Space Tapering of Linear and Planar Arrays

7/29/2019 Space Tapering of Linear and Planar Arrays

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From Fig. 1, the coordinates x an t i y are related tothe coordinates s and y b>-

s= T - XCrJ- su:y =x30 - o . (2.3)

Using theserelationshipsandrearranging erms, 11can be written,

= J-=(C,c - a p ( s / )*

.S-, esp [jk(c-uS+soC- )y]dy.Since

Letting coS +soC =S and C = ( l - s S J j l r = ~ O C - ~ ~ S (26) becomes

Is, hen examined in a similar way, produces a resultidentical with ( 2 7 ) . Thus, from ( 2 2 ) ,

Space Tapering of Linear andPlanar Arrays*ROBERT E. WILLEUT, SENIOR M E ~ E R , RE

Summary-The recent activity in electronically scanne d arrayshas stimulated interest in means of reducing both the quantities ofelements equired ora given sizeapertureand henumber ofdifferent types of transmitt ers whichwould benecessary nanarray using an illumination taper. One method of doing this is pre-sented in this paper. The conventional amplitude tapered array issimulated by varying the spacing of equally excited elements, hencethe name space apering. Space apered arrays with predictablegains, beamwidths, and sidelobe levels can readily be designed usinggraphical techniquesand simple mathematics.Reduction n henumber of elements of from 50 to 90 per cent for moderate and largesize planar arrays are possible while retaining good patte rn charac-teristics.Although only linear and circular planar arrays are dis-cussed, he echniquecanbe applied to otherplanarand hreedimensional arrays.

I . I N T R O D U C T I O NARAl-S OF RAD1,lTISG elementshavebeenused since the early davs of radar to secure highantennaain, low sidelobes, andteerableruary 27,1962. The xo rk describedwas performed for the IT. S.* Received .August 4, 1961: rex-ised manuscript received, Feb-Lax--?.,Bureau of Ships, as a part of Contract SObsr-77641.t Bendis Radio Div., Bendis Corporation, Baltimore, N d .

beams. In some instances arrays were preferable to re-flector and leus antenn as since better control over theaperture illumination w as afforded. These early radarshad a single transmitt er and receiver, and the radiat ingelements were either fed b>r a multibranched feed orcoupled o a single ransmission ine.Steering of thebeam was accomplished b! rotating hearra\-orbymechanicall)-varying hephase x-elocit>- n the rans-missionine. I n these pplicationshere M - ; L S neversufficient complexit!- towarran t nvest igation of th ecapabilities ndrade-offsha twere possiblesingarrays with other than uniform spacing.

The equi rement for higherpowers and gainsandfaste r scann ing rates led to the concept of the phasedarra).. The se arra ys? in general, have a separate trans-mit ter and receiverforeachelement or group of ele-ments and are scanned bl- var).ing the phase of the in-dividual elements. I n order o control he sidelobes ofthese argearrays, t isnecessary tha t rans mit tin gtubesoperating a t various power evelsbeused: thiscould also require the use of different type tubes. If the


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370array consists of M rows and A T columns of elements ,then a to ta l of transmit ters an be used. \T.ith anillunlination taper, the tran smi tte rs would be requiredto operate at approximately ?Lf117/4 power evels. Thiscan be thecause of considerable difficultywhen theradar is required ooperateover a broad requencyband.

Byapplying a stepped llumination unction usingfewer than XiV/4 power levels the ar ras ; would still re-quire MLV transmitters,but henumber of differenttype transm itters wouldbereduced. If this process iscarried o tsultimate, a uniformly lluminatedarraywould be obtained; the number of transm itters is stillX A T , but only one )-pe is required.

If equally powered and equal ly spaced elements areused, the irst idelobe level will be - 1 3 . 2 db for asquare or rectangular array and- 7.6 db for a circulararr ay; thus if lower first sidelobes are o be obtainedfrom anar ra y of equal ly powered elements,uniformspacingcannotbeused.With heconventionaluni-formlyspacedarray, hebeamwidthandsidelobes ofth eradiationpatternare controlled byadjusting hediameter of the array and the excitationsf the elementsin the array. This changes the current densit)- over theaper ture to some desired distribution. Another methodof controlling the aperture current densitys to vary thedensi ty or dist ribut ion of uniformlyexcitedelements.Thus i t would appear possible to use equall>-poweredelements if the element densi ty (or spacing) were con-trolled in a suita ble manner. In the irst case an ampli -tude taper is used; in the second case a ((space taperis used.

