Hertz Vector Research Articles

Time-domain study of half-space transmission problem with vertical and horizontal dipoles

The paper deals with the transient fields transmitted through the interface between the air and ground. Excitation for the electromagnetic fields is produced by a vertically or horizontally oriented impulsive dipole, which may be electric or magnetic. However, only electric sources are discussed, because the analog results for magnetic sources are obtained easily through the duality transform. The ground medium has arbitrary permittivity and permeability, but it is restricted to lossless, i.e. the conductivity is assumed to be zero. In this study simple integral expressions are derived for the time-dependent Hertz vector in the ground from which the propagating transient fields can be obtained through differentiation. In certain specific cases some components of electric or magnetic field have a closed-form solution in terms of elementary functions. All the nontrivial solutions of this type in the ground medium are given. >

Green's functions and magnetized ferrites

Magnetized ferrites play an important role in the field of microwaves as substrates in various electronic elements. The dyadic Green's functions method is used to solve the electromagnetic field problem for such anisotropic (magnetically gyrotropic) materials. The full dyadic problem is reduced to a scalar one where the complete information is contained within a single scalar Green's function. A connection between the Green's functions formalism and that of scalar Hertz potentials is established and some progress is made towards the calculation of the scalar Green's function.

Recent advances in the use of separation of variables methods in general relativity

This review article is a guide to work that uses the method of separation of variables for problems that occur in general relativity. The main emphasis is on recent progress in the solution of important systems of equations such as Dirac’s equation, Maxwell’s equations and the gravitational perturbation (or spin 2) equations. Recent advances and established results for these equations in Kerr black hole and Robertson—Walker space-time backgrounds form the central theme of the discussion. These two important physical examples also illustrate some of the difficulties in a theory of solution by separation of variables methods for systems of equations. Other aspects of this subject such as solutions of the Rarita-Schwinger equation (spin 3 2 ) and the role of generalized Hertz potentials are also discussed.

Polarimetric scattering characteristics of simple radar targets using Hertz's vector potentials

AbstractHertz's vector potential is used to compute the co‐and cross‐polarized scattered fields due to simple targets like an arbitrarily oriented wire, an arbitrarily oriented loop of wire, and an infinite cylinder. The monostatic cross‐polarization properties of these targets are studied with different radar and target parameters. Important conclusions are summarized. It is found that under certain combinations of parameters, the cross‐polarization radiation is significant even for simple targets.

Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability

Transient fields arising from a general impulsive dipole above ground characterized by permittivity and permeability are studied. A detailed analysis for arbitrary electric dipole excitation is given, and the fields arising from the analog magnetic sources can be obtained by applying the duality transform. The mathematical description of this problem is done using the Hertz vector, whose solution in the time domain is derived in a novel way using the exact image theory. From this solution, which involves a definite integral, the actual fields are obtained through simple differentiations. The efficiency of the method is demonstrated by solving 14 previously unknown closed-form results for various field components in certain specific cases. Seven of these are obtained for electric dipole excitation and listed, whereas the corresponding results in the case of magnetic excitation can be easily written by applying the duality transform. >

A full-wave mixed potential mode-matching method for the analysis of planar or quasi-planar transmission lines

A full-wave mixed potential mode-matching method is presented for the analysis of planar and/or quasi-planar transmission lines. The transmission lines studied consist of layered (stratified) and nonlayered dielectric substrates and metal strips of finite thickness. The y-directed hybrid transverse electric (TE) and transverse-magnetic (TM) Hertzian potentials, perpendicular to the interfaces between each layered region, are employed in the layered regions. The nonlayered regions consist of dielectric step discontinuities that destroy the layered configuration in the horizontal plane, allowing a systematic and easy to handle full-wave formulation of the transmission line problem. The relative convergence criterion needs to be satisfied to obtain accurate electromagnetic field solutions. Theoretical results are in very good agreement with published data for various transmission line structures. Applications of the formulation to the proximity effects of microstrip and microslab lines are illustrated. >

