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.