Digital SAT Math Practice Questions Vibrant Publishers
https://ebookmass.com/product/digital-sat-math-practice-questionsvibrant-publishers/
ebookmass.com
Acute Care for Nurses (Student Survival Skills) (Jan 17, 2023)_(1119882451)_(Wiley-Blackwell) 1st Edition Boyd
https://ebookmass.com/product/acute-care-for-nurses-student-survivalskills-jan-17-2023_1119882451_wiley-blackwell-1st-edition-boyd/
ebookmass.com
Muddy Thinking in the Mississippi River Delta 1st Edition Ned Randolph
https://ebookmass.com/product/muddy-thinking-in-the-mississippi-riverdelta-1st-edition-ned-randolph/
ebookmass.com
The AI Delusion Gary Smith
https://ebookmass.com/product/the-ai-delusion-gary-smith/
ebookmass.com
Racial and Ethnic Relations, Census Update 9th Edition –Ebook PDF Version
https://ebookmass.com/product/racial-and-ethnic-relations-censusupdate-9th-edition-ebook-pdf-version/
ebookmass.com
https://ebookmass.com/product/introduction-to-chemical-engineering-cm-van-t-land/
ebookmass.com
ElectronicCircuitswithMATLAB ® , PSpice®,andSmithChart
ElectronicCircuitswithMATLAB® , PSpice®,andSmithChart
WonY.Yang,JaekwonKim,KyungW.Park, DonghyunBaek,SungjoonLim, JingonJoung,SuhyunPark,HanL.Lee, WooJuneChoi,andTaehoIm
Thiseditionfirstpublished2020
©2020JohnWiley&Sons,Inc.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfrom thistitleisavailableathttp://www.wiley.com/go/permissions.
TherightofWonY.Yang,JaekwonKim,KyungW.Park,DonghyunBaek,SungjoonLim,Jingon Joung,SuhyunPark,HanL.Lee,WooJuneChoi,andTaehoImbeidentifiedastheauthorsofthis workhasbeenassertedinaccordancewithlaw.
RegisteredOffice
JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA
EditorialOffice
111RiverStreet,Hoboken,NJ07030,USA
Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationaboutWiley productsvisitusatwww.wiley.com.
Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand.Some contentthatappearsinstandardprintversionsofthisbookmaynotbeavailableinotherformats.
LimitofLiability/DisclaimerofWarranty
MATLAB® isatrademarkofTheMathWorks,Inc.andisusedwithpermission.TheMathWorks doesnotwarranttheaccuracyofthetextorexercisesinthisbook.Thiswork’suseordiscussionof MATLAB® softwareorrelatedproductsdoesnotconstituteendorsementorsponsorshipbyThe MathWorksofaparticularpedagogicalapproachorparticularuseoftheMATLAB® software. Whilethepublisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,theymakeno representationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsof thisworkandspecificallydisclaimallwarranties,includingwithoutlimitationanyimplied warrantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedor extendedbysalesrepresentatives,writtensalesmaterialsorpromotionalstatementsforthiswork. Thefactthatanorganization,website,orproductisreferredtointhisworkasacitationand/or potentialsourceoffurtherinformationdoesnotmeanthatthepublisherandauthorsendorsethe informationorservicestheorganization,website,orproductmayprovideorrecommendations itmaymake.Thisworkissoldwiththeunderstandingthatthepublisherisnotengagedinrendering professionalservices.Theadviceandstrategiescontainedhereinmaynotbesuitableforyour situation.Youshouldconsultwithaspecialistwhereappropriate.Further,readersshouldbeaware thatwebsiteslistedinthisworkmayhavechangedordisappearedbetweenwhenthisworkwas writtenandwhenitisread.Neitherthepublishernorauthorsshallbeliableforanylossofprofit oranyothercommercialdamages,includingbutnotlimitedtospecial,incidental,consequential, orotherdamages.
