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TransferFunctionsandBodePlotsTransferFunctionsForsinusoidaltimevariations,theinputvoltagetoaltercanbewritten)=RejÏtisthephasorinputvoltage,i.e.ithasanamplitudeandaphase,andjÏt Wecanwriteasfollows: isagainconstantandandarepolynomialsincontainingnoreciprocalpowersofTherootsof 2 2+1)(3+1) Thefunctionhasazeroatandpolesat.Notethat)=0.Becauseofthis,sometextswouldsaythathasazeroat.However,thisisnotcorrectbecauseNotethattheconstanttermsinthenumeratoranddenominatorofarebothunity.Thisisoneoftwostandardwaysforwritingtransferfunctions.Anotherwayistomakethecoecientofthehighestpowersofunity.Inthiscase,theabovetransferfunctionwouldbewritten)=6 2 +2)(+3)BecauseitisusuallyeasiertoconstructBodeplotswiththerstform,thatformisusedhere.Becausethecomplexfrequencyistheoperatorwhichrepresentsd/dtinthedierentialequationforasystem,thetransferfunctioncontainsthedierentialequation.Letthetransferfunctionaboverepresentthevoltagegainofacircuit,i.e.,where,respectively,arethephasoroutputandinputvoltages.Itfollowsthat 6+5 6 Whentheoperatorisreplacedwith,thefollowingdierentialequationisobtained: 62 +5 6 = ,respectively,arethetimedomainoutputandinputvoltages.Notethatthepolesarerelatedtothederivativesoftheoutputandthezerosarerelatedtothederivativesoftheinput.HowtoConstructBodePlotsABodeplotisaplotofeitherthemagnitudeorthephaseofatransferfunctionasafunctionofThemagnitudeplotisthemorecommonplotbecauseitrepresentsthegainofthesystem.Therefore,thetermBodeplotusuallyreferstothemagnitudeplot.TherulesformakingBodeplotscanbederivedfromthefollowingtransferfunction: isapositiveinteger.Forastheexponent,thefunctionhaszerosat.For-,ithaspolesat.With,itfollowsthatÏ/ÏÏ/Ïand .Ifisincreasedbyafactorofchangesbyafactorof.Thusaplotversuslogscaleshasaslopeoflog(10decades/decade.TherearedBsinadecade,sotheslopecanalsobeexpressedasdB/decade.Asarstexample,considerthelow-passtransferfunction Thisfunctionhasapoleatandnozeros.ForandÏ/Ïwehave,and .ForÏ/ÏjÏ/ÏÏ/Ï,and .Onscales,themagnitudeplotforthelow-frequencyapproximationhasaslopeofwhilethatforthehigh-frequencyapproximationhasaslopeof.Thelowandhigh-frequency approximationsintersectwhen,orwhen.For and arctan(1)=.Notethatthisistheaveragevalueofthephaseonthetwoadjoiningasymptotes.TheBodemagnitudeandphaseplotsareshowninFig.1.Notethattheslopeoftheasymptoticmagnitudeplotrotatesby.Becauseisthemagnitudeofthepolefrequency,wesaythatthesloperotatesbyatapole.Astraightlinesegmentthatistangenttothephaseplotatwouldintersectthelevelatandthelevelat Figure1:Bodeplots.(a)Magnitude.(b)Phase.Asasecondexample,considerthetransferfunction Thisfunctionhasazeroat.ForÏ/Ïwehaveand .ForÏ/ÏjÏ/ÏÏ/Ïand =90.Onlogscales,themagnitudeplotforthelow-frequencyapproximationhasaslopeofwhilethatforthehigh-frequencyapproximationhasaslopeof.Thelowandhigh-frequencyapproximationsintersectwhenÏ/Ï,orwhen.For and )=arctan(1)=45.Notethatthisistheaverageofthephaseonthetwoadjoiningasymptotes.TheBodemagnitudeandphaseplotsareshowninFig.2.Notethattheslopeoftheasymptoticmagnitudeplotrotatesby.Becauseisthemagnitudeofthezerofrequency,wesaythatthesloperotatesbyatazero.Astraightlinesegmentthatistangenttothephaseplotatwouldintersectlevelatandthelevelat Figure2:Bodeplots.(a)Magnitude.(b)Phase.Fromtheaboveexamples,wecansummarizethebasicrulesformakingBodeplotsasfollows: 1.InanyfrequencybandwhereatransferfunctioncanbeapproximatedbyjÏ/Ï,theslopeoftheBodemagnitudeplotisdec/dec.Thephaseis2.Polescausetheasymptoticslopeofthemagnitudeplottorotateclockwisebyoneunitatthepolefrequency.3.Zeroscausetheasymptoticslopeofthemagnitudeplottorotatecounter-clockwisebyoneunitatthezerofrequency.Asathirdexample,considerthetransferfunction s/ÏThisfunctionhasapoleatandazeroat.ForandÏ/ÏwehaveÏ/Ï .ForÏ/Ïand .Onlogscales,themagnitudeplotforthelow-frequencyapproximationhasaslopeofwhilethatforthehigh-frequencyapproximationhasaslopeof.Thelowandhigh-frequencyapproximationsintersectwhenÏ/Ïorwhen.For 2 )=90arctan(1)=45.TheBodemagnitudeandphaseplotsareshowninFig.3.Notethattheslopeoftheasymptoticmagnitudeplotrotatesbyatthepole.Thetransferfunctioniscalledahigh-passfunctionbecauseitsgainapproacheszeroatlowfrequencies. Figure3:Bodeplots.