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WIDEBAND AMPLIFIERSWideband AmplifiersbyPETER STARIJo ef StefanLjubljana, SloveniaandERIK MARGANLjubljana, Slovenia Institute ,Jo ef Stefan Institute ,A C.I.P. Catalogue record for this book is available from the Library of Congress.ISBN-100-387-28340-4 (HB)ISBN-13978-0-387-28340-1 (HB)ISBN-13978-0-387-28341-8 (e-book)Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.Printed on acid-free paperAll Rights ReservedNo part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.Printed in the Netherlands.ISBN-100-387-28341-2 (e-book) 2006 Springer www.springer.comWe dedicate this book to all our friends and colleagues in the art of electronics.Table of Contents Release Notes xiiiPart 1:The Laplace Transform Contents 3Part 2:Inductive Peaking Circuits Contents 91Introduction 95DevicesContents 209Introduction 213Contents 307Introduction 311Part 5:System Synthesis and Integration Contents 379Introduction 383Part 6:Computer Algorithms for Analysis and Synthesis of Contents 513Part 7:Algorithm Application Examples Contents 579Introduction 581Index. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .ix Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . Introduction 5 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . ..AcknowledgmentsAmplifier Filter SystemscPart 4:Cascading Amplifier Stages, Selection of Poles Part 3:Wideband Amplifier Stages With Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . 625. . . . . . . . . . . . . . . P.Stari, E.Margan Wideband Amplifiers AcknowledgmentsTheauthorsaregratefulto,,,JohnAddis CarlBattjes DennisFeucht BruceHoffer BobRoss and,allformeremployeesTektronix,Inc.,forallowingustousetheirclassnotes,ideas,andpublications,andfortheirhelpwhenwehadrunintoproblems concerning some specific circuits.Wearealsothankfultoprof.oftheFacultyofMathematicsand IvanVidavPhysics in Ljubljana for his help and reviewing of Part 1, and to, a Csaba Szathmaryformer employee of EMG, Budapest, for allowing us to use some of his measurementresults in Part 5.However,if,inspiteofmeticulouslyreviewingthetext,wehaveoverlookedsomeerrors,this,ofcourse,isourownresponsibilityalone;weshallbegratefultoeveryone for bringing such errors to our attention, so that they can be corrected in thenext edition. To report the errors please use one of the e-mail addresses below.Peter Stari & Erik Margan [email protected] [email protected] IX - P.Stari, E.Margan Wideband Amplifiers ForewordWiththeexceptionofthetragedyonSeptember11,theyear2001wasrelativelynormalanduneventful:remember,thisshouldhavebeentheyearoftheClarkes Kubricks andSpace Odyssey, mission to Juiter; it should have been the yearof the HAL-9000 computer.Today,thePersonalComputerisasubiquitousandomnipresentaswasHALontheDiscoveryspaceship.AndtherateoftechnologydevelopmentandmarketgrowthinelectronicsindustrystillfollowsthefamousMooreLaw,almostfourdecades after it has been first formulated: in 1965, of Intel Corporation Gordon Moorepredicted the doubling of the number of transistors on a chip every 2 years, correctedto 18 months in 1967; at that time, the landing on the Moon was in full preparation.Curiouslyenough,todaynoonecarestogototheMoonagain,letaloneJupiter.And,inspiteofalltheeffortindigitalengineering,westilldonothaveanythingcloseto0.1%oftheHALcapacity(fortunately?!).Whilsttherearemanyresearchlabsstrivingtoputartificialintelligenceintoacomputer,therearealsorumors that this has already happened (with Windows-95, of course!).Intheearly1990sitwasfeltthatdigitalelectronicswilleventuallyrenderanalogsystemsobsolete.Thisneverhappened.Notonlyistheanalogsectorvitalasever,thejobmarketdemandsareexpandinginallfields,fromhigh-speedmeasurementinstrumentationanddataacquisition,telecommunicationsandradiofrequencyengineering,high-qualityaudioandvideo,togrounding andshielding,electromagneticinterferencesuppressionandlow-noiseprintedcircuitboarddesign,to name a few. And it looks like this demand will be going on for decades to come.Butwhilsttheproliferationofdigitalsystemsattractedarelativelyhighnumber of hardware and software engineers, analog engineers are still rare birds. So,forcreativeyoungpeople,whowanttopushtheenvelope,therearelotsofopportunities in the analog field.However,analogelectronicsdidnotearnitsBlack-MagicArtattributeinvain.Ifyouhaveeverexperiencedtheproblemsandfrustrationsfromcircuitsfoundintoomanycook-booksandsure-working schemes in electronics magazines, andifyoubecametiredofperformingexorcismoneverycircuityoubuild,thenitisprobablythetimetotryadifferentway:inourownexperience,thehardwayofdoing the correct math first often turns out to be the easy way!Here is the book. The book is intended to serve both Wideband Amplifiersas a design manual to more experienced engineers, as well as a good learning guide tobeginners. It should help you to improve your analog designs, making better and fasteramplifiercircuits,especiallyiftime-domainperformanceisofmajorconcern.Wehavestrievedtoprovidethecompletemathforeverydesignstage.And,tomakelearningajoyfulexperience,weexplainthederivationofimportantmathrelationsfrom a design engineer point of view, in an intuitive and self-evident manner (rigorousmathematiciansmightnotlikeourapproach).Wehaveincludedmanypracticalapplications, schematics, performance plots, and a number of computer routines.- XI - P.Stari, E.Margan Wideband Amplifiers However, as it is with any interesting subject, the greatest problem was neverwhat to include, but rather what to leave out!IntheforewordofhispopularbookABriefHistoryofTime,StevenHawkingwrotethathispublisherwarnedhimnottoincludeanymath,sincethenumberofreaderswouldbehalvedbyeachformula.SoheincludedtheI 7-#and bravely cut out one half of the world population.Wewentfurther:therearesome220formulaeinPart1only.Byestimatingthecurrentworldpopulationtosome610 ,ofwhich0.01%couldbeelectronics9engineers and assuming an average lifetime interest in the subject of, say, 30 years, ifthe publishers rule holds, there ought to be one reader of our book once every:2 6 10 10 30 356 24 3600 3 10seconds220 9 4 51 a bcor something like 6.610 the total age of the Universe!33Now, whatever you might think of it, this book is about math! It is about notgettingyourdesigntorunrightfirsttime!Bewarned,though,thatitwillbenotenoughtojustreadthebook.Tohaveanyvalue,atheorymustbeputintopractice.Althoughthereisnotheoreticalsubstituteforhands-onexperience,thisbook shouldhelp you to significantly shorten the trial-and-error phase.Wehopethatbystudyingthisbookthoroughlyyouwillfindyourselfatthebeginning of a wonderful journey!Peter Stari and Erik Margan,Ljubljana, June 2003Important Note:WewouldliketoreassuretheConcernedEnvironmentaliststhatduringthewritingofthisbook,noanimalorplanthadsufferedanyharmwhatsoever,eitherindirectorindirectform(excludingtheauthors, one computer mouse and countless computation bugs!).- XII - P.Stari, E.Margan Wideband Amplifiers Release NotesThe manuscript of this book appeared first in spring of 1988.Sincethen,thetexthasbeenrevisedsveraltimes,withsomeminorerrorscorrectedandfiguresredrawn,inparticularinPart 2,whereinductivepeakingnetworksareanalyzed.Severaltopicshavebeenupdatedtoreflectthelatestdevelopmentsinthefield,mainlyin Part 5, dealing with modern high-speed circuits.ThePart 6,whereanumberofcomputeralgorithmsaredeveloped,andPart 7,containing several algorithm application examples, were also brought up to date.This is a release version 3 of the book.ThebookalsocommesintheAdobePortableDocumentFormat(PDF),readablebytheAdobeAcrobatReaderprogram(thelatestversioncanbedownloaded free of charge from ). http://www.adobe.com/products/Acrobat/One of the advantages of the book, offered by the PDF format and the Readerprogram,arenumerous,whichenableeasyaccessto links(blueunderlinedtext)related topics by pointing the mouse cursor on the link and clicking the left mousebutton.ReturningtotheoriginalreadingpositionispossiblebyclickingtherightmousebuttonandselectGo Backfromthepop-upmenu(seetheARHELPmenu).Therearealsonumerousrelatingtothe highlights(greenunderlinedtext)content within the same page.The relate to the contents in different PDF cross-file links (red underlined text)files, which open by clicking the link in the same way.Theandlinksareinviolet(darkmagenta)andare Internet World-Wide-Webaccessed by opening the default browser installed on your computer.The book was written and edited using, the Scientific Word Processor, version 5.0, (made by Simon L. Smith, see ). http://www.expswp.com/The computer algorithms developed and described in Part 6 and 7 are intendedastoolsfortheamplifierdesignprocess.WrittenforMatlab,theLanguageofTechnical Computing (The MathWorks, Inc.,), they have http://www.mathworks.com/allbeenrevisedtoconformwiththenewerversionsofMatlab(version 5.3forStudents),butstillretainingdownwardcompatibility(toversion1)asmuchaspossible. The files can be found on the CD in the Matlab folder as *.M files, alongwiththeinformationofhowtoinstallthemandusewithintheMatlabprogram.WehaveusedMatlabtocheckallthecalculationsanddrawmostofthefigures.Beforeimportingtheminto,thefigureswerefinalizedusingtheAdobeIllustrator, version8 (see). http://www.adobe.com/products/Illustrator/AllcircuitdesignswerecheckedusingMicro-CAP ,theMicrocomputerCircuit Analysis Program, v. 5 (Spectrum Software, ). http://www.spectrum-soft.com/Some of the circuits described in the book can be found on the CD in the MicroCAPfolderas*.CIRfiles,whichthereaderswithaccesstotheMicroCAPprogramcanimport and run the simmulations by themselves.- XIII -P. Stari, E. MarganWideband AmplifiersPart 1The Laplace TransformThere is nothing more practical than a good theory!