The Design of Small Slot Arrays

Transcription

1 214 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-26, NO. 2, MARCH ) The 75 percent BW element design over qurter-hemisphere mximum VSWR vlue of 16, nd consequently, to be prcticl, will require n elborte mtching network. 4) It is pprent tht, with creful perture design, dul ridge rectngulr wveguide offers prcticl solution for wide-bnd phsed rry element. This conclusion is confirmed by the results of [ 71. 5) The present study does not consider the question of polriztion. A minor ddition to the computer progrm will yield the xil rtios nd the tilt ngles. REFERENCES [ 11 S. B. Cohn, Properties of ridge wve guide, in Proc. IRE, pp , Aug [2] S. Hopfer, The design of ridged wveguides, IRE Trns. Microwve Theory Tech., pp , Oct [3] J. R.Pyle, The cutoffwvelengthof TElo modeinridged rectngulr wveguide of ny spect rtio, ZEEE Duns. Microwve Theory Tech., pp , Apr [4] W. J. Getsinger, Ridge wveguide field description nd ppliction to directionl couplers, IRE Trns. Microwve Theory Tech., vol. 10, pp , Jn [5] J. P. Montgomery, On the complet eigenvlue solution of ridgedwveguide, IEEE Trns. Microwve Theory Tech., pp , June S. S. Wng, WideAngle wide-bnd elements for phsed rrys, Ph.D. disserttion, Polytechnic Institute of New York, Frmingdle, NY, June [71 J. P. Montgomery, Ridged wveguide-phsed rry elements, IEEE Trns. Antenns Propgt., vol.ap-24,pp.46-53,jn G. V. Borgiotti, Modl nlysis of periodic plnr phsed rrys of pertures, Proc. ZEEE, vol. 56, pp , Nov G. N. Tsndouls, Wide-bnd Limittions of Wveguide Arrys, Microwve J., pp , Sept H. Altschuler nd L. Goldstone, On network representtions of certin obstcles in wveguide regions, IRE Trns. Microwve Theory Tech., pp , Apr N. Mrcuvitz, Wveguide Hndbook. NewYork:Dover, 1959, p. 57, pp J. C. Slter, Microwve Electronics. Princeton, NJ: D. Vn Nostrnd, 1959, pp L. R. Lewis nd A. Hessel, Propgtion chrcteristics of periodic rrys of dielectric slbs, ZEEE Trns. Microwve Theory Tech., pp , Feb B. L.Dimond nd G. H. Knittel: A new procedure for the design of wveguide element for phsed rry ntenn, in Phsed Arry Antenns, A. A. Oliner nd G. H. Knittel, Eds. Dedhm, MA: Artech House, 1972, pp B. L. Dimond, A generlized pproch to nlysis of infinite plnr rry ntenns, Proc. IEEE, pp Nov G. F. Frrell nd D. H. Kuhn Mutul coupling in infinite plnr rrysofrectngulrwveguidehorns IEEE Trns. Antenn Propgt., vol. AP-16, pp , July The Design of Smll Slot Arrys ROBERT S. ELLIOTT, FELLOW, IEEE, AND L. A. KURT2 Alstrrrcf-The differences in mutul coupling for centtl slot nd peripherl slot cnnot be ignored in smll rrys if good ptterns nd impednce re to be obtined. A theory hs been developed whereby the length nd offset of every slot in the rry cn be determined, in the presence of mutul coupling, for specified perture distribution nd impednce mtch. The theory enlrges on Stevenson s method, nd uses modified form of Booker s reltion bsed on Bbinet s principle to tret nonresonnt longitudinl shunt slots in the brod wll of rectngulr wveguide. A generl reltion between slot voltge nd mode voltge is developed, nd then formuls re derived for the ctive, self-, nd mutul dmittnces mong slots. These formuls result in design procedure. Anlogous tretments of inclined series slots in rectngulr guide nd of strip-line-fed slots re possible. Comprison between vrious experiments nd the theory is presented. Tests of the theory include the resonnt length of zero offset slot, resonnt conductnce versus offset nd resonnt conductnce versus frequency for single slot, nd self- nd mutul dmittnces for two stggered slots. The design nd performnce of two-by-four longitudinl shunt slot rry is lso described. Mnuscript