PE MANC

HIGHPER FO R MANCE DUALSHAPEDRE F LECT OR ANTEN NAS FOR. EARTH STATIONS

By

@KaijunGu

Athesis submittedto the SchoolofGraduateStudies inpartialfulfillmentof therequirements forthedegree of

Masterof Engineering

Fac ulty of Engineer ingandApplied Science MemorialUniversity of Newfoundlan d

March,1995

St.John's Newfoundland Canada

.+.

NalionallJbrary ctcanaee

~uisitioosand Bibbogfaphic5efvicesBranch

=~51reel K1A0N4

Direcliondes acquisitionsel des servicesbibiiographiquBs 395,t\,lIIl'IeMglol1

=~O'llariQ)

THE AUTHORHAS GRANfED AN IRREVOCABLENON-EXCLUSIVE LICENCEALLOWINGTHENATIONAL LffiRARYOF CANADATO REPRODUCE,LOAN,DISTRIBUTEOR SELLCOPIESOF mSfHER THESISBY ANYMEANSANDIN ANYFORMOR FORMAT, MAKING THISTHESIS AVAILABLE TOINTERESTED PERSONS.

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Canad~

L'AUTEURA ACCORDE UNELICENCE IRREVOCABLE ETNON EXCLUSIVE PERMETTANT A LA BIBLlOTHEQUE NATIONALE DUCANADADE REPRODUIRE. PRETER,DlSTRIBUER OU VENDREDESCOPIESDE SA THESEDE QUELQUEMANIEREEI SOUSQUELQUEFORMEQUE CESOIT POURMETTRE DESEXEMPLAIRES DE CETTETHESEALA DISPOSITIONDES PERSQNNE INTERESSEES.

L'AUTEUR CONSERVB LAPROPRIETE DU DROITD'AUTEURQUI PROTEGE SA nlESE.NI LATHESENI DES Exr..AITS SUBSTANTIELS DB CELLE· CI NE DOIVENTETRE IMPRlMESau AUTREMEm REPRODUITSSANSSON AUTORISATION.

Abstrac t

Insatellitecommunications, the earthstation antennaplays amajor eoloilltilt·

vitallin k between thesatelliteand the earth station electronicequi pment.[IIthis thesis, dualsh aped reflector antennas for eart h st a t ions which cnn provlde1\pend!

beamtowarda geostatio narysatelliteareint roduced. The design andiwalys;sof both Cassegrain and Gregorian antennasarc presentedalifollows.

(i)The highperformance antenna feeds are studied indetail.Variollsl\p(~cHi{' characteristicsrequired in Cassegrainand Gregoriananten nas are discussedamiit is shown that the corrugated conicalhorn is the best choice.

(ii)The computer aided synthesis of eubreflectcrand mainreflect o r arelip- velopedby using the geometric optics(CO) approach. Civenthe Iecdradlarlon patternand thedesired main reflector aperture power illumination,thesubaud main reflectors can he shaped to obtain optimalreflectorprofiles.

(iii)Afterhaving the sub and mainreflector profiles,the physical optics (PO) are usedtocalculate thescatteredpattern of the subreflcetor, the radiationp<ltll!rn~

as well asthe whole antennaspecifications.

(iv) A numbe rof power distributionson the main reflectorapert urearc iuvesti- gatedto maximizethe boresight gain and at thesame time,minimizetilellirJdolm levels.A pattern controlmethod is also developed,

(v)Perform ance comparisonbetween Cassegrainantennaand(Ircgorienan- tennais also made.

The majorcontributionof thisthesisis the completecomputer aidedul:llign tool fora veryhighperformance antenna,

Acknowledgements

Iwouldliketo expressmy deepestgratitudeto my supervisorDr.SonLeNgoc, for allhisguidance,adviceand assistanceduringthe courseofmy graduate studies andthe writingup oftilethesis.

The fina ncialsup port from theFac ultyofEngineeringandAppliedScienceand NSEHC isgratefullyacknowledged.

TIll:lasthutno t least,Iwouldlike to give my epeclelthanks to my wonderful wife,YiZhang, forher patienceand help duringmy M.Engprogram.

iii

Contents

Abstract Acknowledgements List of Figures 1 Introduction

1,1 Statementof theProblem 1.2 LiteratureReview ..

iii vii

1.3 Scope oftheWork I:'.

1.4 Organizat ion oftheThesis la

2 High-PerformanceAnte nnaFeeds 15

2.1 Corruga tedwaveguidefeeds .. . . ... . .. . . . .... In

2.1.1 PropagationCharact erist ics If;

2.1.2 Radiat ion Charact eristics J:!J

2.2 CorrugatedConicalHorn Feeds 2:1

2.2.1 Propagation Char acterist ics.. ... . .. .. .. . . 2") 2,2.2 Radiation Charecter istice

2.2.3 Radiatio n Patt ern...

2.3 Numerical Results_ 33

3 Dua lSha pedReflector Ant en n a Design 37

:U AxisymmetricDualReflectorAntenn as. 38

:1.2 Geometric a l Optics(GO)Principles _ ... 41

3.4 Gregorian Antenna.. :1.3 Cassegra in Anten na_

3.3.1 DesignTechniques 3.3.2 Numerical Results

3.4.1 DesignTechniques

42 42 48 54 54 3.4.2 Numerica l Results

4 Dua lShaped Reflect or Antenna Analysis

4.1 Theory of PhysicalOptics(PO) , .

4.2 SubrcflectorScatteredPatt ern s 4.2.1 CoordinateSystem Definition 4.2.2 FeedRadiationPattern.

58

62 62 66 66 68 4.2.3 Current Distributio nOntheSubreflectorSurface 69

4.2.4 Scat tered FieldPatt ern s .. . _ . 71

4.3 MainReflectorFieldPatt ern s . 4.3.1 Coordina teSystemDefinition 4.3.2 Curr ent Distribu tio non the Main Reflecto r. 4.:1.3 RadiatedFieldPatt ern s

-1.'1 Dual Shaped Reflector Antenna Perfor manc e..

4.4.1 Gain .

74 74 75 76 78 78

4.4.2 Efficiency

4.5 Numerical Resultsand Comparison 4.5.1 ComparisonwithExperiment al results 4.5.2 Subreflector Sca tte redPatterns 4.5.3 AntennaRadiation Patterns.

so

sa S:l

5 DualSha p edRefle ct orAntenn a Perfor ma nceTrade-of's andCom-

parison 101

5.1 Dual Shaped Reflector AntennaPerformance Trade-Ofls Hl2 5.1.1 GaussianDistri bu tionon theAntennaAperture. 102 5.1.2 Gain and Sidelobe VersusAperture Distributions 104

5.2 Comparison or CasscgrainandGregcrienAntennas 1O(i

6 Conc lusi o nandFuture Work 116

Refe r ences 120

AEx p r essin g

0 ,

in Sub re flectorCoord inateSystem 127 B Expr essing

q::.

inMain reflect or Coord inate System 128 C Calcu lationoffi.x

pm. O .

X

p m

and~.x

pm

130

D Int e gr ati o n Form ulae 132

E PowerTran smit t edby FeedPT 13 4

Li st of Figures

2.1 Corrugatedwaveguidefeed. (a)Structure.(b) Region1.(e)Region2. 17 2.2 Co-ordinate system used for evaluatingtheKirchhoff-Huygen inte-

gration. 22

2.3 Corrugated conicalhorn.(a)Structure.(b)Region 1. [c]Region 2. 24 2.'1 Radiation patternsof the corrugated conicalhorn. Frequency:14.25GHz,

Apertureradius:a.Oin.(a) Flare angle: 20°,(b)Flareangle: 12",

(c) Flareangle: 7° .• • 34

2.5 Radiationpatterns ofthe corrugatedconicalhorn.Frequency:14.25GI-lz, Flareangle:12~. (a)Aperture radius:4.0in ,(b) Apertureradius:

8.0in,(e) Apertureradiu.s: 16.0in.. , .. . . 35

2.6 Radiation patternsofthecorrugated conicalhom.Freq uency:14.25GHz, Axial lengt h:37.0in. (a)Flare Angle:1°, (b)FlareAngle:2°,(c)

Flare Angle: 20",(d) FlareAngle:12° 36

3.1 Cesseg-einantenna conflguratl on. 39

3.2 Gregorianantenna configuration. 40

3.3 GeometricalopticsferCassegrainantenna reflectors shaping. 43 3.4 Radiation patt ern or the corrugatedconical horn.Frequency:14.25GHz,

Apert ure radius: 8.0in,Flare angle:12° .. vii

.. . 50

3.5 Shaped Cassegrain main and subreflectcrprofiles {Example I),Solid line:reflector profiles(Unit: inches),Dottedline: mainrclhx-tor apertureillumination(Unit : dB)

3,6 Shaped Cassegrainmainandsubrcflector profiles (Example 2). Solid line: reflectorprofiles(Unit:inches),Dot ted line:mainreflector apertureillumination(Unit: dB)

!it

3,7 Reshaped Cassegraineubrcfiect orprofile(Unit : inches).Sclid Ilne:

sub rcflectorprofileafterreshaping,Dottedline:subrcllcctor prof iles

beforereshaping. !',:I

3.8 GeometricalopticsforGregorianantenna reflectors shaping. Mi 3,9 Shaped Gregorianmain andeubrcflcctor profiles(Example:I),Solid

line:reflector profiles (Unit: inches),Dott ed line:main reflector apertureilluminat ion(Unit:dB) "

3,10Shaped Gregorian mainand subrcflcctorprofiles(Example 4).Solid line:reflector profiles(Unit:inches), Dott ed line:mainreflector apertu rei11uminatio n (Unit:dB)

4.1 Co-ordinatesystem usedfor evaluatingthe vectorKirchhoffIli/frilt·

tion integration .,... ,,., ..., , ,..

4.2 Co-ordinatesystem used inthe analysisor thescatteredpatt erns

or

thesubretiector.'

4,3 Co-ordina tesystemusedin the ana lysisof theradiation pattern

o r

themainreflector.

viii

{il

4.4 Thegeometry oftheexperimentalset-up.(a)thehyperboloidalre- fleeterandthe coordinatesystem (b) the geometryof the experi- mentalset-up(f

=

^{9.6GHz) .....•...• .. .•. . .} 84

4.5 Themeasuredfeedhorn pattern in153].. 85

4.6 The measured and thecalculatedH-Plane Scatteredpatterns. Solid line:Calculated Dottedline:Envelop ofthe measureddata .. . 86 4.7 Scatteredpatternof subrefledorforCusegrain antenna(the feed

radiation patternshownin Figure3.4, the mainandsubreflector profiles showninFigure3.5.)... •. . ..• ... ... . • ... 88 4.8Scatteredpattern ofsubreflectorforCasaegrainantenna(the feed

radiationpatt ernshowninFigure3.4, themain and subrefledor profilesshown in Figure3.6.) .. ,•... . . .. . .... • . . ..• 89 4.9 Scatteredpattern of subreflcctorforGregorianantenna(thefeed

radiatiO:lpatternshownin Figure 3.4,themainand suhrefledor profiles shownin Figure3.S.). ....•...•. .. ...•..•.. 90 4.10Scatteredpatternof subreflector for Gregorianantenna (the feed.

radiation patternshown in Figure 3.4,themain and subreflector profilesshownin Figure 3.10.) . . . . , .•.. ... .. . . . .... 91 4.11 Close-inradiationpattern ofmainreflectorforOeseegreinantenna

(thescatteredpat tern ofsubreflectorshown inFigure 4.7, themain andsubreflector profiles shownin Figure3.5.)... ... .•. 92 4.12 Far-angle radiat ion patternofmain reflector forCassegrain antenna

(thescattered patternof subrefleclorshownin Figure4.7,the main and subre6edor profiles showninFigure3.5.) •. . ... ... ... 93

4.13Close-in radiation pattern of main reflectorforCassegrain antenna (thescatteredpattern ofsubreflectorshownin Figure4.8,the main and sub reflector profiles shown inFigure3.6.) .. . . .. 94 4.14 Far-an gleradiationpat ter n ofmainreflectorfor Caseegrai nantenna

(thesca tt ered pattern ofsubreflectorshowninFigure <1.8, themnin and eubreflecto rprofilesshownin Figure3.6. )

4.15Close-inradiat ion pattern ofmainreflector for Gregorianantenna (the scatteredpat tern ofeubreflectcrshownin Figure 4.9,the main and aub reflectorprofilesshown inFigure3.9.) .. ... 4.16Far-angleradiationpa tt ern ofmainreflectorfor Gregoria nantenna

(thescattered patt ern ofsubreflec:torshownin Pigurc4.9, the main andsubr efleetceprofilesshown inFigure3.9.)

4.17 Close-in radiationpatt ern ofmain reflectorforGregorianantenna (thescatteredpatternofeubrelleetcrshownin Figure 1.10,the main andsub rof lectcr profiles shown inFigure3.10.) .

4.18 Far-angle radiati on pat ternof mainreflecto r forGregorian antenna (theSCAt tered patte rn of eubreflcctcris shownFigure4.10,the main and subreflectorprofilesshownillFigure3.10.).. . . 4.193-Dimensiona l rad iationpatt ern (TheE-Planeradiatio n pattern plot-

ted inFigure 4.12). . ....•... .

5.1 Powerdistribu tionon the main reflectoraperture

5.2 Radiatio npatter ns.p...u

=

98.5",P""

=

10.0", Pm,

=

94.0", 74.0", 97

98

99

100

103

M.O",34.0", 14.0",61

=

-2OdB ,~=-l OdB 105

5.3 Antenna profile andradiation pallern.PmG'"=98.5",Pm,=10.0", Pm1

=

94.0"(Case1),~=-2OdB,~=-lOdB

5.'1Antenna profileand radiationpallern'PlttG,,"

=

98.5", Pitt,=10.0", 107

110 Pm,

=

34.0"(Case4),~

=

-20dB,~

=

-IOdB ..•... 108 5.5 Cassegrainand Gregorianantennas withthe same feed pattern (Fig-

ure 3.4),phasecenter locat ion(-27" ),main reflector radius(98.5"), subrefleclorradius(9.0"),subrell.eclorsubtended angle (12.7")andthe same main reflector powerdistribution.(a)Cassegrain antenna (b) Gregorianantenna

5.6 Close-inradiationpatternsof(a) Cassegrainantennaand (b) Gre- goriananten naforthe configurations shown inFigure 5.5.. 111 5.7 Far-angleradiation pattern sof(a) Cassegrainantennaand (b) Gre-

gorian antennafor the configurationsshownin Figure 5.5... 112 5.8 Casscgrainand Gregorianantennas withthe same feed pattern(Fig-

ure3.4),mainreflector radius (98.5"),subreBectorradius(9.0"),sub- reflector subt endedangle(12.7") andthesame mainreflector power distribution,butdifferentphase center locati ons(-27"for Cassgrain antenna, 27" for Gregoriana-tenna](a) Cassegrain antenna (b)

Gregorian antenna 113

5.9 Close-in radiation patterns of (a) Cassegrainantennaand (b) Gre- gorianantenna for the configurationsshownin Figure5.8 . 114 5.10 Far-angle radiationpattern sof (a) Cassegrainantennaand (b) Gre-

gorian antennaforthe configurationsshownin Figure5.8 ... 115

xi

List of Principal Symbols and Abbreviations

A vector potentialin conicalhorn

E electrictield

E. eo-polarizedelectric field component Ex cross-polarized electric field component

l' vector potentialin co nical born F(6) feed radiation pauem

G antennagain

h: (x) spherical Hankel functionof 2ndkindand ordern Ii magnetic field

I(x) aperture power illumination j surfacecurrent

Jm(:r:) Besselfunction of13t kindand orderrn k free space wavenumb er PA main reflector aperture power density P,:"(cos8)associated Legendrefu nction oflstkindand order n Q~(cos9) associated Legendrefunction of 2ndkind and order n Y",(z ) Bessel function of 2st kind and orderm

6Z, distance between origins offeedand eubreflector coordinates 6Z", distance between orig insof main and eubrefiector coordi nates

propagationconstant in thelos sless medium tl) free space pennittivity

antennaefficiency

xii

normalized hybrid factor wavelength po freespacepermeability

angular freq uency

GO geometricaloptics PO physical optics

xiii

Chapter 1 Introduct ion

1.1 Statement of t he Problem

Pencil-beam antennas are most widelyused in point-to-pointand point-to- multipointmicrowavecommunication systems.In terrestrialmicrowaverelay~Yll' terns, the beam of one antenna is fixed and directedto anotherantennafor each hop along the path at thetime ofinstallation,whilein satellite communication syst e ms, the beam of the earth station antenna is pointed directlyto thesatdlil(:.

With high gain and creating no noise, the antennaplaysanim por tantroll!inthe vital linkbet ween two repeaters or two earth stations. One can conclude that then:

would be no satellite communicationsif therewereno antenna.

It has become a common practiceto focus microwaveenergyinto a desired beam by theuse of metallic reflecting surface (or surfaces) excitedhyradlurion from a small,relativelynon-directionalmicrowave source.These kindeof antennas arecalled reflector antennas.In thefa mily ofreflector antennas, one distinguished member is the dual reflector antenna,especiallythe axisymmclricaldual-reflect o r antenna.The axisymmetricalduel-reflectorantennahas numerousmechanical and electricaladvantagesover the conventional fecal-pointfed large parabo loidalan-

tcnnas.TheCasscgrainantenna is one kindofdual-reflector antennas derivedfrom th etelesco pedesignof_Willill.~Ceseegrein.Ithas a hyperboloidalsubreflectcrand a paraboloidalmain reflector.Thiskindof antenna. bad along historyand still is themostcommonlyusedearth stat io n antenn atoday.

Becauseof theimportanceand wideapplications oftheCassegrainantenna,the antennadesignis alwaysan interestingtopic insatellit ecommu nications . Owing to thehigh cost ,theseearthstationantennasare normallydesigned and optimized suchthat the verystringentrequirementofCCITT(Interna tio nalTelegraphand Telephone ConsultativeCommittee) ismet andyetthe high gain andthe low sldclcbc levelsare still achieved. In addition, theclose-in sideloberequirementis even more criticalwhenthegeostatio n ary orbi tis getting more and more congested everyday.

