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The Relay Testing HandbookPrinciples and Practice
Chris WerstiukProfessional Engineer
Journeyman Power System ElectricianElectrical Engineering Technologist

The Relay Testing HandbookPrinciples and Practice

The Relay Testing HandbookPrinciples and Practice
Chris WerstiukProfessional Engineer
Journeyman Power System ElectricianElectrical Technologist
Valence Electrical Training Services5005 S Kipling Pkwy Suite A7 PMB 395
Littleton, CO 801271375www.RelayTraining.com

2012 Chris Werstiuk and Valence Electrical Training Services LLC. All rights reserved.
No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by an information storage and retrieval system except by a reviewer who may quote brief passages in a review to be printed in a magazine, newspaper, or on the Internet without permission in writing from the publisher.
Although the author and publisher have exhaustively researched all sources to ensure the accuracy and completeness of the information contained in this book, neither the authors nor the publisher nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. We assume no responsibility for errors, inaccuracies, omissions, or any inconsistency herein. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Any slights of people, places, or organizations are unintentional. Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe.
First printing in 2012.
The Relay Testing Handbook: Principles and Practice
ISBN: 9781934348208
Library of Congress Control Number: 2012934627
Published By: Valence Electrical Training Services LLC5005 S Kipling Pkwy Suite A7 PMB 395, Littleton, CO 801271375, U.S.A.  (303) 2508257 
Distributed by: www.relaytraining.com
Edited by: OneonOne Book Production, West Hills, CA
Cover Art: James Steidl. Image from BigStockPhoto.com
Interior Design and Layout: Adina Cucicov, Flamingo Designs
ATTENTION CORPORATIONS, UNIVERSITIES, COLLEGES, and PROFESSIONAL ORGANIZATIONS: Quantity discounts are available on bulk purchases of this book. Special books or book excerpts can also be created to fit specific needs.
Printed in the United States of America. g

v
Authors Note
This book has grown from a 45minute paper presentation at the 2001 InterNational Electrical Testing Association (NETA) conference into a decadelong project. What started as a simple paper about protective relay logic for microprocessor based relays has blossomed into a comprehensive training manual covering all aspects of relay testing. I am grateful for the countless hours my colleagues spent during the peer review process, and for their invaluable contributions to this endeavor.
Traditional protective relay books are written by engineers as a resource for engineers to use when modeling the electrical system or creating relay settings, and they often have very little practical use for the test technician in the field. The Relay Testing Handbook is a practical resource written by a relay tester for relay testers; it is a comprehensive series of practical instructional manuals that provides the knowledge necessary to test most modern protective relays. The complete handbook combines basic electrical fundamentals, detailed descriptions of protective elements, and generic test plans with examples of realworld applications, enabling you to confidently handle nearly any relay testing situation. Practical examples include a wide variety of relay manufacturers and models to demonstrate that you can apply the same basic fundamentals to most relay testing scenarios.
The Relay Testing Handbook is a ninepart series that covers virtually every aspect of relay testing. Eight books of the series have been compiled into this volume that explain the underlying principles of relay testing, including electrical theory, relay testing philosophies, digital logic and test plan creation and implementation. After the foundation is laid, you will find practical stepbystep procedures for testing the most common protection applications for: voltage, overcurrent, differential, and line distance relays.
Thank you for supporting this major undertaking. I hope you find this and all other installments of The Relay Testing Handbook series to be a useful resource. This project is ongoing and we are constantly seeking to make improvements. Our publishing model allows us to quickly correct errors or omissions and implement suggestions. Please contact us at [email protected] to report a problem. If we implement your suggestion, well send you an updated copy and a prize. You can also go to www.relaytraining.com/updates to see whats changed since the The Relay Testing Handbook was released in 2012.

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Philip B. BakerElectrical Technician
Eric Cameron, B.E.Sc.Ainsworth Power Serviceswww.ainsworth.com
Bob Davis, CET PSENorthern Alberta Institute of TechnologyGET IN GO FARwww.nait.ca
David SnyderHydropower Test and EvaluationBonneville Lock and Damwww.nwp.usace.army.mil/op/b/
John HodsonElectrical Power Systems ConsultantCalgary, AlbertaDo it right the first timewww.leadingedgesales.ca, www.psams.ca, www.powerecosystems.com, www.arcteq.fi, www.magnaiv.com
David Magnan Project ManagerPCA Valence Engineering Technologies Ltd.www.pcavalence.com
Mose Ramieh IIILevel IV NETA TechnicianLevel III NICET Electrical Testing TechnicianVice President of Power & Generation Testing, Inc.Electrical Guru and all around nice guy.
Les Warner, C.E.T.PCA Valence Engineering Technologies Ltd.www.pcavalence.com
Lina Dennison
Ken Gibbs, C.E.T.PCA Valence Engineering Technologies Ltd.www.pcavalence.com
Roger Grylls, CETMagna IV EngineeringSuperior Client Service. Practical Solutionswww.magnaiv.com
Jamie MacLeanElectrical Engineer TransAlta Utilities
Acknowledgments
This book would not be possible without support from these fine people

ix
Table of Contents
Authors Note v
Acknowledgments vii
Chapter 1: Electrical Fundamentals 1
1. The ThreePhase Electrical System 12. Transformers 273. Instrument Transformers 344. Fault Types 405. Grounding 476. Sequence Components 527. Fault Types and Sequence Components 58
Chapter 2: Introduction to Protective Relays 65
1. What are Protective Relays? 652. Time Coordination Curves (TCC) and Coordination 73
Chapter 3: A Brief History of Protective Relays 83
1. Electromechanical Relays 832. SolidState Relays 873. Microprocessor Based Relays 87
Chapter 4: Relay Testing Fundamentals 95
1. Reasons for Relay Testing 952. Relay Testing Equipment 983. Relay Testing Methods 1014. Relay Test Procedures 109
Chapter 5: Test Sheets and Documentation 135
1. Your Company Name and Logo 1362. Project Details 1363. Nameplate Data 1374. CT and PT Ratios 1385. Comments and Notes 1386. Metering Test Data 1387. Input / Output Tests 139

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8. Element Test Results 1409. Element Characteristics 14210. Final Output Checks 14411. Test Sheet Template 14512. Final Report 149
Chapter 6: Testing Overvoltage (59) Protection 151
1. Application 1512. Settings 1523. Pickup Testing 1544. Timing Tests 1585. Tips and Tricks to Overcome Common Obstacles 164
Chapter 7: Undervoltage (27) Protection Testing 165
1. Application 1652. Settings 1663. Pickup Testing 1694. Timing Tests 1755. Tips and Tricks to Overcome Common Obstacles 181
Chapter 8: Over/Under Frequency (81) Protection Testing 183
1. Application 1832. Settings 1853. Pickup Testing 1874. Timing Tests 1925. Tips and Tricks to Overcome Common Obstacles 193
Chapter 9: Instantaneous Overcurrent (50) Element Testing 195
1. Application 1952. Settings 1983. Pickup Testing 1994. Timing Tests 2065. Residual Neutral Instantaneous Overcurrent Protection 2096. Tips and Tricks to Overcome Common Obstacles 209
Chapter 10: Time Overcurrent (51) Element Testing 211
1. Application 2112. Settings 2133. Pickup Testing 2144. Timing Tests 2205. Reset Tests 2286. Residual Neutral Time Overcurrent Protection 2287. Tips and Tricks to Overcome Common Obstacles 228

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Chapter 11: Directional Overcurrent (67) Element Testing 231
1. Application 2312. Operation 2353. Settings 2364. Pickup Testing 2385. Timing Test Procedures 2466. Tips and Tricks to Overcome Common Obstacles 246
Chapter 12: Simple Percent Differential (87) Element Testing 247
1. Application 2472. Settings 2573. RestrainedDifferential Pickup Testing 2594. RestrainedDifferential Timing Test Procedure 2645. RestrainedDifferential Slope Testing 2666. Tips and Tricks to Overcome Common Obstacles 283
Chapter 13: Percent Differential (87) Element Testing 285
1. Application 2862. Settings 3123. Current Transformer Connections 3164. 3Phase RestrainedDifferential Pickup Testing 3215. RestrainedDifferential Timing Test Procedure 3306. 3Phase RestrainedDifferential Slope Testing 3327. 1Phase RestrainedDifferential Slope Testing 3588. Harmonic Restraint Testing 3649. Tips and Tricks to Overcome Common Obstacles 370
Chapter 14: UnrestrainedDifferential Testing 371
1. Settings 3722. TestSet Connections 3733. Simple Pickup Test Procedure 3764. Alternate Pickup Test Procedure 3795. Timing Test Procedure 3826. Tips and Tricks to Overcome Common Obstacles 383
Chapter 15: Line Distance (21) Element Testing 385
1. Impedance Relays 3902. Settings 4013. Preventing Interference in Digital Relays 4054. 3Phase Line Distance Protection Testing 4075. PhasetoPhase Line Distance Protection Testing 4256. PhasetoGround Line Distance Protection Testing 451

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Chapter 16: Understanding Digital Logic 473
1. Understanding Logic 4732. Relay Labels 4793. Internal Relay Control Schemes 4814. Individual Element Schemes 4925. Binary Relays 4946. Arithmetic (Math) Scheme 4997. General Electric FlexLogic 524
Chapter 17: Review the Application 543
1. Introduction 5432. Collecting Information 545
Chapter 18: Preparing to Test 575
1. Prepare to Test 5752. Establish Communication 5753. Apply Settings 5794. Connect Your Relay TestSet 5795. Creating the Test Plan 590
Chapter 19: Testing the Relay 609
1. Perform Relay Selfchecks 6092. Verify Digital Inputs and Outputs 6093. Verify Current and Voltage Inputs 6104. Element Testing 6125. Final Output Tests 6156. Prepare Relay for Service 6217. Final Report 622
Bibliography 623
Index 627

