Optimal Planning for Mesh-Based Power Distribution
H. Chen, C.-K. Cheng, A. B. Kahng, Makoto Mori * and Q. Wang
UCSD CSE Department* Fujitsu Limited
Work partially supported by Cadence Design Systems, Inc., the California MICRO program, the MARCO Gigascale Silicon Research Center, NSFMIP-9987678 and the Semiconductor Research Corporation.
Motivation (I)• Voltage drop in the power distribution is critical to
chip performance and reliability• Power distribution network in early design stages
– nominal wiring pitch and width for each layer need to be locked in
– location and logic content of the blocks are unknown– impossible to obtain the pattern of current drawn by sinks– transient analysis is essentially difficult– design decisions are mostly based on DC analysis of
uniform mesh structures, with current drains modeled using simple area-based calculations
Motivation (II)• Current method in practice
– explore different combinations of wire pitch and width for different layers
– select the best combination based on circuit simulations
– problem: computationally infeasible to explore all possible configurations; the result is hence a sub-optimal solution
• What we need: a new approach to optimize topology for a hierarchical, uniform power distribution
Our Work• Study the worst-case static IR-drop on
hierarchical, uniform power meshes using both analytical and empirical methods
• Propose a novel and efficient method for optimizing worst-case IR-drop on two-level uniform power distribution meshes
• Usage of our resultsplanning of hierarchical power meshes in early design stages
Outline
• Problem Formulation• IR-Drop on Single-Level Power Mesh• IR-Drop on Two-Level Power Mesh• Optimal Planning of Two-Level Power Mesh• IR-Drop on Three-Level Power Mesh• Conclusion and On-Going Work
Problem Statement• Given fixed wire pitch and width for the
bottom-level mesh• Find the optimal wire pitch and width for
each mesh except the bottom-level mesh• Objectives
– for a given total routing area, the power mesh achieves the minimum worst-case IR-drop
– for a given worst-case IR-drop requirement, the power mesh meets the requirement with minimum total routing area
Model of Power Network• Hierarchy of metal layers
– uniform and parallel metal wires at each layer– adjacent metal layers connected at the crossing points
• Via resistance: ignored– much smaller than resistance of mesh segments
• C4 power pads evenly distributed on the top layer• Uniform current sinks on the crossing points of the
bottom layer– before the accurate floorplan, the exact current drain at
different locations is unknown
Representative Area• Area surrounded by adjacent power pads• Power mesh
– # power pads in state-of-art designs: larger than 100– infinite resistive grid – constructed by replicating the representative area
• Worst-case IR-drop appears near the center of the representative area
(b) Representative area(a) Two-level power mesh
Bottom-level mesh
C4 pad
Top-levelmesh
Outline
• Problem Formulation• IR-Drop on Single-Level Power Mesh
– a closed-form approximation for the worst-case IR-drop on a single-level power mesh
• IR-Drop on Two-Level Power Mesh• Optimal Planning of Two-Level Power Mesh• IR-Drop on Three-Level Power Mesh• Conclusion and On-Going Work
IR-Drop in Single-Level Power Mesh• IR-drop on a hierarchical power mesh depends
largely on the top-level mesh• We analyze worst-case IR-drop
on a single-level power mesh– power pads
– supply constant current to the mesh
– regarded as current sources– ground: at infinity– our method: analyze voltage
drops caused by current sources and current sinks separately
IR-Drop by Current Sources• Analysis
– IR-drop caused by a single current source• an approximated close-form formula [Atkinson et al. 1999]
– integrate IR-drop for all current sources• Result: worst-case IR-drop when only current
sources are considered
– N : # stripes in the representative area
– R : edge resistance
