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by John L. Murphy

Subject:  The Potentially Audible Effect of Speaker Cable Inductance

(posted 3Feb01 to Bass List)

R. wrote:

> I was under the impression that
> the DC resistance was fairly unimportant (that is what most people seems
> to be saying as well).

I am of the opinion (and most engineers seem to agree) that the DC resistance of a
speaker cable is the SINGLE MOST IMPORTANT specification of a cable at
audio frequencies and for reasonable (< 10 meters) cable lengths.  Cable
reactance (capacitance and inductance) can generally be ignored under these
conditions. (I realize that many vendors of expensive cable will disagree
with this statement)

The reason we do not use small diameter cables for speaker connections is
that the DC resistance can get to be high enough that the frequency response
of the speaker is affected.  A cable with about 1 Ohm (or more) of series
resistance can begin to have a small (about 1 dB) effect on the frequency
response of a typical loudspeaker.  This assumes that the loudspeaker has
significant peaks in its impedance response.



(posted 6Feb01 to Bass List)

Hu wrote:

> Some numbers are in order. Harnwell (Principles of
> Electricity and Magnetism, McGraw, 1949) and Terman (Radio
> Engineering, 3rd Ed, McGraw 1947) agree that the inductance
> of equal-diameter parallel wires, spaced by b, with radius a
> is given by
> L = 0.281 * log (b / a) microhenries/foot
> Assuming (realistically enough?) that b/a = 10, the
> reactance at 20 Khz for a 10 foot run is 0.35 ohms.
> Significant? Insignificant?

Hu, thanks for locating the formula and for the detailed references.

Luc then wrote:

> A normal b/a is something like 2 to 3 in my opinion. This would bring the
> inductance of 10 feet of cable to 0.85 uH or about 0.1 ohm. b/a = 3 would
> give 1.6 times more inductance (1.4 uH) and impedance (0.16 ohm).

I think Luc's estimate of the b/a is a bit low. I measured a
piece of 16-2 zip cord I had on hand and came up with the following numbers:

conductor radius= a = 0.0275 inches
conductor spacing = b = 0.125 inches (measured center to center)

So b/a for my 16-2 zip cord is .125/.0275 = 4.55

Based on the formula Hu provided I get:

L = 0.281 * log (b / a) microhenries/foot
L = 0.281 * log (4.55) microhenries/foot
L = 0.281 * .658 microhenries/foot
L = 0.185 microhenries/foot

So, for a 10 foot length of 16-2 zip cord L = 1.85 microhenries

At 20 kHz the impedance of this inductance is given by: Z = 2 * pi * f * L

Z = 2 * 3.14 * 20000 * .00000185
Z = 2 * 3.14 * .037
Z = 0.233 Ohms (at 20k Hz)

This is consistent with what Luc got.

So, let's see how this plays out for a 10 foot speaker cable made of 16-2
(16 awg, 2 conductor) lamp cord. Assume we have an 8 Ohm speaker which
actually has an 8 Ohm impedance at 20k Hz. The drop in signal level at 20
kHz due to the inductance of the cable would be:

attenuation = 20 * log ( 8 / 8.233 )
attenuation = -0.25 dB

The most cautious number anyone uses for audibility threshold is about 0.25
dB in the midrange (but considerably higher at the frequency extremes) so the 
level drop at 20 kHz due to this speaker cable is safely below even the most 
cautious audibility threshold at 20kHz.

More realistically, a typical 8 Ohm speaker system will have an impedance
more like 10 Ohms at 20kHz due to the voice coil inductance. This further
reduces the effect of cable inductance. Also, the "just noticeable
difference" for a 1/3 octave wide band at 20kHz is certainly much greater than 
the .25 dB I use as a rule of thumb limit for the midrange. Someone could look
it up but I bet it is more like 1dB or greater.

Now, assuming these more realistic numbers, let's ask how far we could run a
"16-2 lamp cord" before it had enough level loss at 20kHz to have even the
slightest chance of being audible by someone with excellent hearing at this
frequency extreme.

Breaking the problem into parts, let's first see what series impedance is
required to give 1 dB attenuation. For a 10 Ohm load I calculate that 1.22
Ohms of series impedance is required.

Now, what inductance results in 1.22 Ohms at 20kHz.

L = Z / (2 * pi * f ) = 1.22 / ( 2 * 3.14 * 20000 ) Henries
L = 1.22 / 125600 Henries
L = 0.00000971 Henries
L = 9.71 micro Henries

Now, what length of 16-2 cable has 9.71 micro Henries of inductance?

Since one foot of cable has 0.185 micro Henries,

Length = 9.71 / 0.185 = 52.5 feet

In other words, a speaker cable made of 16-2 zip cord would have to be about
52 feet long before someone with excellent hearing at 20kHz would have a
chance of hearing any signal loss. And this is a cautious analysis!
Most of us certainly do not hear to 20kHz and my choice of 1 dB as a just
noticeable difference at this frequency is probably paranoid. I bet most of
us would not even hear the loss due to even longer runs of this cable!

Given that even 52 feet of 16-2 zip cord is very unlikely to have any audible effect 
due to cable inductance, I remain of the opinion that simple cable RESISTANCE is 
the only cable parameter that needs to be considered when specifying a speaker cable. 
Even then, only the longest runs of the lightest cable are likely to ever to break the 
threshold of audibility.

For a look at my previous post where I examined the potential for audibility
of cable DC resistance you can visit:  https://www.trueaudio.com\post_007.htm

For the record, and to maintain perspective on this discussion, I am of the
opinion that just about any reasonable length of any reasonable speaker
cable will contribute no audible effect when used with our typical 4 and 8
Ohm dynamic type loudspeakers. This discussion is just to explore the
extremes so we can be confident that we have a wide margin of freedom from
any audible effects due to our choice of speaker cables. Audio enthusiasts
do occasionally run up against these limits when they run very long
lightweight cables (hundreds of feet of 22-2 ?) around their homes.
Professional audio companies routinely approach the threshold of audible
cable effects in the course of large commercial sound installations. (How
far could you run 16-2 cable to feed the speakers your company installed in
a major sports arena before experiencing a significant loss of signal level
or clearly audible (and measurable) frequency coloration?)



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 Resistance


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