Efficient evaluation of spatial-domain MoM matrix entries in theanalysis of planar stratified geometries

An efficient hybrid method for evaluation of spatial-domain method-of-moments (MoM) matrix entries is presented in this paper. It has already been demonstrated that the introduction of the closed-form Green's functions into the MoM formulation results in a significant computational improvement in filling up MoM matrices and, consequently, in the analysis of planar geometries. To achieve further improvement in the computational efficiency of the MoM matrix entries, a hybrid method is proposed in this paper and, through some examples, it is demonstrated that it provides significant acceleration in filling up MoM matrices while preserving the accuracy of the results     

A numerically efficient technique for the analysis of slots inmultilayer media

A numerically efficient technique for the analysis of slot geometries in multilayer media is presented using closed-form Green's functions in spatial domain in conjunction with the method of moments (MoM). The slot is represented by an equivalent magnetic-current distribution, which is then used to determine the total power crossing through the slot and the input impedance. In order to calculate power and current distribution, spatial-domain closed-form Green's functions are expanded as power series of the radial distance ρ, which makes the analytical evaluation of the spatial-domain integrals possible, saving a considerable amount of computation time     

An efficient method for electromagnetic characterization of 2-Dgeometries in stratified media

A numerically efficient technique, based on the spectral-domain method of moments (MoM) in conjunction with the generalized pencil-of-functions (GPOF) method, is developed for the characterization of two-dimensional geometries in multilayer planar media. This approach provides an analytic expression for all the entries of the MoM matrix, explicitly including the indexes of the basis and testing functions provided that the Galerkin's MoM is employed. This feature facilitates an efficient modification of the geometry without the necessity of recalculating the additional elements in the MoM matrix. To assess the efficiency of the approach, the results and the matrix fill times are compared to those obtained with two other efficient methods, namely, the spatial-domain MoM in conjunction with the closed-form Green's functions, and a fast Fourier transform algorithm to evaluate the MoM matrix entries. Among these, the spectral-domain MoM using the GPOF algorithm is the most efficient approach for printed multilayer geometries     

A Mesh-Adapted Closed-Form Regular Kernel for 3D Singular Integral Equations

The Green's functions employed in the method of moments (MoM) diverge when observation and source points coincide; this is at the origin of the difficulties in computing the MoM matrix entries, and in handling the near-field interactions in fast Fourier transform (FFT)-based fast methods and other sampling-based methods. In this paper, we show that this singularity can be avoided, and a modified regular Green's function can be used instead. This latter is obtained from the spectral representation of the usual Green's function via windowing of its spectrum; the width of the spectral window depends on the size of the mesh employed for discretizing the problem, so that the proposed regular Green's function is a mesh-adapted regular kernel. We address a general 3D problem; we relate the MoM reaction integrals to the 2D Fourier spectrum of the Green's function, that allows to discuss the necessary spectral bandwidth for the windowed Green's function. We employ a tapered window, and present a closed-form expression for the spatial Green's function. Numerical results are presented for 3D antenna and scattering problems discretized with Rao-Wilton-Glisson (RWG) functions, and for uniform and nonuniform meshing. They show that the proposed method yields accurate solutions also for the antenna input impedance. The meaning of the regularized Green's function is also discussed and put in perspective.     

New closed-form Green's functions for microstrip structures theory and results

This paper presents an efficient technique to evaluate the Green's functions of single-layer and multilayer structures. Using the generalized pencil of function method, a Green's function in the spectral domain is accurately approximated by a short series of exponentials, which represent images in spatial domain. New compact closed-form spatial-domain Green's functions are found from these images using several semi-infinite integrals of Bessel functions. With the numerical integration of the Sommerfeld integrals avoided, this method has the advantages of speed and simplicity over numerical techniques, and it leads to closed-form expressions for the method-of-moments matrix coefficients. Numerical examples are given and compared with those from numerical integration     

