The Conical Dipole of Wide Angle

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Abstract
The following report deals with a method of calculating the admittance of dipoles that consist of complete cones of wide angle. The theory makes use of the orthogonal properties of Legendre's functions and of their derivatives in order to make the outside field's tangential component vanish over the spherical end surfaces of the dipoles, and to make the inside and outside fields fit at the boundary sphere. This is a surface of the same radius as the dipole, extending from the upper to the lower end surface. This leads to a set of Eqs. (10) for the infinite set of coefficients bk, and an Eq. (3) that expresses the effective load admittance yt in terms of the coefficients bk. If a sufficient number of these coefficients is considered, there should be no appreciable discontinuities in field at the boundary sphere, and the tangential field should vanish on all the surfaces of the dipole. On the assumption that only the first two outside waves, and the internal T.E.M. and first higher order inside wave are important for dipoles whose radius is less than a half‐wave‐length, an approximate formula for yt is found, and the results are shown in Figs. 4 and 6. It is unnecessary to evaluate the solutions of Legendre's equation Ln(θ), because the integrals that involve them are expressed in terms of n(μ1) and dn(μ1)/dμ1, n being the root of Ln(A)=0, and μ1=cosA. The solutions are transformed to the center of the dipole, and plotted in Fig. 6, while the relative terminating admittances, K·yt, are plotted on an impedance diagram in Fig. 7.