1576
IEEE
JOURNAL
OF
SOLID-STATE CIRCUITS,
VOL.
24,
NO.
6,
DECEMBER
1989
Ope a ional T ansconduc ance Ampli ie -Based
Nonlinea Func ion Syn heses
EDGAR
SANCHEZ-SINENCIO,
SENIOR
MEMBER, IEEE,
JAIME
RAM~REZ-ANGULO, MEMBER,
IEEE,
BERNABE LINARES-BARRANCO,
AND
ANGEL
RODR~GUEZ-VAZQUEZ, MEMBER,
IEEE
Abs uc
-We show ha he ope a ional ansconduc ance ampli ie
(OTA), as he ac i e elemen in basic building
blocks,
can be e icien ly
used
o
p og ammable nonlinea con inuous- ime unc ion syn hesis. Two
e icien nonlinea unc ion syn hesis app oaches a e p esen 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 ab ica ed using a 3-pm pwell
CMOS
p ocess. The lexibili y o he designed and es ed ci cui s was con i med.
We will p esen a numbe o nonlinea OTA ci cui s and
will discuss wo nonlinea analog unc ion syn hesis ech-
niques
based
On
hese
OTA
basic
One syn hesis app oach app oxima ion
unc-
ions and he o he uses a piecewise-linea app oxima ion.
Ac ual ci cui implemen a ions will be p esen ed as well as
he expe imen al esul s om se e al 3-pm p-well CMOS
,
1. INTRODUCTION es p o o ypes.
ATELY, se e al au ho s [1]-[5] ha e been success ully
L
using he ope a ional ansconduc ance ampli ie
(OTA) as he main ac i e elemen in con inuous- ime
ac i e il e s. The OTA is a p og ammable de ice’ and has
only a single high-impedance node, in con as o con en-
ional op amps. This makes he OTA an excellen de ice
candida e o high- equency and ol age (o cu en ) p o-
g ammable analog basic building blocks.
The applicabili y o OTA’s as componen s in he design
o linea ne wo ks has been ex ensi ely discussed else-
whe e
[l],
[6]
and will no be epea ed he e.
The objec i e o his pape is o examine he applicabil-
i y o OTA’s as he basic elemen s in he design o
nonlin-
ea
ne wo ks. The e is no much epo ed in he li e a u e
on he use o OTA’s o designing nonlinea componen s
[7],
[8].
Excellen con ibu ions [9]-[ll], [16] a e epo ed
o nonlinea ci cui s dealing wi h pa icula impo an
nonlinea p oblems. In hs pape , a he han y o ackle
a speci ic p oblem, we ocus
ou
a en ion on a gene al
app oach dealing wi h nonlinea basic building blocks
using OTA’s as he main ac i e elemen s. No hing special
was done o op imize he ci cui pe o mance bu a he o
explo e he po en ial and applicabili y
o
he OTA-based
nonlinea sys em app oach.
Manusc ip ecei ed Feb ua y 4, 1989; e ised Augus
8,
1989.
E. Sinchez-Sinencio and
J.
Rami ez-Angulo a e wi h he Depa men
o
Elec ical Enginee ing, Texas A&M Uni e si y, College S a ion, TX
77843-3128.
11.
BASIC
BUILDING
BLOCKS
In his sec ion we in oduce he OTA-based undamen-
al nonlinea building blocks in ol ed o he syn hesis
p ocedu es.
A.
Mul iplie
Block
A
wo-inpu ou -quad an mul iplie has an ou pu
cu en gi en by
I,
=
KMVIV*
whe e he mul iplie cons an
K,
has uni s o ampe es pe
squa e ol . I
V,
and
V,
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 espond-
ing OTA-based implemen a ions a e shown in Fig. l(b)
and (c). The iangula block labeled
a
ep esen s a signal
a enua o (wi h an a enua ion ac o
a);
i s unc ion is o
equalize he maximum ol age swing o
V,
and
V,.
-
V,,,,
is he usual bias con ol
o
he OTA. An ac i e a enua o
can be implemen ed in CMOS echnology [15]. The signal
le el in he mul iplie is es ic ed by a ew hund ed
milli ol s o
V;,
and
V z.
Al hough no indica ed in Fig. 1,
assume he powe supplies o he OTA’s a e
V,,
and
-Vss.
