Speaker Analysis Using
Thiele-Small Parameters
Designing loudspeakers requires a great deal of research and development
of many factors. The basis for which speaker enclosures are designed today are
specifications called Thiele-Small parameters. There are a number of
performance characteristics of speakers that were developed by Neville Thiele
and Richard Small. These Thiele-Small parameters can be used to predict the
performance of a speaker in different types of speaker enclosures.
In 1961, Neville Thiele first published his work on his study of
loudspeakers in different types of vented speaker enclosures which was entitled
“Loudspeakers in Vented Boxes”, and published in the Proceedings of the IRE
Australia. In 1972, Richard Small published articles that built on the work of
Neville Thiele and covered vented speaker enclosures and covered ones as well.
This work is widely referred to even today for speaker enclosure design.
Analyzing the acoustic output and characteristics of a speaker was made much
simpler by the work of these two men. Simply using electrical circuit analysis
to measure Thiele-Small parameters made designing speaker enclosures for
different speaker setups much easier.
There are many Thiele-Small parameters that can be measured from
loudspeakers, but the three most important ones are: free air resonance (F(s)),
Q(ts), and volume of suspension (V(as)). The free air resonance is the resonant
frequency of the speaker found in free air. Below, a typical impedance
frequency response of a loudspeaker is shown on a log scale.
At the
resonant frequency, there is a clear increase in impedance for the speaker. The
resonant frequency can clearly be seen here at about 25 Hz. After the resonant
frequency, capacitive impedance reduces the magnitude of the total impedance
until the linear region, in which the impedance stays relatively constant.
Then, at high frequencies, the magnitude of the impedance rises due to
inductive impedance, where the voice coil inductance contributes to the
impedance. The resonant frequency is considered the threshold at which a
speaker can be used. A speaker usually is unable to operate at a frequency
lower than the resonant frequency.
The Q(ts) or total Q of the speaker, also known as
total quality factor, is basically describing the resonance curve of the
speaker. The higher and thinner of a peak the curve has, the lower Q it will
have. The rounder a peak the curve has, the higher Q it will have.
There are two parts that make up Q(ts). These are the electrical Q
(Q(es)) and mechanical Q (Q(ms)). Finally, the last main Thiele-Small parameter
is V(as). This refers to the volume of air that has the same compliance as the
speaker. In other words, it is the volume of air that equals the volume of the
suspension of the speaker.
There are many ways to find the values of the
Thiele-Small parameters, even if there is not advanced equipment available for
one to make measurements with. Fortunately, the Physics 398 lab had some very
useful tools that enabled me to make accurate measurements.
Project setup description
By creating a setup to plot various characteristics
of a speaker vs. frequency, I was able to determine many things about the
speaker. Here, the process of carrying out the project will be explained.
(Thanks to Professor Errede for helping set up the data acquisition system).
The equipment used to take these measurements from
the speaker was: a function generator, a personal computer running the National
Instruments LabWindows CVI data acquisition program, and two lock-in
amplifiers. On the next page, a simple diagram of the data acquisition setup is
shown.
Basically,
the two terminals of the speaker are connected to two different lock-in
amplifiers that measure both voltage and current throughout the range of
frequencies determined by the frequency counter. Lock-in amplifiers are used to
measure information from a signal such as phase and amplitude from a signal
that might be obstructed by background noise. All that is needed to know for
this project is that it is able to relate data about the speaker to the
computer running LabWindows, which gathers the information and plots it. The
program used to run this data acquisition program is written in C, and is a slightly
modified version of the program used to measure the response of various guitar
pickups. The original code was written by Steve Errede and Jack Boparai, and
modified to work with speakers by Professor Errede. This program provides a
very friendly user interface and very helpful options for analyzing data. Many
different types of data could be graphed by using this program: total voltage,
real voltage, imaginary voltage, and voltage phase, all vs. frequency, can be
graphed using the code.
The same holds true for current and impedance. Also, there is an option
to save the data and work with it in Microsoft Excel, if the user so desires.
This proved very useful to me, as I could work with the data at home, since I
had Excel, but not all of the equipment needed to take actual measurements.
Another very important ability of the code was to control the frequency range
for which the setup was supposed to keep track of data. The default setting is
to test the speaker from 10 Hz to 20,000 Hz, which is the human frequency range
of hearing. But sometimes it is advantageous to look at a smaller range of
frequencies to better analyze the characteristics of the speaker. Measuring
speaker parameters from about 10 Hz to 150 Hz was also done to get a “close up”
view of the resonant frequency curve. This was necessary to more accurately
find the F(s) of the speaker. Now that we have the information recorded by the
computer, we can do many things with it.
