Load-independen cha ac e iza ion o ade-o on s
o ope a ional ampli ie s
Abs ac —In eme ging design me hodologies o analog
in eg a ed ci cui s, he use o pe o mance ade-o on s, also
known as Pa e o on s, is a keys one o o e come he limi a ions
o he adi ional op-down me hodologies. Howe e , mos
echniques epo ed so a o gene a e he on neglec he e ec
o he su ounding ci cui y (such as he ou pu load impedance)
on he Pa e o- on , he eby making i only alid o he con ex
whe e he on was gene a ed. This s ongly limi s i s use in
hie a chical analog syn hesis because o he hea y dependence o
key pe o mances on he su ounding ci cui y, bu , mo e
impo an ly, because his ci cui y emains unknown un il he
syn hesis p ocess. We will add ess his p oblem by p oposing a
new echnique o gene a e he ade-o on s ha is independen
o he load ha he ci cui has o d i e. This idea is exploi ed o
a commonly used ci cui , he ope a ional ampli ie , and
expe imen al esul s show ha his is a p omising app oach o
sol e he issue.
Keywo ds - Analog syn hesis; Mul i-objec i e op imiza ion;
Pa e o-op imal on s; Hie a chical syn hesis.
I. INTRODUCTION
Analog in eg a ed ci cui s s ill lag behind in compa ison o
hei digi al coun e pa in e ms o Elec onic Design
Au oma ion. 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 same pace o e olu ion in sys ema ic design
me hodologies.
A ho opic in his sense is he sys ema iza ion o
hie a chical design o analog ci cui s. This p ocess begins wi h
he sys em equi emen s and ends a he de ice le el, wi h he
speci ica ion o ansis o sizes, alues o all passi e de ices,
and so on. The ac s ha mos building blocks in any analog
hie a chy ea u e a mul i-dimensional space o pe o mance
cha ac e is ics and ha he mapping be ween design objec i es
(o pe o mances) and design a iables (like he W and L o a
ansis o ) is an in ol ed p oblem, make analog hie a chical
syn hesis a e y complex p oblem.
T adi ionally, his p oblem has been add essed by using a op-
down design app oach [1], whe e he sys em is hie a chically
decomposed in di e en sub-sys em building blocks, down o
he de ice le el. A each hie a chical le el, an app op ia e
a chi ec u e is selec ed o each block and i s speci ica ions a e
ansmi ed in o a sub-se o speci ica ions o each o he sub-
blocks. The op-down speci ica ion ansmission p ocess ends
up when he de ice le el is eached, i.e., speci ica ion
ansmission a ha le el implies ob aining de ice sizes.
Howe e , his app oach has wo impo an laws: i s , i
does no gua an ee he easibili y o he building blocks (as
hei equi emen s a e being de i ed in he speci ica ion
ansmission p ocess) since i is unknown i hese equi emen s
a e ealizable o no a lowe hie a chical le els; second, he e
a e no accu a e es ima es o powe consump ion and a ea
occupa ion a he beginning o he speci ica ion ansmission
p ocess (a any in e media e hie a chical le el) since hese wo
igu es depend also on low-le el de ails, no known a his
ea ly s age o he design.
In he ecen ly p oposed mul i-objec i e bo om-up
(MOBU) app oach [2], he hie a chy is handled in a bo om-
up- i s way by means o he concep o Pa e o-op imal on
(POF), a p omising esou ce o pallia e he d awbacks o
adi ional op-down design me hodologies [3][4][5]. A POF is
he se o di e en ins ances o designs (e.g., di e en sizing)
o a ci cui block ha bes cha ac e izes he ade-o s be ween
compe ing pe o mances, like powe s. speed. Gene a ing he
POF is a mul i-objec i e op imiza ion p oblem, ypically
sol ed by a popula ion-based op imiza ion algo i hm, coupled
o a pe o mance e alua o (such as an elec ical simula o ).
