capacitance
F c 0.04
Figure 5.5.6 Self-excitation region for an
induction motor in series with capacitance
xc 0.07
Figure 5.5.7 Self-excitation region for an
induction motor in series with capacitance
xc 0.108 6
401
AC Machine Systems
5.6
Dual-Stator-Winding Multi-Phase High-Speed
Induction Generator
As an important part of ship’s integrated power systems, the dual-stator-winding
multi-phase high-speed induction generator system with a rectifier load is
characterized by high power density, good electric supply quality, high mechanical
strength, reliable operating performance, simple maintenance, and convenient
adjustment. Its main shortcoming of low compensation ability for the induction
generator may be overcome by setting control winding in the generator. Therefore,
the induction generator system represents the development tendency of the power
generation sector in ship’s integrated power systems.
To meet the requirements of minimizing the machine size, improving the
mechanical strength, reducing the noise, and enhancing the electromagnetic
compatibility, the stator of this new induction generator has two embedded
windings. These windings include a 12-phase power winding connected to a
rectifier load and a 3-phase control winding connected to a static excitation
regulator, respectively. It has a squirrel-cage solid rotor. Also, there are the
capacitors for self-excitation and the inter-phase reactors in the system for
performance improvement.
It is very difficult to analyze the dual-stator winding multi-phase high-speed
induction generator with a rectifier load by general methods owing to the system’s
complicated nonlinear magnetic/electric circuits, multi-loop variable-topology
configurations, existence of capacitors for self-excitation and inter-phase reactors,
and abrupt variation of loop currents. From the point of analyzing the mmf of the
induction generator, a new idea which divides the whole system into two
sub-systems is presented. The circuit method and the field finite element method
are adopted to analyze the two sub-systems which are linked only by appropriate
terminal variables.
5.6.1
Construction of New Induction Generator
Figure 5.6.1(a) is a connection sketch of the 12-phase winding with a rectifier
load of the dual-stator-winding induction generator. The 12-phase power winding
supplies active power to the load through a bridge rectifier. The 12-phase winding
consists of four Y-connection windings shifted by 15eeach, electrical angle with
independent neutral points, which are represented by a1b1c1, a2b2c2, a3b3c3, and
a4b4c4, where phase a2 leads ahead of phase a1 by 15eelectrical angle, the same
is true of phase a3 and a2 or phase a4 and a3. C and LB are self-excitation
capacitance and inter-phase inductance, respectively, and RL is the load
resistance.
402
5 Analyses of Some Operating Modes of Induction Machine Systems
Figure 5.6.1 Dual-stator-winding system of new induction generator
(a) Power winding with a rectifier load; (b) Control winding with an excitation regulator
Figure 5.6.1(b) is a connection sketch of the 3-phase control winding with a
static excitation regulator. The 3-phase Y-connection winding is used to regulate
the reactive power of the generator for keeping the stability of the rectified load
voltage. Control winding phase A leads ahead of power winding phase a1 by 22.5e
electrical degrees.
Solid cage structure is adopted for the high-speed rotor, which also complicates
the structure of the induction generator.
There exist many difficulties in analyzing such a complicated new induction
generator system. Firstly, the phasor cannot be used to analyze the generator
system with nonlinear and unsymmetrical circuits, and complete decoupling of
the system cannot be achieved by coordinates transformation. Secondly, it is
difficult to use the mode classification method in the system analysis because the
system operation modes are more than one hundred under different turn-on and
turn-off conditions of diodes in rectifier bridges. Thirdly, the existence of
capacitors for self-excitation and inter-phase reactors increases the order of the
circuit system that makes the system model more complex. Finally, it needs to
use the electromagnetic field method instead of the equivalent circuit method to
deal with the solid cage structure.
5.6.2
MMF of Multi-Phase Induction Generator
The expression for a desired harmonic air-gap mmf due to a specified harmonic
power winding current is obtained with the current resolved into fundamental
and harmonic components; consequently, the superposition method is employed
to perform the mmf analysis.
If the order of the harmonic current is denoted by P ( P 1, 5, 7, "), the 12-phase
power winding currents decomposed by the Fourier series are written as
403
AC Machine Systems
iak (t )
¦
ibk (t )
¦
ick (t )
¦
ʌº
½
ª
2 I P sin P «Z t (k 1) »
°
12
¬
¼
°
2ʌ
ʌ º°
ª
2 I P sin P «Z t (k 1) » ¾
3
12 ¼ °
¬
4ʌ
ʌ º°
ª
2 I P sin P «Z t (k 1) » °
3
12 ¼ ¿
¬
(5.6.1)
where k 1, 2,3, 4.
With the space coordinates origin located at the axis of phase a1, the pulsating
harmonic mmfs of order Q (Q r1, r 5, r 7, ") due to Q - th harmonic current are
f akQ ( P )
f bkQ ( P )
f ckQ ( P )
ʌº
ʌº
½
ª
ª
FQ ( P ) sin P «Z t (k 1) » cosQ «D (k 1) »
°
12 ¼
12 ¼
¬
¬
°
2ʌ
ʌº
2ʌ
ʌ º°
ª
ª
FQ ( P ) sin P «Z t (k 1) » cosQ « a (k 1) » ¾
3
12 ¼
3
12 ¼ °
¬
¬
4ʌ
ʌº
4ʌ
ʌ º°
ª
ª
FQ ( P ) sin P «Z t (k 1) » cosQ « a (k 1) » °
3
12 ¼
3
12 ¼ ¿
¬
¬
(5.6.2)
Each phase pulsating mmf is resolved into two rotating mmfs by triangular
identity and the sum of 12-phase mmfs yields the resulting rotating mmf as
expressed by
fQ ( P )
6 FQ ( P ) sin( PZ t QD )
(5.6.3)
Let order ȝ be 1for the fundamental component of the current, and order Ȟ be 1,
5, 7, 11, 13, 17, 19, 23, 25, " , respectively, for all mmf components,
substituting them into (5.6.2), the resulting rotating mmfs yield that f1(1), f23(1),
f25(1), " are of determined values with f5(1), f7(1), f11(1), f13(1), f17(1), f19(1), " being
equal to zeros. Repeating the above process for 5, 7, 11, 13, 17, 19, 23, 25 harmonic
components of the current, " , respectively, the corresponding resulting rotating
mmfs are obtained for various harmonic components of the current.
In general, for symmetrical 12 phase winding, the relation between ȝ and Q is
Q
24k P
k
0, r 1, r 2, r 3, "
(5.6.4)
The relation between the harmonic order of current and mmf of the 12-phase
winding is different from that of the 3-phase winding. The space harmonic mmfs
of the 12-phase winding are much less than those of the 3-phase winding when
the same currents flow in them. It is noted that the harmonic mmfs of order 5, 7,
11, 13, 17, 19, " are not produced in the 12-phase winding with only the
fundamental current flowing, no fundamental mmf can be produced with 5 and 7
order currents flowing, etc.
404
5 Analyses of Some Operating Modes of Induction Machine Systems
Under the condition of the power winding current being rectangular wave and
the control winding current being sinusoidal wave, it is known from mmf amplitude
calculation that any other harmonic mmf amplitudes are less than 5% of the
fundamental mmf amplitude produced by fundamental current. Therefore, only
considering the fundamental mmf produced by the fundamental current in analyzing
the generator performance is generally permitted.
With only the fundamental mmf produced by the fundamental current considered,
the induction generator system is divided into two sub-systems, to which the
circuit method and the electromagnetic field method are applied individually. The
system which consists of 12-phase power winding with a rectifier load is calculated
by the circuit method, and then the emf and current of power winding under
constant voltage operation condition are obtained through iterative computation.
The system of dual stator winding and solid cage rotor is analyzed by the
harmonic electromagnetic field finite element method, and the stator frequency
and the control winding current are obtained by the optimization approach.
5.6.3
Circuit Method for the System of Power Winding with
Rectifier Load
The system of the 12-phase power winding with a rectifier load of the induction
generator belongs to circuit network with multi-loop and variable-topology
configuration in essence, and the a, b, c phase coordinates system is used to analyze
it. The circuit transients are calculated by combining circuit topology theory and
loop current method, from which network equations are automatically established
by the systematic approach; finally, the calculation is conducted.
It is easier to deal with inductances and diode branches to form the impedance
matrix by the loop current method; therefore, the method is adopted to analyze
the system owing to the existence of many inductances and diode branches.
It is known from Fig. 5.6.1 that every rectifier bridge system has 16 branches
except load one and 9 independent loops. Since 6 diodes in a rectifier bridge
cannot be turned on simultaneously in ordinary operation, only 0, 2 or 3 diodes
are turned on at the same time; correspondingly, the actual branch numbers are 9,
12, or 13, respectively with the actual loop numbers 3, 4 or 5. The system of
12-phase winding with a rectifier bridge has a total of 36 independent loops with
65 branches. In the actual operation process, the independent loop l and branch b
are less than 36 and 65, respectively, which are relative to the topology configuration.
Suppose that the column vector of loop current is il, and the column vector of
the branch current is i, then the relation of il and i is
i
B T il
(5.6.5)
where the basic loop matrix B is of l u b orders.
405
AC Machine Systems
Based on the relation between voltage and current of inductances and capacitors,
the circuit differential equations are handled by using the backward Euler’s
method. A transient equivalent circuit composed of the resistances, inductances
and capacitances is obtained in the transient analysis.
Suppose that Z is the impedance matrix of b u b orders, us and is are the
independent voltage and current source column vectors of b u 1 orders, respectively,
and u is the branch voltage matrix of b u 1 orders. According to the relation
between variables in the branch and assumed positive directions, there is
u
Zi us Zis
(5.6.6)
Using KVL leads to
Bu
0
(5.6.7)
Based on the above equations, it follows that
BZB T il
BZis Bus
(5.6.8)
The loop current is
il
Zl 1 BZis Z l 1 Bus
(5.6.9)
where loop impedance matrix Zl is of l u l orders.
The loop current il is obtained from equation (5.6.9). Substituting it into
equations (5.6.5) and (5.6.6), the branch current i and branch voltage u will be
obtained.
In system operation, the state variation of diode is determined according to
voltage uD and current iD of the diode. If the voltage uD of a turned off diode is
increased to 0.7V, then the diode becomes to be in a turned on state from its
turned off state, and if the current iD of a turned on diode is decreased to 0A, then
the diode becomes to be in a turned off state from its turned on state. The
variation of the state of the diode in the rectifier bridge changes the system
topology configuration.
Considering the features of the rectifier load, a method combining the fixed
step and varied step is applied in the simulation. The simulation begins from a
topology configuration with fixed step, after every step, the state of the system
topology configuration has to be judged by the calculated result. If no change
takes place, the simulation with fixed step continues, else if the system topology
configuration changes, an accurate time point when the topology configuration
changes has to be found by varying the step.
After getting the values of some variables at different time points, the
fundamental and harmonics of these periodic variables are decomposed by
Fourier transformation. The FFT is adopted in order to save computation time.
Rated power PN of the sample machine is 18.4kW and output dc voltage Udc at
406
5 Analyses of Some Operating Modes of Induction Machine Systems
the rated load is 230V. The rotational speed keeps at 1500rpm, correspondingly, the
normal frequency is 50Hz. Since the stator frequency f1 is very close to normal
frequency in the ordinary operation of the induction generator, f1 is supposed to
be 50Hz in the simulation.
The calculation begins from a supposed Ep, emf of power winding, with known
variables f1 and RL. After the load voltages at different moments are obtained, the
average value Udc' is compared with 230V so that a new Ep is re-supposed. The
final Ep is found when the difference between Udc' and Udc is less than the set
error, from which other variables are also gained.
Figures 5.6.2(a) and (b) are the simulated waves and experimental waves of
phase a1 winding current in a period, respectively, and Figs. 5.6.3(a) and (b) the
corresponding phase voltage waves. The variation trends of simulation waves
and experimental waves in Figs. 5.6.2 and 5.6.3 are basically the same without
distinct difference. This shows that the analysis method is effective.
Figure 5.6.2 Current waves of power winding
(a) Simulated wave; (b) Experimental wave
Figure 5.6.3 Phase voltage waves of power winding
(a) Simulated wave; (b) Experimental wave
407
AC Machine Systems
The calculated value, experimental value and their error of the phase current
fundamental Ip are 16.71A, 16.47A and 1.46%, respectively, those of voltage
fundamental Up 99.76V, 101.1V and 1.33%, and those of power factor angle ijt
19.46e, 19.08eand 1.78%. The calculated fundamental values and the experimental
ones of the variables match well, which validates the method used in calculating
the fundamentals. It also shows that considering only the fundamental mmf
produced by the fundamental current as the analysis foundation is reasonable.
5.6.4
Field Method for the System of Dual-Winding-Stator and
Solid Cage Rotor
According to the positive direction of the power winding variables prescribed by
the generator convention, it is assumed that the phasor of phase a1 emf reaches
its maximum at zero moment, so its initial phase angle is 0 and the initial phase
angle of the current phasor is ijp. Because the excitation regulator is considered
as a pure capacitor when the control winding only sends reactive power to the
system, the phasor of phase A current leads ahead of that of emf by 90ewith the
same generator convention for the positive direction of control winding variables.
Phase A emf lags phase a1 emf by 22.5eowing to the fact that phase A control
winding leads phase a1 power winding by 22.5eelectric degrees in space. The
resultant phasor diagram is shown in Fig. 5.6.4.
Figure 5.6.4 Resultant phasor diagram
Under the condition of known rms value of power winding emf Ep, it is easy to
get that of control winding emf Ec as follows:
Ec
Wc kwc1
Ep
W p k wp1
(5.6.10)
where Wc and kwc1 are the total series turn number per phase and the fundamental
winding factor of the control winding, Wp and kwp1 are those of the power winding.
408
5 Analyses of Some Operating Modes of Induction Machine Systems
The adopted method is carried out in a Cartesian coordinates system. To simplify
the physical model, the assumptions are made as follows: (i) The generator is
infinite in axial length in order to convert the three-dimensional calculation into a
two-dimensional one. (ii) The rotating rotor is replaced by a static one in the
calculation and the current with slip frequency is added on the stator winding.
(iii) The flux density of stator and rotor varies with time sinusoidally. (iv) The
stator slot leakage flux is considered by the stator slot leakage reactance and the
reduction of air-gap fundamental flux is considered by the skew-slot coefficient.
For the two-dimensional sinusoidal electromagnetic field in the rest medium,
the equation with the magnetic vector potential is
w ª 1 wA º w ª 1 wA º
«
» «
»
wx ¬ P wx ¼ wy ¬ P wy ¼
jZV A Js
(5.6.11)
where the magnetic vector potential is of z axial component only and it does not
vary in the axial direction, Ȧ is slip angular frequency, ȝ magnetic permeability,
ı conductivity.
The two terms to the right of equation (5.6.11) are the induced eddy current
density and the source current density, respectively. For the problem to be solved,
the source current only exists in the area of power winding and control winding
of the stator slots, and the eddy current only exists in the area of the conductor in
rotor slots and solid rotor zone.
The boundary condition of the magnetic vector potential is zero around the stator
outer periphery and the rotor inner periphery as well as periodical radial boundary.
The finite element method is used for calculation. At first, the conductivity and
the magnetic permeability of material in different areas are given, and the source
current density is added to the area of power winding and control winding in the
stator slots. In order to produce rotating magnetic field with negative phase
sequence relative to the rotor, the slip frequency of the current should be reverse
to the actual state, e.g. the phase angle of the current in the upper layer of phase
a1 power winding should be – ijp, the phase angle of the current in the upper layer
of phase A control winding should be 67.5e
, then in this condition the calculated
phase angle of emf of phase a1 power winding should be 0. In the calculation, by
using the ANSYS harmonic field, the slip frequency f is a variable to be
determined by iteration.
After the flux density of each node in the air-gap center periphery is gained
through post-treatment of the result by the finite element method, the amplitude
and phase angle of the magnetic density fundamental will be taken out by Fourier
transformation. The amplitude of flux ĭm per pole is calculated from the fundamental
flux density, consequently, the rms value of fundamental emf per phase of the
power winding will be obtained as follows:
E cp
2ʌ f1W p k pw1)m
(5.6.12)
409
AC Machine Systems
On the basis of the waveform of fundamental flux density, the phase angle M 0c
of fundamental emf of power winding phase a1 may be determined as well.
The rms value of control winding current Ic and that of slip frequency f are two
variables to be found. The following two equations need to be satisfied for
finding out the above two variables:
E cp E p ½
¾
M 0c 0 ¿
(5.6.13)
E cp and M 0c are obtained by the finite element method, so it is impossible to
express them by apparent formulae of Ic and f. Since there is no traditional
method to solve the above equations, a multi-variable optimization approach is
used.
This optimization model is without constraint. Selecting Ic and f as optimization
variables, the object function can be written as
f (Ic , f )
( E cp E p ) 2 kM 0c2
(5.6.14)
where k is a positive ratio coefficient.
In reality, the optimization solution of equation (5.6.14) is to get the minimum
of the object function min f(Ic, f). Obviously, the minimum of the object function
is 0, namely min f(Ic, f)) is equivalent to equations (5.6.13).
The direct search method without derivation is applied as a multi-variable
optimization approach; concretely, it is the coordinate alternant method with
good stability and simple programming, and only the axial search motion for the
variables is made. The optimization begins with the initial values of Ic and f. If
the object function no longer decreases at a certain round of the optimization
process, the steps of two variables ǻIc and ǻf should be decreased. If the step is
decreased to reach the prescribed precision, the optimization process ends, and
consequently, Ic and f corresponding to the optimization point will be found.
Final calculated results are obtained by finite element analysis and optimization
approach at the rated load operation with constant voltage. Calculated flux
density amplitude is 0.7092T from which ĭm is calculated, and E cp equals 101.3V
based on equation (5.6.12). Obviously, E cp is equal to Ep, and M 0c to 0, so
equation (5.6.13) is satisfied, which leads to finding Ic and f.
At rated load, the calculated value, experimental value and their error of the
control winding current Ic are 8.38 A, 7.71 A and 8.73%, and those of the slip s
–0.010 96, –0.011 74 and –6.64%, respectively. The calculated results and
experimental ones are in good agreement, which shows that the adopted analysis
method is correct. The reason why the error occurs is due to two points: firstly,
the iron loss has not been calculated. Secondly, the influence of leakage flux
produced by control winding currents on the emf in power winding has not been
considered.
410
5 Analyses of Some Operating Modes of Induction Machine Systems
References
[1] Gao J D (1963) AC machine transients and operating modes analysis, Vol 2 (Chapter 7, 8,
in Chinese). Science Press, Beijing
[2] Gao J D, Zhang L Z, Huang L P (1983) Induction motor reswitching transients and the
methods of restraining the reswitching surge current in an induction motor (in Chinese).
China Symposium on Electric Machines, Shanghai, pp 154 162
[3] Gao J D, Zhang L Z, Huang L P (1984) Study of starting characteristics for induction
motors (in Chinese). J Electrotechnical Journal, Chinese Electrotechnical Society, Beijing
1: 1 6
[4] Gao J D, Wang X H, Li F H (2004) Analysis of ac machines and their systems, 2nd Ed
(Chapter 1, 5, in Chinese). Tsinghua University Press, Beijing
[5] Rustebakke H M, Concordia C (1970) Self-excited oscillations in a transmission system
using series capacitors. J IEEE Trans. PAS-89 (7): 1504 1512
[6] Wang X H, Wu X Z (2008) Research on dual-stator winding multi-phase high-speed
induction generator with rectifier load. J Sci China Ser E-Tech Sci, 51(6): 683 692
[7] Wu X Z,Wang X H (2007) Circuit analysis of power winding with rectifier system for
12-phase induction generator (in Chinese). J CSEE 27(15): 75 82
[8] Wu X Z,Wang X H (2007) Determination of control winding current and stator frequency
for dual stator-winding high-speed induction generator (in Chinese). J CSEE 27(18): 23 29
[9] Wu X Z,Wang X H, Luo C (2005) Relationship between harmonic currents and corresponding
harmonic magnetomotive forces of multi-phase induction machines (in Chinese). J Tsinghua
Univ. (Sci&Techn.) 45(7): 865 868
[10] Zhang B D, Zhang L Z (1988) Some measures to decrease the switching-over surge
current when starting an induction motor and renewal evaluation of several classical starting
mothods (in Chinese). J Beijing Society of Electrical Engineering 3: 27 40
[11] Zhang B D, Zhang L Z, Wang Z R (1988) Analysis and calculation of switching-over
current for starting an induction motor by use of reduction of its stator voltage (in
Chinese). J Beijing Society of Electrical Engineering 3: 11 26
[12] Zhang L Z (1995) Transient theory of induction machines and its uses in some aspects.
