An Electromagnetic Comparison of the Tapered Spherical Helix and the Negative Permittivity Sphere

The tapered spherical helix, originally introduced by Wheeler as an exemplary small antenna, bears a striking electromagnetic resemblance to the negative permittivity sphere. In this paper, the tapered spherical helix is analyzed in detail, and it is shown that the electric field of its resonant mode closely follows that of a sphere composed of material obeying the Drude dispersion relationship. An analytical formula is derived relating the inductance of the spherical helix to the plasma frequency of the spherical material mimicked by the helix. We validate this result through numerical simulation, and show how the tapered spherical helix and the negative permittivity sphere can each be used to impedance match an electrically short stub antenna using nearly identical geometries. This result supports the interpretaion that the negative permittivity material is equivalent to a spatially distributed inductance, tuning out the capacitance of the small stub. The excellent bandwidth performance observed in both negative permittivity antennas and spherical helix antennas is due specifically to the spherical geometry common to both.