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Table of Figures
Figure 11: Simple SinglePhase Generator 2Figure 12: Simple ThreePhase Generator 3Figure 13: Phase Sequence Examples 4Figure 14: Phasor Diagram Example 5Figure 15: Electrical Diagram Comparison 6Figure 16: Electrical Diagram Comparison with Van Reference 7Figure 17: Phasor Diagram 1 7Figure 18: Phasor Diagram 2 8Figure 19: Phasor Diagram 3 9Figure 110: ABC / ACB Phasor Diagrams 9Figure 111: Current and Voltage Phasors 10Figure 112: Vector Diagrams on Different Axis 11Figure 113: Wye Connection 14Figure 114: Phase to Line Conversion 15Figure 115: Delta/Wye Connected Vector Diagram 16Figure 116: Delta Connection 17Figure 117: Delta Connection Vector Diagram for Balanced System with 30 Lag 17Figure 118: Watts  Current and Voltage Are InPhase 18Figure 119: Inductive VARs  Current Lags the Voltage by 30 19Figure 120: Capacitive VARs  Current Leads the Voltage 20Figure 121: ThreePhase Power Formulas 21Figure 122: ThreePhase Power Formulas Assuming Balanced Conditions 21Figure 123: ThreePhase Power Triangle  Watts Only 22Figure 124: Power Triangle  Only Inductive VARs 22Figure 125: Power Triangle  Only Capacitive VARs 23Figure 126: Power Triangle 24Figure 127: Example ThreePhase Power/VARS/VA Formula Calculations 24Figure 128: Simple Transformer Construction 28Figure 129: Transformer Voltage Ratio Calculation 29Figure 130: Transformer Current Ratio Calculation 29Figure 131: WyeDelta Transformer Connections 31Figure 132: WyeDelta Transformer Calculations 31Figure 133: DeltaWye Transformer Connections 32Figure 134: DeltaWye Transformer Calculations 32

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Figure 135: WyeWye Transformer Connections 33Figure 136: AutoTransformer Connections 33Figure 137: DeltaDelta Transformer Connections 34Figure 138: WyeWye and DeltaDelta Transformer Calculations 34Figure 139: CT Ratio Calculations 35Figure 140: CT Polarity Connections 36Figure 141: CT Accuracy Class 37Figure 142: PT Voltage Ratio Calculation 37Figure 143: 3Wire Delta PT Configurations 38Figure 144: 4Wire Wye PT Configurations 38Figure 145: Nominal PT Voltages 39Figure 146: ThreePhase Fault 3Line Drawing 40Figure 147: ThreePhase Fault Phasor Diagram 41Figure 148: PhasePhase Fault 3Line Drawing 42Figure 149: Fault Current vs. Injected Current 42Figure 150: PhasePhase Fault 43Figure 151: PhaseGround Fault 3Line Drawing 44Figure 152: PhaseGround Fault 45Figure 153: PhasePhaseGround Fault 3Line Drawing 46Figure 154: PhasePhaseGround Fault 46Figure 155: Solidly Grounded System 47Figure 156: Resistive Grounded System 48Figure 157: NGR Maximum Ground Fault Current Formula 48Figure 158: NGR Maximum Ground Fault Voltage Formula 48Figure 159: NGR Ground Fault Voltage Formula 48Figure 160: Neutral Grounding Transformer 49Figure 161: NGTX Effective Resistance Formula 49Figure 162: NGTX Ground Fault Voltage Formula 49Figure 163: Open Corner Delta Ground Detector 50Figure 164: Open Corner Delta Normal Phasors 51Figure 165: Open Corner Delta with One Phase Grounded 51Figure 166: Positive Sequence Formulas 52Figure 167: Positive Sequence Calculation 53Figure 168: Negative Sequence Formulas 54Figure 169: Negative Sequence Calculation 1 54Figure 170: Negative Sequence Calculation 2 55Figure 171: Zero Sequence Formulas 56Figure 172: Zero Sequence Calculation 1 56Figure 173: Zero Sequence Calculation 2 57Figure 21: Zones of Protection 67Figure 22: Overlapping Zones of Protection 69Figure 23: NonOverlapping Zone of Protection Example 69Figure 24: Protective Relay Coordination 71Figure 25: Compensation for Normal System Fluctuations 72

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Figure 26: Fault Study Short Circuit Currents 74Figure 27: Calculation to Convert Current to Base Voltage 75Figure 28: Example Time Coordination Curve (TCC) 76Figure 29: Reference Voltage Conversions 77Figure 210: Damage Curves 78Figure 211: Damage Curve TCC 79Figure 212: Single Line Drawing 81Figure 213: TCC Curve #1 81Figure 214: TCC Curve #2 81Figure 215: TCC Curve #3 81Figure 31: Example of Clapper Style Relay 83Figure 32: Typical Electromechanical Relay with Timing Disk 84Figure 33: Example of an Electromechanical Relay Polarizing Element 85Figure 34: Typical Polarizing Element Electrical Schematic 85Figure 35: Typical Electromechanical Overcurrent Trip Schematic 86Figure 36: Simple Microprocessor Operation Flowchart 88Figure 37: Simple Microprocessor Internal Schematic 89Figure 38: Typical Electromechanical Overcurrent Trip Schematic 93Figure 39: Typical Digital Relay Trip Schematic 93Figure 310: Electromechanical Generator Protection Panel 94Figure 311: Digital Generator Protection Panel 94Figure 41: 3Phase Relay Input Connection Using Two Sources 98Figure 42: Steady State Pickup Testing 102Figure 43: Dynamic On/Off Waveform 102Figure 44: Simple Dynamic Test Waveform 103Figure 45: Complex Dynamic Waveform (Compliments of Manta Test Systems) 106Figure 46: System Modeling Waveform (Compliments of Manta Test Systems) 107Figure 47: Graph of Pickup Test 110Figure 48: Percent Error Formula 111Figure 49: Beckwith Electric M3310 Relay Element Specifications 111Figure 410: Example Pickup Percent Error Calculation 111Figure 411: Simple Off/On Timing Test 112Figure 412: Dynamic On/Off Testing 112Figure 413: Beckwith Electric M3310 Relay Element Specifications 113Figure 414: Example Timing Test Percent Error Calculation 113Figure 415: SEL311C Relay Element Specifications 114Figure 416: SEL311C Output Contact Specifications 114Figure 417: Manta Test Systems MTS1710 Technical Specifications 115Figure 418: 50Element Maximum Expected Error 115Figure 419: 50Element Ideal Combination Test 122Figure 420: 50Element Combination Test 123Figure 421: 21Element Combination Test 126Figure 422: 21Element Dual Zone Combination Test 127Figure 423: 21Element Dual Zone Combination Test 128

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Figure 424: System Modeling Waveform (Compliments of Manta Test Systems) 134Figure 51: Example Test Sheet Header 136Figure 52: Example Test Sheet with Project Details 136Figure 53: Example Test Sheet with Nameplate Data 137Figure 54: Example Test Sheet with Notes 138Figure 55: Example Test Sheet with Metering 139Figure 56: Example Test Sheet with Input / Output Verification 139Figure 57: Example Test Sheet for Element Pickup Tests 141Figure 58: Example Test Sheet #2 for Element Pickup Test 142Figure 59: Test Sheet Example Element 143Figure 510: Example Test Sheet with Element Characteristics 143Figure 511: Example Test Sheet with Final Output Check 144Figure 61: Example 59Element Settings and Test Results with PhasetoNeutral Voltages 154Figure 62: Example 59Element Settings and Test Results with PhasetoPhase Voltages 154Figure 63: Substitution Chart for 59Procedures 154Figure 64: Simple PhasetoNeutral Overvoltage Connections 155Figure 65: Simple PhasetoPhase Overvoltage Connections 155Figure 66: PhasetoPhase Overvoltage Connections with 2 Voltage Sources 156Figure 67: GE/Multilin SR750 Overvoltage Relay Specifications 157Figure 68: GE/Multilin SR750 Overvoltage Relay Specifications 161Figure 69: Inverse Curve for Overvoltage Protection 162Figure 610: GE/Multilin SR489 Overvoltage Relay Specifications 163Figure 71: 27Element Example Settings and Test Results 169Figure 72: Substitution Chart for 27Procedures 169Figure 73: Simple PhasetoNeutral Undervoltage Connections 170Figure 74: Simple PhasetoPhase Undervoltage Connections 171Figure 75: PhasetoPhase Undervoltage Connections with 2 Voltage Sources 171Figure 76: GE D60 Undervoltage Relay Specifications 174Figure 77: GE D60 Undervoltage Relay Specifications 179Figure 78: Undervoltage Inverse Curve 180Figure 81: Generator Trip Frequency Requirements 184Figure 82: 81Element Example Settings and Test Results 187Figure 83: Simple PhasetoNeutral Frequency Test Connections 188Figure 84: Simple PhasetoPhase Overvoltage Connections 189Figure 85: PhasetoPhase Overvoltage Connections with 2 Voltage Sources 189Figure 86: SEL300G Frequency Specifications 191Figure 91: Ground Fault Protection SingleLine Drawing 196Figure 92: Ground Protection TCC 196Figure 93: 50/51 TCC #1 197Figure 94: 50/51 TCC #2 197Figure 95: 50/51 TCC #3 197Figure 96: 50/51 TCC #4 197Figure 97: Simple Instantaneous Overcurrent Connections 201Figure 98: High Current Connections #1 201

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Figure 99: High Current Connections #2 202Figure 910: Neutral or Residual Ground Bypass Connection 202Figure 911: Neutral or Residual Ground Bypass Connection via  Connection 203Figure 912: Pickup Test Graph 203Figure 913: Pickup Test Graph via Jogging Method 204Figure 914: 50Element Timing Test 206Figure 915: GE D60 Relay Overcurrent Technical Specifications 207Figure 916: GE D60 Relay Output Contact Technical Specifications 207Figure 917: Manta Test Systems M1710 Technical Specifications 207Figure 918: 50Element Minimum Pickup 208Figure 919: 50Element Alternate Relay Connection 210Figure 101: 51Element North American Curves 212Figure 102: 51Element IEC European Curves 212Figure 103: ANSI Extremely Inverse with Different Pickup Settings 212Figure 104: ANSI Extremely Inverse with Different Timing Settings 212Figure 105: Simple Time Overcurrent Connections 215Figure 106: High Current Connections #1 216Figure 107: High Current Connections #2 216Figure 108: Neutral or Residual Ground Bypass Connection 217Figure 109: Neutral or Residual Ground Bypass Connection via  Connection 217Figure 1010: Pickup Test Graph 218Figure 1011: SEL311C 51 Time Overcurrent Specifications 219Figure 1012: 51Element North American Curves 220Figure 1013: 51Element Timing Test 221Figure 1014: 51Element SEL311C Timing Curve Characteristic Formulas 222Figure 1015: 51Element Example Time Coordination Curve 225Figure 1016: 51Element Timing for GE D60 226Figure 1017: 51Element Time Delay Calculation for Table Method 226Figure 1018: 51Element Alternate Relay Connection 229Figure 111: Parallel Transmission Lines with Standard Overcurrent Protection 232Figure 112: Parallel Transmission Lines with Directional Overcurrent Protection 233Figure 113: Directional Ground Overcurrent Protection for Transmission Lines 234Figure 114: Directional Overcurrent Protection in an Industrial Application 234Figure 115: Standard Phasor Diagram 235Figure 116: Directional Polarizing 235Figure 117: Example SingleLine Drawing 239Figure 118: Typical Directional Polarizing Using SEL Relays 240Figure 119: Directional Polarizing Using GE Relays and a 60 MTA Setting 240Figure 1110: 3Line Drawing for Example TestSet Connection 241Figure 1111: Directional Overcurrent TestSet Connections 241Figure 1112: Normal Phasors 243Figure 1113: Phase A Characteristic Phasor 243Figure 121: Simple Differential Protection 248Figure 122: Simple Differential Protection with External Fault 248