A quasi-one-dimensional integration technique for the analysis ofplanar microstrip circuits via MPIE/MoM

The mixed-potential integral-equation approach, using spatial-domain closed-form Green's functions, and discretized with the method-of-moments, is a state-of-the-art method for the analysis of planar microstrip circuits. One of its most time-demanding tasks is the evaluation of the impedance matrix terms, which typically requires the numerical computation of two-dimensional integrals. A method based on suitable changes of coordinates and domains is introduced in this paper in order to reduce such integrals to a quasi-one-dimensional numerical integration, with a substantial enhancement in the efficiency of the analysis, without affecting the accuracy of the approach. Results are given demonstrating, for practical accuracy values, an improvement of typically one order of magnitude in simulation times     

Numerically efficient analysis of planar microstrip configurationsusing closed-form Green's functions

An efficient technique for the analysis of a general class of microstrip structures with a substrate and a superstrate is investigated in this paper using newly-derived closed-form spatial domain Green's functions employed in conjunction with the Method of Moments (MoM). The computed current distributions on the microstrip structure are used to determine the scattering parameters of microstrip discontinuities and the input impedances of microstrip patch antennas. It is shown that the use of the closed-form Green's functions in the context of the MoM provides a computational advantage in terms of the CPU time by almost two orders of magnitude over the conventional spectral domain approach employing the transformed version of the Green's functions     

Evaluation of layered media Green's functions via rational function fitting

An efficient rational function fitting methodology, called VECTFIT, is utilized toward the closed-form evaluation of the Sommerfeld integrals associated with electromagnetic Green's functions in planar layered media. VECTFIT approximates the component of the spectrum of the Green's function that remains after the extraction of the primary source contribution and the quasistatic part with a rational function, thus enabling a robust and expedient closed-form evaluation of the Sommerfeld integral for electromagnetic potentials and associated field quantities.     

Estimation of spurious radiation from microstrip etches usingclosed-form Green's functions

The problem of spurious radiation from electronic packages is considered by investigating the power radiated from microstrip etches that are excited by arbitrarily located current sources and terminated by complex loads at both ends. The first step in the procedure is to compute the current distribution on the microstrip line by using the method of moments (MoM). Two contributions of this paper are: (i) employing the recently derived closed-form Green's functions in the spatial domain, which permits an efficient computation of the elements of the MoM matrix; and (ii) incorporating complex load terminations in a convenient manner with virtually no increase in the computation time. The computed current distribution is used to calculate the spurious radiated power, and the result is compared with that derived by using an approximate, transmission line analysis     

空域MPIE/MoM全波分析三维微带电路

矩量法在空域分析三维微带电路的关键是闭式格林函数的求解。本文首先介绍谱域格林函数一种新的表达式,使得源点和场点在同一层时,离散复镜像法可提取出与它们无关的闭式,从而避免了插值;不在同一层时,可提取与场点无关的闭式,此时只须对源点进行一维插值,因而提高了计算效率。然后,利用闭式格林函数和RWG基函数,基于混合位积分方程的矩量法就可以在空域精确、有效地分析三维任意形状微带电路。给出了几个典型实例,表明本文方法的有效性。     

Application of a point-matching MoM reduced scheme to scatteringfrom finite cylinders

One of the most common methods for the solution of three-dimensional (3-D) scattering problems is the electric-field volume integral equation numerically solved by the application of the method of moments (MoM)-usually the point-matching version. Although simple to formulate, it shows inherent difficulty and complexity because of the 3-D integrals appearing in the interaction matrix elements and of the singularity of the dyadic Green's function (DGF) present in the computation of the self-cell elements. In this paper, a transformation method is presented, which in the case of the point-matching MoM, both reduces the 3-D integrals to two-dimensional (2-D) ones, and also eliminates the need of separate treatment of the singularity while maintaining the same degree of approximation. Comparison to published results is made for the case of scattering by a finite dielectric cylinder. Further examples are presented for scattering by layered dielectric cylinders and lossy cylindrical shells excited by uniform plane waves     

Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions

The method of moments (MoM) solution of combined field integral equation (CFIE) for electromagnetic scattering problems requires calculation of singular double surface integrals. When Galerkin's method with triangular vector basis functions, Rao-Wilton-Glisson functions, and the CFIE are applied to solve electromagnetic scattering by a dielectric object, both RWG and n/spl times/RWG functions (n is normal unit vector) should be considered as testing functions. Robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the CFIE with these basis and test functions. In computing the impedance matrix elements, including the gradient of the Green's function, we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand. In the developed method, all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions. In addition, we present compact iterative formulas for computing the extracted terms in closed form. By these formulas, we can extract any number of terms from the singular kernels of CFIE formulations with RWG and n/spl times/RWG functions.     

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.      

An Efficient Full-Wave Simulation Algorithm for Multiple Vertical Conductors in Printed Circuits

An efficient and rigorous numerical method, based on the spatial-domain method of moments (MoM) in conjunction with the closed-form Green's functions, is devised for the analysis of multiple vertical conductors in printed circuits. As this combination has already proven to be very efficient for the analysis of printed structures with horizontal and vertical conductors, it is extended to efficiently handle multiple vertical conductors. Some circuits with multiple vertical strips are analyzed using the proposed method, and results are compared either to those presented in the literature or to those obtained from the commercial software em by SONNET Software, North Syracuse, NY. Computational efficiency of the algorithm is assessed in terms of CPU time, and it is observed that the computational cost of the proposed algorithm is an order of magnitude less than that of the straightforward implementation of the underlaying method of the algorithm—the spatial-domain MOM using closed-form Green's functions.     

Capacitance computations in a multilayered dielectric medium usingclosed-form spatial Green's functions

An efficient method to compute the 2-D and 3-D capacitance matrices of multiconductor interconnects in a multilayered dielectric medium is presented. The method is based on an integral equation approach and assumes the quasi-static condition. It is applicable to conductors of arbitrary polygonal shape embedded in a multilayered dielectric medium with possible ground planes on the top or bottom of the dielectric layers. The computation time required to evaluate the space-domain Green's function for the multilayered medium, which involves an infinite summation, has been greatly reduced by obtaining a closed-form expression, which is derived by approximating the Green's function using a finite number of images in the spectral domain. Then the corresponding space-domain Green's functions are obtained using the proper closed-form integrations. In both 2-D and 3-D cases, the unknown surface charge density is represented by pulse basis functions, and the delta testing function (point matching) is used to solve the integral equation. The elements of the resulting matrix are computed using the closed-form formulation, avoiding any numerical integration. The presented method is compared with other published results and showed good agreement. Finally, the equivalent microstrip crossover capacitance is computed to illustrate the use of a combination of 2-D and 3-D Green's functions     

Derivation of closed-form Green's functions for a generalmicrostrip geometry

The derivation of the closed-form spatial domain Green's functions for the vector and scalar potentials is presented for a microstrip geometry with a substrate and a superstrate, whose thicknesses can be arbitrary. The spatial domain Green's functions for printed circuits are typically expressed as Sommerfeld integrals, which are inverse Hankel transforms of the corresponding spectral domain Green's functions and are time-consuming to evaluate. Closed-form representations of these Green's functions in the spatial domains can only be obtained if the integrands are approximated by a linear combination of functions that are analytically integrable. This is accomplished here by approximating the spectral domain Green's functions in terms of complex exponentials by using the least square Prony's method     

