How does a horn work? And why is it needed at all? (6 photos). Radio Liberty: how the main mouthpiece of Western propaganda works The mouthpiece doesn’t work

Definition of expansion loop

The theory of the conical horn was originally developed by Lord Rayleigh, but the first serious attempts to determine a practical formula for the exponential horn were not made until 1919 or even later. The basic formulas for the transmission of sound waves through a horn were given in modern terms by V. Salmon and others. Beranek provides graphs of the acoustic impedance in the throat of a horn as a function of frequency for several “infinite” horns of different profiles, but with the same throat cross-section. These data are shown in Fig. 1.

Rice. 1. Dependences of active and reactive acoustic impedances on frequency in the throat of horns of different circuits having an infinite length​

It can be shown that for optimal loading of a loudspeaker driver, the impedance at the throat of the horn must be fully active and also maintain its value over the operating frequency range. In other words, sound propagation must be a “function of power”. Having studied the curves in Fig. 1, it can be established that curves of exponential and hyperbolic profiles most closely satisfy these conditions.

The next condition that must be satisfied is minimal distortion in the throat of the horn caused by “air overload”. When a sound wave travels through air, a series of harmonics are created that distort the waveform. This occurs because if equal positive and negative changes in pressure act on a mass of air, the changes in volume will not be equal; the change in volume due to an increase in pressure will be less than due to an equal decrease in pressure. The rapid expansion and contraction of air caused by the propagation of sound waves occurs according to the adiabatic law, that is, there is no heat transfer. Thus, the relationship between pressure and volume is described by the formula:

Where:
p = pressure;
V = volume;
γ = adiabatic constant (approximately 1.4 for air under normal conditions).

Rice. 2. Relationship between pressure and volume of air in the case of adiabatic compression/expansion​

The mechanism of distortion occurrence is illustrated graphically in Fig. 2. With equal changes in volume, the change in pressure turns out to be different, which leads to distortions.​

If the horn were a cylindrical tube, the distortion would increase as the wave propagates towards the mouth.​

However, in the case of an expanding horn, the amplitude of the wave pressure decreases as it moves away from the throat. Therefore, for minimal distortion, the horn must expand sharply so that the pressure wave amplitude decreases as quickly as possible after the sound wave leaves the throat. From this point of view, it is obvious that parabolic and conical contours give the least distortion due to air overload, while a hyperbolic horn, on the contrary, will give the greatest distortion, because for an equal reduction in pressure the sound wave will need to travel a greater distance.​

Further study of Fig. Figure 1 shows that the acoustic impedance of a hyperbolic horn is within 10% of its limit value over a wider frequency range than that of an exponential horn. For this reason, a hyperbolic horn provides better conditions for matching the load with the driver.​

However, due to the significantly higher distortion in a hyperbolic horn, the exponential profile (or one of its derivatives) is chosen as the most satisfactory compromise between hyperbolic and conical contours.​

In applications where it is necessary to take advantage of long, slowly expanding horns without the accompanying high distortion, Olson recommends using several different exponential sections, the first of which (near the throat) should be short but expand very sharply to minimize distortion. Next comes a longer section with a lower expansion ratio, after which comes the main part of the horn, which expands very slowly. Klipsch also mentioned this technique in his article on the corner horn, calling it the "rubber throat."

The acoustic impedance of the mouth of each section is calculated to match the impedance of the throat of the next section. This method can calculate almost any relationship between impedances depending on frequency, however, due to the complexity of this procedure, additional calculation efforts are not always justified.​

Determination of the mouth area

Acoustic active and reactance impedances for an exponential horn are presented graphically in rice. 3. It can be seen that the resistance is completely reactive below the frequency, determined by the formula:

Where:
c = speed of sound;
m = expansion constant, which appears in the basic formula for the profile of an exponential horn.

Where:
Sx = area at distance x from the throat;
ST = throat area.

Rice. 3. Active and reactive acoustic impedance of an exponential horn​

Frequency fc, known as the cutoff frequency, is the lowest frequency at which a horn transmits acoustic power, so the expansion constant determines the lowest reproducible frequency of a particular horn. The expansion constant can be calculated for any selected cutoff frequency and the horn profile can then be constructed.​

The above statement is completely valid only for horns of infinite length. In horns, as in cylindrical pipes, wave fronts whose length exceeds the diameter of the mouth tend to be reflected in the opposite direction, resulting in interference with subsequent wave fronts. As well as for the throat of the horn, for the mouth the condition of the active nature of the resistance of the medium in the operating frequency range must be met.​

Beranek showed that in order for the radiation resistance at the mouth to be active, the condition C/λ > 1 must be satisfied, where C = the circumference of the mouth, and = λ = the wavelength at the lowest reproducible frequency. If the horn mouth is non-circular, the condition will be similar, but for an equivalent mouth area.​

That is, if C = 2πrm > λс, then:​

λc = wavelength at cutoff frequency;​

rm = mouth radius;​

Sm = mouth area.​

Thus, for a square horn, the condition must be ensured that the mouth area exceeds:

Hanna and Slepian examined the behavior of sound wave fronts at the mouth of a horn from various perspectives and concluded that minimal reflections were observed at a profile tilt of 45 (i.e., an inscribed angle of 90). This will happen if the circumference of the mouth is equal to the wavelength at the cutoff frequency. This also illustrates the importance of differentiating between the expansion constant value used to calculate exponential area expansion and the one used when drawing the actual profile. Graphs in Fig. 4 (according to Olson) illustrate the effect of shortening the horn length against the ideal value.

