Mixed Potential Integral Equation Research Articles

LOW-FREQUENCY EXCITATION OF LEAKY MODES IN A MICROSTRIP LINE WITH A TOP COVER

LOW-FREQUENCY EXCITATION OF LEAKY MODES INA MICROSTRIP LINE WITH A TOP COVERJ. BernalDepartment of Applied Physics 3University of SevilleETS de Ingenier´ia, Camino de los descubrimientos s/n, 41092-Seville,SpainF. MesaDepartment of Applied Physics 1University of SevilleETS de Ingenier´ia Inform´atica, Avda. Reina Mercedes s/n, 41012-Seville, SpainD. R. JacksonDepartment Electrical and Computer EngineeringUniversity of Houston, Houston, TX 77204-4005, USAAbstract—This paper studies the excitation of a physical leaky modein a covered microstrip structure at low frequencies. We calculate thecurrent excited in the line by a delta-gap voltage source via a full waveanalysis based on a mixed potential integral equation scheme. Thecurrent in the line is decomposed into its bound mode and continuousspectrum components. The bound mode component is associated withthe propagation effects whereas the continuous spectrum componentis associated with reactive and/or radiative effects and contains thecontribution of the leaky mode. Our analysis also includes a detailstudy of the dispersion relations of the bound and leaky modes alongwith their corresponding electric fields. At low frequencies, in thecovered microstripstructurewithalow topcover height, wehavefoundthat theboundmoderole is supersededby the leaky mode, inthe sensethat it is the leaky mode which partially or totally carries the signalenergy. Therefore, the spurious effects associated with the excitationof a leaky mode, which usually appear at high frequencies in open

Singularity Evaluation of the Straight-Wire Mixed-Potential Integral Equation in the Method of Moments Procedure

A rigorous treatment on the computation of the various integrals that arise in the method of moments (MoM) formulation of the straight-wire electric field integral equation is provided. For triangle basis functions along with delta function, pulse or triangle weights, particular attention is given to integrals whose integrands are weakly singular. A singularity extraction technique is employed that splits the integral under question into two parts: one that is numerically integrable and one that is analytically integrable. Closed-form approximations based on Taylor series techniques are also provided for the former. These approximations are very robust resulting in errors less than 0.1% when 2α/δ <; 1 ; here δ is the length of a single MoM segment and α is the wire radius. Results are compared with data from the literature to demonstrate the robustness of the presented approach for fat wires.

Scattering and Radiation from/by 1-D Periodic Metallizations Residing in Layered Media

An efficient technique is proposed to compute the scattering or radiation from/by 1-D periodic structures residing in a layered background medium. The technique is based on a mixed potential integral equation (MPIE) combined with the method of moments (MoM), solving for the unknown current density flowing within a unit cell of the periodic structure. The formalism requires the knowledge of the pertinent layered medium Green's functions with 1-D periodicity. Here, these Green's functions are derived in closed-form by invoking the perfectly matched layer (PML)-paradigm. The stationary phase method is applied to determine the far field of the infinite, periodic structure, leading to a series of Floquet modes. In addition, approximating this series leads to an efficient technique to estimate the scattering or radiation from/by large, but finite, periodic structures with a 1-D periodic character. The theory is illustrated and validated by means of various examples, stemming from scattering and radiation applications from/by antenna arrays residing on microstrip substrates. The efficiency of the method is also demonstrated.

Multiresolution preconditioned multilevel UV method for analysis of planar layered finite frequency selective surface

In this article, the planar layered finite frequency selective surface (FSS) is analyzed by the mixed potential integral equation. The multiresolution (MR) preconditioned multilevel UV method is applied to accelerate the matrix-vector multiplication. The MR preconditioning technique is based on the construction of MR basis functions, which are constructed as linear combinations of Rao-Wilton-Glisson (RWG) basis functions. Therefore, the MR-preconditioner allows direct application to the multilevel UV method with RWG basis functions to speed up the convergence of the generalized minimal residual iterative solver. Numerical results demonstrate that the MR preconditioned multilevel UV method is efficient for the analysis of planar layered finite FSSs. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1530–1536, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.25233

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.

