SymmetricalComponents forPower Systems

Engineering

ELECTRICAL ENGINEERING AND ELECTRONICSA Series of Reference Books and Textbooks

EXECUTIVE EDITORS

Marlin O. ThurstonDepartment of Electrical EngineeringThe Ohio State UniversityColumbus, Ohio

WtttamMtddendorfDepartment of Electricaland Computer EngineeringUniversity of CincinnatiCincinnati, Ohio

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Maurice BellangerTelecommunications, Radiollectriques,et TOMphoniques (TRT)Le Plessis-Robinson, France

J. Lewis BlackburnBothell, Washington

Sing-Tze BowDepartment of Electrical EngineeringNorthern Illinois UniversityDe Kalb, Illinois

Norman B. FuquaReliability Analysis CenterGriffiss Air Force Base, New York

Charles A. HarperWestinghouse Electrical Engineeringand Technology Seminars, Inc.Timonium, Maryland

NaimA.KhcirDepartment of Electrical andSystems EngineeringOakland UniversityRochester, Michigan

Lionel M. LevinsonGeneral Electric CompanySchenectady, New York

V. RajagopalanDepartment of EngineeringUniversity du QuebeckTrois-RivitresTrois-Rivifcres, Quebec, Canada

Earl E. SwartdanderTRW Defense Systems GroupRedondo Beach, California

SpyrosG. TtqfestasDepartment of ElectricalEngineeringNational Technical Universityof AthensAthens, Greece

Central Research Institute ofthe Electric Power IndustryTokyo, Japan

1. Rational Fault Analysis, edited by Richard Seeks and S. a. Liberty2. Nonparametrlc Methods in Communications, edited by P. Papantonl-

Kazakos and Dimitri Kazakos3. Interactive Pattern Recognition, Yi-tzuu Chien4. Solid-State Electronics, Lawrence E. Murr5. Electronic, Magnetic, and Thermal Properties of Solid Materials, Klaus

Schrdder6. Magnetic-Bubble Memory Technology, Hsu Chang7. Transformer and Inductor Design Handbook, Colonel Wm. T. McLyman8. Electromagnetics: Classical and Modern Theory and Applications,

Samuel Seely and Alexander D. Poularikas9. One-Dimensional Digital Signal Processing, Chi-Tsong Chen

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Guide to Practice and Specification, Colonel Wm. 7. McLyman14. Static and Rotating Electromagnetic Devices, Richard H. Engelmann15. Energy-Efficient Electric Motors: Selection and Application, John C.

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Mahmoud, Mohamed F. Hassanf and Mohamed G. Darw/sh27. Microprocessor Software Project Management, Eli T. Fathiand Cedrlc

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Sakae Yamamura31. Ceramic Motors for Electronics: Processing, Properties, and Applica-

tions, edited by Relva C. Buchanan

32. Microcomputer Bus Structures and Bus Interface Design, Arthur LDexter

33. End User's Guide to Innovative Flexible Circuit Packaging, JayJ. Mtolet34. Reliability Engineering for Electronic Design, Norman B. Fugue35. Design Fundamentals for Low-Voltage Distribution and Control, Frank

W. Kussy and Jack L Warren36. Encapsulation of Electronic Devices and Components, Edward R.

Salmon37. Protective Relaying: Principles and Applications, J. Lewis Blackburn38. Testing Active and Passive Electronic Components, Richard F. Powell39. Adaptive Control Systems: Techniques and Applications, V. V. Chalam40. Computer-Aided Analysis of Power Electronic Systems, Venkatachart

Kajagopalan41. Integrated Circuit Quality and Reliability, Eugene It. Hnatek42. Systolic Signal Processing Systems, edited by Earl E. Swartzlander, Jr.43. Adaptive Digital Filters and Signal Analysis, Maurice 6. Betanger44. Electronic Ceramics: Properties, Configuration, and Applications, edited

by Lionel M. Lev/nson45. Computer Systems Engineering Management, Robert S. Afford46. Systems Modeling and Computer Simulation, edited by Nairn A. Khek47. Rigid-Flex Printed Wiring Design for Production Readiness, Walter S.

