Mixed Potential Integral Equation Formulation Research Articles

Calculation of Potential Green’s Functions for a Planar Metasurface Excited by an Electric Dipole Using Homogenization Method

One of the rapid and efficient methods to calculate the interaction between electromagnetic fields and periodic surfaces with periodicity much less than the wavelength in the surrounding medium is the homogenization method. In this paper, the Hertzian potential Green’s functions corresponding to electric dipoles which excite a homogenized metasurface are presented. The metasurface includes a periodic array of planar scatterers and is replaced by a thin sheet with equivalent electric and magnetic surface polarization density dyads. The magnetic vector potentials are calculated using the generalized sheet transition conditions on the plane of the metasurface. Also the scalar electric potential corresponding to the structure is calculated with the aid of the Lorentz gauge. These potentials together are necessary to use in the mixed potential integral equation formulation of the method of moment in an electromagnetic problem. Two metasurfaces are investigated to calculate the electric fields; a perfect electric conductor square patch array whose surface polarization densities are calculated analytically, and a Jerusalem cross array whose surface polarization dyads are retrieved from reflection and transmission coefficients.

Homogenized Dyadic Potential Green’s Function for Planar Metasurface Placed at Two Half Spaces Interface Excited by Vertical and Horizontal Ideal Electric Dipole

In this article, dyadic magnetic vector potential Green’s functions corresponding to electric dipole which illuminates homogenized metasurface have been presented. Metasurface is constructed by a periodic arrangement of planar metallic scatterers and located at the interface of two different dielectric half spaces. The metasurface is replaced by a thin sheet with the dyadic equivalent surface distribution of electric and magnetic susceptibilities. These vector potentials have been derived employing generalized sheet transition condition (GSTC) on the plane of the metasurface. The retrieval relations to obtain equivalent surface susceptibilities of such a surface are calculated by means of the calculated reflection and transmission coefficients due to TE polarized plane waves. Also, the transverse and longitudinal scalar electric potentials relative to the problem have been calculated with the aid of the Lorentz gauge. These vector and scalar potentials together are necessary to use in the mixed potential integral equation (MPIE) formulation of the method of moment (MoM) in an electromagnetic problem.

Coupling Analysis for Wires in a Cable Tray Using Circuit Extraction Based on Mixed-Potential Integral Equation Formulation

The China high-speed trains use cable trays to neatly arrange the cables running through the distributed systems. On one hand, they protect cables against external electromagnetic interference and reduce external radiation from the cables. On the other hand, the cable trays create waveguide structures that affect the coupling among cables inside the cable trays. To simplify coupling analysis for cables going through a cable tray, a lumped circuit model was built using admittance blocks extracted from a mixed-potential integral equation (MPIE) formulation. In the MPIE formulation, either the half-free space or the waveguide dyadic Green's function was used depending on the region where the cables were. A test case was investigated. Results by the proposed circuit model were validated by measurement and full-wave simulation results.

Efficient Computation of 1-D Periodic Layered Mixed Potentials for the Analysis of Leaky-Wave Antennas With Vertical Elements

An efficient mixed-potential integral equation formulation is proposed for the analysis of one-dimensional (1-D) periodic leaky-wave antennas (LWAs) based on planar stratified configurations with inclusions of arbitrarily oriented metallic or dielectric perturbations. Both the transverse and vertical components of the mixed-potential Green’s functions due to a 1-D phased array of dipoles in a layered medium are accelerated using suitable homogeneous-medium asymptotic extractions from the standard spectral series of Floquet harmonics. A novel acceleration procedure is applied for the computation of the vertical potentials whose extracted terms can be expressed as potentials from a 1-D phased array of half-line sources in a homogeneous medium. Their numerical calculation requires a suitable modification of the Ewald method, thus resulting in new modified spectral and spatial series, having Gaussian convergence even in the case of complex modes and improper harmonics. Numerical comparisons for the 1-D periodic potentials, both in the case of bounded and unbounded (e.g., leaky) harmonics, validate the efficiency and accuracy of the proposed acceleration technique. The method is illustrated and verified by determining the dispersion behavior of both bound and leaky modes for several LWA test cases.

