Simple relations derived fom a phased-array antenna made of an infinite current sheet

Abstract

The simplest concept of a phased array is an infinite planar current sheet backed by a reflecting boundary. The electric current sheet, or resistance sheet, is the limiting case of many small electric dipoles, closely spaced, and backed by an open-circuit boundary. If this array is viewed as a receiver, a plane wave incident on the array at some angle ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta</tex> ) meets a boundary resistance varying in proportion to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cos \theta</tex> for angles in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> plane, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/\cos \theta</tex> for angles in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> plane. If the array is matched at broadside ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta=0</tex> ), the corresponding reflection coefficient has the magnitude <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\tan \frac{1}{2}\theta)^{2}</tex> . While the electric current sheet is realizable, the open-circuit boundary is not. However, a magnetic current sheet can be simulated by a conductive sheet with holes utilized as magnetic dipoles, such a sheet providing the backing equivalent to a short-circuit boundary. The latter case is related to the former by electromagnetic inversion or duality. Therefore, an incident plane wave meets a boundary conductance varying in proportion to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cos \theta</tex> for angles in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> plane, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/\cos \theta</tex> for angles in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> plane. The predicted behavior is verified qualitatively by tests of such a model with elements of a practice size. The derivation is based on the principle of dividing the space in front of the array into parallel tubes or waveguides, one for each element cell in the sheet or array. This is one of the principles published by the author in 1948. A related principle enables the simulation of an infinite array by imaging a few elements in the walls of a waveguide. This latter principle is utilized for making tests of the array.