Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Nonresonant Nodes and Arbitrary Frequency-Variant Reactive Couplings

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this article. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary reactive frequency-variant coupling (AFVC) networks is proposed. The engineered formulation is more general than prior-art ones—with the only restriction that the coupling network is a reactive-type two-port circuit—and can be treated as an extension of previous synthesis models since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, which allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded nonresonating nodes (NRNs), frequency-dependent source–load coupling, multiple frequency-variant cross couplings, and/or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, the third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs).