In this article, a new approach to designing a modified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N^{2} \times N^{2}$ </tex-math></inline-formula> Butler matrix (BM) based on an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -way directional coupler (NWDC) is presented. First, by introducing phase shifters and ingeniously arranging the input and output ports, the topologies of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9 \times 9$ </tex-math></inline-formula> BM based on a three-way directional coupler (3WDC) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$16 \times 16$ </tex-math></inline-formula> BM based on a four-way directional coupler (4WDC) are built. Next, according to the design method of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9 \times 9$ </tex-math></inline-formula> BM and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$16 \times 16$ </tex-math></inline-formula> BM, the topology design of modified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N^{2} \times N^{2}$ </tex-math></inline-formula> BM based on NWDC is developed. Considering whether there are progressive phase differences (PPDs) of 0° at output ports, the two cases in which <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is odd and even are discussed. Finally, in order to verify the design method, a prototype of microstrip <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9 \times 9$ </tex-math></inline-formula> BM working at 5.5 GHz is designed, fabricated, and measured. The results show that the designed microstrip <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9 \times 9$ </tex-math></inline-formula> BM achieves equal power division and nine different PPDs among the output signals when the nine input ports are excited, respectively, which meets design expectations. And it is used as a beamforming network (BFN) for a multibeam antenna array (MAA) to generate nine radiation beams. And then, the design method of modified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N^{2} \times N^{2}$ </tex-math></inline-formula> BM is validated by the implementation of microstrip <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9 \times 9$ </tex-math></inline-formula> BM. And the design method of the modified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N^{2} \times N^{2}$ </tex-math></inline-formula> BM provides a novel approach for BFN technology when massive input and output ports are required.
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