HIGH‐FREQUENCY ELECTROMAGNETIC COUPLING BETWEEN SMALL COPLANAR LOOPS OVER AN INHOMOGENEOUS GROUND

Geophysics ◽  
1972 ◽  
Vol 37 (6) ◽  
pp. 997-1004 ◽  
Author(s):  
James A. Fuller ◽  
James R. Wait
Keyword(s):  
Half Space ◽  
High Frequency ◽  
Electromagnetic Coupling ◽  
Current Source ◽  
Vertical Axis ◽  
Loop Current ◽  
Mutual Impedance ◽  
Multiplicative Factor ◽  
The Earth ◽  
Horizontal Electric Dipole

An integral formulation is given for the fields of a loop current source which is located over a horizontally stratified half‐space and has a vertical axis. The electrical properties of the half‐space vary exponentially with the depth into the earth. An asymptotic solution is developed for the case of source and observer on the interface but separated by a large numerical distance. The approximate solution is then used to determine the mutual impedance between two small loops and between the loop and a horizontal electric dipole, when the antennas are on the interface. It is found that the effect of stratification on the mutual impedance is represented approximately by a single multiplicative factor.

On modeling a well casing for resistivity and induced polarization

Geophysics ◽  
1984 ◽  
Vol 49 (11) ◽  
pp. 2061-2063 ◽  
Author(s):  
James R. Wait
Keyword(s):  
Electrical Properties ◽  
Electric Dipole ◽  
Free Space ◽  
Eddy Currents ◽  
Interfacial Polarization ◽  
Induced Polarization ◽  
Mutual Impedance ◽  
The Earth ◽  
Horizontal Electric Dipole ◽  
Displacement Currents

In a previous communication I proposed an analytical model to simulate the electromagnetic (EM) and induced polarization (IP) response of a metal well casing (Wait, 1983). To facilitate the analysis, the earth was idealized as a homogeneous conducting half‐space of electrical properties (σ, ε, μ). The well casing was represented as a filamental vertical conductor of semiinfinite length that was characterized by a series axial impedance to account for eddy currents and interfacial polarization. A further basic simplification was to neglect displacement currents in the air; this was justified when all significant distances were small compared with the free‐space wavelength. Initially, the source was taken to be a horizontal electric dipole or current element I ds on the air‐earth interface. By integration of the results, the mutual impedance between two grounded circuits could be ascertained. In the absence of the vertical conductor (i.e., the well casing) the results reduced to those given by Sunde (1968) and Ward (1967).

ELECTROMAGNETIC FIELDS OF A VERTICAL MAGNETIC DIPOLE PLACED ABOVE THE EARTH’S SURFACE

Geophysics ◽  
1963 ◽  
Vol 28 (3) ◽  
pp. 408-425 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Electromagnetic Fields ◽  
Current Source ◽  
Step Function ◽  
Impedance Function ◽  
Low Frequencies ◽  
Primary Loop ◽  
Mutual Impedance ◽  
The Earth ◽  
Receiving Antenna ◽  
Rate Of Decay

Electromagnetic fields due to a small loop antenna placed above the surface of a homogeneous and isotropic earth have been calculated. The effect of both the conduction and displacement currents are taken into account. Because of the complexity of the functions defining the fields, expressions valid separately for high and low frequencies are developed for the electric and magnetic field components. These expressions are then utilized to determine, for a step‐function current source, (a) the mutual impedance function [Formula: see text] between the primary loop and a small length of wire and (b) the voltage v(t) induced in a secondary loop. Two parameters are used to fix the locations of the primary loop and the receiving antenna with respect to the earth. A number of curves are plotted showing the mutual impedance function and the voltage function against time for different values of the parameters and the conductivity and the permittivity of the earth. With increase in either the conductivity or the permittivity, the amplitude and the rate of decay of the two functions decrease appreciably. However, the amplitudes of both [Formula: see text] and v(t) become smaller and the rate of decay higher as the receiving antenna is gradually lifted vertically from the ground. For all values of permittivity, the amplitude of the mutual impedance rises to a maximum with the horizontal separation between the two antennas before beginning to decrease, but at the same time the rate of decay of the transient becomes faster. With increase in the horizontal separation, the amplitude of the voltage function decreases inversely as the fifth power of the distance between the image of the transmitting dipole and the receiving antenna, but the rate of decay increases markedly.

Corrections to Millett’s table of electromagnetic coupling phase angles

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1554-1555 ◽  
Author(s):  
R. J. Brown
Keyword(s):  
Half Space ◽  
Random Error ◽  
Electromagnetic Coupling ◽  
Inductive Coupling ◽  
Coupling Phase ◽  
Mutual Impedance ◽  
Phase Angles

Millett (1967) published tables of values of the mutual impedance due to inductive coupling between two collinear dipoles on a uniform, nonpolarizable half‐space. In the course of a recent study (Brown, 1984) I have noticed significant errors, of two different kinds, in the phase angles (ϕ) given by Millett (1967). One kind of error is evidently typographical in nature and occurs only twice, in the M = 3 table, for θ = .01 and .02. The tabled values apparently had their decimal points shifted one place. The second and more serious kind of error is an apparently random error within the range ±0.003 degrees. This is not significant for larger |ϕ|, say |ϕ| > 1 degree, but the values of |ϕ| in Millett’s tables go down to 0.006 degrees (down to 0.0045 degrees after correction) where such errors are clearly significant, particlarly if one is working with logarithmic quantities as is common.

Efficient evaluation of the earth return mutual impedance of overhead conductors over a horizontally multilayered soil

2014 ◽  
Vol 33 (4) ◽  
pp. 1379-1395 ◽  
Author(s):  
Jae-bok Lee ◽  
Jun Zou ◽  
Benliang Li ◽  
Munno Ju
Keyword(s):  
Unit Length ◽  
Oscillatory Integral ◽  
Electromagnetic Transients ◽  
Closed Form Expression ◽  
Computational Time ◽  
Horizontal Distance ◽  
Content Type ◽  
Mutual Impedance ◽  
The Earth ◽  
Highly Oscillatory

Purpose – The per-unit-length earth return mutual impedance of the overhead conductors plays an important role for analyzing electromagnetic transients or couplings of multi-conductor systems. It is impossible to have a closed-form expression to evaluate this kind of impedance. The purpose of this paper is to propose an efficient numerical approach to evaluate the earth return mutual impedance of the overhead conductors above horizontally multi-layered soils. Design/methodology/approach – The expression of the earth return mutual impedance, which contains a complex highly oscillatory semi-infinite integral, is divided into two parts intentionally, i.e. the definite and the tail integral, respectively. The definite integral is calculated using the proposed moment functions after fitting the integrand into the piecewise cubic spline functions, and the tail integral is replaced by exponential integrals with newly developed asymptotic integrands. Findings – The numerical examples show the proposed approach has a satisfactory accuracy for different parameter combinations. Compared to the direct quadrature approach, the computational time of the proposed approach is very competitive, especially, for the large horizontal distance and the low height of the conductors. Originality/value – The advantage of the proposed approach is that the calculation of the highly oscillatory integral is completely avoided due to the fact that the moment function can be evaluated analytically. The contribution of the tail integral is well included by means of the exponential integral, though in an asymptotic way. The proposed approach is completely general, and can be applied to calculate the earth return mutual impedance of overhead conductors above a soil structure with an arbitrary number of horizontal layers.

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