Isotropic transmission-line antenna and its toroid-pattern modification

The far-field strength of the
\lambda/4
transmission-line antenna is derived as a complex vector containing amplitude, phase, and direction of the
E
vector. The product of this phasor-vector and its conjugate is constant and proves that the transmission-line antenna is an isotropic radiator with respect to the powerflow density of coherent electromagnetic waves. Folding of the line conductor parallel to the symmetry plane reduces the pattern to toroid shape analogous to radiation of a short dipole. While the unbalanced straight transmission-line antenna is used on rockets and vehicles, the unbalanced folded type is suitable as efficient transmit antenna at VLF, LF, or MF.