Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 3
,
Ju
n
e
201
6, p
p
. 1
223
~ 12
32
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
3.9
869
1
223
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 Numerical Model of Joule He
ating in Piezoresistive
Pressure Sensors
Abdel
a
z
i
z
Beddiaf
1
, F
o
uad
Kerrour
2
, S
a
l
a
h
Kem
o
uche
2
1
F
acul
t
y
of
S
c
i
e
nce
and
T
echno
log
y
, Khen
che
l
a
Univers
i
t
y
,
Alge
ria
2
University
of C
onstantine 1, Mo
DERN
a
Labor
ator
y
,
Constantin
e, Alg
e
ria
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Des 25, 2015
Rev
i
sed
Feb
28
, 20
16
Accepted
Mar 12, 2016
Thermal dr
ift caused b
y
Joule heati
ng
in p
i
ezoresistiv
e pr
essure sensors
affec
t
s greatl
y
th
e results in the shift of
the offset voltag
e
of the such sensors.
The stud
y
of
the thermal b
e
havior of th
ese sens
ors is essential to define th
e
param
e
ters
th
at
caus
e
the outpu
t chara
c
teristic drift. Th
e impact of Joule
heating in a pr
essure sensor has been
studied. Th
e stud
y
involves
the solutio
n
of heat transf
er equation considering
the conduction in Cartesian coordinates
for the tr
ansient
regime using Finite
Differ
e
nce
Method.
We determine how
the temperature
affects the sensor dur
ing the apply
i
ng a supply
v
o
ltag
e
. For
this, the
tem
p
er
ature r
i
se gener
a
ted b
y
Joule h
eating in piezor
esistors has
been calculated
for differ
e
nt geo
m
etrica
l par
a
meters of the sensor
as well
as
for different op
erating time. It is
observed that Joule heatin
g leads to
im
portant r
i
se t
e
m
p
erature
in
the
pie
z
oresistor
an
d, hen
c
e
,
causes
drift
in
the
output voltag
e
v
a
riations in
a s
e
nsor
during its o
p
erated in
a prolonged time.
This pap
e
r put
emphasis on th
e geomet
r
i
c
influence par
a
meter
s
on these
chara
c
t
e
ris
t
i
c
s
t
o
optim
ize
the
s
e
ns
or perfor
m
ance.
The op
tim
izat
ion o
f
geometric parameters of se
nsor allows us to r
e
duc
ing the inte
rnal
he
at
in
g
effect. Results
showed also that lo
w bias voltage should be applied for
reducing
Joule h
eating.
Keyword:
Joul
e
Heat
i
n
g
Th
erm
a
l Drift
Piezoresistive
Press
u
re
Sens
ors
Finite Differe
n
ce Method
Copyright ©
201
6 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
:
Abdelaziz Beddiaf,
Facu
lty of
Scien
ces and
Tech
no
log
y
,
Khe
n
c
h
ela Uni
v
ersity,
Alge
ria,
Depa
rt
m
e
nt
of
El
ect
ro
ni
cs,
U
n
i
v
e
r
si
t
y
of
C
o
nst
a
nt
i
n
e
1
,
Al
geri
a.
Em
a
il: b
e
d
d
i
afaziz@yah
o
o
.
fr
1.
INTRODUCTION
A silico
n
p
i
ezo
r
esistiv
e
p
r
essu
re sen
s
o
r
wh
ich
u
s
e
th
e Wh
eatst
o
n
e
-brid
g
e
h
a
v
e
foun
d
a wid
e
appl
i
cat
i
o
n i
n
vari
ous
dom
ains, su
ch as aut
o
m
o
t
i
v
e engi
neering, aerospa
ce and bi
om
ed
ical instrum
e
nts. They
h
a
v
e
t
h
e adv
a
n
t
ag
es
o
f
h
i
gh
sensitiv
ity, ex
cellen
t
lin
earity o
f
electrical respo
n
se,
g
ood
techn
o
l
og
ical
co
m
p
atib
ilit
y,
sm
a
l
l
size, lo
w p
o
wer, m
a
ss p
r
odu
ctio
n
an
d
so
m
e
o
t
h
e
r adv
a
n
t
ag
es [1
]-[7]. Nev
e
rt
h
e
less, th
ey
ofte
n s
u
ffer
from
the te
m
p
erat
ure
d
r
i
f
t
,
t
h
e st
udy
of
t
h
e t
h
er
m
a
l
behavi
or
o
f
t
h
ese
sen
s
o
r
s
i
s
necessa
ry
so
as t
o
determ
ine the
param
e
ters tha
t
cause t
h
e
output c
h
aract
er
istics d
r
if
t
.
The kn
ow
ledg
e of
th
e ph
en
o
m
en
on
s
causing
its thermal d
r
ift presen
ts
a
p
a
rticu
l
ar in
terest.
I
n
h
i
s wo
rk
,
Y
i
cai Sun
[5
] sh
ow
e
d
the
existence
of t
h
e electri
c d
r
i
f
t
of
bri
dge
of
fset
f
o
r
piezoresistive press
u
re sens
ors as well as therm
a
l drif
t. His study
uses
the electric drift for com
p
ensati
ng
of
fset
t
h
e
r
m
a
l
dri
f
t
.
I
n
a
p
r
ev
i
ous
pa
per
[
8
]
we
have
i
n
vest
i
g
at
ed t
h
e ef
fe
ct
of
t
e
m
p
erat
ure
an
d
d
opi
ng
l
e
vel
on the c
h
aract
eristics of suc
h
sens
ors. T
h
e
approac
h
to t
h
e com
p
ensation
for tem
p
erature drift of offset
vol
t
a
ge
i
n
t
h
i
s
sens
or
t
y
pe
was
pr
o
pose
d
by
U.
Al
ja
nci
c
[
7
]
.
T
h
e
Jo
ul
e heat
i
n
g i
n
pi
ezo
res
i
st
i
v
e
micro
can
tilev
e
r sen
s
ors
was i
n
d
i
cated b
y
M. Zah
i
d [9
].
His app
r
o
ach fo
cuses on
the an
al
ytical an
d
nu
merical
tech
n
i
qu
es to
ch
aracterize th
e Jo
u
l
e h
eating
in
su
ch
m
i
cro
can
tilev
e
rs.
He in
trodu
ced
a theo
retical
m
o
d
e
l for
pre
d
i
c
t
i
n
g
t
h
e
t
e
m
p
erat
ure
p
r
od
uce
d
by
t
h
e
Jo
ul
e
heat
i
n
g.
The
sam
e
aut
h
o
r
s
ha
ve
de
v
e
l
ope
d a
si
m
p
le an
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
12
2
3
– 12
32
1
224
accurate conduction–convect
ion m
odel
to predict the tem
p
erature distri
bution in p-dope
d piezoresistive
microcantileve
rs beca
use
of self-heating
[10]. Recen
tly, we m
a
de a therm
o
m
echanical
m
odeling of
a
pi
ezo
resi
st
i
v
e
press
u
re se
ns
o
r
[
1
1]
. M
o
re
re
cent
l
y
, i
n
a
n
ot
her
st
u
d
y
,
we
m
odel
e
d t
h
e t
h
erm
a
l
behavi
o
r
o
f
t
h
e
cap
acitiv
e p
r
essu
re sen
s
o
r
s, u
s
ing
Fin
ite Ele
m
en
t An
alysis (
FEA
) establish
e
d
in
COMSOL. Th
e th
erm
a
l
effects
due t
o
the tem
p
erature consideri
n
g the m
a
terial
s’ properties, the
geom
etric sha
p
e and als
o
the heat
trans
f
er m
echanism
s
are deve
lope
d [12]. Piezoresistive
pre
ssure
sens
o
r
u
s
i
ng a
Wh
eat
st
one
b
r
i
d
ge wi
t
h
t
h
e
pi
ezo
resi
st
or
s i
s
t
y
pi
cal
l
y
used
wi
t
h
a s
u
ppl
y
vol
t
a
ge
ra
ngi
ng
f
r
om
3 t
o
1
0
V i
n
vol
ves t
h
erm
a
l
dri
f
t
ca
use
d
by
Joul
e
heat
i
n
g.
The
prese
n
t
w
o
r
k
see
k
s t
o
st
udy
t
h
e i
m
pact of
Jo
ul
e
heat
i
n
g
i
n
a
pi
ez
ore
s
i
s
t
i
v
e pr
essu
r
e
sens
or
. T
h
e
st
udy
i
n
v
o
l
v
es
t
h
e sol
u
t
i
on
o
f
heat
t
r
a
n
sfe
r
equat
i
o
n c
o
n
s
i
d
eri
ng t
h
e co
n
duct
i
o
n i
n
C
a
rt
esi
a
n co
or
di
n
a
t
e
s fo
r
t
h
e t
r
ansi
e
n
t
r
e
gi
m
e
usi
ng F
i
ni
t
e
Di
ffer
e
nc
e M
e
t
hod
(
FDM
). It aim
s
to
expl
ore the
geom
etric influence
param
e
ters on
these c
h
aracteristics to
optim
i
ze the se
ns
or perform
a
nce. T
h
e elevati
o
n of te
m
p
erature c
a
use
d
by
t
h
e J
o
ul
e he
at
i
ng i
n
pi
ez
or
esi
s
t
o
rs
has
be
en cal
cul
a
ted for
va
rious
geometrical parameters of the
de
vice as
well as fo
r sev
e
ral op
erating
ti
m
e
;
it
allo
ws u
s
to
op
timizatio
n
th
e sen
s
o
r
d
e
si
g
n
fo
r redu
cing
the Jo
u
l
e
h
eatin
g.
2.
M
ETHOD
OLOGY AND
THEORY
Joule
heating, also known
as oh
m
i
c heating
or electric
a
l-resistiv
e, is th
e pro
c
ess by wh
ich the
passa
ge of a
n
electric curre
nt
through a conduct
o
r
releas
es
heat. T
h
e bul
k
of the t
h
erm
a
l energy is produce
d
due t
o
loss
of
kinetic ene
r
gy
of c
u
rrent ca
rrying elect
rons
by collisions a
m
ong them
selves and
with the lattice
ato
m
s [1
3
]
. In
th
is wo
rk
,
we
co
nsid
er
t
h
at the J
oule
heating effect as
t
h
e
onl
y
en
er
gy
co
nve
rsi
o
n,
ne
gl
ect
i
n
g
t
h
e ot
her
heat
t
r
ans
f
er
m
odes
suc
h
as t
h
e c
o
nvect
i
o
n a
n
d
r
a
di
at
i
on.
T
h
e
v
a
ri
at
i
on
of
t
e
m
p
erat
ure
d
u
e t
o
Jo
ul
e
heating, c
o
nsideri
ng t
h
e
hea
t
transf
er b
y
co
ndu
ctio
n
in
th
e p
i
ezo
resistiv
e p
r
ess
u
re
sens
or
s, i
s
s
h
o
w
n i
n
Fi
gu
re 1.
Fi
gu
re
1.