POF gene a ion ypically in ol es many housands o
simula ions and, hence, qui e long compu a ion imes. The
ul ima e po en ial o he POF concep is ha , once gene a ed, i
could be used whe e e and whene e necessa y, ha is, in any
syn hesis p oblem in ol ing such building blocks. No ice ha
he on es ablishes ully de ined, bi-uni ocal ela ions
be ween pe o mances and de ice sizes. In his way, i is in
p inciple possible o hie a chically compose all he building
blocks’s ade-o on s o ob ain he comple e ade-o on
o an analog sys em. Then, mapping he sys ems equi emen s
becomes a apid, s aigh o wa d p ocess, whe e, bo h
easibili y and accu a e es ima es a e gua an eed.
Howe e , he e s ill lies a undamen al issue: ha 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 on i s su ounding ci cui y. Tha is, he
.
gene a ed Pa e o- on depends on he con ex whe e he analog
building block is being used. Conside , o ins ance, he load
impedance ha a ypical analog block such as he ope a ional
ampli ie has o d i e. Fo ins ance, i he ope a ional ampli ie
is equi ed o ha e a dc gain o 50dB when a 5kΩ-load is a i s
ou pu and he Pa e o on was ob ained wi h a 100kΩ-load,
hen he selec ed designs may u n useless, possibly because
hei ou pu impedances a e much la ge han 5kΩ. This same
si ua ion a ises in hie a chical syn hesis, because, as said
abo e, he su ounding ci cui y o he building block is
unknown (and so is he load impedance). I is essen ial o s ess
he impo ance o his ac , because he use o POFs o sol e
he issues o op-down design and imp o e he sys ema ic
design o analog ci cui s is hea ily comp omised by his
limi a ion.
In he applica ion o he MOBU me hodology o a eal-li e
desi
on s o
ana
imal on s and he
com
II. GENERATION OF PARETO-OPTIMAL FRONTS
o a
ci c
gn p oblem, a ΔΣ A/D con e e [6], he p oblems wi h he
su ounding ci cui y ha e been ci cum en ed by selec ing
only hose designs om he Pa e o on s o he di e en
building blocks ha mee ce ain cons ain s: o ins ance, ha
he ou pu impedance o he DAC block is highe han he
ou pu esis ance o he co esponding in eg a o , o ha he
i s non-dominan pole o he ou pu impedance o he DAC is
signi ican ly highe han he in eg a o gain-bandwid h p oduc .
The p oblem is ha his is an ad-hoc solu ion, ha o e ly
cons ains he design p oblem and limi s he e ec i eness o
POF-based syn hesis as only a small ac ion o designs o he
POF a e e en ually used (imposing he abo e desc ibed
cons ain s leads o an impo an educ ion in he numbe o
alid solu ions o he con e e , o which he e ini ially we e
o e a hund ed alid designs o each building block).
Ou p oposed app oach is o gene a e Pa e o
log ci cui s ha ci cum en he dependence wi h he
su ounding ci cui y. In his pape , we de elop his app oach
o ope a ional ampli ie s, o which a me hodology o
ans o m he POF o small-signal cha ac e is ics among
a bi a y load condi ions is in oduced
Sec ion II desc ibes Pa e o-op
pu a ional echniques o ob aining hem. Sec ion III
in oduces he gene a ion o POFs independen ly o he
su ounding ci cui y. Sec ion IV is de o ed o discuss di e en
s a egies o he applica ion o hese POFs, mainly o
hie a chical syn hesis p oblems. Finally, conclusions a e
p esen ed in Sec ion V.
