Multilayered Green Research Articles

SLIM: A Well-Conditioned Single-Source Boundary Element Method for Modeling Lossy Conductors in Layered Media

The boundary element method (BEM) enables the efficient electromagnetic modelling of lossy conductors with a surface-based discretization. Existing BEM techniques for conductor modelling require either expensive dual basis functions or the use of both single- and double-layer potential operators to obtain a well-conditioned system matrix. The associated computational cost is particularly significant when conductors are embedded in stratified media, and the expensive multilayer Green's function (MGF) must be invoked. In this work, a novel single-source BEM formulation is proposed, which leads to a well-conditioned system matrix without the need for dual basis functions. The proposed single-layer impedance matrix (SLIM) formulation does not require the double-layer potential to model the background medium, which reduces the cost associated with the MGF. The accuracy and efficiency of the proposed method is demonstrated through realistic examples drawn from different applications.

Hyperbolic model of internal solitary waves in a three-layer stratified fluid

We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional instantaneous variables. This allows one to reduce the dispersive multi-layer Green–Naghdi model to a first-order system of evolution equations. The main attention is paid to the study of three-layer flows over an uneven bottom in the Boussinesq approximation with the additional assumption of hydrostatic pressure in the intermediate layer. The hyperbolicity conditions of the obtained equations of three-layer flows are formulated, and solutions in the class of travelling waves are studied. Based on the proposed hyperbolic and dispersive models, numerical calculations of the generation and propagation of internal solitary waves are carried out and their comparison with experimental data is given. Within the framework of the proposed three-layer hyperbolic model, a numerical study of the propagation and interaction of symmetric and non-symmetric soliton-like waves is performed.

Corrugated Waveguides With Stepped Ridges and Stratified Core for High-Power Transmission: Modal Analysis by Radial-Cavity and Concentric Multilayer Green’s Functions

This paper presents a full-wave modal analysis of transverse corrugated circular waveguides with stepped ridges accompanied by segmented grooves and a waveguiding core that is stratified into concentric layers. The method is validated with the known results from literature and those generated by independent solvers. Its strengths over contemporary numerical alternatives such as the finite-element method (FEM) are also portrayed. Such a topology facilitates the study of ways for reducing metal losses and enhancing the aperture efficiency of the operational mode, especially in high-power high-frequency microwave applications, such as by implementing dielectric linings within the grooves or ceramic coatings over the surface of the corrugations, constituting just two cited examples amongst the many other forms of measures for potential performance enhancement which the technique is able to treat.

Leakage Inductance Analytical Calculation for Planar Components With Leakage Layers

Planar magnetic components are promising solutions for the integration of power electronic systems. The leakage inductance of such components plays an essential role in power converters. In this paper, an analytical modeling method for leakage inductance computation is developed for planar components with plastoferrite leakage layers. This method is based on the solution of Poisson's equations for magnetostatic using multilayered Green's functions. The obtained formulations are general and precise and have been validated by numerical tests. Experimental characterizations have been performed on two magnetic components: A planar LLC and planar common-mode choke. The obtained results show that with the described method, the static leakage inductance of planar components can be accurately estimated.

Graphene plasmonics for tuning photon decay rate near metallic split-ring resonator in a multilayered substrate.

Study of photon decay rate is essential to various optical devices, where graphene is an emerging building block due to its electrical tunability. In this paper, we study photon decay rate of a quantum emitter near a metallic split-ring resonator, which is embedded in a multilayered substrate incorporating a graphene layer. Analyzing photon decay rate in such a complex multilayered system is not only computationally challenging but also highly important to experimentally realizable devices. First, the dispersion relation of graphene plasmonics supported at a dieletric/graphene/dielectric structure is investigated systematically. Meanwhile, the dispersion relation of metallic plasmonics supported at a dielectric/metal structure is studied comparatively. According to our investigation, graphene offers several flexible tuning routes for manipulating photon decay rate, including tunable chemical potential and the emitter's position and polarization. Next, considering plasmonic waves in a graphene sheet occur in the infrared regime, we carefully design a metallic split ring resonating around the same frequency range. Consequently, this design enables a mutual interaction between graphene plasmonics and metallic plasmonics. The boundary element method with a multilayered medium Green's function is adopted in the numerical simulation. Blue-shifted and splitting resonance peaks are theoretically observed, which suggests a strong mode coupling. Moreover, the mode coupling has a switch on-off feature via electrostatically doping the graphene sheet. This work is helpful to dynamically manipulate photon decay rate in complex optical devices.

Modal Analysis of Substrate Integrated Waveguides With Rectangular Via-Holes Using Cavity and Multilayer Green's Functions

The substrate integrated waveguide (SIW) comprising rectangular via-holes is herein treated by a rigorous full-wave modal analysis using the moment method entailing Green's functions for rectangular cavities and planar multilayer structures in the spectral domain. Modal dispersion graphs generated by this solution approach are compared with those obtained by an independent commercial full-wave solver. The ability of the modal solution in treating SIW junctions and discontinuities is demonstrated by the treatment of an interconnection between a conventional waveguide and a SIW using the mode-matching technique. Inhomogeneities within SIWs can also be accommodated by the technique, as showcased by a specific example of the so-called hard sidewalled SIW. Three avenues of losses, namely, dielectric, side-leakage, and conductor losses, are also investigated, thereby elucidating a tradeoff between the latter two.

