piezoacoustic transducer

8/16/2019 Piezoacoustic Transducer

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Solved with COMSOL Multiphysics 4.0a. © COPYRIGHT 2010 COMSOL AB.

P I E Z O A C O U S T I C TR A N S D U C E R | 1

P i e z o a c ou s t i c T r a n s du c e r

Introduction

A piezoelectric transducer can be used either to transform an electric current to an

acoustic pressure field or, the opposite, to produce an electric current from an acoustic

field. These devices are generally useful for applications that require the generation of

sound in air and liquids. Examples of such applications include phased array

microphones, ultrasound equipment, inkjet droplet actuators, drug discovery, sonartransducers, bioimaging, and acousto-biotherapeutics.

Model Definition

In a phased-array microphone, the piezoelectric crystal plate fits into the structure

through a series of stacked layers that are divided into rows. The space between these

layers is referred to as the kerf , and the rows are repeated with a periodicity, or pitch .


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2 | P I E Z O A C O U S T I C TR A N S D U C E R

This model simulates a single crystal plate in such a structure. The element is

rotationally symmetric, making it possible to set up the model in axially symmetric 2D.

Figure 1: The model geometry.

In the air domain, the wave equation describes the pressure distribution:

(1)

For this model, assume that the pressure varies harmonically in time as

Hence Equation 1 simplifies to

(2)

Because there are no sources present, Equation 2 simplifies further to

1

ρ0 cs2

--------------

t2

2

∂

∂ p ∇ 1ρ0------– ∇ p q–( )⎝ ⎠

⎛ ⎞⋅+ Q=

p x t,( ) p x( ) eiω t

=

∇1

ρ0------– ∇ p q–( )

⎝ ⎠⎛ ⎞⋅

ω 2 p

ρ0 cs2

--------------– Q=


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The piezoelectric domain is made of the crystal PZT5-H, which is a common material

in piezoelectric transducers. The structural analysis is also time harmonic although, for

historical reasons, in structural-mechanics terminology it is a frequency response

analysis.

The frequency is set to 200 kHz, which is in the ultrasonic range (dolphins and bats,

for example, communicate in the range of 20 Hz to 150 kHz, while humans can only

hear frequencies in the range 20 Hz to 20 kHz).

B O U N D A R Y C O N D I T I O N S

An AC electric potential of 100 V is applied to the upper part of the transducer, while

the bottom part is grounded. At the interface between the air and solid domain, the

boundary condition for the acoustics application mode is that the pressure is equal to

the normal acceleration of the solid domain

where an is the normal acceleration.

This drives the pressure in the air domain. The solid domain is on the other hand

subjected to the acoustic pressure changes in the air domain. Because of the high

voltage applied to the transducer, this load is probably negligible in comparison. Yet

because the model is in 2D, it is possible to include this load and solve the full model

simultaneously on any computer.

∇1

ρ0------– ∇ p( )⎝ ⎠

⎛ ⎞⋅ ω

2 p

ρ0cs

2--------------– 0=

n1

ρ0------ ∇ p( )⎝ ⎠

⎛ ⎞⋅ an=


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Results and Discussion

Figure 2 shows the pressure distribution in the air domain. This plot clearly shows howthe PML (perfectly matched layer) absorbs the wave effectively.

Figure 2: Surface and height plot of the pressure distribution.

Figure 3 shows the pressure distribution along the air-solid interface. The acousticpressure load is small in comparison to the electrical load, which is plotted in Figure 4.


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Figure 3: Acoustic pressure at the air-solid interface.

Figure 4: von Mises Stress along the air-solid interface.


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The results from a far-field analysis appear in Figure 5. This figure shows that the

sound pressure level reaches a maximum right in front of the transducer.

Figure 5: The far-field sound pressure level.

Model Library path: Acoustics_Module/Tutorial_Models/piezoacoustic_transducer

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click the 2D axisymmetric button.

3 Click Next.

4 In the Add Physics tree, select Structural Mechanics>Piezoelectric Devices (pzd).

5 Click Add Selected.

6 In the Add Physics tree, select Acoustics>Pressure Acoustics (acpr).


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7 Click Add Selected.

8 Click Next.

9 In the Studies tree, select Preset Studies for Selected Physics>Frequency Domain.

10 Click Finish.

G E O M E T R Y 1

1 In the Model Builder window, click Model 1>Geometry 1.

2 Go to the Settings window for Geometry.

3Locate the Geometry Settings section. Find the Units subsection. From the Lengthunit list, select mm.

Begin by drawing the acoustic domain, consisting of air and a PML.

