There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The electromagnetic fields within a detailed model of the human eye and its surrounding bony orbit are calculated for two different frequencies of plane-wave irradiation: 750 MHz and 1.5 GHz. The computation is performed with a finite-difference algorithm for the time-dependent Maxwell's equations, carried out to the sinusoidal steady state. The heating potential, derived from the square of the electric field, is used to calculate the temperatures induced within the eyeball of the model. This computation is performed with the implicit alternating-direction (IAD) algorithm for the heat conduction equation. Using an order-of-magnitude estimate of the heat-sinking capacity of the retinal blood supply, it is determined that a hot spot exceeding 40.4/spl deg/C occurs at the center of the model eyeball at an incident power level of 100 mW/cm/sup 2/ at 1.5 GHz.

Computation of the Electromagnetic Fields and Induced Temperatures Within a Model of the Microwave-Irradiated Human Eye

Transcranial magnetic stimulation in basic and clinical neuroscience: A comprehensive review of fundamental principles and novel insights.

We discuss the transmission line (TL) theory and its application to the problem of lightning electromagnetic field coupling to TLs. We start with the derivation of the general field-to-TL coupling equations for the case of a single-wire line above a perfectly conducting ground. The derived equations are solely based on the thin-wire approximation and they do take into account high-frequency radiation effects. Under the TL approximation, the general equations reduce to the Agrawal et al. field-to-TL coupling equations. After a short discussion on the underlying assumptions of the TL theory, three seemingly different but completely equivalent approaches that have been proposed to describe the coupling of electromagnetic fields to TLs are presented. The derived equations are then extended to deal with the presence of losses and multiple conductors and expressions for the line parameters, including the ground impedance and admittance, are presented. The time-domain representation of the field-to-TL coupling equations, which allows for a straightforward treatment of nonlinear phenomena as well as the variation in the line topology, is also described. Solution methods in the frequency domain and in the time domain are given and application examples with reference to lightning-induced voltages are presented and discussed. Specifically, the effects of ground losses and corona are illustrated and discussed. When the traveling voltage and current waves are originated from lumped excitation sources located at a specific location along a TL (direct lightning strike), both the corona phenomenon and ground losses result, in general, in an attenuation and dispersion of propagating surges along TLs. However, when distributed sources representing the action of the electromagnetic field from a nearby lightning illuminating the line are present, ground losses and corona phenomenon could result in important enhancement of the induced voltage magnitude.

A Review of Field-to-Transmission Line Coupling Models With Special Emphasis to Lightning-Induced Voltages on Overhead Lines

The paper deals with an assessment of the influence of finite conductivity to the current induced along the horizontal grounding electrode. Analysis is performed in frequency and time domain, respectively. Current distribution along the grounding electrode buried in a lossy half-space is determined via analytical solution of the corresponding Pocklington equation in the frequency domain. The corresponding time domain response is obtained by means of Inverse Fast Fourier Transform (IFFT). The electrode is excited via an equivalent current source. Presence of the earth-air interface is taken into account via the simplified reflection coefficient arising from the Modified Image Theory (MIT). The electrode current is calculated for the case of perfectly conducting (PEC) electrode and for the electrodes made of copper and aluminum. Comparison of results shows no significant discrepancy between these electrodes, justifying the use of a PEC electrode approximation.

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Frequency and Time Domain Analysis of Influence of the Grounding Electrode Conductivity on Induced Current Distribution

This chapter deals with simplified canonical models of the human body in the frequency and time domains. Frequency domain (FD) analysis involves parallelepiped body model and antenna body models, single straight thick wire body model and multiple wire representation of the body. Time domain (TD) analysis deals with straight thin wire antenna model of the body. FD antenna models are based on the Pocklington integro-differential approach while time domain formulation is based on the Hallen integral equation approach. Finally, transmission line body models in both FD and TD are given.

Simplified Models of the Human Body

In the literature, often the proposed modeling of electromagnetic transients of power network is mainly based on the numerical solution of full-wave Maxwell's equations. The solutions are usually very rigorous, but can be computationally intensive and difficult to use even by a research engineer. The aim of this proposed work is to show that it is possible to treat the same applications using simplified model based on Transmission Line (TL) without affecting the quality of results. The idea of simplifying the modeling is justified by the fact that the spectral content of the transient in power networks does not exceed a few MHz, the large dimensions of power devices (towers, grounding, ….), the presence of ground-air interface and also the open borders.

A simplified apporoach to the study of electromagnetic transients generated by lightning stroke in power network

In this paper, the evaluation of current induced in human the body model exposed to ELF (extremely low frequency) electric and magnetic field generated by a power line is presented. The induced current density in the three-dimensional (3D) model human body is obtained by using the Finite element method (FEM).In this paper, the human body is represented by an inhomogeneous model (rotationally-cylindrical body model) composed of several organs of spheroid shape; brain, heart, lungs, liver and intestines. The numerical model was implemented in COMSOL Multiphysics.

Current density and internal electric field in a model of the human body exposed to ELF electric and magnetic fields

The paper deals with the comparison between the two approaches to account for the presence of a two media configuration when calculating the electric field radiated by the dipole antenna. The first approach features the well-known Fresnel reflection/transmission coefficient approximation which, beside the electric properties of two media, considers also the incidence angle at the interface. On the other hand, the Modified Image Theory (MIT) approach uses the approximation of a normal incidence, thus neglecting the refraction of the wave in the lower medium. The MIT approach thus simplifies the formulation thus providing the analytical solution and the derivation of the time domain computation. In this paper the frequency domain formulation is used. It is based on the Pocklington's integro-differential equation which is solved numerically using the Galerkin Bubnov scheme of Boundary Element Method. The results obtained by both approaches are compared and conclusions are given regarding the limitations and strengths of the MIT approach with respect to the Fresnel coefficients approximation. The comparison undertaken in this work can serve as a benchmark for the analytical solution modeling and time domain modeling.

Dragan Poljak

Papers

Frequency and Time Domain Analysis of Influence of the Grounding Electrode Conductivity on Induced Current Distribution

Simplified Models of the Human Body

A simplified apporoach to the study of electromagnetic transients generated by lightning stroke in power network

Current density and internal electric field in a model of the human body exposed to ELF electric and magnetic fields

Electric field radiated by a dipole antenna above a lossy half space: Comparison of plane wave approximation with the modified image theory approach