OTA-B
ased Non-linea Func ion App oxima
ions
Edga Shchez-Sinencio’, Jaime Rami ez-Angulo’, Be nabe‘ Lina es-Ba anco’*’, Angel Rod iguez-Viizquez2
‘Depa men o Elec ical Enginee ing, Texas A&M Uni e si y
College S a ion, Texas 77843-3128, USA.
2Depa amen o de Elec 6nica y Elec omagne ismo
Uni e sidad de Se illa, 41012-Se illa, Spain
Abs ac
-
The
sui abili y o ope a ional anscon-
duc ance ampli ie s (OTAs)
as
he
main
ac i e ele-
men
o
ob ain
basic building blocks o
he
design
o p og ammable
non-linea
con inuous- ime
ne -
wo ks is
p esen ed.
The main
pu pose
is
o
show
ha
he
OTA,
as
he ac i e
elemen in
basic build-
ing blocks can
be
e icien ly used o non-linea
con inuous- ime unc ions syn hesis.
Two
e icien
non-linea unc ion syn heses app oaches
a e
p e-
sen ed. The i s app oach is
a
a ional app oxima-
ion and he second is
a
piecewise-linea app oach.
Tes ci cui s ha e been in eg a ed using
a
3j1
p-well
CMOS
p ocess. The lexibili y ci
he
designed and
es ed ci cui s is con i med.
I.
INTRODUCTION
La ely, se e al au ho s
[I]
-
[5]
ha e been success ully
using he Ope a ional T ansconduc ance Ampli ie (OTA)
as
he main ac i e elemen in con inuous- ime ac i e il-
e s. The OTA’s p og ammabili y na u e’ and he ac
ha OTAs ha e only a single high impedance node, in
con as o con en ional op amps make he OTA
an
excel-
len de ice candida e o high equency and ol age (o
cu en ) p og ammable analog basic building blocks. The
applicabili y o OTAs
as
componen s o he design o lin-
ea ne wo ks has been ex ensi ely discussed elsewhe e
[l],
[6] and no epea ed he e. The objec i e o his pape
is
o examine he applicabili y
o
OTAs
as
he basic elemen s
o he design o non-linea ne wo ks. The e
is
no much
epo ed in he li e a u e on he use o OTA o designing
non-linea block componen s [7] -[8]. The e a e epo ed
excellen con ibu ions
191
-
[ll], I161 o non-linea ci cui s
dealing wi h pa icula impo an non-linea p oblems. In
ou p oposed app oach a he han y o ackle a speci ic
p oblem, we ocus ou a en ion in
a
gene al app oach deal-
ing wi h non-linea basic building blocks using OTAs
as
he main ac i e elemen s. A his poin no emphasis was
done o op imize he ci cui pe o mance bu o explo e he
po en ial and applicabili y o he OTA-based non-linea
sys em app oach.
’
The ou pu cu en
Io
o
an OTA due o a di e en ial inpu
Vid
is
10
=
gmUid
and
gm
is
a ol age (cu en ) con ollable pa-
ame e ll], [6],
171.
96
ISCAS ’89
11.
BASIC
BUILDING
BLOCKS
Mul iplie Block. A wo inpu ou quad an mul iplie has
an ou pu (cu en ) gi en by
whe e he mul iplie cons an
KM
has uni s o A/V2. I
Vl and
Vz
can ake any posi i e o nega i e sign, he mul-
iplie is called
a
ou -quad an mul iplie . This mul iplie
is ep esen ed in Fig.
l(a).
The co esponding OTA-based
implemen a ions a e shown in Fig. l(b). The block ‘an
ep esen s a signal a enua o , i s unc ion
is
such ha he
maximum ol age swing
o
Vl and V2 a e equalized, and
-Vbb is he usual bias con ol o he OTA. An ac i e a -
enua o can be implemen ed in CMOS echnology [15].
Al hough no indica ed in Fig. 1, assume he powe
sup-
plies o he OTAs a e VDD and -V~S. Fo he ci cui o
Fig: l(b) we ob ain
and
I,=-g
mZ
1
-
-
-K(VI,
+
VSST)Vl
(2b)
whe e
K
is
a p ocess- and geome y-dependen cons an ,
Vss~
=
Vss -V , and V is
a
ansis o h eshold ol age2.
The ou pu cu en becomes
IO
=
-aKVlV2
=
-/KMIVIV~
whe e
IKMI
=
aK.
We ha e he lexibili y o making he
sign o
KM
posi i e o nega i e by injec ing V2 o OTAl
ins ead o o OTA2.
Di ide Block.
A
wo inpu di ide has an ou pu which
is
he a io o he wo inpu s, mul iplied by a dimensional
(in ol s) cons an
KR,
i.e., VO
=
KR~.
A-symbol o
he di ide
is
shown in Fig. 2(a), whe e he no a ion
n
and d s ands o nume a o and denomina o , espec i ely.