Circle 1

1 Right-click Model 1>Geometry 1 and choose Circle.

2 Go to the Settings window for Circle.

3 Locate the Size section. In the Radius edit field, type 4.

Square 1

1 In the Model Builder window, right-click Geometry 1 and choose Square.

2 Go to the Settings window for Square.

3 Locate the Size section. In the Side length edit field, type 4.

Intersection 1

1 In the Model Builder window, right-click Geometry 1 and choose Boolean

Operations>Intersection.

2 Select the objects c1 and sq1 only.

3 In the Model Builder window, right-click Geometry 1 and choose Build All.

Copy 1

1 Right-click Geometry 1 and choose Transforms>Copy.

2 Select the object int1 only.

Scale 1

1 In the Model Builder window, right-click Geometry 1 and choose Transforms>Scale.

2 Select the object int1 only.

3 Go to the Settings window for Scale.

4 Locate the Scale Factor section. In the Factor edit field, type 1.5.


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5 Click the Build All button.

Next, add the transducer, which is just a rectangle.

Rectangle 1

1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Height edit field, type 0.5.

4 Locate the Position section. In the z edit field, type -0.5.

5 Click the Build All button.

6 Click the Zoom Extents button on the Graphics toolbar.

M A T E R I A L S

1 In the Model Builder window, right-click Model 1>Materials and choose Open Material

Browser .

2 Go to the Material Browser window.

3 Locate the Materials section. In the Materials tree, select Built-In>Lead ZirconateTitanate (PZT-5H).

4 Right-click and choose Add Material to Model from the menu.

Lead Zirconate Titanate (PZT-5H)

1 In the Model Builder window, click Lead Zirconate Titanate (PZT-5H).

2 Go to the Settings window for Material.

3 Locate the Geometric Scope section. Click Clear Selection.4 Select Domain 1 only.

In the Piezoelectric Material Properties library, you can find more than 20

additional piezoelectric material properties.

For a piezoelectric material, you can specify the orientation by defining and selecting

a new coordinate system. In this model, you will use the default Global coordinate

system, which gives you a material oriented in the rz-plane.

5 In the Model Builder window, right-click Materials and choose Open Material Browser .

6 Go to the Material Browser window.

7 Locate the Materials section. In the Materials tree, select Built-In>Air .

8 Right-click and choose Add Material to Model from the menu.


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Air

1 In the Model Builder window, click Air .

2 Select Domains 2 and 3 only.

P I E Z O E L E C T R I C D E V I C E S

You have two interfaces in which to specify the physics. Begin with Piezoelectric

Devices.

1 In the Model Builder window, click Model 1>Piezoelectric Devices.

2 Select Domain 1 only.

Roller 1

1 Right-click Model 1>Piezoelectric Devices and choose Structural>Roller .

2 Select Boundary 2 only.

Ground 1

1 In the Model Builder window, right-click Piezoelectric Devices and choose

Electrical>Ground.


Electric Potential 1


Electrical>Electric Potential.


3 Go to the Settings window for Electric Potential.

4 Locate the Electric Potential section. In the V 0 edit field, type 100.

Boundary Load 1


Structural>Boundary Load.


3 Go to the Settings window for Boundary Load.


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4 Locate the Force section. Specify the F vector as

Here, p is the name of the dependent variable for pressure in the Pressure Acoustics

interface. The pressure acts from the air toward the piezo domain (in the negative

z direction), which explains the minus sign.

P R E S S U R E A C O U S T I C S

1 In the Model Builder window, click Model 1>Pressure Acoustics.

2 Go to the Settings window for Pressure Acoustics.

3 Locate the Domains section. Click Clear Selection.

4 Select Domains 2 and 3 only.

Recall that you selected only Domain 1 for the Piezoelectric Devices interface, and

now Domains 2 and 3 for the Pressure Acoustics interface. Each domain is used in

one, and only one, of the interfaces.

Perfectly Matched Layers 1

1 Right-click Model 1>Pressure Acoustics and choose Perfectly Matched Layers.


Normal Acceleration 1

1 In the Model Builder window, right-click Pressure Acoustics and choose Normal

Acceleration.2 Select Boundary 4 only.

3 Go to the Settings window for Normal Acceleration.

4 Locate the Normal Acceleration section. In the an edit field, type pzd.u_ttZ.

Here, pzd.u_ttZ is the name of the second-order time derivative of the structural

displacement. If you look at its definition in the frequency domain, you can see that

it amounts to the z-displacement multiplied by -omega^2.

