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Piezoelectrics as Multifunctional Electromechanical Materials for Space Applications
Stewart Sherrit
Jet Propulsion Laboratory, California Institute of TechnologyPasadena, CA
3545 Lunchtime SeminarMonday, September 8, 2003
AcknowledgementThe NDEAA Lab team consisting of:• Dr. Yoseph Bar-Cohen, Senior Research Scientist and
Group Leader• Dr. Xiaoqi Bao• Dr. Zensheu Chang• Dr. Mike Lih
Outline
• Describe the various actuation and sensing mechanisms under development in the NDEAA Lab at JPL
• Review a variety of Electroactive Material Dielectric and StrainResponse
•Present the various groups of piezoelectric materials(Ceramic, Single Crystal, Polymer) and typical size of effects
• Review the Macroscopic description of Piezoelectricity and look a the limitations. (Nonlinear, Dispersion, Loss)
• Present the techniques used by transducer designers toamplify the limited strain (Cantilevers, Resonance, Stacking)
• Present the sensor examples (Quartz Crystal Microbalance QCM,Surface Acoustic Wave SAW Devices)
Anti-Ferroelectricity•Low fields - counteracting domains stable•Polarization curve opens up at large externalelectric field as domains switch
D vs E S vs E
Electroactive Response
Ferroelectricity•Presence of a spontaneous polarization •Polarization can be switched by anexternal electric field.
Electric Field E (MV/m)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
Stra
in S
(m/m
)
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
EC
D vs E S vs E
Common applications include ferroelectric RAM, electro-optic devices and displays
Electroactive Response
Electrostriction Electrostriction is the name given to the quadratic relationship
between the strain S and the electric displacement D present in most materials. The most fundamental form of this relationship is
kjijkij DDQS =
.......+++= nmlkjijklmnlkjijkljiji EEEEEEEEED εεε
2kijkij ES λ≈
Applications Actuators, sonar projectors, ultrasonic transducers
Electroactive Response
D is electric displacement
E is field
Ds is the Saturation Electric Displacement
±(E1 – Ec) is the field at which the anti ferroelectric becomes stable
±(E1 + Ec) is the field at which the domains switch to parallel orientation.
k is saturation coefficient (k large switching occurs abruptly k = µ/KT)
normalized first time derivative of the electric field
normalized second derivative of the electric field wrt time
ε is a term to account for linear saturation
DDs2
tanh k E E1 R2⋅+ Ec R1⋅−( )⋅[ ]⋅Ds2
tanh k E E1 R2⋅− Ec R1⋅−( )⋅[ ]⋅+ ε E⋅+⎡⎢⎣
⎤⎥⎦
:=
dtdEdtdER
//1 =
22
22
//2dtEddtEdR =
Semi Empirical Modeling (Ising Spin model)Start with antiferroelectric
DDs2
tanh k E E1 R2⋅+ Ec R1⋅−( )⋅[ ]⋅Ds2
tanh k E E1 R2⋅− Ec R1⋅−( )⋅[ ]⋅+ ε E⋅+⎡⎢⎣
⎤⎥⎦
:=
2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4
0.2
0
0.2
0.40.4
0.4−
D
2 106×2− 106× E
E1 = 0 Ferroelectric
2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4
0.2
0
0.2
0.40.4
0.4−
D
2 106×2− 106× E
2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4
0.2
0
0.2
0.40.4
0.4−
D
2 106×2− 106× E
Ec small Electrostrictive
For all the above S = QD2
What about piezoelectrics?
All constants Antiferroelectric
2 .106 1.5.106 1 .106 5 .105 0 5 .105 1 .106 1.5.106 2 .1060.4
0.2
0
0.2
0.40.4
0.4−
D
2 106×2− 106× E
S = Q(D + Dr)2
= (2QDr)D + QD2 + QDr2
= gD + nonlinear term + remanent strain
Piezoelectric coefficient in the poled materials is due to coupling of the electrostrictive coefficient with the remenant electric displacement.
