The simple derivation for the effective permittivities formulae in the finite difference grids

In recently years, the finite difference time domain method (FDTD) has been the most popular numerical method in electromagnetic fields, such as for microwave circuit, antennas, etc. In order to deal with inhomogeneous dielectric in the interface, the usual method is to plot out the mesh along the interface in two kinds of dielectric (J.G. Maloney, and G.S. Smith, IEEE Microwave Guided Wave Lett., vol. 10, pp. 359-361, 2000), but this limits the size of the unit grid which is much smaller, and increases computation time. In this paper, by using analogy with the circuit concept of series-parallel connection of capacitance, the approximate formulae of effective permittivities for various inhomogeneous unit grids are deduced. Under the condition of numerical stability, these formulae are combined with the FDTD method of arbitrary spatial mesh to calculate frequency of the resonant cavity. This proposed method can save computing time and memory than the conventional FDTD method. These effective permittivities formulae will be applied in other difference methods for electromagnetic problem.