Time-harmonic Electromagnetic Fields Research Articles

Dirichlet problem for electromagnetic fields in chiral media with fractal boundary

The main goal of this work is to formulate and solve the Dirichlet problem for time-harmonic electromagnetic fields in case of chiral media with fractal boundary. To this end, a reformulation of the Maxwell system in quaternionic form is taken as a starting point. The Whitney extension theorem as well as some results from fractal geometry are used.

INVERSE SOURCE PROBLEM FOR ACOUSTICALLY-MODULATED ELECTROMAGNETIC WAVES.

We propose a method to reconstruct the electrical current density from acoustically-modulated boundary measurements of time-harmonic electromagnetic fields. We show that the current can be uniquely reconstructed with Lipschitz stability. We also report numerical simulations to illustrate the analytical results.

Micromagnetic study of inertial spin waves in ferromagnetic nanodots

Here we report the possibility to excite ultra-short spin waves in ferromagnetic thin-films by using time-harmonic electromagnetic fields with terahertz frequency. Such ultra-fast excitation requires to include inertial effects in the description of magnetization dynamics. In this respect, we consider the inertial Landau-Lifshitz-Gilbert (iLLG) equation and develop analytical theory for exchange-dominated inertial spin waves. The theory predicts a finite limit for inertial spin wave propagation velocity, as well as spin wave spatial decay and lifetime as function of material parameters. Then, guided by the theory, we perform numerical micromagnetic simulations that demonstrate the excitation of ultra-short inertial spin waves (20 nm long) propagating at finite speed in a confined magnetic nanodot. The results are in agreement with the theory and provide the order of magnitude of quantities observable in realistic ultra-fast dynamics experiments.

Multilevel domain uncertainty quantification in computational electromagnetics

We continue our study [R. Aylwin, C. Jerez-Hanckes, C. Schwab and J. Zech, Domain uncertainty quantification in computational electromagnetics, SIAM/ASA J. Uncertain. Quant. 8 (2020) 301–341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback solutions on the nominal domain is characterized in piecewise Sobolev spaces. We prove error convergence rates and optimize the algorithmic steering of parameters for edge-element discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution mappings. All calculations are performed on the polyhedral nominal domain, which enables the use of standard simplicial finite element meshes. We provide a rigorous fully discrete error analysis and show, in all cases, that dimension-independent algebraic convergence is achieved. For the multilevel sparse-grid quadrature methods, we prove higher order convergence rates free from the so-called curse of dimensionality. Numerical experiments confirm our theoretical results and verify the superiority of the sparse-grid methods.

Electric dipole scattering from a finitely conducting cylinder

The time-harmonic electromagnetic fields for an arbitrarily oriented electric dipole over a cylindrical structure are derived in substantial detail. The problem is formulated by the classical method of separation of variables with an angular mode Fourier–Bessel series. Using boundary conditions on tangential components of E and H, we obtain integral representations for the electromagnetic fields in each of the homogeneous regions. The analysis is simplified by using the axial components of the electric and magnetic fields as potentials. The electromagnetic response of resistive or conductive structures is very important in fields such as hydrogeology, mineral exploration, archeology, civil engineering, geotechnics and hydrocarbon exploration. In many of these applications, numerical modeling is performed to aid in the acquisition and interpretation of data. Validation of these numerical routines involves comparison between numerical results obtained from simplified models and their analytical solutions. For validation by two-dimensional analytical solutions, the electrical response of a cylinder in a homogeneous medium is probably the best choice (if not the only one), however, most computational routines assume that the cylindrical structure is perfectly conductive or resistive, which can be a problem in the computation of some numerical solutions. Thus, we have developed here a procedure to obtain the semianalytic solution for the electromagnetic response of a finite conducting cylinder in the vicinity of an electric dipole, so that it can be reliably used as a form of validation for 2D electromagnetic numerical modeling programs. As part of the code verification, we analyze the modeled fields based on the behavior of the electric currents that form in the regions near the cylindrical heterogeneity, as well as a secondary field validation where we compare the results of a more general frequency-domain EM numerical method.

