First, the assertion:

1 INTRODUCTION

Much ballyhoo surrounds the concept of "damping factor." it's been
suggested that it accounts for the alleged "dramatic differences" in
sound between tube and solid state amplifiers. The claim is made
(and partially cloaked in some physical reality) that a low source
resistance aids in controlling the motion of the cone at resonance
and elsewhere, for example:

    "reducing the output impedance of an amplifier and
    thereby increasing its damping factor will draw more
    energy from the loudspeaker driver as it is oscillating
    under its own inertial power." [1]

This is certainly true, to a point. But many of the claims made,
especially for the need for triple-digit damping factors, are not
based in any reality, be it theoretical, engineering, or acoustical.
This same person even suggested:

    "a damping factor of 5, ..., GROSSLY changes the time/
    amplitude envelope of bass notes, for instance. ... the
    note will start sluggishly and continue to increase in
    volume for a considerable amount of time, perhaps a
    second and a half."

Instead of unbridled hyperbole, there have been attempts at a
reasoned justification for damping factor. Witness a recent
rec.audio.tech post:

    "Since the amplifier source impedance is indeed much
    smaller than the speaker impedance, the latter is almost
    insignificant. In fact, an amplifier with a damping factor
    of 50 will sink twice the current of one with a damping
    factor of 25, and therefore dissipate four times the
    resonant energy." [2]

As intuitive as this analysis might seem, it is quite flawed since,
as we will see, it simply ignores the one major loss factor in the
entire system, throwing it out the window as if the single most
important controlling element over cone motion had no real
relevance.

2 DAMPING FACTOR: A SUMMARY

What is damping factor? Simply stated, it is the ratio between the
nominal load impedance (typically 8 ohms) and the source impedance
of the amplifier. Note that all modern amplifiers (with some
extremely rare exceptions) are, essentially, voltage sources, whose
output impedance is very low. That means their output voltage is
independent, over a wide range, of load impedance.

Many manufacturers trumpet their high damping factors (some claim
figures in the hundreds or thousands) as a figure of some
importance, hinting strongly that those amplifiers with lower
damping factors are decidedly inferior as a result. Historically,
this started in the late '60's and early '70's with the widespread
availability of solid state output stages in amplifiers, where the
effects of high plate resistance and output transformer windings
traditionally found in tube amplifiers could be avoided.

Is damping factor important? Maybe. We'll set out to do an analysis
of what effect damping factor has on what most proponents claim is
the most significant property: controlling the motion of the speaker
where it is at its highest, resonance.

The subject of damping factor and its effects on loudspeaker
response is not some black art or magic science, or even excessively
complex as to prevent its unserstanding by anyone with a reasonable
grasp of high-school level math. It has been exhaustively dealt with
by Thiele [3], Small [4] and many others decades ago.

3 SYSTEM Q AND DAMPING FACTOR

The definitive measurement of such motion is a concept called Q.
Technically, it is the ratio of the motional impedance to losses at
resonance. Another, completely equivalent view is that Q is the
ratio between the amount of energy stored in the system vs the
energy dissipated by losses.

It is a figure of merit that is intimately connected to the response
of the system in both the frequency and the time domains. A loud-
speaker system's response at cutoff is determined by the system's
total Q, designated Qtc, and represents the total resistive losses
in the system.

Two loss components make up Qtc: the combined mechanical and
acoustical losses, designated by Qmc, and the electrical losses,
designated by Qec. The total Qtc is related to each of these
components as follows:

         Qmc * Qec
   Qtc = ---------                                          [Eq 1]
         Qmc + Qec

Qmc is determined by the losses in the driver suspension, absorption
losses in the enclosure, leakage losses, and so on. Qec is
determined by the combination of the electrical resistance from the
DC resistance of the voice coil winding, lead resistance, crossover
components, and amplifier source resistance. Thus, it is the
electrical Q, Qec, that is affected by the amplifier source
resistance, and thus damping factor.

Qec itself is a measure of, simply, the ratio of the energy stored
in the moving system to the energy dissipated electrically by the
losses in the system, that is, in the resistances in the system. The
energy stored in the moving system, the kinetic energy, is dependent
upon the amount of mass and the velocity.