(William Thompson, Lord Kelvin) P. Stari, E.Margan The Laplace Transform -1.2-About TransformsThe Laplace transform can be used as a powerful method of solvinglineardifferentialequations.Byusingatimedomainintegrationtoobtainthefrequencydomaintransferfunctionandafrequencydomainintegrationtoobtainthetimedomainresponse,wearerelievedofafewnuisancesofdifferential equations, such as defining boundary conditions, not to speak ofthe difficulties of solving high order systems of equations.Although Laplace had already used integrals of exponential functionsforthispurposeatthebeginningofthe19 century,themethodwenowthattributetohimwaseffectivelydevelopedsome100yearslaterinHeavisides operational calculus.Themethodisapplicableto a variety of physical systems (and evensomenonphysicalones,too! )involvingtrasportofenergy,storageandtransform,butwearegoingtouseitinarelativelynarrowfieldofcalculatingthetimedomainresponseofamplifierfiltersystems,startingfrom a known frequency domain transfer function.Asforanytool,thetransformtools,betheyFourier,Laplace,Hilbert,etc.,havetheirlimitations.Sincetheparametersofelectronicsystemscanvaryoverthewidestofranges,itisimportanttobeawareofthese limitations in order to use the transform tool correctly. P. Stari, E.Margan The Laplace Transform -1.3-Contents ...........................................................................................................................................1.3List of Tables ...................................................................................................................................1.4List of Figures ..................................................................................................................................1.4Contents:1.0Introduction .............................................................................................................................................1.51.2The Fourier Series .................................................................................................................................1.111.3The Fourier Integral ..............................................................................................................................1.171.4The Laplace Transform .........................................................................................................................1.231.5Examples of Direct Laplace Transform ................................................................................................. 1.251.5.1Example 1 ............................................................................................................................1.251.5.2Example 2 ............................................................................................................................1.251.5.3Example 3 ............................................................................................................................1.261.5.4Example 4 ............................................................................................................................1.261.5.5Example 5 ............................................................................................................................1.271.5.6Example 6 ............................................................................................................................1.271.5.7Example 7 ............................................................................................................................1.281.5.8Example 8 ............................................................................................................................. 1.281.5.9Example 9 ............................................................................................................................1.291.5.10Example 10 ........................................................................................................................1.291.6Important Properties of the Laplace Transform ..................................................................................... 1.311.6.1Linearity (1) .........................................................................................................................1.311.6.2Linearity (2) .........................................................................................................................1.311.6.3Real Differentiation .............................................................................................................. 1.311.6.4Real Integration .................................................................................................................... 1.321.6.5Change of Scale ...................................................................................................................1.341.6.6Impulse( ) ........................................................................................................... $> ............... 1.351.6.7Initial and Final Value Theorems ......................................................................................... 1.361.6.8Convolution .......................................................................................................................... 1.371.7Application of the transform in Network Analysis ........................................................................ _ ....1.411.7.1Inductance ............................................................................................................................1.411.7.2Capacitance ..........................................................................................................................1.411.7.3Resistance ............................................................................................................................1.421.7.4Resistor and capacitor in parallel .........................................................................................1.421.8Complex Line Integrals .........................................................................................................................1.451.8.1Example 1 ............................................................................................................................1.491.8.2Example 2 ............................................................................................................................1.491.8.3Example 3 ............................................................................................................................1.491.8.4Example 4 ............................................................................................................................1.501.8.5Example 5 ............................................................................................................................1.501.8.6Example 6 ............................................................................................................................1.501.9Contour Integrals ................................................................................................................................... 1.531.10.1Example 1 ..........................................................................................................................1.581.10.2Example 2 ..........................................................................................................................1.581.11Residues of Functions with Multiple Poles, the Laurent Series ........................................................... 1.611.11.1Example 1 ..........................................................................................................................1.631.11.2Example 2 ..........................................................................................................................1.631.12Complex Integration Around Many Poles:The CauchyGoursat Theorem ......................................................................................................1.651.13Equality of the Integrals e ande.......................................................... 1 (J= .= J= .==> =>-c4_-b4_(.671.14Application of the Inverse Laplace Transform ....................................................................................1.731.15Convolution .........................................................................................................................................1.81Rsum of Part 1 ..........................................................................................................................................1.85References ....................................................................................................................................................1.87Appendix 1.1:Simple Poles, Complex Spaces ...................................................................................(CD) A1.11.1Three Different Ways of Expressing a Sinusoidal Function .................................................................. 1.71.10Cauchys Way of Expressing Analytic Functions ...............................................................................1.55 P. Stari, E.Margan The Laplace Transform -1.4-List of Tables:Table 1.2.1: Square Wave Fourier Components ...........................................................................................1.15Table 1.5.1: Ten Laplace Transform Examples ............................................................................................ 1.30Table 1.6.1: Laplace Transform Properties ..................................................................................................1.39Table 1.8.1: Differences Between Real and Complex Line Integrals ...........................................................1.48List of Figures:Fig. 1.1.1: Sine wave in three ways................................................................................................................. 1.7Fig. 1.1.2: Amplifier overdrive harmonics ...................................................................................................... 1.9Fig. 1.1.3: Complex phasors ...........................................................................................................................1.9Fig. 1.2.1: Square wave and its phasors ........................................................................................................ 1.11Fig. 1.2.2: Square wave phasors rotating ......................................................................................................1.12Fig. 1.2.3: Waveform with and without DC component ...............................................................................1.13Fig. 1.2.4: Integration of rotating and stationary phasors .............................................................................1.14Fig. 1.2.5: Square wave signal definition ...................................................................................................... 1.14Fig. 1.2.6: Square wave frequency spectrum ................................................................................................1.14Fig. 1.2.7: Gibbs phenomenon ....................................................................................................................1.16Fig. 1.2.8: Periodic waveform example ........................................................................................................1.16Fig. 1.3.1: Square wave with extended period ..............................................................................................1.17Fig. 1.3.2: Complex spectrum of the timely spaced sqare wave ...................................................................1.17Fig. 1.3.3: Complex spectrum of the square pulse with infinite period ......................................................... 