Loudspeaker Thiele/Small
Parameter Extraction
Here
we extract basic free air Theile/Small parameters from the complex impedance
curve of a 10” low frequency
driver. These parameters may be used to verify the manufacturers
published specifications and provide the basis for enclosure design.
Two
external resistors are required to perform this measurement. First a
resistor (R1), to establish a constant current in the circuit is
required. The precision and value of this resistor is not critical.
However, if it is too low (< 30W), its purpose is defeated and if it
is too high (> 5kW) noise immunity of the circuit will be compromised.
Secondly a precision reference resistor (R2), whose value is known to two
decimal points, is required. This provides the software with a known
reference so that it can accurately plot the absolute value of the driver
impedance in the spectrum analyzer.
We
will use "32768_MLS_Impedance_Measurement.process" to perform
the measurement. This process ships with the release version of this
product. It consists of four modules. The first is the signal generator, which
generates a 32768 length MLS stimulus to excite the DUT. Second is the
SoundIO module, which plays the stimulus and records the response of the
driver. Third is the Oscilloscope module, which allows us to view the
time domain response of the driver. Finally is the Spectrum Analyzer,
which performs an FHT/ FFT on the time domain data and allows us to view
impedance vs. frequency and phase vs. frequency graphs.
You
must have a dual channel, duplex sound card to perform this measurement.
1. Measure the value of
the reference resistor using a precision ohmmeter or DC resistance
bridge. Our reference measures in at 10.067W.
2. Measure the DC
resistance of the drivers voice coil (RE) to the nearest 0.1
ohm. Use a precision ohmmeter or bridge. Our voice coil measures in at
6.89W. This application cannot measure the DC resistance of the drivers
voice coil directly. This is due to the fact that most sound cards are AC
coupled, that is they have a capacitor between their internal amplifier
and their output jack, and thus do not pass DC current. Driver voice coil
resistance can be measured using very low frequency sine waves (< 1Hz)
and a series precision resistance but most sound cards cannot reliably
reproduce such frequencies. If you have a 1.5volt battery and voltmeter
available you can wire the loudspeaker and the precision resistor in
series across the battery terminals and calculate the voice coil
resistance using the following equations.
3. Suspend the driver
vertically about half way between the ceiling and the floor using a piece
of wire or twine. Do not place the driver horizontally on a table or
other reflective surface. This will result in cone pre-loading that will
cause errors. Reflections from any nearby surface will cause response
ripples in the impedance curve.
4. Wire the circuit as
shown in Figure 1. Use short, low resistance or shielded wiring.
Note that many sound cards speaker outputs have more swing than their
respective line inputs.
Figure 1: Thiele/Small Extraction Process
Calibration Wiring
5. Open “32768_MLS_Impedance_Measure.process” from the applications File…Open… menu. Press OK if
the “No Compatible Calibration File
Present” message box appears.
6. Open the FFT Options
dialog from the applications Options…FFT… menu and ensure the FFT Size is 32768. Press OK in
the FFT Options dialog box. Press OK when the “No Compatible Calibration File Present” message box appears.
7. Press the Open Mixer
button the SoundIO modules Options group. Select the Options… Properties… menu. Choose the
sound card from the Mixer Device and press the Recording radio button in
the Adjust Volume for group. Press the OK button.
8. Deselect all Record
Control mixer paths except the Line. Adjust the Line mixer slider to its
one-quarter setting and equalize its balance slider.
Figure 2: Recording Mixer Settings
9. Select the Record
Controls Options… Properties… menu. Press the Playback radio button in the Adjust
Volume for group. Press the OK button.
10. Mute all Playback mixer gain
settings except the Master Volume and Wave. Set both volume sliders to one
quarter and equalize their balance sliders.
Figure 3: Playback Mixer Settings
11. Press the applications Run
button. You should be able to see the MLS sequence in the oscilloscope
module as shown in Figure 4. If a SoundIO “No data in record
buffer” message appears first check that
your wiring conforms to Figure 1. If it is correct, increase the mixers
Playback Volume Control and Wave Out sliders or Recording Line controls.
