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(1):

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

   This is  absolutely 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."

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  grasp  by  anyone  with  a
   reasonable  grasp   of  high-school   level  math.   It  has   been
   exhaustively dealt  with by  Thiele(2) and Small(3) and many others
   decades ago.

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. It  is a figure of merit that is intimately connected to
   the response  of the  system in both the frequency and the time do-
   mains. A  loudspeaker 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 = ---------                                          [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.

   The  effect   of  source   resistance  on   Qec   is   simple   and
   straightforward. From Small(3):

                     Re + Rs
          Qec' = Qec -------                                       [2]
                       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.

   It's very  important at  this point  to note  two points. First, in
   nearly every loudspeaker system, and certainly in every loudspeaker
   system that has nay 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. 2,
   above.
   The effect  of the  total Q on response at resonance is also fairly
   straightforward. Again, from Small, we find:

                           4
                        Qtc      1/2
          Gh(max) = (-----------)                                  [3]
                        2
                     Qtc  - 0.25

   This is  valid for  Qtc values  greater than 0.707. Below that, the
   system response is overdamped.

   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.

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  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, 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     Ohms                           dB    time (sec)
     -------------------------------------------------------------
        inf        0       0.9252    0.7071    0.0*      0.0396
       2000        0.004   0.9257    0.7074    0.0*      0.0396
       1000        0.008   0.9263    0.7078    0.0*      0.0396
        500        0.016   0.9274    0.7084    0.0001    0.0396
        200        0.04    0.9309    0.7104    0.0004    0.0397
        100        0.08    0.9366    0.7137    0.0015    0.0400
         50        0.16    0.9479    0.7203    0.0058    0.0403
         20        0.4     0.9821    0.7399    0.0327    0.0414
         10        0.8     1.0390    0.7717    0.1133    0.0432
          5        1.6     1.1529    0.8328    0.3523    0.0466
          2        4       1.4945    0.9976    1.2352    0.0559
          1        8       2.0638    1.2227    2.5411    0.0685
     -------------------------------------------------------------
     * less than 0.0001 dB

   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 Qmc'  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.

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.

   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.

   Further, the  performance of  drivers and systems is dependent upon
   temperature,  humidity   and   barometric   pressure,   and   those
   environmental 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.

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.

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

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

                      Copyright 1994 Dick Pierce
          Permission given for one-time no-charge electronic
  distribution via rec.audio.high-end with subsequent followups. All
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