Tech Design.....
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Integration of front, and front and rear tweeters to a dipole midrange.
Summary:

When designing a dipole speaker with conventional drivers we are faced with the decision of whether we should
use either a single front firing tweeter or both a front and rear tweeter. Assuming flat on axis response is the goal
for either approach we must look to other considerations to see the difference is behavior of these two approaches.
Power response seems a like choice. However, it is not power alone that matters, it is also the distribution of
power, or directivity of the speaker that matters. As will be shown below in excruciating detail, this leads to some
interesting results. To summarize, let us ignore driver directionality for the moment and consider the case of a front
only tweeter. If the speaker’s baffle is sufficiently wide the tweeter only radiates into the front hemisphere, or 2 Pi
space. This would be 3 dB below a monopole with the same on axis response radiating into 4 Pi space. The
monopole into 4 Pi space will be considered the reference, 0 dB power level. A dipole midrange would nominally
radiate 4.8 dB less than the monopole, but into 4 Pi space; ½ the power into the front hemisphere and ½ to the rear.
From an elementary point of view the single front tweeter looks ok as there is only a 1.8 dB power mismatch
between the dipole midrange and the tweeter. But looking closer we see that if we concentrate on only the front
hemisphere the mismatch is actually 4.8 dB in favor of the tweeter. On the back side, there is no high frequency
power at all. The result is a spectral imbalance not only between the direct and reflected sound originating from the
front and rear sides of the speaker, but also an imbalance vs. frequency on the front side alone.

If the baffle is narrower, in the  lower frequency range of the tweeter radiation wraps around the baffle edges into
4Pi space. Thus at low frequency the total power radiated by the tweeter, with flat on axis response, will approach
that of the monopole. At first glance this may appear to make things worst since in the crossover region the tweeter
would radiate a total power approaching 0 dB while the dipole midrange would still radiate at -4.8 dB. However,
when we look again at the front and rear sides of the speaker we find no change in the power radiated to the front,
but now have some additional power radiated to the rear of the speaker improving the spectral balance of the rear
radiation.

Adding a rear tweeter presents an even more interesting picture. Looking at the wide baffle case, the front and rear
tweeter would each act as if isolated from each other. The result is that the speaker becomes symmetric, front to
rear. That is, the same power is radiated form both sides. The power distribution vs. frequency on each side will be
identical to that of the single tweeter on a wide baffle; 4.8 dB more power radiated by the tweeter. However, when
the baffle becomes narrower, careful consideration of the baffle shape and crossover point, in conjunction with the
directional characteristics of real tweeters, results in an interaction of the tweeters at low frequency resulting  in a
dipole like pattern. This actually reduces the radiated power by the tweeters in the crossover region. Combined with
the directional characteristics of the tweeters as the frequency rises the overall effect is a reduction in the power
mismatch between midrange and tweeter in the crossover region and a smoother overall power response of the
speaker while at the same time achieving front the rear symmetry.

In the final analysis, a single front tweeter can result is as much as a 4.8 dB power mismatch in the front
hemisphere and complete lack of radiated power above the crossover point in the rear hemisphere. With the
addition of a rear tweeter, consideration of baffle size and shape, tweeter directionality and crossover point, the
power mismatch through the crossover region can be reduced, the variation of power with frequency made
smoother, and power response of the speaker made symmetric, front and rear, and the directionality of the speaker
is made more uniform over a wider frequency range.  Over all this yields a more uniform distribution of energy
between the direct and reflected sound which seems to have a positive impact on the performance of the
loudspeaker.
Technical  analysis and discussion:
The design of 3 way a dipole, or open backed loudspeaker, like any speaker system, involves a series of trade offs.
Perhaps the most straight forward aspect of the design is that of the woofer system. Aside from the choice of the low
frequency driver(s) the design of a flat baffle or H frame dipole woofer is as close to a text book exercise as will be
found in loudspeaker design. Similarly, the design of the midrange, though critical to the speakers performance, can
also be reduced to a relatively straight forward procedure, taking into account the relationship between baffle size,
dipole response and driver directionality. The crossover between woofer and midrange is also straight forward given
the constant directivity of the woofer and midrange through this crossover region. However, things become
somewhat complicated when the tweeter is considered. With a conventional speaker system a fairly direct choice of
crossover frequency between the midrange and tweeter can be made based on consideration of the directionality
characteristics of the midrange and tweeter. In fact, with a relatively wide baffle, insuring that the tweeter operate in
2Pi space, although somewhat contradictory to current design trends, choosing a crossover point below the
frequency at which the midrange driver exhibits significant directionality will result in good retention of uniform
directivity and power response through the crossover region, other issues aside. This is because through the
crossover region both the midrange and tweeter will be radiating into 2Pi space, the front hemisphere, with similar
directivity. With flat response through the crossover and the correct choice of crossover type, this will result in
uniform power response through the crossover as well.

