Loudspeaker Driver — Frequency-Domain Analysis

In the tree, select .

The Model Wizard lets you select the first one of the study steps you plan to use in the model. Select a stationary study used for solving the static magnetic fields.

In the tree, select .

When working with your own modeling project of an acoustic driver, you will typically either draw the geometry in COMSOL Multiphysics, or import a CAD file of the driver itself and add the surrounding air and PML domains. Here, the entire geometry is imported as a sequence from the geometry file. The instructions to the geometry can be found in the appendix at the end of this document.

Geometry 1

In the toolbar, click .

Browse to the model’s Application Libraries folder and double-click the file loudspeaker_driver_geom_sequence.mph.

In the toolbar, click .

Click the button in the toolbar.

Global Definitions

Parameters 1

Enter the model parameters or load them from the file loudspeaker_driver_parameters.txt. Here, as well as in all following sections, the Description field helps you keep track of what you are doing, but is completely optional.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
N0	100	100	Number of turns in coil
V0	4[V]	4 V	Peak driving voltage
f0	40[Hz]	40 Hz	Frequency at which loss factor is given
omega0	2pif0	251.33 Hz	Angular frequency at which loss factor is given

The loss factor frequency definition will be used when setting up the structural properties.

Definitions

Create a selection for the coil, this will simplify setting up the physics.

Explicit 1

In the toolbar, click .

In the window for , type Coil in the text field.

Select Domain 8 only.

Perfectly Matched Layer 1 (pml1)

In the toolbar, click .

Use the Perfectly Matched Layers (PMLs) to model a nonreflecting condition-like behavior and avoid unphysical reflections (spurious reflections) where the sound leaves the model. This feature is also set up under . This makes them available for any physics interface that needs them.

Select Domains 1 and 5 only.

Materials

While the material properties used in this model are partly made up, they resemble those used in a real driver. The diaphragm and dust cap both consist of a HexaCone®-like material; a light and very stiff composite. The apex has properties representative of glass fiber materials. The spider, acting as a spring, is made of a phenolic cloth with a much lower stiffness. The material used in the coil is taken to be lighter than copper, as the wire is insulated and does not completely fill the coil domain. The surround, finally, is a light resistive foam.

Except for air and soft Iron, the materials you will use all come from a material library created especially for this model (to be loaded from the file loudspeaker_driver_materials.mph). You may notice that some of the materials will report missing properties. For example, the composite does not include any electromagnetic properties. This is fine, as you will not model the magnetic fields in the domains where the composite is used.

Add Material

In the toolbar, click to open the window.

Go to the window.

Click in the window toolbar.

First, add air which will be present everywhere in your geometry. Next, switch to using nonlinear Iron in the pole piece and top plate.

Click in the window toolbar.

Materials

Air (mat1)

Soft Iron (With Losses) (mat2)

In the window, click .

Right-click and choose .

Material Browser

In the window, In the ribbon make sure to select the tab and then click the icon.

The functionality is activated by clicking the small icon at the bottom right, below the Material Browser tree.

Click .

Browse to the model’s Application Libraries folder and double-click the file loudspeaker_driver_materials.mph.

Add Material

Go to the window.

Click in the window toolbar.

Click in the window toolbar.

Click in the window toolbar.

Click in the window toolbar.

Click in the window toolbar.

Select Domains 3 and 10 only.

Click in the window toolbar.

In the toolbar, click to close the window.

Materials

Composite (mat3)

In the window, under click .

Cloth (mat4)

In the window, click .

Select Domain 9 only.

Foam (mat5)

In the window, click .

Select Domain 13 only.

Coil (mat6)

In the window, click .

Select Domain 8 only.

Glass Fiber (mat7)

In the window, click .

Select Domain 7 only.

Generic Ferrite (mat8)

In the window, click .

Select Domain 12 only.

Magnetic Fields (mf)

The Magnetic Fields equation needs to be solved in and around the magnetic motor. To reduce simulation time, make this physics interface active only where it is needed. You can remove all domains where you expect the magnetic field to be negligible.

In the window, under click .

Select Domains 2, 6–9, 11, and 12 only.

