in this paper the following statement is written:
An area A is allotted to each element in the infinite array. This is the maximum area available to each element, and is usually greater than the physical size of the actual element. It is natural to assume that the maximum gain obtainable from an element in the array is related to the area A by the well-known gain formula for apertures large compared with a wavelength, because the entire array is indeed large. Furthermore, since the effective area of an element should be proportional to its projected area in the direction of interest, the element gain should have a cos theta variation with angle. Based on this intuitive reasoning, the maximum element gain would be
This relation gives a fundamental upper limit to the gain obtainable in an element of an infinite planar array. It also implies that the ideal shape of the gain pattern of such an element would approach the cos theta variation. However, there are factors not contained in (1) [the previous equation] which must be considered in any objective analysis of element gain.
Well, this reasoning is not intuitive for me. Precisely:
For hypothesis, mutual coupling is absent. Mutual coupling is a real cause of anisotropic radiation pattern but the author takes it into account in the following part of the article by adding an active reflection coefficient multiplicative term. For what concerns my doubts, mutual coupling does not exist.
For hypothesis, the planar array is infinite. So, there is no reason for why different radiating elements should have different patterns, as they are surrounded each one by the same environment.
Provided these hypotheses, looks like the author says that:
Even if the isolated single element pattern is isotropic, it would become anisotropic in a phased array.
This result comes from the known relationship:
$$D(\theta, \phi) = \frac{4*\pi *A_e}{\lambda^2} $$
This relationship is fine. What I do not understand is why the author does consider the geometric area of the cell (with its corresponding cosine projection term) where the single element is put, instead of the effective area of the single element.
If the radiating element is isotropic, its effective area will be:
$$A_e = \frac{\lambda^2}{4*\pi} $$
This obviously will say that the radiating element inside the array is still isotropic.
My question could be rephrased like that: "Why does the effective area of each radiating element in a phased array equal its available cell and not its isolated effective area?" The space between the single element and the cell A is empty: why should it catch power?
