A VECTOR FINITE-ELEMENT ANALYSIS OF COMPLEX MODES IN SHIELDED MICROSTRIP LINES

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Md. Shah Aiam, Kolchi Hirayama, and Yoshio Hayashi Department of Electronic Engineering Kitami Institute of Technology Kitarni 090, Japan

Masanori Koshiba Department of Electronic Engineering Hokkaido University Sapporo 060, Japan

KEY TERMS Microstrip line, finite-element method, triangular mixed element, complex mode

ABSTRACT The finite-elemenr method is employed to carry out the numerical analysis of complex modes, as well as propagating and evanescent modes in lossless shielded microstrip lines. In this finite-element cal- cularion, rhe triangular mixed element is used. The possible electric field disrributions of the complex modes are also presented. D 1993 Joliri Wdey & Suns. Inc.

1. INTRODUCTION With the growing interest in the study of the discontinuities between planar transmission lines, a number of investigations of complex modes have been reported in fin lines [l-31, shielded microstrip lines [4-71, and shielded suspended micro- strip lines (8-91. Even though the complex modes are not strongly excited, at both sides of a discontinuity in a wave- guide, they can severely affect the modal energy distribution. So their analyses are helpful in the study of the junction discontinuities which are frequently encountered in most prac- tical integrated circuits.

In this article, using the finite-element method (FEM), we present some numerical results on the complex modes in loss- less shielded microstrip lines. The triangular mixed element [lo] is used; it is composed of a constant edge element and a linear nodal (conventional Lagrange) element. In this ap- proach, the tangential continuity of electromagnetic field across material interfaces can be straightforwardly imposed and no spurious solutions appear. In [lo] the propagation characteristics of first two modes (propagating modes) of a shielded microstrip line has been discussed. To show the gen- erality of the FEM [lo], the same approach has been utilized in this article to analyze the complex modes along with the propagating and evanescent modes in shielded microstrip lines. To check the accuracy, computed results of the FEM are compared with those of the spectral domain method (SDM). This article also highlights the possible electric field distribution of the complex modes in shielded microstrip lines.

2. THE FINITE-ELEMENT APPROACH Using the FEM with the triangular mixed element [lo], we obtain the following eigenvalue problem for electric field for- mulation:

where y = a + jj3 is the complex propagation constant and the components of the {E,} vector are the values of the tan- gential component of the electric field on the side of the

4

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hl f 2A

Figure 1 Shielded microstrip line

triangular mixed elements. The eigenvalue problem is solved for propagating and evanescent modes as well as complex modes of the shielded microstrip line. The field distribution of the modes can be obtained from the eigenvectors of the eigenvalue problem.

Figure 1 shows the cross section of the shielded microstrip line, where the strip width is 2W, the thickness and the relative permittivity of the isotropic substrate are hl and E,, respec- tively, and the height and the width of the shielding waveguide are h1 + h2 and 2A, respectively. We assume that the strip consists of all perfect conductors and the strip thickness is very small, i.e., negligible.

In the FEM, using symmetry conditions, we subdivide only one-half of the waveguide cross section into triangular mixed elements. In this work, the number of elements is 450, and the size of the eigenvalue matrix equation in Eq. (1) becomes 657 or 642 for the even or odd modes, respectively, since the boundary conditions are imposed on the strip, the shielding waveguide, and the symmetry plane.

3. NUMERICAL RESULTS Using the FEM and the SDM, we calculate the propagation constants in shielded microstrip lines, as shown in Figure 1.

Figure 2 shows the propagation constant versus frequency, where the broken, dotted, and solid lines represent the results of the SDM for the dominant, higher-order, and complex modes, respectively, and the dark circles represent those of the FEM. The complex modes exist in the frequency range 17.4-20.4 GHz for the even mode and 21.6-22.8 GHz for the odd mode. The results of the SDM are compared with the results of Reference [4] and good agreement is obtained.

Next, Figure 3 shows the propagation constant for the even modes with the shielding waveguide of WRJ-18. We note that changing the structural parameters causes the frequency range to increase to 14.7-20.6 GHz.

In the FEM, since the eigenvalues in Eq. ( l) , i.e., - y2 = (j3 - ja)*, involve the propagation constants of all the modes, the calculation procedure is the same whether the complex modes exist or not. The eigenvalues are either real or complex conjugate pairs. The real positive and negative eigenvalues correspond to the propagating and evanescent modes, respectively, while the complex modes are character- ized by the complex conjugate pairs. The complex propaga- tion constant may be of the form kfl ? ja, and accordingly, four possible modes are obtained. Also, using the SDM with the Muller method, we have confirmed that all four possible modes may be solutions of the complex modes. Two of the four, kj3 - ja, represent the modes propagating in the + z

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 16, December 20 1993 873

odd modes I E ~~=8.875 .

10 15 20 25 frequency (GHz)

(b) Figure 2 Propagation constant versus frequency, where E , = 8.875, W = 0.635 mm, A = 6.35 mm, h , = 1.27 mm, h2 = 11.43 mm

evenmodes

I . .. : 3 1 . 1 ' ' a I ' I ' 1-

Figure 4 The electric field for the complex modes on the waveguide cross section. (a) The complex mode for Figure 2(a) at 18 GHz. (b) The complex mode for Figure 2(b) at 22 GHz. (c) The complex mode for Figure 3 at 18 GHz

10 15 20 25 Ikquency (GHz)

Figure 3 Propagation constant versus frequency, where = 8.875, W = 0.635 mm, A = 6.5 mm, h , = 1.27 mm, h2 = 5.23 mm

874 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 16, December 20 1993

direction, and the other modes propagating in the - 2 direc- tion. In Figures 2 and 3, we have shown one of the four, /3 - ja, as the propagation constant of the complex mode.

