Hello again Erik;
I think you may be confusing the uncertainty in M with the uncertainty in the calibration standard. For example; if the calibration standard is 45 ohms (a substantial 10 percent error), Only the value of Z0 changes in your equation. However; calibration of the VNA with this standard (uncorrected) equates the value of Z0 to 45 ohms, but declares it to be 50 ohms. "IF" there are "NO" other sources of uncertainty in our measurement, this would force a 5 ohms real uncertainty in our measurement results. The mathematical (and graphical) consequence of this (now hidden) error is that it shortens (compresses) the length of the perfectly linear distance to the right from 50 to infinity by 5 units, the entire length which then gets normalized to the length of 1, such that all values remain inside the boundary of the Smith Chart. Conversely; it also lengthens (stretches) the the line representing the perfectly linear distance to the left from 50 to 1/infinity (zero).
Because we have decided that there are no other uncertainties in the measurement, we get a clear picture of the consequence of the 5 ohms of uncertainty as we move away from the center of the chart, and reveal that the uncertainty asymptotically approaches zero as we move toward the outside edge. Hence; it is my assertion that the consequence of errors in the load standard are most relevant and arguably critical for measurements of an impedance at or near Z0.
In an analogous manner, I might even argue that reactance uncertainties may have even less influence on measurements of DUTs exhibiting a real part less of less than Z0; since Zero reactance is established at all three calibration points.
In a practical sense, and given that even DIY calibration standards of high quality are not difficult to manufacture; it would appear that frequency accuracy and the precise definition of the reference plane are perhaps the most sensitive, and thus the most critical calibration parameters to be controlled. Ironically; uncertainties in those parameters diminish as impedance approaches Z0. What also becomes evident from these assertions is that the quality of the standards with respect to establishing the measurement reference plane increases as frequency increases.
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73
Gary, N3GO
