Thiele/Small parameters (commonly abbreviated T/S parameters, or TSP) are a set of electromechanical parameters that define the specified low frequency performance of a loudspeaker driver. These parameters are published in specification sheets by driver manufacturers so that designers have a guide in selecting off-the-shelf drivers for loudspeaker designs. Using these parameters, a loudspeaker designer may simulate the position, velocity and acceleration of the diaphragm, the input impedance and the sound output of a system comprising a loudspeaker and enclosure. Many of the parameters are strictly defined only at the resonant frequency, but the approach is generally applicable in the frequency range where the diaphragm motion is largely pistonic, i.e., when the entire cone moves in and out as a unit without cone breakup.
Rather than purchase off-the-shelf components, loudspeaker design engineers often define desired performance and work backwards to a set of parameters and manufacture a driver with said characteristics or order it from a driver manufacturer. This process of generating parameters from a target response is known as synthesis. Thiele/Small parameters are named after A. Neville Thiele of the Australian Broadcasting Commission, and Richard H. Small of the University of Sydney, who pioneered this line of analysis for loudspeakers. A common use of Thiele/Small parameters is in designing PA system and hi-fi speaker enclosures; the TSP calculations indicate to the speaker design professionals how large a speaker cabinet will need to be and how large and long the bass reflex port (if it is used) should be.
The 1925 paper of Chester W. Rice and Edward W. Kellogg, fueled by advances in radio and electronics, increased interest in direct radiator loudspeakers. In 1930, A. J. Thuras of Bell Labs patented (US Patent No. 1869178) his "Sound Translating Device" (essentially a vented box) which was evidence of the interest in many types of enclosure design at the time.
Progress on loudspeaker enclosure design and analysis using acoustic analogous circuits by academic acousticians like Harry F. Olson continued until 1954 when Leo L. Beranek of the Massachusetts Institute of Technology published Acoustics, a book summarizing and extending the electroacoustics of the era. J. F. Novak used novel simplifying assumptions in an analysis in a 1959 paper which led to a practical solution for the response of a given loudspeaker in sealed and vented boxes, and also established their applicability by empirical measurement. In 1961, leaning heavily on Novak's work, A. N. Thiele described a series of sealed and vented box "alignments" (i.e., enclosure designs based on electrical filter theory with well-characterized behavior, including frequency response, power handling, cone excursion, etc.) in a publication in an Australian journal. This paper remained relatively unknown outside Australia until it was re-published in the Journal of the Audio Engineering Society in 1971. It is important to note that Thiele's work neglected enclosure losses and, although the application of filter theory is still important, his alignment tables now have little real-world utility due to neglecting enclosure losses.
Many others continued to develop various aspects of loudspeaker enclosure design in the 1960s and early 1970s. From 1968 to 1972, J. E. Benson published three articles in an Australian journal that thoroughly analyzed sealed, vented and passive radiator designs, all using the same basic model, which included the effects of enclosure, leakage and port losses. Beginning in June 1972, Richard H. Small published a series of very influential articles on direct radiator loudspeaker system analysis, including closed-box, vented-box, and passive-radiator loudspeaker systems, in the Journal of the Audio Engineering Society, restating and extending Thiele's work. These articles were also originally published in Australia, where he had attended graduate school, and where his thesis supervisor was J. E. Benson. The work of Benson and Small overlapped considerably, but differed in that Benson performed his work using computer programs and Small used analog simulators. Small also analyzed the systems including enclosure losses. Richard H. Small and Garry Margolis, the latter of JBL, published an article in the Journal of the Audio Engineering Society (June 1981), which recast much of the work that had been published up till then into forms suited to the programmable calculators of the time.