Considerable ttentionhas been directed ecentlytoward the design and analys is of nonuniformly spacedarral-s. These works have considered variable spacingsin uniformly illuminated linear a rrays and variable spac-ings and amplitudesn linear and p lanar arrays.l-j Someof the echniques would be difficult t o app ly to la rgelinear arraysorplanararr ays of evenmoderate sizesince they require either matrix inversion or would betoo cumbersome to handle conveniently. Others wouldnot be applicable to our situation since they make useof amplitude tapering in addition to variable elementspacing. The method to be presented here requires onlyth e us e of some elementary mathematics when appliedspacmg, IRE TRANS.N X~-T E N~. ~- . ~SN D PROPAGATIOS,ol. .-1P-9,I.R. F. Harrington , Sidelobe reduction by nonuniform elementpp. 187-192; March, 1961.D.D. King, R. F. Packard, ad R . K. Thomas,Unequally-spaced, broad-band antenna arrays, IRE TR~NS.S AXTENNASXDPROPAGATION, VOI. AP-8, pp. 38&384; July, 1960.Electronics Res Lab., University of California, Berkeley, Rept . Ser.H. Unz, Linear Arrays with Arbitrarily Distributed Elements,No. 60, Issue -\J ,o.68; November 2, 1956.DistributedElements,Electronics Res. Lab., University of Cali-H . Lnz, Multi-DimensionalLatticeArrayswithArbitrarilyfornia, Berkeley, Rept. Ser. o . 60, Issue So . 172; December 19,1956.

6 G. Swenson, Jr., and Y .T. Lo, The University of Illinois radiotelescope, IRE Truss. ox A~TEIWASKD PROPAGATION,ol.AP-9, pp. 9-16: Janu ary , 1961.

to linea r arrays. Although the mathematics would be-come more complex when applied to pla nar arra ys, itis possible to avoid this by using a graphical techniquefor the array design.

The determi nat ion of the maximum number of ele-ments in a space tapered array as a function of arraydiameter, element spacing, and illumination is includeI t will be shown that the maximum number is also theoptimum number as regards dupli cation of the desiredillumination function. As the qua nt ity of elements fallsbelow this number, the correspondence between the patern of the space tapered array and the reference pat -tern (desired pat tern) will become worse. A s might beexpected ntuitively, he arger hearrayaperture inwavelengths, hebetter he greementbetween hespace aperedand eferencepatternswhenusing heoptimum number of elements.

The array factor of the space tapered array can bepredicted with accuracy onlyor the main beam and firfew sidelobes. BelFond this region th e ar ray factor willusually contain an approximately constant peak side-lobe level, the level being dependent on the number ofelements used. Because of this sidelobe deterioration, itis necessary to adj us t theesign sidelobe level accordinto the size of the array and the numbe r of elements tobe used i n order to obtain the best pattern. A means ofapproximating hisparticularsidelobe level isgiven.

Since the reduc tion of the number of elements in thearr ay will affect the gain, empirical dataf th e expectedgainof a space aperedarrayarepresented.Experi-ment al da ta also are presented on space tapered arraypatterns.

In th e following sections, design procedures for bothlinear and planar arrays are discussed. The analysis habeen restricted t o cases where a linear phase variationexistsover heaperture,and hediscussion of two-dimensional arrays is limited to circular planar arrays.Thishas been done o simplifycalculations, but hemethod is not restricted o circular arrays. It can beused with any shape planar array and can be extendedwith modifications, to three-d imensional arrays. In allcases, i t is assumed that the array consists of radiat ingelements over a ground plane, ;.e., the array radiatesinto half-space. The discussion will emphasizeheapplication of space tapering to planar arrays. Althouga section on linear arrays is presented, this method hasno distinct advantage over the other methods given inthe referencesexcept th at t iseasier toapply.Themerit of this method comes fromts adaptabi lity to wo-and three-dimensional arrays.