Proc CICEM’95, International Academic Publishers, Hangzhou, pp 338 342
[13] Zhang L Z, Cheang T S (1999) Further research on double fed induction motors
controlled by inverters. Proc CICEM’99, International Academic Publishers, Xi’an,
pp 29 32
[14] Zhang L Z, Chok S C, Cheang T F (1995) Internal faults of 3-phase induction motors.
Proc CICEM’95, International Academic Publishers, Hangzhou, pp 323 325
411
6 Internal Asymmetric Analysis of AC Machines
Abstract Internal faults in ac machine stator winding will cause severe
damages to the machine. In order to protect electric machine and lighten the
damage owing to internal fault, the speediness and reliability of relay protection
are very important which demand understanding the currents in electric
machine windings under internal faults clearly.
Under the situation of internal fault of electric machine winding, there are
stronger space harmonics in air-gap magnetic field and strtonger time
harmonics in winding currents, which make symmetric component method
be not suitable for analyzing it. At the same time the ideal electric machine
model is no longer suitable, accordingly, adopting d, q, 0 coordinates system
based on the corresponding model will cause very large errors. Therefore it is
needed to adopt Multi-loop Theory and Method based on single coil as
stated in chapter 1 for analyzing it. In this chapter using the presented
method the rotor failures of bar broking and end-ring broking of induction
machine, the internal faults of stator windings of salient pole synchronous
machine and non-salient pole synchronous machine, etc. are discussed. The
short circuit current of internal fault for stator winding is large, so it is
necessary to pay special attention to the protection. In addition, the transient
ac component is a little more than the steady-state current not just as the
short circuit of external terminals. By the way, the short circuit current
depends upon not only the short circuit sorts but also the short circuit turn
ratio and the space position of short circuit turn, etc. The calculation results
are supported by corresponding experiments.
In above Chapters the operation problems of ac machines with symmetric 3-phase
stator windings and their system are analyzed mainly. In this chapter the internal
asymmetric problems in electric machine windings will be analyzed and researched.
The electric machine with internal asymmetric winding can be of normal condition,
or non-normal condition or fault condition.
For example, the synchronous machine is an electric machine with asymmetric
rotor winding, its excitation winding is on direct axis, damping effects of direct
axis and quadrature axis are different too. Single phase induction machine whose
application is very wide is electric machine with asymmetric stator winding, its
main winding and auxiliary winding are asymmetric two phase windings.
6 Internal Asymmetric Analysis of AC Machines
If part coils of stator winding of ac machine are damaged, after removing the
damaged coils the machine could continue operation, in this instance the electric
machine stator winding is of asymmetric condition too.
When internal fault in ac machine stator winding occurs, the electric machine
winding is also of asymmetric condition. Here the electric machine could be
damaged severely. In order to protect electric machine and lighten the damage
owing to internal fault, the speediness and reliability of relay protection are very
important, which demand understanding the currents in electric machine windings
under internal faults clearly. Under the condition of larger unit capacity the normal
current of electric machine can reach thousands of amperes, even several ten
thousands amp., at this time adopting multi-branch configuration for stator winding
is necessary, and higher need will be presented for the relay protection of electric
machine.
The internal faults of stator winding of ac machine are various, including
symmetric 3-phase internal short circuit and asymmetric internal short circuit.
The forms of asymmetric internal short circuit are much more too, for example,
inter turn short circuit (including short circuit between coils of the same branch
and short circuit between different branches of the same phase), short circuit
between branches of different phases, 3-phase asymmetric short circuit and short
circuit of simultaneous two faults and so on.
Besides short circuit, the internal asymmetry of ac machine may be caused owing
to wrong connection of branch coil group. The polarity of coil group is connected
in reverse, coil is connected to wrong branch, etc. which are all actual examples.
Main feature of internal asymmetry of electric machine is of very strong space
harmonics in air-gap magnetic field, even if the mmf is sine distribution, the space
harmonics of air-gap magnetic field produced by it are also stronger. It will cause
very large error to calculate only space fundamental of air-gap magnetic field.
As an example, a salient pole synchronous machine is researched, this electric
machine stator is of 108 slots, and the stator winding can be connected to form 6
parallel branches. The calculated results of self and mutual inductances of stator
single coil, branch and phase winding are showed in Table 6.0.1, including two
conditions—only considering space fundamental magnetic field and considering
space harmonic magnetic field. N.B., in the table only the constant inductance
parts independent of rotor position, that is, L0 and M 0 (refer to formulas (1.3.25),
and (1.3.38), etc.) are listed.
From Table 6.0.1 it can be seen that for stator single coil or branch, if only
consider space fundamental magnetic field, the errors of the inductances are very
large, even the sign is in reverse sometimes. But for the phase winding, the errors
caused by only considering space fundamental magnetic field are not large
(below 8% in the example). The reason is that when the current flows in stator
single coil or single branch, the fractional and lower order harmonics are very
strong, some of them even exceed the fundamental magnetic field, but while the
current flows in phase winding, the basic wave of air-gap magnetic field is
413
AC Machine Systems
fundamental, the fractional harmonics and lower order harmonics of mmf of
various coils composing the phase winding counteract each other, so total mmf of
phase winding is fundamental mainly.
Table 6.0.1 Calculated results of stator winding inductances of a 550kW salient
pole synchronous machine
unit: H
Only considering space
fundamental of air-gap
magnetic field
Considering space
harmonics of air-gap
magnetic field
0.884 u104
0.106 u102
Mutual inductance between
adjacent coils
0.830 u10
4
0.962 u103
Self inductance of single
branch
0.293 u102
0.185 u101
Mutual inductance between
adjacent branches of the same
phase
0.293 u102
0.134 u103
0.127
0.132
0.0561
0.0578
Inductance sort
Self inductance of single coil
Self inductance of phase
A winding
Mutual inductance between
phase A, B windings
This example shows that for internal asymmetric problems in the stator, only
considering space fundamental magnetic field may cause not permitted error. Of
course, magnitudes of the error are different owing to the difference of researched
problems. For example, for the inter-turn short circuit of synchronous machine
with wave winding and two poles turbo-generator, the space harmonics of air-gap
magnetic field is less; but for inter-turn short circuit of synchronous machine
with lap winding, the space harmonics of air-gap magnetic field are very strong,
the more pole-pair number is, the larger harmonic component is.
To sum up, under the situation of internal asymmetry of electric machine
winding, due to strong space harmonics in the air-gap magnetic field and with
different rotation speed and different rotation direction, the current harmonics in
each winding are more and stronger, that is, at this time there are stronger space
harmonics in air-gap magnetic field and stronger time harmonics in winding
currents. Therefore in the face of the feature of internal asymmetry and various
sorts of winding internal asymmetry it is needed to adopt Multi-loop Theory and
Method based on single coil as showed in Chapter 1 for analyzing.
In this Chapter using the method the rotor failures of bar broking and end ring
broking of induction machine, the internal faults of stator winding of salient
pole synchronous machine, calculation of loop parameters, etc. are discussed. As
to another internal asymmetry in the winding, for example the internal faults of
stator winding of induction machine, the operation of synchronous machine after
414
6 Internal Asymmetric Analysis of AC Machines
cutting damaged stator coils, internal faults of excitation winding, etc., they can
be researched by using similar method, here do not give unnecessary details.
6.1 Analysis of Rotor Winding Failures of Squirrel Cage
3-Phase Induction Machines
Rotor bar broking and end ring broking of induction machine are familiar failures
for induction machine, they may be caused by the disfigurements in manufacture
process or in operation. For example, thin bar or broking bar may appear in cast
aluminium rotor of medium-size and small induction machine owing to technology
problems; dummy jointing between bar and end ring in larger-scale induction
machine may appear due to bad welding; improper operation, bad work environment
and frequently starting, etc. can easily cause the failures in squirrel cage rotor.
Therefore realizing the effects of rotor winding failures of induction machine on
operation performance and detecting the rotor failures based on the variation of
electric quantities of the machine after failures are important technical steps for
avoiding the accidents in operation and ensuring safe production[1 3].
For analysis convenience, under the condition of no effect on physical essence,
make the following assumptions:
(i) Surfaces of stator and rotor are smooth, the effects of tooth and slots are
expressed by Carter’s coefficient.
(ii) Neglecting the eddy current, hysteresis and nonlinear, the effect of iron
reluctance is calculated by magnifying the air-gap adequately.
(iii) Air-gap of electric machine is uniform, the slot harmonics are not
considered, and before failure occurring the electric machine is symmetric in
both electric and magnetic aspects.
(iv) Insulation exists between rotor bar and iron.
In the following the rotor failures of squirrel cage induction machine are discussed
according to the electric machine example showed in Fig. 6.1.1.
Figure 6.1.1 Stator windings and rotor loops of squirrel cage 3-phase induction
machine
415
AC Machine Systems
Figure 6.1.1 (a) is 3-phase symmetric stator windings of squirrel cage induction
machine, Y-connection (similarly treating for '-connection); Figure 6.1.1(b) is
rotor loops. If the rotor slot number is n , before the rotor failure occurring the
node number of rotor circuit is 2n , the branch number is 3n , hence the current
number of independent loops is 3n (2n 1) n 1 . The rotor currents are showed
in the figure, the loop currents are i1 , i2 ," , in , ie respectively. So the number of
independent currents in stator and rotor is n 4 .
6.1.1
Basic Equations of Squirrel Cage Three Phase Induction
Machine
Based on above mentioned assumptions and positive direction regulation in
Fig. 6.1.1 the voltage equations and flux-linkage equations before failure occurring
are listed as
U
p\ RI
(6.1.1)
\
LI
(6.1.2)
where U , I , \ all are (n 4) u 1 column matrix; R, L are (n 4) u (n 4) square
matrix. Above two formulas can be spread to
416
ª ua º
«u »
« b»
« uc »
« »
«0»
«0»
« »
«#»
«0»
« »
«¬ 0 »¼
ª\ a º ª r
«\ » «0
« b» «
«\ c » «0
« » «
\
0
«
p 1»«
«\ 2 » «0
« » «
« # » «#
«\ » «0
« n» «
«¬\ e »¼ «¬0
ª\ a º
«\ »
« b»
«\ c »
« »
«\ 1 »
«\ 2 »
« »
«# »
«\ »
« n»
«¬\ e »¼
ª Laa
«M
« ba
« M ca
«
« M 1a
« M 2a
«
« #
«M
« na
«¬ M ea
0
r
0
0
0
#
0 0
0 0
r 0
0 rr
0 rb
# #
0 0 rb
0 0 re
0
0
0
rb
rr
#
0
re
" 0
" 0
" 0
" rb
" 0
#
" rr
" re
"
"
"
"
"
º ªia º
» «i »
»« b»
» « ic »
»« »
» « i1 »
» «i2 »
»« »
»«# »
re » «in »
»« »
nre »¼ «¬ ie »¼
M ab
Lbb
M cb
M 1b
M 2b
#
M ac
M bc
Lcc
M 1c
M 2c
#
M a1
M b1
M c1
L11
M 21
#
M a2
M b2
M c2
M 12
L22
#
M nb
M eb
M nc
M ec
M n1
M e1
M n 2 " Lnn
M e 2 " M en
0
0
0
re
re
#
M an
M bn
M cn
M 1n
M 2n
#
M ae º ªia º
M be »» «« ib »»
M ce » « ic »
»« »
M 1e » « i1 »
M 2e » «i2 »
»« »
# »«# »
M ne » «in »
»« »
Lee »¼ «¬ ie »¼
(6.1.3)
(6.1.4)
6 Internal Asymmetric Analysis of AC Machines
In formula (6.1.3) rr
6.1.2
2(rb re ) is self resistance of rotor loop.
Calculation of Loop Inductances of Squirrel Cage Three
Phase Induction Machine
As is the same as the inductances of single phase induction machine (refer to
Section 1.10), calculation formulas of loop inductances of squirrel cage 3-phase
induction machine can be written as follows.
(1) Self inductances of stator windings (refer to formula (1.10.8)) are
2
Laa
Lbb
Lcc
Ls
2 w12W l
§k ·
O0 ¦ ¨ wk ¸ Lsl
2
PS
k © k ¹
(6.1.5)
Mutual inductances between stator windings (refer to (1.10.10)) are
M ab
M bc
M ca
M ba
M cb
M ac
Ms
2
2 w12W l
§k ·
§ 2S ·
O0 ¦ ¨ wk ¸ cos ¨ K ¸ M sl
2
PS
© 3 ¹
k © k ¹
(6.1.6)
It is the result of changing qT in formula (1.1.10) to 2ʌ / 3 , in which the electric
angle between central lines of phase windings. In above two formulas k 1, 3, 5, " .
Lsl for formula (6.1.5) and M sl in formula (6.1.6) are leakage self inductance and
leakage mutual inductance of phase windings respectively.
Self inductances of rotor loops (refer to formula (1.10.10)) are
L11
L22
" Lnn
Lr
kE S
2W l
1
O0 ¦ 2 sin 2 r 2( Lb Le )
P S2
k
2
k
(6.1.7)
1 2
, ,"; 2( Lb Le ) is leakage self inductance of rotor loop.
P P
Self inductance of end ring loop is sum of leakage inductances of several
segments of the end ring, that is
in which k
Lee
nLe
(6.1.8)
Mutual inductance between i - th loop and j - th loop of rotor (refer to formula
(1.10.12)) is
M ij
where k
kE S
2W l
1
O
sin 2 r cos k ( j i )M
2 0¦ 2
PS
2
k k
(6.1.9)
1 2
, ,"
P P
417
AC Machine Systems
When the two rotor loops are adjacent, that is | i j | 1 or n 1 , the leakage
mutual inductance Lb should be added to above formula.
Mutual inductance between end ring loop and bar loop is
M ek
M ke
k 1, 2, " , n
Le
(6.1.10)
Above formula indicates that the mutual inductance between end ring loop and
bar loop is namely leakage inductance of one segment of end ring, because the
end ring loop and bar loop are perpendicular to each other, the mutual inductance
responding to air-gap magnetic field is zero.
Under uniform air-gap for induction machine formulas (6.1.7) and (6.1.9) can
be simplified (refer to Fig. 6.1.2). mmf produced by the current of rotor i -th
loop is
F ( x)
­n 1
°° n i
®
° 1 i
°̄ n
M
2
x
x
M
2
M
2
or x !
M
2
where the electric angle between rotor adjacent loops is M
2PS
. The air-gap
n
flux density produced by this mmf is
B( x)
F ( x)OG ( x)
F ( x)
O0
2
Hence the self inductance of rotor loop produced by air-gap magnetic field is
Lii
\ ii
i
Wl
M
iS ³
2
M B ( x )dx
2
Wl
M
n 1 O0
dx
n 2
2
M
S³
2
PW l O0
n 1
n2
Considering leakage self inductance, 2( Lb Le ) should be added to above formula.
Figure 6.1.2 Air-gap mmf produced by the rotor loop current of induction machine
Mutual inductance between rotor i - th loop and j - th loop ( i z j ) is
418
6 Internal Asymmetric Analysis of AC Machines
M ij
\ ij
i
Wl
iS ³
( j i )M ( j i )M M
2
M
B( x)dx
2
Wl
S³
( j i )M ( j i )M M
2
M
2
1 O0
dx
n 2
PW l O0
1
n2
If i -th loop and j -th loop are adjacent, the leakage mutual inductance Lb
should be added to the mutual inductance too.
(2) Mutual inductance between stator winding and rotor loop (refer to formula
(1.10.16)) is
M sj
2 w1W l
kE S
1
2S
ª
º
O0 ¦ 2 kwk sin r cos k «J (i 1) jM »
2
PS
2
3
¬
¼
k k
(6.1.11)
in which k 1,3,5,"; J is electric angle by which the central line of rotor loop
leads the stator winding axis; s represents stator winding a, b, c 3-phases
which correspond to i 1, 2,3 respectively; j is sequence number of rotor bar
loop, j 1, 2," , n .
The mutual inductance between stator winding and end ring loop can be
neglected, that is
M ae
M be
M ce
M ea
M eb
M ec
0
(6.1.12)
From above formulas it can be seen that in squirrel cage 3-phase induction
machine the inductances of stator windings and inductances of rotor loops are all
constants independent of rotor position which is easy to be understood due to the
cylindrical form of stator and rotor. Only the mutual inductances between stator
windings and rotor loops are time-variant parameter relative to rotor position.
It is needed to point out that formulas (6.1.5), (6.1.6) and (6.1.11) are all gained
for single branch stator winding. If phase winding is of multi-branches with branch
number a1 per phase, then w1 in the formulas has to be changed to w1 / a1 .
Also attention, if skewed slot is adopted in the electric machine, the skewed
kE § kE ·
slot coefficient ksk sin
¨
¸ , should be added to formula (6.1.11), here
2
© 2 ¹
E is skewed slot space electric angle.
6.1.3
Amending Basic Equations Based on Rotor Failure Condition
After various inductances are substituted into flux-linkage equation (6.1.4) and
voltage equation (6.1.3), a differential equation set is obtained which is an equation
under normal condition, with inductances and resistances as coefficients, and various
loop currents are the unknowns in the equation set. After rotor failures occur, the
equation set can be modified in order to get corresponding equations, without
writing other equations again.
419
AC Machine Systems
(1) Rotor bar broking
Figure 6.1.3 shows the variety condition while rotor bar j broking. Here the
rotor loop current i j 1 i j , which is equivalent to combining of loop j and loop
j 1 to bar j adjacent to become a new loop. Therefore accordingly basic
equations (6.1.3) and (6.1.4) have to be modified as follows:
(i) Elements in ( j 4)-th row of the matrix R and L are added to
corresponding elements in ( j 3)-th row.
(ii) Elements in ( j 4)-th column of the matrix R and L are added to
corresponding elements in ( j 3)-th column.
(iii) Elements in ( j 4)-th row and ( j 4)-th column of the matrix R and
L are canceled.
(iv) Elements in ( j 4)-th row of the column vector U , I and \ are
canceled.
Actually such treatment is canceling original ( j 3)-th and ( j 4)-th
equations (i.e. the equations of loop j and loop j 1 of the rotor), and they are
exchanged to the equation of rotor new “large” loop (it consists of original loop
j and loop j 1 ).
(2) Rotor end ring broking
As showed in Fig. 6.1.4(a), if end ring of squirrel cage rotor is broken, obviously,
there is i j 0 . Accordingly in the basic equations (6.1.3) and (6.1.4),
the ( j 3)-th row and ( j 3)-th column of the matrix R and L , and ( j 3)-th
row of the column vector U , I and \ should be canceled.
