In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
Vo
l
.
1
0
, No
.
2
, Ju
n
e
20
1
9
, p
p
.
1
0
6
4
~
1
071
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
2.pp1064-1071
1064
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
Study and sim
u
lation with
VHDL-AMS of the electrical
imp
e
dance of a piezoelect
ric ultrasonic transducer
T
o
uf
ik
M
erd
jan
a
, A
bde
lhafid
C
h
a
ab
i
H
yper
F
r
eq
uen
c
y
an
d
S
e
mico
nd
uct
o
r
L
a
borat
ory
(LHS
),
U
niv
e
rsit
y
M
e
ntou
ri Co
n
s
ta
nt
in
e
1
,
Alg
e
ria
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
c
e
i
v
e
d
Sep
2
2
,
2
018
Re
vise
d N
ov
1
9
,
201
8
Ac
ce
p
t
ed
M
ar 3
, 2
0
19
Ultras
oni
c
t
r
ansducers
are
a
ke
y
el
em
ent
th
a
t
g
o
v
erns
t
he
p
erfo
rm
an
ces
o
f
bo
th
g
en
eratin
g
and
receivi
ng
u
ltras
oun
d
in
a
n
u
l
t
r
asonic
meas
u
rem
e
nt
sy
st
e
m
.
El
ectric
a
l
im
p
e
dance
i
s
a
p
aramet
er
s
en
sit
i
v
e
t
o
the
en
v
i
ronm
e
n
t
of
th
e
tran
sd
ucer;
it
c
on
tai
n
s
inf
o
rm
ati
o
n
abo
u
t
t
h
e
t
r
ans
ducer
b
u
t
al
so
o
n
th
e
m
e
di
u
m
i
n
w
h
i
c
h
it
is
i
mm
ersed
.
S
everal
p
ract
ic
al
a
pp
li
ca
tio
ns
e
x
ploi
t
th
is
p
r
op
e
r
ty
.
F
o
r
th
i
s
s
tu
dy
,
th
e
m
o
de
l
i
s
i
mp
le
me
n
t
e
d
w
i
t
h
t
h
e
V
HD
L-
A
M
S
beh
a
vi
oral
l
ang
u
age.
T
he
s
im
ulati
o
n
s
a
p
p
roach
es
p
re
s
e
nt
ed
i
n
th
is
w
o
r
k
are
bas
e
d
on
t
he
e
le
ct
rical
R
edw
o
o
d
m
od
e
l
a
nd
its
p
aramet
ers
are
d
e
duce
d
from
th
e tran
sd
ucer elec
t
roaco
us
tic
c
h
a
racteri
s
ti
cs
.
K
eyw
ord
s
:
Im
pedanc
e
Piez
oele
c
tic
Tran
sducer
U
ltraso
n
i
c
VHDL-AM
S
Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
To
ufik
M
er
dj
a
n
a,
De
p
a
rt
men
t
of
El
ect
ro
ni
c
,
Fac
u
lty
o
f
sc
i
e
n
c
e an
d
Te
c
h
no
l
ogy
,
U
n
i
v
ersi
ty
M
e
n
t
o
uri C
o
n
s
ta
n
t
i
n
e
1
,
A
l
geria
Cons
ta
nt
i
n
e,
A
lger
ia
Em
ail:
me
rdjana_
t
ou
fik
@
um
c
.
e
du.dz
1.
I
N
TR
OD
U
C
TI
O
N
U
ltra
s
o
u
n
d
sy
s
t
em
s a
r
e w
i
de
l
y
u
sed.
T
he
y fi
nd m
a
ny a
p
pl
ic
ati
o
ns
i
n en
g
i
nee
r
i
ng, m
edi
c
ine
, bi
o
lo
g
y
,
and othe
r
a
r
ea
s. The one
i
n
d
i
s
pen
s
ib
le par
t
i
n
t
hese sy
s
t
e
m
s
i
s
t
he
t
r
a
n
s
d
u
c
e
r.
T
h
e
se w
il
l
use
t
h
e pro
p
er
t
i
e
s
of
ma
gnet
o
s
t
r
i
c
t
ive
or
p
iez
o
elec
tric
m
ate
r
ia
ls
t
o
co
n
v
ert
elec
tr
ic
al
e
ner
gy
i
n
t
o
u
l
t
raso
nic
m
echa
n
i
c
a
l
e
ner
gy
[1]
.
P
i
ez
oele
c
t
r
i
c
ma
terials
ha
ve
t
he
a
d
v
an
ta
ge
o
ver
o
t
her
syste
m
s
o
f
h
a
v
i
ng
g
ood
p
e
r
f
o
rma
n
ce
a
nd
b
e
i
n
g
ava
ila
b
l
e
i
n
v
e
r
y
d
i
ve
rse
geom
etr
i
es.
The
e
l
ec
trom
ec
han
i
ca
l
i
n
t
e
r
ac
t
i
o
n
o
f
p
i
e
z
o
elec
tric
t
r
a
ns
ducer
,
repr
esente
d
b
y
elec
t
r
i
c
a
l
e
q
u
i
va
le
nt
c
ircu
it
s,
w
as
f
irst
i
ntro
d
uc
e
d
b
y
M
a
son
[2]
.
H
e
prop
ose
d
a
n
e
x
act
equ
i
vale
n
t
c
irc
u
it
tha
t
s
e
p
ar
a
t
ed
t
he
p
iez
o
e
l
e
c
tric
m
ater
ia
l
i
n
t
o
a
n
e
l
e
c
t
r
i
ca
l
p
o
rt
a
nd
t
w
o
ac
o
u
stica
l
por
ts
thr
o
u
g
h
t
he
u
s
e
o
f
a
n
i
dea
l
e
le
ctrom
echa
n
i
c
al
t
r
a
ns
form
er
.
The
prob
l
e
ms
w
it
h
t
h
e
mo
de
l
ar
e
tha
t
i
t
r
e
q
u
i
red
a
nega
t
i
ve
c
a
p
a
c
i
t
a
nce
a
t
t
he
e
lec
t
r
i
c
a
l
p
o
r
t.
R
edw
o
od
[3]
i
m
pro
v
e
d
t
h
i
s
elec
t
r
o
m
e
c
hani
ca
l
mode
l
b
y
i
n
co
rpo
r
at
i
ng
a
t
r
an
smissi
on
l
i
n
e
,
m
a
k
in
g
possi
b
l
e
to
e
xt
rac
t
u
sefu
l
in
formati
o
n
on
t
he
t
e
m
pora
l
r
es
po
n
s
e
o
f
the p
i
ez
oe
lec
t
r
i
c
c
o
mpone
n
t
.
Th
e
El
e
c
t
r
i
cal
i
mp
ed
an
c
e
i
s
a
p
a
ra
me
t
e
r
s
e
n
s
i
t
iv
e
to
t
he
e
n
v
i
ro
nm
e
n
t
of
t
he
t
ra
nsd
u
c
e
r
.
S
e
ve
ral
prac
t
i
c
a
l
a
ppl
i
c
a
t
i
o
ns
e
x
p
lo
i
t
th
i
s
p
ro
pert
y[4,
5
].
T
he
m
ea
sure
me
nt
o
f
th
e
impe
da
nce
m
a
kes
i
t
pos
si
ble,
f
or
exa
m
ple,
t
o
de
tec
t
t
he
p
h
y
sic
a
l
o
r
s
t
ruc
t
ura
l
m
odi
fica
ti
on
s
of
t
h
e
me
d
i
u
m
due
i
n
pa
rt
ic
ular
t
o
da
ma
g
e
.
Thi
s
appr
oa
ch
i
s
u
s
e
d
f
or
n
o
n
-
d
estruc
ti
ve
t
es
t
i
ng
t
o
m
o
n
it
or
t
he
c
o
n
dition
of
s
tructures
suc
h
a
s
aging
and
corr
osio
n
[6,
7
]
.
Fr
equenc
y
a
n
al
ys
i
s
o
f
t
h
e
impeda
nce
m
a
ke
s
i
t
poss
i
ble
t
o
p
re
c
i
se
ly
l
oc
ate
t
h
e
reso
nanc
e
zone
o
f
t
h
e
t
r
a
n
sd
ucer
.
This
l
ocat
i
on
c
a
n
b
e
exp
l
oite
d
t
o
c
o
n
t
r
o
l
a
n
d
s
t
a
b
i
l
i
ze
t
he
o
pera
ting
fre
que
nc
y
o
f
h
ig
h
pow
er
s
ys
tem
s
s
uc
h
a
s
u
l
t
rason
i
c
w
e
ld
in
g
d
e
vice
s
[8]
.
