Paul J. Nahin Transients for Electrical Engineers Elementary Switched-Circuit Analysis in the Time and Laplace Transform Domains (with a touch of MATLAB®) Transients for Electrical Engineers OliverHeaviside(1850–1925),thepatronsaint(amongelectricalengineers)of transientanalysts.(Reproducedfromoneofseveralnegatives,dated1893,foundin anoldcardboardboxwithanoteinHeaviside’shand:“Theonewithhandsinpockets isperhapsthebest,thoughhismotherwouldhavepreferredasmile.”) FrontispiecephotocourtesyoftheInstitutionofElectricalEngineers(London) Paul J. Nahin Transients for Electrical Engineers Elementary Switched-Circuit Analysis in the Time and Laplace Transform Domains ® (with a touch of MATLAB ) Foreword by John I. Molinder PaulJ.Nahin UniversityofNewHampshire Durham,NewHampshire,USA ISBN978-3-319-77597-5 ISBN978-3-319-77598-2 (eBook) https://doi.org/10.1007/978-3-319-77598-2 LibraryofCongressControlNumber:2018940348 ©SpringerInternationalPublishingAG,partofSpringerNature2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpartof SpringerNature. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland “An electrical transient is an outward manifestation of a sudden change in circuit conditions, as when a switch opens or closes ... The transient period is usually very short ... yet these transient periods are extremely important, for it is at such times that the circuit components are subjected to the greatest stresses from excessive currents or voltages ... it is unfortunate that many electrical engineers have only the haziest conception of what is happening in the circuit atsuchtimes.Indeed,someappeartoviewthe subject as bordering on the occult.” and “The study of electrical transients is an investigationofoneofthelessobviousaspects of Nature. They possess that fleeting quality which appeals to the aesthetic sense. Their study treats of the borderland between the broad fields of uniformly flowing events ... only to be sensed by those who are especially attentive.” —wordsfromtwoolderbooks1 on electrical transients,expressingviewsthatarevalidtoday. 1The first quotation is from Allen Greenwood, Electrical Transients in Power Systems, Wiley- Interscience 1971, and the second is from L. A. Ware and G. R. Town, Electrical Transients, Macmillan1954. In support of many of the theoretical calculations performed in this book, software packages developed by The MathWorks, Inc. of Natick, MA, were used (specifically, MATLAB® 8.1 Release 2013a and Symbolic Math Toolbox 5.10), running on a Windows 7 PC. This software is now several releases old, but all the commands used in this book work with the newer versionsandarelikelytocontinuetoworkfornewerversionsforseveralyears more.TheMathWorks,Inc.,doesnotwarranttheaccuracyofthetextofthis book. This book’s use or discussions of MATLAB do not constitute an endorsement or sponsorship by The MathWorks, Inc., of a particular pedagogical approach or particular use of MATLAB, or of the Symbolic MathToolboxsoftware. To the Memory of Sidney Darlington (1906–1997) a pioneer in electrical/electronic circuit analysis,2 who was a colleague and friend for twenty years at the University of New Hampshire. Sidney’s long and creative life spanned the eras of slide rules to electronic computers, and he was pretty darn good at using both. 2Sidneyreceivedthe1945PresidentialMedalofFreedomforhiscontributionstomilitarytechnol- ogyduringWorldWarII,andthe1981I.E.MedalofHonor.Oneofhisminorinventionsisthe famous,nowubiquitousDarlingtonpair(theconnectionoftwo“ordinary”transistorstomakea “super”transistor).WhenIonceaskedhimhowhecametodiscoverhiscircuit,hejustlaughedand said,“Well,itwasn’tthathard—eachtransistorhasjustthreeleads,andsotherereallyaren’talotof differentwaystohooktwotransistorstogether!”I’mstillnotsureifhewassimplyjoking. Foreword Day and night, year after year, all over the world electrical devices are being switched “on” and “off” (either manually or automatically) or plugged in and unplugged.Examplesincludehouselights,streetlights,kitchenappliances,refriger- ators,fans,airconditioners,varioustypesofmotors,and(hopefullynotveryoften) part of the electrical grid. Usually, we are only interested in whether these devices areeither“on”or“off”andarenotconcernedwiththefactthatswitchingfromone state to the other often results in the occurrence of an effect (called a transient) betweenthetimetheswitchisthrownandthedesiredconditionof“on”or“off”(the steady state) is reached. Unless the devices are designed to suppress or withstand them,thesetransientscancausedamagetothedeviceorevendestroyit. Takethecaseofanincandescentlightbulb.Thefilamentiscold(itsresistanceis low)beforetheswitchisturnedonandbecomeshot(itsresistanceismuchhigher)a shorttimeaftertheswitchisturnedon.Assumingthevoltageisconstantthefilament current has an initial surge, called the inrush current, which can be more than ten times the steady state current after the filament becomes hot. Significant inrush current can also occur in LED bulbs depending on the design of the circuitry that converts the alternating voltage and current from the building wiring to the much lowerdirectvoltageandcurrentrequiredbytheLED.Duetothecompressormotor, a refrigerator or air conditioner has an inrush current during startup that can be severaltimesthesteadystatecurrentwhenthemotorisuptospeedandtherotating armatureproducesthebackEMF.It’sveryimportanttotakethisintoaccountwhen purchasing an emergency generator. The inrush current of the starter motor in an automobile explains why it may run properly with a weak battery but requires boostercablesfromanotherbatterytogetitstarted. In this book, Paul Nahin focuses on electrical transients starting with circuits consistingofresistors,capacitors,inductors,andtransformersandculminatingwith transmission lines. He shows how to model and analyze them using differential equations and how to solve these equations in the time domain or the Laplace transform domain. Along the way, he identifies and resolves some interesting apparentparadoxes. ix x Foreword Readersareassumedtohavesomefamiliaritywithsolvingdifferentialequations inthetimedomainbutthosewhodon’tcanlearnagooddealfromtheexamplesthat areworkedoutindetail.Ontheotherhand,thebookcontainsacarefuldevelopment oftheLaplacetransform,itsproperties,derivationsofanumberoftransformpairs, anditsuseinsolvingbothordinaryandpartialdifferentialequations.The“touchof Matlab” shows how modern computer software in conjunction with the Laplace transform makes it easy to solve and visualize the solution of even complicated equations.Ofcourse,aclearunderstandingofthefundamentalprinciplesisrequired touseitcorrectly. As in his other books, in addition to the technical material Nahin includes the fascinatinghistoryofitsdevelopment,includingthekeypeopleinvolved.Inthecase oftheTrans-Atlantictelegraphcable,youwillalsolearnhowtheywereabletosolve thetransmissionlineequationswithoutthebenefitoftheLaplacetransform. Whilethefocusofthebookisonelectricaltransientsandelectricalengineers,the tools and techniques are useful in many other disciplines including signal processing, mechanical and thermal systems, feedback control systems, and com- municationsystems.Studentswillfindthatthisbookprovidesasolidfoundationfor their further studies in these and other areas. Professionals will also learn some things.Icertainlydid! HarveyMuddCollege,Claremont, JohnI.Molinder CA,USA January2018