IEEE JOURNAL
OF
SOLID-STATE CIRCUITS, VOL.
30,
NO.
3,
MARCH
1995
327
E icien Symbolic Compu a ion
o
App oxima ed
Small-Signal Cha ac e is ics
o
Analog In eg a ed Ci cui s
Pie Wambacq, F ancisco V. Fe nBndez, Geo ges Gielen, Willy Sansen, and Angel Rodiguez-VBzquez
Abs ac -A
symbolic analysis ool is p esen ed ha gene a es
simpli ied symbolic exp essions
o
he small-signal cha ac e -
is ics o la ge analog in eg a ed ci cui s. The exp essions a e
app oxima ed while hey a e compu ed,
so
ha only hose e ms
a e gene a ed which emain in he inal exp ession. This p inciple
causes d as ic sa ings in
CPU
ime and memo y, compa ed wi h
p e ious symbolic analysis ools. In his way, he maximum
size
o
ci cui s ha can be analyzed, is la gely inc eased. By
aking in o accoun a ange
o
he alue
o
a ci cui pa ame e
a he han one single numbe , he gene a ed exp essions a e
also
mo e gene ally alid. Misma ch handling
is
explici ly aken in o
accoun in he algo i hm. The capabili ies
o
he new ool a e
illus a ed wi h se e al expe imen al esul s.
I.
INTRODUCTION
URRENT
ools o small-signal symbolic analysis o
C
analog in eg a ed ci cui s, like o ins ance
ISAAC
[
11
and
ASAP
[2],
a e able o e alua e ne wo k unc ions in he
s-domain wi h he complex equency a iable and he ci cui
pa ame e s (capaci ances, esis ances, ansconduc ances, e c.)
kep as symbols. These unc ions a e ypically gi en as an
expanded cancella ion- ee sum o p oduc s,
in which
xT
=
(21,
ZZ,...,XQ}
is he ec o o symbolic
ci cui pa ame e s and he
i
and
gj
a e sums o p oduc s.
Since hese exp essions a e calcula ed au oma ically, analog
designe s a e eleased om he in ol ed calcula ions needed
o ge insigh in o he ac beha io o ci cui s.
Also,
analog
cells can be au oma ically sized o gi en ac speci ica ions
h ough he i e a i e op imiza ion o he symbolic equa ions
gene a ed o hei gain, poles, ze os, e minal impedances,
PSRR, CMRR,
e c. O he po en ial applica ions o symbolic
analyze s, o syn hesis, s a is ical op imiza ion, es abili y,
e c., exploi also he compu a ional ad an ages o pe o m
epe i i e e alua ions o p ecalcula ed models
[3].
Howe e ,
hese applica ions can be ealized a ully only i he au-
oma ic gene a ion o symbolic exp essions uns pa allel o
he au oma ic p uning o insigni ican e ms in hese exp es-
Manusc ip ecei ed July
13,
1994; e ised No embe 4, 1994
P.
Wambacq,
G.
Gielen, and
W.
Sansen
a e
wi h Ka holieke Uni e si ei
Lue en, Dep. Elek o echniek, ESAT-MICAS,
B-3001
He e lee, Belgium.
F.
V.
Fe nandez and A. Rodnguez-Vhquez
a e
wi h Depa men
o
Analog
Ci cui Design, Cen o Nacional de Mic oelec 6nica. Edi . CICA, E-4101 2
Se illa, Spain.
IEEE Log Numbe 9408739.
sions-simila o wha expe analog designe s do when hey
analyze ci cui s by hand.
Al hough exis ing analyze s like
ISAAC
and
ASAP
in-
co po a e such simpli ica ion ea u e, hei algo i hms ha e
wo impo an d awbacks: a) simpli ica ions a e pe o med
only a e he exac symbolic exp ession is gene a ed in an
expanded sum-o -p oduc o ma ; and b) he signi icance o
each e m in he sums-o -p oduc s is assessed on he basis
o nume ical e alua ions using ypical alues o he ci cui
pa ame e s, a a single poin o he design pa ame e space.