These are the physical parameters of a loudspeaker driver, as measured at small signal levels, used in the equivalent electrical circuit models. Some of these values are neither easy nor convenient to measure in a finished loudspeaker driver, so when designing speakers using existing drive units (which is almost always the case), the more easily measured parameters listed under Small Signal Parameters are more practical:
S\rm
M\rm
M\rm
C\rm
R\rm
L\rm
R\rm
Bl
These values can be determined by measuring the input impedance of the driver, near the resonance frequency, at small input levels for which the mechanical behavior of the driver is effectively linear (i.e., proportional to its input). These values are more easily measured than the fundamental ones above. The small signal parameters are:
f\rm
f\rm=
| 1 | |
| 2\pi ⋅ \sqrt{C\rm ⋅ M\rm |
Q\rm
Q
f\rm
R\rm
Q\rm=
| 2\pi ⋅ f\rm ⋅ M\rm ⋅ R\rm | |
| (Bl)2 |
=
| R\rm | \sqrt{ | |
| (Bl)2 |
| M\rm | |
| C\rm |
Q\rm
Q
f\rm
Q\rm=
| 2\pi ⋅ f\rm ⋅ M\rm | |
| R\rm |
=
| 1 | \sqrt{ | |
| R\rm |
| M\rm | |
| C\rm |
Q\rm
Q
f\rm
Q\rm=
| Q\rm ⋅ Q\rm | |
| Q\rm+Q\rm |
V\rm
V\rm=\rho ⋅ c2 ⋅
| 2 | |
| S | |
| \rmd |
⋅ C\rm
where
\rho
c
V\rm
These parameters are useful for predicting the approximate output of a driver at high input levels, though they are harder, sometimes extremely hard or impossible, to accurately measure. In addition, power compression, thermal, and mechanical effects due to high signal levels (e.g., high electric current and voltage, extended mechanical motion, and so on) all change driver behavior, often increasing distortion of several kinds:
X\rm
X\rm
P\rm
V\rm
V\rm
S\rm
X\rm
Z\rm
f\rm
Q\rm
Q\rm
Z\rm=
| R | ||||
|
\right)
EBP
EBP>100
EBP<50
50<EBP<100
EBP=
| f\rm | |
| Q\rm |
Z\rm
η0
η0=\left(
| |||||||||
|
\right) x 100\%
The expression
\rho/2\pic
A version that is more easily calculated with typical published parameters is:
η0=\left(
| |||||||||
| c3 ⋅ Q\rm |
\right) x 100\%
The expression
4\pi2/c3
A speaker with an efficiency of 100% (1.0) would output a watt for every watt of input. Considering the driver as a point source in an infinite baffle, at one metre this would be distributed over a hemisphere with area
2\pi
1/(2\pi)
The SPL at 1 metre for an input of 1 watt is then: dB(1 watt) = 112.02 + 10·log(
η0
The SPL at 1 metre for an input of 2.83 volts is then: dB(2.83 V) = dB(1 watt) + 10·log(8/
Re
η0
Re
f\rm
Resonance frequency of driver, measured in hertz (Hz). The frequency at which the combination of the energy stored in the moving mass and suspension compliance is maximum, and results in maximum cone velocity. A more compliant suspension or a larger moving mass will cause a lower resonance frequency, and vice versa. Usually it is less efficient to produce output at frequencies below
f\rm
f\rm
f\rm
f\rm
f\rm
Q\rm
A unitless measurement, characterizing the combined electric and mechanical damping of the driver. In electronics,
Q
Q\rm
f\rm
Q\rm
Q\rm
A unitless measurement, characterizing the mechanical damping of the driver, that is, the losses in the suspension (surround and spider). It varies roughly between 0.5 and 10, with a typical value around 3. High
Q\rm
Q\rm
Q\rm
Q\rm
Q\rm
Q\rm
Q\rm
Q\rm
A unitless measurement, describing the electrical damping of the loudspeaker. As the coil of wire moves through the magnetic field, it generates a current which opposes the motion of the coil. This so-called "Back-EMF" (proportional to
Bl
Q\rm
Q\rm
R\rm
Q\rm
Bl
Measured in tesla-metres (T·m). Technically this is
B x l
B x lsin(\theta)
sin(\theta)=1
B x l
B x l
B x l
B x l
Q\rm
V\rm
Measured in litres (L) or cubic metres, it is an inverse measure of the 'stiffness' of the suspension with the driver mounted in free air. It represents the volume of air that has the same stiffness as the driver's suspension when acted on by a piston of the same area (
S\rm
V\rm
V\rm
M\rm
Measured in grams (g) or kilograms (kg), this is the mass of the cone, coil, voice-coil former and other moving parts of a driver, including the acoustic load imposed by the air in contact with the driver cone.
