Mixed Potential Integral Equation Research Articles

Mixed-Potential Integral Equation (MPIE) Formulation for Arbitrarily Shaped Conducting Objects in Plane-Stratified Uniaxial Media—A New Look

Mixed-Potential Integral Equation (MPIE) Formulation for Arbitrarily Shaped Conducting Objects in Plane-Stratified Uniaxial Media—A New Look

A Full-Wave Generalized PEEC Model for Periodic Metallic Structure With Arbitrary Shape

In this work, a full-wave partial element equivalent circuit (PEEC) model for 2-D periodic metallic structures is proposed. The mixed potential integral equations (MPIEs) with periodic Green’s function (GF) are interpreted as loop equations following Kirchhoff’s voltage law (KVL) in a circuit domain. The impact of gauge freedom in the MPIE on the corresponding PEEC model is discussed. A tailored gauge function is introduced to form a generalized PEEC model for periodic structures, which demonstrates superior passivity and frequency stability. With the proposed PEEC model, lumped and active elements can be easily integrated with conducting bodies for field-circuit co-simulation. Through the distributed circuit including coupling elements, the PEEC model provides a clear physical insight into the electromagnetic behavior of the physical models for design and optimization. Three numerical examples, including frequency-selective surface (FSS), array antenna, and pixelwise meta-atom, are given to illustrate the accuracy and efficiency of the proposed PEEC model.

FPGA Acceleration of Matrix-Assembly Phase of RWG-Based MoM

In this letter, the field-programmable gate array (FPGA) accelerated implementation of matrix-assembly phase of the method of moments (MoM) is presented. The solution is based on a discretization of the frequency-domain mixed potential integral equation using the Rao–Wilton–Glisson basis functions and their extension to wire-to-surface junctions. To take advantage of the given hardware resources (i.e., Xilinx Alveo U200 accelerator card), nine independent processing paths/runtime efficient compute units are developed and synthesized. Numerical results provided for a quadrifilar spiral antenna mounted on a conductive handset box show that the proposed parallelization scheme performs 9.53× faster than a traditional (i.e., serial) central processing unit (CPU) MoM implementation, and about 1.67× faster than a parallel six-core CPU MoM implementation.

Convergence of the MPIE Galerkin MoM Thin Wire Formulation

Convergence studies of the Method of Moments solution of thin wires have long been handicapped by the issue of conditional convergence: when the segment length approaches the wire radius, the approximations typically made result in exponentially diverging solutions. In this communication, it is shown that it is the approximation of the singularity which is the main cause of this problem. Using the mixed potential integral equation, paralleling the classic RWG formulation for surfaces, the formulation in this communication approximates the surface (as opposed to equivalent filamentary) current, and applies Galerkin testing on the wire surface. The use of contemporary singularity-canceling integral transforms for highly efficient and accurate quadrature is demonstrated. Results are presented for typical thin-wire dipoles for both radiation and scattering problems. It is shown that with a rigorous treatment of the singularity, a far more comprehensive convergence study may be undertaken than previously demonstrated. Furthermore, the results shown justify the many years of successful use in practice of simple source models with modest mesh requirements.

Direct Characterization of Grounding System Wide-Band Input Impedance

In this article, direct characterization of the wide-band input impedance of grounding system buried in multilayer soil is presented. The problem is formulated by mixed-potential integral equations solved by the method of moments (MoM). The method accommodates both delta-gap voltage and impressed-current excitations. The input impedance is then directly obtained from the MoM matrix. The proposed method does not rely on the integration of the electric field on the ground surface. Hence, the calculated input impedance is unique and path-independent. The accuracy of the proposed method is evaluated for the case of vertical ground electrodes buried in homogeneous soil. The cases of vertical grounding electrode buried in multilayer soil as well as square and circle-plates buried in homogeneous soil are studied. It is shown that for plate-shape grounding systems, the total surface area determines the grounding system input impedance and the effect of plate geometry is negligible.

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.

Method of Moments Based on Equivalent Periodic Problem and FFT with NURBS Surfaces for Analysis of Multilayer Periodic Structures

In this paper, an efficient technique of computation of method of moments (MM) matrix entries for multilayer periodic structures with NURBS surface and Bézier patches modelling is proposed. An approximation in terms of constant pulses of generalized rooftop basis functions (BFs) defined on Bézier patches is proposed. This approximation leads discrete convolutions instead of usual continuous convolution between Green’s functions and BFs obtained by the direct mixed potential integral equation (MPIE) approach. An equivalent periodic problem (EPP) which contains the original problem is proposed to transform the discrete convolutions in discrete cyclic convolutions. The resultant discrete cyclic convolutions are computed by efficiently using the Fast Fourier Transform (FFT) procedure. The performance of the proposed method and direct computation of the MM entries are compared for phases of reflection coefficient. The proposed method is between 9 and 50 times faster than the direct computation for phase errors less than 1 deg. The proposed method exhibits a behaviour of CPU time consumption of O(NbLog10Nb) as the number Nb of BFs increases. This behaviour provides significant CPU time savings with respect to the expected behaviour of O(Nb2) provided by the direct computation of the MM matrix entries.