Figure 6.1.3 Variety of rotor currents while rotor bar j broking
(a) General expression of rotor loops; (b) Bar current and end ring current corresponding to Fig. (a);
(c) After bar j broking i j 1 i j ; (d) Expression of loop current corresponding to Fig. (c)
If two end rings are broken simultaneously, failure of one of the two end rings
can be treated according to above mentioned method, and suppose that another
420
6 Internal Asymmetric Analysis of AC Machines
end ring is broken at loop K . Here the loop current ik is equal to negative value
of the end ring loop current ie (refer to Fig. 6.1.4(b)), which corresponds to
combining loop K and end ring loop to form a new loop. Accordingly, basic
equations (6.1.4) and (6.1.3) have to be modified as follows:
Figure 6.1.4 Expression of end ring broking
(i) The negative values of the elements in ( K 3)-th row of the matrix R
and L are added to corresponding elements in last row.
(ii) The negative values of the elements in ( K 3)-th column of the matrix
R and L are added to corresponding elements in last column.
(iii) Elements in ( K 3)-th row and ( K 3)-th column of the matrix R and
L are canceled.
(iv) Elements in ( K 3)-th row of the column vector U , I and \ are
canceled.
Actually such treatment is to cancel original ( K 3)-th and last equations
(namely the equations of rotor loop K and end ring loop), and they are exchanged
to the equation of rotor new end ring loop (it consists of original end ring loop
and loop K ).
If the end ring loop is broken at several places, based on broking condition and
above mentioned method it can be treated one by one. If the bar and end ring are
broken simultaneously, according to above two methods it can be treated
respectively.
6.1.4
Calculation of Steady State Loop Currents
Based on above mentioned method, after modifying the basic equations according
to the failure situations, a new group of differential equations with variant
coefficients and variables of loop currents is obtained. Then corresponding transient
and steady-state conditions of the electric machine may be analyzed and researched.
Calculation of the steady-state loop currents of squirrel cage 3-phase induction
machine is discussed as follows. In steady-state firstly the expression of each
loop current of electric machine is assumed, then the expressions are substituted
into above mentioned differential equations, based on the principles that the same
frequency quantities in two sides of equation are equal to each other, the differential
421
AC Machine Systems
equation group is transformed to linear algebraic equation group. Then the
steady-state values of every various currents can be obtained according to the
linear algebra equation group.
Suppose that the power source is sine wave voltage with fundamental frequency
in the steady-state operation of squirrel cage 3-phase induction machine with
rotor failures, and the voltage expressions are
ua
ub
uc
½
°
°
2ʌ · °
§
U cos ¨ Z 0 t ¸ ¾
3 ¹°
©
2ʌ · °
§
U cos ¨ Z 0 t ¸°
3 ¹¿
©
U cos Z 0 t
(6.1.13)
These stator fundamental voltages will cause fundamental currents in stator
windings, and produce revolving magnetic field with synchronous speed and
positive direction in air-gap, and produce the currents with frequency sZ 0 in rotor
loops. The rotor currents will produce the revolving magnetic field with various
rotation speeds and different rotation directions owing to rotor asymmetry. But
the currents of stator 3-phase symmetric windings produced by fractional
harmonic and high order harmonic magnetic field caused by rotor currents can be
neglected in general, hence only the fundamental magnetic field due to rotor
currents with positive and reverse direction rotation can induce the currents in
stator windings, the former induces fundamental currents, the latter induces the
currents of frequency (1 2 s )Z0 . Since stator three windings are symmetric,
these stator currents of two frequencies will correspond to the rotor currents of
frequency sZ 0 only, and do not cause currents with other frequency in rotor.
Therefore the expressions of stator phase currents can be written as
ia
ib
ic
422
½
I1 cos Z0 t I1c sin Z 0 t I 2 s cos(1 2s )Z0 t I 2cs sin(1 2 s)Z 0 t °
°
°
2S ·
2S ·
§
§
I1 cos ¨ Z0 t °
¸ I1c sin ¨ Z0 t ¸
3 ¹
3 ¹
©
©
°
°°
2S º
2S º
ª
ª
I 2 s cos «(1 2 s )Z0 t » I 2c s sin «(1 2s )Z 0 t »
¾
3¼
3¼
¬
¬
°
°
2S ·
2S ·
§
§
I1 cos ¨ Z0 t °
¸ I1c sin ¨ Z 0 t ¸
3 ¹
3 ¹
©
©
°
°
2
2
S
S
ª
º
ª
º
°
I 2 s cos «(1 2 s )Z0 t » I 2cs sin «(1 2 s )Z0 t »
3¼
3¼
°¿
¬
¬
(6.1.14)
6 Internal Asymmetric Analysis of AC Machines
The expressions of rotor loop currents can be written as
i1
i2
in
ie
I r1 cos sZ 0 t I rc1 sin sZ 0 t ½
°
I r 2 cos sZ0 t I rc2 sin sZ 0 t °
°
#
¾
I rn cos sZ0 t I rnc sin sZ 0 t °
°
I re cos sZ0 t I rec sin sZ0 t °¿
(6.1.15)
The expressions of these currents are substituted into formulas (6.1.4) and
(6.1.5), then we can get the following formulas:
n
Ls ia M s ib M s ic ¦ M aj i j
\a
j 1
n
( Ls M s )ia ¦ M aj i j
j 1
( Ls M s )[ I1 cos Z 0 t I1c sin Z 0 t I 2 s cos(1 2s )Z 0 t
n
I 2cs sin(1 2 s)Z 0 t ] ¦ M rs cos[(1 s )Z0 t jM )]
j 1
u ( I rj cos sZ0 t I rjc sin sZ 0 t )
­
½
M rs n
( I rj cos jM I rjc sin jM ) ¾ cos Z 0 t
®( Ls M s ) I1 ¦
2 j1
¯
¿
­
½
M n
®( Ls M s ) I1c rs ¦ ( I rj sin jM I rjc cos jM ) ¾ sin Z0 t
2 j1
¯
¿
­
M
®( Ls M s ) I 2 s rs
2
¯
­
M
®( Ls M s ) I 2cs rs
2
¯
½
cos jM I rjc sin jM ) ¾ cos(1 2 s )Z0 t
j 1
¿
n
½
( I rj sin jM I rjc cos jM ) ¾ sin(1 2s )Z0 t
¦
j 1
¿
n
¦ (I
rj
where for mutual inductances between stator windings and rotor loops, the
effects of high order harmonics are neglected, only considering the fundamental,
that is (refer to formula (6.1.11))
M aj
2 w1W l
ES
O0 kw1 sin r cos[(1 s )Z 0 t jM ]
2
PS
2
M rs cos[(1 s )Z 0 t jM ]
moreover
423
AC Machine Systems
ua
U cos Z0 t
p\ a ria
­
Z0 M rs
®Z0 ( Ls M s ) I1c 2
¯
n
¦ ( I
j 1
rj
½
sin jM I rjc cos jM ) rI1 ¾ cos Z 0 t
¿
­
½
ZM n
®Z0 ( Ls M s ) I1 0 rs ¦ ( I rj cos jM I rjc sin jM ) rI c¾ sin Z 0t
2 j1
¯
¿
n
­
1
®(1 2 s )Z0 ( Ls M s ) I 2cs (1 2 s )Z0 M rs ¦ ( I rj sin M
2
j 1
¯
½
­
I rjc cos jM ) rI 2 s ¾ cos(1 2s )Z 0 t ® (1 2s )Z 0 ( Ls M s ) I 2 s
¿
¯
n
½
1
(1 2 s )Z0 M rs ¦ ( I rj cos jM I rjc sin jM ) rI 2cs ¾ sin(1 2s )Z 0 t
2
j 1
¿
(6.1.16)
Letting the terms of two sides of the above equation including cos Z0 t , sin Z0 t ,
cos(1 2 s ) Z 0 t and sin(1 2 s )Z 0 t by equal respectively, then
U
Z 0 ( Ls M s ) I1c
½
°
1
°
c
Z 0 M rs ¦ ( I rj sin jM I rj cos jM )u rI1
°
2
j 1
°
0 Z 0 ( Ls M s ) I1
°
°
n
1
°
Z0 M rs ¦ ( I rj cos jM I rjc sin jM ) rI c
2
°
j 1
¾
0 (1 2 s )Z 0 ( Ls M s ) I 2c s
°
n
°
1
(1 2 s )Z 0 M rs ¦ ( I rj sin jM I rjc cos jM ) rI 2 s °
2
j 1
°
°
0 (1 2 s )Z 0 ( Ls M s ) I 2 s
°
n
°
1
(1 2 s)Z 0 M rs ¦ ( I rj cos jM I rjc sin jM ) rI 2c s °
2
j 1
¿
n
(6.1.17)
For phases b and c the voltage equation similar to equation (6.1.17) can be
got.
The flux-linkage of rotor loop j is
\j
M ja ia M jb ib M jc ic Lr i j
M ij ¦ ik Lb (i j 1 i j 1 ) Le ie
k
424
6 Internal Asymmetric Analysis of AC Machines
M rs cos[(1 s )Z0 t jM ][ I1 cos Z0 t I1c sin Z 0 t
I 2 s cos(1 2s )Z 0 t I 2c s sin(1 2s )Z 0 t ]
2S º ­
2S ·
ª
§
M rs cos «(1 s )Z0 t jM » ® I1 cos ¨ Z0 t ¸
3 ¼¯
3 ¹
¬
©
2S ·
2S º
§
ª
I1c sin ¨ Z 0 t ¸ I 2 s cos «(1 2s )Z 0 t »
3 ¹
3¼
©
¬
2S º ½
2S º
ª
ª
I 2cs sin «(1 2 s )Z0 t » ¾ M rs cos «(1 s )Z 0 t jM »
3 ¼¿
3¼
¬
¬
­
2S ·
2S ·
§
§
u ® I1 cos ¨ Z0 t ¸ I1c sin ¨ Z 0 t ¸
3
3 ¹
©
¹
©
¯
2S º
2S º ½
ª
ª
I 2 s cos «(1 2s )Z 0 t » I 2cs sin «(1 2 s)Z 0 t » ¾
3
3 ¼¿
¬
¼
¬
Lr ( I rj cos sZ 0 t I rjc sin sZ 0 t ) M ij ¦ ( I rk cos sZ 0 t I rkc sin sZ 0 t )
K
Lb ( I rj 1 cos sZ 0 t I rjc 1 sin sZ0 t I rj 1 cos sZ 0 t
I rj 1 sin sZ 0 t ) Le ( I re cos sZ0 t I rec sin sZ 0 t )
­3
® M rs ( I1 cos jM I1c sin jM I 2 s cos jM I 2c s sin jM ) Lr I rj
¯2
½
M ij ¦ I rk Lb ( I rj 1 I rj 1 ) Le I re ¾ cos sZ0 t
k
¿
­3
® M rs ( I1 sin jM I1c cos jM I 2 s sin jM I 2c s cos jM ) Lr I rjc
¯2
½
M ij ¦ I rkc Lb ( I rjc 1 I rjc 1 ) Le I rec ¾ sin sZ0 t
k
¿
It is known that the voltage equation is
uj
p\ j rr i j rb i j 1 rb i j 1 re ie
0
Substituting \ j into above formula, then get
0
­
ª3
® sZ0 « M rs ( I1 sin jM I1c cos jM I 2 s sin jM I 2c s cos jM )
¬2
¯
º
Lr I rjc M ij ¦ I rkc Lb ( I rjc 1 I rjc 1 ) Le I rec » rr I rj
k
¼
­
½
ª3
rb I rj 1 rb I rj 1 re I re¾ cos sZ0 t ® sZ0 « M rs ( I1 cos jM
¬2
¿
¯
425
AC Machine Systems
I1c sin jM I 2 s cos jM I 2cs sin jM ) Lr Lrj M ij ¦ I rk
k
º
½
Lb ( I rj 1 I rj 1 ) Le I re » rr I j rb I rjc 1 rb I rjc 1 re I rec ¾ sin sZ0 t
¼
¿
(6.1.18)
where k 1, 2," , j 1, j 1," , n .
Letting the terms of two sides of above formula including cos sZ 0 t and sin sZ 0 t
be equal respectively, then we can obtain the following two equations:
0
0
½
°
°
°
°
°
°
¾
ª3
sZ0 « M rs ( I1 cos jM I1c sin jM I 2 s cos jM I 2cs sin jM ) °
°
¬2
°
º
°
Lr I rj M ij ¦ I rk Lb ( I rj 1 I rj 1 ) Le I re »
°
k
¼
°
rr I rjc rb ( I rjc 1 I rjc 1 ) re I rec
¿
ª3
sZ0 « M rs ( I1 sin jM I1c cos jM I 2 s sin jM I 2c s cos jM )
¬2
º
Lr I rjc M ij ¦ I rkc Lb ( I rjc 1 I rjc 1 ) Le I rec »
k
¼
rr I rj rb ( I rj 1 I rj 1 ) re I re
(6.1.19)
where k 1, 2," , j 1, j 1," , n .
Similarly, for end ring loop there is
n
\e
¦L i
e j
nLe ie
j 1
n
¦ L (I
e
rj
cos sZ 0 t I rjc sin sZ 0t ) nLe ( I re cos sZ 0 t I rec sin sZ0 t )
j 1
n
ue
0
p\ e nre ie ¦ re i j
j 1
n
n
[ sZ 0 ( Le ¦ I rjc nLe I rec ) nre I re re ¦ I rj ]cos sZ 0 t
j 1
j 1
n
n
j 1
j 1
[ sZ 0 ( Le ¦ I rj nLe I re ) nre I rec re ¦ I rjc ] sin sZ 0 t
(6.1.20)
or
0
0
426
n
n
½
sZ0 ( Le ¦ I rjc nLe I rec ) nre I re re ¦ I rj °
j 1
j 1
°
¾
n
n
sZ0 ( Le ¦ I rj nLe I re ) nre I rec re ¦ I rjc °
°¿
j 1
j 1
(6.1.21)
6 Internal Asymmetric Analysis of AC Machines
In the linear algebraic equation set consisting of formulas (6.1.17), (6.1.19) and
(6.1.21), the unknowns are the amplitudes of sine and cosine terms of electric
machine various loop currents, the total number of unknowns is 2( n 3) , the
coefficients are inductances and resistances, the number of independent equations
is also 2(n 3) . Hence the currents of various loops can be obtained by Gauss
elimination method.
Electro-magnetic torque (refer to formula (1.9.8)) is
Te
P wL T
I
I
2 wJ
(6.1.22)
The currents of above formula are divided into stator currents and rotor currents,
and the inductance matrix is written as partition matrices, then
Te
d
ª
º
Lsr » T
«
ªI º
dJ
P
»« s »
Is Ir «
d
2
«
» ¬ I rT ¼
L
« dJ rs
»
¬
¼
·
P§ d
d
Lrs I sT I s
Lsr I rT ¸
¨ Ir
2 © dJ
dJ
¹
PI s
d
Lrs I rT
dJ
PM rs
Z
>ia
ib
ic @
cos[(1 s )Z0 t 2M ]
ªcos[(1 s )Z0 t M ]
«
2S
2S
ª
º
ª
º
M » cos «(1 s )Z 0 t 2M »
d «cos (1 s )Z0 t u « «¬
3
3
¼
¬
¼
dt «
2
S
2
S
º
ª
º
«cos ª(1 s )Z t M » cos «(1 s )Z 0 t 2M »
0
«¬ «¬
3
3
¼
¬
¼
" cos[(1 s )Z 0 t nM ]
2S
ª
º
" cos «(1 s )Z 0 t nM »
3
¬
¼
2
S
ª
º
" cos «(1 s )Z 0 t nM »
3
¬
¼
0 º ªi1 º
» «i »
2
0» « »
» «# »
»« »
i
0» « n »
»¼ «¬ie »¼
The expressions of stator currents and rotor currents (6.1.14) and (6.1.15) are
substituted into above formula, and it is simplified as
Te
Tave Tpul
427
AC Machine Systems
where Tave is average torque, Tpul is pulsating torque, and
n
n
ª
º
3
PM rs «( I1c I 2cs )¦ I rj cos jM ( I1 I 2 s )¦ I rjc cos jM »
4
j 1
j 1
¬
¼
Tave
n
n
ª
º
3
PM rs «( I1c I 2cs )¦ I rjc sin jM ( I1 I 2 s )¦ I rj sin jM »
4
j 1
j 1
¬
¼
n
n
ª
3
PM rs «( I1c I 2cs )¦ I rj cos jM ( I1 I 2 s )¦ I rjc cos jM
4
j 1
j 1
¬
Tpul
n
n
º
( I1c I 2cs )¦ I rjc sin jM ( I1 I 2 s ) ¦ I rj sin jM » cos 2sZ 0 t
j 1
j 1
¼
n
n
ª
3
PM rs «( I1c I 2cs )¦ I rj sin jM ( I1 I 2 s )¦ I rjc sin jM
4
j 1
j 1
¬
n
n
º
( I1c I 2cs )¦ I rjc cos jM ( I1 I 2 s ) ¦ I rj cos jM » sin 2sZ0 t
j 1
j 1
¼
Experiments, analysis and calculation have been carried out for a 7.5kW squirrel
cage 3-phase induction machine under four conditions: rotor symmetry, one bar
broking, rotor series three bars broking and end ring broking, etc. Table 6.1.1
shows stator currents, input power, electro-magnetic torque and power factor under
Table 6.1.1 Comparison between calculated results and experimental data of a
7.5kW squirrel cage 3-phase induction machine
Operation conditions
performances
Stator current
I1 / A
Average electromagnetic torque
Tave / N ˜ m
Input power
P1 / W
Power factor
cos M
428
Calculated
Experimental
Relative error
Calculated
Experimental
Relative error
Calculated
Experimental
Relative error
Calculated
Experimental
Relative error
Rotor
symmetry
9.51
9.45
0.63%
56.6
55.4
2.17%
9444
9500
0.59%
0.871
0.882
1.25%
Rotor one Rotor series
bar
three bars
broking
broking
9.27
8.91
4.04%
54.7
51.9
5.39%
9111
9025
0.95%
0.862
0.889
3.04%
8.53
8.61
0.93%
47.4
47.3
0.21%
7893
8365
5.64%
0.813
0.852
4.60%
Rotor end
ring
broking
8.86
9.18
3.49%
51.8
52.2
0.77%
8624
9008
4.26%
0.854
0.861
0.78%
6 Internal Asymmetric Analysis of AC Machines
above conditions while stator phase voltage is 380V, slip is 0.03 . Therein power
factor angle M is the phase angle between stator phase a voltage and phase a
fundamental current ia1 , and it is gained from the calculated results directly.
Input power is calculated based on formula 3U a I a1 cos M . From the table it can be
seen that calculated results and experimental data are consistent basically.
In order to realize the effect of rotor failures more clearly, based on the
calculated values of rotor currents, the rotor current phasor diagrams under rotor
winding normal condition and failure condition are drawn as showed in Fig. 6.1.5.
Figure 6.1.5(a) is rotor current phasor diagram under rotor winding normal
condition, and Fig. 6.1.5(b) is rotor current phasor diagram under the condition
of bar 10,11,12 broking simultaneously.
A light rotor failure can not bring about large effect on normal operation of
squirrel cage 3-phase induction machine, normally it only causes the slip slightly
to increase at constant load, accordingly the stator current, input power, etc.
increase. But when severe rotor failure occurs, the starting time of the electric
machine maybe obviously increases, even electric machine can not be started,
and acute vibration is accompanied with it. At this time the electric machine
loading capability drops obviously, which will affects the normal operation of the
machine.
From above analysis and calculations it can be seen that while a certain bar
broking, adjacent bar currents maybe increase, and the adjacent bar broking will
be accelerated. Therefore when slight rotor failure of electric machine occurs,
detecting the failure in time and adopting suitable steps are very necessary. It is
very important in both economics and safe production.
Only fundamental component exists in the stator current of induction machine
in normal operation. After rotor failure occurs, addtional component of frequency
(1 2 s ) f appears in stator current, and its magnitude depends on severe degree of
rotor asymmetry. So detecting the stator current component of frequency (1 2 s ) f
can judge rotor failure. The current component has following specific features:
(i) Generally the slip of induction machine under steady-state operation is very
small, so frequency (1 2s ) f is very near to the fundamental frequency f , and
it is very difficult to separate the component of frequency (1 2s ) f from the
frequency f component by using filter;
(ii) The amplitude of the current component with frequency (1 2s ) f is very
small, and it varies with different values of s .
