FDTD and TLM

Numerical integration methods for the transmission line equations and electromagnetic field problems have developed along two important directions. The first approach was pioneered by Yee [214] in the mid 1960s, and has since blossomed into what is now known as the finite difference time domain method, or FDTD [184]. The idea behind the method is a straightforward application of centered differences to (in Yee's case) the defining equations of electromagnetics, namely Maxwell's equations. (2+1)D simplifications of Maxwell's equations which describe the evolution of transverse electric (TE) and transverse magnetic (TM) fields can also be treated as well, and are, with some trivial modifications, equivalent to the (2+1)D parallel-plate problem. The important advantage of Yee's method is that, due to the structure of the system of equations to be modeled, it is not necessary to calculate all the field components simultaneously--the field components are interleaved both temporally and spatially. We will examine FDTD in the (2+1)D case explicitly in §4.4. The literature on FDTD is quite large; we refer to chapters 2 and 3 of [184] for a succinct technical overview.

The transmission line matrix method, or TLM [4,29,90] appeared a bit later, in the early 1970s [97,100]. It (like the wave digital filtering approach) is a descendant of the ground-breaking work of Kron [109], who developed circuit models of electromagnetic field problems before the widespread availability of electronic computers. TLM is very similar to the DWN, in that it employs a network of discrete transmission lines connected at scattering junctions in order to simulate the behavior of a distributed system. The first formulation, known as the expanded node formulation was derived from a lumped (RLC) model of the (2+1)D transmission line equations [90], and is identical to the type III DWN we will present in §4.3.6. TLM has developed in numerous ways since its inception; the most significant thrust has been towards formulations for which the various field components are not staggered, but computed together at larger nodes. The symmetric condensed node [99] and its numerous offspring, such as the hybrid symmetrical condensed node [159] are the results of this work.

FDTD and TLM have been compared and linked in various ways [38,98], most significantly through the use of field expansions [110], and new variants of FDTD have been developed using TLM as a starting point [27]. We will take a different approach here. Beginning from the observations that have been made regarding the equivalence of certain DWNs to difference methods [67,157,198,200] we will show that Yee's algorithm is equivalent to a family of scattering structures, some of which appear to be quite different from those that have been proposed in the TLM literature. The correspondence holds for media with spatially-varying material parameters; numerical integration of the equations defining such materials has not, as yet, been approached using DWNs. We also note that the TLM community appears to be aware neither of the many valuable numerical properties which scattering-based numerical methods possess [46,165], in particular their behavior in finite arithmetic, nor of other useful signal-processing manipulations (such as power-normalization of wave quantities and dynamic range minimization [167]) which have their roots in electrical network theory.