Research on Efficient Numerical Analysis Method for Nonlinear Resistive Circuits and on the Properties of the Solutions of the above Circuits

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We studied on the nonlinear resistive circuits composed of nonlinear resistive one-ports or nonlinear amplifiers and linear resistive elements including linear active elements. Fundamental problems concerning these circuits are, for example, efficient numerical analysis methods, existence and uniqueness of a solution, the number of solutions, and stability analysis of solutions. Many researchers have studied on these problems, but most problems have so far been unsolved completely. In this research we studied on efficient numerical method of piecewise-linear equations and the number of solutions of an important class of nonlinear equations.1. We proposed an efficient method to find all solutions of piecewise-linear equations. This method requires the amount of computation (multiplications) of O (Mn), where the capital letter O means the order of computational complexity, while previous best results so far known required the computatio of O (Mn^2) in the worst case. We presented the res … More ult at the 1989 ISCAS sponsored by IEEE CAS Society. We conjecture that our result is the best in the sense of computational complexity, but have not proven it yet.2. In the previous theoretical research on the nonlinear resistive circuits it has exclusively been assumed thatAssumption 1 : The v-i characteristics i=f (v) of nonlinear resistive one-ports and the amplification functions v_2=f (v_1) of nonlinear amplifiers are monotonically increasing.Under Assumption 1, however, we can obtain only the condition for the unique solution. To obtain more practical results we assume in this research that :Assumption 2 : The second derivative as well as the first derivative of the nonlinear function f (・) are always positive.Assumption 3 : Amplifiers have ideal saturation characteristics.Under these assumptions we investigated the number of solutions and gave some necessary and sufficient conditions concerning them. Some of them are the conditions for equations to have a finite number of solutions and the upper limit of the number of solutions. Less