Antenna Modelling Using the Dgf-fdtd Method

A new discrete Green’s function formulation of the finite difference time domain (DGF-FDTD) method has been developed which expresses the field response as a convolution of the current sources and the impulse response of the FDTD equation system. DGF-FDTD avoids the need for computation of free space nodes and absorbing boundary conditions. The method has been demonstrated by its application to a simple Yagi-Uda antenna where the efficiency over the standard FDTD method is clearly demonstrated. INTRODUCTION The Finite Difference Time Domain (FDTD) method [1] is a general method to solve Maxwell’s partial differential equations numerically in the time domain. It has been used extensively to model all kinds of electromagnetic problems, such as radiation, scattering, and circuit problems. However, due to the limitation of computer resources, it is difficult to apply the FDTD method in the modelling of electrically large objects. At the same time, the implementation of the FDTD method requires absorbing boundary conditions to terminate the spatial grid, which adds an additional burden on computational resources. Moreover, the recursive nature of the classic Yee algorithm for the FDTD method implies that all the cells contained in a given volume must be computed. Therefore, when applied to model scattering problems, the computation of free space nodes between scatterers is required even though this is actually unnecessary. As a result, the FDTD method is characteristically time consuming and memory demanding. To overcome these limitations, the discrete Green’s function formulation of the FDTD method (DGF-FDTD) has been developed. METHOD The DGF-FDTD method is based on the idea that the Yee algorithm may not be the only possible formulation of the FDTD method. Classic theory of discrete systems shows that any linear and invariant system, which the FDTD equations are, can be completely determined by the impulse response of the system. With the impulse response, which is referred to as the discrete Green’s function of the system, the response of any excitation can be represented as a convolution of the input signal with the discrete Green’s function [2]. This is the discrete version of the Green’s function technique extensively used in electromagnetics. Due to its inherently discrete property in both the spatial and temporal domains, this Greens Function method is well suited to be combined with the traditional FDTD method since this does not introduce any problems with interfacing between the two techniques. The FDTD equations can be treated as a linear and invariant discrete system whose ‘inputs’ are the electric and magnetic current sources J and M respectively and the ‘outputs’ are the electric and magnetic fields E and H. The impulse response of the FDTD equations can be determined by using Kronecker delta impulse functions as the sources. This gives a set of 2 by 2 matrices [ ] n n ej em ijk ijk imp n n imp hj hj ijk ijk G G E j H m G G             = ⋅                         ! " ! " (1) that represent the impulse response electric and magnetic fields. According to discrete system theory, an arbitrary current source can be expressed as a convolution sum of Kronecker delta functions (2). n n n n ijk i i j j k k i j k n i j k n n n n ijk i i j j k k i j k n i j k J J M M δ δ ′ ′ − ′ ′ ′ ′ ′ ′ − − − ′ ′ ′ ′ ′ ′ − ′ ′ ′ ′ ′ ′ − − − ′ ′ ′ ′ = ⋅