However, when a dipole midrange is considered things are not so simple. As we shall see, it is not generally possible
to obtain constant directivity and uniform power response when a dipole midrange is combined with a conventional,
front firing tweeter. Things improve with a rear tweeter, but still aren't perfect. To begin to understand the integration
of a tweeter (or tweeters, front and rear) with a dipole midrange we must first understand the radiation characteristics
of the midrange driver. Figure 1 shows the radiation pattern for an ideal dipole composed of two omnidirectional
sources separated by a distance d at various frequencies as a function of d divided by the wave length, w. What we
see is that when the wave
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.
Above the dipole peak, the radiated power is essentially equivalent to that of two uncorrelated sources. Thus, above
on axis peak in the dipole frequency response, the power radiated from an ideal dipole is that of two uncorrelated
sources. The second important observation is shown by the green line. Here we see that at low frequency the
power is -4.8 dB relative to a single, omni-directional source (monopole), as expected. However, above the dipole =
monopole frequency the power radiated by an
equalized dipole begins to increase, slowly at first, but then rapidly
rising to meet the uncorrelated source result. Clearly,Figure 1 and 2 show that if we were truly interested in
constant directivity and power we would need to limit the use of the dipole to an upper frequency range equal to
the dipole = monopole frequency. However, there are other factors at play; namely, driver directionality.

Let us consider what happens when we place a typical midrange driver with effective surface area of 220 sq cm
on a 12" diameter circular baffle. Figure 3 shows the result. Due to the driver directionality the power in the
uncorrelated frequency region roll off at 6dB/octave as the directionality of the driver increases. Thus when the
dipole is formed from this driver and equalized for flat on axis response
over its useful range,
there is very little rise
in power and the polar
response remains
fairly uniform as
shown at the right.
Thus, driver
directionality saves the
day, provided it is
correctly considered,
and provides us with a
reasonable constant
directivity and
constant power
midrange. To provide
the most uniform
response the effective
baffle diameter should
be such that the dipole
peak frequency occurs
at the frequency
where the uncorrelated
power is at a level of
-4.8dB. This generally
would result in a baffle
which was too narrow
to support the
midrange driver. A
reasonable baffle
diameter is twice the
driver effective
diameter.

The same arguments
applied to the
midrange can be
applied to the tweeter.
If we intended to use a
tweeter with effective
diameter of 1", then if
front and rear of the
tweeter were mounted
with a separation of 1"
(a hypothetical 2"
diameter baffle) the
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.
response would appear as shown in Figure 4. where the tweeter power and polar response at 15K hz are overlaid
with the midrange result. This would clearly be an ideal result: reasonable flat power response to 15 KHz and uniform
directionality as well. I am currently researching tweeters which  might be used in such an application.

So much for the future, what about typical current implementations using dome tweeters with dipole midrange
drivers on relatively wide baffles. Four different possible scenarios are considered with the results shown in Figure 5
where they are overlaid with the midrange power response. In all cases a 1" tweeter is assumed. The fist scenario
assumes that only a front tweeter is used and that the baffle is sufficiently large that the tweeter radiates only into 2
Pi space. This result is shown in turquoise. The second scenario is that of a single front firing tweeter mounted in
free space, shown in lime green. The third scenario is that of a front and rear tweeter operating as if isolated from
each other with each radiating into the front or rear 1/2 space, respectively, shown in brown. The fourth scenario
assumes the front and rear sources are mounted on a 12" diameter circular baffle.  In all cases  except the last the
power response is referenced to flat on axis amplitude response. For the dipole result, the response is based the
response being normalized to flat over what was previously referred to as the use range of the dipole. This on axis
response is shown in Figure 6.
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.
Figures 8 and 9 show the results of the integration of the rear tweeter on the NaO II. Figure 8 shows the response of
the front tweeter, measured at 1 meter on axis in red. The green trace shows  the response of the rear measured at 1
meter on axis from the front. When the rear tweeter is switched on, the on axis response at the same measurement
point is as shown in blue. This represents the sum of the red and green responses.  All measurements are made
without the crossover in place. The hint of a dipole peak is apparent in the frequency range centered around 700 Hz
and a slight dip in the response appears at 1.4 K Hz. The small peak and dip near 2.0k  and 2.8k Hz can also be
attributed to the dipole behavior. These variation are, however, small in amplitude generally below the crossover point
used in the NaO II.
Figure 8. Response of front, rear, and summed front and rear tweeters measured on axis from the front side
at 1 meter.
Figure 9 shows the on axis response of the NaO II speaker measured at 1 meter with the rear tweeter switched on in
red. The response at 1M, 90 degrees off axis,  with and without the rear tweeter switch on is shown in blue and gray,
respectively. The benefit of the rear tweeter in cancelling the 90 degree off axis sound radiated from the tweeter is
clearly apparent, yielding more uniform directivity and smoother prower response.
Figure 9. On axis, red, and 90 degree off axis with and without the rear tweeter switched on, for the NaO II
speaker system.
length is large compared to d
(small d/w) the radiation pattern is
fairly constant and appears like a
figure 8. However, as the
frequency rises such that d/w is
greater that 0.25, the pattern starts
to balloon out. This effect
continues until as the frequencies
rises until just before the first
dipole null at d/w = 1. As the
frequency continues to rise, the
polar response breaks down into a
multi-lobed pattern. The useful
limit of an ideal dipole for a
loudspeaker would lie somewhere
between the  dipole peak (d/w =
0.5) and the first dipole on axis
null (d/w=1).

We must next ask ourselves, what
is the power radiated by this ideal
dipole? Let us assume for the
moment that the separation of the
sources, d, is 6". The power
response is shown in Figure 2.
Here the red line indicates the
power response of the unequalized
dipole. The blue line represents the
power of two uncorrelated
sources. The green line represents
the power of the dipole if
equalized to have flat on axis
response over its potentially useful
range form the point of view of
obtaining flat on axis response.

Several important observations
should be made. First, the
unequalized dipole power rises at
6dB/octave finally crossing the
uncorrelated power result at the
dipole peak frequency.