Current and charge Integral equation formulation

A new stable frequency domain surface integral equation formulation is proposed for the three dimensional electromagnetic scattering of composite metallic and dielectric objects. The developed formulation does not suffer from the low frequency breakdown and leads to a well balanced and stable system on a wide frequency band. Surface charge densities are used as unknowns in addition to the traditional surface current densities. The balance of the system is achieved by using normalized field quantities and by enforcing the continuity of the fields across the boundaries with carefully chosen scaling factors. The linear dependence between the currents and charges is taken into account with an integral operator, and the linear dependence in charges is removed with the deflation method. A combined field integral equation form of the formulation is proposed to remove the internal resonance problem associated to the closed metallic objects. Due to the good balance in the new formulation, fast converging iterative solutions on a very wide frequency band can be obtained. The new formulation and its convergence is verified with numerical examples.     

Integral formulation of the measured equation of invariance

A novel integral formulation of the measured equation of invariance is derived from the reciprocity theorem. This formulation leads to a sparse matrix equation for the induced surface current, resulting in great CPU time and memory savings over the conventional approaches. The algorithm has been implemented for two-dimensional perfectly conducting scatterers     

Integral equation formulation for probe corrected far-field reconstruction from measurements on a cylinder

A novel and numerically efficient method of far-field evaluation from measurements taken on a cylinder is based on the representation of both the antenna and the probe fields as superpositions of plane waves. A system of two integral equations are established whose unknown functions are the azimuthal and elevation components of the antenna pattern and whose known terms are the set of measurement data taken with two different probes-the second probe in most practical instances being simply the same probe with a different geometrical orientation. The equations express the known data-for each angular position of the antenna under measurement-as the integrals of the products of the corresponding components of the unknown antenna and known probe patterns multiplied by a phase term. The convolutional nature of the integral equations makes their solutions straightforward. If, as is virtually always the case, the probe has small or moderate size so that the axis of rotation of the antenna mount is in the far field of the probe, the intervention of asymptotic techniques makes the solution numerically very efficient. The agreement of calculated and experimental patterns is excellent.     

An integral equation formulation of the direct scattering problem for an inhomogeneous slab

A numerical technique based on an integral equation scheme is developed for solving the direct scattering problem for an inhomogeneous slab. The integral equation is derived, using the induced current concept and the Green's function technique. The numerical method of solving the integral equation is presented. The method is proved to be numerically satisfactory and is applied to the slab of specific profiles. Numerical results for several cases are also included.     

Integral equation formulation for inhomogeneous anisotropic mediaGreen's dyad with application to microstrip transmission linepropagation and leakage

A straightforward numerical technique based on the equivalence principle is presented to determine the complete spectral Green's dyad for inhomogeneous anisotropic media. This method is relevant to guided-wave problems where propagation characteristics are desired in the axial transform domain. Spectral Green's components are determined from a one-dimensional polarization-type integral equation. This method is very simple and versatile, and can be used to model continuously varying or stratified dielectric media with permittivity dyads of the most general form. As an application, a microstrip transmission line residing on a generally orientated uniaxial and biaxial substrate is considered, and new results for higher-order mode leakage are presented     

Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem

A combined field integral equation (CFIE) formulation is presented for two- and three-dimensional bodies having discrete dielectric and conducting regions. A three-dimensional case is restricted to bodies of revolution (BORs). The two-dimensional case is analogous to the BOR case when the Fourier mode number is zero. The method of moments (MM) is used to solve the CFIE in terms of two integral operators. It is shown that the CFIE formulation yields accurate answers for scattering problems where the scatterer may be internally resonant. The CFIE results were validated using Mie series results and measured data. The junction problem associated with the CFIE-based formulation is explicated, and several geometries with multiple junctions are used to validate the general CFIE formulation. A number of configurations are tested where the penetrable region consisted of a free-space coating. Extensive numerical studies have shown that such limiting cases are sensitive indicators of the stability of MM solutions and allow direct comparison of different configurations     

Integral equation for scattering by a dielectric

The determination of the scattered and transmitted transient electromagnetic waves produced by a uniform dielectric body is reduced to the solution of a singular integral equation of the first kind for one tangential vector field defined on the surface. All derivations are carried out within the heuristic approach to Green functions and delta functions. The electric and magnetic fields are expressed in terms of the sources, initial values, and the boundary values by means of the Green function for the scalar wave equation. The appropriate integral equation is derived, and the integrals for the scattered and transmitted fields are given. The simpler problem of scattering of scalar waves is developed first. Formulas for the scattering of monochromatic fields are also given in the scalar and electromagnetic cases when transmitted fields do not vanish.     

