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HARMONIC EFFECTS IN ROTATING ELECTRICAL MACHINES
Mona Samaha-Fahmy, B.Sc. (Eng.), (Cairo Univ.)
EFFETS HARMONIQUES DANS LES MACHINES TOURNANTES ~LECTRIQUES.
Mona Same:ha-Fahmy, B.Sc. (eng) ,(Univ. du Caire)
RESUME
Cette thèse contient une étude experimentale et
analytique des effets de la r'action d'armature, tenant
compte des differents harmoniques. Grâce à un développement
en sJ{rie d·es fo·nctions trigonometriques rencontre"es, on a pu
etudier l'effet de chaque harmonique d'espace sur le courant
et la tension en r~gime permanant. Ce travail a été possible
par la mécanisation des opérations algébriques permises par . .
l'utilisation du syst~me FORMAC d'IBM. Les résultats obeenus
sont discutés,et confirment les conclusio~ theoriques et ~ , , ,
experimentales deja connues. Cependant la formulation mathe-
matique .. ~ .-
presentee dans cette. etude, et, basee sur la matrice
d'imp~dance harmoniqu~ permet de terminer les celculs cent ou
mille fois plusrapidement que. les mithodes itératives utilisées
jusqu'~' présent. Une attention particulière a été apportée à l'étude des harmoniques de la FMM cre~es dans une machine poly
phasée, connectée à un r;seau equilibré , desequlibré , ou
monophasé. Une ~tude experimental~ ainsi qu'une repr~sentation
graphique des r~sultats théoriques obtenues dans le cas o~ l~ 1 ., ,
machine polyphasee est connectee en monophase~ prouvent que les
effets des harmoniques de la réaction d'armature ne sont pas
négligeable dans les machines polyphas~es de~equilibrées.
1 J 1
HARMONIC EFFECTS IN ROTATING ELECTRICAL MACHINES
by
Mona Samaha-Fahmy, B.Sc. (Eng.), (Cairo Univ.)
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Master of Engineering.
Department of Electrical Engineering,
McGiII University,
Montrea l , Canada,
@)
January, 1973.
\ , .\
Mona Samaha-Fabmy 1973 1 1 !
s'· ~.
ABSTRACT
ln this thesis the phenomena of multiple armature reaction effects is
studied analytically and experimentally. The study of the effect of space harmonies
on the steady state current and voltage waveforms is investigated analytically by the
method of direct algebraic expansion of trigonometric series. The solution depends
for its success on the mechanization of this process by the use of the IBM FORMAC
system. The results obtained are discussed and compared with the existing theoreti-
cal and experimental data and found in good general agreement. Using the harmonic
impedance matrix formulation introduced in this work, a reduction of computing time
between 2 to 3 orders of magnitudes is achieved over earlier numerical iterative
techniques. Special attention is given to the mmf harmonics produced in the windings
of balanced polyphase machines and unbalanced or single phase connections. Experi-
mental study and predicted waveforms for single phase connections show that multiple
armature reaction effects are extremely severe in unbalanced machines.
ii
ACKNOWlEDGEMENTS
The author wishes to express her deep and sincere gratitude to
Dr. T. H. Sarton for his helpful guidance and encouragement throughout this
project.
Special thanks are due to Mr. H.l. Nakra for his helpful suggestions.
My sincere appreciation and thanks are due to Mrs. P. Hyland for the neat lay-out
and excellent typing.
The financial support received from the National Research Council
of Canada is gratefully acknowledged.
. ... ~
"
iii
...,... t,.'
TABLE OF CONTENTS
Page
ABSTRACT
ACKNOWLEDGEMENTS ii , "J ~
TABLE OF CONTENTS iii 1 i
CHAPTER INTRODUCTION i ~1
~ ~
CHAPTER " THE STEADY STATE CURRENT AND VOLTAGE (t
WAVEFORMS IN ELECTRICAl MACHINES 4 ~ 2.1 Introduction 4 ",1
".~
2.2 Multiple Armature Reaction Effects 5 \.~
~
2.2.1 The Basic Armature Reaction Cycle 5 2.3 General Equations of a Machine 6 2.3.1 Steady State Solution of the Machine Equations 8 2.3.2 The Analytical Approach 10 2.4 Introduction to FORMAC 12
CHAPTER III COMPUTER PROGRAM FOR A GENERAL MACHINE 16
3.1 Number of Variables and Equations in the Analytical Solution 16
3.2 General Structure of the Computer program 20 3.3 Trigonometric Analysis Routine 23
CHAPTER IV ANALYTICAL SOLUTION OF THE SYNCHRONOUS MACHINE, STEADY STATE PERFORMANCE 27
4.1 Introduction 27 4.2 Steady State Performance of a 4-Wire Star
Connected Synchronous Machine 27 4.2.1 Computer program 32 4.2.2 Results 37 4.3 Steady State Solution of a 3-Wire Star
Connected Synchronous Machine 41 4.3.1 Computer Program 42 4.3.2 Results 42
< t 4.4 The 3-Wire Versus the 4-Wire Connection 45 4.5 Conclusion 48
t
CHAPTER
CHAPTER
CHAPTER
APPENDIX
v
5.1 5.2 5.2.1 5.3 5.4 5.5
VI
6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.4
VII
APPENDIX Il
A
B
APPENDIX III
APPENDIX IV
APPENDIX V
REFERENCES
iv
Page
ANALYTICAL SOLUTION FOR INDUCTION MOTORS WITH PHASE - WOUND ROTORS 50
Introduction 50 Harmonic Interactions in Induction Machines 52 Harmonic Effects in a Symmetrical Two-Phase Machine 55 Computer Program 56 Results 57 Conclusion 63
SINGLE PHASE MACHINES, EXPERIMENTAL STUDY 64
Introduction 64 Single Phase Synchronous Machine 66 Analysis 66 Results 67 Single Phase / Single Phase Induction Machine 70 Analysis 72 Reliùlts 72 Conclusion 78
SUMMARY AND CONCLUSIONS
VOLTAGES INDUCED IN SYMMETRICAL WINDINGS
3 - PHASE SYNCHRONOUS MACHINE
4-Wire Star-Connection
3-Wire Star-Connection
2-PHASE INDUCTION MOTOR
SINGLE PHASE SYNCHRONOUS MACHINE
SINGLE PHASE - SINGLE PHASE INDUCTION MACHINE
79
81
83
84
92
95
103
106
109
CHAPTER 1
INTRODUCTION
It is now more than forty years since Kron unveiled his generalized electric
machine theory and while interest in it was only slowly aroused, it is now firmly
established as an analytic tool. The original theoretical treatment, although fully
explored by Kron was left by him in a somewhat indigestible state. By the end of the
1950l s and beginning of the 1960Is Gibbs El] , Lynn [2] , White and Woodson [3]
and many others have so clarified the basic theoretical issues that attention was fruit-
fully turned to the closer correlation of the structure of Kron1s primitives with the
nonideal structure of practical machines. Hamdi - Sepen [4] extended the method
of two-axis considerations by ascribing direct axis and quadrature axis saturation
factors as weil as direct axis and quadrature axis saturation coupling factors. The
theoretical premises of the 2 - axis the ory have been shown by Carter et al [5] and
Dunfield and Barton [6] to be invalid (or questionable) to varying degrees in actual
machines. This is due to armature mmf harmonics in addition to the fundamental,
and airgap permeance harmonics higher than the second, these showing up in the form of
inductance coefficients that do not conform to the Park - Kron definition.
Recently, interest in the stea~y state behaviour in terms of machine analysis
has been on the increase. Various transformations are available to simplify the analysis,
Dunfie Id and Barton [7, 8] introduced a three-phase to two phase si ip-ring transforma-
tion. Willems [9] shows that, in sorne cases where space harmonics are taken into
consideration, a linear transformation can be set up to transform the non-stationary
-.~:.;r-- ".
1 } 1
i j j
j
2
equations describing an electrical machine to stationary equations. The paper shows
that an interesting relationship exists between unifjed machine theory and Iinear
system theory. Chalmers [10] puts many of the problems of the analysis of non- ideal
electric machines in practical perspective. He also shows how a. c. machine
windings may be arranged to reduce harmonic content [] 1 ] .
Most of the study of harmonic effects in a. c. machines have been concerned
with balanced polyphase conditions and have not included unbalanced or single-phase
operation. Buchanan [12] specifically treated an equivalent circuit for a single phase
motor having space harmonics in its magnetic field. Davis and Novotny [l3] have
developed the equivalent circuits for single-phase squirrel - cage induction machine
considering the effects of both odd and even mmf harmonics. More recently, they
studied both analytically and experimentally the even order mmf harmonies in squirrel
cage induction motors under transient state conditions [14] •
ln this thesis, a simple formalism for steady state harmonic analysis in electric
machines is presented. The analytical strategy Î:i to apply a transformation which re-
duces the differential equations describing the machine into a linear matrix equation
which is readily solved. This formalism has long been known, but is seldom used be-
cause of the great complexity of the algebraic and trigonometric expressions encountered.
FORMAC standing for "Formula Manipulation Compiler" is demonstrated to be an
efficient and powerful computer aid for solving such problems.
ln Chapter Il, the effect of multiple armature reaction is explained, the
analytical approach is discussed and FORMAC is introduced. Chapter III is con-
~!
j ,j
J. ,
1 i :1 ! .j 1 1 1 ~
1
1 j
1 j
l
J
3
cerned with developing a computer program for a ge ne ra 1 machine. In Chapters IV
and V, two typical examples are worked out, namely a 3-phase synchronous machine
and a 2-phase induction machine. Chapter VI carries the experimental and analyti-
cal results for a single phase s; .. nchronous machine and a single phase/single phase
machine.
ln summary, this thesis as a contribution to the theory of electrical machines,
introcluces a new computer aid which, to the authorls knowledge, has not been used so
far in solving such problems.
, 1
-1 ,
i i i l • î
\ i 1 l l l ,
4
CHAPTER Il
THE STEADY STATE CURRENT AND· VOLTAGE WAVEFORMS
IN ELECTRICAL MACHINES
2. l Introduction
Real electrical machines differ from ideal primitive machines as follows :
1. The primitive machihes have idealized air gap and winding
geometries which is to say that airgap permeance harmonics
higher than the second and ail harmonies of winding mmf 's
are considered negligible. Robinson [15] , proves that the
airgap flux density for an a. c. machine cannot be expressed
as a simple rotating wave function, sinusoidal in space and
time, but contains in general a combination of such rotating
waves, harmonically related in space or in time. The method
of approach is to regard any airgap flux - density distribution
as the superposition of components having this simple form. l
t
2. Sorne of the effects of departures from the ideal machine are 1
1 i
related to magnetic saturation. The technique of using ad just-';
j j
i 1
able parameters in place of the constants in ideal machine
equations can be handled with reasonable accuracy, e.g. a
saturated reactance may give reasonable results for the behaviour
of a machine, in terms of average and fundamental frequency
phenomena, but it intentionally ignores the harmonie phenomena
t
5
taking place within the machine and observable in its external
currents and voltages.
Jones [16J , Carter [5J, and Dunfield and Barton [6J have shown in both
analytical and experimental studies that real electrical machines have windings that
produce significant space harmonics of mmf and also have non-simple air gap geometry
which gives rise to air gap permeance harmonics higher than the second.
2 .2 Mu 1 tip le Armature Reaction Effecfs
When current flows in a normal machine winding, in general both positive
and negative rotating fields having pole numbers that are odd multiples of the funda-
mental number of poles are created. These cause multiple frequency currents to flow
in the secondary which in turn create multiple rotating fields reacting on the primary.
This phenomena has been called multiple armature reaction Cl7J a term which, while
perhaps not ideal, will be employed here.
2.2.1 The Basic Armature Reaction Cycle
If we consider a voltage of frequency f applied to the primary winding, p
this·will create a current of the same frequency, also it will create fields rotating at
a frequency of :1: f / m relative to the primary, where m is an odd integer. These p
fields rotate at frequencies of (:1: f / m - f) relative to the secondary, fr being p r
, .,
l 1 1 1 ~ ~
·l .~ ',:J ~ .J ~ .'1; .. , l .. ~
l , :! l '~
" :J.
i
.,
i ï ~ ~ J .. , , 1 .j
,~ 1 ; .1 '1
1 1 :~ .1 t fi 1 .~ t
i ;l .!
.~ :~
:! J J
.-J .
6
the frequency of rotation of the machine. The term frequency of rotation, synonymous
with speed of rotation in electrical revolutions per second, is used for convenience.
The primary fields induce voltages in the secondary windings at frequencies
(±f lm - f ) m = (±f - m f ). Negative frequency components arising From this p r p r
expression need not cause concern, they merely signify a particular phase sequence.
The resulting secondary currents will praduce fields which rotate at frequencies
± ( ± f - m f ) 1 n , where n is an add integer independent of m. In their turn, p r
these fie Ids rotate re lative to the primary at ± ( ± f - m f ) 1 n + f , inducing voltages p r r
in the primary windings at frequencies ± ( ± f - m f ) + n f = f ± (m ± n ) f , and p r r p r
the cycle is complete, each voltage component producing its corresponding current as
before.
Thus associated with the primary excitation frequency f , is a range of p
primary frequencies f :!: l f where l = m ± n is an even integer, and a range of p r
secondary frequencies f :!: m f where m is an odd integer. In the following, we will p r
use the term harmonic series to define any such series of sinusoidally varying components
no matter whether the frequencies are integral multiples of a fundamental or not.
2.3 General Equations of a Machine
It is we Il known that, when a balanced polyphase winding is excited From
a polyphase voltage source of sinusoidal waveform the general equation of the machine
can be written in a matrix form as :
-,
l ',,' 1 ...,
7
V=(R+DL)I (2-1)
where D is the differential operator d / dt.
Since we are here concerned with steady state response we may assume
currents in the primary and secondary sides of the form.
and
where
and
1 = L 1 l cos [ (Ca) + l Ca) ) t - al] , p L p p r
f p
f r
l
m
1 cos [ (Ca) + m Ca) ) t - a ] , sm p r m
is the supply frequency 1
is the rotation frequency ,
is any even integer positive or negative,
is any odd integer positive or negative.
Equati on (2- 1 ) can be written as :
v p
v 5
=
z pp
z ps
Z sp
Z 55
1 p
1 5
(2-2a)
(2-2b)
(2-3)
The primary and secondary self inductances as weil as the mutual induc-
tance between two primary windings or two secondary windings of a real electrical
machine can be expressed as an even harmonie series of the form
L = I LL cos ( L 1T /2) cos ( lla)r t - Sl) ,
l
8
while the mutual inductances between primary and secondary windings can be expressed
as odd harmonie series
M = )' L.
m
I Mlm
cos ( L1T /2) cos (m la)r t - Slm ) ,
L
where Sl and Slm are angles depending on the relative geometrical configuration of
the windings. In ge ne rai , the magnitude factors L land M lm depend on the permeance
and winding constants of the machine.
Equation (2-3) can now be rewritten
v p
v s
=
ze pp
Zo ps
Zo le sp p
Ze 10
ss s
where the superscripts e and 0 stand for even and odd harmonie series in la) r
respectively. This matrix equation will lead to V being an even harmonie series, p
and V an odd harmonie series in la) . s r
2.3.1 Steady State Solution of the Machine Equations
(2-4)
Dunfield and Sarton [18] obtained the steady state solution for a synchro-
nous machine by the direct application of the Runge-Kutta numerical integration
l "
l 1
1 .\
1 i j 1
i
1 1
9
technique. They found that the integration step size must be small enough to give
at least ten steps per period of the highest significant frequency. Thus if we have
a basically 60 Hz problem in which we feel that the fifth harmonic may be signifi
cant, the step size must be less thon 1 / ( 10 x 5 x 60 ) sec., i.e., 0.33 msec.
For the steady state problems there is no way of knowing the correct
initial values of the state variables. Thus every problem becomes a transient problem
and a number of transient cycles must be followed before the desired steady state is
reached. Even under the best conditions much use less information is generated. This
method although easy to use is inefficient when the harmonic series representing the
winding inductance include a large number of terms.
Since only the steady state solution is desired, another method of solution
was sought with a trade-off between solution time and accuracy being made [19J .
The form of the stator currents and rotor currents was assumed known :
= I 1. cos (Cal. t - a.) . 1 1 1
(2-5)
A set of equalities were established by substituting these currents in the matrix equation
(2-1) and performing the necessary multiplications and differentiations and finally
separating terms of the same frequency and phase on the L. H . S. and R. H. S .. The ex-
panded form yielded an equation of the form :
v = Q ( sin a., cos a.) • 1 1 1
where V is the column vector [V. J whose elements are the in-phase and quadrature 1
.~ ,. ,~
10
'harmonic components, 1 is the column vector [1. ] whose elements are the magni-1
tudes of the harmonic currents, and Q is a rectangular matrix of coefficients depending
on the impedance elements as weil as the phases of the currents. A c10sed form solu-
tion of this equation is not possible since transcendental relationships are involved, but
a solution may be obtained using numerical methods such as the Newton-Raphson
technique [20, 21] .