LibraryofCongressCataloging-in-Publicationdataappliedfor ISBN:9781119598923
Coverdesign:Wiley
Coverimage:©green_01/Shutterstock
Setin10/12ptWarnockbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica 10987654321
Toourparentsandfamilies wholoveandsupportus and toourteachersandstudents whoenrichedourknowledge
Contents
Preface xiii
AbouttheCompanionWebsite xv
1LoadLineAnalysisandFourierSeries 1
1.1LoadLineAnalysis 1
1.1.1 Load LineAnalysisofaNonlinearResistorCircuit 3
1.1.2LoadLineAnalysisofaNonlinear RL circuit 7
1.2Voltage-CurrentSourceTransformation 10
1.3Thevenin/NortonEquivalentCircuits 11
1.4Miller’sTheorem 18
1.5FourierSeries 18
1.5.1ComputationofFourierCoefficientsUsingSymmetry 20
1.5.2CircuitAnalysisUsingFourierSeries 29
1.5.3RMSValueandDistortionFactorofaNon-Sinusoidal PeriodicSignal 35 Problems 36
2DiodeCircuits 43
2.1The v-i Characteristicof Diodes 43
2.1.1Large-SignalDiodeModelforSwitchingOperations 44
2.1.2Small-SignalDiodeModelforAmplifyingOperations 44
2.2Analysis/SimulationofDiodeCircuits 46
2.2.1ExamplesofDiodeCircuits 46
2.2.2Clipper/ClamperCircuits 51
2.2.3Half-waveRectifier 53
2.2.4Half-waveRectifierwithCapacitor – PeakRectifier 53
2.2.5Full-waveRectifier 57
2.2.6Full-waveRectifierwith LC Filter 59
2.2.7PrecisionRectifiers 62
2.2.7.1ImprovedPrecisionHalf-waveRectifier 63
2.2.7.2PrecisionFull-waveRectifier 65
2.2.8Small-Signal(AC)AnalysisofDiodeCircuits 67
2.3ZenderDiodes 75 Problems 85
3BJTCircuits 105
3.1BJT(BipolarJunctionTransistor) 106
3.1.1Ebers-Moll Representation ofBJT 106
3.1.2OperationModes(Regions)ofBJT 109
3.1.3ParametersofBJT 109
3.1.4Common-BaseConfiguration 111
3.1.5Common-EmitterConfiguration 113
3.1.6Large-Signal(DC)ModelofBJT 115
3.1.7Small-Signal(AC)ModelofBJT 142
3.1.8AnalysisofBJTCircuits 143
3.1.9BJTCurrentMirror 156
3.1.10BJTInverter/Switch 161
3.1.11Emitter-CoupledDifferentialPair 165
3.2BJTAmplifierCircuits 168
3.2.1Common-Emitter(CE)Amplifier 169
3.2.2Common-Collector(CC)Amplifier(EmitterFollower) 173
3.2.3Common-Base(CB)Amplifier 180
3.2.4MultistageCascadedBJTAmplifier 187
3.2.5Composite/CompoundMulti-StageBJTAmplifier 199
3.3LogicGatesUsingDiodes/Transistors[C-3,M-1] 209
3.3.1DTLNANDGate 209
3.3.2TTLNANDGate 215
3.3.2.1BasicTTLNANDGateUsingTwoBJTs 215
3.3.2.2TTLNANDGateUsingThreeBJTs 218
3.3.2.3Totem-PoleOutputStage 222
3.3.2.4Open-CollectorOutputandTristateOutput 227
3.3.3ECL(Emitter-CoupledLogic)OR/NORGate 229
3.4DesignofBJTAmplifier 239
3.4.1DesignofCEAmplifierwithSpecifiedVoltageGain 232
3.4.2DesignofCCAmplifier(EmitterFollower)withSpecifiedInput Resistance 239
3.5BJTAmplifierFrequencyResponse 243
3.5.1CEAmplifier 243
3.5.2CCAmplifier(EmitterFollower) 248
3.5.3CBAmplifier 255
3.6BJTInverterTimeResponse 259 Problems 266
4FETCircuits 303
4.1Field-EffectTransistor(FET) 303
4.1.1 JFET (JunctionFET) 304
4.1.2MOSFET(Metal-Oxide-SemiconductorFET) 313
4.1.3MOSFETUsedasaResistor 327
4.1.4FETCurrentMirror 328
4.1.5MOSFETInverter 338
4.1.5.1NMOSInverterUsinganEnhancementNMOSasaLoad 342
4.1.5.2NMOSInverterUsingaDepletionNMOSasaLoad 347
4.1.5.3CMOSInverter 350
4.1.6Source-CoupledDifferentialPair 355
4.1.7CMOSLogicCircuits 359
4.2FETAmplifer 360
4.2.1Common-Source(CS)Amplifier 362
4.2.2CDAmplifier(SourceFollower) 366
4.2.3Common-Gate(CG)Amplifier 370
4.2.4Common-Source(CS)AmplifierwithFETLoad 373
