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     

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     

Numerical solution of integral equations for dielectric objects ofprismatic shapes

A numerical method to investigate scattering from dielectric geometries of prismatic shapes has been developed. The surface integral equations are formulated by Schelkunoff's equivalence principle in terms of equivalent surface electric and magnetic currents. To solve these integral equations for the unknown currents, the object's cross-section is mapped onto a circle. In the transformed space, Fourier type entire-domain basis functions are used in the cross section and triangular subdomain basis functions are selected along the generating curve to represent the currents. A moment method is then used to reduce the integral equations to a matrix equation to compute the current coefficients. It is found that the transformation of the object's surface to a circular shape improves the convergence of the current mode in the cross-section. However, the current modes are coupled on the surface and the matrix equation includes all the modes     

Planar near-field to far-field transformation using an array ofdipole probes

A method is presented for computing far-field antenna patterns from measured near-field data measured by an array of planar dipole probes. The method utilizes the near-field data to determine some equivalent magnetic current sources over a fictitious planar surface which encompasses the antenna. These currents are then used to find the far fields. The near-field measurement is carried out by terminating each dipole with 50 Ω load impedances and measuring the complex voltages across the loads. An electric field integral equation (EFIE) is developed to relate the measured complex voltages to the equivalent magnetic currents. The mutual coupling between the array of probes and the test antenna modeled by magnetic dipoles is taken into account. The method of moments with Galerkin's type solution procedure is used to transform the integral equation into a matrix one. The matrix equation is solved with the conjugate gradient-fast Fourier transformation (CG-FFT) method exploiting the block Toeplitz structure of the matrix. Numerical results are presented for several antenna configurations to show the validity of the method     

Moment method analysis of the magnetic shielding factor of aconducting TM shield at ELF

This paper presents an integral equation and method of moments (MoM) solution to the problem of TM transmission by a metallic conducting shield at extremely low frequencies (ELF). In order to accurately compute the total fields interior to the shield, equivalent problems are formulated which avoid the numerically difficult problem of computing the total fields as the sum of the incident plus scattered fields. In particular, the total electric field on the interior surface of the shield is obtained by a volume current equivalent problem, and then the total magnetic field interior to the shield is formulated in terms of equivalent magnetic surface currents flowing on the interior surface of the shield replaced by a perfect conductor     

Electromagnetic scattering from objects composed of multiplehomogeneous regions using a region-by-region solution

The paper presents an efficient procedure to calculate the electromagnetic field scattered by an inhomogeneous object consisting of N+1 linear isotropic homogeneous regions. The procedure is based on surface integral equation (SIE) formulations and the method of moments. The method of moments (MM) is used to reduce the integral equations for each homogeneous dielectric region into individual matrices. These matrices are each solved for the equivalent electric current in terms of the equivalent magnetic current. A simple algebraic procedure is used to combine these solutions and to solve for the magnetic current on the outer dielectric surfaces of the scatterer. With the magnetic current determined, the electric current on the outer surface of the scatterer is calculated. Because the matrix corresponding to each dielectric region is solved separately, the authors call this procedure the region-by-region method. The procedure is simple and efficient. It requires less computer storage and less execution time than the conventional MM approach, in which all the unknown currents are solved for simultaneously. To illustrate the use of the procedure, the bistatic and monostatic radar cross sections (RCS) of several objects are computed. The computed results are verified by comparison with results obtained numerically using the conventional numerical procedure as well as via the series solution for circular cylindrical structures. The possibility of nonunique solutions has also been investigated     

Higher order impedance boundary conditions applied to scattering bycoated bodies of revolution

In this paper higher order impedance boundary conditions will be employed in the solution of scattering by coated conducting bodies of revolution. The higher order impedance solution reduces the total number of unknowns relative to the exact solution, and produces a system matrix which is less dense than that of the exact solution. The construction of the solution involves two distinct steps. In the first step the body of revolution is replaced by an equivalent set of electric and magnetic currents on its exterior surface which generate the true fields outside the body. An integral equation relating these currents through the free space Green's function is derived. Step two employs the higher order impedance boundary condition to relate the electric and magnetic currents on the surface of the body. This replaces the rigorous solution of the interior problem. The higher order impedance boundary conditions are derived by obtaining an exact impedance boundary condition in the spectral domain for the coated ground plane, approximating the impedances as ratios of polynomials in the transform variables, and employing the Fourier transform. The resulting spatial domain differential equations are solved in conjunction with the integral equation using the method of moments. Several examples of bistatic and monostatic radar cross section for coated bodies of revolution are used to illustrate the accuracy of the higher order impedance boundary condition solution relative to the standard impedance boundary condition solution and the exact solution. The effects of coating thickness, loss, and curvature on the accuracy of the solution are discussed     

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.      

