Transmission line design of clock trees

We investigate appropriate regimes for transmission line propagation of signals on digital integrated circuits. We start from exact solutions to the transmission line equations proposed by Davis and Meindl (IEEE Trans. Electron. Dev., vol. 47, no. 11, pp. 2068-2087, 2000.). We make appropriate modifications due to finite rise time. They affect the delay calculation and hypothesis pertaining to the constancy of the electromagnetic parameters. We study these effects in detail. To find the domain of physical variables where transmission line behavior is feasible, we pose the problem as a nonlinear minimization problem in a space spanned by two continuous variables, with four parameters. From the resulting solutions and employing monotonicity properties of the functional we extract regimes of validity. These regimes of validity happen to be commensurate with what is reachable and doable with current leading technologies. We complete this study with a qualitative analysis of driver insertion in the presence of transmission lines. The resulting configurations are suitable for the development of an improved clock design discipline.