Thedual-reflectorantennaconsists ofthefred, subrefiector andmain reflector.

To achievethedesiredspecifications,suchasthe boresightgain,sldelobelevels,an- tennaefficiency,cross-polarizationlevels,ctc., the feed must be designedto provide a suitableradiation patternas well, thetwo reflectorsshouldbe shapedin such a way that the desiredaperturepower illuminati onis obtained. Forthesereasons, thecompu ter-aided-design(C AD) andthe computer-aided-anal ysis(CAA)of the dualreflectorantennas have beenthe popular topicinrecent years.The CAD and CAAtools arc normallyneed ed forthesatellitecommunicationengineering.

One ofthe variationsof the Cassegrainantenna isthe Gregorianantennain which the hyperbo loidalsu breflectoris replaced bythe ellipsoidaleubeefle ctor.Rev- olutionarysymmet ricalGrogorianan tennashave been designedand usedfor many years byAndrewAntenna Corpora tion.But untilnow,there are neit her CAD

nor CAA programsappearedintheopenlitera ture, Therefore,itis also\I~d1l1 to developthe CAD andCAA programsto designtheshaped reflectorGn~lIriall antennaforspecific requirement as well as for optimizat ion,

In tbis thesis. thedesign and analyses of Ceeeegrelnand Gregorianantennasart"

carriedout in detail.Corrugated conicalbornis studied andshowntheheatchoice- for thedual reflectorantennafeeds. The propaga tionand radiationchnractoriatics are discussedin detail.Onthe assumptionthatthedimenslone ofrdl('Ct()f~aft'

muchlarger than the wavelength ,the geometr icalopt ics(GO)approachisused throughoutin the reflector design. The analysisprogramsare developedba.'iI!i1Oil

physicaloptics (PO)techniqu e. Various performancetrade-offsandcomparison betweenCassegrain and Gregorianante nnasarc carried out.

1.2 Literature Review

Reflecto r antennashave been usedsince theradio pioneering era of I.odge , Hertz.andMarconi. but ittook the exigentdemands of radiocommunications in WorldWar IIto stimulatea real developme ntinthe reflector art.Su hsffllucn t interestin thescience of radioastronomy andthe inceptionof terrestria lmicrowave linkswere res ponsible forafastgrowt hill thefield.solhatin the1910'saud 1950's thedesign principlesand requirementsforprimefocusfed systemwen: well established. With theadventofsatellit e trackingand telecommunicationIll~tworks.

Cassegrain, or secondaryfocussystem, andho rn reflectors came intoprominen ce inthe early 1960's.The desireto maximize the gain, or thegain-te mperat u re ratio . hasled tothe developmentofsophist icatedcomputeranalysistechniqu es. For prop erly imposi ngthepowerilluminationoverthe apertu reofthemain reflector

and hyshaping ofboth reflectors,oneca nminimise spillover and inturn maximize efficien cy. Inthe applicationof radioastronomy and spaceteleco mmunica tions, many kinds ofreflector antennas have beenproposed andstudied based onthe theor eticaland experimental research.Oneof thecommonly used reflector ante nnas isthe revolutionary symmetrica l Cassegrainantennawhich wasconsideredonly for those applicationsrequiring the 3db beamwidt.hofless than aboutone degree.

Today,this antennashows even moreimportant due to therapid development of spacecommunications and other applications.

The lirst reflectorantennawasborn inthe year 1888in thelaboratory ofHclu- richHertz.He used it10experimentallydemon stratedthe existen ce'of theclec- tmma gnctic waves that had been predicted theor eticallyby JamesClerkMaxwell some fifteenyearsearlier,liela uncher! decimeter-waveradiationfrom a parabolic mirroranten n a fedby a dipole. The directiveantennas played a major partin Hertz's system , providing himwitha laboratory microwavelink11].Hisworkstim- ulated a lot of scientiststoward furtherinvestigati on.A.Righididflopionee r wcrk onmicrowaveoptics [l]. Inspiredby Righi'swork, Marconi conceivedtheideaof modulating thc generatorsotha t. the inte lligence couldbetransmittedon the radio frequencyopt.icallink and he proved to the BritishPest Officethat hisidea came trueby usingtwo Hertz-type parabolic reflectors separated by a range of four miles in1897.Itis known that the optical properties of the parabololic reflectors were alwaysused at that time,Marconigot his first patent onthesharp beam cylin- dricalparabola antenna in 1896(11.The first hollow pipe or"waveguide" radiator were developed by Lodge in 1894

III

and in 1897,Bose firstused thepyra midal electromagnetic horn asthe receivingantenna which wasthen called"collecting

funnel"[11. Actually, his antenna was the first !Iare waveguidewhichhl' wouldlike to use to collect more energy by increasingthe cross section. Il shouldbenoted that the major practical advancestended to ariseIrom the microwaveopti(~nud these researchespointed the direction in whichthe new microwaveradiolt~'lltl()l{)p;~' must go.

In1931, the firstdiscovery of extra-terrestrial rad io emission madetlwgl'lLl'sis of the science of radio astronomyand thenin 1937, the firstlarge paraboloidal reflector was constructed and used as a radio telescope antenna byReh er [21.Till' reflector was 9.6min diameterandthe wavelength was LOrn.In World WarII,n vast and intensivedevelopmenteffortwas the microwave physicsfounded outile solid base of physicaloptics and electromagnetictheory.This stimulated thetie- velopment or the unifiedtheory or microwaveantennaswhich was wellI:OVCfC(!hi the now classictext by Silver[3).With the rebirthorthe radio astronomy imme- diately afterthe war,manyIormer wartimeradioscientiststurnedtheirillt(~rcst~

to develop radio telescopes and many reallylarge reflector antennas came011.Bllt thecommonly usedfeed was the dipole and thelimited beam steeringcouldhe achievedby displacingthe reedfromthefocal point. The gain wasonly about :11<111 14).

Itwas Cutlerwho first publishedthe paper dealing with UlCpolarixatiou char- acteristics ofreflectorantennas[51in1947.He showedqualitativelytlI1I.tthe ideal feed shouldradiatea sphericalwavewith thelinearpolarizationand alsoexamined the polarizationcharacteristicsof thethen-po pular dipolefeed and statedlIJll.tit would give rise to cross-polerizetioncomponents in the reflectedfield.Inhis paper in 1954[6],Jones firstshowedhowan electr icdipoleand amagnet ic dipolemay

he combinedto producea unidirectionalHuygenasourcehaving ideal polarization characteristicsfor feeding a paraboloid. La ter, Koffman also did some resear ch on feed polarizationforparallel currentsin reflectorsgeneratedbyconic sections17]

anddrewthe same conclusionas Jones.

No discussion of two reflectorantennaswould be completewithou ta reference to thatfamilywhich is derivedfrom the 17t hcenturyopticaltelescope devised by the Caescgrain,include variantsduetoGregoryand Newton.Cassegrainantennas were being experimentallydevelopedatleast as earlyas the mid19505,and the wholefamilyhas been well described byHan nan in 1961[81.Tworeflectorantenna systems havebeen analyzed fromthepoint of view ofgeometrical opticsby Kinber in1962 {9].Morgangavesome examples ofgeneralizedCassegrain and Gregorian antennas(10J. Theyshowedthatthe Cassegrain antenna can produce a verysharp beam andan extremely high gainfortelecommunications.

Since1960's,with the developmentof satellitecommunications,the large an- tennas are nolonger the exclusive province of radioastronomers.Manyreflector antennaswereusedin the commercialsatellitecommunicationsearth stationsand such anten naswerecostly.In Potter'spape r[11].it was estimatedthatthecost ofIlsinglelargereflector antenna was proportionalto its diameterraised tothe power2.78.Thus it becameimperative to increase the apertureefficiencyandto reducefeedspilloverso as to maximizethe gain. In 1963,Galindo first showed thatwit hatwo-reflectorsystemitis possible to achievearbitraryphase and am- plitudedistribut ionsover themainapert ureusinggeometricoptics (12).In1965, Williamsdescribe dAmodifiedCessegrein system to whichan approachW1J3cleo gantlyapplied,leadingto aimprovementabout25% inapertur e efficiency (13).

Its operation can be explainedquitesimplyin terms ofray optics,Startiul\wit h a conventionalCassegrainsystem withprescribedfeedpattern, the hyperboloidal subreHector 'ssurface is deformedinsuch a way as to increasetheraydensityradi- ally outward from theaxis,Whenproperlydone a goodapproximationtoitunilorrn amplitude distribut ion acrossthemain aperture isrealized,but onlyattheexpense- of a nonuniformphasedistribution. However,thephaseerror canheCOrn'rtf "!h) a relative minorchangein themain reflectorsurfacewithoutsigniGcantlyalr(~ti ng the amplitudetaper.