Pi
ezo
resi
st
i
v
e
pre
s
s
u
re
sen
s
or structure
with
heat
trans
f
er pat
h
The
heat
c
o
nd
uct
i
o
n
e
quat
i
o
n
pre
d
i
c
t
i
n
g
t
h
e J
o
ul
e he
at
i
n
g
o
n
t
h
e
pi
e
z
oresi
s
t
i
v
e
p
r
e
ssure
se
ns
or
con
s
i
d
eri
ng i
t
s
t
h
ree-
di
m
e
nsional
f
o
rm
i
n
C
a
rt
esi
a
n co
or
di
nat
e
s f
o
r t
r
a
n
si
ent
re
gi
m
e
and i
n
cl
u
d
i
n
g t
h
erm
a
l
ener
gy
gene
rat
i
on i
s
gi
ven
by
[
9
]
:
22
2
1
22
2
TT
T
T
q
tk
xy
z
(1)
Whe
r
e
q
i
s
h
e
a
t
f
l
u
x
,
k
i
s
t
h
erm
a
l
conduct
i
vi
t
y
,
α
is th
ermal d
i
ffu
si
v
ity and
t
is th
e time. Neg
l
ecting
th
e
change i
n
tem
p
erature
along t
h
e
perpe
ndic
u
lar
di
rect
i
o
n, e
q
uat
i
o
n
(
1
)
can
be m
odi
fi
ed
as
:
22
1
22
TT
T
q
tk
xy
(2)
The rat
e
o
f
e
n
e
r
gy
ge
nerat
i
o
n i
s
ex
presse
d by
:
2
0
2
V
q
R
da
(3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Numerical
Model of J
o
ule He
ating in
Piezores
istive Pressu
re
S
e
n
s
o
r
(
A
bd
ela
z
iz Bedd
ia
f)
1
225
Whe
r
e
V
0
is applied electrical pote
n
tial,
d
is
th
e th
ickn
ess
of m
e
m
b
ran
e
,
a
i
s
t
h
e l
e
ngt
h o
f
t
h
e s
qua
re-
s
h
a
pe
d
me
m
b
r
a
n
e
an
d
R
i
s
t
h
e
resi
st
a
n
ce
of
t
h
e
di
f
f
u
s
ed
pi
ezo
resi
st
or
,
gi
ve
n
by
t
h
e f
o
l
l
o
wi
ng
ex
pressi
o
n
[
9
]
:
pz
r
e
pzr
L
R
A
(4)
Whe
r
e
L
pzr
is t
h
e length of t
h
e piezoresistor,
A
pzr
the cross-sectional a
r
ea
and
ρ
e
is t
h
e el
ectrical resistivity.
Th
e app
licab
le b
o
u
n
d
a
ry co
nd
itio
n
s
in
cl
u
d
e
th
e ad
iab
a
tic h
eat con
d
ition an
d
m
a
in
tain
i
n
g
the h
e
at
continuity at the edges
which
are:
0
0
0
0
T
x
x
T
y
y
(
5
)
Tak
i
ng
also
i
n
to
acco
u
n
t
that th
e
d
i
ffu
s
ed
p
i
ezo
r
esist
o
r
can
b
e
co
nsid
ered
a
p
e
rfect
con
d
u
c
tor, we will
h
a
v
e
:
()
0
()
0
T
kh
T
T
x
x
a
T
kh
T
T
y
ya
(6)
Whe
r
e
h
is th
e h
eat tran
sfer co
efficien
t that d
e
p
e
nd
s
o
n
th
e
m
a
terial
an
d
th
e su
rro
un
d
i
n
g
air. Th
e in
itial
co
nd
itio
n in the all stru
ct
u
r
e i
s
:
(,
,
)
0
0
Tx
y
t
T
t
(
7
)
3.
FINITE
DIFFERENCE ME
THOD (
FD
M
)
Th
e m
a
terial p
r
op
erties
o
f
Silico
n
used in
t
h
is work
are indicated
in
Tab
l
e 1
.
Tab
l
e
1
.
Material p
r
o
p
e
rty of
Silico
n
[9
],[10]
Para
m
e
ter Values
M
a
ss density
,
ρ
(k
g
/
m
3
) 2320
Heat tr
ansf
er coef
ficient,
h
(Wm
-2
K
-1
) 2.
219
Electri
cal r
e
sistivity,
ρ
e (
Ω
.m
)
T
h
erm
a
l conductivity
,
K
(Wm
-1
K
-1
)
Specif
i
c
heat,
c (J/
K
gK)
T
h
erm
a
l diffusivit
y
,
α
(m
2
/s)
10
-3
150
712
0.
9*10
-4
3.
1.
Finite Di
ffere
nce Me
th
od
v
a
lida
t
ion
The 2
D
heat
con
d
u
ct
i
on e
q
uat
i
on i
n
t
r
an
si
ent
regi
m
e
is di
scret
i
zed
usi
n
g t
h
e Fi
ni
t
e
Di
ffere
nc
e
Meth
od
(
FDM
) and t
h
e o
b
t
a
i
n
ed sy
st
em
of l
i
n
ear eq
uat
i
ons i
s
sol
v
ed
by
t
h
e Thom
as al
gori
t
h
m
usi
ng t
h
e
Matlab
calcu
latio
n
software [1
4
]
. To
v
a
lid
ate th
e Fin
ite
Di
ffe
rence m
odel
,
we c
o
m
p
ared the obtained
results
o
f
ou
r m
o
d
e
l with
th
ose resu
lt
s ob
tain
ed
b
y
oth
e
r au
tho
r
s [15
]
.
C
.
Pram
ani
k
has devel
ope
d a
n
anal
y
t
i
cal
model
f
o
r t
h
e re
sol
u
t
i
o
n o
f
t
h
e
heat
t
r
ansfe
r
equat
i
o
n i
n
cylin
d
r
ical co
ord
i
n
a
tes for ci
rcu
l
ar stru
cture in
t
h
e
tran
si
en
t reg
i
m
e
u
s
in
g th
e m
e
th
o
d
of sup
e
rp
ositio
n of
vari
a
b
l
e
s [
1
5]
.
Using
th
ei
r resu
lts to
valid
ate o
u
r m
o
d
e
l,
we co
m
p
ar
ed t
h
e results
of t
h
e
cha
nge i
n
temperat
ure as
fun
c
tion
o
f
time with
t
h
ose
o
f
th
e an
alytical m
o
d
e
l (Figu
r
e
2).
