Electromagnetic Scattering From Arbitrarily Shaped Dielectric Bodies Using Paired Pulse Vector Basis Functions and Method of Moments

A pair of orthogonal pulse vector basis functions is demonstrated for the calculation of electromagnetic scattering from arbitrarily-shaped material bodies. The basis functions are intended for use with triangular surface patch modeling applied to a method of moments (MoM) solution. For modeling the behavior of dielectric materials, several authors have used the same set of basis functions to represent equivalent electric and magnetic surface currents. This practice can result in zero-valued or very small diagonal terms in the moment matrix and an unstable numerical solution. To provide a more stable solution, we have developed orthogonally placed, pulse basis vectors: one for the electric surface current and one for the magnetic surface current. This combination ensures strongly diagonal moment matrices. The basis functions are suitable for electric field integral equation (EFIE), magnetic field integral equation (HFIE), and combined field formulations. In this work, we describe the implementations for EFIE and HFIE formulations and show example results for canonical figures.      

Computation of TM and TE modes in waveguides based on a surfaceintegral formulation

The surface integral formulation is used for the computation of TM and TE modes propagating in dielectric loaded waveguides. This formulation makes use of the surface equivalence principle whereby the field at any point internal or external to the waveguide can be expressed in terms of equivalent surface currents. This procedure reduces the original problem into a set of integro-differential equations which is then reduced to a matrix equation using the method of moments. The solution of this matrix equation provides the propagation characteristics of the waveguide and the equivalent surface currents existing on the waveguide walls. The equivalent surface currents can be used to compute the fields at all points, both inside and outside the waveguide. The surface integral method has been used to compute the propagation characteristics of waves propagating in dielectric loaded waveguides. The computed results agree very well with analytical and published data. A method that can be used to remove spurious modes is illustrated     

Single integral equation for electromagnetic scattering bythree-dimensional homogeneous dielectric objects

A single integral equation formulation for electromagnetic scattering by three-dimensional (3-D) homogeneous dielectric objects is developed. In this formulation, a single effective electric current on the surface S of a dielectric object is used to generate the scattered fields in the interior region. The equivalent electric and magnetic currents for the exterior region are obtained by enforcing the continuity of the tangential fields across S. A single integral equation for the effective electric current is obtained by enforcing the vanishing of the total field due to the exterior equivalent currents inside S. The single integral equation is solved by the method of moments. Numerical results for a dielectric sphere obtained with this method are in good agreement with the exact results. Furthermore, the convergence speed of the iterative solution of the matrix equation in this formulation is significantly greater than that of the coupled integral equations formulation     

A new method for obtaining the shape sensitivities of planarmicrostrip structures by a full-wave analysis

We present the principles and the derivation of a new mixed potential integral equation for the derivative of the surface current with respect to a geometrical parameter for planar microstrip structures embedded in a multilayered substrate. This new integral equation is solved together with the original integral equation with the method of moments by using the same set of test and basis functions. Expressions for the matrix elements as a function of the basis and test functions are given. From the geometrical derivatives of the surface currents, geometrical derivatives of the S-parameters are obtained. In the examples a geometrical parameter is swept over some interval, and the derivative, obtained with the new integral equation, is compared with estimates calculated by using finite differences. Very good agreement is found between these estimates     

Integral equation solution for analyzing scattering fromone-dimensional periodic coated strips

A set of integral equations based on the surface/surface formulation are developed for analyzing electromagnetic scattering by one-dimensional periodic structures. To compare the accuracy, efficiency, and robustness of the formulation, the electric field integral equation (EFIE), magnetic field integral equation (MFIE), and combined field integral equation (CFIE) are developed for analyzing the same structure for different excitations. Due to the periodicity of the structure, the integral equations are formulated in the spectral domain using the Fourier transform of the integrodifferential operators. The generalized-biconjugate-gradient-fast Fourier transform method with subdomain basis functions is used to solve the matrix equation     

Dynamic analysis of a microstrip line over a perforated groundplane

A full wave analysis is presented to compute the characteristic impedance and propagation constant of a microstrip line over a perforated ground plane. The perforations in the ground plane are modeled by equivalent magnetic currents. The method of moments is applied to solve the coupled integral equations for the unknown electric current on the microstrip line and the unknown magnetic currents in the apertures. The fields are formulated using the space domain Sommerfeld type Green's functions. The matrix pencil technique is used to obtain the amplitude and the propagation constant of the fundamental modes for both current and the voltage on the microstrip line. Typical numerical results are given     

Full wave analysis of microstrip floating line structures bywavelet expansion method

