Determination of the volume and settings of the phase inverter program. A simple way to set up bass reflex speakers. What kind of music is the bass reflex suitable for?
Radio amateurs engaged in the independent manufacture of loudspeakers-phase inverters (hereinafter for brevity - just a phase inverter), often face the fact that their repeated designs do not provide the technical characteristics given in the descriptions. This is due to the significant technological variation in the parameters of the low-frequency drivers, so each manufactured loudspeaker must be tuned.
When adjusting phase inverters, radio amateurs usually use the same technique as when calculating them. As a result, the acoustic losses occurring in a real design, the difference between the equivalent and physical volumes of the box, and a number of other factors affecting the accuracy of tuning are not taken into account. The proposed tuning technique takes these factors into account, so its accuracy is much higher.
Tuning any phase inverter comes down, as you know, to finding a certain combination of values of the frequency of its tuning ff and the output impedance of the amplifier Rout at which a smooth frequency response of the radiation at the lowest sound frequencies is provided. You can find these values using the relationship between the parameters of the bass reflex and the closed box. If you close the tunnel opening in a phase inverter with a smooth frequency response, then the total quality factor of the head-closed box system will be equal to 0.6, and the resonant frequency of the head in such a box will be related to the frequency of the phase inverter by the dependence ff = 0.61 ... 0.65 fр. The proportionality coefficient of the indicated values depends on the ratio of the equivalent volume of the head to the useful volume of the box, and if we take it equal to 0.63, then the error in determining the frequency ff will not exceed 5% for any ratios of the indicated volumes found in real structures.
Setting up the bass reflex should start with determining the optimal amount of sound-absorbing material to be placed in it. To do this, closing the tunnel opening (for example, with a plywood circle smeared with plasticine at the edges), select such an amount of material at which the frequency fр is minimal. Then, fixing the absorbing material on the walls of the box, the resonant frequency of the head-closed box system is measured and, using the relation ff = 0.63 fp, the tuning frequency of the phase inverter is determined, and then the length of its tunnel:
where V is the free volume of the phase inverter box in liters, and S is the area of the phase inverter tunnel opening in sq. cm.
Usually, the equivalent volume of acoustic design, when the optimal amount of sound-absorbing material is placed in it, turns out to be greater than the geometric one, therefore, the length of the tunnel must be reduced when adjusting the phase inverter. To determine the refined value 1 ", the value of the tuning frequency of the phase inverter, obtained with the length of the tunnel 1, is substituted into the above formula and the equivalent volume of the design Ve is found. Then, replacing V with Ve in the same formula, the refined value of the tunnel length is calculated. (Mospagebreak)
The output impedance of the amplifier Rout can be found based on the condition under which the quality factor of the amplifier - closed box system takes on a value equal to 0.6, however, it is preferable to determine the non-value of Rout from the condition under which the quality factor of the amplifier - box phase inverter system takes an optimal value equal to 1 ( in this case, the amplifier tuning technique is simplified and the losses arising in the inverter tunnel are taken into account)
The quality factor of the head-box-phase inverter system is determined by the method adopted for the head-closed box systems, but all the necessary measurements are carried out near the frequency of the high-frequency resonance of the frequency response of the input impedance of the loudspeaker f p (see figure). To improve the accuracy of subsequent calculations, the parameters of the frequency response of the input impedance of the loudspeaker should be measured from the side of the connector for connecting it to the amplifier. In this case, the influence of the active resistance of the connecting wire and the crossover filter coil on the loudspeaker parameters is taken into account.
By calculating the acoustic Q factor
where Uр is the voltage at the frequency fр, Uem is the voltage at the frequency of the electromechanical resonance fem, f1 and f2 are the cutoff frequencies at the voltage level U1,2 = root (UрUem), find the electrical and total quality factor of the system:
if the found value of Qp differs from unity by no more than 10%, then the frequency response of the phase inverter will be quite smooth when working together with almost any transistor amplifier with low output impedance. If Qp> 1.1 (this is the case most often encountered in amateur radio practice), then an amplifier with negative output impedance should be used to work with the phase inverter. To get a smooth frequency response of the loudspeaker radiation,
it is necessary to adjust the feedback loop that forms the negative output impedance of the amplifier. For this, the damping coefficient Kd = Qp / Qp.opt is preliminarily determined, which shows how many times the total quality factor of the head-box-phase inverter system must be reduced in order to obtain optimal damping. Since the condition for optimal damping of the phase inverter assumes Qp opt = 1, then Kd = Qp. Further, by connecting the loudspeaker to the amplifier and applying to the last sound signal with frequency fem, the feedback circuit bridge is balanced and the voltage at the amplifier output is measured. Then, by rebuilding the generator to the frequency fр and changing the transmission coefficient of the feedback circuit, they achieve a decrease in the voltage at the amplifier output by a factor of Kd. As a result of this setting, exactly the value of the output impedance of the amplifier is set, at which a smooth frequency response of the loudspeaker radiation at low frequencies is obtained.
When calculating a power amplifier, it is advisable to determine the required output impedance in advance. It is calculated by the formula
The above technique, without any changes, is also applicable for setting up loudspeakers in which dual or several heads of the same type are installed.
