A Fast OnChip Decoupling Capacitance Allocation Method

1
A Fast On-Chip Decoupling Capacitance Allocation
Method

• Min Zhao, Rajendran Panda,
• Savithri Sundareswaran, Shu Yan, Yuhong Fu
• Freescale Semiconductor Inc.

2
Outline

• Decap allocation problem
• Prior art
• Our solution
• Network reduction through macro-modeling
• Decap optimization using charge-based sequence of
  linear programming (LP)
• Experimental results
• Summary

3
Decap Allocation Overview

• Function
• Suppress dynamic noise by supplying sudden
  current demands from local charge storage
• Side effect of adding too much decap
• Increased leakage
• Increased die area
• Risk of lower yield
• Need to add minimum amount of decap, yet
  sufficient for reducing noise

4
Decap Allocation Problem

• Formulated as a constrained optimization problem
• Objective Identify decap locations and minimize
  the total decap to be added
• Constraints
• Voltage constraints voltage at all
  transistors/gates should be better than a
  specified threshold, at all time
• Circuit system constraints KCL, KVL and circuit
  element equations
• Decap constraints amount of decap allowed at a
  location is limited

• Challenges
• Power network is a multi-million element (RLC)
  circuit
• Based on transient analysis result
• The optimization problem is non-linear

5
Prior Art

• Charge-based decap estimation
• Method 1 (S. Zhao et al., TCAD 2002)
• Sensitivity-based nonlinear optimization
• Method 2 quadratic program (H. Su et al., TCAD
  2003)
• Method 3 conjugate gradient solver(J. Fu et
  al., ASP-DAC 2003)
• Method 4 fast conjugate gradient method(H. Li
  et al., DAC2005)
• Method 5 sequence of linear programs(J. Fan et
  al., ISQED 2006)
• Problem size reduction
• Method 6 multi-grid method (K. Wang et al.,
  DAC2003)
• Method 7 divide and conquer (H. Li et al., DAC
  2005)

6
Prior Art Charge-based Method

• Method 1 (S. Zhao et al., TCAD 2002)
• In initial iteration, decap is estimated for each
  module
• through simple and approximate charge-based
  formula
• Decap C (1 - 1/ ?) Q/Vthre ,
• where ? max(1, V/Vthre) , Q ?0T I(t) dt
• Then iteratively run transient analysis and
  increase decap, until Vthre are satisfied
• Issues
• May substantially over-estimation
• Or may require multiple transient analysis
  iterations

7
Prior Art Sensitivity-based Method

• Method 2-5
• Compute sensitivity of voltage to decap placed at
  different locations and use sensitivity to guide
  the non-linear optimization
• Issues
• Many iterations of sensitivity calculation
• One (adjoint) sensitivity calculation is
  equivalent to one expensive transient analysis

8
Prior Art Problem Size Reduction

• Method 6
• With multi-grid like method
• Issues
• Applied only to regular grid
• Method 7
• Partition the power network
• Each partition is solved by a sensitivity-based
  non-linear optimizer
• Issues
• The assumption that partition boundary voltage is
  not affected by decap may introduce error
• Partition size becomes very large when one hot
  spot cover large area

9
Our Solution
10
Background Macro-modeling

• Matrix equations for the unreduced power network
• G V I
• Conductance matrix voltage vector current
  vector
• Size of matrix/vectors is extremely large
  (multi-million)
• Through macro-modeling, get
• I A V S,
• Relates port currents (I)to port voltages (V)
• A is a much smaller matrix(few thousands)
• Abstract away the internal nodes

11
Violation Region and Sampling Nodes

• Divide the die into small tiles
• Hot spot (yellow) part of network with bad
  supply voltage
• Violation region tiles covering hot spot (yellow
  orange) surrounding tiles within effective
  radius of decap (red)
• Sampling nodes (green)nodes with worst voltage
  drop, sampled one per tile in the violation
  region
• With macro-modeling technique, network is reduced
  to sampling nodes in the violation region All
  other nodes are abstracted away
• If two hot spots overlap or close, they are
  merged into one violation region

Tile
12
Overall Flow
13
Non-uniform Grid

• Definition
• Voltage gradient (n1, n2) (V(n1)-V(n2))/distance
  (n1,n2)
• Usually multiple violation regions in a network
• Some regions are large with a slow voltage
  gradient
• Need coarse grain tile to reduce problem size
• Some regions are small with a large voltage
  gradient
• Need fine grain tile to enable one sampling node
  to represent the entire sampling region
• Use non-uniform grid, so that optimization
  problem size of all violation regions are within
  limitation

