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New techniques for thedetermination of
synchronous machineparameters
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• A Master's Dissertation, submitted in partial fulfilment of the require-ments of the award of the Master Of Science degree of LoughboroughUniversity.
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Publisher: c© Ali Mohammed Baki
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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY
LIBRARY AUTHOR/FILING TITLE
--------~ ~_~L I--A --------------- ----- ----- --'
ACCESSION/COPY NO. . I
--VOL~NO~-------O-~~~S~~-~~~---·----- -----~--
t 3 FEB 1998 , '
I' It t1A'l 1998 I,
, I
I , \
I -
I
I
NEW TECHNIQUES FOR THE DETERMiNATION
OF SYNCHRONOUS MACHINE PARAMETERS
by
Ali Mohammed Baki, B.Sc;
A Master's Thesis
Submitted for the Award of the Degree of
Master of Science
of
Loughborough University of Technology
March 1979
Supervisor:
.. Professor I R Smitn, B.Sc .. Ph.D .. D.Sc., C.Eng., F.I.E.E.
C by Ali Mohammed Baki
•
Loughborough Un!verslty
of T.-.hnolo,".1 I.brary
Dot. J....-.I!. "; "I Clilss
Ar.c. OOlo;tf>/oL No" I
AfKNOWLEDGEMENTS
I would like to take this opportunity to express my
special thanks to my supervisor. Professor I R Smith. for his
invaluable guidance. great advice and patience during the
period of research and preparation of this the~is.
I wish to thank the Head and staff of the Electronic and
Electrical Engineering Department at Loughborough University of
Technology for the facilities provided throughout this work.
My acknowledgements go 'to the University of Technology
of Baghdad for the financial support received while this pro
ject was undertaken.
I am indebted to my parents and my wife for their support
and help through the period of study.
I also wish to thank my colleagues. Mr T M Ali and Mr S
Muslih.for their valuable comments and helpl and also Mrs J
Smith and Mr P LeGood for their considerable help in the typing
and preparing copies of the characteristics in this thesis.
"~
i11
• SYNOPSIS
The ever-increasing complexity of power systems is leading
to more severe problems of system stability, and considerations
of this are requiring a more detailed modelling of the system
generators than has hitherto been undertaken. 1he induced
eddy currents in the iron body of the machine, and the con
siderable a-ctive resistance of this iron, have previously been
regarded as fully accounted for by a single damper circuit on
each axis of the machine. However, recent work has shown that
this is not an adequate representation, and that calculations
on this basis may exhibit considerable discrepancies with prac
tical measurements of the transient performance of the machine.
Measurements of the generator parameters has been based on a
single damper representation, and it is now necessary to re
examine carefully t~e assumptions underlying these tests and
to determine if they can be used to provide a-ri-improwd machine
model.
The main aim of this thesis is to investigate the possibility
of using step-response tests to obtain the parameters of a machine
model with two damper windings on each axis. From an investi-
gation of this form of equivalent circuit, expressions in the
form of transfer functions are developed for the operational
inductance of the machine, and these are correlated with measured
results to provide'~he necessary numerical data. Some of the results
required can also be obtained from well-established test procedures,
and these are therefore used to provide further confirmation of
the accuracy of the new machine representation.
,
iv
CONTENTS
Page Nos.
Title page i
Acknowledgements 11
Synopsis 11i
contents • iv
Nomenclature vii
INTRODUCTION 1
CHAPTER I: EARLY ANALYSES OF THE PARAMETERS OF 5 SALIENT-POLE MACHINES
1.1 Operational Impedances of an Idealised 6 Salient-Pole Synchronous Machine
1.2 Determination of the Mechine Parameters 8 from Current-Decay Test
1.3 Determination of Parameters from Indicial 16 Response Tests
1.4 Summary 21
CHAPTER 2: OPERATIONAL INDUCTANCES OF A MODIFIED 23 SALIENT-POLE SYNCHRONOUS GENERATOR
2.1 The Modified Salient-Pole Synchronous 23 Generator
2.2 Operational Inductances 24
2.2.1 q-axis operational inductance 25
2.2.2 q-axis additional subtransient 29 and subtransient inductances
2.2.3 d-axis operational inductance 31
2.2.4 d-axis reactances 35
CHAPTER 3: ANALYSIS OF A MODIFIED SALIENT-POLE SYNCHRONOUS GENERATOR
3.1 Analyses of the q-axis Circuits
40
40
v
Page Nos
3.1.1 q-axis synchronous and addi- 43 tional subtransient reactances
3.1.2 Determination of the q-axis 44 reactances from the q-axis open and short circuit time constants
3.1.3 q-axis open and short circuit 46 time constants
3.1.4 q-axis magnetic coupling 49 coefficients
3.2 Analysis of the d-axis Circuits 49
CHAPTER 4:
3.2.1 d-axis synchronous and additional subtransient reactances
3.2.2 Derivation of the d-axis reactances from the d-axis open and short-circuit time constants
3.2.3 d-axis open and short circuit time constants
3.2.4 d-axis magnetic coupling coefficients
METHODS OF MEASUREMENT OF PARAMETERS
56
58
59
63
4.1 Indicial Response Test 65
4.2 Determination of the q-axis Synchronous, 67
4.3
4.4
Subtransient and Additional Subtransient Reactances by the Indicial Response Method
d-axis Indicial Response TestJ Field Circuit Open
d-axis Indicial Response TestJ Field Circuit Closed
73
76
4.5 Determination of the Field Time Constant 81 "From the d-axis Field Transient Current with Armature Windings Open
4.6 Conventional Methods of Measurement of 82 Para~eters
4.6.1 Determination of the d-axis syn- 82 chronous reactance from the no-load saturation and sustained 3-phase short circuit characteristics
4.6.2 Determination of d- and q-axis 84 additional subtransient reactances from Oalton and Cameron method
4.7
4.8
4.9
4.10
\f CHAPTER 5, I
~ REFERENCES
APPENDIX 7..1
APPENDIX 7.2
APPENDIX 7.3
vi
Page Nos
4.6.3 Determination of d--and q-axis 86 synchronous re8ctances from a slip test
4.6.4 Determination of q-axis synchro- 8B nous reactance from maximum lagging current method
4.6.5 Determination of q-axis synchro- 90 nous reactance from reluctance motor method
Effect of the Brushes on the Parameters Measured
92
4.7.1 Measurements of q-axis synchronous. 92 subtransient and additional subtransient reactances (without brushes)
4.7.2 Measurements of d-axis synchronous. 93 subtransient and additional subtransient reactancesJ with the field winding open (without brushes)
4.7.3 Measurements of d-axis synchronous. 94 transient. subtransient and additional subtransient reactancesJ with the field circuit closed (without brushes)
'4.7.4 Determination of d and q-axes addi- 95 tiona1 subtransient reactances by Da1ton and Cameron method (without brushes)
Decay Test
Linear Regression (Method of Least Squares)
Comparison of the Results
COMMENTS AND CONCLUSIONS
fER UNIT EQUATIONS
LINEAR REGRESSION (METHOD OF LEAST SQUARES)
COMPUTER PROGRAM
96
98
111
145
147
150
155
159
vii
NOMENCLATURE
Subscript notation:
The subscript notation is as follows:
a,b,c
F
10. IQ
20. 2Q
=
=
=
=
armature windings of a. band c phases respectively
field winding
damper windings of 10. IQ axes res~ectively
damper windings (representing iron effects) of 20. 2Q axes respectively
The quantities given in the following are per-unit values (unless in bold-type as defined in Appendix 7.1. which are in SI units).
Efd
Id' Iq
I 1d·I2d •
R2d ·R1q •
R2q
Lafd
=
=
=
=
=
=
La1d·La2d =
L11d • L22d =
L11q·L22q =
=
=
main field winding voltage
armature current in d. q axes. respectively
currents in damper circuits
armature resistances·in the d and q-axes respectively
resistances of F. 10. 20. IQ and 2Q respectively
mutual inductance between armature and field
mutual inductances between armature and 10 and 20. respectively
mutual inductances between armature and IQ and 2Q. respectively
self-inductances of 10 and 20. respectively
self-inductances of IQ and 2Q. respectively
self-inductance of the main field
mutual inductances between the main field and 10. 20. respecti vely
mutual inductance between 20 and 10
mutual inductance between 2Q and IQ
d-axis flux linkilges with the' armature winding
q-axis flux linkages with the armature winding
Wfd
W1d,W2d
W1q,W2q
Ld,Ld,Ld, L'"
d
L L",L'" q' q q
Ld(p)'
L q(p)
G(p)
X" X'" d' d
x X" q' q'
X'" q
Xd (p)'
X . q(p)
Kafd,Ka1d'
Ka2d,Kf1d'
= flux linkages with the field winding
= flux linkages with ID and 20, respectively
= flux linkages with IQ and 2Q, respectively
d-axis synchronous, transient, subtransient and addi-= tional subtransient inductances, respectively
• = q-axis synchronous, subtransient and additional subtransient inductances, respectively
= operational inductances of the d and q-axes, respectively
= an operator which relates field voltage with armature current and linkages in the d-axis
=
=
d-axis synchronous, transient, subtransient and addi
tional subtransient reactances, respectively
q-axis synchronous, subtransient and additional sub
transient reactances, respectively
= operatio~al impedances of the d and q-axes, respectively
Coupling coefficients between a and F, a and ID,
a and 20, F and ID, F and 20, 20 and ID, a and IQ,
Kf2d,K21d'= a and 2Q, 2Q and IQ, respectively
°af12d
= q-axis total coefficient of dispersion (defined by equation (2.14)
= coefficient of· dispersion between 2Q and IQ (defined by equation (2.15)
= d-axis total ~oefficient of dispersion (defined by equation 2.38)
= coefficient of dispflrsion botw8en F, 20 "nd ID (dofined by o~uiltion 2.30)
a =
wt =
p =
1 =
~ =
s =
E =
I Ao =
·IA1 , I AZ ' =
I A3
IA1s,IAZs' =
IA3s,IA4s
TA1 ,TAZ ' =
TA3
TA1s ' TAZs '
= TA3s,TA4s
Tad
, Taq
, =
Tfd
,T1d
,
TZd ,T1q,
T2q
T' T" dD' dO '
T'" dO
T" T'" qO' qO
=
=
=
electric an81e of rotor
time in electrical radians
d/dt
unit step function
Laplace transform of
Laplace transform variable
applied d.c. voltage on a-phase
steady-state current of a-phase (defined by equations 3.11 and 3.53]
decaying components of I and Id (with the field winding q open for the latter)
decaying components of Id with the field winding
closed
time constants of IA1
, IAZ
and IA3
,
respectively
time constants of I A1s ' IAZs ' I A3s and IA4s '
respectivelt
d and q-axes armature open circuit time constants
time constants of F, ID, 20, IQ and 2Q, respectively
d-axis transient, subtransient and additional sub
transient open-circuit time constants
q-axis subtransient and additional subtransient open circuit time constants
T' Too d' d '
T'" d
Tit T"t q' q
=
=
x
d-axis transient, subtransient and additional
subtransient short circuit time constants
q-axis subtransient and additional subtransient short ·circuit time constants
•
I
INTRODUCTION •
The basis for modern analysis of salient-pole synchronous
machines was established about 40 years ago, when Doherty and
1 Nickle extended the 2-reaction theory of Blondel, and developed
methods for determining the performance under certain steady-state
and transient conditions. 2 •
Subsequently, Park generalized the
approach by using Heaviside's operational calculus to analyze the
behaviour, and he introduced the now familiar characteristic opera-
tional impedances.
4 Waring and Crary explained this impedance in terms of the
machine constants and, by deriving formulae for the important reac-
tances and time constants, threw new light upon the performance of
these machines. Following this early work, some investigators have
presented different techniques for machine analysis and others have
developed methods for determining the machine parameters from test.
Summaries of the test procedures are contained in references (3) and
(5).
Comparatively recent experience has shown that the performance
of a modern salient-pole machine may differ in certain important
respects from that predicted by the idealized theory based on Park's
equations,· and studies of more accurate machine representations have
6-9 been made by several investigators. In recent years, interest has
been growing in the use of static methods1o,11,12 to obtain machine
data, replacing some of the more familiar tests requiring rotation
of the machine. A careful appraisal of the results obtained from
these tests is clearly important, since the conditions under Which
the measurements are taken are artificial and reproduce only to a
2
limited degree the normal-operating conditions of the machine.
Methods10,12 have been developed to obtain the synchronous,transient
and subtransient reactances from the analysis of current transients
caused by the creation or cessation of direct current in the
armature. The presence of the eddy currents produ~es a damping
action which is difficult to calculate, but which is sufficiently
important to cause the behaviour of the machine to differ appreciably
from that predicted by the usual simplified theory.
Oespite the enormous quantity of work reported on measurement
of the parameters of salient-pole machines, no experimentally sound
technique has been developed which accounts for the induced eddy
currents in the rotor iron. Indeed, a careful review of the lite-
10 12 rature reveals that several authors employing static tests '
have disregarded this effect entirely.
12 The indicia I response method proposed by Kaminosono and
6 Uyeda has been used by Yao-nan Yu and H.A.M. Moussa , but their
tests have failed to give reasonable results.
In this thesis, step-response methods are applied to a 5 kVA
salient-pole laboratory generator, and the results obtained indicate
more than one exponential component in the transient response curve.
The experimental work was repeated for different levels of current,
and each time it was found that this extra component was significant
both for the quadrature axis and for the direct axis. Neglecting
this component gives results which are far from accurate when com-
11 14 pared with other tests.' It is thus necessary to re-examine the
assumptions on which the equations for those measurements are based.
The conventional equivalent arrangement with one damper circuit on
3
each rotor axis is inadequate to calculate the transient conditions.
since the actual characteristic differs considerably from that
10 12 often assumed • when examining the transient processes. The
question then naturally arises of how to represent more exactly
the performance of a salient-pole generator. and hence how to
determine more accurately the circuit parameters'using a step-
response test. To obtain the best results from this type of test
requires that the effect of saturation. the frequency dependance
of the parameters and the complex influence of the rotor iron are
taken into account. However. to attempt to do this fully would
unreasonably complicate an initial investigation of the principal
effects involved. With an indicial response method. it is possible
to include saturation to a small extent by additional or bias mag-
netization on the stator or the field. By ignoring the effects of
frequency and by replacing the rotor damping action by two closed
damper circuits on each axis. a similar but more elaborate version
of the primitive machine of Park is obtained.
Scope of the Thesis
The object of this thesis is to show how Park's theory for the
salient-pole synchronous generator can be modified to include the
damping effect of the rotor iron. and to determine how the
machine's parameters of the new mathematical model can easily be
calculated from step response test.
The first step when examining the response of the machine is to
express the machine equations in operational form. In the first part
of this thesis. operational Inductances are derived in terms of the
22 machine inductances and resistances. following the Kirschbaum p.u.
4
system. In this thesis. each axis is analyzed in the same way as
a multi-winding transformer~ and the derivation of these new op-
erational inductances is clearly now cumbersome. because of the
two damper circuits on each axis.
In deriving the parameters stated in Chapters (2) & (3). the direct
and quadrature axis equations ara reprasented in terms of the syn-
chronous inductances. the coupling coefficients and the time con-
stants of the individual circuits. This enables a comparison to
be made between the present matrix forms and those derived in
12 similar form in a previous investigation. and identifies the
inaccuracy in this previous work. To show the damping effect of
the additional rotor circuit and to determine the para-
meters of the machine. measurements were made using the indicial
12 response method on the 5 kVA. 240V, 50 Hz salient-pole synchronous
generator. Measurements were made with armature currents up to l4A.
A least squares curve fitting technique was used to show the inaccu-
racy of disregarding the damping effect of the rotor iron. The
measurement technique required a step voltage to be applied to one
armature phase with the armature stationary, with the resulting
transient current being measured by a U.V. recorder. The curves
obtained consisted of a multi-exponential decay, and their analysis
consisted of resolving them into the sum of three exponential func-
tions for the quadrature axis and three or four exponential functions
for the direct axis when the field winding is open and short circui tedr:~.
respectively. It is demonstrated in the thesis that the magnitudes
and time constants of the exponential components are linked by simple
mathematical relations to the parameters of the machine.
5
CHAPTER I
EARLY ANALYSES OF· THE PARAMETERS
OF SALIENT-POLE MACHINES
The damping effect of the rotor iron plays a considerable
part in determining the performance of a synchronous machine. as
discussed in the Introduction. Although this effect has been
ignored in many previous investigations. it is becoming increasingly
necessary to include it when a rigorous analysis of the performance
is attempted. Nevertheless. since the previous studies form the
basis from which techniques described later in the thesis were
developed. the following sections are included to provide a
summary of two methods lO• 12 used to determine the parameters of
a salient-pole machine with the effect of the rotor iron dis-
regarded. Although these methods involve slightly different
tests these differ only in detail. and since they both originate
from the same basic concept the information provided is funda-
mentally the same.
The first test determines the parameters of the machine from
the decay of a direct current through two series phases of the
armature winding. and,the second uses a similar approach in deter-
mining the parameters from a test involving a step increase of
voltage to one" phase of the armature. The basic principle involved
in the two tests requires knowledge of the operational impedances
of an idealised salient-pole machine. as established by Waring and
4 Crary and discussed below.
6
1.1 Operational Impedances of an Idealised Salient~Pole Synchronous Machine
4 Early attempts to derive the operational impedances of a
salient-pole synchronous machine, in terms of the machine reac-
tances and resistances, were based on considerations of an
idealised machine with one damper circuit on each of the direct
(d) and quadrature (q) axes, and with a field winding on the d-
axis. Based on the 3-winding transformer analogue shown in
Fig. 1.1, the authors obtained expressions for the d---and q-
axis operational impedances from consideration of the rotor
4 circuit voltages and flux linkages as:
O.ll
and 0.2)
respectively. In the above equations, subscripts d and q refer to
the d and q axes of the machine, and subscripts 1, a, f to the damper,
armature and field circuits respectively. Thus Xd is the synchronous
reactance of the d-axis armature circuit, X11d is the self-reactance
of the d-axis damper circuit, and Xafd is the mutual reactance bet
ween the armature and the field circuit along the d-axis. The other
resistance and reactance terms are similarly defined. The d and q
axis subtransient reactances are then found by letting p~ in the
7
above equations, when, from equation (1.1)
X· = X (~) = X -d d d
X11dX!fd-2Xf1dXa1dXafd+XffdX!1d
X11dXffd-X~1d 0.3a)
Alternatively, in tenns of the various coupling 9oefficients,
equation (l.3a) becomes:
X" = d
2 where: Kafd =
Similarly, from equation (1.2):
X· = X (~) = X q q q
or in tenns of t.he coupling coefficient:
X· = Xq (1 - K2 ) q a1q
0.3b)
0.4a)
0.4b)
The d-axis.transient reactance is found by letting p ~~ and
X11d = ~ in equation (1.1), when
O. 4c)
or in tenns of the coupling coefficient
Cl. 4d)
8
1.2 Determination of't~Machina'Parametars'froma'Current'DecaY'Tast
10 In 1962. M.K. Paw1uk described a further method for deter-
mining the parameters of a salient-pole machine from the decay
of a direct-current through two phases in series with the
armature stationary. The resulting transient current is measured
by an oscillograph. and its analysis consists of resolving it into
the sum of two exponential functions for the q-axis and the sum
of two and three exponential functions for the d-axis when the
field is respectively open and short circuited. From the magni-
tudes and time constants of the individual components of the
measuring transient current. the synchronous. transient and sub-
transient reactances included in the machine equations are found.
