Solution to the interconnected wire-scatterer problem

In this paper, it is attempted to approach a fast efficient algorithm for solving the famous Hallen's and Pocklington's integral equations, regarding the current distribution on a finite-length linear thin wire antenna. Here, the conventional moment method in conjunction with wavelet basis functions was used to obtain the current distribution of the antenna. The aim of this work is first to introduce the application of wavelet in electromagnetic scattering, secondly a comparison of the two method of analysis the thin wire antenna. By using the wavelet expansion, wavelets as basis and testing functions, a sparse matrix is generated from the previous moment method dense matrix. A sparsely filled matrix is easier to store and invert. The result extracted from Pocklington's integral equation gives better convergence at the feeding point, though it takes more time to be computed because of the complexity in Pocklington's equation. Results are compared to the previous work done and published, excellent results are obtained.

Several of the integral equations in everyday use for electromagnetic computations are "not well-posed" (e.g. they are ill-posed). In most other fields, solutions occur, but with arbitrarily large errors that cause serious practical problems. On the other hand, the specific ill-posed equations used in electromagnetics seem to work just fine in practice. G.C. Hsiao and R.E. Kleinman (see IEEE Trans. Antennas and Propag., vol.45, no.3, p.316-28, 1997) considered the ill-posed problem of the electric field integral equation (EFIE). It has been suggested that Pocklington's integral equation is even less well behaved. Our approach is two fold. We re-examine a paper (Canning, F.X. and Scholl, J.F., IEEE Trans. Antennas and Propag., vol.44, no.9, p.1239-46, 1996) which shows how wavelets can be used to improve the condition number (or stability) of the EFIE. We show how these results are equivalent to the results of Hsiao and Kleinman. These wavelet results provide a particularly clear physical picture of what is happening. The stability of the EFIE for its intended purpose becomes clear, in spite of its ill-posed nature. Second, we extend the wavelet related results of Canning and Scholl to Pocklington's integral equation, and explicate its stability properties.

In this paper we analyze the numerical aspects of the various methods that have been utilized to analyze thin wire antennas. First, we derive the properties of the operators for Pocklington's and Hallen's integral equation. On the basis of these properties, we discuss the various iterative methods used to find current distribution on thin wire structures. An attempt has been made to resolve the question of numerical stability associated with various entire domain and subdomain expansion functions in Galerkin's method. It has been shown that the sequence of solutions generated by the iterative methods monotonically approaches the exact solution provided the excitations chosen for these problems are in the range of the operator. Such a statement may not hold for Galerkin's methods if the inverse operator is unbounded. Moreover, if the excitation function is not in the range of the operator, then the sequence of solutions forms an asymptotic series. Examples have been presented to illustrate this point.

A hybrid method using Method of Moment (MoM) and Transmission Matrix (TM) is first time applied in the analysis of log-periodic dipole array antennas (LPDA) in this paper. It utilizes the advantages of both MoM and TM. And this method doesn't need to construct an extended matrix which usually has to be used in other methods. Further more, it has very clear physical meaning. In order to solve the Pocklington's integral equation, an iteration technique is adopted. The effectiveness is demonstrated by calculation the radiation pattern, input impedance, current and voltage distribution of LPDA. The results using this method agree very well with the results of other references.

It is shown that the equation for the current flowing on a cylindrical antenna-Pocklington's integral equation-can be put into 'a standard form' of dual integral equations. A general method for solving such dual integral equations is discussed. It consists of a transformation into a single Fredholm integral equation of the second kind with continuous kernel, and then into a linear system of algebraic equations by using the weighted residual method. Plots of the current along the antenna and of the input admittance are finally given.

The Numerical Solution of Integral Equations on the Half-Line

A coaxial line excited monopole probe has attracted many researchers for in situ permittivity measurement of materials. In most of these cases, a delta gap excitation is assumed for the probe in relating the measured input impedance to the permittivity of the material. Since an open-ended coaxial line is more accurately represented by a magnetic frill generator, we employed this model of excitation for the present study. This paper reports the results obtained by solving the Pocklington's integral equation numerically using a magnetic frill excitation and piecewise sinusoidal Galerkin method. A comparison of these results with the available experimental and theoretical data is presented and some limitations of this approach are discussed.

There is a wide class of antennas that can be presented as a set of thin-wire perfect conductors. Current distribution, radiation and scattering problems are usually solved by the method of integral equations (IE). While using antennas, there is the possibility of the formation of different kinds of coating, for instance, ice, that can be approximated by an ideal dielectric. In that case, a model of a system of dielectric-coated thin-wire conductors is used. Numerical analysis of an integral equation based on Pocklington's equation for dielectric-coated thin-wire conductors is given. To obtain solution stability, recommendations are offered for choosing a segment length depending on dielectric layer permittivity and thickness.

