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Piezoacoustic Transducer
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, sonar transducers, 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.
This model simulates a single crystal plate in such a structure. The element is rotationally symmetric, making it possible to set up the model as a 2D axially symmetric problem.
Figure 1: The model geometry.
physics implementation in domains
The model uses the built-in Acoustic-Piezoelectric Interaction, Frequency Domain multiphysics interface, which contains three fundamental physics interfaces: Pressure Acoustics, Solid Mechanics, and Electrostatics. The first one solves for the wave equation in the fluid media surrounding the transducer. The latter two are used to model the piezoelectric effect.
In the air domain, in absence of volumetric sound sources, and assuming the pressure varies harmonically in time, the wave equation describing the acoustic pressure distribution is:
(1)
Equation 1 is solved by the Pressure Acoustics, Frequency Domain interface.
The piezoelectric domain is made of the material PZT-5H, which is a common material in piezoelectric transducers. The piezoelectric material is modeled by solving the Solid Mechanics and Electrostatics interfaces that are coupled via linear constitutive equations that correlate stresses and strains to electric displacement and electric field. These physics interfaces solve for the balance of body forces and volume charge density respectively as shown in Equation 2 and Equation 3.
(2)
(3)
In COMSOL Multiphysics, this coupling is automatically implemented by the Piezoelectric Effect node located under the Multiphysics branch in the Model Builder.
The structural and electrical analyses are also time harmonic. For historical reasons, in structural-mechanics terminology it is called frequency response analysis, whereas in electrical engineering terminology it is called frequency domain analysis.
In this model, the excitation 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 from 20 Hz to 20 kHz).
Boundary Conditions
An AC electric potential of 100 V is applied to the upper surface of the transducer, and the bottom part is grounded. The bottom surface of the piezo transducer is assigned to a roller boundary condition which prevents it to move vertically, that is, along the z-direction.
At the interface between the air and solid domain, the normal component of the structural acceleration of the solid (piezo transducer) boundary is used to drive the air domain. This is described by the following equation:
where an is the normal acceleration.
The acoustic pressure at the interface between the air and solid domain acts as a boundary load on the solid,
The bidirectional coupling at the solid and air interface boundaries is automatically taken care of by the Acoustic-Structure boundary node located under the Multiphysics branch when you use the built-in Acoustic-Piezoelectric Interaction, Frequency Domain interface. The interface boundaries are automatically detected once you assign appropriate parts of the modeling geometry to the Pressure Acoustics, Frequency Domain and Solid Mechanics interfaces, respectively.
A Spherical Wave Radiation boundary condition is used on the outer surface of the air domain. This helps in implementing the idea that the air domain is infinitely extended in reality, and the spherical wavefronts travel outward from the geometric boundary that truncates the air domain with minimal reflection. Additionally, a Far Field Calculation is also set up on the same boundary, which helps to evaluate the pressure and sound pressure level in the far-field limit. Refer to the Acoustics Module User’s Guide for more information on these boundary conditions.
Results and Discussion
Figure 2 shows the pressure distribution in the air domain.
Figure 2: Surface and height plot of the pressure distribution.
Figure 3 shows the pressure distribution along the air-solid interface. Notice that the acoustic pressure is small in comparison to the mechanical stress, which is plotted in Figure 4.
Figure 3: Acoustic pressure at the air-solid interface.
Figure 4: von Mises Stress along the air-solid interface.
The results from the far-field analysis are shown in Figure 5. This figure demonstrates that in the far-field limit, the sound pressure level reaches a maximum right in front of the transducer.
Figure 5: A polar plot of the far-field sound pressure level. The 0 degree axis coincides with the +z-direction of the rz-plane in the 2D axisymmetric model.
Application Library path: Structural_Mechanics_Module/Acoustic-Structure_Interaction/piezoacoustic_transducer
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Acoustics>Acoustic-Structure Interaction>Acoustic-Piezoelectric Interaction, Frequency Domain.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Frequency Domain.
6
Click  Done.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Begin by drawing the acoustic domain.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 4.
4
In the Sector angle text field, type 90.
5
Click  Build Selected.
Next, add the transducer, which is just a rectangle.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Height text field, type 0.5.
4
Locate the Position section. In the z text field, type -0.5.
5
Click  Build All Objects.
6
Click the  Zoom Extents button in the Graphics toolbar.
Before adding materials, select the domains related to each physics.
Pressure Acoustics, Frequency Domain (acpr)
1
In the Model Builder window, under Component 1 (comp1) click Pressure Acoustics, Frequency Domain (acpr).
2
Select Domain 2 only.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
Select Domain 1 only.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
Select Domain 1 only.
