A
Gene al Sub h eshold
MOS
T anslinea Theo em
Te esa Se ano-Go a edona’, Be nab6 Lina es-Ba anco’, and And eas
G.
And eou2
Na ional Mic oelec onics Cen e (CNM), Ed. CICA, AV. Reina Me cedes sln, 41012 Se illa, Spain
*
The Johns Hopkins Uni e si y,
3400
N. Cha les S ee , Ba on Hall, Bal imo e, MD 21218, USA
Abs ac
This pape ou lines he condi ions unde which
he T anslinea P inciple can be ully exploi ed o
MOS
ansis o s ope a ing in sub h eshold. Due o
he exponen ial na u e o sub h eshold
MOS
ansis o s he T anslinea p inciple applies
immedia ely as long as he
Sou ce- o-Bulk
ol ages
a e made equal o ze o
(o
cons an ). This pape
add esses he condi ions unde which sub h eshold
MOS
ansis o s s ill sa is y a T anslinea p inciple
bu wi hou imposing his cons ain . I is ound ha
he T anslinea p inciple esul s in a mo e gene al
o mula ion han he o iginal
o
BJTs since now
mul iple T anslinea loops can be in ol ed. The
cons ain
o
e en numbe o ansis o s is no longe
necessa y.
I. In oduc ion
The anslinea p inciple, in oduced by Ba y
Gilbe in 1975
[I],
is
one o he mos impo an ci cui
heo y con ibu ions in he elec onics e a. In i s o iginal
o mula ion, he anslinea p inciple p o ides a simple
and e icien way o analyze and syn hesize nonlinea
ci cui s based on bipola junc ion ansis o s (BJTs).
Due o hei exponen ial cha ac e is ics, he anslinea
p inciple can be ex ended o
MOS
ansis o s ope a ing
in
weak in e sion [2], [3] wi hou and wi h loa ing-ga e
de ices [4]. Fo MOS ansis o s ope a ing abo e
h eshold he e has also been ound a simila way o
analyze and syn hesize nonlinea ci cui s
[5].
Fo bipola ansis o s one p ac ical p oblem ha
may equi e some a en ion when applying he
anslinea p inciple is he nonze o base cu en
[6].
In
con as , he anslinea p inciple holds o
MOS
sub h eshold ansis o s in an exac manne i sou ce
and bulk a e sho -ci cui ed. Howe e , i has been ound
ha he p inciple holds as well in an exac manne unde
di e en ci cums ances [2]-[3], al hough a gene al
sub h eshold MOS anslinea heo em has no been
de ised un il now. In his pape we p o ide his gene al
heo em and ou line he condi ions unde which
sub h eshold MOS ansis o s, iewed as ou e minal
de ices, sa is y a gene al anslinea p inciple.
The ope a ion o a sub h eshold MOS can be
desc ibed by
he
ollowing
equa ion
[2],
[7]-[9]
whe e
V h
=
KTIq
is he mal ol age,
I,
is a posi i e
cons an cu en ,
S
is he ansis o size ac o
(S
=
WIL
,
whe e
W
is ansis o wid h and
L
is i s
leng h), and
K
is a echnology dependen posi i e
pa ame e . This equa ion holds ue as long as
whe e
@FB
is he de ice’s la band ol age [lo].
Vol age
VBs
can ake ei he posi i e o nega i e alues
as long as he pa allel PN diode junc ion is biased below
i s o wa d conduc ion h eshold ol age. Pa ame e
K
is known o ha e a sligh dependency on ol age
VBS
[2]. Howe e ,
in
his pape we will assume
K
o be
cons an , which is a easonable assump ion i ca e is
aken o make he VBs ol ages simila o all
ansis o s.
Fo ope a ion
in
sa u a ion eq. (1) can be simpli ied
o (i
VDs
>>
V h)
(3)
and be ew i en as
whe e
I
(a
“no malized cu en ”)
is ansis o cu en
no malized wi h espec o ansis o size ac o
S
=
WIL,
and
i,,
i,
a e dimensionless numbe s
called
pseudo-cu en s
and equal o
(5)
GS‘ Ih
.
‘BSI
‘1
h
i,
=
e
iB
=
e
Le us use he symbol in Fig. 1 o ep esen a weak-
in e sion MOS in sa u a ion. Le us call he pa h ha
goes om he ga e e minal
G
o he sou ce e minal
S
he
G
-
b unch
(o
Ga e-b unch),
and he pa h ha
goes om he bulk e minal
B
o e minal
S
he
B
-
b unch
(o
Bulk-b unch).