On the basis of an idealised salient-pole machine. and from
10 the flux linkages and voltage equations • the operational
equations obtained for- the q-axis were:
R + pX q q
i' q
i' Q
= p £1.5)
o
where in the original terminology Rand RQ are the resistances . q
of the armature and the q-axis damper circuit respectivelYJ Xq
and XQ are the q-axis reactances of the armature and damper circuits
respectivelYJ XqQ is the q-axis mutual reactance between the arma
ture and the respective damper circuitJ i~ and iO are the q-axis
currents in the armature and damper circuits, and iqD is the initial
9
value of i. After solving equation (1.5) for the current i'. q q
and introducing the time constants T~Q and TQ q and the dispersion
coefficient crqQ in place of the reactances. and resistances. the
following equation is obtained:
1 p~
i· = ________ Q;o.q;;:.... ___ --- i
q 1 1 crqQ qO P 2
+ P ('f' + "f') + ""T"" -";;,T ..... -qQ Qq qQ Qq
(1. Ba)
where:
and
crqQ = total coefficient of dispersion on the q-axis
(1. Bb)
T' = time constant for the armature on the q-axis. with qQ
the damper circuit closed and its resistance RQ
assumed negligible.
= T cr q qQ (1. Bc)
Tq = time constant for the q-axis armature circuit with the
damper circuit open.
T' Qq Cl. Bd).
The time constants TQ and TQq have similar interpretations relative
to the damper winding Q.
10
C The denominator of equation (1.6a) is a 2nd-order polynomial
in p. with the two roots
1 [- (T' qQ
1 + -) T' Qq
4 ° Q T' T~ 1
qQ Qq O. 7a)
As 0 < 0qQ < 1. the term under the root of equation (1.7a) is
always positive and the denominator of equation (1.6a) has there-
fore two real and negative roots. ,Consequently. the time variation
is the sum of two exponential functions with time constants
0.7b)
and
From the known relationships between the sum and the products of
the roots of 2nd-order algebraic equations. the following relations
are obtained:
•
III 1 --+--=--+--TqQ TQq Tq1 Tq2
By applying the Heaviside theorem. the time variation of the
current i is obtained as: q
0.8)
n.9)
n.lO)
11
i ~·I q q1
-tIT q1 e . -t/T
+ I. e q2 q2
After putting iq1
(1.8) and (l~lO):
I 1 ~ -q-
iqO
T' ~ 1
qQ i 1 i 2 ....9..... + _q_ Tq1 Tq2
and T' ~ 1
Qq i 1 i 2 _q_ +....9..... Tq2 Tq1
I 2 ~ ....9.....
i • qO
and solving equations
In practice. the test procedure required the recording of the
(l~l1)
(1.12)
transient current to be transferred to semi log paper. from which
the two decaying components and their corresponding time constants
14 . could be calculated by the we11- known means These values were
then substituted in equation (1.12). and when the values of T~Q and
TQq thus found were substituted into equation (1.10)' the coefficient
of dispersion aqQ was obtained.
The time constant T follows simply from equation (1.6c) and q
by measuring the q-axis armature re~istance per phase. the q-axis
synchronous reactance is given by:
(1.13)
The subtransient reactance is:
X" = -X' = X (J q qQ . q qQ
12
= R· T er Q q q q (1.14)
The d-axis synchronous reactance (Xd
) may be similarly obtained,
merely by replacing subscript q by subscript d.
The operational equations which apply for the d-axis with
the field circuit closed are:
= o
o
(1.15)
where, again, in the original terminology, id' if and iO are respec
tively the d-axis armature, field and damper currents I Xf
and Xo
the self reactances of the field and damper windingsl Rf
and RO
the field and damper resi~tancesl Xdf
and XdO the mutu~l reactance
between the armature/field and field/damper circuits respectively I
XfO
is the mutual reactance between the field and the damper winding.
On solving equation (1.15) for id:
(1.16a)
13
where:
Cl dfD = total coefficie·nt of dispersion on the d-axis
O.16b)
and TO = (l.l6g)
o .l6h)
14
From the :elationship applying for a 3rd-order polynomial equation,
the relation between the machine time constants and the three
time constants read from an oscillographic recording are:
(1.17)
1 1 1 1 1 1 --+-+-=-+--+--Tdfo TfTO Td3 Td4 Td5
. 0.18)
0.19 )
where Td3 , Td4 and Td5 are the time constants found from replotting
the recording transient current on semilog graph paper. Td and TO
were found previously from the d-axis decay test with the field
winding open circuited, and Tfo and adfo are found from equation
(1.17) and (1.19) respectively.
The d-axis subtransient reactance is given by:
eX" d
a Rd (T ~) =
d afo 0.20)
where afo
is found from the current decay test as applied to the
d-axis field winding with the armature winding open circuit (similar
to the d-ax1s armature current decay test with the field winding
open) as:
afo =
Tf1 Tf2 0.21)
Tf TO
= equation 0.16d).
15
where Tf1 .and Tf2 are the time constants obtained from trans
ferring the recorded transient field current to semilog paper.
By substituting for afD , together with Xd and adfD , into
the equation (1.20), Xd can be found.
If the valu~s of adfO and afO given by equations (1.16b) and
(1.16d) are written in terms of the coupling coefficients of the
machine, equation (1.20) becomes:
or
X" . d
X" = X [1-d d
where:
Cl.22)
etc.
4 Equation (1.22) is similar to that derived by Waring and Crary,
and given by equation (1.3b).
The d-axis transient reactance is given by:
X' = d
Cl.23)
where adf is the coefficient of dispersion between the d-axis
armature winding with the field winding. Since this cannot be
found (since the damper winding cannot be opened) the following
procedure is required. By application of a current decay test on
16
the d-axis armature, with the field winding open, adD can be
found. By substituting adO together with adfo and Tf into equation
.(1.16e), Tf can be found. TO then follows from equation (1.18),
and by substituting into the equation (1.16g), adf can be found.
Finally, the d-axis transient reactance is given by:
X' = d
(1. 23)
Kaminsono and uyeda12 performed indicial response tests, in
which a step,increase of voltage was applied to one phase of an
armature winding. Their analysis follows that of the previous
section, and they obtained the components of the transient phenomena
in exactly the same manner. Using terminology as defined in the
nomenclature, the p.u. differential equations which apply when
their test is performed on a phase aligned with the d-axis arel
with the field winding open circuit:
E1 pL + d rd -p K
a1d Id
(1. 24) =
0 -p Ka1d p + 1 I 1d T1c;
and with the field winding short circuit:
E1 pLd
+ rd -p Ka1d -p Kafd Id
0 = -p Ka1d p 'I + --T1d
-p Kf1d I 1d 0.25)
0 -p Kafd -p Kf1d 1
Ifd P +-Tfd
•
17
The transient current through the armature phase winding is
obtained from the Laplace transform of equation (1.24) as:
-t/TA1 e
where from equation (1.24) and (1.26):
and
I T Ld = rdTA1 Cl + A2 A2)/Cl
IA1 TA1
The transient current with the field winding closed follows
similarly from equation (1.25) as:
(1.26)
(1.27)
(1.28 )
(1.29)
while from solution of equations (1.25) and (1.29), the following
relations were obtained:
1 1 1 1 1 1 -=--+--+--------Tfd TA1s TA2s TA3s Tad T1d
I A1s 1 1 1 . 1 = - (- - -H- - -)
E TA1s lA2s TA3s TA1s
(1.30)
(1.31)
18
and
The q-ax~s quantities may be obtained by a similar procedure,
simply by replacing subscript d by subscript q in equations (1.24), (1.26),
(1.27) and (1.28).
The per unit synchronous reactances of the machine are
calculated from:
Xd "~L 2 d
(1. 33)
and X "~L q 2 q
where equation (1.33) shows the relation between the p.u. synchronous
inductances introduced by Kaminosono and the equivalent impedance
28 conventionally used •
. The p.u. subtransient reactances; of the machine are:
X· "~L d 2 d
and
[1 -
2 Ka1d -
1 (1.34 )
0.35)
Again, equations (1.34) and (1.35) show the relation between quan-
12 tities introduced by Kaminosono and their familiar equivalent
/
19
impedance (for more details see reference 28).
An examination of the matrix equations (1.24) and (1.25),
from the point of view of the p.u. values of the quantities
involved, shO\~s that these p.u. equations were incorrectly
derived by the original authors, and their solution therefore
provides a set of equations which differ from equations (1.27),
0.28),0.30), '(1.31) and (1.32).
Equations (3. 1) and (3.41) in Chapter 3 repre-,
sent the p.u. operational equations derived for the modified
salient-pole machine where the damping effect of the rotor iron
has been taken into consideration. If this effect is ignored, and sub-
script q is replaced by subscri~t d in equation (3.1), then eqns.(3.l)'
and [3.411 reduce to: With the field winding open circuit:
E1 pLd + rd -p L , d , Id
= (1. 36)
0 -p Ka1d p + 1 I1d T1d
and with the field winding short circuit:
E1 pLd + rd -PLd
Ka1d -pL
d Kafd Id
0 = -p Ka1d p + _1_ -p K
f1d I1d
(1.37 ) T1d
0 -p Kafd -p K
f1d p +..!.. Ifd
Tfd
20
Equations (1.36) and (1.37) represent the correct p.u. operational
equations of the idealised salient-pole synchronous machine and
from their solution ~8 obtain:
1 -IA1sTA2sTA3S+IA2STA1sTA3S+IA3sTA1sTA2S
T1d TfiAOs
(1. 38)
Cl.39 )
0.40)
Comparison of the equations (1.36) and (1.37) ~ith equations (1.24)
and (1.25) respectively, sho~s the necessity of correcting the
latter t~o equations by multiplying the second term of the first
ro~ of equation (1.24), and the second and the third terms of the
first ro~ of equation (1.25) all by Ld
•
Comparison of equations (1.38), (1.39) and (1.40) ~ith the pre-
viously derived equations (1.30), (1.31) and (1.32) sho~s the
necessary corrections. Equations (1. 36), Cl. 38), (1. 39) and (1.40)
may also be obtained by replacing the damping circuit of the rotor
iron by the field ~inding (i.e. replacing suffix 2 by suffix f)
and sUQscript q by subscript d in equations (3.1), (3.12), (3.39)
and (3.40), derived later in Chapter 3 (~ith suffix s added
to each current component and its respective time constant). It
.,' '~"" " '?i' ,
21
is necessary to make clear that equation (1.38) was also obtained
by PawluklO and given by equation (1.17). Equation (1.37) may
be checked by comparison with equation (3.41), when the damping
circuit of the rotor iron is ignored.
1.4 Summary
As has been described in this Chapter, two similar step res-
ponse tests have
chronous machine.
been used to determine the parameters of a syn, 10
The first method determines the synchronous,
transient and subtransient reactances from·the decay of a direct
current through two armature phases in series, and requires a
\ high resistance R » r (the armature phase resistance) to be used
as shown in Fig. 1.2 to protect the supply after the armature
terminals have been short circuited. The condition that R » r
will also minimise the voltage drop across the contacts of the switch
S, and thereby reduce the associated measurement errors.
The second method12
determines the synchronous and subtrans-
ient reactances of a machine from tests requiring the application
of a d.c. step voltage to one phase of the armature as shown in
Fig. 1.3 and provided that the impedance of the d.c. source is
negligible, it will provide results identical with those given by
the first method.
oil .... --...J\
g .-e L-
" .. " L-~ .. e e
LL
L-
e e----~---~
FIG 1.1.
F .
rn) FIG. 1.2.
FIG. I. 3.
R
5
fi«ld t«rminalr.
CHAPTER 2
OPERATIONAL INDUCTANCES OF A MOOIFIEO
SALIENT-POLE SYNCHRONOUS GENERATOR
2.1 The Modified Salient~Pole SYnchronous Generator
As explained in Chapter 1. the performance of a salient-pole
synchronous generator differs in some important respects from that
predicted from an idealised theory based on Park's equations. and
to reduce this difference it is necessary to improve the repre-
sentation by re-examining the assumptions on which the derivation
of the performance equations is~based.
The presence of the considerable mass of the (possibly) solid
rotor iron within the magnetic circuits of the machine, needs to
be taken into account when calculating accurately the transient
processes. To alleviate the complexity of the equations derived.
attempts have been made to adopt an improved generator represen-
tation. based on modified Park's equations and with two damper circuits
employed on each axis. Figure 2.1 shows such a'modified d.q circuit
model. with the armature phase windings represented by coils a. b
o and c (mutually separated by 120 electrical). the field winding
by coil f. the damper windings by t'-IO closed circuits 10 and 10.
and the damping effect of the rotor iron by coils 20 and 20 res-
pectively. In analysing this model. the following assumptions are
involved.
1. The flux distribution in the air-gap is sinusoidal. with
slot effects being neglected.
24
2. Saturation and hysteresis effect are neglectedJ this
includes local saturation caused by the skin effect
associated with the eddy currents in the rotor iron.
3. The effect of frequency on all parameters is neglected.
All the eight circuits represented in Fig. 2.1 have indi-
vidual resistances and self-inductances, and a mutual induc-
tance with respect to all other circuits on the same axis.
4. No·mutual resistance between the damper circuits on the
same machine axis is considered. This assumption is valid
because there are no current paths common to both damper
circuits.
5. In the 2-axis representation of the machine shown in Fig. 2.1,
it is assumed that the fluxes in the two axes act independently,
and that there is no mutual inductance between windings on the
d and the q axes.
2.2 . Operational Inductances
In many important operating problems, it is necessary to
determine results viewed from the mach+ne armature terminals as,
- for example, when computing the short-circuit currents or any other
kind of transient change of current and voltage. Early attempts
4 to derive the operational impedances of a synchronous machine
were based on simplified considerations of an idealised machine.
It is now regarded as necessary to show that the damping effect
of the eddy currents in the rotor iron has an influence on the
transient processes, and hence that the familiar equivalent arrange-
25
ment with one damper circuit on each rotor axis, is insufficient.
4 In consequence, the previous theory is sometimes not adequate
and requires modification. It is possible to represent the whole
group of d or q axis circuits by an equivalent static circuit.
Since no mutual resistance exists between these circuits, each
axis may be analysed in exactly the same manner as a multiwinding
transformer4, with the number of windings being the same as the
number of circuits symmetrical about the axis being considered.
Reciprocity of the mutual inductance coefficients is an essential
feature of a static equivalent circuit, and this introduces a
.very helpful simplification in calculating the operational induc-
tances, which b"ecome increaSingly cumbersome as additional damping
circuits are introduced.
For the reason mentioned above, the new operational induc
tances are derived following H.S. Kirschbaum21 , using the same
basic equations. In deriving the operati,onal inductances for
both groups of symmetrical coils. the q-axis is represented by
the 3-winding transformer of Figure 2.2, and the d-axis by the
4-winding transformer of Figure 2.3.
._" 2.2.1 q-axis operational inductance
Using the terminology defined i.n the nomenclature, the
voltage equations for the q-axis rotor circuits may be written
directly from Figure 2.2 as:
(2.1)
and (2.2)
26
a-axl • ..
ID~,
F'IG.2.1. MODIFIED CIRCUIT MODEL OF A 3- PHASE
SALIENT- POLE SYNCHRONOUS GENERATOR
"" .. -~a "'c 0-EE .... 0:
FIG.2.2, A 3-WINDING TRANSFORMER ANALOGOUS TO THE CIRCUITS SYMMETRICAL ABOUT THE g.-AXIS
'" o c 'e .. .. ...
" .. ::J ... C e .. c
fie Id terminals
FIG.2.3. A 4-WINDING TRANSFORMER ANALOGOUS TO THE CIRCUITS SYMMETRICAL ABOUT THE d -AXIS
27
The p.u. equations follow Kirschbaum's21 p.u. system (see equations
7.1.8 - 7.1.10 in Apendix 7.1.1).
i. e. W1q = L I + L q K12q I + L K I (2.3) q 1q 2q q a1q q
I W2q = Lq K21q I 1q + L I + L q Ka2q I (2.4) q 2q q
and the q-axis flux linkage with the armature is:
Wq = L K I 1q + L K I + L I (2.5) q a1 q q a2q 2q q q
Solving equations (2.1) and (2.2) and noting that K12q = K21q ,
we obtain:
and
[p2L~ (K21q Ka2q Ka1q ) - PLq Ka1q R2q ] Iq
[P2L~ (1 - K~1q) + PLq (R1q + R2q ) + R1q R2q]
(K21q Ka1q - Ka2q ) - PLq Ka2q R1q] Iq
(1 - K~1q) + PLq (R1q + R2q ) + R1q R2q ]
(2.6)
(2.7J
After. substitution of equations (2.6) and (2.7) in equation (2.5)
I L [p2L 2 (1-K~1q-K!1q-K!2q + 2Ka1qKa2qK21q) - PLq (K2 R + q q q a1q 2q
K2 R1q - R - R2q ) + R1q R2q] Wq =
a2q 1q
[p2L 2 Cl - K2 ) + PLq (R1q + R2q ) + R1q R2q] q 21q
(2.8)
28
2 The general equation, derived by Park for the q-axis flux
linkages with the armatures can be written as:
"0 = L (p) I "'q q q (2.9)
. I where L (p) = operationali'nductance of the
compar(~g equations (2.8) and (2.9), it is
q-axis circuit.
evident that with
two rotor circuits on the q-axis:
L [p2L2 (1-K2 _K2 -K2 + 2K K K J - pL (K2 R2q q q 21q a1q a2q a1q a2q 21q q a1q
+ K2 R1q - R - R ) + R R2ql L (p)='
a2q 1q 2q 1q q
K~1q) [P2L 2 (1 - + pL (R + R ) + R1q R2ql q q 1q 2q
·(2.10)
The direction of I has been defined as positive when an m.m.f. q
is produced in the same direction as the m.m.f. produced by the
21 damper circuit, and follows that used by Kirschbaum If the
reference direction of I is reversed, the sign of the .last terms q
on the right-hand side of equations (2.3)-(2.5) becom§l's negative.
