Design of microwave networks with broadband directional couplers

Transcription

1 Dsign of microwav ntworks with broadband dirctional couplrs

2 6 Contnts Summary...7 Strszczni...8 List of symbols.... Introduction.... Microwav dirctional couplrs, powr dividrs and phas shiftrs Dirctional couplrs and powr dividrs with dirctly connctd transmission lins 9.. Dirctional couplrs and phas shiftrs utilizing coupld transmission lins Singl-sction coupld-lin dirctional couplrs Multisction coupld-lin dirctional couplrs Broadband phas shiftrs Dsign of coupld-strip-transmission-lin dirctional couplrs and phas shiftrs with improvd frquncy rspons Summary Dirctional couplrs in application to balancd and n-way microwav circuits Balancd and n-way microwav circuits Larg signal charactrization of solid-stat dvics Asymmtric coupld-lin dirctional couplrs as impdanc transformrs in balancd and n-way powr amplifirs Approach to th dsign of asymmtric coupld-lin impdanc transforming dirctional couplrs Broadband asymmtric coupld-lin impdanc transforming dirctional couplrs Summary Broadband Butlr matrics utilizing coupld-lin dirctional couplrs Butlr matrics utilizing singl-sction coupld-lin dirctional couplrs x 4 Butlr matrix x 8 Butlr matrix Butlr matrics utilizing multisction coupld-lin dirctional couplrs Butlr matrics with th us of /8 dirctional couplrs Summary Dsign of miniaturizd broadband dirctional couplrs Miniaturization tchniqus in th dsign of microwav dirctional couplrs Dsign of broadband quasi-lumpd dirctional couplrs Summary Final rmarks...46 Rfrncs...48

3 KRYSTOF WINCA Dsign of microwav ntworks with broadband dirctional couplrs Summary Th monograph focuss on th dsign of complx microwav ntworks with th us of broadband coupld-lin dirctional couplrs. Thr diffrnt aras of th dsign and application of such couplrs hav bn addrssd, which ar:. Dsign of coupld-lin impdanc transforming dirctional couplrs.. Dsign of broadband Butlr matrics utilizing coupld-lin dirctional couplrs. 3. Dsign of miniaturizd dirctional couplrs with th us of quasi-lumpdlmnt tchniqu. All aspcts hav bn comprhnsivly rsarchd by th Author ovr th rcnt yars. In particular, th application of asymmtric coupld-lin dirctional couplrs in balancd and n-way powr amplifirs has bn dscribd, and th dsign of such circuits has bn shown with th mphasis on thir impdanc transforming capability. Th maor advantag of th proposd approach is that it allows to raliz both powr dividing and impdanc matching within a singl componnt. Singl-sction and multisction impdanc transforming dirctional couplrs ar analyzd and th achivabl impdanc transformation ratio for such couplrs is drivd. Th scond issu prsntd in th monograph concrns th dsign of broadband Butlr matrics. A mthod for th dsign of broadband Butlr matrics utilizing coupld-lin dirctional couplrs is prsntd, in which multisction symmtrical coupld-lin dirctional couplrs ar usd. Th mthod proposd by th Author of broadband diffrntial phas shiftrs ralization for applications in such ntworks, is outlind. It is shown, that th dsign of broadband Butlr matrics with th us of multisction dirctional couplrs rquirs applying of Schiffman C -sctions in ordr to minimiz th phas imbalanc of th rsulting ntwork. Morovr, a novl arrangmnt of an 8 x 8 Butlr matrix is proposd, that allows for planar fully intgratd ralization. Finally, th dsign mthod of miniaturizd broadband dirctional couplrs, dvlopd by th Author, is dscribd, in which a quasi-lumpd-lmnt tchniqu is usd. Th proposd approach allows for achiving th frquncy rspons of miniaturizd couplrs comparabl to th rspons of thir distributd countrparts. Morovr, th mthod allows for achiving broad bandwidths, du to its suitability for th dsign of multisction miniaturizd dirctional couplrs. Th prsntd thortical analyss hav bn confirmd by th masurmnts of a numbr of manufacturd coupld-lin ntworks. 7

4 KRYSTOF WINCA Proktowani układów mikrofalowych wykorzystuących szrokopasmow sprzęgacz kirunkow Strszczni W ninisz monografii przdstawion zostały zagadninia związan z proktowanim złoŝonych sici mikrofalowych wykorzystuących szrokopasmow sprzęgacz kirunkow. Przdstawion zostały następuąc trzy róŝn aspkty proktowania i zastosowań szrokopasmowych sprzęgaczy o liniach sprzęŝonych:. Proktowani sprzęgaczy o liniach sprzęŝonych transformuących impdancę.. Proktowani szrokopasmowych macirzy Butlra wykorzystuących sprzęgacz o liniach sprzęŝonych. 3. Proktowani zminiaturyzowanych szrokopasmowych sprzęgaczy kirunkowych z wykorzystanim tchniki lmntów quasi skupionych. Wyminion obszary zostały dogłębni przanalizowan przz Autora monografii na przstrzni ostatnich kilku lat. W szczgólności, przanalizowana została moŝliwość zastosowania asymtrycznych sprzęgaczy kirunkowych w zrównowaŝonych wzmacniaczach mocy mikrofalow oraz w układach z sumowanim mocy, biorąc pod uwagę moŝliwy do pozyskania stosunk impdanci obciąŝnia w wrotach wściowych i wyściowych. Główną zaltą zaproponowanych rozwiązań st moŝliwość dnoczsn ralizaci funkci podziału mocy i dopasowania impdancyngo w dnym układzi. Przanalizowan zostały zarówno układy dnoskcynych ak i wiloskcynych asymtrycznych sprzęgaczy kirunkowych i wyprowadzon zostały zalŝności opisuąc moŝliwy do pozyskania w tych układach stosunk impdanci. Kolnym przdstawionym w pracy zagadninim st proktowani szrokopasmowych macirzy Butlra. Opisana została mtoda proktowania takich układów z wykorzystanim wiloskcynych sprzęgaczy kirunkowych o liniach sprzęŝonych. aprzntowana została takŝ, opracowana przz Autora, mtoda ralizaci szrokopasmowych przsuwników fazy o stałym róŝnicowym przsunięciu fazy wykorzystywanych w układach macirzy Butlra. Przprowadzon analizy wykazały, Ŝ proktowani szrokopasmowych macirzy Butlra wykorzystuących wiloskcyn sprzęgacz o liniach sprzęŝonych wymaga zastosowania skci C przsuwników fazy Schiffmana w clu minimalizaci rozrównowaŝnia charaktrystyk fazy róŝnicow. Dodatkowo, w pracy zaproponowan zostało now rozwiązani układow pozwalaąc na ralizacę w płni planarn, zintgrowan macirzy Butlra 8 x 8. Ostatnim zagadninim przstawionym w monografii st mtoda proktowania szrokopasmowych zminiaturyzowanych sprzęgaczy kirunkowych, zaproponowana przz Autora, w któr znaczn zmniszni rozmiaru ralizowanych układów pozysku się poprzz zastosowani tchniki lmntów quasi skupionych. Opracowana mtoda pozwala na ralizacę miniaturowych sprzęgaczy kirunkowych cchuących się paramtrami po- 8

5 równywalnymi z sprzęgaczami zralizowanymi w tchnic sprzęŝonych linii transmisynych. Ponadto, mtoda ta umoŝliwia pozyskani szrokigo pasma pracy, z względu na fakt, Ŝ pozwala na proktowani wiloskcynych, zminiaturyzowanych sprzęgaczy kirunkowych. Przdstawion w pracy analizy tortyczn zostały potwirdzon poprzz pomiary wykonanych modli zaproktowanych i wykonanych układów mikrofalowych. 9

6 List of symbols β phas constant Γ rflction cofficint ε prmittivity of fr spac µ prmability of fr spac λ fr spac wavlngth λ g guidd wavlngth θ lctrical lngth ν p, ν po vn, odd mod phas vlocity ε r dilctric constant (rlativ prmittivity) ε r, ε ro ffctiv dilctric constant (rlativ prmittivity) for vn, odd mod BW oprational bandwidth C capacitanc matrix c fr spac light vlocity C coupling C, pr unit lngth capacitanc of lin, C, C o pr unit lngth vn, odd mod capacitanc of coupld lins C m pr unit lngth mutual capacitanc of coupld lins D dirctivity I isolation IL insrtion losss k coupling cofficint k L, k C inductiv, capacitiv coupling cofficint L inductanc matrix l lngth of a coupld-lin sction L, pr unit lngth slf inductanc of lin, L m pr unit lngth mutual inductanc of coupld lins R impdanc ratio of th transformr RL rturn losss S scattring matrix S,o scattring matrix for vn and mod xcitation T transmission charactristic impdanc, o vn, odd mod charactristic impdanc T, st, nd lin trminating impdanc VSWR voltag standing wav ratio

7 . Introduction In th dsign of microwav circuits, dirctional couplrs and powr dividrs/combinrs constitut an important class of passiv componnts, du to varity of thir applications. On can nam antnna fding ntworks, balancd mixrs, balancd amplifirs, rflctomtr bridgs, six-port ntworks for complx rflction cofficint masurmnts, tc. Among dirctional couplrs two gnral groups of ntworks can b distinguishd, which ar: dirctly connctd couplrs and powr dividrs/combinrs and coupldlin couplrs and powr dividrs/combinrs. Ths two groups of ntworks diffr by th principl of opration and, thrfor, by thir proprtis. In th first group a circuit is formd by a ntwork consisting of a numbr of dirctly connctd transmission-lin sctions. A wll-known xampl of such a dvic, is a branch-lin dirctional couplr, which is as all th dirctional couplrs in gnral a four-port losslss ntwork, in which a signal dlivrd to th input port is dividd with th spcifid ratio btwn th coupld and dirct ports. At th fourth port calld isolatd port no powr is rcivd. Th most broadly 3- db quadratur couplrs ar usd, in which th input powr is qually split btwn th coupld and transmission ports. Additionally, such couplrs hav uniqu phas proprtis, i.. th signals rcivd at coupld and transmission ports ar 9 out-of-phas. Anothr wll-known dvic blonging to th group of dirctional couplrs having dirctly connctd transmission-lins is a rat-rac couplr, in which th phas rlation is diffrnt from th branch-lin couplr. In this cas th signals rcivd at th coupld and transmission ports ar 8 out-of-phas. Th scond group of dvics utilizs lctromagntic coupling btwn transmission lins, which ar not dirctly connctd, and th powr from th main lin is coupld lctromagntically to th scond lin. Th basic componnt blonging to this group, is a quartrwav-long dirctional couplr having th proprty of 9 diffrntial phas charactristic, similarly to th branch-lin couplr. Th two groups of dvics diffr in thir construction, but what is mor important from th application point of viw, also significantly diffr by thir proprtis. Th dirctly connctd transmission-lin componnts fatur narrow oprational bandwidth, and thrfor, ar not suitd for modrn tlcommunication applications, whr it is rquird to oprat in rlativly larg bandwidths, or altrnativly in multipl bands with th us of singl hardwar. Such a dmand can b mt whn coupld-lin dirctional couplrs ar applid. Apart from allowing for broadband opration, thy also provid for significant miniaturization, sinc th ara ndd for a typical 3-dB branch-lin couplr is much gratr thn th ara of its coupld-lin countrpart.

8 Th dsign of coupld-lin dirctional couplrs has bn a subct of xtnsiv studis ovr th yars, starting from arly works [9, 3, 9, 9, 93, 4,, 37], whr th basic paramtrs of coupld-lin dirctional couplrs hav bn formulatd. Th bginning of TEM coupld-lin couplrs is datd to mid 5s, whn two-wir coupld symmtrical lins and coupld striplins hav bn considrd. Th dvlopmnt of symmtrical coupld-lin couplrs, in which th coupld-lin structur faturs symmtry btwn coupld conductors, ld to th dscription of coupld-lin sction proprtis, in trms of vn- and odd mod charactristic impdancs, which ar dirctly rlatd to th coupling of th couplr and its trminating impdancs. It was also shown in [9, 9, 93, 37], that th bandwidth of such couplrs can b furthr incrasd by a cascad connction of a numbr of coupld-lin sctions having appropriatly chosn coupling cofficints. Anothr class of dirctional couplrs offring vry broad frquncy rspons is th class of taprd-lin couplrs, in which th coupling cofficint continuously changs along th coupld lins. Thortically, taprd-lin asymmtric dirctional couplrs hav high-pass frquncy charactristics, howvr in practic th bandwidth is limitd by th discontinuity ffcts in th rgion of connction of signal and coupld lins and also by th manufacturing tolrancs. Furthr, in mid 7s asymmtric coupld-lin structurs bcam of intrst of rsarch community, sinc such componnts can b asir incorporatd into modrn microwav intgratd circuits and monolithic microwav intgratd circuits. In this class of couplrs, no symmtry btwn coupld conductors xists, and th proprtis of such structurs ar dfind by modal impdancs of and c mods, and by th rspctiv voltag mod numbrs. Th thortical analysis of asymmtric coupld-transmission-lins has bn shown in svral paprs [3, 46, 47, 48, 49, 5]. It has bn shown, that such couplrs fatur nithr prfct isolation nor prfct impdanc match, unlss th two conditions of quasiidal coupld-lin sction ralization ar fulfilld, i.. th inductiv and capacitiv coupling cofficints ar qual and th coupld conductors ar trminatd by appropriat impdancs [7]. A lot of attntion has bn paid to th dvlopmnt of diffrnt mthods that allow for quasi-idal asymmtric coupld-lin dirctional couplr ralization [4, 5, 6, 3]. Th Author has focusd at th arly stag of his rsarch on th dsign mthods of high-prformanc broadband coupl-lin dirctional couplrs, in which capacitiv lmnts ar addd to th coupld-lins in various cross-sctions and to th signal lins. Such an approach allows for significant improvmnt of th couplrs proprtis. Th capacitiv componnts compnsat th parasitic ractancs within th coupld-lin structurs which svrly dtriorat thir prformanc, spcially thir rturn losss and isolation [63, 64, 65, 69, 7]. Scondly, th mthods hav bn proposd which allow for mting th conditions of idal couplr ralization in symmtric and asymmtric structurs with th us of lumpd-lmnt tchniqu [6, 65, 67]. In on of th rcnt co-authord paprs [7] th gnralizd mthods for th dsign of quasi-idal symmtric and asymmtric coupld-lin sctions hav bn proposd. Th prsntd approach allows for dsigning dirctional couplrs with improvd prformanc ovr broad frquncy rang in arbitrary dilctric structurs, in which th inductiv coupling cofficint is ithr gratr or smallr than th capacitiv on. Th dvlopd mthods allowd for th dsign of a numbr of highprformanc singl and multisction dirctional couplrs, in both symmtric and asymmtric coupld-lin structurs [6, 6, 6, 63, 64, 65, 69, 7, 7, 65, 66, 67, 69]. This monograph is focusd on th application of broadband high-prformanc coupld-lin sctions in complx microwav ntworks within th thr aras of applications, that hav bn comprhnsivly invstigatd by th Author, which ar:

9 . high powr amplifirs,. broadband Butlr matrics, 3. broadband miniaturizd dirctional couplrs. Th dsign of high-powr, solid-stat amplifirs oprating in a microwav frquncy rang is most oftn approachd by th us of multipl-transistor powr stags, in which th input powr is appropriatly distributd among th powr amplifirs, ach having output powr bing a fraction of th total output powr. Th total output powr is obtaind by combining th powr from all th amplifir stags with th us of a powr combining ntwork. Typically, th componnts usd for th dsign of powr dividing/combining at microwav frquncis ar th dirctly connctd branch-lin dirctional couplrs, Wilkinson dividrs or Gysl-typ dividrs [54, 75, 7, 75]. Such ntworks fatur rlativly larg dimnsions in a lowr frquncy rang, and also can introduc insrtion losss, limiting th maximum output powr of an amplifir. Th Author has proposd in [79] an altrnativ approach to th dsign of powr dividing/combining ntworks, in which asymmtric coupld-lin dirctional couplrs ar utilizd. As it was shown, such couplrs ar utmost suitabl for th dsign of balancd powr amplifirs and corporat ntworks of N-way amplifirs, allowing for a siz rduction of th rsulting ntwork and, what follows dirctly from th siz rduction, allowing for insrtion losss minimization. Morovr, it has bn shown that such couplrs can provid for impdanc transformation, apart from splitting/combining microwav signals. By th appropriat dsign, an asymmtric coupld-lin dirctional couplr can hav proprtis of an idal dirctional couplr at a cntr frquncy, whn transmission and coupld ports ar trminatd with impdancs lowr (or highr), thn th sourc impdanc. It is thrfor possibl to dsign ntworks having th output impdanc much lowr that th sourc impdanc, which is typically rquird sinc th solid-stat high-powr dvics usually fatur input/output impdancs wll blow Ω. Impdanc transformation within powr dividing/combining ntworks approachs th aspct of miniaturization of amplifir units, and minimization of losss within th ntwork. As it was shown, th coupld-lin powr transforming dirctional couplrs do not fatur th proprtis of an idal dirctional couplr in a wid frquncy rang, but oprat in a frquncy rang sufficint for such applications, sinc thy offr th bandwidth widr that th branch-lin couplrs or singl-sction impdanc transformrs. It has also bn shown that th oprational bandwidth can b incrasd by dsigning multisction impdanctransforming dirctional couplrs. Such dirctional couplrs combin in a singl componnt th ida of broadband asymmtric multisction dirctional couplrs, dscribd in [9, 93], and multisction impdanc transformrs, prsntd in []. Anothr important class of microwav ntworks, in which high-prformanc coupldlin sctions find broad rang of applications ar Butlr matrics. Such ntworks hav bn first introducd in [] and ar widly usd in contmporary microwav lctronics. Gnrally, an N x N Butlr matrix is a multiport ntwork having N inputs and N outputs, in which a signal applid to on of its input ports is qually dividd into all output ports. Morovr, th signals rcivd at output ports fatur phas shifts dpndnt on th matrix ordr and th choic of th xcitation port. Ths uniqu proprtis rsult in a broad rang of thir applications in contmporary communication systms;.g. as bamforming ntworks of multibam antnnas [4, 6,, 3, 7, 8, 6, 35, 36, 37, 45, 48, 49, 5, 6, 67, 68, 79, 8, 87, 94, 5, 36, 58, 59, 6, 63, 64, 69], in dirction finding systms [96, 4], or in multichannl amplifirs [7, 34, 7, 76]. Rcntly, larg attntion has bn paid to th dsign of such ntworks in various structurs. Most of th dscribd Butlr matrics hav 3

10 bn ralizd with th us of branch-lin couplrs dvlopd in diffrnt wavguiding structurs, such as microstrip, striplin coplanar wavguid, slotlin or rctangular wavguid [34, 45, 49, 77, 87,9,,, 7, 56]. As it was alrady mntiond, such ntworks fatur disadvantags of th applid couplrs, which is narrow oprational band and rlativly larg spac. Widr oprational bandwidth can b achivd whn multisction branch-lin couplrs ar applid at th xpns of much largr occupid ara [36, 44, 5]. Thr ar fw xampls of ralization of Butlr matrics with th us of coupld-lin dirctional couplrs. In [] a compact 4 x 4 Butlr matrix is prsntd, in which broadsid coupld-lin dirctional couplrs ar usd. Howvr, in this dsign th bandwidth is limitd by th application of narrow phas shiftrs ralizd as sctions of transmission lins. Th Author has rsarchd th possibility of broadband Butlr matrics ralization for application in multibam antnnas with th us of singl-sction coupld-lin dirctional couplrs and th rsults ar publishd in a numbr of paprs [6, 67, 58, 59, 6, 63, 64, 69]. In [64, 69] a broadband 4 x 4 Butlr matrix, which consists of six singlsction coupld-lin dirctional couplrs dsignd in a striplin tchniqu has bn dscribd. In this solution a tandm connction of two couplrs srvs as a transmission-lin crossovr and provids, togthr with sctions of transmission lins, broadband 45 phas shiftrs. A similar approach of a broadband 4 x 4 Butlr matrix ralization can b applid in an asymmtric microstrip structur, as it was shown in [6]. Apart from communication systm applications, Butlr matrics ar usd as a basic lmnt of instantanous dirction finding systms, primarily for military applications. In such systms it is rquird for th ntworks to oprat in a vry wid frquncy rang. Th Author has also focusd on th dsign and dvlopmnt of ultrabroadband Butlr matrics. In [68] an innovativ approach of broadband multi-octav 4 x 4 Butlr matrics ralization has bn shown, in which multisction symmtrical 3dB/9 dirctional couplrs ar applid. Highr ordr Butlr matrics, such as 8 x 8 or highr, ar rarly rportd, du to thir larg complxity, and thrfor, difficultis with dsigning and manufacturing as intgratd units. Exmplary dsigns can b found in [7, 53, 8], whr diffrnt tchniqus such as LTCC, CMOS or rctangular wavguid hav bn applid. An imprssiv dsign of an 8 x 8 Butlr matrix is prsntd in [35]. In this work a tandm connction of taprd-lin dirctional couplrs having coupling C = 8.34 db hav bn usd as a broadband 3-dB dirctional couplr and discrtly stppd 6-sction F-typ Schiffman ntworks hav bn usd to form th ndd phas shiftrs. Th obtaind bandwidth of th dvlopd 8 x 8 Butlr matrix quals BW = 9:. Howvr, th matrix has bn dsignd in a modular fashion, and th moduls hav bn connctd with th coaxial cabls. A possibility of ralization of fully intgratd broadband 8 x 8 Butlr matrics with th us of coupld-lin dirctional couplrs has bn also invstigatd by th Author. Th third aspct addrssd in this monograph concrns miniaturization of broadband dirctional couplrs. Th problm of siz rduction of passiv componnts is spcially important at lowr frquncy rangs, whr th siz of componnts bcoms significant. Morovr, th siz rduction is a primary obctiv, whn a passiv componnt is to b dirctly usd in advancd tchnologis such as LTCC or MMIC. Miniaturization of microwav circuits is oftn approachd with th us of lumpd or smi-lumpd-lmnt tchniqus [5, 39, 57]. Many authors hav focusd on miniaturization of dirctly connctd transmission-lin dirctional couplrs and powr dividrs. Th principls of th dsign of lumpd-lmnt branch-lin couplrs hav bn comprhnsivly invstigatd, and th dsign procdurs for various ralizations of such ntworks hav bn shown in [57]. In 4

11 [, 3], rat-rac couplrs ar prsntd, in which opn stubs and coupld-lin C -sctions ar usd, rspctivly, for providing significant minaturization. Othr dsigns of miniaturizd dirctly connctd transmission-lin couplrs and powr dividrs can b found in [9, 85, 86, 7, 3, 8, 9, 4, 5, 54, 74, 77]. Exmplary ralizations of similar componnts in LTCC and MMIC tchnology can b found in [4, 76, 88, 98, 38], whr lumpd-lmnt approach is utilizd. In this approach sctions of transmission lins ar substitutd by ntworks consisting of lumpd capacitors and inductors. Sinc much broadr bandwidth can b achivd with th us of coupld-lin dirctional couplrs an ffort has bn mad to dvlop miniaturizd vrsions of such dvics. Th commonly usd approach assum vry simpl quivalnt circuit of coupld lins consisting of only singl-sction approximation [8, 47, 9, 39, 55]. Th obtaind prformanc of such couplrs is limitd by th assumd quivalnt circuit, and is rflctd in th poor amplitud charactristics which narrow th couplr s oprational bandwidth. Th Author has focusd on th dsign of miniaturizd dirctional couplrs having attractiv frquncy rspons comparabl with th distributd coupld-lin dirctional couplrs [66]. Th influnc of th numbr of subsctions usd for approximation of a coupld-lin dirctional couplr on th rsulting prformanc has bn invstigatd. Th prformd analysis rval that thr subsctions ar sufficint for ralization of dirctional couplrs having good proprtis in a wid frquncy rang, which on th othr hand nsurs a significant miniaturization of th circuit. A gnral dscription of microwav passiv componnts with dirctly connctd transmission-lins is prsntd in th scond chaptr followd by th introduction to th coupld-lin dirctional couplrs. Th dscription of modal analysis of symmtrical coupld lins has bn shown togthr with th formulas for inhomognous dilctric stratification. Furthr, multisction coupld-lin dirctional couplrs ar brifly dscribd showing th possibility of broadning th couplrs oprational bandwidth. Th chaptr is concludd with a brif ovrviw of th mthods invstigatd by th Author, which allow for th dsign of coupld-lin sctions and dirctional couplrs with improvd prformanc. Th third chaptr focuss on th dsign of dirctional couplrs in applications to balancd and n-way microwav circuits. A gnral thory of balancd amplifirs is prsntd showing thir main advantags. Subsquntly, a mthod for larg-signal charactrization of microwav powr transistors is dscribd, in which rat-rac couplrs with sctions of shortd transmission lins ar usd, as input and output tuning circuits. It has bn shown, that such a rat-rac couplr allow for ralization of any passiv impdanc, thrfor, can b utilizd as a losslss matching ntwork at th input and output of a powr transistor. Furthr, a novl approach has bn prsntd, in which asymmtric coupld-lin 3-dB dirctional couplrs ar usd as simultanous powr dividrs and impdanc transformrs. It has bn shown, that such a couplr can offr th proprtis of an idal dirctional couplr having qual trminating impdancs at coupld and transmission ports diffrnt than th trminating impdancs of th xcitd and isolatd ports. Furthr analysis rval that physically ralizabl impdanc transformation ratio is limitd, and for th cas of a 3-dB dirctional couplr quals. Thn it is possibl to dsign 3-dB asymmtric dirctional couplrs transforming simultanously 5 Ω signal sourc impdanc into 5 Ω load. Highr transformation ratios can b obtaind whn corporat ntworks consisting of appropriatly dsignd 3-dB impdanc transforming dirctional couplrs ar usd. Th thortical considrations hav bn confirmd by th dsign of a 4-way powr amplifir oprating in L- band. Furthr, an altrnativ approach to th analysis and dsign of 3-dB impdanc trans- 5

