ISCAS
2000
-
IEEE
In e na ional Symposium on Ci cui s and Sys ems,
May
28-31, 2000,
Gene a, Swi ze land
An E o -Con olled Me hodology o App oxima e Hie a chical Symbolic Analysis
0.
Gue a,
J.D.
Rod iguez-Ga cia, E. Roca,
F.
V. Fe nindez and A. Rod iguez-Vizquez
Ins i u o de Mic oelec hica de Se illa, Cen o Nacional de Mic oelec 6nica, Se illa, SPAIN
Phone: +34 955056666,
FAX:
+34 955056686 E-mail:
{
gue a, jda id, eli, paco , angel} @imse.cnm.es
Abs ac *
Limi a ions o exis ing app oaches o symbolic analysis o la ge
analog ci cui s a e discussed.
To
add ess hei solu ion,
a
new
me hodology o hie a chical symbolic analysis is in oduced. 'lke
combina ion o a hie a chical modeling echnique and app oxima-
ion s a egies, comp ising ci cui educ ion, g aph-based symbolic
solu ion o ci cui equa ions and ma ix-based e o con ol, p o-
ides op imum esul s in e ms
o
speed and
quali y
o esul s.
1.
In oduc ion
Symbolic analyze s a e aimed o analyze ci cui s in which pa o
all hei pa ame e s a e symbols. The gene a ed exp essions
p o-
ide he keys o unde s anding he in ica e mechanisms unde -
nea h he ci cui ope a ion. I s applica ions o p o iding insigh in
in e ac i e ci cui design, gene a ing beha io al models o lib a y
cha ac e iza ion, gene a ing design equa ions o syn hesis
o
op i-
miza ion asks, a e well-known [I].
A majo p oblem in he applica ion o hese echniques was he
exponen ial g ow h o he complexi y o he symbolic exp essions
wi h he ci cui s sizes. Di e en solu ions ha e been p oposed o
pallia e his p oblem.
On he one hand, Simpli ica ion Be o e (SBG) and Du ing
(SDG)
Gene a ion app oaches
[2]-[4]
ha e ex ended he analyzable ci -
cui sizes and ha e made he symbolic exp essions in e p e able
bu a e s ill insu icien o e y la ge ci cui s.
Hie a chical analysis echniques cons i u e an al e na i e o ana-
lyze e y la ge ci cui s al hough epo ed echniques do no inco -
po a e app oxima ion capabili ies, making in e p e a ion and as
e alua ion o he esul s
a
p oblem
[5]-[7].
Recen ly, new echniques based on De e minan Decision Dia-
g ams ha e been p oposed, which a e able o ep esen he exac
ci cui beha io in
a
compac , al hough unin e p e able, o m
81
,PI.
The me hodology p esen ed in his pape is buil on he ideas in
[IO]
o
o mula e and implemen
a
hie a chical analysis me hodol-
ogy able o inco po a e he ci cui educ ion and dominan con i-
bu ion s a egies con ained in SBG and SDG echniques.
The pape is o ganized as ollows. Sec ion 2 e iews and compa es
exis ing symbolic analysis echniques. Sec ion
3
desc ibes he hie -
a chical modeling me hodology while Sec ion
4
in oduces he
app oxima e analysis s a egy. Finally, Sec ion
5
p esen s expe i-
men al esul s o assess he quali y o he me hodology.
2.
Re iew
o
p e ious app oaches
2.1.
App oxima e la analysis echniques
The i s app oxima ion app oaches we e based on Simpli ica ion
A e Gene a ion (SAG) echniques, ha p une he leas signi ican
symbolic e ms once he exac exp ession has been compu ed.
*
This
wo k
has
been suppo ed
by
he
ESPRIT
P ojec
#21812
(AMA-
DEUS)
and
he Spanish
C.I.C.Y.T.
unde
con ac
TIC97-0580.
