Introduction to Linear Circuit Analysis and Modelling From DC to RF To our families Introduction to Linear Circuit Analysis and Modelling From DC to RF Luis Moura University of Algarve, Portugal Izzat Darwazeh University College London, UK Newnes AnimprintofElsevier LinacreHouse,JordanHill,OxfordOX28DP 30CorporateDrive,Burlington,MA01803 Firstpublished2005 Copyright c LuisMouraandIzzatDarwazeh,2005. Allrightsreserved. (cid:176) TherightofLuisMouraandIzzatDarwazehtobeidenti(cid:2)edastheauthors ofthisworkhasbeenassertedinaccordancewiththeCopyright,Designsand PatentsAct1988. 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BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN0750659327 Contents Preface ix Acknowledgements xi 1 Elementaryelectricalcircuitanalysis 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Voltageandcurrent . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Voltagesources . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Currentsources . . . . . . . . . . . . . . . . . . . . . 3 1.3 Electricalpassiveelements . . . . . . . . . . . . . . . . . . . 5 1.3.1 Resistanceandconductance . . . . . . . . . . . . . . 5 1.3.2 Capacitance . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Inductance . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Kirchhoff’slaws . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Seriesandparallelcombinationsofpassiveelements . 11 1.4.2 Othertypesofcircuitelementconnections. . . . . . . 17 1.4.3 Electricalnetworkanalysis(cid:150)Nodalanalysis. . . . . . 18 1.4.4 Resistivevoltageandcurrentdividers . . . . . . . . . 20 1.4.5 Controlledsources . . . . . . . . . . . . . . . . . . . 21 1.5 The·venin’stheorem . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Norton’stheorem . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7 Superpositiontheorem . . . . . . . . . . . . . . . . . . . . . 25 1.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Complexnumbers: Anintroduction 32 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 De(cid:2)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Elementaryalgebra . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.2 Subtraction . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.3 Multiplication. . . . . . . . . . . . . . . . . . . . . . 35 2.3.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.5 Complexequations . . . . . . . . . . . . . . . . . . . 38 2.3.6 Quadraticequations . . . . . . . . . . . . . . . . . . 38 2.4 Polarrepresentation . . . . . . . . . . . . . . . . . . . . . . . 39 vi CONTENTS 2.4.1 Multiplicationanddivision . . . . . . . . . . . . . . . 40 2.5 Theexponentialform . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1 Trigonometricfunctionsandtheexponentialform. . . 42 2.6 Powersandroots . . . . . . . . . . . . . . . . . . . . . . . . 43 2.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Frequencydomainelectricalsignalandcircuitanalysis 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 SinusoidalACelectricalanalysis . . . . . . . . . . . . . . . . 48 3.2.1 Effectiveelectricalvalues . . . . . . . . . . . . . . . 49 3.2.2 I(cid:150)V characteristicsforpassiveelements . . . . . . . . 50 3.2.3 Phasoranalysis . . . . . . . . . . . . . . . . . . . . . 54 3.2.4 Thegeneralisedimpedance . . . . . . . . . . . . . . . 56 3.2.5 Maximumpowertransfer . . . . . . . . . . . . . . . . 64 3.3 Generalisedfrequencydomainanalysis . . . . . . . . . . . . 65 3.3.1 TheFourierseries . . . . . . . . . . . . . . . . . . . 65 3.3.2 Fouriercoef(cid:2)cients,phasorsandlinespectra . . . . . 72 3.3.3 Electricalsignalandcircuitbandwidths . . . . . . . . 73 3.3.4 Lineardistortion . . . . . . . . . . . . . . . . . . . . 79 3.3.5 Bodeplots . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.6 TheFouriertransform . . . . . . . . . . . . . . . . . 88 3.3.7 Transferfunctionandimpulseresponse . . . . . . . . 98 3.3.8 Theconvolutionoperation . . . . . . . . . . . . . . . 100 3.