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Figure 123: Simple Differential Protection with External Fault 2 249Figure 124: Simple Differential Protection with Internal Fault 249Figure 125: Simple Differential Protection with Internal Fault 2 250Figure 126: Simple Differential Protection with Worst Case CT Error 251Figure 127: Simple Differential Protection with Worst Case CT Error and External Fault 251Figure 128: Percentage Differential Protection Schematic 252Figure 129: Percentage Differential Protection Operating Mechanism 252Figure 1210: Percentage Differential Protection and External Faults 253Figure 1211: Percentage Differential Protection and Internal Faults 253Figure 1212: Percentage Differential Protection and Internal Faults 2 254Figure 1213: Percentage Differential Protection Characteristic Curve 254Figure 1214: Percentage Differential Protection Characteristic Curve with Minimum Pickup 256Figure 1215: Percentage Differential Protection Dual Slope Characteristic Curve 257Figure 1216: Simple 87Element TestSet Connections 260Figure 1217: Simple 3Phase 87Element TestSet Connections 261Figure 1218: Simple 3Phase 87Element TestSet Connections with Six Current Channels 261Figure 1219: Pickup Test Graph 262Figure 1220: GE Power Management 489 Analog Input Specifications 263Figure 1221: GE Power Management 489 Differential and Output Relay Specifications 264Figure 1222: GE Power Management 489 Minimum Trip Time 265Figure 1223: GE Power Management 489 Specifications 267Figure 1224: GE Power Management 489 Slope Test Connection Table 267Figure 1225: GE Power Management 489 Slope Test Connections Example #1 268Figure 1226: GE Power Management 489 Slope Test Connections Example #2 268Figure 1227: Simple 87Element Slope TestSet Connections 269Figure 1228: Simple 87Element High Current Slope TestSet Connections 269Figure 1229: Percentage Differential Protection Dual Slope Characteristic Curve 270Figure 1230: GE 489 Differential Formulas 271Figure 1231: Example Characteristic Curve in Amps with 1st Transition Defined 272Figure 1232: Percentage Differential Protection Dual Slope Characteristic Curve 272Figure 1233: Percentage Differential Protection Dual Slope Characteristic Curve 274Figure 1234: Percentage Differential Protection Dual Slope Characteristic Curve in Amps 276Figure 1235: Using Graphs to Determine Pickup Settings 277Figure 1236: GE Power Management 489 Specifications 278Figure 1237: Determine Slope by Rise/Run Calculation 281Figure 131: Zones of Protection Example 286Figure 132: 3 Phase Generator Differential Protection 287Figure 133: 3 Phase Transformer Differential Protection 288Figure 134: 3 Phase Transformer Differential Protection Using Tap Settings 289Figure 135: WyeDelta Transformer Differential Protection Using CT Connections 290Figure 136: WyeDelta Transformer Differential Protection Using CT Connections 296Figure 137: Phase Relationship / Clock Position with Leading Angles 299Figure 138: Phase Relationship / Clock Position with Lagging Angles 299Figure 139: WyeWye Transformer 300

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Figure 1310: WyeWye Transformer Phasor Diagram 301Figure 1311: DeltaDelta Transformer Connections 302Figure 1312: DeltaDelta Transformer Phasors 303Figure 1313: WyeDelta Transformer Connections 304Figure 1314: WyeDelta Transformer Phasor Diagrams 304Figure 1315: WyeDelta Alternate Transformer Connections 305Figure 1316: WyeDelta Alternate Transformer Phasor Diagrams 305Figure 1317: DeltaWye Transformer Connections 306Figure 1318: DeltaWye Transformer Connections 307Figure 1319: DeltaWye Alternate Transformer Connections 308Figure 1320: DeltaWye Alternate Transformer Phasor Diagrams 309Figure 1321: Transformer Nameplate Phase Relationships 311Figure 1322: Common PhaseAngle Compensation Settings 314Figure 1323: Zones of Protection Example 317Figure 1324: CT Connections Example #1 318Figure 1325: CT Connections Example #2 318Figure 1326: GE/Multilin SR745 Transformer Protective Relay Connections 319Figure 1327: GE T60 Transformer Protective Relay Connections 319Figure 1328: Beckwith Electric M3310 Transformer Protective Relay Connections 320Figure 1329: Schweitzer Electric SEL587 Transformer Protective Relay Connections 320Figure 1330: Schweitzer Electric SEL387 Transformer Protective Relay Connections 321Figure 1331: Simple 3Phase 87Element TestSet Connections 323Figure 1332: Simple 3Phase 87Element TestSet Connections with Six Current Channels 323Figure 1333: Pickup Test Graph 324Figure 1334: SEL387E Specifications 325Figure 1335: Simple 87Element TestSet Connections 326Figure 1336: Simple 3Phase 87Element TestSet Connections 327Figure 1337: Simple 3Phase 87Element TestSet Connections with Six Current Channels 327Figure 1338: Pickup Test Graph 329Figure 1339: SEL387E Specifications 329Figure 1340: SEL387 Differential and Output Relay Specifications 331Figure 1341: SEL387 Differential Minimum Trip Time 331Figure 1342: Common PhaseAngle Compensation Settings 333Figure 1343: Yy12 or Yy0 3Phase Differential Restraint Test Connections 334Figure 1344: Dd0 3Phase Differential Restraint Test Connections 335Figure 1345: Dy1 3Phase Differential Restraint Test Connections 336Figure 1346: Yd1 3Phase Differential Restraint Test Connections 337Figure 1347: Dy11 3Phase Differential Restraint Test Connections 337Figure 1348: Yd11 3Phase Differential Restraint Test Connections 338Figure 1349: Schweitzer Electric SEL387 Transformer Protective Relay Connections 339Figure 1350: 3Phase RestrainedDifferential Slope TestSet Connections 340Figure 1351: Percentage Differential Protection Dual Slope Characteristic Curve 341Figure 1352: SEL387 Slope1 Differential Formulas 342Figure 1353: SEL387 Definition of IOP and IRT 342

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Figure 1354: SEL387 Slope2 Differential Formulas 346Figure 1355: Percentage Differential Protection Dual Slope Characteristic Curve in Amps 347Figure 1356: Using Graphs to Determine Pickup Settings 348Figure 1357: Determine Slope by Rise/Run Calculation 353Figure 1358: YDac Transformer Connection 358Figure 1359: Transformer Relay Connections for SinglePhase Differential Testing 359Figure 1360: Schweitzer Electric SEL387 Transformer Protective Relay Connections 360Figure 1361: 1Phase RestrainedDifferential Slope TestSet A Yd1 Connections 360Figure 1362: 1Phase RestrainedDifferential Slope TestSet B Yd1 Connections 361Figure 1363: 1Phase RestrainedDifferential Slope TestSet C Yd1 Connections 361Figure 1364: Transformer Inrush Waveform 364Figure 1365: Simple 3Phase 87Element TestSet Connections 365Figure 1366: Simple, Higher Current 3Phase 87Element TestSet Connections 366Figure 141: Simple Differential Protection with Worst Case CT Error and External Fault 371Figure 142: Simple 87UElement TestSet Connections 373Figure 143: Parallel 87UElement TestSet Connections 374Figure 144: Parallel 87UElement TestSet Connections with Equal W_CTC Settings 374Figure 145: Transformer Relay Connections for SinglePhase Differential Testing 375Figure 146: 1Phase Differential TestSet A Yd1 Connections 375Figure 147: SEL387E Specifications 377Figure 148: SEL387 Differential Element Specifications 381Figure 149: Preferred SEL387 Output Contact Specifications 382Figure 151: Radial Transmission System with Overcurrent Protection 385Figure 152: Typical Radial Transmission System with Directional Overcurrent Protection 386Figure 153: Typical Transmission System with Directional Overcurrent Protection 386Figure 154: Typical Transmission System with LineDifferential Protection 387Figure 155: Equivalent Transmission Line Impedance 388Figure 156: Typical Transmission System with Early Line Distance Protection (Primary Ohms) 389Figure 157: Primary to Secondary Impedance Calculation 389Figure 158: Typical Transmission System with Early Line Distance Protection (Secondary Ohms)
390Figure 159: Equivalent Transmission Line Impedance 391Figure 1510: Phasor Diagram vs. Impedance Diagram Under Normal Conditions 391Figure 1511: Phasor Diagram vs. Impedance Diagram Under Fault Conditions 392Figure 1512: Impedance Relay Operating Characteristics 393Figure 1513: Impedance Relay Zone of Protection 393Figure 1514: Directional Impedance Relay Operating Characteristics 394Figure 1515: Directional Impedance Relay Zone of Protection 394Figure 1516: MHO Impedance Relay Operating Characteristics 395Figure 1517: Alternate Impedance Relay Operating Characteristics 396Figure 1518: MHO Impedance Relay Zone of Protection 397Figure 1519: 2 Zone MHO Impedance Relay Zone of Protection 397Figure 1520: 2 Zone MHO Impedance Diagram 398Figure 1521: 3 ForwardZone MHO Impedance Relay Zone of Protection 399