Since he size o he exac symbol exp essions inc eases
exponen ially wi h he numbe o nodes and elemen s
in
he ci cui , he i s d awback pu s an uppe limi on he
complexi y o analyzable ci cui s; a ound en ansis o s i each
ansis o is ep esen ed by a high- equency model con aining
abou nine ci cui elemen s. On he o he hand, app oxima ing
symbolic exp essions by conside ing jus a single poin o he
pa ame e space does no seem o be consis en wi h he e y
na u e o he symbolic analysis p ocedu e, whe e he exac
nume ical alue o he pa ame e s is, by de ini ion, unknown
a
p io i.
E en in he case symbolic analysis is used o s udy
c i ical pa ame e a ia ions
in
an al eady sized schema ic,
simpli ying by using jus in o ma ion abou a nominal poin
may lead o impo an inaccu acies in misma ch-sensi i e
cha ac e is ics, as o ins ance
PSRR
o
CMRR
o ope a ional
ampli ie s.
This pape p esen s a simpli ica ion algo i hm o o e come
bo h d awbacks abo e. Fi s o all, he complexi y limi s
o analyzable ci cui s a e ex ended by gene a ing only he
dominan e ms, wi hou i s compu ing he comple e exac
exp ession.
In
his app oach, which is deno ed as
simplijica ion
du ing gene a ion,
he dominan e ms a e gene a ed un il
he accu acy alls wi hin a gi en use -supplied accu acy.
As
shown in Fig. 1, his is a much mo e e icien app oach,
bo h in e ms o memo y usage and
CPU
ime, han he
classical app oach ollowed in
[l],
[2].
The idea o simpli-
ica ion du ing gene a ion was i s men ioned in [4], and
la e ound also in
[5]
and
[6].
Howe e , simpli ica ions in
[4] and
[6]
a e pe o med a a nominal poin o he design
space, and, consequen ly, any e alua ion o he exp essions
in
ano he ope a ing poin migh cause la ge e o s.
To
educe
hese e o s, his pape u he elabo a es he app oach in [5]
o combine he concep o simpli ica ion du ing gene a ion
wi h he use o anges
[7],
ins ead o single alues, o he
ci cui design pa ame e s. This inc eases he compliance o
gene a ed exp essions, while keeping he compu a ion ime
and he memo y esou ces needed o symbolic analysis o
la ge analog ci cui s bounded.
Also,
he combina ion o bo h
0018-9200/95$04.00
0
1995 IEEE
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328 IEEE
JOURNAL
OF
SOLID-STATE CIRCUITS,
VOL.
30,
NO.
3,
MARCH
1995
.....
. .
.,.........
.
....
.
..
.....
..
...
.......
.
.,
nes ed
-
expanded app oxima ed
(a)
(b)
Fig.
1.
Schema ic ep esen a ion
o
he memo y usage du ing simula ion
ime wi h he classical app oach (a) and wi h he p oposed app oach
o
simpli ica ion du ing gene a ion (b). Classically, a symbolic exp ession is i s
gene a ed in a nes ed o ma . Fo a eliable app oxima ion, he exp ession is
hen expanded and he cancelling e ms a e elabo a ed. This expansion can
lead o a huge numbe o e ms whose s o age exceeds he memo y limi s
(dashed line), while only a ew e ms a e e ained a e app oxima ion. This
p oblem is ci cum en ed i he exp ession
is
simpli ied when
i
is gene a ed:
he memo y usage inc eases wi h he equi ed accu acy o he numbe
o
e ms o he symbolic exp ession.
echniques, simpli ica ion du ing gene a ion and he use o
anges, demons a e be e esul s han p e ious app oaches
o he handling o ma ching be ween symbolic pa ame e s
and misma ches.
Sec ion I1 p esen s he concep and ou lines he algo i hms
he IC- h powe , spanning ees in dec easing o de mus be
gene a ed con aining exac ly
IC
capaci ance b anches. This
can be o mula ed as he ollowing g aph- heo e ical p oblem:
gi en a g aph wi h
n
nodes and wi h ed (co esponding o
( ans)conduc ances) and blue (co esponding o capaci ances)
weigh ed b anches, enume a e in dec easing o de he spanning
ees ha con ain exac ly
IC
blue b anches and
n
-
k
-
1
ed
b anches, in which
IC
can ha e a alue be ween ze o and
n
-
1.
Fo his p oblem, an algo i hm
[9]
has been de eloped whose
ime complexi y and memo y equi emen s inc ease linea ly
wi h he numbe o gene a ed spanning ees.
111.