1.20Fig. 1.3.4: Periodic and aperiodic functions .................................................................................................1.21Fig. 1.4.1: The abscissa of absolute convergence .........................................................................................1.24Fig. 1.5.1: Unit step function ........................................................................................................................1.25Fig. 1.5.2: Unit step delayed .........................................................................................................................1.25Fig. 1.5.3: Exponential function ...................................................................................................................1.26Fig. 1.5.4: Sine function ...............................................................................................................................1.26Fig. 1.5.5: Cosine function ............................................................................................................................ 1.27Fig. 1.5.6: Damped oscillations ....................................................................................................................1.27Fig. 1.5.7: Linear ramp function ...................................................................................................................1.28Fig. 1.5.8: Power function ............................................................................................................................1.28Fig. 1.5.9: Composite linear and exponential function .................................................................................1.30Fig. 1.5.10: Composite power and exponential function ..............................................................................1.30Fig. 1.6.1: The Dirac impulse function .........................................................................................................1.35Fig. 1.7.1: Instantaneous voltage on, and ................................................................................. PG V ......... 1.41Fig. 1.7.2: Step response of an-network ...................................................................................... VG ..........1.43Fig. 1.8.1: Integral of a real inverting function .............................................................................................1.45Fig. 1.8.2: Integral of a complex inverting function .....................................................................................1.47Fig. 1.8.3: Different integration paths of equal result ...................................................................................1.49Fig. 1.8.4: Similar integration paths of different result .................................................................................1.49Fig. 1.8.5: Integration paths about a pole .....................................................................................................1.51Fig. 1.8.6: Integration paths near a pole .......................................................................................................1.51Fig. 1.8.7: Arbitrary integration paths ..........................................................................................................1.51Fig. 1.8.8: Integration path encircling a pole ................................................................................................1.51Fig. 1.9.1: Contour integration path around a pole .......................................................................................1.53Fig. 1.9.2: Contour integration not including a pole .....................................................................................1.53Fig. 1.10.1: Cauchys method of expressing analytical functions .................................................................1.55Fig. 1.12.1: Emmentaler cheese ....................................................................................................................1.65Fig. 1.12.2: Integration path encircling many poles ...................................................................................... 1.65Fig. 1.13.1: Complex line integration of a complex function .......................................................................1.67Fig. 1.13.2: Integration path of Fig.1.13.1 ....................................................................................................1.67Fig. 1.13.3: Integral area is smaller than ................................................................................... QP ............ 1.67Fig. 1.13.4: Cartesian and polar representation of complex numbers ...........................................................1.68Fig. 1.13.5: Integration path for proving of the Laplace transform ............................................................... 1.69Fig. 1.13.6: Integration path for proving of input functions .......................................................................... 1.71Fig. 1.14.1: circuit driven by a current step ............................................................................... VPG ........1.73Fig. 1.14.2: circuit transfer function magnitude ............................................................................ VPG ....1.75Fig. 1.14.3: circuit in time domain ......................................................................................... VPG ...........1.79Fig. 1.15.1: Convolution of two functions ....................................................................................................1.82Fig. 1.15.2: System response calculus in time and frequency domain ..........................................................1.83 P. Stari, E.Margan The Laplace Transform -1.5-1.0IntroductionWith the advent of television and radar during the Second World War, the behaviorof wideband amplifiers in the time domain has become very important [Ref. 1.1]. In todaysdigital world this is even more the case. It is a paradox that designers and troubleshooters ofdigital equipment still depend on oscilloscopes, which at least in their fast and low levelinput part consist of wideband amplifiers. So the calculation of the time domain analogresponseofwidebandamplifiershasbecomeevenmoreimportantthanthefrequency,phase, and time delay response.The emphasis of this book is on the amplifiers time domain response. Therefore athoroughknowledgeoftimerelatedcalculus,explainedinPart 1,isanecessarypre-requisiteforunderstandingallotherpartsofthisbookwherewidebandamplifiernetworks are discussed.The time domain response of an amplifier can be calculated by two main methods:Thefirstone is based on and the second uses thedifferential equations inverse Laplacetransform( transform) _c".Thedifferentialequationmethodrequiresthe calculation ofboundary conditions, which in case of high order equations means an unpleasant andtimeconsumingjob.Anothermethod,whichalsousesdifferentialequations,isthesocalledcalculation,inwhichadifferentialequationoforder is split intostatevariable 8 8differentialequationsofthefirstorder,inordertosimplifythecalculations.Thestatevariable method also allows the calculation of non linear differential equations. We will useneither of these, for the simple reason that the Laplace transform and its inverse are basedonthesystempolesandzeros,whichprovesousefulfornetworkcalculationsinthefrequencydomaininthelaterpartsofthebook.Somostofthedatawhicharecalculatedthere is used further in the time domain analysis, thus saving a great deal of work. Also theuse of the transform does not require the calculation of boundary conditions, giving the _c"result directly in the time domain.In using the transform most engineers depend on tables. Their method consists _c"firstly of splitting the amplifier transfer function into partial fractions and then looking forthecorrespondingtimedomainfunctionsinthetransformtables.Thesumofallthese _functions (as derived from partial fractions) is then the result. The difficulty arises when nocorrespondingfunctioncanbefoundinthetables,orevenatanearlierstage,ifthemathematical knowledge available is insufficient to transform the partial fractions into sucha form as to correspond to the formulae in the tables.In our opinion an amplifier designer should be self-sufficient in calculating the timedomainresponseofawidebandamplifier.Fortunately,thiscanbealmostalwaysderivedfrom simple rational functions and it is relatively easy to learn the transforms for such _c"cases. In Part 1 we show how this is done generally, as well as for a few simple examples.A great deal of effort has been spent on illustrating the less clear relationships by relevantfigures.Sinceengineersseektoobtainafirstglanceinsightoftheirsubjectofstudy,webelieve this approach will be helpful.This part consists of four main sections. In the first, the concept of harmonic (e.g.,sinusoidal) functions, expressed by pairs of counter-rotating complex conjugate phasors, isexplained.ThentheFourierseriesofperiodicwaveformsarediscussedtoobtainthediscretespectraofperiodicwaveforms.ThisisfollowedbytheFourierintegraltoobtaincontinuousspectraofnon-repetitivewaveforms.TheconvergenceproblemoftheFourier P. Stari, E.Margan The Laplace Transform -1.6-integral is solved by introducing the complex frequency variable, thus coming = b4 5 =to direct Laplace transform (transform). _Thesecondsectionshowssomeexamplesofthetransforms.Theresultsare _useful when we seek the inverse transforms of simple functions.Thethirdsectiondealswiththe theory of functions of complex variables, but onlytotheextent that is needed for understanding the inverse Laplace transform. Here the lineand contour integrals (Cauchy integrals), the theory of residues, the Laurent series and the_ _c" c"transformofrationalfunctionsarediscussed.Theexistenceofthetransformforrational functions is proved by means of the Cauchy integral.Finally,theconcludingsectiondealswithsomeaspectsofthetransformsand _c"theconvolutionintegral.Onlytwostandardproblemsofthetransformareshown, _c"because all the transient response calculations (by means of the contour integration and thetheory of residues) of amplifier networks, presented in Parts 25, give enough examples andhelp to acquire the necessary know-how.ItisprobablyimpossibletodiscussLaplacetransforminamannerwhichwouldsatisfybothengineersandmathematicians.Professorsaid: IvanVidav Ifwemathematiciansaresatisfied,youengineerswouldnotbe,andviceversa.Herewehavetriedtoachievethebestpossiblecompromise:tosatisfyelectronicsengineersandatthesametimenottooffendthemathematicians.But,aslatecolleague,thephysicistMarkoKogoj Engineersneverknowenoughofmathematics;onlymathematicians ,usedtosay:know their science to the extent which is satisfactory for an engineer, but they hardly everknowwhattodowithit! Thussuccessfulengineerskeepimprovingtheirgeneralknowledge of mathematics far beyond the text presented here.Afterstudyingthispartthereaderswillhaveenoughknowledgetounderstandallthetimedomaincalculationsinthesubsequentpartsofthebook.Inaddition,thereaderswill acquire the basic knowledge needed to do the time-domain calculations by themselvesandsobecomeindependentoftransformtables.Ofcourse,inordertosavetime,they _willundoubtedlystillusethetablesoccasionally,orevenmaketablesoftheirown.Butthey will be using them with much more understanding and self-confidence, in comparisonwith those who can do transform only via the partial fraction expansion and the tables _c"of basic functions.ThosereaderswhohavealreadymasteredtheLaplacetransform, anditsinversecanskipthispartuptoSec. 