Rice. 4. Behavior of shortened horns. Reflections at the wellhead cause peaks and dips in the frequency response near the cutoff frequency​

When the orifice circumference becomes smaller than the wavelength, reflections at the orifice cause unwanted peaks and valleys in the frequency response around the lower cutoff frequency. Thus, if the size of the orifice in the design is very limited, then, as a rule, it is preferable to increase the lower cutoff frequency to a value corresponding to the size of the orifice, rather than end up with unevenness in the bass region, illustrated in Fig. 4.​

Flat and curved wave fronts

Until recently, it was assumed that successive wave fronts remained plane as they propagated through the horn. In a straight round pipe this is indeed the case: the wave front must be perpendicular to the axis and the walls (if the pulse front were approaching or moving away from the walls, the energy would be absorbed or emitted, respectively; on the other hand, a complex front consisting of the original wave and its reflections from the walls will be perpendicular to the walls). Thus, pulse edges transmitted through a cylindrical tube will be flat, while pulse edges transmitted through a conical horn will be spherical. It is clear that the wavefront emerging from an exponential horn will be curved to some extent, and that ordinary calculations made on the assumption that the wavefront is plane will be obviously erroneous. In practice, the actual lower cutoff frequency will differ slightly from the value obtained theoretically, although the error in the horn profile will not be excessive.

It would not be entirely correct to assume that the areas of successive fronts expand strictly according to an exponential law, since any chosen profile will itself determine the shape of the wave fronts, and in general this shape will change from the original one. Wilson assumed that the fronts were spherical, with their curvature varying from zero (flat front) at the throat of the horn. On this basis, he calculated a modified contour, which is inside the strictly exponential one and is very close to it. If, for example, you make a “truly exponential” horn using the papier-mâché method, then after “drying” the shape will become very close to the modified Wilson profile. However, his main statement that the fronts are spherical and change their curvature does not in any way mean that this is actually the case.

Tractor circuit

Voight, in his 1927 patent, was based on the simpler assumption that the shape of the wave fronts within the horn is spherical, with the radius of the sphere being constant throughout the propagation process. He justified this assumption by reasoning that if the front curvature increases from zero (a plane wave) at the throat to some value at the mouth, then points on the front on the axis will move at a higher speed than points near the walls of the horn. But, since the entire front must mix at the same speed, equal to the speed of sound, the shape of the front can only be spherical and of constant radius. This requires that the horn circuit be a tractrix.

Traxtrix = a flat curve drawn by a weight that is pulled on a rope, with the person pulling moving in a straight line that does not pass through the weight. This = not the curve of a chase method or the trajectory of a missile that tends to a fleeing target, as is often mistakenly believed. The length of the tractrix corresponding to the mouth with a circle Xc can be expressed in terms of the wavelength corresponding to the lower limit frequency:

Where y = radius

Equivalent exponent:

Both of these curves are shown in rice. 5

Rice. 5.Comparison of exponential contour and tractrix​

It can be seen that the tractrix has a dominant exponential component, which becomes less significant as it approaches the mouth. For the first 50% of the length, the exponential contour and the tractrix for the same cutoff frequency and throat area are virtually identical, after which the tractrix begins to expand much faster until it reaches the fully “opened” mouth (inscribed angle 180¦). Due to the complex nature of the formula, the best way to construct a tractor is graphical. The curve thus obtained, after some smoothing (to eliminate irregularities associated with the graphical method of construction), can be used to determine the ordinates of the horn contour points.​

While the tractor ends when the angle between the horn and the axis is 90¦ (inscribed = 180¦), the normal exponential continues to go to infinity in both directions. Thus, the “tractrix” horn turns out to be shorter than the exponential one with the throat and mouth equal in size.​

Efficiency​

The efficiency of an exponential horn is determined by a large number of parameters, which were comprehensively reviewed by Olson. Typical efficiency of bass horns reaches 50%, while mid- and high-frequency horns can have an efficiency of more than 10%. These figures look very advantageous against the background of bass reflexes (2-5%) and closed boxes (usually less than 1%). The exceptionally high efficiency of horn loudspeakers does not mean that their main advantage is the ability to use amplifiers at low power. For example, some amplifiers with Class B output stages with horns may, on the contrary, produce high levels of distortion, since such an amplifier will operate at low output levels when the level of step-type distortion will be relatively high.​

The fundamental advantage resulting from high sensitivity is that the amplitude of movement of the loudspeaker head diaphragm will be significantly less than for all other types of designs. Therefore, the effects caused by the nonlinearity of the magnetic field and suspension are sharply reduced, in addition, the diffuser is less prone to the occurrence of a band effect. In this way, the relatively high distortion inherent in the drivers is minimized, and since the horn itself does not introduce distortion, the sound produced is of very high quality.​

An additional benefit obtained from reducing the amplitude of the cone displacement is that certain types of intermodulation distortion resulting from changes in the volume of air between the cone and the horn throat can also be reduced to negligible amounts.​

