Vented Box Notes

In the interest of giving back to the community I've scanned some notes related to vented boxes which might be of interest to those of you who want to understand what's going on under the hood of a box modelling program. I've posted the same thing to the Parts Express forum as well. There is nothing new here. Its all covered in Benson, but I've tried to distill the calculation to its essence, and have ignored the box, port and leakage losses. Benson goes into so much detail that its easy to miss the forest for the trees. Small improved on this and his presentation is more concise, but I've tried to make the calculation even more compact.

Undamped Vented Box
 
FWIW, Thiele did something very much like this in 1961.

Sure, noting is even remotely new. The equations for a vented system were known before Thiele 1961 (see Novak 1959 for example) but Thiele was the one who made the connection to filter theory and alignments. Small, and Benson independently, wrote all the relevant equations too. Benson's trio of papers is the most complete, but Small's were the ones that really caught on. More accessible. FWIW I have generally used Benson as a starting point.

I believe these notes are useful to illustrate the "essence" of the calculation and to develop intuition, There is a feeling that the calculation is really complicated, but its not. I've also tried to clarify the symmetric relation between the input impedance (from the electrical circuit) and the pressure response (from the mechanical circuit). The relation between the two puzzled me when I first tried to digest the theory.

The calculation of the impedance is left as an exercise.
 
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