received June 4, 1976;revisedMy 13, This work ws supportedbytherntecdivision,emersonelectriccompny. R. S. Elliott iswith the Deprtment of Electricl Sciences, University of Cliforni, Los Angeles,CA 90024, nd Consultnt to the RntecDivision,EmersonElectricCompny,Clbss,CA L. A. Kurtz is with the Rntec Division, Emerson Electric Compny, Clbs, CA, C THEORY ONSIDER the module consisting of the solid lines shown in Fig. 1. This is section of rectngulr wveguide Ag/2 long contining longitudinl slot of length 22 nd displcement x cut in its upper brod wll. One- nd two-dimensionl slot rrys cn be constructed by plcing such modules in tndem nd prllel positions. The module of Fig. 1 is two-port device, the ports being t z = +Ag/4 if the origin is tken in the wveguide cross section which bisects the slot. But no loss in generlity occurs if the ports re tken t the positions z = &Ag, shown dotted in Fig. 1, becuse reltions between the two sets of ports involve simple known liner trnsformtions. It is convenient to choose the ports t z = *Ag; with this convention dopted, the equivlent circuit for the nth module1 is s shown in Fig. 2. This equivlent circuit is subject to the following interpret- tion. It is ssumed tht only the dominnt TElo mode cn propgte in the wveguide. This mode is represented by the voltge/current pir V,, I, t the input port (z = -Az). A lod 1 For nottionl simplicity the single index n is used to identify this module, but it is importnt to remember tht these modules cn be rrnged to form either liner rry or plnr rry. Double subscript nottion could be used in the ltter cse X/78/ $00.75 O 1978 IEEE

2 ELLIOTT AND KURTZ: SMALL SLOT ARRAYS 215 9p;,/*,*,r- ---7,I I ' I In words, (4) sttes tht the ctive dmittnce t the terminls I I,' I, '.' of the nth module equls the self-dmittnce of the nth slot. 5r plus term which ccounts for mutul coupling. This ltter ---,,' 2=h9 I term is summtion which involves not only the mutul dmittnces between ports, but lso the reltive voltges t the,c:,,' bl,," different ports. As the nlysis develops further, it will be seen I z= -4-, z=o c: e,,i tht YnA is decisive in determining the mplitude nd phse of.,.ag I - 1 the electric field in the nth slot. Since the ltter is dictted by I z=-,. I c I, 4., I pttern, the rdition desired point focl YnA the becomes of Y I*. I design. rry z=-a off shunt element is Fig. 1. Wveguide/slot module. symmetricl nd given by It is well known tht the scttering cc\y 1 Y,* B=C=---V (5) 2Go "" in which B nd C re the mplitudes of the bck nd forwrd TElo scttered modes. In the mnner of Stevenson [I], one cn show tht B is relted to the slot voltge Vns by the I I I L = -A 2=0 z=\ eqution Fig. 2. Equivlent circuit of nth module. dmittnce YnL is plced t the output port (z = +X,). B=-j [.' E:;] -sin 112 7rx, - Q (cos pz, This dmittnce (trnsformed through 3hg/4) represent wht could - cos kl,) vns (6) the nth module "sees" looking down its brnch line t ll the modules beyond, or it could be n pproprite termintion, wherein k = 27r/h0, nd 7) is the impednce of free spce; u nd such s n open circuit. Y,A is the ctive dmittnce of the b re the interior dimensions of the rectngulr wveguide. nth slot. It is n importnt prmeter in this nlysis nd its When (5) nd (6) re combined, one obtins mening cn be pprecited by considering the interreltions mong &l the modules. To ccount for mutul coupling, one cn write N XXn sin -(cos Q 02, - cos kl,) in which Ym, is the mutul dmittnce between the input ports m nd n. Y,, is the self-dmittnce of port n; tht is, in which Y, is the vlue tht YnA would ssume if ll other input ports were short-circuited. Y, is commonly clled the self-dmittnce of the nth