Dr
S s T d E
D E d Tp pq
Eq pm m
m mnT
n pm p
= +
= +ε
PiezoelectricityCoupled “Linear” Relationship between the strain S, electric displacement D and the electric field E and the stress T
Derived from thermodynamic potentials
•Superscripts designate boundary conditions•Subscripts designate anisotropy
•Notation: IEEE Standards on Piezoelectricity
Elastic Response – Strain S, Stress TS = sT s is compliance
T
S= ∆x/l
T = F/A
l
∆xslope = s
Dielectric Response – Field E, Electric Displacement DD = ε0E+P = εE ε is permittivity
E =V/l
D
D = Q/A
l
slope = ε
V
+ + + + +
- - - - -
Direct Piezoelectric Response – Electric Displacement D, Stress T
D = dT d is piezoelectric charge coefficient.
T
T = F/A
slope = d
Converse Piezoelectric Response – Field E, Strain SS = dE d is piezoelectric charge coeff.
E =V/l l
slope = d
V
+ + + + +
- - - - -
I
∫ ∂= tIA
D 1
S= ∆x/l ∆x
Coupling constant
lV
∆x
versaor vice Density Energy ElectricalInput
DensityEnergy MechanialOutput 2
22
EsTkε
==
Energy conversion per cycle
sdkε
22 ∝
( )G s T T g D T D DijklD
ij kl nij n ij mnT
m n112
122= − + + β
Sample under isothermal and adiabatic conditions and ignoring higher order effects the elastic Gibbs function may be described by
The linear equations of piezoelectricity for this potential are determined from the derivative of G1 and are
SGT
s T g D
EGD
D g T
ijij
ijklD
kl nij n
mm
mnT
n nij ij
= − = +
= = −
∂∂
∂∂
1
1 β
Thermodynamic Derivation
S s T g D
E D g Tp pq
Dq pm m
m mnT
m pm p
= +
= −β
S s T d E
D E d Tp pq
Eq pm m
m mnT
n pm p
= +
= +ε
T c S e E
D E e Sp pq
Eq pm m
m mnS
n pm p
= −
= +ε
T c S h D
E D h Sp pq
Dq pm m
m mnS
n pm p
= −
= −β
Other Forms of Linear Piezoelectricity
Reduced Matrix Form
ET
x
xd
xd
xs
DS
T
E
•=)33(
)36(
)63(
)66(
ε
T
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
•
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
Τ33
Τ32
Τ31
Τ23
Τ22
Τ21
Τ13
Τ12
Τ11
3
2
1
6
5
4
3
2
1
363534333231
262524232221
161514131211
636261666564636261
535251565554535251
434241464544434241
333231363534333231
232221262524232221
131211161514131211
3
2
1
6
5
4
3
2
1
EEETTTTTT
dddddddddddddddddd
dddssssssdddssssssdddssssssdddssssssdddssssssdddssssss
DDDSSSSSS
EEEEEE
EEEEEE
EEEEEE
EEEEEE
EEEEEE
EEEEEE
εεεεεεεεε
PZT has C∞ symmetry.Therefore in ideal case only 10independent constants are required to describe the response.
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
•
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
000000
−
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
Τ33
Τ11
Τ11
3
2
1
6
5
4
3
2
1
331313
15
15
1211
1555
1555
33331313
13131112
13131211
3
2
1
6
5
4
3
2
1
0000000000000
000)(20000000000000000000
000000000000000
EEETTTTTT
dddd
dss
dsds
dsssdsssdsss
DDDSSSSSS
EE
E
E
EEE
EEE
EEE
εε
ε
Real or Actual Piezoelectrics
( ) ( )tTTEistTTEss ijiEmniiji
Emnr
Emn ,,,,,,,, ωω +=
( ) ( )tTTEidtTTEdd ijimniijimnrmn ,,,,,,,, ωω +=
( ) ( )tTTEitTTE ijiTmniiji
Tmnr
Τmn ,,,,,,,, ωεωεε +=
LossDispersion
Non-LinearitiesTemperature Dependence
AgingALL May be Significant
Linear Loss
where x and y can be the field E, Displacement D, stress T or strain S
x =x0sin(ω0 t)•b real then y and x are in phase.•b complex then y and x are out of phase and depending on the variables a energy loss is present .