Considering these two features it is difficult to analyze the frequency of stator
currents by general harmonic analysis apparatus. One method is to memorize the
current signal of the electric machine by magnetic tape recorder, then analyzing the
signal using signal processor, and getting its frequency spectrum. When analyzing
the signal the “by-pass petal” of main frequency should decrease as best as
possible, in order to prevent the signal (1 2 s ) f from submerging in the by-pass
petal. So it is needed to select suitable window function. The main petal of
429
AC Machine Systems
Figure 6.1.5 Rotor current phasor diagram of induction machine
(a) Rotor current phasor diagram under rotor winding normal condition; (b) Rotor current phasor
diagram under the condition of bar 10,11,12 broking simultaneously.
Hamming Window is wider, but the by-pass petal can be restrained well, hence it
can be selected and used in practice. Moreover, because these two frequencies of
current signals are adjacent, the sampling point number should increase as best as
possible in order to improve frequency differentiation rate. Another method is that
the signal uia A cos(Z t M ) Ac cos > (1 2s )Z t M1 @ , is drawn out from phase
430
6 Internal Asymmetric Analysis of AC Machines
a current, the signal uac A1 cos(Z t 30e
) is drawn out from voltage uac , and
make A | A1 (refer to stator voltage phasor diagram Fig. 6.1.6), then these two
signals are subtracted from each other as
uia uac
A cos(Z t M ) A1 cos(Z t 30e
)
Ac cos[(1 2s )Z t M1 ]
M
§M
· §
·
| 2 A sin ¨ 15e
¸ sin ¨ Z t 15e
¸
2
2
©
¹ ©
¹
Ac cos[(1 2s )Z t M1 ]
Figure 6.1.6 Phasor diagram of stator voltage and current
It can be seen that for the signal uia , ratio of component (1 2s ) f to
ª
§M
·º
component f is Ac / A , but for signal uia uac , the ratio is Ac « 2 A sin ¨ 15e
¸» .
©2
¹¼
¬
The power factor angle M of 3-phase induction machine is near 30e generally,
while analyzing the signal uia uac , the amplitude of component (1 2 s ) f
increases much more relative to the component f , therefore it is more easy to
detect the component (1 2 s ) f .
6.2 Internal Fault Analysis of Stator Winding of Salient
Pole Synchronous Machines
Internal faults of stator winding of synchronous machine are normal faults with
destructive damages. The very large short circuit current of internal fault will
produce destructive electro-magnetic force, and maybe also produce overheat,
cause the winding and iron to be burned. The non asynchronous magnetic field
produced by the fault current maybe exceeds the value permitted greatly, and
causes severe damage of rotor. Therefore it is very important to research the
internal faults of stator winding of synchronous machine[4 7], find the distribution
and variation rules of the electric quantities at internal faults, and design protection
scheme for the internal faults in order to decrease fault damage.
The symmetric component method was used to analyze the internal faults of
431
AC Machine Systems
stator winding of synchronous machine before, but the effect of both air-gap
magnetic field harmonics and the winding space position can not be considered
by the method, and all branch currents of windings can not be obtained by it, so
the results can not reflect the internal fault regulation, and also can not direct the
design of internal fault relay protection.
An important character of stator winding internal fault is existence of very
strong space harmonics in air-gap magnetic field which make phase sequence
components of symmetric component method have dependence relation, so as to
lose the excellences of symmetric component method. At the same time the ideal
electric machine model is no longer suitable, accordingly, adopting d , q, 0
coordinates system based on the model will cause very large errors.
Phase coordinates method can consider the effect of air-gap magnetic field
harmonics on the parameters[8]. Based on the phase coordinates method the
parameter calculation is made by considering the phase winding as a whole, but
the phase winding at stator winding internal fault is no longer a whole one, hence
the phase coordinates method is not suitable for analyzing the winding internal
fault either.
In this section the Multi-loop Method is presented to analyze internal fault
performance both for steady-state and for transients. When the internal faults
occur, the electric machine circuit changes, and very strong harmonics appear in
air-gap magnetic field of the electric machine, it is necessary to consider the
effect of these magnetic field harmonics on inductances which is main character
and difficulty of using Multi-loop Method for researching electric machine[9 13].
6.2.1
Mathematic Model of Stator Winding Internal Fault of
Salient Pole Synchronous Machine
In the mathematic model of stator winding internal fault of synchronous machine
according to Multi-loop Method, the electric machine is considered as the circuit
consisting of many loops having relative motion, the voltage equations and fluxlinkage equations are listed in terms of actual loops of stator and rotor windings.
Stator branch equations and rotor loop equations are discussed respectively in the
following, then the stator equations will be modified according to stator winding
fault sort.
In this section the positive direction selection of electro magnetic quantities is
consistent with Chapter 1: for stator loops, positive current produces negative
flux-linkage, and the relation between voltage and current is defined according to
generator convention, that is, positive directions of current and voltage are
consistent towards load direction; for rotor loops (including excitation loop and
damper loops), positive current produces positive flux-linkage, and the relation
between voltage and current is defined according to motor convention, that is,
positive directions of current and voltage are consistent towards winding direction.
432
6 Internal Asymmetric Analysis of AC Machines
(1) Stator branch equations
Each winding branch without internal fault is considered as a branch and the
branch voltage equation is listed. When internal fault occurs, the fault branch is
divided into two branches from short circuit position. Suppose that the parallel
branch number per phase of the stator winding of salient pole synchronous
machine is a , and the phase number is m , then the total branch number N of
stator windings is
N
For normal winding
­ma
°
®ma 1 For inter-turn short circuit in the same branch
°ma 2 For short circuit between different branches
¯
The voltage equation of any branch in stator winding is
uQ
p\ Q rQ iQ
(6.2.1)
where uQ , \ Q , rQ , iQ are voltage, flux-linkage, resistance and current respectively.
The flux-linkage equation of branch Q is
\Q
N
Nd
S 1
ld 1
¦ M QS iS ¦ M Qld ild M Qfd i fd
(6.2.2)
in which iS , ild and i fd are the currents of stator branch S , damper loop ld and
excitation loop respectively, M QS is mutual inductance between stator branch S
and branch Q , M Qld is mutual inductance between damper loop ld and stator
branch Q , M Qfd is mutual inductance between excitation loop and stator branch
Q , Nd is the total number of damper bars.
It should be noted that any loop (branch) current has its own contribution to the
flux-linkage. Substitute formula (6.2.2) into formula (6.2.1), and obtain differential
equation as follows in which the unknowns are various branch currents
uQ
Nd
ª N
º
p « ¦ M QS iS ¦ M Qld ild M Qfd i fd » rQ iQ
ld 1
¬ S1
¼
(6.2.3)
The voltage equations of stator load side are
ua
ub
uc
LT pia rT ia uac ½
°
LT pib rT ib ubc ¾
LT pic rT ic ucc °¿
(6.2.4)
where rT , LT , uac , ubc , ucc are the resistance, inductance and network phase voltages
of the transformer referred to generator side respectively (refer to Fig. 6.2.2).
433
AC Machine Systems
In such a way the voltage and flux-linkage equations of all stator branches are
obtained.
(2) Rotor loop equations
Excitation loop voltage equation is
u fd
p\ fd rfd i fd
(6.2.5)
where \ fd , rfd are excitation loop flux-linkage and resistance respectively.
Flux-linkage equation of excitation loop is
\ fd
N
Nd
S 1
ld 1
¦ M Sfd iS ¦ M ldfd ild L fd i fd
(6.2.6)
in which M Sfd is mutual inductance between stator loop S and excitation loop,
M ldfd is mutual inductance between damper loop ld and excitation loop, L fd is
self inductance of excitation loop. Substitute formula (6.2.6) into formula (6.2.5),
then obtain the following differential equation in which the unknowns are various
currents:
u fd
Nd
§ N
·
p ¨ ¦ M Sfd iS ¦ M ldfd ild L fd i fd ¸ rfd i fd
ld 1
© S1
¹
(6.2.7)
According to selected loops showed in Fig. 6.2.1, the voltage equation of a
damper loop gd is
0
p\ gd rgd igd rc (igd 1 igd 1 )
(6.2.8)
where \ gd , rgd are flux-linkage and resistance of the damper loop gd , rc is damper
bar resistance.
Figure 6.2.1 Sketch map of damper loops
Flux-linkage equation of damper loop gd is
434
6 Internal Asymmetric Analysis of AC Machines
\ gd
N
Nd
S 1
ld 1
¦ M Sgd iS ¦ M gdld ild M gdfd i fd
(6.2.9)
in which M gdld is mutual inductance of two damper loops gd and ld . Substitute
formula (6.2.9) into formula (6.2.8), then obtain the following differential equation
of damper loop in which unknowns are various currents:
0
Nd
§ N
·
p ¨ ¦ M Sgd iS ¦ M gdld ild M gdfd i fd ¸ rgd igd rc (igd 1 igd 1 )
ld 1
© S1
¹
(6.2.10)
In such a way all voltage equations of rotor loops are gained.
All voltage equations of stator and rotor are written as matrix form:
ª u1 º
« # »
«u »
« Q»
« # »
« uN »
«0»
« »
« # »
«0»
« # »
« »
«0»
«u fd »
«u »
« a»
« ub »
¬« uc ¼»
­ ª L1
°« #
°«
° « M Q1
°« #
°« M
N1
°«
° « M 1d 1
°° « #
p ®«
°« M gd 1
°« #
°« M Nd 1
°«
°« M fd 1
°« 0
°« 0
°«
°¯ ¬ 0
" M 1Q
#
" LQ
#
" M NQ
" M 1dQ
#
" M gdQ
#
" M NdQ
" M fdQ
" "
" "
" "
" M 1N
#
" M QN
#
" LN
" M 1dN
#
" M gdN
#
" M NdN
" M fdN
" "
" "
" "
M 11d
#
M Q1d
#
M N 1d
L1d
#
M gd 1d
#
M Nd 1d
M fd 1d
"
"
"
" M 1gd
#
" M Qgd
#
" M Ngd
" M 1dgd
#
" Lgd
#
" M Ndgd
" M fdgd
" "
" "
" "
" M 1Nd
#
" M QNd
#
" M NNd
" M 1dNd
#
" M gdNd
#
" LNd
" M fdNd
" "
" "
" "
M 1 fd 0
#
#
M Qfd #
#
#
M Nfd #
M 1dfd #
#
#
M gdfd #
#
#
M Ndfd #
L fd
0
LT
0
"
0
" "
0 0º
# # »
»
# # »
# # »
# # »
»
# # »
# # »
# # »»
# # »
# # »
»
# # »
0 # »
LT 0 »
»
0 LT ¼
ª i1 º ½ ª r1
º ª i1 º ª 0 º
« # »° « %
» «# » «0»
« »° «
» « » « »
« iQ » ° «
» « iQ » « 0 »
rQ
« »° «
» « » « »
%
« # »° «
» « # » «0»
« i »° «
» «i » « 0 »
r
N
« N »° «
» « N» « »
r1d rc
rc
« i1d » ° «
» « i1d » « 0 »
« # »° «
» « # » «0»
rc % %
°
»u« » « »
u« »¾ «
« igd » ° «
» « igd » « 0 »
rc rgd rc
« »° «
» « » « »
% % rc
« # »° «
» « # » «0»
«iNd » ° «
» «i » « 0 »
rc
rc rNd
« »° «
» « Nd » « »
i
rfd
« fd » ° «
» « i fd » « 0 »
« i »° «
» « i » «u »
rT
« a »° «
» « a » « ac »
« ib » ° «
rT
» « ib » «ub c »
« »° «
» « » « »
rT ¼ ¬ ic ¼ ¬ ucc ¼
¬ ic ¼ ¿ ¬
(6.2.11)
435
AC Machine Systems
Formula (6.2.11) is re-written as simple form:
U
p ( LI ) RI B c
(6.2.12)
Owing to the relative movement between stator and rotor, the matrix L is
time-variant, hence the voltage equations of stator and rotor are all differential
equations with time-variant coefficients.
(3) Loop selection of synchronous generator with internal fault and forming of
state equations
The rotor voltage in formula (6.2.12) is loop voltage, but the stator voltage in
the equation is branch voltage, for convenience of treatment the loop voltage
should be adopted for both stator voltage and rotor voltage, so it is needed to
treat newly stator loop, that is, stator branch voltage should be converted to stator
loop voltage equation.
Figure 6.2.2 Sketch map of stator circuit at normal operation
Figure 6.2.2 is sketch map of stator circuit at normal operation of electric
machine, based on selected loops in the figure the transformation matrix from
branches to loops is as follows:
436
6 Internal Asymmetric Analysis of AC Machines
H
ª1 1
º
«
»
1 1
1 1
«
»
1 1
«
»
«
1 1
1 1»»
«
«
»
1 1
«
»
1
«
»
«
»
%
«
»
1
«
»
«
»
%
«
»
1
«
»
«¬
»¼
1
(6.2.13)
Formula (6.2.12) pre multiplied by formula (6.2.13) is
HU
HLpI HpLI HRI HBc
(6.2.14)
There is also the transformation formula of stator branch current and rotor loop
current to stator loop current and rotor loop current
I
HTIc
(6.2.15)
where I c is stator and rotor loop currents.
Substitute formula (6.2.15) into formula (6.2.14), then the stator and rotor loop
voltage equations with unknown loop currents are obtained.
Uc
HLH T pI c HpLH T I c HRH T I c HBc
LcpI c RcI c HBc
Lc
where
Rc
(6.2.16)
HLH T
pLc HRH T
The loop currents are considered as state variables, transforming formula (6.2.16),
then state equation of synchronous generator is obtained as
pI c Lc1RcI c Lc1 (U c HB c)
AI c B
(6.2.17)
A Lc1 Rc
where
B
Lc1U c Lc1 HBc
Formula (6.2.17) is the Multi-loop mathematic model of synchronous machine.
When the interturn short circuit in the same branch occurs, the loop selection
is showed as Fig. 6.2.3 (as an example, branch number per phase is 2).
437
AC Machine Systems
Figure 6.2.3 Sketch map of interturn short circuit in the same branch
Here the transformation matrix from branches to loops is
H
ª 1 1
º
«
»
1 1
1 1
«
»
«
»
1 1
«
»
1 1
1 1»
«
«
»
1 1 1
«
»
1
«
»
«
»
1
«
»
%
«
»
«
»
1
«
»
«
»
%
«
»
1
«
»
«¬
»¼
1
(6.2.18)
When the short circuit between different branches occurs, as an example the
loop selection at short circuit between branches of different phases is showed in
Fig. 6.2.4.
438
6 Internal Asymmetric Analysis of AC Machines
Figure 6.2.4 Sketch map of short circuit between branches of different phases
Here the transformation matrix from branches to loops is
H
ª 1 1
º
«
»
1 1
1 1
«
»
«
»
1 1
1
«
»
1 1
1 1
1 1»
«
«
»
1 1
1
«
»
1 1
«
»
«
»
1
«
»
%
«
»
«
»
1
«
»
«
»
%
«
»
1
«
»
«¬
»¼
1
(6.2.19)
The key for solving above state equations is calculation of time-variant
coefficients. After getting the expressions of time-variant inductances using four-th
order Runge-Kutta Method to solve the differential equation set with time-variant
coefficients, then the numerical solutions of various loop currents of stator and rotor
will be obtained, and the steady-state and transient currents of stator branches are
also gained through the transformation of formula (6.2.15).
439
AC Machine Systems
6.2.2
Loop Parameters of Stator Winding Internal Faults of
Synchronous Machine
Calculation of loop parameters (here are loop inductances mainly) is the key of
analyzing stator winding internal faults of synchronous machine. The inductance
parameters are mostly time-variant owing to relative movement between stator
and rotor. In the following the inductances of stator circuit and rotor circuit and
mutual inductances between stator and rotor circuits are calculated respectively.
(1) Inductances of stator loops (branches)
The inductances of electric machine circuits are produced mainly by air-gap
magnetic field, furthermore by leakage field. Basic thought of calculating the
inductances relative to air-gap magnetic field is: according to the basic concept
of inductance M
\
, firstly the air-gap mmf produced by the current i flowing
i
in the loop are resolved into harmonics, and each harmonic magnetic field is
obtained by using the air-gap permeance, then each harmonic flux-linkage of the
loop and total flux-linkage are calculated, finally the ratio of flux-linkage to
current i is the inductance of electric machine.
Calculation of the inductances relative to leakage field includes two parts
owing to end region leakage flux and slot leakage flux. Basic thought of calculating
the inductance produced by end region leakage field is: the coil end part with
current is divided into several current elements, and the magnetic field produced
by the current elements is calculated in terms of Biot-Savart law. The magnetic
field ata certain point of end region is superposition of magnetic fields at that
point produced by all current elements. Finally the flux-linkage of coil end part
and corresponding inductance produced by end region leakage flux are obtained.
As to the inductance owing to slot leakage flux, if the leakage flux is not in
saturation state and the reluctance of iron part is ignored, then mutual inductance
produced by slot leakage flux only exists between coil bars in the same slot, and
mutual inductance produced by slot leakage flux does not exist between coil bars
in different slots. After knowing the connection of stator winding, the inductance
of each stator loop produced by slot leakage flux can be calculated by adopting
the method of searching coil bars in the same slots one by one. The total
inductance is the sum of the inductance produced by air-gap magnetic field and the
inductance produced by leakage magnetic field. As to the concrete calculation
formulas please see Chapter 1.
The inductances of stator loops of salient pole synchronous machine are
time-variant parameters relative to rotor position.
(2) Inductances of rotor loops
Calculation formulas of the inductances of excitation winding and damper loops
are showed in Chapter 1. When the slot effect is not considered, the inductances
of rotor loops are constants independent of rotor position.
440
6 Internal Asymmetric Analysis of AC Machines
(3) Inductances between stator loops (or branches) and rotor loops
Based on the inductances of stator single coil and rotor loops the mutual
inductances between stator loops (or branches) and rotor loops are obtained.
Corresponding calculation formulas can be found in Chapter 1.
The mutual inductances between stator and rotor loops are time-variant
parameters relative to rotor position.
6.2.3
Calculation of Steady State for Internal Fault of Salient
Pole Synchronous Machine
After the stator and rotor voltage equations of synchronous machine are established
by using Multi-loop Model and the loop parameters are calculated, a group of
differential equations with time-variant coefficients is gained as formula (6.2.17),
then using fourth order Runge-Kutta method to solve the differential equation set,
finally get the steady state and transient values of stator and rotor currents,
furthermore other electric quantities (for example, power, etc.).
If only the steady-state of internal faults of electric machine stator winding is
concerned, the following method can be adopted in order to save computing time.
(1) Simplification of internal fault model in steady-state
At first based on physical concept the frequencies of various loop currents at
internal faults of electric machine stator winding are defined. When the electric
machine rotates at synchronous speed, because the configuration of the windings
(excitation winding and damper loops) under each pole of rotor are the same,
only stator winding is internal asymmetric, so the frequencies of stator and rotor
loop currents can be proved to be:
Frequency of stator loop current is mZ , m 1, 3, "
Frequency of excitation loop current is m1Z , m1 2, 4, "
n
Frequency of damper loop current is Z , n 1, 2, 3, " , P is pole-pair number
P
of the electric machine.