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 Naturalandforcedresponsescircuitanalysis 109 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Timedomainanalysis . . . . . . . . . . . . . . . . . . . . . . 109 4.3 TransientanalysisusingFouriertransforms . . . . . . . . . . 113 4.3.1 Differentiationtheorem . . . . . . . . . . . . . . . . . 113 4.3.2 Integrationtheorem . . . . . . . . . . . . . . . . . . . 114 4.3.3 I(cid:150)V characteristicsforpassiveelements . . . . . . . . 114 4.4 TheLaplacetransform . . . . . . . . . . . . . . . . . . . . . 117 4.4.1 TheoremsoftheLaplacetransform . . . . . . . . . . 119 4.4.2 Partial-fractionexpansion . . . . . . . . . . . . . . . 123 4.5 AnalysisusingLaplacetransforms . . . . . . . . . . . . . . . 127 4.5.1 Solvingdifferentialequations . . . . . . . . . . . . . 127 4.5.2 I(cid:150)V characteristicsforpassiveelements . . . . . . . . 128 4.5.3 Naturalresponse . . . . . . . . . . . . . . . . . . . . 130 4.5.4 Responsetothestepfunction . . . . . . . . . . . . . 136 4.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 CONTENTS vii 5 Electricaltwo-portnetworkanalysis 150 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2 Electricalrepresentations . . . . . . . . . . . . . . . . . . . . 150 5.2.1 Electricalimpedancerepresentation . . . . . . . . . . 150 5.2.2 Electricaladmittancerepresentation . . . . . . . . . . 153 5.2.3 Electricalchainrepresentation . . . . . . . . . . . . . 157 5.2.4 Conversionbetweenelectricalrepresentations . . . . . 159 5.2.5 Miller’stheorem . . . . . . . . . . . . . . . . . . . . 161 5.3 Computer-aidedelectricalanalysis . . . . . . . . . . . . . . . 163 5.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6 Basicelectronicampli(cid:2)erbuildingblocks 169 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2 Modellingtheampli(cid:2)cationprocess . . . . . . . . . . . . . . 169 6.3 Operationalampli(cid:2)ers . . . . . . . . . . . . . . . . . . . . . 177 6.3.1 Open-loopandfeedbackconcepts . . . . . . . . . . . 177 6.3.2 Otherexamplesandapplications . . . . . . . . . . . . 179 6.4 Activedevices . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.4.1 Thejunctionorp(cid:150)ndiode . . . . . . . . . . . . . . . 183 6.4.2 Thebipolarjunctiontransistor . . . . . . . . . . . . . 183 6.4.3 Theinsulatedgate(cid:2)eld-effecttransistor . . . . . . . . 192 6.4.4 Thecommon-emitterampli(cid:2)er . . . . . . . . . . . . . 196 6.4.5 Thedifferentialpairampli(cid:2)er . . . . . . . . . . . . . 215 6.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7 RFcircuitanalysistechniques 224 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.2 Lumpedversusdistributed . . . . . . . . . . . . . . . . . . . 224 7.3 Electricalmodelforidealtransmissionlines . . . . . . . . . . 227 7.3.1 VoltageStandingWaveRatio(cid:150)VSWR . . . . . . . . 234 7.3.2 The‚=4transformer . . . . . . . . . . . . . . . . . . 239 7.3.3 Lossytransmissionlines . . . . . . . . . . . . . . . . 242 7.3.4 Microstriptransmissionlines . . . . . . . . . . . . . . 247 7.4 Scatteringparameters . . . . . . . . . . . . . . . . . . . . . . 248 7.4.1 S-parametersandpowerwaves . . . . . . . . . . . . 254 7.4.2 PowerwavesandgeneralisedS-parameters . . . . . . 258 7.4.3 Conversionsbetweendifferenttwo-portparameters . . 263 7.5 TheSmithchart . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.5.1 Theimpedanceandthere(cid:3)ectioncoef(cid:2)cientplanes . . 264 7.5.2 Representationofimpedances . . . . . . . . . . . . . 266 7.5.3 Introductiontoimpedancematching . . . . . . . . . . 272 7.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 276 7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 viii CONTENTS 8 Noiseinelectroniccircuits 279 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.2 Randomvariables . . . . . . . . . . . . . . . . . . . . . . . . 279 8.2.1 Momentsofarandomvariable . . . . . . . . . . . . . 285 8.2.2 Thecharacteristicfunction . . . . . . . . . . . . . . . 