A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this paper. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary frequency-variant coupling (AFVC) coefficients is proposed. The engineered formulation is more general than prior-art ones 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, allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded non-resonating 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, 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)
On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type