Dual GPOF/DCIM for Fast Computation of Sommerfeld Integrals and EM Scattering From an Object Partially Embedded in Dielectric Half-Space

A novel approach of dual GPOF (general pencil of functions) combined with the DCIM (the discrete complex image method) is developed to accelerate the Sommerfeld integral evaluation, when the mixed potential integral equation (MPIE) is applied to numerically calculate scattering from a conductive object partially embedded in a dielectric half-space, where the field and source points are located in different media. Firstly, the factors related with the field point are separated from the spectrum function, and the GPOF is used to find the complex image parameters at finite discrete source points. Secondly, the GPOF is used again to fit the functional relationship of each complex image parameter with the source positions to evaluate the complex image parameters for any source position in a closed functional summation. Comparing numerical results of dual GPOF and direct numerical integration, this dual GPOF process is proved highly effective. Finally, dual GPOF/DCIM is applied to computation of electromagnetic scattering from a PEC spherical object partially embedded in a dielectric half-space medium, and numerical scattering results upon different model parameters are presented and analyzed.

High-Frequency Multiconductor Transmission-Line Theory

This work presents a thorough derivation of the full-wave transmission-line equations on the basis of Maxwell’s theory. The multiconductor system is assumed to be composed of nonuniform thin wires. It is shown that the mixed potential integral equations are equivalent to generalized telegrapher equations. Novel, exact, and compact expressions for the multiconductor transmission-line parameters are derived, and their connection to radiation effects is shown. Iteration and perturbation procedures are proposed for the solution of the generalized transmission-line equations.

Application of Two Mixed Potential Integral Equations to Electromagnetic-circuit Simulation of Three-dimensional Interconnects in Layered Media

Electromagnetic-circuit simulation of three-dimensional (3-D) arbitrary-shaped conducting objects over conducting substrate layers is addressed. The electromagnetic-circuit surface integral equations are presented for two mixed potential integral equation (MPIE) forms of electric field integral equation (EFIE) based on the Michalski-Mosig formulation where different sets of layered kernels are involved. Various new ways of applying external voltage and current sources are discussed in detail. Numerical results are presented for validation of the proposed methods.

Transient analysis of thin-wire structures above a multilayer medium using complex-time Green's functions

The transient response of thin-wire structures in the presence of a multilayer medium is investigated by means of a new formulation of the method of moments in the time domain (MoM-TD). The proposed mathematical model is based on a novel representation of the time-domain mixed potential integral equation (TD-MPIE) for thin-wire structures located above a multilayer medium through recently developed closed-form time-domain Green&apos;s functions called complex-time Green&apos;s functions. The time-domain integral equation is solved by using MoM-TD along with marching-on-in-time (MOT) procedure resulting in the space–time current distribution on the thin-wire structure being stimulated by an arbitrary time-varying source. The numerical results are verified by comparison with other known methods. Accuracy, stability and numerical efficiency are the main advantages of the developed method.

Direct use of discrete complex image method for evaluating electric field expressions in a lossy half space

A modelling technique is proposed for direct use of the discrete complex image method (DCIM) to derive closed-form expressions for electric field components encountered in the electric field integral equation (EFIE) representing a lossy half space problem. The technique circumvents time consuming numerical computation of Sommerfeld integrals by approximating the kernel of the integrals with appropriate mathematical functions. This is done by appropriate use of either the least-square Prony (LS-Prony) method or the matrix pencil method (MPM) to represent electric field expressions in terms of spherical waves and their derivatives. A comparison is made between the two methods based on the computation time and accuracy and it is shown that the LS-Prony method performs two–three times faster than the MPM in approximating the integral kernels depending on the platform. The main feature of the proposed technique is its ability for direct inclusion in the kernel of computational tools based on the method of moments solution of the EFIE. This can be viewed as an advantage over the conventional DCIM approximation of spatial Green&apos;s functions for mixed potential integral equation for cases where the problem in hand can be more efficiently represented by the EFIE (e.g. the thin-wire EFIE). The accuracy of the proposed technique is validated against numerical integration of Sommerfeld integrals for an arbitrary electric dipole inside a lossy half space.

A Hybrid Method for Predicting the Shielding Effectiveness of Rectangular Metallic Enclosures with Thickness Apertures

An efficient hybrid method is presented to predict the shielding effectiveness of rectangular metallic enclosures with the finite thickness apertures on the front panels. The model is mainly divided into three regions: the half free space, slot waveguide and rectangular cavity respectively. The mode-matching technique and mixed potential integral equation based on the method of moments are applied to this model. The numerical results are validated by comparisons with those obtained by means of measurement. Furthermore, this method has been extended to account for the effects of finite thickness apertures and multiple apertures as well. The results of the shielding effectiveness varying with the thickness of apertures and multiple apertures have been given in this paper. Then the conclusions could be claimed that with the thickness of the apertures growing, the shielding effectiveness improves, and when a fixed area is divided into some small apertures, the shielding effectiveness can be improved too.

