Analysis of finite-differencing errors to determine cell size when modeling ferrites and other lossy electric and magnetic materials using FDTD

The focus of this paper is the modeling of materials that have both significant electric and magnetic losses, such as ferrites, using finite-difference time domain (FDTD). The primary contribution is identifying appropriate cell sizes when modeling these types of materials. It is shown that finite-differencing errors increase in lossy media compared to lossless media when sampling at the same number of cells per wavelength. Losses in a medium are defined by the ratio of the attenuation constant, α, to the phase constant, β, since that ratio accounts for all losses, whether they be electric or magnetic. In addition to a detailed finite-differencing error analysis, a simple approximation is given for selecting a cell size in a lossy material that will give the same finite-differencing error as ten cells per wavelength in a lossless material. This paper also presents a means for deriving pure real constitutive parameters from complex constitutive parameters. Being able to make such calculations is useful in cases where complex constitutive parameters are given for a material, and the FDTD model being used only accepts pure real constitutive parameters, as is the case for several contemporary models. Comparisons of theoretical and FDTD-modeled reflection and transmission show that the derived, real constitutive parameters are valid.