Analytical Simulation of Silicon Condenser Microphones

Capacitive silicon microphones are designed for the reception of sound. For optimum performance perforated backplates are required. Characteristics cannot be explained by the behavior of the diaphragms used. In addition to the membrane, the airgap between the electrodes has to be taken into account. Furthermore dynamic effects as the mechanical force of the airgap and flow losses in the airgap as well as the acoustic holes in the backplate has to be considered. This paper describes an extended analytical model for the calculation of the receiving and transmitting sensitivity, membrane deflection and input impedance of silicon microphones. Theory of dynamic modeling Electrostatic models well describe the effect of the bias voltage on the membrane deflection. Existing models [1, 2] used for the calculation of microphones will be adapted. The differential equation for the membrane deflection is solved numerically in the dynamic case employing the finite difference method to get insight into the form of the membrane vibration. For a complete model of condenser microphone it is necessary to include the airgap between membrane and backplate, the acoustic holes and the backvolume into the model. The theory of the transducer model and the employed electromechanical equivalent circuit is described mainely in [3]. If a deflection in the form s(x, y) is assumed, the differential equation for the membrane behavior at homogeneous static pressure P with membrane thickness dm and the tensile stress σ is [4]: ∂s(x, y) ∂x2 + ∂s(x, y) ∂y2 = P σdm (1) In the case of dynamic modeling the equation must be extended for the description of the dynamic membrane behavior by a mass term with ρm as membrane density, p(t) as time variable pressure [4]. ∂s(x, y, t) ∂x2 + ∂s(x, y, t) ∂y2 = p(t) σdm + ρm σ ∂s(x, y, t) ∂t2 (2) All time-dependent parameters, e.g. velocity, deflection or pressure are assumed to be harmonic. All parameters are represented by the product from complex indicators describing the place and frequency dependence. By using the complex indicators the differential equation for the description of the dynamic membrane can be written in the following form: ∂S(x, y, ω) ∂x2 + ∂S(x, y, ω) ∂y2 = P σdm + ρm σ ωS(x, y, ω) (3) This differential equation permits the calculation of the deflection S(x, y, ω) of a square membrane. The equation still has no damping term, which has to be taken into account in order to avoid infinite deflections at resonance. If the acoustic impedance of air Ra, is considered the differential equation becomes: σdm [ ∂S(x, y, ω) ∂x2 + ∂S(x, y, ω) ∂y2 ] = P + ( jωRa − ωρmdm ) · S(x, y, ω) (4) The solution of the differential equation is the behaviorial description of the membrane. Using electromechanical analogies a simplified analytical model was derived. This permits the integration of electrical and mechanical impedances into the same equivalent electrical circuit. Figure 1 shows a comparison of the differential equation an the simplified model. Figure 1: Comparison of differential equation and simplified piston model for membrane description (gold membrane with edge length 1.8mm, tensile stress 30 MPa) After simplification of the membrane description it is possible to add the dynamic effects of the microphone. The following dynamic effects are included in the complete analytical model: . membrane: acoustic impedance of air above membrane Ra, effective membrane mass Mm, compliance of membrane Cm . airgap: compliance of air Cg, mechanical impedance for flow towards acoustic holes Rg, mass of air inside airgap Mg . acoustic holes: impedance for all acoustic holes Rh, inertial force for the mass of air Mh . backvolume: compliance of air inside backchamber Cv CFA/DAGA'04, Strasbourg, 22-25/03/2004