Models for a dipole loudspeaker design
If you consider to build a dipole loudspeaker that uses conventional dynamic drivers, then you need some understanding of the inherent acoustic frequency response of such a speaker and how to design the necessary equalization for it. The following is a collection of models and circuits that I have used in the design of the PHOENIX loudspeaker. The numerical examples and sketches can help you understand the technical details and guide you in the development of your own design.
A - Two point sources
of opposite polarity model
D - Midrange dipole equalization
E - Midrange to tweeter crossover
G - Level adjustment controls
H - Psycho-acoustic 3 kHz dip
I - Voltage sensitivity of an active system
A - Two point sources of opposite polarity
Here is the basic model for dipole radiation and the only one that has a readily calculated closed form solution. Everything beyond this leads to complicated mathematics or approximations with limited range of applicability.
Take two point sources, like two small closed box
speakers. "Small" means that all the speaker dimensions are small
compared to the wavelength radiated. These are monopole, omni-directional
sources. Space them at distance D apart and drive them with opposite polarity
(=180 degrees out-of-phase).
The acoustic path length difference to the two sources is d = D*cos(a) when
listening from far away. Thus, when each source emits a single impulse at the
same time, one would measure two impulses separated by a time interval T = d/v
at far distance. (v is the speed of sound propagation, 343 m/s). The received
impulse response is a "doublet" consisting of a positive and a
negative pulse. The doublet shortens with cos(a) and becomes zero for a=90
Note: The graphs differ
significantly from those, which are calculated for "Radiation from a rigid circular
piston in a finite circular open baffle". Response widening with
increasing D/l is marginal and on-axis nulls are not
observed. But the increase in 6 dB/oct on-axis slope and a dominant peak agree with
observations and required equalization.
The frequency response has a characteristic 6 dB/octave roll-off towards lower frequencies and sharp nulls at higher frequencies. The off-axis response, which is shown for 30, 45 and 60 degrees, is lower in amplitude at low frequencies by 1, 3, 6 dB respectively and follows a figure-eight or cos(a) pattern with nulls at +/-90 degrees. The cos(a) pattern is maintained for D/l < 0.1, down to the lowest frequency. The pattern widens with increasing frequency. There are also deep notches on-axis every time that D is a multiple of a wavelength and the polar pattern looks like made up of flower petals.
When the two opposite polarity point sources are separated by 1/2 wavelength ( f = 0.5*v/D ), then the additional 180 degrees of phase shift j add to a total of 360 degrees and the resulting sound pressure is twice that (+6 dB) of a single source.
A1 - Dipole equalization
The sloping response of the dipole is not very useful and must be equalized. This is accomplished in the frequency domain by boosting the response at a uniform 6 dB/oct rate as frequency is lowered. Thus, the frequency response becomes flat at low frequencies and rolls off at 6 dB/oct in the region where the interference nulls occur. Such equalization corresponds to an integration and turns the doublet impulse response into a single impulse of duration T.
In practice, the dipole
response is made flat by boosting the amplifier output at 6 dB/oct rate, but
this places serious demands on the cone excursion capability of the driver.
f = 1/(6*T) = 0.17*v/D
f = 229 Hz in figure 2 above. The comparable monopole has a flat frequency response (0 dB), whereas the dipole rolls off at 6 dB/oct. To maintain constant sound pressure level (0 dB) the monopole displaces four times (+12 dB/oct) the air volume for every halving of frequency, but the dipole has to have eight times (+18 dB/oct) the displacement. Very quickly the maximum excursion capability of a driver becomes the limiting factor for maximally achievable sound level. Multiple drivers must be used to go beyond this limit. A spreadsheet spl_max1.xls allows you to calculate the excursion limited SPL for known driver piston area and peak excursion. You find the logic behind the numbers in theory.gif . Top
A2 - Point dipole and point monopole in a circular baffle
Two opposite polarity point sources with separation D, as in dipole model A above, only approximate the behavior of a finite size dipole radiator in a flat baffle. A dipole point source in a flat, circular baffle gives better agreement with the measured performance of a finite size driver in a flat or folded baffle as seen, for example, in the graph below.