According
to the
va
riation of tem
p
erature
as a
fu
nct
i
o
n of t
i
m
e i
n
t
h
e curve of Fi
gu
re 2, we can
not
ice that the re
sults are
in good agreem
ent. The
co
m
p
ariso
n
of
th
e ob
tain
ed
resu
lts allo
ws
u
s
to
v
a
lid
ate th
e
FDM
m
odel
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
12
2
3
– 12
32
1
226
0
0.
5
1
1.
5
2
2.
5
3
24
26
28
30
32
34
36
38
40
42
Ti
m
e
[
h
o
u
r
s
]
Tem
p
er
at
u
r
e [
°
C
]
a=
20
00 µ
m
,
d=
20 µ
m
,
R
=
0.
6 O
h
m
,
I
=
2 m
A
A
n
a
l
yt
i
c
al
m
o
d
e
l
[
15]
In
th
i
s
w
o
rk
Fi
gu
re
2.
Va
ri
at
i
on
of
t
e
m
p
erat
ure as
a
fu
nct
i
o
n
o
f
t
i
m
e [15]
3.
2.
Geome
t
ric effect par
a
meter
s in the
gener
a
ti
on
of tem
p
e
r
ature
As p
r
evi
o
u
s
l
y
not
e
d
, t
h
i
s
st
u
d
y
i
s
dev
o
t
e
d t
o
t
h
e t
h
erm
a
l
behavi
or
of t
h
e
sens
ors
.
Fo
r t
h
i
s
, we ad
opt
th
e
m
o
d
e
l o
f
Fin
ite Differen
ce Meth
od
.
Th
e th
erm
a
l d
r
i
f
t
s
pr
o
v
o
k
e
d
by
Joul
e hea
t
i
ng i
n
pi
ezo
r
e
si
st
i
v
e
press
u
re se
ns
ors affect great
ly th
e per
f
o
r
m
ance of
su
c
h
se
ns
ors.
As
we
kn
o
w
, t
h
e ge
om
et
ri
c infl
uence
p
a
r
a
m
e
ter
s
on
th
e r
i
se
of
temp
er
at
u
r
e
h
a
v
e
an
en
or
m
o
u
s
i
m
p
act, w
e
fo
cu
sed
o
n
th
e t
h
er
m
a
l d
r
if
t study o
f
th
e
Jo
u
l
e h
eating
i
n
th
is sensor t
y
p
e
.
In th
e
recen
t
work
[1
2
]
, we
h
a
v
e
st
u
d
i
ed
th
e th
erm
a
l
effects of capacitiv
e
press
u
re sens
or due to the
te
m
p
er
atu
r
e tak
i
ng
in
to
co
nsid
eration
th
e
geom
etric shape, the m
a
te
rials’
properties a
nd also t
h
e
h
eat transfe
r
m
echani
s
m
s
.
We
will stu
dy th
e th
erm
a
l
drift cau
s
ed b
y
Jou
l
e h
e
atin
g
o
f
p
i
ezo
r
esistiv
e pressu
re sen
s
or
characte
r
istics, so the ge
ometric influe
nce
param
e
te
rs on these chara
c
teristics to optim
ize the s
e
ns
or
per
f
o
r
m
a
nce.
The c
h
oi
ce o
f
t
h
ese
geom
et
ri
cal
param
e
t
e
rs fo
r si
m
u
l
a
t
i
o
n
s
i
s
t
a
ke
n f
r
o
m
earl
i
e
r pape
rs [
1
1]
,
[1
6]
.
3.
2.
1.
Effect of
a
p
plied vo
ltag
e in
generation
of temperature
To
see th
e effect o
f
app
lied vo
ltag
e
on
th
e t
e
m
p
er
at
ur
e
ge
nerat
i
o
n i
n
t
h
e
cen
ter of t
h
e
piezoresistor,
we vari
e
d
m
a
ny
param
e
t
e
rs as:
t
h
e t
h
i
c
kness of t
h
e di
a
p
h
r
a
g
m
,
t
h
e l
e
ngt
h of t
h
e m
e
m
b
rane
, ge
o
m
et
ri
cal
piezoresistor a
n
d ope
r
ating time of t
h
e
devi
ce.
3.
2.
1.
1.
Geome
t
ric effect par
a
meter
s of
the
piez
oresistor
Fi
gu
re
3
sh
ow
s t
h
e
va
ri
at
i
ons
i
n
t
e
m
p
erat
ure
as a
f
u
nct
i
o
n
of
ap
pl
i
e
d
v
o
l
t
a
ge i
n
t
h
e
ra
n
g
e
o
f
3 t
o
1
0
V fo
r seve
ral
l
e
ngt
hs o
f
t
h
e pi
ezo
resi
st
or
L
pzr
, where t
h
e
devi
ce i
s
ope
r
a
t
i
ng f
o
r
one hour. In this case, we
vari
e
d
L
pzr
wi
t
h
t
h
e
fi
xi
n
g
o
f
t
h
e ot
he
r ge
o
m
et
ri
cal
para
m
e
ters; we
h
a
v
e
ob
serv
ed
that th
e tem
p
eratu
r
e is
i
n
creasi
n
g
f
unc
t
i
on
of
t
h
e a
p
pl
i
e
d v
o
l
t
a
ge
.
The T
(
V
0
)
is i
n
v
e
r
s
ely pro
por
tio
n
a
l t
o
L
pzr
.
So,
t
h
e se
ns
or
s
wi
t
h
t
h
e l
o
w
Joul
e
heat
i
n
g
i
s
obt
ai
ned
fo
r a g
r
eat
l
e
n
g
t
h of t
h
e pi
ez
o
r
esi
s
t
o
r
w
h
i
c
h
i
s
a dra
w
bac
k
.
