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Newbie Q: Speaker parameters mesurment?

Last response: in Home Audio
July 12, 2005 6:24:25 PM

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I would like to measure the parameters of speakers - T/S, impedance,
inductance, etc. but can't find very much info on the web. I have
electronics experience so I have test equipment (multimeters, signal
generator, frequency counters, etc.). I looked at the sound
card/software approach, but I'm sceptical of it, does anyone have any
info/pointers on doing it the 'old-fashioned' way?

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July 12, 2005 6:51:20 PM

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Nigel Thomson wrote:
> I would like to measure the parameters of speakers - T/S, impedance,
> inductance, etc. but can't find very much info on the web. I have
> electronics experience so I have test equipment (multimeters, signal
> generator, frequency counters, etc.). I looked at the sound
> card/software approach, but I'm sceptical of it,

Well, depending upon what "it" is, "it" can be faster, more
accurate, more repeatable and more error-free than doing it
the "old fashinoned way." I've used a number of said products
and will never measure Thiele/SMall parameters the old fashioned
way ever again. Mind you, it's oftenm the case that when I have
to measure drivers, I'll have to measure a couple of dozen of
them at one time, as a manufacturer will send me a pile of samples
to evaluate, so anything that makes it faster, more accurate, more
repeatable and less error-prone is fine by me.

But, ...

> does anyone have any info/pointers on doing it the 'old-
> fashioned' way?

Below are two articles I have published around these parts in the
past. The first is the basics of measuring impedance, and measuring
impedance is a necessary part of measuring Thiele/Small parameters

You're best reading these using a fixed-space font.

Dick Pierce


An oft-asked question is "What's a simple way I can measure
the impedance of my loudspeaker or driver?" The answer to
that depends upon a lot of things, like what equipment you
might have at your disposal, how much work you want to put
into the enterprise, and so on. I'll present one method here
that can give reasonably accurate results with the bare
minimum of test equipment needed.


Simply stated, it's the obstacle to current flow provided by
an electrical circuit to the imposition of an AC electrical
signal. It is like resistance in that sense, but different in
that it is almost always (in these applications)
frequency-dependent (it's value is different at different
frequencies) and it is "complex" (meaning that,
mathematically, it is a vector quantity, consisting of a
resistive and a reactive part)

The law governing the relationship between DC resistance,
voltage and current, known as Ohm's law, is:

E = I * R

where E is the impressed voltage in volts across the
resistance R in ohms, resulting in a current I in amperes
flowing through that resistance. Simple high school algebra
allows us to rearrance this basic equation:

R = --- and I = ---

AC impedance, voltage and current follow the same basic

E = I * Z

where, now, E is the impressed voltage magnitude in volts
impressed across the impedance magnitude Z in ohms, resulting
in a current magnitude of I in amperes flowing through that
impedance. And, as above, we can rearrange out equations:

Z = --- and I = ---

Now, I use the terms like "impedance magnitude" here. The AC
impedance, as mentioned above, is a complex value: it is
vector sum of the resistive (or "real") and reactive (or
"imaginary') components of the impedance. That vector sum is
computed as (for example):

2 2
Zm = sqrt ( R + X )

where R is the resistive portion and X is the reactive
portion. (In this context, real and imaginary have very
specific mathematical meanings: an imaginary number is not
one that exists only in one's imagination, rather it is a
number that has the square root of negative one as one of its

Because of the energy storage properties of the reactive
portion, the instantaneous current flowing through the
impedance is not in step or in phase with the instantaneous
voltage across it. Rather is precedes or follows the voltage
by some amount dependent upon the ratio of reactance to the
resistance, specifically:

-1 X
P = tan ( --- )

where P (more properly, the Greek letter Phi) is the phase
angle, usually expressed in degrees. It should be noted that
in the grand scheme of things, both the resistance R and the
reactance X can take on any value, positive, negative or

However, in the case of loudspeaker impedance, R will never
be negative, and almost certainly never 0, while X can either
be positive (inductive) or negative (capacitive) or 0.
Looking at the equation for the impedance phase angle, this
means that the phase angle of the impedance will always be
inside the range of -90 to +90 degrees. (Indeed, it is quite
unusual to find the impedance phase to be outside the range
of +- 70 degrees). The fact that the real or resistive
portion of the impedance is always positive ensures that the
impedance phase angle never exceeds these 90 degree limits.
(for those with a more technical inclination, that means that
the entire impedance is confined to the right of the
imaginary axis in the complex s-plane).