Amilestonein the development of reflectorantennaswas thatthedipolefeeds had beenreplacedmostlyby themicrowavehorns in1960's. Nevertheles stil(' do.ninantmode horn's characteristicsarefar from idealforthispurpose, chiefly because its principalEandHplane radiationpatterns are quite different ,InI!)(;:~, Potterdevisedacleversolutionto this problemin thecase cl thcTEllmode excited conical horn[141.He notedthatastep discontinuityindiameterncarthet!lrtm l ofthehornwould cause some ofthe dominant mode to beconverted to thehighcr-

. .

orderTMu mode,He furthershowed that thecorrect combinationoftllcs\:two modes at thehorn aperturewouldlead to a radiationpattern have almostidcl1t i(:al patterns intheEandHplanes,and that thenormallyhighEplane sidclobca wouldbe supp ressedtoa.very lowlevel.ThenPotterand Ludwig[lSIextended the conceptto include additional highermodes(TEI~, TEI3andTMn )inthe conica l horn forpu r posesof beam shap ingandshowed howto obtainafeed patternthal more nearlyapproxima tesa uniformillumina t ionover areflect orapert ure, These multimodchornsare not broadbanddevicesbecause thevariousmodes propagate withdifferent velocities.Thisdifficulty was ove/co,mein thecorrugated110mwhic h

seemstohave beenconceivedintheUSA andinAustralia almost atthe sa me tim e, about1964. InUSA,Kayused groovedwalls in awide flareangleho rn andcalled it the scalar feedbecause its pro pertieswerelargely inde pendentof polarization[161. InAustra lia,an analytical descrip t ionofthe synt hesis ofth e hybridIfEl lmodewasgiven in anearlypape rbyMinneu an dThomas(171 and theyalsost u d iedthefieldsin theimagespaceofsymme tricalfocusingreflectorsand proposedtheguidelineon howtosynth esizethehigh-efficiencylow-noise reedsusin g hy brid-wavesin cor r ugated waveguide s[181.Becausethecorrugat ed hornradia tes apatternwiththe polarization propertiesof aHuygen s source,itresultsin a very lowlevelofcross-po larizationradiat ion inreflector ant ennas.Thestrik ingsuccess of thishorn as a feedhasbeenun importantfactor lead ingto imp roved performan ce illreflector systems and has inspired a grealdea l of investigat ionof somesclentiste 11911201121]12211231.

Inthe design ofpencil-be amante nnas forvariou s applications, themostim- portantpa r ametersare:a) highefficiency,b) lowcross-polariza tion, c) satisfactory eidelobcenvelope.Many scien tistshaveputa tromendous elfor t inimprovingth e se requirement s.In ordertoachieve highefficienc y,bothof thetworeflectorswill be sh a pedbas e d on theclassicalCassegrainanten nato obtain th euniformapert u re illuminationin bot h phaseandamplitude.Galindofirstusedthegeometric op- ticstechniqu e(GO )to designthe reflectors to achie vehigh efficiency [12J.If the an tennaisla rge,a successful designmaybeundert ak en usingGO to satisfy in- cre asingly stringentrequireme nts onefficiency and rad iation patt ern.Indeed, these te chniquesareinwidespreadused and nowform apartof theimporta ntbody ofan- ten na theory. Whennearmax imumgain isrequired,the symme trical dualreflect o rs

III

such asCassegrain and Gregorian conlig uratio nsarcshapedtoprov ide anopt imal solution .This techniquehasbeen extended to offset dual-reflectorconfiguratiolls [24][25)[26].In recent ye ars , the GO isalso widelyusedillthesatellitealllcllllilell"

sign[271128Jandlargerear th sta t ion antennadesign[29][30}. Thecornputor"hll'eI design ofthereflectorantennaswiththe GO isa verypopulartopicillun tcnna theory andtech n ologytoday.However,itmustbe remembered thatthl' GO isonly validwhen the wavelength ismuchsma llercompa redwiththe dim ensions of the reflectors .Ifsignificantefficiencyenhancementis hopedto beobt a inedillallhut the largest sys te m, diffractiontechniquesmustbe used.Daveau[31]lirstapplil!<.l scalar-d iffraction theorytoscattering by the subreflectorinordertocll~igllfor au optimumilluminationpattern.Thephase may the n beequalized bysmallchallgc~

in the mainreflecto rprofile. Thisapproachwasa directextensionof GOall,llysi~

of Green [32}.P.O.Potter useda phase- matchingcriterio n1.0obta inthesuhre- Rector shape basedon theapplicati on ofsphericalwavetheory to Casegrainia n-fed paraboloids [33 ].Asign ificant advance,enablingdirectoptimization ofelliciency to be undertaken,wasmade by Wood[34J1351136].in hisapproach, using a methud based on reciprocity,a conciseexpressionfor efficiency isobtainedwhich on lyin- volves aninteg rationoverthe sub reflector surface. Severalotherauthors[:171[:181 [39Jalsosuccessfully usedtheGTDtechniquesto mo difyGOtodesignthereflector antennas.

The mostsig n ificant cause ofthe cross-polarizationisthe fadtha tfeedB -plane andH-plane patternsarenot identical. Inaddit.ionto the feed, the scattering from thestrut s usedto keepthe feedorsuhreRectorinposi tion isanothe rreason forcross-polari zation.Fortwo-re flectorcasethedieledric cone feed suggestedby

11

Ilartl e tt andMosely [401andlater develo ped by Clarricoa tsand Salema (4 1] was a effectivewaytoreduce thecross-polariza tion.Andthe offsetreflector antennasfed bycarefullydesi gnedfeed havebecomewidespread to eliminatethe struts scattering

I"J·

Acco rding tothe cu rrentCelTTreco mmendati onsforearthstationant ennas, the close-inan dthefar-angle sidc\obessho uldmeetthe specificatio nto minimize possibleinterferenceswithothe rcommunicationsystems,especially inareas of pop- ulatio nconcentration. The overall patt er ncontrol becomes one of the majoran- Lennadesign issues.Thencar-insidelobe s depend mainly on theamplitudetaper altheedge of the mainreflectorand the centralblocka ge.In 1960,Taylorfirst proposedtheTaylor ap e rture dis t ribution and used it toin vestigat e the bea mwidth andthesidelobes[43].Later,Han sen[44J,Ludwig[45]alsopublishedtheirpapers 011howtoredu ce the reflectorantennaeidelcbes basedonthe studieson the aper- ture amplitudedistribu tio ns.Rec ently, theGaussiandistribution and a generalized three-parameter aperture distri b utionwe re used by Galindo[30]andDua n[46]to impro vetherad iation characte ristics.Thefar-ang lesidclobes ofthe two-re flector anten n asmainlycome from thefeed sp illoverpas t the su brellectorand thesub- reflect o rrimdiffraction,main re flector sp illover and diffractio n,energyscattered fromthefeedafterrefle ction.The subre fIectorandmainreflectorspilloversmay 11(' red ucedbyincreasin g theedge tapers, but theante n naefficie ncywillbe de- crease d. Properly selec ti ngthe edge ta p ersand shaping thereflect orprofilesarc veryimportan tin the antennadesign,as statedbyClarrico8.U1and Poulto n(47].

Ontheevaluation ofthe pe r formance of the reflector antennas,theradiation ]Il\tternsarcnormally consideredbyusing theme th odsknown as aeymptot.ic so-

I:!

lutionsofMaxwell's eq uat ions explainedbyKouyoumjian [0\8J.Those~ludie~1l1il)'

bedivided intothree classes.The firstis thegeometricaloptics(GO),illwhich thepropa~ationof electromagneticenergytakesplacealo ng the ray.LllWSof uplin suchas tjOergy conservation,Fermat 'sprinciple ,Fresnellawsetc. an'USt~ 1[.mJ.

The secotd isthegeometrical optics theory of diffraction (GT D),whichb anI'X, tension ofthe GObythe intro ductionofdiffracted n.ysbecauseGO is1101.valitlat the edgesorintheshadowofa reflector. Keller \50) was thefirst who int rmlm:ed this theory,lat er impro vedbyKouyoumjian [5IJ.The thirdis physicalopt.ics (POl app roximatio n,whichhas beenusedbyagreat numberof scient istsslldlIt.~Klnbor [52J,Rusch[53J, andShogen128],Theradiation propertiesofany rc/k'Clorfallhe calculateddirectlyfro mthe electric currentswhicharcphysically rl'Spo llsilJlc:lor init iatingthe radiation,Itis shown that POis valid in calculati ngthemdiatiou patternseven ifthediameter ofthereflector is onlyseveralwavelengths(MI,TIll) PO is almostalways recommended foruse in calculatingthoraclialiollpauems

or

the dual reflector anten nasusedinsatellite communicati ons,Thework_IIiL~been donebyRusch andPotter[55J,butthereis stiUlimit at io nintheiranalysis,Never- theless,both the scatteringpatt rens ofthe subreflectcrandthe radiation pattern s canbeobtainedwith POand thentheentire antenna perfo rmanceCllIIbl~prl:didt~l.

1.3 Scope of the Work

In this thesis,two kinds of revolution ary symmet ricreflectorantennas,Cn!l.~egrai ll andGregorianantennas whicharemostlyusedin satelliterommnnicatkmearth stat ionswill be stud iedin detail. The research is concentrat edalltilefollowing topics:

13

(I)Theantenna feedswillbe analyzedanddesigned to prod uce high-efficiency pencil-beam. Therequ irement s of feedsused in Cassegrainand Gregorianantennas willbe descussed andthe feedselectionwillbe given.

