天線原理第六章

16

LinearArray（線陣）1AntennaTheoryandDesignwangmin@16.1IntroductionSingleelement—relativelywideradiationpattern,lowdirectivity(gain).Todesignantennaswithverydirectivecharacteristics—increasetheelectricalsizeoftheantenna,orformanassemblyofradiatingelementsinanelectricalandgeometricalconfiguration,i.e.,anarray.Thetotalfieldofthearrayisdeterminedbythevectoradditionofthefieldsradiatedbytheindividualelements.Thisassumesthatthecurrentineachelementisthesameasthatoftheisolatedelement(neglectingcoupling).2Shapingofthearraypattern—makingelementfieldsinterfereconstructively(add)inthedesireddirectionsandinterferedestructively(canceleachother)intheremainingspace.Fivecontrolsforanarrayofidenticalelements1.thegeometricalconfigurationoftheoverallarray(linear,circular,rectangular,spherical,etc.)2.therelativedisplacementbetweentheelements3.theexcitationamplitudeoftheindividualelements4.theexcitationphaseoftheindividualelements5.therelativepatternoftheindividualelements36.2Two-elementarray（二元陣列）4Tosimplifythepresentationandgiveabetterphysicalinterpretationofthetechniques,atwo-elementarraywillfirstbeconsidered.twoinfinitesimalhorizontaldipolesalongthez-axis,whereβisthedifferenceinphaseexcitationbetweentheelements.Themagnitudeexcitationoftheradiatorsisidentical.(6.1)5Approximationforfar-fieldobservationselementpatternarrayfactorPatternmultiplicationforarrays（陣列的乘法原理）Patternmultiplicationforarraysofidenticalelements(6.2)(6.5)6Arrayfactor（陣因子）innormalizedformafunctionofthenumberofelements,theirgeometricalarrangement,theirrelativemagnitudes,theirrelativephases,andtheirspacings.notdependonthedirectionalcharacteristicsoftheradiatingelementsthemselves,itcanbeformulatedbyreplacingtheactualelementswithisotropic(point)sources.thetotalfieldoftheactualarrayisobtainedbyPatternmultiplication.

(6.4)7Example6.1（相同的陣列結構，不同的激勵相位，形成不同的零點）GiventhearrayofFigures6.1(a)and(b),findthenullsofthetotalfieldwhend=λ/4anda.β=0;b.β=+π/2;c.β=?π/2.ThenormalizedfieldisSolution:8Theonlynulloccursatθ=90andisduetothepatternoftheindividualelements.Thearrayfactordoesnotcontributeanyadditionalnullsbecausethereisnotenoughseparationbetweentheelementstointroduceaphasedifferenceof180betweentheelements,foranyobservationangle.a.β=09b.β=+π/2;Thenullsofthearrayoccuratθ=90and0.Thenullat0isintroducedbythearrangementoftheelements(arrayfactor).10aninitialphaselagof90;anadditional90phaseretardationduetopropagation.outofphaseinphasePhaseaccumulationfortwo-elementarrayfornullformation（二元陣形成零點的相位疊加關系）11c.β=-π/2;Thenullsoccurat90and180.Theelementatthepositivez-axishasaphaselagof90relativetotheother,andthephasedifferenceis180whentravelisrestrictedtowardthenegativez-axis.12Illustrationtothepatternmultiplicationrule（方向圖乘法原理的圖示）Element,arrayfactor,andtotalfieldpatternsofatwo-elementarrayofinfinitesimalhorizontaldipoleswithidenticalphaseexcitation(β=0,d=λ/4).13(β=90,d=λ/4)(β=-90,d=λ/4)14Generalformulationfornullsofabovearrayoftwoidenticalinfinitesimaldipoles（無窮小偶極子二元陣零點的一般公式）156.3N-ElementLinearArray:UniformAmplitudeandSpacing（等幅等間隔N元線陣）16Uniformarray（均勻陣列）

