The characteristic impedance of two infinite cones of arbitrary cross section

A theoretical method of finding the characteristic impedance of two infinite cones of arbitrary cross section is presented. In this problem, the spherical geometry can be mapped on a plane, thus reducing the problem to finding the solution of Laplace's equation by using conformal mapping techniques. This method is applied to the inclined biconical antenna and several types of fin antennas. The solution for the biconical antenna is
Z_{0}=\eta/2\pi \cosh ^{-1}F
, where
F
is a function of the geometry of the antenna. For the fin, the solution takes the form
Z_{0} =\etaK(k)/K'(k)
where
K
and
K'
are complete elliptic integrals of the first kind and
k
depends on the geometry.