We a e using a diode-
like symbol o ep esen he exponen ial ela ionship
be ween he ol age o he b anch and he cu en
lowing ou o he de ice and a capaci i e-like
e mina ion o each diode symbol o ep esen he
capaci i e coupling na u e o he
Ga e
and
Bulk
e minals. I
Vss
=
0
(o cons an ) he e is an exac
exponen ial ela ionship be ween
VGs
and
ZDs
(see
eq.
(3)) and he o iginal BJT anslinea o mula ion can
be di ec ly and exac ly applied.
11. Gene alized T anslinea Theo em o
Sub h eshold
MOS
T ansis o s
In his sec ion we will conside he condi ions unde
which anslinea p inciples can be applied o ci cui s
PB
S
Fig.
1:
T anslinea Symbol Rep esen a ion o
Sub h eshold
MOS
Tkansis o in Sa u a ion
0-7803-5471
-0/99/$10.0001999
IEEE
11-302
Fig.
2: Example o Coupled
G-loops
using
anslinea symbol ep esen a ion
Fig.
3: Illus a ion o he
G-o de
concep . De ices ‘1-2-3-
6’ o m a
G-loop
which is coupled o
G-loop
o med by
de ices ‘2-4-5’, because de ices ‘1-2-3-4’ o m a
B-loop.
De ices ‘1-2-3-4-5-6’ o m a
Closed T anslinea Se ,
and
so
do de ices 6 and
7.
De ice 2 has a
G-o de
o
nc2
=
2
because i s
G-b anch
belongs o wo
G-loops
o he same
Closed T anslinea Se .
All o he de ices ha e
G-o de
one.
wi h sub h eshold
MOS
ansis o s bu wi hou
imposing he cons ain o making
Vss
=
0.
We will
in oduce i s some p elimina y heo ems and
de ini ions, and hen s a e and p oo he gene alized
anslinea heo em o sub h eshold
MOS
de ices.
The i s concep s o be in oduced a e
G-loop
and
B-loop.
A
G-loop
(o
Ga e-loop)
is a closed loop o
G-
b anches,
and a
B-loop
(o
Bulk-loop)
is a closed
loop
o
B-b anches.
Fo hese loops we can s a e anslinea
heo ems o hei
pseudo-cu en s,
whose p oo is
iden ical as o he o iginal BJT T anslinea Theo em.
Theo em:
In a G-loop con aining an
a bi a y
numbe o G-b anches, he p oduc o pseudo-
cu en s o b anches cpnnec ed in he
Clock-Wis~G(CW) di ec ion
is
equal o he
co esponding p oduc o b anches connec ed
in he Coun e -Clock-Wise (CCW) di ec ion.
No e ha since
pseudo-cu en s
a e dimensionless
en i ies we can ha e an a bi a y numbe o b anches
o ien ed
CW
and ano he a bi a y numbe o b anches
o ien ed CCW (as opposed o he case o eq.
(7)).
A
comple ely equi alen heo em holds di ec ly o
B-
Up o now hings a e simila o classical anslinea
loops, excep ha an a bi a y numbe o b anches a e
allowed. Howe e , he p esence o wo exponen ial
b anch ol ages in eq. (3)
is
wha makes sub h eshold
MOS
anslinea
loops
mo e gene al and complica ed
han he classical ones. A i s consequence
o
his ac is
he ollowing concep o
coupled loops:
De ini ion: Two
G-loops a e said o be
coupled i a leas one
MOS
de ice
o
he
i s G-loop and a leas one o he di e en
de ice o he second
G-loop
sha e hei
espec i e B-b anches in
a
common
B-loop.
loops.
This is illus a ed in Fig.
3.
De ices ‘1-3-5’ o m a
G-
loop
and de ices ‘2-4-6’ o m ano he
G-loop.
Howe e , de ices ‘1-2-3-4’ o m
a
B-loop,
hus making
he p e ious wo
G-loops
o be coupled h ough he
B-
b anches
o de ices
1,2,3
and 4.
An equi alen de ini ion applies o
coupled
B-
loops.
No e ha wo loops may ha e a common b anch
wi hou being necessa ily
coupled loops.