M\rm
M\rm
M\rm
M\rm
R\rm
Units are not usually given for this parameter, but it is in mechanical 'ohms'.
R\rm
Q\rm
R\rm
C\rm
Measured in metre per newton (m/N). Describes the compliance (i.e., the inverse of stiffness) of the suspension. The more compliant a suspension system is, the lower its stiffness, so the higher the
V\rm
C\rm
V\rm
R\rm
Measured in ohms (Ω), this is the DC resistance (DCR) of the voice coil, best measured with the cone blocked, or prevented from moving or vibrating because otherwise the pickup of ambient sounds can cause the measurement to be unreliable.
R\rm
R\rm
R\rm
L\rm
Measured in henries (H), this is the inductance of the voice coil. The voice coil is a lossy inductor, in part due to losses in the pole piece, so the apparent inductance changes with frequency. Large
L\rm
L\rm
S\rm
Measured in square metres (m2). The effective projected area of the cone or diaphragm. It is difficult to measure and depends largely on the shape and properties of the surround. Generally accepted as the cone body diameter plus one third to one half the width of the annulus (surround). Drivers with wide roll surrounds can have significantly less
S\rm
X\rm
Specified in millimetres (mm). In the simplest form, subtract the height of the voice-coil winding from the height of the magnetic gap, take the absolute value and divide by 2. This technique was suggested by JBL's Mark Gander in a 1981 AES paper, as an indicator of a loudspeaker motor's linear range. Although easily determined, it neglects magnetic and mechanical non-linearities and asymmetry, which are substantial for some drivers. Subsequently, a combined mechanical/acoustical measure was suggested, in which a driver is progressively driven to high levels at low frequencies, with
X\rm
P\rm
Specified in watts. Frequently two power ratings are given, an "RMS" rating and a "music" (or "peak", or "system") rating, usually peak is given as ≈2 times the RMS rating. Loudspeakers have complex behavior, and a single number is really unsatisfactory. There are two aspects of power handling: thermal and mechanical. The thermal capacity is related to coil temperature and the point where adhesives and coil insulation melt or change shape. The mechanical limit comes into play at low frequencies, where excursions are largest, and involves mechanical failure of some component. A speaker that can handle 200 watts thermally at 200 Hz, may sometimes be damaged by only a few watts at some very low frequency, like 10 Hz. Power handling specifications are usually generated destructively, by long-term industry standard noise signals (IEC 268, for example) that filter out low frequencies and test only the thermal capability of the driver. Actual mechanical power handling depends greatly on the enclosure in which the driver is installed.
V\rm
Specified in litres (L). The volume displaced by the cone, equal to the cone area (
S\rm
X\rm
X\rm
X\rm
V\rm
X\rm
X\rm
η0
Reference efficiency, specified in percent (%). Comparing drivers by their calculated reference efficiency is often more useful than using 'sensitivity' since manufacturer sensitivity figures are too often optimistic.
Some caution is required when using and interpreting T/S parameters. Individual units may not match manufacturer specifications. Parameters values are almost never individually taken, but are at best averages across a production run, due to inevitable manufacturing variations. Driver characteristics will generally lie within a (sometimes specified) tolerance range.
C\rm
C\rm
It is also important to understand that most T/S parameters are linearized small signal values. An analysis based on them is an idealized view of driver behavior, since the actual values of these parameters vary in all drivers according to drive level, voice coil temperature, over the life of the driver, etc.
