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2 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide x JP Jcobs, J Joubert nd JW Odendl University of Pretori South Afric. Introduction Slot dipole ntenns fed by coplnr wveguide (CPW) hve number of ppeling chrcteristics tht include reltively low feed line dispersion nd losses t millimeter-wve frequencies, ese of integrtion with circuit components due to the uniplnr configurtion, nd considerbly wider impednce bndwidth thn microstrip ptch ntenns on comprble substrtes, e.g., (Mio et l., 2). In the sub-millimeter nd millimeter-wve rnges, CPW-fed slots hve been used to feed dielectric lens ntenns (Focrdi et l., 22); rrys of CPW-fed slots re lso expected to be used for spce pplictions (Neto et l., 23). While vriety of CPW-fed slot rrys hve been reported (see section 3), rigorous design procedures for CPW-fed slot rrys tht incorporte the effects of element interctions in the form of mutul coupling hve only rrely been presented (Hung et l., 999; Jcobs & Joubert, 29b). An itertive design typiclly involves the priori genertion of selfdmittnce dt for representtive slot dimensions, nd requires the clcultion of mutul coupling between ll possible slot pirs in the rry; this is especilly pertinent if nonuniform perture distribution imed t chieving reduced sidelobe levels is to be relized. In this chpter n overview of the current stte-of-the-rt in the design of non-uniform liner CPW-fed ntenns, s well s its subsidiry clcultions, is presented. In section 2, the field distribution in n isolted CPW-fed is shown; this is importnt informtion for the clcultion of mutul coupling. Section 3 focuses on the clcultion of mutul coupling using reciprocity-bsed prdigm. This is done in the context of two importnt prcticl cses. The first involves mutul coupling between slots on electriclly thin substrtes, where simplified yet ccurte method cn be used tht obvites use of the substrte Green s function. Arrys of slots on such substrtes exhibit bi-directionl rdition. The second involves n extended reciprocity pproch for clculting the mutul dmittnce between slots on conductor-bcked two-lyer substrte, configurtion tht would be useful when unidirectionl rdition is required. Section 4 gives detils of two types of CPW-fed liner rry design. First, the itertive design using n pproximte procedure of uniform rry on conductor-bcked two-lyer substrte is summrized; mesured results re presented. The procedure doesn t require the explicit clcultion of the mutul dmittnce between

3 534 Pssive Microwve Components nd Antenns pirs of slots, but does tke such interctions into ccount through full-wve clcultions of the input impednce of the rry s whole. Second, the implementtion of rigorous itertive design procedure for non-uniform rry with reduced sidelobe levels on n electriclly thin substrte is described; mesured sidelobe levels of bout 6 db below the min bem were chieved. Due to the electriclly thin substrte, the design procedure could drw on the simplified method for clculting mutul dmittnce outlined in section 3. Section 5 briefly summrizes some of the min findings. 2. Field properties of isolted CPW-fed slots CPW-fed slots re normlly operted in the vicinity of their second resonnces becuse of the fvourble impednce bndwidth properties here (compred to the first-resonnt region). A CPW-fed slot on single dielectric lyer is shown in Fig.. In order to more ccurtely ccount for its use in liner rry environment (see Fig. 4), the slot is terminted in perfect short-circuit relized by section of trnsmission line of length l s. W Port L w s l s ε r, h Fig.. Top nd side views of CPW-fed slot on single-lyer substrte. L hlf-length nd W width of rditing slot; s slot width nd w center strip width of CPW feed line; h dielectric lyer height; ε r reltive dielectric constnt; l s length of CPW stub implementing short-circuit. When evluting the mutul dmittnce between pirs of slots, the perture electric field distribution is required (see section 3). IE3D (Zelnd Softwre, 2), full-wve electromgnetic simultor tht employs mgnetic current modeling nd ssumes lterlly infinite ground plnes nd dielectrics, cn be used to obtin this informtion. Figs. 2 nd 3 (Jcobs & Joubert, 29) show mgnitudes nd phses t 6 GHz of the tngentil electric fields long longitudinl slot centers of three isolted slots on n electriclly thin substrte with h =.575 mm nd ε r = 2.33 (t 6 GHz, h =.48λ d =.35λ, with λ d nd λ the wvelengths in the dielectric nd free spce respectively). The slots widths W ws mm, their hlf-lengths L were in the vicinity of the second-resonnt hlf-length L res = mm,

4 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 535 nd the length of the short-circuit stub l s ws.48λ CPW = 9.4 mm. The slots were fed by 87 Ω CPW feed line tht hd dimensions w = 3 mm nd s = mm. (The sme rditing slot nd feed line dimensions were used in obtining the mutul dmittnce results of Figs. 6 8.) The perture field vector component shown is the one directed cross the width of the slot; the orthogonl component ws negligible by comprison. 25 Electric field mgnitude (V/m) L res L res.l res x (mm) Fig. 2. Mgnitude of electric field of isolted slots with hlf-lengths in vicinity of secondresonnce hlf-length L res. L res = 2.75 mm, W = mm, l s = 9.4 mm =.48 λ CPW, h =.575 mm, ε r = 2.33, w = 3 mm, s = mm. Verticl lines correspond to position of CPW. 6 Phse (deg.) L res L res.l res x (mm) Fig. 3. Phse of electric field of isolted slots with hlf-lengths in vicinity of secondresonnce hlf-length L res. L res = 2.75 mm, W = mm, l s = 9.4 mm =.48 λ CPW, h =.575 mm, ε r = 2.33, w = 3 mm, s = mm. Verticl lines correspond to position of CPW.

