Time-harmonic Field Research Articles

Full‐wave analytical explicit expressions for the surface fields of an electrically large horizontal circular loop antenna placed on a layered ground

This study presents full-wave analytical explicit expressions for the time-harmonic surface fields of a large horizontal circular loop antenna located on a multilayer Earth. The expressions are derived by applying the following procedure. First, the complete integral representations for the EM field components are decomposed into three parts, that is, the residue contributions from the poles of the integrands and the two branch-cut integrals associated with the wavenumbers in free-space and in the bottom layer of the stratified medium. Next, the square-root term that generates the branch cut is extracted from the integrand of the generic branch-cut integral. Finally, the square-root algorithm-based partial fraction expansion is applied to each extracted square-root term, so as to convert the branch-cut integrals into closed-contour integrals around a set of pole singularities, which can be straightforwardly evaluated. As a result of the developed procedure, the fields are expressed as convergent sums of special functions. Numerical results are presented to validate the proposed formulation. In addition to being accurate, the derived formulas offer advantages in terms of computation time over standard numerical integration techniques.

Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

We find solutions E:Ω→R3 of the problem{∇×(μ(x)−1∇×E)−ω2ε(x)E=∂EF(x,E)in Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3 with exterior normal ν:∂Ω→R3. Here ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in an anisotropic material with a magnetic permeability tensor μ(x)∈R3×3 and a permittivity tensor ε(x)∈R3×3. The boundary conditions are those for Ω surrounded by a perfect conductor. It is required that μ(x) and ε(x) are symmetric and positive definite uniformly for x∈Ω, and that μ,ε∈L∞(Ω,R3×3). The nonlinearity F:Ω×R3→R is superquadratic and subcritical in E, the model nonlinearity being of Kerr-type: F(x,E)=|Γ(x)E|p for some 2<p<6 with Γ(x)∈GL(3) invertible for every x∈Ω and Γ,Γ−1∈L∞(Ω,R3×3). We prove the existence of a ground state solution and of bound states if F is even in E. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.

Unusual energy properties of leaky backward Lamb waves in a submerged plate

It is found that leaky backward Lamb waves, i.e. waves with negative energy-flux velocity, propagating in a plate submerged in a liquid possess extraordinary energy properties distinguishing them from any other type of waves in isotropic media. Namely, the total time-averaged energy flux along the waveguide axis is equal to zero for these waves due to opposite directions of the longitudinal energy fluxes in the adjacent media. This property gives rise to the fundamental question of how to define and calculate correctly the energy velocity in such an unusual case. The procedure of calculation based on incomplete integration of the energy flux density over the plate thickness alone is applied. The derivative of the angular frequency with respect to the wave vector, usually referred to as the group velocity, happens to be close to the energy velocity defined by this mean in that part of the frequency range where the backward mode exists in the free plate. The existence region of the backward mode is formally increased for the submerged plate in comparison to the free plate as a result of the liquid-induced hybridization of propagating and nonpropagating (evanescent) Lamb modes. It is shown that the Rayleigh's principle (i.e. equipartition of total time-averaged kinetic and potential energies for time-harmonic acoustic fields) is violated due to the leakage of Lamb waves, in spite of considering nondissipative media.

Semi-Analytical Approach for Finite-Element Analysis of Multi-Turn Coil Considering Skin and Proximity Effects

Native application of finite-element method (FEM) to the analysis of skin and proximity effects in multi-turn coils results in large equation systems, whose solution needs long computational time. This paper proposes a semi-analytical approach to overcome this problem. For the analysis of the proximity effect, the complex permeability of a round conducting wire immersed in uniform time-harmonic magnetic fields is represented in a closed form. Then, the homogenized complex permeability over the cross section of the multi-turn coil is analytically evaluated using the Ollendorff formula. The magnetoquasistatic problem is thus replaced by the magnetostatic one, in which the multi-turn coil is treated as a uniform material with the homogenized complex permeability. The skin effect is taken into consideration by introducing the corresponding impedance in the circuit equation. The proposed method is shown to give the impedance of multi-turn coils, which is in good agreement with that obtained by the conventional FEM as well as experiments.

