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OW Microspeaker:
Simulation and Correlation with Measurements
Introduction
In this tutorial, the electromagnetic, mechanical, and acoustical characteristics of the Ole Wolff OWS-1943T-8CP (discontinued) microspeaker are analyzed and correlated to measurements performed by Ole Wolff Electronics. In this context, microspeakers are electrodynamic transducers of reduced dimensions that are used in small electronic equipment like smart phones, laptops, hand-held terminals, or scanners to reproduce most of the acoustic frequency range in a single transducer.
Figure 1: The OWS-1943T-8CP microspeaker with model results superposed.
In the first step, starting from the geometry of the speaker, an axisymmetric electromagnetic model is used to characterize the frequency-dependent response of the voice coil and the magnetic circuit. In the second step, the nonlinear mechanical characteristics of the diaphragm are computed and compared to measurements. In the third step, a lumped circuit, representing the electromagnetic physics, is coupled to a 3D model where the mechanical and acoustic response of the speaker is analyzed and compared with measurements. The comparison with measurements shows a good level of correlation for all of the physics and measurements considered.
Note: In this model, the OWS-1943T-8CP (discontinued) speaker geometry and measurement data are copyright by Ole Wolff Elektronik A/S.
Model Definition
The geometry of the microspeaker is shown on Figure 2. The requirement for compactness makes the general layout of the speaker differ from traditional dynamic loudspeakers; in this microspeaker the diaphragm covers the functions of diaphragm, dust cap, and spider used in larger loudspeakers. The dimensions of the speaker are approximately 19 mm in diameter and 2.8 mm in height.
The microspeaker has certain features, like the distribution of the back vents, the front plate holes, and the geometry of the diaphragm, that make it necessary to consider a full 3D geometry of the microspeaker in the vibroacoustic analysis.
Figure 2: Geometry of the OWS-1943T-8CP microspeaker from top and bottom views. The geometry is copyright by Ole Wolff Elektronik A/S.
As shown in Figure 2 and Figure 3, the components that influence the electromagnetic field — that is, the pole piece, the magnets, and the voice coil — show axial symmetry. The axisymmetric assumption seems to be a valid approach for the electromagnetic analysis of the microspeaker.
Figure 3: Schematic cross-section view of the microspeaker.
Due to the reduced dimensions of the microspeaker compared to the wavelengths solved for, it is assumed that all components will behave as acoustically rigid in the vibroacoustic analysis, except for the diaphragm and the voice coil.
Electromagnetic analysis
The aim of this analysis is to obtain the BL factor of the microspeaker and the frequency dependent impedance characteristics of the voice coil and the electromagnetic circuit. The BL factor is the product of the magnetic field flux perpendicular to the coil and the total length of the coil.
The electromagnetic analysis of the microspeaker uses a Small-Signal Analysis, Frequency Domain study which includes two steps (automatically generated); a Stationary step where the stationary magnetic field generated by the magnets is considered, and a Frequency Domain Perturbation, where the voice coil is excited with a harmonic AC voltage and additional currents are induced in the electromagnetic circuit.
Due to the small influence of the voice coil position in the stationary magnetic field, instead of moving the coil and extracting the BL factor, the model uses logical operators and an integrand over the complete air and voice coil domain to account for the different positions of the voice coil as it moves. The BL factor is plotted versus the coil offset, a measurement of the distance between the current position of the voice coil and the resting position (see Figure 6 in the results section).
It is worth highlighting that small imperfections due to surface roughness or thin glue layers between components in the electromagnetic circuit can have large influence on the BL factor. Therefore, this model uses the Thin Low Permeability Gap feature to add a small gap with air permeability to account for this. The areas where this feature have been applied are shown on Figure 4.
Figure 4: Edges of the model where the thin low permeability gap condition is applied.
Structural nonlinear analysis
This step analyses the nonlinear structural response of the diaphragm. In the analysis, the only source of nonlinearity (in the diaphragm) is geometric nonlinearity; as the diaphragm is a relatively thin shell, small displacements perpendicular to the diaphragm will have an influence on the stiffness of the diaphragm.
It is worth noting that a substantial simplification has been done in this analysis regarding the stiffness of the voice coil. As shown in Figure 5, the voice coil presents a nonisotropic structure with varying properties depending on the direction. To avoid specifically modeling each individual wire of the coil and its insulation, the properties of the voice coil have been homogenized to a single material.
Figure 5: Voice coil structure and its homogenized model.
Ideally, this homogenized material should maintain the anisotropy of the voice coil in the different directions. The properties of this orthotropic equivalent material can be derived from a submodel or from testing. For the sake of simplicity in this tutorial, the homogenized voice coil has been assumed to have an isotropic behavior.
The compliance of the glue that attaches the voice coil to the diaphragm has not been specifically included in the model and that could also be a source for the small differences between the model and the measurements (see Figure 7 in the results section).
Vibroacoustic Analysis
During this step, the properties derived from the electromagnetic analysis are included in an electric circuit and coupled to the other physics following the example shown in Ref. 1. For the analysis, the speaker is placed in an infinite baffle test configuration. The Exterior Field Calculation feature is used to compute the response at a given distance in front of the speaker.
In the acoustic analysis, the damping in the diaphragm will be the main mode of dissipation of energy, so it is quite important to capture the correct damping of the diaphragm. In this tutorial, a constant isotropic loss factor is assumed for the diaphragm, but a frequency dependent damping is likely to produce better correlation. Possible ways to identify the diaphragm damping include testing the microspeaker in vacuum, to decouple the damping of the diaphragm from the damping of the air, or a dynamic testing of the diaphragm material.
Results and Discussion
The measured and computed BL factors are shown in Figure 6. The results show good agreement in the complete voice coil offset range. The model is using generic soft iron properties from the COMSOL material library. A possible way to improve the correlation of the BL curve is to measure the B-H curves of the iron used in the microspeaker and use this curve in the model.
Figure 6: Measured and computed BL factor.
The measured and computed mechanical compliance Cms(x) curves are shown in Figure 7. The results show good agreement in the positive part of the voice coil offset. As discussed in the Model Definition section, the simplifications regarding the voice coil structure could be the main source of limited correlation on the negative part of the voice coil offset.
Figure 7: Measured and computed mechanical compliance Cms.
Figure 8 shows the measured and the computed impedance of the electric circuit. The main mechanical mode of the microspeaker, at around 950 Hz, shows a good match with the measurements both in terms of the frequency at which it is produced and the amount of damping of the mode. The response peak around 7500 Hz is caused by a radial mode combined with a rocking mode (see below) induced by the nonsymmetric distribution of the back vents. In the simulation results, the peak is relatively sharp while the measurements are showing a much more damped response.
Figure 8: Measured and computed complex impedance of the electromagnetic system.
Figure 9 shows the displacement at 7500 Hz, highlighting that the resonance is due to the excitation of a radial mode combined with a rocking mode induced by the nonsymmetric distribution of the back vents. The fact that the voice coil is deforming at this frequency and not acting as a solid body, suggest that correctly capturing the stiffness of the voice coil structure could improve the correlation of the frequency at which this resonance is produced.