R
eal-t
ime
k
n
o
wl
e
d
ge
o
f
t
h
e
e
l
ec
t
r
i
c
al
i
mpe
d
a
n
ce
a
l
s
o
ma
kes
it
pos
sible
t
o
d
e
t
e
r
m
i
n
e
a
n
d
o
p
tim
iz
e
the
p
o
w
e
r
em
i
t
te
d
b
y
a
t
r
ans
m
itter
or
t
he
s
e
n
s
iti
vi
ty
i
n
r
e
c
e
p
t
i
on
of
a
piez
o
e
l
ec
tr
i
c
sensor
[9].
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Stu
d
y
an
d sim
u
la
tio
n w
i
t
h
VH
D
L
-
A
MS
o
f
the
e
l
e
c
tr
i
c
a
l
im
p
e
da
nce
o
f
a
p
i
e
z
oe
lectr
i
c
…
(
T
oufik
Merd
j
a
na)
1
065
D
i
ffe
r
e
nt
a
p
p
r
o
a
c
he
s
w
e
r
e
p
r
o
pose
d
t
o
p
r
edic
t
t
h
e
p
i
e
z
oe
lec
t
r
i
c
t
r
a
n
s
du
c
e
r
be
ha
v
i
o
r
s
uc
h
as
a
num
er
i
c
a
l
r
eso
l
u
tio
n
of
p
iez
o
elec
t
r
i
c
e
q
u
a
t
ions.
A
n
o
t
her
a
p
pr
oa
ch is
bas
e
d
o
n
t
he
e
q
u
i
va
l
e
n
t
e
le
ctr
i
ca
l
cir
c
uit
sim
u
la
ti
on
us
ing
an
e
l
e
c
t
ric
simu
lat
o
r
l
i
k
e
SPICE
(
Sim
u
la
tio
n
P
r
ogr
am
w
ith
I
nte
g
r
a
te
d
Ci
r
c
ui
t
Em
p
h
as
is)
[10
,
1
1
]
.
Ho
wev
e
r
SPI
C
E
p
resent
s
so
me
limitation
s
SP
ICE
is
o
n
a
c
ont
inu
o
u
s
b
asi
s
i
t
can
not
s
up
po
rt
d
isc
r
e
t
e
r
e
pr
e
s
e
n
ta
t
i
o
n
s
,
and
as a
r
e
su
l
t
,
i
t
is
no
t su
ita
ble
f
o
r
m
i
xe
d
m
odel
i
ng
[
1
2
,
13]
.
V
H
D
L-
AMS
(
V
e
r
y
H
i
g
h
spe
e
d
in
te
gr
a
t
e
d
c
ir
c
u
i
t
H
a
r
dw
ar
e
D
e
scr
i
pt
i
on
La
ngua
ge
A
na
lo
g
a
n
d
M
i
x
ed
S
i
g
n
a
l
)
i
s
a
h
i
g
h
-
l
e
v
e
l
l
a
ngua
g
e
t
h
a
t
e
n
ab
les
nume
r
ica
l
a
nd
a
n
a
l
o
g
s
imu
l
a
t
i
o
ns,
while
g
i
v
in
g
the
po
ss
ib
il
i
t
y
to
s
imula
t
e
s
y
s
t
em
s
wi
th
d
iffe
ren
t
ph
ys
ica
l
m
ag
n
itu
de
s:
m
ec
ha
nica
l,
t
her
m
a
l
a
nd
e
l
ectr
i
c
a
l
.
The
us
e
of
a
n
an
al
ys
i
s
t
oo
l
,
s
u
c
h
as
t
h
e
V
H
D
L-
A
M
S
beha
vior
al
d
e
s
c
r
ip
tio
n
l
a
ngua
ge,
ca
n
be
a
s
o
l
u
t
i
o
n
t
o
t
he
l
im
i
t
a
tio
ns
c
au
sed
b
y
t
he
u
se
o
f
the
S
P
ICE
sim
u
la
tor
.
2.
THE
O
RY
O
F
P
IE
Z
O
ELE
C
TRIC U
L
T
R
A
SONIC
T
RAN
S
DUCER
The
p
i
ez
oe
lec
t
r
i
c
m
a
teria
l
i
s
the
pr
inc
i
pa
l
elem
en
t
i
n
a
n
ul
tra
so
n
i
c
tra
n
sd
ucer
.
The
pi
e
z
o
e
l
e
c
t
ric
m
a
ter
i
als
ha
ve
t
he
a
d
v
an
ta
ge
c
om
par
e
d
to
t
he
o
t
h
er
s
ys
te
m
s
o
f
p
r
e
sent
i
ng
go
od
per
f
o
r
ma
nces
a
nd
t
o
b
e
a
v
ai
la
b
l
e
in
v
e
r
y
d
i
ver
s
e
ge
o
m
e
t
r
i
e
s
.
These
ma
t
e
rial
s
a
r
e
gener
al
l
y
a
pp
eare
d
a
s
a
di
sc
,
a
rin
g
o
r
pl
ate
.
O
u
r
piez
oe
lec
t
r
i
c
el
em
ent
has
the
cha
r
ac
t
e
ris
tic
t
o
v
i
bra
t
e
i
n
t
hic
kne
ss
mode
,
on
on
ly
o
ne
d
i
r
e
c
ti
on
a
x
is
(
z
)
.
This
m
odel
l
i
n
g
st
u
d
y
i
s
t
hus
l
i
m
i
t
e
d
t
o
one
g
e
o
m
e
tr
ica
l
d
im
ens
i
on
(
1
-D),
w
i
t
h
t
h
e
b
ound
a
r
y
con
d
it
ions
a
t
th
e
a
c
ou
st
i
c
p
or
ts
F
igur
e
1
[1
4]
.
(
1
)
Whe
r
e
A
is
t
he
a
r
e
a
of
t
he
t
r
a
nsd
u
ce
r
a
nd
T
i
s
t
he
i
n
t
er
na
l
str
ess.
F
i
gur
e
1.
P
i
e
z
oele
ctr
i
c
p
l
a
t
e
o
f
t
h
i
c
knes
s
e
a
nd
its
r
epr
e
sen
t
a
ti
on
as
a
t
hr
e
e
p
or
t
sys
t
e
m
Let’
s
c
ons
ide
r
e
t
he
t
h
i
c
kne
ss
o
f
t
h
e
p
i
ez
oel
ectr
i
c
p
l
ate
a
n
d
V
3
i
s
the
ex
cita
t
i
on
v
o
l
t
a
g
e
.
F
1
a
nd
F
2
pr
esen
ted
t
h
e
f
o
r
c
e
s
t
r
a
nsm
i
t
t
e
d
to
t
he
p
r
o
p
a
gat
i
o
n
m
e
di
u
m
on
t
he
f
ro
n
t
a
n
d
b
a
c
k
face
o
f
t
h
e
t
r
ans
d
u
cer
.
ν
1
a
nd
ν
2
a
r
e
the a
c
ou
st
i
c
par
tic
l
e
s vel
o
ci
t
i
es a
t
t
h
e fron
t
an
d
t
h
e
ba
c
k
fac
es of
the tra
n
s
ducer
.
I
n
t
hese
c
on
d
i
t
i
ons,
th
e
tr
ans
f
er
m
a
t
r
i
x
f
o
r
m
(
2)
w
hich
d
esc
r
ibes
t
he
g
l
oba
l
be
ha
vi
or
b
etw
e
e
n
t
h
e
e
l
e
c
tr
ic
e
xc
ita
t
i
on
p
o
r
t
a
n
d
t
he
t
w
o
a
c
ous
tic
p
or
ts.
⎣
⎢
⎢
⎢
⎢
⎢
⎡
1
2
3
⎦
⎥
⎥
⎥
⎥
⎥
⎤
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎡
tan
sin
ℎ
33
si
n
tan
ℎ
33
ℎ
33
ℎ
33
1
0
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎤
⎣
⎢
⎢
⎢
⎢
⎢
⎡
1
2
3
⎦
⎥
⎥
⎥
⎥
⎥
⎤
(
2
)
ν
1
a
nd
ν
2
(
m
/
s)
a
re
t
he
a
c
o
ust
i
c
p
ar
tic
l
e
v
e
l
oc
ities
at
t
he
f
ro
nt
a
n
d
t
h
e
b
a
ck
f
ac
es
o
f
th
e
p
l
at
e
,
F
1
a
n
d
F
2
a
re
t
he
a
cous
t
i
c
for
ces
a
t
the
tra
n
sd
u
c
e
r
f
ace
s,
Z
i
s
the
acous
t
i
c
im
pedanc
e
of
t
he
p
i
e
z
o
ele
c
tr
i
c
ma
t
e
ri
a
l
,,
h
e
ε
⁄
i
s
t
h
e
p
i
e
z
o
e
l
ect
ri
cit
y
c
on
st
ant
and
C
i
s
t
h
e
c
a
pa
c
ita
nce
va
lue
of
t
he
p
la
t
e
,
and
I
i
s
V
3
I
3
ν
1
ν
2
F
2
F
1
Acoustic
P
o
r
t
2
Acoustic
P
ort
1
El
ectrical
P
ort
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
2
, June
20
1
9
:
106
4 –
1
0
71
1
066
the
e
l
ec
trica
l
c
ur
rent.
