From: brian@navaho.ugrad.ee.ufl.edu (Brian Gentry)
Subject: 4th order Bandpass Enclosures 
Organization: UF EE Department (student)
Date: Mon, 13 Apr 92 22:05:34 GMT


      An Introduction to Design of 4th Order Bandpass Enclosures 

This describes how to design a 4th order symetrical bandpass enclosure.
Most of the below equations come from Speaker Builder magazine, issue 6/88.
The article is titled, "The Third Dimension: Symetrically Loaded".  It was
written by Jean Margerand.  I would like to credit Robert Cheng with
helping me to gain a great deal of insight into the equations.  In fact,
Robert found the method that Jean Margerand used to generate the tables in
his article.  The new equations, which do not require the use of the tables,
were derived by Robert.  For those of you who don't know, a 4th order
symetrical bandpass enclosure has two chambers, with the driver mounted on
the common wall between the two enlcosures.  One of the chambers is ported
and one is sealed.  A diagram is below.

 ______________________________
|                 |            |
|              _  |   Vp       |
|             | | /            |
|             | |/       ______|  
|     Vs      | |         PORT
|             | |        ______
|             | |\             |
|             |_| \            |
|                 |            |
|_________________|____________|


Equations:

[1]   Vp = [2 * S * Qts]^2 * Vas
[2]  Fl' = [Fl * Qts] / Fs
[3]  Qt' = [Fl'* (Fl' + b) ]^0.5
[4]   Vs = Vas / [ (Qt'/Qts)^2 - 1 ]
[5]   Fh = (Fl' + b) * Fs / Qts
[6]   Pa = -40 * LOG [ 1/(Qt' * 2 * S) ]  (this is a base 10 logarithm)
[7]   Fo = Qt' * Fs / Qts


Where:

      Vas is a Thiel-Small parameter supplied by the manufacturer of the driver
      Qts is the total Q of the driver (from manufacturer)
      Fs is the resonant frequency of the driver (from manuf.)
      Fl is the lower -3 dB frequency (chosen by you).
      Fh is the upper -3 dB frequency (not chosen by you).
      Vp is the volume of the ported chamber (same units as Vas)
      Vs is the volume of the sealed chamber (same units as Vas) 
      Fl' is an intermediate variable (no real world signifigance)
      Qt' is an intermediate variable (has some signifigance)
      Pa is the relative sensitivity of the system (how it compares to a
         standard single chamber enclosure) units are dB.
      Fo is the tuning frequency of the vent.
      S is the damping factor of the system.  
      b is a variable determined by S.  See the table below.


S	b		ripple (dB)
-----------------------------------
0.4	1.7172		2.70
0.5	1.2712		1.25
0.6	0.9560		0.35
0.7	0.7206		0.0


So, you do have to use a table, but it's a small one.  The tables in the
original article were HUGE by comparison.  The ripple shows up as peaking
at either end of the response curve.  So, the system with S = 0.4 has two
2.7 dB peaks -- one at the high end of the curve, and one at the low end of
the curve.

To design a system, choose a damping factor (S), choose a lower -3 dB
frequency (Fl), and get the 3 Thiel-Small parameters for the driver you
want to use.  Then solve each of the above equations.  The eqation for Fh
will tell you what you upper -3 dB frequency is.  After that, use your
favorite venting formula to find the length of the vent you want to use.
Make sure to let the vent have adequate area.  If the cross-sectional area
of the vent is too small, the port will make a whistling noise at high
volume levels.  The Pa figure tells you how the efficiency of the bandpass
system compares to the raw sensitivity of the driver.  For example, if you
do the calculations for a driver with a "raw" sensitivity of 88 dB @ 1 watt
and 1 meter, and you get a Pa of +3.5, then the system will have a
sensitivity of 88 + 3.5 = 91.5 dB.  Similarly, if you get a negative Pa, the
system will have a lower sensitivity than the 'raw' sensitivity of the
driver.

The other equations in the article (too many too reproduce) can be
manipulated to allow you to choose both the upper and lower -3 dB
frequencies.  When you do this, the alignment (damping factor) becomes a
function of the frequencies that you choose.

Questions and comments are welcome.

                          Brian L. Gentry
                   (brian@sioux.ugrad.ee.ufl.edu)


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