Figure 9: Displacement of the diaphragm and the voice coil at 7500 Hz.
The sound pressure level (SPL) evaluated 39.0 mm in front of the speaker is shown in Figure 10. The figure compares the simulation results and measurements. The COMSOL model presents a good level of correlation from low frequency up to 5000 Hz, with the curve showing a difference of 1.1 dB at the main resonance and 1.5 dB as the maximum difference. The response around 7500 Hz seems to be underdamped in the model, suggesting that thermoviscous damping in the air is relevant at that frequency. This was also indicated in the impedance curve in Figure 8.
To test this, a new study was created where Thermoviscous Acoustics, Frequency Domain was used to model the air domain behind the diaphragm and in the vents. The thermoviscous domain is easily coupled to the diaphragm and voice coil structures, and the pressure acoustics domains using the built-in multiphysics couplings (Acoustic-Thermoviscous Acoustic Boundary and Thermoviscous Acoustic-Structure Boundary). This part of the analysis is not included in the current model.
A finer mesh was used in this part of the model to properly resolve the thermoviscous boundary layers. Due to the additional degrees of freedom required to solve the problem, the resulting model was solved used a high-performance computer (HPC). This full detailed setup requires about 95 GB of RAM while the model that only uses pressure acoustics fits in a machine with 16 GB of RAM. The SPL curve has been imported as a reference and is seen in Figure 10. Using a thermoviscous model that captures all thermoviscous losses will improve the correlation of the damping around the resonance at 7500 Hz.
Figure 10: Measured and computed SPL, 39 mm front of the microspeaker. The result from a model with thermoviscous losses is also shown as a reference.
Both the pure pressure acoustics and the thermoviscous acoustics models still show significant differences in the range between 12 000 Hz and 18 500 Hz, caused by the difference in the circumferential modes. Again, the correlation is likely to improve if a better modeling of the voice coil structure is implemented. The model correctly captures the dip at the end of the acoustic range.
The tutorial has demonstrated that a good level of correlation is achieved with a relatively simple and a computationally affordable model using COMSOL Multiphysics. As with any other model, good material data is mandatory to produce meaningful predictions (both absolute and relative). It is also important to capture the thermoviscous effects in acoustic devices of reduced dimensions like this microspeaker.
Reference
1. Lumped Loudspeaker Driver, from the COMSOL Application Library.
Application Library path: Acoustics_Module/Electroacoustic_Transducers/ow_microspeaker
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 AC/DC>Electromagnetic Fields>Magnetic Fields (mf).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Small-Signal Analysis, Frequency Domain.
6
Click  Done.
In the next steps, import the parameters that will be used in the model and then import the measurements that will be used to validate the model.
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 ow_microspeaker_parameters.txt.
Measured SPL at 39 mm
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Measured SPL at 39 mm in the Label text field.
3
Locate the Definition section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_spl_39mm_test.txt.
5
In the Function name text field, type SPL_Test.
6
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
t
Hz
7
In the Function table, enter the following settings:
Function
Unit
SPL_Test
dB
8
Click  Plot.
This plot shows the measured sound pressure level (SPL) 39 mm in front of the microspeaker.
Measured BL curve
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Measured BL curve in the Label text field.
3
Locate the Definition section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_bl_test.txt.
5
In the Function name text field, type BL_Test.
6
Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Piecewise cubic.
7
From the Extrapolation list, choose Linear.
8
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
t
mm
9
In the Function table, enter the following settings:
Function
Unit
BL_Test
Wb/m
10
Click  Plot.
This plot shows the measured BL factor as a function of the coil displacement.
Measured Z curve
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Measured Z curve in the Label text field.
3
Locate the Definition section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_z_test.txt.
5
In the Function name text field, type Z_Test.
6
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
t
Hz
7
In the Function table, enter the following settings:
Function
Unit
Z_Test
ohm
8
Click  Plot.
This plot shows the measured impedance as a function of the frequency.
Measured CMS curve
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Measured CMS curve in the Label text field.
3
Locate the Definition section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_cms_test.txt.
5
In the Function name text field, type CMS_Test.
6
Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Cubic spline.
7
From the Extrapolation list, choose Linear.
8
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
t
mm
9
In the Function table, enter the following settings:
Function
Unit
CMS_Test
mm/N
10
Click  Plot.
This plot shows the measured compliance (Cms) as a function of the coil displacement.
Electromagnetic analysis
As described in Model Definition, the microspeaker presents axial symmetry for the electromagnetic field. Thus, this part of the analysis will be performed assuming axial symmetry.
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.
Import 1 (imp1)
1
In the Home toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
From the Source list, choose COMSOL Multiphysics file.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_axisymmetric.mphbin.
6
Click  Import.
This 2D geometry has been generated through the use of the Cross Section feature, which uses a 3D geometry and a work plane to generate a 2D section. For the sake of simplicity, the result of some cleanup operations is directly imported as an external file.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_variables_2d.txt.
The first two variables are logical operators that will be used to obtain the BL curve. The logical operators identify only the area that should contribute to the integral that is used to compute the BL factor. Read the documentation on the DEST built-in operator for further information. The third variable is the integrand used in the integral used to obtain the BL factor.
Area to integrate BL
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type Area to integrate BL in the Label text field.
3
In the Operator name text field, type int_BL.
4
Select Domains 1 and 2 only.
5
Locate the Advanced section. In the Integration order text field, type 12.
6
Clear the Compute integral in revolved geometry check box.
As the coil does not have an influence on the static electromagnetic field, the model uses the logic operators to identify the position of the coil in the domain defined by the coil and the air.
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>Air.
4
Click Add to Component in the window toolbar.
5
In the tree, select AC/DC>Soft Iron (With Losses).
6
Click Add to Component in the window toolbar.
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Air (mat1)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Air (mat1).
2
Select Domains 1 and 2 only.
Soft Iron (With Losses) (mat2)
1
In the Model Builder window, click Soft Iron (With Losses) (mat2).
2
Select Domains 3, 5, and 7 only.
The next step defines the domains that will be part of the coil and pole pieces.
Magnet
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
Select Domains 4 and 6 only.
3
In the Settings window for Material, type Magnet in the Label text field.
4
Locate the Material Properties section. In the Material properties tree, select Electromagnetic Models>Remanent Flux Density>Remanent flux density norm (normBr).
5
Click  Add to Material.
6
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Recoil permeability
murec_iso ; murecii = murec_iso, murecij = 0
 
1
Remanent flux density
Remanent flux density norm
normBr
B0
T
Remanent flux density
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1
1
Basic
Electrical conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
0
S/m
Basic
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1
1
Basic
Magnetic Fields (mf)
Coil 1
1
In the Model Builder window, under Component 1 (comp1) right-click Magnetic Fields (mf) and choose the domain setting Coil.
2
Select Domain 2 only.
3
In the Settings window for Coil, locate the Coil section.
4
From the Conductor model list, choose Homogenized multiturn.
5
From the Coil excitation list, choose Voltage.
6
In the Vcoil text field, type linper(V0).