5 536 Pssive Microwve Components nd Antenns Contrry to the nerly constnt phses of CPW-fed slots with hlf-lengths in the vicinity of the first-resonnt hlf-length (Jcobs, 27), the phses of these second-resonnt slots exhibit shrp rise close to the CPW feed line, while chnging little in the outer reches of the slots. Increses in slot length result in offsets of phses with respect to phses of preceding lengths. Similr grphs were obtined for slots on conductor-bcked two-lyer substrtes (Jcobs, 27). 3. Mutul coupling between CPW-fed slots The focus of this section is the clcultion of mutul coupling between slot pirs in liner rrys of slots fed in series by CPW. The geometry of such n rry, designed for brodside min lobe, is shown in Fig. 4. While vriety of types of CPW-fed slot rrys hve been reported, for instnce widebnd liner CPW-fed log-periodic dumb-bell slot rrys (Kim et l., 26), mplifier rrys using CPW-fed folded slot ntenns (Tsi et l., 994), nd CPW-fed plnr (two-dimensionl) slot rrys in multi-chip module-deposition (MCM- D) technology (Solimn et l., 999), the rigorous clcultion of mutul coupling between slot rry elements in the context of itertive rry design procedures hs only rrely been ddressed, e.g., Hung et l. (999). For certin pplictions, such clcultions seem unnecessry: the design of uniform CPW-fed rrys on both single-lyer substrtes (Qui et l., 22) nd conductor-bcked two-lyer substrtes (Jcobs et l., 23) hve been ccomplished without explicitly clculting the mutul coupling between individul pirs of slots, s noted erlier. On the other hnd, the design of non-uniform rrys subject to rigorous sidelobe-level specifictions generlly requires explicit, ccurte ccounts of the mutul coupling between ll possible slot pirs in the rry. Fig. 4. Top view of N-element CPW-fed liner slot rry with equiphse element excittions. L m hlf-length nd W m width of rditing slot m, where m =, 2.. N; s slot width nd w center strip width of CPW feed lines; λ CPW CPW wvelength. 3. Mutul coupling between slots on electriclly thin single-lyer dielectric substrte For electriclly thin substrtes simplified method bsed on well-known reciprocity-bsed expression cn be used to find the mutul dmittnce between two brodside CPW-fed slots (Jcobs & Joubert, 29). The method is eminently suitble for esy incorportion into

6 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 537 itertive rry design lgorithms, nd for fst evlution. The geometry is shown in Fig. 5; slot orienttions correspond to their orienttion in liner rry such s tht of Fig. 4. The slots in Fig. 5 re intended to be ccurte models of slots in n rry environment hence their termintion in CPW sections implementing short-circuits tht extend beyond their rditing portions (this will be more fully described below). The kind of mutul dmittnce clcultion performed here is required when dopting so-clled first-order interction pproch (Amity et l., 972) towrds finding the mutul dmittnce between ny two slots in n rry: the mutul dmittnce between specific pir of slots is, for the ske of simplicity, ssumed to be identicl to the mutul dmittnce between them when the other slots in the rry re removed. Fig. 5. Top nd side views of brodside CPW-fed slots on single-lyer dielectric substrte. L = L 2 hlf-lengths nd W = W 2 widths of rditing slots; d distnce between rditing slots; s slot width nd w center strip width of CPW feed lines; h dielectric lyer height; ε r reltive dielectric constnt; l s length of CPW stub implementing short-circuit. The simplified method ssumes tht, for mutul dmittnce clcultions between CPW-fed slots on electriclly thin single-lyer substrtes, the inhomogeneous ir-dielectric lyer-ir medium cn be sufficiently ccurtely pproximted by homogenous free spce (this is different from the pproch tken by Hung et l. (999)). The mutul dmittnce between CPW-fed slots nd 2 of Fig. 5 cn then be found by dpting results originlly derived for wire dipoles rditing in free spce in (Blnis, 996) to slots in n infinite ground plne rditing in free spce. If it is ssumed tht CPW feed lines hve negligible effect on mutul coupling, only the rditing portions of the CPW-fed slots need to be considered. Hence, where ( w /2s) w /2sL2 Y 2 H z2( z2 ) Im2( z2 ) dz2 H z2( z2 ) Im2( z2 ) dz2 () VV2 ( w /2sL2 ) w /2s H ) H ( z )cos H ( ) sin (b) z2( z2 r 2 z2

7 538 Pssive Microwve Components nd Antenns with H ( z r 2 ) ( w /2s) m ( w /2sL w /2sL I I m w /2s ( z ( z ) L cos ) 2 2r L cos ) 2 2r e jkr e jkr jk jk r r dz dz (c) nd H ( z 2 ) ( w /2s) m ( w /2sL w /2sL I I m w /2s ( z ( z ) k ) j k ) j L sin 4r L sin 4r jk r jk r k k r r 2 2 e e jk jk r r dz dz (d) In the bove, H z2 is the mgnetic field of equivlent mgnetic current I m t the position of slot 2 when slot rdites in isoltion (I m represents slot when rditing in isoltion), I m2 is the equivlent mgnetic current representing slot 2 when rditing in isoltion, V is the terminl voltge of slot when rditing in isoltion, V 2 is the terminl voltge of slot 2 when rditing in isoltion, L nd L 2 re the hlf-lengths of slots nd 2 respectively, s is the slot width nd w the center strip width of the CPW feed line, k is the free-spce wvenumber, nd 2 2 z ) r ( z d, the free-spce intrinsic impednce. r nd re sphericl coordintes nd the dimensions constituting the integrtion boundries re shown in Fig. 5. The bove equtions cn be implemented s follows to find Y 2 ginst slot spcing d for two CPW-fed slots on n electriclly thin substrte: First, ech CPW-fed slot of Fig. 5 is simulted in isoltion using IE3D (in the cse of twin slots, only one slot needs to be simulted). As noted erlier, in order to ccurtely model CPW-fed liner rry environment, ech slot is terminted in perfect short-circuit relized by section of trnsmission line of length l s, with l s mesured from the center of the rditing portion of the slot. For composite slot (feed line, rditing slot nd short-circuit termintion) k, where k = or 2, the simultion yields the tngentil electric field long the center of the rditing slot (see Figs. 2 & 3), from which its equivlent mgnetic current density I mk cn be obtined. It is ssumed tht the rditing slot electric field only hs vector component cross the width of the slot (in other words, the smll longitudinl component predicted by IE3D is neglected), nd tht the field is constnt cross the width of the slot. The ltter ssumption, which enbles clcultion of I m nd I m2 by simply multiplying the corresponding equivlent mgnetic current density by the slot width, cn be enforced in IE3D by dopting discretiztion tht llows for only one cell cross the slot width. In ddition to the slot tngentil field, the isolted-slot moment-method nlysis lso gives the terminl voltges V k, where k = or 2. Excittion ports in IE3D re defined t the ends of feed lines (cf. Fig. 5). The port voltge computed by IE3D t the end of the feed line 2

