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

178 J.B. ~arifio Acebal, P.M. Crespo Bofill, E. de los Reyes Dav6 E. Ma s grau ~6mez. E.T.S. Ingenieros de Telecomunicaci6nBarcelona -SPAIN COHN'S FILTERS WITH PARALLEL-COUPLED TRANSMISSION-LINES AND NO DISCONTINUITIES. ABSTRACT.- A new method for designing Cohn's parallel-coupled transmission-line filters is introduced. This method is particularly well suited to obt~in designs without discontinuities in the junction of the different transmission lines constituting the filter. I. INTRODUCTION.-The main problem facing the designer of micro __ wave filters with transmission lines, are the junction discontinui_ ties between the diferent;s lines. The consideratioas of their effects in The design results very troublesome, for whichreason, different methods, incorporating corrections to the desings made without such considerations,have been proposed(1). Another way to overcome the problem, is to use interdigital structures, even though they do·not advoid the discontinuities with the 50 0. input-output lines and are inadecuate for strip-line design. In this paper we . employ Cohn's structures(2) with open ended couped-lines /Fif.1-a/, which offers a very high number of degrees of freedom, and a desing method yelding configurations whitout inter-lines discontinuities, is proposed. II. OBTENTION OF· EQUIVALENT REALIZATIONS.- When Richard's transformation (3) s= th j~ 1 S being the propagation constant for the T.E.M. mode, is aplied to the Cohn's structure /Fig.1-a/ the s-plane prototype shown in Fi~ure 1-b is obtained. Figure 2-a shows an interdigital structure which is the dual of the Cohn's filter. It is easily shown that the normalized static capacitance matrices of the corresponding sections satisfy - the relationship: i -i -1 - [cri] =[cp] 1= 1, ... ,m+1 with m the order of the filter. The normalized static capacitance matrix CT of the dual interdigital structure is given by C1 D:U 1 CD12 m+.l CD12 cm+1 U22 179 ' In thi s manner, by inverting the normalized static capacitance matrices of each section of de eohn's structure, and compounding the 0 Ltained matrices in the way shown by the preceied i ng expression [eT/ E], ~~e obtain the normalizad static capacitance matrices of an e quivalent interdigital structure in their selective properties. As it is known (4), equivalent interdigital realizations can be der·ived from thi s one using the transformation [e~/£1 = l nl[ eT/EJ[n l /1/ ·: ·1here n is a diagonal matrix. In order for the terminal impedances to r e main unchanged after the transformation, we must have n = n 2=1 1 m+ If, as usual, we work with symetrical structure s , the equality . rm+3' n1= nm+J-i 1=2, ••. N= .~ must hold. Therefore, we have N-1 ~egrees of freedom. Inverting the sequence of operations thad led to ~T/~we obtain from ~~~~an implementation equivalent to the original Cohn's filter Along this process new deg~ees of freedom are introduced ; -- nam. ely, the partitioning · of each of the elements on the main diagonal of th e matrix, . (c~/E].,If we express 1 j:his partitioning as cii = ai eii + ( la) cii the symetry of the implementation is kept if 1 ,ai = am+J-i i=2, •.• ,N /2/ ( e 11 and em+ 2,m+2 are not partitioned). When m is odd, condition /2/ is supplemented by the fact that aN= 0.5 , since the two central coupled-lines must be equal. It is easily sho~n that the number of degrees of freedom now introduced is M=(m/2] , resulting them a total number , MT= N-l+M=m, of degrees of freedom in the obtention of an equivalent realization of the initial design. III.- DESLGN METHOD.- The proposed design method is based in the process of obtention of the equivalent realizations xe.pose?, and the use of the degrees of freedoms that have appeared in the design of a configuration without disc9ntinuities • . • ' ' Once one partition for (CT/£J has been chosen, each pair of coupled lines that set up the first half of the filter, is indepen_ dent of the remaining with the exception of the central one when m is even. This is so because, when we number those transmission lines from the closest to the end to the furthest, the normalized static capacitance matrix ~i/£~for each of them is affected by an element .. J · . .: 180 ni of [nJ not appearing previusly. Thus we can take for each matrix the value of ni that elimi_ nates the discontinuity in the juntion of thr current pair with the previous one. If m is even, to adjust the pair of central coupled lines we must use aN in an iterative process. IV.- PRACTICAL REALIZATION. - A band-pass filter has been des i g_ 1 ned with the following · specifications: ·-central frequency f 0 = 3. 9 GHz. -relative bandwith Bwr 5 % -filter order m=3 -terminal impedances z0= 50 0, The filter is constructed in strip-line te<fuology with 1/16" 1 2200 Rexolite substrat (Er= 2.56). From the Butterwoth (3) apr o ximation a prototype in the s-plane /Fig.2-b/ of the interdigital structure is obteined, leading to the following normaliced static capacitance matrix for the first half of the structure where . 1 , • no 1 fTI~ =7£ l1 t/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 represents the partition that has to be done on c22 and constituts the remaining degree of freedom in the design. Taking a= 0.4825 and using the transformation 111 we obtain [ 11 L•1.osso26s _1 -8.0850265 CP~ = -0.850265 n2 O.o8So265 where n; 1 has to be chosen such that the 1r1idth -1 ] n2-1 n2 of the line of the pair coupled conected to the 500. terminal line has the same value that this one, which for the proposed material is w/b = 0.7366771 with b the separation between strip line planes. Using the expresion for the dimensions of a pair of coupled lines in strip line confi guration as a function of the static capacitance matrix(S), n; 1= 2.646267 is obtain by means of succesive aproximations. -1 In a similar way, n3 = 4.461639 is obtained for the second - line. Table I summarizes the dimensions of the lines forming the - filter. Figure 3 shows the response of the constructed filter. 181 REFERENCES (1) J.Malherbe,A. Steyn " 'l'he compensation of step discontinuities in T.E.M.-Mode trans mision lines" M.T.T/16,pp.889-885,November 1978. ( 2 ) S. B. COHN "Paralled-coupled transmission-line-resonator filters" M.T.T.-6 pp.223 .- 231, April 1958 (3) R.J. Wenzel "Exact design of T.E.M. Microwavw Networks using quarter-wave lines" M.T.T. 12 pp.94-111, January 1964 (4) R.J. Wenzel "Theoretical and practical applications of capacitancematrix trans formations to T.E.M. network design" M.T.T.-14 pp.635-647,December 1966 (5) S.B. Cohn "Shielded coupled- - strip transmission line" M.T.T.-3 pp.29-38, October 1955 AUTHOR'S ADDRESS.- E.T.S.I. de Telecomunicacion. Aptdo. 30.002. BarcelonaSPAIN 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-- -bWB1 WA4 0.19 WB2 _wA3 0.48 --b- =-b- -b- --b- / s/b 0.12 s/b 10.66 1 b= ground planes spacing T A B L E I l 182 a.) [c!/el [c}f] [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 ... ., tn g 0 I I N '