Phase error corrections for Microstrip 8×8 Butler Matrix used in 7 Tesla MRI

This paper describes a method to determine fixed phase shifters of the Microstrip 8×8 Butler Matrix, realized in one substrate board, which is designed and fabricated for our 7-Tesla Magnetic Resonance Imaging (MRI) system. This method is based on a global optimization problem to minimize a cost function including a variety of 64 functions as a function of the fixed phase shifts of the 8×8 Butler Matrix. Finally, after optimizing this cost function, estimating the phase shifts and finding the phase error of each fixed phase shifter, the phase error corrections are realized using three different methods: shorting bends (filling), adding open-circuited stubs to the lines (shunt capacitance) and cutting small areas of lines (series capacitance).