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Piezoacoustic Transducer
Introduction
A piezoelectric transducer can be used to either 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. In this tutorial, a piezoelectric speaker, colloquially known as a “piezo buzzer”, will be analyzed.
Model Definition
A piezo buzzer is a type of piezoelectric transducer that uses a piezoelectric crystal attached to a membrane to induce bending on the membrane and thus generate acoustic pressure. This type of transducers are typically driven at the resonance frequency, where the sound is efficiently radiated.
This model simulates a buzzer intended for ultrasonic applications. The membrane and crystal are rotationally symmetric, making it possible to set up the model as a 2D axisymmetric 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 piezoelectricity and solids.
In the air domain, the Helmholtz equation, that described the distribution of pressure, is solved. 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 1 and Equation 2.
(1)
(2)
In COMSOL Multiphysics, this coupling is automatically implemented by the Piezoelectricity 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 swept from 10 kHz to 60 kHz, which partially overlaps with 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). This sweep helps identify the resonance frequency at which the transducer is likely to be excited.
Boundary Conditions
An AC electric potential of 5 V is applied to the upper surface of the crystal, and the bottom part is grounded. The crystal is attached to a brass membrane which is fixed at its outer edge.
At the interface between the air and solid domain, the normal component of the structural acceleration of the solid (brass membrane) boundary is used to drive the air domain, while the acoustic pressure at the interface 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 in the Model Builder. 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.
The Perfectly Matched Boundary condition is used on the outer surface of the air domain. It is used to model an open infinitely extended domain. The wavefronts travel outward from the geometric boundary that truncates the air domain with minimal reflection. Additionally, an Exterior Field Calculation feature is also set up on the same boundary, it is used to evaluate the pressure and sound pressure level in points exterior to the computational domain. 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 at 60 kHz.
Figure 2: Surface and height plot of the pressure distribution.
Figure 3 shows the on-axis sound pressure level at 1 m as a function of the driving frequency. The clear resonance around 33 kHz indicates that this would be the frequency that will be used to drive the transducer.
Figure 3: On-axis sound pressure level 1 m in front of the transducer.
Figure 4 shows the pressure distribution at the top surface of the brass membrane. Notice that the acoustic pressure is very small in comparison to the mechanical von Mises stress, which is plotted in Figure 5.
Figure 4: Acoustic pressure at the top of the brass membrane.
Figure 5: von Mises Stress along the air-solid interface.
The polar plot of the sound pressure level at 1 m from the transducer at different frequencies is shown in Figure 6. Note how the radiated sound pressure level is higher at the resonance frequency and how the complexity and directionality increase as the frequency increases.
Figure 6: A polar plot of the exterior-field sound pressure level at 1 m. The 0 degree axis coincides with the +z direction of the rz-plane in the 2D axisymmetric model.
The effect of the frequency on the directionality of the transducer can also be seen in Figure 7, which shows the angular width of the source. The beamwidth is defined as the angular arc in front of the transducer where the sound pressure level is above -3 dB from the value in front of the transducer.
The displacement of the membrane at the resonance frequency can be seen in Figure 8.
Figure 7: Beamwidth and null-to-null beam width as a function of the frequency.
Figure 8: Displacement of the membrane at the resonance frequency.
Application Library path: Acoustics_Module/Piezoelectric_Devices/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.
Begin by importing some parameters that will be used during the model definition.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file piezoacoustic_transducer_parameters.txt.
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.
Start by drawing the acoustic domain.
Air
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Air in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type r_air.
4
In the Sector angle text field, type 90.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
6
From the Color list, choose None or — if you are running the cross-platform desktop —Custom. On the cross-platform desktop, click the Color button.
7
Click Define custom colors.
8
Set the RGB values to 177, 221, and 255, respectively.
9
Click Add to custom colors.
10
Click Show color palette only or OK on the cross-platform desktop.
11
Click  Build Selected.
Next, add the brass membrane, which is just a rectangle.
Brass Membrane
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Brass Membrane in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type r_memb.
4
In the Height text field, type th_memb.
5
Locate the Position section. In the z text field, type -th_memb.
6
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
7
From the Color list, choose None or — if you are running the cross-platform desktop —Custom. On the cross-platform desktop, click the Color button.
8
Click Define custom colors.
9
Set the RGB values to 219, 179, and 135, respectively.
10
Click Add to custom colors.
11
Click Show color palette only or OK on the cross-platform desktop.
12
Find the Cumulative selection subsection. Click New.
13
In the New Cumulative Selection dialog, type Structural Domains in the Name text field.
14
Click OK.
15
In the Settings window for Rectangle, click  Build Selected.
Now draw the piezoelectric material.
Piezoelectric material
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Piezoelectric material in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type r_pzt.
4
In the Height text field, type th_pzt.
5
Locate the Position section. In the z text field, type -th_memb-th_pzt.
6
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
7
From the Color list, choose None or — if you are running the cross-platform desktop —Custom. On the cross-platform desktop, click the Color button.
8
Click Define custom colors.
9
Set the RGB values to 114, 119, and 122, respectively.
10
Click Add to custom colors.
11
Click Show color palette only or OK on the cross-platform desktop.