Plane-Wave Propagation in Extreme Magnetoelectric Media

The extreme magnetoelectric medium (EME medium) is defined in terms of two medium dyadics, $\alpha$, producing electric polarization by the magnetic field and $\beta$, producing magnetic polarization by the electric field. Plane-wave propagation of time-harmonic fields of fixed finite frequency in the EME medium is studied. It is shown that (if $\omega\neq 0$) the dispersion equation has a cubic and homogeneous form, whence the wave vector $k$ is either zero or has arbitrary magnitude. In many cases there is no dispersion equation ("NDE medium") to restrict the wave vector in an EME medium. Attention is paid to the case where the two medium dyadics have the same set of eigenvectors. In such a case the $k$ vector is restricted to three eigenplanes defined by the medium dyadics. The emergence of such a result is demonstrated by considering a more regular medium, and taking the limit of zero permittivity and permeability. The special case of uniaxial EME medium is studied in detail. It is shown that an interface of a uniaxial EME medium appears as a DB boundary when the axis of the medium is normal to the interface. More in general, EME media display interesting wave effects that can potentially be realized through metasurface engineering.

Axially uniform magnetic field-modulation excitation for electron paramagnetic resonance in rectangular and cylindrical cavities by slot cutting

In continuous-wave (CW) Electron Paramagnetic Resonance (EPR) a low-frequency time-harmonic magnetic field, called field modulation, is applied parallel to the static magnetic field and incident on the sample. Varying amplitude of the field modulation incident on the sample has consequences on spectral line-shape and line-height over the axis of the sample. Here we present a method of coupling magnetic field into the cavity using slots perpendicular to the sample axis where the slot depths are designed in such a way to produce an axially uniform magnetic field along the sample. Previous literature typically assumes a uniform cross-section and axial excitation due to the wavelength of the field modulation being much larger than the cavity. Through numerical analysis and insights obtained from the eigenfunction expansion of dyadic Green’s functions, it is shown that evanescent standing-wave modes with complex cross-sections are formed within the cavity. From this analysis, a W-band (94GHz) cylindrical cavity is designed where modulation slots are optimized to present a uniform 100kHz field modulation over the length of the sample.

Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains

In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along the integration lines and are called the longitudinal and transverse measurements, respectively. In most cases, however, the transverse measurements cannot be physically acquired. Therefore, the VT methods are typically used to reconstruct divergence-free (or source-free) velocity and flow fields that can be reconstructed solely from the longitudinal measurements. In this paper, we show how vector fields with non-zero divergence in a bounded domain can also be reconstructed from the longitudinal measurements without the need of explicitly evaluating the transverse measurements. To the best of our knowledge, VT has not previously been used for this purpose. In particular, we study low-frequency, time-harmonic electric fields generated by dipole sources in convex bounded domains which arise, for example, in electroencephalography (EEG) source imaging. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modeling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field.

Coil Winding Losses: Decomposition Strategy

Precise modeling of the magnetic field in the coil wire of an electric machine often becomes a major challenge: with the high number of turns and small penetration depth, the number of degrees of freedom exceeds reasonable limits in any standard approximation method. Precise approximation, however, is critical, e.g., to reliable coil loss estimation. Hence, there is a call for specialized approximative methods. This paper presents a method for time-harmonic coil wire field computations in 2-D problems. We replace the coil by a lattice of polygonal plane fillers and span a low-dimensional function space on the polygon boundaries. The reliability of loss estimates requires accurate computations of the responses to the interface excitations of this space. The responses constitute a Dirichlet-to-Neumann map to efficiently couple plane fillers together and to a standard finite-element method (FEM) outside the coil regions. The outcome is significantly faster than the standard FEM alone. The results are still in good agreement.

Broad angle negative refraction in lossless all dielectric or semiconductor based asymmetric anisotropic metamaterial

In this article, it has been theoretically shown that broad angle negative refraction is possible with asymmetric anisotropic metamaterials (AAMs) constructed by only dielectrics or lossless semiconductors at the telecommunication and relative wavelength range. Though natural uniaxial materials can exhibit negative refraction, the maximum angle of negative refraction and critical incident angle lie in a very narrow range. This problem can be overcome by our proposed structure. In our structures, negative refraction originates from the highly asymmetric elliptical iso-frequency. This is artificially created by the rotated multilayer sub-wavelength dielectric or semiconductor stack, which acts as an effective AAM. This negative refraction is achieved without using any negative permittivity materials such as metals. As we are using simple dielectrics, fabrication of such structures would be less complex than that of the metal based metamaterials. By considering the time harmonic field incidence, negative refraction has been demonstrated for two dimensional bi-dielectric structures for TM polarization with realistic parameters. Our proposed ideas have been validated by the full wave simulations considering both the effective medium approach and realistic structure model. This device might find some important applications in photonics and optoelectronics.

