Inductance Effects in RLC Trees

A closed form solution for characterizing voltage-based signals in an RLC tree is presented. This closed form solution is used to derive figures of merit to characterize the effects of inductance at a specific node in an RLC tree. The effective damping factor of the signal at a specific node in an RLC tree is shown to be a useful figure of merit. As the effective damping factor of a signal increases, an RC model is sufficiently accurate to characterize that waveform. The rise time of the input signal driving an RLC tree is another factor characterizing the importance of inductance. As the rise time of the input signal becomes much larger than the effective LC time constant at a specific node within an RLC tree, the signal at this node does not exhibit the effects of inductance. Evidence is provided showing that using a single line analysis to determine the importance of including inductance to characterize a tree structured interconnect line is invalid in many cases and can lead to erroneous conclusions. I. Introduction Inductance is becoming increasingly important with faster on-chip rise times and longer wire lengths. Wide wires are frequently encountered in clock distribution networks and in upper metal layers. These wires are low resistive lines that can exhibit significant inductive effects. Furthermore, performance requirements are pushing the introduction of new materials for low resistance interconnect (1) and new dielectrics to reduce interconnect capacitance. These technological advances significantly reduce the RC time constants of the interconnect which, as will be shown, increases the importance of on-chip inductance. The importance of on-chip inductance for single lines has been characterized in (2)-(5). However, the nets in a VLSI circuit are often structured as trees rather than as single lines. It is shown here that the branches of a tree can not be treated as single lines for the purpose of evaluating the importance of inductance. Rather, the entire tree should be examined for inductance effects as a single entity since there is a large interaction among the different branches.