1
Con ex -dependen ans o ma ion o Pa e o-
op imal pe o mance on s o ope a ional
ampli ie s
Elisenda Roca, Manuel Velasco-Jiménez, Ra ael Cas o-López and F ancisco V. Fe nández
Ins i u e o Mic oelec onics o Se ille, IMSE-CNM, CSIC and Uni e si y o Se ille.
A da. Ame ico Vespucio s/n, Isla de la Ca uja, 41092 Se ille, Spain.
Phone: +34 954 46 66 66
FAX: +34 954 46 66 00
Co esponding Au ho :
Elisenda Roca, [email protected]
Abs ac :
The use o Pa e o-op imal pe o mance on s in eme ging design me hodologies o analog
in eg a ed ci cui s is a keys one o o e come he limi a ions o adi ional design me hodologies.
Howe e , mos echniques o gene a e he on s epo ed so a neglec he e ec ha he
su ounding ci cui y (such as he load impedance) has on he Pa e o- on , he eby making i only
ealis ic o he con ex whe e he on was gene a ed. This s ongly limi s he use o he Pa e o
on because o he s ong dependence be ween he key pe o mances o an analog ci cui and i s
su ounding ci cui y, bu , mo e impo an ly, because his ci cui y emains unknown un il he
Pa e o-op imal on is being used. Since pe o mance on gene a ion is a cos ly p ocess, his
pape p oposes ha pe o mance on s o a new con ex o use o a gi en ci cui can be ob ained
om on s ha we e p e iously gene a ed unde some di e en condi ions. Towa ds his goal, a
ans o ma ion me hodology o pe o mance objec i es o ope a ional ampli ie s has been
de eloped. Expe imen al esul s o a olded-cascode and a Mille -compensa ed ope a ional
ampli ie s show ha his is a p omising app oach o euse he on s in mul iple con ex s.
Keywo ds: analog in eg a ed ci cui s, design au oma ion, ci cui sizing, simula ion-based
op imiza ion, mul i-objec i e op imiza ion
2
1 In oduc ion
Elec onic design au oma ion o analog in eg a ed ci cui s s ill lags behind i s digi al coun e pa .
Beyond any single eason, he inhe en complexi y o designing he simples o he analog sys ems
( he many non-ideal e ec s, he la ge sensi i i y o noise, e c.) has hinde ed he e olu ion o
sys ema ic design me hodologies.
Key asks in any analog design au oma ion low a e opology selec ion and opology sizing [1],[2].
Gi en a se o design speci ica ions, he i s ask e e s o he selec ion o an app op ia e opology,
among a se o unc ionally equi alen ones, ha can mee he speci ica ions. Once a opology has
been selec ed, gi en as an in e connec ion o sub-blocks, he second ask e e s o he
de e mina ion o sub-block speci ica ions o which he highe -le el speci ica ions can be
achie ed. A he ansis o le el, opology sizing educes o se ing he de ice sizes and bias
ol ages and cu en s.
In he pas couple o decades mos app oaches o analog sizing ha e been buil a ound
op imiza ion-based app oaches: some i e a i e op imiza ion algo i hm wi h a pe o mance
e alua o in he loop [3]-[9]. The sizing p oblem has usually been o mula ed as:
min ( )
subjec o () 0
L
H

