Transmission lines (TL) have always exerted a certain fascination in the majority of loudspeaker enthusiasts and some very successful designs from the past have helped to increase the attractiveness of those systems. The IMF loudspeakers, to whom we owe the first successful commercial application of transmission line, are a good example. Subsequently TDL, founded by IMF in the late 80’s, continued this tradition which probably peaked with the Studio series. In this regard, it is worth remembering a very appreciated transmission line system: the TDL Studio 1M, reviewed by Bebo Moroni on AUDIOreview n.129 (July-August 1993).
When a few years ago I decided to build my first speaker systems, not being exempt from this influence, I considered the possibility of using a transmission line to load the low frequency driver. Unfortunately, once I consulted the (few) texts I had available, I realized that I would have to develop the design in an essentially empirical way. For example, Vance Dickason’s popular Loudspeaker Design Cookbook, a reference text for many DIYers, in its 5th edition in 1995, about TLs contains little (often inaccurate) information and some rules of thumb not easily applicable to a specific design. To tell the truth Dickason in those years seemed very prejudiced towards that kind of enclosure: the short chapter dedicated to the TL starts with a quote by Martin Colloms, founder of Monitor Audio, who “believes that the performance of the TL is no better than that of a well built enclosure, and that it is difficult to obtain a uniform low frequency performance without causing line resonances, sources of coloration in the low-mid range”.
Although Colloms’ thinking was much more articulated , Dickason, to confirm this statement, goes on to compare the low frequency response curve of a TL with that of a closed box and that of a bass-reflex system. He does so, however, using LEAP software, which does not simulate the TLs, artificially modifying the system response to make it similar to that of the graphs (also simulated on the computer) present in a previous work by Juha Backman . Dickason could only assume that Backman’s model was accurate, because his software was never made available, but now we know for sure that it was not, because, like other previous models, it was based on wrong assumptions. Some basic TL configurations are then presented, but in general the author tended to discourage the reader from “taking on” the task of building such a “complex” enclosure.
A very similar thought probably had Joseph D’Appolito who in his Testing Loudspeaker (1998) dedicated to TL only a couple of paragraphs that mostly refer the interested reader to the above mentioned chapter 4 of the LDC. However, I like to underline that just two years later, when the Norwegian Seas commissioned him a reference project for the new Excel magnesium series, D’Appolito chose the transmission line for his Thor-Excel system. To be honest, as we will see later, just the previous year Augspurger  had made available the first useful (albeit a little coarse) design tool of TL.
The lack of a mathematical model suitable for the design of a transmission line starting from the parameters of the loudspeakers is also underlined by Vincenzo Landi in an interesting and in-depth article , divided into five episodes, published on AUDIOreview between April 1991 and December 1993.
Needless to say, with the material and resources at my disposal at the time, the intention to design a TL ran aground in the bud. After having studied the subject in depth I feel I can say that, even today, the enthusiast who wanted to design this type of speaker would still have a difficult task.
I have recently developed a simulation model, based on the theory of electrical circuits, using some principles that we will see later. The circuit simulation program used is freeware: LTspice by Linear Technology (now part of Analog Devices).
With this article I aim to explore in a general way the topic and describe the tools used to develop the model. I will also propose the design of a small, single-driver nearfield monitor, with amazing performance compared to its small size, based on Dayton Audio’s RS100-4 wide band speaker.
In order to take stock of the subject a brief introduction is necessary, supported by some historical background.
Historical Notes and Origin of TL Speakers
When approaching the transmission line concept for the first time, the surprise due to the almost total lack of universally accepted design methods or models lasts very little. Unlike a closed or bass-reflex loaded enclosure, predicting the behaviour of a sound wave propagating inside a duct, especially when filled or lined with absorbent material, becomes a very difficult task. Benjamin J. Olney , director of the research department at Stromberg-Carlson Telephone Co., noticed this as early as 1936 when he was looking for a method to improve the low-frequency performance of radio receivers at that time and came up with the idea of the acoustic labyrinth: a long duct, lined internally with absorbent material, with one opening occupied by the loudspeaker and the other open and facing the floor. The duct, once folded back on itself, took on the appearance of a normal loudspeaker. Unlike the rear open enclosures used at the time, the acoustic labyrinth would have made it possible to attenuate the mid-range resonances and to overcome the evident low-frequency acoustic short circuit of the previous solution. While understanding the importance of the damping material, Olney realised that developing an impedance model that took this into account would be very challenging; he explained the behaviour of the acoustic labyrinth assuming that it was empty and then described the alterations due to the damping material.
