Piezoelectric Ceramics Nonlinear Behavior. Application
to Langevin Transducer
D. Guyomar, N. Aurelle, L. Eyraud
To cite this version:
D. Guyomar, N. Aurelle, L. Eyraud. Piezoelectric Ceramics Nonlinear Behavior. Application to
Langevin Transducer. Journal de Physique III, 1997, 7 (6), pp.1197-1208. �10.1051/jp3:1997183�.
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Phys.
J.
III
Hance
(1997)
7
Piezoelectric
Ceramics
Application
D,
Guyomar,
avenue
(Received
#lectrique
G4nie
Albert
2
Einstein,
July 1996,
PACS.43.25.-x
43.38.-p
PACS
Ferrodlectricitd,
et
Villeurbanne
69621
revised
Nonlinear
1997,
PAGE
1197
(*) and L. Eyraud
Aurelle
N.
JUNE
Behavior.
Nonlinear
Transducer
Langevin
to
Laboratoire
20
1197-1208
Bitiment
Cedex,
September1996,
30
acoustics,
504, INSA Lyon,
France
accepted
28
1996)
November
macrosonics
Transduction;
acoustical
devices
reproduction of sound
for
the
generation
and
PACS.43.30.-k
Abstract.
ics.
Drastic
better
A
Underwater
sound
behavior
characterization
changes
high
under
occur
for
excitations
transducers
power
is then required
using
for
PZT type of
cerammaterials.
piezoelectric
nonlinear
behaviors
frequency shift with hysteresis effect
such as
resonance
generation are observed
experimentally. They result in front mass displacement
saturation, drop in performances and
instabilities.
To interpret
nonlinear
such features, a
approach is proposed.
This resulting
nonlinear
model has been applied to the
transducer
and
simulations
given. The comparison between
simulations
numerical
and experimental
results
are
shows good agreement.
Usually, observed
saturation
effects are interpreted as in term of viscous
factor
nonlinear
approach gives an
alternative
explanation that explains also others
increase,
our
observed
phenomena.
Indeed,
and
characteristic
overtones
observds
des
transducDes
changements importants de comportement
sont
sur
des
puissance
des
cdramiques de type PZT; Une
meilleure
caractdrisation
utilisant
matdriaux
pidzo41ectriques sous hauts
s'avAre
alors'ndcessalre.
En effet, des ph#nombnes
niveaux
caractdristiques
d'un
lindaire
tels que le ddcalage de la frdquence de rdsocomportement
non
hystdr4sis
d'harmoniques
la
gdndration
observds
expdrimentalement. Ceci
ont dtd
nance
avec
ou
traduit
du
de ddplacement de la face avant,
limitation
saturation
se
par des effets de
niveau
une
des performances du systkme et un
fonctionnement
Afin d'interprdter de tels comporteinstable.
hn6aire
dtd appliqud h notre
ensuite
ments, une thdorie
structure
est proposde. Ce modkle a
non
de
transducteur.
Une comparaison
simulations
rdsultats
expdrimentaux a permis de vaentre
et
lider cette
approche. Les effets de saturation
nouvelle
observds
toujours
ont dt6 jusqu'h pr6sent
relids h la
variation
coefficient
d'arnortissement,
dtude a permis de
du
ddmontrer
qu'une
notre
explication lide aux non
lin6arit6s
start possible et permettait
mAme temps
d'expliquer
autre
en
d'autres
ph6nomAnes observ6s
R4sum4.
(*)
@
teurs
de
Author
for
Les
tditions
correspondence
de
Physique
(e-mail:
1997
naurelle@ge-serveur.insa-lyon.fr).