Pre-horn camera setup

The cavity inevitably present between the speaker diaphragm and the throat of the horn plays an important role in the design of horn systems, since it can be used to limit the maximum reproducible frequency. The lower cutoff frequency can be set with fairly high accuracy based on the expansion coefficient of the horn in combination with the size of the mouth area. The upper frequency limit is more difficult to determine, since it depends on:

A) unequal distances between different parts of the diaphragm and throat of the horn;
b) internal reflections and diffraction effects inside the horn, especially if it is folded;
c) characteristics of the head itself in the high-frequency region;
d) the effectiveness of the cavity between the diaphragm and the throat, acting as a low-pass filter.
It can be shown that a cavity of a fixed volume represents an acoustic reactance of magnitude

Where:
Sp = diffuser area;
V = volume of the pre-horn chamber;
p = air density;
c = speed of sound;
f = frequency.

When the cavity is located between the diaphragm and the throat, it behaves as a capacitance that shunts the resistance of the throat itself, therefore, when choosing the correct parameters, the combination of the cavity and the throat works as a low-pass filter, the tuning frequency of which is determined by the equality of the complex resistances of the cavity and throat,

Where:
St = throat area;
f = required value of the upper limit frequency.
From here

The volume of the pre-horn chamber can now be designed in such a way as to ensure a roll-off of the response at high frequencies even before those values ​​when the difficult-to-determine effects (a) and (c) described above begin to appear.

An additional benefit obtained by using a pre-horn chamber configured to prevent mid- and high-frequency frequencies from passing into the bass horn is that these frequencies are much better reproduced from the opposite side of the cone, loaded onto a mid/high-frequency horn mounted on the front of the loudspeaker. .

A more detailed discussion of issues related to the practical determination of the upper and lower limits of the reproduced frequency band will be given below.

Acoustic design of the back side of the loudspeaker head

Above, an opinion was expressed about distortions caused by the nonlinearity of the processes of expansion and compression of air. This effect is further emphasized when the speaker is horn-loaded on only one side, since the throat acts as an active acoustic impedance only when the diaphragm moves in one (forward) direction. When the diaphragm moves in the opposite direction, it experiences significantly less resistance, as a result of which its displacement increases. The ideal way to eliminate such distortion is to load the diaphragm on both sides with the same horns, or use a bass horn driving the “rear” side of the speaker, and load the cone with a front mid/high horn at the front.

Rice. 6. The effect of the pre-horn chamber limiting high frequencies​

An alternative method used by many designers is to load the back of the cone with a closed compression chamber, which creates about the same resistance as the horn. Thus, the compression chamber reduces the effects of non-linearity from unequal load on different sides of the diaphragm, and also provides a more convenient load for the diaphragm = a closed chamber on the back side of the diffuser itself gives an inductive nature of resistance, which balances the capacitive resistance, which represents the throat of the horn on low frequencies.​

Klipsch states that the volume of the compression chamber is defined as the throat area multiplied by the speed of sound divided by 2L and the cutoff frequency. This is derived based on the following relationships:​

Compression chamber resistance

Where:
Sp = diaphragm area;
V = air chamber volume.
Throat resistance at cutoff frequency

Where St = throat area.

Equating these two expressions, we get:

However, some experts note that the use of a compression chamber detracts from the realism of the reproduced sound and insist on loading in the form of a horn on both sides of the diaphragm, or on a combination of a horn on one side and direct radiation on the other. In other words, the most realistic sound reproduction occurs when both sides of the diaphragm are allowed to radiate sound into space.

Conclusion

To summarize this part of the article, it should be noted that there are no universal formulas or rules in the design of horn loudspeakers. The main point of listing various alternative approaches is to encourage others to experiment in areas where results must largely be assessed subjectively through careful comparative listening a posteriori.

As Wilson wrote: There is no reason to believe that a horn, when placed in a housing, has the absolutely exact characteristics of any particular type of straight horn, be it exponential, hyperbolic, conical or tractrix, even if its dimensions taken as starting points were followed exactly during manufacture. Repeated changes in direction, combined with reflections, absorptions and internal resonances will always be the factors that lead to discrepancies in characteristics and negate any attempts to make a correct comparison. Each horn design must be evaluated by both objective measurements and subjective evaluation.

The next part of the article will talk about other aspects of horn design, as well as provide recommendations regarding the design of multi-way systems. Particular attention will be paid to the design of the low-frequency section, since the bass horn section is of the greatest practical value to professionals and car audio enthusiasts. This is especially true for those who take part in SPL competitions, where exceptionally high horn output is exactly what is required to win.

Part 2

The previous part of this article highlighted the physics behind how horns work and showed how, by following some basic rules, you can get horns to sound with stunning clarity and realism. However, and this is also obvious, if one is not prepared to build and operate extremely large and expensive structures, trying to reduce the size to a more acceptable one can simply lose many of the potential qualities of the horns. The following discussion focuses on the techniques adopted in the design of horn-shaped enclosures.

It has already been stated that the horn behaves like a transformer, converting acoustic energy from high pressure and low vibrational velocity in the throat area to low pressure and corresponding high velocity at the outlet of the mouth. Similar to an electrical transformer, in which electrical voltage and current correspond to acoustic pressure and velocity, the basic requirements of an acoustic horn are:

(a) the throat must be correctly matched to the signal source (loudspeaker head);
(b) the orifice must be properly matched to the air volume load in the listening room;
(c) the horn must operate within a certain range of input power and frequency.

