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Which TS Parameters are the most important for enclosure design and why?

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The answer here is a quite complicated and technical.  First of all, the parameters are for the Thiele/Small mathematical model of driver behavior.  Without a complete set of parameters, the model is incomplete and not usable.  Now, it's possible to work with simpler models that require fewer parameters, but will leave you without vital information about the operation of the system.  The T/S model is also limited with regard to how it treats things like inductance, which is a particular problem when modeling subs.  Hence, it's often useful to use a model with extended inductance parameters, which can be obtained by data fitting to impedance measurements.

Now there's a second point.  The T/S parameters you see published often are redundant.  A complete set required for the model is only a subset of all the possible parameters you might see defined for a driver.  For example, if you know both the mechanical compliance constant (Cms) and the effective moving mass (Mms), then you can directly calculate resonance frequency (Fs), meaning that Fs is redundant in that case.  Other similar redundancies exist, but to understand them requires delving into a lot of details, and I want to keep this short.

The long answer to your question is the subject of many books.  Good luck!

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If you want to get a better understanding of T/S parameters, I'd recommend finding out how each parameter is measured and understanding what that measurement process tells you about it. That'll give you a pretty good understanding of what's going on and makes it easier to interpret data, instead of just working with numbers using abstract units. I thought I had a pdf about it somewhere, but couldn't find it now.

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Also...To reiterate, most of the parameters are directly linked to, affected by and calculated from others. None of them really exist in a vacuum. 

To further complicate things these are known as small signal parameters and many of them will shift greatly once the driver is subjected to large signals. Subs in particular operate with what would be considered large signals. 

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The Wikipedia page might be helpful: https://en.wikipedia.org/wiki/Thiele/Small_parameters.  It's useful to think of the physical system in terms of what they call the "Fundamental Parameters".  Note that this is only one of many possible sets of parameters that must be specified to complete the model.  A background in physics will help a lot there.

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