slot. Generlly, the input dmittnce t the nth port is N, Vm = Y,, + x - Ym, = Y, + YnL + m=l Vn m=l Vn in which the prime on the summtion sign mens tht the term m = n hs been excluded. It follows tht NI Vm Y,A = Y, + x v, Ymn. m=l It will be seen shortly tht (7) is one of the two principl design equtions which emerge from the nlysis. A study of (7) revels tht the mode voltge nd slot voltge re in phse qudrture if YEA/Go is pure rel. In most slot rry design problems, Vns is governed by the pttern requirements nd V, is common voltge in ny given brnch line. Thus if ll the Vns slot voltges re to be in phse with ech other, nd ll the mode voltges V, re to be in phse with ech other, it follows tht ll the ctive dmittnces YnA should hve common phse. A simple choice is to require tht ll YnA be pure rel. But study of (4) indictes tht, if YnA is to be pure rel, in generl Y,, the self-dmittnce of the nth slot, will not be pure rel. In other words, when mutul coupling is tken :into ccount, one cnnot expect the resonnt selfconductnce dt wil be pertinent in the design. Indeed, in mny prcticl pplictions, the requisite vlue of Y, cn be quite fr off resonnce. The other principl design eqution rises from linking the performnce of the slot rry to tht of n equivlent dipole rry vi Bbinet's principle. Clerly, if the usul ssumption of n infinite perfectly conducting ground plne is mde, nd if the feeding currents of the center-fed strip dipoles mtch the slot voltges of the slots, the ptterns will be essentilly the sme. To get the impednce chrcteristics to mtch lso, one needs to plce lod impednce ZnL in series with the nth

3 216 IEEE TRANSACTIONS ANTENNAS AND PROPAGATION, VOL. AP-26, NO. 2, MARCH 1978 equivlent dipole to ccount for the fct tht the resonnt length of the slot is ffected by its offset, wheres no corresponding effect exists for the dipole. When this is done nd complex powers re equted for corresponding elements in the two rrys, it cn be shown [ 21 tht Eqution (11) leds to the interesting conclusion tht When use is mde of (lo)-( 12), it is importnt to remember tht Y,, Y,, re dmittnces ssocited with the mode voltges in the slot rry, nd tht, wheres Z, is the conventionl mutul impednce between dipoles, Z,, is the loded self-impednce of the nth dipole since it contins ZnL. In (8), Z,* is the ctive impednce of the nth strip dipole, defined by N, Im Z,A = z, + znl + 2 r z, m=l n wherein 2, is the self-impednce of the dipole, ZnL is the lod impednce plced in series with it, Z,, is the conventionl mutul impednce between dipoles clculble from formuls such s those of Bker nd LGrone [3], nd Im/I, is the perture distribution. Thus if the pttern requirement is known (so tht Im/In is known), nd if (Z, -I- ZnL) is known s function of x, nd I, (this reltion will be deduced shortly), then ZnA cn be clculted, plced in (8), nd YnA/Go cn be determined. Eqution (8) permits the interesting interprettion tht the normlized ctive dmittnce of longitudinl shunt slot is equl to Stevenson s expression for the resonnt normlized conductnce (the fctor in curly brckets) divided by the ctive impednce of the corresponding loded dipole normlized to 73. Eqution (8) lso pplies for the cse of n isolted slot, in which cse ZnA reduces to ZD + ZL, with ZD the selfimpednce of the isolted strip dipole nd ZL the lod impednce in series with it whose presence models the rective effects of internl higher order mode scttering off the slot due to its offset. This serves to point up some of the limittions of Stevenson s originl expression. Not only does it pply only for resonnt length slots, but strictly it becomes less ccurte pproximtion s the slot width nd/or its offset is incresed. This is becuse