General Coupling Matrix Synthesis for Cruciform Couplers and Its Applications in AFSIW Nolen Matrix Design

In the advancement of millimeter wave (mmWave) circuits and systems, the air-filled substrate integrated waveguide (AFSIW) stands out thanks to its low loss, low cost, ease of integration, and high performance. Meanwhile, the Nolen matrix represents a beamforming network renowned for its functionalities without the need of cross-overs. In this paper, a 4x4 AFSIW Nolen matrix operating at 28 GHz will be designed using a general synthesis of cruciform couplers with various coupling ratios. Compensating phase shifters are added to mitigate phase errors generated by non-ideal couplers. A good operation was demonstrated through simulation across a 1 GHz bandwidth will all return losses greater than 15 dB and phase differences between two adjacent ports are −45°, 135°, −45°, and −135° with the worst error is 13° at the center frequency. The general synthesis of cruciform couplers simplifies the design of complex beamforming networks with different coupling ratios.