Rigllng48. Analog Methods for Computer-Aided Circuit Analysis and Diagnosis,

edited by Takao Oiawa49. Transformer and Inductor Design Handbook: Second Edition, Revised

and Expanded, Colonel Wm. T. McLyman50. Power System Grounding and Transients: An Introduction, A. P. Sakfs

Mel/opoulos51. Signal Processing Handbook, edited by C. H. Chen52. Electronic Product Design for Automated Manufacturing, H. Richard

Stlllwell53. Dynamic Models and Discrete Event Simulation, William Delaney and

Ermfnia Vaccari54. FET Technology and Application: An Introduction, Edwin S. Oxner55. Digital Speech Processing, Synthesis, and Recognition, Sadaokl Furul56. VLSI RISC Architecture and Organization, Stephen B. Furber57. Surface Mount and Related Technologies, Gerald Ginsberg58. Uninterruptible Power Supplies: Power Conditioners for Critical Equip*

ment, David C. Griffith59. Polyphase Induction Motors: Analysis, Design, and Application, Pau/L

Cochran60. Battery Technology Handbook, edited by H. A. Klehne61. Network Modeling, Simulation, and Analysis, edited by Ricardo F. Garzia

and Mario R. Garzia62. Linear Circuits, Systems, and Signal Processing: Advanced Theory and

Applications, edited by Nobuo Nagal

63. High-Voltage Engineering: Theory and Practice, edited by M. Khalifa64. Large-Scale Systems Control and Decision Making, edited by Hiroyuki

Tamura and Tsuneo Yoshikawa65. Industrial Power Distribution and Illuminating Systems, Kao Chen66. Distributed Computer Control for Industrial Automation, Dobrivoje

Popovic and V/Jay P. Bhatkar67. Computer-Aided Analysis of Active Circuits, Adrian loinovici68. Designing with Analog Switches, Steve Moore69. Contamination Effects on Electronic Products, Carl J. Tautscher70. Computer-Operated Systems Control, Magdi S. Mahmoud71. Integrated Microwave Circuits, edited by Yosh/h/ro Konishi72. Ceramic Materials for Electronics: Processing, Properties, and Appli-

cations, Second Edition, Revised and Expanded, edited by Relva C.Buchanan

73. Electromagnetic Compatibility: Principles and Applications, David A.Weston

74. Intelligent Robotic Systems, edited by Spyros G. Tzafestas75. Switching Phenomena in High-Voltage Circuit Breakers, edited by Kunto

Nakanishi76. Advances in Speech Signal Processing, edited by Sadaoki Furui and M.

Mohan Sondhi77. Pattern Recognition and Image Preprocessing, Sing-Tze Bow78. Energy-Efficient Electric Motors: Selection and Application, Second

Edition, John C. Andreas79. Stochastic Large-Scale Engineering Systems, edited by Spyros G.

Tzafestas and Keigo Watanabe80. Two-Dimensional Digital Filters, Wu-Sheng Lu and Andreas Antoniou81. Computer-Aided Analysis and Design of Switch-Mode Power Supplies,

Y/m-Shu Lee82. Placement and Routing of Electronic Modules, edited by Michael Pecht83. Applied Control: Current Trends and Modern Methodologies, edited by

Spyros G. Tzafestas84. Algorithms for Computer-Aided Design of Multivariable Control

Systems, Stanoje Bingulac and Hugh F. VanLandingham85. Symmetrical Components for Power Systems Engineering, J. Lewis

Blackburn

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SymmetricalComponents forPower Systems

Engineering

J. Lewis BlackburnConsultant

Botheff, Washington

CRC PressTaylor & Francis Croup

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Preface

The method of symmetrical components is a powerful tool for under-standing and determining unbalanced currents and voltages in three-phase electrical power systems. In a sense it is the "language" of thoseassociated with relay protection. For almost all faults, the intolerableconditions that require isolation of the problem area involve unbal-ances. Thus the quantities that operate the protection are directly orindirectly related to symmetrical components.

