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Hello GIN&PEZ;

@ GIN&PEZ - /g/nanovna-users/message/9108

#103 : The Fourth Load *

“It seems that the ratio of these two cross-ratios, the one of the loads and the
other one of their corresponding measurements, is equal to the constant (1,0). “

I agree and find this statement to be true FACUPOV.

Viz.

AnyVNA is physically bound by the limits of its sensitivity (noise floor) and its maximum full scale signal magnitude, and those physical boundaries are calibrated, scaled, and transformed mathematically to precisely define a unit circle of magnitude G = infinity centered on a user defined load of any arbitrary value needed/desired/chosen by the user to be used as the reference Z0 (characteristic impedance) of the measurement system.

Without regard to accuracy, uncertainty, resolution, or absolute value of the chosen Z0, all VNA's can be considered equal on these points, and any measurements made of an identical and arbitrary DUT would be expected to map to precisely the same identical coordinates, when calibrated/characterized with the identical set of reference standards, in an identical test environment.

@ GIN&PEZ - /g/nanovna-users/message/9073

“So, for each triple of (S, L, O) values we think, believe, accept,
or whatever else we like to be consider as "true", this equation
produces a * d i f f e r e n t * value of G.

I do not think this statement fits the definition of G which is constrained to within the limits of 0 and 1 by definition (Gamma = E-reflected / E-forward).

FACUPOV, the statement would be precisely true for the mobius transformation of G to Z (or Y)... i.e. the Smith Chart.

“The Smith chart provides a graphical representation of Γ that permits the determination of quantities such as the VSWR or the terminating impedance of a device under test (DUT). It uses a bilinear Moebius transformation, projecting the complex impedance plane onto the complex Γ plane:. “ (1)


@ GIN&PEZ - [#96]:The Ultimate Criterion for Endless Reliable VNA Comparisons
28 December 2019 - /g/nanovna-users/message/8749

As a criterion for endless reliable VNA comparisons, I think the ratio of cross-ratios falls short of being the Ultimate test, and is probably better described as a fundamental criteria of AnyVNA.

Alternatively; this may be a potentially valid tool for comparison of a known (or assumed) healthy VNA to another of questionable performance. To that end, one would interpret cross- ratio results uncomfortably removed from (1, 0 ) as a confirmation of defective hardware; e.g. gross instability, signal saturation/compression, non- uniformity in frequency step size, incorrectly compensated system errors during calibration, etc.

The cross-ratio results do appear to confirm that:

1) The process referred to as calibration does in fact mitigate all but but the uncertainty of accuracy in the measurements.
2) Accuracy is defined by the known precision of the standards used for calibration, and precision of control over the position of the measurement reference plane.
3) Sources of unavoidable uncertainty in the measurements are exposed and revealed; resolution (number of bits used to quantize the measurement), noise floor (minimum number of bits toggled), and the linear range of the measurement; all of which are measured and remembered through calibration.
4) The Mobius transformation process compresses all values from 0 to infinity into the unit circle, and by so doing, amplifies the accuracy uncertainty exponentially as G approaches zero and infinity. In addition; the resolvable accuracy of the load used and defined as Z0 establishes/defines/bounds, the “Ultimate” accuracy achievable by any appropriately and correctly calibrated AnyVNA measurement system.

I have attempted to summarize the findings of this discussion thread to the best of my understanding, and only from this common user's point of view. It is important to point out that this summary is limited in scope. It does not include any consideration or influence of the DERDEI model (2) used to graphically reveal the upper and lower boundaries of uncertainty defined at the onset of this thread. This is not to suggest, much less conclude, that the boundaries of uncertainty are not important.

I do perceive the findings as evidence that the uncertainties are an independent artifact of the measurement, and not an artifact of the measurement itself nor of the process. In light of that, I also perceive that understanding of the DERDEI models as proposed by GIN&PEZ are very much worthy of scrutiny, validation, and understanding. FACUPOV and amateur radio users in general, they may well be of minimal value or concern. Product developers, and researchers however, would be well served to pursue it with great interest.

72

Gary, N3GO


Ref: (1)

(2) @ GIN&PEZ - /g/nanovna-users/message/2984


73

Gary, N3GO