A combined-source solution for radiation and scattering from a perfectly conducting body

A combined-source solution is developed for electromagnetic radiation and scattering from a perfectly conducting body. In this solution a combination of electric and magnetic currents, called the combined source, is placed on the surfaceSof the conducting body. The combined-source operator equation is obtained from theE-field boundary-value equation. It is shown that the solution to this operator equation is unique at all frequencies. The combined-field operator equation also has a unique solution, but it is not directly applicable to the aperture radiation problem. TheH-field andE-field operator equations fail to give unique solutions at frequencies corresponding to the resonant frequencies of a cavity formed by a hollow conductor of the same shape. The combined-source operator equation is solved by the method of moments. The solution, valid for a three-dimensional closed surfaceS, is then applied to a surface of revolution. Examples of numerical computations are given for a sphere, a cone-sphere, and a finite circular cylinder.     

Known-Plaintext Attack to Two Cryptosystems Based on the BB Equation

Recently, Rama Murthy and Swamy proposed a symmetric cryptosystem based on the Brahmagupta-Bhaskara (BB) equation. The BB equation is the quadratic Diophantine equation nx ² + k = y ², where k is an integer and n is a positive integer such that radic(n) is irrational. For the particular case k=1, the equation is called the Pell equation. The proposed cryptosystem was modified later by the same authors in order to avoid the cryptanalysis given by Youssef. Below, a known-plaintext attack to both cryptosystems is presented.     

Diffraction from an abruptly terminated dielectric slab waveguide in the presence of a cylindrical scatterer

The efficiency of using an adaptor lens in front of an abruptly terminated symmetric slab optical waveguide is examined analytically in this paper. The adaptor lens is assumed to be of cylindrical shape with a constant refractive index. The coupling of the modes propagating inside the slab guide into radiation waves in the presence of the lens is treated by integral equation methods. In the first step an integral equation is derived for the Green's functionG(r/r')of an abruptly terminated slab waveguide. The integral equation is solved approximately by an iterative procedure giving accurate results when the difference of the refractive indices between the slab waveguide and substrate-cover regions is small. The Green's function is then used to formulate another integral equation for the unknown field inside the adaptor lens. The latter integral equation for the cylindrical lens cross section area is solved by adopting a cylindrical partial wave expansion for the unknown interior field. After determining this field, the reflection-coupling coefficients for the guided modes propagating in the opposite direction of the incident wave, are computed by using the Green's functionG(r/r'). Radiation patterns are also derived in the far field region for an incident slab guided mode. Numerical results are computed and presented for several guide dimensions, lens radii, and refractive indices.     

Cryptographic applications of Brahmagupta-Bha/spl tilde/skara equation

The Brahmagupta-Bha/spl tilde/skara (BB) equation is a quadratic Diophantine equation of the form NX/sup 2/+k=Y/sup 2/, where k is an integer (positive or negative) and N is a positive integer such that /spl radic/N is irrational. A particular case of the BB equation with k=1 is also known as Pell equation in literature. This equation in the Galois Field GF(p), where p is an odd prime has some practically useful properties. Application of these properties in two different fields of cryptography, namely, digital encryption and user authentication are discussed in this paper. For those applications, where software computation of the roots of the BB equation is unacceptable for being too slow, a hardware architecture for using the BB equation in GF(p) is given that is useful for implementation in VLSI form.     

Finite horizon H/sub /spl infin// fixed-lag smoothing for time-varying continuous systems

In this paper, we aim to solve the long-standing H/sub /spl infin// fixed-lag smoothing problem for time-varying continuous systems. By applying a novel innovation analysis approach in an indefinite linear space, a sufficient and necessary condition for the existence of an H/sub /spl infin// fixed-lag smoother is derived. The H/sub /spl infin// smoother is calculated by performing the linear matrix differential equation and the integral equation.     