– I : total current drain in the representative area– C = -0.1324
IR-Drop by Current Sinks• Analysis
– uniform resistive lattice: a discrete approximation to a continuous resistive medium
– potential increases with D2
where D = distance from the center, if• a continuous resistive medium• evenly distributed current sinks
– impose a form proportional to D2
• Result: worst-case IR-drop when only current sinks are considered
Verification of IR-Drop Formula (I)• Worst-case IR-drop• HSpice simulations
– fixed total current drain I– fixed edge resistance R
– #stripes between power pads: N = 4 to 12
Verification of IR-Drop Formula (II)
Simulation results for worst-case IR-drop on single-level power meshes, compared to estimated values
N IR Drop Estimated IRDrop Error4 333.33 324.56 8.776 392.86 389.09 3.768 436.97 434.88 2.0910 471.73 470.39 1.3312 500.34 499.41 0.92
100 836.87 836.86 0.01
2 4 6 8 10 12 14300320340360380400420440460480500520540
IR-D
rop
(mV)
# Stripes between Power Pads
Estimated IR-Drop HSpice Simulations
Accuracy within 1% when N > 4
Outline
• Problem Formulation• IR-Drop on Single-Level Power Mesh• IR-Drop on Two-Level Power Mesh
– an accurate empirical expression for the worst-case IR-drop on a two-level power mesh
• Optimal Planning of Two-Level Power Mesh• IR-Drop on Three-Level Power Mesh• Conclusion and On-Going Work
IR-Drop in Two-Level Power Mesh
• Model: two uniform infinite resistive lattices– top-level mesh
• connected to power pads• wider metal lines• coarser grid
– bottom-level mesh• connected to devices• thinner metal lines• finer grid
• Analysis method: consider IR-drop on two meshes separately
IR-Drop in the Coarser Mesh• Assumption: currents flow along an
equivalent single-level coarse mesh– most current flows along the coarser mesh
• IR-drop in the coarser mesh:
–N1 : # stripes of the coarser mesh in the representative area
–Re : equivalent edge resistance
–I : total current drain in the representative area
–c : a constant
Verification
• HSpice simulations of two-level power meshes– fixed total current drain I
– fixed Re
• fixed routing resource of two meshes
• bottom-level mesh is 10 times finer than the top-level one
– # stripes of the coarser mesh N1 = 3 ~ 10
N 1 3 4 5 6IR-Drop 170.15 188.62 206.75 219.79N 1 7 8 9 10IR-Drop 232.42 242.32 251.96 259.91
V ~ ln(N1): nice linearity
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4160
180
200
220
240
260
ln(# Stripes on the Coarse Mesh)
HSpice Simulations
IR-D
rop
(mV)
Equivalent Edge Resistance• Re : slope of the line V ~ ln(N1)• HSpice simulations of two-level power meshes
– fixed total current drain I
– # stripes of the coarser mesh N1 = 19– bottom-level mesh: 10 times finer than the top-level one– routing resource of the finer mesh = 1
fixed edge resistance of the finer mesh R– different total routing resource r
different Re
r 1.667 2 4 6 8R / Re 1.661 1.991 3.953 5.888 7.806
• Empirically, Re R / r
IR-Drop in the Finer Mesh (I)
• Assumption: finer mesh within each cell formed by the coarser mesh has equal voltage on the cell boundary– coarser mesh: much smaller edge resistance
• HSpice simulations of finer mesh– equal voltage on the boundary– fixed edge resistance of the finer mesh R– fixed current drain of each device i– # stripes within each cell: M = 2 ~ 22
IR-Drop in the Finer Mesh (II)M 3 4 5 6 7 8 9 10 11 12
IR-Drop 1.13 1.67 2.60 3.43 4.66 5.79 7.32 8.73 10.55 12.27M 13 14 15 16 17 18 19 20 21 22
IR-Drop 14.38 16.39 18.80 21.11 23.91 26.41 29.41 32.31 35.39 37.58
0 100 200 300 400
0
5
10
15
20
25
30
35 HSpice Simulations
(# Stripes on the Finer Mesh)2
IR-D
rop
(mV)
Vfine ~ M2: nice linearity
IR-Drop Formula (I)• IR-drop
– C1(r), C2(r) are functions of r
• HSpice simulations of two-level meshes– fixed total current drain I
– bottom-level mesh: 10 times finer than the top-level one– routing resource of the finer mesh = 1
