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 a ll or multiple de operating points of nonlinear circuits and systems. Sever al sufficient conditions for the connectivity of all the solutions along a single homotopy path are derived. These conditions offer criteria to determ ine an initial guess from which all the solutions can be obtained by follow ing the single homotopy path. For the class of nonlinear circuits and syste ms in which all of the solutions lie on several homotopy paths, a new syste matic method to explicitly construct a starting point for each homotopy pat h is developed. From a practical viewpoint, the constructive method develop ed does not require the difficult task of finding a good initial guess and is applicable to general nonlinear circuits and systems. From a methodologi cal viewpoint, the constructive method developed is applicable to general h omotopy methods with different homotopy functions. One particularly importa nt feature of the developed constructive method is that when this method is applied to a de problem with box constraints, it is very computationally e fficient because it locates only feasible solutions satisfying the given bo x constraints. The proposed method is applied to several test examples with promising results.