Feasibility and Convex Design of Probe-Position Matching in a Scanning X-Band Radar Array

Abstract

The probe position of a microstrip element is normally chosen from the isolated-element input resistance. This paper replaces that procedure with a decision available before any array optimisation. A single Floquet unit-cell solution at one arbitrary probe position is decomposed into a feed inductance, a transformer ratio carrying the probe position, and an array-loaded resonator. Three closed-form results follow. The set of input impedances reachable by probe position and resonant length is a disk in the impedance plane. An exact match to a real reference Z0 exists if and only if Rp >= Z0 + Xp^2/Z0. When this fails, the best attainable reflection follows from the image of that disk under the bilinear map. Admitting one series reactance as a second variable, the worst-case reflection over a scan sector is shown to be quasiconvex and globally solvable by bisection. The results are applied to a sixteen-element X-band array and the corresponding 16x24 lattice. For the array considered, the isolated interior element is matched to -16.94 dB while the broadside active reflection is -6.6 dB. For the two-dimensional lattice the criterion is violated at every scan angle, and the minimax design improves the worst-case sector reflection from -2.6 to -11.1 dB. Both matching designs are predictions of the extracted circuit model and have not been re-solved in the full-wave model.