Automatic Model Extraction of the Match Standard in Symmetric--Reciprocal--Match Calibration

TL;DR

This work extends symmetric-reciprocal-match (SRM) calibration by enabling automatic frequency-dependent parasitic modeling of the match standard. It formulates a nonlinear, global-optimization approach that fits unknown match parasitics while anchoring the DC resistance, leveraging a nullspace-based framework and the average fourth singular value as the objective. The method is validated through synthetic CPW simulations and PCB measurements, showing accuracy comparable to multiline TRL calibration when measuring a PCB with flip-chip resistors. The results demonstrate SRM's flexibility and practical viability for high-frequency VNA calibration, particularly when explicit characterization of the match standard is challenging.

Abstract

This paper addresses the modeling of parasitics of the match standard in the symmetric-reciprocal-match (SRM) calibration method of vector network analyzers (VNAs). In the general SRM procedure, the match standard is assumed to be fully known. Here, we demonstrate that the match can be modeled with an arbitrary frequency-dependent model using a non-linear global optimization procedure. To highlight the validity of the suggested approach, numerical tests were conducted, demonstrating the ability to recover the match standard parasitic model down to software numerical precision. Additionally, we performed microstrip line measurements to compare the SRM calibration with match modeling to the multiline thru-reflect-line (TRL) calibration one, showing that automatic model extraction can achieve accuracy similar to using a match standard defined through multiline TRL calibration.

Figures (10)

• Figure 1: Two-port VNA error box model illustrating the standards used in SRM calibration. All matrices are provided as T-parameters. The index $i$ indicates the measured standard, where $i=1,2,\ldots, M$, with $M \geq 3$. Note that the last standard could also be measured at port-B.
• Figure 2: Equivalent circuit model of a non-ideal match standard. The values of $l$ and $\rho_\mathrm{dc}$ are assumed to be known. The parameters $L(f)$, $C(f)$, $\gamma(f)$, and $Z_c(f)$ are modeled as frequency-dependent.
• Figure 3: Block diagram illustration of the numerical simulation concept to generate realistic synthetic data.
• Figure 4: Models used to simulate non-ideal load standards (a) $50\,\Omega$ match standard with $L_0=25\,\mathrm{pH}, C_0=1\,\mathrm{fF}$, (b) short standard with $L_0=30\,\mathrm{pH}, L_1=10\times10^{-12}\,\mathrm{pH/Hz}, C_0=0.5\,\mathrm{fF}$, and (c) open standard with $C_0=15\,\mathrm{fF}, C_1=5\times10^{-15}\,\mathrm{fF/Hz}, L_0=1\,\mathrm{pH}$. All standards include a $200\,\mu\mathrm{m}$ CPW line offset. These models are used for automatic parasitic extraction.
• Figure 5: Relative errors in estimated parameters for short and match standards during DE optimization versus iteration count.