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6.10. Measurement of Thiele–Small parameters

Before embarking on a loudspeaker enclosure design, we need to know the six Thiele–Small parameters that will be used to calculate the low-frequency response of our chosen drive unit in the enclosure. Most drive unit manufacturers now provide these on their data sheets, but if they are not available, we have to measure them. Even if they are available, production tolerances are such that we cannot always expect our computed frequency response to match the measured one unless the model used for computation is based on parameters obtained from the measured sample. In this section, it is shown how to obtain the Thiele–Small parameters purely by measuring the electrical input voltage at different frequencies using a multimeter and a calibrated variable-frequency oscillator.

During the tests, the electrodynamic loudspeaker is preferably mounted in a baffle such as the IEC 268-5 baffle [7], which is the standard baffle used by manufacturers. We see from Fig. 12.28 that if the outer diameter of the baffle is at least four times that of the loudspeaker, the radiation mass or reactive load is that of a piston in an infinite baffle. It should be borne in mind that if the loudspeaker is measured without any baffle, the radiation mass is halved which will result in a small error that can be corrected. It is not essential to perform the tests in an anechoic chamber: A large room that is not too noisy or reverberant will suffice. A variable-frequency source of sound with an output impedance R _g greater than 20 times the nominal impedance of the loudspeaker is connected to the loudspeaker terminals. An AC voltmeter is then connected across the terminals. The value of R _g should include both the inherent output impedance of the generator and any external resistor connected to in order to make up the desired impedance. The value of e _g(rms) is that measured by the meter before the loudspeaker is connected. Some AC meters are only designed to work at around 50–60 Hz, so it is worth checking to see if the reading varies with frequency with the loudspeaker disconnected. If it does, the open-circuit readings should be used to calibrate the measurements to the loudspeaker. The parameters are then determined as follows.

Measurement of suspension resonance frequency f _S

To measure f _S, the frequency is varied until a maximum meter reading is obtained (see Fig. 6.9). From Fig. 6.2 and Fig. 6.8, we see that maximum electrical loudspeaker impedance corresponds to maximum mechanical admittance, which in turn occurs at the resonance frequency f _S or ω _S. The reading obtained at this frequency is e _max.

Figure 6.9 Circuit for determining the Thiele–Small parameters of a loudspeaker.

Measurement of Q _MS and Q _ES

The minimum reading e _min in Fig. 6.8 is found by reducing the frequency until the voltage reading no longer changes. Increase the frequency again until the voltage reading gives a value of

e_{mid} = \sqrt{e_{max} e_{\min}}

$e_{mid} = \sqrt{e_{max} e_{\min}}$

. The frequency at this point is f _L. Then, increase the frequency beyond f _S (maximum voltage reading) until the voltage reading gives a value of

\sqrt{e_{max} e_{\min}}

$\sqrt{e_{max} e_{\min}}$

for a second time. The frequency at this point is f _U. Note that

f_{S} = \sqrt{f_{L} f_{U}}

$f_{S} = \sqrt{f_{L} f_{U}}$

. The mechanical Q is then given by

Q_{M S} = \frac{f_{S}}{f_{U} - f_{L}} (\frac{e_{g} - e_{\min}}{e_{g} - e_{\max}}) \sqrt{\frac{e_{\max}}{e_{\min}}}

$Q_{M S} = \frac{f_{S}}{f_{U} - f_{L}} (\frac{e_{g} - e_{\min}}{e_{g} - e_{\max}}) \sqrt{\frac{e_{\max}}{e_{\min}}}$

(6.58)

and the electrical Q by

Q_{E S} = (1 - \frac{e_{\max}}{e_{g}}) \frac{e_{\min} Q_{M S}}{e_{\max} - e_{\min}}

$Q_{E S} = (1 - \frac{e_{\max}}{e_{g}}) \frac{e_{\min} Q_{M S}}{e_{\max} - e_{\min}}$

(6.59)

If the signal generator is a current source, then e _g → ∞ and the bracketed terms become unity. These equations assume that the effect of the coil inductance L _E at around f _S is negligible.

Measurement of R _E

The electrical resistance of the voice coil is measured with a milliohm meter.

Measurement of S _D

The effective area of the diaphragm can be determined by coupling its front side to a closed box. The volume of air V ₀ enclosed in the space bounded by the diaphragm and the sides of the box must be determined accurately. Then a slant manometer for measuring air pressure is connected to the airspace. The cone is then displaced a known distance ξ meters, the manometer is read, and the incremental pressure p is determined. Then,

p = \frac{P_{0}}{V_{0}} ξ S_{D}

$p = \frac{P_{0}}{V_{0}} ξ S_{D}$

(6.60)

S_{D} = \frac{P_{0}}{V_{0}} \frac{p}{ξ}

$S_{D} = \frac{P_{0}}{V_{0}} \frac{p}{ξ}$

(6.61)

where P ₀ is the ambient pressure. The pressures P ₀ and p both must be measured in the same units, and V ₀/ξ should be determined in m².