7
Locate the Homogenized Multiturn Conductor section. In the N text field, type N0.
8
In the σwire text field, type sigma_wire.
9
From the Coil wire cross-section area list, choose From round wire diameter.
10
In the dwire text field, type d_wire.
Ampère’s Law First Magnet
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
In the Settings window for Ampère’s Law, type Ampère's Law First Magnet in the Label text field.
3
Select Domain 6 only.
4
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Remanent flux density.
5
Specify the e vector as
0
r
0
phi
1
z
Ampère’s Law Second Magnet
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
In the Settings window for Ampère’s Law, type Ampère's Law Second Magnet in the Label text field.
3
Select Domain 4 only.
4
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Remanent flux density.
5
Specify the e vector as
0
r
0
phi
-1
z
Please note that the remanent flux of the two magnets faces opposite directions.
Ampère’s Law Soft Iron
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
In the Settings window for Ampère’s Law, type Ampère's Law Soft Iron in the Label text field.
3
Select Domains 3, 5, and 7 only.
4
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose B-H curve.
Thin Low Permeability Gap 1
1
In the Physics toolbar, click  Boundaries and choose Thin Low Permeability Gap.
2
Select Boundaries 13, 15, 21, and 23 only.
3
In the Settings window for Thin Low Permeability Gap, locate the Thin Low Permeability Gap section.
4
From the μr list, choose User defined. In the ds text field, type th_gap.
The Thin Low Permeability Gap feature allows to capture the imperfect magnetic contact between the two domains due to the presence of glue or the surface roughness. The selection should look like that in Figure 4.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size
1
In the Model Builder window, under Component 1 (comp1)>Mesh 1 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 1.
5
In the Maximum element growth rate text field, type 1.1.
6
Click  Build Selected.
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
Select Domains 4 and 6 only.
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 check box. In the associated text field, type 0.1.
6
Click  Build Selected.
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 Boundary.
4
Select Boundaries 18, 19, and 22 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size check box. In the associated text field, type 0.02.
8
Click  Build Selected.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 3, 5, and 7 only.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
Select Boundaries 11, 13, 15, 16, 18, 19, 21–25, 27, and 28 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
From the Thickness specification list, choose First layer.
5
In the Thickness text field, type 0.005.
6
Click  Build All.
The mesh is manually set up to make sure that a finer mesh is present in the iron domains. The resulting mesh should look like this:
Study 1 - Axisymmetric Magnetic Analysis
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - Axisymmetric Magnetic Analysis in the Label text field.
Step 2: Frequency Domain Perturbation
1
In the Model Builder window, under Study 1 - Axisymmetric Magnetic Analysis click Step 2: Frequency Domain Perturbation.
2
In the Settings window for Frequency Domain Perturbation, locate the Study Settings section.
3
Click  Range.
4
In the Range dialog box, choose Logarithmic from the Entry method list.
5
In the Start text field, type 100.
6
In the Stop text field, type 20000.
7
In the Steps per decade text field, type 10.
8
Click Replace.
9
In the Home toolbar, click  Compute.
Results
Coil Displacement
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, type Coil Displacement in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1 - Axisymmetric Magnetic Analysis/Solution Store 1 (sol2).
4
Locate the Line Data section. In row Point 1, set r to (4.5+4.61307)/2.
5
In row Point 1, set z to (-1.53-0.05)/2-0.5.
6
In row Point 2, set r to (4.5+4.61307)/2.
7
In row Point 2, set z to (-1.53-0.05)/2+0.5.
8
Click  Plot.
This line defines the center of the coil as it travels 0.5 mm in both directions from the initial position across the z-coordinate.
Coil properties from COMSOL
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Coil properties from COMSOL in the Label text field.
3
Locate the Plot Settings section. Select the Two y-axes check box.
4
Locate the Axis section. Select the x-axis log scale check box.
5
Select the Secondary y-axis log scale check box.
6
Locate the Legend section. From the Position list, choose Upper left.
Resistance
1
Right-click Coil properties from COMSOL and choose Global.
2
In the Settings window for Global, type Resistance in the Label text field.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
real(mf.VCoil_1/mf.ICoil_1)
Ω
Resistance
Reactance
1
In the Model Builder window, right-click Coil properties from COMSOL and choose Global.
2
In the Settings window for Global, type Reactance in the Label text field.
3
Locate the y-Axis section. Select the Plot on secondary y-axis check box.
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
imag(mf.VCoil_1/mf.ICoil_1)
Ω
Reactance
5
In the Coil properties from COMSOL toolbar, click  Plot.
The plot shows the impedance due to the voice coil and electromagnetic circuit. It will be combined with the other physics to generate a representative model of the microspeaker.
BL Factor
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type BL Factor in the Label text field.
3
Locate the Data section. From the Dataset list, choose Coil Displacement.
4
Click to expand the Title section. Locate the Plot Settings section.
5
Select the x-axis label check box. In the associated text field, type Voice coil offset (mm).
6
Locate the Title section. From the Title type list, choose Manual.
7
In the Title text area, type BL Factor (Wb/m).
8
Locate the Axis section. Select the Manual axis limits check box.
9
In the x minimum text field, type -0.5.
10
In the x maximum text field, type 0.5.
11
In the y minimum text field, type 0.
12
In the y maximum text field, type 0.85.
13
Locate the Legend section. From the Position list, choose Lower right.
BL From COMSOL
1
Right-click BL Factor and choose Line Graph.
2
In the Settings window for Line Graph, type BL From COMSOL in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type -int_BL(BL_integrand*coil_location_r*coil_location_z).
4
Select the Description check box. In the associated text field, type BL Factor.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type z-(-1.53[mm]-0.05[mm])/2.
7
Select the Description check box. In the associated text field, type Coil offset.
8
Click to expand the Coloring and Style section. From the Color list, choose Blue.
9
Click to expand the Legends section. Select the Show legends check box.
10
From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
COMSOL results
Through the use of the logical operators modifying the integrand, only the area of the coil around the point of interest is considered in the integration.
BL From COMSOL - inverted
1
In the Model Builder window, right-click BL Factor and choose Line Graph.
2
In the Settings window for Line Graph, type BL From COMSOL - inverted in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type -int_BL(BL_integrand*coil_location_r*coil_location_z).
4
Select the Description check box. In the associated text field, type BL Factor.
5
Click to expand the Title section. From the Title type list, choose None.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type -(z-(-1.53[mm]-0.05[mm])/2).
8
Select the Description check box. In the associated text field, type Coil Offset.
9
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dotted.
10
From the Color list, choose Blue.
BL From Test
1
Right-click BL Factor and choose Line Graph.
2
In the Settings window for Line Graph, type BL From Test in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type BL_Test(z-(-1.53[mm]-0.05[mm])/2).
4
Select the Description check box. In the associated text field, type BL Factor.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type z-(-1.53[mm]-0.05[mm])/2.
7
Select the Description check box. In the associated text field, type Coil Offset.
8
Locate the Coloring and Style section. From the Color list, choose Red.
9
Locate the Legends section. Select the Show legends check box.
10
From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
Test Results
BL From Test - inverted
1
Right-click BL Factor and choose Line Graph.
