The Trapezoidal Recursive Convolution (TRC) scheme was previously used to model Nth order Lorentz type dispersive media. In this paper, the full derivation of this quasi-trapezoidal-based algorithm is presented and the derivation is expanded to include the Nth order Debye type dispersion as well as Sellmeyer's dispersion equation. In addition, the case of general convolution integrals is considered where any arbitrary integrand or the integral itself is represented as a sum of exponential functions, i.e. Prony's method. The technique is compared to several previously published schemes and it is shown that its performance equals or exceeds various other methods in terms of accuracy, robustness, and computational efficiency. A comparison to the exact application of trapezoidal numerical integration is made and it is shown that, for time increments encountered in typical FDTD analyses, the truncation error due to applying the quasi-trapezoidal approximation is negligible. Finally, it is shown how the skin effect phenomenon, as it applies to multiconductor transmission lines, can be modeled using a rational function approximation to the frequency dependency of the line resistance. This model is obtained by using Levy's method to curve fit the line resistance directly in the frequency domain and then the convolution integral is formulated in a form amenable to the TRC algorithm.
We compare Yee's finite-difference time-domain (FDTD) and symmetric condensed-node transmission-line matrix (SCN-TLM) solutions for a cavity containing a metallic fin. Differential equationbased numerical methods are known to produce inaccurate results for this type of problem due to the rapid spatial variation of the field distribution in the vicinity of the singularity at the edge of the metal fin. This problem is relevant to the analysis of structures of practical interest such as microstrip and coplanar waveguides. Based on simulations, it is determined that for identical discretizations, SCN-TLM is more accurate than FDTD for this problem. We interpret this result as an indication that the symmetric condensed representation of fields (used within the SCN-TLM) lends itself to a more accurate algorithm than the distributed representation used by Yee. We estimate that the FDTD method requires 3.33 times more cells for a given three-dimensional problem than the transmission-line matrix (TLM) method (1.49 times more cells per linear dimension of the problem) in order to achieve the same accuracy. If we consider the requirements to update and store a single TLM or FDTD cell, we find the SCN-TLM algorithm is more efficient than the Yee FDTD algorithm in terms of both computational effort and memory requirements. Our conclusions regarding computational effort and memory requirements are limited to problems with homogeneous material properties.
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