This book presents the fundamental concepts of symmetrical com-ponents along with a review of per unit (percent), phasors, and polarity.Typical examples are solved throughout the text, and an additionalproblem section is included for further studies. The book is intendedas a text for students and as a reference manual for practicing engineersand technicians—all who are involved or associated with relaying andpower system analysis.

The modern computer provides large volumes of fault and relateddata but without any understanding or appreciation. Thus the aim ofthis book is to provide (I) a practical understanding of system unbal-ances, the basic circuits, and calculations, (2) the techniques of makingcalculations when a computer or program is unavailable, (3) an over-view for visualization of faults, unbalances, and the sequence quan-

V

vi Preface

tities, (4) the determination of system parameters for manual calcu-lations or computer programs.

This text has been developed from notes used over many years ofpresenting symmetrical components. Originally they were based on theclassic book by Wagner and Evans, Symmetrical Components (1933).Associates, students and friends within Westinghouse, the IEEE,CIGRE, many utilities, and industrial and consulting companies aroundthe world have directly or indirectly added contributions over the lastfifty-five years.

Special thanks are extended to William M. Strang for his great workon the figures and to Ruth A. Dawe and Lila Harris, Marcel Dekker,Inc., editors, for their wonderful assistance and encouragement.

J. Lewis Blackburn

Contents

Preface v

\. Introduction and Historical Background 1

I. I Introduction and General Aims I1.2 Historical Background 3

2, Per Unit and Percent Values 5

2.1 Introduction 52.2 Per Unit and Percent Definitions 52.3 Advantages of Per Unit and Percent 62.4 General Relationships Between Circuit

Quantities 72.5 Base Quantities 102.6 Per Unit and Percent Impedance Relationships 112.7 Per Unit and Percent Impedances of

Transformer Units 132.8 Changing Per Unit (Percent) Quantities to

Different Bases 16

vii

viii Contents

3. Phasors, Polarity, and System Harmonics 213.1 Introduction 213.2 Phasors 213.3 Circuit and Phasor Diagrams for a Balanced

Three-Phase Power System 293.4 Phasor and Phase Rotation 323.5 Polarity 333.6 Power System Harmonics 37

4. Bask Fundamentals and the Sequence Networks 39

4.1 Introduction 394.2 Positive-Sequence Set 394.3 Nomenclature Convenience 414.4 Negative-Sequence Set 414.5 Zero-Sequence Set 424.6 General Equations 434.7 Sequence Independence 444.8 Sequence Networks 454.9 Positive-Sequence Network 474.10 Negative-Sequence Network 494.11 Zero-Sequence Network 524.12 Impedance and Sequence Connections for

Transformer Banks 554.13 Sequence Phase Shifts Through Wye-Delta

Transformer Banks 594.14 Sequence Network Voltages 644.15 Sequence Network Reduction 654.16 Th£venin Theorem in Network Reduction 664.17 Wye-Delta Network Transformations 674.18 Short-Circuit MVA and Equivalent Impedance 694.19 Equivalent Network from a Previous Fault

Study 714.20 Example: Determining an Equivalent Network

from a Previous Fault Study 774.21 Network Reduction by Simultaneous Equations 834.22 Other Network Reduction Techniques 84

Contents ix

5* Shunt Unbalance Sequence Network Interconnections 85

5.1 Introduction 855.2 General Representation of Power Systems and

Sequence Networks 865.3 Sequence Network Interconnections for Three-

Phase Faults 875.4 Sequence Network Interconnections for Phase-

to-Ground Faults 895.5 Sequence Network Interconnections for Phase-

to-Phase Faults 935.6 Sequence Network Interconnections for Two-

Phase-to-Ground Faults 955.7 Other Sequence Network Interconnections for

Shunt System Conditions 985.8 Fault Impedance 985.9 Substation and Tower Footing Impedance 995.10 Ground Faults on Ungrounded or High