Literature
- Vinogradova E. Designing loudspeakers with smoothed frequency characteristics.- Moscow: Energiya, 1978
- Efrussm M. More about the calculation and manufacture of the loudspeaker. - Radio, 1984, N 10, p. 32-33.
- Popov P., Shorov V. Improving the sound quality of loudspeakers. - Radio, 1983. N 6, p. 50-53.
- EMOS or negative output impedance? - Radio, 1981, N 1, p. 40- 44.
Editor’s Note: An article by an Italian acoustician, reproduced here with the blessing of the author, was originally titled “Teoria e pratica del condotto di accordo”. That is, in a literal translation - "Theory and Practice of Bass Reflex". This title, in our opinion, corresponded to the content of the article only formally. Indeed, we are talking about the ratio of the simplest theoretical model of a phase inverter and those surprises that practice prepares. But this is - if formal and superficial. And in essence, the article contains the answer to the questions that arise, judging by the editorial mail, all the time when calculating and manufacturing a subwoofer-phase inverter. The first question: "If you calculate the phase inverter according to the formula that has been known for a long time, will the finished phase inverter get the design frequency?" Our Italian colleague, who has eaten about a dozen of dogs on phase inverters in his lifetime, answers: "No, it won't work." And then he explains why and, what is most valuable, how much it will not work. Question two: “I calculated the tunnel, but it is so long that it does not fit anywhere. How to be? " And here the signor proposes such original solutions that it is precisely this side of his works that we put in the title. So the key word in the new title should be understood not in New Russian (otherwise we would have written: "in short - phase inverter"), but quite literally. Geometrically. And now Signor Matarazzo has the floor to speak.
Bass reflex: shorter!
About the author: Jean-Piero Matarazzo was born in 1953 in Avellino, Italy. Since the beginning of the 70s he has been working in the field of professional acoustics. For many years he was responsible for testing acoustic systems for the magazine "Suono" ("Sound"). In the 90s, he developed a number of new mathematical models of the process of sound emission by loudspeaker diffusers and several projects of acoustic systems for industry, including the Opera model, which was popular in Italy. Since the late 90s, he has been actively cooperating with the magazines "Audio Review", "Digital Video" and, what is most important for us, "ACS" ("Audio Car Stereo"). In all three, he is in charge of measuring parameters and testing acoustics. What else? .. Married. Two little sons are growing, 7 years old and 10.
Fig 1. Schematic diagram of a Helmholtz resonator. That from which everything comes.
Fig 2. Classical bass reflex design. In this case, the influence of the wall is often not taken into account.
Fig 3. Phase inverter with a tunnel, the ends of which are in free space. There is no wall influence here.
Fig 4. You can bring the tunnel out completely. Here again "virtual lengthening" will occur.
Fig 5. It is possible to obtain a "virtual extension" at both ends of the tunnel by making another flange.
Fig 6. Slotted tunnel located far from the walls of the box.
Fig 7. Slotted tunnel located near the wall. As a result of the influence of the wall, its "acoustic" length is greater than the geometric one.
Fig 8. Tunnel in the form of a truncated cone.
Fig 9. Main dimensions of the conical tunnel.
Fig 10. Dimensions of the slotted version of the conical tunnel.
Fig 11. Exponential tunnel.
Figure 12. Hourglass-shaped tunnel.
Fig 13. Main dimensions of the hourglass tunnel.
Fig 14. Slotted version of the hourglass.
Magic formulas
One of the most common wishes in the author's e-mail is to give a "magic formula" by which the ACS reader could calculate the phase inverter himself. This is, in principle, not difficult. The phase inverter is one of the cases of the implementation of the device called "Helmholtz resonator". The formula for calculating it is not much more complicated than the most common and available model of such a resonator. An empty Coca-Cola bottle (only a bottle, not an aluminum can) is just such a resonator, tuned to a frequency of 185 Hz, it has been verified. However, the Helmholtz resonator is much older than even this, gradually falling out of use, packaging of a popular drink. However, the classical scheme of a Helmholtz resonator is similar to a bottle (Fig. 1). In order for such a resonator to work, it is important that it has a volume V and a tunnel with a cross-sectional area S and a length L. Knowing this, the tuning frequency of the Helmholtz resonator (or phase inverter, which is the same) can now be calculated by the formula:
- Fb- tuning frequency of the phase inverter pipe (Hz)
- With- speed of sound, constant = 344 m / s
- S- the area of the bass reflex tunnel (m 2)
- L- length of the bass reflex tunnel (m)
- V- body volume (m 3)
- P - constant value = 3,14
This formula is truly magical, in the sense that the bass reflex setting does not depend on the parameters of the speaker that will be installed in it. The volume of the box and the dimensions of the tunnel determine the frequency of tuning once and for all. Everything, it would seem, is done. Let's get started. Suppose we have a box with a volume of 50 liters. We want to turn it into a bass reflex box with a 50 Hz setting. It was decided to make the diameter of the tunnel 8 cm. According to the formula just given, the tuning frequency of 50 Hz will be obtained if the length of the tunnel is 12.05 cm. We carefully manufacture all the parts, assemble them into a structure, as in Fig. 2, and for verification we measure the actual resonant frequency of the phase inverter. And we see, to our surprise, that it is not equal to 50 Hz, as it would be expected by the formula, but 41 Hz. What's the matter and where did we go wrong? Yes, nowhere. Our freshly built phase inverter would be tuned to a frequency close to that obtained by the Helmholtz formula if it were made, as shown in Fig. 3. This case is closest to the ideal model described by the formula: here both ends of the tunnel "hang in the air", relatively far from any obstacles. In our design, one of the ends of the tunnel mates with the wall of the box. For the air vibrating in the tunnel, it is not indifferent, because of the influence of the "flange" at the end of the tunnel, its virtual lengthening occurs. The phase inverter will be tuned as if the length of the tunnel was 18 cm, and not 12, as in reality.
