电路基础chapter4

1Chapter4CircuitTheorems4.1Motivation4.2LinearityProperty4.3Superposition4.4SourceTransformation4.5Substitutiontheorem4.6Simplificationofaone-portnetworkcontainsnoindependentsources4.7Thevenin’sTheorem4.8Norton’sTheorem4.9MaximumPowerTransfer2Ifyouaregiventhefollowingcircuit,arethereanyotheralternativetodeterminethevoltageacross2Wresistor?Whatarethey?Andhow?Canyouworkitoutbyinspection?4.1Motivation34.2LinearityProperty(1)Itisthepropertyofanelementdescribingalinearrelationshipbetweencauseandeffect.Alinearcircuitisonewhoseoutputislinearlyrelated(ordirectlyproportional)toitsinput.Homogeneity(scaling)propertyv=iR→kv=kiRAdditivepropertyv1=i1Randv2=i2R→v=(i1+i2)R=v1+v244.2LinearityProperty(2)Example1ByassumeIo=1A,uselinearitytofindtheactualvalueofIointhecircuitshownbelow.AnswerIo=3A54.3SuperpositionTheorem(1)Itstatesthatthevoltageacross(orcurrentthrough)anelementinalinearcircuitisthealgebraicsumofthevoltageacross(orcurrentsthrough)thatelementduetoEACHindependentsourceactingalone.Theprincipleofsuperpositionhelpsustoanalyzealinearcircuitwithmorethanoneindependentsourcebycalculatingthecontributionofeachindependentsourceseparately.6Weconsidertheeffectsof8Aand20Vonebyone,thenaddthetwoeffectstogetherforfinalvo.4.3SuperpositionTheorem(2)7v01(a)v02(b)I1I2Bysimpleanalyzingweobtainv01=4Vv02=8V18I1251020III1I2012vV8TheSuperpositionPrinciple(叠加原理)FindUandI.+=IsUsR1R221212RRIRRRUss)11(/211RRIRUUssssIUIII21ssIUUUU21sURRRU212sIRRRRU211221RRUIssIRRRI21194.3SuperpositionTheorem(3)Stepstoapplysuperpositionprinciple1.Turnoffallindependentsourcesexceptonesource.Findtheoutput(voltageorcurrent)duetothatactivesourceusingnodalormeshanalysis.2.Repeatstep1foreachoftheotherindependentsources.3.Findthetotalcontributionbyaddingalgebraicallyallthecontributionsduetotheindependentsources.104.3SuperpositionTheorem(4)Twothingshavetobekeepinmind:Whenwesayturnoffallotherindependentsources:Independentvoltagesourcesarereplacedby0V(shortcircuit)andIndependentcurrentsourcesarereplacedby0A(opencircuit).Dependentsourcesareleftintactbecausetheyarecontrolledbycircuitvariables.11Note:2)actingalonemeansotherindependentsources“inactive”,“turnedoff”or“zeroedout”,dependentsourcesareingeneralactive.Avoltagesourcesettozeroactslikeashortcircuit0ViiAcurrentsourcesettozeroactslikeanopencircuitu0AuNote:1)Superpositionisbasedonlinearity.Note:4)ThereferencedirectionNote:5)Notapplicabletotheeffectonpower.124.3SuperpositionTheorem(5)Example2Usethesuperpositiontheoremtofindvinthecircuitshownbelow.3Aisdiscardedbyopen-circuit6Visdiscardedbyshort-circuitAnswerv=10V13Example3Usesuperpositiontofindvxinthecircuitbelow.AnswerVx=12.5V2Aisdiscardedbyopen-circuit20Wv14W10V+(a)0.1v14W2A(b)20W0.1v2v210Visdiscardedbyopen-circuitDependantsourcekeepunchanged14Example1.UsesuperpositiontocomputeuandiinthecircuitshowninfollowingFig..Solution:Let28Vsourceactalone,Ai4.181228Vu8.4Let2Asourceactalone,Ai2.1281212Vu46.16ThusAi6.2Vu66.1115Example3.Inthefollowingcircuit,ifUs=1V,Is=1A:U2=0；ifUs=10V,Is=0：U2=1V.IfUs=0,Is=10A,findU2.Solution:SupposethefollowingequationusingthesuperpositionprinciplessUKIKU212Therefore11021KK100121KK1.01.021KKssUIU1.01.02VU12Solvingtheequations,wehaveThusifUs=0,Is=10A,164.4SourceTransformation(1)•Anequivalentcircuitisonewhosev-icharacteristicsareidenticalwiththeoriginalcircuit.•ItistheprocessofreplacingavoltagesourcevSinserieswitharesistorRbyacurrentsourceiSinparallelwitharesistorR,orviceversa.174.4SourceTransformation(2)•Thearrowofthecurrentsourceisdirectedtowardthepositiveterminalofthevoltagesource.•ThesourcetransformationisnotpossiblewhenR=0forvoltagesourceandR=∞forcurrentsource.(a)Independentsourcetransform(b)Dependentsourcetransform++++----18Example4Findiointhecircuitshownbelowusingsourcetransformation.i0=1.78A19•Ifthevoltageacrossandcurrentthroughanybranchofadcnetworkwithtwoterminalsareknown,thisbranchcanbereplacedbyanycombinationofelementsthatwillmaintainthesamevoltageacrossandcurrentthroughthechosenbranch.4.5Substitutiontheorem(替代定理)N1N2usisN1usN1isN1us/is20Example:FindUsandR.I1=0.4ASolution：US=43.6vI=2AU=28vApplySubstitutiontheorem,wehave66.020281U=10vIR=0.6-0.4=0.2AR=50W.US+U1-IRI1214.6Simplificationofaone-portnetworkcontainsnoindependentsourcesCase1:containsnodependentsources(itsequivalenceisaresistor)RiRi=1.5Wi22Case2:containsdependentsources(itsequivalenceisaresistor)Method:Applyeitheratestvoltagesourceoratestcurrentsourcetotheterminalsoftheone-portnetwork,theratioofthevoltageacrossthetestsourcetothecurrentdeliveredbythetestsourceequalstotheequivalentresistance.Example1:Simplifythefollowingcircuit.Solution:Applyatestvoltagesource,32ui21uui21iii23uuuu)2131(iuR21311356ui2i123-2i0+i0i1i3i2abcd244.7Thevenin’sTheorem(1)Itstatesthatalineartwo-terminalcircuit(Fig.a)canbereplacedbyanequivalentcircuit(Fig.b)consistingofavoltagesourceVTHinserieswitharesistorRTH,where•VTHistheopen-circuitvoltageattheterminals.•RTHistheinputorequivalentresistanceatthetermin