[ ])t +(ωy=|b| x δ00 sin
y = bx
• Material Loss - First order correction • Rayleigh’s law Y= (a0 +αXm)X ± α/2(Xm
2 - X2) where the response increases linearly with maximum field/stress and the remanence increases with the square of the field/stress
S,T,D,E
-1.0 -0.5 0.0 0.5 1.0
S,T,D
,E
-0.9
-0.4
0.0
0.4
0.9
Rayleigh Behavior (1st order)
DispersionCausal (response follows input)•Passivity (Can only dissipate) Example PVDF-TRFE
( ) ( )" ' 333333 ωεωεε SSS i+=
( ) ( )" ' ωω ttt ikkk +=
tk
( ) ( )" ' 333333 ωω DDD iccc +=
Non Linearities
• First order corrections – Rayleigh•Higher order expand the elastic Gibbsenergy in field and stress
• Definition of how material constants are measured1. Average response (∆Y/∆X)2. Differential response (dY/dX)3. Biased small signal differential response(∂Y/∂X at X0)
D, S
EEBIAS
Under a large low frequency field domainstructure changes irreversibly
Under a small high frequency field domain walls vibrate reversibly
Piezoelectric Properties - Ceramics
TABLE 5.2-1: Piezoelectric coefficients and permittivity at three temperatures for three typical commercial PZT materials [Based on: Hooker, 1998]. Performance Standard set out by Navy eg, Navy I , Navy II, etc
Property PZT 4 PZT 5A PZT 5H*
-150oC
25oC 250o
C-
150oC25oC 250o
C-
150o
C
25oC 180o
C
d33 (pC/N) 210 225 360 190 350 440 260 585 900
d31 (pC/N) -70 -85 -130 -100 -190 -270 -115 -265 -500
ε (10-9 F/m)
7.1 9.7 27 5.3 10 27 7.5 28 124
*maximum in dielectric constant is at 180oC
Piezoelectric Properties - Polymers
d31 = 28 pC/N, d32 =4 pC/N, d33 =-35 pC/N
PVDF
Relative Permittivity =10-15Mechanical Q = 10Coupling up to 0.15
d31 = 10.7 pC/N, d32 =10.1 pC/N, d33 =-33.5 pC/N
Relative Permittivity =5-10Mechanical Q = 10Coupling up to 0.25
PVDF/TrFE 75/25 Copolymer
Table 2: The material constant for the length extensional resonance mode, at roomtemperature, sample 2.
Material constant sD
33 (m2/V) Re × 10-11 1.73
Im × 10-13 -1.76
sE
33 (m2/V) Re × 10-10
Im × 10-12
1.30 -2.17
d33 (C/N)
Re × 10-9
Im × 10-11
1.97 -3.32
εT
33 (F/m) Re × 10-8
Im × 10-10
3.42 -5.48
k33
Re Im x 10-4
0.93 -5.06
Res
ista
nce
(Ohm
s)
1000.0
10000.0
100000.0
1000000.0
10000000.0
100000000.0
Frequency (kHz)
0 500 1000 1500 2000 2500
Rea
ctan
ce (O
hms)
-2.0e+7
-1.0e+7
0.0e+0
1.0e+7
2.0e+7
Le mode PZN-PT 8%
E l e c t r i c F ie l d ( M V / m )
- 2 . 0 - 1 . 0 0 . 0 1 . 0 2 . 0
Elec
tric
Dis
plac
emen
t (C
/m2 )
- 0 . 4- 0 . 3- 0 . 2- 0 . 1
0 . 00 . 10 . 20 . 30 . 4
E l e c t r i c F i e l d ( M V / m )
- 2 . 0 - 1 . 0 0 . 0 1 . 0 2 . 0
Stra
in (µ
m/m
)
- 1 0 0 0
0
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