In these sections, the terms reference pattern andreference array will occurrequently.Thepacetapered array is an approximation to an array contain-ing amplitude tapered, uniformly spaced elements; thislatter array is referred to as the reference array andits pat tern as the (reference patter n.


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1962 W i l l e y : Spac e T ape r ing qf L i n e a r and P lanar . I rrays 37 1I I . LIKEARRRAYS

Th emethod of synthesizing a space apered ineararray involves approximat ion of a desired current dis-tribution over the aperture. For an antenna possessinga continuous aperture, thear-field pattern canbe calcu-lated from a knowledge of the cu rrents in the ape rtu re.Th e sidelobe structure and beamwidth of the radiationpatternare controlled bl; the use of anappropriateillumination function which tapers the aperture currentdensity.

In an amplitude tapered array, the current sheet ofthecontinuousaperture is approximatedbydiscretecurren t sources, the elements. The illumination taper isobtained by varying the relative amplitudes of the ele-ments i n the array. For arravs in which the elementsare closelJ- spaced (abou t 0.5 wavelength), there is littledifference between the patterns formed by the continu-ous aperture and by the array aperture.

The method of space tapering a linear arrays similarto th at described in that an approximation to a desiredcontinuous current density is made. However, the ap-proximation is obtained by varying the density of uni-form current sources instead of vary ing the cur ren t ofuniformll- spaced sources. Consider a continuous arm).,117ing along the x axis of a rectangular coordinate sl-s-tem, with an illumination taper given by y=f(s). Theamplitude of the current a t any point is y, and I = J y d xis the otalcurrentassociatedwith hearray . If theareaunder he f ( x ) curve is divided nto a numberof equalsegmentsandequalcurrentsources,or ele-ments, assigned to each segment, then we will have anapproximation to the f(x) curve.

The manner in which theelementsareassigned oeachsegment is import ant.Themostdirectmethodwould be to place the elements at t he mid poin t of eachsegment. However, the illumination taper can vary overeachsegment, hesmaller hearrayor he fewer thenumber of elements, the greater the variation. Hence,some form of weighting should be used. This weightingtakes the following form: the centroid of each segmentis found and projected onto thex axis. This point on theaxis det ermines the location f t he element in the spacetapered arm!-. For arrays containing a large number ofelements, selection of the midpoint of each segment tolocate the element is a good approximation.To illustrate he application of this method, i t wasapplied o an aperture 12.75h long and containing 24elements. -4 reference ar ray having a Taylor illumina-tion for 22-db idelobes was assumed.6Thearrayfactor of thisarray is shown n Fig. 1. Sormallya12.75h aperture would contain 25 elements spaced about

Fig. l-Alrray factor of 21-element linear, space taperedarray w i th -22-db Ta!,lor illumination,

In dealingwithplanararrays hecurrentconceptmay still be used, but t~~ ~o- dim ensi onal ins teadf linecurrentsmustbe considered. The discussion of thespace taper method as applied to planar arral-swill con-sider only circular planar arraps with circularlJ- s!m-metric llumination unctions . t will be ob\.ious th atthemethodcanbeextended ootherplanargeome-tri es, bu t none of them can be halldled with the sameease as the circular array.

Consider an arr ay ying i n the sy plane of a rectangu-la rcoordinatesystem.The llumination aper will begiven by z =f(p), where p ' = x 2 + y 2 . The illtegral off ( p ) over the aperture will repres&-ur_r,es. This

x g r a l s also equivalent to the volume under thej(p)curve. If this volume is now divided into R equal seg-ments, where R s the number of elements i n the spacetapered arra y, and equal owered elements are assignedto each segment, then a n approximat ion will have beenmade to thef(p j curve much a s was done for the lineararray.