Gene a ion o he POF o he selec ed pe o mances
ui block can be posed as a mul i-objec i e op imiza ion
p oblem. This p oblem is o mula ed by maximizing o
minimizing, simul aneously, a se o b design objec i es,
{
}
() (), (), , ()
12 b
=L
y
=
xxx x
, whe e x is a ec o o
e o mance cha ac e is ics
o sub-blocks a in e media e hie a chical le els), and each
i(x)is a pe o mance cha ac e is ic o he block (such as dc
subjec o some cons ain s (e.g., slew a e la ge han a
ce ain alue). The p oblem can be ma hema ically posed by
min ( )
x
x
design a iables (e.g. de ice sizes, p
gain),
ollowing he nex o mula ion:
(1)
() 0
subjec o <<
≥
⎧
⎨
⎩LH
XxX
gx
whe e L
X
and
H
X
a e he lowe and uppe bounds
ec o , es co esponds o he use -
de i ons s, delimi ble egion. A design
o he x
p
ned c
ec i ely. Vec o
ain g(x)≥0
ing he easi
poin ,
∈
aX, is d o domina e ano he design poin , sai
∈
bX,
(no ed as pab) i () ()≤
F
aand ()< ()
ii
ab o a
leas one unc ion i
Fb
1. The design poin ais said o be non-
domin he e is no o he design ha domina es e
non-domina known
as he Pa e o-op imal on . All hese concep s a e illus a ed in
Figu e 1 o a wo pe o mance on .
The compu a ion o he Pa e o on is ypically e icien ly
and accu a ely done by using mul i-objec i e e olu iona y
s [7], coupled o an ele
a ed i i . Th
ed se o he en i e easible sea ch space is
c ical simula o (e.g.,
HS
algo i hm
PICE). These algo i hms s a wi h a andom popula ion o
indi iduals ha , a e being e alua ed by he elec ical
simula 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 i ness e alua ions (ci cui simula ions) is he main
d awback o hese algo i hms [8]. E iciency, con e gence o
he ue POF, and di e si y o solu ions a e a eas o in ense and
cu en esea ch. An example is he de ini ion o new quali y
e alua ion me ics sui able o analog design p oblems p oposed
in [9].
Figu e 1. Illus a ing he Pa e o-op imal on concep o a wo-
dimensional on .
1 This o mula ion is alid o minimiza ion p oblems. A simple
change o sign applies o maximiza ion.
In o de o illus a e he po en iali y o POFs, le us conside
he Mille ope a ional ampli ie depic ed in , o which
he pe o mances o in e es a e: dc gain,
Figu e 2
0
A
, uni y-gain
equency, u
, phase ma gin, PM, and ou pu impedance, o
Z
.
The op imiza ion p ocess aims a maximizing he i s h ee
pe o mances ( 0
A
,u
and PM) and minimizing o
Z
. Some o
hese pe o mances depend on he load condi ions, so when he
POF o his block is gene a ed, a load will be included in he
e alua ion o he indi iduals o he POF. The design a iables
in his case a e he wid h and leng h o he ansis o s, plus he
bias cu en and he compensa ion capaci o . Di e en
cons ain s a e imposed o he p oblem 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, and phase ma gin 90 . The
POF wi h a capaci i e load o 1pF 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
º 10º>PM>
[10]. The
popula ion size and numbe o gene a ions we e 1500 and 150,
espec i ely. Gene a ion o his POF ook abou 1 hou 36
minu es o CPU ime on a 2.2 GHz p ocesso . The esul is
ob iously 1500 sample poin s o he 4-dimensional POF. Fo
illus a ion pu poses, shows he p ojec ions o he
1500 poin s o his 4-dimensional hype su ace on he dc gain
s. uni y-gain equency plane, on he phase ma gin s. uni y-
gain- equency plane, and on he dc gain s. ou pu impedance
plane. Each o hese poin s ep esen s a sized ci cui showing
he bes ade-o s among he ou pe o mances conside ed.
