Closed-form Green Research Articles

Closed-Form Green's Function Representations in Cylindrically Stratified Media for Method of Moments Applications

Closed-form Green's function (CFGF) representations for cylindrically stratified media, which can be used as the kernel of an electric field integral equation, are developed. The developed CFGF representations can safely be used in a method of moments solution procedure, as they are valid for almost all possible source and field points that lie on the same radial distance from the axis of the cylinder (such as the air-dielectric and dielectric-dielectric interfaces) including the axial line (rho = rho' and phi = phi' ), which has not been available before. In the course of obtaining these expressions, the conventional spectral domain Green's function representations are rewritten in a different form so that: i) we can attack the axial line problem and ii) the method can handle electrically large cylinders. Available acceleration techniques that exist in the literature are implemented to perform the summation over the cylindrical eigenmodes efficiently. Lastly, the resulting expressions are transformed to the spatial domain using the discrete complex image method with the help of the generalized pencil of function method, where a modified two-level approach is used. Numerical results are presented in the form of mutual coupling between two current modes to assess the accuracy of the final spatial domain CFGF representations.

Two-Dimensional Discrete Complex Image Method (DCIM) for Closed-Form Green's Function of Arbitrary 3D Structures in General Multilayered Media

The traditional discrete complex image method (DCIM) is not efficient when the source and field points are in different layers because all the spatial coordinates can not be analytically included in the image terms in spatial domain. A two dimensional method (2D-DCIM) is introduced in this paper. In the new methodology we reorganize the spectral kernel as a 2D function. We use the 2D matrix pencil method (2D-MPM) to fit this 2D function and to generate a set of complex images independent of any spatial coordinates. The closed-form spatial domain Green's function can be obtained for arbitrary locations of source and field points in general multilayered media. Compared with traditional 1D-DCIM, we do not have to run MPM for each vertical combination of source and field points. The efficiency of the matrix filling kernel of method of moments (MoM) is significantly improved.

마이크로스트립 구조 해석을 위한 Closed-Form 그린 함수

평면형 구조에서 공간 영역 그린 함수를 구하기 위해서는 무한 Sommerfeld 적분을 계산하여야 한다. 무한 적분을 짧은 시간에 계산하기 위한 실수축 적분 방법(real axis integration method)은 주파수와 거리에 대해 넓은 범위에 걸쳐 매우 정확한 방법이지만 관측점의 수지 성분 z가 변하게 되면, 근사의 과정을 반복해야 하는 비효율성과 z가 커질수록 파수 영역 함수의 변화가 심해져 정확한 근사가 어렵다는 단점이 있다. 본 논문에서는 폐루프 적분 경로를 이용하여 파수 영역 함수를 근사화 함으로써 일반적인 마이크로스트립 구조에서 기존의 결과에 비해 정확한 closed-form 그린 함수를 구하는 방법을 제시하였다. In the layered medium, the Sommerfeld integral must be evaluated to calculate a space domain Green's function. The real axis integration method provides stable and accurate results over wide ranges of the observation distance and the singnal frequency. But this method has the in efficiency of approximation when the field point z is changed. Also, as the amplitude of z increases, the change of the spectral domain function is more rapidly. Therefore, the approximation is difficult when z becomes larger. In this paper, we propose a method to calculate an accurate closed-form Green's function for microstrip structure by using the closed-loop integration path.

On the efficient computation of closed-form Green's functions in planar stratified media

The spatial-domain Green's functions for the vector and scalar potentials in planar stratified media are cast into closed forms via a two-level approximation of their spectral-domain counterparts. The proposed methodology begins with the approximation of the spectral-domain Green's functions over large values of the spectral variable by complex exponentials, and continues with the approximation of the remainder by rational functions. Finally, the closed-form Green's functions in terms of spherical and cylindrical waves are derived, making use of some well-known integral identities. A key-feature of the proposed approach is that although it does not call for an analytical extraction of the quasistatic terms and the surface wave poles, it provides the means for the accurate description of both the near-field and far-field physics. Moreover, the rational function spectrum fitting proposed here overcomes the problem of the spurious singular behavior of the spatial-domain Green's functions because of the use of Hankel functions. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

Integral equation-based approach for the analysis of tapered transmission lines

A method for analysing tapered lossy transmission lines is presented. The Telegrapher's equations are solved by using the closed-form Green's function of the corresponding uniform transmission line. The space-dependent characteristic impedance is handled as a forcing term of the Telegrapher's equations whose effect is incorporated into the Fredholm integral equation of the second kind describing the space voltage distribution. The method allows to avoid the segmentation technique which is prone to inaccuracies because of spurious reflections at the boundaries of each section. Several tests are carried out and excellent results are obtained confirming the superior accuracy of the proposed method with respect to the standard segmentation technique.