ZD is ffected by the width of the strip dipole, nd ZL is ffected by the offset of the slot. Eqution (8) cn be prtitioned [2] to yield the first-order results Yn zmm (cos 01, - cos kl,) sin - \--, EXPERIMENT If the foregoing theory is vlid, the proper design of oneor two-dimensionl longitudinl shunt slot rry involves the choice of offsets nd lengths for the vrious slots such tht (7) nd (8) re simultneously stisfied for ll vlues of n. One begins by knowing the desired perture distribution ( Vms/Vns for the slots, or I, /I, for the equivlent dipoles) nd the reltive mode voltges Vm/Vn (these would ll be the sme in stnding wve liner rry, but would depend on the selection of min-line/brnch-line coupling coefficients in plnr rry). Then knowledge of the function ZnA(xl,.-, XN, 11, -*, IN) permits determintion of ll the lengths nd offsets such tht the desired perture distribution is chieved, nd such tht the individul vlues of YnA/Go cuse the brnch line dmittnces nd min line dmittnces to dd up to give the desired mtch. A key ingredient in this process is to find the function ZnA (x1, --, xn, 11, -*, ZN). As mentioned erlier, the mutul prt of ZnA cn be clculted from conventionl formuls if the perture distribution is specified. Now we turn our ttention to the determintion of the self-prt of ZnA, nmely (Z, -t- ZnL). If we ssume tht (Zn + ZnL) is essentilly the sme whether the other dipoles re present nd open circuited, or bsent, then (2, + ZnL) = (ZSELF + ZLOAD); tht is, it equls the loded self-impednce of n isolted dipole (corresponding to n isolted slot). But for this cse (8) becomes - cos kz,)2 sin2 % }. Regrdless of the shpe of the slot (rectngulr, rounded ends, dumbbell, etc.), if one mexurex YSELFIGO s function of offset x nd length 1, (13) cn be used to express (Z, i- ZnL) s function of x, nd I,. This cn then be used in (8) for ll perture distributions nd feeding rrngements. For rectngulr slots, the theoreticl vlues of YSELF/GO obtined by the method of Khc [4] cn be used in lieu of experimentlly obtined informtion. It is desirble to ccumulte the dt on YSELF/GO in the universl form discovered by Stegen [ 51 nd illustrted in Fig. 9-5 of Jsik [6]. This figure shows plots of the rel nd imginry prts of YSELF/GO + GRES/GO versus ~ The ~ ~ rnge. of gretest use in the design of slot rrys is 0.95 <l/lr~s < 1.05 nd the theoreticl work of Khc [ 41 supports the ssumption of universlity in this rnge. Fig. 9-5 of Jsik requires his compnion Figs. 9-6 nd 9-7, in which GRES/GO nd 2lRES/hO re plotted s functions of slot offset. When polyfits re mde

4 ELLIOTT AND KURTZ: SMALL SLOT ARRAYS to the four curves in Fig. 9-5, 6, nd 7 of Jsik, (2,-I- ZnL) 0.60 cn be expressed in form esily hndled by computer Fig. 9-7 of Jsik leds to first test of the theory. Stegen 0.00 delt with round ended slots in wll in thick. The W question rises s to the length of the equivlent strip dipole W' of rectngulr contour in wll of "zero" thickness. This cn 0.20 be determined by the following rgument. As the offset x --f 0, + 0 the mplitudes of ll the modes scttered off the slot tend to 2 n zero. With respect to higher order mode scttering, this hs the z impliction for the complementry dipole tht its loding I impednce tends to zero lso. But in this circumstnce, (13) z indictes tht Z~ELF should be pure rel for the dipole when Y~ELF is pure rel for the slot. Ti hs shown [ 71 tht strip 0.05 dipole of width w nd negligible thickness is equivlent to - cylindricl dipole of dimeter d = w/2. Ti lso provides con- < 0.04 venient formul [ 71 for the impednce of cylindricl dipole B s function of its length 22 nd its rdius = d/2. Since Stegen used slots in wide, if one plces = in in