Network representations of angular regions for electromagnetic scattering

53 Pages·2017·6.28 MB·English

RESEARCHARTICLE
Network representations of angular regions
for electromagnetic scattering
VitoG.Daniele1,2☯,GuidoLombardi1,2☯*,RodolfoS.Zich1,2☯
1PolitecnicodiTorino,CorsoDucadegliAbruzzi24,I-10129Torino,Italy,2IstitutoSuperioreMarioBoella,
viaP.C.Boggio61,I-10138Torino,Italy
☯Theseauthorscontributedequallytothiswork.
*[email protected]
Abstract
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Networkmodelinginelectromagneticsisaneffectivetechniqueintreatingscatteringprob-
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lemsbycanonicalandcomplexstructures.Geometriesconstitutedofangularregions
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a1111111111 (wedges)togetherwithplanarlayerscannowbeapproachedwiththeGeneralizedWiener-
HopfTechniquesupportedbynetworkrepresentationinspectraldomain.Evenifthenet-
workrepresentationsinspectralplanesareofgreatimportancebythemselves,theaimof
thispaperistopresentatheoreticalbaseandageneralprocedurefortheformulationof
complexscatteringproblemsusingnetworkrepresentationfortheGeneralizedWienerHopf
OPENACCESS
Techniquestartingbasicallyfromthewaveequation.Inparticularwhilethespectralnetwork
Citation:DanieleVG,LombardiG,ZichRS(2017)
representationsarerelativelywellknownforplanarlayers,thenetworkmodellingforan
Networkrepresentationsofangularregionsfor
electromagneticscattering.PLoSONE12(8): angularregionrequiresanewtheorythatwillbedevelopedinthispaper.Withthistheorywe
e0182763.https://doi.org/10.1371/journal. completetheformulationofanetworkmethodologywhoseeffectivenessisdemonstrated
pone.0182763
bytheapplicationtoacomplexscatteringproblemwithpracticalsolutionsgivenintermsof
Editor:JunMa,LanzhouUniversityofTechnology, GTD/UTDdiffractioncoefficientsandtotalfarfieldsforengineeringapplications.Themeth-
CHINA
odologycanbeappliedtootherphysicsfields.
Received:December12,2016
Accepted:July24,2017
Published:August17,2017
Copyright:©2017Danieleetal.Thisisanopen
accessarticledistributedunderthetermsofthe
CreativeCommonsAttributionLicense,which 1Introduction
permitsunrestricteduse,distribution,and
Networkmodelinghasbeenwidelyusedinelectromagneticssinceearlydevelopmentsof
reproductioninanymedium,providedtheoriginal
authorandsourcearecredited. microwaves[1]anditisconstitutiveofthegeneraltheoryofradiationandscatteringaspro-
posedbyMarcuvitzandFelsen[2].
DataAvailabilityStatement:Allrelevantdataare
Forwhatconcernsthescatteringbystratifiedplanarmediainpresenceofplanardisconti-
withinthepaper.
nuitiesacompletereferencebasedonnetworkformulationsis[3].
Funding:ThisworkissupportedbytheIstituto
Recently,geometriesconstitutedofcoupledangularandplanarregions[4–10]havebeen
SuperioreMarioBoella(ISMB),Torino,Italy,http://
studiedthroughtheGeneralizedWienerHopftechniqueintroducingLaplacetransform(LT)
www.ismb.it/.Thefunderhadnoroleinstudy
design,datacollectionandanalysis,decisionto inradialdirection,extendingtheclassofsolvableproblemsthroughtheWiener-Hopf(WH)
publish,orpreparationofthemanuscript. technique,see[11]andreferencestherein.MoreovertheWiener-Hopftechniqueiseffectively
appliedtostudyproblemswithabruptdiscontinuitiesindifferentphysics[11]asacoustics,
Competinginterests:Theauthorshavedeclared
thatnocompetinginterestsexist. elasticity,fracturemechanics...