Surface Impedance Modeling of Plasmonic Circuits at Optical Communication Wavelengths

A novel surface impedance model is developed to analyze plasmonic circuits supporting long range surface plasmon polaritons (LR-SPP). The analysis is carried out in two steps. First, a higher order approximation of the surface impedance is obtained for the metal strip through a 2-D analysis of the waveguide cross section. Second, the developed surface impedance boundary condition is incorporated in the mixed potential integral equation formulation of the problem, and the method of moments is employed to find the unknown surface current distributions on the strips carrying LR-SPP. In other words, in this method, the volumetric currents flowing inside the metal are substituted by a surface current model flowing on an infinitely thin strip characterized by the developed surface impedance model. This procedure reduces the number of unknowns significantly, in this way increasing the speed of simulation. Validity and accuracy of the proposed model are demonstrated by analyzing three popular LR-SPP circuits.

Enhanced MoM Analysis of the Scattering by Periodic Strip Gratings in Multilayered Substrates

The method of moments (MoM) is applied to the analysis of the scattering of a multilayered periodic strip grating by a plane wave with oblique incidence and arbitrary polarization. Although this problem has been traditionally solved by means of the MoM in the spectral domain, this is an approach which leads to the computation of slowly convergent infinite summations. In this paper, the problem is solved by means of the mixed potential integral equation (MPIE) formulation of the MoM in the spatial domain. While applying the MoM in the spatial domain, two improvements are introduced which lead to important CPU time savings. First, the multilayered periodic Green's functions are accurately interpolated in terms of Chebyshev polynomials. Second, half the integrals involved in the computation of the MoM matrix entries are obtained in closed form. As a consequence of these two improvements, the spatial domain version of the MoM presented in this paper turns out to be between one and two orders of magnitude faster than the conventional spectral domain version when basis functions that account for edge singularities are used in the modeling of the current density on the metallizations.

Spherical Probe-Fed Antennas with Parasitic Elements

The theory and experiment of a probe-fed conformal antenna with a conducting parasitic element mounted on a spherical multilayer structure are presented in this article. Linear Rao–Wilton–Glisson triangular basis functions are applied in a mixed-potential integral equation formulation. Current distributions on coaxial probe and conformal radiating elements are computed by using the proper spatial domain electrical dyadic Green's function and its asymptotic approximation. A prototype of such an antenna is fabricated and tested. The effect of the parasitic element on the input impedance, gain, and radiation patterns of the antenna is investigated. It is shown that the antenna characteristics are improved significantly with the presence of the stack. Good agreement is achieved between the results obtained from the proposed methods and measurement results.

Quasi 3-D Mixed Potential Integral Equation Formulation for Periodic Structures

This paper introduces the use of the quasi 3-D mixed potential integral equation formulation for structures with a periodicity in two dimensions. The quasi 3-D formulation allows to use both horizontal and vertical conductors, and dielectric inclusions. The key innovation is the use of an effective algorithm to calculate Periodic Green's Functions (PGFs) for vertical currents in the planar multilayered background medium. This algorithm retains the same scheme as for a single source Green's Function (GF). In both cases, the GFs are calculated via an inverse Fourier transform of the GFs in the spectral domain, which are known in closed form. In order to remedy the poor convergence for small source—observation distances, an asymptotic extraction is used, yielding a very compact algorithm, and acceleration techniques are applied. Simple guidelines are proposed to construct asymptotes having the required leading terms and a good convergence. It is shown that many acceleration algorithms widely used in literature are actually able to perform well only in specific circumstances. As far as the authors can see, there is still no general and robust technique, yielding highly accurate results under all circumstances. Several examples illustrate the efficiency of the technique.

