comprehensive analysis for e-plane of horn ant

7/27/2019 Comprehensive Analysis for E-Plane of Horn Ant

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EEE TRANSACIIOKSN AXTEKSAS AND PROPAGATION VOL. AP-14,KO. 9

Comprehensive Analysis for E-Plane of HornAntennas by Edge Diffraction Theorv

Absfracf-Edge diffraction theory is used in analyzing the radia-characteristics of typical horn an tennas . The far-sidelobe andacklobe radiation has been solved without employing field equiv-

alence principles which are impractical in the problem.A corner reflector with a magnetic line source ocated at theis proposed as a model for the principal E-plane radiation ofA complete pattern, including multiple interactions

and images of induced line sources, is obtained in intinite seriesom . Diffraction mechanisms are used for appropriate approxima-ions in the computations. The computed patterns are in excellent

agreement with measured patte rns of typical horn antennas. Radia-ion intensity of the backlobe relative to mainlobe intensity is ob-ained as a back-to-front ratio and plotted as a function of antenna

dimensions.

A I. INTRODUCTIOXNALYTICALLY, hedescr iptions of theprop-agatingmodes nahornare ummarizedbyKraus [ l ] . n general, aperture techniques mustbe used to calculate the radiation pattern, and it is as-sumed that the aperture distribution is tha t of the in-cident wave and is zero outside the aperture. The pat-terns husobtainedgivesatisfactorymain obesandnear sidelobes. As forar idelobes ndbacklobes,Schelkunoffsequivalenceprinciples could beappliedif currentdistributions on theouter urfaces wereknown.For a hornantenna,extreme difficulty s in-volved naccuratelydescribing hecurrentdistribu-tions. Furthermore, difficulty would arise in evaluatingthe consequent surface integrals. The impracticality ofthe equivalence principles had left radiation problemsof the far sidelobes and backlobes still unsolved untiledge diffraction techniques were applied [ 5 ] .

In 1962, Ye Kinber [ 2 ] derived horn patterns and thecoefficient of coupling between two adjacent horns bydiffraction theow. Examples are given for both E-planeandH-planepatterns, inwhich discontinuities repointed out, and emphasis has not been made on sideand backlobes. In 1963, Ohba [3 ] used diffractiontheory to compute the radiation pattern n the N-planeof a corner reflector. Disagreement was noted as a re-sul t of neglecting contributions from the other edges ofthecorner eflector.Recently,Russo,Rudduck,andPeters [j] employeddgeiffractionheorywith

The research reported herein was supported in part by Contract AFManuscript received August 12, 1965; revised November 4, 965.30(602)-3269 between Air Force Systems Command, Research an dTechnology Division, Rome Air Development Center, Griffiss AirForce Base, New York, and heAntenna Laboratory, The OhioStat e University Research Foundation, Columbus, Ohio, Project No.li67.The authors arewith the Antenna Laboratory, Dept.of ElectricalEngineering, The Ohio State University, Columbus, Ohio.

MARCH, 1966

J

AND L. PETERS, p. ,ENIOR ~ M B E R ,EEEproperassumptions oobtainE-planepatterns of athin-edged and a thick-edged horn. Only the first-orderdiffraction terms were used to compute the thin-edgedhorn patterns. The results, n-ith possible discontinuitiesleft in the side and backlobes, are in good agreementn-ith the measured pattern.Even hough hehigher-orderdiffraction ermsand he reflections nside thehorn are neglected, the combination of the employedconcepts and assumptions constitutes a new method ofanalysisforhornantennas. IVe shall follow this newmethod to develop a more complete analysis by includ-ing hepreviously neglectedhigher-orderdiffractionsand the reflections inside the horn antenna.