12 forming dirctional couplrs having th maximum obtainabl impdanc ratio has bn prsntd. In this approach coupld-lins ar rplacd by th appropriat connction of two uncoupld transmission-lin sctions. Also an xmplary ralization is shown, in which th couplr is composd of a coaxial transmission-lin sction in conunction with a microstrip sction. Finally, th possibility of ralization of broadband coupld-lin impdanc transforming dirctional couplrs is prsntd. As an xampl, th dsign of a two-sction dirctional couplr oprating ovr 3: frquncy band and having th impdanc ratio qual.43 is shown. In th fourth chaptr broadband Butlr matrics ar dscribd and thir basic proprtis and applications ar outlind. A brif ovrviw of th dvlopd broadband 4 x 4 Butlr matrics utilizing singl-sction coupld-lin couplrs hav bn prsntd. Two dsigns in diffrnt coupld-lin structurs ar shown, in which a tandm connction of two 3-dB/9 coupld-lin dirctional couplrs in conunction with transmission-lin sctions of appropriat lngth ar usd, for simultanous ralization of a transmission-lin crossovr and broadband 45 phas shiftrs. Subsquntly, a novl arrangmnt of an 8 x 8 Butlr matrix is proposd, that allows for ralization of a fully intgratd, planar circuit oprating ovr a broad frquncy rang. Th proposd mthod has bn xprimntally vrifid by masurmnts of th 8 x 8 Butlr matrix oprating ovr GHz frquncy rang, proving th usfulnss of th prsntd approach. Th chaptr also dscribs th ralization of broadband Butlr matrics in which multisction 3-dB symmtrical dirctional couplrs ar utilizd. It is shown that in ordr to xtnd th ida of broadband 45 phas shiftrs ralization with th us of a tandm connction of 3-dB couplrs, it is rquird to apply phas corrcting ntworks consisting of multisction coupld-lin Schiffman C -sctions in conunction with transmission-lin sctions. A class of 4 x 4 Butlr matrics bing a connction of multisction 3dB/9 symmtrical coupld-lin dirctional couplrs has bn invstigatd, and tabls giving normalizd vn-mod impdancs of th phas corrction ntworks ndd for rspctiv coupld-lin couplrs hav bn providd. Th proposd concpt has bn xprimntally vrifid by two dsigns of 4 x 4 Butlr matrics utilizing thr-sction dirctional couplrs, and an ultrabroadband Butlr matrix oprating within.8 GHz frquncy rang dsignd with th us of fiv-sction dirctional couplrs. Th prsntd concpt has bn also xtndd on 8 x 8 broadband Butlr matrics and confirmd by th dsign of an 8 x 8 Butlr matrix with th us of fiv-sction dirctional couplrs. Finally, Butlr matrics consisting of taprd-lin dirctional couplrs hav bn considrd. Th fifth chaptr prsnts th rsults of Author s invstigation on th dsign of miniaturizd dirctional couplrs having th proprtis of coupld-lin distributd couplrs. Th proposd dsign mthod allows for significant miniaturization for structurs, in which th achivabl inductiv coupling cofficint is much gratr thn th nominal coupling of th dsignd couplr. Th proposd approach is basd on quasi-lumpd tchniqu, in which coupld-inductors ar ralizd as short sctions of tightly-coupld transmission lins, whil th ndd slf and mutual capacitancs ar ralizd as lumpd capacitors. Th prsntd mthod allows on to achiv good proprtis of th dsignd couplrs ovr a wid frquncy rang with significant miniaturization of th circuit. Morovr, th Author has shown that such a tchniqu is suitabl for ralization of broadband multisction dirctional couplrs. 6

13 Th Author prviously publishd works rlatd to th dsign of microwav ntworks utilizing broadband dirctional couplrs ar citd in larg xtnd throughout th scop of th monograph [59, 6, 6, 68, 64, 66, 68, 79]. Th rsarch rsults prsntd in this monograph hav bn in larg part financially supportd by th Polish Ministry of Scinc and Highr Education undr th following rsarch grants:. Rsarch grant no. 464/T//7, Microwav coupld-lin circuits - dsign, optimization and applications in modrn tlcommunication dvics and systms, ralizd at th AGH Univrsity of Scinc and Tchnology undr th suprvision of th Author.. Rsarch grant no. N , Broadband microwav Butlr matrics dsignd in coupld striplin tchniqus, ralizd at th Wroclaw Univrsity of Tchnology with th Author s substantial contribution. 3. Rsarch grant no. 4999/B/T//4, Miniaturizd microwav broadband dirctional couplrs dsignd with th us of quasi-lumpd tchniqu, ralizd at th AGH Univrsity of Scinc and Tchnology undr th suprvision of th Author. 7

14 . Microwav dirctional couplrs, powr dividrs and phas shiftrs Dirctional couplrs and powr dividrs find broad applications in microwav ntworks and systms. Thy prform a varity of functions, such as splitting and combining powr in mixrs, sampling powr of sourcs for lvl control, sparating incidnt and rflctd signals in ntwork analyzrs, or dividing powr among a numbr of loads. Gnrally two typs of ntworks can b distinguishd:. connctd-lin couplrs and powr dividrs,. coupld-lin couplrs and powr dividrs. Th chosn ralization tchniqu implis th proprtis of dsignd componnts, spcially th oprational bandwidth. Th most simpl and wll-known among powr dividrs is th Wilkinson powr dividr, which blongs to th first typ of ntworks. Th basic Wilkinson powr dividr oprats in rlativly narrow band, but a modification in which mor than on sction of transmission lin pairs is applid allows for th bandwidth broadning. Such lmnts can also b dsignd with th us of coupld-lin sctions, which also improv th oprational bandwidth. In cas of dirctional couplrs a significant diffrnc in frquncy rspons is obsrvd btwn th couplrs ralizd with th us of connctdlin tchniqu and th ons dsignd with th us of coupld-lin tchniqu. Th first group of couplrs, among which a branch-lin couplr is th most common xampl, faturs narrow oprational bandwidth which can b also nhancd by applying a multisction branch-lin couplr at th xpns of significant incras of th couplrs dimnsions, and thrfor, insrtion losss. Significant improvmnt of oprational band is achivd whn coupld-lin dirctional couplrs ar usd. A singl-sction of coupld lins can constitut a dirctional couplr having oprational bandwidth up to on frquncy octav, which can b furthr nhancd by dsigning multisction coupld-lin dirctional couplrs. In this Chaptr a brif introduction to th connctd-lin couplrs and powr dividrs has bn shown and th proprtis of both 3-dB Wilkinson powr dividrs and dirctional couplrs hav bn dscribd. Furthr, an analysis of an coupld-lin sction has bn prsntd and its utilization for dsigning singl-sction and multisction symmtrical as wll as asymmtrical dirctional couplrs and phas shiftrs, has bn outlind. Finally, th conditions of idal couplr ralization in symmtric and asymmtric coupld-lin structurs hav bn dscribd and xmplary dsigns of singl-sction 3-dB and 8.34-dB dirctional couplrs ralizd in symmtric striplin tchniqu and a thr-sction 3-dB dirctional couplr dsignd in multilayr microstrip tchniqus, hav bn shown. 8

15 .. Dirctional couplrs and powr dividrs with dirctly connctd transmission lins Powr dividrs and dirctional couplrs ar passiv microwav componnts usd for powr division and powr combining. A signal applid to th input port is dividd by th couplr/dividr into two or mor signals of lowr powr. Typically th powr dividr is a 3- port lossy or losslss componnt and tak th form of T-unction, but also dividrs with gratr numbr of ports can b pointd out for n-way powr splitting, whras, a dirctional couplr is a 4-port componnt. Th simplst typ of powr dividr is a T-unction which is a 3-port ntwork having on input and two output ports. Th scattring matrix of a 3-port ntwork has nin indpndnt lmnts: S [ S] = S S 3 S S S 3 S S S (.) For a passiv circuit that contains no anisotropic matrials its S matrix must b symmtric (S i, = S,i ) and th circuit must b rciprocal. It is also dsirabl that th circuit dos not introduc powr losss, which mans that th T-unction should b losslss and matchd at all ports. Howvr, it can b asily shown that it is impossibl to construct a 3-port ntwork that is losslss, rciprocal and matchd at all ports. If all ports of th ntwork ar matchd thn all diagonal lmnts of its S matrix ar zro, and taking into account rciprocal proprtis, th S matrix taks form [ S] = S S 3 S S 3 S3 S 3 (.) If th circuit is also losslss thn its S matrix has to b unitary [], which lads to th following conditions [] S + S (.3) 3 = S + S (.4) 3 = S + S (.5) 3 3 = * S S (.6) 3 3 = * S S (.7) 3 = * S S (.8) 3 = 9

16 Equations (.6.8) show that at last two of th thr paramtrs (S, S 3, S 3 ) must b zro which is inconsistnt with on of th thr quations (.3.5). It implis that a thrport ntwork cannot b losslss, rciprocal, and matchd at all ports. By rlaxing on of ths thr conditions physically ralizabl circuit can b achivd. If th 3-port ntwork is allowd to b lossy, it can b rciprocal and matchd at all ports. Th most common xampl of such a dvic is th Wilkinson powr dividr, shown in Fig... In this lmnt, output signals ar in-phas and hav qual amplitud, and also th output ports ar idally isolatd at th cntr frquncy. Th circuit is composd of two λ g /4 transmission-lin sctions having charactristic impdanc qual and a lumpd rsistor conncting th output ports and having valu, whr is th trminating impdanc of all thr ports of th dividr. Th frquncy rspons of th Wilkinson powr dividr is shown in Fig... S matrix of such an lmnt has th following form []: [ S] = (.9) Fig... Equivalnt transmission-lin circuit of th Wilkinson powr dividr. Fig... Frquncy rspons of an qual-split Wilkinson powr dividr.

17 In cas of a four-port rciprocal ntwork, which is matchd at all ports, its S matrix has th following form S [ S] = S S 3 4 S S S 3 4 S S S S4 S 4 S 34 (.) If th ntwork is losslss, th quations rsult from th unitarity or nrgy consrvation condition []. Considring th multiplication of row and row, and multiplication of row 3 and row 4: * * S S + S S (.) = * * S S + S S (.) = Whn multiplying (.) by S * 4 and multiplying (.) by S * 3 and subtracting th two, th following xprssion is obtaind * S 4 S3 S4 = (.3) Similarly, th multiplication of row and row 3 and th multiplication of row 4 and row, givs * * S S + S S (.4) = * * S S + S S (.5) = Whn multiplying (.4) by S and multiplying (.5) by S 34 and subtracting th two, th following formula is obtaind S 3 S S34 = (.6) On way for (.3) and (.6) to b mt is if S 4 = S 3 =, which rsults in a dirctional couplr circuit. Thn th slf-products of th rows of th unitary S matrix (.) yild th following quations S + S (.7) 3 = S + S (.8) 4 = S + S (.9) 3 34 =

18 S + S (.) 4 34 = which imply that S 3 = S (using (.7.8)) and 4 S = S (using (.9.)). 34 θ Choosing th phas rfrncs on thr of th four ports as S = S 34 = α, S = β and 3 β φ, whr α and β ar ral and θ and φ ar phas constants to b dtrmind (on of 4 S = which is still fr to choos), furthr simplification is achivd. Th dot product of rows and 3 of th S matrix givs * * S S + S S (.) = which yilds a rlation btwn th phas constants as θ + φ = ± n (.) Thr ar two particular cass that occur in practic []:. Th symmtrical couplr, for which θ = φ = /. Whn th phass of th trms having amplitud β ar chosn qual, thn th scattring matrix has th following form α [ S] = β α β β α β α (.3). Th antisymmtrical couplr, for which θ =, φ =. Whn th phass of th trms having amplitud β ar chosn to b 8 apart, thn th scattring matrix has th following form α [ S] = β α β β α β α (.4) Th two couplrs diffr only in th choic of rfrnc plans and th amplituds α and β ar not indpndnt bcaus (.7.) rquirs that α + β = (.5) Thus, apart from phas rfrncs, an idal dirctional couplr faturs only on dgr of frdom. Th basic opration of an idal dirctional couplr is illustratd in Fig..3, which shows th dirctional couplr s port dfinitions. Powr dlivrd to port # is coupld to

19 port #3 th coupld port, with th coupling factor S =, whil th rmaining input 3 β powr is dlivrd to port # th transmission port, with th cofficint S = α = β, and no powr is dlivrd to port #4 th isolatd port. Th following thr quantitis dscrib th proprtis of th dirctional couplr [39]: P Coupling = C = log = log β [db] (.6) P 3 P3 β Dirctivity = D = log = log [db] (.7) P S 4 P Isolation = I = log = log S4 [db] (.8) P 4 Th coupling factor indicats th part of th input powr that is coupld to th output port, whras, th dirctivity is a masur of a couplr s ability to isolat forward and backward wavs, as is th isolation. Ths thr quantitis ar rlatd as I = D + C [db] (.9) Th idal dirctional couplr has infinit dirctivity and isolation and both amplituds α and β can b dtrmind from th coupling factor C. Hybrid couplrs ar spcial cass of dirctional couplrs, whr th coupling quals 3 db, which implis that α + β =/. Thr ar two typs of hybrids, i.. th quadratur hybrid (th branch-lin couplr) and th magic-t hybrid (th rat-rac couplr). A quadratur couplr consists of four quartr-wav long transmission lins, two of thm having charactristic impdanc qual, th othr two qual / appropriatly connctd, as shown in Fig..3. Th powr ntring port # is vnly dividd btwn ports # and #3, with 9 phas shift btwn ths outputs. No powr appars in port #4 (th isolatd port). Thus, th S matrix has th following form: [ S] = 4 (.3) It is sn that th branch-lin couplr has a high-dgr of symmtry, which mans that any port can b usd as th input port. Th output ports will always b on th opposit sid of th unction from th input port, whras th isolatd port will b th rmaining port on th sam sid as th input port. Th frquncy rspons of th branch-lin couplr is shown in Fig..4. 3

20 Fig..3. Equivalnt transmission-lin circuit of a 3-dB branch-lin couplr. (a) (b) Fig..4. Frquncy rspons of a 3-dB branch-lin couplr. Amplitud charactristics (a) and diffrntial phas charactristics (b). 4

21 Th most common ralization of a magic-t hybrid is a rat-rac couplr, a four-port ntwork consisting of four sctions of transmission lins, having charactristic impdanc qual, thr of thm bing quartr-wav long and th forth having lngth 3λ g /4, as it is shown in Fig..5. A signal applid to port # is vnly split into two in-phas output signals at port # and #3, and port #4 is isolatd. If th input signal is applid to port #4 it will b qually split into two output signal having 8 phas diffrnc at ports # and #3, and port # will b isolatd. Whn th couplr oprats as a combinr with input signals applid at ports # and #3, th sum of th inputs will b formd at port #, whras, th diffrnc will b formd at port #4. Hnc ports # and #4 ar rfrrd to as sum and diffrnc ports, rspctivly. Thus, th scattring matrix of th idal rat-rac couplr, at th cntr frquncy, has th following form []: [ S] = (.3) Th frquncy rspons of a rat-rac couplr is shown in Fig..6. Fig..5. Equivalnt transmission-lin circuit of a 3-dB rat-rac couplr (a) 5

22 (b) Fig..6. Frquncy rspons of a 3-dB rat-rac couplr. Amplitud charactristics (a) and diffrntial phas charactristics (b)... Dirctional couplrs and phas shiftrs utilizing coupld transmission lins... Singl-sction coupld-lin dirctional couplrs Two paralll transmission lins, bing in clos proximity, so that th lctromagntic filds of both lins influnc ach othr, hav th proprtis of a dirctional couplr. Du to th proximity of th two transmission lins som part of th powr of th xcitd lin appars in th coupld lin. Coupling cofficint is a function of physical dimnsions and th proprtis of th coupld-lin structur. Figur.7 prsnts th gnral cas of a thr-conductor coupld-lin systm consisting of two signal conductors placd ovr a common ground plan []. Such a ntwork is dscribd by a capacitanc systm consisting of capacitancs C and C of th first and scond lin rspctivly, and a mutual capacitanc C m. If th gomtrical dimnsions of th signal lins ar idntical th rsulting slf capacitancs ar qual, i.. C = C. In this cas an vn- and odd-mod xcitation mthod can b applid to charactriz th coupld-lin systm. Figur.7b shows an vn-mod xcitation of coupld lins, in which both lins ar xcitd with th in-phas signals of qual amplituds rsulting in an vn symmtry of th lctric fild about th cntr lin. This lads to th quivalnt capacitanc ntwork whr C m is ffctivly opn-circuitd and th rsulting capacitanc of ithr lin for vn mod is C = C = C (.3) Thn th charactristic impdanc for th vn mod is 6

23 L LC = (.33) νc = = C C whr ν is th vlocity of propagation in th lin and L is th inductanc of th lin. Similarly, in th odd-mod xcitation, shown in Fig..7c, th two out-of-phas signals of qual amplituds ar applid to th signal conductors rsulting in virtual short-circuit in th symmtry about th cntr lins. This lads to th quivalnt capacitanc ntwork whr C m is ffctivly in paralll to ach of th slf capacitancs, hnc th rsulting capacitanc of ithr lin for odd mod is C = C C (.34) o + Cm = C + Thn th charactristic impdanc for th odd mod is o m o = (.35) νc (a) (b) (c) Fig..7. A thr-conductor coupld-transmission-lin systm and its quivalnt capacitanc ntwork (a). Evn-mod xcitation of coupld lins and th rsulting capacitanc ntwork (b), and oddmod xcitation of coupld lins and th rsulting capacitanc ntwork (c) []. 7

24 Th proprtis of a coupld-lin sction, having lctrical lngth θ, can b drivd applying th mthod of analysis of symmtrical ntworks, as shown in []. Th rsulting S paramtrs of th coupld-lin sction having vn- and odd-mod charactristic impdancs, o is as follows [3]: S = S = S = S (.36) = S = S = S = S (.37) S = k = S = S34 = S43 = (.38) k cosθ + sinθ whr S 3 k = k sinθ = S3 = S4 = S4 = (.39) k cosθ + sinθ o m (.4) + o = C ( C + C )( C + C ) m m Morovr, vn- and odd mod charactristic impdancs ar rlatd by o = (.4) whr is th trminating impdanc of all ports of th dirctional couplr. Fig..8. Frquncy rspons of a 3-dB coupld-lin dirctional couplr. Th maximum coupling btwn th coupld lins occurs at th frquncy for which θ = 9 and quals k, whras, th powr dlivrd to th dirct port is qual k. Morovr 8

25 th diffrntial phas of signals masurd at coupld and dirct ports is constant and quals 9 at all frquncis. Th xmplary frquncy rspons of an idal 3-dB dirctional couplr is prsntd in Fig..8. Th prsntd proprtis of a coupld-lin sction hav bn drivd assuming that th phas vlocitis of both propagating mods ar qual, which rquirs th homognity of th dilctric structur of th coupld-lin systm. In cas of inhomognous dilctric structur, th vn- and odd-mod phas vlocitis (θ and θ o ) ar diffrnt and th following scattring paramtrs ar drivd [8] S ( z y ) sinθ, o, o, o, o = (.4) M, o S, o = (.43) M, o M, o 4cosθ, o + ( z, o + y, o ) sinθ, o = (.44) θ = θ ( u), θ = θ ( u) (.45) + o o[, o] z, o =, y, o = (.46) z, o u θ θ θ + θ o r ro = = (.47) o ε ε r + ε ε ro = ( S + S ) (.48) S o = ( S S ) (.49) S o = ( S + S ) (.5) S3 o = ( S S ) (.5) S 4 o whr ε r, ε ro ar th vn and odd mod dilctric constants. Symmtrical couplrs rprsnt a vry usful but rstrictd class of coupld lins. In many practical cass, it might b mor usful or vn ncssary to dsign componnts using asymmtrical coupld lins. For asymmtrical coupld lins th two normal mods of propagation ar known as c- and -mods [43, 7, 35, 46, 47, 48]. In such couplrs th inductiv and capacitiv coupling cofficints dfind as Lm kl = (.5) L L 9

26 Cm kc = (.53) C C ar not qual, and th maximum coupling of th asymmtric dirctional couplr quals [5] k L + k k = C (.54) whr L, L, L m ar slf and mutual pr unit lngth inductancs of th coupld lins and similarly C, C, C m ar thir slf and mutual pr unit lngth capacitancs.... Multisction coupld-lin dirctional couplrs Th dscribd in th abov Sction singl-sction coupld-lin dirctional couplrs allow for achiving oprational bandwidths up to on frquncy octav. To achiv a spcifid coupling ovr a widr frquncy bandwidth, than it is possibl with th us of a singlsction couplr, a numbr of coupld-lin sctions can b cascadd, as shown in Fig..9. Each sction is quartr-wav long at th cntr frquncy. By proprly choosing th vnand odd-mod impdancs of ach sction, th bandwidth of th couplr can b incrasd [9, 55, 9, 93, 6, 37]. As it was shown in [45] th multisction dirctional couplr shown in Fig..9 can b rprsntd by a cascadd connction of transmission-lin sctions, and th rflction and transmission cofficints of such a circuit giv, rspctivly, coupling and transmission of th multisction dirctional couplr. Th vn- and odd-mod impdancs of th i th sction of th multisction couplr ar rlatd by (.4). A multisction couplr can b ithr symmtrical or asymmtrical. In multisction couplrs th trm symmtrical is usd to dnot a couplr that has nd-to-nd symmtry. A symmtrical couplr utilizs an odd numbr of sctions and th i th sction is idntical to th N+-i th sction, as shown in Fig..9b. If th couplr dos not hav nd-to-nd symmtry (s Fig..9a), it is rfrrd to as an asymmtrical couplr. An asymmtrical couplr can consist of ithr vn or odd numbr of sctions. Th maor diffrnc btwn th two typs of multisction couplrs is obsrvd whn comparing thir phas proprtis. Th symmtrical couplrs fatur, similar to th singl-sction couplrs, constant 9 diffrntial phas charactristic btwn signals masurd at coupld and transmission ports, which rsults dirctly from th unitary proprtis of th scattring matrix []. Bcaus of this proprty, 3-dB symmtrical dirctional couplrs ar broadly usd in diplxrs, multiplxrs, dirctional filtrs, balancd mixrs, Butlr matrics, balancd amplifirs, and in othr dvics whr th 9 phas diffrnc proprty is rquird. Exmplary coupling rsponss of two- and thrsction asymmtrical dirctional couplrs hav bn prsntd in Fig... Th synthsis of multisction asymmtrical couplrs has bn shown by Lvy in [9, 93], whras, th synthsis of multisction symmtrical couplrs has bn shown by Cristal and Young in [9] and also by Toulios and Todd in [44]. Th analytical dsign xprssions usd for th synthsis of multisction couplrs ar vry cumbrsom vn for a small numbr of sctions. For aiding dsignrs, Lvy has prpard dsign tabls for qual-rippl asymmtrical couplrs for various valus of coupling and bandwidth for up to six sctions 3

27 [93]. For qual-rippl and maximally flat symmtrical couplrs, Cristal and Young prsntd similar tabls for couplrs having up to nin sctions [9]. (a) (b) Fig..9. Multisction coupld-lin dirctional couplrs, asymmtrical (a) and symmtrical (b). Fig... Coupling charactristics of a two-sction (solid lin) and thr-sction (dashd lin) dirctional couplrs having coupling imbalanc qual δc = ±. db. 3

28 ..3. Broadband phas shiftrs It is known that th coupld-lin sctions can b usd for th dsign of all-pass ntworks. In particular, it was shown in [84] that two paralll coupld lins of qual lngth connctd at on nd, for which th condition (.4) is fulfilld, constitut a two trminal ntwork (calld C -sction) bing idally matchd with S =. Th phas charactristic of such a sction is givn by [33] φ = arccos o o tan + tan θ θ (.55) Th phas charactristic of th C -sction has bn shown in Fig... It is sn that th bhavior of th rspons dpnds on th coupling k of th sction, bing linar for k = and bcoming highly non-linar whn k incrass. Figur. prsnts diffrntial phas charactristics btwn a sction of transmission lin having lctrical lngth θ = 7 and C - sctions having coupling k =,.5 and.77. It is sn that for crtain valus of coupling k th diffrntial phas charactristics bcoms qual-rippl with th man valu of 9, thus, such a C -sction togthr with a transmission-lin sction constituts broadband 9 diffrntial phas shiftr [33], with th bandwidth and phas imbalanc dpnding on th coupling k. Fig... Phas charactristic of a C -sction for thr diffrnt valus of coupling cofficint k. Th prsntd C -sction in conunction with a transmission-lin sction may provid any othr diffrntial phas shift ovr broad frquncy rang. It was shown in [33, 78] that th bandwidth of th diffrntial phas shiftr can b furthr incrasd whn a ntwork having mor than on coupld-lin sction is mployd. In particular, in cas of a twosction Schiffman C -sction an qual-rippl diffrntial phas shiftr can b dsignd with th numbr of rippls incrasing into four. This ntwork is suitabl for achiving bandwidths of about 3:. Morovr, in [78] a multisction phas-shift ntwork is analyzd and th following formula is givn, dscribing its phas charactristics 3

29 ' ( tanθ ) ' ( tanθ ) ρ φ = arccos (.56) n ρ + whr θ = θ + arctan ' ' ' n ' n i, i+ θ = θ + arctan M θ = θ θ = θ σ n n = i ( i+ ) ' ( σ tanθ ) ' ( σ tanθ ) 3 + arctan = ' ( σ tanθ ) n, n o( i+ ) oi 3 n Th prsntd xprssion allows for calculation of phas charactristics of a multisction phas shiftr having coupld-lin sctions of unqual lngths. By allowing th lngths of th sctions to b varid, gratr dgr of frdom in phas shaping is achivd. Fig... Diffrntial phas charactristic btwn a transmission-lin sction having lctrical lngth θ = 7 and a C -sction for thr diffrnt valus of coupling factor k...4. Dsign of coupld-strip-transmission-lin dirctional couplrs and phas shiftrs with improvd frquncy rspons Th coupld-lin circuits can b dsignd in tchnology of ithr pur-tem striplins in homognous dilctric mdium, or quasi-tem microstriplins in inhomognous dilctric mdium, in which normal wavs ar propagating with diffrnt phas vlocitis. Rali- 33

