Analysis of leakage in multilayered microstrip lines using complex images

Analysis of leakage in multilayered microstrip lines using complex images J. Bernal, E Mesa , F. Medina GN~O de Microondas. Dept. Electmica y Electromagnetismo Facultad de Fsica Avda. Reina Mercedes s/n, 41012 Sevilla (SPAIN) Fax: + 34 5 4239434 email: [email protected] Abstract The Mixed Potential Integral Equation (MPIE) is combined with the complcx iinagc method Lo analyze the complete spectrum of multilayered printed transmission lines. A relevant contribution of this method IS its ability to study both the bound and leaky regimes in a very simple and efficient way. Since the analysis is carried out in the spatial domain, this work also makes it possible to analyze the leakage phenomenon for structures with nonzero thickness conductors. 1 Introduction The propagation characteristics of guiding structures in layered media are very efficiently computed by means of integral equation methods. For the layered geometry, the required Green’s functions are only known in closed-form in the spectral domain. If a spatial domain formulation were used, the spatial-domain Green’s functions can be obtained from their spectral versions through Fourier transform inversion [I]. Strictly zero-thickness printed transmission lines can be solved both in the spectral and the spatial domains with rather low computational effort. Non-planar conductors are better accounted for by means of the spatial-domain version of the integral equation [Z]. During the last decade, a powerful tool -the Discrete Complex Images Technique (DCITwas developed to obtain the spatial Green’s functions for layered structures [3]. This technique has been predominantly applied to 3D planar circuits and antennas problems. Recently this method has been adapted to deal with 2D guiding problems, where the relevance of a fast generation of the Green’s functions is even more significant than in the 3D context. Thus, planar strip-like [4] and slot-like [5] structures have been solved following that procedure. However, the important question posed by the existence of both surface and space leaky-wave solutions has not been considered in those works where only bound modes were treated. Leaky-waves have been comprehensively studied in the frame of the Spectral Domain Analysis (SDA) [6]. Working in SDA implies to perform numerical integration along a properly chosen path in the spectral variable complex plane. Various physical and mathematical arguments have been employed to determine the features of this path. In this paper we will show how to adapt the method in [4] to include the leaky regime. This new approach has the advantage of being very simple and efficient and the capability of dealing with non-planar conducting structures. 0-7803-7070-8/0 1 /$lO.OO 0200 1 IEEE 580 Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 09,2020 at 14:11:05 UTC from IEEE Xplore. Restrictions apply. 2 Analysis The problem of the determination of the propagation constants of a multistrip system embedded in a layered substrate can be posed in terms of a Mixed Potential Integral Equation iis shown in [4]. The Green'r functions for the scalar and vector potentials, Ji"' and K,; , inust be obtained froin their spectral verbions. This Implies to perform Fourier t~iiisform inversions of the type: where A,J = m, A., bring tlie transverw wavenuinber and I;: = siiined propagation constant. - j.u the asThe DClT can be advantageously uaed in this probletn to avoid the tedious numericiil integration involved in (I ). The underlying idea ia to expand the spectral function ii\i a sum of complex exponential functions whose spatial-domain counterpart is known thanks to a 2D Somtncrfeld identity [4]. However, this technique was only applied to charxterize bound modes. Next we will explain the guidelines ofthe method along with tlie modifications that must be introduced to generalize the approach to account for leaky modes. - It is \vel1 known that the spectral functions G(b:,) show two types of singularities: branch pointa at b., = f and poles. They are respectively related to the radiai~oii iiito fire space and to the inodcs of the background waveguide. Since the complex im,igcs cxlmision ciiii only fit analytical functions, the above singularities must be properly haiidled. This IS done by first introducing a change of variable (ILO = m) that eliminates branch points and second extracting out in analytical form the significmt polcs (in prxtice, thobe poles associated with above cutoff waveguide modes). .tatic contribution IS also explicitly extracted out. Thus, the spectral domain Green'\ f'unctlons are split into three contributions: - -- where G,, ih the quasi-static tcrin (Go = Ii~iia,+~ G), Gp represents the contribution ofthe \urthce wave terms and LG, is the remaining that is suivable for complex exponen~i;iI expansion. /! " The spatial doinain version of Gp turns out to be [4]: where No i\ the number of significant poles, k, is the location of thep-th pole, 4 its reqidue and = k:: - k,$ In the bound regime (k: = fl > k,,) this expression yields exponentially decaying fields in the tranbverse direction. If we are interested in the leaky regime asociated to ii particular surface-wave pole, it can be proved that it suffices to chenge tlie sign of 6,> for that particular pole in (3). 