Determining Thiele-Small Parameters
With the data that was recorded by the computer, a couple of key
Thiele-Small parameters can now be found. Finding F(s) is relatively simple. It
is just the frequency at which the highest point of the peak of the impedance
occurs. The following method of finding Q(ts) was found at www.epanorama.net/documents/audio/speaker_parameters.
Measure the DC resistance of the speaker, and designate it as Re. Now, measure
the voltage of the speaker at this resonant frequency. This will be called Re +
Res. Then find the ratio between Re and Re + Res and designate this ratio as
Rc. Then, find two frequencies f1 and f2, where the impedance is Re*sqrt(Rc) on
both sides of the resonant frequency. From this, we can calculate Q(ms), Q(es), and Q(ts). Below
are the equations to find each of these values:
Q(ms) = Fs * sqrt (Rc)
----------------
f2 – f1
Q(es) = Q(ms)
-------------
(Rc – 1)
Q(ts) = Q(es) * Q(ms)
-------------------
Q(es) + Q(ms)
Now, two of the major Thiele-Small parameters are known. Finding V(as),
however is more difficult, and I did not have enough time to find V(as) for the
different speakers. However, I will mention a crude method to find V(as) that I
came across. To find V(as), repeat all of the steps above, but the speaker has
to be in a sealed box that will not leak. Find the volume of the box Vb, and
use the following equation:
V(as) =
Vb * F(c) * Q(ec)
{ -------------- - 1} F(s) * Q(es)
where
F(c) and Q(ec) are equivalent values of F(s) and Q(es) in the sealed box.
Here are the Thiele-Small parameters that I calculated from the data I
have obtained, by following the steps above.
Speaker
|
F(s)
|
Re
|
Re+Res
|
Rc
|
f1
|
f2
|
Q(ms)
|
Q(es)
|
Q(ts)
|
4" speaker
|
75
|
2.6
|
6.531
|
2.511923
|
57
|
90
|
3.602056
|
2.382434
|
1.433984
|
Jensen
P8R-4
|
133
|
3.1
|
19
|
6.129032
|
112
|
152
|
8.231663
|
1.604916
|
1.343061
|
Eminence 8"
|
150
|
6.7
|
11.1
|
1.656716
|
110
|
200
|
2.145225
|
3.266593
|
1.294866
|
Celestion Vintage 8"
|
170
|
6.2
|
30
|
4.83871
|
140
|
200
|
6.232502
|
1.623593
|
1.28805
|
Celestion Vintage 10"
|
124
|
7
|
110
|
15.71429
|
104
|
145
|
11.98906
|
0.814791
|
0.76294
|
10" in Peavey
|
60
|
3.2
|
10.5
|
3.28125
|
38
|
82
|
2.470121
|
1.082793
|
0.752799
|
12 “ subwoofer
|
25
|
5.2
|
62.18
|
11.95769
|
19
|
32
|
6.64998
|
0.606878
|
0.556126
|
Now that the Thiele-Small parameters can be found, a further analysis of
the Thiele-Small results can be done. First, F(s) will be discussed. From the
data compiled, I was able to see that larger speakers such as the subwoofer I
tested had a much lower F(s) compared to the other speakers. Manufacturers of
speakers will create different types of speakers with a whole range of resonant
frequencies. Basically, the resonant frequency of speakers determines what kind
of speaker it will be. Subwoofers will operate with low resonant frequencies,
while other types of speakers will typically have higher resonant frequencies.
The Q(ts) value of the speaker is an indicator of the “resonant magnification”
of the speaker. This value is important in deciding what type of enclosure a
speaker should go into. Also, a high value for Q(ts) will have a “warmer” tone,
while lower values have a more clear and “shallower” sound, according to The
Loudspeaker Design Cookbook by Vance Dickason. A very high Q(ts) or
a very low value for Q(ts) is not good. The speakers towards the middle of the
table should be of higher quality and should sound better than those at the
edges, since the table is sorted with respect to Q(ts).