4.2.4.1CSAmplifierwithanEnhancementFETLoad 373
4.2.4.2CSAmplifierwithaDepletionFETLoad 376
4.2.5MultistageFETAmplifiers 380
4.3DesignofFETAmplifier 398
4.3.1DesignofCSAmplifier 398
4.3.2DesignofCDAmplifier 405
4.4FETAmplifierFrequencyResponse 409
4.4.1CSAmplifier 410
4.4.2CDAmplifier(SourceFollower) 415
4.4.3CGAmplifier 419
4.5FETInverterTimeResponse 423 Problems 428
5OPAmpCircuits 467
5.1OPAmpBasics [Y-1] 468
5.2OPAmpCircuitswithResistors [Y-1] 471
5.2.1OPAmpCircuitswithNegativeFeedback 471
5.2.1.1InvertingOPAmpCircuit 471
5.2.1.2Non-InvertingOPAmpCircuit 473
5.2.1.3VoltageFollower 476
5.2.1.4LinearCombiner 477
5.2.2OPAmpCircuitswithPositiveFeedback 479
5.2.2.1InvertingPositiveFeedbackOPAmpCircuit 480
5.2.2.2Non-InvertingPositiveFeedbackOPAmpCircuit 481
5.3First-OrderOPAmpCircuits [Y-1] 485
5.3.1First-OrderOPAmpCircuitswithNegativeFeedback 485
5.3.2First-OrderOPAmpCircuitswithPositiveFeedback 487
5.3.2.1Square(Rectangular)-WaveGenerator 487
5.3.2.2Rectangular/Triangular-WaveGenerator 490
5.3.3555TimerUsingOPAmpasComparator 492
5.4Second-OrderOPAmpCircuits [Y-1] 495
5.4.1MFB(Multi-FeedBack)Topology 495
5.4.2Sallen-KeyTopology 496
5.5ActiveFilter [Y-1] 502
5.5.1First-OrderActiveFilter 502
5.5.2Second-OrderActiveLPF/HPF 503
5.5.3Second-OrderActiveBPF 505
5.5.4Second-OrderActiveBSF 507
Problems 512
6AnalogFilter 523
6.1AnalogFilterDesign 523
6.2Passive Filter 533
6.2.1 Low-passFilter(LPF) 533
6.2.1.1Series LR Circuit 533
6.2.1.2Series RC Circuit 535
6.2.2High-passFilter(HPF) 535
6.2.2.1Series CR Circuit 535
6.2.2.2Series RL Circuit 536
6.2.3Band-passFilter(BPF) 537
6.2.3.1SeriesResistor,anInductor,andaCapacitor(RLC) CircuitandSeries Resonance 536
6.2.3.2Parallel RLC CircuitandParallelResonance 539
6.2.4Band-stopFilter(BSF) 541
6.2.4.1Series RLC Circuit 541
6.2.4.2Parallel RLC Circuit 544
6.2.5QualityFactor 545
6.2.6InsertionLoss 549
6.2.7FrequencyScalingandTransformation 549
6.3PassiveFilterRealization 553
6.3.1 LC Ladder 553
6.3.2L-TypeImpedanceMatcher 561
6.3.3T-and П-TypeImpedanceMatchers 565
6.3.4Tapped-C ImpedanceMatchers 571
6.4ActiveFilterRealization 576 Problems 586
7SmithChartandImpedanceMatching 601
7.1TransmissionLine 601
7.2Smith Chart 608
7.3ImpedanceMatchingUsingSmithChart 616
7.3.1ReactanceEffectofaLosslessLine 616
7.3.2Single-StubImpedanceMatching 618
7.3.2.1Shunt-ConnectedSingleStub 618
7.3.2.2Series-ConnectedSingleStub 622
7.3.3Double-StubImpedanceMatching 626
7.3.4TheQuarter-WaveTransformer 631
7.3.4.1BinomialMultisectionQWT 633
7.3.4.2ChebyshevMultisectionQWT 634
7.3.5FilterImplementationUsingStubs [P-1] 635
7.3.6ImpedanceMatchingwithLumpedElements 646 Problems 661
8Two-PortNetworkandParameters 677
8.1Two-PortParameters [Y-1] 677
8.1.1DefinitionsandExamplesofTwo-PortParameters 678
8.1.2RelationshipsAmongTwo-PortParameters 685
8.1.3InterconnectionofTwo-PortNetworks 689
8.1.3.1SeriesConnectionand z-parameters 690
8.1.3.2Parallel(Shunt)Connectionand y-parameters 690
8.1.3.3Series-Parallel(Shunt)Connectionand h-parameters 691
8.1.3.4Parallel(Shunt)-SeriesConnectionand g-parameters 691
8.1.3.5CascadeConnectionand a-parameters 692
8.1.4CurseofPortCondition 692
8.1.5CircuitModelswithGivenParameters 697
8.1.5.1CircuitModelwithGiven z-parameters 697
8.1.5.2CircuitModelwithGiven y-parameters 699
8.1.5.3CircuitModelwithGiven a/b-parameters 699