The signs of I1q
and I 2q in equations (2.6) and (2.7) become
negative, but since the positive signs of Iq in equations (2.8)
and (2.9) become negative, the signs of all the terms in equation
(2.10) remain unchanged. In other words, reversing the direction
of the reference current I has no effect on any of the terms of q
equation (2.10).
29
'. V"
and 'addi t:Ldnal sub transient ",.' " .'
\ ....... ''--" .. "
" '.: .,., ~., '._'.- "'--~ , "
Having deduced the operational form of L (p),in equation q
(2.10), it is easy to find the q-axis additional subtransient
inductance L'" of the machine. (This new definition has been q
I given because of the damping effect of the rotor iron, and
i I differs from that conventionally used and given by Ld when an
idealised machine is considered).
The operator Lq(P) represents the inductance of the q-axis
circuits, and its value at t = 0 is found by letting p~, which
19 in the notation of the operational calculus corresponds to
taking the initial value of a function. Thus, when neglecting
the rotor resistance, we get for t = 0, and p~, the additional
subtransient inductance as:
~ I. q
= L (00) q
K2 + K2 - 2K K K = Lq [l - _a:..:1,::cq _-,a2q a1 q a2q 21 q ]
1 - K~1q
or in terms of the corresponding inductances:
L22q La21q + L11q La
22q - 2L L L
::L--':::.'-"--_---'-.:.:!.--'== __ ..::a:.c1.:!q-=a2q . 2_~q
L11q L22q - L~2q
(2.11)
(2.12)
'Equation (2.11) can be written in terms of the coefficients of
dispersion, in the same manner as described in reference [15],
i.e.
where:
(J aZ1q ,
I
7(J ~ I1 =
_ a::.:Z:..:1..:o.q L • q q (J21 q
= total coefficient
circuits
30
of dispersion from the
= 1 - (K2 + K2 + K~1q - 2Ka1qKa2qK21q) a1q a2q
q-axis
and (J21q = coefficient of dispersion between the first and
second damper circuits
(2.13)
(2.14)
(2.15)
The q-axis subtransient inductance of an idealised machine Ld (with the damping effect of the rotor iron neglected) may be
found from equation (2.10), by letting L22q = 00, and p~ as
L" = L Cl - K2 ) q q a1q.
(2.16) 2
= L _ La1q
q L11q
_ The relation between the quantities shown by equations (2.11),
26 (2.12), (2.13) & (2.16) and the conventional equivalent impe-
dances are:
3 X '" - L q 2 q
X" '" ~ L" q 2 q (2.17)
and ~'J~~ L!" q 2 q
31
, 3 Multiplying equation' (2.16) by a factor 2 thus leads to the
familiar q-axis subtransient reactance (X") derived by Waring q 4 and Crary •
2.2.3 d-axis operational inductance I
JSing the terminology defined in the nomenclature, the
I voltage equations of the d-axis rotor circuits may be written
directly from Figure 2.3 as:
21 The p. u. equations follow those used by Kirschbaum .
(see Appendix 7.1.2)
i.e.
and the d-axis flux linkages with the armature is:
(2.18)
(2.19)
(2.20)
(2.211
.-(2.22)
(2.23)
(2.24)
32
Solving equations (2.18)-(2.20) simultaneously, and noting that
resultj in:
[P2L~ (1-K~1d) + PLd (R1d + R2d ) + R1dR2d1 Efd +
. {p 3K3 (K K2 + K K + K K K K K K d afd 21d f1d a1d f2d a2d- afd- f1d 21d a2d
Mp) ,
I = - Ka1dR2d-Ka1dRfd) - pLd (Ka1dR2dRfd l] Id 1d
Mp) ,
(2.25)
(2.26)
33
(2.271
(A(p) •
where:
A( ) • = 313 (1 K2 K2 -K 2 + 2K K K ) P P d - f1d- f2d 21d f1d f2d 21d
(2.28)
After substitution of equations. (2.25)-(2.27) in equation (2.24)
we obtain:
-(KafdR1dR2d)] Efd/A(p)'
34
f
(2.29)
The general relation between $d' Efd and Id in terms 'of the opera
tors G(p) and Ld(p) is2
(2.30)
By comparing equations (2.29) and (2.30) it is evident that:
(2.31)
35
2K K K 2K K K +K2 +K2 +K2 K2 K2 - f2d a2d afd- a1d a2d 21d afd· a1d a2d a1d f2d ,
I
The direction of Id in equations (2.21)-(2.24) again follows
21 Kirschbaum • If this direction is reversed, the sign of the
last terms on the right hand side of equations (2.21)-(2.24)
become negative, and the sign of the terms multiplied by Id
become negative in equations (2.25)-(2.27). Consequently, the
(2.32)
positive signs of Id in equations (2.29) and (2.30) become
negative, so that equations (2.31) and (2.32) remain unchanged •
. 2.2.4d~axisreactances
The d-axis additional subtransient inductance ~d at time
. t = 0, is found from equation (2.32), by neglecting the rotor
resistances and letting p400. Hence the d-axis additional sub-
,;)0
transient inductance of a salient pole synchronous machine with
two damper circuits is:
L'" = L (00) = L d d d
where:
1- +2Kf1 dKafdK21 dKa2d
I-K~1d-K~2d-K~1d+2Kf1dKf2dK21d
, etc.
(2.331"
or when equation (2.33) is written in terms of the corresponding
inductances:
(2.34)
.""~ " 'if,;' ".
37
By putting p400 in equation -(2.32), the d-axis subtransient
inductance L" is obtained as: d
,
I K2 +K2 2K K K
afd a1d - f1d a1d afd L" = L Cl - -=~-"--"---'--"-~- ) d d
or in terms of the corresponding inductances
L" . d
(2.35)
(2.36)
The d-axis transient inductance Ld may be found by letting
p400 with L11d
= L22d = 00 in equation (2.32), i.e. by considering
no circuits on the d-axis other than the main field. Equation
(2.32) thus leads to:
---,'
(2.37)
The corresponding reactances are obtained when each inductance of
3 equations (2.33)-(2.37) is multiplied by a factor of 2' Thus
equations (2.35) or (2.36) and (2.37), lead to the familiar two
equations (18) and (20) of reference [4] •
. " ..,.
38
The d-axis additional subtransient inductance may also be
written in terms of the coefficient of dispersion. If all
four circuits (the armature, field and two damper circuits) are
considered, the total coefficient of dispersion is defined by:
The coefficient of dispersion for the case when three coils are
considered (the field and the two dampers) is:
hence L'· d
(2.38)"
(2.39)
(2.40)
It will be seen later that these operational inductances are help- :I:"~
ful in developing simple mathematical forms of the machine reac-
tances, that is in terms of the transient current decaying components
and their respective time constants. ;
39
•
These characteristic operators will also be helpful in
investigations of'various transient processes, l.e. the deter
mination of new relations for the resulting armature and field
currents following a sudden 3-phase short circuit.
I
40
CHAPTER 3
ANALYSIS OF A MODIFIED SALIENT-POLE
SYNCHRONOUS GENERATOR
When analysing the behaviour of the modified salient-pole
generator, it is assumed that it can be represented by the model
of Figure 2.1, and that the various reactances involved can be
obtained from the various expressions developed from this model.
3.1 Analysis of the q-axis Circuits
When deriving the voltage equations, it is convenient to
use the time rate-of-change of flux linkages, and this leads to
expressions for the self and mutual inductances of the machine
windings. The voltage in any circuit is given by the alge~
braic sum of the resistive drop and the rate-of-change of the
total flux linkages with the circuit.
The q-axis representation of phase a, together with the
first and the second q-axis damper windings is shown in Figure
2.1. The second damper winding accounts for the damping effect
of .the induced eddy currents in the body of the rotor.
Using terminology as defined in the nomenclature, the per
unit equations, as used by Kirschbaum21 (see Appendix 7.1) are:
E1 pL +r pL K PLla2q I q q q a1q q
0 = pKa1q p + _1_ PK21q 11q T1q
0 PKa2q PK21q 1 12q P + --T2q
(3.1 )
41
If Iq' 11q and 12q
transform of (3.1)
are zero at t = 0, then by taking the Laplace
~I q =
I ! -
and solving by Cramer's rule we obtain: /
E[82(1~K~1;) + 8 (-t- + /f+ T \ ] h. . . .1 1q ;2q 1q/2q
(3.2)
Since the measured q-axis transient armature current contains
three distinct decaying components. as is shown in Chapter 4,
the armature transient current (I ) can be written as: q
(3.3)
where I AO ' I A1 , I A2 , I A3 , TA1 , TA2 and TA3 are defined in the
-~.
nomenclature.
Taking the Laplace transform of equation (3.3) gives:
+
(3.4)
1
42
Dividing both the numerator and the denominator of equation
(3.2) by K = L [(1-K 21 )_K 2 _K 2
+ 2K 1 K 2 K21 ], and com-q 2 q a1q a2q a q a q q
paring the coefficients of the powers of S in equations (3.2)
and (3.4) we obtain:
From the numerator:
I El ( 2 ) K i 1-K21q
I Ell - (- +-) K T1q T2q
. E
and from the denominator: K 2 K 2
L (_1_ + _1_) _ L a1q _ L _a_2_q + (1 K2 )
(3.5)
IA1 IA2 I A3 TA2\3 - TA1 TA3 - TA1 TA2
(3.6)
(3.7)
T T T T rq - 21q q 1q 2q q 2q q 1q 1 1 1 --=--....::.::!--'---=c:c..,;-----.:.::!.------- = - + -- + -K TA1 TA2 TA3
(3.8)
(3.9)
(3.10)
,
43
Dividing equation (3.7) by-equation (3.10) yields:
3/' 1.1 q-axis synchronous and additional subtransient l"eactances
Dividing equation (3.9) by equation (3.10) yields:
(3.11)
(3.12)
and by dividing equation (3.6) by equation (3.7), and subtrac-
ting this .result from equation (3.12), we obtain the q-axis open
circuit time constant of the armature winding as:
T aq
where:
(l+AB+cB) = TA1 1 + A + C (3.13)
Since the self-inductance of the armature winding is r~T , " aq
therefore:
r!j T A1 Cl + AB + CD) L = q 1 + A + C'
\ (
(3.14)
From equations (2.17l and (3.14)
3 Cl + AB .. CD) 2" rq \1 X '" q 1 + A + C
(3.15)
44
The q-axis additional subtransient inductance L'o derived in q
Chapter 2 and given in equation (2.11) is:
K2 +K2 - 2K K K Lilt er L Cl -
a1q a2q a19 a2q 21q) q q 1 2 - K21q
Solving equation (3.5) for (1 -K2 +K 2 - 2K K K
a1q a2q a1q a2q 21 q), 2 1 - K21q
substituting the result in the above equation yields:
L" = q
':'q.TA1 (1 + A +,C)
l AC +-+-
B 0
From equations (2.17) and (3.14)
X''': q'
and
3.1.2 Derivation of the q-exis reactances ~om the q-axis open and short circuit time-constants
(3.16)
(3.17)
The value of Lq(P) is given in equation (2.10). The negative
reciprocalsof the roots of the numerator gives the short-circuit
time constants T~ and T~ .Similarly, the negative reciprocals' of the
roots of the denominator provides the open circuit time constants
T" and T"' • Hence, the q-axis operational inductance of equation qO qO
(2.10) can be written as:
45
Cl + P T") Cl + pT''') L (p) = q q
q (3.18a)
(1 + P T" ) Cl + pT"' I qO qO
3 multiplying Lq in equation (3.1Bal by 2' and rewriting as Laplace
transform yields:
Cl + 8 T") Cl + 8 T'OI X (8) ,, ____ q;;L.-___ -"q_
q Cl + 8 T" I Cl + 8 T'" ) qO qO
X q
If 8 tends to infinity, the limiting value of X (8) defined as q
(3.18b)
X·' is found from equation (3.1Bbl (similar procedure is applied q
in reference 17) as:
X'" ~ q
T" T"' q q
TN T'" qO qO
.X q
(3.191
which is the q-axis reactance effective during the first instants
of a transient condition.
In the absence of any iron damping, the high frequency value
of Xq(81, defined by the high frequency limit of Xq' follows from
(3.1Bb) as:
TOO X" ,,_..-9._ X
q Too q qO
(3.20)
Ouring the steady-state, when 8 = 0, the q-axis operational impe-
dance Xq(81 is approximately equal to the q-axis synchronous reac
tance X. From equations (3.191 and (3.201: q
X" ~ q
T'" qD
T"' q X'"
q
3.1.3 q-axis open and short circuit time constants
(3.21)
The open and short circuit time constants can-be found from the i
magnitudes and time constants of the decaying components. Thus the
transiJnt current of equation (3.4) is:
- 'lAD I (8)= - -q - - 8
(3.22)
or
I I (8)= AD q _ 8
(3.23) -,
'where constants A, B, C and 0 are given in section 3.1.1. The first
of equations (3.1) can also be written in terms of the operational
2 15 impedance' Xq(p) as: -.::::'"
El = I r + p W = I r + p I X (p) qq q qq qq (3.24 )
and if the Lap1ace- transform of this equation is obtained, the trans-
ient current is:
I (8) = EI8 [r + S X (s l] q - q q (3.25 )
By substituting for E in equation (3.11) we obtain:
or
, I
If X (s) q •
I constants
is replaced . N (s)
q by DesT
q
47
(3.26)
(3.27)
the two q-axis open circuit time
T" and T" may be obtained by equating equations qO qO
(3.23) and (3.26) to zero. Thus
... / +A(l+sT ) (1+sT/ ) SO+ST )(l+sT ) = 0
Ai / A3'~ Ai A2. ,
(3.28)
Where the negative reciprocals of the roots (U1q
and U2q ) of
equation (3.28) are the q-axis open circuit time constants, i.e.
- SimilarlY, the q-axis short circuit time constants Tn and T'" are q q
obtained by equating equations (3.22) and (3.27) to zero. Thus
(l-tS T A2) Cl +8 T 1\3 )+A8 (1 +s T Ai) Cl +s \3 )+ccit~S.T Ai) Cl +8\2) = 0 , '. , .
(3.30)
48
where the negative reciprocals of the roots (S1q and S2Q) of
equation (3 .. 30) give the q-axis short circuit time constants
i. e.
T" = 1 q - S1q
and Till = 1 q - S2q
The roots of the denominator of L (p) in equation (2.10) are q
-(T1q+T2q ) ± !(T1q+T2q )2- 4 T1qT2q (I-K~1q)
2 T1qT2q(1-K~1q)
(3.311
(3.32)
Using equation (3.29), the algebraic sum and product of the roots
of equation (3.32) are respectively:
T" + Tn • T + T-qo qo =
1q· 2q (3.33)
T" Till T1qT2q Cl-K2· )
qo qo 21q
and T" T"' = T1qT2q (l-K~1 q) (3.34) qo qo .--'
from equations (3.33) and (3.34)
(3.35)
Similarly, from the roots of the numerator of equation (2.10), and
by using equation (3.31), the algebraic sum and product of the
roots, are respectivelY:
/
49
T~ + T~' = [T1q(l-K~1q) + T2q(1-K~2q) ]
Tq Tq' T1qT2q (1-K~1q-K~2q-K~1q + 2 Ka1qKa2qK21q)
and
hence
3.1.4 q-axis magnetic coupling coefficients
The coupling coefficients K21q
equations (3.5), (3.8) and (3.10).
by equation (3.10) we obtain:
and K 2 may be found from a q
On dividing equation (3.5)
(3.36)
(3.37)
(3.38)
K~1q = [1 - (IA1TA2TA3 + IA2TA1IA3 + IA3TA1TA2)/T1qT2qIAO]
(3.39)
and on'dividing equation (3.8) by equation (3.10):
1 + -:-..:.,1".TaqT2q
3.2 Analysis of the d-axis Circuits
The d-axis representation of phase a, the field, and the
first and second d-axis damper windings is shown in Figure 2.3.
(3.40)
50
lile d-axis quantities with' the field circuit open may be obtained
simply by replabing subscript q by subscript d in equations (3.1 -
3.40) •
Under stationary conditions,. and with the field circuit
closed, the equations for the circuits of Figure 2.3 are (see
Appendix 7.1.2):
El PLd+rd PLd Ka1d PLd Ka2d PLd Kafd
0 P Ka1d '1
p K21d P Kf1d P + --
lId =
0 p Ka2d p K21d P + _1_ p Kf2d l2d
0 p Kafd p K
f1d P Kf2d P + _1_
lfd
By taking Laplace transforms of (3.41), and solving by Cramer's
·rule for Id we obtain:
2 1 + B (-
T1d
Id
I1d
I2d
Ifd
(3.41)
(3.42)
51
+ K2 K2 + K2 K2 a2d f1d 21d afd
2 K2 K21 d a1d
- Tfd - T 2d
3 1 1 1 K~1d K~2d + 8 [L (-- + -- + -- - --- - --
d T1d T2d Tfd T2d T1d
1 1 + r C- + -- + d T
1d T2d
1
Tfd
2Ka1dKa2i21d
Tfd
Since the measured d-axis armature transient current with the field
winding closed contains four distinct decaying component~ as will
" be shown in Chapter 4, the armature transient current Id can
be wri tten as:
(3.43)
,t 'f.r~If..' " ~;f ,
i
52
By taking I;/lplace ·transforms of.cequation (3.43) we obtain:
+ .8
(3.44)
+ K2 K2 + K2 K2 + K2 K2 2K K K K a1d f2d a2d f1d 21d afd - a1d 21d f2d afd
Dividing the numerator and the denominator of equation (3.42)
by K , and comparing the coefficients of powers of 8 with those s
in equation (3.44),we obtain:
from the numerator
IA1s
IA2s
IA3s
"IA4s
= --+--+--+--TA1s TA2s TA3s TA4s
(3.45)
I = A1SC_1_+_1_+_1_) +
TA1s TA2s TA3s TA4s
I A2s C_1 _ + _1 _ + _1 _) +
TA2s TA1s TA3s TA4s
54
and from the denominator
l A3S 1 1 1 -- (-- + -- + -) + TA3s TA1s TA2s TA4s
-~
1 +.:--) TA3s
(3.46)
(3.47)
(3.48)
1 K s
55
2Ka1dKa2dK21d+
Tfd
2K f1 dKafdKa1 d T
2d
1 1 1 1 --+--+--+--TA1s TA2s TA3s TA4s
(3.49)
1 + 1 + 1 TA1sTA2s TA2sTA3s TA3sTA4s
.~
+ 1 1 1 T T +T T +T T
A1s A3s A1s A4s A2s A4s
'(3.50)
56
(3.51 )
(3.52)
Dividing equation (3.48) by equation (3.52), we obtain the steady
state current l AO :
Hence - = E (3.53)
which is the same as given by equation (3.11). Dividing equation
(3.51) by equation (3.52) we obtain:
(3.54)
3.2.1 d-axis synchronous and additional subtransient reactances
Dividing equation (3.47) by equation (3.52), and by using
equation (3.54) we obtain:
TA1s
(1 + A B + C D + E F ) ss ss ss
Ta d = --"-'':'::'''--;--'::'-;;--'''':--r:-=-:-'':=---':'''':::'''' l+A +C +E s s s
(3.55)
·'I:'.'!~' .",
,
57
where:
hence the d-axis synchronous reactance is:
rd TA1s (1 + A B + C 0 + E F ) . ss ss ss Ld = -=.~~--::--:.."..:'--~;;...::---=;....:;.l+A +C +E
5 5 5
From equations (2.17) and (3.56) (replacing subscript q by
subscript d in the former):
+ A B + C 0 + E F ) ss Ss Ss
1 +A +C +E 5 5 5
Solving equation (3.45) and substituting from equation (2.33)
yields:
rd TA1s (1 + A + C + E ).
