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Variable Fractional Order Analysis of Loudspeaker Transducers: Theory, Simulations, Measurements, and Synthesis

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Loudspeaker transducer models with fractional derivatives can accurately approximate the inductive part of the voice coil impedance of a transducer over a wide frequency band, while maintaining the number of fitting parameters to a minimum. Analytical solutions to Maxwell equations in infinite lossy coils can also be interpreted as fractional derivative models. However, they suggest that the fractional order a  cannot be a constant, but rather a function of frequency that takes on values between 1/2 and 1. This paper uses Finite Element (FEM) simulations to bridge the gap between the theoretical first-principles approach and lumped parameter models using fractional derivatives. The study explores the dependence of a on frequency for idealized infinite and finite cores as well as in four real loudspeaker transducers. To better match the measured impedances and frequency-dependent a values we propose to represent the voice coil impedance by a cascade of R-L sections.

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