Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 4
,
A
ugu
st
2014
, pp
. 53
2
~
53
8
I
S
SN
: 208
8-8
7
0
8
5
32
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
A CMOS-based Analog Function
Generator: HSPICE
Modeling and Simulation
Madina H
a
mi
ane
Department o
f
Telecommunication
Engin
eering
,
Ahlia
Univ
ersit
y
,
M
a
nam
a
, B
a
hrai
n
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 14, 2014
Rev
i
sed
Jun
18,
201
4
Accepte
d J
u
l
4, 2014
In many
Eng
i
neering applicatio
ns, anal
og cir
c
uits present man
y
advantag
es
over the
i
r digi
ta
l count
erparts
a
nd have r
ecen
tl
y been p
a
rti
c
ular
l
y
used
in a
wide range of signal processor circuits
. In this paper, an an
alog
non-linear
function s
y
n
t
hes
i
zer is presented
ba
sed on a poly
nomial
expansion model.
The proposed
function s
y
nth
e
sizer model is
based on a
10th order
poly
nomial app
r
oximation of
an
y
of
the required non-linear fun
c
tions. The
poly
nomial app
r
oximations of
th
ese functions can then
b
e
implemented using
basic CMOS circuit blo
c
ks. Th
e propos
ed circuit model can simultan
e
ous
ly
s
y
nthesize and
generate man
y
differen
t
m
a
th
e
m
atica
l
fun
c
tion
s
. The
c
i
rcui
t
m
odel is desig
n
ed and
sim
u
la
ted wi
th HSPICE and
its p
e
r
f
orm
a
nce is
demonstrated
thr
ough the simulation of
a number
of non-lin
ear
fun
c
tions.
Keyword:
C
M
OS T
r
ansi
s
t
ors m
odel
s
Fu
nct
i
o
n
sy
nt
h
e
si
zer
HSPICE Sim
u
latio
n
Po
lyno
m
i
a
l
mo
d
e
l
Si
gnal
p
r
oces
s
o
r
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
M
a
di
na Ham
i
ane
Depa
rt
m
e
nt
of
Tel
ecom
m
uni
cat
i
on E
n
gi
nee
r
i
n
g
,
Ah
lia Un
i
v
ersi
ty,
Gosi
C
o
m
p
l
e
x,
M
a
nam
a
, B
a
hrai
n
Em
a
il: m
h
a
m
ia
n
e
@ah
lia.edu
.b
h
1.
INTRODUCTION
Anal
og
no
nl
i
n
ear
ci
rc
ui
t
s
ha
ve m
a
ny
appl
i
c
at
i
ons, es
peci
al
l
y
i
n
si
gnal
pr
ocessi
n
g
, c
o
m
m
uni
cat
i
on,
i
n
st
rum
e
nt
at
i
on, ne
ural
net
w
or
ks, a
nd m
e
di
cal
equi
pm
ent
.
As a res
u
l
t
,
a l
a
rge num
ber o
f
anal
og
si
gnal
p
r
o
cesso
rs
h
a
ve b
een
d
i
scu
s
sed
in
th
e literatu
re. In
itially,
an
alog
sign
al p
r
o
cesso
rs
were d
e
sign
ed
with
the
use
of
pa
ssi
ve
el
ect
ro
ni
c co
m
ponent
s s
u
c
h
rersi
s
t
o
rs
an
d
sim
p
l
e
sem
i
cond
uct
o
r
devi
ce
s suc
h
as
di
od
es an
d
BJT tra
n
sistors.
W
i
t
h
t
h
e a
d
vant
of J
F
ET
and MOSFET
transist
ors
,
t
h
e non-linea
r c
h
aracteristics
of thes
e
devi
ces
ha
ve t
h
en
bee
n
e
x
pl
oi
t
e
d i
n
t
h
e
de
si
gn
o
f
s
u
ch
p
r
oces
so
rs. M
a
ny
ap
p
r
oac
h
es
i
n
v
o
l
v
i
n
g
t
h
e
use
o
f
piecewise-li
ne
ar function a
p
proxim
at
ions
of
non-line
a
r functions ha
ve
be
en re
ported i
n
the literature [1], [2].