⁄
.
The
de
p
h
asa
g
e
ge
ne
rates
b
y
t
he
p
r
opa
ga
ti
o
n
w
it
h
ω2
π
f
i
s
t
h
e
elec
tr
ical
p
uls
a
ti
o
n
w
h
e
re
f
i
s
the
fre
q
uenc
y,
e
i
s
the
th
ic
kne
ss
of
t
he
p
iez
o
e
l
ec
tric
p
l
a
t
e,
ρ
i
s
the
materia
l
den
s
i
t
y,
is the
e
last
ici
t
y
mod
u
l
u
s w
i
th c
o
n
st
ant
d
i
s
p
lac
e
m
e
nt fie
ld.
Det
erm
i
n
atio
n
of the
e
lec
t
ric
a
l
impeda
nce
o
f
t
he
piezo
electric
transd
ucer
The
tra
n
sduc
e
r
i
s
ass
u
me
d
here
c
har
g
e
d
t
o
t
h
e
a
c
ous
t
i
c
por
ts
by
h
o
m
ogene
o
u
s
m
e
di
ums.
W
e
deter
m
i
n
e
the
i
npu
t
e
l
ec
t
r
ica
l
i
m
p
eda
n
ce
o
f
t
h
e
trans
d
u
c
e
r
b
y
t
h
e
i
mp
ed
an
ce
s
o
f
t
h
e
a
c
o
u
s
t
i
c
lo
ad
s
Z
1
a
n
d
Z
2
respe
c
t
i
ve
l
y
a
c
ous
t
i
c
impe
da
nce
of
t
h
e
f
r
o
n
t
m
ed
ium
an
d
the
ba
c
k
m
e
d
i
um.
A
t
port
s
a
cou
s
t
i
c
a
n
d
g
i
ven
t
h
e
sense
o
f
spe
ed
i
n t
h
e
d
i
agra
m,
we have
:
(3
)
(
4
)
We
r
e
p
o
r
t
t
h
e
s
e
rela
ti
o
n
s
i
n
t
he
f
irst
t
w
o
e
qu
a
t
i
o
ns
o
f
t
h
e
m
a
tri
x
(
2)
w
hic
h
m
akes
i
t
p
o
ss
ible
t
o
d
e
t
e
rmin
e
t
h
e el
ec
t
r
i
c
al
i
mpe
d
an
c
e
1
(
5
)
The
e
xpr
essi
o
n
o
f
electr
i
c
a
l i
m
p
e
danc
e
is g
e
n
era
lly
f
ou
n
d
in t
h
e
li
tera
ture
i
n
the f
o
l
l
ow
i
n
g for
m
:
1
(
6
)
Where
F
or
l
ong
i
t
ud
ina
l
w
a
v
es,
the
pa
ram
e
ter
i
s
often
d
e
fine
d
a
s
t
he
p
iez
o
e
l
e
c
t
ric
cou
p
l
i
ng
co
ns
t
a
nt
f
or
a
t
ransv
e
rsely
clam
pe
d
ma
t
e
rial;
f
o
r
it
i
s
the
ef
fec
t
i
v
e
piez
oelec
t
ric
c
o
n
s
ta
nt
u
se
d
when
t
h
e
r
e i
s
no
mo
ti
on
t
ra
n
s
v
e
rs
e to
t
h
e
e
l
ect
ri
c
fi
el
d
.
3.
ELECT
R
O-A
C
O
U
S
T
IC
M
ODEL
O
F
PIE
Z
O
E
LE
CTRIC ELE
M
E
NT
F
o
r
r
e
a
s
o
n
s
o
f
s
i
m
p
l
i
c
i
t
y
,
w
e
c
h
o
s
e
t
o
u
s
e
R
e
d
w
o
o
d
'
s
e
q
u
i
v
a
l
e
n
t
e
l
e
ct
rica
l
ci
rc
uit
.
T
h
e
e
q
u
i
v
al
e
n
t
circ
u
i
t
o
f
a
t
h
i
c
kne
ss–mo
de
p
ie
z
o
e
l
ec
t
r
i
c
t
rans
duce
r
c
a
n
b
e
re
p
rese
nt
e
d
by
the
Re
dw
o
od
mo
de
l
a
s
s
how
n
i
n
F
i
gure
2.
F
i
gur
e 2.
Equ
i
v
ale
n
t c
i
rcu
it o
f
R
e
d
w
o
o
d
'
s
mode
l
The
mode
l
is
d
i
v
id
e
d
i
n
t
w
o
pa
rts.
T
he
f
irs
t
one
i
s
the
ele
c
t
r
i
c
a
l
p
ort
wh
ich
i
n
cl
u
d
es
t
h
e
c
apac
it
ors
C
a
nd
C
t
ha
t
r
e
pre
s
en
t
t
h
e
c
a
pac
ita
nce
m
o
tio
na
l
effec
t
.
Th
i
s
e
l
e
c
t
r
i
ca
l
p
o
rt
i
s
con
n
ec
ted
to
a
r
esista
nc
e
R
Re
dw
o
od m
o
d
e
l
F
1
V
3
I
3
-C
0
T
Z
back
F
2
Z
f
r
ont
R
C
0
:1
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
St
u
d
y an
d
si
m
u
la
t
i
on
wi
th VH
D
L
-AM
S
of t
h
e
electr
i
ca
l im
p
e
da
nc
e
of
a p
i
e
z
oe
lec
t
r
i
c
… (T
oufik
Me
rd
j
a
na)
1
067
and
a
vo
ltage
s
ource
n
ot
e
d
V
.
T
he
s
ec
o
nd
pa
rt
i
s
com
pose
d
by
the
tw
o
ac
oust
i
c
p
o
rts,
T
i
s
an
i
d
eal
e
l
e
c
t
ro
-
ac
oust
i
c
tra
n
sf
orme
r
w
ith
a
r
a
t
i
o
h
.
P
i
e
z
oce
r
am
ic
l
aye
r
i
s
a
s
si
m
i
l
a
te
d
t
o
a
p
ro
paga
tio
n
l
i
n
e
c
har
a
c
t
eri
z
ed
by
i
t
s
c
h
ara
c
te
rist
ic
i
m
p
eda
n
ce
(
.
),
i
s
t
h
e
a
r
e
a
a
nd
the
propa
ga
t
i
o
n
time
T
d
(
⁄
),
w
i
t
h
i
s
the a
c
o
u
st
i
c
veloc
ity
an
d
is the
t
h
ic
k
n
e
s
s.
On
e
b
r
an
ch
o
f
t
h
e
pi
e
z
o
cera
m
i
c
l
a
y
er
i
s
in
c
o
n
t
act
w
it
h
th
e
b
a
c
k
medi
um
Z
back
and
t
h
e
ot
her
i
s
i
n
con
t
ac
t
w
ith
t
he
propa
gat
i
on
me
dium
Z
f
r
ont.
4.
VHDL
-
A
MS
B
E
H
AVIORAL
M
ODEL
O
F
T
R
A
NDU
CER
Bef
o
re
d
i
s
c
u
s
s
i
ng
the
mo
del
i
ng
of
t
ransd
u
ce
rs
i
n
V
H
D
L-A
M
S
,
w
e
q
u
i
c
k
l
y
p
r
e
s
e
n
t
t
h
i
s
p
r
og
rammi
n
g
l
a
n
g
u
a
g
e
.
Th
e
VHDL-
A
M
S
i
s
an
I
EEE
s
t
a
n
d
a
r
d
[
15
]
.
I
t
w
a
s
d
e
v
e
l
o
p
e
d
a
s
a
n
e
x
t
e
n
s
i
o
n
o
f
VHDL
t
o
en
a
b
l
e
t
h
e
m
o
d
eling
an
d
si
mu
l
a
tio
n
of
c
i
r
cui
t
s an
d
an
a
l
og
an
d di
g
i
t
a
l -
analo
g
m
i
xed
sys
t
em
s.