—Anarrayofidenticalelementsallofidenticalmagnitudeandeachwithaprogressivephase.β—thephasebywhichthecurrentineachelementleadsthecurrentoftheprecedingelement;Assumingtheelementsbepointsources,onlyarrayfactorwillbeconsidered.17Arrayfactor(6.6)(6.7)18(6.10)Referencepointisthephysicalcenterofthearray,SmallψTonormalizethearrayfactors19Nullsofthearray（陣列零點）Forn=N,2N,3N,...,(6-10c)attainsitsmaximumvaluesbecauseitreducestoasin(0)/0form.Thevaluesofndeterminetheorderofthenulls(first,second,etc.).For

azerotoexist,theargumentofthearccosinecannotexceedunity.Thusthenumberofnullsthatcanexistwillbeafunctionoftheelementseparationdandthephaseexcitationdifferenceβ.20Themaximumvaluesofthearray（陣列最大值點）onlyonemaximumoccurswhenm=0.(6-13)The3-dBpointforthearrayfactor（陣因子半功率點）orHalf-powerbeamwidthh（半功率波束寬度）(6-14c)21Secondarymaxima(maximaofminorlobes次級最大值)—occurapproximatelywhensin(N/2)attainsitsmaximumvalue.maximumofthefirstminorlobewhenatthispoint,=-13.46dB22Themaximumradiationofanarrayisdirectednormaltotheaxisofthearray[broadside;θ0=90]6.3.1BroadsideArray（邊射陣）Desirabledesignthemaximaofthesingleelementandofthearrayfactorshouldbothbedirectedtowardθ0=90.Alltheelementshavethesamephaseexcitation(inadditiontothesameamplitudeexcitation).Theseparationbetweentheelementscanbeofanyvalue.(6-18a)Broadside(=0,d=/4)23（柵瓣）—theprincipalmaximainotherdirectionsGratinglobesIfd=nλ,n=1,2,3,...andβ=0,thenThereareadditionalmaximadirectedalongtheaxis(θ0=0,180)ofthearray(endfireradiation).Toavoidanygratinglobe,thelargestspacingbetweentheelementsshouldbelessthanonewavelength(dmax

<

λ).Broadside/end-fire（邊射/端射）(=0,d=)24Ifλ<d<2λ,thenthemaximumtowardθ0=0shiftstowardtheangularregion0

<θ0<90,whilethemaximumtowardθ0=180shiftstoward90<θ0<180.Whend=2λ,therearemaximatoward0,60,90,120and180.(=0,d=)(=0,d=/4)25Formulasforuniformamplitudebroadsidearrays26Formulasforuniformamplitudebroadsidearrays276.3.2OrdinaryEnd-FireArray（普通端射陣）Desirabledesign—Themaximumradiationofanarrayisdirectedalongtheaxisofthearray(end-fire),i.e.,eitherθ0=0or180.(6-20a)-firstmaximumtowardθ0=0,-firstmaximumtowardθ0=180,(6-20b)(N=10,d=λ/4)

