In power electronic as well as in many other applications, the periodic steady-state behavior of circuits is of great interest to circuit designers. Various methods have been developed to find the steady-state solution to circuit equations, and each has its strengths and weaknesses. Our work focuses on shooting methods, which seek an initial condition x*(0) that forces the circuit immediately into its periodic state, bypassing transient effects. Such an initial condition satisfies f(x*(0), 0, T) = x*(0) where f(x(0), 0, T) is the state transition function that maps the initial state of the circuit x(0) into its state after one period x(T). Approaches to implementing the shooting method include using Newton-Raphson and extrapolation techniques. However, these methods may have convergence difficulties if the initial guess is not close enough to the final solution and if the circuit's state-transition function is highly nonlinear.

In this project, we apply artificial parameter homotopy methods to solve the shooting problem. Under certain conditions, the homotopy methods are globally convergent with probability one. While Newton-Raphson and extrapolation based shooting methods are only locally convergent, we expect that this homotopy-based algorithm will possess global convergence. The homotopy methods have been already successfully applied to solving particularly hard DC operating point problems that conventional methods have failed to solve.

Homotopy methods reformulate the original difficult problem into an easy-to-solve problem. By continuously sweeping a parameter, the easy problem is deformed into the original difficult problem. An example of a homotopy is the mapping H(x, l) = l F(x) + (1- l )(x-a), where F(x) = 0 is the nonlinear equation to be solved, ``a'' is a known constant vector, and ``l'' is the homotopy parameter that is swept from 0 to 1. The easy problem is H(x,0) := x-a = 0, while H(x,1) := F(x) = 0 is the problem we want to solve.

At this time, we have successfully connected an algorithm from HOMPACK to the circuit simulator SSPICE. HOMPACK is a public domain software package that uses homotopy methods to solve sets of nonlinear equations. SSPICE is a circuit simulation program, an extension of SPICE3 that includes the Newton-Raphson and extrapolation based shooting methods. Future work includes further testing of the homotopy-based shooting method on various circuits, using different algorithms from HOMPACK with SSPICE, and a more detailed study of the theoretical aspects of these algorithms.