Al hough he in e p e abili y
was
g ea ly imp o ed, he ini ial gen-
e a ion
o
he exac exp ession exhaus ed he compu e esou ces
e en in case o medium size ci cui s
[I].
To sol e bo h, he in e p e abili y and he excessi e consump ion
o compu e esou ces, wo new ideas we e in oduced: SBG ech-
niques, which simpli y he sys em
o
ci cui equa ions (a he
ma ix o he g aph le el) be o e being sol ed; and SDG ech-
niques, which calcula e di ec ly an app oxima ed solu ion o he
sys em o ci cui equa ions, con aining ,only he dominan con ibu-
ions [2]-[4].
2.2.
Hie a chical analysis echniques
T adi ionally,
a
h ee s ep app oach has been used (see M.M. Has-
soun's "Hie a chical Symbolic Analysis
o
La ge Analog Ci cui s"
-
Chap e
5
in
[I]
o
a
de ailed e iew):
Di ision
o
he ci cui in subblocks
(ci cui pa i ioning).
Cha ac e iza ion
o
he lowes le el blocks in e ms
o
hei
inpu s and ou pu s
( e minal block analysis).
I e a i e cha ac e iza ion o blocks in e ms
o
hei inpu s and
ou pu s by pe o ming ope a ions on he cha a e iza ions o he
cons i uen subblocks
(middle block analysis)
Fo e minal and middle block analysis, h ee app oaches ha e
been epo ed: Coa es lowg aph, Mason lowg aph and di ec ne -
wo k me hods.
The Coa es lowg aph me hod equi es o build he Coa es g aph,
which is hen pa i ioned
[5].
Pa i ioning de ined by he use is no
possible. Besides, since he Coa es g aph does no co espond wi h
ci cui nodes and b anches, pa i ioning in o ma ion canno be
di ec ly mapped o he ci cui le el.
Mason signal lowg aph echniques use educ ion echniques on
Mason's g aphs
o
yield
a
desc ip ion
o
each block in e ms o i s
inpu and ou pu nodes only
[6].
Finally,
a
combina ion o he
blocks is made applying he same educ ion echniques
o
ge
a
desc ip ion o he en i e ci cui .
The di ec ne wo k me hod ope a es by educing he
Modi ied
Nodal Analysis (MNA)
ma ix, ep esen ing each subci cui in o
a
Reduced ModiJied Nodal Analysis
ma ix, which only depends on
e minal nodes o he block. A e wa ds,
a
successi e ecombina-
ion o such ma ices is pe o med o ob ain
a
educed ma ix ep-
esen ing he en i e ci cui
271.
2.3.
DDD-based analysis echniques
The echnique is based on he ep esen a ion o he symbolic
exp essions by means
o
De e minan Decision Diag ams, ha a e
signed oo ed acyclic g aphs wi h wo e minal e ices. This ep-
esen aion is buil om he ma ix ha models he ci cui , exploi -
ing he spa si y and sha ing o p oduc e ms
[8],
he e o e i inds
i s main ad an age in case o epe i i e opologies (like ladde -
s uc u ed ne wo ks).
2.4.
Compa a i e discussion
Al hough he e is no
a
p ecise limi o he applicabili y o
app oxima e Ra analysis echniques
(i depends on he ci cui
0-7803-5482-6/99/$10.00
02000
IEEE
111-133
size, model complexi y, ci cui connec i i y, igh ness o e o
speci ica ions), such bounda y exis s. Beyond, al e na i e me hods
a e needed; in pa icula , hose exploi ing he inhe en hie a chy in
he cons uc i e p ocess o la ge ci cui s.
The
hie a chical Coa es lowg aph me hod
will no dese e con-
side a ion due o i s inabili y o handle p e-pa i ioned ci cui s.
Nei he he
di ec ne wo k me hod,
no he
Mason lowg aph
me hod
inco po a e app oxima ion echniques. This exceedingly
hampe s hei applica ion o la ge p ac ical ci cui s which is co -
obo a ed by he ac ha epo ed expe imen al esul s use
ex emely simple block models; i.e. il e s using ideal models o
he opamps.