Eddy-current induction in a uniaxially anisotropic plate

A theoretical model is presented to describe the time-harmonic electromagnetic induction of eddy currents in a uniaxially anisotropic plate. The model assumes that the plate is nonmagnetic and that the axis of uniaxial anisotropy is horizontal (i.e., lying in a plane parallel to the surface of the plate). The fields and currents induced by a general magnetic source are obtained by formulating the problem in terms of reduced electromagnetic potentials (Hertzian potentials) and making extensive use of integral transforms. Closed-form expressions for the reduced potentials are given explicitly, enabling the induced fields to be calculated in all regions provided the free-space source fields are known. The change in probe-coil impedance ΔZ due to eddy-current induction is considered in detail, and an explicit closed-form expression for ΔZ is derived for the general case. This general theory is illustrated by calculating ΔZ for a cylindrical air-cored coil when (i) the coil axis is normal to the surface of the plate and (ii) the coil axis is tangential to the surface of the plate. The validity of these calculations was established by performing a series of measurements of ΔZ using an aluminum-alloy sheet unidirectionally reinforced with boron fibers. The results of the theoretical calculations are in excellent agreement with experiment. The significance of the results for the nondestructive characterization of metal-matrix composite materials is discussed.

Resonant frequency of a rectangular microstrip patch on several uniaxial substrates

A full-wave analysis for determining the resonant frequency of a rectangular microstrip patch on multiple uniaxial anisotropic layers with or without an anisotropic overlay is presented. Two independent methods are used to derive the immittance matrix for the patch, from which the resonant frequency is determined. They are the Hertz vector potentials and the modified spectral domain immittance approach. Numerical results of the resonant frequency are given for several patch configurations, including cases of a patch on a single anisotropic layer, a patch on a double layer with one layer anisotropic and one layer isotropic, a suspended patch resonator with anisotropic substrate, a patch with anisotropic overlay and a patch on two anisotropic substrates with an anisotropic overlay. Changes in the resonant frequency of up to 58% are reported as n/sub x//n/sub y/ is changed from 1.0 (for isotropic substrates) to 2.0. >

Transient electromagnetic field of a vertical magnetic dipole above a plane conducting Earth

The pulsed electromagnetic radiation from a vertical magnetic dipole above a conducting Earth is investigated theoretically. The modified Cagniard method is used to derive closed form expressions for the magnetic Hertzian vector anywhere above the conducting Earth. Numerical results are presented for the magnetic Hertzian vector for different points of excitation and observation above the Earth, as well as for different values of the Earth's material parameters, namely, permittivity and electrical conductivity.

Electromagnetic Field Representation in (Inhomogeneous) Isotropic Chiral Media

Abstract A general theoretical approach to the electromagnetics of isotropic chiral media is presented. An electromagnetic field representation in terms of scalar Hertz potentials is obtained. The validity of this representation encompasses inhomogeneous media where the constitutive parameters of the chiral structures - permittivity, permeability and chirality - are functions of one coordinate. In the homogeneous case the corresponding electric and magnetic scalar Green's functions are calculated analytically thus constituting the field solution for dipole sources.

Spectrum of electromagnetic fluctuations in the Casimir effect

The quantum fluctuations of the electromagnetic field between two parallel plates at zero or finite temperature are analysed. The spectrum of the energy-momentum tensor and the two-point correlations of the field are obtained in analytic form through the use of Hertz potentials in the Lorentz gauge. In particular, the general expression for the transition probability of atomic systems is explicitly worked out.