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Networkrepresentationsofangularregionsforelectromagneticscattering
TheuseoftheWiener-Hopftechniqueincomplexstructureshighlightstherelevanceof
networkrepresentations.Duringthesestudiesitbecameevidentthatmoreefforthastobe
expressedtocompletelymasterthenetworkrepresentationofanangularregionthatcanbe
combinedwiththerepresentationsofplanarregionstomodelcomplexproblems:thisisthe
aimofthepaper.ThelogicalframeworkofthepaperissketchedinFig1wheretheconsidered
geometriesarepresentedtogetherwiththerelationshipamongthespectralrepresentationsof
thescatteringproblem.
Networkmodellingofangularregionswasfirstlyproposedin[2]wheretherepresentations
wereobtainedusingtheclassicalseparationvariablemethod[12]throughthedefinitionof
Green’sfunctionsinnaturaldomain(cylindricalcoordinatesρ,φ,z).However,thismethodis
effectiveonlyforparticularboundaryconditionsasreportedinSec.6.6of[2],wherethecon-
clusionisthatthegeneralangularregionproblem“requires moresophisticatedmathematical
techniques”.
Pureisolatedangulargeometries(wedges)areeffectivelyapproachedinliteraturebyspec-
tralrepresentations.Anexhaustiveanalysisofliteratureshowsthatthemostimportantrepre-
sentationsaretheoneformulatedintermsofSommerfeldMalyuzhinets(SM)functionsand
theonebasedontheKontorovich-Lebedev(KL)transform.TheintroductionoftheGeneral-
izedWienerHopf(GWH)techniqueforangularregion[13]completesthespectralmethods.
Thethreementionedmethodsaredefinedinspecificspectral(transformation)domains,
seeSection3,whichareinmutualrelations:
• theSommerfeld-Malyuzhinets(SM)functionmethodwithspectralcomplexplanew
[14–21]
• theKontorovich-Lebedev(KL)transformmethodwithspectralcomplexplanet[22–25]
• methodsbasedontheLaplacetransform(LT)withspectralcomplexplaneη[11,13,26–31],
thatincludetheGWHtechnique.
Intheaboveitemizationwehavereportedasampleofliteraturerelatedtothethreemeth-
ods:theoriginalworkstogetherwithreviewworks.
Fig1.Conceptofcomplexgeometryidentifiedbygrayfilledboxasscatteringproblemconstitutedof
coupledangularandlayeredregions(boxeswithsolidblackcontours);graphicalinterpretationof
thetechniqueproposedinthepaperbasedontheconnectionamongspectralrepresentationsof
scatteringproblemsthroughWHequationsandnetworkrepresentations(boxeswithdashedgray
contours).
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Networkrepresentationsofangularregionsforelectromagneticscattering
Fig2.Graphicalresumeofnaturalplane(ρ)andcomplexspectralplanes(η,w,t)consideredinthe
paper.
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TheserepresentationsareinterrelatedseeFig2.Inparticulartherelationshipbetweenη
andwplaneisthroughthemappingη=−kcoswwithpropagationconstantkandtherelation-
shipbetweenwandtplanesisthroughtheMalyuzhinets-Fourier(MF),seeSection3fora
detaileddefinitionanddiscussion.
Westatethateachrepresentationpresentsadvantagesanddisadvantageswhichareillus-
tratedinthispaper.
Infactwhilethenetworkrepresentationsintheplanarlayersareobtainableworkingonly
intheframeworkoftheLaplacetransforms(see[3–5]anddeductionofnetworkrepresenta-
tionofplanarlayerregion2inSection7),thegeneralproblemofangularregioncannotbe
easilystudiedbyusingthepropertiesofasinglespectralplane.Wepresentinthispaperapro-
cedurethattakesadvantageoftheformulationsgivenineachoneofthedefinedspectral
planes,althoughouraimistopresentmodelingandsolutionsthatprivilegestheuseofLaplace
Transform.InthefollowingthecomplexplaneofLaplaceTransformislabeledwithη.Inthe
paperwemakealsoreferencetoSMfunctionsandtotheKLtransformwherewelabelthe
respectivespectralplaneswithwandt.Theadoptednotationsarestronglyinfluencedfrom
thefundamentalbookonradiationandscattering[2]whosenotationsaredifferentfromthose
usedbyauthorsspecializedintheSMmethod.
