Analyzing the Multiplicity of Solutions in Filters With Parallel-Connected Structures

Determining the solution for a specific topology in parallel-connected filters might be challenging, especially due to the nonuniqueness of the solutions. This work proposes a method to address the multiplicity of solutions in high-order filters by focusing on the reconfiguration of the coupling matrix (CM). Unlike conventional techniques, which can be complicated by analyzing the network as a whole, the proposed approach examines each branch of the filter independently, where each existing solution depends on the grouping of the eigenvalues in each branch. It is further demonstrated that the analysis can be performed for an inner transversal subnetwork by previously reconfiguring the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N +2$</tex-math> </inline-formula> transversal CM to a transversal CM of lower order than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N +2$</tex-math> </inline-formula>. To validate this analysis, the results are compared with all existing solutions with the help of a verification software based on the numerical interval Newton algorithm (NINA), which guarantees the global convergence of all solutions. To validate the concept, an example of a 7th-order topology is presented.