L'" = 5 5 5
d
Since Xd " ~ L 2 d
hence X·'" ~ d 2
A C E (1 +...2- +...2- 5
B 0 + r)
s 5 5
rd TA1s (1 + As + Cs + Es)
ACE 5 s s
(1 + B + 0 + r) s 5 s
-~.
(3.56 )
(3.57)
(3.58 )
(3.59)
58
,.','
3.2.2' Derivation cif the od-aXis 'rea'ctances from the d-axis 'open and short circuit time constants
Bya procedure similar to that of Section (3.1.2). the d
axis operational inductance Ld(P) given by equation (2.32) in
Chapter 2 is:
3 multiplying Ld (p) by 2"' and rewri,ting in Laplace" transform
notation:
(3.60 )
(3.61)
If 0$-+00, the limiting value of Xd(S) defined as X;;' and found from
equation (3.61) is:
T' Tn T'" X., _ d d d d -T' T" Tt"
dO dO dO (3.62)
which is the d-axis reactance effective during the first instants
" of a transient condition. .-'
In the absence of any iron damping, the high-frequency limit
of Xd(o$) defined by the high-frequency value of X;;. follows from
equation (3.61) as:
(3.63 )
59
In the absence of the two damper circuits, i.e by considering
only the main field on the d-axis, the value of Xd (8) defined
by Xd and found from equation (3.61) is:
T' d Xd' "-T' Xd
dO
3.2.3 d-axis open and short circuit time constants
The transient current of equation (3.44) can be rewritten:
(3.64)
.. ,._--,---------------
(3.65)
or
I Id(S) = :0 [(l+8T A2s)(1 +8TA3s ) (1 +8T A4s)+ \ (1 +8T A1 s)(1 +8 T A3s) (1 +8T A4s)
: Cs (1 +ST A1 ,,) (1 +8T A2s) (1 +8T I\4s )+E~ (1 +8 TA1 s) (1 +8T A2S) (1 +8T A3s l]
(1+As +Cs +Es) (1 +STA1 s) (1+8TA2s ) (1 +8T A3s) (1 +8T Ms)
(3.66 )
60
where constants A , B , C ,',Os' E and F are given in section s s s. s s
(3.2.1).
The d-axis open circuit time constants are obtained by
equating equations (3.66) and (3.26) [with subscript q in the Nd (8)
latter replaced by subscript d, and then Xd (8) by 0 (8) ) to d
zero. Thus:
Cl+8TA2s ) (1 +8TA3s ) (1 +8T A4s) + As (1 +8T A1s) (1 +BTA3s ) (1+BT A4s) ,
where the negative reciprocal of the roots (a1d
, a2d
,and a3d
)
(3.67)
of equation (3.67) are the d-axis open circuit time constants, i.e.
and
\io = 1 a
1d
T" .= 1 dO a2d
1 1''' d = ---
-~
(3.68)
Similarly, the d-axis short circuit time constants are obtained by
equation equations (3.65) and (3.27) (with subscript q in the latter Nd (B)
replaced by subscript d, and then Xd(s) by 0 (B) ) to zero. Thus: d
61
(3.69)
where the negative reciprocal of the roots ·(S1d' S2d and 83d )
of equation (3.69) give the d-axis short circuit time constants,
i.e.
T' 1 d = - 81d
T" . 1 d = - 8
2d (3.70)
Tt!. = 1 d - 8
3d
--
Using equation (3.68), the algebraic sum and product of the roots
of the denominator in equation (2.32) (with the numerator and
denominator being factorised) are respectively:
T' + TH + 1"' = T + T + T dO dO dO 1 d 2d fd ' (3.71 )
62
(3.72b) ,
or
(3.72c)
Replacing \ subscript q for subscript d in equation (3.34) and
substituting into equation (3.72c) yields:
(3.73)
Similarly, from the roots of the numerator in equation (2.32),
and by using equation (3.70), the algebraic sum of the roots
is:
(3.74)
Again replacing subscript q by subscript d in equation (3.38) and
subtracting from equation (3.74J yields:
(3.75)
63
3.2.4 d-axis magnetic coupling coefficients
Dividing equation (3.50) by equation (3.52) yields
(3.76)
Dividing equation (3.46) by equation (3.52) yields
(3.77)
64
It has been shown that the parameters of the modified
machine can be found in simple mathematical forms in terms of
the decaying components and their respective time constants;
these simple relations are used in the next chapter.
65
CHAPTER 4
METHODS OF MEASUREMENT OF PARAMETERS
The methods of measurement used for testing the 5 kVA
experimental salient-pole machine can be divided into two
groups. The first uses standstill measurements and the second
measurements with the machine rotating. The standstill tests
employed an indicial response method, and it will be shown that
this produces an accurate predicti~n of the machine's para
meters, based on the theory developed in Chapter 3. The other
conventional methods of measurement were used to validate the
values obtained from the indicial response test (e.g. open and
short circuit tests, slip test, Oalton arid Cameron method, maxi
mum lagging current method and reluctance motor method).
4.1 Indicial Response Test
The indicia I-response test, as illustrated by Figure (4.1)
required the application of a step function of voltage to one
of the armature phases, and the measurement of the corresponding
transient current in that phase, the remaining two armature
phases being open circuited.
For each test the variation of the armature current is
obtained via an ultra-violet recorder, and an analysis of the
curves obtained yields the sum of three and four exponential
functions for the q and d axes respectively (with the field cir
cuit closed for the latter).
66
From the magnitudes and time constants of the individual
components of the transient current. the reactances included
in the modified Park's equation may be determined.
A program Was run on a 1900 re computer. using the magni-
tudes and time constants of the individual components to con-
firm the measured values of the transient current and a compari-
son between the two sets of results for different currents are ,
given in Appendix 7.3. The q- and a-axis positions of the rotor
14 were obtained by the standard procedure of connecting a volt-
meter to the terminals of the rotor. with an a.c. supply to the
a-phase of the armature. The q-axis'positio~ corresponds to minimum (nearll
zero) voltage induced in the field. and the d-axis position is
obtained when the voltage becomes a maximum. The indicial-
response test requires a constant voltage source. This is
provided by a heavy-duty battery with very low internal resis-
tance., (a l2V. 91 AH Car battery was used).
The applied d.c. voltage, the transient current in the
armature phase winding and the induced voltage in the field
winding. were recorded directly on a multi channel ultra-violet
recorder.
High speed recordings (up to 2 m/s) were made to expand the
time scale of the readings and thereby to increase the accuracy
of the results obtained.
The total resistance pf the circuit shown in Figure 4.1,
includ:l,ng that of the ammeter, the shunt and the connecting leads
was minimised,as any external resistance decreases the time con-
stants and tends to reduce the accuracy with which the parameters
are measured.
01
4.2 Determination of the q-axis Synchronous. Subtransient and Additional Subtransient Reactances by the Indicial Response Method
Indicial-response tests were made on a 3-phase salient-
pole synchronous generator, this machine has armature winding
on the rotor, and the power supply is connected via slip rings
and brushes. The field and damper windings are on the stator.
I The .name plate data of the star connected machine is:
I Output kVA = 5
Line voltage = 24DV
Current = l2A
Frequency = 50 Hz
Speed = 1500 r.p.m.
Number of Poles = 4
As expla~ned in Section 4.1, the rotor of the machine was aligned
with the q-axis, and a step of d.c. voltage was applied to the
a-phase armature winding. Oscillograms of the applied d.c. vol-
tage, the transient current in the armature phase winding and
the induced voltage in the field winding were recorded, as shown
in Figure 4.2. The small induced field voltage shown is due to
the difficulty in aligning an exact q-axis position, but since the
field winding has a large number of turns, this small induced
voltage can be reasonably assumed to be negligible. Figure 4.3
shows the induced voltage in the other two phase windings.
The exact value of the steady-state current in the arma-
ture a-phase winding was measured by the a·mmeter (A), a few
seconds after closure of the switch (S).
To prevent the armature winding from over-heating, the
current was only allowed to flow for the shortest period needed
68
to obtain the measurements required, this being especially
necessary when the current was near the rated value.
The de~aying components were obtained by plotting the
difference of the transient current and the steady-state value
on semilog paper. Consideration of the results of the test,
show that there are three distinct decaying components l A1 .,
lA2 and lA~ in the transient current, as shown in Figure 4.4,
implying clearly that two damper circuits are required on each
axis, as described in Section 2.2. The time constants TA1
, TA2
and TA~ were found by multiplying the magnitude of each decay
ing component by lie. The magnitudes and time constants of the
decaying components obtained with a test current of 9 .72A are:
lAl = 2.0A; lA2 = 4.2A; and l A3 = 3.52A
TAl = 0.026s; TA2 = 0.0045s; TA3
= 0.0016s.
i.e. A = 210x 10- 2 C = 176 X 10- 2
B = 17.3 x 10- 2 o = 6.153 x 10~z
where: A = lA/lA11 B = TA21T~1; C = lA3IIA1 and 0 = \lTA1 •
From equation (3.15), the q-axis synchronous reactance
is:
or X q
'- " .J
TA1 (1 + AB +/Zuf X '" 1.5 r
q q 1 + A + C:
'" 1. 5 (6l'.728xlO- 2 )xO .026x314(l +2rOX.LU -2 x17 .3xlO- 2 +176 ;'xlO- 2 x6 .153xll
11.547(1+210xlO- 2 +176xlO- 2 )
~ 19.825 X 10- 2 p.u.
69
where:
= 2nf = 314 radians/second: rated impedance/phase
I' = q-axis armature circuit resistance q
V h = -E.... =
I
= armature resistance/phase + brush resistance (which
is a function of current) + connection lead resistance
+ ammeter internal resistance
+ shunt resistance + internal battery resistance ! . ",72' • ,,-', The change in resistance of the brushes during the transient
is assumed negligible compared with the total resistance of
the armature circuit. Substituting the data into equation
(3.28) yields:
The roots of this equation are:
Cl1q " -46.888
-_ ....
and Cl2q ~ -344.951
From equation (3.29)
1 1 10- 2 sec T" = = = 2.132 x qO Cl1q
46.888
and r" 1 1 2.899 X 10- 3 sec = = = qO Cl2q 344.951
11.547S"l
70
where: T" = q-axis subtrabsient open circuit time constant qO
and T" = q-axis additional subtransient open circuit time qO
/ constant.
Similarly by substituting the data into equation (3.30) yields:
(1.619 X 10- 6) + 8(1.010 X 10- 3 ) + 7.653 X 10- 2 = 0
The roots of this equation are:
(31q = - 90.533
and (32q = - 464.716
From equation (3.31)
, T" ·1
q = - (31q = 1
= 1.105 X 10- 2 sec 90.533
and T'" = q
1 """4"'"64""."=7'"'"1-::"6 = 2.151 X 10- 3 seo
-~
where: T" = q-axis subtransient short circuit time constant q
and T'" = q-axis additional subtransient short circuit time q
constant.
71
The q-axis subtransient reactance obtained from equation
(3.20) is:
i.8.
T" X"= r.X q. qO q
i
I" 1.105 X 10- 2
2.132 X 10-2 x 19.825 x 10-2
X" = 10.275 X 10- 2 p.u. q
The q-axis additional subtransient reactance is obtained from
equation (3.17):
X'· ~ 1.5 q
rq TA1 (1 + A + C)
A C 1+-+-
B 0
X'" q
~ 7.621 X 10- 2 p.u.
Th~q-axis additional subtransient reactance can also be
obtained from equation (3.19):
T" T'" X'" =. q q X
q T" T"' q qO qO
(1.105xlO- 2) (2.151xlO- 3) x 10-2 = (2.132xlO- 2)(2.899xl0-') x 19.825
= 7.623 X 10-2 p.u.
72
The result obtained from this equation is useful, as it enables
the accuracy of the results obtained from equation (3.17) to
be determined, and in consequence the accuracy of the subtrans-
ient reactance obtained from equation (3.20). The procedure
was repeated for a
I ponding components
as shbLn in Figure !
low value of test current of 2.62A~ The corres-
of the current and their time constants are
4.5. A comparison of the open and short
circuit time constants with the corresponding reactances at
the two different currents are given below in Table 4.1.
Current T" T'" T" Till X (p.u) X"(p.u) X·"(P.u) (Amps) .qo qo q q
(Sec) (Sec) (Sec) (Sec) q q q
9.72 2.132x 2.899x 1.105x 2.151x 19.825 10.275x 7.621x 10- 2 10- 3 10- 2 10-3 10- 2 10- 2 10- 2
2.62 1.826x 2.863x 1.077x 1.543x 24.2x 14 .• 273x 7.696 10- 2 10- 3 10- 2 10- 3 . 10- 2 10- 2 10- 2
TABLE 4.1
/
73
4.3 d-axis Indicial Response Test; Field Circuit Open
This,test was performed with the a-phase of the armature
aligned with the d-axis. using the method described in
Section 4.1. and the test procedure of Section 4.2.
Figure 4.6 shows typicai oscillograms of the applied
voltage. the transient current in the a-phase armature winding
and the induced voltage in the field winding. An analysis of
the transient current shows three decaying components as indi-
cated in Figure 4.7. The magnitudes and time constants of the
decaying components obtained on resolution of the transient
current of 11.09A are:
lAl = 3.31AI
TAl = 0.0627sl
i.e. A = 72.507 X 10- 2
B = 11.164 X 10- 2
C = 162.537 X 10- 2
and 0 = 2.870' X 10- 2
lA2 = 2.4AI and
TA2 = 0.007s and
IA3 = 5.38A
TA3
= 0.0018s.
The d-axis synchronous and additional subtransient reactances.
open and short circuit time constants. and subtransient reactance
are simply obtained by replacing subscrMpt q by subscript d in
equations (3.15) and (3.17). (3.28) and (3.29). (3.30) and (3.3lJ.
and (3.20) respectively. On substituting the above data into
equations (3.15) and (3.17) yields:
" 1.5xO.541x6.27xlO- 2x314(l+72.507x11.164x1o- 4
+162.537x2.87ox1o- 4
11.547x(l+72.507x1o- 2+162.537xlo- 2 )
",46.566 x 10- 2 p.u.
and
and
74
(l+A+C) ,,1.5xO.541x6.27xlO- 2x314(1+72.507xlO- 2
X;;;" 1.5 rd TAl A C 1+-+- +162.537xlO- 2 )
B 0
~ 7.228 X 10- 2 p.u.
11. 547 (lJ2.507xlO- 2
1l.164xlO- 2
Xd = d-axis synchronous reactance
Xd' = d-axis additional subtransient reactance
162.537xlO- 2 + )
2.870xlO- 2
On substituting the data into equation (3.28), and by using
equation (3.29) yields:
TdO = 4.504 x 10-2 sec
and TdO = 5.352 x 10- 3 sec.
where: Too = d-axis open-circuit dO
and TI" = d-axis open-circuit
dO
time constant.
_. Similarly from equations (3.30)
TOO = 1. 226 X 10- 2 sec d
and Td' = 3.052 x 10- 3 sec.
and
su btra ns ient time constant
additional subtransient
-~
(3.31)
where: TOO = d-axis short-circuit subtransient time constant d
and Td' = d-axis short-circuit additional subtransient
time constant.
75
The d-axis subtransient reactance (Xd) is obtained from equation
(3.20)
T" X• - d X d-P"d
d
1.226 X 10- 2 . " x 46.566 X 10-2
4.504 X 10- 2
The procedure was repeated for a test current of 6.8 A. the
corresponding components of this current and their time con-
stants are shown in Figure 4.8. and a comparison between the
two values of d-axis time constants and reactances are given
in Table (4.2).
Current T" T'" T" T'" Xd(pu) X"(pu) X'" (pu) (Amps) dO dO d d
(sec) (sec) (Sec) (sec) d d
11.09 4.504x 5.352x 1. 226x 3.052x 46.566x 12.675x 7.228x
10- 2 10- 3 10- 2 10-3 10-2 10- 2 10- 2
6.8 5.381x 4.586x 1.436x 2.719x 47.506x 12.677x 7.519x 10- 2 10- 3 10- 2 10- 3 10- 2 10-2 10-2
TABLE 4.2
76
4.4 d-axis Indicial Response Testl Field Circuit Closed
The ~est procedure of Section (4.3) was repeated with
the field circuit closed. and Figure 4.9 shows typical oscillo-
grams of the applied d.c. voltage and the transient current
through the phase winding. It was not passible to obtain an
accurate record of the current in the short-circuited field
winding, since inserting the high resistance shunt needed had
an appreciable effect an the time constants involved.
The analysis of the transient armature current given in
Section (3.2) showed the four decaying components indicated in
Figures (4.10) and (4.11), and the magnitudes and time constants
of these for a final current of ll.09A are:
i.e.
I A1s = O.86AI
TA1s " 0.20s1
I A2s a 1.2AI lA3s ~ 4.0AI and I A4s = 5.03A
TA2S a 0.0215s1TA3s = 0.0043s & TA4s = 0.0018s.
A =.139.534 X 10-2 E = 584.883 X 10- 2 s s
B = 10.750 X 10- 2 F = 0.9 X 10- 2 s s
GS
~ 465.i1S X 10-2
0 = 2.150 X 10- 2 s
The d-axis synchronous reactance (Xd) is obtained from equation
(3.57):
(1 + AB' CO. E F ) ss ss ss Xd 0: 1. 5 l'd T A1 s --"'::""'::"--=-""--':;"';::""
l'A 'C 'E s s s
1. 5xO. 54lxO. 2x3l4 (1' 139. 534xlO. 750xlO-~ +465 .1l6x2 .150xlO-~ +
584.883xO.9xlO-~) o: ________________ ~~~~~~~~ ___________ __
11.547(1+139.534x10- 2 .+465.116xlO- 2 +584.883xlO- 2 )
~ 44.583 X 10- 2 p.u.