288 8.2.3 Thecentrallimittheorem . . . . . . . . . . . . . . . . 290 8.2.4 BivariateGaussiandistributions . . . . . . . . . . . . 293 8.3 Stochasticprocesses. . . . . . . . . . . . . . . . . . . . . . . 295 8.3.1 Ensembleaverages . . . . . . . . . . . . . . . . . . . 295 8.3.2 Stationaryrandomprocesses . . . . . . . . . . . . . . 296 8.3.3 Ergodicrandomprocesses . . . . . . . . . . . . . . . 296 8.3.4 Powerspectrum . . . . . . . . . . . . . . . . . . . . . 298 8.3.5 Cross-powerspectrum . . . . . . . . . . . . . . . . . 301 8.3.6 Gaussianrandomprocesses . . . . . . . . . . . . . . 302 8.3.7 Filteredrandomsignals . . . . . . . . . . . . . . . . . 303 8.4 Noiseinelectroniccircuits . . . . . . . . . . . . . . . . . . . 305 8.4.1 Thermalnoise. . . . . . . . . . . . . . . . . . . . . . 305 8.4.2 Electronicshot-noise . . . . . . . . . . . . . . . . . . 306 8.4.3 1=f noise . . . . . . . . . . . . . . . . . . . . . . . . 306 8.4.4 Noisemodelsforpassivedevices . . . . . . . . . . . 307 8.4.5 Noisemodelsforactivedevices . . . . . . . . . . . . 308 8.4.6 Theequivalentinputnoisesources . . . . . . . . . . . 310 8.4.7 Thenoise(cid:2)gure. . . . . . . . . . . . . . . . . . . . . 319 8.5 Computer-aidednoisemodellingandanalysis . . . . . . . . . 323 8.5.1 Noiserepresentations . . . . . . . . . . . . . . . . . . 323 8.5.2 Calculationofthecorrelationmatrices . . . . . . . . . 325 8.5.3 Elementarytwo-portinterconnections . . . . . . . . . 327 8.5.4 Transformationmatrices . . . . . . . . . . . . . . . . 328 8.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 A Mathematicalformulaeforelectricalengineering 342 B Elementarymatrixalgebra 350 C Two-portelectricalparameters 352 Index 356 Preface Themathematicalrepresentationandanalysisofcircuits,signalsandnoiseare key tools for electronic engineers. These tools have changed dramatically in recentyearsbutthetheoreticalbasisremainunchanged. Nowadays, themost complicatedcircuitscanbeanalysedquicklyusingcomputer-basedsimulation. However,goodappreciationofthefundamentalsonwhichsimulationtoolsare basedisessentialtomakethebestuseofthem. In this book we address the theoretical basis of circuit analysis across a broadspectrumofapplicationsencounteredintoday’selectronicsystems,es- peciallyforcommunications. Throughoutthebookwefollowamathematical- based approach to explain the different concepts using plenty of examples to illustratetheseconcepts. Thisbookisaimedatengineeringandsciencesstudentsandotherprofes- sionalswhowantsolidgroundingincircuitanalysis. Thebasicscoveredinthe (cid:2)rstfourchaptersaresuitablefor(cid:2)rstyearundergraduates. Thematerialcov- eredinChapters(cid:2)veandsixismorespecialistandprovidesagoodbackground at an intermediate level, especially for those aiming to learn about electronic circuitsandtheirbuildingblocks.Thelasttwochaptersaremoreadvancedand requiregoodgroundingintheconceptscoveredearlierinthebook. Thesetwo chaptersaresuitedtostudentsinthe(cid:2)nalyearoftheirengineeringdegreeand topost-graduates. Inthe(cid:2)rstchapterwebeginbyreviewingthefundamentallawsandtheo- rems applicable to electrical circuits. In Chapter two we include a review of complexnumbers,crucialfordealingwithACsignalsandcircuits. Thevariedandcomplexnatureofsignalsandelectronicsystemsrequirea thoroughunderstandingofthemathematicaldescriptionofsignalsandthecir- cuitsthatprocessthem.Frequencydomaincircuitandsignalanalysis,basedon theapplicationofFouriertechniques,arediscussedinChapterthreewithspe- cialemphasisontheuseofthesetechniquesinthecontextofcircuits. Chapter four then considers time domain analysis and Laplace techniques, again with similaremphasis. Chapter (cid:2)ve covers the analysis techniques used in two-port circuits and also covers various circuit representations and parameters. These techniques areimportantforcomputer-basedanalysisoflinearelectroniccircuits. Inthis chapterthetreatmentisfrequency-domain-based.
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