M.I. Aksun

A robust approach for the derivation of closed-form Green's functions

Spatial-domain Greens functions for multilayer, planar geometries are cast into closed forms with two-level approximation of the spectral-domain representation of the Greens functions. This approach is very robust and much faster compared to the original one-level approximation. Moreover, it does not require the investigation of the spectral-domain behavior of the Greens functions in advance to decide on the parameters of the approximation technique, and it can be applied to any component of the dyadic Greens function with the same ease.

Closed-form Green's functions for general sources and stratified media

The closed-form Greens functions of the vector and scalar potentials in the spatial domain are presented for the sources of horizontal electric, magnetic, and vertical electric, magnetic dipoles embedded in general, multilayer, planar media. First, the spectral domain Greens functions in an arbitrary layer are derived analytically from the Greens functions in the source layer by using a recursive algorithm. Then, the spatial domain Greens functions are obtained by adding the contributions of the direct terms, surface waves, and complex images approximated by the Generalized Pencil of Functions Method (GPOF). In the derivations, the main emphasis is to put these closed-form representations in a suitable form for the solution of the mixed potential integral equation (MPIE) by the method of moments in a general three-dimensional geometry. The contributions of this paper are: 1) providing the complete set of closed-form Greens functions in spectral and spatial domains for general stratified media; 2) using the GPOF method, which is more robust and less noise sensitive, in the derivation of the closed-form spatial domain Greens functions; and 3) casting the closed-form Greens functions in a form to provide efficient applications of the method of moments. >

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

The derivation of the closed-form spatial domain Greens 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 Greens functions for printed circuits are typically expressed as Sommerfeld integrals, which are inverse Hankel transforms of the corresponding spectral domain Greens functions and are time-consuming to evaluate. Closed-form representations of these Greens 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 Greens functions in terms of complex exponentials by using the least square Pronys method. >

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

The closed-form Greens 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 Greens functions. In addition, the range of validity of these Greens 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 Greens functions by complex exponentials, and continues with applying the Sommerfeld identity to cast these approximated spectral-domain Greens 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 Greens 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.

Comparative study of acceleration techniques for integrals and series in electromagnetic problems

Most electromagnetic problems can be reduced to either integrating oscillatory integrals or summing up complex series. However, limits of the integrals and the series usually extend to infinity, and, in addition, they may be slowly convergent. Therefore numerically efficient techniques for evaluating the integrals or for calculating the sum of an infinite series have to be used to make the numerical solution feasible and attractive. In the literature there are a wide range of applications of such methods to various EM problems. In this paper our main aim is to critically examine the popular series transformation (acceleration) methods which are used in electromagnetic problems and to introduce a new acceleration technique for integrals involving Bessel functions and sinusoidal functions.

Closed-Form Green's Functions in Planar Layered Media for All Ranges and Materials

An important extension of the two-level discrete complex image method is proposed to eliminate any concerns on and shortcomings of the approximations of the spatial-domain Greens functions in closed form in planar multilayered media. The proposed approach has been devised to account for the possible wave constituents of a dipole in layered media, such as spherical, cylindrical, and lateral waves, with the aim of obtaining accurate closed-form approximations of Greens functions over all distances from the source. This goal has been achieved by judiciously introducing an additional level into the two-level approach to pick up the contributions of lateral waves in the spatial domain. As a result, three different three-level algorithms have been proposed, investigated, and shown that they work properly over all ranges of distances from the source. In addition to the accuracy of the results at all distances, these approaches also proved to be robust and computationally efficient as compared to the previous algorithms, which can be attributed to the fact that the sampling of the spectral-domain Greens functions in the proposed approaches gives proper emphasis to the associated singularities of the wave types in the spectral domain. However, the judicious choices of the sampling paths may not be enough to get accurate results from the approximations unless the approximating functions in the spectral domain can provide similar wave natures in the spatial domain. To address this issue, the proposed algorithms employ two different approximations; the rational function fitting methods to capture the cylindrical waves (surface waves), and exponential fitting methods to capture both spherical and lateral waves. It is shown and numerically verified that a linear combination of exponential functions in the spectral domain represent the lateral waves at the interface of the involved layers.

On slot-coupled microstrip antennas and their applications to CP operation-theory and experiment

A simple theory based on the cavity model is developed to analyze microstrip antennas excited by a slot in the ground plane. By using an equivalent magnetic current source at the feed, the electric field under the patch is obtained in terms of a set of cavity modes. In particular, the loci of the slot feed location for achieving the circular polarization and the input impedance are computed and found to be in excellent agreement with the experimentally measured results. Simple but surprisingly accurate formulas for slot-fed circularly polarized microstrip antennas are derived and compared with those for probe-fed counterparts. >

Efficient use of closed-form Green's functions for the analysis of planar geometries with vertical connections

An efficient and rigorous method for the analysis of planarly layered geometries with vertical metallizations is presented. The method is based on the use of the closed-form spatial-domain Greens functions in conjunction with the method of moments (MoM). It has already been demonstrated that the introduction of the closed-form Greens functions into the MoM formulation results in significant computational improvement for the analysis of planar geometries. However, in cases of vertical metallizations, such as shorting pins, via holes, etc., there are some difficulties in incorporating the closed-form Greens functions into the MoM formulation. In this paper, these difficulties are discussed and their remedies are proposed. The proposed approach is compared to traditional approaches from a theoretical point of view, and the numerical implementation is demonstrated through some examples. The results are also compared to those obtained from the commercial software em by SONNET.

Analytical evaluation of the MoM matrix elements

Derivation of the closed-form Greens functions has eliminated the computationally expensive evaluation of the Sommerfeld integrals to obtain the Greens functions in the spatial domain. Therefore, using the closed-form Greens functions in conjunction with the method of moments (MoM) has improved the computational efficiency of the technique significantly. Further improvement can be achieved on the calculation of the matrix elements involved in the MoM, usually double integrals for planar geometries, by eliminating the numerical integration. The contribution of this paper is to present the analytical evaluation of the matrix elements when the closed-form Greens functions are used, and to demonstrate the amount of improvement in computation time.

Choices of expansion and testing functions for the method of moments applied to a class of electromagnetic problems

It is well known that the choice of expansion and testing functions plays an important role in determining the rate of convergence of the integrals associated with the moment method matrix, and that an improper choice can lead to erroneous results. This convergence issue is critically examined, and to criteria for the choice of these expansion and testing functions are provided. The question of whether these functions need to satisfy the Holder condition is also investigated, and the convergence behavior of the integrals involved in the spatial- and spectral-domain moment method is discussed for some representative expansion and testing functions. >