Why Guess when You can Test?

 Rather than simply relying on simulations to design a multiple driver system and hoping things work as predicted, it is possible to
 extract the simulation parameters of a multiple driver system as if all the drivers were combined into one larger driver. Being able
 to measure this data and understanding the principles behind the test will be especially helpful when it comes to fine tuning the
 final system. This paper presents how the Thiele Small test can be performed on a multiple driver system and how this relates to the
 T/S parameters of a single driver.

 Using the Delta Compliance Method

 When two equivalent drivers are placed in a single box, the principle of symmetry tells us the air volume is effectively halved for
 each driver. That is, the system behaves as if there were a physical wall between the two drivers. This principle is useful in
 determining the properties of a multiple driver system. However, finding two drivers that match in every respect may not be that

 Suspension stiffness variation from driver to driver is often the hardest thing for a manufacturer to keep constant. Thankfully, in
 most cases when the driver is put into a box, the box is small and the air spring (Kms) adds to the suspension stiffness
 (Kms=1/Cms). Nevertheless, driver matching may become important in a multiple driver configuration. At the very least, some
 consideration should be given to the fact that drivers should be broken in, and that over time some aging will also occur. Values
 for moving mass, BL, Revc and Q should match reasonably well. The good news is that barring some odd condition or catastrophe,
 these parameters are relatively constant with time. At the least, they will track between similarly built drivers with temperature
 and humidity.

 The results below are for two drivers tested separately, followed by being connected in series (parallel will yield similar
 results). The delta compliance test box method was used as this was more convenient for the two small drivers chosen for this
 experiment. The delta mass test can also be used as long as the added test masses are equally divided between the two drivers.
 In either case, the effective radiating area Sd is doubled making it necessary to multiply the diameter of one driver
 by sqrt(2)=1.414. The table below is for two Tang Band W4-654S mid bass drivers using 0.6L test enclosures (550 mL kitchen glass,
 plus a front side cone volume of ~50mL)

     Param    Driver1    Driver2   Driver1+2


   Revc      5.9586     5.9328    11.9766 ohms

   Fs       78.6429    80.4181    79.9911 Hz

   Zmax     48.5530    50.3548    97.6974 ohms

   Qes       0.6434     0.6360     0.6465

   Qms       4.5994     4.7618     4.6276

   Qts       0.5645     0.5610     0.5673

   Diam     82.5500    82.5500   116.7423 mm

   Sd     5352.0971  5352.0971 10704.0061 mm^2

   Vas       2.6825     2.6547     5.1863 L

   BL        5.3017     5.2896    10.6938 N/A

   Mms       6.1425     5.9359    12.2832 g

   Cms     666.7743   659.8568   322.2894 uM/N

   Kms    1499.7579  1515.4803  3102.8010 N/M

   Rms       0.6599     0.6299     1.3341 R mechanical

   Eff       0.1955     0.2093     0.3958 %

   Sens     84.9111    85.2072    87.9746 dB @1W/1m

   Sens     86.1905    86.5055    86.2222 dB @2.83Vrms/1m

 Multiple Driver Test Results

 The experimental data above shows that by simply testing the two drivers in series and doubling the total radiating area, the
 suspension stiffness, moving mass, BL and Vas are doubled. The overall effect of adding multiple drivers is that as drivers are
 added, each driver will operate into a smaller volume of air. Parallel or series wiring has no effect, but series wiring will
 be used for this derivation.

 Derivation of Equations (verification of Results)

 When the drivers were connected in series, and a constant current was applied, each motor produced an identical force.
 The doubled force was then applied to twice the cone area, and pushed against twice the suspension stiffness. Examination of
 the equations for Qes, Vas and efficiency then reveals the effect on the Thiele-Small parameters. Tabulating these observations
 we get:

                     Series connect Observations (Current in the two drivers will be the same)

   BL   doubles    two motors, each producing BL*I drive

   Sd   doubles    two cone areas

   Revc doubles    series connect

   Kms  doubles    two springs to push agains (spring stiffness=1/cms)

   Mms  doubles    two cone masses

   Fs   constant   double mass, but also double stiffness

   Rms  doubles    double the mechanical loss

   Equations of interest

   Vas = SpeedOfSound^2*AirDensity*Sd^2/Kms; (DOUBLES)

   Qes = Revc*Kms/(BL^2*Fs*2*PI);            (CONSTANT)

   Qms = 2*PI*Fms*Mms/Rms;                   (CONSTANT)

   Eff = 9.64e-10*Fs^3*Vas/Qes;              (DOUBLES)

 Unmatched Drivers

 Predicting the response of unmatched drivers is a bit more complex. Nevertheless, some observations and recommendations can
 be made. Basically, if driver-to-driver variation is possible, it would be desirable to put the drivers in separate boxes or
 sub-enclosures, and wire them in parallel instead of in series. In this case, the only part that interacts is the acoustic
 output. Again, starting from the series connection because it is easier to follow:

   BL  = BL1+BL2         motor drives add

   Sd    doubles         two cones

   R   = R1+R2           resistance adds

   Kms = Kms1+Kms2       But the cones might not be moving in sympathy!

   Mms = MMs1+Mms2       mass adds

   Fs  = (Fs1+Fs2)/2     Average?

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