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Re: [Phys-l] physics of dissipation



That's all fine, John, but RG-174 and, to a lesser extent, RG-223 and -59 can provide massive resonances at 5 MHz and above. Especially when one is using chirped signals in those frequency ranges (and thus cannot definitively impedance-match at both ends), the cable resonances show up so strongly as to make delicate little signals have real problems (we're talking at least orders of magnitudes in the transmitted amplitudes, let alone the requisite phase shifts).
In fact, the resonances I observed ended up giving me a good measure of group velocities in the cables, as I was able to swap out some different lengths and see the resulting resonance peaks squish closer together and farther apart.

All I'm saying is that these effects can be extremely pedagogically useful (and, as you mention, test the ability of physicists to resolve things in terms of their own Aha! moments), but they need to be accounted for in careful work, and particularly in work where one is not restricted to a tied-down, static cable configuration.

/************************************
Down with categorical imperative!
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________________________________
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wed, November 4, 2009 12:45:32 PM
Subject: Re: [Phys-l] physics of dissipation

On 11/04/2009 11:45 AM, curtis osterhoudt wrote:
It's very much frequency dependent. For example, above about 5 MHz,
the "50 Ohm" standard starts to break down, but the exact details are
also dependent on the coax diameters, etc. Larger cables tend to
stick closer to the 50 Ohm standard to higher frequencies.

Particular field work a few years ago was made practically laughable
due to the need for some finely-tuned preamplifiers (tuned to
nearly-oscillating). The combination of that plus dealing with thin
coax cable resonances in the 4 - 10 MHz range made things nigh unto
impossible. A combination of damping resistors at either cable end,
plus inductive loading, made things workable, but it was a bad
oversight to have neglected the cables' effects on the experiment.

Stick any "50 Ohm" cable on a network analyzer and sweep through the
frequencies. Cable manufacturers are well aware of this, and
different cables are recommended when starting to push the boundaries
of what's feasible.

Been there, done that, didn't see anything weird up to a
couple of GHz. I was using "good" coax, and I'm sure you
can find "bad" coax if you set your mind to it ... but
there is no reason to think all coax is bad in principle.

At GHz frequencies, you have to be a bit of an artiste to
properly attach connectors at the ends of the coax. I've
seen lots of problems with connectors, not so much with
the coax per se ... even when the coax was abused e.g. by
having one end at 300 K and the other end at 30 mK.

At reeeally high frequencies, the coax starts to look more
like a waveguide, but at GHz frequencies and below, all
such modes are well beyond cutoff, unless your coax has
the diameter of a sewer pipe.

Even if you have something where the impedance *is* quite
frequency dependent, such as a waveguide, the Nyquist
model still works. Just analyze the system frequency by
frequency. It's more laborious and less elegant, but it
is just a bunch of routine crank-turning. The ideas are
the same. The fluctuation/dissipation theorem applies on
a mode-by-mode basis ... even when you are shining a
flashlight into a sewer pipe.

Impedance is basically the density of modes, i.e. number
of modes per unit k vector. The physics of "inductance
per unit length" and "capacitance per unit length" gives
a constant impedance. Work it out. It's a really good
exercise. I know about five different ways of working
it out, such as
-- Starting from a lumped circuit model and passing to
the continuum limit. This is discussed in Feynman and
probably many other places.
-- Starting by writing down the Lagrangian and doing
the classical physics. Calculus of variations leads
to the Euler-Lagrange equation of motion in the usual
way. This leaves no doubt as to what is the electrical
analog of momentum, canonically conjugate to the
coordinate ... thereby paving the road to the full
quantum mechanical analysis.
-- et cetera.

Also (!) the same techniques can be used to analyze the
physics of piano strings ... which is actually how I got
into this game. I did it just for fun, as an exercise
I posed for myself ... but it laid the foundation for
one of the greatest Aha! moments of my life.

I think it is some kind of rite of passage. I know
at least two of my friends did the same calculation
-- spontaneously, before I met them -- just as an
exercise, testing the unity of physics, to see how
ideas from one area (classical and/or quantum field
theory) applied to another area (electronics and/or
piano strings).
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