ANALYSIS AND SYNTHESIS OF MILLIMETRIC E-PLANE DIPLEXERS IN RECTANGULAR WAVEGUIDE -Antonio Morini, Tullio Rozzi Dipartimento di Elettronica e Automatica, Universitk di Ancona ABSTRACI A novel E-plane diplexer is proposed and discussed. Its feature is the matching section of the three-port junction, obtained by means of an E-plane septum, realized on the same mask as the branching filters and placed in the cavity of an abrupt E- plane junction. The latter is designed-in such-a way that, when combine d with two,reasbnably 7good filters, separately designed and suitably positioned, the resulting diplexer gives good performances without further optimization. This design technique, based on the properties of the scattering parameters of a reciprocal lossless three-portjunction, is , as such, of general validity and easily extended to other structures. INTRQD_UCTION The more expensive part of an E-plane circuit in rectangular waveguide for millimetric application is its housing. In fact, its fabrication requires high precision milling machines and it is difficult to: reduce tolerances below 10 t for each half housing. Therefore, it becomes important to simplify as much as possible the housing geometry, even increasing the complexity of the mask. For this reason, we study an E-plane diplexer based on an abrupt three-portjunction, in lieu of a tapered junction, Dittlof and Andt- (1-2), Vahldieck and Varailhon- de la Filolie(3), requiring more expensive mechanical construction. The junction is tuned by a single inductive post, built on the same mask as the E-plane filters and placed between the junction and the bifurcation, as shown in fig.1, Morini et al. (4). Consequently, the resulting diplexer consists ofjust three parts: a single mask containing filters and tuning post and the two halves of the housing wherefrom junction-and guides are made.The diplexer is then assembled by sandwiching the mask between the two halves of the housing . Modelling the junction, inclusive of the post, involves a rigorous treatment of the interaction between adjacent discontinuities by the concept of accessible modes, Rozzi et al. (5). As contrasted to (5), modes of both TE and TM polarization occur, depending on two indices. b~~~~~~~~~~~~~~~~~F filtler[2 ~~~~~~~~~~~~~p port3 p 2 _____________________ mantching post fig.1 Geometry of the E-plane diplexer (1) E-section of the junction (r) mask containing filtes and matching post Particular attention was also given to the problem of 'optimum' synthesis of the whole diplexer with two given good filters, not amenable to modification. It is a common misconception that optimum results are achieved by utilizing a good power divider closed by the branching filters. In a recent paper we showed this assumption to be groundless, and, by exploiting some general properties of lossless reciprocal three-port junctions endowed with a plane of symmetry, we recovered some conditions to be satisfied in order to obtain an 'optimum' three-port for diplexer application with two given fllters, Morini and Rozzi (6). The technique here described, based on the above conditions, permits to- separate the synthesis of thejunction from that of the filters, giving an explicit formula for the correct positioning of the filters, without any further optimization. 1351 JUNCTION ANALYSIS We shall now focus on the analysis of the junction (4), whereas reference is made to (5) for that of the filters. Symmetry of the junction about its longitudinal mid-plane implies s11 = S22 and 53 = S23, moreover by adopting even and odd excitations the three-port junction reduces to the two two- port situations shown in fig.2. I I I I ; 'd2d2 ' ~electric wallg d2d mgei wall dl d3 dl d3 fig.2 even/odd excitation at ports 1 and 2: resulting structures Both comprize the cascade of three discontinuities dl, d2, d3 separated by short sections of guide and, as such, interacting. These discontinuities are analyzed individually as 2N- port, N being the number of' accessible modes '.These are chosen according to the following considerations: Discontinuities 1 and 3, that is two E-plane steps, under the excitation of an.incident TEB0 -mode, excite in turn just LSE1n -modes with n=l,2,3, due to their uniformity with respect to x. .On the other hand, discontinuity 2, i.e. the metal septum, due to its uniformity with respect to y, when excited by-LSE1I -modes excites in turn the TEmn and TMmn families with m4gl,3,5... Finally, the latter impinging on discontinuities 1 and 3 excite TEmn and TMmn modes with n l,2,3.. This complex mechansm is summarized in table I. TB10TE~~~~~~~~BTd1d3w S in d2 C¶mu 1d3 urn TMmu ~TM