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Re: Light bulb ohmicity (very long)



Although I tend to agree with Barlow Newbolt that we have probably
overdosed on the topic of light bulb (non)ohmicity, it seems that Leigh
wanted me to answer his last post. So I'll gladly once again oblige.
Leigh wrote:
...
First, the ordinary light bulb runs at 60 Hz. This is so far
short of being fast enough to reach thermal steady state, as you
guessed, that I have seen its effects in high shutter speed
videotapes. The illumination brightens and dims slowly as the
field frequency beats with line frequency. (This was done with a
halogen lamp illuminant. The effect is greater for such a bulb
than it would be for a non halogen bulb.)

Since I have not done any actual experiments concerning this problem
(nor have I bothered to look up various material parameters for tungsten
such as its thermal conductivity, heat capacity, etc. over a very large
temperature range and investigate the geometric details of typical bulb
filaments in order to make reasonable calculations of cooling rates,
internal temperature gradients, etc.) I do not know for sure just how
high of an AC frequency is needed before the thermal excursions between
the cycle peaks and the zero crossings become relatively unobjectionable
when investigating the "ohmicity" of a light light bulb. I do want to
observe however that the frequency needed is significantly less than that
needed to eliminate most of the apparent brightness excursions in the
bulb. The reason for this lower sensitivity of the "ohmicity" is
twofold.

First, the (isothermal) resistivity of a metal such as tungsten tends to
be approximately directly proportional to its absolute temperature,
whereas the total bolometric output of the filament tends to be
proportional to the 4th power of the absolute temperaure. Thus the
relative excursions in radiant output will be about 4 times greater than
the relative exursions in the resistivity when these relative excursions
are a relatively small fraction of unity.

Second, when an incadescent bulb is at normal operating temperature its
most intense output is in the near IR slightly more than an octave below
the middle of the visible spectrum. This means that the visible portion
of the filament's spectrum is on the high frequency slope and includes a
relatively small fraction of the spectrum. As the filament's temperature
changes this spectrum alternatively shifts its peak toward the visible
range and shifts its high frequency tail toward that range. This causes
a much larger relative excursion in the visible portion of the light
output than is the excursion in the total output itself.

My guess is that due to a combination of these two effects that the
relative excursions in the bulb's luminous flux is about at least an
order of magnitude greater than the excursions in its resistivity. Thus,
even though the 120 Hz thermal cycling frequency is too low to eliminate
appreciable flicker in the bulb's output, it just may still be high
enough so that the filament's resistivity may be roughly constant enough
so that a log - log plot of |I(t)| vs |V(t)| over the AC phase cycle
would be reasonably straight with a slope close to 1. Thus it is
possible that when the bulb operates in a normal manner when it is
plugged into the usual 60 Hz AC line, it is still reasonably ohmic for
crude calculational purposes. It seems that a simple experiment would
answer the question though.

Of course, since halogen bulbs operate at higher filament temperatures
than ordinary bulbs they would cool relatively faster during the cycle
zero crossings in the interval between the wave peaks, and would,
therefore, have a somewhat greater flicker and resistivity excursions.

Hysteresis is important here because of heating. The temperature
of the filament's surface will vary with a phase lag with
respect to that of the current squared, the reason being that the
thermal conductivity of the filament requires that there be a
higher temperature at the filament's center than at its surface.

It is certain that this effect must be present. I did not mention it
because I did not want to bog down my post with every conceivable
effect. I also did not mention that this effect may be partially
compensated for during the AC conduction process by an opposing
alternating nonuniform radial dependence on the current density. During
the main heating phase of the cycle the temperature gradient between the
central part of the filament and its surface grows. During the zero
crossing phase this difference is reduced. But when this difference
grows the higher resistivity of the central part of the filament causes
the current to preferentially flow along the cooler surface. This would
tend to heat outer parts of the filament at the expense of the central
region, and this would act to reduce the temperature gradient and its
excursions over a larger part of the filament's cross section and the
heating cycle's phases than would be the case if the current density
remained uniform across the filament cross-section.

It should be remembered here that since tungsten is a metal it has a
quite relatively high thermal conductivity, and this would act to help
minimize the the temperature gradient across the cross section of the
filament. The actual amount of the gradient would be the result of a
competition of the internal conduction time scale and the surface cooling
time scale from radiation and convective processes. If the rate limiting
step occurs at the surface, then the internal gradient would be
relatively small. If the internal conduction is the rate limiting step
(not likely for a narrow metal) then the internal gradient and its
hysteretic phase lags would be much higher.

