Study on the stability and numerical dispersion of the FDTD technique including lumped inductors

A study of the stability and numerical dissipation of the finite-difference time-domain technique including passive and active lumped elements has been reported recently by Thiel and Katehi. In particular, three different formulations for lumped inductors were analyzed: the explicit, semi-implicit, and implicit schemes. The implicit scheme was identified to be the same as the formulation previously introduced by Piket-May et al., and its dissipative nature was modeled by a series resistor. In this paper, we perform a numerical dispersion analysis of the above-mentioned schemes, which has not been reported thus far. Other numerical properties of these schemes such as the stability are also discussed. The dispersion analysis allows the numerical impedance of the lumped inductor to be defined or, alternatively, to show that a region of lumped inductors can be interpreted as a dielectric material with a frequency-dependent permittivity. Special attention is given to the implicit scheme. For this scheme, three main conclusions are drawn, i.e., 1) the formulation introduced by Piket-May et al. does not correspond to the implicit scheme, but to the explicit one, 2) the implicit scheme is unconditionally stable for a range of inductance values, and 3) by means of the dispersion analysis, it is shown that the dissipative nature of the implicit scheme is more naturally modeled by a parallel resistor.