Infinite Double Integral Research Articles

Electromagnetic Scattering by a Perfectly Conducting Rectangular Plate Buried in a Lossy Half-Space

The aim of this paper is the accurate and efficient analysis of the electromagnetic scattering by an arbitrary oriented perfectly conducting rectangular plate entirely buried in a lossy half-space. The problem, formulated as an electric field integral equation (EFIE) in the spectral domain for the surface current density on the rectangular plate, is discretized by means of Galerkin's method with a set of orthonormal analytically Fourier transformable basis functions factorizing the behavior of the unknown at the edges. In this way, fast convergence is achieved even for scatterer size of some wavelengths. Unfortunately, this method leads to the numerical evaluation of infinite double integrals of oscillating and slowly decaying functions. To overcome this problem, a new analytical technique that allows to write such integrals as combinations of very quickly converging integrals is introduced.

A New High Efficient Analysis of the Scattering by a Perfectly Conducting Rectangular Plate

The analysis of the scattering by a perfectly conducting rectangular plate by means of Galerkin's method in the spectral domain with products of Chebyshev polynomials of first and second kind multiplied by their orthogonal weights as basis functions is fast convergent even for scatterers size of some wavelengths but leads to the numerical evaluation of infinite double integrals of oscillating and slowly decaying functions. The aim of this paper is the introduction of a new analytical technique that allows to write such integrals as combinations of very quickly converging integrals.

Mutual impedance calculation between buried conductors of finite length

PurposeThe paper aims to develop expressions for calculating the mutual impedance between isolated conductors buried in homogeneous earth. The conductors have finite length and arbitrary position.Design/methodology/approachThe conductors are represented by the use of elementary electric dipoles. Well‐known existing expressions are employed for the electric field of these dipoles. The induced voltages are evaluated and the final expressions for the mutual impedance are derived. The resulting expressions involve infinite double integrals, evaluated by using adaptive quadratures that are, however, time consuming. Therefore, an alternative approach is followed involving Sommerfeld integrals (SI) for representing the electric field of a dipole and a recently devised method for computing the SI, in the spatial domain, by using calculation of discrete complex images.FindingsFinal expressions for parallel and perpendicular conductors were derived and numerical results for several values of frequency, conductors' length and horizontal distance between them, were produced. Comparison to results produced with the well‐known Pollaczek formula showed excellent agreement.Research limitations/implicationsFor future research, it is possible to use the developed expressions for earthing systems study, where the grounding grid is discretized and the moment method is invoked.Originality/valueCurrently, the formulas used for calculating mutual impedance are valid for parallel conductors of infinite length. With the present work, accurate expressions are given for finite length and arbitrary horizontal positioned conductors. In addition, the use of SI and the discrete complex image method results in a rapid and efficient tool for massive impedance calculations.

A closed‐form solution to the asymptotic part of the MOM impedance matrix and the MOM excitation vector for printed structures on planar grounded dielectric slabs

AbstractIn the spectral domain method of moments (MoM) solution of printed structures on planar grounded dielectric slabs, the infinite double integrals which appear in the asymptotic parts of the MoM impedance matrix and the MoM excitation vector elements, have been previously transformed to one‐dimensional finite integrals, which have been numerically computed using the highly specialized “International Mathematics and Statistics Library” subroutines. In this paper, these one‐dimensional integrals are evaluated in closed‐form, resulting in an improved efficiency and accuracy for the rigorous investigation of printed antennas and complex millimeter and microwave integrated circuits. Numerical results in the form of mutual impedance between two expansion functions and input impedance of various microstrip antennas are presented to assess the accuracy of these closed‐form expressions. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 882–886, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22274

Physical asymptotic model for a conducting patch and how this enables the inclusion of dielectric substrate in a free-space moment method code

The asymptotic field expression, in the spectral domain, of a conducting patch on a multilayer dielectric substrate is derived. It is shown that the corresponding field solution in the spatial domain is equal to that of a patch in a homogeneous region with an effective permittivity. This allows the infinite double integral of the asymptotic part of the method of moments matrix elements to be computed in a homogeneous spatial domain. As a result, the computation time can be reduced considerably. It is shown that a spatial domain MoM code for conducting patches and lines in homogeneous regions can easily be modified to include a multilayer dielectric substrate by adding a correction factor to the matrix elements.