 Tim Mellow & Leo Karkkainen, "A dipole loudspeaker with balanced directivity pattern", J. Acoust. Soc. Am. 128 (5), November 2010, pp. 2749 - 2757
A3 - Radiation from a rigid circular piston in a finite circular open baffle
The normalized on-axis response of a plane circular piston of radius a in a flat open circular baffle of radius b was calculated by Tim Mellow for b = a, 2a, 4a, 8a, inf., and also polar responses were plotted in: Leo L. Beranek & Tim J. Mellow, Acoustics - Sound Fields and Transducers, Elsevier (2012), Chapter 13 - Radiation and scattering of sound by the boundary integral solution. It is instructive to see how the calculated responses differ from those predicted by the simple point source and edge diffraction models used here and elsewhere. In particular, the change in slope to >6 dB/oct just below the dipole peak and reduced ripple in the response above the peak confirm experimental observations.
The on-axis response for the point source model deviates significantly from the response of the circular piston (A3-2). The slope is greater than -6 dB below the dipole peak for the piston (A3-1). This is in agreement with the typically required dipole equalization. Above the dipole peak the ripple is minimal for the unbaffled piston. This was also confirmed in the development of the LX521 midrange baffle shape.
The radiation pattern of the point source model agrees with the unbaffled circular piston only for kD << 1, at long wavelengths and low frequencies where it follows a cos(alpha) pattern (A3-3). In general the pattern of the point sources is wider at higher frequencies as seen for kD = 3, 5, 10, while the piston starts to narrow its pattern. The two point source model is not a realistic guide to understanding practical dipole behavior for kD > 1. When a baffle is added to the piston then the pattern narrows even further and more side lobes and interference notches are created (A3-4). Adjacent lobes have opposite polarity.
B - Open-baffle loudspeakers
There is not much sense in building a dipole speaker with two small closed box speakers driven in opposite phase, when one of the objectives is to remove the sound character that boxes impart. The two boxes a can be joined at their backs, though, and the connecting wall removed b.
Since the two cones move back and forth in unison, there
is little air pressure inside the enclosure b at very low frequencies.
When the internal length L becomes half wavelength,
there is a sharp resonance of the transmission line between the cones, causing a
severe dip and peak irregularity in the frequency response. The two drivers in b
can be replaced by a single driver c without loss in volume displacement
capability. The latter arrangement, called H baffle, is very practical for
dipole woofer construction. It too has a severe resonance because the waveguide
of effective length L in front and behind the cone sees large impedance
mismatches at the cone and at the open end of the cabinet. The resonance occurs
when L = l/4 =
The closed dipole baffle b with two drivers can be evolved d into the flat, circular, open baffle e with a single driver, while maintaining the same excursion limited output capability.
The circular baffle with a -|+ point source at its center has nearly the same polar radiation pattern as the two opposite polarity point sources spaced D apart in the model of figure A1 above. The circular baffle's usefulness at high frequencies is limited by the sharp interference nulls when D is a multiple of a wavelength. This behavior can be considerably smoothed by making the baffle f rectangular which gives a variation to the length of D.
For structural and aesthetic reasons you may want to fold
back the baffle g. The baffle depth d must be kept shallow to avoid
forming a cavity which stores acoustic energy and resonates. The exact shape of
the folded baffle is best determined experimentally. Since there are no
significant forces on the baffle you can quickly construct it from heavy
corrugated card board. Only the drivers need to be held solidly in place.
Measure on-axis and off-axis frequency response as you vary width height and
B1 - "Compound dipole" woofer model
Completely open driver arrangements have been used by Celestion and Legacy
Audio. A simple model to describe this case would be given by two drivers mounted on their own small
baffles of effective radius d1 and separated by
2d2 from each other.
The model predicts that the SPL at very low frequencies is merely the sum of two dipoles with spacing D = d1. The separation 2d2 between them has no influence on the total output as long as it is small compared to the wavelength of radiation. I see no compounding effect other than summing two dipoles, but the two baffles might as well be placed next to each other. A single driver in an H-frame would have the same output if the distance D between the openings is 2d1. Even order non-linear distortion can be reduced by reversing one of the drivers in the compound configuration so that the two magnets face each other. The whole arrangement does not strike me as a very effective use of a second driver and cabinet space compared to an H or W frame. I have no data how high in frequency the "compound woofer" can be used, but its radiation pattern will become more lobing than that of the two point source woofer. Top
C - Dipole woofer equalization
If you have build an H baffle woofer, then the first step is to measure the frequency response of the drivers in the cabinet, In general, you can expect that the air loading on the cones will reduce the mechanical resonance frequency Fs and increase Qt. There will also be a response peak due to a l/4 resonance of the waveguides in front and behind the drivers. The measurement is performed right at the opening of the cabinet, so that the microphone sees only one of the two sources that form the dipole. Therefore you will not see the characteristic 6 dB/oct dipole slope in the data.
