In the first case, if you follow the current flows from outer connection to inner termination (assuming the coils are connected in series, so that they share equally, and so there's a well defined starting node that's very localized, e.g. a two pin plug), the currents make full loops, back and forth, of decreasing size, in the same locations.
At DC, these fields will cancel out, leaving little more than the residual from a parallel line case (more on that below).
At AC, there may be frequencies where the individual loop sections radiate, making a perhaps not-terrible antenna. If you drive a bifilar spiral from the center (the ends are open circuit), you get an Archimedes spiral antenna, which is wideband (but has terrible phase response).
If you imagine the path not being folded in on itself as a spiral, but bent out every half turn, so it's stretched out in a line, then it looks something sort of like a chain of loop antennas, of decreasing size and alternate phase. This would probably be analogous to the log periodic dipole, if a logarithmic spiral were used.
When spiraled centrally, the fields may cancel, or propagate. I'm not sure. I expect something would happen, due to the lateral symmetry. It might not be very special.
As for the bifilar case, you simply have a spiraled up parallel-line transmission line. There will be turn-to-turn coupling resulting in an impedance(frequency) different from that of a pure line, and there will be some radiation due to imbalance (one turn is always larger than the other), but it won't be particularly important or useful.
Tim
Seven Transistor Labs, LLC Electrical Engineering Consultation and Contract Design Website:
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I am trying to visualize the EM field geometry of two special cases of panckake (flat spiral) coils.
The first is comprised of two single winding coils, placed one on top of the other (in contact, centers aligned), but with the current running oppositely in each.
The second is a single pancake but with two windings side-by-side (bifilar) with the current moving oppositely in each.
I had initially thought the EM vectors would cancel somehow, but on closer inspection, they seem to reinforce each other in both of these configurations.
This seems strange, particularly in the first example, since like magnetic poles are being converged.
Any comments?
Claus Jensen