Passive modelling of the electrodynamic loudspeaker: from the Thiele–Small model to nonlinear port-Hamiltonian systems

Authors

Antoine Falaize, Thomas Hélie

Abstract

The electrodynamic loudspeaker couples mechanical, magnetic, electric and thermodynamic phenomena. The Thiele/Small (TS) model provides a low frequency approximation, combining passive linear (multiphysical or electric-equivalent) components. This is commonly used by manufacturers as a reference to specify basic parameters and characteristic transfer functions. This paper presents more refined nonlinear models of electric, magnetic and mechanical phenomena, for which fundamental properties such as passivity and causality are guaranteed. More precisely, multiphysical models of the driver are formulated in the core class of port-Hamiltonian systems (PHS), which satisfies a power balance decomposed into conservative, dissipative and source parts. First, the TS model is reformulated as a linear PHS. Then, refinements are introduced, step-by-step, benefiting from the component-based approach allowed by the PHS formalism. Guaranteed-passive simulations are proposed, based on a numerical scheme that preserves the power balance. Numerical experiments that qualitatively comply with measured behaviors available in the literature are presented throughout the paper.

Citation

• Journal: Acta Acustica
• Year: 2020
• Volume: 4
• Issue: 1
• Pages: 1
• Publisher: EDP Sciences
• DOI: 10.1051/aacus/2019001

BibTeX

@article{Falaize_2020,
  title={{Passive modelling of the electrodynamic loudspeaker: from the Thiele–Small model to nonlinear port-Hamiltonian systems}},
  volume={4},
  ISSN={2681-4617},
  DOI={10.1051/aacus/2019001},
  number={1},
  journal={Acta Acustica},
  publisher={EDP Sciences},
  author={Falaize, Antoine and Hélie, Thomas},
  year={2020},
  pages={1}
}

Download the bib file

References

• Klippel, Journal of the Audio Engineering Society (2006)
• Suykens, Journal of the Audio Engineering Society (1995)
• Tassart, S., Valcin, S. & Menu, M. Active Loudspeaker Heat Protection. Journal of the Audio Engineering Society vol. 62 767–775 (2014) – 10.17743/jaes.2014.0041
• Thiele, Journal of the Audio Engineering Society (1971)
• Thiele, Journal of the Audio Engineering Society (1971)
• Small, Journal of the Audio Engineering Society (1972)
• Small, Journal of the Audio Engineering Society (1973)
• Marshall Leach, Journal of the Audio Engineering Society (2002)
• Klippel, Journal of the Audio Engineering Society (2004)
• Thorborg, Journal of the Audio Engineering Society (2010)
• Klippel, Journal of the Audio Engineering Society (1990)
• Bai, M. R. & Huang, C.-M. Expert diagnostic system for moving-coil loudspeakers using nonlinear modeling. The Journal of the Acoustical Society of America vol. 125 819–830 (2009) – 10.1121/1.3058639
• Kaizer, Journal of the Audio Engineering Society (1987)
• Brunet, P. & Shafai, B. State-Space Modeling and Identification of Loudspeaker with Nonlinear Distortion. Computational Intelligence and Bioinformatics / 755: Modelling, Identification, and Simulation (2011) doi:10.2316/p.2011.755-054 – 10.2316/p.2011.755-054
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Falaize, A. & Hélie, T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. Applied Sciences vol. 6 273 (2016) – 10.3390/app6100273
• Knudsen, Journal of the Audio Engineering Society (1993)
• Koeller, R. C. Applications of Fractional Calculus to the Theory of Viscoelasticity. Journal of Applied Mechanics vol. 51 299–307 (1984) – 10.1115/1.3167616
• Lewandowski, R. & Chorążyczewski, B. Identification of the parameters of the Kelvin–Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers. Computers & Structures vol. 88 1–17 (2010) – 10.1016/j.compstruc.2009.09.001
• Wright, Journal of the Audio Engineering Society (1990)
• Kong, X.-P., Agerkvist, F. & Zeng, X.-W. Modeling of Lossy Inductance in Moving-Coil Loudspeakers. Acta Acustica united with Acustica vol. 101 650–656 (2015) – 10.3813/aaa.918860
• Buntenbach, R. A generalized circuit model for multiwinding inductive devices. IEEE Transactions on Magnetics vol. 6 65–65 (1970) – 10.1109/tmag.1970.1066689
• Hamill, D. C. Lumped equivalent circuits of magnetic components: the gyrator-capacitor approach. IEEE Transactions on Power Electronics vol. 8 97–103 (1993) – 10.1109/63.223957
• Advances in Fractional Calculus. (Springer Netherlands, 2007). doi:10.1007/978-1-4020-6042-7 – 10.1007/978-1-4020-6042-7
• Schäfer, I. & Krüger, K. Modelling of lossy coils using fractional derivatives. Journal of Physics D: Applied Physics vol. 41 045001 (2008) – 10.1088/0022-3727/41/4/045001
• Brunet, P. & Shafai, B. Identification of Loudspeakers Using Fractional Derivatives. Journal of the Audio Engineering Society vol. 62 505–515 (2014) – 10.17743/jaes.2014.0028
• Itoh, T. & Abe, K. Hamiltonian-conserving discrete canonical equations based on variational difference quotients. Journal of Computational Physics vol. 76 85–102 (1988) – 10.1016/0021-9991(88)90132-5
• Lopes, N., Hélie, T. & Falaize, A. Explicit second-order accurate method for the passive guaranteed simulation of port-Hamiltonian systems. IFAC-PapersOnLine vol. 48 223–228 (2015) – 10.1016/j.ifacol.2015.10.243