Tech Design.....
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Correcting acoustic center offsets in dipole speakers...
The misalignment of the AC's results in a tilting of the polar
response downward and as the frequency increases the main
lobe of the response becomes narrower. The the AC misalignment
is corrected, the main lobe is aligned with the design axis.

Now let us consider a similar situation between a woofer and a
midrange of a 3-way conventional speaker. The situation is shown
schematically in Figure 2. We shall consider a system with a 141
Hz crossover and a woofer to midrange separation of 2 feet. At
141 Hz the wave length is 8 feet so the driver separation is only
1/4 wave length. We shall also exaggerate the offset, d, for
reasons which will become apparent later, as 8 inches. The polar
response at the crossover frequency of 141 Hz and a distance of
3 meters is shown in Figure 3. The red curve is with the offset
corrected and the green is with no correction. As you can see
there is little effect on the polar response with only very small,
insignificant tilt. The 8" offset corresponds to a phase error of 30
degrees at the crossover frequency. With an LR4 crossover a 30
degree phase error result sin a -0.3 dB error in the on axis
response. This is apparent in Figure 3 along the horizontal axis.
The same can not be said for other types of crossovers. For
example, for Butterworth crossover, where the intended phase
difference between sources is 90 degrees, a 30 degree error
would result in either a -3dB or +1.75 dB error on axis. Thus the
wisdom of using an "in phase" crossover is apparent. There is
little reason to make any effort to correct for this offset with an
LR4 crossover (or any order "in phase" crossover for that matter)
when the crossover frequency is low, or when the driver
separation and the offset is significantly less than a wave length at
the crossover frequency.  However, if it is desired to dot the i's
and cross the t's it certainly doesn't hurt to compensate for the
delay, other than the added complexity of the crossover.
Figure 1. Polar response
for conventional speaker
with driver acoustic
centers aligned (red) and
offset 1.5". Driver
separation is 6".
Crossover frequency =
1100 Hz for the red and
dark blue traces and 1550
for the light blue trace.
Conventional
Speakers
Figure 2. Schematic representation of a conventional
speaker with woofer and midrange separated by a
distance, d, and off set acoustic centers.
Figure 3. Polar response for the conditions
of Figure 2 where d = 2 feet and the AC
offset = 8 inches. Red, with offset
correction; Green, without offset correction
Dipole midrange
and woofer
Figure 4. Speaker system using flat baffle dipole midrange with H-frame dipole woofer system. Baffle aligned to front
plane of H-frame resulting is significant driver AC misalignment.
Solutions to the
problem
The solution to the problem, if you must retain a dipole woofer system, is to either uses a flat baffle woofer system or to
align the midrange baffle with the woofer system, as shown in Figure 5. The resulting AC alignment would result in no
errors for any type crossover. The trade off of figure 5 is the potential for diffraction and reflection from the top, front
side of the woofer H-frame directed towards the listener. However, the dipole radiation patterns tends to minimize this
and if required acoustic damping material could be applied to the surface.
Figure 5. Aligning the woofer and midrange dipole symmetry planes.
Cardioid and U-frame
woofers
If you are not fixed on the
implementation of a dipole woofer
system a solution to the AC offset
problem can be found in the application
of cardioid or U-frame woofer systems.
Figure 6 shows a speaker system using
a cardioid or U-frame woofer system.
Where as the AC of the H-frame dipole is
located at the center of the H-frame the
AC of a U-frame or cardioid woofer
system is essentially located at the front
plane of the woofer. Thus the ACs  are
essentially aligned  when the midrange
baffle is aligned with the front of the
woofer enclosure. Any remaining offset
is very small compared to the wave
length at the crossover point resulting in
insignificant errors in the front on axis
response regardless of the crossover
type. To the rear the result is
inconsequential due to the transition
from dipole to cardioid. However, this
transition is best handled using an in
phase crossover. Since the rear
response ceases to be a factor, allpass
delays, while unnecessary, can again be
applied to compensate any residual
offset, if desired.
Figure 6. Speaker system with dipole midrange and cardioid woofer system.
In conclusion it is apparent that the conventional wisdom of correcting AC offsets using allpass delays or other means of
introducing excess phase in the crossover, as applied to box type direct radiator speakers, is flawed when applied to
speaker systems employing multiple dipole sources. This is because the offset of source 1 relative to source 2 reverses
as we move from the front side of the dipole speaker to the rear side. If the offset is X cm from the front, it is -X cm from
the rear. Thus the delay compensation, while the same magnitude, would have to be applied to the 2nd source rather
than the 1st, or vice versa, situations which can not exist simultaneously. Application of a delay will alway introduce
greater error to the rear response if it corrects the misalignment relative to the front of the speaker. The only correct
solution is to physical align the AC of the two dipole sources. Otherwise it would appear better use an in phase type
crossover when the wave length at the crossover frequency is large relative to the AC offset and source separation, and  
accept the small, symmetric front and rear errors. In reality, under these conditions the errors introduced either with, or
without the presence of the delay circuits is likely to have no audible consequences. This further suggests that
introduction of delay circuits in such application only introduces greater, unnecessary complexity and cost.

By contrast, if a cardioid or U-frame woofer system is employed, allpass delays can be used to correct the AC offset
between the cardioid woofer and dipole midrange since the rear response ceases to be an issue. However, with an in
phase crossover, and even with a Butterworth type crossover, the small offsets encountered relative the wave length at
the crossover typically will not require any such correction.
Closing
Rremarks