Early Verification of the Power Delivery Network in Integrated Circuits

The verification of power grids in modern integrated circuits must start early in the design process when adjustments can be most easily incorporated. We adopt an existing early verification framework. The framework is vectorless, i.e., it does not require input test patterns and does not rely on simulating the power grid subject to these patterns. In this framework, circuit uncertainty is captured via a set of current constraints that capture what may be known or specified from circuit behavior. Grid verification becomes a question of finding the worst-case grid behavior which, in turn, entails the solution of linear programs (LPs) whose size and number is proportional to the size of the grids. The thesis builds on this systematic framework for dealing with circuit uncertainty with the aim of improving efficiency and expanding the capabilities handled within. One contribution introduces an efficient method based on a sparse approximate inverse technique to greatly reduce the size of the required linear programs while ensuring a user-specified over-estimation margin on the exact solution. The application of the method is exhibited under both R and RC grid models. Another contribution first extends grid verification under RC grid models to also check for the worst-case branch currents . This would require as many LPs as there are branches. Then, it shows how to adapt the approximate inverse technique to speed up the branch current verification process. A third contribution proposes a novel approach to reduce the number of LPs in the voltage drop and branch current verification problems. This is achieved by examining dominance relations among node voltage drops and among branch currents. This allows us to replace a group of LPs by one conservative and tight LP. A fourth contribution proposes an efficient verification technique under RLC models. The proposed approach provides tight conservative bounds on the maximum and minimum worst-case voltage drops at every node on the grid.