Infinite Grids Research Articles

A new approach to diffraction analysis of conductor grids. II. Perpendicular-polarized incident plane waves

For pt.I see ibid., vol.37, no.1, p.84-88 (1988). Diffraction analysis is given for infinite planar conducting-strip grids illuminated by normally incident (perpendicular-polarized) plane waves, the electric fields of which are perpendicular to the strip axes. Iris-surface electric field integral equations are used which are based on the equivalent waveguide theory, and then the electric field is solved for using the moment method. This is a universal approach applicable to infinite planar grids made of conducting strips of rectangular cross section, uniform or periodic, dense or sparse, single layer or multilayer. >

A new approach to diffraction analysis of conductor grids. I. Parallel-polarized incident plane waves

Diffraction analysis is presented for infinite planar conducting-cylinder grids illuminated by normally incident (parallel-polarized) plane waves, the electric fields of which are parallel to the cylinder axes. The Green's function kernel integral equations are used for the induced currents, which are based on the equivalent waveguide theory and solved for the currents by the moment method. This is a universal analysis approach, applicable to infinite planar grids made of conducting cylinders of arbitrary cross section, uniform or periodic, dense or sparse, single layer or multilayer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modeling of borehole resistivity measurements using infinite electrical grids : Zemanian, A H; Anderson, B GeophysicsV52, N11, Nov 1987, P1525–1534

Modeling of borehole resistivity measurements using infinite electrical grids : Zemanian, A H; Anderson, B GeophysicsV52, N11, Nov 1987, P1525–1534

Modeling of borehole resistivity measurements using infinite electrical grids

An infinite electrical resistive grid generated by a finite‐difference approximation to Poisson’s equation in cylindrical coordinates with no azimuthal variation has been used to model borehole resistivity measurements. The medium surrounding the borehole is infinite in extent and its resistivity varies only in the radial direction. As a result, the branch resistance values of the infinite grid also vary only in the radial direction. A continued fraction of Laurent operators determines the node voltages along the borehole’s surface. Use of an operational calculus based upon the finite Fourier transformation converts the continued fraction into a readily computed form. This yields new formulas for apparent resistivity, given various resistivity measuring arrays and a resistivity profile in the radial direction. In contrast to classical methods, the need to match boundary conditions at the interface of a cylindrically layered earth is eliminated, so that the procedure is just as easily applied to a continuous resistivity profile as it is to a stepped profile. Moreover, the method is computationally fast by virtue of the fast Fourier transform algorithm and the avoidance of boundary‐condition matching. The speed and versatility of this model make it useful for studying the effects of complex invasion on the response of resistivity tools.

On the electromagnetic scattering from infinite rectangular grids with finite conductivity

-The development of a new numerical technique for the analysis of electromagnetic scattering from a rectangular wire (or strip) mesh. According to this technique the conjugate gradient method (CGM) and the fast Fourier transform (FFT) are used to solved for the aperture fields and the induced current densities. The method is efficient because 1) it makes use of the fast Fourier transform, 2) it converts a set of integrodifferential equations into a set of algebraic equations, and 3) it does not require inversion of any matrices. Unlike the moment method, the spectral iteration approach (SIT), and the averaged boundary conditions method, this technique can be used for arbitrary size and spacing between adjacent strips. The equivalent radius principle is used to determine the reflection coefficient from a wire mesh. Results, examples, and comparisons of this method with other methods support the validity of the algorithm.

Operator-Valued Transmission Lines in the Analysis of Two-Dimensional Anomalies Imbedded in a Horizontally Layered Earth Under Transient Polarized Electromagnetic Excitation

A new method for computing the transient electromagnetic behavior of a horizontally layered earth containing a two-dimensional cylindrical anomaly is presented. The vertical extent of the anomaly is assumed to be finite. Above and below the anomaly the electrical parameters are assumed to vary only in the vertical direction, whereas at the levels of the anomaly the electrical parameters are allowed to vary in two dimensions. The electromagnetic excitation is assumed to be polarized in the axial direction. Also, displacement current is not neglected. Two infinite electrical grids model the finite-difference approximations to Maxwell’s equations, one for a transverse-electric polarization and the other for a transverse-magnetic polarization. Unlike virtually all computational methods for geophysical electromagnetic models, the present work does not truncate the earth except at the levels where the anomaly exists. Instead, the theory of operator-valued transmission lines is used to represent the earth above ...

Realizability and synthesis for infinite grids of positive resistances

A semi-infinite resistive grid can be characterized from its boundary conditions by a certain function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(z)</tex> of the complex variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> . We establish herein both pole-zero and pole-residue realizability conditions that insure that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(z)</tex> does represent a resistive grid. We also show how that grid can be synthesized from a realizable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(z)</tex> . These results are established for both two-dimensional and three-dimensional grids.

A classical puzzle: The driving-point resistances of infinite grids

What is the driving-point resistance across the diagonal of a square face — or more generally, between any two nodes — of a uniform infinite square grid of 1-Ω resistances? This puzzle has been cropping up in the electrical engineering literature for many decades and reappeared in two recent issues of IEEE Circuits and Systems Magazine. Described herein is an analysis that completely resolves the matter. In fact, it does so for the general case of an n-dimensional resistive grid whose resistors are all the same, or perhaps vary in one direction only.