Traditional Transmission Lines

Traditional Transmission Lines 12/31/2002

I'm putting this page up prematurily. Several related topic are covered that should be split out into their own pages. But there is imformation here that would help those not versed in TL theory or practice. So, here it is!

Almost all of the information on this site relates to pipes that use the pipe resonanse to extend the bass down as much as a half octave below what is possible with vented boxes. The advantages of resonant pipes are the obvious low bass and also the virtual elimination of back-wave reflection through the cone. The penalty is very large cabinet size relative to sealed or vented boxes.

The large cabinet size starts to become untenable with drivers around 8" and larger. Happily, these larger drivers also have substantially lower fs values than smaller drivers. This allows for a different approach to cabinet design. Where smaller drivers require the pipe resonance to lower f3, larger drivers go sufficiently deep on their own that the pipe output is not needed at all. We are entering the realm of the infinite baffle speaker.

At this point, I need to digress for a moment and discuss the model I use, the Martin King MathCAD worksheets and the model used by George Augspurger for his AES paper and Speaker Builder articles. You can go to the references and read about the math. Here I only want to show the similarities and differences. The MathCAD verson the the Vifa P17WJ example, figure 19 and 20 from the Speaker Builder article, Part 3:

Red = Combined Frequency Response, Blue = Infinite Baffle Response

Red = Driver Output, Blue = Pipe Output

Note that the wiggles are in the same place in both models. This indicates that the harmonic series is being handled the same by both models. The King model correctly maintains a constant SPL at the upper end, while the Augspurger model rolls off at the high end. Conversly, at the low end, the Augspurger model correctly predicts the low end roll-off, while the King model rolls-off to early and too much. It has been known that a pipe in MDF will get a few Hertz deeper than King predicts. Basically, both models perform the same in the middle where it counts.

(The driver that will be used for this discussion is the Peerless 850146, A 10" driver with f_s = 27, Q_ts = 0.46, and V_as = 68. As you might have quessed, this discussion is part of the preliminary investigation for my DIY 2003 entry.)

First, let's look at the driver mounted on an infinite baffle. This was modeled with a 1000 liter box. This provides both a base line and a sanity check that the model is working correctly.

Infinite Baffle

The red line does indeed follow the blue line, so all is well. The small glitch at 100Hz is the standing wave between the top and bottom of the box. (All sealed and vented boxes modeled on this page have golden ratio dimensions.)f₃ is ~45Hz and f₁₀ is ~20Hz.

The first TL example is a "Rule of Thumb" TL. The length of the pipe is 1/4 wave-length at f_s. The wide end area is 2*S_d and the small end area is S_d. This is a pipe with dimensions 82" long, 100in² at the wide end and 50in² and the small end. The total pipe volume is 100 liters. This pipe has an f₃ of 38Hz and f₁₀ 23Hz.

Rule of Thumb Pipe for Peerless 850146

A caution here. The Peerless 850146 works here because it is a high Q_ts, low V_as driver. What happens with a low Q_ts, high V_as driver? Here is the Vifa M25WO in the same pipe:

Rule of Thumb Pipe for Vifa M25WO

Even though the Vifa is more efficient than the Peerless, the f₃ and f₁₀ of the Vifa are higher then the Peerless. This speaker will be very weak in the bass.

Augspurger addresses the problem of differing driver parameters by providing a table of pipe parameters based on the driver Q_ts. Using the "opimized" values from Table 2, a pipe for the 850146 has dimensions 63" long, 64in² at the wide end and 16in²at the small end. The total pipe volume is 33 liters. This pipe has an f₃ of 42Hz and f₁₀ 28Hz.

Augsburger "Opimized" pipe for Peerless 850146

Augspurger goes on to suggest that extended bass output is possible at the expense of increased pipe volume. He supplies a second set of "extended" tables. Using the values from Table 2 (extended) and extrapolating to a Q_ts of 0.46, a is generated with the dimensions 88" long, 189in² at the wide end and 47in²at the small end. The total pipe volume is 170 liters. This pipe has an f₃ of 32Hz and f₁₀ 20Hz.