A Three-Dimensional Precorrected FFT Algorithm for Fast Method of Moments Solutions of the Mixed-Potential Integral Equation in Layered Media

A three-dimensional pre-corrected fast Fourier transform (PFFT) algorithm for the rapid solution of the full-dyadic Michalski-Zheng's mixed potential integral equation is presented. The integral equation is discretized with the Rao-Wilton-Glisson (RWG) method of moments. Handling the method of moments interactions with the dyadic kernel is simplified via representation of the RWG functions in terms of barycentric shape functions. The proposed three-dimensional precorrected FFT method distributes two-dimensional FFT grids nonuniformly along the direction of stratification according to conductor locations within the layers. For P two-dimensional FFT grids each with an average of Np associated triangular elements the method exhibits O(P 2 Np logNp) computational complexity and O(P Np) memory usage. The low-frequency breakdown of the integral equation is eliminated via loop-tree decomposition. A unique combination of O(N logN) computational complexity, fully three-dimensional boundary-element modeling in layered substrates, and full-wave modeling from dc to multi-gigahertz frequencies makes the algorithm particularly useful for characterizing large interconnect networks embedded in multilayered substrates. The method is implemented as the electromagnetic solver in Cadence's Virtuoso RF Designer software.

Integral Equation Modeling of Waveguide-Fed Planar Antennas

This paper presents a method for the analysis of planar multilayered waveguide-fed antennas. The method combines mixed-potential integral equations (for laterally open regions) and modal-field integral equations (for laterally closed regions) with a seamless transition between the two domains. The method has been implemented in a numerical tool, and the simulation results of two waveguide-fed planar structures are presented. The results are in good agreement with both measurements and simulations obtained with other commercial electromagnetic tools. Comparisons in terms of memory utilization and simulation time have also been performed.

Generalized Partial-Element Equivalent-Circuit Analysis for Planar Circuits With Slotted Ground

A generalized partial-element equivalent-circuit (PEEC) method is proposed for modeling a planar circuit with a thin narrow slot on the ground. The approach is based on the coupled mixed potential integral equations for a problem with mixed electric and magnetic currents. The coupled integral equations are converted into a lumped-element circuit network using Kirchhoff's voltage law and Kirchhoff's current law of the circuit theory. The full-wave Green's functions for a grounded dielectric substrate problem are used. The interactions between electric current on a microstrip line and magnetic current on a slot are taken into account by introducing two kinds of controlled sources. This generalized PEEC model will be very useful in signal-integrity analysis for multilayered circuits. To validate the generalized model, three numerical examples consisting of microstrip lines and slots on the ground are presented. The results obtained by the proposed generalized PEEC model are compared with those obtained by commercial electromagnetic simulation software and published experimental results. Good agreement is obtained.

Analytical Evaluation of the Quadruple Static Potential Integrals on Rectangular Domains to Solve 3-D Electromagnetic Problems

In this communication, original analytical expressions for the quadruple static potential singular integrals on rectangular domains are presented. These integrals can appear when the method of moments is used to solve the mixed potential integral equation but also the electric and the magnetic static integral equations. The analytical expressions have been obtained for both source and test cells integrals when using 3-D rectangular meshes. The reported expressions can be used to solve an extensive range of electrostatic, magnetostatic and full-wave problems.

Efficient implementation of discrete complex images method in spatial domain moment method to analyze microstrip transmission lines

Planar guiding structures are analyzed using a mixed-potential integral-equation (MPIE) formulation solved by the method of moments (MoM). Closed-form two-dimensional Green's functions for the scalar and vector potentials are analytically obtained in the space domain due to the approximation of its spectral domain version with discrete complex images method (DCIM), thus avoiding lengthy numerical evaluations. It will be shown that a suitable choice of the sampling path is key to reducing the number of images and to avoid the necessity of extracting spectral poles. The advantages of this approach will be illustrated by numerical results.