Accordingly, general expressions of various loop currents can be written out,
for example, stator loop Q current can be expressed as
iQ
¦ (I
Qm
c sin mZ t )
cos mZ t I Qm
m 1, 3, "
(6.2.20)
m
Excitation current can be expressed as
i fd
I f 0 ¦ ( I fm1 cos m1Z t I cfm1 sin m1Z t )
m1
2, 4, "
(6.2.21)
m1
The i - th damper loop current of first pole can be expressed as
441
AC Machine Systems
i1,i
§
¦ ¨© I
n
in
n
n ·
cos Z t I inc sin Z t ¸
P
P ¹
(6.2.22)
Accordingly, i - th damper loop current of g - th pole can be expressed as
ig , i
n
n
­
½
(1) g 1 ¦ ® I in cos >Z t ( g 1)S@ I inc sin >Z t ( g 1)S@¾
P
P
¯
¿
n
(6.2.23)
In above two formulas there is n 1, 2, 3, "
The expressions of various loop currents are substituted into differential equation
set (6.2.17), obtaining a transcendental equation set; then based on the equivalent
principle of the components with the same frequency, for each frequency component
the corresponding equation is listed. If two special times are selected in order to
make the sine term or cosine term be zero (for example, for fundamental order
Z t 0 and Z t S / 2 ), then two corresponding linear algebraic equations without
the time t are got, in these equations the unknowns are the amplitudes of sine
term and cosine term of each harmonic of each loop current, the coefficients of
the equations are inductance (including amplitude and phase angle) and resistance
of each loop. Solving these linear algebraic equations, the steady state currents of
all loops of the machine with internal faults, furthermore other electric quantities,
can be obtained.
(2) Comparison between steady state calculations and experiments of stator
winding internal faults of salient pole synchronous machine
While the internal faults of stator winding of salient pole synchronous machine
occur, the loop number of stator and rotor is more, besides fundamental, stator
currents also include 3, 5, " , etc. odd harmonics, besides direct current component,
excitation current also includes 2, 4, " even harmonics, the harmonics of damper
loop currents are much more. But the effect of the harmonics of higher order on
calculation results is less, hence generally calculated harmonic order is defined in
a certain scope. If the harmonic order of stator current is taken as J1 , the harmonic
order of excitation current is taken as J1 1 , the harmonic order of damper loop
current is taken as J 2 , the total number of stator branches is N , damper bar
number per pole of rotor is nc , then the total number of unknowns (i.e. the
amplitudes of sine terms and cosine terms of various loop currents of electric
machine) in the linear algebraic equation set describing the internal fault in steady
state is
M
( J1 1) N J1 1 J 2 u 2 Pnc
(6.2.24)
In general this high order linear algebraic equation set can be solved by Gauss
elimination method.
442
6 Internal Asymmetric Analysis of AC Machines
In the following the calculation results and corresponding experimental data of
various loop currents of stator winding internal faults of two salient pole synchronous
machines are given. Here the internal short circuits of a 630kW salient pole
synchronous machine occur during no-load and separate operation (island operation),
the electric machine stator winding is of 6 parallel branches per phase, 5 coils per
branch. Calculations and experiments are made for two internal faults: the short
circuit of the point 40% of first branch a1 of phase a to the neutral point, and the
short circuit of the point 40% of first branch a1 of phase a to the point 40% of
first branch b1 of phase b . In Table 6.2.1 the calculation results and experimental
data of the short circuit of the point 40% of first branch a1 of phase a to the
point 40% of first branch b1 of phase b are listed. In the table the dc excitation
current of calculation and experiment is I f 0 10.64A , various currents are rms
values, I f 2 is rms value of second harmonic of excitation current.
Another electric machine is a 30kW salient pole synchronous machine for
simulation and experiments, its stator winding is of two parallel branches per
phase. Three representative internal short circuit modes of the machine on line
with load are studied: the short circuit of the point 40% to point 2% of first
branch a1 of phase a , the short circuit of the point 40% of first branch a1 to the
point 10% of second branch a2 of phase a , and the short circuit of the point
20% of first branch a1 of phase a to the point 10% of first branch b1 of phase
b . In Table 6.2.2 the calculation and experiment results of the short circuit of the
point 40% to point 2% of first branch a1 are listed. In the table the dc
excitation current of calculation and experiment is I f 0 4.25A , various currents
and voltages are amplitude values, I f 2 is amplitude value of second harmonic of
excitation current, I 0 0 is current of connection line between two neutral points,
3U 0 is amplitude value of fundamental component of zero sequence voltage.
6.2.4
Calculation of Transients for Internal Fault of Salient Pole
Synchronous Machine
As mentioned above adopting Runge-Kutta method, etc. the differential equation
(6.2.17) is solved, and transient and steady state values of stator and rotor currents
can be obtained.
Simulations and experiments of stator winding internal faults of a 12kW
salient pole synchronous machine are made. Stator of the electric machine is of
two branches per phase, each branch consists of seven coils. The simulations and
experiments of interturn short circuit, the short circuit of two branches of the same
443
AC Machine Systems
phase, and short circuit of two branches of different phases are compared with
each others. Figure 6.2.5 is the simulation and experiment waves of the short
circuit current and excitation current while the short circuit of fifth coil of second
branch of phase a to neutral point occurs. Figure 6.2.6 is the simulation and
experiment waves of the short circuit current and excitation current while the
short circuit of sixth coil of first branch of phase a to third coil of second branch
of the same phase occurs. Figure 6.2.7 is the simulation and experiment waves of
the short circuit current and excitation current while the short circuit of third coil
of second branch of phase a to third coil of second branch of phase c occurs.
Table 6.2.1 Calculation and experiment
results of short circuit of the point a1 40%
to point b1 40% of a 630 kW salient pole
synchronous machine
Experimental
Table 6.2.2 Calculation and experiment
results of interturn short circuit of the
branch a1 of a 30 kW simulation electric
machine
Calculated Relative
J 1 3 J 2 2 error/%
Relative
ExperiCalculated
error/%
mental
I kL
51.0
50.372
1.2
U ab / V
***
330.2
I a1 f
27.0
27.450
1.7
U bc / V
***
377.5
I b1 f
22.5
20.782
7.6
Ia / A
49.6
44.2
10.9
I a1
Ia2
***
22.947
Ib / A
50.5
47.5
5.9
1.75
1.973
12.7
I a3
10.7
13.0
21.5
5.304
4.8
Ic / A
5.57
Ia4
5.02
4.858
3.2
P / kW
14.5
14.7
1.4
I a5
4.83
4.686
3.0
I a1 / A
75.8
64.3
15.2
I a6
I b1
6.70
6.241
6.9
Ia2 / A
30.6
30.0
2.0
***
29.608
I b1 / A
52.6
44.8
14.8
Ib 2
4.75
4.678
1.5
Ib2 / A
9.0
6.6
26.7
Ib3
4.82
5.095
5.7
I c1 / A
34.0
26.1
23.2
Ib 4
***
5.360
Ic2 / A
36.7
33.8
7.9
Ib5
4.31
4.922
14.2
I kL / A
343.0
323.8
Ib6
5.6
9.56
9.657
1.0
I c1
3.478
21.3
If2 /A
4.42
0.36
0.404
Ic 2
2.30
1.553
32.5
I00 / A
8.0
6.6
Ic3
***
0.332
Ic 4
***
Ic5
12.2
17.5
3U 0 / V
***
37.0
0.321
U2 / A
***
30.2
***
0.439
I2 / A
***
20.0
Ic6
3.33
3.289
1.2
P2 / W
***
239.9
If2
2.30
2.260
1.7
Note:***express no experiment results.
Note:***express no experiment results.
444
6 Internal Asymmetric Analysis of AC Machines
6.2.5
Features of Internal Faults of Salient Pole Synchronous
Machine
When internal faults of stator winding of salient pole synchronous machine occur,
besides the fundamental magnetic field with positive rotation synchronous speed
and fundamental magnetic field with reverse rotation synchronous speed (negative
sequence magnetic field), other harmonic magnetic fields with different rotation
speeds and different rotation directions exist in the air-gap.
Except the fundamental positive rotation magnetic field, other negative sequence
magnetic field and harmonic magnetic fields with relative movement to the rotor
induce the currents in rotor damper loops, but only negative sequence magnetic
field and odd harmonic magnetic fields cause the induced currents in the excitation
winding. Therefore the variation of electric quantities of stator winding internal
faults has its own features.
At first, the short circuit current of internal fault is very large, as showed in
Figs. 6.2.5, 6.2.6 and 6.2.7, if converted to no-load rated voltage condition the
excitation current I f 0 8.24A, the steady state internal short circuit currents of
the three states are 78.4A, 59.1A and 46.0A respectively, their values are 7.23,
5.45 and 4.24 times of rated branch current 21.7 / 2 10.85A respectively and
exceed enormously each branch current 7.06A of 3-phase steady state short
circuit. Considering so large internal fault short circuit current it is necessary to
provide suitable protection devices to deduce the damages.
The magnitude of internal short circuit current is relative to short circuit turn
number, winding form, connection fashion and space place of winding short circuit
part etc. Generally speaking, the less the short circuit turn number is, the larger
the short circuit current is, but its effect on asymmetry of other branch currents
without short circuits is smaller. In order to design suitable protection schedule for
internal faults it is necessary to analyze and calculate each internal fault roundly
and consider each factor synthetically.
The harmonics of each current of stator winding internal short circuit are
stronger. Besides dc component, second harmonic, fourth harmonic, etc. exist in
the excitation current; for stator currents, besides fundamental component, third
harmonic, fifth harmonic, etc. exist, as showed in Figs. 6.2.5, 6.2.6 and 6.2.7. In
Fig. 6.2.6 the second and fourth harmonics of excitation current are 50.5% and
14% of its dc component respectively, and third and fifth harmonics of stator
short circuit current are 13.8% and 4% of its fundamental component respectively.
These current harmonic components are caused by air-gap negative sequence
magnetic field and harmonic magnetic fields.
Similar to 3-phase sudden short circuit, the transient currents of internal short
circuits have both ac and dc components, the ac component will decay to steady
445
AC Machine Systems
Figure 6.2.5 Waves of short circuit currents and excitation currents at short circuit
from U a 25 to neutral point (dc excitation current I f 0 1.48A )
(a) Experiments; (b) Simulations
Figure 6.2.6 Waves of short circuit currents and excitation currents at short circuit
from U a16 to U a 23 (dc excitation current I f 0 3.10A )
(a) Experiments; (b) Simulations
446
6 Internal Asymmetric Analysis of AC Machines
Figure 6.2.7 Waves of short circuit currents and excitation currents at short circuit
from U a 23 to U c 23 (dc excitation current I f 0 2.05A )
(a) Experiments; (b) Simulations
value, the dc component will decay to zero, and initial value of dc component is
relative to the short circuit time. But large difference exists between the two short
circuit modes: compared with the steady state current the transient current of
internal short circuit is larger but not too much, which is not like 3-phase sudden
short circuit as showed in Figs. 6.2.5, 6.2.6, and 6.2.7. The reason is that even if
during the internal short circuit in steady-state, the induced currents are also caused
in rotor excitation winding and damper loops (owing to the air-gap negative
sequence magnetic field and harmonic magnetic fields with relative movement to
the rotor), consequently the currents of stator side become larger.
6.3 Internal Fault Analysis of Stator Winding of
Non-Salient Pole Synchronous Machines
Similar to salient pole synchronous machine, the internal faults of non-salient pole
synchronous machine stator winding will cause severe damages to the machine. It
is necessary to analyze deeply the internal faults of non-salient pole synchronous
machine stator winding and design suitable protection devices.
Analysis method of internal faults of non-salient pole synchronous machine is
different compared with the salient pole synchronous machine owing to its
447
AC Machine Systems
configuration features[14,15]. The internal fault analysis begins from mathematic
model of non-salient pole synchronous machine as follows.
6.3.1
Mathematic Model of Stator Winding Internal Fault of
Non-salient Pole Synchronous Machine
Compared with salient pole synchronous machine there are the following features
for the mathematic model of non-salient pole synchronous machine.
(i) The air-gap of non-salient pole electric machine is uniform, without
considering the effect of slot harmonic permeance, its self inductances and mutual
inductances of stator loops are all constants, the expressions of other inductances
are also simpler than salient pole machine (refer to Chapter 1).
(ii) Excitation winding of non-salient pole synchronous machine is distributed
symmetrically relative to the direct axis, so its calculation formula of inductance is
also different from salient pole electric machine.
(iii) The rotor of non-salient pole synchronous machine is whole solid
configuration forged mostly, so it is necessary to consider the eddy current damping
effect of solid magnetic pole.
Considering the features of non-salient pole synchronous machine, the calculation
of corresponding parameters is discussed in the following.
(1) Self inductance of excitation loop
Self inductance of excitation loop consists of two components, namely
L fd
L fdl L fdG
(6.3.1)
in which L fdl is inductance corresponding to end region leakage flux, inter pole
leakage flux and pole face leakage flux. L fdG is the inductance corresponding to
air-gap flux, its calculation method is as follows.
Figure 6.3.1 Excitation loop of non-salient pole electric machine
448
6 Internal Asymmetric Analysis of AC Machines
The excitation winding of non-salient pole electric machine consists of a series
of concentric coils in series, as showed in Fig. 6.3.1. Therein the pitch of i - th
concentric coil is E fdiW , the current in it is i fd / a fd , mmf produced by the current is
Ffd ( x)
¦F
fdk
cos(kx)
(6.3.2)
k
where
i fd
k E fdi S
14
sin
w fd
2
kS
a fd
Ffdk
w fd is turn number of one excitation coil per pole, a fd is parallel branch number
of excitation winding. So the self inductance of excitation winding is
N
L fdG
16 w2fdW lP P 0
N
¦¦ M
fdij
S2 a 2fd G
i 1 j 1
N
N
u ¦¦
i 1 j 1k
§ k E fdi S · § k E dfj S ·
1
sin ¨
¸ sin ¨
¸
2
1, 3, 5, ˜˜˜ k
© 3 ¹ © 2 ¹
¦
(6.3.3)
where N is series coil number per pole.
(2) Mutual inductance between damper loop and excitation loop
The mutual inductance between excitation winding and damper loop 11c is
8P 0 w fd wrW l
N
M 1, fd
¦¦ M
i 1
1, fdik
k
N
u¦
i 1 k
S2 a fd G
§ k E fdi S · § k E1S ·
1
sin ¨
¸ sin ¨
¸ cos kD1
2
1, 3, 5, " k
© 2 ¹ © 2 ¹
¦
(6.3.4)
where wr is damper loop turn number (generally it is 1), E1 is short pitch ratio
of damper loop 11c , D1 is the angle between the central line of damper loop 11c
and rotor direct axis.
(3) Inductances between stator coils and excitation winding
The mutual inductance between stator coil AAc and excitation winding is
M fda
8P 0 w fd wKW l
S2 a fd G
N
u¦
i 1 k
§ k E fdi S · § k E S ·
1
sin ¨
¸ sin ¨
¸ cos kJ
2
1, 3, 5, " k
© 2 ¹ © 2 ¹
¦
(6.3.5)
449
AC Machine Systems
where wK is stator coil turn number, E is short pitch ration of stator coil and J
is electric angle by which the rotor d-axis leads the stator coordinates axis.
(4) Effect of eddy current in rotor iron on parameters of non-salient pole
synchronous machine
Solid rotor is adopted mostly for non-salient pole synchronous machine
(especially for large turbo-generator), the effect of eddy current on electric machine
parameters, especially damper loop parameters, is larger. Once electric machine
configuration and material, etc. are defined, the eddy current effect of solid rotor
will be definite, and some corresponding parameters can be used to represent it,
the relation between parameters and frequencies is nonlinear. Here equivalent
concentrated treatment method is introduced, the eddy current parameters under
Multi-loop system are obtained from frequency response characteristic. For the
internal faults in stator winding of non-salient pole synchronous machine, it can
be considered that the induced eddy current in the rotor is mainly the component
of 100Hz . Without considering rotor iron saturation the calculation formula of
actual penetration depth corresponding to rotor eddy current effect is
2
2Sf sVP
G
Equivalent resistance and inductance are respectively
Red
l
VG b
,
Led
Red
2S fs
(6.3.6)
in which l is rotor length, P is magnetic permeability, V is conductivity, f s is
100Hz, b is equivalent width of calculated eddy region, while adopting full pitch
damper loop equivalent to the eddy current effect of solid rotor there is
SDr
b
, Dr is rotor diameter.
4P
The penetration depth is in reverse proportion to square root of frequency,
when frequency increases, the penetration depth decreases, resistance increases
and inductance decreases. It can be considered that the flux corresponding to the
eddy current only links the rotor itself, hence gained equivalent inductance can
be seen as leakage inductance.
6.3.2
Calculation and Experiments of Internal Faults of
Non-salient Pole Synchronous Machine
The flux-linkage equations and voltage equations in internal faults of stator
winding of non-salient pole synchronous machine are similar to salient pole
450
6 Internal Asymmetric Analysis of AC Machines
synchronous machine. While calculating internal faults of stator winding of nonsalient pole synchronous machine, its parameter expressions are substituted into
corresponding flux-linkage equations and voltage equations, gaining a differential
equation set with time-variant coefficients, the fourth order Runge-Kutta method
or other numerical methods are adopted to solve the differential equation set, then
steady state and transient values of stator and rotor currents, furthermore other
electric quantities (for example, power, etc.), will be obtained. If only analysis
and calculation of steady-state of internal faults are concerned, according to
physical concept the frequencies of loop currents of internal faults of stator
winding of electric machine are defined at first, and the expressions of various
loop currents are substituted into above mentioned differential equation set,
obtaining a transcendental equation set; then based on the equivalent principle of
the components with the same frequency, for each frequency component the
corresponding equation is listed. If two special times are selected in order to
make the sine term or cosine term be zero (for example, for fundamental order
Z t 0 or Z t ʌ / 2 ), then two corresponding linear algebraic equations without
the time t are got, in these equations the unknowns are the amplitudes of sine
term and cosine term of each current, the coefficients of the equations are
inductance (including amplitude and phase angle) and resistance of each loop.
Solving these linear algebraic equations, the steady state currents of all loops of
the machine with internal faults, furthermore other electric quantities, can be
obtained. The solving process is similar to the internal fault calculation of salient
pole synchronous machine, here more describing is not needed.
In order to research internal faults of stator winding of non-salient pole
synchronous machine, a series of simulation calculation and corresponding
experiments for internal faults of a non-salient pole generator are carried out.
Main specifications of the electric machine are:
S N 30kV ˜ A, PN 24kW, U N 400V, I N 43.3A,cos M N 0.8, f N 50Hz,
54, q 3, a1 2 (stator branch number per phase), wk 8
Z
30
(turn number per coil), 2
(rotor slot number/division number of rotor slots),
Z 2c 42
nN
G
1000r / min, Z1
1.1mm, I f 0
0.9576A, N c
7 (damping bar number per pole).
The following three sort internal faults are researched: interturn short circuit of
the same branch; short circuit between two branches in the same phase; short circuit
between two branches in different phases (refer to Fig. 6.3.2). The conditions for
calculation and experiments are: internal short circuits occur under single
machine separate operation and no load, the dc component of excitation
current I f 0 is kept constant. Figures 6.3.3, 6.3.4 and 6.3.5 are the simulation wave
and experimental wave at stator short circuit between 40% of branch a1 and 2% of
451
AC Machine Systems
branch a1. Table 6.3.1 is corresponding simulation and experiment result
comparison between steady state currents and line voltages. In the table ikL is
current of short circuit loop, ia1 , ib1 , ic1 are all the branch currents, uab , ubc are
terminal line voltages, ifd2 is second harmonic of excitation current, i0 0 is current
of connection line between two neutral points.
