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Simulation Models Relevant to the Protection of Synchronous Machines and Transformers
by Dharshana De S. Muthumuni
A dissertation submitted to The Faculty of Graduate Studies in
partial fulfillment of the requirements for the degree of
Doctor of Philosophy
The Department of Electrical and Cornputer Engineering
The University of Manitoba
Winnipeg, Manitoba, Canada
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THE UNIVERSITY OF MANFI'OBA
FACULTY OF GRADUATE STITDIES *****
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SIMULATION MODELS RELEVANT TO THE PROTECTION OF SYNCHRONOUS MACHINES AND TRANSFORMERS
A Thesis/Practicum submitted to the Faculty of Graduate Studies of The Ua~e i s i ty of
Manitoba in partid fiilnllment of the requirement of the degree
of
DOCTOR OF PEILOSOPHY
Permission bas been granted ta the Libray of the University of Manitoba ta tend or seU copies of this thesis/practicum, to the National Library of Canada to microfdm this thais and to lend or seU copies of the fim, and to Univenity MicmCilmb Inc, to publish .a abtract of this tbesislpracticum.
This reproduction or copy of tbis thesis b u ban made avaüabk by authority of the copyright owner solely for the purpose of private stady and raearch, and may oniy bc reproduced and
copied as permitted by copyright hws or with express wntrCn aothorizatbn from the copflght owner.
To my parent8 and family
Acknowledgement s
1 wish to express my deep appreciation to Professor Peter McLaren. 1 consider myself
plivileged to have had the opportunity to work under his guidance. 1 wish to thank
him for his counsel, guidance, patience and encouragement during the course of this
work.
1 wish to thank Professor Aniruddha Gole for all the support and guidance. He was
always willing to share his knowledge of synchronous machine modeling, despite his
busy schedule. Professor M. R. Raghuveer is also thanked for his usehil comments
and suggestions.
Professor Rohan Lucas deserves a special word of thanks for encouraging me to pursue
my doctoral studies a t the University of Manitoba and ananging the opportunity to
do so.
1 am thankfid to the test of the staff and my colleagues at the Power Systems Research
Group. 1 acknowledge with great appreciation, my friend Dr Rohitha Jayasinghe, who
was never too busy whenever 1 needed information. Professor Udaya h a k k a g e , in
addition to his encouraging ways provided expertise on CT modeling. Pradeepa, Va-
jira, Waruna, Namal and Sudath were generous and could be counted upon whenever
1 needed their valuable tirne.
This acknowledgement would not be complete without thanking my family. 1 extend
my heart felt gratitude to my parents. They were aiways there for me and were always
understanding. 1 th& them, my brother and my sister for all the love and support.
My wife, Punya, gave up her career so that 1 could continue my studies. 1 wish to let
her know that she was the source of inspiration which kept me going. Thank you.
Abstract
The purpose of this research is to develop models which can be used to produce
redis tic test waveforms for the evaluat ion of protection syst ems used for generators
and transformers. Software models of generators and transformers which have the
capability to calculate voltage and current waveforms in the presence of interna1
faults are presented in this thesis.
The thesis dso presents accurate models of current transformers used in differential
current protection schemes. These include air gapped current transformers which are
widely used in transformer and generator protection.
The models of generators and transformers can be used with the models of current
transformers to obtain test waveforms to evaluate a protection systern
The models are validated by comparing the results obtained fiom simulations with
recorded waveforms.
Table of contents
... Acknowledgements ln
Abstract iv
1 Introduction 1.1 S u m m q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Synchronous generators 1.3 Protection of synchronous generators . . . . . . . . . . . . . . . . . . 1.4 Protection of the stator winding . . . . . . . . . . . . . . . . . . . . . 1.5 Methods usec! to protect the stator winding . . . . . . . . . . . . . .
1.5.1 Protection against turn to ground faults . . . . . . . . . . . . 1.5.2 Protection against tum to tum and phase to phase faults . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Transfomers 1.7 Methods used to detect interna1 faults in transformers . . . . . . . . . 1.8 Current transfomers in protection schemes . . . . . . . . . . . . . . . 1.9 The need for machine and transformer models to sirnulate internal faults
2 Interna1 fault simulation in synchronous machines 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Siimmary 22
2.2 Overview of the avaiiable machine models . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . 2.3 Some fundamental aspects of machine modeling 23
2.3.1 Transformation of the Phase windings and the Park's transfor- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mation 26
2.4 Drawback of the d-q-O approach in the presence of internal winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fadts 30
. . . . . . . . . . . . 2.5 Phase domain mode1 of a synchronous machine 35
3 Development of a machine mode1 for the analysis of interna1 faults 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Srimmary 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Introduction 40
. . . . . . . . . . . . . . . 3.3 Overview of the available machine models 41 . . . . . . . . . . . . . . . . . . 3.4 Description of the machine windings 42
3.5 Calculation of the inductances involving the fadted coils . . . . . . . 46 3.6 Estimation of the leakage inductances of the windings . . . . . . . . . 51 3.7 Results and cornparisons . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Interna1 fadt simulation ia tramformers 58 4.1 Stimmary . . . . . . . . . . . . . . . . . . . , . O . . . . . . . . . . . 58 4.2 Introduction . . . . . . . . . O . . . . . . . . . . . . . . . . . . . . . - 58 4.3 Simulation of intemal faults in transformers . . . . . . . . . . . . . . 60
4.3.1 General . . . . . . . . . O . . . . . . . . . . . . . . . . . t . . 60 4.3.2 Intemal tum to ground fault . . . . . . . . . . . . . . . . . . . 61
4.4 A method to calculate the leakage inductance of two windings wound on the same leg of a transformer . . . . . . . . . . . . . . . . . . . . . 68
. . . . . . 4.5 Magnetic saturation and hysteresis in the transformer core 72 . . . . . . . . . . . 4.5.1 Modeling saturation and hysteresis effects 73
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Simulation results and observations 80 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Synchronous machine . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . 5.2.1 Consistency of the equations derived for the faulted coiis 80 5.2.2 Simulation results for interna1 faults . . . . . . . . . . . . . . . 83
5.3 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LOO 5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
. . . . . . . . . . . . . . . 5.3.2 Simulation results for interna1 faults 101 . . . . . . . . . . . . . . . . . . . . 5.3.3 Saturation in transformers 111
6 Application of the machine model and the transformer model in protection studies 119 6.1 Siimmary . . . . . . . . . . . . O . . . . . . . . . . . . . . - . . . . . 119 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3 Effects of saturation in current transformers . . . . . . . . . . . . . . 120 6.4 Behavior of air gapped current transformers . . . . . . . . . . . . . . 122 6.5 Behaviour of three current transformers connected in delta for trans-
former difFerential protection . . . . . . . . . . . . . . . . . . . . . . . 128 6.6 Behaviour of several ielaying schemes used in machine and transformer
protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7 Conclusions 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Surnmary 138
7.2 Main contributions of the thesis . . . . . . . . . . . . . . . . . . . . . 138 7.3 Recommendations for futther work . . . . . . . . . . . . . . . . . . . 140
vii
Appendices
A Elements of the inductance matrix of a synchronous machine 141
B Solution to the machine equations using Trapizoidal integration 143
C Conversion of the d-q-O data to the a-b-c domain 145
D Inductances of coils sharing a common flux path 149
E Elements of the inductance mat* of the four pole synchronous machine 152
F Calculation of winding inductance parameters for simulation of intemal faults in synchronous machines 154 F.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.2 Description of the machine windings and the inductances under normal
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.3 Description of the machine windings and the inductances in the pres-
ence of a turn to ground fault . . . . . . . . . . . . . . . . . . . . . . 159 F.4 Inductances between the winding Al and the normal windings. . . . . 162 F.5 Inductances of the coils of phase A. . . . . . . . . . . . . . . . . . . . 163 F.6 Mutual inductances between A l and the other coils in Phase A . . . 170 F.7 Self inductances of coils A2, A3 and A4 . . . . . . . . . . . . . . . . . 173 F.8 Mutual inductances between A2, A3 and A4 . . . . . . . . . . . . . . 173 F.9 Mutual inductance between the phase B winding and a coil in phase A 179 F.10 Mutual inductance between the phase C winding and a coil in phase A 184 F.ll Mutud inductance of the field winding with the coils of phase A . . . 184 F.12 Mutual inductance of the the d-axis damper winding with the coils of
phaseA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 F. 13 Mutual inductance of the the q-axis damper winding with the coils of
phase A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 F.14 The calculated inductances . . . . . . . . . . . . . . . . . . . . . . . . 190
G Leakage inductance of the machine windings 196 G.l Slot leakage calculations . . . . . . . . . . . . . . . . . . . . . . . . . 196 G.2 End leakage calcdations . . . . . . . . . . . . . . . . . . . . . . . . . 200 G.3 Air gap leakage calculations . . . . . . . . . . . . . . . . . . . . . . . 201
H Modeling saturation in the transformer core in fault studies 203 H.1 Turn to turn fault in a single phase transformer . . . . . . . . . . . . 203
1 The simulation mode1 of current transformers in a delta configuration 206
J Inductance of a part of a phase winding on a spiral wound machine 212
K Details of the machine 216
L Equations to show the decay of the de offset, ratio error and the phase &if% in air-gapped CTs 218
M Simulation mode1 of the air gapped CT 224 M . l Derivation of the simulation model of an air gapped CT . . . . . . . . 224 M.2 Derivation of the B-H data for the magnetic material . . . . . . . . . 229 M.3 Equations for the three air gapped CT connection . . . . . . . . . . . 230
N Cornparison of simulation results for the machine mode1 235
O Useful formulae and derivations 241
Bibliography 244
List of Figures
1.1 Fault current path for a turn to ground fault . . . . . . . . . . . . . . 1.2 Fault current path for a phase to phase fault . . . . . . . . . . . . . . 1.3 Fault current path for a turn to turn fault . . . . . . . . . . . . . . . 1.4 Fault current in a grounded generator . . . . . . . . . . . . . . . . . . 1.5 Low impedance grounding of a generator . . . . . . . . . . . . . . . . 1.6 High impedance grounding of a generator through a distribution tram-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . former .... 1.7 Ground differential protection of a Iow impedance grounded generator 1.8 Third harmonic undercurrent det ection scheme . . . . . . . . . . . . . 1.9 Sub-harmonic voltage injection method . . . . . . . . . . . . . . . . . 1.10 Percentage differential relay connection and the operating characteristics 1.11 Self balancing protection scheme . . . . . . . . . . . . . . . . . . . . . 1.12 Split phase protection using six curent transformers . . . . . . . . . 1.13 Split phase protection using three single window current tram formers 1.14 Restricted earth f ad t protection on a transformer star winding . . . . 1.15 A biased difFerentia1 scheme for a delta - star transformer . . . . . . 1.16 A typical bias characteristic . . . . . . . . . . . . . . . . . . . . . . .
Representation of the machine coils and the direction of their magnetic axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the machine as a system of six magnetic* coupled
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coils Representation of the machine coils referred to the direct and quadra- tureaxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position of the phase windings and the referred windings . . . . . . . Winding diagram of a three phase 4 pole machine with 6 slots per pole Phase A of the winding shown in Figure 2.5 . . . . . . . . . . . . . . Position of the phase A conductors and the directions of the magnetic axes of the different coils . . . . . . . . . . . . . . . . . . . . . . . . . Coils of the phase A winding . . . . . . . . . . . . . . . . . . . . . . . Representation of the fault at F1 and the directions of the magnetic axes of the phase A coils . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Representation of the fault at F3 and the directions of the magnetic . . . . . . . . . . . . . . . . . . . . . . . . . axes of the phase A coils
2.11 Representation of the fault at F 2 and the directions of the magnetic . . . . . . . . . . . . . . . . . . . . . . . . . axes of the phase A coils
2.12 Representation of the fault at F4 and the directions of the magnetic axes of the phase A coils . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Representation of a generator connected to a remote source . . . . . . 2.14 Comparison of results derived using the d-q-O domain model and the
a-b-c domain mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Comparison of results derived using the d-q-O domain model and the
a-b-c domain mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Comparison of results derived using the d-q-O domain rnodel and the
a-b-c domain mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Short circuit current envelopes of the phase currents . . . . . . . . . . 2.18 Short circuit current envelopes of the d and q axis winding currents .
3-1 The stator winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Positions of the phase A conductors inside the stator . . . . . . . . . . 44 3.3 Representation of the phase A winding with a fault on one parallel path . 44 3.4 Representation of the machine coils under an interna1 short circuit . . 45 3.5 Two parailel coils sharing the same magnetic path . . . . . . . . . . . 47 3.6 One patallel path of Phase A . . . . . . . . . . . . . . . . . . . . . . 48 3.7 Flux pattern due to current in coi1 X . . . . . . . . . . . . . . . . . . . 49 3.8 Slot leakage and àiEerentia1 leakage fields . . . . . . . . . . . . . . . . 52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 End leakage field 52 3.10 Atumtoturnfaui t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.11 A fault involving the two parallel paths of phase A . . . . . . . . . . 55 3.12 A fault between phases A and B . . . . . . . . . . . . . . . . . . . . . 56 3.13 Phase A to ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.14 Two phase fault between phases A and B . . . . . . . . . . . . . . . . 56
. . . . . . . . 3.15 Phase A to ground fault with a low grounding resistance 57
4.1 Single phase transformer cores . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Three phase transformer cores . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Cross section of a three phase two winding transformer . . . . . . . . . 60 4.4 The transformer represented as six coupled coils . . . . . . . . . . . . 60
. . . . . . . . . . 4.5 Representation of a turn to ground fault in coi1 (1) 61 4.6 Schematic representation of the six coils . . . . . . . . . . . . . . . . 65 4.7 Three coils wound on a common magnetic core . . . . . . . . . . . . . 65
. . . . . . . . 4.8 Internai fault waveforms for a single phase transformer 67 . . . . . . . . . 4.9 Interna1 fault waveforms for a three phase transformer 68
4.10 Leakage flux pattern inside a transformer . . . . . . . . . . . . . . . . 69
4.11 Non-linear characteristics of the core . . . . . . . . . . . . . . . . . . 4.12 B - H loop of a transformer core . . . . . . . . . . . . . . . . . . . . 4.13 M - He loop of a transformer material . . . . . . . . . . . . . . . . . 4.14 Turn to turn fault in a single phase transformer . . . . . . . . . . . . 4.15 Circuit representation of a turn to tum fault in a single phase trans-
former . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - . . 5.1 Cornparison of the extemal fault waveforms derived using the normal
machine model and the machine model developed to simulate intemal faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 A part of phase A of a six pole concentric winding . . . . . . . . . . . 5.3 Two magnetic circuits to represent a portion of a concentric winding . 5.4 A tuni to ground fault on phase A . . . . . . . . . . . . . . . . . . . 5.5 Current in the machine windings for a turn to ground fault at 20% fiom
the neutral with the field de-energized before the machine is isolated from the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Current in the machine windings for a tum to ground fault at 20% fiom the neutral with the field de-energized before the machine is isolated from the system . . . . . . . . . . . . . . . . . . . . . - . . . . . . . .
5.7 Current in the machine windings for a tum to ground fault at 20% fkom the neutral with the field de-energized before the machine is isolated from the system . . . . . . . . . . . . . . . . . . . . . . . . . . . - - -
5.8 Current in the machine windings for a turn to ground fault at 20% fiom the neutral with the field de-energized before the machine is isolat ed from the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Current in the machine windings for a turn to ground fault at 20% fiom the neutral with the field de-energized before the machine is isolated from the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Effect of winding resistance on the fault curent. . . . . . . . . . . . . 5.11 Current in the machine windings for a turn to ground fault at 20% from
the neutral with the machine isolated before the field is de-energized . 5.12 Current in the machine windings for a turn to ground fault at 20% from
the neutral with the machine isolated before the field is de-energized . 5.13 Inauence of the position of the fault and the grounding impedance on
the currents in faulted windings . . . . . . . . . . . . . . . . . . . . . 5.14 Effect of the higher order terms in the self inductance expressions with
L4 set to 30% of L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Effect of the higher order terms in the self inductance expressions with
Lq set to 10% of L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 A turn to tum fault on phase A . . . . . . . . . . . . . . . . . . . . . 5.17 A turn to turn fault involving 7.5% of the winding with Rtit equal to
0.1 Ohms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.18 A tum to turn fault involving 10% of the winding with RIIt equal to 1 Ohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.19 A turn to tuni fault involving 10% of the wïnding with Rtit equal to 1 Ohm, with the machine operating close to its MVA rating of 160 . . 99
5.20 A turn to turn fault involving 0.30% of the winding with Rtrt equal to I Ohm, with the machine operating close to its MVA rating of 160 . 100
5.21 Muence of the load current on the current in the faulted winding . . 101 5.22 Muence of the number of shorted t u n s on the current in the faulted
winding . . - . . . . . . . . . - . . . . . . . . . . . . . - . . . - . . . 101 5.23 Interna1 fault in a star-star connected transformer . . - . . . - . . 102 5.24 A turn to ground fault 5% from the terminal on the phase A winding 102 5.25 A tum to ground fadt 5% from the terminal on the phase A winding 103 5.26 A turn to ground fadt 5% from the neutral on the phase A winding . 103 5.27 A turn to ground fault 5% fiom the neutral on the phase A winding . 104 5.28 A turn to ground fault 5% fkom the neutral on the phase A winding
with the transformer grounded through an impedance of 20 Ohms . . 105 5.29 A turn to ground fault 5% from the neutral on the phase A winding
with the transformer grounded through an impedance of 20 Ohms . . 105 5.30 Interna1 fault in a star-delta connected transformer . . . . . . . . . . 106 5.31 A turn to ground fault 5% fiom the terminal on the phase A winding
with the secondary side connected in delta . . . . . . . . . . . . . . . 106 5.32 A turn to ground fault 5% fiom the terminal on the phase A winding
with the secondary side connected in delta . . . . . . . . . . . . . . . 107 5.33 A turn to ground fadt 5% fiom the neutral on the phase A winding
with the secondary side connected in delta . . . . . . . . . . . . . . . 107 5.34 Intemal fault in a delta-star connected three phase transformer . . . 108 5.35 A turn to ground fault 31% from the phase B terminal on the delta sidel08 5.36 A turn to ground fault 31% from the phase B terminal on the delta sidel09 5.37 A turn to gound fault 31% from the phase A terminal when both sides
are connected in star . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.38 A turn to ground fault 31% from the phase A terminal when both sides
are connected instar . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.39 A turn to turn fault on the delta side of a transformer . . . . . . . . . 111 5.40 A turn to turn fault involving 1% of the winding . . . . . . . . . . . . 112 5.41 A turn to tum fault involving 1% of the winding . . . . . . . . . . . . 112 5.42 A turn to turn fault involving 10% of the winding . . . . . . . . . . . 113 5.43 A turn to turn fault involving 10% of the winding . . . . . . . . . . . 113 5.44 A turn to tum fault involving 10% of the winding with the transformer
supplying a higher load . . . . . . . . . . . . . . - . . . . . . . . . . . 114 5.45 A turn to tuni fault in a single phase transformer . . . . . . . . . . . 114 5.46 The shape of the B- H loop of the transformer core material . . . . . 115 5.47 Magnetizing current when the core is saturated . . . . . . . . . . . . 116
5.48 A tum to turn fault involving 4% of the winding . . . . . . . . . . . . 5.49 The remanent flux in the core . . . . . . . . . . . . . . . . . . . . . . 6.1 A single CT connected to a burden . . . . . . . . . . . . . . . . . . . 6.2 Fault currents with an initial dc exponential component for a fault
occuring close to the generator . . . . . . . . . . . . . . . . . . . . . . 6.3 Secondary currents in the CT under different burdens . . . . . . . . . 6.4 Flux in the CT core under different conditions . . . . . . . . . . . . . 6.5 Secondary currents in the CT when air gapped CTs are employed . . 6.6 Decay of the primary current and the flux in air gapped CTs . . . . . 6.7 Primary and the secondary current in the CT to demonstrate the ratio
error and the phase angle error . . . . . . . . . . . . . . . . . . . . . . 6.8 Flux in the CT core when air gapped CTs are employed . . . . . . . 6.9 Schematic diagram of an air gapped CT . . . . . . . . . . . . . . . . 6.10 Air gapped CT feeding a relay burden . . . . . . . . . . . . . . . . . 6.11 CS with a 0.03% air gap . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 CT with a 0.2% air gap . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 CT with a 0.2% air gap . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Three CTs comected in delta . . . . . . . . . . . . . . . . . . . . . . 6.15 Comparison of the calculated wavefonns with measured data to d i -
date the delta CT mode1 . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Comparison of the calculated waveforms with measured data to vali-
date the delta CT mode1 . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Comparison of the calculated wavefonns with measured data to vali-
date the delta CT mode1 . . . . . . . . . . . . . . . . . . . . . . . . . 6.18 Harmonies present in the secondary and line currents . . . . . . . . . 6.19 Currents during a three phase fault . . . . . . . . . . . . . . . . . . . 6.20 Restricted earth fault protection on a transformer star winding . . . . 6.21 The effect of remanence on the relay current during external faults . . 6.22 Relay curent in the presence of an interna1 turn to ground fault on
the star side of the transformer . . . . . . . . . . . . . . . . . . . . . 6.23 Relay current in a restricted earth fault protection scheme . . . . . . . 6.24 Dinerential relay cments due to a tum to ground fault on phase A .