2
In the Settings window for Line Graph, type BL From Test - inverted in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type BL_Test(z-(-1.53[mm]-0.05[mm])/2).
4
Select the Description check box. In the associated text field, type BL Factor.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type -(z-(-1.53[mm]-0.05[mm])/2).
7
Select the Description check box. In the associated text field, type Coil Offset.
8
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dotted.
9
From the Color list, choose Red.
10
In the BL Factor toolbar, click  Plot.
The plot should look like Figure 6.
BL Factor, Coil properties from COMSOL, Magnetic Flux Density Norm (mf), Magnetic Flux Density Norm, Revolved Geometry (mf)
1
In the Model Builder window, under Results, Ctrl-click to select Magnetic Flux Density Norm (mf), Magnetic Flux Density Norm, Revolved Geometry (mf), Coil properties from COMSOL, and BL Factor.
2
Right-click and choose Group.
Postprocessing - Electromagnetic analysis
1
In the Settings window for Group, type Postprocessing - Electromagnetic analysis in the Label text field.
The electromagnetic results are now under a single group, which makes easier the navigation between results of different studies.
Evaluation Group - Coil Properties
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Evaluation Group - Coil Properties in the Label text field.
Global Evaluation 1
1
Right-click Evaluation Group - Coil Properties and choose Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(mf.VCoil_1/mf.ICoil_1)
Ω
Resistance
imag(mf.VCoil_1/mf.ICoil_1)
Ω
Reactance
4
In the Evaluation Group - Coil Properties toolbar, click  Evaluate.
The evaluation group is used to bring the impedance of the voice coil and the electromagnetic circuit into an electric circuit connected to the 3D model.
Evaluation Group - BL Factor
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Evaluation Group - BL Factor in the Label text field.
3
Locate the Data section. From the Parameter selection (freq) list, choose First.
Surface Average 1
1
Right-click Evaluation Group - BL Factor and choose Average>Surface Average.
2
Select Domain 2 only.
3
In the Settings window for Surface Average, locate the Expressions section.
4
In the table, enter the following settings:
Expression
Unit
Description
-mf.Br*N0*2*pi*r
Wb/m
 
5
From the Expression evaluated for list, choose Static solution.
6
Locate the Integration Settings section. Clear the Compute volume integral check box.
7
In the Evaluation Group - BL Factor toolbar, click  Evaluate.
This is the value of the BL factor predicted by the model that will be used the electric circuit to the other physics.
Global Definitions
Coil inductance from axisymmetric model
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Coil inductance from axisymmetric model in the Label text field.
3
Locate the Definition section. From the Data source list, choose Result table.
4
Find the Functions subsection. In the table, enter the following settings:
Function name
Position in file
Zreal
1
Zimag
2
5
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
Zreal
ohm
Zimag
ohm
6
In the Argument table, enter the following settings:
Argument
Unit
Column 1
Hz
7
Click  Plot.
The plot should look like this:
Parameters 1
1
In the Model Builder window, click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
BL
0.76519[Wb/m]
0.76519 Wb/m
Computed BL Factor
This is the value of the BL factor computed through the evaluation group previously.
Add Component
In the Model Builder window, right-click the root node and choose Add Component>3D.
Geometry 2
1
In the Settings window for Geometry, locate the Units section.
2
From the Length unit list, choose mm.
Import 1 (imp1)
1
In the Home toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
From the Source list, choose COMSOL Multiphysics file.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_geometry.mphbin.
6
Click  Import.
7
Click the  Wireframe Rendering button in the Graphics toolbar.
Definitions (comp2)
Variables 2
1
In the Model Builder window, under Component 2 (comp2) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_variables_3d.txt.
The first and fourth variables are used to update the density of the voice coil to make sure that the mass of the voice coil and the diaphragm match the measured mass. The second variable is the average velocity of the coil, which is used to couple the electric circuit with the other physics. The third variable is just a measurement of the displacement of the coil. The fifth variable is just the total acoustic energy traveling through a surface and will be used to confirm the validity of the PML.
Diaphragm triangular mesh
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Diaphragm triangular mesh in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 31, 32, 36–44, 50, 53, 55, 58–60, 63–65, 73, 74, 77, 83–85, 88–90, 93–98, 122–124, 135, 136, 143, 146–156, 162–164, 171–179, 225, 247, 251–254, 257, 258, 270–281, 283, 285, 286, 291–296, 299–301, 309–314, 316–321, and 323–334 only.
The selection should look like this:
Diaphragm mapped mesh
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Diaphragm mapped mesh in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 22, 23, 110, 117, 134, 191, 204, 206, 211, and 262 only.
Diaphragm
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, type Diaphragm 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 box, in the Selections to add list, choose Diaphragm triangular mesh and Diaphragm mapped mesh.
6
Click OK.
Coil
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Coil in the Label text field.
3
Select Domains 12 and 13 only.
The selection should look like this:
Air
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Air in the Label text field.
3
Select Domains 1–11 and 14–24 only.
The selection should look like this:
PML Top
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type PML Top in the Label text field.
3
Select Domain 2 only.
The selection should look like this:
PML Bottom
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type PML Bottom in the Label text field.
3
Select Domain 1 only.
The selection should look like this:
Exterior Field
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Exterior Field in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 15, 16, 189, and 227 only.
The selection should look like this:
Air Without PML
1
In the Definitions toolbar, click  Difference.
2
In the Settings window for Difference, type Air Without PML in the Label text field.
3
Locate the Input Entities section. Under Selections to add, click  Add.
4
In the Add dialog box, select Air in the Selections to add list.
5
Click OK.
6
In the Settings window for Difference, locate the Input Entities section.
7
Under Selections to subtract, click  Add.
8
In the Add dialog box, in the Selections to subtract list, choose PML Top and PML Bottom.
9
Click OK.
Inner gap
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Inner gap in the Label text field.
3
Select Domains 14, 15, 19, and 20 only.
The selection should look like this:
Outer gap
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Outer gap in the Label text field.
3
Select Domains 10, 11, 18, and 21 only.
The selection should look like this:
Outer channels
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Outer channels in the Label text field.
3
Select Domains 5, 7, 8, 17, and 22–24 only.
The selection should look like this:
The variable utilities menu grant access to the Mass Properties features that will be used to track the mass of the coil and the diaphragm.
Coil Mass
1
Right-click Definitions and choose Physics Utilities>Mass Properties.
2
In the Settings window for Mass Properties, type Coil Mass in the Label text field.
3
In the Name text field, type mass_coil.
4
Locate the Source Selection section. From the Selection list, choose Coil.
Diaphragm Mass
1
Right-click Definitions and choose Physics Utilities>Mass Properties.
2
In the Settings window for Mass Properties, type Diaphragm Mass in the Label text field.
3
In the Name text field, type mass_diaphragm.
4
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
5
From the Selection list, choose Diaphragm.
6
Locate the Density section. From the Density source list, choose From physics interface.
Coil Average
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Average.
2
In the Settings window for Average, type Coil Average in the Label text field.
3
In the Operator name text field, type ave_coil.