(2) Thecomputer -aide dsynthesi sof eubreflectorandmainreflectorwillbecar- riedout based onthegeometricopt ics(GO) approac h. Using the calculated or measuredfeedradia tion patternsand thedesiredmain reflectorapert urepower il- lumination , GO can be employedto obtaina set ofdifferent ialequationsrelatedto thesurfaceprofilesofsubrefiectcr and mainreflector, thenthese differ ential equa- tionscauillturn be solved bynumeri cal methodsto obtainthe reflectorprofiles.

(3) Oncethe profilesof subrcHectorand mainreflector are known, thephysical optics(PO)will be used to evaluate the scattered patternsoftheeubreflector and theradiati on patte rnsof the mainreflector.Thewholeantenna performance,such asaperture efficiency,bcresightgain, sldelcbes andbeamwidth are pred icted.

(<\) Variouspower distri but ionswillbeimposed onthemain reflector apertureto investigatethe trade-offsbetweenantennagain and sidelobes.Comparisonbetween Cesscgralnantennaand Gregorianantennawillbe made.

1. 4 Or ganizat ion of the Thesis

Thisthesis is organizedas follows:

Chapter2 presentshigh perform anceantenna feeds.

Chapter {}brieflyintroducesthedual reflectorantennaconfigrationsandin detailthe reflectorprofile synthesismethod wit h GO.

Chapter .freport sonthe implementati onof POto calculatethesubr eflector scatteredpatterns andthe main reflector radiationpat te rnsofCassegrainantennas

,.,

and Gregorian antennas.

Chapter5cont ains a brier discussion of trade-offapet ween lheapertureillumi- nation andthe performance.The performances of the Cassegrain antennasandtlu- Gregorianantennasarecompared.

Chapter6 givesthe conclusionand the recommendation sforfuture work.

Chapter 2

High-Performance Antenna Feeds

In pencilbeam antennadesigns, majoreffortshavebeen concentratedonthe designoffeeds whichcan efficiently illuminat ethereflectors sothat theantenna performancecan be greatlyincrease d.Forexample,the antenna efficiency is closely relat edto thefeed characte risticsfacto rs;such as the unsymmetricfeedradiation patternsintr oducetheundefinedphasecenters,cause erose-polari zation fieldsand may createthenon-uniform ap ertureilluminationsand spillover losses.Hence, a properfeed designplays a veryimportant role inthe antennadesign.

The selectionof feeds that are most suitablefor reflector antennaswas always a major researchtopic.H.C. Minnel andB.M. Thomas werethe firstoneswho investigatedthefielddistributio ninthefocal plane orthe paraboloidal reflector basedonthe Maxwell'sequatio ns[17][18].Usingonlythe induced primary surface cur rent duetoan incidentplane wave field,theyshowedthat the fieldsin the focal plant:canbe represented by a spectrumofhybridwaves that are simply linear combinationor TEand TM modesand,when satisfyingsomespecial waveguide boundaryconditions,thesehybridmodescanexistin the waveguide.Th isresearch was thetheor etical foundationor thedevelopment ofcorrugatedwaveguidesand

15

IIi

corrugatedconicalhorns.

2.1 Corrugated waveguide feeds

Thecorr ugated waveguide struc turcisshown in Figures2.1a,b,c.The analyses given inthissectionwillbeextended tothe corrugatedconical horns.

2.1.1 PropagationCharact eristi cs (a)Field Components:

Basedonthe Maxwell'sequation s, it can beshownthat thelongit udinalcompo- nents,i.e.E.andH.of transverse electri c and magnetic fieldsin thohOllltl gclll 'OUS waveguide satisfythe Helmholtzequation srespectively:

[V~

+

(P-P' )]E.

=

0 [V~

+

(k'-/1' )J1I.

=

0

whereP

=

^211"/>.,the propagationconstant inthe losslessmedium,..\is wavelength.

f;2=W'/lofo,kisthe freespace wavenumber,w is angular frequency,Itnand (-0are free space permeabili tyand permitt ivity,When bothE.andJI.arcknown, then the transversefieldsin cylindricalcoordin ates(r,t/;,z)arcgiven by

E,

=-

k2

~ ^{p (pl!f;} + w~oa;~.)

EIb

= k'~ fP( -~~~·+wjJoa;:. )

H,=

k2~ p,(7~ - f3lJ;. )

HoJ,=-k'

~(p(WfO¥; ⁺ ~8~.)

(2.1)

(2.2) (2·"1

(2.1)

lal

---- --- _(i) --- -- 1-

r

(hI

~"

- ~ _{~ r~}

Ie)

Figure2.1:Corrugatedwaveguide feed.(a)Structure.(h) Region1.(c) Region 2.

IS

Inregion1(r<rj},E~andHzaregivenh.v:

wherex

=

Kr,K2

=

k2 -/)2,1'0

= vtz;= ..j1!;" r =

-k ~.IIIorderIUl<illil<I:\' the boundaryconditionthat thefieldsat r

=

0 are finite,(,heD"So",,1Ium-tionof the l st kindisused forillE.andH:.Using cqs.(2.l)- (2,6).t.lll' transvers,H,,[,I componentscan be obtained:

(2.7) E",=a",~J"';X)[mtJ

+

rF",(x)]rj",~· ^(2.8) HT

=

-am~l'aJ~xl(,ijrF", (x)+IIlJfjl"~' ^(2.!J) H",=-amj~1'/m;X)[m.Jr+Fm(X)Je-''''.· ^{/2, \(J)}

whereiJ

=

~,Fm(x)=x~,III eqs.{2,7)- (2.10),the factorrj(",'-fl.)il'll1\11['~rl'll,OOfI throughout.

Inregion2(Tl<r<TO),asshowninFi~un'2.l fc),iftill'wuveWtid,'wall i:sperfectlycond ucting,we cons iderthai.onlyT}v!modesillthb regiuu «xist.,i.,'.

H.=0,thenE.must existando/J:-=0 and~

=

0 becuusci, .fl=0at.1,111' perfectl y cond uctingwall. In addition.ifthe slotisrela tively shortwithn~"pl'I~1 tothe waveleng th,theTJv!modesare assumedto hI:z-indupendeut.,Thutl/I'~lUIS

13tends toO.Thesemodes are no tthepropagati on modes. Hence,E.r-unI", expressedas:

whichsatisfiesthe boundar yconditionat tilewaveguidewall.r

=

^1'(1.E:

=

O.where

:r'=1;1'.Eqs.(2.l )-{2.4)arcused to obtain other field componentsinthisregion.

Gut,whatwe arcreally interested inisBolowhichwillheused inthe following udmitt.ancematchingtechnique.H.,can heexpressed as:

(b)Radial Field Ad mit t a nceMat chingTechn iqueatr- r]:

FiPlI,Il('cau~t,!IC'transv erseelectri c (T El modesin tile slots donot.exit. itis rcnscnuhletcassumeE~,

=

0 at r

=

rt.sothat the TE modescannotbesup ported illl,llt' slots.Thenfrom thefieldsin region 1,wehave

(2.11)

AIr="I,the admittance intheradialdirectionH",!E:should he same ill hath rl'/-lioIlS.Usingthe fieldexpressionsabove.we cau gel

where

S",(1',y )

=

;rJ~(X)1~"'(1J)-J",{y )l:;'(x ) Jm(x)lm(Y) Jm(Y)Ym(x )

(2.12)

(2.13)

The cq.(2.12)ill calledthe characteristicequationof

a .

Ittreatstheboundary at r=1'1n.'!ifitis acont inuous impedance surface.

(c) Balanced . Hy bridCondition:

For u givenmode,the pass-baudfor propagationhas lower and upperfrequency limitsgivenh~'

13

=

0-+F",(xll

=

S",{X'I' X~)

i3 =

⁰⁰-+S",(x; ,x~)=0

Whenij

=

00forcutoffmodes,XIsatisfiestheconditionS",{.r't-r;l)

=

n.hu ttill' otherpropag atio nmodesexnibitsspecialcharacteristi~from("<l.t:?!:?).thntis

Refer toeq.(2.11),wehave

Thisis calleel thebalanced-hybridcondi tion,where+sil{lIreferto1,lIt·IIEIIlolh· and-signrefertoEHmode.

(d)HEllmodes:

LetIII=1,fromcqs.(2.5H2.1O)we call expresstheHEI",!IlOII!'fir-ldsillIIII' linearlypolarizedformas:

ForEfields ,use

wecanobtain:

E.=

-jal ~ljjJl (X)

^{+rJI; X)}

I

cos!/' E",

= ja; ~Ii3JI~X} +

r

J;( x)l sill ~:'

H.