8.1.5.4CircuitModelwithGiven h/g-parameters 699
8.1.6PropertiesofTwo-PortNetworkswithSource/Load 700
8.2ScatteringParameters 709
8.2.1DefinitionofScatteringParameters 709
8.2.2Two-PortNetworkwithSource/Load 714
8.3GainandStability 723
8.3.1Two-PortPowerGains [L-1,P-1] 723
8.3.2Stability [E-1,L-1,P-1] 728
8.3.3DesignforMaximumGain [M-2,P-1] 733
8.3.4DesignforSpecifiedGain [M-2,P-1] 740 Problems 746
AppendixALaplaceTransform 761
AppendixBMatrixOperationswithMATLAB 767
AppendixCComplexNumberOperationswithMATLAB 773
AppendixDNonlinear/DifferentialEquationswithMATLAB 775
AppendixESymbolicComputationswithMATLAB 779
AppendixFUsefulFormulas 783
AppendixGStandardValuesofResistors,Capacitors,andInductors 785
AppendixHOrCAD/PSpice® 791
AppendixIMATLAB® Introduction 831
AppendixJDiode/BJT/FET 835
Bibliography 845
Index 849
Preface
Theaimofthisbookisnottoletthereadersdrownedintoaseaofcomputations.Morehopefully,itaimstoinspirethereaderswithmindandstrength tomakefulluseoftheMATLABandPSpicesoftwaressothattheycanfeel comfortablewithmathematicalequationswithoutcaringabouthowtosolve themandfurthercanenjoydevelopingtheirabilitytoanalyze/designelectronic circuits.Itaimsalsotopresentthereaderswithasteppingstonetoradio frequency(RF)circuitdesignfromjunior–seniorleveltosenior-graduate levelbydemonstratinghowMATLABcanbeusedforthedesignandimplementationofmicrostripfilters.Thefeaturesofthisbookcanbesummarized asfollows:
1)Forrepresentativeexamplesofdesigning/analyzingelectroniccircuits,the analyticalsolutionsarepresentedtogetherwiththeresultsofMATLAB designandanalysis(basedonthetheory)andPSpicesimulation(similarto theexperiment)intheformoftrinity.Thisapproachgivesthereadersnot onlyinformationaboutthestateoftheart,butalsoconfidenceinthe legitimacyofthesolutionaslongasthesolutionsobtainedbyusingthetwo softwaretoolsagreewitheachother.
2)Forrepresentativeexamplesofimpedancematchingandfilterdesign,the solutionusingMATLABandthatusingSmithcharthavebeenpresented forcomparison/crosscheck.Thisapproachisexpectedtogivethereaders notonlyconfidenceinthelegitimacyofthesolution,butalsodeeper understandingofthesolution.
3)Thepurposesofthetwosoftwares,MATLABandPSpice,seemtobe overlappedanditispartlytrue.However,theycanbedifferentiatedsince MATLABismainlyusedtodesigncircuitsandperformapreliminary analysisof(designed)circuitswhilePSpiceismainlyusedfordetailedand almostreal-worldsimulationof(designed)circuits.
4)Especially,itpresentshowtouseMATLABandPSpicenotonlyfor designing/analyzingelectronicandRFcircuitsbutalsoforunderstanding theunderlyingprocessesandrelatedequationswithouthavingtostruggle withtime-consuming/error-pronecomputations.
Thecontentsofthisbookarederivedfromtheworksofmany(knownor unknown)greatscientists,scholars,andresearchers,allofwhomaredeeply appreciated.Wewouldliketothankthereviewersfortheirvaluablecomments andsuggestions,whichcontributetoenrichingthisbook.
WealsothankthepeopleoftheSchoolofElectronicandElectricalEngineering,Chung-AngUniversityforgivingusanacademicenvironment.Without affectionsandsupportsofourfamiliesandfriends,thisbookcouldnotbe written.Wegratefullyacknowledgetheeditorial,BrettKurzmanandproductionstaffofJohnWiley&Sons,Inc.includingProjectEditorAntonySami andProductionEditorViniprammiaPremkumarfortheirkind,efficient,and encouragingguide.
