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.
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.
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.
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