Ho
we
ver
,
i
t
can be see
n
i
n
Fi
gu
re 4 t
h
at
t
h
e
Joul
e
heating causes
a raise
of the te
m
p
erature
by inc
r
easing the cross
-
sec
tional area
A
pzr
. No
ting
t
h
at th
e
t
e
m
p
erat
ure
an
d a
ppl
i
e
d
v
o
l
t
a
ge s
h
ows
a
par
a
bol
i
c
dep
e
n
d
e
n
ce.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Numerical
Model of J
o
ule He
ating in
Piezores
istive Pressu
re
S
e
n
s
o
r
(
A
bd
ela
z
iz Bedd
ia
f)
1
227
3
4
5
6
7
8
9
10
35
40
45
50
55
60
65
70
V
o
l
t
ag
e V
0
[
V
]
Te
m
per
at
ur
e
[
°
C
]
a
=
10
00
µ
m
,
d
=
20
µ
m
,
A
p
z
r
=
6
µ
m
2,
t
=
6
0
m
i
n
Lp
z
r
=
6
0 µ
m
Lp
z
r
=
1
20
µ
m
Lp
z
r
=
1
80
µ
m
Lp
z
r
=
2
40
µ
m
Fi
gu
re
3.
Va
ri
at
i
on
of
t
e
m
p
erat
ure as
f
u
nct
i
o
n
of
ap
pl
i
e
d
v
o
l
t
a
ge f
o
r
seve
r
a
l
l
e
ngt
h
s
of t
h
e pi
ezo
resi
st
o
r
L
pzr
3
4
5
6
7
8
9
10
30
40
50
60
70
80
90
10
0
11
0
V
o
l
t
ag
e V
0
[
V
]
Te
m
p
er
a
t
ur
e
[
°
C
]
a=
10
00
µ
m
,
d
=
20
µ
m
,
L
p
z
r
=
1
00
µ
m
,
t
=
60
m
i
n
Apz
r
=
1
µ
m
2
Apz
r
=
4
µ
m
2
A
p
z
r
=
10 µ
m
2
A
p
z
r
=
20 µ
m
2
Fi
gu
re
4.
Va
ri
at
i
on
of
t
e
m
p
erat
ure as
f
u
nct
i
o
n
of
ap
pl
i
e
d
v
o
l
t
a
ge f
o
r
seve
r
a
l
cros
s-sect
i
o
nal
areas
A
pzr
3.
2.
1.
2.
Effect
of the operating
time
of
the
sens
or
t
Fi
gu
re
5
gi
ves
t
h
e ev
ol
ut
i
o
n
o
f
t
h
e
t
e
m
p
erat
ure as a
f
u
nct
i
o
n
o
f
a
ppl
i
e
d
v
o
l
t
a
ge
fo
r se
veral
val
u
es
o
f
ti
m
e
t
. It is clear that t
h
e temperat
ure is
prop
ortion
a
l to
t
h
e op
erating
tim
e
of th
e d
e
v
i
ce.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
12
2
3
– 12
32
1
228
3
4
5
6
7
8
9
10
30
40
50
60
70
80
90
100
V
o
l
t
a
g
e
V
0
[V
]
T
e
m
per
at
ur
e
[
°
C
]
a
=
1
000
µ
m
,
d
=
20 µ
m
,
Lp
z
r
=
4
0 µ
m
,
A
p
z
r
=
6
µ
m
2
t=
3
0
m
i
n
t=
4
5
m
i
n
t=
6
0
m
i
n
t=
7
5
m
i
n
Fi
gu
re
5.
Tem
p
erat
ure
va
ri
at
i
o
n
vs
. a
p
pl
i
e
d
vol
t
a
ge
f
o
r
di
ff
erent
val
u
es
o
f
t
i
m
e
3.
2.
1.
3.
Membr
a
ne
thi
c
kness e
ffec
t
d
To
h
i
gh
ligh
t
th
e effect
o
f
th
e th
ickn
ess
o
f
th
e
d
i
aphrag
m
,
we
h
a
ve shown in
Fig
u
re
6
the
t
e
m
p
erat
ure
va
ri
at
i
on acc
o
r
di
ng
t
o
t
h
e a
p
pl
i
e
d
vol
t
a
g
e
f
o
r
di
ffe
re
nt
val
u
e
s
o
f
t
h
e
t
h
i
c
kn
ess
d
. T
h
e
T(
V
0
) g
o
es
on dec
r
easing
as the m
e
m
b
rane thic
kne
ss is
increase
d
. Ho
weve
r, accordi
n
g
to pre
v
i
ous
studies [11],[12]
this
lead
s to weak
en
ing
th
e pressure
sen
s
itiv
ity of th
e sensors.
3
4
5
6
7
8
9
10
30
40
50
60
70
80
90
100
110
V
o
l
t
ag
e V
0
[
V
]
Tem
p
e
r
at
ur
e
[
°
C
]
a=
1
0
0
0
µ
m
,
L
p
z
r
=
4
0 µ
m
,
A
p
z
r
=
6
µ
m
2,
t
=
60
m
i
n
d=
1
5
µ
m
d=
2
0
µ
m
d=
3
0
µ
m
d=
4
0
µ
m
Fi
gu
re
6.
Tem
p
erat
ure
va
ri
at
i
o
n
vs
. a
p
pl
i
e
d
vol
t
a
ge
f
o
r
se
v
e
ral
m
e
m
b
rane t
h
i
c
k
n
ess
d
3.
2.
1.
4.