x x
XxX
gx (1)
In his equa ion, he ec o () x is he se o objec i e unc ions o be minimized. Vec o
xco esponds o he design a iables, and L
X
and
H
X
a e hei lowe and uppe bounds,
espec i ely. The ec o () 0gx co esponds o he design cons ain s and delimi s he easible
pe o mance egion.
I ec o () x con ains only one objec i e, he design p oblem can be sol ed by using any single-
objec i e op imiza ion algo i hm, e.g., simula ed annealing [10], gene ic algo i hms [11],
di e en ial e olu ion [12], e c. I ec o () xcon ains mo e han one objec i e, he e a e usually
mul iple solu ions o (1), each one co esponding o a di e en se o alues o he objec i es. In
he majo i y o epo ed app oaches, he sizing p oblem has been sol ed by applying a single-
objec i e op imiza ion algo i hm. Fo he case o se e al objec i e unc ions, his en ails he
ans o ma ion o he mul i-objec i e op imiza ion p oblem in o a single-objec i e op imiza ion
p oblem, usually by applying a weigh ed sum o he objec i e unc ions. This equi es selec ing a
p io i one solu ion among he mul iple ones men ioned abo e.
Mo e ecen ly, a wa e o inno a ion has a isen wi h he de elopmen and applica ion o mul i-
objec i e op imiza ion algo i hms o his kind o p oblems. Se e al algo i hms ha e been epo ed
(e.g., NSGAII [13], SPEA2 [14], MOEA/D [15], e c.) and success ully applied o design
au oma ion p oblems [16]-[22]. The ou come o he applica ion o hese algo i hms o p oblems
wi h se e al mu ually con lic ing objec i es is a Pa e o-op imal pe o mance on , which shows
he bes ade-o s among he objec i e unc ions. In a Pa e o-op imal on , one pe o mance
objec i e canno be imp o ed wi hou wo sening a leas one o he o he objec i es. Pa e o on s
allow, o ins ance, he selec ion o a design solu ion a pos e io i, once he comple e se o
solu ions has been gene a ed. Besides, new possibili ies a ise. Fo ins ance, hey can be used as a
esou ce o opology selec ion. Since he bes ade-o s among solu ions a e a ailable, i can be
immedia ely known i a gi en opology can mee a gi en se o speci ica ions. Pe o mance on s
ha e been e en p oposed as a basic ing edien o hie a chical mul i-objec i e bo om-up syn hesis
me hodologies [21],[22].
3
Mos popula algo i hms o pe o mance on gene a ion a e mul i-objec i e e olu iona y
algo i hms [23],[24], whe e a popula ion o candida e solu ions ( ypically in he o de o some
hund eds o a ew housands) e ol es along a numbe o gene a ions ( ypically in he o de o
some hund eds). The e o e, he op imiza ion algo i hm equi es a numbe o execu ions o he
pe o mance e alua o in he o de o housands o some hund ed housands. High accu acy is
usually wished and, hence, he pe o mance e alua o is ela i ely cos ly: elec ical simula ion
(e.g., HSPICE [25]) o cell-le el sizing o de ailed beha io al simula ion a highe le els [26]. I
is easy o conclude ha he gene a ion o a pe o mance on is a cos ly p ocess ha may ake
se e al hou s o compu a ion ime. The eason such a high compu a ional ime may be conside ed
accep able is ha , unlike single-objec i e design p oblems, he ou come o mul i-objec i e
op imiza ion algo i hms can be used o a wide ange o p oblems. Howe e , in his easoning a
undamen al issue has been igno ed in he li e a u e. The objec i e unc ions canno be selec ed
a bi a ily. They mus co espond o he ypical pe o mances ha he designe is in e es ed in, and
he ul ima e alue o many commonly used pe o mance cha ac e is ics o analog ci cui s do no
only depend on he block i sel , bu , also, on i s su ounding ci cui y, i.e., he gene a ed Pa e o-
on depends on he con ex whe e he analog building block is being used.
To add ess his con ex dependency o he on s, a ans o ma ion p ocedu e is p oposed in his
pape 1. The pape is s uc u ed as ollows. Basic concep s o mul i-objec i e op imiza ion and
gene a ion o Pa e o-op imal pe o mance on s a e in oduced in Sec ion 2. Sec ion 3 de ails he
p ac ical p oblems ha mo i a e his wo k and Sec ion 4 p oposes a ans o ma ion p ocedu e ha
sol es he p e ious p oblems. Sec ion 5 illus a es his me hodology wi h wo examples, a olded-
cascode ope a ional ampli ie and a Mille ope a ional ampli ie , and di e en numbe o
objec i es. Finally, conclusions a e p esen ed in Sec ion 6.
2 Gene a ion o Pa e o-op imal pe o mance
on s
As s a ed in equa ion (1), he sizing p oblem is o mula ed as he maximiza ion o minimiza ion o
a se o n design objec i es,


