Horn Loading the Faital Pro 3FE2216
This is not a
suggestion to horn load this cone speaker or any other cone speaker other than a woofer. At higher frequencies where the wavelengths
get very short, resonances and cancellations can happen at the
throat. Back cavity and front cavity volumes can be difficult to
obtain, especially the front cavity. Cone speakers don't easily
lend themselves to the addition of phase plugs to correct this and will
usually result in a ring shaped throat, complicating things further as
the phase plug has to me rigidly installed. The addition of the
phase plug will require increasing the outer diameter of the throat,
hence the ring throat.
The phase plug will usually protrude into the horn. This will affect the initial part of the horn which will have to have a diameter equal to the outer diameter of the ring. Thus, the horn walls will have to expand due to the presence of the tapered plug. That part of the horn will then have a ring taper, the area of which will expand according to that of the flare rate constant, b, which is determined by the low frequency cutoff, f_{c} of the horn. This was done in the early seventies when a Wharfedale Super 3 tweeter was horn loaded with an f_{c} of 400hz. The need for the plug was circumvented by using an initial taper/adapter to go from a circular throat to a rectangular throat of the same aspect ratio of the horn and also maintaining an exponential flare during this length, which was 3/4 inch. Of course, there was no way then to measure actual frequency response but they sure did sound good in conjunction with the horn loaded W15. There are two pages on that for the curious. horng and hornh There is also the handicap of working with wood as, to achieve the above, a lathe is required and the use of very hard woods as hard woods can be machined to better tolerances than soft woods; soft woods will absorb high frequency energy. The pressure exerted on a horn loaded diaphragm is greater than that of a direct radiator. The horn is an acoustical transformer, matching the high impedance imposed at the front of the diaphragm to the much lower impedance of the air in the room. Hence their efficiency. It's much the same as connecting an 4 ohm speaker to the 8 ohm tap of an amplifier; the power transfer is compromised to the tune of 50%. 
Symbols used
A_{m} mouth area; A_{t} or S_{t} throat area; b flare rate constant; c velocity of sound in air at 75^{o}F (13608 i/s, 345.6 m/s); e base of natural logarithms, 2.71828;
f_{c} horn low frequency cutoff; f_{hc} upper rolloff corner frequency 3dB f_{s }open air resonance of speaker;
l length along axis of horn in which the cross sectional area doubles; Q_{ts} total driver Q at resonance, f_{s};
S_{d} effective piston area of the speaker (projected); V_{as} volume of air with same compliance as the moving system of the speaker;
V_{b} rear cavity volume; V_{fc} volume of the front cavity
NOTE. For decades, I've used A_{t} instead of S_{t} for the throat area. So, from this point on I'll use A_{t}
The velocity of sound, c, was taken at 75^{o}F because that seems to reflect a more common temperature found in most homes
A friend had asked me about horns, specifically midrange horns. I was hesitant to go into such a topic due to the complexity of horn design, not to mention the physical size of a bass horn.^{1} His thoughts were to buy a horn and screw in a horn driver. Making a horn would be quite a task, especially a circular one. For the most part, buying a horn could produce satisfactory results, provided a suitable driver could be found. Most drivers are designed for professional use which doesn't necessarily work in a high fidelity system. These drivers have a substantial rise in the band covering about 1khz to 4khz. If the driver is to be used in a 3way system, that problem can easily be overcome with a bandpass section in a 3way crossover. The driver would have to be padded down to match the sensitivity of the woofer, or more accurately, the insensitivity of the woofer. A horn loaded tweeter or even a dome, planar or ribbon tweeter are available with sensitivities above 90db to match a horn loaded midrange. A good horn driver for midrange would have to cover the passband from, at least 400hz to 4khz. Finding a driver that could go to 400hz proves close to impossible unless one was ready to spend a few hundred dollars or more for such a driver. Such drivers will have a sensitivity of well over 100dB and as high as 118dB. (JBL anyone?) The best that could be found, cost not considered, would cover from 400hz/500hz to about 12khz and only a few were to be found. In my possession is a pair of ElectroVoice 1824M drivers that do well to 400hz but are now rare. This proved impractical. The next best option was to find a suitable 3 inch to 5 inch diameter driver that could cover this passband. There are oodles of them to be found. However, the largest horn throat to be found has a 2 inch diameter throat. Couplers are available to bring that down from 2 inches to 1.4 inches diameter and from 1.4 inch diameter to to 1 inch diameter. The trick now was to find such a driver that could couple to a throat of the above mentioned diameters. After testing several of each of 5" and 4" speakers, it became evident that none would work with the above mentioned throat diameters. Also, may just wouldn't perform well to 10khz, let alone 16khz. This left 3" and 3.5", reducing the search considerably. After a couple of days searching for a suitable loudspeaker, one was found, the Faital 3FE22 which comes in 4W, 8W and 16W For this design, the 16W was used due to availability. In a way, this may be a better choice due to the fact that less power would be transferred to it, 3dB less than if it were 8W. This speaker also has quite a linear response from about 150hz to above16khz, allowing a 2way system to be used. This speaker is also available with a ceramic magnet, designated as 3FE25 but the larger magnet would be more difficult to front load. Now, for the technical stuff. First, to find out if any speaker will work through a 2". 1.4" or 1" throat. Well, Don Keele of Klipsch and Associates, in 1977, submitted a paper to the AES (Audio Engineering Society) which was titled, "Low Frequency Horn Design using Thiele/Small Driver Parameters" In it, he gives the throat area, as S_{t}=2pf_{s}Q_{ts}V_{as}/c (Eq1) where f_{s} is the free air resonance, Q_{ts} is the total Q, V_{as} is the volume of air of equal compliance to that of the speaker and c is the velocity of sound, 345.6 m/sec (13608 ips) at 75^{o}F Since these parameters are readily available, throat area is found in short order, depending, of course on how many speakers are being checked.