11. RADIATIONIE C H A K ISM STh e proposed antenna model used inRusso,Rud-

duck, and Peters [S I was a corner reflector formed bytwo perfectly conducting plane walls intersecting at anangle 283 a s shown n Fig. 1. If we let the corner re-flector be nfinite n extent along he z-direction, heproblem is thus reduced to a two-dimensional one. T heprimary source is a magnetic line current assumed a tthe vertex S. This assumption considerably simplifiestreatment in the principal E-planeof a horn antenna edby a waveguide supporting the Elo mode.The angular coordinate, shown in Fig. 1, is the com-mon reference angle. Th e angles 4 are the field anglesreferred to each individual wedge at xhic h diffractionoccurs. The ang les a are called incidentangles of il-lumination. Al l the first subscripts refer to the points a twhichdiffractionoccurs,while the second subscriptsrefer to the pointsof origin of incident rays. This nota-tion will be used th roughout the following discussions.There are fourwedges ( A ,B , S, and W ) o be treated bydiffraction heory. IjTedges A and B have zero wedgeangle, whilewedges S and W have wedge angles of20E and 2 ( r - e E ) , respectively. T he propert y of sym-metry of th e reflector will be used t o simplify the p rob-lem by considering only the upper half of the patte rn.

From the assumed magnetic line source a t S , a uni-form cylindrical wave is radiated in the region - 8 E < 858,. This uniform cylindrical wave s called th e geo-metrical optics wave which illuminates wedges A andB. The diffractionsa t A and B caused by thi s illumina-tionare he irst-orderdiffractions, which are direc-tional cylindrical waves radiated from the wedges. Th egeometrical optics rays from S and the first-order dif-fracted rays from A and B are shown in Fig. 2(a). Th efirst-order radiation pattern in the far-field can now be

138


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W ET U.: -PLLNE OF HORN AXTEhWAS 139FAR-FIELD

x P

L*JWFig. 1. Corner reflector.

V *

I - /.Fig. 2. Radiation mechanism of the antenna model. (a) Direct ragsand the irst-order diffracted rays due tollimination from primarysource at S. (b) The second-order diffractions due t o the first-

due to the first-order diffraction at A . (d) The second-order dif-order illumination from B. (c) The first image in the lower wallfraction due to the first image in the lower wall.

obtained by superposition of t he far-field intensiti es ofthe primary source and the two induced sources. Th efirst-order nalysis spresented nRusso,Rudduck,andPeters [SI .

To consider the diffraction process further, one canobservefromFig.2(b) th at th e inducedsource a t Billuminates wedges A , S , and T.t. to give three second-order diffraction terms. In the same manner, the first-order-induced source a t A illuminates wedges B , S, andW to give hreemoresecond-orderdiffractedwaves.These six second-order-induced sources will continue togive hird-andhigher-orderdiffraction. The inducedintensitybecomessmallerwith ncreasingorder,andthe phase delay of successive illumination can be prop-erly taken into account.

Since the eflector ha s perfectly conducting walls, thediffracted waves fromA and B are part ially reflected bythe walls. Some of the first-order diffracted rays from Aare reflected by the lower wall, as shown in Fig. 2(c) .

Th e reflected rays can be described by the method ofimages.

Figure 2(d) shows th at wedge A is illuminated by oneof the first-order images from t he lower wall. Th e nu m-ber of images is determined by the flare angle of thereflector. T he effects of the reflector walls can then betaken into account by the images and the subsequentdiffraction of t he images.

When he process of diffraction and reflection de-scribed above is completed, the far-field patt erns of thereflector antenna can be obtained by superimposing thcontributions rom heprimarysource a t S; the n-ducedsources a t -4, , S , and W ; and the images nboth walls. Formulation of th epattern, includingallorders of diffraction and reflection, is obtained in Sec-tion 111.

111. FORMULATION OF SOLUTION I NINFINITE SERIES FORM

In his ormulation hediffraction of a cylindricalwave by a perfect ly conducting wedge is employed [4],[SI . Paulis ormulation [ 6 ] of Sommerfeldssolution[7] to wedge diffraction of a plane wave permits the diffractedwave obe expressed in te rms of Fresnel n-tegrals. The far-zonediffractedwaves, both he in-cident and reflected terms, nducedby auniformcy-lindrical wave of unit intensity from a line source, canbe written as

+ ( igher-order terms negligiblefor large k p or FZ very close to 2with

a = 1+ cos f # A ~ . (2)The propagation factor I?-/* Exp ( - j k X ) where R isthe dis tance of the far-field point from the edge is sup-pressed in (1). The quantity p is the distance from t heedge to the source point. The ang les i ~ rre the diffrac-tion-fieldangles of the nciden tand reflected terms,respectively. The value of n is obtained by setting thewedge angle equal to 2 - )n.The solution JBis a direc-tional cylindrical wave radiated from he edge of thewedge. It may , therefore, be considered th at a direc-tional line source is induced a t t he edge of the wedge.The conce pt of induced line source is good in generalexcept when n is qui te different from 2 and Kp is verysmall.


