Thiele/Small

"Thiele/Small" commonly refers to a set of electromechanical parameters that define how a loudspeaker driver performs. They are useful when designing speakers because they are more easily determined experimentally than more fundamental mechanical parameters. They are named after A. N. Thiele of the Australian Broadcasting Commission, and Richard H. Small of the University of Sydney, who pioneered this line of analysis for loudspeakers.

Contents 1 History 2 Fundamental small signal mechanical parameters 3 Small signal parameters 4 Large signal parameters 5 Other parameters 6 Qualitative descriptions 7 References 8 See also 9 External links

History

After the Chester W. Rice and Edward W. Kellog paper in 1925, fueled by advances in radio and electronics, interest in direct radiator loudspeakers increased. In 1930, A. J. Thuras of Bell Labs patented (US Patent No. 1869178) his "Sound Translating Device" -- essentially a vented box, which stimulated interest in many types of enclosure design. Progress on loudspeaker enclosure design using acoustic analogous circuits somewhat continued until 1954 when Leo L. Beranek of the Massachusetts Institute of Technology published "Acoustics", a book summarizing and extending electroacoustics. Simplifying assumptions by J. F. Novak in a 1959 paper led to a practical solution for the response of a given loudspeaker in a box. These simplifications were validated by actual measurements. About the same time, A. N. Thiele described a series of "alignments" (ie, enclosure designs based on electrical filter theory with well characterized behavior, including frequency response, power handling, cone excursion, etc) in papers in an Australian Journal in 1961. This paper remained relatively unknown outside Australia until it was re-published in the Journal of the Audio Engineering Society in 1971. Many others continued to develop various aspects of loudspeaker enclosure design in the 1960's and early 1970's. And, from 1968-1972 J. E. Benson published three articles in an Australian journal that analyzed sealed, vented and passive radiator designs. Beginning in December 1972, Richard Small published a series of very influential articles in the Journal of the Audio Engineering Society covering Thiele's work and extending it. These articles were also originally published in Australia, where he had attended graduate school.

Fundamental small signal mechanical parameters

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_d - Projected area of the driver diaphragm, in square metres.
• M_ms - Mass of the diaphragm, including acoustic load, in kilograms.
• C_ms - Compliance of the driver's suspension, in metres per newton (the reciprocal of its 'stiffness').
• R_ms - The mechanical resistance of a driver's suspension (ie, 'lossiness') in N·s/m
• L_e - Voice coil inductance measured in millihenries (mH).
• R_e - DC resistance of the voice coil, measured in ohms.
• Bl - The product of magnet field strength in the voice coil gap and the length of wire in the magnetic field, in T·m (tesla·metres).

Small signal parameters

These values can be determined by measuring the input impedance of the driver, especially near the resonance frequency, at small input levels for which the mechanical behavior of the driver is effectively linear (ie, proportional to its input). These values are more easily measured than the fundamental ones above, though accuracy and precision require care.

• F_s – Resonance frequency of the driver

<math>F_{\rm s} = \frac{1}{2 \pi\cdot\sqrt{C_{\rm ms}\cdot M_{\rm ms}}}</math>

• Q_es – Electrical Q of the driver at F_s

<math>Q_{\rm es} = \frac{2 \pi\cdot F_{\rm s}\cdot M_{\rm ms} \cdot R_{\rm e}}{(Bl)^2}</math>

• Q_ms – Mechanical Q of the driver at F_s

<math>Q_{\rm ms} = \frac{2 \pi\cdot F_{\rm s}\cdot M_{\rm ms}}{R_{\rm ms}}</math>

• Q_ts – Total Q of the driver at F_s

<math>Q_{\rm ts} = \frac{Q_{\rm ms} \cdot Q_{\rm es}}{Q_{\rm ms} + Q_{\rm es}}</math>

• V_as – Volume of air (in cubic metres) which, when acted upon by a piston of area S_d, has the same compliance as the driver's suspension. To get V_as in litres, multiply the result of the equation below by 1000.

<math>V_{\rm as} = \rho \cdot c^2 \cdot S_{\rm d}^2 \cdot C_{\rm ms}</math>

Where ρ is the density of air (1.184 kg/m³ at 25°C), and c is the speed of sound (346.1 m/s at 25°C).