Ti's formul, one cn deduce tht 2l,/&, = 0.464, wherein 21, is 0.01 the resonnt length of the unloded strip dipole. On the other 0 OX hnd, study of Fig. 9-7 of Jsik revels tht Stegen's symp SLOTOFFSET IN INCHES totic vlue is 21,"lho = 0.483, in which 21,' is the resonnt Fig. 3. Gr/G, for resonnt longitudinl slot versus offset GHz, length of his round ended slot t zero offset. From this it fol- = in, I, = in, slot width = in, wll thickness = lows tht f = 2Zro/2Z, = This length djustment fctor is in. Points re Stegen's mesured vlues; curve is theoreticl. in greement with the findings of Oliner [8], who ttributes 2 percent correction for round ends nd 2 percent correction 0.8 I I for wli thickness in this sitution. - THEORY Q Q When the foregoing theory is used to design slot rrys, the EXPERIMENT procedure just described cn be utilized to determine the length 0.7 djustment fctor f. Experience shows tht fis quite sensitive to the b dimension of the wveguide, s well s to wll thickness. c) '- 0.6 c) A second test involves prediction of resonnt conductnce W' 0 versus offset for n isolted slot. Since the higher order mode z scttering off this slot is nonpropgting nd thus contributes primrily to the storge of rective energy, it seems resonble n z to ssume tht the lod impednce ZL possesses smll resistive 8 component RL. In prcticl circumstnces, the dipole self- e 0.4 IL! impednce ZD hs resistive component in the neighborhood of 73, nd thus one should expect tht RL < RD. For I resonnt slot X, = -XD, nd in this cse (8) cn be pproxi- z 0.3 mted by 0.2 (COS /31r - COS k1,)2 sin2 -. (14) For stndrd X-bnd guide, frequency of GHz, nd length djustment fctor f = 1.04, (14) yields the solid curve found in Fig. 3. Stegen's experimentl points re shown for comprison. It should be recognized tht the greement seen between theory nd experiment in Fig. 3 is not cse of djusting prmeter in the theoreticl formul to get curve fitting. All tht hs been done in (14) is to ignore RL nd ssume tht the equivlent dipole is resonnt. A plot of the originl Stevenson formul would lie 20 percent below the solid curve of Fig. 3 t the low end, nd 10 percent below it t the high end. A third test involving n isolted slot concerns the frequency dependence of resonnt conductnce. Stegen [ 51 found experimentlly tht his curve of resonnt length versus offset for longitudinl shunt slot (Fig. 9-7 of Jsik) is universl in the sense tht if the offset remins constnt, 21,"/ho lso remins essentilly constnt even though the frequency vries. This hs RESONANT FREQUENCY, GHz Fig. 4. Resonnt conductnce versus frequency. the impliction tht if k2, remins constnt in (14), tht is, the slot length is continully djusted s the frequency is chnged so s to mintin resonnce, then for slot of given offset, GJG, is function of.frequency only becuse /3/k vries with frequency. For slot of offset in, (14) yields the solid curve shown in Fig. 4. Stegen's experimentl dt points re shown for comprison. Now let us consider situtions involving more thn just one isolted slot. As first step, n rry of two slots, one ech in two prllel wveguides, with the slots stggered longitudinlly qurter of guide wve-length, ws constructed with the dimensions shown in Fig. 5 nd imbedded in n 8-in by 10-in ground plne. This rry ws used to testhe vlidity of (10)-(12) in the following wy. With one slot covered over

