SYMBOLIC
ANALYSIS TOOLS
-
THE STATE-OF-THE-ART
F ancisco V. Fe n Andez and Angel Rod iguez-Vkzquez
Dep . o Analog Ci cui Design, Cen o Nacional de Mic oelec cjnica
Edi . CICA, A da. Reina Me cedes s/n, E-41012 Se illa, SPAIN
Phone: +34
5
4239923,
FAX:
+34
5
4231832, E-mail: [email p o ec ed]
ABSTRACT
This pape e iews he main las gene a ion symbolic ana-
lyze s, compa ing hem in e ms o unc ionali y, poin ing ou
also hei sho comings. The s a e-o - he-a in his ield is
also s udied, poin ing ou di ec ions o u u e esea ch.
1.
INTRODUCTION
Ci cui analysis is a basic miles one o e icien design o
in eg a ed ci cui s. E e since powe ul compu e s ha e been
a ailable, designe s ha e de eloped p og ams o analyze ci -
cui s au oma ically. Today, all elec ical enginee ing p o es-
sionals and s uden s use
elec ical simula o s
(such as
HSPICE o ELDO). Howe e , elec ical simula o s do no
co e all he analysis asks equi ed o in eg a ed ci cui
design. Essen ially, hey only se e o
e z y
he pe o mance
o p e iously
sized
ci cui s. Among o he hings, designe s
mus be able o
p edic
he beha io o
unsized
ci cui s by
acing ela ionships among pe o mance igu es and design
pa ame e s. These ela ionships may be in he o m
o
ans-
e unc ions, poles and ze os, oo loci, pa ame ic Bode
plo s, ha monic dis o ion coe icien s, e c. Tools used o
de i e hem au oma ically a e called
symbolic analyze s.
The i s gene a ion o symbolic analyze s appea ed in he
la e 60’s
[l],
whe e he basic analysis echniques we e p o-
posed. Mos
o
hese echniques a e compiled in a ecen book
by P.M. Lin
[2].
These ools did no sp ead on he designe
communi y, pa ially due o hei compu a ional cos and
because hey did no ully i he designe equi emen s. On
he o he hand, he lack o in e es on cus om analog ICs du -
ing he 70’s ende ed symbolic analyze s seconda y as com-
pa ed o elec ical simula ions. This scena io changed du ing
he
80’s
wi h he inc eased economical in e es on cus om
analog and mixed-signal ASICs. Symbolic analyze s we e
p omp ly ecognized c ucial ools o: p o ide insigh and
knowledge abou analog ci cui s; suppo au oma ic beha -
io al modeling
o
analog in eg a ed ci cui s; p o ide com-
piled models o au oma ic epe i i e e alua ions equi ed o
analog syn hesis, among o he applica ions
[3],
[4].
I
esul ed in he eme gence o a new gene a ion
o
ools du ing
he la e
80’s.
Main ea u es
o
his new gene a ion ha e been
app oxima ion, pos -p ocessing and he ex ension o weakly
non-linea ci cui analysis. This pape summa izes some
main con ibu ions in hese second gene a ion
ools,
and
compa es he cha ac e is ics o many o hem.
2.
SYMBOLIC EXPRESSION APPROXIMATION
Simpli ica ion in symbolic analysis is de ined as he educ-
ion o o mula complexi y by elimina ing insigni ican e ms
o sub-exp essions, based on nume ical es ima es
o
he sym-
bolic pa ame e s. Simpli ica ion (o app oxima ion) is
equi ed e en o small ci cui s gi en he exponen ial
inc ease
o
he exp ession leng h wi h he ci cui size. Con-
0-7803-3073-0/96/$5
.OO
‘1996
IEEE
798
en ional simpli ica ion app oaches i s calcula e he
com-
ple e
symbolic exp ession, and hen simpli ies i
--
Simpli ica ion A e Gene a ion (SAG). Di e en e sions o
SAG ha e been implemen ed in mode n symbolic analyze s:
ISAAC [5], SSPICE [6] and ASAP
[7]
o
expanded
o ma
exp essions, and
SYNAP [8]
and ASAP [7] o
nes ed
o ma
exp essions. Di e ences a ise in he con ol o e o s and in
he use o ei he a nominal poin o a nominal in e al o es i-
ma e he magni ude o he symbolic e ms.
Due o he necessi y o gene a ing comple e exp essions,
SAG echniques a e cons ained o small and medium com-
plexi y ci cui s (below abou
100
symbols). La ge ci cui s
can be analyzed by using he newes app oaches o simpli y-
ing
du ing,
o
be o e
o, he analysis s age. They a e called
Simpli ica ion Qu ing Gene a ion (SDG) and Simpli ica ion
Be o e Gene a ion (SBG) echniques espec i ely.