Programfilescanbedownloadedfromhttps://wyyang53.wixsite.com/mysite/ publications.Anyquestions,comments,andsuggestionsregardingthisbook arewelcomeandtheyshouldbemailedtowyyang53@hanmail.net. WonY.Yangetal.
AbouttheCompanionWebsite
Donotforgettovisitthecompanionwebsiteforthisbook:
www.wiley.com/go/yang/electroniccircuits
ScanthisQRcodetovisitthecompanionwebsite.
Thereyouwillfindvaluablematerialdesignedtoenhanceyourlearning, includingthefollowing:
• LearningOutcomesforallchapters
• Exercisesforallchapters
• Referencesforallchapters
• Furtherreadingforallchapters
• FiguresforChapters16,22,and30
LoadLineAnalysisandFourierSeries
CHAPTEROUTLINE
1.1LoadLineAnalysis,1
1.1.1LoadLineAnalysisofaNonlinearResistorCircuit,3
1.1.2LoadLineAnalysisofaNonlinear RL circuit,7
1.2Voltage-CurrentSourceTransformation,10
1.3Thevenin/NortonEquivalentCircuits,11
1.4Miller’sTheorem,18
1.5FourierSeries,18
1.5.1ComputationofFourierCoefficientsUsingSymmetry,20
1.5.2CircuitAnalysisUsingFourierSeries,29
1.5.3RMSValueandDistortionFactorofaNon-SinusoidalPeriodicSignal,35 Problems,36
1.1LoadLineAnalysis
The v-i characteristicofanonlinearresistorsuchasadiodeoratransistoris oftendescribedbyacurveonthe v-i planeratherthanbyamathematicalrelation.The v-i characteristiccurvecanbeobtainedbyusingacurvetracerfor nonlinearresistors.Toanalyzecircuitscontaininganonlinearresistor,we shouldusethe loadlineanalysis.Tograsptheconceptoftheloadline,consider thegraphicalanalysisofthecircuitinFigure1.1(a),whichconsistsofalinear resistor R1,anonlinearresistor R2,aDCvoltagesource Vs,andanACvoltage sourceofsmallamplitude vδ Vs.Kirchhoff’svoltagelaw(KVL)canbeapplied aroundthemeshtoyieldthemeshequationas
ElectronicCircuitswithMATLAB®,PSpice®,andSmithChart,FirstEdition.WonY.Yang, JaekwonKim,KyungW.Park,DonghyunBaek,SungjoonLim,JingonJoung,SuhyunPark, HanL.Lee,WooJuneChoi,andTaehoIm.
©2020JohnWiley&Sons,Inc.Published2020byJohnWiley&Sons,Inc. Companionwebsite:www.wiley.com/go/yang/electroniccircuits
Nonlinear resistor circuit
Graphical analysis method 1
(Load line) Q Operating point Slope
The characteristic curve of the nonlinear resistor
Graphical analysis method 2 using load line
wherethe v-i relationshipof R2 isdenotedby v2(i)andrepresentedbythecharacteristiccurveinFigure1.1(b).Wewillconsideragraphicalmethod,which yieldsthe quiescent, operating,or biaspoint Q =(IQ, VQ),thatis,apairof thecurrentthroughandthevoltageacross R2 for vδ =0.
Sincenospecificmathematicalexpressionof v2(i)isgiven,wecannotuseany analyticalmethodtosolvethisequationandthatiswhywearegoingtoresortto agraphicalmethod.First,wemaythinkofplottingthegraphfortheLHS(lefthandside)ofEq.(1.1.1)andfindingitsintersectionwithahorizontallineforthe RHS(right-handside),thatis, v = Vs asdepictedinFigure1.1(b).Anotherwayis toleaveonlythenonlineartermontheLHSandmovetheotherterm(s)intothe RHStorewriteEq.(1.1.1)as
Figure1.1 Graphicalanalysisofalinear/nonlinearresistorcircuit.
