Tech Design.....
___________________________

Dipoles and Open Baffle Design Considerations
This tutorial is directed at the discussion of dipole (DP) and open baffle (OB) speakers system employing flat
baffles, various design considerations and their differences and similarities. More specifically it is directed at
OB/DP midrange response and influencing factors. The discussion begins with point source drivers and moves
onwards to directional drivers with symmetric front and rear response. Consideration of drivers where the front
and rear responses are asymmetric, which may be likened to rear world drivers, is commented on briefly. Only
generalized results are presented yielding an indication of how various parameters influence the response.
Many of these aspects are discussed in other sections of this website, however here the attempt is to focus only
on DP and OB system. No consideration is given to room interaction in the discussion. The information will be
updated and expanded over time.

To begin, consider a two simple systems: a single point source (monopole) radiating into free space, and a
dipole composed ot two point sources separated by a distance, d, as shown in Figure 1. The monopole radiates
uniformly in all directions
at all frequencies. The
dipole has an on axis
response that appears as
shown below in Figure 2.
A dipole peak is present
at the  frequency given by

Fp = C/(2d)

The level of the dipole in
the 1/f region is equal to
that of monopole at a
frequency given by

Feq = C/(6d).

The peak frequency
occurs where the
propagation delay is equal to 1/2 a period (the
separation is 1/2 a wave length) corresponding to a 180
degree phase delay.
Feq occurs at the frequency
where the propagation delay is equal to 1/6 th of a
period, (60 degrees). Under such conditions the vector
sum of the front and rear radiation forms an equilateral
triangle with the resultant response vector. The polar
response of the dipole, far from the center of the dipole,
is as shown in Figure 3 as a function of the separation
divided by wave length, w. For long wavelengths,
(low frequency) d/W is
small and the radiation
pattern is the classical
dipole figure 8. This
pattern remains fairly
constant up to about
d/w = 0.25, or an octave
below the dipole peak.
Above d/w = 0.25 the
radiation pattern initially
broadens  up to about d/w
= 0.66 with an off axis
bulge before degenerating
into a multi-lobed
response. The number of
lobes above d/w = 1.0 will
increase as the value of  
d/w increases. Figure 3
shows the result for d/w =
1. which corresponds the
frequency of the first null
above the dipole peaks in
Figure 2.

One of the common
misconceptions about
dipoles is that they have
constant directivity thus
can be equalized to yield
constant power. As shown in Figure 3,
this is strictly true only for frequencies
for which d/w is less than 0.25 and even
then there is some minor variation with
frequency. However, we can take the
power as constant below this point and
equal to 4.8 dB less than an omni
directional source with the same on axis
response.  At the dipole peak an
unequalized dipole has an on axis
response of +6 dB. At the same
frequency the radiated power is 3 dB
greater than an omni directional source.
When equalized to have the same level
as the omni source at this frequency
the dipole radiates only 3dB less power.
This is 1.8 dB greater than the radiated
power below d/w = 0.25.

Obviously, we can not equalize the
response flat above the dipole peak
due to the nulls and peaks and the
multi-lobed radiation pattern. However,
Figure 4 shows that for the unequalized
dipole the power response degenerates
to that ot two uncorrelated, omni
directional sources at a level 3dB greater than a single source. The power is independent of the format, dipole or
otherwise, in the uncorrelated region. Thus, even before we consider the roll of a baffle in creating a dipole with
a conventional driver it must be recognized that if constant radiated power is an objective there is a need to
proceed with caution. This is of particular concern if it is desired to use the response above the dipole peak.

Circular Baffle Dipoles with Omni Sources.