=-kd

=kd28AvoidingGratinglobesIfd=λ/2,end-fireradiationexistssimultaneouslyinbothdirections(θ0=0andθ0=180).Ifd=nλ,thenthereexistfourmaxima;twointhebroadsidedirectionsandtwoalongtheaxisofthearray.Tohaveonlyoneend-firemaximumandtoavoidanygratinglobes,themaximumspacingbetweentheelementsshouldbelessthandmax<λ/2.Broadside/end-fire(=0,d=)Formulasforuniformamplitudeend-firearrays(omitted)296.3.3Phased(Scanning)Array（相控掃描陣）Desirabledesign—bycontrollingtheprogressivephasedifferencebetweentheelements,themaximumradiationcanbesquintedinanydesireddirectiontoformascanningarray.(6-21)Phaseshifter—continuouslyvaryingtheprogressivephasebetweentheelements.Ferritephaseshifters:thephaseshiftiscontrolledbythemagneticfieldwithintheferrite.Diodephaseshifterusingbalanced,hybrid-coupledvaractors,orPINdiodes.Significantinsertionloss!3010-elementarray,d=λ/4,θ0=6031Half-powerbeamwidthofthescanningarrayValidforbroadside,andscanninguniformarrays.Notvalidforordinaryend-firearrays.SinceN=(L+d)/d,(6-22a)32Figure6.12Half-powerbeamwidthforbroadside,ordinaryend-fire,andscanninguniformlineararrays.336.3.4Hansen-WoodyardEnd-FireArray（漢森-伍德亞德端射陣）Hansen-Woodyardconditionsforend-fireradiationToenhancethedirectivityofanend-firearraywithoutdestroyinganyoftheothercharacteristics,HansenandWoodyard[5]in1938proposedthattherequiredphaseshiftbetweencloselyspacedelementsofaverylongarray.(6-23)Theseconditionsdonotnecessarilyyieldthemaximumpossibledirectivity.Maximaandsidelobelevels,dependonthenumberofarrayelements.34ConditionstorealizetheincreaseindirectivityForanN-elementarray,theconditionof|ψ|πissatisfiedbyusing(6-23a)forθ=0,(6-23b)forθ=180,andchoosingforeachaspacingof(6-25)Foralargeuniformarray,theHansen-Woodyardconditioncanonlyyieldanimproveddirectivityprovidedthespacingbetweentheelementsisapproximatelyλ/4.35ComparisonbetweentheordinaryandHansen-Woodyardend-firearraypatterns(N=10,d=λ/4)a73%increaseindirectivity;approximately1.805times(or2.56dB)greaterThereisatrade-offbetweendirectivity(orhalf-powerbeamwidth)andsidelobelevel.HPBW=74

HPBW=37Higherdirectivity=19Lowersidelobe(-13.5dB)Lowersidelobe(-8.9dB)Directivity=1136Thed=λ/2patternfailstorealizealargerdirectivityduetoitslargerbacklobes.Patternsforthesamearray(N=10)withd=λ/4(β=?3π/5)andd=λ/2(β=?11π/10)thenecessary|ψ|θ=180

πisnotsatisfied.37DerivationofHansen-Woodyardconditions*FromthearrayfactorofanN-elementarrayforsmallψ(ψ=kdcosθ+β),ifwhere(6-29)Theradiationintensitywhere(6-30a)(6-30)Dividing(6-30)by(6-30a),sothatU(=0)=1,BecauseWewanttominimizetheaverageradiationintensityU0!(6-31)38TheaverageradiationintensityU0where(6-33)U0isproportionaltog(υ)!(6-34)39Theminimumvalueofg(υ)occurswhen(6-37)40FormulasforparameterofHansen-Woodyardend-firearraysTheexpressionsforthenulls,maxima,half-powerpoints,minorlobemaxima,andbeamwidths.(omitted）Maximumelementspacingdmaxtomaintaineitheroneortwoamplitudemaximaofalineararray416.4N-ElementLinearArray:Directivity42arrayfactor6.4.1BroadsideArrayradiationintensitydirectivityforalargearray(L>>

d)(6-44)43Example6.3Givenalinear,broadside,uniformarrayof10isotropicelements(N=10)withaseparationofλ/4(d=λ/4)

betweentheelements,findthedirectivityofthearray.Solution:Using(6-44a)44arrayfactorforθ=0

6.4.2OrdinaryEnd-FireArrayradiationintensitydirectivityforalargearray(L>>

d)(6-45)(6-49)Twicethatforthebroadsidearray!45Example6.4Givenalinear,endfire,uniformarrayof10elements(N=10)withaseparationofλ/4(d=λ/4)

betweentheelements,findthedirectivityofthearrayfactor.ThisarrayisidenticaltothebroadsidearrayofExample6.3.Solution:Using(6-49)Thisapproximatevalueforthedirectivity(D0=10)comparesveryfavorablywiththevalueofD0=10.05obtainedbynumericallyintegrating(6-45).466.4.3Hansen-WoodyardEnd-FireArrayMaximumradiationintensityatθ=0