1.14,wherethe transform of a two pole network is dealt _c"with. From there on we discuss the basic examples, which we use later in many parts of thebook;thecontentofSec. 1.14shouldbeunderstoodthoroughly.However,ifthereadernotices any substantial gaps in his/her knowledge, it is better to start at the beginning.Inthelasttwopartsofthisbook,Part 6and7,wederiveasetofcomputeralgorithms which reduce the circuits time domain analysis, performance plotting and polelayoutoptimizationtoapureroutine.Howeverattractivethismayseem,weneverthelessrecommend the study of Part 1: a good engineer must understand the tools he/she is using inorder to use them effectively. P. Stari, E.Margan The Laplace Transform -1.7-1.1Three Different Ways of Expressing a Sinusoidal FunctionWewillfirstshowhowasinusoidalfunctioncanbeexpressedinthreedifferentways.Themostcommonwayistoexpresstheinstantaneousvalueofasinusoidof +amplitudeandangularfrequency,( frequency)bythewellknown E # 0 0 = 1" " "formula:+ 0> E > sin ="(1.1.1)Thereasonthatwehaveappendedtheindex towillbecomeapparentvery " =soonwhenwewilldiscusscomplexsignalscontainingdifferentfrequencycomponents.Theamplitude vs. timerelationofthisfunctionisshowninFig. 1.1.1a.Thisisthemostfamiliar display seen by using any sine-wave oscillator and an oscilloscope. 3a) b)c) d)yyxaAt 021= = 011111= a A = sin t 11=/t A2/ A24224 Fig. 1.1.1:a) b) Three different presentations of a sine wave:amplitude in time domain; a phasorof length, rotating with angular frequency; two complex conjugate phasors of length, E E# ="c)rotating in opposite directions with angular frequency, at; the same as c), except at = =" "> ! d)= 1"> %.In electrical engineering, another presentation of a sinusoidal function is often used,coming from the vertical axis projection of a rotating phasor, as displayed in Fig. 1.1.1b, Efor which the same Eq. 1.1.1 is valid. Here both axes are real, but one of the axes may alsobe imaginary. In this case the corresponding mathematical presentation is:0> E Ese (1.1.2)4 > ="whereisacomplexquantityande isthebasisofnaturallogarithms. E #(") #)"sHowever,wecanalsoobtaintherealquantitybyexpressingthesinusoidalfunctionby +twocomplexconjugatephasorsoflengthwhichrotateinoppositedirections,as E# P. Stari, E.Margan The Laplace Transform -1.8-displayed in a three-dimensional presentation in Fig. 1.1.1c. Here both phasors are shown at= = 1 1 > ! > # % + (or,, ).Thesumofbothphasorshastheinstantaneousvalue,whichis.Thisisensuredbecausebothphasorsrotatewiththesameangular alwaysrealfrequencyand,startingasshowninFig. 1.1.1c,andthereforetheyarealways b c = =" "complex conjugate at any instant. We express by the well-known formula: + Euler+ 0> E > cE#4sin ="4 > c4 > e e (1.1.3) = =" "The in the denominator means that both phasors are imaginary at. The sum of both 4 > !rotating phasors is then zero, because:0! c !E E#4 #4e e (1.1.4)4 ! c4 ! = =" "BothphasorsinFig. 1.1.1cand1.1.1dareplacedonthefrequencyaxisatsuchadistancefromtheoriginastocorrespondtothefrequency.Sincethephasorsrotate ="with time the Fig. 1.1.1d, which shows them at, helps us to acquire the idea : = 1 > %1ofathree-dimensionalpresentation.Theunderstandingofthesesimpletime-frequencyrelations,presentedinFig. 1.1.1cand1.1.1dandexpressedbyEq. 1.1.3,isessentialforunderstanding both the Fourier transform and the Laplace transform.Eq. 1.1.3 can be changed to the function if the phasor with is multiplied cosine b="bye andthephasorwithbye .Thefirstmultiplicationmeansa 4 c c4 4 # c4 #"1 1=counter clockwise clockwise -rotation by and the second a rotation by. This causes *! *!both phasors to become real at time, their sum again equaling: > ! E0> b E >E E#4 #4e e (1.1.5)4 > c4 >"= =" "cos =In general a sinusoidal function with a non-zero phase angle at is expressed as: : > !E > b cE#4sin = : e e (1.1.6) 4 >b c4 >b = : = :TheneedtointroducethefrequencyaxisinFig. 1.1.1cand1.1.1dwillbecomeapparent in the experiment shown in Fig. 1.1.2. Here we have a unity gain amplifier with apoorloopgain,drivenbyasinewavesourcewithfrequencyandamplitude,and =" "Eloadedbytheresistor.Iftheresistorsvalueistoolowandtheamplitudeof the input VLsignal is high the amplifier reaches its maximum output current level, and the output signal0> E becomesdistorted(wehavepurposelykeptthesamenotationasinthepreviousfigure, rather than introducing the sign for voltage). The distorted output signal contains Znotjusttheoriginalsignalwiththesamefundamentalfrequency,butalsoathird ="harmonic component with the amplitude and frequncy: E E $$ " $ "= =0> E > bE $ > E > bE >" " $ " " " $ $sin sin sin sin = = = = (1.1.7) P. Stari, E.Margan The Laplace Transform -1.9-Vt1i334V1V3ViVoVoViAi= sin =tt1V1A1= sin t1V3A3= sin t3VoV1V3= +=1RLAiA1A32t=1Fig. 1.1.2: The amplifier is slightly overdriven by a pure sinusoidal signal,, with a frequencyZi="andamplitude.Theoutputsignalisdistorted,anditcanberepresentedasasumoftwo E Zi osignals,.Thefundamentalfrequencyofisanditsamplitudeissomewhat lower. Z bZ Z E" $ " " "=The frequency of (the third harmonic component) is and its amplitude is. Z $ E$ $ " $= =Now let us draw the output signal in the same way as we did in Fig. 1.1.1c,d. Herewe have two pairs of harmonic components: the first pair of phasors rotating with the E#"fundamentalfrequency,andthesecondpairrotatingwiththethirdharmonic E# =" $frequency, which are three times more distant from the origin than. This is shown = =$ "in Fig. 1.1.3a, where all four phasors are drawn at time. Fig. 1.1.3b shows the phasors > !attime.Becausethethirdharmonicphasorpairrotateswithanangular > % 1 =frequency three times higher, they rotate up to an angle in the same time. $ % 1a)b)A12A12A32A32 = 0311 311 1 311 31 =/4 3 3Fig. 1.1.3: The output signal of the amplifier in Fig. 1.1.2, expressed by two pairs of complexconjugate phasors: at; at. a) b) = = 1" "> ! > %Mathematically Eq. 1.1.7, according to Fig. 1.1.2 and 1.1.3, can be expressed as: E > bE > 0>" " $ $sin sin = =e e e e (1.1.8) c b cE E#4 #4" $4 > c4 > 4 > c4 > = = = =" " $ $ P. Stari, E.Margan The Laplace Transform -1.10-Theamplifieroutputobviouslycannotexceedeitheritssupplyvoltageoritsmaximumoutputcurrent.Soifwekeepincreasingtheinputamplitudetheamplifierwillcliptheupperandlowerpeaksoftheoutputwaveform(someinputprotection,aswellassome internal signal source resistance must be assumed if we want the amplifier to surviveintheseconditions),thusgeneratingmoreharmonics.Iftheinputamplitudeisveryhighandiftheamplifierloopgainishighaswell,theoutputvoltagewouldeventually 0>approachasquarewaveshape,suchasinFig. 1.2.1binthefollowingsection.Atruemathematicalsquarewavehasaninfinitenumber of harmonics; since no amplifier has aninfinite bandwidth, the number of harmonics in the output voltage of any practical amplifierwill always be finite.In the next section we are going to examine a generalized harmonic analysis. P. Stari, E.Margan The Laplace Transform -1.11-1.2The Fourier SeriesIn the experiment shown in Fig. 1.1.2 we have the sinusoidal waveforms composedwiththeamplitudesandtogettheoutputtime-function.Now,ifwehavea E E 0>" $squarewave,asinFig. 1.2.1b,wewouldhavetodealwithmanymorediscretefrequencycomponents.Weintendtocalculatetheamplitudesofthem,assumingthatthetimefunctionofthesquare waveisknown.Thismeansaofthetimefunction decomposition0> into the corresponding harmonic frequency components. To do so we will examine theFourier Jean Baptiste Joseph de Fourier series, following the French mathematician.1Thesquarewavetime function is periodic. A function is periodic if it acquires thesame value after its characteristic period, at any instant: X # >" "1=0> 0> bX"(1.2.1)Consequently the same is true for, where is an integer. According to 0> 0> b8X 8"Fourierthissquarewavecanbeexpressedasasumofharmoniccomponentswithfrequencies.Ifwehavethefundamentalfrequencywithaphasor 0 8X 8 " 08 " "E# 0 E# E #" c" " c",rotatingcounter-clockwise.Thephasorwiththesamelength rotates clockwise and forms a complex conjugate pair with the first one. A true square wavewould have an infinite number of odd-order harmonics (all even order harmonics are zero).t = 0 A12A32A72A521 31 51 711 31 51 71A12A32A52A7222a) b)f t ( ) =1,1,0 ! # 8 8 1 =", where is an integer representing the number of the period.1It is interesting that Fourier developed this method in connection with thermal engineering. As a general inthe Napoleon's army he was concerned with gun deformation by heat. He supposed that one side of a straightmetalbarisheatedandthenbent,joiningtheends,toformathorus.Thenhecalculatedthetemperaturedistributionalongthecirclesoformed,insuchawaythatitwould be the sum of sinusoidal functions, eachhaving a different amplitude and a different angular frequency. P. Stari, E.Margan The Laplace Transform -1.12-InFig. 1.2.1,wehavedrawnthecomplex-conjugatephasorpairsofthefirst4harmonics. Because all the phasor pairs are always complex-conjugate, the sum of any pair,aswellastheirtotalsum,isalwaysreal.Thephasorsrotatewithdifferentspeedsandinoppositedirections.Fig. 1.2.2ashowsthemattimetohelpthereadersimagination. X)"Althoughthisfigurelooksconfusing,thephasorsshownhaveanexactinter-relationship.Lookingatthepositiveaxis,thephasorwiththeamplitudehasrotatedinthe = E#"counter-clockwisedirectionbyanangleof.Duringthesameintervalofthe 1% X)"remainingphasorshaverotated:by;by;by;etc.The E# $ % E# & % E# ( %$ & (1 1 1corresponding complex conjugate phasors on the negative axis rotate likewise, but in the =opposite (clockwise) direction. The sum of all phasors at any instant is the instantaneous >amplitude of the time domain function. In general, the time function with the fundamentalfrequency is expressed as: ="e 0>E#"_8c_84 8 > ="e e e b bb bE E E# # #c8 c# c"c4 8 > c4 # > c4 > = = =" " "e e e (1.2.2) bEb b bb bE E E# # #!" # 84 > 4 # > 4 8 > = = =" " "t = = A121 31 51 711 31 51 714 b) a)tT1111 =2T101 1 /f t ( ) =1,1,0 % # 8 1 1 ="expressed by complex conjugate phasor pairs, corresponds to the instant in. > :="b)Notethatforthesquarewavealltheevenfrequencycomponentsaremissing.Forother types of waveforms the even coefficients can be non-zero. In general may also be Eicomplex,thuscontainingsomenon-zeroinitialphaseangle.InEq. 1.2.2wehavealso :iintroduced, the DC component, which did not exist in our special case. The meaning of E!E! can be understood by examining Fig. 1.2.3a, where the so-called sawtooth waveform isshown,withnoDCcomponent.InFig. 