mapping in the complex plane

5/2/2015 MappingintheComplexPlane

http://lpsa.swarthmore.edu/Nyquist/NyquistMapping.html 1/24

MappingintheComplexPlaneOverview Mapping Stability Examples Bode BodeExamples NyquistGui Printable

ContentsMappinginthecomplexplaneMappingwithasinglezeroMappingwithasinglepoleMappingwithmultiplepolesand/orzerosThepathin"s"neednotbecircularKeyConceptsaboutMappingMovingForward

MappingintheComplexPlaneThemappingoffunctionsinthecomplexplaneisconceptuallysimple,butwillleadustoaverypowerfultechniquefordeterminingsystemstability.Inadditionit

willgiveusinsightintohowtoavoidinstability.Tointroducetheconceptwewillstartwithsomesimpleexamples.Thereareseveralvideosonthispagetheymerelysupportthewrittenmaterial,butarenotabsolutelyvital.

MappingofFunctionswithaSingleZeroThesimplestfunctionstomaparethosewithasinglezero.Severalexamplesfollow.

Example:Mappingwithcirclein"s",zeroattheoriginin"L(s)"

ConsiderthetrivialfunctionL(s)=s(wewilldealwithmorecomplicatedfunctionslater,thissimplefunctionallowsustointroduceconceptsassociatedwithmapping).Weletthevariablestraverseacircularpathcenteredattheoriginwitharadiusof5movingintheclockwisedirection(theleftsidegraph),Inotherwords

WethenplotL(s)ontherighthandgraph.Thisisshownintheimagebelow,followedsoonthereafterbyavideothatbetterdemonstratesthemapping.



NotesaboutimageaboveThereareseveralaspectsabouttheimageabovethatareimportant(manyofthesecomeuplater):

Thepathinthe"s"planeisshownattheleft.Itisacircleofradius5,centeredattheoriginandmovinginaclockwisedirection,asindicatedbythearrows.Thearrowsshowthedirectionofmotion,butthespacingofthearrowsisarbitrary.Thecolorsofthetwoplotsmatch,sothattheportionof"s"thatisyellowisalsoyellowon"L(s)".Thisisobvioushere,butbecomesausefuldistinctioninmorecomplexplotstocome.Thereisapinkcirclecenteredonthezerointhe"s"plane.Thisshowstheanglesubtendedbetweenthezeroandthepathins.Youcanseethegrowthofthiscircle(intheclockwisedirection)inthevideobelow.Thereisalsoapinkcirclecenteredontheorigininthe"L(s)"plane.Thisshowstheanglesubtendedbetweentheoriginandthepathins.Thecolorindicatesthedirectionpinkindicatestheclockwisedirection.Ifthedirectionhadbeencounterclockwiseitwouldbeblue.Notethatasthepathin"s"encirclesthezeroonetimeinthecounterclockwisedirection,thepathin"L(s)"encirclestheoriginonceinthesamedirection.



Video:Mappingwithasinglezero2minutevideocreatedwiththeMatlabscriptNyquistGui)

Example:Mappingwithcirclein"s",zeroats=4

Ifwenowplaceazeroats=4sothatL(s)=s+4,themappingisstillverystraightforward.Everylocationin"s"simplymapstoalocationthatis4unitstotherightin"L(s)".Thepathinsremainsasbefore.(Note:thisexampleisalsoconsideredaspartofthevideoabove.)



NotesaboutimageaboveThepathin"s"isstillacircle,asisthepathin"L(s)",butthepathissimplyoffsetby4(sinceL(s)=s+4.Notethatasthepathin"s"encirclesthezeroonetimeintheclockwisedirection,thepathin"L(s)"encirclestheoriginonceinthesamedirection.Asthepathin"s"getsnearthepolein"L(s)"(wherethepathhasalightbluecolor),thepathin"L(s)"becomesthesmallest(i.e.,theclosesttotheorigin).Thisisbecausetheterminthenumerator,inthiscase(s+4),isminimalwhen"s"isnearthepoleat4.Also,whenthedistancefromthezerotothepathin"s"ismaximal(red),soisthedistancefromL(s)totheorigin.

Example:Mappingwithcirclein"s",zeroats=6

Ifwenowplaceazeroats=6sothatL(s)=s+6,themappingisstillverystraightforward.Everylocationin"s"simplymapstoalocationthatis4unitstotherightin"L(s)."Thepathinsremainsasbefore.(Note:thisexampleisalsoconsideredaspartofthevideoabove.)



