Equivalent Surface Current Densities Research Articles

Strain-Induced Electromagnetic Transmission Modulation via a Reconfigurable Kirigami Metasurface

Tunable three-dimensional (3D) electromagnetic (EM) metasurfaces are critical for dynamic modulation of EM responses but their construction and tuning mechanism are still complex. Here, we report a simple yet effective 3D reconfigurable EM metasurface, which was obtained from a planar kirigami polyimide substrate printed with periodically arranged copper split-ring resonator. Under mechanical stretch, the two-dimensional (2D) planar metasurface can be uniformly deformed into a 3D state, which is effective for tuning its EM transmission characteristic. By combining mechanics and EM simulations as well as experimental measurements, we revealed the deformation mode and active EM transmission modulation capability of the metasurface. It is shown that at the initial state, the planar kirigami metasurface exhibits ideal frequency selective transmission to transverse electric (TE) wave but allows for complete transmission for transverse magnetic (TM) wave. As the applied strain increases from 0% to 20%, the transmission was adjusted from −17.74[Formula: see text]dB to −9.74[Formula: see text]dB for TE wave but merely from 0[Formula: see text]dB to −3.25[Formula: see text]dB for TM wave. Meanwhile, the resonant frequency experienced a visible shift for both TE and TM waves. Finally, the equivalent circuit analysis and simulated surface current density were conducted to reveal the tuning mechanism of the proposed metasurface.

Metasurface-based polarization-insensitive absorber with dual-band tunable absorption intensity

In this paper, a polarization-insensitive absorber with dual-band tunable absorption intensity is proposed. Two absorption bands are generated by introducing external branches into a ring structure. Since the resistance of the loaded PIN diode changes with bias current, the reflectivity of two absorption bands can be dynamically adjusted from almost 0 dB to less than −10 dB at normal incidence. Inductors are loaded on both sides of the metal structure, which guarantees not only that the units are connected during DC feeding, but also the symmetry of the metal structure in RF band to ensure polarization insensitivity. The mechanisms of dual absorption bands and tunable absorption intensity are revealed through the detailed analysis of resonant modes, equivalent circuit model, surface current, and loss density distribution. A sample prototype has also been fabricated, and the measured results are well consistent with the simulated ones. The proposed absorber has the unique advantage of dynamic tuning of dual-band absorption intensity, making it a versatile choice in modern communication, radar, and stealth systems.

Absorptive/transmissive frequency selective surface with multiple transmission zeros for reduced transition bandwidth

An absorptive/transmissive frequency selective surface (ATFSS) with multiple transmission zeros is proposed, which consists of one lossy layer and two lossless layers separated by air space. As the key features, the slot-line loading and interlayer coupling technologies are adopted in the lossy layer and lossless layers, respectively, to introduce the infinite impedance points and four transmission zeros, thereby simultaneously achieving low insertion loss and high selectivity of the transmission band. Benefiting from the ohmic loss of the lumped resistance in the lossy layer, effective absorption can be readily implemented. A detailed qualitative analysis of the proposed structure is presented through the equivalent circuit theory, surface current, and power loss density distributions. Under normal incidence, the overall structure exhibits two discrete absorption bands over the frequency range of 6.34 to 13.40 GHz, whereas a transmission behavior is obtained from 9.34 to 11.40 GHz with a minimum insertion loss of only 0.14 dB. In particular, there are only 3.9% and 3% transition bandwidths between absorption and transmission functions. Then, a sample prototype is fabricated for demonstration, and the measured results are well consistent with the simulated one. As expected, the proposed ATFSS has the advantage of multiple transmission zeros, loss insertion loss, and narrow transition band, which is of great potential in the application of radar stealth radome.