GENERATION
OF
THE
NUMERICAL
REFERENCE
AND
APPROXIMATION
OVER
RANGES
The ee enume a ion p ocedu e desc ibed abo e ob iously
needs a s opping c i e ion o know when enough e ms ha e
been gene a ed. The gene a ion o e ms o a ce ain powe
IC
o
s
in he enume a o o denomina o can s op when
ICnum. e alua ion
o
gene a ed e ms1
(num. alue
o
coe icien
o
sk
I
>
(1
-Ek).
(2)
-
used o simpli ica ion du ing gene a ion. Sec ion 111 explains
how in e als a e inco po a ed in he s opping c i e ion ha
con ols he gene a ion o e ms. In Sec ion IV i is explained
how ma ching elemen s and co esponding misma ches a e
handled in he new app oach. Finally, Sec ion V p esen s
examples ha demons a e he sui abili y o he echniques
p esen ed o analog cells con aining mo e han
20
ansis o s,
which app oaches he size o p ac ical ci cui s used in odays
IC
designs.
In his equa ion, he nume ical e alua ion is pe o med in
a nominal ope a ing poin o he ci cui . The denomina o
in
(2)
ep esen s he nume ical alue o he coe icien o
sk
in ei he he nume a o o denomina o o he ne wo k
unc ion. The comple e coe icien s a e ne e gene a ed and,
hence, hei nume ical alue mus be calcula ed in ad ance
(wi hou knowing he symbolic exp essions). This is e icien ly
pe o med using he polynomial in e pola ion me hod [lo].
Fo he ex ension o he s opping c i e ion o
(2)
o in e als,
11.
SIMPLIFICATION
DURING
GENERATION
The idea o simpli ica ion du ing gene a ion needs a e m by
e m gene a ion mechanism, which inds he e ms in dec eas-
ing o de o magni ude, wi hou skipping any e m. This can
be achie ed wi h he undi ec ed ee enume a ion me hod
[8].
This is a opological me hod ha ope a es on wo weigh ed
g aphs, he ol age g aph and he cu en g aph, which a e
easily de i ed om he gi en (small-signal) ne wo k.
A
e m is
alid only i i s co esponding b anches cons i u e a spanning
ee in bo h g aphs. The symbolic e m is gi en by he p oduc
o he b anch weigh s (admi ances) in any
o
he g aphs.
The sign o a e m is de e mined sepa a ely, using opological
in o ma ion o bo h g aphs. By augmen ing he ne wo k in a
special way wi h ic i ious elemen s, i is possible o gene a e
he e ms o bo h nume a o and denomina o a he same
ime
[8].
The numbe o ees inc eases exponen ially wi h he ci cui
size. Since we a e in e es ed only in he dominan e ms
and he e o e no in all ees, he new algo i hm enume a es
spanning ees in he ol age g aph in dec easing o de . Fo
e e y spanning ee, i is checked whe he he co esponding
b anches in he cu en g aph cons i u e a spanning ee as
well. I
so,
a alid e m is ound and i s sign is de e mined.
This echnique is pe o med o e e y powe o he e-
quency a iable
s
in bo h he nume a o and denomina o
o he ne wo k unc ion. Fo a nonze o powe
o
s,
say
i
is assumed ha a symbolic pa ame e
z
can ake a alue
inside a gi en in e al de e mined by i s lowe bound
XL
and
i s uppe bound
XH.
The in oduc ion
o
in e als o he symbolic ci cui pa-
ame e s gi es ise o mul idimensional in e als o he alue
o he coe icien s
i
and
gj
om
(1).
These a e compu ed by
an in e al ex ension o he polynomial in e pola ion me hod.
The esul ing in e al o a coe icien is usually a pessimis ic
o e es ima e. The e o e, in e als a e na owed using he
algo i hm desc ibed in
[
1
I].
In e als o he small-signal ci cui pa ame e s a e ei he
de e mined by speci ying a ela i e a ia ion a ound a gi en
nominal alue, o hey can be de i ed om in e als o he bias
alues and echnological pa ame e s. In e als o symbolic
e ms o sums o p oduc s a e hen de e mined using he di ec
in e al ex ension
[ll],
o , mo e accu a ely, wi h he mean
alue o m
[ll].