Aircraft panel with sensorless active sound power reduction capabilities through virtual mechanical impedances

This paper deals with an active structural acoustic control approach to reduce the transmission of tonal noise in aircraft cabins. The focus is on the practical implementation of the virtual mechanical impedances method by using sensoriactuators instead of conventional control units composed of separate sensors and actuators. The experimental setup includes two sensoriactuators developed from the electrodynamic inertial exciter and distributed over an aircraft trim panel which is subject to a time-harmonic diffuse sound field. The target mechanical impedances are first defined by solving a linear optimization problem from sound power measurements before being applied to the test panel using a complex envelope controller. Measured data are compared to results obtained with sensor–actuator pairs consisting of an accelerometer and an inertial exciter, particularly as regards sound power reduction. It is shown that the two types of control unit provide similar performance, and that here virtual impedance control stands apart from conventional active damping. In particular, it is clear from this study that extra vibrational energy must be provided by the actuators for optimal sound power reduction, mainly due to the high structural damping in the aircraft trim panel. Concluding remarks on the benefits of using these electrodynamic sensoriactuators to control tonal disturbances are also provided.

Chaotic dynamics and basin erosion in nanomagnets subject to time-harmonic magnetic fields

Magnetization dynamics in uniformly magnetized particles subject to time-harmonic (AC) external fields is considered. The study is focused on the behavior of the AC-driven dynamics close to saddle equilibria. It happens that such dynamics has chaotic nature at moderately low power level, due to the heteroclinic tangle phenomenon which is produced by the combined effect of AC-excitations and saddle type dynamics. By using analytical theory for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle together with numerical simulations, we quantify and show how the tangle produces the erosion of the safe basin around the stable equilibria.

Tunable magnetic resonances of high-permittivity dielectric rings coated with graphene

We consider a planar resonator made of a thin, high-permittivity dielectric ring coated on one side with graphene. It is shown that the resonant magnetic response of the ring to an axially applied time-harmonic magnetic field can be efficiently controlled by electrical gating of the graphene coating. Variations of the resonance frequencies and quality factors with the Fermi energy of graphene are investigated for the micrometer- and centimeter-size rings operating in the terahertz and gigahertz frequency ranges.

A new look at the pushing force of an electromagnetic wave on a classical charged particle

The interaction between a plane wave and a classical charged particle is reviewed in this paper. A matrix formulation is used to derive the relativistic motion of the particle due to the Lorentz force of a spatially homogeneous time-harmonic radiation field. The Landau–Lifshitz approach to the radiation damping force is then investigated for the same field. In conclusion only the radiation damping accelerates the particle in the direction of the Poynting vector for this type of field. The Lorentz force only leads to an extra constant drift velocity, plus oscillations with the frequency of the field.

Heun Functions Describing Fermions Evolving in Parallel Electric and Magnetic Fields

The present work deals with the behavior of fermions moving in a static magnetic induction and a time-harmonic electric field, both oriented along Oz. For the ultra-relativistic particles described by a Heun double confluent equation, we derive the corresponding wave functions and the conserved current density components.

Ground States of Time-Harmonic Semilinear Maxwell Equations in $${\mathbb{R}^3}$$ R 3 with Vanishing Permittivity

We investigate the existence of solutions \({E:\mathbb{R}^3 \to \mathbb{R}^3}\) of the time-harmonic semilinear Maxwell equation $$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$where \({V:\mathbb{R}^3 \to \mathbb{R}}\), \({V(x) \leqq 0}\) almost everywhere on \({\mathbb{R}^3}\), \({\nabla \times}\) denotes the curl operator in \({\mathbb{R}^3}\) and \({F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}\) is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the \({L^{3/2}}\) -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and \({V(x) = -\mu\omega^2 \varepsilon(x)}\) where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field \({\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}\) and \({\varepsilon}\) is the linear part of the permittivity in an inhomogeneous medium.

