Path Finding for 3D Power Distribution Networks A. B. Kahng and C. K. Cheng UC San Diego Feb 18, 2011.
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Power Grid Optimization Based on Rent’s Rule
2
Higher current density in the inner grid
Lowest current density
Highest current density
We consider onequarter of the power grid
Vdd
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-vertical
Power Grid Parameters
Pitch Initial Width
Width Range
Local Density Constraint
Min-max Constraint
M1 2.5um 0.17um N/A N/A N/A
M3 8.0um 0.25um N/A N/A N/A
M6 20um 4.2um 2um-8um 15%-80% 2um-
12um
AP 40um 10um 3um-16um 15%-80% 2um-
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 ;• The total power in an area of d*d is c*d^β where β=(α+2)/2;
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( , )
2)
| | | |( ,
( )*x y S and dx y
x yI I d d
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
branches;6
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. 119-124.
• 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. 252-259.
• S. Boyd, “Convex Optimization of Graph Laplacian Eigenvalues”, Proceedings International Congress of Mathematicians, 2006, 3: pp. 1311-1319.
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(0,0),(50,0),(50,50),(100,0),(100,50),(100,100),(150,0),(150,50),(150,100),(150,150),(200,0),(200,50),(200,100),(200,150).
• 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 origin.
Sizing Results of NLP
Segment, β=0.75, VD=0.2941Wire, β=0.75, VD=0.2936
Segment, β=1.0, VD=0.2945Wire, β=1.0, VD=0.2957
VD for uniform sizing = 0.3054
Sizing Results of NLP
Segment, β=0.25, VD=0.2921Wire, β=0.25, VD=0.2934
Segment, β=0.5, VD=0.2932Wire, β=0.5, VD=0.2945
VD for uniform sizing = 0.3054
Observations
• 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
M6 : 3um-7umAP : 3um-16umVD = 0.2918
M6 : 4um-6umAP : 3um-16umVD = 0.2932
Original Setup M6 : 2um-8umAP : 3um-16umVD = 0.2902
Width Range Adjustment for AP
M6 : 2um-8umAP : 5um-14umVD = 0.2961
M6 : 2um-8umAP : 7um-12umVD = 0.2975
Original Setup M6 : 2um-8umAP : 3um-16umVD = 0.2902
Width Range Adjustment for Both M6 and AP
M6 : 4um-6umAP : 5um-14umVD = 0.2965
M6 : 3um-7umAP : 5um-14umVD = 0.2932 Original Setup
M6 : 2um-8umAP : 3um-16umVD = 0.2902
M6 : 3um-7umAP : 7um-12umVD = 0.2953
M6 : 4um-6umAP : 7um-12umVD = 0.2983
Observations
• 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.
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