Th
e
VHDL-AM
S
i
mp
l
e
mentat
i
o
n
of
t
h
e
p
r
e
v
i
ou
s
mo
d
e
l
as
s
h
o
w
n
in
F
ig
u
r
e
2
i
s
d
ivi
d
ed in
tw
o
par
t
s.
F
irst
is
t
he
d
ec
lar
a
t
i
on
of
t
he
e
nti
t
y
w
h
ich
is
c
o
m
pose
d
o
f
t
h
e
ph
ys
i
c
a
l
c
har
a
cteris
t
i
cs
o
f
t
h
e
t
r
a
n
sd
ucer
a
nd
t
h
e
di
ffe
re
nt
t
e
r
m
i
na
ls
u
se
d
i
n
c
o
nne
c
t
i
o
n.
E
a
c
h
T
E
R
MIN
A
L
de
pe
n
d
s
o
f
t
h
e
ph
ysi
c
al
n
atu
r
e
of
t
h
e
r
e
l
ati
o
n
to
b
e
i
m
p
l
e
m
en
t
e
d
to
d
escri
b
e
th
e
e
l
e
m
e
n
t
.
U
si
n
g
th
e
st
at
emen
t
s
ELEC
T
R
I
C
A
L
i
n
t
h
e
e
l
e
c
t
r
i
c
a
l
d
o
m
a
i
n
a
n
d
K
I
N
E
MA
TI
C_
V
in
t
he
a
c
ous
ti
c
d
o
ma
in.
Th
e
se
co
nd
p
ar
t
of
t
he
m
ode
l
is
t
he
a
rc
hitec
t
ur
e
w
h
ic
h
e
s
t
a
blis
hes
the p
h
y
s
i
c
l
a
ws re
l
ated t
o
the
m
a
them
at
ical
r
elat
i
on b
e
t
w
e
e
n
e
a
c
h
term
i
na
l [1
6].
I
n
t
h
i
s
a
pproac
h
,
electr
i
c
a
l
a
n
d
aco
us
tic
e
le
m
e
nts
ar
e
de
sc
ribe
d.
E
lectric
a
l
ele
m
e
n
t
is
d
efi
n
e
d
w
i
t
h
vo
lta
ge
a
nd
c
u
rrent
q
ua
nt
it
i
e
s
a
nd
a
c
ous
tic
e
le
me
n
t
w
i
t
h
forc
e
an
d
ve
loc
ity
q
ua
n
t
i
t
ies
[1
7,
18].
The
VHDL-AM
S
i
mp
l
e
ment
ati
o
n
o
f
Red
w
o
od'
s eq
ui
v
a
l
e
nt
m
o
d
el
as
sh
o
w
n
i
n
F
igure
2
i
s
g
ive
n
b
y
:
EN
TI
T
Y
Re
d
w
oo
d
IS
GE
N
E
RIC
(C
0
,
k
t
,
Z
0
, Td: r
eal);
PORT
(TER
MI
N
A
L
p,
m
:
ele
ctrica
l;
TERM
I
NAL
t1
1
,
m1
1,
t22,
m
22:
k
i
n
em
ati
c
_
v
)
;
EN
D
E
N
TI
T
Y
Redw
oo
d
;
ARCHIT
EC
TURE
st
r
u
ct
ure
OF
Redw
o
od
IS
TERMINAL
p
1
:
e
l
e
ct
ri
cal
;
TERMINAL
t
1,
t1x,
t
2
x
:
k
in
em
atic
_v
;
QUANT
ITY
v
1
acr
o
ss
i
1
th
r
o
u
g
h
p
T
O
m
;
QUANT
ITY
v
2
acr
o
ss
i
2
th
r
o
u
g
h
p
TO
p1;
QUANT
ITY
vte
acr
o
ss
i
t
e
th
rou
g
h
p1
TO
m
;
QUANT
ITY
p
ti
a
cro
ss
uti
t
h
r
o
ug
h
t1
T
O
kinem
a
t
i
c
_
v_
grou
n
d
;
QUANT
ITY
p
1xr
acr
o
ss
u1
x
r
th
r
o
ug
h
t
1x
TO
t1;
QUANT
ITY
p
2xr
acr
o
ss
u2
x
r
th
r
o
ug
h
t
2x
TO
t1;
QUANT
ITY
p
1x
acr
o
ss
u
2x
through
t1x T
O
t
1
1
;
QUANT
ITY
p
2x
across
u
1x
th
r
o
ug
h
t
2x
T
O
t22
;
QUANT
ITY
p
11
acr
o
ss
t
11
T
O
t
1
;
QUANT
ITY
p
22
acr
o
ss
t
22
T
O
t
1
;
BEGIN
i1
==
C
0
* v
1
'dot;
i2
==-
C
0
*
v
2
'
d
o
t;
pt
i=
=kt
* v
t
e
;
Ut
i
=
=
-
it
e
/
k
t;
p1
xr
==p
2
2
’D
ELA
Y
E
D
(Td)
–
p1x
;
p2
xr
==p
1
1
’D
ELA
Y
E
D
(Td)
–
p2x
;
p
1x=
=
(
u1x+
u2x
’D
ELAYED
(Td
))*
Z
0
/2.
0
;
p
2x=
=
(
u2x+
u1x
’D
ELAYED
(Td
))*
Z
0
/2.
0
;
E
N
D AR
CHITE
C
TURE
str
u
cture;
5.
SIMU
L
A
TION
R
ESULT
S
We
p
re
sen
t
ons
i
n
t
h
is
p
a
r
t
th
e
re
sul
t
s
o
f
s
imu
l
at
i
on
w
ith
V
H
D
L-
AMS
of
t
he
i
npu
t
ele
c
t
ric
a
l
impe
da
nce
of
t
he
t
ran
s
d
u
ce
r.
T
he
s
tu
d
i
e
d
t
rans
duc
er
i
s
bu
il
t
w
i
th
P
ZT
cer
am
i
c
o
f
P
1
-8
8
t
y
p
e
,
prod
uc
ed
b
y
Q
u
ar
t
z
a
nd
S
ilic
e
s
o
c
i
et
y,
v
i
b
r
a
t
i
n
g
i
n
t
h
ic
kne
ss
m
o
d
e
a
t
a
f
re
que
nc
y
o
f
2
.25
MH
z,
1
m
m
th
ick
an
d
are
a
A
=
13
2.
73 m
m
2
.
These
cha
r
ac
te
ristic
s
are
re
c
a
l
l
e
d
i
n Tab
l
e
1.
;
⁄
;
ℎ
⁄
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
2
, June
20
1
9
:
106
4 –
1
0
71
1
068
Tab
l
e
1. Tr
a
nsduce
r
a
co
ustic
c
ha
rac
t
eris
tic
s
[1
9]
P
a
r
a
mè
trs
De
finiti
on
Val
u
e
Unit
ρ
D
e
nsity
7700
Kg
/
m
3
v
Ac
ousti
c
v
e
lo
c
i
t
y
4530
m/s
Z
A
c
oust
i
c
i
m
p
e
d
a
nc
e
34.
9
Mra
y
ls
C
C
a
p
a
c
itor
of
t
he
c
e
r
am
ic
759
PF
K
c
oupling
fac
t
or
0.
49
-
C
Ela
s
tic
c
onsta
nt
15.
8x1
0
10
N/
m
2
ε
d
i
el
ect
r
i
c
co
n
s
t
a
n
t
870
ε
F/m
h
pie
z
o
e
l
ec
tri
c
c
onst
a
nt
1.
49x1
0
9
-
ta
n
D
i
e
l
e
c
t
ri
c
l
o
ss
f
a
ctor
0
.
02
-
Th
e
fre
qu
en
ti
al
t
ra
n
s
d
u
c
ers
respo
n
s
e
st
udy
i
s
e
s
se
nti
a
l
to
p
re
d
ict
the
sen
s
i
tiv
it
y
o
f
t
he
s
ys
t
e
m
for
t
h
e
vari
ous
a
na
lyz
e
d
m
edi
u
ms.
Tab
l
e
2
gi
ves
acou
s
t
i
c
char
a
c
ter
i
s
t
i
c
s
o
f
s
i
m
ulate
d
m
ed
iu
ms.
The
am
plitu
de
o
f
the tra
n
sduc
er
i
s fi
xe
d
at 1V
o
lt
with a f
r
e
quency
o
f
2.25MH
z.
