[SI-LIST] Skin Effect Phase 2
- From: Doug Brooks <dbrooks9@xxxxxxxxxxxxxxx>
- To: rhaller@xxxxxxxxxxxxx,vachkumar@xxxxxxxxx
- Date: Tue, 12 Jun 2012 14:31:54 -0700
Thanks for sending this link.
The following is in no way an indictment re how we handle skin effect
calculations. It's just that as I look further into it, I'm a little
surprised at how many approximations we make.
Consider the following:
1. We hypothesize a thing called the skin depth, which we can define
with some implied precision.
2. Then we assume the current density is uniform from the surface
down to the skin depth. This allows us to then calculate the
cross-sectional area of the conductor where this assumed current is
flowing, allowing us to then infer the increased resistance of the
conductor. Except.......... the current does not flow that way.
3. We assume the current density is following an exponential curve.
We conveniently define this (normalized) curve as J =
(1/sd)*e^(-d/sd). This allows us to calculate the normalized current
density at the skin depth as J=1/(e*sd). Except this doesn't fit, either.
4. We have defined three curves under which we can calculate the
area: (a) a rectangle of uniform current density (J) through out the
conductor, (b) a rectangle of uniform current density (J/(e*sd)) down
to the skin depth, and (c) an exponential curve. If each truly
represents the situation accurately, the (normalized) area under each
of these curves equals 1.0, o0r at least equal each other. But in
fact, the area under the exponential curve does not. It is very close
for sd<<conductor radius, but the error goes up the deeper the skin
depth. (It is still true, however, that this error is small, less
than a couple of percent.)
So the skin effect calculations we make are models of the skin effect
based on assumptions. Probably very good assumptions. But
nevertheless, they are not exact calculations. Is it too much to
expect that some exact calculations exist, or is this the best we can
do with our current knowledge and capabilities?
Doug Brooks
At 05:56 AM 6/12/2012, Robert Haller wrote:
>I had the pleasure if working with Mike Tsuk (PHD) at DEC who did his
>thesis on Skin effect and supported all our EM tools. When I left DEC his
>parting gift to me was a very concise table comparing skin depth versus
>frequency which I find invaluable (and published with his permission). You
>can find this table in a paper I wrote comparing lossy versus lossless
>T-line simulation results.
>I keep his table on the wall of my cube to refresh my memory how dramatic
>the skin effect is at high frequencies. Regards and hopefully you will
>find this helpful.
>
>At 1 MHz ~ skin depth is 2.5 mils
>At 100 Mhz ~ skin depth is .26 mils
>At 1 Ghz ~ skin depth is .08 mils
>At 10G ~ skin depth is 26 uinches
>
>http://www.iec.org/newsletter/aug06_2/design_eng_1.html
>
>Regards
>Bob
>
>
>-----Original Message-----
>From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
>On Behalf Of Vachan
>Sent: Monday, June 04, 2012 1:33 PM
>To: dbrooks7@xxxxxxxxxxxxxxx
>Cc: si-list@xxxxxxxxxxxxx
>Subject: [SI-LIST] Re: Skin Effect question
>
>I think that is the point of defining a skin depth - An exponential
>current density is mathematically equivalent to an approximation where you
>assume uniform current density just below the surface (up to 1 skin
>depth), and then the current density suddenly drops to 0.
>On Sun, Jun 3, 2012 at 7:44 PM, Doug Brooks
><dbrooks7@xxxxxxxxxxxxxxx>wrote:
>
> > This question relates to skin effect.
> > Consider a current density function with (surface density) Io=8 and
> > skin depth = .125, unity radius.
> >
> > consider the current density function y=8*e^(-x/.125) integrate the
> > area under this function between 0=<x=<1. The answer is
> > .9997 (according to the tool I am using!)
> >
> > Consider the rectangle formed by the points 0,0, 0,8, .125,8, .125,0
> > (Note that x=.125 [i.e. at the skin depth] is where the exponent of e
> > in the current density function is -1.) the area under this rectangle
> > is 1.0 (same as the area under the current density function.)
> >
> > Here is my question. Is this a fortunate coincidence or can this
> > identity be proven mathematically?
> >
> >
> > Doug has a new e-mail address dbrooks7@xxxxxxxxxxxxxxx check out the
> > free resources at http://www.ultracad.com
> >
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>
>
>--
>Vachan
>
>
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