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Wax Guard Acoustics:
Transfer Matrix Computation
Introduction
In this tutorial, the acoustic properties of a wax guard are analyzed. A wax guard is a small perforated mesh used to protect the receiver (the miniature loudspeaker in a hearing aid) used for receiver-in-the-ear (RITE) or receiver-in-canal (RIC) hearing aids. Because of the very small dimensions of the structure the thermal and viscous boundary layer losses need to be included in detail, and therefore the Thermoviscous Acoustics, Frequency Domain interface is used.
In the first step, the transfer matrix (or two-port) of the wax guard is computed using the Port Sweep functionality and the Port boundary conditions. The wax guard geometry is imported from a CAD file and prepared for simulation.
In the second step, the response of the wax guard subsystem, when placed in a typical measurement setup, is computed and compared to actual measurements. This is done using the lumped transfer matrix approach. The computed wax guard transfer matrix is used together with other transfer matrix components for a receiver (miniature loudspeaker), a narrow pipe, and a coupler volume.
Note: In this model, the NanoCare wax guard CAD geometry, receiver transfer matrix data, coupler transfer matrix data, microphone impedance data, and measurement data are copyright by Widex1.
Model Definition
Wax Guards
A wax guard is a small replaceable protective mesh that is use for receiver-in-the-ear (RITE) or receiver-in.canal (RIC) hearing aids. The mesh is placed in a small structure that can be removed and replaced using a custom tool. An illustration of the location of the wax guard in the micro loudspeaker assembly is given in Figure 1. The system is located inside an earmold and placed in the ear canal of the hearing aid user. In hearing aids, the miniature loudspeaker is commonly referred to as the receiver. This is why this type of hearing aids is called receiver-in-the-ear (RITE) or receiver-in-canal (RIC). The receiver system is powered through the wire that is also seen in Figure 1. The wire is connected to the main body of the hearing aid that is located behind the ear of the user. The microphones, battery, and electronics are located in the body of the hearing aid.
Figure 1: Illustration of the receiver assembly and location of the wax guard. S0R here stands for the type Small, of length 0, and Right ear. The pictures are copyright by Widex.
The geometry of the wax guard is depicted in Figure 2 in various views. The diameter of the main duct is 1 mm, the length of the wax guard is 1.6 mm, the thickness of the perforated plate (the mesh) is 50 µm, and the diameter of a perforate (mesh hole) is 190 µm.
Figure 2: Wax guard geometry. CAD geometry is copyright by Widex.
Model Setup
In this tutorial the acoustic properties of the wax guard are analyzed. Because the dimensions of the system are small (sub mm), it is important to capture the losses associated with the thermal and viscous acoustic boundary layers. The model is solved in the frequency domain with the Thermoviscous Acoustics, Frequency Domain physics interface. The transfer matrix of the system is computed using the Port boundary condition. Because the ports assume plane wave propagation, they need to be located away from any sudden geometry changes (like the perforated plate in the wax guard). In order to do this, an inlet tube of length 1 mm is added to the geometry. The simulation domain is the air volume inside the wax guard including the inlet tube, as depicted in Figure 3.
Figure 3: Simulation domain consisting of the air domain inside the wax guard as well as the extra inlet tube.
Transfer Matrix
A transfer matrix (also known as a two-port matrix or a transmission matrix) is a lumped representation of a system consisting of an inlet and an outlet. In the case of a pure acoustic system the matrix relates the pressure and volume flow at the inlet (pi,Qi) and outlet (po,Qo), see image below. For an electroacoustic transducer the transfer matrix will relate the voltage and current at the transducer terminal to the pressure and volume flow at the acoustic side. A classical introduction to transfer matrices used in electroacoustic applications is given in Ref. 1. The transfer matrix representation is valid for plane-wave propagation in the acoustic system. Because of the small dimensions in most hearing aid applications, only plane waves propagate in the audible range from 20 Hz to 20 kHz. This makes the use of the transfer matrix analogy very attractive and it is often used for prototyping new designs. It is important to acknowledge that because of the small dimensions, the transfer matrix representation needs to include the thermal and viscous boundary layer losses.
A schematic representation of a two-port system is given in the schematic above (notice the orientation of the volume flow). The associated transfer matrix T and the port values are related through
For a receiver (micro speaker) the input consists of a voltage Vi and current Ii, instead of a pressure and volume flow. For a microphone, the output will have electric units as well. For systems consisting of several components (transducer, ducts, couplers, and so on) the full system transfer matrix is computed by matrix multiplication (from the left) of the individual transfer matrices. Notice that the output of the first component is the input of the next and so forth.
A system is typically terminated with an impedance as in the above schematic. In the acoustic case, the output impedance is an acoustic impedance Zo = po/Qo (SI unit:Pa·s/m3). The impedance can, for example, represent a radiation impedance, a coupler volume compliance, or the mechanical properties of a microphone. Two useful relations for such systems are the associated input impedance and the transfer function. The input impedance is given by
(1)
and the transfer function is given by
(2)
Analytical expressions exist for various transfer matrices and can be found in the literature. One important transfer matrix in micro-acoustics and in hearing aids is that of a narrow cylindrical duct of length L and radius a, see Ref. 2 and Ref. 3. The model that includes thermoviscous boundary layer losses is given by
(3)
This expression is used in the tutorial model and defined under Variables: Narrow Tube Transfer Matrix. The transfer matrix components of transducers are often measured using the so-called two load method.
