From: DPierce@world.std.com (Richard d Pierce)
Subject: Speaker wire and Damping Factor ( Why it's hogwash )
Organization: The World Public Access UNIX, Brookline, MA
Date: Tue, 15 Sep 1992 03:25:52 GMT

I usually trim articles down quite a bit when posting a followup, but I
felt it necessary to include most of this one. Also, this is one of the
rare times when I have already written the followup months ago (in reply
to the article cited below). My real followup will appear in a forthcoming
SPeaker Builder.

In article <lb9jtqINN9hh@exodus.Eng.Sun.COM> tuohy@dune.Eng.Sun.COM (Bill Tuohy) writes:
>With all the recent discussion of cables (again), and no accomanying
>reasonable arguments either way (again), I thought I'd relate the only
>reasonable explanation of cable effects that I have read.  This is from
>Speaker Builder, 3/92, p. 26-27, "Damping Factor Reviewed" by Gary Galo,
>Contributing Editor.
>
>Damping Factor (DF) is the ratio between the speaker impedance and the
>amplifier source impedance, at the speakers terminals.  The amplifier source
>impedance seen by the speakers is the actual amp source PLUS the cable
>resistance.  
>
>Practically speaking, damping factor refers to the amps ability to control
>the motion of the speaker cone.  When the amp sends a signal to the speaker,
>the cone moves, but when the signal changes or disappears, the cone
>may continue to move if not well controlled.  This will distort the sound,
>with ringing or overshoot on transients, etc.  The lower the amp source
>impedance, the closer the cone movement will match the signal.
>Higher source impedance causes the extra movement.
>
>Since amps rarely spec output impedance, you can calculate it from the
>damping factor spec if you know your speakers impedance at the frequency
>where the damping factor is speced (typically 50 Hz).
>
>As an example, Galo uses an imaginary amp with a DF of 400 at 8 ohms. 
>This means this amp has an output impedance of 8/400 = 0.02 ohms.  Now 
>see what happens with 15' of 16 gauge wire attached. Since it's 2-condutcor
>cable, you actually have 30' of wire in the signal path.  30' of 16 ga. is
>about 0.1228 ohms.  Add this to the 0.02 ohms of the amp, and you get 0.1428
>ohms.  Now the cable resistance is quite significant.  The resulting
>DF = 8/0.1428 = 56!  The damping factor is reduced from 400 to 56 by normal
>wire!
>
>With 12 ga. wire, the wire impedance is 0.048 ohms, and the resulting DF
>is about 117.  This is a big difference from 56.  
>
>This means that people who compute wire resistance of around 0.1 ohms, and
>say this is insignificant compared to the 8 ohm speaker load, are looking
>at it from the wrong end of the wire ( the amp end ). You need to look at
>it from the speaker end of the wire, where 0.1 ohms is VERY significant
>compared to amp output impedances that range from 0.1 down to
>0.01 ohms for the some of the better amps.
>
>Series resistance of crossover components also affects
>DF.  This discussion just shows how to keep the
>effects of the wire to a minimum - short runs of
>12 ga. or better wire.  Galo recommends longer
>interconnect from preamp to amp, to shorten speaker
>wire runs, if the preamp can handle the extra capacitcance.
>
Well, Bill, you and Gary fell face first into the big trap of Damping Factor.
In all the caclulations you quote (and Gary made), you all neglected a very
important source of resistance that completely overshadows all else. That is
the DC resistance of the voice coil. In my reply to Gary's article, I take
his example and apply some real math from real world speakers to it. Let's
take his DF=400 amplifier and now acount for the DC resistance of the voice
coil. In a typical 8 ohm system, this is somewher around 6.5 ohms or so. Now,
given that, with an amplifier source resistance of 0.02 ohms, let's presume
a resistance in the cables of 0.000. Add to that the series resistance of
the voice coil, 6.5 ohms, and the total is 6.52 ohms. Now the real damping
factor that the motor really sees is 8/6.52 or 1.227. Now let's add the 0.1228
ohms for our 16 gauge wire where we get a total series resistance of 6.6428
ohms, resulting in a whopping damping factor reduction to 1.2043!

I would argue this difference is hardly as dramatic as Gary or Bill would
believe.

Let's look at it from a different viewpoint. Small states that the total change
in Q of a speaker due to amplifier source resistance is as follows:

              Re + Rg
     Qe' = Qe---------
                 Rg

Where Qt is the original system electrical Q at resonance presuming zero source
resistance, Re is the driver voice coil resistance, Rg is the source
resistance from all source (amplifier, crossover, speaker leads) and
Qe' is the resulting electrical Q at resonance.

Now, the total system Q is calculated by:

          Qe Qm
    Qt = --------
         Qe + Qm

Where Qm is the mechanical Q, and Qt is the total system Q.

Take a maximally flat closed box system, with a Qt of 0.707. Such a system
might have a mechanical Q of about 4, an electrical of 0.859. 

Let's add our 0.1228 ohms of resistance and see what happens to the Q. Well, it
turns out the Qe rises:

               6.5 + .1228
   Qe' = .859 ------------
                  6.5

Which results in a new Qe' of 0.875, and the new QT' then becomes:

          4 * 0.875
    Qt' = ---------
          4 + 0.875

Which equals 0.7179, instead of 0.707. So what? What's, say, the frequency
response effect of this raise in Qt? Well, Small further tells us that the
bump at resonance due related to Q is 

                 4
               Qt     1/2
   Gmax = [----------]
             2
           Qt  - 0.25

PLuggin our numbers in to this equation results in the revelation that our
change of Q due to tha addition of .1228 ohm changes our system response
from dead flat (0.00000.... db peak) to a peak that is 0.00387 dB high.

I would argue that this is probably inaudible.

Now, does this suggest that the wire has no effect on the sound? No, but it
sure does very strongly suggest that the change in damping factor isn't why
there might be change.

>
>Just trying to throw a little gas on the fire...
>
Sorry if I dumped an ocean or two of water on it instead...

.


-- 
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