In this
respect,
BJT a
n
d BiCMOS tra
n
sistors ha
ve bee
n
us
ed
t
o
si
m
u
l
a
t
e
no
n
-
l
i
n
ear f
unc
t
i
ons.
Mo
re recen
tly, CMOS an
alog
circu
its b
a
sed
o
n
th
e exp
o
n
e
n
tial-law
an
d
th
e
squ
a
re-l
aw
characte
r
istics
of a M
O
S t
r
ansistor
op
erating
in
strong
and
weak inv
e
rsio
n resp
ectiv
ely
h
a
v
e
b
e
en
repo
rted
[3], [4]. These
circuit
realizations
prese
n
t
s
o
me
di
sadv
an
tag
e
s, t
h
e two
mo
st im
p
o
r
tan
t
bein
g
t
h
e realizatio
n
of
onl
y
o
n
e f
u
nct
i
on at
a t
i
m
e
and t
h
ei
r
o
p
erat
i
o
n i
n
v
o
l
t
age m
ode or
m
i
xed cur
r
ent
and
vol
t
a
ge
m
ode.
Ho
we
ver
,
i
n
cur
r
ent
-
m
ode
ci
rcui
t
s
wi
der si
gnal
ba
nd
wi
dt
hs
a
n
d l
a
rge
r
d
y
n
am
i
c
ran
g
es
of
ope
rat
i
o
n can b
e
obt
ai
ne
d a
s
op
pos
ed
t
o
v
o
l
t
a
g
e
-m
ode ci
rc
ui
t
s
.
A
num
ber
of
C
M
OS
c
u
r
r
e
n
t
-
m
ode
anal
o
g
p
r
oces
so
rs
have
been
re
po
rt
ed
i
n
t
h
e l
i
t
eart
u
r
e
. H
o
we
ve
r,
t
h
ese ci
rcui
t
s
prese
n
t
m
a
ny
di
sad
v
a
n
t
a
ges
suc
h
as t
h
ei
r r
eal
i
zat
i
on of o
n
l
y
a few fu
n
c
t
i
ons an
d o
n
l
y
one
funtion at a ti
me [5]-[7]. In add
ition, these
circuits are based on pi
ecewise linear approxi
m
a
t
i
ons of the
non-
lin
ear fun
c
tions.
CMOS curre
nt-m
ode analog signalsynt
h
es
izer has
r
ecently been propose
d
[7]. T
h
e
circuit
was
base
d o
n
a t
h
i
r
d
or
der Tay
l
or
’s seri
es ex
pan
s
i
o
ns o
f
n
onl
i
n
ea
r f
u
n
c
t
i
ons
whi
c
h res
t
ri
ct
ed t
h
e nu
m
b
er of
functions
that
can be realized
and
the
acc
ura
c
y of their
realizations.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
53
2
–
53
8
53
3
2.
MO
DEL FO
R
M
UL
ATIO
N
In t
h
i
s
pa
per,
a C
M
OS-
b
ase
d
ci
rc
ui
t
m
ode
l
of a cu
rre
nt
-
m
ode anl
og
fu
nct
i
on
sy
nt
hesi
zer t
h
at
ca
n
real
i
ze a l
a
rge num
ber of n
o
n
-
l
i
n
ear f
u
nct
i
ons i
s
prese
n
t
e
d. The ci
rc
ui
t
m
odel
i
s
based o
n
a 10t
h
-
or
der
p
o
l
yno
m
i
al ap
p
r
ox
im
a
tio
n
of an
y
no
n-lin
ear fu
n
c
tion and
i
s
co
m
p
atib
le with
th
e CM
OS
tech
no
log
y
cu
rren
tly
use
d
i
n
di
gi
t
a
l
si
gnal
pr
ocess
i
ng.
An
ot
he
r a
d
ava
n
t
a
ge
of t
h
e p
r
o
p
o
sed m
odel
i
s
t
h
e o
p
e
r
at
i
on
of t
h
e C
M
OS
t
r
ansi
st
o
r
s i
n
t
h
e st
r
o
n
g
i
n
ver
s
i
on
regi
on
, l
eadi
n
g t
o
t
h
e
p
o
ssi
bl
e ci
rcui
t
o
p
erat
i
o
n at
hi
g
h
f
r
eq
ue
nci
e
s.
Ot
her
adva
nt
age
s
o
f
t
h
e pr
o
pose
d
c
i
rcui
t
m
odel
ar
e t
h
e sim
u
l
a
t
n
eou
s
real
i
zat
i
on o
f
m
a
ny
non
l
i
n
ear f
unct
i
o
n
s
at
a
tim
e
that do
not nee
d
the
us
e of
piece linear appr
oxim
a
tion.
In t
h
e propose
d
circ
uit
m
odel, a
10
th
or
der
pol
y
n
o
m
i
al of t
h
e fo
rm
gi
ven i
n
equat
i
o
n (
1
) i
s
used t
o
ap
pr
o
x
i
m
at
e non
-l
i
n
ear f
u
nct
i
o
i
n
s wi
t
h
a hi
gh
deg
r
e
e
of accuracy.
(1
)
|
|
1
3.
PROP
OSE
D
CIRCUIT MODEL
Eq
uat
i
on (
1
)
c
a
n
b
e
real
i
zed
by
t
a
ki
n
g
t
h
e
sum
of
t
h
e
w
ei
g
h
t
e
d
out
put
c
u
rre
nt
s
o
f
a
num
ber
o
f
building blocks that
consist of the tr
ad
itional
class-AB current
m
i
rror ci
rcu
it
to provide bo
th power-raisin
g and
a
m
p
l
i
f
i
c
a
t
i
o
n
o
f
t
h
e
c
u
r
r
e
n
t
i
n
p
u
t
,
and a
d
di
n
g
i
t
t
o
a
co
nst
a
nt
cu
rre
nt
.
O
n
e
suc
h
b
u
i
l
d
i
n
g
bl
oc
k i
s
t
h
e s
q
uar
r
i
n
g
uni
t
s
h
ow
n i
n
F
i
g
u
r
e
1
.
Fi
gu
re
1.
M
o
di
fi
ed c
u
r
r
e
n
t
m
i
rr
or
t
o
pr
o
v
i
d
e
out
put
c
u
rre
nt
s
p
r
op
or
tio
n
a
l t
o
th
e squ
a
r
e
of
t
h
e i
n
pu
t cu
rr
en
t
The Tra
n
sisitors
T
1
and
T
2
as
well as
T
3
an
d
T
4
are
ass
u
m
e
d t
o
be
wel
l
m
a
t
c
hed
an
d
Tra
n
si
st
o
r
ss
T
1
th
ro
ugh
T
8
are
assum
e
d t
o
h
a
ve t
h
e sam
e
val
u
e
of
t
h
e transc
onductanc
e
param
e
ter i.e.,
n
=
p
an
d
are
ope
rat
i
n
g i
n
t
h
ei
r st
aurat
i
on
r
e
gi
o
n
. T
h
e a
s
p
ect
rat
i
o
s
(
W
/
L
)
of
t
r
a
n
si
st
o
r
s
T
1
–
T
8
of
Fi
gu
re
1 ar
e g
i
ven i
n
Tabl
e 1.
Tabl
e
1.