For any subsystem, the transfer matrix components can also be computed using the port boundary conditions and port sweep functionality available in the Acoustics Module (both in pressure acoustics and in thermoviscous acoustics). If two port boundary conditions are added to a system and the port sweep option is enabled, the transfer matrix between the inlet (first port added) and the outlet (second port added) is automatically computed. The values of the transfer matrix can be postprocessed, in thermoviscous acoustics the four components are ta.T11, ta.T12, ta.T21, and ta.T22, respectively. Transfer matrix parameters are only computed if two and only two ports are added, while the scattering matrix of the system is computed also for multiport configurations.
In this tutorial, the simulated system consists of an inlet duct of length 1 mm and the wax guard. Their total transfer matrix (which is the output of the model) is given by
(4)
which implies that the wax guard transfer matrix is computed as
(5)
The tube transfer matrix and its inverse is defined in the model as we set up a matrix inverse variable (matinv1). The wax guard transfer matrix is defined by a normal matrix variable (Twg).
Test Setup
In one set of measurements the wax guard is placed in a test setup consisting of a receiver (micro loudspeaker), a receiver tube of length Lrt = 1.19 mm and radius art = 0.725 mm, the wax guard, a coupler (artificial ear canal simulator), and a measurement microphone. The system is represented schematically below.
In the tutorial model the wax guard transfer matrix Twg is computed as described above. The receiver-tube matrix Trt is defined using Equation 3, while values for the receiver matrix Trec (a typical receiver used in hearing aids), the coupler matrix Tcp (a typical measurement coupler of the 711-type), and the microphone impedance Zmic are imported as interpolation functions. These are provided by Widex and are copyright by Widex. Note that the coupler transfer matrix could be computed by modifying the tutorial model Generic 711 Coupler — An Occluded Ear-Canal Simulator placing a port at the inlet and outlet, and doing a port sweep, just as in this model.
Results and Discussion
The acoustic pressure (using a nonsymmetric color scale) is depicted at 10 kHz in Figure 4. The instantaneous velocity and the temperature variations at 1 kHz are depicted in Figure 5 and Figure 6, respectively. In the model, the frequency parameter can be changed in order to visualize the extend of the viscous and thermal boundary layers. The port exciting the system can also be changed (either the inlet or the outlet).
Figure 4: Pressure distribution at 10 kHz, for the port excitation the outlet.
Figure 5: Velocity distribution at 1 kHz, with the port excitation at the outlet.
Figure 6: Temperature variation distribution at 1 kHz, with the port excitation at the outlet.
The values of the full system transfer matrix components are depicted in Figure 7 in four different plots. The plots contain the real and imaginary parts of the four matrix components. The computed transfer matrix components of the wax guard, using Equation 5, are depicted in Figure 8. The real and imaginary values are also computed in the Evaluation Group: Wax Guard, T Matrix (real/imag) node and represented in a table. The values can be exported to a text file using the Export button. In this way the transfer matrix of the wax guard subsystem can be imported and used in a lumped system simulation tool.
Figure 7: Transfer matrix components of full system (wax guard and inlet tube).
Figure 8: Computed transfer matrix components of wax guard.
Figure 9: Comparison of simulation and measurements of the SPL response measured at the microphone in the coupler. The measurements are copyright by Widex.
The response of a typical measurement setup that includes the wax guard is depicted in Figure 9. The graph shows the sound pressure level at the measurement microphone in a coupler volume, when the driver has a 0.1 V peak input harmonic signal. The graph shows a comparison between measurements and the system modeled using the transfer matrix approach. The transfer matrix used for the wax guard is the one computed in the model. The system response is computed using Equation 2 (with the input pressure replaced by an input voltage). The measurements and simulation show good agreement up to about 6 kHz. At high frequencies the lumped representation of the receiver (miniature loudspeaker) is not fully valid.
References
1. M. Lampton, “Transmission Matrices in electroacoustics,” Acoustica, vol. 39, pp  239–251, 1978.
2. H. Tijdeman, “On the propagation of sound waves in cylindrical tubes,” J. Sound Vib., vol. 39, pp. 1–33, 1975.
3. M.R. Stinson, “The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross sectional shapes,” J. Acoust. Soc. Am., vol. 89, pp. 550–558, 1991.
Application Library path: Acoustics_Module/Tutorials,_Thermoviscous_Acoustics/wax_guard_acoustics
Modeling Instructions
The following shows the modeling instructions. The geometry is set up in the Detailed Geometry Instructions section at the end of this document.
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Acoustics > Thermoviscous Acoustics > Thermoviscous Acoustics, Frequency Domain (ta).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Frequency Domain.
6
Click  Done.
Set up and import two sets of parameters and variables, then set up a global material. The material values are used in the variables that define the transfer matrix of a narrow tube and it will be used in the model (using a material link). For postprocessing the results of the model you will get back to the Global Definitions node and import data as well as define various transfer matrices.
Global Definitions
Parameters: Model
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Parameters: Model in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_parameters.txt.
Parameters: Tube Segments
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, type Parameters: Tube Segments in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_parameters_tubes.txt.
Variables: Narrow Tube Transfer Matrix
1
In the Model Builder window, right-click Global Definitions and choose Variables.
2
In the Settings window for Variables, type Variables: Narrow Tube Transfer Matrix in the Label text field.
