A.N. Thiele and R. H. Small developed a set of parameters and one-dimensional equations defining the relationship between a loudspeaker, a particular enclosure and the radiation of sound waves. Over time there have been a lot of developments to extend these early 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) should be mentioned here („Diagnosis and Remedy of Nonlinearities in Electrodynamical Transducers”, 109th AES Convention, 2000).
However, all these lumped parameter models have one significant drawback; they are based on one-dimensional, scalar equations to describe a physical domain. Whereas the usage of a lumped parameter model for voice coil electromagnetics turned out to be very useful, for the mechanical domain and for the acoustical domain one can see a lot of limitations. E.g. the mechanical system is described in a lumped parameter model by a set of scalar values of stiffness, mass and damping. This leads to the fact that this model is only capable of describing the pistonic motion of a loudspeaker’s vibration system. This will yield to significant limitations in the frequency range where the lumped parameter model is applicable. At higher frequencies the motion pattern is not pistonic anymore and a multidimensional approach is necessary to model this behavior.
Development of models based on Matrix Methods as Consequence
This fact has led to the development of models based on matrix methods. A fully coupled electrical-mechanical-acoustical simulation model for loudspeakers is now presented, which uses a lumped parameter model for voice coil electromagnetics, a finite element model for the mechanical domain and a finite or boundary element model for the acoustical domain.
1.1 Voice Coil Electromagnetics
In the lumped parameter model for voice coil electromagnetics the electrical force fe acting on the coil under the assumption of a constant voltage source U is defined as:
Whereas B is the flux density, l the length of the wire in B, R the DC resistance of the coil, L the inductance of the coil and ve is the velocity of the moving coil. fe,L is the Lorentz force and fe,EMF is the back electromagnetic force.
1.2 Structural Dynamics
The governing equation for the mechanical vibrations in the frequency domain, discretized by means of finite elements, can be written as follows:
At first glance there seems to be only a little difference in the governing equations by matrix methods and by lumped parameter models. However, the big difference is the dimension of the system. In the finite element governing equation stiffness, mass and damping are being described via matrices. Km is the stiffness matrix, Dm is the damping matrix and Mm is the mass matrix. Furthermore, um is the vector of displacements and fm is the vector of mechanical forces exciting the system. ω is the angular frequency. Typically the dimension is of several of thousands degrees of freedom. In fact the governing equation is a system of equations describing the mechanical vibrations with respect to a detailed definition of the geometry (CAD model) discretized via finite elements. Thus it is possible to use these models for the whole audible frequency range which is typically from 20 Hz up to 20 kHz where a lot of non-pistonic and non-axisymmetric motion patterns occur, which cannot be described via a one-dimensional lumped parameter model.
1.3 The Acoustical Domain
The principle of using matrices for describing a physical domain is also used for the acoustical domain where the Helmholtz equation is discretized by means of finite or boundary elements to describe the three-dimensional propagation of sound waves in the frequency domain:
Here Ba is an acoustic coefficient matrix which defines the relationship between the vector of the sound pressure pa and a vector including effects of incident sound waves fa. Thus it is possible to describe the sound radiation with respect to a detailed definition of the geometry (CAD model) including reflectional as well as diffractional effects.
1.4 The Coupled Multiphysical System
If we now tie together all the governing equations of the different physical domains, electrical, mechanical and acoustical domain, we get the following coupled system of equations describing the multiphysics of an electroacoustical transducer in the frequency domain:
Cma and Cam are coupling matrices connecting the mechanical and acoustical domain. These coupling matrices arise from the assumption of continuity of the velocity and of the pressure in the mechanical as well as the acoustical domain in the direction normal to the coupling surface. De is an electromagnetic damping which derived from the back electromagnetic force.
It must be mentioned here that the coupled model alone will not automatically lead to realistic simulations. Additionally, we need to accurately describe our material properties in the electrical and structural domain. Thus a key aspect here are material measurement procedures specifically designed to measure electrical and mechanical parameters as well.
2. An industrial example for a realistic loudspeaker simulation
While in a “real” product development process a “real” loudspeaker is typically being measured in an anechoic chamber, we follow a similar procedure in the virtual world. An example of a typical woofer loudspeaker for the reproduction of low frequencies can be seen in figure 3.
In figure 4 a comparison of measured and simulated frequency response of the radiated sound pressure is given. The accuracy of the simulation based on the previously presented theory is within the manufacturing tolerances of the loudspeaker, and thus can be entitled as a realistic simulation.
3. Major challenges in each physical domain
Actually the most important challenge is the strong coupling of all physical domains involved. Strong coupling within that context means that each physical domain interacts bi-directionally with other domains.
While typically the motor system can be treated as an axisymmetric device, and thus simplified 2D models can be applied for a majority of applications, its strong coupling to the structural domain (the loudspeaker’s vibration system) via the voice coil acting in the magnet’s air gap must be accounted for. For some motor structures also the variation of the flux field in axial direction is of crucial importance. Thus, typically finite element models for detailed motor design and optimization are being used.
At large excursions of the voice coil (when the loudspeaker is driven in the region of nominal power), a significant portion of the voice coil moves out of the main flux field, and thus less mechanical force is being induced. This nonlinear effect is very essential and causes unwanted distortion in the radiated sound. Additionally, voice coil inductance is also dependent on voice coil excursion and also on current. This leads to the need of nonlinear models to predict the loudspeaker behavior at large signals.
For system or subsystem level simulations (without the goal of designing a motor) 1D lumped models (additionally including nonlinearities to predict large signal behavior) as presented in chapter 1 are highly efficient.
As discussed in chapter 1 the structural domain has to be modeled via finite elements to account for non-pistonic effects. The existence of circumferential bending waves can only be accounted for by 3D models. Thus 2D models have to be used with care. Additionally challenging are the thin-walled structures of cone and dust cap (and also surround and spider). By simply using 3D solid elements we would end up in a very large and unhandy model. Thus shell finite elements are typically being used to model the vibration system.
At large signals (and thus large excursions) major nonlinear effects arise from the material behavior and the change of the geometry, leading to a change in stiffness of the vibration system and thus generating distortion in sound radiation. Even if we would have a super-linear material, its change in geometric stiffness would lead to distortion. On top of this, due to the change in geometric stiffness instabilities may occur (snap-through and bifurcation), defining an additional source of heavy distortion.
The major challenge here is the strong coupling to the structural domain. I.e. the movement of the loudspeaker’s vibration system has an effect on the surrounding acoustic medium (air, sound waves are being generated) and the surrounding air has vice versa an effect on the movement of the structural domain (typically called added mass and stiffness effect).
However, for some applications, e.g. small transducers as used in mobile devices and horns for professional applications, viscous effects of the air have to be accounted for. Sound energy is being transformed to thermal energy (viscothermal effect) and heavily influences the radiated sound field – another source of nonlinearity.
The above mentioned explanation illustrates the particular challenges for realistic speaker simulations. A speaker is a complex device and requires advanced CAE technology to predict the performance. Our answer is clearly YES – Multiphysical simulation methods for speakers are an endless story due to the various challenges in each physical domain. The good news is that for all challenges solutions are available. However, the successful implementation of the simulation requires a comprehensive simulation and modeling knowledge. The key factor is the proper “modeling strategy”.