Overview: Piezoelectricity

Manfred Kaltenbacher

Department of Sensor Technology, University Erlangen-Nuremberg

manfred@lse.eei.uni-erlangen.dewww.lse.uni-erlangen.de

Outline• Department of Sensor Technology• Basics

• Piezoelectric effect• Piezoelectric equations• Piezoelectric materials

• Finite Element Formulation• Weak form• Spatial and time discretization• Solvers for the algebraic system of equations

• Material parameter determination• Experimental methods• Invers scheme

Location

DepartmentProf. Dr.-Ing. Reinhard Lerch

SecretaryC. Salley-Sippel

Simulation/DesignPD Dr. M. Kaltenbacher

AdministrationB. Melberg

MeasurementPD Dr. M. Kaltenbacher

Dr. A. Sutor

G. Link

A. Sutor

K. BauderM. Pelz

J. StrobelL. BahrM. GüntherM. Meiler

B. Baffoun

TechnologyDr. A. Sutor

N. BretzM. Bezdek

T. HegewaldA. Streicher

U. BollertM. Escobar

C. Hahn

M. MohrT. Lahmer

Research GroupPD Dr. B. Kaltenbacher

A. Hauck

E. Leder

Overview: Piezoelectricity

• Basics

• Finite Element Formulation

• Material Parameter Determination

Piezoelectric Effect

• Direct effect (sensor)

• Indirect effect (actuator)

Governing Equations

• Constituitive equations

• Piezoelectric materials can be subdivided in the followingthree categories

• Single crystals, like quartz

• Piezoelectric ceramics like barium titanate or lead zirconate

• Polymers like PVDF (polyvinylidenfluoride)

Tensors for 6mm Crystal Class

Polarization / Strain versus Electric Field Intensity

Overview: Piezoelectricity

• Basics

• Finite Element Formulation

• Material Parameter Determination

Coupled Equations (I)

• Mechanical field (Navier‘s equation)

• Electrostatic field (Maxwell‘s equations)

Coupled Equations (II)

• Constituitive equations

• Coupled equations

• Boundary conditions

Finite Element Formulation (I)• Weak form: Find such that

for any

• Spatial approximation Nodal shape function(local support)

Finite Element Formulation (II)

• Semidiscrete Galerkin formulation:

• Introduced damping via Rayleigh model:

Finite Element Formulation (III)

• Time discretization: Newmark method

• Choice of integration parameters:• Explicit: Stability depends on mesh size, material parameters

makes just sense for piezoelectic-structure interactionexplicit/implicit splitting

• Implicit: 2nd order accurate

Algebraic Solver (I)

• Static case

• Harmonic case

Neagtiveeigenvalues

Algebraic Solver (II)

• GMRES-solver (General Minimal Residual)

• ILDL(k)-preconditioner (Incomplete LDL-decomposition with fill-in level k)

7

6

5

4

3

2

1

1

0

0no precond

k = 1k = 2k = 3k = 4k = 5

0 5 10 15 20 25 30Iteration number

1001011E-11E-21E-31E-41E-51E-61E-7

2-no

rm o

f res

idua

l

Algebraic Solver (III)

• Dependency of iteration number on problem size

10

30

50

70

90

A B C D

n

k = 1k = 2

A: 36.519 B: 77.696 C: 300.243 D: 618.403

Overview: Piezoelectricity

• Basics

• Finite Element Formulation

• Material Parameter Determination

Simplification to the one-dimensional case, direct relation betweenresonance frequencies and coefficients

State of the Art

• Test samples with special geometries: (IEEE Standard)

• Example: thickness resonantor

Identification of Material Parametersby Simulation of the Full System

Find material tensors

c , e , εfrom measured impedance

E S

Equipment for characterization of linear and nonlinear material properties

Combined measurement methods for accuratepiezoelectric material parameters

linear nonlinearelectricalimpedance

pulse response

mechanicaldisplacement

force

acousticalpulse echoe

Impedance / gain-phase analyzer

Ritec SNAP: two 5kW-pulse amplifiers with receivers

measurements

differential fibreinterferometer

500W CW RF-amplifier

100kN materialstesting machine

Measurement setupAutomated setup for impedance measurement

Contacting phase

- X-positioning of the spring contactpair Lc-Lp for contact force

- Testing contact resistance withconstant current

sample

Hc

Hp

Lc

Lplinear stage

Measuring sequence

- Contacting phase for theshort/open circuit

- Measurement for calibration

- Contacting phase for thepiezo sample

- Material measurement(V0=1Veff)

Auto balancing bridge

virtual ground

Material configuration

7 mm

7 mm32.5 mm

layers:200

material: the same as the bulk samples

not within the IEEE range because width > 5 * thickness is not fulfilled

Finite element model

Bulk samples– PZT disc (radial and thickness mode)thickness: 0.43 mm, radius: 6.35 mm– PZT strip (transversal mode)thickness: 0.43 mm, length: 11.03 mm, width: 2.44 mm

Device sample: stack actuator of a CR-injector application

12.70 mm

thickness: 0.43 mm

Results (1)

PZT disc – fitted impedance curves

2 3 4 5 6 7 8

x 106

100

101

102

|Z|

Hz

MessungApproximationAusgang

0 1 2 3 4 5 6

x 105

100

101

102

103

104

105

Hz

|Z|

MessungApproximationAusgang

measurementforward simulation with fitted material parametersforward simulation with IEEE material parametersdata for the vector of measurements

radial mode thickness mode

100

101

102

(MHz)8765432

|Z| (Ω)

100 200 300 400 500(kHz)

104

105

100

101

102

103

|Z| (Ω)

y

0.8 1 1.2 1.4 1.6 1.8 2

x 105

102

103

104

MessungApproximationAusgang

Hz

|Z|

|Z| (Ω)

104

103

102

(MHz)1.00.8 1.2 1.4 1.6 1.8

Results (2)

PZT strip – fitted impedance curves

y

transversal mode 11.03 mm

2.44 mm

thickness: 0.43 mm

measurementforward simulation with fitted material parametersforward simulation with IEEE material parametersdata for the vector of measurements

Results (3)

Stack actuator (based on results of raw material)

2 3 4 5 6 7 8

x 104

10-2

10-1

100

101

102

Frequency (Hz)

|Z| (

Ohm

)Impedance Stack, 200 Layers

Measurement

Simulation

Applications

• Medical diagnostic, nondestructive testing

Acoustic lens

Focal areaUltrasonic beam

Transducer element

Backing

Scanning direction

Matching layer

Applications

• Surface acoustic wave devices (mobile phones, television)

Puls compression filter (43 MHz)

Resonator (1 GHz)

Origin: Siemens

Sensor for wheel pressure

WLAN

Applications

• High intensity ultrasound: e.g. lithotripsy

Applications

• High intensity ultrasound: e.g. cleaning

Applications

• Stack actuators (common-rail injection systems)

Inner electrodes

Single ceramic layer

Thickness: 30-200µmNumber : till 1000 layers

Thank you foryour attention