C.1 Typical voltage and current wavefonn recordings from a slip test . . .
D.1 Three coils wound on the same core . . . . . . . . . . . . . . . . . . . D.2 Three coils wound on the same core with two of them comected in
parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.l Winding diagram of the four pole machine . . . . . . . . . . . . . . . . F.2 Rotor arrangement of a four-pole synchronous machine with salient
poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.3 Schematic of the winding arrangement . . . . . . . . . . . . . . . . . 156 F.4 Schematic diagram of the six coupled coils of the machine . . . . . . . 157 F.5 Placement of conductors inside the stator slots . . . . . . . . . . . . . 158
. . . . . . . . . . . . . . . . . . . . . F.6 The coils of the phase A winding 159 F.7 Representation of the phase A winding with a fault on one parallel path.159 F.8 Representation of the machine coils under an interna1 short circuit . . 160
. . . . . . . . . . . . . . . . . . . . . . . . F.9 Winding X and arinding Y 163 . . . . . . . . . . . . . . . . . . F.10 Flux pattern when coi1 X is energized 167 . . . . . . . . . . . . . . . . . F.l l Flux pattern when coi1 A l is energized 171
F.12 Flux pattern when coi1 A3 is energized . . . . . . . . . . . . . . . . . 177 F.13 Flux pattern when Phase B is energized with the rotor displaced by an
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . angle of 15O 180 F.14 Flux pattem when the field winding is energized with all other coils
. . . . . . . . . . . . . . . . . . . . . . . . . . . . kept open circuited 185
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.l Slot leakage flux 197 . . . . . . . . . . . G.2 Slot leakage flux in a double layer wound machine 199
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.3 End leakage flux 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4 Air gap leakage flux 202
H . 1 Representation of a tum to turn fault on a single phase transformer . 203
1.1 Three delta connected current transformers feeding a star connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . burden 206
1.2 Simplified schematic diagram of the three delta connected CTs . . . . 207
. . . . . . . . . . . J.1 Part of a spiral winding of a synchronous machine 212 J.2 A simple magnetic circuit to represent the two spiral wound coils . . 215
L.l Schematic diagram of an air gapped CT . . . . . . . . . . . . . . . . 219 . . . . . . . . . . . . . . . . . L.2 Air gapped CT feeding a relay burden 219
M.l Air gapped CT feeding a relay burden . . . . . . . . . . . . . . . . . 224 M.2 Schematic diagram of an air gapped CT . . . . . . . . . . . . . . . . . 225 M.3 flux-mmf cuve of an air gapped CT . . . . . . . . . . . . . . . . . . 229 M.4 Three CTs connected to a relay . . . . . . . . . . . . . . . . . . . . . 231
N.l The 6 pole machine with tappings . . . . . . . . . . . . . . . . . . . . 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . N.2 A tum to ground fault 236
. . . . . N.3 A turn to ground fault with a 2.5 Ohm grounding resistance 236 N.4 A turn to ground fault with a 15 Ohm gounding resistance . . . . . 237 N.5 A turn to ground fault with a 3 Ohm grounding resistance and with
the field current set to 110 % of its rated value . . . . . . . . . . . . . 237 N.6 A fault between phases A and B . . . . . . . . . . . . . . . . . . . . . 238
N.7 A fault between phases A and B . . . . . . . . . . . . . . . . . . . . . 238 N.8 A fault between phases A and B . . . . . . . . . . . . . . . . . . . . . 239 N.9 A fault between phases A and B . . . . . . . . . . . . . . . . . . . . . 240
0.1 Coils wound on a magnetic core . . . . . . . . . . . . . . . . . . . . . 241
List of Tables
4.1 Comparison of the measured and calculated leakage inductances . . . 73
List of Symbols
Inductance matrix of the synchronous machine
Inductance mat& of the faulted synchronous machine
Flux linkage vectot of the synchronous machine
Flux linkage vector of the faulted syncbronous machine
Position of the rotor in electrical radians with respect
to the axis of the phase A winding
Position of the rotor in radians
Effective number of tums per phase of a synchronous machine
Instantaneous current in a winding
Instantaneous voltage across a winding
Park transformation matrix of a synchronous machine
constants used to describe the modulation of coi1
inductance with the rotor position
Leakage inductance of a phase winding of a synchronous machine
Self inductance of winding k
Mutual inductance between windings j and k
Number of poles in a synchronous machine
Leakage inductance of winding k
Magnetizing component of the self inductance of winding k
Stator dot angle of a synchronous machine
Number of tums in winding k
Number of series tunis per phase of a synchronous machine
Pole pitch of a synchronous machine
Winding pitch of a synchronous machine
Diameter of the air gap of a synchronous machine
Leakage factor between windings x and y in a transformer
Leakage inductauce of windings x and y referred to winding x,
in a transformer
Slope of the M-H curve of a magnetic material
Magnetization of a magnetic material
Magnetic flux density
Magnetic field intensity
Magnetic flux
Reluctance of the air gap between the pole face and the stator
of a synchronous machine
Inductance matrix of the transformer
Resistance matrix of the transformer
Chapter 1
Introduction
The purpose of this research is to develop models which will allow the engineer to
produce realistic test waveforms for the evaluation of protection systems used for
generators and transformers. Models of generators and transformers capable of r e p
resenting internal faults are the most important elements of the project but an accu-
rate mode1 for a current transformer used in dinerential current schemes also requires
attention. Chapter 1 outlines the motivation and the reason for undertaking such
a project despite the fact that accurate models of synchronous machines and power
transformers are already avaiiable for system studies. This chapter also describes the
importance of protecting these two important elements in an electrical power system
and examines the various protection schemes employed to detect faults when a prob-
Iem occurs inside the unit or in the system to which it is comected. The causes for
internal faults and the resulting adverse effects on the power system and the faulted
unit are aIso briefly addressed. The current transformer plays an important role in
the performance of a protection scheme and should be accurately represented when
studying a protection scheme. It is very cornmon in machine and transformer protec-
tion to employ air gapped current transformers. Models of solid core and air gapped
current transformers are presented in Chapter 6.
Chap ter 1
1.2 Synchronous generators
The synchronous generator can be described as one of the most important pieces of
equipment in an electrical power system. In an interconnected system, the proper
functioning of the generators is critical in maintainhg an uninterrupted power s u p
ply to the customer. The ever increasing demand for electrical energy has made it
essential for most systems to be operated close to their capacity limits. In such a
situation, the maloperation of a generator can cause the system to become unstable
leading to possible supply interruptions. Undesirable conditions can occur inside the
generator due to fadts in the extemal system to which it is connected and also due to
faults inside the generator itself. It must be properly protected so that any abnormal
condition is detected quickly, enabling corrective measures to be implemented. An
interna1 fault usually means that the machine is already damaged. Protection will
limit further damage by de-energising the machine.
The protection of synchronous generators involves the consideration of more harmhil
abnormal operating conditions than the protection of any other power system ele-
ment. As a result, the protection scheme of a generator connected to the system is
complicated. The concern that this complicated scherne will operate when it should
not is quite valid. However, even though an unnecessary tripping of the generator
is not desirable, the consequences of not tripping it in the presence of a fault and
damaging the machine are far worse [Il. The generators in a system are not all iden-
tical. They ciiffer in size, type, winding design and many other features. Thus, any
given generator must be studied with due attention to its design before the protection
engineer can corne up with the appropriate protection. The engineer needs a variety
of information about the generator, the system to which it is connected, the method
used to ground the generator, the equipment to be used in the protection scheme and
the type of loads to which the generator is expected to be connected.
Chap ter f
1.3 Protection of synchronous generators
A generat or protection scheme has t O consider many possible abnormal conditions.
Some typical situations requiring protection are listed below [4].
1. Fauits in the stator winding
These are caused by the failure of the insulation between the conductors
or by the faiiure of the insulation between a conductor and the iron core.
2. Faults in the rotor windings
The insulation of the rotor windings can break down giving rise to ground
faults. Open circuit faults can occur in the rotor circuit due to damage to
the structural parts of the rotor.This is a consequence of heating due to
unbalanced stator currents and over-speed.
3. Abnormal currents in the stator windings due to faults in the extemal system
Extemal short circuits can cause large currents through the stator. Gen-
erators which are grounded through a low impedaace are the ones at a
higher risk.
This is mainly due to a sudden loss of load caused by a transmission line
tripping.
This is also caused by a sudden loss of load.
6. Motoring
This is caused by the la& of sutncient prime mover energy.
7. Mechanical faults
Chapter 1 4
These include malfunctioning of the cooling system, vibration and bearing
pro blems.
1.4 Protection of the stator winding
In order to set the protective relays to recognize winding faults, it is necessary to have
a prior knowledge of the current and voltage waveforms which would occur under such
situations. This is especiaiiy tme in the case of turn to ground faults as the fault
current depends on the location of the fault as well as on the method used to ground
the generator. If the f ad t is close to the neutral or if the grounding impedance is high,
the fault current would be small and usually less than the sensitivity of the differential
relay. Turn to turn faults in the same phase are not detected by dinerential relays
and split phase relaying is generaily used to detect them.
The dielectric strength of the insulation gets weakened due to factors such as age,
presence of corona inside the machine, presence of moisture, baking of the insulation
and accumulation of dirt. This can ultimately lead to interna1 short circuit faults.
Over-voltage too can seriously damage the winding insdation and give rise to internai
faults. The over-voltages are caused by lightning surges, switching surges, or over
speed and over voltage due to a sudden loss of load.
A wlliding fault in the stator is considered serious as it causes severe damage to the
winding itself and possibly to the shafts and couplings of the machine [2]. When
a fault is detected inside the windings, it is necessary to isolate the machine from
the rest of the system and to de-energise the field supply. If the field is not shut
off, the fault current will continue to flow inside the machine as can be seen fkom
the schematic diagrams shown in Figures 1.1, 1.2 and 1.3. The fault current wil l
continue to flow for a kw cycles even after the machine is isolated and the field shut
off, because of the energy stored in the magnetic field inside the machine [2].
Heavy fault currents can badly weaken the insulation of the stator. If the fault can
be cleared before the laminated core is affected, then the repair cost and the repair
Chapter 1
FiIed current Phase A winding
I 1
Phase 0 winding I 1 1 I I 1 1
Phase C winding I I I
_L Fault current Breaker fault
-
Figure 1.1: Fault
- current path for a tum to ground fault
Field winding
d d Filed current
Phase A winding
I I 1 I
Fault cumnt 1 -
Figure 1.2: Fault
Phase to phase fault - - - - I Breaker
current path for a phase to phase fault
Field winding
Filed current Phase A winding - - - - -
I , I I
Phase B winding I 1 I I I 1 I I I I
Phase C winding I I I I I
l - - - - l
Fault currenc Breaker T m to mm fault fauIt
-
Figure 1.3: Fault current path for a turn to turn fadt
Chapter 1
1 Fault current
O Grounding impedance t i f Fault current fmm the remote system
Figure 1.4: Fauit current in a grounded generator
t h e will be relatively low. However, if the laminations are dected, the consequences
are much more severe. The asymmetric nature of the winding faults gives rise to
unbalanced currents which, in-turn, exert unbalanced magnetic forces. These forces
cause the machine to vibrate and, as a result, the shaft and couplings of the machine
can be damaged. Also, the windings can be displaced fiom their original location
inside the slots. The possibility of a f i e is another concern and the fault should be
cleared before there is a chance for this to occur. To minimize the damage arising
Fom interna1 faults they are detected and cleared in the least possible tirne. Several
protection strategies are employed to achieve this objective and to protect the stator.
1.5 Methods used to protect the stator winding
1.5.1 Protection agaznst turn to ground faults
The fault current in the event of a turn to ground fault is influenced by the method
used to ground the generator, as can be seen Fom Figure 1.4.
If the generator is solidly grounded, an external line to ground fault will produce
a very high magnitude fault current in the machine windings. In an ungrounded
system, the fault current is generdy extremely small. However, the neutral point
voltage can shift to a high value in the presence of an external line to ground fault
and, as a result, the winding insulation may experience a large stress that may lead
Chap ter 1
H-V system
Generator transformer
Figure 1.5: Low impedance grounding of a generator
to an insulation failure. Both the above situations have the potential to extensively
harm the machine and, thus, these two grounding methods are almost never used in
practice.
The grounding impedance should ideally be sufficient to limit any fault curent to a
moderate level and a t the same time it should not be too large to cause severe over
voltages. Two methods widely used in practice are the low impedance grounding
shown in Figure 1.5 and high impedance grounding, shown in Figure 1.6.
Neutra1 over-voltage det ection method
This met hod is shown in Figure 1.6 and it is used in high impedance giounded systerns.
The relay is connected across the grounding resistance and can detect the fundamental
voltage developed due to the fault current in the neutral lead. Since the grounding
resistance is much higher than the generator leakage reactance and resistance of the
stator windings, the voltage of the faulted phase will be impressed across it. The
magnitude of the voltage available for detection will get smaller as the fault gets
closer to the generator neutral. As a consequence, faults that are very close to the
H.V system I
T Generator transformer
Generator
Dism%ution transformer
-Tirne delayed over voltage relay
Grounding resistance
Figure 1.6: High impedance grounding of a generator through a distribution trans- former
neutral, typicdy up to 4% nom it, cannot be successfully detected by this scheme
[5] . The t h e delay of the relay should be coordinated with the other relays in the
system t O minimize the possibility of mwanted operation.
DifFerential protection of the stator winding
Phase diaerential relays can provide ground fault protection for faults close to the
machine terminais. However they will not detect phase faults that are closer to the
neutral. The ground dinerential scheme shown in Figure 1.7 is used in low impedance
grounded systems to detect tum to ground faults. The current in the neutral CT is
fed to the restraining coi1 and the differential current is fed into the operating coii of
the relay. This scherne is extremely sensitive to intemal ground faults.
Chapter 1
Generator CT- 1 n-
I l I l
Cr-2
relays
OC - Operating coi1
RC - Restraining coi1
Figure 1.7: Ground differential protection of a low impedance grounded generator
Third harmonic detection method
The two methods stated previously cannot be reliably used to detect ground faults
which are very close to the neutrai. The third harrnonic detection method uses the
fact that, for most machine designs, the third harmonic current in the neutral wire for
such faults is very small. Thus, a third harmonic under-voltage detection relay is used
with filters tuned to the appropriate fiequency to detect faults which are very close
t O the neutral. The conventional fundamental over-voltage detect ion relay provides
protection for the rest of the winding. Thus 100% protection c a n be attained for the
winding using a combination of these methods. This technique is illustrated in Figure
1.8.
Sub-harmonic voltage injection met hod
This is another method used to provide 100% winding protection against stator
ground faults. It is used in situations where the machine design does not give rise to
sufficient third harmonic voltages to use the third harmonic detection method. In this
scheme a sub-harmonic voltage signai, synchronized with the generator fkequency, is
injected into the neutrd of the machine. This signal is provided by a separate source
Chap ter 1
Grounding unit
# Generator
RI - Over voltage relay R2 - Third harmonic undervoltage relay
Figure 1.8: Third harmonie undercurent detection scheme
Generator
li Grounding resistance
G - Sub-hannonic source LP - Lowpass filter R3 - Relay to detect sub-barmonic c w e n t
Figure 1.9: Sub-harmonic voltage injection method
Chapter 1 11
and this is shown in Figure 1.9. A 15 Hz signal in a 60 EIz system is typical. The
resulting sub-harmonic current in the neutral is increased in the presence of a ground
fault and this is used to operate the relay. This scheme can provide protection to the
entire winding and its performance is independent of the operating conditions of the
generator. The disadvantage of the method is the need to provide a sub-hannonic
source. This method can be used to detect open circuits in the neutral circuit because
an open circuit wiU reduce the harmonic current to zero.
1.5.2 Protection against tvrn to t u m and phase to phase faults
The fault currents in the cases of tuni to turn fauits and phase to phase faults are
limited only by the resistance and the leakage reactance of the tums linked to the fault.
The number of t u s involved aiso determines the induced voltage across the faulted
section which will drive this fauit current. Since these faults can cause severe damage
to the machine, high speed protection with no intentional time delay is generally
employed to detect them. High speed dinerential relays are used to detect phase to
phase faults. They can detect three phase and two phase to ground faults as well.
However, differential relay schemes cannot detect turn to turn short circuits on the
same phase since the curent at the two ends of the winding would be the same despite
the presence of the fault. Split phase relaying is used to detect turn to tum faults in
machines where there are two or more coils in parallel in each phase winding.
The current tramformers make up a very important portion of almost all protection
schemes. Their behaviour influences the performance of the relays and therefore
this must be carefdiy taken into consideration at the design stage of the protection
and when setting the relays. This is especially tme in Merential and split phase
protection since external fault currents with a sigxdicant dc exponential and high
magnitude, caused by faults close to the generator, can saturate the CTs leading to
malfunct ion.
The following methods can be used to protect the machine h m phase to phase and
turn to turn faults.
Chapter 1
OC - Operanng coi1 RC - Reshaining coi1
Figure 1.10: Percentage diEerentia1 relay conneetion and the operating characteristics
Percent age differential protection
This is outlined in Figure 1.10. This scheme is more tolerant to current transformer
errors and is widely used with larger machines.
High impedance differential protection
This is similar to the percentage differential relay conneetion. However there are no
restraining coils in the relay and the operating coil has a high impedance. The relay
responds to the voltage across the operating coil. Identical CTs with very low leakage
inductances should be used to feed the relays.
Chapter 1
Generator
w Relay Grounding
resistor
Figure 1.11: Self balancing protection scheme
Self balancing differential protection
This scheme is outlined in Figure 1.11. Under normal conditions the flux in the CT
core is zero and thus the relay current will be zero as well. Only three CTs are used
here and this scheme can detect phase to phase and turn to ground faults inside the
winding .
Split phase protection
Split phase protection is used to detect tum to turn faults in the same winding. Two
variations of this method are illustrated in Figures 1.12 and 1.13. The method can
only be employed in generators with two or more pardel windings per phase. The
parallel windings of each phase are grouped into pairs and the currents are compared.
A turn to turn fault WU resdt in unequal currents flowing in the coils of a group. An
instantaneous over-curent relay with a very inverse characteristic is generally used
here. This relay should be carefully set so that it can discriminate between interna1
faults and normal unbdanced conditions. The cunent transformer behaviour must be
considered carefidly when the relay settings are made as non-identical CT behaviour
WU cause the relay to operate for external faults.
Chap ter I
Figure 1.12: Split phase protection using six curent transformers
The method shown in Figure 1.12 uses two current transformers per phase while the
one shown in Figure 1.13 uses only one current transformer per phase.
1.6 Transforrners
The power transformer is an important apparatus in a power system. This has a
simpler construction compared to the synchronous machine and it is generally a very
reliable piece of equipment. This reliability is achieved through proper design, proper
construction, maintenance and the provision of an adequate protection system [6],
[7] . Like in a generator, the interna1 faults in transformers are considered serious. In
addition to damaging the winhgs and insulation, there is the risk of fire. System
instability due to prolonged voltage dips is another concern resulting fkom transformer
fadts not being detected and dealt with quickly.
The current due to a turn to ground fault in a transformer is determined by the
location of the fault and the grounding impedance, and aiso by the leakage reactance
of the windings. It is also dected by the three phase transformer connection. The
current in the case of turn to turn faults or phase fadts is determined by the fault
Chap ter 1
Figure 1.13: Split phase protection using t hree single window current t ransformers
location, the number of tums involved, the winding resistance and the leakage re-
actance of the faulted sections of the windings. These faults are caused mainly by
voltage surges arising due to switching or lightning. The citculating current within
the faulted loop can be very high. However, this current would not be seen at the
terminais of the transformer due to the high turns ratio between the wlliding and the
fadted section of the coil. This makes turn to turn faults hard to detect.
Although the transformer hasi a simple construction, it presents certain challenges to
the protection engineer. Some challenges which are unique to transformers are listed
below.
1. Different voltage levels
The voltages and currents on the two sides of the transformer are not
equal as they are related by the turns ratio. This is further complicated by
the taps available on most power transformers. The current transformer
ratios stiould be selected accordingly.
2. Mismatch among current transformers
Chap ter 1 16
The mismatch introduced by non-identical curent transfomers in a dif-
ferential protection scheme must be carefully considered to avoid false trip-
P ~ W S
3. Magnetizing in-rush current
The differential relays see the magnetizing in-rush curent as an internal
fault. This occurs when a transformer is switched on to the power system.
The harmonic content present in the in-rush cwrent is used to distinguish
it h m internal fadt currents.
4. Phase shift at the two ends of the transformer
When the transformers are connected in a delta-star configuration, it
introduces a phase shiR of 30° between the voltages at the two ends. The
current transformers shouid be connected in a manner which accounts for
this phase shift.
Three phase banks made out of single phase transformer units, zig-zag transformer
connections, auto-tramformers and multiple winding transformers make the task of
protecting a transformer more complicated.
1.7 Methods used to detect internal faults in transformers
Restricted earth fault protection
This is a relatively simple, economical way to protect the windings against internal
faults involving the ground. The method is outlined in Figure 1.14.