While the Runge-Kutta method is easily programmed and has a reasonable
efficiency, solving the sa me system using Newton-Raphson method gives more insight
into significant harmonic interactions and yields computed results in less time. A dis-
advantage of this method is that the elements of the matrix Q are functions of the
phase angles a. , which necessitates the repetition of the Newton- Raphson Search at 1
every load angle desired. This inefficient utilization of computer time provided the
incentive to search for an alternative method of solution.
2.3.2 The Analytical Approach
The current harmonie series of equation (2-5) can be analyzed into the
in-phase and quadrature current components and rewritten
where
O. = Co). t + /3 1 1
/3 being a constant
of integration depending on the initial rotor position.
(2-6)
1
1
) ~ ~ ~
! 1 j
1 1 1
,1
"
1
11
From equation (2-4) the product of a row of the inductance matrix with
the current column vector involves the products of elements having inductance
harmonics and current harmonies. This in general yields a large number of elements
of the type
(2-7)
This cosine product is analyzed into a cosine sum
E = A / 2 [cos ( (0 1 + °2 ) - (" 1 + "2) J + cos ( ( ° 1 - °2 ) - (" 1 - "2) 1]
(2-8)
which can be further analyzed into in-phase and quadrature elements,
+ cos (')11 - ')12) cos (01 - 02)
+ sin ( ')11 - ')12) sin (°1 - °2 ) ]
which if differentiated with respect to time yield
- (1.)1 - 1.)2) cos (')11 - "2) sin (01 - 02)
+ (1.)1 - 1.)2) sin ("1 - ')12) cos (01 - 02) ]
(2-9)
(2-10)
1 i
f i j 1 1
j l
1 1
1 , 1
1
12
The very large number of terms thus generated (4 times the original number), is
scanned to group together terms of the same frequency and phase which are then
separately equated to the corresponding voltage terms on the L.H.S. yielding,
v = AI (2-11 )
where A is a square matrix whose elements are only dependent on the machine
constants. This matrix relates the harmonie components in the voltage waveforms to
the corresponding harmonie components in the current waveforms, hence the name
harmonie impedance matrix.
Performing such analysis requires in general weeks or months of hand mani-
pulation of algebraic and trigonometric expressions. As in the case of numerical
computations, the human is error-prone when manipulating long complicated formulae
like these. FORMAC, standing for 1 FORMULA MANIPULATION COMPILER 1 is - - -a powerful computer aid for solving such problems.
2.4 Introduction to FORMAC
IFORMAC is an experimental programming system which was designed to
permit the engineer to use, in a practical way, analytic as weil as numeric techniques
on a digital computer [22, 23]. .The advantages of using a digital computer for
numeric computation apply in almost equally large measure to the use of a computer for
non-numeric work.
13
The basic concepts of IFORMAC 1 were deve loped by Jean E. Sammet
[24, 25J (assisted by Robert G. Tobey) at 1 B MiS Boston Advanced programming
Department in July 1962. At first FORMAC was developed as an extension of
FORTRAN IV on the 1 B M 7090 /94 Computer. Consideration of language and
implementation for the 1 B M System 360 started in the fall of 1964, with the intent
to provide a better capability and associate it with PL /1 rather than FORTRAN .
The PL / 1 - FORMAC system was released in November 1967.
FORMAC expressions can contain variables, user defined functions,
rational constants with up to 2295 digits, and symbolic constants representing 1f ,
and e , as weil as trigonometric, logarithmic and exponential functions.
To demonstrate how FORMAC applies to the analysis of an expression
term by term in every detail we shall discuss sorne of its powerful features by considering
the application of the FORMAC operators NARGS, ARG and LOP to the simple
expression of equation (2-7)
The operator NARGS (E) evaluates the ~umber of argument~ in the
expression E. In our example, since in the FORMAC language, the expression E
is regarded as the product of three arguments A, cos (Q1 - "1) and cos (Q2 - ')'2) ,
therefore NARGS (E) returns the integer 3 to the main program.
The operator ARG ( 1, E) is used to separate the arguments contained
in an expression. Thus the statement
i i
.~ 1
14
G = ARG (2, E )
yields
G = cos (01 - ')1 1) .
ln general ARG (l, E) returns the 1 th argument in the expression E .
ln order that the y may be manipulated, the arguments are identified
by code numbers by means of the operator LOP. The function LOP (G) returns an
integer code for the !ead operator in the expression G. The code for the subtrac
tion sign (-) for instance is 25, while that for an exponential is 31. 1 cos l ,
1 sin 1 and constants are also recognized by LOP and given the codes S, 4 and 37
respectively. Applying the above functions to the subexpressions in E:
G (1 ) = ARG ( l, E) --.. G (1) = A
G (2) = cos (01 - ')Il )
G (3) = cos (°2 - jl2 )
and
L ( 1 ) = LOP (G (1» -... L (1) = 37
L (2) = 5
L (3) = 5
To extract the argument of the cosine term of G (2) we reapply the function ARG:
ARG [l, G (2)J = ARG [1, ARG (2,E) J = 01 - ')11 .
15
Now that the expression E is dissected almost completely, the various
components within it, in our case harmonies, can be identified and the expression
can be reformed as desired. In our case, it is desirable to convert the cosine product
into a sum. This process is described in de ta il in Chapter "1 .
Many other functions and routines are available in FORMAC and can
easily be combined to treat almost any expression. For a more complete understanding
of FORMAC the reader is referred to Reference [23] •
16
CHAPTER III
COMPUTER· PRO GRAM FOR A GENERAL MACHINE
3.1 Number of Variables and Equations in the Analytical Solution
ln the previous chapter, we found that the machine equations can be
written according to equation (2-4). 1 and 1 as expressed byequations (2-2a) p s
and (2-2b) can be rewritten in a form more suitable for machine computations using
the in-phase and quadrature current components and putting
Q = Co) t o P
and
where ~ is a constant of integration depending on the initial rotor position,
1 p =
=
1 L cos [ (Co) + l Co) ) t - al] p p r
where IpDl
= Ipl
cos (a l + l ~) is the in-phase current compone nt and
IpQl = IpL
sin (al + L ~) is the quadrature current compone nt • Similarly
1 s
= 1 D cos (Q + m Q) + 1 Q sin (Q + m Q ) s m 0 s m 0
where l is any even integer and m any ocId integer,positive or negative.
.~
17
ln most practical cases convergence is so rapid that it is only necessary
to consider a relatively small number of terms in the harmonie series. Restricting
our problem to the Nth harmonie, we proceed to estimate the number of unknowns
and the required equations for a complete solution.
ln the primary side the number of unknowns in the range - N :s: l :s: N is
N =4(N:2)+2 p
per primary winding.
Here ( : ) indicates an integer division with no round off. Thus if N is 9 then
the quotient N : 2 is returned as 4 •
On the secondary side the number of unknowns in the range
- N :s: m :s: N is N = 4[(N+1):2J s
persecondarywinding.
Thus, for a machine having W primary windings and W secondary windings, the p 5
solution up to the NtÏl harmonie requires the evaluation of a total of
W [4 (N : 2) + 2 ] + 4 W [( N + 1) : 2 J unknowns • p s
To solve for this large number of currents the harmonie series of the volt-
ages evaluated on the right hand side of the machine equation should be equated term
by term to the left hand side up to the Nth harmonie providing just enough equations
for the complete solution.
Many cases of great practical interest have symmetries which will
drastically reduce the number of independent unknowns. Thus the number of unknowns
and equations may be reduced in some specifie cases by three factors:
\.'
18
r: 1. Many components will have the same frequency and thus can
he combi ned .
2. Many components are zero and can be eliminated in the initial
machine equations.
3. Many components form balanced spatial systems, thus implying
specific phase differences in both voltages and currents (e.g.
2-phase or 3-phase balanced systems). A more complete
discussion of this effect is given in Appendix 1 •
As an application of the above-mentioned principles we consider a
balancecJ, 3-phase synchronous machine having 2 damper windings such as that depicted
in Figure [3- 1] •
If the harmonie series of the primary and secondary currents are terminated
at the 7th harmonic, this machine having 3 primary windings and 3 secondary wind-
ings will he represented by 42 current components on the primary side and 48 current
components on the secondary side, a total of 90 unknowns.
However, in this special case, we can take advantage of the balanced
three secondary windings and make the justifiable assumption that :
1 = 1 (la) t ) a s
lb = 1 (Ia)t- 21(/3) s .
.{~ and 1 = 1 (la)t+21(/3) , c s
thus reducing the book-keeping labour on the secondary side by a factor of 3 •
_J
19
DA
a
FIGURE 3-1. 3 - PHASE SYNCHRONOUS MACHINE WITH 2 DAMPER WINDINGS.
--)
.~ '"
j .;
ï , ~
. ~. , ., '"
"
20
On the primary side, the two damper windings being isolated from the
field winding enable us to make the assumption that the harmonic of order zero (d. c.)
in these windings is identical to zero, reducing the primary unknowns to 38.
Further, in a synchronous machine having the field winding connected to
a d.c. supply, Co) = 0 while Co) = Co) , the synchronous frequency. 1 and 1 p r s p s
will therefore have terms of frequencies ± L Co) and ± m Co) respectively where s s
o ~ l ~ N and 1 ~ m ~ N. Terms having the same frequency and different signs
can be combined with proper adiustment for the phases thus reducing the total burden by
a factor of 2. The total number of unknowns will therefore be reduced from 90 to
27 out of which only 8 belong to the secondary side.
3.2 General Structure of the Computer Program
ln this section, an outline of the computer program used in this study is
given. In Appendices Il and III computer listings of two typical versions of the pro-
gram are shown.
The general structure of the program is depicted in the flow graph of
Figure [3-2]. The program starts by reading the input data. This includes the number
of windings NW, number of stator windings NSW, number of unknown current components
NU , the highest harmonic order considered NHAR, and the number of equations to be
solved N.
-r-.....
(
NT = 2
DIFFERENTIATE
No
FORM (A)
READ DATA
INITIALIZATION NT = 1
Q=WL * CU Q = R * CU
NQ = NARGS (Q)
E = ARG (K , Q )
TRIGONOMETRIC
Yes
PRINT AND
PUNCH
ANALYSIS ROUTINE
No
" Z
0 1-
Il
~
STOP
FIGURE ~2. GENERAL STRUCTURE OF THE ANALYSIS PROGRAM.
21
Z
0 1-
Il
I=f
22
Next we read as alphameric data the reactance matrix Wl, the resis-
tance matrix R, the current vector CU as harmonic series, and the current
components vector CUDQ containing the in-phase and quadrature current compo-
nents arrange~ in a suitable order. A typical data set is shown with each program
listing.
After initializing the control variables, the program proceeds by
multiplying one row of the reactance matrix by the current vector resulting in the
variable named Q. In general Q has the form :
sin Q = L\ 1 X . [ } (a 0 + a)
cos
sin [ cos} (b 0 + ~ )
The analytical expression Q is broken into sub-expressions E which are analyzed in
the trigonometric analysis routine explained in the following section. The output of
this routine consists of the coefficients of the 1 th harmonic compone nt assigned to the
variables C (1) in case they are associated with a cosine function or to S (1) in case
the}; are associated with a sine function. The differentiation of the expressions will be
equivalent to an exchange between C (1) and S (1) , with the proper adjustment for
the sign and coefficient of the independent variable.
The coefficients thus obtained are stored and we proceed by forming the
product R 1 which is then analyzed using the same procedure. The resulting
coefficients are added to th ose obtained previously forming the complete analytical ex-
pressions for the voltage components. Each of these expressions is further broken into
a row vector of coefficients multiplied by the column current vector CUDQ, thus
forming the harmonie impedance matrix A •
23
3.3 Trigonometric Analysis Routine
The trigonometric analysis routine is depicted in Figure [ 3-3]. In
the most general case every subexpression E consists of the product of several terms
not more than two of which are the trigonometric functions sine and cosine,
sin sin E = IX· [ } (aO+a). [ l (bO+~)
cos cos
The problem is then to recognize these terms, and replace them with the proper expansion
or alternatively rebuild the coefficients of the expanded form and label them correctly
for further manipulations.
This is done using the following procedure :
First, E is broken into terms G. ; i = l, NE. Since an exponent might be screened 1
by the leading product operator, we first check whether G is operated upon by an ex-
ponent, and if so, what is the argument of the SIN or COS function. An index is
associated with each case to facilitate future labeling of the different coefficients. In
case the term is free from an exponential element, it is tested for SIN or COS and
accordingly an index is associated. Then the remaining terms are tested for another SIN
or COS in which case the index is changed again.
Table [3-1 ] shows in detail the indices and arguments in al\ possible
cases.
INPUT
J=J+l
No
NE = NARGS (E)
INDEX = 0
G (J) = ARG (J,E) J = l, NE
A = ARG (l, G) B=O
B = ARG (l, G)
FORM
Yes
RECONSTRUCT E C AND S
A = ARG (l, G) B=A
EXIT
FIGURE 3-3. TRIGONOMETRIC ANAlYSIS ROUTINE.
24
25
TABLE r 3 - 1 ]
---Index Case Argument A Argument B
i· 0 No sin or cos - -
.'. sin
2 (A) ag + a B = A
1
sin (A) sin ( B) ag + a bg + ~
sin (A) ag + a 0 2
sin (A ) cos ( B) ag + a bg + ~
3 cos (A) sin ( B ) ag + a b g + ~
2 ag A cos (A) + a B =
4 cos (A) cos ( B ) ag + a b g + ~
cos (A) ag + a 0
To complete the analysis, we need to reconstruct the product of the non-
trigonometric terms and also to find out the order of the harmonic elements associated
with the expressions. This is done by forming the sum and difference of the c·rguments
x = A + B and Z = A - B ,
and extracting the coefficients of g in both X and Z, 1 X = a + band
.,: î
J j
26
1 Z = a - b which are the harmonie orders of concern. Finally, we allocate the
reconstructed expression, preceded by the proper coefficient as implied by the index-
ing system, to the SIN and COS terms having the same harmonic order as computed.
Table [3-2] summarizes the coefficients attributed to each case ..
TABLE [3 - 2 ]
Index sin (a + b) 0 cos (a + b) 0 sin (a - b) 0 cos (a - b) 0
l 0 .. 5 sin (a +~) -0.5 cos (a + ~ ) -0.5 sin (a - ~) 0.5 cos (a - ~)
2 o .5 cos (a + ~ ) 0.5sin (a+~) o .5 cos (a - ~) 0.5 sin (a - ~)
3 0.5 cos (a + ~) 0.5 sin (a + ~ ) -0.5cos(a-~) -0.5 sin (a - ~)
4 -0.5 sin (a + ~ ) o . 5 cos (a + ~ ) -0.5 sin (a - ~) 0.5 cos (a - ~)
The output of this procedure is as mentioned before the coefficients C (1) and S (1)
associated with the 1 th order cosine and sine harmonies respectively, where 1
extends from - NHAR to NHAR. (Note: Negative subscripts are allowed in
PL / 1 - FORMAC, an advantage which we make use of) .
27
CHAPTER IV
ANALYTICAL SOLUTION OF THE SYNCHRONOUS MACHINE
STEADY STATE PERFORMANCE
4.1 1 ntroducti on
ln this chapter we studya special model of a salient pole rotating field
synchronous machine. The configuration of the machine is defined in Figure [4-1] ,
where a, band c represent the three stator windings, 1 and 2 the direct and
quadrature axis damper windings respectively and f the field winding.
This sa me example has been treated by Dunfield [19] using the Runge-
Kutta numerical integration method and a modified Newton - Raphson method.
Although both techniques give the accuracy required, the computation time involved
is considerable. In the following the results obtained using the analytical solution are
compared to those obtained numerically for both the 4 - wire and 3 - wire connections.
4.2 Steady State Performance of a 4 - Wire Star Connected Synchronous Machine
Under steady state condition the relationship between the voltages and the
currents in the different windings of the machine introduced in Figure [4-1] , con be
written in a matrix form
28
DA
FIGURE 4-1. 3 - PHASE SYNCHRONOUS MACHINE WITH 2 DAMPER WINDINGS.
(
o
o
v a
v c
=
Z aa
Z ac
~c
Z ac
z cc
x
i a
i c
To determine the solution of equation (4- 1) in the state - space form we rewrite
v = (R +DL) 1
where D is the differential operator.