As
pec
t
R
a
t
i
o
s (
W
/
L
)
fo
r t
h
e
t
r
a
n
si
st
ors
o
f
Fi
g
u
re
1
Transistor
T
1
T
2
T
3
T
4
T
5
T
6
T
7
T
8
W/L
1/ 1
1/ 1
1
/ 1
1
/ 1
1/ 1
1
/ 1
1
/ 1
1
/ 1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A C
M
OS
-b
ase
d
A
n
al
o
g
F
unc
t
i
on
Gene
rat
o
r
:
H
S
P
I
C
E
M
o
d
e
l
i
ng
an
d
Si
m
u
l
a
t
i
on (
M
adi
na
H
a
mi
ane)
53
4
W
i
t
h
these ass
u
m
p
tions, the translinea
r
pri
n
ci
pl
e i
s
appl
i
e
d t
o
pr
o
duce
t
h
e
ou
tpu
t
cu
rren
t
I
out
whi
c
h can be
then e
x
pres
sed as [7]
(2
)
In
o
r
d
e
r t
o
ob
t
a
in
an
o
t
h
e
r
ou
tpu
t
cu
rren
t propo
rtio
n
a
l to
th
e inp
u
t
cu
rren
t, two
add
ition
a
l tran
sistors
T
9
and
T
10
are a
d
ded
with as
pect rati
os
1/2 an
d
1/
1
respect
i
v
el
y
as
sh
ow
n i
n
Fi
g
u
r
e
2.
Fro
m
th
is circuit, o
u
t
p
u
t
cu
rren
ts of
v
a
lu
e
a
1
x
or
a
2
x
2
, can
b
e
ob
tain
ed
by u
s
ing
add
ition
a
l curren
t
m
i
rrors
o
f
dif
f
e
rent as
pect
ratio val
u
es
(
W
/L
) .
Fi
gu
re
2.
M
o
d
i
fi
ed s
qua
ri
n
g
ci
rcui
t
o
f
fi
g
u
r
e
1 t
o
pr
o
v
i
d
e
out
put
c
u
r
r
ent
s
propo
rtion
a
l t
o
th
e inp
u
t
curren
t and
its
square.
App
l
yin
g
th
e t
r
an
slin
ear prin
cip
l
e, th
e
no
rm
alized
ou
tpu
t
curren
t
in
Figu
re
2
will b
e
g
i
v
e
n b
y
:
or
(3
)
whe
r
e
x
=
I
in
/
I
b
represen
ts t
h
e
n
o
rm
alized
in
pu
t cu
rren
t.
Equ
a
tion
(2) can also b
e
re-written
u
s
i
n
g
th
e
norm
alized input curre
nt as:
or
(4
)
An
d i
n
or
der
t
o
o
b
t
a
i
n
a c
u
rre
nt
p
r
op
ort
i
o
nal
t
o
x
3
, th
e
fo
ll
owing
r
elation
is u
s
ed
:
(5
)
Th
e co
rrespond
ing
circu
it w
ill th
erefo
r
e requ
ires two
m
o
d
i
fied
squ
a
ri
n
g
circu
its w
ith
inp
u
t
s
p
r
op
ortio
n
a
l to th
e d
i
fferen
ce an
d
t
h
e su
m
o
f
th
e in
pu
t cu
rren
t and
its sq
u
a
re. Th
e requ
ired th
ird
ord
e
r term in
eq
u
a
tion
(1
)
can
th
en
b
e
ob
tain
ed
b
y
selectin
g
a
p
propriate values
of t
h
e as
pect ratios
(W/
L
).
There
f
ore, a
n
d
i
n
or
der t
o
o
b
t
a
i
n
o
u
t
p
ut
cu
rre
nt
s p
r
o
p
o
rt
i
onal
t
o
e
v
e
n
a
nd
o
dd
po
we
rs
of t
h
e i
n
p
u
t
current, the m
odified
squari
ng circuit
of
Fi
g
u
r
e
2 alon
g w
ith equ
a
tio
n (5
)
ar
e re
peatedly
used.