NotesaboutimageaboveThepathin"s"isstillacircle,asisthepathin"L(s)",butthepathissimplyoffsetby4(sinceL(s)=s+4.Notethatasthepathin"s"nolongerencirclesthezero,andthepathin"L(s)"nolongerencirclestheorigin.

Commentsaboutmappingwithapolein"L(s)"Therearethreeimportant,andgeneral,statementswecannowmakeaboutmappingfrom"s"to"L(s)"whenthereisazeroin"L(s)":

1. Ifthepathin"s"isintheclockwisedirection,thenthepathin"L(s)"isintheclockwisedirection.2. Asthepathin"s"getsclosetothezeroin"L(s)"thepathin"L(s)"goestoitssmallestvalue.3. Ifthepathin"s"encirclesthepoleof"L(s),"thenthepathin"L(s)"encirclestheoriginonceinthesamedirection.

Mappingwithasinglepole



Mappingoffunctionswithasinglepoleisnotmuchmoredifficultthanmappingwithasinglezero.TherearetwoimportantfactsaboutencirclementsofpolesthatcanbeshownbyconsideringL(s)withasinglepoleattheorigin,andthepathin"s"beingaclockwisecircleofradius'r'aroundtheorigin:1. Thefirstcharacteristictoberealizedisthatasthepathin"s"comesclosetoapole,thepathin"L(s)"getslarge

Clearlyastheradiusoftheencirclement,r,becomessmall,themagnitudeofL(s)becomeslarge.2. Thesecondcharacteristicisthatthepathin"L(s)"isintheoppositedirectionofthepathin"s."Inthisexample,thepathin"s"isclockwise,sothepathin"L(s)"

iscounterclockwise.

Themappingaroundvariousfunctions,L(s),withasinglepolesareshowninthediagramsintheexamplesbelow,followedbyavideothatshowsseveralfunctions.Eachexampleinthevideoisalsoincludedintheexamplesthatfollow.Itisusefultoreadtheexamplesbeforeviewingthevideo.

Example:Mappingwithcirclein"s",poleatorigin

IfwechooseL(s)suchthatithasapoleattheorigin,

(note:theconstantmultipliermakestheplotslooksnicer,butisn'tnecessaryforthemathematicstowork)

andweletsfollowthesamepathasbefore

Weget



Becausetheejwasoriginallyinthedenominator,itssignischangedwhenitmovestothenumerator.Inotherwords,thepathin"L(s)"isacircleofradius2thatencirclestheoriginonceintheclockwisedirection.

NotesaboutimageaboveThepathinthe"s"planeisshownattheleft.Itisacircleofradius5,centeredattheoriginandmovinginaclockwisedirection.Thereisabluecirclecenteredonthepoleinthe"s"plane(apinkcirclewillbeusedforzeros,asinthepreviousangles).Thisshowstheanglesubtendedbetweenthepoleandthepathins.Youcanseethegrowthofthiscircleinthevideobelow.Thereisagreycirclecenteredontheorigininthe"L(s)"plane.Thisshowstheanglesubtendedbetweentheoriginandthepathins.Notethatasthepathin"s"encirclesthepoleonetimeintheclockwisedirection,thepathin"L(s)"encirclestheoriginonceintheopposite(counterclockwise)direction.

Example:Mappingwithcirclein"s",poleats=4

Ifwenowplaceapoleats=4sothat



Note:thisexampleisalsoconsideredaspartofthevideoabove.)

Thepathinsremainsasbefore,butthepathin"L(s)"haschanged.Thecenterandextentofthepathin"L(s)"havebothchanged.(note:ifyoucanshowthatthepathin"L(s)"isalsoacircleandderiveequationsfortheradiusandcenter,I'llincludeithere,withanacknowledgementforthefirstpersonwhosentittome)

NotesaboutimageaboveThepathin"s"stillencirclesthepoleinaclockwisedirection,andthepathin"L(s)"stillencirclestheorigininacounterclockwise(opposite)direction.Asthepathin"s"getsnearthepolein"L(s)"(wherethepathhasalightbluecolor),thepathin"L(s)"becomesthelargest(i.e.,thefarthestfromtheorigin).Thisisbecausetheterminthedenominator,inthiscase(s+4),isminimalwhen"s"isnearthepoleat4.Also,whenthedistancefromthepoletothepathin"s"ismaximal(red),thedistancefromL(s)totheoriginisminimal

Example:Mappingwithcirclein"s",poleats=6



Ifwenowplaceapoleats=6sothat


Thepathinsremainsasbefore,butthepathin"L(s)"haschanged.Thecenterandextentofthepathin"L(s)"havebothchanged.