= -jfl, Yo~[br

^{J;(x )+}

JI~x) ]Sillt/, H~. = _jflll'o*ljjr JI~X)

+J;(.z)]crtH,"

(2.15)

(2.1G) (2.li)

12.18) (2.1!)) (2.2fl)

21

(2.22)

IIIthebaJHll(:~d-hy)'ridcondit ionforHEllmode,r

=

^{+1.Ifthe radius"Iis large}

,mOlll{h.~fIthatkr

»

1,thenQ-+IamiE~

=

O.This istheHEllmode's Jill- l'arlypolarizat ionsituation.Onthe other hand. sincethe longitudinalcomponents ofelectric andmagnetic fieldsofHEllmode are inthe ratiooffreespace wave

illljlt!11Ullt'C.itCUllhe predict thattheradiation patternofthe waveguide exhibits symtuetrvandzero cross-polarization.

IIIt.J1t'waveguidestructure.whenTil>'is large,5,(x;,x~)-+x;cot(x~-.r;),011

tIlt'halnneed-hybridcondition51

=

0, so the slot depth isnearlyTO- Tl

=

>'/4.

2.1.2 Radia ti o n Characteri st ics

ForIIII'HEllmode, to obtain the radiationpatternof an open-ended corm- gntod\\'.1.vllgui dc.the Kirclrhoff-Huygenintegra tion[3jover theapertu refieldill the region r<rl,assumingallfieldsto vanishforr>ri ,isused.RefertoFigure2.2, till'electricradia tionfieldcanheexpressedas:

E,,(R' , f)' , ¢' )

= j~(':~~n'

^in'x

J.~<TIU..ti"

^x

^£1-

^Z0

⁷

R,x

(7..

xHI)}eik'·'{n' dS (2.23) Ul'Iiugthe fieldexpression illHEllmode,it canhe shownthatthefieldsHE ll urculve nlUIfollows:

II'l lt'rt '

E~,((l,¢')⁼F(e') 2a~~~, e-jkn'coso'

E~(O',¢') = -F(8')2~~.~,e-j·R'

^sin¢'

(2.24) (2.20)

x

y

Figure 2.2:Co-ordina tesystem usedforevaluati ngtheKirchhcll-Huygenilltl~Arii' tion.

L(B)= JI(udJ1(x d UIXI

Q(O)

=

xIJI(xdJ;(ud-uIJl!uj)J;(.r.l

xi u 't

23

(2.2 il

(2.28)

where111=

x ;

^{SillS',Xl=Jo:rl'ij}can be calculate d fromiJ

=

FI{x I).

IIIE-plllllC(tI,'

=

~),ErA

=

F(B'}ff!tre-i kR' ,

inH-plallc(¢'=0).EM>=F(B' )~e-ikR',the E-plall eandH-phuw patternaft' hk-ntical.

Since

Er

=

E8(8'.4/ )cos~'-E~(O', ~') sin ¢' E~=E6(0',¢i )sin¢'

+

E.(6',(,')1,'081/>'

itfollows that then'nrc110crosspolar ized component sinthe radiation fieldifthe Iil'lli illtill'apertureii'llinoarlvpolarized .

2.2 Corrugated Conical Horn Feeds

Clflrl'ic mJls(231(-Ii]showstlia l,HEl lmodecorrugated waveguideIeeda are lIotgenerallysuitablefor theCassegrainantenna s becausetheirbeamwid the tire too broad.IfWI'wauttousea narrowhcamwidth circularlysymmetric radiati on pattern10Illuminat e tilesubreflector, tilecorrugated conicalhornisIIfavorable choice Alt houghcorrugatedwaveguides havefound ratherlimitedapplication as feeds forlargereflectorantennas, themethod ofana lysisprovidessubstantialinsight intoth('dcehm ofcorrugate dconicalhorns.

" / ' n

,-,- ,~:~-,-

~~ ~

~~ 'n

I.I:""",~~::::;:"

(el

Figure2.3;Corruga tedconicalhorn.(a)Struct ure.(h)Re~iollI.[c]Ill'~i(J1l2.

2.2.1 Pro p agat ionChara ct e ri stics

The structureof the corrugated conical horn isshown iuFig ure 2.3(a). If 11U'lmlf Hareangle does notexceed ahout5°,the cylindrica lmode analysisof the mrruKaledwaveguidefeedpresentedabove may be applied directlytothCSf!horns.

For the largeangle,it.isusuallymore convenient to analyzethiskindof horn based onsplll'rklilhyhrid modesandwe alsousetheimpedancematchingtechnique011 tIl(' boundarybetweenregio n1and region 2,asshown inFigur e2.3hand Figure z.Sc rcspecnvclv.

(8)Field s Exp ressi ons:

Themodesillthe conical hornca n be derived from thevector po tentials ..r and

r,

givenhy

F=ii.F.

when'ii.istileunit vectorillrdirection. The fieldsexpressedbyJandf are

E=-'Vxf+ ~vxVxA_JW€o ii=VxA+ ~Vx vxf_JWllo .4.,amiF.sat.isfyHelm holtz equation:

V^2A.+k^2A.=0

(2.29 ) (2.30)

(2.31) (2.32) IIIl\sphericalcoordinate system[r,O.¢).we definethe fields correspo nding to

.·r

^nn'calledTAlmocles and thatcorresponding tofare calledTEmodes. Sofor

:?ti

TAlmodes.thefield expressions are:

1 EP 2 •

E^T=j ...fo(a;:;+klA, E9=~!D1AT

jW forur80 E=~---.!.-02Ar

." j Wfo l'si n0 8I'u¢

HT= O H9=

rs~n(}OO~T

H.

= -~ %f

ForT E modes,thefield expressionsare:

E,=0

E9 =-rs~no~

E~ =~~

H, =

j~'io ( .g;+ k2 )F,

Hu

= j~JI)~;~;1

1 1 [PF,

Ha,= jwpo /'siIl80rD,j

(:!.a:1l

(:!.:n)

1' ·"1 (:!.:Iri) (:!.:17)

f:!.:I!J)

f:!AII}

12AI}

(:U 2)

(:!.HI

Forthe hybrid mode.thefieldsarcthecombinatio nof T MawlTEmod,'S.

Inspherical coordinatesyst em,tilesolutionsfortileHuhuholtx 1'(11:11)-(32)ran beobtainedhy themethod ofseparat io nof varluhies.

III rCl;ioll1(-Of)<()<(0),the hybridmodeexista.Ifkr

»

1.theradiation rnndltioushould he satisfied,we ha ve:

F;

=

Bkrh~2](kr)P:(cos 8),i"'o

(2.-15)

(2.46)

TIll'roefficlents A and[Jarcdefined hy the rela tionDIA

=

-j Zor,Zo-=~.In 1'i1.(2..1b )ami(2.4G) Q::'(cosO)isomitted because0

=

0isinclu ded. Inthe allow tWlIequations,h~,~)(~'r)isaspher ical Hankel funct ionof the 2nd kind .P;'(cosO) allliq~~(('nsO)an'the asso c iatedLegendre funct ion of the first.kindandthe second kill/I.respectively.

The fieldcompo nentscallhe oht ained fro m eqs.(2 .33).(2.44).For convenience.

II'!fin=krh~~f {krl.P,~'kosOI=P,":,and~"'oisunders tood.

E.=AZolI'; ; ;iTn?;:,

H,

=

-Ar n n

k~~

¹IInP;:' Ee=

-ZoA(r:;~8iInP: ⁺ ~iln d~m

^I

E"=

ZoA(-~iln

^d:J

⁺ rS:loil~ Pn"')

Yo

=

A(jT'l~ifnpm_r~dP::')

rSllIOn r dB

H~ = _A(~d~'" +j~:~i p:)

(2.47)

(2.48) (2.49) (2.50) (2.51)

(2.52)

In n-giou 2.the groovesillthefeed areperpe ndicu lar tothewall ofthehorn, as shownillFigure 2.3(a).the comput a tionofthe field sisvery difficul tbecause the boundariesdonot coincide withthe spher icalcoordinate.Ther efore.for agroove

not too close to the apexofthehorn,itisassumedtoIll'the(lilt'SIIIl\\'l1illFiv;url' 2,3(c),ItisalsoassumedthatonlyTH modescanexist.illIIU'I(rnlJ\·~'.$ollul.1"..I, is usedto givethe fieldex p ressio ns .Wt'have:

From the boundary conditio ns:E4>

=

^{0atT=TIandT=T'l'wehave:}

I

J..(krl) + kr IJ~(kl'l) Y..(krl)+ krl l·~(krl)

1 -0

J..(kI',)+!T,J~(kr,l ^{Y..(kl', )}+h,l;;(kr, ) - (::l.n·l) If thisequati onhas solut io ns,thenTMmodesexis t.Usingtln-bonndarvI'ulllli·

tious:E.

=

0 andE;

=

0 at (}

=

82,we can gelATillthe formof:

X(emcosm¢+Imsin m¢ )

Nowle t's consid ereq.(2.54 ) toderive thecondit ion forTill1II1l111~I'" blalln'.WI'

I~(x)=;I..(I)- In+l(x )=1.._l(X)-~/,,(I) In(I+h)='In(I )+h/~ (x)-i-"(/l'l)

whereIn(x )standsforIn(I )andYIt(I ).we alsoaSSUIll ('kfJ='k(,''l-TI )

«

Iand krl

»

1, then Irom eq.(2.54),wecanobt a in [56):

29

Sf>tileT:\I modesdo existeven ifthe groove width is verv small.

