Study on the Stability and Numerical Error of the Four-stages Split-step Fdtd Method including Lumped Inductors

The stability and numerical error of the extended fourstages split-step finite-difference time-domain (SS4-FDTD) method including lumped inductors are systematically studied. In particular, three different formulations for the lumped inductor are analyzed: the explicit, the semi-implicit, and the implicit schemes. Then, the numerical stability of the extended SS4-FDTD method is analyzed by using the von Neumann method, and the results show that the proposed method is unconditionally-stable in the semi-implicit and the implicit schemes, whereas it is conditionally stable in the explicit scheme, which its stability is related to both the mesh size and the values of the element. Moreover, the analysis of the numerical error of the extended SS4-FDTD is studied, which is based on the Norton equivalent circuit. Theoretical results show that: 1) the numerical impedance is a pure imaginary for the explicit scheme; 2) the numerical equivalent circuit of the lumped inductor is an inductor in parallel with a resistor for the semi-implicit and implicit schemes. Finally, a simple microstrip circuit including a lumped inductor is simulated to demonstrate the validity of the theoretical results.