JOURNAL
1198
PHYSIQUE
DE
III
N°6
Introduction
1.
ceramics,
Piezoelectric
used
systems,
are
domains
are
subjected
rapidly reached
sufficient
to
interpret
Actually,
in
a
is
no
often
in
not
prediction
on
(P189
and
paper,
standard
the
has
achieved
solved for
power
be
must
induced
in
to
test
we
compare
transducer.
power
structure.
model
for
our
theory
nonlinear
our
Phenomena
the
a
been
a
linear
which
apply to our
lumped
mechanical
that
model
rather
motion
that
nonlinearities
nonlinear
but
a
are
hypothesis
parameters
two
small
problems
nonlinear
the
model
Mason
Nonlinear
classical
given between
relations
present
we
piezoelectric
theoretical
a
use
first
drives.
to be valid in
motions [1-4].
for
this
the
welding or cleaning
Thus, nonlinear working
linear approach is no longer
ultrasonic
sonars,
electrical
assumed
is
piezoelectric ceramic
behavior
prestressed symmetrical Langevin
study
work
materials
these
linearization
The
Experimental
2.
and
Finally, a simulation
work
with
experimental results.
transducer.
To
In
propose
we
will
mechanical
large amplitude
piezoelectricity has to cope with
Nowadays,
applications.
Then
for
for
transducers
power
high
experimental results.
problems of elasticity, the
convenient
more
We
most
approach.
first
in
to
under
high
transducer,
excitations,
mechanical
made
of four
up
power
we
chose
ceramic
to
rings
France) and two tail masses.
Experimentation has been
company
frequency of the transducer, consequently low electrical fields have
been
cf. Fig. I). A low distortion signal is generated by the HP3577A
analyzer and then amplified with the NF4505.
This
equipment presents high rejection level.
However to make sure that the
nonlinearities
really
generated in the ceramics, we have run
are
experiments on pure resistive load instead of the
On the pure
transducer.
resistive load, under
driving levels equivalent to the ones used for the transducer, the harmonics to
fundamental
made
from
Quartz & Silice
in air at the
resonant
applied to the ceramics
ratio
below
was
opposite poling,
One of the
first
Besides,
40 dB.
can
be
inserted
observation
is
a
in
stress
sensor,
the
middle
the
resonance
made
out
of two
disks of P189
thin
increasing the
level. An hysteresis effect
excitation
appears
frequency is swept in increasing or decreasing mode [5,6].
when
also
is
noticed
current1
~l
Analyser
HP3
577A
Amplifier
NF
4505
Transducer
(~~~d P~~)
(-
voltage
U
ii
Vibration
velocity
Spectrum
Analyser
Fig.
1
Experimental
set-up
ceramic
with
piezostack to record the stress
waveform.
frequency shift towards the low frequencies that
of the
whether
the
NONLINEARITIES
N°6
Moreover,
pears
This
electrical
due
the
to
power signals (cf. Figs.
in the Fourier domain, to
a
(cf. Fig. 5).
subharmonics
1199
distinct
a
distortion
ap-
2, 3, 4)
or
and
harmonics
enough,
sufficient
level is
excitation
velocity, stress
effect corresponds,
current,
distortion
structure
levels,
the
when
the
on
CERAMICS
PIEZOELECTRIC
IN
spectrum
note,
We
(f/2).
period doubling
appearance
transfer
from the
increasing the input voltage, the energy
of the displacement and
results in an amplitude
saturation
that
for
presents
a ray
excitation
high
sudden
a
When
monics
fundamental
the
to
har-
current.
Model
3.
piezoelectric ceramics, we extend up to the second
the prestressed
around
static
equations,
state (To> Do)nonlinear
piezoelectric coefficients [7, 8]. Then, thanks to thermointroduce
Indeed
new
we
dynamic relationships, the number of independent coefficients has been reduced, subsequently
describe
To
the
stress
nonlinear
the
the
order
T and
of the
behavior
piezoelectric
constitutive
induction
D
T
To
=
expressed
be
can
+ cS
as:
~
eE +
p E2
s2
/s2
D
S is the
where
Do
=
strain, E the
d~2
+ eS + SE +
electrical
field,
c, e,
'ySE
+
e
classical
the
(1)
fISE
+
linear
piezoelectric
coefficients
coupling
and weakly
dynamic and static states, the terms To and Do will not be considered
nonlinear
equations will be obtained.
frequency in the air, the electrical fields applied are low. ThereAs we work at the
resonant
nonlinearities
due to electrical
fore, in the following calculations, we do not take into account the
field. Our
transducer
be regarded as a mass-spring system in a low frequency regime (the
can
wavelength is such larger than the piezostack length), so we can write:
and
o,
fl,
y, d the
new
nonlinear
ones.