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.

Singularity Expansion Method for thin wires and the Method of Modal Parameters

Abstract. Here, we describe a technique to define the Singularity Expansion Method (SEM) poles for short-circuited thin-wire structures developed using the Method of Modal Parameters (MoMP). The MoMP method consists of in the expansion of the system of mixed-potential integral equations (MPIE) into the Fourier series, including the kernels containing Green's function. Corresponding equations for Fourier modes contain infinite matrices of p.u.l. inductance and capacitance, and the solution for current can be obtained using the infinity matrix of p.u.l. impedance. The SEM poles are given by the zeros of the determinant of this matrix. For the case of the symmetrical circular loop, this equation transforms to one well-know from the literature. Numerical investigation of solutions for the poles of the first layer has shown good agreement with previously obtained analytical and numerical results for different wire configurations.

Novel two‐dimensional periodic Green's function for the efficient analysis of structures made of elements perpendicular to the dielectric substrate by complex images

Here, a simple, accurate and rapidly convergent method based on complex images (CI) technique is developed to derive a closed-form Green's function for periodic structures perpendicular to a substrate. The Green's function is represented in a two criteria form, based on the distance of the field point from the source point. That is, for the small field point to source point distances, it is given in terms of a CI representation, while for the larger field point to source point distances, the contribution of the poles in the spectral domain function is used through the application of the residue theorem. The main advantage of the derived Green's function lies on its accuracy, ease of computation, and fast convergence. The usage of the derived closed-form Green's functions in the mixed potential integral equation (MPIE), an efficient method for analysing periodic structures, has been introduced. The proposed method is applied to several examples, and the results are in good agreement with the simulations in commercial software.

A Generalized Multiple-Scattering Method for Modeling a Cable Harness With Ground Connections to a Nearby Metal Surface

This paper proposes a generalized multiple-scattering (GMS) method to evaluate the current distribution on a cable harness with ground connections to a nearby metal surface. The GMS method is a hybrid method combining the transmission line theory and the method of moments. The GMS method uses the generalized multiconductor transmission line (GMTL) solver for the cable harness part and the mixed-potential integral equation (MPIE) solver for the rest of the structure including the metal surface and the grounding wires. Neither the GMTL nor the MPIE solver alone takes into account the mutual interactions between the cable harness and the rest of the structure. Therefore, an iterative scheme is arranged in the GMS method to compensate the above-mentioned interactions. These interactions occur via not only field couplings, but also current conducting through the grounding points on the cable harness. A numerical test case is provided to benchmark the proposed GMS method.

A full wave description for thin wire structures with TLST and perturbation theory

Abstract. A full wave description of a thin wire structure, that includes mutual interactions and radiation, can be obtained in closed form with the so-called Transmission Line Super Theory or a refined variant of this method that utilises perturbation theory. In either procedure, a set of mixed potential integral equations is solved for the currents that propagate along a wire. With the perturbation approach, no iteration is required to approximate the initial current distribution on the wire. This procedure will be applied to solve multi-wire problems. The theory will be derived and computed results will be shown to be in good agreement with method of moment computations.

Harmonic Impedance of Grounding Electrodes Buried in a Horizontally Stratified Multilayer Ground: A Full-Wave Approach

The effect of multilayer stratified ground on the harmonic impedance of grounding systems is evaluated. To this aim, the mixed-potential integral equation is used. The method is general as it allows taking into account 1) stratified soil structure with any number of layers and 2) grounding electrodes with arbitrary shapes. Moreover, the method has no limitation in terms of the maximum working frequency. The impressed-current model is used as the excitation and the harmonic impedance is directly obtained from the method of moments matrix. This is in contrast with the conventional methods where the harmonic impedance is calculated from the integration of the electric field at ground surface. The cases of horizontal and vertical electrodes buried in soils with different layers are analyzed. The accuracy of the proposed approach is validated by comparing the results with those obtained by independent numerical methods.

Transmission Line Reflection Coefficient Including High-Frequency Effects

This paper deals with the analytic determination of the current reflection coefficient for a wire above ground. The trajectory of the wire at the wire terminal is arbitrary and the wire is connected to the ground with an impedance. The reflection coefficient is approximated by an iterative approach for the solution of the so-called mixed potential integral equations. The iteration is initialized by the classical transmission line solution. The tangential electric field on the surface of the thin wire corresponding to the classical solution is then used as a distributed source to determine a correction term for the current. By this way, the scattering of the transverse electromagnetic modes at the discontinuity is included. Very good agreement between the analytic solution and a numerical one is observed. It is shown that broadband matching is not possible with a frequency-independent load. The importance of the incorporation of the high-frequency effects is highlighted by the evaluation of the natural frequencies for two specific wire configurations.

