TransienTs in elecTrical sysTemsAnAlysis, Recognition, And MitigAtionJ. C. DasNew York Chicago San Francisco Lisbon London MadridMexico City Milan New Delhi San Juan SeoulSingapore Sydney TorontoCopyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permis-sion of the publisher.ISBN: 978-0-07-162603-3MHID: 0-07-162603-4The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-162248-6, MHID: 0-07-162248-9.All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benet of the trademark owner, with no intention of infringement of the trademark. 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Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.To My ParentsJ. C. Das is currently Staff Consultant, Electrical Power Systems, AMEC Inc., Tucker, Georgia, USA. He has varied experience in the utility industry, industrial establishments, hydroelectric gen-eration, and atomic energy. He is responsible for power system studies, including short-circuit, load flow, harmonics, stability, arc-flash hazard, grounding, switching transients, and also, protective relaying. He conducts courses for continuing education in power systems and has authored or coauthored about 60 technical publica-tions. He is author of the book Power System Analysis, Short-Circuit, Load Flow and Harmonics (New York, Marcel Dekker, 2002); its second edition is forthcoming. His interests include power system transients, EMTP simulations, harmonics, power quality, protection, and relaying. He has published 185 electrical power systems study reports for his clients.He is a Life Fellow of the Institute of Electrical and Electronics Engineers, IEEE (USA), a member of the IEEE Industry Applications and IEEE Power Engineering societies, a Fellow of Institution of Engi-neering Technology (UK), a Life Fellow of the Institution of Engineers (India), a member of the Federation of European Engineers (France), and a member of CIGRE (France). He is a registered Professional Engineer in the states of Georgia and Oklahoma, a Chartered Engineer (C. Eng.) in the UK, and a European Engineer (Eur. Ing.).He received a MSEE degree from Tulsa University, Tulsa, Oklahoma in 1982 and BA (mathematics) and BEE degrees in India.About the AuthorPreface xiiichapTer 1 inTroducTion To TransienTs1-1 Classification of Transients 11-2 Classification with Respect to Frequency Groups 11-3 Frequency-Dependent Modeling 21-4 Other Sources of Transients 31-5 Study of Transients 31-6 TNAsAnalog Computers 31-7 Digital Simulations, EMTP/ATP, and Similar Programs 3References 4Further Reading 4chapTer 2 TransienTs in lumped circuiTs2-1 Lumped and Distributed Parameters 52-2 Time Invariance 52-3 Linear and Nonlinear Systems 52-4 Property of Decomposition 62-5 Time Domain Analysis of Linear Systems 62-6 Static and Dynamic Systems 62-7 Fundamental Concepts 62-8 First-Order Transients 112-9 Second-Order Transients 152-10 Parallel RLC Circuit 182-11 Second-Order Step Response 212-12 Resonance in Series and Parallel RLC Circuits 212-13 Loop and Nodal Matrix Methods for Transient Analysis 242-14 State Variable Representation 252-15 Discrete-Time Systems 282-16 State Variable Model of a Discrete System 302-17 Linear Approximation 30Problems 31Reference 32Further Reading 32chapTer 3 conTrol sysTems3-1 Transfer Function 333-2 General Feedback Theory 353-3 Continuous System Frequency Response 383-4 Transfer Function of a Discrete-Time System 383-5 Stability 393-6 Block Diagrams 413-7 Signal-Flow Graphs 413-8 Block Diagrams of State Models 443-9 State Diagrams of Differential Equations 453-10 Steady-State Errors 473-11 Frequency-Domain Response Specifications 493-12 Time-Domain Response Specifications 493-13 Root-Locus Analysis 503-14 Bode Plot 553-15 Relative Stability 583-16 The Nyquist Diagram 603-17 TACS in EMTP 61Problems 61References 63Further Reading 63chapTer 4 Modeling of Transmission Lines and Cables for TransienT STudies4-1 ABCD Parameters 654-2 ABCD Parameters of Transmission Line Models 674-3 Long Transmission Line Model-Wave Equation 674-4 Reflection and Transmission at Transition Points 704-5 Lattice Diagrams 714-6 Behavior with Unit Step Functions at Transition Points 724-7 Infinite Line 744-8 Tuned Power Line 744-9 Ferranti Effect 744-10 Symmetrical Line at No Load 754-11 Lossless Line 774-12 Generalized Wave Equations 774-13 Modal Analysis 77Contentsvvi contents 6-12 Interruptions of Capacitance Currents 1446-13 Control of Switching Transients 1476-14 Shunt Capacitor Bank Arrangements 150Problems 152References 153Further Reading 153chapTer 7 swiTching TransienTs and Temporary overvolTages7-1 Classification of Voltage Stresses 1557-2 Maximum System Voltage 1557-3 Temporary Overvoltages 1567-4 Switching Surges 1577-5 Switching Surges and System Voltage 1577-6 Closing and Reclosing of Transmission Lines 1587-7 Overvoltages Due to Resonance 1647-8 Switching Overvoltages of Overhead Lines and Underground Cables 1657-9 Cable Models 1667-10 Overvoltages Due to Load Rejection 1687-11 Ferroresonance 1697-12 Compensation of Transmission Lines 1697-13 Out-of-Phase Closing 1737-14 Overvoltage Control 1737-15 Statistical Studies 175Problems 179References 180Further Reading 180chapTer 8 currenT inTerrupTion in ac circuiTs8-1 Arc Interruption 1818-2 Arc Interruption Theories 1828-3 Current-Zero Breaker 1828-4 Transient Recovery Voltage 1838-5 Single-Frequency TRV Terminal Fault 1868-6 Double-Frequency TRV 1898-7 ANSI/IEEE Standards for TRV 1918-8 IEC TRV Profiles 1938-9 Short-Line Fault 1958-10 Interruption of Low Inductive Currents 1978-11 Interruption of Capacitive Currents 2008-12 Prestrikes in Circuit Breakers 2008-13 Breakdown in Gases 2004-14 Damping and Attenuation 794-15 Corona 794-16 Transmission Line Models for Transient Analysis 814-17 Cable Types 85Problems 89References 89Further Reading 89chapTer 5 lighTning sTrokes, shielding, and backflashovers5-1 Formation of Clouds 915-2 Lightning Discharge Types 925-3 The Ground Flash 925-4 Lightning Parameters 945-5 Ground Flash Density and Keraunic Level 985-6 Lightning Strikes on Overhead lines 995-7 BIL/CFO of Electrical Equipment 1005-8 Frequency of Direct Strokes to Transmission Lines 1025-9 Direct Lightning Strokes 1045-10 Lightning Strokes to Towers 1045-11 Lightning Stroke to Ground Wire 1075-12 Strokes to Ground in Vicinity of Transmission Lines 1075-13 Shielding 1085-14 Shielding Designs 1105-15 Backflashovers 113Problems 117References 121Further Reading 121chapTer 6 TransienTs of shunT capaciTor banks6-1 Origin of Switching Transients 1236-2 Transients on Energizing a Single Capacitor Bank 1236-3 Application of Power Capacitors with Nonlinear Loads 1266-4 Back-to-Back Switching 1336-5 Switching Devices for Capacitor Banks 1346-6 Inrush Current Limiting Reactors 1356-7 Discharge Currents Through Parallel Banks 1366-8 Secondary Resonance 1366-9 Phase-to-Phase Overvoltages 1396-10 Capacitor Switching Impact on Drive Systems 1406-11 Switching of Capacitors with Motors 140 contents viichapTer 11 TransienT behavior of inducTion and synchronous moTors11-1 Transient and Steady-State Models of Induction Machines 26511-2 Induction Machine Model with Saturation 27011-3 Induction Generator 27111-4 Stability of Induction Motors on Voltage Dips 27111-5 Short-Circuit Transients of an Induction Motor 27411-6 Starting Methods 27411-7 Study of Starting Transients 27811-8 Synchronous Motors 28011-9 Stability of Synchronous Motors 284Problems 288References 291Further Reading 291chapTer 12 power sysTem sTabiliTy12-1 Classification of Power System Stability 29312-2 Equal Area Concept of Stability 29512-3 Factors Affecting Stability 29712-4 Swing Equation of a Generator 29812-5 Classical Stability Model 29912-6 Data Required to Run a Transient Stability Study 30112-7 State Equations 30212-8 Numerical Techniques 30212-9 Synchronous Generator Models for Stability 30412-10 Small-Signal Stability 31712-11 Eigenvalues and Stability 31712-12 Voltage Stability 32112-13 Load Models 32412-14 Direct Stability Methods 328Problems 331References 331Further Reading 332chapTer 13 exciTaTion sysTems and power sysTem sTabilizers13-1 Reactive Capability Curve (Operating Chart) of a Synchronous Generator 33313-2 Steady-State Stability Curves 33613-3 Short-Circuit Ratio 33613-4 Per Unit Systems 33713-5 Nominal Response of the Excitation System 3378-14 Stresses in Circuit Breakers 204Problems 205References 206Further Reading 206chapTer 9 symmeTrical and unsymmeTrical shorT-circuiT currenTs9-1 Symmetrical and Unsymmetrical Faults 2079-2 Symmetrical Components 2089-3 Sequence Impedance of Network Components 2109-4 Fault Analysis Using Symmetrical Components 2119-5 Matrix Methods of Short-Circuit Current Calculations 2219-6 Computer-Based Calculations 2249-7 Overvoltages Due to Ground Faults 224Problems 232References 233Further Reading 233chapTer 10 TransienT behavior of synchronous generaTors10-1 Three-Phase Terminal Fault 23510-2 Reactances of a Synchronous Generator 23710-3 Saturation of Reactances 23810-4 Time Constants of Synchronous Generators 23810-5 Synchronous Generator Behavior on Terminal Short-Circuit 23910-6 Circuit Equations of Unit Machines 24410-7 Parks Transformation 24610-8 Parks Voltage Equation 24710-9 Circuit Model of Synchronous Generators 24810-10 Calculation Procedure and Examples 24910-11 Steady-State Model of Synchronous Generator 25210-12 Symmetrical Short Circuit of a Generator at No Load 25310-13 Manufacturers Data 25510-14 Interruption of Currents with Delayed Current Zeros 25510-15 Synchronous Generator on Infinite Bus 257Problems 263References 264Further Reading 264viii contents 15-9 Static Series Synchronous Compensator 41615-10 Unified Power Flow Controller 41915-11 NGH-SSR Damper 42215-12 Displacement Power Factor 42315-13 Instantaneous Power Theory 42415-14 Active Filters 425Problems 425References 426Further Reading 426chapTer 16 flicker, bus Transfer, Torsional dynamics, and oTher TransienTs16-1 Flicker 42916-2 Autotransfer of Loads 43216-3 Static Transfer Switches and Solid-State Breakers 43816-4 