Directivityor(6-51)(6-50)1.805timesgreaterthanthatofordinaryend-firearray!47Example6.5Givenalinear,end-fire(withimproveddirectivity)Hansen-Woodyard,uniformarrayof10elements(N=10)withaseparationofλ/4(d=λ/4)betweentheelements,findthedirectivityofthearrayfactor.ThisarrayisidenticaltothatofExamples6.3(broadside)and6.4(ordinaryend-fire),anditisusedforcomparison.Solution:Using(6-51)Thevalueofthisdirectivity(D0=18.05)is1.805timesgreaterthanthatofExample6.4(ordinaryend-fire)and3.61timesgreaterthanthatfoundinExample6.3(broadside).48Summarize:DirectivitiesforBroadsideandEnd-FireArrays6.5DesignProcedure49Designparameters—elementnumber,element,excitation(amplitudeandphase),half-powerbeamwidth,directivity,andsidelobelevel.Analysisprocedure:

equationsandsomegraphscanbeusedtodeterminethehalf-powerbeamwidthanddirectivity,oncethenumberofelementsandspacingarespecified.Designapproach:tospecifythehalf-powerbeamwidthordirectivityandtodeterminemostoftheotherparameters.Moreexactvaluescanbeobtained,ifnecessary,usingiterativeornumericalmethods.Someoftheseparametersarespecifiedandtheothersarethendetermined.50Example6.6Designauniformlinearscanningarraywhosemaximumofthearrayfactoris30fromtheaxisofthearray(θ0=30).Thedesiredhalf-powerbeamwidthis2whilethespacingbetweentheelementsisλ/4.Determinetheexcitationoftheelements(amplitudeandphase),lengthofthearray(inwavelengths),numberofelements,anddirectivity(indB).Solution:(1)calculatetheprogressivephase(2)Determinethelengthofthearraybyusinganiterativeprocedureof(6-22)oritsgraphicalsolutionofFigure6.12.L+d=50λ51(3)Determinethenumberofelements(4)Calculatethedirectivity.ThedirectivityofthearrayisobtainedusingtheradiationintensityandthecomputerprogramDIRECTIVITYofChapter2,anditisequalto100.72or20.03dB.6.6

N-ElementLinearArray:There-DimensionalCharacteristics52Nisotropicelements,spacingd,amplitudeexcitationanandaprogressivephaseexcitationβ.Arrayfactor6.6.1N-ElementsAlongZ-Axis(6-52)(6-52a)γistheanglebetweentheaxisofthearray(z-axis)andtheradialvectorfromtheorigintotheobservationpoint.(6-53)536.6.2N-ElementsAlongX-AxisorY-Axis(6-52)(6-54)ForanarrayalongX-axisForanarrayalongY-axis(6-55)Numericallytheyyieldidenticalpatternseventhoughtheirmathematicalformsaredifferent.54Example6.7Twohalf-wavelengthdipole(l=λ/2)arepositionedalongthex-axisandareseparatedbyadistanced,asshowninFigure6.17.Thelengthsofthedipolesareparalleltothez-axis.Findthetotalfieldofthearray.Assumeuniformamplitudeexcitationandaprogressivephasedifferenceofβ.Solution:

(1)elementpattern(2)Arrayfactor(3)Totalfield55β=0:thenullsintheθ=0direction,providedbytheindividualelementsofthearray,additionalnullsalongthex-axis(θ=π/2,

=0and=π)providedbytheformationofthearray.The180?phasedifferencerequiredtoformthenullsalongthex-axisisaresultoftheSeparationoftheelements[kd=(2π/λ)(λ/2)

=π].

β=180:thenullsalongthez-axis(θ=0?)areprovidedbytheindividualelements,nullsalongthey-axisareformedbythe180?excitationphasedifference.6.7

Rectangular-to-PolarGraphicalSolution56Generalsolutionformforfieldpattern(6-56)whereCandζareconstantsandγisavariable.ApproximatearrayfactorofanN-element,uniformamplitudelineararrayisthatofasin(ζ)/ζ

