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1、Solutionsformodule1313.4(P268)Solution:(b)GH(s)=105(1+45)10qJ1+166!?0=_90_tan"(4Q)0T—9O0T—180Hence:Frequencyresponseforw<0isreflectionof69>0.NowplottheNyquistcontourintheS-plane,modifiedduetopoleattheorigin,completethecontourinthel+GH(s)plane.DfisconnectedtoAbyccosemi-circle.Noencircle
2、mentsof(-1jO)/.stablesystemFromtabic12.19dtOMt40t069TCOA/—>0(/)—>-180Alsocon=IradIsg=0.5PlotregularNyquistcontourNoencirclementsofcriticalpoint/.systemisstable(e)GH(s)=S(S+1)(5+2)2.5s(l+s)(l+0.5s)2.50j(l+/)(l+”/4),(f)=-90-tan-1(co)-tan-1(o)/2)69—>0M—g0T-90®fgMtO-270Hencesketchfrequency
3、response(Nyquist)Whereisintersectiononnegativerealaxis?-180=-90一tan-1(q)-tan-1(q/2)90=tan1(69)+tarT”(69/2)Taketangents:oo=少+e2henceco=41=lA4rad/s1S/2magnitudeatthisfrequency=M=5/6InplottingtheS-plancNyquistcontour,adetouraroundthepoleattheoriginisnecessary.Thel+GH(s)contourfollowsdire
4、ctly.Sincetherearenoencirclementsof(-1JO)Thesystemisstable.Notehowever,thatifK>5,thesystemmaybecomeunstable,infactthemaxA5gainconsistentwithstabilityisK==6J5/613.5Solution:Sincetherearethreeopen-looppoles(-3j,0f+3/)ontheimaginaryaxis,TheNyquistcontourrsshouldbemodifiedasshowninFig.l3.5
5、-1toavoidthem:ImG=0)Fig.13.5-2Let'sbeginfromco=0+.Atthisfrequency,Mtoo,(p=-902、so「ghstartsfromthelowerendoftheimaginaryaxis.Aso)increases,Mdecreasestoafinitevalue,butwhencogetsclosetoMstopsdecreasingandgoesbacktoinfinityagainduetotheinfiniteresonancepeakoftheundamped2nd-orderelement.It
6、isimportanttonotethatuptonow,cp=-90(Why?SeeFig.5.3onpage78).ShowninFig.l3.5-2istheNyquistdiagramforthefrequencyrangesg(0+,3-).Fromco=3-toe=3+，(pjumpsfrom-90°to-270Accordingly,ranjumpsfromthenegativeimaginaryaxistothepositiveimaginaryaxis.Next,A/willdecreasefrominfinitytozerowhencocont
7、inuestoincreasefrom3+to+oo.Fig.13-5-3showstheNyquistdiagramforthefrequencyrange(0+,3—)and(3+,+oo)・Now,wcneedtodeterminetheimagefortheinfinitesimalsemicircle(无穷彳、半圆)aroundtheopen-looppoleats=+3j.Sincethereisonlyoneopen-looppoleatthispoint,thisinfinitesimalsemicircleinCCWdirectionmapsi