NotesaboutimageaboveNotethatasthepathin"s"nolongerencirclesthepoleinaclockwisedirection,andthepathin"L(s)"nolongerencirclestheorigin.

Example:Mappingwithcirclein"s",poleats=4.8

Ifwenowplaceapoleats=4.8sothat




Thepathinsremainsasbefore,butthepathin"L(s)"haschanged.Thecenterandextentofthepathin"L(s)"havebothchanged.Theshape(note:ifyoucanshowthatthepathin"L(s)"isalsoacircleandderiveequationsfortheradiusandcenter,I'llincludeithere,withanacknowledgementforthefirstonetosendittome)

NotesaboutimageaboveNotethatasthepathin"s"encirclesthepoleinaclockwisedirection,andthepathin"L(s)"stillencirclestheorigininacounterclockwisedirection.Asthepathin"s"comesveryclosetothepolein"L(s)"(wherethepathsarelightblue),thepathin"L(s)"becomesverylarge.Inthesediagramsthisisshownasgoingtowardsinfinity.Intheseplots,anythingwitharadiusofmorethantwelveistruncatedandshownatinfinity.

Video:Mappingwithasinglepole(2minutevideocreatedwiththeMatlabscriptNyquistGui)



Commentsaboutmappingwithapolein"L(s)"Therearetheeimportant,andgeneral,statementswecannowmakeaboutmappingfrom"s"to"L(s)"whenthereisapolein"L(s)":

1. Ifthepathin"s"isintheclockwisedirection,thenthepathin"L(s)"isinthecounterclockwisedirection.2. Asthepathin"s"getsclosetothepolein"L(s)"thepathin"L(s)"goestoitslargestvalue.3. Ifthepathin"s"encirclesthepoleof"L(s),"thenthepathin"L(s)"encirclestheoriginonceintheoppositedirection.

Mappingwithmultiplepolesand/orzerosIfyouunderstandtheconceptofmappingoffunctionswithindividualpolesandzeros,itisnotmuchhardertounderstandmappingoffunctionswithmultiplepoles

andzeros.Afewexampleswillillustratethis.Youshouldreadthroughthefirstexamplecarefully,ithasalotofimportantinformation.

Example:L(s)hastwopoles,onezeroallareencircled

Considermappingofthetransferfunction



wheresfollowsaclockwisecircularpathofradius5aroundtheorigin,asbefore.

Togetanideaofwhatthemappingwilllooklike,let'sexpressthefunctioninpolarnotation.

AtthispointwearemostlyinterestedintheangleofL(s),soletsexamineitmoreclosely.

Recallthat,ingeneral,theangle

isdeterminedbydrawingalinefroms0tos,andfindingtheanglebetweenthatlineandthehorizontal(describedhere).Soifwelets0=2,thentheangle

wouldbedeterminedbydrawingalinefroms=2tosandfindingtheangletothehorizontal.Thisisshowninthediagrambelowontheleftforanglesbetweenthelocations=5jtothezeroat2,andthepolesat1and3.



SincetheangleofL(s)isgivenby

thenthisistheangleshownintheimageaboveontheright.SinceweknowthefirsttermintheangleofL(s)goesfrom02*,andwesubtracttheothertwoterms,thentheangleofL(s)mustgofrom02*,thatisitencirclestheoriginonceinthecounterclockwisedirection.Thiscanbeseenintheimagebelow,andinthevideothatfollows.



NotesaboutimageaboveThepathin"s"encirclesthepolesandzerointheclockwisedirection.Theanglebetweenthepathandthezeroisshowninred,andtheanglebetweenthepathandthepoleisshowninblue.Thepathin"L(s)",whichisshowningray,isthesumoftheanglesfromthezeros(red)minustheanglesfromthepoles(blue).Thisismoreobviousinthevideo.Ifapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=1,P=2soN=1,andwehaveoneencirclementoftheoriginin"L(s)"inthecounterclockwisedirection.

Aside:MagnitudeofL(s)

ThemagnitudeofL(s)isgivenby

Thismeansthatifsisveryclosetoazero(i.e.,nears=2)thatthemagnitudeofL(s)becomesverysmall,andifsisveryclosetoapole(i.e.,nears=1ors=3)thatthemagnitudeofL(s)becomesverylarge.



KeyConcept:MagnitudeandphaseofL(s)

ThephaseofL(s)issimplythesumoftheanglesfromthezerosofL(s)tos,minustheanglesfromthepolesofL(s)tos.

ThemagnitudeofL(s)issmallnearzerosofL(s)andlargenearpolesofL(s).