1.2.3b,thewaveformhasaDCcomponentofmagnitude. E! P. Stari, E.Margan The Laplace Transform -1.13-Eq. 1.2.2representstheofthefunction,whileFig. 1.2.1 complexspectrum 0>representsthecorrespondingmostsignificantpartofthecomplexspectrumofasquarewave. The next step is the calculation of the magnitudes of the rotating phasors.a) b)( )= 00f t ( ) f tA0At tFig. 1.2.3: a)b) A waveform without a DC component; with a DC component. E!Ifwewanttomeasuresafelyandaccuratelythediameterofawheelofaworkingmachine, we must first stop the machine. Something similar can be done with our Eq. 1.2.2,except that here we can mathematically stop the rotation of any single phasor. Suppose wehaveaphasor,rotatingcounter-clockwisewithfrequencyk withaninitial E# k k= ="phase angle (at), which is expressed as: :k> ! e ee (1.2.3)E E# #k k4 > b 4 > 4 = : = :k k k kNowwemultiplythisexpressionbyaunitamplitude,clockwiserotatingphasorec4 =k(having the same angular frequency) to cancel the eterm, [Ref. 1.2]: =k4 =k eee e (1.2.4)E E# #k k4 4 > c4 > 4 : = = :k k k kand obtain a non-rotating component which has the magnitude and phase angleE#k k: atany time. With this in mind let us attack the whole time function. The duration of the 0>multiplication must last exactly one whole period and the corresponding expression is: e (1.2.5)E "# X 0> .>k(X#cX#c4 > =kSince we have integrated over the whole period in order to get the average value of that Xharmonic component, the result of the integration must be divided by, as in Eq. 1.2.5. If Xthere is a DC component (with) in the spectrum, the calculation of it is simply: = !E 0> .>"X!X#cX# (1.2.6)(ToreturntoEq. 1.2.5,letusexplainthemeaningoftheintegrationEq. 1.2.5bymeans of Fig. 1.2.4. P. Stari, E.Margan The Laplace Transform -1.14-Bymultiplyingthefunctionbye wehavestoppedtherotatingphasor 0>c4 > =kE#k,whileduringthetimeintervalofintegrationalltheotherphasorshaverotatedthrough an angle of (where is an integer), the DC phasor, because it is 8# 8 E 1 including!now multiplied by e . The result of the integration for all these rotating phasors is zero,c4 > =kas indicated in Fig. 1.2.4a, while the phasor has stopped, integrating eventually to its E#kfull amplitude; the integration for this phasor only is shown in Fig. 1.2.4b.Theunderstandingoftheeffectdescribedofthemultiplicatione is 0>c4 > =essentialtounderstandingthebasicprinciplesoftheFourierseries,theFourierintegral and the Laplace transform.a) b)2kAdk2kAd2kA ikd = ik=Fig. 1.2.4: a) b) The integral over the full periodof a rotating phasor is zero; the integral Xover a full periodof a non-rotating phasor, gives its amplitude,,the symbol X . E# E# k k. .>X .> p > > % standsforinthisfigures such that. Note that a stationary J J= 1kphasor retains its initial angle. :kFor us the Fourier series represents only a transitional station on the journey towardstheLaplacetransform.SowewilldrivethroughitwithamoderatespeedviatheMainStreet, without investigating some interesting things in the side streets. Nevertheless, it isusefultomakeapracticalexample.Sincewehavestartedwithasquarewave,showninFig. 1.2.5,letuscalculateitscomplexspectrumcomponents,assumingthatthe E#8square wave amplitude is. E "1 31 51 71A1A3A5A7tT111=2T104A9 A11 A13 91 111 131AkA1=kk1k =f t ( ) =1,1,0 c" cX# > !b" ! > X# P. Stari, E.Margan The Laplace Transform -1.15-According to Eq. 1.2.5 we calculate:e e c" .> b b" .>E "# X8!cX#c4 # 8>X c4 # 8>XX#! ( (1 1e e b" X XX 4 # 8 c4 # 8: ; 1 1c4 # 8>X c4 # 8>X! X#cX# !1 1e e e e " c c b" c"" c" b4 # 8 4 8 # 1 1 94 8 c4 84 8 c4 81 11 1(1.2.7) 8 c"c"4 8 11 cosTheresultiszerofor(theDCcomponent)andforanyeven.Forany 8 ! E 8!odd the value of, and for such cases the result is: 8 8 c" cos 1 (1.2.8)E # c#4# 4 8 8 81 1The factor in the numerator means that for any positive (and for,,, c4 8 > ! # % 1 1' 8 1, )thephasorisnegativeandimaginary,whilstfornegativeitispositiveandimaginary. This is evident from Fig. 1.2.1a.Let us calculate the first eight phasors by using Eq. 1.2.8. The lengths of phasors inFig. 1.2.1aand1.2.2.bcorrespondtothevaluesreportedinTable 1.2.1.Allthephasorsform complex conjugate pairs and their total sum. always gives a real valueTable 1.2.1: The first few harmonics of a square wave8 ! " $ & ( * "" "$E# ! #4 #4$ #4& #4( #4* #4"" #4"$81 1 1 1 1 1 1However, a spectrum can also be shown with real values only, e.g., as it appears onthecathoderaytubescreenofaspectrumanalyzer.Toobtainthis,wesimplysumthecorrespondingcomplexconjugatephasorpairs(e.g.,)andplace lE#l blE #l E8 c8 8them on the abscissa of a two-dimensional coordinate system, as shown in Fig. 1.2.6. Sucha non-rotating spectrum has only the positive frequency axis. Although such a presentationofspectraisveryusefulintheanalysisofsignalscontainingseveral(ormany)frequencycomponents,wewillcontinuecalculatingwiththecomplexspectra,becausethephaseinformationisalsoimportant.And,ofcourse,theLaplacetransform,whichisourmaingoal, is based on a complex variable.Nowletusrecomposethewaveformusingonlytheharmonicfrequencycomponents from Table 1.2.1, as shown in Fig. 1.2.7a. The waveform resembles the squarewave but it has an exaggerated overshoot18 % of the nominal amplitude. $The reason for the overshoot is that we have abruptly cut off the higher harmonic $componentsfromacertainfrequencyupwards.Wouldthisovershootbelowerifwetake P. Stari, E.Margan The Laplace Transform -1.16-moreharmonics?InFig. 1.2.7bwehaveincreasedthenumberofharmoniccomponentsthree times, but the overshoot remained the same. No matter how many, yet for any finitenumberofharmoniccomponents,usedtorecomposethewaveform,theovershootwouldstay the same (only its duration becomes shorter if the number of harmonic components isincreased, as is evident from Fig. 1.2.7a and 1.2.7b ).This is the phenomenon. It tells us that we should not cut off the frequency Gibbsresponse of an amplifier abruptly if we do not wish to add an undesirably high overshoot totheamplifiedpulse.Fortunately,realamplifierscannothaveaninfinitelysteephighfrequencyrolloff,soagradualdecayofhighfrequencyresponseisalwaysensured.However, as we will explain in Part 2 and 4, the overshoot may increase as a result of othereffects.0110.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2a)7 harmonicstTb)21 harmonics 0.2 tTFig. 1.2.7: a) TheGibbsphenomenon;Asignalcomposedofthefirstsevenharmonicsofasquare wave spectrum from Table 1.2.1. The overshoot is18 % of the nominal amplitude; $b) Even if we take three times more harmonics the overshoot is nearly equal in both cases. $Inasimilarwaytothatforthesquarewave,anyperiodicsignaloffiniteamplitudeandwithafinitenumberofdiscontinuitieswithinoneperiod,canbedecomposedintoitsfrequencycomponents.AsanexamplethewaveforminFig. 1.2.8couldalsobedecomposed,butwewillnotdoithere. Instead in the following section wewillanalyze another waveform which will allow us to generalize the method of frequencyanalysis.0 20 40 100 60 80[ s] 05101520()[V]fttFig. 1.2.8: An example of a periodic waveform (a typical flyback switching power supply), havinga finite number of discontinuities within one period. Its frequency spectrum can also be calculatedusingtheFouriertransform,ifneeded(e.g.,toanalyzethepossibilityofelectromagneticinterference at various frequencies), in the same way as we did for the square wave. P. Stari, E.Margan The Laplace Transform -1.17-1.3The Fourier IntegralSuppose we have a function composed of square waves with the duration and 0> 7repeating with a period, as shown in Fig. 1.3.1. For this function we can also calculate the XFourierseries(thecorrespondingspectrumisshowninFig. 1.3.2)inthesamewayasforthe continuous square wave case in the previous section.f tTt101( )The difference between the continuous square wave spectrum and the spaced squarewaveinFig. 1.3.1isthattheintegralofthisfunctioncanbebrokenintotwoparts,onecomprisingthelengthofthepulse,,andthezero-valuedpartbetweentwopulsesofa 7length. The reader can do this integration for himself, because it is fairly simple. We Xc7will only write the result: (1.3.1)E 8 %# 8 % c48 "#"7= 7= 7sin c dwhere,assumingthatthepulseamplitudeis(iftheamplitudewereit = 1" # X " Ewould simply multiply the right hand side of the equation). For the conditions in Fig. 1.3.1,where and, the spectrum has the form shown in Fig. 1.3.2, with. X & E " # 7 = 17740 211=24 2 T=2Fig. 1.3.2: Complex spectrum of the waveform in Fig. 1.3.1.Fig. 1.3.1: A square wave with duration 7 7 and period X & . P. Stari, E.Margan The Laplace Transform -1.18-Averyinterestingquestionisthatofwhatwouldhappentothespectrumifweletthe period? In general a function can be recomposed by adding all its harmonic X p_ 0>components:0> E#"_8c_84 8 > e (1.3.2)="where may also be complex, thus containing the initial phase angle. Again, as in the E8:iprevious section, each discrete harmonic component can be calculated with the integral: e (1.3.3)E "# X 0> .>8X#cX#c4 8 >(="For the case in Fig. 1.3.1 the integration should start at and the integral has the form: > ! e (1.3.4)E "# X 0> .>8X!c4 8 >(="Insert this into Eq. 1.3.2:0> 0 ."X" (_8c_X!c4 8 4 8 > e e (1.3.5) 7 7=7 =" "Here we have introduced a dummy variable in the integral, in order to distinguish it from 7the variable outside the brackets. Now we express the integral inside the brackets as: >e e (1.3.6)( ( ! !X Xc4 8 c4 # 8 X"0 . 0 . J J8# 8X7 7 7 7 =1=7 1 7" Thus:e e J8 J8 0>" " #X # X" "_ _8c_ 8c_" "4 8 > 4 >= =11= =" "e (1.3.7) J8 "# 1= ="_8c_" "4 > ="where. If we let then becomes infinitesimal, and we call it. Also # X X p_ . 1 = = =" "8= =" becomes a continuous variable. So in Eq. 1.3.7 the following changes take place:
"(_8c__c_" " . 8 = = = =With all these changes Eq. 1.3.7 is transformed into Eq. 1.3.8:0> J ."#e (1.3.8)1= =(_c_4 > = P. Stari, E.Margan The Laplace Transform -1.19-Consequently Eq. 1.3.6 also changes, obtaining the form:J 0> .> =(_!c4 >e (1.3.9)=In Eq. 1.3.9 has no discrete frequency components but it forms aJ = continuousspectrum.SincetheDCpartvanishes(asitwouldforpulseshape,notjust X p_ anysymmetrical shapes), according to Eq. 1.2.6:E 0> .> !"X!Xp_X!lim (1.3.10)(Eq. 1.3.8and1.3.9arecalled.Undercertain(usuallyrather Fourierintegralslimited)conditions,whichwewilldiscusslater,itispossibletousethemforthecalculationoftransientphenomena.Thesecondintegral( Eq. 1.3.9 )iscalledthedirectFourier transform, which we express in a shorter way:Y = 0> J (1.3.11)Thefirstintegral(Eq. 1.3.8)representstheanditis inverseFouriertransformusually written as:Y =c" J 0> (1.3.12)In Eq. 1.3.9, means a spectrum and the factor emeans the rotation of J = firm4 > =each of the corresponding infinite spectrum components contained in with its angular J =frequency,whichisacontinuousvariable.InEq. 1.3.8meansthecompletetime = 0>function, containing an infinite number of phasors and the factor emeans the rotatingc4 > =rotation in the opposite direction to stop the rotation of the corresponding rotating phasorecontained in, at its particular frequency.4 > =0> =Letusnowselectasuitabletimefunctionandcalculateitscontinuous 0>spectrum.Sincewehavealreadycalculatedthespectrumofaperiodicsquarewave,itwouldbeinterestingtodisplaythespectrumofasinglesquarewaveasshowninFig. 1.3.3b. We use Eq. 1.3.9: e e e (1.3.13) J 0> .> c" .> b b" .> =( ( (_ !c# c#c4 > c4 > c4 >#! 7 7= = =7Herewehaveasinglesquarewavewithaperiodfromto. X > c# b_ 7However,weneedtointegrateonlyfromto,becauseiszero > c# > # 0> 7 7outsidethisinterval.Itisimportanttonotethatatthediscontinuity where, we have > !started the second integral. For a function with more discontinuities, between each of themwemustwriteaseparateintegral.Thusitisobviousthatthefunctionmusthavea 0>finite number of discontinuities for it to be possible to calculate its spectrum. P. Stari, E.Margan The Laplace Transform -1.20-The result of the above integration is:e e e e c" b b c" " c J " # bc4 4 #== = : ;4 # c4 #4 # c4 #=7 =7=7 =7 " c # c# 4 c# 4 c% 4# % % = = ==7 =7 =7 cos sin sin# #(1.3.14) c4%%7=7=7sin#A three-dimensional display of a spectrum, corresponding to this result, is shown inFig. 1.3.3a. Here the frequency scale has been altered with respect to Fig. 1.2.1a in order todisplay the spectrum better.40 2=24 2 T= 066= 22011tb)a)Fig. 1.3.3:a)Thefrequencyspectrumofasinglesquarewaveisexpressedbycomplexconjugate phasors. Since the phasors are infinitely many, they merge in a continuous planarform. Also the spectrum extends to. The corresponding waveform is shown in. = _ b)Note that all the even frequency components are missing (is an integer). # 8 8 1 7By comparing Fig. 1.2.1a and 1.3.3a we may draw the following conclusions:1.Bothspectracontainnoevenfrequencycomponents,e.g.,at,, # % = =7 7etc., where; = 177 # 2. In both spectra there is no DC component; E!3. By comparing Fig. 1.3.2 and 1.3.3 we note that the envelope of both spectra canbe expressed by Eq. 1.3.14;4. By comparing Eq. 1.3.1 and 1.3.14 we note that the discrete frequency from 8="the first equation is replaced by the continuous variable in the second equation. =Everything else has remained the same. P. Stari, E.Margan The Laplace Transform -1.21-Intheaboveexamplewehavedecomposedanaperiodicwaveform(alsocalledatransient),expressedas,intoaspectrum.Before 0> J continuous complex =discussing the functions which are suitable for the application of the Fourier integral let usseesomecommonperiodicandnon-periodicsignals.Asustained tone from a trumpet weconsider to be a periodic signal, whilst a beat on a drum is a non-periodic signal (in a strictmathematical sense, both signals are non-periodic, because the first one also started out ofsilence). The transition from silence to sound we call the. In accordance with this transientdefinition, of the waveforms in Fig. 1.3.4 only a) and b) show a periodic waveform, whilstc) and d) display transients.a)b)c)d)( ) f t ( ) f t( ) f t( ) f tt tt tT kThe question arises of whether it is possible to calculate the spectra of the transientsin Fig. 1.3.4c and 1.3.4d by means of the Fourier integral using Eq. 1.3.8?Theansweris,becausetheintegralinEq. 1.3.8doesnotconvergeforanyof nothesetwofunctions.Theintegralisalsonon-convergentforthemostsimplestepsignal,whichweintendtouseextensivelyforthecalculationofthestepresponseofamplifiernetworks.Thisinconveniencecanbeavoidedifwemultiplythefunctionbyasuitable 0>convergence factor, e.g., e , where and its magnitude is selected so that the integralc->- !in Eq. 1.3.2 remais finite when. In this way, the problem is solved for. In doing > p_ >!so, however, the integral becomes divergent for, because for negative time the factor > !e hasapositiveexponent,causingarapidincreasetoinfinity.Butthis,too,canbec->avoided, if we assume that the function is zero for. In electrical engineering and 0> > !electronics we can always assume that a circuit is dead until we switch the power on or weapplyastepvoltagesignaltoitsinputandthusgenerateatransient.Thetransformwhere0> > ! must be zero for is called a. unilateral transformForfunctionswhicharesuitablefortheunilateralFouriertransformthefollowingrelation must hold [Ref. 1.3]:limXpX!c->_e (1.3.15)(l0>l .> _where is a single-valued function of and is positive and real. 0> > -Fig. 1.3.4: a) and b) periodic functions, c) and d) aperiodic functions. P. Stari, E.Margan The Laplace Transform -1.22-If so, we can write the direct transform:J- 0> .> ,e e (1.3.16) =( < _!c-> c4 > =If we want this integral to converge to some finite value for, the real constant > p_mustbe,whereisthe.Themagnitudeof- 5 5 5a a aabscissaofabsoluteconvergencedepends on the nature of the function. I.e., if, then, and ife 0> 0> " ! 0> 5ac> !then,where.By applying the convergence factor e , the inverse Fourier 5 ! !a c !c->transform obtains the form:0> J- . >!"#e,e for(1.3.17) c-> 4 >_c_1= =(=Herewemustaddallthecomplex-conjugatephasorswithfrequenciesfrom= c_ b_ to.AlthoughthedirectFouriertransforminourcasewasunilateral,theinversetransformisalwaysbilateral.BecauseinEq. 1.3.16wehavedeliberatelyintroducedtheconvergencefactore wemustlimitaftertheintegralissolvedinc->- p!order to get the required. J =SinceourfinalgoalistheLaplacetransformwewillstopthediscussionoftheFourier transform here. We will, however, return to this topic later in Part 6, where we willdiscussthesolvingofsystemtransferfunctionsandtransientresponsesusingnumericalmethods,suitableformachinecomputation.TherewewilldiscusstheapplicationoftheveryefficientFastFourierTransform(FFT)algorithmtoboth frequency and time domainrelated problems. P. Stari, E.Margan The Laplace Transform -1.23-1.4The Laplace TransformByaslightchangeofEq. 1.3.16and1.3.17wemayarriveatageneralcomplexFouriertransform[Ref. 1.3].Thisisdonesothatwejointhekernele andthec4 > =convergence factor e . In this way Eq. 1.3.16 is transformed into:c->J- b4 0> .> - = 5( < _!c- b4 >e where(1.4.1)=aThe formula for an inverse transform is derived from Eq. 1.3.17 if both sides of theequation are multiplied by e . In addition, the simple variable is now replaced by a new->=one:. By doing so we obtain: - b4 = eforand (1.4.2) J- b4 .- b4 >! - 0>"# 4 1= = 5(-b4_-c4_- b4 > =aIf in Eq. 1.4.1 and 1.4.2 the becomes a, both equations are constant real variable - 5transformedintotheformcalled.Thenameisfullyjustified,sincethe LaplacetransformFrenchmathematicianhadalreadyintroducedthistransformin PierreSimondeLaplace1779, whilst Fourier published his transform 43 years later.Itisacustomtodenotethecomplexvariablebyasinglesymbol,which 5 = b4 =we also call the (in some, mostly mathematical, literature this variable is complex frequencyalso denoted as). With this new variable Eq. 1.4.1 can be rewritten: :J= 0> 0> .> _e f(_!c=>e (1.4.3)andthisiscalledthe,or.Itrepresentsthecomplex directLaplacetransform transform _spectrum.Theaboveintegralisvalidforfunctionssuchthatthefactore J= 0>c=>keeps the integral convergent. If we now insert the variable in Eq. 1.4.2, we have: =0> J= J= .="# 4_1c" =>-c4_-b4_e f( e (1.4.4)This integral is called the, or. inverse Laplace transform transform _c"LiketheinverseFouriertransform,Eq. 1.4.4isatoo.Inthe bilateraltransformintegralEq. 1.4.3itisassumedthatfor,thusthatequationrepresentsthe 0> ! > !unilateraltransform.Inaddition,therealpartofthevariablesatisfies, = = de f 5 5awhereisthe,aswehavealreadydiscussedfor 5aabscissaofabsoluteconvergenceEq. 1.3.16 and 1.3.17[seealsoRef. 1.23].IntheintegralEq. 1.4.4,soheretoowe >!must have. 5 5 a P. Stari, E.Margan The Laplace Transform -1.24-Thepathofintegrationisparallelwiththeimaginaryaxis,asshowninFig. 1.4.1.Theconstantintheintegrationlimitsmustbeproperlychosen,inordertoensurethe -convergence of the integral.direction of integration0cj+c jc j Fig. 1.4.1: The abscissa of absolute convergence the integration path for Eq. 1.4.4.Thefactore inEq. 1.4.3isneededtostoptherotationofthecorrespondingc=>phasor e ; there are infinitely many such phasors in the time function. As our variable=>0>is now complex,, the factor edoes not mean a simple rotation, but a= b4 5 =c=>spiralrotation decreasing in which the radius is exponentially with because of, the real part > 5of.Thisisnecessarytocancelthecorrespondingrotatione ,containedin,witha = 0>=>radius, which, in an exactly equal manner, with [Ref. 1.23]. increases >Since in Eq. 1.4.4 the factor ebecomes divergent if the exponent, the=>de f = > "aboveconditionsforthevariable(andfortheconstant)mustbemettoensurethe 5 -convergence of the integral. In the analysis of passive networks these conditions can alwaysbe met, as we will show in many examples in the subsequent sections.Now, because we have reached our goal, the Laplace transform and its inverse, wemay ask ourselves what we have accomplished by doing all this hard work.Forthetimebeingwecanclaimthatwehavetransformedthefunctionofarealvariable complexvariable intoafunctionofa.Thisallowsustocalculate,usingthe > =_ transform, the spectrum function of a finite transient, defined by the function. J= 0>Or, more important for us, by means of the transform we can calculate the time domain _c"function, if the frequency domain function is known. J=Later we will show how we can transformin the time linear differential equationsdomain, by means of the transform, into in the domain. Since the _ algebraic equations =algebraic equations are much easier to solve than the differential ones, this means one has agreatfacility.Onceourcalculationsinthedomainarecompleted,thenbymeansofthe =_c" transformweobtainthecorrespondingtimedomainfunction.Inthiswayweavoidsolving directly the differential equations and the calculation of boundary conditions. P. Stari, E.Margan The Laplace Transform -1.25-1.5Examples of Direct Laplace TransformNow let us put our new tools to use and calculate the transform of several simple _functions. The results may also be used for the transform and the reader is encouraged _c"to learn the most basic of them by heart, because they are used extensively in the other partsof the book and, of course, in the analysis of the most common electronics circuits.1.5.1Example 1Most of our calculations will deal with the step response of a network. To do so ourexcitationfunctionwillbeasimpleunitstep,orthefunction(after2> a b Heaviside OliverHeaviside, 18501925) as is shown in Fig. 1.5.1. This function is defined as:0> 2> ! > !" > ! for for ( ) f t ( ) f t1 10 0t taFig. 