4
Locate the Source Selection section. From the Selection list, choose Coil.
PML Top Integral
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type PML Top Integral in the Label text field.
3
In the Operator name text field, type int_PML_Top.
4
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
5
Select Boundaries 15, 16, 189, and 227 only.
The selection should look like this:
PML Bottom Integral
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type PML Bottom Integral in the Label text field.
3
In the Operator name text field, type int_PML_Bottom.
4
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
5
Select Boundaries 11, 12, 187, and 220 only.
The selection should look like this:
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>Air.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
PEN 38 um
1
In the Model Builder window, under Component 2 (comp2) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type PEN 38 um in the Label text field.
3
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Diaphragm.
5
Locate the Material Properties section. In the Material properties tree, select Basic Properties>Density.
6
Click  Add to Material.
7
In the Material properties tree, select Basic Properties>Isotropic Structural Loss Factor.
8
Click  Add to Material.
9
In the Material properties tree, select Basic Properties>Poisson’s Ratio.
10
Click  Add to Material.
11
In the Material properties tree, select Basic Properties>Young’s Modulus.
12
Click  Add to Material.
13
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Density
rho
1360
kg/m³
Basic
Isotropic structural loss factor
eta_s
0.06
1
Basic
Poisson’s ratio
nu
0.3
1
Basic
Young’s modulus
E
7.20[GPa]
Pa
Basic
As described in Model Definition, it is critical to capture the correct properties of the diaphragm in order to have a representative model.
Coil
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Coil in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Coil.
4
Locate the Material Properties section. In the Material properties tree, select Basic Properties>Density.
5
Click  Add to Material.
6
In the Material properties tree, select Basic Properties>Poisson’s Ratio.
7
Click  Add to Material.
8
In the Material properties tree, select Basic Properties>Young’s Modulus.
9
Click  Add to Material.
10
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Density
rho
rho_coil
kg/m³
Basic
Poisson’s ratio
nu
0.3
1
Basic
Young’s modulus
E
0.5[GPa]
Pa
Basic
As described in Model Definition, the assumption that the voice coil presents isotropic properties is an oversimplification. An orthotropic material is probably a better approach to model the homogenized properties of the voice coil.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Study 1 - Axisymmetric Magnetic Analysis.
4
In the tree, select Acoustics>Pressure Acoustics>Pressure Acoustics, Frequency Domain (acpr).
5
Click Add to Component 2 in the window toolbar.
6
In the table, clear the Solve check box for Study 1 - Axisymmetric Magnetic Analysis.
7
In the tree, select Structural Mechanics>Solid Mechanics (solid).
8
Click Add to Component 2 in the window toolbar.
9
In the table, clear the Solve check box for Study 1 - Axisymmetric Magnetic Analysis.
10
In the tree, select Structural Mechanics>Shell (shell).
11
Click Add to Component 2 in the window toolbar.
12
In the table, clear the Solve check box for Study 1 - Axisymmetric Magnetic Analysis.
13
In the tree, select AC/DC>Electrical Circuit (cir).
14
Click Add to Component 2 in the window toolbar.
15
In the Home toolbar, click  Add Physics to close the Add Physics window.
With this step, the model has all the physics that will be used in the analysis.
Pressure Acoustics, Frequency Domain (acpr)
1
In the Model Builder window, under Component 2 (comp2) 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.
Exterior Field Calculation 1
1
In the Physics toolbar, click  Boundaries and choose Exterior Field Calculation.
2
In the Settings window for Exterior Field Calculation, locate the Boundary Selection section.
3
From the Selection list, choose Exterior Field.
4
Locate the Exterior Field Calculation section. From the Condition in the z = z0 plane list, choose Symmetric/Infinite sound hard boundary.
Narrow Region Acoustics 1
1
In the Physics toolbar, click  Domains and choose Narrow Region Acoustics.
2
In the Settings window for Narrow Region Acoustics, locate the Domain Selection section.
3
From the Selection list, choose Inner gap.
4
Locate the Duct Properties section. From the Duct type list, choose Slit.
5
In the h text field, type voice_coil_gap1.
Narrow Region Acoustics 2
1
In the Physics toolbar, click  Domains and choose Narrow Region Acoustics.
2
In the Settings window for Narrow Region Acoustics, locate the Domain Selection section.
3
From the Selection list, choose Outer gap.
4
Locate the Duct Properties section. From the Duct type list, choose Slit.
5
In the h text field, type voice_coil_gap2.
Narrow Region Acoustics 3
1
In the Physics toolbar, click  Domains and choose Narrow Region Acoustics.
2
In the Settings window for Narrow Region Acoustics, locate the Domain Selection section.
3
From the Selection list, choose Outer channels.
4
Locate the Duct Properties section. From the Duct type list, choose Rectangular duct.
5
In the W text field, type back_slits_w.
6
In the H text field, type back_slits_h.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 2 (comp2) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, locate the Domain Selection section.
3
From the Selection list, choose Coil.
Body Load - From Circuit
1
In the Physics toolbar, click  Domains and choose Body Load.
2
In the Settings window for Body Load, type Body Load - From Circuit in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Coil.
4
Locate the Force section. From the Load type list, choose Total force.
5
Specify the Ftot vector as
0
x
0
y
BL*cir.R1_i
z
This force is used to connect the electric circuit with the other physics.
Body Load - Applied Force
1
In the Physics toolbar, click  Domains and choose Body Load.
2
In the Settings window for Body Load, type Body Load - Applied Force in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Coil.
4
Locate the Force section. From the Load type list, choose Total force.
5
Specify the Ftot vector as
0
x
0
y
applied_force
z
This force is used to move the diaphragm around in the Cms analysis.
Shell (shell)
1
In the Model Builder window, under Component 2 (comp2) click Shell (shell).
2
In the Settings window for Shell, locate the Boundary Selection section.
3
From the Selection list, choose Diaphragm.
Linear Elastic Material 1
In the Model Builder window, under Component 2 (comp2)>Shell (shell) click Linear Elastic Material 1.
Damping 1
1
In the Physics toolbar, click  Attributes and choose Damping.
2
In the Settings window for Damping, locate the Damping Settings section.
3
From the Damping type list, choose Isotropic loss factor.
Thickness and Offset 1
1
In the Model Builder window, under Component 2 (comp2)>Shell (shell) click Thickness and Offset 1.
2
In the Settings window for Thickness and Offset, locate the Thickness and Offset section.
3
In the d0 text field, type d_diag.
4
From the Position list, choose User defined.
5
In the zreloffset text field, type 0.5.
The geometry of the diaphragm represents the bottom face of the solid, thus the model uses an offset in the shell definition.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundaries 22, 23, 191, and 262 only.
Electrical Circuit (cir)
In the Model Builder window, under Component 2 (comp2) click Electrical Circuit (cir).
Voltage Source 1 (V1)
1
In the Electrical Circuit toolbar, click  Voltage Source.
2
In the Settings window for Voltage Source, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
n
0
4
Locate the Device Parameters section. In the vsrc text field, type V0.
Resistor 1 (R1)
1
In the Electrical Circuit toolbar, click  Resistor.