77
The d-axis additional subtransient reactance is obtained from
equation (3.59):
1.5xO.541xO.2x314 (l+139.534xl0- 2 +465.116xl0- 2 +584.883xl0- 2 )
11.547(1 + 139.534 + 10.750
465.116 + 584.883) 2.150 0.9
'" 6.466 X 10- 2 p.u.
Substituting the data into equation (3.67) yields:
Using -8 Newton and Raphson method, the approximate roots of this
equation are found as:
.-".'
Q1d = - 5.4288
= -49.751 --~
= -326.051
From equation (3.68)
T~1O 1 1 ID-I = - -- = - = 1.842 X sec
Q1d 5.4288
T" = 1 1 2.01 10- 2 sec - er- = = x dO 2d 49.751
and T'" = dO
78
where: TdO = d-axis open circuit transient time constant
.substi tuting the data into equation (3.69) yields:
The roots of this equation are:
fl 1d = -17.513
fl 2d = -86.206
and fl3d = - 402.068
From equation (3.70)
1 1 0.571 10- 1 T' =
- fl1 d = = x
d 17.513
1 1 1.160 10- 2 T" = - fl2d
= = x d 86.208
1 1 2.487 10- 3 and T"'= - fl3d
= = x d 402.068
where:, T' = d-axis short circuit transient time constant. d
:'I'·.'~'
'" "
79
The d-axis subtransient reactance [Xd) is obtained from equation
(3.63)
~ 5.71 x 1.16 x 10-1
x 44.583 x'10-2 1.842 x 2.01
~ 7.975 X 10- 2 p.u.
and the d-axis transient reactance (Xd) is obtained from equation
(3.64)
5.71 X 10- 1 x 10-2 ~ 1.842 x 44.583
~ 13.82 X 10-2 p.u.
The values of the open and short circuit time constants,
and the corresponding reactances obtained from Figures (4.10)
and (4.11) are shown in Table 4.3.
It is necessary to plot the curve accurately to obtain accurate
values of the various transient components and their respective
time constants from the ultra-violet readings.
80
Current T,jO T" TIl' T' T" Tit' Xd X' X" X'"
(A) dO dO d d d .d d d (5) (5) (5) (5) (5) (5) (pu) (pu) (pu) (pu
11.09 1.842 2.01 3.067 0.571 1.160 2.487 44.583 13.82 7.975 6.464 x lO-1 x10-2 xlO- 3 xlO- 1 xlO- 2 xlO- "lO-2 xlO- 2 xlO- 2 xlO- 2 .
6.8 18.514 1.67 2.951 5.54 1.lO 2.33 46.17E 13.82 9.1 7.207 x lO-2 xlO- 2 xlO- 3 xlO- 2 xlO- 2 xlO- . xlO- xlO-2 xlO- 2 xlO- 2
TABLE 4.3
There is a small discrepancy between values obtained for TdO '
TdO and Xd from tests with the field winding open and short
circuited. but this unfortunately still remains unexplained.
4.5 Determination of the Field Time Constant from the d-axis Field Transient Current with Armature Windings Open
As with the previous test. the present test was performed with
the a-phase winding aligned with the d-axis. and with the armature
phase winding open. A low d.c. voltage was applied to the field
winding and oscillograms of this and the corresponding field trans-
ient current were recorded as shown in Figure 4.12. The difference
betweer. the steady state and the transient currents was resolved by
plotting on semilog graph paper.
The test indicated three distinct decaying components. as
shown in Figure 4.13. implying that for each axis. two damper cir-
cuits need to be considered.
This test is similar to that described in Section 4.3. with
the supply voltage applied to the field rather than the armature
phase winding. Formulae similar to those of equations (3.12) and
81
(3.13) will apply (with suffix a replaced by suffix f and sub-
script q replaced by subscript d).
Thus, for the d-axis circuits comprising the f, 10 and 20
windings only:
(4.1)
and (4.2)
where:
IF2 A =
F IF1 and
Applying equation (4.2), the field time constant (Tf ) obtained
from Figure 4.15 is
Tf
= 0.19 seconds
4.6 Conventional Methods of Measurement of Parameters
4.6.1 Determination of the d-axis synchronous reactance from the no-load saturation and sustained 3-phase short circuit characteristics
The open-circuit characteristic of the test machine was obtained
from the variation of the, no load voltage with the excitation current
_ at rated speed. Due to a residual voltage of 2.8V (Figure 4.14), the
characteristic does not pass through the origin, and a correction
was therefore introduced by extending the straight portion of the
no-load curve to the point of intersection with the abscissa (point
C in Figure 4.14), the length of the abscissa cut by the projected
curve (DC) representing the correction value. Following this tech-
nique, a value df D.D2A was added to all measured values of the
excitation current, and the corrected no-load saturated curve is
82
shown in Figure 4.15.
The short-circuit characteristic was obtained by driving
the test machine at rated speed, and recording the armature
current as the excitation current was varied. The resulting
straight-line curve is shown in Figure 4.15.
3 14 The d-axis synchronous reactance (Xd
) is given' from
this test by:
= oa = 0.5 = Xd Ob 0.967
0.517 p. u. (according to AIEE definition 14)
or
= £ = 138.56 Xd d 23.2
= S.972fl
= 0.517 p.u. (according to reference 3).
83
4.6.2' Determination of d- andq-axi's addition'al subtransient
reactances from Dalton arid Camercin method 11
The test is performed with the rotor stationary in any arbi-
trary position, and the field winding short-circuited through an
ammeter. A low single-phase voltage (D-24V) of rated frequency
is applied to two armature terminals a and b, with the third
terminal c left open (see Figure 4.16). Following measurement of
the armature voltage and current, the test voltage is transferred
to terminals band c (with terminal a open), and thereafter to term-
inals c and a (with terminal b open). The armature voltage and
I current are measured and recorded for both of the re-connections.
The results of the three measurements gives three stator
voltage/current ratios, that is three additional subtransient
reactances between the stator terminals, corresponding to three
different positions of the rotor. The voltage/current ratios
11 14 may be designated' as A , Band C respectively. It is Ss ss ss '
frequently considered that the angular variation of the reactance
of a single-phase rotor winding comprises a constant term plus a
sinusoidal function varying at twice the frequency of the rotor
d T tt i · by11,14 isplacement. he cons tan erm s gl.ven
K = (A + B + C )/3 ss ss ss ss
while the amplitude of the sinusoidal component is:
(C - A )2 ss Ss
3
(4.3)
(4.4)
84
The ohmic values of the per phase d- and q-axis additional sub-
transient reactances are:
X'" = (K - M )12 d ss ss
(4.5)
and X"' = (K + M )12 (4.6) q ss ss
and the p.u. values are obtained by dividing these values by the
base impedance/phase.
At a given voltage of 6.63V and a current of 2.9A, the
additional subtransient reactances were calculated as:
A = 6.63/2.9 = 2.286n ss
B = 6.5/3 = 2.l67n ss
and C = 6.712.94 = 2 •. 283n I ss
so that from equation (4.3) K = 2.245n ss
and from equation (4.4) M = 0.078n ss
Hence: X'" = 1. 083n = 1.083/11. 547 p.u. d
• • X"' = d
9.39 X 10- 2 p.u.
and X'" = q 1.16ln = 0.100 p.u.
--'
When the test was repeated with 12.lV and 6.02A, the constants
obtained were:
A = 12.1/6.02 = 2.0l0n Ss
and Cs~= 12.15/6.08 1.998n
B = 10.95/5.97 = 1.834n Ss
85
from which K = 1.947>2 and M = O.114n ss . ss
hence: X· ' = 7.95 X 10-2 p.u. d
and Xf" q = 8.93 X 10- 2 p.u.
The values of the d- and q-axis additional subtransient reactances
obtained for different currents are given in Table 4.4.
Armature.current (A) 2.9 6.02 8.9 12.05 -
Xd' (p. u) 9.39 x 10- 2 7.95 X 10- 2 7.5 X 10- 2 7.11 X
X· , (p.u) 10 x 10-2 8.93 x q
10- 2 8.64 X 10- 2 8.41 X
TABLE 4.4
4.6.3 Determination of d- ·and q-axis synchronous reactance from a slip test
A slip test was performed by driving the test generator by a
d.c. machine and applying to the armature (with the field open cir-
cuit) 3-phase positive sequence voltages of rated frequency. The
armature voltage was sufficiently low for the ma·chine to be working
14 below the knee of the open-circuit characteristic •
The test was made at a slip sufficiently small for the induced
currents in the damper winding and in the rotor iron not to become
excessive.
Figure (4.17) shows oscillograms of the armature line voltage
and current. together with the e.m.f. induced in the field winding
at a slip of 0.34%. Although the armature was energised by symme-
trical 3-phase voltages from a 3-phase transformer. the oscillogram
I 10- 2
10- 2
86
of field induced vol tags exhibi ts a series of hannonics, which
make it difficult to determine the actual fundamental component
of the induced e.m.f. Further, as Figure (4.17) shows, the
maximum and,minimum annature voltages obtained do not coincide
with the minimum and maximum armature currents, and in calcu-
lating the reactances the minimum and maximum armature currents
were divided into the voltages at the corresponding points on
the oscillogram of the terminal voltage.
However, as shown in Figure (4.17) the maximum and minimum
annature voltages differ only by a small amount, and the above
considerations do not much affect the actual results. From
Figure (4.17) the corresponding reactances are calculated as:
I = 0.86 x 0.1/0.015 = 5.733A (peak min
V = 2.2 x 25 = 55V (peak max
I max = 2.13 x 0.1/0.015 = 14.2A (peak
and Vmin = 2.12 x 25 = 53V (peak
, where 0.015Q is the shunt resistance in the
According
, X d
to
V
the s ta ndard 14 test ,
55 55 5 • 7 33 Q ~-;:-5"""'. 7;-;:3"'3'::'x=---"2"""0 p. u •
to peak)
to peak)
to peak)
to peak)
armature circuit.
= 0.479 p.u.
X q = ( min)
I max
53 53 = 14.2 Q = -::-147""':.2,,=X=-....,,2::::'0 p.u. = 0.187 p.u.
where:
20Q = base impedance in ohms.
X q
14 can also be found from:
87
(4.7J
where Vmin and Vmax ' Imin ~nd Imax are the minimum and maximum
values of voltage and current obtained from Fig. (4.17), and Xd
14 is found from Figure (4.15).
Henc'e:
x = 0.517 (53.25) (5.733) = 0.198 p.u •. q 56.25 14.2
It will be seen later that these results are very close to those
obtained from indicial response methods.
4.6.4 Determination'ofq-axis synchronous reactance from maximum lagging current method
14 The test was performed with the machine operating as a
synchronous motor with an initial applied test voltage of 107V/
phase and a corresponding excitation of 0.B02A. (It is recommended
that the test should be carried out at a voltage not greater than
about 75% of rated voltage, with approximately normal no load
excitation). The excitation current was gradually reduced to
zero, following which its polarity was reversed and then gradually
increased until instability occurred (i.e. the machine began to
pole slip). The supply voltage and current and the excitation
current were recorded up to the maximum stable negative excita-
tion, as given in Table 4.5a. It can be seen that the line voltage
and current corresponding to the maximum stable negative excitation
of O.B65A are 161V and 39.9A respectively. The q-axis synchronous
reactance is14
(4.8)
Line Voltage 185.3 184.4 183.5 181.8 180.4 179.7 178 177 175.6 173.3 172.5 170.8 169.7 168.7 (Volts)
Input Power 1.14 1.75 3.44 5.48 7.55 9.57 11.74 14 16.95 19.32 21.63 24.22 26.35 28 (Watts)
Exci tation 0.802 0.7 0.6 0.5 0.4 0.3 0.2 0.094 0 -0.1 -0.2 -0.3 -0.4 -0.46 Current (Amperes)
Line Voltage 168.3 168 167.9 167.1 166.6 166.4 165.9 165.4 164.1 163.1 -*161 (Volts) -
Input Power 28.6 28.75 29.4 3D 3D .76 31. 5 32.55 33.6 34.65 36.75 "39.9 (Watts) -- TABLE 4. Sa
Excitation -0.48 -0.494 -0.514 -0.538 -0.565 -0.6 -0.642 -0.675 -0.715 -0.715 *-0.865 Current (Amperes)
Line Voltage 165.5 153.4 151.7 149.9 148.6 146.8 146.1 145.4 144.6 144 142 141.1 139.4 '138.5 (Volts)
Input Power 2 14.07 16.59 19.2 21. 5 23 23.9 24.57 25.46 26.25 28.9 30.45 31.6 *33.6 (~latt9 ) --Excitation 0.7 0 -0.1 -0.2 -0.3 -0.36 -0.4 -0.44 -0.454 -0.488 -0.58 -0.63 -0.655 '-0.86 Current (Amperes)
TABLE 4.5b
'Point at which the machine begins to pole slip
where
Hence:
89
v = terminal voltage, p.u.
I1= armature current at stability limit, p.u.
x- = q
1611240 39.9/12 = 0.202 p.u.
The test was repeated at an initial line voltage of 165.5V and an
initial excitation of D.7A. As shown in Table 4.5b, the supply
voltage and current corresponding to the maximum negative stable
excitation of D.66A are 138.5V and. 33.6A, so that:
138.51240 33.6/12 = 0.206 p.u.
It is clear that the tests described provide results which agree
very closely.
4.6.5 -Determination -of -q~axis -synchronous -react-ance from reluctance-motor method
This test was performed with the machine operating on no-load
27 and as a reluctance motor ,with the terminal voltage being
gradually reduced until the machine fell out of step. The no-.--
load input power and the terminal voltage up to the moment at which
- loss of synchronism occurred were measured and recorded, as listed
in Table 4.6a.
From the final stable readings, -the q-axis synchronous reac-
27 tance can be calculated as:
x q =
V2 (Xd
+ 2r1
V2 + 2pX d
(4.91
Line Voltage 240 235.7 225.1 207.8 (Volts)
Input Power 402 1386 1128 888 (Watts)
Excitation Current 0.97·· - - -(Amperes)
Line Voltage 69.2 *66.7 (Volts)
Input Power 696 *702 (Watts) -
Excitation Current - -(Amperes)
Line Voltage 197.4 188.8 173.2 155.9 (Volts)
Input Power 300 876 726 702 (Watts)
Exci tation Current 0.83 - - -(Amperes)
* Point at Which loss of synchronism occurred.
190.5 173.2 155.9 138.5 121.3
816 744 699 687 660
- - - - -
TABLE 4.6a
138.6 121.3 103.9 86.6 76.2
. 690 636 642 633 648
- - - - -
TABLE 4.6b
103.9 86.6
666 636
- -
72.75 69.3
660 672
- -
76.2
660
-
67;5
687
-
72.7
666
-
*65.8 --
*714 -
-
o (
where:
v =
Xd =
p =
r =
hence:
i. e.
91
terminal .line voltage, V
d-axis synchronous reactance found from Figure 4.17
by well known methods3 •14 , n
total no-load input power. W
resistance per phase. n
(66.68)2 (5.97 + 2 x 0.5)
(66.68)2 +(2 x 702 x 5.97)
x = q
2.41 11.547
:; 0.209 p.u.
2.4lDn
The test was repeated at a voltage below the normal level. with
the results shown in Table 4.6b. From the readings at the stable
limit of the no-load input power. the q-axis synchronous reac-
tance is:
(65.81)2 (6.97) X = = 2.348n = 2.348/11.547 p.u.
q (65.81)2 +(2 x 714 x 5.97)
Le. X = 0.203 p.u. q
It is evident from these two sets of measurements, that pull-out
occurred at a terminal voltage of approximately 0.25 of rated
voltage. The magnetic circuit of the machine was therefore un-
saturated. and X obtained by this method is an unsaturated value. q
,
92
4.7 Effect of the Brushes on the Parameter Measurement
The tests of Sections 4.2, 4.3, 4.4 and 4.6.2 were repeated
with a direct connection to the armature winding, i.e. to the
slip rings, so that current did not flow through the brushes,
these tests are classified as follows.
4.7.1 Measurements of q-axis synchronous, subtransient and additional subtransient reactances (without brushes)
The tests were carried out using the procedure of Section
4.2. Figure 4.18 shows typical oscillograms of the applied
voltage, the transient current through the a-phase of the arma-
ture winding, and the induced field voltage for a test current
of 9.22 A. Figures 4.19. and 4.20 present analyses of transient
armature currents, which have a steady-state value of 9.22 A and
4.71 A respectively, and the corresponding time constants and
reactances are shown in Table 4.7.
Current TOO T'" TOO TnI X X" X'" (Amps) qO qO q q q q q
(Sec) (Sec) (Seol (Sec) (pu) (pu) (pu)
9.22 2.63x 6.57x 1.574x 4.23x 18.7x 11.2x 7.20x
10- 2 10- 3 10- 2 10- 3 10- 2 10- 2 10- 2
4.71 2.31x 4.27x 1. 377 x 2.7x 19.8x lr.8x 7.46x
10- 2 10- 3 10- 2 10- 3 10- 2 10- 2 10- 2
TABLE 4.7
4.7.2 Measurements of d-axis synchronous, subtransient and additional subtransient reactances; with the field winding open (without brushes)
The test procedures of Section 4.3 were repeated with the
phase a aligned with the d-axis, and the field winding open-
circuited. Figure 4.21 shows typical oscillograms for a test
current of 13.52 A. Figures 4.22 and 4.23 show the
developme"nt of the transient components for test currents of
13.52 and 9.15 amperes respectively. The corresponding
time constants and reactances are shown in Table 4.8, from" which ,
it can be seen that the reactances are" affected slightly by iron
saturation, which is dependent on the flux density which is a
function of the armature current.
Current TelO T'" T" TU, Xd X" X"'
(Amps) dO d d d d (s) (s) (s) (s) (pu) (pu) (pu)
13.52 3.885 4.452 1.098 7.484 43.258 12.227 6.823
xlD- 2 xlO- 3 xlO- 2 10- 3 xl0- 2 xlO- 2 xlO- 2
9.15 3.935 4.457 1.096 .585 44.549 12.406 7.194
xlO- 2 xlO- 3 10- 2 X . xlO- 3 xlO- 2 xl0- 2 xl0- 2 ..-
TABLE 4.8
94
4.7.3 Measurements of d-axis synchronous. transient. subtransient and additional subtransient reactancesl with the field circuit closed (without brushes)
The test procedures of Section 4.4 were again carried out
with the a-phase of the rotor aligned with the d-axis. and the
field winding short circuited. Figure 4.24 shows typical
oscillograms of the applied voltage and the transient armature
current. The short-circuited field current was not recorded
for the reason explained previously in Section 4.4. The analy-
ses of two different transient current tests are shown in
Figures 4.25 and 4.26 respectively. and the corresponding time
constants and reactances are shown in Table 4.9. The effect of
saturation due to armature current is to reduce the value of
reactances. as is apparent from a comparison of the values
obtained for armature currents of 13.52A and 9.15A.