8 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 539 is in fct the terminl voltge, nd is found by integrting the trnsverse electric field in one of the two CPW slots over the slot width. In the mutul dmittnce clcultions described below, terminl voltges (nd isolted slot self-dmittnces) were referred to centers of rditing slots. Second, Eqution () ws evluted for ech instnce of slot seprtion d. In ccordnce with the definition of the quntities tht constitute eqution (), I m, I m2, V nd V 2 were kept the sme for ll vlues of d. Curves of Y 2 ginst d obtined with the reciprocity-bsed method outlined bove were compred with curves computed using IE3D. Using moment-method pproch towrds this end implies tht the entire two-slot structure of Fig. 5 needs to be solved for ech instnce of d. Ports were defined t the ends of the CPW feed lines, nd from the full-wve solution IE3D clculted the two-port Y prmeters of the structure with respect to the bove ports. Y prmeters (including Y 2 ) were fterwrds referred to centers of rditing slots. Y 2 ginst d with.9λ CPW d 2λ CPW ws computed for three pirs of identicl brodside (twin) slots on the electriclly thin substrte with h =.575 mm nd ε r = 2.33 (t 6 GHz, h =.48λ d =.35λ, with λ d nd λ the wvelengths in the dielectric nd free spce respectively). The slots widths W ws mm, their hlf-lengths L = L 2 were.9l res, L res nd.l res respectively, with L res = mm, nd the length of the short-circuit stub l s ws.48λ CPW = 9.4 mm. The slots were fed by 87 Ω CPW feed line tht hd dimensions w = 3 mm nd s = mm. The resonnt slot hd self-impednce of 7 Ω, nd hence cn be considered typicl slot from liner rry perspective, given tht the input impednce of brodside CPW-fed liner rry is simply the sum of the slots ctive impednces, nd since slot self-impednce is often considered first-order pproximtion to slot ctive impednce (Elliott, 98). The self-impednces of the.9l res nd.l res slots were 2-j6 Ω nd 6+j9 Ω respectively. The rel nd imginry prts of Y 2 ginst normlized slot seprtion d/λ CPW for the twin slots with L = L 2 = L res re shown in Fig. 6; results from both the reciprocity-bsed method (the curves lbeled rec. ) nd IE3D re given. Likewise, Y 2 ginst d/λ CPW for twin slots with L = L 2 =.9L res nd L = L 2 =.L res re shown in Figs. 7 nd 8 respectively. The third set of curves (lbeled improved rec. ) in ech of Figs. 6 8 correspond to n improved version of the reciprocity-bsed method tht used, insted of the terminl voltges V k obtined from IE3D, new terminl voltges V k, new tht were computed s follows. The most ccurte results for Y 2 in liner rry with brodside min lobe re required when the slot spcing d = λ CPW ; this is the minimum spcing between slot elements, nd the effect of mutul coupling would be the gretest here. Thus, for ech of the twin slot cses represented in Figs. 6 8, the vlue of Y 2 ws computed t d = λ CPW using IE3D; so ws the rection integrl in eqution (). The product V, new V 2, new ws then obtined from the quotient of IE3D s Y 2 nd the rection integrl vlue. Since twin slots were involved, V, new nd V 2, new re equl. These voltges were then used in new evlution of eqution () over the rnge of d. Figs. 6 8 revel generlly good greement between Y 2 computed using the originl reciprocity-bsed method nd Y 2 computed using IE3D, with greement being best for the.9l res twin slots nd lest good for the longest (i.e.,.l res ) twin slots. For ech of the three cses, the mgnitude of the difference between the Y 2 -ginst-d curves obtined with the originl reciprocity-bsed method nd IE3D were computed, nd normlized to the mgnitude of the isolted slot self-dmittnce, Y self. Averged over the rnge of d, the normlized mgnitude of the difference ws found to be.2% for the L res slots,.8% for the

9 54 Pssive Microwve Components nd Antenns.9L res slots, nd 2% for the.l res slots. These verge errors decresed to 3%,.4%, nd 2.4% respectively when the improved reciprocity-bsed method ws used. Hence excellent results cn be obtined t the cost of further pre-processing required to compute the terminl voltges V, new nd V 2, new in the mnner described bove. In n ctul liner rry design, this would need to be done once only, presumbly for the sme mtrix of slot lengths nd widths tht self-dmittnce dt will be generted for (Hung et l., 999). In order to verify the ccurcy of IE3D s clcultions, Y 2 ginst frequency ws computed using tht simultor for twin slot configurtion with fixed lyout identicl to tht shown in Fig. 5, except for one of the slots being rotted by 8 o in order to enble feeding it from the edge of the substrte vi coxil luncher. The twin slots were designed to be t their second resonnces t 6 GHz, nd hd L = L 2 = 2.95 mm = L res, 6 GHz, nd W = W 2 = mm on n electriclly thin substrte with h =.83 mm, ε r = 3.38±.5 nd tn δ =.27 (i.e., Rogers RO43C lminte). The feed line chrcteristic impednce ws 83 Ω (w = 3 mm nd s = mm). Fig. 9 shows good greement between mesured nd simulted Y 2 (referenced to the centers of the slots) dt. Mutul dmittnce Y 2 (ms) Re: IE3D Im: IE3D Re: rec. Im: rec. Re: improved rec. Im: improved rec d/ CPW Fig. 6. Mutul dmittnce Y 2 ginst brodside distnce d/λ CPW t 6 GHz for CPW-fed twin slots with L = L 2 = L res = 2.75 mm. W = W 2 = mm, l s = 9.4 mm =.48 λ CPW, h =.575 mm, ε r = 2.33, w = 3 mm, s = mm.

10 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 54 Mutul dmittnce Y 2 (ms) Re: IE3D Im: IE3D Re: rec. Im: rec. Re: improved rec. Im: improved rec d/ CPW Fig. 7. Mutul dmittnce Y 2 ginst brodside distnce d/λ CPW t 6 GHz for CPW-fed twin slots with L = L 2 =.9L res = mm. W = W 2 = mm, l s = 9.4 mm =.48 λ CPW, h =.575 mm, ε r = 2.33, w = 3 mm, s = mm. Mutul dmittnce Y 2 (ms) d/ CPW Re: IE3D Im: IE3D Re: rec. Im: rec. Re: improved rec. Im: improved rec. Fig. 8. Mutul dmittnce Y 2 ginst brodside distnce d/λ CPW t 6 GHz for CPW-fed twin slots with L = L 2 =.L res = mm. W = W 2 = mm, l s = 9.4 mm =.48 λ CPW, h =.575 mm, ε r = 2.33, w = 3 mm, s = mm.