30 zation of idal, symmtrical coupld-lin sctions rquirs that two conditions hav to b fulfilld, which ar: (i) th product of vn and odd mod charactristic impdancs of coupld lins quals th squar of th trminating impdanc (.4) and (ii) th modal phas vlocitis ar qual. In asymmtric, coupld-lin sctions two mods, th c mod and mod, propagat in th structur. It was shown in [7] that for such a coupld-lin sction th conditions for an idal couplr ralization ar: (i) th ratio of slf inductanc and slf capacitanc of th lin i quals th squar of th trminating impdanc Lii Ti = (.57) Cii whr Ti, i =, charactristic impdancs of trminating lins, and (ii) inductiv and capacitiv cofficints ar qual k L = k C. (.58) It is important to rmmbr that symmtric coupld-lins ar th spcial cas of mor gnral asymmtric coupld lins, thrfor, th conditions of idal asymmtric coupldlin sction ralization ar mor gnral, and apply also for th cas of symmtric coupldlin sctions. Th physical ralization of coupld-lin dirctional couplrs rquirs th connction of signal lins to th coupld-lins and in cas of multisction couplrs th connction of coupld-lin sctions having diffrnt gomtrical dimnsions. Such connctions introduc parasitic ractancs that svrly dtriorat frquncy charactristics of dirctional couplrs, spcially thir rturn losss and isolations btwn rspctiv ports. To summariz, thr ar two important aspcts that hav to b approachd whn dirctional couplrs having suprior prformanc ar to b dsignd:. Equalization of modal phas vlocitis or inductiv and capacitiv coupling cofficints.. Taking into account parasitic ractancs of th transition rgions within th dirctional couplrs gomtry. A numbr of tchniqus hav bn rportd ovr th yars to approach th first aspct. Th known tchniqus can b dividd into four groups:. Th compnsation tchniqus in which qualization of coupling cofficints is achivd by a propr choic of dilctric layrs and coupld-lin gomtry [46, 5, 6, 65, 6, 3, 67].. Th compnsation tchniqus in which lumpd lmnts ar connctd to coupld lins [4, 46,, 67]. Th prsntd mthods assum th qualization of vn and odd mod phas vlocitis for th cntr frquncy only, which causs that th frquncy charactristics dtriorat in widr frquncy rang. A numbr of paprs follows th ida of a singl or multipl lmnt compnsation tchniqu prsntd in [4] with modifications on a typ, placmnt and a numbr of compnsating lmnts (i.. lumpd capacitors, lumpd inductors, stp-impdanc transformrs, complx impdancs, coupld spur-lins, intrdigital capacitors, fdback transmission-lin sctions) [73]. 3. Th compnsation tchniqus in which coupld lins ar modifid priodically rsulting in modal phas vlocitis qualization or coupling cofficints qualization [73]. 4. Othr compnsation tchniqus such as compnsation with th us of dlay lins, stppd impdanc sctions, or compnsation using corrugatd ground plans [73]. 34

31 Th Author has rsarchd th two aspcts concrning th ralization of dirctional couplrs having improvd prformanc, and th rsults can b found in [6, 6, 63, 64, 65, 69, 7, 7, 65, 66, 67, 69]. In [65] th Author has invstigatd a capacitiv compnsation mthod that allows for coupling cofficints qualization in a suspndd striplin tchniqu. In this mthod, compnsating lmnts ar qually distributd along coupld lins allowing for appropriat dcras of th capacitiv coupling cofficint to th valu of th inductiv on. Th proposd procdur is itrativ sinc it is also dsird to achiv appropriat impdanc of coupld lins. Th mthod has bn also applid for th dsign of a broadband thr-sction dirctional couplr shown in [6]. On of th rcnt co-authord paprs proposs th gnralizd mthods to dsign quasi-idal symmtric and asymmtric coupld-lin sctions [7]. Th scond aspct, i.. th problm of discontinuitis in coupld-striplin dirctional couplrs and phas shiftrs has bn comprhnsivly invstigatd by th Author in [63, 64, 69, 7, 67]. In [63, 64] a mthod of capacitiv compnsation of parasitic ractancs has bn proposd, in which a numbr of lumpd capacitors connctd to coupld and signal lins in various cross-sctions allow for significant improvmnt of coupld-lin circuits prformanc. Th proposd mthod has bn succssfully applid in various singl and multisction dirctional couplrs dsignd in various dilctric structurs. Th usfulnss of th proposd in [63] mthod of parasitic ractancs compnsation can b illustratd with th dsign of 3-dB and 8.34-dB dirctional couplrs in homognous striplin tchniqu. Fig..3 shows th chosn, dilctrically homognous (ε r = 3.38) structur, in which th dirctional couplrs hav bn dsignd, and consists of two.54 mm thick laminats btwn which a thin laminat layr has bn placd, having thicknss h =.5 mm. Th initial dimnsions of th coupld lins of both typs of couplrs hav bn calculatd using Linpar softwar [38] which ar: w =.6 mm and offst =.8 mm for th 3-dB dirctional couplr and w =.36 mm and offst =.66 mm for th 8.34-dB dirctional couplr. Fig..3. Cross-sctional viw of th striplin coupld lins usd for th dsign of high-prformanc dirctional couplrs. Th final layouts of th couplrs hav bn found with th us of lctromagntic optimization, whr th capacitiv lmnts dimnsions hav bn stablishd. Masurd rsults of th dvlopd dirctional couplrs ar prsntd in Fig..4 and Fig..5. Excptionally good paramtrs hav bn obtaind in trms of rturn losss, isolations and coupling-transmission imbalanc. Figur.6 prsnts a pictur of th innr laminat layr on which th tracs of th couplrs ar tchd. In both cass th capacitiv lmnts, which improv th couplrs prformanc, ar visibl. An xmplary dsign, whr th mthods allowing for coupling cofficints qualization hav bn utilizd is shown in [67], in which a thr-sction, 3-dB dirctional couplr is ralizd in a multilayr microstrip tchniqu. In th prsntd dsign thr diffrnt compnsating tchniqus hav bn simultanously usd, i.. qualization of inductiv and capacitiv coupling cofficints by th propr slction of dilctric structur and coupld- 35

32 lin gomtry and by adding capacitiv lmnts btwn coupld lins, and finally th capacitiv compnsation of parasitic ractancs introducd by th transition rgions. Th applid tchniqus allow for achiving an xcllnt prformanc of th couplr, as it is sn in Fig..7. Th dsignd couplr faturs coupling imbalanc δc = ±. db ovr twooctav frquncy rang having isolations and rturn losss as low as -8 db. Figur.8 prsnts a pictur of th manufacturd couplr. Fig..4. Masurd frquncy rspons of th dsignd 3-dB dirctional couplr. Fig..5. Masurd frquncy rspons of th dsignd 8.34-dB dirctional couplr. (a) (b) Fig..6. Picturs of th innr laminat layr on which th tracs of th couplrs ar tchd. 3-dB dirctional couplr (a) and 8.34-dB dirctional couplr (b). 36

33 Fig..7. Masurd amplitud charactristics of th dsignd 3-dB dirctional couplr [67]. Fig..8. Photograph of th dvlopd 3-dB dirctional couplr with compnsating capacitancs visibl [67]..3. Summary Th chaptr prsnts a brif introduction to basic passiv microwav componnts such as dirctional couplrs, powr dividr and phas shiftrs. At first connctd-lin dvics ar prsntd, followd by circuits utilizing sctions of coupld lins. As it was shown, th dvics dsignd in a connctd-lin tchniqu fatur narrowr oprational bandwidth than th coupld-lin circuits. A basic thory of coupld-lins has bn outlind from which conditions to raliz an idal coupld-lin sction, ar drivd. Finally, th two main aspcts in th dsign of high-prformanc coupld-lin circuits hav bn prsntd:. Equalization of modal phas vlocitis or inductiv and capacitiv coupling cofficints in cas of couplrs dsignd in symmtric and asymmtric structurs, rspctivly.. Compnsation of parasitic ractancs of transition rgions within coupldlin circuits. Th Author has invstigatd both phnomna and as a rsult a compnsation mthod of parasitic ractancs of th transition rgions, broadly dscribd in co-authord paprs [63, 64] hav bn proposd. Also mthods of inductiv and capacitiv coupling cofficints qualization hav bn prsntd in th following co-authord paprs [7, 65]. 37

34 3. Dirctional couplrs in application to balancd and n-way microwav circuits On of th primary applications of 3-dB dirctional couplrs and powr dividrs is th signal distribution in balancd circuits and n-way ntworks. Th xampls of high-powr balancd amplifirs can b found in [54, 75, 95, 8, 43, 7, 75]. In [7] th most convntional approach has bn prsntd, whr branch-lin 3-dB dirctional couplrs hav bn usd as powr splittr and combinr. Similar dsign has bn shown in [8], whr a Wilkinson powr dividr has bn utilizd, togthr with mandrd transmission lin sctions for propr phas adustmnt. Anothr typ of powr dividrs, i.. Gysl powr dividrs [54, 75] togthr with a 9 -long sctions of transmission lins hav bn utilizd for ralization of th.5 kw amplifir oprating in L-band, in which ight 3W transistors ar usd in th final powr stag. An xampl of a balancd amplifir in which coupldlin 3-dB dirctional couplrs ar usd has bn prsntd in [75], whr commrcially availabl smi-rigid cabl with two intrnal wirs is utilizd to raliz a coupld-lin sction. This sction shows th mthods of th dsign of balancd and n-way powr amplifirs with th us of asymmtric coupld-lin dirctional couplrs, in which coupld-lin sctions ar usd to simultanously provid qual powr division btwn dirct and coupld ports, and to provid impdanc transformation in both dirct and coupld ports [79]. Such a solution allows to dsign high-powr amplifirs with rducd both siz and losss within th powr splitting/combining ntwork. Whil th impdanc transformation is achivd at th first and vry following stag of th n-way powr dividr, th microwav powr is distributd by a ntwork having lowr (typically) impdanc. In cas of striplin ralization it provids for widr conducting strips and, thrfor, for lowr attnuation. Morovr, th dcrasd impdanc can b brought clos to th input/output impdanc lvl of th applid smiconductors, and thrfor, simplr matching ntworks can b dsignd. Th larg contnt of th Author s work publishd in [79] is citd in this sction. In sction 3. th basic proprtis of balancd circuits hav bn dscribd, followd by th xmplary application in balancd mixrs and switchs. Furthr, n-way powr dividing/combining ntworks hav bn brifly outlind and a multichannl ntwork has bn prsntd. In th nxt sction, a mthod for larg signal masurmnts of transistor rflction cofficints, with th us of 3-dB//8 dirctional couplrs has bn shown [59]. Th utilization of asymmtric coupld-lin dirctional couplrs in application to balancd circuits has bn thortically invstigatd in Sctions 3.3 and 3.4 and a limitation for impdanc ratio transformation / has bn drivd. This limitation can b ovrcom by 38

35 xpanding a simpl balancd circuit into a largr n-way ntwork, in which th transformation ratio can b incrasd up to / n. Th thortical invstigations hav bn vrifid by masurmnts of th dsignd impdanc-transforming dirctional couplrs, balancd, and 4-way ntworks. Also, th masurd rsults of a 4-way powr amplifir ar shown, in which th dvlopd powr splittrs/combinrs ar succssfully utilizd. Morovr, th concpt of impdanc-transforming dirctional couplrs has bn xtndd in sction 3.5 into multisction dirctional couplrs, which rsults in significant nhancmnt of th oprational bandwidth. 3.. Balancd and n-way microwav circuits A balancd circuit, shown schmatically in Fig. 3., consists of two 3-dB dirctional couplrs/powr dividrs btwn which two idntical two-port ntworks ar insrtd. Such a circuit if oftn usd in microwav tchniqu, du to its advantagous faturs, for th dsign of amplifirs, voltag-controlld PIN-diod attnuators, modulators, powr switchs and diplxrs [58]. In gnral, th principal advantag of such a circuit is an idal impdanc match of its input port in th oprating frquncy rang of th dirctional couplr/powr dividrs. Th impdanc match is indpndnt from th insrtd two-port ntworks, undr th condition howvr, that th two-port ntworks ar idntical. Such a balancd circuit can b rgardd as a cascad connction of thr four-port ntworks, in which th middl ntwork is a compound of two idntical two-port ntworks. Having th scattring matrix of a two-port ntwork as [66]: N N N S S [ S ] = (3.) N N S S and assuming th idntity of both two-port ntworks, th following scattring matrix of th four-port ntwork can b writtn: [ S NN N S ] = N S S S N N S S N N N S N S (3.) Th scattring matrix of a complt balancd circuit can b calculatd as a cascad connction of th four-port ntworks. Gnrally, th scattring matrix of th connction of two n- port ntworks can b calculatd as follows [66]: I II I II I I II I II ( ) ( ) ( ) ( ) SS S S S SS S I II I II II I II I II SS S S + S SS SS I S + S [ S] = (3.3) II S whr th quantitis in th formula ar block S matrics of n-port ntworks as shown in Fig

36 Fig. 3.. Schmatic diagram of a balancd circuit. Fig. 3.. Schmatic diagram of a cascad connction of two n-port ntworks. Balancd circuits can b dsignd with th us of diffrnt typs of dirctional couplrs and powr dividrs, and thir combination. Figur 3.3 prsnts basic configurations with th us of branch-lin dirctional couplrs (Fig. 3.3a), coupld-lin dirctional couplrs (Fig. 3.3b), a coupld-lin dirctional couplr in conunction with a rat-rac couplr (Fig. 3.3c) and a coupld-lin dirctional couplr in conunction with a Wilkinson powr dividr (Fig. 3.3d). In th last two prsntd circuits 9 constant phas shiftrs ar rquird for propr opration. Fig Balancd ntworks composd of (a) two branch-lin dirctional couplrs, (b) two coupldlin dirctional couplrs, (c) a coupld-lin dirctional couplr and a rat-rac couplr in conunction with a 9 phas shiftr, and (d) a coupld-lin dirctional couplr and a Wilkinson powr dividr in conunction with a 9 phas shiftr [66]. Th principl of opration of a balancd circuit has bn outlind in Fig Th prsntd circuit consists of two coupld-lin 3-dB/9 dirctional couplrs and two idntical -port ntworks. Th input and output ports hav bn distinguishd, howvr, taking into account th rciprocity of th circuit, any port can play a rol of an input port, as long as th -port 4

37 ntworks ar also rciprocal. Th incidnt and rflctd wavs hav bn schmatically shown in th pictur with th spcial attntion to thir rspctiv phas rlations, which play a crucial rol in th principl of balancd circuits opration. A wav of magnitud a is incidnt upon th input port of th circuit and is bing split btwn coupld and transmittd ports of th input dirctional couplr. Wavs with qual amplitud and with th phas diffrnc of 9 ar incidnt upon th -port ntworks and rflctd with additional phas shift α. Both rflctd wavs ar again incidnt upon th ports of th input dirctional couplr and ar dirctd to th load and th input port. Th dirctional couplr introducs additional 9 phas shift of th rflctd signals rsulting in thir out-of-phas canclation at th input port, and thir in-phas addition at th port whr a load is applid, hnc, no rflctd powr appars at th input port. Th wavs propagating through th -port ntworks acquir som additional, but idntical in both channls, phas shifts β. Th wavs c ar, similarly as th wavs b, incidnt upon th ports of th output dirctional couplr, whr an additional 9 phas shift is introducd. Du to similar mchanism as dscribd for th input dirctional couplr, th wavs ar addd at th output port, and thy cancl out in th port whr th load is connctd. In th prsntd circuit, assuming idal dirctional couplrs and losslss -port ntworks, no insrtion losss ar introducd and all powr dlivrd into th input port is bing split btwn th output port and th load connctd to th input dirctional couplr []. Fig Schmatic diagram of a balancd circuit showing th principls of opration. An xampl of coupld-lin couplrs applications in th dsign of balancd modulators with th us of monolithic mixrs and switchs is prsntd in [6]. Figur 3.5 prsnts schmatic diagrams of th dvlopd modulators. Two typs of modulators hav bn considrd, i.. th modulator A which utilizs microwav switchs and th modulator B with monolithic microwav mixrs. Both circuits hav bn dvlopd on a cost-ffctiv FR4 laminat. Rsults of masurmnts of th A modulator ar shown in Fig As it is sn th circuit faturs low insrtion losss IL = 85 MHz and high isolation I > 3 db. Th rturn losss in both stats ar bttr than 5 db. 4

38 (a) (b) Fig Balancd circuit modulators; with microwav switchs (a) and microwav mixrs (b) [6]. Fig Insrtion losss IL and rturn losss RL of th dsignd modulator with th us of microwav switchs. Rsults of masurmnts for on and off stat of th modulator [6]. (a) (b) Fig Rsults of masurmnts of th modulator dsignd with th us of microwav mixrs. Insrtion losss (a) and rturn losss (b) for various valus of modulation currnt [6]. 4

39 (a) (b) Fig Spctrum of th modulatd signals obtaind with th us of th modulator consisting of microwav switchs (a) and microwav mixrs (b) [6]. Masurmnt rsults of th modulator B ar shown in Fig. 3.7 for various valus of modulation currnt. Th lowst insrtion losss on th lvl of MHz ar achivd with a total currnt (takn by both mixrs) raching I = 3 ma, whras isolation xcding 4 db can b achivd for currnt I = ma. As in th prvious cas th dvlopd modulator faturs rturn losss bttr than 5 db. (a) (b) Fig Balancd circuit modulators with microwav switchs (a) and microwav mixrs (b) [6]. Fig. 3.8 prsnts frquncy spctrum of a modulatd signal obtaind with th us of both typs of modulators. In cas of th modulator A multi-componnt spctrum rsults from a switching opration of th modulator, whras in cas of th modulator B classic singl ton AM modulation is sn. Picturs of th dvlopd modulators ar shown in Fig Th dscribd balancd circuits ar oftn usd for th dsign of balancd amplifirs. In a microwav amplifir dsign, th primary goal is to achiv a constant gain and good input matching ovr th dsird frquncy rang. As it is shown in [] th conugat matching givs th maximum gain only ovr th rlativly narrow band, whras dsigning for lss thn th maximum gain improvs th gain bandwidth, at th xpns of poor impdanc match of input and output ports of th amplifir. Th application of a balancd circuit solvs th problm of a broadband amplifir dsign offring a numbr of advantags []: 43

40 . Th individual amplifir stags can b optimizd for gain flatnss or nois figur without th concrn for input and output matching.. Th stability of th amplifir is improvd whil th rflctions ar absorbd in th couplrs trminations. 3. Dgradation of gain by only 6 db is obsrvd whn a singl sction of th amplifir fails. 4. Th amplifir can oprat in a broad bandwidth limitd only by th bandwidth of th usd dirctional couplr. 5. Th maximum output powr incrass by 3 db in comparison with th maximum powr of a singl stag. Following th ida of balancd circuit amplifirs mor complx ntworks consisting of a numbr of dirctional couplrs or powr dividrs can b ralizd. Such circuits allow to dsign high-powr amplifirs, in which th maximum output powr is incrasd by N, whr N is th numbr of usd output stags. Th xmplary topologis of 8-way powr amplifirs ar shown in Fig. 3.. (a) (b) Fig way powr amplifirs, uniform topology powr stag (a), and noniuniform topology powr stag (b). Onc dsigning an n-way powr dividrs/combinrs a car has to b takn to th phas rlation btwn th signals within th structur, in ordr to nsur propr combining th microwav powr at th output port. Thr ar two possibl approachs in th dsign of n- way powr dividrs and combinrs. Th first rlis dirctly on th principl of a balancd circuit shown in Fig. 3.4, i.. th sam schm is rpatd at vry stag of dividing and combining th powr, rsulting in idntical dividing and combing ntworks (Fig. 3.a). Th othr approach is shown in Fig. 3.b. In this topology two typs of couplrs ar usd, on bing a mirror imag of th othr. Th goal is to minimiz th insrtion losss of th combining ntwork at th xpns of incrasing th insrtion losss of th powr dividr. Such a modification allows to obtain th maximum output powr at th xpns of th gain 44

41 ndd to nsur it, but in practic highr gain is lss troublsom than highr output powr. Anothr typ of an n-way amplifir that utilizs dirctional couplrs is a multichannl amplifir. Figur 3. prsnts an xmplary four-channl amplifir with th input and output ntworks consisting of four dirctional couplrs connctd in a Butlrmatrix-typ arrangmnt. Such a circuit is wll-suitd for applications in low-nois rcivrs of focal arrays in multibam rflctor antnnas. Th powr introducd to any on of its input ports is dividd qually btwn th output ports of th input ntwork, but with various phas dlays. If th dividd output powr is fd into th idntical output ntwork attachd back-to-back th powr is rcombind in th scond matrix and th total powr appars at a singl port diagonally opposit th input port. Figur 3. prsnts a pictur of th dvlopd four-channl amplifir with th us of monolithic intgratd circuits, th input and output matrics ar dsignd in a multilayr microstrip tchnology [76]. Fig. 3.. Schmatic diagram of a four-channl amplifir. Fig. 3.. Pictur of th dsignd four-channl amplifir. Courtsy of prof. Krzysztof Sachs, Microwav Thory and Tchniqus Division, Wroclaw Univrsity of Tchnology, Poland. 45

42 3.. Larg signal charactrization of solid-stat dvics Th prsntd in prvious sction concpts of n-way powr amplifirs rquir introduction of amplifirs into th powr dividing/combining ntworks. Morovr, it is dsird to minimiz rflctions within th amplifir ntwork and to maximiz th output powr of ach singl amplifir unit. In ordr to achiv that, howvr, it is rquird to masur input and output impdancs of th dvic usd for th amplifir dsign. Typically, in th dsign of a low-powr amplifir, th small-signal transistors paramtrs ar masurd with th us of a vctor ntwork analyzr. Such an approach is inadquat in th dsign of a high-powr amplifir, sinc th activ dvic oprats in a highly nonlinar rgim. Th commonly usd tchniqu for masuring transistors impdanc undr larg-signal opration is a load pull tchniqu. In this procdur th high-powr transistor s input and output ar simultanously matchd in an itrativ procss, whras, th output powr is controlld aiming to its maximization. Such a class of tchniqus is widly dscribd in litratur [3, 3, 34]. In most cass th ky lmnts usd for tuning th systm, i.. changing th input and output impdanc of th ntwork tak a form of a manual or automatd tunr, in which on or mor probs is/ar coupld to a transmission lin and both position and coupling of probs can b changd [5, 4]. In [59] th Author proposd a concpt of applying two 3dB/8º dirctional couplrs, which togthr with two shortd transmission lins srv as a sourc and load pull tunrs. Th contnt of [59] is citd in this sction in a larg xtnt. Th prsntd concpt has its origin in a rctangular wavguid tchniqu, whr a magic-t ntwork with two shortd wavguid sctions (E-H tunr) is usd to obtain impdanc match at a cntr frquncy. Th proposd in [59] masurmnt st-up, which can b usd for dtrmining input and output impdancs of a transistor undr larg signal conditions is prsntd in Fig. 3.. It consists of a signal sourc, input and output matching ntworks, an activ lmnt to b matchd (DUT) and a powr mtr. Th input and output matching ntworks consist of ratrac couplrs in conunction with two sctions of shortd transmission lins connctd to ports # and #4, as it is schmatically shown in Fig By adusting th lngth of shortd transmission lins th impdanc sn at th matching ntworks inputs can b changd. Fig Schmatic diagram of a st-up for sourc and load pull masurmnt tchniqu with ratrac couplrs as matching ntworks [59]. 46

43 47 Fig A rat-rac couplr with two shortd transmission lins as a matching ntwork [59]. Th bhavior of such a matching ntwork can b dscribd by xamining S-paramtrs of an idal rat-rac couplr. Taking into account th ports ordr shown in Fig. 3.4, th scattring matrix (.3) has th following form = S (3.4) Trmination of port # with 5 Ω matchd load lads to th following S matrix = S (3.5) By conncting shortd sctions of 5 Ω transmission lins having th rflction cofficint,, Θ = Γ l (3.6) to ports # and #3, th S matrix is rducd in th following mannr Γ = S l (3.7) l l T S Γ Γ = Γ = (3.8) + Θ = Θ Θ (3.9) Finally assuming (3.9) th rflction cofficint of a matching ntwork can b xprssd as:

44 Θ Θ Θ Θ ( + ) = ( + ) ΓT = (3.) Th analysis of (3.) shows that any impdanc of a passiv ntwork can b ralizd for Θ, Θ (, ). Th displacmnt of a rflction cofficint in th Smith chart as a function of transmission lin lngths Θ and Θ is shown in Fig It can b sn that by changing on of th variabls, for instanc Θ, on can chang th rflction cofficint valu along th circl markd O, whras, th othr variabl corrsponds to th shift of th O circl cntr. Within th rang of to th rflction cofficint travls full circl for on of th variabls ( Θ) and th circl cntr prforms full turn around th Smith chart for th othr (Θ ), allowing to raliz any dsird rflction cofficint. Fig Rflction cofficint of a rat-rac couplr matching ntwork with rflction cofficint displacmnt as a function of shortd lin lngths Θ and Θ [59]. Th prsntd concpt of larg-signal S-paramtrs masurmnts can b illustratd with th dsign of a 4W-powr amplifir oprating at.43 GHz [59]. Th rat-rac couplr has bn dsignd on.8mm thick FR4 laminat and transmission lin sctions hav bn manufacturd as finit ground microstrip lins allowing for applying sliding lmnts shorting th transmission lin strips to th ground. Fig. 3.6 prsnts th masurd minimum and maximum VSWR of th manufacturd ntwork. Th VSWR ratio for th manufacturd ntwork quals 8: and allows for achiving impdanc on th lvl of =.7 Ω, which is sufficint for maority of powr transistors. Th limitation of VSWR ratio in th prsntd cas is causd by rlativly high losss of th chosn FR4 laminat and can b improvd by carful dsign of th matching ntwork with th us of low-loss dilctric substrats. 48

