Introduction to Linear Circuit Analysis and Modelling: From DC to RF

374 Pages·2005·8.94 MB·English

by  Luis Moura

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.
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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.