Nonlinear fractional order derivative models of components and materials in hearing aids and transducers

Abstract

Hearing aids and transducers have become an integral part of our everyday lives, with the desire to make these devices smaller and more efficient pushing the limits of the materials and components used in their construction. This thesis focuses on two effects which complicate the design and control of these transducers, namely, viscoelasticity of the rubber polymers, and lossy inductance due to eddy currents. While these effects have been studied before, the models commonly used in electroacoustics are either inaccurate or difficult to incorporate into the nonlinear case. It is therefore relevant to develop methods that accurately describe the linear frequency dependencies of viscoelasticity and lossy inductance, but that can also be used in the nonlinear case. To this end, fractional derivative based models, which incorporate a form of memory into the system, are shown to accurately describe measurements of the viscoelastic and lossy inductance effects. In the case of viscoelasticity, this was done by characterizing the material properties of two viscoelastic rubber polymers over a wide frequency range using dynamic mechanical analysis. Different fractional order viscoelastic model structures were then compared with these results. Similar component level fractional viscoelastic models are used to predict the long-term viscoelastic creep of a loudspeaker suspension, over a much longer time-scale than is commonly analyzed. In terms of lossy inductance, measurements of the impedance for four different loudspeaker voice coils showed that the fractional model parameters change nonlinearly with position. These findings motivated the development of a novel variable order lossy inductance model, where the order of the fractional derivative is no longer constant. The time domain versions of these fractional viscoelasticity and lossy inductance are then incorporated into an electromechanical loudspeaker model with two nonlinear effects: a nonlinear position dependent suspension stiffness, and a nonlinear position dependent voice coil impedance. Simulations of this nonlinear fractional order state-space model produce the expected harmonic and intermodulation distortion components, as known from loudspeaker measurements. Further simulations using a nonlinear variable order state-space model of the nonlinear inductance also show this characteristic distortion pattern. The newly developed nonlinear fractional order and variable order statespace models of a loudspeaker are applicable to both design and control applications, where the new method may form the basis for better nonlinear compensation schemes, based on the improved prediction of the transducer response. Advances in technology are likely to make these techniques for improving sound quality even more widespread, but the results will only ever be as good as the underlying model.