The PHOENIX woofer has D=19" (0.48 m) separation between its openings. The peak should be at f = 0.5*v/D = 357 Hz, but the cabinet layout is too complicated for such simple calculation to apply exactly. The woofer will be crossed over at 100 Hz with a 12 dB/oct L-R low-pass filter. The resonance peak will not be attenuated sufficiently by the low-pass and must be removed first with a notch filter. This usually takes some trial and error to find the best trade-off. Top
C1 - Notch filter design
From the graph, the 270 Hz peak rises about 11 dB. This requires the notch to dip down to -11 dB or to a = 10^(-11/20) = 0.28. The Q of the peak is about (270 Hz)/(100 Hz) = 2.7 and determines the width of the notch.
Select R1 = 5.11k, then R = 5110*0.28/(1-0.28) = 1987 ohm.
C2 - 6 dB/oct dipole slope correction
This part of the equalization is easy and only involves the decision at
which frequency to start and to end. A circuit that is suitable for the task is
the shelving low-pass. The woofer
dipole with D=19" has its first peak ideally at 357 Hz and transitions into
the 6 dB/oct slope somewhat below this frequency. Thus I chose f2 = 300 Hz. From
the measured woofer response above, you can see that it is quite flat and 3 dB
down at 13 Hz. Extending the equalization down to f1 = 10 Hz makes for a gradual
transition into the ultimately 18 dB/oct roll-off region of the dipole below the
C3 - Equalization of low Qt woofer
You may be using a woofer which rolls off over a wider frequency range than
the one in the PHOENIX because its Qt in the cabinet is less than 0.7, or the
resonance frequency Fs is too high. In such cases you can equalize what you have
into a more desirable response by using a special biquad circuit f0Q0fpQp.gif.
To apply the circuit you must determine f0 and Q0 for the drivers in the cabinet
from a measurement of their terminal impedance according to f0Q0.gif.
D - Midrange dipole equalization
When you place a driver (e.g. SS 21W/8554) on a circular
open baffle, then you measure a response that differs somewhat from the model under
Very important to note is the first response peak. It is a function of the driver used and almost all drivers exhibit it to varying degrees. The peak is caused by an acoustic filter formed by the basket openings and trapped air between cone and basket. This filter is the reason for the differences in high frequency response between front and rear.
A driver becomes directional of its own, when its effective piston diameter becomes larger than 1/3 of a wavelength. For an 8" driver this would be above 558 Hz. Still, the expected +6 dB dipole peak, when the rear wave adds fully to the front radiation at 838 Hz for the D = 8" circular baffle, can be seen in the measured data above. The expected null at 1675 Hz, though, is only partially formed, because not enough energy comes around the baffle edge.
The combined response of the two 8" drivers mounted on the PHOENIX baffle also exhibits a peak due to the basket resonators. The peak must first be removed with a notch-filter, so that a shelving low-pass filter equalization can give the proper transition from the 6 dB/oct region into the flat region of the dipole response.
The off-axis response at 30, 45 and 60 degrees exhibits
some interesting characteristics. At low frequencies it follows the cos(a)
pattern as in the dipole model under A above. Around 700 Hz and 1500 Hz the
horizontal pattern actually widens, and only above 2 kHz becomes the pattern
progressively narrower, as indicated by the separation of the response curves
for different angles. The pattern would become narrow at much lower frequency,
if you closed the back of the baffle.
With the above as background in mind, the midrange dipole
equalization follows the steps outlined for the woofer under C. The only
difference is the choice of frequency f1 for the shelving
low-pass. Since the chosen crossover to the PHOENIX woofer is of 2nd-order,
one of the two 1st-order high-pass sections can be realized by placing f1 at the
crossover frequency of 100 Hz. The second high-pass filter is realized with the
90HP stage of the crossover/eq.
E - Midrange to tweeter crossover
We start the crossover design with on-axis and horizontal off-axis frequency response measurements of the tweeter and the two midrange drivers on the PHOENIX main panel. It is important to know the off-axis behavior in order to choose the crossover frequency such that the combined midrange and tweeter response is as smooth as possible off-axis. The on axis ripples in the tweeter response are a mixture of diffraction off the midrange driver cones in the M-T-M arrangement, primarily at lower frequencies, and diffraction off the panel edges at higher frequencies. No attempt has been made to further reduce these effects by equalization, because they are dominant only on-axis and smoothed out off-axis.
The drivers have a 13 dB difference in sensitivity. The
midrange channel gain will be set 13 dB lower than the tweeter channel to
correct for this (x1.gif).