() (), (), , ()
12 n

xxx x
, whe e x is he ec o o design
a iables (e.g., de ice sizes, pe o mance cha ac e is ics o sub-blocks a in e media e hie a chical
le els), and each ()
i
xis a pe o mance cha ac e is ic o he block (such as dc gain), subjec o
some cons ain s (e.g., slew a e la ge han a ce ain alue o an ope a ional ampli ie ).
In he single-objec i e case, design candida es, each one ep esen ed by a ec o o design
a iables x, can be easily o de ed by conside ing he objec i e unc ion alue () x. Howe e , in
he mul i-objec i e case, a di e en o de ing me hod is equi ed: Pa e o dominance. A design
poin , aX, is said o domina e ano he design poin ,

bX, (no ed as ab) i () ()

a b
and ()< ()
ii
ab o a leas one unc ion i 2. The design poin
a
is said o be non-domina ed i
he e is no o he design poin ha domina es i . The non-domina ed se o he en i e easible sea ch
space is known as he Pa e o-op imal on . All hese concep s a e illus a ed in Figu e 1 o a 2-
dimensional objec i e space.
1 A p elimina y e sion o his pape appea ed a he XI In . Wo kshop on Symbolic and
Nume ical Me hods, Modeling and Applica ions o Ci cui Design [27].
2 This o mula ion is alid o minimiza ion p oblems. A simple change o sign applies o
maximiza ion.
4
The compu a ion o an app oxima ion o he ideal Pa e o-op imal on is ypically pe o med by
using mul i-objec i e e olu iona y algo i hms [23]. O de ing o solu ions acco ding o he Pa e o
dominance concep equi es a pe o mance e alua o , which assesses objec i e and cons ain
alues o each design solu ion, ypically named indi idual in he li e a u e on e olu iona y
algo i hms. These mul i-objec i e e olu iona y algo i hms ypically s a wi h a andom popula ion
o indi iduals ha , a e being e alua ed by he pe o mance e alua o , is modi ied in such a way
ha a e n i e a ions (called gene a ions) a popula ion o non-domina ed indi iduals is ob ained:
he Pa e o-op imal on . Being o s ochas ic na u e, he compu a ional cos due o he high numbe
o equi ed pe o mance e alua ions is he main d awback o hese algo i hms. E iciency,
con e gence o he ideal Pa e o-op imal on , and di e si y o solu ions a e a eas o in ense and
cu en esea ch. An illus a i e example o ecen esea ch in hese a eas is he de ini ion o
quali y me ics ha , p ope ly acked, can con ol when he op imiza ion algo i hm should be
s opped [28].
In o de o illus a e he po en ial o Pa e o-op imal on s, le us conside he olded-cascode
ope a ional ampli ie o Figu e 2, whe e a capaci i e load o 1pF has been conside ed a he
ou pu . The ci cui is op imized o ob ain he maximum alues o wo con lic ing objec i es: he dc
gain, 0
A
, and he uni y-gain equency, u
. The design a iables a e he wid h and leng h o he
ansis o s, as well as he bias cu en s. Di e en cons ain s a e imposed in o de o ob ain co ec
and use ul sized ci cui s; o example, dc gain is se o be la ge han 20dB, phase ma gin is se o
lie be ween 10º and 90º, and ansis o s a e en o ced o ope a e in he sa u a ion egion. The
Pa e o-op imal on was gene a ed by coupling he elec ical simula o HSPICE o he mul i-
objec i e e olu iona y op imiza ion algo i hm NSGAII. The popula ion size and numbe o
gene a ions we e se a 1500 and 100, espec i ely. The gene a ion o his Pa e o-op imal
pe o mance on ook 1 hou o CPU ime on a 2.2-GHz p ocesso . The esul is a 1500-poin s
app oxima ion o he 2-dimensional Pa e o on , which can be seen as black poin s in Figu e 3.
Each poin ep esen s a sized ci cui showing a bes ade-o among he wo pe o mances being
conside ed. To illus a e he e olu ion o he popula ion o design solu ions owa ds he Pa e o-
op imal on , Figu e 3 shows he popula ion a e 8 gene a ions ( ed poin s), 25 gene a ions (blue
poin s) and 100 gene a ions (black poin s).
Figu e 1. Illus a ing Pa e o dominance and Pa e o-op imal on concep s o a 2-
dimensional pe o mance space.
5
3 Po en ial p oblems in using pe o mance on s
As discussed in p e ious sec ions, he gene a ion o Pa e o-op imal pe o mance on s may be a
compu a ionally expensi e p ocess. Ye , hei g ea po en ial bene i is ha mul iple pe o mance
ade-o s a e a ailable. Ei he i hey a e used in e ac i ely o wi hin some design au oma ion
low, once hey a e gene a ed hey can be eused many imes by selec ing he design solu ion wi h
he app op ia e pe o mance ade-o in each case. Howe e , he e is a undamen al law in his
easoning: some ci cui pe o mances do no depend only on he ci cui i sel bu also on he o he
ci cui y o which i is connec ed. Conside , o ins ance, he load impedance ha a ypical analog
block such as he Mille -compensa ed ope a ional ampli ie in Figu e 4 has o d i e. Al hough,
M2
M1
M
0
Vip
Mbn
Ibn
Vdd
Vin
M3M4
M5M6
M8
M7
M10
M9
Mbp
Ibp
Vss
Vdd
Vdd
Vss
Vss
Vss
Vcp Vcn Vou
ZL
Figu e 2. Folded-cascode ampli ie .
Figu e 3. Illus a ing he Pa e o-op imal on gene a ion o he olded-cascode ampli ie o
Figu e 2 whe e dc gain and uni y gain equency ha e been maximized.

