S-parameter Extrapolation for Improving Near-f_max Accuracy in 2x-thru Calibration

The 2x-thru calibration often faces the problem that, at the highest measurement frequency f_max, the calibrated results exhibit spurious ripples or spikes that are clearly unphysical, such as violating passivity. Although being a common phenomenon, it has attracted little attention in the literature. In this article, the cause of this problem is identified, and a solution is proposed, which is to pre-extrapolate the S-parameters before carrying out the 2x-thru algorithm. Meanwhile, a novel extrapolation method, called the Sun-Earth-Moon (SEM) fitting, is proposed, which is shown to be able to provide a smooth extension of S-parameters for several GHz beyond f_max. The effectiveness of SEM extrapolation in reducing the spurious response at f_max is demonstrated by simulation and measurement examples. The results also indicate that with extrapolation, the calibration accuracy can be slightly improved even at lower frequencies. Comparison with three commercial 2x-thru tools reveals that the near-f_max problem shows up also in the industry-standard tools, indicating the potential of the proposed method for advancing the state-of-the-art.