Application of the integral equation method of smoothing to random surface scattering

The integral equation method of smoothing (IEMS) is applied to the magnetic field integral equation (MFIE) weighted by the exponentialexp (jk_{1}zeta)wherezetais the stochastic surface height. An integral equation in coordinate space for the average of the product of the surface current and the exponential factor is developed. The exact closed-form solution of this integral equation is obtained based on the specularity of the average scattered field. The complex amplitude of the average scattered field is thus determined by an algebraic equation which clearly shows the effects of multiple scattering on the surface. In addition, it is shown how the incoherent scattered power can be obtained using this method. Comparisons with the Kirchhoff approximation and the dishonest approach are presented, and the first-order smoothing result is shown to be superior to both.     

A sparse-matrix/canonical grid method for analyzing densely packedinterconnects

In this paper, a fast numerical method called the sparse-matrix/canonical-grid (SM/CG) method is employed to analyze densely packed microstrip interconnects that involve a large number of unknowns. The mixed-potential integral equation is solved by using the method of moments in the spatial domain. The closed-form expressions of the spatial Green's functions of microstrip structures are obtained from the combination of the fast Hankel transform and the matrix pencil method. The Rao-Wilton-Glisson triangular basis functions are used to convert the integral equation into a matrix equation. The matrix equation is then solved by using the SM/CG method, in which the far-interaction portion of the matrix-vector multiplication in the iterative solution is performed by the fast Fourier transforms (FFTs). This is achieved by the Taylor series expansions of the spatial Green's functions about the uniformly spaced canonical grid points overlaying the triangular discretization. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed method. The SM/CG method has computational complexity of O(NlogN). Furthermore, being FFT-based facilitates the implementation for parallel computation     

TM scattering by an impedance sheet extension of a paraboliccylinder

An integral equation and method of moments (MM) solution are presented for the two-dimensional (2-D) problem of transverse magnetic (TM) scattering by an impedance-sheet extension of a perfectly conducting parabolic cylinder. An integral equation is formulated for a dielectric cylinder of general cross section in the presence of a perfectly conducting parabolic cylinder. It is then shown that the solution for a general dielectric cylinder considerably simplifies for the special case of TM scattering by a thin multilayered dielectric strip that can be represented as an impedance sheet. The solution is termed an MM/Green's function solution, where the unknowns in the integral equation are the electric surface currents flowing in the impedance sheet; the presence of the parabolic cylinder is accounted for by including its Green's function in the kernel of the integral equation. The MM solution is briefly reviewed, and expressions for the elements in the matrix equation and the scattered fields are given. Sample numerical results are provided     

Propagation in a two-dimensional periodic random medium withinhomogeneous particle distribution

The behavior of electromagnetic waves when propagating in a periodic random medium, such as a row-structured canopy, is considered. The semideterministic character of the particle distributions is represented by nonuniform extinction and phase matrices and the problem is formulated by the radiative transfer equation. Solution of the radiative transfer equation is pursued both iteratively and by using a numerical technique, based on the discrete-ordinate approximation and Taylor series expansion. It is shown that the numerical solution for the periodic canopy is computationally efficient, and a closed-form for the first-order solution (iterative approach) of the radiative transfer equation is obtained for periodic cases. The analytical and numerical results are compared with transmission measurements at L- and C-band frequencies for a corn canopy for a variety of canopy conditions, with good agreement     

A Closed-Form Equation for Estimating Capacitance of Signal Vias in Arbitrarily Multilayered PCBs

This paper provides a significant improvement to the coaxial-line equation that is used for analytical estimation of via capacitance in multilayered printed circuit boards (PCBs) Previously, a pure coaxial-line equation is used for estimating via capacitance with a maximized and minimized number of pads in high-density multilayered PCBs. Now, in the via capacitance equation, both the capacitances associated to via body and via pads in multilayered PCBs with arbitrary stackups are included. Laboratory measurements by a vector network analyzer and digital inductance–capacitance–resistance (LCR) meter, as well as quasi-static and full-wave computational simulations, confirm solid accuracy of the derived equation used for estimating the capacitance of arbitrary signal via geometry common in practice.      

