Jim Lux wrote:
Richard Fry wrote:
[...]
Note that the attenuation is the same whether the inner conductor is
solid or tubular. This is the result of "skin effect," which for r-f
frequencies 1.8 MHz and higher confines the r-f current on the inner
conductor from its outer surface to a depth of less than 0.18 mm.


One should be aware that this formula applies only to "large" coaxial
transmission lines, where the skin depth is a small fraction of the
conductor thickness.

It's not like the current is confined in a uniform band of the skin
depth, and zero elsewhere. The skin depth is a convenient mathematical
fiction.. it's the depth at which the current density is 1/e, so you
can calculate things like voltage drop by assuming a uniform current
density in a layer that thick, instead of actually integrating it.

On a smallish round conductor, where the circumference isn't many, many
skin depths, there's a broken assumption in the skin depth formula of
an infinite flat plane. Actually solving for the true AC resistance (or
current distribution) involves elliptic integrals which only have
infinite series solutions.


Which is why there are nifty tables and empirical formulas for AC
resistance of round conductors (solid and tubular) that get you
arbitrarily close. See, e.g., NBS Circular 75 or Grover or Reference
Data for Radio Engineers.


Lest you think I am nit picking here.. take a piece of venerable RG-8
style coax, with the AWG13 inner conductor (0.072" diameter, 1.83 mm).
The skin depth at 1.8 MHz (per the above post) is 0.18mm, so the wire
is 10 skin depths across, so it's probably a reasonable assumption.


No, that wasn't nit picking; those are all fair points.

The underlying point is that engineering is ultimately about numbers. We
all like to think in words and mental images if we can, but in marginal
cases these simple slogans and cartoons won't work.

On the other hand, the marginal cases don't invalidate the point that
the skin effect *will* be present. If there isn't enough conductor depth
to allow the skin effect to develop unhindered, it only affects our
estimates of the AC/RF resistance.

If the available depth of conductor is too small, the inside boundary
will push the current density profile outward towards the surface. For a
round conductor, we can think of it as 'current crowding' along the
centreline. A closely related case is copper-plated steel, where the
magnetic nature of the steel increases its AC/RF resistance by a further
factor of sqrt(mu), which squeezes a much higher fraction of the total
current into the thin layer of copper.

However, let's take something a bit smaller, like RG-8X or RG-58 type
coaxes, which have a inner conductor on the order of 0.9mm. Now,
you're talking only 4-5 skin depths, and the assumption of an infinite
plane probably doesn't hold.

We can see a little further into this without the need for detailed
math. The radius of the conductor is 2.5 skin depths (again using
0.18mm) so the current density at this depth would normally be 1/e^2.5
or about 1/12 of its surface value. That suggests that the perturbation
in RF resistance due to insufficient depth is only taking place at
around the 10% level. In the context of *estimating* the RF resistance
to help us decide whether to buy a drum of cable, that wouldn't be a
serious error.

However, it warns of a very serious error if the centre conductor was
made of copper-plated steel instead of solid copper.


So.. Terman's equation probably holds for coax where the inner
conductor is 20 skin depths,

Sorry, Jim, you lost me: why such a large number as 20?

At 2.5 skin depths, the current density is 10% of the surface value;
at 5 skin depths, 1%. If at least 5 skin depths are available, we can
be confident in the accuracy of the standard, uncorrected equation for
most purposes.

A more serious effect of insufficient conductor depth may be in
estimating the effectiveness of shielding. The residual fields at the
opposite side of an extremely thin shield can be very significant if
we're looking for attenuations of 40dB, 60dB or more.



--

73 from Ian GM3SEK 'In Practice' columnist for RadCom (RSGB)
http://www.ifwtech.co.uk/g3sek