The wo op ions o Fig. l(b) and (c) allow us o
change he sign o
K,.
Thus o he ci cui o Fig. l(b) we
ob ain
B. Lina es-Ba anco is wi h he Depa men o Elec ical Enginee ing,
amen o de Elec bnica y Elec omagne ismo, Uni e sidad de Se illa,
Texas A&M Uni e si y, College S a ion, TX 77843 and wi h he Depa -
(24
41012 Se illa. Spain.
Elec omagne ismo, Uni e sidad de Se illa, 41012 Se illa, Spain.
A. Rod iguez-Vbquez is wi h he Depa amen o de Elec bnica y
’The ou pu cu en
I,,
o an OTA due o
a
di e en ial inpu
,~
is
I,,
=
g,,,c>,,.
and
g,,,
is a ol age (cu en ) con ollable pa ame e [l], [6],
and
IEEE Log Numbe 8931198.
JO2
=
-
s,,V1=
-
K(
VIz
+
VSST)
Vl
(2b)
whe e
K
is a p ocess- and geome y-dependen cons an ,
[71.
0018-9200/89/1200-1576$01.00
01989 IEEE
SANCHEZ-SINENCIO
e
al.
:
OTA-BASED
NONLINEAR
FUNCTION
SYNTHESES
Vl
-VBIAS
1577
2
The ou pu cu en
Io
becomes
1,
=
lo,
+
zO2
=
[
K(
-
BIAS
+
&si-)
-
K(
u 2
-
'BIAS
+
'SST)]
'1
(4)
(5)
Z,
=
-
aKV1V2
=
KMVlV2,
K,
=
-
uK.
A simila analysis
o
he ci cui o Fig. l(c) yields
Zo
=
aKViV2
=
KMViV2, KM
=
uK.
(6)
The e o e, we can make he sign
o
KM
posi i e o nega-
i e.
B. Di ide Block
A wo-inpu di ide has an ou pu whxh is he a io o
he wo inpu s, mul iplied by a cons an
K,
wi h dimen-
sions in ol s:
Vl
V,
=
K,-
2
(7)
A symbol o he di ide is shown in Fig. 2(a), whe e
n
and
*We ha e assumed equal
K's
and h eshold ol ages
y's
o
he
OTAs.
(a) (b)
Fig.
2.
Di ide :
(a)
symbol and (b)
OTA
implemen a ion
d
s and o nume a o and denomina o , espec i ely. The
co esponding ci cui implemen a ion using he mul iplie
symbol is shown in Fig. 2(b). Analysis yields
11
=
g,V1
(84
I2
=
K,VoV2.
(W
(94
By Ki chho s cu en law (KCL) we ob ain
Il
+
Z2
=
gmV,
+
K,VoV2
=
0.
Thus, he esul ing ou pu signal is p opo ional o he
a io o he inpu signals
Obse e ha
KR
can also be ei he nega i e o posi i e
and
V2
=
0
is no allowed o a oid ou pu sa u a ion
(V,).
A close look a he ci cui o Fig. 2(b) is equi ed o s udy
s abili y. Assuming he dominan dynamic elemen is a
pa asi ic capaci ance
C'
a he ou pu , (9a) is modi ied as
z1
+
I,
=
SCPV,
(loa)
which yields
Thus,
V,
has a pole loca ed a
K,
s,
=
7
V2.
LP
Hence, in o de o he ci cui o be s able, i s pole mus be
in he le -hand plane (LHP), which dic a es ha 3
K,V2
<
0.
(104
A summa y o he s abili y condi ions a e pic o ially
indica ed by he hype bolas o Fig. 3.
C.
Squa ing and High-Powe s (Exponen ia ion) Blocks
A one-inpu squa e has an ou pu p opo ional o he
squa e
o
he inpu :
30bse e ha we a e applying linea ea men o a nonlinea ci cui .
This
linea analysis is alid i o a ce ain ins an
x,
V,( ,)
and
b(
)
a e conside ed cons an ; hen
V,
will
e en ually each
i s
solu ion in&-
penden
o
he ini ial condi ions.
1578
IEEE
JOURNAL
OF
SOLID-STATE CIRCUITS, VOL.
24,
NO.
6,
DECEMBER
1989
VO
4,
Kn<
0
(a)
(b)
Fig.
