Space-domain Green's function approach to the capacitancecalculation of multiconductor lines in multilayered dielectrics withimproved surface charge modeling

An integral equation method for the calculation of capacitance and inductance matrices is presented. The method is suitable for multiconductor transmission lines embedded in a multilayered dielectric medium on top of a ground plane. Conductors of arbitrary polygonal cross section can be handled, as well as infinitely thin conductors. The method is new in two respects. The kernel of the integral equation is the space-domain Green's function of the layered medium. The accuracy of the solution is enhanced by using basis functions that exactly model the singular behavior of the charge density in the neighborhood of a conductor edge. Numerical examples show the accuracy of the calculations and the complexity of the configurations that can be treated     

一种新的闭式空域格林函数及其在三维互连结构电容参数提取中的应用

该文提出了一种提取分层介质中三维静态闭式空域格林函数的通用算法,并成功地运用于三维多层多导体互连结构的电容参数提取。该算法在谱域等效传输线模型的基础上,应用Krylov子空间维数缩减技术求出谱域格林函数的有理逼近表达式,再由留数定理得到闭式空域格林函数。所得闭式空域格林函数结合矩量法可以方便地用于提取三维互连结构的电容参数。数值结果验证了方法的准确性和有效性。     

Computation of excess capacitances of various strip discontinuitiesusing closed-form Green's functions

An efficient quasi-static method to compute excess (equivalent) capacitances of various strip discontinuities in a multilayered dielectric medium is presented. The excess charge distribution on the surface of a conductor is obtained by solving an integral equation in conjunction with closed-form Green's functions. A complete list of expressions of the closed-form Green's functions for a point charge, a line charge, and a semi-infinite line charge is presented. An open end, a bend, a step junction, and a T junction are considered as numerical examples     

Computation of the equivalent capacitance of a via in amultilayered board using the closed-form Green's function

A method based on the quasi-static approximation for computing the equivalent capacitance of a via is presented in this paper. The geometry of a via consists of traces, pads and a perfectly conducting cylindrical rod; the via is buried in a multilayered dielectric medium with optional reference (ground) planes. The total number of traces, pads, and ground planes can be arbitrary, as well as the angles and cross sections. The method is based on the excess charge formulation of an integral equation applied in conjunction with the recently developed closed-form Green's function     

Quick quasi-TEM analysis of multiconductor transmission lines withrectangular cross section

This paper presents an efficient and accurate procedure for computing the quasi-static matrix parameters ([C], [L], [G], and [R]) of rectangular-shaped conductors embedded in a multilayered dielectric medium over an infinite ground plane. An additional top ground plane can also be considered., The problem is formulated in terms of the space-domain integral equation for the free-charge distribution on the slab conductor surfaces. The spatial Green's function is computed from its spectral counterpart using system identification techniques [Prony's method or matrix pencil method (MPM)]. The integral equation is solved by means of a Galerkin scheme employing entire domain basis functions. This results in a small matrix size. In addition, the quasi-analytical evaluation of the entries of the Galerkin matrix leads to a very efficient and accurate computer code. A detailed study on the convergence and accuracy of the method has been included     

高速IC互连线分布参数提取的一种新方法:采用Pade逼近的介质格林函数法

本文提出采用Ｐａｄｅ逼近的介质格林函数方法计算多导体互连线分布参数。由于采用了分层介质的格林函数，只需在导体表面进行剖分，大大降低了未知变量的规模，本文首先提出利用Ｐａｄｅ逼近对Ｌａｐｌａｃｅ方程的级数形式的格林函数进行收敛性加速。文中给出了加速收敛的结果，表明这种方法是很有效的。     

多导体互连线分布电容矩阵的一种近似格林函数算法

本文提出了一种计算多层介质多导体互连线系统分布电容，分布电感矩阵的简单有效的方法，本方法能适用于横截形状为矩形或无限薄的多导体系统，主要是利用谱域格林函数直接求出空域格林函数的近似表达式，采用一定的加速方法后，在保持必要的计算精度的基础上能明显提高了计算速度，计算结果与文献符合很好。     

Efficient 3-d capacitance extraction considering lossy substrate with multilayered green's function

An efficient algorithm for extraction of three-dimensional (3-D) capacitance on multilayered and lossy substrates is presented. The new algorithm presents a major improvement over the quasi-3-D approach used in a Green's function-based solver and takes into consideration the sidewalls of 3-D conductors. To improve the efficiency of the computation and the transformation of the Green's function, a nonuniform grid is adopted. The most computationally intensive part in the transformation of the Green's function is computed separately as technology-independent matrices Tk foremost. Once computed, Tk can be stored and used for any technology, thus the storage requirement and computational complexity are reduced from O (S/sup 2/) and O (S/sup 2/ log S/sup 2/), respectively, to just O [(log S/sub max/)/sup 2/]. Extensive tests have been performed to verify the new algorithm, and its accuracy has been established by comparing with other programs.     

