Acoustic transmission line
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An acoustic transmission line is the acoustic analog of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it. The term "acoustic transmission line" is synonymous with "acoustic waveguide". The now mostly obsolete speaking tube served to transmit sounds to a remote location with minimal loss and distortion, as a simple coaxial cable or waveguide does for electrical signals. Musical wind instruments such as pipe organs, woodwinds and brass instruments can be also be modeled in part as transmission lines, though their job also includes generating the sound, controlling its spectrum, and coupling it efficiently to the open air, functions analogous to those of electronic oscillators, filters and antennas.
The comparison between an acoustic duct and an electrical transmission line is useful in "lumped-element" modeling of acoustical systems, in which acoustic elements like volumes, tubes, pistons, and screens can be modeled as single elements in a circuit. With the substitution of pressure for voltage, and volume particle velocity for current, the equations are essentially the same. Electrical transmission lines can be used to describe acoustic tubes and ducts, provided the frequency of the waves in the tube is below the critical frequency, such that they are purely planar.
The rest of this article will describe a particular use of the term, "acoustic transmission line" as the name of a specific audio speaker enclosure topology, in which sound from the back of the bass speaker chassis passes along a long (generally convoluted) path within the speaker enclosure. The energy is absorbed on this path, or emerges from the open end in phase with the sound radiated from the front of the driver, enhancing the output level at low frequencies.
Proper transmission line loudspeakers employ a tube-like resonant cavity whose length is set between 1/6 and 1/2 the wavelength of the fundamental resonant frequency of the loudspeaker driver being used. The cross-sectional area of the tube is typically comparable to the cross-sectional area of the driver's radiating surface area. This cross section is typically tapered down to approximately 1/4 of the starting area at the terminus or open end of the line. While not all lines use a taper, the standard classical transmission line employs a taper from 1/3 to 1/4 area (ratio of terminus area to starting area directly behind driver). This taper serves to dampen the buildup of standing waves within the line, which can create sharp nulls in response at the terminus output at even multiples of the driver's Fs.
Essentially, the goal of the transmission line is to minimize acoustical or mechanical impedance at frequencies corresponding to the driver's fundamental free air resonance. This simultaneously reduces stored energy in the driver's motion, reduces distortion, and critically damps the driver by maximizing acoustic output (maximal acoustical loading or coupling) at the terminus. This also minimizes the negative effects of acoustic energy that would otherwise (as with a sealed enclosure) be reflected back to the driver in a sealed cavity.
Older acoustical models discuss transmission lines in terms of "impedance mismatch" or pressure waves "reflected" off the terminus opening back into the cavity. In fact, there is no "reflection". The driver mounted in a resonant cavity exhibits behavior akin to "cavitation" in which a series of gas pressurizations and rarefactions oscillate back and forth in a captive state. As the driver propagates this alternating train of weak adjacent pressure and vacuum pulses down the transmission line - waves that fit neatly within the cavity (anti node at terminus) remain largely captive (low acoustic output) while waves that do not (node or peak pressure at the terminus) exhibit high levels of energy transfer. Those that meet neither condition exactly produce output that is neither maximum nor minimum. There is no physical phenomenon that can cause "reflection". The electrical circuit analogy upon which the concept of "reflection" is based has no physical embodiment in an acoustical transmission line. As discussed below, the degree of acoustical coupling achieved and hence, loading, is determined by the difference between the distance from the driver to the terminus and the length of the quarter-wave peak of the fundamental wavefront (Fs) and its odd-ordered harmonics. The greater the difference, the lower the acoustical coupling. The smaller the difference, the greater the acoustical coupling and hence the lower the acoustical impedance.
History of transmission line loudspeakers
The concept was termed "acoustical labyrinth" by Stromberg-Carlson Co. when used in their console radios beginning in 1936.http://www.radiomuseum.org/r/stromberg_acoustical_labyrinth_837.html This type of loudspeaker enclosure was proposed in October 1965 by Dr A.R. Bailey and A.H. Radford in Wireless World (p483-486) magazine. The article postulated that energy from the rear of a driver unit could be essentially absorbed, without damping the cone's motion or superimposing internal reflections and resonance, so Bailey and Radford reasoned that the rear wave could be channeled down a long pipe. If the acoustic energy was absorbed, it would not be available to excite resonances. A pipe of sufficient length could be tapered, and stuffed so that the energy loss was almost complete, minimizing output from the open end. No broad consensus on the ideal taper (expanding, uniform cross-section, or contracting) has been established.
