Thiele/Small parameters

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 breakups.

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.

History

The 1925 paper

:The expression <math>\rho/2\pi c</math> can be replaced by the value 5.445×10−4 m2·s/kg for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 5.365×10−4 m2·s/kg.

:A version that is more easily calculated with typical published parameters is:

:<math>\eta_0 = \left(\frac{4 \cdot \pi^2 \cdot f_{\rm s}^3 \cdot V_{\rm as{c^3 \cdot Q_{\rm es\right)\times100\%</math>

:The expression <math>4\pi^2/c^3</math> 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 s3/m3.

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 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 <math>2\pi</math> m2 for an intensity of <math>1/(2\pi)</math> = 0.159155 W/m2. The auditory threshold is taken to be 10–12 W/m2 (which corresponds to a pressure level of 20×10−6 Pa). Therefore a speaker with 100% efficiency would produce an SPL equal to 10log(0.159155/10–12), which is 112.02 dB.

:The SPL at 1 metre for an input of 1 watt is then: dB(1 watt) = 112.02 + 10·log(<math>\eta_0</math>)

:The SPL at 1 metre for an input of 2.83 volts is then: dB(2.83 V) = dB(1 watt) + 10·log(8/<math>R_e</math>) = 112.02 + 10·log(<math>\eta_0</math>) + 10·log(8/<math>R_e</math>)

Qualitative descriptions

thumb|right|450px|Cross-section of a dynamic cone loudspeaker. Image not to scale.

; <math>f_{\rm s}</math>

: 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 <math>f_{\rm s}</math>, and input signals significantly below <math>f_{\rm s}</math> can cause large excursions, mechanically endangering the driver. Woofers typically have an <math>f_{\rm s}</math> in the range of 13–60 Hz. Midranges usually have an <math>f_{\rm s}</math> in the range of 60–500 Hz and tweeters between 500 Hz and 4 kHz. A typical factory tolerance for the value of <math>f_{\rm s}</math> is ±15%.

; <math>Q_{\rm ts}</math>

: A unitless measurement, characterizing the combined electric and mechanical damping of the driver. In electronics, <math>Q</math> is the inverse of the damping ratio. The value of <math>Q_{\rm ts}</math> is proportional to the energy stored, divided by the energy dissipated, and is defined at resonance (<math>f_{\rm s}</math>). Most drivers have <math>Q_{\rm ts}</math> values between 0.2 and 0.5, but there are valid (if unusual) reasons to have a value outside this range.

; <math>Q_{\rm ms}</math>

: 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 <math>Q_{\rm ms}</math> indicates lower mechanical losses, and low <math>Q_{\rm ms}</math> indicates higher losses. The main effect of <math>Q_{\rm ms}</math> is on the impedance of the driver, with high <math>Q_{\rm ms}</math> drivers displaying a higher impedance peak. One predictor for low <math>Q_{\rm ms}</math> is a metallic voice-coil former. These act as eddy-current brakes and increase damping, reducing <math>Q_{\rm ms}</math>. 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. High <math>Q_{\rm ms}</math> drivers are often built with nonconductive formers made from paper or various plastics.

; <math>Q_{\rm es}</math>

: 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 <math>Bl</math> × velocity) decreases the total current through the coil near the resonance frequency, reducing cone movement and increasing impedance. In most drivers, <math>Q_{\rm es}</math> is the dominant factor in the voice coil damping. <math>Q_{\rm es}</math> 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 <math>R_{\rm e}</math> for calculations involving <math>Q_{\rm es}</math>.

; <math>Bl</math>

: Measured in tesla-metres (T·m). Technically this is <math>B \times l</math> or <math>B \times l sin(\theta)</math> (a vector cross product), but the standard geometry of a circular coil in an annular voice-coil gap gives <math>sin(\theta)=1</math>. <math>B \times l</math> is also known as the 'force factor' because the force on the coil imposed by the magnet is <math>B \times l</math> multiplied by the current through the coil. The higher the <math>B \times l</math> product, the larger the force that is generated by a given current flowing through the voice coil. <math>B \times l</math> has a very strong effect on <math>Q_{\rm es}</math>.

; <math>V_{\rm as}</math>

: 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 (<math>S_{\rm d}</math>) as the cone. Larger values mean lower stiffness, and generally require larger enclosures. <math>V_{\rm as}</math> varies with the square of the diameter. A typical factory measurement tolerance for <math>V_{\rm as}</math> is ±20–30%.

; <math>M_{\rm ms}</math>

: 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. <math>M_{\rm md}</math> is the cone/coil mass without the acoustic load, and the two should not be confused. Some simulation software calculates <math>M_{\rm ms}</math> when <math>M_{\rm md}</math> is entered. <math>M_{\rm md}</math> can be very closely controlled by the manufacturer.

; <math>R_{\rm ms}</math>

: Units are not usually given for this parameter, but it is in mechanical 'ohms'. <math>R_{\rm ms}</math> is a measurement of the losses, or damping, in a driver's suspension and moving system. It is the main factor in determining <math>Q_{\rm ms}</math>. <math>R_{\rm ms}</math> is influenced by suspension topology, materials, and by the voice-coil former (bobbin) material.

; <math>C_{\rm ms}</math>

: 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 <math>V_{\rm as}</math> will be. <math>C_{\rm ms}</math> is proportional to <math>V_{\rm as}</math> and thus has the same tolerance ranges.

; <math>R_{\rm e}</math>

: 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. <math>R_{\rm e}</math> should not be confused with the rated driver impedance, <math>R_{\rm e}</math> can be tightly controlled by the manufacturer, while rated impedance values are often approximate at best. American EIA standard RS-299A specifies that <math>R_{\rm e}</math> (or DCR) should be at least 80% of the rated driver impedance, so an 8-ohm rated driver should have a DC resistance of at least 6.4 ohms, and a 4-ohm unit should measure 3.2 ohms minimum. This standard is voluntary, and many 8-ohm drivers have resistances of ≈5.5 ohms, and proportionally lower for lower rated impedances.

; <math>L_{\rm e}</math>

: 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.

External links

Measuring speaker element parameters
Fast Bass, Slow Bass - Myth vs. Fact
Understanding Power Compression
Acoustic Analogous Circuits - the method behind the formulas
The Thiele-Small Loudspeaker Database