Tech Design.....
___________________________

Integration of front, and front and rear tweeters to a dipole
midrange.
Technical  analysis and discussion:
Figure 1. Polar response of ideal dipole as a function of separation,d,
divided by wave length, w.
Figure 2. Power vs frequency for an ideal dipole with 6" separation.
Figure 3. Power and polar response of an equalized dipole based on a directional
driver with 200 sq cm effective cone area.
Figure 4. Power response of dipole tweeter overlaid with the midrange result
of                   Figure 3.
Figure 6. On axis response of dipole tweeter.
Figure 5. Tweeter power response for 4 different tweeter scenarios.
Clearly the results presented in Figure 5 and 6 leave plenty to be discussed. First, in the cases where
there is only a front tweeter it is obvious that actual result will lie somewhere between the infinite baffle
(2Pi) and free space (4Pi) results. The exact nature of the total radiated power will depend on the baffle
dimensions and shape and at what frequency the response undergoes a transition from 4Pi to 2Pi
radiation. This is just the typical baffle step result  as seen for a conventional speaker. After all, with a
single front firing tweeter, the tweeter behavior is identical to a conventional speaker. However, it should
be recognized that response represents the power radiated to the front hemisphere only and the
difference between the turquoise and lime green traces represents the power radiated to the rear
hemisphere due to the "leakage" around the baffle. The infinite baffle result seems to indicate that there
would be nominal a 1.8dB power mismatch in the crossover region before the tweeter power begins to roll
off. But the distribution of the power is also important. For the 2Pi case all the tweeter power is delivered
to the front side of the speaker where as for the midrange 1/2 the power is radiated to the front and 1/2 to
the rear. Thus, with regard to the front radiation there is a full 4.8 dB power mismatch. If the baffle step
frequency is high enough that the single front tweeter
does contribute radiation to the rear at lower
frequencies, then it is the differences between the turquoise and green limes that represents the power  
radiated to the rear and the actual power radiated to the front  side is unaltered.

Adding a rear firing tweeter which is isolated from the front, as shown by the brown trace, increases the
total power radiated at high frequency by 3dB over the single tweeter result. But at low frequency the total
radiated power would equal to that of the single tweeter radiating into 4Pi. The result would be a full 4.8
dB difference in power between the tweeter and the midrange in both the front and rear hemispheres, but
the front and rear would be identical.

When the front and rear tweeter pair act as a dipole things get very interesting. The case plotted is a
worst case scenario of a circular tweeter baffle. If the directionality model is reasonable and the dipole
tweeter response on axis could be equalized to flat over the useful range (limited to a little less than an
octave above the dipole peak frequency)  and remain unequalized above it (about 2k Hz in this example)
the power response would looks at shown by the violet line. The tweeter on axis response would appear
as shown in Figure 6 with 6dB peaks and nulls which decay as the driver becomes more directional. It's
makes no sense to normalize the power response by the on axis response in the region above 2K due to
the extreme undulations. However, the spatially averaged response would be at + 3dB and if the power
were normalized by the spatially averaged response it would drop to the 0dB level here. But, obviously
this would not be a reasonable or useful result for the tweeter on axis response. Either the directionality
model is grossly in error, or the circular baffle is an extreme case.

A simple measurement is all that is required to check the directionality model. Figure 7 (below) shows the
measured result for a dipole tweeter constructed as shown in the lower right hand of the response plot
using two Seas 27TDFC tweeters. The similarity between Figure 6 and 7 gives an indication that the
directionality model is reasonable. The difference in the frequency of the first null is simply an indication of
a different effective path length around the baffle edge to the measurement point compared to that
specified in the simulation. The result indicates that if we are to have a front and rear tweeter we must
some how control the directivity of the tweeters so as not to show the severity of influence in the response
above the useful range. This can be accomplished in much the same way we design a baffle and select
the tweeter position for a conventional speaker. After all, the dipole behavior is just a variant of the baffle
diffraction problem. This suggests that we can obtain smooth on axis response at high frequency by
designing the baffle for the dipole tweeter so as to minimize the effects of diffraction just as we would for a
conventional speaker.

Assuming that we do this then the most likely response for the dipole tweeter would be something as
depicted by the yellow line in Figure 5 where the tweeter would under go a smooth transition from dipole
behavior at frequencies below the dipole peak to (1130 Hz for the example) to behavior of two isolated
sources above it. The verification of the directionality model also suggests that for a single front tweeter
the lime green line may be closer to the true power response. Thus, with a single front firing tweeter we
would expect a polar response of the tweeter to be more like the green polar response shown in Figure 5.
With a rear tweeter the power response would follow the yellow line and the polar response would appear
dipole like, reducing the radiated power at this frequency.
Thus through the addition of a rear
tweeter it is possible provide a smoother transition in the power response through the
crossover between the midrange and tweeter, provided it is an inherent aspect of the design.
The addition of the rear tweeter also provides uniform spectral balance between the direct sound and the
reflected sound potentially yields a more natural sound over all. This approach has been part of the NaO
II speaker design since its inception.
Figure 7. On axis response of a dipole tweeter configuration on a  baffle providing
conformation that the driver directionality model is reasonable.
Figure 8. Response of front, rear, and summed front and rear tweeters measured on axis from the
front side at 1 meter.
Figure 9. On axis, red, and 90 degree off axis with and without the rear tweeter switched on, for
the NaO II speaker system.