Time-harmonic Field Research Articles

Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations.

We consider numerical modeling of the optical properties of devices typical of beam-steering devices based on liquid-crystal materials: two-dimensional, anisotropic and inhomogeneous dielectric properties, periodic in one dimension. A mathematical formulation of the system of second-order partial differential equations for the components of the time-harmonic electric field is discretized by using a finite-element method based on curl-conforming edge elements. The discrete equations are also interpreted as equivalent finite-difference equations. It is shown how the resulting large sparse complex system of linear algebraic equations can be solved by an iterative method with convergence accelerated by a preconditioner based on fast Fourier transforms. Benchmarking results and the application to a realistic problem are reported. The practical limitations of the approach and its advantages and disadvantages compared with other approaches are discussed.

WAVE PROPAGATION IN A 3-D OPTICAL WAVEGUIDE

In this paper we study the problem of electromagnetic wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.3 The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavior of the solutions of the eigenvalue problem. Moreover, the derivation of the transform formulas needed to solve the wave propagation problem involves nontrivial calculations. The paper provides a complete description on how to construct the solution to the wave propagation problem in a 3-D optical waveguide with cylindrical symmetry. A follow-up article will study the particular cases of a step-index fiber and of a coaxial waveguide. In those cases we will obtain concrete formulas for the field and numerical examples.

Stability of time-modulated electroosmotic flow

We present a linear stability analysis of parallel electroosmotic flow in a slot geometry. A spatially uniform time harmonic electric field is applied to a dilute electrolyte solution contained between two infinite parallel plates. The base state ion concentrations and electric potential are determined using the Poisson–Boltzmann equation in the Debye–Hückel approximation. The base velocity field is found to be time harmonic and parallel. It is shown that the original system can be replaced by an equivalent one consisting of an electrically neutral fluid enclosed between oscillating parallel plates, whose speed and frequency of oscillation depend on the modulated electric field. Further, the system of linearized disturbance equations can be decoupled into two stability problems: The first, called the electrokinetic problem, describes the evolution of disturbance ion concentrations and electric potential and is independent of the disturbance velocity components. The second, called the Stokes layer problem describes an oscillatory Stokes layer forced by an electrical body force. The stability of each system is determined by Floquet analysis of a dynamical system obtained from a truncated Galerkin expansion of the perturbation quantities. Our calculations show the system to be linearly stable over a wide range of parameters, with damping rates that become quite small for certain combinations of Stokes and Reynolds numbers.

Rapid calculations of time-harmonic nearfield pressures produced by rectangular pistons

A rapid method for calculating the nearfield pressure distribution generated by a rectangular piston is derived for time-harmonic excitations. This rapid approach improves the numerical performance relative to the impulse response with an equivalent integral expression that removes the numerical singularities caused by inverse trigonometric functions. The resulting errors are demonstrated in pressure field calculations using the time-harmonic impulse response solution for a rectangular source 5 wavelengths wide by 7.5 wavelengths high. Simulations using this source geometry show that the rapid method eliminates the singularities introduced by the impulse response. The results of pressure field computations are then evaluated in terms of relative errors and computational speeds. The results show that, when the same number of Gauss abscissas are applied to both approaches for time-harmonic pressure field calculations, the rapid method is consistently faster than the impulse response, and the rapid method consistently produces smaller maximum errors than the impulse response. For specified maximum error values of 10% and 1%, the rapid method is 2.6 times faster than the impulse response for pressure field calculations performed on a 61 by 101 point grid. The rapid approach achieves even greater reductions in the computation time for smaller errors and larger grids.

The Helmholtz equation least-squares method and Rayleigh hypothesis in near-field acoustical holography

In this paper we present a numerical investigation of reconstructing time-harmonic acoustic pressure field in two dimensional space by using a series expansion—the so-called Helmholtz equation least-squares (HELS) method. Series expansion methods (or the Rayleigh methods) have been widely used in predicting the scattered acoustic pressure. With regularization, they can also be applied to reconstruction of acoustic pressure on the source surface from the measurements taken in the field, and HELS is the first such attempt for these problems. In this paper, we establish HELS in the framework of the Rayleigh methods and reveal its interrelationship with the Rayleigh hypothesis. In particular, to regularize a reconstruction problem, we use the method of quasisolutions, i.e., a Tikhonov regularization with an a posteriori choice of the regularization parameter. It is shown that without regularization HELS can still yield a satisfactory reconstruction of acoustic radiation from an arbitrary object when enough measurements are taken at sufficiently close range to the source. With regularization the number of measurements can be reduced and reconstruction accuracy be enhanced. It is concluded that HELS can be used to reconstruct acoustic radiation from a convex arbitrarily shaped vibrating object regardless of the validity of the Rayleigh hypothesis, although in practice the results will depend on the rate of convergence of the approximating sequence.

