an accurate pll behavioral model for fast

NC

U Electrical Engineering Electronic D

esign Autom

ation Laboratory

Department ofDepartment of ElectricalElectrical Engineering Engineering EElectronic lectronic DDesignesign AAutomation Laboratoryutomation LaboratoryNational Central University

2007/10/12

An Accurate PLL Behavioral Model for Fast An Accurate PLL Behavioral Model for Fast Monte Carlo Analysis under Process VariationMonte Carlo Analysis under Process Variation

Authors : *Chin-Cheng Kuo, Meng-Jung Lee, I-Ching Tsai,Chien-Nan Jimmy Liu, and Ching-Ji Huang

Department of Electrical Engineering National Central University, Taiwan (R.O.C.)

* : speaker

NC


esign Autom

ation Laboratory

Chin-Cheng Kuo2007/10/12 2

OutlineOutline

IntroductionBottom-up Behavioral ModelingModified Sensitivity AnalysisExperimental ResultsConclusions

NC


esign Autom

ation Laboratory


Process Variation EffectsProcess Variation Effects

In deep-submicron technology, impacts of device parameter variation become major factors limiting circuit performance

Process variation aware analysis is necessary in the early design stage (re-design↓)

Monte Carlo Simulation (MCS) is often usedStatistical analysisMuch random data for analysisMany simulation times → time-consuming

NC


esign Autom

ation Laboratory


Traditional Statistical AnalysisTraditional Statistical AnalysisTransistor-level Monte Carlo Simulation (MCS)Based on the statistical models of transistor parameters from IC foundry

High accuracyLong simulation time

HSPICE Monte Carlo

Analysis(circuit performance)

Device variation

NC


esign Autom

ation Laboratory


Hierarchical Statistical AnalysisHierarchical Statistical AnalysisSolve the speed issue of traditional MC analysis Response Surface Methodology (RSM) technique

Regression-based methodNumeric statistical results

HSPICE MCS

complicated

simple

NC


esign Autom

ation Laboratory


Regression Cost of RSMRegression Cost of RSM1st-order RSM :

2nd-order RSM :

Number of training samples is at least 4 times greater than the number of unknown coefficients

Even if a simplified algorithm is used

Example: If k =4 1st RSM : 5 unknown coefficients �20 training samples 2nd RSM : 15 unknown coefficients � 60 training samples

kk XaXaXaaY ++++= ...22110

ref: Xin Li, Jiayong Le, Pileggi, L.T., Strojwas, A., “Projection-based performance modeling for inter/intra-die variations”, ICCAD, 2005.

∑∑∑= ==

++=k

i

k

jjiij

k

iii XXaXaaY

1 110

NC


esign Autom

ation Laboratory


Our TargetsOur Targets

HSPICE MCS

Pure RSM-based

Our *BMCS

Regression cost N/A Worse

Accuracy Better Based on complexity Good

Better

Worse

Good

Good

Good

Simulation time Worse

Observability(waveforms)

Better

Hierarchical Statistical Analysis

*BMCS: Behavioral Monte Carlo Simulation

*SE - like analysis

efficient PLL behavioral model

Developed methods

*SE: Sensitivity Analysis

NC


esign Autom

ation Laboratory


Sensitivity Analysis (SSensitivity Analysis (SEE))Delay under process variation :

Use sensitivity analysis (SE) to reflect the process variation effects in behavioral parameters

Can save considerable regression time for complicated curve fitting

Disadvantage: Traditional sensitivity analysis may have larger error at analog blocks

We propose modified sensitivity analysis for analog circuits, without extra simulation cost

ii

ddidd x

xTTxTT Δ×∂∂

≈−Δ=Δ 0)(Td0 : Nominal delay

: Delay sensitivity to deviceparameter xii

d

xT∂∂

NC


esign Autom

ation Laboratory


Behavioral MCSBehavioral MCS

Behavioral Monte Carlo Simulation (BMCS) Analyze process variation effects at behavioral levelSimulation results include detailed output waveforms and performance shift

The accuracy of the behavioral models is the most critical issue in BMCS-based approaches