1.5.1: Fig. 1.5.2: Unit step function. Unit step function starting at. > +Asweagreedintheprevioussection,forforallthefollowing 0> ! > !functions, and we will not repeat this statement in further examples. At the same time let usmention that for our calculations of transform it is not important what is the actual value _of, providing it is finite [Ref. 1.3]. 0!The transform for the unit step functionis: _ 0 > 2> a b a bJ= 0> " .> " "c= =_e f c d( _!c=> c=>> _> !e e (1.5.1) 1.5.2Example 2The function is the same as in Example 1, except that the step does not start at > !but at ( Fig. 1.5.2 ): > + !0> ! > +" > + for for Solution:J= " .> " "c= =( c d_+c=> c=> c+=> _> +e e e (1.5.2) P. Stari, E.Margan The Laplace Transform -1.26-1.5.3Example 3The exponential decay function is shown in Fig. 1.5.3; its mathematical expression:0> ec > 5"is defined for, as agreed, and is a constant. > ! 5"Solution:e e e .> .> J=( (_ _! !c > c=> c b=> 5 5" "e (1.5.3) c" "b= b= 5 5" "c b=>> _> !5"Later, we shall meet this and the following function and also their product very often.( ) f t ( ) f t1 10ttt0TTTtsin (2 )e111/1=eFig. 1.5.3: Fig. 1.5.4:Exponential function.Sinusoidal function.1.5.4Example 4WehaveasinusoidalfunctionasinFig. 1.5.4;itscorrespondingmathematicalexpression is:0> > sin ="where the constant. = 1" # XSolution: its transform is: _J= > .>( a b_!"c=>sin = e (1.5.4)To integrate this function we substitute it using Eulers formula:sin ="4 > c4 >> c"# 4 e e (1.5.5) = =" "Then we have:e e e e .> c .> J="# 4 : ;( (_ _! !4 > c=> c4 > c=> = =" "e e (1.5.6) .> c .>"# 4 ( (_ _! !c=c4 > c=b4 > = =" " P. Stari, E.Margan The Laplace Transform -1.27-The solution of this integral is, in a way, similar to that in the previous example: c J=" " "# 4 = c4 = b4 9= =" "(1.5.7) " = b4 c= b4# 4 = b = b= = == =" " "# # # #" "Thisisatypicalfunctionofacontinuouswave(CW)sinusoidaloscillator,withafrequency. ="1.5.5Example 5Here we have the cosine function as in Fig. 1.5.5, expressed as:0> > cos ="Solution:thetransformofthisfunctioniscalculatedinasimilarwayasforthesine. _According to Eulers formula:cos ="4 > c4 >> b"# e e (1.5.8) = =" "Thus we obtain:e e .> b .> b J=" " " "# # = c4 = b4 ( ( 9_ _! !c= c4 > c= b4 >" "= =" "= =(1.5.9) " = b4 b= c4 =# = b = b= == =" "# # # #" "( ) f t( ) f t110ttt0TTTtcos (2 )e1/te1t T sin (2 ) /Fig. 1.5.5:Fig. 1.5.6: Cosine function. Damped oscillations.1.5.6Example 6In Fig. 1.5.6 we have a damped oscillation, expressed by the formula:0> > ec >"5"sin =Solution: we again substitute the sine function, according to Eulers formula: P. Stari, E.Margan The Laplace Transform -1.28-e e e c .> J="# 4 ( _!c= b > 4 > c4 > 5 = =" " "e e c .>"# 4 ( _!c=b c4 > c=b b4 > 5 = 5 =" " " "(1.5.10) c " " "# 4 = b c4 = b b4 = b b 95 = 5 = 5 ==" " " " ""# #"AninterestingsimilarityisfoundifthisformulaiscomparedwiththeresultofExample 4. There, for a CW we have in the denominator alone, whilst here, because the =#oscillations are damped, we have instead, and is the damping factor. = b5 5" "#( ) f t ( ) f t0 0tt ttnFig. 1.5.7:Fig. 1.5.8: Linear ramp Power function0> > 0> > ..81.5.7Example 7A linear ramp, as shown in Fig. 1.5.7, is expressed as:0> > Solution: we integrate by parts according to the known relation:( (?.@ ?@ c @ .?and we assign and eto obtain: > ? .> .@c= >e ee > .> b .> J=> "c= =( ( _ _! !c=> c=>c=>>_>!e (1.5.11) ! c! c " "= =# #c=>>_>!1.5.8Example 8Fig. 1.5.8 displays a function which has a general analytical form:0> > 8 P. Stari, E.Margan The Laplace Transform -1.29-Solution: again we integrate by parts, decomposing the integrandeinto: >8 c=>e e ? > .? 8> .> @ .@ .>"c=8 8c" c=> c=>With these substitutions we obtain:e ee > .> b > .> J=> 8c= =( (_ _! !8 c=> 8c" c=>8 c=>>_>!e (1.5.12) > .>8= (_!8c" c=>Again integrating by parts:e e e b > .>8 > 88 c"= c= => .>( (_ _! !8c" c=> 8c# c=>8c" c=>>_>!#e (1.5.13) > .>88 c"=#_!8c# c=>(By repeating this procedure times we finally arrive at: 8e e (1.5.14) > .> > .> J=88 c"8 c# $ # " 8x= =( (_ _! !8 c=> ! c=>8 8b"1.5.9Example 9The function shown in Fig. 1.5.9 corresponds to the expression:0> > ec > 5"Solution: by integrating by parts we obtain:e e e (1.5.15) > .> > .> J=" b=( (_ _! !c > c=> c b=>"#5 5" "51.5.10Example 10Similarly to Example 9, except that here we have, as in Fig. 1.5.10: >80> >8 c >e5"Solution: we apply the procedure from Example 8 and Example 9:e e e (1.5.16) > .> > .> J=8x b=( (_ _! !8 c > c=> 8 c b=>"8b"5 5" "5 P. Stari, E.Margan The Laplace Transform -1.30-e( ) f t( ) f t ( ) f t10.10.20.10.22120ttte1te1tte1tte11 2t tt01 201 2012tt 1ne tt 1n( ) f ttnFig. 1.5.9: Fig. 1.5.10: Function Function0> > 0> > e .e .c > 8 c > 5 5" "Thesetenexamples,whichwefrequentlymeetinpractice,demonstratethatthecalculationofantransformisnotdifficult.Sincetheresultsderivedareusedoften,we _have collected them in Table 1.5.1.Table 1.5.1 Ten frequently met transform examples : _No. No. (for) e (for) e e 0> J= 0> J=" " > ! ' >"= = b b# " > + ( >" "= =$ ) >" 8xb= =%c >""#" "# #c+ >#c > 8"8b"55""sinsin==5 =5=""# #"#c >""# 8b""#8 c >"> * >= b b="& > "! >= 8x= b b=e e == 5== 555""cos P. Stari, E.Margan The Laplace Transform -1.31-1.6Important Properties of the Laplace TransformIt is useful to know some of the most important properties of the transform: _1.6.1Linearity (1)_ _ _ e f e f e f 0> > 0> > g g (1.6.1)Example:_ = _ _ ===e f e f e f > b > > b > b"= = bsin sin" "# #"#" (1.6.2)1.6.2Linearity (2)_ _ e f e f O 0> O 0>(1.6.3)where is a real constant. OExample:_ = _ ==5 = % > % > %= b be e (1.6.4) c > c >" """# #"5 5" "sin sin1.6.3Real Differentiation_ I (1.6.5).0>.> = J= c0! bThetransformofaderivativeofthefunctionisobtainedifwemultiply0> J=byandsubtractthevalueofiffromtheside,denotedbythe+signat = 0> ! o> right0! 0! 0! b c b (the direction is important because the values and can be different). Wewill prove this statement by deriving it from the definition of the transform: _J= 0> 0> .> _e f(_!c=>e (1.6.6)We will integrate by parts by making and e . The result is: 0> ? .> .@c=>e e e 0> b .> 0> .>" " . 0>c= = .>( ( 9_ _! !c=> c=> c=>> _> !e b .>0! " . 0>= = .>b_!c=>( 9(1.6.7) b0! " .0>= = .>b_ IBy rearranging, we prove the statement expressed in Eq. 1.6.6:e = J= c0! = 0> .> c0! (_!c=> b be (1.6.8) .> . 0> . 0>.> .>( I_!c=>_ P. Stari, E.Margan The Laplace Transform -1.32-Example:_5 55 Ie (1.6.9). " c.> = b = b = J= c0! = c" c >b" ""5"We may also check the result by first differentiating the function e :c > 5"ee (1.6.10). .> cc >"c >55""5and then applying the transform: __ 5 5 _55 c c c= b" "c > c >""e e (1.6.11) 5 5" "The result is the same.By now the advantage of the transform against differential equations should have _becomeobvious.Inthedomainthederivativeofthefunctioncorrespondsto= 0> J=multiplied by and subtracting the value. The reason that must approach zero from = 0! >btheright(+)sideisourprescribingtobezerofor.Inotherwords,wehavea 0> > !unilateral transform.The higher derivatives are obtained by repeating the above procedure. If for the firstderivative we have obtained:_e f 0> = J= c0! w b(1.6.12)then the transform of the second derivative is: _ = 0> c0! 0 > _ _ e f e f ww w w bs (1.6.13) = = J= c0! c0! = J c= 0! c0! b w b # b w bBy a similar procedure the transform of the third derivative is: _ = = J= c= 0! c0! c0 ! 0 > _e f www # b w b ww b(1.6.14) = J= c= 0! c= 0! c0 ! $ # b w b ww bThus the transform of theth derivative is simply: _ 8_ 0 > = J= c= 0! c= 0! c= 0 ! c8 8 8c" b 8c# w b 8c$ ww b(1.6.15) c=0 ! c= 0 ! c0 ! # 8c$ b 8c# b 8c" b a ba b1.6.4Real IntegrationWe intend to prove that:_ 7 7 (>!0 . J== (1.6.16) P. Stari, E.Margan The Laplace Transform -1.33-We will derive the proof from the basic definition of the transform: __ 7 7 7 7 ( ( ( > _ >! ! !c=>0 . 0 . .> e (1.6.17)For the integration by parts we assign:? 0 . .? 0 . @ .@ .>"c=(>!c=> c=>7 7 7 7e e (1.6.18)By considering all this we may write the integral:e e e (1.6.19) 0 . c 0> .> 0 . .>" "c= c=( ( ( ( _ > > _! ! ! !c=> c=> c=>>_>!7 7 7 7Thetermbetweenbothlimitsiszeroforbecause,andforaswell > ! ! > _'!!because the exponential function e . Thus only the last integral remains, from whichc_ !we can factor out the term. The result is: "=_ 7 7 ( (> _! !c=>0 . 0> .> " J== =e (1.6.20) and thus the statement expressed by Eq. 1.6.16 is proved.Example:0> > e (1.6.21)c >"5"sin =We have already calculated the transform of this function ( Eq. 1.5.10 ) and it is:J= = b b=5 =""##"a bLetusnowcalculatetheintegralofthisfunctionaccordingtoEq. 1.6.20byintroducingadummy variable: 7_ =7 7=5 = (c d>!c"""# #"e (1.6.22) 57"sin . J== == b bThisexpressiondescribesthestepresponseofanetwork,havingacomplexconjugate pole pair. We meet such functions very often in the analysis of inductive peakingcircuits or in calculating the step response of an amplifier with negative feedback.Wemayobtainthetransformofmultipleintegralsbyrepeatingtheprocedureexpressed by Eq. 1.6.16. P. Stari, E.Margan The Laplace Transform -1.34-By doing so we obtain for thesingle integral: _ 7 7 (>!0 . J==double integral: _ 7 7 ( (>! !#7"0 . J==triple integral: _ 7 7 ( ( (>! ! !$7 7" #0 . J==-th integral:(1.6.23) integrals 8 0 . 8J==_ 7 7J ( (>! !878c"Thetransformoftheintegralofthefunctiongivesthecomplexfunction _ 0>J== J= = . The function must be divided by as many times as we integrate.Hereagainweseeagreatadvantageofthetransform,for_ wecanreplacetheintegrationinthetimedomain byasimple (oftenaratherdemandingprocedure) division by in the (complex) frequency domain. =1.6.5Change of ScaleWe have the function:_e f(0+ > 0+ > .>_!c=>e (1.6.24)We introduce a new variable, and for this and also. Thus @ + > .@ + .> > @+we obtain:_e f( 0+ > 0@ .@ J" " =+ + +e (1.6.25)_@!c=>+Example: we have the function:0> > e (1.6.26)c$ >WehavealreadycalculatedthetransformofasimilarfunctionbyEq. 1.5.15.Forthe _function above the result is:J= "= b$(1.6.27)a b# P. Stari, E.Margan The Laplace Transform -1.35-Now let us change the scale tenfold. The new function is:g> 0"! > "! > e (1.6.28)c$! >According to Eq. 1.6.25 it follows that:_e f a bg> J " = " "!"! "!"! b$="!= b$! (1.6.29)# #1.6.6Impulse( ) $>In Fig. 1.6.1a we have a square pulse with amplitude and duration. The E " > ""areaunderthepulse,equaltothetimeintegralofthispulse,is and thus amplitude time equalto.Itisobviousthatwemayobtainthesametimeintegralifthedurationofthe "pulseishalvedanditsamplitudedoubled( ).Thepulsehasafourtimeshigher E Eamplitude and its duration is only and still has the same time-integral. > !