Finite antenna arrays : an eigencurrent approach

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Finite antenna arrays : an eigencurrent approach
Citation for published version (APA):
Bekers, D. J. (2004). Finite antenna arrays : an eigencurrent approach. [Phd Thesis 1 (Research TU/e /
Graduation TU/e), Mathematics and Computer Science]. Technische Universiteit Eindhoven.
https://doi.org/10.6100/IR582264
DOI:
10.6100/IR582264
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Published: 01/01/2004
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Finite Antenna Arrays:
An Eigencurrent Approach
DaveBekers
ThisPhDthesisistheresultofaprojectcarriedoutundersponsorshipofThalesNederland,the
Netherlands, and the Stan Ackermans Institute of the Technische Universiteit Eindhoven, the
Netherlands. TheprojectwascarriedoutatThalesNederland.
Finite Antenna Arrays:
An Eigencurrent Approach
PROEFONTWERP
terverkrijgingvandegraadvandoctoraande
TechnischeUniversiteitEindhoven,opgezagvande
RectorMagnificus,prof.dr. R.A.vanSanten,vooreen
commissieaangewezendoorhetCollegevoor
Promotiesinhetopenbaarteverdedigen
opmaandag13december2004om16.00uur
door
DaveJohannesBekers
geborenteBreda
Dedocumentatievanhetproefontwerpisgoedgekeurddoordepromotoren:
prof.dr. A.G.Tijhuis
en
prof.dr.ir. C.J.vanDuijn
Copromotor:
dr.ir. S.J.L.vanEijndhoven
CIP-DATALIBRARYTECHNISCHEUNIVERSITEITEINDHOVEN
Bekers,DaveJohannes
Finiteantennaarrays: aneigencurrentapproach/byDaveJohannesBekers.
Eindhoven: TechnischeUniversiteitEindhoven,2004.
Proefontwerp. -ISBN90-386-1012-2
NUR919
Subject headings: antenna arrays / antennas / electromagnetic waves / mathematical models
/ mathematical moment problems / eigenvalue problems / numerical simulation / sensitivity
analysis
2000MathematicsSubjectClassification: 35Q60,78M05,47N99,31B10,00A73,35B34
Press: Universiteitsdrukkerij,TechnischeUniversiteitEindhoven
Coverdesign: JeroenWillekens
PaulVerspaget&CarinBruinink,GrafischeVormgeving – Communicatie
Copyright c 2004byDaveJohannesBekers
°
v
Preface
”Whydoweneeda(n)(industrial)mathematician?”Itisafrequentlyaskedquestioninindus-
try. Probablytheanswersareevenmorenumerous: tocarryoutaspecificcalculationalstep,to
developa(n)(numerical)algorithm,tofindanoptimalstrategy,andtotestahypothesisareonly
someexamples.Amoreprofoundansweristhatanabstractlookatacertainproblemmaygivea
deeperinsightandmayestablishlinkswithotherfields,whereasolutiontotheproblemisavail-
able. One of the strongest unifying concepts in mathematics is the concept of eigenvalue. As
L.N.Trefethen[115]wrote: “They[Eigenvalues]giveanoperatorapersonality”. Represented
inthecomplexplane,eigenvaluesaremucheasiertodigestbythehumanbrainthantheabstract
notion of an operator that describes a certain process or phenomenon. Moreover, eigenvalues
may provide insight into physical phenomena like resonance, stability, and rate of increase or
decay. More specifically, in mechanics, eigenvalues may determine under which conditions a
bridgewillcollapseoranmusicinstrumentwillgiveapropersound. Inelectromagnetism,they
may determine whether a certain signal is propagating. In ecology, they may predict whether
layersofsaltbecomeunstable.Inheattransfer,theymaydeterminethecoolingtimeofamolded
compactdiscortheheatingtimeofacopyingmachine.
Inthisthesisanapproachbasedontheconceptofeigenvalueisproposedfortheanalysisof
antennaarrays. Examplesrevealedthateigenvalues,andtherelatedeigenfunctionsoreigencur-
rents, are one-to-one related to the specific array functions like scanning and the technique of
monopulse.Moreover,theexcitationofspecific,resonant,eigencurrentsexplainsvariouseffects
observedinpractice,likevariationsofelementimpedancesattributedtoarraysurfacewavesand
modulationsofelementimpedances. Thevisualpowerofeigenvaluesisexploitedaswellinthe
sensethattheirdistributioninthecomplexplanemayrevealsuitable(surface)loadingtoreduce
resonantbehavior.
The preceding paragraph illustrates the strong relation between the concept of eigenvalue
and antenna-array design. Moreover, it illustrates how an abstract look at antenna arrays may
providepracticalinformationfordesign. Inthepastfouryears,theserelationswerenotalways
asclearformeastheyarenow. Thehardestpartoftheprojectwasprobablytokeepbelieving
thattheapproachbasedoneigencurrentswasappropriateandtoexplainwhysuchanapproach
vi PREFACE
wasneeded. Anadditionaldifficultywasthatelectricalengineersandmathematicianstalkdif-
ferent ‘languages’. Moreover, both ‘languages’ consist of many ‘sublanguages’. To write one
thesisforseverallanguageswasnotaneasytask. Toputitdifferently,tryingtobemathemati-
callystrictandindustriallyappliedatthesametimeisforamathematicianlikeclimbingoneof
theridgesofamountain: thedangeristodisappearintothedeepravinesoneitherofthesides.
Inthisrespect,joiningtheprogramMathematicsforIndustrybeforecarryingoutaPhDproject
wasveryuseful.