The relay operates only for faults in the star winding of the transformer. In the event
of a fault, the relay sees the whole fault current and this increases the reliability
of operation. The line currents flowing into delta connected windings or unearthed,
star connected windings will always add to zero unless there is a turn to ground
fault. Thus, normal earth fault protection can be applied to these windings. Here
Chapter 1
Transformer winding
- Figure 1.14: Restncted earth fault protection on a transformer star winding
the fourth CT on the neutral can be omitted. The restricted earth fault protection
can be applied separately to both sides of the transformer.
Difkrential protection
Differential protection is used almost universally on Iarger transformer banks to detect
winding faults. Many factors must be carefully assessed before the relays are set. The
current transformers must be selected with special attention given to their saturation
characteristics. Heavy external fault currents can saturate the CTs to dinerent leveis
and this can drive a Uerential current through the relay. Measures must be provided
to distinguish these fiom internal fault currents for which the relay is set to operate
Pl
Magnetic in-rush is another consideration in dinerential relay schemes for transformers
[IO]. This problem does not occur in generators since the generator voltage is built-up
gradually during initial startup. The in-rush current can be present when the unit is
energized, when an external fault is cleared or when it is connected in parailel with
a second bank. Since the relay sees this as an internal fault a suitable method must
Chap ter 1
- Biascoiis ' I l
Figure 1.15: A biased Merential scheme for a delta - star transformer
be provided to avoid misoperation.
Figure 1.15 shows the connection of the CTs in a differentid scheme of a delta - star connected three phase transformer. The CTs on the star side are connected in a
delta and those on the delta side are connected in a star. This accounts for the phase
shift between the currents on the two sides of the transformer. Since zero sequence
currents cannot flow in the iines on the delta side of the transformer, the CTs on the
star side must be connected in delta to eliminate any zero sequence currents being
diverted to the relay fkom this side.
The relays should be properly biased to account for tap changes and also for CT
mismatches. A typical biasing characteristic is shown in Figure 1.16 where more
current is required to operate the relay as the through fault current increases [7].
1.8 Current transformers in protection schemes
The behaviour of the current transformers influences the performance of protection
schemes in synchronous machine and transformers [2], [9]. Thus caretul attention
must be given to the selection of current transformers. Their characteristics should
be measured before they are connected. Some of the key factors which must be
addressed with regard to current transformers are listed below.
Chapter 1
Through fault current (pu)
Figure 1.16: A typicd b i s characteristic
1. Saturation
The flux-magnetizing current relationship of the current transformer core
is non-linear. As a result, heavy fault currents can drive the device into the
saturation region. This gives rise to ratio errors. Since the magnetizing
current required now is much larger than in the linear region, the primary
current will not be accurately reproduced at the secondary. This problem
is further complicated when the faults are close to the generator or the
transformer. The fault current in a case iike this is likely to be limited by
a highly inductive impedance, and, as a result, an initial exponential with
a long time constant can be superimposed on the fault current. This will
result in the CT being heavily saturated on one haIf cycle [51].
2. Hysteresis of the B-H loop
The B-H characteristic of the CT core displays hysteresis. This leads to
a remanent flux in the core once a fault is removed from the system. The
amount of remanent flux present on the core depends on the point on the
Chapter 1 20
wave of the CT secondary voltage when the primary current is interrupted.
The remanent flux could cause the relays to malfunction upon re-closure
of the breakers leading to false tripping.
3. Mismatch of current transformers in Merential protection
The current transformers on either side of the Merential scheme must
have identical characteristics. However two CTs with the same design will
show dissimilarities when their characteristics are measured [2]. These
dissimilarities must be measured and taken into consideration to avoid
misoperation. This is fkther complicated in transformer protection where
the two sides are at ditlFerent voltages and carry different currents.
Thus, in protection studies, the accurate mathematicai representation of the current
transformer behaviour is important. Models that use curve fitting techniques to
approximate the B -H characteristics give reasonable results but do not represent the
remanent flux accurately [12], [13]. Current transformer models based on the physics
of magnetic materials have been shown to produce very accurate results[ll], [3]. In
these the remanence is accurately represented and, as a result, can be used in relay
studies where a transformer or a generator is successively re-closed into a permanent
external fault [l4].
1.9 The need for machine and transformer models to simu-
late internd faults
The discussion thus far indicates the harmful effects of interna1 faults in machines and
transformers and the need to provide adequate protection to minimize the harmful
effects. It also indicates the complexities that arise when designing protection for
these equipment. The design and the features of large synchronous machines and
transformers are unique and thus each case must be studied independently. Also the
fault currents and voltages depend not only on the design but also on the location of
the fault, the external system which is connected to the device, and other factors. A
Chapter 1 21
prior knowledge of fault current and fault voltage waveforms is very usehi to the relay
engineer when designing a me thod t O provide protection for machines or transfonners.
Although a prior knowledge of current and voltage waveforms is extremely usehl
when designing a protection scheme and setting the relays, such waveforms are not
readily available. It is not practically viable to generate these waveforms using the
machine or the transformer concemed. Suitable machine and transformer models
which can simulate internal faults and produce such waveforms would thus be very
usefid to the relay engineer. These models should be able to sirnulate internal faults
taking into account the particular design, the fault type and location and the features
of the system to which it is connected.
Such models are not readily available at present and even the ones available do not
have the capability of taking into account most of the constructional features of the
devices rnentioned above and, as a result, have limitations. This research is aimed at
developing synchronous machine models and transformer models which c m accurately
simulat e int ernal fault s under different conditions. The waveforms derived from such
a mode1 can be used, dong with accurate current transformer models, to design
protection schemes and to properly set the relays.
Chapter 2
Interna1 fault simulation in
synchronous machines
This chapter investigates the existing synchronous machine models which are widely
used in power system studies. The limitations of these models in interna1 winding
fault studies are outlined. The need to develop the machine mode1 in the phase
domain is explained and a method to calculate the inductances involving the ma-
chine windings is presented. The machine equations are then solved using a suitable
numerical technique. To show the vaiidity of the direct phase domain approach, com-
parisons are made between the simulated waveforms obtained using this method and
those obtained using standard machine models based on the d-q-O transformation.
These cornparisons were done on external faults.
2.2 Overview of the available machine models
The machine models available on most of the electro-magnetic transient simulation
programs [17] are based on the two reaction theory and the resulting Park transfor-
mations [39] [18] [19]. This transformation makes use of the symmetrical nature of
the windings inside the stator. However, an interna1 fault divides the faulted winding
Chapter 2 23
into a number of sections. The symmetry which existed earlier is no-longer present
between these faulted sections and the rest of the machine windings. Thus, the above
models cannot be used in internal fault studies and the transformations cannot be
readily applied in such a situation. A machine model in the direct phase domain,
which is capable of extemal fault simulation, is presented in [20], [21]. The data
supplied by the manufacturer can be readily converted tu a form which can be used
in this model [22], [18].
2.3 Some fundamental aspects of machine modeling
A synchronous machine can be represented as a system of six coupled coils [18] as
shown in Figures 2.1 and 2.2. Here the damper winding is represented by two hy-
pothetical windings whose magnetic axes are at right angles to each other. Figure
2.1 shows the directions of the magnetic axes of the windings. The voltage cunent
relationship is governed by the followïng equation [18].
of rotation
/ C - Axis B - A x i s
Figure 2.1: Representation of the machine coils and the direction of their magnetic axes
Chap ter 2
where
The elements in the inductance matrix [Lsynl], and the matrix [Rsyni] are known
h m the data supplied by the manufacturer. [Rsynl] is a diagonal matrix and the
diagonal elements are the resistances of the six windings. In salient pole machines,
the elements of [Lsynl] depend on the position of the rotor and, hence, they are time
varying. The diagonal elements of the matrix give the self inductances of the machine
windings and the off diagonal elements give the mutual inductance between two given
windings. These elements take the form shown in Appendix A. The angle 8. is in
electrical radians.
Equations 2.1 and 2.2 should be solved using a suitable numericd integration tech-
nique to get the winding currents. If trapezoidal integration is used as the numerical
method, the curent vector is given by the following equation. The complete deriva-
tion is shown in Appendix B.
[Il(t)]sxl = [ G 1 ] 6 ~ 6 [ ~ 1 ( ~ - At) ]~x l - [Hl ]6r6 [~]6x l
The matrices G1 and Hl are given by the following equations.
Figure 2.2: Representation of the machine as a system of six magneticaily coupled coils
Chap ter 2
At -l [.II = [I - 7j- [Al] [I + $ [Al]
1 is the identity matrix and the matrices [A] and [BI are given by,
The matrices [Gi] and [Hl] are funetions of the inductance matrix [Lsynl]. Since the
elements of the matrix [Lsynl] depend on the position of the rotor, they are time
dependent. As a result the two matrices [G1] and [Hl] have to be evaluated at each
time step in order to determine the current vector [Il]. This is a heavy computational
burden, especially since the matrix [Lsynl] has to be inverted at every calculation
time step.
This drawback is overcome by representing the machine windings with equivalent
fictitious windings placed on two perpendicular axes as shown in Figure 2.3 [38]. Here
the phase windings A, B and C of Figure 2.1 are represented by the windings D and Q.
The two axes are named the direct axis and the quadrature &S. This representation
arises fkom the two reaction theory of altemathg current machines. The d-axis is
chosen to be dong the direction of the magnetic axis of the fieId winding. As a result,
the field winding and the two windings representing the dampers can be placed on
the d-axis and the q-axis without any transformations.
Chapter 2 26
2.3.1 Transformation of the Phase windings and the Park's tmnsfor-
mation
The two transformed windings D and Q in Figure 2.4 are assigned ,/&~ number of
turns where Neff is the eEective number of turns per phase. These two hypothetical
windings should produce the same mmf that the three phase windings would in any
given direction. Thus if we consider the mmf in the direction of the d-axis and the
q-axis we will get the following equations [18].
iq = & (i, sin (S.) + is sin ( 8. - - ~ ) + i c s i . ( B e + % ) )
A third variable zo, which is proportional to the zero sequence current is defined as
follows.
( Quadrature (q - a i s )
hh
Direct s i s (d - ais)
Figure 2.3: Representation of the machine coils referred to the direct and quadrature axes
Thus, the three phase currents, z., ib, and ic shown in Figure 2.4 can be transformed
to the d-q-O domain using the transformation matrix [Pl.
where [iOdp] =
[Pl = fi
= - [io , id, i,]' , [i&] = [i., is, i,]' and
COS (e.) cos (e. - F) cos (0. + $) sin (O.) sin (B. - ajT) sin (0, + $)
Figure 2.4: Position of the phase windings and the referred windings
This approach of analyzing the syndvonous machine was first proposed by Park
and the transformation matrix P is tenned the Park Tkansformation- The above
definition for P is slightb dinerent fiom that used by Park [39] [40], but the new
de finition further simplifies the numerical calculat ions. The transformation mat* is
also valid for voltages and flux linkages. Thus in general,
Chapter 2
[foc41 = Pl[foscl
where [f ] represents curent, flux iinkage or voltage. Thus,
The inductance mat& in the a-b-c phase domain is given by [L&]. The elements of
[L&] are shown in Appendix A.
Chapter 2
where
The quantity [L&o] can be considered to be the transformed inductance matrix of
the machine in the d-q-O domain. If the machine inductances are assumed to take
the form shown in Appendix A, this becomes a diagonal matrix and the elements are
independent of the position of the rotor. Hence
where
Chapter 2
and
La and L, are calied the direct axis and the quadrature axis inductances of the ma-
chine respectively, and Lo is c d e d the zero sequence inductance of the machine. The
transformed inductance matrix is time independent and, as a result, if the machine
equations are solved in the d-q-O domain, the matrut need not be inverted at each
calculation time step. This is the major advantage of the Park Transformation. Most
of the existing machine models are based on this approach. Since faults extemal to
the machine do not alter the direction of the magnetic axes of the phase windings,
this approach can be readily used to analyze extemal faults. However, faults on the
machine windings do affect the direction of the magnetic axis of the faulted phase
winding. In addition, the fault would effectively break the faulted winding into a
number of sections. As a result, in the presence of an interna1 fault, the machine in
general cannot be represented by six coupled coils. The geometrical symmetry that
existed between the phase windings in the un-faulted machine would not be present
between the faulted sections of the winding and the other phase windings. This is
described in detail in the following section.
2.4 Drawback of the d-q-O approach in the presence of inter-
nal winding faults
The representation of the three phase coiis of a synchronous machine is shown in
Figure 2.4. The coils are symmetrically placed with the magnetic axis of any one
winding placed 120° fiom the other two. This representation holds tme for all types
of winding configurations. The inductances involving these windings take the form
shown in Appendix A.
Now consider a machine of which the winding diagram is shown in Figure 2.5. Phase
A of this winding is shown in Figure 2.6 for clarity. Each coi1 bas N number of turns
Chap ter 2 31
and the coils belonging to a particular phase are connected in series to form the phase
winding.
Figure 2.5: Winding diagram of a three phase 4 pole machine with 6 slots per pole
A A
Figure 2.6: Phase A of the winding shown in Figure 2.5
The phase A winding shown in Figure 2.6 can be considered to have 8 coils connected
in series. Coils Pl1 and Pl2 have their sides under poles 1 and 2. The directions
of the magnetic axes of these two coils and the direction of the magnetic axis when
the two coils are in series are shown in Figure 2.7. The two coiis Pl1 and P12,
when connected in series, form coil Pl . Coils P2 , P3 and P 4 are dehed in a similar
manner and are depicted in Figure 2.8. The magnetic axes of the other coils are also
shown in Figure 2.8. These axes are 90° apart nom each other and the resulting effect
of the 4 sets of coils can be represented by an equivalent coil A as shown in Figure
Chap ter 2 32
2.4. The representation is further simplified by the use of the 'elect~ical angle', a., as opposed to the actud mechanical angle, 0 [33]. If the machine has p number of poles
As described in the previous section this representation leads to the d-q-O approach
of analyzing the machine.
Figure 2.7: Position of the phase A conductors and the directions of the magnetic axes of the different coils
CntbPI 1 d PI2 CoilrEI &PZ2 CoikP3l d P 3 Z COIL PI1 d PI?
Figure 2.8: Coils of the phase A winding
Now consider a tum to ground fault occurring at point F1 which is the end of coi1
P l . This breaks the phase A winding into 4 parts as shown in Figure 2.9.
Figure 2.9 shows the directions of the magnetic axes of different coils. The self
inductances of the coils P l , P2, P3 and P4 will take the following form
Chapter 2
where i = 1, 2, 3, 4. Ll, Li and C2 are constants which define the elements of
the inductance matrix [Lsyni]. These elements are as shown in Appendix A. The
constants kl and k2 depend on the position of the fault and they are described in
Chapter 3. The inductance matrix of the machine representing the fault will indude
the above constants kl and k2 in four of the diagonal elements. The mutual inductance
tenns involving the faulted winding wi l l also consist of similar constants.
The coii 'al' shown in Figure 2.10 for a fault at F3 will have a self inductance of the
form
where the constants k3, k4 and 0 depend on the position of the fault and the slot
angle.
While it is possible to diagonalize the inductance matrix of the faulted machine, due
to the presence of these additional constants the diagonal elements would no longer be
Figure 2.9: Representation of the fault at F1 and the directions of the magnetic axes of the phase A coils
Chapter 2 34
time independent. This would again require that the transformed inductance matrix
be inverted at every time step when the fault current is calculated. In addition,
the representation of the faulted machine depends on the fault type, the location of
the fault and the design of the stator winding. The representation of the machine
wiadings for faults at different locations are iliustrated in the Figures 2.9, 2.11, 2.10
and 2.12.
PsrrofPIIdoxrm R r r o f P l I c ~ o s a ~ ~ r m i d A (mi l 31) coi1 Pl2 (coi1 a3
1 I PI2 P2 PI Pa
Figure 2.10: Representation of the fauit at F3 and the directions of the magnetic axes of the phase A coils
Figure 2.11: Representation of the fault at F 2 and the directions of the magnetic axes of the phase A coils
It is clear from the above discussion that each fault has to be treated as an individual
case with attention given to the fault type and location and the winding details. It
was also mentioned that diagonalizing the inductance matrix of the fadted machine
would not guarantee a transformed matrix with non tirne varying elements. This
Chapter 2 35
takes away the main advantage of performing such a transformation. For this reason,
it was decided to develop the machine model in the direct phase domain. As the
first step, a machine model was developed in the direct phase domain where extemal
faults could be simulated. This is described briefly in the next section.
2.5 Phase domain model of a synchronous machine
The phase domain model of the machine is very straightforward as it involves directly
solving the machine equations described in Equations 2.1 and 2.2 in Section 2.3. The
same equations are listed below.
where
The above equations were solved using the trapezoidal d e of numerical integration
and the resulting equations are shown in Appendix B. A suitable calculation tirne step
5 Fault at Fa
Figure 2.12: Representation of the fault at F 4 and the directions of the magnetic axes of the phase A coils
Chapter 2 36
must be selected to minimize the errors. The components of the inductance matrix
are held constant over the calculation time interval. This leads to inaccuracies if
the time step used is not sutticiently smaii. Simulations carried out on a number of
identical cases using t h e steps of l p s and 20ps gave matching results. Thus a time
step of 20ps was chosen to be adequate.
The machine data supplied by the manufacturer or those computed by performing
the standard tests on the machine are in a form that can be readily used in standard
d-q-O models. The information needed to d e h e the inductance matrix [Lsynl] must
be derived fiom this information. The method to convert the data to a form that can
be applied to the phase domain model is outlined in Appendix C.
Generator r------------------r------------------'---'-r------------------'---'-r------------------'---'
1
I - - - - - - - - - - - - - - - - - - - - - - - ,
Figure 2.13: Representation of a generator connected to a remote source
Simulation results using the phase domain model are shown in Figures 2.14 through
2.16. The system in Figure 2.13 was considered and the results show the phase A
current and the phase A voltage foliowing a three phase short circuit at the machine
terminals. The point on the voltage waveform at which the fault occurs is different
in the four cases shown. These results are compared with wavefom s i d a t e d using
a d-q-O based model of the machine and the close agreement of the cornparisons
Chapter 2
Figure 2.14: Comparison of results derived using the d-q-O domah model and the a-b-c domain model
Figure 2.15: Comparison of results derived using the d-q-O domain model and the a-b-c domain model
indicate the validity of the a-b-c domain approach and the method used to convert
the machine data fkom the d-q-O domain to the a-b-c domain. The current envelopes
following a three phase short circuit at the terminah are shown in Figures 2.17 and
2.18. These envelopes show the same characteristics described in publications and
standard texts [38], [55].
The next step was to extend this machine model so that it can be used to simulate
interna1 faults in the windings. The main challenge here is to derive the elements of
the inductance rnatrix that wodd represent the faulted machine. The method used
to derive these elements is described in the following chapter.
Figure 2.16: Cornparison of results derived using the d-q-O domain model and the a-b-c domain model
Chapter 2
Figure
- 7
O 05 1 15 i 25 i 1 5 4 45 i rune (SI
Figure 2.17: Short circuit current envelopes of the phase currents
2.18: Short circuit current envelopes of the d and q axis winding currents
Chapter 3
Development of a machine model
for the analysis of internal faults
3.1 Summary
The need to have synchronous machine models for internai fault studies was discussed
in Chapter 1. This chapter outlines the method used to calculate the winding induc-
tance parameters necessary for the fault current caiculations. The machine rnodel is
developed in the phase domain. A method to calculate the inductances involving the
faulted windings is outlined using a four pole, lap wound machine. The machine equa-
tions are then solved using a suitable numerical technique. Comparisons are made
between the simulated waveforms and recorded waveforms to ver* the accuracy of
the model.
3.2 Introduction
There has been a need for a machine model to simulate internal faults for a long
time. The main use of such a model would be in power system protection studies as
indicated in Chapter 1. However, models or methods which can be generalized and
applied to any type of machine are not readily available. Due to the nature of the
problem, the models based on d-q-O transformation [39] canmt be used in internal
Chap ter 3 41
fault studies.
It is essential that the models be validated by comparing the results with recorded
waveforms before these can be used confidently. The absence of recorded data is an-
other major drawback to the development of a mathematical mode1 to study internal
faults [16].
3.3 Overview of the available machine models
The machine models available on most of the electro-magnetic transient simulation
programs [17] are based on the two reaction theory and the resulting Parks transfor-
mations [39] [18] [19]. It was explained that these models, based on a transformation
which reduces the computational requirements, are not suitable to study internal
faults. The model described in this section is derived in the phase domain [20], [21],
[22] for this reason. A machine model in the phase domain which is capable of extemal
fault simulation was presented in Chapter 2. The data supplied by the manufacturer
can be readily converted to a form which can be used in this model as can be seen
from Appendix C.
The performance of a machine under internal faults has not been widely published
and recordings of internal fault wavefonns are very hard to corne by. There are
few machine models available for internal fault analysis [23], [24], [25], [26]. Most
of these methods do not consider the placement of the conductors inside the stator
in an effort to simplify the andysis. However, the fault current depends on the
winding arrangement and any model that does not consider this will lead to errors.
The method used in [23] and [24] neglects the higher order harmonics and this leads
to errors since intemal faults give rise to stronger harmonics. The rnethod used in
[25] does not consider the winduig arrangement inside the machine and, hence, it is
Limited in application. The method described in [26],[46], and [47] considers a two
pole, sinusoiddy distributed winding. Such winciings are hardly ever found in power
systems and, hence, it is also iimited in application.
