1 Magnitude 1 22 Phase 2 3 Combining Poles and Zeroes 1 Introduction Although you should have learned about Bode plots in previou s courses such as EE40 t ID: 22584
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UNIVERSITYOFCALIFORNIAATBERKELEYCollegeofEngineeringDepartmentofElectricalEngineeringandComputerSciencesEE105LabExperimentsBodePlotTutorialContents1Introduction12BodePlotsBasics12.1Magnitude...............................................12.2Phase.................................................23CombiningPolesandZeroes31IntroductionAlthoughyoushouldhavelearnedaboutBodeplotsinpreviouscourses(suchasEE40),thistutorialwillgiveyouabriefreviewofthematerialincaseyourmemoryisrusty.2BodePlotsBasicsMakingtheBodeplotsforatransferfunctioninvolvedrawingboththemagnitudeandphaseplots.Themagnitudeisplottedindecibels(dB)andthephaseisplottedindegrees.Forbothplots,thehorizontalaxisiseitherfrequency(f)orangularfrequency(!),measuredinHzandrad=s,respectively.Thehorizontalaxisshouldbelogarithmic(i.e.increasingbypowersof10).Mostofthetransferfunctionswe'lldealwithinthisclasscanbeseparatedintoageneralthatresemblesthefollowing:H(j!)=Aj!=!z1(1+j!=!z2)(1+j!=!z3)::: j!=!p1(1+j!=!p2)(1+j!=!p3):::(1)Aisanarbitraryconstantandjisp 1.Asyoucansee,thebasiccomponentofthistransferfunctionappearstobe1+j!=!c,where!cissomeconstant(withtheslightvariationj!=!c).Let'sanalyzethisbasiccomponent rstbeforeweanalyzetheentiretransferfunction.2.1MagnitudeRecallthatthede nitionofmagnitude(measuredindB)isasfollows:20logjH(j!)j=20logq [H(j!)]2+=[H(j!)]2Let'sapplythisde nitiontoourbasictransferfunctioncomponent(thisiscalledazerowhenitappearsinthenumeratorofatransferfunction):20logj1+j!=!cj=20logq 1+(!=!c)2Forsmall!,wehave20logj1+j!=!cj0dB.Forlarge!,20logj1+j!=!cj!1.When!=!c,themagnitudeofthetransferfunctionisapproximately3dB.Sincethere'ssolittlechangefrom!=0to!=!c,weapproximatethemagnitudeinthisregionasaconstant0dB.1 2BODEPLOTSBASICS2For!�!c,the(!=!c)2dominatesthemagnitudeexpression,allowingustoapproximatethemagnitudeas20log!=!c.Fromthisexpressionit'sclearthatifweincrease!byafactorof10,weincreasethemagnitudeby20dB.Thus,ourBodeplotapproximationforthezeroisaconstant0dBfor!!candalineconstantlyincreaseby20dB=decadefor!&#x-318;&#x.308;!c,illustratedinFigure1.Figure1alsoillustratestheBodeplotforaDCzerooftheformj!=!c.Thisdi ersonlyslightlyfromthenormalzerointhatislackstheadditional1.Thus,insteadofhavingtheconstantmagnituderegionfor!!c,issimplyalwaysincreasesat20dB=decade.Wedrawitsintersectionwiththefrequencyaxiswhere!=!c,sincethat'swherethemagnitudeis0dB. 40 2002040Magnitude(dB)103104105106107!(rad/s) 20logj1+j!=!cj 20logjj!=!cj Figure1:Bodeplots(magnitude)foranormalzeroandaDCzerofor!c=105rad=s(theplotsoverlapfor!�!c)Thebasictransferfunctioncomponent1+j!=!ccanalsoappearinthedenominator(inwhichcaseitiscalledapole).Althoughthismayseemlikeanentirelydi erentproblem,recallthatwetakethelogarithmofourtransferfunctionbecauseourresultisexpressedindecibels.Takingthelogarithmoftheinverseofafunctionsimplygivesthenegativelogarithmofthefunction,meaningwesimplyhavetonegatetheresultsofourzeroanalysistogettheappropriateexpressionsforpoles.ThesameargumentapplieswithDCpolesoftheformj!