Mathematics of the Transmission Line Speaker Enclosure:
Derivation, Simulations & Recommendations
The Transmission Line is a seldom-used speaker design which is preferred by many audiophiles. Fans say TL’s sound less resonant, less boxy, more open. I’ve built several, one very recently. I don’t think TL’s are necessarily more “accurate” – they undoubtedly have flaws. However when well-designed have a rich, dimensional sound in the lower registers which carries a depth and ease that sealed and ported boxes lack. I prefer the sound of a TL to a ported box.
In 1990, when I was a senior in college, I was frustrated that the kind of mathematical models that are available for acoustic suspension and bass reflex boxes were unavailable for Transmission Lines. I was taking an acoustics class, so for my semester project, decided to derive the math myself.
This is that paper, very slightly cleaned up and digitized, otherwise unchanged. I did not find this project to be at all easy. However when I was done I greatly enjoyed the satisfaction of knowing that I had started with basic physical properties like the mass of a speaker cone and the density of air and derived the entire response of a complex system. And that I had done something that nobody to my awareness at the time had done.
The experience of doing this left the feeling of “diving to the bottom of the swamp” in my muscle memory and has served me in many subsequent projects.
As you will see, TL’s are complicated. They can’t be reduced to simple differential equations like normal boxes can. In this paper, I ran all my simulations on a Hewlett-Packard 28S calculator. I would have loved to have been able to run them on a computer, but I didn’t have access to the right software.
I came to several conclusions that reinforce my personal experience from building transmission lines. I also explore closed TL’s, which have received very little attention in the literature.
I welcome anyone who wants to apply my models on a computer, math program or online calculator. Enjoy this paper.
A Derivation and Analysis of the Transmission Line Speaker Enclosure
I posted this in Martin King’s Transmission Line Discussion Group and Martin responded:
Your paper brought back a lot of memories of my early TL studies. Your paper was well written and laid out the TL theory as it stood when I started in the mid 80’s and continued working with it up to the late 90’s as computers and measurement systems started to become more accessible to the speaker builders.
The math and theory set forth by Bradbury, based on Bailey’s tests data, was so elegant and attractive I spent years working out the details and programming the solution. I worked on computer codes written in Basic on an old PC XT clone and produced curves similar to yours, unfortunately I could never get the models to correlate with measured designs I found in magazine articles and AES papers or my own measurement data.
By the late 90’s and early 2000’s independent TL developers had built and tested TL speakers (in my case a cardboard tube with stuffing) that showed Bradbury and Bailey were wrong. The fibers did not move and the speed of sound was not reduced significantly. New computer models were developed that correlated well with test data and more builders starting making TL derived enclosures.
The calculations were very accurate and designs were built with some confidence that they would work as predicted. The old school moving fiber theorists fought against the newer computer models on the discussion forums but as the years passed they have become less and less vocal and I believe now most speaker builders have moved past this incorrect theory.
If you are still interested in TLs, I think your computer model could be updated with a better fiber damping correlation and be pretty accurate for a constant cross-sectional area TL.
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Martin is correct, my assumptions about absorption have been replaced by a better understanding. The best source I know of for modeling transmission lines is Martin’s site http://quarter-wave.com/. Martin offers a free MathCad computer model that accurately predicts system response.
Perry Marshall’s Speaker Design Page
Archived Comments
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December 30, 2014 at 6:05 pm
Just finished reading your “Mathematics of the Transmission Line Speaker Enclosure”. I didn’t go through the math, stayed with your thoughts and conclusions. Interestingly, when Martin first started posting his transmission line information we started to email each other. After much time had passed I asked him how he became interested in transmission lines. He replied after hearing and being impressed with a pair at an Albany State professors home. I turns out that the transmission lines were a pair that I had built and sold to the professor and that, in fact, Martin lived near me. Many years before I had been building transmission lines from Sonotube and passed that information along to Martin. We were already using Sonotube for our experiments about 10 years before, with lines as long as 14 ft. Martin had asked me if I thought or observed the fibers moving and I did as I think you described. What is critical is when using wool as a stuffing that it is well teased, using a hacksaw blade. At that time my thinking was to build the line as long as was practical while focusing on the resonant frequency. This always worked well and the lines were always turned out well. Eleven feet seemed to be ideal for the Focal drivers that I was using. I built much shorter lines that others were happy with, but the longer lines I thought always performed better. Years later Martin designed a line for me based on another set of Focal drivers and I was not happy with the performance compared to magic of my longer lines. Before emailing with Martin I had built an 11 ft line with compound loaded drivers, the drivers were facing each other, and that was clearly superior to the single driver in the same line. At that point I was only using this in a subwoofer application. Since then I have built a few full range systems. With those speakers I use sealed box woofers to capture the bottom end, in particular the low end of the old Celestion SL6s. While the bass/midrange drivers in the Celestions are small, the bass extension is quite noticeable compared to running the Celestions full range. Perhaps comparable to the idea from Magnepan where they market a bass driver to go with their full range speakers rather than a subwoofer. Incidentally, years ago when Jim Winey from magnepan was contemplating building a tonearm he and I discussed it over the phone and he incorporated my ideas for the counterweight when the arm went into production. My observation is that using a woofer/midrange on the bottom gives a better defined and cleaner low end. My experience is that I can’t mate an open ended midrange transmission line to an open ended bottom. IMF with their Monitor and Studio speaker had their midrange in a closed end transmission line and the bottom in an open ended transmission line. As I recall, IMF used some fiberglass in their line, but weaved it in strips through dowels, essentially extending the length of the line for certain frequencies. If I use the Celestions (sealed cabinet) full range with a transmission line bottom it works out as well, provided that I don’t cross over too high. Thanks… Faustin
January 5, 2015 at 8:01 am
Thanks for your note. Martin has made a tremendous contribution and I LOVE his work. Can you pass the following comment to him, since I don’t know him personally? I suggest he transfer his transmission line math program into an entirely web-based application and make it a membership site that costs $9.95 per month – and allow people who use it to use the designs commercially. I was not willing to use it because in order to do the whole TL math program thing I had to use a Windows machine. I hate Windows. Plus I would have had to learn the math program too. I think it would be a small but viable business, it would make him a few thousand dollars a month. My 2 cents as a very experienced marketer.
April 15, 2015 at 5:05 pm
Dear Perry, Looking to make some speakers for our school – see url and came across your article on the two system using the BG RD 75 planars. Do you have any more details. Very nice impulse response – I thought line arrays were supposed to suffer from non-linear impulse responses? See Design Guide for Practical Near Field Line Arrays by Griffin…
Love to hear more,
Thanks
William
April 22, 2015 at 4:04 am
Hallo, where could I buy the B&G RD 75 ribbons ????, Parts express does not sell them anymore, I have been considering for 8 years to make a two way elec x overed two amped system—thanks
Paolo
April 23, 2015 at 12:15 pm
Contact the manufacturer?
June 3, 2015 at 5:31 pm
Hi,
I worked with David Graebener as a mechanical engineer for BG back in the day and developed that 75? driver with him. BG is now out of business. I may know of a small supply of the 50? and possibly a couple 75? drivers plus many various sizes of neo planars. I am not a bussiness and this comment is meant not to be self promotional so not sure how you are to contact me if interested. After working for 3 planar driver manufactures I have a number of planar drivers/knowledge hanging around doing nothing.