DDD-based me hods
ep esen ci cui beha io in a compac
o m. Al hough, such compac s uc u e is unin e p e able and
CPU-in ensi e o build, hei main ad an age is hei as nume ical
e alua ion. When hey a e used o gene a e app oxima e (in e p e -
able) symbolic exp essions
[8]
hey a e no compe i i e o exis ing
SBG
and SDG app oaches. They ha e also been applied o hie a -
chical analysis in a simila ashion o he Di ec Ne wo k me hod
and using DDDs o ep esen ma ix de e minan s
[9].
Howe e , no
e o -con olled app oxima ion o symbolic exp essions ex ac ed
om such DDD has been epo ed.
3.
Hie a chical modeling me hodology
In
ou
me hodology, blocks a e pa i ioned (de ined by he use
o
au oma ically pe o med) in se e al hie a chical le els. This hie a -
chical s uc u e can be ep esen ed as an in e ed ee, as he exam-
ple in Fig.
1
shows.
Figu e
1.
Example
o
in e ed ee ep esen a ion.
Lea nodes ( e minal blocks) a e modeled by he subs i u ion o
cons i uen de ices by hei co esponding models. Non-lea nodes
(middle blocks) a e modeled using a ( ans)admi ance desc ip ion
as
Fig.
2
illus a es o a h ee- e minal block. Concep ually, each
( ans)admi ance is a unc ion o he models a he immedia e hie -
a chical le el.
An analogous desc ip ion esul s when subblock ma ices a e com-
bined in he Di ec Ne wo k app oach
[7],
bu such me hodology
p e en s he applica ion o e o -con olled app oxima ion s a e-
gies.
Figu e
2.
(T ans)admi ance desc ip ion
o
blocks.
The me hodology p esen ed he ein gene a es app oxima ed exp es-
sions o he needed ( ans)admi ances o each middle block as a
unc ion
o
he ( ans)admi ances o he sub-blocks a he ollow-
ing le el down he hie a chy (de ice models in case o eiminal
blocks).
4.
App oxima ion s a egy
Ou
app oxima e analysis me hodology ollows he low diag am
in
Fig.
3.
I s a s om a hie a chical ci cui desc ip ion and in o -
ma ion on he ne wo k unc ion o calcula e, magni udelphase e o
cons ain s and equency in e als.
Hie a chically-decomposed Ci cui
In e nal
Block
Size Checking
Pa i ion
Figu e
3.
Module s uc u e.
The
Ci cui Reduc ion
module pe o ms node con ac ions and
de ice emo als whose con ibu ion o he global ci cui beha io
is negligible. The e o in oduced by hese ci cui ans o ma ions
is ca e ully con olled by using algo i hms based on in e al analy-
sis echniques o gua an ee ha e o speci ica ions a e no iola ed
wi hin he speci ied equency ange
[4].
Ci cui equa ions mus be
sol ed o con ol magni ude/phase e o s. This is a nume ical p o-
cess and
is
he e o e mo e e icien ly pe o med by using an A4NA
ma ix o mula ion and spa se ma ix echniques o i s solu ion.
This ci cui educ ion echnique is applied o he comple e la ci -
cui , al hough he p ede ined pa i ions a e o mally kep ,
so
ha
hey can be ebuil when he educ ion p ocess is inished. Since
e y e icien spa se ma ix echniques a e used in he e o e aJua-
ion, no signi ican ad an age is gained om applying he educ ion
echnique o he componen blocks sepa a ely. Mo eo e , a sepa a e
applica ion o each block would equi e an e o p opaga ion mech-
anism a his ea ly s age o he analysis p ocess. This necessa ily
yields mo e conse a i e esul s (less educed ci cui s) and, come-
quen ly, has a nega i e impac on he global pe o mance
o
he
analysis me hodology.
