General solutions of Maxwell's equations for signals in a lossymedium. I. Electric and magnetic field strengths due to electricexponential ramp function excitation

Modification of Maxwell's equations to obtain general solutions for a lossy medium is reviewed. It is done by adding an extra term, referred to as the fictitious magnetic charge density. The solutions, which are in integral form, are solved numerically by computer for an exponential ramp function excitation. Computer plots for the electric and magnetic field strengths as functions of time at different locations in a lossy transmission medium are presented     

General solutions of Maxwell's equations for signals in a lossymedium. III. Electric and magnetic field strengths due to electric andmagnetic sinusoidal pulse excitation

For pt.II see ibid., vol.30, no.1, p.37-40 (1988). The representation of a function with a general time variation by a series expansion of time-shifted transients is discussed. On the basis of this representation, numerical solutions of Maxwell's equations are presented for the electric and magnetic field strengths in a lossy medium due to electric and magnetic excitation functions consisting of a finite number of sinusoidal cycles. The solutions are derived by means of a time-series expansion of the available solutions for the electric and magnetic exponential ramp function excitations     

A comparison of transient solutions of Maxwell's equations to thatof the modified Maxwell's equations

In 1986 H.F. Harmuth introduced a modification of Maxwell's equations to study the propagation of transient electric and magnetic field strengths in lossy media. Opponents of this modification of Maxwell's equations have claimed and attempted to demonstrate that Maxwell's equations in their known forms can correctly be solved, for example by the Laplace transformation method, to obtain solutions of transient electric and associated magnetic field strengths in lossy media without encountering any difficulties. This work presents detailed computer plots of Harmuth's transient solutions of the modified Maxwell's equations and that of Maxwell's equations solved by the Laplace transformation characteristic for the two solutions, which indicate that they are not the same. It is shown that Harmuth's procedure results in physically more plausible solutions     

Comments on `Riemann-Green function solution of transientelectromagnetic plane waves in lossy media'

In commenting on the above-named work by O.R. Asfar (see ibid., vol.EMC-32, no.3, p.228-31, Aug. 1990), the commenter notes that one can write infinitely many solutions for the associated magnetic field strength that will all satisfy Maxwell's equations, but Maxwell's equations cannot tell which one of these infinitely many solutions is the right one. It is further pointed out that the physical significance of the magnetic current density term used became clear when transients in lossy media were investigated with Lorentz's equations of electron theory, which allow for the fact that electric charges are always connected with particles having a mass, whereas Maxwell's original equations do not contain the concept of mass. A physical explanation for this is offered, and attention is given to the creation of the singularity in Maxwell's equations that make sit impossible to obtain the associated magnetic field strength without some limit process     

Correction of Maxwell's Equations for Signals II

In the first of two companion papers it was shown that the addition of a magnetic current density to Maxwell's equations is a sufficient condition to obtain solutions in lossy propagation media for waves that are not infinitely extended periodic waves. The solutions obtained represented transients that may be used to represent signals having a beginning and an end. This second paper shows that the addition of a magnetic current density is also a necessary condition for the existence of transient solutions in lossy media. The modification of Maxwell's equations is thus necessary and sufficient for the study of the propagation of signals in lossy media.     

The surface impedance anomaly resulting from a subsurface,infinitely long conductive wire

The change in surface impedance caused by the fields generated by an infinitely long, subsurface wire located in a homogeneous lossy medium (e.g., the Earth) is investigated experimentally and numerically. Sets of curves illustrate the variation in the horizontal electric field, the horizontal magnetic field, and the surface impedance as a function of conductor depth. The half-width at half-amplitude (HWHA) can be used as a measure of the lateral dimensions of the anomaly. While the horizontal magnetic field HWHA is equal to the conductor depth, the horizontal electric field HWHA (and so the surface impedance HWHA) varies not only with depth but also with the conductivity of the medium     

A Nonresonant Perturbation Theory

This paper presents a theory for a nonresonant perturbation technique for the measurement of electric and magnetic field strengths within a device. Most presently employed perturbation field strength measurements require the use of a resonance technique. In the technique discussed here, reflection coefficient measurements are made at the same frequency with, and without, a perturbing object placed at the point at which the field strength is to be measured. By these data, and by the equations derived and presented in this paper, the desired field strength can be calculated. The technique can be used for cavities that are too lossy to support resonance, and is suitable for cavities for which the resonant field configuration differs from the field configuration to be measured. In addition, this technique has the advantage that it permits the measurement of the phase, as well as the amplitude of the field.     