#&If we keep narrowing the pulse and adjusting the amplitude accordingly to keep thevalueofthetimeintegral,weeventuallyarriveatasituationwherethedurationofthe "pulsebecomesinfinitelysmall,,anditsamplitudeinfinitelylarge, > p! &E " p_ & , as shown in Fig. 1.6.1b.This impulse is denoted and it is called the. Mathematically we $ > Diracfunction2express this function as:$& && > 0> " ! > ! > &&J when when(1.6.30)p!f tA( )0112244A2A11 141ttf t ( = ) t ( ) A1A 0t=A1= = A2A40=,0,0 tt=0 =Fig. 1.6.1:TheDiracfunctionasthelimitingcaseofnarrowingthepulsewidth,whilekeepingthetimeintegralconstant:Ifthepulselengthisdecreased,itsamplitudemustincrease a)accordingly. When the pulse length the amplitude is. b) & & p! " p_ a bLet us calculate the transform of this function: _e e e 0> .> .> b ! .> 0>"_&e f( ( ( & &&& ! !_ _c=> c=> c=>e (1.6.31)e c " " c= = & &c=>> > !c=&&2, 19021984, English physicist, Nobel Prize winner in 1933 (together with Erwin Schrdinger). Paul Dirac# 4 P. Stari, E.Margan The Laplace Transform -1.36-Now we express the function ein this result by the following series:c= &e " c " c" c= b= #x c= $x b = = = = #x $x " c b cc= # $ # && && & & & & c dand by letting we obtain: & p!(1.6.32)e_$&& &e f > " c b c "" c = = = #x $xlim lim& &&p! p!c= #Thereforethemagnitudeofthespectrumenvelopeofthisfunctionisanditis oneindependentoffrequency.ThismeansthattheDiracimpulsecontains$> allfrequencycomponents, the amplitude of each component being. E "1.6.7Initial and Final Value TheoremsThe is expressed as: initial value theoremlim lim! o>= p_0> = J= (1.6.33)Wehavewrittenthenotationinordertoemphasizethatapproacheszero ! o> >from the right of the coordinate system. From real differentiation we know that:_ _ Ie f( e (1.6.34). 0>.> 0> 0> .> = J= c0! w w c=> b_!The limit of this integral when is zero: = p_lim= p__!w c=>e (1.6.35)(0> .> !Ifweassumethatiscontinuousatwemaywritethelimitoftheright 0> > !hand side of Eq. 1.6.33:! = J= c0! lim= pb_(1.6.36)or, in a form more useful for practical calculations:lim lim lim= pb! o> ! o>_= J= 0! b 0> 0> (1.6.37)Even if is not continuous at, this relation is still valid, although the proof 0> > !isslightlymoredifficult[Ref. 1.10].Theexpressionisintroducedbecauseweare 0! bdealingwithaunilateraltransform,inwhichitisassumedthatfor,soto 0> ! > !calculate the actual initial value we must approach it from the positive side of the time axis.For the functions which we will discuss in the rest of the book we can, in a similarway, prove the, which is stated as: final value theoremlim lim> p_ = p!0> = J= (1.6.38)(note that for some functions, such as or or the squarewave, this limit does sin cos = =" "> >not exist, since the value oscillates with the time integral of the function!). P. Stari, E.Margan The Laplace Transform -1.37-We repeat the statement from Eq. 1.6.34:_e f(0> 0> .> = J= c0! w w c=> b_!e (1.6.39)Now let (using as an intermediate dummy variable): = p! ;e 0> .> 0> .> 0> .> lim lim= p!_ _! ! !w c=> w w; p_;( ( ((1.6.40) 0; c0! 0> c0! lim lim; p_b b> p_c dAlthoughthelowerlimitoftheintegralisa(simple)zerowehaveneverthelesswrittenintheresult,toemphasizetheunilateraltransform.Thelimitofthe right hand !bside of Eq. 1.6.39, when is: = p!lim= p!b= J= c0! (1.6.41)By comparing the results of Eq. 1.6.34, 1.6.39 and 1.6.41 we may write:lim lim> pb b= p _ !0> c0! = J= c0! (1.6.42)or as stated initially in Eq. 1.6.38: lim lim> p = p! _0> = J=TheEq. 1.6.37and1.6.38areextremelyusefulforcheckingtheresultsofcomplicatedcalculationsbythedirectortheinverseLaplacetransform,aswewillencounter in the following parts of the book. Should the check by these two equations fail,then we have obviously made a mistake somewhere.However, this is a necessary, but not a sufficient condition: if the check was passedwearenotguarantiedthatothersneakymistakeswillnotexist,whichmaybecomeobvious when we plot the resulting function.1.6.8ConvolutionWeneedaprocessbywhichwecancalculatetheresponseoftwosystemsconnectedsothattheoutputofthefirstoneistheinputofthesecondoneandtheirindividual responses are known. We have two functions [Ref. 1.19]:and(1.6.43) 0> J= > K= _ _c" c"e f e f gand we are looking for the inverse transform of the product:C> J= K= _c"e f (1.6.44)The product of functions is equal to the product of their Laplace transforms:J= K= 0 . @ .@( (_ _! !c= c=@7 7 e e (1.6.45)7g P. Stari, E.Margan The Laplace Transform -1.38-Inordertodistinguishbetterbetweenand,weassigntheletterforthe 0> > ? gargument of and for the argument of; thus, and. Since both 0 @ 0 > 0 ? > @ g g g a b a b a b a bvariables are now well separated we may write the above integral also in the form:J= K= 0? @ .@ .?( ( _ _! !c=?b@g e (1.6.46)Let us integrate the expression inside the brackets to the variable. To do so we introduce a @new variable: 77 7 7 ? b@ @ c? .@ . soand(1.6.47)Weconsiderthevariableintheinnerintegraltobea(variable)parameter.Fromthe 7above expressions it follows that if. By considering all this we may transform @ ! ? 7Eq. 1.6.46 into:J= K= 0? c? . .?( ( _ _!c=?bc?e (1.6.48)77g7 7a bWemayalsochangethesequenceofintegration.Thuswemaychooseafixed>"and first integrate from to. In the second integration we integrate from 7 7 ! > ? !"to. Then the above expression obtains the form: ? _J= K= 0? c? .? .( ( _! !>c="g7 7 e (1.6.49)7Now we can return from back to the usual time variable: ? >J= K= 0> c> .> .C>( ( _! !>c="g7 7 e (1.6.50)7Theexpressioninsidethebracketsisthefunctionwhichwearelookingfor, C>whilsttheouterintegralistheusualLaplacetransform.Thuswedefinetheconvolutionprocess, denoted by, as: g> 0> C> J= K= > 0> 0> c> .> _ 7c">!e f(g g "(1.6.51)The operator symbolized by the asterisk (means convolved with. is ) Convolutiothe Latin word for folding. The German name for convolution is and this also die Faltungmeans folding. Obviously:g g c> c> 7 7 = !andg g c> ! 7 7> P. Stari, E.Margan The Laplace Transform -1.39-This means that the function is folded in time around the ordinate, from the rightto the left side of the coordinate system. At the end of this part, after we master the networkanalysisinLaplacespace,wewillmakeanexample(Fig. 1.15.1)inwhichthisfoldingand the convolution process will be explicitly shown, step by step.Ingeneralweconvolvewhicheverofthetwofunctionsissimpler.Wemaydosobecause the convolution is commutative:g g > 0> 0> > (1.6.52)The main properties of the Laplace transform are listed in Table 1.6.1.Table 1.6.1 The main properties of Laplace transform :PropertyReal Differentiation Real Integration Time-Scale Change Impuls0> J=. 0>.>= J= c0! 0> .>J==0+> J" =+ +b>!( "e functionInitial ValueFinal ValueConvolution$7> "0> = J=0> = J=0> c> .> J= K=lim limlim lim! o>= p_> p_ = p!>!("g P. Stari, E.Margan The Laplace Transform -1.41-1.7Application of the transform in Network Analysis _1.7.1InductanceAs we have discussed the fundamentals of transform, we will now apply it in the _network analysis. From basic electrical engineering we know that the instantaneous voltage@> P 3> across an inductance, through which a current flows, as in Fig. 1.7.1a, is:@ > P.3.> (1.7.1)By assuming time and, then, according to Eq. 1.6.5, the transform of the > ! 3! !b_above equation is the voltage across the inductance in the domain: =Z = = PM= (1.7.2)where is the current in the domain. The inductive reactance is then: M = = (1.7.3)Z =M= = PHereandthusitiscomplex;itcanlieanywhereinthe plane. In the = b4 = 5 =special case when, and considering only the positive axis, degenerates into. 5 = = ! 4 = 4Thentheinductivereactancebecomesthefamiliar,asisknownfromtheusual 4 P =phasor analysis of networks.a) c) b)L CRi i iittittitt++ +Fig. 1.7.1: a)The instantaneous voltage as a function of the instantaneous current: @ 3on an inductance; on a capacitance; on a resistanceP G V b) c)1.7.2CapacitanceFrom basic electrical engineering we also know that the instantaneous voltage @>across a capacitance through which a current flows during a time is: 3> >!@> 3 .>;> "G G (1.7.4)(!>asshowninFig. 1.7.1b.Hereistheinstantaneouschargeonthecapacitor.By ;> Gapplying Eq. 1.6.20 we may calculate the voltage on the capacitor in the domain: =Z = M="= G (1.7.5) P. Stari, E.Margan The Laplace Transform -1.42-The capacitive reactance in the domain is: = (1.7.6)Z = "M= = GHere,too,degeneratestoif.Inthiscasethecapacitivereactance = 4 ! = 5becomes simply. "4 G =1.7.3ResistanceFor a resistor ( Fig. 1.7.1c ) the instantaneous voltage is simply:@> V3> (1.7.7)and,astherearenotime-derivativesthesameholdsinthedomain,withthe =corresponding values and: Z = M=Z = VM= (1.7.8)yielding: (1.7.9)Z =M= V1.7.4Resistor and capacitor in parallelBy applying the Eq. 1.7.3, 1.7.6 and 1.7.9 we may transform a differential equationinthedomainintoanalgebraicequationinthedomain.Thuswemayexpressan > =impedance or an admittance of more complicated networks by simple algebraic ^= ] = equations. Let us express a parallel combination of a resistor and a capacitor in domain as =shown in the upper part of Fig. 1.7.2. The impedance is:(1.7.10) ^= V V VK=" c== Gb"Vc c"VG= c c"VG= c= 9 9"""where the (real) pole is at and represents the frequency dependence = c"VG K " " "5Thepoleofafunctionisthatparticularvalueoftheargumentforwhichthefunctiondenominator is equal to zero and, consequently, the function value goes to infinity.Nowletusapplyacurrentstep,V ,toournetworkexpressedintheM= " V =domainas,accordingtoEq. 1.5.1.We introduced the factor in order to get a "=V "Vvoltage ofV on our combination when. The corresponding function is then: " VG > p_ M= ^= VK= K= J= Z =" "=V =" "(1.7.11) " " "VG == c c"VG 9 P. Stari, E.Margan The Laplace Transform -1.43-From a second pole at is introduced, as is drawn in Fig. 1.7.2b and c. To "= = !obtainthetimedomainfunctionofthevoltageacrossourimpedance,weapplyEq. 1.5.3and 1.6.20. First we discuss only the function. According to Eq. 1.5.3: K ="a b_5 e (1.7.12) 5">""= cor inversely: e (1.7.13) _5c" >" I"= c5"By comparing Eq. 1.7.11 with Eq. 1.7.13 we see that: 5" c"VG_c" c>VG Ia b"= cc"VG e (1.7.14)From Eq. 1.6.20 we concluded that the division in the domain corresponds to the =real integration in the domain. By considering this together with Eq. 1.5.1, we obtain: >e 0> J= .> @>"VG "== cc"VG VGo_ _c" c" c>VG>!e f Ic d a b(e e e (1.7.15) cVG c cc" " c"VG a bc>VG c>VG c>VG>!= 0R CtttR Cii oa)b)c)1111000oi1) e = g ttRC(i2) g t (Rh t) = 1 f t (= ( )1 V/R0,> 0< 0ttsRC1=1s1s0s0jjj= 1VetRC=etRC,Fig. 1.7.2:Thecourseofmathematicaloperationsforaparallelnetworkexcitedbyaunitstep VGcurrentThedomainfunctionsareontheleft,thedomainfunctionsareontheright. Theself- 3 >