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The theoretical background of this work is covered in a recent blog post I made (https://www.comsol.com/blogs/exploring-the-partial-fraction-fit-functionality-in-comsol-multiphysics), but the topic is Partial Fractal Fitting, and this method has some characteristics that makes it interesting for transducer analysis and synthesis.
A transfer function can generally be written in three forms: A rational form with a fraction of polynomials, a pole-zero fraction, and finally a partial fraction expansion:
The latter is often the bane of the life of engineering students, as it typically is introduces as a means of doing analytical inverse Laplace transformations, which is not necessarily the most intuitive process. However, in this form you have a nice summation of parts, and each part tells you something about the total system, with an asymptotic part, first order poles and residues, and complex second order poles (typically complex conjugate) and their residues. This form is convenient for a transformation to the time domain, which may be its main feature. Also, having the transfer function in this form makes for insightful visualizations, but it requires having the transfer function in analytical form.
If the transfer function instead is known in numerical form, as from a simulation or a measurement, a partial fraction fitting scheme can find the poles and resides, which basically amounts to the 'resonances' in a system. Knowing them and their individual 'strength' is of great interest, as for example you can relate the found poles with resonances and anti-resonances, and see that some peaks in your response may be the result of multiple poles that are closely spaced. I have recently made a COMSOL app for my own use that does just that, and visualizes the poles for any numerical input, be it from my simulations or from client measurements. Below is an example for a simulated lumped Balanced Armature Receiver model (https://audioxpress.com/article/simulation-techniques-lumped-element-modeling-of-transducers), where a lot of complex poles line up with the resonances in the pressure output for a constant voltage input, but also some poles are lower while still present:
It would be difficult to deduce this directly from an inspection of the frequency response, but the fitting scheme finds the poles, to within a tolerance set by the user, and the user can see how well the set tolerance and the resulting poles approximate the input values. The input does not care if the input is minimum-phase or not, or if there is a constant delay in the frequency response, as it just finds the associated poles, but it might be sensible to still think about what is being input, and how it should be interpreted.
There is also a synthesis aspect here, as complex systems can be directly converted to a lumped system from the PFF description, making for nice reduced order model, but this will be covered in some coming audioXpress and JASA articles.
Have a great weekend!
René, Acculution ApS
A transfer function can generally be written in three forms: A rational form with a fraction of polynomials, a pole-zero fraction, and finally a partial fraction expansion:
The latter is often the bane of the life of engineering students, as it typically is introduces as a means of doing analytical inverse Laplace transformations, which is not necessarily the most intuitive process. However, in this form you have a nice summation of parts, and each part tells you something about the total system, with an asymptotic part, first order poles and residues, and complex second order poles (typically complex conjugate) and their residues. This form is convenient for a transformation to the time domain, which may be its main feature. Also, having the transfer function in this form makes for insightful visualizations, but it requires having the transfer function in analytical form.
If the transfer function instead is known in numerical form, as from a simulation or a measurement, a partial fraction fitting scheme can find the poles and resides, which basically amounts to the 'resonances' in a system. Knowing them and their individual 'strength' is of great interest, as for example you can relate the found poles with resonances and anti-resonances, and see that some peaks in your response may be the result of multiple poles that are closely spaced. I have recently made a COMSOL app for my own use that does just that, and visualizes the poles for any numerical input, be it from my simulations or from client measurements. Below is an example for a simulated lumped Balanced Armature Receiver model (https://audioxpress.com/article/simulation-techniques-lumped-element-modeling-of-transducers), where a lot of complex poles line up with the resonances in the pressure output for a constant voltage input, but also some poles are lower while still present:
It would be difficult to deduce this directly from an inspection of the frequency response, but the fitting scheme finds the poles, to within a tolerance set by the user, and the user can see how well the set tolerance and the resulting poles approximate the input values. The input does not care if the input is minimum-phase or not, or if there is a constant delay in the frequency response, as it just finds the associated poles, but it might be sensible to still think about what is being input, and how it should be interpreted.
There is also a synthesis aspect here, as complex systems can be directly converted to a lumped system from the PFF description, making for nice reduced order model, but this will be covered in some coming audioXpress and JASA articles.
Have a great weekend!
René, Acculution ApS