Efficient Radiation Pattern Estimation of Slot Array Antennas Covered With Frequency Selective Surfaces Using Deterministic Ray Tracing

In this paper, we propose a novel algorithm to compute the radiation pattern of a slot array antenna covered with frequency selective surfaces (FSS) based on ray tracing technique. We observed that the proposed algorithm could greatly reduce computational resources while yielding accurate numerical solution, compared with cases taking standard full-wave and conventional high-frequency approaches (summarized in Table I). The underlying principle is to combine the image method and physical optics technique. More specifically, the proposed algorithm (i) employs a boundary box that encloses the slot array antenna with FSS to correct phase errors and then (ii) calculates equivalent surface currents flowing on the boundary box. The surface of the boundary box is discretized by regular grids (or boundary meshes) and a ray path reaching each square cell in the boundary meshes is determined based on the image method. After collecting all the rays hitting the boundary meshes, equivalent surface current densities can be computed for the resulting radiation pattern. The proposed algorithm is suited best for dealing with electrically large structures much more efficiently and can advance the existing design process.

Construction of the differential surface admittance operator with an extended Fokas method for electromagnetic scattering at polygonal objects with arbitrary material parameters

This article presents a novel method to accurately simulate electromagnetic scattering at homogeneous polygonal cylinders with arbitrary material properties. A single source equivalence approach is invoked, allowing to substitute the background medium for the inner material of the scatterer, provided an equivalent surface current density is introduced. We construct the pertinent differential surface admittance operator by means of the Fokas method, establishing a map between the known Dirichlet boundary values and their unknown Neumann counterparts. However, to allow for lossy materials, we extend the Fokas method to complex wavenumbers. The novel formalism, employing pulse-shaped local basis functions, natively supports combined magnetic and dielectric contrast, accurately captures the skin effect, and is conveniently integrated in traditional boundary integral equation formulations. The correctness and versatility of our technique are verified for various examples by means of analytical validation and through comparison with a Poggio-Miller-Chan-Harrington-Wu-Tsai approach, a volume integral equation method and a commercial solver.

Integral Modeling of AC Losses in HTS Tapes With Magnetic Substrates

This article presents an integral modeling approach for computing ac losses in high-temperature superconductors near magnetic materials. The current density and magnetic field distributions are calculated by solving Maxwell's equations in integral form in the active parts. Equivalent surface and interface current densities are considered in the magnetic material to take into account its magnetic field-dependent relative permeability. The proposed approach is applied to compute ac losses in a second-generation high-temperature superconducting tape with a magnetic substrate. The obtained results are compared with the finite element method using H formulation to check the validity and performances of the proposed integral approach.

Accurate Time-Domain Modeling of Arbitrarily Shaped Graphene Layers Utilizing Unstructured Triangular Grids

The accurate modeling of curved graphene layers for time-domain electromagnetic simulations is discussed in the present work. Initially, the advanced properties of graphene are presented, focusing on the propagation of strongly confined surface plasmon polariton waves at the far-infrared regime. Then, the implementation of an unstructured triangular grid was examined, based on the Delaunay triangulation method. The electric-field components were placed at the edges of the triangles, while two different techniques were proposed for the sampling of the magnetic ones. Specifically, the first one suggests that the magnetic component is placed at the triangle’s circumcenter providing more accurate results, although instability may occur for nonacute triangles. On the other hand, the magnetic field was sampled at the triangle’s centroid, considering the second technique, ensuring the algorithm’s stability, but further approximations were required, leading to a slight accuracy reduction. Moreover, the updating equations in the time-domain were extracted via an appropriate approximation of Maxwell equations in their integral form. Finally, graphene was introduced in the computational domain as an equivalent surface current density, whose location matches the corresponding electric components. The validity of our methodology was successfully performed via the comparison of graphene surface wave propagation properties to their theoretical values, whereas the global error determination indicates the minimal triangle dimensions. Additionally, an instructive setup comprising a circular graphene scatterer was analyzed thoroughly, to reveal our technique’s advantages compared to the conventional staircase discretization.

A 2-D differential surface admittance operator for combined magnetic and dielectric contrast

In this paper, we present a novel technique to accurately model scattering phenomena at two-dimensional circular and rectangular structures consisting of arbitrary homogeneous materials, including magnetic media in particular. The proposed formalism utilizes a differential surface admittance operator, which invokes a single source equivalence theorem to replace the inside material by its surrounding medium, while introducing an equivalent surface current density. The arbitrary magnetic contrast can be combined with an arbitrary electrical conductivity. As such, the skin effect is rigorously taken into account, making our method ideally suited for broadband modeling of good conductors as well. It is demonstrated that an appropriate choice of the basis functions for the discretized problem is critical to obtain a convergent result when magnetic contrast is introduced. The method is analytically validated for the case of a circular cylinder and additional numerical results illustrate the correctness of the technique for (combinations of) rectangular cylinders, through comparison with a Poggio-Miller-Chan-Harrington-Wu-Tsai approach, a volume integral equation method and a commercial solver.

