Prof Lei He

UCLA

LHE@ee.ucla.edu

EE 201C Modeling of VLSI Circuits and Systems

EE 201C Modeling of VLSI Circuits and Systems

2

Chapter 5 Signal and Power IntegrityChapter 5 Signal and Power Integrity

On-chip signal integrityRC and RLC coupling noise

Power integrity

Static noise: IR dropDynamic noise: L di/dt noise

Chapter 6: Beyond die noiseIn-package decap insertionLow frequency P/G resonanceNoise for High-speed signaling

3

Reading on Signal IntegrityReading on Signal Integrity

RC couplingJ. Cong, Z. Pan and P. V. Srinivas, "Improved Crosstalk

Modeling for Noise Constrained Interconnect Optimization", ASPDAC, 2001.

RLC couplingJun Chen and Lei He, "Worst-Case Crosstalk Noise for

Non-Switching Victims in High-speed Buses", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Volume 24, Issue 8, Aug. 2005.

To be covered by student presentation on May 14

4

Reading on Power IntegrityReading on Power Integrity

Static noise: IR drop S. Tan and R. Shi, “Optimization of VLSI Power/Ground (P/G)

Networks Via Sequence of Linear Programmings”, DAC’09

Dynamic noise: L di/dt noise Yiyu Shi, Jinjun Xiong, Chunchen Liu and Lei He, "Efficient

Decoupling Capacitance Budgeting Considering Current Correlation Including Process Variation", ICCAD, San Jose, CA, Nov. 2007.

Supplementary reading: H. Qian, S. R. Nassif, and S. S. Sapatnekar, “Power Grid

Analysis Using Random Walks,” IEEE Trans. on CAD, 2005. Yiyu Shi, Wei Yao, Jinjun Xiong, and Lei He, "Incremental and

On-demand Random Walk for Iterative Power Distribution Network Analysis", ASPDAC 2009

DAC 2009 Best Paper AwardSlides provided by X.D. Tan

Xiang-Dong Tan* and C.-J. Richard Shi

Optimization of VLSI Power/Ground (P/G) Networks Via Sequence of Linear

Programmings

Optimization of VLSI Power/Ground (P/G) Networks Via Sequence of Linear

Programmings

6

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New P/G optimization algorithm

Experimental results

Summary and future work

7

IntroductionIntroduction

...

...

Pad

IR drops:

Voltage difference between power supply pads and individual cell instances.

Electro-migration:

Metal ion mass transport along the grain boundaries when a metallic interconnect is stressed at high current density. Mean Time to Failure (MTF) (Black’s equation):MTF A w l J E kTa ( , ) exp( / )2

8

IntroductionIntroduction

A real industrial chip

#cell instances: 0.5M

#P/G resistors: 0.6M

9

IntroductionIntroduction

Unrestricted IR drops and current densities in power / ground (P/G) network will cause malfunction and reliability problems in deep sub-micron IC chips. Increased cell delays (timing problem) increased resistance and even opens of P/G wires

Most of P/G designs are done manually.An aggressive design will cause more design iterations and

thus lead to increased design costs. Over conservative P/G network design wastes a lot of

important chip areas.

10

MotivationMotivation

Two steps in P/G network design:P/G network construction (P/G routing).Determination of wire segment widths.

Determination of wire segment widths is hard to solve. The problem of determining wire segment widths in a P/G

network subject to reliability constraints is a constrained non-linear optimization problem.

Existing methods are not very efficient.based on the constrained nonlinear programming, can not handle large industrial P/G networks containing

millions of wire segments.

11

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New P/G optimization algorithm

Experimental results

Summary and future work

12

Algorithm ReviewAlgorithm Review

Assumption:Currents (average or maximum) of each individual cell

instance are known a priori before the optimization. computed by using power models of cells in a design

Existing optimization methods differ in the selection of variables.