Theuseofηplaneisofgreatimportance:1)itallowstodefinetheentireproblemina
uniquespectralplanewhendifferentmaterialscoexistand2)itallowstohandleangular
regionswithanyapertureanglesdirectlyintermsofintegralequations.Inparticularproperty
1)yieldsthattheunknownsaredefinedinthesamplecomplexplaneηsinceηplanedoesnot
dependonthematerialandthusthereisnoneedtodefinecumbersomeruletocouplethe
unknownsdefinedindifferentcomplexplanesatthematerialinterfaces.Onthecontraryw
planedependsonthematerialthroughthelocalpropagationconstant(seenextSection3for
definition).
Inspectraldomainthenetworkmodellingisaneffectivetechniquetorepresentcomplex
problemsandinthispaperweproposeanexampleofproceduretoobtainnetworkmodelling
forthescatteringbywedgestructuresalsoinpresenceoflayers.
Networkmodelingmathematicallyrephrasesspectralrepresentationsandityieldsseveral
advantages:
1. Equivalentnetworkmodelingofacomplexproblemallowsadirectandquickpictorial
interpretationoftheinteractionamongstructuresthatconstitutetheoriginalproblem.
2. Theunknownsoftheelectromagneticproblemaredirectlyrelatedtocurrentsandvoltages
ofthenetworkmodel.
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Networkrepresentationsofangularregionsforelectromagneticscattering
Fig3.ExampleofWHgeometriesmodelledbynetworkrepresentations:(a)angularregionsin
presenceofastratification,(b)truncatedwedgewithasemi-infinitelayer,(c)wedgeoverapenetrable
halfspace.
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3. Eachpartofacomplexelectromagneticstructurecanbeidentifiedandmodeledasacir-
cuitalcomponent/networkwhoseparametersareevaluatedonceandforall.
4. Theinteractionbydifferentpartoftheproblemsisenforcedbycurrentandvoltageconti-
nuityattheportsoftheequivalentnetworks.
5. Circuitanalysishelpstoorderandtosystematizethestepsnecessarytoobtaintheequations
thatmodelthecomplexproblemavoidingredundancy.
Evenifthenetworkrepresentationsinspectralplanesisofgreatimportancebyitself,the
aimofthispaperistopresentatheoreticalbaseandageneralprocedurefortheformulationof
complexproblemsusingnetworkrepresentationfortheWienerHopf(WH)Technique.Fig3
reportsalistofexamplesofproblemsthatcanbeformulatedandstudiedusingGeneralized
WienerHopf(GWH)technique.
ThesolutionisusuallyobtainedviathereductionoftheGWHEstoasystemofcoupled
Fredholmintegralequations(FIEs)[32,33]viatheeliminationofoneoftheunknownsinthe
networkmodel.WiththispaperwestatethattheFIEsderivedforGWHEscanbereframed
usingnetworkrepresentationsandinvolvekernelsexpressedintermsofelementaryfunctions.
AnovelpracticalexampleisoutlinedinSection7.
Theaimofthispaperistosimplifyandtosystematicallypresenttheproceduretomodel
electromagneticscatteringproblemsinangularregionsbyresortingtonetworkmodelinginw
andηplanesstartingfromGWHformulationsinηplane.
However,theprocedurereportedinthispaperstartsfromthewaveequationsincylindrical
coordinates(Section4)andfollowingtherotatingwavesmethod[34]yieldstothenetwork
modelingofangularregionsint,wplanes(Sections4and5)andfinallyinηplane(Section6).