A recursive Green's function method for boundary integral analysisof inhomogeneous domains

The recursive Green's function method (RGFM) for computation of fields scattered by two-dimensional (2-D) inhomogeneous dielectric bodies is presented. The algorithm efficiently constructs the Green's function for the inhomogeneous region by recursively combining known Green's functions from smaller subdomains. The fields on the scatterer surface are then computed using a boundary integral formulation. Proper implementation of the RGFM results in computational and storage complexities which scale as N^1.5 and N, respectively, where N is the total number of discrete cells in a domain. Comparisons of results obtained using the RGFM with those computed from moment method and exact solutions show the efficiency and accuracy of the technique     

TM scattering by an impedance sheet extension of a paraboliccylinder

An integral equation and method of moments (MM) solution are presented for the two-dimensional (2-D) problem of transverse magnetic (TM) scattering by an impedance-sheet extension of a perfectly conducting parabolic cylinder. An integral equation is formulated for a dielectric cylinder of general cross section in the presence of a perfectly conducting parabolic cylinder. It is then shown that the solution for a general dielectric cylinder considerably simplifies for the special case of TM scattering by a thin multilayered dielectric strip that can be represented as an impedance sheet. The solution is termed an MM/Green's function solution, where the unknowns in the integral equation are the electric surface currents flowing in the impedance sheet; the presence of the parabolic cylinder is accounted for by including its Green's function in the kernel of the integral equation. The MM solution is briefly reviewed, and expressions for the elements in the matrix equation and the scattered fields are given. Sample numerical results are provided     

Integral-Equation Analysis of 3-D Metallic Objects Arranged in 2-D Lattices Using the Ewald Transformation

We present a space-domain integral-equation method for the analysis of periodic structures formed by three-dimensional (3-D) metallic objects arranged in a general skewed two-dimensional lattice. The computation of the space-domain Green's function is accelerated using the Ewald transformation. The method is validated on several periodic structures ranging from planar frequency-selective surfaces to 3-D photonic crystals and metamaterials. For these structures, our technique shows a clear advantage in terms of computational speed when compared with available commercial softwares.     

Broadband vertical interconnects using slot-coupled shieldedmicrostrip lines

A full-wave space-domain integral equation analysis of aperture coupled shielded microstrip lines is presented based on the Equivalence Principle. The formulation includes the capability to model multilayered substrates through the derivation of the associated dyadic Green's functions which represent the layers through impedance boundary conditions. The method of moments, is used to solve for the line currents and slot voltage with even and odd mode excitations which are then interpreted through transmission line analysis to determine the two-port scattering parameters. A parametric study together with experimental data is presented which demonstrates the behavior of the coupler and the accuracy of the technique     

Dyadic Green's function of multilayer cylindrical closed and sector-structures for waveguide, microstrip-antenna, and network analysis

A clear and systematic method to derive the spectral- and space-domain dyadic Green's function of arbitrary cylindrical multilayer and multiconductor structures is proposed. The derivation is either done for a circumferentially closed or a cylindrical sector structure, which is bounded by electric or magnetic walls in an azimuthal direction. The solution for the dyadic Green's function in the spectral domain is obtained via an equivalent circuit. Relations between the spectral and space domains for the dyadic Green's functions are derived using eigensolution expansions. Finally, the dyadic Green's function is applied to the problem of finding the propagation constants of the two-layer dielectric rod.     

Symbolic derivation and numeric computation of dyadic Green'sfunctions using Mathematica

This paper presents a mathematical-software functional package that is capable of performing symbolic derivation and numeric computation of dyadic Green's functions for certain multilayered structures: a planar stratified multilayered medium, a spherical multilayered medium, a cylindrical multilayered medium, and a conducting rectangular waveguide with a multilayered dielectric load. The algorithms of this software package are based on the eigenfunction-expansion method. Using Mathematica^TM, two packages were written to fulfill the aforementioned objectives. Upon completion of the software development, dyadic Green's functions for three-layered media were generated. A comparison of these outputs with published results showed good agreement. This demonstrated the applicability of the symbolic package. For the numeric package, the Green's dyadics for a particular three-layered spherical isotropic multilayered medium were generated as an illustration. These packages have been successfully implemented, and future derivation of dyadic Green's functions for these media may be performed     

Full-wave analysis of nonplanar transmission lines on layeredmedium by means of MPIE and complex image theory