ANALYTICAL EVALUATION OF THE ASYMPTOTIC IMPEDANCE MATRIX OF A UNIAXIAL SUBSTRATE WITH ROOF-TOP FUNCTIONS

An asymptotic technique is presented to solve the asymptotic part of the impedance matrix in the spectral domain for printed circuit structures on a uniaxial substrate by using roof-top subdomain expansions. The infinite double integral of the asymptotic part of the impedance matrix is reduced to a finite one-dimensional integral by performing the integral transform technique. The efficiency and accuracy of the proposed method are compared with those of the conventional method, and it is demonstrated that the proposed method reduce the computation and improve the accuracy for the anisotropic uniaxial case.

Analytical Technique To Evaluate the Asymptotic Part of the Impedance Matrix of Microstrip Dipole On a Uniaxial Substrate - Abstract

The integral transform method with the asymptotic extraction technique is formulated to evaluate a Sommerfeld type integral for the analysis microstrip dipole on a uniaxial substrate. The infinite double integral of the asymptotic part of the impedance matrix with triangular subdomain basis function with edge condition can be reduced to a finite one-dimensional integral. This finite one-dimensional integral can be easily evaluated numerically after the singular part of the integral is treated analytically. It is demonstrated the efficiency and accuracy of the proposed method to evaluate the asymptotic part of impedance matrix.

Electrostatic force density for a scanned probe above a charged surface

The Coulomb interaction of a dielectric probe tip with a uniform field existing above a semi-infinite, homogeneous dielectric substrate is studied. The induced polarization surface charge density and the field distribution at the bounding surface of the dielectric medium with the geometry of half of a two sheeted hyperboloid of revolution located above the dielectric half space interfaced with a uniform surface charge density is calculated. The force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The calculation is based on solving Laplace’s equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions.

SCATTERING OF A SPHERICAL WAVE BY A THIN HARD STRIP

This paper is concerned with a theoretical solution to the problem of scattering of a spherical wave by a strip. The strip is infinitely thin, infinite in length and of width 2a. The problem is first brought into the wave space through a spatial Fourier transform of the wave equation and of the boundary conditions on the strip. The Fourier transform is taken with respect to the co-ordinate axis parallel to the edges of the strip. Using the boundary conditions on the strip leads to an integral equation of the first kind, the unknown of which is the discontinuous potential jump across the strip. This latter is expanded into some suitable functions and the coefficients of the series expansion are thereafter determined from an infinite system of equations. The system's matrix is found to be mainly diagonal and tests on the stability of the numerical calculations suggest the significant number of equations in the system be limited to approximatelyka+5, withkbeing the wavenumber. Finally, after solving the system of equations and going back to the scattered field, the expression of this latter is made from an infinite series over some infinite double integrals whose approximate evaluation is made with the help of the two-dimensional stationary phase method. This treatment corresponds to the far field case. A further consideration of the right side of the system of equations leads to an improved value of the scattered field. Comparisons are made to an approximated prediction of the scattered field by using the Biot and Tolstoy exact theory of diffraction of a spherical wave by a hard wedge. The implementation of this approach to the strip requires the further consideration of the multiple diffraction between its edges for improving the calculated value of the scattered field. Some numerical examples are treated with discussions on their results.

The 𝑑₂-transformation for infinite double series and the 𝐷₂-transformation for infinite double integrals

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate d d - and D D -transformations. The D 2 D_2 -transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the d 2 d_2 -transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.