The matrices Rand Lare defined byequations (4-3) and (4-4)
R = R a
R a
R a
29
(4-1)
(4-2)
(4-3)
L =
'-.. ( f
Lf
Mfl
M9f cos g
+Maf3 cos 30
Maf
co:. (O-2n/3)
+Maf3
cos 3 g
Mof cos (0+2n/3)
+Mof3
cos J g
Mfl
Ll
L2
Mal cos 0 -Ma2 sin 0
Mal COS(Q-21T/3) -Ma2 sin(O-2n/3)
Ma 1 cos (Q+2n/3) -Mo2 sin (Q+2n/3)
~~~~~~,:....;.;A~,~~;;,:,_~~ ......... ~.-... ~ ..... ;.~~---•. -._~_.,,~ .• _-- -- - .,"
Maf cos 0 Maf cos (Q - 2'11/3)
+ Maf3
cos 3 Q + Maf3
cos 30
Mal cos Q Mal cos (Q - 2n/3)
- Ma2
sin 0 -Ma2 sin (Q - 2n/3)
L -Mab aa
+ L 2 cos 20 aa +Mab2 cos 2(Q..n/3)
+ L 4 cos 4 g aa -Mab4 cos 4(O-n/3)
- Mab L aa
+ Mab2 cos 2 (O-n/3) +L 2 cos 2 (0+n/3) aa
- Mab4 cos 4 (O-n/3) +L 4 cos 4 (Q+n/3) 00
- Mob -Mob
+ Mob2 cos2(Q+n/3) +Mab2 cos 20
- Mob4 cos4{Q+rr/3) - Mob4 cos 4 0
:. ;. ~ ... ,;;._~~:.,.._ .. :l.:.~ ..... ~).!,-:;.;:,..;.;.;, ., ..... ~.:;.;.:.·.:/t:...:.·...,.-io,~.'.;;~!.w ... ".~., •. oI. .. ,.,:...~--_ •• ~:.~.-."'" ••• ~ - : ... ~
~, \
Maf cos (Q + 2n/3)
+ Maf3 cos 30
Mal cos (Q + 2n/3)
-Ma2 sin (0 + 2n/3)
-Mab
+Mab2 cos 2(0 + n/3)
-Mab4 cos 4(0 + n/3)
-Mab
+Mab2 cos 20
-Mob4 cos 4 Q
L 00
+L cos 2 (0-n/3 002
+L oa4 cos4 (0 - n/3)
(4-4) ~
.-1
31
During normal operation the field voltage is constant while the stator voltages form
a balanced three phase set.
Thus
vf
= Vf
v = V cos (Co) t ) a
(4-5) v
b = V cos (Co) t - 21T/3) ,
and v = V cos (Co) t + 2 1T / 3 ) .
c
The instantaneous angular position 0 of the rotor is the time integral of the speed of
rotation,
o = Co) t + ~ (4-6)
where Co) is the angular frequency of the supply and ~ is a constant of integration
chosen so that
= 31T/2-5 (4-60)
5 being the load angle defined as the angle between the direct axis and an axis based
upon the three phase terminal voltages.
The currents in the field winding and the two damper windings are assumed
= 'fO + I (If Dl cos lO + 'fQl sin lO) L
(4-7a)
32
f~ I il = ('1 Dl cos lQ + 'lQl sin lO) (4-7b)
l
i2 = I ('2Dl cos lQ + '2QL sin lO ) 1 (4-7c)
l
:
wh i le the stator currents are
i = I ( 'Dm cos m 0 + 'Qm sin m 0) 1 (4-7d) a
m
ib = i (0- 21T/3) (4-7e) a
i = i (0 + 21T/3) (4-7f) c a
where
L is any positive even integer 1
and
m is any positive odd integer
4 .2. l Computer Program
'n Appendix JI the computer program used to obtain the analytical solution
is listed together with a typical data set. We make use of the knowledge that the
various harmonie series quoted in equations (4-7) converge very rapidly 1 to terminate
the current and inductance harmonie series at the seventh harmonie. More or less terms
can be used depending on the accuracy required and the time and cost of the computer
program.
33
The first four equations of (4-1) are then expanded in detai!, the
fifth and sixth being unneeessary sinee they repeat the information eontained in the
fourth equation due to the assumption made in equations (4-5) and (4-7 e and f)
about the balaneed three phase set of voltages and eurrents. The output is in the
form
v = AI (4-8)
where V is the voltage eolumn vector eonsisting of 27 elements which are the result
of analyzing the harmonie eomponents of each voltage in the direct and quadrature
axes. A is a 27 x 27 harmonie impedanee matrix and 1 is a current eolumn veetor
of 27 elements eonstrueted in a similar way to the voltage vector. Equation (4-8)
is depieted more explieitly in the fold - out in Appendix Il .
A numerieal solution is now straightforward since it only involves the
solution of a set of Iinear equations. However, inspection of the matrix equation sug-
gests some results. The first row of this matrix equation yields
1 j
34
Rows 2, 3, 8 and 9 read :
il
Rf 2 Xf 2 Xf1 'fD2 ~
j !
1.
-2 Xf Rf -2 Xf1
'(
'fQ2 "'.~ :y.
.;.~
,il 0 (4-9)
.~
= ~ ,~
2 Xf1 R1 2 Xl '1D2 ',~
:~ :" ;.;>~
-2 Xf1 -2 Xl Rl I1Q2 .~
:., ·1
.' ,
which, since the square matrix is not singular, imply that
IfD2 = 'fQ2 = '102 = 'lQ2 = 0
Similarly, rows 4, 5, 10 and 11 give
, = 'fQ4 = '104 = 'lQ4 = 0 'fD4
The four rows 14, 15, 16 and 17 also yield
'2D2 = '2Q2 = '204 = '2Q4 = 0
Actually, these results are expected since in a balanced three phase
synchronous machine the induced stator mmf waves that couple to the rotor are the
fifth, seventh, e'eventh ...• harmonics only [15J. The fifth harmonic mmf wave
induced in the stator rotates at a speed w /5 in a forward direction, thus the relative
35
r: velocity with respect to the rotor is (Co) + Co) /5) = 6 Co) / 5, and since it has five
times the 'basic number of poles, the induced field in the rotor side has a frequency
of 6 Co). Similarly, the seventh harmonic stator mmf wave rotates backward at a
relative velocity (Co) - Co) /7) = 6 Co) /7 and has seven times the basic number of
poles th us contributing to the six harmonic rotor mmf waves only.
To solve for the remaining current components we use the measured
parameters shown in Tables [4-1 J and [4-2J quoted from Reference [19J for the
TABLE [4 - 1 J
WINDING RESISTANCE IN OHMS
Rf R1 R2
R a
38.0 3.65 11.97 1.47
synchronous machine under consideration. Substituting these values and solving the
matrix equation (4-8) using a standard gaussian elimination routine (GELG, IBM
scientific subroutine package) , the elements of the current vector are readily obtained.
ii
f-
j
1 J
.1
36
TABLE [4 - 2J
WIN DING INDUCTANCE
Harmonic Order 0 1 2 3 4
Lf (H) 10.2
II (mH) 100.3
l2 (mH) 141.2
l (mH) 60.1* 10.7 3.0 aa
Mfl (mH) 770.0
Mal (mH) 65.8
Ma2 (mH) 49.5
Maf
(mH) 812.0 ** -
Mab (mH) 26.3 19.8 2.0
* This value include the leakage inductance La' which is equal to 4.35 mH •
** The value of Maf3
is negligibly small in the experimental machine.
--
., \ {
37
4.2.2 Results
The results obtained from solving (4-8) at a stator phase voltage of
120 volts, and an average field voltage of 19 volts are compared with the results
obtained by Dunfield in Table [4-3]. The agreement between the different methods
is good. The results obtained using the analytical solution are close to th ose obtained
using the Newton - Raphson technique, which is an indication that the analytical solu
tion aeeuraey with the harmonie expansion limited to the seventh harmonie is acceptable.
Figure [4-2] shows the waveform of the stator line current at two load
angles 5 = - 100
and 5 = - 200
• We notice that the third harmonie eomponent
in the stator eurrent is as large as the fundamental white the effects of the fifth and
seventh harmonics are hardly notieed. The variations of the fundamental, third and
fifth harmonic stator current with the load angle are shown in Figure [4-3J. It is c1ear
that the amplitude of the third harmonie is always comparable to the fundamental while
the amplitude of the fifth harmonie is within about 21 %. The highest value of the
seventh harmonie is 0.09 ampere reaehed at a load angle of 900
, whieh represents
about 1.65 % of the fundamental. Typieal values of the ratios 13 /11 , 15/11
and 17/11 are shown in Table [4-4].
The variation of the sixth harmonie eurrent in the fjeld winding and the
two damper windings is shown in Figure [4-4]. The sixth harmonie eurrent in the
quadrature axis damper winding is higher than that in the direct axis damper winding
especially at large load angles.
1 (A) a
6.0 6 = -20 o
3.0
360 620 o.o~~----~-+----~~----~--~--~
-6.0
1 (A) a
6.
3.
o 6 = -10
360 620 o.~~~ __ ~ __ +-__ ~ __ +-______ ~ __ ~
-3.
-6.
38
loi t ( degree )
loi t (degree )
"j i
:1
~ .~
4 1 ~ 1 1 i "{
i !
il i "~ ;!
j
FIGURE 4-2. STATOR UNE CURRENT FOR A LOAD ANGLE 6 = _200
AND 6 = _100 .j
1
15
'-.
-90 -60 -30
6.0 r (A)
4.0
fi !
1 ~ l ~ \~ .. ··c !
o 30 60 90 5 (degree)
39
FIGURE 4-3. FUNDAMENTAL, 3rd AND 5th HARMONIC UNE CURRENTS, 4- WIRE CONNECTION.
-90 -60 -30 o 30 60 90 5 (degree )
FIGURE 4-.4. 6th HARMONIC FIELD AND DAMPER WINDINGS CURRENTS, 4 - WIRE CONNECTION.
r~
~9 TABLE [4 - 3 J
~--~ , '.
COMPARISON OF RESULTS FOR FORMAC, RUNGE - KUTTA AND NEWTON - RAPHSON METHODS
AT 5 = - 100
, 4 - WIRE CONNECTION
Magnitude (RMS) Phase (degree )
Parameter Harmonie FORMAC Runge - Newton - FORMAC Runge - Newton -Order Kutta Raphson Kutta Raphson
Stator Current 1 1.046 1.047 1.047 154.6 154.7 154.6
3 1.086 1.096 1.086 314.0 314.0 314.0
(A) 5 0.223 0.229 0.224 110.7 110.5 110.7
7 0.017 0.037 0.016 97.1 89.8 98.6
Field Current (A) 6 0.022 0.024 0.022 10.2 10.8 10.2
Direct Axis
Damper Current (A) 6 0.033 0.035 0.038 9.1 9.6 9.0
Quadrature Axis
Damper Current (A) 6 0.126 0.105 0.126 279.1 278.6 279.1
,.-.:. ''''-''L.':'<:..:i.t.~~i.,;:u~~~'''''''''''';''~'~''~.\·':'''~_;'-'_:':' ....,:-~;;.-'.,...~-'Ol....;; ..... ..:,::_~.; ..,,_.~,,;:.-, '~~''-''''C .. ':':;~;~~~~~)~W~.iK.~(~):i~~:"~;i..~~'~~~:;:'..r';:.,;.:~;:!~; ... ;.~..,.,- tr\;\:~".-..• ",-:;:--".J ••• ,., .. :~ .• ~.: ••
1
1
i
1
1
1
1
1 1
1
~
- . --::J
41
TABLE [4 - 4 ]
THE RATIOS 13/ 11 1 15/ 11 AND 17 / Il
FOR THE 4 - WIRE CONNECTION
13/ 11 % 15/ 11 % 17/ 11 %
107.2 21.8 1.84
The second and fourth harmonies in the field current and the two damper
winding currents are identically zero as was discussed previously .
4.3 Steady State Solution of a Three-Wire Star Connected Synchronous Machine
Since for the three-wire star connection the sum of the stator currents is
zero, ia + ib + i c = o. Equation (4-1) must be transformed so that a solution
takes cognizance of this facto The relation between the voltages and currents is therefore
vf Zf Zn Zfa - Zfc Zfb - Zfc
Zn Zll ZIa - Zlc Zlb - Zlc
= Z22 Z - ~ 2a e ~b- ~c
Zaa + Zee Zc;c+ Zab
v -v a c Z -; Zla - Zle ~a- ~e - 2 Z -Z - Z fa c ac ac bc i
Zce+ Zab ~b+ Zee
v - v b c Zfb- Zfc Zlb- Zlc Z - Z -Z -Z -2Z 2b 2e be ac be
a
(4-10)
1
1 j j
42
The resistances and inductances are determined from equations (4-3) and (4-4)
with the proper re-arrangement for the solution.
4.3. 1 Computer program
A similar computer program to that used for the four wire conneetion is
used to solve this problem. Again the first four rows of (4-10) are expanded to
yield the complete analytieal solution
v = AI, (4-11)
whieh is shown explicitly in the fold-out in Appendix Il. A similar analysis to that
outlined in the previous case follows .:j
4.3.2 Results
Sinee for the three-wire star eonneetion the sum of the stator eurrents is
zero, the third harmonie in the stator eurrent does not exist. In Figure [4-5 ] the
fundamental and fifth harmonie eornponents of the stator eurrent for various load angles
are shown, the seventh harmonie being very small, with a maximum value of 0.088 ampere
o at a load angle of 90 •
The sixth harmonie eomponents in the two damper windings, and the field
winding eurrents are depieted in Figure [4-6]. It is clear that the harmonie eurrent
.,
î ',. ~
43
6.0 (A)
-90 -60 -30 o 30 60 90 S (degree )
FIGURE 4-5. FUNDAMENTAL AND 5th HARMONIC lINE CURRENTS, 3 - WIRE CONNECTION.
0.06 (A)
-90 -60 -30 o 30 60 90 S (degree )
FIGURE 4-6. 6th HARMONIC FIELD AND DAMPER WINDINGS CURRENTS, 3 - WIRE CONNECTION.
(
44
in the quadrature axis damper winding is greater than that in the direct axis damper
winding at the sorne load angle, and both reduce to zero at zero load angle. The
a. c. compone nt of field current is very small, typically 0.031 ampere at a load
1 f 900 • ange 0
'n addition we notice that this connection of the machine causes the
generation of identical third harmonie neutral voltages in each phase of the stator
windings. These voltages appear when the voltage is measured between line terminal
and stator neutral or between stator neutral and source neutral. The neutral voltage
may be readily calculated as the variation between source neutral and machine neutral.
The relationships are those of equation (4-8) with 'D3 = lQ3 = o. Therefore,
the neutral voltage VI' where n denotes the source neutral and ni the stator n n
neutral is
= V D3 cos 30 + V Q3 sin 3 Q (4-12)
where
=
r: 45
=
- 1.5 (Xaa2 - Xab2 ) 'D5
-1.5 (Xaa4
+ Xab4
) ID7
Thus, substituting in equation (4-12) the values of the currents obtained
from the solution of equation (4-11), V 1 is obtained. The variation of the neutral n n
voltage with respect to the load angle is depicted in Figure [4-7 J from which we
notice that the neutral voltage increases with the load angle reading V 1 = 46 volts n n
o at a load angle 5 = 90 •
Table [4-5 J compares the magnitudes and phases of the different
harmonic currents and neutral voltage obtained using different computational techniques.
As in Table [4-3] the agreement between the different results indicates that the
accuracy obtained using the series expansion limited to the seventh harmonic is acceptable.
4.4 The 3 - Wire Versus the 4 - Wire Connection
Comparing the results obtained for the 3 - wire connection with those
previously discussed for the 4 - wire connection, we easily notice that the former is
to be preferred, as with this connection the harmonic interactions are greatly reduced
46
60 N)
-90 -60 -30 o 30 60 90 S (degree )
FIGURE 4-7. THIRD HARMONie NEUTRAL VOLTAGE VI· nn
l
I~ (
....... ~
... "
TABLE [4 - 5 J
COMPARISON OF RESULTS FOR· FORMAC, RUNGE - KUTTA AND NEWTON - RAPHSON METHODS
AT S = - 100
1 3 - WIRE CONNECTION
Magnitude ( RMS ) Phase (degree ) ,
1
1
Parame ter Harmonie FORMAC Runge - Newton - FORMAC Runge - Newton - 1
Order Kutta Raphson Kutta Raphson 1
Stator Current 1 0.925 0.926 0.925 153.4 153.1 153.2
3 - - - - - -(A) 5 0.033 0.033 0.033 109.7 109.6 109.7
7 0.014 0.015 0.015 269.3 269.0 268.8
Fie Id Current (A ) 6 0.005 0.005 0.005 9.6 9.4 9.4 1
1
Direct Axis ,
Damper Current (A ) 6 0.009 0.009 0.009 8.5 8.2 8.4 1
Quadrature Axis Damper Current (A) 6 0.010 0.007 0.010 277.9 277.8 277.9
Stator Neutral Voltage (V) 3 7.400 7.313 7.311 235.0 236.1 235.0
tihni "Y"tel' .. ~ ·· •. ...;.% ... 5 "d"';"r: <oH' -,~.'t-1y .~'. --'tG' "~H ··lp'·"»tt' H~.ij~Ü~a.:.~ .. \:.:...u....ç"...t .. ;..:.",~~~~ ...... , .• .,. __ .:".:: .....
~
48
and the third harmonie of the stator current is eliminated. Table [4-6 ] compares
between both connections at a load angle of 200•
TABLE [4 - 6 .J
Connection 13/ 110/0 15/11 0/0 17/ 11 % V 1 (volt) nn
4 wire 107.2 21.8 1 1.48 -3 wire - 3.5 1.5 7.3
A drawback of the 3-wire connection is the neutral voltage which appears
between the line terminal and stator neutral. Although this voltage is small it can be
troublesome because it gives rise to a winding voltage gradient higher than that expected
on the basis of the applied stator voltage and therebyreduces the stator insulation safety
factor.
