1、MMMMThis page intentionally left blankFast analytical techniques forelectricaland electronic circuitsToday,the only method of circuit analysis known to most engineers and students is nodal orloopanalysis.Althoughthis workswellfor obtainingnumericalsolutions,itis almostuseless forobtaining analytical
2、 solutions in all but the simplest cases.In this unique book,Vorpe rian describes remarkable alternative techniques to solve,almostby inspection,complicated linear circuits in symbolic form and obtain meaningful analyticalanswers for any transfer function or impedance.Although not intended to replac
3、e traditional computer-based methods,these techniquesprovide engineers with a powerful set of tools for tackling circuit design problems.They alsohave great value in enhancing students understanding of circuit operation.The numerousproblems and worked examples in this book make it an ideal textbook
4、for senior/graduatecourses or a reference book.This book will show you how to:use less algebra and do most of it directly on the circuit diagram,obtain meaningfulanalyticalsolutions tocomplexcircuitswith reactiveelementsand depend-ent sources by reducing them to a set of simple and purely resistive
5、circuits which can beanalyzed by inspection,analyze feedback amplifiers easily using the simplest and most natural formulation,analyze PWM converters easily using the model of the PWM switch.Originally developed and taught at institutions and companies around the world by ProfessorDavid Middlebrook
6、at Caltech,the extended and new techniques described in this book are anindispensable set of tools for linear electronic circuit analysis and design.Vatche Vorpe rianreceived his PhD in ElectricalEngineeringin 1984fromtheCalifornia InstituteofTechnologyand joinedthefacultyofElectricalEngineeringatVi
7、rginiaTech in thesame year.In 1991 he joined the Jet Propulsion Laboratory where he is currently a senior member of thetechnical staff.He has published over 35 conference and journal papers in the field of powerelectronics and has taught many professional advancement courses to industry.MMMMFast ana
8、lyticaltechniques for electricaland electroniccircuitsVatche Vorpe rianJet Propulsion LaboratoryCalifornia Institute of Technology The Pitt Building,Trumpington Street,Cambridge,United Kingdom The Edinburgh Building,Cambridge CB2 2RU,UK40 West 20th Street,New York,NY 10011-4211,USA477 Williamstown R
9、oad,Port Melbourne,VIC 3207,AustraliaRuiz de Alarcn 13,28014 Madrid,SpainDock House,The Waterfront,Cape Town 8001,South Africahttp:/www.cambridge.orgFirst published in printed format ISBN 0-521-62442-8 hardbackISBN 0-511-01637-9 eBookCambridge University Press 20042002(netLibrary)To my parents Edwar
10、d and Azadouhi Vorpe rianMMMMContentsPrefacexi1Introduction11.1Fast analytical methods11.2Input impedance of a bridge circuit21.3Input impedance of a bridge circuit with a dependent source41.4Input impedance of a reactive bridge circuit with a dependent source81.5Review11Problems11References142Trans
11、fer functions152.1Definition of a transfer function152.2The six types of transfer functions of an electrical circuit172.3Determination of the poles of a network192.4Determination of the zeros of a transfer function242.5The complete response,stability and transfer functions342.6Magnitude and phase re
12、sponse412.7First-order transfer functions432.8Second-order transfer functions482.9Review52Problems533The extra element theorem613.1Introduction613.2Null double injection62vii3.3The EET for impedance elements743.4The EET for dependent sources883.5Review98Problems99References1064TheN-extra element the
13、orem1074.1Introduction1074.2The 2-EET for impedance elements1084.3The 2-EET for dependent sources1304.4The NEET1374.5A proof of the NEET1474.6Review153Problems154References1625Electronic negative feedback1635.1Introduction1635.2The EET for dependent sources and formulation of electronic feedback 164
14、5.2.1 Gain analysis1645.2.2 Driving-point analysis1705.2.3 Loop gain1755.3Does this circuit have feedback or not?This is not the question1795.4Gain analysis of feedback amplifiers1805.5Driving-point analysis of feedback amplifiers1955.5.1 Input impedance for current mixing1965.5.2 Output impedance f
15、or voltage sensing2005.5.3 Input admittance for voltage mixing2045.5.4 Output admittance for current sensing2095.6Loop gain:a more detailed look2135.7Stability2185.8Phase and gain margins2265.9Review233Problems234References251viiiContents6High-frequency and microwave circuits2526.1Introduction2526.2
16、Cascode MOS amplifier2526.3Fifth-order Chebyshev low-pass filter2616.4MESFET amplifier2656.5Review310Problems311References3167Passive filters3177.1Introduction3177.2RC filters with gain3177.3Lattice filters3277.4Resonant filters3357.4.1 Parallel resonant filters3367.4.2 Tapped parallel resonant filt
17、er3397.4.3 The three-winding transformer3447.5Infinite scaling networks3497.5.1 Infinite grid3497.5.2 Infinite scaling networks3517.5.3 A generalized linear element and a unified R,L and C model3567.6Review358Problems358References3648PWM switching dc-to-dc converters3658.1Introduction3658.2Basic cha