There are four main parameters of a horn, namely = mouth area, throat area, expansion profile characteristics and length. Any three of them will determine the fourth, and therefore directly the characteristics of the horn. Once a non-circular cross-section and an axis other than straight are chosen, the problem becomes much more complex, and mathematical and physical concepts are no longer sufficient for design. However, the basic characteristics of even folded horns are largely determined by known acoustic principles, and the most effective design method is to start from these principles. Moreover, any deviation from the theory, if possible, should be scientifically substantiated.

Extension profile

The previous section discussed the most common horn shapes and showed that a design that gives an exponential increase in wave front area from throat to mouth provides the best compromise between extremely smooth hyperbola expansion (optimal speaker loading but excessive distortion at the throat) and fast expansion parabolic and conical horns (minimal distortion, but insufficient load for the driver).

Since the exact shape of the wave front within the horn is unknown, one will have to take as a starting point something between a modified Wilson exponential profile (close to strictly exponential) and a Voight tract rix (which is close to exponential at the beginning, but differs significantly from it at the mouth ). The choice of any particular circuit is largely a matter of personal preference, based primarily on one's own listening experience.

Mouth geometry

The mouth connects the horn itself to the surrounding space = the listening room. One of the most frequently cited disadvantages of horns is that they require a very large mouth area to reproduce full bass. To some extent this is true: you cannot get a double bass from a piccolo flute. However, there are many ways in which you can reduce the mouth area to a manageable size without sacrificing bass response.

As long as the sound waves travel inside the gradually enlarging horn, they do not encounter any inhomogeneities along the way. Obviously, unless the length and diameter of the horn are infinite, there comes a point when the sound wave leaves the mouth. Although the cutoff frequency of an exponential horn is determined by the expansion constant, the linearity of the acoustic impedance versus frequency is determined by the throat area, which (for a chosen throat area and expansion constant) determines the overall length of the horn. Strictly speaking, for there to be no heterogeneity, the mouth must have an infinitely large area. However, Olson showed that if the perimeter of the mouth is four times the wavelength at the lowest operating frequency, then the acoustic impedance of the mouth will not differ significantly from the case of an infinitely large horn.

The more important consequence is that if one accepts a small reduction in acoustic impedance (by 6dB), the perimeter of the orifice can be made equal to the wavelength at the cutoff frequency, i.e. the area of ​​the orifice will be equal to

Where λc = wavelength at cutoff frequency.

As the area decreases below this value, the nonlinearity of the acoustic impedance will increase.
These values ​​refer to the situation where the horn is in free space, i.e. the emission angle is 4π steradians. In practice, this situation never occurs. Even if the horn were placed at the center of an infinite plane, the radiation would occur only in half the space, that is, the solid angle would be 2π steradians; when located in the center of a wall, the angle will be π steradians, and in a corner formed by two walls and a floor, the mouth will radiate only in the region of π/2 steradians. The conclusion is that the minimum mouth area for a circular horn turns out to be

When emitting into a solid angle, there are 4π steradians, and this value can be halved each time the solid angle is divided by two. In this way, the orifice area can be reduced to a more manageable size. For example, a horn with a cutoff frequency of 56 Hz (wavelength 6.1 m) would require a mouth area of ​​3 square meters. meters in case of free space, 0.74 sq. meters when placed opposite a wall, and only 0.37 square meters. meters = in a corner, and the deviation of acoustic impedance will be less than 6dB.

Fig.8. Solid angles into which the horn radiates at different locations​

The situation, which is illustrated in rice. 8, can be compared to the mouth of a single horn placed at the intersection of eight rooms: four on one floor and four on the other. Even though the listener in each room will see only an eighth of the full horn area, the bass response will remain at the level of a “full size” horn. It's rare that anything comes for free, and those who choose a corner speaker placement to maximize bass range and use the smallest possible cabinet size will likely have to come to terms with the overtones that may result from such placement.​

When looking at the floor plan of a room with a corner horn, it is clear that the room itself provides a natural extension of the mouth of the horn. Many listeners note that shortened corner horns reproduce bass notes well below the theoretical limit due to the size of the mouth. This encourages the user to reduce the mouth area beyond the previously established 3dB limit, and instead rely on placement directly in the corner to “virtually” increase the area and length of the horn. But this method cannot be recommended because, although the response in the bass region is indeed preserved, close listening reveals that in the region of the first two octaves above the cutoff frequency there is an unevenness that often negates the realism inherent in horns. Therefore, in cases where cabinet size is limited, a properly designed horn with a higher cutoff frequency, say 80 Hz, is recommended. It will provide greater linearity and listening pleasure than a shortened horn, where the expansion constant is chosen based on a cutoff frequency of 40 Hz, but the length is limited so that the mouth area corresponds to a cutoff frequency of 80 Hz.​

Rice. 9. Distortion caused by air congestion in the throat​

Most horns for home use have a rectangular cross-section for ease and cheapness of manufacture. The previous comments regarding circular horns also apply to rectangular horns, although it is clear that at corners the wave front must behave in a more complex manner, and therefore the effective area in the case of a rectangular cross-section is slightly reduced. Provided the aspect ratio of the mouth does not exceed 4:3, rectangular horns can produce good results.