Cogging and Crawling of Induction Motors 43916-5 Synchronous Motor-Driven Reciprocating Compressors 44116-6 Torsional Dynamics 44616-7 Out-of-Phase Synchronization 449Problems 451References 451Further Reading 452chapTer 17 insulaTion coordinaTion17-1 Insulating Materials 45317-2 Atmospheric Effects and Pollution 45317-3 Dielectrics 45517-4 Insulation Breakdown 45617-5 Insulation CharacteristicsBIL and BSL 45917-6 Volt-Time Characteristics 46117-7 Nonstandard Wave Forms 46117-8 Probabilistic Concepts 46217-9 Minimum Time to Breakdown 46517-10 Weibull Probability Distribution 46517-11 Air Clearances 46517-12 Insulation Coordination 46617-13 Representation of Slow Front Overvoltages (SFOV) 46917-14 Risk of Failure 47017-15 Coordination for Fast-Front Surges 47217-16 Switching Surge Flashover Rate 47317-17 Open Breaker Position 47413-6 Building Blocks of Excitation Systems 33913-7 Saturation Characteristics of Exciter 34013-8 Types of Excitation Systems 34313-9 Power System Stabilizers 35213-10 Tuning a PSS 35513-11 Models of Prime Movers 35813-12 Automatic Generation Control 35813-13 On-Line Security Assessments 361Problems 362References 362Further Reading 363chapTer 14 TransienT behavior of Transformers14-1 Frequency-Dependent Models 36514-2 Model of a Two-Winding Transformer 36514-3 Equivalent Circuits for Tap Changing 36714-4 Inrush Current Transients 36814-5 Transient Voltages Impacts on Transformers 36814-6 Matrix Representations 37114-7 Extended Models of Transformers 37314-8 EMTP Model FDBIT 38014-9 Sympathetic Inrush 38214-10 High-Frequency Models 38314-11 Surge Transference Through Transformers 38414-12 Surge Voltage Distribution Across Windings 38914-13 Duality Models 38914-14 GIC Models 39114-15 Ferroresonance 39114-16 Transformer Reliability 394Problems 395References 396Further Reading 396chapTer 15 power elecTronic equipmenT and FaCTS15-1 The Three-Phase Bridge Circuits 39715-2 Voltage Source Three-Phase Bridge 40115-3 Three-Level Converter 40215-4 Static VAR Compensator (SVC) 40515-5 Series Capacitors 40815-6 FACTS 41415-7 Synchronous Voltage Source 41415-8 Static Synchronous Compensator 415 contents ixchapTer 20 surge arresTers20-1 Ideal Surge Arrester 52520-2 Rod Gaps 52520-3 Expulsion-Type Arresters 52620-4 Valve-Type Silicon Carbide Arresters 52620-5 Metal-Oxide Surge Arresters 52920-6 Response to Lightning Surges 53420-7 Switching Surge Durability 53720-8 Arrester Lead Length and Separation Distance 53920-9 Application Considerations 54120-10 Surge Arrester Models 54420-11 Surge Protection of AC Motors 54520-12 Surge Protection of Generators 54720-13 Surge Protection of Capacitor Banks 54820-14 Current-Limiting Fuses 551Problems 554References 555Further Reading 555chapTer 21 TransienTs in grounding sysTems21-1 Solid Grounding 55721-2 Resistance Grounding 56021-3 Ungrounded Systems 56321-4 Reactance Grounding 56421-5 Grounding of Variable-Speed Drive Systems 56721-6 Grounding for Electrical Safety 56921-7 Finite Element Methods 57721-8 Grounding and Bonding 57921-9 Fall of Potential Outside the Grid 58121-10 Influence on Buried Pipelines 58321-11 Behavior Under Lightning Impulse Current 583Problems 585References 585Further Reading 586chapTer 22 lighTning proTecTion of sTrucTures22-1 Parameters of Lightning Current 58722-2 Types of Structures 58722-3 Risk Assessment According to IEC 58822-4 Criteria for Protection 58922-5 Protection Measures 59222-6 Transient Behavior of Grounding System 59417-18 Monte Carlo Method 47417-19 Simplified Approach 47417-20 Summary of Steps in Insulation Coordination 475Problems 475References 476Further Reading 476chapTer 18 gas-insulaTed subsTaTionsvery fasT TransienTs18-1 Categorization of VFT 47718-2 Disconnector-Induced Transients 47718-3 Breakdown in GISFree Particles 48018-4 External Transients 48118-5 Effect of Lumped Capacitance at Entrance to GIS 48218-6 Transient Electromagnetic Fields 48318-7 Breakdown in SF6 48318-8 Modeling of Transients in GIS 48418-9 Insulation Coordination 48718-10 Surge Arresters for GIS 488Problems 493References 493Further Reading 494chapTer 19 TransienTs and surge proTecTion in low-volTage sysTems19-1 Modes of Protection 49519-2 Multiple-Grounded Distribution Systems 49519-3 High-Frequency Cross Interference 49819-4 Surge Voltages 49919-5 Exposure Levels 49919-6 Test Wave Shapes 50019-7 Location Categories 50219-8 Surge Protection Devices 50519-9 SPD Components 50819-10 Connection of SPD Devices 51219-11 Power Quality Problems 51619-12 Surge Protection of Computers 51719-13 Power Quality for Computers 52019-14 Typical Application of SPDs 520Problems 523References 523Further Reading 524x contents A-5 Clairauts Equation 649A-6 Complementary Function and Particular Integral 649A-7 Forced and Free Response 649A-8 Linear Differential Equations of the Second Order (With Constant Coefficients) 650A-9 Calculation of Complementary Function 650A-10 Higher-Order Equations 651A-11 Calculations of Particular Integrals 651A-12 Solved Examples 653A-13 Homogeneous Linear Differential Equations 654A-14 Simultaneous Differential Equations 655A-15 Partial Differential Equations 655Further Reading 658appendix b laplace TransformB-1 Method of Partial Fractions 659B-2 Laplace Transform of a Derivative of f (t ) 661B-3 Laplace Transform of an Integral 661B-4 Laplace Transform of tf (t ) 662B-5 Laplace Transform of (1/t ) f (t ) 662B-6 Initial-Value Theorem 662B-7 Final-Value Theorem 662B-8 Solution of Differential Equations 662B-9 Solution of Simultaneous Differential Equations 662B-10 Unit-Step Function 663B-11 Impulse Function 663B-12 Gate Function 663B-13 Second Shifting Theorem 663B-14 Periodic Functions 665B-15 Convolution Theorem 666B-16 Inverse Laplace Transform by Residue Method 666B-17 Correspondence with Fourier Transform 667Further Reading 667appendix c z-TransformC-1 Properties of z-Transform 670C-2 Initial-Value Theorem 671C-3 Final-Value Theorem 672C-4 Partial Sum 672C-5 Convolution 672C-6 Inverse z-Transform 672C-7 Inversion by Partial Fractions 674C-8 Inversion by Residue Method 67422-7 Internal LPS Systems According to IEC 59422-8 Lightning Protection According to NFPA Standard 780 59422-9 Lightning Risk Assessment According to NFPA 780 59522-10 Protection of Ordinary Structures 59622-11 NFPA Rolling Sphere Model 59722-12 Alternate Lightning Protection Technologies 59822-13 Is EMF Harmful to Humans? 602Problems 602References 603Further Reading 603chapTer 23 dc sysTems, shorT circuiTs, disTribuTions, and hvdc23-1 Short-Circuit Transients 60523-2 Current Interruption in DC Circuits 61523-3 DC Industrial and Commercial Distribution Systems 61723-4 HVDC Transmission 618Problems 627References 628Further Reading 629chapTer 24 smarT grids and wind power generaTion24-1 WAMS and Phasor Measurement Devices 63124-2 System Integrity Protection Schemes 63224-3 Adaptive Protection 63324-4 Wind-Power Stations 63424-5 Wind-Energy Conversion 63524-6 The Cube Law 63624-7 Operation 63824-8 Wind Generators 63924-9 Power Electronics 64024-10 Computer Modeling 64224-11 Floating Wind Turbines 645References 645Further Reading 645appendix a differenTial equaTionsA-1 Homogeneous Differential Equations 647A-2 Linear Differential Equations 648A-3 Bernoullis Equation 648A-4 Exact Differential Equations 648 contents xiappendix f sTaTisTics and probabiliTyF-1 Mean, Mode, and Median 695F-2 Mean and Standard Deviation 695F-3 Skewness and Kurtosis 696F-4 Curve Fitting and Regression 696F-5 Probability 698F-6 Binomial Distribution 699F-7 Poisson Distribution 699F-8 Normal or Gaussian Distribution 699F-9 Weibull Distribution 701Reference 702Further Reading 702appendix g numerical TechniquesG-1 Network Equations 703G-2 Compensation Methods 703G-3 Nonlinear Inductance 704G-4 Piecewise Linear Inductance 704G-5 Newton-Raphson Method 704G-6 Numerical Solution of Linear Differential Equations 706G-7 Laplace Transform 706G-8 Taylor Series 706G-9 Trapezoidal Rule of Integration 706G-10 Runge-Kutta Methods 707G-11 Predictor-Corrector Methods 708G-12 Richardson Extrapolation and Romberg Integration 708References 709Further Reading 709Index 711C-9 Solution of Difference Equations 675C-10 State Variable Form 676Further Reading 676appendix d sequence impedances of Transmission lines and cablesD-1 AC Resistance of Conductors 677D-2 Inductance of Transmission Lines 678D-3 Transposed Line 678D-4 Composite Conductors 679D-5 Impedance Matrix 680D-6 Three-Phase Line with Ground Conductors 680D-7 Bundle Conductors 681D-8 Carsons Formula 682D-9 Capacitance of Lines 684D-10 Cable Constants 685D-11 Frequency-Dependent Transmission Line Models 688References 688appendix e energy funcTions and sTabiliTyE-1 Dynamic Elements 691E-2 Passivity 691E-3 Equilibrium Points 691E-4 State Equations 692E-5 Stability of Equilibrium Points 692E-6 Hartman-Grobman Linearization Theorem 692E-7 Lyapunov Function 692E-8 LaSalles Invariant Principle 692E-9 Asymptotic Behavior 692E-10 Periodic Inputs 693References 693Further Reading 693This page intentionally left blank The book aims to serve as a textbook for upper undergraduate and graduate level students in the universities, a practical and analytical guide for practicing engineers, and a standard reference book on tran-sients. At the undergraduate level, the subject of transients is covered under circuit theory, which does not go very far for understanding the nature and impact of transients. The transient analyses must account for special modeling and frequency-dependent behavior and are important in the context of modern power systems of increasing complexity.Often, it is difficult to predict intuitively that a transient problem exists in a certain section of the system. Dynamic modeling in the planning stage of the systems may not be fully investigated. The book addresses analyses, recognition, and mitigation. Chapters on surge protection, TVSS (transient voltage surge suppression), and insulation coordination are included to meet this objective.The book is a harmonious combination of theory and practice. The theory