Some comments on Harmuth and his critics [propagation of pulses ina lossy medium]

A series of controversial papers on the propagation of pulses in a lossy medium by H.F. Harmuth (1986) have appeared in Transactions on Electromagnetic Compatibility. Many negative comments have appeared subsequently that betray a limited understanding of the points Harmuth is trying to make. Part of this difficulty lies in the method Harmuth chooses to present his results, namely the Fourier transform instead of the Laplace transform. The authors develop the formalism necessary to solve the propagation of pulses in a lossy medium for both the magnetic and electric fields using the Laplace transformation and generalized functions in order to clarify the strengths and weaknesses of the points Harmuth is trying to make     

Scattering by two-dimensional lossy, inhomogeneous dielectric andmagnetic cylinders using linear pyramid basis functions and pointmatching

A volume integral equation approach is used to calculate the scattering characteristics of lossy, inhomogeneous, arbitrarily shaped, two-dimensional dielectric and magnetic bodies. The scatterer is divided into triangular patches, which simulate curved and piecewise linear boundaries more closely than circular cylinder cells. Linear pyramid basis functions are employed to expand the unknown total electric field at the triangle nodes. The enforcement of the boundary conditions by point matching at the nodes converts the electric field integral equation to a matrix equation. Example cases are run and compared to previous moment methods and exact solutions, and this method shows good agreement. This method requires only one unknown per node in dielectric and magnetic material, which is a significant reduction in unknowns and matrix storage compared to traditional methods. By duality, this method can be used at either transverse electric or transverse magnetic polarization     

Analysis of lossy inhomogeneous planar layers using Taylor's series expansion

A general method is introduced to frequency domain analysis of arbitrary lossy inhomogeneous planar layers. In this method, all electric and magnetic parameters and also all components of the electric and magnetic fields are expanded in a Taylor's series. The field solutions are obtained after finding unknown coefficients of the series. The validity of the method is verified using analysis of some special types of planar layers.     

Gaussian beam representation of aperture fields in layered, lossymedia: simulation and experiment

It is demonstrated that a three-dimensional electromagnetic field of a given linear polarization, emanating from an aperture source and propagating in a lossy medium, can be represented by an astigmatic Gaussian beam with complex source coefficients. The values of the coefficients can be determined experimentally by scans of the phase and amplitude of the field in the electric and magnetic principal planes near the aperture by means of a monopole probe and a liquid phantom (a phantom being a device that simulates the conditions encountered when radiation (e.g. microwaves) is deposited in biological tissues (e.g. human muscles) and permits a quantitative estimation of its effects). Once the source parameters are obtained, computations of the field everywhere else can be achieved rapidly. The theory is verified experimentally for bounded, homogeneous, and layered lossy media. Agreement is within 3% (relative to the maximum field at the aperture) over the entire scanned area     

A set of integral equations for electromagnetic wave radiation from an arbitrary source in a linear, lossy, inhomogeneous, cold magnetoionic medium

By decomposing the permittivity tensor into its isotropic, longitudinal, and transverse parts (with respect to the static magnetic field), a set of simultaneous integral equations are derived for the electric field components in a linear, lossy, inhomogeneous, cold magneto-plasma. The developed integral equations are useful to obtain an approximate solution for electromagnetic radiation as well as scattering problems in such a medium.     

Solutions of Maxwell's equations for general nonperiodic waves inlossy media

Solutions of Maxwell's equations in lossy media for signals excited by a general applied source at the boundary plane are given. The excitation at the boundary plane can be through either electric or magnetic functions of any general time variation. No additional terms need be added to Maxwell's equations to obtain the solutions. Excitations by an electric step, exponential, and finite duration sinusoidal; functions of time are given as examples     

Electromagnetic coupling of coplanar waveguides and microstriplines to highly lossy dielectric media

The spectral-domain technique is utilized to analyze the coupling characteristics of coplanar waveguides and microstrip lines coupled with multilayer lossy dielectric media. Numerical results illustrating the dispersion characteristics of coplanar and microstrip lines, as well as the various electric field components coupled to highly lossy dielectric media, are presented. It is shown that the presence of a superstrate of lossless dielectric between the coplanar waveguide and the lossy medium plays a key role in setting up an axial electric field component that facilitates leaky-wave-type coupling to the lossy medium. The thickness of the superstrate relative to the gap width in the coplanar waveguide is important in controlling the magnitude of this axial electric field component. The coupling characteristics of the microstrip and coplanar lines are compared, and results generally show improved coupling if coplanar waveguides are utilized. Values of the attenuation constant α are higher for coplanar waveguide than for microstrip line, and for both structures α decreases with frequency     

