COMSOL 6.4 - Lumped Loudspeaker Driver Transient Analysis with Nonlinear Large-Signal Parameters

Lumped Loudspeaker Driver Transient Analysis with Nonlinear Large-Signal Parameters

This model shows how to include the nonlinear (large signal) behavior of certain lumped components in a simplified loudspeaker analysis in the time domain. The mechanical and electrical system is modeled using an equivalent electrical circuit (a Thiele–Small representation). The large signal compliance CMS(x) and force factor BL(x) are here nonlinear functions of the speaker location. Moreover, the mechanical damping RMS(v) is a function of the speaker velocity.

The nonlinear effects associated with the compliance and BL factor are especially important at lower frequencies. This is also where a lumped modeling approach of the electromechanical system has its main application. At higher frequencies the piston assumption, of the lumped model, is no longer valid and breakup can occur in the membrane and other mechanical components. The approach is in particular relevant for microtransducer modeling (and their implementation in larger system designs) as the lumped representation is valid in a larger frequency range, due to the small size.

The model uses the built-in coupling between the (or , if the back volume is also lumped) feature in the interface and an interface. The axial speaker location x and speaker membrane velocity v exist as predefined built-in variables.

Model Definition

The model definitions, geometry, and most details of the lumped model follow the setup in the frequency domain variant; the Lumped Loudspeaker Driver also found in the Acoustics Module Application Library. In the present model, only the open back volume configuration is modeled.

This model is solved in the time domain which changes the allowed structure of the circuit. All values of the components need to be real valued and cannot be a function of frequency. This limits the possibility to, for example, use frequency dependent component values to model certain loss types. The components values can, on the other hand, be given by time dependent quantities which allows the introduction of the large signal values: compliance CMS(x), force factor BL(x), and the mechanical damping RMS(v), where the time dependency is through x = x(t) and v = v(t). The axial location of the diaphragm (the mechanical system) x is defined through a predefined (global) variable actd.ilsb1.x_ax and the diaphragm velocity v is defined through a predefined (global) variable actd.ilsb1.v_ax. When using these, add the appropriate physics and feature tags.

The large signal parameters are depicted in Figure 1. The values of the curves are taken to roughly correspond to properties extracted from the Loudspeaker Driver — Frequency-Domain Analysis and the Loudspeaker Driver — Transient Analysis tutorial models. The data is just an example for the purpose of demonstration in this model.

Figure 1: Large signal parameters used in the model, CMS(x), BL(x), and RMS(v).

In this model, the applied voltage (to the Voltage Source in the Electrical Circuit model) is a single frequency harmonic signal of the type

where the frequency f0 is set to 50 Hz and the signal amplitude V0 is 15 V. Both values can be found in the Parameters list and modified as necessary. The current input signal will allow for the evaluation of the total harmonic distortion (THD); the signal can be modified to any desired signal, for example, if inter modulation distortion (IMD) is of interest. Note that in the model a ramp is added to the signal to increase it smoothly over the first period. In the analysis of the results, the first two periods are skipped in order to avoid inclusion of any transients.

Results and Discussion

The model contains several plots that analyze the response of the speaker as well as the behavior and contribution of the large signal parameters. The contribution of each large signal parameter, to the overall nonlinear behavior of the speaker, can of course be studied by setting one or more of the parameters to constants.

The pressure is evaluated in a point 20 cm in front of the speaker. The time signal is depicted in Figure 2. The frequency spectrum of the nonlinear periodic part of the signal (skipping the first two periods) is depicted in Figure 3. The peak values of the generated harmonics are also depicted. These can be used to evaluate the THD.

The speaker axial position and velocity as function of time are depicted in Figure 4. Finally, the variation of the large signal parameters over time is depicted in Figure 5.

Figure 2: Pressure as function of time.

Figure 3: Frequency spectrum of the nonlinear periodic part of the signal. Peak values shown for the generated harmonics.

Figure 4: Axial position and velocity as function of time.

Figure 5: The variation of the large signal parameters as function of time.

