CMOSOp-ampDesignandOptimizationvia

GeometricProgramming

MarHershenson,StephenBoyd,ThomasLee

ElectricalEngineeringDepartment

StanfordUniversity

UCSB10/24/97

CMOSanalogampli�erdesign

problem:choosetransistordimensions,biascurrents,componentvalues

�criticalpartofmixed-mode(digital-analog)ICs

�fortypicalmixed-modeIC,

{1:10analog:digitalarea

{10:1analog:digitaldesigntime

thistalk:anewmethodforCMOSop-ampdesign,basedongeometric

programming

�globallyoptimalandextremelyfast

�handleswidevarietyofpracticalconstraints&specs

UCSB10/24/97

1

Outline

�Geometricprogramming

�Two-stageop-amp

�MOSmodels

�Constraints&specs

�Designexamples&trade-o�curves

�Extensions

�Conclusions

UCSB10/24/97

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Monomial&

posynomialfunctions

x=(x1 ;:::;xn):vectorofpositivevariables

functiongofform

g(x)=x�1

1

x�2

2

���x�n

n

;

with�i2R,iscalledmonomial

functionfofform

f(x)=

tXk

=

1

ck x�1k

1

x�2k

2

���x�nk

n

;

withck�0,�ik2R,iscalledposynomial

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�posynomialsclosedundersums,products,nonnegativescaling

�monomialsclosedunderproducts,division,nonnegativescaling

�if1=fisposynomialwesayfisinverseposynomial

examples:

�0:1x1 x�0

:5

3

+x1

:5

2

x0

:7

3

isposynomial

�1=(1+x1 x1

:3

2

)isinverse-posynomial

�2x3 px1 =x2

ismonomial(hencealsoposy.&inv-posy.)

UCSB10/24/97

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Geometricprogramming

aspecialformofoptimizationproblem:

minimize

f0 (x)

subjectto

fi (x)�1;

i=1;:::;m

gi (x)=1;

i=1;:::;p

xi>0;

i=1;:::;n

wherefiareposynomialandgiaremonomial

UCSB10/24/97

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moregenerallywithgeometricprogrammingwecan

�minimizeanyposynomialormonomialfunction,or

�maximizeanyinverse-posynomialormonomialfunction

subjecttoanycombinationof

�upperboundsonposynomialormonomialfunctions

�lowerboundsoninverse-posynomialormonomialfunctions

�equalityconstraintsbetweenmonomialfunctions

UCSB10/24/97

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Geometricprogramming:history&

methods

�usedinengineeringsince1967(Du�n,Peterson,Zener)

�usedfordigitalcircuittransistorsizingwithElmoredelaysince1980

(Fishburn&Dunlap'sTILOS)

new(interior-point)methodsforGP(e.g.,Kortaneketal)

�areextremelyfast

�handlemediumandlarge-scaleproblems

(100svbles,1000sconstraintseasilysolvedonPCinminutes)

�either�ndglobaloptimalsolution,orprovideproofofinfeasibility

UCSB10/24/97

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Two-stageop-amp

M1

M2

M3

M4

M5

M6

M7

M8

Ibias

Vdd

Vss

CL

Cc

Rc

Vin+

Vin�

�commonop-amparchitecture

�19designvariables:W1 ;:::;W8 ,L1 ;:::;L8 ,Rc ,Cc ,Ib

ia

s

UCSB10/24/97

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LargesignalMOSmodel

PMOS

NMOS

D

D

G

G

S

S

ID

ID

NMOSsaturationcondition:VD

S

�VG

S

�VT

N

square-lawmodelID

=k1 (W=L)(VG

S

�VT

N

)2

similarcondition&modelforPMOS

(moreaccuratemodelpossible,e.g.,forshortchannel)

UCSB10/24/97

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SmallsignaldynamicMOSmodel

Cgb

Cgs

gm

vgs

go

Cdb

Cgd

Bulk

S

D

G

transconductanceandoutputconductance,

gm

=k2 pIDW=L;

go=k3 ID

aremonomialinW,L,ID

capacitancesareall(approximately)posynomialinW,L,ID

UCSB10/24/97

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Dimensionconstraints

limitsondevicesizes:

Lm

in

�Li�Lm

a

x ;

Wm

in

�Wi�Wm

a

x

(expressasLi =Lm

a

x

�1,etc.)

symmetryconstraints:W1

=W2 ,L1

=L2 ,W3

=W4 ,L3

=L4

biastransistormatching:L5

=L7

=L8

toreducesystematicinputo�setvoltage:

W3 =L3

W6 =L6

=W4 =L4

W6 =L6

=

W5 =L5

2W7 =L7

area=�1 Cc

+�2 PiWi Liisposynomial,hencecanimposeupperlimit

UCSB10/24/97

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Biasconstraints

eachtransistormustremaininsaturationoverspeci�ed

�common-modeinputrange[Vcm

;m

in ;Vcm

;m

a

x ]

�outputvoltageswing[Vo

u

t;m

in ;Vo

u

t;m

a

x ]

leadstofourposynomialinequalities

e.g.,forM5

weget

k4 rI1 L1

W1

+k5 rI5 L5

W1

�Vd

d

�Vcm

;m

a

x

+VT

P

(everydraincurrentismonomialinthedesignvariables)

UCSB10/24/97

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Quiescentpower&

slewratespecs

quiescentpowerisposynomial:

P=(Vd

d

�Vss )(Ib

ia

s

+I5

+I7 )

hencecanimposeupperlimitonpower(orminimizeit)

slewrateis

min �2I1

Cc

;