3.
S abili y egions
o
di ide .
(a)
S able
o
K,,,
>
0
and
V2
<
0.
(b)
S able o
K,
<
0
and
V2
>
0.
Fig.
4.
Exponen ia ion ( aising o a powe ) ope a ion: (a) squa e ,
(b)
cubic, and (c) p h.
(a) (b)
Fig.
5.
Squa e oo e : (a) implemen a ion and
(b)
OTA
implemen a ion.
The implemen a ion o he squa e is ob ained by simply
using a mul iplie wi h equal inpu s, as shown in Fig. 4(a).
To ob ain an
exponen ia ion
( aising o a powe ) ci cui
wi h an inpu
V,
and an ou pu o be p opo ional o
VIP,
whe e
p
is an in ege g ea e han
2,
we equi e
(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 use as he inpu s o subsequen mul iplie s.
This can be done 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 . An
example o
p
=
3 is shown in Fig. 4(b). I should be
e iden ha a simila p ocedu e can be ollowed o ob ain
an exponen ia ion o any o de
p;
his is symbolically
illus a ed in Fig. 4(c).
D.
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.
Fig. 5(a) shows he implemen a ion o he squa e oo e ,
whe e he ou pu
V,
is gi en by
,
VO
o
=
K,-
(134
whch yields
A mo e de ailed desc ip ion o he implemen a ion is shown
in Fig. 5(b). The ci cui will be s able i , a e a pe u ba-
ion, he ou pu e ol es owa ds he desi ed ou pu alue.
Assume he inpu is ixed a
=
U,.
To s udy he s abili y
and he dynamics o he ci cui , a pa asi ic capaci ance
Cp
a he ou pu is again conside ed. Using he
KCL
a he
ou pu node esul s in he ollowing nonlinea di e en ial
equa ion:
c
-
dV0
=
g,U,
+
K, ,'
d
which can be ew i en as
1
dVn
d
-.L eg a ing bo h sides o (14b) and sol ing o
Vo(
)
(when
(
g,
/KM
1
us
<
0)
yields
1-
whe e
K.
=
hence
Fo
(g,/K,)U,
>
0
he solu ion yields an unbounded
ou pu . I is concluded ha a s able squa e- oo e ci cui is
ob ained when
(gm/K,)y
<
0,
i.e.,
KRV,
>
0.
Fu he -
mo e, he pola i y o
Vo
can be de e mined acco ding o
(15b)
o (1%). Fig.
6
shows he condi ions o s able
ope a ion
o
he squa e oo e .
E.
Piecewise-Linea Func ion Gene a o s
Diodes in e connec ed wi h OTA's can simula e ideal
diodes, hence allowing he c ea ion
o
a piecewise-linea
app oxima ion o any desi ed nonlinea unc ion. The ideal
basic building blocks o a piecewise-linea unc ion ap-
p oxima ion a e shown in Fig.
7.
High- equency imp o e-
-.
SANCHEZ-SINENCIO
e
01.
:
OTA-BASED
NONLINEAR
FUNCTION
SYNTHESES
1579
*.."
Fig.
6.
S abili y egions
o
squa e oo e
men s
o
his basic block a e discussed in he Appendix.
No e ha
I,
=
0
un il he b eaking poin ( ol age e e -
ence
V,)
is eached. The slopes o he linea segmen s a e
p opo ional o he
g,'s.
The diodes can be implemen ed
wi h
MOS
ansis o s wi h hei ga e and d ain ied o-
ge 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 o Fig.
7
can be subs i u ed by an
OTA
compa a o which ideally simula es a la ge
g,
and a
sa u a ion (ou pu ) cu en o
IBIAS.
111. NONLINEAR FUNCTION SYNTHESES
We p esen wo app oaches o nonlinea unc ion syn-
heses. The i s app oxima ion uses a polynomial ap-
p oach, and he second app oxima ion a piecewise-linea
app oach.
The i s is a
a ional app oxima 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. Assume an elemen
K,xP/q
needs o be implemen ed.
This
is ob ained as
shown in Fig. 8(a). The exponen ia ion blocks a e o he
ype o Fig 4(c). I a nega i e
-
p/q
is needed, an addi-
ional di ide is used as shown in Fig. 8(b).
