Re: Impedance &c.

[John Denker <jsd@AV8N.COM>]

Chuck Britton <britton@NCSSM.EDU>:asked:

> are other impedances also going to be expressed as the ratio of an
> intensive variable to an extensive variable - with the product of
> these two variables being power???

Well, yes, power *is* going to be extensive.  If energy is
extensive, so is power.  (Nitpickers note: We're being
three-dimensional i.e. non-relativistic here.)

But I wouldn't have said that (V I) is a product of an
intensive quantity with an extensive quantity.  If you
have a constant electric field, then voltage-drop is
extensive in one direction (along the field) and intensive
in the other two directions.  Similarly if you have a
constant current density, the current will be extensive
in two dimensions and intensive in the third.  It all
works out because the product has 1+2 degrees of extensivity.

You could doll this up in the language of Clifford algebra
(based on what Grassmann called his Algebra of Extension)
but it is easy enough to see as scaling law so we might as
well leave it at that:  an extensive quantity has to scale
like volume, so it has to pick up three factors of length
from somewhere.

=======================================

In general, asking about "impedance matching" in general is
like asking about dogs in general.
 A) Puppies are cute and cuddly, suitable for children to play with.
 B) Adult timber-wolves are not particularly cuddly and children
run the risk of being eaten.

 A) When dealing with lumped circuits, matching is almost trivial.
   A1) For non-reactive circuits, the source impedances should match
the load impedance.  Draw the load-line.  Open-circuit voltage,
short-circuit current, Thevenin, Norton, two points determina a line,
yada-yada.  Max voltage times no current equals no power.  Max
current times no voltage equals no power.  Finding the max-power
point is trivial using calculus and almost trivial otherwise.  If
you don't have a "natural" match, fix it with a transformer. This
is a beautiful thing:  very useful and very easy to explain.  Some
students, alas, will underestimate its importance because the
explanation took so little time.
  A2) If the load is reactive *and* it's a narrowband situation,
i.e. we are working at or near a single frequency, then the right
strategy is to get rid of the imaginary part.  Use a capacitor
to cancel an inductor and vice versa.  C and L in parallel makes
infinite impedance, which is a good thing to have in parallel
with your load.  Meanwhile C and L in series makes zero impedance,
which is a good thing to have
in series with your load.  Then match the real parts as in (A1)
and you're done.  For extra credit mumble something about the
triangle inequality to show that this strategy is not just a
rule of thumb but is in fact provably optimal.
  A3) For non-lumped circuits (e.g. transmission lines) in the
narrowband case, things start to get interesting, but still not
too bad if you have a Smith chart and know how to use it.  See
anecdote below.

  B) In the general case, where you have broadband non-lumped
elements, you aren't going to get by with a few simple rules
of thumb.  Entire books have been written on the subject.
Magic tees for waveguides.  Anti-reflective dielectric coatings
for lenses, and reflective dielectric coatings for mirrors.
Antennas.  This is serious physics and serious engineering.
(No, I'm not going to explain all that via email.)

A standard reference (not for the fainthearted) is
  Simon Ramo, John R. Whinnery, and Theodore Van Duzer
  _Fields and Waves in Communication Electronics_
Yes, it's THAT Simon Ramo.  Quite an interesting character.

=================================

Once upon a time I wanted to send/receive narrowband 1GHz signals
to/from a sample that was at a temperature of a few milliKelvin.
A 1GHz waveguide was clearly too big.  Coax carries the waves
just fine, but it also carries heat, orders of magnitude more
heat than any reasonable fridge can handle at mK temperatures,
unless you heat-sink the coax at some intermediate point(s)
where the heat-handling capability is greater.  Heat-sinking
the outside of the coax is straightforward ... but how can we
heat-sink the inner conductor?

The solution was to splice a "stub" onto the coax.  The stub
was 1/4 wavelength long.  It was dead-shorted inner-to-outer at
the end, thereby heat-sinking the inner.  From the point of
view of the main cable, a zero-impedance short 1/4 wavelength
away is an infinite impedance, which is a good thing to have
teed in parallel with the cable.  It worked great.  I thought
it was an amusing piece of out-of-the-box thinking.  Not earth-
shaking ... just one of the thousand little bits of cleverness
that makes up a piece of physics apparatus.

(BTW we were careful to *not* use tin/lead solder to make
the splice to the inner, because tin/lead is a superconductor.
Superconductors constitute a huuuge exception to the
Wiedemann-Franz law: they conduct electricity but not heat.
Remember the whole point was to suck heat through the stub.)