Thesis (M.S., Electrical and Computer Engineering) -- University of Idaho, 2016 | Vector Fitting (VF) is one of the most well-known techniques in calculating a rational approximation based on given frequency responses, and VF has been used in modeling different electrical systems. This thesis focuses on three main areas. First, the VF algorithm will be studied in depth regarding its advantages, weaknesses and strengths. Second, the implementation of the VF algorithm using python scripting language, and validation of the performance and the functionality of VF in python through numerical examples; this includes the original VF and the relaxed VF. Third, the analysis of the stability and the passivity of the results of each numerical example that is generated by VF. Semi-Analytic and Cellular method is used to analyze the stability and the passivity of each RLC circuit branch, while the Hamiltonian Matrix method is used to verify the passivity of the network. Also, one of applications of VF discussed in this thesis is using VF to approximate poles and residues for Scattering (S-) and Admittance (Y-) parameters of a two-port network to synthesize the RLC circuit based on the approximated poles and residues.
The results show that the VF method has demonstrated accurate approximation with very small error between actual frequency response and fitted frequency response. The relaxed VF (RVF) method produces a more accurate approximation when the values of responses are small compared to non-relaxed VF. The VF feature to enforce stability is confirmed and its result is in agreement with the result from stability theory analysis, the real part of all poles is less than zero. Passivity analysis shows that even though there might be no passivity at one or more RLRC circuit branches in the network, the network could still be passive. Also, ability of the VF algorithm to detect unstable systems is investigated and verified. Finally, the frequency responses of S-parameters of synthesized equivalent RLC circuits, which are approximated by VF, are well matched against the given tabulated frequency responses.