OPERA: OPtimization with Ellipsoidal uncertainty for Robust Analog IC design

OPERA: OPERA: OPtimizationOPtimization with Ellipsoidal with Ellipsoidal uncertainty for Robust Analog IC designuncertainty for Robust Analog IC design

Yang Xu, KanYang Xu, Kan--Lin Lin HsiungHsiung, , XinXin LiLiIvan Ivan NausiedaNausieda, , Stephen Boyd, Larry Stephen Boyd, Larry PileggiPileggi

June 16, 2005June 16, 2005

ContentsContents

IntroductionIntroductionBackgroundBackgroundProposed ApproachProposed ApproachImplementation IssuesImplementation IssuesExamples and Preliminary ResultsExamples and Preliminary ResultsDiscussions and ConclusionsDiscussions and Conclusions

IntroductionIntroduction

DesignDesign--manufacturing manufacturing interface is interface is becoming more becoming more and more and more complexcomplexRF and analog RF and analog integrated integrated circuits are very circuits are very sensitive to sensitive to process variationprocess variation

Trends of variabilityTrends of variabilityVariability in DSM technologies is increasingVariability in DSM technologies is increasingLargeLarge--scale variation results in lowerscale variation results in lowerproduct yieldproduct yieldControl performance variability in early Control performance variability in early design stagesdesign stages

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

250 180 130 100 70LEFF (nm)

Varia

tion

VTH

TOX

WLHρ

[Nassif 01]

Traditional Traditional CornerCorner--Enumeration OptimizationEnumeration Optimization

Corner-enumeration worst-case optimizationMost widely used robust design techniqueUncertain parameters are often assumed to have independent uniform distributionsDesign is optimized for all corners ofa ±3σ tolerance box

Problems with corner-enumeration optimizationOften ignores correlation between process parameters Problem size increases exponentially in number of uncertain parametersNo guarantee for parameter points inside the ±3σtolerance boxDesign for all corners could result in over-design

Design Optimizationon Corners

ContentsContents

IntroductionIntroductionBackgroundBackgroundProposed ApproachProposed Approach

Capturing process variationsCapturing process variationsRobust optimization approachRobust optimization approach

Implementation IssuesImplementation IssuesExamples and Preliminary ResultsExamples and Preliminary ResultsDiscussions and ConclusionsDiscussions and Conclusions

Sources and Statistics of VariabilitySources and Statistics of VariabilityEnvironment variationsEnvironment variations

Power supply voltagePower supply voltageTemperatureTemperatureNoise couplingNoise coupling

Model environmental Model environmental variations by Uniform variations by Uniform distributions and capture distributions and capture their variability by their variability by polyhedronpolyhedronManufacturing variationsManufacturing variations

DeviceDeviceInterconnectInterconnect

Model manufacturing Model manufacturing variations by Normal variations by Normal distributions and capture distributions and capture their variability by their variability by ellipsoidellipsoid

-4-2

02

4

-4-2

0

24

-4

-2

0

2

4

Parameter #1

Correlated Parameter Variations Captured by Ellipsoid

Parameter #2

Par

amet

er #

3

Capturing the process variationCapturing the process variation

Process variation statistics are characterizedProcess variation statistics are characterizedby jointby joint--pdfpdf [Nassif’01][Nassif’01]

Independent Gaussian random variablesIndependent Gaussian random variablesCorrelated Gaussian random variablesCorrelated Gaussian random variables

Multivariate Gaussian Distribution (Multivariate Gaussian Distribution (µµ, , ΣΣ))

EquidensityEquidensity contour is concentric ellipsoidcontour is concentric ellipsoid

Probability has a chiProbability has a chi--square distribution with square distribution with degree degree nn

−Σ−−

Σ= − )()(

21exp

)2(1)( 1

2/12/µµ

πXXXp T

nX

{ } 21 )()(),,( rXXRXrfU Tn ≤−Σ−∈=Σ= − µµµ

( ) )(Prob 22 rFUunχ

=∈

OPERA ConceptOPERA Concept

Robust GP with Ellipsoidal Uncertainty

Stochastic GP with Joint Probability

Process Variations Variance linked to meanVariance not linked to mean

GP Modeling of Analog Circuit +

Capture Process Variations by Ellipsoid(Chi-square distribution)

Specification on Yield

Robust Design with Guaranteed YieldRobust Design with Guaranteed Yield

Robust Optimization ApproachRobust Optimization Approach

Robust geometric programmingRobust geometric programming††Robust GP incorporates a model of data uncertainty and Robust GP incorporates a model of data uncertainty and optimizes for optimizes for the worstthe worst--case scenariocase scenario under the modelunder the model

Computation time increases Computation time increases linearlylinearly in number of uncertain parametersin number of uncertain parameters

Design for variability via robust GP:Design for variability via robust GP:Many analog IC design for variability problems can be cast asMany analog IC design for variability problems can be cast asrobust GPsrobust GPs

