Dimension Expansion Method and Implementation of Broadband Planar 2-D 3 × 3 Nolen Matrix

Traditional two-dimensional (2-D) beamforming network (BFN) topology is constructed by cross-connecting two stacks of one-dimension (1-D) BFNs, which results in a nonplanar topology, limited bandwidth, and bulky size. To overcome these limitations, a dimension expansion method is proposed in this article. By using the dimension expansion method, the traditional 2-D BFN topology can be planarized effectively. To validate the effectiveness of the dimension expansion method, a broadband planar 2-D beamforming topology constructed by six
$3\times3$
sub-Nolen matrices and a novel
$9\times9$
crossed phase shift network (CPSN) is designed. Vertically installed planar (VIP) couplers and broadband differential phase shifters are used to construct the broadband sub-Nolen matrices. A modified CPSN consisting of nine subcrossovers is designed to skillfully circumvent the intersection of different paths between the two expanded BFN groups. For verification, a broadband planar 2-D Nolen matrix operating at 1.8 GHz is designed, fabricated, and measured. Measured results match well with simulation. A bandwidth of 50.79% based on a stable generated array factor is realized, which further highlights the effectiveness of the dimension expansion method and its broadband characteristics.