The s opping c i e ion gi en in
(2)
can now be e o mula ed
as:
(3)
In his equa ion
(SL, SH]
ep esen s he in e al o he coe i-
cien o
sk
ob ained as desc ibed abo e. The in e al
[GL, GH]
deno es he in e al o he sum o he signi ican e ms ha
ha e al eady been gene a ed. The symbols
L
and
U
deno e
he lowe and uppe bound o an in e al, espec i ely.
L(l[GL,
GHll)
>
-
k),
U(I[SL
SHII)
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VOL.
30, NO. 3,
MARCH
1995
329
Fig.
2.
CPU
ime on a SUN
SPARC
10
o he symbolic compu a ion wi h a
25%
e o
o
he ol age gain
Vou /K,
o he esis i e ladde ne wo k. The
do ed line co esponds o imes measu ed wi h
ASAP
(con en ional symbolic
analyze ). The solid line co esponds o he new app oach.
The use o in e als p o ides a e y good ade-o be ween
accu acy and complexi y. Ob iously, mo e e ms appea in he
inal esul han when using ixed alues, and i he in e als
a e aken oo wide, hen he in e p e a ion o esul s can
become complica ed again.
IV.
MATCHING ELEMENTS
Ma ching elemen s play an impo an ole in analog and
especially in di e en ial in eg a ed ci cui s. In symbolic cal-
cula ions hey a e ep esen ed by he same nominal symbol.
A e doing
so,
p oduc e ms can occu wi h exac ly he same
symbols,
so
ha hey cancel
o
add, depending on hei sign.
The de ec ion o ma ching e ms equi es a lo o o e head in
CPU ime and memo y consump ion in con en ional symbolic
analyze s
[
11,
[2].
Wi h he new echnique, howe e , ma ching
e ms a e easily de ec ed: since hey a e equal in magni ude,
hey a e gene a ed one immedia ely a e he o he . Hence,
he cancella ions can be elabo a ed by looking only a he las
ew gene a ed e ms ha ha e he same magni ude as he las
gene a ed e m.
Misma ches a e modeled explici ly by adding a small sym-
bolic misma ch elemen in pa allel wi h he nominal elemen .
F om ha momen , bo h elemen s a e handled independen ly,
and wi h hei own nume ical magni ude. Fo example, he
ansconduc ances o wo ma ching ansis o s
MIA
and
M~B
a e w i en as
gmM1
and
gmM1
+
ASmMleA,
espec i ely.
In his way, p oduc e ms con aining misma ch symbols a e
gene a ed much la e han he co esponding nominal e ms
and only when necessa y.
In echniques p e iously used
[
11,
[2]
in symbolic analyze s,
misma ch e ms we e always gi en a magni ude ( he maximum
de ia ion) and a sign. This is no ealis ic, since hei sign is
no known
in
ad ance. This p oblem is o e come he e by
ep esen ing a misma ch e m by a symme ic in e al a ound
ze o.
V.
EXAMPLES
The new echnique no only exceeds he limi s o a con-
en ional symbolic analyze , i can also- o smalle ci -
cui s-gene a e an app oxima e exp ession in a CPU ime ha
is up
o
se e al o de s o magni ude smalle han wi h con-
en ional analysis. This is shown wi h he symbolic analysis
o he esis i e ladde ne wo k shown in Fig.
2,
which is o en
9-
-1
mD
.-
. .
GND
..
Fig.
3.
A
illy di e en ial
BiCMOS
ope a ional ansconduc ance ampli ie
wi h common-mode eedback.
1
0‘
!Io3
1
o2
/
IO’
ela l eem
10’
IO-’
IO-*
IO”
loJ
IO”
#
e ms
32 182
664
1898
1606
9334
#di e en
e ms
7
36
120
339
788
1606
Fig.
4.
CPU
ime (in seconds on a
SUN
SPARC
10
wo ks a ion) e sus he
ela i e e o o
he
gene a ed symbolic exp ession
o
he denomina o o he
low- equency di e en ial-mode gain
o
he
BiCMOS
ampli ie (Fig.
3).
The
numbe o enns ha co esponds o he accu acy is indica ed as well.
used as a benchma k ci cui o symbolic analyze s. The CPU
ime is shown as a unc ion o he numbe o s ages o he
symbolic compu a ion o he ol age gain wi h a
25%
e o .
The d ama ic inc ease in CPU ime wi h he numbe o s ages
o con en ional symbolic analysis is due o he ac ha he
exac exp ession mus be gene a ed.
