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%