11 542 Pssive Microwve Components nd Antenns Mutul dmittnce Y 2 (ms) Re: IE3D Im: IE3D Re: mesured Im: mesured Frequency (GHz) Fig. 9. Computed nd mesured mutul dmittnce Y 2 ginst frequency for CPW-fed twin slots. L = L 2 = L res, 6 GHz = 2.95 mm, W = W 2 = mm, l s = 8.3 mm = λ CPW, 6 GHz, d = λ CPW, 6 GHz = 38. mm, h =.83 mm, ε r = 3.38±.5, tn δ =.27, w = 3 mm, s = mm. 3.2 Slots on conductor-bcked two-lyer substrte Fig. shows twin CPW-fed slots on two-lyer conductor-bcked substrte with n ir bottom lyer (i.e., ε r2 = ). For pplictions tht require unidirectionl rdition, this prticulr structure is of interest s the rdition efficiency ttinble with ppropritelyspced twin slots on such substrte is comprble to tht of twin slots on λ d /4 single-lyer substrtes with reflector positioned λ o /4 wy from the CPW ground plnes side (Qiu et l., 2) (λ d is the wvelength in the dielectric; λ o is the free-spce wvelength). This is true even though the substrte of Fig. is of lesser electricl height thn the twin slots on single-lyer substrte potentilly n dvntge t microwve frequencies Effect of conducting bck plne The ntenn input impednce of liner slot rry on λ d /4 single dielectric substrte is unffected by bck reflector plced λ o /4 on the side of the CPW ground plnes (Qiu et l., 22). This is n indiction tht the bck reflector hs miniml influence on mutul coupling between the rry slots. However, in this section the presence of bck reflector plced below two-lyer substrte (Fig. ) is shown to chnge the mutul coupling between slots. This implies chnge in internl coupling (i.e., coupling on the dielectric side) due to the presence of the bck plne. Thus, the pproprite prllel-plte two-lyer Green's function would be needed in order to ccount for mutul coupling with sufficient ccurcy in n itertive rry design procedure (Jcobs et l., 25).

12 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 543 Fig.. Top nd side views of brodside CPW-fed twin slot ntenns on conductor-bcked two-lyer substrte. L hlf-length nd W width of rditing slots; d distnce between rditing slots; s slot width nd w centre strip width of feed lines; l f length of feed lines; h nd h 2 dielectric lyer heights; ε r nd ε r2 reltive dielectric constnts. Assuming infinite top nd bottom conducting plnes nd dielectric lyers with dielectric constnts ε r = 3.38 nd ε r2 =, with the top substrte lyer height h =.83 mm, simultions were crried out t GHz to demonstrte the effect of bck plne height on mutul coupling between the twin slots of Fig. (Jcobs et l., 25). Three vlues of bottom lyer height, or bck plne distnce, were considered, nmely h 2 =, λ o /4, nd λ o /6 (t GHz, λ o /4 = 7.5 mm nd λ o /6 = 5 mm). The cse h 2 = is equivlent to the bsence of bck plne. A CPW feed line with chrcteristic impednce of bout 5 Ω for the cse h 2 = is used (w = 3.7 mm nd s =.2 mm). For ech of the cses h 2 =, λ o /4, nd λ o /6, n isolted (rditing) slot with width W =.4 mm ws designed to be resonnt t GHz by djusting its hlflength L. The resulting resonnt slot hlf-lengths were.36 mm,.6 mm nd.67 mm respectively, with corresponding self-impednces of 2.5,.2, nd 3.9. Fig. shows the rel nd imginry prts g 2 nd b 2 of the normlized mutul dmittnce y 2 s function of normlized distnce d/λ CPW for ech of the three cses h 2 =, λ o /4, nd λ o /6. y 2 is the mutul dmittnce Y 2 normlized with respect to the relevnt isolted resonnt slot self-dmittnce. The reson for normlizing Y 2 is tht, in liner rry context, the reltive size of mutul dmittnce mgnitudes with respect to the slots' selfdmittnce mgnitudes is n indictor of the extent of the effect of mutul coupling on the rry input impednce. The normlized curves indicte tht, in the bsence of bck plne, the mximum vlues of the mgnitudes of the rel prt (G 2 ) nd imginry prt (B 2 ) of the mutul dmittnce Y 2 re bout 4% nd 27% respectively of the resonnt slot self-dmittnce; the mximum of g 2 occurs t d = λ CPW nd the mximum of b 2 t d =.4 λ CPW.

13 544 Pssive Microwve Components nd Antenns Fig.. Rel nd imginry prts g 2 nd b 2 of normlized mutul dmittnce y 2 vs. interslot distnce d/λ CPW t GHz. Adding bck plne t h 2 = λ o /4 results in significntly higher reltive mximum vlues of bout 68% nd 4% for G 2 nd B 2 respectively; other extrem lso show notble increses compred to the h 2 = cse. The bck plne hs the effect of shifting the curves for g 2 nd b 2 so tht their extrem in generl re not ligned with those of the curves for h 2 =. These effects cn be scribed to internl mutul coupling (the externl equivlent problem, concerned with fields in the hlf-spce djcent to the CPW ground plnes, is unchnged by the ddition of bck plne), indicting tht the input impednce of n rry on the singlelyer substrte would be significntly ffected when bck plne is plced λ o /4 wy. Decresing the bck plne distnce to h 2 = λ o /6 results in curves quite similr to, nd more or less "in phse" with, the curves for the cse h 2 = λ o /4 with somewht higher mximum vlues for g 2 nd b 2 (.7 nd.45 respectively vs..68 nd.4), suggesting tht the effect of mutul coupling increses s bck plne distnce decreses. At d = λ CPW the mgnitude of the rel prt of the mutul dmittnce ws bout 4% of the relevnt resonnt slot selfdmittnce for ll cses of h An extended reciprocity-bsed method for clculting mutul dmittnce In this section, computtionl strtegy bsed on the well-known expression derived from reciprocity is proposed to clculte the mutul dmittnce between two CPW-fed slots on conductor-bcked two-lyer substrtes (Jcobs et l, 29). The formultion itertively updtes the field distribution in both slots to ccount for the coupling interction between the slots. The clssicl reciprocity-bsed pproximtion ssumes tht the field in the slot is the sme whether the slot is rditing in isoltion or in the presence of second short