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Figure 1522: 3ForwardZone MHO Impedance Diagram 399Figure 1523: Forward1Reverse 3 Zone MHO Impedance Relay Zone of Protection 400Figure 1524: 2Forward1Reverse 3 Zone MHO Impedance Diagram 401Figure 1525: Directional MHO Distance Characteristic 408Figure 1526: MHO Distance Characteristic Sample Shapes 408Figure 1527: 3Phase Example MHO Distance Characteristic 409Figure 1528: 3Phase TestSet Connections 409Figure 1529: 3Phase Example MHO Distance MTA Test 410Figure 1530: 3Phase Example MHO Distance MTA Test Configuration 412Figure 1531: 3Phase MTA Test Procedure on Phasor Diagram 413Figure 1532: 3Phase MTA Test Procedure on Impedance Diagram 414Figure 1533: GE D60 Phase Distance Specification 415Figure 1534: 3Phase Example MHO Distance MTA Test Configuration 418Figure 1535: 3Phase Example MHO Distance Reach Test Procedure 419Figure 1536: GE D60 Phase Distance Specification 419Figure 1537: 3Phase Example MHO Distance MTA Test Configuration 422Figure 1538: GE D60 Phase Distance Timing Specification 423Figure 1539: GE D60 Phase Element (21P) Timing Specification 423Figure 1540: PhasetoPhase Fault Characteristic 426Figure 1541: AB Fault TestSet Configuration 431Figure 1542: BC Fault TestSet Configuration 431Figure 1543: CA Fault TestSet Configuration 432Figure 1544: AB PhasetoPhase TestSet Connections 433Figure 1545: BC PhasetoPhase TestSet Connections 434Figure 1546: CA PhasetoPhase TestSet Connections 434Figure 1547: GE D60 Phase Distance Specification 438Figure 1548: GE D60 Phase Distance Specification 443Figure 1549: GE D60 Phase Distance Timing Specification 449Figure 1550: GE D60 Phase Element (21P) Timing Specification 449Figure 1551: PhasetoGround Fault Characteristic 451Figure 1552: PhasetoNeutral Example MHO Distance MTA Test Configuration 458Figure 1553: SEL311C Ground Distance Specification 459Figure 1554: PhasetoNeutral Example MHO Distance MTA Test Configuration 463Figure 1555: SEL311C Ground Distance Specification 464Figure 1556: PhasetoNeutral Example MHO Distance MTA Test Configuration 469Figure 1557: SEL311C Timer and Output Specifications 470Figure 1558: SEL311C Timer and Output Graph (SELFigure 3.18) 470Figure 161: OR Gate Logic 473Figure 162: AND Gate Logic 474Figure 163: NOT Logic 474Figure 164: NOR Gate Logic 475Figure 165: NAND Gate Logic 475Figure 166: XOR Gate Logic 476Figure 167: XNOR Gate Logic 476

Principles and Practice
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Figure 168: Comparator Logic 477Figure 169: Timer Logic 477Figure 1610: Summary of Logic Element Operation 478Figure 1611: SEL311C Relay Labels 479Figure 1612: SEL311C Relay Label Definitions 480Figure 1613: GE D60 Relay Labels and Definitions 480Figure 1614: SEL311C Voltage Elements Settings and Setting Ranges 481Figure 1615: SEL311C Relay Word Bits 482Figure 1616: SEL311C Relay Word Bit Definitions 482Figure 1617: SEL311C SinglePhase and ThreePhase Undervoltage (27) Logic 483Figure 1618: SEL311C Undervoltage (27) Logic Electrical Schematic 483Figure 1619: SEL311C Undervoltage (27) Scenario #1 Logic 484Figure 1620: SEL311C Undervoltage (27) Scenario #2 Logic 485Figure 1621: GE D60 Phase Undervoltage (27) Settings 486Figure 1622: GE D60 Undervoltage (27) Relay Labels and Definitions 486Figure 1623: GE D60 Simplified Undervoltage (27) Logic 488Figure 1624: GE D60 Electrical Schematic of Undervoltage (27) Logic 489Figure 1625: GE D60 Undervoltage (27) Scenario #1 Logic 490Figure 1626: GE D60 Undervoltage (27) Scenario #2 Logic 491Figure 1627: Example of Individual Element Scheme 492Figure 1628: Summary Example for Individual Scheme 493Figure 1629: Example of Simple Binary Logic Scheme 494Figure 1630: Simple Binary Logic Relay Labels 495Figure 1631: Simple Binary Logic Diagram 495Figure 1632: Simple Binary Logic Schematic 495Figure 1633: Example of Complex Binary Scheme 496Figure 1634: Complex Binary Example Logic Diagram 497Figure 1635: Complex Binary Logic Electrical Schematic Example 498Figure 1636: Math Scheme Symbol Definitions in Order 499Figure 1637: SEL Relay Label Definitions 502Figure 1638: Relay Input/Output Schematic 502Figure 1639: Example #1 Initial Logic 503Figure 1640: Arithmetic NOT Example Final Logic 504Figure 1641: Evaluating Math Logic Example 505Figure 1642: Evaluation of Example #1, Scenario #1 Logic 506Figure 1643: Evaluation of Example #1, Scenario #2 Logic 507Figure 1644: Evaluation of Example #1, Scenario #3 Logic 508Figure 1645: GE D60 Simplified Undervoltage (27) Logic 509Figure 1646: GE D60 Undervoltage (27) Logic Electrical Schematic 510Figure 1647: SEL311C Undervoltage (27) and Overvoltage (59) Logic 511Figure 1648: SEL311C Voltage Elements Settings and Setting Ranges 511Figure 1649: SEL311C Relay Word Bits 512Figure 1650: SEL311C Relay Word Bit Definitions 512Figure 1651: Example #2 Logic Diagram 514

Principles and Practice
xxiii
Figure 1652: Example #2 Revised Logic Diagram 515Figure 1653: Evaluation of Example #2, Scenario #1 Logic 518Figure 1654: Evaluation of Example #2, Scenario #1 Logic 521Figure 1655: SEL311C Relay Word Bits 522Figure 1656: Electrical Schematic of Latching Relay 522Figure 1657: Logic Schemes for Latching Relays 523Figure 1658: Illustration of Rising Edge Operation 523Figure 1659: Illustration of Falling Edge Operation 523Figure 1660: Description of FlexLogic Functions 527Figure 1661: FlexLogic Example Relay Schematic 528Figure 1662: GE  UR FlexLogic Operand Types 529Figure 1663: GE D60 FlexLogic Operands 531Figure 1664: GE D60 Logic Example Schematic Output H1 532Figure 1665: GE D60 Logic Example Output H1 532Figure 1666: GE D60 H1 Output Logic Entry Example 534Figure 1667: GE D60 H1 Output Logic Display Example 535Figure 1668: GE D60 H2 Output Logic Example Schematic 536Figure 1669: GE D60 H2 Output Logic Example 536Figure 1670: GE D60 H2 Output Logic Entry Example 536Figure 1671: GE D60 H2 Output Logic Display Example 537Figure 1672: GE D60 H3 Output Logic Example Schematic 537Figure 1673: GE D60 H3 Output Logic Example 538Figure 1674: GE D60 H3 Output Logic Entry Example 538Figure 1675: GE D60 H3 Output Logic Display Example 538Figure 1676: GE D60 H4 Output Logic Example Schematic 539Figure 1677: GE D60 H4 Output Logic Example 540Figure 1678: GE D60 H4 Output Logic Entry Example 540Figure 1679: GE D60 H4 Output Logic Display Example 540Figure 1680: GE D60 Virtual Output Labels 541Figure 1681: GE D60 Contact Output Assignments 541Figure 171: Complex SingleLine Drawings 545Figure 172: Simple SingleLine Drawings 546Figure 173: SingleLine Drawings with DC Logic and Settings 548Figure 174: Example Relay Information Checklist #1 549Figure 175: Example ThreeLine Drawing 550Figure 176: Example Relay Information Checklist #2 551Figure 177: GE/Multilin Manufacturers Typical Wiring Diagram 552Figure 178: Example Relay Connection Checklist 553Figure 179: Alternate CT Connection #1 554Figure 1710: Alternate CT Connection #2 554Figure 1711: Alternate CT Connection #3 555Figure 1712: Alternate CT Connection #4 555Figure 1713: Using ACB Rotation with an ACB Relay 556Figure 1714: Alternate CT Connection 557

Principles and Practice
xxiv
Figure 1715: Voltage FT Test Switches 558Figure 1716: Current FT Test Switches 559Figure 1717: Current FT Test Switch Connection Drawing 560Figure 1718: Common FT Test Switch Configurations 560Figure 1719: General Electric PK Style Test Block 561Figure 1720: Example Relay Schematic Drawing 562Figure 1721: Example DC Output Connections 563Figure 1722: Example DC Input Connections 563Figure 1723: Example Relay Information Checklist #3 564Figure 1724: Example Manufacturer's Connection Diagram 565Figure 1725: Example Trip / Close Schematic 566Figure 1726: Example Setting Printout 570Figure 1727: Example Relay Information Checklist #4 571Figure 1728: Example Relay Settings 572Figure 1729: Example TCC Drawing 573Figure 181: Simple TestSet Input Connections 580Figure 182: TestSet Input Connections with Contact in Parallel 581Figure 183: TestSet Input Connections in DC Circuit 583Figure 184: Dangerous TestSet Input Connection in Trip Circuit 584Figure 185: M3310 Relay Input Connections 585Figure 186: SEL311C Input Connections 585Figure 187: GE/Multilin SR750 Input Connections 585Figure 188: TestSet Output Connections in DC Circuit 586Figure 189: Relay TestSet Connections 587Figure 1810: 3Phase TestSet Connections Using Two Phases 588Figure 1811: Example AC TestSet Connections 589Figure 1812: Example Relay Settings 592Figure 1813: GE Multilin SR750 Specifications 594Figure 191: Example Digital Input/Output Test Sheet 610Figure 192: Phase Angle Relationships 611Figure 193: Example Metering Test Sheet 612Figure 194: Example Overcurrent Test Sheet 613Figure 195: Example Undervoltage Test Sheet 614Figure 196: Example Output Logic 616Figure 197: Example Test Sheet for Output Logic 617Figure 198: Example #2 Settings 618Figure 199: Example BreakerFailure Logic 618Figure 1910: Example Test Sheet for BreakerFailure Logic 619

1
Chapter 1
Electrical Fundamentals
Every relay technician should thoroughly understand the basics of electricity and power systems so they can apply this knowledge to their relay testing tasks. This first chapter is dedicated to these topics.
1. The ThreePhase Electrical System
Many people have difficulty understanding or visualizing a threephase electrical system. Understanding the relationships between phases is not necessary for simple current, voltage, and/or frequency relay testing. However, it is imperative when testing more complex protective elements. The following explanations use shortcuts to demonstrate the principles described and may not be technically correct. We want you to focus on the information necessary to understand the threephase electrical system and encourage you to seek out other material to better understand the details for any given subject.
A) Generation
Almost all electrical systems are supplied by generators of one sort or another. Although generators can operate using gas, diesel, steam, water, wind, etc.; actual generator construction is basically the same despite the input fuel. A generator rotor is inserted within the center of stator poles that are installed on the outside edge of a circular generator. The rotor is a large electromagnet rotated within the stator poles via a prime mover which can be turned by any of the fuels described previously. The generator voltage is determined by the number of coils wrapped around each iron core within the stator poles and the strength of the rotors magnetic field.