Basically, all we need to do is then to put a voltage across
the unknown impedance, measure the current going through it,
plug the numbers into the following equation (from above):

Z = ---

And out pops the impedance, Z.

In principle, this is absolutely correct, but in practice, it
is more difficult. The main reason for this is the range of
typical values for the impedance of most loudspeakers and
drivers (from a few ohms to a few dozen ohms) combined with
the sensitivity of most common measurement instruments.

Imagine putting a voltage of 10 volts across an 8 ohm
loudspeaker. Ohms law says that the current going through
that speaker will be:

E 10
I = --- = ---- = 1.25 amps
R 8

While 1.25 amps is a convenient current to measure (it's
large enough to ensure reasonable accuracy with many comment
meters) it is a LOT of current to put through the voice coil,
and that poor speaker and the people near it will be
subjected to a rather deafening level of sound. Additionally
it does pose some risk of damage to some drivers.


A common assumption is that one needs two meters: one to
measure voltage placed across the impedance and one to
measure current placed in series with the impedance. Then, by
Ohms law:

Z = ---

However, this poses some problems. As mentioned above, it
requires a hefty amount of current to get enough of a reading
to be dependable. Most commonly available meters that measure
AC current at all well aren't very sensitive. There is also
the issue of having to go through the calculation for each
and every frequency being measured.

Another method that seems to have escaped many peoples'
attention is the "impedometer." This is nothing more that a
calibrated constant current source. When properly set up, no
calculation is required and it is reasonably accurate over a
wide range of impedances. Another advantage is that it
requires less equipment than other methods. It is the
impedometer method that we will discuss here.


Very little is required for a properly working impedometer.
We'll enumerate the requirements here.

1. AC sine wave generator

This can either a function generator (usually meaning an
instrument that has the capability of sine, square, and
triangle waves, and often has pulse output as well) or a
Wein-bridge or twin-T audio oscillator. The major
requirements are stable AC output, stable frequency,
reasonably low distortion (less than 1%), flat frequency
response over the audio bandwidth, and reasonable voltage
output (10 volts or more into 1 kOhm is good).

There are a lot of new instruments that are acceptable,
funtions generators by B&K, Tenma, Leader and others can
be had, but often cost several hundred dollars new. Their
performance is generally more than good enough, and they
are versatile instruments for other purposes as well.
Often they have frequency ranges far in excess of what's
needed, like 0.02 Hz to 2 MHz, but that's okay, too.

On the other hand, you can often find used equipment that
is very serviceably as well as inexpensive. I have seen
excellent units from the likes of Wavetek an Krohn-Hite
for under $100. In working order, they have superb
specificationa and are ideal for this sort of use. Their
distortion is not the lowest (because, like other
function generators, they synthesize the square wave frm
the triangle output), but, for impedance and frequency
response meaasurements, they are superbly accurate for
audio use.

One of the all-time best sine generators is the venerable
HP 200 audio oscillator. I have seen them at swap meets
and even at yard sales for as low as 5 dollars. The have
good frequency response, good stability and high output
voltage (25 volts into 600 ohms). There are several
variants, the 200 AB and 200 CD are the most common and
both are equally good. Look for examples from General
Radio or GR as well. The GR1309 can often be had for $50
and can be tuned to have very low distortion, under
0.05%, while 1304 will do 20-20 kHz without range
sweeping and has high ouput voltage as well. Be prepared
for a little tune-up work, like cleaning and lubing dial
shafts, maybe replacing a tube and an electrolytic
capacitor or two. Otherwise, these units last absolutely
forever. I cannot recommend them too highly.

2. AC voltmeter

This can either be an analog or digital unit. Ideally, it
must be capable of reading down to about about 10 mV full
scale with reasonable accuracy. It must also have flat
frequency response over the audio range.

