The following article is based on news articles posted to rec.audio.tech newsgroup by Richard Pierce, Dan Marshall and John Woodgate at 1998 and 1999- The article is compiled and edited by Tomi Engdahl at 1999.
The following things have effect on speaker impedance:
There is a convention to the use of the term "nominal impedance", and if the impedance over the majority of the bandwidth, specifically covering the range in spectrum where majority of the musical spectral power occurs, it's 8 ohms. A single number cannot tell all there is to tell about an impedance that varies with frequency.
You must keep in mind that 'nominal impedacnce' is not defined in IEC. Indeed, the electronics industry was advised when the Trade Descriptions Act was introduced, that the word 'nominal' should no longer be used in specifications. That is why the IEC concept of 'rated value' is so useful. There is a very detailed definition and explanation of this term in IEC60268-2.
The IEC standard (IEC60268-3) allows any "increase" above the rated value, but limits the "decrease". The standard does not allow the impedance to fall below the 80 % of the nominal value at any frequency, including DC.
In practice all loudspeakers are a compromise, the designer is therefore free to allow the speaker to suck more power from the amp in order to optimise other parameters. Most high-quality loudspeakers do dip well below 80% of their nominal impedance at one or more points in the audio band. Speakers which attempt to present a flat impedance load using conjugate techniques have sometimes been described as 'flat and boring', which may or may not be connected to their excessively complex crossovers. Speaker design is non-trivial!
Remeber that a specification is only of relevance when a product is claimed to meet it. A specification is only of value when it lays down a minimum standard which is of relevance to the intended purpose of the product. A high-quality speaker may reasonably be assumed to be intended to be driven by a high quality amplifier, hence minimum impedance is not an important criterion in establishing sonic performance.
If you just want to find out the nominal impedence of the speaker e.g. ist it 4, 8 or 15 ohms then there is a rough & ready way. Just use your multimeter to measure the DC resistance of the voice coil i.e. across the speaker terminals (with nothing else connected) and multiply the answer by 1.3. So if the DC resistance is say 6 ohms then the speaker is nominally 8 ohm impedance.
More complete analysis with minimal equipments:
The single most dominant branch of the model is the voice coil DC resistance, Re. It's going to be in series with everything else we will look at (you mentioned "stray capacitance". Yes, there is some, but it's magnitude is absolutely miniscule compared to all other components so it can be ignored).
Next we have the voice coil inductance (we'll call it Lvc). Now, it, too, is in series with everything else, but it's no simple inductance.
So far, we have the two real electrical components, and they look like:
o-----Re------Lvc----oNow, the next major set of components are the electrical equivalents of the major mechanical components of suspension compliance, cone mass and suspension losses. The suspension compliance is modelled as an inductor, Lces. The cone mass is modelled as a capacitance, Cmes, and the suspension losses are modeled as a resistor, Res. These three are in parallel and form a damped, parallel resonant branch called the drivere mechanical branch.
Finally, in series with that, is the radiation impedance. No single lumped-parameter synthesis comes close to approximating this. Also the magnitude of the impedance of this branch is small compared to the others, so for simulating the ELECTRICAL characteristics, it can be safely eliminated.
The driver electrical model, then, looks like this:
o------Re------Lvc------+
|
+------+------+
| | |
Lces Cmes Res
| | |
+------+------+
|
Xrs
|
o-----------------------+
Now, the relative values of these components depends upon the magnitudes
of the physical values times a transformation factor. That transformation
factor is the electromagnteic transduction factor, proportional to the Bl
product (the product of the length of the wire l immersed in the magnetic
field B), measured in N/A (or T/M, if you will). So, IF we know the
magnitudes of the physical components, we can easily calculate their
electrical equivalents:
Re - don't calculate it, just measure it with a good ohmmeter!
Lvc - measure it, but see below!
Lces - depends upon the suspension compliance:
2
Lces = Cms (Bl)
where Cms is the mechanical compliance in m/N, and the resulting
inductance is in henries.
