The grating-lobe series for the impedance variation in a planar phased-array antenna

In an infinite planar array of elements with periodic spacing, the element active impedance varies with phasing for beam steering. This impedance variation may be expressed as the sum of a double Fourier series. This series is identified with the periodic grating-lobe pattern on the " plane" which is also the plane of two-dimensional phasing coordinates. An "impedance crater," with contours peculiar to the kind of element, is placed on every grating-lobe center. The inside of the central crater, which coincides with the unit circle of real space on this plane, determines the resistance variation with scan angle of the main lobe. The central crater and the skirts of the surrounding craters overlap in this circle; their sum determines the accompanying reactance variation. All craters together form the "grating-lobe series," which gives a picture of the entire impedance variation with scan angle. In a simple example, the reactance variation associated with half-wave spacing of the elements is found to be nearly equal to the resistance variation associated with the kind of element.