Large signal parameters

These parameters are useful for predicting the approximate output of a driver at high input levels.

• X_max - Maximum linear peak (or sometimes peak-to-peak) excursion (in mm) of the cone. Note that, because of mechanical issues, the motion of a driver cone becomes non-linear with large enough inputs, ie those in excess of this parameter.
• X_mech - Maximum physical excursion of the driver before physical damage. With a sufficiently large input, the voice coil and cone will cause voice coil damage or to some mechanical part of the driver.
• P_e - Thermal power handling capacity of the driver, in watts. This value is difficult to characterize and is often overestimated, by manufacturers and others.
• V_d - Peak displacement volume, calculated by V_d = S_d·X_max

Other parameters

• Z_max - The impedance of the driver at F_s, used when measuring Q_es and Q_ms.

<math>Z_{max} = R_e(1+\frac{Q_{ms}}{Q_{es}})</math>

• EBP - The Efficiency Bandwidth Product, an indicator measure. For certain values, a driver is best used in a vented enclosure, while other values suggest a sealed enclosure.

<math>EBP = \frac{F_s}{Q_{es}}</math>

• Z_nom - The nominal impedance of the loudspeaker, typically 4, 8 or 16 ohms.
• η₀ - The reference or "power available" efficiency of the driver, in percent.

<math>\eta_0 = \left(\frac{4 . \pi^2 . F_s^3 . V_{as}}{c^3 . Q_{es}}\right)\times100\ %</math>

The expression 4*pi²/c³ can be replaced by the value 9.506×10^–7 s³/m³ for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 9.424×10^–7.

