Metrics and bounds for phase delay and signal attenuation in RC(L) clock trees

As IC clock frequencies approach the GHz range, the distribution of the clock signals becomes more critical in terms of controlling both skew and signal attenuation. Moreover, inductance effects are evident since RC transmission lines will overly attenuate these high-frequency clock signals. To facilitate accurate optimization of clock tree performance and skew requires simple metrics which capture these high-frequency effects. In this paper, we derive simple metrics and bounds for the phase delay and the attenuation of a periodic [RC(L)] tree response as a function of the fundamental frequency of the clock signal. These metrics are based on the first two moments of the impulse response, and are shown to further provide a mechanism for control of underdamped responses (reflections). An important result of this work is the clear demonstration that once the attenuation of the clock signal is controlled, the phase delay can be accurately captured in terms of the first-moment. Furthermore, the form of these metrics and their relationship to one another provides an excellent foundation for various forms of clock tree optimization.