Full-wave analysis of a strip crossover

A full-wave analysis of a strip crossover above a conducting plane is carried out. Higher-order modes are excited in the form of evanescent waves in the vicinity of the discontinuity, while further away only the dominant (TEM) modes exist. The higher-order mode currents are modeled by triangle functions and the dominant modes by outgoing traveling waves. The method of moments is used to reduce the integral equations on the surface of each strip to matrix equations whose solution determines the currents on each strip. The impedance and scattering matrices of the four-port network and the equivalent circuit were determined. At low frequencies, the equivalent circuit agrees very well with that which was obtained previously using a quasi-static analysis. The two approaches begin to disagree when the cross-sectional dimensions of the crossover become comparable to a tenth of the wavelength. At that point the quasi-static analysis becomes inaccurate, while the full-wave analysis presented here remains valid     

Signal solutions of Maxwell's equations for charge carriers withnon-negligible mass

Discusses signal solutions to Maxwell's equations for charge carriers with non-negligible mass. In order to find solutions the authors add information to Maxwell's equations by means of a physical assumption to obtain a defined solution. The authors' assumption is that magnetic dipoles and magnetic dipole currents should be represented by a magnetic (dipole) current density term just as electric dipoles and electric dipole currents-or electric polarization currents-have always been represented by an electric current density term. It is perfectly possible that other physical assumptions can be made that yield defined solutions and that will withstand public scrutiny     

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     

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     

Geometrical Effects on Shielding Effectiveness at Low Frequencies

A frequent approach to computing the magnetic shielding effectiveness of enclosures is to consider the effect of a plane wave impinging on a sheet of infinite extent. This permits an analysis based on a transmissionline characterization. However, when the wavelength is large compared to the dimensions of the enclosure, other analytical approaches provide better results. It has been shown that the current distribution on a box-like object scattering in the Rayleigh region tends to concentrate at the edges and corners of the box. This leads to concentrations of the magnetic field in the vicinity of edges and corners both inside and outside the enclosure. Since the effects of the current concentrations are localized, the magnetic shielding problem can be simplified by assuming a uniform current distribution on the exterior of the enclosure. Under this assumption the socalled ?circuit approach? can be applied. The box-like enclosure is characterized as a series of shorted turns which shield a sensor within the enclosure. Based on the geometry, the mutual and leakage impedances between the source and sensor are used to compute the magnetic shielding effectiveness. This approach yields valid results for shields constructed of either wire mesh or sheet metal. It can also be extended to account for degradation due to bad bonds. A comparison of results, both transient and steady state, of the circuit approach and scattering theory show close agreement for spherical enclosures.     

A mathematical technique for an exact small-signal field analysis of multiple-stream interaction in a finite longitudinal magnetic field

A mathematical technique for solving Maxwell's equations and the Lorentz force equation with no approximations except the small-signal approximation is presented. A finite dc magnetic field parallel to the dc velocity of the charges is included. Polarization variables are used, and the boundary conditions include ac surface charge density and surface current density. The advantages of the method are that both fast waves and slow waves are included without a quasi-static approximation, and only the determinantal equation requires computer solution. The partial differential equations are solved directly and need not be solved by computer.     

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.     

Review of Circuit Approach to Calculate Shielding Effectiveness

The shielding effectiveness of an enclosure at low frequencies can be readily computed using a circuit approach. Not only does this technique include the effects of the properties of the shield material, but it also includes the details of the geometry of the enclosure. Furthermore, this approach allows a nonempirical consideration of mesh enclosures and the effects of resistive seams in enclosure walls. By working with the circuit analogue, penetration by transient fields can also be computed. Essentially the enclosure is viewed as an antenna. In the case of magnetic shielding effectiveness, the enclosure is viewed as a short circuited loop antenna. In the case of electric field penetration, the enclosure is viewed as a fat electric dipole. Using this characterization and exact solutions where available, the current distribution on the outside of the enclosure is first determined. Then, based on the current distribution, the penetrating fields are computed. The equations are developed in such a way as to preserve a lumped circuit analogue for the low-frequency region. The basic circuit equations for magnetic field penetration are rederived from a rigorous solution. Rules to estimate the rise-time, fall-time, and peak magnitudes of transient penetrating fields are developed. The electric shielding effectiveness is developed in a similar manner. In both cases the results of the circuit approach agree well with those based on rigorous solutions of the electromagnetic boundary conditions. The results also agree with published experimental data on both large and small enclosures.     