must lead to solutions of practical importance and real world situations.A specialist or a beginner will find the book equally engrossing and interesting because, starting from the fundamentals, gradually, the subjects are developed to a higher level of understanding. In this process, enough material is provided to sustain a readers interest and motivate him to explore further and deeper into an aspect of his/her liking.The comprehensive nature of the book is its foremost asset. All the transient frequencies, in the frequency range from 0.1 Hz to 50 MHz, which are classified into four groups: (1) low frequency oscillations, (2) slow front surges, (3) fast front surges, and (4) very fast front surges, are discussed. Transients that affect power system stability and transients in transmission lines, transformers, rotat-ing machines, electronic equipment, FACTs, bus transfer schemes, grounding systems, gas insulated substations, and dc systems are covered. A review of the contents will provide further details of the subject matter covered and the organization of the book.An aspect of importance is the practical and real world feel of the transients. Computer and EMTP simulations provide a vivid visual impact. Many illustrative examples at each stage of the devel-opment of a subject provide deeper understanding. The author is thankful to Taisuke Soda of McGraw-Hill for his help and suggestions in the preparation of the manuscript and subsequent printing.J. C. DasPrefACexiiiThis page intentionally left blank Electrical power systems are highly nonlinear and dynamic in nature: circuit breakers are closing and opening, faults are being cleared, generation is varying in response to load demand, and the power systems are subjected to atmospheric disturbances, that is, light-ning. Assuming a given steady state, the system must settle to a new acceptable steady state in a short duration. Thus, the electromagnetic and electromechanical energy is constantly being redistributed in the power systems, among the system components. These energy exchanges cannot take place instantaneously, but take some time period which brings about the transient state. The energy statuses of the sources can also undergo changes and may subject the system to higher stresses resulting from increased currents and voltages. The analysis of these excursions, for example, currents, volt-ages, speeds, frequency, torques, in the electrical systems is the main objective of transient analysis and simulation of transients in power systems.1-1 ClassifiCation of transientsBroadly, the transients are studied in two categories, based upon their origin:1. Of atmospheric origin, that is, lightning2. Of switching origin, that is, all switching operations, load rejection, and faultsAnother classification can be done based upon the mode of gen-eration of transients:1. Electromagnetic transients. Generated predominantly by the interaction between the electrical fields of capacitance and magnetic fields of inductances in the power systems. The electromagnetic phenomena may appear as traveling waves on transmission lines, cables, bus sections, and oscillations between inductance and capacitance.2. Electromechanical transients. Interaction between the elec-trical energy stored in the system and the mechanical energy stored in the inertia of the rotating machines, that is, genera-tors and motors.As an example, in transient stability analysis, both these effects are present. The term transient, synonymous with surges, is used loosely to describe a wide range of frequencies and magnitudes. Table 1-1 shows the power system transients with respect to the time duration of the phenomena.1-2 ClassifiCation with respeCt to frequenCy GroupsThe study of transients in power systems involves frequency range from dc to about 50 MHz and in specific cases even more. Table 1-2 gives the origin of transients and most common frequency ranges. Usually, transients above power frequency involve electromagnetic phenomena. Below power frequency, electromechanical transients in rotating machines occur.Table 1-3 shows the division into four groups, and also the phe-nomena giving rise to transients in a certain group is indicated. This classification is more appropriate from system modeling consider-ations and is proposed by CIGRE Working Group 33.02.1Transients in the frequency range of 100 kHz to 50 MHz are termed very fast transients (VFT), also called very fast front transients. These belong to the highest range of transients in power systems. According to IEC 60071-1,2 the shape of a very fast transient is usually unidirectional with time to peak less than 0.1 s, total duration less than 3 ms, and with superimposed oscillation at a frequency of 30 kHz < f < 100 MHz. Generally, the term is applied to transients of frequencies above 1 MHz. These transients can originate in gas-insulated substations (GIS), by switching of motors and transformers with short connections to the switchgear, by certain lightning con-ditions, as per IEC 60071-2.2Lightning is the fastest disturbance, from nanoseconds to micro-seconds. The peak currents can approach 100 kA in the first stroke and even higher in the subsequent strokes.Nonpermanent departures form the normal line voltage, and frequency can be classified as power system disturbances. These deviations can be in wave shape, frequency, phase relationship, voltage unbalance, outages and interruptions, surges and sags, and impulses and noise. The phenomena shown in italics may loosely be called transients.3 A stricter definition is that a transient is a subcycle disturbance in the ac waveform that is evidenced by a sharp, brief discontinuity in the waveform, which may be additive or subtractive IntroductIon to transIents1C h a p t e r 12 chapter onefrom the original waveform. Yet, in common use, the term tran-sients embraces overvoltages of various origins, transients in the control systems, transient and dynamic stability of power systems, and dynamics of the power system on short circuits, starting of motors, operation of current limiting fuses, grounding systems, and the like.The switching and fault events give rise to overvoltages, up to three times the rated voltage for phase-to-ground transients, and up to four times for phase-to-phase transients. The rise time varies from 50 s to some thousands of microseconds. The simulation time may be in several cycles, if system recovery from disturbance is required to be investigated.The physical characteristics of a specific network element, which affect a certain transient phenomena, must receive detailed considerations. Specimen examples are: The saturation characteristics of transformers and reactors can be of importance in case of fault clearing, transformer energization, and if significant temporary overvoltages are expected. Temporary overvoltages originate from transformer energization, fault overvoltages, and overvoltages due to load rejection and resonance. On transmission line switching, not only the characteristics of the line itself, but also of the feeding and terminating net-works will be of interest. If details of initial rate of rise of over-voltages are of importance, the substation details, capacitances of measuring transformers, and the number of outgoing lines and their surge impedances also become equally important. When studying phenomena above 1 MHz, for example, in GIS caused by a disconnector strike, the small capacitances and inductances of each section of GIS become important.These are some representative statements. The system con-figuration under study and the component models of the system are of major importance. Therefore, the importance of frequency- dependent models cannot be overstated. Referring to Table 1-3, note that the groups assigned are not hard and fast with respect to the phenomena described, that is, faults of switching origin may also create steep fronted surges in the local vicinity.1-3 frequenCy-DepenDent MoDelinGThe power system components have frequency-dependent behav-ior, and the development of models that are accurate enough for a wide range of frequencies is a difficult task. The mathematical representation of each power system component can, generally, be developed for a specific frequency range. This means that one model cannot be applied to every type of transient study. This can lead to considerable errors and results far removed from the real-world situations. This leads to the importance of correct model-ing for each specific study type, which is not so straightforward in every case.41-3-1 softSoFTTM (Swiss Technology Award, 2006) is a new approach that measures the true and full frequency-dependent behavior of the electrical equipment. This reveals the interplay between the three phases of an ac system, equipment interaction, and system reso-nances to achieve the most accurate frequency-dependent models of electrical components. The three-step process is:1. On-site measurements2. Determining the frequency dependent models3. Simulation and modelingThe modeling fits a state-space model to the measured data, based upon vector fitting techniques. Five frequency-independent matrices representing the state-space are generated, and in the fre-quency domain, the matrix techniques are used to eliminate the state vector x.. An admittance matrix is then generated. The matrices of state-space can be directly imported into programs like EMTP-RV. Thus, a highly accurate simulation can be performed.Apart from the reference here, this book does not discuss the field measurement techniques for ascertaining system data for modeling.tabl e 1- 3 Classification of frequency ranges1 FrequenCy range Shape repreSentation group For repreSentation DeSignation Mainly ForI 0.1 Hz3 kHz Low-frequency Temporary oscillations overvoltagesII 50/60 Hz20 kHz Slow front surges Switching overvoltagesIII 10 kHz3 MHz Fast front surges Lightning overvoltagesIV 100 kHz50 MHz Very fast front Restrike overvoltages, surges GIStabl e 1- 2 frequency