The e ec o he ci cui educ ion is no only he size educ ion o
he e minal blocks bu also he elimina ion o many
o
he model-
ing ( ans)admi ances o he middle blocks.
A e he ci cui educ ion p ocess, he hie a chical s uc u e is
econs uc ed and
Block Checking
s ep s a s. I a simpli ied block
con ains a oo small numbe o de ices
o
in e nal nodes, analyzing
i as an independen block becomes e y ine icien . Then, i is
ad isable o join he block o i s bes neighbo o inco po a e i in o
he immedia ely uppe hie a chical le el. On he con a y, e en
a e he ci cui educ ion, some block may s ill con ain oo many
nodes and de ices o an e icien symbolic exp ession gene a ion.
In his case, an in e nal pa i ioning is p o ided which inds op imal
blocks o he subsequen exp ession gene a ion module.
The in e nal pa i ioning mechanism p o ides he solu ion o he
case in which no p e-de ined blocks
a e
gi en. A e he ci cui
111-
134
educ ion s ep, he ci cui a hand is in e nally pa i ioned o gene -
a e a numbe o blocks ha enables an op imal esul in e ms o
compu a ional ime and exp ession complexi y.
To p ese e use equi emen s, bo h p ocesses: in e connec ion and
pa i ioning, can be con olled by he use .
Once he hie a chical block s uc u e has been ebuil and checked,
a ci cui whe e blocks a e modeled in e ms o he ( ans)admi -
ances is buil up. Then, app op ia e analysis algo i hms gene a e
app oxima e symbolic exp essions o each ( ans)admi ance o
each block in he s uc u e as a unc ion o he componen de ices
o ha block. Analogous analysis algo i hms a e applied o ob ain
he desi ed ne wo k unc ion (de ined by he global inpu /ou pu
signals o he ci cui ) in e ms
o
he ( ans)admi ances modeling
he blocks a he uppe mos hie a chical le el.
When symbolically analyzing a block, a se o ne wo k equa ions
( opological and cons i u i e ela ionships) has o be sol ed.
Fo
he app oxima e symbolic solu ion o a se o linea equa ions,
g aph me hods ha e p o en o be supe io
[4].
E icien echniques
a ailable o la ci cui s (based on he wo-g aph me hod) can be
used a each hie a chical le el, as e minal blocks a e la in e con-
nec ions
o
basic ci cui elemen s, and he same happens in middle
blocks once he componen subblocks
a e
eplaced by he co e-
sponding ( ans)admi ances.
The ope a ion o he
e o -con olled e m gene a ion
is shown
in Fig.
4.
Ini ially, a equency alue is chosen. Each elemen has
i s admi ance as associa ed weigh . The weigh o each
( ans)admi ance is a complex numbe because i unc ionally
depends on
all
he de ices composing such block.
The con ibu ion o each ( ans)admi ance o he global ci cui
beha io
is
hen nume ically e alua ed. This is e icien ly done
using a hie a chical
MNA
o mula ion and spa se echniques o
sol e he ma ices. The magni ude o hese con ibu ions indica es
which e m gene a o mus become ac i e.
The gene a ion p ocess con inues i e a i ely un il he e o c i e ion
is me . Ob iously, his is gua an eed only a he selec ed equency
sample. An algo i hm o maximum
e o
de ec ion, which elays
in a obus nume ical e e ence gene a o and in e al analysis
O iginal
Ci cui
No
Gene a o
Figu e 4. E o -con olled exp ession gene a ion
echniques is used o de ec equency alues whe e he e o s a e
exceeded
[4].
Then, he p ocess is epea ed un il he e o c i e ia
a e me in he equi ed equency ange.
5.
Expe imen al esul s
Two examples a e analyzed using he p oposed echnique. Each is
.
ep esen a i e o di e en applica ion scena ios: a ci cui com-
posed o blocks ( oo la ge o be analyzed using la analysis algo-
i hms) and a building block desc ibed a he ansis o le el.