AlthoughthesameequationsareobtainablefromGWHformulation[13]byusingthe
Fredholmfactorization[4,32,33,35–37],herewehaveprivilegedtheuseofrotatingwaves
methodbecausethetransmissionlinemodelandthusthenetworkmodelisstraightforwardin
thisparadigm.Moreoverrotatingwavesmethodworksdirectlyinwplaneandasstatedin[2]
thiscomplexplanedoesnothideanyspectralpropertiesoftheproblemwithrespecttoηplane
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(properandimpropersheetsdefinedwithrespecttothex¼ k2(cid:0) Z2function).
ThepaperisfocusedontheNortontwo-portrepresentation,butothertwo-portrepresenta-
tionsarepossible(forinstancetheTheveninmodel)followingthesameprocedure.Forspace
reasonthisitemisnotconsideredinthispaper.HowevertheNortonrepresentationsofangu-
larregionsandlayeredregionsprovidetheformulationintermsofFIEsofelectromagnetic
problemswithgeometrieslikethoseillustratedinFig3.
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Networkrepresentationsofangularregionsforelectromagneticscattering
WewillshowinSection4.3thatthenetworkparametersofthetwoportint-planeare
expressedbyelementaryfunctions.Adifficulttaskistogettherepresentationsinη-plane,
startingfromthesesimpleexpressions.
ThepaperisdividedintosevenSections,includingtheIntroduction.Section2presents
howtomodelangularregionsthroughtwoportNortonnetworksinabstractspaceandintro-
ducesthespectralplanesandthenotationusedinthispaper.Section3presentstheprincipal
spectraldomainsinusetodealwithscatteringprobleminelectromagnetics.Section4esti-
matestheparameteroftheNortonnetworkinabsenceofsource,inpresenceofconcentrated
sourcesandforanincidentplanewave.Section5analyzesthecaseofanangularregiondelim-
itedbyanimpenetrablesurfaceandpresentstheoneportNortonnetworkinparticularfor
regionsterminatedbyaperfectelectricconducting(PEC)surface.Section6presentsthepro-
ceduretoobtainFIEsforangularregionsinηplane,inparticularforthePECwedge.The
methodologyisgeneralandtheequationsarefullyvalidatedbynumerousexamples.
Throughoutthepaper,wefocustheimplementationonproblemsconstitutedofPEC
wedgesurroundedbydifferentobjects/materialswheretheNortonrepresentationisparticu-
larlyeffective.Extensiontosurfaceimpedancestructuresandpenetrablestructuresispossible
andofpracticalinterest.Moreoverothertwoportnetworkmodellingcanbederived(for
instancetheTheveninmodel).
Finally,wevalidatetheproposedmethodinSection7bypresentinganovelcomplexprob-
lem.TheproblemisconstitutedbythescatteringofanE polarizedplanewavebyaPEC
z
wedgeaboveaPECplanethatrecallsahornantenna.Theresultsshowthecapabilityofnet-
workmodellingtodealwithcomplexproblemsandshowtheconvergence,theefficiencyand
theefficacyoftheproposedmethodestimatingtheelectromagneticfarfieldintermsofGeo-
metricalOptics(GO)componentsandUniformTheoryofDiffraction(UTD)components.
Thesolutionprocedureisbasedoncouplingnetworkrepresentationsofangularandplanar
layeredregionsinηplane.Thenetworkrepresentationoftheangularregioninηplaneis
obtainedinthispaperstartingfromthewaveequation(Section4)andfollowingtheprocedure
proposedinSections5and6.Theplanarlayeredregionisdescribeddirectlythroughanexam-
pleinSection7sinceitisderivedfromthetheoryofplanarstratifiedregiondescribedin[3].
ToimprovethereadabilityofthemanuscriptwehaveaddedanAppendixandalistofcon-
ventions/notations/symbolsattheendofthemanuscript.
2Nortontwo-portnetworkforanangularregioninabstractspace
Fig4presentsanexampleofspacesubdividedintoangularregionswitharbitraryaperture
angles.Eachcylindricalsectorisfilledbyalinearmedium.Withoutlossofgenerality,letus
focustheattentionontheregiondelimitedbythehalfplanesφ=φ andφ=φ whereadistrib-
1 2
utedsourceSislocated.Cartesiancoordinates(x,y,z)aswellaspolarcoordinates(ρ,φ,z)will
beusedtodescribetheproblem.