3
Locate the Variables section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_variables.txt.
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 > Air.
4
Click the Add to Global Materials button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Next, import the CAD model of the wax guard. Define the air domain and perform some virtual operations to prepare the geometry for modeling and meshing. The geometry is of an actual NanoCare wax guard and is copyright by Widex.
Geometry 1
Check to see if your Geometry representation setting is set to the CAD kernel. This is required to import the model geometry and requires the CAD Import Module. If the COMSOL kernel is selected proceed with the next step.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Advanced section.
3
From the Geometry representation list, choose CAD kernel.
Import the geometry as a sequence from a file. To see the detailed instructions go to the last section in this document.
The imported wax guard CAD geometry, used in the imported geometry sequence, is depicted in Figure 2. The imported CAD file is of the device (the wax guard). The air domain inside (the simulation domain of the acoustics) is created together with a 1 mm inlet section.
4
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
5
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_geom_sequence.mph.
6
In the Geometry toolbar, click  Build All.
The finalized geometry should look like the one in Figure 3.
Materials
Material Link: Air
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, type Material Link: Air in the Label text field.
Set up selections used in physics and meshing.
Definitions
All domains
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type All domains in the Label text field.
3
Locate the Input Entities section. Select the All domains checkbox.
Inlet (port 1)
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Inlet (port 1) in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 3 only.
Outlet (port 2)
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Outlet (port 2) in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 14 only.
All exterior boundaries
1
In the Definitions toolbar, click  Adjacent.
2
In the Settings window for Adjacent, type All exterior boundaries in the Label text field.
3
Locate the Input Entities section. Under Input selections, click  Add.
4
In the Add dialog, select All domains in the Input selections list.
5
Click OK.
Walls (boundary layer mesh)
1
In the Definitions toolbar, click  Difference.
2
In the Settings window for Difference, type Walls (boundary layer mesh) in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click  Add.
5
In the Add dialog, select All exterior boundaries in the Selections to add list.
6
Click OK.
7
In the Settings window for Difference, locate the Input Entities section.
8
Under Selections to subtract, click  Add.
9
In the Add dialog, in the Selections to subtract list, choose Inlet (port 1) and Outlet (port 2).
10
Click OK.
Thermoviscous Acoustics, Frequency Domain (ta)
Thermoviscous Acoustics Model 1
1
In the Model Builder window, under Component 1 (comp1) > Thermoviscous Acoustics, Frequency Domain (ta) click Thermoviscous Acoustics Model 1.
2
In the Settings window for Thermoviscous Acoustics Model, locate the Model Input section.
3
In the p0 text field, type pA.
4
In the T0 text field, type T0.
Port 1
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet (port 1).
4
Locate the Port Properties section. From the Type of port list, choose Circular (0,0)-mode.
5
Locate the Incident Mode Settings section. In the Apin text field, type 1.
Port 2
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet (port 2).
4
Locate the Port Properties section. From the Type of port list, choose Circular (0,0)-mode.
5
In the Model Builder window, click Thermoviscous Acoustics, Frequency Domain (ta).
6
In the Settings window for Thermoviscous Acoustics, Frequency Domain, locate the Global Port Settings section.
7
From the Port sweep settings list, choose Activate port sweep.
Notice the parameter PortName shown in the Sweep parameter name field, it is also present in the Parameters: Model list. The parameter was added when loading the parameter file. In general, when selecting Activate port sweep the parameter has to be manually added. This parameter is used in a parametric sweep in the study.
Proceed and create the mesh. Use a well resolved tetrahedral mesh near the perforated plate and a swept mesh for the remaining domains. Then add a boundary layer mesh to resolve the viscous and thermal boundary layers. The parameter dvisc gives a measure of the layer thickness at the largest frequency studied.
Mesh 1
Free Tetrahedral 1
In the Mesh toolbar, click  Free Tetrahedral.
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 100[um].
5
In the Minimum element size text field, type 40[um].
6
In the Curvature factor text field, type 0.5.
7
In the Resolution of narrow regions text field, type 1.
Free Tetrahedral 1
1
In the Model Builder window, click Free Tetrahedral 1.
2
In the Settings window for Free Tetrahedral, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
5
Click  Build Selected.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
5
Click to expand the Corner Settings section. From the Handling of sharp edges list, choose Trimming.
6
Click to expand the Transition section. Clear the Smooth transition to interior mesh checkbox.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.
3
From the Selection list, choose Walls (boundary layer mesh).
4
Locate the Layers section. In the Number of layers text field, type 5.
5
From the Thickness specification list, choose First layer.
6
In the Thickness text field, type 0.15*dvisc.
Swept 1
1
In the Mesh toolbar, click  Swept.
2
In the Model Builder window, right-click Mesh 1 and choose Build All.
The mesh should look like the image below.
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
Click  Range.
4
In the Range dialog, choose ISO preferred frequencies from the Entry method list.
5
In the Start frequency text field, type 100.
6
In the Stop frequency text field, type 10000.
7
From the Interval list, choose 1/3 octave.
8
Click Replace.
Add a parametric sweep over the port parameter PortName in order to compute the full scattering and transfer matrix (port 1 and 2).
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
PortName (Port name)
1 2
 