Tables
2-a and
2-
b
gi
ve
t
h
e
de
t
a
i
l
s
of t
h
e
i
n
p
u
t
s
t
h
at
a
r
e
use
d
t
o
p
r
od
uce
o
u
t
p
ut
cu
rre
nt
s
pr
o
p
o
r
t
i
onal
t
o
x
3
t
h
r
o
ug
h
x
10
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
53
2
–
53
8
53
5
Tabl
e
2-a.
O
u
t
put
c
u
rre
nt
s
pr
op
o
r
t
i
onal
t
o
o
d
d
p
o
w
ers
o
f
i
n
p
u
t
c
u
r
r
ent
s
I
in
x + x
2
and
x - x
2
x + x
4
and
x - x
4
x + x
6
and
x – x
6
x + x
8
and
x – x
8
I
1
x
3
/2
x
5
/2
x
7
/2
x
9
/2
Tabl
e2
-b
.
Out
p
ut
cu
rre
nt
s
pr
o
p
o
r
t
i
o
nal
t
o
e
v
en
of
i
n
put
c
u
r
r
ent
s
I
in
x x
2
x
3
x
4
x
5
I
2
x
2
/8
x
4
/8
x
6
/8
x
8
/8
x
10
/8
It can
th
erefore b
e
seen
th
at h
i
gh
er-ord
er term
s o
f
eq
u
a
tion
(1
) can
b
e
ob
tain
ed
b
y
repetitiv
e u
s
e of
th
e circu
it m
o
d
e
l of Figu
re
2
withou
t th
e
n
eed
fo
r
d
e
d
i
cated
curren
t
mu
ltip
liers.
W
i
t
h
th
is
d
e
sign
an
d
t
h
e
ad
d
ition
of a no
rm
alized
DC
cu
rren
t, an
y n
o
n
lin
ear
fun
c
tion
can
b
e
realized
u
s
ing
MOSFET curre
n
t
-mirrors
with the a
p
propriate aspect
r
a
t
i
o
s (
W
/
L
). Fi
gu
re
3 sh
o
w
s t
h
e ba
si
c ci
rcui
t
m
odel
of t
h
e
fu
nct
i
o
n sy
nt
h
e
si
zer
whe
r
e B
re
fer
s
t
o
t
h
e s
qua
ri
n
g
ci
rc
ui
t
m
ode
l
of Fi
gu
re
2
.
Th
e circu
it sho
w
s on
ly ou
tpu
t
s propo
rtional to
x
th
ro
ugh
x
6
.
Fi
gu
re
3.
B
a
si
c ci
rcui
t
m
odel
f
o
r
t
h
e
fu
nct
i
o
n
sy
nt
hesi
zer
sh
o
w
i
n
g
out
put
s
p
r
o
p
o
rt
i
o
nal
to
th
e
first
6
term
s o
f
th
e po
lyn
o
m
ial ex
p
a
n
s
io
n
4.
SIMULATION RESULTS
The basi
c ci
rc
ui
t
m
odel
s
of
Fi
gu
re 3
was u
s
ed i
n
t
h
e si
m
u
l
a
t
i
on of a
nu
m
b
er of n
onl
i
n
ear fu
nct
i
o
ns
.
Th
e co
rr
espond
ing
po
lyno
m
i
al ex
p
a
n
s
ion
co
eff
i
cien
ts
a
i
,
i =
1,
… 1
0
f
o
r sel
ect
e
d
f
u
nct
i
o
n
s
are
gi
ven i
n
Tables 3-a and 3-b, and the transist
ors
’
as
pe
cts ratios
were
selected accordingly.
HSPIC
E
circuit sim
u
lation
envi
ro
nm
ent
was use
d
an
d
t
h
e sim
u
l
a
t
i
on was carri
e
d
o
u
t
usi
n
g t
h
e B
S
IM
2 l
e
vel
3
9
M
O
SFET t
r
a
n
si
st
o
r
m
o
d
e
ls with
L=
0.
1
μ
m
, bias
curre
nt
I
b
=
1
μ
A
a
n
d s
u
ppl
y
vol
t
a
ges
V
DD
= -
V
SS
=
2
V
.
