Optimal Planning for Mesh-Based Power Distribution

1. Optimal Planning for Mesh-Based Power Distribution H. Chen, C.-K. Cheng, A. B. Kahng, Makoto Mori * and Q. Wang UCSD CSE Department * Fujitsu Limited Work partially supported by Cadence Design Systems, Inc., the California MICRO program, the MARCO Gigascale Silicon Research Center, NSFMIP-9987678 and the Semiconductor Research Corporation.

2. Motivation (I) • Voltage drop in the power distribution is critical to chip performance and reliability • Power distribution network in early design stages • nominal wiring pitch and width for each layer need to be locked in • location and logic content of the blocks are unknown • impossible to obtain the pattern of current drawn by sinks • transient analysis is essentially difficult • design decisions are mostly based on DC analysis of uniform mesh structures, with current drains modeled using simple area-based calculations

3. Motivation (II) • Current method in practice • explore different combinations of wire pitch and width for different layers • select the best combination based on circuit simulations • problem: computationally infeasible to explore all possible configurations; the result is hence a sub-optimal solution • What we need: a new approach to optimize topology for a hierarchical, uniform power distribution

4. Our Work • Study the worst-case static IR-drop on hierarchical, uniform power meshes using both analytical and empirical methods • Propose a novel and efficient method for optimizing worst-case IR-drop on two-level uniform power distribution meshes • Usage of our results planning of hierarchical power meshes in early design stages

5. Outline • Problem Formulation • IR-Drop on Single-Level Power Mesh • IR-Drop on Two-Level Power Mesh • Optimal Planning of Two-Level Power Mesh • IR-Drop on Three-Level Power Mesh • Conclusion and On-Going Work

6. Problem Statement • Given fixed wire pitch and width for the bottom-level mesh • Find the optimal wire pitch and width for each mesh except the bottom-level mesh • Objectives • for a given total routing area, the power mesh achieves the minimum worst-case IR-drop • for a given worst-case IR-drop requirement, the power mesh meets the requirement with minimum total routing area

7. Model of Power Network • Hierarchy of metal layers • uniform and parallel metal wires at each layer • adjacent metal layers connected at the crossing points • Via resistance: ignored • much smaller than resistance of mesh segments • C4 power pads evenly distributed on the top layer • Uniform current sinks on the crossing points of the bottom layer • before the accurate floorplan, the exact current drain at different locations is unknown

8. Representative Area • Area surrounded by adjacent power pads • Power mesh • # power pads in state-of-art designs: larger than 100 • infinite resistive grid • constructed by replicating the representative area • Worst-case IR-drop appears near the center of the representative area

9. Outline • Problem Formulation • IR-Drop on Single-Level Power Mesh • a closed-form approximation for the worst-case IR-drop on a single-level power mesh • IR-Drop on Two-Level Power Mesh • Optimal Planning of Two-Level Power Mesh • IR-Drop on Three-Level Power Mesh • Conclusion and On-Going Work

10. IR-Drop in Single-Level Power Mesh • IR-drop on a hierarchical power mesh depends largely on the top-level mesh • We analyze worst-case IR-drop on a single-level power mesh • power pads • supply constant current to the mesh • regarded as current sources • ground: at infinity • our method: analyze voltage drops caused by current sources and current sinks separately

11. IR-Drop by Current Sources • Analysis • IR-drop caused by a single current source • an approximated close-form formula [Atkinson et al. 1999] • integrate IR-drop for all current sources • Result: worst-case IR-drop when only current sources are considered • N: # stripes in the representative area • R: edge resistance • I: total current drain in the representative area • C = -0.1324

12. IR-Drop by Current Sinks • Analysis • uniform resistive lattice: a discrete approximation to a continuous resistive medium • potential increases with D2 where D = distance from the center, if • a continuous resistive medium • evenly distributed current sinks • impose a form proportional to D2 • Result: worst-case IR-drop when only current sinks are considered

13. Verification of IR-Drop Formula (I) • Worst-case IR-drop • HSpice simulations • fixedtotal current drain I • fixededge resistance R • #stripes between power pads: N= 4 to 12

14. Verification of IR-Drop Formula (II) Simulation results for worst-case IR-drop on single-level power meshes, compared to estimated values Accuracy within 1% when N > 4

15. Outline • Problem Formulation • IR-Drop on Single-Level Power Mesh • IR-Drop on Two-Level Power Mesh • an accurate empirical expression for the worst-case IR-drop on a two-level power mesh • Optimal Planning of Two-Level Power Mesh • IR-Drop on Three-Level Power Mesh • Conclusion and On-Going Work

16. IR-Drop in Two-Level Power Mesh • Model: two uniform infinite resistive lattices • top-level mesh • connected to power pads • wider metal lines • coarser grid • bottom-level mesh • connected to devices • thinner metal lines • finer grid • Analysis method: consider IR-drop on two meshes separately