Resortingtotheuniquenesstheorem,westatethattworelationsmustconnecttheelectro-
magneticfieldcomponentstangenttotheinterfaces.Theseto-bedeterminedrelationsare
constitutiveofatwoportnetworkrepresentinganarbitraryangularregion.Fig5agraphically
representsthetwo-portnetworkinanabstractspace[38]wheretheelectromagneticfieldcom-
ponentstangenttotheinterfacesφ=φ andφ=φ arelabeledrespectivelyE ;H and
1 2 t1 t1
E ;H .
t2 t2
Sincetheproblemislinear(linearmedia),two-portnetworkrepresentationsarepossible.
Fig5bshows,inparticular,theNortonrepresentationwheretheequivalentNortoncurrents
I ;I representH ;H whentwoPEChalf-planesarelocatedatφ=φ andφ=φ .
N1 N2 t1 t2 1 2
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Networkrepresentationsofangularregionsforelectromagneticscattering
Fig4.Arbitraryangularregionsfilledbylinearmedia.InSection3,4,5wemakereferencetotheangular
region1delimitedbyφ=φ andφ=φ withadistributedsourceS.
1 2
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Fig5.Networkrepresentationoftheangularregion1ofFig4inanabstractspace.(a)Undefined
representation,(b)Nortonrepresentation.
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Inabstractoperatoralgebra,theparameteroftheNortonrepresentationy areabstract
ij
operators[38].
Theexplicitdeterminationofparametersinatwo-portnetworkmodelispossiblebyselect-
inganexplicitfunctionspace(seeSection3)torepresenttheunknowns,i.e.thephysicalquan-
tities.InparticularweanticipatetheresultthatintheKLt-spacetheNortonrepresentationis
algebraic,althoughinotherspacestherepresentationisdefinedthroughintegraloperators.
Asstatedintheintroduction,thenaturaldomaindoesnotallowtomodelthegeneralangu-
larregionproblemand,inparticular,eventheevaluationofNortoncurrentsconstitutesa
cumbersomeelectromagneticproblem.Theproblemsimplifiesifoneusesspectralrepresenta-
tionsoftheelectromagneticfields.WithreferencetoFig4,forthesakeofsimplicitytohigh-
lighttheprocedure,weconsiderinthispaperthecasewithcylindricalsymmetry,i.e.thetwo
dimensionalcase,withaE-polarizedelectricfield,forexampleexcitedbyadistributedsource
z
SconstitutedbyanelectriccurrentdensityJ(ρ,φ).Sincethegeometryoftheproblemis
z
invariantwithrespecttothezdirection,thethreedimensionalcasedoesnotintroducenew
conceptualdifficultiesbutsimplyincreasesthenumberoftheunknownsandequations.
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Networkrepresentationsofangularregionsforelectromagneticscattering
3Spectraldomainsforscatteringprobleminelectromagnetics
Consideringtimeharmonicelectromagneticfieldswithatimedependencespecifiedbythe
factorejωt(whichisomitted)andwithreferencetoFig4,theformulationoftheangularregion
problemforaE-polarizedelectricfieldinspectraldomainstartsfromthedefinitionofthe
z
Laplacetransformsofelectromagneticfieldcomponents:
Z Z
1 1
V_ ðs;φÞ¼ E ðr;φÞe(cid:0) srdr; I_ ðs;φÞ¼ H ðr;φÞe(cid:0) srdr ð1Þ
þ z þ r
0 0
InthefollowingwewillmakeextensiveuseoftheWiener-Hopfquantitiesdefinedintheη=js
plane,i.e.FðZ;φÞ¼F_ðs¼(cid:0) jZ;φÞ.
