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

2. Ferzli/NajmECE Graduate Symposium2 Uncertainty: A Critical Constraint n The crux of the problem: key decisions for the power grid need to be made even before design/layout l How does one design robust grids that meet IR drop budgets? n The backbone: a solid “grid verifier” that incorporates uncertainty in the design…and the process! n Uncertainty, ascertained: l Switching currents: not known pre-placement l Leakage currents: very sensitive to on-chip process variations l Grid parasitics

3. Ferzli/NajmECE Graduate Symposium3 Grid Verification Methodology

4. Ferzli/NajmECE Graduate Symposium4 Switching Currents n Incremental constraint-based approach (work done with Dionysios Kouroussis) l Preserves vectorless, conservative characteristics of constraint- driven power grid verification l Combines a “divide-and-conquer” approach with “local” grid properties l Is to power grid analysis what static timing is to timing analysis n DC analysis done, transient analysis in progress

5. Ferzli/NajmECE Graduate Symposium5 Leakage Currents n Power grid verification formulated as a probability that IR drop on one or several nodes exceeds a certain threshold l Analysis involves statistical and numerical Monte Carlo techniques V Iswitching V Ileak,nom V Ileak,dd V Ileak,wd Nominal supply voltage (Vdd) Allowable voltage level (e.g. 90%Vdd)

6. Ferzli/NajmECE Graduate Symposium6 Grid Parasitics n Higher chip currents, tighter noise margins make parasitic (R, L, C) variations an important concern n Complicated problem: l Size of the system to be solved l Statistical correlations l No trivial worst case n Framework: l Decorrelation of physical variations l Expansion of grid voltages in terms of orthogonal polynomials (“polynomial chaos”) l Solution of a stochastic system