The Approximate Representation of the Distant Field of Linear Radiators

After a brief résumé of the general theory of the distant field of arrays of linear radiators in terms of the vector potential (including the definition of a radiation function, radiation resistance, and directivity), approximate representations in terms of a sine distribution of current and in terms of an "equivalent" uniform distribution with an "effective length" are considered. It is shown that the conventional definition of an "effective height or length" 2h
e
for a symmetrical, center-driven antenna of length 2h, viz., 2Ih
e
= f^ I
z
dz, does not give the best approximation. This is obtained by expanding the field characteristic in a Fourier series and defining a new "effective length" to be the coefficient of the leading term. It is shown that a reasonably good approximate representation in terms of an "effective length" is possible for antenna half lengths h which are less than λ/π, but not for longer ones. The case of two crossed antennas is discussed as an illustration of the simplification obtained using the approximate formulas.