Piezoelectric Transducer

NTU50235100

PIEZOELECTRIC TRANSDUCERS 11/29

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NTU50235100

•Introduction •Electrostatics •Piezoelectricity •Signal conditioning Pierre and Jacques Curie (1880)

•Applications 11/29

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Introduction •1880 Pierre and Jacques Curie: discovered the piezoelectric effect - mechanical stress induces surface electric charges •1881 Lippman: predicted the converse piezoelectric effect - electric field induces mechanical deformation •Lord Kelvin, Pockels, Duhem, Voigt, R. E. Gibbs, Max Born: established theories and models •Langevin: device for detecting submarines in World War I •1917 A. M. Nicolson: loud speakers, microphones, phonograph pickups, crystal oscillator •1921 Cady: Quartz crystal oscillators (GT cut) •1942 W. P. Mason: frequency filters •1950s~1960s R. D. Mindlin, H. F. Tiersten: vibration of piezoelectric plates 11/29

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Eve

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• Comparison of sensing principles

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Principle

Strain Sensitivity (V/µ*)

Threshold (µ*)

Span to threshold ratio

Piezoelectric

5.0

0.00001

100.000.000

Piezoresistive

0.0001

0.0001

2.500.000

Inductive

0.001

0.0005

2.000.000

Capacitive

0.005

0.0001

750.000

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Applications High-voltage sources Spark source Transformer

Sensors Microphone Contact microphone Microbalance Accelerometer Hydrophone

Actuators Loudspeaker Ultrasonic Acousto-optic modulator Inkjet head Fuel injector

Frequency standard Quartz resonator 11/29

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Electrostatics •Electric field

F: force q: charge E: electric field

F = qE

- Coulomb’s law F = kq1q2

x1 − x 2

3

x1 − x 2

- Electric field of a point charge E = kq1

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x − x1 x − x1

3

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- Electric field due to a charge density E = k ∫ ρ (x′)

x − x′ x − x′

3

ρ ( x)

dx ′

N ⋅ m2 ⎞ k≡ = 10 c = 8.988 × 10 ⎜ 2⎟ 4πε 0 ⎝ Coul ⎠ 1

9⎛

−7 2

ε 0 = 8.8542 × 10

−12 ⎛

Coul 2 ⎞ ⎜ ⎟ : dielectric constant in vacuum ⎝ N ⋅ m2 ⎠

- Gauss’s law ∫ E ⋅ nds =

1

ε0

∇⋅E =

∫V ρ (x)dx

ρ ε0

- Scalar potential E = −∇ϕ

,

ϕ (x) =

1

ρ (x′)

4πε 0 ∫ x − x ′

dx ′

,

∇×E=0

ϕ : electrostatic potential 11/29

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- Poisson and Laplace equations ∇⋅E =

ρ ε0

∇ ⋅ ( −∇ϕ ) = ∇ 2ϕ = −

ρ ε0

In regions of space where there is no charge density

ρ : Poisson equation ε0

∇ 2ϕ = 0 : Laplace equation

- Boundary conditions Dirichlet B.C.

ϕ

Neumann B.C.

∂ϕ specified ∂n

specified

- Discontinuities in the field and potential ( E2 − E1 ) ⋅ n =

σ ε0

;

ϕ1 = ϕ 2

σ : surface - charge density ( Coul/m2 ) 11/29

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- Conductors Equipotential region, equipotential surface E1 = 0 ; E 2 = −∇ϕ ∇ϕ ⋅ n =

∂ϕ ∂ϕ σ ; =− ∂n ∂n ε0

- Parallel-plate condenser ϕ1 − ϕ 2 = V , V : " voltage" Qd σ V = Ed = d = A ε0 ε0 Q Aε 0 C=

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V

=

d

Coul ⎞ ⎛ , C: capatance ⎜ Farad= ⎟ ⎝ Volt ⎠

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- Electric dipole p = qd : dipole moment p=qd , d: points from − q to +q

ϕ (R ) =

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1 p⋅R : dipole potential 4πε 0 R 3

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- Potential for lump of charges Far field approximation, di << R

ϕ (R ) =

1 ⎛ Q p⋅R ⎞ ⎜ + 3 +L⎟ ⎠ 4πε 0 ⎝ R R

Q = ∑ qi ; p = ∑ qi di

For a neutral object, Q = 0

ϕ (R ) =

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1 p⋅R : dipole potential 4πε 0 R 3

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- Dielectrics V= C=

σ ( d − b) ε0 ε0 A

d (1 − b / d )

The capacitance is increased.

Faraday discovered that the capacitance ⇒ is increased when an insulator is put between the plates.

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The net charge inside the surface must be lower than it woiuld be without the dielectric.

