The E-parallel polarization response of a two-dimensionalheterogeneous layer modeled by two thin sheets

The flexibility of the thin-sheet technique for modeling two-dimensional structures is extended by analyzing the response to E-parallel to the strike of the anomaly polarization, also known as the transverse electric (TE) mode, when a heterogeneous Earth is modeled by two thin sheets over a horizontally stratified half-space. The electromagnetic fields induced by the heterogeneous layer when a uniform plane wave is normally incident on the Earth are expressed as integral equations, which are then evaluated using the method of moments. The resulting matrix equations are solved by the Gauss-Seidel method, and the magnetotelluric impedance is calculated     

Electromagnetic induction in thin sheet conductivity anomalies at the surface of the earth

Lateral variations in the Earth's conductivity complicate considerably the calculation of the electromagnetic response of the Earth to an external inducing field which is uniform and horizontal. Although analytic solutions have been found for a few simple two-dimensional models in which the conductivity varies in one horizontal direction only, it is necessary, in general, to resort to numerical methods. If the conductivity variations of interest are confined to a surface layer it is often possible to represent the Earth mathematically as a uniform conducting half-space covered by an infinitely thin sheet of variable surface conductance. This simplification effectively reduces by one the number of dimensions over which the field equations need to be integrated numerically. It is shown that for a two-dimensional model the horizontal component of the electric field satisfies an integral equation on the surface of the thin sheet, which can be solved numerically for arbitrary sheet conductance. The accuracy of the numerical procedure is confined by applying it to E- and B-polarization induction in two adjacent half-sheets and then comparing the solution obtained with known analytic solutions of the same problem. In three-dimensions the two horizontal components of the surface electric field satisfy a coupled pair of double integral equations which can also be solved numerically for an arbitrarily varying conductance of the surface sheet.     

Electromagnetic scattering from perfectly conducting rough surfaces(a new full wave method)

A method that does not make use of the telegraphist's equations and takes into account the two-dimensional roughness of the surface from the start is developed. It is shown that the scattering coefficients obtained agree with those given in earlier work by E. Bahar (1973, 1987). The method is based on reducing the three-dimensional scattering problem to a two-dimensional problem by expanding each rectangular component of Maxwell's equations in terms of local basis functions along the perpendicular direction to the mean surface. The transformed two-dimensional field equations are solved using Fourier transforms. The full wave solutions are also compared with the first-order perturbation solutions, the Kirchhoff-type solutions, and integral equation results     

Surface electric fields and impedance matrix elements of stratifiedmedia

For the various geometrical configurations of waves in stratified media, we consider the important case when both source and field points are located on the same interface separating two different dielectric media. We denote this configuration as surface electric field case. In this paper, the electric fields are calculated numerically without using potentials. For the surface electric field case the integrand of the electric field grows with k_ρ^3/2 for large κ_ρ making the Sommerfeld integral singular. To calculate the surface electric fields in the spatial domain, we previously applied a technique of higher order asymptotic extraction. In the higher order asymptotic extraction, the higher order asymptotic parts were calculated analytically. The remainder, which has an integrand decays as κ_ρ^-3/2 was calculated numerically along the Sommerfeld contour path of integration. In this paper, we use a different extraction technique, the half-space extraction. After the half-space extraction, the integrand of the Sommerfeld integral of stratified media decays exponentially and the integral is calculated along the Sommerfeld integration path. The half-space extraction part is calculated by numerical integration along the vertical branch cuts. The surface electric fields for stratified media using half-space extraction and higher order asymptotic extraction are in good agreement. To validate the accuracy of the solution, we also compute the impedance matrix elements using surface electric fields, testing, and basis functions all in the spatial domain. The results are then compared with the results of the spectral domain method. The comparisons of the complex impedance matrix elements are tabulated and show that the difference is less than 2%     

Response of a source on top of a vertically stratified half-space

The solution of the response of a source on top of a horizontally stratified half-space is well-known. However, when the half-space is vertically stratified, the problem can only be solved with numerical methods like the finite element method. Here a semi-analytic approach to solve such a problem is presented. The three-dimensional variation of the problem is reduced to two-dimensional variation by Fourier transform in one coordinate variable. The remaining two-dimensional problem is solved by finding the eigensolution in each of the half-spaces. The eigensolutions of each region are found from the partial differential equation directly using the same basis set of expansion functions. This makes the calculation of the reflection and transmission operators very efficient. The reflection and transmission operators account for the mode-conversion, reflection, and transmission of the waves. With the reflection and transmission operators, the field everywhere can be calculated. The solution reduces to that of the Sommerfeld's half-space problem when the two half-spaces are homogeneous.     