How large should a tile be?
14
Outline

• Decap allocation problem
• Prior art
• Our solution
• Network reduction through macro-modeling
• Decap optimization using charged-based sequence
  of LP
• Charge-based nodal equations
• Charge-based decap optimization formulation
• Sequence of LP
• Experimental results
• Summary

15
Charge-based Nodal Equations

• Violation time window ts, te
• Current-based nodal equations of a macro-model
• I A V S, for each time point
• I is 0 if no external elements

• During decap optimization, using AYB0
  to approximate the
  transient analysis

16
Original Decap Allocation Problem

• Objective Identify decap locations and minimize
  the total decap to be added
• Constraints
• Voltage constraints voltage at all
  transistors/gates should be better than a
  specified threshold, at all time
• Circuit system constraints KCL, KVL and circuit
  element equations
• Decap constraints amount of decap allowed at a
  location is limited

17
Charge-Based Voltage Constraints

• Voltage should be ? Vthre at all time points
• Vi,t ? Vthre , ?t ?ts,te
• In charge-based formula,
• Approximated to a trapezoid area Yi ?
  (Vo,iVthre) (te-ts)/2
• ts, te is violation time window
• Vo,I is the voltage at time ts
• Charge-based voltage constraints in vector formY
  ? (te ts) . (Vo,1Vthre)/2, (Vo,2Vthre)/2,
  ., (Vo,iVthre)/2 T

18
Charge-based Circuit System Constraints

• Charge released from decap during ts,te
• Qi (Vo,i-Vi) Ci
• Its influence on voltage (Y) is governed by
• Q A Y B
• Charge from decap should be large enough to pull
  the voltage above Vthre
• (Vo,1 V1 ).C1, (Vo,2 V2 ).C2, , (V o,m-Vm
  ).Cm T ? A Y B
• Y ? (te ts) . (Vo,1Vthre)/2,
  (Vo,2Vthre)/2, ., (Vo,iVthre)/2 T

19
Charge-based Decap Allocation

• Minimize ? Ci
• Subject to
• Y ? (te ts) . (Vo,1Vthre)/2,
  (Vo,2Vthre)/2, ., (Vo,iVthre)/2 T
• // voltage constraints
• (Vo,1 V1 ).C1, (Vo,2 V2 ).C2, , (Vo,m-Vm
  ).Cm T ? A Y B
• //
  circuit system constraints
• Ci ? C max,I // decap constraints
• Yi (Vo,iVi) (te-ts)/2

• Circuit system constraints are non-linear
• The objective and the other constraints are linear

20
Sequence of Linear Programming

• In (Vo,1 V1 ).C1, (Vo,2 V2 ).C2, , (Vo,m-
  Vm).Cm T ? A Y B, voltages Vi change with
  decap to be added, Ci
• Run iterations of LP to account for this
  non-linearity

21
Experimental Results

• Benchmark circuits information
• Benchmarked against a greedy method
• Place decap uniformly across the core violation
  region
• For each core violation region, the amount of
  decap placed is decided through binary search

Analysis CPU(s)
time points
Worst voltage(v)
Supply voltage(v)
nodes (million)
Chip
7
42
2.14
2.50
0.2
Circuit 1
345
62
1.42
1.65
4.9
Circuit 2
22
Experimental Results - Circuit 1
23
Experimental Results - Circuit 2
24
Advantages

• Our method comes up with an accurate decap
  allocation within 1-2 transient analysis
  iterations, compared with many iterations of
  transient analysis (sensitivity calculation) of
  the previous methods
• Our method scales very well with p/g network size
• Our charge-based SLP optimization method obtains
  the decap budgeting through several iterations of
  linear programming and linear equations solution

25
Summary

• Solved the decap allocation problem optimize the
  amount of decap to reduce power supply noise
• Where to add?
• How much to add at each location?
• Proposed the efficient optimization methods
• Several techniques to reduce problem size
  (macro-modeling, sampling, and consideration of
  effective radius for decap)
• Charge-based sequence of linear programming
  optimization
• Iterations of linear programming and linear
  equations solution

26

• Thank You!