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by John L. Murphy

Subject:  The Potentially Audible Effect of Speaker Cable Resistance

(posted 31Jan01 to Bass List)

C. asked:

> To keep from curling up the speaker wires at one of my speakers, I'd like to
> use unequal speaker wire lengths.  One will be about 10' shorter than the
> other (20'v.30').  Does this really matter?  Am I better off using equal
> lengths and folding up the extra wire?

M. wrote:

> We are talking 0.02 ohms in the 10 foot run and 0.04 ohms for the 20 foot. 
> Into an 8 ohm resistance that would only be a L/R difference of 0.02 dB! Now 
> we can bring inductance, skin effect, phase of the moon, and so forth into 
> it, but hundredths of an ohm are swamped by inductor R, amp/cable connection, 
> and much more.

M. has a good point.

Such low cable resistance is unlikely to ever have an audible effect as other sources of series resistance are likely to be much greater. Sources of unwanted series resistance might be the connectors at the amp and at the speakers...and series inductors in the crossover, etc.

It would take about 0.25 Ohms of total source resistance to begin even approaching a super conservative audibility threshold of, say, .25 dB. In this case the total series resistance might result in swells of .25 dB where the speaker impedance peaks at various subsystem resonances. (woofer resonance, tweeter resonance...)

Let's ACTUALLY PERFORM an engineering analysis of a hypothetical (but realistic IMHO) loudspeaker system.

Here are the system details:

Amp output impedance: 0.02 Ohms (for damping factor = 400 @ 8 Ohms) 
Amp connector resistance: .05 Ohms (just a guess)
Cable Resistance: 0.02 Ohms (as discussed)
Speaker connector resistance: .07 Ohms (again, just a guess)
Leads from speaker connector plate to the crossover: .01 (small gage hookup wire!)
Series inductor in the crossover: .4 Ohms (the builder cheaped out on the inductor)
Leads from crossover to the woofer: .01 Ohms ( a guess)
Quick connects used to connect to woofer: .05 Ohms ( a guess)
Funky flex leads used within the woofer: .05 Ohms ( a guess)

(from the amps output devices to the voice coil)

This would appear to make any concern over the intrinsic speaker cable resistance vanish!

Now, how significant is this hypothetical source resistance of .68 Ohms??

Let's assume that our 8 Ohm speaker actually has an impedance of 6.5 Ohms at the woofer's minimum impedance and that the (closed box) woofer has an impedance of 50 Ohms at system resonance f(c).

At the frequency range where the woofer is near minimum impedance we have:

V(spk terminals) = V(amp out) * (6.5 / (6.5 + 0.68) ) = 6.5 / 7.18 = .9053 * V(amp out) 
( this is just the well known "voltage divider" equation)

=> - 0.864 dB , level drop in minimum impedance range

At the woofer resonance we would have:

V(spk terminals) = V(amp out) * (50 / (50 + 0.68) ) = 50 / 50.68 = .9866 * V(amp out) 

=> - 0.117 dB , level drop at woofer system resonance

What we would hear would be the DIFFERENCE between these levels:

Apparent peak heard at woofer resonance f(c): + .747 dB

We would also have a swell in the upper range of the woofer as the impedance increases, but it would definitely be less than the .747 dB of coloration at woofer resonance. There could also be a small peak at tweeter resonance.

Now, we might ask, would this woofer peak at f(c) be audible?

My OPINION is that in a carefully conducted ABX listening test with exactly the right source material (a sine wave at f(c) ) we would probably all be able to just hear this .75 dB difference.

Using pink noise for the test some of us might just barely hear the difference (just my estimated OPINION)

Using complex program material (again in a strictly controlled ABX test), I doubt any of us would hear a difference, but I could be wrong.

In any of our living rooms, with typical music source material, where the total series resistance changed from .68 Ohms one day to 0.00 Ohms the next day I really doubt that even one us would ever notice the difference. (again, just my OPINION)

Some additional questions to consider based on this analysis:

In the above example, If we were to switch from 12 awg speaker cable to 24 awg cable AND replace the .4 Ohm inductor with a .10 Ohm inductor would the DC resistance in the conductor path from the amp to the woofer be increased or decreased? Would we hear the difference (considering only DC resistance effects)?

Should any of us be concerned about audible effects resulting from a difference of 10 feet in our speaker cables?

What adjectives might a listener use to describe the .75 dB peak at woofer resonance? ...at tweeter resonance? 

How much of a difference in loudness at f(c) might result from production variations between two different drivers from two different production batches?

Would 18 awg zip cord have adequately low DC resistance for this system?

With this example I'm trying to show the value of careful engineering analysis when it comes to questions of audio system performance. In my experience using a combination of good analysis, careful measurement and well designed listening tests audio mysteries just melt away. I've never needed to resort to "magic" or "mysticism" to resolve an audio design problem. 



John L. Murphy
Physicist/Audio Engineer
True Audio
Check out my recent book "Introduction to Loudspeaker Design" at Amazon.com

Also See: The Potentially Audible Effect of Speaker Cable Inductance


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