The crossover frequency at 1400 Hz is quite low and
therefore it is important to assure that the tweeter has adequately low
distortion at the volume displacements required.
The crossover design is not complete yet, because the
tweeter's high-pass behavior and the midrange's roll-off cause phase shifts that
are part of the crossover. In addition, the physical offset between driver voice
coils causes delays and associated phase shift that must be corrected. This is
best done experimentally and started with the driver offset.
The delay of the circuit changes with frequency, but the
1400 Hz crossover should fall into the flat region of the delay and therefore f0
> 1400 Hz. You can estimate the number of stages required from this
F - Midrange to woofer crossover
From C above you can see that the dipole woofer's useable frequency range after equalization extends to 250 Hz at the most. If the response should follow the crossover low-pass filter closely over the first octave of roll-off, then the crossover frequency cannot be higher than 100 Hz.
Initially, I used a 24 dB/oct L-R crossover for a similar speaker, the Audio Artistry Dvorak, but we found after extensive listening that a 12 dB/oct L-R gave a slight improvement in bass realism. I think this is due to the more gradual transition of the group delay response of the 1st order all-pass formed by the lower order crossover vs. the 2nd order all-pass for the 24 dB/oct crossover. I had found earlier (Ref. 17) that phase distortion is audible at low frequencies even with a 1st order allpass. At high frequencies it must be much more severe than the phase distortion of a 24 dB/oct L-R crossover before it becomes audible.
I have recently investigated my previous assumption about the audibility of reduced phase distortion from a 12 dB/oct crossover between woofer and midrange, with the conclusion that the difference in phase shift between a 12 dB/oct and a 24 dB/oct crossover is not audible, not even when compared to a zero distortion 6 dB/oct crossover. Thus it would seem reasonable to use the higher order crossover which reduces the potential for non-linear distortion in the midrange driver and attenuates any higher frequency contribution from the woofer. I invite anyone interested to test for themselves the validity of my observation.
A 12 dB/oct crossover places much greater demands on the excursion capability of the midrange drivers. When you apply a constant voltage to the driver terminals, then the cone excursion X1 increases at a 12 dB/oct rate going down in frequency, figure a. Below the driver resonance Fs the excursion X1 becomes constant. Dipole equalization Veq boosts the excursion at +6 dB/oct up to the crossover frequency Fxo to give a flat SPL response. The terminal voltage decreases at 6 dB/oct rate below Fxo and, in conjunction with the dipole roll-off of 6 dB/oct, gives the 12 dB/oct acoustic high-pass filter response for the L-R crossover.
The net effect of equalization for flat response and of the 12 dB/oct crossover is an increase of excursion X2 at 18 dB/oct above Fxo, 6 dB/oct below it, and a decrease at 6 dB/oct below Fs as in figure b.
The SS 21W/8554 8" driver on the PHOENIX main panel
has 200 cm^2 piston area, 6.5 mm linear excursion and 10 mm Xmax. With an
effective back to front path length D of 9.75" (248 mm) the dipole SPL at
100 Hz, 1 m, becomes 97 dB. The second driver
increases the value by 6 dB and the woofer contributes another 6 dB for a total
of 109 dB SPL. Driven to Xmax adds 4 dB more.
You can begin to see why I added two 10" drivers to the AA Beethoven and four 10" drivers to the Beethoven-Grand to cover the range below 200 Hz. The transition between 10" and 8" drivers is at 6 dB/oct for group delay reasons, but requires phase compensation to steer the vertical polar pattern towards the listener. Such 4-way system is a little tricky to design and measure.
The 12 dB/oct 100 Hz crossover for two 8" drivers is thus a trade-off between maximum sound level and sound quality. Circuit implementation is straight forward with a 2nd order low-pass (99LP) in the woofer channel, the midrange dipole boost ending at 100 Hz (90-500LP) and a 100 Hz high-pass filter (90HP) in the midrange channel. The actual filter corner frequencies differ from the nominal values due to standard component value selection and trimming of the measured crossover response (x1.gif). Top
F1 - Woofer level setting
The crossover is not complete without setting the proper woofer level relative to the midrange. It is difficult to set by listening to program material because recordings vary greatly and room resonances can change the perceived level. It is best to start with outdoor measurements and to refine the result by indoor listening if a technically justifiable reason exists for it. (I am assuming that you want to build a transducer and not a musical instrument)
For the outdoor measurement you might might raise woofer and main panel some large distance Y above ground to minimize and delay the effect of reflections, figure a. Assume you adjust the system to be flat through the crossover region and beyond. Now, when you set the woofer on the ground and the panel at normal listening height, figure b, the woofer output will receive a uniform 6 dB boost because it radiates into half-space.