SPATIAL-SPECTRAL FORMULATION OF METHOD OF MOMENT FOR RIGOROUS ANALYSIS OF MICROSTRIP STRUCTURES

In this paper, we present an efficient hybrid spatial- spectral formulation of the method of moment (MoM) in conjunction with the Mixed-Potential Integral Equation (MPIE) for planar circuit analysis. This method is based on the decomposition of the Green's functions in two parts: quasi-static in the near field region and the dynamic contribution in the far field region. Using this decomposition of Green's functions, the method of moment matrix entries can be reduced to a sum of two integrals. The first one is expressed in the spatial field and corresponds to the quasi-static contribution. It is analytically evaluated after a development in Taylor series of the exponential terms in the function to be integrated. The integrals expressed in the spectral field and corresponding to the dynamic part have the advantage of being calculated on a finite range and this is independent of the choice of the basis and test functions. The integrals expressed in the spectral field are performed by using numerical integration. It is also demonstrated that this hybrid method has accelerated the matrix fill in time by using a Fast Fourier Transform (FFT) algorithm. In order to validate the proposed method, numerical results are presented.

A novel approach of dual GPOF /DCIM for fast computation of the sommerfeld integrals and electromagnetic scattering from an object partially embedded in dielectric half-space

The mixed-potential integral equation （MPIE） has been usually employed in numerical approach of electromagnetic scattering of the object, such as the method of moment （MoM）, due to its low-level singularity of vector and scalar potential Green functions. When an object is embedded partially in dielectric half-space medium, the Green’s function contains the Sommerfeld-type integrals, which embody the effect of the dielectric interface on scattering fields. Using the discrete complex image method （DCIM） and the Sommerfeld identity, the Sommerfeld integrals can be evaluated as the summation of finite complex image functions without directly numerical integration which always consumes large CPU time. As the points of the field and source are co-located in the same side of the interface, the spectrum function g（kzp） is not related with the field or source positions, and the complex image parameters fitted with the general pencil of functions （GPOF） method are approximate for all positions. However, if the points of the field and source are located, separately, in different sides of the interface, the spectral function is now related to z and z′. Generally, the GPOF is repetitiously used to find the complex image parameters for every z and z′, which consumes large CPU time and memory. This paper presents a novel method of Dual GPOF combining with DCIM for fast computation of the Sommerfeld integral. Firstly, the factors related with the variable z are separated, and the GPOF is used to find the complex image parameters for finite discrete source points z′l. Secondly, GPOF is used again to fit the relationship of each complex image parameter with variable z′. Then, the complex image parameter of any z′ can be evaluated as the direct function summation, and there is no need to perform GPOF for all z′ points. Comparing numerical values of Dual GPOF, point-by-point GPOF, and direct numerical integration, the Dual GPOF method is proved effective and efficient. Finally, Dual GPOF is applied to computation of electromagnetic scattering from a P.E.C. sphere object partially embedded in dielectric half-space, and the scattering patterns are presented and analyzed.

Regularization of Mixed-Potential Layered-Media Green's Functions for Efficient Interpolation Procedures in Planar Periodic Structures

The problem of improving the computational efficiency in the numerical analysis of planar periodic structures is investigated here using the mixed-potential integral-equation (MPIE) approach. A new regularization of the periodic Green's functions (PGFs) that are involved in the analysis of multilayered structures is introduced, based on the effective-medium concept. This regularization involves extracting the singularities of the PGFs up to second-order terms. The resulting regularized PGF is very smooth and amenable to interpolation. Thus, optimized interpolation procedures for the PGFs can be applied, resulting in a considerable reduction of computation time without any significant effect on the accuracy. Another benefit of the regularization is that it significantly enhances the convergence of the series for both the vector- and scalar-potential PGFs. The theoretical formulation is fully validated with various numerical results for both two-dimensional (2-D) and one-dimensional (1-D) layered-media periodic structures.

PEEC Formulation Based on Dyadic Green's Functions for Layered Media in the Time and Frequency Domains

This paper presents a novel time- and frequency-domain concept of modeling with the partial element equivalent circuit (PEEC) method, which applies the mixed potential integral equation (MPIE) with dyadic Green's functions for layered media (DGFLM-PEEC). On the one hand, it represents an exact full-wave semianalytical solution for an arbitrary configuration of traces and via holes in multilayered printed circuit boards. On the other hand, the DGFLM-PEEC model is represented in a circuit form, and thus, may be included in general-purpose circuit simulators. The paper derives a general DGFLM-PEEC formulation, which may be applied to all types of the MPIE with dyadic Green's functions. Using this concept, a particular type of layered media, namely a lossy dielectric between two grounds (stripline region), is thoroughly investigated and used to set up a particular DGFLM-PEEC model. The closed-form expressions for partial inductances and potential coefficients have been derived for this case. The time- and frequency-domain DGFLM-PEEC models for the stripline region have been validated using the measurements and the simulation by the method of moments.