The met hod present ed in t his t hesis t akes int O account the winding arrangement inside
the stator and, hence, it can be extended and used for any type of winding design
[41]. A turn to ground fault is considered here but any fault type can be analyzed in
a similar manner. The machine is represented as a system of coupled coils and the
number of coils in this system is determined by the nature of the machine winding
and the type of fault. The main challenge is to derive the self inductances of these
coils and the mutual inductances between any two of them. The following sections
describe the metbod used in this thesis to compute these elements. A machine with
four poles is considered to outline the method.
3.4 Description of the machine windings
The two equations below give the voltage-current relationship of a synchronous ma-
chine, represented as six coupled coils [18].
where
The elements in the inductance matrix [Lsyni], and the matrix [Ri] are known fiom
the data supplied by the manufacturer. In salient pole machines, the elements of
[Lsynl] depend on the position of the rotor and hence are time varying.
Consider the four pole machine of which the winding diagram is shown in Figure 3.1.
It has two parallel paths per phase and each phase coil occupies two dots per pole,
as shown in Figure 3.2. This winding arrangement is used to outiine the procedure
used in this thesis to calculate the inductances involving the faulted windings. In this
example, the two sides of any given coil are placed either at the top or at the bottom of
the respective slots. Whde this does not represent the practical arrangement (one top,
one bottom conductor) it ailows us to express the magnetic axis positions in multiples
Chap ter 3 43
of the dot angle, 6. The actual displacement of the magnetic axis of any single coii
will M e r fkom the values used in this thesis by a very small angle, the tangent of
which is the slot depth over the coil pitch. Neither of the latter measurements are
readily available.
- - Top layer No. of paralle1 paths = 3 No- of Pales = 3
- - - - - - - - Bottomiayer No. of Slots = 21
Figure 3.1: The stator winding.
Consider a short circuit fault to ground on coil (6) which is on one parallel path of
phase A. This breaks coil (6) into two parts, A3 and A4 as shown in Figure 3.3. The
phase A winding can now be considered to be made up of five parts, Al , A2, A3, A4
and A5.
Winding A l consists of coils (1) - (4) connected in series. A2 is coil (5). A4 is the
portion of coil (6) that is connected to coil (7). A3 is the other portion of coil (6) and
it is connected to coil (5). A5 is the combination of the coils (7) and(8), connected
in series. This arrangement is shown in Figure 3.3. The machine can now be viewed
as a system of 10 coupled coils.
Figure 3.2: Positions of the phase A conductors inside the stator.
Figure 3.3: Representation of the phase A winding with a fault on one pardel path.
Chap ter 3 45
The matrix equation governing the voltage-curent relationship of the machine in
the presence of an intemal turn to ground fault can be expressed as follows. The
directions of the currents and voltages are shown in Figure 3.4.
Figure 3.4: Representation of the machine coils under an interna1 short circuit.
where
[A2110xl = [LsY%] 10x 10 [~2110x1
The inductance matrix of the system is symmetric and hence there are 55 independent
elements of which 10 are self inductance terms. The self inductance of a coi1 and the
mutual inductance between two coils for a given position of the rotor depend on the
following features and parameters.
1. Geometry of the winding or the windings concemed
2. How the windings are placed on the stator or on the rotor.
3. Length of the air gap.
4. Permeability of the iron core.
If the machine is assumed to be operating near the knee point of the B - H cuve
then the effects of saturation can be negiected. The permeability of the iron core
of the stator and the rotor would be constant. Thus, under this assumption, the
self inductances of the normal windings are not affected by the fault. The mutual
inductance between any two normal windings is also not affected by the presence of
the fa&. Such elements in [Lsyna] are directly known from the inductance matrix
[Ls ynl] of the normal machine. Thus, 40 elements involving the faulted coils need to
be evaluated to define the inductance matrix [Lsynz] and to solve the equations to
h d the fault currents.
3.5 Calculat ion of the inductances involving the fault ed coiis
The winding A l in Figure 3.3 makes up one parallel path of the phase A winding.
The self inductance of the phase A winding is given by the foilowing equation where
Li, LI and L2 are constants.
The angle 0 in this case is the actual mechanical angle of the reference pole on the
rotor shown in Figure F.2, with respect to the magnetic axis of coils (1) and (2) or
coils (5) and (6) shown in Figure F.5. This is explained in detail in Appendix F. The
relationship between the 'electrical angle', O., and the actual mechanical angle, 0, is
given by
where p is the number of poles in the machine.
The winding of the machine shown in Figure 3.1 has four poles. The inductances of
the windings of this machine are given in Appendix E with the actual mechanical
angle used to describe the position of the rotor. Using the actual mechanical angle
Chap ter 3 47
was found to make the analysis far less complicated and th% is the reason for using
this rather than the more common approach of using the 'eleetrical' angle. Eiuther,
in intemal fault studies, there is no real advantage of transforming the angle to an
equivalent 'electrical' angle as was pointed out in Chapter 2.
The magnetizing flux due to a current flowing in any one pardel winding of phase A
shares a common path. This situation c m be depicted by the simple magnetic circuit
shown in Figure 3.5. Thus, it c m be shown that the magnetizing component of the
self inductance of the phase A winding is equal to the magnetizing component of the
self inductance of any one parallel path of this winding. This is proved in Appendix
D. As a result, the self inductance of the winding A l can be written as
Figure 3.5: Two parailel coils sharing the same magnetic path
Lal, is the leakage component of the self inductance and can be computed using
the methods explained in [27], [33]. These methods are described in Section 3.6 and
Chapter 3 48
Appendix G .
Since the two parallel windings have a common flux path, it can be shown that the
mutual inductance between A l and any other unfauited winding is equal to the mutual
inductance between the phase A winding and that unfaulted winding. This can be
seen from the results derived in Appendix D. This leads to the following inductance
terms involving the un-faulted paralle1 path Al.
Now consider any one parallel path of phase A. This can be thought of as being made
up of two windings, X and Y, as shown in Figures 3.6 and 3.7. The winding X is made up of coils (5) and (6) and the winding Y is made up of coils (7) and (8). If
this parallel path is energized with a voltage, V I , with the rotor kept stationary at the
position shown in Figure 3.7 and if the current drawn is il, then the voltage current
relationship can be written in terms of the self inductance of the coi1 as follows.
Figure 3.6: One paraliel path of Phase A
The two windings X and Y can be viewed as two identical windings, placed 180°
apart on the stator. Thus they will have the same self inductance, L,. Let the
mutual inductance between them be M,. Using the fact that X and Y make up the
the faulted paralie1 path we can show that
As the windings X and Y are 180' apart the leakage flux of any one coil does not
couple the other coil. Using this fact and combining Equations 3.13 and 3.14 the
following relationship can be denved. The details of the derivations are shown in
Appendix F.
Lpma, is the magnetizing component of L,.
Figure 3.7: Flux pattern due to current in coil X.
Chap ter 3 50
Now consider a case where a current, .i,, is passed through the coil X with all other
windings kept open circuited. The resulting flux pattern can be approximated as
shown in Figure 3.7. Appiying Amperes circuital law to these flux paths we can show
that
Lpms 361 -- -- = 3 M P 41
From Equations 3.15 and 3.16
Equations (3.19) and (3.20) give the magnetizing portion of Lx, and LX,Y when the
rotor angle 19 = 0'. For any other position of the rotor, with the leakage inductance
also taken into account, Lx and LXVY can be expressed as foilows.
The winding A5, which consists of the coils (7) and (8) is identical to winding (Y). Thus the self inductance of this coil can be written as
Chapter 3
The axis of the coil A2 is shifted by an angle (6) fkom the axis of the coiI X. The
angle (6) is the slot angle as shown in Figure 3.2. Using the relationship between the
inductance and the number of tu- in the coil, the mutual inductance between the
coils A2 and A5 can be written as follows. The detailed description of this is given
in Appendix F.
The other unknown elements in the inductance matrix can be derived in a similar
manner. These are shown in Section F.14. The details of these calculations are
presented in Appendix F.
3.6 Estimation of the leakage inductances of the windings
The mutuai flux that crosses the air gap and couples both the stator and the rotor
windings causes the energy transfer from one point to the other. In addition to the
mutual flux, there are flux lines that link only one winding. Such a flux is called the
leakage flux. In synchronous machine studies, a flux line that would couple two stator
windings but would not couple any rotor windings would also be considered part of
the leakage flux. The effect of the leahge flux is to distort the shape of the main or
the mutual flux and hence to reduce its effectiveness in generating an electro motive
Chapter 3 52
force in the armature windings 1331, [35], [42]. Thus, the effect due to the leakage flux
c m be treated as an interna1 reactance and it is termed the leakage reactance. The
correspondhg inductance is the leakage inductance.
Figure 3.8: Slot leakage and differential leakage fields
Figure 3.9: End leakage field
The leakage reactance of a generator is signiscantly small compared to the grounding
impedance. Thus the effect of the lealcage inductance on the fault current in turn
Chap ter 3 53
to ground faults is negligible. However, the fault current in the case of turn to turn
faults is limited by the leakage inductance of the faulted section of the coil and the
resistance of that section. The fault current in phase to phase faults is limited by the
resis tances and the lealcage inductances of the involved coil sections.
Machine designers treat the estimation of the leakage inductance in many difEerent
ways. The Ieakage flux is divided into Merent categories which are considered sepa-
rately. The resultant leakage inductance is the s u m of all these components [35], [54].
The following categorization of leakage fields is commody used for design purposes.
These fields are f i s t ra ted in Figures 3-8 and 3.9.
1. Slot leakage
2. End winding leakage
3. DXerentiaI or air gap leakage
Slot leakage accounts for the flux that crosses the dot where the coil is embedded.
In the computation of this portion of the leakage inductance, the reluctance of the
stator core can be neglected compared to that of the air path. A method to estimate
the slot leakage of a phase winding is given in [35]. This method can be used, with
modifications, to compute the slot lealrage inductance of part of a winding. Since
the slot dimensions , the way the coils in the slot are electrically connected and the
position of the conductors inside the slot must all be considered in the calculations,
each case must be considered separately. Appendix G shows how the method is
applied to the machine considered in this chapter.
End winding leakage fields are shown in Figure 3.9. These fields are formed around
the coil ends protruding fiom the dots at the two ends. The estimation of this part is
mainly based on formulas derived empiricaliy. The formula described in [42] and [35]
is widely used. This is used to compute the end leakage of the entire phase winding
and takes the following form.
D is the diameter of the air gap and y is the winding pitch. The other parameters
are explained in Appendix G. This formula is used with suitable modifications in
Appendix G to calculate the end leakage of a coi1 of the example machine considered
in t his chapt er.
The dinerential or the air gap leakage is due to the flux that crosses the air gap but
which does not couple the rotor windings. This flux accounts for the space harmonies
present in the air gap magnetic field. In [42], this leakage is considered in two parts
named zig zag leakage and belt leakage and an empirical formula is presented to
estimate it.
The total leakage inductance of a coiI is the sum of these three components.
3.7 Results and cornparisons
Once the inductance matrix of the system is known, the system in Figure
be solved. The machine described in Section 3.4 gave rise to a system of 10
coils. This number would be' different for another machine with a different
3.4 C a R
coupled
winding
design or a dinerent type of fault. However, in the case of a tum to ground fault, the
final system can be reduced and be viewed as a system of 8 coupled coils and hence
is described by a matrix equation of the order 8. It has the following form where [RI is the diagonal resistance matrix of the system.
This reduces to
Chap ter 3 55
This equation was solved numericaily using the trapezoidal rule of integration.
Figure 3.10: A turn to turn fault
Figure 3.11: A fault involving the two parallel paths of phase A
Faults between phases and turn to turn faults can be analyzed in a similar manner.
Figures 3.10, 3.11 and 3.12 illustrate a few possible fault situations inside the machine
of Figure 3.1.
The method outlined in the earlier section was applied to obtain the inductance pa-
rameters of a 4 KVA salient pole type machine with six poles. A number of faults
were applied on the stator winding and the resulting waveform recordings were com-
pared with those obtained fkom simulations. Three cases are shown in Figures 3.13,
3.14 and 3.15. More cornparisons and the description of the test system are given
Chapter 3
Figure 3.12: A fault between phases A and B
094 0.96 0- 1 1 1 .a2 1 .M 1 .a6 1 .O8
Time (s) A G FAULT r 50.. or&c &g
Figure 3.13: Phase A to ground fault.
098
Figure 3.14:
in Appendix N. The close
Two phase fault between
1.04
phases A and
match observed between the cdculated
recorded currents is a good indication of the validity of the method
the inductance parameters.
currents and the
used to compute
Chapter 3
O O S 0.97 009 1.01 1 .O3 1 .Q5
Time ( 5 )
Figure 3.15: Phase A to ground fault with a low grounding resistance.
Chapter 4
Interna1 fault simulation in
t ransformers
4.1 Summary
This chapter deals with calcdation of interna1 faults in transformers. Single phase
transformer units and three phase , three limbed two winding transformers are con-
sidered here. The method used to estimate the leakage inductances of faulted winding
sections is described and it is validated by comparing the calculations with measured
data.
4.2 Introduction
The transformer is an important element in a power system. It has a relatively simpler
construction compared to the synchronous machine . The operating characteristics of
the transformer depend on the construction and the design, and the way the windings
are connected. A three phase transformer made from three single phase units will
behave quite differently under fault situations fkom one which is made up of a single
core. The following figures show some commody used core types [29].
The fault current in the case of a turn to ground fault depends on the grounding
58
Chapter 4
Core type Sheli type
Figure 4.1: Single phase transformer cores
Threee phase shell type Threee phaK five Iimbed
Thme phase three lirnbed
Figure 4.2: Three phase transformer cores
practice employed to ground the star point of the transformer comection. Unlike in
synchronous generators, it is common to have solidly grounded transformers. In such
a case the fault current is limited only by the resistance and the leakage inductance
of the faulted part of the winding. The voltage driving this current is the induced
voltage across this faulted section.
There could be some magnetic flux trapped in the core once a transformer is de-
energized upon the detection of a fault. If the transformer is re-closed onto the fault,
the current is influenced by the trapped flux in its core. Thus, it is important to
accurately represent the saturation of the core and the remanent flux. The presence
of the air gap in the synchronous machine causes the trapped flux to decay at a much
faster rate. Thus this is not a major concem in synchronous machine simulation.
1441 -
Chapter 4
4.3 Simulation of interna1 faults in tramformers
4.3.1 GenemI
Phase B
Figure 4.3: Cross section of a three phase two winding transformer.
Figure 4.4: The transformer represented as six coupled coils
Consider the three phase two whding transformer shown in Figure 4.3. The six coils
can be schematically represented as in Figure 4.4. This arrangement cari be viewed
as six coupled coils whose behavior is govemed by the following dinerential equation.
[W.
Chapter 4
where
The element Li represents the self inductance of the coil i and the element Lia rep
resents the mutual inductance between the two coils i and j .
The components of the [Rt] matrix and the [L,] matrix are known from the data
supplied by the manufacturer or they can be computed using standard excitation and
short circuit tests[l9].
4.3.2 Interna1 turn to ground fault
X Y
Faul t
Figure 4.5: Representation of a turn to ground fault in coil (1)
Chapter 4 62
Consider a turn to ground fault in coil(1) of Figure 4.4 which divides coil(1) into two
parts (x) and (y) as shoam in Figure 4.5. Now the voltage current relationship is
governed by the equation
where
If the matrices IRt] and [Lt] are known, then in [R'], R, and R, are the only unlmown
element S.
N, = number of tums in coil x
N, = number of tums in coil y
N, + N, = NI = N3 = N5 = number of tums in coil (1)
Chapter 4
N, and N, determine the position of the fault .
In the matrix [R'];
It is important to accurately calculate the leakage inductances of the faulted parts (x)
and (y) since the fault current wili largely depend on this quantity in low impedance
and solidly grounded transfomers [31].
The leakage factor between any two coils (a) and (b) is defined as follows.
Llab is the leakage inductance of coils (a) and (b) referred to coii (a). Thus for coils
(4 and (Y)
&,, is the leakage inductance of the coils (x) and (y) referred to coi1 (x).
A method to calculate Lm when the coiis a and b are wound on the same leg of the
transformer is shown in Section 4-4.
The self inductance of any winding (k) can be expressed as foilows.
Chapter 4 64
L?q is the leakage component of Lk and Lk,, is the magnetizing component [33]
Since the leakage part of the inductance of the coils (x) and (y) is very small compared
t O their magnetizing part,
where Nz and N, are the number of turns in coils x and y respectively.
If (x) and (y) were connected in series and a current, i, is passed through them, the
total flux Iinkage produced, (A, + A,), should be equal to the flux linkage of coil (1)
when the same current i is passed through it.
Once LF., is known, fkom Equations 4.3, 4.4 and 4.5, L,, L, and L,, can be found.
Consider any coil (j) other than coil (1) where the fault is located. The location of
the coils on the transformer core are schematicdy depicted in Figure 4.6
Consider the simple magnetic circuit shown in Figure 4.7. If S is the reluctance of
Chapter 4
I 1
Figure 4.6: Schematic representation of the six coils
the magnetic path 1, m, n, O , then the mutual inductances LP,,, L,, and Lb+,), are
given by the foilowing equations. The complete derivation of the results is shown in
Appendix O.
Figure 4.7: Three coils wound on a common magnetic core
Chapter 4
If the coils p and q are connected in series to form the coi1 (p + q) then
Extending the above results to the situation in Figure 4.6 we can mite
For coi1 (2) which is on the same leg as coil(1) let
LFZ2 and LF12 can be calculated using the method shown in Section 4.4. L2, and
LZy can then be calculated using the Equations 4.6 and 4.7.
For coils (3), (4), (5), and (6) which are not wound on the same leg as (1) ,
Using Equations 4.6 and 4.8, hc, L*,=, L5,Zl Lecr and Le,, can be
computed. Now Equation 4.2 can be solved to find the internal fault cwrents.
The above method can be easily extended to analyze turn to turn faults. Also, the
same method can be used to analyze intemal faults in single phase tramformers and
in three phase banks made of single phase units.
The calculated inductances can be used to s i d a t e internal faults on the electro-
Chapter 4 67
magnetic transient simulation program PSCAD/EMTDC [VI. Simulat ed current s
and voltages for an internal fault occurring at 33% fkom the high voltage side neutral,
of a 415/11000 Volt, single phase transfomer are shown in Figure 4.8.
Simulated currents and voltages for an internal fault to ground occurring at 33% from
terminal B on the delta side of a 33/11 kV, deltalwye, three phase transformer are
shown in Figure 4.9.
TEILMINAL VOLTAGE ON HIGH VOLTAGE SIDE
UNE CüRRENT ON KIGH VOLTAGE SIDE
FAff LT CURROVT TO GROUSD
Figure 4.8: Internai fault waveforms for a single phase transformer
Chap ter 4
Star sidc linc cumne DI,, o m a r c
Figure 4.9: Interna1 fault waveforms for a three phase transformer
4.4 A method to calculate the leakage inductance of two
windings wound on the same leg of a transformer
Consider two windings (a) and (b ) , wound on the same leg of a transformer as shown
in Figure (4.10).
The leakage flux pattern in Figure 4.10.(1) can be used to estimate the leakage induc-
tance of the windings [3l], [34], [35], [37]. To find the leakage inductance of wuidings
(a) and (b) , referred to winding (a) , consider that the Mnding (a) is energized and
Chapter 4
Figure 4.10: Leakage flux pattern inside a transformer
that the winding (b) is short circuited.
If i, and ib are the currents fiowing in the two windings and if the relative permeability
of the core is assumed to be very high compared to that of air then,
The mmf in the space between the winding (a) and the transformer leg, mmfin, is
given by
mmfin = Ki, - Nbàb = O (4.10)
The mmf in the space between the winding (a) and the winding (b), mrnf2, is
Chap ter 4
The mmf outside the winding (b), mm fat, is
Since the permeability of the core is much greater than that of air, the magnetic field
intensity, Hy in the inter-winding space, H2 can be approximated as
where h is the window height of the transformer.
Since mm fh and mmf,, are both zero, the magnetic field in the space between the
winding (a) and the leg and that beyond the winding (b) are zero.
If the curent density inside the taro windings is assumed to be uniform then H will
vary linearly inside the two windings as shown in the Figure 4.lO.(2).
The energy stored in a magnetic field, spanning a volume V is given by the volume
integr al
where H is the magnetic field intensity inside the incremental volume W . For the
case considered here and shown in Figure 4.10
Chapter 4 71
If the magnetic field is assumed to be symmetrical about the axis of the cote, then
dV c m be written as
where the distance r is measured
dV = (2mh)dr
from the axis of symmetry. Then,
If the leakage inductance of the windings (a) and (b) referred to the winding (a) is
Ltab thefl
Hence we can find Llob as
Hl, 6 and & can be expressed in the foIlowing form with k = 1,2 ,3
In the above equations, Hd is independent of the current
Thus LIab can be found as
The above expression gives the value of the leakage inductances of coils (a) and (b)
referred to coi1 (a).
The method described above was used to calculate the leakage inductance of the
windings of a nurnber of transformers. The cornparisons are shown in Table 4.1.
The measured d u e s were obtained fiom the manufacturer dong with the details of
the transformer necessary to do the calculations. The measured and the calculated
inductances agree very closely and this is a good indication of the validity of the
method used.