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- The polarization vector P Q=0 pi ≠ 0

Q=0 p=0

∑ pi = 0

Q=0 pi ≠ 0

Q=0 p≠0

∑ pi Polarization P = Δv

∑ pi ≠ 0

: dipole moment per unit volume

If the field is not too enormous P ∝ E 11/29

(or Pi = aij E j for anisotropic materials) 周元昉

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- Polarization charges p=ql

: dipole moment per molecule

l : effective movement of positive charges w.r.t. negative charges

q: number of positive charges per molecule number of molecules N= unit volume Nq l ⋅ d S = N p ⋅ d S = P ⋅ d S : positive charges leave through dS

1. Surface polarization charge density

σp =

P ⋅ dS = n ⋅ P : surface polarization charge density dS

On the interface of two materials

σ p = n ⋅ ( P1 − P2 ) 11/29

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2. Polarization charge density

∫ P ⋅ dS

: total positive charges leave volume V = net negative charges in volume V = − ∫V ρ p dV

ρ p : polarization charge density

∫ P ⋅ d S + ∫V ρ p dV = 0 ∫V (∇ ⋅ P + ρ p )dV = 0

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ρ p = −∇ ⋅ P

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- Inside dielectric ∇⋅E =

ρ f + ρp ε0

,

ρ f : free charge density

ε 0∇ ⋅ E = ρ f + ρ p = ρ f − ∇ ⋅ P or

∇ ⋅ ( ε 0E + P ) = ρ f

Define “electric displacement” D = ε 0E + P

Governing equations ∇⋅D = ρf

and

∇×E=0 11/29

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- Constitutive equation Isotropic materials

Anisotropic materials

P = χ e ε 0E

Pi = ε 0 χij E j

D = (1 + χ e )ε 0E = ε r ε 0E = ε E

Di = ε 0 (δ ij + χ ij ) E j = ε ij E j

χe ε0 εr ε

: susceptibility : permittivity of empty space : dielectric constant (relative permittivity) : permittivity

Di = ε ij E j = − ε ijϕ , j

Governing equation Di ,i = ρ f

εijϕ , ji = − ρ f

- Boundary conditions σ f +σp ( E 2 − E1 ) ⋅ n = ε0 ε 0 ( E 2 − E1 ) ⋅ n = σ f + (P1 − P2 ) ⋅ n (D 2 − D1 ) ⋅ n = σ f

ε1 11/29

∂ϕ1 ∂ϕ − ε2 2 = σ f ∂n ∂n

Boundary condition ( Di ni ) 2 − ( Di ni )1 = σ f (εijϕ , j ni )1 − (εijϕ , j ni ) 2 = σ f

and and

ϕ1 = ϕ 2

ϕ1 = ϕ 2 周元昉

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Piezoelectricity

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• Quartz

Photo Copyright © 2000 by John H. Betts.

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Potential is created Quartz in rest state

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Force is applied

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Temperature effect

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•Ferroelectric material: PZT PbTiO3, PbZrO3

PZT unit cell above the Curie temperature

PZT unit cell below the Curie temperature

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Poling of piezoelectric ceramic

Electric dipoles in domains: (1) unpoled ferroelectric ceramic, (2) during and (3) after poling 11/29

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• Constitutive equations E Tij = cijkl Skl − ekij E k

Tij : stress

Di = eikl Skl + εijS E j

Skl : strain

• Alternate forms of constitutive equations E D Sij = sijkl Tkl + d kij E k Sij = sijkl Tkl + g kij Dk Di = dikl Tkl + εikT E k

Ei = − gikl Tkl + βikT Dk

D Tij = cijkl Skl − hkij Dk

Ei = − hikl Skl + βikS Dk

• Governing equations Tij , i = ρ u&&j Di ,i = 0

⇒

E cijkl uk ,li + ekijϕ ,ki = ρ u&& j

ekij ui , jk − εijSϕ ,ij = 0

• Boundary conditions For a surface of discontinuity ni TijI = ni TijII u Ij = u IIj ni DiI = ni DiII 11/29

ϕ I = ϕ II

ni Tij = 0 for a traction free surface uj = 0 for a fixed surface ni Di = 0 at an air-dielectric interface ϕ =0 short-circuited electrodes 周元昉

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Material constants for PZT

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Signal Conditioning • Piezoelectric transducers

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g33 ≡

e /t field produced in direction 3 = 0 stress applied in direction 3 f i / ( wl )

d 33 ≡

charge generated in direction 3 Q = fi force applied in direction 3

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Charge generated by the crystal q = Kq xi

,

xi :deflection

Current generated by the crystal dq dx icr = = Kq i dt dt icr = iC + i R

eo = eC

iC dt ∫ (icr − iR ) dt ∫ = = C

C

de dx e C ⎛⎜ o ⎞⎟ = icr − iR = Kq ⎛⎜ i ⎞⎟ − o ⎝ dt ⎠ ⎝ dt ⎠ R

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Frequency response function xi = X i exp(iω t ) eo = Vo exp(iω t )

( K q / C )i ω iτω ( Kq / C ) V0 = = X i iω + (1 / CR ) iτω + 1

τ = CR Step response xi = A for 0 < t < T xi = 0 for T < t eo = eo =

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Kq A

Kq A C

C

exp( − t / τ )