4.5 Conclusion
ln this chapter, the machine equations for a 3-phase synchronous machine
are expanded to yield the harmonie impedance matrix relating the voltage components
to the current components. The matrix equation th us obtained gives a c10sed form solu-
i .) li 1 j
,l 1 1 1
~ '1 .)
·1
1 .;
:1 il
1 1
,1
J
1 J
"'
j
{'
49
tion for the eurrent eomponents. The results are in good agreement with those ob
tained experimentally and analytically in Reference [l9]. This indicates that
limiting the current and impedance harmonie series to the seventh harmonie in the
analytical solution leads to a reasonable accuracy. Considering computation time,
typically a runof 3 minutes might be required for a poor choice of initial conditions
for the Runge- Kutta program to evaluate the currents at a typical load angle. The
sa me results are obtained in less than 0.3 seconds using the harmonie impedance
matrix, an improvement of about 2 to 3 orders of magnitude.
50
CHAPTER V
ANALYTICAL SOLUTION FOR INDUCTION MOTORS
WITHPHASE-WOUND ROTORS
5. 1 Introduction
As shown in Chapter Il due to the multiple armature reaction effects, when
current flows in a machine winding it will create fields in the primary which will cause
multiple frequency currents to flow in the secondary, in turn these currents create multiple
fields that will react back in the primary.
Oberretl [17] and Robinson [15 J indicated that the assumption of si nu-
soidal mmf's is invalid for induction machines in which multiple armature reaction can
give rise to appreciable harmonic effects. Dunfield and Barton [26] discussed the
theoretical background and reported the experimental results for such phenomena in a
2-phase induction machine. In this chapter we will consider the same simple example,
namelya machine with two primary and two secondary windings as shown in Figure [5-1],
for which the relationship between the voltages and currents can be written in a matrix
form as :
v z Z 1 s ss sr s
= (5-1)
0 Z Z 1 rs rr r
f.~
51
FIGURE 5-1. WINDING CONFIGURATION OF A 2 - PHASE INDUCTION MACHINE.
:[
52
The impedance matrix of such a machine is
R + 0 L Dr Mh
cos h (,)t - Dr Mh s s
sin h n/2 sin h t.) t
R + DL Dr Mh
Dr Mh
cos h (,) t s s
sin h n/2 sin h (,)t
z =
D r ~ cos h (,)t Dr Mh
R + D L r r
sin h n;2 sin h (,)t
- Dr Mh
Dr Mh
cos h (,)t R + D L r r
sin h n;2 sin h (,)t
(5-2)
where h is any positive odd number and D the differential operator.
5.2 Harmonic Interactions in Induction Machines
If the rotor rotates at an angular frequency (,) electrical radians per r
second, and the stator is excited from a 2-phase supply of frequency f ,producing a o
fundamental synchronous speed (,) radians / sec., the fundamental component of the o
stator mmf induces voltages of frequency ((,) - (,) in the rotor windings. The o r
53
resulting rotorClJrrentsproc:luce a fundamental field which rotates in a forward direction
relative to the rotor at (la) - Col ) and in synchronism with the originating field. o r
These rotor currents also proc:luce harmonie fields which move, relative to the rotor,
with the corresponding fractional speed, and whieh then induce voltages in the stator
which are, in general, not at the supply frequency. As an example consider the effect
of the third harmonic component of rotor mmf. This rotates backwards with respect to
the rotor at an angular frequency (Col - LI ) /3, and induces voltages in the stator o r
windings of angular frequency (Col - 4 LI ) • o r
Tables [.> 1 a] and [5-1b] show respectively the non-supply stator and
rotor frequencies of interest which we ean express as
(_ 1 ) k+l k LIS = LI +[m+(-l) ] Col
0 r (5-3)
and
LlR = (_ 1 ) k+l
LI - mLl 0 r (5-4)
where
m = 2 k - 1 , k being any positive integer.
We conclude that the currents in the stator and rotor windings can be
expressed as harmonie series of the above frequencies only, while other components
are expected to be zero.
. .~
;'-
':~
"
,1:.'
if.'
i.'
m
3
5
7
9
m
1
3
5
7
9
TABLE [5 - 1 a ]
STATOR FREQUENCIES OF INTEREST
Induced Rotor Rotation mmf Frequency Frequency
- «,) - (,) }/3 (,) o r r
«,) - (,) }/5 (,) o r r
- «,) - (,) )/7 (,) o r r
«,) - (,) )/9 Co) o r r
TABLE [5 - 1 b ]
ROTOR FREQUENCIES OF INTEREST
Induced Stator Rotation mmf Frequency Frequency
Co) Co) 0 r
- Co) /3 Co) 0 r
Co) /5 Co) 0 r
- Co) /7 Co) 0 r
Co) /9 Co) 0 r
54
Stator Voltage Frequency
- «,) - 4 (,)) o r
«,) + 4 (,) ) o r
- «,) - 8 (,) ) o r
(Co) + 8 Co) o r
Rotor Voltage Frequency
(Co) - Co) ) o r
- «,) + 3 Co) o r
(Co) - 5 Co)} o r
- (Co) + 7 Co) o r
(Co) - 9 Co) o r
55
5.2.1 Harmonie Effeets in a Symmetrieal Two Phase Machine
Under steady state conditions, we can assume the current in the stator
windings
1 1 cos [( la) :i: L la) ) t - al ] s" 0 r
or using the quadrature and direct axis representation
1 Dt cos (la) + l la) ) t + 1 QL sin (la) + L la) ) t sor sor (5-50)
and rotor currents are
\' 1 = L 1 D cos (la) + m la) ) t + 1 Q sin (la) + m la) ) t r rm 0 r rm 0 r
(5-5b)
m
where
is the' rotation frequency rad / sec,
l is an even integer positive or negative,
m is an odd integer positive or negative.
ln general, solving equation (5-1) analytically will lead to a large
number of 1 inear equations as explained before. If we terminate the harmonie current
series at the ninth harmonie and using the same procedure explained in Chapter III ,
the number of equations to be solved will be 76 equations in 76 unknowns. This
large number of equations will be difficult to handle from a computational point of view 1
::
~
, :l
since the array area core storage needed for the FORMAC exit routines will be very large.
j -.J
56
To reduce the number of equations and make them in a suitable form
for computation one can take advantage of the following :
1 . The interaction between the stator and rotor mmf's will
be such that sorne harmonies will be zero as shown in the
previous section.
2. The symmetry of the supply voltage and the configuration of
the machine windings help us to solve the problem for one
stator and one rotor windings only, since the currents in the
two other windings will be of the sa me amplitude but shifted
in phase by 1T / 2 •
5.3 Computer . Program
A typical data set is Iisted in Appendix III with the FORMAC computer
program. In the data, both currents in the stator and rotor are expressed as in equation
(5-5) in complete harmonie series to get the complete solution and a better insight into
the results. The solution is only for one stator winding and one rotor winding thus in-
volving only 38 unknowns and is of the form :
v = A 1 (5-6)
where V is the voltage vector of 38 elements, A is the harmonie impedance matrix
(38 x 38) and 1 is the current vector. This matrix equation is shown explicitly in
(
57
the fold-out in Appendix 111*. The first 18 rows of A are related to the stator
winding and the remaining 20 rows to the rotor winding.
5.4 Results
For the machine we are considering the values of the stator and rotor re-
actances at 60 Hz are
X = 179.24 ohm s
X = 179.47 ohm r
and the 60 Hz magnetizing reactances are
Xl = 174.0 ohm
X3 = 0.695 ohm
X5 = 0.365 ohm
~ = 0.043 ohm
X9 = 0.03 ohm
while the stator and rotor resistances are
R = 5.74 ohm s and R = 5.89 ohm. r
* ln the computer print-out, the current components are indexed with the letter P for
m or l positive and with the letter N for m or L negative.
58
The numerical solution of equation (5-6) confirmed the expectation
that the stator and rotor current series contain the frequencies obtained in
Tables [5-1 a J and [5-1 b J only. Using the computed results we plot the wave-
forms of the stator current at various frequencies of rotation. These are shown in
Figure [5-2 J. The upper curve of each pair is the current waveform, while the
lower curve is the same waveform with the supply frequency component removed and
with the vertical current scale increased five times. The magnitude of the harmonie
content is surprisingly large in view of the very low harmonie content of the inductances.
The excellent agreement between predicted and measured [26J waveforms gives con-
fidence in the theory.
The waveforms of the rotor current at various frequencies of rotation are
shown in Figure [5-3J. These waveforms are distorted from the sinusoidal shape due
to the harmonie effects.
The amplitudes and phases of the stator harmonie currents are drawn in
Figure [5-4 J as a function of the ratio f / f . r 0
From this figure we notice the
trivial reuslt that the amplitude of the l th harmonie current vanishes when
f 0 + l f r = 0 je. g . the current component of frequency f 0 + 4 fris zero at a
re lative frequency f / f = - 0.25 , whi le that of frequency f - 8 f reduces r 0 0 r
to zero at f / f = + 0.125. In general, the harmonie contents represent a small r 0
percentage of the supply current. At a rotational frequency of 0.6 f typical values o
for the fourth and eighth harmonies are 9 percent and 1.6 percent of the supply
current respectively, a result which agrees with the experimental data [26J.
(
(0)
f If = -0.8 r 0
(b)
f If = -0.4 r 0
(c)
f If = 0.6 r 0
'\/\(\[\/ VVVV
", 1\ f\ f\ C\ f\ f\!\ !\ {\ f\ V\}VVV\JVVVV'
59
FIGURE 5-2. STATOR CURRENT WAVEFORMS FOR VARIOUS VALUES OF RELATIVE ROTATIONAL FREQUENCY f If 1 (VERTICAL SCALE 2 AMP./INCH).
r 0 THE LOWER TRACE OF EACH PAIR IS THE SAME WAVEFORM WITH THE FUNDAMENTAL REMOVED AND THE SCALE INCREASED 5 TIMES.
.~
1 1
60
(a)
f 1 f = -0.8 r 0
(b)
f 1 f = -0.4 r 0
\.
(c)
f 1 f = 0.0 r 0
FIGURE 5-3. ROTOR CURRENT WAVEFORM FOR VARIOUS VALUES OF RELATIVE ROTATIONALFREQUENCY f If , (SUPPLY CURRENT 2 AMP.).
r 0
i
1
1
1
j
-0.4
FIGURE 5-4.
(A)
14-
o 0.4 f If r 0
-0.4
(a)
(degree)
90
1,4
0.4 f If r 0
0.12 (A)
o
(b)
(degree)
61
0.4 f If r. 0
0.4
1" 8+
f If r 0
MAGNITUDE AND PHASE OF THE STATOR HARMONIC CURRENTS AT FREQUENCIES f ±4 f AND f ±8 f AS FUNCTION OF THE oro . r RElATIVE ROTATIONAL FREOUENCY f
r 1 f 0 •
l
-0.4
-0.4
FIGURE 5-5.
(
0.12 (A)
o -0.4 o f If r 0
(a) (b)
360 (degree)
360 (degree)
270
13+
62
0.4 f If r 0
7+
)
'x ~ ':f . ~q ;; ~
i ,1
i
1 1
O.4f If -0.4 0 0.4 f Ifl r 0 r 0
1
o
MAGNITUDE AND PHASE OF THE ROTOR HARMONIC CURRENTS AT FREOUENCIES f + 3 f , f - 5 f , f + 7 f AND f - 9 f AS oro r 0 r 0 r FUNCTION OF THE RELATIVE ROTATIONAL FREQUENCY f / f .
r 0
,1 ! ~
ln Figure [5-5Jthe magnitudes and phases of the rotor currents are
drawn with respect to the ratio f / f . Comparing Figures [5-4 J and [5-5 J , r 0
63
we c1early see, the similarity and the dependence of the interacting stator and rotor
currents. We also notice a shift of approximately 180 degrees in the phase of the
rotor current with respect to the phase of the corresponding interacting stator current,
a result which implies the transformer-like nature of the induction machine.
5.5. Conclusion
ln this chapter, the machine equations for an induction machine are
solved analytically. The analysis shows that not ail the harmonic orders expected in
the stator and rotor of a general machine are excited in the induction machine. Further,
in the example considered, the symmetry of the 2 phase machine reduces the initial
problem to a simpler one where only the interaction between one stator and one rotor
windings are considered.
The analytical solution yields the harmonies impedance matrix of the
machine. Numerical results are in excellent agreement with those obtained experimen-
tally. This merely shows the reasonable aecuracy that ean be achieved by truncating
the current and impedance harmonic series at a not-so-high order harmonic, say the
ninth.
" ., 3 "
64
CHAPTER VI
SINGLE PHASE MACHINE, EXPERIMENTAL STUDY
6. 1 Introduction
The purpose of the study in the previous two chapters was to check the
validity of the analytical solution and to evaluate the accuracy of the predicted
results by comparing them to those obtained experimentally.
ln this chapter a more searching test of the validity of the proposed
technique than those balanced three phase situations is provided by sorne single-phase
machines. The single phase synchronous generator is a device of sorne practical
significance in which extreme precautions must be taken to cornbat the harmonic gene-
rating effect of the negative sequence field [15J. Accordingly the synchronous
machine described earlier has also been tested in a single-phase configuration in which,
since it is only equipped with a modest damper winding, multiple armature reaction
effects are severe.
ln addition a single-phase, single-phase induction machine was tested. Such
a machine, while of no practical significance, shows in the most extreme degree the
multiple armature reaction effects under investigation.
For both machines analytical results are also obtained and compared to the
experimental results enhancing our confidence in the analytical method.
"
.~ t
·1 .;
f
1 :~ J
DA
v
FIGURE 6-1. SINGLE PHASE CONNECTION FOR THE EXPERIMENTAL STUDY OF THE SYNCHRONOUS MACHINE.
65
(-
66
6.2 Single Phase Synchronous Machine
The three phase machine used as an·example in the analytical study
in Chapter IV is used here in a special connection to simulate a single phase
machine. This is a salient pole alternator rated at 220 volt, 3 kW, 0.8 PF
lagging. The field is on the rotor and there are four salient poles each having a
600 turn exciting winding, a 2 turn search coil at each end of the main field
winding and a damping cage in the pole face. The 48 slots stator carries a
balanced, three phase, double layer winding of 5/6 pitch and 52 turns per pole
and phase.
The test was carried by connecting the phases a and b of the stator
windings together to a single phase supply of 120 volt, while phase c was left un-
connected. The field winding was connected to a d.c. source adjusted to supply
an average field current of 0.5 ampere. The configuration of the machine is shown
in Figure [6- 1 ] .
6.2.1 Analysis
The machine equations are derived from the 3 phase synchronous machine
matrix equation (4-1) by putting ia = - ib , ic = 0 and v = va - vb • The
resulting matrix equation is :
i :~
'1 , '~
1 :~ J
i ·'.1:
,
,;
67
Zf zn Zfa - Zfb if
Zf1 Zll Zla - Zlb i 1
~2 ~a- ~b i2
v Zfa - Zfb Zla - Zlb Z2a - ~b Zaa + Zbb i a
- 2 Zab
(6-1)
The harmonie impedance matrix is obtained using a slightly modified
version of the program used for the 3 phase machine. The data used as weil as the
harmonies impedance matrix are shown in Appendix IV .
6. 2 .2 Resu Its
Photographs of line current, field current and stator voltage waveforms
were taken with the synchronous machine operating bath as a motor and as a generator
at various load angles. The load angle ô was measured using a line frequency
synchronized strobosc~e, by observing the shaft position at zero load angle, i.e.
when the output current is virtually zero (or minimum) and noticing the change in
angular position as the load is varied. Due to a slight hunting the accuracy in
measuring the load angle is considered to be :1: 2 degrees.
:~
-'~i ., 5 ~::l
~~ " ~ .~ :~
'l 1 ~~ :f :.;
.:
68
60 Hz VOLTAGE REFERENCE
********
Il /1 ~ : 1\ ' 1/ V f V
-l- ./ / ( ( (
\ \t: 1 \ ~. \,1 \
~ '4 \ '
(a)
********
(h)
FIGURE 6-2. PREDICTED AND EXPERIMENTAL WAVEFORMS Of Tt1E STATOR, CURRENT (UPPER TRACE) AND fIELD CURRENT (L0WER TRACE) AT LOAD ANGLES (a) 6 = 400 AND ~) 6 = 200
, Continued ..... .
