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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
A1 - Dipole equalization

B   - Open baffle loudspeakers
- "Compound dipole" woofer model

C   - Dipole woofer equalization
C1 - Notch filter design
C2 - 6 dB/oct slope correction
C3 - Equalization of low Qt woofer

D   - Midrange dipole equalization

E   - Midrange to tweeter crossover

F   - Midrange to woofer crossover
F1 - Woofer level setting

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 degrees off-axis. 
The magnitude response 20*log[ H(f) ] and the phase response (not shown) describe the doublet in the frequency domain.

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 down to the lowest 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. 
A very important number for comparison purposes is the frequency for which the dipole output is the same as that of a single monopole, i.e. when  |H(f)| = 1 = 2*sin(180*f*T) = 2*sin(30), which yields

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


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 imparts. 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 = 0.25*v/F. 
For a baffle of D = 20" (0.5 m) length and with L = 10" (0.25 m) estimated, the resonance peak in the dipole output is at F = 0.25*v/L = 343 Hz. Even when the peak is removed by equalization, the H baffle should only be operated below this frequency. It is a compact baffle for woofer applications and I use it with slightly different driver arrangement for the PHOENIX. 
Output equal to a closed box occurs at Fequal = 0.17*v/D = 117 Hz with the 20" path difference D between the positive and negative polarity sources at the H baffle openings. 
For a deeper analysis of the H-frame see the Issues in speaker design page.

I saw the H baffle concept for the first time, constructed as two woofer towers with six 12" drivers each, D=16", and placed against the side walls, at the home of Brian J. Elliott, Ph.D, Consultant in Electro-Acoustics, in Palo Alto, 1988. I had never heard bass reproduced so naturally before.

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 depth. 
This type of baffle can give you smoother response at all angles than is possible with a closed box, because the rear radiation coming around to the front can be used to advantage. With a closed box you only have the option to make the baffle either very narrow or very wide for smooth horizontal dispersion. For a discussion of diffraction and how it relates to open baffles look at FAQ8
As frequency increases and the driver becomes more directional of its own, the polar pattern still has an approximately cos(a) shape, because the driver radiates front and rear and little to the sides of the open baffle.   Top


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.
L = 2.7*1987/(2*pi*270) = 3.16 H and 
C = 1/(3.16*(2*pi*270)^2) = 110 nF
The input impedance R2 of the following stage is assumed to be large, so that R2 can be neglected for the filter calculation.
The large size inductor L is best realized with an active circuit inductr2.gif.
The inductive reactance is X = 2*pi*270*3.16 = 5360 ohm. For the inductor to have a Q of at least ten times the Q of the notch, the parallel resistor must be greater than Rp = 10*2.7*5360 = 145 kohm. Select 147k as standard value.
Rs needs to be smaller than R (1987 ohm). Select Rs = 511 ohm to give some adjustment room for R.
Now find C = 3.16/(511*(147000-511)) = 42.2 nF.
Finally, reduce R because the inductor contributes already 511 ohm. R = 1987-511 = 1476 ohm.
Next step is to build a circuit with these values, insert it ahead of the power amplifier and measure the woofer response to see if the peak has been removed to your satisfaction or if the circuit values need further trimming. 
The 290NF stage in the PHOENIX crossover/eq woofer channel has the final values and you can see that I had to do some experimentation with R, L and C to obtain the equalized response in the graph above. It also helps to use a SPICE based circuit analysis program for the notch filter circuit to find the inverse of the peak response more readily than via the approximate calculations above.    

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 driver resonance. 
With those values we calculate from f2/f1 = 1+R2/R1 that 
R2 = R1*((f2/f1) -1) = R1*((300/10)-1) = 29*R1.
Select R1 = 4.64k, then R2 = 134.6k or 133k as closest standard value.
Calculate C = 1/(2*pi*f1*R2) = 1/( 2*pi*10*133000) = 120 nF.
Check that f2 is indeed close to 300 Hz for the chosen values.
The circuit has a dc gain of 1+133/4.64 or 29.4 dB, which is high. This stage should come after the level setting circuit to avoid clipping distortion. 
The woofer equalization is now complete (x1.gif).  

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.
Keep in mind that with the biquad you are setting the two complex poles of a 3rd order high-pass filter. The 3rd, real axis pole is created by selecting  f1 for the dipole slope correction. There is no transient optimum high-pass filter like the Bessel filter for the low-pass case. The 3rd order Butterworth high-pass has no other merits than to look pretty. Optimum for time response is to place all three poles on the real axis, i.e. Qt = 0.5 for the dual poles, and then the frequency response is 9 dB down at Fs. Better yet, drivers with very low Qts < 0.5 make it possible to place two real axis poles at 20 Hz and the 3rd pole around 5 Hz, but require more elaborate equalization, because they roll off  with a pole around 80 Hz in addition to the 6 dB/oct dipole roll-off..
In general, I prefer the more gradual roll-off for any woofer, because it corresponds to a more linear phase response (more uniform group delay) and that is audible. The trade-off is in low end extension.   Top


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 A
At low frequencies the response follows the 6 dB/oct slope, but the first peak and null are not clearly formed. Also, in the rear of the baffle, the response rolls off more than in front.

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.
The vertical polar response is dominated by the separation of the 8" drivers and therefore much narrower.
On and off-axis response are very much a function of baffle shape and the amount of fold-back. Computer models are not available and the optimum shape must be determined experimentally.