Tab
l
e
2. A
coust
i
c
cha
r
ac
t
e
ris
t
i
c
of
differ
en
t m
e
dium
s
[2
0]
Me
diu
m
s
Ac
ousti
c
i
m
p
e
d
a
nce
(
M
ra
yl
s)
A
c
o
us
ti
c
v
e
lo
c
i
t
y
(
kW
)
Wa
t
e
r
1.
4
9
1
469
E
t
ha
nol
0
.
9
1
1
158
P
a
raffi
n o
i
l
1
.
8
6
1
420
5.1.
Stud
y
of
th
e e
l
ect
rical i
m
p
e
d
a
n
ce of th
e
R
ed
w
ood
m
od
e
l
The
com
p
lex
im
peda
nc
e
o
f
t
he
R
edw
o
o
d
m
odel
is
o
f
t
h
e
form
:
Z
jw
R
j
X
|
Z
|
e
.
We
d
e
not
e
f
1
a
n
d
f
2
t
h
e
fre
qu
enc
i
e
s
f
o
r
w
h
i
ch
t
h
e
i
mp
e
d
an
c
e
m
o
d
u
l
e
r
ea
ch
e
s
r
es
p
ect
iv
ely
it
s
mi
ni
mum
a
n
d
ma
ximum
va
lu
e
s
,
the
fr
eque
n
c
i
e
s
o
f
r
esona
nce
fr
a
nd
an
t
i
r
e
sona
n
ce
f
a
a
r
e
the
val
u
es
f
or
w
hic
h
t
he
g
lo
ba
l
r
e
a
c
t
a
n
c
e
X
o
f
t
h
e
t
r
a
n
s
d
u
c
e
r
i
s
z
e
r
o
(
θ
=
0
)
.
T
h
e
s
i
m
u
l
a
t
i
o
n
c
o
d
e
us
ed
t
o
ob
t
a
in
t
he
t
ra
n
s
d
u
c
e
r
im
put
elec
tr
ical im
p
e
d
anc
e
i
s
give
n by
:
EN
TI
T
Y
Im
peda
nce
_
S
i
mu
la
t
i
o
n
IS
EN
D
Im
peda
n
ce_S
i
mula
t
i
on
;
ARCHIT
EC
TURE
st
r
u
ct
OF
Impe
danc
e
_
S
i
mula
ti
o
n
IS
TERMINAL
n
1
,
n2: ELE
CT
RICAL
;
TERMINAL
n3
,
n4
: K
I
N
E
MA
TIC
_
V
;
C
O
NSTANT
e
:
real
:=1
.
0
e
-
3
C
O
NSTANT
A:
real
:=
132
.
7
3
e
-3
C
O
NSTANT
:
re
al
:=4
530
.0
;
C
O
NSTANT
f
o
:
r
eal:=
2
.
25
e6
C
O
NSTANT
Z
c
: re
al:
=
34.9e
6;
C
O
NSTANT
k
t
:
r
eal:
=
0
.49;
C
O
NSTANT
e
p
s
i
0
: re
al:
=
8.
8
5
42e-
1
2
;
C
O
NSTANT
e
p
s
i
33
: r
eal:
=
65
0.
0;
C
O
NSTANT
r
o
:
re
al
:=
330
0
.
0;
C
O
NSTANT
h
:
real
:=
k
t
*
*s
qrt(ro/(e
p
s
i
0
*eps
i
33
));
CO
N
S
TA
N
T
Co:
r
ea
l
:
=A
*e
ps
i
33
/e
;
QUANT
ITY
v
a
c
: r
eal
s
pe
ctru
m
1.
0, 0.
0
;
QUANT
ITY
v
i
n
p
u
t a
c
ross ie
t
hr
ou
g
h
n1 t
o
e
lectr
i
ca
l
_gr
o
und;
BEGIN
Vi
np
ut
==
v
a
c
;
R:
EN
T
I
TY
re
s
is
tanc
(
b
hv)
G
EN
ERIC
M
A
P
(
50.0)
PORT
MAP
(n1,
n2);
T1
:
e
n
tit
y
Re
d
w
oo
d
(
bhv)
g
e
n
er
i
c
m
a
p
(
Co,
K,
A
*Z
c
,
⁄
)
P
O
RT
M
AP
(
n3
,
ki
ne
ma
t
i
c
_
v
_
g
r
ou
nd,
n4,
ki
nem
a
t
i
c
_
v
_
g
r
o
u
n
d
,
n2,
e
l
e
c
t
rica
l
_
gro
u
n
d
);
Rf
r
o
n
t
:
EN
TI
T
Y
re
s
i
s
ta
nc
(bh
v)
G
EN
ERIC MA
P
(
1.5e
6*A
)
PO
RT MAP
(
n4, gr
ound)
;
Rba
c
k:
EN
TITY
r
esist
a
nc
(
bh
v) G
EN
ERIC MA
P
(
44
5.
0
A
*
)
PORT
M
AP
(
n3,
e
lec
t
r
i
c
a
l gr
o
und)
;
E
N
D AR
CHITE
C
TURE
st
ru
c
t;
F
i
gure
3
r
e
pre
s
e
n
ts
d
iffe
ren
t
c
har
acter
i
s
t
i
c
cu
rve
s
t
ha
t
can
b
e
e
xtra
cted
f
ro
m
the
im
peda
n
c
e
.
In
p
ra
ct
ice
t
h
e
fre
qu
en
ci
e
s
f
1
a
n
d
fr
o
n
t
h
e
one
h
a
n
d
an
d
th
e
fre
q
ue
ncie
s
fa
a
nd
f
2
o
n
t
h
e
ot
h
e
r
h
a
nd
a
re
v
e
r
y
c
lo
s
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Stu
d
y
an
d sim
u
la
tio
n w
i
t
h
VH
D
L
-
A
MS
o
f
the
e
l
e
c
tr
i
c
a
l
im
p
e
da
nce
o
f
a
p
i
e
z
oe
lectr
i
c
…
(
T
oufik
Merd
j
a
na)
1
069
F
i
g
u
r
e
3
.
C
har
acte
r
ist
i
c
cur
v
e
s
o
f
t
h
e
im
pe
da
nc
e
of
t
he
R
e
d
w
ood
m
ode
l
a
nd
I
m
peda
nc
e
pha
se
5.
2.
I
n
f
lu
e
n
c
e
o
f
t
h
e
p
r
op
agat
i
o
n
med
i
u
m
o
n
t
h
e
i
mp
e
d
an
c
e
m
od
u
l
us
The
vibr
at
i
o
n
mode
s
of
t
he
p
iez
o
e
l
ec
tr
ic
t
r
a
nsd
u
c
e
r
s
a
r
e
s
tr
o
n
g
ly
i
n
f
lu
e
n
ced
b
y
t
h
e
me
di
um
i
n
w
h
ic
h
the
y
a
r
e
i
m
m
e
r
sed
or
b
y
t
h
e
s
t
r
u
c
t
ur
e
of
w
h
i
ch
t
he
y
ar
e
i
nte
g
r
a
l.
I
n
a
h
o
mo
ge
ne
ous
l
i
q
u
i
d
or
g
aseo
us
f
l
u
i
d,
t
he
c
or
r
e
sp
on
d
i
n
g
a
c
o
ustic
l
oa
d
m
o
dif
i
e
s
t
he
v
i
b
r
a
t
i
o
n
v
e
l
oc
it
y
of
t
he
t
r
a
ns
du
c
e
r
and
da
mp
e
n
s
i
t
s
r
e
sona
nce
[2
1]
.
R
e
t
r
o
d
if
f
u
se
d
ec
hoe
s
t
o
t
h
e
t
r
a
nsd
u
c
e
r
g
e
ner
a
t
e
a
r
e
cept
i
o
n
curr
en
t.
I
n
t
h
e
c
a
se
o
f
a
th
i
n
piez
oe
lec
t
r
i
c
b
l
a
d
e
b
o
nde
d
t
o
a
s
o
l
id
s
t
r
uct
u
r
e
u
n
d
er
s
tr
e
s
s,
t
he
i
n
f
l
u
e
n
ce
of
t
he
l
a
tter
i
s
p
r
e
p
onde
r
a
n
t
.
T
h
e
s
e
c
o
n
s
i
d
er
at
io
ns
s
how
t
hat
t
h
e
elec
tr
i
c
al
i
mpe
d
ance
o
f
the
tr
ans
d
uce
r
i
s
ve
r
y
s
e
n
s
iti
ve
t
o
i
t
s
e
n
v
i
r
o
nme
n
t
[2
2]
.