58 1 Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 09,2020 at 14:11:05 UTC from IEEE Xplore. Restrictions apply. - The G, function can be appropriately expanded as a siini of coinplcx exponential functions: !\I,:, G;, c u*c-y~I(o (4) 1=I The space domain contribution of the terms in the above expansion can be analytically obtained by using the following identity: where ( = ,/(:I; - :I:')~ + y2, A'" ib the zero order modified Bessel function of the second kind and Hi2) thc zero order Haenkel function of the wcond kind. The first option give5 place to growing fields \o accounting for leaky regime whereas tlic second option provides the bound solutions. In [6, 71 the space-wavr leaky modes arc obtained by imposing an alternative integration path in the I;, complex plane for the numerical evaluation of :he spectral integrals (I). This ciilculation is superseded by the present approach. The term in (I) can be considered a particular case of(5). To summarize, the proposed method allow us to analytically obtain the spacc doinair kernel of the integral equation in bound and lcaky regimes. Once the space domain kernels have been obtained, the integral equation is solved by using the Galerkin method with adequate basis functions leading to quasi-analytical evilluation of matrix entries 141. 3 Numerical results Our method has been validated by comparing our results with the ones obtained by means of a well established SDA code [7] developed for zero-thickiie~h mip-like striictures. The agreement in both bound and leaky regimes is excellent fbtid1 the cascs considered. As an example, Figure 1 shows propagation constant data for a space-wave leaky mode supported by a microstrip line. For the zero-thickness case, this figure coinpares SDA results versus our results for B/k0 and o/kU. An excellent agreeinent is observed. This structure was also analyzed in [I], showing our results good agreement with the data reported in [I]. Although the extension of our approach to nonzero thickness conductors requires further mathematical steps than those presented in this paper, this task has been done and some numerical results have been include in the figure. 4 Conclusions The MPIE approach combined with the complex iinage method has been adapted for analyzing microstrip transmission lines in leaky regime hy the first time. Both surfacewave and space-wave leaky modes are incorporated to the analysis in a very simple way. The complex image technique provides high accuracy and efficiency to the analysis. Working in the spatial domain allows for an extension to nonplanar conductors. Although this generalization is not straightforward, the theory reported in this paper offers the necessary background. 582 Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 09,2020 at 14:11:05 UTC from IEEE Xplore. Restrictions apply. I2l 1 .o 06 ACO , 4 6 8 io 12 i4 Frequency (GHz) Figure 1 : Normalized propagation (P/ko) and attenuation (culko) constants for a spacewave leaky mode in a microstrip line: E, = 9.8, w = 3 mm, h = 0.635 mm, t = strip thickness. References [I] K. A. Michalrky and D. Zheng, "Rigurous analysis of open microstrip lines of arbitrary cross section in hound and leaky regimes," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 2005-2010, Dec. 1989. [2] C-I.G.Hsu, R.F.Harrington, K.A.Michalski and D.Zheng "Analysis of multiconductor transmission lines of arbitrary cross section in multilayered uniaxial media," IEEE Trans. Microwave Theory Tech., vol. 41, pp. 10-18, Jan. 1993. [3] D.G. Fang, J.J. Yang and G.Y. Delisle, "Discrete image theory for horizontal electric dipoles in a multilayered medium,'' Proc. Inst. Eh. Eng., vol. 135, pp, 297-302, Oct. 1988. [4] J.Bcrnal, EMedina, R.R.Boix and M.Horno, "Fast full wave analysis of multistrip transmission lines based on MPIE and complex images,'' IEEE Trans. Microwave Theory Tech., vol. 48, pp.445452, Mar. 2000. [51 E. A. Soliman, P. Pieters, E. Beyne and G.A.E. Vandenbosch, "Numerically efficient spatial-domain moment method for multislot transmission lines in layered media - aplication to multislot in MCM-D technology," IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1782-1787, Sep. 1999. [61 N.K. Das and D.M. Pozar, "Full-wave spectral-domain computation of material, radiation and guided wave losses in infinite multilayered printed transmission lines," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 54-63, Jan. 2000. [71 R. MarquCs, F. Mesa, "Spectral domain analysis of higher order leaky modes in microstrip lines: a new spectral-gap effect," Journal of Elecfromagneric Waves and Applications, vol. II, pp. 1367-1384.1997. 583 Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 09,2020 at 14:11:05 UTC from IEEE Xplore. Restrictions apply.