Virtual Measurements of Transducer Nonlinearities
April 2, 2019:
Continuing our series of informative articles on acoustics, electro-acoustics and Mvoid’s Virtual Product Development simulation technologies, in this article we will review the fundamentals of loudspeaker modeling with a brief review of small signal dynamic analogies. From there we will introduce the concepts of Multiphysics modeling where we will discuss aspects of non-linear analysis of loudspeakers operated with large signals. Following articles in our series will delve deeper into Mvoid’s complete loudspeaker and audio system Multiphysics modeling methodologies and tools.
Loudspeaker Simulation as a Multiphysical Device
Loudspeakers (transducers), are multiphysical devices that are comprise electro-mechanical, solid and acoustical physics. Modeling and simulating loudspeakers was first made possible with the use of dynamic analogies where mechanical and acoustical systems can be solved by electrical circuit analysis. A large step in modelling loudspeakers occurred in the 1960’s and 1970’s when technical papers by A.N. Thiele  and R.H. Small ,” were published. These related concepts for small signal linear analysis were presented separately, first by Thiele and later, without knowledge of Thiele’s work, by Small. Only later after a colleague suggested that Small would appreciate Thiele’s work did the two meet, exchange ideas and become lifelong friends. The content of these studies was eventually combined into models that came to be known as a loudspeaker’s “Thiele-Small Parameters”. This set of one-dimensional lumped-parameter equations can be used to define the relationship between a loudspeaker, a particular enclosure and the radiation of sound waves. This allows a designer to model the linear acoustic capabilities of a loudspeaker enclosure system in a concise manner. Over time there have been many developments to extend these lumped parameter models to more sophisticated models with improved accuracy. Especially the recent work of W. Klippel on the large signal behavior (i.e. the nonlinear behavior of the loudspeaker) .
To examine a loudspeaker’s non-linear behavior, it is useful to look at the loudspeaker components, how they interact and their relationship with the physics domain, electrical domain constituted by the voice coil immersed in the magnetic field, and mechanical domain comprising the moving assembly and acoustics which includes the air mass moved during vibration.
Each of these systems admit an equivalent circuit where every symbol has its physical meaning. In the electrical are defined the resistance Re, the inductance Le; in the mechanical are defined the moving mass Mm, the stiffness of suspensions Kmsand its resistance Rm; the acoustical by the radiation impedance Za.
However, all these lumped parameters models have one significant drawback: they are based on one-dimensional, scalar equations to describe a physical domain. This fact led to the development of models based on matrix methods.
Moreover, the linearity of the lumped parameters is limited to small excursions of the voice coil. As soon as the coil moves from the uniform magnetic gap there is an attenuation of the Bl(x) force factor. The elastic suspensions increase their Kms(x)stiffness when stretched to their extremes. The inductance of the coil Le(x)depends as well on the position due to the different magnetic permeability experienced by the coil turns exiting the magnetic gap.
The electromagnetic force factor, Bl(x),inductance Le(x), and stiffness Kms(x)vary with coil position in the gap. A significant portion of the voice coil can move out of the main flux field, where less mechanical force induced. This nonlinear effect is very critical and causes unwanted distortion in the radiated sound. Voice coil inductance is also dependent on voice coil displacement and on current. Even if we would have a super-linear material for suspensions, the change in geometric stiffness leads to non-linear behavior. Non-linear solving of the governing equations in time domain is required to accurately simulate that behavior.
Mvoid Virtual Measurements
Mvoid Technologies develops simulation methodologies which span from 2D/3D FEA , equation based 1D and a combination of both.
The following are simulations results for a given 6” woofer that was built in a sample. It is shown a comparison to real world measurements using Klippel measurement system LSI and DIS for the nonlinear force factor Bl(x),nonlinear inductance Le(x), stiffness Kms(x)and resulting distortion results of voice coil excursion, current, acoustic pressure both in free air and coupled to an enclosure.
This particular sample has an unintended -1.5 mm offset in the coil position and dishing of the suspensions from the design intent. As with most axisymmetric loudspeakers this device can be treated as an axisymmetric system where simplified 2D geometries can be used as it is shown here.
