The Radiation Resistance of End-Fire and Collinear Arrays

Expressions for the radiation resistances of end-fire and collinear arrays of half-wave dipoles are obtained in terms of circular functions in a form convenient for computation. No mathematical approximations except for a Fourier representation of the field of a single half-wave dipole are used. The first integral theorem of Sonine and an integral representation of the Bessel function due to Hansen are involved in the integration of the normal component of Poynting's vector. Results computed from the new formula for the radiation resistance of an n-element parallel array in which the spacings and successive phasings of the dipole elements are 180 degrees (bilateral end-fire) agree closely with those of Pistolkors, who used Brillouin's e.m.f. method; they are a little less than the figures of Bontsch-Bruewitsch, who numerically integrated Poynting's vector. Calculations for the radiation resistance of an n-element collinear array using the new formula are compared with those of Bontsch-Bruewitsch, with which they are in satisfactory agreement. The new formula is also used to compute the radiation resistance of an n-element unilateral end-fire array (i.e., an n-element parallel array in which the spacings and successive phasings of the dipole elements are 90 degrees).