C\rm
Bl
X\rm
R\rm
As an example,
f\rm
V\rm
C\rm
f\rm
V\rm
V\rm
Q\rm
Q\rm
Q\rm
Bl
V\rm
f\rm
M\rm
η0
Level shifts caused by resistive heating of the voice coil are termed power compression. Design techniques which reduce nonlinearities may also reduce power compression, and possibly distortions not caused by power compression. There have been several commercial designs that have included cooling arrangements for driver magnetic structures, which are intended to mitigate voice coil temperature rise, and the attendant rise in resistance that is the cause of the power compression. Elegant magnet and coil designs have been used to linearize
Bl
L\rm
C\rm
Bl
C\rm
The mechanical components in typical speaker drivers may change over time. Paper, a popular material in cone fabrication, absorbs moisture easily and unless treated may lose some structural rigidity over time. This may be reduced by coating with water-impregnable material such as various plastic resins. Cracks compromise structural rigidity and if large enough are generally non-repairable. Temperature has a strong, generally reversible effect; typical suspension materials become stiffer at lower temperatures. The suspension experiences fatigue, and also undergoes changes from chemical and environmental effects associated with aging such as exposure to ultraviolet light, and oxidation which affect foam and natural rubber components badly, though butyl, nitrile, SBR rubber, and rubber-plastic alloys (such as Santoprene) are more stable. The polyester type of polyurethane foam is highly prone to disintegration after 10 to 15 years. The changes in behavior from aging may often be positive, though since the environment that they are used in is a major factor the effects are not easily predicted. Gilbert Briggs, founder of Wharfedale Loudspeakers in the UK, undertook several studies of aging effects in speaker drivers in the 1950s and 1960s, publishing some of the data in his books, notably Loudspeakers: The Why and How of Good Reproduction.
There are also mechanical changes which occur in the moving components during use. In this case, however, most of the changes seem to occur early in the life of the driver, and are almost certainly due to relaxation in flexing mechanical parts of the driver (e.g., surround, spider, etc.). Several studies have been published documenting substantial changes in the T/S parameters over the first few hours of use, some parameters changing by as much as 15% or more over these initial periods. The proprietor of the firm GR Research has publicly reported several such investigations of several manufacturers' drivers. Other studies suggest little change, or reversible changes after only the first few minutes. This variability is largely related to the particular characteristics of specific materials, and reputable manufacturers attempt to take them into account. While there are a great many anecdotal reports of the audible effects of such changes in published speaker reviews, the relationship of such early changes to subjective sound quality reports is not completely clear. Some changes early in driver life are complementary (such as a reduction in
f\rm
V\rm
There are numerous methods to measure Thiele-Small parameters, but the simplest use the input impedance of the driver, measured near resonance. The impedance may be measured in free air (with the driver unhoused and either clamped to a fixture or hanging from a wire, or sometimes resting on the magnet on a surface) and/or in test baffles, sealed or vented boxes or with varying amounts of mass added to the diaphragm. Noise in the measurement environment can have an effect on the measurement, so one should measure parameters in a quiet acoustic environment.
The most common (DIY-friendly) method before the advent of computer-controlled measurement techniques is the classic free air constant current method, described by Thiele in 1961. This method uses a large resistance (e.g.,
R\rm
Q
Z\rm
Z\rm
V
I=V/(R\rm+Z\rm)
Z\rm
Z\rm x R\rm/I
A second method is the constant voltage measurement, where the driver is excited by a constant voltage, and the current passing through the coil is measured. The excitation voltage divided by the measured current equals the impedance.
A common source of error using these first two methods is the use of inexpensive AC metres. Most inexpensive metres are designed to measure residential power frequencies (50–60 Hz) and are increasingly inaccurate at other frequencies (e.g., below 40 Hz or above a few hundred hertz). In addition, distorted or non–sine wave signals can cause measurement inaccuracies. Inexpensive voltmeters are also not very accurate or precise at measuring current and can introduce appreciable series resistance, which causes measurement errors.
A third method is a response to the deficiencies of the first two methods. It uses a smaller (e.g., 10 ohm) series resistor and measurements are made of the voltage across the driver, the signal generator, and/or series resistor for frequencies around resonance. Although tedious, and not often used in manual measurements, simple calculations exist which allow the true impedance magnitude and phase to be determined. This is the method used by many computer loudspeaker measurement systems. When this method is used manually, the result of taking the three measurements is that their ratios are more important than their actual value, removing the effect of poor meter frequency response.