Transient scattering from two-dimensional dielectric cylinders ofarbitrary shape

The problem of transient scattering by arbitrarily shaped two-dimensional dielectric cylinders is solved using the marching-on-in-time technique. The dielectric problem is approached via the surface equivalence principle. Two coupled integral equations are derived by enforcing the continuity of the electric and magnetic fields which are solved by using the method of moments. Numerical results are presented for two cross sections, viz. a circle and a square, and compared with inverse discrete Fourier transform (IDFT) techniques. In each case, good agreement is obtained with the IDFT solution     

Parameters Extraction of Millimeter Wave Circuits by Hybrid Vector Finite Element and Moment Method Combined with SOC Technique

This paper presents a hybrid method, which couples the vector finite element method (FEM) and method of moment (MOM) for analyzing the field and current distribution of the millimeter wave circuits. The FEM is applied to handle the interior region of dielectric bodies and MOM is used to solve surface integral equations. Then, These integral expressions are coupled into the FEM equations through the continuity of the tangential fields across the connection boundaries. Simultaneously, the short-open calibration (SOC) technique is used for predicting accurately the scattering parameters of the circuits. Numerical results are well compared with those published in the previous literatures.     

Finite element analysis of electromagnetic scattering from a cavity

A finite element method (FEM) is implemented to compute the radar cross section of a two-dimensional (2D) cavity embedded in an infinite ground plane. The method is based on the variational formulation which uses the Fourier transform to couple the fields outside the cavity and those inside the cavity; hence, the scattering problem can be reduced to a bounded domain. The convergence of the discrete finite element problem is analyzed. Numerical results are presented and compared with those obtained by the standard finite element-Green function method and by the 2D integral equation method.     

A hybrid finite-element method for axisymmetric waveguide-fed horns

A new method for finding radiation patterns and the reflection coefficients associated with an axisymmetric waveguide fed horn is presented. The approach is based on a hybrid finite element method (FEM) wherein the electromagnetic fields in the FEM region are coupled to the fields outside by two surface integral equations. Because of the local nature of the FEM, this formalism allows for the presence of inhomogeneities to be included in the problem domain. The matrix equation which results from the application of this method is shown to be complex-symmetric. Comparisons of calculated and measured data for two different horns show good agreement     

Numerical mode-matching method for the multiregion verticallystratified media [EM wave propagation]

The response of a source in the presence of an N-region, vertically stratified media is solved using the numerical mode-matching method. By treating the fields propagating in the direction parallel to the subboundaries of the stratified media in terms of the propagators, and by introducing the concepts of reflection operators, transmission operators, and generalized reflection operators, the two-dimensional problem is reduced to several one-dimensional problems, which are solved by the one-dimensional finite-element method (FEM). The one-dimensional method saves computer storage and computation time compared to the two-dimensional version. A formulation valid for a general N-region vertically stratified medium is derived. When there are only three regions, the results compare well with those in the literature. Some typical numerical results for N>3 are also shown     

Electromagnetic scattering from composite objects using a mixtureof exact and impedance-boundary conditions

Different surface integral equations for characterizing the electromagnetic scattering from a surface impedance object partially coated with dielectric materials are presented. The impedance boundary condition (IBC) is applied on the impedance surface and the exact boundary condition is applied on the dielectric surface. The resulting integral equations are solved for bodies of revolution using the method of moments. The numerical results are compared with the exact solution for a sphere. Other geometries are considered, and their results are verified by comparing results of the numerical solutions which were obtained using different formulations. The internal resonance problem is examined. It is found that the combined field integral equation (CFIE) can be used at any frequency and with any surface impedance     

An accurate and efficient analysis for transient plane wavesobliquely incident on a conductive half space (TE case)

A method which allows for the analytical evaluation of the inverse Laplace transform representations for a transient TE plane wave, obliquely incident on a conductive half-space, is discussed. We assume that the permittivity and conductivity of the dispersive half-space are independent of frequency. Starting with the equations for the transmitted wave in the Laplace domain, the corresponding time-domain expressions are first represented as inverse Laplace transforms. The transient fields are shown to consist of two canonical integrals f(β) and c(β). The canonical integrals, in turn, are solved analytically, thereby yielding solutions involving incomplete Lipschitz-Hankel integrals (ILHTs). The ILHIs are computed numerically using efficient convergent and asymptotic series expansions, thus enabling the efficient computation of the transient fields. The solutions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast Fourier transform (FFT) algorithms     

On the limits of applicability of an approximate formula for the surface impedance of a rectangular groove

The problem of determination of the equivalent surface impedance of a rectangular groove is considered. The solution is obtained by the integral equations method. The Krylov–Bogoliubov method is used to solve integral equations numerically. The obtained dependence of the impedance value on the groove depth is compared to the dependence calculated using an approximate formula for different groove widths. The limits of applicability of the approximate formula are determined.     