Now generate the default solver, expand the solver nodes, and select one of the automatically generated iterative solver suggestions. For this model size the iterative option using domain decomposition (DD) is a good choice. If the model gets larger this solver will continue to scale and perform well.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 node.
4
Right-click Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 > Suggested Iterative Solver (GMRES with DD) (ta) and choose Enable.
The model requires about 17 GB of RAM to solve.
5
In the Study toolbar, click  Compute.
Results
Start by inspecting the default plots of the acoustic pressure, velocity, and temperature. You can change the frequency studied or the port used to excite the system.
Multislice
1
In the Model Builder window, expand the Acoustic Pressure (ta) node, then click Multislice.
2
In the Settings window for Multislice, locate the Coloring and Style section.
3
From the Scale list, choose Linear.
4
In the Acoustic Pressure (ta) toolbar, click  Plot.
The image should look like Figure 4.
Acoustic Velocity (ta)
1
In the Model Builder window, under Results click Acoustic Velocity (ta).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 1000.
4
In the Acoustic Velocity (ta) toolbar, click  Plot.
The image should look like Figure 5.
Temperature Variation (ta)
1
In the Model Builder window, click Temperature Variation (ta).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 1000.
4
In the Temperature Variation (ta) toolbar, click  Plot.
The image should look like Figure 6.
Full System T-matrix
1
In the Model Builder window, right-click Results and choose Node Group.
2
In the Settings window for Group, type Full System T-matrix in the Label text field.
Full System T11
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Full System T11 in the Label text field.
Global 1
1
Right-click Full System T11 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
real(ta.T11)
1
real(T11)
imag(ta.T11)
1
imag(T11)
4
Click the  y-Axis Log Scale button in the Graphics toolbar.
5
Click the  x-Axis Log Scale button in the Graphics toolbar.
6
In the Full System T11 toolbar, click  Plot.
Now, duplicate the plot and edit it, in order to depict the other three components of the transfer matrix of the system.
Full System T12
1
In the Model Builder window, right-click Full System T11 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Full System T12 in the Label text field.
Global 1
1
In the Model Builder window, expand the Full System T12 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(ta.T12)
kg/(m^4*s)
real(T12)
imag(ta.T12)
kg/(m^4*s)
imag(T12)
4
In the Full System T12 toolbar, click  Plot.
Full System T21
1
In the Model Builder window, right-click Full System T12 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Full System T21 in the Label text field.
Global 1
1
In the Model Builder window, expand the Full System T21 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(ta.T21)
m^4*s/kg
real(T21)
imag(ta.T21)
m^4*s/kg
imag(T21)
4
In the Full System T21 toolbar, click  Plot.
Full System T22
1
In the Model Builder window, right-click Full System T21 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Full System T22 in the Label text field.
Global 1
1
In the Model Builder window, expand the Full System T22 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(ta.T22)
1
real(T22)
imag(ta.T22)
1
imag(T22)
4
In the Full System T22 toolbar, click  Plot.
The four graphs are depicted in Figure 7.
Proceed and set up the matrix variables necessary to compute and postprocess the transfer matrix of the wax guard subsystem. Also define matrix variables necessary to compute the response of the wax guard when placed in a typical measurement setup (using a lumped approach). The steps are explained in the main model documentation. This also involves initially defining several interpolation functions to import the transfer matrix components of the transducer, coupler, and microphone impedance, as well as the measurement results.
Global Definitions
Interpolation: Receiver T-matrix
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, type Interpolation: Receiver T-matrix in the Label text field.
3
Locate the Definition section. From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_T_receiver.csv.
6
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
Column 2
Function values
Function name=int1a
Column 3
Function values
Function name=int1b
Column 5
Function values
Function name=int1c
Column 6
Function values
Function name=int1d
Column 7
Function values
Function name=int1e
Column 8
Function values
Function name=int1f
Column 9
Function values
Function name=int1g
7