Manypeoplehaveguidedorhelpedmeinthisproject. Firstofall,Iwouldliketomention
dr.ir. StefvanEijndhoven,dr.ir. FonsvandeVen,andprof.dr. AntonTijhuis. Stef,many,many
thanks,notonlyforreadingthisthesisuptothe‘milimeter’,butalsoforallyoursuggestionsand
advicewithrespecttotheinterpretationofthemanygenerated(numerical)results. Moreover,
your mental support encouraged me a lot. Fons, thank you for all your support and advice
over the past years and for always having a listening ear. Anton, I appreciate all your help
and support very much and I would like to thank you especially for the extensive time you
took to read my thesis and to give suggestions for improvement. Through your advice, a lot
of ‘language difficulties’ of the nature mentioned above were resolved, although during our
enthusiastic discussions, we sometimes ran ourselves into such a ‘difficulty’. Many thanks go
also to my present and previous supervisors at Thales Nederland: dr.ir. Peter-Paul Borsboom
and ir. Evert Kolk. Peter-Paul, thank you for all your advice and support in the last three
years, especially for all the effort you took to find application areas for my work, both inside
and outsideThales. Evert, thank you foryour guidance inthefirstyear and for giving methe
opportunitytocontinuemyfinalprojectofMathematicsforIndustryasaPhDproject.
Iwouldliketoexpressmygratitudetoprof.dr.ir.HansvanDuijnandprof.dr.ir.GuyVanden-
boschfortheircommentsonthefirstversionsofmythesis. Iamalsothankfultothemembers
andseveralformermembersofthegroupJRS-TUantennaatThalesNederlandfortheirinter-
estandmanyworthwhilediscussionsfromwhichIlearnedalot. Inparticular, Iwouldliketo
thank Joris Buijnsters, Eddy van Ewijk, and Bertus ter Heijde, Bart Morsink, and Gertjan van
Werkhoven for the discussions about the development of radar systems and the relation with
mywork. Moreover,IwouldliketothankKimanVeltfortheHFSSsimulationsthatcouldnot
bedescribedinthisthesisanymoreunfortunately. WarmthanksalsogotoEmielStolp, Frank
Leferink, Hans Schurer, Hans Driessen, and Monique Kedde, for the stimulating discussions
about work and my work in particular. Last but not least, I would like to thank Geert Vulink,
Dolf Boompaal, and Rein Eggens for the pleasant atmosphere in our cubical throughout the
years.
FromtheLaboratoired’Electromagne´tismeetd’Acoustique,Iwouldliketothankprof.Juan
Mosig for giving me the opportunity to work in his group from April till June 2000 and to
presentmyworkinJune2001. Moreover,IwouldliketothankMichaelMattesforthepleasant
cooperationduringmythree-monthsstayandforthewarmwelcomeatmyvisitinJune2001.
PREFACE vii
From the Technische Universiteit Eindhoven, I would like to thank the (former) students
ofMathematicsforIndustry, themembersoftheElectromagneticsgroupofthedepartmentof
electricalengineering,andthemembersoftheappliedanalysisgroup,nowadaysCASA,ofthe
department of mathematics and computer science. In particular, I would like to thank Friso
Hagman, Martijn van Beurden, Tom Gierstberg, Kamyar Malakpoor, Gertjan Pieters, and Jan
Krootfortheirmentalsupportandencouragingdiscussions.
Finally,Iwouldliketothankallmyfriendsandrelativesfortheirfriendshipandsupport. In
particular, I would like to thank Jeroen Willekens for designing the cover. Moreover, I would
like to thank my mother Yvonne, my father Frans, and my brother Colin for their patience,
understanding, and support. Last, but definitely not least, I would like to thank my girlfriend
Shirley for all her love, patience, understanding, support, and of course for drawing several
picturesinthisthesisandtypingseveralpartsofthetext.
DaveBekers,Eindhoven,1november2004

ix
Glossary of Notation
Generalremarks:
Ifmoreequationscorrespondtothesameequationnumber, theyareindicatedbysuper-
•
script numbers at the equation number. For example, (2.1)2 is the second equation in
Equation(2.1).
Asuperscriptsymbolconnectedtoawordindicatesafootnote,e.g.,representation∗.
•
Except for the time-domain quantities in Section 2.1, vectors and vector functions are
•
indicatedbyboldfacecharacters,e.g.,Eandw.Matricesandcolumnvectorsaredenoted
byRomancapitals,e.g.,Z andW. Operatorsandvectorspacesareingeneraldenotedby
calligraphiccharacters,e.g., and .
A Z
If a super- or subscript of a mathematical symbol is typeset in the normal Roman font,
•
the script indicates the abbreviation of a word or word group, e.g., N . If a super- or
sub
subscriptistypesetintheitalicRomanfont,thescriptindicatesamathematicalsymbol(a
variableoracoordinate-axislabel),e.g.,e andu .
x nq
Adotinanargumentofafunctionoranoperatorindicatesthatthecorrespondingvariable
•
isfree. Forexample,ifgisafunctionoftwovariables,thenf = g( ,η)isafunctionof
·
onevariable,wheregisevaluatedwithrespecttoitssecondargumentonly. Thefunction
f evaluatedatξequalsg(ξ,η).
ThedBscaleisingeneraldefinedas1010log . Thedefinition2010log isadopted
• | · | | · |
for(electric)far-fieldcomponentsonly.
Thewordgroup‘absolutevalue(s)ofthe...’ isoftenabbreviatedto’theabsolute...’,e.g.,
•
‘absoluteeigenvalue’insteadof‘absolutevalueoftheeigenvalue’.
Theword‘element’isusedforbothelementsofsetsandelementsofanantenna.
•
Normalizedquantitiesorvariablesaredenotedbyhats,e.g.,ξˆanAˆ .
pq
•