(a)Magnitude.(b)Phase.Ashelvingtransferfunctionhastheforms/Ï s/ÏThefunctionhasapoleatandazeroat.Wewillconsiderthelow-passshelvingfunctionforwhich.ForÏ/Ï,wehaveand .Asisincreased,thepolecausestheasymptoticslopetorotatefrom.Thezerocausestheasymptoticslopetorotatefrombackto.ForÏ/Ï.TheBodemagnitudeplotisshowninFig.4(a).Ifthetransferfunctiondidnothavethezero,theactualgainatwouldbe Thezerocausesthegaintobebetween and.Similarly,thepolecausestheactualgainatbebetweenand .Theactualplotintersectstheasymptoticplotatthegeometricmeanfrequency 12.4 Thephaseplothasaslopethatapproachesatverylowfrequenciesandatveryhighfrequencies.Atthegeometricmeanfrequency ,thephaseisapproaching.Ifthefunctiononlyhadapole,thephaseatwouldbe,approachingathigherfrequencies.However,thezerocausesthehigh-frequencyphasetoapproach.Thusthephaseatismorepositivethan.Atthegeometricmeanfrequency ,theslopeofthephasefunctioniszero.TheBodephaseplotisshowninFig.4(b). Figure4:Bodeplots.(a)Magnitude.(b)Phase.ImpedanceTransferFunctionsRCNetworkTheimpedancetransferfunctionforatwo-terminalnetworkwhichcontainsonlyonecapacitorandisnotanopencircuitatdccanbewritten isthedcresistanceofthenetwork,isthepoletimeconstant,andisthezerotimeconstant.Thepoletimeconstantisthetimeconstantofthenetworkwiththeterminalsopencircuited.Thezerotimeconstantisthetimeconstantofthenetworkwiththeterminalsshortcircuited.Figure5(a)showsthecircuitdiagramofanexampletwo-terminalnetwork.Theimpedancetransferfunctioncanbewrittenbyinspectiontoobtain 1+2) Figure5:Exampleandimpedancenetworks. RLNetworkTheimpedancetransferfunctionforatwo-terminalnetworkwhichcontainsonlyoneinductorandisnotashortcircuitatdccanbewritten isthedcresistanceofthenetwork,isthepoletimeconstant,andisthezerotimeconstant.Thepoletimeconstantisthetimeconstantofthenetworkwiththeterminalsopencircuited.Thezerotimeconstantisthetimeconstantofthenetworkwiththeterminalsshortcircuited.Figure5(b)showsthecircuitdiagramofanexampletwo-terminalnetwork.TheimpedancetransferfunctioncanbewrittenbyinspectiontoobtainO/R 1+[VoltageDividerTransferFunctionsRCNetworkThevoltage-gaintransferfunctionofavoltage-dividernetworkcontainingonlyonecapacitorandhavinganon-zerogainatdccanbewritten = isthedcgain(anopencircuit),isthepoletimeconstant,andisthezerotimeconstant.Thepoletimeconstantisthetimeconstantofthenetworkwithandopencircuited.Thezerotimeconstantisthetimeconstantofthenetworkwithandopencircuited.Figure6(a)showsthecircuitdiagramofanexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =2+3 1+2+3×2k3) 1+[(Figure6(b)showsthecircuitdiagramofasecondexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =3 1+3×1+2) 1+[(High-PassRCNetworkThevoltage-gaintransferfunctionofahigh-passvoltage-dividernetworkcontainingonlyonecapacitorcanbewritten = istheinnitefrequencygain(ashortcircuit)andisthepoletimeconstant.Thepoletimeconstantiscalculatedwithandopencircuited.Figure6(c)showsthecircuitdiagramofathirdexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =2 1+2×(1+2) 1+2) Figure6:Examplevoltagedividernetworks.RLNetworkThevoltage-gaintransferfunctionofavoltage-dividernetworkcontainingonlyoneinductorandhavinganon-zerogainatdccanbewritten = isthezerofrequencygain(ashortcircuit),isthepoletimeconstant,andisthezerotimeconstant.Thepoletimeconstantisthetimeconstantofthenetworkwithopencircuited.Thezerotimeconstantisthetimeconstantofthenetworkwithandopencircuited.Figure7(a)showsthecircuitdiagramofanexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =2 1+[ (R1+R2)kR3])sFigure7(b)showsthecircuitdiagramofasecondexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =3 O/R 1k(2+3 Figure7:Examplevoltagedividercircuits. High-PassRLNetworkThevoltage-gaintransferfunctionofahigh-passvoltage-dividernetworkcontainingonlyoneinductorcanbewritten = istheinnitefrequencygain(anopencircuit)andisthepoletimeconstant.Thepoletimeconstantiscalculatedwithandopencircuited.Figure7(c)showsthecircuitdiagramofathirdexamplenetwork.Thevoltage-gaintransferfunctioncanbewrittenbyinspectiontoobtain =2 1+2×[1k2 1+[