# Q: Speaker inductance measurement

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Anonymous
July 14, 2005 1:11:09 AM

I found a formula for calculating speaker inductance but its a little
unclear to me, is this right?

\ /---------------
\/ M * M - Re * Re / 2Pi
------------------------
frequency

so its the root of (M*M-Re*Re),
divided by (2*Pi),
divided by the frequency?

Where (M) is the impedance Magnitude.

It says, obtain (M) like this:

* Connect an amplifier and frequency generator set to 1000Hz and put a
10 ohm resistor on the output.

* Adjust the gain on the amplifier and frequency generator until you
get 10 V.A.C.

* Remove the resistor and connect the speaker.

* Set the frequency at the drivers highest usable frequency.

* The voltmeter reading is the driver's impedance magnitude (M) for
this frequency.

What is the "drivers highest usable frequency" anyway? ZMax?

Its my understanding that the inductance should be obtained at 1000Hz,
is this what I want?

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Anonymous
July 14, 2005 1:31:41 AM

"Nigel Thomson" <unknown@unknown.net> wrote in message
news:5nebd11of377ararokh0r5aqblebv2o29i@4ax.com...
> I found a formula for calculating speaker inductance but its a little
> unclear to me, is this right?
>
> \ /---------------
> \/ M * M - Re * Re / 2Pi
> ------------------------
> frequency
>
> so its the root of (M*M-Re*Re),
> divided by (2*Pi),
> divided by the frequency?
>
> Where (M) is the impedance Magnitude.
>
> It says, obtain (M) like this:
>
> * Connect an amplifier and frequency generator set to 1000Hz and put a
> 10 ohm resistor on the output.
>
> * Adjust the gain on the amplifier and frequency generator until you
> get 10 V.A.C.
>

By doing these steps you have created a (somewhat) calibrated measuring
system. You know that 1KHz into a 10 ohm, purely resistive load will yield a

> * Remove the resistor and connect the speaker.

Replace a resistive load with a load that is mostly resistance, but some
reactance; the speaker.

>
> * Set the frequency at the drivers highest usable frequency.

If you think of the voice coil as a simple coil, it should have a constant
value for its inductive reactance in Henries. Voice coils are not simple
coils. They are suspended in a strong magnetic fields and they move. But,
you are looking for a single target value for the inductive reactance of the
coil. So, why not take the measurement where it is likely to have more
effect on the whole system? That should be the top end of the portion of the
audio spectrum for which this device is going to be used. Try putting in the
frequency of the crossover point to the next driver.

>
> * The voltmeter reading is the driver's impedance magnitude (M) for
> this frequency.
>

That means that M is just a numerical factor, that is a voltage reading that
is used to derive the real value in Henries.

> What is the "drivers highest usable frequency" anyway? ZMax?
>

No. Zmax is the maximum linear piston motion of the coil before it leaves
the magnetic influence of the pole plate and plug.

> Its my understanding that the inductance should be obtained at 1000Hz,
> is this what I want?
>

I don't think you will be able to measure the difference between 1000Hz and
5000Hz. Only 5000Hz will give you a different reading on the meter. Try it!

http://www.akrobiz.com/speakers/

~James.  )

>
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Anonymous
July 14, 2005 1:43:40 AM

Ooops.

Zmax is the maximum impedance at the resonant frequency of the driver. Not
where you want to measure the coil's inductive reactance!

Xmax is the voice coil throw thingie...

~James.  )
Anonymous
July 17, 2005 3:21:54 PM

Nigel Thomson wrote:
> I found a formula for calculating speaker inductance but its a little
> unclear to me, is this right?
>
> \ /---------------
> \/ M * M - Re * Re / 2Pi
> ------------------------
> frequency
>
> so its the root of (M*M-Re*Re),
> divided by (2*Pi),
> divided by the frequency?

First, a suggestion: don't try to "draw" equations, write them
out using understandable notation. That way, what you're sayinng
isn't so dependent upon your ability to draw or someone's ability

Using that principle, your equation restated might look like:

L = sqrt(M^2 - Re^2)/(2 pi F)

Let's reverse that to understand where it came from.

The impedance of a series inductor and resistor at any given
frequency the vector sum of the resistance and the inductive
reactance (wL, where w is radian frequency, 2 pi F). As a
vector sum, the Pythagorean equation rules:

Zt = sqrt(R^2 + wL^2)

or

Zt = sqrt(R^2 + (2 pi F L)^2)

Square both sides and expand terms:

Zt^2 = R^2 + 4 pi^2 F^2 L^2

Subtract R_2 from both sides:

Zt^2 - R^2 = 4 pi^2 F^2 L^2

Divide both sides by the frequency terms:

(Zt^2 - R^2)/(4 pi^2 F^2) = L^2

And take the square root of both sides:

sqrt(Zt^2 - R^2)/(2 pi F) = L

Or, rearranged:

L = sqrt(Zt^2 - R^2)/(2 pi F)

which matches your equation, and illustrates how it was obtained.

> It says, obtain (M) like this:

Look for a recent posty of mine describing a simple, reliable way
of measuring impedance magnitude.

> What is the "drivers highest usable frequency" anyway? ZMax?
>
> Its my understanding that the inductance should be obtained at 1000Hz,
> is this what I want?

All of what you say is based on the assumption that the voice coil
inductance is essentially a perfect resistor in series with a perfect
inductor, and the physical reality if FAR from that.