Analytical technique to evaluate the asymptotic part of the impedance matrix of Sommerfeld-type integrals

The propagation of electromagnetic waves in a grounded dielectric slab has numerous applications in printed antenna technology and in the analysis of microwave- and millimeter-wave integrated circuits. For the accurate analysis of microstrip dipoles and circuits based on the moment of method (MoM), a crucial step is the precise evaluation of the impedance matrix elements which contain the integration of Sommerfeld-type integrals. The integral transform method with the asymptotic extraction technique is formulated for calculating a Sommerfeld-type integral problem. This formulation allows the infinite double integral of the asymptotic part of the impedance matrix to be transformed into a finite one-dimensional (1-D) integral. This finite 1-D integral contains a spherical Legendre function and can be easily evaluated numerically after the singular part of the integral is performed analytically. It is shown that the proposed method dramatically reduces the computation time and improves the accuracy over the conventional method to evaluate the asymptotic part of impedance matrix.

A unified study on the roundoff noise in 2-D state space digital filters

This paper studies the roundoff noise and the scaling in 2-D state space digital filters. The noise matrix and the covariance matrix in the 2-D case are derived. They are expressed with double infinite sums or double integral. A formulation is presented to synthesize minimum roundoff noise 2-D state space digital filters. A practical synthesis procedure is also provided. A numerical example is given to show the analysis method and synthesis method presented here.

Diffusion and sorption in particles and two‐dimensional dispersion in a porous medium

A solution of the two‐dimensional differential equation of dispersion from a disk source, coupled with a differential equation of diffusion and sorption in particles, is developed. The solution is obtained by the successive use of the Laplace and the Hankel transforms and is given in the form of an infinite double integral. If the lateral dispersion is negligible, the solution is shown to simplify to a solution presented earlier. Dimensionless quantities are introduced. A steady state condition is obtained after long times. This is investigated in some detail. An expression is derived for the highest concentration which may be expected at a point in space. An important relation is obtained when longitudinal dispersion is neglected. The solution for any value of the lateral dispersion coefficient and radial distance from the source is then obtained by simple multiplication of a solution for no lateral dispersion with the steady state value. A method for integrating the infinite double integral is given. Some typical examples are shown.

On Accelerating the Convergence of Infinite Double Series and Integrals

The generalization of Shanks’ e-transformation to double series is discussed and a class of nonlinear transformations, the ${[A/S]_R}$ transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ e-transformation or its equivalent Wynn’s $\varepsilon$-algorithm. A generalization of the ${[A/S]_R}$ transformation to N-dimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent $\varepsilon$-algorithm of Wynn and the G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.

On accelerating the convergence of infinite double series and integrals

The generalization of Shanks’ e-transformation to double series is discussed and a class of nonlinear transformations, the [ A / S ] R {[A/S]_R} transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ e-transformation or its equivalent Wynn’s ε \varepsilon -algorithm. A generalization of the [ A / S ] R {[A/S]_R} transformation to N-dimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent ε \varepsilon -algorithm of Wynn and the G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.

Driving point impedance of linear antennas in the presence of a stratified dielectric

A linear flat strip antenna lies between several dielectric layers in a direction parallel to the interfaces. Its impedance is formulated variationally as an infinite double integral. For thin half-wave antennas in a homogeneous medium the double integral is evaluated analytically and it gives standard impedance expressions. For antennas of finite width the integrals are evaluated numerically and for a homogeneous medium the impedance has been correlated with the theories of King and Middleton, and Wu, with the impedance computed from a complementary slot antenna and with available measurements. The impedance is also computed for antennas in a dielectric layer, and the decrease of the radiation efficiency observed with increasing electrical thickness of the layer is explained with the increased amounts of surface wave power and in part by local dielectric losses. For insulated antennas located in a dissipative medium the theory presented here yields results in better agreement with measurements than the conventional transmission line theory. The transmission line theory is shown to give too large a resistance and reactance at the full-wave resonant peaks of the impedance and too low a resistance for short antennas.