The a ionale behind SDG echniques is ha e ms can be
gene a ed s ic ly in dec easing o de o magni ude s a ing
wi h he la ges . The classical undi ec ed
ee enume a ion
me hod has demons a ed o be he mos e icien o sol e he
unde lying
ma oid
in e sec ion p oblem. This app oach has
been implemen ed wi h sligh di e ences in RAINIER
[9],[10]
and ADAGIO
[11],[12],
he new ool e ol ed om
ASAP and ISAAC.
SBG can ake place in he ne wo k unde analysis, eplac-
ing hose elemen s (o subci cui s), whose con ibu ion
(app op ia ely measu ed) o he ne wo k unc ion is negligi-
ble, wi h a ze o-admi ance o ze o-impedance elemen .
These app oxima ions a e usually done when de i ing ne -
wo k equa ions manually. Howe e , excep o he mos i -
ial eplacemen s, no au oma ic p ocedu e has been epo ed
ye . Repo ed app oaches pe o m app oxima ions di ec ly on
he ne wo k equa ions, ei he in g aph
[SI
o ma ix o m [13],
~41.
3.
POSTPROCESSING
TOOLS
These a e basically in ended o easie in e p e a ion and
manipula ions
o
symbolic o mulae, o o ex ac in o ma-
ion om hem.
3.1.
Symbolic pole-ze o ex ac ion
Poles and ze os a e commonly used by analog designe s.
[
151 in oduced a echnique o symbolic ex ac ion
o
poles
and ze os, based on heu is ic app oxima ions, simila o hose
pe o med by expe analog designe s, e.g.: he
oo -spli ing
echnique. This app oach is also used in [16] which subse-
quen ly applies a symbolic New on i e a ion o e ine he
polelze o loca ions ob ained wi h he oo -spli ing echnique.
Conside , o example, he ac i e eed o wa d compen-
sa ed ampli ie o Fig. 1. A symbolic ex ac ion
o
he i s
pole/ze o pai gi es
[
151
:
3.4. Fac o iza ion and complexi y educ ion
E en app oxima ed exp essions can be di icul o in e p e
due o he conside able size hey ha e when calcula ed wi h
enough accu acy in medium o la ge ci cui s. Fac o iza ion o
common ac o s and cancella ions o common ac o s in
nume a o and denomina o a e some o he needed capabili-
ies o imp o e he in e p e abili y wi hou dec easing he
accu acy. Many mode n ools inco po a e some p imi i e
o m o ac o iza ion. Bu mo e sophis ica ed mechanisms a e
s ill needed, o ins ance
o
elimina e he con ibu ion o bias
ci cui y which usually a ec equally nume a o and denom-
ina o o mos ans e unc ions.
?
Figu e
1
:
Ac i e eed o wa d compensa ed ampli ie
The in e es o symbolic poles and ze os can be seen h ough
his example. F om
(1)
one can in e s he alue o he com-
pensa ion capaci o ha annuls he pole/ze o misma ch and
hence imp o es he high equency beha io o he ampli ie ,
Ano he ex ac ion echnique, epo ed in
[
171, ex ends he
oo spli ing echnique o clus e o poles o ze os which a e
qui e apa om o he clus e s and calcula es app oxima e
exp essions o he ne wo k unc ion o a equency ange
app op ia e o he oo s being calcula ed.
3.2. G aphical in e aces
Some ools inco po a e g aphical in e aces o pa ame ic
ep esen a ions o Bode diag ams o oo locus. Fo illus a-
ion, Fig. 2 shows he g aphical ep esen a ion o he eal pa
o poles and ze os o he ol age gain o Fig.
1
e sus he
alue o
Cpz.
The alue o which he pole/ze o pai cancel is
clea ly seen.
-io*
-106
-104
-102
0
102
io4 io6
108
io10
Re(s)
( adls)
(log)
Figu e
2:
Pole/ze o loca ion ( eal pa ) e sus Cpz.
3.3.
Sensi i i y analysis
Sensi i i y analysis also plays a undamen al ole in analog
ci cui design, specially in he calcula ion o he in luence o
componen misma ches. The sensi i i y
o
a ci cui cha ac e -
ized by a symbolic sys em unc ion
H(s,
x)
can be ob ained
by simply de i ing he sys em unc ion wi h espec o he ci -
cui pa ame e
[
181,
[
191, [20].