Fo
r eac
h
fu
ncti
o
n
sim
u
l
a
t
i
on, t
h
e
i
n
p
u
t
cu
rre
nt
was c
h
an
ge
d f
r
o
m
0
μ
A
to
1
μ
A
, an
d t
h
e
o
u
t
put
c
u
r
r
e
n
t
s
t
h
ro
u
ghl
oad
resi
st
ances
of 1=
M
Ω
wa
s obtaine
d obtained. A DC
current
s
o
urce
= 1
μ
A
wasadd
ed to
th
e ou
tpu
t
n
o
d
e
t
o
represen
t th
e
constant term
in e
quation
(1)
whic
h e
quals
,
accordin
g t
o
T
a
bles
3-a a
n
d
3-b, either to
1 or zero.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A C
M
OS
-b
ase
d
A
n
al
o
g
F
unc
t
i
on
Gene
rat
o
r
:
H
S
P
I
C
E
M
o
d
e
l
i
ng
an
d
Si
m
u
l
a
t
i
on (
M
adi
na
H
a
mi
ane)
53
6
Tabl
e
3-a.
P
o
l
y
nom
i
a
l
expa
nsi
o
n
co
e
fficie
n
ts for s
e
lected functions
Function
a
0
a
1
a
2
a
3
a
4
a
5
sin(
x
) 0
1
0
-
1
/6
0
1/120
1
√
1
1
-
1
/2
3/8
-
5
/16
35/ 128
-
0.
2461
tanh(
x
) 0
1
0
-
1
/3
0
2/15
ln
(1
-
x
)
0
-1
-1
/2
-1
/3
-1
/4
-1
/5
e
x
1 1
1/2
1/6
1/24
1/120
J
1
(
x
)
0 1/2
0
-
1
/16
0
1/384
I
0
(
x
)
1
0 1/4
0
1/64
0
1
1
0 -
1
/2
0 -
1
/8
0
Tabl
e
3-
b.
Pol
y
nom
i
a
l
expan
s
i
o
n
coe
ffi
ci
en
t
s
fo
r sel
ect
ed
f
unct
i
o
ns
Function
a
6
a
7
a
8
a
9
a
10
sin(
x
)
0 -
1
/5040
0
1/362
880
0
1
√
1
0.
2256
-
0
.
2095
0.
1964
-
0
.
1855
0.
1762
tanh (
x
) 0
-
17/315
0
0.
0219
0
ln(1-
x
)
-1
/6
-1
/7
-1
/8
-1
/9
-1
/1
0
e
x
1/720
1/504
0
1/403
20
1/362
8
80
1/362
880
0
J
1
(
x
)
0 -
1
/1843
2
0
1/147
45
60
0
I
0
(
x
)
1/230
4
0
1/147
45 6
0
1/147
45 60
0
1
-
1
/16
0
-
5
/128
0
-
7
/256
The exact nonlinear
functions
were calc
u
lated an
d their gra
p
hs com
p
ared with those
of the
si
m
u
lated
fu
n
c
tio
n
s
as illu
strated
in Figu
re
4
.
In
sp
ecti
o
n
of t
h
is fi
g
u
re clearly sh
ows t
h
at t
h
e sim
u
lated
resu
lts
ar
e in
ex
cellent ag
r
eem
en
t w
ith
th
e calcu
lated
o
n
e
s. Tab
l
e 4
sh
ow
s th
e r
a
n
g
e o
f
input cu
r
r
e
n
t
v
a
lu
es f
o
r
whic
h the error betwee
n corresponding functions is le
ss than
1% which furt
her
re
flect
s the accuracy
of the
pr
o
pose
d
f
unct
i
on sy
nt
hesi
ze
r
ci
rcui
t
m
odel
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
53
2
–
53
8
53
7
Fi
gu
re
4.