The e iciency o he simpli ica ion du ing gene a ion ech-
nique in e ms o CPU ime is illus a ed wi h he symbolic
compu a ion
o
he sys em de e minan o he BiCMOS am-
pli ie o Fig.
3.
This
ci cui , con aining wen y ansis o s,
is a oo complex o be analyzed wi h classical symbolic
analyze s. Fig.
4
indica es how wi h he new echnique he
CPU ime inc eases wi h he accu acy o he gene a ed sym-
bolic exp ession and hence wi h he numbe o e ms, jus
as wi h he p inciple idea shown in Fig.
1.
In con as wi h
he con en ional symbolic analysis app oaches, he less e ms
a e gene a ed ( he la ge he e o ), he less CPU ime is
equi ed, which is a e y “na u al” way o gene a ing e ms
ha cons i u e a la ge exp ession.
Fo la ge ci cui s complica ed exp essions may be gene -
a ed. This
is
illus a ed in Fig.
5,
which shows he symbolic
exp ession o he low- equency di e en ial-mode gain o he
ampli ie
o
Fig.
3.
This exp ession has been gene a ed
in
58 s
on a SUN Spa c
10.
The exp ession, howe e , can be
u he simpli ied wi hou inc easing he e o by a symbolic
pos p ocessing p ocedu e, ha ac o izes he exp essions and
ha akes ad an age o he ac ha he e o on a a io o
wo coe icien s o a ne wo k unc ion is o en much smalle
han he e o on he coe icien s indi idually. Doing
so,
he
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330
IEEE
JOURNAL
OF
SOLID-STATE
CIRCUITS,
VOL.
30,
NO.
3,
MARCH
1995
Fig.
5.
App oxima ed exp ession
(E
=
20%) o he low- equency di e -
en ial-mode gain o he ci cui
o
Fig. 3. The e ms a e so ed in dec easing
o de . Due o ma ching, se e al p oduc e ms occu mo e han once, which
explains he occu ance o in ege coe icien s 4 and 8. The elemen
geq2
is a
lumped elemen 111 consis ing o he pa allel conduc ances
goM9
and
gOQp.
Elemen s om he bias ci cui y (like
gmM6)
o om he common-mode
eedback ci cui y (like
gm~4)
ha don’ in luence he di e en ial-mode gain
a all, disappea a e ac o iza ion.
di e en ial-mode gain educes o
(4)
I is ound ha he ou pu conduc ance o he bipola cascode
is oo small, e en wi h he inclusion o in e als, o con ibu e
signi ican ly o he conduc ance seen a he ou pu node.
An
e en mo e complex ci cui is he comme cial
pA741
opamp. This ci cui con ains
23
nodes,
22
ansis o s, and
13
esis o s. The gene a ion o a symbolic exp ession o he
ampli ie ’s ans e unc ion wi h an e o o
0.1%
(1
10
e ms)
equi es
38
seconds on a SUN Spa c
10
wo ks a ion.
VI.
CONCLUSION
A
new p og am has been p esen ed ha gene a es app oxi-
ma ed symbolic exp essions o small-signal cha ac e is ics o
analog ci cui s. The app oxima ion is pe o med du ing he
gene a ion
o
he exp ession. In his way, only he necessa y
e ms
o
he simpli ied symbolic exp essions a e gene a ed,
which con as s o app oaches o con en ional symbolic an-
alyze s which equi e a lo o o e head o he gene a ion
o he exac symbolic exp ession, which is hen p uned. The
new app oxima ion echnique also akes in o accoun a ange
o he alue
o
he symbolic ci cui pa ame e s a he han
one single alue. This ex ends he ange o alidi y o he
gene a ed symbolic exp essions. Mo eo e , he new echnique
allows an accu a e con ol o he app oxima ion e o . The
in e p e abili y o he exp essions can be enhanced by u he
pos p ocessing. Se e al examples ha e demons a ed ha his
app oach enables he symbolic analysis o la ge analog in e-
g a ed ci cui s o he size o p ac ical analog cells, which we e
impossible o analyze p ope ly be o e.
ACKNOWLEDGMENT
The au ho s wish o hank
P.
Eindho en, The Ne he lands,
and he Human Capi al and Mobili y P og am o he
CEC
o
hei suppo .
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Gielen and W. M. Sansen,
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o
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J.
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