Principles and Practice
2
As the rotor rotates within the stator poles, the magnetic interaction between the rotor and stator creates a voltage. The rotor/stator magnetic interaction, and subsequent voltage, varies as the rotors alignment to the poles change. Figure 11 shows a simplified generator sine wave. A real AC generator would have additional stator poles and rotor magnets. Notice how the horizontal scale of the sine wave graph corresponds to the rotor position inside the stator. When the magnetic north interacts with the stator pole, the induced voltage is positive and the magnetic south creates negative voltage. In our example, a cycle is equivalent to one full rotation of the rotor.
V
S N180 022
5
45135
315270
90
0 45 90 135 180 225 270 315 360
VOLTS
+

ROTOR POSITION
NS270
90
0 45 90 135 180 225 270 315 360
VOLTS
+

ROTOR POSITION
S
N
270
90
0 45 90 135 180 225 270 315 360
VOLTS
+

ROTOR POSITION
SN270
90
0 45 90 135 180 225 270 315 360
VOLTS
+

ROTOR POSITION
S N
270
90
0 45 90 135 180 225 270 315 360
VOLTS
+

ROTOR POSITION
180 0
225
45135
315
180 0
225
45135
315
180 0
225
45135
315
180 0
225
45135
315
V
V
V
V
Figure 11: Simple SinglePhase Generator

Chapter 1: Electrical Fundamentals
3
Most electrical systems use threephase voltages which are created by adding additional poles around the rotor with overlapping edges to ensure smooth transitions. The three poles in this simplified example (real generators have many more poles and rotor magnetic fields) are arranged 120 apart and create the sine wave shown in Figure 12. The rotor interacts with more than one pole simultaneously. Pay attention to the north and south poles. Notice how the horizontal axis is marked in degrees, but the axis units could also be time when the rotor speed is used for reference instead of rotor position. Most applications in North America rotate at 60 cycles per second or 60 hertz as shown in Figure 12.
VOLTS
+

VOLTS
+

VOLTS
+

VOLTS
+

VOLTS
+

A
B
A
B
C
0 45 90 135 180 225 270 315 360
0 2.083 4.166 6.250 8.333 10.42 12.50 14.58 16.67Degrees
ms0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00cycles
390
19.47
1.125
A
B
C
A
B
C
A B C
A
N
S180 0
225
45135
315270
90
B C
A
N
S
180 0
225
45135
315270
90
B C
A
N
S
180 0
225
45135
315270
90
B C
A
N
S
180 0
225
45135
315270
90
B C
A
N
S180 0
225
45135
315270
90
B C
0 45 90 135 180 225 270 315 360
0 2.083 4.166 6.250 8.333 10.42 12.50 14.58 16.67Degrees
ms0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00cycles
0 45 90 135 180 225 270 315 360
0 2.083 4.166 6.250 8.333 10.42 12.50 14.58 16.67Degrees
ms0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00cycles
0 45 90 135 180 225 270 315 360
0 2.083 4.166 6.250 8.333 10.42 12.50 14.58 16.67Degrees
ms0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00cycles
0 45 90 135 180 225 270 315 360
0 2.083 4.166 6.250 8.333 10.42 12.50 14.58 16.67Degrees
ms0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00cycles
Figure 12: Simple ThreePhase Generator

Principles and Practice
4
Threephase systems can be either clockwise (ABC) or counterclockwise (ACB) rotation. The phase rotation of the system can be determined by the rotor direction of rotation, or the position of the windings. To correctly determine phase rotation, imagine that the sine waves are constantly moving from right to left and pick a vertical line as a reference. Notice which order the positive peaks pass through your reference. The A wave will cross first followed by B , and then C which is ABC rotation. We always write ABC and ACB as a standard even though BCA and CAB are technically possible in an ABC system: and CBA and BAC are technically correct for an ACB system. Use the examples in Figure 13 to help understand phase rotation.
VOLTS
+

VOLTS
+

A B C A B
A B C
CLOCKWISE ROTATION EQUALS ABC ROTATION
VOLTS
+

A C B
COUNTERCLOCKWISE ROTATION EQUALS ACB ROTATION
VOLTS
+

A C B
CHANGING ANY TWO POLES EQUALS ACB ROTATION
A
N
S180 0
225
45135
315270
90
C B
A
N
S180 0
225
45135
315270
90
B C
A
N
S180 0
225
45135
315270
90
C B
Figure 13: Phase Sequence Examples

Chapter 1: Electrical Fundamentals
5
B) Frequency
Frequency is generically defined as the rate at which something occurs or is repeated over a particular period of time or in a given sample. The electrical frequency is defined as the number of cycles that occur in 1 second. In North America, the typical system frequency is 60 Hz or 60 cycles per second. In Europe, the typical frequency is 50 Hz or 50 cycles per second. There are more cycles per second in North America, which means that each cycle is generated in 16.666ms. In Europe, each cycle is generated in 20ms.
The difference between these standards is shown in Figure 14 below. Notice that while the time reference remains the same regardless of the 50 or 60Hz waveform, the cycle and degrees references are relative to the generated cycle.
VOLTS
+

0 90 180 270 360
0 4.166 8.333 12.50 16.67
Degrees
ms0 0.25 0.50 0.75 1.00cycles
0 90 180 270 360Degrees0 0.25 0.50 0.75 1.00cycles
0 5.000 10.00 15.00 20.00ms
60Hz
50Hz
50Hz
60Hz
Figure 14: Phasor Diagram Example
The formulas to convert cyclestoseconds and secondstocycles are:
= cyclesFrequencyseconds
cyclesseconds=Frequency
1 cycleseconds=60 Hz
seconds=0.0166
=
cyclesFrequencyseconds
cycles=Frequency seconds
cycles=60 0.0166
cycles=1

Principles and Practice
6
C) Phasor Diagrams
While sinewave drawings display the actual output of a generator, they are hard to draw and take up a lot of space. They also become more difficult to read as more information is added. We need an easier way to display the relationships between phases as well as the relationships between voltages and currents. Phasor diagrams were designed to quickly understand the relationships in an electrical system. A properly constructed phasor diagram displays all of the pertinent information we need to visually analyze an electrical system and can also be called a vector diagram. Each phasor (vector) on a phasor diagram is composed of the following:
A line drawn to scale representing the rms voltage or current of the positivehalfcycle of the waveform. The line length is usually referred to as the magnitude.
Phasors are typically drawn starting at the center of the diagram (the origin). The phasors angle is determined by the location of the peak voltage or current in the positivehalfcycle of the waveform.
An arrow is drawn on the tip of the phasor. There is no official standard for phasor arrows but whatever standard that is chosen should be consistent. Typically a voltage phasor has an open arrow and a current phasor has a closed arrow.
The phasor is labeled with the electrical parameter (e.g. Volts = V, E) it represents, and the connection (e.g. AN phase = AN, an). The first letter of the connection represents the arrow tip and the second letter represents the origin. In Figure 15 the n designations of Van, Vbn, and Vcn are all connected together at the origin of the phasor diagram.
See Figure 15 to help understand how phasor diagrams relate to the electrical system. Those of you familiar with phasor diagrams may be quick to point out the phasor diagram does not appear to be correct at first glance, but pay attention to the angle reference in each drawing.
VOLTS
+

AN BN CN
90DEGREES 210 330
N
Van
VcnVbn
A
N
S180 0
225
45135
315270
90
B C
+

180 0
240
60150
33027090
210
30
120
300
Figure 15: Electrical Diagram Comparison

Chapter 1: Electrical Fundamentals
7
The phasor diagram looks static on the page but actually represents values rotating in the counterclockwise direction. You can visualize phasor diagrams by imagining a tire turning at 60 revolutions per second. If a strobe light shines onto the tire 60 times per second, the tire will appear to be standing still. If we mark three points on the tire 120 degrees apart and start it spinning again with the strobe light, the three marks are the points on a phasor diagram. The marks appear to be standing still because of the strobe light but they are actually turning in the direction of phase rotation.
Phasor diagrams need a reference and the Aphase voltage is usually used as the reference which is typically the Van phasor. We can modify Figure 15 to use Van as the reference by making the Vanrmsmaximumofpositivehalfcycle 0 on all diagrams and plotting the other phasors relative to Van as shown on Figure 16. Phasor diagram purists, pay attention to the angles again.
VOLTS
+

AN BN CN
0DEGREES 120 240
N
Van
Vcn
VbnA
N
S 180
0
225
45
135
315
27090
B C
+

180 0
240
60150
33027090
210
30
120
300
Figure 16: Electrical Diagram Comparison with Van Reference
Phasor diagrams need to be consistent in order to be effective so we will modify the previous examples to create a more traditional phasor diagram. Remember that the generator examples have been simplified to make generator operation easier to understand and we will only use the sinewave and phasor diagrams in the future. Phasor diagrams typically rotate with counterclockwise direction. Figure 17 demonstrates a more realistic phasor diagram with Van as the reference at 0, counterclockwise rotation, ABC rotation, and a lagging angle reference (Where the phasor angles are displayed 0360 clockwise).
VOLTS
+

AN BN CN
0DEGREES 120 240
Van
Vbn
Vcn
+

180 0
120
300
210
3090
270
150
330
240
60
Figure 17: Phasor Diagram 1

Principles and Practice
8
The big differences between Figure 17 and the previous examples are the reference angles inside the diagram and the location of the Vbn and Vcn phasors. Remember that you can determine the phase rotation on a sinewave drawing by selecting a reference and pretending that the waves are flowing towards the left. The phase rotation for this example is ABC. You can tell the phase rotation on a phasor diagram using the same technique. Use 0 as the reference and rotate the phasors in a counterclockwise direction. The Van phasor crosses first followed by the Vbn phasor and finally the Vcn phasor. Both diagrams indicate ABC rotation.
The angles on the Phasor Diagram are relative and depend on your background. The angles shown in Figure 17 make a lot of sense if you decide that your reference is 0 and all angles are considered to be lagging the reference. GE/Multilin relays use this system and a typical, threephase electrical system is defined as [email protected], [email protected], and [email protected] as shown in Figure 17.
Many organizations and engineers define the exact same electrical system as [email protected], Vbn[email protected], and [email protected] as shown in Figure 18. Notice that the only difference between Figures 17 and 18 is the angle references. The actual phasors have not changed position because counterclockwise rotation is the standard for phasor diagrams unless defined otherwise.
VOLTS
+

AN BN CN
0DEGREES 240 120 Vbn
Vcn
+

180 0
240
60150
330270
90
210
30
120
300
Van
Figure 18: Phasor Diagram 2
Because having two standards for defining was not confusing enough, we can define the exact same electrical system as [email protected], [email protected], and [email protected] as shown in Figure 19. This is the standard that was used by my instructors and will be the standard used for the majority of this book. Remember that there is no right or wrong standard as long as they are applied consistently and you are aware what standard is inuse.