45 ,5 7,4 VSWR 3 9,3, 5,,4,4,44,46,48,5 f [GHz] max VSWR min. VSWR Fig Masurd min. and max. VSWR of a rat-rac couplr matching ntwork shown schmatically in Fig. 3.4 [59]. Th matching ntwork has bn tstd whil dsigning a powr amplifir oprating at.43 GHz with th us of a MW6S4NT N-channl nhancmnt-mod latral MOSFET transistor from Frscal Smiconductor. By th procdur of manual tuning, th input and output impdancs of a transistor (togthr with a mounting fixtur) hav bn dtrmind and output powr raching 6 W has bn masurd. Th manufacturd masurmnt st-up is shown in Fig Th masurd valu of th input and output rflction cofficints (impdancs) hav bn shown in Tabl 3., in comparison with th small signal rflction cofficints givn in th transistor spcification. Basd on th masurd impdanc valus th powr amplifir shown in Fig. 3.8 has bn dsignd and manufacturd. Figur 3.9 prsnts th input and output matching ntworks of th dvlopd amplifir and masurd rsults ar shown in Fig. 3. and Fig. 3. for diffrnt bias conditions. It is sn that th output powr raching 6 W has bn achiv with th gain qual 5 db. Fig Pictur of a manufacturd masurmnt stup with powr transistor, input and output ratrac couplr matching ntworks and bias circuitry [59]. 49

46 Fig Pictur of th manufacturd singl-stag powr amplifir [59]. Fig Diagram schmatic of input and output matching ntworks of th dsignd singl-stag powr amplifir [59]. Th proposd tchniqu has bn also applid in th dsign of a two-stag powr amplifir. Th input and output matching ntworks rmaind unchangd and a nw intrmdiat matching ntwork has bn found. Figur 3. shows a schmatic diagram of th dsignd amplifir and th obtaind masurmnt rsults ar prsntd in Fig Th obtaind output powr quals 6 W with th gain xcding 3 db. Tabl 3. Masurd input and output rflction cofficints (impdancs) in comparison with small signal rflction cofficints givn in th transistor spcification [59] small signal masurd * input rflction cofficint Γ =.95 Ang Γ = -79º =.5-.65Ω Γ =.66 Ang Γ = -67.º =.-5.34Ω output rflction cofficint Γ =.866 Ang Γ = -3.4º = Ω * Valus masurd togthr with th transistor fixtur. Γ =.65 Ang Γ = +45º =.6+4.9Ω Fig. 3.. Masurd output powr vs. input powr of th manufacturd powr amplifir. Thr curvs ar shown for thr diffrnt valus of transistor s drain quiscnt currnt [59]. 5

47 Fig. 3.. Masurd gain vs. input powr of th manufacturd powr amplifir. Thr curvs ar shown for thr diffrnt valus of transistor s drain quiscnt currnt [59]. Fig. 3.. Diagram schmatic of input, intrmdiat and output matching ntworks of th dsignd two-stag powr amplifir [59]. 4 Uds = 8 V; Idq = 4. ma (for singl transistor); Idmax = 7 ma (for singl transistor); Duty = 6 %, t = 33 us Pout [dbm] Pin [dbm] Fig Masurd output powr vs. input powr of th manufacturd two-stag [59]. 5

48 3.3. Asymmtric coupld-lin dirctional couplrs as impdanc transformrs in balancd and n-way powr amplifirs As it was shown in sction 3., coupld-lin dirctional couplrs ar th ky componnts of balancd circuits and n-way powr dividr/combinrs. On th othr hand th impdancs of high-powr microwav activ lmnts ar small in comparison with typically qual 5 Ω - as shown in sction 3.. It would b, thrfor, dsirabl to dvlop dirctional couplrs having th proprtis of both powr dividrs and impdanc transformrs. As it was shown in [3] an asymmtric coupld-lin sction offrs th possibility of impdanc transformation in a sns that, whn appropriatly dsignd, ach lin of th sction can b trminatd with th diffrnt impdancs dfind by (.57). Thrfor, th asymmtric coupld-lin sction can b usd as an impdanc transformr, whr partial powr dfind by coupling C of th asymmtric coupld-lin sction (dirctional couplr) is dlivrd to th load having diffrnt impdanc than th sourc impdanc. Such a sction cannot b straightforwardly applid into th balancd circuit, sinc th impdancs at th coupld and dirct ports would hav diffrnt valus. On th othr hand both symmtrical and asymmtric coupld-lin sctions can b usd for th dsign of two-port impdanc transformrs as it was thortically prsntd in [9]. In [] a simpl solution has bn prsntd allowing for siz rduction of a multisction stppd quartr-wav transformr by rplacing sctions of quartr-wav-long transmission lins by mandrd sctions, in which coupling btwn appropriat arms has bn considrd. In [5, 8, 7] impdanc transformrs hav bn considrd, in which mandrd coupld-lin sctions hav bn utilizd. A coupld-lin sction can also b usd in th dsign of matching circuits of microwav powr amplifirs. In [] an xmplary dsign of a millimtr-wav monolithic amplifir dsign is prsntd, in which coupld-lin sctions hav bn usd to dsign matching circuits for both gat and drain, utilizing th ida prsntd in []. A mthod of utilization of asymmtric coupld-lin sctions having th proprtis of simultanous powr division and impdanc transformation in th dirct and coupld ports, dvlopd by th Author in [79], is dscribd blow. Lt us considr, th most simpl circuit that allows for impdanc transformation, which is a quartr-wav-long sction of a transmission lin, as shown in Fig. 3.4a. (a) Fig Schmatic diagram of a singl-sction quartr-wav transformr (a) and an asymmtric coupld-lin quartr-wav transforming dirctional couplr (b) [79]. (b) 5

49 Th impdanc of th lin dpnds on th valus of trminating impdancs and as follows L = (3.) Similarly, a sction of asymmtric coupld lins, shown in Fig. 3.4b, can b usd as an impdanc transformr. In this cas a similar rlation for th main-lin impdanc L should hold = (3.) L T T Lt us dfin th impdanc ratio of th transformr as T = R (3.3) T Sinc it is dsird to obtain powr split and th impdanc transformation simultanously, and morovr, th impdanc of th loads to which th powr is transmittd should b qual, i.. T = T3, th impdanc L of th coupld lin is calculatd as follows T = T T (3.4) 3 4 T R L = Th abov rlation has bn drivd assuming, that in ordr to obtain idal match of all ports (at th cntr frquncy) it is ncssary to provid th sam impdanc transformation R in both lins of th coupld-lin sction. Th trminating impdanc of th isolatd ports T4 is thn calculatd in th following mannr = T (3.5) T 4 R Lt us xamin th capacitanc matrix of an idal asymmtric coupld-lin sction in homognous dilctric mdium having ε r = (th phas vlocity of th propagating wavs quals c fr spac light vlocity). Th valus of th capacitanc matrix lmnts for th coupld-lin sction dscribd by L, L, k, c ar dfind as follows C = (3.6) c L C = (3.7) c L C = k C C (3.8) m Th capacitancs of th coupld conductors, shown in Fig. 3.5a ar xprssd as C = C Cm, C = C C m (3.9) 53

50 (a) (b) Fig Schmatic diagram of th cross-sction of an asymmtric coupld-lin systm (a) and th sam systm in which C = (b) [79]. Valus of capacitancs C, C and C m for th idal asymmtric coupld-lin sction vs. load impdanc at dirct and coupld ports T = T3 hav bn plottd in Fig. 3.6a, assuming sourc impdanc T = 5 Ω, C = 3 db (k =.77) and ε r =. It is important to notic that th capacitanc C of th dirct lin dcrass with th incras of impdanc ratio R taking ngativ valus for larg valus of R. Fig. 3.6b shows similar calculations, in which th coupling C = 6 db of th coupld-lin sction has bn assumd. As in th prvious cas th capacitanc of th dirct lin C taks ngativ valus for gratr R, diffrnt howvr, than in th first cas. Th rlation btwn th coupling C (k) and th impdanc ratio can b drivd for th simpl cas of homognous coupld lins. (a) (b) Fig Calculatd slf and mutual capacitancs of an asymmtric coupld-lin sction assuming T = 5 Ω and uniform dilctric mdium (ε r = ) vrsus T = T3 for th cas of coupling C = 3 db (a) and C = 6 db (b) [79]. 54

51 To driv th rlation lt us first not that th gratst valu of R is obtaind whn C =, which is th boundary of physical ralization of th circuit. Substituting that condition into (.57) th following xprssions for st and nd lin impdanc ar obtaind L C T = C Cm = L (3.) T = L C = L C + C m (3.) Lt us now calculat th inductanc matrix also assuming C = L = ε µ C = ε µ C C ε µ C C C m m C C m C Cm C = C m Cm C C C m CCm C = ε µ C m C C (3.) Applying calculatd L and L into (3.) and (3.), th following xprssions for lin impdancs ar obtaind: L ε µ C T = = (3.3) Cm CCm T = L C = C ε µ (3.4) ( C + C ) m Calculating th impdanc ratio R from (3.3), (3.3) and (3.4) on obtains: C C = (3.5) ( C + C ) T R = C m = T CCm Cm On th othr hand th coupling of coupld lins k in this cas quals: k = k C m C m C = = CC CmC = C Cm (3.6) C Thrfor, th limitation on th impdanc ratio is xprssd by th simpl formula, drivd by comparing (3.5) and (3.6): R = (3.7) k Th abov prsntd limitation on impdanc transformation with th us of coupld-lin sctions stats that, for th cas of an qual-split (C = 3 db) dirctional couplr and th sam trminating impdancs of th transmission and coupld ports, th maximum physically ralizabl impdanc ratio quals R =. Calculatd frquncy charactristics of such a 55

52 3-dB idal coupld-lin sction with impdanc transformation R = ar prsntd in Fig It is sn that th sction is idally matchd at th cntr frquncy and has qual powr split. Th rturn losss of th coupld-lin sction hav bn additionally compard to th rturn losss of a singl-sction quartr-wav-long transmission-lin transformr having R =, shown in Fig. 3.4a, and it is sn that th coupld-lin circuit oprats in a broadr frquncy rang than th singl sction transformr. Finally, it is intrsting to tst th asymmtric coupld-lin sction in application to th balancd circuit. Fig. 3.8 prsnts a schmatic diagram of a balancd circuit consisting of two idntical 3-dB asymmtric coupld-lin sctions for th cas of maximum attainabl impdanc transformation R =. Th impdanc within th balancd circuit quals T,3 = 5 Ω (assuming sourc impdanc T = 5 Ω) and th impdanc of th trmination of th isolatd port T4 =.5 Ω. Calculatd frquncy charactristics of th rgardd balancd circuit ar prsntd in Fig As it is sn th circuit is matchd in a widr bandwidth than a singl-sction transmissionlin transformr and has thortically no losss from input to output. It is important to not that in n-way powr dividrs th impdanc ratio can b asily incrasd to th valu R total = n by th appropriat connction of svral 3-dB dirctional couplrs, ach transforming th impdanc with th ratio R =. Fig Calculatd frquncy charactristics of an asymmtric coupld-lin sction for which C = 3 db, R = (solid lins), in comparison with th rturn losss of a singl-sction quartr-wav transmission-lin transformr for which R = [79]. Fig Schmatic diagram of a balancd circuit consisting of two asymmtric 3-dB dirctional couplrs providing impdanc transformation with th ratio R = [79]. 56

53 Fig Calculatd frquncy charactristics of a balancd circuit shown schmatically in Fig. 3.8 [79]. To vrify th thortical analysis a 3-dB coupld-lin dirctional couplr has bn dsignd and manufacturd. To achiv th maximum ratio of th transformd impdancs R = th coupld-lin structur shown in Fig. 3.3 has bn chosn which allows to minimiz th capacitanc C of th uppr striplin, whil it is shildd from th ground plan by th wid lowr striplin. It is important to undrlin that th minimization of th C capacitanc nsurs maximum impdanc ratio R. Th assumd input impdanc is qual T = 5 Ω, whras, th impdancs at th coupld and transmission ports ar qual T,3 = 5 Ω. Th chosn structur has bn analyzd with th us of a spctral domain mthod [3]. Th obtaind rsults ar shown in Tabl 3.. Not, that a vry low valu of C =. pf/m has bn obtaind providing for impdanc ratio R =.96. Th dsignd couplr has bn manufacturd and masurd with th us of quartr-wav impdanc transformrs applid to th coupld and transmission ports, as shown schmatically in Fig Figur 3.3 prsnts th masurd S paramtrs in comparison with th simulatd ons. It is sn that a vry good agrmnt has bn achivd in trms of powr splitting and impdanc match. Figur 3.33a shows a photograph of th manufacturd dirctional couplr. Th application of th dsignd couplr in balancd circuits has bn tstd, as shown in Fig Th manufacturd ntwork is shown in Fig. 3.33b and its masurd rsults ar shown in Fig Th achivd insrtion losss ar qual.6 db. Fig Cross-sctional viw of th dilctric structur usd for th dsign of 3-dB asymmtric coupld-lin impdanc-transforming dirctional couplrs [79]. 57

54 Tabl 3. Paramtrs of th coupld-lin sction for which R =, dsignd in th structur shown in Fig. 3.3, having h =.5 mm, ε r = 3.4, h =.4 mm, ε r = 3.38, h 3 =.58 mm, ε r3 = 3.38 and w =.36 mm, w = 3.5 mm [79] paramtr valu C [pf/m].9 C [pf/m] C m [pf/m] 9.7 C [pf/m]. C [pf/m] 8.9 L [nh/m] 8.4 L [nh/m] 48.4 L m [nh/m] 45.9 k L.7 k C.75 k.77 T [Ω] 5 T,3 [Ω] 5.5 T4 [Ω] 3.7 L [Ω] 35.7 L [Ω] 8.4 ε ffc.94 ε ff.75 Fig Schmatic diagram of a 3-dB asymmtric coupld-lin 5/5 Ω impdanc-transforming dirctional couplr with additional singl-sction impdanc transformrs connctd to th dirct and coupld ports [79]. Similarly, a 3-dB dirctional couplr has bn dsignd for impdanc transformation from = 5 Ω to =.5 Ω (Fig. 3.33c). Th paramtrs of th dsignd couplr ar shown in Tabl 3.3. Figur 3.35 prsnts a schmatic diagram of th circuit dvlopd for masurmnts of th dirctional couplr. Masurd rsults in comparison with th thortical ons ar prsntd in Fig Th commnt is ndd, that th condition for th maximum impdanc ratio has bn drivd for th homognous mdium, which is not th cas in th chosn multilayr microstrip structur. In th following sction a thortical analysis is givn, which shows that, whn th highr-impdanc lin is dcoupld from th ground plan, th condition (3.7) convrts into th condition stating that th impdanctransforming ratio quals th invrs product of th inductiv and capacitiv coupling cofficints. 58

55 Fig Calculatd (dashd lin) and masurd (solid lin) frquncy charactristics of a 3-dB asymmtric coupld-lin 5/5 Ω impdanc-transforming dirctional couplr with additional singlsction impdanc transformrs connctd to th dirct and coupld ports, as shown in Fig. 3.3 [79]. (a) (b) (c) Fig Picturs of th manufacturd circuits. (a) th 3-dB dirctional couplr for 5 to 5 Ω impdanc transformation, (b) th balancd circuit consisting of two such 3-dB dirctional couplrs and (c) th 3-dB dirctional couplr for 5 to.5 Ω impdanc transformation [79]. 59

56 Fig Calculatd (dashd lin) and masurd (solid lin) frquncy charactristics of th balancd ntwork (shown in Fig. 3.8) consisting of two 3-dB asymmtric coupld-lin impdanctransforming dirctional couplrs prsntd in Fig. 3.3 [79]. Tabl 3.3 Paramtrs of th coupld-lin sction for which R =, dsignd in th structur shown in Fig. 3.3, having h =.5 mm, ε r = 3.4, h =.4 mm, ε r = 3.38, h 3 =.58 mm, ε r3 = 3.38 and w =.95 mm, w = 7. mm [79] paramtr valu C [pf/m] 44 C [pf/m] C m [pf/m] C [pf/m].7 C [pf/m] L [nh/m] 45.9 L [nh/m] 77 L m [nh/m] 76.3 k L.7 k C.7 k.7 T [Ω] 5.5 T,3 [Ω].75 T4 [Ω] 6.7 L [Ω] 8. L [Ω] 9.3 ε ffc 3.8 ε ff.78 Fig Schmatic diagram of a 3-dB asymmtric coupld-lin 5/.5 Ω impdanc-transforming dirctional couplr with additional singl-sction impdanc transformrs connctd to th input, dirct and coupld ports [79]. 6

57 Fig Calculatd (dashd lin) and masurd (solid lin) frquncy charactristics of th 3-dB asymmtric coupld-lin 5/.5 Ω impdanc-transforming dirctional couplr with additional singl sction impdanc transformrs connctd to th input, dirct and coupld ports, as shown in Fig [79]. (a) Fig Photograph of th dvlopd 4-way powr dividr/combinr in a back-to-back connction (a), and th 4-way powr amplifir (b) [79]. Both dsignd coupls hav bn us to dsign a 4-way powr dividr, th pictur of which in a back-to-back connction is prsntd in Fig. 3.37a. Such a ntwork allows for qual powr split and impdanc transformation with R total = 4. Th calculatd and masurd frquncy rsponss of th back-to-back connction ar shown in Fig. 3.38, whr a vry good agrmnt btwn th obtaind rsults is obsrvd. (b) 6

58 Th dsignd 4-way powr splittr/combinr has bn applid in a 4-way powr amplifir, shown in Fig. 3.37b, in which a 4W N-channl MOSFET transistor (MW6S4NT from Frscal Smiconductor) as an activ lmnt has bn usd. Th larg signal input and output impdancs hav bn masurd with th us of th mthod prsntd in sction 3. and th rsults ar: in =.8.83 Ω, out =. 6.5 Ω. As it is sn, th masurd output impdanc is vry clos to th output impdanc of th dsignd powr splitting/combining ntwork. Th input impdanc of th transistor is lowr than th fding ntwork, but a simpl matching circuit has bn dsignd for th cntr oprating frquncy f =.9 GHz. Th masurd output powr vrsus input powr is prsntd in Fig. 3.39, and on can noticd that th xpctd powr xcding 4 x 4 = 6 W has bn obtaind. Fig Calculatd (dashd lin) and masurd (solid lin) frquncy charactristics of th 4-way 5/.5 Ω impdanc transforming powr dividr/combinr in a back-to-back connction consisting of th dvlopd asymmtric 3-dB dirctional couplrs [79]. Fig Masurd output powr (solid lin) and gain (dashd lin) charactristics vs. input powr of th dvlopd 4-way powr amplifir [79]. 6

59 3.4. Approach to th dsign of asymmtric coupld-lin impdanc transforming dirctional couplrs Th prsntd in prvious sction analysis of impdanc transforming coupld-lin couplrs has shown that th maximum impdanc transformation ratio quals R =. In this cas th slf-capacitanc of on of th coupld lins quals zro. Sinc in practical applications th maximum valu of th impdanc transformation is of th gratst intrst, it is worth to furthr xamin th mthods of analysis and dsign of such transformrs. Figur 3.4 shows an xmplary cross-sction of coupld-transmission lins, in which th innr conductor is compltly nclosd by th outr conductor, satisfying thrfor, th condition C =, and allowing to obtain R =. Such a coupld-lin sction can b rgardd as a connction of two transmission lins, as shown schmatically in Fig Fig Two coupld lins having innr conductor shildd by outr conductor. Fig Schmatic diagram of an impdanc transforming dirctional couplr ralizd as a connction of two transmission lins. In ordr to calculat th impdancs and of th transmission-lin connction having th proprtis corrsponding to th proprtis of a 3-dB coupld-lin sction with impdanc transformation ratio R =, pr-unit-lngth capacitanc and inductanc matrics hav bn considrd. It is vidnt, from Fig. 3.4, that th capacitanc matrix of th coupld transmission lins has th following form Cm C = C m C m C + C m (3.8) 63

60 Lt us assum, that th mdium of propagation is homognous, i.. ε r = ε r = ε r, and th phas vlocity of th propagating wavs quals ν. Th slf and mutual capacitancs can b dfind by lin impdancs as follows: C m ε r = = (3.9) ν c ε r C = = (3.3) ν c C = = (3.3) ν c ε r C ε r ε = + = + r (3.3) ν ν c c Th inductanc matrix can b calculatd from (3.): L = C (3.33) ν whr C is th is th capacitanc matrix of th coupld lins for which th phas vlocitis of propagating wavs qual c, i.. th conductors ar placd in an homognous fr-spac dilctric mdium. In such a cas th componnts of C ar calculatd as follows: C m = (3.34) c ε r C = (3.35) c ε r C = (3.36) c ε r C = + (3.37) c c ε ε r r and componnts of L matrix as: C ε r + ε r L = = (3.38) c C C c m 64

61 L ε r = Lm = = (3.39) c C c Having found th L and C matrix componnts of th coupld lins dscribd by th impdancs of quivalnt circuit composd of two uncoupld lins, th two sts of impdancs, i.., and L, L can b rlatd using (.5,.53) and (.57): L L = (3.4) ( ) L = = + C Cm L L + = = (3.4) C From (3.4, 3.4) and can b found: ( ) = (3.4) L L L L = (3.43) L L L From (3.4, 3.43) th impdancs and of uncoupld transmission lins ar found to b qual = = 5Ω. Th prsntd considrations assumd th homognous dilctric mdium. Not that in such a cas th impdancs and ar not affctd by th chang of th mdium, i.. by th chang of its dilctric constant. In cas of inhomognous dilctric mdia whn ε r ε r, rlations (3.4, 3.4) modify as follows: L = + ε r (3.44) ε r L = ε ε ε ε r r r r + (3.45) In this cas th impdanc can b xprssd as: ε r L = (3.45) ε r and can b found from th following quation: 65

62 = + + L L L L L r r r ε ε ε (3.46) Fig Impdancs and of th transmission lins shown in Fig. 3.4 assuming inhomognous dilctric mdia vs. dilctric constant ratio calculatd using (3.45) and (3,46). Also th limitation on th impdanc transformation ratio R can b drivd for inhomognous coupld lins with th us of th uncoupld-lin approach. Th drivation is similar to th prviously shown ( ) and (3.7), bing th final conclusion that for homognous coupld lins th maximum impdanc transformation ratio R is qual th invrs of squard coupling cofficint k. In this cas th capacitanc and inductanc matrics can b writtn as follows: + = m m m m C C C C C C (3.47) + = r m r r m r m r m C C C C C C ε ε ε ε ε (3.48) [ ] + = + = = C C C C C c C C C C C C C c C c L r r r m r r r m r m r m r m r m r r ε ε ε ε ε ε ε ε ε ε ε ε (3.49) m r m r T C C C c ε ε + = (3.5)

63 T ε r (3.5) = c C ( C + C ) m k C = Cm C + C m (3.5) k ε C r m L = (3.53) ε rc + ε r Cm R = ( C + ε C )( C + C ) ε (3.54) k k T r r m m = = T ε r Cm Th drivd limitation (3.54) stats that in cas of inhomognous coupld lins, in which th highr-impdanc lin is compltly dcoupld from th ground plan, th maximum impdanc-transformation ratio quals th invrs product of inductiv and capacitiv coupling cofficints. Th drivd limitation also provs th prsntd in sction 3.3 dsign approach, in which th dsignd impdanc-transforming dirctional couplrs hav bn dsignd in a multilayr microstrip tchniqu. In ach of th two couplrs, th highrimpdanc coupld-lins wr practically dcoupld from th ground plan, and th impdanc transformation ratios found from (3.54) quals R =.997 and R =.984 in cas of th couplr shown in Tabl 3. and Tabl 3.3, rspctivly. Th analysis of th circuit prsntd in 3.4 can b prformd with th us of th mthod of analysis of symmtrical n-port ntworks with th us of in-phas and out-of-phas xcitations [, 3]. Th convrsion of th circuit into two vn- and odd-mod xcitd subcircuits is prsntd in Fig In ordr to apply this mthod of analysis, it is rquird to assum that all trminating impdancs ar qual, so that th symmtry plan can b constitutd. Th 4-port ntwork simplifis into two -port ntworks with opn-ndd and shortndd transmission-lin stubs. Th impdancs of th stubs s and s ar dfind as: L C o s = θ tan (3.55) o s = θ tan for th ntwork with odd-mod xcitation, and s s = = θ cot θ cot for th ntwork with vn-mod xcitation. (3.55) (3.56) (3.57) 67

64 68 Fig Schmatic diagram of an impdanc transforming dirctional couplr with th symmtry plan indicatd (a), vn- and odd-mod xcitation quivalnt sub-circuits (b) and (c) rspctivly, and thir rconfigurd vrsions (d) - (g). Th rsulting scattring matrics for vn- and odd-mod xcitations ar = o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o s o S,,,,,,,,,,,,,,,,,,,,,,,,, (3.58)

65 Th scattring matrix of th 4-port ntwork can b found from [3]: S S S S S S = = = = = = o ( S + S ) o ( S S ) o ( S + S ) o ( S S ) o ( S + S ) o ( S S ) (3.59) and th rmaining paramtrs ar found from th symmtry condition as S S 4 3 = S = S 4 4 S S 3 3 = S = S 3 4 S S 3 = S = S 4 S S 33 3 = S = S 44 4 S S 34 = S = S 43 (3.6) Th scattring matrix of th ntwork shown in Fig. 3.43a dscribd by ( ) has bn drivd assuming qual trminating impdancs of th 4-port ntwork. In ordr to xamin th proprtis of th impdanc transforming ntwork, th scattring matrix nds to b rnormalizd, so that ach port is trminatd with th propr impdanc, as shown in Fig. 3.4b. Th rnormalization of S matrix trminating impdancs has bn shown in [4, 89] S + + ( S Γ )( I ΓS) ' = A A (3.6) ( i ) i i i whr Γ and A ar th diagonal matrics with thir i th diagonal componnts bing r i and * * r rr / r, rspctivly. Th + indicats th complx conugat transposd matrix and r i ar rflction cofficints of th nw trminating impdancs Ti with rspct to i *. Th calculatd frquncy rsponss of th ntwork for diffrnt dilctric constants ε r and ε r ar shown in Fig. 3.44, whras, Fig shows th valus of transmission, coupling and isolation of th considrd ntwork vs. dilctric constant ratio ε r /ε r, calculatd at th cntr frquncy. 69