Afterwards,
assumption
the
on
that
there
is
no
between
a2~
MM
(~)
-TE
=
ring surface and u the front mass displacement. We make
piezostack and the displacement on the mass are equivalent
the front
consider
to
mass.
mass
as a rigid
so
as
Substituting in this last equation the nonlinear expression of the stress T, the strain S by
the ratio u Ii (I: half length of the piezostack) and adding up a damping term in I, we get the
M is the
where
that
sure
nonlinear
mass,
E the
displacement
the
oscillator
equation:
ii
where
w
is the
ceramic
of the
+ W~U +
eigenpulsation
of the
(
=
In
the
remains
first
experimental part,
consequently
periodic,
w~,
we
the
£UE +
«~U~
2>W
II [E>
13)
=
transducer:
)
=
noticed
the
that
displacement
>
and
al,
u
can
~'j[.
=
speed signal is
expressed as a Fourier
vibration
be
distorted
but
sum:
n=+«
11(t)
~j
=
n=-«
crier"~~,
(4)
JOURNAL
1200
.j
PHYSIQUE
DE
N°6
III
.[
.(
,.,
",'
Voltage
j
....j....
I.
I
.I
..i
;
..,
'j''
'>'
'j''t'
''j''>
..1
j'<
''
'
j'
"l't'
>'
-..I..-...
i
..~
''
',
>'
~
I-
.j
Current
I.
-,
Fig.
2
Voltage
and
signals (Transducer
current
P189
/40 MPa/65 Vpeak).
oos
;
....I
'.
-,--.;
Velocity
..]
,,.
,,-,).
i
f,
,I.
I
,I
(.
I,
t
;
I-("
..,
,'4
'
,
'
'
'
,,,'
,_,j
(-
'
'
,(
t
i
I'
'.
i
Fig.
3
Stress
~~~~~(
j:
,,'
.,--'---
,-,
'S
~.%O,0i7q~a
,.(
,
velocity signals (Transducer
and
..I."
I
P189
/40 MPa/65 Vpeak).
$
~~OOW
.-S.
.--.-..-I
;
(
;
..I
'j'""
t
Fig.
4.
Electrical
i
power
i
signal (Transducer
i
P189
i
i
/40 MPa/65 Vpeak).
.,
NONLINEARITIES
N°6
CERAMICS
PIEZOELECTRIC
IN
1201
Mag
dB
~02.
Fig.
Current
5.
where
Q is the
(Transducer P189/40 MPa/75 Vpeak).
spectrum
variable
signal pulsation,
excitation
E(t)
where
These
n=+m
~j
n=-m
iii°
This
(eJ~~
=
Eo is the electrical field peak value.
previous expressions can be inserted
Q~
equation
of the
holds
coefficients
resolution
The
has
n~cneJ"~~+w~
I[(
that
made
calculations
same
in
n=+m
~j
that
+
is to
+
(5)
equation (3), it gives:
n=+mm=+m
~j
crier"~~+oi
[f[ neJin-i>Qtj
~j
cncmeJ("+'~~~~
n=-mm=-m
frequency ha (k: positive integer), it gives the
correspond to the displacement amplitudes for
until the frequency 3Q (n
3).
Simulation
Simulations
made
and
of
have
Results
T > 0 and S > 0 for
from
I is
evaluated
nonlinear
static
coefficients
regime.