In the context of a speaker, the Qe is (from Small[4]):

                           2 2
   Qec = 2 pi Fc Mmc Re / B l                               [Eq 2]

where Fc is the resonant frequency of the system, Mac is the
equivalent moving mass of the system, and Re is the DC resistance of
the voice coil (and this assumes 0 source impedance or "infinite"
damping factor). Further, B represents the magnetic flux density in
the gap and l the length of wire in the magnetic field. (We will
assume that we are using the same driver for all considerations
here, thus, Fc, Mmc B and l remain the same as well.)

The effect of source resistance on Qec is simple and straight-
forward. From Small again [4]:

               Re + Rs
   Qec' = Qec ---------                                     [Eq 3]
                 Re

where Qec' is the new electrical Q with the effect of source
resistance, Qec is the electrical Q assuming 0 source resistance
(infinite damping factor), Re is the voice coil DC resistance, and
Rs is the combined source resistance.

The factor

       Re + Rs
      ---------                                             [Eq 4]
         Re

comes from the fact that Re is built into the original derivation
for Qec includes Re in it. The correction simply calculates the
incremental increase in Qe with the incremental increase in the
total electrical resistance. Reconciling [Eq 4] with [Eq 2], we see
that:

                                2 2
   Qec = 2 pi Fc Mmc (Re+Rs) / B l                          [Eq 6]

Thus it becomes obvious that the electrical Q of the speaker or,
more generally, the electrical damping of the speaker, is NOT
dependent upon the source resistance Rs alone (as the proponents of
damping factor erroneously claim), but on the TOTAL series
resistance seen by the driver, including the DC resistance of the
voice coil, Re. This mistake, as commonly as it is made, the the
fatal flaw in the entire damping factor argument.

It's very important at this juncture to note two points. First, in
nearly every loudspeaker system, and certainly in every loudspeaker
system that has any pretenses of high-fidelity, the majority of the
losses are electrical in nature, usually by a factor of 3 to 1 or
greater. Secondly, of those electrical losses, the largest part, by
far, is the DC resistance of the voice coil.

Now, once we know the new Qec' due to non-zero source resistances,
we can then recalculate the total system Q as needed using [Eq 3],
above.

The effect of the total Q on response at resonance is also fairly
straightforward. Again, from Small [4], we find:

                         4
                      Qtc
  Gh(max) = sqrt(-------------)                             [Eq 7]
                     2
                  Qtc  - 0.25

This is valid for Qtc values greater than 0.707. Below that, the
system response is overdamped and there is no response peak.

We can also calculated how long it takes for the system to damp
itself out under these various conditions. The scope of this article
precludes a detailed description of the method, but the figures
we'll look at later on are based on both simulations and
measurements of real systems, and the resulting decay times are
based on well-established principles of the audibility of
reverberation times at the frequencies of interest.

4 PRACTICAL EFFECTS OF DAMPING FACTOR ON SYSTEM RESPONSE

With this information in hand, we can now set out to examine what
the exact effect of source resistance and damping factor are on real
loudspeaker systems. Let's take an example of a closed-box, acoustic
suspension system, once that has been optimized for an amplifier
with an infinite damping factor. This system, let's say, has a
system resonance of 40 Hz and a system Qtc of 0.707 which leads to a
maximally flat response with no peak at system resonance. The
mechanical Qmc (i.e. the mechanical contributions to system losses
and thus damping) of such a system is typically about 3, we'll take
that for our model.

Rearranging [Eq 1] to derive the electrical Q of the system:

         Qtc * Qmc
   Qec = ---------                                          [Eq 8]
         Qtc - Qmc

we find that the electrical Q of the system, with an infinite
damping factor, is 0.925.

The DC resistance of the voice coil is typical at about 6.5 ohms.

Let's generate a table that shows the effects of progressively lower
damping factors on the system performance:

      --------------------------------------------------------
      Damping     Rs      Qec'     Qtc'    Gh(max)    Decay
      factor                                           time
      --------------------------------------------------------
       inf.     0 ohms   0.925    0.707    0.0 dB     0.04 sec
       2000     0.004    0.926    0.707    0.0        0.04
       1000     0.008    0.926    0.708    0.0        0.04
        500     0.016    0.927    0.708    0.0001     0.04
        200     0.04     0.931    0.71     0.0004     0.04
        100     0.08     0.936    0.714    0.0015     0.04
         50     0.16     0.948    0.72     0.0058     0.04
         20     0.4      0.982    0.74     0.033      0.041
         10     0.8      1.04     0.77     0.11       0.043
          5     1.6      1.15     0.83     0.35       0.047
          2     4        1.49     0.99     1.24       0.056
          1     8        2.06     1.22     2.54       0.069
      --------------------------------------------------------
                              Table 1

