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 a derivation of simple and efficient analytical solution for the calculation of transient impedance of the horizontal grounding electrode. The Generalized Telegrapher's Equation for spatial distribution of the scattered voltage in the frequency domain is used to derive the transient voltage at the feed point of the grounding electrode. Solution is obtained by using the Inverse Laplace Transform and Cauchy residue theorem. Transient impedance is determined as a ratio of time domain voltage and (lightning) current at the feeding point. Some illustrative numerical results are given.

Direct Time Domain Analytical Solution for the Transient Impedance of the Horizontal Grounding Electrode

Antenna design for ground penetrating radar (GPR) is very complex and depends on specific application. In typical civil engineering GPR applications, the antennas need to be low cost, small size, compact and simple while operating in close vicinity to real ground and in wide frequency band. In this paper we propose an ultra-wideband parabolic bicone antenna. Except for its compactness and small size, the bicone antenna design parameters and the given procedure can be used for a specific GPR application. The proposed antenna achieves VSWR > 2 in ultra-wideband from 0.67 – 4.3 GHz with the total height of 11.8 cm. Time domain response of the proposed antenna shows a negligible excitation signal distortion in the direction of maximum radiation. Also, very good VSWR is obtained when the antenna is very close to the typical real ground found in civil engineering GPR applications.

Ultra-wideband parabolic bicone antenna for ground penetrating radar

This paper aims to couple analytical solution of the Pocklington integro-differential equation with stochastic techniques for the statistics of transient current flowing along horizontal grounding electrode. The electrode is excited with a current source representing a lightning strike current. This problem, as it is the case for electromagnetic compatibility (EMC) issues, is governed by many uncertain parameters (lightning source profile and location, ground properties, electrode position and length, etc.). The proposed results yield additional statistical information which enable more accurate and efficient design of a grounding systems.

Transient statistics from the lightning strike current flowing along grounding electrode

The paper deals with an analytical/numerical assessment of the absorbed power density (S ab ) in the planar model of the human tissue due to radiation of dipole antenna. The effect of two media interface (air-body) is taken into account via the Fresnel plane wave reflection coefficient. The antenna is analyzed by solving the Pocklington integrodifferential equation and related integral expressions for the radiated electric and magnetic field, respectively. Some illustrative numerical results are given for the radiated field on the model surface and S ab pertaining to the frequency range of the fifth generation (5G) mobile communication systems.

Absorbed Power Density at the Surface of Planar Tissue due to Radiation of Dipole Antenna

The paper reviews the frequency domain modeling of dipole antenna using both the Pocklington integro-differential equation and the Hallen integral equation formulation. The details pertaining to the numerical solution procedures are outlined as well. Some strengths and weaknesses of both approaches are presented. A typical computational example for the current distribution induced along the dipole radiating over a lossy ground illustrating both methods is given, as well.

On the Frequency Domain Analysis of Straight Thin Wire Radiating Above a Lossy Half-Space : Pocklington equation versus Hallén equation revisited: 80th Anniversary of the Hallén Integral Equation

Dragan Poljak

Papers

Direct Time Domain Analytical Solution for the Transient Impedance of the Horizontal Grounding Electrode

Ultra-wideband parabolic bicone antenna for ground penetrating radar

Transient statistics from the lightning strike current flowing along grounding electrode

Absorbed Power Density at the Surface of Planar Tissue due to Radiation of Dipole Antenna

On the Frequency Domain Analysis of Straight Thin Wire Radiating Above a Lossy Half-Space : Pocklington equation versus Hallén equation revisited: 80th Anniversary of the Hallén Integral Equation