**Disclaimer:** I’m sorry that I am unable to attribute this written paper on transmission line design to the writer, *but would be more than happy to do so!* Neither can I remember where I attained this paper. I believe it may have been somewhere on the* quarter-wave.com* site, but I am struggling to find it there. However, here is the paper on transmission line design that has helped me the most.

**Transmission Line Design – Fundamentals**

## Intro.

The most common definition (and understanding) of a transmission line relates to it’s electrical derivation. That determining the quality of electromagnetic signal transmission: the matching of transmitting and receiving impedances, and the minimisation of line-derived losses. Curiously, an acoustic transmission line clings to the definition from the point of view of what happens if the electrical equivalent is ineffective – i.e. signal reflections and line losses. So it’s a tenuous definition at most.

## Fundamentals of TL Design.

Right, choose a speaker/subwoofer with a Qts between 0.4 – 0.6, and an fs of reasonable depth. From fs derive ¼ wavelength, chop 5% off that figure – this is your line length. Find the Sd of the target speaker/subwoofer, times it by 2, and this will be the CSA of your line. Build, wad, listen. That’s it, you’re a transmission line designer.

Well, perhaps. For those with a little more time, here’s the long-hand version…

## Fundamentals of TL Design – Line Length.

The oft-quoted ¼ wavelength. Remember that and you’re a reasonable part of the way into designing a transmission line enclosure. But a ¼ wavelength of what?.

Well, ¼ wavelength of the required operating frequency of the line (or pipe – TL’s often being referred to as ‘resonant pipes’) The operating frequency being the frequency to which the line is tuned (not dissimilar to that of a ported enclosure). With the most common transmission line tuning frequency chosen to match the resonant frequency (denoted in specifications as fs, in Hertz) of the target speaker/subwoofer. For example, an fs of 40Hz:

Wavelength = 1120/(4f) =1120/(4*40) = 7ft (84”).

Wavelength = 340/(4f) = 340/(4*40) = 2.125m (212.5cm).

Where 1120 is the speed of sound in feet per second (ft/s), and for the metrically inclined, 340 the equivalent in metres (m/s). 4f is 4 x the tuning frequency.

So, if you have a speaker/subwoofer with an fs of 40Hz, and have a line based around an 84” length, you have the beginnings of a transmission line enclosure. But why ¼ wavelength, what is it’s function?.

A speaker/subwoofer has two playing surfaces. The front of the cone has an output that is relative to, and in phase with the input signal. The rear of the cone has an output that is relative to, but out of phase with the front of the cone. Notwithstanding frequency considerations, should the outputs from the front and rear of the cone meet, then they will cancel each other out. Hence the need for an enclosure (or baffle of sufficient dimension) to control this rearward output. A ported (bass reflex) enclosure goes one step further and ‘recycles’ rearward output using the Helmholtz principle of interdependent air masses to reinforce output from the front of the cone.

Transmission lines do not work on the same principle. The length of the line (¼ wavelength of the chosen tuning frequency) is such that the time taken to travel along the length of the line means that the rear wave emerging from the line terminus is in phase with that from the front of the speaker/subwoofer. If the line has little or no damping (design loss from the addition of wadding), then output from the line is maximal, and summed output would increase in a similar way to a ported enclosure. The drive unit would also tend to unload, and output would drop very steeply (at a theoretical maximum of 24dB/oct) below tuning, again in a similar way to a ported enclosure. This form operation, however, does not typify a classic transmission line.

To understand why, you have to realise that a transmission line is essentially an open-ended pipe, akin to an organ pipe. It has a primary (fundamental) resonant frequency (the tuning frequency) that according to open-end pipe acoustics, will set up (as a function of reflection-derived standing waves) odd-order harmonics of the resonant (fundamental) frequency. If the primary resonant frequency is considered the fundamental, then the line will generate harmonic multiples of that frequency – 3^{rd}, 5^{th}, 7^{th}, etc, etc. So, if the line is tuned to 40Hz (the fundamental resonant frequency) then the line will generate harmonics at 120Hz (3 x 40), 200Hz (5 x 40), 280Hz (7 x 40), etc, etc. This content, along with a peak at the tuning frequency will exit the line at it’s terminus, and sum (constructively and destructively) with the output from the speaker/subwoofer. The resultant frequency response will be peaked at the low end, then sag to the middle ground, which will be heavily comb-filtered. The following illustrations show the individual responses from an un-damped line terminus and the speaker/subwoofer, and their summed response:

In the first one, the red line is the response of the speaker/subwoofer, and the blue line that of the line. In the second one the red line is the summed response against a comparative infinite baffle response in blue.