In he example o Fig. 3, we can w i e o he wo
G-loops
(
‘SS4
+
‘LIS6
-
‘BSZ)
=
Ioe
whe e all
pseudo-cu en s
i,
ha e cancelled ou by
applying he abo e Theo em. Howe e , due o
B-loop
‘1-2-3-4’,
‘BSI
+
‘BS4
=
‘LIS2
+
‘BS3
(7)
which in oduces a coupling be ween he wo equa ions
in
(Il),
and makes
G-loops
‘1-3-5’
and ‘2-4-6’ o be
coupled loops.
When de ices o m mul iple ouching loops i is no
clea which ones o choose o how many o choose. Fo
example, in Fig. 4 one can choose
G-loops
‘1-2-3-6’, ‘6-
7’,
and ‘2-4-5’. Bu why no conside ‘1-3-4-56’,
‘6-7’,
and ‘2-4-5’, o ‘1-3-4-5-7’ and ‘1-2-3-6’. One can y all
possible op ions as long as one chooses a se o
Non-
Redundan
(NR) loops:
De ini ion:
A
se
o
loops
is
said o be Non-
Redundan
(NR)
i he sum
o
b anch ol ages
o any loop canno be exp essed as a linea
combina ion
o
he sum
o
b anch ol ages
o
o he loops in he se .
Fo example, in Fig. 4, o
G-loops
‘1-2-3-6’, ‘2-4-
5’,
and ‘1-3-4-5-6’ hei espec i e sum
o
b anch
ol ages is,
‘GS6
+
‘GS,
=
‘CS2
+
‘GS3
“GS2
+
VGS4
=
VGSS
VGS6
+
VCSl+
VGS4
=
VGS3
+
VGSS
(8)
Any
o
hese h ee equa ions can be exp essed as a
linea combina ion o he o he wo. Thus he h ee
G-
loops
do no o m a
Non-Redundan se
o
G-loops.
Howe e , any wo o hese h ee
G-loops
do
o m a
Non-Redundan se
o
G-loops.
The ac ha sub h eshold
MOS
ansis o s can
o m
coupled loops
yields na u ally o he ollowing
concep o
Closed T anslinea Se ,
De ini ion:
Gi en a se
o
MOS
de ices, and
once a Non-Redundan se
o
loops has been
chosen, a Closed T anslinea Se
(CTS)
is
a
se
o
de ices such ha all loops hey o m
a e only coupled among hem, bu no
.o
loops
11-303
-I
A
+ L
BHCL
8
1V3TH
BF l
8T&J T-
Fig.
4
Example
o
Closed
3’5
T anslinea Se s.
MOS
De ices
7
and
8
o m
a
Closed T anslinea Se ,
and
so
do
MOS
De ices
1-7.
whe e b anches o o he de ices (no
belonging o he
CTS)
a e p esen .
This is illus a ed in Fig.
5.
Le us selec he NR se o
G-loops
‘1-2-3-7’, ‘4-5-6’, and ‘7-8’, and he NR se o
B-loops
‘1-2-3-4-56’, ‘7’, and
‘8’.
G-loops
‘1-2-3-7’
and ‘4-5-6’ a e coupled because he e is a e
B-b anches
o de ices o bo h
G-loops
ha a e sha ed in he
common
B-loop
‘1-2-3-4-5-6’. The wo
G-loops
‘1-2-3-
7’ and ‘4-5-6’, and he wo
B-loops
‘1-2-3-4-5-6’ and
‘7’ a e no coupled o o he loops (ei he
G-loop
‘7-8’
no
B-loop
‘8’), hus ( o he chosen NR se o loops)
de ices ‘1-2-3-4-5-6-7’ o m a
Closed T anslinea Se .
On he o he hand,
G-loop
‘7-8’ is no coupled o any
o he loop, no a e
B-loops
‘7’ and
‘8’.
The e o e,
de ices ‘7-8’ o m ano he
Closed T anslinea Se .