Sensing Model for Detecting Ferromagnetic Debris Based on a High-Gradient Magnetostatic Field

Wear debris sensing is an important approach for machine condition monitoring. To date, different types of debris sensors have been developed, among which the magnetostatic inductive debris sensor is preferable for monitoring ferromagnetic debris, but the existing induced voltage model cannot be used to quantify wear debris in lube oil. Accordingly, a modeling method was proposed for the debris sensor based on a high-gradient magnetostatic field. First, a Gaussian curve is used to fit the magnetic field along the axis of the sensor. Then, time-harmonic electromagnetic field analysis and Fourier series decomposition are combined to establish an induced voltage model. Furthermore, the influence of the relative permeability, size, and velocity of wear debris and induction coil parameters on the output voltage is analyzed. The results show a linear relationship between the theoretical calculations and experimental results. Therefore, the proposed model can be used to realize quantitative analysis of wear debris, laying a foundation for optimizing the sensor structure, and extracting features from output voltage signals.

On the Electromagnetic Field of an Overhead Line Current Source

This work presents an analytical series-form solution for the time-harmonic electromagnetic (EM) field components produced by an overhead current line source. The solution arises from casting the integral term of the complete representation for the generated axial electric field into a form where the non-analytic part of the integrand is expanded into a power series of the vertical propagation coefficient in the air space. This makes it possible to express the electric field as a sum of derivatives of the Sommerfeld integral describing the primary field, whose explicit form is known. As a result, the electric field is given as a sum of cylindrical Hankel functions, with coefficients depending on the position of the field point relative to the line source and its ideal image. Analogous explicit expressions for the magnetic field components are obtained by applying Faraday’s law. The results from numerical simulations show that the derived analytical solution offers advantages in terms of time cost with respect to conventional numerical schemes used for computing Sommerfeld-type integrals.

Dipole excitation of collective modes in viscous two-dimensional electron systems

We describe the structure of the time-harmonic electromagnetic field of a vertical Hertzian electric dipole source radiating over an infinite, translation invariant two-dimensional electron system. Our model for the electron flow takes into account the effects of shear and Hall viscosities as well as an external static magnetic field perpendicular to the sheet. We identify two wave modes, namely, a surface plasmon and a diffusive mode. In the presence of an external static magnetic field, the diffusive mode combines the features of both the conventional and Hall diffusion and may exhibit a negative group velocity. In our analysis, we solve exactly a boundary value problem for the time-harmonic Maxwell equations coupled with linearized hydrodynamic equations for the flat, two-dimensional material. By numerically evaluating the integrals for the electromagnetic field on the sheet, we find that the plasmon contribution dominates in the intermediate-field region of the dipole source. In contrast, the amplitude of the diffusive mode reaches its maximum value in the near-field region, and quickly decays with the distance from the source. We demonstrate that the diffusive mode can be distinguished from the plasmon in the presence of the static magnetic field, when the highly oscillatory plasmon is gapped and tends to disappear.

Analyses of the coupling model combining field and circuit equations based on solenoidal coils for wireless power transfer

A method using the coupling model combining the field and circuit equations is put forward to describe the wireless power transfer (WPT) system in this study. Hollow solenoidal resonant coils are usually excited by alternating currents in practice, doing studies related to the solutions to the solenoidal coils in alternating electromagnetic fields is important. In this study, the analytic expressions of the alternating magnetic vector potential are obtained from Maxwell's formulas for time-harmonic electromagnetic fields, and the mathematical expressions of the electric field intensity, the magnetic flux density, the induced voltage, and the mutual inductance are established for the coaxially symmetric solenoidal coils. The field-circuit coupling model is built for WPT, and the relationship between the input and output variables is obtained. The equations contain the dielectric information of the transmission space and the coils' radii and turn numbers, and can be used for analysing the dielectric effects of the space at lower or higher frequencies. The results are verified by experiments of 2-coil and 4-coil structures. The advantages of being able to analyse the distribution of the fields are illustrated by a comparison experiment with a laterally displaced coil, which is valuable to WPT design.

