Beha io al Modeling
o
PVVL
Analog Ci cui s using Symbolic Analysis
F ancisco
V
Fe ndndez, Bel& Pe' ez-Ve du' and Angel Rod iguez-Vdzquez
Ins i u o de Mic oelec dnica de Se illa
Cen o Nacional de
Mic oelec 6nica-C.S.I.C.
Edi icio CICA-CNM, A da. Reina Me cedes sln, 41
0
12-Se illa, SPAIN
FAX:: 34
5
4231832 Phone: 34
5
4239923 email: [email p o ec ed]
ABSTRACT^
Beha io al models a e used bo h o op-down design and
o bo om-up e i ica ion. Du ing op-down design, models
a e c ea ed ha e lec he nominal beha io o he di e en
analog unc ions,
as
well
as
he cons ain s imposed by he
pa asi ics. In his sce la io, he a ailabili y o symbolic mod-
eling exp essions enable designe s o ge insigh on he ci -
cui s, and educes he compu a ional cos o design space ex-
plo a ion. Du ing bo om-up e i ica ion, models a e c ea ed
ha cap u e he opological and cons i u i e equa ions o he
unde lying de ices id o beha io al desc ip ions. In his sce-
na io symbolic analykis is use ul because i enables o au o-
ma ically ob ain hesi desc ip ions in he o m o equa ions.
This pape includes an example o illus a e he use o sym-
bolic analysis o op-down design.
1.
INTRODUCTION
Ci cui analysis
is
he co ne s one o elec onic ci cui en-
ginee ing. On he on1 hand, i p o ides he keys o unde -
s anding he in ica e mechanisms unde nea h he ci cui
op-
e a ion. On he
o he ^
hand, designe s use analysis o ob ain
models o he ci cui lbeha io
-
he basis on which his be-
ha io can be p edicded. Howe e , manual analysis o e en
he simples ci cui s kncoun e ed in p ac ical applica ions is
a
complica ed, ime-consuming, and e o -p one ask. Sym-
bolic analyze s a e in ended o elie e designe s o sys ema -
ic manual analysis hsks, hus le ing hem concen a e on
c ea i e issues.
I
The las gene a io? symbolic analyze s a e able o handle
up o a ound 400 di b en symbols
[
11.
This pe mi s, o ex-
ample, o analyze di cui s
as
complex
as
he ail- o- ail
CMOS opamp
o
Fig!l(a) 121 wi h he small-signal ansis o
model o Fig. l(b) [3]1-[5]. The analyze s capabili ies include
he calcula ion o s-dbmain exp essions o all ypes o d i -
ing-poin and ans e cha ac e is ics, he simpli ica ion o
hese exp essions o e ain only he dominan e ms, he
ex-
ac ion o hei
poles^
and ze oes, e c.
Recen ly, di e en
au ho s
ha e ocused also on he sym-
hose weak nonlinea -
I
bolic analysis o no ci cui s
[6][7].
The mos impo -
a
ew e ms ( ypically
1.
This wo k
has
been pa ially suppo ed by he Spanish C.I.C.Y.T. unde
con ac TIC97-0580 ind he EEC in he p ojec ESPRIT AMADEUS.
I
I
I
I
B
Figu e
1.
(a) Rail- o- ail opamp;
(b)
small-signal
MOS
ansis o
model.
3) o
a
powe se ies expansion. In [6] sys ema ic echniques
a e p oposed o he au oma ic calcula ion o he Vol e a
ke nels [8] ha cha ac e ize he beha io o hese weakly-
nonlinea ci cui s in he equency domain. Howe e , he
symbolic analysis
o
ha dly-nonlinea ci cui s is ye in ex-
plo a o y phase
171.
To da e, no sys ema ic echnique is
a ailable o au oma e he calcula ion o
a
se o symbolic
equa ions go e ning he la ge signal nonlinea beha io , ei-
he in he s a ic o in he dynamic case.
This lack o nonlinea analysis limi s he modeling capa-
bili ies
o
s a e-o - he-a symbolic analyze s. Howe e , in
many p ac ical applica ions hese limi a ions can be o e -
come by eso ing o piecewise-linea (PWL) ep esen a-
ions. Fo ins ance, hese ep esen a ions su ice o s udy he
quali a i e beha io o many p ac ical dynamical ci cui s
such
as
oscilla o s
and
compa a o s,
as
well
as
o
app oxi-
ma e he ansien esponse o opamps and compa a o s.
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0~7803-4455-3/98/%10.~0
0
1998
lEEE
2.
BEHAVIORAL MODELING
A beha io al model is a se o equa ions ha cap u e he
ope a ion o a ci cui om i s e minals. Beha io al models
can also be gi en as ci cui al ep esen a ions o hese equa-
ions. Depending on he na u e on he exci a ions and e-
sponses, he se
o
equa ions and i s associa ed ci cui al ep-
esen a ion can be s a ic o dynamic, linea o nonlinea . Fo
example, Fig.2 shows di e en beha io al models o an
opamp. The i s one is s a ic and he o he s dynamic; he
i s wo a e linea and he o he s nonlinea . Ci cui al ep e-
sen a ions a e included o each model.