12
Find the Cumulative selection subsection. From the Contribute to list, choose Structural Domains.
13
Click  Build Selected.
14
Click the  Zoom Extents button in the Graphics toolbar.
The geometry should look like the one in Figure 1.
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
In the Settings window for Pressure Acoustics, Frequency Domain, locate the Domain Selection section.
3
From the Selection list, choose Air.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, locate the Domain Selection section.
3
From the Selection list, choose Structural Domains.
Piezoelectric Material 1
1
In the Model Builder window, under Component 1 (comp1) > Solid Mechanics (solid) click Piezoelectric Material 1.
2
In the Settings window for Piezoelectric Material, locate the Domain Selection section.
3
From the Selection list, choose Piezoelectric material.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
In the Settings window for Electrostatics, locate the Domain Selection section.
3
From the Selection list, choose Piezoelectric material.
Add Material
1
In the Materials 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 the Add to Component button in the window toolbar.
Materials
Lead Zirconate Titanate (PZT-5H) (mat1)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Piezoelectric material.
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 the Add to Component button in the window toolbar.
4
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Air (mat2)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Air.
Brass
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Brass in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Brass Membrane.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Young’s modulus
E
100[GPa]
Pa
Young’s modulus and Poisson’s ratio
Poisson’s ratio
nu
0.3
1
Young’s modulus and Poisson’s ratio
Density
rho
8900[kg/m^3]
kg/m³
Basic
Add Atmosphere attenuation with standard temperature, pressure and humidity values. This effect should be taken into account given the frequency range and the distance for far-field evaluation.
Pressure Acoustics, Frequency Domain (acpr)
Pressure Acoustics 1
1
In the Model Builder window, under Component 1 (comp1) > Pressure Acoustics, Frequency Domain (acpr) click Pressure Acoustics 1.
2
In the Settings window for Pressure Acoustics, locate the Pressure Acoustics Model section.
3
From the Fluid model list, choose Atmosphere attenuation.
4
Locate the Model Input section. In the ϕw text field, type 0.5.
Add the Perfectly Matched Boundary condition on the outer boundary of the air domain.
Perfectly Matched Boundary 1
1
In the Physics toolbar, click  Boundaries and choose Perfectly Matched Boundary.
2
Select Boundary 11 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 11 only.
3
In the Settings window for Exterior Field Calculation, locate the Exterior Field Calculation section.
4
From the Condition in the z = z0 plane list, choose Symmetric/Infinite sound hard boundary.
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).
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundary 9 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 V0.
Mesh
Proceed and generate the mesh based on the Physics-controlled mesh suggestion for Pressure Acoustics, Frequency Domain. This is done by only selecting Pressure Acoustics, Frequency Domain as Contributor and then switching to User-controlled mesh on the main mesh node. The frequency controlling the maximum element size is per default taken From study. Set the desired Frequencies in the study step. In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. In this model, use the default Automatic option, which gives 5 elements per wavelength. A Mapped mesh is then added in the structural domain to resolve the thin geometry.
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
From the Frequency unit list, choose kHz.
4
Click  Range.
5
In the Range dialog, type fmin in the Start text field.
6
In the Step text field, type fstep.
7
In the Stop text field, type fmax.
8
Click Replace.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use checkboxes for Solid Mechanics (solid), Electrostatics (es), Acoustic–Structure Boundary 1 (asb1), and Piezoelectricity 1 (pze1).
4
Locate the Sequence Type section. From the list, choose User-controlled mesh.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Structural Domains.
5
Click to expand the Reduce Element Skewness section. Select the Adjust edge mesh checkbox.
Size 1
1
Right-click Mapped 1 and choose 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.
5
Select the Maximum element size checkbox. In the associated text field, type min(th_memb,th_pzt).
Distribution 1
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundaries 7 and 9 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 2.
Proceed and move the Mapped mesh above Free Triangular in the Model Builder tree to resolve the structural domain correctly.
Mapped 1
1
In the Model Builder window, click Mapped 1.
2
Drag and drop below Size Expression 1.
3
In the Settings window for Mapped, click  Build All.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots checkbox.
4
In the Study toolbar, click  Compute.
The first plot from the Result Templates shows a surface plot of the pressure distribution. Add a height plot in order to have a plot similar to that shown in Figure 2.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
2
Go to the Result Templates window.
3
In the tree, select Study 1/Solution 1 (sol1) > Pressure Acoustics, Frequency Domain > Acoustic Pressure (acpr).
4
Click the Add Result Template button in the window toolbar.
Results
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.
The second plot from the Result Templates shows the sound pressure level in the air domain.
Result Templates
1
Go to the Result Templates window.
2
In the tree, select Study 1/Solution 1 (sol1) > Pressure Acoustics, Frequency Domain > Sound Pressure Level (acpr).
3
Click the Add Result Template button in the window toolbar.
Results
Sound Pressure Level (acpr)
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.
Result Templates
1
Go to the Result Templates window.
2
In the tree, select Study 1/Solution 1 (sol1) > Pressure Acoustics, Frequency Domain > Acoustic Pressure, 3D (acpr).
