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

Abstract

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.

Figures (23)

• Figure 1: Error box model of a two-port VNA for the measurement of calibration standards.
• Figure 2: Example illustrating the evolution of the eigenvalues in classical TRL calibration as a function of frequency. (a) Complex plane representation, (b) the eigengap response. Parameters used in this example: length difference of 1 cm, frequency range of 0-25 GHz, and $\epsilon_\mathrm{r,eff} = 2.6$ for the lossless case, while $\epsilon_\mathrm{r,eff} = 2.6(1-0.06j)$ for the lossy case.
• Figure 3: Example illustrating the eigenvalue behavior in multiline TRL calibration for three and four-line configurations. The figure demonstrates how eigenvalues increase with the number of lines used. Relative effective permittivity assumed lossless $\epsilon_\mathrm{r,eff} = 2.6-0j$.
• Figure 4: Comparison of effective phase for three and four lines configurations under both lossless $\epsilon_\mathrm{r,eff} = 2.6-0j$ and lossy $\epsilon_\mathrm{r,eff} = 2.6(1-0.06j)$ conditions across the 0-25 GHz frequency range.
• Figure 5: Comparison of effective phase between four and six-line configurations, where in the six-line configuration, the fourth line is repeated three times. Results are shown for both lossless $\epsilon_\mathrm{r,eff} = 2.6-0j$ and lossy $\epsilon_\mathrm{r,eff} = 2.6(1-0.06j)$ conditions across the 0-25 GHz frequency range.