An Efficient Approach for Extracting Poles of Green's Functions in General Multilayered Media

A fast approach of closed-form Green's functions for mixed-potential integral equation analysis of both near- field and far-field in planar multilayered media is presented in this paper. Since all components of the Green's functions in spectral-domain are restructured concisely by four elementary functions, the surface wave pole extraction is very effective in terms of a function related to the generalized reflection coefficients. Meanwhile, the number of required two-level discrete complex image method (DCIM) with high-order Sommerfeld identities is also minimized under this scheme. The numerical results show that this approach is more efficient and accurate for the electromagnetic scattering by, and radiation in the presence of, electrically large three dimensional (3-D) objects in multilayered media. Although lossless examples are considered in the numerical examples, the method presented here works also for the lossy materials where the poles move away from the real axis in the complex plane.

Application of Euler–Maclaurin sum formula to obtain an approximate closed-form Green's function for a two-dimensional acoustical space

The Green's function solution for the acoustic wave equation in a two-dimensional rectangular space is expressed as an infinite series of terms based on the cross modes of the duct. An approximate closed-form solution is obtained by applying the Euler–Maclaurin sum formula. The procedure provides a closed-form expression in both the space–time and Laplace domains along with an upper bound for a remainder. Plane wave and higher-order waves components are identified. A numerical example for an exponential input gives comparisons of the transient response for the approximate closed form and series Green's function solutions. The time response and analytical transfer function frequency spectrum of the series and Euler–Maclaurin closed-form Green's function are computed. Lastly, the approximate closed-form Green's transfer function expression is used in model-based designed feedforward and feedback control schemes to reduce peaks in the frequency response and provide system damping.

Green's function for a harmonic acoustic point source within seawater overlying a saturated poroelastic seabed

In this paper, Green's function for an acoustic source within half-space seawater overlying a porous seabed is established. The half-space seawater is described using the Helmholtz equation, while the seabed is considered as a saturated porous medium and characterized by Biot's theory. Green's function of the half-space seawater has two components: the principal and complementary parts. The principal part of Green's function is the solution for a conventional acoustic point source in an infinite medium, while the complementary part corresponds to the general solution of the Helmholtz equation for the half-space. Green's function of the half-space seabed is obtained by solving Biot's dynamic equations via the Hankel transform method. Using the continuity conditions at the interface between the seawater and the seabed, the closed-form Green's functions for the seawater and the seabed in the frequency–wavenumber domain are determined. The frequency domain Green's function is recovered by the inversion of the Hankel transform. A parametric study is carried out for Green's function in the frequency domain, and a time domain example is presented in terms of frequency domain Green's function. Numerical results show that the permeability of the porous seabed has very little influence on the response of the seawater, while Biot's modulus has a pronounced influence on the wave field.

A Robust Method for the Computation of Green's Functions in Stratified Media

Closed-form Green's functions for unbounded planar stratified media are derived in terms of cylindrical and spherical waves. The methodology is based on a two-level approximation of the spectral-domain representation of Green's functions. This robust, efficient, and fully numerical approach does not call for an analytical extraction of ldquoproblematicrdquo behaviors, such as the quasi-static terms and the surface wave poles, prior to the spectrum fitting. Instead, the spatial-domain Green's functions derived in this paper provide an accurate description of both the near-field singularity in the vicinity of the source and the far-field dominant behavior of the surface waves.

Application of Total Least Squares to the Derivation of Closed-Form Green's Functions for Planar Layered Media

A new technique is presented for the numerical derivation of closed-form expressions of spatial-domain Green's functions for multilayered media. In the new technique, the spectral-domain Green's functions are approximated by an asymptotic term plus a ratio of two polynomials, the coefficients of these two polynomials being determined via the method of total least squares. The approximation makes it possible to obtain closed-form expressions of the spatial-domain Green's functions consisting of a term containing the near-field singularities plus a finite sum of Hankel functions. A judicious choice of the coefficients of the spectral-domain polynomials prevents the Hankel functions from introducing nonphysical singularities as the horizontal separation between source and field points goes to zero. The new numerical technique requires very few computational resources, and it has the merit of providing single closed-form approximations for the Green's functions that are accurate both in the near and far fields. A very good agreement has been found when comparing the results obtained with the new technique with those obtained via a numerically intensive computation of Sommerfeld integrals

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

A Closed-Form Expression of the Electromagnetic Tensor Green's Functions for a Homogeneous TI-Anisotropic Medium

In this letter, the electromagnetic fields from electric and magnetic current source distributions located in an unbounded homogeneous transverse isotropic (TI)-anisotropic medium are determined. The expressions provide the closed-form tensor Green's functions needed in any integral equation formulation of the forward as well as the inverse scattering problems associated with scatterers embedded in a homogeneous TI-anisotropic medium

Alternative derivations for the Poisson integral formula

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson integral formula are also derived. Two and three-dimensional cases are considered. Also, interior and exterior problems are both solved. Even though the image concept is required, the location of image point can be determined straightforward through the degenerate kernels instead of the method of reciprocal radii.

