Axisymmetric Condenser Microphone

This is a model of a condenser microphone with a simple axisymmetric geometry. The model aims to give a precise description of the physical working principles of such a microphone.

The condenser microphone is considered to be the microphone with highest quality when performing precise acoustical measurements and with high-fidelity reproduction properties when performing sound recordings; see Ref. 2. This electro-mechanical acoustic transducer works by transforming the mechanical deformation of a thin membrane (diaphragm) into an AC voltage signal.

Figure 1: Sketch of the condenser microphone system including variables and coordinate system. The red box indicates the modeled region.

Models for describing condenser microphones have classically been of the equivalent network type; see Ref. 2. Analytical models exist for simpler geometries, but there are also highly advanced analytical models for more complex geometries; see for example Ref. 1 or Ref. 4. In the present detailed finite-element model the thermoviscous acoustic, electric, and structural problem is solved fully coupled using the frequency-domain linear perturbation solver. This includes the DC charging (prepolarization) and deformation of the membrane which makes out the zeroth order linearization point. A small external circuit model is added to model the pre-amplifier. In some cases, lumping of the electrical part is a good approximation, especially for simple geometries such as this one, where the back electrode is flat and has no perforations. During the initial design steps, lumped models are an important tool. In more complex geometries, solving the full set of equations is necessary to get the correct response.

Theoretical Membrane Model

In order to compare the results of the simulation, the model uses the analytical solution derived for an undamped axisymmetric membrane. The displacement U of a thin axisymmetric membrane of thickness tm, under constant tension Tm, and with a density ρm is governed by the following equation

(1)

where r is the radial coordinate, t is time, ρms = ρm/tm is the surface density, and Fs is the sum of surface forces; see for example Ref. 3. In the present model, the surface force is the sum of the external incident pressure pin (it is assumed to be uniform over the microphone membrane), the internal pressure p = p(r) (given by the thermoviscous acoustics model), and the electrostatic force which is the sum of the quiescent Maxwell surface stress

(given by the electrostatic model) and the small-signal force fes. The variation of the deformation U is assumed to be small and harmonic on top of the static contribution U0 from the DC polarization, such that

(2)

Using these expressions, Equation 1 is reformulated into a static and a time-harmonic equation as

(3)

The latter equation may be rewritten in terms of the axial velocity, um = iωU, of the membrane in the form of a Helmholtz equation:

(4)

Here km is the membrane wave number. In this model you disregard the change in tension due to the movement of the membrane, which is a nonlinear effect that is small compared to the tension Tm.

The current model represents a true multiphysics problem that involves several physics interfaces: Thermoviscous Acoustics, Electrostatics, Electrical Circuit, a Membrane model and the Moving Mesh feature.

This application requires the AC/DC Module and the Structural Mechanics Module in addition to the Acoustics Module.

Model Definition

The geometry and model definitions are shown in Figure 1. In many microphones there is a back-volume below the electrode. In this model a simplified approach is taken and a pressure release condition is applied with p0 = 0 Pa. The membrane is deformed due to the electrostatic forces from charging the capacitor and because of the pressure variation from the external incoming uniform acoustic signal pin. The chosen dimensions of the microphone are typical generic dimensions. Dimensions and parameters are given in Table 1.

Table 1: Microphone dimensions and parameters.
Symbol	size & unit	Description
Hm	18 μm	Air gap thickness
Rmem	2 mm	Membrane radius
G	54 μm	Slit gap width
Tm0	3150 N/m	Membrane static tension
Em	221 GPa	Membrane elastic modulus
tm	7 μm	Membrane thickness
ρm	8300 kg/m3	Membrane density
Vpol	100 V	Target polarization voltage
Rpreamp	1 GΩ	Pre-amplifier output impedance
νm	0.4	Poisson’s ratio for the membrane

The membrane is backed by a thin air gap of thickness Hm and a back electrode. Because the gap is so small, the inclusion of thermal and viscous losses in the acoustic model is essential, thus using the thermoviscous acoustics interface. The membrane and back electrode make up a capacitor that is polarized by an external DC voltage source through the pre-amplifier resistance Rpreamp. This will give rise to a surface charge Qm. The air gap acts as a damping layer for the membrane vibrations. As the gap between the membrane and the back electrode varies, a voltage change is induced. This AC voltage is the output of the microphone.

The sensitivity of the condenser microphone L, is measured in the unit dB (relative to 1 V/Pa). It is defined as the ratio of the open circuit output voltage Vout to the input pressure pin and is given by

(5)

In the present model, the membrane (or diaphragm) is modeled using the dedicated Membrane interface from the Structural Mechanics Module. The membrane is subject to a surface load that is the sum of the external incident pressure pin, the internal pressure p = p(r) (given by the thermoviscous acoustics model), and the electrostatic force given by the Maxwell surface stress, n·τ. The incident pressure is here assumed to be uniform over the microphone membrane, which is only an approximation. At the highest frequencies modeled, the acoustic wavelength becomes comparable with the membrane radius.

Results and Discussion

This model involves a detailed description of the physical effects in a simple condenser microphone. The sensitivity of the microphone L, is directly determined from the model (voltage on the terminal divided by the incident pressure) and is shown in Figure 2 below. Notice the slight roll-off at the low frequencies, this is due to the interaction with the pre-amplifier circuit. By increasing the output impedance Rpreamt this effect will disappear, if the value is decreased the roll-off will be more significant.

Figure 2: Sensitivity curve of the microphone measured in dB relative to 1 V/Pa.