18、racteristics of dc-to-dc converters3668.3The buck converter3708.4The boost converter3868.5The buck-boost converter3928.6The Cuk converter3978.7The PWM switch and its invariant terminal characteristics400ixContents8.8Average large-signal and small-signal equivalent circuit models of thePWM switch4028
19、.9The PWM switch in other converter topologies4118.10 The effect of parasitic elements on the model of the PWM switch4268.11 Feedback control of dc-to-dc converters4328.11.1 Single-loop voltage feedback control4338.11.2 Current feedback control4408.11.3 Voltage feedback control with peak current con
20、trol4538.12 Review460Problems461References470Index472xContentsPrefaceThe title of this book could easily have been called Variations on a Theme byMiddlebrook,or Applications of The Extra Element Theorem and its Extensions.Neither title,however,would have captured the unique message of this book that
21、one can solve very complicated linear circuits in symbolic form almost by inspec-tion and obtain more than one meaningful analytical answer for any transferfunction or impedance.The well-known and universally practiced method ofnodal or loop analysis not only becomes intractable when applied to a co
22、mpli-catedlinear circuit in symbolicform,but also yields unintelligibleanswers consist-ing of a massive collection of symbols.In a meaningful analytical answer,thesymbols must be grouped together in low-entropy form a term coined by R.D.Middlebrook clearly indicating series and parallel combination
23、of circuit ele-ments,and sums and products of time constants.The illustrative examples inChapter1 serve as a quick and informal introduction to the basic concepts behindthe radically different approach to network analysis presented in this book.Today,the only method of circuit analysis known to most
24、 engineers,studentsandprofessorsis themethodof nodalor loopanalysis.Althoughthis methodis anexcellent general tool for obtaining numerical solutions,it is almost useless forobtaining analytical solutions in all but the simplest cases.Anyone who hasattempted inverting a matrix with symbolic entries s
25、ometimes as low as second-order knows how tedious the algebra can get and how ridiculous the resultinghigh-entropyexpressionscanlook.Thepurposeofthisbookisnottoeliminatethelinear algebra approach to network analysis,but instead to provide additionalnew and efficient tools for obtaining analytical so
26、lutions with great ease andwithout letting the algebra run into a brick wall.Among the most important techniques discussed in this book are the extraelement theorem(EET)and its extension the N-extra element theorem(NEET).These two theorems are discussed in Chapters 3 and 4 after a brief and essentia
27、lreview of transfer functions given in Chapter 2.The EET and its proof were givenby R.D.Middlebrook.The NEET was given without proof by Sarabjit Sabhar-wal,an undergraduate at Caltech in 1979.In Chapter 4,a completely originaltreatment of the NEET is given,where it is stated in its most general form
28、 using anewcompactnotationand,forthefirsttime,provendirectlyusingmatrixanalysis.The subject of electronic feedback is treated in Chapter 5 using the EET forxidependentsources,and anothertheorem by R.D.Middlebrookcalledsimply thefeedback theorem.Both methods lead to a much more natural formulation of
29、electronic feedback than the well-known block diagram approach found in mosttextbooks.Blockdiagramsareusefultoolsinlinearsystemtheorytohelpvisualizeabstractconcepts,buttheytendtobeveryawkwardtoolsinnetworkanalysis.Forinstance,in an electronic feedback circuit neither the impedance loading nor thebi-
30、directional transmission of the feedback network are easily captured by thesingle-loopfeedbackblock diagramunlessthe feedbacknetworkand theamplifiercircuit are both manipulated and forced to fit the block diagram.The fact is blockdiagrams bear little resemblance to circuits and their use in network
31、analysismainly results in loss of time and insight.The examples presented in Chapters 6 and 7 are a tour de force in analysis ofcomplicated circuits which demonstrate the efficacy of the fast analytical tech-niques developedin the previous chapters.Amongthe examplesdiscussedin thesechapters are high
32、er-order passive filters and a MESFET amplifier.Some infinitenetworks,includingfractalnetworks,arediscussedinChapter7whereaninterest-ing,and possibly new,result is presented.It is shown that a resistor,an inductoranda capacitorare all special casesof a single,two-terminal,linear elementwhosevoltage
33、and current are related by a fractional derivative or its inverse,theRiemannLiouville fractional integral.Pulse-width-modulated(PWM)switching dc-to-dc power converters are intro-duced in Chapter 8 to illustrate further the applications of the fast analyticaltechniques presented in this book.The anal