Ampère’s Law is per default solved in all domains where the physics interface is active. Add a second instance of it to apply to the permanent magnet, where you need a different constitutive relation.

Ampère’s Law 2

In the toolbar, click and choose .

Select Domain 12 only.

In the window for , locate the section.

From the list, choose .

Specify the e vector as


0	r
0	phi
1	z

This setting gives a static remanent flux density equal to 0.4 T in the z direction. This will create a static magnetic field distribution in the model, providing the linearization point for the frequency domain study.

Ampère’s Law 3

In the toolbar, click and choose .

The B-H curve is provided by the soft iron material.

In the window for , locate the section.

From the list, choose .

Coil 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the N text field, type N0.

In the acoil text field, type 3.5e-8[m^2].

With N0 = 100 turns, the total cross-sectional area covered by the wires will be 3.5e-6 m2. The area of the coil domain is 6e-6 m2, making the fill factor approximately 60 %.

Locate the section. From the list, choose .

In the Vcoil text field, type linper(V0).

This is the driving voltage. Because the linper() operator is used it will kick in only in the study.

Pressure Acoustics, Frequency Domain (acpr)

Select the air domains above and under the speaker as well as the PML regions.

In the window, under click .

Select Domains 1, 2, 4, and 5 only.

Exterior Field Calculation 1

In the toolbar, click and choose .

The exterior-field calculation requires a source boundary encompassing all local sound sources, and with a symmetry plane to account for the infinite baffle. After computing the solution, you can evaluate the pressure in any point (r,z) outside the domain by entering pext(r,z).

Select Boundary 70 only.

In the window for , locate the section.

From the list, choose .

Solid Mechanics (solid)

In the window, under click .

Select Domains 3, 7–10, and 13 only.

With the above selection, you leave out the magnet, pole piece, and top plate. You will consider these domains as perfectly rigid, by using the default sound hard wall condition on their surfaces.

Add damping to some of the solid material.

Linear Elastic Material 1

In the window, under click .

Damping 1

In the toolbar, click and choose .

In the window for , locate the section.

Click .

Select Domains 3, 7, and 10 only.

Locate the section. From the list, choose .

Linear Elastic Material 1

In the window, click .

Damping 2

In the toolbar, click and choose .

In the window for , locate the section.

Click .

Select Domain 9 only.

Locate the section. In the βdK text field, type 0.14/omega0.

Linear Elastic Material 1

In the window, click .

Damping 3

In the toolbar, click and choose .

In the window for , locate the section.

Click .

Select Domain 13 only.

Locate the section. In the βdK text field, type 0.46/omega0.

The spider and the surround are attached to the case.

Fixed Constraint 1

In the toolbar, click and choose .

Select Boundaries 58 and 62 only.

Now is a good time to inspect multiphysics coupling under the node. When using a predefined multiphysics interface the coupling is automatically applied to all acoustic-solid boundaries.

Now, the multiphysics feature is added to handle Lorentz force on the coil (it represents the product of the time-harmonic current and the static magnetic field in which it is traveling). For details, see Notes About the COMSOL Implementation.

Multiphysics

Lorentz Coupling 1 (ltzc1)

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

The mesh used in computing the impedance needs to resolve the induced eddy currents in the pole piece and the top plate. For the results to be accurate, the skin depth needs to be resolved by at least 1, preferably 2 quadratic elements.

With a conductivity of 1.12e7 S/m and a peak relative permeability of 1200, the skin depth in the iron at the maximum frequency of 3.5 kHz does not go below 0.07 mm. In practice, most of the induced currents will run in regions of the pole piece where the biased relative permeability is much less than 1200, which makes the skin depth greater. In this model, it is therefore sufficient to use a mesh size of 0.2 mm along the iron surfaces that are closest to the voice coil.

For the acoustic-structure interaction, the air domain and the thin moving structures also need to be well resolved. The Extra fine setting gives a maximum element size of 6 mm which is smaller than the suggested 6 elements per wavelength (this only requires a 16 mm mesh size). The PML is preferably meshed with mapped elements, use 8 elements for the default polynomial scaling.

Mapped 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 1 and 5 only.

Distribution 1

Right-click and choose .

Select Boundaries 1 and 5 only.