At thispoint heproblembecomesmorecomplex.The cause or the additional complexitys the definitionof a volume segment, i.e., what are the shape s of thevarious segments and how are they located with respectto each oth er? The elem ent positions have txvo degreesof freedom: they must be defined b\r a radial distancefrom thecenter of thearral-andbJ -anangular dis-tance from some reference ine in the arr ay (.elementsare defined bycircularcoordinates ather hanCar-tesian coordinates i n order to conform with he circu-Inrlp symmetric llumination unction). To locate heelements represented b\T thevolume egmentsj, hefollowing procedure is used. The array is divided intoannular rings, the width of each ring being equal to th eelement spacing i n the reference arra y. The reason forthis is discussed later. Sext the illunlination function ,f ( p ) , is integrated over each annular ring and over thetotal perture.Elemen ts re hen assigned to eachannular ring , the nulnber of elements per ring being de-fined bs-;

integral over ring number of elements in ring-0.5 wavelengthapart: hus i t is seen t hat pace taDer- integral Over aperture otalnumber of elementsing can also be used for sidelobe reduction. In order to retain the circular symmetry of the refer-lvidth and low side lobes," I R E TRANS.X ANTEXSAS A K D PROPA-T.T. Taylor, "Design of line-source antennas for narrow beam- i l l u m i n a t i o n f u n c t i o n , the elenleIlts should be dis-GATION . vol. -1P-3, pp. 1628 ; January, 1955. tributedniforml\-boutheings. I n detail,hero-


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cedure is to draw a number of equally spaced radialsover he array apertu re, he number of radials obeequal to or greater than the maximum numbe r of ele-ments in any ing,butno t o exceed themaximumquantitS- of elements which is capable of being placedin the outermost ring of the array. The elemen ts arethen distributed about each ringn a s uniform a manneras possible, each element lying on a radial.

After all element positions have been defined in thismanner, their locations can be approximated. For someapplications i t may be desirable to have the elementslocated in the same numberof rows and c olumns a s theelements n he eferencearray.By uperposing hecircle and radial grid containing the elements onto thegrid representing the element locationsof the referencearray, the space tapered array elements cane assignedpositions corresponding to the neares t elemen t positionin the reference arr ay towhich they occur. For relativelylarge arrays, this will produce no significant changes i nthe final pattern.

Thisapproximation is the reasonwhy heannularring width in the space tapered arral- grid as chosen asbeingequal o heelement pacing i n the referencearr ay. If the reference arra). is composed of equiphasedelements spaced less than 1 Xvavelength apart, a mainbeam is formed perpendicular to the array. If t he ele-ment spacing is greater than 1 wavelength, secondar!.maxima will form. These secondart: lobes ma\- also beformed for spacings less than 1 wavelength if the beamis steered too far from boresight.ments m i l l often .exceed.-1. ya ye l~ gt th i u_t bv proper-place-mznt of theelements hegrating obescanbe-- IvoidecL. The pat te rn -of a planar arra y in any p lanethrough the array is the same as th at of an equivalentlinear array . The equi vale nt linear arr ay is the arrayobtained when all elements in the pla nar ar ray ar e pro-jected perpendicularly ontoa line tha t is common to thearray and the pla ne of int erest. If t he linear arr ay thusformed does not have a grating obe, hen he planararr ay will not have a grating obe in that plane. I n Fig. 2is shown asimplified method of placing elements so tha tgrating lobes will be avoided.As an exampleof the application of this method, con-sidera eference arraymeasuring 90 elements in di-ameter, having a row and column separation of 0.54Xand a Taylor illumination taper for -30-db side lobe^.^The tot al num ber of elements will be about 6350. Usingthespace aper echnique, henumber of active ele-ments to be used is 760; this will be referred to as a 1 2per cent space taper (1 2 per cent of the tota l availableelements are utilized).

--n . --..pace tapered array, the spacings betqeenle-.c

7 T. T. Taylor, Design of circular apertu res for narrow beam-width and low sidelobes, I R E T R 4 N S . o s .%~TEK.NASAXD PROP.I-GATIOS, vol. AP-8, pp. 17-22; Jan uar y, 1960.

Fig. 3 represents the space tapered array designed bthis method. The array factor is shown in Fig. 4. A risei n the envelope of the sidelobe peaks is evident here, indicatingeither hatmoreelementsshouldhavebeenused for the -30-dbsidelobedesign or that a highersidelobe design should have been used w ith the 760 ele-ments. This is discussed more fully in Section V .