Scattering from a cylindrically conformal slotted waveguide arrayantenna

A numerical method is developed to investigate electromagnetic scattering by a cylindrically conformal waveguide-fed slot array. The problem is first formulated in terms of integral equations using the equivalence principle. The integral equations are then solved using the method of moments (MoM) in conjunction with global sinusoidal basis functions and Galerkin's testing procedure. The MoM solution requires the evaluation of the generalized admittance matrices involving various dyadic Green's functions. The slow convergence of the series associated with the summation of waveguide modes is accelerated using the Kummer transformation and the slow convergence of the series associated with the summation of the exterior modes is avoided by using the asymptotic solutions with proper treatment for singular integrals. The evaluation of the excitation vector and scattered field is also accelerated using Watson's transformation and asymptotic solutions. Numerical results are presented to illustrate the scattering characteristics of the cylindrically conformal waveguide-fed slot arrays, such as the effects of curvature, slot thickness, and waveguide termination on the radar cross section (RCS) of the arrays     

An introduction to spectral domain method-of-moments formulations

The capacitance per unit length of a microstrip transmission line is obtained using a spectral-domain method-of-moments (MoM) formulation. The paper emphasizes this problem as a teaching tool to introduce students of electromagnetics to this technique. Firstly, the derivation of the spectral-domain Green's function is outlined. Using this, the relevant integral equation is derived to which the Galerkin MoM approach is then applied. The MoM problem is solved in the spectral domain by also transforming the expansion and weighting functions. The inverse Fourier transform is then applied to find the spatial-domain charge distribution, and, hence, capacitance. The issues that arise here - both of selecting how much of the spectrum to include, and how to choose the number of integration points - are discussed, and the results of typical numerical experiments are presented. The time required to compute the elements of the immittance matrix is shown to be 0(N/sup 3/); the use of translational symmetry (and thus Toeplitz matrix structure) to reduce this is outlined. Classroom experience with this material is discussed. Finally, a hybrid spectral/spatial-domain formulation, introducing asymptotics, is outlined to accelerate the evaluation of the immittance matrix.     

Some convergence considerations in space-domain moment-methodanalysis of a class of wide-band microstrip antennas

The method of moments (MoM) analysis of probe-fed rectangular microstrip patches requires the inclusion of a probe-to-patch attachment mode-expansion function when the substrate thickness d⩾0.02λ, where λ is the free-space wavelength. The results for the input impedance showed increased divergence with measurements when the attachment mode was omitted from the full-wave analysis. The attachment mode can be expressed as an infinite eigenfunction series that increases the fill time of the impedance matrix in an MoM analysis. In an earlier investigation, the infinite eigenfunction series was reduced to a residue series that required one or two terms compared to about 55 terms for the eigenfunction series. In this paper, the convergence properties of the eigenfunction and residue series are investigated in view of rigorous MoM analysis. The relative errors resulting from replacing the eigenfunction by the residue series for the attachment mode, are compared by numerically evaluating a class of two-dimensional (2-D) spatial integrals shown to be closely related to the elements of an MoM impedance matrix. Additionally, the computation times for the evaluation of these integrals for the two forms of the attachment mode-expansion function are also included. Based on the superior convergence properties of the residue series for the attachment mode-expansion function, it is mathematically justified that this form can readily be used for analytic reduction of the spatial, reaction integrals from four to 2-D forms. This feature allows further reduction of the fill time of the MoM impedance matrix, suggesting the possibility of developing an efficient space-domain MoM technique for modeling of wide-band microstrip antennas     

Analysis and measurements of conformal patch array antennas on multilayer circular cylinder

In this paper, we present a method of moments (MoM) program to analyze different configurations of arrays of cylindrical-rectangular patches. The patches can be located inside or on the surface of a multilayer circular-cylindrical structure and can be arbitrarily rotated. The antenna structure is rigorously taken into account by using proper Green's functions, and the array is analyzed by using an element-by-element approach. The elements of the MoM matrix are calculated in the spectral domain, and special attention is given to their numerical treatment when analyzing cylindrical antennas with large radii. A laboratory model is built to test different configurations of patches and to validate calculated results. The agreement between measurements and calculations is very good.