A Synthesis Framework for Finding All Solutions of Microwave Filter Topologies Based on Homotopy Continuation

This article proposes a numerical filter synthesis framework based on homotopy continuation (HC) to find all solutions of target topologies reliably. First, the “base model” is defined and exhaustively solved via perturbation starting from one solution that can be calculated analytically. Although the base model features nonideal filtering characteristics, it can act as the starting system of the following homotopy path-tracking process. In particular, the base model can bring the computation burden that is less sensitive to the increase of filter orders, which makes the synthesis of high-order topologies feasible. Afterward, the “target models” with satisfactory filtering responses can be synthesized exhaustively and efficiently via the adaptive-step path tracking from the stored base model. All the solved base models and target models constitute a database, in which the accumulated pre-calculated information can significantly accelerate the synthesis of filters with the same topology but different return loss and transmission zeros (TZs). In addition, the parallel computation technique is exploited to further speed up the presented synthesis framework. Multiple synthesis examples, including higher order filters, are presented to verify the proposed theories. One combline filter is fabricated, and the measurement results exhibit a good agreement with the theoretical response.