/\ ($> ,,, IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY L’@ The Microwave Theory and Techmques Society isanorganization, within the framework of the IEEE, of members with principal professional interest mthe field of microwave theorv and techniques. All members of the IEEE are eligible for membership in the Society and will receive this TRANSACTIONS upon paymmt of the annual Society membership fee of $8.00. Affiliate memb&hip is avadable upon payment 6f the annual affdiate fee of $24.00, plus ‘the SocieLy fee of $8.00. Forinformation enjoining write tothe IEEE atthe address below. ADMINISTRATIVE COMMITTEE R. H. KNERR, President D. N. MCQUIDDY, JR., Vtce Pre$tdenz K. AGARW.aL, Secretary N. w Cox T. ITOH H. J. KLiNO D, N. MCQUIDDY, JR. B. E. SPIELMAN V’. G. GELNOVATCH F. IVANEK R. LEVY E. C. NIEHENKE P. W. STAECKER F’.T. GREILING R. S KAGIWADA S. L. MARCH J. E. RAUE S. J. TEMPLE R. H. KNERR M. A. MAURY, JR, M V. SCHNEIDER Honorary Life Members Dmtinguished Lecwrers A. C. BECK A. A. 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Second-class postage paid at New York, NY and at additional mailing offices. Postmaste~ Send address changes to IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, 445 Hoes Lane, Piscataway, NJ 08854. IFEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-34, NO. 3, MARCH 1986 321 Excitation of Waveguide by Stripline- and Microstrip-Line-Fed Slots B. N. DAS, K. V. S.V. R. PRASAD, AND K. V. SESHAGIRI RAO, MEMBER, IEEE Ab.vtract—This paper presents investigations on coupling between strip- tion required for the analysis of both stripline-fed and fhre/microstrip fine and arectangular ‘wavegnidecoupled through aslot in microstrip-fed slots is determined from one common gen- the ground plane which is fixed in the cross-sectionaf plane of the wave- eral formulation. The impedance seen by the line exciting guide. A closed-form expression for the impedance loading on stripliue/ microstip line is evahrated from knowledge of the complex power flowing the waveguide is determined from the complex power down the rectangular waveguide supporting the dominant modeanddiseoq- flowing down the waveguide and the discontinuity in modal tinuity in the modal voltage in stripline/microstrip line. The reactance voltage in the stripline or rnicrosttip line. The analysis is cancellation is obtained by terminating the stripline/microstrip tine excit- then extended to the case of a waveguide supporting TEIO, ing the slot in a short-circuited stub. The structure under this condition TE20, TE30 modes for slot locations corresponding to eigen forms atransition between stripline/microstrip fine and awavegnide. The design curves on slot length versusfrequency are presented for different excitations of the form (1/2,1,1/2), (fi/2, O,– &/2) values of dielectric constants. The variation of coupling asa fnnction of and (1, – 1,1), when used in the construction of waveguide frequency and also the location of the slot is evaluated. Numerical results simulators [7]. Numerical results on the normalized resis- for slot coupling useful for the design of wavegnide simulators are also tance and the reactance are presented. Further, the effect presented. of a reactive stub in ‘the feedline terminating the slot is L INTRODUCTION investigated. The results on coupling between the s stripline/microstrip line and waveguide have also been OME INVESTIGATIONS on aperture coupling be- evaluated. Finally, the design curves on slot length L for tween stripline/microstrip line and waveguide through which the normalized resistance becomes unity are an aperture in the common wall have been reported [1]–[4]. presented as a function of frequency for various values of Design of a multi-aperture directional coupler has also dielectric constants. been carried out by Kumar and Das [5]. It is found that the coupling between the stripline and the waveguide attained II. GENERAL ANALYSIS in this process is limited to low values. In order to obtain The method of exciting a rectangular waveguide by a higher coupling, the length of the coupling region becomes slot, the center of which is located at (xO, yO) on the very large. The method of exciting a multimode guide ground plane of a, stripline, is shown in Fig. l(a). The through a stripline-fed slot in the transverse section of length of the slot is parallel to the x-axis. The analysis is waveguide has recently been discussed in the literature [6]. carried out for the general case of a stripline having its The electric-field distribution required for the analysis has center strip arbitrarily located parallel to the ground planes been obtained from conformal transformation in the case and embedded at the interface between the two different of a symmetric stripline and from the equivalent parallel- dielectric media. plate configuration in the case of a rnicrostrip line. When The electric field in the cross section of the stripline/ an equivalent parallel-plate configuration is used in the microstrip line can be expressed as [9] case of microstrip line, the maximum dimension of the aperture is restricted. Evaluation of the electric-field distri- E=d” (1) bution by the method of conformal transformation is quite where 2 is the modal vector and V is the modal voltage involved in the cases of asymmetric stripline, symmetric satisfying the orthonormality condition and asymmetric striplines with partial dielectric filling, and microstrip line. e2ds=l / In the present work, excitation of a waveguide by stripline- and microstrip-line-fed slots in the transverse cross where ds represents the area of an infinitesimally small section of the waveguide is investigated. The field distribu- element in the cross section of the waveguide. In Fig. l(a), the coupling of the slot is transverse to the axis of the stripline. Using the formulas by Marcuvitz and Manuscript received May 29, 1985; revised October 28, 1985. Schwinger [13], it can be shown easily that such a slot B. N. Das and K.V.S.V.R. Prasad are with the Department of Elec- tronics and Electrical Communication Engineering, Indian Institute of produces discontinuity in the modal voltage, and the equiv- Technology, Klmragpur 721302, India. alent network of the slot appears as a series element in the K. V. Seshagiri Rao is with the Radar and Communication Centre, transmission-line representation of Fig. l(c). From a Indian Institute of Technology, Kbaragpur 721302, India. IEEE Log Number 8406848. knowledge of this discontinuity in modal voltage AV, the 0018-9480/86/0300-0321 $01.00 01986 IEEE 322 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 3, MARCH 1986 The electric field in the coupling aperture given by (3) excites the TE to x mode in the waveguide, which can be expressed as the superposition of TE and TM modes [9]. The modal amplitudes of the TE~.-mode fields excited in the rectangular waveguide of Fig. l(a) can be obtained from the formula [9] ‘z where &fl=l “for rt=O (a) . 2 for n >0. The complex power in the waveguide is given by [9] (5) where Z= is the unit vector along the axis of the waveguide. Substituting (3) in (4) and using (5), an expression for the complex power in the waveguide is obtained as (b) r mrd 12 --+=+&l v )$( STUB I nTyO mtrxO .cos2— .sin2 — e I b a (c) “KL[ ‘mrL 12 Fig, 1. Configuration of the excitation of a waveguide by a slot in the ground plane of the stnpline. (b) Configuration of the stripline used for CosT – Cos 2a (6) the determination of modal vector function. (c) Equivalent circuit of the mn 2 stub-terminated feed arrangement. – K2 (-)a with normalized admittance loading on the stripline resulting from the power coupled to the waveguide is obtained from the relation P F= (2) YO(AV)2 where YOis the characteristic admittance of the stripline, where d is the width of the slot, V. is the maximum voltage AV is the discontinuity in modal voltage in the stripline, across the slot, EOd, a and b are the broad and narrow and P is the complex power flowing down the waveguide. dimensions of the waveguide, and p and c are, respectively, The electric-field distribution in the aperture plane of a the permeabi~liy and permittivity of the medium inside the slot exciting the guide is assumed to be of the form waveguide. Slrice the waveguide is airfilled, the values of p and c are assumed to be the same as those of free space. (L EY=EOsin K ~–lx–xO\ (3) The expression for the modal vector function required for ) the evaluation of discontinuity in the modal voltage in the where E. is the maximum value of the electric field in the structure of Fig. l(b) having the slot in its ground plane is derived from the potential function. Assuming that the aperture plane of the slot, L is the length of the slot, K = (27T/A o)~, and the expression for c: is given by [8] charge distribution in the center strip is of the form p(x) = *’ where c; is the effective dielectric constant and c, is the relative dielectric constant of the medium filling the region (=)<’<($+:)=0 ‘thertise between the ground planes of the stripline. DAS @f(d.: STRIPLINE- AND MICROSTIUP-LINE-FED SLOTS 323 Following the method suggested by Yamashita and Atsuki [10], the expression for the potential function O(X, z) for the TEM-mode field is obtained as @(x, z) =“=~,,? .JO(%)sin(~) .sirih(~)sin(~) sinh(~) O<.zl. (1) approximated by a series of homogeneous regions. In each Here, R = r/a, r is the radial variable in cylindrical such region, linear combinations of analytic field distribu- coordinates, a is the core radius of the fiber, and P(R) is tions (the Bessel or modified Bessel functions) are related an arbitrary function whose value never exceeds unity such to those in the neighboring regions. In the second method, that the relative permittivity never exceeds c~ (~~> (l). the four tangential components of the electromagnetic fields Along such a fiber, a time-harmonic electromagnetic wave are related by four coupled first-order differential equa- of anular frequency Q propagates with fixed variations tions. The propagation problem is solved by matching at with respect to the axial (z) and the azimuthal (O) direc- the boundary two independent sets of solutions, which in tions as exp (– jm+ – j~z), where the azimuthal mode turn are obtained numerically by direct integrations of the number m is an integer. From Maxwell’s equations, one “ four first-order differential equations. The finite-element obtaines method in [7] is formulated in the axial fields, which may suffer from spurious modes when the node points are not VC(R) chosen carefully [8]. k;f(R)fi+v2H+— Xvx%=o. (2) c(R) Now let us turn to the methods of determining cutoff From such a relation, the propagation problem of the frequency. As early as 1973, Dil and Blok [3] proposed radially inhomogeneous fiber can be described in terms of associated equations, from the four coupled first-order differential equations, for treating the cutoff frequencies of the transverse magnetic fields, M@ and H,, as in the following two coupled second-order differential equations: guided modes, but no serious data was presented at that time. Perhaps, the first detailed evaluation of ‘cutoff frequencies in the vector form is due to Bianciardi and Rizzoli H(’(R)+;H:(R) [1], who have developed a method based on the stair-case approximation for dealing with the cutoff frequency aswell 1 mz 1 + a2(k;c(R)–~2)–z–~ H+(R) as propagation constant. [ E’(R) Manuscript received July 16, 1985; revised October 28, 1985, –#Hr(R)– — C-C. Su is with the Department of Electrical Engineenng, National c(R) Tsing Hua University, Hsinchu, Taiwan, -%@;)K+(&R@) )=+0 C. H. Chen is with the Department of Electrical Engineering, National 1 Taiwan University, Tarpei, Taiwan. (3a) IEEE Log Number 8406850. [ 0018-9480/86/0300-0328$01.00 01986 IEEE

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