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.
It is easiest to model the equalizer with a circuit CAD program and to compare response curves to the measured data. The low Q of the notch filter (x1.gif) makes it difficult to predict circuit values with sufficient accuracy and requires iteration.   Top


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).
Selecting a 1400 Hz crossover frequency based on dispersion gives wide overlap between the drivers. The tweeter goes almost 2 octaves lower and the midrange 2 octaves higher with useable flatness. Thus, when a 24 dB/oct L-R electrical crossover  filter at 1400 Hz is used, the combined driver and filter magnitude (dB) response will be mainly that of the electrical circuit. This specific situation reduces the design of the desired 24 dB/oct L-R acoustical crossover to the selection of components for a well known electrical circuit. Standard component values result in 1440 Hz high-pass filters in the tweeter channel and 1440 Hz low-pass filters in the midrange channel of the crossover/eq.

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. 
From the SS D2905/9700 specification of Xmax = 0.5 mm linear and 8.5 cm^2 cone area you estimate an SPL = 101 dB at 1400 Hz, 1 m, free space, closed box. Add to this 6 dB lift from the baffle and 6 dB from the midrange contribution for a total of 113 dB SPL. Maximum displacement is 1.5 mm or 10 dB higher. The tweeter is within the desired range, but harmonic distortion, intermodulation distortion, stored energy measurements and listening tests must be additional criteria for choosing it.

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 tweeter is approximately 2" (0.05 m) in front of the midranges and its input signal must be delayed by T = d/v = 0.05/343 = 146 us to be in phase with the midranges. The group delay of an all-pass circuit is used for the offset compensation. 

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 inequality. 
If f0 = 1/(2*pi*R*C) > 1400 and thus 
R*C < 1/(2*pi*1400) = 114 us,
then a single stage would have to be operated in its sloping region to obtain 146 us of delay. Thus take two stages. The actual implementation in the tweeter channel provides 85 us and 104 us for a total of 189 us. The value is larger than estimated, because the low-pass behavior of the midranges moves their voice coils effectively further behind the tweeter than the physical distance measurement. 
The final component values for the all-pass circuits have been determined experimentally from an optimization of the combined midrange and tweeter frequency response. The depth of the interference notch when the tweeter polarity is reversed, is an indication of how closely the two outputs are to being 180 degrees out-of-phase and of equal magnitude.   Top


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.
Note, that If you wanted to maintain 113 dB SPL down to 25 Hz, then the displacement of the two 8" drivers would have to increase another 12 dB (X2 figure b) to 4*10 mm = 40 mm peak, even though its contribution to the total SPL output would be 24 dB below the woofer output. In reality the 8" drivers can maximally support 113 -12 = 101 dB SPL at 25 Hz. 

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. 
The transition region between full-space and half-space radiation is equalized with a shelving high-pass filter that gives a 6 dB cut to very low frequencies and has a gradual transition with corner frequencies at 100 Hz and 200 Hz. I determined the frequency values empirically. The filter (100-200HP) is placed ahead of midrange and woofer channels to affect both (x1.gif).

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. 
The woofer channel requires about 8 dB (0.40) of attenuation to match up with the midrange. A resistive ladder is used for this with an output impedance target Rout = R3 = 3.55k ohm to drive the variable gain stage (+/-2.5 dB).   Top


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. 
Diffuse field equalization of dummy-head recordings is discussed in J. Blauert, Spatial Hearing, pp. 363, and headphone diffuse field equalization by G. Theile in JAES, Vol. 34, No. 12. 
Reference to a slight dip in the 1 to 3 kHz region for loudspeaker equalization is made in H. D. Harwood (BBC Research Department), Some factors in loudspeaker quality, Wireless World, May 1976, p.48.

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 introduced. 
This applies primarily to recordings of large orchestral pieces in concert halls where the microphones are much closer to the instruments than any listener. At most listening positions in the hall the sound field has strong diffuse components. 
I use a dip of 4 dB (x1.gif, 2760NF) to equalize for this. The circuit consists of R, C and L in series, forming a frequency dependent ladder attenuator in conjunction with the 5.11k ohm source resistor. You may choose to make the notch filter selectable with a switch for different types of recordings.

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. 
Sensitivity of the two midranges, wired in parallel, and mounted in the open baffle measures about 13 dB higher than the tweeter or 103 dB SPL. The SS specification is 90 dB SPL for a single 21W/8554 driver. For two drivers in parallel add 6 dB and for rear radiation reinforcement from the open baffle add another 6 dB, which explains the high measured value. The crossover circuit attenuates the signal to the midrange power amplifier by 13 dB except towards the lower frequencies where the 6 dB/oct dipole equalization and the crossover roll-off come in. The 7 dB boost at 100 Hz makes up for the loss in midrange sensitivity from 103 dB above 250 Hz  to 103 - 7 = 96 dB SPL at 100 Hz.
The 1252DVC woofer driver has a 91 dB SPL specification which applies at Fequal = 0.17*v/D = 121 Hz. With two drivers in parallel the sensitivity becomes 97 dB, but rolls off at 6 dB/oct rate below 121 Hz. Thus at 60 Hz the sensitivity is 91 dB SPL and goes down to 85 dB SPL at 30 Hz.

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. 

For an estimate of the power required to obtain a desired SPL level in an enclosed space at listening position see the discussion of "Room reverberation time T60" on the Room acoustics page.   Top



The Frequently Asked Questions page will show you answers to inquiries which I received about the PHOENIX project and related subjects.



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What you hear is not the air pressure variation in itself 
but what has drawn your attention
in the streams of superimposed air pressure variations 
at your eardrums

An acoustic event has dimensions of Time, Tone, Loudness and Space
Have they been recorded and rendered sensibly?

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