Directly affect the statistical resultsIdeal top-down model is not suitableUse bottom-up modeling method to extract actual circuit properties to improve the accuracy

NC


esign Autom

ation Laboratory


Our BMCS FlowOur BMCS Flow

NC


esign Autom

ation Laboratory


OutlineOutline


NC


esign Autom

ation Laboratory


Characterization ModeCharacterization ModeCharge pump PLLOnly one extraction pattern Automatically consider parasitic and loading effectsCan extract all required characteristic parameters from simulation results

NC


esign Autom

ation Laboratory


PFD & FDPFD & FDPhase frequency detector and Frequency dividerCharacteristic parameters:

delay time� transition time and reset time for PFD

typical PFD structure PFD output responses

NC


esign Autom

ation Laboratory


CP & LFCP & LFCharge Pump and Loop FilterCharacteristic parameters:

Source current (IUp) & current mismatch (IUp - IDn)Impedance of LF Equivalent switch on/off time

C1

C2

Extract from Vctrloutput waveform

NC


esign Autom

ation Laboratory


VCOVCOVoltage-controlled oscillatorLinear VCO model

Simpler approachCharacteristic parameters: Vmin , fmin ,Vmax , fmax , KVCO

(V2, f2)

Kvco(V1, f1)

(Vmin, fmin)

(Vmax, fmax)

Vctrl [V]

fout [Hz]

NC


esign Autom

ation Laboratory


SSEE Analysis for Process VariationAnalysis for Process Variation

Find the relationship of behavioral parameters under device parameter variation

Just for timing parameters in our behavioral model

Delay change under width variation :

Extract the sensitivity value ( ):

Extend to be a process variation aware model:

WdT ΔΔ ,

WTE dS Δ_,

constantW

TW

TS WdWd

WTE d=

ΔΔ

≈∂

∂= ΔΔ

Δ,,

_,

oxdtd

dd

TTEoxVTEt

LTEWTEdoxtd

STSV

SLSWTTVLWT

ΔΔ

ΔΔ

×Δ+×Δ+

×Δ+×Δ+=ΔΔΔΔ

_,_,

_,_,0

),,,(

NC


esign Autom

ation Laboratory


OutlineOutline


NC


esign Autom

ation Laboratory


Current Variation in CPCurrent Variation in CPDrain current of MOS in saturation region:

Current variation ratio under Vt variation:

2)(2 tGS

ox

nD VV

LW

TI −=

εμ

Sensitive analysis (SE)

SE may not suitable

[ ]

22

2

2

2 '

11

)()(

)()()(

⎟⎠⎞

⎜⎝⎛ Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

Δ−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

Δ−−=

−Δ+−

≅Δ

=Δ

kV

VVV

VVVVV

VVVVV

IVIVratio

t

tGS

t

tGS

ttGS

tGS

ttGS

D

tDt

Only need to find k

NC


esign Autom

ation Laboratory


Our Method vs. Traditional SOur Method vs. Traditional SEE

Same extraction timeMore accurate: more similar to the HSPICE results

NC


esign Autom

ation Laboratory


Modeling Strategy for VCO (1/3)Modeling Strategy for VCO (1/3)VCO transfer curve under length variationTraditional SE method:

, _ΔΔ

= =ΔE f L L

fS constant

NC


esign Autom

ation Laboratory


Modeling Strategy for VCO (2/3)Modeling Strategy for VCO (2/3)

Consider Vctrl effects:

VCOLL Kslope

ffVV+

−−=

ΔΔ

'min

'1

1min,

VCOLL Kslope

ffVV+−

+=Δ

Δ

'2

'max

2max,

minf

L = 0Δ

outf

ctrlV

1fΔ

1V

maxf

L < 0 Δ

minfΔ'

minf

'maxf

2fΔ

2V 0max,V0min,V

LV Δmax,

LV Δmin,

'1f

'2f

1min

max2

min 1, _ , _

max, _ , _

2

;

;