Resistance Grounded Systems 100

6. Fault Calculation Examples for Shunt-Type Faults 109

6.1 Introduction 1096.2 Faults on a Loop-Type Power System 1096.3 Basic Assumptions 1116.4 Fault Calculation 1116.5 Summary of Fault Current 1216.6 Voltages During Faults 1246.7 Summary of Fault Voltages 1276.8 Fault Calculations With and Without Load 1286.9 Solution by Th£venin's Theorem 1316.10 Solution by Network Reduction 1336.11 Solution Without Load 1366.12 Summary 1366.13 Neutral Inversion 1376.14 Example: Ground Fault on an Ungrounded

System 1406.15 Example: Ground Fault with High Resistance

Across Three Distribution Transformers 142

x Contents

6.16 Example: Ground Fault with High Resistance inNeutral 145

6.17 Example: Phase-a-to-Ground Fault Currentsand Voltages on Both Sides of a Wye-DeltaTransformer 152

7. Series and Simultaneous Unbalance Sequence NetworkInterconnections 157

7.1 Introduction 1577.2 Series Unbalance Sequence Interconnections 1577.3 One Phase Open: Broken Conductor or Blown

Fuse 1607.4 Example: Open Phase Calculation 1607.5 Simultaneous Unbalance Sequence

Interconnections 1677.6 Example: Broken Conductor Falling to Ground

on Bus Side 1697.7 Example: Broken Conductor Falling to Ground

on Line Side 1787.8 Example: Open Conductor on High Side and

Ground Fault on Low Side of a Delta-WyeTransformer 180

7.9 Ground Fault on Low Side of a Delta-WyeTransformer 186

7.10 Example: Open Conductor on High Side andGround Fault on Low Side of a Wye-Grounded/Delta-Wye-Grounded Transformer 188

7.11 Ground Fault Calculation for a Mid-TappedGrounded Delta Secondary Transformer 193

7.12 Summary 204

8. Overview of Sequence Currents and Voltages DuringFaults 207

8.1 Introduction 2078.2 Voltage and Current Phasors for Shunt Faults 2078.3 System Voltage Profiles During Shunt Faults 2118.4 Voltage and Current Phasors for All

Combinations of the Four Shunt Faults 2148.5 Summary 218

Contents xi

9. Transformer, Reactor, and Capacitor Characteristics 219

9. t Transformer Fundamentals 2199.2 Example: Impedances of Single-Phase

Transformers in Three-Phase Power Systems 2229.3 Polarity, Standard Terminal Marking, and Phase

Shifts 2259.4 Two-Winding Transformer Banks: Sequence

Impedance and Connections 2269.5 Three-Winding Transformer Banks 2279.6 Three-Winding Transformers: Sequence

Impedance and Connections 2299.7 Example: Three-Winding Transformer

Equivalent 2299.8 Example: Three-Winding Transformer Fault

Calculation 2319.9 Autotransformers 2369.10 Example: Autotransformer Fault Calculation 2369.11 Ungrounded Autotransformers with Tertiary and

Grounded Autotransformers Without Tertiary 2439.12 Test Measurements for Transformer Impedance 2439.13 Determination of the Equivalent Zero-Sequence

Impedances for Three-Winding Three-PhaseTransformers Where the Tertiary Delta WindingIs Not Available 248

9.14 Distribution Transformers with TappedSecondary 250

9.15 Zig-Zag Connected Transformers 2539.16 Reactors 2549.17 Capacitors 255

10. Generator and Motor Characteristics 257

10.1 Introduction 25710.2 Transient in Resistance-Inductance Series

Circuits 25710.3 Transient Generator Currents 26110.4 Negative-Sequence Component 26710.5 Zero-Sequence Component 26910.6 Total RMS Armature Component 269

xii Contents

10.7 Rotating Machine Reactance Factors for FaultCalculations 270

10.8 Time Constants for Various Faults 27010.9 Induction Machines 27310.10 Summary 27S

Appendix: Typical Constants of Three-PhaseSynchronous Machines 278

11. Overhead Line Characteristics: Inductive Impedance 281

11.1 Introduction 28111.2 Reactance of Overhead Conductors 28111.3 GMR and GMD Values 28311.4 The Xtl and Xd Line Constants 28511.5 Positive- and Negative-Sequence Impedance 28611.6 Example 28811.7 Lines with Bundled Conductors 28911.8 Zero-Sequence Impedance 29511.9 Zero-Sequence Impedances of Various Lines 29911.10 Summary for Zero-Sequence Impedance

