A Novel Design of $4 \times 4$ Butler Matrix With Relatively Flexible Phase Differences

Abstract

This letter presents a novel topology of a 4 ×4 Butler matrix, which can realize relatively flexible phase differences at the output ports. The proposed Butler matrix employs couplers with arbitrary phase-differences to replace quadrature couplers in the conventional Butler matrix. By controlling the phase differences of the applied couplers, the progressive phase differences among output ports of the proposed Butler matrix can be relatively flexible. To facilitate the design, closed-form design equations are derived and presented. For verifying the design concept, a planar 4×4 Butler matrix with four unique progressive phase differences ( - 30 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , + 150 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , - 120 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , and + 60 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> ) is designed and fabricated. At the operating frequency, the amplitude imbalance is less than 0.75 dB, and the phase mismatch is within ±6 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> . The measured return loss is better than 16 dB, and the isolation is better than 18 dB. The bandwidth with 10 dB return loss is about 15%.