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 results
planning 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
Bottomlevel mesh
C4 pad
Top-level
mesh
(a) Two-level power mesh
(b) Representative area
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
Simulation results for worst-case IRdrop on single-level power meshes,
compared to estimated values
N
4
6
8
10
12
100
IR Drop Estimated IRDrop
333.33
324.56
392.86
389.09
436.97
434.88
471.73
470.39
500.34
499.41
836.87
836.86
Error
8.77
3.76
2.09
1.33
0.92
0.01
IR-Drop (mV)
Verification of IR-Drop Formula (II)
Estimated IR-Drop
HSpice Simulations
540
520
500
480
460
440
420
400
380
360
340
320
300
2
4
6
8
10
12
# Stripes between Power Pads
Accuracy within
1% when N > 4
14
Outline
• Problem Formulation
• IR-Drop on Single-Level Power Mesh
• IR-Drop on Two-Level Power Mesh
– an accurate empirical expression for the worstcase 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
– fixed total current drain I
– fixed Re
• fixed routing resource of
two meshes
• bottom-level mesh is 10
times finer than the toplevel one
– # stripes of the coarser
mesh N1 = 3 ~ 10
N1
3
4
5
6
IR-Drop 170.15 188.62 206.75 219.79
N1
7
8
9
10
IR-Drop 232.42 242.32 251.96 259.91
HSpice Simulations
260
240
IR-Drop (mV)
• HSpice simulations of
two-level power meshes
220
200
180
160
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
ln(# Stripes on the Coarse Mesh)
V ~ ln(N1): nice linearity
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
8
R / 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 12.27
M
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
HSpice Simulations
35
Vfine ~ M2: nice linearity
30
IR-Drop (mV)
25
20
15
10
5
0
0
100
200
300
400
2
(# Stripes on the Finer Mesh)
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)
r N 1 IR-Drop Estimated IR-Drop Error
16
16
16
16
16
16
16
16
16
1
2
3
4
5
6
7
8
9
77.15
29.37
26.04
24.67
25.76
26.42
27.62
28.46
29.51
82.34
32.52
26.05
25.24
25.75
26.64
27.62
28.58
29.51
5.19
3.15
0.01
0.56
-0.01
0.23
0.00
0.12
0.00
Estimated IR-Drop
HSpice Simulations
90
80
IR-Drop (mV)
Simulation results for worst-case
IR-drop on two-level power meshes
with fixed total routing area,
compared to estimated values
70
60
50
40
30
20
0
2
4
6
8
# Stripes in the Coarser Mesh
Accuracy within
1% when N > 4
10
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 twolevel 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
r
10
11
12
13
14
15
16
C 1 (r)
C 2 (r)
0.07679
0.07663
0.07648
0.07633
0.07618
0.07605
0.07592
0.010986
0.009934
0.009066
0.008338
0.007718
0.007184
0.006719
N*(r)
3.1
3.3
3.4
3.5
3.7
3.8
3.9
V*(r)
37.0
34.2
31.9
29.9
28.2
26.6
25.2
• Optimal r : between 12 and 13
Optimal N1 : 3 or 4
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 IRdrop 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 bottomlevel 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 IRdrop to a relatively small extent
(about 5%)
N 1 N 2 IR-Drop
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
6
10
15
20
40
60
5
6
10
15
20
40
60
35.8
35.2
34.3
34.3
34.7
35.3
36.1
36.4
35.9
35.2
35.1
35.8
36.4
37.1
r1
r2
6.43
6.54
6.76
6.88
6.97
7.07
8.05
5.57
6.13
6.44
6.51
6.77
6.99
7.48
3.57
3.46
3.24
3.12
3.03
2.93
2.95
4.43
3.87
3.56
3.49
3.23
3.01
2.52
Conclusions
• Obtained accurate expression for worstcase 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
Thank You !