Efficient Design and Analysis of Robust Power Distribution Meshes

1
Efficient Design and Analysis of Robust Power
Distribution Meshes

• Puneet Gupta (puneet_at_blaze-dfm.com)
• Blaze DFM Inc.
• Andrew B. Kahng
• ECE CSE UCSD
• Presented by Swamy Muddu

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Motivation

• Current density and hence IR drop is increasing
  with every technology generation
• IR drop
• Static IR drop resistive, driven by peak
  currents
• Transients E.g.,
• IR drop analysis and optimization is very slow
• Optimization is usually formulated as Non-Linear
  Program
• Analysis too slow to be used during Place Route
• Typical global power distribution
• Horizonatal/Vertical stripes
• Power meshes
• Peripheral i/o or flipchip
• This work Static IR drop, peripheral i/o

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The 1D Case

• Consider a power stripe with single power source
  at one end. Order power tap-points 1..N starting
  from the one closest to the power source
• IR drop increases away from the power source
• Wire segments between current tap-points closer
  to Vdd carry higher current
• ?A tapered power stripe is more bang for buck
• Closed form solution for optimal sizing can be
  calculated using Lagrangian multipliers.
• Assume area of a wire segment is proportional to
  its conductance

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The 1D Case contd..

• Solution to the minimum area IR drop (MAIC)
  constrained stripe sizing (rresistance,
  icurrent)
• Solution to the minimum IR drop area constrained
  (MIAC) problem is given by (Gtotal conductance
  constraint)

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IR Drop in Power Grids
Outer Power Ring
Bulls Eye
An example IR drop map for a uniform 25 X 25
power mesh with uniform current requirements
Equipotential ring

• IR Drop increases going in from the power ring
• Radial nature of IR drop variation in a power
  grid ? notice the similarity to the 1D case
• Equipotential contours take the form of rings
  diamond shaped at the center of layout and square
  shaped at the periphery (shape of power ring)

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Analysis of the 2D Case

• Assume square equipotential rings
• Divide power mesh into radial and tangential
  segments
• nXn mesh ? rings
• Intuition Current flows only in the radial
  segments

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Radial vs. Tangential Segments
Peak IR Drop vs. conductance Of wire segments in
a 25x25 power grid

• Variation of peak IR drop with varying widths of
  radial and tangential segments is shown
• Radial segments impact IR drop much more than
  tangential segments ? close to zero current flow
  in the tangential segments
• Assume radial current flow

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Power Grid Optimization

• Three phase optimization of power grids
• Radial Sizing. Size radial segments assuming
  uniform distribution of currents around rings.
• Tangential Sizing. Account for nonuniformity of
  current distribution. Divide rings into sectors
  and redistribute metal among sectors.
• Circumference Correction. Relax square ring
  assumption. Account for diamond to square ring
  shape transition
• All phases are based on closed form sizing
  expressions

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Radial Sizing

• Assume no current flow along the equipotential
  ring ? tangential segments sized to minimum width
  
• MAIC Sizing Solution
• i(p,p1) current from ring p to p1
• MIAC Sizing Solution
• GR total allocated radial conductance
• All radial segments originating from an
  equipotential ring are sized equally

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Tangential Sizing

• Redistribute metal between the radial segments
  along the equipotential ring to account for
  non-uniform current distribution
• Divide power grid into quadrants
• Assume tap-points within a quadrant draw current
  from segments contained in the quadrant
• Enforce the radial-sizing IR drop from
  ring-to-ring
• Tangential Sizing Solution
• GP total conductance of ring p
• iqp current in quadrant q of ring p

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Circumference Correction

• Equipotential contour progresses from a diamond
  at the center of chip to square at the power ring
• Heuristically size tangential segments
• r(x,y) resistance of the segment at location
  (x,y)
• r resistance of the corresponding radial segment
• a constant. na10 gives good tradeoff between
  area and IR drop
• Total Conductance after three phases
• GR Allocated total radial conductance

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Experiments

• Testcases
• T1, T2 uniform current requirements
• T3 derived from an industry flip-chip testcase
• T4 derived from T3
• MATLAB used as numerical solver to compute exact
  IR drop.

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Results
Results of MAIC Sizing
Results of MIAC Sizing

• Upto 33 peak IR drop reduction with same area
• Upto 32 area reduction with same IR drop

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Zero Time IR Drop Analysis

• VQqp IR drop in quadrant q of ring p
• Simple measure with high correlation to actual IR
  drop at any point on the mesh useful for
  optimization and quick, early prediction

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Perturbations and Robustness

• Variation in IR drop can occur due to current
  and/or resistance variation
• Perturbation result
• 8 norm used for peak IR drop
• G power distribution network conductance matrix
• I Current matrix
• E Perturbation in G
• e Perturbation in I
• GXG-18 condition number measure of
  robustness
• Uniform meshes tend to have lower condition
  numbers ? more robust

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Conclusions and Future Work

• Contributions
• Closed form power mesh optimization achieving up
  to 33 peak IR drop reduction
• Closed form IR drop analyses
• Proposed a measure of robustness of power
  distribution networks
• Simple incremental power mesh optimization
  technique (not discussed here)
• Future Work
• Extensions to flipchips, hard macros
• Use in IR-drop aware placement ? place high power
  cells closer to power source (power ring)