T
h
e
t
r
a
n
sd
u
c
er
i
s
l
o
ad
ed
o
n
th
e
f
r
ont
f
ac
e
by
t
h
r
e
e
d
i
f
f
e
ren
t
m
ed
i
u
ms:
w
a
t
e
r,
e
tha
n
o
l
a
nd
pa
raf
f
i
n
o
il.
The
i
r
a
c
ou
st
i
c
c
har
a
cter
is
tic
s
a
r
e show
n
i
n
T
a
b
le 2.
The
s
i
m
u
la
ti
on r
es
ul
t
is
s
h
o
w
n
i
n
the
f
oll
o
w
i
n
g
f
i
g
ur
e
.
F
igur
e
4
cl
ea
rly
sho
w
s
t
h
e
i
n
fl
u
e
n
c
e
of
t
h
e
m
edi
u
m
on
t
h
e
e
l
e
ct
r
i
ca
l
i
m
p
e
d
a
n
c
e
m
o
d
u
l
u
s
a
n
d
p
r
e
c
i
s
e
l
y
o
n
Z
m
a
x
.
W
e
no
te
a
c
onc
or
da
nce
be
t
w
e
e
n
t
he
r
e
s
ul
t
o
b
ta
ine
d
b
y
V
H
D
L
-
A
MS
a
nd
t
h
e
exp
e
ri
me
nt
al
r
e
s
u
lt
p
r
e
s
en
ted
in
[23
]
,
s
o
a
good m
odeling
by
VHDL-AMS.
F
i
gur
e
4.
T
r
a
nsduce
r
i
m
p
e
d
a
n
ce
modul
us
a
n
d
t
he
pha
se
f
or
d
if
f
e
r
e
n
t
pr
op
aga
t
i
on
me
d
i
u
m
s
5
.
3
.
I
nfluence
o
f co
upling
fa
ct
o
r
o
n
t
h
e impeda
n
ce mo
dulus
F
r
o
m
t
h
e
f
o
r
m
u
l
a
(
6
)
i
t
i
s
f
o
u
n
d
t
h
a
t
t
h
e
e
l
e
c
t
r
i
c
a
l
i
n
p
u
t
i
m
p
e
da
nce
of
t
he
t
r
a
n
s
d
u
cer
i
s
d
i
r
e
c
t
l
y
r
e
late
d
t
o
t
he
c
ou
p
l
i
n
g
f
a
c
t
o
r
K
t.
T
he
f
ol
l
o
w
i
ng
f
i
g
u
r
e
s
how
s
t
h
e
v
a
r
i
a
t
i
on
o
f
th
e
i
m
p
e
d
a
n
c
e
modul
us
f
o
r
di
ffere
n
t
co
u
p
l
i
n
g
f
ac
tors.
The tra
n
sd
ucer
is loa
d
e
d
on t
h
e
fro
n
t
f
ace
by
w
a
ter
as
p
r
opaga
ti
o
n
m
edium
.
A
s
s
how
n
in
F
i
g
ur
e
5,
t
he
d
if
f
e
r
e
nce
betw
e
e
n
t
he
r
es
ona
n
t
a
nd
a
n
tir
e
s
o
n
a
n
ce
f
r
e
que
n
c
ie
s
is
s
t
r
on
gl
y
in
f
l
ue
nc
e
d
by
K
t
.
T
h
e
i
nf
lu
en
c
e
o
f
K
t
i
s
i
m
por
ta
nt
on
Zm
ax
b
u
t
ne
gli
g
ib
le
on
Zmin.
0.
5
1.
0
1.
5
2.
0
2.5
3.
0
3.
5
4.0
0
10
20
30
40
50
60
70
80
Fre
q
ue
nc
y
(
MH
z
)
|
Z
T
|
(
Ω
)
Et
hano
l
Wa
t
e
r
Pa
ra
ffi
n
oi
l
5.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.5
4.
0
4.
5
0
20
40
60
80
10
0
12
0
14
0
16
0
Fr
e
que
ncy
(MH
z
)
Phas
e
( °)
+
90°
Et
hano
l
Wa
te
r
Pa
ra
ffi
n
oi
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N: 2
0
8
8
-
86
94
I
n
t
J Po
w
Elec
&
Dr
i
Sy
st,
Vo
l. 1
0
,
No
. 2
,
Ju
n
e
2
019
:
1
0
64
–
1
071
1
070
F
i
gur
e
5.
T
r
a
nsduce
r
i
m
p
e
d
a
n
ce
modul
us
a
n
d
pha
se
f
or
d
i
f
f
e
r
e
nt
va
lu
e
s
o
f
K
t
5
.
4
.
I
nfluence
o
f int
r
insic
di
electri
c
lo
sses
o
n
t
h
e
i
mpeda
n
ce
modu
lus
I
t
s
h
o
u
l
d
be
n
o
t
e
d
t
ha
t
the
f
o
r
m
(
6)
o
f
the
electr
i
c
a
l
impe
da
nc
e
,
d
oes
no
t
sh
ow
t
he
d
ie
l
ectr
i
c
los
s
es
in
t
he
c
e
r
am
ic.
T
he
se
c
an
b
e
t
a
ke
n
i
n
t
o
a
c
c
o
un
t
w
h
en
c
o
n
s
i
der
i
n
g
a
tr
a
n
sm
issi
on
li
ne
w
it
h
l
o
sse
s
.
Elec
tric
a
l
l
oss
e
s
is
w
e
ll
k
n
o
wn
t
o
be
t
he
r
esi
s
ta
n
c
e
lea
k
o
f
th
e
ce
rami
c
c
a
p
a
c
i
t
a
n
c
e
co
nsi
d
ered
a
s
a
r
e
sista
n
c
e
w
hic
h
d
e
p
e
n
ds
o
n
t
h
e
fr
eque
nc
y
d
e
scr
i
be
d
by
r
e
l
a
ti
o
n
(
7)
[
24,
25]
.
(
7
Wi
t
h
k
t
the
co
up
l
i
ng
fac
t
or
i
n
thic
k
n
e
s
s
mo
de,
C
i
s
the
ca
p
a
c
i
ta
nc
e
o
f
t
he
c
e
r
a
m
ic,
tan
δ
i
s
t
h
e
losses factor
a
nd
ω
t
he
p
u
l
sa
tio
n.
F
i
gur
e
6
show
s
a
gr
ow
th
w
it
h
o
f
the
maximum
val
u
e
of
t
he
e
l
e
c
t
r
i
ca
l
i
m
pe
dance
a
t
t
he
a
n
tir
e
s
o
n
a
n
ce
f
r
e
que
nc
y,
l
i
k
e
w
e
c
a
n
n
ot
ice
a
r
e
du
c
tio
n
o
f
t
he
p
h
ase
(t
he
i
n
d
u
ct
ive
e
f
fe
ct
d
e
c
re
ases
i
n
the
r
e
sona
nce
z
o
n
e
w
he
n
i
ncr
ease
s
)
.
T
he
a
nt
i
r
eso
n
anc
e
a
nd
r
e
sona
nce
fr
eq
uenc
i
e
s
a
s
w
el
l
a
s
t
he
m
in
imum
va
lue
a
r
e
i
nde
pen
d
e
n
t
o
f
.
F
i
g
u
r
e
6
.
V
a
r
ia
ti
o
n
s
of
t
he
i
m
p
eda
n
c
e
m
od
u
l
us
a
n
d
t
he
p
has
e
a
s
a
func
ti
o
n
of
the
d
i
elec
tri
c
losses
6.
CONCLUSION
I
n
t
h
i
s
p
a
p
e
r
,
a
n
e
w
a
p
p
r
o
a
c
h
t
o
t
h
e
u
l
t
r
a
s
o
n
i
c
t
r
a
n
s
d
u
c
e
r
m
o
d
e
l
i
ng
syste
m
i
s
pr
ese
n
t
e
d
.
O
ur
s
tud
y
be
g
i
ns
w
it
h
t
h
e
pr
e
s
e
n
ta
t
i
o
n
o
f
an
e
le
ctr
i
c
a
nd
e
l
ec
t
r
o
a
co
u
s
t
i
c
m
od
e
l
o
f
t
h
e
tr
a
n
sd
uc
er
.
F
r
om
t
hi
s
mod
e
l
w
e
de
t
e
rm
ine
the
anal
y
tic
al
e
x
p
ressi
on
o
f
t
h
e
i
npu
t
e
l
ec
tric
al
i
mp
ed
an
c
e
.