2
In the Settings window for Resistor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
1
n
2
4
Locate the Device Parameters section. In the R text field, type Zreal(freq) + i*Zimag(freq).
This resistor uses the complex impedance from the electromagnetic axisymmetric model.
Voltage Source 2 (V2)
1
In the Electrical Circuit toolbar, click  Voltage Source.
2
In the Settings window for Voltage Source, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
2
n
0
4
Locate the Device Parameters section. In the vsrc text field, type BL*v0.
This voltage source represents the feedback from the other physics into the electric circuit.
Definitions (comp2)
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
In the Settings window for Perfectly Matched Layer, locate the Domain Selection section.
3
From the Selection list, choose PML Top.
4
Locate the Geometry section. From the Type list, choose Spherical.
5
Locate the Scaling section. In the PML scaling curvature parameter text field, type 3.
The scaling curvature is updated to make sure that the PML absorbs the incident waves in the complete range of analysis.
Perfectly Matched Layer 2 (pml2)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
In the Settings window for Perfectly Matched Layer, locate the Domain Selection section.
3
From the Selection list, choose PML Bottom.
4
Locate the Geometry section. From the Type list, choose Spherical.
5
Find the Center coordinate subsection. In the table, enter the following settings:
Xm (m)
Ym (m)
Zm (m)
0
0
-2.75 [mm]
6
Locate the Scaling section. In the PML scaling curvature parameter text field, type 3.
The scaling curvature is updated to make sure that the PML absorbs the incident waves in the complete range of analysis.
Multiphysics
Solid-Thin Structure Connection 1 (sshc1)
1
In the Model Builder window, under Component 2 (comp2) right-click Multiphysics and choose Solid-Thin Structure Connection.
2
In the Settings window for Solid-Thin Structure Connection, locate the Connection Settings section.
3
From the Connection type list, choose Shared boundaries.
Acoustic-Structure Boundary - Solid
1
In the Model Builder window, right-click Multiphysics and choose Acoustic-Structure Boundary.
2
In the Settings window for Acoustic-Structure Boundary, type Acoustic-Structure Boundary - Solid in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose All boundaries.
Acoustic-Structure Boundary - Shell
1
Right-click Multiphysics and choose Acoustic-Structure Boundary.
2
In the Settings window for Acoustic-Structure Boundary, type Acoustic-Structure Boundary - Shell in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose All boundaries.
4
Locate the Coupled Interfaces section. From the Structure list, choose Shell (shell).
Mesh 2
This mesh is set up manually to reduce the number of elements in the model and to control the mesh density in the diaphragm. Proceed by directly adding the desired mesh component. 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. Here we use 6 elements per wavelength.
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 lam0/6.
5
In the Minimum element size text field, type 1[mm].
6
In the Curvature factor text field, type 0.5.
7
Click  Build Selected.
Mapped 1
1
In the Mesh toolbar, click  Boundary and choose Mapped.
2
In the Settings window for Mapped, locate the Boundary Selection section.
3
From the Selection list, choose Diaphragm mapped mesh.
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 check box. In the associated text field, type 0.6[mm].
Distribution 1
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Edge 467 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 2.
Free Triangular 1
1
In the Mesh toolbar, click  Boundary and choose Free Triangular.
2
In the Settings window for Free Triangular, locate the Boundary Selection section.
3
From the Selection list, choose Diaphragm triangular mesh.
Size 1
1
Right-click Free Triangular 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 check box. In the associated text field, type 0.6[mm].
6
Select the Minimum element size check box. In the associated text field, type 0.2[mm].
Swept 1
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Coil.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
Swept 2
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Inner gap.
Distribution 1
1
Right-click Swept 2 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
Swept 3
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Outer gap.
Distribution 1
1
Right-click Swept 3 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
Free Tetrahedral 1
1
In the Model Builder window, under Component 2 (comp2)>Mesh 2 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 Domains 3–9, 16, 17, and 22–24 only.
5
Click  Build All.
Swept 4
In the Mesh toolbar, click  Swept.
Distribution 1
1
Right-click Swept 4 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 8.
4
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 Domains 3 and 4 only.
5
Click to expand the Transition section. Clear the Smooth transition to interior mesh check box.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
Select Boundaries 11, 12, 15, 16, 187, 189, 220, and 227 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
In the Number of layers text field, type 1.
5
Click  Build Selected.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check boxes for Magnetic Fields (mf), Pressure Acoustics, Frequency Domain (acpr), and Electrical Circuit (cir).
4
Find the Multiphysics couplings in study subsection. In the table, clear the Solve check boxes for Acoustic-Structure Boundary - Solid (asb1) and Acoustic-Structure Boundary - Shell (asb2).
5
Find the Studies subsection. In the Select Study tree, select General Studies>Stationary.
6
Click Add Study in the window toolbar.
7
In the Model Builder window, click the root node.
8
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2 - CMS Extraction
1
In the Settings window for Study, type Study 2 - CMS Extraction in the Label text field.
This step applies a varying force on the voice coil and measures the displacement to obtain the nonlinear Cms curve. Only the Solid Mechanics and the Shell interface will be considered in this step as the other physics do not modify the Cms curve.
Step 1: Stationary
1
In the Model Builder window, under Study 2 - CMS Extraction click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Study Settings section.
3
Select the Include geometric nonlinearity check box.
4
Click to expand the Mesh Selection section. In the table, enter the following settings:
Component
Mesh
Component 1
No mesh
5
Click to expand the Study Extensions section. Select the Auxiliary sweep check box.
6
Click  Add.
7
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
applied_force (Applied force)
range(-1,0.1,1)
N
8
Right-click Study 2 - CMS Extraction>Step 1: Stationary and choose Get Initial Value for Step.
Solution 3 (sol3)
1
In the Settings window for Stationary Solver, locate the General section.
2
In the Relative tolerance text field, type 1e-5.
The tolerance is manually reduced to assure that numerical noise will not affect the Cms curve obtained.
3
In the Home toolbar, click  Compute.
Results
Evaluation Group - Force Displacement Curve
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Evaluation Group - Force Displacement Curve in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 - CMS Extraction/Solution 3 (4) (sol3).
4
Click to expand the Format section. From the Include parameters list, choose Off.
Global Evaluation 1
1
Right-click Evaluation Group - Force Displacement Curve and choose Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
ave_coil(w)
 
Average coil displacement
applied_force
N
Applied force
4
In the Evaluation Group - Force Displacement Curve toolbar, click  Evaluate.
Through this evaluation group, the force/displacement characteristics of the diaphragm are exported in a table that will be used in further steps.
Global Definitions
Force displacement from COMSOL
1
In the Home toolbar, click  Functions and choose Global>Interpolation.
2
In the Settings window for Interpolation, type Force displacement from COMSOL in the Label text field.
3
Locate the Definition section. From the Data source list, choose Result table.
4
From the Table from list, choose Evaluation Group - Force Displacement Curve.
5
Find the Functions subsection. In the table, enter the following settings:
Function name
Position in file
Force
1
6
Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Piecewise cubic.
