Constructive homotopy methods for finding all or multiple DC operating points of nonlinear circuits and systems

This paper develops a constructive homotopy-based methodology for finding all or multiple dc operating points of nonlinear circuits and systems. Several sufficient conditions for the connectivity of all the solutions along a single homotopy path are derived. These conditions offer criteria to determine an initial guess from which all the solutions can be obtained by following the single homotopy path. For the class of nonlinear circuits and systems in which all of the solutions lie on several homotopy paths, a new systematic method to explicitly construct a starting point for each homotopy path is developed. From a practical viewpoint, the constructive method developed does not require the difficult task of finding a good initial guess and is applicable to general nonlinear circuits and systems. From a methodological viewpoint, the constructive method developed is applicable to general homotopy methods with different homotopy functions. One particularly important feature of the developed constructive method is that when this method is applied to a dc problem with box constraints, it is very computationally efficient because it locates only feasible solutions satisfying the given box constraints. The proposed method is applied to several test examples with promising results