n=1,2,3,... m=1,3,5,.. n4,2,3,... table I scheme of the modal interactions occurring between discontinuities For this reason, we preferred to work with the TE and TM families for both types of discontinuity. Modes .with attenuation larger than 20 dB between successive discontinuities weretconsidered 'localized'. Applying the above criteria we found that in the case of interest it was -sufficient to consider the first five modes as accessible. The six discontinoities occurmng are analyzed by deriving variational expressions for the multiport impedance matrix 4k and Z4 (k=l, 2, 3 ) of each. The discontinuity fields are expanded in polynomials with weight functions singular at the sharp edges, Collin (7), leading to singular integral equations. Having computed 4 and 4k (k=l, 2, 3), the behavior of the whole junction is determined by cascading the corresponding transmission matrices: T and k .In fact, denoting by Ue(l) e UO (1) the standard transmission matrices of a waveguide section of length 1, for the even/odd excitations respectively , the global transmission matrix becomes, in the even case, ;~~~~~ Uellf. e TE(1l,12; s,b,t,f) = T (b,t,f) TUe(li,f) T(s;2b+t,f) .Ue(12,f) *T3e. (b,t,f) : dim 3x3 (la) and, in the odd case, 1352 TO(11,12; s,b,t,f) = T (b,t,Qf)*U°(l,QTf)*TO(s;2b+t,f) *U0(12,f) *TO(b,t,f) : dim 2x2 (lb) When the interaction between junction and filters via modes below-cutoff is negligible, the matrices just obtained can be reduced by closing the ports corresponding to those modes on their characteristic impedances. From the latter circuits it is easy to derive the scattering parameters and then the response of the diplexer by standard network methods. DIPLEXER SYNTHESIS As mentioned in the introduction, a common misconception is that an "optimum" diplexer need be designed starting from a good power divider. Really, this assumption yields good results only when geometries of the branching filters, and consequently their electrical parameters, are substantially modified in such a way that the whole diplexer satisfies specifications, H. Yao et al. (8). In that case, the computer optimization (manipulation) of the filters is so intensive that the original choice of a power divider does not seem to be particularly significant. The method we will follow permits instead to design separately junction and filters. In fact we showed (6) that a three-port symmetrical junction in conjunction with two pass-band filters gives good diplexer performance if thejunction satisfies the following specifications:. Is111 Is221 IS331 f=fi, f=f2 (2a) IS331 . 1/3 (2b) s 11~ s1l2 s22 0 f=f , f=f2 (2c) Where sij(u) is a scattering parameter of the junction, s. (u) its derivative with: respect to the effective frequency v = Oa ; I1 is the propagation constant of the fundamental mode for the waveguide feeds, a is the guide wider wall; finally f1 and f2 are the midband frequencies of the two branching filters. We can consider the performance of the three-portjunction, by closing one.ofits arm on a filter Fi (i=1,2) and by evaluating the bandwidth of the resulting two-port, centered on the midband frequency of the second filter, ft, over which the reflection magni"ude is lower than a fixed e. The distance lFi at which filter Fi need be located is given by the formula (6): 2j1(fj) L[As S33 PFi(fj)] when i=l, j=2 and when i=2 j=l, PFi(t) is the reflection of the filter Fi. It is found that the maximum bandwidth 2 Av of the two-port so obtained, and then of each channel of the diplexer, depends on the reflection magnitude at the common port of thejunction, 1 - 1S33 2A. 2 12e)j(4) ISMI1 Fig 3 shows a plot of eq.4, for a given reflection s = -30 dB and for different IS331 comprised between -9.5 dB an -3 dB, representing the maximum banwidth of the diplexer versus the normalized distance lFi/a. It is seen that Is331 decreases as the ban4width increases. 1353 I \ - s3 = -9.5 dB dte\ _ -;,~~~~~~~33_-7 cf 0.. 0. ~ ~ ~ ~ ~ ~ ~ ~ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 < FJa fig. 3 Maximum nomalized bandwidth vs.- filter positionilg for a given = -30 dB RESULTS In the present case, we had to design a diplexer with the following specifications: bl 37.482 - 37.782 GHz b2 38.742 - 39.042 GHz mrl in the pass-band 26 dB 5 poles filters This corresponds to a normalized bandwidth for each channel given by 2 Au 0.06 (considering a WR 28 rectangular waveguide). By comparison with fig.3 we see that, once 2a-c are verified, IS331 < -5 dB. Therefore we design&d'first tit branchmig fllters (minimum return loss 26 dB) by using a classical synthesis technique, Rhodes (9), and by E-plane program developed by ourselves (5).