IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 12, DECEMBER 1997 2131 Analysis of Coplanar-Waveguide Discontinuities with Finite-Metallization Thickness and Nonrectangular Edge Proﬁle Fang-Lih Lin and Ruey-Beei Wu, Member, IEEE Abstract— In this paper, the hybrid ﬁnite-element method (FEM) is proposed to analyze the coplanar-waveguide (CPW) discontinuities with ﬁnite-metallization thickness. A variational formula for the electric ﬁeld in the slot region between upper and lower half-space is derived by applying the variational-reaction theory and solved by the FEM. In the limiting case of zero metallization thickness, this ﬁnite-element analysis is reduced to a moment-method analysis using Galerkin’s approach with rooftop basis functions. The edge proﬁle effects of a trapezoidal slot cross resulting from the etching or sputtering process can also be easily considered by this approach. Some numerical results are presented for short- and open-ended CPW discontinuities for different conductor thicknesses. It has been shown that not only the metallization thickness, but also the conductor-edge proﬁle, can produce noticeable effects on circuit performance and should be taken into account for accurately modeling the CPW discontinuities. Index Terms— Coplanar waveguides (CPW’s), ﬁnite-element method (FEM), transmission-line discontinuities. I. INTRODUCTION T HE uniplanar transmission-line structure based on the coplanar waveguide (CPW) has been developed as a circuit element for monolithic microwave integrated circuits (MMIC’s). Accurate analysis and characterization of CPW discontinuity is important in designing the MMIC because tuning or trimming of MMIC’s is infeasible. As the line size in the metallization plane shrinks, the ﬁnite-metallization thickness may play a signiﬁcant role in determining the circuit performance and should be taken into account. Several papers have been presented to characterize shielded CPW discontinuity structures with ﬁnite-metallization thickness based on the mode-matching technique [1]–[3], the transverse resonance technique (TRT) [4], [5], or the ﬁnite- difference method in frequency domain [6]. In applying these methods, the structures to be analyzed must be closed by the electric or magnetic walls and, hence, the radiation effects cannot be accounted for. On the other hand, Tran et al. [7] utilized the extended spectral-domain approach to analyze the unshielded CPW discontinuities with ﬁnite-metallization Manuscript received December 9, 1996; revised May 26, 1997. This work was supported in part by the National Science Council, Republic of China, under Grant NSC86-2221-E002-043. The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. Publisher Item Identiﬁer S 0018-9480(97)08246-X. thickness. This method needs to compute the ﬁeld patterns of uniform CPW lines on both sides of discontinuities in advance, which is by itself a considerable task. All the methods mentioned above only dealt with slots of rectangular cross section. In practice, the cross section of the strip is likely to be better approximated by a trapezoid than by a rectangle due to the nonideal underetching or elec- trolytical growth in the integrated circuit (IC) manufacturing process. Recent studies have shown that not only the ﬁnite- metallization thickness, but also the conductor-edge proﬁle will affect the electrical characteristics of uniform microstrip [8]–[12] and CPW lines [13], [14]. It is then necessary to account for the effects of conductor-edge proﬁle for accurately modeling the CPW discontinuities. For this purpose, the ﬁnite-element method (FEM) hy- bridized with suitable integral equations for the exterior ﬁeld was utilized to calculate the capacitance or inductance of CPW discontinuities with ﬁnite metallization in the quasi-static approximation [15], [16]. However, the full-wave analysis is required to account for the high-frequency effects. In this paper, a variational formula in terms of the electric ﬁeld in the slot region is derived in Section II. The effects of surface- and space-wave radiation are included in the integral-equation formulation by the Green’s functions for the exterior ﬁeld. The variational formula is solved in Section III by applying the FEM with edge vector elements as the basis functions. In the limiting case of zero thickness, the formula reduces to an integral equation which is commonly employed in the moment-method analysis. Section IV presents some numerical results for the short- and open-ended discontinuities with ﬁnite-metallization thickness. Effects of conductor-edge proﬁle are considered in Section V. Finally, brief conclusions are drawn in Section VI. II. VARIATIONAL-REACTION FORMULATION The cross section of a CPW with ﬁnite-metallization thick- ness is shown in Fig. 1(a). The solution region lies between the upper and lower slot surfaces, denoted by and , respectively. The variational equation for the unknown electric ﬁeld in can be derived by applying the variational-reaction theory [17], [18]. First, consider the reaction form (1) 0018–9480/97$10.00  1997 IEEE Authorized licensed use limited to: National Taiwan University. Downloaded on February 25, 2009 at 01:05 from IEEE Xplore. Restrictions apply.