Current TdO T" T'" T' T" T" I
(Amps) dO dO d d d (s) (s) (s) (s) (s) (s)
13.52 18.22 2.020 3.239 5.735 1.095 2.512 xlO- 2 xlO- 2 xlO- 3 xlO- 2 xlO- 2 xlO- 3
9.15 18.41 1.832 2.812 5.79 1xlO- 2 2.213 xlO- 2 xlO- 2 xlO- 3 xlO- 2 xlO- 3
Current Xd X' X" X'" (Amps) d d d
(p.u) (p.u) (p.u) (p.u)
13.52 42.90 13.49 7.311 5.669 xlO- 2 xlO- 2 xlO- 2 xlO- 2
9.15 43.96 13.825 7.546 5.94 xlO- 2 xlO- 2 xlO- 2 xlO- 2
-
TABLE 4.9
i
4.7.4 Determination of d and q-axis additional subtransient reactances by Dalton and Cameron method (without brushes)
The test procedure described in Section 4.6.2 was performed.
and the following values were obtained for a test current of
5.46 A:
A = 9.39 V/5.46 A = l.72D(l sc
B = 9.26 V/5.92 A = 1.564(l sc
C = 9.36 V/5.47 A 1.710n sc
hence from equations 4.3 - 4.6:
K = 1. 665(l sc
and M = 0.1000 sc
Thus: X'" = (K -M )/2 d sc sc Q
and X'" = (K +M )/2 q sc sc n
= (1.665 - 0.100) = 6.7B X 2 x 11. 547
= 0.665 + 0.100) = 7.642 X 2 x 11. 547
For a test current of B.91 A. the values obtained were:
A = 15.05 V/B.91 A = 1. 690(l sc
B = 14.9 V/9.B9 A = 1.510 (l sc
and C = 15.04 V/B.96 A = 1.6BD(l sc
hence:
K = 1. 624(l and M = D.llB(l sc sc
i. e. X'" = d
6.55 X 10- 2 p.u.
and < X'" = 7.55 x 10- 2 p.u. q
10- 2
10- 2
p.u.
p.u.
,I'~' \,_1
96
The test was repeated for test currents of 10.91 A, and 13.29 A,
the corresponding additional sub transient reactances are as
shown in Table 4.10
Armature Current 5.46 8.91 10.91 13.29 (A)
XIII (p.u) 7.64 x 10- 2 7.55 X 10- 2 7.38 X 10- 2 7.11 X 10- 2 q
X" d (p.u) 6.78 x 10-2 6.55 X 10- 2 6.49 X 10- 2 6.25 X 10- 2
, .
TABLE. 4.10
4.8 Decay Test
Although a heavy duty battery was used as the source for the
ID indicial response tests, a decay test was carried out on the
q-axis windings to ensure that a good step of voltage had been
provided by the battery. With the phase-a aligned with the q-
axis, the d.c. supply voltage from an external source was applied
directly to the phase winding (i.e. not through the brllshes),
_ through an L-C filter circuit to remove any ripple from the
voltage, and a high resistance R1
, cS shown in Figure 4.27.
The phase winding was short circuited by closing switch 51 and
the decay current recorded as shown in Figure 4.28.
The high resistance R1 avoids errors due to an increase in
the current from the supply as R1 is very much larger than the
resistance of winding plus shunt (i.e. an increase in the current
97
would tend to increase the voltage drop across the terminals
of the armature short circuit switch S1 which introduces an
appreciable error in the decay current). and also' protects
the supply from the effect of the short circuit current. To
determine the exact starting point of the decaying current. the
voltage applied to the phase winding was recorded across the
terminals of the switch S1'
Figure 4.29 shows the analysis of the transient decay of ,
the armature current. The q-axis synchronous. subtransient and
additional subtransient reactances are given by equations (3.15).
(3.2~ and (3.17) respectively. (The formula for the transient
decay of the armature current is the same as that of equations
(3.15). (3.20) and (3.17) ).
The corresponding time constants and reactances obtained
from Figure 4.29 are shown in Table 4.11. It can be .seen that
the results obtained agree well with those from the indicial-
. response tests of Table 4.7. it is considered therefore that the
heavy-duty battery provides an acceptable step function of
voltage. . ,
Current T" T'· T" T'" X X· X"' (Amps) qO qO q q q q q
(Sec) (Sec) (Sec) (Sec) (pu) (pu) (pu)
4.9 2.88x 4.969x 1.437x 3.516x 20.25x 10.lOx 7.151x 10- 2 . 10- 3 10- 2 10- 3 10- 2 10- 2 10- 2
•
TABLE 4.11
98
4.9 Linear Regression (Method of Least Squares)
In order to show the influence of ignoring the damping
effect of the iron on the machine parameters, the linear regression
method is applied, using one of the two techniques of Appendix 7.2.
1. Figure 4.4
Method (a): On applying method (a) of Appendix 7.2 (see
Figure 7,2.1) to Figure 4.4, the second curve is replaced by a
straight line, starting from 7.72A and passing through one point
only.
For the first ten points on the 2nd curve, i.e. from O.OOls
to O.Ols:
Ltj = 5.5 X 10-2 s
Lt 2 R 3.85 x 1O-~ 6 2 j
Lij = 18.617 A
Lijt{ 6.37 x 1O-2A•s
where: i j • tj = the coordinates of the point on the curve
Lij = the algebraic sum of the instantaneous values ., of the current at the first ten points
Ltj = the algebraic sum of the time intervals at the
first ten points.
From equation (7.2.1):
'\
99
where m is the slope of the line.
-3 = -1.06 x 10
The time constant of the line (T) is obtained from equation
(7.2.2) as:
. 1
T = Op (- - 1)
e = 7.72 (-0.632)(-1.06 x 10- 3 ) = 5.204 x 10- 3 s' m
i.e. the second curve of Figure 4.4 is replaced by a straight line,
starting from 7.72A and with a time constant of 5.204 x 10- 3 s.
According to this technique, Figure 4.4 consists of two decaying
components, their magnitudes and the corresponding time constants
being:
i.e.
and
IA2L 7.72 AL = IA1 = -2- = 3.86
5.204 X 10- 3
2.6 X 10- 2
.IA2L= 7.72A
TA2L = 5.204 x 10-3 5
= 2.001 X 10- 1
The new suffix L is used here and is to indicate that a least
square method has been used, and to be distinguished from those
used in Section 4.2 and 4.3. According to the two decaying com-
pontns and their respective time constants given above, equation
(3.15) thus reduces to:
100
(4.10)
, .. " 1.5 X 0.617 X 0.026 X 314 (1 + 3.88 X 2.001 X 10-1)
. ~~~~~~~~11~.~5~4'7~(1~+~3~.8u6~)~~~~~~~~~
23.86 X 10-2 p.u.
and equation (3.17) reduces to:
X" ~ q
~ 1.5 X 0.617 X 0.026 X 314 (1 + 3.86)
= 15.673 X 10- 2 p.U. ,
(4.11)
The value of q-axis reactances measured in Section 4.2 and given
in Table 4.1 are:
x ~ 19.825 X 10- 2 p.u. q
X· ~ 10.275 X 10-2 p.u. q
X"'= 7.621 X 10- 2 p.u. q
101
and the value of q-axis additional subtransient reactance
measured of a test current of l2.05A by the static test (Oalton
and Cameron) in Section 4.6.2 and given in Table 4.4 is:
I
I Comparison of these values with the corresponding values obtained
I by using method (a), shows the considerable difference especially
with sub transient reactance values.
Method (b): On applying method (b) of Appendix 7.2, (see
Figure 7.2.2) to the second curve of Figure 4.4, the slope of
the line is given by equation (7.2.3):
})j tj Ltj 6.37 X 10- 2 5.5 X 10- 2
Ltj 10 18.617 10 3869.35 x 10-" i.e. m = = =
10-" Lt 2 Ltj 3.85 x 10-" 5.5 X 10-2 8.25 x
j
Ltj 10 5.5 X 10-2 10
the initial value of the line is given by equation (7.2.4):
_/
Lt2 j Lij tj 3.85 x 10-" 6.37 X 10-2
Cl Ltj Lij 5.5 x 10-2 18.617 3.664 x 10- 3
= = = -" 8.25 x 10-"
Lt 2 Ltj
8.25 x 10 j
Ltj 10
= 4.441A
102
and from equation (7.2.5), the time constant of the line is
T = = 4.441 (-0.632) x 8.25 -3869.35
Here the second curve of Figure 4.4 is replaced by a straight
line with an initial value of 4.441A and a time constant of
5.984 X 10- 3 s. According to this technique, Figure 4.4 also
has two components, but their magnitudes and the corresponding
time constants are:
and IA2L = 4.441A
TAl = 2.6 x 10- 2 s TA2L = 5.984 x 10- 3 s
hence
AL = 4.441 2.220 =
2
BL 5.984 x 10- 3
2.301 10- 1 = = x 2.6 X 10- 2
From equation (4.10):
x q
1.5 x 0.617 x 0.026 x 314 (1 + 2.22 x 2.301 x 10-1) ~~~~~~~~1~1~.r5747~(~1-+~2'.~22~)~~~~~~~
~ 30.702 X 10- 2 p.u.
j'~ ;.,
•
103
and from equa tion (4.11):
X" !::::: q
1.5 x 0.617 x 0.026 x 314 (1 + 2.22)
11.547 (1 + 2.3;i~210-1)
-2 !::::: 19.787 x 10 p.u.
It can be seen that the results obtained from both methods (a) ,
and (b), especially the subtransient reactance value, do not
agree with those given in Tables 4.1 and 4.4. However method
(a) is more appropriate than method (b), and so method (a) will
be applied to the following figures.
2. Figure 4.7
By applying the method (a) to the second curve of Figure 4.7
we obtain:
1 - = m
3.85 X 10 -~ ~'"""'~:.:..;-=--',;.....::..=.."--=-;:---=,.....,,,= = -1. 0 50 x 10 - 3 6.142 X 10- 2 - 7.78 x 0.055
---
and T = 7.78 (0.632) x 1.050 x 18- 3 = 5.163 X 10- 3 s
Hence the second curve is replaced by a straight line, with an
initial value of 7.78A and a time constant of 5.163 x 10-3 s,
7.78 2.350 =-- = 3.31
5.163 X 10- 3
62.7 X 10- 2 = 8.235 X 10-2
104
The d-axis synchronous (X ) and subtransient reactances (Xd) are d.
obtained from equations (4.10) and (4.11) (with subscript q replaced
by subscript d) as:
Xd '" 1. 5 x 0.541 x 0.0627: x 314 (1 + 2.350 x 8.235 x 10- 2 )
11.547 (1 + 2.350) I
I - 49.294 X 10-2 p.u.
and X" '" 1.5 x 0.541 x 0.0627 x 314 (l + 2:350)
,d 2.350 11.547 (1 + 8.235 x 10-2 )
~ 15.692 X 10- 2 p.u.
The value of d-axis reactances measured in Section 4.3 and given
in Table 4.2 are:
Xd " 46.566 x 10- 2 p.u •
. Xd " 12.675 x 10-2 p.u.
Xd' '" 7.228 x 10- 2 p.u.
and the value of d-axis additional subtransient reactance, measured
at a test current of 12.05A by the static test described in Section
4.6.2, and given in Table 4.4 is:
XI" ~ d
2 . 7.11 x 10- p.u.
105
Comparison of these values with the corresponding values obtained
by using method (a), shows the considerable difference, especially
with the value of subtransient reactance.
3. Figure 4.22 , I
By
result !
1 - = m
applying the method to the Figure 4.22, the following
are obtained:
-9.7502 x 10-4
T = 5.2B7 X 10-3 s
AL = 1.737
BL = B.96l x 10- 2
From equations (4.10) and (4.11)
_ and Xd"14.347 x 10- 2 p.u.
The value of the d-axis reactances measured in Section 4.7.2 and
given in Table 4.8 are:
Xd " 43.258 x 10-2 p.u.
Xd " 12.227 x 10- 2 p.u.
Xd'; 6.B23 x 10- 2 p.u.
\
106
and the d-axis additional subtransient reactance measured at a
test current of 13.29A in Section 4.7.4 and given in Table 4.10
is:
Xd' = 6.25 x 10-2 p.u.
Again it can be seen that the value of sub transient reactance
obtained by using method (a), is much different from those obtained
in Sections 4.7.2. and 4.7.4.
4. Figure 4.25
Applying method (a) to this figure, the following results are
obtained:
\
Hence:
1 3.85 x 1O-~ iii = "'8-..... 15"'8".-x.;:.;.1""0;.-2":':"'-~1"'0-. ""5 8"-x-"5~."'5-x--'1"'0:":-"2
= -7.695 x 1O-~
T = 10.58 (-0.632) (-7.695 x 10- 4)
= 5.145 X 10- 3 S
107
i.e. the third curve of Figure 4.25 is replaced by a straight line.
with an initial value of 10.58A and a time constant of 5.145 x 10-3 s •
According to this technique. Figure 4.25. consists of three decaying
components. their magnitudes and the corresponding time constants
being:
lAls
= 1.24A • lA2s = 1.7A and IA3sL = 10. 58A
TAls = 0.28 TA2s
= 0.0238 and TA3sL = 5.145 x 10- 3
Le.
A = lA2s
= 1.37 s IAls
Bs TA 2s = 1.15 x 10-1 = TA
ls
C = lA3sL = 8.532 sL lA ls
o = TA3SL = 2.572 X 10- 2
sL TAls
Again the new suffix L is used here to indicate that a least
square method has been used. and to be distinguished from those
used in Section 4.4. According to the three decaying components
and their corresponding time constants given above. .
eqns. (3.57). (3.59). (3.62), (3.67). (3.68), (3.69) and (3.70)
thus reduce. respectively. to:,
+ C s.L
(4.12) ,
106
1.5 rd TA1s (1 + As + CsL ) X" " ---'=---;c=---,,---"--':;'::;'-
d A C L (1 + ....§. + _s __ )
B 0 L 6 S
. T' T" X''· :::: d d
Xd d T~JD T" . ; dO
I from w~ich:
I
T' X' d • Xd d '" l' dO
T~JD 1
= !lld
1 TdO = ._-
!l2d
-.
;
(1+8TA1s ) (1+8TAzs 1 = 0
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
r
(4.16 )
109
On substituting AsL' BsL CsL and 0sL into equations (4.12), (4.13),
(4.16) and (4.18) respectively, we 9btein:
x " d 45.728 X 10-2 p.u.
X" " 11.456 X 10-2 p.u. d
aId = -5.485
a 2d = -48.780
Sld = -15.385
S2d = -69.930
from equations (4.17) and (4.19):
\io =" 0.1823 s
TdO = 0.0205 s
Tei = 0.065 5
Td = 0.0143 s
Hence from equation (4.15):
Xci " 0.163
The value of the reactances found in Section 4.73 and given in
Table 4.9 are:
Xd " 42.76 x 10- 2 p.~.
X' " 13.492 X 10-2 p.U. d
X· " 7.311 X 10-2 p.U. d
, X·'" 5.669 X 10-2 p•u • id
I
110
" , )
The value of additional subtransient reactance measured at a
test current of l3.2A in Section 4.7.4 and given in Table 4.10
is:
Xd' = 6.25 x 10- 2 p.u.
Comparison of these reaetances with those determined by using
method (a) above, shows that ignoring the damping effect of the
rotor iron has:
1. had no considerable influence on the measured value of
synchronous reactance.
2. a small influence on the measured value of transient reac-
tance. -' 3. a big influence on the measured value of subtransient reac-
tance.
From what is explained and proved in this section, it can be
said that ignoring the damping effect of the rotor iron mainly
introduces a large error into the value of the measured subtrans-
ient reactance. Therefore the damping effect of the rotor iron
cannot be neglected.
III
4.10 Comparison of the Results
In order to show the accuracy of this approach, the measured
parameters are compared with those measured by static test and
conventional methods as given in Tables 4.12a and 4.12b. From
these two tables it will be seen clearly that the new approach
produces satisfactory results, more so when the brushes are
removed from the armature circuit (Table 4.l2bl •
. -,- .-
.,- .
,/
x X" X", Xd "- X' Methods .q q q -d
(p. u) (p.u) (p.u) (p.u) (p.u) .
No load aOO sustained short-circuit test - - - 51.7 -xlO- 2
A.C. supply to two armature phases with the rotor stationary (oal ton & Cameron) - .- 8.41 - -I = l2.05A xlO- 2
q-axis iOOicial-respons8 test on one 19.825 10.275 7.621 - -armature phase I = 9.72A xlO-2 xlO- 2 xlO- 2
d-axis iOOicial-response test on one armature phase (field open) - - - 46.566 -I = 11.09A xlO- 2
d-axis ind1cial~response .test on one armature phase (field closed) - - - 44.583 13.82 I = l1.09A .xlO- 2 xlO- 2
Slip test (at lA = 62.5 volts)' 19.8 - - I 47.9 -xlO- 2 xlO- 2
Maximum lagging current method 120.4 - - - -(at V = 0.75 Vnormal) xlO- 2
Reluctance motor method 120.6 - - - -xlO- 2
,
TABLE 4.l2a
Measurement of Parameters with Brushes in the Armature Circuit .1 n. ..................... ..... ,. '"
X" d (p.u)
-
-
-
12.675 xlO-2
7.975 xlO- 2
-,
-
-
X", d
(p. u)
-
7.11 xlO- 2
-
7.228 xlO- 2
6.464 xlO- 2
-
-
-
.....
.... N
,
Methods·
q-exis indiciel-response test on one armature phase I = 9.22A ~
d-exis indiciel-response test on one erme ture phese (field open)
I = 13.52A
d-axis indiciel-response test on one armature phase (field closed)
I = 13.52A
A.C. supply to two ermature phases with the rotor Dtetionery (Oalton and Cameron) I = 13.29A
q-axis decay test I = 4.9A
. x X" X"' Xd q q q
(p.u) (p.u) (p.u) (p.u)
IB.7 11.2 7.20 -. xlO- 2 xlO- 2 xlO- 2
- - - 43.25B xlO- 2
- - - 42.90 xlO- 2
- - 7.11 -xlO- 2
20.25 10.10 7.15 -xlO- 2 xlO- 2 xlO- 2
TABLE 4.12b
Measurement of Parameters with
No Brushes in the Armature Circuit
X' d X" d (p.u) (p.u)
- -
- 12.227 xlO- 2
·13.492 7.311 xlO- 2 xlO- 2
- M
- -
X'" d (p.u)
-
6.B23 xlO- 2
5.p69 xlO- 2
6.25 xlO- 2
-
... .....
