Harmonic Time Variation Research Articles

Harmonic model validation of power generation units

This study addresses the validation of harmonic models for power generation units considering practical aspects associated with test-bench and on-site measurements. A new method is proposed for the harmonic model validation. This method has been developed within the joint research project ‘NetzHarmonie,’ which deals with the study of harmonic emissions of renewable energy sources in the power system. The main idea for model validation is to compare the harmonic voltage–current characteristic of the model with the measured harmonic voltage–current pairs. This idea is implemented in a practical process by considering measurement uncertainties as well as the time variation of harmonics. In the validation process, measurement data should meet the given requirements to verify whether they are suitable for the task of validation. The application of the proposed model validation process to photovoltaic and wind power generation units with different topologies and power classes is thoroughly addressed as well. The application of the proposed process is illustrated in detail for selected harmonic orders from these measurement campaigns. This study aims to provide a scientific foundation for the consideration of the harmonic model validation of power generation units in future standards.

High-frequency response of subwavelength-structured metals in the petahertz domain

Electromagnetic plane waves, incident on and reflecting from a dielectric-conductor interface, set up a standing wave in the dielectric with the B-field adjacent to the conductor. It is shown here how the harmonic time variation of this B-field induces an E-field and a conduction current J (c) within the skin depth of a real metal; and that at frequencies in the visible and near-infrared range, the imaginary term sigmai of the complex conductivity sigma = sigma(r) + isigma(i) dominates the optical response. Continuity conditions of the E-field through the surface together with the in-quadrature response of the conductivity determine the phase relation between the incident E-M field and J(c). If slits or grooves are milled into the metal surface, a displacement current in the dielectric gap and oscillating charge dipoles at the structure edges are established in quadrature phase with incident field. These dipoles radiate into the aperture and launch surface waves from the edges. They are the principle source of light transmission through the apertures.

Convergence of Krylov Solvers and Choice of Basis and Weighting Set of Functions in the Moment Method Solution of Electrical Field Integral Equation

In Computational Electromagnetics, iterative techniques for solving algebraic linear system of equations are of fundamental importance, since actual problems give rise to linear systems too large to be practically solved by direct methods. In this work we investigate as performances of the major Krylov subspace iterative solver (i.e., GMRES), is afiected by difierent choice of these set of functions. Speciflcally, we consider the algebraic linear system of equations obtained by reducing the electrical fleld integral equation (EFIE) from the TMz scattering of a plane wave by a metallic strip. It can be observed that exists a critical threshold ¢0 such that, whenever either the basis or the weight pulses are given with an amplitude greater than ¢0, then the total number of internal loops necessary for taking the relative residual under a deflnite tolerance † > 0 increases all of a sudden, in such a dramatic way that it can even prevent the process at all from convergence. We try to explain this numerical behavior by inquiring the relationship between the MoM matrix condition number and the number of overall iterations necessary to numerical convergence. 1. INTRODUCTION It is not easy to give a comprehensive treatment of so many issues involved by analytical and numerical methods in the fleld of Computational Electromagnetics, and speciflcally by the method of moments, for solving integral equations relevant to electromagnetic theory. These include a very large amount of mathematical elements and capabilities, ranging from numerical integration and difierentiation to the feasibility of converting operator equations to a discrete system of algebraic equations, from the choice of suitable sets of basis and weight functions to a deep knowledge of linear algebra, matrix manipulations, algorithms and actual implementations of iterative and direct solvers and so on going (1). The lack of analytical solutions provides a further item of di-culty when trying in formal terms to expound mathematical clues intended to be potentially notable from a practical point of view. Thus it is common to consider simple (i.e., typically scalar) problems for preliminary tests and then to extrapolate general ideas to be applied to more complex cases. In fact, this is the approach we follow in the present paper, where we deal with the scalar problem of computing the current density induced on the surface of an ideal metallic strip when struck by a planar wave of a given wavelength ‚ which is supposed to be small or large with respect to the width 2w of the strip section. A harmonic time variation exp(i!t) is assumed and suppressed throughout. The problem serves as an useful and somehow simple example in electromagnetics for tackling basic concepts and discussing practical issues for numerical methods. Speciflcally, we investigate and try to characterize the behavior of iterative techniques for solving algebraic linear system of equations. Nowadays these are of basic importance, since actual problems give rise to linear systems that are too large to be practically solved by direct methods. There exists much study regarding the numerical aspects of such subjects, but it's not clear at all how the e-ciency of iterative solvers are conditioned by the choosing of the basis and weighting functions in the Moment Method (MM) (2,3). In this sense, we study as performances of the Generalized Minimun Residual (GMRES) Krylov iterative solver (4) is afiected by difierent choice of these set of functions. 2. A WIDTH-VARYING PULSE-PULSE MOMENT METHOD SCHEME An implementation of the moment method exploiting width-varying rectangular pulses for both the basis and the weight sets of functions is adopted for solving the EFIE

Inertial effects in small-amplitude swimming of a finite body

We study the mechanism of small-amplitude swimming of a deformable body of finite size in a viscous incompressible fluid described by the Navier-Stokes equations. The theory is based on a perturbation expansion in powers of the amplitude of surface displacements. A nonvanishing swimming velocity is found in second order perturbation theory. The average motion may include both a translational and a rotational contribution. For harmonic time variation of the first order flow we are led to a natural definition of the efficiency of swimming.

Electromagnetic field calculation inside a conducting cylinder of finite length using the T-Ω method

The solution of a 3-dimensional problem is, almost in all cases, a quite difficult task, since a great computational effort is required. AT-Ω method accompanied by a boundary element technique is presented for the evaluation of the field inside a conducting cylinder of finite length. The excitation is a current dipole with harmonic time variation (electric Hertzian dipole), located somewhere inside the cylinder. According to the method the problem is split into smaller independent successive problems, requiring much less computational effort.