ranges of transients origin oF tranSient FrequenCy rangeRestrikes on disconnectors and faults in GIS 100 kHz50 MHzLightning surges 10 kHz3 MHzMultiple restrikes in circuit breakers 10 kHz1 MHzTransient recovery voltage: Terminal faults 50/60 Hz20 kHz Short-line faults 50/60 kHz100 kHzFault clearing 50/60 Hz3 kHzFault initiation 50/60 Hz20 kHzFault energization 50/60 Hz3 kHzLoad rejection 0.1 Hz3 kHzTransformer energization (dc) 0.1 Hz1 kHzFerroresonance (dc) 0.1 Hz1 kHztabl e 1- 1 time Duration of transient phenomena in electrical systemsnature oF the tranSient phenoMena tiMe DurationLightning 0.1 s1.0 msSwitching 10 s to less than a secondSubsynchronous resonance 0.1 ms5 sTransient stability 1 ms10 sDynamic stability, long-term dynamics 0.51000 sTie line regulation 101000 sDaily load management, operator actions Up to 24 hIntroductIon to transIents 31-4 other sourCes of transientsDetonation of nuclear devices at high altitudes, 40 km and higher, gives rise to transients called high-altitude electromagnetic pulse (HEMP). These are not discussed in this book.Strong geomagnetic storms are caused by sunspot activity every 11 years or so, and this can induce dc currents in the transmission lines and magnetize the cores of the transformers connected to the end of transmission lines. This can result in much saturation of the iron core. In 1989 a large blackout was reported in the U.S. and Canadian electrical utilities due to geomagnetic storms (Chap. 14).Extremely low magnetic fields (ELF), with a frequency of 60 Hz with higher harmonics up to 300 Hz and lower harmonics up to 5 Hz, are created by alternating current, and associations have been made between various cancers and leukemia in some epide-miological studies (Chap. 22).1-5 stuDy of transientsThe transients can be studied from the following angles: Recognition Prediction MitigationThis study of transients is fairly involved, as it must consider the behavior of the equipment and amplification or attenuation of the transient in the equipment itself. The transient voltage excitation can produce equipment responses that may not be easy to decipher intuitively or at first glance. Again coming back to the models of the power system equip-ment, these can be generated on two precepts: (1) based on lumped parameters, that is, motors, capacitors, and reactors (though wave propagation can be applied to transient studies in motor windings) and (2) based on distributed system parameters, that is, overhead lines and underground cables (though simplifying techniques and lumped parameters can be used with certain assumptions). It is important that transient simulations and models must reproduce frequency variations, nonlinearity, magnetic saturation, surge- arresters characteristics, circuit breaker, and power fuse operation.The transient waveforms may contain one or more oscillatory components and can be characterized by the natural frequencies of these oscillations, which are dependent upon the nature of the power system equipment.Transients are generated due to phenomena internal to the equipment, or of atmospheric origin. Therefore, the transients are inherent in the electrical systems. Mitigation through surge arrest-ers, transient voltage surge suppressors (TVSS), active and passive filters, chokes, coils and capacitors, snubber and damping circuits requires knowledge of the characteristics of these devices for appro-priate analyses. Standards establish the surge performance of the electrical equipment by application of a number of test wave shapes and rigorous testing, yet to apply proper strategies and devices for a certain design configuration of a large system, for example, high-voltage transmission networks, detailed modeling and analysis are required. Thus, for mitigation of transients we get back to analysis and recognition of the transient problem.This shows that all the three aspects, analysis, recognition, and mitigation, are interdependent, the share of analysis being more than 75 percent. After all, a mitigation strategy must again be analyzed and its effectiveness be proven by modeling before implementation.It should not, however, be construed that we need to start from the very beginning every time. Much work has been done. Over the past 100 years at least 1000 papers have been written on the subject and ANSI/IEEE and IEC standards provide guidelines.1-6 tnasanaloG CoMputersThe term TNA stands for transient network analyzer. The power system can be modeled by discrete scaled down components of the power system and their interconnections. Low voltage and current levels are used. The analog computer basically solves dif-ferential equations, with several units for specific functions, like adders, integrators, multipliers, CRT displays, and the like. The TNAs work in real time; many runs can be performed quickly and the system data changed, though the setting up of the base sys-tem model may be fairly time-consuming. The behavior of actual control hardware can be studied and validated before field appli-cations. The advancement in digital computation and simulation is somewhat overshadowing the TNA models, yet these remain a powerful analog research tool. It is obvious that these simulators could be relied upon to solve relatively simple problems. The digi-tal computers provide more accurate and general solutions for large complex systems.1-7 DiGital siMulations, eMtp/atp, anD siMilar proGraMsThe electrical power systems parameters and variables to be studied are continuous functions, while digital simulation, by its nature, is discrete. Therefore, the development of algorithms to solve digitally the differ-ential and algebraic equations of the power system was the starting point. H.W. Dommel of Bonneville Power Administration (BPA) pub-lished a paper in 1969,5 enumerating digital solution of power system electromagnetic transients based on difference equations (App. C). The method was called Electromagnetic Transients Program (EMTP). It immediately became an industrial standard all over the world. Many research projects and the Electrical Power Research Institute (EPRI) contributed to it. EMTP was made available to the worldwide community as the Alternate Transient Program (ATP), developed with W. S. Meyer of BPA as the coordinator.6 A major contribution, Tran-sient Analysis of Control Systems (TACS), was added by L. Dub in 1976.A mention of the state variable method seems appropriate here. It is a popular technique for numerical integration of differential equations that will not give rise to a numerical instability problem inherent in numerical integration (App. G). This can be circum-vented by proper modeling techniques.The versatility of EMTP lies in the component models, which can be freely assembled to represent a system under study. Non-linear resistances, inductances, time-varying resistances, switches, lumped elements (single or three-phase), two or three winding transformers, transposed and untransposed transmission lines, detailed generator models according to Parks transformation, con-verter circuits, and surge arresters can be modeled. The insulation coordination, transient stability, fault currents, overvoltages due to switching surges, circuit breaker operations, transient behavior of power system under electronic control subsynchronous resonance, and ferroresonance phenomena can all be studied.Electromagnetic Transients Program for DC (EMTDC) was designed by D. A. Woodford of Manitoba Hydro and A. Gole and R. Menzies in 1970. The original program ran on mainframe comput-ers. The EMTDC version for PC use was released in the 1980s. Manitoba HVDC Research Center developed a comprehensive graphic user interface called Power System Computer Aided Design (PSCAD), and PSCAD/EMTDC version 2 was released in the 1990s for UNIX work stations, followed by a Windows/PC-based version in 1998. EMTP-RV is the restructured version of EMTP.7,8Other EMTP type programs are: MicroTran by Micro Tran Power System Analysis Corporation, Transient Program Analyzer (TPA) based upon MATLAB; NETOMAC by Siemens; SABER by Avant.8It seems that in a large number of cases dynamic analyses are carried out occasionally in the planning stage and in some situations 4 chapter onedynamic analysis is not carried at all.9 The reasons of lack of analy-sis were identified as: A resource problem Lack of data Lack of experienceFurther, the following problems were identified as the most cru-cial, in the order of priority: Lack of models for wind farms Lack of models for new network equipment Lack of models for dispersed generation (equivalent dynamic models for transmission studies) Lack of verified models (specially dynamic models) and data for loads Lack of field verifications and manufacturers data to ensure that generator parameters are correct Lack of open cycle and combined cycle (CC) gas turbine models in some casesThus, data gathering and verifications of the correct data is of great importance for dynamic analysis.EMTP/ATP is used to simulate illustrative examples of transient phenomena discussed in this book. In all simulations it is necessary that the system has a ground node. Consider, for example, the delta winding of a transformer or a three-phase ungrounded capacitor bank. These do have some capacitance to ground. This may not have been shown in the circuit diagrams of configurations for sim-plicity, but the ground node is always implied in all simulations using EMTP. This book also uses both SI and FPS units, the latter being still in practical use in the United States.RefeRences 1. CIGRE joint WG 33.02, Guidelines for Representation of Networks Elements when Calculating Transients, CIGRE Brochure, 1990. 2. IEC 60071-1, ed. 8, Insulation Coordination, Definitions, Principles, and Rules, 2006; IEC 60071-2, 3rd ed., Application Guide, 1996. 3. ANSI/IEEE Std. 446, IEEE Recommended Practice for Emer-gency and Standby Power Systems for Industrial and Commer-cial Applications, 1987. 