A New Green's Function Formulation for Modeling Homogeneous Objects in Layered Medium

A new Green's function formulation is developed systematically for modeling general homogeneous (dielectric or magnetic) objects in a layered medium. The dyadic form of the Green's function is first derived based on the pilot vector potential approach. The matrix representation in the moment method implementation is then derived by applying integration by parts and vector identities. The line integral issue in the matrix representation is investigated, based on the continuity property of the propagation factor and the consistency of the primary term and the secondary term. The extinction theorem is then revisited in the inhomogeneous background and a surface integral equation for general homogeneous objects is set up. Different from the popular mixed potential integral equation formulation, this method avoids the artificial definition of scalar potential. The singularity of the matrix representation of the Green's function can be made as weak as possible. Several numerical results are demonstrated to validate the formulation developed in this paper. Finally, the duality principle of the layered medium Green's function is discussed in the appendix to make the formulation succinct.

MPIE/MoM Acceleration With a General-Purpose Graphics Processing Unit

In this paper, we describe an accelerated implementation of the Method of Moments (MoM). A framework is proposed, exploiting the graphics processing unit (GPU) computing power by means of the software platform compute unified device architecture (CUDA). The mixed-potential integral-equation formulation, applied to microstrip circuit modeling, is adopted, and both the impedance matrix computation and the linear system solution of the MoM are accelerated. The latter point has been handled by considering both a direct dense (LU-based) and an iterative sparse (bi-conjugate gradient stabilized) solver. The best suited method is selected automatically during a thresholding phase, which renders the impedance matrix as sparse as possible. The effectiveness of the GPU-enabled MoM has been extensively tested in the analysis of planar circuits and components. The results achieved confirm the validity, accuracy, and high performance of the proposed approach.

An enhanced integral-equation formulation for accurate analysis of frequency-selective structures

In this work, a very efficient mixed-potential integral-equation formulation is implemented for the rigorous analysis of multilayered structures with arbitrarily shaped two-dimensional periodic metallic and/or dielectric inclusions. Original acceleration techniques have been developed for the computation of the components of the scalar and dyadic Green's functions, based on different types of asymptotic extractions according to the potential considered. The theoretical approach and its computational convenience have been validated through different full-wave analyses concerning both scattering problems and complex-mode dispersive behaviors in various frequency-selective structures for microwave applications.

Considerations on Double Exponential-Based Cubatures for the Computation of Weakly Singular Galerkin Inner Products

Highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in Galerkin mixed potential integral equation formulations. Due to their unique ability to handle non-smooth kernels, the proposed integration schemes can safely replace (in a “plug-n-play” sense) the traditional Gauss-Legendre rules in the existing singularity cancellation and singularity subtraction methods. Numerical examples using RWG basis functions confirm the excellent performance of the proposed method.

Low-Frequency Surface Integral Equation Solution by Multilevel Green's Function Interpolation With Fast Fourier Transform Acceleration

A fast low-frequency solver for surface integral equations is presented. Multilevel Lagrange interpolation of the pertinent homogenous space Green's functions is employed for the factorization of the integral operators. For high computational efficiency, the resulting multilevel convolution/translation operators are diagonalized by fast Fourier transform. A fast adaptive multilevel scheme of the interpolation procedure makes the approach also suitable for broadband applications. The method is investigated for the solution of electric field and combined field integral equations. For the electric field integral equation, the dyadic and mixed-potential integral equation formulations are considered and discussed. Furthermore, the impact of extrapolation errors on the approximation accuracy of the respective Green's functions is analyzed. In several numerical examples, excellent efficiency with respect to computation time and memory requirements as well as good accuracy of the obtained results is demonstrated.

Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula

Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integration schemes utilize the so-called double exponential quadrature rule, originally developed for the integration of functions with singularities at the endpoints of the associated integration interval. The final formulas can easily be incorporated in the context of singularity subtraction, singularity cancellation and fully-numerical methods, often used for the evaluation of multidimensional singular integrals. The performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods.

On the Direct Evaluation of Weakly Singular Integrals in Galerkin Mixed Potential Integral Equation Formulations

Weakly singular integrals over coincident triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, are calculated using a direct evaluation method. The proposed method utilizes a series of coordinate transformations, together with a re-ordering of the integrations, in order to reduce the dimensionality of the original four-dimensional (4D) weakly singular integrals into 1D numerical integrations of smooth functions. The final formulas can be easily evaluated with a standard Gaussian quadrature rule, resulting in a scheme with great accuracy and efficiency properties. Numerical results for the comparison of the proposed method with both singularity subtraction and singularity cancellation methods, often used for the evaluation of multidimensional singular integrals, are presented, indicating the superior overall performance of the direct evaluation scheme.

