Extended (FD)²TD formulations for lumped elements

Currently extended finite-difference time domain (FDTD) formulations require the permittivity and conductivity to be independent of frequency. However, for many real materials of interest, these parameters vary significantly with frequency. The extended frequency-dependent finite-difference time domain (FD)²TD method makes it possible to analyze hybrid integrated circuits with lumped element within dispersive medium by combining lumped parameters (R, C, or L) and frequency-dependent dielectric substrate in the algorithm. This method allows the full-wave analysis of virtually hybrid integrated circuits and gets parameters with a wide frequency range with one simulation.