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IEEE TRANSACTIONS ON ANTENNASNDROPAGATION MARCHThe method of images is also needed in the following

n. For our antenna model, he reflection ofmalls can be described by image-waves [4].

ipt ions are stric tly of geometrical considera-?e shall use only the results for th is paper.

First,referring to Figs. 1 and 2(a), he geometricalradiated from he primary source S is a

aso * (e ) = I, -e , 5 e 5 + BE, (3 )

v* has pointS as phase-reference, and outside the* is identically ero. The cylindrical-waveR-*/2Ex p ( - j k R ) to the far-field is

( 3 ) because only the angular dependenceof interest.Wedges A an d B are illuminated by the cylindrical

S with zero incident-angle. Since there is .nodiffracted waves fromA and B have

Excluding he port ion of wavesdiffracted nto heas

A P = Z B ( ~ E , - E + e, 2), -- i P+ e,), (4a)P2(1) = 7rBS C B ( ~ E ,P - E - , 2 ) ,- 8 2 - (P+@E),4b)2 -DAs and DES designate diffraction a t A and B

of illumination rom 5'. The uperscript (1)first-order diffraction. Th e expressions of 4 ' s in

ms of f? can be obtained from Fig. 1.The argument zs equal to 2 for both A and B because they have zeroangle. I t shouldbenoted that the notati on z B

lows the form of the original solution and the sub-B has no connection with thewedge B.Th e first-order radiation patterns, neglecting the re-

flections inside the corner reflector, canow be obtainedy simply superimposing the terms in (3) and (4). Theiscontinuities in v* a t f?=_+BE in (3) are eliminated byAs ( I )and DES" ) , espectivel>-. 4ltho ugh the patter n is

continuous a t f?= fdE, two sets of new discontinuitiesa t 0 = f7r-/2 and (a+dE)are observed n(4).There-fore, he irst-orderpattern, ngeneral,hasdiscon-tinuities a t these directions.

Let us next examine the reflections of the first-orderrays diffracted into he reflector.Since hediffractedwaves from A and B are symmetrical with respect to8 = 0, ascanbe seen rom (4), he reatment of theimage-waves from th e reflector can be simplified. Theimages are formed symmetrically in theower and upperwalls. The image waves from the two walls can be ob-tainedby replacing f? of DAs( l ) nd DBs") in (4) by( - 2iBE - 6 ) and (2iBE-f?), respectively, giving( f L ( * ) ) i = "B ( pE , 7r - (2i+ 1)eE - 8, 2) ,P P- -2 (i + l)e, I5 - 8,; (Sa)2

i = 1, 2, 3 . . . h ; ( jb )and

h(the largest integer) 5-28EP

where the subscripts L and U ndicate that the imageterms are from the lower and the upp er walls, respec-tively . The number of images in each wall is equal tothe argest nteger h which s ess than orequal on/2f?~.N?hen the ratio 7r/28E is not an integer, the alidregion for the last images should be modified toP--2 (h+ )OE 5 0 5 T - 2 h+ 1)OE for the ower wall,and

for the upper wall.Each term in (5) has its properly defined regions an d isset zero outside the region. The first image in the lowerwall, caused by the first-order-diffracted rays from A ,is shown in Fig. 2(c). Figure 3 shows the images in theIon-er wall for h = 4 . It is .noted hat he rue mag ewaves of the diffracted waves from A are those with iodd in the lower wall and i even in the upperwall.