Note that the same will happen if the tunnel is completely placed outside the box, again aligning one end of it with the wall (Fig. 4). There is an empirical dependence of the "virtual lengthening" of the tunnel depending on its size. For a round tunnel, one cut of which is located far enough from the walls of the box (or other obstacles), and the other is in the plane of the wall, this elongation is approximately 0.85D.
Now, if we substitute all the constants in the Helmholtz formula, introduce a correction for the "virtual elongation", and express all dimensions in the usual units, the final formula for the length of a tunnel with a diameter D, which ensures the tuning of a box of volume V to the frequency Fb, will look like this:
- Fb- the frequency to which the bass reflex is tuned (Hz)
- V- body volume (l)
- D- bass reflex pipe diameter (mm)
- L- the length of the phase inverter pipe (mm)
The obtained result is valuable not only because it allows at the stage of calculation to obtain a length value close to the final one, which gives the required value of the tuning frequency, but also because it opens up certain reserves for shortening the tunnel. We have already won almost one diameter. It is possible to shorten the tunnel even further while maintaining the same tuning frequency by making flanges at both ends, as shown in fig. 5.
Now, it seems, everything has been taken into account, and, armed with this formula, we seem to be omnipotent. This is where difficulties await us.
First difficulties
The first (and main) difficulty is as follows: if a relatively small box needs to be tuned to a fairly low frequency, then, substituting a large diameter in the formula for the length of the tunnel, we will get a large length. Let's try to substitute a smaller diameter - and everything turns out fine. A large diameter requires a long length, and a small one is just small. What's wrong with that? Here's what. Moving, the speaker diffuser with its back side "pushes" practically incompressible air through the bass reflex tunnel. Since the volume of the oscillating air is constant, the air speed in the tunnel will be as many times greater than the oscillatory speed of the diffuser, as many times the cross-sectional area of the tunnel is less than the area of the diffuser. If you make a tunnel dozens of times smaller than the diffuser, the flow velocity in it will be high, and when it reaches 25 - 27 meters per second, the appearance of turbulence and jet noise is inevitable. The great researcher of acoustic systems R. Small showed that the minimum section of the tunnel depends on the diameter of the speaker, the maximum stroke of its diffuser and the frequency of the phase inverter tuning. Small proposed a completely empirical but trouble-free formula for calculating the minimum tunnel size:
Small derived his formula in his usual units, so that the speaker diameter Ds, the maximum diffuser travel Xmax and the minimum tunnel diameter Dmin are expressed in inches. Bass reflex tuning frequency - as usual, in hertz.
Now everything does not look as rosy as before. Very often it turns out that if you choose the right diameter of the tunnel, it comes out incredibly long. And if you reduce the diameter, there is a chance that the tunnel will "whistle" even at medium power. In addition to the actual jet noises, tunnels of small diameter also have a tendency to the so-called "organ resonances", the frequency of which is much higher than the frequency of the phase inverter tuning and which are excited in the tunnel by turbulence at high flow rates.
Faced with such a dilemma, ACS readers usually call the editorial office and ask for a solution. I have three of them: simple, medium and extreme.
Simple solution for small problems
When the calculated length of the tunnel turns out to be such that it almost fits in the body and it is only required to slightly reduce its length with the same setting and cross-sectional area, I recommend using a slot tunnel instead of a round one, and placing it not in the middle of the front wall of the body (as in Fig. 6 ), but close to one of the side walls (as in Fig. 7). Then at the end of the tunnel, which is inside the box, the effect of "virtual lengthening" will be felt due to the wall next to it. Experiments show that with a constant cross-sectional area and tuning frequency, the tunnel shown in Fig. 7, it turns out to be about 15% shorter than with a design like in fig. 6. A slotted bass reflex, in principle, is less prone to organ resonances than a round one, but to protect yourself even more, I recommend installing sound-absorbing elements inside the tunnel, in the form of narrow felt strips glued to the inner surface of the tunnel in the region of a third of its length. This is a simple solution. If it is not enough, you will have to move to the middle one.
Medium solution for bigger problems
An intermediate solution is to use a frustoconical tunnel, as in Fig. 8. My experiments with such tunnels have shown that here it is possible to reduce the cross-sectional area of the inlet in comparison with the minimum allowable according to Small's formula without the danger of jet noise. In addition, a tapered tunnel is much less prone to organ resonances than a cylindrical one.