4. AN OVERVIEW
OF
CURRENT
TOOLS
Many symbolic analyze s ha e e ol ed as membe s o his
second gene a ion. Exac compa ison o hei pe o mance is
di icul due o he lack o s anda d benchma ks o es ci -
cui s and pla o ms. Some key cha ac e is ics o main
epo ed symbolic analyze s a e compa ed
in
Table 1. A b ie
desc ip ion o hose analyze s ollows.
SCAPP [22] has been implemen ed a Iowa S a e Uni e -
si y, in he amewo k o he de elopmen o a new me hod-
ology o symbolic analysis o la ge ci cui s. The p og am
pa i ions he ci cui s hie a chically and symbolically ana-
lyzes each sub-ci cui sepa a ely. Once he e minal blocks
a e analyzed, an in e se un o he pa i ion p ocess econ-
s uc s he global sys em unc ion. Since his analyze uses an
app oach o la ge ci cui s based on hie a chical analysis, he
sys em unc ions a e gi en as a se ies o exp essions a he
han only one exp ession. Thus, he numbe o symbolic
e ms g ows linea ly, ins ead o exponen ially, wi h he ci cui
size. SCAPP can be ob ained o ee ia anonymous p a
cp e
1
.ee.ias a e.edu:/pub/scapp
SCYMBAL and SYBILIN [23] a e wo simula o s de el-
oped in CNET (Bagneux, F ance). A dis inc i e ea u e o
SCYMBAL is ha i was de eloped speci ically o swi ched-
capaci o ci cui s. I allows no only con inuous a iables
such as conduc ances, capaci o s, e c., as symbolic pa ame-
e s, bu also boolean a iables such as clock phases. On he
o he hand, SYBILIN is a con inuous- ime analyze , specially
concei ed o he analysis o mic owa e ci cui s. These ools
can be ob ained om he au ho s on eques . The p ice
depends on he ype o ins i u ion.
SC
[24]
was de eloped a he Uni e si y o A izona, and
concei ed o class oom use, and hus de eloped o pe sonal
compu e s. I has a se o pos -p ocessing ools o ease in e -
p e a ion o symbolic da a: in e ac i e nume ic e alua ion,
nume ic calcula ion o poles and ze os, and g aphic ep esen-
a ion.
SC
can be ob ained om he au ho s wi h a
$6
han-
dling cha ge.
SAPEC [25] was implemen ed a he Uni e si y o
Flo-
ence, also o PCs. One o i s main cha ac e is ics is he
g oup o applica ions de eloped a ound i , such as symbolic
sensi i i y analysis, ansien analysis o elec onic powe ci -
cui s, calcula ion o es abili y index and aul diagnosis.
SAPEC can be ob ained o ee om he au ho (e-mail:
mane i@ ing i 1 .ing.uni i.i )
SSPICE
[6]
was de eloped a Michigan S a e Uni e si y
o bo h pe sonal compu e s and main ames. I inco po a es
a as albei
no
e y accu a e
SAG
echnique.
I
is also
capa-
ble o pe o ming sensi i i y analysis [19]. SSPICE is a ail-
able om Ins uc ional Media Cen e (Michigan S a e
799
Uni e si y) wi h a cos o $5OO/copy and a license ee o
$250/yea . The e is a limi ed s uden e sion o $25.
SYNAP [8] was de eloped a ETH in Zu ich (Swi ze -
land). Toge he wi h ISAAC and ASAP, i was especially
concei ed o he modelling and design o analog in eg a ed
ci cui s and, mo e speci ically, o be included on CAD ame-
wo ks o au oma ic design o analog in eg a ed ci cui s. I
inco po a es encapsula ed models o semiconduc o de ices
and analog unc ional blocks, and can handle misma ches. I s
newes e sion is capable o analyzing linea con inuous- ime
and sample-da a ci cui s, and also inco po a es a p imi i e
DC analysis capabili y. SAG echniques implemen ed in
SYNAP use he concep o "lazy expansions", which pe -
o ms app oxima ions di ec ly on he nes ed exp essions.
SYNAP is a ailable om he au ho only o uni e si ies and
non-p o i esea ch ins i u ions.
The i s e sion o ISAAC
[5]
was de eloped in he
Ka holieke Uni e si ei Leu en. I was he i s ool o
inco po a e he analysis o weakly nonlinea cha ac e is ics
[26]. Like ASAP, SYNAP, and SSPICE
i
ob ained simpli ied
exp essions using SAG echniques. I uses exac exp essions
in expanded o ma and elimina es hose leas signi ican
e ms whose accumula ed sum emains small. ISAAC o ms
a undamen al pa o he au oma ic design sys em
ARIADNE [27]. ISAAC
is
a ailable om he au ho s o a
$400
handling cha ge.
