For analysing the 3D periodic structures at the oblique incident, a weakly conditionally stable finite-difference time-domain method is proposed by applying the Crank-Nicolson (CN) scheme to the field transformation technique. By splitting the space operator matrix in a special way, the transformed Maxwell's equations can be subdivided into two sub-steps for an efficient implementation. The proposed method does not need to introduce additional field components to handle the extra time terms and the time step size is only determined by one space discretisation.

In this work, a new lumped-network hybrid implicit-explicit finite-difference time-domain (LN-HIE-FDTD) method has been proposed. This method is derived from combining the 3D HIE-FDTD method with the Z-transform technique for the hybrid system including linear lumped networks. It possesses the advantage of deriving the updating equations immediately for complex circuits. The accuracy of the proposed method is verified from the comparison of calculated results given by the proposed method and the conventional lumped-network FDTD (LN-FDTD) method. The efficiency of the proposed method is verified from the comparison of the CPU time by the two methods. It is demonstrated that the proposed method is numerically efficient while time-saving. IntroductionThe finite-difference time-domain (FDTD) method has been widely applied in solving problems related to electromagnetism [1][2][3][4][5][6][7][8][9]. With the miniaturization of the hybrid microwave circuits, especially for thin-film structures [10,11], the thickness of the structures decreases to the micron level. It would be difficult for the conventional FDTD to analyze these problems for the time-step size is strictly limited by the stability condition. The small time-step size will significantly call for more calculating time and computational resources when the analysis object has fine scale dimensions compared with wavelength.To cope with this problem, the hybrid implicit-explicit FDTD algorithm (HIE-FDTD) has been proposed [12,13]. In the HIE-FDTD method, the maximum time-step size of the HIE-FDTD method is not strictly limited by the minimum space size. Thus, the stability condition is weaker than that of the conventional FDTD.Recently, the HIE-FDTD method has been used to calculate the hybrid system with lumped elements (LE-HIE-FDTD) and conductive media [14,15]. However, the LE-HIE-FDTD method is not good at calculating networks consisting of the arbitrary connection of several lumped elements [16]. In [14], a large number of schemes are needed to simulate the lumped network with multiple lumped elements. The Z-transform technique has been introduced to deal with this problem in the conventional FDTD method. In [17,18], the bilinear Z-transform is used to derive the transform, but its computational cost is higher because of the increased complexity. In [19], the direct Z-transform is used to deal with lumped loads,

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A new thin-slot formalism is presented for the finite-difference time-domain (FDTD) analysis of the coupling of narrow apertures having depth. To derive the formalism, the equivalence principle is used to divide the slot coupling problem into three decoupled parts. The field distribution near the aperture is gained from the equivalent magnetic current together with a constant field distribution assumption. By incorporating the field distribution into the Ampere's law and Faraday's Law, the updated equations are derived to modify the electromagnetic field components near the apertures. Two numerical examples are adopted to verify the validity of the proposed formalism, and both the high-resolution standard FDTD results and FEKO (FEldberechnung Bei Körpern mit beliebiger Oberfläche) result are presented as references. The proposed formalism is verified to be a stable and accurate model for thin-slot coupling.

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