In 1995 I wrote a program for calculating tapered tunnels. It replaces a tapered tunnel with a sequence of cylindrical ones and, by successive approximations, calculates the length required to replace a conventional tunnel of constant cross-section. This program is made for everyone, and you can take it on the website of the ACS magazine audioreview.it For more information, see the ACS Software section. A small program that works under DOS, you can download and calculate it yourself. And you can do it differently. When preparing the Russian edition of this article, the results of calculations using the CONICO program were summarized in a table, from which you can take the finished version. The table is compiled for a tunnel with a diameter of 80 mm. This diameter value is suitable for most subwoofers with a cone diameter of 250 mm. After calculating the required tunnel length using the formula, find this value in the first column. For example, according to your calculations, it turned out that you need a tunnel with a length of 400 mm, for example, to tune a box with a volume of 30 liters to a frequency of 33 Hz. The project is not trivial, and it will not be easy to place such a tunnel inside such a box. Now we look at the next three columns. There are given the dimensions of the equivalent conical tunnel calculated by the program, the length of which will no longer be 400, but only 250 mm. Quite another matter. What the dimensions mean in the table is shown in Fig. 9.
Table 1. Dimensions of a conical tunnel equivalent to a cylindrical diameter of 80 mm and a length of Lo.
| Lo | L | d | D | h | Win | Wout |
| 160 | 120 | 67 | 84 | 60 | 59 | 92 |
| 200 | 150 | 64 | 85 | 60 | 53 | 95 |
| 260 | 180 | 60 | 85 | 60 | 48 | 95 |
| 330 | 200 | 54 | 86 | 60 | 39 | 98 |
| 400 | 250 | 52 | 87 | 60 | 35 | 99 |
| 500 | 350 | 50 | 99 | 60 | 33 | 129 |
| 630 | 450 | 46 | 109 | 60 | 28 | 155 |
| 750 | 500 | 42 | 112 | 60 | 24 | 164 |
Table 2.
| Lo | L | d | D | h | Win | Wout |
| 270 | 200 | 79 | 107 | 70 | 71 | 129 |
| 330 | 220 | 73 | 108 | 70 | 60 | 131 |
| 420 | 280 | 70 | 109 | 70 | 54 | 133 |
| 530 | 350 | 65 | 114 | 70 | 47 | 143 |
| 650 | 450 | 62 | 124 | 70 | 43 | 174 |
| 800 | 550 | 57 | 134 | 70 | 36 | 200 |
| 1000 | 650 | 50 | 141 | 70 | 29 | 224 |
| 1180 | 750 | 46 | 151 | 70 | 24 | 257 |
Lo- long original cylindrical tunnel
L- the length of the conical tunnel
Table 2 is compiled for the original tunnel with a diameter of 100 mm. This will fit most subwoofers with a 300mm head diameter.
If you decide to use the program yourself, remember: a truncated cone-shaped tunnel is made with an angle of inclination of the generatrix a from 2 to 4 degrees. It is not recommended to make this angle more than 6 - 8 degrees, in this case, the occurrence of turbulence and jet noises at the entrance (narrow) end of the tunnel is possible. However, even with a small taper, the reduction in the length of the tunnel turns out to be quite significant.
A frusto-conical tunnel does not need to have a circular cross-section. Like the usual cylindrical one, it is sometimes more convenient to make it in the form of a slotted one. Even, as a rule, it is more convenient, because then it is assembled from flat parts. The dimensions of the slotted version of the conical tunnel are given in the following columns of the table, and what these dimensions mean is shown in Fig. 10.
Replacing a conventional tunnel with a conical one can solve many problems. But not all. Sometimes the length of the tunnel turns out to be so long that shortening it even by 30 - 35% is not enough. For such severe cases, there is ...
... an extreme solution for big problems
An extreme solution is to use a tunnel with exponential contours, as shown in Fig. 11. At such a tunnel, the cross-sectional area at first gradually decreases, and then also smoothly increases to the maximum. From the point of view of compactness for a given tuning frequency, resistance to jet noises and organ resonances, the exponential tunnel is unmatched. But it is unmatched in the complexity of manufacturing, even if its contours are calculated according to the same principle as was done in the case of a conical tunnel. In order to still take advantage of the exponential tunnel in practice, I came up with a modification of it: the tunnel, which I called the "hourglass" (Fig. 12). The hourglass tunnel consists of a cylindrical section and two conical sections, whence the external resemblance to an ancient device for measuring time. This geometry makes it possible to shorten the tunnel in comparison with the original, constant cross-section, at least one and a half times, or even more. To calculate the hourglass, I also wrote a program, it can be found there, on the ACS website. And just like for the conical tunnel, here is a table with ready-made calculation options.
Table 3. The dimensions of the hourglass-shaped tunnel are equivalent to a cylindrical one with a diameter of 80 mm and a length of Lo.