For the case of the simple geometry used in this model, an analytical solution exists for the dynamics of the undamped membrane; see Ref. 3. The axial displacement is given by

(6)

where km is the wave number defined in Equation 4. The analytical approximation is compared to the model results in Figure 3, which shows the average deformation versus frequency. The results agree well below the resonance frequency of the system. The average behavior above the first resonance (in between resonances) is also well captured by the approximate theoretical model. In the real system the damping introduced by the thermal and viscous losses in the air gap is important, especially at the resonances. This is also seen from the figure, where the resonance of the full (real) system is damped and shifted in frequency. The comparison of the two is used as an extra indicator for the correctness of the COMSOL model.

Figure 3: Comparison of the average membrane deformation given by the COMSOL model and by the theoretical approximation for the undamped membrane.

The shape of the deformed membrane is plotted for f = 300 kHz as a 3D surface in Figure 4, using a revolution 2D dataset. At this frequency it is clear to see how higher order modes in the membrane are the cause for the poor sensitivity.

Figure 4: 3D representation of the harmonic membrane deformation at 300 kHz.

The principles described in this model can be extended to 3D models with more complex geometries. Because the full set of equations is solved, such a model includes all physical effects to a high degree of detail. For example, as in The Brüel & Kjær 4134 Condenser Microphone. It can be used to optimize the performance of microphones, to make virtual tests of new geometries, or to investigate the relative importance of different parameters.

Notes About the COMSOL Implementation

Coupled Static and Frequency-Domain Model Using the Frequency Domain, Prestressed STUDY

The current model solves a fully coupled problem using the Frequency Domain, Prestressed study. A stationary study determines the linearization point and the full system of equations is then linearized and solved around this point to determine the harmonic small-signal response (the Frequency Domain Perturbation study step).

The first step is to determine the linearization point for the problem which requires solving a static model that determines the shape of the membrane after the polarization voltage and the membrane tension are applied. The first step solves the electrostatic model (using the Electrostatics interface in the AC/DC Module) coupled to the membrane model. The acoustic model is automatically deactivated as it is, per construction, a small perturbation and thus has no contribution to the static solver step. To determine the correct capacitance (and electric fields) a Moving Mesh feature is needed. The capacitance is a geometry-dependent quantity.

The second step is to solve the linear perturbation frequency-domain model that describes the time-harmonic small-signal deformation of the membrane and the interaction with the fluid (described by a Thermoviscous Acoustics, Frequency Domain interface) within the microphone.

References

1. T. Lavergne, S. Durand, M. Bruneau, N. Joly, and D. Rodrigues, “Dynamic behavior of the circular membrane of an electrostatic microphone: Effect of holes in the backing electrode,” J. Acoust. Soc. Am., vol. 128, p. 3459, 2010.

2. W. Marshall Leach, Jr., Introduction to Electroacoustics and Audio Amplifier Design, 3rd ed., Kendall/Hunt Publishing Company, 2003.

3. P.M. Morse and K. Uno Ignard, Theoretical Acoustics, Princeton University Press, 1968.

4. V.C. Henriquez, Numerical Transducer Modelling, PhD Thesis, DTU, November 2001.

Acoustics_Module/Electroacoustic_Transducers/condenser_microphone

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

This is an appropriate choice because it is a good assumption that the electric processes in this model are quasistatic.

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

In the text field, type um.

In the table, enter the following settings:


um
vm
wm

The displacement field of the membrane is (um,vm,wm) while the velocity field in the fluid is (u,v,w).

Click

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 condenser_microphone_parameters.txt.

These are the parameters specifying the geometry and the physical properties of the microphone and membrane.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rmem.

In the text field, type Hm.

Click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type G.

In the text field, type Hm.

Locate the section. In the text field, type Rmem-G.

Click

Definitions

Create a selection corresponding to the membrane for use when adding features to the membrane edge.

Membrane

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundaries 3 and 6 only.

In the text field, type Membrane.

Load the variables that define the theoretical response of an undamped membrane.

Variables 1

In the toolbar, click

In the window for , locate the section.

Click

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

Integration 1 (intop1)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Boundary 3 only.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Having set up the materials, proceed to setting up and defining the physics. Begin with the electric problem: Electrostatics and Circuit.

Electrostatics (es)

Terminal 1

In the window, under right-click and choose the boundary condition .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

Ground 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Now, set up the small external electrical circuit that is depicted in Figure 1.

Electrical Circuit (cir)

In the window, under click .

External I vs. U 1 (IvsU1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	1
n	0

Locate the section. From the V list, choose .

Resistor 1 (R1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	1
n	2

Locate the section. In the R text field, type Rpreamp.

Voltage Source 1 (V1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	2
n	0

Locate the section. In the vsrc text field, type Vpol.

Next, set up the acoustics before turning to the membrane model and the moving mesh interface.

Thermoviscous Acoustics, Frequency Domain (ta)

In the window, under click .

Pressure (Adiabatic) 1

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

In the pbnd text field, type p0.

Multiphysics

Thermoviscous Acoustic-Structure Boundary 1 (tsb1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The last step couples the membrane to the thermoviscous acoustic domain. The multiphysics coupling feature sets the acoustic velocity equal to iω times the deformation of the membrane (this is the time derivative in the frequency domain). This condition is a bidirectional constraint, meaning that a reaction force is added to the membrane equation, ensuring a two-way coupling.

Now, set up the membrane model, constrain it at the outer perimeter where it is fixed and add the forces that act on it. They are the electrostatic forces given by the Maxwell stress tensor (es.dnTer,es.dnTez) and the incident pressure field pin. Use the linper() operator to tell COMSOL that the incident pressure is only a harmonic frequency dependent quantity (not a static load).

Membrane (mbrn)