Critical Near-Field Impedance Matrices

TL;DR

This paper analyzes the conditions under which near-field antenna impedance matrices become singular, revealing configurations and approximations where the impedance matrix determinant is zero, impacting the ability to determine antenna behavior.

Contribution

It identifies specific antenna configurations and approximations that lead to singular impedance matrices, highlighting limitations in current impedance modeling methods.

Findings

Zero determinant in far-field approximation for certain configurations.

Singular impedance matrix for a 15-antenna triangular configuration.

Determinant magnitude decreases with increasing grid size at specific spacing.

Abstract

We investigate the theoretical impedance equations for several near-field antenna positions. In the standard model one computes the currents at the antennas for given voltages using the impedance matrix of the antennas, which is only possible if the determinant of the impedance matrix is non-zero. We consider Hertzian group antennas, its relative corresponding impedance and two approximations (mid and far) of it. For the approximations we show that for many situations the determinant is zero. We find three antenna configurations for three antennas, i.e., on a line, on a right triangle, and an isosceles triangle, which result in a zero determinant of the impedance for the far-field approximation. This means that with existing methods, one cannot determine the behavior of this antenna system. For the better mid approximation, we find a configuration of 15 triangular-positioned antennas resulting in a singular impedance matrix. Furthermore, we investigate $n\times n$ grid placed antennas in the more accurate Hertzian impedance model and find that for $d \approx 0.65$ wavelengths of grid distance for n = 2, ..., 8 the absolute value of the determinant of the corresponding impedance matrix decreases by an order of magnitude with each increased grid size.