The Application of the Supercondensed Tlm Node to Propagation Problems in Inhomogeneous Anisotropic Materials

A supercondensed TLM node for modelling inhomogeneous anisotropic media on an arbitrarily graded mesh without using stubs is developed. It requires less storage, computationally is more efficient and can operate on a higher time step then existing TLM nodes. Complete derivation, implementation and validation of the new node are presented in the paper. Introduction Transmission-Line Modelling (TLM) method [1] has been applied successfully to many electromagnetic wave propagation problems including complex anisotropic media [2, 3]. Accurate modelling of these problems usually requires finer mesh within the anisotropic substrate and use of cells with arbitrary aspect ratio suitable for modelling particular geometrical features. In a previous publication [4] it was shown that the hybrid symmetrical condensed node (HSCN) offers significant improvements in efficiency and accuracy over conventional stub-loaded symmetrical condensed node (SCN) [5] used in TLM. More recently, we developed a novel node, referred to as the symmetrical super-condensed node (SSCN) [6], capable of modelling inhomogeneous isotropic media on an arbitrarily graded mesh without using stubs. This decreases computer storage requirements by 33% compared to the stub-loaded SCN, whilst increasing flexibility and efficiency. In the present paper, we extend the capabilities of the SSCN to include modelling of anisotropic materials. Complete derivation of the node parameters and implementation in a TLM mesh are presented and validated by numerical examples of modelling microstrip lines on an anisotropic substrate. Theoretical development of anisotropic SSCN The derivation that follows concerns an anisotropic material described by diagonal permittivity and permeability tensors, where "k = "0"rk and k = 0 rk are components in the k direction. Following the definitions used in the derivation of the SSCN for isotropic cases [6], we write the total capacitance and the total inductance in the k-direction as: Cik i+ Cjk j = "k i j k Lij i+ Lji j = k i j k (1) where i; j; k 2 fx; y; zg and i 6= j 6= k. By introducing normalized quantities as Ĉij = Cij j=("j k) and L̂ij = Lij k=( k j), eqns. (1) are rewritten as: