Hi Ion,
Hmm, it doesn't make any sense to me now that he would use the basket weave coil to reduce parasitic capacitance, because with that big mass of dielectric nearby there would still be plenty of C. So why use a basket weave? So I went and looked.
De Forest, attached, shows the use of a Lecher line wrapped around a tube, as a highly Q transmission line stub. He discusses detecting both electromagnetic and electrostatic nodes on the wire, similar to the use of Lecher wire to measure wavelength.
With the dielectric nearby in Hendershot, you have a distinct slow line structure. With two such slow lines, you might be able to create some large delay effect, that, for instance delays CEMF from the clapper until it becomes a positive force to drive the clapper? Would need a big mass of dielectric, and a coil with a large cross section (=high L).
Pure speculation-- but resonance effects have been used in permanent magnet motors (which the clapper basically is) to apply CEMF to the next motor cycle. (patents by Goddard, Macdonald, Rupp, Subieta-Garron). In this case, the Hendershot machine becomes a permanent magnet motor where CEMF does not impede motor action but supports it.
I note that the iron Cook Coils are long enough that there could be a substantial domain motion delay, with the secondary induction on one coil delayed enough to support the primary on the other coil (or something like that). Tesla also had transformers with a thin layer of metal between the windings to cause a delay, which could reach 90 degrees I think.
The patent from 1925 by Willy shows serpentine coils used as retardation coils, and also right angle coils intended to reduce mutual L and C. It's the only patent in this class besides De Forest that seems relevant. Retardation coils seem to be often discussed in these patents as a way to make voices clearer in radio reception. It was a time of extreme levels of experimentation in radio, with all sorts of weird antennas, including underground ones (Rogers) and miscellaneous even stranger things. In fact, I recall a patent (but can't remember who did it right now) for a 'radiant energy detector' that had a clapper or something like it.
These are more like free associations than research, but maybe there are some clues...There's also a patent class for coil/capacitor combinations, maybe there is something there...
Regards, Fred
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