4.5 Magnet ic saturation and hysteresis in the transformer
core
It was pointed out in an earlier section that the fault current will be iduenced by
the trapped flux in the transformer core if a breaker is re-closed while the fault is
still present. Hysteresis and saturation are considered difficult to be included in
the transformer models of electro-magnetic transient simulation programs [l?] . In
the program E M T D ~ ~ saturation is accounted for by placing a non-linear current
source in parailel with one of the windings. The non-hear characteristics are modeled
by a simple anhysteretic cuve which has the form shown in Figure 4.11. In this
model, hysteretic effects are not considered. Difnculties encountered when modeling
Transformer type
Table 4.1: Comparison of the measured and calculated leakage inductances
Single phase, cote type, IOOKVA, l l k V / 415 V Single phase, core type, 250KVA, l lkV / 415 V
Three phase, core type, Wye /Wye, lOOKVA, l l k v / 415 V
Three phase, core type, Delta/Wye, lOOKVA, l lkV / 415 V
Three phase, core type, Delta/Wye, lOOKVA, 33kV / 415 V
Three phase, core type, Delta/Wye, lOOKVA, l lkV / 415 V
Three phase, core type, Delta/Wye, IGOKVA, 33kV / 415 V
Three phase, core type, Delta/Wye, 250KVll, 33kV 415 V
Three phase, core type, Delta/Wye, 630KVA, 33kV / 415 V
Three phase, core type, Delta/Wye, 2500KVA, 33kV / 400 V
saturation and hysteresis are pointed out in[17].
Measured inductance (H)
Mathematics based on the physics of ferromagnetic hysteresis presented in [Il] and [3]
can be applied to mode1 hysteresis and saturation in tramformers [36]. The process is
outlined in the following section with a turn to tuni fault in a single phase transformer
considered as the example case.
Calculated inductance (H)
O. 127
0.052
0.119
1.855
1.800
1.741
1.119
0.815
0.423
0.122
4.5.1 Modelfng satumtion and h ystemsis effects
0.130
0.052
0.123
1.797
1.814
1.770
1.142
0.809
0.427
0.121
The B - H relationship of the magnetic material of the core takes the form of a
hysteresis loop like the one shown in Figure 4.12.
Cbap ter 4
Magnetizing current
Figure 4.11: Non-linear characteristics of the core
Figure 4.12: B - H loop of a transformer core
The relationship between the magnetic moment, M, and the magnetic field intensity,
H, is given by the following equation [Il].
Chapter 4 75
The effective magnetic field intensity, He, is defined as follows. The parameter a
accounts for the inter-domain coupling inside the material and is a constant.
Using the above relationships, the B - H loop can be converted to a loop between M
and He. The anhysteretic magnetization curve, La,, can be derived fkom this loop as
shown in Figure 4.13.
Figure 4.13: M - He loop of a transformer material
The function f (He) could be any function that can represent the anhysteretic mag-
netization. The foliowing form is used in [14], [43] and [44]. It was shown to give
accurate results. The constants al, an, a3 and b are estimated using a suitable curve
fitting technique such as the non-Iinear least square method [45].
Chapter 4 76
The derivations in [Il] lead to the slope of the M-H m e in terms of the anhysteretic
magne tization curve.
The function Mm can be derived from the B-H loop of the transformer core material.
The parameters c, a and k are constants for a given material. 7 takes the value +1
or -1 depending on the sign of $.
Load %
-- - - -
Figure 4.14: Tum to turn fault in a single phase transformer
Consider a single phase transformer connected to a load as shown in Figure 4.14. The
turn to turn fault on one of the windings has broken this winding into three segments.
The circuit representation of this scheme is given in Figure 4.15. The flwces passing
through the windings (a), (l), (2) and (3) are all equal since they are on the same
core. The leakage fluxes are neglected in the calculations, assuming that they are not
significant compared to the main flux. However, the voltage drop across the leakage
inductances cannot be neglected and these inductances, represented by La, LI, Lz and L3 should be calculated using the method given in Section 4.4. The following
equations describe the behavior of the system shown in Figures 4.14 a d 4.15.
Chap ter 4
Figure 4.15: Circuit representation of a tum to turn fault in a single phase transformer
A is the area of the core. Ni is the number of turns in coi1 j for j = 1, 2 or 3. Since
the flux density, B, c m be expressed in terms of M and H ,
If the dope of the M-H cuve, is equal to s, then
1 is the mean length of the core. Moreover, from Kirchhoff's voltage Law,
Chapter 4
Since ali coils in Figure 4.14 experience the sanie flux,
for j = 1, 2 or 3. Equations 4.23, 4.24, 4.25, 4.26 and 4.27 can be solved to find
the fault currents. The complete derivation is shown in Appendix H. Simulation
results are shown in Chapter 5. Other faults can be analyzed in a similar manner.
Dimensions of the core and the B - H loop for the core material are the essential
data for this analysis. The parameters a, k, and c are generally not supplied by the
manufacturer, In the absence of such information, these parameters can be estimated.
This is explained in [14]. This is a suitable mode1 to study the in-rush phenornena
in transformers as well. The same theory can be applied to three phase transformers
and other core configurations. Here, the problem becomes more involved since the
flux density is not equal in all the limbs. This is the main focus of another ongoing
research in the power systems group at the University of Manitoba.
4.6 Conclusions
A method to calculate intemal fault currents in transformers has been presented.
InternaI faults were simulated and some results are presented. Inclusion of hysteresis
Chap ter 4 79
and saturation was discussed briefly. The calculated leakage inductances were com-
pared with measured values to verify the method used to do the calculations. The
main drawback here is the unavaïlability of measured fault waveforms to compare
with the calculated ones.
Chapter 5
Simulation results and observations
5.1 Summary
The developed models of transfomers and machines were used to simulate diBetent
types of interna1 faults. Some results are presented in this chapter. In addition,
results from tests performed on the models to verify their consistency are presented.
5.2 Synchronous machine
5.2.1 Consistency of the equations derived for the faulted coils
The relationships between the inductances of the coils of the faulted machine should
be consistent with those of the normal machine. Thus the elements in the inductance
matrix [Ls yn2] described in Equation 3.4 must satisfy the following constraints.
with
Chap ter 5
Li and Lilj are the elements in mat* [Lsy*] and L(l+2+--,) is the self inductance
when all n coils in a faulted path are connected in series. LOfZf can be regarded as
a diagonal element in matrix [Lsynl] described in Appendix A. The second constraht
involves any two coiis of the faulted machine and is shown in Equation 5.4. If this
constraint is not satisfied the inductance matrix cannot be inverted and hence the
equations cannot be solved. The shift in magnetic axes of the faulted coils mwt
be given careful consideration in order to satisfy this condition. Equation 5.5 gives
the third condition to be satisfied for the matrix [Ls y*] to be consistent aith the
matrix[Lsynl]. Lc1+2+...n),k is the mutual inductance of coil k with n number of coils
of the faulted path connected in series.
Ü the constraints Listed above are satisfied, then the inductance matrix [Lsyn2] which represents the fault shown in Figure 3.4, should reduce to matrix [Ls ynl] under the
conditions given in Equations 5.6, 5.7 and 5.8. e, is the voltage across the phase A
winding when no intemal faults are present. vj is the induced voltage across any coil
j as illustrated in Figure 3.4.
Chapter 5
The above test was performed on the inductance matrix [Lsynz] , descnbed in A p pendix F. Equations 5.9, 5.11, 5.12 and 5.13 shown below are the proof for element
Laxf in [Lsynl]. Once the conditions in Equations 5.6, 5.7 and 5.8 are applied to
[Lsyn2], Laxf should be given by the following equation.
Substituting the equations for &ai,t from Section F.14 (page 190) in AppendIx F,
5 MF x k t = MFcos28t ( - ) co~2(B-%) + i d 4 cos 6
3 2
(*z6) [2 cos 26 cos 61 = -M'cos~@+ -
Chapter 5
Equation 5.9 is satisfied. A similar analysis can be used for ail other elements. Thus,
if al1 elements in [Lsyn*] meet the above conditions the system shown in Figure 5.4
should behave exactly the same way for extemal faults as a normal model derived
on the a-b-c domain or on the d-q-O domain. The fault resistance Rtit in Figure 5.4
was set to a very large value so that conditions in Equations 5.6, 5.7 and 5.8 are
met. An external fault involving h e A and ground was simuiated using the n o d
machine model described in Section 2.5 and the model described in Section 3.7. The
comparison of the results obtained fiom the two are shown in Figure 5.1. The line
currents, the field current and the cment in the damper windings indicated in Figure
5.4 are shown in Figure 5.1.
The agreement shown by this comparison is further evidence that the inductances
derived in Appendix F are accurate and consistent.
5.2.2 Simulation results for interna1 faults
The results shown in this section were derived using methods outlined in Chapter 3
and Appendix F. A six pole machine was selected and the winding diagram in Figure
5.2 shows the phase A of the stator winding. The data for the machine is given in
Appendix K. As can be seen from Figure 5.2, this machine has a concentric winding
in the stator. This type of winding, also known as a spiral winding, is very cornmonly
Chapter 5 84
0 ia O ia L i e A ro ground fault o i b o i b
-1 -5 4 480 500 520 540 560 580 600
O ikq O ikq x i o - ~
a ikd O ikd XI o - ~
x 1 o9 Time (Sec) XI o4
Figure 5.1: Cornparison of the external fault waveforms derived using the normal machine mode1 and the machine mode1 developed to simulate interna1 faults
Figure 5.2: A part of phase A of a six pole concentric winding
used in synchronous machines. This wïnding is different from the winding considered
in Chapter 3 and, as a result, the derivation of the inductances of the faulted coils
are not exactly the same-as explained in Appendix F. It was stated earlier that each
machine must be considered separately when analyzing internai faults. However the
Chapter 5
Figure 5.3: Two magnetic circuits to represent a portion of a concentric winding
sarne principles used in Appendix F can be used here and the general approach is the
same. The derivations in Appendix J indicate how a spiral winding is t aded . There,
it is shown how a spiral wound coii can be represented by two simple un coupled
magnetic circuits as in Figure 5.3. The areas Ai and A2 are defined in Appendix J.
Field
Figure 5.4: A turn to ground
Source
fault on phase
Waveforms in Figures 5.5 and 5.6 show the current in Merent windings of the machine
when an intemal tum to ground fault occurs in one of the parallel paths in phase A.
The currents are as indicated in Figure 5.4. The grounding impedance is taken as 10
Ohms. The fault was initiated after 1 second. The field current is interrupted, 0.3
seconds after the fault and the machine is isolated fiom the rest of the system 0.7
seconds after the fault. The line currents increase once the field is de energized. This
results from the remote source feeding more current to the machine windings as the
Chapter 5 86
0.8 1 -35 iflt 2.45 3
2.45 3 Time (Sec)
Figure 5.5: Current in the machine windings for a turn to ground fault at 20% from the neutral with the field de-energized before the machine is isolated from the system
induced voltage across them drops when the field current is interrupted as can be seen
from the 6rst graph of Figure 5.6. This can be seen clearly from the third graph of
Figure 5.7 where the waveforms are expanded for clarity. The line currents ia, ib and
ic undergo a phase shift when the field is removed. Once the generator is isolated,
0.7 seconds after the fault, there is still a circulating current in the faulted phase.
The fault causes a current to flow in the damper windings. Sime the magnetic field
in the air gap is now distorted, the damper current does not die out when the fault
current reaches a steady state. This was not the case with balanced extemal fadts
where the damper currents died out after a few cycles from the inception of the fauit.
Chapter 5
0.8 1 -35 ikd 2.45 3
xl03 7 - 5 - 3 - 1 .
-1 T
0.8 1 -35 ikq 2.45 3
-400 .c I 0.8 1 -35 2.45 3
Tirne (Sec)
Figure 5.6: Current in the machine windings for a tum to ground fault at 20% fiom the neutral with the field de-energized before the machine is isolated fiom the system
The removal of the field current causes the direct axis damper to be severely afFected.
The effect on the quadrature axis damper is not so prominent as this winding does
not " see" the direct effect of losing the field current. The waveforms in Figures 5.5
and 5.6 are expanded in Figures 5.8 and 5.9 to show details.
It was mentioned in Chapter 1 that the fault current will continue to flow in the
windings even after the machine is isolated and the field de-energized. This can be
clearly seen from the above simulation. The magnitude and the duration of the current
Chap ter 5
0.95 0.98 ifit 1 .O7 1.1
1.28 1 -29 ikq 1.31 1 -32
1.2 1 -3 1 -4 1 -5 1 -6 1.7 1 -8 Time (Sec)
Figure 5.7: Current in the machine windings for a turn to gound fault at 20% fiom the neutral with the field de-energized before the machine is isolated from the system
Chap ter 5
1.8 1 -9 Time (Sec)
Figure 5.8: Current in the machine windings for a turn to ground fault at 20% from the neutral with the field de-energized before the machine is isolated from the system
Chap ter 5
0.9 0.95 1 ikd 1.2 1 -25 1.3
0.9 0.95 1 ikq 1 -2
1.25 1.3 Time (Sec)
Figure 5.9: Curent in the machine windings for a turn to ground fault at 20% fkom the neutral with the field de-energized before the machine is isolated fiom the system
Chap ter 5 91
after the removal of the field depends to a large extent on the winding resistance. The
waveforms in Figure 5.10 ülustrate this situation where the phase resistance is changed
from the original value of 0.0015 Ohms to 0.003 Ohms. The damper currents are also
dependent on the damper resistance.
400 0 i2 - 0.0015 Ohms O i2 - 0.0030 Ohms 700
400 i3 - 0.00 15 Ohms O i3 - 0.0030 Ohms 700
Time (Sec)
Figure 5.10: Effect of winding resistance on the fault current.
The results in Figures 5.11 and 5.12 are obtained by isolating the machine before the
field is de-energized after 0.7 seconds. This would prevent the fault being fed by the
remote source. Such analysis can be used to determine the protection strategy for
synchronous machines.
The fault current and the current in the windings of the faulted phase are dected by
the position of the fault and the grounding impedance. The graph shown in Figure
Chapter 5
2 2.5 3 3.5
Time (Sec)
Figure 5.11: Current in the machine windings for a turn to ground fault a t 20% from the neutrd with the machine isolated before the field is de-energized
5.13 shows the variation of the current 12 in one of the faulted coils for three different
grounding impedances. The rated current of this machine is 6.15 kA. If the machine
is solidly grounded, the fault current is limited only by the winding tesistances and
the leakage impedances of the respective windings. The current i2 for a fault at 1%
from the neutral when the machine is solidly grounded is 4368 A peak. This is more
than 100 times the value if the grounding impedance was 5 Ohms. This explains why
synchronous machines are almost never solidly grounded in practice.
It is necessary to meet the conditions described in Section 5.2.1 and maintain the
Chap ter 5
Tirne (Sec)
3
Figure 5.12: Current in the machine windings for a turn to ground fault at 20% from the neutral with the machine isolated before the field is de-energized
h O - ,
< Y -3 - - E -6 z -9 -
consistency of the equations if faults very close to the neutral are to be analyzed.
The machine mode1 presented here can be used to simulate faults at any location on
the winding since those conditions are satisfied.
- - - c: U V Li - - C h
U u L V U
Certain relaying schemes use the harmonic content of the line currents to identify
6 -12- O 0.5 1 if 2 2.5 3
400 -r
300 -
200
100 - 0
O 0.5 1 ikd 2 2.5 3
xl03 7 r 5
3 - 1 -
-1 .r
O 0.5 1 ika 2 2.5 3
the presence of a ground fault inside the machine. Such a scheme which employs the
third harmonic content is described in Chapter 1. Due to the distributed nature of
the windings, the inductance terms are not purely sinusoidd. The expressions for
Chapter 5
4 8 12 16 20 24 % turns from neutral to the fault
Figure 5.13: Influence of the position of the fault and the grounding impedance on the currents in faulted windings
the inductances, which are shown in Appendix A, neglect the higher order sine and
cosine terms to keep the analysis simple. When the higher order terms are taken into
account, the inductance of phase R would take the following form [22].
Similar expressions c m be written for other inductance terms and are shown in 1221.
The data required to estimate the additional parameters are not readily adab le
and thus are generaiiy not considered in machine analysis. However [22] explains a
method to estimate them by performing tests on the machine. The waveforms in
Figures 5.14 and 5.15 show the results when the fourth harmonic term in the self
inductance of the phase windings is considered. The fault is at 2.5% from the neutral
Chapter 5
200 3 300 350
V - 150
k' 200 if 300 350
410 - 5 400 -
390 - 380 - 370 . 360 4
150 200 a ikd 0 ikq 300 350 x103 25
1.25 O
-1.25 -25
Time 6) 'loJ
Figure 5.14: Effect of the higher order terms in the self inductance expressions with L4 set to 30% of L2
150 200 0 ikd 0 ikq 300 350
v v v v v v v Y v V v v V V v v v v 150 200 300 350
Time (s) XI O"
Figure 5.15: Effect of the higher order terms in the self inductance expressions with L4 set to 10% of L2
Chap ter 5 96
with the groundùig resistance is set to zero. The winding currents show the presence
of hannonics.
I
Source
f 'F
Figure 5.16: A turo to turn fault on phase A
The diagram in Figure 5.16 shows a turn to turn fauit in phase A. Turn to turn faults
can occur when the minor insulation separating the turns breaks d o m . These fauits
are generally hard to detect and if left undetected for a pïolonged period could lead
to more damage to the insulation. In machines where more than one parallel path is
present, split phase relaying is employed to detect turn to turn faults. The foilowing
figures show the currents in dinerent windings when such faults are present. Point X in Figure 5.16 is at 12.5% from the neutral.
The waveforms in Figure 5.17 result when the fault is at X and involves 7.5% of the
winding. The current i3 is very high The currents in the other windings of phase A
are higher than the currents in phases B and C. If the fault resistance Rtit is high
the current in the faulted section would be smaller and this effect is shown in Figure
5.18
There is no noticeable difference in the line currents before and after the fault. Cur-
rents il and i2 are almost 180' out of phase indicating a circulating current. Since
these two currents are monitored in spüt phase relaying, this fault would be easily
de tected by the relay.
Chap ter 5
-80 4 100 150 200 250 300
ikd XI o - ~ ikq XI 0-3 1.5 1
x1 Tirne (Sec) XI o ‘ ~
Figure 5.17: A turn to tum fault involving 7.5% of the winding with Rfit equal to 0.1 Ohms
Chap ter 5
%
1 O0 1 50 200 300 350 400 Time (Sec) XI o - ~
Figure 5.18: A t m to tum fadt involving 10% of the winding with Rfit equal to 1 Ohm
Chapter 5
x103 4 h
S. 2 c. E 2 O
5 -2
-4 1 O0 130 160 1 9 0 220 250
i3 10 1
XI o - ~ x103
5 - O -
-5 Current in the faulty section
-10 - 100 1 30 160 190 220 250
U ia O ib a ic 6
XI o - ~ XI 03
4 2 O -2 -4 -6
1 O0 1 60 190 220 250 Time (Sec) XI o4
Figure 5.19: A tum to tum fauit involving 10% of the winding with Rfit equd to 1 Ohm, with the machine operating close to its MVA rating of 160
If the fault occurs while the machine is supplying a higher current to the system,
the circulating current tends to get smaller. This can be see fiom the waveforms in
Figures 5.19 and 5.20.
The graph in Figure 5.21 shows the influence of the load current on the current in
the shorted section of the winding. Figure 5.22 shows the influence of the number of
shorted turns on the current in that part of the winduig. In both cases the extemal
fault resistance Rfit is set to 0.1 Ohms.
Chapter 5
2.5 -
0 - -2.5 -
Currcnt in the fauIty section -5 1
1 O 0 130 1 6 0 190 220 250 n ia O ib A ic XI o3
1 0 0 130 1 6 0 1 9 0 220 250 Time (Sec) XI O~
Figure 5.20: A turn to turn fault involving 0.30% of the winding with Rtir equal to 1 Ohm, with the machine operating close to its M V A rating of 160
5.3.1 General
The simulations were done using the transient simulation program E M T D C ~ ~ . Pa-
rameters of the faulted coils were calculated using the methods outlined in the pre-
vious chapter and these values were fed into the program.
' Current i3 (kA)
I
15% of tums shorted/
5% of tums shorted
Rfl t = 0.1 Ohms
Figure 5.21: Muence of the load current on the current in the faulted winding
'Current i3 &A) 24-
8 -
/''? The load supplied kept constant 4 - / Mt = 0-1 Ohms
,/
Figure 5.22: Influence of the number of shorted tums on the current in the faulted winding
5.3.2 Simulation results for interna1 faults
The waveforms presented in Figures 5.24,5.25,5.26 and 5.27 show current and voltage
waveforms during turn to ground faults in the phase A winding. The currents and
voltages are as indicated in Figure 5.23.
Chap ter 5
cal
d HV Side LV Side
Figure 5.23: Interna1 fault in a star-star connected transformer
The three phase transformer is made up of three single phase units and hence there
is no magnetir. coupling between the phases. In the first case the fault occrus closer
to the terminal and in the other case it is closer to the neutral. The fault current to
ground, if, and the current i3 in the winding are larger in the second case whereas
the line current is larger in the first case. The currents would be much less if the
transformer is grounded through an impedance. This can be seen fiom the waveforms
Figure 5.24: A turn to gound fault 5% fiom the terminal on the phase A winding
Chapter 5
0.4 0.5 O .6 0.7 0.8 0.9 1 Time (s)
0 ibl O icl 1 -
Figure 5.25: A tuni to ground fadt 5% fiom the terminal on the phase A winding
0.5 - o .
