IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 17, NO. 4, APRIL 2007 277 A Vector-Fitting Formulation for Parameter Extraction of Lossy Microwave Filters Ching-Ku Liao, Chi-Yang Chang, Member, IEEE, and Jenshan Lin, Senior Member, IEEE Abstract—In this letter, a formulation based on the vector fitting is applied to extract the equivalent circuit model from the frequency response of lossy cross-coupled microwave filters. By approximating the lossy response with short-circuit admittance parameters in partial fractional expansion form, the proposed method can evaluate the unloaded quality factor of resonators and extract the transversal coupling matrix simultaneously. The methodology of the vector fitting can identify the poles and residues of the short-circuit admittance parameters even when the poles are on the complex plane. And the extracted transversal coupling matrix can further transform into the prescribed form corresponding to the physical layout. The proposed method can be used in the tuning process of filter designs where the extraction of a coupling matrix is essential. To verify the method, a cross-coupled quadruplet filter is used as an example. Index Terms—Bandpass filter (BPF), coupling matrix, vector fitting. I. INTRODUCTION T HE cross-coupled filters based on the model proposed in [1] and [2] have found wide applications in wireless communication systems since they can provide the general- ized Chebyshev response which exhibits the optimal in-band response and selectivity. However, the tuning of the filters based on cross-coupled topologies is time-consuming. In order to tune the cross-coupled filters more efficiently, therefore, diagnosis methods are needed to guide the process of the filter tuning [3]–[5]. Since the model proposed in [1] and [2] can be expressed by a coupling matrix, most diagnosis methods in the literature are focused on extracting a coupling matrix from the simulated or measured response. By comparing the extracted coupling matrix to the desired coupling matrix, one can determine how to adjust the filter [3], [5]. Most parameter extraction methods are only valid for lossless filters since this is the assumption in their formulations. Thus, getting a coupling matrix from a lossy filter response is still an important research topic. Recently, a modified formulation of the Cauchy method which can extract the parameters of a loss- less model from the response of a lossy bandpass filter (BPF) is proposed [6]. The formulation in [6] can generate character- Manuscript received October 9, 2006; revised December 15, 2006. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2752-E-009-003-PAE and the NSC Graduate Student Study Abroad Program (GSSAP). C.-K. Liao and C.-Y. Chang are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: ching.cm92g@nctu.edu.tw; mhchang@cc.nctu.edu.tw). J. Lin is with the Electrical and Computer Engineering Department, Univer- sity of Florida, Gainesville, FL 32611 USA (e-mail: jenshan@ufl.edu). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LMWC.2007.892970 Fig. 1. Canonical transversal array. (a) -resonator transversal array including direct source–load coupling . (b) Equivalent circuit of the th “low-pass resonator” in the transversal array. istic polynomials suitable for the synthesis of a low pass pro- totype associated with the lossless model of the filter, which is not feasible in the formulation proposed in [5] and [7]. Strictly speaking, the methods in [5] would require lossless measured data and can not give a measure of how lossy a filter is. In this letter, to take the loss of a filter into consideration, we propose to use the model in Fig. 1. The model in Fig. 1 was mod- ified from the model first proposed in [8] for filter synthesis and known as transversal network. The only difference between the model used in [8] and here is that we added the conductance, , in each branch of the transversal network to model the loss, as shown in Fig. 1(b). As the formulation in [8], the short-circuit ad- mittance parameters, also know as -parameters, of the model in Fig. 1, can be expressed by a polynomial in partial fractional expansion form. Here, the introduction of the loss positions the poles of the -parameters on the complex plane instead of on the imaginary axis as in the lossless case. To effectively get the short-circuit admittance parameters in the partial fraction expan- sion form, the technique of vector fitting [9] is applied. The for- mulation based on the vector fitting can identify the positions of poles and calculate the residue of the -parameters. The poles of the -parameters contain the information of how lossy a filter is. Thus, the proposed method allows: 1) the evaluation of how lossy a filter is from the simulated or measured data; 2) the generation of the -parameters in the partial fraction expansion form, which is suitable for the synthesis of a low-pass prototype by the method in [8]. 1531-1309/$25.00 © 2007 IEEE