Chapter 1: Electrical Fundamentals
9
VOLTS
+

AN BN CN
0DEGREES 120 120 Vbn
Vcn
+

180 0
120
60150
30
90
90
150
30
120
60
Van
Figure 19: Phasor Diagram 3
Once again phase rotation can be changed from ABC to ACB by changing the rotation or changing any two phases. ABC systems rotate counter clockwise and can be verified on the phasor diagram by using zero degrees as a reference and imagining the phasors rotating counterclockwise. When rotating counter clockwise, AN phasor crosses first followed by BN and finally CN. The phasors are always rotating. Any phasor diagram represents one moment in time. Review Figure 110 to see the differences between ABC and ACB systems.
VOLTS
+

AN CN BN
0DEGREES 120 120
Van
Vcn
Vbn
+

180 0
120
60150
3090
90
150
30
120
60
VOLTS
+

AN BN CN
0DEGREES 120 120
Van
Vbn
Vcn
+

180 0
120
60150
3090
90
150
30
120
60
ACB via TwoPhase Switch
ACB via Reverse Rotation
Figure 110: ABC / ACB Phasor Diagrams

Principles and Practice
10
Up until now, we have exclusively focused on the voltage waveforms and phasors that represent the potential energy across two points as measured by a voltmeter. Current waveforms represent the current flow through a device as measured by an ammeter in series with a load. The waveforms and phasor diagrams are drawn in the same manner as the voltage waveforms and phasors but use Amperes (Amps / A) as their scale. The current phasors are drawn with closed arrows to differentiate them from voltage phasors. Review Figure 111 see current and voltage values plotted simultaneously.
VOLTS
+

AN BN CN
0DEGREES 120 120
180 0
30
90
90
60
150120
30
60
1501
20
Van
Vbn
Vcn
ABC ROTATION
Ia
Ib
Ic
Van
Vbn
Vcn
Ia IcIb
Ic
Ia
Ib
C
BA
N
C
N
A
B
Figure 111: Current and Voltage Phasors
I like to place zero degrees at the 3 oclock position, but it can easily be placed at the 12 oclock position if it helps you visualize the diagram. It is simply a matter of personal preference.

Chapter 1: Electrical Fundamentals
11
180 0
30
90
90
60
150120
30
60
1501
20
Van
Vbn
Vcn
0 Degrees on Horizontal Axis
Ic
Ia
Ib
0 Degrees on Vertical Axis
180
0 30
9090
60
150 
120
30
60
150
120
Van
VbnVcn
Ic
Ia
Ib
Figure 112: Vector Diagrams on Different Axis
i) Adding or Subtracting Phasors
There will be times when phasors need to be added together to help understand the properties of the electrical system such as DeltaWye conversions in the next section or Sequence Components later in the chapter.
Phasor addition is probably the most difficult electrical calculation and there are three methods that you can employ:
a. Use a calculator Some calculators will allow you to add [email protected] and [email protected] using the interface. This functionality is rare and is sometimes found on scientific calculators that can add complex numbers using polar quantities.
b. Convert the Phasor (Polar) to Rectangular Adding and subtracting phasors in polar format (magnitude and angle) is very difficult. But if we can convert the phasors into rectangular format (real and imaginary numbers or horizontal and vertical magnitudes) first, the addition or subtraction is quite easy.

Principles and Practice
12
Those who thought that trigonometry would never be useful in the real world may be surprised because we can apply the Pythagorean Theorem to any phasor using the following formulas:
60.00
60
0.8
66
@9
0
=
=
=
=
Real Magnitude cos(Angle)
Real 1 cos(0 )
Real 1 1
Real 1
=
=
=
=
Real Magnitude cos(Angle)
Real 1 cos(60 )
Real 1 0.5
Real 0.5
=
=
=
=
Imaginary Magnitude sin(Angle)
Imaginary 1 sin(0 )
Imaginary 1 0
Imaginary 0
=
=
=
=
Imaginary Magnitude sin(Angle)
Imaginary 1 sin(60 )
Imaginary 1 0.866
Imaginary 0.866
= + = +
= +
Rectangular Real " "Imaginary
Rectangular 1 0
j x jy
j
= + = +
= +
Rectangular Real " "Imaginary
Rectangular 0.5 0.866
j x jy
j
Now we can add the real (xaxis) values together, then the imaginary numbers (yaxis or j in formulas), and then use the Pythagorean Theorem to convert the value back to a phasor:

Chapter 1: Electrical Fundamentals
13
30.00
1.73
0.8
66
@9
0
= +
= +
=
Real(Sum) Real([email protected] ) Real([email protected] )
Real(Sum) 1 0.5
Real(Sum) 1.5
= +
= +
=
Imaginary(Sum) " "Imaginary([email protected] ) " "Imaginary([email protected] )
Imaginary(Sum) 0 0.866
Imaginary(Sum) 0.866
j j
j j
j
= +Sum 1.5 0.866j
( )
( )
( )
= +
= +
= +
=
=
2 2
2 2
Magnitude(Sum) ( ) ( )
Magnitude(Sum) 1.5 0.866
Magnitude(Sum) 2.25 0.749956
Magnitude(Sum) 2.999956
Magnitude(Sum) 1.732
Real Sum Imaginary Sum
( )
=
=
=
=
Imaginary(Sum)Angle(Sum) arctanReal(Sum)
0.866Angle(Sum) arctan1.5
Angle(Sum) arctan 0.577
Angle(Sum) 30

Principles and Practice
14
c. Add the Vectors Graphically If you draw the vectors to scale with a CAD program or a ruler and protractor, you can draw one vector on the tip on the other and measure the distance and angle from the first vector's source as shown in the following section.
D) ThreePhase Connections
There are two main connections when using threephase systems, Delta and Wye (Star). Until now, we have been dealing with Wye connections where all three phases have a common point. With this connection, the line and phase currents are the same, but the phase and line voltages are different as shown in Figure 113:
Ia
Ib
Ic
Van
Vbn
Vcn
Vca
Vab
Vbc
Ia & Ib & Ic = Phase or Line CurrentVan & Vbn & Vcn = Phase VoltageVab & Vbc & Vca = Line Voltage
Figure 113: Wye Connection
In order to draw phasors of the three new line voltages (Vab, Vbc, Vca), we need to follow their connections through the Wye connection. Vca is the vector sum of Vcn and Vna using the vector conventions we discussed earlier. Vcn is still drawn at 120 because it has not changed. Vna is drawn at 180 because it is the opposite of Van at 0 degrees. When performing vector addition, the first vector is drawn (Vna) because a is the reference of a Vca phasor. The second vector (Vcn) is drawn at the correct angle starting on the tip of the first vector. The sum of the two vectors is Vcn + Vna or Vcnna. We can remove the nn and the final phasor designation is Vca.

Chapter 1: Electrical Fundamentals
15
Van
Vcn
Vca
N
A
C
VbnVna
Vbn
Vcn
Van
Vcn
Vcn + Vna = Vcnna = Vca
Vca
180 0
30
90
90
60
150120
30
60
1501
20
Figure 114: Phase to Line Conversion
Experience has shown that when two vectors with the same magnitude are added when they are 60 degrees apart, the sum will be 3 larger at the midpoint between the two vectors. If the vector diagram was drawn to scale, we could measure the resultant vector. (Something to remember when you forget your fancy calculator.) Therefore, Vca is [email protected]
= =
= =
= +
= +
=
Vna 180 , Vcn 120
Vna Vcn 180 120 60
60Vca Vcn2
Vca 120 30
Vca 150.
Based on experience, any conversion between Delta and Wye balanced systems (identical magnitudes 120 degrees apart) will increase or decrease by 3 and be displaced by 30 degrees.

Principles and Practice
16
The vector diagram for a balanced Wye connected threephase system would look like Figure 115 where:
WyE To DElTA DElTA To WyEDElTA WyE DElTA WyE
Van @ 0 Vab = 3Van @ 30 (Van+30) Vab @ 30
= Van @03
Vab
(Vab30)
Vbn @ 120 Vbc = 3Vbn @ 90 (Vbn+30) Vbc @ 90
= Vbn @ 1203
Vbc (Vbc30)
Vcn @ 120 Vca = 3Vcn @ 150 (Vcn+30) Vca @ 150
= Vcn @1203
Vca(Vca30)
180 0
30
90
90
60
150120
30
60
1501
20
Vna
Vbn
Vcn
Van
Vca
Vbc
Vab
Ib
Ic
Ia
Vnb
Vnc
Vbn
Van
Vcn
Figure 115: Delta/Wye Connected Vector Diagram
A Delta connected system is shown in Figure 116. Using this connection, the line and phase voltages are the same, but there is a difference between the phase and line currents.