=!c,sowecannegateourDCzeroanalysistogettheDCpoleresults.Anormalpolewillhaveaconstant0dBvaluefor!!candwilldropby20dB=decadefor!&#x-282;&#x.199;!c.ADCpolewilldropby20dB=decadeforany!andwillintersectthefrequencyaxis(0dB)at!=!c.TheresultsareshowninFigure2.2.2PhaseLet'stakealookatthephaseofazero,DCzero,pole,andDCpole.Recallthede nitionofphase:Arg(H(j!))=tan 1[H(j!)] =[H(j!)] 3COMBININGPOLESANDZEROES3 40 2002040Magnitude(dB)103104105106107!(rad/s) 20log 1 1+j!=!c 20log 1 j!=!c Figure2:Bodeplots(magnitude)foranormalpoleandaDCpolefor!c=105rad=s(theplotsoverlapfor!�!c)Let'sapplythistothenormalzero rst.Arg(1+j!=!c)=tan 1! !cFor!=0,Arg(1+j!=!c)=0.For!!1,Arg(1+j!=!c)!90.For!=!c,Arg(1+j!=!c)!45.Thus,ourapproximationforthephaseofazerois0for!0:1!c,45for!=!c,and90for!&#x-5.1;䡣10!cwithastraightlineconnectingthesepoints.WecanalsolookatthephaseofaDCzero,whichisalways90.TheseresultsareshowninFigure3.Similartoouranalysisofthemagnitude,wecanalsoconsiderpolesandDCpolesinourphaseplots.Itcanbeshownthattan 1 = tan 1,meaningourphaseplotsforpolesandDCpoleswillsimplybenegatedversionsofthezeroplots.TheseareshowninFigure4.3CombiningPolesandZeroesGenerally,atransferfunctionmayinvolvemanypolesandzeroes(andtheirDCcounterparts).InordertomakeiteasiertodrawBodeplots,your rststepshouldbetofactorthetransferfunctionintothecanonicalformshowninEquation1.Thismakesiteasytoidentifyallofthepolesandzeroes.First,you'llhavetohandletheconstantAinfront(ifpresent).ThemagnitudeofAwilla ectyourmagnitudeplot,andthesignofAwilla ectyourphaseplot.Yourmagnitudeplotmustbeshiftedupby20logjAj.Forexample,ifA=10,thenyourmagnitudeplotmustbeshiftedupby20dB.Similarly,ifA=1=10,thenyourmagnitudeplotmustbeshifteddownby20dB.IfA0,thenyourphaseplotmustbeshiftedup(ordown|it'sthesameinthiscase)by180.Second,youneedtodraweachpoleandzeroindividuallyonthesamesetofaxes(whetheryou'remakingamagnitudeorphaseplot). 3COMBININGPOLESANDZEROES4 020406080100Phase(degrees)103104105106107!(rad/s) Arg(1+j!=!c) Arg(j!=!c) Figure3:Bodeplots(phase)foranormalzeroandaDCzerofor!c=105rad=s(theplotsoverlapfor!�10!c)Third,yousimplyaddthecurvesthatyou'vedrawnateachpointtoobtainthe nalBodeplot.RemembertoshiftyourplotsaccordinglybasedontheconstantAasmentionedpreviously.Thissuperpositionprincipleispossiblebecauseofthedecompositionofthetransferfunctionintozeroesandpoles.Whenaddingthepolesandzeroesinthe nalplot,rememberthatinareaswheretwocurvesareconstant,theresultwilljustbethesumoftheconstantvalues.Whenoneisacontantandoneislinear,thentheresultwillstartattheconstantvalueandhavetheslopeofthelinearcurve.Finally,whenbotharelinear,thesumwillhaveaslopeequaltothesumoftheslopesoftheindividalcurves. 3COMBININGPOLESANDZEROES5 100 80 60 40 200Phase(degrees)103104105106107!(rad/s) Arg1 1+j!=!c Arg1 j!=!c Figure4:Bodeplots(phase)foranormalpoleandaDCpolefor!c=105rad=s(theplotsoverlapfor!�10!c)