Si
m
u
l
a
t
e
d an
d cal
c
u
l
a
t
e
d f
unct
i
o
ns
fr
om
Tabl
es 3
-
a an
d
3-
b
Tabl
e
4. R
a
n
g
e
o
f
i
n
p
u
t
cu
rre
nt
val
u
es
Function
sin(
x
)
1
√
1
tanh (
x
) ln(
1
-
x
)
e
x
J
1
(
x
)
I
0
(
x
)
1
Range of
x
< 1
A <
0.
8
A
< 1
A
< 0.
8
A
< 1
A <
1
A
< 1
A
< 0.
9
A
5.
CO
NCL
USI
O
N
Desi
g
n
of a si
m
p
l
e
funct
i
o
n
sy
nt
hesi
zer
u
s
i
ng M
O
SF
ET
t
r
ansi
st
o
r
m
odel
s
avai
l
a
bl
e
i
n
HS
PIC
E
sim
u
l
a
t
i
on env
i
ro
nm
ent
has
been
prese
n
t
e
d
.
The ci
rcui
t
m
odel
was based o
n
ap
pr
oxi
m
a
t
i
ng any
no
nl
i
n
e
a
r
fun
c
tion
with
th
e first 10
term
s in
its p
o
l
yn
o
m
ial ex
p
a
n
s
io
n
.
Th
e circuit
m
o
d
e
l th
at realizes an
y of th
ese
fun
c
tion
s
con
s
ists o
f
power-facto
r
raising
circu
its
b
u
ilt aro
und
a b
a
sic cu
rren
t
squ
a
rer
circu
it, a wei
g
h
t
ed
current am
plifier and a dc
current s
o
urce
. The
propose
d
synt
hesi
zer
m
odel can be
easily
m
odified to
im
ple
m
ent
ma
ny functions by proper sel
ection of th
e
transistors’ a
s
pect ra
tios.
The accuracy
of the
syn
t
h
e
sized fun
c
tio
n will b
e
p
r
im
arily d
ecid
e
d
b
y
th
e
n
u
m
b
e
r
o
f
term
s u
s
ed
in
t
h
e po
wer ex
pan
s
ion
app
r
oxi
m
a
t
i
on and
t
h
e e
ffec
t
s of m
i
sm
at
ch bet
w
ee
n t
r
a
n
sistors used i
n
practical implem
entation of t
h
e
req
u
ire
d
c
u
r
r
e
n
t-m
i
rrors
. E
x
pan
d
in
g
f
u
rt
he
r the
a
p
p
r
o
x
imatio
n
requ
ires th
e u
s
e o
f
a
ddi
t
i
onal
si
m
i
l
a
r po
we
r-
rai
s
i
ng ci
rcui
t
bl
oc
ks.
HSP
I
C
E
Si
m
u
l
a
ti
on o
f
a n
u
m
b
er o
f
no
nl
i
n
ea
r f
unct
i
ons s
u
pp
ort
e
d
by
t
h
e eval
uat
i
on
o
f
t
h
e m
ean squa
re er
ro
r
bet
w
e
e
n exa
c
t
an
d s
i
m
u
l
a
t
e
d fu
nct
i
ons
val
u
es
ve
ri
fi
ed t
h
e val
i
di
t
y
of t
h
e
p
r
o
pos
e
d
fu
nct
i
o
n sy
nt
h
e
si
zer ci
rc
ui
t
m
odel
.
REFERE
NC
ES
[1]
M.
Be
na
mma
r,
“Pre
c
i
se
,
wide
-range approximation to a sine function suita
b
l
e for
analog
im
plem
entation
in s
e
ns
or
s
and instrumentation applications
”,
I
E
EE Transactions on Circu
its
and Systems-
I:
Regular
Papers,
Vol. 52, pp. 262
-
270, 2005
.
[2]
B. Maud
y
and
S. Gift, “Novel
pseudo-exponential cir
c
uits”,
I
EEE Transactio
ns on Circuits
and Systems
-II
:
Express Brie
fs
,
Vol. 52
, pp
. 675
-679, 2005
.