6
usually, a highe numbe o pe o mance objec i es has o be conside ed, o illus a ion’s sake we
will conside only wo: dc gain and uni y-gain equency. The plo wi h black ci cles in Figu e 5
shows he pe o mance on o he Mille opamp exhibi ing he dc gain s. uni y-gain equency
ade-o o a phase ma gin cons ain PM>60º and a capaci i e load o 1pF. I he load
impedance is changed o he pa allel connec ion o a 2-pF capaci ance and a 10-k esis ance, he
pe o mance on ob ained is he one shown in he same igu e wi h g ey ci cles. The e o e, i
becomes clea ha he in o ma ion ha he pe o mance on o e s abou he ade-o s ha a
opology can achie e, depends on he con ex whe e he ci cui is going o be used. The
conclusions d awn om he pe o mance on ob ained o some gi en condi ions canno be hus
di ec ly ex apola ed o some o he condi ions.
M4
M3
M2
M1
M6
Cc
M7
Vss
Vdd
M5
Vin
Mbn
Ib
Vdd
Vip
Vo
ZL
Figu e 4 Mille wo-s age ope a ional ampli ie .
Figu e 5. Dc gain s. uni y-gain equency ade-o o he Mille ope a ional ampli ie o a load
impedance equal o 1pF (black ci cles) and he pa allel connec ion o a 2-pF capaci ance and a
10-k esis ance.
7
As discussed in Sec ion 1, Pa e o-op imal pe o mance on s o di e en opologies can be used
o selec he mos app op ia e a chi ec u e o some gi en speci ica ions. Fo ins ance, Figu e 6
shows he bes dc gain s uni y-gain equency ade-o ha can be ob ained o he Mille -
compensa ed ope a ional ampli ie ( ed do s) in Figu e 4 and he olded-cascode ope a ional
ampli ie (blue do s) in Figu e 2, espec i ely, when bo h a e loaded wi h a 1-pF capaci ance.
Acco ding o hese pe o mance on s, and assuming ha only hese wo pe o mances a e o
in e es , he e is a ange o pe o mance alues o which a Mille ope a ional ampli ie would be
p e e able. A olded-cascode opamp, on he o he hand, would be he p e e ed op ion o o he
pe o mance alues. Assume now ha he ampli ie load impedance is gi en by he pa allel
connec ion o a 2-pF capaci ance and a 50-k esis ance. In his case, he Pa e o-op imal
pe o mance on o he Mille ampli ie is plo ed wi h g een do s and ha o he olded-cascode
ampli ie is plo ed in magen a do s. In his second case, all poin s o he olded-cascode ampli ie
on a e domina ed by poin s o he Mille ampli ie on , hence, he Mille ampli ie would be
he p e e ed op ion wha e e he equi ed pe o mances a e. The e o e, hese expe imen s
exempli y ha knowledge o he pe o mance on s a he igh load condi ions is also essen ial
o p ope selec ion o ci cui opology.
4 Me hodology o con ex -dependen
ans o ma ion o Pa e o-op imal pe o mance
on s o ope a ional ampli ie s
Some o he pe o mances usually conside ed in an op imiza ion p ocess o an ope a ional
ampli ie depend on he load condi ions, o ins ance dc gain 0
A
, uni y-gain equency u
, o phase
ma gin PM. Howe e , he load condi ions a e no known un il he design p ocess is being
pe o med, ha is, un il any o he ci cui y a ound he ampli ie is known. Since he Pa e o on
gene a ion p ocess is a leng hy p ocess, he objec i e o ou esea ch is o gene a e ade-o
Figu e 6. Dc gain s. uni y-gain equency ade-o o he Mille and olded-cascode ope a ional
ampli ie s o a load impedance equal o 1pF and he pa allel connec ion o a 2-
p
F capaci ance and
a 10-k esis ance.
8
in o ma ion a p io i and easily and e icien ly ans o m his in o ma ion in o he Pa e o on s o
he pe o mances when he load condi ions a e known.
The ope a ional ampli ie can be iewed as a wo-po , like he one shown in Figu e 7. Vol age 1
and cu en i1 ep esen he di e en ial inpu ol age and cu en , espec i ely. Vol age 2 and
cu en i2 ep esen he ou pu ol age and cu en , espec i ely. A p io i, any wo-po ma ix
cha ac e iza ion o he ampli ie can be conside ed. Howe e , he wo-po ma ix pa ame e s mus
be selec ed in elligen ly, so ha a load-independen cha ac e iza ion o he ope a ional ampli ie
can be ob ained mo e easily, acco ding o he pe o mances o in e es o he block. Le us
conside he hyb id-2 pa ame e s [29] o cha ac e ize he wo-po :
1111122
2211222
ih hi
h hi