Simulation of Motor Force, Bl(x,I)
The first model solves for a static solution of the magnetic circuit in the woofer motor structure. The iron in the pole piece and top plate is modeled as a nonlinear magnetic material, with the relation between the B and H fields coming from measured data stored in a data base. The static solution provides the magnetic field density at any point in the model.
To calculate the force factor Blversus coil excursion xand electrical current Ia moving mesh for the voice coil is used to obtain a static solution for each voice coil position. The results are the nonlinear behavior of the voice-coil in a nonlinear field, dependent on position and electrical current. This last term is the component responsible for the modulation of the flux.
The design intent virtual measurements (i.e. zero offset of suspensions and voice-coil position due to manufacturing misalignment) are as follows:
The misaligned Bl(x)factor indicates an offset center position of the voice-coil in the physical sample curve and it’s mirrored curve with respect to zero displacement :
Simulation of Suspension (Spider and Surround) Stiffness, Kms(x)
In the simulation for Kms, a frequency of zero (0) Hz for Kms(x)is used. And a stepwise solution for Kmsis performed: a frequency response linearized around the static solution. That is to say: The frequency domain solution is solved in every position of the coil, moving it statically, with no dynamics considered or estimated. It is a direct calculation at every step.
To simulate the stiffness vs displacement of the spider and surround we run two different sweeps that start from zero force. The first sweep is in the positive direction (z direction) ranging from 0 to a reasonable extreme force value with small force steps. The second sweep is in the negative direction. The geometric non linearity is included in the solution.
The physical sample showed a static deformation in the suspensions as well , and this can be easily taken into account in the Kms(x)simulation model:
During exercise the increase in temperature due to friction softens the suspension plastic matrix materials and this can be as well computed:
Simulation of Le(x), L2(x) and R2(x)
The voice coil total electrical impedance depends on the position in the gap and varying with frequency. This is due to the interaction between the voice coil current and the induced AC magnetic field penetrating the steel plates, magnet and air surround the coil.
To describe the losses generated in the steel plates and demodulating rings the widely used LR2 model has been adopted.
In the same simulation which calculates the electrical impedance it is straightforward to extract Le(x) L2(x) R2(x), independent of frequency, as well as to verify the fitting validity against the Reff(x,w) Leff(x,w), effective resistance and inductance dependent on x and frequency.
Time Domain Simulation of Excursion xand Current Iand I2for Large Signals
After having computed the nonlinear parameters , they can be imported as inputs to a simulation based on a system of differential equations that represent the physics involved and their couplings. It doesn’t solve finite elements.
Its strength being its flexibility in defining different transducer constant and non linear parameters (which can be a result of a FEM simulation , a measurement or a mathematical function) to explore different design scenarios in a very short computation time. The variables coil displacement x, current Ion Leand current I2on L2are solved in time domain and the results can be as well represented in frequency. The transducer under exam suffered offsets in coil and suspensions. They can be also defined through a specific x0offset parameter in the equations. The voice coil excursion positive and negative Peak and offset are calculated for the free air condition vs frequency.
Time Domain Simulation of Excursion xand Current Iand I2for Large Signal when the Transducer is Coupled to an Enclosure
This simulation model couples the transducer’s model system of equations to an acoustic finite element model of an enclosure. The results are the virtual microphone’s sound pressure and harmonics, excursion and electrical currents in time and frequency domain.
The advantage in this case is the assessment of the coupled system with the fully geometrical defined enclosure.
Below are the measurements of the transducer under test:
 A. N. Thiele, “Loudspeakers in vented boxes”, Proc. IREE (Australia), 22, 487 (1961): republished in JAES, 19, 382 and 471 (1971)
 R. H. Small, “Direct-Radiator Loudspeaker System Analysis,” J. Audio Eng. Soc., vol. 20, pp. 383-395 (June 1972)
 W. Klippel, “Diagnosis and Remedy of Nonlinearities in Electrodynamical Transducers”, 109th AES Convention, 2000
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