When wo king wi h mul iple
G-loops
and
B-loops,
wi h some o hem being coupled, i is no e y
con enien o classi y each b anch as being CW o
CCW o ien ed,
as
will become appa en la e . Le us
ins ead classi y all b anches in o wo o ien a ion g oups,
he
a-wise
o ien ed b anches and he
p-wise
o ien ed
b anches. Two b anches a e classi ied in o he same
g oup (ei he he
a-wise
o he
p-wise)
i hey appea in
he same loop wi h he same o ien a ion. On he
con a y, wo b anches a e classi ied each in o a
di e en g oup (one in o he
a-wise,
he o he in o he
p-
wise)
i hey appea
in
he same loop wi h opposi e
o ien a ion. No e ha now a CW b anch in one loop and
a CCW b anch
in
ano he loop can be classi ied in o he
same
a-wise
o
p-wise
g oup. I a b anch is sho -
ci cui ed, i o ms a one b anch loop and can be
classi ied as ei he
a-wise
o
p-wise.
Ano he concep ha is use ul o s a ing he
gene alized anslinea sub h eshold
MOS
heo em is
ha o
G-o de
and
B-o de
o
a
MOS
de ice in
a
Closed T anslinea Se :
De ini ion
4:
Once a
NR
se
o
loops
has been
chosen, a sub h eshold
MOS
ansis o which
is pa
o
a
Closed T anslinea Se
is said
o ha e a
G-o de
o
alue
nc,
i i s
G-
b anch
belongs o
nc
G-loops
o he gi en
Closed T anslinea Se .
An equi alen de ini ion o
B-o de
can be s a ed o
B-
loops.
The concep is illus a ed in Fig. 4. Le
us
choose
he NR se o
G-loops
‘1-2-3-6’, ‘2-4-5’, and ‘6-7’ and
o
B-loops
‘1-2-3-4’,
‘5’,
‘6’, and
‘7’.
G-loops
‘1-2-3-6’
and ‘2-4-5’ a e coupled because de ices ‘1-2-3-4’ o m
a
B-loop.
The e a e no o he couplings among he
chosen loops. Consequen ly, de ices
‘
1-2-3-4-5-6’ o m
a
Closed T anslinea Se
which consis s o
G-loops
‘1-
2-3-6’ and ‘2-4-5’ and
B-loops
‘1-2-3-4’,
‘5’,
and ‘6’.
De ices
6
and
7
o m one
G-loop
(‘6-7’) and wo
B-
loops
(‘6’
and
‘7’)
which a e no coupled o any o he
loop. The e o e de ices 6 and 7 o m ano he
Closed
T anslinea Se .
MOS
de ice 2 has
G-o de
nG2
=
2
because i s
G-b anch
appea s in wo
G-loops
o he
same
Closed T anslinea Se .
MOS
de ice 6 does no
ha e
G-o de
2 because, al hough i s
G-b anch
belongs
o wo di e en
G-loops,
hese wo loops do no belong
o he same
Closed T anslinea Se .
All
MOS
de ices
ha e
B-o de
one because hei
B-b anches
appea only
in one
B-loop.
When a
G-b anch
has
G-o de
g ea e han one i
belongs o mo e han one
G-loop
o
he same
Closed
T anslinea Se .
In such cases i is possible ha he
b anch be classi ied as
a-wise
in some
G-loops
and as
p-
wise
in o he
G-loops.
Unde hese ci cums ances i is
con enien o di ide i s
G-o de
in o wo pa s
(9)
nc
=
na.,+n
B,
G
whe e
na
(le
us
call i
G-a-o de )
deno es he imes
his
G-b hnch
is classi ied as
a-wise
in a CTS, and
n
(le us call i
G-p-o de )
deno es he imes i is
clhied as
p-wise
in CTS. Simila ly, o
B-b anches,
he
B-o de
can be sepa a ed in o he
B-a-o de
(
na,
)
and he
B-p-o de
(
nP,
B).
Using he concep s and p elimina y heo ems
in oduced un il now, i is possible o s a e and p oo he
gene alized anslinea heo em o sub h eshold
MOS
ansis o s’:
Theo em:
Gi en a se o sub h eshold
MOS
de ices and choosing o hem a se o
Non-
Redundan G-loops
and
B-loops,
o each
Closed T anslinea Se
he ollowing can be
s a ed:
I
i is possible
o
ind an
a-
and
p-wise
classi ica ion
o
hei
G-loops and B-loops
such ha
a) he
sum
o
G-a-o de s
equals he
sum
o
G-
P-o de s
Zna,,,
=
Cnp,cl
(10)
J
E
{
a-w se)
Is
{p-wise}
bland, e e y ime a de ice’s
G-b anch
is
classi ied as
a-wise
in a
G-loop
i s
B-b anch
can be classi ied as
a-wise
in some
B-loop,
and e e y ime a de ice‘s
G-b anch
is
classi ied as
P-wise
in a
G-loop
i s
B-b anch
can be classi ied as
p-wise
in some
B-loop,
hen he p oduc o no malized cu en s
aised o he powe o hei
G-a-o de
o
all ansis o s in he
CTS
whose
G-b anches
ha e been classi ied
a-wise
equals he
p oduc o no malized cu en s aised o he
powe o hei
G-P-o de
o all ansis o s
whose
G-b anches
ha e been classi ied
P-wise.