Figure 6.3.2 Three sort internal faults
Figure 6.3.3 Transient current of branch a1 during interturn short circuit between
40% of branch a1 and 2% of the same branch
(a) Experiment wave; (b) Simulation wave
Figure 6.3.4 Transient current of branch b1 during interturn short circuit between
40% of branch a1 and 2% of the same branch
(a) Experimental wave; (b) Simulation wave
452
6 Internal Asymmetric Analysis of AC Machines
Figure 6.3.5 Transient current ikL during interturn short circuit between 40% of
branch a1 and 2% of the same branch
(a) Experimental wave; (b) Simulation wave
Table 6.3.1 Comparison between simulation and experimental results of steady
state quantities during interturn short circuit between 40% of branch a1 and 2% of
the same branch
ikL /A
ia 1 /A
ib1 /A
ic1 /A
uab /V
ubc /V
i fd 2 /A
i0 0 /A
Simulation results
(effective value)
126.79
21.92
6.36
9.2
165.3
215.7
0.106
4.1
Experimental data
(rms value)
126.53
19.96
6.61
10.84
152
204
***
3.33
0.2
9.8
3.8
15
8.7
5.7
Error/ˁ
23
Note: ***indicates no experiment result.
Similar to salient pole synchronous machine, for internal faults of stator
winding of non-salient pole synchronous machine the following features exist:
(i) The short circuit current of internal fault is very large, so it is necessary to
pay special attention to the protection for internal faults. As for as the internal
fault current is concerned, the difference between branch currents for the same
phase is some times quite small, hence the difficulty in designing the internal fault
protection increases.
(ii) As the short circuit of external terminals, decaying dc component also exists
in the internal fault current. But there is larger difference between ac components
of both, the transient ac component of internal short circuit is a little more than
the steady state current not just as the short circuit of external terminals, because
at steady state internal short circuit the larger induced current is also caused in
rotor winding, which affects the stator current.
(iii) When internal fault occurs, the current and other electric quantities are
relative to not only the short circuit sorts but also the short circuit turn ratio and
the space position of short circuit turn, etc.
453
AC Machine Systems
References
[1] Qiu A R, Zhang L Zo (1987) Steady Performance Analysis of Rotor Failures of Bar Broking
and End Ring Broking of Squirrel Asynchronous Machine (in Chinese). Transactions of
China Electrotechnical Society, 3: 7 12
[2] Zhang L Z, Qiu A R (1987) Detecting Rotor Failure of Asynchronous Machine by
Frequency Spectrum Analysis (in Chinese). Transactions of China Electrotechnical Society,
4: 46 50
[3] Williamson S, Smith A C (1982) Steady-state analysis of 3-phase cage motors with rotor —
bar and end-ring faults. IEE Proc. B, 3: 93 100
[4] Kinitsky V A (1965) Calculation of Internal Fault Currents in Synchronous Machines.
IEEE PAS, 5: 381 389
[5] Kinitsky V A (1968) Digital Computer Calculation of Internal Fault Currents in a Synchronous
Machines, .IEEE PAS, 8: 1675 1679
[6] Kazofusk E Y i, etc. ( 1969) Abnormal Operation of Large Scale Synchronous Machine (in
Russian). Science Press
[7] Wang Xiangheng (1987) The Internal Fault Analysis of Stator Winding of Synchronous
Machine with Multi Branches on No-load (in Chinese). Proceeding of the CSEE, 5: 1 10
[8] Brameller A, Pandy R S (1974) General Fault Analysis using Phase Frame of Reference.
PIEE, 5: 366 368
[9] Zhang L Z, Wang X H, Gao J D (1991) A New Approach to Analysis of Synchronous
Machines with Asymmetrical Armature Windings (ĉ), Fundamental and Approach. Science
in China, Series A, 7: 866 874
[10] Zhang L Z, Wang X H, Gao J D (1991) A New Approach to Analysis of Synchronous
Machines with Asymmetrical Armature Windings (Ċ), Parameter Calculation. Science in
China, Series A, 8: 998 1004
[11] Wang X H, Sun Y G., Ouyang B, Wang W J, Zhu Z Q ,Howe D (2002) Transient behavior
of salient-pole synchronous machines with internal stator winding faults. IEE proc. -Electr.
Power Appl, 2: 143 151
[12] Gao J D, Wang X H, Li F H (1993) Analysis of AC Machines and Their Systems, 1st
Edition (in Chinese). Tsinghua University Press, Beijing
[13] Gao J D, Wang X H, Li F H (2005) Analysis of AC Machines and Their Systems, 2nd
Edition (in Chinese). Tsinghua University Press, Beijing
[14] Wang X H, Chen S L, Wang W J, Sun Y G, Xu L Y (2002) A Study of Armature Winding
Internal Faults for Turbogenerators. IEEE Trans. on IA, 3: 625 631
[15] Kulig T S, Buckly G W, Lambreicht D and Liese M (1990) A new approach to determine
transient generator winding and damper currents in case of internal and external faults and
abnormal operation. IEEE Trans. Energy Conversion, 3: 70 78
454
Appendix A Air-Gap Permeance Coefficients in
Consideration of Slot Effect
If the stator inner-surface is smooth, then during consideration of rotor salientpole effect the air-gap permeance coefficient is
Or ( x)
O0
2
¦ O2l cos 2lx
(A1)
l
where l 1, 2, " and the origin of abscissa x is taken on the rotor d-axis.
Supposing the rotor surface is smooth, considering the stator slot effect the
air-gap permeance coefficient is
Os (D1 ) a0 ¦ aQ cosQ
Q
Z
D1
P
(A2)
in which v 1, 2, " , Z is number of stator slots, D1 is expressed in electric degree
and the coordinate origin is taken on the central line of stator slot as shown in
Fig. A1. The coordinate origin is also taken on the central line of stator tooth as
shown in D 2 , and the distance between two coordinate origins is half a toothpitch D t / 2, where
Dt
Figure A1
Substituting D1 D 2 Dt
2
P u 2S
Z
Stator teeth and their coordinates
into expression (A2) exists
(A3)
AC Machine Systems
§ Z
·
D 2 Q S¸
© P
¹
Q
Z
a0 ¦ (1)Q aQ cosQ D 2
P
Q
Os (D 2 ) a0 ¦ aQ cos ¨Q
(A4)
where Q 1, 2, "
From the above formula we can see, no matter how taking the origin on the
central line of stator slot or tooth, the form of Os is unchanged but the harmonic
terms change their sign. In the following we take D to express the electric angle
of stator coordinates to unify them.
The whole air-gap permeance coefficient can be expressed as
O ( x, D ) Or ( x)Os (D )
§ Z ·
a0 Or ( x) ¦ aQ cos ¨Q D ¸Or ( x)
© P ¹
Q
Q 1, 2, "
(A5)
in which the first term a0 Or ( x) indicates the decrease of air-gap permeance due
to stator slot effect, and the second term represents the slot harmonics of air-gap
permeance due to stator slot effect. If Ot indicates the slot harmonics, then there is
Ot ( x, D )
§ Z
·
aQ cos ¨Q D ¸O ( x)
¦
© P ¹
Q
r
Q 1, 2, "
(A6)
Formula (A6) is the product of two infinite series, in which the first several terms
are primary. If taking the first two terms for the two series, then exists
§ O0
·
O2 cos 2 x ¸
© 2
¹
Ot ( x, D ) (a1 cos tD a2 cos 2tD ) ¨
(A7)
Z
and the relation of stator coordinates x to rotor coordinates D
P
(see Fig. A2) is
in which t
D
x T
Figure A2 Relation between stator and rotor coordinates
456
(A8)
Appendix A Air-Gap Permeance Coefficients in Consideration of Slot Effect
Substituting formula (A8) into (A7) exists
Ot ( x)
1
1
a1O0 cos t ( x T ) a2 O0 cos 2t ( x T )
2
2
1
1
a1O2 cos[(t 2) x tT ] a1O2 cos[(t 2) x tT ]
2
2
1
1
a2 O2 cos[(2t 2) x 2tT ] a2 O2 cos[(2t 2) x 2tT ]
2
2
(A9)
If expressing the slot harmonics of air-gap permeance coefficient according to
stator coordinates D, then there is
Ot (D )
1
1
1
a1O0 cos tD a2 O0 cos 2tD a1O2 cos[(t 2)D 2T ]
2
2
2
1
1
a1O2 cos[(t 2)D 2T ] a2 O2 cos[(2t 2)D 2T ]
2
2
1
a2 O2 cos[(2t 2)D 2T ]
2
(A10)
Based on the slot harmonics of air-gap permeance coefficient, it is not difficult to
get the inductances of stator and rotor various loops corresponding to the slot
harmonics of air-gap permeance.
457
Appendix B Expressions of Effective Air-Gap
Air-gap and relevant dimensions are shown in Fig. B1 for salient-pole synchronous
machines. Because the effective air-gap under pole-shoe is quite different from
that between adjacent poles, the air-gap can be divided into two parts, namely
DS ·
§
the part under pole-shoe ¨ 0 x ¸ and the part between adjacent poles
¹
©
S·
§ DS
x ¸ for convenience, and then get the corresponding calculating formulas.
¨
¹
© Figure B1
Air-gap and relevant dimensions for salient-pole synchronous machines
(1) The part under pole-shoe
The air-gap length under pole-shoe is much less than a pole-pitch in practical
machines, so it can be considered that the flux passes through air-gap in radial
direction and the air-gap permeance coefficient O2lP is inversely proportional to
G P ( x) only. Thus, according to formula (1.2.5) write down the calculating formula
of O2lP the air-gap permeance coefficient under pole-shoe as follows:
OG ( x)
O2lP
P0
G P ( x)
4P 0 DS cos 2lx
dx
S ³ G P ( x)
(B1)
Appendix B
Expressions of Effective Air-Gap
Based upon Fig. B1, G P ( x) can be expressed as
G P ( x) R H cos
x
§ x·
rP2 H 2 sin 2 ¨ ¸
P
© P¹
(B2)
For convenience, the above formula of G P ( x) can be approximate to
G P ( x) G min (G max
§ x·
sin 2 ¨ ¸
© P¹
G min )
DS ·
§
sin 2 ¨
¸
© 2P ¹
If using relative value G P ( x), then there is
G P ( x)
1 ( U 1)
G P ( x)
G min
O2lP
P0
G min
§ x·
sin 2 ¨ ¸
©P¹
§ DS ·
sin 2 ¨
¸
© 2P ¹
ª 4 D2S cos 2lx º
dx »
« ³0
G P ( x) ¼
¬S
(B3)
P0
O 2lP
G min
where
O 2lP
U
4 D2S cos 2lx
dx
S ³ 0 G P ( x)
G max
G min
(2) The part between adjacent poles
The effective length of air-gap between adjacent poles is large, so the flux
distribution depends upon not only the air-gap length but also the order of mmf
harmonics and the position the mmf acts on. In the light of graphic and analytical
results, we can get the approximate formula of O2li the air-gap permeance
coefficient between adjacent poles.
(i) For the mmf, whose zero-value is at the central line between two adjacent
poles, such as the mmf Fdk ( x) during k odd number and mmf Fqk ( x) during
k even number, the relative value of the coefficient is
O d 2li
4 S2 cos 2lx
dx
DS
S ³ 2 G di ( x)
(B4)
in which
459
AC Machine Systems
U sinh K
G di ( x)
K
ª S x º
sinh « K
»
¬ D S ¼
ª1 D W º
0.768 «
»
¬ 2 G max ¼
0.617
(B5)
ª1 D W º
arctan «
»
¬ 2 G max ¼
(ii) For the mmf, whose maximum is at the central line between two adjacent
poles, such as the mmf Fqk ( x) during k odd number and mmf Fdk ( x) during
k even number, the relative value of the coefficient is
4 S2 cos 2lx
dx
DS
S ³ G qi ( x)
O q 2li
(B6)
where
G qi ( x)
ª
U «1 ¬
DS ·
S º
§
»
¨x
¸ cos
G max S ©
2 ¹
2P ¼
W
(B7)
Having the permeance coefficient under pole-shoe and that between two adjacent
poles on two conditions, the relative values of air-gap permeance coefficient
O 2l on two conditions can be got and indicated respectively by O d 2l and O q 2l
(i) When the zero-value of mmf is at the central line between two adjacent
poles, there is
O d 2l
O 2lP O d 2li
(B8)
(ii) When the maximum value of mmf is at the central line between two
adjacent poles exists
O q 2l
O 2lP O q 2li
(B9)
In practical terms, the difference between Od 2li and Oq 2li for the part between
adjacent poles is small. In addition, neither the zero-value nor the maximum
value of mmf is at the central line between two adjacent poles during harmonic
order k fraction, so during calculation of air-gap permeance coefficient take
the average value of both O d 2li and O q 2li , namely
460
O 2li
1
(O d 2li O q 2li )
2
O 2l
1
(O d 2 l O q 2 l )
2
(B10)
O 2lP O 2li
(B11)
Appendix C
Inductances Due to Leakage Flux of
Stator Winding End
Calculating the leakage flux of stator winding end for 3-phase symmetrical
winding, the symmetrical 3-phase currents can be processed usually by use of
equivalent current sheets, but this method is unsuitable for studying inner
unsymmetrical problems of synchronous machine with salient-poles based on the
single coil, so the discrete method can be used to analyse those problems. When
a coil end carrying current, it can be divided into several current elements, the
corresponding magnetic field for one current element is calculated according to
Biot-Savart law, and the magnetic field at a certain point of coil end can be got
by superposition of the corresponding magnetic fields for various current elements
to obtain flux-linkages of various coil ends and the corresponding inductances.
The conception of ferromagnetic image and air-gap current is adopted for evaluation.
The inductances due to leakage magnetic field of stator winding end are discussed
respectively on Cartesian coordinates and cylindrical coordinates as follows.
(1) Calculation of inductances due to leakage magnetic field of stator winding
end on cartesian coordinates
For convenience, the evolute part and straight-line part of stator winding end
are considered to be on the same cylindrical surface, which can be developed into
a plane and then the inductances due to leakage magnetic field of stator winding
end are discussed on this plane.
Cartesian coordinates are built as shown in Fig. C1, in which CDEFG indicates
the reduced coil end on the plane xOy. The leakage magnetic field of stator winding
end is of three dimensions and in the magnetic field exist various media such as
air, iron, etc. The media in the magnetic field can be reduced in the following.
Firstly the air-gap action is superseded by air-gap current. In air-gap there is
imaginary air-gap current and the air-gap doesn’t exist any longer, so the analysed
problem is changed to the problem that one side is iron and another is air. Then,
according to the conception of ferromagnetic image method the ferromagnetic
medium action can be replaced by image current, so the leakage magnetic field
of stator coil end is simplified as the field caused by coil-end current, air-gap
current and their image in the uniform air medium. Supposing current i passes
2P E
through the coil end CDEFG, then the air-gap current of section AB is
i
2P
Ei
for the outer part of coil end. BC and AG
for the inner part of coil end and
2P
are imaginary radial current to make current continuous, and CDcE cF cG is the
AC Machine Systems
image of coil end current CDEFG. Similarly, air-gap current and radial current
have their own images, too.
Figure C1 The reduced stator coil end on Cartesian coordinates
After so simplification, the leakage magnetic field of stator winding end can
be solved by use of Biot-Savart law. The passages are all short for radial currents
BC and AG and then have opposite direction, so their effect can be neglected. In
addition, the air-gap current and coil-end current are not on the same plane, so
they can be processed separately.
In the following analyse the magnetic field produced by the coil end current and
its image on the coil end plane. The plane xOy is plotted separately as shown in
Fig. C2, where the coordinate positions are indicated for various points of coil end.
Figure C2
462
Magnetic field produced by coil end current and its image on the plane xOy
Appendix C
Inductances Due to Leakage Flux of Stator Winding End
According to Biot-Savart law, for the flux density at any point N ( x0 , y0 ) exists
dB
P 0 idl u r0
4S
(C1)
r2
Considering the action of each section current separately, the current element of
section FG is idx, and there is
P0 idx sin T
dBFG
4S
r2
P0
§a
·
i ¨ y0 ¸
©2
¹
2 3/ 2
4S ª
§a
· º
2
«( x0 x) ¨ y0 ¸ »
©2
¹ ¼»
¬«
dx
where dBFG is the flux density in z-direction produced by the current element
idx for the section FG at point N assuming the positive direction of flux density
B to be out of the paper.
The flux density in z-direction caused by the section FG current at point N is
BFG
³
0
§a
·
i ¨ y0 ¸
2
©
¹
P0
b
2 3/ 2
4S ª
§a
· º
2
«( x0 x) ¨ y0 ¸ »
©2
¹ »¼
«¬
P0
dx
x0
x0 b
ª
«
2
2
4S a y
§a
·
§a
·
2
0 « x2 ( x0 b) ¨ y0 ¸
y
2
« 0 ¨© 2 0 ¸¹
©2
¹
¬
i
º
»
»
»
¼
(C2)
The flux density in z-direction produced by the section F cG current at point N is
BF cG
³
0
b
P0
P0
§a
·
i ¨ y0 ¸
2
©
¹
2 3/ 2
4S ª
§a
· º
2
«( x0 x) ¨ y0 ¸ »
©2
¹ »¼
«¬
dx
x0
x0 b
ª
«
2
2
4S a y
§a
·
§a
·
2
0 « x2 ( x0 b) ¨ y0 ¸
y
2
« 0 ¨© 2 0 ¸¹
©2
¹
¬
i
º
»
»
»
¼
(C3)
The flux density in z-direction caused by the section CD current at point N is
463
AC Machine Systems
³
BCD
b
0
P0
§a
·
i ¨ y0 ¸
©2
¹
2 3/ 2
4S ª
§a
· º
2
«( x0 x) ¨ y0 ¸ »
©2
¹ »¼
¬«
dx
P0
x0
x0 b
i ª
º
«
»
2
2
a
4S y
§a
·
§a
· »
2
0 « x2 (
)
y
x
b
y
¨
0¸
0
0¸ »
2
« 0 ¨© 2
¹
©2
¹ ¼
¬
(C4)
The flux density in z-direction produced by the section CDc current at point N is
BCDc
³
0
b
P0
§a
·
i ¨ y0 ¸
©2
¹
2 3/ 2
4S ª
§a
· º
2
«( x0 x) ¨ y0 ¸ »
©2
¹ »¼
¬«
dx
P0
x0
x0 b
i ª
º
«
»
2
2
a
4S y
§a
·
§a
· »
2
0 « x2 (
)
y
x
b
y
¨
0¸
0
0¸ »
2
« 0 ©¨ 2
¹
©2
¹ ¼
¬
(C5)
The flux density in z-direction caused by the section EF current at point N is
BEF
a
ac
x0 (b c) y0 2
2
i
3/ 2
4S
ª a 2 4(b c) 2 º
(c b) «
»
2
¬ 4(b c)
¼
P0
­°
½°
c P1
b P1
®
¾
2
2
¯° Q1 (c P1 ) Q1 Q1 (b P1 ) Q1 ¿°
(C6)
The flux density in z-direction produced by the section E cF c current at point N is
BE cF c
a
ac
x0 (c b) y0 2
2
i
3/ 2
4S
ª a 2 4(c b) 2 º
(b c) «
»
2
¬ 4(c b)
¼
P0
­°
c P2
b P2
®
2
2
°¯ Q2 (c P2 ) Q2 Q2 (b P2 ) Q2
½°
¾
¿°
(C7)
The flux density in z-direction caused by the section DE current at point N is
464
Appendix C
BDE
Inductances Due to Leakage Flux of Stator Winding End
a
ac
x0 (b c) y0 2
i 2
3/ 2
4S
ª a 2 4(b c) 2 º
(c b ) «
»
2
¬ 4(b c)
¼
P0
­°
½°
b P3
c P3
®
¾
2
2
¯° Q3 (b P3 ) Q3 Q3 (c P3 ) Q3 ¿°
(C8)
The flux density in z-direction produced by the section DcE c current at point N is
BDcE c
a
ac
x0 (c b) y0 2
2
i
3/ 2
2
2
4S
ª a 4(c b) º
(b c) «
»
2
¬ 4(c b)
¼
P0
­°
c P4
b P4
®
2
2
°¯ Q4 (c P4 ) Q4 Q4 (b P4 ) Q4
½°
¾
°¿
(C9)
P1 and Q1 in formula (C6) are expressed respectively as
x0 P1
·
a § ac
y0 ¸
¨
2(b c) © 2(b c)
¹
2
2
a 4(b c)
4(b c) 2
2
Q1
(C10)
§ ac
· ª
·º
a § ac
y0 ¸ « x0 y0 ¸ »
x02 ¨
¨
2(b c) © 2(b c)
© 2(b c)
¹ «
¹»
2
2
«
»
a 4(b c)
a 2 4(b c) 2
«
»
2
2
4(b c)
4(b c)
¬
¼
2
(C11)
In formulas (C10) and (C11), b and c are replaced respectively by ( b) and ( c)
to get P2 and Q2 in formula (C7); a is superseded by ( a) to obtain P3 and Q3 in
formula (C8); a, b and c are replaced respectively by ( a), ( b) and ( c) to
get P4 and Q4 in formula (C9).