'.:-r
,\
.? :J j J f j
i l
j j
1 l
j 1 )
60 Hz VOLTAGE REFERENCE
********
(c)
********
(d)
FIGURE 6-2 (CONTINUED) - STATOR AND FIELD CURRENT WAVEFORMS AT (c) Ô = _200 AND (d) Ô = 0 •
69
70
Figure [6-2] shows oscillograms of actual waveforms of the supply
voltage, the stator induced current and the field current compared with the predicted
waveforms at different load angles 0 These waveforms show that the a oC 0 component
of the field current contains ail the even harmonics, thus confirming the results dis-
cussed in Reference [15] 0
Measured and computed waveforms of line current at a load angle of
20 degrees are shown superimposed in Figure [6-3]. This figure shows the reason-
able agreement between the predicted waveform and the experimental one 0 However,
a smalt discrepancy points out a source of error either in truncating the harmonic series
at the seventh harmonic or in the measured machine parameters 0 At the same load
angle the frequency spectrum of the stator current waveform with a logarithmic vertical
scale is shown in Figure [6-4] 0 The ratio of the third harmonic to the fundamental
is 'J7 % while that of the fifth is 15 %. The seventh harmonic is only 2.8 % of
the fundamental which suggests a reasonable accuracy by terminating the harmonie
series after the seventh order 0
6.3 Single-Phase / Single-Pha~ Induction Machine
The harmonic interactions in the case of a simple model of a single phase
induction machine is studied experimentally and analytically. The stator of the
machine used in this study has a single layer 2 pole a oc. winding in 24 slots. The
rotor winding is of the double layer lap type, wound in 36 slots, with a coil pitch of
-, -,
1 ; ! :1 1
1 i 1 :1
1
4
2
o
-2
-4
(A)
90
Predicted
A..A.A Experimenta 1
270
71
(,,) t (degree)
FIGURE 6-3. PREDICTED AND EXPERIMENTAL STATOR CURRENT WAVEFORMS AT A LOAD ANGLE ô = 20° •
FIGURE 6-4.
,
~ f,
f' Ji
\ f 1 ft :1 , f, j-'
)~ f' n ~ ::---- t '\ '1 ~ , \" ."' \, -""
. , 1 1
60 180 300 420 540 Frequency (Hz)
FREQUENCY SPECTRUM OF THE STATOR CURRENT AT A LOAD ANGLE ô = 20°. THE VERTICAL SCALE IS LOGARITHMIC.
1
", f
r: 72
1 - 19 slots. In the single phase configuration the machine is connected to a single
phase supply of 120 volts. A d.c. dynamometer is used todrive the machine at
the desired speed.
6.3.1 Analysis
The relationship between the currents and the voltages in the single phase
machine is
v s
o
=
R + DL s s
Dr Mh
cos h Cal t
Dr Mh
cos h Cal t
(6-2)
R + DL r r
which is derived from the 2 - phase case, equation (5-1) , by forcing is2
and ir2
to be identically zero. The data used and the output matrix are listed in Appendix V .
6.3.2 Results
To obtain the machine constants, an open-circuit and a short-circuit
tests were performed. Ignoring the effect of harmonies, the equivalent circuit shown
in Figure [6-5] is obtained. The total stator and rotor winding reactances at the
su pp Iy frequency are X = 154.3 ohm and X = 158.6 ohm respective Iy . The s r
\,
J
1 •
FIGURE 6-5.
73
1.8 5.3 9.6 1.24/5
SIMPLIFIED EQUIVALENT CIRCUIT OF EXPERIMENTAL INDUCTION MACHINE. RESISTANCES AND REACTANCES ARE IN OHMS AT 60 Hz.
"1 ,
::.
74
mutual reactance between the stator and rotor is Xl = 149 ohm while the stator
and rotor resistances are R = 1.8 ohm and R = 1.24 ohm respectively. , The r s
60 Hz - harmonic magnetizing reactances are calculated from equation (6) of
Reference [6] :
X3 = 0.596 ohm
X5 = 0.310 ohm
X7 = 0.037 ohm
X9 = 0.025 ohm
The excellence of the predictions of the rotor current waveforms is
demonstrated in Figure [6-6] where oscillograms are compared to calculated plots
at different speeds of rotation. At a slip s = l - (LJ / LJ ), the significant fre-r 0 i
quencies are obtained by putting m = 1 in the set [LJ ± m LJ ] = [1 ± m ( 1 - s ) ] LJ , ,1 oro 1
which is the frequency spectrum of the rotor current. We focus attention on one of
the waveforms, sayat a slip of o. l • Compared to the 60 Hz reference waveform,
the frequencies of interest are the' 1.9 LJ spiky waveform and the 0.1 LJ envelope. o 0
Figure [6-7] shows the complete frequency spectrum of the rotor and stator current
waveforms at o. 1 slip. The magnitudes of the current components are Iisted in
Tables [6..1] and [6-2], from which we notice that in the analytical solution trun-
cating the harmonic series at the ninth order is a justifiable approximation. A similar
analysis and deduction can be performed on any of the waveforms shown in Figure [6-6].
1
\ ~ 3 1
1 1 ~ ~ 'j .~
l 1
~ ~ 1
';,:
·0
.:~
:~
'f ,~ .. , li
.~
.75
60 Hz VOLTAGE REFERENCE
• 1
S = 0.1 ... t': ~ i\~
; , A "'1 ~" ... ~ -1
1"0.. k J \ 1 ) 1 'J v
S = 0.3
,.. .~ )~ A. tJ ?tA ~~ j~ r ~ , '1
..
il i
-1+1+
~ ,... ~r ~ '. ~M ~. ;1:' l/1 t ...."
S = 0.7 V
- + - + 1\ :t:
11 t
S = 0.9 i"\.
,. A. -...~.~
. _T ... ... ~~ , . ,1 T
t
f
.FIGURE 6-6. PREDICTED AND EXPERIMENTAL ROTOR CURRENT WAVEFORMS AT VARIOUS VALUES OF SLIP.
-::7 /1
1 j 1 1 1
1
• l ,
-500
~:,-, . .....
. , 1 . .',
-500
. v :["
F.IGURE 6-7.
(A)
5 1-
4 r-
3 1-
2
1. 1.
-300 -100
(A)
5 r-
4 r-
3 r-
2
1 -
1 • • -300 -100
(0)
1 1
100 300
, 100 300
76
t 1 1
500
Frequency (Hz)
(b)
, 1
500 Frequency (Hz)
FREQUENCY SPECTRUM OF : (0) THE ROTOR CURRENT 1
(b) THE STATOR CURRENT AT A SLIP OF 0.1 •
77
TABLE [6-1]
ROTOR HARMONIC CURRENTS AT A SLIP OF 0.1
Harmonie Current Percent from Compone nt Frequeney Compo'1ent
JA)_ Supply Current
Cal :1: Cal 3.5 75 0 r
Cal :1: 3 Cal 2.2 47 0 r
Cal :1: 5 Cal 1.35 29 0 r
Cal :1: 7 Cal 0.73 15 0 r
Cal :1: 9 Cal 0.23 4.9 0 r
TABLE [6-2J
STATOR HARMONIC CURRENTS AT A SLIP OF 0.1
Harmonie Current Percent from Compone nt Frequeney Component Supply Current
(A)
Cal :1: 2 Cal 2.9 60 0 r
Cal :1: 4 Cal 1.8 38 0 r
Cal :1: 6 Cal 1.04 22 0 r
Cal :1: 8 Cal 0.48 10 0 r
t
78
6.4 Conclusion
Experimental verification and confirmation of the analytical formalism
are done in this chap~r using as an example a single phase synchronous machine and
a single phase induction machine. It is demonstrated that in single phase machines,
a wide frequency spectrum of harmonics exists.
Using the measured parameters for the induction machine, the predicted
results are found to be in very good agreement with the experimental results. For the
synchronous machine the parameters were obtained from Reference [19], analytical
and experimental results agree within an error margin which is thought to be due to
the inaccuracy of measurement of sorne of the machine parameters.
79
CHAPTER VII
SUMMARY AND CONCLUSIONS
The main concern of this thesis is the development of a new analytical
approach to the study of the effects of space harmonics on the steady state current
and voltage waveforms in electrical rotating machines. The analytical approach in-
volves the manipulation of trigonometric and algebraic functions of moderate complexity.
The completion of this work was only made possible by the use of the facilities of the
IBM - FORMAC language available at McGiII University Computing Centre. Thus,
computer programs have been written in FORMAC and used to obtain a matrix formula-
tion relating the harmonic current components to the applied voltages for some typical
examples such as a 3 - phase synchronous machine, a 2 ~ phase lnduction machine and
single phase synchronous and induction machines. The harmonic impedance matrix
obtained using any one of the se programs is then used in a conventional FORTRAN
program to obtain numerical values for the current components under prescribed condi-
tions.
Using the analytically. computed matrices, a reduction of canputing time
of between 2 to 3 orders of magnitudes is achieved over earlier numerical iterative
techniques. The results obtained using different methods are compared and found to be
in very good agreement.
Experimental results for the single phase synchronous and induction machines
show the excellence of the predictions of the analytical solution. The experimental and
analytical studies for single phase machines show also that if single phase windings, or
80
their equivalent exist on both sides of the airgap, an infinite series of harmonie
components field is set up. A similar situation is expected to arise when un-
balanced loading or unbalanced faults occur in a 3 - phase alternator without a
complete damping winding on the rotor.
~. ~.' Th!:! areas of the future investigation based on this thesis are :
1. To develop a general computer program in FORMAC, or
any similar language, to 'be available for expanding the
equations of any machine in the generalized harmonie i~
pedance matrix form.
2. To make use of such programs in predicting harmonies induced
in machine windings due to different design considerations, and,
3. To develop programs able to optimize a certain machine design
or connection in view of minimizing the harmonie effects.
A saturated mach ine, because of its large magnetizing current, has a
low power factor and a slightly lower efficiency. A study of space harmonies as the
result of magnetic saturation in the machine is also another field of application of the
techniques described in this thesis. These space harmonies are of great practical signi-
ficance especially at high saturation levels and their effect must be included to obtain
greater accuracy in performance calculations.
'(" .~
81
APPENDIX 1
VOLTAGES INDUCED IN SYMMETRICAL WINDINGS
If we consider a wave of m pole pairs and (01 as the basic electrical
frequency in rad / sec, the general form of the flux density compone nt at an angle
cp in the airgap is
B CD = B sin (m cp- (01 t )
where B is the peak flux density (webers / m 2
) •
The flux linkage with a winding whose axis is at an angle p is
P+'Ir/2 X = J B sin (m cp- (01 t) d cp
P - 'Ir /2
= (2 B / m) sin (m 'Ir /2) • sin (m p - (01 t )
The voltage induced in the winding is given by
v = -dX/dt
= (2 B / m ) (01 sin (m 'Ir /2) cos (m p - (01 t )
This can be rewritten as
V(m,p) =
t This general formula is applied to spatially symmetrical windings to
determine the relations between their induced voltages and currents.
j
{~.
82
For example in symmetrical 3-phase windings , the voltages induced
can be expressed as :
v = l V (m, p) a m
m
Vb = l V (m, p + 2 'If /3)
m m
and
V = l V ( m, p - 2 1f / 3) c m
m
while in symmetrical 2-phase windings
V = \' Vm
(m, p) 1 a t....> m
and
Vb = l V (m, p + 'If /2)
m m
=
=
=
=
=
L V cos [ Co) t - m p ] m
m
)' V cos [ Co) t - m (p + 2 1f / 3 ) ] I....J m . m
~ V cos [ Co) t - m (p - 2 1f / 3 ) ] L.J m m
~ V m cos [ Co) t - m p ] m
\--. 1 V cos [Co) t - m (p + 1f / 2 ) ]
L.J m m
It is also deduced that the currents induced in the windings will have the same phase
relations as the voltages.
j
83
APPENDIX Il
3 - PHASE SYNCHRONOUS MACHINE
ln the following sections, a conventional notation is used for the magni-
tu des of the different components of the voltages, currents, resistances and reactances.
These are denoted by V, 1, R and X followed by an alphameric string indicating
the particular compone nt • F denotes the field winding, land 2 stand for the direct
and quadrature axis damper windings, while A, Band C denote the different stator
windings. The letters 0 and Q point to the direct and quadrature axes respectively.
A stator related quantity is also denoted by S while R denotes the rotor related
quantities. The letter T is used in the computer print-outs for 0 and the FORMAC
symbol # P for 'Ir. Sorne other abbreviations are used to COl'ltract the notations,
and these are pointed out wherever felt necessary.
\ 'i
J 1 •
1
Note :
A - 3 - PHASE SYNCHRONOUS MACHINE,
4 - WIRE STAR - CONNECTION
1. Program listing .
2. A Typical Data Set.
3. The Harmonie Impedance Matrix.
VD and VQ are derived from :
v = v = V cos '" t a
= VD cosO + V Q sin 0
where
0 = '" t + P ,
and
P = 3 'Ir /2 - 6 ,
6 being the load angle.
84
1* THIS PRnGRAH ANAlYZES THE M.M.F. HARMONIC EFFECTS IN 3 PHASE SVNCHRONOUS MACHINES.
