Multiparameter homotopy methods for finding DC operating points of nonlinear circuits

This paper introduces multiparameter homotopy methods for finding dc operating points. The question of whether adding extra real or complex parameters to a single-parameter homotopy function can lead to improved solution paths is investigated. It is shown that no number of added real parameters can lead to local fold avoidance, but that generic folds may be efficiently avoided by complexifying the homotopy parameter and tracing a closed curve in complex parameter space around the critical fold value. A combination of real 2-parameter homotopy and complex parameter homotopy is shown to be sufficient for avoiding real fork bifurcations and enumerating all real, locally connected branches. Additionally, the potential of complex parameter homotopy methods for finding all circuit solutions is explored. Results from algebraic geometry indicate that if an analytic homotopy function with a single complex parameter is irreducible, then there exist regular paths through the complex parameter plane connecting any solution of H(x,λ´)=0 to any other solution of H(x,λ´)=0. Thus, in principle at least, complex parameter homotopy can be used to find all circuit solutions