A three-wave FDTD approach to surface scattering with applicationsto remote sensing of geophysical surfaces

A major difficulty in physical interpretation of radio wave scattering from geophysical surfaces is the lack of detailed information on the signatures of geologically plausible discrete objects. Although the aggregate response will never be dominated by any single object, differences in the population of discrete objects on or near the surface (their sizes and shapes, for example) can change the character of a radio echo markedly. When the average surface is modelled as a flat, homogeneous half-space, the field that “drives” the scattering process is a composite consisting of the incident plane wave and the reflected and transmitted plane waves, all of which are known quantities; the total field can then be defined as the sum of the driving field and the scattered field. When a discrete object is near the surface, the total field can be calculated using finite-difference time-domain (FDTD) techniques, and the scattered near field can be calculated accordingly. The Green's functions for electric and magnetic currents above and below the surface, obtained by Sommerfeld theory and employed in conjunction with Huygens' principle, transform the local scattered fields to the far field. The FDTD implementation accommodates discrete lossy dielectric and magnetic scatterers in the vicinity of a dielectric surface; extension to a lossy half-space is straightforward. Two-dimensional results for scattering from perfectly conducting circular cylinders above and below a dielectric surface agree with moment method solutions within a few percent. Results for scattering from a dielectric wedge exhibit expected forward diffraction and internal reflection phenomena     

Comments on `In defense of J.A. Stratton' by J.R. Wait

Comments are made on the above-named work (see ibid., vol.30, no.4, p.590, Nov. 1988). Attention is given to whether Stratton considered the problem of obtaining solutions of the complete electromagnetic fields for transients in lossy media trivial     

Time-domain study of half-space transmission problem with verticaland 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     

Generalized formulations for electromagnetic scattering fromperfectly conducting and homogeneous material bodies-theory andnumerical solution

A generalized E-field formulation for three-dimensional scattering from perfectly conducting bodies and generalized coupled operator equations for three-dimensional scattering from material bodies are introduced. A fictitious electric current flowing on a mathematical surface enclosed inside the body is used to simulate the scattered field, and, in the material case, a fictitious electric current flowing on a mathematical surface enclosing the body is used to simulate the diffracted field inside the body. Application of the respective boundary conditions lead to operator equations to be solved for the unknown fictitious currents, which facilitates calculation of the fields in the various regions, using the magnetic vector potential integral. The existence and uniqueness of the solution are discussed. These alternative operator equations are solvable using the method of moments. The numerical solution is simple to execute, rapidly converging, and general in that bodies of smooth but otherwise arbitrary surface, both lossless and lossy, can be handled effectively. Comparison of the results with available analytic solutions demonstrates the accuracy of the moment procedure     

Comments on `Some comments on Harmuth and his critics' by J.E. Grayand S.P. Bowen

J.E. Gray and S.P. Bowen (see ibid., vol.30, no.4, p.586-9, Nov. 1988) claim to have developed the formalism necessary to solve the propagation of pulses in a lossy medium for both the magnetic and electric fields using the Laplace transformation and generalized functions. They claim that Harmuth's introduction of the magnetic current s, is neither necessary nor a sufficient reason to insure consistency and that their method permits calculating both the electric and magnetic fields uniquely for a wide variety of pulses. The commenter claims, however, that Gray and Bowen introduce covert assumptions that deny their claim     

A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform

A general three-dimensional tensor finite-difference time-domain (TFDTD) formulation is derived to model electrically inhomogeneous lossy media of arbitrary shapes. The time domain representation of electric losses is achieved using Z-transforms. The regular cubical grid structure is maintained everywhere in the calculation domain by defining a 3-D face-fraction based 3 x 3 permittivity tensor on the interfaces that describes the relationship between the (known) average flux density vector and the (unknown) local electric field vector. For electrically lossy media, this tensor is complex in the frequency domain. However, it can be modified for use with the Z-transform. Only this modified real form is inverted, then transformed from the frequency into the Z-domain, and finally into the time domain. Furthermore, a local interface matrix is used to describe the relationship between the local electric field in the grid node and its counterpart on the other side of the interface. This matrix is complex in the frequency domain for lossy media. By applying the Z-transform, this matrix can also be transformed into the time domain using only real modified matrix elements. The accuracy of the method is confirmed by comparisons with analytical solutions.