Fast Simulation of Complicated 3-D Structures Above Lossy Magnetic Media

A fast numerical method is presented for the simulation of complicated 3-D structures, such as inductors constructed from Litz or stranded wires, above or sandwiched between the planar lossy magnetic media. Making use of its smoothness, the quasi-static multilayer Green's function is numerically computed using finite differences, and its source height dependence is computed using an optimal Toeplitz-plus-Hankel decomposition. We show that a modified precorrected fast Fourier transform method can be applied to reduce the dense linear algebra problem to near-linear time, and that frequency-dependent setups can be avoided to result in a considerable speed-up. Experimental verifications are made for a 16-strand Litz wire coil realistically modeled down to each individual strand. Results are obtained in 2-3 h, showing an excellent agreement to measurements, and can be used to study the impact of transposition patterns in Litz wire construction.

Singularity extraction for reflected Sommerfeld integrals over multilayered media

A new analytical-numerical method based on singularity extraction is presented for efficient and accurate evaluation of one of the reflected Sommerfeld integrals over multilayered media backed by a conducting (PEC) plane. Method is based on extraction of all the poles of the integrand. The method is demonstrated on one element of the tensor multilayered Green's function, where only the singular part of the integrand, i.e. the reflection coefficient, is taken into consideration, as the cause of the largest calculation difficulties. It was also shown that the method practically eliminates all the branch cut singularities except for one.

An Accurate and Efficient Evaluation of Planar Multilayered Green's Functions Using Modified Fast Hankel Transform Method

When the fast Hankel transform (FHT) filter technique is used to calculate the spatial-domain Green's functions for planar multilayered media, it can be difficult to obtain accurate numerical results because of branch-cut singularities and the surface-wave pole singularities. Although the singularities can be efficiently avoided through deforming the integration path of the Hankel transform from the real axis to the first quadrant in the complex plane, the argument of the integral kernel becomes a complex number so that the FHT filter algorithm cannot be directly applied. The modified FHT filter algorithm is proposed to overcome this problem by expressing the Bessel function with a complex argument as a sum of terms of product of Bessel function with the real part of the argument and Bessel function with the imaginary part of the argument. The FHT filter technique can then be applied to each expansion term. Numerical results confirm that the proposed approach has high accuracy and good efficiency in the near and intermediate fields. More importantly, it successfully extends the applicability of the conventional FHT method for general multilayered geometries.

Hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of cavities with multilayer stratified dielectric coating

This work presents a hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with multilayer stratified dielectric coating. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing a closed-form multilayer Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. The closed-form Green's function in the spatial domain for multilayer planar coating is expressed in terms of complex images using the generalized pencil-of-function method in conjunction with a two-level sampling approach. Placing the truncation boundary at the opening of the cavities and inside the dielectric coating results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for optimization problems involving scattering from grating surfaces. The near fields are generated for array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.

Susceptibility Analysis of Printed Circuit Boards Within Cavity Enclosures

This article considers the susceptibility of enclosed printed circuit board structures subject to external electromagnetic fields using a simple and efficient analysis. The method employs the magnetic currents on the nonmetallic surface of the cavity enclosing the multi-layer printed circuit boards. By using magnetic currents, coupling from one layer of the cavity to the other is conducted. In this manner, a single cavity Green's function is needed, avoiding the time-consuming modeling of junctions and vertical interconnects associated with multi-layer Green's functions. To validate the proposed method, numerical results and parametric studies are presented.

Efficient Computation of the Off-Diagonal Elements of the Vector-Potential Multilayered Periodic Dyadic Green's Function

The authors focus on the efficient computation of the slowly convergent infinite series that lead to the off-diagonal elements of the vector potential multilayered periodic dyadic Green's function. Two different approaches based on Kummer's transformation are applied to the evaluation of these series. The well-known approach that makes use of the generalized pencil of functions (GPoF) and Ewald's method is the fastest approach, but it does not provide accurate results when the distance between the field point and any of the source points is close to zero. To avoid this problem, we present a novel approach based on the GPoF and the spectral Kummer-Poisson's method with higher-order asymptotic extraction. This latter approach is slightly slower than the former one, but it is accurate in the whole range of distances between the field point and the sources.

Fast Computation of Sommerfeld Integral Tails via Direct Integration Based on Double Exponential-Type Quadrature Formulas

A direct integration algorithm, based on double exponential-type quadrature rules, is presented for the efficient computation of the Sommerfeld integral tails, arising in the evaluation of multilayered Green's functions. The proposed scheme maintains the error controllable nature of the so-called partition-extrapolation methods, often used to tackle this problem, whereas it requires substantially reduced computational time. Moreover, the proposed method is very easy to implement, since the associated weights and abscissas can be precomputed. The overall behavior of the proposed method both in terms of accuracy and efficiency is demonstrated through a series of representative numerical experiments, where compared with one of the most proven methods available in the literature.

