The Dynavox LW6004/BR-1 cabinet with high pass filter allowed 50 watts but there was a hump in the bass response (see the Good bass from a low Xmax driver page).  Depending on how much bass you like this would be OK, but if you want a flat response lowering the Qes to 0.5 in Winisd gives the desired output. We can electronically change Qes and hence Qts by means of a negative output resistance amplifier. I figured a Qts of 0.7 meant a Qes of  0.78, see below to see how.

Thanks to Brian Steele on this link there is a formula for the relationship source resistance vs Qes:

https://www.diysubwoofers.org/dipole/

Rs = Re*(Qes’-Qes)/Qes where Qes’ is the desired new Qes

-1.86 = 5.2 (0.5-0.78)/0.78

As Thiele pointed out in his 1961 paper positive current feedback can provide a negative amplifier output resistance and the circuit shown uses this to generate -2 ohms. Positive feedback is used in oscillators and if you are too greedy with the amount of negative resistance the circuit will oscillate. If you try and eliminate all the voice coil resistance the circuit will hoot. Just in case you think this is a new feature,  the Pye HF10 and HF25 amplifiers of the mid fifties had an adjustable negative resistance control! Warner Clements original 1951 article on negative resistance is on page 22 here

Op-amp U1 is a audio power amp like the LM1875, TDA2030 or LM3886 and U2 is an audio op-amp like the TL072 or NE5532. U1 and U2 need the usual power supplies and decoupling and U1 needs a zobel network to make it work. R8 is the loudspeaker.

Fig1 shows the gain suitable for the power amps mentioned above. How to prove the circuit is generating negative resistance. Making R8 2.05 ohms gives a  big peak in the output showing that the net resistance is zero and the gain is more or less infinite (Fig2). It also shows where the feedback is 360 or 0degrees which by luck (or design) happens to be at the box tuning frequency.

Fig 1 Gain response

Fig 2 Gain response with R8 as 2.05 ohms

Given a random drive unit and a random vented box enclosure,  can we match them? Bass reflex enclosures usually have to be closely matched to the drive unit. The drive unit has 3  parameters needed for vented box design, the Resonant frequency Fs,  the compliance Vas  and the Qts. Since the cabinet size is approximately a function of the Qts cubed, it is very dependent on the value of Qts.  This technique effectively lowers the Qts  to  suit a smaller cabinet than normal.  The other way of using a smaller cabinet is a variovent but in my opinion that technique is difficult to control.

To go back to the old ways of doing things, Hoge came up with some formulas to make vented box design easier these are:

Vb = 15 X Vas X Qts^2.87

F-3 = 0.26 X Fs/Qts^1.4

Fb =  0.42 X Fs/Qts^0.9

If you take my Mission CP168  with its 0.42 Qts,  0.445 Qes, 9.2  Qms, 23.8 litre Vas , Fs of 47.4Hz, Re of 7.5 and plug it into the equations you get a box size of 29.6 litres which is bigger than the 11 litre box I want to use.  What Qts is required for the 11 litre cabinet ?

Rearranging the first equation

Vb/(15 X Vas)= log(base Qts)2.87 which can be done on the calculator by pressing inv before x^y key

Hence required Qts’=0.3

The new Qts and Qes are represented by Qts’ and Qes’

Another  formula from the Brian Steele page

Qes’=Qts’*Qms/(Qms-Qts’)
Qes’ = 0.3*9.2/(9.2-0.3)
Qes’ = 2.76/8.9
Qes’ = 0.31

Rs = Re*(Qes’-Qes)/Qes
Rs = 7.5*(0.31-0.445)/0.445
Rs = 7.5*(-0.135/0.445)
Rs = -2.27 ohms

With  a negative resistance amplifier  generating -2.27 ohms and an 11 litre cabinet  the above equations give a box tuning of 58.8Hz and  a -3db point of 65.5Hz. More on the Hoge formulae:

http://www.mh-audio.nl/Calculators/rbc.html

http://www.audioheritage.org/vbulletin/archive/index.php/t-39807.html

The above link shows that the renowned speaker engineer Keele was the originator of the Hoge formulae. I did a similar analysis to the CP168 for the LW6004 and the Qts came out at 0.4 Fb came out at 50Hz . Winisd showed  the excursion would only take a 15W amp and needed a 44Hz 3rd order HPF Butterworth which from my table in Xmax problems is this:

So the negative resistance amp can be as above. A suitable device would be the LM1875 and the other half of the op-amp could be used for this filter which gives a 54Hz -3dB response for the overall system.

Typical domestic programme material will make the voice coil temperature rise by 10%, so the negative resistance should be increased to compensate. This effect is mitigated by using drivers with large diameter voice coils which have large surface area therefore good cooling. Hence the recommendation for the LW6004. If the voice coil heating bothers you read this by Francis Brooke (translate from French)