Toavoidthepresenceofsingularitiesontherealaxisofη=jsplane,thepropagationcon-
stantskareassumedwithanegative(vanishing)imaginarypart.UsingtheframeworkofWie-
ner-Hopftechnique,thesubscript+inEq(1)indicatesplusfunctions,i.e.functionshaving
regularityhalf-planesthatareupperhalf-planesintheη-plane.Converselythesubscript—indi-
catesminusfunctions,i.e.functionshavingregularityhalf-planesthatarelowerhalf-planesin
theη-plane.InthefollowingwewillmakeextensiveuseofthespectralquantitiesEq(1)
definedforφ=φ andφ=φ ,thatwelabel
1 2
V ðZÞ¼V ðZ;φ Þ¼V_ ðs;φ Þ
qþ þ q þ q
; q¼1;2 ð2Þ
I ðZÞ¼I ðZ;φ Þ¼I_ ðs;φ Þ
qþ þ q þ q
Inthispaper,wefirstderiveageneralproceduretoobtainthenetworkmodelbyusingthe
angularcomplexplanew
Z¼(cid:0) kcosw ð3Þ
startingfromthecomplexplanetdefinedthroughtheMalyuzhinets-Fourier(MF)transform
andtheKontorovich-Lebedev(KL)transform,seebelowforacompletedefinition.
Thebenefitsofthecomplexplanewaredescribedinseveralworks,seeforexample[2,3,
13,34].Inparticular,weremarkthefollowingproperties:
• aplusfunctionintheη-planeisanevenfunctionofw
• transmissionlineequationsholdfortheunknownsinw-plane
• theSommerfeldfunctionsaredefinedinw-plane
• inw-planetheNortonparametersareoperatorsthatdonotinvolvemultivaluedfunctions,
onthecontrarymultivaluedfunctionsaretobedefinedinηplane
Inthefollowingweapplythenotationswherewelabelthequantitiesinthespectraldomain
withoutanysignontopofthevariablewhenthequantityisdefinedinηplane,with(cid:1)(dot),^
(hat)or−(overline)ontopofthevariablewhenthequantityisdefinedrespectivelyins,wort
plane,andwith*(tilde)ontopofthevariableifwemakeabstractlyreferencetoageneral
spectraldomain(η,s,wort).
Inwplanewedefine
V^ ðw;φÞ¼sinwV^ ðw;φÞ¼sinwV ð(cid:0) kcosw;φÞ; ^I ðw;φÞ¼I ð(cid:0) kcosw;φÞ
d þ þ þ þ
ð4Þ
V^ ðwÞ¼sinwV^ ðwÞ¼sinwV ð(cid:0) kcoswÞ; ^I ðwÞ¼I ð(cid:0) kcoswÞ
qd qþ qþ qþ qþ
withq=1,2.
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Inparticularwedefinethetwo-portnetworkinwplaneintermsofthevoltagesV^ ðwÞand
qd
thecurrents^I ðwÞwithq=1,2.
qþ
WeintroducetheMalyuzhinets-Fourier(MF)transformthatrelatesquantitiesdefinedin
thecomplexplanewtoquantitiesinthecomplexplanet[34]
Z
j1
MfF^ðwÞg¼F(cid:22)ðtÞ¼(cid:0) j F^ðwÞeþjtwdw; Re½t(cid:138)¼0 ð5Þ
(cid:0) j1
Z
j j1
F^ðwÞ¼M(cid:0) 1fF(cid:22)ðtÞg¼(cid:0) F(cid:22)ðtÞe(cid:0) jtwdt; Re½w(cid:138)¼0 ð6Þ
2p
(cid:0) j1
Thistransformmustbedefinedinthedistributionspace,thustheintegralsinEqs(5)and(6)
areCauchyprincipalvalues.ThemainbenefitofthecomplexplanetisthattheNortonmodel
isconstitutedofalgebraicparameters,asdemonstratedintheSection4.
Relationsamongnaturaldomaininρ,LaplaceTransform(LT)inη,thecomplexplanew(η
=−kcosw),theMalyuzhinets-Fourier(MF)transformint,theKontorovich-Lebedev(KL)
transformandtheSommerfeld-Malyuzhinets(SM)functionaredeeplystudiedin[34]andwe
graphicallyresumetheserelationsinFig2.
