Cohn's filters with parallel-coupled transmission lines and no discontinuities

178 J.B. ~a i io Acebal, P.M. C espo Bo ill, E. de los Reyes Da 6 E. Ma s g au ~6mez. E.T.S. Ingenie os de Telecomunicaci6n- Ba celona -SPAIN COHN'S FILTERS WITH PARALLEL-COUPLED TRANSMISSION-LINES AND NO DISCONTINUITIES. ABSTRACT.- A new me hod o designing Cohn's pa allel-coupled ansmission-line il e s is in oduced. This me hod is pa icula ly well sui ed o ob ~in designs wi hou discon inui ies in he junc- ion o he di e en ansmission lines cons i u ing he il e . I. INTRODUCTION.-The main p oblem acing he designe o mic o __ wa e il e s wi h ansmission lines, a e he junc ion discon inui_ ies be ween he di e en ;s lines. The conside a ioas o hei e ec s in The design esul s e y oublesome, o which eason, di e en me hods, inco po a ing co ec ions o he desings made wi hou such conside a ions,ha e been p oposed(1). Ano he way o o e come he p oblem, is o use in e digi al s uc u es, e en hough hey do·no ad oid he discon inui ies wi h he 50 0. inpu -ou pu lines and a e inadecua e o s ip-line design. In his pape we . employ Cohn's s uc u es(2) wi h open ended couped-lines /Fi .1-a/, which o e s a e y high numbe o deg ees o eedom, and a desing me hod yelding con igu a ions whi ou in e -lines discon inui ies, is p oposed. II. OBTENTION OF· EQUIVALENT REALIZATIONS.- When Richa d's ans o ma ion (3) s= h j~ 1 S being he p opaga ion cons an o he T.E.M. mode, is aplied o he Cohn's s uc u e /Fig.1-a/ he s-plane p o o ype shown in Fi~u e 1-b is ob ained. Figu e 2-a shows an in e digi al s uc u e which is he dual o he Cohn's il e . I is easily shown ha he no malized s a ic capaci ance ma ices o he co esponding sec ions sa is y - he ela ionship: i -i -1 - [c i] =[cp] 1= 1, ... ,m+1 wi h m he o de o he il e . The no malized s a ic capaci ance ma ix CT o he dual in e digi al s uc u e is gi en by C1 D:U 1 CD12 m+.l CD12 cm+1 U22 179 ' In hi s manne , by in e ing he no malized s a ic capaci ance ma ices o each sec ion o de eohn's s uc u e, and compounding he 0 L ained ma ices in he way shown by he p eceied i ng exp ession [eT/ E], ~~e ob ain he no malizad s a ic capaci ance ma ices o an e qui alen in e digi al s uc u e in hei selec i e p ope ies. As i is known (4), equi alen in e digi al ealiza ions can be de ·i ed om hi s one using he ans o ma ion [e~/£1 = l nl[ eT/EJ[n l /1/ ·: ·1he e n is a diagonal ma ix. In o de o he e minal impedances o e main unchanged a e he ans o ma ion, we mus ha e n = n 2=1 1 m+ I , as usual, we wo k wi h syme ical s uc u e s , he equali y . m+3' n1= nm+J-i 1=2, ••. N= .~ mus hold. The e o e, we ha e N-1 ~eg ees o eedom. In e ing he sequence o ope a ions had led o ~T/~we ob ain om ~~~~an implemen a ion equi alen o he o iginal Cohn's il e Along his p ocess new deg~ees o eedom a e in oduced ; -- nam. ely, he pa i ioning · o each o he elemen s on he main diagonal o h e ma ix, . (c~/E].,I we exp ess 1 j:his pa i ioning as cii = ai eii + ( l- a) cii he syme y o he implemen a ion is kep i 1 ,ai = am+J-i i=2, •.• ,N /2/ ( e 11 and em+ 2,m+2 a e no pa i ioned). When m is odd, condi ion /2/ is supplemen ed by he ac ha aN= 0.5 , since he wo cen al coupled-lines mus be equal. I is easily sho~n ha he numbe o deg ees o eedom now in oduced is M=(m/2] , esul ing hem a o al numbe , MT= N-l+M=m, o deg ees o eedom in he ob en ion o an equi alen ealiza ion o he ini ial design. III.