Time-harmonic Frequency Research Articles

A linearised inverse conductivity problem for the Maxwell system at a high frequency

We consider a linearised inverse conductivity problem for electromagnetic waves in a three dimensional bounded domain at a high time-harmonic frequency. Increasing stability bounds for the conductivity coefficient in the full Maxwell system and in a simplified transverse electric mode are derived. These bounds contain a Lipschitz term with a factor growing polynomially in terms of the frequency, a Hölder term, and a logarithmic term which decays with respect to the frequency as a power. To validate this increasing stability numerically, we propose a reconstruction algorithm aiming at the recovery of sufficiently many Fourier modes of the conductivity. A numerical evidence shed light on the influence of the growing frequency and confirms the improved resolution.

Deformation in transversely isotropic thermoelastic thin circular plate due to multi-dual-phase-lag heat transfer and time-harmonic sources

In this paper, the distortion in homogeneous transversely isotropic thermoelastic (TIT) axisymmetric 2-D thin circular plate with two temperature due to time-harmonic sources have been analysed. The mathematical model has been formed for the rotating thin circular plate with multi-dual-phase-lag generalized heat transfer and has been converted to dimensionless form. The Hankel transform method has been employed to solve the mathematical model. A bicubic equation obtained for nontrivial solution of mathematical model so obtained is solved using MATLAB software. A normal force of unit magnitude and a cubical thermal source and vanishing tangential stress at the boundaries has been taken as boundary conditions. As an application a constant load and a decaying heat source in the radial and axial direction of the thin circular plate has been taken. The displacement components, conductive temperature, and stress components are computed in the transformed domain and numerical inversion techniques are used to find the expressions in the physical domain. The effects of time harmonic frequency and rotation with two temperatures have been represented graphically using MATLAB and analysed for multi-dual phase lag model of heat transfer. Some particular cases have also been discussed.

Tensor Train Accelerated Solution of Volume Integral Equation for 2-D Scattering Problems and Magneto-Quasi-Static Characterization of Multiconductor Transmission Lines

The method of moments (MoM) discretization of volume integral equation (VIE) results in dense $N \times N$ matrix, $N$ being the number of MoM basis functions. Naive solution of such a system of linear algebraic equations (SLAE) is expensive when problems become large scale. Recently, tensor decomposition has been introduced for solving the SLAE by folding its matrix and its vectors into high-dimensional tensors. In this paper, we present detailed explanations for tensor train (TT) decomposition of the SLAE matrices and vectors resulting from MoM discretization of VIE for scalar 2-D scattering problems under TM-polarization and magneto-quasi-static characterization of multiconductor transmission lines. For Toeplitz matrices resulted from MoM discretization on structured meshes, the extraordinary performance of TT with scaling of $\log (N)$ in CPU time and memory is shown to directly solve SLAE with millions of unknowns within a few minutes and few megabytes of memory. Such $\log (N)$ performance is limited, however, to the SLAE with purely Toeplitz matrices corresponding to the scattering problems on homogeneous dielectric scatterers of the rectangular cross section. To overcome this limitation and solve the problems with arbitrarily shaped inhomogeneous objects, we propose an iterative conjugate gradient-TT (CG-TT) scheme for solving MoM discretized VIE, which utilizes TT decomposition for the fast evaluation of the matrix-vector products. The CG-TT shows CPU and memory scaling in the low- and high-frequency regimes with $O(N)$ and $O(r^{2}~N \log N)$ , respectively, $r$ being the highest rank in the TT carriages. Detailed analysis of memory and CPU time scaling with respect to a number of unknowns and time-harmonic frequency is presented.

Evaluating Harmonic Distortions in Brushless Doubly Fed Induction Machines

The brushless doubly fed induction machine (DFIM) is a special machine type that is of interest as a generator in wind turbine drive trains. The brushless DFIM has a significant space-harmonic content compared with conventional machine types, due to its construction. This results in additional harmonic distortions, such as torque-ripple and time harmonics. This paper studies the causes and origins of harmonic distortions in the brushless DFIM. Using time- and space-harmonic analysis, an analytical evaluation method is derived to predict the torque ripple frequencies and time-harmonic frequencies in stator and rotor voltages and currents. The evaluation method is applied to a prototype brushless DFIM to demonstrate the severity of harmonic related distortions that can be present in the brushless DFIMs. The evaluation is validated by additional finite-element analysis and measurements. Furthermore, measures are proposed to reduce harmonic distortions. The insight into harmonic distortions in the brushless DFIM and the ability to predict and prevent them must lead to the improved brushless DFIM designs in the future.