The TS parameters in use here are: f_{s}=110hz; Q_{ts}=0.56; V_{as}=69.12 cu in; f_{c}=500hz (f_{c} is the low frequency cutoff of the horn)
The throat area required for this 3FE22 is 1.96 sq.in. The smaller circular throat area of the 1.4 inch to 2 inch adapter is1.54 sq.in. This is 79% of the required throat area but that was accepted, hoping for the best. Besides, it was the closest one to be found and this is only an experiment to determine if horn loading such a speaker is
Keele, in that same paper, gives the volume of the rear chamber as V_{b}=V_{as} / [(f_{c}/f_{s}Q_{ts})1] (Eq2) and if V_{as}>>V_{b} then V_{b}=V_{as}f_{s}Q_{ts}/f_{c} (Eq3)
Eq 2 comes out to 9.71 cubic inches and eq 3 comes out to 8.52 cubic inches. Since V_{as}>>V_{b} the value obtained from eq 3 was used. This is also corroborated by Klipsch in his paper titled "A Low Frequency Horn of Small Dimensions", submitted to the JASA (Journal of the Acoustical Society of America), Oct. 1941 in which he gives the back chamber volume as V_{b}=2.9A_{t}l, (Eq4 ) where A_{t} is the throat area and l is the length along the horn axis in which the crosssectional area doubles (for an exponential horn) For clarification, it should be noted here that "l" is a lower case letter "L", pronounced EL.
Now, to find the length in which the area doubles. We'll begin with the mouth area. C.R.Hanna, in his paper, "Loudspeakers of High Efficiency and Load Capacity", presented to the AIEE (American Institute of Electrical Engineers, Feb. 1928, states that the horn mouth diameter, D be equal to 4/b, D=4/b (Eq5) expressed also as D=l/p (Eq6). I've always used the equation D=l/2, (Eq7), which gives a slightly larger mouth. Using (Eq7), we get a diameter of 13.6 inches which equates to a mouth area of 145 sq.in. Calculating the flare rate constant, b, in inverse inches, Hanna, in that same paper mentioned above gives b=4pf_{c}/c (Eq8)
Hanna also gives the general equation for an exponential horn as A_{x}/A_{t}=e^{bx} (Eq9) The total horn length is then found by this equation. Dividing A_{x} which is now the mouth area, A_{m} by A_{t} and taking the natural log of that number, we get the value of bx. The total horn length, x, is found by dividing that last obtained result by b, which for this horn is 9.85 inches.
The length, l, in which the cross sectional area doubles is found by setting e^{bx} equal to 2 and again, taking the natural log of both sides of the equation, we get bx=0.693147. Dividing this by b, we get the area doubling length, which, for this horn is 1.5 inches.
Referring back to eq4, the back chamber volume, V_{b} as given by Klipsch is V_{b}=2.9A_{t}l Since we now have the throat area and the length in which the area doubles, eq4 gives a volume of 8.537 cubic inches, which is very close to the result obtained by Keele's eq3, which is 8.52 cubic inches.
The front cavity volume, V_{fc} as given by Keele in his 1977 paper is f_{hc} = 2Q_{ts}f_{s}V_{as} / V_{fc } (Eq10) which, when solved for V_{fc} is V_{fc} = 2Q_{ts}f_{s}V_{as} / f_{hc} (Eq11)
Setting f_{hc}=15000hz, we get a front cavity volume of 0.567 cubic inches, about 3 times smaller that the actual volume here. Solving eq10 for the upper limit, we get 5677hz. That was disappointing to say the least but it was decided to accept that and see what happens. The result was astonishing and taking a SWAG, (Scientific Wild Ass Guess), it was thought that the beaming of high frequencies was the reason. SEE photo 9 for details.
PHOTO 8
This is the current rear chamber. The red rectangle is the front baffle, onto which is front loaded the speaker. The blue outlined rectangle above it is the spacer. The speaker gasket rises above this by about 0.5mm to provide a seal to the horn flange, which is the top rectangle 

