### Monday, May 13, 2013

## The Relationship Between Amplifier Damping Factor, Impedance & Cable

What it really means within the context of a systemEver have one of your friendly amplifier reps walk in your office to present their new mondo-gazillion-watt beast and point out the damping factor spec of greater than a bazillion? Why, gee-whiz! That’s like 10 times more than the other guy! It must be awesome! Right?

Well, as we have seen before, it depends on how you are going to use it. Let’s start with defining damping factor and see what it means to us.

Amplifier damping factor is defined as “the ratio of the load impedance (loudspeaker plus wire resistance) to the amplifier internal output impedance.”

This basically indicates the amplifier’s ability to control overshoot of the loudspeaker, i.e., to stop the cone from moving. It is most evident at frequencies below 150 Hz or so where the size and weight of the cones become significant.

A system where the damping factor of the entire loudspeaker/wire/amplifier circuit is very low will exhibit poor definition in the low frequency range. Low frequency transients such as kick drum hits will sound “muddy” instead of that crisp “punch” we would ideally want from the system.

The formula for calculating damping factor:

**Where:
ZL = The impedance of the loudspeaker(s)
ZAMP = The output impedance of the amplifier
RW = The resistance of the wire times 2 for the total loop resistance**

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Very few amplifier spec sheets state the output impedance, but you can generally call the manufacturer for this spec or you can calculate it by dividing the minimum rated load impedance by the damping factor rating.

For example, if we are using amplifier with a damping factor rating of 400 and it requires a minimum load of 2 ohms, then its output impedance would be calculated as being 0.005 ohms.

So let’s look at several examples and figure out what we can control in the design of our system to achieve the best results. Say we have two 8 ohm subwoofers connected to an amplifier with a damping factor of 400 with 100 feet of 12 ga. wire with a resistance of 0.00159 Ohms/foot times 100 feet gives us a total resistance of 0.159 Ohms.

Plugging the numbers into our formula, we get:

In this case, our system damping factor is just 12. Most experts agree that a reasonable minimum target damping factor (DF) for a live sound reinforcement system would be 20, so we need to consider changing something to get this up.

The critical element in this definition is the “loudspeaker plus wire resistance” part. In this case, the resistance in 100 feet of 12 ga. wire with a 4 ohm load results in around 0.7 dB of loss, much greater than the maximum target of 0.4 dB of loss, so let’s try bigger wire. 10 ga. wire has a resistance of .000999 ohms/foot times 100 feet equals .0999 ohms and will get us to the 0.4 dB target.

What will it do for DF?

OK, now we’re pretty close to the 20 we were looking for. Notice that the loudspeaker impedance can also give us a big change.

The higher the circuit impedance, the less loss we have due to wire resistance.

What if we change our wiring so we have one 8 ohm loudspeaker connected instead of two? Going back to our 12 ga. wire, we calculate:

Even better! In fact, if you run the numbers a few times, you will see that in a system with some significant length of wire, we will find that damping factor will generally be 20 or higher as long as our total wire loss is 0.4 dB or less.

What if we have a self-powered subwoofer? In this case, our loudspeaker wire is probably around 14 ga. and since the amplifier is in the loudspeaker enclosure, it is probably less than a couple feet long.

Assuming the manufacturer is connecting two 8 ohm loudspeakers to the amplifier, and 14 ga. wire has a resistance of .00256 ohms/foot times 2 feet equals 0.00506 Ohms of resistance, and our amplifier has a damping factor spec of 400, what do we get?

Wow! Now that’s a significant difference! Kind of supports the idea of using self-powered subwoofers, or at least putting the subwoofer amps as close as possible to the subs.

So we’ve looked at the differences in the size and length of our wire and the differences in hanging one loudspeaker on the line vs. two to change the impedance of the line.

What if we choose an amplifier with a higher damping factor spec., say 3,000? That’s a big difference, so we should see a much higher damping factor in our circuit, right?

Assuming this amplifier can drive a minimum 2 ohm load, we find the output impedance would be 0.001 ohms. Plugging the numbers into our single loudspeaker with 12 ga. wire system, we get:

Hmm, not such a big deal.

That higher amplifier damping factor only improved our system damping factor by 0.31 over the amplifier with a DF spec of only 400.

What if we use the amplifier with the 3,000 DF spec in our self-powered sub with 2 feet of 14 ga. wire?

Remember our calculation using the 400 DF amplifier was 264.55, so now we start to see when the amplifier spec becomes significant.

Essentially, in sound reinforcement systems where we have some significant length of wire between the amplifier and the loudspeaker, the amplifier DF spec has little affect on the performance of the system.

So what have we learned? In live sound reinforcement systems, damping factor is really driven by the length and size of our wire and the impedance of the loudspeakers we connect at the other end.

Since damping factor is mostly affects low frequency, we should endeavor to keep our subwoofer loudspeaker lines as short as possible and/or use larger gauge wire. We should keep the impedance of the connected load as high as possible by connecting only one transducer per wire instead of two.

So is more amplifier damping factor better? As one of my colleagues recently said, “Sure! If the loudspeaker terminals are welded to the amplifier output terminals!” Well, maybe he overstated it a little bit, but yes, as long as the loudspeaker wire is really short, then by all means!

**Jerrold Stevens has more than 25 years of experience in the audio industry, including contracting, independent sales representative, live sound and studio engineering, and audio system consulting and design. He now works with Eastern Acoustic Works (EAW).**

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