Then,
Comparison
and
displacement
permitted
for
equation
induction
the
an
current
or
evaluation
nonlinear
of the
Experimental
with
equations,
nonlinear
(D).
harmonics
a
made
better
with
fit
the
around
curves
this
described,
following
The damping
previously
coefficients.
representing the
initial
guess
is
They give the
frequencies,
Results
The
compression, although not standard, was adopted.
the
A first
transducer
quality factor
measurement.
is
and phase
frequency. The
modulus
each
=
transposed
be
can
amplitudes evolution, corresponding to the
fundamental
and
current
for different driving levels (Eo) or different driving frequencies (Q).
4.
pulsation:
field
electrical
e~J~~),
n=-m
crier<n+i>Qt
the
say
for each
cn,
been
4kHz
strain
made
to
ver8n8
adjust
have
been
convention:
coefficient
evaluation
of the
the
in the
the
stress
displacement
JOURNAL
1202
PHYSIQUE
DE
N°6
III
Displacement (microns)
__-------~jj)~-"#18i
---'~~'
__
~,,.
,,/
,,
,,
P189
,'
,
II
,/
II
4
2
6
8
10
Fig
and
displacement
Curve:
6.
dimensions
whose
are
=
have
w
over
the
with
27r
=
L
mm,
curves
obtained
been
f(static stress).
=
6.35
experimental
current
below
#
27000
x
15
=
(Pal
Experimentation
a large
following
Hz
mode
given
are
stress
in
then
can
between
6 [9]. The
deduced.
We
be
1
N s~ m~l
0.0015
=
ceramic
on
rods
y
=
fl
=
from
see
-2.8
6
displayed
results
kg~l
101~ N m~~ kg~~
x
10~ N m~l V~l
-1.55
=
x
10~~ N V~~.
x
in
stress
of the
~>ariation
can
made
simulation
set:
and
strain
Figure
The
parameters
al
=
relationship
107
been
has
voltage.
of
range
=
The
x
mm
E
8 x 10~~ m~
1= 6.6 x 10~~ m
M
20 x 10~~ kg
Note.
16
14
12
Stress
these
the
regime
static
s(
compliance
static
that
curves
the
s(
compression
the
in
of the
function
coefficient
applied
increases
and
With our sign
compressive stress, or that the cl coefficient
decreases.
convention
coefficient a has to be
between
nonlinear
and dynamic modes, the
static
to keep a continuity
negative to explain a decrease of the c(~ coefficient in compression. Indeed, it can be written:
with
the
T=(c+I)S-eE-ySE
First,
simulation
nonlinear
term
a
allow
runs
that
driving levels,
evolution is a symmetrical
(cf. Fig. 7).
For
low
When
the
(cf. Fig. 7),
appears.
It can
zone.
be
It
consequence
frequency
These
shown
in
the
that
an
of
the
the AC
is
~
co.
part is unstable,
exhibits
therefore
phase jumps
a
different
In
and
is
due
the
to
displacement amplitude
eigenfrequency (w)
system
the
case,
the
the
that
level.
equation (6) when
forj
frequency
particular
a
for
this
shift
strain
at
obtained
solutions
~~~~~
and
the
located
solution
real
~~
amplitude
with
influence
of low
maximum
three
condition
transducer
~the hysteresis
amplitude
anj C to
from A to B
resonant
frequency( for
increase
with
a
cannot
D.
n
=
phenomenon
venture
in
this
cf. Fig. 8). As
upgoing
or
frequency
experimental
resonance
the
shift
measurement
of the
the
driving signal.
qualiti
factor Qm
I
domain
or
a
downgoing
sweep.
hysteresis
high driving levels,
and badly defined.