In any event, if the thermal cycling frequency is high enough for the
temperature profile (whatever it happens to be) to remain relatively time
independent then this hysteretic effect, along with all the other details
of the thermal transport processes becomes insignificant. My guess of
an AC frequency of 1 kHz being sufficiently high for this to be the case
seems reasonable to me. In light of what I said above, it is conceivable
that an ordinary 60 Hz line rate may be good enough for crude
calculations (at least for non-halogen bulbs).

As I recall, Ham radio operators often connect the RF output of their
rigs to a metal-shielded light bulb, (or an array of multiple bulbs if
the power is very high) while aligning, calibrating, and testing their
systems to avoid broadcasting an illegal or otherwise unwanted signal
into space. Once such a setup is properly trimmed for an impedence match
between the bulbs and the RF output stage the bulb(s) operate(s) quite
ohmically. At RF frequencies there is no danger of any measurable
temperature excursion-induced non-ohmicity.

Since heat is produced at a not inconsiderable rate (power
density = j E = j^2 / sigma) and there is no phase change taking
place, thermal equilibrium would require the magical removal of
energy, since finite thermal conductivity limits the rate at
which it can be removed by other means. In the AC case the
specific heat of the metal also matters.

Quite true. Fortunately, true global thermal equilibrium for the
whole system is not required, only a reasonable approximation to a steady
state in the local temperature and resistivity field profiles are needed.
I said "fortunately" here because the very concept of conduction is not
an equilibrium concept. It instead a transport process involving a
linear response (linear that is if we are talking about ohmic behavior)
to a steady perturbation away from equilibrium.

In consideration of this energy disposal problem, do you still
feel that a requirement of isothermality serves any physically
useful end in discussion of the ohmicity of a material?

Yes. This is especially true for any material or device (e.g. ordinary
low power resistors) whose range of operating temperatures permits an
inconsequential change in the material's resistivity and the device's
resistance. As I said above, it is even conceivable (but I haven't
checked it) that it may be reasonably useful for an ordinary bulb
connected to the local 60 Hz utility. The "iso" part of the
isothermality condition only needs to apply (in a practical sense) to
the time parameter, not space, in order for a resistive device to act
(device-)ohmically in operation. If different parts of a resistive
device are at different temperatures, then those parts may have different
resistivities, but the device as a whole would still have an overall
device-ohmic behavior as long as the temperature-dependent resitivity
profile remains reasonably time independent as the voltages and currents
vary in time during that operation.

Energy disposal is not an insurmountable problem in all cases. It may be
often a nuisance though.

Do you
feel that some materials would be found to be ohmic and some
not?

Yes.

If your answer to that question is "Yes", please give me
some examples (I can only think of one). If your answer is "No"
(or if my example is the only one extant) then I would say the
concept is not useful.

All kinds of metals are ohmic and they act that way in many usages.
It seems that many substances and devices where the conduction is via
some sort of tunnelling process would tend to be intrinsically non-ohmic
due intrinsic nonlinearities in tunnelling transport processes. For
instance, I would think that homogeneous narrow-gap semiconductors (held
at a temperature cold enough so that the equilibrium carrier
concentration in the conduction band is insignificant) would tend to be
quite non-ohmic over a large range of electric field strengths because
because the field-induced tunnelling rate from the valence band to the
conduction band (across the field-tilted band-gap) would be nonlinear in
the field strength. Certainly lots of various tunnelling junctions
especially involving superconductors and semiconductors (e.g. squids,
tunnel diodes, etc.) would be intrinsically non-ohmic. Even ordinary
(non-tunnelling) doped semiconductor junctions are notoriously non-ohmic
at a fixed temperature. This second class of examples (junctions) are
not homogeneous materials though. (Did you want only want examples of
homogeneous materials?) How about polarizable electrolytes? Consider
the case of pure liquid water. In its equilibrium state at any given
temperature there is a naturally occuring concentration of OH- and H3O+
ions in a highly polarizable background. As the electric field increases
the fluid polarizes greatly which would, I assume, affect the diffusion
properties of the ionic carriers present, which would, presumably,
produce an intrinsic nonlinear conductive response and, consequently, a
nonohmic I vs V relation at a fixed temperature. I expect that there are
other examples of intrinsically non-ohmic materials whose nonohmicity is
manifest at a fixed temperature, but I do not feel like investing the
time and effort to think of them (or try to look them up).

David Bowman
dbowman@gtc.georgetown.ky.us