| Lo | Lmax | d | D | L1 | L2 | h | Wmin | Wmax |
| 160 | 100 | 58 | 81 | 60 | 20 | 50 | 52 | 103 |
| 200 | 125 | 58 | 81 | 75 | 25 | 50 | 52 | 103 |
| 260 | 175 | 58 | 82 | 105 | 35 | 50 | 52 | 104 |
| 330 | 200 | 55 | 82 | 120 | 40 | 50 | 48 | 104 |
| 400 | 250 | 55 | 83 | 150 | 50 | 50 | 48 | 105 |
| 500 | 300 | 54 | 83 | 180 | 60 | 50 | 45 | 105 |
| 630 | 400 | 54 | 84 | 240 | 80 | 50 | 45 | 106 |
| 750 | 450 | 54 | 84 | 270 | 90 | 50 | 45 | 106 |
Table 4. The same for the original tunnel with a diameter of 100 mm
| Lo | Lmax | d | D | L1 | L2 | h | Wmin | Wmax |
| 270 | 175 | 71 | 100 | 105 | 35 | 60 | 69 | 130 |
| 330 | 200 | 71 | 100 | 120 | 40 | 60 | 69 | 130 |
| 420 | 250 | 71 | 100 | 150 | 50 | 60 | 69 | 130 |
| 530 | 300 | 69 | 102 | 180 | 60 | 60 | 66 | 133 |
| 650 | 400 | 69 | 102 | 240 | 80 | 60 | 66 | 133 |
| 800 | 500 | 68 | 103 | 300 | 100 | 60 | 63 | 135 |
| 1000 | 600 | 68 | 103 | 360 | 120 | 60 | 63 | 135 |
| 1180 | 750 | 68 | 103 | 450 | 150 | 60 | 63 | 135 |
What the dimensions in tables 3 and 4 mean will become clear from fig. 13. D and d are the diameter of the cylindrical section and the largest diameter of the conical section, respectively, L1 and L2 are the lengths of the sections. Lmax - the full length of the hourglass-shaped tunnel, just for comparison, how much shorter it was possible to make, but in general, it is L1 + 2L2.
Technologically, an hourglass of round cross-section is not always easy and convenient to make. Therefore, here you can also make it in the form of a profiled slot, it will turn out, as in Fig. 14. For replacing a tunnel with a diameter of 80 mm, I recommend choosing a slot height equal to 50 mm, and for replacing a 100 mm cylindrical tunnel - equal to 60 mm. Then the width of the section of constant cross-section Wmin and the maximum width at the entrance and exit of the tunnel Wmax will be the same as in the table (the lengths of the sections L1 and L2 - as in the case of a circular cross-section, nothing changes here). If necessary, the height of the slotted tunnel h can be changed by simultaneously adjusting Wmin and Wmax so that the values of the cross-sectional area (h.Wmin, h.Wmax) remain unchanged.
I used a variant of the bass reflex with an hourglass-shaped tunnel, for example, when I was making a subwoofer for a home theater with a tuning frequency of 17 Hz. The estimated length of the tunnel turned out to be more than a meter, and by calculating the "hourglass", I was able to cut it almost in half, while there was no noise even at a power of about 100 watts. Hope it helps you too ...
Translated from Italian by Zhurkova E., based on materials cxem.net
From the Editor: An article by an Italian acoustician, reproduced here with the author's blessing, was originally titled Teoria e pratica del condotto di accordo. That is, in a literal translation - "Theory and Practice of Bass Reflex". This title, in our opinion, corresponded to the content of the article only formally. Indeed, we are talking about the ratio of the simplest theoretical model of a phase inverter and those surprises that practice prepares. But this is - if formal and superficial. And in essence, the article contains the answer to the questions that arise, judging by the editorial mail, all the time when calculating and manufacturing a subwoofer-phase inverter. The first question: "If you calculate the phase inverter according to the formula that has been known for a long time, will the finished phase inverter get the design frequency?" Our Italian colleague, who has eaten about a dozen of dogs on phase inverters in his lifetime, answers: "No, it won't work." And then he explains why and, what is most valuable, how much it will not work. Question two: “I calculated the tunnel, but it is so long that it does not fit anywhere. How to be? " And here the signor proposes such original solutions that it is precisely this side of his works that we put in the title. So the key word in the new title should be understood not in New Russian (otherwise we would have written: "in short - phase inverter"), but quite literally. Geometrically. And now Signor Matarazzo has the floor to speak.
Bass reflex: shorter!
Jean-Pierrot MATARAZZO Translated from Italian by E. Zhurkova
About the author: Jean-Piero Matarazzo was born in 1953 in Avellino, Italy. Since the beginning of the 70s he has been working in the field of professional acoustics. For many years he was responsible for testing acoustic systems for the magazine "Suono" ("Sound"). In the 90s, he developed a number of new mathematical models of the process of sound emission by loudspeaker diffusers and several projects of acoustic systems for industry, including the Opera model, which was popular in Italy. Since the late 90s, he has been actively cooperating with the magazines "Audio Review", "Digital Video" and, what is most important for us, "ACS" ("Audio Car Stereo"). In all three, he is in charge of measuring parameters and testing acoustics. What else? .. Married. Two little sons are growing, 7 years old and 10.
Fig 1. Schematic diagram of a Helmholtz resonator. That from which everything comes.
Fig 2. Classical bass reflex design. In this case, the influence of the wall is often not taken into account.
Fig 3. Phase inverter with a tunnel, the ends of which are in free space. There is no wall influence here.
Fig 4. You can bring the tunnel out completely. Here again "virtual lengthening" will occur.
Fig 5. It is possible to obtain a "virtual extension" at both ends of the tunnel by making another flange.
Fig 6. Slotted tunnel located far from the walls of the box.
Fig 7. Slotted tunnel located near the wall. As a result of the influence of the wall, its "acoustic" length is greater than the geometric one.
Fig 8. Tunnel in the form of a truncated cone.
Fig 9. Main dimensions of the conical tunnel.
Fig 10. Dimensions of the slotted version of the conical tunnel.
Fig 11. Exponential tunnel.
Figure 12. Hourglass-shaped tunnel.
Fig 13. Main dimensions of the hourglass tunnel.
