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 scattering approach for the calculation of currents induced on the multiple overhead wires illuminated by a plane wave with an arbitrary angle of incidence is presented In this work. The influence of a lossy ground is taken into account via the Fresnel reflection coefficient appearing within the kernel of the thin wire electric field integral equations (EFIE). The set of EFIE is solved by means of the finite element integral equation method (FEIEM). Various numerical results are presented.

Electromagnetic field coupling to multiple finite length transmission lines above an imperfect ground

The circular loop antenna is analysed by using the electric field integro-differential equation in the frequency domain. The weak form of the integro-differential is derived and then the current distribution along the circular loop antenna is calculated by solving the resulting equation via the finite element technique. Accurate results are obtained using the linear shape and test functions. © 1998 John Wiley & Sons, Ltd.

Finite element integral equation modelling of a thin wire loop antenna

The paper deals with a comparison of numerical and analytical approaches to study the fields radiated from dipole antenna radiating above a homogeneous ground. The frequency domain formulation is based on the integro differential equation of the Pocklington type. The influence of the earth-air interface is taken into account via the Fresnel reflection coefficient. The Pocklington equation is solved via the Galerkin-Bubnov variant of the Indirect Boundary Element Method (GB-IBEM). The related reflected field is determined by numerically evaluating field integrals. Analytical approach involves the use of the approximate sinusoidal current distribution and analytical evaluation of field integrals. Some illustrative numerical results for the electric field reflected from the interface and propagated through the lossy medium obtained by both approximations are given in the paper.

Analytical versus Numerical Approach to the Analysis of Dipole Antenna Radiated Field over a Lossy Ground

The paper deals with the comparison of novel experimental and computational method for obtaining the impulse impedance of the horizontal grounding electrode defined as a ratio of maximum values of induced voltage and current at the injection point on the electrode. Impedance is obtained in-situ based on injection of the 1.5 A current pulses (4/10 μs and 8/20 μs) and measurement of corresponding voltage drop. On the other hand, the current distribution along the electrode is calculated using the antenna theory based model featuring the thin-wire approximation and corresponding Pocklington integro-differential equation in the frequency domain. The equation is solved using analytical technique. The results obtained via different methods agree satisfactorily, thus validating antenna model of the grounding electrode. Obtained results are applicable for any grounding system based on horizontal grounding electrode.

Dragan Poljak

Papers

Electromagnetic field coupling to multiple finite length transmission lines above an imperfect ground

Finite element integral equation modelling of a thin wire loop antenna

Analytical versus Numerical Approach to the Analysis of Dipole Antenna Radiated Field over a Lossy Ground

Impulse impedance of the horizontal grounding electrode: Experimental analysis versus full-wave computational model

Exposure of Humans to Electromagnetic Radiation