Usually, S _D can be determined accurately enough for most calculations from Fig. 6.1, that is, S _D = πa ². To determine the effective radius a more accurately, we assume that the displacement of the surround (8) decreases linearly between its inner and outer edges which have radii a ₁ and a ₂, respectively. Then, the effective radius is given by

a = \sqrt{\frac{a_{1}^{2} + a_{1} a_{2} + a_{2}^{2}}{3}}

$a = \sqrt{\frac{a_{1}^{2} + a_{1} a_{2} + a_{2}^{2}}{3}}$

(6.62)

and, hence, the effective area by

S_{D} = \frac{π}{3} (a_{1}^{2} + a_{1} a_{2} + a_{2}^{2})

$S_{D} = \frac{π}{3} (a_{1}^{2} + a_{1} a_{2} + a_{2}^{2})$

(6.63)

Measurement of V _AS

The equivalent suspension volume V _AS (see Eq. 6.26) can be obtained in two ways. Either we add mass to the diaphragm and observe the change in resonance frequency or we add stiffness in the form of a sealed enclosure. The first method is simplest and is suitable for most loudspeakers. However, in the case of microspeakers used in mobile devices, it is impractical to attach masses as the risk of destabilizing the diaphragm is too great.

The added mass is usually a rod of nonferrous metal (e.g., enameled copper wire or solder) bent into a circle or spiral and attached to the diaphragm using tape or blu tack so that it does not bounce.

If the original resonance frequency was f _S and the resonance frequency after addition of a mass M _x kg is f′ _S, then

f_{S} = \frac{1}{2 π \sqrt{M_{M S} C_{M S}}}

$f_{S} = \frac{1}{2 π \sqrt{M_{M S} C_{M S}}}$

(6.64)

and

{f^{'}}_{S} = \frac{1}{2 π \sqrt{(M_{M S} + M_{x}) C_{M S}}}

${f^{'}}_{S} = \frac{1}{2 π \sqrt{(M_{M S} + M_{x}) C_{M S}}}$

(6.65)

where C _MS is mechanical compliance of the suspension in m/N. Simultaneous solution of Eqs. (6.64) and (6.65) yields

M_{M S} = \frac{M_{x}}{{(f_{S} / {f^{'}}_{S})}^{2} - 1}

$M_{M S} = \frac{M_{x}}{{(f_{S} / {f^{'}}_{S})}^{2} - 1}$

(6.66)

Combining this with Eqs. (6.27) and (6.28) yields

V_{A S} = (1 - \frac{{f^{'}}_{S}^{2}}{f_{S}^{2}}) \frac{S_{D}^{2} ρ_{0} c^{2}}{{(2 π {f^{'}}_{S})}^{2} M_{x}}

$V_{A S} = (1 - \frac{{f^{'}}_{S}^{2}}{f_{S}^{2}}) \frac{S_{D}^{2} ρ_{0} c^{2}}{{(2 π {f^{'}}_{S})}^{2} M_{x}}$

(6.67)

Alternatively, if the drive unit is mounted in a sealed enclosure of known volume V _B, then the new resonance frequency is given by

f_{C} = \frac{1}{2 π} \sqrt{\frac{C_{M S} + C_{M B}}{M_{M C} C_{M S} C_{M B}}}

$f_{C} = \frac{1}{2 π} \sqrt{\frac{C_{M S} + C_{M B}}{M_{M C} C_{M S} C_{M B}}}$

(6.68)

where C _MB is the mechanical stiffness due to the air in the enclosure given by

C_{M B} = \frac{V_{B}}{S_{D}^{2} ρ_{0} c^{2}}

$C_{M B} = \frac{V_{B}}{S_{D}^{2} ρ_{0} c^{2}}$

(6.69)

Because of the air mass loading within the box, the total moving mass may be modified slightly, in which case we denote a new value M _MC and a new electrical Q _EC:

Q_{E C} = \frac{R_{E}}{B^{2} l^{2}} \sqrt{\frac{M_{M C} (C_{M S} + C_{M B})}{C_{M S} C_{M B}}}

$Q_{E C} = \frac{R_{E}}{B^{2} l^{2}} \sqrt{\frac{M_{M C} (C_{M S} + C_{M B})}{C_{M S} C_{M B}}}$

(6.70)

Simultaneous solution of Eqs. (6.11), (6.64), (6.68), and (6.70) yields

C_{M S} = (\frac{f_{C} Q_{E C}}{f_{S} Q_{E S}} - 1) C_{M B}

$C_{M S} = (\frac{f_{C} Q_{E C}}{f_{S} Q_{E S}} - 1) C_{M B}$

(6.71)

Hence, from Eq. (6.27)

V_{A S} = (\frac{f_{C} Q_{E C}}{f_{S} Q_{E S}} - 1) V_{B}

$V_{A S} = (\frac{f_{C} Q_{E C}}{f_{S} Q_{E S}} - 1) V_{B}$

(6.72)

In Chapter 7 of this book, on Loudspeaker Enclosures, design charts are presented from which it is possible to determine, without laborious computation, the sound pressure from a direct-radiator loudspeaker as a function of frequency including the directivity characteristics. Methods for determining the constants of box and bass-reflex enclosures are also presented. If the reader is interested only in learning how to choose a baffle for a loudspeaker, he or she may proceed directly to Chapter 7. The next part deals with the factors in design that determine the overall response and efficiency of the loudspeaker.

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Table of Contents for Acoustics

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6.10. Measurement of Thiele–Small parameters

Measurement of suspension resonance frequency f S

Measurement of Q MS and Q ES

Measurement of R E

Measurement of S D

Measurement of V AS

Table of Contents for
Acoustics

Measurement of suspension resonance frequency f _S

Measurement of Q _MS and Q _ES

Measurement of R _E

Measurement of S _D

Measurement of V _AS