4EvaluationoftheNortonnetworkinspectraldomains
Letusconsiderthehomogenousangularregionφ <φ<φ withadistributedsourcemod-
1 2
elledbyJ(ρ,φ),seeFig4.Asalreadystatedinthenaturaldomain,takingintoaccountthe
z
uniquenesstheorem,twoequationsmustrelatethefourquantitiesV~ ;~I (q=1,2)alsoin
qd qþ
spectraldomains.Fig6agraphicallyrepresentsthetwo-portnetworkinanabstractspectral
domain.Fig6billustratestheNortonrepresentation
0 1 2 30 1 0 1
~I Y~e Y~e V~ ~I
@ 1þ A¼4 11 125@ 1dAþ@ 1Nþ A ð7Þ
(cid:0) ~I Y~e Y~e V~ (cid:0) ~I
2þ 21 22 2d 2Nþ
wherethevectors~I ;~I aretheshort-circuitcurrentsonthetwo-portnetworkandthe
1Nþ 2Nþ
matrixistheadmittancematrixofthetwo-portnetworkthatisingeneralconstitutedofopera-
torelements.TheproceduretoobtaintheNortonparametersisillustratedinFig7anditis
similartoandbasedontheclassicalapproachofcircuittheory.
Asalreadydiscussedfortheabstractspace,weobtaintheNortonequivalentcurrentsby
introducingtwoPEChalfplanesatφ=φ andφ=φ .WithreferencetoFig7a,accordingto
1 2
Eqs(1)and(2),ingeneralinaspectraldomain,thePECconstraintforE-polarizationis
z
Fig6.Networkrepresentationoftheangularregioninaspectraldomain.(a)Undefinedrepresentation,
(b)Nortonrepresentation.
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Fig7.EvaluationoftheNortonparametersofthetwo-portnetworkrepresentingtheangularregionin
aspectraldomain.(a)Nortoncurrents,(b)Nortonadmittances.
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equivalenttoshort-circuitthetwo-portoftheequivalentnetwork(i.e.V~ ¼0andV~ ¼0),
1d 2d
yieldingtherequiredequivalentNortoncurrents~I and~I .Asclassicallydoneincircuit
1Nþ 2Nþ
theory,theadmittanceparametersareobtainedbyremovinginternalsourcesandbyfeeding
thetwo-portbytwoarbitraryvoltagegeneratorV~ andV~ .WithreferencetoFig7b,we
1d 2d
definetheadmittanceparameters:
~I ¼Y~e V~ j ; ~I ¼Y~e V~ j ; ~I ¼(cid:0) Y~e V~ j ; ~I ¼(cid:0) Y~e V~ j ð8Þ
1þ 11 1d V~2d¼0 1þ 12 2d V~1d¼0 2þ 21 1d V~2d¼0 2þ 22 2d V~1d¼0
Whentheproblemisreciprocalandsymmetrical(Y~e ¼Y~e andY~e ¼Y~e ),theNorton
12 21 22 11
representationcanberepresentedthroughaπnetwork,seeFig8,whereY~e ¼Y~e þY~e and
1 11 12
Y~e ¼(cid:0) Y~e .WerecallthatingeneralY~e withi,j=1,2aremathematicaloperator.
3 12 ij
Fortheactualcomputationofthenetwork,wefirstconsiderthewaveequationincylindri-
calcoordinateswithacurrentdensitysourceJ(ρ,φ)thatdoesnotdependonz:
z
(cid:18) (cid:19)
@ @E @E @2E
r2 z (cid:0) r zþ zþk2r2E ¼jkZ r2J ðr;φÞ ð9Þ
@r @r @r @φ2 z o z
whereZ andkaretheimpedanceandthepropagationconstantofthefreespace.