Augsburger "Extended" pipe for Peerless 850146

As with any alignment table there is an infinite number of possible alignment between the table entries. The designer must choose between the cabinet size he can stand and the bass extension he wants. A possibility is a pipe 80" long, 132in² at the wide end and 25in² at the small end for a pipe volume of 119 liters.

An optimized pipe for Peerless 850146

Another digression. The ultimate roll-off rate of a driver on an IB is 12dB/octave. The driver in a TL or a vented box behaves as if it is on an IB. There are dips where the pipe (box) resonates, but basically, the driver output follows the IB curve, i.e., 12dB/octave. The pipe also rolls off at 12dB/octave. That cannot be seen with a large driver/pipe combination as on this page, but can be seen clearly in the examples on the "Nature" page. Since the pipe and driver are a coupled system and out of phase below the pipe resonance, they add destructively to and ultimate roll-off of 24dB/octave.

The Augspurger alignment for the Vifa M25WO is a much smaller pipe. The length is 54", 168in² at the wide end and 63in²at the small end. The total pipe volume is 89 liters. This pipe has an f₃ of 53Hz and f₁₀ 33Hz.

The moral of the story is that while you can shoe-horn a low Q_ts into a TL, the best results are obtained with a moderately high Q_ts of 0.4-0.5.

Augsburger pipe for Vifa M25WO

Now let's look at other solutions for woofer boxes. You can pick f₃ and f₁₀ points off for yourself.

38 liter sealed box tuned to 44Hz Q_c = 0.707

70 liter vented box tuned to 23Hz ~BB4

100 liter vented box tuned to 23Hz QB3

If I were to choose something other than a TL, the 38 liter sealed box would by my choice for this driver. While the f₃ is in the mid 40's, the f10 is in upper 20's. This box would be a good performer. The two vented boxes have some serious problems, which is caused by room gain.

Room gain is hard to pin down. It depends on room size and shape, and is virtually unpredictable if the room is irregular, with lots of knooks and crannies, such as dining rooms and hallways. Anyway, looking over the literature and from my own experience, I declare room gain to be 3dB ad 40Hz and 6dB at 20Hz. YMMV. The point of this is that all models are anechoic. Once the speaker is put in a real room, there will be more bass than expected from the model. Applying room gain to the models of the 850146:

	Anechoic		With Room Gain
	f₃	f₁₀	f₃	f₁₀
Infinite Baffle	45	20	29	12
Rule-of-Thumb	38	23	29	16
Augspurger Optimized	42	28	35	18
Augspurger Extended	36	22	26	15
"Proposed"	33	23	27	17
Sealed	42	28	33	19
BB4	23	18	18	15
QB3	20	17	17	15

All of the TL's and the sealed box give in-room f₃ around 30Hz, which is perfect for classical music and acoustic instruments. With f₁₀ below 20Hz, you can even expect to listen to some serious organ music. The two vented alignments go very low, but are going to have elevated bass, which is a bad thing for music, but a good thing for HT. The QB3 box will probably be unlistenable for music. The BB4 box, however, is going to be very good for HT, and, with the port plugged, will be a very good alignment for music. Just some off-topic thoughts.

I must admit that if I had the resources, I would go with IB using multiple 15" drivers with a large room or garage as a back chamber, or vented into an attic through a manifold. This is the route to the best, clean, uncolored bass. But, since I don't have the resouces, I built TL's. A TL does remove the back wave from the driver almost as effectively as an IB, giving the TL better, cleaner bass than any other type of box. The TL is, however, a much smaller package, being as it is not infinite, and additionally, it provides damping for the driver that the IB cannot.

The matter of driver damping is the reason that we can get much more SPL out of a TL, or sealed box for that matter, than we can out of the same driver on an IB. Here is the 850146 on an IB. The red line for the driver is exactly on the blue line, which defines the blue line as the IB condition in the remaining plots. The power level is 1 watt.

Infinite Baffle

The Augspurger "Extended" TL

The TL maintains much better control of the driver down to 20Hz.

The "Proposed" TL

38 liter sealed box tuned to 44Hz

Sealed boxes provide the best damping of all the choices presented here.

70l vented box tuned to 23Hz

100l vented box tuned to 23Hz

Vented boxes provide good damping down to box resonance, but then they unload rapidly and completely.

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