Nex we discuss a
piecewise-linea app oxima ion
syn-
hesis app oach. Ths app oach consis s o adding ( ans-
conduc ance) gain segmen s ha ha e null con ibu ion
un il a e e ence ( h eshold) ol age is eached. A simple
bu illus a i e example is shown in Fig.
9
whe e
a
con ex
cu e is app oxima ed. The b eakpoin s occu a
yl,
yz,
and
V,?.
The slopes a e gi en by
so
=
0,
I,
=
0,
Fig. 7. Piecewise-linea
(PL)
unc ion gene a o building blocks
'i~l~~ ol
Exponen la lon
Block
Exponen la lon
Block
(a)
(b)
Fig.
8.
F ac ion powe exponen ia ion: (a)
KJ"4
and
(b)
V'-P/Y.
3
>
V 2
>
l
(a)
(b)
Fig.
9.
Con ex cu e piecewise-linea app oxima ion: (a)
,
e sus
]
cha ac e is ics,
and
(b)
ci cui implemen a ion.
No e he con inuing inc ease in magni ude o he slope as
he inpu
ui
inc eases, hus o ming a con ex cu e. The
app oxima ion will imp o e as he numbe o segmen
lines inc eases.
Obse e ha by combining he basic building blocks o
Fig.
7,
a bi a y unc ions wi h a iable posi i e and nega-
i e slopes and b eakpoin s can be app oxima ed. Fu he -
mo e, he slopes and b eakpoin s a e
ol age p o-
g ammable,
which gi es
an
addi ional lexibili y in he
unc ion app oxima ion design p oblem. No e ha 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 ides a e y
o
V,]
>
1580
IEEE
JOURNAL
OF
SOLID-STATE CIRCUITS,
VOL.
24,
NO.
6,
DECEMBER
1989
Fig. 10. Ci cui diag am
o
he
OTA
[3]
used
Fig. 12 Nonlinea i y mul iplie e o : ixed
VI
=1
V
and a iable ian-
gula wa e o
V .
Fig. 11. La ge-signal cha ac e is ics o mul iplie .
V,
=
k
{0.75, 0.50,
0.25.0.0)
V.
good empe a u e compensa ion
[18]
and accu acy im-
p o emen . In s ingen applica ions whe e minimum em-
pe a u e dependence is equi ed, he use o a esis i e-
load OTA
is
needed. 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 gi en in he Appendix.
Fig. 13. Nonlinea i y mul iplie e o : ixed
=
1
V
and a iable
VI.
IV.
EXPERIMENTAL
RESULTS
Se e al es ci cui s con aining OTA's and ansis o s
connec ed as diodes we e ab ica ed using a 3-pm p-well
CMOS p ocess h ough (and hanks
o)
MOSIS. The lin-
ea ized OTA used
o
syn hesize he di e en nonlinea
analog unc ions is epo ed elsewhe e
[3].
I s schema ic is
shown in Fig.
10.
The OTA has an a ea o
220x700
pm2
and consumes
10
mW o
5-V
supply ol ages. In all he
examples (unless o he wise indica ed) he ou pu cu en
was measu ed ac oss a lOO-kQ load esis o .
A.
T ansconduc ance Mul iplie
The s uc u e used is as shown in Fig.
1.
The measu ed
alue o
IKMI
is 3.3
pA/V2.
The ou pu cu en was
measu ed ac oss a 100-kQ load esis o . The la ge-signal
cha ac e is ics o he mul iplie a e shown in Fig.
11.
V,
was held cons an (a
0.0,
50.25,
0.50,
and
kO.75
V),
while he inpu
V,
a ied be ween
&
1
V.
The nonlinea i y
e o is shown in Fig.
12.
Fo
V,,
a iangula
2-V
peak-
o-peak signal was applied, while keeping
V,
equal
o
1
V.

SANCHEZ-SINENCIO
e
d.
:
OTA-BASED
NONLINEAR
FUNCTION
SYNTHESES
1581
4250 4750
250 1250 2250 3250
750 1750 2750 3750
quency
Spec um
o
he mul iplie ou pu ol age:
V,
=1
V
and
Vz
=
2sin?
X
103 .
Fig.
14.
Fig.
15.
Modula ion
o
wo-inpu sinusoidal signals. Fig.
16.
