Computing uncertainties of S-parameters by means of Monte Carlo Simulation

The uncertainty of a Vector Network Analyzer (VNA) measurement is a consequence of the uncertainties both of calibration standards and the VNA. How the latter uncertainties move forward to the final measurement uncertainties is a complicated multistage process. The first stage (calibration of the VNA) essentially results in the error box terms. The nonlinear relationship between the parameters of the standards and the error box terms poses in this stage a problem. In the second stage (measurement of Device Under Test (DUT)) the uncertain error box terms are used to calculate the S-parameters of the DUT and again mix up with uncertainties of the VNA and the connectors. In this stage the correlation of error box terms amongst each other poses a second problem. It is important to note that only uncertainty distribution functions (with assumed parameters such as shape and variance) of the VNA and the standards are used to obtain predictions about the final measurement uncertainties. One common simplification of conventional approaches in VNA uncertainty computations is that stage one formulas (which express error box terms as a function of the standards' S-parameters) are replaced by their respective Taylor expansions. Another commonly applied approximation is that computed distributions of error terms are subsequently applied to the DUT without accounting for correlation of error terms among themselves. The goal of a good approach must be that the effects of different types of input uncertainties (e.g., transmission phase or reflection coefficient of standards) can be studied separately. Unfortunately, as a consequence of the above approximations and in contradiction to more rigorous approaches nearly the same measurement uncertainty is calculated for different types of uncertainties in the standards. Unlike traditional approaches the method presented in this paper is (1) based on Monte Carlo Simulation (MCS) and (2) does not use any simplifying assumptions to compute the measurement uncertainty. Essentially the whole calibration and subsequent measurement process is simulated a couple of ten thousand times by starting with random input values for the standards. The resulting distribution of the S-parameters of a given DUT is then analyzed using a statistics software. This approach was particularly useful for studying different calibration techniques in conjunction with snap on connectors which show large phase and small reflection coefficient variances. The MCS makes it possible to clearly distinguish between effects of transmission phase deviations and reflection coefficient deviations. We found that MCS is a well suited method for the computation of uncertainties in VNA calibration. In particular we make comparisons between different calibration strategies in conjunction with snap on connectors. The main outcome of the study is that it is favorable to use a calibration for 1.85 mm connectors and de-embed the used 1.85 mm to snap on connector adapters, rather than making a calibration using snap on standards. Experienced measurement engineers mostly would have proposed this kind of calibration for a snap on connector. Our results do not only confirm these heuristic approaches but give additional hard facts and quantify the differences between selected calibration strategies.