14 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 545 circuited slot. In cses where mutul coupling is high, such s for resonnt-length twin slots, this ssumption becomes invlid. The two CPW-fed slots on conductor-bcked two-lyer substrte, s shown in Fig., re center-fed by n infinitesiml voltge source tht is plced in the centre of short section of slotline connecting the two CPW feed line slots t their ends. Employing the stndrd formultion derived from reciprocity the mutul dmittnce Y 2 between them is (Nuwelers & Vn de Cpelle, 988) Y 2 V V 2 S2 H 2 M 2 ds 2 (2) where M 2 is the equivlent mgnetic current density representing slot 2, V nd V 2 re the terminl voltges of slots nd 2, respectively nd, ssuming tht the effects of mutul coupling between CPW feed lines, nd between feed lines nd rditing slots re negligible, S 2 is the surfce re of slot 2. H 2 is the mgnetic field t the position of slot 2 due to the equivlent mgnetic current density, M, representing slot. H 2 is clculted s the sum of the internl field (using the Green s function for the conductor-bcked two-lyer substrte), nd the externl field (using the free-spce Green s function). The initil estimtes for M nd M 2 re determined by clculting the electric field distributions in the isolted slots using moment-method solver e.g. IE3D. The isolted-slot moment-method nlysis lso yielded the terminl voltges V nd V 2. A schemtic representtion of CPW-fed twin slots is shown in Fig. 2. In sitution, slot is ssumed to be excited such tht its terminl voltge V equls V (i.e., the terminl voltge of slot when rditing in isoltion), while slot 2 is short-circuited. Hence I YV Yself V (3) where I is the terminl current entering port. The current I 2 entering port 2 cn then be found using (Blnis, 982) dpted for slots (from its originl form for wire dipoles): I 2 V 2 S2 H 2x M 2x ds 2 (4) where ll quntities on the right-hnd side re similr in mening to the corresponding quntities in eqution (2). For the first interction, H 2x is the longitudinl component of the mgnetic field t the position of slot 2 when slot rdites in isoltion with terminl voltge V (nd terminl current I ), M 2x nd V 2 re the equivlent mgnetic current density nd terminl voltge respectively of slot 2 when rditing in isoltion, nd S 2 is the surfce re of slot 2. From equtions (2) nd (4) it follows tht

15 546 Pssive Microwve Components nd Antenns I V 2 Y2 (5) where Y 2 is the first itertion of the mutul dmittnce, computed using eqution (2) with the field distribution for the slots rditing in isoltion. Fig. 2. Schemtic representtion of CPW-fed twin slots with terminl voltges nd currents, nd port termintions used in higher-order itertive procedure. Sitution b is set up to ccount for the effect tht the current I 2 induced by slot t the terminls of slot 2 (cf. sitution ) in turn hs on slot. Hence slot 2 is ssumed to be excited by terminl current b I2 I2 (6) while slot is short-circuited. The short-circuit current b I t port cn then be found from I b V S H b 2 x M x ds (7) In the bove, b H 2 x is the mgnetic field t the position of slot when slot 2 rdites in isoltion (its terminl current is b I 2 ), while M x nd V re the equivlent mgnetic current density nd terminl voltge respectively of slot when rditing in isoltion. Since b b H2 x / I2 H2x / I, it follows from equtions (2) nd (7) tht with Y 2 = Y 2. I b b I2 V2Y 2 I (8)

16 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 547 The finl step in the itertive procedure is to revisit sitution nd compute updted versions updte I, nd updte V, of the current I nd voltge V t port :, updte b I I I (9) V, updte I, updte Y self () An updted version of the mutul dmittnce Y 2 = Y 2 tht includes the effect of the bove higher-order interctions between the slots cn then be found from Y 2, updte Y2, updte V I2, updte () while n estimte Y, est of the two-port self-dmittnce Y (t this point only Y self is vilble) cn be computed s follows: Y, est V I, updte (2) The process of updting the two-port dmittnce prmeters cn be repeted itertively until the effect of the next higher-order interction is negligible. Fig. 3 shows the Y 2 -ginst-d curves for the clssicl reciprocity pproch, the itertive reciprocity-bsed formultions nd moment-method solver (IE3D) for the cse of two identicl (twin) resonnt slots. Y 2 ginst brodside slot seprtion d ws computed t GHz for CPW-fed slots on conductor-bcked two-lyer substrte configured for high rdition efficiency. The substrte hd h =.83 mm =.5λ d, h 2 = 5 mm = λ /6, ε r = 3.38, nd ε r2 = (λ d is the dielectric wvelength in the top lyer nd λ the free-spce wvelength). The substrte llowed for propgtion of the TM two-lyer prllel-plte mode only. Slot dimensions were W =.4 mm nd L = L res =.87 mm, yielding resonnt self-resistnce of 4 Ω; dimensions of the 5 Ω feed line were w = 3.7 mm nd s =.2 mm. The extended reciprocity-bsed curves resemble the shpe of the moment-method curves much closer thn the clssicl reciprocity-bsed curves. The reciprocity-bsed pproch for mutul dmittnce clcultions between CPW-fed slots on conductor-bcked two-lyer substrtes modified to ccount for higher-order interctions is vible lterntive to moment-method-bsed pproch, offering comprble ccurcy nd the dvntges of simplicity of implementtion within itertive rry design procedures.

17 548 Pssive Microwve Components nd Antenns Fig. 3. Mutul dmittnce Y 2 ginst brodside distnce d/λ CPW for CPW-fed twin slots with L = L 2 = L res =.87 mm computed using the clssicl nd extended reciprocity pproches, nd IE3D. W = W 2 =.4 mm; h =.83 mm; h 2 = 5 mm; ε r = 3.38; ε r2 = ; l f =.5λ CPW. 4. Liner rrys of CPW-fed slots 4. Uniform rry on conductor-bcked two-lyer substrte This section describes how n pproximte itertive design procedure for uniformly-excited CPW-fed liner slot rrys on single-lyer substrtes (Qiu et l., 22) cn be extended to the cse of n 8-element CPW-fed slot rry on conductor-bcked higher-lower permittivity two-lyer substrte (Jcobs et l., 23). When designing series-fed brodside liner slot rrys, it is desirble to hve slots excited in-phse nd with equl mgnitude for mximum gin. This implies tht slots should be spced λ CPW rther thn λ g /2 prt, even though the ltter the spcing would seem preferble due to the ensuing phse cncelltion of the substrte mode which would result in improved rdition efficiency (λ CPW is the wvelength of the CPW feed line; λ g is the wvelength of the dominnt substrte mode). Previously reported investigtions of liner rrys on single-lyer substrtes, nd λ d /4 substrtes with bck reflector positioned λ o /4 wy, however indicted tht λ CPW spcing (s opposed to λ g /2 spcing) does not ffect gin dversely (Qiu et l., 22). The top view lyout of generic CPW-fed liner rry is shown in Fig. 4; given the present context, ll slot hlf-lengths were ssumed to be equl nd likewise for slot widths. For simultion purposes, substrte lyer heights nd dielectric constnts were selected (s before) to ensure tht non-leky CPW feed line would result this requires higher-lower permittivity substrte, the higher permittivity lyer being djcent to the slots (Hung & Kuo, 998). The selected vlues were h =.83 mm, ε r = 3.38, h 2 = 5 mm (λ o /6 t GHz), nd ε r2 =. (see side view in Fig. ). Using IE3D, 5 Ω CPW feed line ws designed tht hd w = 3.7 mm nd s =.2 mm.

