Path Finding for 3D Power Distribution Networks
A. B. Kahng and C. K. ChengUC San DiegoFeb 18, 2011
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;
5
( , )
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.
Area Budget Adjustment between M6 and AP
The sizing results of both methods achieve smaller voltage drop when more area resources are allocated from AP to M6.
M6 Initial Width
AP InitialWidth
4.2um-90%
SatisfyingAreaConstraints
4.2um-75%
4.2um-60%…
4.2um+75%
4.2um+90%