Tabular design data is provided for both rectangular and circular horns, calculated for angular (π/2 steradian) locations, as well as near a wall (π steradian).

Throat geometry

The throat of the horn serves to transmit wave fronts from the loudspeaker, which are ideally flat in the throat, directly into the horn. It was shown above that the horn is an acoustic transformer, converting acoustic radiation with high pressure and low vibrational velocity in the throat into low pressure and high vibrational velocity at the mouth. The obvious advantage of high pressure and, accordingly, low oscillatory speed at the mouth is that at low speed the amplitude of the diffuser displacement is reduced, which in turn reduces distortions caused by non-linearity of the magnetic field and suspension. One of the ways to increase the pressure, as well as the greatest flattening of the shape of the sound wave front, is to choose a throat area that is significantly smaller than the area of ​​the loudspeaker diffuser. Tests carried out on many loudspeakers show that the effective cone area is approximately 70% of the radiating area of ​​the cone, that is, a loudspeaker cone shaped as a truncated cone has the same output as a flat cone with 70% of the cone area. area of ​​the cone diffuser.

There are many reasons why modern loudspeaker cones are not made flat. One of the undesirable consequences of using cone diffusers is that the waves they emit have a non-planar shape. However, it has been empirically established that with a throat area of ​​1/4 to 1/2 the effective area of ​​the diffuser, it is possible to ensure satisfactory matching between the loudspeaker and the horn, as well as to ensure an approximately flat shape of sound waves if their lengths significantly exceed the dimensions of the throat. It should be emphasized that for higher frequencies, when the wavelength is comparable to the physical dimensions of the loudspeaker cone, the throat area must be chosen of the same size; Moreover, to eliminate standing waves, the horn must have a circular cross-section, at least in the throat area.

The phenomenon of air overload distortion is caused by the nonlinear relationship between the pressure and volume of air in the throat of the horn due to the fact that the process of expansion and contraction occurs according to the adiabatic law. Beranek derived the relations for the second harmonic coefficient at the throat of an infinite exponential horn:

% 2nd harmonic = 1.73(f / fc)It x 10-2

Where
f = frequency
fc = cutoff frequency
It = specific power (watts/square inch) at the throat.

This expression also gives values ​​close to the truth for horns of finite length, because distortion mainly occurs near the throat. This expression is presented graphically in Fig. 9, from where the throat area can be determined for the selected power value and distortion factor.

It is important to understand that the acoustic power emitted by musical instruments is extremely low, and that the higher the frequency, the lower the acoustic power required to produce the same subjective loudness as perceived by the human ear. With the exception of large symphony orchestras and organs (which are generally futile to try to reproduce at home anywhere near their normal volume level), acoustic power levels are extremely low. If, say, you set the value to 3 W at 1% distortion for the cutoff frequency, then at a frequency four times higher this will give values ​​of 0.05 W and 0.5% distortion, which is quite enough for normal everyday listening.

The above power and distortion proposals according to Fig. 9 gives a throat area of ​​approximately 10 sq. cm, which is not bad for a 3 1/2-inch speaker, which has an effective area of ​​43 sq. cm (its quarter is just over 10 sq. cm). Of course, if the throat area is increased, as is the case with larger loudspeakers, then the permissible power for a given level of distortion will also increase.

By establishing values ​​for the throat area, mouth area, and expansion constant, the length of the horn (and therefore its area at any point) can be calculated mathematically or graphically.

Horn as a filter

The previous sections show how a horn can act as a bandpass filter = the lower cutoff frequency is determined by the spreading factor and the high cutoff = the volume of the chamber between the loudspeaker and the throat. It is important that in this frequency band the horn characteristics are very linear. In addition, by carefully choosing the lower cut-off frequency and throat area, taking into account the future location, it can be ensured that the non-linearity and distortion generated by the horn at low frequencies will be at a very low level.

At higher frequencies, approximately four times the cutoff frequency, an increase in the amplitude of nonlinear distortion within the horn becomes apparent due to internal reflections and standing waves. Nonlinearities will be even higher if the material from which the horn is made tends to resonate, and also in the case of folded horns, where the wavefronts at higher frequencies are distorted into bends. In fact, there is a certain limit above which the use of a folded horn becomes undesirable: there should be no bends beyond the point at which the wavelength of the highest reproducible frequency exceeds 0.6 of the current diameter. This limitation will be discussed in more detail during the discussion of folding methods, but the practical limit to the highest frequency that a horn can accurately reproduce is now clear.

A further limitation becomes apparent from the graph of throat distortion versus frequency ( rice. 9). As frequency increases, the percentage of throat distortion for a given power will also increase, and although it is known that in most complex musical sounds the energy level decreases with increasing frequency, still, beyond a certain frequency, throat distortion will become unacceptable.

A simple but very adequate rule is usually used: “on the fingers” = the horn should not reproduce more than four octaves above its lowest cutoff frequency. However, purists sometimes prefer to limit the range to only three octaves to guarantee even lower distortion levels.

