No, the electrical damping factor determines the degree of "control" the amp has over the drivers and it is true for any type of load.
No, "damping factor" has everything to do with
reonance, and comes from the world of dynamic loudspeakers.
If you look at the impedance plot of a typical dynamic loudspeaker, you'll see that it has a peak at the low end of its frequency range (dynamic headphone drivers also exhibit this same basic characteristic). Where this curve peaks represents the driver's resonant frequency and relates to the driver's low frequency response.
This resonance is the result of the reactive elements of a loudspeaker driver, which are capacitance and inductance in the electrical domain and mass and compliance ("springiness") in the mechanical domain.
The character of this resonance is described by "Q," which in driver specs is given as "Qts." Qts is a composite of the driver's electrical resonance (Qes) and its mechanical resonance (Qms).
A Q of 0.5 is what's called "critically damped," meaning you'll get the flattest low frequency response without any ringing or overshoot. A Q below 0.5 is called "overdamped," and while there is no overshoot or ringing, low frequency response will be diminished. A Q above 0.5 is called "underdamped," and results in varying degrees of ringing and overshoot as well a a peaking in the low frequency response.
There is a common compromise that shoots for a damping factor of 0.707. The "give" is that there will be a small amount of ringing and overshoot, but the "take" is that you get the flattest, lowest extending low frequency response.
If you put a driver of a given Qts in a sealed box, you end up with a higher overall Q (called Qtc). How much higher depends on the size of the box. Smaller boxes will raise Qtc to a greater degree than a larger box.
When when designing loudspeakers, it's assumed that the loudspeaker will be driven by a voltage source, which ideally means an output impedance of zero.
And now here is where damping factor comes in.
If you have a loudspeaker that you have designed to have a Qtc when driven by a zero or very low output impedance, as you increase the output impedance, you raise the overall Q, just as you would if you decreased the size of the box, and this means you will begin to get more and more ringing and overshoot, as well as a peak in the low frequency response.
Basically what happens is the higher output impedance diminishes electrical Q and the resonant behavior begins to resemble the driver's mechanical Q which is typically rather high.
Anyway, the salient point I'm trying to make here is that damping (and hence "damping factor") has everything to do with resonant behavior. And if you're driving a purely resistive load with no resonance, "damping factor" is meaningless.
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