Figure 4: Thiele/Small Extraction Process MLS Sequence Setup
12. If all three controls are at
maximum you may reduce the level at which the sound card triggers. When
in Record/Play mode, the SoundIO module sends a record buffer to the
sound card that is 1.4 longer than required. This is to compensate for
various system delays. It then scans the buffer for the first level that
is greater than the trigger level. It then marks this point as the
beginning of the record and returns the remainder of the record (up to
the number of samples required for the selected FFT size) to the
application. This is the record that the modules processes and sends to
subsequent modules. Trigger level is expressed in terms of percentage
full scale. Check the value in the Trig. Level (%F.S.) in the SoundIO
Trigger Parameters group. If it is greater than 20 select 10 in the combo
box. Press the Run button and check the oscilloscope display again. You
can reduce this value to as low as 1%. This corresponds to 1% of the
sound card full-scale output. You can estimate the length of the buffer
that is sent to the sound card for a given FFT Size from the equation
below.
If you know the full scale output voltage of your sound
card, you can estimate the level that causes the SoundIO module to
trigger from the equation below. Sound cards have a typical input swing
ranging from +0.5 to +2.0 volts.
Once you have a valid trigger, adjust the Play Control and
the Wave sliders so that the signal in the oscilloscope display is not
clipped (as in Figure 4).
13. Press the Calibration button in
the SoundIO Options group. The Calibration dialog box will open.
14. Input Signal Levels calibration
is not required for this process because the spectrum analyzer |Z| scale
measures impedance relatively. Select Auto from the Calibration Type
Select: combo box. Select Vpeak from the Input Cal. Meas. Type Sel: combo
box. Select MLS in the Freq. Cal. Type Sel. combo box.
15. Press the Calibration Run button
and wait for the hour glass cursor to disappear. If a “No data in record buffer” message appears
increase the mixers Playback Master Volume, Wave or Line In sliders.
16. Check both Apply check boxes in
Frequency Response Calibration group box. When the process is fully
calibrated, the calibration dialog should look like Figure 5. Press the
OK button in the calibration dialog and select Yes when the “Calibration Parameter Has Changed. Save To Process
File?” message box appears.
Figure 5: Thiele/Small Extraction Process
Calibration Dialog after Auto Calibration
17. Now rewire the circuit as shown
in Figure 6.
Figure 6:
Thiele/Small Extraction Process Test Wiring
18. Change the Spectrum Analyzers
XAxis Selection to Log10. Set the Stop Frequency to 689Hz using the Stop:
Dn button in the XAxis group. Change the Y-Axis Selection and scale to
|Z| and 10 ohms/div respectively. Enter the exact value of the reference
resistor wired between Ch1 Line-In and Ch2 Line-In in the Ref1: edit box
in the spectrum analyzer. Check the Apply Freq Cal checkbox. Change
the Averaging factor to 4 in the Avg: combo box of in the spectrum
analyzer. This will help smooth the low frequency results.
Figure 7: Thiele/Small Extraction Process
Spectrum Analyzer Module Settings
19. Press the Run button on the
application toolbar and observe the trace in the spectrum analyzer. The
impedance and phase appear in channels one and two respectively.
Figure 8:
Thiele/Small Extraction Process Spectrum Analyzer Module With Driver
Impedance and Phase Plots
20. Find the maximum impedance in
the spectrum analyzer and record it (Rmax) and the frequency
at which it occurred (fSA). Pressing the right mouse button
and dragging it past the impedance peak will show the driver resonant
frequency and peak impedance. Our Rmax = 54.93 ohms and our fSA
= 40.8 Hz.
21. Calculate r0 using
the following equation.
21. Calculate Rx using the following
equation. Use the markers to find the frequencies f1 and f2
on the impedance curve (above and below fSA) that corresponds
exactly to the impedance value of RX. Our frequencies are f1
= 32.9Hz and f2 = 50.7Hz respectively.
22. Check that the square roots of
the product of the frequencies (f1 and f2) are within 1 Hz of the
measured fSA. Use the following equation.
23. Now calculate the mechanical Q
of the driver at its free air resonant frequency (QMS) as
follows:
24. Now calculate the electrical Q
of the driver at its free air resonant frequency (QES) as
follows:
25. Now calculate the total Q of the
driver at its free air resonant frequency (QTS) as follows:
Below
is a table showing the measured and calculated Thiele/Small parameters
using two methods. First the constant current method above (column 2) and
second the voltage divider method (see setup in Figure 9) using three
different values of reference resistance (columns 3 to 5). They are
10.07W, 100.77W and 994.43W respectively. When using the Voltage Divider
method accuracy depends strongly upon the assumption that RREF
is much larger than the driver impedance. As can be seen from Table 1 the
measured value of the driver peak impedance at its resonant frequency (Rmax)decreases
as the value of RREF increases. Hence the calculated values of
QMS, QES and QT (which are inversely
proportional to RMAX) will decrease as RREF
increases.
Figure 9: Thiele/Small Extraction Process
Voltage Divider Method Test Wiring
|
Reference
Resistor
|
10.067W
Const.
Current
|
10.07W
Voltage
Divider
|
100.77W
Voltage
Divider
|
994.43W
Voltage
Divider
|
|
fSA(meas)
|
40.8Hz
|
40.8Hz
|
40.8Hz
|
40.8Hz
|
|
RMAX(meas)
|
54.93 W
|
62.41 W
|
55.41 W
|
54.63 W
|
|
RE(meas)
|
6.89 W
|
6.89 W
|
6.89 W
|
6.89 W
|
|
r0(calc)
|
7.97 W
|
9.05 W
|
8.04 W
|
7.93 W
|
|
RX(calc)
|
19.45 W
|
20.72 W
|
19.53 W
|
19.40 W
|
|
f1(meas)
|
32.9Hz
|
33.8Hz
|
32.4Hz
|
33.3Hz
|
|
f2(meas)
|
50.7Hz
|
48.7Hz
|
50.7Hz
|
50.0Hz
|
|
QMS(calc)
|
6.47
|
8.23
|
6.32
|
6.87
|
|
QES(calc)
|
0.92
|
1.23
|
0.90
|
0.99
|
|
QTS(calc)
|
0.81
|
0.91
|
0.79
|
0.86
|
Table 1: Thiele/Small Extraction Process
Constant Current Verses Voltage Divider Method Test Results
21. Now add a mass (such as back to back
ceramic disk magnets or blobs of silly putty or talc) symmetrically about
the cone apex. This will lower the drivers resonant frequency by
increasing its cone mass. The resonant frequency of the driver must be
lowered by at least 25% in order for this to work. In our case this would
be a reduction of about 10 Hz. Press the Run button on the application
toolbar and observe the trace in the spectrum analyzer. If the resonant
frequency is not lowered by about 25% add more mass and repeat. Never add
a mass greater than the cone mass. Remove the mass and weigh it to
0.1gram accuracy. You will need a good scale to do this. The total mass
of our silly putty is 13.50 grams Note that paper cones may be damaged
when removing silly putty. Notice the response ripples in the curves due
to cone interaction with the added mass. The better distributed the mass
the lower amplitude of the ripples.
Figure 10: Impedance
and Phase Plots of Driver with Added Mass
22. Find the drivers new resonant
frequency on the impedance curve. Ours is now about 31.1Hz. Then compute
the mechanical mass of the driver cone assembly (including air load) (MMS
in kilograms) as follows:
23. The mechanical compliance of a
drivers suspension is the inverse of how much force (Force = mass x acceleration)
it takes to push the cone per unit distance (per meter). The lower the
compliance the harder it is to push the cone back into the drivers
magnetic assembly. Compliance is inversely proportional to drivers cone
mass and it free air resonant frequency. This makes sense because the
greater a given mass, the lower its resonant frequency and the harder it
is to push around. Compute the drivers mechanical compliance (CMS)
as follows:
30. Measure the driver cone
effective surface area (SD in meters2). Do this by
measuring the diameter of the driver cone including one half of the
surround. Ours is a 10” driver so our
effective diameter is about 8.25”. Thus our radius
(effective diameter / 2) is 4.125”. Converting to
metric we get 4.125” x 1m / 39.37” = 0.105m. Thus:
29. The free air resonant frequency
of a driver is inversely proportional to the square root of its
compliance and its total mass. This mass includes the mass of the cone,
the voice coil assembly, the dust cap and about half of the surround and
spider. The surrounding air has mass as well and so it exerts a pressure
on the cone. This lowers the resonant frequency. The mass is proportional
to air density (p = 1.18kg/m3 @ 20C @ sea level) and the radius of the
cone (0.105m). It ranges from 0.001kg for a 3" driver to about
0.027kg for an 18" driver. When the driver is suspended in
free air this effective mass (MM1) can be calculated as
follows.
30. The mass of the driver cone
assembly(MMD) excluding air load is:
31. We can now check the free air
driver resonant frequency (fSA). The result is very close to
our measured value of 40.8Hz.
32. A driver in an enclosed box is a
piston pushing against a volume of compressible air. The larger the cone
and the smaller the air volume the harder it is for the cone to move back
and forth against the enclosed volume of air. The enclosed volume of air
that has the same stiffness as the drivers suspension system when
compressed by a piston the same diameter as the driver cone is known as VAS.
It is usually measured in manufacturers data sheets as liters of air at
standard temperature (20C) and pressure (sea level). However 1 liter of
air at standard temperature and pressure occupies 1000 liters The volume
of air having the same acoustic compliance as the driver suspension (VAS
in meters3) is calculated as follows:
where: p = the density of air (1.18
kg/m3 at 20°C @ sea level)
and: c = speed of sound (344.5 m/s
at 20°C @ sea level)
33. From these basic parameters we
can determine an enclosure type (baffle, closed or vented box).
Calculating the drivers Efficiency Band-width Product (EBP) will give us
an idea of the type of enclosure to design. If the EBP is less than 50 a
sealed enclosure might be more suitable. If greater a vented enclosure
should be designed. As can be seen our driver should go into a closed
box.
34. When a driver is placed in a
closed box it behaves like another driver with a heavier cone and a
stiffer suspension. Its resonant frequency is lowered and its Q is
raised. In order to calculate the driver Bl product (Bl), efficiency (n0)
and power sensitivity (Sp) we need to know a little about the intended
enclosure. Calculation of these parameters requires that we know the
resonant frequency of the driver (fSB) when it is mounted in
the enclosure. We will design a sealed enclosure for the driver with a
total system QTC of 1.0. This system will have warm robust
lower end. The system response will be boosted by about 1.25 dB near the
cut-off frequency. First we calculate system alpha as follows:
35. The estimated system resonance
may the be calculated as follows:
36. The required box volume may the
be calculated as follows:
37. In order to calculate the
resonant frequency of the driver (fSB) when it is mounted in
the enclosure we need to know the mass reactance loading on both the
front and rear of the cone. To calculate the mass reactance loading for
the front of the driver cone (MMR[front]) in a box less than
8ft3:
38. The equation for the effective
mass loading on the rear of the cone is
39. To solve this equation we need
to know the value of Km For rectangular baffles Km
approximately equals:
40. To solve this equation we need
to know the value of B. B is the ratio of the cone area (SD)
to the front baffle area (height x width). We will design a rectangular
box with side dimensions (height, width and depth) in the ratio of
1.6:1.2:1. Thus to calculate the depth of the box from which all other
dimensions can be derived from:
41. Thus the ratio (B) of the cone
area (SD) to the front baffle area (h x w).
 
40. So Km equates to:
30. So the effective mass, reactance
loading on the rear of the cone (MMR[rear]) is:
31. In order to find the mechanical
mass of the driver excluding air load (MMD) we subtract the
mass, reactance loading of the drivers piston mounted in free air (MM1)
from the total mass, reactance of the driver cone assembly including air
load (MMS)
41. Now we add the mechanical mass of
the driver excluding air load (MMD) plus the mass, reactance
loading for the front of the driver cone (MMR[front]) plus the
effective mass, reactance loading on the rear of the cone (MMR[rear])
to get the mechanical mass of the driver cone assembly excluding air load
(M'MS) when it is mounted in the box.
42. Now we can find the resonant
frequency (fSB) and Q (QTB) of the driver mounted
in the enclosure. Notice that the driver resonant frequency is lowered
(from fSA = 40.8 Hz to fSB = 38.4 Hz) and its Q is
raised (from QTS = 0.81 to QTB = 0.86), just as
predicted.
43. The Bl product is a measure of
the driver motor strength. It is equal to the field strength of the
driver magnet multiplied by the length of the voice coil wire in the
field. Large, heavy cones with large excursion requirements (Xmax)
need strong motor systems to control harmonic distortion. As more of the
voice coil moves out of the pole piece gap it takes more electrical
energy to move the cone a given distance. From the corrected resonant
frequency (fSB) the Bl product (Bl) of the 10” low frequency driver may be calculated as follows:
44. Efficiency defines how much
acoustic power per electrical watt of input the driver can produce. It is
used to match components in multi driver systems. Then also from the
corrected resonant frequency (fSB) the half plane, mid-band
efficiency (n0) (full-space efficiency is 3dB less) of the 10” low frequency driver may be calculated as follows:
45. Reference power sensitivity is
the sound pressure level produced by the driver when mounted on a large
baffle at 1 meter with an electrical input of 1 Watt. From the corrected
mid-band efficiency (n0) the power sensitivity of the 10” low frequency driver can be calculated as follows:
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