© S.P. Skobelev

The general antenna theory is applied to derivation of the expressions connecting the gain of a radiating element in an infinite planar dual-periodic phased array antenna with the radiation pattern of the element. As the first step, expressions connecting the amplitudes of the Floquet modes formed above the array at its quasi-periodic excitation with the electric and magnetic currents in the array cells are derived. Then, the excitation of one array input (aperiodic excitation) is considered, at which the element pattern and element gain are determined, and it is shown how the currents corresponding to the aperiodic excitation are expressed via the currents corresponding to the quasi-periodic excitation and vice versa. The relations obtained are used for derivation of the electric and magnetic vector potential components in far zone, the electric and magnetic field strength components , , and, respectively, the components and of the vector array element pattern corresponding to the power Pin supplied to the element input. In particular, it is shown that the indicated components are determined by the amplitudes of the zero order Floquet modes corresponding to the quasi-periodic excitation multiplied by the cosine of the angle measured from the array broadside. It is also shown that the indicated element pattern components determine the element gain as follows , where 0 is the wave resistance of the free space. The ideal element pattern is determined as the pattern corresponding to radiation of all the power supplied to the array inputs at its quasi-periodic excitation into the main lobe of the array. It is shown that the power element pattern is determined by formula . The last two expressions are used for derivation of the well known formula for the ideal array element gain. It is thereby shown that the ideal element gain is the secondary parameter determined by the ideal element pattern though, historically, the ideal element gain was was written earlier that the element pattern was on the basis of some elementary reasoning.