Complete multi-horn system

The maximum frequency range that a high-quality full-range loudspeaker can reproduce is approximately 9 octaves, that is, from 40 Hz to 20 kHz. Clearly, for the reasons noted above, this is too wide a range for a single horn loudspeaker. But it can be conveniently divided into three sub-bands, that is: 40 Hz - 320 Hz, 320 Hz - 2.5 kHz and 2.5 kHz = 20 kHz. In practice, a 10% frequency overlap between the subbands should be allowed to avoid any anomalies in the interface areas. A four-horn system can be used to reproduce an even wider range of frequencies.

It is worth considering a more modest option. If the frequency band is limited to 80 Hz to 18 kHz and we consider a system with two horns, each operating in a bandwidth of just under four octaves, then the frequency sub-bands become 80 Hz to 1.2 kHz and 1.2 kHz to 18 kHz . Again, it is necessary to ensure 10% overlap of subbands in frequency.

The big appeal of this two-horn system is that a single loudspeaker is required: the bass horn will drive the rear of the cone, while the front will drive the mid/high frequency horn. This eliminates interference and diffraction effects caused by the basket and magnet. It has already been emphasized that at higher frequencies the throat of the horn must closely match the dimensions of the loudspeaker, and this is especially attractive in the case of a double-cone driver. A pre-horn chamber is used to prevent high-frequency sound waves from entering the bass horn.

Maximalist listeners might argue that the frequency range from 80 Hz to 18 kHz is insufficient. However, from experience, the linear reproduction of this range with little distortion, as well as the presence effect provided by horn systems, make such a mini-horn system very attractive in terms of sound quality, as well as relative simplicity and low cost.

Once a multiband approach is adopted, a variety of crossover frequency options immediately arise. For example, 320 Hz and 2.5 kHz in the case of a three-way system, and 1.2 kHz for a two-horn system. It is essential that the radiation from a pair of horns in the region of the crossover frequency must be phase matched, otherwise the amplitude-frequency response in these regions will be uneven. This is especially true for the bass horn, since it is folded in such a way that its mouth is located adjacent to the rest of the horns (folding mid- and high-frequency horns is usually not necessary, and indeed generally undesirable). This requirement imposes an additional limitation on the length of the horn, which until now has been determined solely on the basis of the throat and mouth diameters, as well as the expansion constant. It is now evident that the length of the lower frequency horn of each pair must be either an even or an odd number of wavelengths at the crossover frequency, depending on whether the drivers of the two horns are respectively switched in phase or out of phase.

Thus, if separate loudspeakers are used and the voice coils are in phase, then the total length of the horn from the loudspeaker to the orifice plane must be an even number of half-wavelengths. In contrast, if a single loudspeaker loaded by two horns is used, the emissions from the front and rear sides of the cone will be out of phase, so the total length of the two horns must equal an odd number of half-wavelengths.

In practice, the overall dimensions of the design will be determined mainly by the lower frequency horn, since it is significantly longer.

Rolling, enclosures and indoor placement

Until recently, the discussion was limited to consideration of ideal horns: circular, rectilinear, and made of a very stiff material. Although the typical dimensions of practical horns have not yet been formally calculated, it will be clear from numerous tables and diagrams that the dimensions of bass horns will almost certainly be too large to fit comfortably in an ordinary room. Therefore, the design procedure must include
two additional stages have been added: bringing the horn cross-section to a rectangular one, as well as collapsing it to a compact size.

Rayleigh showed that bends in pipes of constant cross-section will have no effect on transmitted sounds if the wavelength is greater than the diameter, but any mutual vibrations occurring within the pipe will have a fundamental frequency corresponding to a wavelength equal to 1.7 times the diameter of the pipe. Wilson formulated three basic rules for folded horns:

The wave fronts should not bend across the horn;
the diameter of the horn (width for a rectangular one) must be less than 0.6 of the wavelength of the lowest frequency sounds that will pass through this horn;
The wave front must pass through rounded bends to maintain its shape.

As soon as there is a deviation from straightness and a circular cross-section, all of the above-described scientific principles of design cease to be the ultimate truth and become rather advisory in nature; although the three basic rules indicated above, in combination with a competent choice of the appropriate (primarily in terms of rigidity) material for construction, provide very decent results.

The wrapping method, which involves bending the wave front in different planes, is extremely difficult to implement in practice, so its use is preferably avoided by using wrapping in only one plane. The requirement to accelerate the wave front around a bend to maintain its shape is difficult to achieve when more than one bend is present, since it requires the rectangular section before the bend to become trapezoidal immediately around the bend, after which it returns to a rectangular section of a different shape (and area) . In reality, for horns that bend repeatedly, this is impractical and, moreover, unnecessary, because subsequent bends correct the waveform. But if there is only one bend, this approach may be quite applicable.

Rice. 10. Methods for folding horns:​

(a) Olson; (b) Mass; (c) Lowther; (d) Newcomb; (f) Klipsch.​

The Patent Office records of folded horn designs filed during the 1920s and 1930s are a wonderful monument to the ingenuity of loudspeaker designers and are truly fascinating to study.