4. IEEE, Modeling and Analysis of System Transients Using Digital Programs, Document TP-133-0, 1998. (This document provides 985 further references). 5. H. W. Dommel, Digital Computer Solution of Electromagnetic Transients in Single and Multiphase Networks, IEEE Trans. Power Apparatus and Systems, vol. PAS-88, no. 4, pp. 388399, Apr. 1969. 6. ATP Rule Book, ATP User Group, Portland, OR, 1992. 7. J. Mahseredjian, S. Dennetiere, L. Dub, B. Khodabakhehian, and L. Gerin-Lajoie, A New Approach for the Simulation of Transients in Power Systems, International Conference on Power System Transients, Montreal, Canada, June 2005. 8. EMTP, www.emtp.org; NETOMAC, www.ev.siemens.de/en/pages; EMTAP, www.edsa.com; TPA, www.mpr.com; PSCAD/EMTDC, www.hvdc.ca; EMTP-RV, www.emtp.com. 9. CIGRE WG C1.04, Application and Required Developments of Dynamic Models to Support Practical Planning, Electra, no. 230, pp. 1832, Feb. 2007.fuRtheR ReadingA. Clerici, Analogue and Digital Simulation for Transient Voltage Determinations, Electra, no. 22, pp. 111138, 1972.H. W. Dommel and W. S. Meyer, Computation of Electromagnetic Transients, IEE Proc. no. 62, pp. 983993, 1974.L. Dube and H. W. Dommel, Simulation of Control Systems in An Electromagnetic Transients Program with TACS, Proc. IEEE PICA, pp. 266271, 1977. M. Erche, Network Analyzer for Study of Electromagnetic Tran-sients in High-Voltage Networks, Siemens Power Engineering and Automation, no. 7, pp. 285290, 1985.B. Gustavsen and A. Semlyen, Rational Approximation of Fre-quency Domain Responses by Vector Fitting, IEEE Trans. PD, vol. 14, no. 3, pp. 10521061, July 1999.B. Gustavsen and A. Semlyen, Enforcing Passivity of Admittance Matrices Approximated by Rational Functions, IEEE Trans. PS, vol. 16, no. 1, pp. 97104, Feb 2001.M. Zitnik, Numerical Modeling of Transients in Electrical Systems, Uppsal Dissertations from the Faculty of Science and Technology (35), Elanders Gutab, Stockholm, 2001.In this chapter the transients in lumped, passive, linear circuits are studied. Complex electrical systems can be modeled with certain constraints and interconnections of passive system components, which can be excited from a variety of sources. A familiarity with basic circuit concepts, circuit theorems, and matrices is assumed. A reader may like to pursue the synopsis of differential equations, Laplace transform, and z-transform in Apps. A, B, and C, respec-tively, before proceeding with this chapter. Fourier transform can also be used for transient analysis; while Laplace transform con-verts a time domain function into complex frequency (s s + w), the Fourier transform converts it into imaginary frequency of jw. We will confine our discussion to Laplace transform in this chapter.2-1 Lumped and distributed parametersA lumped parameter system is that in which the disturbance origi-nating at one point of the system is propagated instantaneously to every other point in the system. The assumption is valid if the largest physical dimension of the system is small compared to the wavelength of the highest significant frequency. These systems can be modeled by ordinary differential equations.In a distributed parameter system, it takes a finite time for a dis-turbance at one point to be transmitted to the other point. Thus, we deal with space variable in addition to independent time variable. The equations describing distributed parameter systems are partial differential equations.All systems are in fact, to an extent, distributed parameter sys-tems. The power transmission line models are an example. Each elemental section of the line has resistance, inductance, shunt con-ductance, and shunt capacitance. For short lines we ignore shunt capacitance and conductance all together, for medium long lines we approximate with lumped T and models, and for long lines we use distributed parameter models (see Chap. 4).2-2 time invarianceWhen the characteristics of the system do not change with time it is said to be a time invariant or stationary system.Mathematically, if the state of the system at t t0 is x(t) and for a delayed input it is w(t) then the system changes its state in the station-ary or time invariant manner if: w t x tw t x t( ) ( )( ) ( )+ (2-1)This is shown in Fig. 2-1. A shift in waveform by t will have no effect on the waveform of the state variables except for a shift by t. This suggests that in time invariant systems the time origin t0 is not important. The reference time for a time invariant system can be chosen as zero, Therefore: xx xx rr ( ) [ ( ), ( , )] t t 0 0 (2-2)To some extent physical systems do vary with time, for example, due to aging and tolerances in component values. A time invariant system is, thus, an idealization of a practical system or, in other words, we say that the changes are very slow with respect to the input.2-3 Linear and nonLinear systemsLinearity implies two conditions:1. Homogeneity2. SuperpositionConsider the state of a system defined by (see Sec. 2-14 on state equations): xx ff xx rr [ ( ), ( ), ] t t t (2-3)If x (t) is the solution to this differential equation with initial condi-tions x(t0) at t t0 and input r(t), t > t0: xx xx rr ( ) [ ( ), ( )] t t t 0 (2-4)Then homogeneity implies that: [ ( ), ( )] [ ( ), ( )] x t t t t0 0 r x r (2-5)where a is a scalar constant. This means that x(t) with input ar(t) is equal to a times x(t) with input r(t) for any scalar a.TransienTs in Lumped CirCuiTs5C h a p t e r 26 ChapTer Two Superposition implies that: [ ( ), ( ) ( )] [ ( ), ( )] [ ( ), ( x r r x r x r1 2 1 2t t t t t t0 0 0+ + tt)] (2-6)That is, x(t) with inputs r1(t) + r2(t) is sum of x(t) with input r1(t) and x(t) with input r2(t). Thus linearity is superimposition plus homogeneity.2-4 property of decompositionA system is said to be linear if it satisfies the decomposition prop-erty and the decomposed components are linear.If x(t) is solution of Eq. (2-3) when system is in zero state for all inputs r(t), that is: x 0 r ( ) [ , ( )] t t (2-7)And x(t) is the solution when for all states x(t0), the input r(t) is zero, that is: x x 0 ( ) [ ( ), ] t t 0 (2-8)Then, the system is said to have the decomposition property if: x x x ( ) ( ) ( ) t t t + (2-9)The zero input response and zero state response satisfy the properties of homogeneity and superimposition with respect to ini-tial states and initial inputs, respectively. If this is not true, then the system is nonlinear.Electrical power systems are perhaps the most nonlinear systems in the physical world. For nonlinear systems, general methods of solutions are not available and each system must be studied spe-cifically. Yet, we apply linear techniques of solution to nonlinear systems over a certain time interval. Perhaps the system is not changing so fast, and for a certain range of applications linearity can be applied. Thus, the linear system analysis forms the very fun-damental aspect of the study.2-5 time domain anaLysis of Linear systemsWe can study the behavior of an electrical system in the time domain. A linear system can be described by a set of linear dif-ferential or difference equations (App. C). The output of the system for some given inputs can be studied. If the behavior of the system at all points in the system is to be studied, then a mathematical description of the system can be obtained in state variable form.A transform of the time signals in another form can often express the problem in a more simplified way. Examples of transform tech-niques are Laplace transform, Fourier transform, z-transform, and integral transform, which are powerful analytical tools. There are inherently three steps in applying a transform:1. The original problem is transformed into a simpler form for solution using a transform.2. The problem is solved, and possibly the transformed form is mathematically easy to manipulate and solve.3. Inverse transform is applied to get to the original solution.As we will see, all three steps may not be necessary, and some-times a direct solution can be more easily obtained.2-6 static and dynamic systemsConsider a time-invariant, linear resistor element across a voltage source. The output, that is, the voltage across the resistor, is solely dependent upon the input voltage at that instant. We may say that the resistor does not have a memory, and is a static system. On the other hand the voltage across a capacitor depends not only upon the input, but also upon its initial charge, that is, the past history of current flow. We say that the capacitor has a memory and is a dynamic system. The state of the system with memory is determined by state vari-ables that vary with time. The state transition from x(t1) at time t1 to x(t2) at time t2 is a dynamic process that can be described by differ-ential equations. For a capacitor connected to a voltage source, the dynamics of the state variable x(t) e(t) can be described by: xC r t r t i t 1( ) ( ) ( ) (2-10)We will revert to the state variable form in Sec. 2-14. By this definition, practically all electrical systems are dynamic in nature.2-7 fundamentaL conceptsSome basic concepts are outlined for the solution of transients, which are discussed in many texts.2-7-1 representation of sourcesWe will represent the independent and dependent current and volt-age sources as shown in Fig. 2-2a, b, and c. Recall that in a depen-dent controlled source the controlling physical parameter may be current, voltage, light intensity, temperature, and the like. An ideal voltage source will have a Thvenin impedance of zero, that is, any amount of current can be taken from the source without altering the source voltage. An ideal current source (Norton equivalent) Fi g u r e 2- 1 A time invariant system, effect of shifted input by t. TransienTs in Lumped CirCuiTs 7resistor and, the charging current assuming no source resistance and ignoring resistance of connections, will be theoretically infinite. Practically some resistance in the circuit, shown dotted as R1, will limit the current. Note that the symbol t 0+ signifies the time after the switch is closed: i VRCs( ) 01+