Finally. in Figure 4, we illustrate the electric field for the complex modes on the waveguide half cross section. We ob- serve that the field expands over the whole of the waveguide cross section.

4. CONCLUSIONS A numerical analysis of the complex modes in shielded micro- strip lines was carried out by using two methods, namely, the FEM and the SDM. The triangular mixed element was used in the FEM, where no spurious solutions appeared anywhere. Also, by using the FEM, the complex modes in shielded microstrip lines were analyzed for the first time.

The present approach of the FEM can be applied easily to microstrips having finite thickness and nonrectangular cross section, which may result from underetching or electrolytic growth during fabrication. Also, more accurate results may be obtained by using high-order triangular mixed elements. In the future we would like to report on them.

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REFERENCES A. S. Omar and K. Schunemann, “Formulation of the Singular Integral Equation Technique for Planar Transmission Lines,” I E E E Trans. Microwave Theory Tech., Vol. MTT-33, Dec. 1985, DD. 1313-1321. . . A. S. Omar and K. Schunemann, “The Effect of Complex Modes a t Finline Discontinuities,” IEEE Trans. Microwave Theory Tech.. Vol. MTT-34. Dec. 1986. pp. 1508-1514. W. K. Wang. C. K. C. Tzuang. J . S. Chang, and T. H. Wang, ”Investigations of Complex Modes in a Generalized Bilateral Finline with Mounting Grooves and Finite Conductor Thick- ness.” lEEE Trans. Microwave Theory Tech., Vol. MTT-37, Dec.

W. X . Huang and T. Itoh. “Complex Modes in Lossless Shielded Microstrip Lines.” IEEE Trans. Microwave Theory Tech., Vol. MTT-36. Jan. 1988. pp. 363-165. C. J . Railton and T. Rozzi. “Complex Modes in Boxed Micro- strip.” IEEE Trans. Microwave Theory Tech., Vol. MTT-36, May IYXX. pp. 865-874. C . K. C. Tzuang. J . T. Kuo. C. C. Tien, J . S. Jang, and T. H. Wang, “Complex Modes in Shielded Planar Microstrip Lines,” I Y 8 Y IEEE MTT-S International Symposium Digest, Vol. I , June

F. Hurct. P. Kcnnis, and P. Pribetich, “Combined Effects of Microstrip Width and Substrate Height on Complex Mode Fre- quency Range of Box Microstrip Lines.” Microwave Opt. Tech- nol. Lett.. Vol. 5. Nov. 1992, pp, 635-638. J. T. Kuo and C. K. C. Tzuang. “Complex Modes in Shielded Suspended Microstrip Lines.” lEEE Trans. Microwave Theory Tech.. Vol. MTT-38. Sept. 1990, pp. 1278-1286. J. W. Tao. “A Modified Transverse Resonance Method for the Analysis of Multilayered. Multiconductor Quasiplanar Structures with Finite Conductor Thickness and Mounting Grooves,” IEEE Trans. Microwave Theory Tech., Vol. MTT-40, Oct. 1992. pp.

M. Koshiba and K. Inoue, “Simple and Efficient Finite-Element Analysis of Microwave and Optical Waveguides.” IEEE Trans. Microwave Theory Tech., Vol. MTT-40, Feb. 1992, pp. 371-377.

19x9. pp. IXYl-18Y7.

19x9. pp. 495-498.

1966-1970.

Received 8-5-93

SINGLE AND DOUBLE SQUARE-LOOP FREQUENCY-SELECTIVE SURFACES WITH THICK METALLIZATION David 8. Webb, Eric Michielssen, and Raj Mittra Department of Electrical and Computer Engineering University of Illinois Urbana, Illinois 61801

KEY TERMS Frequency-selective surfaces, layered medium, spatial filters

ABSTRACT Single and double square-loop frequency-selective surfaces (FSSs) are investigated to determine the effects of metallization thickness on the scattering characteristics. Results presented are computed via a spatial-domain method of moments technique using triangular patch basis functions. To isolate the effects of metallization thickness, free- standing screens of infinite extent are comidered. 0 l9W J o l i r i Wiler CG Sons. Inc.

1. INTRODUCTION Frequency-selective surfaces (FSS), which consist of a peri- odic arrangement of metallic elements embedded in a layered medium, find widespread applications as spatial filters over much of the electromagnetic spectrum. Techniques for con- trolling the reflection and transmission characteristics of the FSS include choice of lattice spacing for the periodic screen [l], use of a particular type of element in each unit cell of the periodic structure, which can have an effect on the passband- stopband transition sharpness 121, cascading of multiple screens [3], and use of dielectric layers. This study focuses on metallization thickness as an additional means of controlling bandwidth. To isolate the effects of metallization thickness on the scattering characteristics, the present study is limited to freestanding structures.

A parametric study of the reflection and transmission prop- erties of freestanding FSSs composed of single and double concentric square-loop elements [Figures l (a) and 2(a)] has been performed. The frequency response of the single square- loop structure of Figure l (a) with thin metallization appears in Figure 3 for lattice packing parameter W ’ = 0.6875. This response has a reflection band centered at fO, the (funda- mental) resonant frequency of the loop. At f(,, electric current

“;J ILI 7 Unitcellboundary

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Microwave and Optical Technology Letters, 6/ 16, 873-875 0 l9Y3 John Wiley & Sons, Inc. CCC 089.5-2477193

Figure 1 Single-loop FSS screens: (a) single-loop structure, (b) unit cell, reflective filter with thick metallization, and (c) unit cell, trans- missive filter with thick metallization

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