The cons uc ion o a beha io al model does no s ic ly
equi e o know he de ailed ci cui schema ic. Nei he he
model ci cui al ep esen a ion mus e lec he ci cui opolo-
gy. Models can be buil by, i s , applying a sui able se o ex-
ci a ion- esponse pai s and, hen, inding and uning a ma h
s uc u e ha ep oduce hese pai s wi hin some p esc ibed
e o ma gin. Conside o example he OTA shown in
Fig.3(a) and assume ha we a e in e es ed only in i s linea
AC beha io . This can be modeled by
h
-pa ame e s, which
can be expe imen ally calcula ed in he lab by using a ne -
wo k analyze in he con igu a ions o Fig.3(c).
Al e na i ely, beha io al models can be cons uc ed
h ough he elec ical analysis o he in e nal ci cui schema -
ic. Fo ins ance, assuming ha he
OTA
schema ic o
A dd
.
Figu e
3.
(a) Simple
OTA;
(b) simpli ied small-signal
MOS
an-
sis o model; (c) black box modeling p ocedu e o a wo-po .
Fig.3(b) and he ansis o model o Fig.3(c), a las gene a-
ion symbolic analyze such as
SYMBA
[9]
would e u n he
ollowing
h
-pa ame e app oxima ed exp essions,
hi(s)
=
"b
-
+
-
gm3(gml+ gm2)
2
-
-
scg.~2gm1gm3
Cgs2[gml(Cgs3
'gs4)
+
gm3Cgsl]
s[gm3(cgsl
+
'gs2)
+
(gml
+
gm2)(Cgs3
+
Cg. dI
sCgs2gmlgm3
+S
Cgs2[gml(cgs3
+
Cg34)
+
gm3CgslI
h, W
=
+
2
+
n-;
+
FVO
Vi-+
u
VIP
'"b
-
gdslgm3 +sgd.~l(Cg.sl
+
cgs3
+
Cg.J
gmlgml+
S[gml(Cg. 3
+
Cgs4)
+
gm3Cgsll
gds2
gm1
+sCgsl
gds1gm3
+
Sgdsl(Cgs1
+
cgs3
+
Cgs4)
gmlgm3 +s[gin1(cgs3
+
cg.~4)
+~m3Cg. 1]
-
-
-
h,(s)
=
Vi-+
,+
i;
h,(s)
=
(1)
In p ac ice, designe s use hei expe ience o build modu-
la models, whe e unc ional subs uc u es a e used o ep e-
sen he nominal ci cui beha io as well as he pa asi ics.
Thus, o example, Fig.2(a) cap u es he nominal opamp be-
-
-
Figu e
2.
Opamp beha io al models.
VI-
18
ha io ; ha o Fig.2
o he dynamics ass1
and Fig.2(c) models
y. These h ee mod<
o he hand, Fig.2(d)
ing-poin beha io s
The di e en un
ha io al model can
su emen s o h oug
he case o
PWL
mc
nonlinea subs uc u
ed h ough hei syi
p esen ed in he sec
in e es
o
hese
PW
3.
A PWL MOD
OPA
Conside he SC
opamp ep esen ed
I
ansien obse ed ii
he in eg a ing capac
pa . The linea pa
gml(
V,(S)
=
-
C,(C
VJS)
=
-
whe e,
yc
=
cic2.
yielding,
whe e he coe iciei
pa ame e s
[lo].
Th
lowing condi ion is
1)
includes a i s -o de app oxima ion
:ia ed o any ol age gain mechanism;
ie i s s age ansconduc o nonlinea i-
;
ocus on he ans e beha io , On he
ip u es also o a i s app oach he d i -
he ci cui e minals.
onal subs uc u es pe aining
o
a
be-
e
uned ei he h ough blackbox mea-
analysis o he unde lying ci cui . In
els,
measu emen s a e used o une he
s
while hose linea ones a e ep esen -
bolic ans e unc ions. The example
)n
below illus a es abou he p ac ical
models.
L
FOR TOP-DOWN TRANSIENT
[P OOPTIMIZATION
i eg a o o Fig.4(a) and assume he
he
PWL
model o Fig.4(b)
[lo].
The
he ans e ing o he inpu ol age o
o con ains a linea pa an a nonlinea
in
be calcula ed symbolically om,
ic,
+
C2C,
+
c2c,
+
c,c,
,
=
ci
+
c,
+
c,
(3)
cp(-a )cos@
+
C,, exp(-a )sinp
up(-a )cosp
+
Cllexp(-a )sinp
(4)
cp(-a )cosp
+
CUI exp(-a )sinp
s
a e gi en as unc ions o he model
e equa ions emain alid while he ol-
Hilled,
8mzI *
(q
I,
(5)
VI-
19
Figu e
4.