Integral equation formulations for imperfectly conducting scatterers

Integral equation formulations are presented for characterizing the electromagnetic (EM) scattering interaction for nonmetallic surfaced bodies. Three different boundary conditions are considered for the surfaces: namely, the impedance (Leontovich), the resistive sheet, and its dual, the magnetically conducting sheet boundary. The integral equation formulations presented for a general geometry are specialized for bodies of revolution and solved with the method of moments (MM). The current expansion functions, which are chosen, result in a symmetric system of equations. This system is expressed in terms of two Galerkin matrix operators that have special properties. The solutions of the integral equation for the impedance boundary at internal resonances of the associated perfectly conducting scatterer are examined. The results are compared with the Mie solution for impedance-coated spheres and with the MM solutions of the electric, magnetic, and combined field formulations for impedance-coated bodies.     

Integral equation solution to the skin effect problem in conductorstrips of finite thickness

The skin effect of single and coupled conductor strips of finite thickness is analyzed using the dyadic Green's function and the integral equation formulation. Galerkin's method is used to solve the integral equation for the dispersion characteristics. The effects of the geometrical and electrical parameters on the conductor loss are investigated. Results are compared with previously published data and shown to be in good agreement. This approach is very useful for analyzing the electrical properties of interconnects in high-performance computer circuitries     

Doppler spread for Gaussian scatter density environments employing smart antennas

In this paper, the effects of actual smart antennas on the Power Doppler Spectrum (PDS) are studied when smart antennas are deployed at the base station (BS), employing a statistical Gaussian Scatter Density Model (GSDM). This proposed approach is general and it can be applied to omnidirectional and sectorized antennas as well, for beam patterns of linear and circular arrays, the results obtained are compared with those for omnidirectional and sectorized antennas. The characterization of PDS is presented for channel scenarios including different motion direction, scatterer spread, and a variety of beam patterns.     

Integral equation method for computer evaluation of electron optics

A Green's function technique for the computation of electric potentials in electrostatic systems of arbitrary configuration, especially with cylindrical symmetry, is developed. The system may include open boundaries and dielectric media. The integral equation is numerically solved by iteration. Its convergence is very fast. Computer calculated field and potential values are of high accuracy (usually better than 10^-4). Errors in electron optical data, derived from trajectory computation, were found to be within fabrication tolerances of experimental systems. The computerized method proved to be an excellent tool for the design of electron optical systems, as electron image intensifiers.     

A new formulation for state equation representation for Petri nets

A new algorithm for determination of state equations for Petri nets has been proposed. The proposed algorithm results in state equations similar to the state equations for linear sequential machines. All Petri nets may not be represented in the form of linear sequential machines. The resulting state equations are different from Petri net state equations and include output equations used in control theory literature.     

A practical tropospheric scatter model using the parabolic equation

A simple method that accounts for tropospheric scatter using the parabolic equation portion of the radio physical optics (RPO) model is described. RPO is a hybrid propagation model that combines ray-optics and parabolic-equation methods to assess realistic range-dependent tropospheric effects at frequencies from 100 MHz to 20 GHz. A semiempirical scatter model adds a random refractive-index fluctuation to the mean refractive-index value at each height considered by the parabolic equation method. The results of this scatter model are compared with those of another scatter model and with a few sample radio measurements     

Integral equation analysis of artificial media

This paper presents an integral equation and method of moments (MM) solution to determine the effective permittivity and permeability of an artificial medium. The artificial medium is modeled as a triple infinite periodic array of identical scattering elements. A plane wave of unknown phase constant is assumed to propagate in the periodic medium in a given direction, and the periodic moment method (PMM) is used to set up a matrix equation for the currents on the center element of the periodic array. By setting the determinant of the PMM impedance matrix to zero, one can determine the phase constant of the plane wave, and then the effective permittivity and permeability of the artificial medium     

A new well-conditioned Integral formulation for Maxwell equations in three dimensions

We present a new boundary integral equation dedicated to the solution of the boundary problem of a perfectly electrically conducting surface for the harmonic Maxwell equations in unbounded domains. Any solution of the harmonic Maxwell equations is represented as the electromagnetic field generated by a combination of electric and magnetic potentials. These potentials are those appearing in the classical combined field integral equation (CFIE), but their coupling is realized by an operator Y/spl tilde//sup +/ instead of a coefficient. Therefore, the integral equation obtained can be viewed as a generalization of the CFIE. In this paper, we propose an explicit construction of the coupling operator Y/spl tilde//sup +/ which is designed to approximate the exterior admittance operator of the scattering obstacle. A local approximation by the admittance operator of the tangential plane seems to be relevant thanks to the localization effects related to high-frequency phenomena. The provided numerical simulations show that this formulation leads to linear systems that are better conditioned compared to more classical integral equations, which speeds up the resolution when solved with iterative techniques.     

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.     

Integral equation for the potential distribution in a Hall generator

It is proved that the potential distribution in a Hall generator is also given as the electrostatic potential caused by a fictitious charge distribution along the boundary. The problem is then reduced to an integral equation, which is easily solved on a digital computer.     

Integral equation solution for traveling–wave tube parameters

An integral equation method for evaluating the electronic behavior of traveling-wave amplifiers is given. This method is utilized to obtain the effects of a thick beam and variations in dc density and dc velocity across the beam. Curves of the space charge parameter 4QC and impedance K are obtained for thick beams, which compare favorably with those of Pierce.