fixed edge resistance of the finer mesh R– fixed total routing resource r = 16– # stripes of the coarser mesh N1 = 1 ~ 9– C1, C2 obtained by simulation results for N1 = 7 and 9
IR-Drop Formula (II)Simulation results for worst-case IR-drop on two-level power meshes with fixed total routing area, compared to estimated values
Accuracy within 1% when N > 4
0 2 4 6 8 1020
30
40
50
60
70
80
90 Estimated IR-Drop HSpice Simulations
# Stripes in the Coarser Mesh
IR-D
rop
(mV)
r N 1 IR-Drop Estimated IR-Drop Error16 1 77.15 82.34 5.1916 2 29.37 32.52 3.1516 3 26.04 26.05 0.0116 4 24.67 25.24 0.5616 5 25.76 25.75 -0.0116 6 26.42 26.64 0.2316 7 27.62 27.62 0.0016 8 28.46 28.58 0.1216 9 29.51 29.51 0.00
Outline• Problem Formulation• IR-Drop on Single-Level Power Mesh• IR-Drop on Two-Level Power Mesh• Optimal Planning of Two-Level Power Mesh
– a new approach to optimize the topology of two-level power mesh
• IR-Drop on Three-Level Power Mesh• Conclusion and On-Going Work
Optimizing Topology with a Given Total Routing Area• Problem Statement
– given fixed total routing area r– find optimal # stripes in the coarser mesh N1
– objective = min worst-case IR-drop• Optimization Method
– based on the IR-drop formula
• E.g., when r = 16, N1* = 3.9
Optimizing Topology with a Given Worst-Case IR-Drop Requirement• Problem Statement
– given worst-case IR-drop requirement– find optimal # stripes in the coarser mesh N1
– objective = min total routing area r• Optimization Method
– for each value of r• simulate two-level power meshes for a few values of N1
• calculate the values of C1(r), C2(r) • compute the optimal worst-case IR-drop V*(r)
– find minimum total routing area r with V*(r) meets given requirement
Example• Requirement: worst-case IR-drop < 30mV• Compute optimal IR-drop V*(r) for each value of r
• Optimal r : between 12 and 13Optimal N1 : 3 or 4
r C 1 (r) C 2 (r) N*(r) V*(r)10 0.07679 0.010986 3.1 37.011 0.07663 0.009934 3.3 34.212 0.07648 0.009066 3.4 31.913 0.07633 0.008338 3.5 29.914 0.07618 0.007718 3.7 28.215 0.07605 0.007184 3.8 26.616 0.07592 0.006719 3.9 25.2
Outline
• Problem Formulation• IR-Drop on Single-Level Power Mesh• IR-Drop on Two-Level Power Mesh• Optimal Planning of Two-Level Power Mesh• IR-Drop on Three-Level Power Mesh
– a third, middle-level mesh helps to reduce IR-drop by only a relatively small extent (about 5%, according to our experiments)
• Conclusion and On-Going Work
Optimal Resource Distribution• Problem
– given topology of three-level mesh# stripes of three grids
– given total routing area – find optimal resource distribution
• Method– a simplified power network wire sizing technique
Sequential LP method [Tan et al. DAC99]– for a given width assignment, find the voltage at each node
by solving a set of linear equations– fix the node voltages and find the optimal width assignment
to maximize current drain at the center– repeat this process iteratively until the solution converges
IR-Drop in Three-Level Power Mesh• Analysis method
– fix # stripes in the top- and bottom-level meshes
– explore different # stripes for the middle-level mesh
– find optimal resource allocation and IR-drop
• Top, middle and bottom meshes– # stripes: N1 ,N2 and 120– wiring resource: r1 , r2 and 1
(1 + r1 + r2 = 10)• Middle-level mesh reduces IR-
drop to a relatively small extent (about 5%)
N 1 N 2 IR-Drop r 1 r 2
3 4 35.8 6.43 3.573 6 35.2 6.54 3.463 10 34.3 6.76 3.243 15 34.3 6.88 3.123 20 34.7 6.97 3.033 40 35.3 7.07 2.933 60 36.1 8.05 2.954 5 36.4 5.57 4.434 6 35.9 6.13 3.874 10 35.2 6.44 3.564 15 35.1 6.51 3.494 20 35.8 6.77 3.234 40 36.4 6.99 3.014 60 37.1 7.48 2.52
Conclusions
• Obtained accurate expression for worst-case IR-drop in two-level uniform meshes
• Proposed a new method of optimizing topology of two-level uniform power mesh– used to decide nominal wire width and pitch
for power networks in early design stages• Ongoing work:
– optimization of non-uniform power meshes– interactions with layout or detailed current
analysis