最新-電路分析英文Lecture8-課件

ECTE170–Lecture8/111ECTE170–Lecture8/1111Chapter14–BoylestadSinusoidalresponseofresistor,inductorandcapacitorLowandhighfrequencyresponseofinductorsandcapacitorsAveragepowerandPowerFactorBasicElementsandPhasors2Chapter14–BoylestadBasicEl2IntroductionTheresponseofthebasicR,L,andCelementstoasinusoidalvoltageandcurrentwillbeexaminedwithaspecialnoteofhowfrequencywillaffectthe“opposing”characteristicofeachelement.Phasornotationwillthenbeintroducedtoestablishamethodofanalysis.3IntroductionTheresponseoft3TheDerivativeTounderstandtheresponseofthebasicR,L,andCelementstoasinusoidalsignal,youneedtoexaminetheconceptofthederivative.Thederivativedx/dtisdefinedastherateofchangeofxwithrespecttotime.Ifxfailstochangeataparticularinstant,dx=0,andthederivativeiszero.Forthesinusoidalwaveform,dx/dtiszeroonlyatthepositiveandnegativepeaks(wt=p/2and3p/2)sincexfailstochangeattheseinstantsoftime.4TheDerivativeTounderstandt4555TheDerivativeThederivativedx/dtisactuallytheslopeofthegraphatanyinstantoftime.Thegreatestchangeinxwilloccurattheinstantswt=0,p,and2p.Forvariousvaluesofwtbetweenthesemaximaandminima,thederivativewillexistandwillhavevaluesfromtheminimumtothemaximuminclusive.Thederivativeofasinewaveisacosinewave;ithasthesameperiodandfrequencyastheoriginalsinusoidalwaveform.6TheDerivativeThederivative6777SinusoidalResponse:ResistorForaresistorthevoltageandcurrentareinphaseandarerelatedbyOhm’slaw8SinusoidalResponse:ResistorF8Thevoltageandcurrentofaresistiveelementareinphase.9Thevoltageandcurrentofar9SinusoidalResponse:InductorsForaninductorthecurrentlagsthevoltageby90degreesXL=Liscalledtheinductivereactance-unit10SinusoidalResponse:Inductors10SinusoidalResponse:WaveformsInductor11SinusoidalResponse:Waveforms11SinusoidalResponse:CapacitorsForacapacitorthecurrentleadsthevoltageby90degreesXc=1/Ciscalledthecapacitivereactance–unit12SinusoidalResponse:Capacitor12SinusoidalResponse:WaveformsCapacitor13SinusoidalResponse:Waveforms13SinusoidalResponseThecurrentthrougha5ohmresistorisi=40sin(377t+30°)A.Findtheexpressionforvoltageacrossit.14SinusoidalResponseThecurrent14SinusoidalResponseThecurrentthrougha0.1Hcoilisi=7sin(377t–70°)A.Findthevoltageacrossit.15SinusoidalResponseThecurrent15161616CapacitorExample17CapacitorExample1717181818Morequestions19Morequestions1919202020212121LowandHighFrequencyResponseforInductorsInductors:XL=ωLAtlowfrequencies,andespeciallyDC,thereactanceofainductorisverylow(zeroforDC)Henceatverylowfrequencies,aninductormaybeconsideredasashortcircuitAsinputfrequenciesbecomeveryhigh,thereactanceofaninductorapproachesinfinityHenceatveryhighfrequencies,aninductormaybeconsideredasanopencircuitBoylestad,PrenticeHall201922LowandHighFrequencyRespons22LowandHighFrequencyResponseforCapacitorsCapacitors:Xc=1/ωCAtlowfrequencies,andespeciallyDC,thereactanceofacapacitorisveryhigh(infiniteforDC)Henceatverylowfrequencies,acapacitormaybeconsideredasanopencircuitAsinputfrequenciesbecomeveryhigh,thereactanceofacapacitorapproaches0ΩHenceatveryhighfrequencies,acapacitormaybeconsideredasashortcircuit23LowandHighFrequencyRespons23AveragePowerandPowerFactorSecondtermhasazeroaveragevalueoveracycleandcausesnoaveragepowerFirsttermisindependentof(a)timeandisconstant(b)whethervleadsorlagsi,andwillbetheAveragePowerortheRealPowerIngeneralv=VmsintandI=Imsin(t-)

24AveragePowerandPowerFactor24AveragePowerandPowerFactorBoylestad,PrenticeHall201925AveragePowerandPowerFactor25AveragePowerandPowerFactorwhereVandIarermsvaluesofthesinusoidalvoltageandcurrentrespectivelyThefactor(cos)whichcontrolstheaveragepowerflowiscalledthePowerFactor.ForaresistorthePowerFactorisunityForaninductororcapacitorPowerFactoriszeroAnotherwayoffindingthePowerFactoristousetheexpression26AveragePowerandPowerFactor26AveragePowerandPowerFactor