E le c t r i c F i e l d ( M V / m )
0 . 0 1 . 0 2 . 0
Elec
tric
Dis
plac
emen
t (C
/m2 )
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
0 . 2 0
0 . 2 5
E l e c t r i c F i e l d ( M V / m )
0 . 0 1 . 0 2 . 0
Stra
in (µ
m/m
)
- 5 0 00
5 0 01 0 0 01 5 0 02 0 0 02 5 0 03 0 0 03 5 0 04 0 0 0
High field studies PZN-PT 4.5%Bipolar Excitation Unipolar Excitation
Piezoelectric Properties – Single Crystals
Piezoelectric Properties – Single Crystals
AT Cut Quartz
SεDc
kh
5.46x10-11(1 - 0.024i)2.96x1010(1 + 0.00012i)
0.072(1 - 0.0088i)1.67x109(1 + 0.021 i)
Density = 2650 kg/m3
Q’s up to 100,000106 in vacuum
Temperature stability extremely high about RTWhich is the reason they are used timing
Limitation of Piezoelectrics
Consider Motorola 3203 HD PZTwhich has a fairly large d33 constant
S = d33E T = e33E
For E = 105 V/md33 = 6x10-10 m/V e33 = 25 C/m2
S = 6x10-5T = 2.5 MPa
Q. What do you do now?A. Design around this. → Resonators or
other Strain Amplification techniques
Stacks and Bimorphs
3333 ndd eff =L
AnCT33
2ε= 2
2313tVLdy =∆
2
231
23
tVLdy =∆
Uni-morphs
Example – Clemson’s RAINBOW- NASA’s THUNDER
Flextensional Devices
Cantilevers
Resonator Example Thickness Extensional
T c S h D3 33 3 33 3= −D
E h S D3 33 3 33 3= − + β s
∂∂ ρ
∂∂
23
233
23
32
ut
ux
D
=c
Wave Equation
Linear EquationsOf Piezoelectricity
Also S = du/dx and
B.C. T3 = 0 at x3 = 0T3 = 0 at x3 = t
vc
D
D
= 33
ρ
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎟⎠
⎞⎜⎜⎝
⎛=
D
D
D
DD
xt
tx
vv
vvc
DhS 3333
33
33
sinsin
1coscos ω
ω
ωω
( ) 30
33333330
3 dxdxt
st
∫∫ −=−= DShEV ε
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−−=
D
DD
Ds
t
t
l
v
vc
vDhDV 3
ω
ω
ωsin
2cos2
33
233
333ε
ID
D= = −Ad
d ti A3
3ω
Ai
ttD
DDs
ω
ωω ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+
=vc
vh
Z2
tan233
233
33ε
Strain from Wave Equationand Boundary Conditions
Calculate Voltage
Calculate Current
Finally Determinethe complex ImpedanceVERY IMPORTANT
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
p
p
f
fk
Z
4
4tan
1
2
33ω
ω
ω
t
SAitε
SSD
pDSt l 3333
3333
2332 1 24
εβ
β===
vc
hk f
Simple Measurement Method to determine the ElasticDielectric and Piezoelectric Constants and loss by measuring the complex impedance as a function of frequency
Impedance Analyzer
Sample
Common Modes of Sensing
Select crystal properties, axis to monitor ∆f due tospecific environmental variable, Temperature, Pressure, Field, Viscosity or Mass change due to thermal or electrochemical deposition.
Select interface layer axis to monitor ∆f due tospecific reactant, adsorbent in enviroment
Integrate heater/cooler for thermal adsorption-desorption curves.