- DESLGN METHOD.- The p oposed design me hod is based in he p ocess o ob en ion o he equi alen ealiza ions xe.pose?, and he use o he deg ees o eedoms ha ha e appea ed in he design o a con igu a ion wi hou disc9n inui ies • . • ' ' Once one pa i ion o (CT/£J has been chosen, each pai o coupled lines ha se up he i s hal o he il e , is indepen_ den o he emaining wi h he excep ion o he cen al one when m is e en. This is so because, when we numbe hose ansmission lines om he closes o he end o he u hes , he no malized s a ic capaci ance ma ix ~i/£~ o each o hem is a ec ed by an elemen .. J · . .: 180 ni o [nJ no appea ing p e iusly. Thus we can ake o each ma ix he alue o ni ha elimi_ na es he discon inui y in he jun ion o h cu en pai wi h he p e ious one. I m is e en, o adjus he pai o cen al coupled lines we mus use aN in an i e a i e p ocess. IV.- PRACTICAL REALIZATION. - A band-pass il e has been des i g_ 1 ned wi h he ollowing · speci ica ions: ·-cen al equency 0 = 3. 9 GHz. - ela i e bandwi h Bw 5 % - il e o de m=3 - e minal impedances z0= 50 0, The il e is cons uc ed in s ip-line e< uology wi h 1/16" 1 2200 Rexoli e subs a (E = 2.56). F om he Bu e wo h (3) ap o xima ion a p o o ype in he s-plane /Fig.2-b/ o he in e digi al s uc u e is ob eined, leading o he ollowing no maliced s a ic capaci ance ma ix o he i s hal o he s uc u e whe e . 1 , • no 1 TI~ =7£ l1 /2 I' 0 o<a< 1 1 0 2 6 . 4 4 7 7 3a + 2 6 • 4 4 7 7 3 ( 1-a ) 1 j. 1 25.47734 ep esen s he pa i ion ha has o be done on c22 and cons i u s he emaining deg ee o eedom in he design. Taking a= 0.4825 and using he ans o ma ion 111 we ob ain [ 11 L•1.osso26s _1 -8.0850265 CP~ = -0.850265 n2 O.o8So265 whe e n; 1 has o be chosen such ha he 1 1id h -1 ] n2-1 n2 o he line o he pai coupled conec ed o he 500. e minal line has he same alue ha his one, which o he p oposed ma e ial is w/b = 0.7366771 wi h b he sepa a ion be ween s ip line planes. Using he exp esion o he dimensions o a pai o coupled lines in s ip line con i gu a ion as a unc ion o he s a ic capaci ance ma ix(S), n; 1= 2.646267 is ob ain by means o succesi e ap oxima ions. -1 In a simila way, n3 = 4.461639 is ob ained o he second - line. Table I summa izes he dimensions o he lines o ming he - il e . Figu e 3 shows he esponse o he cons uc ed il e . 181 REFERENCES (1) J.Malhe be,A. S eyn " 'l'he compensa ion o s ep discon inui ies in T.E.M.-Mode ans mision lines" M.T.T/16,pp.889-885,No embe 1978. ( 2 ) S. B. COHN "Pa alled-coupled ansmission-line- esona o il e s" M.T.T.-6 pp.223 .- 231, Ap il 1958 (3) R.J. Wenzel "Exac design o T.E.M. Mic owa w Ne wo ks using qua e -wa e lines" M.T.T. 12 pp.94-111, Janua y 1964 (4) R.J. Wenzel "Theo e ical and p ac ical applica ions o capaci ancema ix ans o ma ions o T.E.M. ne wo k design" M.T.T.-14 pp.635-647,Decembe 1966 (5) S.B. Cohn "Shielded coupled- - s ip ansmission line" M.T.T.-3 pp.29-38, Oc obe 1955 AUTHOR'S ADDRESS.- E.T.S.I. de Telecomunicacion. Ap do. 30.002. Ba celona- SPAIN Lines 1 and 4 Lines 2 and 3 _W A1 W.B4 ~- . i I- . -WA_2 ____ W_B_3 ---.-0-. ~ :·- • -b-= -b-- I 0. 74 -b-- -b- WB1 WA4 0.19 WB2 _wA3 0.48 --b- =-b- -b- --b- / s/b 0.12 s/b 10.66 1 b= g ound planes spacing T A B L E I l 182 a.) [c!/el [c} ] [c'!E}·~ ~~1 J c'l 1~ VzL c!ll . c! Yza c: IJ 18} en --------------------------- ~ I ~ I I I I I I I I I I CO I ~ I 1$ 0 ... ., n g 0 I I N '