Electromagnetic Sensor for Detecting Wear Debris in Lubricating Oil

The detection technology on wear debris in lubricating oil is critical to ensure the safe operation of the engines in the aerospace industry. In this article, a triple-coil reverse double-excitation solenoid sensor is discussed. Inside the sensor, electromagnetic field generated by two excitation coils is sensitive to the occurrence of wear debris, and its changes can be captured by the induction coil. Thus, the impedance and inductive voltage of the induction coil can be used to detect the wear debris. Assuming the debris is spherical, an electromagnetic model is established. Via time-harmonic electromagnetic field analysis method, the expressions of the induction coil’s impedance increment and inductive voltage are derived. The relationships between the impedance increment and the parameters of the debris and the coils are investigated. The theoretical results are validated by Maxwell from Ansys, Canonsburg, PA, USA, simulations and experiments. In order to get an accurate voltage output signal, a passband filter amplifier is designed to connect to the output of the induction coil and empirical mode decomposition is presented to process the digital voltage signal. The derivation method on the coil impedance increment and inductive voltage and the signal processing method can be extended to design sensors measuring more complicated wear debris.

Mutual inductance calculation of the one-primary multiple-secondary coil system

It is of great significance to accurately calculate the mutual inductance of one-primary multiple-secondary coil system (OPMSC) in order to improve energy transmission efficiency, measurement accuracy of magnetic sensor, etc. In this paper, a method based on time harmonic field to calculate such mutual inductance is proposed. Both the high frequency alternating current model and the time-harmonic electromagnetic field calculation method are used to ensure engineering practicability of the method. The current of each coil and the relative position between coils are considered in the proposed method. Moreover, according to the full derivation of a new analytical method. the extremum phenomenon of mutual inductance is described. And this method is verified by both finite element simulations and experiments, which proves that the method is effective for optimizing the relative position of the multiple coils when mutual influence among all the coils is taken into account.

In silico evaluation of the thermal stress induced by MRI switched gradient fields in patients with metallic hip implant

This work focuses on the in silico evaluation of the energy deposed by MRI switched gradient fields in bulk metallic implants and the consequent temperature increase in the surrounding tissues. An original computational strategy, based on the subdivision of the gradient coil switching sequences into sub-signals and on the time-harmonic electromagnetic field solution, allows to realistically simulate the evolution of the phenomena produced by the gradient coils fed according to any MRI sequence. Then, Pennes’ bioheat equation is solved through a Douglas–Gunn time split scheme to compute the time-dependent temperature increase. The procedure is validated by comparison with laboratory results, using a component of a realistic hip implant embedded within a phantom, obtaining an agreement on the temperature increase better than 5%, lower than the overall measurement uncertainty. The heating generated inside the body of a patient with a unilateral hip implant when undergoing an Echo-Planar Imaging (EPI) MRI sequence is evaluated and the role of the parameters affecting the thermal results (body position, coil performing the frequency encoding, effects of thermoregulation) is discussed. The results show that the gradient coils can generate local increases of temperature up to some kelvin when acting without radiofrequency excitation. Hence, their contribution in general should not be disregarded when evaluating patients’ safety.

Enhanced FANO Structure Based on Tip-Field-Enhancement Theory

High-Q metasurfaces have attracted much interest owing to their potential application in biological sensors. FANO is a type of high-Q factor metasurface. However, it is difficult to achieve large resonant intensity and a high-Q factor at the same time. In this paper, by sharpening the tips of the asymmetrical split-ring FANO structure and letting more charges stack at the tips to enhance tip coupling, the Q factor was significantly improved without sacrificing too much resonant intensity. Simulation results showed that the Q factor increased up to 2.4 times, while the resonant intensity stayed higher than 20 dB, and the experiment results agreed with the simulations. This indicated that the tip-field-enhancement theory can be applied in time-harmonic electromagnetic-fields, and the method proposed here can be used to increase the sensitivity and accuracy of microfluidic sensors. Additionally, other types of research, such as on antenna design, could benefit from this theory.