Fig 14. Slotted version of the hourglass.
Magic formulas
One of the most common wishes in the author's e-mail is to give a "magic formula" by which the ACS reader could calculate the phase inverter himself. This is, in principle, not difficult. The phase inverter is one of the cases of the implementation of the device called "Helmholtz resonator". The formula for calculating it is not much more complicated than the most common and available model of such a resonator. An empty Coca-Cola bottle (only a bottle, not an aluminum can) is just such a resonator, tuned to a frequency of 185 Hz, it has been verified. However, the Helmholtz resonator is much older than even this, gradually falling out of use, packaging of a popular drink. However, the classical scheme of a Helmholtz resonator is similar to a bottle (Fig. 1). In order for such a resonator to work, it is important that it has a volume V and a tunnel with a cross-sectional area S and a length L. Knowing this, the tuning frequency of the Helmholtz resonator (or phase inverter, which is the same) can now be calculated by the formula:
where Fb is the tuning frequency in Hz, s is the speed of sound equal to 344 m / s, S is the area of the tunnel in sq. m, L is the length of the tunnel in m, V is the volume of the box in cubic meters. m = 3.14, it goes without saying.
This formula is truly magical, in the sense that the bass reflex setting does not depend on the parameters of the speaker that will be installed in it. The volume of the box and the dimensions of the tunnel determine the frequency of tuning once and for all. Everything, it would seem, is done. Let's get started. Suppose we have a box with a volume of 50 liters. We want to turn it into a bass reflex box with a 50 Hz setting. It was decided to make the diameter of the tunnel 8 cm. According to the formula just given, the tuning frequency of 50 Hz will be obtained if the length of the tunnel is 12.05 cm. We carefully manufacture all the parts, assemble them into a structure, as in Fig. 2, and for verification we measure the actual resonant frequency of the phase inverter. And we see, to our surprise, that it is not equal to 50 Hz, as it would be expected by the formula, but 41 Hz. What's the matter and where did we go wrong? Yes, nowhere. Our freshly built phase inverter would be tuned to a frequency close to that obtained by the Helmholtz formula if it were made, as shown in Fig. 3. This case is closest to the ideal model described by the formula: here both ends of the tunnel "hang in the air", relatively far from any obstacles. In our design, one of the ends of the tunnel mates with the wall of the box. For the air vibrating in the tunnel, it is not indifferent, because of the influence of the "flange" at the end of the tunnel, its virtual lengthening occurs. The phase inverter will be tuned as if the length of the tunnel was 18 cm, and not 12, as in reality.
Note that the same will happen if the tunnel is completely placed outside the box, again aligning one end of it with the wall (Fig. 4). There is an empirical dependence of the "virtual lengthening" of the tunnel depending on its size. For a round tunnel, one cut of which is located far enough from the walls of the box (or other obstacles), and the other is in the plane of the wall, this elongation is approximately 0.85D.
Now, if we substitute all the constants in the Helmholtz formula, introduce a correction for the "virtual elongation", and express all dimensions in the usual units, the final formula for the length of a tunnel with a diameter D, which ensures the tuning of a box of volume V to the frequency Fb, will look like this:
Here the frequency is in hertz, the volume is in liters, and the length and diameter of the tunnel are in millimeters, as we are used to.
The obtained result is valuable not only because it allows at the stage of calculation to obtain a length value close to the final one, which gives the required value of the tuning frequency, but also because it opens up certain reserves for shortening the tunnel. We have already won almost one diameter. It is possible to shorten the tunnel even further while maintaining the same tuning frequency by making flanges at both ends, as shown in fig. 5.
Now, it seems, everything has been taken into account, and, armed with this formula, we seem to be omnipotent. This is where difficulties await us.
First difficulties
The first (and main) difficulty is as follows: if a relatively small box needs to be tuned to a fairly low frequency, then, substituting a large diameter in the formula for the length of the tunnel, we will get a large length. Let's try to substitute a smaller diameter - and everything turns out fine. A large diameter requires a long length, and a small one is just small. What's wrong with that? Here's what. Moving, the speaker diffuser with its back side "pushes" practically incompressible air through the bass reflex tunnel. Since the volume of the oscillating air is constant, the air speed in the tunnel will be as many times greater than the oscillatory speed of the diffuser, as many times the cross-sectional area of the tunnel is less than the area of the diffuser. If you make a tunnel dozens of times smaller than the diffuser, the flow velocity in it will be high, and when it reaches 25 - 27 meters per second, the appearance of turbulence and jet noise is inevitable. The great researcher of acoustic systems R. Small showed that the minimum section of the tunnel depends on the diameter of the speaker, the maximum stroke of its diffuser and the frequency of the phase inverter tuning. Small proposed a completely empirical but trouble-free formula for calculating the minimum tunnel size:
Small derived his formula in his usual units, so that the speaker diameter Ds, the maximum diffuser travel Xmax and the minimum tunnel diameter Dmin are expressed in inches. Bass reflex tuning frequency - as usual, in hertz.
Now everything does not look as rosy as before. Very often it turns out that if you choose the right diameter of the tunnel, it comes out incredibly long. And if you reduce the diameter, there is a chance that the tunnel will "whistle" even at medium power. In addition to the actual jet noises, tunnels of small diameter also have a tendency to the so-called "organ resonances", the frequency of which is much higher than the frequency of the phase inverter tuning and which are excited in the tunnel by turbulence at high flow rates.