THE FOllDWING IS A LIST OF INPUT VARIABLES
VARiABLE MEANING -_._------------------~-------------~--~----~--. NW NUMBER OF WINDINGS IN THE MACHINE
N NUMBER OF EQUATIONS TD BF. SDLVED NSW HUMBER OF STATOR WINOINGS NU NUMBER OF UNKNOWNS (CURRF.NT COHPONENTS) NHAR HIGHEST HARMONIC OROER Wl THE REACTANCE HATRIX R 'THE RESISTANCE DIAGONAL MATRIX CU THE CURRENT VECTOR CUDQ 'THE VARIABLE ARRANGE~'ENT VEtTOR *'
WVFRH: PROCEDURE OPTIeNS (MAIN' ; FORMAC_OPTIONS ; DECLARE PUNCH FILE; Del STRING CHAR (66) VARYING ; DCl Wllb,6)CHAR(SOl ; Del CU(6) CHAR(240); Del R (6) CHARUOl; Del CUDQ (21) CHAR(lO);
/·* •••••• *.* •••• *1 1. 1 NPUT DATA *1 /***.****.·*.*.*·1
GET LIST (NW,N,NSW,Nu,NHAR); GET LIST (WL,R,CU,CUOQ):
/*** ••• *****.**.** ••• , /* INITIALIZATION *' ,* •• ****.**.**.*.**.*/
lu=O; IEI=O; MNHAR ."NHAR ;
,.*.********.***.*.***.**.*.**.***./ ,* MAIN 00 LOOP FOR N EQUATIONS ., ,**.**.****************** •••• *.****,
DO Il= lTn N; ,.**··*.· ••• ****.***.1 1* INITIALIZATION *1 1********************1
LET (ll-"II";Q-O); NT=1; DO I=HNHAR TD NHAR ;
LET (1="1"; 5(1)-0; C(I)-O); END :
,.* ••• * •• * •• *.***** •••• *** ••• *** ••••• *, 1* FOR NT_l ANAlVZE g-O(l.I)/OCT) ,*1 ,. NT.Z ANAlYZE g=R*I ./ ,*.* •• * •• *********.*** ••• **.** ••••••• */
START: IF NT -2 THEN DO ; LET(Q="RtIl)"."CIHU," )J
END; ElSE 00 .;
DO JJ-l Ta NW .; LET C JJ="JJ";
85
SYNC0010 SVNC0020 SVNC0030 SVNC0040 SVNCOOSO SVNC0060 SVNCo070 SVNCo080 SVNCo090 SVNColoo SVNCollO SVNC012Q SVNCOUO
·SVNco140 SVNCo150 SVNCo160 SVNC0l70 SVNColBO SVNCo190 SVNC0200 SVNC0210 SVNC0220 SVNC0230 SVNC0240 SVNC025C1 SVNC0260 SVNC0270 SVNcn280 SVNC0290 SVNC0300 SVNC0310 SVNC0320 SVNC0330 SVNC0340 SVNC0350 SVNC0360 SVNC0370 SVNC03BO SVNC0390 SVNC0400 SVNC0410 SVNC0420 SVNC0430 SVNC0440 SVNC0450 SVNC0460 SVNC0470 SVNC0480 SVNC0490 SVNC0500 SVNC0510 SVNC0520 SVNC0530 SVNC0540 SVNCO'SO SVNCO'60 SYNCO'70 SVNCO'8C1
n
END ; END J
LET (Q=EXPAND (Q) ,;
NQ =NARGS (Q);
'************************************************' '* ANALVZe EACH SUBEXPRESSION (E, DUT OF (NQ) *' '************************************************'
DO K =1 TO Na ; LET(Ka"K"; EcARG (K~Q); Elo al';
1*************************1 1* 10ENTIFV NEGATIVE (E) *1 1*************************1
ID cLOP (E); IF ID -25 THEN DO ;
LET(E-ARG(l~E' JEIOc-l) J END l
1******************************' 1* BREAK (E) INTO NE TERHS (G)*, 1******************************'
NE cNARGS (E); DO Jal TO NE ;
LET (J="J"; G(J)a ARG(J~E'}J END; 10-0;
1******************************************************1 1* TEST EVERV G~INDEX wILL TAKF THE FOLLOHING VALUeS *1 1* *1 1* 0 NO SIN DR CDS *1 1* 1 SIN**2 DR SIN*SIN *1 1* 2 SIN OR SIN*COS *1 1* 3 COS*SIN *1 1* 4 COS**2 DR COs*COS OR cos *1 1******************************************************1
INDEX -a ,; DO 1=1 TO NE ; LET (la"I");
JO-O J 1*******************************1 1* TEST FOR SIN**2 OR COS**2 *1 1*******************************1
LI -LOP (G(I»),; IF Lt a 31 THEN DO J
LET(G(I)=ARG(llG(I'" J LI-LDP(G(I»; IF LI =4 THrN INDEX =ll
ELSE It-lDEX =4 ,; 10=1 ; LET (A=ARG(i~GCI)'J BaA) ,; GD TO lBLlJ
EtJD .i 1**************************1 1* TEST FOR SIN DR CDS *1 1**************************1
IF Lr-4ILI=S THEN DO .i IF LI -4 THEN INDEX =2 ,;
EI.SE INDEX -4 ,; 10=1 .i LET (AaARG (l~G~I)',; B-O ;
86
SVNC0590 . SVNCo600 SVNCo610 SVNCo620 SYNC0630 SVNCo640 S'INC0650 S'INC066(\ SYNC0670 SYNC06B('\ SYNC0690 S'INCo7no SVNCOnO SVNco'720 SVNCo130 SVNCo740 SVNCo150 SVNCo760 SVNCo770 S'INCo780 SVNCo790 S'INC0800 SVNC08l0 SVNeo820 SVNeo830 SYNC0840 SVNC0850 SVNCoB60 SVNCo870 SVNC0880 SVNco89n SVNCo900 SVNCo910 SVNC0920 SVNCo930 SVNCo940 SVNCo950 SVNC0960 SVNCo970 SVNC0980 SVNC0990 SVNCIOOO SVNCIOIO SVNCI020 S'iNCl030 SVNCI040 SVNCIOSO SVNCI060 SVNCI070 SVNCl080 sVNci09n SVNCll00 SVNCUln S'INCl12n S'iNClUO S'INCl140 S'INCUSO SVNCl160
r
-_.J
87
IF I=NE THEN GD TD LBL1; SVNC1170 Ile 1+1 ; SVNCllao
1**.* •• * •• * •• ** •••• ****.********.*.**.*.*** •• ** •• *1 SVNCl190 1* TEST REMAINING TERMS FOR At:OTHER SIN nR COS *' SVNC12no ,**** •••• * ••• ** ••••• **.******* ••• * ••• ******** ••••• , sVNC1210
DO Jall TO NE J SVNC1220 LET eJ-"J")J SVNCl230 LJa LnP (G(J,,; SVNC1240 IF LJ .4ILJ ., SYNCl250 THEN nO ~ SVNCl260
IF LJ=4 THEN INOex=INDEXwlj SVNC127Q jOaJ ; SVNC1280 LETCBcARG(l,GeJ,»; SVNCl290 GO TO LBLI J SVNCl3no END; SYNC1310
END; SVNC1320 END ; SVNC1330
END; SVNC1340 ,* •• **.* •••• * •••••••••• , SVNC135n 1* INDEX =0 RETURN E ., SVNC1360 ,.*** •••••••••••••••••• , SVNC1370
lBll :IF INDEX cO THEN 00 ; SVNC13BO LET ( ceo). CCO) +e.elo )J SVNCl390 GO TO CONT J SVNC1400
END; SVNC1410 , •• * •••• ** •••••••••••••••• *.* •••••••••••• , SVNCl420 ,. RECONSTRUCT " E" ExCEPT FOR SIN&COS ., SVNC1430 , ••• * •••••••• * ••••••••••••••••••••••••••• , SVNC1440
LET eE=ElD ); SVNC1450 DO J cl TD NE J SVNCl460
IF J aJOIJ=IO THEN GD TO SKIPl; SVNC1470 LETCJ="J"; E-E.GeJ»; SVNC14BO
SKIPl:ENO; SYNC149n , •••••••• * ••••••• ** ••••• * •• **.* ••• *** •••• ** •••••• , SVNC1500 ,. FORM THE ARGUH~NTS OF SlNeA.e) t COS(A+B).' SYNC1510 , ••• ** ••••••••••• * ••••••••••••••• ** •••• *.** •• *.*., SYNClS20
LET (X=A+B ; SVNCl530 ZcA-B ; S·YNC 1540 IX=COEFF(X,T) ; SYNC1550 IZ=COEFF(Z,T) ; SYNC1560. X=X-IX*T ; SVNC1570 Z=Z-IZ.T J SYNC15aQ
·M=0.5 ); SVNCl'9n ,***.*** ••••• * •• ** ••••• ***** ••••• * •••• * ••••••••••• , SVNC1600 ,. ACCDRDING TO INDEX BREAK E INTO COEFFICIENTS *' SVNC161n '* FOR SIN & COS ., SVNCl620 , •••••••••••• **.****.* ••••••••••••••••• * •• * •••• ***, SVNC1630
IF INDEX-31 INDEX=4 THEN 00 J SVNCl640 LET CH-·M); SVNC16,O
END; SVNC1660 IF INDEX-li !NDEX-4 T~EN DO J SVNC1670
LETrSC1X). M.E*SIN(X)+Sf!X)j SVNCl6ao (eIX). .H*e*COSex'+ce!X)J SVNCl690 SeIZ).·O.5.E*SIN(Z)+S(IZ)J SVNC1700 CelZ)a O.5*E*COS(Z)+CeIZ)';SVNC1710
GD TO CONT ; SVNCl720 END.; SVNCl730
!F INDEX-21 INDEX-3 TH EN DO J SYNC1740
J
1 1
1 1
i,:,.
f -
88
LETCS(!X)CClX)S(IZ)CClZ)-
END ;
O.5*e*CCS(X)+S(IX)J SVNC1750 O.5.e*SINeX)+C(IX)J SVNC17bO
H*e*CCSeZ)+SCIZ)J SVNC177n H*e*SINCZ)+C(!Z»;SVNC178n
SVNC1790 CONT: END .:
IF NT-2 THEN GO TO CLCT; 1*****·***.*·***.*****·******·********·*1 1* IF NT~l ,DIFFERENTIATE EXPRESSIONS *1 1****************.·**···*********·*·***·1
LET (CCO)=O); DO I-! TD NHAR':
LETCi-"I"; J=-I .:
END .:
E~(SCI)-S(J»*IJ SCI)= EXPAND c~I.(C(I)+ eCJ»); S (J) =0; C(J)·O; e(I). EXP AND (E»;
NT -NT+1 .: GO TO START ;
1*·.*.***** •• ****··**··*·********···.·' 1. COLLEeT 51MILAR TERHSI " ., 1* SINC~T)=-5INCT) t COse-T).CoSeT) *1 1.·**.**.*** •• ·*.··**··********·**··**1
ClCT: 00 l=lTO NHAR ; LET (!="I";J--I .:
S ( Il cS <1 ) -5 (J).: C'I)=C(I)+CCJ»;
END ; 1* •• *.· •• * •••• *.** •• *·· •• *********·.*****··.·**.*····*1 1* ~RINT & PUNCH OUTPUT IN A PROPERLV SELECTED FORH *1 1*.*.*** •• ****.****** •• *.***** ••• *******.* ••• *·.**·***1
IF 11 >1 THEN IU=Z J IF II >( NW -NSW) THEN IU.l j DO J -1 TO NU .:
LET C J-"J" .: N-"CUDQ,J)" ) j lE -iEI .: DO 1 - lU TO NHAR BV Z .:
lE • lE + 1 J LET ( lE ="IE" J 1="1";
AC lE ,J ) • COEFF( ec 1)/N )U PRINT_OUT C A(!EIJ»; CHAREX (STRING. ACIE,J»': PUT FllE (PUNCH) EDITcSTRtNG' (SKIPC1),XC6)IA(66».: IF I,~O THEN 00.: lE .IE+1 .: lET C lE • " lE " .:
A( lE ,J)- COEFFc Sct),N»; PRINT_OUTCACIE,J»; CHAREX ( STRING- A(IEIJ»J
SVNC1BOO SVNC1810 SVNC1820 SVNC1830 SVNClB40 SVNC1850 SVNC1860 SVNC1870 SVNC1880 SVNC1890 SVNci900 SVNC1910 SVNC1920 SVNC1930 SVNci940 SVNC1950 SVNC1960 SVNC1970 SVNC19BO SVNC1990 SVNCZOOO SVNCzOlO SVNCZOZO SVNCZ030 SVNCz040 SVNCZ050 SVNCz060 SVNCZ070 SVNCZOBO SVNCZ090 SVNCzlo0 SVNCZ1l0 SVNC2120 SVNCZUO sVNCz140 SVNCz150 SVNC2160 SVNC2170 SVNC21BO SVNCZ190 SVNCZ200 SVNC2"UO SVNCZ2Z0 SVNCZ230 SVNC2240
PUT FILE (PUNCH) EDITcSTRtNG) END;
CSKIP(1),XC6)IAC66,); SVNCZZ50
END; END ; lEI- lE ;
END; END wVFRH ;
SVNCZ260 SVNCZ270 SVNCZZ80 SVNC2290 SVNCZ300 SVNCZ310
"":...J "
t·
1**.*.** •• *.* ••• *.** ••• * •• ******* •• ** •• ****** •• *********./ ,. n'PICAL ClATA FOR A 4 WIRE , STAR cOtINeCTeD, :3 PHASE ./ ,. SYNCHRONOuS MACHINE , WITH Z DAI1PER WINDINGS. ./ / ••• ** •• ** •••• * ••• * ••••••••••• * ••• * ••• * ••• * •••••••••• * ••• ,
6 4 3 27 7
'XF' 'XH' '0' 'XAr-.COSCT)+XAF3.COSC3.T), 'XAF*COSCT-Z.#P/3'+XAF3*COSC:3*T)1 'XAF*CDSCT+2*#P/3'+XAF3.COSC3*T)1 'XF1' 'Xl' '0' 'XAI*CDS C 1) , 'XA1*COScT-Z.#P/3" 'XA1.COSCT+Z.,P/3)' '0' '0' 'X2' ' .. XAZ·SINCT) , ' .. XAZ·SINCT-Z.#P/3" '~XAZ·SINCT+2.#P/3" 'XAF.COSCT)+XAF3·COSC3.T), 'XA1.COScT) , ... XA2*SINCT)' 'XAA+XAA2*CDS(Z.T)+XAA4.COSC4*T)' '-XA8+XA8Z·COS(2.T~~.NP/3)-XAB4·COSC4.T+2*_P/3" '~XA8+XA8Z*COS(2.T+2.#P/3)-XAB4*COS(4.T-2*jP/3" 'XAF.COS(T-Z.#P/3)+XAF3.CDS(3*T)1 . ' 'XA1.COSCT-Z.#P/3), '-XAZ·SINCT-Z*8P/3)' ' .. XAB+XASZ*COS(Z.T-2.HP/3'-XAB4*COSC4.T+Z*#P/3), 'XAA+XAAZ·COS(2*T+Z*8P/3)+XAA4.COSC4.T-Z*tP/3)' '-XAB+XABZ·COSCZ.T).XA84*COS(4.T), 'XAF.COSCT+2*tP/3)+xAF3*COSC3-T" 'XA1.CCSCT+Z*NP/3), '-XAZ*SIN(T+Z*'P/3" '-XAB+XAsZ*caS(7.*T+2*NP/3)-XAB4*CDSt4.T-Z*8P/3" '~XA8+XA82*COS(2*T)+XAB4·COS(4*T" 'XAA+XAAZ*COSCZ*T-2*#P/3)+XAA4*COS(4*T+Z.#P/3)'
'RF' 'Ri' 'R2' 'RA' 'RA' 'RA'
'IFO+IFDZ*COSl2*T'+tFa2*SINcZ.T)+IF04*COSl4*T)+IFQ4*srN(4*T) +IFD6*COSl6.T)+IF06*SIN(6*T) • 'IIOZ*COSCZ*T)+I102*SIN(Z*T,+11D4.COSC4*T)+!lQ4*SINC4*T' +J1D6*COS(6*T)+II06*SIN(6.T" 'IZ02*COSCZ*T)+11Q2*SIN(1*T)+12D4.COS(4.T)+!2Q4*SINC4*T) +1206*COS(6*T'+I206*SIN(6.T) , 'I01.CQSCT'+IQ1*SIN,TI+ID3*COS(3.T'+IQ3*SINI3.T)+IDS*tOS(5*T' +IQS*SIN(5*T)+I07*COS(7*T)+IQ7.SIN(7*T) , 'ID1.COS(T-Z*#P/3)+IQ1.SIN(T-2*NP/31+I03*CO~(3*T)+IQ3*SIN(3*T' +ID5*CQSc5*T+Z.HP/3)+IQS*SIN(5*T+2.NP/31+ID,.CDS(7.T-Z*HP/3, +IQ7*Slilc7*T-Z*liP/3,'
..
89
l j 1
'101*COSC T+2*NP/3)+IQ1*SIN(T+2*NP/3)+I03*CDS(3*T)+IQ3*SIN(3*T) +ID5*CDS(5*T-2*NP/3)+IQS*SINC5*T~2*NP/3)+10'*CDS(7*T+2*~P/3) +IQ7*SIN(7*T+2*NP/3),
'IFO'
'101'
'IF02' 11102 ' 11202' 'IQ1'
'IFQ1.' 'IF04 1 'IFQ4' 'IF06' 'IFQ6' 'I1Q2' '1104' 111Q4' '11D6' '11Q6' 'I2Q2' '1204' IJ2Q4' 'I2D6' '12Q6'
'ID3' 'IQ3 1 '105' IIQS' '!07' 'IQ7'
90
. - ------------
'IF
2
Il
b
1
10
12
14
u.
11 --.--- ---.---
ZO VDI
"Ql
------
--.---
--.---Z4
--~---
2 .. • RF
.--------.--------.----.------.----.-.-.------------------------------------_.--~ , .
. -----------------.----.------.-.--.-.-.-------------------.---.----------.-.--.. I-X".) 1 Rr I-XF1~Z
-----~-----------------------------------------------------.---.----------.----.. 1 RF .--------.--------.----.------.----.-.-.--------------... ------.----------.----.-
I-XF·4 1 RF .--------.-------------.------.----.-.-.--------.-------.------.----------.----.-
1 RF XFU , .
. --------.---.-.--------------.----.-.-.----------.------------.----------.-._-.-1 RF
.------------.----------------.---.. ---.------------.------_.--------------------1 xrl*2 1 Rl
-----------------------.------.--------.-----------------------------------------I-X"1.2 I-Xl.'
--._-----.---_._-----------------------.----------------.-----------------.------1 )(Fl*4 . 1 • 1
------------------------------.--------.-----------------------------------------.--._-------------.-----------.----.---.-----------------------------------------.--------------------------------------.------------------------------------------------------------------------------.---._-----~--------------.----------.------
-------------------------------------.-----_.-----------_._----.----------.-----------------------------.---------------------------------------.----------.----.-
. . --------------------------------------------------.------------.--------------------------.----------------------------------------------------------------.-----_. ',. 1 1 1 1 1 1 1
-----------------------.---------------.--------------._-----------------~.-_.---,
-------------._--------.------.--------.-----------------------_.---------.------~ 1 XA' •• ' I-Xf,F3-.' 1 1
------------------------~-------------------------------------------------.----_ .. 1-l(AF •• ' l-xAn*., 1 1
I-XAU., 1 1 1
---------.-.-----------.---------------.-------------------._------------------_.-XA"-l.5I
1 1 1
-------------------------------------------------------------------------.. --------XA').) l-xAF.l.' I-XAF.l.' I-XAF).l.'1 I-XAI-l.' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ 1 •
---------------.----------------------------------------------------------.-------1 XA".z.51 1 1
1 XAF.Z.' 1 1 1
l'F-Z.' 1 • 1
----_.---_ .. -----------------------_._-~-----~--.-_._.-----_.--.. ---------.-------l-xAF3d." 1 1
1 1
---------------------------------------.-----------------------.----------.-------1 Un-J.,I l',-J.' 1 1 1 1 1 1. 1 - ,-1 • 1
----------------_._----.--------_ ... -.-.-._.------... ------~--------~---_ .. -------.
-----------------------------------.-.-.----------------------------------.-------
• 10 tZ 1" lb 18
1 ,_.e _______________ -____________ .-________________ .-__ ~ __________ . __ -_. ________________ -____ -_-_-____ -______ . ______ . __ _ 1 Xr:l-Z 1
-._----------------------------.--.-----------------------------------.--------------.-.--.-.-.--------.--------------, z 1 -------------------.-------.---.-------------------------------_._----.----------------.--.-.-.--------.----------.---,
1 XFl-" 1 -._----------------------.-.---.-----------------.--------------------.--.-----------.-.--.-.-.--------.--------------. I-XFl*.. 1 -------------_._-------------------------------------~---------_._----.--.-----------.----.-.-.-----------------------. 'WF1*6 -----------------------------------------------------~----------------.----------------.----.-.-----------------------. I-XFl*6 ._--------------------------------._-----------------~----------.-----.----------------.------------------------------.