Eigenmodes for electromagnetic waves propagating in a toroidal cavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time-domain analysis of horizontal dipoles in front of planar dielectric interface

Time-domain solutions for fields due to an impulsive horizontal magnetic and electric dipole above a planar dielectric interface are studied analytically. General time-domain expressions for the Hertzian potentials corresponding to the electromagnetic fields in the upper medium are analyzed in a novel way by exploiting exact image theory. Analytic closed-form solutions are derived for certain special locations of the source and observation points. Some of the results are new, while the rest agree with the solutions in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Hertz potentials in higher dimensions

The author generalises the Hertz potentials to all spacetime dimensions D>or=4 and to all antisymmetric tensor fields. He shows that the Whittaker-Debye-Bromwich reduction of the Hertz potential can be successfully exploited in all dimensions and for all antisymmetric tensor fields if the underlying spacetime is a warped product manifold. He illustrates the Hertzian approach by constructing the multipole expansions of all antisymmetric tensor fields on a D-dimensional Schwarzschild solution. A no-hair theorem for the D-dimensional black holes follows from these multipole expansions.

Solutions to time-harmonic Maxwell equations with a Hertz vector

The physical solutions to time-harmonic Maxwell equations for arbitrary sources in vacuo and for arbitrary media are formulated with a single Hertz vector. It is shown that in the special case of magnetic dielectrics the formulation is equivalent to the traditional approach with an electric and a magnetic Hertz vector. The formalism is illustrated with an integral-equation analysis of reflection from a semi-infinite conducting dielectric.

Full-wave analysis of microstrip lines on ferrite and anisotropic dielectric substrates

Microstrip lines on ferrimagnetic and uniaxial anisotropic dielectric substrates are examined. Hertz vector potentials oriented along the magnetization direction for the ferrimagnetic substrate and along the optical axis for the uniaxial anisotropic dielectric substrate are used to obtain the dyadic Green's functions in the Fourier transform domain. Galerkin's method is used to obtain the frequency behavior of the propagation constant. Curves for a few cases are shown as examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An analytic method to study open thick microstrips

In this paper an analytic method of study the dispersion characteristics of an open thick microstrip is presented. By the help of the hertzian vector it was found the integral equation for the current distribution and the transmission coefficient. From the numerical results it is shown that the attenuation constant due to the conductor loss is less in the rounded edge microstrip than in the rectangular one. For the current distribution the transverse current was at least ten times less the longitudinal one. So to study an open thick microstrip we have good approximation by using only the longitudinal current.

Analysis of an arbitrarily excited – loaded dipole antenna with a derived expression for the field in the feed gap

Expressions are derived for the aperture time-harmonic electromagnetic fields in the feed gap of a solid infinite antenna along the z axis. This is done by matching the Fourier representation of the external electromagnetic fields to the Fourier representation of the fields inside the feed gap. The Fourier expansion is built up by superposition of basic sets of [Formula: see text]-independent cylindrical waves that are obtained from the solution of Helmholtz equations governing z-directed electric Hertzian potentials. The z-directed electric field, which is in the form of a Fourier–Bessel series, is then used to calculate the input admittance of an infinitely long antenna excited across a practically wide feed gap. This is done using the axial field discontinuity method, which is based on the Wiener–Hopf type analysis. The method treats the dipole antenna as a series combination of longitudinal electric-field surface functions that may exist owing to feeding and (or) loading. A comparison is made between the radiation characteristics based on the Fourier-series field inside a wide gap and those obtained using the delta-function excitation for a finite-length dipole antenna with various excitation and loading conditions.

Isotropic chiral media and scalar Hertz potentials

A new approach to electromagnetic field representations in isotropic chiral media is presented. It is shown that by a simple transformation of the fields the constitutive relations of the isotropic chiral medium (which include differential operators) can be transformed into those of a special form of bianisotropic media. It is finally found that the electromagnetic field can be represented by two scalar Hertz potentials which are solutions of a system of second-order differential equations. The Green functions of this system are calculated.

Conduit reaction to moving charge

An electron moves along the axis of a conduit; the primary potential as well as the Hertz vector of the secondary field are obtained. The “characteristic velocity” of the conduit is of basic importance for the performed calculation.