24.5
Figure 5.26: A turn to gound fauit 5% from the neutral on the phase‘^ winding
=: " - - w fi - k A " - d -. d - - m "
s -1 4 E
0.4 0.5 0.6 0.7 0.8 0.9 1 .? i3
Chapter 5
in Figures 5.28 and 5.29.
The waveforms shown in Figures 5.31, 5.32 and 5.33 result when the low voltage
side of the transformer is connected in delta as shown in Figure 5.30. The fault is
located on the high voltage side. Line currents on the delta side after the fault behave
Werently than when it was connected in star. The waveforms presented so far are
for three phase banks consisting of three single phase, 11 kV/415 V, transformers.
The results shown in Figures 5.35 and 5.36 are for a three phase, three limbed, 33
kV/ l l kV, transformer. The 33 kV side is connected in delta as shown in Figure
5.34. Since all the phase coils are magneticaily coupled to each other, the behavior
under intemal faults c m be different from the cases discussed previously. The fault
is doser to the terminal of iine B and it can be seen that the fault is fed mainly from
O ibl 0 icl
O -4 0.5 0.6 0.7 0.8 0.9 1 Tirne (s)
Figure 5.27: A turn to ground fault 5% fkom the neutral on the phase A winding
Chap ter 5
0 cal 0 ebl A cc1
Figure 5.28: A tum to ground fault 5% from the neutral on the phase A winding with the transformer grounded through an impedance of 20 Ohms
n ibi O icl
1 O -4 0.5 0 -6 0 -7 0.8 0.9 1
Time (s)
Figure 5.29: A turn to ground fault 5% fiom the neutral on the phase A winding with the transformer grounded through an impedance of 20 Ohms
Chap ter 5
Figure 5.30: Interna1 fault in a star-delta connected transformer
line B. This is not reflected to the star side since the current in the phase A winding
of the delta side carries the fadt current.
Figures 5.37 and 5.38 show the results when the windings are connected in star on
both sides.
- 0' O .4 0.5 0.6 O.? 0.8 0.9 1
ia 1
Figure 5.31: A turn to ground fault 5% from the terminal on the phase A winding with the secondary side c o ~ e c t e d in delta
Chapter 5
0.4 0.5 0.6 0 -7 0.8 0.9 1 Time (s)
Figure 5.32: A turn to ground fault 5% fiom the terminal on the phase A winding with the secondary side connected in delta
a 0 1 0 ebl 4 ecI
Figure 5.33: A turn to ground fault 5% fiom the neutral on the phase A winding with the secondary side connected in delta
Chapter 5
Figure 5.34: Internai fault in a delta-star connected three phase transformer
eah 0 ebh A ech
Time (s) XI om3 Figure 5.35: A tum to ground fault 31% from the phase B terminal on the delta side
Chapter 5
4 - O - - - -4 - c:
- 8 7 400
L- 450 500 550 600 650 700
iad XI o3
O 1 M O icd x 1 O"
1 7 1
-1 1 J 400 450 500 550 600 650 700
Time (s) XI o3 Figure 5.36: A tum to ground fault 31% fkom the phase B terminal on the delta side
Chap ter 5
eah 0 ebh Cr ech
Time (s) x i o3 Figure 5.37: A turn to ground fault 31% fkom the phase A terminal when both sides are connected in star
Figure 5.38: A turn to ground fault 31% fkom the phase A terminal when both sides are connected in star
Chapter 5
eah
Figue 5.39: A turn to turn fault on the delta side of a transformer
Tuni to turn fadts can be simulated using the models and Figures 5.40, 5.41, 5.42,
5.43 and 5.44 show simulation results for faults on the delta side of a transformer.
The transformer connection is schematically shown in Figure 5.39. The fault curent
depends on the number of tums in the faulted part. It can be seen that the fault
current is not reflected onto the currents in other sections of the winding or on the
other side of the transformer. This is because the turns ratio between the faulted
section and any other winding is very small when only a few turns are shorted. The
influence of the load current on the fault current is not s i w c a n t as can be seen from
Figures 5.43 and 5.44.
5.3.3 Saturation in transformers
A method to include saturation and hysteresis was outlined in Sections 4.5 and 4.5.1.
This method was used to simulate the turn to turn fault on a single phase transformer
shown in Figure 5.45. The complete derivation of the mode1 is shown in Appendix H.
The B-H loop for the transformer material is shown in Figure 5.46. The input
voltage, Ea, was increased above the rated value in this simulation to drive the core
n eal 0 ebl A eci
400 450 500 550 600 650 700 O iaI 0 131 A icl -. -4
5 400 450 500 550 600 650 700 L, n z O i 3
XI o9 xl04
200
O
-200
-400 V - V W V V V V - V - V - V - V - V - 1
400 450 500 550 600 650 700
Time (s) x1 o4
Figure 5.40: A tuni to turn fault involving 1% of the winding
400 450 500 550 600 650 700 Time (s) x10-3
Figure 5.41: A turn to tum fault involving 1% of the winding
Chap ter 5
400 450 500 550 600 650 700 O ial O ibl A icl
X I U
*IO-3 600 - 400 . 200 .
O - -200 - -400 . -600 4
Time (s) XI o3
Figure 5.42: A tum to turn fault involving 10% of the winding
Figure 5.43: A tum to turn fault invoIving 10% of the winding
Figure 5.44: A turn to tum fault involving 10% of the wincling with the transformer supplying a higher load
Figure 5.45: A turn to turn fault in a single phase transformer
Chap ter 5 115
deeply into the saturation region. The B-H loop at the rated voltage can be seen in
the fkst graph of Figure 5.48. If the supply voltage Ea is maintained sinusoidal, then
the flux density B too will show a sinusoidal variation even in the saturation region
and the current wil l be distorted. This can be seen from the waveforms in Figure
5.47.
The waveforms shown in Figure 5.48 display the situation when there is a turn to
turn fault in the high voltage winding. The important thing to note is that the core
does not go into saturation as a result of the fault.
Figure 5.49 shows a case where the supply was interrupted at a current zero. The flux
in the core does not drop to zero but maintains its remanent value. This feature makes
this mode1 suitable to analyze the effects of in-rush currents when a transformer is
energized or when the breakers are re-closed after the detection of an external fault .
B-H IOOP
H (Am XI o3 Figure 5.46: The shape of the B- H loop of the transformer core material
Chapter 5
-300 -
H(Am) 100 300 0 Tuneta) 100 150
1 200
Magnetic Field Intensity x1 o3 Magnetising current on the LV side
x i 0 3 4
2 2 z O 5 4
- 4 . . i i l . . . l l l l
-6 J O 100 50 Tme(s ) 150 200 O =O Erne@) 100 1 50 200
if 0 iL xl O" ia XI o9
Figure 5.47: Magnetizing curent when the core is saturated
Chapter 5
Figure 5.48: A turn to turn fault involving 4% of the winding
Cbap ter 5
Timc (s) U
Erne (s) r d
Figure 5.49: The remarient flux in the core
Chapter 6
Application of the machine model
and the transformer mode1 in
protection st udies
6.1 Summary
The models developed in the previous chapters can to be used to design protection
schemes for transformers and synchronous machines and to study how they would be-
have in difFerent situations. Current transformer models are developed in this chapter
to be used in such studies. Behaviour of certain protection schemes are presented with
the fault waveforms obtained fkom the machine model and the transformer model be-
ing used as the input to the CTs.
6.2 Introduction
The fault wavefonns obtained kom the machine and transformer models can be used
to analyze the behaviour of protection schemes. The current transformer plays a very
important role in the protection schemes and must be properly represented in such
studies. Zkipping of the breakers due to signals fkom Merential relays is treated
with utmost precaution and the unit is never re-closed. Thus, it is important to set
Chapter 6 120
the differential relays so that they would not operate for external faults. Extemal
faults with a high initial exponential component and a large t h e constant tend to
saturate the current transformers giving rise to a number of p r o b h . Such fault
currents are common if the fault occuts close to the transformer or the machine. It is
very common for utilities to use air gapped current transformers in transformer and
machine protection to overcome t hese problems. However, the higher magnetizing
current drawn by air gapped CTs leads to 1! arger ratio errors and larger phase angle
errors. These must be carefully studied before such CTs are used. Models of solid
core current transfonners which have been tested for accuracy are available and can
be used in protection studies [14], [48]. However analytical methods to predict the
behaviour of air gapped CTs are not readily available [50]. An analytical method to
predict the behaviour of gapped current transformers is presented in this chapter 1511.
Sransformers connected in star-delta configuration require that the CTs on the star
side be connected in delta to account for the phase shift in the luie currents on the two
sides of the transformer. If the current harmonies produced in the delta connected
secondary windings are not considered carefidly false tripping could occur. To study
such situations, a delta configuration of three current transformers is modeled and
tested [49].
6.3 Effects of saturation in current transformers
Figure 6.1: A single CT connected to a burden
Chapter 6 121
Transmission line faults occurring ciose to a generator give rise to a fault current of
very high magnitude. In addition, since the fault wili experience a high reactance
to resistance ratio X/R, there will be an initial dc exponential component in the
fault current which wil l decay very slowly. Figure 6.2 shows typical fault currents
during a line to ground fault for two different values of XIR. If the X/R is high
the dc exponential component will decay slowly. Now consider the simple single
CT connection shown in Figure 6.1. The resulting secondary currents for different
burdens are shown in Figure 6.3. The CT in this case is made of a solid core with no
air gaps. Methods described in [l4] and [48] were used to simulate these secondary
current S.
x10-3
n < 25 Y E
f O
50 100 150 200 250
Time ( s ) XI o - ~ Figure 6.2: Fault currents with an initial dc exponential component for a fault occur- ing close to the generator.
The secondary current is distorted in the high burden case. The amount of distortion
depends on the burden as weU as the remanent flux trapped in the core at the instant
of the occurrence of the fault. Figure 6.4 shows the flux in the core during and
Chapter 6
0 Iprimary-(scaled) 0 [sec-l - High irnpedance A [sec-2 - Low impedance
-12 J 1 80 1 20 160 200 240
Time (s ) XI o - ~ Figure 6.3: Secondary currents in the CT under different burdens
after the fault. The remanent flux depends on the current magnitude, the initial
exponential in the primary current, the burden and the instant at which the fault
is cleared. This can be seen fkom Figure 6.4. The remanent flux does not decay to
a lower level upon the clearance of the fault. If the breakers are re-closed while the
fault is still present, the secondary curent then will be influenced by this remanent
flux. It is very important to understand the effects of remanence on the performance
of the relays. It is also very necessary in generator protection to employ methods to
reduce the levels of remanence [53],[52]. This is achieved by employing CTs with an
air gap of suitable len! gth.
6.4 Behavior of air gapped current transformers
Figure 6.5 shows the secondary currents if the solid core CT in Figure 6.1 is replaced
by an air gapped CT. The primary current, referred to the secondary side, is also
Chapter 6
t~ XIR = 40, High impedance (B 1 ) O X R =100, High impedance (B2) A X/R = 40, High impedance (B3) V XIR = 40, Low impedance (B4)
1
50 100 150 200 250
Time (s) XI o4 Figure 6.4: Flux in the CT core under different conditions
depicted in this figure. DiEerent gap lengths were used in the simulations. The shape
and the magnitude of the secondary current depend on the gap length.
The initial exponential component in the primary current waveform is not as promi-
nent in the secondary currents in the cases where the gap length is larger. This is
an important feature of air gapped CTs. Another feature to note is the ratio error
between the primary and secondary waveforms. This is caused by the magnetising
current which builds up the flux in the core and the air gap. The larger the is air
gap, the more current is required to maintain the flux and, as a resdt, the ratio
error increases with the gap length. This error is also infiuenced by the relay burden,
the secondary impedance of the CTs and the lead impedances since these determine
the voltage across the CT secondary coi1 and this voltage, in turn, determines the
flux density in the core. The slow decay of the primary current and the rapid decay
of the remanent flux upon the removal of the fault can be seen in Figure 6.6. The
higher magnetising currents also give rise to a larger phase angle error between the
Chap ter 6
0 Iprimary - scaled 0 Isec - 0.03% gap A Isec - 125% gap v Isec - 2% gap
1 O0 1 20 140 160 1 80 200 220
Time (s) XI o9 Figure 6.5: Secondary currents in the CT when air gapped CTs are employed
primary and the secondary currents. Figure 6.7 illustrates these two errors that must
be accounted for before the relay settings are determined. These effects get very
complicated to visualize in protection schemes where multiple CTs are connected to-
gether i . different configurations and accurate simulation models become an absolute
necessity.
The flux in the core for the cases presented in Figure 6.5 is shown in Figure 6.8. It is
clear from this diagram that the problems caused by saturation can be controlled by
having an air gap of suitable length in the CT core. Upon the removal of the fault
the trapped flux decays a t a much faster rate. This would result in a lower level of
remanence when the breakers are te-closed and hence less chance for the protection
system to malfunction. The features of the gapped CTs highlighted in this section
can be verified analytically using simple electric and magnetic circuits. If the primary
current with an initial exponential is expressed as follows
Chap ter 6
n Ipri-(scaled) 0 Isec m-, 1
60 100 140 180 220 260 300
Time (s) XI O" Isec
20 1
60 100 140 180 220 260 300
Tirne (s) x i o9 B
5 o . . m
-1
-2 +
Time (s) XI o9
Figure 6.6: Decay of the prirnary current and the flux in air gapped CTs
the secondary current can be shown to be equal to i,.. . The currents, voltages and
the flux are shown in Figures 6.9 and 6.10.
i,,, = A l
The exponential component is attenuated by a factor of f i The phase angle k1-$ -
Chapter 6
0 Iprimary - scaled 0 Isec - 0-7% gap 8
50 100 150 200 250
Tirne (s) x i o9 Figure 6.7: Primary and the secondary current in the CT to demonstrate the ratio error and the phase angle error.
error is 6 and the ratio error is due to the term . The constants KI, K2,
K3, and K4 depend on the magnetising inductance, La==, of the secondary coi1 and
the mutual inductance, MW, between the two coils. The details of this derivation are
given in Appendix L.
The simulation resdts in Figures 6.6, 6.7, and 6.8 display the characteristics, the
above equations describe. The derivation of the model, based on Figures 6.9 and 6.10
U B-1 - 1.25%gap O B-2 - 0.03% gap A B-3 - 2% gap 0.2 1
-
4
Time (s) XI o - ~ Figure 6.8: Flux in the CT core when air gapped CTs are employed
Air gap
Figure 6.9: Schematic diagram of an air gapped CT
and the three Equations 6.5, 6.6 and 6.7 is shown in Appendix M.
Chapter 6
Figure 6.10: Air gapped CT feeding a relay burden
The model of the air gapped CT was validated by comparing the simulated waveforms
with those recorded from a relay manufacturer's synthetic test plant in Stafford, UK.
Three air gapped CTs were connected in a star and the currents in the CT secondary
windings were recorded. Figures 6.11, 6.12, and 6.13 show the calcdated and the
measured secondary current for two different cases. The close agreement of the two
waveforms indicates the accuracy of the model. The high fiequencies observed in the
measured waveforms are due to noise introduced by the measuring equipment.
6.5 Behaviour of three current transformers connected in
delta for transformer differential protection
Delta connected CTs are still very cornmon in differential protection schemes of
star -delta connected transformers. Modem relays based on microprocessors do not
Chapter 6
Phase A current (CalcuIated) 0 Measured
Time(s)
Figure 6.11: CT with a 0.03% air gap
0 Phase A current(Calculated) 0 Measured
-10 J 1.25 1.3 1.35 1.4 1 -45 1.5 1.55 1 -6
Time(s)
Figure 6.12: CT with a 0.2% air gap
0 Phase B current(Calculated) 0 Measured 8
Figure 6.13: CT with a 0.2% air gap
require such a connection since the relay can be programmed to account for the phase
shiR between the line currents on the two ends of the transformer. However, most
utilities still use this method and the delta connected CTs have to be matched with
the star connected CTs on the other side of the transformer. The mode1 developed
Chap ter 6
Figure 6.14: Three CTs comected in delta
here is based on methods similar to those presented in [48] and is outhed in A p pendix 1. Figures 6.15, 6.16 and 6.17 show the cornparisons of the simulations with
data measured at a relay manufacturer's test plant in Stafford, UK. The connection
is illustrated in Figure 6.14.
a Calculated 0 Measured 30
I Line A 1
2.8 2-85 2.9 2.95 3 3.05 3.1 0 Calculated 0 Measured
15
h 6
S. 4 -3 c Q
-12 O
-21 1 10 Ohm lead . A-N fault with flux dnvcn in the opposiie direction
-30 2.8 2.85 2.9 2.95 3 3.05 3.1
Time(s)
Figure 6.15: Comparison of the calculated waveforms with measured data to validate the delta CT mode1
Chapter 6
0 Calculated 0 Measured
Line A
n 5 s = -5 ' -15
10 Ohm lead, A-N huit wiih fluir driven in the same direction -25 4 I
5.7 5.75 5.8 5.85 5.9 5.95 6 O Calculated 0 Measured
Line B 1
Figure 6.16: Cornparison of the calculated waveforms with measured data to validate the delta CT mode1
An interesting feature observed in the delta connection is the presence of a circulating
current in the delta windings. This effect is amplified when the CTs are driven into
saturation by larger fault currents with a slowly decaying initial exponential. A sub-
stantial proportion of second and third harmonics are present in the CT secondaries.
In addition, the line currents feeding the relays contain a significant amount of second
and third harmonics. The CTs on both sides must be selected with due care to avoid
unnecessary trippings due to these harmonics that are seen only on the delta side of
the differential scheme. These effects can be seen fiom the simulation results shown
in Figures 6.18 and 6.19. The secondary currents in the fkst graph of Figure 6.19 are
almost in phase which indicates a circulating current. It s W d be noted that the
third harmonics in the three lines are unbalanced and that they do add to zero.
Chapter 6
30 1 Line A
I l
-30 f 13.5 Ohm lead. A-BC fklt
3.3 3.35 3 -4 3.45 3.5 3.55 3.6 a Calculateai 0 MeaswPd
30 Line B
Figure 6.17: Cornparison of the calculated waveforms with measured data to validate the delta CT mode1
6.6 Behaviour of several relaying schemes used in machine
and transformer protection
Interna1 faults were simulated using the methods presented in Chapters 3 and 4. The
fault current waveforms were fed to the CTs as their primary current. This wodd
enable the calculation of the relay currents under a given situation. This information
can be used
1. to decide on the proper setting for the relay
2. to decide on the size and class of the CTs to be used
3. to take steps to minimise false trippings etc.
Chap ter 6
C'ï secondary current in Phase A - is l
1 O0 200~urrent in the phase A relay - IL1 300 400
CI 50 'Oo i ~ 1 200 Frequency (Hz)
5 1 4 - GI
10.5- Z - - 7 - a 3.5- 2 7 7 0 C 2
0 + O 50 100 1 50 200 Frequency (Hz)
Figure 6.18: Harmonies present in the secondary and line currents
The following figures show simulation results where the fault currents calculated using
the simulation models were used as the primary current input to the CTs in the
protection scheme. Figures 6.21 and 6.22 show the behaviour of the restricted earth
fault protection relay connected to the 33 kV side of a transformer. Figure 6.20 shows
the way the four CTs are connected in this configuration.
Figure 6.2 1 shows the relay current during an extemal fault close to the transformer.
In the two cases considered, the same primary current was applied to the CTs. The
remarient flux on the CT of phase A was set at different levels and this has an effect on
the relay current as can be seen fiom the waveforms. The relay curent is duenced
by the relay impedance and the lead impedances connecting the CTs to the relay. The
Chap ter 6 134
40 70 100 330 160 190 220 Time (Sec)
XI 04
Figure 6.19: Currents during a three phase fault
CTs tend to saturate when the relay impedance is high and this too influences the
current passed to the relay. The e s t two relay current waveforms in Figure 6.22 are
for a turn to ground fault on phase A, at 31% from the neutral. The relay, whether
high impedance type or low impedance type, would not have a problem detecting
this fault. However, if a fault occurs very close to the neutral, the relay current can
get very small. The current into a relay, with a low impedance of 5 Ohms, due to a
fault at 2.8% fiom the neutral is shown in Figure 6.22. This current is small and is
comparable with the relay currents during external faults. Fault discrimination thus
becomes a problem when the fault is close to the neutral.
Chapter 6
Transformer wiaduig
-
Figure 6.20: Restricted earth fault protection on a transformer star winding
The performance of a restricted earth fault protection scheme is shown in Figure
6.23. The scheme is used to protect the star connected windgins of a three phase
transformer as shown in Figure 6.20. The transformer mode1 is used to derive the
fault currents and these currents are fed to the primary windings of the CTs in the
protection scheme. The relay current depends on the relay impedence and this can
be seen Tom the results shown in Figure 6.23. The CTs tend to saturate when the
relay impedence is high and as a result the the current to the relay gets distorted.
The dinerential relay current due to a tuni to ground fault on a synchronous machine
is shown in Figure 6.24. Six CTs are employed in this scheme as shown in Figure 1.7.
The relay in this case had a low impedence.