Wave energy transfer in elastic half-spaces with soft interlayers.

The paper deals with guided waves generated by a surface load in a coated elastic half-space. The analysis is based on the explicit integral and asymptotic expressions derived in terms of Green's matrix and given loads for both laminate and functionally graded substrates. To perform the energy analysis, explicit expressions for the time-averaged amount of energy transferred in the time-harmonic wave field by every excited guided or body wave through horizontal planes and lateral cylindrical surfaces have been also derived. The study is focused on the peculiarities of wave energy transmission in substrates with soft interlayers that serve as internal channels for the excited guided waves. The notable features of the source energy partitioning in such media are the domination of a single emerging mode in each consecutive frequency subrange and the appearance of reverse energy fluxes at certain frequencies. These effects as well as modal and spatial distribution of the wave energy coming from the source into the substructure are numerically analyzed and discussed.

Heteroclinic tangle phenomena in nanomagnets subject to time-harmonic excitations

Magnetization dynamics in uniformly magnetized nanomagnets excited by time-harmonic (AC) external fields or spin-polarized injected currents is considered. The analysis is focused on the behaviour of the AC-excited dynamics near saddle equilibria. It turns out that this dynamics has a chaotic character at moderately low power level. This chaotic and fractal nature is due to the phenomenon of heteroclinic tangle which is produced by the combined effect of AC-excitations and saddle type dynamics. By using the perturbation technique based on Melnikov function, analytical formulas for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle are derived. Both the cases of AC applied fields and AC spin-polarized injected currents are treated. Then, by means of numerical simulations, we show how heteroclinic tangle is accompanied by the erosion of the safe basin around the stable regimes.

Thermodynamical Bending of FGM Sandwich Plates Resting on Pasternak’s Elastic Foundations

AbstractThe analysis of thermoelastic deformations of a simply supported functionally graded material (FGM) sandwich plates subjected to a time harmonic sinusoidal temperature field on the top surface and varying through-the-thickness is illustrated in this paper. The FGM sandwich plates are assumed to be made of three layers and resting on Pasternak’s elastic foundations. The volume fractions of the constituents of the upper and lower layers and, hence, the effective material properties of them are assumed to vary in the thickness direction only whereas the core layer is still homogeneous. When in-plane sinusoidal variations of the displacements and the temperature that identically satisfy the boundary conditions at the edges, the governing equations of motion are solved analytically by using various shear deformation theories as well as the classical one. The influences of the time parameter, power law index, temperature exponent, top-to-bottom surface temperature ratio, side-to-thickness ratio and the foundation parameters on the dynamic bending are investigated.

On the Surface Fields of a Small Circular Loop Antenna Placed on Plane Stratified Earth

An analytical method is presented which makes it possible to derive exact explicit expressions for the time-harmonic surface fields excited by a small circular loop antenna placed on the top surface of plane layered earth. The developed procedure leads to casting the complete integral representations for the EM field components into forms suitable for application of Cauchy’s integral formula. As a result, the surface fields are expressed as sums of Hankel functions. Numerical simulations are performed to show the validity and accuracy of the proposed solution.

Spectral Stochastic Finite Element Method for Electromagnetic Problems with Random Geometry

Abstract In electromagnetic problems, the problem geometry may not always be exactly known. One example of such a case is a rotating machine with random-wound windings. While spectral stochastic finite element methods have been used to solve statistical electromagnetic problems such as this, their use has been mainly limited to problems with uncertainties in material parameters only. This paper presents a simple method to solve both static and time-harmonic magnetic field problems with source currents in random positions. By using an indicator function, the geometric uncertainties are effectively reduced to material uncertainties, and the problem can be solved using the established spectral stochastic procedures. The proposed method is used to solve a demonstrative single-conductor problem, and the results are compared to the Monte Carlo method. Based on these simulations, the method appears to yield accurate mean values and variances both for the vector potential and current, converging close to the results obtained by time-consuming Monte Carlo analysis. However, further study may be needed to use the method for more complicated multi-conductor problems and to reduce the sensitivity of the method on the mesh used.