In the table, click to select the cell at row number 1 and column number 1.
8
In the Unit text field, type Hz.
9
In the table, click to select the cell at row number 2 and column number 2.
10
In the Name text field, type Trec11_real.
11
In the Unit text field, type 1.
12
In the table, click to select the cell at row number 3 and column number 2.
13
In the Name text field, type Trec11_imag.
14
In the Unit text field, type 1.
15
In the table, click to select the cell at row number 4 and column number 2.
16
In the Name text field, type Trec12_real.
17
In the Unit text field, type 1.
18
In the table, click to select the cell at row number 5 and column number 2.
19
In the Name text field, type Trec12_imag.
20
In the Unit text field, type 1.
21
In the table, click to select the cell at row number 6 and column number 2.
22
In the Name text field, type Trec21_real.
23
In the Unit text field, type 1.
24
In the table, click to select the cell at row number 7 and column number 2.
25
In the Name text field, type Trec21_imag.
26
In the Unit text field, type 1.
27
In the table, click to select the cell at row number 8 and column number 2.
28
In the Name text field, type Trec22_real.
29
In the Unit text field, type 1.
30
In the table, click to select the cell at row number 9 and column number 2.
31
In the Name text field, type Trec22_imag.
32
In the Unit text field, type 1.
33
Locate the Definition section. Click  Import.
Interpolation: Coupler T-matrix
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, type Interpolation: Coupler T-matrix in the Label text field.
3
Locate the Definition section. From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_T_coupler.csv.
6
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
Column 2
Function values
Function name=int2a
Column 3
Function values
Function name=int2b
Column 5
Function values
Function name=int2c
Column 6
Function values
Function name=int2d
Column 7
Function values
Function name=int2e
Column 8
Function values
Function name=int2f
Column 9
Function values
Function name=int2g
7
In the table, click to select the cell at row number 1 and column number 1.
8
In the Unit text field, type Hz.
9
In the table, click to select the cell at row number 2 and column number 2.
10
In the Name text field, type Tcp11_real.
11
In the Unit text field, type 1.
12
In the table, click to select the cell at row number 3 and column number 2.
13
In the Name text field, type Tcp11_imag.
14
In the Unit text field, type 1.
15
In the table, click to select the cell at row number 4 and column number 2.
16
In the Name text field, type Tcp12_real.
17
In the Unit text field, type 1.
18
In the table, click to select the cell at row number 5 and column number 2.
19
In the Name text field, type Tcp12_imag.
20
In the Unit text field, type 1.
21
In the table, click to select the cell at row number 6 and column number 2.
22
In the Name text field, type Tcp21_real.
23
In the Unit text field, type 1.
24
In the table, click to select the cell at row number 7 and column number 2.
25
In the Name text field, type Tcp21_imag.
26
In the Unit text field, type 1.
27
In the table, click to select the cell at row number 8 and column number 2.
28
In the Name text field, type Tcp22_real.
29
In the Unit text field, type 1.
30
In the table, click to select the cell at row number 9 and column number 2.
31
In the Name text field, type Tcp22_imag.
32
In the Unit text field, type 1.
33
Locate the Definition section. Click  Import.
Interpolation: Microphone Impedance
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, type Interpolation: Microphone Impedance in the Label text field.
3
Locate the Definition section. From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_mic_impedance.csv.
6
Locate the Data Column Settings section. In the table, click to select the cell at row number 1 and column number 1.
7
In the Unit text field, type Hz.
8
In the table, enter the following settings:
Columns
Type
Settings
Column 2
Function values
Function name=Zmic_real
9
In the Name text field, type Zmic_real.
10
In the Unit text field, type 1.
11
In the table, click to select the cell at row number 3 and column number 1.
12
In the Name text field, type Zmic_imag.
13
In the Unit text field, type 1.
14
Locate the Definition section. Click  Import.
Interpolation: Measurements
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, type Interpolation: Measurements in the Label text field.
3
Locate the Definition section. From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_measurements.csv.
6
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
Column 2
Function values
Function name=int4a
Column 3
Function values
Function name=int4b
Column 4
Ignored column
 
