A Novel Method for Estimating Residual Model Parameters to Evaluate Uncertainty in Scattering Parameter Measurements

In this article, we present a novel method for estimating the parameters of residual models, which are widely used to evaluate the measurement uncertainty of the scattering parameters measured by a vector network analyzer (VNA). Conventionally, the parameters of the residual model are obtained using the ripple method with an air line, where the insulator that distinguishes the inner and outer lines of the coaxial line is air. However, due to inaccuracies caused by losses from air lines, the usable range of frequencies is limited. In addition, methods using the invariance property of the cross ratio of complex numbers result in overestimation because the maximum error boundary is calculated. Above all, since both methods estimate only the magnitude uncertainty, the phase uncertainty estimation is not rigorous, and the correlation between the magnitude and the phase cannot be estimated. In contrast, the proposed method calculates the parameters of the residual model directly from the calibration standards uncertainty, including the phase as well as magnitude, achieving the same result as the VNA error model. The proposed parameter estimation for residual models can also easily propagate to the uncertainty of other measurements since it can obtain the correlation between magnitude and phase as well as the cross frequency. In this article, we compared the parameters of the residual model using the proposed method, the conventional ripple method, and the invariance property of the cross ratio of complex numbers. For validation, the uncertainty obtained using the residual model with the proposed parameter estimation was compared to the results from the VNA error model. The two uncertainties showed excellent agreement for highly reflective and matched devices. Finally, we discuss how to effectively manage the calibration and measurement capabilities of scattering parameters using the proposed uncertainty evaluation method.