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
density
We consider one
quarter of the
power grid
Highest current
density
2
Power Grid Topology
• Quarter of Die: 200um
X 200um
• Four Metal Layers: M1,
M3, M6, AP
• Wire Direction: M1horizontal, M3-vertical,
M6-Horizontal, APvertical
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
AP
40um
10um
2um8um
3um16um
15%-80%
15%-80%
2um12um
2um35um
• “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
I
 I (d ) * d ;

such that
• The total power in an area of d*d is c*d^β where β=(α+2)/2;
2
( x , y )S and |x|| y|  d
( x, y )
5
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
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
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
Original Setup
M6 : 2um-8um
AP : 3um-16um
VD = 0.2902
M6 : 3um-7um
AP : 3um-16um
VD = 0.2918
M6 : 4um-6um
AP : 3um-16um
VD = 0.2932
Width Range Adjustment for AP
Original Setup
M6 : 2um-8um
AP : 3um-16um
VD = 0.2902
M6 : 2um-8um
AP : 5um-14um
VD = 0.2961
M6 : 2um-8um
AP : 7um-12um
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
VD = 0.2902
M6 : 4um-6um
AP : 5um-14um
VD = 0.2965
M6 : 4um-6um
AP : 7um-12um
VD = 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
M6 Initial
Width
AP Initial
Width
4.2um-90%
4.2um-75%
4.2um-60%
…
Satisfying
Area
Constraints
4.2um+75%
4.2um+90%
The sizing results of both methods achieve
smaller voltage drop when more area
resources are allocated from AP to M6.