Calibration Error Modeling

Even with perfectly known calibration standards, there are two sources of error that can lead to significant errors in calibration: measurement noise and connection non-repeatability. The library provides mechanisms to model these errors and reduce their effects. For example, the error-modeling.py script in the examples directory runs 10,000 simulated calibration runs of a 16-term error model with and without error modeling, and compares the quality of the resulting error terms.

Not only does error modeling move the CDF to the left in this example, but maybe more importantly, it eliminates the long tail. Thus, error modeling gives both more accurate and more repeatable results. The table below gives another view of the same results.

Comparison of Error Term RMS Error With and Without Error Modeling

Error Modeling	Mean	25 Pctile	Median	75 Pctile	Success Rate
No	413.37891	0.01538	0.01948	0.02602	99.88%
Yes	0.01772	0.01294	0.01598	0.02011	92.87%

Without error modeling, a few calibration solutions are so extremely wrong that the mean is pushed orders of magnitude over the median. With error modeling, the extreme values are eliminated.

A price we pay with error modeling, however, is that significantly more calibration attempts fail in the solve step and must be repeated. In our example, the number of calibration attempts that fail increases from 0.12% to over 7%. Arguably, though, the runs that failed would have produced lower quality results.

Libvna provides two mechanisms for error modeling: measurement error modeling and connection non-repeatability modeling. Either can be used alone, or both together. The next sections describe these mechanisms.

Measurement Error Modeling

Measurement error modeling is enabled through the Solver.set_m_error method:

Solver.set_m_error(frequency_vector, noise_floor_vector, tracking_noise_vector=None)

where frequency_vector is a vector of frequency points in Hz, noise_floor_vector is a vector of standard deviations of complex noise detected by the VNA at each frequency when no signal from the DUT is applied, and tracking_noise_vector is an optional vector of standard deviations of noise detected by the VNA at each frequency, proportional the amplitude of the received signal. The two noise sources are assumed to be Gaussian and independent.

Calling this method does several things. First, it alters the calibration equations such that the residual in each equation is equal to the error in the measurement primarily associated with the equation. It next weights the equations such that all have equal expected residuals. Then, after the system of calibration equations has been solved, it runs a chi-square test on the final residuals, where the null hypothesis is that the magnitudes of the residuals are explained by the measurement errors, and alternative is that they are not. The Solver.pvalue_limit attribute controls the threshold needed to reject the null. By default, the value is 0.001.

Without measurement error modeling, the solved error terms are made to fit the calibration measurements in a least squares manner, but no attempt is made to reject solutions with poor fit. With error modeling, calibration solutions in which the measurements are self-contradictory and resulting solution has a poor fit to the measurements are rejected.

Note that enabling measurement error modeling always makes the system non-linear, even if it wasn’t before, and an iterative minimization solution will be used.

Connection Non-repeatability Modeling

Each time a calibration standard or DUT is connected to the VNA, a certain amount of error results from connection non-repeatability. To model this, the library provides a special type of unknown calibration parameter, the CorrelatedParameter:

calset.correlated_parameter(other, frequency_vector, sigma_vector)

where calset is the calibration set used for calibration, other is another calibration parameter (possibly also unknown) to which this parameter is correlated, frequency_vector is a vector of frequency points, and sigma_vector is the square root of the covariance between this parameter and the correlate at each frequency.

When a new calibration standard is connected, we can model each of its parameters as a CorrelatedParameter, where the correlate is the known or unknown true parameter of the standard, and the sigma value is the expected error due to non-repeatability.

Some care is required in where to use correlated parameters. Each introduces a new variable and a new equation, which on the surface sounds free, but introducing more variables that must be solved in a non-linear system can make calibration more finicky. Keep in mind that many complex minimization problems are fractals, and even small errors in initial conditions can result in converging to an unwanted solution. Our advice is to make many measurements of your system and accurately characterize your measurement errors, then before using measurement error modeling, run simulations as in our example to validate that the model you intend to use genuinely improves the results.