A Novel Skin-Effect Based Surface Impedance Formulation for Broadband Modeling of 3-D Interconnects With Electric Field Integral Equation

The problem of interconnect modeling embedded in multilayered substrate is initially formulated in terms of the volume integral equation (IE) with respect to 3-D conduction current density. One out of three degrees of freedom in volumetric current variation is then eliminated by approximating the current behavior over the coordinate normal to the conductor surface according to the skin-effect. The remaining two degrees of freedom in the volumetric current variation constitute the unknown current distribution on the conductor surface for which a governing surface electric field integral equation is obtained directly from the volume IE via restriction of the volumetric operator's range to the conductor surface. The resultant surface IE features a global to the conductor cross section surface impedance operator, which is shown to approximate the relationship between tangential electric and magnetic field components on the conductor surface. The proposed novel surface IE featuring multilayered media dyadic Green's function is amenable to various discretization schemes including the Rao-Wilton-Glisson method of moments. In this paper the latter is implemented by casting the scattered field operator into Michalski-Zheng's mixed-potential form in conjunction with enforcement of global relationships between the basis and testing functions in conductor cross sections according to the surface impedance operator. Numerical comparisons to alternative conductor loss models show that the method achieves volumetric solution accuracy within the framework of a boundary-element formulation.

The square root depth wave equations

We introduce a set of coupled equations for multi-layer water waves that removes the ill-posedness of the multi-layer Green–Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear, they retain the same travelling wave solutions as MGN. We call the new model the square root depth ( ) equations from the modified form of their kinetic energy of vertical motion. Our numerical results show how the equations model the effects of multi-layer wave propagation and interaction, with and without shear.

Integral Equation Analysis of Scattering From Multilayered Periodic Array Using Equivalence Principle and Connection Scheme

An integral equation approach is presented to analyze the two-dimensional (2-D) scattering from multilayered periodic array. The proposed approach is capable of handling scattering from the array filled with different media in different layers. Combining the equivalence principle algorithm (EPA) and connection scheme (EPACS), it can be avoided to find and evaluate the multilayered periodic Green's functions. For 2N identical layers, the elimination of the unknowns between top and bottom surfaces can be accelerated using the logarithm algorithm. More importantly, based on EPACS, an approach is proposed to effectively handle the semi-infinitely layered case in which a unit consisting of several layers is repeated infinitely in one direction.

A Novel Autofocusing Approach for Real-Time Through-Wall Imaging Under Unknown Wall Characteristics

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A novel real-time through-wall imaging (TWI) algorithm with autofocusing ability in the presence of wall ambiguities is proposed in this paper. The spectrum Green's function is employed to formulate the TWI algorithm, where the fast Fourier transform can be used to reconstruct the image in a very short computation time. The complex scattering process due to the presence of the wall is automatically included in the imaging formulation through the multilayer Green's function. The autofocusing is achieved by introducing a time factor in the TWI formulation to get a dynamic image at different focusing time. The image at the time instant when the defined entropy is minimized is stored as the output of the TWI result. Simulation results show that the proposed method can provide high-quality focused image in a short computation time regardless of the estimated value of the wall parameters. </para>

A Technique for Efficient Evaluation of the Multilayered Green's Function for Frequency Sweep Analysis of Planar Microstrip Structures

A novel procedure for the evaluation of the multilayered Green's function and the related method of moment (MoM) matrix is presented in this letter. The proposed technique yields accurate results and reduces the computation time in comparison to the conventional approach by using spline-polynomials for the frequency-domain interpolation of planar wideband devices. Illustrative examples that demonstrate the reliability and efficiency of the described procedure are reported.

A Euler–Poincaré framework for the multilayer Green–Nagdhi equations

The Green–Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler–Poincaré theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.

AN IMPROVEMENT OF THE GE-ESSELLE'S METHOD FOR THE EVALUATION OF THE GREEN'S FUNCTIONS IN THE SHIELDED MULTILAYERED STRUCTURES

In this paper,we improve the Ge-Esselle's (GE's) method and apply it to calculate the multilayered Green's functions in the shielded structures. In the improved GE's method,1) the poles are first extracted using a recursive contour integration method; 2) then the general pencil-of-function (GPOF) is applied to approximate the part of the spectral-domain Green's functions (with the poles contributions being extracted) just along the real axis of the kρ plane instead of the rooftop shaped path defined in the original GE's method; 3) Subsequently,an analytical identity is employed to obtain the spatial- domain Green's functions. In step 2),a smoothing procedure is also applied here to eliminate the abrupt peak of the sampled spectral- domain Green's function caused by the finite machine accuracy of the poles locations. The numerical results in this paper show that the improved GE's method can accurately and efficiently calculate the Green's functions in the shielded multilayered structure.