PHOTO 9
The red area is the proposed additional spacer onto which would be placed a ring, shown by the inverted triangles. the blue area is the same size as the horn throat The idea was to decrease the front cavity volume but that idea went to hell in the proverbial handcart, PDQ. (Pretty Darn Quick) The volume would have been reduced by a about only 20% because the additional volume of the blue throat would have partially offset the volume reduction of the ring A phase plug (green) can be added to further reduce the volume but this would have to be designed in conjunction with the red ring encircling the throat to obtain the correct throat area, meaning a circular ring throat. This method was used in some ring radiators. One that comes to mind because I had a pair and swapped them with a musician for the ElectroVoice T350 units is the JBL075 tweeter (aka ring radiator) Considering this, it was deemed to be a waste of effort. It was decided to carry on with the original design and let my ears do the critique followed by the CLIO, in that order so as not to be biased by a response curve. 

PHOTO 10
The final crossover. A second order was considered but this worked very well. Besides, it was easier to juggle the component values. Juggle?? After seeing the initial horn response with a wide peak of several dB between 400hz and 800hz, and knowing the woofer output would exacerbate that, it was decided to spread the crossover points (Juggle) After several attempts, a low pass of 5mh in the woofer and a high pass of 5.6uf in the upper section, the red response curve of fig 1 was obtained. Initially, the system honked. So the crossover points were spread apart, first starting with the woofer, then the tweeter. When the honking stopped, a response curve was run and tweaked a little further in an attempt to smooth the response between 500 and 2khz. See fig 3 This didn't have the effect I desired so the tweeter polarity was reversed, going from the black trace to the red one in fig 1 The grey trace at the bottom is the THD, less than 1% above 100hz 

PHOTO 11
The added flange which made no difference The purpose of this is described in the section titled "Mouth Reflection Dip" at the bottom of this page

The CLIO Results
Conclusion OK. So it looks good on paper but how does it sound and is there much, if any, difference between direct radiating and horn loaded? Keep in mind that the following may contain some subjectivity regardless of how much effort is put into maintaining objectivity A higher output than that of the direct radiator was expected but not realized.^{2} This is thought to be due to the very wide bandwidth for which the horn was designed. One way to determine the suitability of a speaker for horn loading is by the EBP, )Efficiency Bandwidth Product) which is stated as, EBP=f_{s}/Q_{es} (Eq12) The general consensus is that for sealed enclosures, EBP should be less then or equal to 50 and for vented enclosures, greater that or equal to 100. This particular unit has an EBP of 17, making it suitable for a horn since a horn uses a sealed enclosure behind the speaker, V_{b} Also, the slightly smaller throat will have a negative effect on the lower part of the band, imposing a heavier load on the diaphragm. The response in fig 3 was obtained without the throat adapter, using the 2 inch throat of the horn. Initially, it was thought that this would negatively affect the extreme high end but that proved to be incorrect, (red trace, fig.1). The bump in the band between 400hz and 1300hz (fig 3) was flattened by spreading the crossover points, now 300hz and 2khz, first order. A smaller front cavity and maybe a phase plug (photo 9) may have helped increase the efficiency of the highs. That modification hasn't yet been shelved. The larger back chamber volume may also have contributed to this lower midrange rise. The current back chamber volume will be reduced by adding wood blocks inside to get to the ideal 7 cubic inches from the current 15 cubic inches. It was originally thought that the highly packed wool in that chamber would achieve this. Replacing the wool with wood will verify that Despite the lack of increase in efficiency, what was noticed was the presence of the mids. It should be noted that efficiency and bandwidth are inversely proportional. Since the response is relatively linear, (red trace, fig 1) it was noticed that the horizontal dispersion widened. Listening tests showed that the listener's position can move as much as 45^{o} off axis before an loss of the very high frequencies could be noticed. This wasn't tested because of having to move the coffee table plus the stuff stashed under it A couple of friends gave it a listen and were astounded at how well they sounded upon entering the room. They were, at that point, about 12 feet from the speakers and about 45^{o} +/5^{o} off axis. Both agreed with my feeling that it sounded a little bright on axis but that was at better than 95dB SPL at several feet So, in short, better horizontal dispersion was obtained with no loss or gain in efficiency. A satisfactory tradeoff

FIGURE 6 These response curves are those calculated by Bas Box Pro, v6, re: the 3FE22. They show the normalized response in an optimum sealed enclosure of 63 cubic inches (red) and the box for the horn, (orange) of 15 cubic inches and (yellow) 7 cubic inches. This comparison was run to find out if such a reduction in volume would produce a bump. There appears to be a 3dB to 4dB rise around 400hz, (orange) which may be a problem. The yellow shows a 6dB rise in that region. With a first or second order filter at 500hz, the rise at that frequency would be lowered by 3dB, the knee of the filter