Effect
of side
le
ngth
of the membrane
a
Based
on
th
e
resu
lts sh
own in
Figu
res
7 an
d
8
,
it can b
e
ob
serv
ed
th
at th
e te
m
p
eratu
r
e
rise
gene
rat
e
d
by
t
h
e Jo
ul
e heat
i
n
g i
n
pi
ezo
resi
s
t
i
v
e press
u
re s
e
ns
ors i
s
a dec
r
easi
n
g f
unct
i
o
n of t
h
e si
de l
e
ngt
h
of
t
h
e
di
ap
hr
ag
m
a
. So
, to
redu
ce th
is effect, it is requ
ired
t
o
h
a
v
e
a larg
e
sid
e
leng
th. This so
lu
tion
is easy to
estab
lish
an
d
do
es
n
o
t
affect t
h
e
p
r
essure sen
s
itiv
ity. Neverth
e
less, it lead
s to th
e en
larg
em
en
t o
f
th
e
size o
f
t
h
e sens
ors
,
w
h
i
c
h i
s
a di
sad
v
ant
a
ge. I
n
i
t
s
expe
ri
m
e
nt
al
st
udy
o
f
T. L.
Yo
u
ng
[1
6]
, o
b
t
a
i
n
ed
fr
om
a sup
p
l
y
vol
t
a
ge
of
7 V
,
t
h
at
t
h
e vari
a
t
i
on o
f
t
h
e o
ffs
et
vol
t
a
ge ve
rs
us t
e
m
p
erat
ure
for t
h
e di
ap
h
r
agm
si
ze i
s
800 µm
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Numerical
Model of J
o
ule He
ating in
Piezores
istive Pressu
re
S
e
n
s
o
r
(
A
bd
ela
z
iz Bedd
ia
f)
1
229
X8
0
0
µm
and 20 µm
i
n
t
h
i
c
kness:
i
t
i
s
0.7 m
V
/
°
C
.
In t
h
i
s
case, t
h
e Jo
ul
e heat
i
ng
pr
ov
o
k
es a t
e
m
p
erat
ure
o
f
90 °C. T
h
e
r
efore, the
val
u
e
of the offs
et voltage create
d
by
this effect is
12
1.9 m
V
, co
nsid
er
ing
the of
fset o
f
t
h
e b
r
i
d
ge at
3
0
°C
t
h
at
was a
p
p
r
oxi
m
a
t
e
ly
79.
9 m
V
[
16]
.
3
4
5
6
7
8
9
10
20
40
60
80
10
0
12
0
14
0
16
0
V
o
l
t
ag
e V
0
[
V
]
Te
m
p
e
r
at
ur
e [
°
C
]
d=
2
0
µ
m
,
Lp
z
r
=
4
0 µ
m
,
A
p
z
r
=
6
µ
m
2
,
t
=
6
0
m
i
n
a
=
80
0 µ
m
a
=
10
00
µ
m
a
=
14
00
µ
m
a
=
18
00
µ
m
a
=
22
00
µ
m
Fi
gu
re
7.
Tem
p
erat
ure
va
ri
at
i
o
n
vs
a
ppl
i
e
d
v
o
l
t
a
ge
fo
r se
ve
ral
si
de l
e
ngt
hs
o
f
t
h
e
di
a
p
h
r
a
g
m
a
80
0
10
00
1
200
14
00
16
00
18
00
20
00
22
00
20
40
60
80
10
0
12
0
14
0
16
0
L
e
n
g
th
o
f
t
h
e
m
e
m
b
ra
n
e
[µ
m
]
Tem
p
e
r
at
ur
e
[
°
C
]
d
=
2
0
µ
m
, L
p
z
r
=4
0
µ
m
,
A
p
z
r
=6
µ
m
2
,
t=6
0
m
i
n
V0=
3
V
V0=
5
V
V0=
7
V
V0=
9
V
V0=
1
0 V
Fi
gu
re
8.
Va
ri
at
i
on t
e
m
p
erat
ur
e as a
fu
nct
i
o
n
of
si
de l
e
ngt
h
f
o
r
di
f
f
ere
n
t
val
u
es
of
v
o
l
t
a
ge
V
0
3.
2.
2.
Effect
of the operating
time of
th
e
tem
p
er
ature rise in
the sens
or
In
o
r
d
e
r to
ackn
owledg
e t
h
e
ev
o
l
u
tio
n of the te
m
p
er
ature
created
by joule heating for a
peri
od of 3
hours,
we a
n
alyze the te
m
p
erature
ri
se by
v
a
ry
i
ng
seve
ral
geom
et
ri
cal
param
e
t
e
rs. As
we ca
n see
fr
o
m
t
h
e
bel
o
w Fi
g
u
res
,
t
h
at
aft
e
r oper
a
t
i
ng t
h
e de
vi
ce for a pe
ri
o
d
r
a
ngi
ng f
r
o
m
0
t
o
18
0 m
i
n by
appl
y
i
n
g
a vol
t
a
ge of
5
V, t
h
e
t
e
m
p
erat
ure
t
a
kes a
st
eady
st
at
e va
l
u
e bey
o
n
d
1
0
0
m
i
n and m
a
ke i
t
i
n
depe
n
d
e
nt
at
t
h
e t
i
m
e. It
has
been
o
b
se
rve
d
in Fig
u
re
9 t
h
a
t
the ri
se in te
m
p
erature goe
s
on dec
r
easi
n
g as
the leng
th o
f
t
h
e
p
i
ezoresisto
r
is
increase
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
12
2
3
– 12
32
1
230
0
50
10
0
15
0
200
24
26
28
30
32
34
36
38
40
42
T
i
m
e
[m
i
n
]
Te
m
p
er
at
ur
e
[
°
C]
a
=
1
0
0
0
µ
m
,
d
=
2
0
µ
m
, A
p
z
r
=
6
µ
m
2
,
V
0
=
5
V
Lpz
r
=
6
0 µ
m
Lpz
r
=
1
20
µ
m
Lpz
r
=
1
80
µ
m
Lpz
r
=
2
40
µ
m
Fi
gu
re 9.