E L E L

E L E L

f f

f f

L L

L L

f fS S

f fS S

Δ Δ

Δ Δ

Δ Δ= =

Δ ΔΔ Δ

= =Δ Δ

_ _2 1, ,

2 1

( )E f L E f LL

L S S

V Vε Δ Δ

Δ

Δ × −=

−

NC


esign Autom

ation Laboratory


Modeling Strategy for VCO (3/3)Modeling Strategy for VCO (3/3)

W L V Toxtφ φ φ φ φΔ Δ Δ ΔΔ = + + +=>

Our modified SE

_ min,

_ max,

_ min,

min

max

min

, _

,

,

,

,

, [ ]

( )

)

(

( ) -

( , )L

L

L L

E f L ctrl

E f L ctrl E f L ctrl

E f L ctrl

if

if

V t V

L V V t V

S

S S

S V t V ε

Δ

Δ

Δ Δ

Δ

Δ Δ

Δ

≤

Δ ≥=

+ ×

, otherwise

⎧⎪⎪⎨⎪⎪⎩

min

min,

max,

, _

min

max

, _

, _

2

2

2

( )

, ( )

[ ( ) -

( ) ,

( )

( )

ctrl

L

L

E f L

E f L ctrl

L E f L ctrl

L S if V t V

L S if V t V

L S V tt

t

t

π

π

π

φΔ

Δ

Δ

Δ

Δ Δ

Δ ≤

Δ ≥

Δ

×

= ×

× +

( ){ }min, ] , L L otherwiseV dtεΔ Δ×

⎧⎪⎪⎨⎪⎪⎩ ∫

NC


esign Autom

ation Laboratory


Experimental ResultsExperimental ResultsCharge pump PLL with TSMC RF 0.18μm processUse Verilog-A language to describe our PLL modelSimulation environment: Analog Artist (Cadence)

HSPICEOur model

No process variation: Vctrl waveform

NC


esign Autom

ation Laboratory


Process Variation ExperimentsProcess Variation Experiments

HSPICE Monte Carlo

Analysis(output performance)

Device variationBehavioral

Monte Carlo Simulation

(output performance)

WΔ

LΔ

tVΔ

oxTΔ

Comparison100 runs

(Any Distribution)

NC


esign Autom

ation Laboratory


Our BMCS vs. HSPICE MCSOur BMCS vs. HSPICE MCS

*Corr. Coe. : Correlation Coefficient

Corr. Coe. = 0.999

VVlocklock

Corr. Coe. = 0.991

TTsettlesettle

*Perfect match: points on the red line(slope = 1)

NC


esign Autom

ation Laboratory


100 Runs MC Simulation Results100 Runs MC Simulation Results

Standard Deviation (St. Dev.)

1st RSM+ BMCS

Trad. SE+ BMCS

Modified SE+ BMCS HSPICE

Mean 0.993 -0.1% -0.1% 0.1%32.4% 2.9%

1.9%-0.8%-6.1%0.7%-1.8%

2.0%-6.1%-6.1%63.6%-3.5%

5.9%2.2%-0.5%-7.6%-2.9%-2.4%

0.0363.4490.57312.21.36

Worst 16.6 16.4 16.7 17.0�34.27

2.43

0.993 0.995 0.994St. Dev. 0.045 0.035 0.034

Mean 3.441 3.438 3.374St. Dev. 0.541 0.572 0.576

Mean 12.4 12.4 13.2St. Dev. 2.29 1.41 1.40

Extraction time (hrs) 8.57 8.57 N/ASimulation time (hrs) 2.51 2.64 598.54

Jitterpk-pk(ps)

Vlock(V)

Tsettle(μs)

Use same PLL behavioral model (ours)

NC


esign Autom

ation Laboratory


ConclusionsConclusions

Use accurate behavioral model to perform fast MC analysis for process variation (BMCS)

Our modified sensitivity analysis saves considerable regression cost for complicated curve fitting

Handle any distribution of device parameter variation

Include detailed output waveforms and behavior shift, not some statistical numbers only (well observability)

Reduce the simulation time for MC analysis from several days to several hours

Retain high accuracy

NC


esign Autom

ation Laboratory


Thanks for your attention !!!

an accurate pll behavioral model for fast

Documents