A sensor for the direct measurement of curvature based on flexoelectricity

A direct curvature sensing measurement based on the flexoelectricity of Ba0.64Sr0.36TiO3 (BST) material through electromechanical coupling is proposed and developed in this paper. The curvature sensing was demonstrated in four point bending tests of a beam with bonded BST curvature sensors under different applied loads with low time-harmonic frequencies from 0.5 to 3 Hz. A shear lag concept which describes the efficiency of the loading transfer from the epoxy bonding layer was taken into account in extracting the actual curvature from the sensor measurement. A finite element analysis has been performed to estimate the curvature transfer efficiency and the bonding layer thickness is found to be a critical parameter in determining the curvature transfer. Experimental results showed a good linearity of charge output dependence on curvature inputs in a limited frequency range and showed a curvature sensitivity of 30.78 pC m, in comparison with 32.48 pC m from theoretical predictions. Using the measured curvature, the bending stiffness of the beam was then obtained from the experimentally obtained moment–curvature curve. This work demonstrated that the flexoelectric BST sensor provides a direct curvature measurement instead of using a traditional strain gage sensor through interpolation, and thus offers an important avenue for on-line and in situ structural health monitoring.

Application of Slope PWM Strategies to Reduce Acoustic Noise Radiated by Inverter-Fed Induction Motors

In this paper, two strategies based on the slope pulsewidth modulation (PWM) technique are applied to acoustic noise reduction in inverter-fed induction motors. This generated technique uses a trapezoidal modulator wave with a triangular or a sinusoidal carrier signal. The main advantage of the proposed method is that only two control parameters are necessary to achieve a voltage spectrum configuration that helps reduce radial electromagnetic forces. One of them modifies the amplitude of the modulated output of the inverter, and the other controls the frequency spectrum distribution. Small changes to these parameters can yield a wide range of electrical spectra that allow for selective acoustic harmonic reduction. Thus, it is possible to avoid the overlap of radial forces produced by the MMF time harmonic frequencies of the inverter and the radial forces produced by the MMF space harmonic frequencies due to the discretized distribution of stator winding in slots of the motor, maintaining the number of pulses per period chosen. Also, it can help to avoid activating the resonances due to the mechanical natural frequencies. Results are checked experimentally and compared to those of other PWM techniques.

Multi-frequency topological derivative for approximate shape acquisition of curve-like thin electromagnetic inhomogeneities

In this paper, we investigate a non-iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and/or magnetic permeability differ from those in the embedding (homogeneous) space. The main objective is the imaging of crack-like thin inclusions, but the algorithm can be applied to arbitrarily shaped inclusions. For this purpose, we apply multiple time-harmonic frequencies and normalize the topological derivative imaging function by its maximum value. In order to verify its validity, we apply it for the imaging of two-dimensional crack-like thin electromagnetic inhomogeneities completely hidden in a homogeneous material. Corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm.

Multiphysics modelling of the separation of suspended particles via frequency ramping of ultrasonic standing waves

A model was developed to determine the local changes of concentration of particles and the formations of bands induced by a standing acoustic wave field subjected to a sawtooth frequency ramping pattern. The mass transport equation was modified to incorporate the effect of acoustic forces on the concentration of particles. This was achieved by balancing the forces acting on particles. The frequency ramping was implemented as a parametric sweep for the time harmonic frequency response in time steps of 0.1s. The physics phenomena of piezoelectricity, acoustic fields and diffusion of particles were coupled and solved in COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden) following a three step approach. The first step solves the governing partial differential equations describing the acoustic field by assuming that the pressure field achieves a pseudo steady state. In the second step, the acoustic radiation force is calculated from the pressure field. The final step allows calculating the locally changing concentration of particles as a function of time by solving the modified equation of particle transport. The diffusivity was calculated as function of concentration following the Garg and Ruthven [1] equation which describes the steep increase of diffusivity when the concentration approaches saturation. However, it was found that this steep increase creates numerical instabilities at high voltages (in the piezoelectricity equations) and high initial particle concentration. The model was simplified to a pseudo one-dimensional case due to computation power limitations. The predicted particle distribution calculated with the model is in good agreement with the experimental data as it follows accurately the movement of the bands in the centre of the chamber.