7
In the table, click to select the cell at row number 1 and column number 1.
8
In the Unit text field, type Hz.
9
In the table, click to select the cell at row number 2 and column number 1.
10
In the Name text field, type pWGon_real.
11
In the Unit text field, type Pa.
12
In the table, click to select the cell at row number 3 and column number 1.
13
In the Name text field, type pWGon_imag.
14
In the Unit text field, type Pa.
15
Locate the Definition section. Click  Import.
In order to use the matrix variables turn on the show variables utilities (if not already selected). Click the small eye at the top of the Model Builder tree.
16
Click the  Show More Options button in the Model Builder toolbar.
17
In the Show More Options dialog, in the tree, select the checkbox for the node General > Variable Utilities.
18
Click OK.
Matrix Inverse: Inlet Tube
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix Inverse.
2
In the Settings window for Matrix Inverse, type Matrix Inverse: Inlet Tube in the Label text field.
3
Locate the Input Matrix section. From the Matrix size list, choose 2X2.
4
In the table, enter the following settings:
cos(kc_in*L_in)
-Zc_in/(i*S_in)*sin(kc_in*L_in)
i*S_in/Zc_in*sin(kc_in*L_in)
cos(kc_in*L_in)
Matrix: Wax Guard (model)
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, type Matrix: Wax Guard (model) in the Label text field.
3
In the Name text field, type Twg.
4
Locate the Input Matrix section. From the Matrix size list, choose 2X2.
The expression entered is the matrix product of the inverse of the inlet tube T-matrix and the total system matrix.
5
In the table, enter the following settings:
matinv1.invT11*comp1.ta.T11+matinv1.invT12*comp1.ta.T21
matinv1.invT11*comp1.ta.T12+matinv1.invT12*comp1.ta.T22
matinv1.invT21*comp1.ta.T11+matinv1.invT22*comp1.ta.T21
matinv1.invT21*comp1.ta.T12+matinv1.invT22*comp1.ta.T22
Matrix: Receiver
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, type Matrix: Receiver in the Label text field.
3
In the Name text field, type Trec.
4
Locate the Input Matrix section. From the Matrix size list, choose 2X2.
5
In the table, enter the following settings:
Trec11_real(freq)+i*Trec11_imag(freq)
Trec12_real(freq)+i*Trec12_imag(freq)
Trec21_real(freq)+i*Trec21_imag(freq)
Trec22_real(freq)+i*Trec22_imag(freq)
Matrix: Receiver Tube
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, type Matrix: Receiver Tube in the Label text field.
3
In the Name text field, type Trt.
4
Locate the Input Matrix section. From the Matrix size list, choose 2X2.
5
In the table, enter the following settings:
cos(kc_rt*L_rt)
-Zc_rt/(i*S_rt)*sin(kc_rt*L_rt)
i*S_rt/Zc_rt*sin(kc_rt*L_rt)
cos(kc_rt*L_rt)
Matrix: Coupler
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, type Matrix: Coupler in the Label text field.
3
In the Name text field, type Tcp.
4
Locate the Input Matrix section. From the Matrix size list, choose 2X2.
5
In the table, enter the following settings:
Tcp11_real(freq)+i*Tcp11_imag(freq)
Tcp12_real(freq)+i*Tcp12_imag(freq)
Tcp21_real(freq)+i*Tcp21_imag(freq)
Tcp22_real(freq)+i*Tcp22_imag(freq)
Group: Matrix Products
1
In the Model Builder window, right-click Global Definitions and choose Node Group.
2
In the Settings window for Group, type Group: Matrix Products in the Label text field.
Matrix 5 (T5)
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, locate the Input Matrix section.
3
From the Matrix size list, choose 2X2.
4
In the table, enter the following settings:
Trec11*Trt11+Trec12*Trt21
Trec11*Trt12+Trec12*Trt22
Trec21*Trt11+Trec22*Trt21
Trec21*Trt12+Trec22*Trt22
Matrix 6 (T6)
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, locate the Input Matrix section.
3
From the Matrix size list, choose 2X2.
4
In the table, enter the following settings:
T511*Twg11+T512*Twg21
T511*Twg12+T512*Twg22
T521*Twg11+T522*Twg21
T521*Twg12+T522*Twg22
Matrix 7 (T7)
1
In the Home toolbar, click  Variable Utilities and choose Global > Matrix.
2
In the Settings window for Matrix, locate the Input Matrix section.
3
From the Matrix size list, choose 2X2.
4
In the table, enter the following settings:
T611*Tcp11+T612*Tcp21
T611*Tcp12+T612*Tcp22
T621*Tcp11+T622*Tcp21
T621*Tcp12+T622*Tcp22
All the necessary matrix variables have now been created. In order to use them in postprocessing update the solution (it is not necessary to re-solve the model).
Study 1
In the Study toolbar, click  Update Solution.
Results
Wax Guard T-matrix
1
In the Model Builder window, right-click Full System T-matrix and choose Duplicate.
2
In the Settings window for Group, type Wax Guard T-matrix in the Label text field.
Wax Guard T11
1
In the Model Builder window, expand the Wax Guard T-matrix node, then click Full System T11.1.
2
In the Settings window for 1D Plot Group, type Wax Guard T11 in the Label text field.
Global 1
1
In the Model Builder window, expand the Wax Guard T11 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(Twg11)
 