In the case of an empty tube with circular cross-section S, the acoustic impedance seen from the speaker is:
where ZA is the acoustic impedance of the tube at the opening in Nsm−5, LT is the length of the tube in meters, ρ0 is the density of the air in kgm-3, c is the speed of sound in air in ms-1 and k is the wavenumber with frequency expressed in terms of pulsation or angular frequency: k = ω/c = 2π/λ.
After assigning an arbitrary value of 10 mechanical ohms (Nsm−5) per cm2 to ZA, he drew the graph of the equation and observed that, when the frequency (f1) was such that the tube length coincided with a quarter of its wavelength (λ), the impedance curve assumed very high and purely resistive values. He also noticed that by increasing the frequency to a value (f2), so that L = λ/2 the tube impedance dropped to a value of 10 ohm determined by the opening alone. We can easily verify this by entering the impedance equation in a calculation software (e.g. Mathcad) and plotting the function graph (Graph 1).
It was immediately evident that these two peculiarities of the acoustic labyrinth could be exploited advantageously. In particular, by matching the resonance frequency of the loudspeaker with the high resistance zone (λ/4), it was possible to control the diaphragm excursion by considerably increasing the power handling of the system.
Olney also enthusiastically pointed out that by sizing the duct properly, one could take advantage of the f2 condition to greatly extend the system’s low frequency response. This is because, at the above mentioned frequency, the acoustic wave emitted frontally by the loudspeaker is constructively added to the acoustic wave emitted by the opening: the wave emitted by the rear cone, 180° delayed with respect to the one emitted frontally, undergoes a further 180° delay (equal to half a wavelength) due to the path inside the duct and reoccurs at the opening in phase with the loudspeaker. Both these conditions, so far exploited favourably, repeat cyclically as the frequency increases, producing undesirable dips and peaks in the response. Looking at the graph we notice in particular that the peaks coincide with the duct’s resonance modes, i.e. with its fundamental frequency and its odd harmonics. To better understand this concept we think of a string of length L tied at both ends. If plucked, this string vibrates as shown in Figure 1.
The points that remain stationary are called nodes while the points of maximum wave amplitude are called antinoids.
This vibration is the first harmonic (or fundamental mode) and corresponds to the minimum frequency reachable by the string. To determine this frequency we observe Figure 1 and establish that:
the corresponding frequency will then be:
Where v is the speed of the waves on the string. In addition to the first harmonic, the string can also generate frequencies that are integer multiples of the fundamental. From Figure 2 we observe that each harmonic always contains half a wavelength more than the previous one.
For the second harmonic the wavelength will then be:
so its frequency will be:
In the case of the third harmonic L is divided into three half wavelengths:
hence the frequency of the third harmonic:
We can write:
In the case of the string we can therefore say that the harmonics are the integer multiples of the fundamental frequency. These waves are also called stationary because they oscillate in time remaining stationary in their position.
Let’s now examine the behaviour of a duct with a closed end (in our case by the loudspeaker): in this system, for all harmonics, the closed side acts as a node (air cannot oscillate freely) while the open side, where air can move swirling, represents the antinode.
We call c the speed of sound. From Figure 3 we immediately observe that the first harmonic is represented by a quarter of the wavelength:
We find the next harmonic by adding half a wavelength and, helping us again with Figure 3, we observe that L corresponds to ¾λ:
Being three times the fundamental, this frequency corresponds to the third harmonic. Adding half a wavelength we find the next harmonic:
We note that the succession of harmonics is different from that of the string because there are no even harmonics:
In the case of the duct, since the oscillations of the air column are longitudinal, sine waves should be considered as an abstract representation of air velocity.
Olney demonstrated with a series of measurements that the amplitude of unwanted standing waves could be practically zeroed by lining the tube with damping material. He also noted that the effect extended to the low frequency region, flattening the resistive peak at f1. Looking for a rigorous method to predict the behaviour of the acoustic labyrinth, he thought at first to apply the proven theory of losses in power lines, recognizing the analogy between the behaviour of the absorbent and that of a shunt resistance between the conductors of the line. However, as soon as he tried to apply this model, he had difficulty in assigning the correct value to the resistance as a function of a given frequency. The data referring to the absorption coefficients of commercial products available at the time were poor and totally unsuitable for the desired application. Olney also suspected that the mechanism of transmission of sound waves through the lined tube could not be fully explained by the classical theory of plane wave propagation: the absorbent could reasonably influence the speed of sound near walls, modifying the waveform and making it become more and more convex as it went along its path. Despite the theoretical difficulties, Olney was able to demonstrate, with a series of measurements on the impedance curve of the loudspeaker, that the acoustic labyrinth brought a considerable improvement in quality compared to previous open rear speakers.
