General line-line method for propagation constant measurement of non-reciprocal networks

A general “line-line” (LL) method is proposed for propagation constant determination (γ+ and γ−) of nonreciprocal networks using multiple reference networks. The new formalism provides flexibility in implementation of LL methods so that not only more accurate γ+ (and γ−) could be retrieved but also the effect of inaccurate length information of the non-reciprocal network on γ+ (and γ−) evaluation could be suppressed. For validation of the method, γ+ and γ− of a microwave phase shifter (a nonreciprocal network) are determined using uncalibrated scattering parameter measurements at X-band (8.2–12.4 GHz). Two different statistical analyses based on normalized root-mean-square-error and goodness of fit and based on coefficient of variation and confidence interval are individually performed to evaluate the performance and accuracy of our method.