Descr iptions of the first-order diffracted waves from-4an d B and the ir reflected waves have been completedabo ve. The higher-order terms to be treated in the fol-loxx*ing discussion are necessary for cases in which smalldimensions are encountered orhigh accuracy is desired.Physicalll-, the higher-order terms describe the effectsof illumination of edges by he lower-order-inducedsources and their images. llathemati cally, they are re-quired to overcome thediscontinuit ies of the lower-order erms n he radiation pattern. Taking he wofirst-orderdiffraction erms n 4), or nstance, hediscontinuities mentioned earlier can only be eliminatedby tak ing into accoun t the second-order diffraction inthe specified directions.

At f? =7r/2, 11-edgeA is illuminated by the first-order-induced source a t B in (4b). This intensity of illumina-tion from B to A is called t he first-order coupling co-efficient,

as shown in Fig. 2(b). Because of symmetry, the first-order coupling coefficient from A to B in the direction,g=- /2 is equal t o CAB(^). Therefore, using (4), wehave


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1966 W ET AL.: E-PLANE OF HOR? ANTENNAS 141. 9eE= n

Fig. 3. Fou r images in th e lower wall.

Similarly , wedge W islluminatedyoth A and B at , i = 1 , 2 , 3 . . . ( 1 ~ - 1 ) ,0= f r + d E ) , respectively. T he couplingcoefficientscan be obtained in the same manner as a t which the wedges A and B are illuminated by the

ar e slowly var).ing f u n c - tively. Figure 3 shows the geometry of the images withtions in the neighborhood of a certain angle, it is a good h=4 in the lower wall. The coupling coefficients fromnatedy uniformylindricalaves of thentensities ?I-ond-order diffracted waves can be obtained as

CWA ( l ) = CWB ' " = z B ( p ~ , ?1-,2) . (7) raysrommages of lower and pper walls,espec-since the diffracted

approximation that wedges A , B , and iv ar e illumi- the images to \\'edge A can be obtained from (5) asshownn ( 6 ) and (7). Underhisssumption,he set- C A , ' ~ ) I L " ) , i = 1, 2, . * * , 12 - 1).

+ zlB b ,- 2eE + e, 2 ,(: )By symmet ry, the coupling coefficients from the imagesin the upper wall to wedge B are equal to C-l i ( l )as

i = 1 , 2 , . . . (12 - 1). (9)( s a ) The second-order diffraction a t A and B illuminated by

the images cannow be written as

and28 En w = 2 - -

37r1- VE i ( i,- (i+ 2 )eE - , 22 )


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142 IEEE TIWNSACTIONS ON ANTEXNAS AND PROPAGATION MARCHand

where the arguments can be obtained by using Fig. 3as reference. The second-orderdiffraction terms ob-tained in (10) are appropriately arranged for each boun-da ry of the defined regions in ( S ) , except t hat the la stboundary is given by 8= f 7r/2 - h+ 1)eE) if T/28E isan integer, or 8 = -t (r-(2h+l)BE) if a/289 is not aninteger. The boundarie s, in either case, correspond othe directions in xvhich wedge S is illuminated by theinducedsources a t A and B. Therefore, hecouplingcoefficients from A and B to S can be obtained from(5) by symmetry as

CSA) = CSB = *dB(PE,0, 2), (11)which gives rise to th e econd-order diffraction a t wedgeS as

and(12c)

Now, me have completed the descript ions of all sec-ond-orderdiffractions whichphysically take ntoac-count he effects of illumination by he first-order-inducedsources and mathematically eliminate all thefirst-order iscontinuities.Summing phe econd-order diffraction a t A , B , W , nd S gives

i= 1

i= l

where (8), ( lo) , and (12) can be used for computation.Following the same procedure used to obtai n (S), thesecond-order magewaves rom he lower and upperwalls can be obtained s

and

where the 8 of DA(?)nd D B ( ~ )n (13) have been replacedby (T 2 8 E - 8 ) , respectively. Kot e tha t the bou ndaryof t he l as t image i=k should follow (4), if 7r/2BE is n otexac tly an integer.

lye hav e observedabove th a t while the first-orderdiscontinui ties are eliminated by the inclusion of sec-ond-order diffractions, new discontinuities occur againat the boundariesf the regions defined in (13) and (14).Thesesecond-orderdiscontinuitiescanbeeliminatedonly by introducing third-order diffraction. The higherthe orderof diffraction the smaller will be the magni tudeof discontinuities. I t is heoretically possible to con-sider theorder of diffraction as high as desired. Inother words, the magnitude of each discontinuity canbe made negligibly small if sufficient orders of diffrac-tion are included. For completeness, the ite rat ive for-mulas for all possible higher-order diffractions ar e pre-sented in the Appendix.