MUSIC-type imaging of a thin penetrable inclusion from its multi-static response matrix

The imaging of a thin inclusion, with dielectric and/or magnetic contrasts with respect to the embedding homogeneous medium, is investigated. A MUSIC-type algorithm operating at a single time-harmonic frequency is developed in order to map the inclusion (that is, to retrieve its supporting curve) from scattered field data collected within the multi-static response matrix. Numerical experiments carried out for several types of inclusions (dielectric and/or magnetic ones, straight or curved ones), mostly single inclusions and also two of them close by as a straightforward extension, illustrate the pros and cons of the proposed imaging method.

Harmonic Electromagnetic Forces in Induction Motors

Recently, there has been increasing demand for quiet motors, and the same trend has been observed in the case of induction motors. In induction motors, electromagnetic noise is sometimes the predominant acoustic noise. In small motors, the major cause of vibration and noise is electromagnetic forces resulting from the combination of harmonic fluxes in the air gap. In this study, the spatial distribution of fundamental and harmonic time electromagnetic forces was studied by using search coils, by performing FEM analysis, and by using conventional equations. In a four-pole 2.2kW motor, harmonic electromagnetic forces were measured using 36 search coils on the inner surface of the stator teeth, and the spatial distribution of electromagnetic forces was obtained at each time harmonic frequency. Spatial distribution was also analyzed by FEM, and the results were analytically validated by using conventional equations. On the basis of these analyses, the spatial distribution of electromagnetic forces for various time harmonics was confirmed. These results can be used in the design and development of quiet motors.

High-frequency elastodynamic boundary integral inversion using asymptotic phase transformation

Use of conventional finite element or boundary element computational methods for problems of time-harmonic elastic wave transmission, diffraction, and surface wave generation at a nonplanar fluid–solid interface become inefficient at high frequencies due to the fineness of discretization required for numerical convergence. A more efficient computational procedure has been developed that exploits the field phase as predicted by high-frequency asymptotic analysis (i.e., the geometrical theory of diffraction, or GTD). In this method, the boundary integral equation which governs elastic wave transmission is transformed by assuming a solution in the form of the GTD ansatz A(x)exp[iωp(x)], where p(x) is the field phase, ω is time harmonic frequency, and A(x) is the field amplitude. The phase p(x) is prescribed analytically through solution of the leading term eikonal equations of GTD. The transformed boundary integral equation is then solved numerically for the amplitudes A(x). The amplitudes A(x) will have a narrow spatial frequency spectral bandwidth, and hence can be efficiently calculated. This approach exploits the strengths of the GTD eikonal equation solutions while avoiding the analytical pitfalls encountered in the GTD transport equations.

An unsteady wind-driven ocean circulation model

This model of a continuously stratified ocean, consisting of surface Ekman and thermal layers and a deep abyssal region, is driven by an unsteady surface wind stress. The model yields a nonlinear planetary wave equation for the surface temperature which is solved by the method of characteristics. Horizontal distributions of the surface temperature, the heat content and the potential energy are computed. For a model zonal wind stress relevant to the North Atlantic, a parameter sensitivity study is made.Long time scale observations of the heat content in the North Atlantic north of 45°N show that periods of heating and cooling of a few decades duration frequently occur. Using a zonal wind stress consisting of the superposition of a mean and two time harmonic components, secular variations in the surface temperature field manifest themselves as a beat period, provided the time harmonic frequencies are close together. Finally, to investigate the effects on the circulation produced by the changing intensity of the westerly winds, numerical results are presented for the case when the wind stress distribution oscillates in the north-south direction with time. Results from the study show that the heating and cooling periods of a few decades duration in the North Atlantic can be reproduced.