Transmission Zeros of Trisection and Quadruplet Bandpass Filters With Mixed Cross Coupling

This article solved the inverse problem for trisection and quadruple bandpass filters (BPFs) with mixed cross coupling. It allows for a given placement of transmission zeros and known main coupling coefficients to determine mixed cross coupling K = K_m + K_e, containing magnetic and electrical components. Based on the obtained solution, it was established that trisection BPF has ten different options for placing two transmission zeros on the complex plane S = σ + jQ. It is shown that the considered trisection and quadruplet BPFs can have a second-order transmission zero on the jQ axis, which provides a deeper attenuation pole at insertion loss curve. With the help of the obtained inverse problem solution, some restrictions are established on the possible options for the placement of three transmission zeros of quadruplet BPF with mixed cross coupling K₁₄: transmission zeros cannot be placed on the o-axis and two of the three transmission zeros on the jQ-axis cannot be equidistantly relative to S = 0. It is found that to obtain a flat group delay, the transmission zeros must be located in the S-plane at the corners of an isosceles triangle, the vertex of which lies on the jQ axis, and the sides intersect the o-axis. In this case, in addition to the flat group delay, the insertion loss curve has an attenuation pole. Theoretical results are validated with two microstrip quasi-inline trisection BPFs and one stripline quadruplet BPF.