Comparing speaker data in free air vs. in cabinet
Other than Thie le-Small parameters, other things
can be observed from looking at the impedance curves from a speaker. I decided
to look at the results of a speaker measured in free air vs. a speaker measured
in an open back cabinet, which served to simulate an infinite baffle board. I
was surprised to see some differences. First, here are two graphs of the
impedance curve of a 10” speaker found in a Peavey amp. The first
measured
in a cabinet:
Total Impedence vs. Frequency for Peavey
10" in free air zoom
|
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There are
many interesting observations that can be drawn from looking at these two
graphs. One of the biggest things that you can see is that the resonant
frequency is actually shifted down when the speaker is in the cabinet as
opposed to in free air. In the graphs, above, the F(s) of the speaker in free
air is about 58 Hz, while the F(s) of the speaker in the cabinet is about 53
Hz. From these observations, we can draw the conclusion that speakers work
better in a cabinet or infinite baffle rather than in free air. This is because
when a speaker is in free air, the low frequency sound waves emanating from
behind and in front of the speaker cone may interfere with each other and
cancel out. If the speaker is mounted on an infinite baffle, the waves behind
the cone will not interfere with the audio signals produced by the speaker in
front of the baffle. The other obvious difference we can see from a speaker in
free air and a speaker in the cabinet is that the in the free air situation,
the curve is much more smooth than in the cabinet. The true characteristic of
the impedance curve is amplified in the cabinet. For example, at about 100 Hz,
a jagged discontinuity in the curve can be seen in the free air graph. Looking
at the cabinet graph, it is much larger and pronounced. There are many more
discontinuities in the graph of the speaker in the cabinet. I believe that this
is a much truer picture of the impedance, because the speaker will always be in
a cabinet when it is being used, so the data received when testing a speaker in
a cabinet is more useful. Overall, it was very important to look at and compare
the effects of a speaker tested in free air and mounted in a cabinet.
Of course, the total impedance curve is not the
only important thing to look at when viewing the data plots from the speakers.
Both voltage and current were also measured from each speaker. In each set of
graphs, these curves were very similar. All of the data can be viewed in the
graphs that are printed out separately from this report. On some of the
speakers I tested, there are 4 sets of data. Either the data was taken from the
normal range of frequency (10 Hz – 20,000 Hz) or in a smaller range of
frequency (10 Hz – 200 Hz) to better view the resonant area of the impedance
curve. Also, for each range, there was a set of data for the speaker in free
air and for the speaker in a cabinet. Real voltage generally followed the same
shape as the total impedance, as did total voltage. For the imaginary voltage,
there was a sharp change of sign at the resonance. This is due to the
capacitative resistance taking over and reducing the total resistance after
resonance. The voltage phase behaved the same way. Real and imaginary current
generally stayed constant throughout. Of course, this contributes to a constant
total current. There might be some dips seen in the graphs, but keep in mind
that it is a very small scale. Real, imaginary, and total impedance mimic the
voltage data described above for the same reasons. An interesting graph to look
at is the Re(Voltage) vs. Im(Voltage) and similarly the Re(Impedance) vs.
Im(Impedance):
Z real vs. Z imag for Peavey 10" in
free air zoom
|
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4
|
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3
|
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2
|
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1
|
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(ohms)
|
0
|
2
|
4
|
6
|
8
|
10
|
12
|
0
|
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Zimag
|
Z real vs. Z imag
|
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-1
|
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-2
|
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-3
|
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-4
|
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-5
|
11
|
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Z
real (ohms)
|
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This graph shows the resonance associated with the speaker. The circular
shape suggests perfect resonance with the speaker.
Possibilities for future experiments
There is much more to study and research on this
topic. Even though I made some steps towards understanding speaker
characteristics, there is much more to do.
First of
all, I was able to calculate F(s) and Q(ts) of certain speakers, but to take it
further, one could try to calculate V(as) by doing the measurements I did,
except in a sealed box. Perhaps after finding all three of these important
Thiele-Small parameters, there can be a better understanding of how these
parameters can be used to design enclosures. Values for exact dimension
projections of a cabinet and what type of cabinet, such as closed back, vented
back, etc. can be researched. Possibly actually creating different cabinets
with different sizes and types and testing how certain speakers sound and
respond would be interesting to see. Further studying which types of speakers
and enclosure would work in different types of purposes would also be
interesting. How would a speaker cabinet designed for a home theater be
different for a setup for a rock concert? There is an endless amount of
extensions from this topic to pursue.
12
Sources used
Introduction to Loudspeaker Design
- John L.
Murphy
The Loudspeaker Design Cookbook
- Vance
Dickason
www.epanorama.net/documents/audio/speaker_parameters.html
www.sound.au.com/tsp.htm
www.silicom.com
www.webervst.com/spkrcalc/para.htm
Professor Errede
(Thanks for brining in speakers
for me to test and for the help along the way.)
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