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     

Nonlinear diffusion and internal voltages in conductingferromagnetic enclosures subjected to lightning currents

The voltage inside a conducting ferromagnetic enclosure, excited by direct attachment of lightning, is determined. A nonlinear approximate model is constructed and solved analytically to model the magnetic field diffusion through the enclosure wall excited by a filament current. This model gives the internal voltage waveform (at very early times it forms a rough approximation to the solution). The solution (including the early time behavior) for an attached current carrying conductor of large radius is also given. The model is constructed by use of known nonlinear saturation front theory. The paper also considers nonnegligible magnetic flux behind the saturation front (in the saturated region), and shows that this saturated flux results in a simple additive time shift to the penetration of the front     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.     

Combined field solution for TM scattering from multiple conducting and dielectric cylinders of arbitrary cross section

A simple moment solution is given for the problem of electromagnetic scattering from multiple conducting and dielectric cylinders of arbitrary cross section. The system of conducting and dielectric cylinders is excited by a plane-wave polarized transverse magnetic to the axis of the cylinders. The equivalence principle is used to obtain three coupled integral equations for the induced electric current on the conducting cylinders and the equivalent electric and magnetic currents on the surface of dielectric cylinders. The combined field integral equation (CFIE) formulation is used. Sample numerical results are presented. The agreement with available published data is excellent.     

A technique for solution of Maxwell's equations in a moving dielectric medium

A simple technique is presented for converting a known solution for the electric and magnetic vector fields in a dielectric medium at rest into the corresponding fields in a moving dielectric medium. The technique combines methods presented by Tai [1] with a scaling procedure developed by Clemmow [2]. Tai's work reduces the moving medium problem to the solution of Maxwell's equations in a uniaxial medium, and Clemmow's procedure enables one to convert a known solution in an isotropic medium to the corresponding solution in a uniaxial medium. Thus by first solving for the fields in the medium at rest, then following Clemmow's procedure to obtain the fields in Tai's uniaxial medium, and finally applying Tai's reasoning, one may easily obtain the solution of Maxwell's equations in the moving medium.     

A comparison of integral equations with unique solution in theresonance region for scattering by conducting bodies

A comparison of integral equations, for problems involving scattering by arbitrary-shape conducting bodies, having a unique solution in the resonance region is presented. The augmented electric and magnetic field integral equations and the combined field integral equation, in their exact and approximate versions, are considered. The integral equations and the basis and test functions used in the method of moments to solve them are reviewed. Their implementation in a computer code is analyzed, mainly the relation between the matrix properties and the CPU time and memory. Numerical results (condition number and backscattering cross section) are presented for the cube. It is shown that the combined field integral equation, and the approximate (symmetric) combined field integral equation, are the most efficient equations to use in the neighborhood of resonant frequencies, because the overdetermined augmented integral equations require an extra matrix multiplication     

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.     

Modeling of magnetizing processes

Quasi-static magnetic processes such as those found in magnetic recording are examined. The hysteresis in medium-hard magnetic materials of the type used in recording media and the magnetizability of soft magnetic materials such as those used in recording heads are discussed. Two general modeling techniques are used to describe these processes: physical modeling and phenomenological modeling. In physical modeling, the basic processes involved are simulated in order to be able to describe the basic magnetizing modes. In phenomenological models, the gross behavior of the material is described mathematically by Preisach-type models in order to couple the material properties to Maxwell's equations so as to obtain solutions of field problems. The latter models are computationally more efficient than the former, but they do not give any insight into the physical principles involved     

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     

Low-Frequency Shielding Effectiveness of a Double Cylinder Enclosure

The low-frequency shielding effectiveness of a long double cylinder shield is determined through a solution of Maxwell's field equations. The shielding expression obtained is then compared with the results obtained by both the circuit approach and the transmission-line analogy. The findings of the present paper are also compared with the analysis by previous authors of the multishield problem. A digitalcomputer program for numerical evaluation of the effectiveness of adouble cylinder shield is developed and used to study the influence of the shield dimensions and material constants.     

The Maxwell's equations in the sense of distributions

It is well known that there is no direct one-to-one correspondence between the electromagnetic theory based on the physical laws and that based on the Maxwell's differential equations. For example, in order to derive the boundary conditions from the Maxwell's differential equations, one assumes that some integral identities derived from them are valid even when the field components (or material parameters) are discontinuous. This assumption violates, in a sense, the completeness of the theory of electromagnetism based on the Maxwell's differential equations. We will prove that if one postulates that the Maxwell's equations are valid in the sense of distributions, then this incompleteness will be removed and the boundary conditions will appear implicitly in the basic differential equations.     

Interaction of Magnetic Fields and Ferromagnetic Shields

An analytic procedure is developed for the purpose of determining the effectiveness of ferromagnetic shields in shielding against magnetic fields. The basic approach is to separate the magnetization relation of a ferromagnetic material into regions in which each region is characterized by a constant permeability. Maxwell's equations are then solved in each time-varying geometric region (which correspond to the regions of the magnetization relation) and the solutions are matched at interfaces. This procedure permits solutions for nonlinear shielding problems to be readily obtained using linear techniques.