Accelerating the solving of nonlinear equations using the homotopy method: application on finding the operating point of complex circuits

Abstract

Analog circuits with nonlinear elements (e.g., diode, BJT, and CMOS) and integrated circuits are very complex systems. As a result, they are very difficult to analyze because of the need to generate a nonlinear equation system solution in order to do so. Solving nonlinear equations is still a challenging problem. Iterative methods, however, are frequently used to solve them. This paper describes a method that can be used to both accelerate the solving of nonlinear equations and find the operating point in various integrated circuits by construction of the global homotopy equation of the analog circuit. This is done by converting the elements in the circuit to their equivalent dependent sources. The proposed method is based on three steps. The first step is the use of the homotopy method to get a continuous function. The second step consists of finding the best direction of the solution by applying prediction and the correction procedures. The third step consists of the control of step size to accelerate the solution search. In order to demonstrate the effectiveness of the proposed method, a new type of software called PyAMS (Python Language for Analog and Mixed Signal) was created by the authors based on the proposed method. A comparison was done between the proposed method and two other methods and, in doing so, many types of integrated circuits were used. Based on the proposed method, an analysis of many universal circuits was carried out to verify the correct function of the circuit.