1.1LoadLineAnalysis 3
andfindtheintersection,calledthe operatingpoint anddenotedby Q (quiescent point),ofthegraphsforbothsidesasdepictedinFigure1.1(c).Thestraightline withtheslopeof R1 iscalledthe loadline.Thisgraphicalmethodisbetterthan thefirstoneintheaspectthatitdoesnotrequireustoplotanewcurvefor v2(i) + R1i.Thatiswhyitiswidelyusedtoanalyzenonlinearresistorcircuitsinthe nameof ‘loadlineanalysis’.Notethefollowing:
• Mostresistorsappearinginthisbookarelinearinthesensethattheirvoltages arelinearlyproportionaltotheircurrentssothattheirvoltage-current relationships(VCRs)aredescribedbyOhm’slaw
andconsequently,their v-i characteristicsaredescribedbystraightlinespassingthroughtheoriginwiththeslopescorrespondingtotheirresistanceson the i-v plane.However,theymayhavebeenmodeledorapproximatedtobe linearjustforsimplicityandconvenience,becauseallphysicalresistorsmore orlessexhibitsomenonlinearcharacteristic.Theproblemiswhetherornot themodelingisvalidintherangeofpracticaloperationsothatitmayyieldthe solutionwithsufficientaccuracytoservetheobjectiveofanalysisanddesign.
• Acurvetracerisaninstrumentthatdisplaysthe v-i characteristiccurveofan electricelementonacathode-raytube(CRT)whentheelementisinserted intoanappropriatereceptacle.
1.1.1LoadLineAnalysisofaNonlinearResistorCircuit
ConsiderthecircuitinFigure1.1(a),wherealinearloadresistor R1 = RL anda nonlinearresistor R connectedinseriesaredrivenbyaDCvoltagesource Vs inserieswithasmall-amplitudeACvoltagesourceproducingthevirtualvoltage as
TheVCR v(i)ofthenonlinearresistor R isdescribedbythecharacteristiccurve inFigure1.2.
AsdepictedinFigure1.2,theupper/lowerlimitsaswellastheequilibrium valueofthecurrent i throughthecircuitcanbeobtainedfromthethreeoperatingpoints,thatis,theintersections(Q1, Q,and Q2)ofthecharacteristiccurve withthefollowingthreeloadlines.
Figure1.2 Variationofthevoltageandcurrentofanonlinearresistoraroundtheoperating point Q.
Althoughthisapproachgivestheexactsolution,wegainnoinsightintothe solutionfromit.Instead,wetakearatherapproximateapproach,whichconsists ofthefollowingtwosteps.
• Findtheequilibrium(IQ, VQ)atthemajoroperatingpoint Q,whichisthe intersectionofthecharacteristiccurvewiththeDCloadline(1.1.5b).
• Findthetwoapproximateminoroperatingpoints Q1 and Q2 fromtheintersectionsofthetangenttothecharacteristiccurveat Q withthetwominor loadlines(1.1.5a)and(1.1.5c).
Thenwewillhavethecurrentas
Withthe dynamic, small-signal,or ACresistancerd definedtobetheslopeof thetangenttothecharacteristiccurveat Q as
letusfindtheanalyticalexpressionsof IQ and iδ intermsof Vs and vδ,respectively.ReferringtotheencircledareaaroundtheoperatingpointinFigure1.2, wecanexpress iδ intermsof vδ as
1.1LoadLineAnalysis
Thiscorrespondstoapproximatingthecharacteristiccurveintheoperation rangebyitstangentattheoperatingpoint.Notingthat
• theloadlineandthetangenttothecharacteristiccurveat Q areatanglesof (180 θ L)and θ tothepositive i-axis,
• theslopeoftheloadlineistan(180 θ L)= tan θ L anditmustbe RL,which istheproportionalitycoefficientin i oftheloadlineEq.(1.1.2);tan θ L =RL,and
• theslopeofthetangenttothecharacteristiccurveat Q isthedynamicresistance rd definedbyEq.(1.1.7);tan θ = rd, wecanwriteEq.(1.1.8)as
Nowwedefinethe static or DCresistance ofthenonlinearresistor R tobethe ratioofthevoltage VQ tothecurrent IQ attheoperatingpoint Q as
sothattheDCcomponentofthecurrent, IQ,canbewrittenas
Finally,wecombinetheaboveresultstowritethecurrentthroughandthe voltageacrossthenonlinearresistor R asfollows.
Thisresultimpliesthatthenonlinearresistorexhibitstwofoldresistance,thatis, the staticresistanceRs toaDCinputandthe dynamicresistancerd toanAC inputofsmallamplitude.Thatiswhy rd isalsocalledthe(small-signal) AC resistance,while Rs iscalledthe DCresistance.
Remark1.1OperatingPointandStatic/DynamicResistances ofaNonlinearResistor
1) Foranonlinearresistor R2 connectedwithlinearresistorsinacircuitexcited byaDCsourceandasmall-amplitudeACsource,its operatingpointQ = (VQ, IQ)istheintersectionofitscharacteristiccurve v(i)andtheloadline.