Transmission line speakers fall into essentially two categories: closed or vented.
Closed type transmission lines typically have negligible acoustic output from the enclosure except from the driver. Open ended lines exploit the low-pass filter effect of the line, and the resultant low bass energy emerges to reinforce the output from the driver at low frequencies. Well designed transmission line enclosures have smooth impedance curves, possibly from a lack of frequency-specific resonances, but can have low efficiency if not properly designed.
One key advantage of transmission lines is their ability to conduct the back wave behind the transducer more effectively away from it - reducing the chance for reflected energy permeating back through the diaphragm out of phase with the primary signal. Not all transmission lines designs do this effectively. Most offset transmission line speakers place a reflective wall fairly close behind the transducer within the enclosure - posing a problem for internal reflections emanating back through the transducer diaphragm.
The birth of the modern Transmission Line speaker design came about in 1965 with the publication of A R Bailey’s article in Wireless World, “A Non-resonant Loudspeaker Enclosure Design”, detailing a working Transmission Line. Radford Audio took up this innovative design and briefly manufactured the first commercial Transmission Line loudspeaker. Shortly thereafter John Wright of IMF Electronics designed a range of Transmission Line designs and made them popular through his refinement and development of Bailey’s theory. Although acknowledged as the father of the Transmission Line, Bailey’s work drew on the work on labyrinth design, dating back as early as the 1930s. His design, however, differed significantly in the way in which he filled the cabinet with absorbent materials. Bailey hit upon the idea of absorbing all the energy generated by the bass unit inside the cabinet, providing an inert platform for the drive unit to work from; unchecked, this energy produces spurious resonances in the cabinet and its structure, adding distortion to the original signal.
The Transmission Line (TL) is the theoretical ideal and most complex construction with which to load a moving coil drive unit. The most practical implementation is to fit a drive unit to the end of a long duct that is open ended. In practice, the duct is folded inside a conventional shaped cabinet with the open end of the duct usually appearing as a vent on the front of the cabinet. There are many ways in which the duct can be folded and the line is often tapered in crossection to avoid parallel internal surfaces that encourage standing waves. Depending upon the drive unit and quantity – and various physical properties – of absorbent material, the amount of taper will be adjusted during the design process to tune the duct to remove irregularities in its response. The internal partitioning provides substantial bracing for the entire structure, reducing cabinet flexing and colouration. The inside faces of the duct or line, are treated with an absorbent material to provide the correct termination with frequency to load the drive unit as a TL. A theoretically perfect TL would absorb all frequencies entering the line from the rear of the drive unit but remains theoretical, as it would have to be infinitely long. The physical constraints of the real world, demand that the length of the line must often be less than 4 meters before the cabinet becomes too large for any practical applications, so not all the rear energy can be absorbed by the line. In a realized TL, only the upper bass is TL loaded in the true sense of the term (i.e. fully absorbed); the low bass is allowed to freely radiate from the vent in the cabinet. The line therefore effectively works as a low pass filter, another crossover point in fact, achieved acoustically by the line and its absorbent filling. Below this “crossover point” the low bass is loaded by the column of air formed by the length of the line. The length is specified to reverse the phase of the rear output of the drive unit as it exits the vent. This energy combines with the output of the bass unit, extending its response and effectively creating a second driver.
Phase inversion is achieved by selecting a length of line that is equal to the quarter wavelength of the target lowest frequency. The effect is illustrated in
Fig x, which shows a hard boundary at one end (the speaker) and the open-ended line vent at the other. The phase relationship between the bass driver and vent is in phase in the pass band until the frequency approaches the quarter wavelength, when the relationship reaches 90 degrees as shown. However by this time the vent is producing most of the output – Fig y.
Because the line is operating over several octaves with the drive unit, cone excursion is reduced, providing higher SPL’s and lower distortion levels, compared with reflex and infinite baffle designs.
