There is a class of antennas whose pattern as well as impedance is practically independent of frequency for all frequencies above a certain value. The general formula for their shape is

where

are the usual spherical coordinates, a and

are constants and

is any function of

. Assuming a to be positive,

ranges from -

to some finite value which determines the low frequency limit. For such antennas a change of frequency is equivalent to a rotation of the antenna about

. It appears that the pattern converges to the characteristic pattern as the frequency is raised, if a is not

, and that the impedance converges to the characteristic impedance for all

.