Skin Effect in Speaker Cables Calculations
The Skin Effect was calculated on the last page and simulated in MathCad to be:
Rac = 1.34 * Rdc @ 20kHz (worst case frequency)
For 12AWG wire the Rdc is about 1.6 ohms/1000 feet. (Source: Belden Cables )
For 10ft of cable, we get a Rdc of: (1.6 / 1000)*20 = 32 mohms.
Thus total Rcable Resistance (Rac) @ 20kHz = 1.34*32 = 42.9 mohms .
As you recall, we calculated that Skin Effect does attribute about a 34% increase in cable resistance at 20kHz. This again assumes solid core wire. In actuality, multistranded wire has less of an issue with skin effect because each individual strand has a smaller cross sectional area than the skin depth at the particular frequency. Although un-insulated multi stranded wire is not considered true Litz wire, it still behaves somewhat as such and thus reduces the overall skin effect problem. However, for argument sake we will ignore this factor and thus we will ultimately yield a more conservate estimate of the skin effect problem. Recall that actual measured increase in AC resistance at 20kHz due to skin effect was only about 3% from our Cable Face Off article.
But what does this mean in terms of real world losses?
First we have to make some additional basic assumptions. For it we fail to identify all of the important variables, while not eliminating the non essentials ones, we set ourselves up for defeat and the situation becomes hopelessly complex and too convoluted to accurately analyze.
Speaker Load Considerations
Ok, here is the tricky part. Since loudspeakers are reactive loads, and there are a wide assortment of products on the market all with their own individual impedance characteristics, we have to make some assumptions. Usually a good loudspeaker design maintains a stable impedance vs frequency within the audio band. Here is a plot from a loudspeaker previously reviewed on Audioholics.com.
As you can see, the magnitude of impedance is centered around the nominal impedance rating of the speaker (8ohms), with dips down to 6ohms.
This speaker load is a relatively easy load for an amplifier to drive. Most modern day loudspeakers are relatively benign loads to drive compared to some of the more difficult ESL designs which typically have impedance dips down to about an ohm at 20kHz. We consider the ESL case an outlier as it is not representitive of a typical consumer product and usually requires cabling with low DC resistance and inductance for optimal performance.
Lets take a worst case typical modern speaker design load and assume we have a dip down to 2 ohms at 20kHz (very rare case).
We can now determine the voltage divider ratio between the speaker cable and load.
Rcable = 42.9 mohms @ 20kHz
Rspeaker (magnitude) = 2ohms @ 20kHz
Loss(dB) = 20*log ( Rspeaker / (Rspeaker + Rcable) ) = -0.18dB loss.
Again, recall our measured increase of AC Resistance due to skin effect was only about 3% at 20kHz resulting in a total cable resistance of 32.96 mohms for a 10ft length.
Actual Measured Loss
Loss(dB) (act) = -0.142dB
As you can see, the resultant calculated loss due to Rdc and Skin Effect (Rac) in the cable is negligable. In fact, if we neglected the Skin Effect losses, the resultant loss would be -0.14dB just from the DC Resistance of the speaker cables alone.
Thus the calculated Skin Effect losses in this example only account for -.04dB of total loss at 20kHz while the measured Skin Effect losses would be a mere 0.002dB! Also don't forget this is assuming a 2 ohm load, which most modern speakers do not dip down that low at 20kHz. In reality, a real world speaker load would make this loss almost immeasurable.