Chapter 1: Electrical Fundamentals
17
Ia
Ib
Ic
Ia & Ib & Ic = Line CurrentIab & Ibc & Ica = Phase Current
Vca
Vab
Vbc
Vab & Vbc & Vca = Line or Phase Voltage
Ica
Iab
Ibc
Figure 116: Delta Connection
Figure 117 shows the same phasor diagram from Figure 115 with a Delta connection.We cannot use the Van voltage as a reference (as shown in Figure 115) because Van does not exixt in a Delta system, so we must pick a different reference voltage. Most people choose Vab as the reference in a delta system, which means that all phasors must rotate 30 clockwise. That can be confusing when you are used to looking at PN references because it now looks like the current is lagging by 60 instead of 30 , but if you look at the relationship between Vab and Ia in Figure 115 and compare it to the relationship in Figure 117, you'll see that they are the same difference apart.
180 0
30
90
90
60
150120
30
60
1501
20
Vca
Vbc
Ib
Ic
Ia
Vab
Figure 117: Delta Connection Vector Diagram for Balanced System with 30 Lag

Principles and Practice
18
E) Watts
Generators are turned by prime movers that are controlled by a governor. The governor operates like the cruise control in your car and tries to maintain the same speed (usually 60Hz). If the load increases or decreases, the generator will momentarily slow down or speed up before the governor can respond to compensate for the load change. The generator's speed controls the real power output of the generator that is measured in Watts when connected to the grid. Watts are a measure of the real power consumed by a load like actual torque in a motor, heat from a heater, or light and heat from an incandescent light bulb. Watts do all of the work in an electrical system.
A system that is generating 100% Watts are displayed on sinewave and phasor diagrams with the phase voltage and phase current at the same phase angle as displayed in Figure 118.
VOLTS
+

AN BN CN
0DEGREES 120 120
180 0
30
90
90
60
150120
30
60
1501
20
Van
Vbn
Ia IcIb
Ic
Ia
Ib
Vcn
Figure 118: Watts  Current and Voltage Are InPhase

Chapter 1: Electrical Fundamentals
19
F) Inductive VARS
In the early years of electricity, a major debate was raised between the widespread use of AC or DC in electrical systems. Ultimately AC won the war and is the standard throughout most of the world because of its ability to transform lower voltages into higher voltages. This was an important distinction because the higher voltages require less current to transmit the same amount of power, which drastically reduced transmission costs. This transformation ability uses induction to create a magnetic field when conductors are coiled around a magnetic core.
Inductive VARs represent the energy created by the magnetic fields and is created by the interaction of electrical coils and magnetic cores in equipment such as AC motors, transformers, and generators.
Ohms Law in DC systems explains the relationship of Volts, Amps, and Resistance in a circuit and it is updated in AC systems to include Reactance. Inductive reactance is a component of the Inductive VARs and will resist any change in current as opposed to resistance which resists the entire flow of current. The Inductive Reactance causes the current to lag the voltage due to the opposing force in inductive systems. Most systems are inductive due to electric motors and current usually lags the voltage in most systems.
Figure 119 simplifies the AC waveform by only showing one phase to help you visualize the lag between voltage and current. The other two phases will change accordingly in a balanced system. Remember that the axis remains in place while the waveform moves from right to left. The voltage peak will cross the yaxis before the current peak and, therefore, the current lags the voltage (or the voltage leads the current). In the phasor diagram, the vectors rotate counterclockwise and the voltage will cross the 0 degree axis before the current phasor.
VOLTS
+

Van
DEGREES 0
180 0
30
90
90
60
150120
30
60
1501
20 Van
Vbn
Vcn
ABC ROTATION WITH 30 DEGREE LAG
IaIc
IaIb
30
Figure 119: Inductive VARs  Current Lags the Voltage by 30

Principles and Practice
20
G) Capacitive VARS
Capacitive VARs represent the reactive power stored in capacitors, overexcited generators, cables or power lines, or other sources that cause the current to lead the voltage. While inductive VARs resist any change in current, capacitive VARs give the voltage an extra boost. Inductive and capacitive VARs operate in opposition and can cancel each other out. (See The Power Triangle in the next section for details.) In a system where the capacitive VARs are greater than the inductive VARs, the current will lead the voltage.
While the generator governor controls the power output of the generator as described previously, the generator exciter controls the generator output voltage. If the generator is not connected to the grid, the exciter can raise or lower the rotor current that, in turn, raises or lowers the generator voltage in direct proportion. However, when the generator is connected to a grid the voltage is relatively fixed by the grid and raising and lowering the rotor current causes the generator to input or export VARs. Generators supply most of the VARs required by inductive machines like transformers or motors.
Figure 120 simplifies the AC waveform by showing only one phase to help you visualize how the current leads the voltage. The other two phases will change accordingly in a balanced system. Remember, the axis remains in place while the waveform moves from right to left. The current peak will cross the yaxis before the voltage peak and, therefore, the current leads the voltage. In the phasor diagram, the vectors rotate counterclockwise and the current will cross the 0 degree axis before the voltage phasor.
180 0
30
90
90
60
150120
30
60
1501
20 Van
Vbn
Vcn
ABC ROTATION WITH 30 DEGREE LEAD
Ic Ia
Ib
VOLTS
+

Van
DEGREES 0
Ia
30
Figure 120: Capacitive VARs  Current Leads the Voltage

Chapter 1: Electrical Fundamentals
21
H) The Power Triangle
In an AC electrical system, all power can be broken down into three separate components. VA is the apparent power supplied and is calculated by multiplying the voltage and current. The real energy produced by an electrical system is measured in Watts and is the amount of current in phase with the voltage and multiplied by the voltage. Reactive VARs is the combination of capacitive and inductive VARs produced by capacitors and inductive machines and are +/90 out of phase with the voltage respectively. Watts and VARs are combined to create VA. It can be hard to grasp how these electrical measurements interact, but the power triangle helps put it into perspective. The standard formulas for threephase power are shown in the following figure:
( ) ( ) ( )= + +
= + +
= + +
VA
WATTS cos( ) cos( ) cos( )
VARS sin( ) sin( ) sin( )
Van Ia Vbn Ib Vcn Ic
Van Ia Van Ia Vbn Ib Vbn Ib Vcn Ic Vcn Ic
Van Ia Van Ia Vbn Ib Vbn Ib Vcn Ic Vcn Ic
Figure 121: ThreePhase Power Formulas
All of the formulas in Figure 122 can be used to calculate threephase power assuming that all threephase currents and voltages are balanced. While no system is perfectly balanced, these equations will put you in the ballpark. Make sure you use the correct phasors when determining relationships. ANGLE in the VAR and Watt formulas are the difference between the phasetoneutral voltage and the line current (ANGLE = VLN angle ILN angle) phasetophase voltages can be used to determine ANGLE in balanced systems only by assuming that the VLL leads VLN by 30 degrees (ANGLE = VLL angle  30  ILN angle).
=
=
=
VA 3 LineVolts LineAmps
VARS 3 LineVolts LineAmps sin( )
WATTS 3 LineVolts LineAmps cos( )
ANGLE
ANGLE
Figure 122: ThreePhase Power Formulas Assuming Balanced Conditions

Principles and Practice
22
In a perfect electrical system where only Watts exist, the power triangle is a straight line as no VARs are produced. See Figure 123 to see a power triangle where only pure power is produced.
180 0
30
90
90
60
150120
30
60
1501
20
Van = 1V
Vbn = 1 V
Vcn = 1 V
Ic = 1A
Ia = 1 A
Ib = 1A
Vab = 1.732VVca = 1.732V
Vbc = 1.732V
=
=
=
VA 3 Vab Ia
VA 3 1.732V 1.000A
VA 3VA
=
=
=
=
WATTS 3 Vab Ia cos(Vangle 30 Iangle)
WATTS 3 1.732V 1.000A cos(30 30 0 )
WATTS 3 VA cos(0 )
WATTS 3 WATTS
Figure 123: ThreePhase Power Triangle  Watts Only
When the electrical system supplies a purely inductive source, the current will lag the voltage by 90 degrees, and the power triangle is a vertical line as shown in Figure 124:
180 0
30
90
90
60
150120
30
60
1501
20
Van = 1V
Vbn = 1 V
Vcn = 1 V
Ic = 1AIb = 1A
Vab = 1.732VVca = 1.732V
Vbc = 1.732V
Ia = 1A
=
=
=
VA 3 Vab Ia
VA 3 1.732V 1.000A
VA 3VA
=
=
=
=
VARS 3 Vab Ia sine(Vangle 30 Iangle)
VARS 3 1.732V 1.000A sine(30 30 90 )
VARS 3 VA sine(90 )
VARS 3 VARS
Figure 124: Power Triangle  Only Inductive VARs
When the electrical system supplies a purely capacitive source, the current will lead the voltage by 90 degrees, and the power triangle is a vertical line as shown in Figure 125:

Chapter 1: Electrical Fundamentals
23
180 0
30
90
90
60
150120
30
60
1501
20
Van = 1V
Vbn = 1 V
Vcn = 1 V
Ic = 1A Ib = 1A
Vab = 1.732VVca = 1.732V
Vbc = 1.732V
Ia = 1A =
=
=
VA 3 Vab Ia
VA 3 1.732V 1.000A
VA 3VA
VARS 3 VA sine (90 )=
VARS 3 1.732V 1.000A sine(30 30 90 )
=
=
=
VARS 3 Vab Ia sine(Vangle 30 Iangle)
VARS 3 VARS
Figure 125: Power Triangle  Only Capacitive VARs

Principles and Practice
24
Our next example is closer to real life with the current lagging the voltage by 30 degrees. Most power systems have all three parts of the power triangle with Watts, VARs, and VA as shown in Figure 126.
180 03
0
90
90
60
150120
30
60
1501
20
Van = 1V
Vbn = 1 V
Vcn = 1 VIc = 1A
Ia = 1 AIb = 1A
Vab = 1.732VVca = 1.732V
Vbc = 1.732V
POWER (P)Watts = 2.60 Watts
REA
CTIV
E P
OW
ER
VA
RS
= 1
.5 V
AR
S
30.00APPARENT POWER (S)
VA = 3 VA
Figure 126: Power Triangle
=
=
=
=
=
WATTS 3 Vab Ia cos(Vangle 30 Iangle)
WATTS 3 1.732V 1.0A cos(30 30 30 )
WATTS 3 cos(30 )
WATTS 3 0.866
WATTS 2.60 WATTS
=
=
=
VA 3 Vab Ia
VA 3 1.732V 1.0A
VA 3VA
=
=
=
=
=
VARS 3 Vab Ia sine(Vangle 30 Iangle)
VARS 3 1.732V 1.0A sine(30 30 30 )
VARS 3 sine(30 )
VARS 3 0.5
VARS 1.5 VARS
Figure 127: Example ThreePhase Power/VARS/VA Formula Calculations