Three-dimensional finite element computation of laser cavity eigenmodes [corrected

A new method for computing eigenmodes of a laser resonator by the use of finite element analysis is presented. For this purpose, the scalar wave equation (delta + k2)E(x, y, z) = 0 is transformed into a solvable three-dimensional eigenvalue problem by the separation of the propagation factor exp(-ikz) from the phasor amplitude E(x, y, z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements, the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a diode-pumped solid-state laser with a thermally lensing crystal inside the cavity, the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.

Optimization of power in the problems of active control of sound

We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulation of the problem. We have also obtained optimal solutions that minimize the L 1 or L 2 norm of the control sources; the physical interpretation of the former being the overall absolute acoustic source strength. In the current paper, we minimize the power required for the operation of the active control system. It turns out that the corresponding analysis necessarily involves interaction between the sources of sound and the surrounding acoustic field, which was not the case for either L 1 or L 2 . Even though it may first seem counterintuitive, one can build a control system (a particular combination of surface monopoles and dipoles) that would require no power input for operation and would even produce a net power gain while providing the exact noise cancellation. This usually comes at the expense of having the original sources of noise produce even more energy.

Loss calculation of induction motors considering harmonic electromagnetic field in stator and rotor

AbstractA method of loss calculation for induction motors is proposed. The combined 3D–2D time‐stepping finite element analysis is carried out to obtain the copper loss and the time variation of the magnetic field in the motor. The iron loss is calculated approximately considering the time variation of the magnetic field direction and the minor hysteresis loops caused by the time‐harmonic fields using practical computer resources. The proposed method is applied to four types of induction motors, which are the solid rotor induction motors with/without slot and the cage induction motors with/without skew. The measured and the calculated total losses and the iron losses agree well in all cases. The differences of the loss distributions of each motor are also compared and investigated. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 147(2): 63–73, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10305

Optical fields inside a conical waveguide with a subwavelength-sized exit hole

Studies of the spatial distribution of time-harmonic optical fields inside a metallized cone, tapered to a subwavelength diameter, are reported. We consider the electric-type (TMmn)wave with the lowest-order indices m = 0 and n = 1. For these waves we obtain exact analytical results for electromagnetic fields inside a conical waveguide with a loss-free dielectric core and perfectly conducting metallic walls. The contributions of different field components to the energy density distributions are explicitly expressed as functions of the radial and angular coordinates. We outline a technique for the evaluation of the optical fields in the near-field zone beyond the exit aperture. This provides tools for the calculation of the amplitude reflection coefficient taking into account the influence of a plane interface between the truncated cone and free space. Particular attention is paid to the calculations of the energy density at the exit aperture and the near-field transmission coefficient of an optical probe with a glass core. We present a detailed analysis of the optical transmission of such a waveguide as a function of the wavelength for large taper angles and various values of the aperture diameter reaching . The results obtained yield the range of the cone parameters and wavelengths in which one can achieve a high spatial resolution capability and sufficiently high transmission efficiency.

The Determination of the Surface Impedance of a Partially Coated Obstacle from Far Field Data

A variational method is given for determining the essential supremum of the surface impedance of a partially coated perfect conductor from a knowledge of the far field pattern of the time-harmonic electric field at fixed frequency. It is assumed that the shape of the scatterer has been determined (e.g., by solving the far field equation and using the linear sampling method). Numerical examples are given for the scalar case with constant surface impedance.

A UNIQUENESS THEOREM OF THE 3-DIMENSIONAL ACOUSTIC SCATTERING PROBLEM IN A SHALLOW OCEAN WITH A FLUID-LIKE SEABED

This paper shows that under the assumption of the out-going radiation conditions at infinity, the time-harmonic acoustic scattered field off a sound-soft solid in a shallow ocean with a fluid-like seabed is unique in [Formula: see text]. Here M1 is the water part, M2 the seabed, [Formula: see text] the waveguide and Ω the solid object. The associated modal problem is studied and a representation formula for the solution in terms of the Green's function is derived.

Thermal dose optimization method for ultrasound surgery

In this paper, a model-based optimization method is derived to control the thermal dose in biological tissues for ultrasound surgery. The optimization method uses the bioheat equation as a system model and quadratic cost criteria for the desired thermal dose. Time-harmonic quasi-stationary ultrasound fields are used as the heat source. In this method the optimal phase and the amplitude trajectories are found directly by minimizing the associated cost function. The approach also allows for maximum input amplitude constraints. The method is based on the Hamiltonian form of the system and results in a large dimensional nonlinear optimization problem which is solved with a gradient-type iterative scheme. The performance of the optimization method is tested with 2D simulations and it is shown that the approach is able to yield a feasible nominal solution. This nominal evolution would then eventually be sought to be maintained with the help of a feedback controller during the actual sonication.