Contribution to the Theory of Planar Wire Loop Antennas Used for Reception

Using “electromagnetic field” to designate an electric field and a magnetic field which satisfy Maxwell's equations, we define a decomposition of an arbitrary incident time-harmonic electromagnetic field into four elementary time-harmonic electromagnetic fields F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> , F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> , F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</sub> , and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> . A formula, which gives the response of an arbitrary planar wire loop antenna used for reception, is based on this decomposition. This formula is applicable to any incident field configuration, and valid at any frequency at which the thin wire approximation applies. It separates the response of the antenna into three parts, one of which may be viewed as the intended response of the antenna. Our analysis teaches that F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</sub> and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> have no effect on the antenna and how F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> excite the antenna. Thus, it allows us to better understand the characteristics and limitations of a planar wire loop antenna used as a measuring antenna or as a direction finder.

Electromagnetic time-harmonic and static field polygonal rotator with homogeneous materials

We propose a scheme of designing polygonal rotator with homogenous materials by using linear coordinate transformation. Our strategy is available for both time-harmonic electromagnetic field case and static field case. In particular, we found that only one anisotropic material is needed in static field case, and the density of field in the central region can be altered to be denser or sparser, or stay the same. The magnetostatic field rotator can be realized by multilayered structure composed of ferromagnetic materials and superconductor, and the direct current rotator can be realized by metals with different conductivity. Numerical results verify the effectiveness of our strategy in both time-harmonic field case and static field case.

Nonlinear interaction of whistler waves in a magnetized plasma with density ducts

The nonlinear resonant interactions between whistler waves guided by density ducts surrounded by a uniform magnetized plasma are studied. It is shown that, under specific conditions that are determined, a time-harmonic external electromagnetic field can drive the parametric instability of guided whistlers. Both cases of cylindrical and planar ducts are considered, of either decreased or increased plasma density. The frequency interval where the magnetized plasma may be resonant is analyzed. The growth rate and the threshold of the parametric instability are determined. Numerical calculations are presented for parameters typical of “space plasmas and” modeling laboratory experiments where guided whistler propagation was observed.

A Systematic and Extensive Determination of Far Field Time Harmonic Electromagnetic Field of a Radiating Element

A Systematic and Extensive Determination of Far Field Time Harmonic Electromagnetic Field of a Radiating Element

Stored electromagnetic field energies in general materials

The most general expressions of the stored energies for time-harmonic electromagnetic fields are derived from the time-domain Poynting theorem, and are valuable in characterizing the energy storage and transport properties of complex media. A new energy conservation law for the time-harmonic electromagnetic fields, which involves the derived general expressions of the stored energies, is introduced. In contrast to the well-established Poynting theorem for time-harmonic fields, the real part of the new energy conservation law gives an equation for the sum of stored electric and magnetic field energies; the imaginary part involves an equation related to the difference between the dissipated electric and magnetic field energies. In a lossless isotropic and homogeneous medium, the new energy conservation law has a clear physical implication: the stored electromagnetic field energy of a radiating system enclosed by a surface is equal to the total field energy inside the surface subtracted by the field energy flowing out of the surface.

A high-order wideband direct solver for electromagnetic scattering from bodies of revolution

The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as ω→0 in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nyström discretization of a one-dimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples.

Eddy Current Loss and Detuning Effect of Seawater on Wireless Power Transfer

Due to the conductivity of the seawater, the traditional mutual inductance circuit model in the air cannot be used directly to describe wireless power transfer (WPT) systems in seawater applications. This paper proposes a modified mutual inductance circuit model of an underwater WPT system to analyze the eddy current loss (ECL) and the detuning effect caused by the seawater. The time-harmonic electromagnetic field in the seawater and the air near the coil that carries a sinusoidal alternating current is analyzed. The root-mean-square (rms) value and phase angle of the induced voltage on the secondary coil can be obtained by the integral of the electric field intensity along the coil path. By introducing the equivalent ECL impedance at both the primary and secondary sides, a modified mutual inductance circuit model of an underwater WPT system was obtained. Through adding a compensation inductance to the primary circuit, the detuned system in the seawater is turned back to be resonant at the same frequency as in the air. A seawater WPT prototype was built and the experimental results verified the theoretical analysis.