5 218 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-26, NO. 2, MARCH 1978 INPUT PORT NO. 2 INPUT rc2y-i L.J. PORT NO. 1 - t-- I DIMENSIONS(lnches1 2iy = = x, = b = f 0 = t = x: = w = Fig. 5. f = GHr Two-slot rry. OUTPUT PORT NO. 2 OUTPUT PORT NO. 1 TABLE I MEASURED AND COMPUTED DATA FOR TWO-SLOT ARRAY Qurntity YIIG, (Isolted) V2/Go (Isolred) Yl GO YZIG, Y12/GO Mesured Vlue 0.52 t j j j j & Computed Vlue j j j j j with conducting tpe, short circuit ws plced 3hg/4 beyond the other slot, nd mesurement ws tken of its input dmittnce. This resulted in the dt shown in the first two rows of Tble I. A length contrction fctor f = 1.03 ws found to pply for this configurtion nd used to determine I, nd Ip. Eqution (8) then gve = z22 = (15) YJG, (isolted) Y2/Go (isolted) from which the entries in the third nd fourth rows of Tble I were obtined. Zll nd Z22 s they pper in (15) re the loded selfimpednces of the strip dipoles equivlent to ech isolted slot. Strictly speking, they re not the sme s the quntities one should use when other dipoles re present but open-circuited; however, t this slot spcing the pproximtion is good one, nd therefore the entries for Zll nd Z22 in Tble I will be used in ( lo)-( 12). The clcultion of mutul dipole impednce ws mde using the formuls of Bker nd LGrone [3], nd provides the fifth row entry in Tble I. Eqution (10) predicts tht if the conducting tpe covering the second slot is removed nd replced by short circuit Ag from the center of the second slot, nd then the input dmittnce of the first slot is mesured, the result should stisfy _-- _-- Yl y Go Zll -(z122/z22) Go z22 -(Z122/z11) (16) When the vlues listed in Tble I for Zll, Z,,, nd Z12 re used in (16), computed vlues of Y1/Go nd Y2/GO cn be scertined. These vlues hve been entered in the sixth nd seventh rows of Tble I. The mesured vlues re listed longside for comprison. Finlly, (12) of the theory cn be put in the form -- YI2-[ (17) Go Go Go y1 y2 -G22 1 When the vrious computed vlues found in Tble I re inserted in (17), the prediction is tht Y12/Go = L-3.1. The ccurte mesurement of Y12/Go is difficult. After some experimenttion, the following procedure ws dopted. Short circuits were plced 3Ag/4 beyond ech slot. Slotted lines were put in tndem with both input ports. A vrible ttenutor ws plced before one slotted line, nd vrible phse shifter before the other. The two brnches were fed through conventionl T junction. But from (4), It follows tht one cn select the ttenutor setting such tht I Vl/V2 I = 1. If then the phse of Y1/Y2 is vried, YIA/Go nd Y2A/Go will hve loci which re circles of the sme size, centered round Yl/Go nd Y2/Go, respectively. The rdius of these two equl circles is I Y12/Go 1. The phse of the mutul dmittnce cn be determined from corresponding points on the two loci. This experimentl procedure resulted in the Smith chrt shown in Fig. 6. From this dt, the verge mesured vlue of mutul dmittnce ws deduced to be Y12/Go =0.146L-4.2O. Lstly, let us consider the ppliction of (7) nd (8) to the design of two-dimensionl rry. The procedure cn be outlined s follows. ) Select the frequency of opertion nd the wveguide dimensions. b) Deduce the length contrction fctor f. This cn be done s in the erlier discussion of resonnt length for zero offset. c) Specify the slot voitge distribution needed to get the desired pttern nd the sum of the ctive dmittnces desired in ech brnch line wveguide. d) Solve (7) nd (8) simultneously to give those vlues of x, nd I, which stisfy the required perture distribution nd dmittnce level. When the bove procedure ws pplied to the design of two-by-four rry, the results were s shown in Fig. 7. The specified dmittnce level ws A 2 Y,A/G, X = 2 Y,A/G, = 2 +io. n= 1 n=5 The mesured vlues were YnA/Go = j0 YnA/G0 = io. n=l n=5 8

