Nov
19
Speaker impedance
Filed Under Speakers |
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:
- Voice coil’s electrical impedance (resistance, inductance, stray capacitance)
- Driver’s mechanical impedance (stiffness, mass, damping)
- Driver’s acoustic radiation impedance (resistance, reactance)
Spaker nominal 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.
Practical case
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.
Measuring speaker nominal impedance
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:
- a) Measure the DC resistance across the voice coil with the driver
disconnected. The ohm value you get, is the lower impedance bound of
the driver. Add an ohm or two to this value, and you should be at the
nominal (rated) impedance of the driver. - b) Connect a pot in series with the driver voice coil and then to the
amp. - c) Connect frequency generator or CD player with suitable test CD in
it to the amp input, and start walking up the frequency scale in
steps. For each step, measure the voltage
across the voice coil terminals and then across the pot. Adjust the
pot until both voltages match. Now shut down the amp. and measure the
resistance across the pot. This is the driver impedance for the
current drive frequency.
This approach is not the most accurate, but it needs minimal set of
measuring equipments: multimeter, signal generator and a potentiometer
of 50 ohms 5-10 watt. The clear advantage of this approach is that the
accuracy of the measurement is not affected by the multimeter frequency
response (their AC range is designed to show right values at around 50 Hz
range and at higher frequencies the accuracy can drop noticably depensing
on the meter construction, but this does not affect in this measurement
because the absolutely correct AC voltage values are not needed).
Warning, KEEP DRIVE LEVELS TO SPEAKER VERY LOW. High levels of
sustained sine wave can up the driver voice coil.
Speaker model
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----o
Now, 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.
Typical characteristics
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
Effect of enclosure
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.
Typical impedance characteristics of speaker element at different frequencies
Let’s look at the impedance of a very typical driver. It has the following
characteristics:
- At DC, the impedance is completely dominated by the DC resistance of the voice coil
- As you increase in frequency towards the fundamantal mechanical
resonance, the reflected motional impedance begins to dominate
and is inductive in nature. However, the total phase angle of the
impedance RARELY exceeds 45 degrees and thus the resistive and
reactive (inductive) parts of the impedance are just about equal. - At fundamental resonance, the impedance is purely resistive, its
phase angle is 0, and is determined by the effective series
combination of the voice coil DC resistance and the reflected
mechanical losses of (primarily) the suspension (Re + Res in
standard Thiele/Small notation). - Above fundamental resonance, the impedancs drops, has a negative
phase angle (rarely exceeding 45 degrees) and is, surprise,
capacitive in nature. The impedance drops until… - In the midrange, it approaches the DC resistance of the voice coil
it is SLIGHTLY higher than that DC resistance for a variety of
reasons, typically about 10-20% (and THIS is the region that is used
by MOST reasonably responsible manufacturers for specifying the
nominal impedance). The impedance at these frequencies is
predominantly resistive in nature and is dominated by the DC
resistance of the voice coil. - Above this region, the inductance of the voice coil begins to
influence the impedance. However, it NEVER becomes purely inductive,
or even remotely close. First, over the majority of the range of
operation, the voice coil resistance still dominates. Second, eddy
current losses in the pole piece (see Vaderkooy, et al) dominate
quickly, such that the phase angle of the impedance asymtotically
approaches about 45 degrees, and NEVER 90 degrees, which would be
necessary if your assertioon that the impedance was almost a pure
inductance were true.
Inmpedance of a speaker IS
NOT ALMOST A PURE INDUCTANCE. It is NOWHERE NEAR a pure inductance.
The impedance of a speaker is only a rough average of the impedance and
that the the voice coil dc resistance of most normal cone type dynamic
speaker is roughly 75% of its “rated” impedance as the industry rates
impedance. Most 8 ohm speakers will measure somewhere around 6+ ohms dc
give or take a bit. (When horn loaded, the impedance increases).
Impedance and effiency
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
- k is a constant determined by the properties of air
- B is the magnetic flux density in the gap
- l is the length of wire in the magnetic field
- Re is the DC resistance
- Sd is the radiating area of the cone
- Mas is the effective total moving acoustical mass of the driver.
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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