Faced with such a dilemma, ACS readers usually call the editorial office and ask for a solution. I have three of them: simple, medium and extreme.
Simple solution for small problems
When the calculated length of the tunnel turns out to be such that it almost fits in the body and it is only required to slightly reduce its length with the same setting and cross-sectional area, I recommend using a slot tunnel instead of a round one, and placing it not in the middle of the front wall of the body (as in Fig. 6 ), but close to one of the side walls (as in Fig. 7). Then at the end of the tunnel, which is inside the box, the effect of "virtual lengthening" will be felt due to the wall next to it. Experiments show that with a constant cross-sectional area and tuning frequency, the tunnel shown in Fig. 7, it turns out to be about 15% shorter than with a design like in fig. 6. A slotted bass reflex, in principle, is less prone to organ resonances than a round one, but to protect yourself even more, I recommend installing sound-absorbing elements inside the tunnel, in the form of narrow felt strips glued to the inner surface of the tunnel in the region of a third of its length. This is a simple solution. If it is not enough, you will have to move to the middle one.
Medium solution for bigger problems
An intermediate solution is to use a frustoconical tunnel, as in Fig. 8. My experiments with such tunnels have shown that here it is possible to reduce the cross-sectional area of the inlet in comparison with the minimum allowable according to Small's formula without the danger of jet noise. In addition, a tapered tunnel is much less prone to organ resonances than a cylindrical one.
In 1995 I wrote a program for calculating tapered tunnels. It replaces a tapered tunnel with a sequence of cylindrical ones and, by successive approximations, calculates the length required to replace a conventional tunnel of constant cross-section. This program is made for everyone, and it can be downloaded from the ACS magazine website http://www.audiocarstereo.it/ in the ACS Software section. A small program that works under DOS, you can download and calculate it yourself. And you can do it differently. When preparing the Russian edition of this article, the results of calculations using the CONICO program were summarized in a table, from which you can take the finished version. The table is compiled for a tunnel with a diameter of 80 mm. This diameter value is suitable for most subwoofers with a cone diameter of 250 mm. After calculating the required tunnel length using the formula, find this value in the first column. For example, according to your calculations, it turned out that you need a tunnel with a length of 400 mm, for example, to tune a box with a volume of 30 liters to a frequency of 33 Hz. The project is not trivial, and it will not be easy to place such a tunnel inside such a box. Now we look at the next three columns. There are given the dimensions of the equivalent conical tunnel calculated by the program, the length of which will no longer be 400, but only 250 mm. Quite another matter. What the dimensions mean in the table is shown in Fig. 9.
Table 2 is compiled for the original tunnel with a diameter of 100 mm. This will fit most subwoofers with a 300mm head diameter.
If you decide to use the program yourself, remember: a truncated cone-shaped tunnel is made with an angle of inclination of the generatrix a from 2 to 4 degrees. It is not recommended to make this angle more than 6 - 8 degrees, in this case, the occurrence of turbulence and jet noises at the entrance (narrow) end of the tunnel is possible. However, even with a small taper, the reduction in the length of the tunnel turns out to be quite significant.
A frusto-conical tunnel does not need to have a circular cross-section. Like the usual cylindrical one, it is sometimes more convenient to make it in the form of a slotted one. Even, as a rule, it is more convenient, because then it is assembled from flat parts. The dimensions of the slotted version of the conical tunnel are given in the following columns of the table, and what these dimensions mean is shown in Fig. 10.
Replacing a conventional tunnel with a conical one can solve many problems. But not all. Sometimes the length of the tunnel turns out to be so long that shortening it even by 30 - 35% is not enough. For such severe cases, there is ...
An extreme solution to big problems
An extreme solution is to use a tunnel with exponential contours, as shown in Fig. 11. At such a tunnel, the cross-sectional area at first gradually decreases, and then also smoothly increases to the maximum. From the point of view of compactness for a given tuning frequency, resistance to jet noises and organ resonances, the exponential tunnel is unmatched. But it is unmatched in the complexity of manufacturing, even if its contours are calculated according to the same principle as was done in the case of a conical tunnel. In order to still take advantage of the exponential tunnel in practice, I came up with a modification of it: the tunnel, which I called the "hourglass" (Fig. 12). The hourglass tunnel consists of a cylindrical section and two conical sections, whence the external resemblance to an ancient device for measuring time. This geometry makes it possible to shorten the tunnel in comparison with the original, constant cross-section, at least one and a half times, or even more. To calculate the hourglass, I also wrote a program, it can be found there, on the ACS website. And just like for the conical tunnel, here is a table with ready-made calculation options.
What the dimensions in tables 3 and 4 mean will become clear from fig. 13. D and d are the diameter of the cylindrical section and the largest diameter of the conical section, respectively, L1 and L2 are the lengths of the sections. Lmax - the full length of the hourglass-shaped tunnel, just for comparison, how much shorter it was possible to make, but in general, it is L1 + 2L2.