66 (a) (b) (c) Fig Frquncy charactristics of th impdanc-transforming ntwork shown in Fig. 3.4 and Fig. 3.4 vs. lctrical lngth in fr spac, for thr diffrnt dilctric proprtis, i..: ε r = ε r = (a), ε r =, ε r = (b) and ε r =, ε r = (c). 7

67 Fig Transmission, coupling and isolation of th impdanc-transforming ntwork shown in Fig. 3.4 and Fig. 3.4 vs. dilctric constant ratio, calculatd for θ = 9 (in fr spac). From th prsntd in Fig 3.44 and 3.45 calculation rsults of paramtrs of th impdanc transforming ntwork, it is sn that, similarly, as in th cas of dirctional couplrs, th inhomognity dgrads th isolation and th transmission and coupling imbalanc of th rsulting couplr at th cntr frquncy. It is important to not, howvr, that sinc th lins ar not coupld lctromagntically but connctd dirctly, th compnsation of dilctric constant inquality my b achivd by adusting th lins physical lngth, so that both lins hav qual lctrical lngth at th cntr frquncy. Th maor advantag of th modifid ntwork having two uncoupld lins is its dsign simplicity. Figur 3.46 shows two possibl ways of physical ralization of th impdanctransforming ntwork with two uncoupld lins, which ar: a shildd striplin on a dilctric sht ovr a ground layr (Fig. 3.46a), and a coaxial lin placd on a dilctric sht ovr a ground layr (Fig. 3.46b). Th dsign procdur rducs to th calculation of impdancs and ffctiv dilctric prmittivity of wir microstrip lin, symmtric striplin and finit thicknss microstrip lin. For all cass ithr formulas can b found [7], or thy can b calculatd numrically using standard commrcially availabl numrical softwar [38]. Fig Cross-sctional viw of a shildd striplin on a dilctric sht ovr a ground layr and a coaxial lin placd on a dilctric sht ovr a ground layr. 7

68 Th approach to th dsign of impdanc transforming dirctional couplrs with th us of th ida of two-uncoupld lins, has bn vrifid xprimntally with th cross-sction shown in Fig. 3.46b, in which a coaxial transmission lin is placd on a dilctric sht. Th impdancs of both lins hav to b qual = = 5 Ω as found from ( ). In ordr to raliz a sction of a coaxial transmission lin having th charactristic impdanc qual 5 Ω, a commrcially availabl smi-rigid cabl can b utilizd. In th dsign, a cabl having th impdanc of 5 Ω, ε r =.5, and outr diamtr D =.3 mm has bn chosn. Th scond transmission lin has bn dsignd on ARLON 5 laminat with th dilctric constant ε r =.5 and thicknss h =.5 mm. In ordr to obtain th impdanc of th lin qual 5 Ω, th cross-sction has to b modifid by introducing additional strip as shown in Fig. 3.47, to dcras th impdanc of th wir-microstrip-lin. Th width of a standard striplin conductor having impdanc = 5 Ω on th chosn dilctric substrat quals w = 9.3, which is by far widr than th outr diamtr of th coaxial lin. This causs that th formr can b nglctd whil dsigning th gomtry, hnc, th standard formulas for microstrip lin calculation can b applid. Th assmbld modl of th coaxial-microstrip impdanc transforming dirctional couplr is shown in Fig Th couplr has bn ralizd togthr with singl-sction impdanc transformrs, as shown in Fig As it is sn th coaxial lin has bn mandrd in ordr to compnsat ffctiv dilctric constant inquality (ε r =.5, ε r =.9). Th masurd frquncy charactristics ar shown in Fig Fig Cross-sctional viw of th coaxial-microstrip lin. Fig Pictur of th dvlopd coaxial-microstrip lin impdanc-transforming dirctional couplr. 7

69 Fig Masurd frquncy charactristics of th 3-dB coaxial-microstrip coupld-lin 5/5 Ω impdanc-transforming dirctional couplr with additional singl-sction impdanc transformrs connctd to th dirct and coupld ports, as shown in Fig Th masurd rsults ar in a good agrmnt with th thortical ons. Th poor isolation and imbalanc of coupling-transmission charactristic ar causd by th mismatch at th isolatd ports, whr th trminating impdanc =.5 Ω has bn connctd, composd of six SMD rsistors having rsistanc R = 75 Ω Broadband asymmtric coupld-lin impdanc transforming dirctional couplrs Th prsntd in sctions 3.3 and 3.4 singl-sction impdanc transforming dirctional couplrs fatur rlativly narrow oprational bandwidth. As it was shown in sction.., th oprational bandwidth of a coupld-lin dirctional couplr can b incrasd whn multisction dirctional couplrs ar dsignd [9, 9, 93, 37]. Similarly, in ordr to incras th oprational bandwidth of an impdanc transformr, a cascad connction of quartr-wav-long sctions can b usd instad of a singl-sction quartr-wav-long transmission lin which constituts a multisction transformr [5, 7, 8, ]. In this sction th dsign of broadband two-sction impdanc-transforming dirctional couplrs is prsntd, and appropriat formulas allowing for calculation of pr-unitlngth impdancs and capacitancs of ach sction ar drivd. Also, th appropriat formulas for th dsign of two-sction impdanc transformrs and asymmtrical dirctional couplrs ar brifly prsntd aftr [] and [9], rspctivly. Th ida of a two-sction broadband impdanc transforming dirctional couplr follows dirctly th mthod dvlopd for th dsign of a two-sction quartr-wav impdanc transformr, so that th impdancs of dirct-fd lins of coupld-lin sctions ar qual to th impdancs of th corrsponding two-sction transformr. Figur 3.5 shows schmati- 73

70 cally, a two-sction transformr in which th impdancs L and L of th quartr-wav transmission lins ar calculatd from th corrsponding valus of VSWR: T L L =, L = (3.6) v v Th VSWR valus for a Chbyshv transformr ar found from []: R v = C + R + C, v = (3.63) v whr: ( ) µ C = R (3.64) ( µ ) w q µ = sin (3.65) 4 ( f f) ( f f ) w = (3.66) q + Fig Schmatic diagram of a two-sction impdanc transformr []. Fig Schmatic diagram of a two-sction impdanc-transforming dirctional couplr. 74

71 A schmatic diagram of a two-sction impdanc-transforming dirctional couplr has bn prsntd in Fig Th impdancs of th dirctly-fd lin L and L qual to th impdancs found from (3.6). Sinc th limiting condition (3.7) holds for ach sction of th dirctional couplr sparatly, it is obvious, that in cas of a two-sction 3-dB dirctional couplr, th impdanc transforming ratio R nds to b lowr than, sinc in such a dirctional couplr coupling k >.77 (> 3dB) is rquird for th tight-coupld sction. Th coupld-lin sctions impdancs L and L ar calculatd for ach sction individually, as shown in sction 3.3 for a singl-sction impdanc-transforming dirctional couplr. Th coupling cofficints for both sctions of th dirctional couplr, having qual-rippl rspons of coupling frquncy charactristic ar xprssd as [9]: z, k, = (3.67) z +, whr: z a + = z = z d (3.68) c, a = c = + d = + β h + β + β + + β h ( β + h) ( β + h) + β + βh β ( β + h) + β h + β + βh + β h (3.69) β = cosh(j )h (3.7) h = (3.7) cos( H ) cosh( J ) C( db) / ( sinh(j )) H =.5sinh (3.7) J = cosh (3.73) cos( θ ) C is th man coupling of th dirctional couplr and θ is th lctrical lngth of th sction at th lowr cut-off frquncy of th dirctional couplr. Having found th impdanc valus for both sctions from ( ) and coupling cofficints from ( ) th pr-unit-lngth paramtrs of ach sction can b calculatd as: C, = ck, L, k, (3.74) 75

72 C, = ck, L, k, (3.75) C, = c L, L, k, (3.76) [ ] = [ C] L (3.77) c Figur 3.5 shows th calculatd frquncy charactristics of a two-sction impdanctransforming dirctional couplr dsignd with th prsntd procdur. In th prsntd xampl th input impdanc of th dirctional couplr has bn chosn to b qual T = 5 Ω and th output impdanc T = 35 Ω, th isolatd port has bn trminatd with th impdanc T4 = 4.5 Ω. Th bandwidth for th impdanc transformr has bn chosn f/f =.8., for which th following valus of impdancs of transmission-lin sctions hav bn found L = Ω and L = Ω. Th corrsponding impdancs for th coupld lins ar L = 3.85 Ω and L = 6.88 Ω. Th coupling cofficints for both sctions hav bn chosn k =.85 and k =.7. From ( ) th following valus of pr-unit-lngth inductancs and capacitanc hav bn found: C = 6.3 pf/m, C = 8.5 pf/m, C = -3 pf/m, L = 6.5 nh/m, L = 83. nh/m, L = 78.3 nh/m. Fig Calculatd S-paramtrs of a two-sction impdanc-transforming dirctional couplr with impdanc ratio R =.43. Th dsignd dirctional couplr faturs qual-rippl coupling charactristic, in which th rturn losss at th input port and isolation ar bttr than 5 db. Th application of such a 76

73 couplr into balancd circuit rquirs phas compnsation, sinc as it was mntiond in sction., asymmtrical dirctional couplrs do not fatur th constant 9 diffrntial phas charactristic. Th mthod for compnsation of diffrntial phas charactristic has bn shown in Fig. 3.53, whr a 9 -long sction of a transmission lin has bn addd to th coupld port. Th rsulting diffrntial phas charactristic is shown in Fig. 3.54, and th phas imbalanc for th dsignd couplr dos not xcd δφ = ±5. Th frquncy rspons of th rsulting balancd circuit ar shown in Fig. 3.55, as it is sn propr transmission has bn obtaind in a broad frquncy rang of ovr two octavs! Fig Schmatic diagram of a two-sction impdanc-transforming dirctional couplr with a transmission-lin sction usd for phas compnsation (a), and a balancd circuit composd of two such dirctional couplrs (b). Fig Diffrntial phas charactristic of a two-sction impdanc-transforming dirctional couplr with a transmission-lin sction usd for phas compnsation. 77

74 Fig Calculatd S-paramtrs of a balancd circuit composd of two two-sction impdanctransforming dirctional couplrs, shown in Fig Summary Th Chaptr prsnts a novl mthod for th dsign of balancd circuits and n-way powr splittr/dividrs, in which a nw class of impdanc-transforming coupld-lin dirctional couplrs is usd to provid simultanous powr splitting/combing and impdanc transformation. At first, a mthod of larg-signal rflction cofficint masurmnt with th us of 3-dB//8 dirctional couplrs, rcntly proposd by th Author [59] has bn dscribd. Scondly, it was shown that, by th application of asymmtric 3-dB dirctional couplrs, it is possibl to simultanously provid powr split and impdanc transformation, in such a way that th impdancs sn at dirct and coupld ports ar qual [79]. It was also shown that to achiv idal match at all ports both lins nd to srv as transforming sctions with th sam impdanc ratio R. Morovr, it is provd that th maximum achivabl impdanc transforming ratio R is dpndant on th coupling C of th usd dirctional couplr and in cas of a 3-dB dirctional couplr R max = for homognous dilctric mdium [79]. Th thortical invstigation has bn supportd by xprimnts, in which two diffrnt 3-dB dirctional couplrs having R = hav bn dsignd and masurd. Th first couplr has bn dsignd to transform th impdanc from 5 to 5 Ω and th scond from 5 to.5 Ω. Ths two couplrs hav bn usd in a dsign of a 4- way powr splitting/combining ntwork, which has bn succssfully tstd in a 4-way powr amplifir [79]. Morovr, in sction 3.4, a novl approach to th dsign of asymmtric impdanc-transforming dirctional couplrs has bn prsntd, in which th couplr has bn rprsntd as a connction of two uncoupld transmission-lin sctions. Th thortical analysis has bn vrifid by an xmplary ralization of a coaxial microstrip impdanc-transforming dirctional couplr. Finally, th concpt of impdanctransforming dirctional couplrs has also bn usd in th dsign of broadband two-sction dirctional couplrs having th impdanc transforming ratio qual R =.43. Th dsign couplr provids both qual powr split, with amplitud imbalanc lss than.5 db, and impdanc transformation within ovr two-octav frquncy rang.

75 4. Broadband Butlr matrics utilizing coupldlin dirctional couplrs Th Butlr matrics ar wll-known and commonly usd ntworks that allow for gnration of multipl bams from a linar array [, 36]. It is du to th fact that bams gnratd with th us of such ntworks fatur an adquat lvl of ovrlapping and th ntworks thmslvs ar losslss and rlativly asy in ralization. In gnral an N x N Butlr matrix is a ntwork having N input and N output ports, in which a signal applid to on of th input ports is qually dividd into N output ports. Morovr, th rsulting phas offst of output signals btwn th rspctiv output ports is constant and dpnds on th choic of th input ports. Ths uniqu proprtis of Butlr matrics rsult in a broad rang of thir applications in modrn-day communications;.g. as bamforming ntworks of multibam antnnas [4,, 7, 8, 6, 35, 37, 45, 5, 53, 8, 36, 59], in dirction finding systms [96, 4], or in multichannl amplifirs [7, 34, 7, 76]. Among othr ntworks allowing for multipl-bam gnration ar: Blass matrics [9, 6], Noln [97] matrics and Rotman matrics [4]. Th most commonly usd ar 4 x 4 Butlr matrics, which ar composd of four 3-dB/9 dirctional couplrs in conunction with two 45 phas shiftrs. Thr has bn significant ffort dvotd to ralization of 4 x 4 Butlr matrics dsignd as a connction of 3-dB branch-lin couplrs in a microstrip tchniqu [33, 36, 48, 49, 77, 87, 56]. In [49] a dual-polarization, multibam antnna intgratd with Butlr matrix has bn shown, in which ight branch-lin couplrs hav bn utilizd, two of which connctd in a tandm connction, raliz a transmission-lin crossovr. As dscribd in sction branch-lin couplrs fatur narrow bandwidth, so do th Butlr matrics composd of such couplrs. Th rsulting bandwidth of a 4 x 4 Butlr matrix composd of a basic branch-lin couplr prsntd in Fig..3 dos not xcd 6.6%, in which rturn losss ar bttr than db. Th amplitud imbalanc of such a matrix quals ±. db and th phas imbalanc is lss than ±. On way to incras th bandwidth of a Butlr matrix it to mploy multisction branch-lin couplrs, as it was shown,.g. in [5], howvr, in such a solution th siz of th rsulting ntwork radically incrass. In [77] it was shown that th oprational bandwidth can b incrasd by adding sctions of transmission-lins half-wav-long at th cntr frquncy. Also ralizations of Butlr matrics oprating in two sparat frquncy rangs ar rportd [6]. Apart from microstrip tchniqu, Butlr matrics can b ralizd in a coplanar-wavguid tchniqu [, ]. In [] a Butlr matrix ralizd as a connction of two branch-lin couplrs and two coplanar-wavguid couplrs has bn prsntd. Th coplanar-wavguid couplrs utiliz lctromagntic coupling via slot in a common ground plan, thrfor, in th rsulting circuit th transmission-lin crossovrs 79

76 8 ar avoidd. Fw ralization of Butlr matrics in rctangular wavguid [45, 7] and slotlin [9] ar rportd. Th bandwidth of Butlr matrics can b significantly incrasd togthr with th siz rduction, whn coupld-lin dirctional couplrs ar applid. An xmplary ralization has bn shown in [], whr a connction of four coupld-lin dirctional couplrs ralizd in a symmtric striplin tchniqu has bn considrd. In th prsntd solution, a significant siz rduction of th circuit has bn obtaind, howvr, th bandwidth is limitd by th fact that th phas shiftrs ar ralizd as sctions of transmission lins having appropriat lngths. Butlr matrics can also b utilizd in multichannl amplifirs, as it was shown in sction 3. [6, 7, 76]. In [34] th application of a Butlr matrix for multichannl switching of high-powr microwav signal has bn prsntd, in which N phas shiftrs hav bn connctd btwn two Butlr matrics. Th most simpl Butlr matrix is constitutd by a singl 3-dB/9 dirctional couplr. By conncting radiating lmnts into th coupld and transmission ports two radiating bams can b gnratd simultanously, placd symmtrically along th lin prpndicular to th lin conncting radiating lmnts [69]. Phas progrssion of th signals xciting radiating lmnts quals ±9 and th scattring matrix of th x Butlr matrix is rprsntd by (.3). Highr ordr Butlr matrics ar cratd by th appropriat connction of two Butlr matrics of ordr N- and by th addition of N/ phas shiftrs and N/ dirctional couplrs. A 4 x 4 Butlr matrix has bn prsntd in Fig. 4. and consists of four 3- db/9 dirctional couplrs and two 45 phas shiftrs. Scattring matrix of such a ntwork can b xprssd as [69]: = S (4.) Fig. 4.. Schmatic diagram of a 4 x 4 Butlr matrix.

77 8 Similarly an 8 x 8 Butlr matrix is cratd from a 4 x 4 Butlr matrix, as shown in Fig. 4.. Th matrix consists of two 4 x 4 Butlr matrics, in which 45 phas shiftrs hav bn rplacd with.5 and 67.5 phas shiftrs, and additional four 45 phas shiftrs and four 3-dB dirctional couplrs ar addd. Th scattring matrix of an 8 x 8 Butlr matrix taks th form: = 8 T A A S (4.) whr: = A (4.3) Fig. 4.. Schmatic diagram of an 8 x 8 Butlr matrix. Th procdur of crating Butlr matrics of highr ordr has bn dscribd in [8,, 5].

78 Th chaptr prsnts th Author s thortical and xprimntal invstigations rgarding th dsign of 4 x 4 and 8 x 8 broadband Butlr matrics. At first, th prviously publishd dsigns of broadband 4 x 4 Butlr matrics utilizing singl-sction coupld-lin dirctional couplrs and ralizd in symmtric striplin and asymmtric multilayr microstrip tchniqus hav bn prsntd. Aftrwards, a novl approach to th dsign of an 8 x 8 Butlr matrix is shown, allowing for ralization of a fully planar intgratd ntwork. Sction 4. dscribs th mthod of broadband 4 x 4 Butlr matrics ralization with th us of multisction symmtrical 3-dB/9 dirctional couplrs. Th larg contnt of th Author s work publishd in [6, 68, 64] is citd in this sction. Th nw concpt of broadband 4 x 4 Butlr matrics ralization has bn xprimntally vrifid by th dsign of thr diffrnt 4 x 4 Butlr matrics, th masurmnts of which ar also prsntd. Morovr, th mthod dvlopd for th dsign of broadband 4 x 4 Butlr matrics has bn xtndd on th dsign of 8 x 8 Butlr matrics, which has also bn xprimntally vrifid. Finally, 4 x 4 Butlr matrics utilizing 3-dB//8 dirctional couplrs ar considrd in th last sction. 4.. Butlr matrics utilizing singl-sction coupld-lin dirctional couplrs x 4 Butlr matrix It is known that singl-sction coupld-lin dirctional couplrs allow for achiving oprational bandwidth up to on frquncy octav. In [6] and [64] th Author has shown th dsigns of on-octav 4 x 4 Butlr matrics ralizd in homognous symmtric striplin and asymmtric microstrip tchnologis. In both cass th prsntd in Fig. 4. Butlr matrix has bn dsignd as a connction of six 3-dB/9 singl-sction coupld-lin dirctional couplrs. In ths dsigns two of th couplrs constitut a -db couplr, which togthr with two sctions of 35 -long transmission lins ralizs a transmission lin crossovr and two broadband 45 diffrntial phas shiftrs [64]. Th dvlopd Butlr matrics hav bn dsignd without th last crossovrs at th outputs (#6 and #7 in Fig. 4.), which nd to b sparatly dsignd for intgration in multibam antnnas [6, 6]. Morovr, th approach of utilizing a tandm connction of two 3-dB/9 couplrs allows to avoid th intr-layr connctions, which can b ralizd as platd-trough-hols or with th us of lctromagntic coupling. Th compnsatd 3-dB coupld-lin dirctional couplrs hav bn usd as basic lmnts of th dsignd Butlr matrics. First of th couplrs has bn dvlopd with th us of th dilctric structur shown in Fig..3. Figur 4.3 shows th masurd rsults of th manufacturd couplr [64]. Th scond on has bn dvlopd in a dilctric structur shown in Fig Th thr-strip multilayr microstip structur has bn slctd to allow for compnsation of inductiv and capacitiv coupling cofficints ncssary for ralization of a quasi-idal coupld-lin sction [65, 5]. Th masurd frquncy charactristics ar prsntd in Fig Excptionally good lctrical prformanc of th couplr has bn achivd with th us of th compnsation mthod of parasitic ractancs connctd with th transition rgions in coupld-lins, dvlopd by th Author [63]. 8

79 Fig Masurd amplitud charactristics of th high-prformanc 3dB/9 coupld-lin dirctional couplr dsignd in a dilctric structur shown in Fig..3 [64]. Fig Cross-sctional viw of multilayr-microstrip coupld-lins usd for th dsign of th 3-dB dirctional couplr and th 4 x 4 Butlr matrix [65]. Fig Masurd amplitud charactristics of th high-prformanc 3dB/9 coupld-lin dirctional couplr dsignd in a dilctric structur shown in Fig. 4.4 [65]. Figur 4.6 prsnts th masurd frquncy rsponss of th 4 x 4 Butlr matrix dvlopd in a symmtric striplin tchniqu. Th manufacturd matrix oprats within 3 GHz frquncy rang, in which th amplitud imbalanc is lss than ±.4 db and th diffrntial phas rippl dos not xcd ±4. Morovr, th matrix faturs vry good impdanc match and high isolation btwn th rspctiv ports, which has bn achivd by th us of th applid tchniqu of parasitic ractancs compnsation [63]. Figurs 4.7 and 4.8 show th photographs of th dvlopd matrix. 83

80 (a) (b) (c) (d) () (f) Fig Masurd frquncy charactristics of th dvlopd 4 x 4 Butlr matrix dsignd in a symmtric striplin tchniqu with th us of th dirctional couplr shown in Fig Transmissions whn port # is fd (a), transmissions whn port # is fd (b), isolations (c), rturn losss (d) and diffrntial phas charactristics whn port # is fd () and whn port # is fd (f) [69]. 84

81 Fig A photograph of th innr layr of th dvlopd 4 x 4 Butlr matrix in a symmtric striplin tchniqu. [69]. Fig A photograph of th assmbld modl of th dvlopd 4 x 4 Butlr matrix in a symmtric striplin tchniqu [69]. Th 4 x 4 Butlr matrix dsignd in a multilayr microstrip tchniqu with th us of th dirctional couplr shown in Fig. 4.4 is prsntd in Fig. 4.9, whras, its masurd frquncy rsponss ar plottd in Fig. 4.. On can s that th matrix faturs slightly largr amplitud imbalanc du to th largr imbalanc of th applid dirctional couplr. Th matrix oprats within.5-3. GHz frquncy rang, in which th amplitud imbalanc is lss than ±db and th diffrntial phas rippl dos not xcd ±6. Also in this cas th matrix faturs xcptionally good rturn losss and isolations btwn rspctiv ports which ar bttr than 5 db. Fig A photograph of th dvlopd 4 x 4 Butlr matrix in an asymmtric multilayr microstrip tchniqu [6]. 85

82 (a) (b) (c) (d) () (f) Fig. 4.. Masurd frquncy charactristics of th dvlopd 4 x 4 Butlr matrix dsignd in an asymmtric multilayr microstrip tchniqu with th us of th dirctional couplr shown in Fig Transmissions whn port # is fd (a), transmissions whn port # is fd (b), isolations (c), rturn losss (d) and diffrntial phas charactristics whn port # is fd () and whn port # is fd (f) [69]. 86

83 x 8 Butlr matrix 8 x 8 and highr ordr Butlr matrics ar vry sldom rportd, du to thir larg complxity, and thrfor, difficulty in dsigning and manufacturing as intgratd units. An xmplary ralization of an 8 x 8 Butlr matrix can b found in [53], whr a compact multilayr dsign has bn shown in LTCC tchnology, having 4 layrs appropriatly stackd and foldd. Anothr xampl of an 8 x 8 Butlr matrix ralization can b found in [7], whr a planar matrix oprating within Ka-band is prsntd, dsignd in a rctangular wavguid tchniqu. On of th rcnt paprs [7] rports th ralization of an 8 x 8 Butlr matrix in CMOS tchnology oprating at 5.5 GHz, in which branch-lin couplrs dsignd with th us of a lumpd-lmnt tchniqu, ar applid. Among rportd broadband 8 x 8 Butlr matrics imprssiv rsults hav bn prsntd in [35], whr a Butlr matrix utilizing taprd-lin dirctional couplrs and oprating in a frquncy rang of f u /f l = 9: has bn prsntd. Th matrix has bn dsignd in a modular fashion, and th moduls hav bn connctd with coaxial cabls. Th ndd broadband phas shiftrs hav bn ralizd with th us of multisction Schiffman phas shiftrs. In this Sction a novl arrangmnt of an 8 x 8 Butlr matrix is prsntd, which allows for a planar fully intgratd dsign. A schmatic diagram of an 8 x 8 Butlr matrix is shown, aftr [36], in Fig. 4.a with th xcption of transmission-lin crossings at th output of th matrix (th numration of th output ports corrsponds to th numration of th original Butlr matrix). Th matrix consists of twlv 3-dB/9 dirctional couplrs, two 67.5, two.5 and four 45 phas shiftrs, additionally multipl crossings of innr conncting lins btwn componnts ar ndd, which complicats th physical ralization. Th ida of ralization of a fully planar intgratd Butlr matrix is in dtail xplaind in Fig. 4.b Fig. 4.d. (a) 87

