Optimal Planning for Mesh-Based Power Distribution

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Verification of IR-Drop Formula (II)
Simulation results for worst-case IR-drop on
single-level power meshes, compared to estimated
values
Accuracy within 1 when N gt 4
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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

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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

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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

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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

V ln(N1) nice linearity
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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

• Empirically, Re ? R / r

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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

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IR-Drop in the Finer Mesh (II)
Vfine M2 nice linearity
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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

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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 gt 4
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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

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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

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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

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Example

• Requirement worst-case IR-drop lt 30mV
• Compute optimal IR-drop V(r) for each value of r
• Optimal r between 12 and 13
• Optimal N1 3 or 4

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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

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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

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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)

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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

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Thank You !