We
m
o
d
e
l
e
d
a
nd
s
i
mul
a
t
e
d
th
e
i
n
pu
t
el
ec
t
r
i
c
al
i
mp
e
d
a
n
ce
o
f
t
h
e
pie
z
o
el
ec
t
r
i
c
t
ra
n
s
du
ce
r
wi
t
h
V
HDL
-
AM
S,
as
w
ell
as
t
he
i
nflue
n
c
e
o
f
t
h
e v
a
r
i
o
u
s
pa
r
a
m
e
ter
s
s
uc
h
as
t
he
c
ou
p
l
i
ng
f
a
c
t
or
,
the
pr
o
p
a
g
a
t
i
o
n
m
e
di
um
an
d
t
h
e
e
l
ectr
i
c
a
l
losse
s
on
the
mod
u
l
us
o
f
the
la
t
t
er
.
We
h
ave
sh
ow
n
th
a
t
t
he
f
r
e
q
u
e
n
cy
a
na
lys
i
s
of
t
he
e
l
ec
trica
l
i
m
p
eda
n
ce
c
a
n
p
r
ecise
l
y
l
oc
a
t
e
the
Fr
e
que
ncy
(MH
z
)
1.
8
1.9
2.0
2.
1
2.2
2.
3
2.4
2.
5
2.6
0
20
40
60
80
100
12
0
14
0
16
0
Phase
( °) +
90°
K
t=0.
55
Kt
=
0.
50
K
t=0.
45
K
t=0.
40
4.
0
400
0.
5
1.
0
1.
5
2.0
2.
5
3.0
3.
5
0
50
100
150
200
250
300
350
Fr
e
que
ncy
(MH
z
)
|
Z
T
|
(
Ω
)
Re=100Ω
Re
=
400
Ω
Re=800Ω
5.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.
5
4.
0
4.
5
0
20
40
60
80
10
0
12
0
14
0
16
0
F
r
e
que
ncy
(MH
z
)
Ph
a
s
e
( °) +
90°
Re=8
00Ω
Re
=
40
0Ω
Re=100Ω
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
St
u
d
y an
d
si
m
u
la
t
i
on
wi
th VH
D
L
-AM
S
of t
h
e
electr
i
ca
l im
p
e
da
nc
e
of
a p
i
e
z
oe
lec
t
r
i
c
… (T
oufik
Me
rd
j
a
na)
1
071
resona
nce
z
o
n
e
t
o
co
n
t
ro
l
a
nd
sta
b
ilize
th
e
ope
ra
ti
n
g
f
re
que
nc
y
o
f
the
system
.
I
n
a
dd
i
t
i
on,
t
he
u
se
o
f
t
h
e
VHDL-AM
S
l
an
gu
ag
e
h
a
s
th
e
adv
a
n
t
a
g
e
o
f
c
o
m
b
i
nin
g
m
u
l
ti
-phy
s
i
cal
d
oma
i
ns,
an
d
i
n
dica
tes
th
at
t
h
e
si
m
u
lat
i
on
of
a
n
u
l
traso
n
ic
d
e
t
ec
ti
on
de
v
i
ce,
c
om
prisin
g
bo
th
e
le
c
t
ro
n
i
c
co
mpone
n
t
s
a
nd
(e
l
e
ctrom
echa
n
ica
l
)
t
r
an
sd
u
c
ers,
i
s
p
o
ssibl
e
with
VHD
L-AMS.
F
i
n
al
l
y
,
thi
s
s
t
udy
m
ak
e
s
i
t
p
o
ss
ib
le
t
o
obta
i
n
a
go
o
d
repr
esenta
t
i
o
n
of
t
h
e sta
t
e,
the
b
eha
v
i
o
r
and
the
per
f
orm
a
nc
e
s
in r
eal
t
im
e
of t
he
t
rans
d
u
c
e
r
.
REFE
RENCES
[1]
I. Iswan
t
o
,
W. S
. Agu
st
i
n
ing
s
ih
,
F. M
ujaahid
, R
.
Roh
m
an
sy
ah, an
d
A
.
B
udim
a
n
"A
ccum
u
l
a
to
r
Ch
arg
i
ng
C
on
trol
with
P
i
ezoelect
ric
Based
o
n
F
u
zzy
A
l
g
o
r
it
h
m
S
chedu
l
i
ng"
,
T
E
LKOMNIK
A
Te
le
c
o
mmun
ic
at
io
n,
Compu
tin
g,
Ele
c
t
ron
i
c
s
an
d
Co
ntro
l
,
Vol.
16
,
N
o
.
2
,
pp.
6
35-640
,
2
018
.
[2]
W.
P
.
Mason
, "
E
l
ectromechani
cal T
ransduce
rs and
W
ave F
ilters",
New
York, D. Van N
os
trand, 19
4
2.
[3]
R.
M
.
Redw
oo
d,
"
Tran
scien
t
p
erf
o
rm
ance
o
f
a
p
iezo
electri
c
trans
du
cer
"
,
A
c
ou
stical
So
ciety Amer
ica
,
Vol
33,
p
p
.
52
7-5
36,
1
9
6
1
.
[4]
S
.
R
.
Gh
orayeb
,
E.
M
ai
on
e
an
d
V.
L
a
M
a
gna,
"M
od
eli
n
g
of
u
ltraso
ni
c
wave
p
ropag
a
ti
on
i
n
t
e
e
t
h
u
sin
g
P
S
p
ice:
a
c
o
m
p
a
r
i
s
o
n
w
i
t
h
f
i
n
i
t
e
e
l
e
m
e
n
t
m
o
d
e
l
s
,
"
i
n
IEE
E
Tr
an
sa
ctio
n
s
on
Ul
tr
as
on
i
c
s,
Ferr
oe
l
ectrics,
an
d F
r
equ
e
ncy
Control
, v
ol
.
4
8
,
n
o. 4
, pp
. 11
2
4
-
11
31
, Ju
l
y
20
01
.
[5]
A.S
a
f
a
ri
,
E.
K
o
r
ay
A
kdo
gan
"
P
i
ezoelect
ric
a
n
d
A
c
ous
ti
c
Materi
a
l
s
fo
r
Tran
sd
ucer
A
p
p
l
i
cat
ion
s
",
S
p
r
in
ger,
Hei
d
elberg,
200
8
[6]
S
.
B
h
a
ll
a
an
d
C.
K
.
S
o
h
,
"
P
r
og
ress
i
n
Stru
ctural
H
ealt
h
M
on
ito
r
in
g
and
N
on-
D
es
tru
c
ti
ve
E
va
l
u
atio
n
Usi
ng
Pi
ezo-
Im
ped
a
n
ce
Tran
s
ducers
"
,
Smart M
a
te
ria
l
s an
d
Stru
c
t
u
r
e
s
:
N
e
w
Research
,
N
o
v
a
S
ci
ence
P
ubl
is
hers
,
In
c
,
N
e
w
Yo
rk,
p
p
.
1
7
7
-228,
2
0
0
7
.
[7]
D.
M
.
Peai
rs,
P.
A
.
T
a
razag
a,
a
nd
D
.
J
.
I
nm
an
,
"Freq
u
en
cy
r
ang
e
s
elect
io
n
f
o
r
im
ped
a
nce-b
a
s
e
d
st
ruc
t
ural
h
ealth
mo
ni
tor
i
ng
"
,
Jo
ur
nal o
f
Vib
ratio
n
and
Aco
u
s
t
i
c
s
,
Vol
.
1
29
,
No.
6
,
p
p
.
7
01
–7
09,
2007
.
[8]
I
.
I
b
r
a
h
i
m
,
A
.
N
.
N
o
r
d
i
n
,
A
.
F
.
M
.
M
a
n
s
o
r
,
Y
.
Z
.
H
.
H
a
s
h
i
m
,
a
n
d
I
.
Vo
icu
l
es
cu
"
Co
nt
ent
Cy
to
to
xicity
S
tudies
o
f
Col
o
rectal
C
arcinom
a
Cell
s
Using
P
r
i
n
t
e
d
Im
pedan
ce
S
e
n
s
ors
"
,
B
u
ll
e
t
in
o
f
E
l
e
c
t
r
i
c
a
l
E
n
g
i
n
e
e
r
in
g
an
d
I
n
f
o
r
m
at
ic
s
(B
E
E
I
)
Vo
l. 6
,
N
o
. 4,
p
p
.
31
7~
32
6, 20
1
7
.
[9]
F
.