HandlesHandles correlatedcorrelated statistical variations in both process parameters and statistical variations in both process parameters and design variablesdesign variables

Can carry out robust designs with required yield bound Can carry out robust designs with required yield bound

Results in less overResults in less over--design (compared with cornerdesign (compared with corner--enumeration enumeration optimization)optimization)

† More details can be found in “Tractable Approximate Robust Geometric Programming”, revised for publication, May 2005

ContentsContents

IntroductionIntroductionBackgroundBackgroundProposed ApproachProposed ApproachImplementation IssuesImplementation Issues

VarianceVariance--linkedlinked--toto--mean uncertaintymean uncertaintyVarianceVariance--notnot--linkedlinked--toto--mean uncertaintymean uncertainty

Examples and Preliminary ResultsExamples and Preliminary ResultsDiscussions and ConclusionsDiscussions and Conclusions

Modeling Process VariationsModeling Process Variations

VarianceVariance--linkedlinked--toto--mean variationmean variationRelative variations Relative variations (e.g. (e.g. ∆∆R/R, R/R, ∆∆C/C),C/C),

i.e. i.e. variance is proportional to meanvariance is proportional to meanModel the variations in Model the variations in process parametersprocess parameters byby

VarianceVariance--notnot--linkedlinked--toto--mean variationmean variationAbsolute variations Absolute variations (e.g. (e.g. ∆∆W, W, ∆∆L, L, ∆∆VVthth),),

i.e. i.e. variance is independent of meanvariance is independent of meanModel the variations in bothModel the variations in both design variablesdesign variables andandprocess parameters process parameters asas

n,jxNxq,ipNip jji,1),,0(~,,1),,0(~ 22 KK == σδσδ

q,iNpp iii ,1),2

,0(~ K= σδ

Lognormal approximationLognormal approximationNarrow normal distribution can be Narrow normal distribution can be approximated as lognormal distributionapproximated as lognormal distribution

Most process parameter variation satisfy this Most process parameter variation satisfy this condition (e.g. condition (e.g. toxtox))

µ=4.45nm, σ=0.1nm

4 4 . 1 4 . 2 4 . 3 4 . 4 4 . 5 4 . 6 4 . 7 4 . 8 4 . 9 50

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4F i t t i n g n o rm a l d i s tr i b u t i o n w i th lo g n o rm a l d i s tr i b u t i o n (T o x)

E ffe c t i v e T o x ( n m )

prob

abili

ty d

ensi

ty fu

nctio

n

T o x (4 . 4 5 n m , 0 . 1 n m )F i t t in g e r ro r 1 . 3 %

N o rm a l d is t r i b u t io n(µ , σ )

L o g n o rm a l d is t r i b u t io n ( l o g (µ ) , σ /µ )

ContentsContents

IntroductionIntroductionBackgroundBackgroundProposed ApproachProposed ApproachImplementation IssuesImplementation IssuesExamples and Preliminary ResultsExamples and Preliminary Results

Ring Oscillator ExampleRing Oscillator ExampleLC VoltageLC Voltage--Controlled Oscillator ExampleControlled Oscillator Example

Discussions and ConclusionsDiscussions and Conclusions

Ring Oscillator Design ExampleRing Oscillator Design Example

5GHz Ring Oscillator design example5GHz Ring Oscillator design exampleIBM 7HP 1.8V 0.18IBM 7HP 1.8V 0.18µµm BiCMOS processm BiCMOS processDesign Variables: Design Variables:

WWeffeff, L, , L, ∆∆VVDesign objective and constraints:Design objective and constraints:

Minimize PowerSubject to Phase Noise ≤ PN_max

f _resonant = f0

Optimization results (Optimization results (90%90% yield bound for robust yield bound for robust GP, Freq: 5GP, Freq: 5±±1 GHz):1 GHz):

Robust design achieve better yield with higher Robust design achieve better yield with higher design costdesign cost

Robust Ring Oscillator Design ResultsRobust Ring Oscillator Design Results

0.387V0.387V0.42V0.42V∆∆VV0.240.24µµmm0.260.26µµmmLengthLength6.686.68µµmm4.534.53µµmmWeffWeff

Robust GP Robust GP DesignDesign

GP DesignGP DesignDesign VariablesDesign Variables

≥≥ 90%90%50%50%YieldYield

4.85GHz4.85GHz5GHz5GHzFrequencyFrequency--101dBc/Hz101dBc/Hz--100dBc/Hz100dBc/HzPhase NoisePhase Noise

2.59mW2.59mW1.87mW1.87mWPowerPower

Robust GP Robust GP DesignDesign

GP DesignGP DesignSpecificationsSpecifications

Monte Carlo VerificationMonte Carlo VerificationActual yield is estimated by Monte Carlo analysis Actual yield is estimated by Monte Carlo analysis with 10K sampleswith 10K samples