This paper presents a basic theory for modeling printed circuit antennas for millimeter wave applications. Printed circuit dipoles are considered and the Pocklington integral equation is solved for the antenna current distribution. Substrates of arbitrary thickness and dielectric constant can be taken into account in this model and the precise contribution of surface wave modes to antenna parameters is discussed.

The distribution of induced electric current in thin metallic shields with holes is determined by using the current sheet integral equation. A polyhedral surface with triangular elements is chosen to model the shield surface. The surface density of current is taken to be uniform over each element and is described by using specialized vector functions whose coefficients are scalar quantities associated with the interior nodes and with the holes in the shield. A Galerkin method is used to solve the integral equation. Numerical experiments show that the proposed method is more efficient than other methods applied so far to thin shields with holes. Once the current distribution is computed, the magnetic induction, the shielding efficiency and the forces exerted on the shield can rapidly be evaluated.

A numerical method to investigate the diffraction of a surface wave at the edges of a truncated dielectric structure is presented. The method is based on the equivalence principle. The current distribution is found from the relevant integral equation obtained from the condition of continuity of the tangential magnetic field. The diffraction pattern and the reflection coefficient of the surface wave are computed after the solution of the integral equation by the Galerkin method. The numerical results show the influence of the substrate's permittivity and thickness on the radiation parameters.

Multiscalets in the multiwavelet family are used as the basis and testing functions in Galerkin's method. Since the multiscalets are orthogonal to their translations under the Sobolev inner product, the resulting Galerkin's method behaves like a collocation method but possesses the ability of derivative tracking for unknown functions in solving integral equations. The former makes the method simple in implementation and the latter allows to use coarse meshes in discretization. These robust features have been demonstrated in solving two‐dimensional (2D) electromagnetic (EM) problems, but have not been exploited in three‐dimensional (3D) scenarios. For 3D problems, the unknown functions in the integral equations are dependent on two coordinate variables. In order to preserve the use of coarse meshes for 3D cases, we realize the omnidirectional derivative tracking by tracking the directional derivatives along two orthogonal directions, or equivalently tracking the gradient. This process yields a nonsquare matrix equation and we use the least‐squares method (LSM) to solve it. Numerical examples show that the multiscalet‐based Galerkin's method is also robust in solving for 3D EM integral equations with a minor cost increase from LSM. Copyright © 2007 John Wiley & Sons, Ltd.

1. The essence of the problem. Thin electric dipoles have been widely applied both as antennas in their own right and as components of complex antenna systems. As a rule, the design of a thin electric dipole is based on solving an integral equation describing the current distribution in the dipole. By now, the Pocklington and Harrington integrodifferential equations and also the Hullen integral equation are known [1‐4]. The kernels of these equations are of the Fredholm type, and the form of these kernels stems from using the integral relations for the vector electrodynamic potential. The kernel in the formula for the latter is the Green’s function for an inhomogeneous Helmholtz equation written in a spherical coordinate system. Most often, integral equations are solved by the method of moments [1‐3]. In essence, this method reduces to transforming of the aforementioned equations into a system of linear algebraic equations for N unknowns. Typically, the latter are the coefficients in the expansion for current in a certain appropriate system of basis functions. Depending on the choice of the basis and weighting functions, there exists a fairly great diversity of specific implementations for the integral-equation method: the Galerkin method, sewing at discrete points, matching the responses, etc. In our opinion, the main drawback of such an approach consists in the fact that the aforementioned integral equations with the Fredholm kernel belong to the integral equations of the first kind; finding a solution to these equations is an ill-posed problem [5]. As a result, the question of how to check whether the solution is proper and establish its adequacy to the physical problem under consideration remains open. 2. Integral equation of the first kind. In this paper,

Minimum continuity requirements for basis functions used with the pocklington integral equation

In papers [1], [2], the authors extend the so-called effective current method to the Hallen's and Pocklington's equations with the finite-gap generator (FGG). The indicated method considered earlier in [3] with a reference to [4] is associated with application of the method of auxiliary sources to the analysis of cylindrical dipole antennas [5]-[7]. The purpose of these comments is to draw the authors' and readers' attention to the existence ofpaper [8] where the Pocklington's equation with the FGG is solved for the axial current considered there to be auxiliary. The equation is solved by the Galerkin method using the piecewise sinusoidal basis and testing functions. The auxiliary axial current is used further for calculation of the surface current (which is referred to as the effective current in [1]-[4]), and the surface current itself is used for calculation of the input impedance. So, in some respects papers [1]-[7] just repeat the main idea and technique of paper [8], although of course papers [1]-[7] also contain a lot of their own new, interesting, and useful analytical and numerical results.