Re: [Phys-l] Frequency dependence of resistance

[John Denker <jsd@av8n.com>]

On 09/18/2008 12:43 PM, Dan Crowe wrote:
> > Resistance is defined to be the component of the impedance that is
> > due to the component of the current that is in phase with the
> > potential difference.  The definition of resistance does not imply
> > that resistance is independent of frequency. 

On 09/18/2008 02:04 PM, I responded:

> OK, I'll bite.
>
> Can you give an example of a circuit that has a frequency-dependent
> resistance with a frequency-independent I/V phase relationship?
> A lumped-circuit example would be especially helpful.
>
> I have never seen such a critter.  If you can exhibit one, I'll be 
> impressed.

There's a deep theoretical angle to this story.  There's a reason
why I don't expect to see such a critter.

There's a Kramers-Kronig relation at work here.  

  Digression:  Terminology (or not):

  a) I'm not 1000% sure that Kramers-Kronig is the exact right 
  terminology, but at the very least the math and physics here
  is the same as for a Kramers-Kronig relation.  

  b) Some folks call this a "dispersion relation", but this is  
  one of those many terms in math and physics that have multiple 
  meanings, and this is not my favorite meaning for the term
  "dispersion relation".

  c) Some people seem to think that the Kramers-Kronig term 
  applies narrowly to optics ... but I care more about ideas
  than about terminology.  For sure the *idea* applies much
  more broadly.

Anyway, getting back to physics:  I think it is safe to assume 
that the electrical response function (the I/V relationship) is 
integrable and causal.

That is sufficient to give us a Kramers-Kronig relation.  That is,
the real part of the complex impedance at a given frequency can
be expressed as an integral over the imaginary part (at all
frequencies).  Similarly the imaginary part at a given frequency
can be expressed as an integral over the real part.

The structure of the integral is such that a tiny blip in the
real part (such as a very narrow resonance) at omega=3 will 
produce an imaginary part that extends for miles in both 
directions, roughly of the form 1/(omega-3).  I've used this
time and time again in the lab.  Hint:  If you're looking 
for a narrow resonance, don't look for the on-resonance dip;
look for the phase shift everywhere else!

Applying this to the problem at hand:  If you start by saying
that your resistor has a frequency-independent impedance, you
automatically know it has an in-phase I/V relationship.  This
is an immediate corollary, via the Kramers-Kronig relation.

I suspect the converse is true also, although I don't immediately 
see how to prove that.  If so, that would make it quite pointless
to argue over which of the two expressions is the best way to 
"define" resistance.

Reference:  http://www.hep.phys.soton.ac.uk/~evans/Lab/ci.ps