Electromagnetic scattering and radiation from finite microstripstructures

Two techniques are presented for the analysis of electromagnetic radiation and scattering from finite microstrip structures. The two techniques are based on two different formulations, viz. the volume-surface and surface-surface formulations. In the volume-surface formulation the finite-sized dielectric is replaced by an equivalent volume polarization current whereas the conducting plates are replaced by equivalent surface currents. For the surface-surface formulation the surface covering the dielectric volume is replaced by equivalent electric and magnetic currents and the conducting plates by surface electric currents. Both techniques can be utilized for the analysis of arbitrarily shaped finite microstrip structures. The techniques are quite accurate, and they are utilized to validate each other. Typical numerical results are presented to demonstrate the agreement between these two solution techniques     

Scattering from multiple conducting and dielectric bodies ofarbitrary shape

A simple moment-method solution is presented for the problem of electromagnetic scattering from structures consisting of multiple perfectly conducting and dielectric bodies of arbitrary shape. The system is excited by a plane wave. The surface equivalence principle is used to replace the bodies by equivalent electric and magnetic surface currents, radiating into an unbounded medium. A set of coupled integral equations, involving the surface currents, is obtained by enforcing the boundary conditions on the tangential components of the total electric and magnetic fields. The method of moments is used to solve the integral equations. The surfaces of the bodies are approximated by planar triangular patches, and linearly varying vector functions are used for both expansion and testing functions. Some of the limitations of the method are briefly discussed. Results for the scattering cross sections are presented. The computed results are in very good agreement with the exact solutions and with published data     

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 an arbitrarily shaped three-dimensional homogeneous chiral body

The method of moments technique for analyzing electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body is presented based on the combined field integral equations. The body is assumed to be illuminated by a plane wave. The surface equivalence principle is used to replace the body by equivalent electric and magnetic surface currents. These currents radiating in unbounded free space produce the correct scattered field outside. The negatives of these currents produce the correct total internal field, when radiating in an unbounded chiral medium. By enforcing the continuity of the tangential components of the total electric and magnetic fields on the surface of the body, a set of coupled integral equations is obtained for the equivalent surface currents. The surface of the body is modeled using triangular patches. The triangular rooftop vector expansion functions are used for both equivalent surface currents. The coefficients of these expansion functions are obtained using the method of moments. The mixed potential formulation for a chiral medium is developed and used to obtain explicit expressions for the electric and magnetic fields produced by surface currents. Numerical results for bistatic radar cross sections are presented for three chiral scatterers - a sphere, a finite circular cylinder, and a cube.     

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     

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     

Combined field solution for TM scattering from multiple conducting and dielectric cylinders of arbitrary cross section

A simple moment solution is given for the problem of electromagnetic scattering from multiple conducting and dielectric cylinders of arbitrary cross section. The system of conducting and dielectric cylinders is excited by a plane-wave polarized transverse magnetic to the axis of the cylinders. The equivalence principle is used to obtain three coupled integral equations for the induced electric current on the conducting cylinders and the equivalent electric and magnetic currents on the surface of dielectric cylinders. The combined field integral equation (CFIE) formulation is used. Sample numerical results are presented. The agreement with available published data is excellent.     

RCS of two-dimensional structures consisting of both dielectricsand conductors of arbitrary cross section

The problem of determining the electromagnetic field scattered by two-dimensional structures consisting of both dielectric and conducting cylinders of arbitrary cross section is considered. The conductors may be in the form of strips and the dielectrics may be in the form of shells. The conductors may be partially or fully covered by dielectric layers, while the dielectrics may be partially covered by conductors. Only homogeneous dielectrics are studied. Both the transverse electric (TE) and the transverse magnetic (TM) cases are considered. The problem is formulated in terms of a set of coupled integral equations involving equivalent electric and magnetic surface currents radiating in unbounded media. The method of moments is used to solve the integral equations. Simple expansion and testing procedures are used. Numerical results for scattering cross sections are given for various structures     

Moment method analysis of the electric shielding factor of aconducting TM shield at ELF

This paper presents an integral equation and method of moments (MM) solution to the problem of TM transmission by a metallic conducting shield at extremely low frequencies (ELF). To obtain an accurate and efficient solution, the equivalent volume polarization currents representing the shield are expanded in terms of physical basis functions, corresponding to two planewaves propagating normal to the surface of the shield. ELF approximations are used to obtain closed form expressions for certain crucial elements in the MM matrix equation where extreme accuracy between the relative magnitudes of self and mutual impedance terms are required. Numerical data will illustrate that despite the fact that the equivalent polarization currents are being computed very accurately, the method is not capable of computing the extreme near zone electric fields of these currents with sufficient accuracy to compute the electric shielding factor. An alternate method, based upon the use of the volume equivalence theorem to directly compute the total electric field in the shield is devised, and is found to accurately compute the electric shielding factor     

Scattering by an infinite wedge with tensor impedance boundaryconditions-a moment method/physical optics solution for thecurrents

Plane wave scattering by an infinite, two-dimensional wedge whose faces are characterized by impedance tensors is discussed. A combination of the moment method (MM) and physical optics (PO) is used to obtain a solution for the equivalent electric currents. The currents near the edge on each face are expanded with a set of basis functions consisting of pulse functions, defined on a meshed region, plus a function spanning the whole face. The currents outside the meshed region are taken to be the sum of physical optics currents, taken to be known, plus the whole-face basis function current. Expressing the equivalent magnetic currents in terms of the electric currents through the impedance tensors, the expansion coefficients for the electric current expansion are determined through an MM solution of the magnetic field integral equation. Sample results for wedges with isotropic and anisotropic face impedances are presented     

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     

A sheet impedance approximation for electrically thick multilayered shields

This paper describes an extension of the sheet impedance concept to treat inhomogeneous or multilayered shields that may be thick in terms of material shield wavelengths. For shields with magnetic materials, a simple relation between the equivalent electric and magnetic currents representing the shield is obtained. This allows the magnetic current to be treated as a dependent unknown and the electric current to be found as the solution of a single surface integral equation shown to be a perturbation of that for a perfect electric conducting (PEC) surface. By using the proper interior equivalent problem, it is shown that the method produces accurate and stable results for shielding by a rectangular box.