Thiele/Small

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"Thiele/Small" commonly refers to 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. Many of the parameters are strictly defined only at the resonance frequency, but they are generally applicable in the frequency range where the diaphragm motion is purely pistonic, or without cone breakup. 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. 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. 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.

Contents

[edit] History

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 also representative of a keen 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. Simplifying assumptions by J. F. Novak in a 1959 paper (validated by actual measurements) led to a practical solution for the response of a given loudspeaker in a box. In 1961, leaning heavily on Novak, 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 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.

Many others continued to develop various aspects of loudspeaker enclosure design in the 1960's and early 1970's. From 1968-1972 J. E. Benson published three articles in an Australian journal that thoroughly analyzed sealed, vented and passive radiator designs. Beginning in June 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. J.E. Benson was the examiner for Richard Small's graduate thesis.

[edit] 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.

  • Sd - Projected area of the driver diaphragm, in square metres.
  • Mms - Mass of the diaphragm, including acoustic load, in kilograms.
  • Cms - Compliance of the driver's suspension, in metres per newton (the reciprocal of its 'stiffness').
  • Rms - The mechanical resistance of a driver's suspension (ie, 'lossiness') in N·s/m
  • Le - Voice coil inductance measured in millihenries (mH).
  • Re - 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).

[edit] Small signal parameters

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 (ie, proportional to its input). These values are more easily measured than the fundamental ones above.

  • Fs – Resonance frequency of the driver
F_{\rm s} = \frac{1}{2 \pi\cdot\sqrt{C_{\rm ms}\cdot M_{\rm ms}}}
Q_{\rm es} = \frac{2 \pi\cdot F_{\rm s}\cdot M_{\rm ms} \cdot R_{\rm e}}{(Bl)^2}
  • Qms – Mechanical Q of the driver at Fs
Q_{\rm ms} = \frac{2 \pi\cdot F_{\rm s}\cdot M_{\rm ms}}{R_{\rm ms}}
  • Qts – Total Q of the driver at Fs
Q_{\rm ts} = \frac{Q_{\rm ms} \cdot Q_{\rm es}}{Q_{\rm ms} + Q_{\rm es}}
  • Vas – Volume of air (in cubic metres) which, when acted upon by a piston of area Sd, has the same compliance as the driver's suspension. To get Vas in litres, multiply the result of the equation below by 1000.
V_{\rm as} = \rho \cdot c^2 \cdot S_{\rm d}^2 \cdot C_{\rm ms}

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).

[edit] Large signal parameters

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

  • Xmax - 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 excursions, especially those in excess of this parameter.
  • Xmech - Maximum physical excursion of the driver before physical damage. With a sufficiently large input, the excursion will cause damage to the voice coil or other moving part of the driver.
  • Pe - Thermal power handling capacity of the driver, in watts. This value is difficult to characterize and is often overestimated, by manufacturers and others.
  • Vd - Peak displacement volume, calculated by Vd = Sd·Xmax

[edit] Other parameters

  • Zmax - The impedance of the driver at Fs, used when measuring Qes and Qms.
Z_{max} = R_e(1+\frac{Q_{ms}}{Q_{es}})
  • 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.
EBP = \frac{F_s}{Q_{es}}
  • Znom - The nominal impedance of the loudspeaker, typically 4, 8 or 16 ohms.
  • η0 - The reference or "power available" efficiency of the driver, in percent.
\eta_0 = \left(\frac{\rho . Bl^2 . S_{d}^2}{2 . \pi . c . M_{ms}^2 . R_{e}}\right)\times100\ %
The expression ρ/(2*pi*c) can be replaced by the value 5.445×10-4 m^2*s/kg for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 5.365×10-4.
  • A version more easily calculated with typical published parameters is:
\eta_0 = \left(\frac{4 . \pi^2 . F_s^3 . V_{as}}{c^3 . Q_{es}}\right)\times100\ %
The expression 4*pi2/c3 can be replaced by the value 9.523×10–7 s³/m³ for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 9.438×10–7.
  • From the efficiency, we may calculate sensitivity, which is the sound pressure level a speaker produces for a given input:
A speaker with an efficiency of 100% (1.0) would output a watt of energy for every watt input. Considering the driver as a point source in an infinite baffle, at one meter this would be distributed over a hemisphere with area 2*pi m^2 for an intensity of (1/(2*pi))=0.159154 W/m^2, which gives an SPL of 112.02 dB.
SPL at 1 meter for an input of 1 watt is then:
dB(1 Watt) = 112.02+ 10*log(η0)
SPL at 1 meter for an input of 2.83 Volts is then:
dB(2.83V) = dB(1 Watt) + 10*log(8/Re)