real(T11)
imag(Twg11)
 
imag(T11)
4
In the Wax Guard T11 toolbar, click  Plot.
Wax Guard T12
1
In the Model Builder window, under Results > Wax Guard T-matrix click Full System T12.1.
2
In the Settings window for 1D Plot Group, type Wax Guard T12 in the Label text field.
Global 1
1
In the Model Builder window, expand the Wax Guard T12 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(Twg12)
 
real(T12)
imag(Twg12)
 
imag(T12)
4
In the Wax Guard T12 toolbar, click  Plot.
Wax Guard T21
1
In the Model Builder window, under Results > Wax Guard T-matrix click Full System T21.1.
2
In the Settings window for 1D Plot Group, type Wax Guard T21 in the Label text field.
Global 1
1
In the Model Builder window, expand the Wax Guard T21 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(Twg21)
 
real(T21)
imag(Twg21)
 
imag(T21)
4
In the Wax Guard T21 toolbar, click  Plot.
Wax Guard T22
1
In the Model Builder window, under Results > Wax Guard T-matrix click Full System T22.1.
2
In the Settings window for 1D Plot Group, type Wax Guard T22 in the Label text field.
Global 1
1
In the Model Builder window, expand the Wax Guard T22 node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(Twg22)
 
real(T22)
imag(Twg22)
 
imag(T22)
4
In the Wax Guard T22 toolbar, click  Plot.
The four graphs are depicted in Figure 8.
Mesh
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Mesh in the Label text field.
Mesh 1
1
Right-click Mesh and choose Mesh.
2
In the Settings window for Mesh, locate the Level section.
3
From the Level list, choose Volume.
4
Click to expand the Element Filter section. Select the Enable filter checkbox.
5
In the Expression text field, type x>0.
6
In the Mesh toolbar, click  Plot.
This plot makes it possible to inspect the inside of the mesh. It should look like the image below.
Full System Response
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Full System Response 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 Microphone (dB SPL).
6
Locate the Axis section. Select the x-axis log scale checkbox.
7
Locate the Legend section. From the Position list, choose Upper left.
Global 1
1
Right-click Full System Response 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
20*log10(abs(pWGon_real(freq)+i*pWGon_imag(freq))/20e-6)
 
Measurements
20*log10(abs(Vrec/(T711+T712/Zmic))/20e-6)
 
Transfer matrix (model)
4
In the Full System Response toolbar, click  Plot.
This plot shows the response of the wax guard when placed in a typical measurement setup. The results should look like Figure 9.
Next create an evaluation group and evaluate the wax guard transfer matrix components (real and imaginary parts). This will create a table that can be used to export the data into a text file for further use.
Evaluation Group: Wax Guard, T-Matrix (real/imag)
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Evaluation Group: Wax Guard, T-Matrix (real/imag) in the Label text field.
Global Evaluation 1
1
Right-click Evaluation Group: Wax Guard, T-Matrix (real/imag) and choose Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (sol1).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
real(Twg11)
 
 
imag(Twg11)
 
 
real(Twg12)
 
 
imag(Twg12)
 
 
real(Twg21)
 
 
imag(Twg21)
 
 
real(Twg22)
 
 
imag(Twg22)
 