The co esponding OTA-based ci cui implemen a ion us-
ing he mul iplie symbol is shown in Fig. 2(b).
Squa ing and High
Powe s
(Ezponen ia ion) Blocks.
A
one
inpu squa e has an ou pu p opo ional o he squa e o
We ha e Msumed equal
K’s
and h eshold ol ages
’s
o
he
OTAs.
CH2692-2/89/0000-00%
$1.00
0
1989
IEEE
Ucpa mn
o
Elcc ncal
and
Compu e
EngLneenng
and
The
coo dina ed
Science
Labo a o y
he inpu ,
10
=
KMV:. The implemen a ion o he squa e
is ob ained by simply using a mul iplie wi h equal inpu s.
To ob ain an
ezponen ia ion
( aising o a powe ) block
op-
e a o wi h an inpu V; and an ou pu o be p opo ional
o
Yp
whe e
p
>
2,
i is equi ed
(p
+
1)/2
mul iplie s o
p
odd and
p/2
mul iplie s o
p
e en. Fu he mo e, since
he p oposed mul iplie s a e o he ansconduc ance ype,
he ou pu s mus be con e ed in o ol ages o be able o
use hem
as
he inpu s o ollowing mul iplie s. This can
be easily ob ained by connec ing an equi alen esis o a
he ou pu . An equi alen esis o using an OTA [5]
-
[6]
is implemen ed by connec ing he ou pu o he nega i e
OTA inpu and g ounding he posi i e OTA inpu .
Squa e-Roo e Block.
A one inpu squa e- oo e has an
ou pu wi h he nega i e o posi i e squa e oo o an inpu
ol age mul iplied by a cons an o a p ope pola i y, e.g.,
o=*lGl
,
V,>Oo Vo=+IJqI
,
i<
0.
Fig. 3(a) shows he implemen a ion o he squa e-
oo e , whe e he ou pu VO is gi en by VO
=
KR~
which
yields V,
=
1-1.
A mo e de ailed desc ip ion o he
implemen a ion is shown in Fig. 3(b).
Piecewise-Linea Func ion Gene a o s.
Diodes in e con-
nec ed wi h OTAs can simula e ideal diodes, hus allow-
ing a piecewise-linea app oxima ion o any desi ed non-
linea unc ion. The accu acy, na u ally, imp o es wi h he
numbe o line segmen s in ol ed. The ideal basic building
block o piecewise-linea unc ion app oxima ion
is
shown
in Fig. 4. No e ha
ID
=
0
un il he b eaking poin
( ol age e e ence
V,)
is eached. .The
slopes
o he lin-
ea segmen s a e p opo ional o he
gm’s.
The diodes can
be implemen ed wi h MOS ansis o s wi h hei ga e and
d ain ied oge he . I a s ep ype inpu -ou pu cha ac e -
is ic is needed o implemen discon inui ies in he unc ion
app oxima ion, he linea OTA can be subs i u ed by an
OTA compa a o which ideally simula es a la ge
gm
and a
sa u a ion (ou pu ) cu en o
* biaS.
111.
NONLINEAR
FUNCTION
SYNTHESES
A
a ional app ozima ion
ha has he gene al o m
o
a polynomial unc ion o o a a io o polynomials,i.e.,
whe e
i
is a posi i e in ege numbe . In ac , he exponen
i
can be a ac ional exponen
o
he o m
p/q,
whe e
p
and
q
a e nega i e o posi i e in ege s. The exponen a ion
blocks a e o he ype o Fig. 5. I a nega i e
-p/q
is
needed, an addi ional di ide has o be used.
A
piecewise linea app ozima ion
can be ob ained by
using he basic building block
o
Fig.
4.
Changing he
PO-
la i y o diodes and inpu e minals o OTA’s allow he
ob
en ion o nega i e and posi i e slopes. A bi a y unc ions
wi h a iable posi i e and nega i e slopes can be app oxi-
ma ed. Fu he mo e, he
slopes
a e
ol age p og ammable3
which gi es an addi ional lexibili y in he unc ion app ox-
ima ion design p oblem. One example
o
an a bi a y unc-
ion app oxima ion con aining nega i e and posi i e slopes
is discussed in he nex sec ion. De ails on he p ac ical
conside a ions o he OTA-based piecewise-linea ci cui s
a e unde conside a ion.
I .
EXPERIMENTAL
RESULTS
Se e al es -ci cui s con aining OTAs and ansis o s
connec ed
as
diodes we e ab ica ed using a 3pm p-we11
CMOS p ocess by MOSIS. The linea ized OTA used o
syn hesize he di e en non-linea analog unc ions is e-
po ed in (31. The die a ea o each OTA is
220x700p nz
and i consumes 10
mW
o powe wi h 5V supply ol -
ages. A chip pho omic og aph showing wo comple e OTAs
o
he es ci cui
is
depic ed in Fig 6.
A.
Pansconduc ance Mul iplie .