Quartz shear resonator examples
Network RepresentationSolution to the wave equationwith open boundary conditions
Current/Voltage←Transformer→Force/Velocity
-C0
C0
1
N
ZTZT
ZS
tASε
=0Ch0CN =
DD AA cv ρρ ==0Z
Dcρωω
== DvΓ
( )2/tan0 tiZ T ΓZ=
( )tiZ S Γcsc0Z−=
Mason’s Model
Acoustic Layer NetworksSolution to the wave equation with open boundary conditions
Force = VoltageVelocity = Current
Layer ParametersL thicknessA areav velocityρ density
Frequency (kHz)
0 500 1000 1500 2000
log|
Vel
ocity
| (1
m/s
)
-6
-5
-4
-3
-2
-1
0
1
2
Rigidly Fixed Free
= F0 / mω
= F0 ωL/c33A
Combination of Piezoelectric and 1 layer
Acoustic Load Regimes
Depending on the acoustic length in the layer compared to the acoustic length of the piezoelectric. Three distinct loadingregimes can load the piezoelectric differently. These include:
• Mass loading•Elastic loading•Radiation loading
Mass Loading
ωωρωρ imLAiLAil
llll
lllL =⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
vv
vvZ tan
Elastic Loading
⎟⎟⎠
⎞⎜⎜⎝
⎛=
llllL
LAiv
vZ ωρ tan
Radiation Loading
( ) lllllll
lllL AiAiLAi vvv
vZ ρρωρ =−=⎟⎟⎠
⎞⎜⎜⎝
⎛= tan
The constants (including loss) of the piezoelectriccan be determine using curve fitting techniques
prior to the addition of the layer
Known prior to deposition by analysis of free resonator
⎟⎟⎠
⎞⎜⎜⎝
⎛=
llllL
LAiv
vZ ωρ tan
( )TSA
SATTL ZZZ
ZZZZZ
−−−−
=22
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
C
CA
ZZ
NZZ
1
2
Since ZC, ZT, ZS, N can be determined fromthe free spectra prior to deposition or attachment of the layer then we can determine the acousticload directly for the layer using
Where Z is the impedance spectra of the compositeresonator
Transform applied to mass loading - QCM
ZL
m* =ZL/iω
( ) ( )( )iff
iffiffmf
ss
ssssS
11
221102
* )(∆+
∆+−∆+=m
Complex version of Sauerbrey equation
m* theory= (8.48-1.02x10-5i) mg
m* transform = (8.48-1.03x10-5i) mg
Error Propagation in Transform
Add 0.1 % Error to the composite resonator Impedance
Error does not propagateinto fs
Error Propagation in Transform – Cont.
Add Stray capacitance to the composite resonator Impedance
Again slope is
constant about
fs
Transform applied to water load
SεDc
kh
AT Cut Quartz Resonator in air
(F/m)(N/m2)
(#)(V/m)
ρ (kg/m3)
5.45933x10-11(1 - 0.02388i)2.94472x1010(1 + 0.00012i)
0.0717744(1 - 0.00884i)1.67x109(1 + 0.02084 i)
2650t (m) 0.000333
A (m2) 4.195x10-5
m0 (kg) 3.701x10-5
fsq (Hz) (quartz) 4.99477x106 + 110.9i
fs w(Hz) (quartz + water) 4.99390x106 + 990.8i
∆m (kg) (Sauerbrey eq.) 6.45x10-9-6.52x10-9i
∆m (kg) (m*(Re(fSq )) Figure 13.
6.44x10-9-6.03x10-9i
( ) ( ) ωηωρρρωρ *2/1tan imiAAiAiLAi llllllll
lllL ===−=⎟⎟⎠
⎞⎜⎜⎝
⎛= vv
vvZ
Water Load is special case of Radiation Loading
m*= 6.44x10-9- 6.03x10-9i kg with ω= 3.138x107 rads/s, A= 4.195x10-5 m2, ρ=1000 kg/m3
From previous table
η=1.38(1+0.066i) x10-3 Ns/m2
Book Value of η=1.12 x10-3 Ns/m2 at 60 oF
complex part of η determined above is a measure of the experimental error (no correction for parasitic electrical component), surface roughness anddeviations from pure Newtonian fluid
Curing/evaporation studies
Combination of Piezoelectric and 1 layer
Zp
Also present is a parasiticelectrical impedance that shiftsthe anti-resonance frequency fa
Free SAW Resonance Equation
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛++⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
000
20
2
00
20 2sin
2tan4
2tan
2sin
2tan
22
ωπω
ωπω
ωπω
πω
ωωπω
ωπω
πω
ω NNKci
NciNKc
Y ss
s
K = coupling factorω0 = resonance frequency ∝ Rayleigh velocitycs = capacitance of IDT pairN = Number of electrode Pairs