Th e to ta l far-field pat tern of the corner reflector cannow be obtained by superposition f all terms presentedin the -Appendix. Taking wedge A as a common phasereference, and considering only the upper-half region,058_


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1966 W ET AL.: E-PLANE OF HORN ANTENNAS 143

D s ( ~ ) A S + D A ( ~ )m 1 c,, 1a

i=l

f [ =l I r ' " ) ) h ] y B h y A B , ( 1 3where the ast mages n he upper wall ar e includedbecause n general they may contr ibute to the upper -half region. Th e local phase factors refe rred to A canbe written fromFigs. 1and 3 as

yas = EXP. [ - j2 . rrpE COS (-ez + e ) ] ,y - 4 ~ Exp. [ - j 2ab sin e ] ,)JAW = Exp. [ - j 2 7 r p ~ COS (0, - ) ] = Y A S ,y A i = Exp. [ j 2 n p i sin (io, + e ) ] , (16)

andy B h = Exp. [- j2rp i sin ( M E - e ) ] .

11,'. THE PPROXIMATED SOLUTIOKSFor the idealized corner reflector in Fig. 1, the pat-

tern can generally be calculated by (15) as accuratelyas desired. The fundamen tal limitation of the edge dif-fraction method s he approximation of the mult iplediffraction as omnidirectional ine ources.However,this limitation will rarely be encountered for the sym-metricalcorner eflector of typicaldimensions.Sincethe contributions of the higher-order terms to th e pa t-tern decrease with increasing order, computations maybe made by including only those terms which are sig-nificant in their defined regions. The following approxi-matio ns are made to obtain a patter n includingonlysignificant higher-order terms.

The total contr ibution from wedge S is given by thefirst term of (15) which contains diffraction terms equalor higher than the second-order. For typical dimensions,thesediffraction ermsare negligibly mall as com-pared to v* of unit intensity. Therefore, for ll practicalpurposes, the first term of (15) may simply be approxi-mated by v* alone. In the following three terms in (15),the second-order diffraction should be retained for cal-culation because their regions include side- and back-lobes nwhich o* is absent. The mage erms n (15)contr ibute both to the regions of main- and sidelobes.Therefore, approximation of images should, in theory,be made individually. We shall instead treat the imagesof the same order as a group and assume that only thefirst-order mages have significant contribution o hepattern.

Based on the appro ximat ions mentioned above hepat tern of the corner reflector can now be obtained ap-proximately from (15) as:

L i= 1

+++

Since this is an approximated pattern, discontinuitiesare expected tobe increasinglynoticeablewithde-creasingsize of the corner eflector. Let u s examine,term by term, the cont inuity of (17) in the upper-halfregion O


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144 B E E TRANSACTIONS ON AhTEhnTASAND PROPAGATION MARCH

where the property of symmet ry is used an d C A B A andCAB^ are given from (18). Th e cont inui ty of the tota l

pattern a t 8 =a/2 is now ensured by using C A B insteadof CAB(1) for DAB(?)nd D,A@) in (17) as

an d

I "F0_


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1966 kTJ ET AL.: EPLAATE OF HORN LLWEATNAS 145When he requency s ncreasedcorresponding to

pE=24.8X for the same horn, the three patterns are asshown in Fig. 6. The same conclusions drawn for Fig. 5remain true, except that the interference from the wave-guide and the associated structure becomes larger be-cause the physical size is larger in terms of wavelength.I n Fig. 7 , threepatternsareshownfor a smallhornantenna of pE=5.61X and 28E=21.2" . A41though theoverall lobe structure is till in good agreement with themeasured pattern, a larger deviation in intensityevel isobserved around 0 = 80" of the pattern by 2 2 ) . This dis-agreement results because the second-order image termsneglected are no t egligibly small for mall horns. There-

fore, better patterns can be obtained for small horns byincluding the second-order image terms and their sub-sequent effects on the total pattern.

