Extracting Coupling Matrix and Unload Q from Scattering Parameters of Lossy Filters

This paper presents a method for extracting the coupling matrix and the unloaded Q from the measured (or electromagnetic simulated) S-parameters of a narrow band cross-coupled resonator bandpass filter with losses. The Cauchy method is applied to determine the characteristic polynomials of the S-parameters of a filter in the normalized low-pass frequency domain. A five-parameter optimization method is proposed to obtain the unloaded Q and to remove the phase shift of the measured S-parameters, which is caused by the phase loading and the transmission lines at the input/output ports of a filter. Once the characteristic polynomials of the S-parameters with the phase shift removed have been determined, the coupling matrix of a filter with a given topology can be extracted using well established techniques. Two application examples are given to illustrate the validity of the proposed method.