## DEFINING "DAMPING FACTOR"

by John L. MurphyPhysicist/Audio Engineer

True Audio

Check out my recent book "Introduction to Loudspeaker Design"

*(Part 2 of 2 parts)*

Go To Part 1

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An audio power amplifier′s damping factor is defined as the ratio of the load impedance to the output impedance of the amplifier.

**Example 1:**

Amp output impedance at 1kHz is known to be: 0.25 Ohms

Impedance of the test load is 8 Ohms (at 1kHz)

Damping Factor = (Load Impedance) / (Output Impedance) = 8 / 0.25 = 32 (dimensionless ratio)

Now, add a 0.25 Ohm speaker cable between the amp and the speaker and measure the damping factor at the speaker terminals and you would get: Damping Factor = 16 (Note that damping factor varies with frequency)

**Example 2:**

What if you started with an amp with output impedance of 0.0025 Ohms?

DF = 8 / 0.0025 = 3200

WOW! What a spec!

Now, add your .25 Ohm speaker cable and evaluate the damping factor at the speaker terminals:

New source impedance = 0.0025 + 0.25 = 0.2525 Ohms (at spkr terminals)

DF = 8 / .2525 = 31.7

Where did the DF=3000 go! I paid extra for that number!

**Example 3:**

Now determine the damping factor at the actual woofer terminals:

( Hint: Internally the speaker has a 0.5 Ohm inductor in series with the woofer)

Source Impedance = 0.0025 + 0.25 + 0.5 = 0.7525 Ohms (amp+cable+inductor)

DF = 8 / .7525 = 10.6

The point I′m trying to make is that the actual amplifier damping factor specification has little to do with the damping factor seen by a typical woofer...unless the woofer is welded directly to the output terminals of the amplifier ... there could be a patent here. :-)

Many audio engineers are of the opinion that an amplifier damping factor of 10 or greater is adequate. Those sky high damping factors seen on the spec sheets of some amps are frequently just inventions of the marketing department and are irrelevant to actual system performance. The effect of higher source impedances (lower damping factors) is the same as adding series resistance in the speaker cable. Ultimately, the effect is a micro equalization of the frequency response as the voltage drive to the speaker becomes non-flat due to the frequency dependant impedance of the speaker. (adding series resistance creates a small peak at the speaker′s own impedance peak...often on the order of 0.25 dB or so) The effect of the series resistance of the "damping" of the speaker is difficult to see when the problem is viewed this way.

The Q_{tc} of a closed box speaker is increased by the addition of a series resistance. Here is the formula for this increase in system Q:

Q_{tc} = Q_{tco} ( (R_{e} + R_{g})/ R_{e} ) where:

Q_{tc} is the final Q of the speaker system

Q_{tco} is the Q of the speaker with zero Ohms source impedance

R_{e} is the DC resistance of the speaker

R_{g} is the added series resistance

**Example 4:**

Say we have a speaker system with Q_{tco} = 0.707 and DC resistance R_{e} = 6.5 Ohms. We add 0.25 Ohms of series resistance by way of our amp, speaker cable and crossover. The net Q of the speaker then becomes:

Q_{tc} = 0.707 ( (6.5 + .25) / 6.5 ) = .707 (6.75/6.5) = .734

So the effect of 0.25 Ohms series resistance is really to raise the Q of the speaker from .707 to .734. We could calculate the damping factor...but who cares! We are really only concerned with our net system response. Yes, you could say the "lower damping factor" has affected the transient response of the speaker for the worse. We′ve all heard the mysterious explanation that "the cone keeps moving after the signal has stopped". But I prefer to look at the problem in terms of the speaker′s Q_{tc}. We can all relate to the speaker Q much better than "the cone keeps moving...". So I prefer to move any discussion of amplifier damping factor away from the mysterious "cone keeps moving..." and into the much better understood arena of speaker system Q.

As you can see there is much more to the issue of "speaker damping" than just the amplifier′s damping factor. In many systems the amp′s DF will be irrelevant to the final system response because of the series resistance added by the speaker cable and the passive crossover components (see Example 3 above). Speaker designers should always be aware of the source impedance from which their speakers will be driven so that they can compensate for the source impedance in their design. If in fact your goal is to design a speaker system that will have a net response Q_{tc} = .707 then you will need to anticipate the R_{g} (source impedance) the driver will "see" and design the enclosure for some lower Q_{tco} such that the R_{g} will raise the NET Q_{tc} to the targeted 0.707.

Regards,

John