Re: [Phys-l] Frequency dependence of resistance

[John Denker <jsd@av8n.com>]

On 09/18/2008 12:38 AM, Savinainen Antti wrote:

> I was wondering how to explain qualitatively why the resistance of a resistor
> stays constant (at least in the first approximation, that is in the HS physics
> :-)) when frequency of current increases. The frequency dependence in the case
> of reactance of coil or capacitor is quite easy to explain in terms of
> self-induction (coil) and charging/discharging (capacitor). These results can
> also be derived using simple calculus which is understandable by (good) HS
> students.
>
> One possible explanation might use the idea of storing energy: resistor just
> dissipates energy to thermal energy which cannot be returned to the power
> supply.  Capacitor and coil can store energy in electric/magnetic fields and
> then give it back to the power supply when the cycle proceeds. Can these
> considerations lead to a plausible explanation why the resistance of a resistor
> stays constant in an AC-circuit, no matter what frequency? Or is so that energy
> has nothing to do with this?
>
> It would also be interesting to hear to what extent resistance is actually
> independent of frequency.

Answer #1)  I assume the intent was to ask about the _impedance_ of a
resistor.  If the impedance is observed to be frequency independent,
that's a nontrivial observation.  In contrast, the "resistance" is by
definition the frequency-independent part of the impedance.  So asking
about a frequency-independent "resistance" would be circular and 
pointless.

Answer #2)  If you want the real _reason why_ commercial off-the-shelf 
resistors are independent of frequency, it's because people who wanted 
them to be independent of frequency have spent the last 100 years or so 
figuring out how to make them independent of frequency.
 2a) Partly I'm making the philosophical point that people have reasons
  for doing things ... in contrast to electrons, which do not have 
  reasons and do not need reasons for doing things.  They just do
  what they do.
 2b) Partly I'm making the physics point that if you take some random 
  piece of junk, it will have a impedance that is not particularly
  independent of frequency.
 2c) Partly related to item (1) and partly related to item (2b), if
  an object doesn't have a reasonably frequency-independent impedance,
  we don't call it a resistor.

Answer #3)  All real resistors have some parasitic inductance in series 
and some parasitic capacitance in parallel.  The practical question is, 
how high can you go in frequency before these parasitic effects become
significant.

As you can imagine, this depends on a lot of things, including the
nominal value of the resistor.  Parasitic capacitance is much more of
a problem for 100 gigohm resistors than it is for 1 ohm resistors.
And vice versa for parasitic inductance.

For low-value resistors and/or at high frequencies, the inductance
of the _leads_ is significant.  Cut the leads short, or buy leadless
resistor chips.

Answer #4)  The physics of electrons in resistors is not Newtonian 
physics but rather Aristotelian physics.  That is: objects at rest 
tend to remain at rest, and objects in motion tend to come to rest.  
The velocity (not the acceleration) is proportional to the applied
force.  (As an interesting and important analogy, the same words 
apply to bacteria swimming in a viscous fluid such as water.)

This is achieved by having 
 a) a high concentrations of electrons, and 
 b) giving them a really short mean-free-path.

That means
 a) you can carry a big current without any particular electron
  needing to move very fast, and
 b) the electron doesn't spend very much time accelerating before
  it hits something and goes back to velocity = 0 or thereabouts.

Basically I'm invoking the "Drude model" of electron transport.  
You can read about it in any solid-state physics texbook e.g. 
Ashcroft & Mermin.
  http://ipv6.google.com/search?q=drude-model