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
0.
2
0.
4
0.
6
0.
8
x
si
n
(
x)
s
imu
l
a
t
e
d
ex
a
c
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0.
8
0.
9
1
x
1
/
s
q
rt
(1
+
x
)
s
imu
la
t
e
d
ex
ac
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
0.
2
0.
4
0.
6
0.
8
x
ta
n
h
(
x
)
s
i
mu
la
t
e
d
ex
a
c
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-3
-2
-1
0
l
n
(1
-x
)
s
imu
l
a
t
e
d
ex
ac
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
1
1.
5
2
2.
5
x
e
x
sim
u
l
a
t
e
d
ex
a
c
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0.
2
0.
4
0.
6
0.
8
1
x
(1
-x
2
)
si
m
u
l
a
t
e
d
ex
ac
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
0.
2
0.
4
x
J1
(
x
)
sim
u
la
t
e
d
ex
a
c
t
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
0.
2
0.
4
0.
6
Io
(
x
)
si
m
u
l
a
t
e
d
ex
a
c
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A C
M
OS
-b
ase
d
A
n
al
o
g
F
unc
t
i
on
Gene
rat
o
r
:
H
S
P
I
C
E
M
o
d
e
l
i
ng
an
d
Si
m
u
l
a
t
i
on (
M
adi
na
H
a
mi
ane)
53
8
[3]
M. Tavako
li
and
R. Sarpeshkar
,
“A sinh resistor and its app
lic
ati
on to tanh
line
a
r
i
za
tion”
,
IEEE
Journal of Solid-
State Circuits
, V
o
l. 40
, pp
. 536-5
43, 2005
.
[4]
C.A. De La Cruz-Blas, A.J. Lopez-
Martin and
J. Ramirez-Angulo, “Com
pact power-efficient class-AB CMOS
exponential voltage
conver
t
er”,
Ele
c
tronics Le
tte
rs
, Vol. 42
, pp
. 1
27-128, 2006
.
[5]
T. Arthansir
i
an
d V. Kasensuwan, “curren
t
-mode pse
udo-expo
nential-con
t
rol v
a
riab
le-gain am
plifier usning 4
th
-
order Tay
l
or
series approximatio
n
”,
El
ectr
oni
cs
L
e
tt
er
s
,
Vol. 42, p
p
. 379-380
, 200
6.
[6]
M.
A.
Ha
shie
sh, S.
A.
Ma
hmoud a
nd A.
M.
Solima
n,
“Ne
w
4
th
-quadrant CMOS curren
t
-mode and voltag
e
-mode
m
u
ltipliers”
,
An
alog Integrated
Circuits and
Sig
nal Processing
,
Vol. 45
, pp
. 295
-307, 2005
.
[7]
M.T.Abuelma'atti, “Universal C
M
OS cu
rrent-mode analog function s
y
nth
e
sizer
”,
IEEE Transa
c
tions on Circuits
and Systems
-I: Fundamental Th
eor
y
and App
licat
ions, Vol. 49, 20
02, pp
. 1468-14
74
BI
O
G
R
A
P
HY
OF
A
U
T
HO
R
M
a
dina Hamiane
receiv
e
d her
BS
c in Electr
onics
from
Uni
v
ers
ite des
S
c
ie
nces
et de l
a
Techno
logie Ho
uari Boumedien
n
e (USTHB), Alge
ria; and her
Master’s and PhD degrees in
C
y
bern
etics and
Control Engin
e
ering from the Univer
sity
of Reading, UK, and the
University
of
S
h
effield
,
UK, res
p
ect
ivel
y.
S
h
e
is
now with the
College of Eng
i
neering a
t
Ahlia
Universit
y
in
the Kingdom of Bahrain
.
Dr. Hamiane’s current
res
earch
interests span signal processing, pattern
recognition, bio
m
edical signal a
nd image
analy
s
is, computer
simu
lation
of electr
onic and control
s
y
ste
m
s.
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