Figu e 3
III. LOAD-INDEPENDENT PARETO-OPTIMAL FRONTS
A. POF Gene a ion Me hodogy
Le us conside he same ampli ie and he same
pe o mances used in Sec ion II. As al eady men ioned, some
o hese pe o mances depend on he load condi ions. Howe e ,
he load condi ions a e no known un il he syn hesis 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 POF gene a ion p ocess is a
compu a ionally expensi e p ocess, he objec i e o ou
esea ch is o gene a e ade-o 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 POFs o
pe o mances ( 0
A
,u
, PM, and o
Z
) when he load condi ions
a e known.
The ope a ional ampli ie can be conside ed a wo-po , like
ha shown in Figu e 4. 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 (in case o single ou pu ) and he di e en ial
ou pu ol age and cu en (in case o ully di e en ial
ampli ie ). As we need a load-independen cha ac e iza ion o
he ampli ie , we may conside , a p io i, any ma ix
cha ac e iza ion o wo-po s [11]. Howe e , he wo-po
ma ix pa ame e s mus be selec ed in elligen ly, 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 [11] o cha ac e ize he wo-po :
Figu e 3. P ojec ions o he POF gene a ed by he mul iobjec i e
op imiza ion algo i hm o a capaci i e load o 1pF.
1
i
2
i
2
1
+
+
−
−
L
Z
Figu e 4. Two-po wi h a bi a y load.
Figu e 2. Mille ope a ional ampli ie used in his wo k.
M4
M3
M2
M1
M6
Cc
M7
Vss
Vdd
M5
Vin
Mbn
Vdd
Ib
Vo
Vip CL
(2)
1111122
221122
ih hi
h hi
′′
=⋅+⋅
′′
=⋅+⋅
2
In his equa ion, pa ame e ep esen s he inpu
impedance, is he in e se cu en gain, ep esen s he
ol age gain o he ampli ie wi hou any load and
11
h′
12
h′21
h′
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:
(3)
2L
Z=− ⋅2
i
yielding:
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 om he ol age gain,
22
h′()
A
s, and ou pu
impedance, ()
o
Z
s, o some known load condi ions, and ice
e sa, ob ain he ol age gain ()
A
sand ou pu
impedance ()
o
Z
s
22
′
o some load om he hyb id-2 pa ame e s
and . Mo eo e , a pa icula case is ha in which
. In his case, and a e iden ical o
21
h′
L
Z
h
→∞ 21
h′22
h′()
A
sand
(
o)
Z
s, espec i ely. This is he key o de eloping he POF
gene a ion and ans o ma ion me hodology ha is p oposed in
his pape o ans o m a POF o some known loading
condi ions o a bi a y new loading condi ions.
Le us assume ha we wish o gene a e he POF o he
ou pu impedance, dc ol age gain, uni y-gain equency, and
phase ma gin o some a bi a y loading condi ions. The
gene a ion me hodology p oceeds as ollows:
1) Gene a e he POF o he pe o mances o in e es o
some known loading 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 .
2) Fo each sample o indi idual o his POF, 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. This
in o ma ion can be e ie ed om common elec ical
simula o s. I his in o ma ion is no a ailable, a educed wo-
pole model can be easily ex ac ed om he dc gain, uni y-gain
equency, and phase ma gin alues.
3) Use equa ion (4) o ex ac he hyb id pa ame e s 21()hs
′
and o each sample. No ice ha , om basic ci cui
heo y, he poles o and a e iden ical.
22 ()hs
′
21()hs
′22 ()hs
′
4) Apply equa ion (4) o ob ain he ol age gain o he new
a bi a y loading 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, phase ma gin, and ou pu impedance by simple
p ocessing o he ne wo k unc ions.
No ice ha his p ocedu e can be applied o any ini ial
known loading condi ions. This includes he case in which he
POFs a e gene a ed o he ci cui wi hou any load. In his
case, s eps 2 and 3 a e uni ied in a single s ep.
The i s s ep o his me hodology has he hea ies
compu a ional e o by a , bu no ice ha he esul s o s ep 3
a e independen o he applica ion, i.e., independen o he inal
loading condi ions. The e o e, he i s h ee s eps can be
pe o med be o ehand, and he esul s s o ed and used
whene e and whe e e necessa y.