A
Lucid Common Lisp license is
equi ed. Dis ibu ion o companies is done on a case by case
basis. Cu en ly i is being imp o ed o include SDG and
SBG, among o he new ea u es
--
a coope a i e p ojec
sha ed wi h ASAP.
ASAP [28] was de eloped a he Cen o Nacional de
Mic oelec hica in Se illa (Spain). Like SSPICE, i uns on
wo ks a ions and PCs. I sha es wi h ISAAC he capabili y o
pe o ming g oupings o elemen s and oge he wi h SYNAP,
he explici exp ession o misma ching. I calcula es app ox-
ima e poles and ze os symbolically. ASAP also inco po a es
a new SAG c i e ia which educe e o s by conside ing in e -
als o he symbolic pa ame e s ins ead o ixed nume ic al-
ues. App oxima ion wi h a ia ion anges has also been used
o de elop a eliable and p ecise c i e ion o simpli y nes ed
exp essions esul ing om hie a chical analysis. ASAP has
been comme cialized wi hin he ELDO amewo k and is
a ailable h ough ANACAD-EES.
SIFTER
[
141
and RAINIER [9] ha e been de eloped a he
Uni e si y o Washing on a Sea le, and ha e mainly
add essed he implemen a ion o SBG and SDG echniques,
as discussed in Sec ion 2. RAINIER is a ailable o ee ia
anonymous p a
wol 8.ee.washing on.edu.
5.
FUTURE DEVELOPMENTS
One o he majo challenges o u u e symbolic analyze s
is inc easing he complexi y o ci cui s, while keeping he
in e p e abili y o esul s.
A
possible solu ion has been shown
o be he applica ion o hie a chical analysis echniques.
These echniques a e specially adequa e o la ge ci cui s
consis ing o loosely coupled sub-blocks. Fo s ongly con-
nec ed ci cui s new SBG and SDG echniques a e a ising.
The symbolic analysis o la ge ne wo ks has been he ocus
o in ense a en ion in he las yea s,
is
s ill o in e es oday,
and doub less will con inue o be
o
in e es in he u u e.
Table
1
shows ha mos ools a e limi ed o he analysis o
linea (ized) ci cui s. The i s a emp o nonlinea analysis
has been epo ed in [26] al hough limi ed o weak non-linea -
i ies. Reducing he ac ual symbolic analysis echniques o he
s udy o linea o weakly nonlinea cha ac e is ics limi s i s
p ac ical applica ions. I can be assu ed ha he possible
u u e a ailabili y o la ge signal symbolic analysis ech-
niques will suppose he ull in eg a ion o hese ools in au o-
ma ic design and es sys ems, and in gene al, wi hin any
CAD en i onmen . Rega ding he ime domain, he i s
a emp s a ime-domain symbolic simula ion ha e been
epo ed in [291.
Toge he wi h he de elopmen o symbolic analysis ech-
niques o non- adi ional cha ac e is ics, and he new, mo e
e icien algo i hms, applica ions a e being ampli ied wi hin
ields such as au oma ic design o analog in eg a ed ci cui s.
REFERENCES
[
11
P.
M.
Lin, "A Su ey o Applica ions o Symbolic Ne wo k
Func ions,"
IEEE T ans.
on
Ci cui Theo y,
Vol. CT-20, pp.
732-737,
NOV.
1973.
[2] P.M. Lin,
Symbolic Ne wo k Analysis.
Else ie , 1991.
[3] G. E. Gielen and W. Sansen,
Symbolic Analysis o Au oma ed
Design o Analog In eg a ed Ci cui s.
Kluwe Academic
Publishe s, 1991.
[4]
A.
Rod iguez-VBzquez e al. (eds.),
Analog CAD Tools:
Symbolic Techniques and Applica ions.
IEEE
P ess, 1996.
[5] G. Gielen,
H.
Walscha s and
W.
Sansen, "ISAAC: A
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No.
6,
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[6] G. Wie zba, A. S i as aba,
V.
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-
A Symbolic
SPICE
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171 F.V. Fe n indez,
A.
Rod iguez-V izquez, J.D. Ma in and J.L.
Hue as, "Fo mula App oxima ion o Fla and Hie a chical
Symbolic Analysis,"
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P ocessing,
Vo1.3, pp. 43-58, Kluwe , Jan. 1993.
81
S.
Seda,
M.
