Symbolic computation tuning method for the evaluation of algorithms in small signal parameters extraction

The math problem associated to the problem of parameter extraction of a circuit can be reduced to the problem of finding the global minimum of an error function obtained with the least squares method. In the case of the small signal analysis of a circuit (when a linear model is considered), the circuit's associated function is a rational function and as a consequence the error function is of the same form too. This makes possible to take into account solving the resulting equation system, formed with the partial derivatives of the error function, in order to find, in the end, the global minimum of the error function. The method, unlike any pure numeric method, no longer requires start values for the parameters being extracted and also guarantees that the final result corresponds to the global minimum. This approach leads to the mathematical problem of finding the solutions of an equation system formed with rational functions, which can be transformed in an equivalent polynomial system. The paper highlights the dependency of the degree of this initial polynomial equation system, with the number of measured values of a linear circuit small signal function which is considered in a particular extraction problem. It is shown that the total execution time (which depends on the degree of the initial polynomial system and also on the algorithm for reducing the system to an equivalent quasi-triangular, numerically solvable form) can be tuned by adjusting this number. Using this property/dependency which makes possible to generate several mathematical problems (initial polynomial equation systems) having different complexities, for the same extraction problem, we analyze, using a particular circuit, the possibility of getting a solution in a reasonable amount of time, with the algorithms implemented in two different Math-CAD systems. The conclusions identify which of the mathematical instruments used, could be used to implement a standalone program for extraction, which should not necessarily be based on a CAD system.