7
From the Extrapolation list, choose Nearest function.
8
Locate the Units section. In the Argument table, enter the following settings:
Argument
Unit
Column 1
mm
9
In the Function table, enter the following settings:
Function
Unit
Force
N
10
Click  Plot.
This function is declared to facilitate the derivation of the force-displacement curve, the stiffness. The plot should look like this:
Study 2 - CMS Extraction
1
In the Study toolbar, click  Update Solution.
With the study update, the function previously declared will be available in the results of the CMS Extraction study.
Results
Grid 1D 1
1
In the Results toolbar, click  More Datasets and choose Grid>Grid 1D.
2
In the Settings window for Grid 1D, locate the Data section.
3
From the Dataset list, choose Study 2 - CMS Extraction/Solution 3 (4) (sol3).
4
Locate the Parameter Bounds section. In the Minimum text field, type -0.5.
5
In the Maximum text field, type 0.5.
CMS vs Displacement
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type CMS vs Displacement in the Label text field.
3
Locate the Data section. From the Dataset list, choose Grid 1D 1.
4
From the Parameter selection (applied_force) list, choose Last.
5
Locate the Title section. From the Title type list, choose Manual.
6
In the Title text area, type CMS (mm/N) vs Displacement (mm).
7
Locate the Plot Settings section.
8
Select the x-axis label check box. In the associated text field, type Voice coil offset (mm).
9
Locate the Axis section. Select the Manual axis limits check box.
10
In the x minimum text field, type -0.5.
11
In the x maximum text field, type 0.5.
12
In the y minimum text field, type 0.
13
In the y maximum text field, type 0.7.
14
Locate the Legend section. From the Position list, choose Lower left.
CMS curve from test
1
Right-click CMS vs Displacement and choose Line Graph.
2
In the Settings window for Line Graph, type CMS curve from test in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type CMS_Test(x).
4
In the Unit field, type mm/N.
5
Locate the Title section. From the Title type list, choose None.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type x.
8
Locate the Legends section. Select the Show legends check box.
9
From the Legends list, choose Manual.
10
In the table, enter the following settings:
Legends
CMS Test
CMS curve from test (inverted)
1
Right-click CMS curve from test and choose Duplicate.
2
In the Settings window for Line Graph, type CMS curve from test (inverted) in the Label text field.
3
Locate the x-Axis Data section. In the Expression text field, type -x.
4
Locate the Legends section. Clear the Show legends check box.
5
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dotted.
6
From the Color list, choose Blue.
CMS curve from COMSOL
1
In the Model Builder window, right-click CMS vs Displacement and choose Line Graph.
2
In the Settings window for Line Graph, type CMS curve from COMSOL in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type 1/(d(Force(x),x)).
4
In the Unit field, type mm/N.
5
Locate the Title section. From the Title type list, choose None.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type x.
8
Locate the Coloring and Style section. From the Color list, choose Red.
9
Locate the Legends section. Select the Show legends check box.
10
From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
CMS COMSOL
CMS curve from COMSOL (inverted)
1
Right-click CMS curve from COMSOL and choose Duplicate.
2
In the Settings window for Line Graph, type CMS curve from COMSOL (inverted) in the Label text field.
3
Locate the x-Axis Data section. In the Expression text field, type -x.
4
Locate the Legends section. Clear the Show legends check box.
5
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dotted.
6
In the CMS vs Displacement toolbar, click  Plot.
The plot should look like Figure 7.
CMS vs Displacement, Stress (shell), Stress (solid)
1
In the Model Builder window, under Results, Ctrl-click to select Stress (solid), Stress (shell), and CMS vs Displacement.
2
Right-click and choose Group.
Postprocessing - CMS Analysis
1
In the Settings window for Group, type Postprocessing - CMS Analysis in the Label text field.
Grouping the plots facilitates the navigation through the different output in the file.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Magnetic Fields (mf).
4
Find the Studies subsection. In the Select Study tree, select General Studies>Frequency Domain.
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
The magnetic study is turned off as its contribution is included in the electric circuit.
Study 3 - Acoustic Analysis
1
In the Model Builder window, click Study 3.
2
In the Settings window for Study, type Study 3 - Acoustic Analysis in the Label text field.
Step 1: Frequency Domain
1
In the Model Builder window, under Study 3 - Acoustic Analysis 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 box, 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 20000.
7
From the Interval list, choose 1/12 octave.
8
Click Replace.
9
In the Settings window for Frequency Domain, locate the Physics and Variables Selection section.
10
Select the Modify model configuration for study step check box.
11
In the tree, select Component 2 (comp2)>Solid Mechanics (solid)>Body Load - Applied Force.
12
Right-click and choose Disable.
13
Click to expand the Mesh Selection section. In the table, enter the following settings:
Component
Mesh
Component 1
No mesh
14
Right-click Study 3 - Acoustic Analysis>Step 1: Frequency Domain and choose Get Initial Value for Step.
Solution 4 (sol4)
1
In the Model Builder window, under Study 3 - Acoustic Analysis>Solver Configurations>Solution 4 (sol4) right-click Stationary Solver 1 and choose Fully Coupled.
2
Right-click Study 3 - Acoustic Analysis>Solver Configurations>Solution 4 (sol4)>Stationary Solver 1>Suggested Iterative Solver (GMRES with DirectPrecond.) (asb1_sshc1_asb2) and choose Enable.
3
In the Model Builder window, expand the Study 3 - Acoustic Analysis>Solver Configurations>Solution 4 (sol4)>Stationary Solver 1>Suggested Iterative Solver (GMRES with DirectPrecond.) (asb1_sshc1_asb2) node, then click Direct Preconditioner 2.
4
In the Settings window for Direct Preconditioner, click to expand the Hybridization section.
5
Under Preconditioner variables, click  Add.
6
In the Add dialog box, select Current through device R1 (comp2.currents) in the Preconditioner variables list.
7
Click OK.
8
In the Settings window for Direct Preconditioner, click  Compute.
Results
Thermoviscous results
1
In the Results toolbar, click  Table.
2
In the Settings window for Table, locate the Data section.
3
Click Import.
4
Browse to the model’s Application Libraries folder and double-click the file ow_microspeaker_spl_39mm_thermoviscous.txt.
5
In the Label text field, type Thermoviscous results.
As described in Model Definition, this table contains the results of a thermoviscous model that will be compared to the tutorial results and the measurements.
Outgoing acoustic energy
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Outgoing acoustic energy in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3 - Acoustic Analysis/Solution 4 (6) (sol4).
4
Locate the Plot Settings section.
5
Select the x-axis label check box. In the associated text field, type freq (Hz).
6
Select the y-axis label check box. In the associated text field, type Energy (W).
7
Locate the Axis section. Select the x-axis log scale check box.
8
Select the y-axis log scale check box.
9
Locate the Legend section. From the Position list, choose Lower middle.
Global 1
1
Right-click Outgoing acoustic energy 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
int_PML_Top(I_int)
W
PML Top Integral
int_PML_Bottom(I_int)
W
PML Bottom Integral
4
In the Outgoing acoustic energy toolbar, click  Plot.
