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1、OvershootasaFunctionofPhaseMarginJ.C.DalyElectricalandComputerEngineeringUniversityofRhodeIsland4/19/03Figure1月3)Figure2Amplifierfrequencyresponse.PMWt/WeqQ%0S55°0.7000.92513.3%60°0.5800.8178.7%65°0.4700.7174.7%70°0.3600.6221.4%75°0.2700.5270.008%TableI•PMisthephasemargin.
2、•wtistheunitygainfrequency(rad/sec)•WhenanamplifierwithagainA(QisputinafeedbackloopasshowninFigure1,theclosedloopgain,Vo/Vin=AcL•weqisthefrequencyoftheequivalenthigherorderpole(rad/sec)••QisthesystemQualityfactor.Thesystemisunstable•OSistheOverShoot.whentheloopgain,力A(s),e
3、quals-1.Thatis,力A(s)hasamagnitudeofoneandaphaseof-180degrees.Anunstablesystemoscillates.Asystemclosetobeingunstablehasalargeringingovershootinresponsetoastepinput.Thephasemarginisameasureofhowclosethephaseoftheloopgainisto-180degrees,whenthemagnitudeoftheloopgainisone.Thep
4、hasemarginistheadditionalphaserequiredtobringthephaseoftheloopgainto-180degrees.PhaseMargin=Phaseofloopgain・(-180).TheloopgainhasadominantpoleatHigherorderpolescanberepresentedbyanequivalentpoleat"旳・TheamplifierisapproximatedbyafunctionwithtwopolesasshowninEquation2.Sincef
5、orfrequenciesofinterestwheretheloopgainmagnitudeisclosetounity,专>Up、And,Also,itcanbeshown.DefiningForfrequenciesofinterest(frequenciesclosetotheunitygainfrequency),theamplifiergaincanbewritten,PluggingEquation7intoEquation1resultsinthefollowingexpressionfortheclosedloopgai
6、n.A宓何=*:⑻丹丄十瓦盲十飾片Equation8isthetransferfunctionforasecondordersystem.Thegeneralformfortheresponseofasecondordersystem,wheresystempropertiesaredescribedbyitsQandresonantfrequencew0,isshowninEquation10.A化A)=
7、f(9)BycomparingEquation8toEquation9wecangetanexpressionfortheresona
8、ntfrequencyandQoftheamplifierclosedloopgain・(Equatecoefficientsoflikepowersofsinthedominators.)闷=作忑监(10)(11)TheloopgainisthefeedbackfactorBmultipliedbytheamplifiergainA(s).叫=乔m(⑵Thephasemarginisafunctionofthephaseoftheloopgainatthefrequencywherethemagnitudeoftheloopgainisu
9、nity.(13)where5istheloopgainunitygainfrequency.ItfollowsfromEquations12and13that,Also,sol
