Electromagnetic response of a discretely grounded circuit—An integral equation solution

Abstract

We derive an integral equation to describe the electromagnetic response of a discretely grounded circuit. This investigation is relevant to the study of man‐made structures such as metallic fences, grounded powerlines, and pipelines, all of which may fall into the class of discretely grounded conductors. The solution developed here is an extension to existing circuit theory and takes into account the self and mutual interaction of the circuit elements. It is possible to ignore these interactions at low frequencies where the grounding impedances dominate the effective impedance of the circuit. However, at frequencies where the electromagnetic skin depth is comparable to the length between adjacent grounding points, the effective impedance of the circuit is proportional to frequency, and the inductance of the circuit dominates its electromagnetic response. Within the quasi‐static limit (i.e., where displacement currents can be neglected) electromagnetic excitation by either horizontal electric or vertical magnetic dipoles produces a constant primary electric field at high frequencies (far‐field). Thus, the electric current in the discretely grounded circuit will always be inversely proportional to frequency for these types of sources. Horizontal magnetic dipole or vertical electric dipole sources generate primary electric fields that are proportional to the inverse square root of frequency in the high frequency limit of the quasi‐static domain, and thus the current in a circuit excited by such sources will decrease as the inverse of square root of frequency. The integral equation solution derived here can be used to investigate the influence from cultural conductors on actual electromagnetic surveys and also provides further insights into the current channeling response of surficial conductors.