Calculations 315

12. Overhead Line Characteristics: Mutual Impedance 321

12.1 Introduction 32112.2 Mutual Coupling Fundamentals 32112.3 Positive- and Negative-Sequence Mutual

Impedance 32312.4 Zero-Sequence Mutual Impedance 33012.5 Mutual Impedances Between Lines of Different

Voltages 33012.6 Power System-Induced Voltages in Wire

Communication Lines 33112.7 Summary 339

13. Overhead Line Characteristics: Capacitive Reactance 341

13.1 Introduction 34113.2 Capacitance of Overhead Conductors 34113.3 Positive- and Negative-Sequence Capacitance 34413.4 Example: Three-Phase Circuit Capacitive

Reactance 345

Contents xiii

13.5 Example: Double-Three-Phase-CircuitCapacitive Reactance 347

13.6 Zero-Sequence Capacitance 34813.7 Zero-Sequence Capacitance: Transposed Three-

Phase Line 34913.8 Example: Zero-Sequence Capacitance,

Transposed Three-Phase Line 35013.9 Summary 350

14. Cable Characteristics 353

14.1 Introduction 35314.2 Positive- and Negative-Sequence Constants 35514.3 Three-Conductor Cables 36214.4 Zero-Sequence Constants of Cables 364

Problems 377

Appendix: Overhead Line Conductor Characteristics 405

Table A.I All-Aluminum Concentric-Lay Class AA andA Stranded Bare Conductors 407

Table A.2 All-Aluminum Concentric-Lay Class AA andA Bare Stranded Conductors I350-H19ASTMB231 408

Table A.3 All-Aluminum Shaped-Wire Concentric-LayCompact Conductors AAC/TW 409

Table A.4 All-Aluminum Shaped-Wire Concentric-LayCompact Conductors AAC/TW 410

Table A.5 Bare Aluminum Conductors, Steel-Reinforced (ACSR) Electrical Properties ofSingle-Layer Sizes 411

Table A.6 Bare Aluminum Conductors, Steel-Reinforced (ACSR) Electrical Properties ofMultilayer Sizes 412

Table A.7 Shaped-Wire Concentric-Lay CompactAluminum Conductors Steel-Reinforced(ACSR/TW) 414

Table A.8 Shaped-Wire Concentric-Lay CompactAluminum Conductors Steel-Reinforced(ACSR/TW) 415

xiv Contents

Table A.9 Bare Aluminum Conductors, 1350-H19 WiresStranded with Aluminum-Clad Steel Wires(Alumoweld) as Reinforcement (AWAC) inDistribution and Neutral-Messenger Sizes 416

Bibliography 417

Index 419

1Introduction and HistoricalBackground

1.1 INTRODUCTION AND GENERAL AIMS

The method of symmetrical components provides a practicaltechnology for understanding and analyzing electric power sys-tem operation during unbalanced conditions. Typical unbalancesare those caused by faults between the phases and/or ground(phase to phase, double phase to ground, phase to ground), openphases, unbalanced impedances, and combinations of these. Bal-anced three-phase faults are included. Also, many protectiverelays operate from symmetrical component quantities. For ex-ample, all ground relays operate from zero-sequence quantities,which are normally not present in the power system. Therefore,a good understanding of this subject is of great importance andis a very important "tool11 in system protection.

This discussion is for three-phase electric systems, which areassumed from a practical standpoint to be balanced or sym-metrical up to a point or area of unbalance. A normal area orpoint of unbalance in a power system will usually be down in

1

2 Chapter 1

the low-voltage or distribution area, where single-phase loadsare connected or where nonsymmetrical equipment is used.