(b) FieldImped anceMat chingTechnique at8=80:

Inregion2. SUPIKlSetherearemanygroovesinone,.\andwe formulat etilt·

IHJlllld llrycoudhious at 8

=

80in terms oftwoimpeda nceZ.ant!Z~definedI~':

fromtheTMfieldexpressions in thisregion,underthe condi tio nskr

»

1.kb

«

1.

i.e..rhoI;TOOVCSarc farawayfromapexandthegroovewidthismuch smallerthan

"\,llil'lIwe have

z. =o

z~:::::l-jZotan(,h)

!iL'ldefinedin Figure2.3(c).Whens-+>'/4,Z.-+00.

(2.56)

111regionI,ill order to supportonlytileTAlmodesin thegrooves .theboundary hnpcdanceaat6=80must hetilesameasthosein region 2.Fromeqs.(2Ai) -(2.52).

\\ '1'1I;\\'e

z,

⁼

~ ⁼ n~~:;)(P:(8)+ ~::9) z, = !It: =- n:~:;)[P:(8)

+

j~i:;~ 1

p; = dP:'~~"' O) /P; I""

0) h _"; (h )

"-"" Ih)

12.;8)

(2.;9 )

12.60) (2.6 1)

IfI!I('fields we areinterest edinare farawayfrom the apex, i.e., kr

»

I,then 11_-+-j.Imposing theim pedance matchingtech nique,wehave

(2.62)

30

(2.6:1)

(2.G-I) where

f"--=l....n(TI+lj - Zrl'iJ kr (c)Balanced-HybridCondition:

Ifineq.(2.57},wechooses

=

^>./4.thenf-

=

^{O.From c<1.(2 (2)lind}('<I.(:.l.G:n.

we obta in:

r=±l (2.Gl:l)

This is calledbalanced-hybridcondition. The+signreferstoHEllInt!I'liullll- sign referstotheEH modes. Under this condition,the field component»E,allll H,are theninthe ratio of free spaceim pedance.

(d )HEI ...mode:

IfIn=1under balanced-hybridcondition withr=+1.from('<1.(2 .62),ill- tr aducinga functionf~,we can get the churactcristicequation ofHElmJllllll,,:>

f~

⁼

dP~~~seo)

⁺

P~s\~~:O) =

^(J (2.GG) In eq.(2.66),ncanbe determi nedandthe firstroot correspondst.oHEllllIude.

IIIthe meantime, using eqs.(2.49).(2 .52),wehave the fieldexpressions forHEI",

modes as:

E8

=

-Zo

A~" (s:~(1

⁺

~)(,OS4~

E4>=

ZoA~"(S:~8 + ~) sill¢

H9=-YoE4I

(2.G7)

(2.G8) 12.G9) f2.7fJ)

31

2.2.2 Radiation Characteristics

Fo rHElmmOlII$.thefieldexpressions as statedineqs(2.67)-(2,70 )utIIH' surface,r

=

RushownillFigure2.3(a )are known. According toFigure 2.2,'he radia tion fieldEpcanbe determi nedbyKirchhoff-Huygeu integrationoverthe nperturchoun dedbyr

=

RO.-O_l< ()<0₁.0< 4><2lT:

, " jke-jHl'~

J. .... .. .. ..

^'Id'·

Ep(R ,O, ¢ ) = 47TR' in'x cpcrllou{inXE,-Zoin,X(inxH,)}e-'",l' dS (2.il) wherr-in"r..art'nuitvectors. andH/,E,are tangentialfields to thesurfaceof r=Ro, as shown in Figure2.3fa).The filialradiation fields ofHEI ",modecan he ohta i lU..las:

E~i" ,¢')~(F,(B')

+

jF,(B')jo".

Ep~(B' ,q/)=-[FrfO')

+

jF,.(O'l]siJl/;/

p,_jkn' /0"

F,.18')~DR ' }, f~(')G,(',B')d' i

G

=

((1+(058)(1+coss')sinO~~:(acos8)Jo(bsin8)]

-[(1-cos8)(1-cosll)si n 8:~(acos8)J2{bsin8)]

=f12siu8'sin28

~~:

^{Jdbsin 8)J}

(2.72) (2.73)

(2.74)

where(I=~·Rocosll'.b=krosin 8',Dis a constant independent, ofR',8',1J' . Itcan heseenthat. the E-pla ne and H·plancpattern areidentica land there Is uocrosspolarlzedcomponentsinthe rad iation field when theaper turefield is linearlypolarize d.

(2.7G) 2.2.3 Radi ationPattern

Ohviously,theexpressionsfortheradiated fleld.jn\'(IS.(2.72)-(2.7.1)an'St,

complicatedthattheyare notquitesuitable forthecomp ute r uualyscs .Using theseparationofvariables.D.\'. Rao[571hils simplifiedrhoHelmhol tz"'1l1al inllill sphericalcoordina tesystem amitheresults arcgiven here:

1

a.

80 ",1

8sin888(sIll8

N )

+n(lI+I) -Sill10

=

U (2. 7~) Thisequationcanhe furtherapproximatedby replacingsinOh.\·(Jfor II<:lU", resultingin:

~:~ + ~%~+(l -~)0 =O

whichhasthesolutionof

e =

DmJm(v).I'

= J"( » +

l)·().

Then followthesame procedure described nbovoand expresst.11I'tall~" lIl i1l1 electricfieldon the circularaperture011thehorn forHEllmode:

E,~ EoJo(2.~05p)e-J~P'ej1j>(p +jl/~) (2.77)

wherep,l;~aretheunitvectorsat apointP(p,~IJ)011ther-irculurupnrtun-{Iftilt·

hornexpressedillcylindricalcoordinatesystem.n=LIT/An ,Listil{'axiallml~th ofthehorn,Eois a constant. Usingthe vectorrliffrac tiou formula[3],UlI' rwlialioll electricfield canbeobtained:

(2.781

where

12.70) and rr=27rusin8f>"o.r=p/ (l.V=rra'l/(AnL),aistheapertureradius.[ftIll' apert urefieldislinearlypolarized, then

33

E~~(P)

=

C(l+cosO)Mcos¢B E.~(P)

=

-C(l+cosO)Msin¢i¢

The radiated fields are linearly polarized .This far field expressionis culv valid for Ou<3(JQ.Itcallmeet. therequiremen tofengineeringdesign ofCessegrain antennas.

2.3 Numer ica l Result s

Bused on theanalyses above,espec iallytheapproximated radiationpatt erns givenillScr:. !!.!!. 3.a computer analysisprogramhas been written.Thecomputed resultsarcshownillFig-nrc2.4·Figur e2.6.The E-planeandH-pla nepatterns nrc identical.Several casesare given h,vvaryingtheflare angle (fixed apertureradius inFigure2.4 and fixedaxiallength in Figure2.6) or changing theapert ure radius (fixedflareanglein Figure2.5).Itca n beseen from Figure2.6 that. whentheflare llll/(It,isvery sma ll. l.e.,the horn callhetaken asthecorrugatedwaveguide.tho benmwid t.hisverywidelindtile corrugatedwaveguidecan not.beof practicalusc insuper p;aillantennas.Proper choiceofthe dimensionsofthe corrugated conical horn will lendtotile desiredhorn radiationpatterns.

-80

10 ~ ~ ~ ~ W ro

thela (deg) 80 90

Figure2.4: Radiationpatternsof thecorrugated l:Ollka!horn. Fn-queney:

14.25G Hz. Aperture radius:B.Oin. (a) Flareallgll~:20".(h) Flnr«Hllg ll':1:.1", (c) Flareangle:"to

35

·50

·60

·70

·60

·90

10 20 30 40 50 60

lhela(deg)

70 80 90

Fi/(UTl' 2.5: Radiation patterns ofthecorrugated conical horn. Frequency:

]-l,25GH'l. Flare ang le:12°, (a) Aperture radius:4.0in, (h) Apertureradius:

S,Uill_(c)Apertureradius:16.0in.

-10

-20

-70

-80

.

'~"""

" ,

:\ ,,,.~ ,",:",,,,,, .

, ,

" ,

, ,

'.'

3G

10 20 ~ 40 ~ ~ ro M 00

ttlela(deg)

Figure2.6: Radi ationpatterns ofthecorrugated eonlculhom. Fn~llwlII·.\I:

14.25GHz.Axiallength:37.0in.(n)Flare Angle: l",(iJ)FlareAIl~le:2~ .(e) FlareAngle:20°,(d)FlareAngle:12°

Chapter 3

Dual Shaped R eflector Antenna D esign

TIlt'rotat ionallysy mmetrical dualsha ped reflectorantennasusedillearth sta - linnsnredescrihed.The classical Cassegrainand Grego rianant ennas havea low l'flicil'lle,\'cluetutheIactthattheapertureillumination can nothe pre-sp ecified.