Rice. 10 illustrates several of the most well-known folding methods. Based on the limitation on the width of the horn, which in bending should not be more than 0.6 of the longest wavelength of the transmitted sound, it is initially assumed that folding can only be used at the very beginning of the horn during the first few tens of centimeters of its length; from a certain point the width reaches the above limit. However, this limitation can be overcome in the following way: after each point where a width limitation occurs, the horn bifurcates into two identical tunnels. This design is called bifurcating (branched, forked). Thus, the mouth of the horn may consist, for example, of four equal parts, connected for convenience and to preserve sound realism. Each of these four “quarter horns” can be folded much closer to the mouth than in the original case of a single large horn. Rayleigh in Art. 264 showed that a bifurcating tunnel has no effect on sound transmission if the lengths of each of the two parts are equal, and also if the sum of their areas at the corresponding points is equal to the area of ​​the original tunnel.

In many cases, the front side of a loudspeaker whose back side is horn-loaded will be physically located in close proximity to the mouth of that horn. This raises the concern that some frequencies will be suppressed due to interference between two signals emitted out of phase. However, since the direct radiation from the front of the cone is only a few percent of the radiation from the horn, the amount of such suppression will be negligible.

Frequency processing

As has already been shown, each horn operates as an acoustic bandpass filter, the lower cutoff frequency of which is determined by the expansion constant, and the upper cutoff frequency is determined by the volume of the pre-horn chamber. However, there are important reasons why a wideband audio signal should not be applied directly to all horns, regardless of the operating bandwidth of each. Study of the low-frequency part of the spectrum on rice. 3(see previous part of the article) shows that below the cutoff frequency there is no active component in the acoustic load provided by the horn. Thus, any signals below the cut-off frequency will cause excessive displacement of the speaker cone, the magnitude of which will be limited only by mechanical and electromagnetic factors. Excessive movement means the speaker is operating outside its linear range. This can cause high intermodulation as well as other types of nonlinear distortion. At the higher end of the frequency spectrum, signals with excessive power can also cause distortion due to the interaction between the pre-horn chamber and the throat. It is therefore advantageous to limit the bandwidth of the electrical signal arriving at each loudspeaker to match the acoustic bandwidth of the corresponding horn.

Rice. eleven. Diagram of an active filter chain. Values​

capacitances and resistances of the elements are given in the appendix​

Although most multi-band commercial systems use passive LC crossover chains between the power amplifier and loudspeaker to split the frequency band reaching each speaker, careful comparative listening shows that these devices clearly introduce additional dullness or loss of luster to the resulting sound. There are many explanations and assumptions about the causes of such phenomena; The most likely of these is the loss of direct coupling to the amplifier output, accompanied by a significant reduction in the amount of electrical damping provided by the low output impedance of the amplifier.​

Another well-known approach is to divide the input signal frequency band at a low level, then use a separate power amplifier for each frequency range, directly coupled to its loudspeaker. The crossover block consists of three (or four) frequency-dependent channels connected in parallel, including active Sallen-Key filters that provide the specified characteristics of the low- and high-pass filters. Each channel uses a slight gain adjustment to account for the inevitable differences in sensitivity of each speaker/horn combination. Active filters provide the characteristic of a 2nd order Butterworth filter, which seems to have the least severe effects in the crossover frequency region. (Any filter chain inevitably causes phase shifts, the effect of which on transient processes leads to noticeable differences in the nature of their sound.)​

Thus, in general, in addition to the acoustic crossover provided by the horn itself, an electrical one, in one form or another, is also needed. An exception occurs when a single loudspeaker is loaded onto two horns: one on the front side of the diffuser, the other on the rear. In this situation, some compromise will have to be made regarding the acceptable level of distortion and system bandwidth.​

Directional Horns​

The article has already noted the exceptional ability of the horn to radiate wave fronts that are almost flat at its mouth. However, it is sometimes desirable for the wavefront to have different directivity characteristics in the vertical and horizontal planes, especially if mid- and high-frequency horns are used in stereo systems. It is often necessary to expand the radiation pattern in the vertical plane, while maintaining as much as possible the characteristic of a point source in the horizontal plane. To achieve this, there are many different methods based on the phenomena of diffraction and refraction that occur at the mouth of the horn with the relatively short wavelengths of sound waves (a few inches or less) that these high-frequency horns deal with.​

The design and manufacture of horns using the above effects is beyond the scope of this article, and quite possibly beyond the capabilities of most amateur designers. Those interested may refer to the publications of Smith, Winslow, and the corresponding chapters of the monographs of Olson and Cohen.​

In the design of his high-frequency horn described by Klipsch, the length/width ratio of the rectangular mouth takes on a value of more than 4:1. Optimal measurements, length-to-width ratio, and proportional expansion along the long and short axis depend on a number of complex factors. It was, however, found that good practical results are obtained with an aspect ratio of the mouth from 2:1 to 4:1, while the opening along the axes occurs in a similar ratio.​

Design process: step by step​

The previous sections examined in some detail the basic theory of horn design, and also highlighted the essential aspects of designing different types of horns that can cover the full audio range. In the final part, as an illustration, the minihorn project will be examined in detail.​

Since all horns are designed with slightly different requirements, many readers will likely want to change the specifications more or less to suit their needs. Therefore, project information is presented in tabular form so that it is possible to calculate horns for the widest range of applications.​

Bass Horn Calculation

First of all, the bass horn is designed, starting from the mouth.

