Thiele/Small
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Thiele/Small commonly refers to a set of electromechanical parameters that define how a loudspeaker driver performs. These serve as useful quantities for designing speakers because they are more easily determined experimentally than the fundamental mechanical parameters. They are named after A. N. Thiele of the Australian Broadcasting Commission, and Richard H. Small from the University of Sydney, who are two prominent developers of loudspeaker design theory.
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[edit] History
After the Chester W. Rice and Edward W. Kellog paper in 1925, fueled by developments 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 continued to develop until 1954 when Leo L. Beranek of the Massachusetts Institute of Technology published "Acoustics", his seminal work focused especially on electroacoustics. Simplifying assumptions by J. F. Novak in a 1959 paper led to a simple solution for the response of a given loudspeaker in a box. These simplifications were validated by actual measurements. A. N. Thiele described a series of "alignments" (designs based on filter theory having known frequency responses) in his paper in an Australian Journal in 1961. This paper was relatively unknown in the USA until it was re-published in the Journal of the Audio Engineering Society in 1971. Many others continued the developments on 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 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. These articles were also originally published in Australia, where he attended graduate school.
[edit] Fundamental small signal mechanical parameters
These are the linearized physical parameters of a loudspeaker driver, as defined at small signal levels and modeled in the equivalent circuit. Some of these are not convenient to measure in a finished loudspeaker, so when designing speakers with off the shelf drive units, the more easily measured parameters below are more useful.
- 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 (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 strength and the length of wire in the magnetic field, in T·m (tesla·metres).
[edit] Small signal parameters
These parameters are determined by measuring the input impedance of the loudspeaker (especially near the resonance frequency) at small input levels where the mechanical behavior of the driver is largely linear - or proportional to the input.
- Fs – Resonant frequency of the driver
- Qes – Electrical Q of the driver at Fs
- Qms – Mechanical Q of the driver at Fs
- Qts – Total Q of the driver at Fs
- 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.
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 capability of a driver in a particular configuration.
- Xmax - Maximum linear peak (or sometimes peak-to-peak) excursion (in mm) of the cone
- Xmech - Maximum physical excursion of the driver before damage
- Pe - Thermal power handling capacity of the driver, in watts
- Vd - Peak displacement volume, calculated by Vd = Sd·Xmax
[edit] Other parameters
- Zmax - The impedance of the loudspeaker at Fs, used when measuring Qes and Qms.
- EBP - The Efficiency Bandwidth Product, an indicator of whether a driver should be in a vented or sealed enclosure.
- 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.
- The expression 4*pi^2/c^3 can be replaced by the value 9.506*10^-7 for dry air at 25C. For 25C air with 50% relative humidity the expression evaluates to 9.424*10^-7.
[edit] Qualitative descriptions
Fs
Also called F0, measured in hertz (Hz). The frequency where the combination of the moving mass and suspension compliance allows the driver to resonate. A more compliant suspension or a larger moving mass will cause a lower resonant frequency, and vice versa. Usually it is less efficient to produce frequencies below Fs. Woofers typically have an Fs in the range of 13–60 Hz. Midranges may have an Fs in the range of 60–500 Hz and tweeters have an Fs in between 500 Hz and 4 kHz.
Qts
A unitless measurement, describing the total 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, resonance (Fs). Most drivers have Qts values between 0.2 and 0.8.
Qms
A unitless measurement, describing 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 less losses, and low Qms indicates more. The main effect of Qms is on the impedance of the driver, with high Qms drivers displaying a higher impedance peak. One predictor for Qms is an aluminium voice coil former, which acts as an eddy-current brake and increases damping, reducing Qms.
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, reducing cone movement. In most drivers, Qes is the dominant damping force.
Bl
Measured in tesla-metres (T·m). Also known as the force factor. This is the cross product of the magnetic field (B) and the length of conductor in the gap (L). The higher the Bl value, the larger the force generated by a given current flowing through the voice coil. The force on the coil imposed by the magnet is Bl multiplied by the current through the coil.
Vas
Measured in litres (L), describes the stiffness of the suspension with the driver 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.
Mms
Measured in grams (g), it is the mass of the cone, coil and other moving parts of a driver, including the acoustic load. 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
Describes the amount of 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 and materials and by voice coil former (bobbin) material.
Cms
Measured in litres (L). Describes the compliance (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 80% of the rated driver impedance, so an 8-ohm unit should have a DC resistance of 6.4 ohms, and a 4-ohm unit should measure 3.2 ohms.
Le
Measured in millihenries (mH), it is the approximate inductance of the coil. The coil is a lossy inductor 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 modelling software neglects the effects of Le, so cannot predict the effects.
Sd
Measured in square metres (m²). The effective area of the cone or diaphragm. Generally accepted as the cone body diameter plus half the width of the annulus (surround). Wide roll surrounds can have signifcicantly 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, taking the absolute value and dividing 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 ignores 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 driver performance.
Vd
Specified in litres (L). The volume displaced by the cone, equal to the cone area (Sd) multiplied by Xmax. The same Vd value may be obtained 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 drivers may be inefficient, since much of the voice coil winding is outside the magnetic gap at any time and therefore contributes little or nothing to cone motion.
η0
Specified in percent (%). Comparing drivers by their reference efficiency is more useful than using 'sensitivity' since sensitivity figures may be inflated by manufacturers in order to make their product appear louder.
Measurement notes
Some caution is required in measuring and interpreting T/S parameters. Fs varies considerably with excitation level. A typical 110 mm full-range driver with an Fs of 95 Hz at 0.5 V signal level, drops to 64 Hz when excited with a 5 V signal. The higher voltage reading is preferred, as more typical of operating conditions. A typical Asian factory's Fs spec is ±15%. Vsub>as drops with increasing excitation level. A driver showing a Vas of 7 L at 0.5 V, may show 13 L when tested at 4 V. Qms is stable ±2%, regardless of excitation level, but Qes and Qts drop >13% as the signal level rises from 0.5 V to 4 V. Because Vas rises >80%, and Fs drops >30%, with only 3% change in measured Mms, calculated sensitivity (η0) can drop by >30% as the test signal level rises from 0.5 V to 4 V. Of course, the driver's actual sensitivity has not changed at all.
Notes about large signal behavior
It is important to note that T/S parameters are linearized small signal values. This is an idealized view of the performance of a loudspeaker, and the values of parameters vary with drive level. Fs generally decreases as power level increases. Bl is generally maximum at rest, and drops as the voice coil approaches Xmax. Re increases as the coil heats and the value will double by 270 °C, the point of rapid driver failure. The result of these level-dependent nonlinearities is some complication in computer modelling, usually predicting higher SPL than can actually be achieved. Sophisticated magnet or coil designs attempt to linearize Bl and reduce the value and modulation of Le.
[edit] References
- (1954) Beranek, Leo L., "Acoustics", New York : McGraw-Hill, ISBN 0-88318-494-X
- (1996) Benson, J.E., "Theory and Design of Loudspeaker Enclosures", Indianapolis, Howard Sams & Company ISBN 0-7906-1093-0
- Thiele, A.N., "Loudspeakers in Vented boxes" Journal of the Audio Engineering Society, 1971 May and June
- Small, Richard "Closed Box Loudspeaker Systems" Journal of the Audio Engineering Society, Part I: "Analysis" December 1972, Part 2 : "Synthesis" Jan/Feb 1973.