Loss Calculation of Induction Motors Considering Harmonic Electromagnetic Fields in Stator and Rotor

A method of loss calculation for induction motors is proposed. The combined 3D-2D time stepping finite element analysis is carried out to obtain the copper loss and the time-variation of the magnetic field in the motor. The iron loss is calculated approximately considering the time-variation of the magnetic field direction and the minor hysteresis loops caused by the time-harmonic fields using practical computer resources. The proposed method is applied to 4 types of induction motors, which are the solid rotor induction motors with/without slot and the cage induction motors with/without skew. The measured and the calculated total losses and the iron losses agree well in all cases. The differences of the loss distributions of each motor are also compared and investigated.

An optimal control approach for ultrasound induced heating

The absorbed energy of ultrasound and microwaves is changed into heat in a dissipative medium. This wave field induced heating has numerous applications in industry and medicine. In these applications the ability to control the resulting heat distribution is essential. In this paper we design a controller for quadratic cost criteria in the ultrasound induced heating problem in inhomogeneous media. In this approach the time-harmonic pressure fields are computed from individual wave sources by solving the Helmholtz equation with the ultra-weak variational formulation. The equations that define the controller are derived from the Hamiltonian form of the system, which results in a large dimension non-linear control problem. As an example we study a numerical simulation that resembles an ultrasound therapy problem in 2D. The simulation shows that feasible solutions can be obtained with this approach.

A complete lumped equivalent circuit of three-phase squirrel-cage induction motors using two-dimensional finite-elements technique

This paper presents the use of the finite element technique for determining the parameters of a three-phase squirrel-cage induction motor. The common parameters, in addition to the core losses and ratio of number of tums, are obtained from the finite element field solutions. The magnetizing characteristic and core losses curve are used to determine the flux distribution within the motor structure. The linear time harmonic vector potential field solution is utilized for the inductances computation. The accuracy of the finite element application is verified using the available precise results.

A Complete Lumped Equivalent Circuit of Three-Phase Squirrel-Cage Induction Motors Using the Two-Dinensional Finite Elements Technique

This paper presents the use of the finite element technique for determining the parameters of a three-phase squirrel-cage induction motor. The common parameters, in addition to the core losses and ratio of number of tums, are obtained from the finite element field solutions. The magnetizing characteristic and core losses curve are used to determine the flux distribution within the motor structure. The linear time harmonic vector potential field solution is utilized for the inductances computation. The accuracy of the finite element application is verified using the available precise results.

Time-harmonic field analysis of electric logging systems

A time-harmonic field analysis procedure, based on an integral equation approach, is implemented to study electric logging systems. The tools are modeled as the linear superposition of basic current elements. A new method for computing the transform-type integrals was developed.

Quasi-magnetostatic solution for a conducting and permeable spheroid with arbitrary excitation

Broad-band electromagnetic induction (EMI) methods are promising in the detection and discrimination of subsurface metallic targets. In this paper, the quasi-magnetostatic solution for a conducting and permeable prolate spheroid under arbitrary excitation by a time-harmonic primary field is obtained by using the separation of variables method with vector spheroidal wave functions. Numerical results for the induced dipole moments are presented for uniform axial and transverse excitations, where the primary field is oriented along the major and minor axis of the prolate spheroid, respectively. They show that the EMI frequency responses are sensitive to the orientation and permeability of the spheroid. An approximation is also developed that aims to extend the exact solution to higher frequencies by assuming slight penetration of the primary field into the spheroid. Under this approximation, a system of equations that refers only to the external field expansions is derived. It is shown that, for spheroids with high relative permeability, this approximation is in fact capable of yielding an accurate broad-band response even for highly elongated spheroids.

On the statistics of a sum of harmonic waveforms

In this paper, we address certain aspects of the problem of statistically characterizing the electromagnetic field inside an enclosure. The field that we are interested in describing is time-harmonic and a three-dimensional spatial vector; therefore, two random variables are required for each vector component at each location in the enclosure. We can describe either the magnitude (or intensity) and phase, or the real (in-phase) and imaginary (quadrature) parts, of each spatial component. It is the relationship between these two modes of description that is addressed in this paper. We show that this relationship is given by the Blanc-Lapierre transform and when there is a sum of more than one time-harmonic field, by equations first derived by Kluyver. The relationships are derived for any form of distribution taken on by any of the random variable. We also address issues related to the approximation of the probability density function (pdf) of the amplitude of an electromagnetic field given a known pdf of the intensity of this field. The work presented herein fills in some of the gaps which were left in some literature wherein the independence of the variables to each other was assumed, that is, the independence of the in-phase to the quadrature variables.

Numerical Modelling of Time-Harmonic Seismic Wave Fields in Simple Structures by the Gaussian Beam Method. Part I.

Numerical Modelling of Time-Harmonic Seismic Wave Fields in Simple Structures by the Gaussian Beam Method. Part I.