'"
,
s JL~-------~--~~
TO -
T U.V.R.
... ~--_~ 1
TO U.V.R.
-- - --
FIG.4.1. SCHEME OF MEASUREMENT OF THE TRANSIENT CURRENT
I
FIELD U.V.R.
.... ~ . "
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'.,
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. ' .... _ t.l--nu-~--_--.-n-- __ n_.Iq.." T!~n_sIen:. (ur~~nt of armature winding phase a. VF = Induced voltage of field winding.
0.01 V/cm ------· ________ .. \_e
FIG.4.2. OSCILLOGRAM FOR INDICIAL RESPONSE METHOD AT 8 =90 DEGREES
2·5 V/cm
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145
CHAPTER 5
COMMENTS AND CONCLUSIONS
The analyses of the stability of a power system and the study
of its short-circuit behaviour are very important, more so as the
system gets bigger and more complex.
Previous attempts to establish operational impedances and
hence the parameters of the machine have been entirely based on
elementary analyses using very simple machine models lO ,12.
In practice a salient-pole synchronous machine has a signi-
ficant part of its rotor made of solid iron .. which exercises a
major influence on the operational performance. The results
obtained may differ greatly from the theoretical results if
this influence is neglected. In order to include this effect
and to provide a more accurate representation, an additional
rotor circuit in the machine model is required. Accordingly,
a new approach has been attempted, with the damping action of the
iron represented by an additional damper winding along each axis.
This naturally leads to an important modification to the familiar
form of Park's equations.
In this study a step-by-step derivation of the operational
inductances and parameters for the new model of a salient-pole
synchronous generator were developed from fundamental principles. .,.',~];. , ,
As shown in Chapter 3, the parameters are derived from the basic
equations applied to the new model. with two damper w1ndings in
each of the d- and q-axes. •
From the experimental work presented in this thesis, the
following conclusions can be drawn:
146
1. It suffices to consider both the q- and d-axes as each
having two damper windings.
2. The results obtained from the new analysis agree closely
with those obtained from static and conventional methods
as shown in Tables 4.12a and 4.12b.
3. Saturation effects are small as is clear from Tables 4.1-
4.3 and 4.7-4.9.
4. By removing the brushes from the armature circuit. the
measured parameters change considerably. as is clear from
the comparison of the tables 4.1 and 4.7. 4.2 and 4.8.
4.3 and 4.9. and finally 4.4 and 4.10 • •
The effect of the brushes is to introduce a non linear
resistance in series with the measuring circuit. This
introduces errors in the measured time constants during
the step response test. The result is that the measured
values of reactances are different when measurements are made
with and without brushes. Hence to obtain accurate and con-
stant values of parameters. it is necessary to make measure-
ments. if possible. without brushes. (This applies particu-
larly to step-response tests as the nonlinear effect of brush
resistance affects the transient current).
5. The damping effect of the iron cannot be neglected. otherwise
. the derived parameters will be considerably in error.
147
REFERENCES
-I 1. R.E. Doherty and C.A. Nick1e, Synchronous Machines I and
11, An Extension of B1onde1's Two-Reaction Theory,
AIEE, Trans, Vol. XLV, pp.912-47,1926.
"2. R.H. Park, Two Reaction Theory of Synchronous Machines,
Part I, AIEE, Trans, Vol. 48, July 1929, pp.716-27.
~3. S.H. Wright, Determination of Synchronous Machine Constants
by Test, AIEE Tans, Vol. 50, pp.133l-5l, December 1931.
4. M.L. Waring and S.B. Crary, The Operational Impedances of
a Synchronous Machine, General Electric Review, Vol. 35,
No. 11, November 1932, pp.578-82.
~5. W. Nurnberg, Die prufung elecktrischen maschinen, Springer
Verlag, Berlin 1939.
6. Yao Nan Yu and H.A.M. Moussa, Experimental Determination of
Exact Equivalent Circuit Parameters of 'Synchronous
Machines, IEEE, Trans, pp.2555-59, 1971.
7. G.R. Slemon, Analytical Models for Saturated Synchronous
Machines, IEEE Trans, Vol. Pas-9D, pp.4D7-l7, March 1971.
8. L.A. Snider and I.R. Smith, Measurement of Inductance Coeffi-
cients of Saturated .Synchronous Machines, Proc. lEE,
Vol. 119, No. 5, pp.597-6D2, May 1972.
- 9. Y. Takeda and B. Adkins, Determination of Synchronous Machine
Parameters Allowing Unequal Mutual Inductances, Proc.
lEE, Vol. 121, No. 12, pp.150l-l504, December 1974.
148
, 10. Pawluk, M.K. Methode.Statique de Measure des Constantes
. , de temps et des reactances d'une machine Synchrone,
, Rev. gen electr. Bd-71 (1962) S.303-312.
+11. F.K. Dalton and W.W. Cameron. Simplified Measurement of
Subtransient and Negative Sequence Reactances in
Salient-Pclle Synchronous Machines, AIEE Trans, Vol. 71,
pp.752-57. October 1952.
k 12. H. Kaminosono and K. Uyeda, New Measurement of Synchronous
Machine Quantities, IEEE Trans, Pas-87,' No. 11,
November 1968, pp.1908-1918.
"13. C. Concordia and F.J. Maginniss, Inherent Errors in the
Determination of Synchronous Machine Reactances by Tests,
Trans. AIEE,·Vol. 64, pp.288-294, June 1945.
'14. Test Procedure for Synchronous Machines, IEEE Publ. 115, 1965.
15. T. Laible, Die Theorie der Synchronmachine in nichtstationaren
Betrieb, Springer Verlag, Berlin 1952.
16. A.W. Rankin, The Equations of the Idealised Synchronous' Machines,
General Electric Review, Vol. 47, June 1944, pp.31-36.
-17. B. Adkins and R.G. Harley, The General Theory of Alternating
Current Machines, Chapman and Hall 1975.
18. R.E. Steven. An Experimental Effective Value of the Quadrature-"-
axis Synchronous Reactance of a Synchronous Machine, lEE
Proc. Paper No. 375DS, December 1961.
19. E.J. Berg. Heaviside's Operational Calculus, McGraw-Hi 11. 1927.
20. A.E. Fitzgerald and Charles Kingsley, Jr. Electric Machinery,
McGraw-Hi11, 1961".
.' -'
149
21. H.S. Kirschbaum, Per Unit Inductances of Synchronous Machine,
Trans. AIEE, Vol. 69, 1950, pp.231-234.
22. R •. Rudenberg, Transient Performance of Electric Power Systems -
New York, McGraw-Hill 1950.
23. Paul D. Agerwal, Eddy Current Losses in Solid and Laminated
Iron, Trans AIEE, May 1959, pp.169-81.
"24. J.C. Prescott and A.K. El-Kharashi, A Method of Measuring
Self-inductances Applicable to Large. Electrical Machines.
Proc. lEE, Part A, April 1959, Vol. 1, 106, No. 26.
25. R.H. Park, Definition of an Ideal Synchronous Machine and
Formula for the Armature Flux Linkages, General Electric
Review, Vol. 31, June 1928.
26. R.E. Doherty, A Simplified Method of Analysing Short Circuit
Problems, Trans AIEE, Vol. 42, pp.841-848, 1923.
27. Charles Kingsley, Measurement of Quadrature-axis Synchronous
Reactance (letter to the Editor), AIEE Trans. Vol. 541,
1935·, p. 572.
,28. C. Concordia, Synchronous Machines. Theory and Performance,
New York. Wiley, 1951.
/
150
APPENDIX 7.1
PER UNIT EQUATIONS
7.1.1 O-axis voltage equations with the field circuit open
U~der stationary condition. and with the field circuit
open aJ in Figure 2.3. the voltage equations in terms of the
time I of the flux linkages are: rate-of-change ,
(7.1.11
(7.1.2)
(7.1.3)
where the bold letters indicate quantities ill normal (SI) units.
To transform these equations to the p.u. system. and at the same
21 time to preserve the reciprocity of the mutual inductances • the
d-axis equivalent self-inductance Ld must be multiplied by 2/3
and both sides of each equation must be divided by the base stator
d-axis voltage (i.e. edo ' where edo = Wo idOLdO' i dO = base stator
d-axis current).
are:
where:
21 The relation between the stator and rotor base currents
= IfdO/iffd. and "3 Ld
=1 ~ Indole
3 d
IfdO = base field current in the d-axis
IndO = base damper current in the d-axis
151
Lffd = self inductance of field
and L d = self inductance of any direct-axis damper circuit. nn
Hence equation (7.1.1) becomes:
E1
or 1 11
where.
=
=
rd1d + p(Ldld+LdKa1dI1d+LdKa2dI2d)
(pLd+rd)Id + pLdKa1dI1d + pLdKa2dI2d
and· K 2d . a
LdO = base stator inductance of d-axis circuit.
(7.1.4a)
(7.1.4b)
Similarly, on dividing equation (7.1.2) by edo and substituting
for the corresponding rotor currents, and multiplying by
we obtain:
-~(7.1.5a)
or (7.1.5b)
Similarly, by multiplying equation (7.1.3)
and substituting for the corresponding damper currents, we obtain:
152
(7.1.6a)
(7.1.6b)
Hence from equations (7.1.4b), (7.1.5b) and (7.1.6b), the p.u.
matrix form of these equations is:
E1 pLd+rd pLla1d pLdKa2d Id
0 = pKa1d p + _1_ pK21d I 1d (7.1.7)
T1d
0 pKa2d pK21d 1 12d P +-
T2Gl
Similar expressions for the q-axis circuits can be obtained by
replacing subscript d in equations (7.1.1-7.1.7) by subscript q.
Hence from equations (7.1.4a), (7.1.5a) and (7.1.6a) the q-axis
flux-linkages in p.u. are:
= L K I + L K I + L I q a1 q 1 q q a2q 2q q q (7.1.8)
~
- (7.1.9)
and (7.1.10)
and from equations (7.1.4b), (7.1.5b) and (7.1.6b)' the p.u. \
matrix form of the q-axis voltage equations is:
153
. E1 pL +r pL K . pL K I q q q a1q q a2q q
0 = pKa1q p + ...2- pK21q 11q (7.1.11) T1q
0 pKa2q pK21q p + _1_ T2q
12q
7.1.2 d-axis voltage equations with the field circuit closed
Under stationary condition, and with the field circuit of
Fig. 2.3 'closed, the voltage equations are:
(7.1.12)
(7.1.13)
(7.1.14)
(7.1.15)
By dividing each equation by edO and multiplying equations
(7.1.13)' (7.1.14) and (7.1.15) by )
and respectively, we obtain
the p.u. matrix form of these equations:
154
E1 pLd+rd pL
dKa1d pLla2d pLdKafd Id
0 pKa1d p + _1_ pK21d pK
f1d 1
1d T1d =
0 pKa2d pK21d p + _1_
T2d pKf2d 12d
I 0 I pKafd pKf1d
pKf2d 1
Ifd P +-
I Tfd
(7.1.16)
. From equation (7.1.16) the p.u. flux linkage equations are:
(7.1.17)
(7.1.18)
(7.1.19)
(7.1.20)
where:
L K = afd afd A 2
ffd 3" Ld
etc.
155
APPENDIX 7.2
LINEAR REGRESSION (METHOD OF LEAST SQUARES)
The decrement curve of Fig. 7.2.1 can be replaced by a
straight line, by either of the following methods:
a) A straight line starting from point p and passing through
one point on the curve, such as CA the coordinates of which are
t j , i j • If the equation for the curve is:
~A = op + mt m < 0
where m is a slope, we must minimise the function n
4>(m) = I j=1
d~ J
where n is the number of first points on the decay curve.
Hence for the first ten points:
n=1o 4> = I [(Op + mtj ) - i ] 2
j=1 j
For 4> to be a minimum d4> _
0, or dm -
m I t 2 = I i j t. - Op I tj j J . ../
1 1. t 2
j i. e. - = 1 i j tj- Op I tj (7.2.1l
m
where:
I ij
= the algebraic sum of the currents at the first ten
points on the curve.
I tj = the algebraic sum of the time intervals at the first
ten points.
156
The time constant of the line can be found from:
Op (lIe - 1) T =---::--
m
b) The method of least squares. I
(7.2.2)
In this method the curve is represented by a straight line
w1iCh does not need to start from p. but from some other
point like C1
as shown in Fig. 7.2.2. Hence:
i = mt + C1
where m is a slope. and C1
a constant.
We must minimise the function $ = L dj
and
n $ = L (mt
j j=1 + C - i )2
1 j
i.e. m and C1
are given from these equations.
Hence. for the first ten points:
157
i.e.
~ij tj ~tj
" 10 l,ij m = (7.2.3.)
~t2 j ~tj
I, tj 10
~t2 ~ij . ' :j j
and C1
= ~tj ~ij
(7.2.4) ~t2 })j j
~tj 10
and the time constant of the line is:
.. C (1/e - 1) T = .....:..1 ___ _ (7.2.5)
m
•
158
.. i
•
t
FlU .• 7.1.
o t
. fiG. 7.2.
159
APPENDIX 7.3
COMPUTER PROGRAM
A computer program for calculating the transient currents
using measured values of time constants and their corresponding
magnitude values of current is given in the next page. The
program reads the values of time constants and their magnitudes
of the current corresponding to the test under consideration.
The X- and V-scales for plotting the results are adjusted to
correspond with the actual speed of the U.V. ·paper and the
deflection of the current trace (depending on shunt resistance
used and the sensitivity of the galvanometer). This enables
direct comparison of calculated plot of transient current and
the measured trace obtained on the U.V. recording paper. The
shunt resistances used were 0.01 and O.OOW in series. The
variation in contact resistances between tests was between two
limits (0.0005 to 0.0015) resulting in the effective shunt
resistance variation from q.OllS to 0.012SQ.
160
lIB~A~Y(ED,)UAij~~UPr.INO) PROGRA"I(ALIS, OllTPUT Z::LI'(j INPIIT 1 =CRO COMPRESS l~rEGER AND LOGICAL TRACE 0 E"O I',UrER(ALI8)
C THE nO C IH~ E '" T 0 A T4 4 pES U BST IT lJ T E DIN TOE ~ 1/ A TI 0 N ( 3 , J) C T~E r.ALCUL,H ED r.URVFS AH SHOWN IN FIGURES 7,3 AND 7,4 C R~SPECTIVEL1
RFAL IA1,1A~,IA3,IAn,1 ~IMENSION X(1000),Y(10QO) STP::O·0001 r.ALL LU19.54 CHL U~lTS(10.) C.uL DEV PA D (30, ,25.,1) CA L I. A j( I pO S ( 1 , .5 , ,3 .. ? 0 . , 1> CHI. AXIPOS(1,S.,3 .. 18 .. Z)
CALL AXISCA(3,Zn,a.,.1.1) CALL AXISCA(3,lR,0 •• o.18,2) D010 K"1,2 T=O. R f 4 0 (1 , 5) TA I , TA 2 , T.o\ ~, I A 1 , I A 2, I A 3 '; I A 0
5 FuRtIAT(9FO,Q) WRlrF.(l,7)T~1,TA2,TA3,IA1,IA2,IA3,IAO
7 FOR '1 AT( 1 X , 9 r, 11 , :51 DO <'0 J~1"OOO
T=T+STP 1:IAO-IA1*eXp(·T/TA1).lA2*EXP(~T/TA2)-IA3*EXP(-T/TA3) x(J) .. T y(J,.I*O.01 l 5
?n COtjT I NllF. r.AlL AXlDRA(1,1,1) CALL AXIOR4(-1'-1,21 r.All GRAPOL(X,Y,1DOn)
tAU. PICCLE 10 CONTIIIUe
CAlL DEVENO STOP no FINI~II
* •• * nllr.IJ"ltNT D~TA
* •• *
n.026 O.OO~5 U.~01~ 2.0 4.2 3,5<, ~;72 0.02?6 0,0052 ~.n011 0.6 1.12 0.9 2;62
161
LIBRARY(F.D,SUBGRnUPr.INO, PROGRAM(AlIB) OUTPUT 2.LPO INPUT hCRO COMPRESs INTEGER ANn LOGICAl. TR4CE 0 END MAsn R (ALIB)
C FOR THE TEST CURRENT OF 9.15 A,THE TEST IS CARRIEO OUT ~ITH C A DIRECT CONNECTION TO THE ~llP RINGS(WITH ~UT BRUSHES) C THE DOCUMENT DATA hE SUBSTITUTEI'I INTO EQIIATJON(3:3) c (WITH SUBSCRIBT q BF.ING REPlACED BV- SUBSCRlllT cl) C THE CALCULATED CURVF~ ARE SHOWN IN FIGUREs(7.5) AND(7.12)
REAL lA1,IA2,IA3.IAn.1 DIMENSION X(1000),Y/1000) STP=0.U001 CALL LU1934 CAlL UNI TS(1 0.) CALL DEv PAP(30 •• 25 .• 1 ) CALL AXIPOS(1,3 •• 3 .. 20 .. 1) CALL AXIPOS(1 ,3 •• 3 •• 18 •• 2)
CALL AXISCA(3,20,0 ••• 1.1) CALL AK[SCA(3,1S.0 •• 0.1R,2) 00101(=1,2 T"O. READ(1,5)TA1,TA2.TA~.IA1,IA~.IA3.IAU
5 FORMAT<9FO.O) WRITE(l,7)TA1,TA2,T43,IA1,IA2,IA~,'AO
7 FORMAT(1X,9G11.3) DO 20 J=1" 000
T=T+~TP I=IAO-IA1*EXP(-T/TA,'-142*Exp(·T/TA2)-IA5*FXP(-T/TA3) x(J).T y(JI"I*O.011S
20 CONTINUE CALL AXIORI\<1,1.n CALL AXIDRA(-1 ,-1 ,:11 CALL GRAPOL(X,Y,100n)
CALl PICCLE 10 CONTlNU~
CALL ~EVEND HOP END FlNI~H
**** DOCUMENT I)ATII
****
0.067' n.007 O.On18 ~.31 2.4 5.41 1'.O~ D.060~ 0.0065 U.n01R 3.42 2:42 3.31 9.15
162
LIBRARY(ED,SUBijR~UPGINO) PRO(jR\'1CALIIl) OUTPUT 7,_LPO I i4P;JT 1 =CRO COHPRESS l~rEGER ANn LOGICAL TRACF V , E~D ' MASrF~CALIB)
C TilE nIJCIJI~Ellr OAH APF SUBSTITUTED INTO EQIIATION(3.3) C CV/I,TH SUKSCHIR r q I!F ING REPLACED BV SU8SCRIlH d) C THE r~LCIJLArED CURVF Is SHOWN IN FIGURf 7,6
R~A~ IA1,IA~,IA3,IAn,1 DIME~SION X(20 UO),Y,2000) STP=1·0001 CALL LU1934 CALL LI~ITS(lD.) CALL llEv P4P(3V. ,25.,1) CALL AXIPOS(1,J .. 3 .. 20 .. 1) CALL AXIPOS(1 ,J, ,3, ,18, ,2) CALL AXISCA(3,~o,o •• ,2,,> CALL AXISCA(3,1~,O"n.18,2) T"O. R FAD ( 1 , 5) TAl, TA 7. , T A ~ , I A 1 , I A 2, I A3; I 40
"i FORtlAI(9FOo\» W RI TF (, ,7) r A, , TA 2 , TA 3 , t A 1 , I A 2 , t A 3 , I A 0
? FORI·lAT(lX, 9G 1,.3l 00 20 J=1,Zilon T.T+STP l=tAO-IA1*EXP(-T/TA1l-IA2.EXP(.T/TA2)-1~3*EXP(-T/TA3 x(J)=' y(J):I*O.Oll
7n COrlTl'~UE
****
CALL A(t DRA(1, 1,') CAlc AXtORA(-1 ,-1 ,2\ CAL~ GRAPOLlX,Y,~ODol CHL DEVFND STOP eND FINIsH
[)nrUME'IT ;JAlA (l.O? \1,0065 n.u01? 1.66 2.0? 3.12 6,8
,,'
163
LIBRARY(EO,SUBGROUPGINO) PROGRA~(ALIB) ~ OUTPUT 2.I,PO INPUT heRO COMPRESS INTEGER AND LOGICAL TRACE 0 END MASTERtALIB)
C THE OOCUME~T DATA ARF. SUBSTITUTED INTO EQUA r rON(3.43) C FOR TH~ TEST CURRENT OF 9.15 A,THE TEST Is CARRIED OUT WITH C A DlRter CONNECTION TO THE sLIP RINCl,.SCWITH UUT BRUSHES) C THE CALCULATED CURVFS ARE SHOWN IN FIGUREs(7.7) ANDC7.14)
REAL lA1S,IA2S,I.3S.IA4s,IAo.I DIMENSION X(1000),YI10nO) STP_O.U001 CALL L01934 CALL UMITS(10.) CALL DEV PAP(30 •• Z5 •• 1) CALL AKIPOSC1.3 .. 3 .. 20 .. 1) CALl, AKIPOS(1 ,3 •• 3, .18. ,Z) CALL AKISCA(3,lO,O, •• 1,1) CAU AKISCA(3,18,O •• 0.18,2) DO 10 b1,2 T-O. . READC1,5)TA1S,TA2S,TA3S,TA 4s,IA1s,IAZs,IA3S,IA4S,IAO
5 FORMAT(9FO.O) WRIT~(l,7)TA1S'TAZS.TA3s,TA4S.IA1S,IA2S,IA3S,IA4S;IAO
7 FORtlAT(1X,\I~11,3) DO 2U J.1,1000
T·T+sTP I.IAO~IA1S*EXP(_T/TA1S).IA2s*EXPC.T/TA2S).IA3S*EXP(~TITA3S)·
2IA4S*EKpC-T/TA4S) X(J).T Y(J).'*O.Ol1s
20 CONTINUe" CALL AKIDRA(1,'.1) CALL AKIDRA(-1,-1,ZI CALL GRAPOL(x,Y.,Oon)
CALL PICCLE 10 r.ONTINUE
CALL DEVEND STOP END FlNISI!