6 ELLIOTT AND KURTZ: SMALL SLOT ARRAYS 219 Fig. 6. Active dmittnce loci for two-slot rry. Fig Slot No. n Offset x (1nches)n M.O, x x1.122 OFFSETS (Inches) = b = t = w = f = GHz 8 by 10 inch Grwnd Plne Two-by-fourslotrry. Length 21: (Inches) The specified perture distribution ws uniform mplitude/ uniform phse, so the predicted pttern hs brodside bem, symmetricl sidelobes, nd 13.5-dB sidelobe level. The experimentl H-plne pttern is shown in Fig. 8. A study of the tble of slot lengths nd offsets (Fig. 7) revels severl interesting nd surprising things. First, there is 2: 1 rnge in slot offsets. (Were one to ignore mutul coupling, or include it but ignore its vribility from slot to slot, ll offsets would be the sme.) Second, no slot in this rry is selfresonnt; ech slot is detuned ppropritely to mke the individul ctive dmittnce resonnt. Third, there is qudrnt I/ qudrnt I11 nd qudrnt II/qudrnt IV symmetry to the lengths nd offsets, but no symmetry round the X xis nor round the Y xis. This cn be trced to nonsymmetricl effects cused by stggering the offsets. The rnge of lengths nd offsets found for this two-by-four rry illustrtes the generl observtion tht smll rrys pre- sent more difficult design problem thn do lrge rrys. In the ltter, only elements ner n edge see different mutul coupling environment, so chieving the proper ctive dmittnce becomesimpler. Further, mechnicl nd electricl tolernces ese off s the rry gets lrger [ 91. Though the detils re not being reported here, the bove procedure hs been used successfully to design 12-slot liner rry for 30 db side lobe level, 19-slot liner rry for symmetric side lobes (ll t 20 db except the inner three on ANGLE FROM BROADSIDE, DEGREES Fig. 8. H-plne pttern of two-by-four rry described in Fig. 7, f = GHz. one side of the min bem t 30 db) nd 52element twodimensionl slot rry with uniform perture distribution. CONCLUSIONS A theory hs been presented which cn ccount for the rry behvior of longitudinl shunt slots in terms of the chrcteristics of complementry dipoles. Formuls for ctive, self-, nd mutul dmittnces of longitudinl slots hve been derived. Slot rrys cn be designed by choosing the lengths nd offsets of individul slots such tht (7) yields slot voltge distribution consistent with the desired pttern, nd such tht (8) yields n ctive dmittnce distribution consistent with the feed nd mtch requirements of the rry. The nlysis cn be repeted, in step-by-step nlog, for the cse of inclined series slots in the brod wll of rectngulr wveguides. It cn lso be extended to rrys of strip-line-fed slots, The theory hs been tested experimentlly in vriety of situtions involving single slot, pir of slots, nd smll two-dimensionl rry. In generl, the greement hs been found to be quite stisfctory. REFERENCES [l] A. F. Stevenson, Theory of slots in rectngulr wveguides, J. Appl. Phy~., VO~. 19, pp ;.In (21 R. S. Elliott, Longitudinl shunt slots in rectngulr wveguide: Prt I, theory, Rntec, Clbss, CA, Rntec Report No TN- 1. [3] H. C. Bkernd A. H. LGrone, Digitl computtion of the mutul impednce between thin dipoles, ZEEE Trns. Antenns Propgt.. vol. AP-10, pp , Mr [4] T. Vu Khc, A study of some slot discontinuities in rectngulr wveguides, Ph.D. disserttion, Monsh University, Austrli, Nov [5] R. J.Stegen, Longitudinl Shunt SlotChrcteristics, Hughes TechniclMemorndumNo. 261, Culver City, CA.Nov., [6] H. Jsik, Antenn Engineering Hndbook. New York:McGrw- Hill, 1961, Chpter 9. [ 71 C. T. Ti, Thrcteristics of liner ntenn elements, in Antenn Engineering Hndbook, H. Jsik,Ed. New York:McCrw-Hill, 1961, Chpter 3. [8] A. A. Oliner, The impednce properties of nrrow rditing dots in the brod fce of rectngulr wveguide, IEEE Trns. AntennsPropgt., vol. AP-5, pp. 4-20, Jn., [9] R. S. Elliott, Mechnicl nd electricl tolernces for twodimensionl scnning ntennrrys, IRE Trns. Antenns Propgt., vol AP-6, pp ; 1958.

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