The first column is the damping factor using a nominal 8 ohm load.
The second is the effective amplifier source resistance that yields
that damping factor. The third column is the resulting Qec' caused
by the non-zero source resistance, the fourth is the new total
system Qtc' that results. The fifth column is the resulting peak
that is the direct result of the loss of damping control because of
the non- zero source resistance, and the last column is the decay
time to below audibility in seconds.

5 ANALYSIS

Several things are apparent from this table. First and foremost, any
notion of severe overhang or extended "time amplitude envelopes)
resulting from low damping factors simple does not exist. We see, at
most, a doubling of decay time (this doubling is true no matter WHAT
criteria is selected for decay time). The figure we see here of 70
milliseconds is well over an order of magnitude lower than that
suggested by one person, and this represents what I think we all
agree is an absolute worst-case scenario of a damping factor of 1.

Secondly, the effects of this loss of damping on system frequency
response is non-existent in most cases, and minimal in all but the
worst case scenario. If we select a criteria that 0.1 dB is the
absolute best in terms of the audibility of such a peak (and this is
probably overly optimistic by at least a factor of 2 to 5), then the
data in the table suggests that ANY damping factor over 10 is going
to result in  inaudible differences between such a damping factor
and one equal to infinity. It's highly doubtful that a response peak
of 1/3 dB is going to be identifiable reliably, thus extending the
limit another factor of two lower to a damping factor of 5.

Further, we simply do not observe the "factor-of-four" increase in
energy dissipation with a factor of two reduction in source
resistance as claimed in [2]. The statement that it's all about
energy dissipation is quite correct: remember that what damping is
doing is removing energy from a resonant system, and that the
measure of damping is Q, the ratio of energy stored to energy
dissipated. Look, for example, at the difference in Qt between a
damping factor of 50 and 20: the actual difference in the energy
dissipated is less than 3%. According to the theory expounded in
[2], the difference in energy dissipation should be around a factor
of 6!

All this is well and good, but the argument suggesting that these
minute changes may be audible suffers from even more fatal flaws.
The differences that we see in Q figures up to the point where the
damping factor is less than 10 are far less than the variations seen
in normal driver-to-driver parameters in single-lot productions.
Even those manufacturers who deliberately sort and match drivers are
not likely to match a Qt figure to better than 5%, and those numbers
will swamp any differences in damping factor greater than 20.

It is well known that the performance of drivers and systems is
dependent upon temperature, humidity and barometric pressure, and
those environ- mental variables will introduce performance changes
on the order of those presented by damping factors of 20 or less.
And we have completely ignored the effects presented by the
crossover and lead resistances, which will be a constant in any of
these figures, and further diminish the effects of non-zero source
resistance.

6 CONCLUSIONS

There may be audible differences that are caused by non-zero source
resistance. However, this analysis and any mode of measurement and
listening demonstrates conclusively that it is not due to the
changes in damping the motion of the cone at the point where it's at
it's most uncontrolled: system resonances. We have not looked at the
frequency- dependent attenuative effects of the source resistance,
but that's not what the strident claims are about.

Rather, the people advocating the importance of high damping factors
must look elsewhere for a culprit: motion control at resonance
simply fails utterly to explain the claimed differences.

7 REFERENCES

[1] James Kraft, reply to "Amplifier Damping Factor,
    Another  Useless Spec," rec.audio.high-end article
    2rcccn$...@introl.introl.com, 24 May 1994.

[2] Steve (aq...@lafn.org), reply to "How can 2 amps
    sound so different?," rec.audio.tech article
    7go6da$b8...@nnrp1.dejanews.com, 04 May 1999.

[3] A. Neville Thiele, "Loudspeakers in Vented Boxes,"
    Proc. IRE Australia, 1961 Aug., reprinted J. Audio
    Eng. Soc., 1971 May and June.

[4] Richard H. Small, "Closed-Box Loudspeaker Systems,"
    J. Audio Eng. Soc., Part I: "Analysis," 1972 Dec,
    Part II, "Synthesis," 1973 Jan/Feb.