A full wave analysis of microstrip floating line structures by wavelet expansion method is presented. The surface integral equation developed from a dyadic Green's function is solved by Galerkin's method, with the integral kernel and the unknown current expanded in terms of orthogonal wavelets. Using the orthonormal wavelets (and scaling functions) with compact support as basis functions and weighting functions, the integral equation is converted into a set of linear algebraic equations, with the matrices nearly diagonal or block-diagonal due to the localization, orthogonality, and cancellation properties of the orthogonal wavelets. Limitations inherited in the traditional orthogonal basis systems are released: The problem-dependent normal modes have been replaced by the problem-independent wavelets, preserving the orthogonality; the trade-off between orthogonality and continuity (e.g. subsectional basis functions including pulse functions, roof-top functions, piecewise sinusoidal functions, etc.) is well balanced by the orthogonal wavelets. Numerical results are compared with measurements and previous published data with good agreement     

Conductor loss in hollow waveguides using surface integralformulation

The power-loss method along with a surface integral formulation is used to compute the attenuation constant in hollow waveguides of arbitrary cross-section. An E-field integral equation is developed for the surface electric currents which is transformed into a matrix equation using the method of moments. An iterative technique, i.e. Muller's method, is used to obtain the relation between the propagation constant and frequency. The attenuation constants have been calculated and formulated for various waveguides and are in good agreement with published data     

Electromagnetic scattering from a chiral cylinder of arbitrarycross section

A simple moment solution is presented to the problem of electromagnetic scattering from a homogeneous chiral cylinder of arbitrary cross-section. The cylinder is assumed to be illuminated by either a TE or a TM wave. The surface equivalence principle is used to replace the cylinder by equivalent and magnetic-surface currents. These currents radiating in unbounded external medium produce the correct scattered field outside. When radiating in an unbounded chiral medium, they produce the correct total internal field. By enforcing the continuity of the tangential components of the total electric field on the surface of the cylinder, a set of coupled integral equations is obtained for the equivalent surface currents. Unlike a regular dielectric, the chiral scatterer produces both copolarized and cross-polarized scattered fields. Hence, both the electric and magnetic current each have a longitudinal and a circumferential component. These four components of the currents are obtained by using the method of moments (MoM) to solve the coupled set of integral equations. Pulses are used as expansion functions and point matching is used. The Green's dyads are used to develop explicit expressions for the electric field produced by two-dimensional surface currents radiating in an unbounded chiral medium. Some of the advantages and limitations of the method are discussed. The computed results include the internal field and the bistatic and monostatic echo widths. The results for a circular cylinder are in very good agreement with the exact eigenfunction solution     

Planar near-field to far-field transformation using an equivalentmagnetic current approach

An alternative method is presented for computing far-field antenna patterns from near-field measurements. The method utilizes the near-field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna, and these currents are used to ascertain the far fields. Under certain approximations, the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near-field measurements are made. An electric field integral equation (EFIE) is developed to relate the near fields to the equivalent magnetic currents. The method of moments is used to transform the integral equation into a matrix one. The matrix equation is solved with the conjugate gradient method, and in the case of a rectangular matrix, a least-squares solution for the currents is found without explicitly computing the normal form of the equation. Near-field to far-field transformation for planar scanning may be efficiently performed under certain conditions. Numerical results are presented for several antenna configurations     

Integral equation solution of Maxwell's equations from zerofrequency to microwave frequencies

We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies     

An efficient curved-wire integral equation solution technique

Computation of the currents on curved-wires by integral equation methods is often inefficient when the structure is tortuous but the length of wire is not large relative to the wavelength at the frequency of operation. The number of terms needed in an accurate piecewise straight model of a highly curved-wire can be large, yet, if the total length of wire is small relative to the wavelength, the current can be accurately represented by a simple linear function. A new solution method for the cured-wire integral equation is introduced. It is amenable to uncoupling of the number of segments required to accurately model the wire structure from the number of basis functions needed to represent the current. This feature lends itself to high efficiency. The principles set forth can be used to improve the efficiency of most solution techniques applied to the curved-wire integral equation. New composite basis and testing functions are defined and constructed as linear combinations of other commonly used basis and testing functions. We show how the composite basis and testing functions can lead to a reduced-rank matrix, which can be computed via a transformation of a system matrix created from traditional basis and testing functions. Supporting data demonstrate the accuracy of the technique and its effectiveness in decreasing matrix rank and solution time for curved-wire structures     

Full-wave analysis of a strip crossover

A full-wave analysis of a strip crossover above a conducting plane is carried out. Higher-order modes are excited in the form of evanescent waves in the vicinity of the discontinuity, while further away only the dominant (TEM) modes exist. The higher-order mode currents are modeled by triangle functions and the dominant modes by outgoing traveling waves. The method of moments is used to reduce the integral equations on the surface of each strip to matrix equations whose solution determines the currents on each strip. The impedance and scattering matrices of the four-port network and the equivalent circuit were determined. At low frequencies, the equivalent circuit agrees very well with that which was obtained previously using a quasi-static analysis. The two approaches begin to disagree when the cross-sectional dimensions of the crossover become comparable to a tenth of the wavelength. At that point the quasi-static analysis becomes inaccurate, while the full-wave analysis presented here remains valid     

Surface currents and RCS of a spherical shell with a circular aperture

The electromagnetic scattering behavior of a perfectly conducting, infinitesimally thin, spherical shell with a circular aperture is studied. A time-harmonic plane wave is symmetrically incident upon the aperture. The problem is formulated in terms of theE-field integral equation. This produces two coupled integral equations for the tangential components of the currents on the scatterer surface. The equations are cast into matrix form by application of the method of moments, and expressions for the matrix elements are derived. Calculated values of the surface currents and radar cross sections, not previously available in the open literature, are presented and discussed for several cases of interest.     