Ohm’s Law

RV V

I

l

wii i

i

i

i

1 2 wi

li

Vi1 Vi2I i

13

Problem FormulationProblem Formulation

Min-area objective function: (Problem P)

IR drop constraints:

Electro-migration constraints:

Minimum width constraints:

f l wI l

V Vi ii B

i i

i ii B

( , )w V, I

2

1 2

V V V Vi i min maxor

| | | |I w V V li i i i i 1 2

wl I

V Vwi

i i

i i

1 2

min

14

Algorithm ReviewAlgorithm Review

Resistance values and branch currents are variables (Chowdhury and Breuer’87)

Both objective function and IR drop constraints are nonlinear

Solution: augmented Lagrangian method

Resistance values are variables (Dutta and Marek-Sadowska’89)

All the constraints are nonlinear Solution: feasible direction method

Nodal voltage and branch current are variables (Chowdhury’89)

Only objective function is nonlinear Solution: linear programming & conjugate gradient

Topology construction (Mitsuhashi and Kuh’92)

15

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New P/G optimization algorithm

Experimental results

Summary and future work

16

Relaxed P/G Optimization AlgorithmRelaxed P/G Optimization Algorithm

Relaxation: current directions are fixed. Nodal voltages and branch currents can be selected as

variables and be optimized separately.

Two optimization steps to solve P (Chowdhury’89) Solve for nodal voltages under fixed branch currents

(problem P1) Solve for branch current under fixed nodal voltages

(problem P2)

Advantage: All the constraints become linear and P2 is a linear

programming problem. P1 is a convex programming problem.

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Problem P1Problem P1

Nonlinear Optimization Problem (P1)Objective function:

Subject to

fV V

i

i ii Bi( ) ,V

1 2

Iili2

V

Vi

i

V

Vmin

max

V Vi i1 2

I

l

wi

i

min

V Vi i1 2 0

I i

IR drop: Electro-migration: Minimum width:

| |V Vi i1 2 l i

18

Problem P2Problem P2

Linear Optimization Problem (P2)Objective function:

Subject to

f I ii

i B

( ) ,I

li2

V Vi1 i2

Minimum width: KCL law:

l

V Vw

i1 i2min

i iI

s Ii i

i B j ( )

0

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ObservationsObservations

Solution to P1: First transform P1 into a unconstrained nonlinear

optimization problem by adding a penalty function to the objective function.

Conjugate gradient method was used to solve the unconstrained nonlinear problem.

Disadvantage:Very slow convergence (almost linear)Conjugate gradient directions may deteriorate

20

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New P/G optimization algorithm

Experimental results

Summary and future work

21

New Optimization AlgorithmNew Optimization Algorithm

Basic idea ---- linearize the nonlinear objective function in P1. Define

Linearized objective function:

v sign I V Vi i i i ( )( )1 2 0

fvi

ii B

( )| |

v

g ff

v vvi

ii B

i

ii Bi( ) ( )

( )( )

| | | |v v

v

vv v

0

00

0 02

2

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New Optimization AlgorithmNew Optimization Algorithm

The g(v) makes sense only if

Each product term, h(x) = c/x, in f(x) is a monotonic decreasing function.

g g f f( ) ( ) ( ) ( )v v v v0 0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-50

0

50

100

x

1/x and its first order Taylor's expansion at 0.04

h(x) = 1/x

H(x) = 1/x0 -1/(x0^2)*(x-x0)

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Two Optimization ScenariosTwo Optimization Scenarios

(1) All the branch voltage drops increase.

(2) Only some branch voltage drops increase while others decrease or stay unchanged.