Optimization without considering process variation Optimization without considering process variation (i.e. nominal design) might have very low yield(i.e. nominal design) might have very low yield((50%50% in this example)in this example)Robust GP design achieved guaranteed yield Robust GP design achieved guaranteed yield ≥≥ 90%90%

55 60 65 70 75 80 85 90 95 1002.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2D esign cost (power consum ption) versus yie ld requirem ent

Yie ld requirement (% )

Pow

er c

onsu

mpt

ion

(mW

)TradeTrade--off curve of off curve of power consumptionpower consumption(design cost ) versus (design cost ) versus yield requirementyield requirement

Design cost increases when yield requirement Design cost increases when yield requirement increasesincreases

Design Cost vs. Yield RequirementDesign Cost vs. Yield Requirement

Very high Very high design cost to design cost to achieve yield achieve yield approaching approaching 100%100%

VoltageVoltage--Controlled OscillatorControlled OscillatorDesign ExampleDesign Example

2.1GHz LC VCO design 2.1GHz LC VCO design exampleexampleHitachi SiGe BiCMOS Hitachi SiGe BiCMOS process using 90GHz process using 90GHz ffTTNPNNPNDifferential VCO is Differential VCO is equivalent to a tank equivalent to a tank modelmodel

VCO Experiment SetupVCO Experiment Setup

Design Variables: Design Variables: I_bias, g_tank, C_tank, L, I_bias, g_tank, C_tank, L, VswVsw

Design objective and constraints:Design objective and constraints:

Design Uncertainty:Design Uncertainty:

),0(~,,23

232

231

223

22

221

213

212

21

∆∆∆

σσσσσσσσσ

NLL

gg

CC

tank

tank

tank

tank

Minimize PowerSubject to Phase Noise ≤ PN_max

Loop Gain ≥ LG_minL_tank*C_tank*ω2 = 1Vsw ≤ VddVsw ≤ I_bias/g_tank

VCO optimization resultsVCO optimization results

Robust GP yield bound:Robust GP yield bound:Yield bound can be set by adjusting the ellipsoid radiusYield bound can be set by adjusting the ellipsoid radius

Corner selection Corner selection vertices of polyhedron vertices of polyhedron Confidence ellipsoid in the robust optimization is Confidence ellipsoid in the robust optimization is inscribed in this polyhedroninscribed in this polyhedron

Optimization results (for 90% yield bound, Freq: Optimization results (for 90% yield bound, Freq: 2.1±0.4GHz)2.1±0.4GHz)

2.5V2.5V2.5V2.5VVswVsw2.82nH2.82nH2.83nH2.83nHLL1.018m1.018mSS0.8940.894mmSSg_tankg_tank1.26pF1.26pF1.33pF1.33pFC_tankC_tank2.72mA2.72mA2.41mA2.41mAI_biasI_bias

Corner Corner -- enumeration enumeration optimizationoptimization

Robust optimizationRobust optimization

LC Oscillator design cost vs. yield LC Oscillator design cost vs. yield bound and actual yield bound and actual yield

Yield is estimated by 10K Monte Carlo analysisYield is estimated by 10K Monte Carlo analysisDesign cost increase when yield requirement increaseDesign cost increase when yield requirement increase20% over design for 20% over design for ±±33σσ actual actual yield in corneryield in corner--based based optimization compared to robust optimizationoptimization compared to robust optimization

60 65 70 75 80 85 90 95 1002

2.5

3

3.5Design cost (Power consumption) versus yield requirement

Yield requirement (%)

Pow

er c

onsu

mpt

ion

(mW

)

Design cost for robust optimizationDesign cost for corner-based optimization

85 90 95 1002

2.5

3

3.5Design cost (Power consumption) versus actual yield

Actual yield by Monte Carlo analysis(%)

Pow

er c

onsu

mpt

ion

(mW

))

Design cost for robust optimizationDesign cost for corner-based optimization

ContentsContents

IntroductionIntroductionBackgroundBackgroundProposed ApproachProposed ApproachImplementation IssuesImplementation IssuesExamples and Preliminary ResultsExamples and Preliminary ResultsDiscussions and ConclusionsDiscussions and Conclusions

Discussions and ConclusionsDiscussions and Conclusions

Ellipsoidal uncertainty captures both independent Ellipsoidal uncertainty captures both independent and and correlatedcorrelated process variationsprocess variationsYield requirement can be explicitly incorporated as Yield requirement can be explicitly incorporated as a design constrainta design constraintRobust optimization using Robust optimization using posynomialposynomial equations equations (requires fewer simulations)(requires fewer simulations)Guaranteed yield bound by assuring all parameters Guaranteed yield bound by assuring all parameters within the ellipsoid instead of sampling the within the ellipsoid instead of sampling the process variationprocess variationHandles both parameter and Handles both parameter and design variabledesign variableuncertaintyuncertaintyAchieve the same yield with much less overAchieve the same yield with much less over--designdesign(compared with corner(compared with corner--enumeration optimization)enumeration optimization)