Define the Acoustic-Structure boundary.
Multiphysics
Acoustic-Structure Boundary 1 (asb1)
1
In the Model Builder window, under Component 1 (comp1)>Multiphysics click Acoustic-Structure Boundary 1 (asb1).
2
Select Boundary 4 only.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Lead Zirconate Titanate (PZT-5H).
4
Click Add to Component in the window toolbar.
Materials
Lead Zirconate Titanate (PZT-5H) (mat1)
Select Domain 1 only.
Note that in the Piezoelectric Material Properties library, you can find more than 20 additional piezoelectric materials. 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 that is poled along the z direction in the rz-plane.
Add Material
1
Go to the Add Material window.
2
In the tree, select Built-in>Air.
3
Click Add to Component in the window toolbar.
4
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Air (mat2)
Select Domain 2 only.
Pressure Acoustics, Frequency Domain (acpr)
Add the Spherical Wave Radiation boundary condition on the outer boundary of the air domain.
1
In the Model Builder window, under Component 1 (comp1) click Pressure Acoustics, Frequency Domain (acpr).
Spherical Wave Radiation 1
1
In the Physics toolbar, click  Boundaries and choose Spherical Wave Radiation.
2
Select Boundary 7 only.
Finally, add the exterior-field calculation feature. This feature adds variables to evaluate the pressure and sound pressure level outside the computational domain.
Exterior Field Calculation 1
1
In the Physics toolbar, click  Boundaries and choose Exterior Field Calculation.
2
Select Boundary 7 only.
3
In the Settings window for Exterior Field Calculation, locate the Exterior Field Calculation section.
4
From the Condition in the z = z^0 plane list, choose Symmetric/Infinite sound hard boundary.
5
From the Type of integral list, choose Far-field integral approximation for r-> infinity.
For more information on exterior-field calculation click the Help button in the toolbar or press F1.
Solid Mechanics (solid)
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundary 2 only.
Electrostatics (es)
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 2 only.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundary 4 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type 100.
Mesh 1
It is important to use a mesh size sufficiently small to resolve the wavelength using 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 shear waves makes it somewhat more difficult to define and compute. Because this is a small model, you can afford to use a very fine mesh.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type 0.2.
Size 1
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
In the associated text field, type 0.05.
8
Click  Build All.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type 200[kHz].
4
In the Home toolbar, click  Compute.
Results
Acoustic Pressure (acpr)
The first default plot shows a surface plot of the pressure distribution. Change the color table and add a height plot in order to have a plot similar to that shown in Figure 2.
Height Expression 1
1
In the Model Builder window, expand the Acoustic Pressure (acpr) node.
2
Right-click Surface 1 and choose Height Expression.
3
Click the  Zoom Extents button in the Graphics toolbar.
Sound Pressure Level (acpr)
The second default plot shows the sound pressure level in the air domain.
1
Click the  Zoom Extents button in the Graphics toolbar.
The third and fourth plots are the 3D revolved plots of the acoustic pressure and the sound pressure level.
Exterior-Field Sound Pressure Level (acpr)
The fifth default plot shows the exterior-field sound pressure level. To reproduce Figure 5, you need to adjust the default settings. Note that 0 degree in the polar plot corresponds to the z-axis direction.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Sound Pressure Level (acpr) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the φ start text field, type -90.
5
In the φ range text field, type 180.
6
In the Exterior-Field Sound Pressure Level (acpr) toolbar, click  Plot.
Exterior-Field Pressure (acpr)
Next, do the same with the sixth default plot, which is the exterior-field pressure.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Pressure (acpr) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the φ start text field, type -90.
5
In the φ range text field, type 180.
6
In the Exterior-Field Pressure (acpr) toolbar, click  Plot.
Add a plot of the displacement in the piezoelectric transducer.
Displacement
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Displacement in the Label text field.
Surface 1
1
Right-click Displacement and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Solid Mechanics>Displacement>solid.disp - Displacement magnitude - m.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Displacement toolbar, click  Plot.
Next, create 1D plot groups to recreate Figure 3 and Figure 4.
von Mises Stress
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type von Mises Stress in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click von Mises Stress and choose Line Graph.
2
Select Boundary 4 only.
3
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Solid Mechanics>Stress>solid.mises - von Mises stress - N/m².
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type r.
6
In the von Mises Stress toolbar, click  Plot.
Pressure
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pressure in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click Pressure and choose Line Graph.
2
Select Boundary 4 only.
3
In the Settings window for Line Graph, locate the x-Axis Data section.
4
From the Parameter list, choose Expression.
5
In the Expression text field, type r.
6
In the Pressure toolbar, click  Plot.