84 (b) (c) 88

85 Fig. 4.. Schmatic diagram of th 8 x 8 Butlr matrix. Original arrangmnt a), ntwork with phas shiftrs as a tandm connction of 3-dB/9 dirctional couplrs and sctions of transmission lins b), th rarrangd ntwork with only two crossovrs c), th final matrix arrangmnt suitabl for intgratd planar ralization d). As a first stp all th phas shiftrs hav bn ralizd using th mthod prsntd in [64]. Th ndd 67.5 and.5 phas shiftrs togthr with crossovrs of transmission lins hav bn ralizd as a tandm connction of two 3-dB/9 dirctional couplrs and two transmission lins having lctrical lngths θ = 57.5 and θ =.5 (s Fig. 4.b). Th phas imbalanc of such a phas shiftr dos not xcd ±7.5 in an octav frquncy rang. Th four 45 phas shiftrs hav bn ralizd in a similar mannr. A subcircuit with two 45 phas shiftrs and a corrsponding transmission lin crossovr, markd with boldd lins in Fig. 4.a, is rplacd by a tandm connction of two 3-dB/9 dirctional couplrs and two transmission lins having lctrical lngths θ 3 = 35, as shown in Fig. 4.b. Th phas diffrnc of such 45 phas shiftrs dos not xcd ±3. In th rsulting circuit four transmission-lin crossovrs rmain to b rsolvd (s Fig. 4.b). This is don by th rarrangmnt of th four subsctions of th matrix, as shown in Fig. 4.c. Such a rarrangmnt rducs th numbr of transmission-lin crossovrs into two, as it is sn in Fig. 4.c. Th last rmaining crossovrs can b liminatd by rplacing ach of th two 3-dB/9 dirctional couplrs with a tandm connction of two db/9 dirctional couplrs. Such a tandm connction is narly quivalnt to th original 3-dB/9 dirctional couplr, having, howvr, intrchangd coupling and transmission ports and slightly rducd bandwidth. In ordr to nsur propr phas rlations of th ntir ntwork four sctions of transmission lins having lctrical lngth qual θ 4 = 9 hav bn introducd in th corrsponding channls. Th final arrangmnt of th 8 x 8 Butlr matrix, prsntd in Fig. 4.d, allows for its fully intgratd planar ralization, at th xpns of having th input and output ports scattrd around th ntwork. (d) 89

86 (a) (b) (c) (d) Fig. 4.. Frquncy rspons of an idal 8 x 8 Butlr matrix shown in Fig. 4.d. Amplitud charactristics whn port # is fd a), amplitud charactristics whn port # is fd b), diffrntial phas charactristics whn port # is fd c), diffrntial phas charactristics whn port # is fd d). Th thortical prformanc of th 8x8 Butlr matrix shown in Fig. 4.d, has bn calculatd and th rsults ar prsntd in Fig. 4.. Idal coupld-lin dirctional couplrs with =.7 Ω (3-dB dirctional couplr with th amplitud imbalanc of δc = ±. db) and = 75.4 Ω (8.34-dB dirctional couplr giving th amplitud imbalanc of its tandm connction qual δc = ±. db) hav bn assumd for th calculations. Th rsulting Butlr matrix oprats in a frquncy rang of.5: (f/f =.79.) with th maximum amplitud imbalanc ±.45 db and th maximum phas rippl lss than ±5. Th 8 x 8 Butlr matrix has bn xprimntally tstd with th us of th dvlopd 3-dB and 8.34-dB dirctional couplrs shown in Fig Th masurd rsults of th ntir 8 x 8 Butlr matrix ar prsntd in Fig Th obtaind rsults ar in a vry good agrmnt with th thortical ons. Th masurd amplitud imbalanc quals ±.45 db and th maximum phas rror is lss than ±7.5 in th frquncy rang of GHz. Th rturn losss ar bttr than db and isolations bttr than 5 db within th oprational bandwidth. Figur 4.4 shows a pictur of th cntr laminat layr with th tracs of th ntwork tchd. As it is sn th matrix is fully planar with no nd of intrlayr connctions. 9

87 (a) (b) (c) (d) () (f) Fig Masurd frquncy rspons of th dsignd broadband fully intgratd 8 x 8 Butlr matrix. Amplitud charactristics whn port # is fd a), amplitud charactristics whn port # is fd b), diffrntial phas charactristics whn port # is fd c), diffrntial phas charactristics whn port # is fd d), rturn losss ) and isolations f). 9

88 Fig Pictur of th innr laminat layr on which th tracs of th 8 x 8 Butlr matrix wr tchd. 4.. Butlr matrics utilizing multisction coupld-lin dirctional couplrs Th prsntd in sction 4. Butlr matrics offr oprational bandwidths up to on frquncy octav. Th limitation of th achivabl bandwidth is a rsult of th mploymnt of singl-sction coupld-lin dirctional couplrs. As it was mntiond in sction.., widr bandwidths can b obtaind with th us of multisction coupld-lin dirctional couplrs. In [68], th Author has proposd a nw class of broadband 4 x 4 Butlr matrics, which consists of multisction symmtrical 3-dB/9 coupld-lin dirctional couplrs and phas corrction ntworks. A larg contnt of [68] has bn citd in this sction. In sction 4. two xmplary ralizations of singl-octav 4 x 4 Butlr matrics hav bn prsntd, ach composd of six 3dB/9 dirctional couplrs [64]. A tandm connction of two of th six dirctional couplrs srvs as a transmission-lin crossovr, and togthr with 35 -long transmission lins, as two broadband 45 phas shiftrs. This configuration works wll in cas of singl-sction 3dB/9 dirctional couplrs but cannot b straightforwardly applid for ralization of broadband 4 x 4 Butlr matrics consisting of multisction 3dB/9 dirctional couplrs. This is du to th fact that th tandm connction of two multisction 3dB/9 dirctional couplrs togthr with th rfrnc lins dos not xhibit broadband 45 diffrntial phas charactristics. Figur 4.5 prsnts a diffrntial phas charactristic btwn a tandm connction of two thr-sction symmtrical 3dB/9 dirctional couplrs and a 35 -long rfrnc lin (dashd lin). It can b sn, that th charactristic inclins rsulting in a larg ovrall diffrntial phas imbalanc. To ovrcom th dscribd difficulty, th Author has proposd in [68] a modifid circuit in which a coupld-lin C -sction 9 long at f (introducd by Schiffman in [33]) is addd. This modification allows to obtain qual-rippl diffrntial phas charactristics in a wid frquncy rang. Th modifid schmatic diagram of th proposd broadband 4 x 4 Butlr 9

89 matrix is prsntd in Fig Th ntwork consists of six thr-sction symmtrical 3- db/9 coupld-lin dirctional couplrs. Th middl ntwork (as shown in Fig. 4.6) consists of a tandm connction of two dirctional couplrs and two coupld-lin C -sctions togthr with sctions of transmission lins. Th calculatd amplitud and diffrntial phas charactristics of th Butlr matrix shown in Fig. 4.6, ar prsntd in Fig For ths calculations a thr-sction coupld-lin couplr with coupling rippl δc = ±.9 db has bn considrd and th achivd paramtrs of th Butlr matrix ar: maximum amplitud imbalanc δc max = db and phas rippl δϕ = ± 6. Th normalizd modal charactristic impdancs of th coupld-lin C -sction ar: =.8 Ω and o =.8 Ω, whras, transmission-lin sctions ar 35 -long (at f ). Th obtaind bandwidth of th Butlr matrix quals BW = 4.38 and is qual to th bandwidth of th applid dirctional couplr. Fig Diffrntial phas charactristics btwn transmission of a tandm connction of two thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs and a sction of transmission lin with (solid lin) and without (dashd lin) a C -sction of coupld-lins [68]. Fig Gnric schmatic of a broadband 4x4 Butlr matrix consisting of thr-sction symmtrical coupld-lin dirctional couplrs. Two of th couplrs constitut a broadband transmission lin crossovr and, togthr with coupld-lin C -sctions and rfrnc lins, broadband 45 phas shiftrs [68]. 93

90 (a) (b) Fig Frquncy charactristics of a 4x4 Butlr matrix bing a connction of six thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs having coupling imbalanc δc = ±.9 db, two phas corrction ntworks and two 35 -long transmission lins. Transmissions whn port # is fd (a), diffrntial phas charactristics whn port # is fd (b) and diffrntial phas charactristics whn port # is fd (c) [68]. (c) 94

91 Figur 4.8 prsnts th comparison btwn th bandwidth of th middl ntwork (dashd lin) and th bandwidth of th usd dirctional couplrs alon (solid lins). Morovr, it should b undrlind, that th amplitud imbalanc of th couplrs has a dirct influnc on th amplitud charactristic of a crossovr bing thir tandm connction as prsntd in Fig. 4.9 and dtrmins th isolation lvl of th crossovr (isolation I has bn schmatically markd in Fig. 4.6). Th analysis of such broadband Butlr matrics shows, that th limitd isolation I of th crossovr dtriorats ovrall amplitud and phas charactristics of th ntworks [68]. Figur 4. shows th diffrntial phas charactristics of th Butlr matrix shown in Fig. 4.6, in which th isolation I is assumd idal, i.. I =. It is sn, that th rsulting diffrntial phas charactristics hav an qual-rippl charactr with smallr imbalanc and two of th thr charactristics ar idntical. This phnomna has bn utilizd in a concpt of 4 x 4 Butlr matrix ralization having a modifid middl ntwork also proposd by th Author in [68]. Fig Transmission and coupling (solid lins) of a thr-sction symmtrical 3-dB/9 dirctional couplr and diffrntial phas charactristic (dashd lin) btwn transmission of a tandm connction of two thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs and a sction of transmission lin togthr with a C -sction of coupld-lins [68]. Fig Transmission and isolation of a crossovr bing a connction of six thr-sction symmtrical 3dB/9 coupld-lin dirctional couplrs having coupling rippl δc = ±.9 db [68]. 95

92 Fig. 4.. Diffrntial phas charactristics of a 4 x 4 Butlr matrix bing a connction of six thrsction symmtrical 3-dB/9 coupld-lin dirctional couplrs having coupling rippl δc = ±.9 db and two phas corrction ntworks in which th isolation I (Fig. 4.6) is assumd idal (I = ) [68]. Fig. 4.. Gnric schmatic of a middl ntwork consisting of N-sction symmtrical coupld-lin dirctional couplrs, two (N-)/-sction coupld-lin phas corrction ntworks and rfrnc lins [68]. Th ida of broadband 4 x 4 Butlr matrix ralization with th us of thr-sction symmtrical 3-dB/9 dirctional couplrs can b xtndd on broadband 4 x 4 Butlr matrics consisting of multisction symmtrical dirctional couplrs. In such ntworks, if th numbr of coupld-lin sctions of a dirctional couplr applid in a crossovr changs, thn an appropriat chang of th numbr of coupld-lin sctions has to b introducd to th phas corrction ntwork. A schmatic diagram of a middl ntwork, in which an N-sction dirctional couplr is applid, is shown in Fig. 4.. Th numbr of C -sctions usd within th phas corrction ntwork incrass with th incras of th coupld-lin sctions of th dirctional couplr and can b xprssd as follows: 96 Nc N p = (4.4) whr: N p numbr of coupld-lin sctions of th phas corrction ntwork, N c numbr of sctions of th dirctional couplr. Th lctrical lngth of th rfrnc lin can b found from: Θ o o c = N p (4.5)

93 Fig. 4.. Diffrntial phas charactristics of a middl ntwork (Fig. 4.) for th cas of thr-, fiv-, svn- and nin-sction dirctional couplrs (δc = ±. db). Th phas corrction ntworks numbr of sctions and normalizd impdanc for ach sction wr slctd to obtain qual-rippl charactristics [68]. (a) Fig Frquncy charactristics of a 4 x 4 Butlr matrix bing a connction of six nin-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs having coupling rippl δc = ±. db, two phas corrction ntworks, and two 855 -long transmission lins. Transmissions (a) and diffrntial phas charactristics (b) [68]. (b) 97

94 Tabl 4. Normalizd vn-mod charactristic impdancs of phas-compnsation ntworks, paramtrs of middl ntworks and paramtrs of 4 x 4 Butlr matrics dsignd with th us of 3-, 5-, 7-, and 9-sction 3-dB symmtrical dirctional couplrs [68] n = 3 Θ c = 35 n = 5 Θ c = 495 n = 7 Θ c = 675 n = 9 Θ c = 855 I δc = ±. db I = 3.6 db δϕ = ±. δc max =.34 db δϕ max =. BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork II δc = ±. db I = 6.5 db δϕ = ±. δc max =.7 db δϕ max = 4. BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork III δc = ±.4 db I =.3 db δϕ = ± 3.5 δc max =.4 db δϕ max = 8. BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork IV δc = ±.6 db I = 6.6 db δϕ = ± 5. δc max =. db δϕ max =.5 BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork V δc = ±.8 db I = 3.9 db δϕ = ± 6.5 δc max = 3. db δϕ max = 5.5 BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork VI δc = ± db I =.7 db δϕ = ± 8.5 δc max = 3.9 db δϕ max =

95 BW n [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork [Ω] * couplr [Ω] phas comp. ntwork * couplr valus ar takn from [9], whr: δc dirctional couplrs coupling charactristic rippl, BW bandwidth of dirctional couplrs and Butlr matrics, I crossovr isolation, δϕ - diffrntial phas rippl of middl ntworks, δc max Butlr matrics maximum amplitud imbalanc, δϕ max - Butlr matrics diffrntial phas maximum imbalanc. Fig A broadband 4 x 4 Butlr matrix consisting of thr-sction symmtrical coupld-lin dirctional couplrs with improvd amplitud and phas charactristics. Th middl ntwork consists of a tandm connction of two fiv-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs, two two-sction coupld-lin phas corrction ntworks and rfrnc lins [68]. Th comparison of diffrntial phas charactristics for th cas of 3-, 5-, 7- and 9- sction dirctional couplrs with th coupling rippl δc = ±. db is prsntd in Fig. 4.. It is sn that in ach cas qual-rippl charactristics hav bn obtaind. Figur 4.3 prsnts th amplitud and diffrntial phas charactristics of a 4 x 4 Butlr matrix consisting of nin-sction coupld-lin couplrs having coupling rippl δc = ±. db. Th normalizd vn-mod charactristic impdancs of th phas corrction ntworks ar =.3 Ω and =. Ω, 3 =.7 Ω, 4 =.795 Ω and th transmission lin sctions ar 855 -long (at f ). It is sn that th obtaind bandwidth of th Butlr matrix xcds on frquncy dcad which confirms th possibility of ultrabroadband 4 x 4 Butlr matrix ralization with th us of th mthod proposd in [68]. Th following paramtrs of th 4 x 4 Butlr matrix hav bn found: maximum amplitud imbalanc δc max =.7 db and phas imbalanc δϕ = ± 4.. In Tabl 3. th rsults of numrical analysis of broadband multisction 4 x 4 Butlr matrics ar shown [68]. Th normalizd vn-mod charactristics impdancs ar givn for on-, two-, thr- and four-sction phas corrction ntworks ndd rspctivly for thr-, fiv-, svn- and nin-sction coupld-lin couplrs. Valus of th couplrs normalizd vn-mod charactristics impdancs ar takn from [9]. Th tabl also provid th obtaind paramtrs of rspctiv middl ntworks such as: lngth of th ncssary sction of th rfrnc transmission lin Θ c, th crossovr 99

96 isolation I and th diffrntial phas imbalanc δϕ. Morovr, th rsulting Butlr matrix paramtrs for ach cas ar givn, which ar: th maximum amplitud imbalanc δc max and th maximum diffrntial phas imbalanc δϕ max. Tabl 4. Normalizd vn-mod charactristic impdancs of phas-compnsation ntworks, paramtrs of middl ntworks and paramtrs of 4 x 4 Butlr matrics dsignd with th us of 3-sction 3-dB symmtrical dirctional couplrs and 5-sction 3-dB symmtrical dirctional couplrs usd in th middl ntwork [68] couplr applid in a middl ntwork phas compnsation ntwork I δc = ±. db BW = 3. n n [Ω] n [Ω] I δc = ±.54 db BW = 4.5 n n [Ω] n [Ω] I δc = ±. db BW = 5.54 n n [Ω] n [Ω] I δc = ±. db BW = n n [Ω] n [Ω] I δc = ±.38 db BW = 8.45 n n [Ω] n [Ω] I δc = ±.5 db BW = 9.5 n n [Ω] n [Ω] middl ntwork Θ = 495 I = 46.7 db δϕ = ±.4 Θ = 495 I = 38. db δϕ = ±.7 Θ = 495 I = 3. db δϕ = ±.3 Θ = 495 I = 5.9 db δϕ = ±. Θ = 495 I =. db δϕ = ± 3. Θ = 495 I = 8.4 db δϕ = ± 4. Butlr matrix δc max =. db δϕ max =.5 δc max =.3 db δϕ max =.8 δc max =.9 db δϕ max =.8 δc max =.54 db δϕ max = 6. δc max =.8 db δϕ max =.6 δc max =.8 db δϕ max = 4.4 As it was shown in Fig. 4., th paramtrs of broadband 4 x 4 Butlr matrics dpnd on th isolation of th crossovr dsignd as a tandm connction of two multisction 3- db/9 coupld-lin dirctional couplrs. Th limitd isolation of such a crossovr dtriorats th amplitud and diffrntial phas imbalanc of th rsulting 4 x 4 Butlr matrix. Th Butlr matrix s charactristics can b improvd with th us of a modifid structur, in

97 which th middl ntwork is dsignd with th us of dirctional couplrs having gratr numbr of sctions than th numbr of sctions of th rmaining couplrs. This solution is schmatically shown in Fig. 4.4, whr a Butlr matrix dsignd with th us of thrsction 3-dB/9 coupld-lin dirctional couplrs is shown. As it is sn 5-sction coupld-lin couplrs ar usd within th middl ntwork, and to achiv 45 qual-rippl diffrntial phas charactristics th phas corrction ntworks consisting of two coupldlin sctions ar usd. Th amplitud imbalanc of th couplrs applid within th middl ntworks is proprly slctd in such a way that th couplrs oprat within th sam frquncy rang as th rmaining couplrs. Such ntworks hav bn also analyzd by th Author in [68]. Th obtaind numrical rsults of such 4 x 4 Butlr matrics ar prsntd in Tabl 4.. Gnrally, such ntworks xhibit smallr amplitud and diffrntial phas charactristic imbalanc. For xampl in cas of a 4 x 4 Butlr matrix dsignd with th us of dirctional couplrs having coupling rippl δc = ±. db th achivd amplitud and diffrntial phas charactristic imbalanc quals δc max =.7 db, δϕ max = 4. for th cas of an unmodifid Butlr matrix and δc max =.3 db, δϕ max =.8 for th cas of a modifid Butlr matrix. Th prsntd concpt of broadband 4 x 4 Butlr matrix ralization has bn xprimntally vrifid by thr diffrnt dsigns, namly:. A broadband 4 x 4 Butlr matrix dsignd in a symmtric striplin tchniqu in which as basic lmnts thr-sction symmtrical 3-dB dirctional couplrs ar usd [68].. A broadband 4 x 4 Butlr matrix dsignd in an asymmtric multilayr microstrip tchniqu also with th us of thr-sction symmtrical 3-dB dirctional couplrs [68]. 3. An ultrabroadband 4 x 4 Butlr matrix dsignd in a symmtric striplin tchniqu, in which fiv-sction symmtrical 3-dB dirctional couplrs ar usd. Th first 4 x 4 Butlr matrix has bn ralizd with th us of th thr-sction 3-dB/9 coupld-lin dirctional couplr dscribd by th Author in [65]. As a middl ntwork a tandm connction of two such couplrs hav bn usd togthr with two 35 -long transmission lins and two coupld-lin C -sctions having normalizd vn-mod charactristic impdanc qual =.8 Ω. Th C -sctions hav bn compnsatd with th us of th mthod dvlopd by th Author [63], which allowd for significant improvmnt of its rturn losss. Layout of th dsignd 4 x 4 Butlr matrix is shown in Fig Masurd amplitud charactristics of th manufacturd 4 x 4 Butlr matrix in comparison with th charactristics of an idal Butlr matrix ar shown in Fig Th Butlr matrix xhibits wid oprational bandwidth qual BW = 5. Th obtaind amplitud imbalanc dos not xcd ± db, whras, both rturn losss and isolations ar bttr than db within th ntir bandwidth. Fig. 4.7 shows th masurd diffrntial phas charactristics in comparison with th idal ons. Th obtaind diffrntial phas imbalanc dos not xcd ±8 and is comparabl with th calculatd on. Fig. 4.8 prsnts a photograph of th dvlopd broadband 4 x 4 Butlr matrix.

98 Fig Layout of th dsignd broadband 4 x 4 Butlr matrix consisting of six thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs, two C -sctions of coupld-lins and rfrnc lins [68]. Fig Calculatd (dashd lins) and masurd (solid lins) amplitud charactristics of th dvlopd broadband 4 x 4 Butlr matrix consisting of six thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs, two C -sctions of coupld-lins and rfrnc lins [68]. Fig Calculatd (dashd lins) and masurd (solid lins) diffrntial phas charactristics of th dvlopd broadband 4 x 4 Butlr matrix consisting of six thr-sction symmtrical 3-dB/9 coupld-lin dirctional couplrs, two C -sctions of coupld-lins and rfrnc lins [68].

99 (a) (b) Fig Photograph of th dvlopd broadband 4 x 4 Butlr matrix. Th cntr laminat in which tracs ar tchd (a) and th assmbld modl (b) [68]. Fig Schmatic diagram of th broadband 4x4 Butlr matrix dsignd in a microstrip multilayr structur [68]. Th scond xampl of a broadband 4 x 4 Butlr matrix ralization has bn prsntd in [68]. A schmatic diagram of th dsignd Butlr matrix is shown in Fig It utilizs th sam concpt, howvr, in this cas Schiffman C -sctions hav to b dsignd sparatly, du to th fact that th chosn microstrip structur is asymmtric, as shown in Fig It consists of a thick laminat layr (h =.787 mm ε r = 3.) on which a vry thin layr (h =.5 mm ε r = 3.4) has bn bondd using prprg matrial (h =.99 mm ε r = 3.38). Th tracs ar tchd on both sids of th uppr thin dilctric. As a ky lmnt of th dvlopd broadband Butlr matrix th high-prformanc thr-sction 3-dB multilayr 3

100 dirctional couplr dsignd with th us of multi-tchniqu compnsation has bn usd [67]. Taking into account th couplr s proprtis th rquird Schiffman C -sctions hav bn optimizd for which th normalizd vn-mod impdanc has bn found to b qual =.3 Ω. Th rquird C -sctions hav bn dsignd and compnsatd with th us of shunt capacitancs connctd to coupld-lins, as it was shown in [63]. It is worth mntioning that in this cas th compnsation tchniqu has bn usd not only for compnsation of parasitic ractancs of th transition rgion btwn coupld and signal lins, but also, to improv th rturn losss which dtriorat du to inquality of modal phas vlocitis in microstrip coupld lins [3]. Th calculatd rsults of th broadband 4 x 4 Butlr matrix ar shown in Fig Th achivd amplitud imbalanc quals ±.7 db and th diffrntial phas imbalanc quals ±4 within two-octav frquncy band (. 4.5 GHz). Fig Cross-sctional viw of a multilayr microstrip structur in which th broadband 4x4 Butlr matrix has bn dsignd [68]. (a) (b) (c) (d) Fig Calculatd rsponss of 4x4 Butlr matrix. (a) transmissions whn port # is fd, (b) transmissions whn port # is fd, (c) diffrntial phas charactristics whn port # is fd and (d) diffrntial phas charactristics whn port # is fd [68]. 4

101 (a) (b) (c) (d) () (f) Fig Masurd rsponss of broadband 4 x 4 Butlr matrix: (a) transmissions whn port # is fd, (b) transmissions whn port # is fd, (c) rturn losss, (d) isolations, () diffrntial phas charactristics whn port # is fd and (f) ovrall dissipation losss [68]. Masurd rsults of th manufacturd Butlr matrix ar shown in Fig Th obtaind amplitud and phas imbalanc quals ±.9 db and ±, rspctivly, which is in a good agrmnt with th calculatd rsults, shown in Fig Morovr, th matrix xhibits good rturn losss and isolation which ar bttr than 6 db and db, rspctivly. Fig. 4.3f shows th masurd ovrall dissipation losss of th manufacturd broadband 4 x 4 Butlr matrix, which do not xcd db for th cntr frquncy of 3 GHz. Layout of th dsignd Butlr matrix is shown in Fig. 4.33, in which two diffrnt dsigns of th C - sctions ar sn. Fig prsnts a pictur of th assmbld modl of th manufacturd broadband Butlr matrix. 5

102 Fig Layout of dvlopd broadband 4 x 4 Butlr matrix dsignd in a microstrip multilayr structur [68]. Fig Photograph of assmbld modl of th dvlopd broadband 4 x 4 Butlr matrix [68]. Th third xampl shows th dsign of ultrabroadband 4 x 4 Butlr matrix oprating from.8 up to GHz, which consists of six 5-sction 3-dB/9 symmtrical coupld-lin dirctional couplrs and two -sction Schiffman C -sctions. Th utilizd compnsatd 5- sction symmtrical 3-dB/9 coupld-lin dirctional couplr is prsntd in [56]. Th couplr has bn compnsatd with th mthod dscribd in [63, 64], which allowd for significant improvmnt of its amplitud charactristics. Howvr, th applid lumpd capacitors, that ar usd for compnsation of parasitic ractancs of th transition rgions btwn coupld-lin sctions modify phas charactristics of th couplr. It is important to strss that du to th fact that th couplr faturs full symmtry th diffrntial phas charactristics ar unchangd and ar qual 9 at all frquncis. Howvr, th phas charactristics of th transmission and coupling ar changd spcially at high frquncy rang, which influncs th diffrntial phas charactristics of th dsignd Butlr matrix. Fig. 4.35a shows th calculatd diffrntial phas charactristics of th 4 x 4 Butlr matrix consisting of six 5-sction coupld-lin dirctional couplrs, for which according to [68], th rquird vn-mod charactristic impdancs of th coupld-lin sctions of Schiffman C -sctions qual: = 7.5 Ω, = Ω. Each of th sctions has 9 lctrical lngth at th cntr frquncy of th couplr. Th 4 x 4 Butlr matrix faturs good diffrntial phas charactristics at lowr frquncy rang, whras, at highr frquncis larg imbalanc is obsrvd. Furthr rsarch showd that this dtrioration is causd du to th capacitiv compnsation applid to th dirctional couplrs, and to improv ths charactristics modifid Schiffman C -sctions hav to b applid. 6