A direct discrete complex image method from the closed-form Green's functions in multilayered media

Sommerfeld integration is introduced to calculate the spatial-domain Green's functions (GF) for the method of moments in multilayered media. To avoid time-consuming numerical integration, the discrete complex image method (DCIM) was introduced by approximating the spectral-domain GF by a sum of exponentials. However, traditional DCIM is not accurate in the far- and/or near-field region. Quasi-static and surface-wave terms need to be extracted before the approximation and it is complicated to extract the surface-wave terms. In this paper, some features of the matrix pencil method (MPM) are clarified. A new direct DCIM without any quasi-static and surface-wave extraction is introduced. Instead of avoiding large variations of the spectral kernel, we introduce a novel path to include more variation before we apply the MPM. The spatial-domain GF obtained by the new DCIM is accurate both in the near- and far-field regions. The CPU time used to perform the new DCIM is less than 1 s for computing the fields with a horizontal source-field separation from 1.6/spl times/10/sup -4//spl lambda/ to 16/spl lambda/. The new DCIM can be even accurate up to 160/spl lambda/ provided the variation of the spectral kernel is large enough and we have accounted for a sufficient number of complex images.

An automated and efficient substrate noise analysis tool

This paper presents a methodology for the efficient modeling of substrate noise coupling. A closed-form Green's function for nonuniformly doped substrates is employed with the correct singular characteristics. A novel global surface impedance matrix scheme is introduced to efficiently model nonuniformly doped wells, channel stop implants, and buried layers. Layout, device, and netlist extractors are developed so as to integrate our boundary element method (BEM) solver with commercial layout and circuit simulation tools.

Green's function for multilayer arbitrarily biased anisotropic structures-application to phase shifters, transducers, and magnetization angle effect

This paper presents the closed-form Green's function for an arbitrarily biased magnetically anisotropic slab. The Green's function formulated using the transmission matrix is compared with the previously published data for two special cases: the transversely and normally biased ferrite slab. An excellent agreement is achieved. Three microwave devices are investigated using one numerical model that utilizes the derived Green's function by changing the magnetization angle. In the normal magnetization case, the ferrite substrate supports the edge-mode isolators. In the transversal magnetization case, the same structure supports the phase shifters. In the longitudinal magnetization case, the same structure supports magnetic surface wave transducers. The propagation direction in the three cases is kept in the longitudinal direction. A good agreement in all cases with previously published results is achieved. Here, we show only our analysis and results for the phase shifters and transducers, since edge isolator analysis was presented by Elshafiey et al. in 1996. In addition, the effect of the deviation of the magnetization direction from the assumed one is studied, and the improvement of the predicted results compared with experimental ones is shown.

Clarification of issues on the closed-form Green's functions in stratified media

The closed-form Green's functions (CFGF), derived for the vector and scalar potentials in planar multilayer media, have been revisited to clarify some issues and misunderstandings on the derivation of these Green's functions. In addition, the range of validity of these Green's functions is assessed with and without explicit evaluation of the surface wave contributions. As it is well-known, the derivation of the CFGF begins with the approximation of the spectral-domain Green's functions by complex exponentials, and continues with applying the Sommerfeld identity to cast these approximated spectral-domain Green's functions into the space domain in closed forms. Questions and misunderstandings of this derivation, which have mainly originated from the approximation process of the spectral-domain Green's functions in terms of complex exponentials, can be categorized and discussed under the topics of: 1) branch-point contributions; 2) surface wave pole contributions; and 3) the accuracy of the obtained CFGF. When these issues are clarified, the region of validity of the CFGF so obtained may be defined better. Therefore, in this paper, these issues will be addressed first, and then their origins and possible remedies will be provided with solid analysis and numerical demonstrations.

Closed-Form Green's Functions for Finite Grounded Dielectric Substrate

The approximate closed-form Green's functions of the vector and scalar potentials in the spatial domain are derived for a horizontal electric dipole placed over a finite grounded dielectric medium. The effects of the discontinuity at the edges are considered by including the surface wave reflections from the edges, which are obtained as a function of the incident angle by using the edge admittance concept. The closed-form expressions of the reflection coefficients are then derived by means of the generalized pencil of functions method. The resulting Green's functions are the sum of the contributions of discrete complex images, the incident, and the reflected surface wave contributions.

A simple approach to closed-form Green's functions of microstrip structures

The spectral-domain Green's functions for open microstrip structures are expressed in terms of pole-residue expressions by a vector-fitting technique after extracting the quasi-dynamic images. After the inverse Hankel transformation, the spatial Green's function takes the form of summing up a series of Hankel functions plus the quasi-dynamic exponential terms. This simple approach avoids the difficulty of manually extracting the surface-wave poles required in the traditional discrete complex image method (DCIM), while retaining the high accuracy of the closed-form expressions in the near-field region. The numerical results for the Green's functions of one-layer and two-layer structures verify the validity and efficiency of this approach. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 435–438, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20846

A robust approach for derivation of new class of closed‐form Green's functions for microstrip structures

AbstractBased upon three‐level approximation and the steepest descent path (SDP) method, we consider a robust approach for the derivation of a new class of closed‐form Green's functions for microstrip structures. Some numerical results are presented for the validation of the present method. It is observed that this method can provide a more accurate evaluation of the spatial Green's functions than the previous method, even without the advance investigation of the spectral functions, over a wide frequency range. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 235–238, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20428