Efficient Design and Analysis of Robust Power Distribution Meshes Puneet Gupta Blaze DFM Inc. Andrew B. Kahng ECE & CSE UCSD Presented by Swamy Muddu

Motivation Current density and hence IR drop is increasing with every technology generation IR drop Static IR drop: resistive, driven by peak currents Transients: E.g., IR drop analysis and optimization is very slow Optimization is usually formulated as Non-Linear Program Analysis too slow to be used during Place & Route Typical global power distribution Horizonatal/Vertical stripes Power meshes Peripheral i/o or flipchip This work: Static IR drop, peripheral i/o

The 1D Case Consider a power stripe with single power source at one end. Order power tap-points 1..N starting from the one closest to the power source IR drop increases away from the power source Wire segments between current tap-points closer to Vdd carry higher current  A tapered power stripe is more bang for buck Closed form solution for optimal sizing can be calculated using Lagrangian multipliers. Assume area of a wire segment is proportional to its conductance V dd Power tap points r 45 i 23

The 1D Case contd.. Solution to the minimum area IR drop (MAIC) constrained stripe sizing: (r=resistance, i=current) Solution to the minimum IR drop area constrained (MIAC) problem is given by: (G=total conductance constraint)

IR Drop in Power Grids IR Drop increases going in from the power ring “Radial” nature of IR drop variation in a power grid  notice the similarity to the 1D case Equipotential contours take the form of “rings” : diamond shaped at the center of layout and square shaped at the periphery (shape of power ring) Bull’s Eye Equipotential ring Outer Power Ring An example IR drop map for a uniform 25 X 25 power mesh with uniform current requirements

Analysis of the 2D Case Assume square equipotential rings Divide power mesh into radial and tangential segments nXn mesh  rings Intuition: Current flows only in the radial segments

Radial vs. Tangential Segments Variation of peak IR drop with varying widths of radial and tangential segments is shown Radial segments impact IR drop much more than tangential segments  close to zero current flow in the tangential segments Assume radial current flow Peak IR Drop vs. conductance Of wire segments in a 25x25 power grid

Power Grid Optimization Three phase optimization of power grids: Radial Sizing. Size radial segments assuming uniform distribution of currents around rings. Tangential Sizing. Account for nonuniformity of current distribution. Divide rings into sectors and redistribute metal among sectors. Circumference Correction. Relax square ring assumption. Account for diamond to square ring shape transition All phases are based on closed form sizing expressions

Radial Sizing Assume no current flow along the equipotential ring  tangential segments sized to minimum width MAIC Sizing Solution i(p,p+1): current from ring p to p+1 MIAC Sizing Solution G R : total allocated radial conductance All radial segments originating from an equipotential ring are sized equally

Tangential Sizing Redistribute metal between the radial segments along the equipotential ring to account for non-uniform current distribution Divide power grid into quadrants Assume tap-points within a quadrant draw current from segments contained in the quadrant Enforce the radial-sizing IR drop from ring-to-ring Tangential Sizing Solution G P : total conductance of ring p i q p : current in quadrant q of ring p

Circumference Correction Equipotential contour progresses from a diamond at the center of chip to square at the power ring Heuristically size tangential segments r(x,y): resistance of the segment at location (x,y) r: resistance of the corresponding radial segment α: constant. nα=10 gives good tradeoff between area and IR drop Total Conductance after three phases G R : Allocated total radial conductance

Experiments Testcases: T1, T2: uniform current requirements T3: derived from an industry flip-chip testcase T4: derived from T3 MATLAB used as numerical solver to compute exact IR drop.

Results Upto 33% peak IR drop reduction with same area Upto 32% area reduction with same IR drop Results of MAIC Sizing Results of MIAC Sizing

Zero Time IR Drop Analysis VQ q p : IR drop in quadrant q of ring p Simple measure with high correlation to actual IR drop at any point on the mesh useful for optimization and quick, early prediction

Perturbations and Robustness Variation in IR drop can occur due to current and/or resistance variation Perturbation result: ∞ norm used for peak IR drop G: power distribution network conductance matrix I: Current matrix E: Perturbation in G e: Perturbation in I ||G||X||G -1 || ∞ : condition number = measure of robustness Uniform meshes tend to have lower condition numbers  more robust

Conclusions and Future Work Contributions: Closed form power mesh optimization achieving up to 33% peak IR drop reduction Closed form IR drop analyses Proposed a measure of robustness of power distribution networks Simple incremental power mesh optimization technique (not discussed here) Future Work Extensions to flipchips, hard macros Use in IR-drop aware placement  place high power cells closer to power source (power ring)