Fast Time-Domain Analysis of a Metallic Enclosure With Arbitrary-Shaped Apertures by Marching-on-in-Degree

In this paper, an efficient time-domain technique is introduced for transient analysis of a rectangular metallic enclosure with arbitrary-shaped apertures. The surface equivalent theorem is applied to divide the problem into two separated regions. Then, assuming the magnetic current density on the apertures surface as the unknown of the problem, the governing mixed potential integral equation (MPIE) is derived. Here, to solve the resultant MPIE, we use a marching-on-in-degree (MOD) scheme with the weighted Laguerre polynomials as the entire-domain temporal basis functions. As well, in order to treat the arbitrary-shaped apertures, Rao–Wilton–Glisson spatial basis function is applied. In addition, to improve the efficiency of the proposed method, we apply some accelerating techniques that make the developed MOD method more efficient than the well-known Computer Simulation Technology finite integration technique, even for highly resonating enclosures.

A Fast IE-FFT Algorithm to Analyze Electrically Large Planar Microstrip Antenna Arrays

A fast integral-equation fast Fourier transform (IE-FFT) algorithm is developed, analyzed, and applied to the electromagnetic solution of radiation problems. The solution begins by the method-of-moments solution of a mixed potential integral equation on electrically large antenna arrays. Similar to other grid-based algorithms, the IE-FFT uses a Cartesian grid to significantly reduce memory requirement and speed up CPU time for both matrix assembly and matrix-vector multiplication when used with an iterative solver. The IE-FFT algorithm employs two discretizations, one for unknown current on an unstructured triangular mesh and the other on a uniform Cartesian grid for interpolating the layered medium's Green's functions. The uniform interpolation of the Green's functions allows us a global FFT for well-separated far-interaction terms. However, the near-interaction terms should be effectively corrected. A simple and efficient Lagrangian interpolation of the Green's functions is proposed. For a 2.5-D planar layered structure problem, the complexity of an IE-FFT is found to approximately $O(N)$ and $O({N\,{\rm{log}}\,N})$ for memory and CPU time, respectively.

Notice of Retraction: Direct Characterization of Grounding System Wide-Band Input Impedance

In this article, direct characterization of the wide-band input impedance of grounding system buried in multilayer soil is presented. The problem is formulated by mixed-potential integral equations solved by the method of moments (MoM). The method accommodates both delta-gap voltage and impressed-current excitations. The input impedance is then directly obtained from the MoM matrix. The proposed method does not rely on the integration of the electric field on the ground surface. Hence, the calculated input impedance is unique and path-independent. The accuracy of the proposed method is evaluated for the case of vertical ground electrodes buried in homogeneous soil. The cases of vertical grounding electrode buried in multilayer soil as well as square and circle-plates buried in homogeneous soil are studied. It is shown that for plate-shape grounding systems, the total surface area determines the grounding system input impedance and the effect of plate geometry is negligible.

Equivalent Surface Impedance-Based Mixed Potential Integral Equation Accelerated by Optimized $\cal {H}$ -Matrix for 3-D Interconnects

The equivalent surface impedance (ESI)-based mixed potential integral equation (MPIE) is proposed in this paper for parameter extraction of 3-D interconnects. Boundary integral equations (BIEs) describing the conductor region and the nonconductor region are utilized to derive the ESI model, which incorporates with an MPIE to simplify the electromagnetic simulation. For large-scale problems, the solution of MPIE is accelerated by the hierarchical matrix ( $\mathcal {H}$ -matrix) algorithm. Since the interconnect problems usually have multiple ports, the method of moments discretization of MPIE leads to the matrix equation with multiple right-hand sides, which is efficiently solved by the $\mathcal {H}$ -LU-based direct solution. Procedures for $\mathcal {H}$ -matrix are optimized to improve the overall efficiency. The proposed method to optimize $\mathcal {H}$ -matrix benefits both the $\mathcal {H}$ -matrix construction and the $\mathcal {H}$ -LU procedures. The complexities of the CPU time and memory cost for the construction of the optimized $\mathcal {H}$ -matrix are of $\mathcal {O}(N\log N)$ , and the complexity for the direct $\mathcal {H}$ -LU solution is of $\mathcal {O}(N\log ^{2}N)$ . Numerical results demonstrate that the proposed method is both accurate and efficient in a broadband frequency, which is suitable for modeling of 3-D interconnects and on-chip passive structures.

Mixed-Potential Integral Equation for Full-Wave Modeling of Grounding Systems Buried in a Lossy Multilayer Stratified Ground

The paper presents a comprehensive approach for full-wave modeling of grounding systems buried in a multilayer stratified ground. The dyadic spectral domain Green's function associated with the multilayer stratified ground is first derived. This Green's function is then evaluated by numerical solution of Sommerfeld's integrals along the deformed path in the complex plane. The proposed method is general as it can deal with arbitrarily shaped grounding electrodes buried in soils with any number of layers. The case of horizontal, vertical, and oblique electrodes buried in soils with different layers is analyzed. The accuracy of the proposed approach is first validated by comparing the results with those obtained by NEC-4 for a homogeneous soil, and with those obtained by finite difference time domain method for a multilayer soil.

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.