The flux-density normal component perpendicular to the coil-end plane
produced by air-gap current of section AB at point N is
Bz1
a
2
a
2
³
P0 2P E
4S
2P
ix0
dy
[ f ( y0 y )2 x02 ]3 / 2
2
ª
a
«
y0
P0 2P E
1 «
2
ix0 2
2
4S 2 P
f x02 «
« f 2 x2 § a y ·
¨
0
0¸
«
©2
¹
¬
º
»
»
2 »
§a
·
f 2 x02 ¨ y0 ¸ »
©2
¹ »¼
(C12)
a
y0
2
465
AC Machine Systems
The air-gap current for the outer part of coil end is divided into two sections,
and the flux-density normal components perpendicular to the coil-end plane
produced by them at point N respectively are
Bz 2
³
a
2
Pa
E
P0 E
ix0
dy
4S 2 P [ f x ( y0 y )2 ]3 / 2
2
2
0
ª
a
y0
1 «
2
«
ix0 2
2
4ʌ 2 P
f x02 «
§a
·
2
2
« f x0 ¨ y0 ¸
©2
¹
«¬
P0 E
Pa
Bz 3
³
E
a
2
Pa
º
»
»
2
»
ª
º
§
·
Pa
f 2 x02 Ǭ
y
¸ 0» »
¬© E ¹
¼ »¼
(C13)
E
y0
P0 E
ix0
dy
4S 2 P [ f 2 x02 ( y0 y )2 ]3 / 2
Pa
ª
y0
1 «
E
«
ix0 2
2
f x02 «
4ʌ 2 P
ª
º
§
·
Pa
« f 2 x02 «¨
¸ y0 »
«¬
¬© E ¹
¼
P0 E
º
»
»
2
»
ª
º
a
§
·
f 2 x02 «¨ ¸ y0 » »
¬© 2 ¹
¼ »¼
(C14)
a
y0
2
In formulas (C12), (C13) and (C14), f is the perpendicular distance between the
air-gap current and coil-end plane. Counting in the image of air-gap current, the
values of Bz1 , Bz 2 and Bz 3 obtained above should be doubled.
Summarizing the flux-density normal components produced by various section
currents of coil end, air-gap current and their images at point N on the coil-end
plane, we can get the total flux density at this point as follows:
B
BFG BFG c BCD BCDc BEF BE cF c BDE BDcE c
2( Bz1 Bz 2 Bz 3 )
(C15)
Dividing the coil-end plane into several domains, finding out the flux-density
and flux for each domain, adding the flux values of all domains and then
multiplying the flux by corresponding number of turns to get flux-linkage, the
flux-linkage is divided by current i to obtain self inductance corresponding to
leakage flux of coil end. In a similar way, the mutual inductance due to leakage
flux of coil end can also be got.
During evaluation the conductor current is assumed to be concentrated on the
central line, so the flux density of conductor position can’t be calculated or the
result is infinite. Calculating the leakage self inductance of coil end, the conductor
is considered to have circular cross-section and only the flux passing through the
466
Appendix C
Inductances Due to Leakage Flux of Stator Winding End
area C1 D1 E1 F1G1 is counted in, the area being formed by conductor inner-surface
of coil end as shown in Fig. C3. As for the inner self inductance due to inner flux-
P0
le where le is the length of coil end.
8S
After finding out the inductance of single coil end, the inductances due to
coil-end leakage flux can also be obtained for various stator loops. N.B. each
stator coil has two ends.
linkage of the conductor itself, its value is
Figure C3 Outer self inductance and inner self inductance for the coil-end conductor
(2) Calculation of inductances due to leakage magnetic field of stator winding
end on cylindrical coordinates
Although it is feasible to study inductances due to leakage flux of stator coil
end on Cartesian coordinates, the coil end is in fact a cylindrical surface so it is
more suitable to use cylindrical coordinates to study the topic. The evolute part and
straight-line part of the coil end are assumed to be on the same cylindrical surface.
On cylindrical coordinates the flux densities caused by the end straight-line
part current, end evolute part current and air-gap current are analysed respectively
as follows.
(a) Flux density produced by the end straight-line part current
Firstly analyse the normal flux density produced by axial current element idl
for point A at point K on the same end cylindrical surface, referring to Fig. C4.
Figure C4 Analysis of the flux density caused by the end straight-line part current
on cylindrical coordinates
467
AC Machine Systems
In the light of Biot-Savart law exists
dB
P 0 dl u r0
4S
i
r2
The dB and dl u r0 have the same direction. Since dB is perpendicular to dl, the
dB at point K sits in the lateral plane and is perpendicular to the plane of Ƹ$BK.
Thus, dB is perpendicular to BK, too. The normal component of dB at point K is
(dB) n
dB cos
D
2
where D is the difference between spacial angles for points A and K along the
circumference and dB is
dB
P0
sin T
idl 2
4S
r
P0
BK
idl 3
4S
r
2 R sin
P0
4S
id l
D
2
2
ª 2 §
D· º
« AB ¨ 2 R sin ¸ »
2 ¹ ¼»
©
¬«
3/ 2
If the axial coordinate position for calculating point K is z0 and the position for
current element point A is Z, then there is AB z0 z and dl dz , so exists
(dB) n
P0
4S
iR
sin D
2
ª
D· º
§
2
«( z0 z ) ¨ 2 R sin ¸ »
2 ¹ ¼»
©
¬«
3/ 2
dz
(C16)
Assuming the overhanging length to be b for the end straight-line part, then the
normal flux density caused by its current at point K of the end is
Bn
³
b
P0
0
4S
iR
sin D
2
ª
D· º
§
2
«( z0 z ) ¨ 2 R sin ¸ »
2 ¹ ¼»
©
¬«
3/ 2
dz
ª
«
P0
z0
z0 b
i
«
«
2
2
D
4S 2 R tan
D·
§
2
« z 2 § 2 R sin D ·
(
)
2
sin
z
b
R
¸
¨
¸
0
2« 0 ¨
2¹
2¹
©
©
¬
º
»
»
»
»
»
¼
(C17)
In order to obtain the normal flux density produced by the end straight-line
part currents and their images, totalling 4 section currents, at any point K on the
end cylindrical surface, we build the cylindrical coordinates as shown in Fig. C5,
in which the coordinate position at point K is ( R, D 0 , z0 ), CD and FG are the end
468
Appendix C
Inductances Due to Leakage Flux of Stator Winding End
straight-line parts, CDc and FG c are their images respectively. The normal flux
densities caused by various section currents at point K are studied respectively in
the following.
Figure C5 Analysis of normal flux density produced by the end straight-line part
currents on the end cylindrical surface according to cylindrical coordinates
§T
·
For section CD, the angle D in formula (C17) is replaced by ¨ D 0 ¸ ;
©2
¹
§T
·
For section FG, the angle D in formula (C17) is superseded by ¨ D 0 ¸ ;
©2
¹
§T
·
For section CDc, in formula (C17) the angle D is replaced by ¨ D 0 ¸
2
©
¹
and z0 is replaced by ( z0 b);
§T
·
For section F cG, in formula (C17) the angle D is superseded by ¨ D 0 ¸ and
2
©
¹
z0 is superseded by ( z0 b).
Thus, various normal flux densities caused by the end straight-line part currents
and their images at any point K on the cylindrical surface are
BFG BF cG BCD BCDc
­
°
P0 i °
1
®
(T / 2) D 0
4S 2 R °
tan
°
2
¯
ª
«
2 z0
«
«
2
(T / 2) D 0 º
« 2 ª
»
« z0 «¬ 2 R sin
2
¼
¬
º
»
z0 b
z0 b
»
2
2 »
(T / 2) D 0 º
(T / 2) D 0 º »
ª
ª
( z0 b)2 « 2 R sin
( z0 b) 2 « 2 R sin
»
» »
2
2
¬
¼
¬
¼ ¼
469
AC Machine Systems
ª
«
2 z0
z0 b
1
«
«
2
2
(T / 2) D 0
(T / 2) D 0 º
(T / 2) D 0 º
ª
« 2 ª
tan
2
( z0 b) « 2 R sin
2
»
»
« z0 «¬ 2 R sin
2
2
¼
¬
¼
¬
º½
»°
z0 b
»°
¾
2 »
(T / 2) D 0 º » °
ª
2
( z0 b) « 2 R sin
» »°
2
¬
¼ ¼¿
(C18)
(b) Flux density produced by the end evolute part current
The coil end and its image are unfolded on the end cylindrical surface as
shown in Fig. C6, in which E is the angle included between the evolute part and
straight-line part. If the current element i'l of the end evolute part is divided
into axial and circumferential components, then there is
i'l
i ('l ) z e z i ('l )M eM
i'l cos E e z i'l sin E eM
(C19)
where e z and eM are axial and circumferential unit vectors respectively.
Figure C6
The map of coil end and its image unfolded on the end cylindrical surface
The actions of axial and circumferential currents will be discussed respectively
as follows. The normal flux density caused by the axial current of evolute section
DE at point K on the end is
470
Appendix C
Inductances Due to Leakage Flux of Stator Winding End
P0
¦ 4S iR
Bna1
z
sin D
2
ª
D· º
§
2
z
z
R
(
)
2
sin
« 0
¨
¸ »
2 ¹ »¼
©
«¬
3/ 2
'z
(C20)
cz T
D 0 and z is from b to c for summation.
cb 2
The normal flux density produced by the axial components of sections DE and
EF currents and their images, totalling 4 section currents, at point K is
where D
Bna
­
°
P0 °°
sin D1
¦z 4S iR ®
2 3/ 2
°ª
D1 · º
§
2
° «( z0 z ) ¨ 2 R sin ¸ »
2 ¹ »¼
©
¯° «¬
sin D 3
sin D 2
2 3/ 2
2 3/ 2
ª
ª
D2 · º
D3 · º
§
§
2
2
«( z0 z ) ¨ 2 R sin ¸ »
« ( z0 z ) ¨ 2 R sin ¸ »
2 ¹ »¼
2 ¹ ¼»
©
©
«¬
¬«
½
°
°°
sin D 4
¾'z
3
/
2
2
ª
º
°
D ·
§
2
«( z0 z ) ¨ 2 R sin 4 ¸ » °
2 ¹ ¼» °¿
©
¬«
(C21)
where
D1
D3
czT
D0 ,
cb 2
czT
D0 ,
cb 2
D2
D4
czT
D0
cb 2
czT
D0
cb 2
The normal flux density caused by the circumferential current of evolute section
DE at point K on the end is
Bnt1
P0
¦ 4S i tan E cosD
z
z0 z
2
ª
D· º
§
2
«( z0 z ) ¨ 2 R sin ¸ »
2 ¹ ¼»
©
¬«
3/ 2
'z
(C22)
where
D
cz T
D0
cb 2
471
AC Machine Systems
The normal flux density produced by the circumferential components of evolute
sections DE and EF currents and their images, totalling 4 section currents, at
point K on the end is
Bnt
­
°
°°
P0
( z z0 ) cos D1
¦z 4S i tan E ®
2 3/ 2
°ª
D1 · º
§
2
° «( z z0 ) ¨ 2 R sin ¸ »
2 ¹ »¼
©
°¯ «¬
( z z0 ) cos D 2
( z z0 ) cos D 3
3
/
2
2
2 3/ 2
ª
ª
D2 · º
D3 · º
§
§
2
2
«( z z0 ) ¨ 2 R sin ¸ »
«( z z0 ) ¨ 2 R sin ¸ »
2 ¹ ¼»
2 ¹ ¼»
©
©
¬«
¬«
½
°
°°
( z z0 ) cos D 4
¾' z
3
/
2
2
ª
º
°
D
§
·
2
«( z z4 ) ¨ 2 R sin 4 ¸ » °
2 ¹ ¼» ¿°
©
«¬
(C23)
where
D1
D3
czT
D0 , D2
cb 2
czT
D0 , D4
cb 2
czT
D0
cb 2
czT
D0
cb 2
From formula (C21) to (C23), z is from b to c for summation.
(c) The end flux density produced by air-gap current
Air-gap current and its image total two sections, and the normal flux density
produced by air-gap current of coil inner-part at point K is
BnG1
P0
2i
¦
D 4S
z0 R c
2P E
'D
cos D 2
2
2
2P
( z0 R Rc 2 RRc cos D )3 / 2
(C24)
where Rc is the radius for the position the air-gap current sits at, and D is from
T·
T·
§
§
¨ D 0 ¸ to ¨ D 0 ¸ .
2¹
2¹
©
©
The normal flux density produced by air-gap current of coil outer-part at point
K is
BnG 2
472
P0
E
2i
cos D
¦
4S 2 P
(z
D
2
0
z0 R c
'D
2
c
R R 2 RRc cos D )3 / 2
2
(C25)
Appendix C
Inductances Due to Leakage Flux of Stator Winding End
T·
T
§
§
·
where D is from ¨ D 0 ¸ to ¨ D 0 2S ¸ .
2
2¹
©
©
¹
Summarizing the results of formulas (C18), (C21), (C23), (C24) and (C25),
the total normal flux density at point K on the end is
Bn
BFG BF cG BCD BCDc Bna Bnt BnG1 BnG 2
(C26)
After obtaining the normal flux density of coil end, the flux-linkage and
corresponding inductance can be got for the coil end.
As mentioned above, calculation of coil-end inductance has been discussed on
the basis of Cartesian coordinates and cylindrical coordinates. When evaluating
the normal flux density of coil end, the analytical expression can be got
according to Cartesian coordinates, but on cylindrical coordinates the analytical
expression can’t be sometimes obtained due to difficulty of integration, which
obliges us to adopt summation of various pieces to bring about a tedious
calculation. The calculating example also demonstrates that the evaluation results
according to the two coordinates above are approximate to each other.
473
Appendix D Heaviside’s Operational Calculus and
Its Application to Transients Analysis
In Laplace’s operational calculus, the transformation functions of voltages and
currents are all the functions of p, which is sometimes named the pure operational
calculus and usually used in analysis of linear circuits and control systems.
Besides the method has existed another operational calculus that is known as
Heaviside’s operational calculus. If taking a R-L series circuit as an example and
writing the voltage equation as Heavisides’s operational form, we can get
u (t )
Lpi (t ) Ri (t )
( Lp R)i (t )
or
i(t )
1
u (t )
Lp R
1
u (t )
Z ( p)
d
is termed the operator. In this method, the operational impedance
dt
Z ( p) is a function of p but the voltage and current are still time functions. The
mathematical basis of this method is not perfect enough but it is often convenient
to use it. In analysis of machine transients, the basic relations are nonlinear
differential equations during a variable speed, which can be described easily by
Heaviside’s operational method. However, we have to look out for neglect
during application of this method. Furthermore, the method is often used only on
the zero initial condition, so the unit function [1] is sometimes put behind the
expression but is omitted here for convenience. When this method is used to
analyse a circuit of the nonzero initial condition, it is necessary to adopt the
superposition theorem to give a special treatment.
Supposing that the initial condition of the circuit is zero, the applied voltage is
u (t ), the transient current is i (t ), and the operational impedance is Z ( p ), we
can write Heaviside’s operational equation as
in which p
Z ( p )i (t ) u (t )
or
i(t )
1
u (t )
Z ( p)
(D1)
Appendix D Heaviside’s Operational Calculus and Its Application to Transients Analysis
It should be noted that in the formulas above Z ( p ) is considered to be applied
1
applied on u (t ). For example, the latter is considered to be not
on i (t ) or
Z ( p)
1
that the original function f (t ) of
is multiplied by u (t ), i.e.
Z ( p)
1
i (t ) z f (t ) ˜ u (t ), but that
operates on u (t ). If the equation form, however,
Z ( p)
1
1
is i(t ) u (t )
is not applied on u (t ) but i (t ) is
, it will imply that
Z ( p)
Z ( p)
1
equal to u (t ) times the original function f (t ) of
. Therefore, during
Z ( p)
application of this operational method, the sequence of various terms can’t
change arbitrarily, which is also a weakness of this operational method.
Supposing that the voltage u (t ) in equation (D1) is converted into the
corresponding image function u ( p), the current can be expressed as the image
function as well, namely
i( p)
1
u ( p)
Z ( p)
(D2)
which is the pure operational form, where the sequence of various terms can
change at will because it has been a pure algebraic equation.
After determination of the Heaviside’s operational expression of the current,
the corresponding original function may be looked up from the Heaviside’s
operational collating table which is found in some references and omitted here.
As for several ordinary voltages, however, it is convenient to use the Heaviside’s
shifting theorem and decomposition one to get the solutions of the currents,
which will be introduced as follows.
(a) Heaviside’s shifting theorem
F ( p )u (t ) AeD t
AeD t F ( p D )u (t )
(D3)
in which A and D are all constants (real or complex).
(b) Heaviside’s operation and its decomposition theorem when the applied
voltage is an exponential voltage UeD t
Under this condition, the Heaviside’s operational expression with reference to
equation (D1) can be written as
i (t )
1
UeD t
Z ( p)
(D4)
In the light of Heaviside’s shifting theorem exists
475
AC Machine Systems
i (t ) UeD t
1
Z( p D)
Now the problem is how to find the original function of
(D5)
1
, through
Z(p D)
which times UeD t we can get the transient current i (t ). The method of deciding
1
is the same as the pure operational calculus.
the original function of
Z(p D)
Generally speaking, Z ( p) may be a complicated rational fraction and letting
Z ( p)
Q( p)
H ( p)
there will be
Z(p D)
Q( p D )
H(p D)
1
Z(p D)
H(p D)
Q( p D )
so
Supposing that the roots of Q( p) 0 are p1 , p2 ," , pn , the roots of
Q( p D ) 0 can be written as p1 D , p2 D ," , pn D , so
H ( pk )
1
H (D ) n
e( pk D ) t
̯
¦
Z ( p D ) Q(D ) k 1 ( pk D )Qc( pk )
n
1
1
e( pk D ) t
¦
Z (D ) k 1 ( pk D ) Z c( pk )
(D6)
where pk is each root of Q( p) 0.
After substituting equation (D6) into (D5), there will be
ª H (D ) D t n
º
H ( pk )
i(t ) U «
e ¦
e pk t »
k 1 ( pk D )Q c( pk )
¬ Q(D )
¼
ª 1 Dt n
º
1
e ¦
e pk t »
U«
k 1 ( pk D ) Z c( pk )
¬ Z (D )
¼
(D7a)
which is sometimes called the Heaviside’s decomposition theorem during an
exponential voltage. Taking D 0 in equation (D7a), we can get the decomposition
theorem for a dc voltage as follows:
476
Appendix D Heaviside’s Operational Calculus and Its Application to Transients Analysis
ª H (0) n H ( pk ) pk t º
i(t ) U «
e »
¦
¬ Q(0) k 1 pk Qc( pk )
¼
n
ª 1
º
1
e pk t »
U«
¦
¬ Z (0) k 1 pk Z c( pk )
¼
(D7b)
U
is the dc constraint component and the second is a free
Z (0)
Q (0)
is the total resistance corresponding to the dc steadycomponent; Z (0)
H (0)
state component which is a special condition of the operational impedance and
Q( p)
will be equal to Z ( p)
during p 0.
