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Damping Factor: Effects On System Response

by Dick Pierce August 30, 2004

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."

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."

Damping Factor: A Summary
What is damping factor? Simply stated, it is the ratio between the nominal load impedance (typically 8W ) 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 and Small and many others decades ago.

System Q and Damping Factor
The definitive measurement of such motion is a concept called . 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 domains. A loudspeaker system's response at cutoff is determined by the system's total , designated , and represents the total resistive losses in the system. Two loss components make up : the combined mechanical and acoustical losses, designated by , and the electrical losses, designated by . The total is related to each of these components as follows:

is determined by the losses in the driver suspension, absorption losses in the enclosure, leakage losses, and so on. 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 , , that is affected by the amplifier source resistance, and thus damping factor.

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

where is the new electrical with the effect of source resistance, is the electrical assuming 0W source resistance (infinite damping factor), is the voice coil DC resistance, and 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 due to non-zero source resistances, we can then recalculate the total system as needed using eq. 2, above.

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

This is valid for values greater than 0.707. Below that, the system response is over-damped 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.

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, one 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 of 0.707 which leads to a maximally flat response with no peak at system resonance. The mechanical of such a system is typically about 3, we'll take that for our model. Rearranging Eq. 1 to derive the electrical of the system, we find that the electrical of the system, with an infinite damping factor, is 0.925. The DC resistance of the voice coil is typical at about 6.5 W . From this data and the equations above, let's generate a table that shows the effects of progressively lower damping factors on the system performance

Damping
factor

R S

Q E '

Q T '

G H(MAX)

Decay
time

¥

0 W

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


The first column is the damping factor using a nominal 8W load. The second is the effective amplifier source resistance that yields that damping factor. The third column is the resulting caused by the non-zero source resistance, the fourth is the new total system 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.

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By Dick Pierce