In 1965 Arthur R. Bailey, an engineer at the University of Bradford, published an article  in Wireless World describing the design of a loudspeaker he called “non-resonant”. Unlike the acoustic labyrinth, only lined, the duct was completely filled with damping material and the opening faced the same plane as the loudspeaker. This configuration was the first referred to as a transmission line. In his work Bailey first highlights the great advances in loudspeaker technology over the years and argues that the design technique of the enclosures “has not kept pace with these developement”. The bass-reflex was already the most popular loading method at the time, but its performance was still very poor. We should remember that although Thiele introduced a modern approach to bass-reflex design as early as 1961, his work only became popular with designers ten years later when it was taken up and completed by Richard Small. It is therefore not surprising that Bailey devalued the sound characteristics of these systems by calling them “boomy” and “ringing”, both of which refer to irregularities in response due to resonance phenomena. Failing to highlight these anomalies with classical methods, he resorted to the explosive wire technique (EWM): a 1000 uF capacitor, charged at 250 V, was discharged through a 1 cm long copper wire with a diameter of just over one millimeter causing a real detonation.The latter, which took place inside the cabinet near the loudspeaker hole (previously closed), was picked up by a microphone also placed inside the cabinet. The results of the test showed that, with respect to the transmission line, the bass-reflex loudspeakers stored a much higher amount of energy that originated inevitable and annoying queues in the response, moreover, the amount of absorbent material needed to attenuate the phenomenon would have made the action of the opening no longer occur with a consequent reduction of the emission at the tuning frequency. In the transmission line, instead, the duct acted as a low-pass filter, strongly attenuating the higher frequencies thanks to the damping material. Bailey identified natural long fiber wool as the material with the best characteristics and gave few other practical tips. In 1972 he presented an updated design , with a new geometry that further improved impulse response. In this review Bailey also mentions the property of slowing the speed of sound (by a factor of 0.7 to 0.8) that long-fibre wool would have if properly placed along the duct. However, the method of development remained completely empirical and no attempt to dimension the line according to the driver parameters had yet been made. However Bailey had the merit of bringing to light a type of enclosure almost completely abandoned by the designers and taking a step towards the development of the modern transmission line.
In 1976, Leslie J. S. Bradbury , Engineer in the Department of Mechanics at the University of Surrey, drew on Bailey’s work to explore the acoustic properties of fibrous absorbent materials. Of particular interest was the effect that long-fibre wool could have on the speed of sound, as it could positively affect the size of bulky rear horn-loaded or transmission-line speakers. He identified a value of 8 kg/m3 as the ideal absorbent density capable of strongly attenuating frequencies above 100 Hz, without practically affecting the acoustic load of the driver. He also noted that this density reduced the speed of sound at lower frequencies by up to 50%, making it possible to halve the length of the tube. He explained this phenomenon by assuming that at low frequencies the fibres were moved, literally dragged, by the airflow generated by the loudspeaker. At high frequencies the fibres would not be able to follow the air in its rapid movement and would not move as a result. He then obtained an equation which, through a parameter called flow resistance, correlated the speed of sound in the line with the absorbent density. From these premises derived what, for several years, was the only mathematical tool available for the study of acoustic transmission lines. In 1980, however, Robert Bullock  developed a software for the simulation of transmission line loudspeaker systems based on the Bradbury model, but the concordance with the measurements, according to the same developer, proved “unsatisfactory”. As we will see, even in subsequent investigations, the Bradbury equations did not prove to work properly.
The first useful work for the practical realization of a transmission line is a document  presented in 1999 by George L. Augspurger at the 107th Audio Engineering Society convention. The author, an engineer very active in the field of electroacoustics (he was also manager and technical director of JBL before devoting himself full-time to his consulting firm, Perception Inc.), assimilated the transmission line to horn loading with the addition of losses due to absorbent material. He then developed a similar electrical model, based on a previous work by Bart N. Locanthi . Augspurger’s model consists of 32 sections, each composed by an LC network, where inductors and capacitors represent respectively the air compliance and the air mass; a resistor for each section is added to represent the losses due to the damping material (Fig. 4).