 
5
In the Evaluation Group: Wax Guard, T-Matrix (real/imag) toolbar, click  Evaluate.
Finally, create the image that is used as the model thumbnail. This step can be skipped.
Thumbnail
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Thumbnail in the Label text field.
3
Click to expand the Title section. From the Title type list, choose None.
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
Slice 1
1
Right-click Thumbnail and choose Slice.
2
In the Settings window for Slice, locate the Expression section.
3
In the Expression text field, type ta.v_rms.
4
From the Unit list, choose mm/s.
5
Locate the Plane Data section. In the Planes text field, type 1.
Surface 1
1
In the Model Builder window, right-click Thumbnail and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Gray.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 7-10, 15-54 in the Selection text field.
5
Click OK.
Line 1
1
In the Model Builder window, right-click Thumbnail and choose Line.
2
In the Settings window for Line, locate the Expression section.
3
In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Black.
Selection 1
1
Right-click Line 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 7, 8, 10, 11, 13, 14, 17, 19, 31, 32, 34, 36, 42, 43, 89-92, 94, 96, 103, 104, 137-140 in the Selection text field.
5
Click OK.
Thumbnail
1
In the Model Builder window, under Results click Thumbnail.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 1000.
4
Locate the Color Legend section. Select the Show units checkbox.
5
Select the Show maximum and minimum values checkbox.
6
In the Graphics window toolbar, clicknext to  Scene Light, then choose Ambient Occlusion.
7
In the Thumbnail toolbar, click  Plot.
Detailed Geometry Instructions
If you want to create the geometry yourself, follow these steps.
Geometry 1
Check to see if your Geometry representation setting is set to the CAD kernel. This is required to import the model geometry and requires the CAD Import Module. If the COMSOL kernel is select proceed with the next step.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Advanced section.
3
From the Geometry representation list, choose CAD kernel.
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file wax_guard_acoustics_cad_geometry.stp.
5
Click  null.
6
Locate the Length Unit section. From the Length unit list, choose From CAD document.
7
Click  Build Selected.
The CAD geometry of the wax guard is depicted in Figure 2. Use the graphics window tools to rotate, move, and zoom. The CAD file is of the device, now create the air domain inside as well as a 1 mm inlet section.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
3
In the Settings window for Union, locate the Union section.
4
From the Repair tolerance list, choose Absolute.
5
In the Absolute repair tolerance text field, type 1.0E-4.
If the tolerance is set too low, for example, at 1E-6, the CAD will not union correctly because of tolerances used in production.
6
Click  Build All Objects.
Cap Faces 1 (cap1)
1
In the Geometry toolbar, click  Defeaturing and Repair and choose Cap Faces.
2
On the object uni1, select Edges 57 and 58 only.
Extrude 1 (ext1)
1
In the Geometry toolbar, click  Extrude.
2
On the object cap1, select Boundary 38 only.
3
In the Settings window for Extrude, locate the Distances section.
4
From the Specify list, choose Vertices to extrude to.
5
On the object cap1, select Point 200 only.
6
Click  Build Selected.
Now delete all the domains that do not represent the air inside the system. Do it in two steps, to simplify selecting the domains.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object ext1, select Domain 1 only.
5
Click  Build Selected.
Delete Entities 2 (del2)
1
Right-click Geometry 1 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object del1, select Domains 1 and 3–6 only.
5
Click  Build Selected.
Extrude 2 (ext2)
1
In the Geometry toolbar, click  Extrude.
2
On the object del2, select Boundary 3 only.
3
In the Settings window for Extrude, locate the Distances section.
4
In the table, enter the following settings:
Distances (mm)
1[mm]
The value 1[mm] should correspond to the parameter L_in defined in the model.
5
Click  Build Selected.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type -0.75.
Partition Domains 1 (pard1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Domains.
2
On the object ext2, select Domain 2 only.
3
In the Settings window for Partition Domains, click  Build Selected.
Form Union (fin)
1
In the Geometry toolbar, click  Build All.
2
In the Model Builder window, click Form Union (fin).
Now, perform some virtual geometry operations to prepare the geometry for modeling.
Form Composite Faces 1 (cmf1)
1
In the Geometry toolbar, click  Virtual Operations and choose Form Composite Faces.
2
On the object fin, select Boundaries 35, 39, 45, 47, 64, 65, 69, and 75 only.
3
In the Settings window for Form Composite Faces, click  Build Selected.
Ignore Edges 1 (ige1)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
3
On the object cmf1, select Edges 40, 45, 110, 112, 123, 128, 185, and 187 only.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
In the Settings window for Ignore Edges, click  Build Selected.
Ignore Edges 2 (ige2)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
On the object ige1, select Edges 21, 23, 27, 45, 47, 61, 63, 80, 82, 85, 86, 95, 99, 130–134, 136, 140, 141, 174, 176, 184, and 186 only.
3
In the Settings window for Ignore Edges, click  Build Selected.
4
Click the  Wireframe Rendering button in the Graphics toolbar.
The finalized geometry should look like the one below.
1
“Widex” covers Widex A/S and Affiliates. “Affiliate” means a legal entity, now or hereafter, directly or indirectly, owned or controlled by, or owning or controlling, or under common control with by one of the Parties, but such legal entity shall be deemed to be an Affiliate only so long as such ownership or control exists. For purposes of this definition “control” of a legal entity shall mean to have, directly or indirectly, the power to direct or cause the direction of the management and policies of a legal entity, whether (a) through the ownership of voting securities entitling to the right to elect or appoint, directly or indirectly, the majority of the board of directors, or a similar managing authority; (b) by contract; or (c) otherwise. For purposes of this definition, Widex A/S, Sivantos Pte Ltd, together with their Affiliates, are Affiliates of each other.