I I / I

:s%iE

Fig. 2-Method of placing elements to avoid grating lobes.

Fig. 3-760-element space tapered array, -3 0d bTaylor illumination.

Fig. 4-.4rray fac tor of 760-element array,-30-db Taylor illumination.1 1 2 . b h X I M U M X L X B E R OF ELEMENTS

IN A SPACE APEREDRRAYIn a space tapere d a rray that s derived from a refer-

ence arra y using an illumination aperwhichhas amaximum in t h e center of the array, the minimum ac-tive ele,ment spacing will occur at the cen te r. As th enumber of elements increases, the minimum spacing de-creases; onversely, as thenumber of elementsde-creases, theminimumspacing ncreases. I t is usuallydesirable tohaveasmanyelementsas possible,con-sistent with he minimum allowable spacing, n orderthat as accurate an approximat ion as possible be madeto the reference illumination function.

The minimum allowable spacing will be determinedprincipally bl- the physical size of th e element and ,per-haps,by he effects of mutual couplingbetween ele-


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rnents. 1 1 1 this ectionmethod of deternlining hen~a x im un~ermissible number of elements in an ar ray,consistent with the criterion for the n~ in i~num spaci ng ,will be developed. I t \Till beseen that his maxim umnumber of elements will provide the best approximat ionto he desired illumination unction. This method isbased on the fact t hat the i llumination taper can be ex-pressed as a function of its moments. T o illustrate this,consider t he field pa tte rn expression ior a sl-mmetricallinear array of 2i\T+l elements spaced d,'h a m r t :

SE @ )= x A ,, exp j2a- Z sin e .n=-.\- [ x ] (1)

Rewriting (1 ) as

where

we can now express the exponential in ( 2 ) bs- its 1,Jac-laurinexpansion

Substituting ( 3 ) nto ( 2 ) , and rearranging terms,we have

where pi; is the kth momentof the illumination function.sinlilarprocedure is followed todetermine he

monlents of thespace apered llumination unction.Th e field pa tte rn of the space tapered arm>- s

where d, is the di stance from the center of the arra y tothe -rth element. Expressing d, as

d , =Y Jwhere d is the element spacing in the reference ar rayand 7 is not restricted to integral values, (3 ) can be ex-pressed as

Pl i l =y$)"where pR ' is the kth moment of the space tapered llumi-nation function.

To have complete agreement between the uniformlyspaced array and the pace tapered array, t is necessarytha t

Let us now examine the zproth order moments. For theuniformly spaced ar ra y

.\-PO = x -4,;

n=-S

for the space tapered array,Ro

Pal = =Ro.I n order to satisfy the relation O = p 0 ' as defined by (5 ) ,i t is seen th at th e e-roth monlent of the uniformly spacedarray will determine henumber of elements i n thespace tapered arras;.

For the pla nar array, a similar procedure will showthat

m ,'where pOO is the zeroth order moment of the referencearrayand X is thenumber of elements in thespaceLapered array.

For rrays of many lements, uch svery onglinear arrays or planar arrays of moderate size, po maybecalculated b\r a simplermethod.Consider a lineararra y as ying along thex axis of a Cartesian coordinatesystem. They axis will represent the ill umination func-tion. The area under he llumination function is giv-en b>-

-. . . . . .

.4 =sl ( s ) d s (6)where 2a is the length of the linear array.Approximating (6 ) by a summation, we have

This can be regarded a s representing the summation ofthe excitations of a uniformly spaced ar ray of elementswith a spacing of A x . The excitations. y ( x ; ) ,can now beexpressed as -.li. The area under the curve s now

-4 =as :li.1here thenumber of elements in the pace apered


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374 IR E TRAATSACTIONS O X A:\.TELV~VAS A N D PROPAGATION J d YThis expression is equivalent o po multiplied by Ax.Hence ,uo can be expressed as

Ap O = - = R 0Axwhere A x is now the minimum element spacing in thespace tapered array. A , the area under the illuminationcurve,canbe ntegrateddire ctly , if it is ananalpticfunction, or by standard numerical methods.For a planar array, a similar analysis will show that

Vpoo =- RAzAywhere V is the volume under he llumination curve,and A x and Ay represent the minimum allowable rowand column spacings in the space tapered array.

