A NONLINEAR LEAST-SQUARES SOLUTION WITH CAUSALITY CONSTRAINTS APPLIED TO TRANSMISSION-LINE PERMITTIVITY AND PERMEABILITY DETERMINATION

A technique for the solution of 1-port and 2-port scattering equations for complex permittivity and permeability determination is presented. Using a nonlinear regression procedure, the model determines parameters for the specification of the spectral functional form of complex permittivity and permeability. The method is based on a nonlinear regression technique using the fact that a causal, analytic function can be represented by poles and zeros. The technique allows the accurate determination of many low- and high-permittivity dielectric or magnetic materials in either the low- or high-loss range. Permeability and permittivity can be obtained either from fitting all scattering-parameter data or by fitting S21 or S11 or \S11\ and \S21\ data by itself. The model allows for small adjustments, consistent with the physics of the problem, to independent variable data such as angular frequency, sample length, sample position, and cutoff wavelength. The model can determine permittivity and permeability for samples where sample length, sample position, and sample holder length are not known precisely. The problem of local minima is discussed.