The output from the panel will not increase 6 dB except at
low frequencies where the sounds via the direct ray and the ground reflected ray
to the listener L are almost in phase. At higher frequencies the ground
reflection subtracts and adds periodically. The panel essentially radiates into
full-space as in figure a, but with floor reflection superimposed to the
response at L. If we left the woofer level adjustment as found in a, then
the woofer contribution at the listening position in b would be 6 dB too
high relative to the high frequencies from the panel. At low frequencies the
panel receives a similar ground boost and its output becomes too large relative
to its high frequencies.
Instead of measuring the combined frequency response at large elevation as in figure a, I measure it with the microphone on the ground, figure b. Under these conditions woofer and panel receive the same ground boost and no shelving filter is used. Crossover filters are trimmed and woofer level set for flat response. In my case this is not a completely reflection free measurement because of adjacent buildings and structures and I use some mental averaging of the wiggles. Ambient noise tends to limit the low frequency dynamic range and I therefore pre-emphasize the low frequencies of the MLS stimulus signal to increase measurement range.
When I now measure at L in figure b I find that the low frequency response is too high, but correct when the shelving filter is in the circuit.
Unfiltered and octave smoothed curves are shown above.
We usually do not listen outdoors, but even in a room we have the same floor effects. In addition there are the effects due to modes, if excited, and they become superimposed to the above response and should be dealt with separately, for example by room equalization with notch filters. The PHOENIX printed circuit board provides layouts for this. Top
G - Level adjustment controls
The gain of the midrange channel is held constant. Tweeter and woofer outputs are adjusted relative to it. Since the midrange driver voltage sensitivity is 12 dB higher than the tweeter (see E above) I use a 12 dB attenuator ahead of the dipole equalization (90-500LP). The 400NF notch filter, which is driven from the attenuator, must see an impedance of 5k ohm for proper equalization of the midrange peak as determined by experiment. Thus it becomes necessary to design a resistive ladder with an attenuation factor a = 10^(-12/20) = 0.25 and 5000 ohm output impedance.
For the variable gain in the woofer and tweeter channel I chose a 5 dB adjustment range which also matches with the10 tick marks on the trim potentiometer. The linear dB scale is generated with a circuit that I saw used by my former colleague Russ Riley. I have found many applications for it.
R4 is a linear 1k ohm potentiometer. The values for R3 and
R5 are determined from gmax = 0 dB (1.0) and gmin = -5 dB (0.56) with a little
algebra as R3 = 3.55k and R5 = 2.55k. The closest standard series values of
3.48k and 2.61k are used for the tweeter channel gain adjustment.
H - Psycho-acoustic 3 kHz dip
Our perception of loudness is slightly different for
sounds arriving frontally versus sounds arriving from random directions at our
ears. The difference between equal-loudness-level contours in frontal
free-fields and diffuse sound fields is documented, for example, in ISO
Recommendation 454 and in E. Zwicker, H. Fastl, Psycho-acoustics, p. 205.
Around 3 kHz our hearing is less sensitive to diffuse
fields. Recording microphones, though, are usually flat in frequency response
even under diffuse field conditions. When such recordings are played back over
loudspeakers, there is more energy in the 3 kHz region than we would have
perceived if present at the recording venue and a degree of unnaturalness is
I have found through my own head-related recordings of symphonic music that the dip adds greater realism, especially to large chorus and to soprano voice and allows for higher playback levels. Top
I - Voltage sensitivity of an active system
Normally the equivalent 1 m sound pressure level of a speaker is specified for a 2.83 V signal at the crossover input terminals. This would correspond to 1 W into 8 ohm if the terminal impedance was actually 8 ohm. Defining the 2.83 V, 1 m sensitivity for a system with electronic crossover/eq like the PHOENIX is not as straight forward.
Starting with the SS D2904/9800 tweeter, the
manufacturer's specification is 90 dB SPL, 2.83 V, 1 m.
The above SPL values are for 2.83 V across the driver terminals. With a tweeter impedance of about 4.5 ohm over its used frequency range this corresponds to (2.83)^2 / 4.5 = 1.8 W. For the paralleled midranges with 3.5 ohm impedance the power becomes 2.3 W and <1.2 W for the woofer drivers at >6.5 ohm total.
The Frequently Asked Questions page will show you answers to inquiries which I received about the PHOENIX project and related subjects.