Showing that vented-box loudspeakers act as 4th order high-pass filters

It is commonly stated in loudspeaker design literature that, at low frequencies, vented loudspeakers behave as 4th order high-pass filters.

The acoustical equivalent circuit below is used to determine the frequency response. Although not labelled here, Uc is defined as the current provided by the source.

Where Rat is the total series resistance and p is the laplacian vairable.

This looked a bit unusual to me so I tried doing the circuit analysis myself. The circuit is essentailly a current divider so I started out with the ratio of branch to total impedance by putting all the components in their impedance form, applying a factor of: $$sCab*Cas(s^2 + \frac{1}{Cab*Map})Map$$ To clean up and fully expand the denomiator expression, which gives:

The denominator seems to be in the correct form but I can't reduce the numerator to a single term and I was expecting a 4th order polynomial. Even if I apply the factor that appears in the final answer (1/s*Mas) that doesn't solve either problem.

So my question is have I made a mistake in my analysis or are there further steps to put this into the correct form?

• Without defining where Uc is in the picture there is NO solution to check. If Uc is the twin circle current source to the left of the top picture then it won't be a fourth order equation because the series elements play NO part in the solution. – Andy aka Aug 10 '17 at 16:11
• I'm voting to close this question as off-topic because it isn't about loudspeakers (which is probably off topic given the title) and it isn't about EE. It does appear to be about math and algebra but that doesn't, on its own, make it on-topic here. – Andy aka Aug 10 '17 at 16:17
• @Andyaka Yes Uc is the current source on the left. I'm afraid I don't understand how this is off topic. It is a circuit analysis problem directly related to the equivalent circuit of a loudspeaker. I'm unsure whether my analysis is correct which is why I consider it a circuit analysis problem rather than an algebra problem. – GWright Aug 10 '17 at 16:26
• You're not listening. Your analysis is flawed because it uses a current source therefore all the series elements are not part of the solution. You haven't stated what the output is and your picture doesn't adequately tell me what the input is. Your title implies an off-topic question and the starting point of your analysis is a mystery. Simplify back to the electrical starting point and use sensible electrical letters like L1, C1 etc. and don't use p instead of s. No need to mention loudspeakers or any of the complex terms in the picture. – Andy aka Aug 10 '17 at 16:38
• The circuit is not really clear to me: you have a current source in the left side (the stimulus?) and this current feeds the network made of several energy-storing elements. Where do you observe the response: across capacitor $C_{ab}$? If you use brute-force analysis, I am afraid you will have difficulty to factor the result in a usable form. You should try the FACTs : ) – Verbal Kint Aug 10 '17 at 16:38