In the window for , locate the section.

In the text field, type 8.

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Free Triangular 1

In the window, right-click and choose .

Size 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 1[mm].

Boundary Layers 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Boundary Layer Properties

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type 3.

From the list, choose .

In the text field, type 0.2[mm].

Click .

Study 1

Your Study node already contains the Stationary study that you picked from the Model Wizard. Disable the Pressure Acoustics and Solid Mechanics interfaces.

Add a Frequency-Domain, Perturbation study.

In the window, click .

In the window for , locate the section.

Clear the check box.

Step 1: Stationary

In the window, under click .

In the window for , locate the section.

In the table, clear the check box for and .

Frequency Domain Perturbation

In the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 10^(range(1,(log10(3500)-1)/39,log10(3500))).

This generates 40 exponentially distributed frequencies between 10 Hz and 3.5 kHz.

Locate the section. In the table, clear the check box for and .

NOTE: The first solution of this model is only of the electromagnetic part of the problem, with the coil assumed to be fixed in order to extract the BL factor and the blocked coil impedance. If you would like to skip ahead to the solution of the full electroacoustic problem, you can do so by proceeding from here to the instructions starting at the section , further below. Otherwise, continue by computing the solution.

In the toolbar, click .

Results

First, create a 2D plot to view the magnetic field distribution from the permanent magnet.

2D Plot Group 1

In the toolbar, click and choose .

In the window for , type Static Magnetic Field in the text field.

Locate the section. From the list, choose .

Each step in a study creates its own data set. The numbering of the data sets begins from the last step. Hence in this model Solution 2 contains the stationary solution and Solution 1 the frequency domain perturbation.

You are now looking at the magnetic field norm created by the permanent magnet. Note that it has a distinct maximum in the gap where the voice coil is moving. If you zoom in a little, the plot should look like Figure 3.

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the toolbar, click .

Another interesting result is the permeability distribution in the iron. Begin by duplicating the existing plot.

Static Magnetic Field 1

In the window, right-click and choose .

In the window for , type Effective Permeability in the text field.

This is the effective relative permeability at the linearization point, as given by the ratio of the flux density and the field multiplied by the permeability of vacuum. The plot should look like Figure 4.

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type mf.normB/(mu0_const*mf.normH).

In the toolbar, click .

Derived Values

Next, evaluate the BL force factor.

Surface Average 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
-mf.BrN02pir	Wb/m

Locate the section. Clear the check box.

The BL factor evaluates to 7.53 Wb/m.

Next, study the induced current density at a few different frequencies.

2D Plot Group 3

In the toolbar, click .

In the window for , type Induced Current Density in the text field.

At 3500 Hz, the induced currents are highly localized to the surfaces of the top plate and pole piece.

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

The check box, if turned on, wraps the expression in a lindev() operator. For more information, search for lindev in the COMSOL Multiphysics Documentation.

In the toolbar, click .

Induced Current Density

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click .

At 10 Hz, the induced currents distribute throughout the material. The distribution depends on the vicinity to the voice coil, but also very much on the local effective permeability from the static study. To reproduce Figure 5, try two frequencies in between.

From the list, choose .

Before proceeding to the full electroacoustic analysis, plot the blocked coil inductance as a function of the frequency.

1D Plot Group 4

In the toolbar, click and choose .

In the window for , type Blocked Coil Inductance in the text field.

Global 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click .

The result should look like Figure 6.

Having extracted the force factor and the voice coil inductance, you now have most of the electromagnetic data required for a separate model of the acoustic parts of a boxed loudspeaker driver. See the Loudspeaker Driver in a Vented Enclosure model also in the Application Library. What remains is the real part of the blocked coil impedance. This is available as mf.RCoil_1, the Coil Resistance. If you would like to extract this too, you can set it up in a Global plot just like the inductance, and export the results as described in the following steps.

Plot 1

Right-click and choose .

If you want to export the data, you can now enter a file name and click the button.

Component 1 (comp1)

It is now time to compute and evaluate the solution of the entire model, including the acoustic-structure interaction. In order to this, you will set up a new study. Although you could technically reuse the static magnetic fields solution from the first study, it can be handy to include this step in the new study too. This allows you to make changes anywhere in the model, including such that affect the static magnetic fields, and run only the new study again to get correctly updated results.

