Re: Alternating Current

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

I thought I would comment on this thread as a small addition to Roger
Haar's important comments:
> ....
>     Because surface of the filament cools radiatively and the interior
> cools conductivity to the surface there are still more complications.
>
>        I think the end effect is that the detected light output is a
> constant plus a time varying piece.  The time varying part is at 120 Hz and
> is blurred to look like a sign wave.

I'm not certain just what cooling mechanism is the predominates in
cooling the filament's exterior, but I suspect that convective cooling
involving the gas inside the bulb is at least comparable in importance
with radiative cooling from the filament surface if not outright
predominant in its own right.  So I would attribute this convection to
another of the "more complications" that Roger mentions.

The time it takes for the filament to cool a significant way from its
mean operating temperature down to near room temperature is much longer
than 1/120 s.  Since the AC heating cycling time is at 120 Hz we see that
the amplitude of the temperature (and hence luminosity) excursions are a
relatively small ripple amplitude on top of the base average
temperature/luminosity.  The 120 Hz heating cycle is thus effectively
averaged over an effective time base interval which is long compared to
1/120 s.  The complicated cooling mechanisms and the nonlinearities and
non-constancies in the temperature dependence of the tungsten's specific
heat, electrical conductivity, interior temperature profile, Stefan-
Boltzmann emission function, cooling mechanisms, etc. (& other
complications that Roger alluded to) all conspire to make the time
dependence of the heating power a complicated strictly *non*-sinusoidal
yet periodic function whose period is 1/120 Hz.  Because of the various
(internal and external to the tungsten) heat transport mechanisms
involved with the cooling processes tend to have a long effective
averaging time (relative to 1/120 s) this means that the output
luminosity function of time (for the wavelength band over which Dave's
detector is sensitive) has effectively been put through a *low-pass
filter* relative to the time dependence of the input heating power
function.  The averaging low-pass filter significantly attenuates all of
the harmonics of the non-sinusoidal input function relative to the
average DC level.  But it attenuates the higher harmonics much more
effectively than than it does the lowest frequency 120 Hz fundamental
which is attenuated the least.  The resulting output luminosity function
is thus an excellent approximation to a low amplitude 120 Hz sine wave
ripple riding on top of the mean DC luminosity.  Once the average
background DC level is discarded the remaining 120 Hz AC luminosity
signal *is* thus well-approximated by a sine wave as seen by Dave
Abineri's fits to the data.  The sluggish luminosity response to the
heating power fluctuations automatically gives a sine wave output no
matter how complicated the heating power wave form actually is.

I suspect, however, that even in the presence of all those complications
the heating power function is still reasonably well approximated by a
function of the form A*sin^2([omega]*t) = (A/2)*(1 - cos(2*[omega]*t))
(with [omega] = 2*[pi]*60 1/s) so there probably is not much in the way
of power in harmonics of frequencies higher than 120 Hz present in the
heating power function to be filtered out anyway.

Btw, I, too, strongly suspect that the discrepant 124 Hz frequency that
Dave measured is due to a mis-calibration of the sampling rate.

David Bowman
dbowman@georgetowncollege.edu