Tech Design.....
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Distortion in Isobaric Woofer Systems.
Figure 1
Figure 2
half the compliance, half the impedance for which it draws twice the input current and hence double the power."
This sounds very straight forward and indeed, if the drivers were connected by an incompressible medium or solid
rod then the arguments of how distortion is reduced would hold some validity. However, the compressibility of air
is not dependent on the volume of the air in the chamber or frequency although the
coupling between the drivers
is, and the drivers are not connected by a light weight, rigid rod.

When the isobaric system is correctly analyzed it becomes readily apparent that the distortion canceling
mechanism of a push pull system does not translate directly to the isobaric system. Before going into any detail
some observations about the cancellation mechanism are in order. In the standard push-pull system the drivers
are independently driven and it is their acoustic output that is
summed. As noted above, even order HD
components will be 180 degrees out of phase and cancel in the
sum. However, in the isobaric system the
individual acoustic output from each driver do not sum so the same distortion mechanism can not be at work. On
the other hand, if the drivers were connected by an infinitely stiff rod they would be forced to move with the same
motion. The forces generated by the motor and suspension would be the
sum of the forces generated by each
motor and  suspension; and with the motors wired out of phase and with a push-pull mounting, the even order
distortion in the
summed forces would cancel. The resulting driver motion and radiated SPL should, therefore, be
void of even order HD provided that no additional distortion is introduced by the air compliance of the rear box.
Note that the key here is the summation of SPL (in the standard push-pull configuration) or forces (in the isobaric
case) which leads to the cancellation. But is this summation of forces appropriate for an isobaric system? To
answer this question a very simple analysis is all that is required.

To start we must accept that for the distortion cancellation to occur in the isobaric system the motion of both
drivers must be the same. Since this is a mechanical system, if the drivers are to have identical motion then the
forces acting on each driver must also be identical. If it can be shown that these two conditions are mutually
exclusive then it will suffice to show that the cancellation mechanism doesn't stand. We begin by accounting for
the forces acting on the drivers. The figure below shows these forces. Fd1 is the force generated by the
suspension and motor of driver 1. Fd2 is the force from the suspension and motor of driver 2. K1 and K2 are
spring constants and represent the compliance of the air in the rear box and isobaric chamber. X1 and X2 are the
positions of the respective drivers relative to their rest positions. K1 and K2 are taken to be finite. If the drivers
are to undergo the same motion, Newton's Law tells us forces acting on each driver must be the same. The total
force on each driver is:

(1)   Ft1 =Fd1 - K1 X1 - K2 (X1 - X2)

(2)   Ft2 = Fd2 + K2 (X1 - X2)

In Equations (1&2) the term K2(X1-X2) is the coupling between the drivers by the isobaric chamber. Since for
identical motion Ft1 = Ft2,

(3)   Fd1 - K1 X1 - K2 (X1 -X2) = Fd2 + K2 (X1 - X2)

with a little manipulation we find that

(4)   X2 = [1 + K1
/K2/2] X1 + (Fd2 - Fd1)/K2/2

We can also note that the spring constants are proportional to 1/V, where V is the volume of the box, Vbox,  or
isobaric chamber, Viso. Thus,

(5)   X2 ~ [1 + Viso
/(2Vbox)] X1 + (Fd2 - Fd1) Viso/2

If we normalize the volumes by Vas we can write,

(6)   X2 ~ [1 +
α/(2β)] X1 + (Fd2 - Fd1) Vas/(2β)

where  α = Vas/Vbox and β = Vas/Viso. If the system is linear (Fd2 - Fd1)=0. Equation (4) tells us that only as
K2
tends to infinity (a perfectly ridgid rod) will X1 = X2. If K2 were indeed infinite, then we would need to rewrite
Eqs(1&2) as FT1 + FT2 = Fd1 + Fd2 - K1 X1, and X1 would equal X2 by definition. That X1 can not equal X2 for
finite K2 becomes even more obvious when we refer back to Eqs (1&2). If X1 = X2 then there would be no
coupling between the drivers, with any finite value of K2, by the isobaric chamber. Thus the forces on the drivers
would have to be different. We therefore have a contradiction; if we assume the total forces on each driver are
the same then, for any finite K2, the displacements are different; if we assume identical displacements, the
isobaric coupling is zero for any finite K2, and  the forces are different. Equation (6) also gives us a hint of
where the idea that Vas for the isobaric system is 1/2 that of the single driver system since
α/2 = (Vas/2)/Vbox.