103 (a) (b) (c) Fig Diffrntial phas charactristics whn port # is fd of th 4 x 4 Butlr matrix composd of six compnsatd 5-sction coupld-lin dirctional couplrs and Schiffman phas shiftrs having - = 7.5 Ω, = Ω, Θ = 9 (a), = 79 Ω, = 54.3 Ω, Θ = 9 (b) and = 8 Ω, = 55 Ω, Θ = 7.3 (c). Rsults of circuit analysis. 7

104 (a) (b) Fig Rsults of lctromagntic calculations th 4 x 4 Butlr matrix composd of compnsatd 5-sction coupld-lin dirctional couplrs and th modifid middl ntwork. Transmissions whn port # is fd (a), transmissions whn port # is fd (b). Fig. 4.35b shows th calculation rsults of th diffrntial phas of th 4 x 4 Butlr matrix, in which th vn-mod charactristic impdancs of th coupld-lin Schiffman C sctions hav bn modifid to = 79 Ω, = 54.3 Ω and lctrical lngths of th rfrnc lins ar qual 5. It is sn that th crtain improvmnt has bn achivd, howvr, th diffrntial phas charactristics of th Butlr matrix still fatur larg imbalanc. Furthr improvmnt can b achivd whn th lctrical lngths of th Schiffman C -sctions ar modifid. Fig. 4.35c shows th diffrntial phas of th 4 x 4 Butlr matrix, in which th paramtrs of th coupld-lin sctions of Schiffman phas shiftrs ar qual = 8 Ω, = 55 Ω, th sctions hav lctrical lngth qual 7.3 at th cntr frquncy of th applid couplr, and th rfrnc lins hav lctrical lngths qual 588. Such a modification allows for significant improvmnt of th diffrntial phas charactristics, which fatur total imbalanc lss than ±5. Th dsignd 4 x 4 Butlr matrix with th modifid middl ntwork has bn analyzd lctromagntically. Th calculatd amplitud charactristics ar shown in Fig and 4.37, and th achivd amplitud imbalanc dos not xcd ± db, whras, th rturn losss ar bttr than 5 db. Th calculatd diffrntial phass ar shown in Fig and fatur imbalanc qual ±9. 8

105 Fig Rturn losss th 4 x 4 Butlr matrix composd of compnsatd 5-sction coupld-lin dirctional couplrs and th modifid middl ntwork. Rsults of lctromagntic calculations. (a) (b) Fig Rsults of lctromagntic calculations th 4 x 4 Butlr matrix composd of compnsatd 5-sction coupld-lin dirctional couplrs and th modifid middl ntwork. Diffrntial phas charactristics whn port # is fd (a), diffrntial phas charactristics whn port # is fd (b). 9

106 Th masurd amplitud charactristics of th manufacturd matrix ar shown in Fig and 4.4. Th obtaind rturn losss ar bttr than db at all input ports which is consistnt with th simulatd rsults. Th achivd amplitud imbalanc is qual ± db which corrsponds to th simulation rsults xcpt for th transmissions masurd at port # within th frquncy rang btwn.5 - GHz, whr th amplitud imbalanc is qual ± db, du to high rflctions within th circuit. Masurd diffrntial phas charactristics ar shown in Fig. 4.4, and th achivd phas imbalanc is qual ±5. Fig. 4.4 shows a pictur of th cntr laminat layr, in which th tracs of th manufacturd Butlr matrix ar visibl. (a) (b) Fig Masurmnt rsults of th manufacturd 4 x 4 Butlr matrix. Transmissions whn port # is fd (a), transmissions whn port # is fd (b).

107 Fig Masurd rturn losss of th manufacturd 4 x 4 Butlr matrix. (a) (b) Fig Masurmnt rsults of th manufacturd 4 x 4 Butlr matrix. Diffrntial phas charactristics whn port is fd (a), diffrntial phas charactristics whn port is fd (b).

108 Fig Photograph of th cntr laminat of th ultrabroadband 4x4 Butlr matrix. Th concpt of th dsign of broadband 4 x 4 Butlr matrics with th us of multisction symmtrical dirctional couplrs and thir tandm connctions can b xtndd on ralization of highr ordr Butlr matrics. Figur 4.43 shows a schmatic diagram of a broadband 8 x 8 Butlr matrix, in which four submatrics ar distinguishd, similarly to th prsntd in Fig. 4. concpt of th 8 x 8 Butlr matrix ralizd with th us of singlsction coupld-lin couplrs. Th input of th broadband 8 x 8 Butlr matrix is constitutd by two idntical A submatrics which ar similar to th prsntd arlir broadband 4 x 4 Butlr matrics. Howvr, in this cas th phas corrcting ntworks hav to b modifid to nsur constant broadband.5 and 67.5 phas shifts. This can b achivd by an appropriat modification applid to th middl ntwork, as shown in Fig Th.5 phas shift can b ralizd with application of two-sction Schiffman C -sction (C phas shiftr in Fig. 4.43) and a transmission lin having lctrical lngth Θ, whras th 67.5 phas shift can b ralizd with th us of a singl-sction Schiffman C -sction (C phas shiftr in Fig. 4.43) and a transmission lin having lctrical lngth Θ. Th calculatd rsults of th diffrntial phas charactristics of such middl ntworks of th submatrix A ar shown in Fig. 4.44, in which 5-sction symmtrical coupld-lin dirctional couplrs having = 8. Ω, = 7.36 Ω, 3 = 55.7 Ω ar usd. Th vn-mod charactristic impdanc of th C Schiffman C -sction quals = 53.9 Ω and th lctrical lngth of th transmission lin sction Θ = 697.5, whras th vn-mod charactristic impdancs of th C Schiffman C -sction qual = 94.9 Ω, = 6. Ω and th lctrical lngth of th transmission lin sction Θ = Th achivd diffrntial phas imbalanc quals ±.5 and ±6.5 for th.5 and 67.5 phas shiftrs, rspctivly. Th corrsponding diffrntial phas charactristics of th A submatrix with th dscribd middl ntwork ar shown in Fig Th obtaind diffrntial phas imbalanc for all ports dos not xcd th phas imbalanc of th 67.5 phas shiftr ralizd with th us of th dscribd middl ntwork, i.. ±6.5. Th output part of th broadband 8 x 8 Butlr matrix prsntd in Fig is constitutd by two idntical B submatrics, in which 45 phas shiftrs ar ralizd in an idntical mannr as in th cas of broadband 4 x 4 Butlr matrics. Th concpt of a broadband 8 x 8 Butlr matrix with th us of 5-sction symmtrical 3- db/9 dirctional couplrs has bn vrifid xprimntally. In th dsign th dirctional couplr dscribd in [56] has bn usd. As th B matrix, th 4 x 4 Butlr matrix prsntd in Fig. 4.4 has bn usd in which th two input dirctional couplrs hav bn rmovd. In ordr to dsign th A matrix two diffrnt Schiffman C -sctions hav bn dvlopd. Du to th fact that th rquird coupling of th tight coupld sction within th

109 C Schiffman C -sction is not ralizabl in a symmtric striplin dg-coupld tchniqu, and th broadsid coupld-lin tchniqu is not suitabl (th ncssity of intr-layr connctions), th tight coupld sction of th C Schiffman C -sction has bn ralizd as rntrant coupld lins. Th analysis of such coupld lins can b found in [4]. Fig Schmatic diagram of a broadband 8 x 8 Butlr matrix utilizing 5-sction symmtrical 3- db/9 dirctional couplrs. 3

110 Fig Calculatd diffrntial phas charactristics btwn a tandm connction of two 5-sction symmtrical coupld-lin dirctional couplrs and phas corrcting ntworks showing th ralization of broadband.5 and 67.5 phas shiftrs. (a) (b) (c) (d) Fig Calculatd diffrntial phas charactristics of th A submatrix of th 8 x 8 Butlr matrix shown in Fig Diffrntial phass whn port # is fd (a), port # is fd (b), port #3 is fd (c), port #4 is fd (d). Solid lins rfr to lft axs, dashd lins rfr to right axs. 4

111 (a) (b) (c) (d) Fig Frquncy rsponss of th ultrabroadband 8 x 8 Butlr matrix calculatd with th us of th masurmnt rsults of th A and B submatrics. Amplitud charactristics whn port # is fd a), port # is fd b), port #3 is fd c), and port #4 is fd d). Th final coupling cofficints of all phas compnsating coupld-lin sctions and thir lctrical lngths hav bn modifid taking into account th actual paramtrs of th dsignd 5-sction 3-dB dirctional couplrs obtaind from lctromagntic analysis, similarly to th mthod dscribd in th cas of 4 x 4 Butlr matrix prsntd in Fig Th following paramtrs hav bn obtaind: = 63.4 Ω and Θ = 7.8 for th C Schiffman C -sction and = 6. Ω, Θ = 83.9, = 59.6 Ω, Θ = for th C Schiffman C -sction. Th lctrical lngths of th transmission lin sctions Θ and Θ rmain unchangd, i.. Θ = and Θ = Both submatrics A and B hav bn manufacturd, and thir masurd rsults hav bn usd to calculat frquncy rspons of th broadband 8 x 8 Butlr matrix. Th obtaind rsults ar shown in Fig and Th achivd amplitud imbalanc dos not xcd ±.5 db for all ports, and th phas imbalanc is lss than ±4. Figur 4.48 prsnts picturs of th cntr laminats of both manufacturd submatrics. 5

112 (a) (b) (c) (d) Fig Frquncy rsponss of th ultrabroadband 8 x 8 Butlr matrix calculatd with th us of th masurmnt rsults of th A and B submatrics. Diffrntial phas charactristics whn port # is fd a), port # is fd b), port #3 is fd c), and port #4 is fd d). (a) (b) Fig Picturs of th cntr laminats of th manufacturd submatrics of th ultrabroadband 8 x 8 Butlr matrix. Submatrix A (a) and submatrix B (b). 6

113 4.3. Butlr matrics with th us of /8 dirctional couplrs Th dscribd in prvious sctions 4 x 4 Butlr matrics produc ±45 and ±35 diffrntial phass btwn adacnt output ports. Anothr way of achiving constant phas shifts btwn adacnt outputs of th Butlr matrix has bn dscribd in [4], in which four 3- db//8 dirctional couplrs ar usd togthr with a 9 phas shiftr. Th rsulting phas offsts of such a matrix ar qual, 8 and ±9, thus whn connctd to a linar antnna array allow for gnration four bams: broadsid, ndfir and two bams symmtrically locatd on both sids with rspct to th broadsid bam. An quivalnt way of ralization of such a ntwork rquirs th connction of thr 3-dB/ /8 dirctional couplrs and on 3-dB/9 dirctional couplr, as proposd in []. In this cas no fixd phas shiftrs ar rquird. Thr ar fw paprs dscribing ralizations of such ntworks. In [7], a mthod for broadband dirctional couplr ralization with any dsird frquncy insnsitiv phas shift is prsntd, as a connction of a wak coupling dirctional couplr with 3-dB /8 hybrid couplr. In [44], a Butlr matrix is proposd in which ach hybrid was ralizd using a thr branch-lin 9 hybrid augmntd by two 45 Schiffman phas shiftrs at th input sid and two rvrsd 45 Schiffman phas shiftrs at th output sid. Th achivd bandwidth quals 6%. In this sction th attntion is paid to th dsign of broadband 4 x 4 Butlr matrics utilizing 3-dB//8 dirctional couplrs. In ordr to nsur broadband opration asymmtric taprd-coupld-lin couplrs ar usd sinc thy offr thortically high-pass frquncy rspons [57, 73] and /8 diffrntial phas proprtis whn rfrncd to transmission lin sctions of appropriat lngth [4, 74]. Figur 4.49 shows a schmatic diagram of a 4 x 4 Butlr matrix prsntd in [], in which taprd-coupld-lin 3-dB /8 dirctional couplrs hav bn applid as hybrid couplrs. For furthr analysis th following formula dscribing normalizd vn-mod charactristic impdanc distribution of th dirctional couplr has bn considrd [64]: 3 ( x) x = x (4.6) l l Th calculatd amplitud and phas charactristics of such an idal dirctional couplr ar shown in Fig. 4.5 and 5.5, rspctivly. Th couplr xhibits high-pass frquncy rsponss allowing for ralization of ntworks with thortically infinit bandwidth, and faturs constant /8 diffrntial phas charactristics. Figurs 4.5 and 4.53 show th calculatd amplitud and diffrntial phas charactristics of a 4 x 4 Butlr matrix consisting of four such couplrs and an idal 9 phas shiftr. Th amplitud imbalanc quals ±.6 db, whras, th achivd phas rippl quals ±3. For th purpos of th matrix s physical ralization a broadband 9 phas shiftr nds to b dsignd. On of th possibl solutions is to mploy a multi-sction Schiffman phas shiftr [33], as shown in Fig To valuat th matrix s proprtis a two-sction Schiffman phas shiftr, having th vnmod impdancs of coupld lins qual = 35 Ω, = 68. Ω, has bn considrd, for which th diffrntial phas imbalanc quals ±5. Th chosn phas shiftr allows for opration of th matrix in th frquncy rang of 8 GHz. Th diffrntial phas charactristic of th Butlr matrix ar prsntd in Fig. 4.55, whr it is sn, that th applid phas shiftr modifis ±9 diffrntial phas charactristics. 7

114 Fig Schmatic diagram of a broadband 4 x 4 Butlr matrix utilizing taprd-coupld-lin dirctional couplrs. Fig Coupling and transmission charactristics of an idal taprd-coupld-lin dirctional couplr. Fig Diffrntial phas charactristics of an idal taprd-coupld-lin dirctional couplr. 8

115 Fig Transmissions S 5, S 6, S 7, S 8 of an idal 4 x 4 Butlr matrix composd of four taprdcoupld-lin dirctional couplrs and an idal 9 phas shiftr. Fig Diffrntial phas charactristics of an idal 4 x 4 Butlr matrix composd of four taprdcoupld-lin dirctional couplrs and an idal 9 phas shiftr showing th dsird, 8, ±9 diffrntial phas shifts. Fig Schmatic diagram of a broadband 4 x 4 Butlr matrix utilizing taprd-coupld-lin dirctional couplrs and a two-sction typ-f Schiffman phas shiftr. 9

116 Fig Diffrntial phas charactristics of an idal 4 x 4 Butlr matrix composd of four taprdcoupld-lin dirctional couplrs and a two-sction 9 Schiffman phas shiftr showing th dsird, 8, ±9 diffrntial phas shifts. Anothr way to raliz a 4 x 4 Butlr matrix that allows for achiving, 8 and ±9 diffrntial phass is shown in Fig [], whr on of th taprd-lin couplrs is substitutd by a broadband 3-dB/9 dirctional couplr. Th proprtis of such a ntwork ar valuatd assuming th taprd-lin dirctional couplrs with th normalizd vn-mod charactristic impdanc distribution givn by (4.6) and a 5-sction symmtrical 3-dB coupld-lin dirctional couplr. As a taprd-lin dirctional couplr th couplr prsntd in [7] and dsignd in a broadsid coupld-striplin tchnology, has bn usd. Th coupld lins hav bn tchd on a thin CuFlon laminat having thicknss h =.5 mm and dilctric constant ε r =.5, which was placd btwn two thick h =.575 mm layrs with th sam dilctric constant. Figur 4.57 shows th masurd frquncy charactristics of th taprd-coupld-lin dirctional couplr [7], which xhibits good proprtis in th frquncy rang up to 8 GHz. Th amplitud imbalanc is lss than.5 db and isolation and rturn losss ar bttr than 5 db. Diffrntial phas charactristics of th manufacturd couplr ar shown in Fig Th achivd phas rippl quals ±8 within -8 GHz frquncy rang. Figur 4.59 prsnts th pictur of th cntr laminat of th manufacturd dirctional couplr. Fig Schmatic diagram of a broadband 4 x 4 Butlr matrix utilizing taprd-coupld-lin dirctional couplrs and a two-sction typ-f Schiffman phas shiftr.

117 Fig Coupling, transmission, isolation and rturn loss charactristics of th manufacturd taprd-coupld-lin dirctional couplr [7]. Fig Diffrntial phas charactristics of th manufacturd taprd-coupld-lin dirctional couplr [7]. Fig Photograph of th cntr laminat of th taprd-lin couplr [7]. Th ndd 5-sction 3-dB/9 coupld-lin dirctional couplr has bn dsignd using dilctric structur consisting of a thin CuFlon laminat having thicknss h =.5 mm and dilctric constant ε r =.5 placd btwn two thick h =.787 mm layrs with th sam dilctric constant. In this dsign, a novl approach has bn proposd, in which th strongst coupld sction is ralizd as a taprd-lin sction. Th coupling of th taprd-lin sction changs linarly from th strongst achivabl coupling to th nominal coupling of th nxt sction, i.. from = 6 Ω to = 69.9 Ω. Such an approach allows to minimiz th parasitic ractancs of th transition rgion btwn coupld-lin sctions. In practic, th taprd-lin sction has bn dividd into lctrically short subsctions. Similarly, th transition rgions btwn wakly coupld sctions hav bn ralizd as lctrically short 5 subsctions, in which th coupling cofficint is changing linarly from = 69.9 Ω to = 5.5 Ω. Masurmnt rsults of th manufacturd dirctional couplr ar shown in Fig. 4.6, and a photograph of th cntr laminat, whr th taprd intrsction connctions ar visibl, is shown in Fig. 4.6.

118 Th masurd frquncy charactristics of both couplrs (th taprd-lin couplr and th 5-sction 3-dB/9 couplr) hav bn usd for calculating th charactristics of th 4 x 4 Butlr matrix. Th calculatd amplitud and diffrntial phas charactristics ar shown in Fig. 4.6 and 4.63, rspctivly. Th achivd amplitud imbalanc is lss than ±.5 db, whras, th diffrntial phas rippl quals ±. Fig Coupling, transmission, isolation and rturn loss charactristics of th manufacturd 5- sction 3-dB/9 coupld-lin dirctional couplr with taprd intr-sction connctions. Fig Photograph of th cntr laminat of th 3-dB/9 coupld-lin dirctional couplr with taprd intr-sction connctions. Fig Transmissions S 5, S 6, S 7, S 8 of a 4 x 4 Butlr matrix calculatd with th us of th masurd rsponss of th manufacturd taprd-coupld-lin dirctional couplr and th 5-sction 3dB/9 coupld-lin dirctional couplr with taprd intr-sction connctions.

119 Fig Diffrntial phas charactristics calculatd with th us of th masurd rsponss of th manufacturd taprd-coupld-lin dirctional couplr and th 5-sction 3dB/9 coupld-lin dirctional couplr with taprd intr-sction connctions Summary Th chaptr dscribs mthods for th dsign of broadband Butlr matrics utilizing coupld-lin dirctional couplrs. Firstly, Butlr matrics consisting of singl-sction coupld-lin dirctional couplrs ar considrd, which allow for achiving th oprational bandwidth up to on frquncy octav. Two diffrnt dsigns of such 4 x 4 Butlr matrics ar prsntd, ralizd in symmtric striplin and asymmtric microstrip tchniqus [6, 64]. Furthrmor, a novl arrangmnt of th 8 x 8 Butlr matrix is proposd, which allows for ralization of a fully planar intgratd ntwork having broadband frquncy rspons. In this concpt, th broadband opration in trms of diffrntial phas charactristics is nsurd by th ralization of th slctd crossovrs as tandm connctions of two 3- db/9 coupld-lin dirctional couplrs and rfrnc lins having appropriat lctrical lngths. Apart from ralization of broadband phas shiftrs such an approach allows for achiving four crossovrs within th cntr part of th matrix, which can b furthr rducd into two crossovrs by rarranging th circuit, as it has bn shown. An arrangmnt has bn proposd in which two final crossovrs, to b ralizd, hav bn avoidd by rplacmnt of typical coupld-lin dirctional couplrs with a tandm connction of two 8.34-dB dirctional couplrs. Th approach proposd by th Author in [68] to raliz broadband multi-octav 4 x 4 Butlr matrics is outlind in Sction 4.. Th prsntd Butlr matrics consist of multisction 3-dB/9 symmtrical coupld-lin dirctional couplrs and middl ntworks, which nsur simultanously a transmission-lin crossovr and broadband 45 phas shiftrs. A gnral analysis of 4 x 4 Butlr matrics bing a connction of multisction 3dB/9 symmtrical coupld-lin dirctional couplrs is prsntd and tabls giving normalizd vn-mod impdancs of th phas corrction ntworks ndd for rspctiv coupld-lin couplrs ar providd aftr [9]. Morovr, th problm of limitd isolation of a tandm connction of two dirctional couplrs is outlind and a modifid middl ntwork for im- 3

120 proving Butlr matrics charactristics proposd by th Author in [68], is shown. In such a ntwork, dirctional couplrs constituting a tandm connction hav mor sctions than th rmaining couplrs and phas corrction ntworks consist of a gratr numbr of coupldlin sctions. This approach allows for significant improvmnt of both amplitud and diffrntial phas charactristics of th dsignd Butlr matrics. Th prsntd approach for broadband 4 x 4 Butlr matrics ralization has bn xprimntally vrifid by thr diffrnt dsigns, i.. th 4 x 4 Butlr matrix dsignd in a symmtric striplin tchniqu with th us of thr-sction coupld-lin dirctional couplrs, similar dsign in an asymmtric microstrip tchniqu, and th ultrabroadband 4 x 4 Butlr matrix dsignd in a symmtric striplin tchniqu in which 5-sction coupld-lin dirctional couplrs ar usd. Morovr, th concpt of broadband 4 x 4 Butlr matrix ralization has bn applid to th dsign of broadband 8 x 8 Butlr matrics. An xmplary dsign with th us of 5-sction coupld-lin dirctional couplrs is shown, in which a modular approach, basd on th division of th ntwork into four submatrics, is applid. Finally, broadband 4 x 4 Butlr matrics utilizing 3-dB//8 dirctional couplrs ar outlind. Th broadband opration of th considrd ntworks is nsurd by th application of taprd-lin dirctional couplrs having thortically high-pass frquncy rspons. An xmplary dsign of such a Butlr matrix is shown, which has bn ralizd with th us of thr taprd-lin dirctional couplrs and a 5-sction 3-dB/9 symmtric coupld-lin dirctional couplr. 4

121 5. Dsign of miniaturizd broadband dirctional couplrs In lowr microwav frquncy rang th convntional distributd passiv componnts such as branch-lin couplrs, Wilkinson powr dividrs, Gysl powr dividrs, coupld-lin dirctional couplrs, tc. consum larg ara of a microwav intgratd circuit (MIC) or a monolithic microwav intgratd circuit (MMIC). Thrfor, siz rduction tchniqus of microwav passiv componnts hav bn th subct of intnsiv rsarch ovr th last yars. In applications, whr siz rduction is rquird, lumpd or smi-lumpd lmnt dvics that rquir only a small ara ar vry attractiv [3, 8, 6, 4, 47, 76, 83, 88, 99, 57]. In addition, a rcnt rapid dvlopmnt of micromachining tchnology maks utmost miniaturization of microwav componnts possibl. Th dvlopmnt of lumpd-lmnt dvics is also important in this sns. In th following sction a brif introduction to th miniaturization tchniqus is prsntd in application to microwav componnts utilizing dirctly-connctd transmission lins. Th prsntd tchniqus allow for significant miniaturization, howvr, th dsignd componnts fatur by principl narrow oprational bandwidth. Th Author has focusd on th dsign of coupld-lin dirctional couplrs, which allow for broadband and ultrabroadband opration, as it was shown in th prvious chaptrs. Th sction 5. prsnts th rsults of Author s invstigation aiming at th dvlopmnt of miniaturization tchniqus of broadband coupld-lin dirctional couplrs. Th dsign tchniqu of quasi-lumpd broadband dirctional couplrs proposd in [66] is citd in larg xtnd. Th influnc of numbr of subsctions in a lumpd-lmnt quivalnt circuit on th couplrs ovrall prformanc has bn outlind. Th proposd miniaturization tchniqu of coupld-lin dirctional couplrs has bn xprimntally vrifid by th dsign of miniatur singl-sction and multisction broadband dirctional couplrs. 5.. Miniaturization tchniqus in th dsign of microwav dirctional couplrs In th dsign of passiv microwav dirctional couplrs and powr splittrs in microwav frquncy rang, th common approach is to utiliz distributd ntworks bing a connction of transmission lins in various tchnologis, or lctromagntically coupld-lin sctions 5

