"
Well, to a first approximation, you're correct, but in the first
approximation, the details of X are QUITE important. one issue
before proceeding further:
There can be a large difference between the relationship between
the DC resistance and the impedance of a single driver and that
of a multi-way system, so we need to be careful in specifying
which we are discussing.
Let's explore the contributors to the impedance of a single
driver, let's say (and this is derived from a quick reading
of the context of the discussion, as I just looked it up)
we are loking at a woofer.
The following are the major contributors to the impedance of
such a driver:
1. The electrical impedance of the vooice coil itself
2. The mechanical impedance of the driver, as reflected through
the transduction trasnformation of the voice coil/magnet. The
transformation ratio here is B^2 l^2, where B is the flux
density in the gap in Tesla, and l is the length of wire immersed
in the flux.
3. The acoustical impedance of the driver, as reflected through
to the mechanical domain by the driver radiating area, then
reflected through to the electrical domain by the voice coil
and magnet. The acoustical-to-mechanical transformation goes
as Sd^2, where Sd is the emissive area of the cone.
Now, assume, for the moment, we are talking a typical direct-radiator
driver of the kind found in most domestic music system loudspeakers.
The total contribution of the acoustical impedance is actually quite
small, usually less than 1% of the total reflected electrical impedance,
and, no suprise, corresponds to the reference efficiency of the
driver. Since in order to produce sound, the speaker must do work into
an impedance, and the efficiency is small, then the contribution of'that
impedance into which the work is being done is similarily small. We'll
see that come up again and again, though it's not often discussed in
texts.
By the time one sees then acoustical impedance reflected back into the
electrical domain, it is a VERY small portion of the total that has
the property of going, roughly as frequency squared up to a limit
that's determined by the emissive area of the diaphragm. The problem is
no simple electrical analog has a f^2 impedance analog, but we're safe
becase the effect is, as we said, a very minor part of the total.
Then there is the mechanical contribution to the impedance. By far
the single largest contributor is the bulk motional impedance of
the speaker due to the fundamental mechanical resonance of the driver
at some low frequency. You have a mechanically resonant circuit
which consists of the reactive components of the mass of the diaphragm
and the compliance of the mechanical suspension, and a loss element
due to friction in the suspension (some have mentioned the air-load mass
and the air compliance, but these are small compared to the simple
mechanical mass and compliance of the system). As reflected back
through the voice coil, these look all the world like an ordinary,
everyday parallel RLC resonant tank circuit. To the amplifier or
crossover, it could just as well be and RLC tank circuit.
Now, add a box and a port and other things, and this makes this
circuit a little more complex, but we can get a pretty damned
good approximation to within a couple of % of reality with less
than 10 equivalent passive components, i.e., inductors, capacitors
and resistors.
And last we have the electrical portion. To a reasonable first-order
approximation, this looks like a simple RL series combination, where
R is the DC resistance of the voice coil, and L is the inductance.
So, a first stab at the electrical properties of the impedance nets
a circuit that looks like this SPICE net list (values taken from
an actual set of measurements for a SPICE model was developed):
Re 1 2 6.25 * DC voice coil resistance
Le 2 3 0.67MH * voice coil inductance
Lces 3 0 57.8MH * inductive equivalent of compliance
Cmes 3 0 320.9UF * capacitive equivalent of mass
Rces 3 0 26.8 * resistive equivalent of suspension losses
This is for a 6 1/2" driver that has the following T/S and
electromechanical parameters:
Fs 36 Hz
Vas 32 L
Qms 2.06
Qes 0.48
Qts 0.39
Re 6.25 Ohms
Mms 20.1 g
Cms 0.97 mm/N
Rms 2.21 kg/s
Bl 7.70 N/A
Sd 1.54 x 10^-2 m^2
But, upon comparing this to the measured impedance, while we find
the fit is very good in the low frequency region (up to 200 Hz),
we find a widening gap between model and physical reality above 500
Hz. The model's impedance, no surprisingly asymotically reaches a
slope of 2 (the impedance doubles every octave), while the reality
only reaches a slope of approximately sqrt(2).
The reason is one someone else in the thread alluded to in reference
the Lipschitz/Vanderkooy article. There is a very important mechanism
at work at higher frequencies: the metal part of the magnet assembly
surrounding the voice coil are electrically conductive. The time-varying
field generated by the voice coil induces eddy currents in these
metal parts in a manner that is, to a first approximation, freuqency
dependent.
The net result is a rather surprising (at first, until you think
about it) effect: the effective resistive part of the voice coil's
contribution to the impedance INCREASES iwith increasing frequency,
while the effective inductive part of of the voice coils impedance
DECREASES with increasing frequency. More and more of the magnetic
field generated by the voice oicl goes into the production of these
eddy currents as you go higher and higher in frequency. The real
electrical resistance seen by these eddy current causes real work
to be done: it heats up the pole piece and front plate of the magnet!
These metal parts look like the seondary turns of a transformer whose
coupling increases with increasing frequency.
Above 500 Hz, we see THIS phenonenon as THE dominating factor of the
electrical impedance of woofers.
Now, to answer the original bait, uh, I mean, question: the impedance
seen at around 200 Hz, which is higher than the DC resistance of the
voice coil, is the result of several contributors:
1. The DC resistance of the voice coil
2. A small portion of the the mechanical impedance of the
driver's fundamental resonance
3. The voice coil inductance
4. Eddy current losses in the conductive metal parts of the
magnet assembly
5. Lots of other small particpants, the sum of which has a
very minor effect on the total impedance magnitude, such
as the radiation impedance of the driver, mechanical
resonances in the cone, as so forth.
And the notion that it is 20% greater that the DC resistance is
simply one possible figure out of a fairly wide range of typical
values. I have seen it as low as a couple of percent, seldom higher
than 20%.
Now, the question of what is the "nominal" impedance is, in fact,
a matter of convention and, in some cases, standardization. TO the
latter point, one finds such documents as IEC 60268 which has specific
recommendations of how one determines the "nominal" impedance of
a driver or speaker. There are various reasonable justifications for
the recommendations given, and I would direct anyone interested in
learning these rationalizations to the relevant documents.
+---------------------------------------+
| Dick Pierce |
| Professional Audio Development |
| Acoustics and Digital Audio |
| (1) 781/826-4953 Voice and FAX |
|
[email protected] |
+---------------------------------------+
"