SAW Resonator with reflectors (Simplest model BVD circuit)
L1
R1
C1
C0
L1, C1, R1 = Motional componentsC0 = Clamped capacitance
Saw Example
SAW Example (55.2 MHz)
Q = 22000
SAW Example (62.2 MHz)
Q = 27000
SAW Example (49.6 MHz) : Poor Quality resonance
Q = 1000
SAW Stability (55.2 MHz)
90 Hz over 18 hrs = 1.7 ppm
4:30 pm
10:30 am
Example of SAW Sensor - Dust
Clean SAW Resonator
SAW Resonator withLimestone Dust
Passive Saw Interrogating SensorsTable 1: SAW Properties of Common Materials (Yili Wu et al. “Principals of Surface Acoustic Waves and its application in Electronic Technology” Defense Industrial Press, PRC, 1983, (In Chinese)
Material SAW velocity v (m/s)
Coupling K2(%)
ε ( pF/m) αT(ppm/oC)
Quartz ST-X 3158 0.16 55 ≈0
LiNbO3 (YZ) 3485 4.5 460 91
LiNbO3 (128o) 3921 5.7 - 57
ZnO 2715 1 - 40
Bi12GeO20(100) (011)
1681 1.5 400 130
Correction Techniques for cross sensitivities
σσ
γγ
ββ
σγ∆
+∆
+∆
+∆
+∆
=∆
≈∆−
=∆ kk
TTk
cck
mmk
ff
vv
Tcm
v= velocityβ= propagation factorm = massc = elastic stiffnessT = Temperatureγ = Surface tensionσ = film stress
The k factors above are material parameters so chose 5 sensors with different materials, cuts, or frequencies to get cross sensitivity matrix ( 1-5 designate specific sensor) (k normalized by dividing through by initial condition) (Hietala et al. IEEE Trans UFFC 48, pp. 262-267)
[ ] [ ]
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆∆∆∆∆
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆∆∆∆∆
⇒∆=∆⇒
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆∆∆∆∆
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆∆∆∆∆
−
5
4
3
2
11
55555
44444
33333
22222
11111
55555
44444
33333
22222
11111
5
4
3
2
1
fffff
kkkkkkkkkkkkkkkkkkkkkkkkk
Tcm
xkfTcm
kkkkkkkkkkkkkkkkkkkkkkkkk
fffff
Tcm
Tcm
Tcm
Tcm
Tcm
i
Tcm
Tcm
Tcm
Tcm
Tcm
σγ
σγ
σγ
σγ
σγ
σγ
σγ
σγ
σγ
σγ
σγ
σγ
jjii Xkf ,=∆
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆
∆∆∆
nnnnnn
ijiii
nj
nj
nj
n X
XXX
kkkkkkkk
kkkkkkkkkkkkkkk
f
fff
M
K
MM3
2
1
321
321
33333231
22232221
11131211
3
2
1
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∆
∆∆∆
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡−
nnnnnn
ijiii
nj
nj
nj
n f
fff
kkkkkkkk
kkkkkkkkkkkkkkk
X
XXX
M
K
MM3
2
11
321
321
33333231
22232221
11131211
3
2
1
⇒
General Solution
Example
Passive Saw Interrogating Sensors
Transmitter/Receiver
Antennae
SAW
Reflector
eg. Bao et al. IEEE Ultrasonics 1987, pp. 583-585
Phase difference in reflected pulse is a measure of the strain due to thermal expansion
USDC Technology
Ultrasonic Actuator(Horn/Stack/Backing)
Free Mass
Drill StemRock
Device is drivenat ultrasonic frequenciesbut producessonic and ultrasonic Impacts.
Various Applications of the USDC TechnologyURATUltrasonic Rock Abrasion Tool
Smart USDC with Integrated Sensors
2 cm.
Folded Horns
Various Applications of the USDC Technology
Deep Drill
Rock Crusher
Powdering Tool
Surface Acoustic Wave (SAW) Motors
128o Y-cut LiNiO3
V up to 10 m/s
Force proportional to µN
Signal On
IDT excite a surface wave on piezoelectricsubstrate causes a mass to surf towards
the source
Piezoelectric Motors (Early Versions)
Ultrasonic Motors
Stator surface
Friction layer
Rotor ω
v =velocity of traveling wave in stator
F
τ
Elliptical motion of
contact point
-
V0cosωt V0sinωt
High Torque DensityLow RPMSelf Braking
Piezoelectric Pump
Current Specifications4-5 cc/min1100 PaNo Moving PartsPeristaltic
For More Information see
Devices -http://ndeaa.jpl.nasa.gov/
Materials –http://www.rmc.ca/academic/physics/ferroelectrics/Publications_e.html