Pillow Deformation of a Mesh Reflector.

We present in this paper an analytic solution for the pillow deformation of the deployable mesh antenna which is segmented into a regular convex polygon by the cable net. We treat this problem by using both the methods of power series expansion of the complex variables and of Fourier analysis, in which we take into account the self-consistent boundary condition of the mesh on the supporting cable. Using this solution, we find that the magnitude of the pillow deformation of the regular polygonal facets is directly proportional to the square of the side length of the facet and the reflector curvature, and is an increasing function of the ratio of the tension of the mesh and cable. We also derive the approximate formulas for the pillow deformation and the best-fit surface error from the ideal parabola for the case of triangular and rectangular facets. We performed the numerical evaluation of the solution, and discuss the applicability of above mentioned approximate formulas. The analytic solution here derived gives us the basic and quantitative ideas of the pillow deformation and is useful for the quick numerical evaluation of the pillow deformation.