17. IR-Drop in the Coarser Mesh • Assumption: currents flow along an equivalent single-level coarse mesh • most current flows along the coarser mesh • IR-drop in the coarser mesh: • N1: # stripes of the coarser mesh in the representative area • Re : equivalent edge resistance • I: total current drain in the representative area • c : a constant

18. Verification • HSpice simulations of two-level power meshes • fixed total current drain I • fixed Re • fixed routing resource of two meshes • bottom-level mesh is 10 times finer than the top-level one • # stripes of the coarser mesh N1 = 3 ~ 10 V ~ ln(N1): nice linearity

19. Equivalent Edge Resistance • Re : slope of the line V ~ ln(N1) • HSpice simulations of two-level power meshes • fixed total current drain I • # stripes of the coarser mesh N1 = 19 • bottom-level mesh: 10 times finer than the top-level one • routing resource of the finer mesh = 1  fixed edge resistance of the finer mesh R • different total routing resource r  different Re • Empirically, Re  R / r

20. IR-Drop in the Finer Mesh (I) • Assumption: finer mesh within each cell formed by the coarser mesh has equal voltage on the cell boundary • coarser mesh: much smaller edge resistance • HSpice simulations of finer mesh • equal voltage on the boundary • fixed edge resistance of the finer mesh R • fixed current drain of each device i • # stripes within each cell: M = 2 ~ 22

21. IR-Drop in the Finer Mesh (II) Vfine ~ M2: nice linearity

22. IR-Drop Formula (I) • IR-drop • C1(r), C2(r) are functions of r • HSpice simulations of two-level meshes • fixed total current drain I • bottom-level mesh: 10 times finer than the top-level one • routing resource of the finer mesh = 1 •  fixed edge resistance of the finer mesh R • fixed total routing resource r = 16 • # stripes of the coarser mesh N1 = 1 ~ 9 • C1, C2 obtained by simulation results for N1 = 7 and 9

23. IR-Drop Formula (II) Simulation results for worst-case IR-drop on two-level power meshes with fixed total routing area, compared to estimated values Accuracy within 1% when N > 4

24. Outline • Problem Formulation • IR-Drop on Single-Level Power Mesh • IR-Drop on Two-Level Power Mesh • Optimal Planning of Two-Level Power Mesh • a new approach to optimize the topology of two-level power mesh • IR-Drop on Three-Level Power Mesh • Conclusion and On-Going Work

25. Optimizing Topology with a Given Total Routing Area • Problem Statement • given fixed total routing area r • find optimal # stripes in the coarser mesh N1 • objective = min worst-case IR-drop • Optimization Method • based on the IR-drop formula • E.g., when r = 16, N1* = 3.9

26. Optimizing Topology with a Given Worst-Case IR-Drop Requirement • Problem Statement • given worst-case IR-drop requirement • find optimal # stripes in the coarser mesh N1 • objective = min total routing area r • Optimization Method • for each value of r • simulate two-level power meshes for a few values of N1 • calculate the values of C1(r), C2(r) • compute the optimal worst-case IR-drop V*(r) • find minimum total routing area r with V*(r) meets given requirement

27. Example • Requirement: worst-case IR-drop < 30mV • Compute optimal IR-drop V*(r) for each value of r • Optimal r :between 12 and 13 Optimal N1 :3 or 4

28. Outline • Problem Formulation • IR-Drop on Single-Level Power Mesh • IR-Drop on Two-Level Power Mesh • Optimal Planning of Two-Level Power Mesh • IR-Drop on Three-Level Power Mesh • a third, middle-level mesh helps to reduce IR-drop by only a relatively small extent (about 5%, according to our experiments) • Conclusion and On-Going Work

29. Optimal Resource Distribution • Problem • given topology of three-level mesh # stripes of three grids • given total routing area • find optimal resource distribution • Method • a simplified power network wire sizing technique Sequential LP method [Tan et al. DAC99] • for a given width assignment, find the voltage at each node by solving a set of linear equations • fix the node voltages and find the optimal width assignment to maximize current drain at the center • repeat this process iteratively until the solution converges

30. IR-Drop in Three-Level Power Mesh • Analysis method • fix # stripes in the top- and bottom-level meshes • explore different # stripes for the middle-level mesh • find optimal resource allocation and IR-drop • Top, middle and bottom meshes • # stripes: N1 ,N2 and 120 • wiring resource: r1 , r2 and 1 (1 + r1 + r2 = 10) • Middle-level mesh reduces IR-drop to a relatively small extent (about 5%)

31. Conclusions • Obtained accurate expression for worst-case IR-drop in two-level uniform meshes • Proposed a new method of optimizing topology of two-level uniform power mesh • used to decide nominal wire width and pitch for power networks in early design stages • Ongoing work: • optimization of non-uniform power meshes • interactions with layout or detailed current analysis

32. Thank You !