By "perfect," I mean that the resitive part of the model remains
constant with both current and frequency, as does the inductive part,
and the only variable is the inductive reactance, which is directly
proportional to frequency. However, what actually happens in a driver
far from that perfect model.

We can largely ignore the effects of the motional impedance in most
cases, as the peak in the motional impedance, occuring at the driver's
fundamental mechanical resonance, is far enough separated in frequency
so as to have negligable influence. Additonally, since the velocity of
the voice coil goes as the reciprocal of frequency above resonance,
the raw effects of simple voice coil motion simply becomes
insignificant
at higher frequencies, contrary to the implications made by another
respondant.

Instead, what has a SUBSTANTIAL confounding influence is that
the voice coil is immersed in a large amount of (relatively)
poor electrically conductive material, notably the pole piece
and front plate of the magnet structure. This material is

The generation of time-variant magnetic fields by the signal
passing through the voice coil generates secondary eddy currents
in these metallic structures, and these metals are lossy. The
degree of coupling is frequency dependent as well.

The result is that when, in fact, you analyze the rising impedance
of the voice coil, you find that it does NOT behave as a simple
series resistor-inductor model. INstead, you find, curiously, that
the VALUE of the resistive part increases with frequency, and the
VALUE of the inductive part DECRESES with frequency.

This can be readily observed in the deviation from the ideal
model. In the ideal model, the impedance rise should approach
and, eventually, reach a rate where the impedance doubles with
each octave. Further, you'll find that the phase angle of the
impedance will approach 90 degrees.

In actually measuring the impedance over frequency, you will
of the impedance doubling with each octave, that it increases
only by about 40% or so each octave. And you'll find that the
impedance, sintead of approaching 90 degrees, only approaches
and never exceeds about 45 degrees.

Analyzing the impedance characteristics reveals a curious thing:
that the resistive part is NOT constant, equal to Re, but there
is a second resistive part in addition taht's negligable at low
frequencies and significant at higher frequencies, and increases
roughly as the square root of frequency.

Equally curious is the fact that the INDUCTANACE is high at low
frequencies and decreases as frequency goes up, roughly as the
inverse square root of frequency, rather than staying constant.

This means that once you get to a certain frequency, and that's
only about an octave above the resistive portion of the mid range
impedance trough, the inductuive REACTANCE, rather than doubling
with each octave, climbs at a slower rate, but the resistive
component, which should remain constant with frequency, increases
itself at about the square root of the frequency.

For example, a typical 8" woofer might be "rated" by the manu-
facturer as having an inductance at 1 kHz of .8 mH and a DC
resistance of 6.5 ohms. That would indicate that at 1 kHz,
the driver's impedance is sqrt(6.5^2 + (2 pi 1e3 .0008)^2)
or about 8.2 ohms. In fact, many manufacturers simply measure
the impedance at 1 kHz and "assume" the equation you stated is
correct and thus derive the impedance from that.

If the simple model were true, then we would expect that the
impedance at, say 2 kHz would be sqrt(6.5^2 + (2 pi 2e3 .0008)^2)
or 12 ohms, at 4 kHz about 21 ohms, and so forth.

In fact, when we actually MEASURE the impedance, we find that at
2 kHz, the imepdance is only about 10 Ohms, and at 4 kHz it's

What went wrong?

Well, the issue is that instead of storing the energy in a
magnetic field, it's because the generated eddy current
flowing through the metallic front plate and pole piece is
dissipating the energy by slightly heating these elements
through simple ohmic losses.

The basic principle is that you CANNOT assume the voice coil
model at high frequencies is a simple series inductance-
resistance: it's more complex that that and not in any subtle
way. The assumption is sufficiently at variance with the actual
physical reality such that the predicted vs real impedance is
WAY off, often by a factor of two or more within the audio band.

One of the consequences is that if you attempt to calculate the
values for a complex conjugate circuit correcting for driver
inductance based on the simple model, the values you derive will
most assuredly NOT be correct, and be off by a substantial amount,
enough to screw up the response of your network based on such
assumptions.

For more details, you might want to check a number of articles
on the topic, the most notable of which is Vanderkooy, "A Model
of Loudspeaker Impedance Incorporating Eddy Currents in the
Pole Structure" (J. Audio Eng. Soc., vol 37, no 3, pp 119-128,
1983 March).

The fact that the coil moves and is suspended in a strong
magnetic field, as suggested by another poster, is largely
irrelevant to the problem, as Vaderkooy clearly demonstrates
in his article. He performed experiements, for example, using
a demagnetized structure and with the voice coil locked firmly
in place and found no substantial differences in the high-
frequency impedance function of the driver., Rather, as I
describe above, the major source of deviation from ideal
behavior is the presence of the conductive metalic structure
surrounding the voice coil and the effects of generated eddy
currents in that structure.

Without, then, the ability to precisely measure the true
characteristics of the impedance, notable, the actuall resistive
and inductive portions of the impedance at the frequency of
interest, the best you can do is measure the actual impedance
at the frequency of interest. If it's significantly up the
impedance curve, assume the phase angle is near 45 degrees
and then dervice both the resistive and inductive portions
from that and proceed accordingly. In the transition region
between the midrange torugh and the point where the impedance
is steadily climbing in frequency, assume the phase angle is
somweherte in between 0 and 45 degrees and proceed from there.

Ideally, the best procedure is to measure both the imepdance
magnitude AND phase (as I allude to in my impedometer article)
and, with those in hand, you CAN derive accurate values for the
resistive and inductive portions of the impedance.
!