Lumped Model of Condenser Microphone with both Sound and Vibration Sensitivities
Condenser microphones will have an electrical output for any displacement of the membrane relative to the back plate, which is usually bound to the microphone housing. For the typical use case, a pressure input will create a pressure difference across the membrane, which causes a displacement and hence an electrical output corresponding to the pressure. However, vibrating the entire microphone will also lead to a membrane displacement due to inertia and compressional effects of the vibroacoustic setup, so an electrical output can be had that does not relate to the input pressure. Microphones placed in a hearing aid will inevitably vibrate as the user moves, and feedback stability will be an issue, as the electrical output does not uniquely stem from the pressure input.
A new lumped model has been developed, which simultaneously includes acoustic pressure and mechanical vibration, such that both pressure sensitivity and vibration ditto can be calculated a priori for a given acoustical and mechanical setup. The lumped components include the inertial and compressional effects of the mechanical parts with for example the membrane and housing considered stiff but connected via a suspension component. Also importantly, the inertial and compressional effects of the acoustics are included as the air in the front and rear volumes coupled to the membrane does not only add stiffness as in the typical pressure input only use case, but now also adds effective mass, as the air volumes are physically displaced in space. This two-way coupled model outputs a voltage dependent on both pressure and vibration, and as superposition holds, the effects can be investigated separately and combined as desired. An equivalent mechanical schematic is shown in Figure 1, but more details are added in the final lumped model network with inclusion of damping effects. In the present work, an axisymmetric model is assumed such that only axial vibration is included but the lumped model can be expanded upon to include sideways vibration for a more general geometry.
The model has been validated against finite element simulations in COMSOL Multiphysics using a generic condenser microphone, see Figure 2, with excellent matching between them. For example, in Figure 3, the resulting displacements of both the membrane and the housing for a given input pressure, and all relevant resonance characteristics are captured in either model. Thus, any combination of pressure and vibration input can be studied and linked to feedback stability strategies in the hearing aid in question.