The Mouth
Reflection Dip
Paul Klipsch describes this in very great detail, along with the solution in his paper titled "A New High Frequency Horn" ^{3} Briefly, a dip in the response of about 7dB was noticed at 480hz. The solution was to install the horn onto a flange of area three times the horn mouth area. This also resulted in being able to halve the mouth of the horn, thus shortening it by close to 3 inches. The flange (baffle) maintained a close aspect ratio of the horn mouth dimensions. The dip noticed in my design may be the result of that or caused by the horn's proximity to the woofer. Fig 6 shows no dip at 600hz with the woofer disconnected. Adding a baffle of 3 times the mouth area and the same aspect ratio to this horn made no difference. The baffle measures 23" wide by 15" high Photo 11

Footnotes 1 Consider a straight axis bass horn for the Faital 15PR400 with a modest low frequency cutoff of 47hz, like the Klipschorn. It would have a mouth diameter of 24 feet and have a total length of 13 feet. One must look with admiration upon the Klipschorn with its total length of about 4 feet 2 Suitability of Low frequency Drivers for Horn Loaded Loudspeaker Systems  Richard Small Audio Engineering Society (AES) May 1977 3 Submitted, June 24, 1963, IEEE Transactions On Audio, pages 201206 
Deviations
From Generally Accepted Horn Theory
FIGURE 7 (click for larger image) Many horns designed for midrange or full range cone speakers seem to deviate from horn theory, especially at the throat. For a circular horn, it seems to be common practice to fit the loudspeaker at the throat which has a diameter equal to the inside diameter of the speaker's front gasket. If square or rectangular, the square may have an area equal to that of the above or the effective cone area. If the corners of this rectangle or square protrude beyond the circumference of the speaker, the rectangle or square will be adjusted to fit, resulting in a smaller area which, in actuality, me be closer to the correct throat area. The crude figure below gives an example of such a design; it assumes a 6" speaker. The length and mouth diameter is approximated from such designs seen online. It assumes an exponential horn although they could be tractrix, which will result in a shorter length but the mouth area will be much the same as that is determined by f_{c} the low frequency cutoff. This example uses inches. The following is more readable than the chicken scratch in the figure below, from the top We are given the throat diameter as per C.R.Hanna, D as 4/b or l/p and b=4pf_{c}/c To the right of that is stated a 6" speaker being used with a 5" horn throat diameter. Below that, is Keele's equation for throat area expressed in TS parameters, S_{t}=2pf_{s}Q_{ts}V_{as}/c Under Parameters we have the throat diameter and mouth diameter, D_{t} and D_{m} respectively and below them we have throat area and mouth area, A_{t} and A_{m} respectively, followed by the horn length, L_{h} (which, to maintain the nomenclature previously used, should have been written as H_{L} ) Next we divide the mouth area by the throat area and get 16.028, which is equal to e^{bx} . Taking the natural log of both sides, we get bx=2.774 and since the horn length, x=24", we get b=0.1156. Now, using the equation given above for b, by transposition, we get f_{c}=bc/4p=125hz. (The wavelength of 125hz is shown under that equation). From this we can calculate the mouth diameter to get 34.6" which is 1.73 times larger than originally estimated. This method is similar to a process known as reducto ad absurdum, whereby an initial statement being assumed to be true is proven false, assuming, of course, that the logic between is without fault.
Despite the deviation from accepted throat area, not to mention the cavity volume in front of the c one, these deviant horns can sound damn good.. The horn used for the Faital is also a deviant from horn theory d sounds damn good. Admittedly, these circular horns look a whole lot better and if they sound as good as they look, which I don't doubt, they have to sound nothing less than awesome. The biggest problem with circular horns is their construction, not to imply that a rectangular horn is easy. 
The following figures
are spectra taken at various frequencies. The left column are
those of the HF speaker in its back chamber of 7 in^3 without the horn
and the right column are with the horn. All are taken at 1w1m, the
power being calculated at the impedance of the system at that
frequency. All are measured with the woofer and through the
crossover.
These spectra were placed here in an attempt to "see" one of the reasons the horn sounds different than the direct radiator. The hash at the left is ambient noise, most likely from tha AC unit. Note in figs 9 & 16 the missing bump in the octave between 50hz and 100hz, probably due to the AC unit shutting down. 

FIGURE 8

FIGURE
9

FIGURE
10

FIGURE
11

FIGURE
12

FIGURE
13

FIGURE
14

FIGURE
15

FIGURE
16

FIGURE
17

FIGURE
18

FIGURE
19

Back to the loudspeaker main page