The machine models, the transformer models and the CT models presented here pro-
vide the relay engineer with the necessary tools to simulate a protection scheme and
decide on the relay settings. Such analysis would enable the engineer to enhance the
performance of the protection scheme such as the speed of operation, fault discrimi-
nation and security.
Chapter 6
Rernanent flux = 2 T 0 Rernanent flux = O
-1 O 50 1 O0 150 200 250
Relay current when initial flux = 2 T 30 1 XI o3
O 50 100 150 200 250 Relay current when initial flux = O T
x1 O" *
1
O 50 1 O0 200 250
Time (s) XI om3 Figure 6.21: The effect of remanence on the relay current during extemal faults
Figure 6.22: Relay curent in the presence of an interna1 tuni to ground fault on the star side of the transformer
Chapter 6
50 Ir
Relay cwrent (Low irnœdance case, 1
Y) k
Relay current (High irnpcduice case)
-50 + 150 200 250 300 350 400 4sa
Time (s) xifl
Figure 6.23: Relay curent in a restncted earth fault protection scheme.
Figure 6.24: Dserential relay currents due to a turn to ground fault on phase A
Chapter 7
Conclusions
The main aim of this thesis has been to develop models of synchronous machines,
transformers and current transformers which c m be used in power system protection
studies. Models with the capability to accommodate faults inside the windings have
long been sought after in the area of power system relaying because there is no recouse
to experimentation on the actual machine or the transformer to gather the necessary
information. Current transformers, considered a key element in protection systems,
were also modeled as a part of this thesis.
7.2 Main contributions of the thesis
The following are considered contributions of this thesis.
1. The synchronous machine model
This model takes into account the way conductors are actually placed
inside the winding. This is necessary because the fault divides the faulted
coil into a number of sections. The method presented here is very general
and can be easily extended to study any type of fault on any type of
winding configuration. Interna1 fault current was shown to be dependent
138
Chapter 7 139
on several factors such as the grounding method, position of the fault,
number of tums involved, the winding design and the loading condition.
2. Transformer models and inclusion of saturation
Interna1 fault current in solidly grounded transfomers is ümited only by
the winding resistance and the leakage inductance. The method used to
calculate the leakage inductance was tested by comparing the calculations
with those measured by a transformer manufacturer. Like in the machine,
details of the winding and the core are necessary to calculate the param-
eters needed to simulate internai faults. A method to include saturation
and hysteresis was outlined and this is based on theories of ferro-magnetic
hy st er esis.
3. Model of an air gapped CT
Air gapped CTs are widely used by utilities in transformer and machine
protection to overcome problems caused by CT saturation. Hysteresis and
saturation are accurately modeled and the simulations were tested with
recorded waveforrns to validate the model. A configuration in which three
CTs were comected in star was considered in these tests. There is hardly
any published material covering this topic and this model would enable
an engineer to select a gapped CT with the desired characteristics. It is
important to match the gapped CT with the other CTs, gapped or solid
core, to make sure of satisfactory operation of the protection. The air gap
is considered as a series reluctance in the iron core. This allows the model
to display accurately key features of the component such as the slow decay
of the secondary current once the fault is removed and the rapid decay of
the trapped flux in the core.
4. Model of three CTs connected in a delta configuration
Delta connected CTs are common in tramformer differential protection.
Since the delta connected side shows characteristics that are not seen on
the star connected side of a differential relay connection, steps must be
taken to avoid maloperation of the relays. In most cases problems occur
when the CTs are driven into saturation by heavy fault currents. The
presence of a large circulating current in the delta windings was observed
in the simulation results. The presence of third harmonies in the delta
winding as well as in the lines connecting the CTs to the relays was another
observation. The model was validated by comparing recorded waveforms
with the simulations.
It was shown how aiI the components modeled here are integrated to study the per-
formance of a particular relay connection.
7.3 Recornmendations for h t h e r work
The transformer models should be tes ted with recorded waveforms during interna1
faults. Inchsion of saturation in the a-b-c domain machine models was not considered
in this thesis. The coefficients in the inductance terms that were treated as constants
would change in the presence of saturation and this change has to be calculated and
included in the model. Saturation and hysteresis in three phase transformers can be
treated in a fashion siniilar to how the saturation in a single phase transformer was
treated.
Appendix A
Elements of the inductance rnatrix
of a synchronous machine
The elements of the inductance matrix, [Lsyni], can be expressed in the following
forms [l8]. The angle 0. is in electrical radians. Ll, &, L2, M., LF , Lo , Lp, MR, MF,
MD and Mq can be derived nom the data supplied by the manufacturer, as shown
in Appendix C. Li is the leakage inductance of a phase winding. Depending on the
position of the rotor, the self inductance of a phase winding will have a maximum
value of ( 4 + LI + L2 ) and a minimum value of ( Ll + L1 - L2 ).
Appendix B
Solution to the machine equations
using Trapizoidal int egrat ion
Equations 2.1 and 2.2 in Chapter 2 describe the behaviour of the syndvonous ma-
chine. These two equations can be combined to form the following equation.
d d [ [ ~ s m l + di [ L S Y ~ ~ I ] (4 = - [ [ ~ s m l [I~I] - [Vil 03.4)
If two matrices A and B are defined as follows then equation (B.4) can be rearranged
and written as show in equation (B.5).
Appendix B
If a time step of At is used to numericaiiy calculate the vector [Il] at any time t, the vector [Il] is given by the following expression.
When the trapizoidai rule of integration is applied to the above equation, the following
equation will result.
Thus [Il] is given by the following equation with the matrices [G1], [Hl] and the vector
[V;] defined as show below. [Il is the identity m a t e
Appendix C
Conversion of the d-q-O data to the
a-b-c domain
The machine data supplied by the manufacturers or those derived fiom standard
tests, are in a form applicable in dqO based models. These must be converted so that
they can be used with abc domain models.
The leakage inductance of the machine is supplied in data sheets. It can also be
estimated by performing an open circuit test and the zero power factor test and then
using the Poitier triangle approach [33].
The parameters Ld, Lq and Lo are again supplied by the manufacturer. They can
be estimated with reasonable accuracy by performing a slip test on the machine [59].
The following derivations show that the results of the slip test give the values of Ld
and L, directly. A zero sequence test, where al1 three phase windings are connected
in series and a current passed through them can be used to estimate the value of Lo.
In the slip test a balanced voltage is supplied to the three phase windings with the
field winding kept open. The machine is tumed a t a speed slightly below or above
the rated speed. The current in the phase A winding would be a minimum when
the direct axis is in line with the axis of Phase A. Similarly it WU be a maximum
when the two axes are 90° apart. Thus, maximum current occurs at 0 = 90" and the
minimum current is at 0 = oO. The currents in the three windings would be balanced.
Hence at any given instant the currents are out of phase by 120" from each other.
Typical voltage and current waveforms for Phase A, obtained from this test are shown
in Figure C.1.
Figure C.l: Typical voltage and current waveform recordings fkom a slip test
At 0 = oO, assume that the current is a minimum in Phase A. If this has the magnitude
I,m'" then phases B and C will carry a current -(Tl. The voltage across Phase A
would be a m d m u m given by Vo-. The inductance seen at the Phase A terminais,
Lm,, is given by
Similarly, at 0 = go0,
For any position of the rotor, if the same current is passed through
resulting inductance Lo is,
(C.4)
the windings, the
Once the leakage inductance is known Li, L2 and M, can be calculated. The value of
MF is est imat ed using the open circuit characterist ics supplied by the manufacturer.
For a field current if if the induced peak voltage across the phase winding is VMPak,
The phase resistance and the field resistance are normally given by the manufacturer.
They can also be estimated fiom simple dc resistance tests. The resistance of the two
dampers are estimated using the direct-axis and the quadrature-axis time constants.
The leakage inductances of theses windings and that of the field winding too can be
calculated using the same time constants [56] [57]
The equations presented in [la] and summarized below, can be used to calculate the
remaining iinknown parameten Lq, MQ, LF, CD, MD, and MR
(C. 1 1)
(C.12)
Appendix D
Inductances of coils sharing a
common flux path
Consider the magnetic circuit shown in Figure D.1. The coils (a) and (b) are identical
and each has N number of turns. Let the number of turns in coil (c) be Nc.
-- .-
Figure D.1: Three coils wound on the same core.
When coi1 (a) is energized with the other two kept open circuited, the following
equations can be written. 4 is the flux in the core, LI is the self inductance of coil
(a) and S is reluctance of the magnetic path. The flux is assumed to be contined to
the core. In Figure D.2, the two coits (a) and (b) are connected in pardel. The same
voltage V is applied across the coils. Since the voltage is the same, the flux in the
core has to be the same as in the earlier case where only coi1 (a) was excited.
From equations D.1 and D.4,
-- - - ----- - --
,/Y> Flux
Figure D.2: Three coils wound on the same core with two of them connected in parallel.
Appendix D
If Le, is the inductance of the parallel combination of coils (a) and (b) then,
The self inductance of the parallel combination is equal to the self inductance of any
one winding. This is tme when the two CO& are tightly coupled.
If the mutual inductance between eoils (1) and (3) is Ml> and if the mutual inductance
between coils (3) and the parallel combination is M(1C2p, then,
(D. 10)
This shows that the mutual inductance between coil (c) and the pardel combination
is the same as the mutuai inductance between coil (c) and any one coii in the parallel
combination.
Appendix E
Elements of the inductance matrix
of the four pole synchronous
machine
The elements of the inductance matrix, [Lsynl], of a four pole machine can be ex-
pressed in the following foms. The angle 0 is the actual mechanical angle in radians.
Appendix F
Calculat ion of winding inductance
parameters for simulation of
interna1 faults in synchronous
machines
Inductances involving the faulted CO& are derived here to enable winding fault simula-
tions. A four pole, lap wound machine with two parallel paths per phase is considered
as the example case. However the method described can be easi-y extended for other
types of windings. It is assumed that the inductances of the normal windings are
known from the data supplied by the manufacturer.
F.2 Description of the machine windings and the inductances
under normal conditions
A 3 phase, 4 pole synchronous machine with a lap winding in the stator is shown in
Figure F.1. The rotor is assumed to be of the salient pole type.
Each phase winding has two pardel paths with the number of series tums per phase
being equal to Nph. Thus, each dot carries conductors.
Sub windings 1 (In slots 1 and 7), 2 (in dots 2 and 8), 3 (in slots 13 and 19), and 4 (in
slots 14 and 20) when connected in series as shown in Figured F.l form one parallel
path of the phase A winding and coils 5 ,6 ,7 and 8 fonn the other parallel path. Each
coi1 consists of 4 tums. The two parallel paths when combined, form the phase A
winding of the machine. The remaining slots carry the windings of phases B and C.
- - Top layer No- of piuailel paths = 2 No- o f Poles = 4
----_-- - Bottom layer No. of Slors = 21
Figure F.1: Winding diagram of the four pole machine.
The field winding F is on the salient pales of the rotor and the four coils are connected
in series as shown in Figure F.2.
The damper windiag can be represented by two short circuited coils, one with its axis
along the d-axis and the other with its axis along the q-axis as shown in Figure F.3.
The system can now be viewed as a system of six magneticaliy coupled coils. This
Figure F.2: Rotor
-- - - - Scuorsiois
arrangement of a four-pole synchronous machine with
C-Axis 6 - Axis
of rotation
salient poles.
Figure F.3: Schematic of the winding arrangement
Appendix F 157
arrangement is shown in Figure F.4, and the voltage current relationship is given by
the foUowing set of equations [l8].
where
The elements of the inductance mat* [Lsynl] depend on the position of the rotor
and, hence, they are time varying.
Since the windings are placed symrnetricaiiy along the circderence of the stator,
we can make use of the "electrical angle" of displacement of the rotor axis from a
reference direction in order to define the position of the rotor at any given instant.
The elements of the inductance matrix [Lsynl] can be expressed as in Appendix A
[Ml. The axis of the phase A winding is taken as the reference direction.
When an interna1 fault occurs in one of the stator windings, then that divides the
Figure F.4: Schematic diagram of the six coupled coils of the machine.
Appendix F 158
faulted phase winding into a number of parts. Under such conditions, generally, the
geometrical symmetry which existed between the normal phase windings would no
longer be present. Hence, to simplifjr the derivation of the elements of the inductance
matrix under winding fault situations, the actual mechanicd angle of displacement
of the rotor as opposed to the "electrical angle " wïll be used. However, it should be
noted that for most types of interna1 faults, it is still possible to use the "electricd
angle". The elements of the inductance matrix [Lsyni] as a h c t i o n of the actual
mechanical angle measured fkom the reference direction is given in Appendix E. The
magnetic axis when coiis 1 and 2 in Figures F.1 and F.5 are connected in series is
taken as the reference direction. 0 is the angle between the reference direction and
the pole 1 of the rotor shown in Figure F.2.
Slot angle
Figure F.5: Placement of conductors inside the stator slots
F .3 Description of the machine windings and the inductances
in the presence of a turn to ground fault
The positions of the coils 1, 2, . . . and 8 of the machine are shown in Figure F.5. The
two parallel paths of the phase A winding, 4,i and 42 are shown in Figure F.6.
Consider a short circuit fault to ground on coil6 which is on the parallel path 4~ in
phase A. This breaks the coil 6 into two parts, A3 and A4 as shown in Figure F.7-
The phase A winding can now be viewed as made up of five parts, Al , A2, A3, A4
and A5.
A l consists of the coils 1 - 4 connected in series and is the same as 41. A2 is the
coil 5. A3 is the portion of coil6 that is connected to coil 5. A4 is the other portion
of the coil 6 and it is connected to coil 7. A5 is the combination of the coils 7 and 8,
connected in series. This arrangement is shown in Figure F.7. The axes of coils A2,
A3 and A4 are located under one pole and the axis of A5 is located under a pole 180'
apart.
The matrix equation governing the voltage-current relationship of the machine in
Figure F.6: The coils of the phase A winding.
Figure F.7: Representation of the phase A winding with a fault on one paraiiel path.
Appendix F 160
the presence of an internal tum to ground fault can be expressed as follows. The
directions of the currents and voltages are shown in Figure F.8.
7 Fault
m- 'fld Field
Figure F.8: Representation of the machine coils under an internal short circuit.
where
and
Lsyn* is a symmetric matrix and
Li = self inductance of coi1 i
Li = mutual inductance between the coils i and j
Since [Lsyn2] is a symmetric matrix, we need to evaluate the 10 diagonal elements
and 45 off-diagonal elements, in order to determine [Lsynz]. SeIf inductances of the
normal windings and the mutual inductances between any two normal windings are
not affected by the fault. As a result, of the 55 elements mentioned above, 15 are
directly known fkom the inductance matrix [LsynJ. Thus, to fully define the matrix
[Ls ynz], only 40 elements need be deterrnined. A method to e d u a t e these self and
the mutual inductances involving the faulted winding is discussed in the following
sections.
Appendix F 162
F.4 Inductances between the winding Al and the normal
windings.
Winding A l makes up one parallel path of the phase A winding. The self inductance
of the phase A winding is given by the foIIowing equation.
Here, 8 is the displacement of the rotor pole 1 fiom the reference axis which is the
axis of the coîls 1 and 2 when they are connected in series. La can be further written
as
where
LI = the leakage inductance of the phase A winding
Lu-,g = the magnetising inductance of the phase A winding
The magnetic flux which is associated with the magnetizing inductance Lamg crosses
the air gap between the stator and the rotor and links all windings on the rotor as
well as the other windings on the stator. LI is constant where-as
depends on the position of the rotor.
The two parallel paths of phase A are placed inside the same slots as shown in
Figure F.5. The magnetizing flux due to each winding shares a common path. Thus the magnetizing inductance of any one parailel path is equal to the magnetizing
inductance of the phase A winding. This is shown in Appendix D. Thus the self
Appendix F
inductance of the parallel path Al , Lal can be written as foliows.
Lall is the leakage component of Lal and can be found using the forms shown in
Appendix G. The results in Appendix D show that the mutual inductance between
A l and any other normal winding is equal to the mutual inductance between the
phase A winding and the normal winding concerned. Thus,
F.5 Inductances of the coils of phase A.
Figure F.9: Winding X and winding Y .
Consider the parailel path 42 of phase A which is shown in Figure F.6. This consists
of the coils 5, 6, 7 and 8. If we break this into two parts X and Y, as in Figure F.9,
where the winding X is made up of coils 5 and 6 and the winding Y is made up of
coils 7 and 8, then X and Y can be viewed as h o identical windings placed 180' apart
on the stator. This can be seen in Figure F.5. Let Lx and Ly be the self inductances
of these two windings and let LxgY be the mutuai inductance between them. Due to
the symmetry mentioned above
Since the magnetizing part of the self inductance of the parallel path A2 is equal to
the magnetizing part of the self inductance of the phase A winding it can be written
as
Lap2i consists of the leakage inductances Lix and Lw of the windings X and Y.
The leakage flux of one of these windings does not couple with the other winding.
Assiiming each parallel path has the same leakage inductance, we can mite
Since the two windings are identical
(F. 15)
Inductance Lap2 is maximum when the salient pole is in line with the reference axis
where 8 = O. Then
Appendix F
When the rotor is stationary at this position, if a voltage ul is applied to the winding
ApZ with all the other windings open circuited and if the current 00-g is il then
Since coils X and Y make up Ap2
LXO, LYO, LX,YO and LY.x0 are the d u e s of Lx, LY, and LuIx when 0 = 0.
Equation F.19 can be simplified and written as
where
and
The sign of LX,Y should be chosen correctly by considering the directions of the
respective windings. If this is not done properly, the inductance mat* of the faulted
system wili end up being singular or will lead to an unstable system of equations.
From Equations F.18 and F.20
Appendix F
I;, takes the form
Lait LP = -j- + Lpmg
where LpWg is the magnetizing part of 4. Then
(F. 24)
Consider the case where coil X is energized with coii Y and ail the other coils are
kept open circuited. Since hF, » Pa+ the flux pattern due to a current i, in winding
Y when 8 = O can be approximated as shown in Figure F.10.
The reluctance of the magnetic materiai is very smail compared to the that of the air
gap. Let the reluctance of the air gap between the stator and the pole face be Sa.
For the path O-a-b-O in Figure F.10,
For the path O-a-b-c-O,
For the path O-a-d-O,
The flux Linking the coi1 X i s 41 + q52 + 43 + t4 and the flux linking the coii Y is 42 + $3. Thus Lpmg and Mp can be written as follows.
Equations F.27, F.28 and F.29 can be simplified to show that
and
Fkom Equations F-26 and F.35
Equations F.38 and F.39 give the magnetizing portion of Lx, and, &,y when the
rotor angle 0 = O. For any other position of the rotor,with the leakage inductance
too taken into account, Lx and LxVu can be expressed as follows.
The winding A5, which consists of the coils 7 and 8, is identical to winding Y. Thus
the self inductance of this winding can be written as
For coils X and Y,
For any other coil, 2, in the same slots as coi1 X and, with Nz turns,
Thus for A2 and A5
The axis of the winding A2 is shifted by an angle 6 fkom that of the winding X. The
angle b is the slot angle of the stator of the machine. If n is the number of slots in
the stator then,
Considering this shifk caused by the distributed nature of the windings, the mutual
inductance between the windings A5 and A2 for any other rotor position 0, can be
expressed as follows.
(F. 47)
Then mutud inductances L(a3zas) and L(04,05) can be derived in a similar manner.
F.6 Mutual inductances between A l and the other coils in
Phase A
To estirnate the mutual inductances between A l and the windiogs A2, A3, A4 and
A5, consider the flux pattern shown in Figure F.11. This occurs when 6 set to zero
and the winding A l is energized with all other coils kept opened. If the applied
voltage to this coi1 is 712 and if the current flowing is i2, then
For the path O-a-b-O,
But
Therefore
Figure F.ll: Flux pattern when coil Al is energized
NAS~Z = 4Sa&
The total flux passing through the coil A5 is 2&,. Therefore
From Equations F.27, F.33 and F.34
From Equation F.30
Therefor e
By substituting this in Equation F.54
For any rotor position 8,
Using a sirnilar approach and considering the respective shift in their magnetizing
ax is due to the slot angle 6, the mutual inductances, La3,ai and La4,ai can be expressed as follows.
Appendix F
F.7 Self inductances of coils A2, A3 and A4
The axis of the coil A2 is shifted by an angle f from that of winding x. Thus the self
inductance of A2 can be written as
NA^ LaIl L, = (%), ( T + : (L1+L2cos4 (et;)))
F.8 Mutual inductances between A2, A3 and A4
A2 is coil 5. A3 and A4 when connected in series is coil 6
The inductance of coils 5 and 6, L5+6, when connected in series would be identical to
that of coil A5.
Therefore
when N5 and 2V6 are the number of turns of coils 5 and 6
Thus, considering the winding arrangement, the self inductance of coil 6 can be
written as fouows.
(F. 73)
Let NA2 = a and NA5 = b.
If the mutual inductance between coils 5 and 6 is LSY6 then
L(5 + 6),, L(5) , and L(6)mg are the magnetizing component of the self induc-
tance of the respective coils. Substituting the expressions in Equations F.42, F.67
and F.73 we can find L5,6
(Cl +L2cos4B) - (;) ' (i) [Xi + L* (cos 4 (e - ;) + cos 4 (0 + f))]] (F.75)
But in the case considered here
Therefore
Since A3 and A4 forms the coi1 6
and
(F. 76)
Appendix F
The coils A3 and A4 occupy the same dots on the stator. As a result the magnetic
coupling between those two is very strong. The mutual inductance between the two
windings takes the following form.
