How do you evaluate/measure skin effect?
It would take me several pages to go through the detailed calculations, but this is a summary of the methodology:
Alternative I:
A)If the cable consists of a single conductor for each leg (plus and minus):
1)Start with a wire gauge table such as
this one, which shows, among other things, resistance per unit length for the various gauges.
2)Calculate the total resistance for the length of the cable multiplied by 2, reflecting the two-way round-trip that the signal has to make.
3)Calculate the corresponding signal attenuation at 20kHz (that would occur in the absence of skin effect), which (as a worst case approximation) will be dependent on the ratio of total cable resistance to the sum of total cable resistance and the impedance of the speaker at 20kHz.
4)As explained in
this reference, make the simplifying approximation that the skin depth that will be utilized by high frequency signal components is equal to the depth at which the current density of the highest frequency of interest (20kHz) is attenuated to 37% of its value at the surface, which for copper at an exaggeratedly worst case temperature of 70 degC is in turn equal to 2837/(square root of 20000 Hz) = 0.02 inches.
5)Calculate the cross-sectional area of the conductor.
6)Calculate the cross-sectional area of the central part of the conductor that is bounded by the effective skin depth.
7)Subtract no. 6 from no. 5 to get the approximate cross-sectional area that will be utilized by the 20kHz signal.
8)Using the wire gauge table, calculate the resistance of the gauge that corresponds to the cross-sectional area calculated in no. 7.
9)Calculate the resulting signal attenuation at 20kHz, which (as a worst case approximation) will be dependent on the ratio of the cable resistance calculated in no. 8 to the total of that figure and the impedance of the speaker at 20kHz.
10)Subtract the number of db calculated in no. 3 from the number of db calculated in no. 9. The result will be a reasonable approximation of the high frequency loss due to skin effect.
B)If the cable consists of multiple individually insulated conductors, that are all equal in gauge:
The calculations are similar to those described above, except that resistances are first calculated for one of the individually insulated conductors, and the two resulting resistance numbers are then divided by the total number of conductors to get the overall resistances.
C)If the cable consists of multiple individually insulated conductors, that are not equal in gauge:
The calculation is similar to (B) above, except that the resistances of the multiple conductors combine as the reciprocal of the sum of the reciprocals of the individual resistances.
Alternative II:
Don't bother with any of the above and accept various published statements, by people who should know, that under typical circumstances skin effect losses at 20kHz will be on the order of a fraction of a db, and considerably less at lower treble frequencies.
Regards,
-- Al