3
Click the Add Result Template button in the window toolbar.
4
In the tree, select Study 1/Solution 1 (sol1) > Pressure Acoustics, Frequency Domain > Sound Pressure Level, 3D (acpr).
5
Click the Add Result Template button in the window toolbar.
The fifth plot from the Result Templates shows the exterior-field sound pressure level. To reproduce Figure 6, you need to adjust the default settings. Note that 0 degree in the polar plot corresponds to the z-axis direction.
6
In the tree, select Study 1/Solution 1 (sol1) > Pressure Acoustics, Frequency Domain > Exterior-Field Sound Pressure Level (acpr).
7
Click the Add Result Template button in the window toolbar.
8
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Exterior-Field Sound Pressure Level - Selected Frequencies
1
In the Settings window for Polar Plot Group, type Exterior-Field Sound Pressure Level - Selected Frequencies in the Label text field.
2
Locate the Data section. From the Parameter selection (freq) list, choose Manual.
3
In the Parameter indices (1-51) text field, type 1 11 24 31 41 51.
4
In the Exterior-Field Sound Pressure Level - Selected Frequencies toolbar, click  Plot.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Sound Pressure Level - Selected Frequencies 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
Find the Evaluation distance subsection. In the Radius text field, type 1[m].
7
In the Exterior-Field Sound Pressure Level - Selected Frequencies toolbar, click  Plot.
Exterior-Field Sound Pressure Level - All Frequencies
1
In the Model Builder window, right-click Exterior-Field Sound Pressure Level - Selected Frequencies and choose Duplicate.
2
In the Settings window for Polar Plot Group, type Exterior-Field Sound Pressure Level - All Frequencies in the Label text field.
3
Locate the Data section. From the Parameter selection (freq) list, choose All.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Sound Pressure Level - All Frequencies 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 Compute beamwidth list, choose On.
4
In the Level down text field, type 3.
5
In the Exterior-Field Sound Pressure Level - All Frequencies toolbar, click  Plot.
Beamwidth
1
Go to the Beamwidth window.
2
Click the Table Graph button in the window toolbar.
Results
Table Graph 1
1
In the Settings window for Table Graph, click to expand the Legends section.
2
Select the Show legends checkbox.
Beamwidth
1
In the Model Builder window, under Results click 1D Plot Group 7.
2
In the Settings window for 1D Plot Group, type Beamwidth in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Angle (deg).
6
Locate the Legend section. From the Position list, choose Lower left.
7
In the Beamwidth toolbar, click  Plot.
Annotation 1
1
In the Model Builder window, right-click Beamwidth and choose Annotation.
2
In the Settings window for Annotation, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (sol1).
4
Locate the Position section. In the r text field, type 10.
5
Locate the Coloring and Style section. Clear the Show point checkbox.
6
In the Beamwidth toolbar, click  Plot.
The image should look like the one in Figure 7.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
Add a plot of the displacement in the piezoelectric transducer.
2
Go to the Result Templates window.
3
In the tree, select Study 1/Solution 1 (sol1) > Solid Mechanics > Displacement (solid).
4
Click the Add Result Template button in the window toolbar.
5
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Displacement (solid)
1
In the Settings window for 2D Plot Group, locate the Data section.
2
From the Parameter value (freq (kHz)) list, choose 33.
3
Locate the Color Legend section. Select the Show units checkbox.
Surface 1
1
In the Model Builder window, expand the Displacement (solid) node, then click Surface 1.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Rainbow.
4
In the Displacement (solid) toolbar, click  Plot.
The image should look like the one in Figure 8.
Next, create 1D plot groups to recreate Figure 4 and Figure 5.
Stress at the Top of the Membrane
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Stress at the Top of the Membrane in the Label text field.
3
Locate the Data section. From the Parameter selection (freq) list, choose From list.
4
In the Parameter values (freq (kHz)) list box, select 33.
5
Click to expand the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click Stress at the Top of the Membrane and choose Line Graph.
2
Select Boundary 6 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.misesGp - von Mises stress - N/m².
4
Locate the y-Axis Data section. From the Unit list, choose MPa.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type r.
7
In the Stress at the Top of the Membrane toolbar, click  Plot.
The image should look like the one in Figure 5.
Pressure at the Top of the Membrane
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pressure at the Top of the Membrane in the Label text field.
3
Locate the Data section. From the Parameter selection (freq) list, choose From list.
4
In the Parameter values (freq (kHz)) list box, select 33.
5
Locate the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click Pressure at the Top of the Membrane and choose Line Graph.
2
Select Boundary 6 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 at the Top of the Membrane toolbar, click  Plot.
The image should look like the one in Figure 4.
On-axis Sound Pressure Level at 1 m
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type On-axis Sound Pressure Level at 1 m in the Label text field.
Global 1
1
Right-click On-axis Sound Pressure Level at 1 m and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
subst(acpr.efc1.Lp_pext,r,0,z,1[m])
dB
On-axis sound pressure level at 1 m
4
Click to expand the Legends section. Clear the Show legends checkbox.
5
In the On-axis Sound Pressure Level at 1 m toolbar, click  Plot.
The image should look like the one in Figure 3.