In a symmetrical or balanced system the source voltages (gen-erators) are equal in magnitude and are in phase, with their threephases displaced 120° in phase relations. Also, the impedancesof the three-phase circuits and equipment are of equal magnitudeand phase angle.

In a sense, symmetrical components can be called the "lan-guage" of the protection engineer or technician. Its great valueis both in thinking or visualizing system unbalances, and as ameans of detailed analysis of them from the system parameters.In this simile it is like a language, in that it requires experienceand practice for easy access and application. Fortunately, faultsand unbalances occur infrequently, and many do not require de-tailed analysis. Thus it becomes difficult to "practice the lan-guage." This difficulty has increased significantly with the readyavailability of fault studies via computers. These provide rapidaccess to voluminous data, frequently with little user under-standing of the information, the background, or the method thatprovides the data.

The goal of this book is to provide (1) a practical understandingand appreciation of the fundamentals, basic circuits, and cal-culations; (2) an overview directed toward a clear visualizationof faults and system unbalances; (3) where access to a computeror a proper fault program may not be available, the means ofmaking fault and unbalanced calculations "by hand91; and (4)determination of the necessary system parameters for calcula-tions or computer programs. Throughout, the math is kept assimple as possible.

Although computers and hand calculators provide great ac-curacy, fault and unbalance calculations will not be the same aswill be experienced for real-life occurrences. The principal rea-sons are (1) approximations involved in the determination ofmany of the system parameters, especially for lines; (2) param-eter changes resulting from temperature variations; (3) highlyvariable and relatively unknown fault resistance; and (4) variablegenerator impedances with time. Thus high accuracy for fault

Introduction and Historical Background 3

and unbalance calculations is not possible and should not beexpected, and from a practical standpoint is not really needed.

Surprisingly, the calculated values often are quite close to ac-tual values, and in general the calculations are very practical forsystem design and equipment selection and for the applicationand setting of protective relays. The general practice is to cal-culate only solid faults. This provides maximum values impor-tant for system design. Fault resistance will reduce the currentvalues. However, as indicated above, it is highly variable andrelatively unknown. Thus it becomes impossible practically toselect the "correct" value for fault and unbalance studies.

In protective relaying, using solid fault data, overcurrent re-lays should be set so that their minimum pickup is at least one-half of the minimum fault currents and, hopefully, more sensitiveas long as the phase relays do not operate on the maximum short-time load current and ground relays do not operate on themaximum tolerable zero-sequence unbalance. This generallyprovides good protection for system arcing faults, except forpossible very high resistance faults in low-voltage distribution.

The method of symmetrical components is applicable to mul-tiphase systems, but only symmetrical components for three-phase systems are covered in this book.

1.2 HISTORICAL BACKGROUND

The method of symmetrical components was developed late in1913 by Charles L. Fortescue of Westinghouse when investi-gating mathematically the operation of induction motors underunbalanced conditions. At the 34th Annual Convention of theAIEE on June 28, 1918, in Atlantic City, he presented a paperentitled "Method of Symmetrical Co-ordinates Applied to theSolution of Polyphase Networks." This was published in theAIEE Transactions, Volume 37, Part II, pages 1027-1140. Thispublished paper was 89 pages (5 by 8 in.) in length, with 25 pagesof discussion by six well-known "giants" of electric power en-gineering: J. Slepian, C. P. Steinmetz, V. Karapetoff, A. M.Dudley, Charles F. Scott, and C. O. Mailloux.

4 Chapter 1

Application of the method "to the study of short circuits andsystem disturbances/9 and the method as we know it today, wasmade by C. F. Wagner and R. D. Evans. They began a seriesof articles in the Westinghouse magazine The Electric Journalthat ran for 10 issues, from March 1928 through November 1931.This series was enlarged by the two authors and published inthe classic and still very useful book Symmetrical Components,published by McGraw-Hill Book Company, New York, 1933.

