Marshall Leach Jr Copyright 19942009 All rights reserved Abstract Many introductory circuits texts state or imply that superposition of dependent sources cannot be used in linear circuit analysis Although the use of superposition of only indepen den ID: 26637
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OntheApplicationofSuperpositiontoDependentSourcesinCircuitAnalysisW.MarshallLeach,Jr.Copyright1994-2009.Allrightsreserved.AbstractManyintroductorycircuitstextsstateorimplythatsuperpositionofdependentsourcescannotbeusedinlinearcircuitanalysis.Althoughtheuseofsuperpositionofonlyindependentsourcesleadstothecorrectsolution,itdoesnotmakeuseofthefullpowerofsuperposition.Theuseofsuperpositionofdependentsourcesoftenleadstoasimplersolutionthanothertechniquesofcircuitanalysis.Aformalproofispresentedthatsuperpositionofdependentsourcesisvalidprovidedthecontrollingvariableisnotsettozerowhenthesourceisdeactivated.Severalexamplesaregivenwhichillustratethetechnique.IndexTermsCircuitanalysis,superposition,dependentsources,controlledsourcesI.PREFACEWhenhewasasophomoreincollege,theauthorwasre-quiredtotaketwosemestersofcircuittheory.ThetextwasareproductionofasetoftypewrittennoteswrittenbyRonalE.Scottwhichwerepublishedlaterthatyearasahardboundtext[18].Thechapterthatcoveredsuperpositionhadanum-berofproblemswherethestudentwasinstructedtowritebyinspectionthesolutionforavoltageoracurrentbyusingonlysuperposition,Ohmslaw,voltagedivision,andcurrentdivi-sion.Theauthorfoundtheseproblemstobefascinatingandhspenthoursmasteringthem.Indescribingtheprincipleofsuperposition,Scottonlyhaindependentsourcesinhiscircuits.However,theauthorfoundthathecouldalsosolvecircuitscontainingcontrolledsourcesusingsuperposition.Later,afterbecomingateacher,hewachallengedbystudentswhenheusedthesetechniquesinteach-ingelectronicscourses.Thestudentssaidthattheyhadbeetaughtthatsuperpositionwithcontrolledsourceswasnotal-lowed.Thestudentswereeasilysatisedwhentheywereshownthatanodevoltageanalysisyieldsthesamesolution.Afterencounteringsomanychallengesbystudents,theau-thorresearchedcircuitsbooksinthelibraryathisschoolandatastoreofalargebookstorechain.Noneofthebookssaidthatsuperpositioncanbeusedwithcontrolledsources.Indeed,themajoritystatedclearlythatitcouldnot.Thinkingthatothereducatorsmightndthetopictobeinterestinganduseful,theauthorsubmittedapaperonthetopicttheIEEETrans.onEducationinearly1994.Itwasrejectedaf-terthereviewerconceivedacircuitthatitcouldnotbeappliedto.ThereviewerscircuitcontainedaoatingnodeatwhichTheauthoriswiththeSchoolofElectricalandComputerEngineeringattheGeorgiaInstituteofTechnology,Atlanta,GA30332-0250USA.Email:mleach@ece.gatech.edu.thevoltagewasindeterminate.Thecircuitcouldnotbeana-lyzedbyanyconventionaltechnique.Arequesttotheeditortohaveitreviewedagainbyanotherreviewerwentunanswered.ThepaperwasthensubmittedtotheIEEETrans.onCir-cuitsandSystemsin1995.Thereviewerrecommendedseveralchanges.Thepaperwasrevisedandsubmittedasecondtimein1996.ThereviewerthenrecommendedthatitbesubmittedinsteadtotheIEEETrans.onEducation.Itwassubmittedasecondtimetothelatterjournal.Theeditorrejecteditwithouthavingitpeerreviewed.Theauthorconcludedthatthepaperwasofnointeresttoeducators,sohepublisheditontheinter-netinlate1996.Acopyoftherstreviewisincludedattheendofthisdocument.II.INTRODUCTIONTheauthorhasinvestigatedthepresentationofsuperposi-tionincircuitstextsbysurveyingtwentyintroductorybooksoncircuitanalysis[1]-[20].Fourteenexplicitlystatethatifadependentsourceispresent,itisneverdeactivatedandmustre-mainactive(unaltered)duringthesuperpositionprocess.Theremainingsixspecicallyrefertothesourcesasbeingindependentinstatingtheprincipleofsuperposition.Threeofthesepresentanexamplecircuitcontainingadependentsourcewhichisneverdeactivated.Theotherthreedonotpresentanexam-pleinwhichdependentsourcesarepresent.Fromthislimitesurvey,itisclearthatcircuitstextseitherstateorimplythatsuperpositionofdependentsourcesisnotallowed.Theauthorcontendsthatthisisamisconception.Asimpleargumentcanbeusedtojustifythispremise.Sup-posealinearcircuitcontainingindependentanddependentsourcesisanalyzedbyanymeansotherthansuperposition.Boththeoutputofthecircuitandthevalueofeachdependentsourcearesolvedfor.Solvethecircuitasecondtimeusingsu-perpositionoftheindependentanddependentsources,treatingthedependentsourcesasindependentsourceshavingthevaluesfoundintherstsolution.Itisclearthatthesameresulmustbeobtained.Thussuperpositionmustholdwiththede-pendentsourcesiftheirvaluesareknown.Itcanthenbearguedthatsuperpositionmustholdevenifthevaluesofthedependentsourcesarenotknown,providedtheyaretreatedasindepen-dentsources.Ofcourse,theoutputofthecircuitcannotbedetermineduntiltheirvaluesareknown,butthesevaluescabecalculatedaspartofthesuperposition.Toapplysuperpositiontodependentsources,thecontrollingvariablesmustnotbesettozerowhenasourceisdeactivatedThisisillustratedinthefollowingwithseveralexamples.When allsourcesbutonearedeactivated,thecircuitmustcontainnonodesatwhichthevoltageisindeterminateandnobranchesinwhichthecurrentisindeterminate.Thisspecicallyrulesoutcaseswherecurrentsourcesareconnectedinseriesorvolt-agesourcesareconnectedinparallel.Insuchcases,superpo-sitioncannotbeusedevenifallthesourcesareindependentForexample,iftwocurrentsourceshavingthesamecurrentareconnectedinseries,thevoltageatthecommonnodebe-tweenthesourcesisindeterminate.Iftwovoltagesourceshav-ingthesamevoltageareconnectedinparallel,thecurrentieachsourceisindeterminate.Becausethetechniquesthataredescribedinthispaperarecountertothosepresentedinallofthetextsthattheauthorhasexamined,thequestionarisesastowhichtechniqueisthemosteffectiveasateachingtool.Theauthorhasnoquantitativewayofevaluatingthis.However,hehasusedthemethodssuccessfullyoveraperiodofmanyyearsinteachingjuniorandseniolevelelectronicscoursesattheGeorgiaInstituteofTechnology.Hehasreceivednothingbutpositiveresponsesfromstudents.III.THERINCIPLEOFUPERPOSITIONAproofoftheprincipleofsuperpositionispresentedin[18],whereallsourcesareconsideredtobeindependent.Withsommodications,theproofpresentedherefollowstheonein[18],butitassumesthatdependentsourcesarepresent.Thestartingpointistoassumeageneralsetofmeshornodeequationsforanygivenlinearcircuit.Nodeequationsareassumedhere.Ithecircuitcontainsvoltagesources,theymustrstbeconvertedintoequivalentcurrentsourcesbymakingNortonequivalencircuits.Suchatransformationdoesnotchangetheresponsofthenetworkexternaltothesource.Inthecaseofavolt-agesourcewithnoseriesresistance,atransformationwhichin[18]iscalledpushingavoltagesourcethroughanodemustrstbeperformed.Thisreplacesavoltagesourcehavingnose-riesresistancewithseveralvoltagesources,oneineachofthebranchesradiatingfromthenodetowhichtheoriginalsourcconnects.Thegeneralnodeequationsforalinearcircuitcontainingnodescanbewritten(1)whereisthesumofthecurrentsdeliveredtonodebybothindependentanddependentsources,isthevoltageatnodeisthetotaladmittanceradiatingfromnode,andistheadmittancebetweennodesand.Theequationscanbewrit-teninthematrixform,whereisanadmit-tancematrix.Becausethedependentsourcesarecontainedithecurrentvector,thematrixcorrespondstowhatiscalledthebranchadmittancematrix.Thisistheadmittancematrixwithalldependentsourcesdeactivated.Itissymmetricalaboutthemaindiagonal,themaindiagonaltermsareallpositive,andalloffdiagonaltermsarenegative.Adeterminantsolutionforcanbewritten (2)whereisthedeterminant(3)Acofactorexpansionof(2)yields (4)whereisthedeterminantformedbydeletingrowandcol-umnin.Similarsolutionsfollowfortheothernodevolt-ages.Eachtermin(4)isidenticaltothetermwhichwouldbewrit-tenifonlythecurrentisactive,thusprovingtheprincipleofsuperposition.Buteachcanbeasumofbothindependentanddependentcurrentsources.Itfollowsthatsuperpositionap-pliestoeachofthesesources.Ifasourceconnectsfromnodetothedatumnode,itscurrentappearsinthesuperpositionaonlyonenode.Ifasourceconnectsbetweentwonodes,neitheofwhichisthedatumnode,itscurrentappearsinthesuperpositionattwonodes.Inthiscase,superpositioncanbeappliedtoeachsideofthesourcebytreatingthetwosidesasseparatcurrentsources,eachofwhichcanbeturnedonandoffinde-pendently.Theaboveproofdoesnotimplythecontrollingvariablesofadependentsourcearedeactivatedwhenapplyingsuperpositiontothesource.Onlytheoutputofthesourceissettozero.Thiproceduremakesitispossibletowritecircuitequationsbycon-sideringonlyonesourceatatimeortoonesideofasourceatatime.Thisconsiderablysimpliestheuseofsuperpositionwithdependentsourcescomparedtothewaythatitispresentedinmostcircuitstexts.IV.EXAMPLESThefollowingexamplesillustratetheproperuseofsuper-positionofdependentsources.Allsuperpositionequationsarewrittenbyinspectionusingvoltagedivision,currentdivision,series-parallelcombinations,andOhmslaw.Ineachcase,itissimplernottousesuperpositionifthedependentsourcesre-mainactive.Someoftheexamplesaretakenfromtextscitedithereferences.A.Example1Thisexamplecomesfrom[7].TheobjectistosolveforthecurrentinthecircuitofFig.1.Bysuperposition,onecanwrite \n 3 Fig.1.Circuitforexample1.Solutionforyields Ifthecontrolledsourceisnotzeroedinthesuperposition,twosolutionsmustbefound.Let,whereisthesolutionwiththe sourcezeroedandisthesolutionwiththe sourcezeroed.Wecanwritetwoloopequations.\rwhichyields \r whichyields Thetotalsolutionis Thisisthesameanswerobtainedbyusingsuperpositionofthcontrolledsource.B.Example2Thisexamplecomesfrom[9].TheobjectistosolveforthevoltagesandacrossthecurrentsourcesinFig.2,wherethedatumnodeisthelowerbranch.Bysuperposition,thecurrentisgivenby \n\n \n\n \n \n\n Solutionforyields Althoughsuperpositioncanbeusedtosolveforand,itissimplertowrite\n \r\n Ifsuperpositionisnotusedatall,thecircuitequationsar \n \n \n Thesethreeequationscanbesolvedsimultaneouslyforand.Ifsuperpositionoftheindependentsourcesisused,twosolutionsmustbefound.Letand,wherethesubscriptdenotesthesolutionwiththe sourcezeroedandthesubscriptdenotesthesolutionwiththe