----------------------------------.---_.-------------~--------_ .. _------_.----------------------------------------.---. 1 It \
-----------------------------------------------------~---------- ---.-.-------------------.---------------------------. lU 1 Xl-" ._---------------------------------------------------~---------_.--------------------.-.------------------------------. 1 RI
._------------------------------------.--------------~---------- .-----.--.-----------.----.-.-.-._------------------.--1 Itl
----------------._-------------.------.-------~------~---------._-----.--.-.----------------.---------------------.---. 1 I-Xl.~ Pl ------------------------------------------------------._--------.. ---_ .. _--... ----------------.------------.-------------1 •
---------------------------------------------------.------------- _.- .. ,---.------------------.-----_.-------------------1 R2
----.--------------------------.--.----------------------------- .. _----.-----------------------_._-----------------.----l , Rl 1 xZ-"
,----------------------------------------------------------------------.-----------------------.------------.---.--.----1 I-X2*" 'RZ
,---------------------------------.-------------------_._-------- --------------------.----------------------------.----1 R2 1 X2*b .. ,
._-------------.------------------------------------------------- --------------------.-.---------------.--.----.-------1 1t2
,--------------.--------------------.------------.----._--------_._---------------------------------------.-------------ICU-.' XAZ*.' 1 !tA 1 1 1
--------------------------.----------_.--------------'--------------------------_.---.------.--------------------------I-UA '-UA I·UI I-XAI:
._----------------------------------------------------.------------... ----------------.--------------------------.----, I-XAZ*1.5 1 1
d
._-----.----------------------.--.-------------------'--------------------------------------.-------------------------.' ., '1 1 1 1
I-XAl*1.5 1 1 ,
I-UA' '·UAj IUAI' I-UI·
._------------------------._--.--------------------------------------.----------------.------------------------------_. 1 1
1 XAl-Z.' 1 1 1
I-XAZ*2.' 1
XAZ*Z.,
._-----------------------------------------------------------._------.----------------------------------.-----.------.. 1 1
I-Ul*Z.' 1 1 1
1 XAZ*Z.' 1 .' . . _----------------------------.-----------------------.----------.------------------.----------------------------------1
1 I-XAZU.' 1
._----------_._-~-------.-_.-------.-._-------------.---------------------------------.----.-.--._---------------.-----1 1
I-Ul*!., 1 . . • l, l,
I-XAZU.' 1
----------------.----------------------------------------------------------------------------.-----------------------.-
20 24 26
----.----------.-------------------------.-.--------------.-~-------------------· . ----.----------.-._----------------------------.-----------------.--------------· .
----.----------.-._------------------------.---.-----------------------------------------------.----------------------------------------------------------------· .. -----------------------------~---------------------------------.-.-.------------
----.----------.-------------------------------.-----------------.------------------.------.---.---------------------------.------.--.--------------------------
· . --------------~-----------------------------------------------------------------
---------------.--------------------------------------.-------------------------· .. ---------------.--------------------------------------.-------------------------------------------------------------------------------.-------------------------XAl-9.
.--------------.----------------------------------.-_.--------------------------
.---------------~---------------------------------------------------------------
---------------.------------------------------------------------------------------------------------------------------------------------------------------------._------------------------------------------------------------------------------._--.--------------------------------------.--------------------------------_ .. -1 x,\2.9.
._--_._--------.----------------------------------------------------------------XU ...
.---.--.------------_._---------------------------------------------------------1 UA I-UA:!_.' I+XA8 I-XAB~
1 UA2-., I-UA,,-., I-UU-.' l-xU,...,
1 !tA"-.!I I-XU4 1 ,
--------------.------------__ e _________________ -- ____ • ____________________ • ___ _
-XAA -XAU-.' -XA8 -XA8Z
I-XUZ-.5 ,-XAA'-.5 ,.XlI2-.5 '-XA8'-.'
,-uu-.' 'dAl4 , ,
--------------.----.-----------------------------------------------------------, XUZ-1.', RA ,-XU.-1.', '-XU7-1.51 '-U8~-lt 51
1 Uh, I-U"6 1
, , , , , UAZ-1." ,-U"?-l.!I'
, UA4-1.!I ,+UI4-1., , , ,
1 , 1 , --------------.--------______ e ___ - __________ --_______ • __________ .-____________ _
-UU-l.!l1 -XAA'Ul.5I UAU-l.!II -XAI'-l.!II
I-XU-' ,.X&I-6 , 1
I .. XU2-l"1 '.UU-1,5t l , , 1 · .
I-XAU-l.'1 '-)(,\84-1.'1 1 1 , 1
--------------.----------------------------------------------------._----_ .. ---1 XAA'-Z.!l1 1 XAA2-Z.5' RA , XAA-' '-XA8~-' l-xUZ-Z.' 1 I.U~-'
1 XUZ-Z., I+UU-'
----------------------------------------------.------.----------------_._------~XAA4-2.!l1 .XAI'_' 1
I-XU2-Z.'1 I·XUUZ.'I
IAl 1 · .
I-UA2·Z.JI I-XAU.' 1
-------------------------------------------------------------------------------1 XUU3.,1 '+Uh.3.!l1
, UU-,.,I RA I+UU." 1
--------------.-----------p-------------------.--------------------------------I-XAA'.'.5' I-XAA2."'1 • I-XlI"'. 5 , I-X4I2.1 1 ..
IAl 1
--------------.-------._--------.----------------------------------------------
X
IFO ----.----IFD' ----.---- i:~
IFa~
----.----IfD4 ----.--_. .,
IFa~ .,
----.----IFD~
----.----IFO~
.---~----I1D' ----.----Ila~
----.----Iln4 ------._-Ila~
----.----I1D~
----.---- .'~.
lla6 ' J}~~ .,,.,.
----.---- ,~~
UD! ~;: ---------ua, ';
----.. ---- :~ IZD~ -
j~ ----.. ----ua. ----.---- }É unb ,11 ----.----12C1~ ;'~
----.---- , 101
'~ .~~~
;.;<
--------- .. ~ 101
.. ~ ,.
1 . 'li. -------_. . '.! ,
ID' ~?~ :i .-1
.--~----.- ·if lOI 'i~
. ----.-._-IDS
----.----105 . ----.---- ~. ID'
' .
----.--.- . ": 10' ::
----.-._. .. Z! i~ " ;: . . ;. :':
L B - 3 - PHASE SYNCHRONOUS MACHINE,
Notes:
3 - WIRE STAR - CONNECTION
J • A TypicaJ Data Set.
2. The Harmonie Impedance Matrix •
1. The following abbreviations are used :
xx = XAA + XAB
XX2 = XAA 2 + 2 * XAB2
XX4 = XAA4 - 2' '* XAB4
2. VD and VQ are derived from
v = v - v a e
= V cos rd t - V eos (rd t + 2 'Ir / 3 )
= VD cos Q + VQ sin Q
92
1*****·*********.*************************.**********1 1* TYPICAL DATA FOR 3 WIRE ~STAR CDNNECTEO~ 3 PHASE *1 1* SYNCHRDNOUS MACHINE WITH Z DAMPER WINnINGS *1 1*****·**.********.*.*****.**************************1
5 4 2 27 7
')CF' 'XH' '0' '2.0*XAF*SINCT+IP/3)*SQRT(3.0)/2.0' '2.0*XAF*SINCT)*SORTC3.0)/2.0' 'XF1' 'Xl' '0 ' 'Z.O*XA1*SIN(T+'P/3)*SQRT(3.0)/2.0' , Z.O*XA1*SIN CT)*SORT(3.0)/Z.O' '0 ' '0' 'XZ' '2.0*XAZ*SQRTC3.0)/2.0*CDS(T.NP/3) , '2.0*XAZ*SQRTC3.0)/Z.O*CDSCTl' '2.0*XAF*SINCT+'P/3)*SQRT(3.0)/2.0' '2.0*XA1*SINCT+~P/3)*SQRT(3.0)/2.0' '2.0*XAZ*SQRTC3.0)/2.0*CDSCT+NP/3) , '2.0*XX+XXZ*COS(Z*T-IP/3)+XX4*CDS(4*T+8P/3) , 'XX-XXZ*COS(2*T+IP/3)-XX4*CDSC4*T·,P/3) , 'Z.O*XAF*SINCT)*SQRTC3.0)/Z.O' , 2.0*XA1*SIN CT)*SQRTC3.0)/Z.O' 'Z.O*XA2*SQRT(3.0)/2.0*CDSCT), 'XX-XXZ*CDS(Z*T+IP/3)-XX4*COSC4*T-NP/3) , '2.0*XX-XX2*COS(2*T)~XX4*CDS(4*T) ,
'IFO+IFDZ*COSCZ*T)+iFQZ*SIN(2*T)+IFD4*CDS(4*T)+IFQ4*SIN(4*T) +IF06*COSC6*T)+IF06~SINC6*T) , '11DZ*COS(Z*T)+I10Z*SINCZ*T)+I104*COS(4*T)+tlQ4*SINC4*T) +tlD6*COS(6*T)+I1Q6*SIN(6*T), 'IZDZ*COS(2*T)+IZ02*SIN(Z*T)+I204*COS(4*T)+tZQ4*SIN(4*T) +IZ06*COS(6*T)+I2Q6*SIN(6*T) , 'I01*COSCT)+IQ1*SINCT1+ID3*CDS(3*T)+IQ3*SINf3*T)+IDS*COS(S*T) +IQS*StN(5*Tl+I07*CnS(7*T)+IQ7*SINC7*T) , 'I01*CDS(T-2*#P/3l+IQ1*SIN(T-2*HP/3)+ID3*CDSC3*T)+IQ3*SIN(3*T) +IOS*COSC5*T+2*#P/3)+IQS*SINCS*T+2*NP/3l+ID7*COSC7*T.Z .HP/3) +IQ7*SINC7*T-2*~P/a)'
'IFO'
'101'
'IF02' 'IlD2 ' '1202' 'IQl '
'IFQ2' 'IFD4' 'IFQ4' 'IFOh' 'IFQ6' 'IlQ?, , 1104' 'I1Q4' 'IlDh' 'IlQ6' 'IzQ2' '12D4' '12Q4' '12n6' '12Q6'
'103' '103' 'IDS' 'lQ5' '107' 'IQ7'
93
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Zlt
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.------------------------------------.--------------------------------------._--------------, 1 RF
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J •
-~-----------------------------------------------------------------------~------------------' . 1 1 XF1." 1 R
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x~F*2.Z51-XAFIIII·291 , 1 1 . 1 1 1·
XAI*1.Z91 1 1 1
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1 1 1 1
1 1 1 1
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1 1 1
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10 u II 10
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1 1 I-XF1.6 . , 1 -i----------------------------------------------------------________ - _____________ --______________________________ - 1 1 1 1 1
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'.
_. APPENDIX III
2 - PHASE INDUCTION MOTOR
1 • program Listing.
2. A Typieal Data Set.
3. The Harmonie Impedance Matrix.
Notes:
1 • The abbreviation
F = Rotation Frequeney = Supply Frequeney
is used .
f If r 0
2. The letters P and N are added to the harmonie eurrents
to denote the positive and negative frequeney eomponents
respeetively.
95
/. THIS pROGRAH ANAlVZES THE M.H.F. HARHONIC EFFECTS IN 2 PHASE INDUCTION MOTOR WITH PHASE WOUND ROTORS •
THe FOLlOwING IS A llST OF INPUT VARIABLES
MEANING . -~------~-----------~------_.---------------~---" NW NUMBER OF WINDINGS IN THr: MACHINE
N NUMBER OF EQUATIONS TO BF SOlVEO NU NUMBER OF UNKNOWNS (CURRFNT COMPONENTS) NHAR HIGHEST HARMONIe OROER Wl THE REACTANCE MATRIX R THE RESISTANCE DIAGONAL HATRIX CU THE CURRENT VECTOR CUDQ THE VARIABLE ARRANGEMENT VECTOR ./
WVFRH: PROCEDURE OPTIONS (MAIN) ; FORMAt_OPTIONS ; DECLARE PUNCH FILE; DCl STRING CHAR (66) VARVING ; DCl WL(4 1 4)CHAR(80) J Del CU(4) CHAR(400); Del R (4) CHAR( 10) ; Del CUoQ (38) CHAR(lO);
/· •••• • •• *·* •• ***1 /. INPUT DATA *1 /· •• * •• *.***.*.**1 Ger LIST (NWIN,NU,NHAR); GE-T- LIST- (WL1RlCU,CUDQ);
/ •••• *****.* •••• **.**/ /. INITIAlI7.ATION */ /·.·********.*****·**1
IU=O; 1 E 1 -0; MNHAR --NHAR ; lETnERo .. a;;
/* •• **.*****.*****.**.** ••• *.*.***./ /* HAIN na LOOP FOR N EQUATIONS ./ / ••• *******.**.*.****.** •• **.*****./ 00 II- 1 TO N; , •• ****************.*1 1* INITIALIZATION *1 1*.****************.*1
lET (II-"II";Q=O); NT=l; 00 I=MNHAR TO NHAR ;
LET (1."1"; 5(1)-0; C(I)=O); END ;
1*********************.*********··****/ ,. FOR NT=l ANALV!E Q-OCl*l)/DfT) ,*1 1. NT=2 ANAlVZE QcR*1 *1 1*****.************.******************1
START; IF NT =2 THEN DO '; lET(Q-"RCII)"*-CUCII)" )J
END ; El SE DO ;
DO JJ-l TO Nw .:
96
INDC0010 INDC0020 INDC0030 INDC004n INDCo050 INDC0060 INDC0070 INDC008n INDC0090 INDColOO INDColl0 iNDCo120 INDCOUO INDC0140 JNDC0150 INDCo160 JNDC0170 INDC0180 INDC0190 JNDC0200 iNDCo210 INDC0220 INDC0230 INDC0240 iNDC0250 INDe0260 INDC0270 INDC0280 INDeo290 JNDC0300 INDC0310 JNDC0320 INDC0330 INDC0340 INDeo350 INDC0360 INDC0370 INDC0380 INDC0390 INDC0400 tNDC0410 INDC0420 INDC0430 INDC044(') INDC04S0 INDC0460 INDC0470 INDC0480 INDC0490 INDC0500 INDC0510 IND~0520 INDCo53n INDC0540 INDCn55n INDC0560 INDC057/') INDC058n