C
ou
tard,
"Mo
d
éli
s
a
t
io
n
et
op
timisat
i
on
de
s
tructures
élect
r
o
ni
qu
es
p
o
u
r
l
e
c
o
n
d
iti
on
ne
m
e
nt
t
ran
s
du
cteu
rs
pi
ézoélect
riq
u
es" P
h
D. in
Elect
ro
n
i
cs
,
th
esi
s
,
U
nivers
it
y
of
N
a
n
c
y,
F
ran
ce,
2
007.
[10]
C
.
G
.
H
u
t
c
h
e
n
s
a
n
d
S
.
A
.
M
o
r
r
i
s
,
"
A
T
h
r
e
e
P
o
r
t
M
o
d
e
l
f
o
r
T
h
i
c
k
n
ess
Mode
T
ra
ns
duc
e
rs
U
s
i
ng
SPICE
II,"
IE
E
E
19
84
Ultr
aso
n
i
c
s S
y
mp
osium
, Dall
a
s,
T
ex
as, U
S
A
, 19
8
4
, pp
.
8
97
-9
02
.
[11]
E.
M
ari
o
n
e
,
P
.
T
o
r
toli
,
G
.
L
y
p
acewi
cz,
A
.
Nowick
i,
J
.
M.
R
ei
d
a
n
d
L
.
Fe
llo
w
"
PSPIC
E
m
o
d
e
l
li
ng
o
f
u
ltra
s
ou
nd
tran
sdu
cers"
U
l
tr
as
on
.,
F
e
rr
oele
ct
ri
c Fr
equen
cy
Con
t
r
o
l
,
vo
l.
46,
N
o.
2
,
p
p
.
399
-4
0
6
,
199
9.
[12]
S
.
J
e
m
m
a
li,
"
Con
t
ri
buti
on
à
l’élaborat
io
n
d
e
m
étho
do
lo
g
i
e
s
e
t
d
’outils
d
’ai
d
e
à
l
a
c
once
pti
o
n
de
s
ystèmes
multi-
tech
no
log
i
q
u
es
"
P
h
D t
h
esis
o
f th
e
Nati
on
a
l
S
ch
o
o
l of Tel
ecommun
icati
o
n
s
o
f
Paris, Fran
ce,
2
00
3
.
[13]
T.
M
e
r
d
j
a
n
a
a
n
d
A.
C
h
a
a
b
i
"
C
omp
a
riso
n
be
t
w
e
e
n
V
HDL-A
M
S
a
n
d
PSPI
CE
m
o
d
elin
g
o
f
u
ltrasoni
c
t
r
ans
ducer
f
o
r
bi
ol
ogical
m
edium
s
"
6th
Inter
n
a
t
i
onal Confer
e
n
ce
o
n
Sci
e
nc
es of
Elect
ronics
,
Tech
no
lo
gies
of
Inf
o
r
m
a
tio
n
and
T
e
leco
mmu
ni
cati
on
s
(
S
ET
IT)
,
Sous
se,
T
unisia,
pp.
1
8
3
-1
88
,
20
12
[14]
G.
S
.
Kino
,
"
.
A
c
ou
stic
W
a
v
e
s
:
De
vic
e
s
,
I
m
a
g
in
g
,
a
nd
A
na
log
Sign
al
P
ro
cessing
",
E
d.
R
enti
c
e
-Hall
,
I
nc.
N
e
w
Jers
ey.
1
98
7.
[15]
IEEE
Standard
V
HD
L
Ana
l
og
a
nd
M
ix
ed-S
ign
a
l
Extensions
,
I
EEE
St
d
1
076
.
1
-199
9,
S
H94
7
3
1
,
IE
EE
P
ress,
L
o
s
Ala
m
i
t
os, 1
99
9.
[16]
R.
G
ue
l
az,
D
.
K
o
urt
i
che,
M
.
Nad
i
a
nd
Y
.
H
e
rve,
"
Ult
r
aso
n
i
c
p
ie
z
ocer
am
ic
t
ran
s
d
u
cer
m
odel
i
n
g
w
ith
VHD
L
-
AM
S
IEEE 1076
.1,"
S
E
NSOR
S, 20
04 IEEE
,
Vien
na,
2
0
0
4,
pp.
8
7
-
90
vol.
1
.
[17]
Y. Herv
e
,
"
VHDL-AMS Applic
ati
o
ns
e
t enj
e
ux indust
r
iels
"
, Paris: Dunod,
2002.
[18]
T.
M
erdjan
a,
A
.
Ch
aabi and
S
.
R
o
u
ab
ah
"
VH
DL
-A
M
S
a
nd
P
SP
ICE
mo
de
lin
g
of
u
lt
ra
s
o
n
i
c
piezo
elect
ri
c
t
r
ansd
ucer
f
o
r biolo
g
ical medi
u
m
s
app
li
ca
t
i
o
n
",
P
r
o
cess
i
ng
- A
T
SI
P'
,
S
ous
se
,
Tu
nisia
pp
:
5
23-5
2
8
,
2
014
.
[19]
Qu
artz & Sili
ce. P
iezo
electri
c cer
a
m
i
c
s
(
C
om
pl
e
t
e
t
e
n
s
o
r
c
h
a
r
a
c
ter
i
zat
io
n
of
P
18
8).
[20]
Han
d
b
ook
o
f
Ch
em
istry
and
P
hysi
c
s,
4
5
t
h
ed
n.
C
hem
i
cal
R
ub
ber Co
.
,
C
l
e
velan
d
O
hi
o,
196
4.
[21]
M
.
F
.
A
.
S
h
a
i
b
,
R
.
A
.
R
a
h
i
m
,
a
n
d
S
.
Z
.
M
.
M
u
j
i
,
"
Devel
o
p
m
ent
of
N
on
-In
v
asiv
e
U
l
t
r
aso
n
ic
M
easu
r
in
g
S
y
stem
f
o
r
M
o
n
i
to
rin
g
M
ulti
phas
e
F
l
o
w
i
n
L
i
quid
M
e
dia
withi
n
C
om
p
o
s
i
t
e
P
i
p
eli
n
e",
I
n
te
r
n
at
io
na
l J
o
u
r
n
a
l o
f
E
l
e
c
t
r
i
c
a
l
a
n
d
Com
put
er Engineeri
ng (
I
JECE)
,
V
ol
.
7
,
N
o
.
6
,
p
p
.
3
0
76-3
0
8
7
,
2017.
[22]
T.
M
erdjana
and
A.
C
haa
b
i
"
M
odelling
o
f
P
Z
T
/
P
VDF
Ul
t
r
a
s
oni
c
T
ra
n
s
du
cer
w
ith
V
H
DL
-AM
S
f
o
r
M
edical
Ap
plicati
ons",
Sens
ors
&
Tr
an
s
d
u
cers
, Vo
l
. 2
21
,
Issue 3
, p
p.
23
-
2
9
, 20
1
8
.
[23]
L.
A
l
lies,
"
U
l
t
r
ason
ic
N
o
n
-Lineari
t
y
P
h
eno
m
e
n
a
S
t
ud
y
"
,
do
cto
r
al
t
h
e
sis,
H
e
n
ri
P
o
i
ncaré
Un
ivers
i
t
y
,
Nan
c
y
I,
F
r
ance, 2
00
8.
[24]
R.
G
u
e
laz
and
D
.
K
o
u
rtiche.
"D
oub
le
e
lem
e
n
t
u
l
t
ras
onic
tran
sd
uc
e
r
m
o
d
e
ll
in
g
w
i
th
V
H
D
L
-
A
M
S
f
o
r
l
os
sy
pi
ezocer
am
ic",
1st Int
e
rnati
o
n
a
l
Con
f
er
e
n
ce on
S
e
nsi
n
g
T
echn
o
l
ogy
,
P
a
lm
erst
on No
rth
,
New Zealand
, 20
0
5
.
[25]
R.
M
artin
ez,
A
.
V
e
ra
a
n
d
L
.
Leij
a.
"
H
eat
T
h
e
rapy
H
I
F
U
T
rans
du
ce
r
E
l
ect
rica
l
Im
p
e
d
a
nce
M
o
d
e
li
ng
b
y
u
s
i
ng
FEM"
.
in
Pr
oc
eed
ing
s
of
t
h
e
con
f
eren
ce
o
n
I
n
s
t
rumen
t
atio
n
M
e
a
s
u
r
emen
t Techn
o
l
ogy Con
f
er
e
n
ce
(
I
2M
T
C
)
,
Mo
ntev
i
d
eo,
Ur
u
g
u
ay
,
p
p
. 2
99
–
30
3
,
2
01
4
.
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