Unfortunately, the sensitivity requirement eliminates
most "passive" VOMs (volt-ohm-milliammeters), including
the ubiquitus and venerable Simpson 260 (which is truly
unfortunate, because the 3 I have here of different
vintages all have excellent frequency response to well
beyond 50 kHz on the 2.5 volt AC scale, sigh).

Equally unfortunate is the fact that many hand-held DVMs
(Digital Volt Meters) have poor high-frequency response,
often showing significant errors as low as 500 Hz.
Generally, most meters that advertise themselves as "true
RMS" have adequate frequency response.

Again, turning to the used or surplus markets, there are
treasures to be had. The Hewlett-Packard 400D has all the
needed sensitivity (1 millivolt full scale), wide
frequency response below 10 Hz to 1 MHZ), excellent
linearity and are plentiful and easy to find. Again, I
have seem them for as low as $25 in serviceable
condition, and even arrived 30 seconds too late one day
as I saw 20 of them being crushed at a local landfill!
Again, they may need new capacitors here and there and
occasional new tube, but little else is needed to keep
them going. Most of the HP 400 series are equally useful.
Look also for meters from GR, Ballantine, B&K and others.
Heath made an AC millivoltmeter that is quite useful.

3. 1 KOhm resistor

This need not be anything fancy. A noninductive carbon or
metal film, 1/2 watt 5% resistor is really all that's
needed. This will turn our oscillator into a current

4. 4, 8, or 10 ohm precision non-inductive resistor

This is used to calibrate the impedometer. It can be any
value that's close to the impedance you expect to
measure. Just make sure that it's non-inductive (film
resistors work here) and that you know its resistance
accurately (a 1% or better tolerance is ideal). You'll
only need a small resistor, 1/2 watt is probably fine.

5. Frequency counter

Not essential, but considering that the frequency dial
calibration on many oscillators and function generators
can be considerably off, it's a useful thing to have.
Digital frequency counters can be had both on the new and
used market for not a lot of money. Remember that the
accuracy is directly proportional to the reciprocal of
the needed accuracy: if you want 1/10 Hz accuracy, you'll
have to wait 10 seconds to get there.

6. Oscilloscope

Not essential, but useful for several things: it can help
you verify that nothing is being distorted. Connected as
an X-Y scope, it can help you unambiguously find the
exact resonant frequency and also enable you to (via a
rather laborious procedure) estimate the phase angle of
the impedance. If you're going to get a scope, get one
that has X-Y capability with no less than 10 mV/cm on
both axes. Scopes fitting the bill can be found for
anywhere from $50 used to many tens of thousands of
dollars. Look for an old HP 130, the best audio scope for
the cheapest money around. There are some big-ass
Tektronix 500 series that are huge and cheap, also look
for 400 series, and scopes by Phillips and others.

7. Miscellaneous

If you're going to be doing this a lot, buy a metal box,
some good 5-way binding posts and some high-quality
switches, and make your life easier. Use good sized wire,
because a 1/2 ohm of parasitic resistance in your test
harness is a 1/2 ohm that WON'T be there when you connect
your crossover.


The actual connection is very simple. Let's refer to the
diagram below:

+---- 1 kOhm ----+------------+
| | |
+-----+ +------+ +---+---+
| | | | | switch
Sine wave | | AC V
Oscillator | | voltmeter o o
| | | | | |
| | Calibrated +------o
| 'X' 'Y' | resistor Driver
| | | +------o
| | | | | |

If you need more signal level, you can insert an amplifier
between the oscillator and the 1 kOhm resistor.

Calibration is simple: connect the calibration resistor to
the output (via the switch, if you've constructed it that
way, or just hook the resistor where the speaker would be
connected). Adjust the output of the oscillator and the gain
of the meter until you get a reading in some convenient units
that is the same as the resistance of the calibrator. For
example, if the oscillator was putting out 1 volt into the 1
kOhm resistor, you'd probably find that the voltage across an
8 ohm calibrator was almost exactly 8 millivolts. Fine, now
you know that your impedomoter has a calibration factor of 1
mV/ohm. This is because we have calibrated our AC current
source for 1 mA output. Remember Ohms law:

Z = --- I = 1 mA, so Z (ohms) = E (millivolts)

You might want to adjust it for a higher level, like 10
mV/ohm. You see here why you might want an oscillator with a
nice high output voltage, because you might want to measure
the impedance at several different current levels. (I have a
laboratory amplifier that's capable of more than 100 volts at
100 mA into a 1 kohm load: this is very useful for measuring
drivers at reasonably high current levels).

It's a good idea to check the calibration across the entire
frequency range.


To actually measure the impedance, make sure your setup is
calibrated, then disconnect your calibration resistor and
connect your speaker. Dial the oscillator to the desired
frequency and then read the impedance. It's that simple.

If your want to know the impedance across the whole frequency
range, it's good to measure it at 1/3 octave intervals. This
will be enough to plot an pretty accurate graph of the
impedance curve. Here are the standard 1/3 octave

20 200 2000
25 250 2500
31.5 315 3100
40 400 4000
50 500 5000
63 630 6300
80 800 8000
100 1000 10000
126 1260 12600
159 1590 16900

To find a resonance, look for a frequency where the impedance
is at a maximum. In a typical loudspeaker system, or a
bass-reflex enclosure system, you'll find several such
maxima. Record them all.

Look for other "critical points" such as minima in impedance.

An oscilloscope can be useful here. Connect the X axis to the
oscillator output (shown as 'X' in the above diagram) and the
Y axis to the same place as the AC voltmeter ('Y'). Adjust
the gain of the X axis so that the trace takes up nearly the
whole width of the screen. The Y axis gain can be similarily
adjusted, but you'll have to keep changing it as the
impedance changes.

You'll notice that, over most of the range, the trace is an
ellipse aligned along a line going from the lower left to the
upper right (if it goes in the opposite direction look for a
switch on the scope called "phase invert" and push it). The
elliptical shape indicates that the impedance has both a
reactive and a resistive component. In fact, you can measure
the phase by measuring the relative "openness" of the
ellipse, though we won't go into that here.

What IS important is that at some frequencies, the ellipse
closes up into a line. This indicates the impedance is purely
reistive, and this will occur at the exact center of a
resonance, and is a reliable way of finding the resonant

The trace can also tell you other things. If the ends of the
ellipse are flattened or distorted, it's likely that you've
exceeded the output voltage capability of your oscialltor or
amplifier. Sorry, only one way to fix it: turn it down and
recalibrate your impedometer. If the traces shows a figure-8
shape, especially near and at resonance, you're likely
looking at some non-linearity in the driver itself. Finally,
if your trace looks fuzzy or has lots of little wiggles on
it, you have an electrical inteference problem that you'll
have to cure.


The impedometer method provides a simple, inexpensive,
reliable, repeatable and reasonably accurate way of measuring
loudspeaker impedance, assuming you use reasonable
instruments and take care to check and maintain calibration.

There are certainly more steamlined methods, including new
computer based applications that are fast, very detailed and
accurate. Not everyone can afford such a solution, not
everyone has the time, and not everyone needs that level of
sophistication. The impedometer method is useful for
occasional measurements, and the equipment needed is quite
useful for an entire array of audio measurements.
Measuring Driver Thiele-Small Parameters
Dick Pierce

Assuming you have the means to measure the impedance magnitude
of a driver with reasonable accurcy, determining the
Thiele-Small smnall signal parameters of a driver requires but a
few simple measurements and some straightforward calculations.

Warning: make sure that you use consistent units throughout. I
highly recommend you use straightforward metric units of meters,
kilograms, seconds and derived units of newtons and so forth.

Here's how to proceed:

* Measure the DC resistance of the driver to test. This gives
you Re. [Let's say it's 6.5 ohms]

* Replace the calibration resistor with the driver to test. Do
not change the voltage from the generator!

* Adjust the frequency in the region of the specified
resonance until the voltage across the driver is at a
MAXIMUM. Record the frequency. This is Fs, the resonant
frequency [let's say it's 32 Hz]. Also, measure the voltage
across the driver. This is Re+Res. [let's say it's 42 ohms].

(using an oscilloscope set for phase measurement, Fs will
also be where the phase is 0).