Hence the antennagain, eldelobelevelsca llnot hesynthesized.Although Silver [3]

husproposedthe methodof shaping tilereflectorsurfaceshyspeci f~'i ll gthenper- turc illuminationsince1040,Thisproblemhasbeen unsol ved untilthecompute r power\\ 'USavailablein1970's.Generally, the problem can he tackledilltwo ways : (n)Thefeed radiationpatt ernandthe main reflectoraper tureilluminationare r.:ivcll.The profiles of mainandeuhreflectora aretohe optimized.Thisscenariois ronunonlyencounteredin the desiglloCanew antenna.

(il)The ft'Cdradiationpattern ant!themainreflect orprofileare given,The subre- Ik'l.'lurprofileistobefoundh~'usingphaseuniform apertureillumination s,This is Ihe mostuse fulappr oach for upgrading those existingantennasystemsdesigned ii,\'usingtheclosed formsolutions.

Pinllt'l!riugworksas to(a)and(b)havebeencarried0111.hy Galindo[12]and

37

Potter (331.etc.In thlsrhapter,GaJimlo'smet hod ofC'flkulnt illltth.·",1I:t]",1r.' I1., "

torsurfacesispresentedandtheresu lts areused 10.1('Si~1lCns.'>t~r.\illuurlGtI~'lrilln antennasfnrtilelargeeart hstations.

3.1 Axisymmetric Dual Reflector Antennas

Forthedua l reflector antenna. themain reflt'elmisfonlll,lh.\"wlal i u~a parabolaaboutitsownaxis and thesubretlcctori",lumlt·sill1ilarl,vh.1"mt.u!illl;

eit her an ellipse orhyper bo la. The subrcflcctorlmst\\~1£ucllll'ulnl.s.(hi.'ftll'ill pointis also the phasecenterof th efeed,andthesecondfu,'ul l'Clili1is11I:\<1 ..III coincide with tilefo cal pointof theparaboloidalmai u rrfit'I·f( ,r.

AsshowninFigure 3.1.the Cassegrainantenna has ah.VPNIH:~uillalsnltn·lh'Ctllr.

Hence.the secondfocalpo in tis avirtual one.Till'Gregorianantenna. IllIth.' llllll'r ha nd.hasaellipso idalsuh reflectorwith a~aljocalIKlillt. astd llM'1Iinfil:llll'3.2.

Paraboloid

Figure 3.1:Caesegminantennaconfigura t ion.

39

Figure3,2:GregorianantennucOllli~llratiulI ,

41

3.2 Geo metrical Opt ic s (G O) P r inciples

IIItlll'ballisuctheory of light, tile propaga tionofelectromagnet icenergyis (~xrlailJl..>(1interms of thekinematicsofphoto particles. Whetheror not this 11.\·- pothesisis Hearer 1,0physicalre ali tythanthatof wave propagation,itdoes lead In a particularly useful concept,i.e.anopticalray as thetrajectory ofaphotonics particle.The tangent to aray at anypoint represents equallythe localdirection nfprop aga tion, orthe normalto the wavefront . Alsoassociatedwiththera~'at lillypoint isitsinte nsity. Proportionaltothelatt eraretherayfieldvectorE andHillmagnitude. Geometricalopticshas a theoreticalfoundationbased on Maxwell's equationsandresultsfrom anasymptotic solutiontothese equationsas

!.IwfrequencyWtends to infinity1551.

IIIthr- nun-closedform antennadesigns,as longas the dimensionsof the refiec- turs arc verylargecompared wit h thewavelength,till'methodis successful.The haljil'principles of GO are:

(1) Snell'5Law:Atapoint on thereflector surface,the incident. and reflected rays as well asthesurface normalvector are coplanar,then the angleof incidence equalsto theangle of reflection.

(2)Conservationof energy law:Energyalongeach differenttubeof ray remains const ant.even when thetubeundergoesreflection.

(3) Maluslaw: Thesurfacesof the consta ntphase arenormalto'heray trajecto riesevenaftera numberof reflections. The ray lengthbetween two equal- phasesurfaces keeps constant .

3.3 Cassegrain Antenna

3.3.1 Design Techniques

(i)Design ofbothmain reflector andsubre ftect orgiventhe main reftecto r apert ure illumination and the feed rad iati onpattern

The mainreflector and subreflectorof the Caesegrnin antennawithrevolution- ally symmetricstruc tureare shown in Figure3.3. The profileof maiurdh'(~',C1riii describedin rectangularcoordinate (x, .:) andthat ofenbrcflcctoriii(bnibl'dill hothrectangularcoordinate(x' , ;;')and polarcoordluutcs(",0).TIll'originIXlilli of(x', :;' )isthe feed phase centerF.0.isthedistance betweenthe' two rectangular coordinate systems.The relationsh ipamong the three systems are:

x' psinO

pcosB

Ifthesubreflector surface isexpressedas:

p=f(8)

5(8)

=

p- f(8)

=

^0,

the nthe unit vectornormal to thesubreflectorsurfaceis

n.

175 (8)/1175 (8)1

i[± (-~COSt1

^+fsinB)J+

il±(~Sint1

^{+!{;os 811}

x

--- --- -- ---- - - - -~- - -

---

Figu re3.3: GeometricalopticsforCassegrain antenna reflectorsshaping.

43

where

The incidentvectoris:

i. =p=xsiIlO+ ':cosO

and the reflectionvector is:

e,=isinl1t+':cos ~"

FromSnell' slaw,i.e.,II'r

=

-i' ll,the followingexpression ca ll 1)('oln.ained:

~

⁼

~

⁼pcot (ti" ; O) Ifthe profileof mainreflector surface is expressedas:

,~g(xl

M(x )

=

z-g (x )=0 The unitvector normaltothe main reflectorsurfaceis:

where

TILeincident vectoris:

(3.1)

i

=

isillf/'+=COs ~"

HllIlt1wreflec tionvectoris:

r ==

From Snell'slaw, the differentia lequation ahoutthe main reflectorcallhedescribed

~=cot(~)

^(3.2)

If die feed patternis known,the totalradiated powerwithinHIe incrementdB of the patternF{B)will be

F(B)211"sin8dB TIll"total rudiutcdpower from

e

=0tolingle()m~rwill then he

2:Ir

fo'mu

F(8)sinOdB

Similarly,if thedesiredilIumination l(x ) isknown,thepowerwithinthe incremen t tiT ofthemainreflectoraperture is

1(:c)2rrxd;t· Again.t1JC totalpowerwithin theregion(O,Xm Br)is

From tIlt!conservation of energy law. the followingequa tioncanbe derived:

~=A F(8) sin B

dO I(x)x ^(3.3)

where

A

= .M

ⁱ- ,I(J:)rdr

.r: .... .

F(O) siIl OdO

So thethree differential equat ions ,i.e.,eq.(3.1),l'<1.(3.2 )and''(I,(a .3)lUI'ohlai11l,d with threeunknowns,p.:;andr.,Thcvart>sufficienttu.h-tcnuincllll' n-Hertur profileswiththeinitialvaluesofPmoZ",O", or'.\",',r' Tilt'lIllI/flf'/I IIt/1I1111'11",,1is employedtosolvethesedifferentialequat ions. Thisapproachisrunuuuulvu-a-dIII design a new antenna.

(ii)Design oftheon lysu breftect or whenthe mai nrefle ct orprofileand feedra di a ti onpattern aregiv e n

Thereexist a number ofexistingeart hstationswhichdolint1Il1'PIt.ln-CelT "!"

specification,especiallythe close-Inandthe far-anglesidelolx-levels.The anll'nUIl needtohemod ifiedto upgradeitsperforma ncesat the least (;(Jsl-,TIIPr\'l li 'si~1lIOf thefeed or/ andthe mainreflectoris normally me re costlythantill 'mndifk-ntion of thesubre flcct oronly, It.is worthwhileto presenttill'uumerlcnlapl'TOlld lfor thisparticulardesign.Thefeedpatternis commonlyassumedt,1lln-rl'l'lllulionally symmet ricandcall berepresented asF(OJ,This assumptionisoftenvalid hmwd 011thefact thatthesuhtentedangleof thesubreflecturis usuallv small.TIlt'muiu reflector profileis givenandcallhe expressedas

,=

^9(X)

Assumethe powerilluminationon themain reflectoraperture,/(;r),isIlll;~,;,' unifor m,thisim pliesthat the pathlengthfromthe feed phasecunterto1,lu,npertun-

mus t he constant hllliedon Maluslaw,usingthe same coordinate system shownill Figure3.3,cue has

p+p'+p~=c

The cons tantlengthCcanhe obtainedfrom theconditionattileedges

C=Pm.,+P~d>: ^(whenp"=0) AccordingtoFigure3.3,

, XM-XS

e

vr-r-r-_sin11'

p=J[x~+;;~]

:s

=

Q

+

Xs-~M

+

2M tauto' Thefollowi ngexpression holds

TIII'IltheXscan hederived.

x l,

+2XM(~-~ )

+

(C-n)(C+0:+2:"tl

:rs- 2(XM +~ -~)

(3AI

(3.5)

Becnuso tilt>mainreflector profileis given,applySnell'slawon the mainreflector, it is casH.\'to obtainthefollowing expressio nfrom e<j.(3.2)

til

= 2arctan(~)

Thesubreflcctorprofilecallhe unique lydefined hyeq.(3.4 )andeq.(3.5) .