The 2x-thru calibration often faces the problem that at the highest measurement frequency fmax, the calibrated results exhibit spurious ripples or spikes that are clearly unphysical such as violating passivity. Though being a common phenomenon, it has attracted little attention in the literature. In this paper, the cause of this problem is identified, and a solution is proposed which is to pre-extrapolate the S parameters before carrying out 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 smooth extension of S parameters for several GHz beyond fmax. The effectiveness of SEM extrapolation on reducing the spurious response at fmax is demonstrated by 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 this near-fmax problem shows up also in the industry-standard tools, indicating that the proposed method has the potential of advancing the state-of-the-art.Â

The 2x-thru calibration has emerged as an attractive alternative to the classical TRL for S parameter measurement of printed circuit board and packaging devices, due to its simplicity that only one THRU standard suffices to fully characterize the test fixtures, thus eliminating the LINE measurement in TRL and also overcoming the bandwidth limit. Although various 2x-thru calibration tools are available nowadays, there are still some issues that have not been studied in detail. This paper addresses the following two fundamental questions: (i) what is the reference impedance (Zref) of the S parameters after 2x-thru calibration? (ii) what is the time-domain reflectometry (TDR) response of the THRU for high conductor-loss transmission lines, and how to estimate Zref given such TDR? For problem (i), we rigorously show that the Zref is equal to the characteristic impedance of the center transmission line in the THRU standard, and provide the important interpretation for the S parameters in terms of the traveling-wave and pseudowave concepts. For problem (ii), we show that the TDR impedance is a monotonically increasing function instead of constant, and we propose a new method to estimate the characteristic impedance by fitting the TDR with a causal impedance model. Simulation and measurement examples are provided to validate the proposed theory and techniques. In the last example, we also compare five commercial/open-source 2x-thru calibration tools; the results reveal that with the proposed method of impedance estimation, better accuracy can be achieved.

The 2x-thru calibration is frequently used in the characterization of printed circuit board (PCB) transmission lines. Since the reference impedance of the calibrated S parameters are exactly given by the characteristic impedance, ZLINE, of the trace at the center of THRU, accurate estimation of this impedance becomes crucial to the accuracy of the renormalized S parameters. In this paper, we compare three different methods of estimating ZLINE: the mid-point TDR method, the initial-value TDR method, and the analytic model fitting method. Through a simulation test case, it is demonstrated that (i) the initial-value method gives accurate high-frequency responses, but inaccurate low-frequency and time-domain responses, (ii) the mid-point TDR gives inaccurate results in both time and frequency domains, and (iii) the analytic model method produces the most accurate results.