3-D BEM Formulations for Eddy-Current Problems With Multiply-Connected Domains and Circuit Coupling

Quasi-static linear problems can be solved efficiently with Boundary Element Method (BEM). This method is based on surface integral equations dealing with equivalent magnetic and electric surface current densities. Many works have shown the potentiality of BEM especially for the modeling of non-destructive testing devices. In this paper, after selecting formulations enabling the modeling of multiply-connected regions, an original coupling is proposed in order to take into account external electric circuit in the problem.

A 3-D Differential Surface Admittance Operator for Lossy Dipole Antenna Analysis

This letter presents a novel approach to simulate materials with arbitrary properties, in particular good conductors, in a boundary integral equation context. The advocated differential surface admittance operator permits the replacement of the material by the background medium through the introduction of an equivalent surface current density. A formulation based on the eigenfunctions of the volume is constructed and successfully demonstrated through scattering at a conducting cylinder and through the analysis of a lossy dipole antenna.

New Single-Source Surface Integral Equation for Magneto-Quasi-Static Characterization of Transmission Lines Situated in Multilayered Media

We recently proposed a novel single-source integral equation (SSIE) for accurate broadband resistance and inductance extraction and current flow modeling in 2-D conductors. The new surface integral equation is advantageous compared with the traditional volume electric field integral equation (V-EFIE) used for the inductance extraction, since the unknown function is defined on the surface of conductors as opposed to the volumetric unknown current density in V-EFIE. The new SSIE is also more suitable for the solution of inductance extraction problems than the traditional surface integral equation formulations, as it features only a single unknown surface function as opposed to having the unknown equivalent electric and magnetic surface current densities. The new equation also features only the electric field Green’s functions unlike the previously known SSIE formulations. The latter property makes the new SSIE equation particularly suitable to the inclusion of the multilayered substrate effect into the inductance extraction model. This paper describes the generalization of the new SSIE formulation to the case of transmission line models embedded into the multilayered lossy substrates. This paper also shows how the matrix sparsity in the method of moments discretization of the novel integral equation can be exploited to accelerate its numerical solution and reduce associated memory use. This sparsity arises due to the skin-effect-based attenuation of the fields in conductors’ cross sections leading to vanishing levels of the matrix elements corresponding to the distant interactions. Typical examples of inductance extraction in complex interconnects situated in lossy substrate are considered to validate the proposed techniques against traditional approaches.

Explicit Torque and Back EMF Expressions for Slotless Surface Permanent Magnet Machines With Different Magnetization Patterns

Slotless permanent magnet machines are attractive in some modern drive and power generation fields, where the cogging torque and additional losses need to be minimized or removed. The stator slotless design can be combined with different surface permanent magnet (SPM) rotor topologies. In this paper, explicit analytical expressions are derived to analytically compute the slotless machine torque and no-load back Electro-Motive Force in the case of segmented SPM rotor with parallel, radial, or Halbach-array magnetization patterns. The expressions are found by solving the magnetic field due to the slotless stator winding and to the permanent magnet blocks; the latter modeled through equivalent surface current densities. The accuracy of the method is successfully assessed by comparison with the finite-element analysis (FEA). The proposed formulas are an effective alternative to the FEA to quickly compare different design solutions as well as to optimize them. Application examples are provided in which the presented method is adopted to define the machine cross section that maximizes the torque density.