 



 
(2)
In his equa ion, pa ame e 11
h ep esen s he inpu impedance, 12
h

is he in e se cu en gain, 21
h

ep esen s he ol age gain o he ampli ie wi hou any load and 22
h

is he ou pu impedance. The
ol age gain and ou pu impedance when a ce ain load
L
Z
is added can be ob ained om (2) and
he cons i u i e equa ion:
22L
Zi

 (3)
yielding he ollowing exp essions:
21
22
22
()
() ()
1()
() ()
L
o
hs
As hs
Z
s
Z
shs





(4)
Equa ion (4) allows o ob ain he hyb id-2 pa ame e s 21
h

and 22
h

om he ol age gain, ()
A
s,
and ou pu impedance, ()
o
Z
s, o some known load condi ions, and ice e sa, o ob ain he
ol age gain ()
A
sand ou pu impedance ()
o
Z
s o some load om he hyb id-2 pa ame e s 21
h

and 22
h. Mo eo e , a pa icula case is when L
Z. In his case, 21
h

is iden ical o ()
A
s. This
equa ion is he key o de eloping he me hodology ha is p oposed in his pape o ans o m a
Pa e o on o some known load condi ions o a bi a y new load condi ions.
Le us assume ha we wish o gene a e he Pa e o-op imal pe o mance on o he ou pu
impedance, dc ol age gain, uni y-gain equency, and phase ma gin o some a bi a y load
condi ions. The gene a ion me hodology would p oceed as ollows:
1) Gene a e he Pa e o-op imal on o he pe o mances o in e es o some known load
condi ions by using a mul i-objec i e op imiza ion algo i hm wi h a nes ed elec ical simula o as
pe o mance e alua o . The pe o mances abo e (ou pu impedance, gain, uni y-gain equency
1
i
2
i
2
1