34、ysis of PWM converters has been oneof the hot topics of nearly every conference in power electronics since the early1970s,and many specialized analytical techniques have been developed since.Thesimplest and fastest of these techniques is based on the equivalent circuit model ofthePWM switch,whichis
35、introducedafter a discussionof basicPWMconverters.The PWM switch is a three-terminal nonlinear device which is solely responsiblefor the dc-to-dc conversion function inside a PWM converter.Hence,the PWMswitch and its equivalent circuit model are to a PWM converter what the transis-tor and its equiva
36、lent circuit are to an amplifier.To analyze the dynamics of aPWM converter,one simply replaces the PWM switch with its equivalent circuitmodel and proceeds in exactly the same way as in an amplifier circuit analysis.This book is based on my experience in electronic circuit analysis as a student,desi
37、gn engineer,teacher and researcher.The limitations of the standard circuitanalysis I studied as an undergraduate soon became apparent on my first job as apowersupplydesignengineeratDigitalEquipmentCorporation,Maynard,MA.Ispent inordinate amounts of time deriving various small-signal transfer functio
38、nsof switching converters in order to understand and improve their stability anddynamicbehavior.Most of the senior engineers around me had acquired excellentdesign skills mostly by experience and did not rely too much on analysis.When IxiiPrefacereturned to graduate school at Caltech,I took Middlebr
39、ooks course whichengendered a complete turn around:I learned how to handle complicated linearnetworks and obtain transfer functions,in low-entropy form,using very simpleand elegant techniques.I gradually adopted these techniques in my seven years ofteaching at Virginia Polytechnic Institute and Stat
40、e University confirming theadage,the best way to learn something is to teach it.Logically,Middlebrooks book,which is still in preparation,should have pre-ceded mine.I began writing this book in the summer of 1996 with the intention ofcompletingit by thewinterof 1997.Clearly,I did notrealize thatwrit
41、inga bookatnightsand on weekendswould be considerablymore difficultandtime consumingthan I had ever imagined.Fortunately,I had the constantsupport and encourage-ment of family,friends and colleagues.I would especially like to thank GeneWester and Dave Rogers,both at the Jet Propulsion Laboratory,for
42、 their carefulreview and corrections of some of the chapters of this book.I would also like tothankmyformersupervisorRobertDetwiler;mycurrentsupervisorMarkUnder-wood;my colleagues Chris Stell,Tony Tang,Roman Gutierrez,Avo Demirjian,Dan Karmon,Mario Matal,Joseph Toczylowsky,Karl Yee,James Gittens,Mik
43、eNewell,David Hykes,Chuck Derksen and Tien Nguyen for making JPL anenjoyable place to work.Although this book is dedicated to my parents for theircountless sacrifices,I would not have been able to write it without the enduringsupport,loveandcareofmyfavoritemezzo-soprano,bestfriendandwifeShoghig.Vatc
44、he Vorpe rianJune 2000 xiiiPrefaceMMMM1IntroductionThe joys of network analysis1.1Fast analytical methodsThe universally adopted method of teaching network theory is the formal andsystematic method of nodal or loop analysis.Although the matrix algebra offormal network analysis is ideal for obtaining
45、 numerical answers by a computer,itfails hopelessly for obtaining analytical answers which provide physical insightintotheoperationofthecircuit.It is nothardtoseethat,whennumericalvaluesofcircuit components are not given,inverting a 3?3,or higher-order,matrix withsymbolic entries can be very time co
46、nsuming.This is only part of the problem ofmatrix analysis because even if one were to survive the algebra of inverting amatrix symbolically,the answer could be an unintelligible and lengthy symbolicexpression.It is important to realize that an analytical answer is not merely asymbolic expression,bu
47、t an expression in which various circuit elements aregrouped together in one or more of the following ways:(a)series and parallel combinations of resistancesExample:R?R?(R?R?)(b)ratios of resistances,time constants and gainsExample:1?RR?R?,1?g?R?A?,A?1?(c)polynomials in the frequency variable,s,with
48、 a unity leading term and coeffi-cients in terms of sums and products of time constantsExample:1?s(?)?s?Such analytical expressions have been called low-entropy expressions by R.D.Middlebrook?because they reveal useful and recognizable information(low noiseor entropy)about the performance of the cir
49、cuit.Another extremely importantadvantageof low-entropyexpressionsis that they can be easily approximatedintosimpler expressions which are useful for design purposes.For instance,a series-parallel combination of resistances,as in(a),can be simplified by ignoring thesmallerof two resistances in a ser
50、ies combination and the larger of two resistances1in a parallel combination.When ratios are used as in(b),they can be simplifieddepending on their relative magnitude to unity.Depending on the relative magni-tudeoftimeconstants,frequencyresponsecharacteristicsasin(c)canbesimplifiedand either factored
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