Whenthepowerfactorisstateditisimportanttostatewhetheritisleadingorlagginginadditiontoitsvalue(notethatitliesbetween0and1.0)27AveragePowerandPowerFactor27AveragePowerandPowerFactor28AveragePowerandPowerFactor28AveragePowerandPowerFactor29AveragePowerandPowerFactor29303030PowerFactorFpPowerFactor=Fp=cosThetermleadingorlaggingisoftenwritteninconjunctionwiththepowerfactor.Theyaredefinedbythecurrentthroughtheload.IfthecurrentleadthevoltagethenitsaleadingpowerfactorIfthecurrentlagsthevoltagethenitsalaggingpowerfactorCapacitivecircuitshaveleadingpowerfactors,whileInductivecircuitshavelaggingpowerfactors31PowerFactorFpPowerFactor=31ExamplePowerfactor32ExamplePowerfactor3232333333ComplexNumbersAsanessentialtoolcomplexnumberswillbeusedinsolvingaccircuitsRectangularformZ=a+jbwhere‘j’isanoperatorwhichturnstherealnumber‘b’by90°intheanti-clockwisedirectiononthecomplexplanePolarformZ=Z/°

AdditionisconvenientinrectangularformDivision/multiplicationisconvenientinpolarformajb34ComplexNumbersAsanessential34ComplexNumbersDefiningtherectangularform.35ComplexNumbersDefiningthere35PolarformDefiningthepolarform.Demonstratingtheeffectofanegativesignonthepolarform.36PolarformDefiningthepolarf36373737ComplexConjugate38ComplexConjugate3838ConversionBetweenForms39ConversionBetweenForms3939ComplexNumbers40ComplexNumbers4040ComplexNumbers41ComplexNumbers4141MathsOperations42MathsOperations4242434343Polarformoperations44Polarformoperations4444PhasorsandPhasorDiagramsOftenaddition/subtractionofsinusoidalvoltagesandcurrentsisrequiredinaccircuitanalysis.Forexampleconsidertheadditionoftwosinusoidalsignalsv1=Vm1sin(wt+)andv2=Vm2sinwtWecanaddthetwowaveformsonapoint-by-pointbasisasshown(atediousprocess!!)toobtainvT.Boylestad45PhasorsandPhasorDiagramsOft45PhasorsandPhasorDiagramsHoweveritisveryconvenienttorepresentthetwosinusoidalwaveformsv1=Vm1sin(wt+)andv2=Vm2sinwt

asvectorsandthenaddthevectors.Thesevectorsarecalledphasors.V2=Vm2/0VmTPhasordiagramofV1,V2andVTNotethephasorswithunderscore–inthebookshowninboldV1=Vm1/46PhasorsandPhasorDiagramsHo46PhasorsandPhasorDiagramsPhasorsarecharacterisedbyamagnitudeandaphaseangleItiscommontorepresentthemagnitudeofaphasorasanrmsquantityratherthanapeakvalueTheyrepresentasnapshotsoftherotatingvectorsatt=0Theydonotcarryinformationaboutthefrequency

V2=V2/0°VTV1=V1/°47PhasorsandPhasorDiagramsPha47PhasorAddition48PhasorAddition4848494949505050