Chapter 1: Electrical Fundamentals
25
You can apply the Pythagorean Theorem in nearly every aspect of phasor diagrams and other electrical properties as shown in the following example:
POWER (P)Watts = y Watts
REA
CTIV
E P
OW
ER
VA
RS
= x
VA
RS25.00APPARENT POWER (S)
VA = 3 VA
=
=
=
=
=
WATTS 3 Vab Ia cos(Vangle 30 Iangle)
WATTS VA cos(30 30 25 )
WATTS 3 cos(25 )
WATTS 3 0.906
WATTS 2.72 WATTS
=
=
=
=
ycosVA
ycos(25)3VA
cos(25) 3VA=0.906 3VA
2.72 WATTS
y
y
=
=
=
=
=
VARS 3 Vab Ia sine(Vangle 30 Iangle)
VARS VA sine(30 30 25 )
VARS 3 sine(25 )
VARS 3 0.423
VARS 1.268 VARS
=
=
=
=
xsineVA
xsine(25)3VA
sine(25) 3VA=0.423 3VA
1.268 VARS
x
x
POWER (P)Watts = 2.72 Watts
REA
CTIV
E P
OW
ER
VA
RS
= 1
.26
8 V
AR
S
25.00APPARENT POWER (S)
VA = z VA
( )
( )
( )
( )
= +
= +
= +
= =
2 2
2 2
VA WATTS VARS
VA 2.72 1.268
VA 7.3984 1.6078
VA 9.00 3 VA

Principles and Practice
26
I) Power Factor
An electrical systems power factor represents the angle between the Watts and VA of the electrical system. It can also be the angle between voltage and current in any phase when calculated on a phasebyphase basis. Power factor is expressed as the Cosine of the angle, which can be immediately converted into efficiency by multiplying the power factor by 100.
The meaning of a positive and negative power factor is another one of those situations where the correct answer depends on the location and background of the measuring device. The following statements are usually true in North America (unless you are in a generator plant) and it is always a good idea to represent the power factor as importing or exporting VARS if you are communicating with someone who might use a different standard.
When the current leads the voltage or the system has capacitive VARs, the power factor is considered leading, negative, or exporting. When the system is primarily inductive and the current lags the voltage, the power factor is positive, lagging, or importing. Review the following examples of power factor calculations:
i) Unity Power Factor Calculation (Voltage and Current in Phase)
=
=
=
PowerFactor cos(Angle)
PowerFactor cos(0)
PowerFactor 1.00
=
=
=
Efficiency PowerFactor 100
Efficiency 1.00 100
Efficiency 100%
(All of the energy supplied is real power or Watts)

Chapter 1: Electrical Fundamentals
27
ii) Inductive System (Current Lags Voltage by 30)
=
=
= +
PowerFactor cos(Angle)
PowerFactor cos( 30)
PowerFactor 0.867 lag or 0.867
=
=
=
Efficiency PowerFactor 100
Efficiency 0.867 100
Efficiency 86.7%
(86.7% of the supplied energy is real power or Watts)
iii) Capacitive System (Current Leads Voltage by 30)
=
=
=
PowerFactor cos(Angle)
PowerFactor cos(30)
PowerFactor 0.867 lead or 0.867
=
=
=
Efficiency PowerFactor 100
Efficiency 0.867 100
Efficiency 86.7%
(86.7% of the supplied energy is real power or Watts)
2. Transformers
As we discussed previously, the ability to transform smaller voltages to higher voltages is the primary reason why we use AC electricity today. Using higher voltages to transfer electricity over long distances saves money, because less current is required at higher voltages to transfer the same amount of energy. Less current means smaller conductors which cost less than large conductors and weigh less so the support structures are also smaller and cheaper. Online savings are realized through reduced line losses as less current flows through the conductors. Transformers are nearly perfect(ideal) machines that require very little energy to create the transformation.

Principles and Practice
28
The simplest transformers are constructed around a core made of magnetic material. Two different sets of copper or aluminum coils are wound around the core, and the number of turns determines the rated voltages of the transformer. The set of coils with the most turns is called the High side because it will have a higher voltage than the Low side. Some people automatically refer to the high side as the transformer primary even though the primary of a transformer is defined by normal load flow. A stepup transformer steps a lower voltage to a higher voltage and the low side is the transformer primary. The opposite is true with a step down transformer.
TRANSFORMERSIMPLE STEPDOWN
VHIGH
AHIGH
VLOWALOW
HIGHSIDE
LOWSIDE
6 TURNS 2 TURNS
SOURCE LOAD
PRIMARY SECONDARY
6 TurnsTransformer Ratio(x:1)= =3:1Ratio2 Turns
TRANSFORMERSIMPLE STEPUP
VHIGH
AHIGH
VLOWALOW
LOWSIDE
HIGHSIDE
2 TURNS 6 TURNS
LOAD
PRIMARY SECONDARY
SOURCE
Figure 128: Simple Transformer Construction
There are two main principals that apply to all ideal transformers:
The transformer ratio is absolute Powerin equals Powerout

Chapter 1: Electrical Fundamentals
29
The transformer ratio is predefined when the transformer is built and is based on the transformer turns ratio. Transformer ratios are defined on the transformer nameplate by high and low side voltages. For example, a 3:1 transformer ratio could be listed on the transformer nameplate as
345,000V (345kV): 115,000V (115kV).
= =345,000VTransformer Ratio (x:1) 3 :1115,000V
=High Side VoltageTransformer Ratio (x:1)Low Side Voltage OR
= High Side Voltage Transformer Ratio (x:1) Low Side Voltage OR
=High Side VoltageLow Side Voltage
Transformer Ratio (x:1)
Figure 129: Transformer Voltage Ratio Calculation
Threephase power is calculated by multiplying the system voltage, current, and power factor.
= LINE LINE3 Phase Power (watts) 3 V I Power Factor
If a transformer has two different voltage levels and the powerin equals the powerout, the current must also be transformed. Transformer current is inversely proportional to the voltage. Therefore, the highvoltage side of the transformer has less current flowing through its winding and more current will flow through the lowvoltage windings. If the highside current of our example transformer was 35A, the lowside would be rated for 105A.
( = = ):
=Low Side CurrentTransformer Ratio(x:1)High Side Current OR
=Low Side CurrentHigh Side Current
Transformer Ratio(x:1) OR
= Low Side Current High Side Current Transformer Ratio(x:1)
Figure 130: Transformer Current Ratio Calculation

Principles and Practice
30
Transformers can be connected Wye, Delta or with a combination of Wye and Delta windings. Remember that typical Delta systems only have ratings for Line (linetoline) voltages and currents. Wye systems have Phase (phasetoneutral) and Line voltage and line current ratings. The transformer nameplate will display the line and phase (if applicable) voltages for each winding along with the apparent (VA) power rating of the transformer. The transformer ratio will always use the LineLine voltages.
There are often several VA (E.g. 25/35/45/50 MVA) ratings listed and each VA rating corresponds to a different cooling stage. The smallest VA rating applies when the transformer is air cooled only. Each higher rating applies as another cooling method is applied. The following items might be true if four VA ratings are listed on a transformers nameplate:
Smallest VA = Air Cooled Next largest VA = Stage 1 Fans Next Largest VA = Stage 1 and Stage 2 Fans Largest VA = Stage 1 and Stage 2 Fans and Recirculating Pump
There are two schools of thought to determine which rating is used. A conservative approach uses the minimum VA rating because this is the transformer rating if all cooling methods failed for whatever reason. The more common approach uses the largest VA rating because this is the actual transformer rating under normal conditions.
The most important information for transformer related relay testing includes:
Rated voltage for all windings, Rated current for all windings, Winding connection (Delta or Wye), And VA.
The rated current isnt always shown on a transformers nameplate but it is easily calculated using the following formula. Review the following figures for examples of the most typical transformer applications and calculations that apply to threephase balanced systems. Always remember to use only line voltages in calculations and the following unit modifiers apply.
V, A, VA = Listed Value x 1 (e.g. 2A = 2A)
kV, kA, kVA = Listed Value x 1,000 (e.g. 5kV = 5,000 V)
MVA, MW, MVARS = Listed Value x 1,000,000 (e.g. 10MVA = 10,000,000 VA)
LINELINE IV3AV = OR
LINE
LINEV3
VAI
= OR
LINE
LINEI3
VAV
=

Chapter 1: Electrical Fundamentals
31
H31
H23
H12
H30
H20H10
H1
H3
H2
NAMEPLATEDIAGRAM
X1
H0
X2
A
B
C
H1A
H2B
H3C
H0
X1a
X2b
X3c
CONNECTIONDIAGRAM
PHASORDIAGRAM
X23
X31
X12
A
C B
ENGINEERINGDIAGRAM
a
c
b
Yd11
X3
H31
H23
H12
H30
H20H10
H1
H3
H2
NAMEPLATEDIAGRAM
X1
H0
X2
A
B
C
H1A
H2B
H3C
H0
X1a
X2b
X3c
CONNECTIONDIAGRAM
PHASORDIAGRAM
X23
X31 X12
A
C B
ENGINEERINGDIAGRAM
a
c
b
Yd1
X3
Figure 131: WyeDelta Transformer Connections
VA = 1 MVAVoltage: 4160/2400 y : 600V
= = =
LINE(High)LINE
VA 1,000kVAI 138.79A3 V 3 4.16kV
= = =
LINE(Low)LINE
VA 1,000,000VAI 962.28A3 V 3 600V
= = =High Side Voltage 4160VTransformer Ratio (x:1) 6.9333 :1Low Side Voltage 600V
= = =Low Side Current 962.28ATransformer Ratio(x:1) 6.9333 :1High Side Current 138.79A
Figure 132: WyeDelta Transformer Calculations

Principles and Practice
32
H31
H23
H12
X30
X20X10
NAMEPLATEDIAGRAM
H1
H3
H1A
H2B
H3C
X1a
X2b
X3c
CONNECTIONDIAGRAM
PHASORDIAGRAM
X23
X31 X12A
C B
ENGINEERINGDIAGRAM
a
c
b
Dy11
H2
A
B
C
X0
X1
X3
X2
X0
H31
H23
H12
X30
X20
X10
NAMEPLATEDIAGRAM
H1 H3
H1A
H2B
H3C
X1a
X2b
X3c
CONNECTIONDIAGRAM
PHASORDIAGRAM
X23
X31
X12