M E S H 1

It is important to use a mesh size sufficiently small to resolve the wavelength, by at least

5–6 elements per wavelength. At 200 kHz, the wavelength in air is 1.7 mm. In the

piezo material, the presence of both pressure and sheer waves makes it somewhat more

difficult to define and compute. In any case, this is a small model and you can afford

to use a very fine mesh.

0 r-p z


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Size

1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Triangular .

2 Go to the Settings window for Size.

3 Click to expand the Element Size Parameters section.

4 In the Maximum element size edit field, type 0.2.

Size 1

1 In the Model Builder window, right-click Free Triangular 1 and choose Size.


3 Locate the Geometric Scope section. From the Geometric entity level list, select

Domain.



6 Click to expand the section.

7 Locate the Element Size Parameters section. In the Maximum element size edit field,

type 0.05.8 Click the Build All button.

S T U D Y 1

Step 1: Frequency Domain

1 In the Model Builder window, expand the Study 1 node, then click Step 1: Frequency

Domain.

2 Go to the Settings window for Frequency Domain.

3 Locate the Study Settings section. In the Frequencies edit field, type 200[kHz].

4 In the Model Builder window, right-click Study 1 and choose Compute.


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R E S U L T S

2D Plot Group 1

The default plot shows the radial displacement in the transducer. Follow the

instructions below to show the pressure distribution and reproduce Figure 2.

1 In the Model Builder window, expand the 2D Plot Group 1 node, then click Surface 1.

2 Go to the Settings window for Surface.

3 In the upper-right corner of the Expression section, click Replace Expression.

4 From the menu, choose Pressure Acoustics>Total acoustic pressure field (acpr.p_t).

5 Click the Plot button.

6 Right-click Surface 1 and choose Height Expression.

Next, create 1D plot groups to recreate Figure 3 and Figure 4.

1D Plot Group 2

1 Right-click Results and choose 1D Plot Group.

2 Go to the Settings window for 1D Plot Group.

3 Locate the Plot Settings section. Select the Title check box.

4 In the associated edit field, type von Mises stress.


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5 In the x-axis label edit field, type r (mm).

6 In the y-axis label edit field, type Stress (Pa).

7 Right-click Results>1D Plot Group 2 and choose Line Graph.

8 Go to the Settings window for Line Graph.

9 In the upper-right corner of the Selection section, click Activate Selection.


11 In the upper-right corner of the Y-Axis Data section, click Replace Expression.

12 From the menu, choose Piezoelectric Devices>von Mises stress (pzd.mises).

13 In the upper-right corner of the X-Axis Data section, click Replace Expression.

14 From the menu, choose Geometry and Mesh>Coordinate>r-Coordinate (r).


1D Plot Group 3

1 In the Model Builder window, right-click Results and choose 1D Plot Group.


3 Locate the Plot Settings section. Select the Title check box.

4 In the associated edit field, type Pressure.

5 In the x-axis label edit field, type r (mm).

6 In the y-axis label edit field, type p (Pa).



9 In the upper-right corner of the Selection section, click Activate Selection.



12 From the menu, choose Pressure Acoustics>Total acoustic pressure field (acpr.p_t).

13 In the upper-right corner of the X-Axis Data section, click Replace Expression.

14 From the menu, choose r-Coordinate (r).


To reproduce Figure 5, you need to add a far field calculation.


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P R E S S U R E A C O U S T I C S

Far-Field Calculation 1

1 In the Model Builder window, right-click Model 1>Pressure Acoustics and choose

Far-Field Calculation.


3 Go to the Settings window for Far-Field Calculation.

4 Locate the Far-Field Calculation section. Select the Symmetry in the z=0 plane check

box.

S T U D Y 1

Update the solution to make the far-field postprocessing variable you just defined

available.

Solver 1

1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1

node.

2 Right-click Solver 1 and choose Solution>Update.

R E S U L T S

1D Plot Group 4

1 In the Model Builder window, right-click Results and choose 1D Plot Group.


3Locate the Plot Settings section. Select the Title check box.

4 In the associated edit field, type Far field sound pressure level.

5 In the x-axis label edit field, type Polar angle (deg).

6 In the y-axis label edit field, type Sound pressure level (dB).



9 In the upper-right corner of the Selection section, click Activate Selection.10 Select Boundary 9 only.


12 From the menu, choose Pressure Acoustics>Far-field variable (dB) (acpr.Lp_pfar).

13 Locate the X-Axis Data section. From the Parameter list, select Expression.


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14 In the Expression edit field, type 180/pi*atan2(z,r).

The atan2 operator returns the arctangent of the ratio between the first and the

second argument.



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