On the effects of boundary conditions at the cathode on the equations of steady space-charge flow

The steady, zero-temperature, space-charge flow problem has a unique solution if formulated in the differential equation form. However, if the differential equations are approximated by a difference equation formulation, it is shown that the solution is no longer unique. The effect is demonstrated by the example of the Langmuir planar diode. It is proved theoretically for the planar diode, and demonstrated numerically for a two-dimensional example, that the ambiguity in the perveance of a gun due to this effect is of orderh/d; herehis the mesh size of the difference mesh, anddis the mean length of the particle trajectories.     

High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell's equations based on a nonstandard finite-difference model

We previously introduced high-accuracy finite-difference time-domain (FDTD) algorithms based on nonstandard finite differences (NSFD) to solve the nonabsorbing wave equation and the nonconducting Maxwell equations. We now extend our methodology to the absorbing wave equation and the conducting Maxwell equations. We first derive an exact NSFD model of the one-dimensional wave equation, and extend it to construct high-accuracy FDTD algorithms to solve the absorbing wave equation, and the conducting Maxwell's Equations in two and three dimensions. For grid spacing h, and wavelength /spl lambda/, the NSFD solution error is /spl epsiv//spl sim/(h//spl lambda/)/sup 6/ compared with (h//spl lambda/)/sup 2/ for ordinary FDTD algorithms using second-order central finite-differences. This high accuracy is achieved not by using higher-order finite differences but by exploiting the analytical properties of the decaying-harmonic solution basis functions. Besides higher accuracy, in the NSFD algorithms the maximum time step can be somewhat longer than for the ordinary second-order FDTD algorithms.     

Diffraction of an H-polarized electromagnetic wave by a circular cylinder with an infinite axial slot

A comparison of three methods of solution for the problem of scattering and diffraction of a transverse electric (TE) polarized plane wave by an infinite circular cylinder having an infinite axial slot is presented. In one method of solution, the aperture field integral equation (AFIE) method, the fields in and around the cylinder are found from the apertureE-field and the Green's functions for the interior and exterior of a cylinder. In the other two methods, the fields are determined from the surface current, which is obtained by solution of theH-field integral equation (HFIE) or theE-field integral equation (EFIE). The field in the aperture of the cylinder is found from the three methods, and the advantages and disadvantages of each method of solution are discussed. In addition, it is also shown that for shell thickness less than 1/20 of a wavelength, the aperture fields do not differ signifcantly from those of an infinitely thin shell cylinder.     

The accuracy of numerical solutions for electron gun design

The solution of Poisson's equation and the equation of motion by numerical methods is discussed. By considering the truncation error due to the use of difference equations in the one-dimensional case, it is shown, on the basis of a perturbation theory, that the error in the solution is proportional to the mesh sizeh. This theory is shown to be in good agreement with numerical solutions obtained on a computer. A correction formula is then derived which makes it possible, by obtaining sointions for two different mesh sizes haand hb, to form a space-charge-flow solution accurate to second order inh. It is also shown that more accurate forms of the difference equation may be obtained to represent the steady space-charge flow. By these various means, it is possible to increase the accuracy of numerical solutions by an order of magnitude or more. Thus, by using a coarse mesh, but better accuracy, the speed of computation may be considerably increased.     

