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     

The near-zone field characteristics of an E-polarization plane wavepenetrating through cylindrical multiple apertures (non) coated withlossy and lossless media

The direct integral equation is formulated for describing the current on the multiple perfectly conducting strips in cylindrical geometries for an E-polarization plane wave of normal incidence. By using the Galerkin's method, the surface currents on the conducting strips are expanded in the form of a series of Chebyshev polynomials of the first kind, while the unknown expanding coefficients are solved by a set of matrix equations of finite order with a fast convergence rate and a high accuracy. Furthermore, numerical results are presented to demonstrate the variation of the penetrated near-zone field in the presence of one, two, three, four and six cylindrical apertures, and the hybrid effects of both aperture number and aperture angular widths on the penetrated fields are investigated in detail     

Electromagnetic diffraction from a dielectric-filled slit-cylinderenclosing a cylinder of arbitrary cross section: TM case

A simple moment solution to the problem of the diffraction of a TM plane wave from an infinite, perfectly conducting slotted cylinder of an arbitrary cross section is summarized. The slit cylinder encloses a smaller perfectly conducting cylinder of an arbitrary cross section, and the space between the cylinders is filled with a dielectric material. The equivalence principle is used to obtain a set of coupled integral equations for the induced/equivalent surface currents on the cylinders, and the method of moments is used to solve numerically the integral equations. The electric field integral equation formulation is used. The advantages and the limitations of the method are discussed. Sample results for the induced current, aperture field, internal field, and scattering cross sections are given. These are in good agreement with some of the available published data     

A Nonmodal Formulation for Electromagnetic Transmission through a Filled Slot of Arbitrary Cross Section in a Thick Conducting Screen

This paper considers the two-dimensional problem of electromagnetic transmission through a filled slot of arbitrary cross section in a thick perfectly conducting screen. The equivalence principle is used to divide the original problem into three isolated parts where postulated equivalent sources radiate into unbounded, homogeneous media. These equivalent electric and magnetic currents are chosen to ensure continuity of the tangential components of electric and magnetic fields at each aperture. An integral equation is written for each of the three parts with the equivalent currents as unknowns. The resulting set of coupled integral equations is solved by the method of moments. It is shown in the Appendix that this set of equations has a unique solution. The primary quantities computed are the equivalent magnetic and electric currents on each aperture and the electric current on the remaining portions of the slot cross section. These results are compared with those obtained from a modal solution, where the fields in the slot cross section are expressed in terms of parallel-plate waveguide modes.     

Analysis of a cylindrical antenna containing an aperture coupledload

In this paper we present the basis for the analysis of shielded tuning networks coupled to wire antenna elements. The structure analyzed comprises three conducting cylindrical tubes that form an aperture-fed circular coaxial waveguide. Two methods are presented for formulating and solving integral equations for the structure. The first method is based on the mixed potential electric field integral equation enforced on the three cylindrical tubes, and the second is based on aperture theory. An end correction capacitance is used to adjust the reflection coefficient in the eigenmode expressions of the aperture method. The data obtained by the two methods are in close agreement. Differences in actual currents and equivalent currents obtained from solutions are discussed and reconciled     

Analysis of cavity-backed aperture antennas with a dielectricoverlay

A full-wave analysis of cavity-backed aperture antennas with a dielectric overlay is presented. The theoretical approach uses a closed-form dyadic Green's function in the spectral domain. The aperture equivalent magnetic currents are obtained using the surface equivalence theorem and an integral equation is obtained by matching the fields across the aperture. The moment method applied in spectral domain analysis is employed to solve the integral equation for the equivalent magnetic currents with proper combination of subdomain or entire domain expansion functions. Numerical results include the aperture field distribution and antenna parameters such as input impedance, bandwidth, and efficiency. A set of measurements data is compared with results based on the theoretical work     

Matching of equivalent field regions

In aperture problems, integral equations for equivalent currents are often found by enforcing matching of equivalent fields. The enforcement is made in the aperture surface region adjoining the two volumes on each side of the aperture. In the case of an aperture in a planar perfectly conducting screen, having the same homogeneous medium on both sides and an impressed current on one side, an alternative procedure is relevant. We make use of the fact that in the aperture the tangential component of the magnetic field due to the induced currents in the screen is zero. The use of such a procedure shows that equivalent currents can be found by a consideration of only one of the two volumes into which the aperture plane divides the space. Furthermore, from a consideration of an automatic matching at the aperture, additional information about tangential as well as normal field components is obtained. We compare the two procedures in this tutorial article.     