Numerical simulation of scattering from three-dimensional randomlyrough surfaces

Randomly rough surface patches in three dimensions are generated on the computer. The FD-TD method is used to compute scattering from surface patches by converting the Maxwell's equations into difference equations using a central difference approximation for the space and time derivatives. The volume of grids above the rough surface is divided into the total field and the scattered field regions. In between these two regions, obliquely incident waves are generated. To reduce computation, the volume of grids is chosen to be small, and a transformation is used to convert the scattered field into far zone fields for bistatic scattering coefficient calculations. Possible errors near the edge of the surface due to the use of a relatively small volume are suppressed by introducing a windowing function. Very good agreements are obtained between the results obtained by this method and those calculated by an integral equation method (IEM) for scattering from randomly rough perfectly conducting and dielectric surfaces     

Fringe integral equation method for a truncated grounded dielectricslab

The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TM_z polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes occurring at the truncation. The F-IEs are obtained by subtracting from the surface/surface integral equations pertinent to the truncated slab, an auxiliary set of equations obtained for the canonical problem of an infinite grounded slab illuminated by the same source. The F-IEs are solved by the method of moments by using a set of subdomain basis functions close to the truncation and semi-infinite domain basis functions far from it. These latter functions are properly shaped to reproduce the asymptotic behavior of the diffracted waves, which is obtained by physical inspection. The present solution is applied to the case of an electric line source located at the air-dielectric interface of the slab. Numerical results are compared with those calculated by a physical optics approach and by an alternative solution, in which the integral equation is constructed from the field continuity through an aperture orthogonal to the slab. The applications of the solution to an array of line currents are also presented and discussed     

Volume integral equations for analysis of dielectric branchingwaveguides

Novel forms of volume integral equations are developed for the exact treatment of wave propagation in two-dimensional dielectric branching waveguides. The integral equations can be obtained by considering the condition at a point far away from the junction section. An approximate solution by the Born approximation and a numerical solution by the moment method establish the validity of the new volume integral equations. The numerical results are discussed from the viewpoint of energy conservation and reciprocity. The solution is exact if sufficiently large computer memory and computational time are used. The method can be extended to problems of a more general nature (i.e. the incident TM mode), and complex configurations of branching waveguides. The basic idea is also applicable to techniques using boundary (surface) integral equations which are applicable to three-dimensional problems     

Equivalent Surface Impedance of an Infinite Periodic Array of Slot Impedance Loads with Dielectric Coating

The problem of plane-wave excitation of an infinite array of slot impedance loads with a dielectric layer is considered in order to determine the equivalent surface impedance. The solution of the problem is found by the method of integral equations, solved numerically by the Krylov–Bogolyubov method, which reduces the integral equation to a system of linear algebraic equations. The numerical results are presented in the form of dependences of the equivalent surface impedance on the geometric dimensions of the structure at fixed values of the period of the array, the angle of incidence, and the thickness of the dielectric layer. It is shown that the equivalent surface impedance can be controlled by varying the width of the slot and the strip conductor. The dependences obtained are compared with the characteristics of an array of slot impedance loads without a dielectric layer.     

Transient response of nonlinearly loaded wires in a two media configuration

The transient behavior of thin wires with nonlinear resistive load in the presence of a dielectric half-space is analyzed directly in the time domain. The nonlinear wire problem is formulated by the space-time Hallen integral equation. The effect of a two-media configuration is taken into account via the space-time reflection coefficient appearing within the Green function. The resulting integral equation is handled by the space-time boundary integral equation method. The transient response for the case of a thin wire isolated in free space computed by this direct time-domain approach is compared with results obtained by another method solved using data derived from frequency-domain analysis. Results for various other configurations are also presented.     

Electromagnetic scattering from arbitrary shaped conducting bodiescoated with lossy materials of arbitrary thickness

A simple and efficient numerical technique is presented to solve the electromagnetic scattering problem of coated conducting bodies of arbitrary shape. The surface equivalence principle is used to formulate the problem in terms of a set of coupled integral equations involving equivalent electric and magnetic surface currents which represent boundary fields. The conducting structures and the dielectric materials are modeled by planar triangular patches, and the method of moments is used to solve the integral equations. Numerical results for scattering cross sections are given for various structures and compared with other available data. These results are proved accurate by a number of representative examples     

Ground effects for VHF/HF antennas on helicopter airframes

In this paper, the finite element method (FEM) is used to predict the space and surface wave radiation patterns of VHF/HF antennas mounted on a helicopter in the presence of a lossy ground. The equivalent sources of the radiation system are obtained by solving an FEM problem in conjunction with an absorbing boundary condition (ABC) or an impedance boundary condition (IBC). From the equivalent sources, the total radiated field is calculated using the equivalence principle and superposition; the original problem is converted into a set of properly combined Hertzian dipoles referred to as the Sommerfeld problem. Instead of evaluating the Sommerfeld integral rigorously, Norton's approximation is used to improve the overall computational efficiency. The validation of this method is accomplished in two steps: first, the FEM is compared with the finite-difference time-domain method (FDTD) in the absence of a lossy ground; second, the Hertzian dipole problem is solved in the presence of a lossy ground and the results are compared with analytic solutions. Finally, this technique is extended to analyze an antenna on a helicopter above a lossy ground