Combine two scenarios, we have

f f g g( ) ( ) ( ) ( )v v v v0 0 and

We always have

v v vi i i0 02 0 1 ( ) ,

v vi i0

g g f f( ) ( ) ( ) ( )v v v v0 0

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New Optimization Problem P3New Optimization Problem P3

Minimize Linearized objective function

Subject to

g VV V V V

V Vi

i ii B

i

i ii i

i B

( )( )

( )

2

10

20

10

20 1 2

V

Vi

i

V

Vmin

max

V Vi i1 2

I

l

wi

i

min

IR drop: Electro-migration: Minimum width:

| |V Vi i1 2 l i

Extra constraint:

sign I V V sign I V Vi i i i i i( )( ) ( )( )10

20

1 2

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Sequence of Linear ProgrammingsSequence of Linear Programmings

New P/G Optimization Algorithm

1. Obtain an initial solution for a given P/G network

2. Build all the constraints for Problem P3

3. Solve P3 by sequence of linear programmings and record the result as

4. Build all the constraints based on of step (3) for the problem P2

5. Solve P2 by a linear programming and record the result as

6. Stop if improvement over previous result is small. Otherwise, goto step (2)

V , Ik k

V k 1

Ik 1

V k 1

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Theoretical ResultTheoretical Result

Theorem: There exists a so that step (3) always converges to the global minimum in the convex problem space of P1.

xmin

x0x1

x2

x3

x4

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Practical ConsiderationsPractical Considerations

Selection of At the beginning, solution space in P3 should be as

large as possible, so should be small. It should be close to 1 in later course of sequence of

linear programmings.

Numerical Stability Power networks are converted ground networks to

improve the numerical stability. Voltage drop in a ground network has to be

represented by by a power network with 5V supply voltage.

Power networks can be easily converted into ground networks.

2 5 10 5.

4 999975.

28

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New optimization algorithm

Experimental results

Summary and future work

29

Experimental ResultsExperimental Results

Sequence of linear programming and Conjugate gradient method were performed on 4 oversized p/g networks on SUN Untra-I with 169 MHz.

Ckt #node #bch New Algorithm Conjugate Gradient SP#it time rd% #it time rd%

p4x4 17 23 4 0.43 95.1 21 78.7 94.2 183.0p20x20 402 439 3 12.6 91.8 255 36147.1 90.8 2868.8p3x500 1502 1505 2 37.6 47.8 67 2135.4 26.8 >56.8g300x10 3002 3599 2 609.9 93.7 137 15192.1 78.4 >25.0

p100X100 10002 10199 4 1325.6 80.7 117 41716.8 48.9 >31.5

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Comparison in CPU timeComparison in CPU time

p4x4 p20x20 p3x500 p300x10 p100x100

01

00

00

20

00

03

00

00

40

00

05

00

00

p4x4 p20x20 p3x500 p300x10 p100x100

CPU time bynew algorithm

CPU time byCG

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Comparison in PerformanceComparison in Performance

p4x4 p20x20 3x500 g300x10 p100x1000

10

20

30

40

50

60

70

80

p4x4 p20x20 3x500 g300x10 p100x100

reduced to (%)by newalgorithm

reduced to (%)by CG

32

Experimental ResultsExperimental Results

1 2 3 4 5 61

1.005

1.01

1.015

1.02

1.025

1.03

# iteration

f(x)

/f_m

in(x

)scaled cost vs #iterations for new algorithm

The cost reduction versus the number of iterations (p4x4)

33

Experimental ResultsExperimental Results

0.4 0.6 0.8 13

3.5

4

4.5

5

5.5

6

6.5

7

xi

# o

f lin

ea

r p

rog

ram

min

gs

# of linear programmings vs xi

0.4 0.6 0.8 1424

426

428

430

432

434

436

438

440

442

xi

co

st

cost vs xi

Effect of on the performance of the new algorithm ( )xi

34

Outline of PresentationOutline of Presentation

Introduction and motivation

Review of existing algorithms

Relaxed P/G optimization procedure

New optimization algorithm

Experimental results

Summary and future work

35

SummarySummary

A new method based sequence of linear programmings was proposed to determine the widths of P/G network segments subject to reliability constraints.

We showed theoretically that new method is capable of finding solution as good as that by the best-known method.

Experimental results demonstrated that new method is orders-of-magnitude faster than the best-known method with constantly better solution quality.