One final point with regard to deriving, indeed optimising, line length is that of boundary end-correction. This, like port end-correction, accounts for the apparent increase in line length (port length) due to the interaction of the air mass just outside of the line terminus. It is calculable, but that will be dealt with as part of design optimisation – for non-optimised designs experience suggests that if you reduce your ¼ wavelength-derived line length by 5%, then you aren’t far off. So, returning to our 40Hz design:

Wavelength = 1120/(4f) =1120/(4*40) = 7ft (84”) 84-(84*(5/100) = 80” app.

So, we have our ¼ wavelength-derived line length, which we have amended for boundary end-correction. If the intention is to use a single line length, then one can simply move on to derive line area. However, given the length of full-range and bass-specific lines, it is common, perhaps inevitable, to ‘fold’ the line into two or more sections.

Now, whilst this does add complexity to the design and build, it also adds design flexibility, not least in the placement of the line terminus. Under normal operating conditions a damped line may have little terminus output. However, and more so if the design/intent is for increased terminus output, then placement becomes more of an issue – not only that the terminus can be close to and/or on the same face as the speaker/subwoofer, but also positioned in terms of room-placement re boundary effect. Perhaps inevitably, however, folding the line has a knock-on effect on line length.

This effect has to do with calculating the reduction in line length, which occurs as the line travels through 90 degree and 180 degree bends. It is calculable, but the calculations are not exactly ‘visible’ in terms of their effect on the line. A preferable means of calculating, and showing line length, is to run a centre-line throughout your on-paper design**, making measurements to suit. The following illustration shows one of my designs with such a centre line – marked in blue, with dots marking the intersections.

**Given the potential complexity of transmission line design, and particularly fabrication, it is always worthwhile committing the design to paper (or screen). The large number of measurements within a folded line, and the need for them to remain accurate into fabrication, virtually necessitates it. Further, doing so allows you to ‘visualise’ your design, which I believe to be vital. Not least from the point of view of the design itself, but also in highlighting any potential pitfalls – especially the simple things that are often overshadowed by the more complex: line area that is too shallow for the rear of the speaker/subwoofer; a front panel that isn’t wide enough for the target speaker/subwoofer; and perfect dimensions that translate into a flawed enclosure simply because material thickness was not taken into consideration. Simple things that will affect/potentially ruin enclosure performance. Commit to paper before you commit to wood.

## Fundamentals of TL Design – Line Area.

Simply, the Cross-Sectional Area (CSA) of the line – width and depth times by the line length. If we know line length from our ¼ wavelength calculations above, that just leaves width and depth. So what do we base our requirements for line CSA on?. Well, primarily it is a function of the surface area of the target speaker/subwoofer, a Thiele/Small parameter known as Sd, most often measured in cm^{2}. Let’s say our target speaker has an Sd of 132 cm^{2}, which for personal ease of use equates to 20in^{2}.

For our line to equate to Sd, then, the simplest width and depth dimensions would be 5” x 4”. Of course, this doesn’t take into account such things as the width of the target speaker, nor it’s mounting depth, which you have to. And, in doing so, you may run into problems quite quickly – although they are as easily remedied by increasing area (either to suit, or from optimising the design). Indeed, most basic design criteria suggest a line area of between 1.5 and 2.5 x Sd. Further, it is possible, and popular, to have an increased CSA at the closed (speaker/subwoofer) end, and taper it down by some factor at the closed end. Naturally, altering line CSA and/or line taper has an effect on the ultimate response of the line.

Considering first the effect of increasing line CSA alone. Theoretically, there are no limits to how far you can exceed Sd with line CSA. Thankfully, however, practicality imparts limits, with overall enclosure size increasing rapidly alongside line CSA. Naturally, Sd, and thus line CSA, are determined by the target speaker/subwoofer. If most fit between 1.5 and 2.5 x Sd what is the effect of choosing a CSA at either extreme?. Well, basically, as you increase line CSA, you increase efficiency at the resonant (tuning) frequency, which itself may change, but only fractionally. The effect is similar to, and could be described as, altering system Q (Qtc) of a sealed enclosure. As system Q of a sealed enclosure increases, so does the peak at resonance. Transient response is compromised, and response becomes increasing ‘one-note’. That said, it is unlikely for designs between 1.5 and 2.5 x Sd. Indeed, it is perhaps more practical to consider the increase in CSA as a means of facilitating different specification speakers/subwoofers.

