Synthesis of Underdetermined Filter Topologies with Nonresonating Nodes within a Limited Range

This paper presents the synthesis of underdetermined filter topologies incorporating nonresonating nodes (NRN), which have an infinite number of solutions. To fit the physical constraints and fabrication technology, slack variables are introduced to only synthesize in a prescribed limited range. The NRNs are inserted into the coupling matrix directly as the independent nodes, so the synthesis problem is converted to the solving of underdetermined nonlinear equations based on the congruent transformation. To handle the singular Jacobian matrix, an efficient generalized continuation Newton method (GCNM) with the trust-region updating strategy is introduced. One fourth-order NRN filter with all positive couplings is synthesized and manufactured to validate the proposed synthesis theories.