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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
2
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
UCSB10/24/97
3
�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
4
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
5
moregenerallywithgeometricprogrammingwecan
�minimizeanyposynomialormonomialfunction,or
�maximizeanyinverse-posynomialormonomialfunction
subjecttoanycombinationof
�upperboundsonposynomialormonomialfunctions
�lowerboundsoninverse-posynomialormonomialfunctions
�equalityconstraintsbetweenmonomialfunctions
UCSB10/24/97
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
UCSB10/24/97
15
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
16
Otherspecs
�mincommon-moderejectionratio
�min(pos.&neg.)powersupplyrejectionratios
�maxspotnoiseatanyfrequency
�maxtotalRMSnoiseoveranyfrequencyband
�mingateoverdrive
canallbehandledbygeometricprogramming
UCSB10/24/97
17
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
18
(Globally)optimaltrade-o�curves
��xallspecsexceptone(e.g.,power)
�optimizeobjective(e.g.,maximizecrossoverfrequency)fordi�erent
valuesofspec
�yieldsgloballyoptimaltrade-o�curvebetweenobjectiveandspec
(withothers�xed)
UCSB10/24/97
19
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
20
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
25
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
26
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