Scattering of surface plasmons on graphene by a discontinuity in surface conductivity

The scattering of graphene surface plasmons from an arbitrary, one-dimensional discontinuity in graphene surface conductivity is treated analytically by an exact solution of the quasi-static integral equation for surface current density in the spectral domain. It is found that the reflection and transmission coefficients are not governed by the Fresnel formulas obtained by means of the effective medium approach. Furthermore, the reflection coefficient generally exhibits an anomalous reflection phase, which has so far only been reported for the particular case of reflection from abrupt edges. This anomalous phase becomes frequency-independent in the regime where the effect of inter-band transitions on graphene conductivity is negligible. The results are in excellent agreement with full-wave electromagnetic simulations, and can serve as a basis for the analysis of inhomogeneous graphene layers with a piecewise-constant conductivity profile.

Solution of 3D inverse scattering problems by combined inverse equivalent current and finite element methods

An approach combining boundary integral and finite element methods is introduced for the solution of three-dimensional inverse electromagnetic medium scattering problems. Based on the equivalence principle, unknown equivalent electric and magnetic surface current densities on a closed surface are utilized to decompose the inverse medium problem into two parts: a linear radiation problem and a nonlinear cavity problem. The first problem is formulated by a boundary integral equation, the computational burden of which is reduced by employing the multilevel fast multipole method (MLFMM). Reconstructed Cauchy data on the surface allows the utilization of the Lorentz reciprocity and the Poynting's theorems. Exploiting these theorems, the noise level and an initial guess are estimated for the cavity problem. Moreover, it is possible to determine whether the material is lossy or not. In the second problem, the estimated surface currents form inhomogeneous boundary conditions of the cavity problem. The cavity problem is formulated by the finite element technique and solved iteratively by the Gauss–Newton method to reconstruct the properties of the object. Regularization for both the first and the second problems is achieved by a Krylov subspace method. The proposed method is tested against both synthetic and experimental data and promising reconstruction results are obtained.

ELECTROMAGNETIC FIELD TRANSFORMATIONS FOR MEASUREMENTS AND SIMULATIONS (Invited Paper)

Electromagnetic field transformations are important for electromagnetic simulations and for measurements. Especially for field measurements, the influence of the measurement probe must be considered, and this can be achieved by working with weighted field transformations. This paper is a review paper on weighted field transformations, where new information on algorithmic properties and new results are also included. Starting from the spatial domain weighted radiation integral involving free space Green's functions, properties such as uniqueness and the meaning of the weighting function are discussed. Several spectral domain formulations of the weighted field transformation integrals are reviewed. The focus of the paper is on hierarchical multilevel representations of irregular field transformations with propagating plane waves on the Ewald sphere. The resulting Fast Irregular Antenna Field Transformation Algorithm (FIAFTA) is a versatile and efficient transformation technique for arbitrary antenna and scattering fields. The fields can be sampled at arbitrary irregular locations and with arbitrary measurement probes without compromising the accuracy and the efficiency of the algorithm. FIAFTA supports different equivalent sources representations of the radiation or scattering object: 1) equivalent surface current densities discretized on triangular meshes, 2) plane wave representations, 3) spherical harmonics representations. The current densities provide for excellent spatial localization and deliver most diagnostics information about the test object. A priori information about the test object can easily be incorporated, too. Using plane wave and spherical harmonics representations, the spatial localization is not as good as with spatial current densities, but still much better than in the case of conventional modal expansions. Both far-field based expansions lead to faster transformations than the equivalent currents and in particular the orthogonal spherical harmonics expansion is a very attractive and robust choice. All three expansions are well-suited for efficient echo suppression by spatial filtering. Various new field transformation and new computational performance results are shown in order to illustrate some capabilities of the algorithm.

Analytical finite element matrix elements and global matrix assembly for hierarchical 3-D vector basis functions within the hybrid finite element boundary integral method

Abstract. A hybrid higher-order finite element boundary integral (FE-BI) technique is discussed where the higher-order FE matrix elements are computed by a fully analytical procedure and where the gobal matrix assembly is organized by a self-identifying procedure of the local to global transformation. This assembly procedure applys to both, the FE part as well as the BI part of the algorithm. The geometry is meshed into three-dimensional tetrahedra as finite elements and nearly orthogonal hierarchical basis functions are employed. The boundary conditions are implemented in a strong sense such that the boundary values of the volume basis functions are directly utilized within the BI, either for the tangential electric and magnetic fields or for the asssociated equivalent surface current densities by applying a cross product with the unit surface normals. The self-identified method for the global matrix assembly automatically discerns the global order of the basis functions for generating the matrix elements. Higher order basis functions do need more unknowns for each single FE, however, fewer FEs are needed to achieve the same satisfiable accuracy. This improvement provides a lot more flexibility for meshing and allows the mesh size to raise up to λ/3. The performance of the implemented system is evaluated in terms of computation time, accuracy and memory occupation, where excellent results with respect to precision and computation times of large scale simulations are found.