The first step is to understand the roll of a circular baffle in the formation of a "dipole like" response. Rather than
separating two omni sources by a distance, d, to form a dipole a circular baffle of radius d can be substituted.
On
axis
the propagation of the wave from the rear source to the baffle edge results in a delay equal to that which
would occur if the sources were separated. Thus the on axis response will appear identical to that shown in
Figure 2. However, off axis there will
be significant differences. For the
true dipole, as we move off axis the
effective separation between the
sources varies as d' = d cos(theta)
where theta is the off axis angle.
Thus, when theta = 90 degrees the
effective separation goes to zero and
the sources, being of inverted
polarity, will cancel exactly yielding
the 90 degree null. At 45 degrees off
axis the effective separation is
reduced to 0.707d and the result is a
shifting to the dipole peak 1/2 octave
higher, at 60 degrees off axis the
effective separation is reduced to
0.5d moving the dipole peak a full
octave higher. Figure 6 shows the
ramification of this behavior.
The lowering of the response level in the 1/f region is apparent, but as we move off axis and the dipole peak
moves to a higher frequency the effect is obvious. Conversely, as shown in Figure 7, moving off axis for the
circular baffle dipole has the same effect at low frequency while resulting in smoother response at higher
frequency. This is a consequence of the fact that the circular baffle dipole operates very much like a point source
surrounded by a ring radiator where the integrated strength of the ring radiator around the baffle edge is equal
to that of the point source in the center, but of opposite polarity. This analogy is reasonable on axis and for
moderate off axis angles but will ultimately fail as the off axis angle approaches 90 degrees. The reason for the
observed behavior should be obvious: the path length form all points on the ring radiator to any point
on the
dipole axis
is the same. However, when we move off axis the path length from different positions on ring radiator
to the observation point will vary. Since the on axis case is the worst case, resulting in 6dB peaks and nulls above
the dipole peak, moving off axis can only improve the situation, with a reduction in the peaks and a filling of the
nulls. If we change our thinking to the time domain, then the impulse response of the unequalized, circular baffle
dipole looks like the blue trace in Figure 8 on axis. At 45 degrees off axis it appears as shown by the red trace
where the inverted impulse form the ring radiator is scattered or smeared over time. While on the subject
of the impulse response we should also look briefly at the impulse
response of the dipole on axis, with and w/o equalization for flat
response, as shown in Figure 9. Equalization of the dipole for flat
response in the 1/f region results in turning the double impulse of
the unequalized dipole (blue) into a finite width pulse with the pulse
width equal to the delay between sources. Thus the equalized
dipole time smears the impulse response. This is a characteristic
of the equalized dipole and is not related to room effects. Figures 8
and 9 represent what would be measured on and off axis for unequalized and equalized circular baffle dipoles
using omni sources. Obviously equalization above the 1/f region is not possible due to the nature of the
response.

Non Circular Baffles with Omni Sources:

The behavior of the circular baffle response off axis suggests that the on axis response of a dipole composed of
omni sources could be smoothed somewhat if the rear wave could be scattered relative to the on axis position.
The use of an irregularly shaped baffle can accomplish this task since relative to a position on axis the delay will
vary around the baffle circumference. Additionally, as discussed
elsewhere on this web site, below the dipole
peak it can be shown that irregularly shaped baffles are equivalent to circular baffles at low frequency with
respect to frequency response and level. Thus, an irregularly shaped baffle can help smooth the on axis
response above the dipole peak without loss of low frequency sensitivity, compared to the equivalent circular
baffle. We need not go to extremes here. A rectangular baffle can provide a significant improvement, and
positioning the driver so that it is centered horizontally will assure left-right symmetry in the response. Figure 10
shows such a case for omni front and rear sources. The sources are centered on a rectangular baffle with
dimensions chosen such that at low frequency the equivalent diameter is that of the circular baffle considered
above while the height and width were selected to yield a smoother response between 500 and 2 K Hz, both on
axis (blue), and out to 25 degrees off axis (green). The red line is the circular baffle result. In this example the
baffle was 20 cm wide and 52 cm high. The plot represents the result far from the baffle.
(The lack of the deep
nulls in the circular baffle result of this figure is due to the frequency resolution of the simulation.)
Consideration of Directional Drivers:

Thus far in the discussion we have considered only omni directional sources on circular and rectangular
baffles. We have seen that using an irregularly shaped baffle can yield smoother response above the dipole
peak, both on and off axis, compared to an circular baffle with the same equivalent radius. The reason for this
is straight forward, as discussed, and is due to the varying delays from the different positions around the baffle
edge. However, there is another factor which must be considered when using real drivers on an opened baffle;  
driver directionality. As the radiated frequency increases a driver becomes directional. As a result, since the
radiation from the rear of the baffle is dependent on the 90 degree off axis response of the driver, the
response of the "ring radiator" will generally weaken as the frequency rises. Figure 11 shown the theoretical 90
degree off axis response for a 6" flat piston in blue and a crude approximation to the response in red. The on
axis response of a dipole using such a driver on a
30 cm circular baffle is shown in Figure 12. As can
be observed, the response is much smoother above
the dipole peak than the omni directional driver
shown in red in Figure 10 and is even smoother
than the omni directional driver mounted on a
rectangular baffle as shown in blue in Figure 10.
This is because the rear response reaching the on
axis position is effectively low pass filtered and there
is a loss of the dipole response above the on axis
peak for this driver/baffle size combination.

The impulse response for this system is shown in
Figure 13. At the left of Figure 13 the impulse for the
unequalized response is presented. The negative
pulse from the rear reflects the low pass filtering
effect. This should be compared with the blue trace
in Figure 9. The impulse when the response of
Figure 12 is equalized flat is presented at the right
of Figure 13. This should be compared to the red
trace of Figure 9. This is clearly an
improvement over the characteristic of an omni source. The
reason is that the response ceases to be of a dipole nature
above the frequency where the driver becomes directional
provided that the baffle is sized such that the dipole peak is at a
frequency where directionality is becoming significant.

As a further example, consider the same driver model on a 60
cm diameter circular baffle. This places the dipole peak one
octave lower with an on axis frequency response that appears
as shown in red in Figure 14. While we observe an improvement
than the corresponding dip (at 2k Hz) for the 30 cm baffle. This is because the driver is closer to an omni
directional source at 1k Hz, as is apparent from Figure 11.The impulse response for the unequalized 60 cm
baffle is shown in Figure 15. It is identical to that at the left of Figure 13 except the rear pulse is delayed twice
as long. The greater irregularity in the response would likely prevent use of this baffle /driver configuration
even if suitable equalization to flatten the response were applied. It just isn't a good starting point. However
several important observations can be made. First, the directional characteristics of the driver determine the
frequency limit above which true dipole operation can be obtained from a specific driver. Above that
frequency the characteristics of the front and rear radiation are dominated by the directional characteristics
of the driver and diffraction from the baffle edge is much more like that for a conventional speaker. Second,
with a circular baffle, the baffle diameter will set the frequency of the dipole peak, and more importantly, the
frequency of that peak in relation to the frequency above which the driver becomes directional will determine
the characteristic of the response at higher frequency. Third, while a larger baffle will increase the low
frequency sensitivity the greater delay will translate a smearing of the impulse response over a longer period
of time.

Since a larger circular baffle results in lowering the dipole peak to a frequency where the response just
above the peak behaves more like that for an omni source the same approach to smoothing the response is
appropriate; smear the rear response over time by using a rectangular baffle. Figure 16 shows the response
of the 60 cm circular baffle in blue and that for a rectangular baffle of the same equivalent radius in red. The
driver was centered on the rectangular baffle which was 40 cm wide and 76 cm high. As can be seen,
in  the low
frequency
sensitivity, which
is the main reason
for selecting a
larger baffle, we
also note that the
dip in the
response at 1k Hz
is much deeper
the rectangular baffle has the same low frequency
characteristic but results in a relatively smooth
response above the dipole peak, thus requiring
minimal equalization consisting of 6dB/octave boost
below the peak and a Q bump filter to flatten the peak.