As the current in the capacitor is given by: i C dvdtC We can write: dvdtVR Cs

1 When the capacitor is fully charged, dv/dt and iC are zero and the cur-rent through R2 is given by: i VR Rs21 2

+ And the voltage across the capacitor as well as the resistor R2 is: v i R V RR RCs( ) +2 221 2 We have not calculated the time-charging current transient pro-file and have arrived at the initial and final value results by elemen-tary circuit conditions. Now consider that the capacitor is replaced by an inductor as shown in Fig. 2-3b. Again consider that there is no stored energy in the reactor prior to closing the switch. Inductance acts like an open circuit on closing the switch, therefore all the current flows through R2.Thus, the voltage across resistor or inductor is: v V RR R L didtdidtV RL R RLs LL s

+

+21 221 2( ) In steady state diL /dt 0, there is no voltage drop across the inductor. It acts like a short circuit across R2, and the current will be limited only by R1. It is equal to V/R1.2-7-3 coupled coilsTwo coupled coils are shown in Fig. 2-4. We can write the following equations relating current and voltages: v L didt M didtv M didt L didt1 11 22122 + + (2-11)where M is given by the coefficient of coupling: M k L L 1 2 (2-12)In an ideal transformer, k 1. These equations can be treated as two loop equations; the voltage generated in loop 1 is due to cur-rent in loop 2 and vice versa. This is the example of a bilateral will have an infinite admittance across it. In practice a large generator approximates to an ideal current and voltage source. Sometimes util-ity systems are modeled as ideal sources, but this can lead to appre-ciable errors depending upon the problem under study.2-7-2 inductance and capacitance excited by dc sourceConsider that an ideal dc voltage source is connected through a switch, normally open (t 0) to a parallel combination of a capaci-tor and resistors shown in Fig. 2-3a. Further consider that there is no prior charge on the capacitor. When the switch is suddenly closed at t 0+, the capacitor acts like a short circuit across the Fi g u r e 2- 2 (a) Independent voltage and current source. (b) Voltage and current controlled voltage source. (c) Voltage and current controlled current source.Fi g u r e 2- 3 (a) Switching of a capacitor on a dc voltage source. (b) Switching of a reactor on a dc voltage source.8 ChapTer Two circuit, which can be represented by an equivalent circuit of con-trolled sources (Fig. 2-5).In a three-terminal device, with voltages measured to common third terminal (Fig. 2-6), a matrix equation of the following form can be written: vvz zz ziii rf12 012

(2-13)where zi is input impedance, zr is reverse impedance, zf is forward impedance, and z0 is output impedance. The z parameters result in current-controlled voltage sources in series with impedances. These can be converted to voltage-controlled current sources in parallel with admittancesy-parameter formation. For example, consider a three-terminal device, with following y parameters: y y y yi r f 0 012 0 001 0 0067 0 0020. . . .All the above values are in mhos. Then the following equations can be written: i v vi v v1 1 22 1 20 012 0 0010 0067 0 002 +. .. . Each of these equations describes connection to one node, and the voltages are measured with respect to the reference node. These can be represented by the equivalent circuit shown in Fig. 2-7.2-7-4 two-port networksTwo-port networks such as transformers, transistors, and transmission lines may be three- or four-terminal devices. They are assumed to be linear. A representation of such a network is shown in Fig. 2-8, with four variables which are related with the following matrix equation: vvz zz zii1211 1221 2212

(2-14)Note the convention used for the current flow and the voltage polarity. The subscript 1 pertains to input port and the input termi-nals; the subscript 2 indicates output port and output terminals. The four port variables can be dependent or independent, that is, the independent variables may be currents and the dependent vari-ables may be voltages. By choosing voltage as the independent variable, y parameters are obtained, and by choosing the input current and the output voltage as independent variable h, parameters are obtained.2-7-5 network reductionsCircuit reductions; loop and mesh equations; and Thvenin, Norton, Miller, maximum power transfer, and superposition theorems, which are fundamental to circuit concepts, are not discussed, and a knowledge of these basic concepts is assumed. A network for study of transients can be simplified using these theorems. The following simple example illustrates this.Example 2-1 Consider the circuit configuration shown in Fig. 2-9a. It is required to write the differential equation for the voltage across the capacitor.We could write three loop equations and then solve these simulta-neous equations for the current in the capacitor. However, this can be much simplified, using a basic circuit transformation. It is seen from Fig. 2-9a that the capacitor and inductor are neither in a series or a parallel configuration. A step-wise reduction of the system is shown in Figs. 2-9b, c, and d. In Fig. 2-9b the voltage source is converted to a current source, and 20 O and 40 O resistances in parallel are com-bined. In Fig. 2-9c, the current source is converted back to the volt-age source. Finally, we can write the following differential equation: 0 02823 33..v vC dvdt isc cL +

_, + Fi g u r e 2- 4 To illustrate mutually coupled coils.Fi g u r e 2- 5 Equivalent circuit model of coupled coils using controlled sources.Fi g u r e 2- 6 Equivalent circuit, voltage of controlled sources.Fi g u r e 2- 7 Equivalent circuit of admittances, y-parameter representation.Fi g u r e 2- 8 Two-port network, showing defined directions of currents and polarity of voltages. TransienTs in Lumped CirCuiTs 92-7-6 impedance formsFor transient and stability analysis, the following impedance forms of simple combination of circuit elements are useful:Inductance v L didt sLi z sLL Capacitance i C dvdt sCv zsCC 1 Series RL z R sLR +

_,

1 (2-17)Series RC z sCRsC

+ 1 (2-18)A simpler reduction could be obtained by wye-delta transforma-tion. Consider the impedances shown dotted in Fig. 2-9a, then: Z Z Z Z Z Z ZZZ Z Z Z Z Z ZZZ Z Z121 2 1 3 2 33131 2 1 3 2 32231

+ +

+ +

22 1 3 2 31+ + Z Z Z ZZ (2-15)Conversely: Z Z ZZ Z ZZ Z ZZ Z ZZ Z Z112 1312 13 23212 2312 13 23313

+ +

+ +

22312 13 23Z Z Z + + (2-16)Fi g u r e 2-9 (a), (b), (c), and (d) Progressive reduction of a network by source transformations/Dotted lines in Fig. 2-9a show wye-delta and delta-wye impedance transformations.10 ChapTer Two Parallel RL z sLsL R

+ 1 / (2-19)Parallel RC z RsCR

+ 1 (2-20)Parallel LC z sLs LC

+ 1 2 (2-21)Series LC z s LCSC

+ 1 2 (2-22)Series RLC z sCR s LCsC

+ + 1 2 (2-23)Parallel RLC z sLsL R s LC

+ + 1 2( ) / (2-24)Response to the application of a voltage V and the resulting cur-rent flow can simply be found by the expression: i s Vs z( ) 1 (2-25)Equation (2-25) is applicable if there is no initial charge on the capacitors and there is no prior stored energy in the reactors. The capacitance voltage does not change instantaneously, and therefore, v vC C( ) ( ) 0 0+ . The capacitance voltage and current are trans-formed according to the equations: [ ( )] ( )[ ( )] ( )( )v t V si t C dv tdt sCV sC CCCC

1]1 CVC( ) 0 (2-26)Figure 2-10a shows the equivalent capacitor circuit. Note that the two-source current model is transformed into an impedance and current source model.In an inductance the transformation is: [ ( )] ( )[ ( )] ( )( )i t I sv t L di tdt sLI sL LLLL

_, IiL( ) 0 (2-27)This two-source model and the equivalent impedance and source model are shown in Fig. 2-10b.Fi g u r e 2- 10 (a) Transformed equivalent circuit for the initial conditions of voltage on a capacitor. (b) Transformed equivalent circuit for the current in an inductor. (c) Circuit diagram for Example 2-2. TransienTs in Lumped CirCuiTs 11Example 2-2 Consider a circuit of Fig. 2-10c. Initially the state vari-ables are 50 V across the capacitor and a current of 500 mA flows in the inductor. It is required to find the voltage across the capacitor for t > 0.We can write two differential equations for the left-hand and right-hand loops, using Kirchoffs voltage laws: 100 10 1010 1 04 +

_, + +i dVdt VV i didtLccc LL. Now apply Laplace transform: 10010 10 010 03s i s V V VV i si IL c c cc L L L + + + [ ( ) ( )]( )II VL c( ) . ( ) 0 0 5 0 50 A V These values can be substituted and the equations solved for capac-itor voltage (App. B).2-8 first-order transientsThe energy storage elements in electrical circuits are inductors and capacitors. The first-order transients occur when the circuit con-tains only one energy storage element, that is, inductance or capaci-tance. This results in a first-order differential equation which can be easily solved. The circuit may be excited by: DC source, giving rise to dc transients AC source, giving rise to ac transients.When switching devices operate, they change the topology of the circuit. Some parts of the system may be connected or disconnected. Hence these may be called switching transients. On the other hand, pulse transients do not change the topology of the circuit, as only the current or voltage waveforms of a source are changed.Example 2-3 Consider Fig. 2-11a, with the following values. R1 1 O R2 10 O L 0.15 H V 10 V With the given parameters and the switch closed, we reduce the circuit to an equivalent Thvenin voltage Vth 9.09 V and series Thvenin resistance Rth 0.909 O (Fig. 2-11b). Therefore we can write the following differential equation: 9 09 0 909 0 15 . . . + i didt When steady-state condition is reached, di/dt 0, and the steady-state current is 10 A, the reactor acts as an open circuit.When the switch is opened, Vth 0 V and Rth 10 O. Therefore: 0 10 0 15 + i didt. di/dt 0 in steady state and therefore, current 0.Example 2-4 In Example 2-3, we replace the inductor with a capacitor of 1 F and solve for the capacitor current, with resis-tances and voltage remaining unchanged. The Thvenin voltage and impedance on closing the switch are the same as calculated in Example 2-3. Therefore we can write the following differential equation: v iR vc + th th The current in the capacitor is given by: i C dvdtCC