In this paper, a multiconductor transmission line consisting of arbitrary cross-sectional perfect conductors printed on a layered isotropic or uniaxial anisotropic dielectric medium is analyzed by solving the mixed-potential integral equation for the free-surface currents. Closed-form expressions of the two-dimensional space-domain Green's functions for the electrodynamic potentials are used. These expressions are obtained by applying the complex image technique to the spectral functions remaining after removing the asymptotic and pole contributions from the original Green's functions. A single set of complex images is obtained for any guess value of the unknown propagation constant and for any pair of source/field points. In addition, the reaction integrals involved in the application of the method of moments are worked out in a quasi-analytical way. The final result is an accurate and highly efficient computation code for analyzing multiconductor structures printed on a layered medium     

Integral equation analysis of vertical current sheets in planarcircuits and antennas

The electric field integral equation (EFIE) for planar circuits and antennas is usually solved with a spectral computation of the Green's functions. The method of moments solution of the EFIE becomes cumbersome for three-dimensional (3D) metallic surfaces in an arbitrary multilayered medium. Here, it is demonstrated how some of the numerical work can be replaced by analytical computations for 3D planar structures by using the closed form expressions for the spectral Green's functions. An example of a two-conductor line short circuited with a vertical metallic plate is presented     

Accurate approximation of Green's functions in planar stratified media in terms of a finite sum of spherical and cylindrical waves

A robust and computationally-expedient methodology is presented for accurate, closed-form approximation of the Green's functions used in the mixed-potential integral equation statement of the electromagnetic boundary value problem in planar stratified media. The proposed methodology is based on the fitting of the spectrum of the Green's function, after the extraction of the quasistatic part, making use of rational functions. The effectiveness and robustness of the proposed methodology rely upon the proper sampling of the spectrum in order to improve the accuracy of the rational function fit. The resulting closed-form approximation is in terms of both spherical and cylindrical waves. Thus, high accuracy is obtained in the approximation of the Green's function irrespective of the distance of the observation point from the source. The methodology is validated through its application to the approximation of the Green's function for a multi-layered, planar dielectric stack.     

Revisiting the partial-capacitance approach to the analysis of coplanar transmission lines on multilayered substrates

A theoretical justification is presented of the partial-capacitance (PC) approach, widely exploited in the modeling of coplanar waveguides on finite-thickness and multilayered substrates. The analysis is based on the static spectral-domain approach to the computation of the capacitance of a set of planar conductors embedded into a multilayered substrate. It is shown that the PC method can be derived from approximating the static Green's function of a multilayered dielectric as the sum of partial contributions; such a decomposition can be applied to the Green's function either in parallel (admittance) or series (impedance) form. The resulting (parallel or series) PC approaches are shown to be accurate with substrates having layers of decreasing or increasing permittivity, respectively. Lines backed by magnetic or electric walls are introduced as limiting cases of multilayered structures.     

Closed-Form Expressions of Multilayered Planar Green's Functions That Account for the Continuous Spectrum in the Far Field

The rational function fitting method has been found useful in the derivation of closed-form expressions of spatial-domain Green's functions for multilayered media. However, former implementations of the rational function fitting method lead to Green's functions expressions that are not accurate in the far field when this far field is dominated by the continuous spectrum instead of being dominated by surface waves (as it happens, for instance, in the case of lossy multilayered media). In this paper, the authors introduce a novel implementation of the rational function fitting method, which leads to Green's functions expressions that are accurate in the far field when this is dominated either by the continuous spectrum or by surface waves. In the new approach, the far-field contribution of the continuous spectrum to the Green's functions is numerically fitted in terms of functions with closed-form Hankel transforms, and this far-field contribution is explicitly added to the total least squares approximations of the Green's functions. The numerical results obtained for the Green's functions with the new approach have been compared with numerical results obtained via direct numerical integration of Sommerfeld integrals, and excellent agreement has been found despite the contribution—continuous spectrum or surface waves—dominating the far field.      

Electromagnetic analysis of an antenna embedded in a compositeenvironment

This paper addresses the problem of an antenna embedded in a hole dug in the ground. The composite medium configuration consists of a half-space dielectric (representing the Earth-air interface) containing a cylindrical hole filled with a different dielectric medium. The wire antenna resides within this hole, on the axis. The solution strategy is based on decomposing the problem into simpler subproblems, which are treated sequentially. First we calculate a numerical dyadic Green's function for the composite medium by solving an integral equation formulated over a background consisting of the unperturbed dielectric half space (for which the Green's functions are known in a spectral integral form). This integral equation is solved via the fictitious currents method, which is a special case of the method of moments. We then solve the integral equation for the antenna currents using this numerical Green's function and determine the input impedance and radiation pattern