V. DESIGN IDELOBEEVELThe firs t sidelobe level and beam wid th of the space

tapered array are obtained by varying the density ofthe elements over the aperture aspreviously described.However, as can be seen from Fig.4 the sidelobe en-velope ten ds to rise above the level of the first sidelobewhen oo few elementsare used orwhen hedesignsidelobe level is too low. Obviously, there is-some opti-g u m value of first.sidelobe level~whi.&will be primaril).aJg_nction of the. numbe r of elements.

The met hod used to determine the value of the opti-mum designsidelobe evelusessomeapproximations,bu t it will be seen that the results are surprisin gly ac-curate . Fi rs t he expression :optimum sidelobe evelmust be defined. For our purposes, it will be defined asthe designsidelobe evelforwhich no sidelobe n hear ray factor will exceed th e level of the first(design)sidelobe.

It has been show n that the peak sidelobe level of anonuniformly pacedinear arr ay isgiven pproxi-mately by8

L VSL(db) =- 10 log-+10 log 1 --( 2 2where Ni s t he nu mb er f elements in the arra!. and d,,is th e average element spacing. This equation uses thenotation of Honey, but can be rewritten in term s of t hespace taper in the following form:

RSL(db) = - 10 log-+10 log 1 - -( :> ( 7 )where L is the n u n t s n the referencea y nd&is related to the element spacing n th e ref-erence ar ray by .,, ~ f r / : - 0.5-.T- - (,xonSRI Project 1954,Stanford Res. Inst., Stanford, Calif.; May,1961.8 R. C. Honey, et al., Xntenna Design Parameters, Final Report

Eq. (7 ) can be used to appr oxim ate the sidelobe leveof a planar space tapered array in theollowing modifieform :

These equations take into account the rise n sidelobelevel as the average element spacing increases. For in-stance, a 900-element space tapered planar array wit19.9 per cent space taper and K =1 would have a peaksidelobe level of -24.5 db as cal culate d by ( 8 ) . Figs.and 6 show a space tapered array containing 00 activ

Fig. 5-900-element space tapered array,-25-db Taylor illumination.

Fig. 6-Array factor of 900-element arr ay ,-25-db Taylor illumination.

elementsan da ypicalpatterncut.The lluminationused was a Taylor distribution for -25 db first sidelobeand f i =10. The m axim um sidelobe found in four pat-tern cuts was -24 db , whichshowedexcellentagree-mentwith hecalculatedvalue of -24.5 db . If ( 7 )is applied to the linear arrayf Section I , the calculatepeak sidelobe is - 2 3 db; the max imu m sidelobe of t hepattern on Fig. 1 is -22 d b , again showing theexcellentagreement between theoretical and calculated sidelobelevels.

\TI. GAIN F SPACE APEREDRRAYSThe gain of space tapered arrays can be calculat ed

quite easilyonce theelementpatternan dgainar eknown. An element pattern is usually taken to be thefree-space patt ern of a n isolate d element, but n h iscase element pattern means the f ree-space pat tern of anelementsurroundedbyotherelementsexactly as i twould be i n the array. Doing this takes into accountall


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effects oi mutual coupling a s long as there are enoughparasiticelementspresent orepresent accuratelL- theelectrical environment the element \vi11 see i n the array.1 he arraL- gain will then be simplJ- the element gaintimes the number of act ive elements in the ar ray. Sincethe conditions inlplied by the reciprocity theorem havenot been violated. this x i 1 1 be true for either the receiv-ing or transmit ting cond ition. If an element i n an arra)'\\-ere capable oi capturing a l l energ)- ncidentupon tfromanydirection, hen hepattern of such an idealelement would be a cosine unction,since he powerdensity incident on the arraywill be proportioual to thecosine of the incident angle . Since ll energy is captur edthis implies that the element is matched for all anglesi n space. Practically this will not be true, and the ele-ment will generally be mismatched; hon-ever, its possi-ble to obtain match a t a n y one angle.AA t h i s angle, theelement gsin v d l equal the gain of the ideal element. Xtall other angles there will be reflection osses and th eelement gain will be less tha n th at of the ideal element.Thus the variat ion of ana> - gain with scan angle willdependupon how theelement is matched. The arraygain may be a maximum a t boresight and fall off duringscanning at a morerapidrate han hecosine of thescan angle, or t may be less than maximum a t bore-sight and fall off less rapidly than the cosine when theelement is matched a t some angle other than boresight.