The calculation of the length of the line required for a certain bass extension appears to be straightforward, based on a simple formula:
λ = 344/4 × f
f is the quarter wavelength frequency
344 ms is the speed of sound in air at 20 degrees C
λ is the length of the transmission line
However the introduction of the absorption materials reduces the velocity of sound through the line, as discovered by Bailey in his original work. L Bradbury published his extensive tests to determine this effect in an AES Journal in 1976  and his results agreed that heavily damped lines could reduce the velocity of sound by as much as 50%, although 35% is typical in medium damped lines. Bradbury’s tests were carried out using fibrous materials, typically longhaired wool and glass fibre. These kinds of materials however produce highly variable effects that are not consistently repeatable for production purposes. They are also liable to produce inconsistencies due to movement, climatic factors and effects over time. High specification acoustic foams, developed by manufacturers such as PMC, with similar characteristics to longhaired wool, provide repeatable results for consistent production. The density of the polymer, the diameter of the pores and the sculptured profiling are all specified to provide the correct absorption for each speaker model. Quantity and position of the foam is critical to engineer a low pass acoustic filter that provides adequate attenuation of the upper bass frequencies, whilst allowing an unimpeded path for the low bass frequencies.
There are therefore two distinct forms of bass loading employed in a TL, which historically and confusingly have been amalgamated in the TL description. Separating the upper and lower bass analysis reveals why the TL has so many advantages over reflex and infinite baffle designs. The upper bass is completely absorbed by the line allowing a clean and neutral response. The lower bass is extended effortlessly and distortion is lowered by the line’s control over the drive unit’s excursion. One of the exclusive benefits of the TL design is its ability to produce very low frequencies even at low monitoring levels.
The complex loading of the bass drive unit demands specific Small-Thiele driver parameters to realise the full benefits of a TL design. Most drive units in the marketplace are developed for the more common reflex and infinite baffle designs and are usually not suitable for TL loading. To design a high efficiency woofer with extended low frequency ability, cones are usually extremely light and flexible with very compliant suspensions. Whilst performing well in a reflex design, these characteristics do not match the demands of a TL design. The drive unit is effectively coupled to a long column of air which has mass. This lowers the resonant frequency of the drive unit, negating the need for a highly compliant device. Furthermore, the control of this column of air requires an extremely rigid cone, to avoid deformation and consequent distortion.
A duct for sound propagation also behaves like a transmission line (e.g. air conditioning duct, car muffler, ...). Its length may be similar to the wavelength of the sound passing through it, but the dimensions of its cross-section are normally smaller than one quarter the wavelength. Sound is introduced at one end of the tube by forcing the pressure across the whole cross-section to vary with time. An almost planar wavefront travels down the line at the speed of sound. When the wave reaches the end of the transmission line, behaviour depends on what is present at the end of the line. There are three possible scenarios:
- The frequency of the pulse generated at the transducer results in a pressure peak at the terminus exit (odd ordered harmonic open pipe resonance) resulting in effectively low acoustic impedance of the duct and high level of energy transfer.
- The frequency of the pulse generated at the transducer results in a pressure null at the terminus exit (even ordered harmonic open pipe anti -resonance) resulting in effectively high acoustic impedance of the duct and low level of energy transfer.
- The frequency of the pulse generated at the transducer results in neither a peak or null in which energy transfer is nominal or in keeping with typical energy dissipation with distance from the source.
|Wikimedia Commons has media related to Loudspeaker.|
- Beranek, Leo (1954) Acoustics. Amer Inst of Physics. ISBN 978-0883184943
- "Hi Fi Heretic - #13 - The Virtuous Transmission Line". Scribd.com. Retrieved 2013-03-13.
- P.NEWELL, K.HOLLAND (2007). Loudspeakers: for music recording and reproduction. UK: Elsevier Ltd. pp. 78–81.
- A R Bailey, “A Non-resonant Loudspeaker Enclosure Design” Wireless World October 1965 P483-486
- L J S Bradbury “The Use of Fibrous Materials in Loudspeaker Enclosures” Journal of the Audio Engineering Society April 1976 P404-412
- Quarterwave loudspeakers – Martin J King, developer of TL modeling software
- Transmission Line Speakers Pages – TL projects, history & more
- Brines Acoustics Articles (Archived 2009-10-24) – Application, tips, essays
- Quarter Wave Tube - DiracDelta.co.uk - description of operation, equation and online calculation