H ( p)
The result above is used to solve the transient process of a R-L series circuit as
follows. Because
where the first term
i (t )
1
UeD t
Z ( p)
1
UeD t
R pL
and the root of the characteristic equation R pL
0 is p
(D8)
R
, according to
L
the decomposition theorem in equation (D7) we can get
ª
º
R »
« 1
t
1
eD t e L »
i (t ) U «
§ R
·
«R DL
»
¨ D ¸ L
«¬
»¼
© L
¹
§R
·
¨ D ¸ t º
UeD t ª
«1 e © L ¹ »
R D L ¬«
¼»
(D9)
(c) Heaviside’s operation and its decomposition theorem when the applied
voltage is a sinusoidal voltage U m sin(Z t \ )
On that condition, the Heaviside’s operational expression will become
i (t )
1
U m sin(Z t \ )
Z ( p)
H ( p ) U m j(Z t \ )
e j(Z t \ ) ]
[e
Q( p) 2 j
(D10)
By virtue of Heaviside’s shifting theorem, there will be
477
AC Machine Systems
i(t )
ª j(Z t \ ) H ( p jZ ) j(Z t \ ) H ( p jZ ) º
e
«e
Q ( p jZ )
Q( p jZ ) »¼
¬
ª
H ( p jZ ) º
U m Im «e j(Z t \ )
Q ( p jZ ) »¼
¬
Um
2j
(D11)
Assuming that the roots of Q( p) 0 are p1 , p2 ," , pn , the roots of Q( p jZ ) 0
can be written as p1 jZ , p2 jZ ," , pn jZ , and according to equation (D6) we
may get
ª H ( jZ ) j(Z t \ ) n
º
e j\ H ( pk )
i(t ) U m Im «
e
e pk t »
¦
k 1 ( pk jZ )Q c( pk )
¬ Q ( jZ )
¼
j(Z t \ )
j\
n
ªe
º
e
1
e pk t »
U m Im «
¦
¬ Z(jZ ) k 1 pk jZ Z c( pk )
¼
(D12)
in which pk is each root of Q( p) 0.
The expression above is also referred to as Heaviside’s decomposition theorem
for the sinusoidal voltage, in which the first term is the ac constraint component
Q ( jZ )
is the complex impedance
and the second the free component; Z ( jZ )
H ( jZ )
corresponding to the ac steady-state component, i.e. the complex impedance is a
special condition of the operational impedance and will be equal to the
operational impedance Z ( p) during p jZ .
For example, in a R-L series circuit there is
i (t )
and the root of R pL
1
U m sin(Z t \ )
Z ( p)
1
U m sin(Z t \ )
R pL
0 is p
(D13)
R
, so according to equation (D12) we can
L
obtain
ª
º
R »
j\
«
t
1
e
e j(Z t \ ) e L »
i (t ) U m Im «
R
j
R
L
Z
§
·
«
»
¨ jZ ¸ L
«¬
»¼
© L
¹
R
ª
tº
1
e j(\ M )
e j(Z t \ M ) e L »
U m Im «
«¬ R 2 (Z L)2
»¼
R 2 (Z L)2
478
(D14)
Appendix D Heaviside’s Operational Calculus and Its Application to Transients Analysis
or
i (t )
Um
R 2 (Z L) 2
[sin(Z t \ M ) sin(\ M )e
R
t
L
]
(D15)
in which
M
arctan
ZL
R
479
Appendix E
Basic Relations and Equivalent Circuits
of Transformers
Firstly discuss a single phase transformer as shown in Fig. E1, in which r1 , r2 ,
L1L , L2L , L1 , L2 , W1 , W2 are resistances, leakage inductances, main inductances
and number of turns for the primary and secondary windings respectively, and M
is the mutual inductance between the primary and secondary.
Letting the transformation ratio of transformer be K W1 / W2 , then exists
L1 K 2 L2 . If neglecting the magnetization current, then there is i1 i2 / K
or i2 Ki1 . The voltage equations of the primary and secondary are
u1
u2
di1
di
di ½
L1 1 M 2 °
dt
dt
dt °
¾
di
di
di
r2 i2 L2 L 2 L2 2 M 1 °
dt
dt
dt °¿
r1i1 L1L
Figure E1
Substituting i2
(E1)
Single phase transformer
Ki1 into equation (E1) exist
u1
u2
di1
di
di
L1 1 KM 1
dt
dt
dt
di
di
di
Kr2 i1 KL2 L 1 KL2 1 M 1
dt
dt
dt
r1i1 L1L
From the two equations above, we can get
u1 Ku2
di1
di
( L1 2 KM K 2 L2 ) 1
dt
dt
di
(r1 K 2 r2 )i1 ( L1L K 2 L2 L ) 1
dt
(r1 K 2 r2 )i1 ( L1L K 2 L2 L )
(E2)
Appendix E
Basic Relations and Equivalent Circuits of Transformers
In a similar way exists
u2 1
u1
K
§ 1
·
§ 1
· di2
¨ 2 r1 r2 ¸ i2 ¨ 2 L1L L2 L ¸
©K
¹
©K
¹ dt
(E3)
For the 3-phase transformer of Y / ' 1 connection referring to Fig. E2, letting
the transformation ratio be K3 U AB / U ab , then the corresponding transformation
ratio of single phase transformer is K1 K3 / 3.
Figure E2
Three-phase transformer of Y / ' 1 connection
According to Fig. E2 and equation (E2) exists
u AB
U A0 U B 0
K1 (uab ubc ) (rA K12 ra )iA ( LAL K12 LaL )
(rB K12 rb )iB ( LBL K12 LbL )
diA
dt
diB
dt
(E4)
In a similar way, the expressions can be obtained for uBC and uCA , so the corresponding equivalent circuit referred to Y-connection side is got as shown in Fig. E3.
Figure E3 Equivalent circuit referred to Y-connection side for 3-phase
transformer of Y / ' 1 connection
If the 3-phase transformer of Y / ' 1 connection is referred to the secondary
referring to Fig. E2, then there is
iD
ia iJ ,
iE
ib iD ,
iJ
ic iE
481
AC Machine Systems
Supposing there is no zero-sequence current for the secondary, i.e. iD iE iJ
0,
then exist
1
(ia ib ),
3
iD
1
(ib ic ),
3
iE
1
(ic ia )
3
iJ
and
iA
1
(ia ib ),
3K1
iB
1
(ib ic ),
3K1
iC
1
(ic ia )
3K1
In addition, referring to Fig. E3 also exists
u AB u BC
If there is rA rB
then exists
3K1ubc (rA K12 ra )iA 2(rB K12 rb )iB
di
(rC K12 rc )iC ( LAL K12 LaL ) A
dt
di
d
i
2( LBL K12 LbL ) B ( LCL K12 LcL ) C
dt
dt
1
ia
3K1ubc [(rC K12 rc ) (rA K12 ra )]
3K1
1
ib
[2(rB K12 rb ) (rA K12 ra )]
3K1
1
ic
[2(rB K12 rb ) (rC K12 rc )]
3K1
1 dia
[( LCL K12 LcL ) ( LAL K12 LaL )]
3K1 dt
1 dib
[2( LBL K12 LbL ) ( LAL K12 LaL )]
3K1 dt
1 dic
[2( LBL K12 LbL ) ( LCL K12 LcL )]
3K1 dt
rC r1 , ra
ubc
rb
rc r2 , LAL LBL LCL L1L , LaL
LbL LcL L2 L ,
·
1
1§ 1
(u BC u AB ) ¨ 2 r1 r2 ¸ (ib ic )
3K1
3 © K1
¹
· d(i i )
1§ 1
¨ 2 L1L L2 L ¸ b c
3 © K1
¹ dt
Similarly, the expressions of uca and uab are obtained, so the corresponding
equivalent circuit referred to the secondary can be got as shown in Fig. E4.
482
Appendix E
Basic Relations and Equivalent Circuits of Transformers
Figure E4 Equivalent circuit referred to the seconday for 3-phase transformer of
Y / ' 1 connection
483
Index
A
Acceleration area 331-333
Accidents 224,268,269,301,315,415
ac-dc generator 167,249-251,253,257, 259,
260,261,263,265
Air-gap flux 11,14,23,24,27,29-33,35,49,
61,64,65,67,74,75,79-81,83,92,93,190,
341,347-349,384,448
Air-gap permeance coefficient 6,8,11,13,
31,455-458,460
Another expression for electromagnetic torque
321
Area rule 267,332,336
Asymmetrical rotor 181,182,219,234
Asymmetrical stator 168,181,182
Asynchronous and repulsion synchronous
self-excitation region 166
Asynchronous operation of synchronous
machines 203
Asynchronously starting of synchronous
motors 214
Automatic voltage regulator (AVR) 258
Average operational reactance 174
Average torque after 3-phase sudden short
circuit 156
B
Bad welding 415
Base value 43,361
Basic equations of induction machines when
counting inertia effect in 386
Basic-frequency current 147,150,181,188,
201,208
Biot-Savart law 39,440,461-463,468
Broad sense displacement 41
Broad sense force 41
By-pass petal 429,430
C
Calculation example of dynamic stability 326
Carter’s coefficient 6,415
Cartesian coordinates system 409
Characteristic equation 118-121,124,129131,161,232,234,287,304,305,307,311,
396,397,399,477
Clockwise along the elliptical locus 298
Commutation instant of diodes 261
Complete decoupling of the system 403
Complex-number axes 173
Concentrated winding 3,23
Connection transformation matrix 254-256,
258
Control winding 339,402,403,405,408-411
Cosine-intersection method 238
Coupled circuit 2
Crawling phenomenon 285,287
Critical resistance 290,301
Current base 90,92,93,96-98,363,366
Cut-off angle 267,332
Cut-off of fault line 332-334
Cut-off time limit 267,333,336
Cycloconverter with 6 pulses and without
circulating current 241
Cycloconverter-fed synchronous motor system
with field-oriented control 236,242,243
Index
Cylindrical coordinates system 39
D
Damper loops 4-6,34,252,254,432,434,
435, 440,441,445,447
Damper winding 2,3,23,24,40,251,252
Damping bar 24-26,38,166,236,242-244,
248
Damping ring 24
Damping torque coefficient 267,275,277,
278,280,282,286,288,322
d-axis operational equivalent circuit 103,
106,107
d-axis operational reactance xd ( p) 280
d-axis speed emf 321
d-axis transformer emf 321
D-domain partition boundary curve 398,
399
D-domain partition method 339,397
Dead-region stage 241
Debugging of program 251
Decay time-constant 124,151,267,285
Deceleration area 331-333
Deep-slot rotor 366
Determination of self-excitation region for
induction machines 395
Direct axis (d-axis) 4,11,12,14,24-26,28,
181,412,448,449
Direct search method without derivation
410
Disconnection process due to source fault
339,371
Dispatching loads 269
Dividing intervals calculation 333,336
Double transmission lines 327
Double-cage 366
Dual-stator-winding multi-phase high-speed
induction generator 340,402
Dynamic stability 267-269,315,316,325-327,
329,331-333,337
E
Eddy current and hysteresis effect 61
Eddy current density 409
Electromagnetic compatibility 250
Electromagnetic torque 40,41,58,59,100,
109,110,112,114,116,117,152-159,171,
174,176,177,179,199-202,208,210,219,
221,222,238,243-245,247,259,282,310,
311,317,321-323,339,344,350,362,365,
371,374,377,378,380,381,383-385
Electromechanical parameter resonance 269
Elliptical equation 294
End leakage flux 15,64,67,104,467
Equivalent concentrated treatment method
450
Equivalent current sheet 39,461
Euler’s method 406
Extreme value 67
F
Fc , Bc ,0 axes 166,167,178,179
F, B, 0 axes 166-168,176,177
Ferro-magnetic saturation 61
Fictitious d-axis winding 80,92
Fictitious q-axis winding 80
Final value theorem of operational calculus
127
Five suits of 3-phase windings 167
Flow chart of simulation of ac-dc generator
systems 260
Flux-extinguishing resistors 384,386
Flux-linkage base 90,96
Fly-wheel moment 370
Forced oscillation 267,270,285,287
Four 3-phase bridge rectifiers 167,250
Fourier series 8,403
Fourier transformation 406
Fourth order Runge-kutta method 441,451
Fractional harmonics 7,9,11,15,19,21,74,
414
Fractional-pitch and distributed stator winding
59
Fractional-slot winding 59
Free oscillation 270
Frequency differentiation rate 430
485
AC Machine Systems
G
Low order harmonics
19
Graphic method 8,331
M
H
H complex-plane 398
Half-difference operational reactance 174
Hamming window 430
Harmonic spectrum 243
Heaviside’s or Laplace’s operational calculus
102
Heaviside’s shift theorem 174
Herwitz’s criteria 267,307
Hydraulic generator 146,311
I
Ideal machine pattern 78
Inertia constant 43,227,259,286,290,291,
362,386,397,398,399
Influence of excitation requlation on static
stability 267,268,311
Inner asymmetric condition 3
Inner power characteristics 314
Inner-overvoltage 193
Instantaneous slip 271
Internal asymmetric analysis of ac machines
412
Internal faults 412-415,431,432,440,441
Inter-turn short circuit 414,433
Iterative computation 405
J
Judge the commutation
251
L
Large disturbance 267,269,315
Leakage mutual inductance 67,84,342,
349,351,417,419
Leakage self inductance 37,417,418,466
Left-upper-angle square matrix 54
Loading capability 429
Long-distance transmission 233,337
Loop current method 405
Lose synchronism 287-289,300,305,316,
329,331
486
Magnetic vector potential 409
Main petal of Hamming window 429
Mathematical patterns corresponding to
analysis of dynamic stability 325
Mihainov’s method 397
Mode classifying method 258
Multi-harmonics 55
Multi-loop method 1,44,56,251,340,354,
432
Multi-loop model and parameters 1
Multi-variable optimization approach 410
Mutual flux-linkages \md and \mq 323
N
Negative sequence reactance 120,124,129,
166,188,213
No-decaying natural oscillation 284
No-load characteristic 34,35,323,324
Nonlinear problem 215,316
Non-salient pole synchronous machine
108,113,116,136,291,412,447,448,450,
451,453
Normal two thyristors conducting mode
238-240
Numerical calculation of starting characteristics 365
Numerical integral 365
Numerical method 215,242,269,305,334,
365,378,380,381,383,385,386,451
O
Object function 410
Operating overvoltage 193
Operational conductance G( p) 171,318
Operational reactance of stator q-axis
xq ( p) 108
Optimization approach 405,410
Oscillation amplification 286
Out of step 203,268,285,287-289,301,305
Outer power characteristic 314
Index
Output reactive power
276
P
P- G curves for steady-state and transients
153
Park’s formulas (Park’s equations) 89
Park’s per-unit system of irreversible mutual
in ductances 96
Per-unit system of reversible mutual
inductances 90,98,99
Per-unit systems in synchronous machines
89
Per-unit value 43,44,91,98,100,354,355,
362
Phase-belt 74
Phase-splitting capacitor 56
Physical pattern of 3-phase ac machine after
axis conversion 102
Positive and negative sets interchanging
mode 239,241
Positive direction regulation of stator and
rotor 4
Power base 90,109
Power winding 339,340,402,403,405,407,
408-411
Power-angle characteristic 113
Preserved kinetic energy 300
Pull-in torque 215,219
Pulsating mmf 11
Pulsating torque after 3-phase sudden short
circuit 154,156
Q
q-axis operational equivalent circuit 108,
109
q-axis speed emf 321
q-axis transformer emf 321
Quadrature axis (q-axis) 4,13,14,24-26,
28,412
Quantities with bottom-bars 91
R
Reactance base 90
Reactive power Q 114
Reciprocity theorem 160
Reclosing transients 381,382
Record file 260
Rectangular mmf 9
Rectified load 403
Rectiformer 235,238-240
Reluctance variation 6
Resonant curves 267,286
Right-down-angle square matrix 54
Ring-segment inductance 353
Ripple of dc voltage 250
Rotor bases 93
Rotor extra-aperiodic current 192
Rotor motion equation 1,40,42-44,238,
317,322,325,333,350,362,365,374,390
Routh’s criterion 267,308,309
Runge-Kutta method 365,378,379,441,
443,451
S
Salient pole synchronous machine 7,8,10,
11,13,15,23-26,36-40,50,108,113,116,
136,291,412,413,432,440,442,443,445,
447,448,450,451,453
Salient pole torque Tr 116
Salient-pole synchronous self-excitation
region 166
Saturated-parameters 323
Self-excitation of Synchronous machines
227,229
Self-excitation when connecting induction
motor in series with capacitance 386
Self-oscilation out of step 288,289
Shannon’s sampling theorem 261
Short circuit between branches of different
phases 438,439
Short circuit torque Tk 115,156,157,210
Short pitch ratio of damper loop 27,30,50
Simulative waveform 243-245
Single phase induction motor 56,59
Skew-slot 341,409
Skew-slot coefficient 409
487
AC Machine Systems
Slot leakage flux 15,38,39,67,409,440
Small increment 273,295,299,303,306,
393,394
Small oscillation 268,270,271,291,296-298,
300,310,
Small perturbation 267-271,302,303,305,
306,310,386,393-396
Small perturbation and linearization of ac
machine systems 268
Solid cage structure 403
Source current density 409
Specific leakage permeance 37
Speed and torque oscillation 370
Speed-up with weak field 244,247-249
Squirrel-cage rotor induction machine 347,
350
Stable limit 311,315
Starting characteristic 215,223,339,340,
363,365,370,371
Starting process for induction motors 339,
363
Static circuits 1,2,103
Static stability 267-270,301-309,311
Statistical data about accidents 315
Stator bases 90,92,98
Stator rated amplitudes 89
Storage coefficient of dynamic stability 331
Storage kinetic-energy 330
Sub-synchronous resonance 266,338
Sudden load 244-246,248,249
Sudden reversal-rotation 244-246,249
Summation term 62,71
Superposition method 403
Surplus kinetic energy 297
Surplus torque 42,44,296,297,330,331
Synchronization torque coefficient 275,
277-279,287,288,310
Synchronous torque Ts 287
T
T-form electric circuit 329
Time base 90,98
488
Ta
x2
time constant corresponding to the
r
aperiodic and second harmonic components
in stator current 128
Tdcc time constant of d-axis damping winding
when stator winding and excitation winding
are all short-circuited in which r 0 and
R fd 0 131
Tdc time constant of excitation winding when
stator winding is short-circuited whose
resistance r is equal to zero 132
Tqcc time constant of q-axis damping winding
when stator winding is short-circuited
whose resistance is equal to zero 130
Time-variant coefficients 1,2,32,436,441,
451
Time-variant parameter 54
Topology of circuit 257
Torque coefficients 268,270,275,279,280,
282,287
Transcendental function 55
Transient emfs of synchronous machines
58,59,147,320
Transient parameters of synchronous
machines 111
Transient stability 301
Transients of reswitching on induction motors
Triggering time 371
Turbogenerator 57,136,146,155,233,266,
311,338,454
Two phases earthed 315
Two thyristors converting mode 240,241
U
Unsymmetrical short circuit 315,327,328
V
Variable-step Runge-Kutta integration method
259
Variable-topolog configurations 402
Virtual displacement principle 41
Voltage base 90,94-97,366
Index
Voltage dip 58,59,142-144,146,150,515,
315,371
W
Weakly linking power system 337
Window function 429
Wound-rotor induction machine 341,344,
345
Z
Zero-axis component 83,85,86,89,356,364
Zero-axis inductance 84
489