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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
easy.
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
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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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