Technologically, an hourglass of round cross-section is not always easy and convenient to make. Therefore, here you can also make it in the form of a profiled slot, it will turn out, as in Fig. 14. For replacing a tunnel with a diameter of 80 mm, I recommend choosing a slot height equal to 50 mm, and for replacing a 100 mm cylindrical tunnel - equal to 60 mm. Then the width of the section of constant cross-section Wmin and the maximum width at the entrance and exit of the tunnel Wmax will be the same as in the table (the lengths of the sections L1 and L2 - as in the case of a circular cross-section, nothing changes here). If necessary, the height of the slotted tunnel h can be changed by simultaneously adjusting Wmin and Wmax so that the values of the cross-sectional area (h.Wmin, h.Wmax) remain unchanged.
I used a variant of the bass reflex with an hourglass-shaped tunnel, for example, when I was making a subwoofer for a home theater with a tuning frequency of 17 Hz. The estimated length of the tunnel turned out to be more than a meter, and by calculating the "hourglass", I was able to cut it almost in half, while there was no noise even at a power of about 100 watts. Hope it helps you too ...
Subwoofer enclosure - bass reflex (FI)
As part of the discussion of the choice of a subwoofer, consider such a case as a bass reflex.
The phase inverter, in contrast to, has a port with which it unfolds the phase of the signal from the rear side of the speaker, thus increasing the efficiency by 2 times.
The principle of operation of the bass reflex
What kind of music is the bass reflex suitable for?
features powerful and spacious bass, and in the region of the tuning frequency it has a hump (a significant increase in the sound volume).
An example of the frequency response of a phase inverter
Therefore, FI suitable for music, in which there is a lot of not fast bass, where low frequencies are the basis of compositions... Choose a bass reflex if you like dubstep, trip-hop, other slow electronics, rap, R&B, etc.
Note: the bass reflex setting is the frequency at which the peak occurs, it is regulated by changing the length and area of the port, as well as the ratio of the port volume to the volume of the body.
Which speaker is suitable for the bass reflex
To choose a subwoofer for a bass reflex, you need to start from. Usually, these data are in the documents, but if you do not have them, then the parameters can be found on the Internet.
In order to understand whether the speaker is suitable for FI, do not make tricky calculations. Divide the value on the value and if the answer is from 60 to 100, then such a sub will be optimal for the phase inverter.
For example - at the speaker SUNDOWN AUDIO E-12 V3 Fs = 32.4 Hz, and Qts = 0.37.
Fs / Qts = 32.4 / 0.37 = 87,6 - such a subwoofer is quite suitable for FI.
If the value for your speaker is outside the range of 60-100, it may be worth looking for another design for it using. Please note that the above table does not prohibit the use of enclosures for speakers that do not match meaning Fs / Qts. It shows options that will definitely work well.
Types of bass reflexes
Bass reflex port- the main element of the body, it can be round (pipe) or rectangular (slot).
Slotted port
Round port (pipe)
It is impossible to say with certainty which of these ports is better. They do what is more convenient or what they like best. The only moment that In sports(sound pressure competition) pipes are used more often, since with their use it is easier to change the setting of the bass reflex, by changing the length of the port.
We should also mention this type as a passive emitter. (more correct - passive reflector) there is the same phase inverter and the principle of its operation is the same. It is used in cases when the desired port for the FI does not suit the dimensions. In a passive radiator instead of a port used by speaker without magnet system.
How a passive radiator works
Advantages and disadvantages of FI
Pros:
- High efficiency (roughly - 2 times louder than ZA);
- Can give a lot of loud bass;
- Can be customized to suit your musical preferences.
Minuses:
- Large dimensions (compared to ZY);
- The relative complexity of the calculation.
Peculiarities
Materials (edit)
Material and assembly requirements are standard. The phase inverter box must be strong, sealed and vibration-free. Material - plywood or MDF from 18 mm. and thicker.
Please note that all wire entry channels, terminal blocks, etc. must be reliably sealed, internal partitions(port walls) should not have gaps.
Rounding the bass reflex port
If the slotted port is long and has turns, then stagnant zones may appear in the corners, to avoid this bends are smoothed out- as a result, the efficiency increases, since reduced resistance to air movement... It is quite difficult to determine the quality improvement by ear, but this solution works for the struggle for a high result in sound pressure.
Smoothing ports options
BassPort software, created specifically for calculating the parameters bass reflex... Capable of performing payment various types of ports: hourglass, funnel, round, round with flanges, slotted, etc. The BassPort software is also equipped with a calculator that allows you to preset the port already in the real subwoofer box.
The program is an invaluable tool for calculating and creating a bass reflex housing. Knowing the required volume for a specific speaker, and entering the required indicators, BassPort will calculate: how long the bass reflex port should have, indicate the air speed in the subwoofer, as well as the volume of air that it displaces.
Description of BassPort
The program has a simple and intuitive interface, all the necessary fields for data entry are clearly indicated. Let's consider the interface of this program.
The first thing that catches your eye is the value of the speed of sound, which by default is 344 meters per second. This field is editable. The next step in calculating the FI is to record the incoming data in the following windows:
- Bass reflex tuning frequency, indicated in Hz;
- Volume of the subwoofer box, liters;
- Speaker cone diameter (measured at the center of the cone corrugation);
- The number of woofers;
- Diffuser stroke (indicator in the speaker's passport);
- Number of ports in the case;
- Port section (circle or rectangle).
After entering all the necessary data, click recalculate. Then he presses the next button below, and we get a drawing of the future subwoofer.
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