Analysis of Stability and Numerical Dispersion of 3-D HIE-FDTD Method Including Lumped Elements

The analyses of the numerical stability and dispersion of the 3-D hybrid implicit–explicit finite-difference time domain (3-D HIE-FDTD), including the lumped elements, are systematically researched. Specifically, three lumped elements of the resistor, inductor, and capacitor are discussed, and the explicit, semi-implicit, and implicit schemes are discussed. First, according to the techniques of von Neumann and Jury criterion, the numerical stability of the extended 3-D HIE-FDTD method is studied. Research results demonstrate that: 1) as to the lumped resistor and inductor under the semi-implicit and implicit schemes, and capacitor under the implicit scheme, the numerical stability condition of the extended 3-D HIE-FDTD method is consistent with the 3-D HIE-FDTD method unloaded lumped elements and 2) for the lumped resistor and inductor under the explicit scheme, the numerical stability condition becomes more stringent than that of the 3-D HIE-FDTD method unloaded case, which is dependent on the value of the resistance or inductance. Second, based on the Norton equivalent circuit, the numerical dispersion characteristics of the extended 3-D HIE-FDTD method are analyzed, and the numerical impedance for the lumped elements under different difference schemes is deduced. Finally, to testify the validity of the theoretical results, the numerical results of microstrip circuits are provided.