Numerical Computation of the Space-Domain Mixed Potential Green's Functions for Planar Layered Structures With Arbitrarily Magnetized Ferrites

This work reports a numerical method to generate the space-domain Green's functions used in the mixed potential integral equation (MPIE) formulation for planar structures when arbitrarily magnetized ferrite slabs are present. The presence of ferrite slabs in planar layered structures has been previously accounted for in the framework of the spectral-domain electric field integral equation (EFIE) formulation of the problem. The proposed numerical approach will allow the MPIE formulation to deal this kind of substrates. Asymptotic extraction and interpolation techniques have been used in order to keep the computational effort within reasonable limits.

A fast multipole method for layered media based on the application of perfectly matched layers - the 2-D case

An efficient fast multipole method (FMM) formalism to model scattering from two-dimensional (2-D) microstrip structures is presented. The technique relies on a mixed potential integral equation (MPIE) formulation and a series expression for the Green functions, based on the use of perfectly matched layers (PML). In this way, a new FMM algorithm is developed to evaluate matrix-vector multiplications arising in the iterative solution of the scattering problem. Novel iteration schemes have been implemented and a computational complexity of order O(N) is achieved. The theory is validated by means of several illustrative, numerical examples. This paper aims at elucidating the PML-FMM-MPIE concept and can be seen as a first step toward a PML based multilevel fast multipole algorithm (MLFMA) for 3-D microstrip structures embedded in layered media.

Crosstalk Between Two Microstrip Lines Excited by a Gap Voltage Source

The crosstalk between two microstrip lines is studied when one of the lines (the source line) is excited by a gap voltage source. In particular, the current induced on the passive line (the victim line) due to electromagnetic coupling to the source line is studied as a function of line separation and frequency for different permittivities. Results are also presented for the case of the source line excited by a vertical electric dipole to explore the effect of different source excitations on the crosstalk current. The current is calculated using a semianalytical method, which allows for an examination of the constituent current components on the lines (the bound-mode and continuous-spectrum currents) so that the physical mechanisms of coupling can be explored. The calculation is performed in an efficient manner using a mixed-potential integral-equation formulation with complex images.

Numerical and Experimental Analysis of Circular-End Wire Bow-Tie Antennas Over a Lossy Ground

This paper presents a numerical and experimental analysis of a wire bow-tie antenna situated horizontally near a lossy ground. The antenna is basically an array of identical wire dipoles having a common feed point and equal angular separation between two neighboring wires. This antenna is particularly suitable for ultrawide-band applications as it possesses a wideband property and allows a simple realization of resistive loading. It also provides a possibility to easily vary the antenna flare angle for the purpose of antenna matching or to adjust its footprint. In this work a theoretical model of the antenna is developed and verified experimentally. The wires are modeled as a tape-like structure by triangular patches and the antenna is analyzed using a mixed-potential integral equation formulation. Moreover, time-domain solutions are computed using the Fourier transformation, and a time-window technique is employed to calculate the antenna characteristic impedance. Using the model a comprehensive analysis of the antenna performance is carried out.

A fast technique based on perfectly matched layers for the full-wave solution of 2-d dispersive microstrip lines

A two-dimensional mixed-potential integral equation formulation is used to analyze the eigenmodes of microstrip lines. The new method involves a fast evaluation scheme for the space domain Green's functions of the substrate by using perfectly matched layers to obtain closed-form expressions as a series of leaky and Berenger modes. Efficient summation of these series is performed by means of the Shanks transform. The modal series can be applied to calculate all field interactions analytically, except for the selfpatch and the nearest neighbor contributions. Examples show an important reduction in CPU time for the new perfectly matched layers approach, as compared to the classical evaluation of the continuous Sommerfeld integrals.