The factor g takes into account the leakage flux between the coils A3 and A4 and is
very close to unity. To calculate g, slot dimensions, arrangement of the conductors
inside the slot, and the geometry of the end windings are required and the methods
outlined in Appendix G c m be used.
In the absence of such information, Ldna4 may be approximated using the following
Appendix F
method.
Figure F.12: Flux pattern when coil A3 is energized
Let the flux coupling the coil A3 when a current i3 is passed through it be 9A3. This is shown in Figure F.12. Assume ail other coils to be open circuited. The self
inductance of this coil can be written as,
Fkom Equation F.68
AppendUc F
where
and
The flux ( 4 ~ 3 ) ~ passes through the winding A4 where-as only a portion of (&A3)1,
taken into account by another constant gl, (and greater than g) passes through this
winding.
The total flux passing through A4, due to a current i3 flowing in A3, (bA3,ar, is given
by
where gl < 1.
Typically, the leakage inductance of a phase winding is much less than the magnetiing
inductance. Therefore
As a result,
Appendix F
L&,a4 N NA^ ( 4 ~ 3 ) 2
i 3
Under the above approximations, L03,a4 wi l l take the foilowing fom.
A ' (LI + L* cos 4 (B + ;)) La3,a4 = - 4 ( N d 2
(F. 100)
F.9 Mutual inductance between the phase B winding and a
coi1 in phase A
The mutua1 inductance between phase A and phase B is given by
The magnitude of L a , is maximum when û = -6. If this is expressed by (La,b)-,
The approximate flux pattern when the phase B winding is energized with B = -5 is shown in Figure F.13. The positions of the conductors of phases A and B are also
shown in the diagram.
If vq is the induced voltage across any one parallel path of the phase A winding when
phase B is energized with aU other windings open,
where i4 is the current in the phase B winding. The total flux passing through the
coils 5 and 6 is the same as that passing through the CO& 7 and 8. Thus the voltage
across windings 5 and 6 and the voltage across 7 and 8 are identical. Coils 7 and 8
Appendix F 180
make up winding A5. The voltage across this must be equal to 7. If (Lb,a5)- is
the mutual inductance between A5 and Phase B, when the rotor is at this position,
t herefore
This relationship is d i d for any position of the rotor angle and LAS,L can be written
as follows
~ & , b = - = - M , C O S ~ e t - 2 2 l [ ( 31 When coils 5 and 6 are connected in series, the mutua1 inductance of this combination
Figure F.13: Flux pattern when Phase B is energized with the rotor displaced by an angle of 15O
Appendix F
with the phase B winding, L(5t6),), is the same as LoSgb
If the induced voltage across these two due to current il is V S + ~ , then
If V5 and V6 are the voltages induced in coils 5 and 6 respectively,
t herefore
hence
Where L5,b and LBC are the mutual inductances of the two coils 5 and 6 with the
phase B winding.
Considering the shift of their axes from that when the two coils 5 and 6 are in series,
L5,$ and LsYb c m be expressed in the fouowing form.
Mis, LmS, Ma6 and are positive constants. Let be the angle between the axis
of coil5 and the axis of the Phase B winding and let & be the angle between the axis
of coil 6 and the axis of the Phase B winding respectivety. Then,
Coils 5 and 6 are identical except that they are displaced by an angle 6. If 6 << 60
then we could make the foliowing approximation.
Thus, fiom Equation F.111,
Appendix F
Using Equations F-116 and F.117
Ma L2 -+-= 2 2
2Mas + 2Lm5 cos 6
Equations F.120 and F.121 give the values for the two constants Mas and LmS. NOW
L5$ and L6,b are given by the fouowing equations.
These two expressions for LS,L and L6$ are consistent with the conditions of equation
F.111
The mutual inductance of the coils A2, A3 and A4 with the Phase B windïng can be
derived using the above results.
Appendix F 184
F.10 Mutual inductance between the phase C winding and
a coi1 in phase A
The mutual induction between Phase A and Phase C is given by
(F. 121)
Using an approach similar to that in section F.9, the mutual inductances of the phase
C with the coils A2, A3, A4 and A5 can be expressed as follows.
F.ll Mutual inductance of the field winding with the coils
of phase A
The mutual inductance between the phase A winding and the field winding is given
by
It takes a maximum value of MF. If the field winding is excited with a current i5, and
if all other windings are kept open, the flux pattern when B = O is shown in Figure
F. 14
Figure F.14: Flux pattern when the field winding is energized with ali other coils kept open circuited
When 8 = O
24f (La,f)r=o = MF = Nph- (F. 135) i5
Coi1 A5 has turns and its mutual inductance with the field winding is thus,
For any angle 8,
L(5+6)1f is the mutual inductance of the series combination of coils 5 and 6 wïth the
field winding. The foilowing equations, which are similar to those in Section F.9 can
be written for inductances involving coils 5 and 6.
(F. 140)
Mg and Mg are constants. Since coils 5 and 6 are identical except for the displacement
in position by an angle 6,
Therefore
(F. 141)
(F. 142)
Appendix F
Therefore
MF M5 = Ms = ---- 4 cos 6
(F. 144)
(F.145)
(F .l46)
(F. 147)
F.12 Mutual inductance of the the d-axis damper winding
with the coils of phase A
Mutual inductance between the d-axis damper winding and the Phase A winding is
given by
LkdVa = MD COS 28 (F. 148)
Mutual inductance between the d-axis damper winding and the faulted coils of phase
A can be derived in a manner similar to that in Section F. l l and are given by the
following equations.
4 cos 6
M~ cos 2 (8 + 4) La3cd = (iy;;) 4;usb
F.13 Mutual inductance of the the q-axis damper winding
with the coils of phase A
The mutual inductance between the q-émis damper winding and the phase A winding
is given by
Lrrp = MQ sin 29 (F.153)
It can be shown, as in section F.12, that the mutual inductance between the q-axis
damper winding and the coi1 A5 , MA5, is given by
MQ = - sin 28 2
(F.154)
Considering the coils 5 and 6 and their series combination as was done in Section
F.11, the following equations can be derived.
and
Hence
When 0 = 2 ,
-- MQ - MSQ [sin (i - 6) + sin (g + 6)] 2
(F.159)
(F. 160)
-- R 7r 7r 7r - MSp [ S ~ ~ ~ ~ O ~ ~ - C ~ - ~ ~ ~ ~ + S Y L - C O I ~ + C O S - S U ~ ~ ]
2 (F. 162) 2 2 2
MQ 7r -= 2
M ~ Q (2 sin 5 cos 6)
Mi? - = 2M& COS 6 2
MQ M5Q = MsQ = - 4 cos 6
(F. 163)
(F.164)
(F.165)
Appendix F
Therefore
F. 14 The calculated inductances
Inductances involving the faulted windings.
(F. 166)
(F.167)
(F.168)
NA4 MD cos 2 (B + ;) = (N;;)
La11 3 Lo5 = -
3 2
+ - L ~ + -L~COS(~O) 4 4
La,b L ~ ~ , , = - = 1 [-M. cos 4 (B + s)] 2 2
MQ L,,, = - sin 28 2
Ni is the number of turns in the coi1 i. Angle 6 is the dot angle. AU other constants
are known fkom the inductance matrix of the un-faulted machine, [Lsynl].
Appendix G
Leakage inductance of the machine
windings
The different components of the total leakage flux were pointed out in Section 3.6. It
was said that any flux h e that would not aid the energy conversion process is treated
as leakage flux. The correspondhg inductance is seen as an impedance causing an
interna1 voltage chop in the winding.
G. 1 Slot leakage calculat ions
Figure G.l shows the slot leakage flux of coil 6 of the winding shown in Figures 3.1
and F.5. In the discussion in Chapter 3, the fadt is assumed to be located on this
winding. Let the total number of tums of coil 6 be Ng and let coil A3, described in
Chapter 3 consist of the ATtlt turns closest to the bottom of the slot.
Let m be the axial length of the stator. Let
where
The incremental flux in an element of thickness dy is given by d& when the height
y is less than Z. The reluctance of the iron path is assumed to be small compared to
that of the air path. A current i is flowing in the faulted part of the coil.
Appendix G
by the following equation.
The inductance
inductance - due to the flm lines above the faulted tunis would correspond to an
A similar analysis would yield the following equation for La2.
The total leakage inductance, due to slot leakage flux for this case, considering
both sides of the coil, is
Figure G.2: Slot leakage flux in a double layer wound machine
The machine described in Section 5.2.2 of Chapter 5 had a double layer winding.
Figure G.2 shows the dot leakage flux when a fault occurs at a point on the top layer.
The two windings are connected in series. The total slot leakage inductance,
for this case is calculated using a similar approaeh.
where,
~6 PO^ 7 Cd3 = - -- 2 w 6 (hl - h2)
End leakage calculations
Figure G.3: End leakage flux
The following empirieal equation is used to calculate the end leakage inductance due to
winding ends protmding from the slots as illustrated in Figure G.3[35]. The diameter
of the air gap is D and the number of poles in the machine is p. The machines
described in Chapters 3 and 5 are both fully pitched and hence the winding pitch is
equal to the pole pitch.
Appendix G
number of dots Y =
P
6 . 3 Air gap leakage calculations
Flux lines that cross the air gap but do not couple the rotor windings are treated as
a part of the leakage field. If the faulted part of the coü is confined to a single slot on
each side, the flux pattern would look like that shown in Figure G.4. It can be seen
fiom the diagram that this flux is not constant but depends on the rotor position even
for cylindrical rotor machines. Using the forms given in [58], the leakage inductance
due to fiux crossing the air gap can be approlcimated as follows. The average air gap
length is g and w is the dot width.
Figure G.4: Air gap lealcage flux
Appendix H
Modeling saturation in the
transformer core in fault studies
H.1 Turn to turn fault in a single phase transformer
Appendix H
Ni is the number of tums in coil i and vi is the voltage across the coil i.
s is the dope of the M - H curve of the core materiai [Il].
3 d d vi = (R.1 + Raz + R.3 + R) ii + (&1 + La1 + Li3 + L) -iL + RIZif + La2z+
i=l dt (H.7)
The above equations will Iead to the following equations.
Appendix H
(H. 10)
The fault curent can be calculated by solving the above equations using a suitable
numericd method such as the Tkapezoidal d e .
Appendix 1
The simulation mode1 of current
transformers in a delta
configuration
The arrangement of the three current transformers, connected in delta and feeding
a burden that is connected in star is shown in Figure 1.1. The schematic of this
arrangement is shown in Figure 1.2.
Figure 1.1: Three delta connected current transformers feeding a star connected b u - den.
Assume that the CTs are identical and the three lines connecting the CTs to the
relays have the same impedance. The three relays are assumeci to have the same
impedance as well. Thus,
Figure 1.2: SimpiSed schematic diagram of the three delta connected CTs.
Rs is the resistance of the secondary winding of any current transformer and La is
the leakage inductance of the secondary winding.
Consider the circuit shown in Figure 1.2. Applying Kirchhoff's voltage law we can
obtain the following equations.
Similarly ua2 and vas can be written as,
But
Appendix 1
where
is the induced voltage across the secondary winding if CT (1) and $1 is the flux
linking the secondary winding of this CT.
= NalAlpo(l+ sl)% where sl = 2. sl gives the dope of the M - H curve of the magnetic material [Il]. & and Ii are
the cross section and the length of the core of CT i respectively.
Using Equation 1.5 and Equation 1.4 we cari express val as foiiows.
Similar equations can be written for v.2 and vas.
Applying the Kirchhoff's current law to the circuit shown in Figure 1.2 , i ~ i , i ~ 2 and
iL3 can be written as foilows.
Substituting Equation 1.9 into Equation 1.1 and applying the trapezoidal d e , we can
obtain the folIowing clifference equations.
AV Z V" + - = (if? - i $ f ) ~ + 2(2Aial - Air2 - Ai,,).
2
Applying the trapezoidal d e to Equation (I.6), we get
old AV Aipl Ais1 Azal val + - - - &dl+ ~ 1 ) ~ - + R, - A G 1
2 K 2 1 0 + si)= - 2 + L.=
Fkom Equations 1.12 and 1.13 we can obtain the foilowing equation.
Similarly we can show that
and
The Equations 1-14, 1.15 and 1.16 can be expressed in the matrix form,
where
[Ai,] =
(2 + S ( 1 +SI) - 3)
Equation 1.17 can be solved to îmd the secondary currents in the current transformers.
Appendix J
Inductance of a part of a phase
winding on a spiral wound machine
Part of the spiral winding in Figure 5.2 in Chapter 5 is shown in Figure J.1.
Figure J.1: Part of a spiral winding of a synchronous machine
Let the inductance of this part of the winding be LaP and let the magnetising part
Appendix J 213
of L, be Lm,. The value of Lm, can be derived using the methods described in
Appendix F. The flux h h g e of the two coils, coil 1 and coil 2, can be approximated
by the following equation. & and i2 are the currents flowing in the two coils. Lm, and L,1 are the magnetizing parts of the inductances of coils 1 and 2.
The total flux linking coil 2 when it is energized will aiso link coil 1. If they have the
same number of t m s , then, Lm* = M. Since the induced voltage across any coil is
the rate of change of flux linhge seen by that coil, the voltage across the two coils,
v, when they are in series is given by,
Rom the above equations,
Let n = number of slots in the machine, p = number of poles, pp = pole pitch of the
machine, D = stator diameter.
If SI is the reluctance seen by the flux due to curent in coil 1 and S2 is the reluctance
seen by the flux due to current in cos 2, then
Appendix J
The areas Al and A2 are given by,
The results in Appendix O can be used to show that,
From equations J.l and 5.3,
The two independent magnetic circuits shown in Figure 5.2 represent the coi1 arrange-
ment in Figure J.1
Air gap Air gap , j r e a (A - A - ,)
Figure J.2: A simple magnetic circuit to represent the two spiral wound coils
Appendix K
Details of the machine
Machine ratings
Rated MVA = 160
Rated Voltage = 15.0 kV
Rated current = 6.15 kA
Fully pit ched Spiral (concent ric) wound stator
Nurnber of poles = 6
Stator slots = 36
Parallel paths per phase = 2
Winding Paramet ers
Li = 0.5595 mH
LI = 3.78 mH
Ls = 0.076 mH
Md = 1.889 mH
LF = 2.189 H
LD = 5.989 mH
LQ = 1.423 mH
& = 0.1079 mH
MF = 89.006 mH
MD = 4.7209 mH
Mq = 2.2690 mH
Appendix L
Equations to show the decay of the
dc offset, ratio error and the phase
shift in air-gapped CTs
Consider the air gapped current transformer shown in Figure L.1. The self inductance,
Lgq2, of the secondary wiii have a leakage component, L a , and a magnetising
component Lrngop. The mutual inductance between the taro coIls is given by Mm.
Using the results shown in Appendix O, Lm, and M,, can be expressed in the
foilowing form.
The CT shown in Figure L.2 is carrying a current il in the primary coii and the
secondary is connected to a relay with a resistive burden. The following equations
can be written for this configuration.
Figure L.1: Schematic diagram of an air gapped CS
Figure L.2: Air gapped CT feeding a relay burden
e = i 2 R
R is the total resistance of the secondary circuit. Thus
Appendix L
The primary current with an initial exponential component will take the following
form.
The steady ac component is given by il., and the dc component in the primary side
is given by il& where
Al, Az and T are constants.
d - Al -t -zl = -- dt e T - A2w sin wt
7
From Equations L.3 and L.4,
R )i2=( ) (%? + A2u sin
L i g a p + &ap 7
d -iz + kli2 = k2 (k& + k4 sin w t ) dt (L-5)
kl, k2, ka, and k4 are constants. The secondary current ia is given by the solution of
Appendix L
Equation L.5.
22 = 1
cos (w t + 4)
sin q5 = 1
The initial exponential component, f2&, and the steady state component, i2.., of the
secondary current can be expressed in the following form.
The angle 6 gives the phase shift between the primary current and the secondary
current.
The primary current was expressed as,
with
il,, = A2 COS w t
The constants kl , k2, k3 and k4 are as follows.
The equation i2,. for the steady state current shows the ratio error and the phase
angle error. The equation iad, for the initial exponential component shows that this
gets attenuated when it is transformed fkom the primary to the secondary. A larger
air gap in the core would result in a greater attenuation. In addition, a larger air
gap would aiso result in a larger ratio error and a larger phase angle error. This is
because, the presence of a larger air gap requires a heavier magnetising current to
produce the required flux in the core. Thus, deciding on the length of the air gap
leads to a compromise between these errors and the attenuation in the exponential
component.
Appendix M
Simulation model of the air gapped
M. 1 Derivation of the simulation model of an air gapped CT
Sec
bur Rbur
Figure M.1: Air gapped CT feeding a relay burden
The arrangement of an air gapped current transformer comected to a burden is shown
in Figure M.1. Figure M.2 shows the schematic of the air gapped current transformer.
Figure M.2: Schematic diagram of an air gapped CT.
Applying Ampere's law to the CT in Figure M.2,
Bai, = PoH,,
Biron = lira &on
and
where qb is the magnetic flux in the iron core and the air gap.
From Equations M.2 and M.8
(M-9)
(M. 10)
The parameter s,, gives the dope of the M-H curve of the magnetic material ( irm).
F'rom equations M.7 and M.8
Fkom equation M.4
F'rom equations M.15 and M.3
(M. 14)
(M.15)
R o m equations M.12 and M.19
Equations M.1 and M.20 can be solved numericaliy to find the current i8,. Applying
the trapezoidal ruie to these we get the Equations M.22 andM.23 where the constant
K is defined by the Equation M.21.
Appendix M
d d b e c A i , Aiaec + - = KNH- - KNaec- 2 At At
From Equations M.22 and M.23
where
and
The new value of the secondary current i., is
(M. 27)
Appenciix M 229
M.2 Derivation of the B-A data for the magnetic material
The M-H characteristic of the magnetic material of the air gapped CT is needed to
solve the equations in Section M.1. If the flux-mmf relationship of the CT shown in
Figure M.3 is available fiom measurements, the B-H characteristic of the magnetic
material can be derived using the following equations. The fringing of the flux near
the air gap causes AaiT to be clinerent from hm. This effect is considered by using
the derivations given in [32].
Figure M.3: flux-mmf curve of an air gapped CT
- 4 1 Miron - (M.38)
Aron PO Lon PO Aair
Equations M.35 and M.38 give the M - H relationship of the magnetic material.
M.3 Equations for the three air gapped CT connection
The schemetic of three CTs connected in a star configuration is shown in Figure
M.4. The following equations can be written and the system can be solved for the
secondary currents.
Figure M.4: Three CTs connected to a relay
Applying the Trapezoidal rule to the above equations with Xdd denotjng the value
of any quantity X at time t - At,
Appendix M
Where
s1 = dMiT, d&'m
Using the results for the single CT case in Section M.1,
The above equations lead to the matrix equation shown below. is the vector
of the incremental secondary currents.
Appendix N
Cornparison of simulation results
for the machine mode1
The machine used for the tests is rated at 4 kVA, 220 Volts, and 21 A. The rated
field current is 2.9 A and the field voltage is 125 Volts. The machine has 6 saiient
poles. The phase windings are tapped at 50% and the tappings were available to be
connected in a desired manner. The windings are as depicted in Figure N.1. A tum
to ground fault on this machine is as shown in Figure N.2.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - 1.. - - - - - - - -_- : Phase windings with a tap at the centre I
1 1 A2
C2
Figure N.l: The 6 pole machine with tappings.
The following figures show the cornparison between the recorded and the calculated
235
Appendur N
results for a number of different cases.
Faut 1 1 4
f 'fld
Figure N.2: A turn to ground fault.
3
field ~ u m n t 25-
Figure N.3: A turn to ground fault Mth a 2.5 Ohm grounding resistance
The third graph in Figure N.4 resulted when the field was suddenly switched on while
the fault was present.
-2 - -54 was nvircfmi on at I scc with the
Figure N.4: A tum to ground fault with a 15 Ohm grounding resistance
Figure N.5: A tuni to ground fault with a 3 Ohm grounding mistance and +th the field curent set to 110 % of its rated value
The waveforms in Figure N.7 resulted when the field was suddenly switched on while
the fault was present.
0.5 l l Fidd cumnr
Figure N.6: A fault between phases A and B
Figure N.7: A fault between phases A and B
Appendix N 239
The waveforms in Figures N.8 and N.9 resulted when the field current was set to 110
% of its rated value. The third and fourth graphs in Figure N.9 resdted when the
field was suddenly switched on while the fault was present.
Figure N.8: A fault between phases A and B
The synchronous machine was driven close to its rated speed of 1200 rpm using an
induction motor. This was a drawback of the test system because the speed of the
induction motor would not remain constant once the fault is switched on.
Figure N.9: A fault between phases A and B
Appendix O
Useful formulae and derivat ions
The induced
Figure 0.1:
voltage
Coüs wound on a magnetic
the coi1
core
The flux, 9, in the core is
Appendix O
The reluctance of the magnetic path is S. When 1 is the length of the core and A is
its area of cross section,
Also
Flux linkage of the coi1 Ap is,
L, is the self inductance of coii P
Appendix O
Induced voltage in coi1 Q is
Mutual inductance M, between the coils is,
When both coils are excited,
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