[edit] Qualitative descriptions

Cross-section of a dynamic cone loudspeaker. Image not to scale.
Cross-section of a dynamic cone loudspeaker. Image not to scale.

Fs
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 Fs, and input signals significantly below Fs can cause uncontrolled motion, mechanically endangering the driver. Woofers typically have an Fs in the range of 13–60 Hz. Midranges usually have an Fs in the range of 60–500 Hz and tweeters between 500 Hz and 4 kHz. A typical factory tolerance for Fs spec is ±15%.

Qts
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 Qts is proportional to the energy stored, divided by the energy dissipated, and is defined at resonance (Fs). Most drivers have Qts values between 0.2 and 0.8.

Qms
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 Qms indicates lower damping losses, and low Qms indicates higher. The main effect of Qms is on the impedance of the driver, with high Qms drivers displaying a higher impedance peak. One predictor for low Qms is a metallic voice coil former. These act as eddy-current brakes and increase damping, reducing Qms. They must be designed with an electrical break in the cylinder (so no conducting loop). Some speaker manufacturers have placed shorted turns at the top and bottom of the voice coil to prevent it leaving the gap, but the sharp noise created by this device when the driver is overdriven is alarming and was perceived as a problem by owners.

Qes
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, Qes is the dominant factor in the voice coil damping. Qes depends on amplifier output impedance. The formula above assumes zero output impedance. When an amplifier with nonzero output impedance is used, its output impedance should be added to Re for calculations involving Qes.

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 Qes.

Vas
Measured in litres (L) or cubic metres, 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 (Sd) as the cone. Larger values mean lower stiffness, and generally require larger enclosures. Vas varies with the square of the diameter. A typical factory tolerance for Vas spec is ±20-30%.

Mms
Measured in grams (g) or kilograms (kg), 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. Mmd is the cone mass without the acoustic load, and the two should not be confused. Some simulation software calculates Mms when Mmd is entered.

Rms
Units are not usually given for this parameter, but it is in mechanical 'ohms'. Rms is a measurement of the losses, or damping, in a driver's suspension and moving system. It is the main factor in determining Qms. Rms is influenced by suspension topology, materials, and by the voice coil former (bobbin) material.

Cms
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 Vas will be.

Re
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.

Le
Measured in millihenries (mH), this is the inductance of the voice coil. The coil is a lossy inductor, in part due to losses in the pole piece, so the apparent inductance changes with frequency. Large Le values limit the high frequency output of the driver and cause response changes near cutoff. Simple modeling software often neglects Le, and so does not include its consequences. Building a copper cap or copper shorting rings near the pole piece can reduce this effect.

Sd
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 Sd than conventional types.

Xmax
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 Xmax determined at 10% THD. This method better represents actual driver performance, but is harder and more time-consuming to determine.

Vd
Specified in litres (L). The volume displaced by the cone, equal to the cone area (Sd) multiplied by Xmax. 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 Xmax. Comparing Vd values will give an indication of the maximum output of a driver at low frequencies. High Xmax, 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 Xmax 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.