The hreeexamplespresentedabovehavedemon-strated the accuracy f ( 2 2 ) for pattern computationsoftypicalhornantennas.Theaccuracy of th e experi-mental measurements is assumed to have 1 dB fluctua-tionwhen he ntensity saround 40 dB below thereference intensity. In view of this , ( 2 2 ) is sufficient forhor n antennas of typical dimensions. When p E and eEbecome smaller, t is easily observed from Fig.7 that thesecond-order image terms in (14) should be included t oensure good prediction around the region0 = 9O0-0E.

CALIBRATED ATTENUATOR

Fig. 4. The E-plane of a horn antenna.

A

__PC : 13 5 xe,: 1750

- E A S U R E DA R - F I E L DA T T E 9 N- -- C O M PU T EDP A T T E R N B Y EO 2 2 ( - 5 d b ) -OMPUTED PATTERN B Y RE F E RE NCE 5 ( + 5 d b l -

1 d b l .20 -- _--- 33--- 4 0 . -

C 3 0 90 12 0 I50 1801 e :

Fig. 5 A comparison of patterns.


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146 IEEE TRAKTSAlcTIONSON AhpTEhWAS AhTl PROPAGATION

- 30

- 40

- E A S U R E OAR - F IEL OAT T ER N- - _ _ OMPUTEDATTERN BY EO. 22. - 5 d b l- . - OMPUTE0 ATTERN B Y REFERENCE 5. l t 5 d b l

43

0 . -60 t e I 90 120 I5 0 I80Fig. 6. A comparison of patterns.

pE ' 5 6 1 A -eE = 10.6"M E A S U R E 0A R - F I E L OA T T E R NC O M PU T EDP A T T E R NBY EQ 2 2 ( - 5 d b l-. - C O M P U T E DP A T T E R NB YR E F E R E N C E 5 ( + 5 d b k -

--

( d b )-20--30-- 1

/ff4 - - x p \ - d', /-

' /I 1 \ 1 1I

J-40- -

0 30 60 90 120 150( e )

Fig. 7. A comparison of patterns.


10/12

1966

( d b

#EJ'Fig. 8. Back-to-front ratios of th e antenna model.

VI. RELATIVE ACK EVELSThe radiation patterns, eithermeasured or computed

alwayshaveabacklobemaximum at B = 180, eventhough this maximum value is not necessarily the largestmaximum in the region 18Oo- -B~


11/12

148 IEEE TRANSACTIONS ON ANTEhTAS AND PROPAGATIONT he sets of p~ an d BE giving rise to the minima n the

cur ves do no t necessarily imp ly tha t u(0) is maximumand that U ( T ) is minimum. The reason for this is themain lobeof the patternbegins to bifurcate t the pointswhere minimum back-to-front ratios occur. For a hornantenna with 0 0=25", and PE = 6 3 , 17X, 2 i h , 38X, and48X, where minima take place in Fig. 8, the main beamof the patte rn is split into two, four, six, eight, and tenlobes, respectively. However, ig. 8 can beused to evalu-ate approximately the representative back-radiation in-tensity zt(7r) relative o front-radiation ntensity ~ ( 0 ) .If the horn antenna is large enough, it is generally safeto expect that the radiation intensity, on the average inthe region 90"


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IEEE TRSNS.4CTIONS ON 4XTENNAS AN D PROPAGATION

used toobtainsymmetrical erms.Thecouplingco-efficients of different orders can also be formulated ininfiniteseries orm [4].The coupling coefficients areobtained as

CAB(m)= CBA(m)= D,(m, e =-3CFA(m)= D,(m) (e = 5T + BE )

S A ( e = - e E)( m ) = DA4(m)

C ( m ) = Dw(m)C ( m ) = DS(m)A T (e = e E )A S (8 = e E ) (25)

in which th e process of itera tion may beused t o includeasmanyorders of diffraction as desired. As soon ascoupling coefficients areproperlyevaluated, hedif-fraction of any order can then be obta ined by makinguse of (23) and (24) .

