Power Grid Verification in the Presence of Uncertainty

1. Power Grid Verification in the Presence of Uncertainty Imad A. Ferzli University of Toronto ferzli@eecg.utoronto.ca Farid N. Najm University of Toronto f.najm@utoronto.Ca

2. Uncertainty: A Critical Constraint • The crux of the problem: key decisions for the power grid need to be made even before design/layout • How does one design robust grids that meet IR drop budgets? • The backbone: a solid “grid verifier” that incorporates uncertainty in the design…and the process! • Uncertainty, ascertained: • Switching currents: not known pre-placement • Leakage currents: very sensitive to on-chip process variations • Grid parasitics ECE Graduate Symposium

3. Grid Verification Methodology ECE Graduate Symposium

4. Switching Currents • Incremental constraint-based approach (work done with Dionysios Kouroussis) • Preserves vectorless, conservative characteristics of constraint-driven power grid verification • Combines a “divide-and-conquer” approach with “local” grid properties • Is to power grid analysis what static timing is to timing analysis • DC analysis done, transient analysis in progress ECE Graduate Symposium

5. Leakage Currents • Power grid verification formulated as a probability that IR drop on one or several nodes exceeds a certain threshold • Analysis involves statistical and numerical Monte Carlo techniques Nominal supply voltage (Vdd) VIswitching VIleak,nom VIleak,dd VIleak,wd Allowable voltage level (e.g. 90%Vdd) ECE Graduate Symposium

6. Grid Parasitics • Higher chip currents, tighter noise margins make parasitic (R, L, C) variations an important concern • Complicated problem: • Size of the system to be solved • Statistical correlations • No trivial worst case • Framework: • Decorrelation of physical variations • Expansion of grid voltages in terms of orthogonal polynomials (“polynomial chaos”) • Solution of a stochastic system ECE Graduate Symposium