2) The v-interceptoftheloadline(v = Vs RLi)isdeterminedbytheDCcomponent(Vs)ofthevoltagesource.Theslopeoftheloadlineisdeterminedby theequivalentresistance(RL)ofthelinearpartseenfromthepairofterminalsofthenonlinearresistor(seeProblem1.2).
3) The static or DCresistance (Rs)istheratioofthevoltage VQ tothecurrent IQ attheoperatingpoint Q.
4) The dynamic, small-signal, AC,or incrementalresistance (rd)istheslopeof thetangenttothecharacteristiccurveat Q.
5) Oncewehave RL, Rs,and rd,wecanuseFormulas(1.1.12)and(1.1.13)tofind approximateexpressionsforthevoltageandcurrentofthenonlinear resistor.
6) Asforlinearresistors,wedonotsaythestaticordynamicresistance,since theyareidentical.
7) TherelationshipbetweentheAC(small-signal)componentsofvoltage acrossandcurrentthroughthenonlinearresistorcanbeattributedtothe TaylorseriesexpansionofitsVCR v(i)uptothefirst-ordertermaround theoperatingpoint Q =(VQ, IQ).
Remark1.2DCAnalysisandSmall-Signal(AC)Analysis
1) Theproceduretoanalyzeacircuit(whichcontainsnonlinearresistorslikea diodeoratransistorandisdrivenbyahighDCvoltagesource Vs [forbiasing] andalowACvoltagesource vδ sin ωt [foramplification])consistsoftwo steps.Thefirststep,calledDCanalysis,istoremovetheACvoltagesource vδ sin ωt andfindtheoperatingpoint Q =(VQ, IQ)ofthenonlinearresistor, whichcorrespondstotheloadlineanalysis.Thesecondstep,calledsmallsignal(AC)analysis,istofindthedynamicresistance rd ofthenonlinear resistor(fromtheslopeofits i-v characteristiccurveorthederivativeof itsVCRequationat Q-point),removetheDCvoltagesource Vs,regard thenonlinearresistorasalinearresistor rd (correspondingtoalinear 1LoadLineAnalysisandFourierSeries
approximationofthecharacteristiccurve),andfindtheACcomponents(iδ sin ωt, rdiδ sin ωt)ofthecurrentthroughandvoltageacrossthenonlinear resistor.TheDCsolutionandACsolutioncanbeaddeduptoyieldthecompletesolution.
2) Asthemagnitude vδ oftheACvoltagebecomeslarge,thelarge-signalmodel (seeSection2.1.1foradiode)orthecharacteristiccurveitselfmighthaveto beusedforanalysissincethenonlinearbehaviorofthenonlinearresistor maybecomeconspicuous,leadingtounignorabledistortionofthevoltage/currentwaveformsobtainedusingthesmall-signalanalysis.
1.1.2LoadLineAnalysisofaNonlinear RL circuit
Asanexampleofapplyingtheloadlineanalysisforanonlinearfirst-ordercircuit,considerthecircuitofFigure1.3.1(a),whichconsistsofanonlinearresistor,alinearresistor R =2 Ω,andaninductor L =14H,andisdrivenbyaDC voltagesourceof Vs =12VandanACvoltagesource vδ sin ωt =2.8sin t[V].
The v-i relationshipofthenonlinearresistoris v(i)= i3 anddescribedbythe characteristiccurveinFigure1.3.1(b).ApplyingKVLyieldsthefollowingmesh equation:
(1.1.15)
First,wecandrawtheloadlineonthegraphofFigure1.3.1(b)tofindthe operatingpoint Q fromtheintersectionoftheloadlineandthe v-i characteristic curveofthenonlinearresistor.Ifthefunction v(i)= i3 ofthecharacteristiccurve isavailable,wecanalsofindtheoperatingpoint Q astheDCsolutionto Eq.(1.1.15)byremovingtheACsourceandthetimederivativetermtowrite
+ I 3
=12 (1.1.16) andsolvingitas IQ =2A, VQ = v2 IQ = I 3 Q =8V Q = IQ , VQ =2A,8V (1.1.17)
Eq.(1.1.16)canbesolvedbyrunningthefollowingMATLABstatements: >>eq_dc=@(i)2*i+i.^3-12;I0=0;IQ=fsolve(eq_dc,I0)