sourcezeroed.Thenodevoltageequationsare \n \n \n \n Fig.2.Circuitforexample2. \n \n \n Itshouldbeobviouswithoutgoingfurtherthatnotusingsuper-positionatallleadstoasolutionwithlesswork.However,therstsolutionthatincludedthedependentsourceinthesuperpo-sitionissimpler.C.Example3Thisexamplecomesfrom[4].TheobjectistosolveforthecurrentinthecircuitofFig.3.Bysuperposition,onecanwrite Solutionforyields Fig.3.Circuitforexample3.D.Example4Thisexamplecomesfrom[4].TheobjecttosolvefortheThéveninequivalentcircuitseenlookingintotheterminalinthecircuitofFig.4.Bysuperposition,thevoltageisgivenby\r\r\n \r whereisthecurrentdrawnbyanyexternalloadandthesym-boldenotesaparallelcombination.Solutionforyields \r\n\r 4 Fig.4.Circuitforexample4.Althoughsuperpositioncanbeusedtosolvefor,itissim-plertowriteItfollowsthattheThéveninequivalentcircuitconsistsof sourceinserieswitharesistor.ThecircuitisshowninFig.5. Fig.5.Théveninequivalentcircuit.E.Example5Thisexamplecomesfrom[15].TheobjectistosolveforthevoltageinthecircuitofFig.6.Bysuperposition,thecurrentisgivenby \n \n \n Solutionforyields\n\n Althoughsuperpositioncanbeusedtosolvefor,itissimplertowrite F.Example6Thisexamplecomesfrom[15].TheobjectistosolveforthevoltageinthecircuitofFig.7.Bysuperposition,thevoltageisgivenby\n Fig.6.Circuitforexample5.Thiscanbesolvedfortoobtain Bysuperposition,isgivenby \n \n \n \n \nThusisgivenby Fig.7.Circuitforexample6.G.Example7Thisexamplecomesfrom[13]inwhichitisstatedinbold,redletters,Sourcesuperpositioncannotbeusedfordependentsources.TheobjectistosolveforthevoltageasafunctionofandinthecircuitinFig.8.Bysuperposition,thecurrentisgivenby \n \n Thiscanbesolvedfortoobtain Bysuperposition,thevoltageisgivenby \n \n \n \n \n \n 5 Fig.8.CircuitforExample7.H.Example8ThisexampleillustratestheuseofsuperpositioninsolvinforthedcbiascurrentsinaBJT.TheobjectistosolveforthecollectorcurrentinthecircuitofFig.9.Althoughnoex-plicitdependentsourcesareshown,thethreeBJTcurrentsarerelatedby \r ,whereisthecurrentgainand \r.Ifanyoneofthecurrentsiszero,theothertwomustalsobezero.However,thecurrentscanbetreatedasindependentvariablesinusingsuperposition. Fig.9.Circuitforexample8.Bysuperpositionof ,and,thevoltageisgivenby \r Anode-voltagesolutionforrequiresthesolutionoftwosi-multaneousequationstoobtainthesameanswerwhichsuper-positionyieldsbyinspection.Thisequationandtheequation\n canbesolvedfortoobtain \n \r Inmostcontemporaryelectronicstexts,thevalue\n isassumedinBJTbiascalculations.I.Example9Thisexampleillustratestheuseofsuperpositiontosolveforthesmall-signalbaseinputresistanceofaBJT.Fig.10showthesmall-signalBJThybrid-pimodelwithacurrentsourceconnectedfromthebasetoground,aresistorfromemittertoground,andaresistorfromcollectortoground.Inthemodel, and\r ,whereisthethermalvoltage,isthedcbasecurrent,istheEarlyvoltage, isthedccollector-emittervoltage,andisthedccollectorcurrent.Thebaseinputresistanceisgivenby Fig.10.Circuitforexample9.Bysuperpositionofand ,thebasevoltageisgivenby Thiscanbesolvedforthebaseinputresistancetoobtain whichsimpliesto\r Anode-voltagesolutionforrequiresthesolutionofthreesi-multaneousequationstoobtainthesameanswerwhichfollowalmosttriviallybysuperposition.J.Example10Thisexampleillustratestheuseofsuperpositiontosolveforthesmall-signalcollectorinputresistanceofaBJT.Fig.1showsthesmall-signalBJThybrid-pimodelwithacurrentsourceconnectedfromcollectortoground,aresistorfrombasetoground,andaresistorfromemittertoground.Inthemodel, and\r ,whereisthethermalvoltage,isthedcbasecurrent,istheEarlyvoltage, isthedccollector-emittervoltage,andisthedccollectorcurrent.Thecollectorinputresistanceisgivenby 6 Fig.11.CircuitforExample10.Bysuperpositionofand ,thecollectorvoltageisgivenby\r Currentdivisioncanbeusedtoexpressinthisequationintermsofasfollows: Substitutionofthisequationforintothetheequationforyields\r Itfollowsthatthecollectorresistanceisgivenby \rThedrainresistanceofaFETcanbeobtainedfromthisexpressionbyusingtherelation andtakingthelimitastoobtain\rInthisequation,theresistanceinserieswiththeFETsourcereplacestheresistanceinserieswiththeBJTemitter.K.Example11Thisexampleillustratestheuseofsuperpositiontosolveforthesmall-signalemitterinputresistanceofaBJT.Fig.12showsthesmall-signalBJThybrid-pimodelwithavoltagesourceconnectedfromemittertoground,aresistorfrombasetoground,andaresistorfromcollectortoground.Inthemodel, and\r ,whereisthethermalvoltage,isthedcbasecurrent,istheEarlyvoltage, isthedccollector-emittervoltage,andisthedccollectorcurrent.TheemitterinputresistanceisgivenbyBysuperpositionofand ,theemittercurrentisgivenby Thebasecurrentisafunctionofonlyandisgivenby \n Fig.12.CircuitforExample11.Substitutionofthisequationforintothetheequationforyields \r\n\r \n Itfollowsthattheemitterresistanceisgivenby \r\n\r If,theequationfortheemitterresistancebecomes\r\rThesourceresistanceofaFETcanbeobtainedfromthisexpressionbyusingtherelation andtakingthelimitastoobtain Inthisequation,theresistanceinserieswiththeFETdrainreplacestheresistanceinserieswiththeBJTcollector.If,theequationforthesourceresistancebecomesL.Example12Thisexampleillustratestheuseofsuperpositionwithanopampcircuit.ThecircuitisshowninFig.13.Theobjectistosolvefor.With,itfollowsthatand.Bysuperpositionofandcanbewritten \n \n With,itfollowsthat,and.Bysuperpositionofandcanbewrit-ten \n \n \n Thusthetotalexpressionforis \n \n \n Fig.13.CircuitforExample12.M.Example13Figure14showsacircuitthatmightbeencounteredinthenoiseanalysisofampliers.Theamplierismodeledbyaparametermodel.Thesquaresourcesrepresentnoisesources.and,respectively,modelthethermalnoisegeneratedbyandandmodelthenoisegeneratedbytheamplier.TheamplierloadisanopencircuitsothatTheopen-circuitoutputvoltageisgivenbyBysuperposition,thecurrentsandaregivenby Notethatwhen,thesources,andareinseriesandcanbeconsideredtobeonesourceequaltothesumofthethree.Whenthesearesubstitutedintotheequationforandtheequationissimplied,weobtain Fig.14.CircuitforExample13.N.Example14Itiscommonlybelievedthatsuperpositioncanonlybeusedwithcircuitsthathavemorethanonesource.Thisexampleillustrateshowitcanbeusewithacircuithavingone.Considertherst-orderall-passltershowninFig.15(a).AnequivalentcircuitisshowninFig.15(b)inwhichsuperpositioncanbeusedtowritebyinspection Fig.15.CircuitforExample14.V.CONCLUSIONSuperpositionofdependentsourcesisvalidinwritingcircuitequationsifitisappliedcorrectly.Often,itcanbeusedtoob-tainsolutionsalmosttriviallybyinspection.Whenallsourcesbutonearedeactivated,thecircuitmustnotcontainanodea 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