The Choice of Line Lengths in Multiline Thru-Reflect-Line Calibration

This paper presents an analysis and rigorous procedure for determining the optimal lengths of line standards in multiline thru-reflect-line (TRL) calibration of vector network analyzers (VNAs). The solution is obtained through nonlinear constrained optimization of the eigenvalue problem in multiline TRL calibration. Additionally, we propose a simplified approach for near-optimal length selection based on predefined sparse rulers. Alongside the length calculation, we discuss the required number of lines to meet bandwidth requirements. The sensitivity of the proposed procedure is evaluated numerically via Monte Carlo simulations, demonstrating that the derived lengths have lower uncertainty than those from existing industry standards. Practical examples are provided for various applications, including lossy and dispersive lines, as well as banded solutions for waveguides.

Propagation of Measurement and Model Uncertainties through Multiline TRL Calibration

In this work, we present a linear uncertainty (LU) propagation treatment of measurement and model uncertainties in multiline thru-reflect-line (TRL) calibration. The proposed method is in accordance with the ISO Guide to the Expression of Uncertainty in Measurement (GUM). We demonstrate with numerical simulation based on the Monte Carlo (MC) method that our proposed LU method delivers identical uncertainty as the MC method but in a more efficient way.