Sensitivity
The sound pressure, in dB, produced by a speaker in response to a specified stimulus. Usually this is specified at an input of 1 Watt or 2.83Volts (2.83 Volts = 1 Watt into an 8 Ohms load) at a distance of one metre.

[edit] Measurement notes - large signal behavior

Some caution is required in measuring and interpreting T/S parameters. It is 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 varies: with with drive level, with voice coil temperature, over the life of the driver, etc. Cms increases the farther the coil moves from rest. Bl is generally maximum at rest, and drops as the voice coil approaches Xmax. 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) thermal failure.

As an example, Fs and Vas may vary considerably with input level, due to nonlinear changes in Cms. A typical 110 mm diameter full-range driver with an Fs of 95 Hz at 0.5 V signal level, might drop to 64 Hz when fed a 5 V input. A driver with a measured Vas of 7 L at 0.5 V, may show a Vas increase to 13 L when tested at 4 V. Qms is typically stable ±2%, regardless of drive level, but Qes and Qts drop >13% as the signal level rises from 0.5 V to 4 V, due to the changes in Bl. Because Vas may rise substantially (eg, >80%) and Fs drop considerably (eg, >30%), with only 3% change in measured Mms, the 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 sensitivity has not changed at all, only a modeling variable shorthand for it. From this example then, the measurement voltage to be preferred whilst designing an enclosure or system is the one likely to represent typical operating conditions which is difficult to determine in a continuously changing operating environment.

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; examples include mor4e heavily load diaphragms more heavily thus increasing efficiency, and there have been several attempts at designing cooling arrangements for driver magnetic structures, which will in turn lower voice coil temperatures and reduce resistance increases. Sophisticated magnet or coil designs attempt to linearize Bl and reduce the value and modulation of Le. Larger, more linear spiders can increase the linear range of Cms, but the large signal values of Bl and Cms must be balanced to avoid a phenomenon called dynamic offset.

[edit] Lifetime changes in driver behavior

The mechanical components in typical speaker drivers may change over time. Temperature has a strong, generally reversible effect, drivers are stiffer at lower temperatures. The elastic properties of surrounds and spiders change with temperature and with chemical changes associated with aging (ie, oxidation in foam and natural rubber; note: some synthetic rubbers are more stable). The changes in behavior from these causes are not all in the same direction, so the net effects of them are not easily predicted. 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 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 (eg, 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 as much as 15%+ over these initial periods. Other studies and reports show little change, or reversible changes after only the first few minutes. 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. Many changes early in driver life are complementary (such as a reduction in Fs accompanied by a rise in Vas) and result in minimal changes (small fractions of a dB) in frequency response. If the performance of loudspeaker system being designed is critical, as in high order or heavily equalized systems, it is advisable to measure T/S parameters after a period of run-in, and to model the effects of parameter changes to judge the sensitivity of the design to these variations.

[edit] Measurement techniques

There are numerous methods to measure T/S 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 resting on a surface) and/or in sealed or vented boxes or with varying amounts of mass added to the diaphragm. The measurement environment has an effect on the parameters, and to be strictly correct one should measure parameters with the driver in an acoustic environment similar to that in which they will be in when housed in an enclosure.<! ref. Benson/>

The most common (DIY-friendly) method before the advent of computer-controlled measurement techniques is the free air constant current method. The classic constant current method uses a large (200 to 1000 ohm) resistance in series with the driver and a signal generator is used to vary the excitation frequency. The voltage across the loudspeaker terminals is measured and considered proportional to the impedance. It is assumed that variations in loudspeaker impedance will have little effect on the current through the loudspeaker. This is an approximation, and the method results in Q measurement errors for drivers with a high Zmax. Inexpensive AC voltmeters are designed to measure house current and may introduce additional errors below 40Hz or above several hundred Hz.

Another method is to use a smaller (eg. 10 ohm) series resistor and measure the voltage across the driver and across 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.

[edit] 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).

[edit] See also

[edit] External links