The next step in understanding the behavior of the isobaric system is to develop a suitable model. Without
presenting the details, I have developed a model from the ordinary differential equations which describe the
driver's motions as influenced by the sealed rear box and the isobaric chamber. The details can be
downloaded
here. This model was solved numerically. The model includes a rigorous accounting for nonlinearity due to the
compression/expansion of air in the rear box and in the isobaric chamber. It also includes a simple nonlinear
compliance model as shown below. The suspension compliance can be symmetrical
about x/xmax = 0.0, or asymmetry can be introduced through an offset. A symmetric nonlinearity introduces only
odd order harmonics. The asymmetry introduced by the offset generates even order distortion components and
reversed polarity/push-pull can be mimicked by introducing a positive offset in one driver and an equal
magnitude, negative offset in the other. Consideration of the nonlinear air compliance is always asymmetric.
Both the nonlinearity of the air compliance and that of the suspension model can be switched on and off to
investigate the effects of each separately or in combination.  The code can also be run as two isolated drivers to
model a true push-pull woofer (first figure).

To demonstrate the effects nonlinear distortion a number of simulations were performed. These were all based
on a driver with the T/S parameters similar to those of a 10" Peerless XLS woofer. The woofer was assumed to
be in a box suitable for a Qtc = 0.707. All calculations were performed at fc, approximately 40 Hz. A suitable
input signal level was used to push the drivers well into the nonlinear excursion region to emphasize the
distortion levels. Results are presented as plots of distortion.

Testing the Code: Isolated drivers and standard push-pull configurations (Figure 1)

First a series of test were performed to verify the operation of the code. To begin, the code was run in separate
driver mode. In separate driver mode
Front Driver data  corresponds to either driver in isolation and its distortion
characteristics. Rear Driver data corresponds
to the summed acoustic output of the two
drivers.

The figure to the immediate left is the result
for the case where no nonlinearities are
included. No distortion is present in either the
individual driver or summed responses.

The result presented at the right is for the
case where the suspension nonlinearity is
considered, but is taken as symmetric about
zero displacement. Note that only odd order
distortion is present. Also note that the  
summed response is identical to the individual
driver's response. There is no reduction in
odd order distortion.

The third result, presented to the left, shows
the rise of even order distortion when a 1mm
offset is introduce in the suspension
compliance nonlinearity. This case represents
the distortion behavior when the drivers are
NOT in a push-pull configuration.

The result at the right is for identical
conditions as those at the left except the
drivers are now in push-pull configuration.
The even order distortion components are
eliminated. (The residual 2nd and 4th order
components are a result of loss of significant
figures due to importing and post processing
the generated simulation data in Excel).



This last result for the isolated driver case
(left) shows what happens to the classical
push-pull configuration when the nonlinearity
of the box air compliance is included. The
even order HD is reduced, but not eliminated
because the nonlinear air compliance
generated distortion is not dependent on how
the driver is mounted.

These results show that the simulation code is
operating correctly.
Application to Isobaric Woofer Systems (Figure 2)

The next series of results are for application of the simulation code to an isobaric woofer system with the same
Qtc and Fc. The rear box volume is 29L, 1/2 that of required for the individual driver case and the volume of the
isobaric chamber was specified as 4L which is reasonable for a face to face, push-pull configuration allowing for
surround clearance and cone motion. For the first simulation, presented directly below, the nonlinear
suspension and air compliance were switched off. Here I have also presented the front and rear driver
excursion and the pressure variation in the isobaric chamber. They are typical. The front driver is that which
actually radiates the sound.  As with the isolated driver case, there is no distortion generated when the model is
linear. However, it is observed that the front driver excursion exceeds that of the rear driver. This is consistent
with the contradiction we observed in the simplified analysis, above. We also see that the pressure in the
isobaric chamber is anything but constant. In fact, it is approximately 1/2 the amplitude of the pressure variation
in the rear box. Again, this makes sense since if the driver motions are close to the same, which they are, then
since the motor and suspension forces are identical for both drivers the pressure forces must  be about the
same too. The front driver is subject to a pressure force of 1/2. The rear driver has a pressure force of 1 acting
on the back face and -1/2 on the front face for a net of 1/2. Note that these are approximate and while the
excursions and pressure forces are close to the same, they are different, and as will become apparent these
small differences are important when considering nonlinear distortion and cancelation.
At the left the result for nonlinear but
symmetric suspension compliance is
presented. Only odd order components are
present but note the difference between this
isobaric case and the separate, isolated
drivers. Here the distortion is different for the
front and rear drivers and the magnitude of
the higher order components is actually
higher for the front driver.

