Numerical Stability and Dispersion Analysis of the 2-D FDTD Method Including Lumped Elements

The numerical stability and dispersion analysis of the extended 2-D finite-difference time-domain (2-D-FDTD) method are systematically studied. Particularly, three different passive linear lumped elements including the resistor, inductor, and capacitor are analyzed, respectively. Moreover, three different formulations of the explicit, semi-implicit, and implicit schemes are discussed, respectively. Furthermore, by combining the von Neumann technique and Jury criterion, the numerical stability of the extended 2-D-FDTD method is analyzed, which has not been reported thus far. Theoretical results show that: 1) for the resistor, the stability condition is same as the FDTD method unloaded case; 2) for the inductor, in the explicit and implicit schemes, the stability is connected with the value of the inductance; 3) for the semi-implicit scheme, the stability is independent of the value of the inductance; and 4) for the capacitor, the stability relationship is related to both the mesh size and the value of the capacitance. On the other hand, based on the Norton equivalent circuit, the analysis of the numerical dispersion of the extended 2-D-FDTD is presented; and some interesting theoretical results are deduced. Finally, the microstrip circuits including the three lumped elements are simulated to demonstrate the validity of the theoretical results.