Thus, we can write: 9 09 0 909 10 6. . + dvdt vCc For steady state, dvC /dt 0 and the capacitor is charged to 9.09 V. When the switch is opened: 0 10 5 + dvdt vCC Again for steady state, dvC/dt is zero and the capacitor voltage is zero. The above examples show the reduction of the system to a simple RL or RC in series excited by a dc source. The differential equation for series RL circuit is: Ri L didt V + (2-28)This is a first-order linear differential equation, and from App. A its solution is: ie VLLR e Ai VR AeRLt RLtRLt + + F i g u r e 2 - 1 1 (a) Circuit with switch open, Example 2-3. (b) Circuit with switch closed, Example 2-3.12 ChapTer Two The coefficient A can be found from the initial conditions, at t 0 i 0. Thus, A V/R: i VR e RLt

1]11 (2-29)The current increases with time and attains a maximum value of V/R, as before. The voltage across the reactor is L di/dt: V VeLtRL

(2-30)That is, the inductor is an open circuit at the instant of switching and is a short circuit ultimately.We could also arrive at the same results by Laplace transform. Taking Laplace transform of Eq. (2-28): Ri s Lsi s LI Vsi s VL s s R L( ) ( ) ( )( )( )+

+01/ The inverse Laplace transform gives the same result as Eq. (2-29) (see App. B). For solution of differential equations, it is not always necessary to solve using Laplace transform. A direct solution can, sometimes, be straightforward.For the series RC circuit excited by a dc voltage, we can write the following general equation: didtiRCV VRC+ ( ) 0 The solution of this equation gives: i V VR eC t RC

( ) 0/ (2-31)VC(0) is the initial voltage on the capacitor.2-8-1 rL series circuit excited by an ac sinusoidal sourceShort circuit of a passive reactor and resistor in series excited by a sinusoidal source is rather an important transient. This depicts the basics of short circuit of a synchronous generator, except that the reactances of a synchronous generator are not time-invariant: L didt Ri V t + + sin( ) The integrating factor is: e eR L dt Rt L( ) / / The solution is: ie VL e t dt AVLeRLtRLt RLtRLt + +

++ sin( )sin 222

_,

+tan 1 LR A Thus: i VR Lt AeRLt

+ + + 2 2 2 sin( ) where:

tan 1 LR The constant A can be found from the initial conditions at t 0, i 0.This gives: i VR Lt eIRt Lm

+ +

2 2( )[sin( ) sin( ) ]sin( /tt I emRt L+ ) sin( ) / (2-32)where Im is the peak value of the steady-state current (V is the peak value of the applied sinusoidal voltage).The same result can be obtained using Laplace transform though more steps are required. Taking Lapalce transform: Ri s Lsi s LI V t tV( ) ( ) ( ) (sin cos cos sin )cos+ +

0 sssi s VL s R L s2 2 2 221+ ++

_,

+ +sin( )( )cos/ 2 2 2++

1]1sssin This can be resolved into partial fractions: 1 1 12 2 2 2 2 2 2 2( )( ) s s sss s + + + + + ++

1]1 (2-33)where a R/L, therefore: i s VL s R Lss sV( ) cos( )

+ + + ++

1]1+ 2 2 2 2 2 21/ssin( ) L s R Lss s2 2 2 2 2 21+ + + ++

1]1/ (2-34)The inverse Laplace transform of function in Eq. (2-33) is: + + + +

112 2 2 21 1( )( ) ( ) cos sins s e t tt ]]1 Therefore: + + + + +12 2 2 21 ss s e t tt( )( ) ( )[ sin cos ] Substituting these results in Eq. (2-34): i VL e t tt

+ +

_,

+( ) cos cos sinsin 2 2 ( cos sin )( )[( cos sit t eVLt+ 1]1

+ 2 2 nn )( cos sin )cos ( cos sin )sin ] et tt + + (2-35) TransienTs in Lumped CirCuiTs 13Remembering that:tan sin ( ) cos ( )/ / + + L R / / /2 2 1 2 2 2 1 2 i VR L t e t

+ + ( ) [sin( ) sin( ) ]2 2 2 1 2 / (2-36)This is the same result as obtained in Eq. (2-32). Again, the deri-vation shows that direct solution of differential equation is simpler.In power systems X/R ratio is high. A 100-MVA, 0.85 power fac-tor generator may have an X/R of 110 and a transformer of the same rating, an X/R 45. The X/R ratios for low-voltage systems may be of the order of 28. Consider that f 90.If the short circuit occurs when the switch is closed at an instant at t 0, q 0, that is, when the voltage wave is crossing through zero amplitude on x axis, the instantaneous value of the short circuit cur-rent from Eq. (2-36) is 2 Im. This is sometimes called the doubling effect.If the short circuit occurs when the switch is closed at an instant at t 0, q p/2, that is, when the voltage wave peaks, the second term in Eq. (2-36) is zero and there is no transient component. This is illus-trated in Fig. 2-12a and b.A physical explanation of the dc transient is that the inductance component of the impedance is high. If the short circuit occurs at the peak of the voltage, the current is zero. No dc component is required to satisfy the physical law that the current in an induc-tance cannot change suddenly. When the fault occurs at an instant when q f 0, there has to be a transient current whose value is equal and opposite to the instantaneous value of the ac short- circuit current. This transient current can be called a dc compo-nent and it decays at an exponential rate. Equation (2-36) can be simply written as: i I t I emRt L + sin dc/ (2-37)This circuit transient is important in power systems from short-circuit considerations. The following inferences are of interest.There are two distinct components of the short-circuit current: (1) an ac component and (2) a decaying dc component at an Fi g u r e 2- 1 2 (a) Short circuit of a passive RL circuit on ac source, switch closed at zero crossing of the voltage wave. (b) Short circuit of a passive RL circuit on ac source, switch closed at the crest of the voltage wave.14 ChapTer Two exponential rate, the initial value of the dc component being equal to the maximum value of the ac component.Factor L/R can be termed as a time constant. The exponen-tial then becomes Idc et/t, where t L/R. In this equation making t t will result in approximately 62.3 percent decay from its original value, that is, the transient current is reduced to a value of 0.368 per unit after a elapsed time equal to the time constant (Fig. 2-13). The dc component always decays to zero in a short time. Consider a modest X/R of 15; the dc component decays to 88 percent of its initial value in five cycles. This phenomenon is important for the rating structure of circuit breakers. Higher is the X/R ratio, slower is the decay.The presence of the dc component makes the total short-circuit current wave asymmetrical about zero line; Fig. 2-12a clearly shows this. The total asymmetrical rms current is given by: it( ) ( ) ( ) rms,asym ac component dc component +2 2 (2-38)In a three-phase system, the phases are displaced from each other by 120. If a fault occurs when the dc component in phase a is zero, the phase b component will be positive and the phase com-ponent will be equal in magnitude but negative. As the fault is sym-metrical, the identity, Ia + Ib + Ic 0, is approximately maintained.Example 2-5 We will illustrate the transient in an RL circuit with EMTP simulation. Consider R 3.76 O and X 37.6 O, the source voltage is 13.8 kV, three phase, 60 Hz. The simulated transients in the three phases are shown in Fig. 2-14. The switch is closed when the voltage wave in phase a crests. The steady-state short-circuit cur-rent is 211A rms < 84.28. This figure clearly shows the asymmetry in phases b and c and in phase a there is no transient. In phases b and c the current does not reach double the steady-state value, here the angle 84 28 . . The lower is the resistance, higher is the asymmetry.The short circuit of synchronous machines is more complex and is discussed in Chaps. 10 and 11.2-8-2 first-order pulse transientsPulse transients are not caused by switching, but by the pulses gen-erated in the sources. Pulses are represented by unit steps (App. B). The unit step is defined as: u t t t tt t( ) < >1 1101 (239)Any pulse train can be constructed from a series of unit steps and the response determined by superimposition (Fig. 2-15). As the network is linear, the step responses are all the same, only shifted in time by the appropriate amount.Consider the response of inductance and capacitance to a pulse transient. The response to a unit step function is determined by the state variable of the energy storage device. When the unit step Fi g u r e 2- 13 To illustrate concept of time constant of an RL circuit.Fi g u r e 2- 14 EMTP simulation of transients in passive RL circuit on three-phase, 13.8 kV, 60 Hz source, transient initiated at peak of the voltage in phase a. TransienTs in Lumped CirCuiTs 15function is zero, there is no voltage on the capacitor and no current in the inductance. At t t1: v t u t t ei tR u t tCt tL( ) ( )( )( ) ( )(( )/ 111111 thee t t ( )/)1 (2-40)where R Cth for the capacitance and L/Rth for the inductance and Rth is the Thvenin resistance.Example 2-6 Consider a pulse source, which puts out two pulses of 5 V, 5 ms in length spaced by 10 ms. Find the voltage on a capacitor of 100 F, Rth 1000 O.The voltage source can be written as: v u t u t u t u t + 5 5 0 05 5 0 015 5 0 02 ( ) ( . ) ( . ) ( . ) The time constant is given by: 10 100 10 103 6 1 Thus, the voltage on the capacitor is: 5 5 0 05 15 0 0155 5 0 05u t e u t eu tt t( ) ( . )( )( .( . ) + ))( ) ( . )( )( . ) ( . )1 5 0 02 15 0 015 5 0 02 e u t et t 2-9 second-order transientsThere are two energy storage elements, and a second-order differen-tial equation describes these systems.2-9-1 dc excitationWe will first consider dc excitation. Consider resistance, induc-tance, and capacitance in series excited by a dc source. There is no charge on the capacitor and no energy stored in the inductance prior to closing the switch.In App. A we studied the second-order differential equation of the form: f t a d qdt b dqdt cq ( ) + +22 (2-41)The solution of the complementary function (CF) depends upon whether the roots are equal, imaginary or real, and different. In a series RLC circuit, the current equation is written as: L didti dtC Ri f t + + ( ) (2-42)Differentiating: d idtRLdidtiLC f t22 + + ( ) (2-43)Case 1 (Overdamped)If R L C > 4 / (2-44)The system is overdamped. The solution can be written as: q t Ae Be qs t s t( ) + +1 2ss (2-45)where qss is the steady-state solution and s1 and s2 are given by: s b b acas b b aca12224242