Di ide expe imen al esul s: cons an
VI
and a ying
V2
The ou pu cu en p oduced a iangula ol age signal o
660-mV peak o peak. Sub ac ing hs signal om an ideal
iangula wa e, he esul ing peak- o-peak e o signal
was 17 mV, which yields a nonlinea i y e o o nea ly 2
pe cen . (The ideal iangula wa e is a scaled e sion o
he inpu in such a way ha he ampli ude o he e o
signal is minimum.) Repea ing he measu emen bu in e -
changing
Vl
and
V2
(VI
is a iangula signal o 2-V peak
o peak), he esul ob ained is shown in Fig. 13. The
peak- o-peak e o signal o
23
mV co esponds o
a
3.5-
pe cen nonlinea i y e o . The asymme y o he mul i-
plie (see inpu s in Fig.
1)
yields hs dis o ion di e ence
when in e changing he inpu s. Making
Vl
=
+
1
V and
V2
a 2-V peak- o-peak sinusoidal wa e o m o
1
kHz, he
spec um o he mul iplie ou pu shown in Fig.
14
was
measu ed. Obse e ha only he second ha monic, 33 dB
below he undamen al, is p esen . Fig. 15 shows he
mul iplie being used as
a
modula o whe e bo h inpu
signals a e sinusoidal.
B.
Vol age Di ide
The es ed ci cui has he s uc u e shown in Fig. 2(b)
wi h
IK,I
as be o e in Sec ion IV-A and
K,
>
0.
The
expe imen al esul shown in Fig. 16 was ob ained by
swi ching
VI
be ween wo symme ical cons an alues
Fig. 17. Squa e expe imen al esul s.
(+
1
V) while a ying
V, (V,
<
0).
This esul ma ches wi h
he heo e ical esul s o Fig. 3(a).
C.
Squa e
The squa e is ob ained by simply making
VI
=
V,
in he
mul iplie discussed in Sec ion
IV-A.
In his pa icula
case,
K,
is
nega i e esul ing in he in e ed pa abola
shown in Fig. 17. The inpu ange was
1
V.
D.
Squa e Roo e
The basic a chi ec u e used
is
he one shown in Fig.
5(b). The inpu signal
V,
is gi en by
V,
=
A
+
A
COS
u
and
1582
IEEE
JOURNAL
OF SOLID-STATE CIRCUITS,
VOL.
24,
NO.
6,
DECEMBER
1989
esis o load, empe a u e a ia ions a e minimized. The
p og ammabili y and lexibili y o he OTA p o ide he
po en ial o design ime- a ying nonlinea ci cui s. The
expe imen al esul s e i ied heo e ical p edic ions. Imple-
men a ions o o he nonlinea syn hesis app oaches [20]
a e easible using he basic blocks he e in oduced. The e
a e many impo an a eas o applica ion o nonlinea
unc ions [22]. One o hem is in neu al ne wo ks [19],
[21]
as shown by Mead
[17,
ch. 61. 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.
Fig.
18.
Squa e oo e expe imen al esul s
he ou pu , ob ained in he i s quad an , has he o m APPENDIX
PRACTICAL CONSIDERATIONS
OF
THE
OTA-BASED
PIECEWISE-LINEAR CIRCUITS
In his Appendix some p ac ical conside a ions o he
OTA-based Diecewise-linea ci cui s a e discussed aking
in o accoun; some nonideali ies o he OTA and o
he
MOS
ansis o used as a diode. A simple modi ica ion o
he ci cui s o Fig.
7
ha leads o a d as ic imp o emen in
hei high- equency pe o mance is also p esen ed.
whe e
g,,
>
0,
K,
<
0,
A
=1
V,
=11
kHz,
g,
=
3.2
pmhos, and
K,
=
3.3 pA/V2. The expe imen al esul s
a e shown in Fig.
18
whe e he lowe ace signal is he
inpu and he ou pu is shown in he uppe ace.
E.
Piecewise-Linea App oxima ion Nonideal Ci cui Elemen s
The in ended ans e cha ac e is ic is shown in Fig.
19(a) and consis s o h ee linea segmen s. The indi idual
slopes due o each OTA a e indica ed in he lowe pa o
Fig. 19(a), and he composed esul ing ans e cha ac e is-
ics a e shown in he uppe pa o Fig. 19(a). The ac ual
OTA ci cui implemen a ion is shown in Fig. 19(b), whe e
an op ional diode and ol age sou ce ha e been added a
he OTA (2 and 3) o imp o e he high- equency pe o -
mance o he ci cui (see he Appendix o mo e de ails).