L
Z
Figu e 7. Two-po wi h a bi a y load.
9
and phase ma gin) can be easily ob ained by p ocessing he ou pu o he AC ci cui analysis
pe o med by he elec ical simula o .
2) Fo each sample o indi idual o his Pa e o on , s o e pole and ze o loca ions o he ou pu
impedance ()
o
Z
s and he ol age gain ()
A
s, bo h being equency dependen unc ions, as well
as hei dc alues. This in o ma ion can be e ie ed om common elec ical simula o s.
3) Use (4) o ex ac he hyb id pa ame e s 21()hs

and 22 ()hs

o each sample; his means
ex ac ing dc alues and bo h, poles and ze os. No e ha , om basic ci cui heo y, he poles o
21()hs
 and 22 ()hs

a e iden ical.
4) Apply (4) o ob ain he ol age gain ()
A
s o he new a bi a y load condi ions (new ()
L
Z
s)
om he p e iously calcula ed hyb id pa ame e s.
5) Ob ain he pe o mance pa ame e s (dc gain, uni y-gain equency and phase ma gin) by simple
p ocessing o he magni ude and phase o he ne wo k unc ion ()
A
s.
No e ha his p ocedu e can be applied o any ini ial known load condi ions.
The i s s ep o his me hodology has he hea ies compu a ional e o by a , bu no e ha he
esul s o s ep 3 a e independen o he applica ion, i.e., independen o he inal load condi ions.
The e o e, he i s h ee s eps can be pe o med be o ehand, as only he ci cui opology and he
pe o mances o in e es a e equi ed. The esul s can be s o ed and used whene e and whe e e
necessa y.
5 Expe imen al Resul s
5.1 Folded-cascode ope a ional ampli ie
The p oposed me hodology will be i s applied o a single-s age ope a ional ampli ie , he olded-
cascode ampli ie in Figu e 2. In his case, he op imiza ion p ocess will be ca ied ou o h ee
di e en pe o mances: dc gain, 0
A
, uni y-gain equency, u
, and phase ma gin, PM. Acco ding o
s ep 1 o he me hodology in Sec ion 4, a Pa e o-op imal on wi h a capaci i e load o 1pF was
i s gene a ed o he h ee objec i es. The design a iables in his case a e he wid h and leng h o
he ansis o s, as well as he bias cu en s. Se e al cons ain s a e imposed in o de o ob ain alid
ci cui s, while a he same ime explo e all he easibili y ange o he pe o mances ( o example,
dc gain is se o be la ge han 20dB, and phase ma gin 90º >PM >10º). The Pa e o on was
gene a ed by using he elec ical simula o HSPICE as pe o mance e alua o wi hin he mul i-
objec i e e olu iona y op imiza ion algo i hm NSGAII 3. The popula ion size and he numbe o
gene a ions we e 1500 and 100, espec i ely. The gene a ion o his Pa e o on ook 1 hou o
CPU ime on a 2.2 GHz p ocesso . The esul is 1500 sample poin s o he 3-dimensional Pa e o
on , which can be seen in Figu e 8 (2 dimensional p ojec ions o his on a e included in Figu e
3 Al hough all expe imen al esul s in his pape ha e been ob ained by applying he mul i-
objec i e op imiza ion algo i hm NSGAII, i should be highligh ed ha he ans o ma ion
p ocedu e in oduced in Sec ion 4 is comple ely independen o he op imiza ion algo i hm used o
ob ain he ini ial Pa e o-op imal on .
16
Fu u e wo k will add ess he gene a ion o pe o mance on s o se e al ini ial load condi ions o
imp o e he di e si y o he ans o med pe o mance on s.
7 Acknowledgemen s
This esea ch wo k has been suppo ed by he TIC-2532 P ojec , unded by Conseje ía de
Inno ación, Ciencia y Emp esa, Jun a de Andalucía, and he TEC2010-14825 P ojec , unded by
he Spanish Minis y o Science and Inno a ion and ERDF.
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