122 in various tchnologis. Th mthods allow for ralization of many diffrnt passiv circuits, howvr, at lowr rgion of microwav frquncis, th rsulting circuits suffr from larg dimnsions. Thrfor, ovr th yars tchniqus for miniaturization of microwav circuits hav bn dvlopd, which allow for significant rduction of th dsignd passiv componnts. A convntional approach of branch-lin dirctional couplrs miniaturization has bn prsntd in [85], whr a capacitiv loadd transmission-lin sction of a shortr lngth θ E has bn usd to rplac th original transmission-lin sction having lngth θ A, as it is shown schmatically in Fig. 5.. Th following quations rlat paramtrs of th two circuits sin θ = sinθ (5.) A A A E E E cosθ = cosθ ωc sinθ (5.) E E E In [85] an application of th mthod for th dsign of a 3-dB thr-branch branch-lin dirctional couplr has bn shown. Th mthod allowd for 68% siz rduction of th miniaturizd couplr ovr th convntional on. A similar approach for th dsign of a branchlin couplr has bn shown in [5], whr a furthr modification of th distributd capacitors in th innr ara of th couplr allowd for slight improvmnt of th siz rduction ratio. In this dsign fingr-shapd distributd capacitors hav bn proposd. Furthr rduction of th siz of th branch-lin couplr has bn rportd in [4], whr th distributd capacitiv lmnts C E hav bn rplacd by sctions of long mandrd transmission lins. Th achivd siz rduction rachs 89%. Th opn-stab loadd transmission-lin-sction tchniqu can b also applid in th dsign of rat-rac couplrs, as shown in [3], whr opn-ndd transmission-lin stubs hav bn qually distributd on th circumfrnc of th couplr s ring. Anothr approach to th miniaturization of a rat-rac couplr has bn prsntd in [], whr a quartr-wav long transmission-lin sction is rplacd by a cascad connction of two sctions of transmission lins and a coupld-lin C sction. Th dvlopd rat-rac couplr has bn miniaturizd to occupy 3% of th ara of a convntional rat-rac couplr, morovr, it has bn optimizd to oprat in dual frquncy rang (.45 GHz and 5. GHz). Exmplary applications of th rducd-siz branchlin couplrs can b found in [5], whr th dsignd miniaturizd couplrs hav bn utilizd for ralization of 4 x 4 Butlr matrics oprating at.8 GHz. Capacitiv loading can also b applid in miniaturization of th forward-wav coupld-lin dirctional couplrs, as it was shown in [4], whr almost 9% miniaturization rat has bn rportd. Furthr miniaturization can b achivd whn th distributd lmnt ntworks ar transfrrd and ralizd with th us of a lumpd-lmnt tchniqu. Th principls of th mthod bas on th wll-known rprsntation of distributd lmnts with thir quivalnt circuits consisting a connction of lumpd capacitors and inductors (losslss transmission lin). Thrfor, a short sction of a transmission lin can b rplacd by th quivalnt circuit consisting of an inductor and two shunt capacitors. Following that ida mor complx circuits can b dsignd. Figur 5.3 shows schmatic diagrams of th quivalnt lumpd-lmnt circuits of th wll-known rat-rac couplrs and quadratur couplrs [39]. 6

123 (a) (b) Fig. 5.. Equivalnt circuits of an arbitrary transmission lin. Convntional transmission lin (a) and quivalnt circuit with distributd componnts (b) [85]. (a) Fig. 5.. Concpt of miniaturizd circuits with th us of th tchniqu dscribd in [85]. Layout of a convntional branch-lin couplr (a) and its rducd-siz vrsion (b). (b) (a) (b) (c) (d) Fig Lumpd-lmnt quivalnt circuits of distributd microwav componnts. A rat-rac couplr (a) and (b), a quadratur couplr (c) and (d) [39]. Th circuits consist of appropriatly connctd LC -ntworks in ach of thir branchs rplacing sctions of transmission lins. In cas of rat-rac couplrs shown in Fig. 5.3a and 5.3b th brachs having lctrical lngth gratr than 9 consist of multipl LC sctions quivalnt to transmission lins. 7

124 (a) (b) (c) (d) Fig Frquncy rsponss of th circuits prsntd in Fig Th rat-rac couplr (a) and (b), th quadratur couplr (c) and (d). Th prformanc of ths circuits rsmbls th prformanc of thir distributd countrparts. Th valus of th L and C lmnts for th circuits shown in Fig. 5.4 can b calculatd with th wll-known formulas, i.. for th rat-rac couplr (Fig. 5.4a and 5.4b) as follows: L = (5.3) ω = (5.4) C ω and for th quadratur couplr shown in Fig. 5.4c as follows: C = (5.5) ω L = (5.6) ω 8

125 C = C (5.7) ω L and finally for th quadratur couplr shown in Fig. 5.4d as follows: + C = (5.8) ω L = (5.9) ω L = (5.) ω Th tchniqus of miniaturization utilizing th lumpd-lmnt approach ar wll suitd to th modrn tchnologis of circuit s ralization such as LTCC (Low Tmpratur Co-fird Cramic) [88, 98, 39, 55] or monolithic tchnologis [6, 4, 76, 57]. Th LTCC tchnology allows for manufacturing multilayr lctronic moduls with th us of a cramic tap (grn tap). Th typical thicknsss of th taps offrd on th markd rang from µm to µm, but also thinnr foils with th thicknss blow µm can b found. Th printing of th tracs on th layrs of cramic taps is mad with th us of scrn printing tchniqu. Th layrs aftr printing ar stackd and laminatd undr high prssur of atm. and fird in th tmpratur of 85 C. Th usag of LTCC tchnology for fabrication of microwav circuits has bn notd for svral yars now. Th succss of th tchnology coms from th fact that, dspit thir rlativly high costs, it offrs good lctrical proprtis togthr with th possibility of manufacturing multilayr structurs having diffrnt microwav componnts placd on diffrnt layrs. Th main advantags of LTCC tchnology can b summarizd as follows []: - low dilctric constant of th grn taps, ε r as low as 3.9, - low dissipation factor tgδ -3 ( GHz), - tmpratur stability of rsonant frquncy T f < ppm/k, - grn tap thicknss stability bttr than 5%, - low rsistivity of th conducting tracs, up to mω/sq., - possibility of manufacturing prcis conducting tracks with th thicknss typically µm, - possibility of manufacturing conductiv connctions btwn layrs with th diamtr of µm. Thr ar currntly numbr of companis on th markt offring commrcial LTCC tchnology, among which on can nam: Barry LTCC, Murata, Schott, Koyocra, SiP Tchnology, Hraus, ColdFusion, VTT Elctronics ach having thir own dsign ruls, which ar summarizd in Tabl 5.. 9

126 Company Minimum thicknss [µm] Dilctric constant Dissipation factor Thrmal xpansion [ppm/ C] Minimum track width [µm] Minimum via diamtr [µm] Rsistivity mω/sq. Tabl 5. Exmplary dsign ruls and matrial proprtis offrd currntly on th markt Barry LTCC Hraus Murrata Schott Koyocra SiP Tchnology Cold- Fusion VTT Elctronics x -3-5x -4 - x -3 - x -3 x -3 - x -3.5% 6x -3 5x -3.5x -3 6x < <3.5 < Figur 5.5 shows a 3-D viw of an xmplary dsign of a branch-lin dirctional couplr having th lctrical quivalnt circuit shown in Fig 5.3d. Th couplr has bn dsignd with th us of AWR Microwav Offic lctromagntic simulator with th assumption of th following ruls: layr thicknss 87 µm, ε r = 7.7, minimum track width 5 µm, minimum via diamtr 4 µm (th dsign ruls of th Institut of Elctron Tchnology, Krakow, Poland). As it is sn th circuit consists of four planar inductors having valus of 3.3 nh and.3 nh and four multilayr capacitors having valus 3. pf. Th couplr has bn dsignd with th us of 5 layrs of a grn tap and th ovrall dimnsions of th componnt ar lss than 5 x 5 mm for th couplr oprating at.4 GHz. Th frquncy rspons of th dsignd couplr, calculatd lctromagntically, ar shown in Fig. 5.6 and ar in full agrmnt with th thortical ons. Fig A 3-D viw of th dsignd branch-lin dirctional couplr for applications in LTCC tchnology. 3

127 Fig Frquncy rspons of th dsignd branch-lin dirctional couplr for applications in LTCC tchnology. Rsults of lctromagntic calculations. 5.. Dsign of broadband quasi-lumpd dirctional couplrs Th prsntd in sction. dirctional couplrs having dirctly-connctd transmission lins, apart from narrow oprational bandwidth, occupy rlativly larg ara, du to larg dimnsions, i.. ach sction of transmission lin bing at last quartr-wav long, in both planar circuits dimnsions. In trm of miniaturization th application of coupld-lin dirctional couplrs is a stp forward, sinc such a couplr faturs xtnsiv dimnsions only in on planar dirction, whil bing rlativly small in th othr dirction. Thrfor, th occupid ara is significantly rducd in comparison to dirctly-connctd dirctional couplrs. Furthr miniaturization of coupld-lin dirctional couplrs can b achivd with th us of th prsntd in sction 5. mthods applid to dirctly-connctd dirctional couplrs, in which distributd lmnts ar substitutd by ntworks having lumpd lmnts. A numbr of paprs rport invstigation rsults rgarding th dsign of lumpdlmnt coupld-lin dirctional couplrs. In [8] and [47] th dsigns of 3-dB coupld-lin dirctional couplrs ar prsntd, whr a dirct singl-sction quivalnt circuit of coupld lins has bn ralizd in MMIC tchnology. Howvr, in both prsntd dsigns, th simplicity of th assumd quivalnt circuits is rflctd in poor prformanc of th transmission-coupling charactristics, which limit th bandwidth of th rsulting couplrs. In [39] coupld-lin dirctional couplrs dsignd in LTCC tchnology wr prsntd, in which coupld inductors hav bn rplacd by th connction of uncoupld ons. Also in this cas, th proposd modification limits th oprational bandwidth of th dsignd couplrs. In [78] a broadband 3-dB dirctional couplr has bn prsntd, in which transmission-coupling charactristics ar in a good agrmnt with th ons obtaind with th standard distributd coupld transmission lin tchniqu. Th prsntd approach involvs a two subsction quivalnt circuit and th couplr has bn dsignd with th us of a planar 3

128 spiral transformr tchniqu. Similarly, in [] a two-subsction quivalnt circuit has bn mployd for th dsign of a 3-dB dirctional couplr in GaAs monolithic tchnology for applications in phas shiftrs, down-convrtrs and I-Q convrtrs. In [9] an xmplary dsign of a wak coupling 4 db dirctional couplr manufacturd using multilayrd organic packaging structur is prsntd. In th prsntd dsign, th achivd dirctivity is low, du to th fact that th obtaind isolation is about 5 db. In [66] th Author has prsntd a comprhnsiv ovrviw rgarding th dsign of quasi-lumpd dirctional couplrs having th proprtis of coupld-lin dirctional couplrs, which is citd in a larg xtnd blow. Th lumpd-lmnt quivalnt circuit of a coupld-lin dirctional couplr is shown in Fig. 5.7a. Th valus of lumpd lmnts can b found onc th coupling k of th dirctional couplr, th cntr frquncy f and th numbr of subsctions chosn for th ralization n ar spcifid. It is assumd that th conditions of idal couplr ralization (.57) and (.58) ar mt. At first th xprssions for capacitanc and inductanc lmnts of th [C] and [L] matrics ar drivd, which dscrib th proprtis of coupld lins. Ths paramtrs can b found as [66]: C' = (5.) ν C m ' ( k ) = C ' k (5.) C ' = C ' C ' (5.3) m C ' L ' = (5.4) ( C ' ') ν C m Having found th lmnts of [C] and [L] matrics, th valus of th dirctional couplr s lmnts, shown in Fig. 5.7a can b calculatd as whr x C = C ' (5.5) n C m x = Cm' (5.6) n x L = L ' (5.7) n ν x = is th lngth of th couplr, ν = c is th fr spac vlocity of light and 4 f coupling of coupld inductors k quals th initially chosn coupling k of th dirctional couplr k = k. coupld inductors dirctional couplr 3

129 (a) Fig Gnric schmatic of a lumpd-lmnt coupld-lin dirctional couplr dividd into n subsctions (a) and a quasi-lumpd ralization of a subsction with th us of th symmtrical coupld-lin modl (b) [66]. Th quivalnt circuit of th lumpd-lmnt dirctional couplr s subsction can b also rprsntd in a form shown in Fig. 5.7b, whr coupld inductors hav bn rplacd by o an lctrically short sction of coupld lins θ << 9 at f. Having assumd that th vlocitis of normal propagating mods qual ν o =ν oo = c and th physical lngth of th cou- pld-lin sction quals l th vn and odd mod charactristic impdancs of th coupldlin sction can b found as c = c o = ( L + L ) l m ( L L ) whr L m = kl, and th lctrical lngth of th sction quals l m (b) (5.8) (5.9) f θ = l (5.) c Application of such an lctrically short coupld-lin sction provids th rquird coupld inductors but also contributs to th slf and mutual capacitancs. Thrfor, th cor- 33

130 rcting capacitancs C cl and C m cl nd to b subtractd from original valus C and C m. Ths valus can b xprssd as C cl = c l ( L + L ) m (5.) C cl m L l = (5.) c m ( L L ) m For ths calculations an homognous air-filld structur is assumd, sinc th valus of slf and mutual inductancs of coupld lins ar invariant with rgard to th dilctric proprtis. Onc th gomtry of coupld lins is found in homognous mdium, it is possibl to includ dilctric proprtis, which will modify only corrcting capacitancs. Th nw valus C cl and C m cl can b obtaind, knowing th vn and odd mod phas vlocitis (ν o and ν oo ) for th chosn structur C cl = ν o l ( L + L ) m (5.3) C l L + L L L cl m = 4 o ( ) ( ) ( ) m m L Lm ν oo ν (5.4) Th ncssary numrical calculations of th rquird coupld-lin sction in a chosn dilctric structur can b prformd using numrical softwar, such as th on prsntd in [38]. Th proprtis of a lumpd-lmnt coupld-lin couplr hav bn invstigatd by th Author with th us of a schmatic diagram shown Fig 5.7a [66]. Figur 5.8 prsnts th normalizd frquncy charactristics of a dirctional couplr for which n = 4 and k =.77 (C = 3 db) ar chosn. Idntical rsults ar obtaind, whn ach subsction from Fig. 5.7a is substitutd with an lctrically short coupld-lin sction and lumpd capacitors, as shown in Fig. 5.7b. Th valus of lumpd lmnts and th coupld-lin sction paramtrs hav bn listd in Tabl 5. (for k =.77, n = 4, l = 6.6 mm, f = GHz). Tabl 5. Valus of lumpd slf and mutual capacitancs, slf inductancs, vn and odd mod impdancs, and lctrical lngth of th coupld-lin sction and corrcting capacitancs for th dirctional couplr shown in Fig. 5.8 for k =.77, n = 4, l = 6.6 mm, f = GHz [66] paramtr valu C [pf].58 C m [pf].66 L [nh] 4.4 [Ω] 34.5 o [Ω] θ [dg] 7.93 C cl [pf].3 C cl m [pf]

131 Such an approach allows for ralization of dirctional couplrs having proprtis of distributd coupld-lin couplrs in a wid frquncy rang. Howvr, th proprtis of th dirctional couplr dsignd with th us of quasi-lumpd lmnts dpnd on th chosn numbr of subsctions n, and th highr numbr of subsctions th bttr approximation of th distributd circuit. It is thn worth invstigating th influnc of th chosn numbr of subsctions n on th couplr s proprtis. Figur 5.9 and Fig. 5. show rspctivly, th calculatd rturn losss and isolation of th couplr, in which th numbr of subsctions changs from to 5. Th proprtis of th couplr ar vry poor for n = but improv rapidly for largr n. Thrfor, broadband opration is nsurd by th choic of n for which rturn losss and isolation of th dirctional couplr ar accptabl in a wid frquncy rang (i.. to f ). For n as low as 3 th rturn losss and isolation of th 3-dB couplr ar bttr than 5 db, which is sufficint for typical applications. Th isolation and rturn losss can b slightly improvd whn th C m and C paramtrs ar tund from th thortical valus. Th miniaturization with th us of th prsntd tchniqu is fficint only, whn a dilctric structur is appropriatly chosn, so that it is possibl to raliz a coupld-lin sction having th inductiv coupling k L qual to th chosn nominal coupling k (C in db) and having th charactristic impdanc givn by (.4) much highr thn of th rsulting couplr. In practical application th limitations on th couplr s miniaturization com from th minimum thicknss of th dilctric layr and th minimum width of th ralizabl strip. To nsur significant miniaturization of th quasi-lumpd coupld-lin couplrs it is rquird to slct a dilctric structur and tchnology having: - larg layr thicknss ratio, i.. th thicknss of th dilctric layr on which th tracs of coupld lins ar placd nds to b much smallr thn th ground-plan sparating layr(s), - possibility of ralization of narrow mtallization strips. To vrify xprimntally th prsntd thortical invstigation a singl-sction 3-dB dirctional couplr with th coupling imbalanc of δc = ±.6 db has bn considrd [66], and dsignd having n = 4 and f =. GHz. Thortical frquncy charactristics of th idal lumpd-lmnt couplr ar shown in Fig. 5.. For furthr lctromagntic analysis a dilctric structur shown in Fig. 5. has bn assumd. Th structur consists of a thin h = 5 µm laminat layr (ε r =3.38) on which lmnts of th couplr ar tchd. Fig Frquncy charactristics of th 3-dB dirctional couplr analyzd using th schmatic shown in Fig. 5.7a for which n = 4 [66]. 35

132 Fig Rturn losss of th 3-dB dirctional couplr analyzd using th schmatic shown in Fig. 5.7a for n =,, 3, 4 and 5 [66]. Fig. 5.. Isolation of th 3-dB dirctional couplr analyzd using th schmatic shown in Fig. 5.7a for n =,, 3, 4 and 5 [66]. Fig. 5.. Frquncy charactristics of th 3 ±.6 db dirctional couplr analyzd using th schmatic shown in Fig. 5.7a for which n = 4 [66]. 36

133 In ordr to maximally miniaturiz th circuit, th coupld strips width has bn chosn as narrow as possibl w =.9 mm (tchnological limitations) to obtain th highst slfinductanc pr unit lngth, which givs th shortst lngth l of th coupld lins. Th dilctric thicknss has bn chosn to obtain coupling k pr unit lngth gratr than dsird. Th strip offst was tund in a way to achiv appropriat coupling. Fig. 5.3 shows th layout of th dsignd couplr, in which all lmnts ar clarly markd. Th lctromagntically calculatd frquncy rspons of th dsignd couplr is shown in Fig. 5.4, whras, th masurd rsults of th manufacturd couplr ar prsntd in Fig. 5.5, Fig 5.6 and Fig 5.7. Th obtaind rturn losss of th dsignd couplr ar bttr thn 3 db and isolation is bttr than db in a wid frquncy rang. Th masurd phas charactristic diffrs from th thortical valu of 9 by lss than 7, and th dissipation losss ar about.5 db masurd at th cntr frquncy. Figur 5.8 prsnts a pictur of th tchd structur of th quasi-lumpd dirctional couplr. A vry important advantag of th dsign of quasi-lumpd coupl-lin dirctional couplrs is that th condition of idal couplr ralization i.. qualization of inductiv and capacitiv coupling cofficints can always b fulfilld! Sinc th capacitiv and inductiv lmnts ar indpndntly dsignd, arbitrary dilctric structur can b chosn. This is opposit to th solutions prsntd in,.g. [6, 65, 67], whr a spcial car has had to b takn whil choosing th dilctric structur togthr with coupld-lin gomtry, in ordr to achiv good proprtis of th dirctional couplr. Fig. 5.. Cross-sction of th dilctric structur usd for th dsign of quasi-lumpd coupld-lin dirctional couplrs [66]. Fig Layout of th dsignd singl-sction 3-dB dirctional couplr, th structur of which has bn dividd into n = 4 subsctions [66]. 37

134 Fig Frquncy charactristics of th dsignd 3 ±.6 db dirctional couplr. Rsults of lctromagntic analysis [66]. Fig Frquncy charactristics of th dsignd 3 ±.6 db dirctional couplr [66]. Fig Diffrntial phas charactristics of th dsignd 3 ±.6 db dirctional couplr [66]. 38

135 Fig Masurd dissipation losss of th dsignd 3 ±.6 db dirctional couplr [66]. Fig Photograph of th thin laminat on which th tracs of th dsignd couplr ar tchd [66]. Th prsntd procdur has also bn vrifid by th dsign of a.9-db dirctional couplr for application in LTCC tchnology and having th cntr frquncy f = GHz. Th dsign ruls offrd by Murata hav bn assumd, bing th most attractiv, sinc th minimum layr thicknss quals.5 µm (s Tabl 5.). Th tracs wr assumd on both sids of th thin h =.5 µm layr sparatd from a ground plan by a thick layr h =.9 mm. Th dilctric constant of th grn tap is qual ε r = 7.7. Assuming n = 7 th following valus of inductanc and capacitanc matrics hav bn found from (5. 5.4): L ' = [nh/m], C ' = [pf/m], and th valus of lumpd lmnts found from ( ) ar L = 9.55 nh, L m =.83 nh, C =.4 pf, C m =.36 pf. In th nxt stp th inductanc and capacitanc matrics (dnotd as L s and C s, s Fig. 5.9) of th coupld lins hav bn found. Th coupld lins hav bn dsignd to hav th minimum ralizabl width and th ndd inductiv coupling cofficint k =.76. Th following valus hav bn found using [38] for trac width w = µm and offst = µm: s L = [nh/m], s C = [pf/m], Sinc th coupld lins in th chosn gomtry ar asymmtrical, thrfor, th rsulting valus L s is not qual L s, and it is dsird to qualiz th slf inductancs, which can b 39

136 don by a slight chang of th width of on of th lins. This has littl practical maning sinc in th prsntd xampl a chang of µm provids for such an qualization. Having found th L s matrix valus, th lngth of an lctrically short coupld-lin subsction can b found as: L l = (5.5) L s and is qual to 3.9 mm. Such a sction ralizs coupld inductors and also contributs to th slf and mutual capacitancs, thrfor, th final capacitancs from Fig. 5.9 ar found as: C C C s s ( C C ) f = C l (5.6) m s s ( C C ) f = C l (5.7) m f m = C lc (5.8) m s m It is important to not that in this cas lumpd capacitancs connctd to both lins hav diffrnt valus, du to th asymmtry of th chosn structur. In th prsntd xampl ths valus ar qual: C f =.78 pf, C f =.9, C f m =. pf. Fig Gnric schmatic of a quasi-lumpd coupld-lin subsction for asymmtric coupld-lin gomtry. Fig. 5.. Frquncy charactristics of th.9-db dirctional couplr dsignd using th schmatic shown in Fig. 5.7a for which n = 7. 4

137 Th calculatd thortical frquncy rspons of th couplr consisting of svn sctions shown in Fig. 5.9, ar shown in Fig. 5., whras, th rsults of lctromagntic calculations of th.9-db dirctional couplr dsignd for ralization in LTCC tchnology ar shown in Fig. 5.. Th dsignd dirctional couplr faturs good lctrical proprtis within a wid frquncy rang. Figur 5. prsnts a 3-D viw of th dsignd couplr. Fig. 5.. Frquncy charactristics of th.9-db dirctional couplr dsignd for ralization in LTCC tchnology. Rsults of lctromagntic analysis. Fig. 5.. A 3-D viw of th.9-db coupld-lin dirctional couplr dsignd for ralization in LTCC tchnology. Th prsntd miniaturization mthod can also b applid for ralization of dirctional couplrs having frquncy rspons of multisction coupld-lin dirctional couplrs prsntd in sction... In this cas ach sction of a multisction dirctional couplr is 4

138 indpndntly dsignd with th us of th dscribd mthod. Th coupling cofficints k for ach coupld-lin sction can b takn dirctly from [9] or [37]. In [66] th Author has shown th dsign of a two-sction asymmtric dirctional couplr for which a cntr frquncy f =.4 GHz, th coupling imbalanc of δc = ±.5 db and th numbr of subsctions for ach sction of th couplr n = 3 hav bn assumd. Th thortical charactristics of th couplr ar prsntd in Fig. 5.3 and ar in agrmnt with th ons obtaind using distributd coupld-lin sctions rgarding th transmission and coupling. Th prformd analysis of th couplr rvals that th frquncy charactristics of th distributd multisction dirctional couplr diffrs from th charactristics of th lumpd-lmnt multisction dirctional couplr in a sns that zro-coupling frquncy of th formr is lowr. Furthr analysis has shown, that th diffrnc dpnds on th numbr of applid subsctions n. Th largr n th smallr diffrnc in zro-coupling frquncy. For th cas of n =, 3, 4, 5 th zro-coupling frquncy is rducd by th factor of.9,.955,.975,.985, rspctivly. Th two-sction 3-dB dirctional couplr has bn dsignd in th dilctric structur shown in Fig. 5.. Layout of th dsignd couplr is prsntd in Fig. 5.4, in which wakly and strongly coupld sctions ar indicatd. Th masurd rsults of th manufacturd couplr ar prsntd in Fig Th coupling transmission imbalanc of th couplr diffrs from th thortical on, sinc no qual-rippl charactr is obtaind. This can b causd by th manufacturing inaccuracy. To illustrat th influnc of manufacturing accuracy on th dirctional couplr frquncy charactristics, a snsitivity analysis has bn prformd, th rsults of which ar shown in Fig. 5.6 [66]. For th purpos of analysis, th idal lumpd-lmnt dirctional couplr - shown in Fig has bn assumd with its lumpd-lmnt valus having normal distribution and dviation qual %. Fig Frquncy charactristics of a two-sction asymmtric 3 ±.5 db dirctional couplr analyzd using th schmatic shown in Fig. 5.7a, in which ach of th two sctions has bn dividd into n = 3 subsctions [66]. ro-coupling frquncy th frquncy for which coupling quals and transmission quals, which corrsponds to th frquncy at which th lctrical lngth of ach sction is qual 8 - s Fig Th valu has bn valuatd basd on th manufacturing accuracy qual ±µm and actual dimnsions of th manufacturd couplr. 4

139 Fig Layout of th dsignd two-sction asymmtric 3 ±.5 db dirctional couplr in which ach of th two sctions has bn dividd into n = 3 subsctions [66]. Fig Frquncy charactristics of th dsignd two-sction asymmtric 3 ±.5 db dirctional couplr [66]. Th masurd rturn losss of th manufacturd couplr ar bttr than db and isolation is bttr than 5 db within th frquncy rang of.5.5 GHz. Th phas charactristic in comparison to th thortical on is shown in Fig. 5.7 and is in a clos agrmnt. Figur 8 shows th masurd dissipation losss, which do not xcd.4 db at th cntr frquncy. It is important to undrlin that th conditions of idal couplr ralization in cas of multisction couplrs ar vn mor difficult to b fulfilld in a classic distributd-coupld-lin couplr dsign [67], whras, thy ar asily obtaind using th prsntd tchniqu, which is its main advantag nxt to th possibility of miniaturization [66]. 43