When asymmetry in introduced in the
suspension compliance by a 1 mm offset the
result is as shown to the right. This is for a
front to back, not a push-pull configuration so
no cancellation would be expected. The even
order distortion components are now present
and again, the higher order components are
greater for the front driver, which radiates the
sound.

Directly to the left is the result when the
drivers are mounted in a push-pull
configuration. In the isolated driver case the
even order distortion canceled completely
(except for small numerical error). Here, while
we do see a reduction in the 2nd order HD
the 4th order HD actually increased over the
non push-pull case.

To the right is the result when nonlinear air
compliance for both the rear box and the
isobaric chamber is considered as well. A
small but significant increase in distortion is
observed. This last figure should be
compared to the last figure for the
standard,
isolated driver, push-pull configuration above.
The dark blue, large data points to the left
should be compared to the smaller pink data
points for the standard push-pull case. It will be observed that compared to the standard push-pull configuration the isobaric push-pull
configuration offers very little in terms of reduced even order distortion. Finally, below to the right is the result for the same isobaric
woofer system with asymmetric suspension
compliance and nonlinear air compliance when not in a push-pull configuration.
Compared to the result directly above very little change in distortion is observed further
demonstrating the lack of significant cancellation of even order distortion in isobaric
woofer system using a push-pull driver format.

One last simulation is presented below to show the effect of reducing the isobaric
chamber volume to 1L from the previous 4L. The lower left plot shows the complete
nonlinear air and compliance result in standard format. At the lower right is the result
when the drivers are placed in push-pull configuration. Improvement in even order
distortion is apparent, however reduction of the isobaric volume to 1L or less is unrealistic.
The calculation presented here are not
intended to be exact representations of a real
drivers or woofer systems. Rather they are
intended as qualitative representations of the
behavior of these types of woofer systems.
The principles upon which the models are
founded are reasonable and the results are
consistent with reasonable expectation of the
trends of real systems.
1. V. Dickason, Loudspeaker Design Cookbook, 5th edition, Audio Amateur Press, Peterborough, NH.

2. M. Colloms,
High Performance Loudspeakers, 5th edition, john Wiley and Son, New York.
One commonly used method to reduce harmonic distortion in woofer systems is through
the use of two woofers mounted in a push-pull format as shown in the figure to the right.
When one driver is connected with inverted polarity in push-pull format the result is that
even order distortion components will be 180 degrees out of phase and should cancel in
the summed
acoustic response. This is fundamentally true and can be realized for dipole
or infinite baffle woofers, and other woofer systems provided the air mass load on both
sides of the woofers is identical. However, in boxed woofers there is an additional
distortion generating element which is unaffected by the driver mounting. This is the
nonlinearity of the compliance due to the air in the box. If the box is large, or the
excursion of the woofers is small, this source of distortion may also be small, thus
cancellation of the majority of even order distortion is possible.

This distortion canceling approach has been routinely adopted to isobaric or compound
woofer systems as well. The lower figure to the right shows an isobaric woofer system
with the drivers mounted face to face. It has been stated [1], [2] that this configuration
benefits from the same even order distortion canceling mechanisms. However, this is not
the case. The premise behind the isobaric configuration is that  such an arrangement
allows the box volume to be 1/2 that of a single woofer configuration, and that the air in
the isobaric chamber remains at constant pressure. If such were the case, then the outer
woofer would have no air spring effect on it and it would operate as if mounted in free
air. We know this is doesn't happen. In actuality the outer woofer moves, approximately,
as if it were mounted by itself in a box of twice the volume. In [2] the following
explanation is offered:
"The small air chamber between the drivers is essentially
incompressible at frequencies below 150 Hz. Hence the diaphragms may be regarded
as closely coupled, as if by a lightweight  rod. Now the analysis is simple. Conventionally
connected, with the motor coils connected in parallel, the composite dual driver has the
following characteristics if compared with the single device: twice the moving mass;