Analysis of dipole antennas using moment methods and haar wavelet

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.

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.

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 new method for treating the interconnections of arbitrarily located straight thin wires irradiated by a linearlypolarised plane wave is presented. The resulting Pocklington's integral equation is solved by the Bubnov-Galerkin projective method. The current distribution along the wires is approximated by polynomials with complex coefficients. This formulation is valid for wires forming any angle, with special provision made for the collinear case. Also, a new formulation for finding the self and mutual impedances of the wire configuration has been obtained. Results show excellent agreement with those of other methods.

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 complex natural frequencies for the thin-wire antenna or scatterer are computed from Hallen's integral equation and are compared with those obtained from Pocklington's equation. The good agreement between the two methods shows that the concepts of the singularity-expansion method are independent of the type of integral equation employed in the formulation of the antenna or scattering problem.

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.

In this paper, the semi-orthogonal compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire electric field integral equation (EFIE) in frequency domain. The method of moments (MoM) is used via the Galerkin procedure. Conventional MoM directly applied to the EFIE, leads to dense matrix which often becomes computationally intractable when large-scale problems are approached. To overcome these difficulties, wavelets can be used as a basis set so obtaining the generation of a sparse matrix; this is due to the local supports and the vanishing moments properties of the wavelets. In the paper, this technique is applied to analyze electromagnetic transients in a lightning protection systems schematized as a thin-wire structure. The study is carried out in frequency domain; a discrete fast Fourier transform algorithm can be used to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method. Results for linear spline wavelets along with their comparison with conventional MoM that uses triangular basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity.

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.

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.

A theoretical and numerical analysis of the wave propagation along a long thin wire antenna is presented. The wire antenna is assumed to be placed in a horizontally layered medium, where each layer is characterized by an individual conductivity. The current distribution along the wire antenna is derived as the solution of an electric field integral equation which is solved by the method of moments. Numerical results are presented and discussed.

Electromagnetic (EM) telemetry is one of the key research subjects that has increased application in transmitting real-time data between the borehole and surface during drilling. There are several methods that can be used to model an EM telemetry system, including the method of moments (MoM) and the electric field integral equation method. We have developed a pulse basis function for the MoM, which assists in the analytical estimation of thin wire kernel integrations and avoids singular problems when simulating current distributions. The scattering field is solved using the current distribution. We examine the method using a simple example of vertical drilling. In the model, a vertically placed drill string is driven through a layered earth model. The drill string represents a long, thin wire antenna with a source located near the drill bit. The validity of the method is confirmed by comparing our results to the analytical solution in a half-space model as well as the current distributions along the drill string derived from the proposed method with those derived from a commercial finite-element solver, COMSOL-Multiphysics. Furthermore, the efficiency of this method is demonstrated by comparing its computational costs with those of COMSOL. The simulated EM telemetry in a known geologic setting is then compared with a field data example. These comparisons show that the proposed method is accurate and efficient. Moreover, we develop insights into how the parameters needed for calculations alter current distributions along the drill string and the signal received at the surface. The proposed method is therefore useful for obtaining the optimal design parameters for EM telemetry systems.

A parallelized version of multilevel Characteristic Basis Function method (ML-CBFM) is applied in this paper to the problem of electromagnetic scattering in the VHF and UHF bands from trees in large forest areas that are modeled as three dimensional, finite-length, dielectric cylinders. The use of this method enables us to achieve a significant reduction in terms of computing time and memory consumption, as compared to the conventional Method of Moments (MoM), and this, in turn, enables us to handle very large forest areas, well beyond the reach of conventional MoM. Furthermore, we take advantage of the fact that the ML-CBFM algorithm is naturally parallelizable, which provides us an added advantage over the conventional MoM. The ML-CBFM results are shown to be in good agreement with those obtained via the conventional MoM, which confirms the fact that the proposed method is not only computationally efficient, but is accurate as well.

This work investigates numerically two different methods of moments applied to Hermite derived probability distribution model and variations of Weibull distribution fitted to the short-term time series peaks sample of stochastic response parameters of a simplified jack-up platform model which represents a source of high non-Gaussian responses. The main focus of the work is to compare the results of short-term extreme response statistics obtained by the so-called linear method of moments (L-moments) and the conventional method of moments using either Hermite or Weibull models as the distribution model for the peaks. A simplified mass-spring system representing a three-legged jack-up platform is initially employed in order to observe directly impacts of the linear method of moments (L-moments) in extreme analysis results. Afterward, the stochastic response of the three-legged jack-up platform is analyzed by means of 3D finite element model. Bias and statistical uncertainty in the estimated extreme statistics parameters are computed considering as the “theoretical” estimates those evaluated by fitting a Gumbel to a sample of episodical extreme values obtained from distinct short-term realizations (or simulations). Results show that the variability of the extreme results, as a function of the simulation length, determined by the linear method of moments (L-moments) is smaller than their corresponding ones derived from the conventional method of moments and the biases are more or less the same.

Efficient electromagnetic (EM) analysis for curved open thin-wire structures is developed by solving integral equations with intervallic wavelets in the method of moments (MoM). The electrical field integral equation for curved thin-wire structures is formulated based on the generalized Pocklington integral equation and the unknown current is expanded using the intervallic Coifman father wavelets as basis functions. In the current representation, an appropriate geometric map is established between thin-wire segments and bounded intervallic wavelets to match different domains. Since the Coifman wavelets (Coiflets) possess a unique feature, i.e. the vanishing moments which can yield a Dirac-δ-like sampling property, they are used to construct the intervallic wavelets. Also, the corresponding edge basis functions are built to represent the unknown current in the edge intervals for open structures. By using the intervallic wavelets in the MoM and performing a fast wavelet transform, sparse impedance matrices can be obtained and the drawbacks of dense matrices are overcome. Numerical examples for analyzing the scattering or radiation properties in different thin-wire structures are presented to demonstrate the robustness of the intervallic wavelets.