18 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 549 The rry (N = 8) ws designed for opertion t GHz using the following steps in n itertive fshion; the procedure relies on the ssumption tht slot ctive impednces re lrgely determined by the corresponding slot self-impednces (nd to lesser extent by mutul coupling).. The self-impednce Z in,slot of single slot dipole with hlf-length L nd (rditing slot) width W ws computed using IE3D (infinite ground plnes were ssumed throughout). For the first itertion, convenient vlues of W nd L were chosen. 2. An 8-element rry ws constructed of identicl slots spced λ CPW prt, ech with the W nd L determined in the previous step. The self-impednce Z in,rry of the rry ws determined using IE3D. 3. The difference Z diff between the rry input impednce nd the desired input impednce of 5 Ω ws found. 4. Using IE3D, new L nd W vlues were found tht would yield new slot selfimpednce of Z in,slot Z diff /N. 5. Steps (2) nd onwrds were repeted until Z in,rry ws within n cceptble mrgin from 5 Ω. The itertive procedure converged to L nd W vlues of 2.4 mm nd.7 mm respectively. In Fig. 4 mesured return loss ginst frequency is shown for the bove design mnufctured on 2 x8 Rogers RO43C substrte. The devition of the mesured operting frequency of 9.7 GHz from the design frequency of GHz cn be ttributed to mnufcturing nd mteril considertions tht likely included devitions from specified dielectric constnts, nd inconsistencies in the thickness of the polystyrene bottom lyer. The E-plne co-polrized rdition pttern mesured t 9.7 GHz is shown in Fig. 5. Good greement with the predicted pttern ws exhibited, indicting insignificnt lekge of the dominnt two-lyer prllel-plte mode from the sides of the ntenn. Sidelobe levels were below -3 db s expected except for sidelobe t 85º tht hd level of bout - db; this could hve been cused by n indvertently-excited higher-order mode on the CPW feed. Return loss (db) Frequency (GHz) Fig. 4. Mesured return loss ginst frequency for 8-element CPW-fed slot rry on conductor-bcked two-lyer substrte.

19 55 Pssive Microwve Components nd Antenns Mgnitude of electric field (db) Elevtion ngle (degrees) Fig. 5. E-plne co-polriztion pttern mesured t 9.7 GHz for 8-element CPW-fed slot rry on conductor-bcked two-lyer substrte. 4.2 Non-uniform rry on single-lyer substrte This section describes design of liner non-uniform slot rry series-fed by CPW, on n electriclly thin substrte for computtionl simplifiction, with sidelobe levels substntilly reduced beyond tht of uniform rry (Jcobs & Joubert, 29b). The geometry of such liner rry is s shown in Fig. 4. A low-sidelobe level specifiction typiclly requires explicit, ccurte ccounts of the mutul coupling between ll possible slot pirs in the rry, nd hence the bility to enforce suitbly tpered slot excittion. In wht follows, the itertive design of 5-element non-uniform liner CPW-fed rry with reduced sidelobe levels on n electriclly thin substrte is described. Becuse of the electriclly thin substrte, mutul coupling cn be ccounted for using the reciprocity-bsed method involving the ssumption of homogeneous free-spce medium, s discussed in section 3. (Jcobs & Joubert, 29) Arry design procedure The first step ws to synthesize 5-element Dolph-Tschebychev rry of isotropic rditors to hve mximum sidelobe level of -2 db below the min bem mximum (Blnis, 996). The spcing between the rditors ws d =.78λ, with λ the free-spce wvelength t the design frequency of 5.6 GHz (this frequency belongs to the upper WLAN bnd)..78λ equls λ CPW, the wvelength of 7 Ω CPW feed line on substrte with height h =.575 mm nd reltive permittivity ε r = 2.33, the substrte of choice for the physicl rry; the CPW centre strip nd slot widths were w = 3 mm nd s =.5 mm respectively (see Fig. 4). The im of this prticulr spcing ws to ensure in-phse excittion of the slots. The resulting excittion vlues were [V V 2 V 3 V 4 V 5 ] = [ ].

20 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 55 The second step entiled setting up slot self-dmittnce dtbse, s well s corresponding field distribution nd terminl voltge dtbses for the clcultion of mutul dmittnces. In prticulr, self-dmittnce dt ws generted using IE3D for grid of hlf-lengths nd widths of isolted CPW-fed slots: hlf-length vlues were between 8.5 nd 27.5 mm (t.75 mm intervl), nd widths were between.25 nd 3.25 mm (t.25 mm intervl). Aperture field nd terminl voltge dt ws collected concurrently with the selfdmittnce dt. The steps followed in itertively designing the rry re now outlined (see lso Hung et l. (999)) consider in this regrd the generic liner CPW-fed slot rry of Fig. 4 tht hs N slots spced λ CPW prt with slot hlf-lengths nd widths L m nd W m (m =, 2,..., N). In network terms, the rry cn be viewed s n equivlent trnsmission line circuit with the slots represented by series ctive impednces seprted by λ CPW -long sections of trnsmission line (Hung et l., 999; Meide et l., 22). An ctive voltge V exists cross the series ctive impednce representing slot m, while current I is common to ll slot impednces due to the series nture of the equivlent circuit. The corresponding ctive dmittnce of slot m is Y m = I/V m. For n N-element rry, the itertive design proceeds s follows:. A desired input impednce Z in,des is chosen (5 Ω for the purposes of this letter). The desired slot ctive dmittnces equtions: Y m, des cn then be found using the following two m Z in, des N = (3) Y m= m, des Y, des V... Y N, des V (4) N 2. Strting vlues for the lengths nd widths L, ) cn then be ssumed or ( m Wm clculted. Good strting vlues cn be obtined by initilly ssuming zero mutul coupling between the slots nd then clculting slot dimensions L, ) from the self-dmittnce dtbse tht would relize Y mm Y m, des. ( m Wm 3. These dimensions L, ) re then used to clculte the mutul dmittnces Y mn ( m Wm between ll possible slot pirs m nd n ccording to section 3. (Jcobs & Joubert, 29). A liner interpoltion scheme is used to clculte the updted mutul dmittnces Y mn during this step for specific L, ) dimensions (from the discreet ( m Wm field distribution nd terminl voltge dtbse compiled previously). 4. A set of 2N non-liner equtions cn then be solved to enforce the desired reltive slot excittions, the chosen mtching criteri, nd of course resonnce of the N slots in the rry. The mutul dmittnces Y mn re kept constnt s clculted in step (3), nd new set of L, ) re sought (in effect new set of self-dmittnces Y mm re ( m Wm determined) tht will stisfy the design equtions. The equtions re of the form:

21 552 Pssive Microwve Components nd Antenns Ym VN, for m =, 2,, N (5) YN Vm N Zin, des (6) m= Ym m Img( Y ) =, for m =,, N (7) Throughout the non-liner eqution solving process, bivrite spline interpoltion scheme (McNmr & Joubert, 994) is used to clculte new self-dmittnces Y mm for rbitry L, ) dimensions, using the previously compiled discreet self-dmittnce ( m Wm dtbse. 5. Updted mutul dmittnces for the new slot dimensions cn then be clculted, nd the true ctive dmittnces determined using N Vn Ym = Y mn, from which the ctul V input impednce of the rry cn then be found. If Z in is not close enough to the desired vlue of, steps 4 nd onwrds re repeted until convergence is obtined. Z in, des n= m Results nd discussion Appliction of the bove procedure yielded the following dimensions for the N = 5 rry: L = L 5 = 23.9 mm, W = W 5 =.3 mm, L 2 = L 4 = 23. mm, W 2 = W 4 =.5 mm, nd L 3 = 23. mm, W 3 = 2.2 mm. For the purpose of simulting the full rry in IE3D, short-circuit stub termintion of length l s = 2 mm λ CPW /2 ws dded (see Figure 4), s well s front-end CPW feed section of length λ CPW /2. The simulted rry ws etched on Rogers RT/duroid 587 lminte (h =.575 mm, ε r = 2.33±.2, tn δ =.9) of dimensions 23 mm 35 mm (see Figure 6), nd its reflection coefficient, rdition ptterns nd gin were mesured. Fig. 6. The mnufctured 5-element CPW-fed rry.

22 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 553 Fig. 7 shows the simulted nd mesured rry reflection coefficient ginst frequency. The simulted nd mesured resonnt frequencies were 5.6 GHz nd 5.59 GHz respectively, which re both very close to the design frequency of 5.6 GHz. The simulted frctionl bndwidth ws.5% (VSWR < 2), while the corresponding mesured bndwidth ws 2.%, confirming the inherently nrrow-bnd nture of the feeding scheme. Discrepncies between simultion nd mesurement cn likely be ttributed to mnufcturing errors. Fig. 8 displys the simulted nd mesured normlized E-plne rdition ptterns (the rry lies in the xy plne). The simulted co-polriztion pttern hs mximum sidelobe level of bout -7 db, which is higher thn the designed-for Dolph-Tschebychev sidelobe level of -2 db. This cn be ttributed to the ccurcy of mutul dmittnce clcultions. The mesured co-polrized pttern on the whole grees well with the simulted pttern (the discontinuity in the simulted pttern t Thet = ±9 is due to IE3D setting the electric field component perpendiculr to the infinite ground plne, i.e., the co-polrized component, to zero t fr-field observtion points coinciding with the ground plne; this pprently is n rtifct of the simultor). The somewht higher mximum sidelobe levels of bout -6 db in the mesurement cn possibly be ttributed to the finite substrte (simultions were crried out for lterlly infinite substrte). Fig. 8 lso gives the mesured cross-polriztion which is significntly below co-polriztion levels (simulted cross-polriztion ws too smll to show on this grph). Fig. 7. Simulted nd mesured reflection coefficient vs. frequency.

23 554 Pssive Microwve Components nd Antenns Fig. 8. Simulted nd mesured E-plne rdition ptterns. The mximum boresight gin of the rry is t 5.62 GHz, i.e., 4 dbi (predicted) nd 3.5 dbi (mesured). The corresponding rdition efficiency ws clculted using IE3D to be 97%, indicting negligible power loss to substrte modes due to the electriclly thin substrte. 5. Finl remrks The design of non-uniform liner CPW-fed slot rrys with sidelobe levels significntly reduced below tht of uniform rrys ws presented; efficient methods for ccomplishing importnt prts of the design, such s the clcultion of mutul coupling subject to representtive substrte configurtions, were summrized. Results of prcticl implementtion on n electriclly thin single-lyer dielectric substrte ws shown; mesured sidelobe levels of -6 db below could be chieved. 6. References Amity, N.; Glindo, V. & Wu, C. P. (972). Theory nd nlysis of phsed rry ntenns, Wiley-Interscience, ISBN : , New York. Blnis, C. A. (996). Antenn theory: nlysis nd design, 2 nd ed., ISBN-: , Wiley, New York. Chen, S.-Y.; Ln, I.-C. & Hsu, P. (27). In-Line Series-Feed Colliner Slot Arry Fed by Coplnr Wveguide, IEEE Trnsctions on Antenns nd Propgtion, Vol. 55, No. 6, pp , ISSN: Chen, W.-S. & Wong, K.-L. (2). A dul-frequency coplnr wveguide-fed slot ntenn, Microwve nd Opticl Technology Letters, Vol. 25, No. 3, pp , ISSN:

24 Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide 555 Elliott, R. S. (98). Antenn theory nd design, Prentice-Hll, ISBN: , Englewood Cliffs, New Jersey. Focrdi, P.; Neto, A. & McGrth, W. R. (22). Coplnr-wveguide-bsed tetrherts hotelectron-bolometer mixers improved embedding circuit description, IEEE Trnsctions on Microwve Theory nd Techniques, Vol. 5, pp , ISSN: Hung, J.-F. & Kuo, C.-W. (998). More investigtions of lekge nd nonlekge conductorbcked coplnr wveguide. IEEE Trnsctions on Electromgnetic Comptibility, Vol. 4, No. 3, pp , ISSN: Hung, T.-F.; Lu, S.-W. & Hsu, P. (999). Anlysis nd design of coplnr wveguide-fed slot ntenn rry, IEEE Trnsctions Antenns nd Propgtion, Vol. 47, No., pp , ISSN: 8-926X. Jcobs, J. P. (27). Self nd mutul dmittnce of CPW-fed slots on conductor-bcked twolyer substrtes, Microwve nd Opticl Technology Letters, Vol. 49, No., pp , ISSN: Jcobs J. P. & Joubert, J. (29). Fst nd efficient clcultion of mutul dmittnce between CPW-fed slots on electriclly thin substrtes, Interntionl Journl of RF nd Microwve Computer-Aided Engineering, Vol. 9, No. 2, pp , ISSN: Jcobs J. P. & Joubert, J. (29b). Design of liner nonuniform CPW-fed slot rry with reduced sidelobe levels, Microwve nd Opticl Technology Letters, Vol. 5, No. 9, pp , ISSN: Jcobs, J. P.; Joubert, J. & Odendl, J. W. (23). Conductor-bcked CPW-fed brodside liner slot ntenn rry on two-lyer dielectric substrte, Proceedings of Interntionl ITG-Conference on Antenns, pp. 6 63, Berlin, Sep. 23, Germny. Jcobs, J. P.; Joubert, J. & Odendl, J. W. (25). Effect of bck plne distnce on mutul coupling between CPW-fed slots on conductor-bcked two-lyer substrtes. Microwve nd Opticl Technology Letters, Vol. 47, No. 5, pp , ISSN: Jcobs, J. P.; Joubert, J. & Odendl, J. W. (29). Extended reciprocity-bsed computtion of mutul dmittnce between CPW-fed slots on conductor-bcked two-lyer substrtes. Microwve nd Opticl Technology Letters, Vol. 5, No., pp. 9-94, ISSN: Kim, S. H.; Choi, J. H.; Bik, J.W. & Kim, Y. S. (26). CPW-fed log-periodic dumb-bell slot ntenn rry, Electronics Letters, Vol. 42, No. 8, pp , ISSN: McNmr D. A. & Joubert, J. (994). On the use of bivrite spline interpoltion of slot dt in the design of slotted wveguide rrys, ACES Journl, Vol. 9, No., pp. 6-9, ISSN: Mio, M.; Ooi, B. L. & Kooi, P. S. (2). Brodbnd CPW-fed wide slot ntenn, Microwve nd Opticl Technology Letters, Vol. 25, No. 3, pp. 26 2, ISSN: Nuwelers, B. K. J. C. & Vn de Cpelle, A. R. (988). Integrls for the mutul coupling between dipoles or between slots: with or without complex conjugte. IEEE Trnsctions on Antenns nd Propgtion, Vol. 36, No.,, pp , ISSN: 8-926X. Neto, A.; De Mgt, P. & Mci, S. (23). Optimized bsis functions for slot ntenns excited by coplnr wveguides, IEEE Trnsctions on Antenns nd Propgtion, Vol. 5, No. 7, pp , ISNN:

25 556 Pssive Microwve Components nd Antenns Qiu, M.; Simcoe, M. & Eleftherides, G. V. (2). Rdition efficiency of printed slot ntenns bcked by ground reflector, Proceedings of 2 IEEE Antenns nd Propgtion Society Interntionl Symposium, pp , ISBN: , Slt Lke City, July 2, USA. Qiu, M.; Simcoe, M. & Eleftherides, G. V. (22). High-gin menderless slot rrys on electriclly thick substrtes t millimeter-wve frequencies, IEEE Trnsctions Microwve Theory nd Techniques, Vol. 5, No. 2, pp , ISSN: Solimn, E. A.; Brebels, S.; Beyne, E. & Vndenbosch, G. A. E. (999). 2x2 nd 4x4 rrys of nnulr slot ntenns in MCM-D technology fed by coplnr CPW networks, IEE Proceedings Microwves, Antenns nd Propgtion, Vol. 46, No. 5, pp , ISSN: Tsi, H. S.; Rodwell, M. J. W. & York, R. A. (994). Plnr mplifier rry with improved bndwidth using folded-slots, IEEE Microwve nd Guided Wve Letters, Vol. 4, No. 4 pp. 2 4, ISSN: Zelnd Softwre (2). IE3D User s Mnul, Relese 8.

26 Pssive Microwve Components nd Antenns Edited by Vitliy Zhurbenko ISBN Hrd cover, 556 pges Publisher InTech Published online, April, 2 Published in print edition April, 2 Modelling nd computtions in electromgnetics is quite fst-growing reserch re. The recent interest in this field is cused by the incresed demnd for designing complex microwve components, modeling electromgnetic mterils, nd rpid increse in computtionl power for clcultion of complex electromgnetic problems. The first prt of this book is devoted to the dvnces in the nlysis techniques such s method of moments, finite-difference time- domin method, boundry perturbtion theory, Fourier nlysis, mode-mtching method, nd nlysis bsed on circuit theory. These techniques re considered with regrd to severl chllenging technologicl pplictions such s those relted to electriclly lrge devices, scttering in lyered structures, photonic crystls, nd rtificil mterils. The second prt of the book dels with wveguides, trnsmission lines nd trnsitions. This includes microstrip lines (MSL), slot wveguides, substrte integrted wveguides (SIW), verticl trnsmission lines in multilyer medi s well s MSL to SIW nd MSL to slot line trnsitions. How to reference In order to correctly reference this scholrly work, feel free to copy nd pste the following: JP Jcobs, J Joubert nd JW Odendl (2). Design of Non-Uniformly Excited Liner Slot Arrys Fed by Coplnr Wveguide, Pssive Microwve Components nd Antenns, Vitliy Zhurbenko (Ed.), ISBN: , InTech, Avilble from: InTech Europe University Cmpus STeP Ri Slvk Krutzek 83/A 5 Rijek, Croti Phone: +385 (5) Fx: +385 (5) InTech Chin Unit 45, Office Block, Hotel Equtoril Shnghi No.65, Yn An Rod (West), Shnghi, 24, Chin Phone: Fx:

27 2 The Author(s). Licensee IntechOpen. This chpter is distributed under the terms of the Cretive Commons Attribution-NonCommercil- ShreAlike-3. License, which permits use, distribution nd reproduction for non-commercil purposes, provided the originl is properly cited nd derivtive works building on this content re distributed under the sme license.

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