(a)
SC
in eg a o and,
(b)
opamp model.
A e wa ds, he second model s age en e s in o sa u a ion and
he ansien is desc ibed by he ollowing equa ions,
whe e.
and he emaining coe icien s a e gi en
as
unc ions o he
model pa ame e s
[
101.
This second ansien pe sis s while
he condi ion
(5)
holds. A e wa ds, he ansien is gi en by,
yielding,
,( )
=
A,,
+B,,exp(-a )cosp + C,,exp(-a )sinp
(10)
whe e he coe icien s a e, as in he p e ious exp essions,
gi en
as
unc ions o he
model
pa ame e s.
The model has been alida ed by compa ing i s ou pu
wa e o m o ha o ob ained h ough de ailed elec ical sim-
ula ion. The ampli ie consis ed o a ully-di e en ial old-
ed-cascode OTA whose co e schema ic and summa ized pe -
o mance a e gi en in Fig.5. Said ampli ie was designed o
ha e small phase ma gin (a ound 4Sdeg) o educe powe
dissipa ion. Fig.6(a) shows he in eg a o ou pu ol age du -
ing he in eg a ion phase ob ained h ough elec ical simula-
ion (HSPICE) and ha ob ained using he model. Model pa-
ame e s a e also enclosed in Fig5 A good conco dance be-
ween bo h app oxima ions is obse ed.
A
mo e gene al
esul is gi en in Fig.6(b) whe e he di e ence be ween he
inal alue o he in eg a o ou pu ol age and i s ideal alue
is shown as a unc ion o he inpu le el.
V-
--I
GBW(3.2pF)
=
106.7MHz
PM
(3.2pF)
=
42.2O
I,
=
500pA
Figu e
5.
Fully
di e en ial
OTA.
0.4
.
.
1
0.3
2
0.2
2
0.1
2
0.0
c
c
0
0)
+-
-0.1
-0.2
-
-0.40
5.0e-09
1
.Oe-08 13-08 2.0e-08
2.5e-08
Time
(s)
(4
-
HSPICE
1
--Model
0.5
1
.o
1.5
2.0
-0.1
‘0
Vol age
s o ed
in
Ci
(V)
Figu e
6.
Simula ed and calcula ed esponses.
(b)
4.
REFERENCES
11 F.V. Fe nsindez, A. Rod iguez-Vsizquez, J.L. Hue as
and G. Gielen, eds.,
Symbolic Analysis Techniques. Ap-
plica ions o Analog Design Au oma ion.
IEEE P ess,
1998.
21
W.S. Wu, W. Helms, J.A. Kuhn and B.E. By ke , “Dig-
i al-compa ible high-pe o mance ope a ional ampli ie
wi h ail- o- ail inpu and ou pu anges,”
IEEE
J.
Sol-
id-s a e Ci cui s,
Vol. 29, pp. 63-66, Jan. 1994.
[3] P. Wambacq, EV. Fe nsindez, G. Gielen,
W.
Sansen and
A. Rod iguez-VBzquez, “E icien symbolic compu a-
ion o app oxima ed small-signal cha ac e is ics o ana-
log in eg a ed ci cui s,”
IEEE
J.
Solid-s a e Ci cui s,
Vol.
30,
pp. 327-330, Ma ch 1995.
[4]
Q.
Yu and
C.
Sechen, “A uni ied app oach o he app ox-
ima e symbolic analysis o la ge analog in eg a ed ci -
cui s,”
IEEE T ans. Ci cui s and Sys ems
I,
Vol. 43,
No.
[5] F.V. Fe nsindez,
0.
Gue a, J.D. Ga cia and A.
Rod iguez-VBzquez, “Symbolic analysis o analog
in eg a ed ci cui s: he nume ical e e ence gene a ion
p oblem,”
IEEE T ans. Ci cui s and Sys ems,
1998, o
appea .
[6] P. Wambacq,
Symbolic Analysis
o
La ge and Weakly
Nonlinea Analog In eg a ed Ci cui s.
Ph.D. Disse a-
ion, Ka holieke Uni e si ei Leu en, Belgium, 1996.
[7]
C.
Bo che s,
L.
Hed ich and E. Ba ke, “Equa ion-based
model gene a ion o nonlinea analog ci cui s,”
P oc.
33 d Design Au oma ion Con$,
pp. 236-239, 1996.
[8]
S.
A. Maas,
Nonlinea Mic owa e Ci cui s,
A ech
House 1988.
[9] IMSE-CNM,
Me hods o
SBG
and
SDG.
ESPRIT
P ojec 21812 AMADEUS, In e nal Repo , Ma ch
1997.
[
101
E
Medei o,
XA-modula o In e~aces o Mixed-Signal
CMOS
IC’s:
Modeling, Simula ion and Au oma ed De-
sign.
Ph.D. disse a ion, Uni e si y
o
Se illa, Spain,
1997.
8,
pp.
656-669, Aug. 1996.
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