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More info?)

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

to "read" a picture.

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

typically made of low-carbon steel.

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

instead that it does neother of these. You find that instead

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

about 16 ohms.

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.