Formulation for Propagation Factor Extractions in Thru-Reflect-Line/Line-Reflect-Line Calibrations and Related Applications

This study reformulated the eigenvalue problems associated with the thru-reflect-line (TRL)/line-reflect-line (LRL) calibration and other related applications without presuming that the line standards or test lines are reciprocal or symmetrical and that they have identical transitions explicitly defined by fixed physical reference planes. The new formulation, therefore, can identify the components of the eigenvalues by using an atypical diagonalization model and explain why the propagation constant extracted from uncalibrated scattering parameters is always accompanied with unwanted ripples; furthermore, this formulation can evaluate the consistency of transitions by regrouping the originally measured calibrated scattering parameters. To enhance numerical stability and avoid the problem of the square-root choice during eigenvalue calculation, the eigenvalues were first transformed into a pair of intermediate eigenvalues that were true reciprocals of each other and could be continuously calculated. The true eigenvalues were then recovered from the transformed eigenvalues. The formulation and experimental data show that the measurement planes cannot be accurately determined if the transitions of the test lines are inconsistent; moreover, the calculated eigenvalues are not equal to the propagation factors of the test lines if the left and right transition discrepancies cannot be completely cancelled out.