# 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
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: M1-
horizontal, M3-vertical,
M6-Horizontal, AP-
vertical
Power Grid Parameters
Local
Initial     Width                      Min-max
Pitch                              Density
Width       Range                      Constraint
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 )
2

• The total power in an area of d*d is c*d^β where β=(α+2)/2;
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             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
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             AP Initial
Width                  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.

```
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