* Calculate the ratio between the DC resistance (Re) and the
maximum impedance (Re+Res), call it Rc. [In this case, it
will be 42/6.5 or 6.46]

* Find the two frequencies on either side of the resonant
frequency f1 and f2 where the impedance is Re * sqrt(Rc) [in
this example, that impedance will be 6.5 * sqrt(6.46) = 16.5
ohms, and let's say that occurs at f1 = 22.6 Hz and f2 =
45.3 Hz].

* Calculate Qms as:

Fs sqrt(Rc)
Qms = -----------
f2 - f1

[in the example, it will be:

32 sqrt(6.46) 32 * 2.54 81.3
Qms = ------------- = ----------- = ---- = 3.58
45.3 - 22.6 22.7 22.7

* Calculate Qes as:

Qes = --------
(Rc - 1)

[in this example, it will be:

3.58 3.58
Qes = --------- = ------ = 0.66
6.46 - 1 5.46

* Calculate Qts as:

Qes * Qms
Qts = ---------
Qes + Qms

[here, it would be:

0.66 * 3.58 2.36
Qts = ----------- = ------ = 0.56
0.66 + 3.58 4.24

So, you have derived Fs, Res, Qms, Qes, Qts for the driver.

* Repeat the measurements in a sealed, leak-free, unlined test
box, and determine the equivalent values of Fc, Qmc, Qec,
and Qtc (use a box whose volume, Vb, is close to the
expected Vas for maximum accuracy). [In our example, Vb =
20L, Fc = 80 Hz, Qec = 0.95]

* calculate Vas as follows:

Fc Qec
Vas = Vb [ -------- - 1 ]
Fs Qes

[In our example:

80 * 0.95
Vas = 20 [ ----------- - 1 ]
32 * 0.66

= 20 [ ---- - 1 ]

= 20 * 2.62 = 52L

Our Vas is 52 liters].

You now have Fs, Re, Qms, Qes, Qts and Vas for the driver. From
these figures, with knowledge of the effective emissive area of
the driver (Sd), you can derive the electromechanical parameters
Mms (moving mass), Cms (mechanical compliance), Rms (mechanical
suspension losses) and Bl (transduction ratio).

Sd, as mentioned, is the emissive area of the driver: that part
of the driver which actually contributes to the total volume
velocity of the driver. You can get a very good approximation of
Sd by taking the diameter d of the driver, including about half
the width of the suspension:

Sd = pi (d/2)^2

The last bit of information we need is about the air around us:

p0 = 1.18 kg/m^3 density of air at STP
c = 341 m/s speed of sound at STP

>From there, the electromechanical parameters can be derived:

Mechanical compliance Cms can be derived directly from the
equivalent valume of compliance, Vas:

Cms = ------------
p0 c^2 Sd^2

Mechanical mass Mms is derived knowing the compliance and the
resonant frequency. Since:

Fs = ------------------
2 pi sqrt(Mms Cas)


Mms = ---------------
4 pi^2 Fs^2 Cms

The mechanical resistance, Rms, is what determines the
mechanical damping, Qms, thus:

Rms = -----------------
2 pi Fs Cms Qms

Finally, the Bl product can be derived using Fs, Qes, Mms and

2 pi Fs Mms Re
Bl = sqrt( ---------------- )

Finally, the reference efficiency, n0 is:

4 pi^2 Fs^3 Vas
n0 = -------- ----------
c^3 Qes

Copyright (c) 1993-2005 by Dick Pierce.
Permission given for one-time no-charge electronic
distribution with subsequent followups.
All other rights reserved.
July 26, 2005 12:36:45 PM

Archived from groups: (More info?)

"Nigel Thomson" <> wrote in message
> I would like to measure the parameters of speakers - T/S, impedance,
> inductance, etc. but can't find very much info on the web. I have
> electronics experience so I have test equipment (multimeters, signal
> generator, frequency counters, etc.). I looked at the sound
> card/software approach, but I'm sceptical of it, does anyone have any
> info/pointers on doing it the 'old-fashioned' way?

The Loudspeaker Design Cookbook by Vance Dickason has a section on measuring
speaker parameters.