B. Resul s
In his sec ion, he p oposed me hodology will be applied o
he gene a ion o he POF o he Mille ope a ional ampli ie ,
when a esis i e-capaci i e load is applied. Following s ep 1, a
POF wi h a capaci i e load o 1pF was i s gene a ed o he
ou objec i es (see Sec ion II): dc ol age gain, uni y-gain
equency, phase ma gin, and ou pu impedance.
F om he elec ical simula ion o he 1500 poin s o he
POF, he ne wo k unc ions ()
A
sand ()
o
Z
s
h
o each design
poin can be easily ob ained. And by applying equa ion (4) o
each o hese poin s, he hyb id-2 pa ame e s 21
′
and 22
h
′
o he
1500 poin s a e calcula ed. These wo s eps a e pe o med in
less han 5 minu es. No ice ha all he s eps pe o med so a
a e independen o he inal load condi ions. The e o e,
al hough compu a ionally cos ly, hey a e pe o med long
be o e he load is known and he o he s eps ha e o be
pe o med.
Assume ha we need now o gene a e he POF o a load o
2pF and 50kΩ. Using he poin s p e iously s o ed, he new
POF is gene a ed by applying s eps 4 and 5 in Sec ion III.
Figu e 5 shows he h ee p ojec ions o he 4-dimensional on .
The applica ion o hese wo s eps akes only 20 seconds.
To assess he p ocedu e, he POF was also gene a ed by
coupling he op imize wi h he elec ical simula o o his new
2pF-50kΩ load ( ha is, no ollowing he ans o ma ion
p ocedu e p oposed he e). A se o samples o he POF wi h
simila quali y cha ac e is ics is ob ained (see Figu e 6).
Howe e , as in he gene a ion o he ini ial POF, 1 hou and 36
minu es o CPU a e employed ins ead o he 20 seconds ha
we e equi ed by he ans o ma ion p ocedu e desc ibed he e.
This is an accep able ime in o de o inco po a e his echnique
in o a hie a chical syn hesis low whe e i e a i e e alua ions o
ci cui pe o mances a e necessa y. No ice ha he POFs in
Figu es 5 and 6 a e simila bu he samples a e di e en due o
he ini e popula ion size and he s ochas ic na u e o he mul i-
objec i e e olu iona y algo i hm. When compa ing Figu e 3
and Figu e 5, i can be obse ed ha he ans o ma ion
p ocedu e implies an impo an mo emen o poin s in he
design objec i e space. Densi y o poin s in Figu e 5 migh
Figu e 5. P ojec ions o he POF ob ained by he ans o ma ion
p ocedu e o a load o 2pF-50kΩ.
Figu e 6. P ojec ions o he POF gene a ed by he mul iobjec i e
op imiza ion algo i hm o a load o 2pF-50KΩ.
become smalle because in he ans o ma ion some poin s may
mo e o egions ha a e no o in e es ( o example, e y low
phase ma gin o dc gain), o because he ans o ma ion
p ocedu e can mo e a poin o a posi ion in he
objec i e/pe o mance space whe e i becomes domina ed by
o he poin s. Al hough some poin s may become domina ed
a e he ans o ma ion, he p ocedu e gua an ees ha a poin
o he ans o med POF may no o igina e om a domina ed
poin o he ini ial pe o mance space.
IV. CONCLUSIONS
This pape in oduces a POF ans o ma ion p ocedu e o
ope a ional ampli ie s based on a hyb id-2 pa ame e
cha ac e iza ion. This app oach enables he e icien and apid
gene a ion o POFs o a bi a y in e connec ion condi ions,
which can be used in hie a chical syn hesis based on POFs.
Fu u e wo k will add ess he gene a ion o dense ans o med
POFs by simul aneously conside ing he POFs gene a ed o
di e en load condi ions.
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