Measuring Thiele/Small parameters
Technical note
Measuring Thiele/Small parameters
A lot can be said about how to measure loudspeaker Thiele/Small parameters. There are
different methods of measurement and they are not all equally good (this will not be
discussed here). Due to drive unit non-linearities, there is not one final set of parameters
for any drive unit. Several of the parameters are a function of cone/voice coil excursion,
hence they depend on the input voltage/current level. At SB Acoustics we use the delta
mass method (as opposed to delta compliance). We use a constant voltage source and a
150 Ω resistor in series with the drive unit. Hence, this is neither a constant voltage nor a
constant current measurement (the value of the resistor is not nearly large enough to
approximate constant current, which is not the intention either).
Prior to measuring T/S parameters, the drive unit should be broken in – for two reasons. If
the suspension of the drive unit is not broken in, its compliance will slightly increase during
the measurement, thus affecting/distorting the results. Furthermore, eventually the drive
unit will end up being broken in (it actually does not take that long), which is why it is
recommended to use T/S parameters that apply for a broken-in drive unit when calculating
box volumes and tuning frequencies. The free air resonance frequency typically drops
about 10-15% during break-in, as the suspension compliance increases. This directly
affects the Q-factors and the equivalent volume.
To break in a drive unit, you are going to need a sine wave or a noise generator with
adjustable output voltage (the former is recommendable) and a power amplifier. Using a
sine wave generator, adjust the frequency somewhere below the expected free air
resonance frequency of the drive unit (typically about 80% of this value). Slowly turn up
the voltage until the suspension reaches maximum displacement. Keep it below clipping
level. Usually you can hear when the suspension goes into clipping mode - there will be
some kind of mechanical noise. Adjust the voltage slightly below this level. During the
process of breaking in the drive unit, it might be necessary to turn down the voltage a little
to keep it from clipping, as the suspension softens. It needs to run for about ten minutes.
Before measuring, let the drive unit cool to ambient temperature.
You are now ready to measure the T/S parameters. First measure the voice coil DCresistance, Re. It is recommended to use a 4-wire ohm-meter (which is quite expensive), in
which case the lead wire resistance is eliminated. You can also use a really good
multimeter, but be advised that many low- and mid-priced multimeters will not do the job
very well. It needs to have a low resistance range (like 0-20 Ω). It is advisable not to use
the lead wires that come with the multimeter when using the lower range, as the quality of
these wires is often too poor. Accuracy should be better than ±0.1 Ω, which requires offset
correction. Resistance is dependent on temperature. A copper voice coil that measures
6.2 Ω at 25 °C will only measure 6.07 Ω at 20 °C. SB Acoustics specifies voice coil DCresistance at 25 °C and our tolerance is ±0.15 Ω or ±2% (larger value applies).
1
In order to avoid unwanted mechanical resonances that may affect the results, the drive
unit should be held firmly in free air in a vertical position.
Make sure not to seal any vent in the pole piece. Therefore, do not place the drive unit on
the floor (which would also make it resonate at some frequency). Furthermore, the drive
unit should not be placed near a reflective surface, such as a wall, as this will change the
radiation impedance – and we want this to be a free air measurement.
The next step is to measure the impedance curve (i.e. modulus of the impedance). To do
this, you need some kind of measurement system. At SB Acoustics we use a stepped sine
sweep with narrow frequency spacing (at least 1/24 oct. is recommended). High frequency
resolution is crucial in order to be able to accurately determine the free air resonance
frequency and the maximum impedance at this frequency – especially with high-Qms drive
units. If it looks like the top of the impedance peak has been chopped off, you need more
measurement points. The voltage that should be applied to the terminals of the drive unit
depends on its size/type. For a typical mid-woofer, the voltage should be about 1 V (rms)
at the resonance frequency.
Notice, only SI-units are used in the following equations. Results may be converted into
other units afterwards.
Determine the free air resonance frequency, fs [Hz].
Determine the maximum impedance level (at the free air resonance frequency), Zmax [Ω].
2
Calculate the impedance level, Z1,2 [Ω], at the side frequencies.
Z 1, 2 = Re ⋅ Z max
Determine the side frequencies (these are not the quadrant frequencies!), f1 and f2 [Hz].
Verify:
f1 ⋅ f 2 = f s
Deviation from the measured value indicates drive unit non-linearities (or an inaccurate
measurement).
Calculate the mechanical Q-factor, Qms.
Qms =
fs
⋅ R0
f 2 − f1
, where
R0 =
Z max
Re
Qes =
Qms
R0 − 1
Calculate the electrical Q-factor, Qes.
Calculate the total Q-factor, Qts.
Qts =
Qms ⋅ Qes
Qms + Qes
3
Now attach an accurately weighed amount of sticky material (e.g. plasticine), ∆m [kg], to
the centre part of the cone or to the dust cap (be careful not to damage the dust cap). Do
not be tempted to use small magnets on either side of the cone as added mass, as this
will affect the measurement results. Add about 70% of the expected moving mass of the
drive unit. Make sure that the entire added mass sticks to the cone, so that no part of it
can vibrate freely. Notice that removal of sticky material may cause certain types of paper
cones to delaminate in the top layers.
Once again measure the impedance curve - do not adjust the voltage (or current).
Determine the new/shifted resonance frequency, fo [Hz].
Calculate the moving mass (incl. air), mms [kg].
mms =
∆m
2
 fs 
  − 1
 fo 
Calculate the mechanical (loss) resistance, Rms [kg/s].
Rms =
2 ⋅ π ⋅ f s ⋅ mms
Qms
Calculate the force factor, BL [Tm].
BL =
(Z max − Re ) ⋅ Rms
Calculate the compliance of the suspension, Cms [m/N].
Cms =
1
(2 ⋅ π ⋅ f s )2 ⋅ mms
4
Measure the piston diameter (as shown in the figure below), Dd [m].
2
Calculate the piston area, Sd [m ].
D 
Sd = π ⋅  d 
 2 
2
The effective piston area is approximately 95% of the above calculated value in most
cases. Hence, a correction should be made. A correction must also be made if the drive
unit uses a porous/vented dust cap or a phase plug, obviously.
Calculate the equivalent volume, Vas [m3].
Vas = C ms ⋅ ρ ⋅ c 2 ⋅ S d2
ρ is the density of air, ρ = 1.2 kg/m3 (20 °C, 50% RH, 1 atm)
c is the speed of sound in air, c = 344 m/s (20 °C, 50% RH)
Determine the minimum impedance level above the resonance frequency, Zmin [Ω].
Determine the frequency, f3 [Hz], at which the impedance level is 3 dB above Zmin (3 dB is
a factor of √2).
5
Calculate the voice coil inductance, Le [H]. An empirical equation is used, as a voice coil
sitting in a motor system does not behave like a true inductor. This model for voice coil
inductance is rather simple – better lumped models have been made.
 Re ⋅ 20 ⋅ 10 3


 ⋅ 10 −3
+
0
.
5
 2 ⋅π ⋅ f

3


Le =
20
You now have a complete set of Thiele/Small parameters.
6
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