l , 7 1 1 . ESPERIMENT.ILESULTS

,.

Inorder o est hearray s)-nthesisprocedurede-scribed previously, a planar array using the space tapertechnique was built . The arraqr measured 24 elementsby 22 elements across the major diameters. The row andcolumnpacingswere 0.500 and 0.557 wavelength,respectively, resulting i n a nearly circular aperture.

The re were 408 elements in the reference arr ay; thespace tapered array conta ined 60 elements, correspond-ing to a 39 per cent space taper. A photograph of thearray is shown in Fig. 7. Fig. 8 shows the locations ofthe active elements in the array. The i llumination usedwas a Taylordistribution or 20-dbsidelobes.Thetheoreticalarray actorandmeasuredpat tern of th earray are shown n Fig. 9. I t can be seen that good cor-relation between the patter ns has een obtained.

hIention might be made here of some factors whichcontr ibuted o he differencesbetween themeasuredand computed patterns. -4 corporate feed was used tofeed the elements for the array. There are 2 %availableoutputs from a corporate feed; since 160 outputs wereto be used, there had to be z8or 256 out put s axd abl e,the unused outputs being terminated in matched loads.Due to the mutual couplingn the arral-, small amountof energy \vi11 be coupled into the corporate feeds. Sincesome of the corporate feed outputs are connec ted to le-ments and some loaded, this coupled energy will see amismatch, be reflected, and reradiate. The mpedance

20m T A W LUlMiNllTlON

Fig. 8-Locatio~~of active elements i n test array.

Fig. 9-Co~nputed a n d nleasured patterns of test array.

looking into each element of the array is not the sameclue to durnm). loads terminating part of each corporatefeed, so the phase of the reradiated energy will differ a tevery element. This will amount essentially to a phaseerror over he face of the array, resulting in a deteri-orated sidelobe structure. 4,'hen this effect is superposeduponhe rrors aused yineengths nd owerdividers, i t is felt tha t th e measured patterns are nottoo different from what was expected.

Pattern measurements were also taken for beam scanangles of 11.5, 23.6 and 53.0degrees romboresight.Thepatterncharacteristics beamwidthand idelobelevel) behaved as \ vas expected: the bearnwidth broad-


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ened by the secant of the scan angle and the averagepeak sidelobe level rose to ab ou t -20 db . Pat ter ns ofthe scanned beam were taken on an array which wasdesigned for -30-dbidelobes.These atternsreshown in Fig. 10. The \TSlA.R beams indicated i n Fig.10werecaused b)? mutual coupled energ!- being e-flected from the junctions of the corporate feeds and re-radiated. Thepositions of these beams were predictable,and a n indication of the mutual coupling can be foundfrom the relative ampli tudes of the main and l-S\TRbeams.

Dummy elemen ts were used t o fill out the array i norder that all active elemen t patte rns would be nearlyidentical.Tests were madewith heactiveelementsonly n he array and with absorbent material i n thevacant spaces, but thegain and sidelobe structure weredefinitely better with the dummy elements present.

(e)Fig. 10-Patterns of scanned beams. (a ) H plane boresight pattern.(b) E plane boresight pattern. (c ) Pattern for 11.5 scan angle.(d) Pattern for 23.6 scan angle. (e) Pattern for 53.0 scan angle.

1:III. COXCLL~SIOKSX simplified theor). of space tape red arr ays has been

presented along with methods of designing arra ys for agiven ain, eamwidth, ndidelobe level. Experi-mentaldatahave shown thatpredictable esultsarepossible. B y using the results presen ted, array param-eterscanbechosenwith ittleeffort. Th e design oflinear and planar arrays can e achieved using graphicaltechniques. Other techniques, such as combining spaceand amplitude tapering,j should offer even better corre-lation between space tapered and reference array pat-terns.

