LAUSR

In this article, an arbitrary-phase-difference (A-PD) transdirectional (TRD) coupler is proposed and applied in the design of a flexible Butler matrix. The A-PD TRD coupler, which also features the characteristics of arbitrary power division and miniaturized size, is composed of four sections of coupled lines and four types of capacitors. Explicit equations for designing the A-PD TRD coupler were derived utilizing the even–odd mode decomposition analysis, and guidelines to obtain the circuit parameters were also given. To demonstrate the design procedure, examples of the proposed coupler are provided and the relevant studies on the coupler’s electrical parameters are conducted. By combining the proposed coupler with the TRD coupled-line-based crossovers and the −45° phase shifters, a miniaturized $4 \times 4$ Butler matrix with flexible output ports phase differences (−25°, 155°, −115°, and 65°) is implemented. It inherits the features of the TRD coupler including A-PD and compact size. At the center frequency, the measured amplitude and phase imbalances are less than 1.4 dB and 3.2°, respectively, between the output signals. Compared to state-of-the-art A-PD-coupler-based Butler matrix, the proposed structure shows a smaller size ( $1.35 \lambda _{g} \times 1.36 \lambda _{g}$ ) and a wider bandwidth (20%).