Just as the Wagner-Evans book was about to be printed, an-other Westinghouse engineer, W. A. Lewis, developed the con-cept of splitting the line reactance into components: one asso-ciated with the conductor reactance (Xa), one associated withthe spacing to the return conductor(s) (Xd)9 and one associatedwith the depth of the earth return (Xf). This was added to thebook as an appendix (VII) and is covered in this book in Chapter11.

Another Westinghouse engineer, E. L. Harder, provided veryuseful tables of fault and unbalance connections that were pre-sented in his paper "Sequence Network Connections" publishedin the December 1937 issue of The Electric Journal. This is cov-ered in Chapter 5.

During this time, Edith Clarke of General Electric was de-veloping notes and lecturing in this area. However, formal pub-lication of her work did not occur until 1943.

On a personal note it was my privilege to meet Dr. Fortescueand Mr. Evans several times during my student assignment onthe AC Network Analyzer in 1937.1 knew C. F. Wagner throughhis son Chuck (C. L. Wagner). Chuck, past President of theIEEE Power Engineering Society, and I have had a long asso-ciation together in protective relaying.

I took my first course in symmetrical components under Dr.Lewis and then applied it while working for Dr. Harder. In fact,Ed Harder was instrumental in my obtaining a permanent job inrelaying. At that point I was totally unfamiliar with relays, butit was a job and in the New York area where I wanted to be.And 55 years have quickly slipped by!

References Anderson, P. M. , Analysis of Faulted Power Systems , Iowa State University Press,Ames, Iowa, 1973. ANSI/IEEE C37.5 , Guide for Calculation of Fault Current for Application of AC High-Voltage Circuit Breakers Rated on a Total Current Basis , 1979. ANSI/IEEE C37.010 , Application Guide for AC High-Voltage Circuit Breakers Rated on aSymmetrical Current Basis. ANSI/IEEE C37.013 , Standard for AC High Voltage Generator Circuit Breakers Rated ona Symmetrical Current Basis , 1990. ANSI/IEEE C37.13 , Standard for Low Voltage AC Power , 1990. Blackburn, J. L. , Protective Relaying: Principles and Applications , Marcel Dekker, Inc.,New York, 1987. Blackburn, J. L. , Voltage Induction in Paralleled Transmission Circuits, IEEETransactions , Vol. 81, Part III, Feb. 1963, pp. 921–929. Clarke, Edith , Circuit Analysis of A-C Power Systems , Vols. I and II, General ElectricCo., Schenectady, N.Y., 1943, 1950. Electrical Transmission and Distribution Reference Book , 4th ed., Westinghouse ElectricCo., Schenectady, N.Y., 1964. Electrical Utility Engineering Reference Book, Vol. 3, Distribution Systems ,Westinghouse Electric Co., Schenectady, N.Y., 1965. Fitzgerald, A. E. , Kingsley, C. , and Umans, S. D. , Electric Machinery , McGraw-HillBook Company, New York, 1983. Harder, E. L. , Sequence Network Connections for Unbalanced Load and FaultConditions, The Electric Journal , Dec. 1937, pp. 481–488. IEEE Red Book, Standard 141 , Recommended Practice for Electric Power Distributionfor Industrial Plants , 1976. Seidman, A. H. , Mahrous, H. , and Hicks, T. G. , Handbook of Electric PowerCalculations , McGraw-Hill Book Company, New York, 1983. Stevenson, W. D., Jr. , Elements of Power Systems Analysis , 4th ed., McGraw-Hill BookCompany, New York, 1982. Wagner, C. F. , and Evans, R. D. , Symmetrical Components , McGraw-Hill BookCompany, New York, 1933. (Available in reprint from R. E. Kreiger Publishing Co., Inc.,Melbourne, Fla., 1982.) Weedy, B. M. , Electric Power Systems , 3rd rev., John Wiley & Sons, Inc., New York,1987. ANSI and IEEE documents are available from IEEE Service Center , P.O. Box 1331,Piscataway, NJ 08855–1331.