The impulse response for this baffle/driver combination
for the unequalized (red)and equalized to flat (blue)
cases is shown in Figure 17. The unequalized
impulse (red) has a characteristic similar to that of the circular baffle shown
in Figure 15 however the negative pulse is lower in amplitude and extends
over a longer time period because of the variable delay around the
perimeter of the baffle. Considering the 6dB increase in low frequency
sensitivity compared to the 30 cm circular baffle of Figure 15 which also had
acceptably smooth response above the dipole peak, it might be expected
that such a configuration would be superior. However, there is another
factor to consider; power response.
Consideration of Power Response:

Consideration of power response is where we can separate two different configurations form each other and
consider one dipole and the other simply an open baffle design. The desire to go to an open baffle is usually
motivated by a desire to eliminate box colorations. However, when power response is considered it is possible
to differentiate further by considering the variation of radiated power with frequency. As discussed at the
beginning of this tutorial the widely accepted view is that a dipole radiates 4.8dB less acoustic power than an
omni directional source with the same on axis response. However, this only applies from approximately one
octave below the dipole peak and lower, providing the driver is effectively omni directional over that  frequency
range. As the frequency goes higher the power response may begin to rise.

Consider the power radiated by the configuration of Figure 12, a directional driver on a 30 cm circular baffle,
and that of Figure 16, the same driver on a rectangular baffle with 60 cm equivalent radius. If constant power
vs frequency is desired which configuration is superior? The answer is obviously dependent on the crossover
frequency and driver directional characteristics, but the baffle configuration must also be considered. The
radiated power for the larger rectangular baffle would appear something like that shown in
blue in Figure 18.
The power is constant at
low frequency but then
rises to a peak at about
1k Hz before falling off
due to the driver's
directional characteristics.
The power response for
the smaller circular baffle
is shown in red. It shows a
relatively constant power
level until beginning to fall
off at higher frequency.
The corresponding
horizontal polar
responses at 1k Hz are
shown at the right. Not
only is the polar response
broadening for the larger
baffle, the off axis
response is actually at a
greater level than the on
axis response. All of this
results because with the
larger baffle the
frequency of the dipole
peak is pushed down to a
frequency where the
driver is still radiating in
an omni directional manner and the radiation pattern balloons as it would for a true dipole (see Figures 2 and
3 above) before driver directionality counters this effect. This may result in an over aggressive sounding
upper midrange when large baffles are implemented to improve low frequency sensitivity and at the same time
taking the observed smoothness of the on axis response as an indication that the crossover can be pushed to
a higher frequency.  With the smaller baffle the tendency for the polar response to balloon at, and initially
above the dipole peak, is countered by the directional characteristics of the driver resulting in a more uniform
power response. Thus, while the larger baffle results in a 6dB increase in low frequency sensitivity, extending
the useful low frequency limit for a given maximum SPL by an octave, this is offset by the requirement that the
useful high frequency limit is reduced by an octave
or more if constant radiated power is desired. If we take
the power response plots above as reasonable, then for the two baffles considered the high frequency
limitation would be of the order of 1.5K Hz for the smaller baffle and around 400 Hz for the larger baffle. Over
all this would mean that form the power response point of view the same driver mounted on a the smaller
baffle would have useful bandwidth about one octave wider than when mounted on the larger baffle in spite of
the apparent smoothness of the on axis response.

Closing Comments:

The discussion presented above is based on theoretical considerations only. As such it provides insight into
the behavior of open baffle and dipole speaker designs based on conventional drivers which should be taken
into consideration for any such design. Other factors also enter the picture which have not been discussed.
hopefully these will be expanded upon in the future.  For example, all the cases above assume that the
radiation from the front and rear of the driver is identical in amplitude and inverted in phase. When real drivers
are considered any asymmetry between the front and rear response must be considered as well. Since such
effect are largely dependent on the specific driver and are generally unpredictable, there can be no substitute
for testing and measurement of any given design both on and off axis.