P oo :
Fo each
G-loop
in he
Closed T anslinea Se
he
ollowing holds:
11
iG1
le
{p-wise)
1.
The heo em will be s a ed using
G-b unches
as
p ima y b anches
and making
B-b anches
o depend on hem. Howe e , because
o
he symme y be ween
G-b unches
and
B-b unches
(due
o
he
symme y be ween
V,,
and
V,,
ol ages in eq.
(3)),
he heo em
can be s a ed
as
well by in e changing
G-b unches
and
B-
b unches.
11-304
Since his is ue o e e y single
G-loop
we can
mul iply hese equa ions o he chosen se
o
Non-
Redundan
G-loops
and hei p oduc will s ill be equal
o uni y,
Fu he mo e, we can aise i o he powe o
K,
and i
s ill be equal o uni y,
No e ha , since he de ices
o m
a CTS, all b anches
will be p esen and no b anch
o
ano he CTS appea s.
Consequen ly, eq.
(18)
includes all
G-b anches
o
he
CTS and only he b anches
o
his CTS. Equi alen ly,
he same can be s a ed o all
B-loops,
bu aising now o he powe o 1
-
IC,
o con enience.
No e ha , due o s a emen
b)
in Theo em 3, e e y ime
a de ice has i s
pseudo-cu en
iGj
in
he nume a o
o
eq.
(1
8), i s
pseudo-cu en
iBj
will also appea in he
nume a o
o
eq. (19), and bo h will appea
na,
cj
imes. And he same holds o
pseudo-cu en s
in he
denomina o s
o
eqs. (18) and (19). The e o e, le us
de ine
na,
and
n
such ha
1
Also, since he de ices o m a
Closed T anslinea Se ,
eq. (1 8) includes all de ices o he CTS, and
so
does eq.
(1
9). Consequen ly, we can mul iply eqs. (1 8) and
(1
9)
and index he
iG
and
i pseudo-cu en s
o
he same
de ice wi h he same su&c ip and use his subsc ip o
index he MOS de ice,
On he o he hand, due o s a emen a) in he
heo em he ollowing is sa is ied
By mul iplying eqs.
(21)
and
(22),
and using eq.
(4)
we
ob ain
which concludes he p oo
o
he
Gene alized
Sub h eshold MOS T anslinea Theo em.
0
111.
Re e ences
B. Gilbe , “T anslinea Ci cui s: A P oposed
Classi ica ion,”
Elec onics Le e s,
ol.
1
I,
No.
1,
pp. 14-16, 1975; e a a, ol.
11,
No.
1.
A. G. And eou and
K.
A. Boahen, “T anslinea
Ci cui s
in
Sub h eshold MOS,”
Jou nal
o
Analog
In eg a ed Ci cui s and Signal P ocessing,
ol. 9,
pp. 141-166, 1996.
E. A. Vi oz, “Analog VLSI Implemen a ion
o
Neu al Ne wo ks,” in
Handbook o Neu al
Compu a ion,
Ins i u e
o
Physics Publishing and
Ox o d Uni e si y P ess, USA.
B. A. Minch, C. Dio io, P. Hasle and
C.
Mead,
“T anslinea Ci cui s using Sub h eshold Floa ing-
Ga e
MOS
T ansis o s,”
Jou nal
o
Analog
In eg a ed Ci cui s and Signal P ocessing,
ol. 9,
pp. 167-179, 1996.
E. See inck and
R.
J. Wiege ink, “Gene alized
T anslinea Ci cui P inciple,”
IEEE Jou nal o
Solid-s a e Ci cui s,
SC-26, ol. 8, pp. 1198-1102,
Augus 1991.
D.
R.
F ey, “Log-Domain Fil e ing: An App oach o
Cu en -Mode Fil e ing,”
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