Add Study

In the toolbar, click to open the window.

Go to the window.

Select an empty study so that you can copy and modify the study steps from your previous study.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Study 1

In the window, click .

In the window for , type Magnetic Fields Study in the text field.

Step 1: Stationary, Step 2: Frequency Domain Perturbation

In the window, under , Ctrl-click to select and .

Right-click and choose .

Study 2

Step 1: Stationary

In the window, right-click and choose .

In the window for , type Complete Study in the text field.

Locate the section. Clear the check box.

Step 2: Frequency Domain Perturbation

Make the Frequency Domain, Perturbed study solve for the acoustics and the solid physics too.

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Discretization
Pressure Acoustics, Frequency Domain (acpr)	√	Physics settings
Solid Mechanics (solid)	√	Physics settings

In the toolbar, click .

Results

You will now have received two more data sets: Solution 4 containing the same static magnetic fields as Solution 2, and Solution 3 with the frequency domain electromagnetic and acoustic-structure interaction results. In order to get a good overview of the latter, begin by adding a selection of all domains except the PMLs. You will then create a Revolution data set in order to enable a 3D view of the model.

Selection

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 2–4 and 6–13 only.

Revolution 2D 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Click to expand the section. In the text field, type -80.

In the text field, type 250.

Create a 3D plot of the instantaneous pressure distribution at zero phase in and around the speaker.

3D Plot Group 5

In the toolbar, click .

In the window for , type Instantaneous Pressure in the text field.

Right-click and choose .

In the window for , locate the section.

In the text field, type acpr.p_t.

Locate the section. From the list, choose .

Select the check box.

Contour 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type acpr.p_t.

Locate the section. From the list, choose .

From the list, choose .

Clear the check box.

Next, reproduce Figure 7 with a plot of the local sound pressure level.

3D Plot Group 6

In the toolbar, click .

In the window for , type Sound Pressure Level in the text field.

Create a 1D plot of the sensitivity versus the frequency, as in Figure 8.

Right-click and choose .

In the window for , locate the section.

In the text field, type acpr.Lp.

In the toolbar, click .

1D Plot Group 7

In the toolbar, click and choose .

In the window for , type Sensitivity in the text field.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

In the text area, type Sensitivity.

Locate the section. Select the check box.

In the associated text field, type Frequency (Hz).

Select the check box.

In the associated text field, type SPL (dB).

Octave Band 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Use the Octave Band Plot to depict the sensitivity. Evaluate the pressure 1 m in front using pext() operator. The reference pressure is the default for an SPL evaluation. Plot the sensitivity both as a continuous curve and in 1/3 octave bands.

Locate the section. In the text field, type pext(0,1).

In the toolbar, click .

Locate the section. From the list, choose .

Octave Band 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click .

Click to expand the section. In the text field, type 2.

Sensitivity

In the window, click .

In the window for , locate the section.

Select the check box.

In the text field, type 50.

In the text field, type 100.

In the toolbar, click .

Finally, plot the total electric impedance versus the frequency.

1D Plot Group 8

In the toolbar, click and choose .

In the window for , type Total Electric Impedance in the text field.

Locate the section. From the list, choose .

In the text area, type Electric Impedance.

Locate the section. Select the check box.

In the associated text field, type Frequency (Hz).

Select the check box.

In the associated text field, type Z (\Omega).

Global 1

Right-click and choose .

The automatic computation of the coil impedance uses the entire applied voltage. Because this now includes a contribution from the motion of the coil, you need to define it manually as only the driving voltage V0 divided by the time-harmonic current through the coil. The results are depicted in Figure 9.

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
abs(V0/mf.ICoil_1)		abs(Z)
real(V0/mf.ICoil_1)		real(Z)
imag(V0/mf.ICoil_1)		imag(Z)

In the toolbar, click .

Click the button in the toolbar.

Next create a plot of the deformation of the loudspeaker. This can in general be used to visualize breakups in the cone and surround.

3D Plot Group 9

In the toolbar, click and choose .

In the window for , type Deformation in the text field.