Qualitative descriptions

F_s
Also called F₀, measured in hertz (Hz). The frequency at which the combination of the moving mass and suspension compliance maximally reinforces cone motion. 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_s, though motion below F_s can cause uncontrolled motion, mechanically endangering the driver. Woofers typically have an F_s in the range of 13–60 Hz. Midranges usually have an F_s in the range of 60–500 Hz and tweeters between 500 Hz and 4 kHz. Q_ts
A unitless measurement, characterizing the combined electric and mechanical damping of the driver. In electronics, Q is the inverse of the damping ratio. The value of Q_ts is proportional to the energy stored, divided by the energy dissipated, and is defined at resonance (F_s). Most drivers have Q_ts values between 0.2 and 0.8. Q_ms
A unitless measurement, characterizing the mechanical damping of the driver, that is, the losses in the suspension (surround and spider.) A typical value is around 3. High Q_ms indicates lower damping losses, and low Q_ms indicates higher. The main effect of Q_ms is on the impedance of the driver, with high Q_ms drivers displaying a higher impedance peak. One predictor for low Q_ms is a metallic voice coil former of a particular configuration. These act as eddy-current brakes and increase damping, reducing Q_ms. The same former, with an electrical break in the cylinder (so no conducting loop) avoids these losses. Q_es
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" decreases the total current through the coil near the resonance frequency, reducing cone movement and increasing impedance. In most drivers, Q_es is the dominant factor in the voice coil damping. Bl
Measured in tesla-metres (T·m). Technically this is B x l (vector cross product or B * l * sin(θ)), but the standard geometry of a circular coil in an annular voice coil gap gives sin(θ)=1. Bl is also known as the 'force factor' because the force on the coil imposed by the magnet is Bl multiplied by the current through the coil. The higher the Bl value, the larger the force generated by a given current flowing through the voice coil. Bl has a very strong effect on Q_es. V_as
Measured in litres (L), is a measure of the free air 'stiffness' of the suspension -- the driver must be 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_d) as the cone. Larger values mean lower stiffness, and generally require larger enclosures. V_as varies with the square of the diameter. M_ms
Measured in grams (g), this is the mass of the cone, coil and other moving parts of a driver, including the acoustic load imposed by the air in contact with the driver cone. M_md is the cone mass without the acoustic load, and the two should not be confused. Some simulation software calculates M_ms when M_md is entered. R_ms
Units are not usually given for this parameter, but it is in mechanical 'ohms'. R_ms is a measurement of the losses, or damping, in a driver's suspension and moving system. It is the main factor in determining Q_ms. R_ms is influenced by suspension topology, materials, and by the voice coil former (bobbin) material. C_ms
Measured in metres per newton (m/N). Describes the compliance (ie, the inverse of stiffness) of the suspension. The more compliant a suspension system is, the lower its stiffness, so the higher the V_as will be. R_e
Measured in ohms (Ω), this is the DC resistance of the voice coil. American EIA standard RS-299A specifies that DCR should be at least 80% of the rated driver impedance, so an 8-ohm rated driver will have a DC resistance of at least 6.4 ohms, and a 4-ohm unit should measure 3.2 ohms minimum. Advertised values are often approximate at best. L_e
Measured in millihenries (mH), this is the inductance of the voice coil. The coil is an inductor in part due to losses in the pole piece, so the apparent inductance changes with frequency. Large L_e values limit the high frequency output of the driver and cause response changes near cutoff. Simple modeling software often neglects the effects of L_e, and so does not include its consequences. Building a copper cap into the magnet structure can reduce this effect. S_d
Measured in square metres (m²). The effective area of the cone or diaphragm. It varies with the conformation of the cone, and details of the surround. Generally accepted as the cone body diameter plus half the width of the annulus (surround). Wide roll surrounds can have significantly less S_d than conventional types. X_max
Specified in millimeters (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 non-linearities and limitations introduced by the suspension. Subsequently, a combined mechanical/acoustical measure was suggested, in which a driver is progressively driven to high levels at low frequencies, with X_max determined at 10% THD. This method better represents actual driver performance, but is harder and more time-consuming to determine. V_d
Specified in litres (L). The volume displaced by the cone, equal to the cone area (S_d) multiplied by X_max. Any particular value may be achieved in any of several ways. For instance, by having a small cone with a large Xmax, or a large cone with a small X_max. Comparing V_d values will give an indication of the maximum output of a driver at low frequencies. High X_max, small cone diameter drivers are likely to be inefficient, since much of the voice coil winding will be outside the magnetic gap at any one time and will therefore contribute little or nothing to cone motion. Likewise, large cone diameter, small X_max drivers are likely to be more efficient as they will not need, and so may not have, long voice coils. η₀
Specified in percent (%). Comparing drivers by their reference efficiency is more useful than using 'sensitivity' since manufacturer sensitivity figures are too often overly optimistic. Measurement notes
Some caution is required in measuring and interpreting T/S parameters. F_s varies considerably with input level. A typical 110 mm diameter full-range driver with an F_s of 95 Hz at 0.5 V signal level, might drop to 64 Hz when fed a 5 V input. The higher voltage value is to be preferred whilst planning an enclosure or system design, as more typical of operating conditions. A typical factory tolerance for F_s spec is ±15%. V_as also changes with increasing excitation level. A driver suggesting a V_as of 7 L at 0.5 V, may suggest 13 L when tested at 4 V. Q_ms is typically stable ±2%, regardless of drive level, but Q_es and Q_ts drop >13% as the signal level rises from 0.5 V to 4 V. Because V_as rises substantially (eg, >80%) and F_s drops considerably (eg, >30%), with only 3% change in measured M_ms, calculated sensitivity (η₀) can drop by >30% as the test signal level changes from 0.5 V to 4 V. Of course, the driver's actual behavior has not changed at all, only a modeling variable shorthand for it. Large signal driver behavior changes
It is important to understand that the T/S parameters are essentially linearized small signal values. An analysis based on them, is an idealized view of the behavior of a driver. In any case, the actual values of these parameters vary with drive level. F_s generally decreases as power level increases. Bl is generally maximum at rest, and drops as the voice coil approaches X_max. C_ms increases the farther the coil moves from rest. Re increases as the coil heats and the value will typically double by 270 °C, a point at which many voice coils are approaching (or have already reached) overheating failure. The result of most of these level-dependent nonlinearities is distortion and lower than predicted output. The level shifts caused by these nonlinearities are often collectively called power compression. Design techniques which reduce nonlinearities will generally reduce both power compression and distortion. Sophisticated magnet or coil designs attempt to linearize Bl and reduce the value and modulation of L_e. Larger, more linear spiders can increase the linear range of C_ms, but the large signal values of Bl and C_ms must be balanced to avoid a phenomenon called dynamic offset. Lifetime changes in driver behavior
The electro-mechanical structures in typical speaker drivers change over time. For some drivers, humidity effects (absorption and evaporation of water molecules) will change behavior, due for example, to changes in moving mass. The elastic properties of surrounds and spiders change for similar reasons and with chemical changes associated with aging (eg, in natural rubber, though many synthetics are more stable) and with exposure to atmospheric pollutants (foam surrounds are particularly prone to deterioration -- there are several firms specializing in replacing such surrounds). The changes in behavior from these causes are not all in the same direction, so the net effects of them are not easily predicted for any driver. Gilbert Briggs, the founder of Wharfedale Loudspeakers in the UK, undertook several studies of aging effects in speaker drivers in the 50s and 60s, publishing some of the data in his book, Loudspeakers. There are also inherent mechanical changes which occur during use. This was also studied at Wharfedale. 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 strictures occurring in the mechanical parts of the driver (eg, surround, spider, etc). Several studies have been published, for selected drivers, documenting substantial changes in the T/S parameters over the first 20 / 30 / 50 hours of use. Some parameters change as much as 15%+ over these initial periods. There are also a great many reports of the audible effects of such changes when playing music or voice in published speaker reviews. Because of these changes, designing a loudspeaker enclosure using the T/S parameters requires good judgement. The performance expected from a particular driver's T/S calculations may not be attained otherwise. A design based on a naive application of a T/S analysis is likely to fail to perform as expected; with luck, the divergence may be acceptably small. In addition, because of the changes noted, designing a crossover for best performance must take them into account. Some understanding and compensation for these changes must be included at several points during a speaker system design. In addition, there is always variation between samples of the same driver model, and those differences can also be large enough to cause significant performance between examples of a speaker system design. Some driver manufacturers are more careful during manufacture and quality control than others. In addition, some loudspeaker system producers burn-in all the drivers they use, to more closely determine their eventual T/S parameters in practice, and individually match them for T/S parameters. This can compensate, in each system produced, for driver performance variations discovered during testing by slight variations in the enclosures (eg, differing amounts of absorbent stuffing, etc).

