A method for improving numerical stability of definition for calibration standards

Abstract

Investigation on numerical fitting treatment for definition of calibration standards is presented. Due to hardware imperfections, calibration is indispensable for vector network analyzers (VNAs), which are accomplished by measuring known quantities called standards. Short and open standards are widely used in the calibration of network analyzers. In fact, the measurement accuracy of VNA depends to a great extent on how accurately the standards are “known”. Thus, calibration standards have to be characterized carefully and precisely beforehand, which is so-called standard definition, so as to assure the high measurement accuracy of VNAs. Short and open standards may be defined based on coefficient model and is utilized most widely. The coefficient model is usually defined by a third order polynomial model. However, by lots of definition experiments with different types of calibration kits, we do have found that the third order polynomial model yields less accuracy for the definition of short standards though it works well for the definition of open standards. A natural solution to this problem maybe is to employ higher order polynomial models to define the short standards. Further research demonstrates that better fitting accuracy can be achieved by using higher order polynomial models. Yet, good accuracy is not always achieved as higher order polynomial models are employed. To ascertain this phenomenon, a careful study is conducted, and it leads the phenomenon to a problem of stability. By analysis of the condition number of the matrix involved in the definition, it is shown that the condition number becomes larger as higher order polynomial model is used, which can account for the above phenomenon well. That is the too large condition number of matrix involved in the definition would contaminate the solutions of equation and lead to worse fitting accuracy. In this paper, a method to reduce the condition number of matrix is also presented and an example is given to verify it.