The experimental data have also shown that beamsfromspace aperedarrayscanbescanned hrough alarge angle without serious deteriorationof the pattern.The beamwidth varied as array theor y would predict,and the increase nsidelobe evel salsopredicted byarray theory.

Space tapering can also permit another innovation inantenna design. I t allows separate transmittin g and re-ceiving elements to be placed in a single aperture. Obvously the sumof the tw o space taper s canno txceed 100per cent. Other han his he only consideration hatneed be made is that the two arrays mesh withou te-quiring an excessive amount of elementrelocation sotha t no transm itti ng and receiving element occupy th esame elenlent position. Fig. 11 shows an array- in whichthis has been done successfully. This technique can beused where space considerations do not permit separatantennas or where igh owerequirements wouldmake duplexing undesirable. Although some type of re-ceiver protection may still be necessary, i t would be atreduced owerevels, the power requi rementbeinglessened by a n amount equal to the element couplingfactor.

Fig. 11-Combined transmit and receive arrays.


9/9

Space tapering call also be extended to planar array sof other shapes and to three-dilnensional arrays such ascylinders an d sp!leres. 111 drawing the annular ring gridwhich was used to locate the elements, we were essen-tiall>- making a contour plot of the llumination f u n c -tion over the aperture. This same procedure can he fol-lowed for otherplanararra>.sand llunlination f u n c -tions. Obviousl\-, howe\ler, the construct ion of the con-tours will not be as str;lightfornmd as for the circular;ma>-vir11 a circular1)- s>'mmetric illunli~~ation function.

\\Then ;Ippl>.ing thismethod to three-dimensionalarra!.s, theonlyextrastep equired is toproject heelements i n the reference arra y ont oa plane that is tan-gent to the center of the arra?.; this will give a "new"

referellce a rray with which to work. Since the new refer-ence arra?; is planar, the procedure from this point on!vi11 be the same as t h a t described previously.

IS. ~ ( . I ~ ~ o \ ~ - L ~ ; I ~ G ~ ~ I ~ ~ T'I'he author is indebted to many who aided i n the

gathering of data for this report: to \Y. chneider, whoperfomled man!.of the calculations and made severalsignificant contributions; to J . Best and H. Dantzig,who offered mall>-helpful uggestions: to L. Ehudin,wh o performed the graphical design of the nrral-s; andto E. Reed, X. Hart , H. R y n a n , E. Crizer, a n d R . Sel-lcrs, who \\-ere responsible for the construction and test-ing of the te st arr ay .

A Quasi-"Isotropic" Antenna in the Microwave Spectrum"

Summary-Thegeneralproblem of radiation by apertures onspherical urfaceswas first considered theoretically by Strat tonand Chu in 1941.' More recently, the problem was again consideredby Bailin and Silver,? and Mushi ake and Webster.3

Al l numericalandexperimental efforts to datehavebeenre-stricted to the special case of a very small sphere as measur ed inwavelengths. This paper is in effect an extens ion of the previousresul ts to the case of large sphere s uniformly excited by an equa-torial slot of varying size. The detailed numerical and exp erimentalresults are restricted to the fi s t TM and TE modes.In general, t s possible oshow that he arge spherical slotantenna can be made unusually "isotropic" with a far-field coveragefactor of over 98 per cent.

If the antenna s to be characterized by a n ornnidirec-tional broad-beam radiation pattern, i t is eviden t thatthesupportingstructuremightbespherical i n shape.hope full^., antennas of this kind c o u l d be made quasi-"isotropic" b\. properl~r exciting he spherical shell.

I he generalproblem oi radiation t)!, apertures onspherical surfaces was first considered heoreticall>- bStratton and C h u i n 1941.' >lo re recentl?., the problem\vas again considered b,. L3ailin and SillTer i n 19.56? andby i'ebster and 1Iushinke i n ls) j i .3

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Documents

Space Tapering of Linear and Planar Arrays