Notes About The COMSOL Implementation

In the interface the capacitor in the mechanical system (representing the compliance of the suspension) is modeled using a device. This is in order to ensure that the time varying capacitance (the compliance) is taken fully into account. The compliance is time dependent in the model through the displacement: CMS = CMS(x) = CMS(x(t)). The setup ensures that the current–voltage relation for the device follows

(1)

such that the full time derivative is taken (chain-rule is automatically applied). Here, Q is the charge. The voltage V is described using the built-in variable sens.v, which measures the voltage difference between the two nodes defined in the section.

The built-in capacitor device uses the simpler relation

(2)

Acoustics_Module/Electroacoustic_Transducers/lumped_loudspeaker_driver_transient

Modeling Instructions

From the menu, choose .

New

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Model Wizard

In the window, click

In the tree, select > > .

Click .

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In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_transient_parameters.txt.

Interpolation 1 (int1)

In the toolbar, click

and choose > .

In the window for , locate the section.

From the list, choose .

Click

Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_transient_bl.txt.

Click

In the text field, type BL.

Locate the section. From the list, choose .

From the list, choose .

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


Function	Unit
BL	Wb/m

In the table, enter the following settings:


Argument	Unit
t	mm

Interpolation 2 (int2)

In the toolbar, click

and choose > .

In the window for , locate the section.

From the list, choose .

Click

Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_transient_c_ms.txt.

Click

In the text field, type C_MS.

Locate the section. From the list, choose .

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


Function	Unit
C_MS	mm/N

In the table, enter the following settings:


Argument	Unit
t	mm

Interpolation 3 (int3)

In the toolbar, click

and choose > .

In the window for , locate the section.

From the list, choose .

Click

Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_transient_r_ms.txt.

Click

In the text field, type R_MS.

Locate the section. From the list, choose .

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


Function	Unit
R_MS	N*s/m

In the table, enter the following settings:


Argument	Unit
t	m/s

Definitions

Ramp 1 (rm1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 0.1*T0.

In the text field, type 1/T0.

Select the checkbox.

Click to expand the section.

Select the checkbox. In the associated text field, type 0.2*T0.

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1[cm].

In the text field, type 180.

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

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rair+Rpml.

In the text field, type 180.

Locate the section. In the text field, type -90.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	Rpml

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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


r (m)	z (m)
3[cm]	-4[cm]
a-1[cm]	0

Polygon 2 (pol2)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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


r (m)	z (m)
a+1[cm]	0
Rair	0

Click

Click the

button in the toolbar.

Quadratic Bézier 1 (qb1)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to 1.8[cm].

In row , set to 3[cm].

In row , set to -3[cm].

In row , set to -3.1[cm].

In row , set to -4[cm].

Locate the section. In the text field, type 1.5.

Click

Delete Entities 1 (del1)

In the window, right-click and choose .

On the object , select Boundaries 3 and 4 only.

In the window for , click

Polygon 3 (pol3)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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


r (m)	z (m)
0.15	0
0.15	-0.1
0	-0.1

Click

Definitions

Perfectly Matched Layer 1 (pml1)

In the toolbar, click

Select Domains 1 and 5 only.

In the window for , locate the section.

In the text field, type 3.

Baffle (interior wall)

In the toolbar, click

In the window for , type Baffle (interior wall) in the text field.

Locate the section. From the list, choose .

Select Boundaries 9, 11, 12, and 19 only.

Speaker

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundaries 8, 15, and 18 only.

Pressure Acoustics, Transient (actd)

Interior Sound Hard Boundary (Wall) 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Interior Lumped Speaker Boundary 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Proceed and add the electrical circuit model for the speaker (the Thiele-Small representation). For the lumped mechanical part of the model, the units will be transformed from mechanical to electric; this is only to avoid the unit warning, the model solves nicely without the unit transform.

It is in the lumped model that the nonlinear parameters are used. The defines variables for the axial position and the axial velocity; they are actd.ilsb1.x_ax and actd.ilsb1.v_ax, respectively.

Add Physics

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