Deg auwe and W. Fich ne , "Lazy-Expansion
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P oc. IEEE
In . Con$ Compu e -Aided Design,
pp. 3 10-3 17, 1992.
91 Q.
Yu
and C. Sechen, "App oxima e Symbolic Analysis o
La ge Analog In eg a ed Ci cui s,"
P oc. IEEE In .
Con$
Compu e -Aided Design,
pp. 664-67 1, 1994.
lO]Q.
Yu
and C. Sechen, "E icien App oxima ion o Symbolic
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pp. 2088-2091,
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[11]F.V. Fe nBndez, P. Wambacq, G. Gielen, A. Rod iguez-
V izquez and W. Sansen, "Symbolic Analysis o La ge
Analog In eg a ed Ci cui s by App oxima ion Du ing
Exp ession Gene a ion,"
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pp.25-28, 1994.
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A.
Rod iguez-VBzquez, "E icien Symbolic Compu a ion
o
App oxima ed Small-Signal Cha ac- e is ics
o
Analog
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IEEE
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Ma 1995
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G.
D oge and E.-H. Ho nebe ,
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wi h
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and
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--
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C dC,
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457-460, 1994.
[
15lF.V. Fe nBndez,
A.
Rod iguez-VBzquez and J.L. Hue as,
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Tool o Symbolic Analysis o Analog In eg a ed Ci cui s
Including PoleIZe o Ex ac ion,"
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Ci cui Theo y and Design,
pp. 752-761, 1991.
800
TABLE
1:
COMPARISON
OF
SYMBOLIC ANALYZERS
SC
MNA
S
no
no
no
SCYMBAL
/SYBILIN SAPEC SSPICE
MNA MNA
S S
no Yes
no no
no no
SYNAP
MNA
s&z
Yes
Yes
ISAAC
CMNA
s&z
Yes
no
ASAP
SFG
S
Yes
Yes
SIFTER RAINIER
nodal ee
analysis enum.
S
S
no no
no no
Ez=pT
&
SFG
Fo mula ion
Analysis domain
SAG
(expanded o ma )
SAG
(nes ed o ma )
SBG no
I
no no
I
yes
I
yes
SDG
no
I
no no
I
no
I
yes no
I
yes yes
I
no
I
yes
Misma chings no
I
no yes
I
no
I
no
Elemen lumping no
I
no no
I
yes yes
I
no
I
no
Nonlinea analysis nonlin.
-I-
Hie a chical
analysis
P/Z ex ac ion no
I
no no
I
no
I
no
G aphical in e ace no
1
no yes
I
yes
I
no
Pla o ms
WS
=
wo ks a ions)
ws
I
ws
Implemen a ion
language
c
I
LISP/
I
c
LISP/ LISP/
c++
I
c
C++
[
16lG. Nebel
e
al., "Symbolic Pole/Ze o Calcula ion Using
SANTAFE,"
P oc. Eu opean Solid-s a e Ci cui s Conj
[17]J.
Hsu
and C. Sechen, "Accu a e Ex ac ion o Simpli ied
Symbolic Pole/Ze o Exp essions o La ge Analog IC's,''
P oc. IEEE In . Symp. Ci cui s and Sys ems,
pp.2083-2087,
1995.
[
18]A. Libe a o e and
S.
Mane i, "Ne wo k Sensi i i y Analysis
Via Symbolic Fo mula ion,"
P oc. IEEE In . Symp. Ci cui s
and Sys ems,
pp. 705-708, 1989.
[19]S. Chang and G. Wie zba, "Symbolic Sensi i i y Analysis
Using Sspice,"
P oc. IEEE Midwes CAS,
pp. 1043-1046,
1991.
[20]P.M. Lin, "Sensi i i y Analysis
o
La ge Linea Ne wo ks
Using Symbolic P og ams,"
P oc. IEEE In . Symp. Ci cui s
and Sys ems,
pp. 1145-1 148, 1992.
[21]J.A. Ech enkamp, M.M. Hassoun, G.M. P abhu and C.T.
W igh , "Hie a chical Sensi i i y Analysis
o
Sequence
o
Exp essions Me hod,"
P oc. Eu opean
Con&
Ci cui Theo y
and Design,
pp. 75-78, 1995.
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o Symbolic Simula ion o La ge-scale Ne wo ks,"
P oc.
lEEE
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Symp. Ci cui s and Sys ems,
pp.
806-809,
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E icien Repe i i e Analysis and A i icial In elligence
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P oc. IEEE In . Symp. Ci cui s and
Sys ems,
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[24]L.P. Huelsman, "Pe sonal Compu e Symbolic Analysis
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