Cmes - depends upon the cone mass:
2
Cmes = Mms/(Bl)
where Mms is the mechanical compliance in kg, and the resulting
capacitance is in farads
Res - depends upon the suspension losses:
2
Res = (Bl) /Rms
where Rms is the mechanical losses in 1/s, and the resulting
resistance is in ohms.
Xrs - depends upon the air, the driver diameter, the baffle dimen-
sions, position of the driver on the baffle, etc., but has
little effect on the electrical impedance.
For example, a typical 8" woofer with an Fs=30 Hz, Vas=60L, Qms=2.40, Qes=0.42, Qts=0.36, Re=6.25 ohms, might have the following mechanical parameters:
-3
Cms = 1.01 x 10 m/N,
-3
Mms = 27.9 x 10 kg,
Rms = 2.19 kg/s
Bl = 8.84 N/A
Then, the electrical equivalents would be:
Lces = 78.9 mH
Cmes = 356 uF
Res = 35.7 ohms
Re = 6.25 ohms
One can construct a similar branch for the enclosure, using the lumped parameters of a capacitive equivalent Cmep for the port mass Mmp, amd inductive equivalent Lceb for the enclosure compliance Cmb a resistive equivalent Reb for the system losses Rmb and the port radiation impedance Xrp (which is, again, small). That branch looks like:
o------+
|
Lceb
|
Cmep
|
Rmb
|
Xrp
|
o------+
The complete driver+enclosure+electrical model looks like:
o------Re------Lvc------+------------+
| |
+------+------+ Lceb
| | | |
Lces Cmes Res Cmep
| | | |
+------+------+ Reb
| |
Xrs Xrp
| |
o-----------------------+------------+
Now there are some other complicating elements that would make for a
complete mechanical and acoustical model, such as the mutual coupling of
the driver and port, etc., but for the electrical model
the above suffices quite well for predicting reality.
Let's look at the impedance of a very typical driver. It has the following characteristics:
Let's look at the following situation: Take an 8 ohm speaker and wind twice the length of wire onto the voice coil. The resistance woul go up, for sure, but because there is no more wire in thegap, the electromagnetic couping coefficient, the Bl product, would also go up. And you would have, as a result, a 16 ohm speaker with essentially the same efficiency as the 8 ohm speaker, all other things being equal.
Or you could design a speaker with both a higher impedance (longer wire in the voice coil) AND a larger magnet assembly with higher flux density in the gap and get a higher impedance driver with higher electro-acoustic efficiency.
Or you could design a higher impedance driver with a stronger magnet and a lighter cone and get even more efficiency.
The point is, the rated impedance IS NOT the same as the efficiency, nor is there any direct correlation between the two. Efficiency of a given direct readiator driver is determined by the folowing relationship:
2 2
B l
n0 = k * ------------
2 2
Re Sd Mas
where
So, we can see that by doubling the length of the wire that's in the gap (doubling l) will, by itself, increase the efficiency by a factor of 4, but since Re also doubles, it drops it by half, meaning that, all other things being equal, lengthening the voice coil winding in the gap increases BOTH impedance AND efficiency. Now, there ARE tradeoffs, and everything CAN'T be equal. Lengthening the wire ALSO increases the mass, though the voice coil is only part of a larger mass (it includes the vouice coil former, the cone, and so on) so there is not a direct relation. Also, the gap may need to be widened to accomadte the greater winding diameter of the voice coil, and that may reduce B.
Add resistance certainly reduces efficiency all by itself. You could, for example, just simply solder a resistor in series and, lo and behold, the impedance goes up and the efficiency goes down. But we already have a case where the efficiency goes up as the impedance goes up.
You could wind the voice coil with the same length of finer gauge wire. The result would be the imepdance goes up, and so does the restistance, but since l remains about the same, l^2, remains the same and the efficiency goes down. But wait!, finer wire means less mass, so we can gain some efficiency back from that and the finer wire means a smaller thickness to the voice coil, and the designer may be able to close up the gapand increase B.
Or, the designer may just design a TOTALLY difference driver with a different B, a different l, a different cone diameter (changes Sd), a different moving mass and a different resistance and get something totally different efficiency wise.
The point being is that a statement like "The higher the impedance, the lower then efficiency," as a generalization has NO basis in physical fact.
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