The plot shows the acoustic energy traveling to the PML. This plot shows positive values for the entire frequency range, as expected for a working PML.
Speaker sensitivity at 39 mm
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Speaker sensitivity at 39 mm in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3 - Acoustic Analysis/Solution 4 (6) (sol4).
4
Locate the Title section. From the Title type list, choose Manual.
5
In the Title text area, type SPL at 39 mm (dB).
6
Locate the Plot Settings section.
7
Select the x-axis label check box. In the associated text field, type freq(Hz).
8
Select the y-axis label check box. In the associated text field, type SPL(dB).
9
Locate the Axis section. Select the Manual axis limits check box.
10
In the x minimum text field, type 100.
11
In the x maximum text field, type 20000.
12
In the y minimum text field, type 50.
13
In the y maximum text field, type 100.
14
Select the x-axis log scale check box.
15
Locate the Legend section. From the Position list, choose Lower right.
Global 1
1
Right-click Speaker sensitivity at 39 mm 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
SPL_Test(freq)
dB
Measured SPL at 39 mm
4
Click to expand the Coloring and Style section. From the Width list, choose 3.
Speaker sensitivity at 39 mm
In the Model Builder window, click Speaker sensitivity at 39 mm.
Octave Band 1
1
Right-click Speaker sensitivity at 39 mm and choose More Plots>Octave Band.
2
In the Settings window for Octave Band, locate the Selection section.
3
From the Geometric entity level list, choose Global.
4
Locate the y-Axis Data section. In the Expression text field, type pext(0,0,39[mm]).
5
Locate the Plot section. From the Quantity list, choose Continuous power spectral density.
6
Click to expand the Legends section. Select the Show legends check box.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
COMSOL results
Table Graph 1
1
Right-click Speaker sensitivity at 39 mm and choose Table Graph.
2
In the Speaker sensitivity at 39 mm toolbar, click  Plot.
3
In the Model Builder window, click Table Graph 1.
4
In the Settings window for Table Graph, click to expand the Legends section.
5
Select the Show legends check box.
6
From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
COMSOL results with thermoviscous
8
In the Speaker sensitivity at 39 mm toolbar, click  Plot.
The plot should look like the one in Figure 10.
Total impedance
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Total impedance in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3 - Acoustic Analysis/Solution 4 (6) (sol4).
4
Locate the Plot Settings section.
5
Select the x-axis label check box. In the associated text field, type freq (Hz).
6
Select the y-axis label check box. In the associated text field, type Impedance (ohm).
7
Locate the Axis section. Select the x-axis log scale check box.
Global 1
1
Right-click Total impedance 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
Z_Test(freq)
Ω
Measured Impedance
abs(-cir.V1_v/cir.V1_i)
Ω
COMSOL results
4
In the Total impedance toolbar, click  Plot.
The plot should look like the one in Figure 8.
Acoustic Pressure (acpr), Acoustic Pressure, Isosurfaces (acpr), Exterior-Field Pressure (acpr), Exterior-Field Sound Pressure Level (acpr), Exterior-Field Sound Pressure Level xy-plane (acpr), Outgoing acoustic energy, Sound Pressure Level (acpr), Speaker sensitivity at 39 mm, Stress (shell) 1, Stress (solid) 1, Total impedance
1
In the Model Builder window, under Results, Ctrl-click to select Acoustic Pressure (acpr), Sound Pressure Level (acpr), Acoustic Pressure, Isosurfaces (acpr), Exterior-Field Sound Pressure Level (acpr), Exterior-Field Pressure (acpr), Exterior-Field Sound Pressure Level xy-plane (acpr), Stress (solid) 1, Stress (shell) 1, Outgoing acoustic energy, Speaker sensitivity at 39 mm, and Total impedance.
2
Right-click and choose Group.
Postprocessing - Acoustic Analysis
1
In the Settings window for Group, type Postprocessing - Acoustic Analysis in the Label text field.
The acoustics results are now under a single group, which simplifies the navigation between results of different studies.
Now, proceed to generate a plot showing the displacement of the diaphragm and the voice coil, similar to the plots in Figure 9.
Thumbnail
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Thumbnail in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3 - Acoustic Analysis/Solution 4 (6) (sol4).
4
From the Parameter value (freq (Hz)) list, choose 7500.
5
Locate the Color Legend section. Select the Show units check box.
6
Click to expand the Title section. From the Title type list, choose Manual.
7
In the Title text area, type Surface: Displacement magnitude (mm).
8
In the Parameter indicator text field, type freq = eval(acpr.freq) Hz.
9
Locate the Plot Settings section. Clear the Plot dataset edges check box.
Shell displacement
1
Right-click Thumbnail and choose Surface.
2
In the Settings window for Surface, type Shell displacement in the Label text field.
3
Locate the Expression section. In the Expression text field, type shell.disp.
4
Locate the Coloring and Style section. Click  Change Color Table.
5
In the Color Table dialog box, select Rainbow>SpectrumLight in the tree.
6
Click OK.
Deformation 1
1
Right-click Shell displacement and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the x-component text field, type u2.
4
In the y-component text field, type v2.
5
In the z-component text field, type w2.
Filter 1
1
In the Model Builder window, right-click Shell displacement and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type x>0.
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.
6
Click to expand the Inherit Style section. From the Plot list, choose Shell displacement.
7
Clear the Color check box.
8
Clear the Color and data range check box.
9
Clear the Tube radius scale factor check box.
Deformation 1
1
Right-click Line 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the x-component text field, type u2.
4
In the y-component text field, type v2.
5
In the z-component text field, type w2.
Filter 1
1
In the Model Builder window, right-click Line 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type x>0.
Solid Displacement
1
In the Model Builder window, right-click Thumbnail and choose Surface.
2
In the Settings window for Surface, type Solid Displacement in the Label text field.
3
Locate the Expression section. In the Expression text field, type solid.disp.
4
Click to expand the Inherit Style section. From the Plot list, choose Shell displacement.
Deformation 1
Right-click Solid Displacement and choose Deformation.
Filter 1
1
In the Model Builder window, right-click Solid Displacement and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type x>0.
Line 2
In the Model Builder window, under Results>Thumbnail right-click Line 1 and choose Duplicate.
Deformation 1
1
In the Model Builder window, expand the Line 2 node, then click Deformation 1.
2
In the Settings window for Deformation, locate the Expression section.
3
In the x-component text field, type u.
4
In the y-component text field, type v.
5
In the z-component text field, type w.
Solid Displacement Section
1
In the Model Builder window, right-click Thumbnail and choose Slice.
2
In the Settings window for Slice, type Solid Displacement Section in the Label text field.
3
Locate the Expression section. In the Expression text field, type solid.disp.
4
Locate the Plane Data section. From the Entry method list, choose Coordinates.
5
Click to expand the Inherit Style section. From the Plot list, choose Shell displacement.
Deformation 1
1
Right-click Solid Displacement Section and choose Deformation.
2
In the Thumbnail toolbar, click  Plot.
The plot should look like Figure 9. You can cycle through the phase of the solution by updating the phase in the dataset options.