ECTE170–Lecture8/1151ECTE170–Lecture8/11151Chapter14–BoylestadSinusoidalresponseofresistor,inductorandcapacitorLowandhighfrequencyresponseofinductorsandcapacitorsAveragepowerandPowerFactorBasicElementsandPhasors52Chapter14–BoylestadBasicEl52IntroductionTheresponseofthebasicR,L,andCelementstoasinusoidalvoltageandcurrentwillbeexaminedwithaspecialnoteofhowfrequencywillaffectthe“opposing”characteristicofeachelement.Phasornotationwillthenbeintroducedtoestablishamethodofanalysis.53IntroductionTheresponseoft53TheDerivativeTounderstandtheresponseofthebasicR,L,andCelementstoasinusoidalsignal,youneedtoexaminetheconceptofthederivative.Thederivativedx/dtisdefinedastherateofchangeofxwithrespecttotime.Ifxfailstochangeataparticularinstant,dx=0,andthederivativeiszero.Forthesinusoidalwaveform,dx/dtiszeroonlyatthepositiveandnegativepeaks(wt=p/2and3p/2)sincexfailstochangeattheseinstantsoftime.54TheDerivativeTounderstandt5455555TheDerivativeThederivativedx/dtisactuallytheslopeofthegraphatanyinstantoftime.Thegreatestchangeinxwilloccurattheinstantswt=0,p,and2p.Forvariousvaluesofwtbetweenthesemaximaandminima,thederivativewillexistandwillhavevaluesfromtheminimumtothemaximuminclusive.Thederivativeofasinewaveisacosinewave;ithasthesameperiodandfrequencyastheoriginalsinusoidalwaveform.56TheDerivativeThederivative5657757SinusoidalResponse:ResistorForaresistorthevoltageandcurrentareinphaseandarerelatedbyOhm’slaw58SinusoidalResponse:ResistorF58Thevoltageandcurrentofaresistiveelementareinphase.59Thevoltageandcurrentofar59SinusoidalResponse:InductorsForaninductorthecurrentlagsthevoltageby90degreesXL=Liscalledtheinductivereactance-unit60SinusoidalResponse:Inductors60SinusoidalResponse:WaveformsInductor61SinusoidalResponse:Waveforms61SinusoidalResponse:CapacitorsForacapacitorthecurrentleadsthevoltageby90degreesXc=1/Ciscalledthecapacitivereactance–unit62SinusoidalResponse:Capacitor62SinusoidalResponse:WaveformsCapacitor63SinusoidalResponse:Waveforms63SinusoidalResponseThecurrentthrougha5ohmresistorisi=40sin(377t+30°)A.Findtheexpressionforvoltageacrossit.64SinusoidalResponseThecurrent64SinusoidalResponseThecurrentthrougha0.1Hcoilisi=7sin(377t–70°)A.Findthevoltageacrossit.65SinusoidalResponseThecurrent65661666CapacitorExample67CapacitorExample1767681868Morequestions69Morequestions1969702070712171LowandHighFrequencyResponseforInductorsInductors:XL=ωLAtlowfrequencies,andespeciallyDC,thereactanceofainductorisverylow(zeroforDC)Henceatverylowfrequencies,aninductormaybeconsideredasashortcircuitAsinputfrequenciesbecomeveryhigh,thereactanceofaninductorapproachesinfinityHenceatveryhighfrequencies,aninductormaybeconsideredasanopencircuitBoylestad,PrenticeHall201972LowandHighFrequencyRespons72LowandHighFrequencyResponseforCapacitorsCapacitors:Xc=1/ωCAtlowfrequencies,andespeciallyDC,thereactanceofacapacitorisveryhigh(infiniteforDC)Henceatverylowfrequencies,acapacitormaybeconsideredasanopencircuitAsinputfrequenciesbecomeveryhigh,thereactanceofacapacitorapproaches0ΩHenceatveryhighfrequencies,acapacitormaybeconsideredasashortcircuit73LowandHighFrequencyRespons73AveragePowerandPowerFactorSecondtermhasazeroaveragevalueoveracycleandcausesnoaveragepowerFirsttermisindependentof(a)timeandisconstant(b)whethervleadsorlagsi,andwillbetheAveragePowerortheRealPowerIngeneralv=VmsintandI=Imsin(t-)

74AveragePowerandPowerFactor74AveragePowerandPowerFactorBoylestad,PrenticeHall201975AveragePowerandPowerFactor75AveragePowerandPowerFactorwhereVandIarermsvaluesofthesinusoidalvoltageandcurrentrespectivelyThefactor(cos)whichcontrolstheaveragepowerflowiscalledthePowerFactor.ForaresistorthePowerFactorisunityForaninductororcapacitorPowerFactoriszeroAnotherwayoffindingthePowerFactoristousetheexpression76AveragePowerandPowerFactor76AveragePowerandPowerFactor

Whenthepowerfactorisstateditisimportanttostatewhetheritisleadingorlagginginadditiontoitsvalue(notethatitliesbetween0and1.0)77AveragePowerandPowerFactor77AveragePowerandPowerFactor78AveragePowerandPowerFactor78AveragePowerandPowerFactor79AveragePowerandPowerFactor79803080PowerFactorFpPowerFactor=Fp=cosThetermleadingorlaggingisoftenwritteninconjunctionwiththepowerfactor.Theyaredefinedbythecurrentthroughtheload.IfthecurrentleadthevoltagethenitsaleadingpowerfactorIfthecurrentlagsthevoltagethenitsalaggingpowerfactorCapacitivecircuitshaveleadingpowerfactors,whileInductivecircuitshavelaggingpowerfactors81PowerFactorFpPowerFactor=81ExamplePowerfactor82ExamplePowerfactor3282833383ComplexNumbersAsanessentialtoolcomplexnumberswillbeusedinsolvingaccircuitsRectangularformZ=a+jbwhere‘j’isanoperatorwhichturnstherealnumber‘b’by90°intheanti-clockwisedirectiononthecomplexplanePolarformZ=Z/°

AdditionisconvenientinrectangularformDivision/multiplicationisconvenientinpolarformajb84ComplexNumbersAsanessential84ComplexNumbersDefiningtherectangularform.85ComplexNumbersDefiningthere85PolarformDefiningthepolarform.Demonstratingtheeffectofanegativesignonthepolarform.86PolarformDefiningthepolarf86