I7

Cc

+CL �

minslewratespeccanbeexpressedasposynomialinequalities

Cc SRm

in

2I1

�1;

(Cc

+CL

)SRm

in

I7

�1

UCSB10/24/97

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Transferfunction

withstandardvalueRc=1=gm

6 ,TFisaccuratelygivenby

H(s)=

Av

(1+s=p1 )(1+s=p2 )(1+s=p3 )(1+s=p4 )

�open-loopgainismonomial:Av=k6 pW2 W6 =L2 L6 I1 I7

�dominantpolep1

ismonomial:p1

=gm

1 =Av Cc

�parasiticpolesp2 ;p3 ;p4

areinverseposynomial

hencecan�xtheopen-loopgainanddominantpole,andlowerboundthe

parasiticpoles

UCSB10/24/97

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3dBbandwidthandunitygaincrossoverspecs

�bandwidthconstraints:jH(j!)j�afor!�

,

jH(j)j2

=

A2v

(1+2=p21 )(1+2=p22 )(1+2=p23 )(1+2=p24 )�a2

,

(a2=A2v )(1+2=p21 )(1+2=p22 )(1+2=p23 )(1+2=p24 )�1

...aposynomialinequality(sincepiareinv.-pos.)

�unitygaincrossoveris(veryaccurately)monomial:!c

=gm

1 =Cc

�hencecan�x(orupperorlowerbound)crossoverfrequency

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Phasemarginspecs

minphasemarginspecis:

�6H(j!c )=

4

Xi=

1

arctan(!c =pi )���PMm

in

extremelygoodapproximation:

4

Xi=

2

!c =pi��=2�PMm

in

(sincep1

contributes90�,andarctan(x)�xforx�50�)

...aposynomialinequalitysinceparasiticpolesareinverseposynomial

UCSB10/24/97

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Otherspecs

�mincommon-moderejectionratio

�min(pos.&neg.)powersupplyrejectionratios

�maxspotnoiseatanyfrequency

�maxtotalRMSnoiseoveranyfrequencyband

�mingateoverdrive

canallbehandledbygeometricprogramming

UCSB10/24/97

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Summary

usinggeometricprogrammingwecangloballyoptimizeadesign

involvingallthespecsdescribedabove:

�dimensionconstraints,area

�biasconstraints,power,slewrate

�bandwidth,crossoverfrequencies,phasemargin

�CMRR,nPSRR,pPSRR

�spot&totalnoise

typicalproblem:

�approx20vbles,10equality&20inequalityconstraints

�solutiontime�1sec(ine�cientMatlabimplementation!)

UCSB10/24/97

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(Globally)optimaltrade-o�curves

��xallspecsexceptone(e.g.,power)

�optimizeobjective(e.g.,maximizecrossoverfrequency)fordi�erent

valuesofspec

�yieldsgloballyoptimaltrade-o�curvebetweenobjectiveandspec

(withothers�xed)

UCSB10/24/97

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Defaultspecs

ourexampleswillmaximizecrossoverBW

withdefaultspecs

�Vd

d

=5V,Vss

=0V,1:2�mprocess

�Li�0:8�m,Wi�2�m,area�10000�m2

�CMinput�xedatmid-supply;outputrangeis10%{90%ofsupply

�power�5mW

�open-loopgain�80dB,PM�60�

�slewrate�10V=�sec

�CMRR�60dB

�input-referredspotnoise(1kHz)�300nV= pHz

(we'llvaryoneormoretogettrade-o�curves)

UCSB10/24/97

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Maximum BW versus power & supply voltage

0 5 10 150

50

100

150

Max

imum

uni

ty−

gain

ban

dwid

th in

MH

z

Power in mW

Vdd=5V Vdd=3.3VVdd=2.5V

UCSB 10/24/97 21

Minimum noise versus power & BW

0 5 10 15100

150

200

250

300

350

400

Min

imum

noi

se in

nV

/Hz0.

5

Power in mW

wc=30MHz

wc=60MHz

wc=90MHz

UCSB 10/24/97 22

Maximum BW versus power & load capacitance

0 5 10 150

20

40

60

80

100

120

140

160

180

200

Max

imum

uni

ty−

gain

ban

dwid

th in

MH

z

Power in mW

CL=1pFCL=3pFCL=9pF

UCSB 10/24/97 23

Maximum BW versus area & power

0 1000 2000 3000 4000 5000 600020

40

60

80

100

120

140

Max

imum

uni

ty−

gain

ban

wid

th in

MH

z

Area in µm2

Pmax=1mW Pmax=5mW Pmax=10mW

UCSB 10/24/97 24

Extensions

�cansolvelargecoupledproblems

(e.g.,totalarea,powerforICwith100op-amps)

�candorobustdesignthatworkswithseveralprocessconditions

�getsensitivitiesforfree

�methodextendstowidevarietyofampli�erarchitectures,BJTs,etc.

�canusefarbetter(monomial)MOSmodels,e.g.,forshort-channel

designs

UCSB10/24/97

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Conclusions

�usinggeometricprogrammingwecangloballyande�cientlysolve

CMOSop-ampdesignproblems

�allowsdesignertospendmoretimedesigning,i.e.,exploringtrade-o�s

betweencompetingobjectives(power,area,bandwidth,...)

�yieldscompletelyautomatedsynthesisofCMOSop-ampsdirectly

from

speci�cations

�hugereductioninanalogdesigntime

(cf.methodsbasedonsimulatedannealing,expertsystems,general

nonlinearprogramming,...)

UCSB10/24/97

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