Volume II, Chapter 3, More on the "skin effect" has a table showing "k" factors for various wire sizes:
http://www.allaboutcircuits.com/vol_2/chpt_3/6.html
Then a formula is provided, Rac = (Rdc)(k)*sqrt(f).
This formula is supposed to give the AC resistance of a wire at some frequency where skin effect is coming into play.
The formula can be re-arranged to give Rac/Rdc = k*sqrt(f), where Rac/Rdc is the high frequency resistance ratio. But, k*sqrt(f) is not the correct Rac/Rdc, it's the ratio of the wire radius to the skin depth. To get the correct value of Rac/Rdc, the value k*sqrt(f) must be used as an entry into a table of values of Rac/Rdc, such as this one found in Terman's "Radio Engineer's Handbook":
For example, consider a 10 gauge wire used at .1 MHz. The table gives a "k" value of 27.6. At .1 MHz the formula gives a value k*sqrt(.1) = 8.73--is this the correct value for Rac/Rdc? If we look up the Rac/Rdc value from Terman's table, we get a value of Rac/Rdc = 3.35.
Which is it--8.73 or 3.35?
Here's a measurement of the AC resistance of a piece of 10 gauge wire taken with an impedance analyzer:
The analyzer swept the frequency from 20 Hz to 100 kHz. The measured resistance at 100 kHz was 19.929 milliohms, and at 10 Hz it was 5.334 milliohms for an Rac/Rdc ratio of 3.74. This compares favorably with the theoretical value of 3.36. The value 8.73 returned by the eBook formula is clearly wrong.
There's a calculator on the web that gives the correct result:
http://chemandy.com/calculators/round-wire-ac-resistance-calculator.htm
http://www.allaboutcircuits.com/vol_2/chpt_3/6.html
Then a formula is provided, Rac = (Rdc)(k)*sqrt(f).
This formula is supposed to give the AC resistance of a wire at some frequency where skin effect is coming into play.
The formula can be re-arranged to give Rac/Rdc = k*sqrt(f), where Rac/Rdc is the high frequency resistance ratio. But, k*sqrt(f) is not the correct Rac/Rdc, it's the ratio of the wire radius to the skin depth. To get the correct value of Rac/Rdc, the value k*sqrt(f) must be used as an entry into a table of values of Rac/Rdc, such as this one found in Terman's "Radio Engineer's Handbook":
For example, consider a 10 gauge wire used at .1 MHz. The table gives a "k" value of 27.6. At .1 MHz the formula gives a value k*sqrt(.1) = 8.73--is this the correct value for Rac/Rdc? If we look up the Rac/Rdc value from Terman's table, we get a value of Rac/Rdc = 3.35.
Which is it--8.73 or 3.35?
Here's a measurement of the AC resistance of a piece of 10 gauge wire taken with an impedance analyzer:
The analyzer swept the frequency from 20 Hz to 100 kHz. The measured resistance at 100 kHz was 19.929 milliohms, and at 10 Hz it was 5.334 milliohms for an Rac/Rdc ratio of 3.74. This compares favorably with the theoretical value of 3.36. The value 8.73 returned by the eBook formula is clearly wrong.
There's a calculator on the web that gives the correct result:
http://chemandy.com/calculators/round-wire-ac-resistance-calculator.htm
