max@eros.uucp (Max Hauser) (09/23/87)
This started out with several topics but I decided to chop it up as it was getting even longer than usual. I don't want my friend Graham from Jamaica to accuse me of long-windedness again. I have omitted sci.physics as this is beneath their dignity. I post this also with some trepidation, as others will no doubt beat me to it, and do a better job, what with the Time Warp and all, and this information being anything but secret. But here goes. In <1549@culdev1.UUCP>, drw@culdev1.UUCP (Dale Worley) writes: > As long as we're debunking things, note that the "skin effect" (if > it's the skin effect I know and love) involves a "penetration depth" > measured in fractions of a wavelength. Since the wavelength at 20kHz > is ... hmmm ... 5 kilometers, it doesn't seem too significant. At high frequencies, current tends to flow at the surface of a conductor, and the better the conductor, the shallower is the characteristic depth (or "skin depth"), at which (roughly) current density falls by a factor of e from the surface. As I and others have posted before, skin effect is (surprisingly, to me) an important matter at audio frequencies. The skin depth is certainly not a fraction of a wavelength [argument third pph following], but rather, inversely proportional to the square roots of frequency and conductivity [1], and for copper, it is about 0.5 mm at 20 kHz [2]. Since the falloff of current with depth is smooth rather than abrupt, this does not of course mean that the metal conducts in only a 0.5 mm layer, but rather that the effect significantly changes resistance for wires on that order of radius -- and speaker wires, solid or heavy stranded, are on that order of radius. Skin depth also depends on magnetic permeability (mu), but this is substantially material-independent, and indeed the the same as for vacuum, with common conductors [2]. Also, the phenomenon affects inductance along with resistance. Both become (weakly) frequency - dependent. Note that with more-resistive materials (like mercury), the skin depth becomes *larger* (not smaller, as an earlier poster mentioned, no doubt in a moment of distraction), but the drop in conductivity more than compensates for this to yield a larger net resistance in the wire, at frequencies where skin depth is important. Skin depths for metals consistently follow the form d / sqrt(freq), where the constant of proportionality, d, is 66 mm for copper; 83 mm for aluminum; 64 mm for silver, and 18.5 mm for typical solder alloy [2]. Therefore the 0.5 mm skin depth of copper wire at 20 kHz becomes only 0.13 mm for typical solder. Finally, those with an applied-math bent will appreciate quickly that not only does skin depth happen not to be proportional to free-space wavelength, it moreover *cannot* be so. Skin effect is a classic example of a linear boundary-layer problem, which is to say a one-dimensional problem arising from a reduction of a partial differential equation near a boundary (in this case the electromagnetic wave-propagation equation when it impinges on an imperfectly conductive plane -- another good example is the velocity of a fluid flow near a parallel wall). As such, it forms a Sturm-Liouville system [3], and the associated characteristic boundary depth (skin depth, in this case) arises as the dissipative (real) part of an eigenvalue of this system. Characteristically in such situations (or it can be shown from dimensions alone), this depth is not determined by wavelength. This article appears as a point of information and should not be misconstrued as a wholesale defense of Luddite audiophilia. References: [1] Any good undergraduate EM text; I suggest Ramo, Whinnery and van Duzer (Wiley, 1965), p. 252. Engineering EM texts treat this in more detail than equivalent physics texts (like Loraine and Corson), because of its engineering importance. Ramo is of course the same as in Thomson-Ramo-Wooldridge (and therefore the "R" in "TRW"). [2] Ramo, Whinnery and van Duzer, table, p. 289 [3] For example, Pearson, editor, _Handbook of Applied Mathematics_, Van Nostrand Reinhold, 1974, pp. 315-323 Max W. Hauser, engineer enthusiastic and curmudgeon extraordinary UUCP: ...{!decvax}!ucbvax!eros!max Internet (domain style): max@eros.berkeley.edu Internet (old style): max%eros@berkeley (415) 642-6666 Some more numbers (numbers are big on the net, in signature files): 6926323; 25653; P1-12-20075; 4,435,655
max@eros.uucp (Max Hauser) (09/23/87)
What did I say about moments of distraction? I blew a decimal in my associated article. >Skin depths for metals consistently follow the form d / sqrt(freq), >where the constant of proportionality, d, is 66 mm for copper; >83 mm for aluminum; 64 mm for silver, and 18.5 mm for typical >solder alloy [2]. ... The last should be d = 185 mm for solder, so at 20 kHz, the depth is 1.3 mm for solder, not 0.13mm as in my original, which contradicted the previous pph as well.