Tables 1, 2, and 3 indicate the relationships between the minimum frequency and mouth sizes for horns placed in free space (solid angle 4π steradians), against a wall (π steradians), and in a corner (π /2 steradians). In Table 1, the speed of sound is assumed to be 343 m/s, and the perimeter of the mouth = equal to the wavelength. The orifice areas in Tables 2 and 3 are 1/4 and 1/8 of the corresponding free space orifice area, and the measurements for round, square and rectangular orifices are derived from these areas. It is tempting to reduce the area of ​​square and rectangular horns to ensure that the perimeter is equal to the wavelength (corresponding to a location near a wall or in a corner), but this is not recommended. However, the shorter side of a rectangular horn is obtained in this way (that is, a square horn with sides of this size will have the corresponding perimeter).

Once the throat dimensions have been established, the throat can be calculated based on the selected loudspeaker. Table 5 offers throat area options for five common speaker sizes. In some projects, loudspeaker selection will be based on overall size (the smallest loudspeaker will have the longest horn length) and whether the loudspeaker needs to reproduce all other frequencies in addition to bass using two separate horns on either side . The effective area of ​​the diffuser was taken to be 0.7 of its area calculated from the average diameter, and the throat area was taken to be 0.3 of the effective area. The indicated dimensions should give quite acceptable results, although, of course, there is ample scope for experimentation.​

Knowing the areas of the throat and mouth, using Tables 6 and 7, one can determine the total lengths of the horns of the true exponential and tractrix profiles located near the wall and in the corner. These lengths are given for different expansion constants as well as several cutoff frequencies (as listed in Table 1). To ensure linearity of the horn over the entire operating range, a correction factor of 1.2 is used when calculating the spreading factor in Table 4 for the lower cut-off frequency. Thus, the expansion constant

where c = speed of sound (343 m/s), and f = lowest reproducible frequency.​

N.B. Tractorx profile horn lengths listed in Tables 6 and 7 = approximate, based on the full Traxtrix contour calculated from the expansion constant and orifice radius determined from the lowest reproducible frequency.​

Mid/High Frequency Horn Design​

Now let's turn our attention to horns designed to reproduce mid and high frequencies. The perimeter of the mouth should not be less than the wavelength at the lowest operating frequency; in practice, to obtain good results, the perimeter is chosen to be one and a half times larger. Table 8 is based on this factor and gives the recommended minimum orifice dimensions for the free space horn case. Unlike the case of a bass horn, at higher frequencies it is safest to assume that the horn is loaded into free space. The effects of diffraction and reflection at short wavelengths are not so pronounced as to consider placing the horn against a wall or in a corner, and it is for this reason that the perimeter is taken to be 1.5 times the wavelength at the lowest operating frequency. The dimensions of the square and rectangular horns were obtained in the same manner as those in Tables 2 and 3. The throat dimensions of the mid- and high-frequency horns should correspond to the size of the loudspeaker itself, and can be determined from the average diameter and area of ​​the selected loudspeaker, as shown in Table 7. Tables 9 and 10 give the expansion constant and length of exponential horns suitable for horns with the throat and mouth dimensions given in Tables 5 and 8 respectively.​

Combining multi-way horns​

It was previously emphasized that the radiation from the mouths of each pair of horns at their common crossover frequency must be in phase. Taking into account that the mouths of all horns will be in the same plane, the total length of each pair of horns must be a multiple of half the wavelength at the crossover frequency. If the speakers in both horns are in phase, the total length must be a multiple of an even number of half-wavelengths; if the radiation at the throats is out of phase (as in the case of a single speaker loaded on both sides by two horns), then the total length must be equal to an odd number of half-wavelengths. If necessary, to ensure optimal matching conditions, the crossover frequency can be changed within small limits, followed by modification of a higher frequency horn.​

Final Project​

In general, the bass horn will be collapsed. The original plan was to provide a table indicating the maximum permissible horn length, beyond which there should be no bending, because the horn diameter became equal to 0.6 of the wavelength of the lowest frequency sound reproduced. However, testing has shown that up to frequencies five times higher than the cutoff frequency (that is, in a bandwidth equal to four octaves), if the horn is located in a corner, this limitation is not achieved (due to the small size of the mouth); for the case of location near a wall, the limitation occurs in the region of 92-95% of the total length of the exponential horn. Therefore, it can be assumed that if a horn placed near a wall is not folded within the last 10% of its length, then the problem of mutual reflections will not arise at all.

Finally, the pre-horn chamber for a low-frequency horn must be calculated using the formula already given, while it is necessary to take into account the loss of chamber volume due to the basket, diffuser holder and magnet of the loudspeaker itself.

Values ​​of the circuit elements of a tunable active bandpass filter for use in a three-band system (see Fig. 11). All microcircuits, for example, are of the N5741V type. R5 - 10 kOhm, R6 - 22 kOhm, R7 - 100 kOhm.

Low pass filter

R1 and R2 are a series-connected constant resistor with a resistance of 12 kOhm and a variable resistor with a resistance of 47 kOhm.​

High pass filter​

R3 is a series-connected constant resistor with a resistance of 6.8 kOhm, and a variable resistor with a resistance of 22 kOhm. R4 = a constant resistor with a resistance of 12 kOhm and a variable resistor with a resistance of 47 kOhm connected in series.​