0< t < T

[exp( − T / τ ) − 1]exp[ − ( t − T ) / τ ]

T< t

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Use series resistor - sacrifices sensitivity

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• Charge Amplifiers

Output voltage change

− Vc ΔC Δvo = Cf or

Sensor

− ΔQ Δvo = Cf Lower cutoff frequency (-3dB)

f cp1

1 = 2πR f C f

Upper cutoff frequency (-3dB)

f cp 2 11/29

1 = 2πR1C 周元昉

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Example 1: Piezoelectric Transducer Charge Amplifier

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Spice simulation 1

Sensor

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Spice simulation 2

Sensor

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Example 2

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Example 3

Alessandro Gandelli & Roberto Ottoboni, 1993 11/29

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• Impedance converter

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Applications • Piezoelectric accelerometers

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Model (Single axis or triaxial)

Single axis linear

Range

g

±2000

Sensitivity

pC/g

-10.000

Frequency Range

Hz

5...10000

Resolution, Threshold

mgrms

1

Transverse Sensitivity

%

1.5

Non linearity

% FSO

±1

Shock

g

5000

Temp. coef. of sensitivity

%/°C

0.13

Operating temperature range

°C

-70...250

Housing/Base

stainless steel

Sealing

hermetic (IP68)

Ground isolation

No

Mass

g

Connector

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14.5 10-32 neg.

Diameter

mm

16

Height

mm

12.19

Mounting

stud/wax

Mounting thread

10-32 UNF x 3,3 周元昉

NTU50235100

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• Quartz torque sensor The torque sensor consists of two steel disks, between which a ring is fitted which contains several shear-sensitive quartz plates. The crystal axes of the quartz plates are oriented tangentially to the peripheral direction and therefore yield a charge exactly proportional to the applied torque.

Application examples • Adjusting torques of pneumatic screw-drivers • Testing of screw connections • Calibration measurements of manual torque wrenches • Testing torsion of springs • Measurements of friction clutches • Measuring starting torques, variations in synchronization and torsional vibrations of fractional horsepower and stepping motors. • Testing of rotary switch 11/29

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The torque sensor must be mounted under elastic preload as the torque must be transmitted by static friction onto the front parts of the sensor.

Testing of rotary switch 11/29

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• Quartz Load Washers The force to be measured acts through the cover and base of the tightly welded steel housing on the quartz sensing elements. Quartz yields an electric charge proportional to the mechanical load. Application examples • forces in spot welding • forces in presses • force variations in bolted connections under high static preload • shock and fatigue resistance • cutting and forming forces • forces in railroad brakes • impact forces

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Mounting The load washers must be installed between two plane-parallel, rigid and fine-machined (preferably ground) faces. This is necessary to achieve a good load distribution on one hand and a wide frequency response on the other hand. The load washers should always be installed under preload.

Preloading screw

Centering clip

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• 3-Component Quartz Crash Force Elements

Specification Measuring Range

Fx

kN

0...500

Fy, Fz

kN

±100

% FSO

<±0.5

fnx

kHz

≈4

fny, fnz

kHz

≈1.7

°C

0...50

Non linearity Natural Frequency

Operating temperature range Sealing

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IP65

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Application

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•

Piezoelectric pressure sensors Specification Type 6005 Measuring range

bar

0...1000

Overload

bar

1500.0

Sensitivity

pC/bar

10

Natural Frequency

kHz

≈140

Non linearity

% FSO

<±0.8

Operating temperature range

°C

-196...200

Acceleration sensitivity

bar/g

<0.001

Thread

Without thread

Cooling

not cooled

Diameter

mm

5.5

Length

mm

6

Connector

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M4x0,35 neg.

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• Dynamometer - Cutting force measurement

Calibration

calibrated Fx, Fy Fz

kN kN

±20 ±30

Mz

N·m

±1100

1/min

max.5000

Fx, Fy Fz

mV/N mV/N

≈-0.5 ≈-0.33

Mz

mV/N· m kHz

≈9

°C

0...60

mm

156

Height

mm

55

Mass

kg

7.5

Measuring Range

Speed Sensitivity

Natural Frequency Operating temperature range Diameter

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≈1

Connection

Non-contacting

Sealing

welded/epoxy (IP67)

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• Torque

Wheel-Sensor

Model

Piezoelectric

Measuring Range

My

kN·m

±3

Natural Frequency

fny

kHz

1.1 Natural Frequency fny: free

Mass

kg

4.4

Maximum r.p.m.

1/min

2200 Max. speed ≈250 km/h

Crosstalk

Fy → My

N·m/kN

<±2

Offset/Variation

Fz → My

N·m/kN

<±2

Non linearity

% FSO

<±1

Diameter

mm

289

Hysteresis

% FSO

≤1

Sealing

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IP65

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• Hydrophone

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