Example:L(s)hastwopoles,onezero,onepoleonezeroisencircled

Nowifwechangethetransferfunctionto

thenthepathinsencirclesthezero,butonlyoneofthepoles.



NotesaboutimageaboveIfapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=1,P=1soN=0,andwehavenoencirclementsoftheoriginin"L(s)".

Example:L(s)hastwopoles,onezeroonezeroisencircled

Nowif

thenthepathin"s"encirclesonlythezero

NotesaboutimageaboveIfapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=1,P=0soN=1,andwehaveoneencirclementoftheoriginin"L(s)"intheclockwisedirection.



Example:L(s)hastwopoles,onezerotwopolesareencircled

Nowif

thenthepathin"s"encirclesonlythezero

NotesaboutimageaboveIfapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=0,P=2soN=2,andwehavetwoencirclementsoftheoriginin"L(s)"intheclockwisedirection.Youcantelltherearetwoencirclementsbecausethecirclein"L(s)"istwiceasdark.Thisismoreclearinthevideo.



Example:L(s)hastwocomplexconjugatepolesencircledbys

Consider

Polesat24jwhichareinsideacirclewithradius5.

NotesaboutimageaboveIfapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=0,P=2soN=2,andwehavetwoencirclementsoftheoriginin"L(s)"intheclockwisedirection.

Example:L(s)hastwocomplexconjugatepolesnotencircledbys



Consider

Polesat44j,whichareoutsideacircleofradius5.

NotesaboutimageaboveIfapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.InthiscaseZ=0,P=0soN=0andwehavenoencirclementsoftheoriginin"L(s)".

Video:Mappingwithamultiplepolesandzeros(3minutevideocreatedwiththeMatlabscriptNyquistGui)



Thepathin"s"neednotbecircularNothingthatwehavedonesofardependsonthefactthatthepathin"s"becircular,whichisimportanttothedevelopmentoftheNyquiststabilitycriteriononthe

nextwebpage.Twoexamplesbelow(andavideo)demonstratethis.Inthefirstexample,immediatelybelow,thepathin"s"encirclesazerointheclockwisedirection,andthepathin"L(s)"encirclestheorigininthesamedirection.InthisfirstexampleL(s)=s,i.e.,thereisazeroattheorigin.



Inthesecondexample,below,thepathin"s"encirclesapoleintheclockwisedirection,andthepathin"L(s)"encirclestheoriginintheoppositedirection,Thoughthepathisaverydifferentshape.Notealsothatwherethedistancefromthepathin"s"tothepoleisminimal(i.e.,wherethepathislightblue),thenthedistanceofpathin"L(s)"totheoriginismaximal(andwherethedistancein"s"ismaximal(wherethepathisred),thedistancein"L(s)"isminimal).InthisexampleL(s)=10/(s+6),i.e.,thereisapoleats=6.



Video:Mappingwithamultiplepolesandzeros(2minutevideocreatedwiththeMatlabscriptNyquistGui))



KeyConcept:Mappingfrom"s"to"L(s)"

Thekeypointstokeepinmindasyoumovetothenextpage:Ifapathin"s"encirclesazeroofL(s)intheclockwisedirection,thenthiscontributes360intheclockwisedirectiontothepathin"L(s)"asitmovesaroundtheorigin.Ifapathin"s"encirclesapoleofL(s)intheclockwisedirection,thenthiscontributes360inthecounterclockwisedirectiontothepathin"L(s)"asitmovesaroundtheorigin.Ifapathin"s"encirclesZzerosandPpolesofL(s)intheclockwisedirection,thenthepathin"L(s)"encirclestheoriginN=ZPtimesintheclockwisedirection.IfNisnegative,thiscorrespondstoencirclementsinthecounterclockwisedirection.Thepathin"s"neednotbecircular.

MovingForwardAfterreadingthroughthematerialabove,thequestionarises"Sowhat?".Whatwehavedonehereisintroduceatechniquethatgivesuseinformationaboutthe



numberofpolesandzerosinaclosedcontour.Todeterminethestabilityofasystem,wewanttodetermineifasystem'stransferfunctionhasanyofpolesintherighthalfplane.Withjustalittlemorework,wecandefineourcontourin"s"astheentirerighthalfplanethenwecanusethistodetermineifthereareanypolesintherighthalfplane.

References

Copyright2005to2015ErikCheeverThispagemaybefreelyusedforeducationalpurposes.Comments?Questions?Suggestions?Corrections?

ErikCheeverDepartmentofEngineeringSwarthmoreCollege

mapping in the complex plane

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