Then,asapreparationforanalyticalapproach,welinearizethenonlinear differentialequation(1.1.15)aroundtheoperatingpoint Q bysubstituting i = IQ + δi =2+ δi intoitandneglectingthesecondorhigherdegreetermsin δi as
Notethatwecansettheslopeofthecharacteristiccurveasthedynamicresistance rd ofthenonlinearresistor:
(1.1.19)
andapplyKVLtothecircuitwiththeDCsource Vs removedandthenonlinear resistorreplacedby rd towritethesamelinearizedequationasEq.(1.1.18):
Wesolvethefirst-orderlineardifferentialequationwithzeroinitialcondition δi(0)=0toget δi(t)byrunningthefollowingMATLABstatements: >>symss
dIs=2.8/(s^2+1)/14/(s+1);dit_linearized=ilaplace(dIs) dit1_linearized=dsolve('Dx=-x+0.2*sin(t)','x(0)=0')%Alternatively
Thisyields
dit_linearized=exp(-t)/10-cos(t)/10+sin(t)/10
whichmeans
WeaddthisACsolutiontotheDCsolution IQ towritetheapproximateanalyticalsolutionfor i(t)as
Now,referringtoAppendixD,weusetheMATLABnumericaldifferential equation(DE)solver ‘ode45()’ tosolvethefirst-ordernonlineardifferential equation(1.1.15)bydefiningitasananonymousfunctionhandle:
>>di=@(t,i)(12+2.8*sin(t)-2*i-i.^3)/14;%Eq.(1.1.15) andthenrunningthefollowingMATLABstatements:
>>i0=IQ;tspan=[010];%InitialvalueandTimespan [t,i_numerical]=ode45(di,tspan,i0);%Numericalsolution Wecanalsoplotthenumericalsolutiontogetherwiththeanalyticalsolution asblackandredlines,respectively,byrunningthefollowingMATLAB statements:
>>i_linearized=eval(IQ+dit_linearized);%Analyticalsolution(1.1.22) plot(t,i_numerical,'k',t,i_linearized,'r')
Nonlinear solution i(t) Linearized solution i(t)
Figure1.3.2 Thelinearizedsolutionandnonlinearsolutionforthenonlinear RL circuitof Figure1.3.1.
%elec01f03.m – fortheanalysisofanonlinearRLCircuit clear,clf
globalRLLVsvd
N=1000;i_step=0.003;i=[0:N]*i_step;%Rangeonthecurrentaxis RL=2;L=14;Vs=12;vd=2.8; eq_dc=@(i,RL,Vs)Vs-RL*i-i.^3;%Eq.(1.1.16):DCpartofEq.(1.1.20) IQ=fsolve(eq_dc,0,optimset('fsolve'),RL,Vs)%CurrentatoperatingpointQ VQ=IQ^3;%VoltageattheQ-point v=Vs-RL*i;%Loadline v2=i.^3;%Characteristiccurve
i1=1.7:i_step:2.3;%Rangeonwhichtoplotthetangentline v4=3*IQ^2*(i1-IQ)+VQ;%TangentlinetothecharacteristiccurveatQ subplot(211),plot(i,v,'k',i,v2,'b',i1,v4,'r',IQ,VQ,'mo') %UsetheLaplacetransformtosolvethelinearizeddifferentialeq. symss
dIs=0.2/(s^2+1)/(s+1);dit_linearized=ilaplace(dIs) %Thisyields0.1*(exp(-t)-cos(t)+sin(t))+2;Eq.(1.1.22). %Alternatively,usethesymbolicdifferentialsolverdsolve()as dit1_linearized=dsolve('Dx=-x+0.2*sin(t)','x(0)=0') %UsenonlinearODEsolverode45()tosolveEq.(1.1.15). di=@(t,i)(12+2.8*sin(t)-2*i-i.^3)/14;%Eq.(1.1.15) [t,i]=ode45(di,[010],IQ);%NumericalsoltoEq.(1.1.15) i_linearized=eval(IQ+dit_linearized);%Eq.(1.1.22)fortimeranget %PlottheAnalytical(linearized)andNumerical(nonlinear)solutions. subplot(212),plot(t,i,'k',t,i_linearized,'r'),ylabel('i(t)') legend('Nonlinearsolutioni(t)','Linearizedsolutioni(t)') title('Analytical(linearized)andNumerical(nonlinear)solutions')
Thiswillyieldtheplotsofthenumericalsolution i(t)andtheapproximate analyticalsolution(1.1.22)forthetimeinterval[0,10s]asdepictedin Figure1.3.2.
Overall,wecanruntheaboveMATLABscript “elec01f03.m” toget Figure1.3.1(b)(theloadlinetogetherwiththecharacteristiccurve)and Figure1.3.2(thenumericalnonlinearsolutiontogetherwiththeanalyticallinearizedsolution)together.
