Equiphase lobe condition of radiation pattern

Voltage radiation pattern of an antenna expressed in the spherical coordinate system is where overrightarpoonup{??}(theta, phi) is the amplitude (vector real function) and is the phase (real function).It is well known that is dependent upon the phase reference point (at which its phase is assigned to zero) of the antenna system. Usually, we say the antenna system possesses a phase center, if we could find such a phase reference point in the system that is equal to zero or in equation (1). In this case, the fields at any points of a lobe in the radiation pattern are inphase. It is termed equiphase lobe in our discussion. R.C.Spencer has discussed the dependence of real lobe ( ) on the current distribution of the linear array (RL Report No. 762-1, Jan. 21, 1946) and A.R. Velipert has given the condition of the phase center of the linear array (Radiotekhnika (Russia), Vol. 16 No.3 1961). In this paper, the condition of equiphase lobe is discussed in general case, certain theorem is proposed and proved and some lemmas are deduced.