Since the behavior of the absorbing material varies with frequency, the resistors used are variable, but the method used to vary the resistance as a function of frequency is not revealed. Unlike Bradbury, Augspurger does not give any importance to the possibility that the fibers are set in motion by the acoustic waves generated by the loudspeaker, and it decisively reduces the quantity and the role of the variation in the speed of sound in relation to the response of the TL at low frequencies. To model the characteristics of the absorbent he refers to four empirical parameters that he considers necessary to describe with good approximation the behaviour of the system; we only know that the first three act on the resistance value while the fourth regulates the capacitance value. Thanks to his computerized model Augspurger virtually varies the parameters of the line and the loudspeaker, obtaining alignment tables that he defines as “optimized”: high pass slope of 12 db/oct, regular midrange response and sensitivity comparable to that of a closed box loudspeaker. These tables for the first time relate the speaker parameters (fs, Qts and Vas) to the resonance frequency and the volume of the pipe (fp and Vp). The tables refer to the three most common geometries: tapered line, with the loudspeaker shifted one fifth from the beginning of the line (offset speaker) and with coupling chamber. Each of these solutions promises advantages over the classic straight line, with a small price to pay in terms of efficiency (Fig. 5).
In 2000 the magazine Speaker Builder published an article by Augspurger, divided into three parts   , containing some updates. In this work updated tables propose new alignments, defined “extended”, which allow improved performance in low frequency extension.
Although it does not provide a complete and developed theory on the transmission line, and its tables do not assign a unique correspondence to the values, Augspurger’s work certainly represents a good starting point for the designer.
Also Vance Dickason was evidently influenced by Augspurger’s work and detailed it in the seventh edition (2006) of his LDC where we finally find a large chapter dedicated to TL. Dickason, recounting an anecdote from his experience of listening to a TL from the 70s, goes so far as to admit that “this often misunderstood device still provides a fruitful pursuit for those questing to have the very best loudspeaker ever produced”.
The publication of Augspurger’s paper is more or less contemporary to the appearance of Martin J. King’s work on the web. A passionate DIY speaker builder, King listened to a pair of transmission line speakers at a local audiophile club member’s home and was strongly impressed by the reproduction quality of the bass range. A few years later, after studying all the literature available on the subject, he seriously considered building a transmission line system. Knowing that no acceptable design method was available, he decided to write the necessary software himself based on spreadsheets in the MathCad environment. After spending a lot of time experimenting with Bradbury equations, King came to the conclusion that the fibers of the absorbent material were stationary. He then developed a model based on the one-dimensional acoustic wave equation with the addition of a viscous damping factor due to the absorbent material.
The software, constantly improved over the years, makes it possible to model different TL geometries satisfactorily. Recent changes to the calculation algorithm have made the system so flexible that it can also be used to design closed boxes, bass-reflex, isobaric speakers and horns (loaded both front and rear). Its website www.quarter-wave.com contains a considerable amount of information on the subject and enjoys great popularity among transmission line enthusiasts. Its “sheets”, also used by some professionals, have given rise to well-sounding and highly appreciated designs. For some years now, King’s files are no longer available for free download and are provided upon purchase of a license. (Update: at the moment the model is no longer available).
 M. Colloms, “High Performance Loudspeakers”, Pentech Press Ltd., Fourth Edition; 1991
 J. Backman, “A Computational Model of Transmission Line Loudspeakers”, 92nd AES Convention, 1992, preprint no. 3326.
 G. L. Augspurger, “Loudspeaker on Damped Pipes”, JAES Volume 48 Issue 5 pp. 424-436; May 2000.
 B. Olney, “A Method of Eliminating Cavity Resonance, Extending Low Frequency Responce and Increasing Acoustical Damping in Cabinet Type Loudspeakers”, J. Acoust. Soc. Amer. Volume 8; October 1936.
 A. R. Bailey, “A Non-resonant Loudspeaker Enclosure”, Wireless World; October 1965.
 A. R. Bailey, “The Transmission Line Loudspeaker Enclosure”, Wireless World; May 1972.
 L. J. S. Bradbury, “The Use of Fibrous Materials in Loudspeaker Enclosures”, JAES Volume 24 Issue 3 pp.162-170; April 1976.
 R. M. Bullock and P. E. Hillman, “A Transmission-line Woofer Model”, 81st AES Convention, 1986, Preprint no. 2384.
 B. N. Locanthi, “Application of Electric Circuit Analogies to Loudspeaker Design Problems”, JAES Volume 19 Issue 9 pp. 778-785; October 1971.
 G. L. Augspurger, “Transmission Lines Updated, Part 1”, Speaker Builder; 2/00.
 G. L. Augspurger, “Transmission Lines Updated, Part 2”, Speaker Builder; 3/00.
 G. L. Augspurger, “Transmission Lines Updated, Part 3”, Speaker Builder; 4/00.
 L. L. Beranek, “Acoustics”, New York: McGraw-Hill; 1954.
 W. M. Leach, Jr., “Computer-Aided Electroacustic Design with SPICE”, JAES Volume 39 Issue 12; December 1991.
 V. Landi, “La linea di trasmissione”, AUDIOreview, n.104-106-112-132-133; 1991-1993.