References

• (1954) Beranek, Leo L., "Acoustics", New York : McGraw-Hill, ISBN 0-88318-494-X
• Briggs, Gilbert, "Loudspeakers", Wharfedale Ltd.
• J.F. Novak, "Performance of Enclosures for Low-Resonance High-Compliance Loudspeakers," J. Audio Eng. Soc., vol. 7, p 29 (Jan. 1959)
• (1996) Benson, J.E., "Theory and Design of Loudspeaker Enclosures", Indianapolis, Howard Sams & Company ISBN 0-7906-1093-0 ( book is a collection of three papers originally published in Australia, 1968-1971)
• Thiele, A.N., "Loudspeakers in Vented Boxes, Parts I and II," J. Audio Eng. Soc., vol. 19, pp. 382-392 (May 1971); pp. 471-483 (June 1971).
• Small, R.H., "Direct-Radiator Loudspeaker System Analysis," J. Audio Eng. Soc., vol. 20, pp. 383-395 (June 1972).
• Small, R.H., "Closed-Box Loudspeaker Systems," J. Audio Eng. Soc., vol. 20, pp. 798-808 (Dec. 1972); vol. 21, pp. 11-18 (Jan./Feb. 1973).
• Small, R.H., "Vented-Box Loudspeaker Systems," J. Audio Eng. Soc., vol. 21, pp. 363-372 (June 1973); pp. 438-444 (July/Aug. 1973); pp. 549-554 (Sept. 1973); pp. 635-639 (Oct. 1973).

See also

• Electrical characteristics of a dynamic loudspeaker
• Acoustical engineering

External links

• Measuring Thiele Small Parameters (method 1 - the classic method)
• Measuring parameters (method 2)
• Fast Bass, Slow Bass - Myth vs. Fact
• Understanding Power Compression
• Acoustic Analogous Circuits - the method behind the formulas