+

(2-46)The constants A and B can be found from the initial conditions. Consider that the transient is initiated at t 0+, the initial values of q and q are found: q A B qq s A s B q( )( )001 2++ + + + + ssss (2-47)These equations give the values of A and B.Case 2 (Critically Damped)If both the roots are equal: b ac R L C24 4 ( ) / (2-48)the system is critically damped. The solution is given by: q t Ae Bte qst st( ) + + ss (2-49)where s is given by: s ba

2 (2-50)The constants A and B are found from the initial conditions.Fi g u r e 2- 1 5 (a) Unit step u (t ). (b) Unit step u (t t1). (c) Pulse u (t ) u (t t1).16 ChapTer Two Case 3 (Underdamped)If the roots are imaginary, that is, b ac R L C24 4 < < ( ) / (2-51)The system is underdamped and the response will be oscillatory. This is the most common situation in the electrical systems as the resistance component of the impedance is small. The solution is given by: q t Ae t Be t qt t( ) cos sin + + ss (2-52)where:

baac ba2422 (2-53)Again, the constants A and B are found from the initial conditions. The role of resistance in the switching circuit is obvious; it will damp the transients. This principle is employed in resistance switching as we will examine further in the chapters to follow.Example 2-7 A series RLC circuit with R 1 O, L 0.2 H, and C 1.25 F, excited by a 10 V dc source. The initial conditions are that at t 0, i 0, and di/dt 0. In the steady state, di/dt and d i dt2 2/ must both be zero, irrespective of the initial conditions. These val-ues are rather hypothetical for the purpose of the example and are not representative of a practical power system. We can write the differential equation: d idtdidt i22 5 4 10 + + Taking Laplace transform: s i s si I si s I i sss i s220 0 5 0 4 10( ) ( ) ( ) [ ( ) ( )] ( )( + + )) ( ) ( )( )( )+ +

+ +5 4 10105 42si s i ssi ss s s Resolve into partial fractions: i ss s s( ) .( ).( ) + +2 5 210 54 Taking inverse transform, the solution is: i e et t + 2 0 5 2 54. . This is an overdamped circuit. Alternatively, the solution can be found as follows: iissssss

+

104 2 505 25 162 15 25 162 412.Therefore the solution is: i Ae Bet t + + 42 5 . A and B can be found from the initial conditions: i A Bi A B( ) .( ) ( ) ( )0 2 5 00 1 4 0 0++ + + + + Given the initial conditions that i i ( ) ( ) 0 0+ + and are both zero, A 2, B 1/2 and the complete solution is: i e et t + 2 0 5 2 54. . It may not be obvious that the solution represents a damped response. The equation can be evaluated at small time intervals and the results plotted.Example 2-8 Consider now the same circuit, but let us change the inductance to 2.5 H. This gives the equation: d idtdidt22 4 4 10 + + Here the roots are equal and the system is critically damped. iisssss 104 2 502. Therefore the solution is: i Ae Btet t + + 2 22 5 . From initial conditions: i Ai A B i( ) .( ) ( )0 0 2 5 00 2++ + + + + ss Thus, A 2.5, B 5The solution is: i e tet t 2 5 1 52 2. ( ) Example 2-9 Next consider an underdamped circuit. R 1 O, L 0.2 H, and C 0.5 F, again excited by 10 V dc source; initial conditions are the same. This gives the equation: d idtdidt i22 5 10 10 + + Therefore from Eq. (2-53):

baac ba2 2 542 1 942.. Therefore the solution can be written as: i Ae t Be t iiit t + +

+2 5 2 51 94 1 9410. .cos . sin .( )ssss + + + ++A i Ai A B issss0 10 2 5 1 1 94 ( ) ( . )( ) . BB B 0 1 29 .Therefore, from Eq. (2-52) the solution is: i e t e tt t + 2 5 2 51 94 1 29 1 94 1. .cos . . sin . TransienTs in Lumped CirCuiTs 17Alternatively, the solution can be written as: i Ce t it + cos( )ss where: C A BBA +

2 2tan This gives: i e tt +1 63 1 94 52 2 12 5. cos( . . ). 2-9-2 rLc circuit on ac sinusoidal excitationWe studied the response with a dc forcing function. These examples will be repeated with a sinusoidal function, 10 cos 2t.Example 2-10: Overdamped Circuit The differential equation is: d idtdidt i t22 5 4 10 2 + + cos Given the same initial conditions that i 0, and di/dt 0 at t 0, as before, we can write the Particular integral (PI) as: PI + + + +

15 410 2 10 12 5 4 210 15 22 2D D tD tDcos( ) coscos tt D tt 10 25 2 22cos( ) sin We can write the CF of the equation as before in Example 2-7: CF + Be Ct t 4 The complete solution is: i Be Ce tt t + + 42 sin The constants B and C are found from the initial conditions: i B Ci B C( )( )0 0 00 4 2++ + + + + This gives B 2/3 and C 2/3. The solution is: i e e tt t + 2323 24sin Alternatively, we can arrive at the same result by using Laplace transform: [ ( ) ( ) ( )] [ ( ) ( )] ( ) s i s si I si s I i ss220 0 5 0 4 204 + + + This gives: i ss s ss s s( )( )( )( )( ) ( )

+ + +

+ + ++204 4 12423 423 122 Taking the inverse Laplace transform, we get the same result.Example 2-11: Critically Damped Circuit The differential equation is: d idtdidt t22 4 4 10 2 + + cos The PI can be calculated as 1.25 sin 2t (see App. A). The complete solution is: i Be Cte tt t + + 2 21 25 2 . sin Again the constants B and C are found from the initial conditions: i Bi B C( )( ) ( ) .0 00 2 2 5 0++ + + Therefore the solution is: i t te t 1 25 2 2 5 2. sin . Example 2-12 Lastly, we will consider the underdamped circuit, excited by an ac source. As before the differential equation is: d idtdidt i t22 5 10 10 2 + cos The PI, that is, the steady-state solution is: PI + + +

10 15 10 2 10 25 610 5 6 22D D t tDD tcos cos( )( )cos225 2 360 44 2 0 74 22( ). cos . sin + i t tss The complete solution is: i Ae t Be ttt t ++ + 2 5 2 51 94 1 940 44 2 0 7. .cos . sin .. cos . 44 2 sin t The constants A and B are found as before by differentiating and substituting the initial values, A 0.44, B 1.22. The solution is therefore: i e t e tt t ++ 1 22 1 94 0 44 1 940 442 5 2 5. cos . . sin .. cos. .22 0 74 2 t t + . sin In general, we can write the roots of the characteristic equation of a series RLC circuit as: s sdn1 22 22, (2-54)where: R L LCd/ / 2 1 (2-55)We can term a as the exponential damping coefficient, wd as the resonant frequency of the circuit and wn is the natural frequency. Figure 2-16a, b, and c shows underdamped, critically damped, and overdamped responses, respectively.We can write the natural response of RLC circuit in the general form: i I e tmtn + sin( ) (2-56)The voltage across the capacitor is: vC i dt I LCe tC mtn + + 1900 sin[ ( )] (2-57)18 ChapTer Two where:

_, + +tan ( )1 2 2 2 1nn dLCand (2-58)The voltage across the inductor is: v L didt I LCe tL mtn + + + sin[ ( )] 900 (2-59)Note that voltage across the capacitor is lagging more than 90 with respect to the current. Also the voltage across the inductor is leading slightly more than 90 with respect to current. In the steady-state solution, these are exactly in phase quadrature. The difference is expressed by angle d, due to exponential damping. This angle denotes the displacement of the deviation of the dis-placement angle. As the resistance is generally small, we can write: nLCRLC 12and tan (2-60)In most power systems, d can be neglected.In the preceding examples we have calculated the constants of integration from the initial conditions. In general, to find n constants, we need: The transient response f(0) and its (n 1) derivatives The initial value of the forced response ff (0) and its n 1 derivatives.Consider for example a second-order system: i t A e A ei t s A e s A ens t s tns t s t( )( ) + +1 21 1 2 21 21 2 Then i i i A Ai i i s An fn f( ) ( ) ( )( ) ( ) ( )0 0 00 0 01 21 1 + + + +ss A2 2 (2-61)From Eq. (2.61) the constants A1 and A2 can be calculated.From the above examples we observe that the end results do not immediately indicate what type of response can be expected. The response can be calculated in small increments of time and plotted to show a graphical representation.Example 2-13 A 500-kvar capacitor bank is switched through an impedance of 0.0069 + j0.067 O, representing the impedance of cable circuit and bus work. The supply system voltage is 480 V, three phase, 60 Hz, and the supply source has an available three-phase short- circuit current of 35 kA at < 80.5. The switch is closed when the phase a voltage peaks in the negative direction.The result of EMTP simulated transients of inrush current and voltage for 50 ms are shown in Fig. 2-17a. This shows that the maximum switching current is 2866 A peak ( 2027 A rms), and that the voltage transient, shown in Fig. 2-17b results in approximately twice the normal system voltage. The calculated steady-state current is 507 A rms. This example is a practical case of capacitor switching transients in low-voltage systems, and we will further revert to this subject in greater detail in the following chapters.To continue with this example, the resistance is changed to corre-sp