5.1.
A
band-pass
il e
The i s example is a decision band-pass il e used in an FSK
modem and shown in Fig. 5(a), whe e he OTA ansis o -le el
schema ics in Fig. 5(c)-(d) and he ansis o model in Fig. 5(b)
we e used. The magni ude/phase e o cons ain s a e
]AMag
5
1 dB
,
]APhs(
I5
deg ees in 10 Hz
I
5
lO'Hz.
4
..
Figu e 5. (a) Band-pass il e ;
(b)
small-signal model;
(c) biasing
OTA;
(d)
OTA
schema ics.
P e iously exis ing hie a chical app oaches did no inco po a e
app oxima ion s a egies and, he e o e, could analyze he ci cui in
Fig. 5(a) only i e y simple mac omodels ins ead o ansis o -
le el desc ip ions o he OTAs we e used.
The small-signal expansion o he ci cui yields a ci cui model
wi h
618
de ices and
45
nodes. A e he ci cui educ ion s ep, he
expanded model con ains
67
de ices and
26
nodes, which means a
la ge educ ion, bu no enough
o
la analysis algo i hms.
Applying
ou
hie a chical app oach, he ollowing ans e unc ion
is ob ained in
100
seconds o
CPU
ime:
(1)
aB9.bp.in
.
(aB8.lp.lp
+
C,
.
s)
-
aB7.bp.l~ aB8.lp.bp
+
aB8.lp.lp. C,
.
s
+
C,
.
C,
.
s2
T
=
whe e he ( ans)admi ances a e he ollowing app oxima e sym-
bolic exp essions:
(3)
111-135
(-gm
I
I
g ~
14g ~
(gds,
+
gds,)
-
gm
,2giii
13gm
lBgd~~2
aB8.lp.bp=
gin,lgin1,(g n12
+
gds2
+
gds,)
(5)
The e o be ween he magni ude and phase beha io p edic ed by
eqs.
(1)-(5)
and he magni ude
and
phase beha io o he o iginal
ci cui
is
shown
in
Fig. 6.
1.0
U
0.5
1
?
1
o5
lo6
equency
(HZ)
io7
5.0
1
I
-3
U
1
o4
1
o5
lo6
equency
(HZ)
io7
Figu e
6.
Magni ude and phase e o s.
5.2.
pa741
ampli ie
Now, he p741 ope a ional ampli ie
in
Fig. 7 will be analyzed.
The magni ude/phase e o cons ain s ‘a e
(AMng(
I
3 dB
,
(Apk l
I
10
deg ees
in
1
Hz
I
I
106Hz, o include he comple e
gain-bandwid h p oduc o he ampli ie . Al hough he ne lis is
inpu wi h
a
p ede ined pa i ioning, which
is
shown
in
Fig.
7, he
in e nal pa i ion size checking de ec s ha he block s uc u e a e
he ci cui educ ion s ep
is
no adequa e o be e icien ly handled.
The e o e, i p o ides some pa i ioning sugges ions (joining Bias,
SC-p o and Ou pu s ages o he uppe le el). Te m gene a ion
yields, hen, he
ollowing
ol age gain:
(6)
uSec 17.9
.
ulnp9.4
go,3B’
(ulnp9.9
+
aSec9.9)
-
uSec9.17. uSecl7.9
H(s)
=
whe e he ( ans)admi ances o he blocks a e
gm3
’
g!n6
’
(gml
+
s
’
CZI)
alnp9.4
=
-
uInp9.9
=
go4
gm5
.
(g nl
+
gi 3)
gg.
5’16.
$‘I7
uSec9.9
=
911116.
S“117
These esul s a e ob ained
in
4.8
seconds
o
CPU ime.
A la analysis ool wi h he same e o speci ica ions p o ides
a
ne wo k unc ion con aining 53 symbolic e ms.
