电路基础chapter6

1Chapter6CapacitorsandInductorsInthischapter,weareparticularinterestedinthefollowingobjectivesFindingthevoltage-currentrelationshipofidealcapacitorsFindingthevoltage-currentrelationshipofidealinductorsCalculatingtheenergystoredininductorsandcapacitors2Methodsforreducingseries/parallelcombinationsofinductorsMethodsforreducingseries/parallelcapacitorcombinationsPredictingthebehaviorofopampcircuitswithcapacitors3Chapter6CapacitorsandInductors6.1Capacitors6.2SeriesandParallelCapacitors6.3Inductors6.4SeriesandParallelInductors6.5Applications46.1Capacitors•Acapacitorisapassiveelementdesignedtostoreenergyinitselectricfield.•Acapacitorconsistsoftwoconductingplatesseparatedbyaninsulator(ordielectric).5•CapacitanceCistheratioofthechargeqononeplateofacapacitortothevoltagedifferencevbetweenthetwoplates,measuredinfarads(F).•Whereisthepermittivityofthedielectricmaterialbetweentheplates,Aisthesurfaceareaofeachplate,disthedistancebetweentheplates.•Unit:F,pF(10–12),nF(10–9),andvCqdACandF(10–6)6•Ifiisflowingintothe+veterminalofC–Charging=iis+ve–Discharging=iis–ve•Thecurrent-voltagerelationshipofcapacitoraccordingtoaboveconventionistdvdCi)(100tvtdiCvttand7Theenergy,w,storedinthecapacitorisAcapacitoris–anopencircuittodc(dv/dt=0).–itsvoltagecannotchangeabruptly.221vCwFig.6.1Current-Voltagerelationshipofacapacitor8Example1Thecurrentthrougha100-Fcapacitorisi(t)=50sin(120t)mA.Calculatethevoltageacrossitatt=1msandt=5ms.Takev(0)=0.Answer:v(1ms)=93.14mVv(5ms)=1.7361V9Example2Aninitiallyuncharged1-mFcapacitorhasthecurrentshownbelowacrossit.Calculatethevoltageacrossitatt=2msandt=5ms.Answer:v(2ms)=100mVv(5ms)=400mVFig.6.110Example3Determinethecurrentthrougha200-μFcapacitorwhosevoltageisshowninFig.6.2Fig.6.2Fig.6.3Thevoltagewavecanbedescribedmathematicallyas50011005013()20050340tVttVtvttVtotherwisesolution11Example3FindtheenergystoredineachcapacitorFig.6.4(a)underdcconditionsFig.6.4126.2SeriesandParallelCapacitorsTheequivalentcapacitanceofNparallel-connectedcapacitorsisthesumoftheindividualcapacitances.aNeqCCCC...21Fig.6.513TheequivalentcapacitanceofNseries-connectedcapacitorsisthereciprocalofthesumofthereciprocalsoftheindividualcapacitances.NeqCCCC1...11121Fig.6.614Example3Findtheequivalentcapacitanceseenattheterminalsofthecircuitinthecircuitshownbelow:Answer:Ceq=40F15Example4Findthevoltageacrosseachofthecapacitorsinthecircuitshownbelow:Answer:v1=30Vv2=30Vv3=10Vv4=20V166.3InductorsAninductorisapassiveelementdesignedtostoreenergyinitsmagneticfield.Aninductorconsistsofacoilofconductingwire.17Inductanceisthepropertywherebyaninductorexhibitsoppositiontothechangeofcurrentflowingthroughit,measuredinhenrys(H).TheunitofinductorsisHenry(H),mH(10–3)andH(10–6).tdidLvlANL2and18Thecurrent-voltagerelationshipofaninductor:Thepowerstoredbyaninductor:)()(100titdtvLitt221iLwAninductoractslikeashortcircuittodc(di/dt=0)anditscurrentcannotchangeabruptly.19Animportantpropertyoftheinductorisitsoppositiontothechangeincurrentflowingthroughit.Thecurrentaninductorcannotchangeinstantaneously.Liketheidealcapacitor,theidealinductordoesnotdissipateenergy.Theenergystoredinitcanberetrievedatalatertime.Theinductortakespowerfromthecircuitwhenstoringenergyanddeliverspowertothecircuitwhenreturningpreviouslystoredenergy.20Apractical,nonidealinductorhasasignificantresistivecomponent.TheresistanceiscalledthewindingresistanceRw.Athighfrequencies,nonidealinductoralsohasawindingcapacitanceCw21Example5Theterminalvoltageofa2-Hinductorisv=10(1-t)VFindthecurrentflowingthroughitatt=4sandtheenergystoredinitwithin0t4s.Assumei(0)=2A.Answer:i(4s)=-18Vw(4s)=320J22Example6Determinevc,iL,andtheenergystoredinthecapacitorandinductorinthecircuitofcircuitshownbelowunderdcconditions.Answer:iL=3AvC=3VwL=1.125JwC=9J236.4SeriesandParallelInductorsTheequivalentinductanceofseries-connectedinductorsisthesumoftheindividualinductances.NeqLLLL...2124Theequivalentcapacitanceofparallelinductorsisthereciprocalofthesumofthereciprocalsoftheindividualinductances.NeqLLLL1...11121a25Example7Calculatetheequivalentinductancefortheinductiveladdernetworkinthecircuitshownbelow:Answer:Leq=25mH26CurrentandvoltagerelationshipforR,L,C+++276.5ApplicationsThecapacitytostoreenergymakesthemusefulastemporaryvoltageorcurrentsources.Thus,theycanbeusedforgeneratingalargeamountofcurrentorvoltageforashortperiodoftime.Capacitorsopposeanyabruptchangeinvoltage,whileinductorsopposeanyabruptchangeincurrent.Thispropertymakesinductorsusefulforsparkoraresuppressionandforconvertingpulsatingdcvoltageintorelativelysmoothdcvoltage.Capacitorsandinductorspossessthefollowingthreespecialpropertiesthatmakethemveryusefulinelectriccircuits:28Capacitorsandinductorsarefrequencysensitive.ThispropertymakesthemusefulforfrequencydiscriminationIntegratorAnintegratorisanopampcircuitwhoseoutputisproportionaltotheintegraloftheinputsignal.Performingnodalanalysisattheinvertinginput,1sviR29fCoutdvdviCCdtdtThus1soutvdvCRdtIntegratingbothsidesgives011()(0)toutoutsfvtvvdtRC