The s uc u e used
is
shown in Fig. 6. The measu ed alue
o
KM
is
2.4pA/V2. In all measu emen s desc ibed he e a LOOKS?
load esis o was used. The la ge-signal cha ac e -
is ics o he mul iplie a e shown in Fig
7.
Vi was
held cons an (a
O.OV,
&0.33V, 0.66V, l.OOV),
while he inpu
Vz
a ied be ween klV. The non-
linea i y e o
is
shown in Fig
8.
Fo Vz, a iangula
2
ol s peak- o-peak signal was applied, while keeping
Vi
=
1V. The ou pu cu en p oduced a iangula
ol age signal o 660 mV peak- o-peak. Subs ac ing
his signal om an ideal iangula wa e, he esul ing
peak- o-peak e o signal was
17
mV which yields a
non-linea i y e o o nea ly
2%.
Repea ing he mea-
su emen bu in e changing
VI
and Vz and being Vi a
iangula signal o
2
ol s peak- wpeak.
A
peak- o-
peak e o signal wi h
an
ampli ude o
23
mV co e-
sponding
o
a
3.5% non-linea i y e o was measu ed.
Fig.
9
shows he mul iplie being used
as
a
modula o
o he case whe e bo h inpu signals a e sinusoidal.
B.
Piecewise Linea App ozima ion.
The in ended ans-
e cha ac e is ic o be conside ed is shown in Fig.
lO(a) and consis s o h ee linea segmen s. The in-
di idual slopes due o each OTA a e indica ed in he
lowe pa
o
Fig. lO(a), and he composed esul ing
ans e cha ac e is ics a e shown in he uppe pa o
Fig.
lO(a).
The ac ual OTA ci cui implemen a ion
is
shown in Fig. 10(b) whe e an op ional diode and ol -
age sou ces ha e been added a he OTA
(2
and
3)
and
shown wi h b oken lines o imp o e high equency
pe o mance o he ci cui . No e ha he
slopes
o
he ans e cha ac e is ics can be easily modi ied by
changing he OTA ol age-dependen ansconduc-
ances. The expe imen al esul s a e shown in Fig.
lO(c).
V.
CONCLUSIONS
The sui abili y
o
OTAs
as
he main ac i e elemen o
ob ain basic building blocks o he design
o
non-linea
ne wo ks was es ablished. Me hods
o
implemen p ac-
Addi ionally,
i
a esis i e load simula ed wi h
an
OTA
is
used,
he slopes become a ios
o
ansconduc ances which p o ida a e y
good empe a u e compensa ion and accu acy imp o emen .
97
ical non-linea ci cui s in a sys ema ic design app oach
we e de eloped. Two p ac ical syn hesis app oaches we e
in oduced. The p og ammabili y and lexibili y o he
OTA
p o ides he po en ial o design adap i e non-linea
ci cui s. Implemen a ions o o he non-linea syn hesis
app oaches a e easible using he basic blocks he e in o-
duced. The p oposed OTA-based building blocks can be
inco po a ed
in
a CAD so wa e
[12]
o ully exploi hei .
unc ionali y and e sa ili y. The es -in eg a ed ci cui
expe imen al esul s e i ied he heo e ical p edic ions.
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E.
Sbnches-Sinencio, ‘Ac i e
Fd m
Ddgn
U&
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A
Tu mizJ*,
IEEE
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20-32,
Ma ch 1985.
[2]
C. Ple , M. A. Copeland and
R.
A. Hadaway, ‘Ton inuow-Time
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Jae,
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May
1986.
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R.
L.
Geige , ‘High-F equency Vol age-
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E.
Shche -Sinencio, R. L. Geige and H. Ne a e -Losano, ‘Gen-
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o
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S uc u es”,
IEEE
Zh cms.
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936-946, Augus 1988.
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o
All
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Si-
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G. Sodini, P.
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I.
Fig.
1
Mul iplie
(a)
Symbol, (b) OTA Implemen a ion
1.
.
A
Fig.
2
Di ide (a) Symbol, (b)
OTA
Implemen a ion.
“D ,
I I
T’
Fig.
3
Squa e-Roo e , (a) Implemen a ion,
(b)
OTA
Implemen a ion.
Fig.
4
Piecewise-Linea (PL) Func ion Gene a o Building
Block
.
I
(0)
,
**.:
(b)
Fig.
5
Exponen ia ion
(C)
(Raising o a powe ) Ope a ion,
(a) Squa e , (b) Cubi , (c) p- h.
98
Depa men
o
ELecmcal
and
Compu e
Enpeenng
and
The
coo dina ed
Science
Labo a o y
~
Fig.
6
A
Chip Pho o Mic og aph
o
Two
Comple e
OTAs.
Fig.
7
La ge-Signal Cha ac e is ics
o
Mul iplie .
VI
=
*{1.00,
0.66,
0.33,
0.O)V.
(a) T ans e Cha ac e is ic
Va iable T iangula Wa e o
Vz.
(e) Expe imen al Resul s.
99