*.*. DOCUMENT DATA
****
0.2 0.U215 0.004] 0.0018 0.R6 1.2 4.0 5,03 11.09 0.2u2 U.OO!] 0.004 0.0017 0,88 1.1 4.5 2,67 9.15
C C
20
.***
164
LIBRARYCED,SUBGRPUPnINO) PROGRAMCAL18) . OU1PUT Z.LPO INPUT heRO COMPRESS INTEGER ANn LOGICAL TRACE 0 END MASTER (MHOS) THE DUCUME~T DATA AR~ SUBSTITUTEn INTO EQUATION(3.43) THE CALCuLATED CURVF 1$ SHOWN IN FIGUREC7.8) REAL IA1S"A2S'1A3S.IA4~.IAO.1 • DIMENSION X(2000),VC2000) STP=O.~001 . CALL LU1934 CALL UNITSC10.) CALL DEV PAPC30 •• 25 .• 1> CALL IIKIPOS(1 ,3 •• 3 •• 20 •• 1) CALL AlIIPOS(1 ,3 •• 3, .18, .2) CALL AlIISCA(],20.0, •. 2,1) CALL AKISCA(3,18.0,.n.18,2) T-O. . READ(1,S)TA1S,TA2S,TA3S,TA 4s.IA1s,IA2s.IA3S,IA4S7IAO
5 FORMAT(9FO.O) WRITE(l,7)TA1S,TA2S.TA3s,TA4S,IA1S.1A2S,IA3S,IA4s,IAO
7 FORllAH 1 X,'}G11,;S) DO 20 Jo:1, 2000 T·T+STP 1.IAO~IA1S*EXP(_T/TA1S)-IA2s*EXP(-T/TA2S)~IA3S*EXPC-TITA35)·
2IA4S.EKpC-T/TA4S) X(J).T V(J)·1*O.01i!
CONTINUE CALL AXIDRA(1,1." CALL AKIORAC-' ,-1 ,2) CALL GRAPOLCx,V.~OOO) CALL DEVF.ND STOP FND FINISH'
DOCUMENT DATA 0.2 O.U17 0.0045 0.no17 0.5 0.75 3.1 2.45 6.8
••• *
165
LIBRARY(EO,5uaGRQU~GINO) ~RO(,RAM(ALIB) OUTPUT 2=LPO INPUT 1=CRO COMPRESS INTeGtR ANn LOGICAL TRA(;F. 0 ENO MASTER(ALIB)
C THE oOCUME~T DATA ARE SUBSTITUTED INTO EQUATION(3~3) C THE CALCULATED ~ CURVES ARE SHOWN IN FIGURFS 7.9 AND 7.10 C RESpeCTIVELY
REAL IA1,IAl,IA3,IAo,l DIMENSION X(10 0 0),Y(1000) STP::O.U001 CALL LU1934 CALL UNITSClO.) CALL DEv PAP(3 U.,25 .• 1) CALL AXIPOS(1,.s .. 3 .. 20.,1) CALL AXIPOS(1 ,j .. 3 .. 18.,2)
CALL AXISCA(3,2n'0.,.1," CALL AXISCA(3,,8,O •• Q.18,2) D01 0 1<::1,2 T=O. READ (1 , 5 ) TA 1 , TA 2 • TA 3 , I A 1 , I A 2, I A 3; I A 0
5 FORIIAT<9FO.0) WRIT E ( , , .,) TA 1 , TA 2 , TA 3 , I A 1 , I A 2 , I A 3 , I A 0
7 FORIIA T<1 x,9G1 1.3) DO 20 J=1,1 000
T"T+sTP I=IAO-IA,*eXp(-T/TA1)·lA2·EXP(~T/TA2)-IA3.EXPC-T/TA3) x(J)=T ' Y(J)=I*0.012
20 CONTINUE CAll. AXIORAC1,1,n CALL AXIDRI\C-1'·, ,21 CALL GRAPOL(X,Y,100n)
CALL PICClE 10 CONTINUE
CALL DEveNO -~ STOP , END FINisH
**** DOCUHENT DATA
****
0.0322) 0.0087j 0.0025 2.0 2.4 4.82 9.22 0.029 5 0.00(25 0.002 1.25 1.88 1.58 4.71
,
c c c c
20
****
166
LIBRARYCED,SuaGROUPGINO) PROGRA'1CALIB) OUTPUT 2=LPO INPUT l:::CRO COMPRtSS INTEGER ANo LOGICAL TRA'CE U END HASTERCALlB) THE Tf-ST IS CARRIED OUT WITH A DIRECT CONNECTIO~ TO THE SLIP RINGS (UITH OUT BRUSHES) THE DUCUHE~T DATA ARE SUBSTITUTED INTO EQUATIONC3.3) (WiTH SUBSCRIBT q BEING REPLACED ay SUBSCRIBT ~) TtlE CALCULATED" CURVE Is SHOWN IN FIGURE (7.11) REAL IA1,IAZ,IA3.IAn,I DIMENSION X(10QO)/V(1000) STP:::0.OQ01 CALL LU19..s4 CALL UNI ,S (1 0.) CALL Dtv PAPC30 •• 25 .• 1) CALL AXIPOS(l ,3. ,3 .. 20 .. 1) CALL AXjPOS(l d • .3 .. 18 .. Z)
CALL AXISCAC3,20/0 .•• 1.1) CALL AXISCA(3,18.0 •• 0.45,Z) T=O. READ(1'5)TA1,TA2.TA3,IA1,IA2,IA3.IAO
5 FORIIATC9FO.0) WRIrECl,7) TAl, TA2,h3, IA1, IA2, IA3, IAO
7 FOR!IATC1X,?Gll ,3) DO 2U J=1,1000
T=T+STP I=IAO-IA1*EXPC-T/TA1>-IA2*Exp(-T/TA2)-IA3*EXPC-T/TA3) X(J)=T V(J)=I*O.0124 CO~JTI NUE
CALL AXIDRA(1,1,1> CALL AXIDRA(-1 /-1 ,2) CALL GRAPOLCx,Y,100n> CALL DEVEND STOP END FINisH
_ DocurlENT [lATA 0;059 0.0065 0,0017 4.94 3.55 5.03 13.52
167
LI8RARY(EO,SU8GROUPr.INO) PROGRA'1 (ALl B) OUTPUT 2.LPO INPUT hCRO COMPRESS INTEGER ANn LOGICAL TRACE Q END MASTeR(SAMB)
C THE DO~UMENT DATA ApE SUBSTITUTED INTO EQUATION(3.43) C THE-reST IS CARRIED OUT WITH A DIRECT CONNECTION C TO THE SLIP RING~ (WITH OUT BRUSHES) C THE CALCULATED CURV~ IS SHOWN IN FIGURE(7.13)
REAL 11I1S,IA2S,I.35,IA45,IAO,1 DIMENSION X(1000),V(1000) STP,.O,0001 CAL~ LU1934 CALL U~ITS(10,) CAll. DEV PAP(3Q •• ii!5 .• 1) CALL AKIPOS(1 ,3 •• 3, .20 •• 1) CALL AKIPOS(1 ,3 •• ], .18 •• 2)
CALL AXISCA(3,20,O.,.1,1) CALL AXISCA(3,1a,O"O.45,2) T-O. . REA0(1,5)TA1S,TA2S,TA3S,TA4S,IA19, IA25,IA3S, IA45,IAO
5 FORllAT(9FO.O) WRITE(Z,7)TA1S,TAii!S.TA3S,TA4S,IA1S,IA2S,IA3S,IA4SjIAO
7 FORllAT(1X,9G11,l) DO 20 J.' .1000
TaT+sTP I.IAO~IA1S*EXP(.T/TA1S).IA2s*EXP(.T/TA25).IA3S*EXP(·TITAlS)
2IA4S*EKp(-T/TA4S) X(J)·r VU)_1 *0 .0124
20 CONTINUE
***."
CALL AKIDRA(1,1,1) CALL AKIDRA(·1 ,., ,2, CALL GRAPOL(X,V,'OOO) CAll. DEVENO STO;.> END FIN1Sti
DOCUMENT DATA 0,2 O.O?3 0.0045 0,0018 1.24 1.7 4.9 5,68 13.52
** ••
c c C c C C
20
****
168
LIBRARV(ED,SUIGROUPr,INO) PR 0<; R A'I( A L re) ~-OUTPUT 2_LPO INPUT l-CRO COMPRESs INTEGER ANn LOGICAL TRAC e 0 END MASTeR(AQSB) THE DOCUMENT DAT_ ARE SUBSTITUTED INTO EQUATION(3.3), wiTH IAO REPLACED Bv ZERO AND ALL THE MINus SIGNS CHANGED TO poSITIVe SIGNS • THE TEST 'IS CARRIED OUT WITH A DIRECT CONNECTION TO THE SLIP RINGS (WITH OUT-RRUSHES) THE,CALCULATED CURVp IS SHOWN IN FIGURE(7,lS) REAL IA1,IAZ,IA3,IAn,I DIMENSION X(1000),V(1000) STP a O,U001 CALL LiJ1934 CALL U~ITS(10.) CALL OEV PAPOO. ,25.,1) CALL AKIPOS(1,3.,3 .. 20 .. 1) CALL AXIPOS<1 ,3. ,3, .18. ,2) CALL AKISCA(3,ZO.O ••• 1,,,CALL A)(ISCA(3.18.0 •• 0.18,2) T-O. REA D (1 ,5)T A 1 , TA2 • TAl. lA 1 , I A:! • I A 3, HO
5 FORtiA T(9 FO. 0) WRITF.(Z,'7)TA1 ,TH,TA], IA1 ,IA2,IA',IAO
'7 FORllAT(1x,9G11.3) DO 2U J.1" 000
TaT+s-rP I_IAO-IA1*EXP(-T/TA1'-IA2*ExPC-T/T_Z)-IA3*EXPC-T/'A3) X(J).r y (J ) _ I *.0 • 01 1 S
CONTINUE CALl. AXIDRA(1",') CAt.~ Al(l DRA(-1 ,-, .2) CALL GRAPOLCX,Y,100n> CAL~ DEVEND STOP END nN!<;H
DI)r.Ur~ENT uATA O.Ol6 J.007 0.0074 -1.18 -1,6 -2.12 0.0
•• **
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I;H,l .! 'l; t;
( I ! ;
/" //
./
---------.--.--~--.. ----.. ---._---.-------
v v v v MEASURED BY U.V.R.
CALCULATED BY EQUATION 3.3.
.--;-------~----;-----.--- .. ---r---.. - ........ -,--.-.---... - ....... - .. _--_._-.-.. _," .... __ _ . , t) nCI \...) I <. 'I..
e I,.. ., ..,
,.... r- r\' I~ \,\;,1 ,,' :: i 'C
, , q c,n ; .. /\.
r, [1 f,! .• .' . ~ i:
r\
V;fY-/
1 ,40_
1 ,30. i .,20_ , ." Vl I , I LJ.
,. (i)0 I J:1 _
,30 . • 80_ ,70 .. ,60_
.40_
.30 .
.20_ ., f)
f'I I () r
v v Y MEASURED BY U.V.R.
--- CALCULATED BY EQUATION 3.3
. 00 -L---r---,-----,r--~_._____:____r--..___:_____._-___r-:__-r---___,___, ,
,00
I-' ...., o
-,. r n x10 I .,\)~I_
• c;n I .... 11{}.
1 ,40_
i J0. 1 .20_ • . n I .."10.
1 .,00
,80 ,70
.50 q)
,JI{}.
,40 in
, J'O.
ir) ~LVJ_
,19
--- CALCULATED BY EQUATION 3.3
1\ x J\ MEASURED BY U.V.R.
,00 -'--.....---:--r----:--~~---.--~~__:_------r--___:_____.___:_-----r____:-r-----___:_..____, I 1 I I
.00 i .50 3,00 4.50 5.00 7,50 9,00 FlG.7.S.
-1 x10 1,50
1 ..50 .. i .,40_ CALCULATED BY EQUATION 3.3.
1 ,,30. 1,20 /\ /\ 1\ MEASURED BY U.V.R.
1 .' '10. . i ,00_
,90. ,8B_ ,...
" N '10 ,/ .
, G0_ [0 ,J .
,40_ 030. ,20_
, 1 0 .. ,00
I I I , , , , , , , .,O0 ,20 ,40 ,G0 ,80 1,00 i ..20 1, 40 i .,50 j,80 2,00
FIG.7.6. y.lrr1
1.l;g. 1 ,5 g. 1 , 4Q. i , ]g ..
1 ,2g.
i .' I g .. 1 ,gg ..
jQ ..
, SQ ..
,IQ ..
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.2Q.
, I g ..
A X Y Y Y Y MEASURED BY U.V.R.
CALCULATED BY EQUATION 3.43.
, g g .J.--,---,---.---_-,-_---,-_-. __ .---_-,--_--,-_-,----,-----,----, I I • j
gg l,Sg j,gg 4,5g £'GG 7,509,00 1=1r. 7 7 V1 rr2
j .1 (o(J'-1
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if) I \'I'U._
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,
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CALCULATED BY EQUATION 3.43.
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-~ '"t ,,~'( .,\
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CALCULATED BY EQUATION 3.3.
. ...,._. __ ...... ~. . ........ _ .. .,.. __ ......... ---... .._-0", --~-- .. --.~--.- .• ",-.. +---.. ' .• - ----"'-"')'""'-- ·---~---···~r---·-··~··· ._. __ ._ .. ",- '.-'~ .... ·-·--~"·-r-····--- .'. -'-'-'''--T ..•• "",,_ -- - •.. -,_, ___ . . ,
. : r. C:C'I ...... (' '. --
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.90.
. 80_
.70. ,60_ .50_ ,10_
030 "0 ,L _
, '10.
1/ Y Y MEASURED BY U.V.R .
CALCULATED BY EQUATION 3.3 •
,00_ r,. ~----~~~~I--~~-'~~~~~--~~--~~--~~ I ! I I
,00 1,50 3.00 4,50 6,00 7,50 9,00 FIG. 7.10. X10-2
X10- 11 .!J· 1
~ ':~ 'I , .. /v·1 J .. ~0,
~ '2 r: L;I J.
, ., C; 1 .• 1 v . •
~ C:O I .,.jl(J,
, '2<" I" J. I Qn I .,:(jI(J_
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CALCULATED BY EQUATION 3.3
A/\j\Al\ MEASURED BY U.V.R.
7 C;C\ , : .1 V.J
I./I(J.
x10 -1 1 ,f;0 1 ,50. 1 ,40. i .30 i ,20 1 ,'] 0. i .00 . . 90 . . 80 . . 70 . . 50. . 50 . . 40_ .30 ,20 . . 10 .
F/lr-7r/\ --"y_V,,-- MEASURED BY U.V.R .
CALCULATED BY EQUATION 3.3 •
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.00 1,50 3.00 4.50 6.00 7,50 9.00 ._')
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CALCULATED BY EQUATION 3.43.
vvVV\I yvy yV\t ,,\{ V"v VV \t \I y v
,98L_~--_____ r-____ ~ _____ -' __ ~ __ '-____ ~ ______ 'r-~ __ ~ ____ ~ ____ ~-.
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