A hybrid method for the calculation of the resistance andinductance of transmission lines with arbitrary cross sections

The frequency-dependent resistance and inductance of uniform transmission lines are calculated with a hybrid technique that combines a cross-section coupled circuit method with a surface integral equation approach. The coupled circuit approach is most applicable for low-frequency calculations, while the integral equation approach is best for high frequencies. The low-frequency method consists in subdividing the cross section of each conductor into triangular filaments, each with an assumed uniform current distribution. The high-frequency method expresses the resistance and inductance of each conductor in terms of the current normal to the surface. An interpolation between the results of these two methods gives very good results over the entire frequency range, even when few basis functions are used. Results for a variety of configurations are shown and are compared with experimental data and other numerical techniques     

A comparison of integral equations with unique solution in theresonance region for scattering by conducting bodies

A comparison of integral equations, for problems involving scattering by arbitrary-shape conducting bodies, having a unique solution in the resonance region is presented. The augmented electric and magnetic field integral equations and the combined field integral equation, in their exact and approximate versions, are considered. The integral equations and the basis and test functions used in the method of moments to solve them are reviewed. Their implementation in a computer code is analyzed, mainly the relation between the matrix properties and the CPU time and memory. Numerical results (condition number and backscattering cross section) are presented for the cube. It is shown that the combined field integral equation, and the approximate (symmetric) combined field integral equation, are the most efficient equations to use in the neighborhood of resonant frequencies, because the overdetermined augmented integral equations require an extra matrix multiplication     

Application of on-surface MEI method on wire antennas

The formulas of the on-surface measured equation of invariance (OSMEI) for wire antennas are derived. The unknowns of each node on the antenna surface are expressed by the vector potential function and surface current density. The unknowns in the vicinity of each node are coupled in a linear equation and the coefficients of the linear equation are determined by the measured equation of invariance (MEI) method. The final impedance matrix obtained by the OSMEI is a highly sparse matrix. It demonstrates that the currents on thin wire antennas may also be solved by a differential equation-based formulation in addition to the conventional integral equations     

AIM Analysis of Scattering and Radiation by Arbitrary Surface-Wire Configurations

The adaptive integral method is utilized to solve electromagnetic scattering and radiation problems of conducting surface-wire configurations. The method of moments (MoM) is applied to establish the integral equations where triangular type basis functions are used to represent the currents on surfaces and wires. Attachment mode has been used to model the surface-wire junction to ensure the current continuity. The resultant matrix system is then solved by an iterative solver where the adaptive integral method (AIM) is employed to reduce memory requirements and to accelerate the matrix-vector multiplications. Numerical results are presented to demonstrate the accuracy and efficiency of the present technique for the arbitrary surface-wire configurations     

On the determination of resonant modes of dielectric objects using surface Integral equations

Although surface integral equations have been extensively used for solving the scattering problem of arbitrarily shaped dielectric objects, when applied to the resonance problem, there are still some issues not fully addressed by the literature. In this paper, the method of moments with Rao-Wilton-Glisson basis functions is applied to the electric field integral equation (EFIE) for solving the resonance problem of dielectric objects. The resonant frequency is obtained by searching for the minimum of the reciprocal of the condition number of the impedance matrix in the complex frequency plane, and the modal field distribution is obtained through singular value decomposition (SVD). The determinant of the impedance matrix is not used since it is difficult to find its roots. For the exterior EFIE, the original basis functions are used as testing functions; for the interior EFIE, the basis functions rotated by 90/spl deg/ are used as testing functions. To obtain an accurate modal field solution, the impedance matrix needs to be reduced by half before SVD is applied to it. Numerical results are given and compared with those obtained by using the volume integral equation.     

Surface integral formulation for calculating conductor anddielectric losses of various transmission structures

The power-loss method, along with a surface integral formulation, has been used to compute the attenuation constant in microstrip and coplanar structures. This method can be used for the analysis of both open and closed structures. Using the surface equivalence principle, the waveguide walls are replaced by equivalent electric surface currents and dielectric surfaces are replaced by equivalent electric and magnetic surface currents. Enforcing the appropriate boundary condition, and E-field integral equation (EFIE) is developed for these currents. Method of moments with pulse expansion and point matching testing procedure is used to transform the integral equation into a matrix one. The relationship between the propagation constant and frequency is found from the minimum eigenvalue of the moment matrix. The eigenvector pertaining to the minimum eigenvalue gives the unknown electric and magnetic surface currents