Analytical Design Method and Implementation of Broadband 4 × 4 Nolen Matrix

Abstract

Beamforming techniques are widely used in antenna array systems. One of the most popular beamforming networks (BFNs) is Nolen matrix, which can achieve flexible phase progression. However, due to the lack of analysis on phase slope alignments between internal components, the conventional topology-based Nolen matrix exhibits narrow bandwidth. To overcome this shortage, an analytical design method based on signal flow graphs (SFGs) and complex exponential signal is proposed. With the use of this analytical design method, rigorous relationships of phase slope alignments between couplers and phase shifters within the operating band in the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix topology can be derived. Based on this, a novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix topology composed of couplers and differential phase shifters (DPSs) with specific phase difference slopes is proposed. For verification, a broadband <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula> 4 Nolen matrix based on the proposed topology centered at 1.8 GHz, is designed, fabricated, and measured. Simulated and measured results are in good agreement. The measured bandwidths of ports 1–4 defined by return losses and isolations (15 dB for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{i1}$ </tex-math></inline-formula> , 10 dB for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{ij,i = 1-4, j = 2-4}$ </tex-math></inline-formula> ) and 1 dB amplitude imbalance are 41.49%, 46.21%, 40%, and 48.24%, respectively. Phase errors are within ±5° for 0° and ±90° phase progressions and vary within the range from −10° to 5° for 180° phase progression. The proposed Nolen matrix exhibits the widest bandwidth and the flattest phase progression among Nolen matrices in existence, which verifies the validity of the proposed analytical design method.