Transmission through dielectric-filled slots in a conductingcylindrical shell of arbitrary cross section: TE case

A simple moment solution is summarized for the problem of electromagnetic transmission through dielectric-filled slots in a conducting cylindrical shell of arbitrary cross section. The system is excited by a plane-wave polarized transverse electric (TE) to the axis of the shell. The equivalence principle is used to replace the shell and the dielectric by equivalent electric and magnetic surface currents radiating into an unbounded medium. Two different sets of coupled integral equations involving the surface currents are 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. Pulses are used for both expansion and testing functions. Special attention is paid to circular and rectangular shells. Results for shell surface current, the internal field, and the aperture field are presented. For the case of air dielectric filling, the results computed using the electric field and/or the magnetic field formulation are in very good agreement with published data. In general, it is observed that the effect of filling a slot with a dielectric is not predictable from a simple theory     

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     

Feed models for coax-driven monopole and dipole antennas

A method is presented for accurately modeling a monopole or dipole antenna fed by a coaxial line. The base of the monopole is attached to a conducting plane through which the coaxial feed line extends to the feed. The feed structures considered are easily adaptable to physically rugged forms and are simple to construct. Equivalent models for the three regions of the structure are devised and coupled integral equations for aperture fields and surface currents are formulated to enforce the boundary conditions. Three variations of the feed configuration are discussed and the reflection coefficient of the antenna feed is determined from the data obtained from the solutions of the coupled integral equations. Computed reflection coefficient values are shown to agree well with values measured on laboratory models.     

Electromagnetic Simulation of Missile Exhaust

A simple moment solution is given for the problem of electromagnetic transmission through dielectric-filled slots in a conducting cylindrical shell of arbitrary cross section. The exciting source is assumed to be either a TM plane wave (receive mode) or an electric line current placed inside the shell (transmit mode). The surface-equivalence principle is used to replace the surfaces of the shell and the dielectrics by equivalent surface currents radiating in an unbounded medium. The application of the appropriate boundary conditions yields a set of coupled integral equations for the surface currents. The moment method with pulse expansion and point-matching testing procedures is used to solve the integral equations. Shells of different cross-sectional shapes are considered. Special attention is paid to circular and rectangular shells. In the transmit mode, the total far field transmitted through the slot is computed. In the receive mode, the aperture field and the field at the center of the shell are computed. For the case of air dielectric filling the slots, the computed results are in very good agreement with available published data.     

An efficient method for the analysis of a structure comprising an appendage attached to a planar surface of a conducting body

A technique is presented to efficiently solve for the currents on an appendage, e.g., antenna, attached to a planar surface of a conducting body. The appendage may be embedded in a homogeneous, dielectric material. The technique presented alleviates the complications associated with the point where the appendage is attached to the body. To illustrate the method, a wire antenna attached to an axisymmetric body is analyzed in detail. A set of coupled integral equations are formulated, appropriate quantities are expanded into Fourier modes, and coupled integral equations are derived for the Fourier coefficients of the unknowns. These equations are solved and the input admittance of the wire antenna is determined from the computed currents and is corroborated by measurements.     

On the electromagnetic field in a cavity fed by a tangentialelectric field in an aperture in its wall

Heretofore, the electromagnetic field produced by a specified tangential electric field in an aperture in the wall of an arbitrarily shaped cavity has most often been expanded in terms of cavity modes. An alternative approach, that of the electric field integral equation is presented. In this approach, the cavity field is expressed as the field of a surface density of tangential electric current, or a surface density of tangential magnetic current, or a combination of surface densities of tangential electric and magnetic currents on the boundary of the cavity. Each surface density is characterized by a single tangential vector function which is determined by the integral equation requiring that the part of the electric field tangent to the boundary of the cavity must reduce to the specified tangential electric field in the aperture and zero elsewhere on the boundary of the cavity. The electric field integral equation method is specialized to more easily determine the field inside an arbitrary cylindrical cavity excited by a tangential electric field in an aperture in its lateral wall. The method is further specialized to a circular cavity     

Scattering from narrow rectangular filled grooves

The solution of the integral equation for a small width rectangular groove is considered. It is shown that by retaining the dominant mode supported by the rectangular groove, the resulting quasi-static integral equations are comparable to those associated with the perfectly conducting narrow strip. They are, therefore, amenable to analytic solution yielding the exact field distribution or equivalent currents across the groove's aperture. The derived currents exhibit the same edge behavior as that associated with the currents of a perfectly conducting half-plane. The corresponding current behavior based on a (numerical) impedance simulation of the groove is quite different. However the resulting echowidths are comparable. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated     

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     

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     

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     

The Variational Treatment of Thick Interacting Inductive Irises

The problem of two thick interacting inductive irises in waveguide is treated with a variational approach. Using the appropriate Green's functions in the continuity equations of the transverse magnetic fields yields two coupled integral equations for the magnetic currents on the apertures. Solving one equation by Fourier expansion and introducing in the remaining equation, a variational expression for the driving-point admittance is obtained. This is treated with a Rayleigh-Ritz procedure and matrix methods, avoiding the explicit computation of field amplitudes. The analysis is carried out in terms of an eigenmode expansion, as well as in terms of an expansion a la Schwinger on the aperture and the features of the two methods are contrasted. In spite of its somewhat greater mathematical complexity, the latter generally provides a superior solution for a given order of the trial field. In both cases the solutions are very accurate, uniformly convergent to their common limit value, and require manipulations with small-order matrices only. The agreement with the experiment is excellent.     

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     

A new approach to the thin scatterer problem using the hybrid equations

A new set of integral equations for electromagnetic scattering problems, the "hybrid" equations, is presented. The advantages of these equations for thin perfect conductors are discussed in comparison to the magnetic and electric field integral equations. Specific comparisons are made with the solution of the electric field integral equation for a finite hollow cylinder. It is demonstrated that the primary advantage of these equations is obtained by minimizing the coupling between component equations for the two surface currents.