6.
Conclusions
This pape has in oduced
a
me hodology o he inco po a ion o
app oxima ion s a egies in o
a
hie a chical analysis echnique.
On
he one hand, his o e comes he p oblems o app oxima e la
analysis echniques when add essing e y la ge ci cui s.
On
he
o he , he inhe en hie a chy o la ge ci cui s
is
espec ed bu he
in oduc ion o app oxima ion echniques makes he esul s mo e
in e p e able and mo e e icien ly e alua ed han wi h con en ional
hie a chical analysis echniques.
Figu e 7. pa741 ope a ional ampli ie wi h explici
block
pa i ioning and small-signal model o ansis o s.
7.
Re e ences
F. V. Femindez, A. Rod iguez-Vizquez, J. L. Hue as and
G.
Gielen, Eds.,
Symbolic Analysis Techniques. Applica ions o
Analog Design Au oma ion.
Pisca away, NJ: IEEE P ess, 1998.
P. Wambacq, F.V. Femindez,
G.
Gielen, W. Sansen and A.
Rod iguez-Vizquez, ”E icien symbolic compu a ion
o
app oxima ed small-signal cha ac e is ics o analog in eg a ed
ci cui s”
IEEE
J.
Solid-S a e Ci cui s,
ol. 30,
No.
3, pp. 327-
330, Ma ch 1995.
Q.
Yu
and C. Sechen, “A uni ied app oach o he app oxima ed
symbolic analysis o la ge analog in eg a ed ci cui s”
IEEE
T ans. Ci cui s and Sys .-I,
Vol. 43, No. 8, pp. 656-669, 1996.
0.
Gue a, J.D. Rod iguez-Ga cia, E. Roca, F.V. Fe nlndez and
A. Rod iguez-Vhquez, “A simpli ica ion be o e and du ing
gene a ion me hodology o symbolic la ge-ci cui s analysis”
P oc. IEEE Inc. Con$ Elec onics, Ci cui s andSys ems,
Vol.
3,
pp. 81-84, Lisbon, Sep embe 1998.
J.
A.
S a zyk and A. Konczykowska, “Flowg aph analysi:s
o
la ge elec onic ne wo ks,”
IEEE T ans.
on
Ci cui s and Sys .,
M.
M.
Hassoun and K.
S.
McCa ille, “Symbolic analysis o
la ge-scale ne wo ks
using
a
hie a chical signal lowg aph
app oach,”
Analog In . Ci cui s and Signal P oc.,
ol. 3, pp.
3
1-
42, Kluwe , Bos on, 1993.
M.
M.
Hassoun and
P.
M.
Lin,
“A hie a chical ne wo k
app oach o symbolic analysis o la ge-scale ne wo ks,”
IEEE
T ans. on Ci cui s and Sys .4,
ol. 42,
No.
4,
pp. 201-211,
Ap il 1995.
X.D. Tan
and
C.J.R. Shi, “In e p e able symbolic small-signal
cha ac e iza ion o la ge analog ci cui s
using
de e minan
decision diag ams“,
P oc.
o
he Design Au oma ion Con$,
pp.
X.D. Tan and C.J.R. Shi, “Hie a chical symbolic analysis o
la ge analog ci cui s wi h de e minan decision diag ams”,
P oc. IEEE In . Symp. Ci cui s and Sys ems,
pp. 3 18-32]. June
1998.
ol.
CAS-33,
NO.
3, pp. 302-315, Ma ch 1986.
448-453,
Ma ch 1999.
[lo]
0.
Gue a, J.D. Rod iguez-Ga cia and A. Rod iguez-V izquez,
“T ue Hie a chical Symbolic Analysis o La ge-scale Analog
In eg a ed Ci cui s”
P oc. In . Wo kshop on Symbolic Me hods
&Applica ions o Ci cui Design,
pp. 164-167, Oc obe 1998.
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136