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. 19(c).
To
show u he lexibili y, a chip es ci cui
was ep og ammed o implemen a i e-segmen cha ac e -
is ic o con e a iangula o a sinusoidal wa e o m. The
expe imen al cha ac e is ic is shown in Fig. 30(a), and he
inpu and ou pu wa e o ms a e shown in Fig. 20(b). The
measu ed THD, a e op imally adjus ing b eakpoin s and
slopes, was
1.5
pe cen . The chip pho omic og aph o
hese expe imen al esul s is shown in Fig. 21.
V.
CONCLUSIONS
The sui abili y o OTA's as he main ac i e elemen o
ob ain basic building blocks o he design o nonlinea
ne wo ks was es ablished. Me hods o implemen p ac ical
nonlinea 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 o-
duced. Obse e ha o bo h app oaches, e e y ime he
ou pu cu en is con e ed in o a ol age by an OTA
Simple equi alen ci cui s o an MOS OTA and o a
diode-connec ed
MOS
ansis o a e shown in Fig. 22. The
OTA is cha ac e ized by i s ou pu impedance (ou pu
capaci ance
CO
and ou pu esis ance
R,)
and inpu capac-
i ance
(C,).
The diode-connec ed MOS ansis o , as shown
in Fig. 22(b), is modeled by a esis ance4
R,
in se ies wi h
a ol age sou ce
V,
( h eshold yol age o he
MOS
ansis-
o ) and an ideal diode, whe e
I,
is he peak cu en in he
MOS
diode. Fig. 22(c) shows he equi alen ci cui o he
OTA when used as a wo- e minal esis i e elemen o
simula e a g ounded load esis ance [6]. The simpli ied
analysis ha ollows assumes ha he ansconduc ance
g,
o he OTA is equency independen and neglec s he
pa asi ic capaci ance o he
MOS
an~is o .~ Also, e ec s
due o OTA and diode esis ance nonlinea i ies a e no
conside ed.
Low-F equency Conside a ions
The low- equency equi alen ci cui o he OTA p eci-
sion ec i ie
o
Fig. 23(a) is shown in Fig. 23(b).
gwZ2
and
RO2
a e he ansconduc ance and he ou pu esis ance o
an OTA used as a load. Pa asi ic capaci ances ha e been
neglec ed o hls low- equency analysis. S anda d ci cui
41n ac , alung in o accoun he mobili y deg ada ion
[14],
R,
can be
mo e accu a ely e alua ed, i.e.,
R,
'In mos p ac ical cases he ime cons an
R,C,
associa ed wi h he
MOS
ansis o is negligible compa ed wi h o he ime cons an s in he
ci cui .
(28/p0COx)(
L/
W).
SANCHEZ-SINENCIO
e
d.
:
OTA-BASED NONLINEAR FUNCTION SYNTHESES
1583
1.2
A
4
I
"c2
(4
esul s.
Fig.
19.
Piecewise-linea app oxima ion unc ion: (a) ans e cha ac e is ic,
(b)
ci cui implemen a ion,
and
(c) expe imen al
analysis shows ha o
U,
>
0:
Equa ion
(17)
can be simpli ied by assuming
l/gm2<<
RO2,
Rol.
Then
The ac o
RJ(
R,
+
RO1)
in he i s e m o he igh
side
o
(18)
ep esen s an undesi ed a enua ion. The sec-
ond e m in
(18)
ep esen s an o se added o he ou pu
signal whch, added o he o se o he OTA, limi s he
minimum ampli ude o he signal ha can be ec i ied.
Assuming
R,
<<
R,,
and
R,,
=
Ro2,
an es ima ed alue
o he second e m is
V,/A 2
whe e
A,=
gm2RO1
is he
ol age gain o he OTA
2.
Typical alues
V,
=
1
V and
A,
=
200
esul in app oxima ely
5
mV o hs second
e m. This is also a ypical alue o he o se ol age o
an
MOS
OTA.
To educe
(18)
o he ideal case, i.e.,
conside he ollowing p ac ical design conside a ions o
low- equency applica ions.