Tem
p
erat
ure ri
se fo
r vari
o
u
s
l
e
n
g
t
h
s of
t
h
e pi
ezo
resi
st
or
L
pzr
On t
h
e
ot
her
h
a
nd
, we
can e
a
si
l
y
obser
ve i
n
Fi
g
u
r
es 10 a
nd
11,
respectively, that the t
e
m
p
erature
rise is
d
ecreasi
n
g fun
c
tio
n of
th
e m
e
m
b
ran
e
sid
e
leng
th
an
d
di
ap
h
r
agm
t
h
i
c
kne
ss.
Acc
o
r
d
i
ng t
o
t
h
e
s
e Fi
gu
res,
the two
param
e
ters have
an
enorm
ous im
pact on the
ef
fe
ct
of
J
oul
e hea
t
i
ng. So
, w
h
en
t
h
ese pa
ram
e
ters
a
r
e
g
r
eat, t
h
is is lead
ing
to
lessen th
e Jou
l
e
h
eatin
g
.
Ho
we
ve
r,
these pa
ram
e
te
rs are t
h
em
selves limited by othe
r
tech
no
log
i
cal facto
r
s
of m
a
n
u
factu
r
e su
ch as: th
e d
i
m
e
n
s
ion
s
o
f
th
e
d
e
v
i
ce, th
e
precision and
reliab
ility.
We m
a
y o
b
s
erv
e
th
at th
ese
fig
u
res are an
al
og
ou
s to
t
h
e circu
it co
nstitu
tin
g
th
e lo
ad
ing
of a cap
acitor
connected t
o
a voltage s
o
urce
th
ro
ugh
a resisto
r
. In
fact, th
i
s
circu
it
m
a
y
b
e
ex
p
l
o
ited
in
th
e fu
tu
re
works fo
r
obt
ai
ni
ng
t
h
e
t
e
m
p
erat
ur
e ri
s
e
due to J
oule
heating.
0
50
10
0
15
0
200
24
26
28
30
32
34
36
38
40
42
44
T
i
m
e
[m
i
n
]
T
e
m
p
er
at
ur
e
[
°
C
]
d=
20 µ
m
,
L
p
z
r
=
4
0 µ
m
,
Apz
r
=
6
µ
m
2
,
V
0
=
5
V
a=
10
00 µ
m
a=
14
00 µ
m
a=
18
00 µ
m
a=
22
00 µ
m
Fi
gu
re
1
0
. Te
m
p
erat
ure ri
se
fo
r
di
ffe
re
nt
m
e
m
b
rane
si
de l
e
ngt
hs
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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ECE
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8-8
7
0
8
A Numerical
Model of J
o
ule He
ating in
Piezores
istive Pressu
re
S
e
n
s
o
r
(
A
bd
ela
z
iz Bedd
ia
f)
1
231
0
50
100
15
0
20
0
25
30
35
40
45
50
T
i
m
e
[m
i
n
]
Tem
per
a
t
u
r
e [
°
C
]
a=
10
00 µ
m
,
Lpz
r
=
40 µ
m
,
A
p
z
r
=
6
µ
m
2,
V
0
=
5
V
d=
10
µ
m
d=
20
µ
m
d=
30
µ
m
d=
40
µ
m
Figu
re
1
1
. Te
m
p
erature rise
fo
r se
veral m
e
m
b
rane thick
n
e
sses
d
4.
CO
NCL
USI
O
N
Prov
i
d
e th
e mo
d
e
ling
o
f
Jou
l
e h
eatin
g
i
n
p
i
ezo
resistiv
e
p
r
essu
re sen
s
ors
is essen
tial to
p
r
ed
ict th
eir
th
erm
a
l d
r
ift.
To
t
h
is en
d, t
h
e
p
r
esen
t
stud
y inv
e
stig
ated
the effect
o
f
app
lied
vo
ltag
e
and
th
e
g
e
o
m
etric
param
e
t
e
rs
o
f
t
e
m
p
erat
ur
e. In
t
h
i
s
pa
pe
r we h
a
ve de
ve
l
o
ped
a n
u
m
e
ri
cal
m
odel
t
o
re
duce
t
h
e J
o
ul
e
heat
i
n
g
i
n
suc
h
se
ns
ors
t
y
pes
usi
n
g
Fi
n
i
t
e
Di
ffe
rence
M
e
t
h
o
d
(
FDM
). Th
e m
o
d
e
l estab
lish
e
d
g
i
ves an opp
ortunity to
st
udy
t
h
e t
e
m
p
erat
ure ri
se c
a
use
d
by
Jo
ul
e heat
i
ng
of s
e
ns
ors c
h
aract
eri
s
t
i
c
s.
W
e
at
t
e
m
p
t
e
d t
o
st
u
d
y
t
h
e
geom
etric influence
pa
ram
e
ters on these c
h
aracteristic
s t
o
optim
ize
the sens
or
perform
ance. The results
co
nfirm
e
d
th
at lo
w b
i
as
v
o
ltag
e
shou
ld
be ap
p
lied
t
o
re
duc
e t
h
e Joul
e
hea
t
i
ng. R
e
s
u
l
t
s
sho
w
e
d
al
so t
h
a
t
t
h
e
ch
ang
e
i
n
temp
erat
u
r
e is in
creasin
g fu
n
c
ti
on
o
f
th
e app
lied
v
o
ltag
e
.
In ad
d
ition
,
t
h
e Jou
l
e h
eati
n
g is in
v
e
rsely
pr
o
p
o
r
t
i
onal
t
o
l
e
ngt
h o
f
t
h
e
pi
ezo
resi
st
or
a
nd i
t
'
s an i
n
c
r
easi
ng
f
unct
i
o
n o
f
t
h
e cr
oss
-
sect
i
onal
area
.
So,
a
sens
ors
wi
t
h
t
h
e l
o
w
J
oul
e
he
at
i
ng, i
s
o
b
t
a
i
n
ed
fo
r a
g
r
eat
l
e
ngt
h
of
t
h
e
pi
ezoresi
st
or
w
h
i
c
h i
s
a
d
r
aw
ba
ck.
