Electromagnetic Response of a Large Circular Loop Source on a Layered Earth: A New Computation Method

Abstract

Integral expressions of electromagnetic (EM) field components due to a large circular loop source carrying an alternating (ac) current and placed on or above the surface of a layered earth model are transformed to such suitable forms that facilitate numerical computation of field response in quasi-static as well as non-quasi-static regions. The improper integrals occurring in expressions of EM field components are evaluated by converting these integrals into the convergent integrals using the process of subtraction or addition of an integral expression inside the integral sign and subsequently adjusting it or its equivalent analytic expression outside the integral sign. The adjusted integral expressions, in turn, are evaluated using the functional relationships described in this paper. The computation method based on this formulation takes into consideration the effects of both conduction as well as displacement currents, and is well suitable for any position of the source loop either in the air or on the surface of the model, in contrary to the earlier methods which face convergence problem. Moreover, the formulation is equally efficient for computing the EM response at any arbitrary receiver position either inside or outside the source loop. For illustrating the accuracy and applicability of the method and studying the nature of EM response of a loop source over a layered earth model, we have applied it for the computation of amplitude and phase of H z field over the various 2-layer and 3-layer models. Results show their characteristic variations, and depict good resolution for the subsurface layering. The results are in agreement with those of the published results for the quasi-static region, and are new extension of quasi-static variation in the non-quasi-static region. The agreement of computed results with published results demonstrates the accuracy of the method. Moreover, this is the initial presentation of numerical results for an arbitrary in-loop point (other than the center point) inside a large circular loop source.

Exact Electromagnetic Field Excited by a Vertical Magnetic Dipole on the Surface of a Lossy Half-Space

“…The time-harmonic analytical expressions for the non-null EM field components E ϕ , H ρ , and H z generated on the surface of the medium can be obtained from those corresponding to source and field points located respectively at heights h and −z above the medium by setting h = 0 and z = 0. Thus, with the time-harmonic factor e jωt suppressed for better clarity, the EM field components may be expressed as [2,8,[15][16][17] …”

“…The small loop assumption is reasonable, since the diameter of the source is in most cases small if compared to both the source-receiver distance and the free-space wavelength. A number of analytical and semianalytical techniques have been recently developed for evaluating the time-and frequency-domain complete integral expressions for the EM field components produced by a VMD placed above the surface of a material half-space [1][2][3][4][5][6][8][9][10][11][12][13].…”

“…Such solutions constitute the preferred choice when the measurement data to be interpreted are in terms of mutual impedance between source and receiver loops (that is the voltage induced in the receiver loop per unit current flowing in the source loop [1,5,6,8,14]), but unfortunately are valid only in the lowfrequency range.…”

High-Order Electromagnetic Modeling of Shortwave Inductive Diathermy Effects

“…The time-harmonic factor e jωt is suppressed throughout analysis. The exact integral representation for the nonzero electric field component induced in the tissue is given by [14]:…”

“…The latter requirement ensures minimum time consumption when it is required to optimize the coil to produce a desired heating pattern. The proposed model is based on the exact integral representation for the electric field produced by a circular turn placed above a stratified medium [14]. At first, such field integral is cast into a form suitable for the application of the Cauchy's residue theorem.…”