for
increases,
frequency
resonance
elasticity
are
whose
consists
by the
defined
results
nonlinearities
curve
the
material
the
level
excitation
(fundamental frequency)
describe
to
modifies
Consequently,
becomes
difficult
NONLINEARITIES
N°6
PIEZOELECTRIC
IN
CERAMICS
1203
lo"~
x
'
75
O
Vp~~~
-i~
'
j
'
10Vpeak
*
'
,
4
'
o
'
'
o,
,
o
-°-
-'-
3
-'
'
,o
,
(i
lo
Q
~
i
~--$-i
,
1---
-_-i---
ix
0
2
2 55
2 65
2 6
2 75
2 7
2 85
2 8
Frequencies (lIz)
Fig
displacement
Simulated
7.
x
10
~
curves.
x10~~
4
._
B
_1_
2
_'_
_'
A
D
~2
2.55
5
2.6
2 65
2.7
Frequencies
Fig.
Hysteresis
8.
Figure 9,
In
from
structed
to
of
are
on
75
the
the
observed
amplitudes
current
Vpeak/3
ones
curves.
mm,
x
lo
~
simulation
waveforms
series, the
of
apparent
(cf. Fig. 9
corresponding
and
current,
displacement,
distortion
on
these
velocity
simulated
and
signals
stress
are
are
recon-
quite similar
Figs. 2-3).
frequency f (where a maximum
resonant
2f
of
the
and displacement signals
to
current
effect
for different
levels (cf. Figs. 10, 11). A
saturation
excitation
appears
The
displacement amplitude shows a drop of 40%, for a driving signal of
with regard to the linear
behavior.
reached)
represented
these
time
Fourier
The
is
phenomenon
2 8
2.75
Hz
and
to the
the first
fundamental
harmonic
JOURNAL
1204
j~4
~
PHYSIQUE
DE
Electrical
III
N°6
(V/m)
field
3
3
~
io-3
Straln
-'-
I
08
06
L
-j-
~
~
-~-
~~
j
~
j
0
-02
.04
~
.06
.08
-'
~
J
'-
L
~
,-
~
-l
Vibrations
1-
~
eed
nvs
i-
~
Cunent(A)
05
04
03
~
1-
02
_i
-,_
L
-----L[j~
-'-
01
0
---1---
~-----
~
~>
'---
~2
___j
'
~3
--_-----____'~~
~4
~5
8
jo
~
Stress
05
-'
~5
0
Fig.
(50
Simulated
9.
dots
time
period/the
per
1
50
signals
resonant
for
_'-
>00
>50
working
resonance
frequency
is
around
200
c<ndition
25
'_
_'
000
Hz
250
75
for
300
Vpeak/time step
highest levels).
=
1/(50.25000)
s
amplitude:
influence of the
nonlinear
also the
term y on the displacement
deformation
amplitude is. Besides, the
nonlinearity y in SE is, the smaller the
model as the
transformer
electromechanical
factor ~'N", defined in the Mason
ratio, decreases
the applied
field is
when
electrical
increased
(cf. Fig. 12). This effect explains that the
Computing from
electromechanical
has a smaller efficiency for high
conversion
excitations
Simulations
the
the
higher
show
the
nonlinear
mode
electromechanical
the
conversion
coefficient
N
shows
that
this
coefficient
is- lower at high driving
conversion
drops with the applied electrical field. The piezoelectric
effects.
amplitude, thus explains the saturation
f(S) shows a different
of the
working in the
behavior
ceramic
simulated
plot T
The
mode
cf. Fig. 13). The piezoelectric model chosen at the beginning
compressive or extension
implies a material hardening in the expansion mode and a material softening in the compressive
=
NONLINEARITIES
N°6
x
7
PIEZOELECTRIC
IN
CERAMICS
1205
Displacement
lo ~~
!
6
fundamental
'
harmonic
'
'-
-'-
'
~
f
2f
-'-
'-
0
20
lo
40
3lJ
60
50
80
70
Volta9e (Vpeak)
Fig
10.
tion.
(*).
Displacement
amplitude
(-)
measurements,
applied voltage for the
of the
function
resonance
working
condi-
simulations.