VOL. AP-14,NO. 2 WCH, 1966

ACKNOWLEDGMEKTThanks are due to . H . Davis for the measurements

and M. L. Tripp for the computations.REFERENCES

J . D. Kraus, Antennas. New York: McGraw-Hill, 1950, pp. 375-380.Radio Engrg. Electron. Phys., vol. 7-10, pp. 1620-1632, OctoberB. Ye Kinber, Diffraction at the open end of a sectoral horn,4 nr?Y.Ohba, On the radiation patte rn of a corner reflector finite inwidth, IEEE Trans. on Antennas and Propagation, vol. AP-11,J . S. Yu and R. C. Rudduck, The E-plane radiation patte rn of anpp. 127-132, March 1963.antenna model for horn antennas, Antenna Lab., The OhioState University Research Foundation, Columbus, Rept. 1767-3,April 1, 1965, prepared under contract AF 30(602)-3269, RomeAlr Development Center, Griffiss Air Force Base, N. Y .computing E-plane pattern of horn antennas, IEEE Trans. mP. M. Russo, R. C. Rudduck, and L. Peters, Jr., A method forAntennas and Propagation, vol. AP-13, pp. 219-224, March 1965.\fT. Pauli, On asymptotic series for functions in the theory ofdiffraction of light, Phys. Rev., vol. 54, pp. 924-931, December1938.A. Sommerfeld, Optics. New York: Academic, 1954, pp. 245-265.H. Jasik, Antenna EngineeyingHandbook. New York: McGraw-Hill, 1961, ch. 10, p. 8.

IYUL.

Numerical Solutions for an Infinite Phased Array of RectangularWaveguides with Thick Walls

Abstract-A numerical analysis ofan infinite phased array ofopenrectangular waveguides has beenmade which includes theeffects of wall thickness. Two planes of scan, the H and quasi-Eplanes, have been considered. In these cases, the general vectorproblem can be expressed in the form of a scalar one-dimensionalFredholm integral equation of the first kind. The approxmate fieldsobtained numerically from the integral equation are used for theevaluation of the input complex reflection coefficient. A variationalexpression for the reflection coefficient is developed and used forimproving the accuracy. Numerical results for the H and quasi4plane scans are presented as a function of wall thickness and scanangle. Agreement with experimental results is very good.

1 INTRODUCTIOSIX LARGE planarphasedarrays, he nput m-pedance to a given element varies with scan angle.Thi s is due to mutual coupling between elements[ l ] . Since he ouplingbetween lementsdecreasesrapidlywithdistance as l / r 2 for ectangularwave-guide arrays [l ] ) , the assumption of an infinite arr ay isvalid for centrally located elements. With this assump-tion, a large class of such problems becomes tractable

by the U. S. Army under contract DA-30-069-AMC-333(Y).pany, N. .

Manuscript received October 18, 1965. This work was supportedThe authors are with Bell Telephone Laboratories, Inc., Whip-

analytically. I n particular, for those cases which can bereduced to a one-dimensional problem, the analytic for-mulations for the tangential aperture fields and inputimpedancearereadilysolvedby use of a high-speeddigital computer.A numerical analysis of infinite phased ar rays of rec-tangular waveguides, as shown in Fig. 1, was made. 4Fredholm ntegralequation of the irstkind or heaperture field becomesscalarone-dimensional n wospecialcases. Th e firstcasehas c = d withscanningtaking place in the H-plane ( X - 2 plane). In the othercase, a =b and the fields vary sinusoidally with period2b in X. nder this condition, scanning in the E-planeis called quas i-E plane scanning. When the waveguidewalls are thin, =b and c =d , these two cases have exactsolutions [ l ] . However,evenamoderateguide wallthickness does have an appreciable effect on the resul tsfor the reflectioncoefficient. Apertureand near-fieldsolutions for these cases, when the walls are thick , areobtained numerically as a function of scan angle andwall thickness in the respective planes of scan.

The approximate ield solutions that are obtain ed nu-merically from the integral equat ion are thensed in theevaluation of a variational expression for the complex

comprehensive analysis for e-plane of horn ant

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