1)
Use
OTAs wi h high- ol age gain in o de o mini-
mize he o se e m in
(18)
and design he OTA yielding
he lowes possible o se ol age. Then limi he minimum
inpu signal ampli ude acco ding
o
(18).
2)
Selec he
W/L
a io o he diode-connec ed
MOS
ansis o such ha
R,
<<
R,
is sa is ied. Ths will make
ROl/( RO1
+
R,)
=
1.
Since he ou pu impedance
R,
o an
OTA is ypically e y la ge, his condi ion in gene al does
no equi e an excessi ely la ge alue o he
W/L
a io o
he
MOS
ansis o .
1584
IEEE JOURNAL
OF
SOLID-STATE CIRCUITS, VOL.
24,
NO.
6,
DECEMBER
1989
he signal, and o he pa icula case
gm,
=
gm9.
(b)
high- equency ci cui; pe o mance) and/o design a “be -
e OTA” wi h Smalle
CO,.
Fig.
20.
(a) Fi e-segmen ans e cha ac e is ic,
(b)
T iangula and
sinusoidal wa e o ms a
1
MHz.
High-F equency Analysis Conside a ions
The mos impo an p ac ical ac o limi ing he high-
equency ope a ion o he ci cui o Fig. 23 (see also Fig.
7)
is he delay ime
,
equi ed o discha ge he ou pu
capaci ance
CO,
om he nega i e peak ol age
-V.
(which is cha ged du ing nega i e hal -cycles) o he alue
V,
equi ed o he diode o s a conduc ing. Assuming,
o simplici y, ha he cu en du ing he nega i e hal -
cycle is en i ely supplied o
CO,
(wi h he cu en in
R,
conside ed negligible), he ollowing ela ionship
o
he
inpu signal
V,
=
V,
sino in ol ing i s equency
and he
del:y ime
,
can be de i ed om
CJdV
/d )
=
g,V,
sin
o
as6:
c.,
whe e
IMAX
=
g,
is he peak ou pu cu en . Fig. 24
illus a es he ypical wa e o m ha is obse ed in he
p ecision ec i ie ci cui o Fig. 23 o high- equency
ope a ion when
,
becomes compa able o he pe iod o
Modi ica ion
o
he Basic P ecision Rec i ie Ci cui o
Imp o e High-F equency Ope a ion
Fig.
25
shows high- equency imp o ed e sions o he
OTA- ec i ie ci cui s including a second diode8
D,
and a
sou ce
V,
=
(
K,
+
V,J
whe e
V,,
and
V,*
a e he h eshold
ol ages o
D,
and
D,,
espec i ely. In hs ci cui he
OTAl ou pu ol age does no go in o sa u a ion, bu is
limi ed o a minimum (maximum) alue
(-
V,,)
du ing
nega i e (posi i e) hal -cycles. A he beginning o he
posi i e (nega i e) hal -cycles, he diode
D,
will be eady
o conduc ion
so
ha no delay due o cha ging o dis-
cha ging o he ou pu capaci ance
c,,
akes place, hence
The addi ion o he diode
D,
and he bias
V,
allows a
d as ic imp o emen in he high- equency pe o mance o
he ci cui wi hou excessi ely inc easing he powe con-
sump ion. Addi ional powe dissipa ed s ill needs o be
supplied by he ba e y
V,
which abso bs he ou pu
cu en o he OTA h ough
D,
du ing he nega i e hal -
cycles, bu his does no inc ease quiescen powe dissipa-
ion.
The emaining high- equency limi a ion o he im-
p o ed ci cui s is ela ed o he low-pass beha io o he
ec i ie . The ci cui
o
Fig. 26 shows a high- equency
equi alen ci cui o he OTA ec i ie (assumed linea o
,
=
0.
’Whe e
lb‘<
I
=
Al,C
o
IV-
I
<
Vss,
o he wise
l c
I
IVssl and
Ss
is
* n
wo ks
simila
o
he ca ch diode
o
he ou-amu e sion
1131
o he
6An addi ional delay,
no
conside ed in he simpli ied analysis p e-
sen ed he e bu ha can be obse ed in Fig.
24,
is he ime aken
o
he he nega i e powe supply o age.
.*
.~
-2
ou pu signal o each he inpu signal once he diode
is
conduc ing. same ci cui .