This is a half-wave dipole for which each arm has been rolled up into a
spiral, so the overall antenna dimensions are much smaller than a
wavelength.
In what sense is such a configuration better than a linear dipole antenna of
the same small size?
Perhaps members of the List can explain what the answer might be.
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To get some idea of possible answers, I ran a NEC4 model of a Tak-Tenna.
Each arm had a total length of 19 m, so the nominal frequency is 7.87 MHz.
Each arm ran straight out from the feedpoint for 30 cm, then was wound in a
7-turn spiral. The outer diameter of the spiral was 60 cm. The wire radius
was 0.8 mm.
NEC4 reports the impedance of this device as
Z = 0.018 - 358 j Ohms @ 7.87 MHz
For comparison, NEC4 calculated that a linear dipole whose arms are each 60
cm long has impedance
Z = 0.018 - 7000 j Ohms @ 7.87 MHz.
Both of these configuration have the same (low) radiation resistance of
0.018 Ohms. Both have large capacitive reactance, tho the linear dipole has
considerably larger reactance.
To use either of these antennas with a 50 Ohm feedline, a tuner/matcher must
be added (preferably close to the antenna feedpoint).
To my thinking, matching to 50 Ohms implies that we must add a 49.982 Ohm
resistor (=> 50 Ohm in practice) in series with the antenna. This resistor
would dissipate most of the feed power, and the antenna efficiency would be
0.018 / 50 = 0.00036.
This seems rather small, and hardly an advantageous feature.
Perhaps it is easier to cancel the capacitive reactance of the Tak-Tenna
than that of the linear dipole, and this is considered a big advantage.
The Tak-Tenna website has plots of VSWR vs. frequency -- after
the matching is done. A good match appears to haven been obtained at the
nominal frequency, where VSWR = 1. To me, this means that no power is
being reflected off the antenna (good), but to achieve this, more than 99.9%
of the power is dumped in the matching resistor (not good).
