Wavelets explain the stability of Pocklington's equation

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.

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.

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.

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 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.

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.

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 this paper we demonstrate that residual error estimators can be used to detect internal resonance problems arising from the electric and magnetic field integral equations for perfectly conducting targets. In addition, a new approach to constructing a “combined” field equation is introduced. When solved, the resulting over-determined system produces a different type of residual that yields insight into the solution accuracy.

Accurate electromagnetic (EM) modeling for packaging structures with lossy interconnects requires to consider the finite conductivity of the conducting interconnects. In the integral equation approach, the conducting loss is usually accounted for through an approximate surface impedance when the skin depth of current is small, but this approximation may not be valid for large skin depth caused by low frequencies or small conductivities. In this paper, we treat the lossy interconnects as dielectric-like media and use the two-region electric field integral equations (EFIEs) to characterize the EM feature of the interconnects. The EFIEs are solved with the method of moments in which the Rao-Wilton-Glisson (RWG) basis function and dual basis function (DBF) are used to represent the electric and magnetic current densities, respectively. The joint use of RWG and DBF can remove the need of wisely formulating integral equations and selecting testing schemes and also alleviate the low-frequency effect in a wideband modeling for multiscale interconnect structures. Numerical examples are presented to demonstrate the approach and its robustness has been verified.

In this paper, shielding effectiveness (SE) of a perforated enclosure with imperfectly conducting walls is evaluated. To this end, first, an accurate numerical technique based on method of Moments (MoM) is presented. In this method, lossy metallic walls of the enclosure are replaced by equivalent electric surface current sources. Then, the impedance boundary condition on the imperfectly conducting surfaces is applied and an electric field integral equation is extracted. At the end, the integral equation is solved numerically by Galerkin method. In addition to the mentioned numerical method, an extremely fast analytical technique based on transmission line model(TLM) is proposed which is able to predict the SE with high level of accuracy over a large frequency bandwidth just in a few seconds. For validation of both methods, other commercial softwares (FEKO and CST) are employed and several enclosures with different conductivities are studied. Lossy MoM method shows accurate results for conductivities down to 10S/m, while efficient TLM method proves its accuracy for conductivities down to 250S/m.

The electric field integral equation (EFIE) is one of simpl formulations of boundary integral equations for Maxwell's equations. It is widely known that the discretised EFIE is ill-conditioned, namely iteration methods to solve the discretised EFIE require a lot of iteration numbers, which results in much computational time. One of effective solutions to this problem is the Caldern preconditioning The Caldern preconditioning constructs a preconditioner based on the so-called Caldern formula, which implies that the square of the electric field integral operator (EFIO) is well-conditioned. Unfortunately simple discretisation of the squared EFIO is known to break down. Hence, in many previous studies of the Caldern preconditioning for the EFIE, the dual basis function is used in addition to the standard basis function, such as the Buffa-Christiansen (BC) basis function Although the Caldern preconditioning can be successfully implemented with the BC and RWG functions, discretisation of the EFIO with the BC basis function takes much more computational time than that with the RWG function. On the other hand our group has found that the dual basis function for the roof-top basis function defined on a quadrilateral mesh is also the roof-top basis function

Abstract. The linear equation systems which arise from the discretization of surface integral equations are conveniently solved with iterative methods because of the possibility to employ fast integral methods like the Multilevel Fast Multipole Method. However, especially integral equations of the first kind often lead to very ill-conditioned systems, which require the usage of effective preconditioners. In this paper, the regularization property of near-zone preconditioning operators on the Electric Field Integral Equation is demonstrated and investigated for problems of different size. Furthermore, comparisons are drawn to second-kind integral equations such as the Combined Field Integral Equation.

Characteristic mode (CM) analysis of multi-scale and complex targets has attracted much attention in computational electromagnetics. However, it is hard to calculate the multi-scale problems in terms of desired accuracy and stability by using conventional methods. What is more, it is a challenging work to generate a good quality mesh for the multi-scale structures including very fine features. In this paper, we present Discontinuous Galerkin (DG) surface integral equations-based theory to analyze the CMs. The multi-scale objects are firstly divided into several subdomains, and we generate the meshes of every parts independently. HRWG basis functions are adopted in our work and the characteristic currents are expanded by HRWG basis functions. In many applications, only characteristic currents with small eigenvalues are desired because they are corresponding to the modes that radiate efficiently and have large modal coefficients. Special singularity extraction schemes are chosen to treat the infinite term in the resulting integral equations since HRWG basis functions are not divergence-conforming and there is divergence operator in electric field integral equation (EFIF) and magnetic field integral equation (MFIE). In DG method, they are no requirements for normal continuity of currents across the contour of between every two subdomains, hence, the proposed method can handle both non-conformal mesh and conformal mesh. Numerical results are presented to validate the propose formulation. This work may offer great flexibility to CM analysis of multi-scale and complex structures.

The integral equation (IE) method is one of the efficient approaches for solving electromagnetic problems, where dyadic Green’s function (DGF) plays an important role as the Kernel of the integrals. In general, a layered medium with planar, cylindrical, or spherical geometry can be used to model different biomedical media such as human skin, body, or head. Therefore, in this chapter, different approaches for the derivation of Green’s function for these structures will be introduced. Due to the recent great interest in two-dimensional (2D) materials, the chapter will also discuss the generalization of the technique to the same structures with interfaces made of isotropic and anisotropic surface impedances. To this end, general formulas for the dyadic Green’s function of the aforementioned structures are extracted based on the scattering superposition method by considering field and source points in the arbitrary locations. Apparently, by setting the surface conductivity of the interfaces equal to zero, the formulations will turn into the associated problem with dielectric boundaries. This section will also aid in the design of various biomedical devices such as sensors, cloaks, and spectrometers, with improved functionality. Finally, the Purcell factor of a dipole emitter in the presence of the layered structures will be discussed as another biomedical application of the formulation.