The
J
oule hea
ting goes on decreasi
n
g
as the
m
e
m
b
ra
n
e
th
ickn
ess is i
n
creased
. Bu
t
th
is affects
greater th
e
p
r
essu
re sen
s
itivity o
f
th
e d
e
v
i
ce. Th
e tem
p
er
atu
r
e rise created
b
y
Jo
u
l
e
h
e
atin
g
in
a sen
s
o
r
is
a
decreasi
ng
fu
n
c
t
i
on o
f
t
h
e si
d
e
l
e
ngt
h o
f
t
h
e
m
e
m
b
rane. Th
ere
f
ore, to m
i
nimize this
effect, it
is necessary to
h
a
v
e
a larg
e sid
e
leng
th
.
Th
i
s
op
tio
n is easy to
i
m
p
l
e
m
en
t an
d
do
es
n
o
t
affect the sensitiv
ity to
p
r
essu
re.
Ho
we
ver
,
i
t
gi
ves l
a
r
g
e si
ze defect
s,
w
h
i
c
h
i
s
a draw
bac
k
. O
n
t
h
e ot
her
han
d
, t
o
hi
ghl
i
ght
t
h
e e
ffect
of t
h
e
ope
rat
i
n
g t
i
m
e o
f
t
h
e
devi
ce,
vari
o
u
s
ge
o
m
et
ri
cal
param
e
t
e
rs
have
bee
n
use
d
.
The
re
sul
t
s
s
h
o
w
e
d
t
h
at
t
h
e
Joul
e
heat
i
ng i
s
red
u
ce
d su
bs
t
a
nt
i
a
l
l
y
for a sho
r
t
o
p
era
ting
tim
e
. Besides, the te
m
p
erature takes a steady state
v
a
lu
e b
e
y
o
nd
1
0
0
m
i
n
an
d
will b
e
in
d
e
p
e
n
d
e
n
t
of th
e time. Fin
a
lly, th
is stu
d
y
allo
ws u
s
to
op
ti
m
i
ze th
e
sens
or
pe
rf
orm
a
nce i
n
f
u
nct
i
o
n t
o
t
h
e a
p
pl
i
cat
i
on
fo
r
whi
c
h
i
t
i
s
dedi
cat
ed
.
ACKNOWLE
DGE
M
ENTS
Th
e au
t
h
ors than
k
Dr. Merouan
i
Lazh
er and Mr. Zah
r
ou
ri
Ah
m
e
d
for th
eir in
v
a
lu
ab
le help
for th
is
pr
o
j
ect
,
whi
c
h
was
f
u
l
l
y
d
one
at
t
h
e
El
ect
ro
ni
cs’
De
part
m
e
nt
, M
o
DER
N
a
La
bo
r
a
t
o
ry
,
U
n
i
v
er
s
i
t
y
of
C
onst
a
nt
i
n
e 1, Al
ge
ri
a. We
al
so
t
h
an
k
a
ll th
e staff m
e
m
b
ers of MoDER
N
a
Labo
ratory.
REFERE
NC
ES
[1]
D. Pitta
ya
,
et a
l
., “Al Microh
eater and Ni Temp
eratur
e Sensor Se
t based-on
Photolithogr
aph
y
with Closed-Loo
p
Control,”
Intern
ational Journal of
Electrical
and
Computer Eng
i
neering (
I
JEC
E
)
, vol. 5
,
pp
. 849-
858, 2015
.
[2]
Y.
Huiy
ang,
et al.
, “
T
he int
e
ll
ig
ent com
p
ensatio
n calibr
a
tion a
l
g
o
r
ithm for 3D p
o
ly
h
e
dron of the temperature drif
t
of the silicon
piezores
istiv
e press
u
re sensor,”
201
5 IEEE International Conferen
ce on Electron D
e
vices and Solid
-
State Circuits (
E
DSSC)
,
pp. 744 -
747, 2015.
[3]
H. Jianqiu, “
D
e
s
ign and Applicat
ion of a High Sensitiv
it
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ssure
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et al.
, “
A
dvance
d
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r
e
e
,
P
i
ezores
i
s
tiv
e, S
O
I-Bas
ed P
r
es
s
u
re S
e
ns
ors
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eas
urem
ents
in
Harsh Environments,”
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, v
o
l.15, pp. 20305
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.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
12
2
3
– 12
32
1
232
[5]
S.
Yic
a
i
, et al.
,
“
E
le
ctri
c dr
ift of
t
h
e bridg
e
offse
t
fo
r pre
ssure
se
ns
ors a
nd it
s ut
il
iza
t
i
o
n,
”
S
e
nsors and Actuators A
,
vol. 58
,
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–256, 1997
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[6]
X. Dacheng
,
et al.
, “A temperature
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e
z
o
re
sistive
pre
ssure
se
nsor a
nd softwa
re
im
plem
entat
i
on,
”
Mecha
t
ronics and
Automation
,
IEEE ICM
A
20
13 Conferenc
e
I
n
ternational S
c
i
e
ntif
ic
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pp
. 1738
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[7]
U. Aljanc
ic
,
et a
l
.
, “
T
em
per
a
tur
e
effec
t
s m
odelin
g in silicon p
i
ez
oresistive pr
essure sensor,”
IEEE Melecon
, Eg
yp
t
,
pp. 7-9
,
2002
.
[8]
B.
Abde
l
a
zi
z,
et al.
, “
T
he Eff
e
c
t
of Tem
p
eratur
e and Doping Lev
e
l on the Chara
c
t
eris
ti
cs
of P
i
ezo
res
i
s
tive P
r
es
s
u
re
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e
nsor Techn
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. 59-65
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.
[9]
M. Z. Ansari
an
d C. Chongdu, “An Analy
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odel of
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e
si
stive Micro
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