Critical Near-Field Impedance Matrices

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, and two approximations (mid and far) of the corresponding relative impedance and 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. These findings show that the usage of approximate Hertz models is not viable for the analysis of the antenna behavior.