Axisymmetric Condenser Microphone with Electrical Lumping

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, using lumped descriptions for some of the physical phenomena. Lumping certain parts of a model can give additional insight to physical phenomena using a simplified description. A fully coupled approach (without lumping internal physics) can be seen in the Axisymmetric Condenser Microphone model, also located in the Application Library of the Acoustics Module.

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 electromechanical 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. In the present detailed finite-element (FE) model, you model the microphone including a static (quiescent) analysis of the DC charging (prepolarization) and deformation of the membrane. You then perform time-harmonic (small-signal) finite element (FE) analysis of the dynamics of the membrane coupled to thermoviscous acoustics. The small-signal electric model for the system is solved as a lumped (electric equivalent) model coupled to the FE model. The model is a true multiphysics problem that involves several physics interfaces: Thermoviscous Acoustics, Electrostatics, Moving Mesh, two user-defined PDE interfaces, a Global ODEs and DAEs interface, as well as an Electrical Circuit model.

This application requires the AC/DC Module.

Model Definition

The geometry and model definitions are shown in Figure 1. 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.
Parameter	Value	Description
Hm	18 μm	Air gap thickness
Rmem	2 mm	Membrane radius
G	54 μm	Slit gap width
Tm	3150 N/m	Membrane tension
tm	7 μm	Membrane thickness
ρm	8300 kg/m3	Membrane density
Vpol	100 V	Polarization voltage

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 makes up a capacitor that is polarized by an external DC voltage source. 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 and is coupled to the capacitor (the quiescent DC capacitance of the system C0) and an external very large resistive load RL.

The electric circuit for this coupling is shown in Figure 2.

Figure 2: Analogous circuit for the electrical part of the condenser microphone.

The sensitivity L of the condenser microphone 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

Next, consider a small-signal (linearized) analysis of the electric part of the condenser microphone. The total voltage is the sum of the quiescent polarization voltage Vpol and the small-signal output voltage Vout. The total charge on the condenser is the sum of the quiescent charge Q and the small-signal charge q. The distance between the back electrode and the membrane is the sum of the initial distance Hm, the quiescent average deformation U0,av and the small-signal deformation Uav.

The capacitance C of a parallel plate capacitor, with a fixed air gap distance h and area A is given by

If the air gap h is varied with an average deformation Uav around the initial static gap distance, Hm + U0,av, the expression becomes

(1)

where the expression has been expanded to first order. Again, the distance U0,av stems from the initial deformation from equilibrium due to the electrostatic forces of the prepolarization of the condenser. A first approximation for the average deformation in axisymmetric coordinates is given as

(2)

The capacitance is by its definition (see Ref. 4) given by

(3)

where Q is the charge on and V the potential across the capacitor. Inserting Equation 1 into Equation 3 and retaining only first-order terms yields

(4)

Differentiating Equation 4 with respect to time, switching to frequency domain, and using Vpol = Q/C0 yields

(5)

where Icap is the induced current through the capacitor, the second term on the right is the electromechanical coupling Vcap, and the average membrane velocity is

(6)

where um is the axial membrane velocity. The circuit model equivalent to equation Equation 5 is shown in Figure 2.The governing equation of the membrane is described in the next section, Membrane Model. The final element necessary to couple the lumped small parameter model of the electric model to the mechanical FE model is the back coupling via the small parameter electrostatic force fes. The force is approximated by the spatial derivative (z direction) of the electric energy stored in a parallel plate condenser; the small parameter component is (see Ref. 2)

The force is applied evenly over the membrane as a surface normal stress

Note that the integrals in Equation 2 and Equation 6 are over the area of the membrane that is backed by the back electrode plus a possible small correction for edge effects. This is especially important if the back electrode has holes. This is the case in many commercials condenser microphones where the holes are placed in order to produce a special sensitivity characteristic of the microphone. In this model the back electrode is flat and uniform.

Membrane Model

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

(7)

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 variations of the deformation U is assumed to be small and harmonic on top of the static contribution U0 from the DC polarization, such that

Using these expressions, Equation 7 is reformulated into a static and a time-harmonic equation as

(8)

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

(9)

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.

Results and Discussion

This model involves a detailed description of the physical effects at play in a simple condenser microphone. The lumping of the small-signal analysis of the electrical part is a good approximation for this simple geometry, where the back electrode is flat and has no perforations. The sensitivity, L, of the microphone is directly determined from the model (voltage across the load resistance divided by the incident pressure) and is shown in Figure 3.

Figure 3: 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. 5. The axial displacement is given by

where km is the wave number defined in Equation 9. The analytical approximation is compared to the model results in Figure 4, 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 models is used as an extra indicator for the correctness of the FE model.

Figure 4: 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 = 0.3 GHz as a 3D surface in Figure 5 using a revolution 2D dataset. At this frequency it is clear to see how higher order modes in the membrane are the cause of the poor sensitivity.

Figure 5: 3D representation of the harmonic membrane deformation at 0.3 GHz.

The principles described in this model may be extended to 3D models with more complex geometries where, for example, the back electrode is perforated or has a convex shape. Such a model may 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

The implementation couples a static model that determines the quiescent shape of the membrane after the polarization voltage is applied to the time harmonic small-signal response. The current model does not consider the transient charging of the condenser. The first step requires solving an electrostatic model (AC/DC interface) coupled to the static membrane model (user defined PDE model). In order for the model to determine the correct quiescent capacitance C0, a Moving Mesh interface is necessary, as the capacitance is a geometric dependent quantity.

The second step is to solve the frequency domain model that describes the time harmonic small-signal deformation of the membrane (user defined PDE) and the interaction with the fluid (thermoviscous acoustics model) within the microphone. This solution is superposed to the static solution. The small-signal electric components of the microphone and the sensitivity is determined by a small lumped AC/DC circuit model.

Stationary Surface Charge and Capacitance

In the variables list the static capacitance variable C0 is defined as:

es.term1.int(es.nD*2*pi*r)/es.V0_1

This term is equal to the stationary terminal charge Q0 divided bu the stationary voltage V0. The stationary charge needs to be calculated as a surface integral of the D field and not evaluated using the variable es.Q0_1. This variable evaluates to 0 in the frequency domain step due to the current way reaction forces are transferred between study steps.

Weak Form of the Membrane Equation

The membrane equations are implemented in COMSOL using the general weak form formulation of a partial differential equation. This is an integral form of the strong formulation of Equation 8. The equation is multiplied by a test function and integration by parts is performed (using Green’s first identity). The resulting equation for the static deformation becomes