(mApeak)
Current
0 4
------~-
0.35
------)fundamental
'-
03
-(......
~
f
~-
-'~
0.25
.w
'
.--'..
.'-
o~
o.15
'-
'-
-'
-'
2f
harmonic
'
0.I
-------i
--r
~
o.05
~
-'-
-,-
-[
0
lo
30
20
40
50
60
70
80
Volta9e (Vpeak)
Fig.
amplitude
Current
11.
(*):
(-):
measurements,
mode.
We
relaxes
the
could
say
elasticity,
function
the
previous
applied
the
that
the
compressive
while
the
extension
hardening [10].
It is commonly accepted that the
with
the driving level ill,12].
crease
Indeed,
of
voltage
for
the
working
resonance
condition.
simulations.
nonlinear
stress
stress
saturation
simulations
Our
have
creates
tends
effect
nonlinear
shown
to
a
align
is
dipole opening and consequently
all dipoles resulting in a material
related
model
that
the
to
the
proposes
saturation
losses
mechanical
a
new
is
in-
interpretation.
linked
to
material
1206
JOURNAL
PHYSIQUE
DE
III
N°6
(A.s/m)
N
-~-
0.22
0.21
~
-
,-
02
~
-'-
-'-
j--
o19
-I
-I-
---1--
---.---
10
20
30
50
40
60
70
80
Volta9e (Vpeak)
Fig.
Simulated
12.
N
curve
x
~
io~
f(voltage)
=
T
o
-10
-1
-0.5
0
0.5
s
Fig.
Simulatedcurve:
13.
nonlinearities.
been
conducted
To
by
compare
numerical
x
10"~
stress=f(strain).
these
two
possible
simulations
in
interpretations,
both
cases
and
a
signal analysis
power
compared
after
with
has
experimental
results.
The
electrical
all
the
variation
power
of the
have
damping
been
coefficient
with
by
simulations.
obtained
the
applied
Nonlinear
equations and the damping coefficient I adjusted for each
displacement amplitudes observed experimentally.
electrical
field
and
terms
have
been
excitation
level
associated
the
so
suppressed
as
to
in
retrieve
NONLINEARITIES
N°6
CERAMICS
PIEZOELECTRIC
IN
1207
~°~~~
20
18
16
p~
~
14
__,wnablc
w
12
lo
~
~
P~
8
6
4
Pf
2
P3f
"
~
80
70
60
50
40
30
20
lo
Voltage (Vpeak)
Fig
Power
14.
Figure
.
displays
14
the
measured
the
simulated
.
resented
the
.
the
as
Figure 14
The
dots),
as
with
obtained
power
damping,
variable
the
and
(rep-
nonlinearities
without
the
nonlinear
with
variable
damping
appearance
of odd
harmonics
have
terms
model,
nonlinear
with
with
a
damping
constant
coefficient
lines).
that
shows
Nonlinear
obtained
curves:
lines),
solid
model
explain the
power
(represented
power
simulated
as
different
power
dashed
(represented
data.
curves
leads
coefficient
the
on
considered
be
to
approach
if
one
to
power
wants
with
fit
better
a
overestimates
the
experimental
the
values
power
and
cannot
signal.
precisely
estimate
to
the
transducer
performances.
Conclusion
5.
For
heavy duty Langevin sources, the nonlinear
It explains especially the
behavior.
and displacement
saturation, the
current
ducer
the
model gives
resonance
acoustical
a
good interpretation
frequency
power
shift
saturation
of the
trans-
hysteresis effects,
and
and
the
overtones
generation.
simulations
Numerical
special working
conditions
lead
of
to
our
nonlinear
coefficients
estimation
that
for
the
Boucher,
for
important
are
transducer.
Acknowledgments
The
authors
supporting
are
this
grateful
research.
to
the
CERDSM
of
Toulon,
and
especially
to
Dr
D.
JOURNAL
1208
DE
PHYSIQUE
III
N°6
References
[1]
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