Tokamak equilibria with strong toroidal current density reversal

The equilibrium of large magnetic islands in the core of a tokamak under conditions of strong toroidal current density reversal is investigated by a new method. The method uses distinct spectral representations to describe each simply connected region as well as the containing shell geometry. This ideal conducting shell may substitute for the plasma edge region or take a virtual character representing the external equilibrium field effect. The internal equilibrium of the islands is solved within the framework of the variational moment method. Equivalent surface current densities are defined on the boundaries of the islands and on the thin containing shell, giving a straightforward formulation to the interaction between regions. The equilibrium of the island–shell system is determined by matching moments of the Dirichlet boundary conditions. Finally, the macroscopic stability against a class of tilting displacements is examined by means of an energy principle. It is found out that the up–down symmetric islands are stable to this particular perturbation and geometry but the asymmetric system presents a bifurcation in the equilibrium.

Time-Domain Analysis of a Rectangular Reflector Antenna

Rectangular reflector antennas have motivated the time-domain analysis of electromagnetic scattering problems. The asymptotic time domain physical-optics (TDPO) is applied to the analysis of a rectangular reflector illuminated by a Gaussian-impulse. The effects of time-delayed mutual coupling between points on the surface will be ignored as a result of utilizing the TDPO method for determining the equivalent surface-current density on the reflector. Finally, in this work the scattered signals at the specular reflection point, at the edges, and at the corners can be clearly distinguished.

Determination of Surface Currents on Circular Microstrip Antennas

This work aims to present a theoretical analysis of the electric and magnetic surface current densities of a circular microstrip antenna (CMSA) as a body of revolution. The rigorous analysis of these problems begins with the application of the equivalence principle, which introduces an unknown electric current density on the conducting surface and both unknown equivalent electric and magnetic surface current densities on the dielectric surface. These current densities satisfy the integral equations (IEs) obtained by canceling the tangential components of the electric field on the conducting surface and enforcing the continuity of the tangential components of the fields across the dielectric surface. The formulation of the radiation problems is based on the combined field integral equation. This formulation is coupled with the method of moments (MoMs) as a numerical solution for this equation. The numerical results of the electric and magnetic surface current densities on the outside boundary of a CMSA excited by <TEX>$TM_{11^-}$</TEX> and <TEX>$TM_{21^-}$</TEX> modes are presented. The radiation pattern is calculated numerically in the two principle planes for a CMSA and gives a good results compared with measured results published by other research workers.

Mixed discretization schemes for electromagnetic surface integral equations

In developing accurate and stable discretization schemes for integral equations, important steps are the choices of the basis and testing functions. In the electromagnetic surface integral equations, the unknown quantities, the equivalent electric and magnetic surface current densities, are by default divergence conforming and therefore should be expanded with the divergence‐conforming basis functions. The choice of the testing functions however is less obvious and depends on the choice of the basis functions and the properties of the integral operators. We develop stable and well‐defined discretization procedures where the operators are tested either their range or on the dual of their range. This requires the use of non‐conventional finite‐elements spaces, called ‘dual spaces’, and non‐traditional testing procedures. We show with numerical examples that in most cases, testing the operators on the dual of their range leads to improved solution accuracy compared with testing the operators on their range. For the integral equations of the first kind, this agrees with the conventional Galerkin's scheme, but for the integral equations of the second kind, this requires the use of Petrov–Galerkin type discretization schemes with appropriate curl‐conforming (dual) testing functions. Copyright © 2012 John Wiley &amp; Sons, Ltd.