o
ByapplyingtheLaplacetransformEq(1)anditsproperties(i.e.L[f(ρ)]=F[s],
h i
L @fðrÞ ¼sF½s(cid:138)(cid:0) fð0Þ,L½rfðrÞ(cid:138)¼(cid:0) @F½s(cid:138))toEq(9)andtakingintoaccountMeixnercondi-
@r @s
tions[39]r2@Ezðr;φÞ!0and@Ezð0;φÞ¼0forρ!0,weobtain
@r @s
@2 @ @2V_ ðs;φÞ @2 @2
s ðsV_ ðs;φÞÞþ ðsV_ ðs;φÞÞþ þ þk2 V_ ðs;φÞ¼jkZ J_ ðs;φÞ ð10Þ
@s2 þ @s þ @φ2 @s2 þ o@s2 z
Fig8.Nortonπnetworkinaspectraldomainfortheangularregionφ <φ<φ .
1 2
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Networkrepresentationsofangularregionsforelectromagneticscattering
andthus:
@2 @ @2V_ ðs;φÞ @2
ðk2þs2Þ V_ ðs;φÞþ3s V_ ðs;φÞþV_ ðs;φÞþ þ ¼jkZ J_ ðs;φÞ ð11Þ
@s2 þ @s þ þ @φ2 o@s2 z
i.e.
(cid:20) (cid:21)
@ @ @ @2V_ ðs;φÞ @2
ðk2þs2Þ V_ ðs;φÞ þ ðsV_ ðs;φÞÞþ þ ¼jkZ J_ ðs;φÞ ð12Þ
@s @s þ @s þ @φ2 o@s2 z
Recallingthats=−jηandη=−kcoswandtakingintoaccountthat @ ¼ 1 @ andk2+s2=k2
@s (cid:0) jksinw@w
sin2w,Eq(12)becomes
(cid:18) (cid:19)
(cid:0) 1 @ @ (cid:0) 1 @ @2V^ ðw;φÞ @2
sinw V^ ðw;φÞ þ ðcoswV^ ðw;φÞÞþ þ ¼jkZ J ðs;φÞ ð13Þ
sinw@w @w þ sinw@w þ @φ2 o@s2 z
IntroducingV^ ðw;φÞEq(4)andaftersomemathematicalmanipulationEq(13)reducesto:
d
(cid:18) (cid:19)
@2V^ ðw;φÞ @2V^ ðw;φÞ @ 1 @
(cid:0) d þ d ¼Z ^J ðw;φÞ ¼S^ðw;φÞ ð14Þ
@2w @2φ o@w jksinw@w z
TheproceduretoobtainEq(14)hasbeeninspiredby[40,41],althoughin[40,41]nosource
ispresent.ByapplyingtheLaplacetransformEq(1)totheMaxwell-FaradayEq(15)
@E
(cid:0) jrkZ H ¼ z ð15Þ
o r @φ
andintroducingthewplane(η=−kcosw)wehave
@^I ðw;φÞ @V^ ðw;φÞ
(cid:0) Z þ ¼ d ð16Þ
o @w @φ
TheapplicationoftheMFtransformEqs(5)to(16)andtoEq(14)yields:
@V(cid:22) ðt;φÞ
d ¼jtZ(cid:22)I ðt;φÞ ð17Þ
@φ o þ
@2V(cid:22) ðt;φÞ
d þt2V(cid:22) ðt;φÞ¼S(cid:22)ðt;φÞ ð18Þ
@φ2 d
SubstitutingEq(17)intoEq(18)yields
@(cid:22)I ðt;φÞ (cid:22)Sðt;φÞ
þ ¼jtY V(cid:22) ðt;φÞþ ð19Þ
@φ o d jtZ
o
whereS(cid:22)ðt;φÞistheMFtransformoftherightendsideofEq(14).
Eqs(17)and(19)areclassicalequationsoftransmissionlinesinfrequencydomainwhere
theazimuthalangleφandthespectralvariablettakerespectivelytheroleofthedirectionof
propagationandthepropagationconstant.
PLOSONE|https://doi.org/10.1371/journal.pone.0182763 August17,2017 10/53