Thermionic saturation of diffusion currents in transistors

Macroscopic diffusion theory is not applicable to large percentage variations in carrier concentration over distances comparable to a scattering path length. It is plausible that the transport velocity for thermionic emission over a barrier of zero height constitutes a limit on the velocity of diffusive current flow at any point in a bulk semiconductor. Not incorporating a thermionic limit, the drift-diffusion current transport equation cannot account for thermionic emission. While a full remedy to these difficulties can only lie in a complete investigation of the relevant statistical physics, we present herein an empirical modification of the current transport equation based on the elementary concept of thermionic saturation of the diffusion current. The modified equation appears capable of accounting, at least in a crude way, for thermionically limited current flow, while permitting a unified treatment of diffusion-drift-thermionic transport. For narrow base transistors (w_b 1000 Å), much larger ratios of stored base charge to collector current are predicted than from diffusion theory. A thermionic emission-diffusion equation is derived for barrier injection in floating base transistors biased into punch-through. It is found for silicon structures that transport is diffusion-drift dominated at base impurity concentrations < 10¹⁷ cm⁻³, and thermionically controlled at higher base doping. A possible small reduction in the thermionic Richardson constant by a factor e is also predicted from the diffusion saturation equation.     

A New Method for the Determination of the Area Under a Cardiac Output Curve

An equation of a continuous curve having the general form of the cardiac output curve has been derived in a logical fashion. The resulting equation of this curve originating at the origin is y = yp(x/xp)m exp (1 - (x/xp)m) where y and x are the vertical and horizontal components of the curve, yp and xp are the vertical and horizontal displacements of the curve at its peak, and m is a positive constant to be determined for each curve. This equation is integrated to determine the area under the curve it represents and appropriate substitutions are employed, with the result that the equation for the area is reduced to area = ypxpW where W is determined from a table included in the text.     

A two-dimensional analysis for MOSFET's fabricated on buried SiO2layer

An accurate numerical analysis for predicting characteristics of MOSFET's Fabricated on an epitaxial Si layer on top of a buried SiO2layer formed by oxygen ion implantation is presented. Basic equations, that is, the current continuity equation for minority carriers, Poisson's equation, and the equation for the majority-carrier concentration including carrier multiplication due to impact ionization, are iteratively calculated for obtaining a self-consistent solution. Good agreement between theoretical and experimental results was obtained over a wide range of device parameters and terminal biases. Threshold dependence upon Si/buried SiO2interface charge as well as Si film thickness is shown. As the Si film and buried SiO2thicknesses are reduced, the threshold voltage shift with channel shortening becomes smaller. Finally, theoretical analysis shows that the short-channel effect in MOSFET fabricated on an epitaxial layer with an underlining buried SiO2layer is smaller than those in bulk Si MOSFET and MOSFET/SOS.     

Transverse Electric Resonances in a Coaxial Line Containing Two Cylinders of Different Dielectric Constant

A coaxial line containing a medium of propagation consisting of two coaxial cylindrical cylinders of different dielectric constant is considered for the special case of TE/sub nm/ resonances, and numerical calculations are carried out for a few cases of the TE/sub 11/ type resonance. A reference paper called to the attention of the author by the reviewer treats the general condition of propagation in such a line. The numerical solutions to the cases of interest in this application were not performed, however, presumably since the interest was in propagating modes and since the general characteristic equation is quite complicated. It is shown here that consideration of transverse boundary conditions only leads to an equation which is much less complicated and which is equivalent to the reduction of the general characteristic equation (of the reference paper) when cutoff is approached.     

TM modes guided by nonlinear dielectric slabs

Propagation of a TM₀ mode in a Kerr-type nonlinear dielectric slab is investigated. The exact analytical expression of E ²(x), E²=E_x²+E_Z ², the decoupled linear differential equation and its equivalent Volterra integral equation of E_x, and the eigenvalue equation of propagation constant β are presented. Finally, an example of approximate analytical solutions obtained by the method of successive approximation is given     

Eikonal equation for moving media and its relation to dynamic programming

The eikonal equation for moving media (first order in V/c) is derived for the electromagnetic case. Maxwell's equations with Minkowski's constitutive relations, and the dynamic programming approach, lead to essentially the same result. The acoustical eikonal equation for moving media is derived by means of the dynamic programming method. A result obtained by Kritikos, believed to be wrong, is shown to be valid as an approximation.