~' l,>
LET JJ""JJ";
END ; END J
LET (QaEXPANO (Q) ); NQ aNARGS (Q);
Qs"WLCII;JJ)"*"CUCJJ)"+Q)J
,*.***.***.********.** •• ********* •• *.***.** •••• *., ,. ANAlV7.E EACH SUBEXPRESSION CE' DUT OF (NQ) ., , ••• *** •• *.***** •• ****.*.*** •••• ******* •••••• ****,
DO K 111 TO NQ ; lETCKII"K"J EcARG CK1Q); EID "1);
, •• *.*********************, ,. IDENTIFY NEGATIVE CE) *' ,* •••••• ** •• * •••••• * •• ****,
ID "lOP cEU IF ID =25 TH EN DO ;
LETCE=ARG(lIE; ;EIDII·l) J END ;
1***·*·.*.**·*·**·******·*·*···' 1* BREAK (E) INTO NE TERHS eG)*, 1***.***.*****· •••• *.··***··*··,
NE eNARGS (E); DO Jill TO NE ;
LET ~J .. "J"; G(J)a ARG(JIE»; END ; 10110;
1* ••• **.*****.*.**** ••• ***.* •••••••• * ••• *.******* •• * ••• , 1* TEST EVERY GIINDEX WILL TAKE THE FDLLOWING VALUES ., ,. *' ,.. 0 NO SIN OR COS *1 ,. 1 SIN*.2 OR SIN*SIN *1 1* 2 SIN DR SIN.COS *1 1* ~ COS.SIN *1 1* 4 COS.*2 OR CDS*COS DR CDS *1 1***.*·***** ••• ***.*.*.******··*****.···*·*··*****····.,
INDEX aO ; DO 1-1 TO NE ; lET (le"I");
JO_O J ,* ••••• ********.**.* •• **.*** •••• , '* TEST FOR SIN.*2 OR COS*.2 *' 1*.******.**.** •• *.*·******·***.,
LI IILOP CGC!»': IF Lyll31 THEN DO .:
LETeG(Il=ARr,(l,GCI)') J L1eLOP(GeIl); IF II "4 THEN INDEX alJ
ElSE INDEX a4 .: 10al .: LET (A=ARG(l,GCI'); a-A) ; GO TO LBll.:
END ; , ••••••• *.******** ••• ******, 1* TEST FOR SIN DR COS *1 1**** •• ***********.*.******1
IF LI1I4ILI=' THEN DO ; IF LI 114 THEN INDEX =2 ;
ELSE HIOEX =4 ; 10=1 ;
iNDC0590 INDCo600 INDC0610 INoco62n !NDC0631) !NDC0640 INDCo650 INDCo660 INDC0670 INDCo680 !NDC0690 !NDC0100 !NDCo71n iNDCo120 !NDC0130 !NDeo140 INDCo750 INDCo160 INDCo170 INDCo780 INDCo790 iNDCo800 INDC0810 INDCo820 INDcn830 INDCo840 INDC0850 INDCo860 INDeo870 INDC0880 INDCo890 INDeo90o INDC0910 INDCo920 INDCo930 INDcn940 INDcn950
'INDC0960 INDC0970 iNDCo9811 INDCo990 INDCIOOO INDC1010 INDCI020 INDCI03n INDCI040 INDCI05C1 INDCI060 INDCI070 INDCI080 INDCI090 INDe 1100 INDC 1110 INDC 1120 INDC 1130 INDC1140 INDC1150 INDCllbn
97
LET (A-ARG n,Gin).: B-O ) ; IF IaNE THEN GD TO LBLI; Ill: 1+1 ;
,*************************************************1 1* TEST REMAINING TERHS FOR ~~OTHER SIN OR COS *1 1*************************************************1
DO JeU TO NE J LET CJ-"~")J LJa LOP (GCJJ); IF lJ -4ILJ -, THEN 00 J
IF LJ-4 THEN INDEX-INDEX-lj JO-" ; CETCB-ARGel,GCJ'»; GD TO LBLl J
END .: END J
END; ,**********************, 1* INDEX =0 RETURN E *1 '**********************'
LBLl :IF INDEX =0 THEN DO J
END .:
LET 1 CeO)~ C(O) .e*E!D )J GD Ta CONT J
END; 1************.***************************' 1* RECONSTRUCT " E" ExCEPT FOR SINtCOS *1 1****************.***********************1
LET eE=EID-); DO J -1 TC NE .:
IF J =JOIJ=IO THEN GO Ta SKIP1J LEtCJI:"J"J e=E*GCJ»; ,
SKIP1:ENOJ 1************************************************1 1* FORM TH~ ARGUMENTS OF SI~eA+B) & cnS(A+B) *1 1******.*****************************************1
LET CX=A+B .: Z=A-B J Jx=coeFFCX"TO); JZ-COEFFCZ,TO) ; IZ=-1;
'M=O.S ); IF IDENTCJZ':IZ) THEN DO ;
LET(JZ=-JZ; Z--Z);
IF INDEX=21 INDFX.3 THEN 00 .:
LETe !X-COEFFeX"T) .: JZ=CDEFFCZ"TJ ; X=X-IX*T-JX*TO ; Z=Z-IZ*T-JZ*TO);
END;
LET'rH=-H'J END.:
1****.*.*****************.************************1 1* ACCORDING TO ItiDEX BREAK E INTO COEFFICIENTS *1 1* FOR SIN & cos *1 1*************************************************1
98
INOCU70 INDC1180 INOC119/) INDC1200 INDC1210 INDC1220 INDC1230 INDC1240 INDC1250 INDCU60 INDCl21() INDC1280 INDC1290 INDC1300 INDC1310 INDC1320 INDC1330 INDC 1340 INDC1350 INDC1360 INDC1370 INDC1380 INDC1390 INDC14t)O INDC1410 INDC1420 INDC1430 INDCi440 INDC1450 INDC1460 INDC1470 INDC1480 INDC1490 INDC1500 INDC1510 INDC1520 INDC1'3n INDC1540 INDC1S50 INDC 1'60' INOe1!170 INOC1580 INDC1590 INDClbOO INDC1 610 INOC1620 INDC1630 INOC1640 INDC1650 INOC1660 INDC1670 INDC1680 INDC1690 INDC1700 rNOCl71o INDC172", INDC 1730' INDC1740
99
IF INDEX-31 INDEX-It THEN DO J INDC1150 lET (H=·H,; INDC1760
END .: INDC 1770 IF INDEX=ll INDEx-1t THEN DO j INDC1780
LETrSCIX)- H*e.SINeX)+sr!X)J INOC1790 CCIX)= .H*E*COS(X)+C(!X)J INOC1 8no SCII) •• O.5*e.sIN(Z)+S(IZ)J INOC1810 CCII)- o.,*e.cOseZ)+C(!Z»;INDC1820
GO TO CONT ; END ;
INDC1830 INDC1840 INoci850 IF INDEX-ZI INDEXe3 THEN DO j
LETrSClx)= CCIX)· Sell)e CCII)·
END ;
O.5*e.cOsex)+sr!X)J INOC1860 O.5.E*SINeX)+C(IX)J INOC1870
H*e*cOsez)+se!Z)J INDC1880 H*E*sINeZ)+C(!Z»);INDC1 890
. INDC19no CONT: END .: INDC1910
INDC1920 INDC1930 INDC1940 INDC1950 INDC1960
IF NT-2 THEN GD TO OUTP ; ,******.* •• ** •• * •• ***.* ••• **** •••• ** •••• , 1* IF NT.l ,DIFFeRENTIATE EXPRESSIONS ., ,* •• *.* •••• * ••• * ••• * •••••• ** •••••• ** •••• ,
DO I-MNHAR TO NHAR .: LET (1="1";
E=SCI) .:
END .:
Srl)=EXPANO(-(l.O+I.F ).Cil»; C(I)eEXPANO«l.O+I.F ).E);;
NT -NT+l .: GO TO STAR1'':
1******"***********************::'*********************·/ 1* PRIMT & PUNCH OUTPUT IN A P~OPERLY SELECTED FORM */ 1*·*************·****·**·****·****·*······**···*******/
INDC1970 INDC1980 INDC1990 INDC2000 INDC2010 INDC2020
DUTPi IF 11&2 THEN lU=1 .:
INnC2030 INDC2040 INDC2050 INDC2060 INDC2070
NEXT1:
NEXTZ:
DO J -1 TO NU ; LET ( J="J" ; W="CUDQeJ)" ) J lE -tEl.: DO 1 • lU TO NHAR ev 2 ;
lE • lE + 1 .: LET ( lE -"lE" J 1="1";
A( lE ,J ) • COEFF( C(-I),W »J IF !OENT(A(IE,J); ZERO) THrN GD TO NEXTl J PRINT_OUT C A(IE,J»; cHAREX (STRING = A(IE,J»;
INDC2080 INDC2090 INDC2100 INDC2110 INDC2120 INoC2130 INDC2140 INDC2150 INDC2160 INDC2170
PUT FILE (PUNCH) EDIT(STRING) IEaIE+l ; LET (IE="le" .:
eSKIP(1),XC6),A(66»; INOCZ180 INDCz190 INDC2200
A(IE,J)·COEFF(S(~I),W »; IF IOENT(A(IE,J); ZERO) THEN GO TO NEXT2 ~ PRINT_OUT (A(IE,J»': CHAREX (STRING. A(IE,J»': PUT FILE (PUNCH' EDIT(STRTNG) (SKIPC1),XC6),A(66,,; IF I~·O THEN DO; lE Qle+l ; LET ( lE • " lE " ;
A( lE ,J). COEFF( ClT',W)); IF IDENT(A(IE,J); ZERO) THrN GD TO NEXT3 J PRINT_oUTCArIE,J»;
INDC2210 INDC2220 INOCZ230 INDC2240 INOC2250 INDC2260 INDC2270 INocz28n INDC2290 INOC2300
CHAREX ( STRING: AtIE'J»; PUT FILE (PUNCH) EDIT(STRJNG)
INDC2310 (SKIP(1),X(6)IA(66»; INDC2320
NEXT4:
IE=IE+l ; lET (IE="!E" .:
A(IE,J).COEFFCS( I),W ))J IF IOENT(ACIE,J); ZERO) THFN GO PRrNT_OUT (ACIE,J»; CHAREX (STRING • ACIE,J»': PUT FllE (PUNCH) EDITcSTRtNG)
END .: END .:
ENO ; lE II: lE .:
END': END WVFRH .:
TD NEXT4 J
100
iNDC2330 INDC2340 INDC2350 INDC236(') INDCZ370 INDCZ380 INDC2390 INDC24nO INDCZ410 INDC2420 INDCZ430 INDC2440 INDC2450
1*********************************************/ 1* TYPICAL DATA FOR 2 PHASE INDUCTION MnTOR */ /*********************************************/
,. 2 3S 9
'xs' 'Xl*CDS(T)+X3*COS(~*T)+X5*COS(5*T)+X7*COS(7*T)+X9*COSC9*T)' '0' '.Xl*SIN(T)+X3*SINC3*T)-XS*SIN(5*T)+X7*SINC7*T)-X9*SINC9*T" Xl*CDS(T)+X3*CnSC~*T'+X5*COSCS*T)+X7*COSC7*T).X9*COSC9*T)' XRI . Xl*SIN(T)"X3*SINC3*T'+X5*SINCS*T)~X7*SINC7.T)+X9*SINC9*T)' 0' 0' Xl*SIN(T)-X3*SIN(3*T)+X'*SINCS*T)-x7*SINC7*T)+X9*SINC9*T)' XS' Xl*COS(T)+X3*COSC3*Tl+XS*COSCS*T)+X7*COSC7*T,+X9*COSC9*T)I .Xl*SIN(T'+X3*SINC3*T)-XS*SINC5*T)+X7*SINC?*T'-X9*SINC9*T" 0' Xl*COS(T'+X3*COSC3*T)+X5*COSCS*T)+X7*COS(7*T'+X9*COSC9*T)' XRI
'RS' 'RR' 'RS' 'RR'
101
'ISOO*COS(TO,+ISQO*SIN(TO) +ISnZP*COS(TO+2*T'+ISQ2P*SIN(TO+l*T)+ISD2N*COSCTO-Z*T'+ISQZN*SINCTO-2*T) +ISD4P*COS(TO+4*T'+ISQ4P*SIN(TO+4*T,+IS04N*r.OSCTO-4*T)+ISQ4N*SINCTO-4*T) +ISD6P*COSITO+6*T'+ISQ6P*SIN(TO+6*T)+ISD6N*COSCTO-6*T)+ISQ6N*SINCTO-6*T) +ISD8P*COS(TO+8*T'+tSQ8P*SIN(TO+8*T)+ISD8N*~OS(TO-B*T).ISQSN*SINCTO-8*T) , 'IRD1P*COS(TO+ T'+IRQIP*SIN(TO+ T)+IRDIN*~OSCTO- T'.IRQ1N*SIN(TO- T) +IRD3P*COS(TO+3*T'+IRQ3P*SIN(TO+3.T'+IRD3N*r.OS(TO~3*T)+IRQ3N*SIN(TO-3*T) +IRDSP*COS(TO+S*TI+JRQSP*SINCTO+S*T)+IRDSN*COSCTO-S*T)+IRQSN*SIN(TO-'*T) ~ +IRD7P*COS(TO+7*T)+IR07P*SIN(TO+7*T)+IRD7N*tOSCTO-7*T)~IRQ7N*SINCTO-7*T) +IRD9P*COS(TO+9*TI+IRQ9P*SINCTO+9*T'+IRD9N*COS(TO-9*T).IRQ9N*SINCTO-9*T) , 'ISOO*SIN(TO,-ISQO*COS(TO) +ISD2P*SIN(TO+2*T)-tSQ2P*COS(TO+2*T)+ISD2N*5INCTO-Z*T)-ISQlN*CDSCTO-Z*T) +ISD4P*SINITO+4*T)-IS04P*CDS(TO+4*T,+ISD4N*SIN(TO-4*T)~ISQ4N*CDSCTn-4*T) +IS06P*SIN(TO+6*T,wiSQ6P*COSITO+6*T'+ISD6N*~INCTO-6*T)-ISQ6N*COSCTO~6*T) +ISD8P*STNITO+8*T)-ISQ8P*CDS(TO+8*T)+IS08N*~INCTO-8*T)-I5Q8N*CDSCTO-S*T) 'IROIP*SINITO+ T)-IR01P*COSITO+ Tl+IROIN*5IN(TO- T)~IRQ1N*CDS(Tn- T) +IRD3P*SIN(TO+3*TI-IR03P*COS(TO+3*T)+IR03N*~IN(TO-3*r)-IR03N*COS(TO-3*T, +IRD5P*SrN(TO+5*T)-tRQ5P*COS(TO+,*T)+IR05N*5INCTO-,*r,-IRQ5N*CDSCTn-5*T) +IRD7P*SIN(TO+7*T)-IRQ7P*COSITO+7*T)+IR07N*~IN(TO-1*T)~IRQ7N*CDSCTn-7*T) +IRD9P*SIN(TO+9*T)-!RQ9P*CDS(TO+9*T)+IR09N*~IN(TO~9*T'wIRQ9N*COSCTO-9*T)
'ISOO' 'ISOO' 'IS02N' 'IS02N' 'ISnlp' 'ISQ2P' 'ISn4N' 'IS04N' '1SC4P' 'IS04P' 'IS06N' 'IS06N' 'ISD6P' 'ISQ6P' 'IS08N' 'ISr.8N' '1S08P' 'ISQSP' 'IRD1N' 'IRQ1N' 'IRnlPI tIRQ1P' tIRD3N' 'IRo3N' 'IRD3P' 'lR03P' 'IR05N' tIR05N' 'IRn5P' 'IRQ5P' 'IR07N' 'IR~7N' 'IRD7P" 'IRQ7P' 'IR09N' IIR09N' 'IRn9P' , IRQ9P'
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APPENDIX IV
SINGLE PHASE SYNCHRONOUS MACHINE
1 • A Typical Data Set.
2. The Harmonie Impedance Matrix.
Notes:
1. The following abbreyiations are used :
XX = XAA + XAB
XX2 = XAA2 + 2 * XAB2
XX4 = XAA4 - 2 * XAB4
2. VD and VQ are deriyed From
y = y - Yb a
= V cos (,) t - V cos ((,) t - 21(/3)
= VD cos 0 + VQ sin 0
where
0 = (,)t + P ,
and
P = 31(/2 - S .
f'": ._-
.{ •.. ".-----.
1*****************************************************1 1* TVPICAL DATA FOR SINGLE PHASE SVNCHORnNOUS MACHINE*I 1*****************************************************1
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APPENDIX V
SINGLE PHASE - SINGLE PHASE INDUCTION MACHINE
Note:
1 • A Typica; Data Set.
2. The Harmonie Impedance Matrix.
The abbreviation
F = Rotation Frequency = Supply Frequency
is used .
f If r 0
106
,.*.·.·.*******· ••• ****** •• ***·.********** •• * •• ***1 ,. TVPICAL DATA FOR A SINGLE PHASE'SINGLE PHASE ./ ,. MACHINE ./ , •• *** •• *.***.* ••• *****.**.**.*.***.***.*.*.** ••• */
2 . 2 38 9
'XS' 'Xl*COS(TI+X3*COS{3*T)+XS*COSt5.T)+X7*COS(7*T)+X9.COSt9*T)1 'Xl*COS(TI+X3.COS(3.TI+X5*COSC'*T)+X7*COS(1*T)+X9.COSC9*T)' 'XRI
'ISDO*COS(TOI+ISQO*SIN(TO) .
107
+ISD2P*CQS(TO+2*T)+ISQZP*SIN(TO+2*TI+ISD2N*COSCTO-2.T'+ISQZN*SINtTO-Z*T) +ISD4P*CQS(TO+4.T)+tSQ4P*SIN(TO+4*TI+IS04N*COSCTO·4.T'.ISQ4N*SIN(Tn~4*T) +ISD6P*COS(TO+b*T)+ISQbP*SIN(TO+6*T)+ISObN*COStTO·6*T,+ISQ6N*SINtTO-6*T) +IS08P*COS(TO+8*T)+is08P*SIN(TO+8.T)+ISD8N*CDStTO~8.T'+ISQ8N*SIN(TO-8*T) , 'IR01P*COS(TO+ T)+[R01P*SIN(TO+ TI+IRD1N*CDStTO· T'+IR01N*SIN(TO- T) +IRD3P*COS(TO+3*T)+IRQ3P*SIN(TO+3*TI+IR03N*CDSCTO.3*T'+IR03N*SIN(TO-3.T) +IROSP.COS(TO+S*TI+IRQSP*SIN(TO+,*TI+[RD5N*COS(TOp'*T'+IRQSN*SINtTO-S*T) +IRD7P*COS(TO+7*TI+IRQ7P*SIN(TO+7*T)+IR07N*r.DS(TO-7*T)+IRQ7N*SINtTO-7.T) +IR09P*COS(TO+9*T)+IR09P*SIN(TO+9*TI+IR09N*COSCTO-9.T'+IRQ9N*SINtTO-9*T) ,
'1500' 'ISOO' 'ISD2N' 'ISOZN' 'ISOZP' 'IS02P' 'ISD4N' 'IS04N' 'IS04P' 'IS04P' 'IS06NI 'ISObN' 'ISObP' '15QbP • 'ISDeN' 'IS08N' tIS08P' 'ISOSP' 'IR01N' 'lRQ1N' 'IR01P' 'IR01P • 'IR03N' IJR03N' 'IR03P' 'IRQ3P' 'IRDSN' 'lROSN' , IRn5P' 'IROSP' 'IR07N' '1R07N' 'IRD7P' 'IRQ7P' 'IRD9N' 'IR09N' 'IR09P' 'IRQ9P'
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