A BEM code for 3-D eddy current calculations

A software package for solving time-harmonic 3-D eddy current problems using the boundary-element method (BEM) is described. The package solves coupled boundary integral equations for the magnetic vector potential and the electric scalar potential within the conductors, and for the magnetic vector potential in non-eddy-current regions. The authors describe the boundary integral formulation and its BEM realization. As an example, the eddy current field of a nonmagnetic conducting cube immersed in a homogeneous magnetic field with harmonic time variation is calculated. Additional numerical results are presented for a multiply connected eddy-current problem (benchmark problem No. 7 defined by the International TEAM workshops). The boundary-element solutions are compared with finite-element results and show excellent agreement.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimum Shunt Capacitor for Power Factor Correction at Busses with Lightly Distorted Voltage

The goal of this paper is to determine the effect of time variation of system impedances and voltage harmonics on the value of optimum capacitor for power factor correction at busses with nonsinusoidal voltage. Two types of 24 hours time-variation of voltage harmonices and Thevenin impedance are assumed. The equivalent load impedance is also considered time variable and assumed to contain a large proportion of induction motors. The daily energy losses are computed and graphed in function of the shunt capacitance used for power factor correction. The results of this study indicate that in order to avoid resonances and to find the optimum capacitor the time variation of harmonics and system impedances must be known as precise as possible.

Low-Frequency Electromagnetic Shielding in a System of Two Coaxial Cylindrical Shells

In this work, the problem of shielding is examined for a system of two coaxial cylindrical shells, when the excitation is a current filament placed parallel to the axis of the cylinders either inside or outside them. For the calculation of the current density within the conducting material of the shells and the magnetic flux density in the different regions of the system, the diffusion equation for the magnetic vector potential is used. The incident field of the filament has a harmonic time variation, but the method can be applied to any time variation of the excitation by using a Laplace or Fourier transform.

The Doppler effect: Now you see it, now you don’t

Two main classes of problems are identified in the theory of electromagnetic scattering in velocity-dependent systems. The first involves transformation of space and time coordinates and field components from the laboratory system of reference to the comoving system of the scatterer, solution of the scattering problem, and inverse transformations. In general, this method displays the Doppler frequency shifts. The second class involves the substitution of Minkowski’s constitutive relations into Maxwell’s equations for harmonic time variation, heuristically stipulating the absence of Doppler frequency shifts. The interrelation between the two methods is investigated here. It is argued that the second method is a limiting case for very low, as well as very high frequencies, and provided the mean square fluctuation of the dielectric constant is small, and the geometrical boundaries defining the scatterers are fixed. Canonical problems of plane, cylindrical, and spherical stratification are discussed and analytical results for the scattered fields are derived. If the parameters of the problem do not meet the above conditions, the first method should be used, giving rise, in general, to a whole spectrum of frequencies due to the Doppler effect.

Integration of near-resonant systems in slow-fluctuation approximation

Autonomous, conservative, nonlinear oscillatory systems of many degrees of freedom with an ’’internal resonance,’’ i.e., with the linearized (’’normal’’) frequencies tied in a near-resonant relation 𝒥giωi=−, can be completely integrated in a transparent approximation which is a variant of the ’’method of slowly variable amplitude and phase.’’ It consists in developing the nonlinear coupling into a trigonometric polynomial and dropping all terms except the most slowly varying ones on the grounds that their perturbative effects are oscillatory but essentially not cumulative. In this ’’slow-fluctuation’’ technique, the amplitudes separate off the phases; they exhibit joint extrema governed by simple conservation laws and all follow from one single quadrature. The phases also require no more than quadratures. There are various possibilities of motion with all amplitudes constant and harmonic time variation at (generally) incommensurate frequencies. The approximation does not apply if a degree of freedom which occurs to the first power in the nonlinear coupling happens to dip down towards zero amplitude; abrupt phase changes can then take place and call for a separate, accurate integration covering the critical time interval. With this sole restriction the method is entirely general. It is exact in the small-amplitude limit; at finite amplitudes it differs from the exact solution for practical purposes by no more than minor high-frequency ripples; and it yields more physical insight than any other known method for less effort.

Generalized Theodorsen solution for singular integral equations of the airfoil class

A class of singular integral equations is considered which arise in various two-dimensional mixed boundary-value problems with simple harmonic time variation. A problem typical of this class is that of determining the lifting pressure distribution on an oscillating airfoil in an unbounded incompressible potential flow. It is shown that Theodorsen's (1935) solution to this problem, with some modification, is valid for a general class of unsteady kernel functions. The technique employed is to consider an equivalent steady problem and then show that the unsteady resolvent and unsteady homogeneous solution can be written directly in terms of the steady solutions and a single frequency-dependent function which reduces to the Theodorsen function for the steady kernel.

Direct Integration of Time-Dependent Maxwell Equations

Maxwell's equations for time-dependent fields are integrated directly by using the scalar-vector analog of Green's theorem and the Dirac delta function. The fields are expressed in terms of volume and surface integrals over the time varying sources. For the particular case of harmonic time variation, the results agree with previous solutions.

Wave Propagation in Drifting Isotropic Warm Plasma

The electro‐magneto‐plasma‐gas‐dynamics (EMPGD) equations are given for the warm, compressible electron gas, with no magnetostatic field, drifting with a constant velocity, by taking into account the fluid dynamics variables of the electron gas and using small‐signal theory. Assuming harmonic time variation, an algebraic equation is found for the refractive index of the plasma characteristic waves. Particular cases are discussed and compared with previous results.