Path Finding for 3D Power Distribution Networks - UC San Diego

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					Path Finding for 3D Power
  Distribution Networks
  A. B. Kahng and C. K. Cheng
         UC San Diego
          Feb 18, 2011
      Power Grid Optimization Based on Rent’s Rule

            Vdd                            Higher current
                                           density in the
                                           inner grid

                                              Lowest current

We consider one
quarter of the
power grid                                   Highest current

Power Grid Topology

            • Quarter of Die: 200um
              X 200um
            • Four Metal Layers: M1,
              M3, M6, AP
            • Wire Direction: M1-
              horizontal, M3-vertical,
              M6-Horizontal, AP-
                     Power Grid Parameters
                        Initial     Width                      Min-max
             Pitch                              Density
                        Width       Range                      Constraint
   M1        2.5um      0.17um      N/A         N/A            N/A

   M3        8.0um      0.25um      N/A         N/A            N/A
                                    2um-                       2um-
   M6        20um       4.2um                   15%-80%
                                    8um                        12um
                                    3um-                       2um-
   AP        40um       10um                    15%-80%
                                    16um                       35um

• “Local Density “ is defined as (2*width)/pitch.
• “Width Range” is determined by intersection of “Local Density Constraint”
  and “Min-max Constraint”.
• Total metal area for M6 and AP layers are fixed.
Current Sources Based on Rent’s Rule

• Current source density function: I(d) =c*d^α ;
• S={(x, y)| (x, y) is the position of a node in M1} ;
• We put a input source I(x,y) for every (x,y) in S
  such that             I    I (d )* d ;
             ( x , y )S and |x|| y| d
                                           ( x, y )

• The total power in an area of d*d is c*d^β where β=(α+2)/2;
                Problem Formulation
• Inputs from the user:
  – Topology of power grid;
  – Default resistances of branches;
  – Possible current distributions satisfying Rent’s rule;
• Optimization for static voltage drop:
  Minimize (Maximum IR drop for all possible
  current distributions)
  Subject to
  – Total wire areas for M6 and AP are fixed;
  – Lower bound and upper bound for resistances of

                    Previous Work
• P. Gupta and A.B. Kahng, "Efficient Design and Analysis of Robust
  Power Distribution Meshes", Proc. International Conference on VLSI
  Design, Jan. 2006, pp. 337-342.
• W. Zhang, L. Zhang, etc, “On-chip power network optimization with
  decoupling capacitors and controlled-ESRs”, ASP-DAC, 2010, pp.
• A. Ghosh, S. Boyd and A. Saberi, “Minimizing effective resistance of
  a graph”, SIAM Review, problems and techniques section, Feb. 2008,
  50(1): pp. 37-66.
• L. Vandenberghe, S. Boyd and A. El Gamal, “Optimal Wire and
  Transistor Sizing for Circuits with Non-Tree Topology”, IEEE/ACM
  International Conference on Computer-Aided Design, Nov 1997, pp.
• S. Boyd, “Convex Optimization of Graph Laplacian Eigenvalues”,
  Proceedings International Congress of Mathematicians, 2006, 3: pp.
Design of Experiments

              •   Two optimization methods
                    – Nonlinear programming
                    – Heuristic search
              •   Fourteen current peak positions (red
                  dots in the left figure) and four β
                  values 0.25,0.5,0.75,1.0 for testing.
              •   The coordinates of the fourteen peak
                  positions are
              •   VD = worst voltage drop of the power
                  grid over all locations and all current
                  distributions satisfying power law.
Method 1: nonlinear programming (NLP)

                       The whole flow of NLP
                       for wire sizing
                       optimization with fixed
                       current distribution. The
                       current peak locates at
           Sizing Results of NLP

Wire, β=1.0, VD=0.2957            Segment, β=1.0, VD=0.2945

Wire, β=0.75, VD=0.2936           Segment, β=0.75, VD=0.2941

               VD for uniform sizing = 0.3054
          Sizing Results of NLP

Wire, β=0.5, VD=0.2945            Segment, β=0.5, VD=0.2932

Wire, β=0.25, VD=0.2934           Segment, β=0.25, VD=0.2921

               VD for uniform sizing = 0.3054
• When β is large (i.e. current sources distribute
  uniformly), the results suggest putting most of
  wire resources near the voltage source.
• When β is small (i.e. most of current sources
  gather at origin), we should give some wire
  resources to segments near the origin.
• “Segment” optimization results are more
  stable than “Wire” optimization results
  relative to change of β.
                       Method 2: Heuristic search

•   The candidates include all combinations of wl,wh,pl,ph.
•   The curve part is fitted by a polynomial function satisfying area constraints.
•   The best wire sizing result is chosen to minimize the worst voltage drop over all locations and all
    possible current distributions with different peaks and β value.
   Sizing Results of Heuristic Search

• Each wire is assumed to have the same width.
• VD for uniform sizing = 0.3054.
• VD for optimized sizing = 0.2902.
  Width Range Adjustment for M6

                         Original Setup
                         M6 : 2um-8um
                         AP : 3um-16um
                         VD = 0.2902

M6 : 3um-7um             M6 : 4um-6um
AP : 3um-16um            AP : 3um-16um
VD = 0.2918              VD = 0.2932
   Width Range Adjustment for AP

                          Original Setup
                          M6 : 2um-8um
                          AP : 3um-16um
                          VD = 0.2902

M6 : 2um-8um              M6 : 2um-8um
AP : 5um-14um             AP : 7um-12um
VD = 0.2961               VD = 0.2975
Width Range Adjustment for Both M6 and AP

                   M6 : 3um-7um
                   AP : 7um-12um
                   VD = 0.2953

  M6 : 3um-7um
  AP : 5um-14um
  VD = 0.2932                      Original Setup
                                   M6 : 2um-8um
                                   AP : 3um-16um
  M6 : 4um-6um                     VD = 0.2902
  AP : 5um-14um
  VD = 0.2965

                   M6 : 4um-6um
                   AP : 7um-12um
                   VD = 0.2983
• The heuristic search method performs better
  than NLP methods on the objective of
  minimizing maximum voltage drop over all
  locations and current distributions.
• Adjustment of width range of AP has more
  effect on performance of optimized sizing
  results than adjustment of width range of M6.
Area Budget Adjustment between M6 and AP

                    M6 Initial             AP Initial
                    Width                  Width
                    4.2um-60%              Satisfying
                    …                      Constraints

                  The sizing results of both methods achieve
                  smaller voltage drop when more area
                  resources are allocated from AP to M6.

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