International ARENA Workshop at Newcastle

Karsten Salomon, June 2006

Measurements and Simulation Studies of Piezoceramics for Acoustic Detection

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Motivation

• Development and simulation of calibration sources for acoustic detection

• Microscopic understanding of acoustic sensors

– The Equation of Piezos

• Macroscopic understanding of acoustic sensors

– Impedance– Displacement– Pressure Amplitude– Direction Characteristics

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Microscopic Description:The Equation of Piezos

• Equation of motion of piezos is complicated - coupled Partial Differential Equations (PDE) of an anisotropic material:– Hooke’s law + electrical coupling– Gauss law + mechanical coupling

– T,S,E,D and u are stress, strain, electric field, electric displacement and mechanical displacement.

– c,e,rho and epsilon are material specific parameters

• Finite Element Method chosen to solve these PDE– Programs used: CAPA and Femlab

0

2

2

klkljkjkjjj

kkijklijkljijj

SeED

udt

dEeScT

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Simple Macroscopic Model:Equivalent Circuit Diagram

• Resonances and antiresonances of a piezo can be described with an equivalent circuit diagram:

• L,C and R are equivalent to mass, stiffness and damping

• With these parameters one gets a simplified piezo model with:– mechanical Force ~ Voltage– mech. Velocity ~ Current

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Impedance of the Piezo: Simulation and Measurement

• Important to know Nyquist noise of piezo

• Important for preamplifier design

• Measured with a capacitor in series– Impedance can be

calculated from the Voltage response at capacitor

– -> talk: C. Naumann

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Displacement of the Piezo : Simulation and Measurement

• Direct measurement of displacment with custom-built fibre coupled interferometer

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Displacement of the Piezo : Simulation and Measurement

• Simulation and Measurement in very good agreement• Averaged displacement over surface can be derived from

equivalent circuit diagram

FE Simulation

Equiv. Circuit

0.

1nm

1nm

10

nm 20 40 60 80 100kHz

displacement averaged over surface

20 40 60 80 100kHz

0.

1nm

1nm

10

nm

FE SimulationMeasurement

displacement of disc centre

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Simulation of Piezo with Coupling to Water

• Problems:– low Frequencies:

• Radius of simulated volume larger than 2 !• FE size smaller than 1/4 of piezo -> MEMORY!!!

– High frequencies• Finite Element size smaller than 0.5 -> MEMORY!!!

• Solutions:• Solve Kirchhoff integration formula

– one must know solution for all previous times– time and MEMORY consuming.

• Add damping elements.• Add non reflecting boundary condition.

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Non Reflecting Boundary conditions 1D

• 1D: exact solution!

• Wave equation

• Boundary Condition

0

uxt

02

2

2

2

uxt

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Non Reflecting Boundary conditions 1D

u=0

Non ref. Boundary condition

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Non Reflecting Boundary conditions 2D

• To supress reflections with angle ”” to the boundary apply:

• Higdon Math. Comput. 47 (1986)

• Far away from piezo sound field looks like point source.– ->

Water boundary as a sphere around piezo with Non reflecting boundary condition of 1st order

uxtxtp

1cos...cos

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Impedance and Displacement with coupling

• Boundary for water: second time deriv. of displacement

• Boundary for piezo: pressure couples back to piezo

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Pressure Amplitude

~f2

• Mean calculated pressure at face of piezo

– Calculated with Femlab

– Exept one factor calculated from equivalent circuit diagram

• Pressure amplitude at certain distance drops with 1/r

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Direction characteristics

• Different for different frequencies.• Take mean pressure in size of

receiver. • Example: pressure field at 20kHz• Cannot be derived from

equivalent circuit diagram

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Direction characteristics

• Example: pressure field at 100kHz

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Summary

– Simulation in good agreement with measurement of piezos

– Signal propagation in water applied to simulation

– Simple Macroscopic equivalent circuit diagram in agreement with microscopic behavior

– Relation between displacement and impedance of piezo shown

K. Salomon, Universität Erlangen-Nürnberg International ARENA Workshop, June 2006

Thanks for your attention