formwithwhere(6-57)f(ζ)—afunctionofζinrectilinearcoordinates.Pattern|f(ζ)|—afunctionofthephysicallyobservableangleθ.57Rectangular-to-polarplotgraphicalsolution1.Plot,usingrectilinearcoordinates,thefunction|f(ζ)|.2.a.DrawacirclewithradiusCandwithitscenterontheabscissaatζ=δ.b.Drawverticallinestotheabscissasothattheywillintersectthecircle.c.Fromthecenterofthecircle,drawradiallinesthroughthepointsonthecircleintersectedbytheverticallines.d.Alongtheradiallines,markoffcorrespondingmagnitudesfromthelinearplot.e.Connectallpointstoformacontinuousgraph.58Example:Uniformlineararray:N=10,d=λ/4,β=?π/4visibleregioninvisibleregionOnlythevisibleregionofthelineargraphisrelatedtothephysicallyobservableangleθ。6.8N–ElementLinearArray:UniformSpacing,NonuniformAmplitude59Broadsidearrayswithuniformspacingbutnonuniformamplitudedistribution:BinomialandDolph-TschebyscheffbroadsidearraysBeamwidthfromthesmallestinorder:uniform,Tschebyscheff,andbinomial;Sidelobesfromthesmallestinorder:binomial,Tschebyscheffanduniformarrays.Acriterionthatcanbeusedtojudgetherelativebeamwidthandsidelobelevelofonedesigntoanotheristheamplitudedistribution(tapering)alongthesource.Tschebyscheffarrayproducesthesmallestbeamwidthforagivensidelobelevel,andleadstothesmallestpossiblesidelobelevelforagivenbeamwidth.606.8.1ArrayFactorAnarrayofanevennumberofisotropicelements2Mpositionedsymmetricallyalongthez-axis,(6-59)Normalizedform61Anarrayofanoddnumberofisotropicelements2M+1(6-60)Normalizedform(6-61)626.8.2BinomialArrayA.ExcitationCoefficients(6-62)Binomialarraysdonotexhibitanyminorlobesprovidedthespacingbetweentheelementsisequalorlessthanone-halfofawavelength.63Pascal’striangle:

positivecoefficientsoftheseriesexpansionfordifferentvaluesofm.Theamplitudecoefficientsforanarraywithmelements.1.Twoelements(2M=2)a1

=12.Threeelements(2M+1=3)2a1=2?a1=1a2=13.Fourelements(2M=4)a1

=3a2=12.Fiveelements(2M+1=5)2a1=6?a1=3a2=4a3=164B.ExcitationCoefficientsApproximateclosed-formexpressionsforHPBWandmaximumdirectivityforthed=λ/2spacing65Patternsofa10-elementbinomialarray(2M=10)withelementspacingsofλ/4,λ/2,3λ/4,andλ.

a1=126,a2=84,a3=36,a4=9,anda5=1.Themaximumelementspacingtomaintaineitheronemaxima:dmax<λ/2.Majorpracticaldisadvantage:widevariationsbetweentheamplitudesofthedifferentelements.66676.8.3Dolph-TschebyscheffArray（道爾夫-切比雪夫陣）(6-66)Acompromisebetweenuniformandbinomialarrays.A.ArrayFactorChebyshevpolynomialsaseriesofcosinefunctionswiththefundamentalfrequencyastheargument68(6-68)Letz=cosu(6-69)RecursionformulaChebyshevpolynomialsTm(z)69PropertiesofChebyshevpolynomials1.Allpolynomials,ofanyorder,passthroughthepoint(1,1).2.Withintherange?1≤z≤1,thepolynomialshavevalueswithin?1to+1.3.Allrootsoccurwithin?1≤z≤1,andallmaximaandminimahavevaluesof+1and?1,respectively.ChebyshevpolynomialsoforderszerothroughfiveSincethearrayfactorisasummationofcosinetermswhoseformisthesameastheChebyshevpolynomials,theunknowncoefficientsofthearrayfactorcanbedeterminedbyequatingtheseriesrepresentingthecosinetermstotheappropriateChebyshevpolynomial.70B.ArrayDesignStatement.DesignabroadsideDolph-Chebyshevarrayof2Mor2M+1elementswithspacingdbetweentheelements.ThesidelobesareR0dBbelowthemaximumofthemajorlobe.Findtheexcitationcoefficientsandformthearrayfactor.ProcedureSelecttheappropriatearrayfactorasgivenby(6-61a)or(6-61b).Expandthearrayfactor.Replaceeachcos(mu)function(m=0,1,2,3,...)byitsapp