Line taper is somewhat different. But first, let’s discuss taper style, and their incidental benefits. Looking at the illustration linked to below you will see two types of taper. 1. is the traditional continuous taper, where CSA is constantly changing throughout the length of the line (in this case a common single-fold design). This design maximises the line-taper effect, with the incidental benefit that the line does not have parallel sides, and thus negates spurious inter-panel and inter-section resonances. 2. is what I refer to as a linear taper. The CSA of a section remains the same throughout it’s length, but the CSA of proceeding sections is reduced in comparison. This is far simpler to execute, but has reduced effect of the constant taper, especially in resonance control.

As with line CSA, there are theoretically no limits to the ratio of line taper you can design – practicality again being the determining factor. 2:1 is popular, and feasible without obviating optimisation of the design. Bear in mind that this is a ratio, and so covers a variety of dimensional designs: i.e. 2 x Sd at the closed end, and Sd at the terminus. Or 3 x Sd at the closed end, and 1.5 x Sd at the terminus. Above 2:1, perhaps 3:1, really requires optimisation, perhaps modelling, and certainly a nod towards driver suitability.

The effect of increasing taper ratio, primarily, is to lower the tuning of the line, by a marked degree, even for modest tapers. If the line is suitably damped, roll-off is shallower (at a theoretical minimum of 12dB/oct), allowing for extension into the sub bass. Frequency-wise, the harmonics do not change, so that the effective distance between the first one (actually the 3^{rd} harmonic), and the fundamental (tuning frequency) increases – a potential benefit to bandwidth-limited designs. For those of a wider bandwidth of operation, the 3^{rd} harmonic is reduced in level, as energy is spread into the low-bass by virtue of the lowering of resonant (tuning) frequency. Imagine an old-style tent: pegged out normally, it can sleep two, with enough room to manoeuvre. However, if you spread the base out, lowering the peak of the tent, you may be able to fit another person in, albeit with far less room for manoeuvre.

The moral?. Everything’s a compromise. As energy is spread into the low bass, increasing bandwidth, there is a corresponding drop in efficiency at the (lowered) tuning frequency (the tent is not so peaked). This becomes more apparent as taper ratio increases. Fortunately, the lack of practicality of increasing taper ratios means that excessive effect is limited. However, even a modest taper may be sufficient, allied to a low Q (Qts) speaker/subwoofer to cause a drop in efficiency that relates sonically to a weak bass response.

There is a flip side to this taper issue. If we know that tapering a line lowers the resonant frequency of the line, and that resonant frequency of the line is otherwise controlled by line length – why not reduce the length of the line?. Why not indeed. In doing so, retaining the desired tuning frequency, allied to the other benefits of tapering, with the additional benefit of reduced line length. Which, given the potential size of a low-tuned transmission line, is a considerable bonus.

The problem is that this really is a function of line calculation and optimisation. That said, it shouldn’t deter the ardent experimenter – to whom I give, derived from experience, a ‘fudge factor’ of 0.8. For tapers of 2:1, perhaps even 3:1, reduce the ¼ wavelength-derived line length by a factor of 0.8 (this also takes care of boundary end-correction). So, our 40Hz example would become:

Wavelength = 1120/(4f) =1120/(4*40) = 7ft (84”) 84*0.8 = 67” app.

## Fundamentals of TL Design – Line Damping.

Right, let’s get one thing out of the way first – adding wadding to a transmission line **does not** increase it’s apparent length. This is a long-held misconception that has been experimentally proven beyond reproach. You can not design for a long line, deliberately build it short, and expect wadding to make up the short-fall, it does not work like that.

What it does do, and should be used for, however, is to damp the rear wave, reflections of the fundamental and generated harmonics, as well as any spurious resonances. At which it is extremely effective. Looking back to the earlier section on line length, and the linked illustrations of the individual and summed responses of an un-damped line – compare those to the same design that has optimised damping:

In the first one, the red line is the response of the speaker/subwoofer, and the blue line that of the line. In the second one the red line is the summed response against a comparative infinite baffle response in blue.

Wadding placement and density are not hard and fast, even for an optimised line. You can measure and model response to different types, densities, and location of wadding within the line, but that does not guarantee good aural results – the ears do that, and should be thus used. If there are guidelines, then experience suggests that the first ¾ of the line can be successfully treated with a good quality acoustic foam, perhaps of the pyramidal variety. The last ¼, which includes the open-end terminus (an antinode and point of maximal air particle motion), may benefit from increased damping. Foam lining allied to loose BAF (Dacron) wadding, say. Ultimately, experiment, and let your ears decide.

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