A fundamental result concerning the topology of transistor circuits with multiple equilibria

It is shown that a necessary and sufficient condition for a transistor network to possess more than one solution to its dc equations, for some choice of network parameter values, is that a certain simple set of connections, involving some pair of transistors, is present in the circuit´s topological structure. The presence of this set of connections, prescribing the presence of a certain special two-transistor substructure, referred to as a feedback structure, is thus a necessary condition that any transistor network must satisfy in order to possess the property of bistability. The proof of this result is based upon a consideration of two topological properties of transistor networks possessing unique solutions to their dc equations. In addition, it is shown how the amount of computation required to verify a certain well-known set of necessary and sufficient conditions which guarantees the uniqueness of solutions to the dc equations of transistor networks can be reduced by determining the location of all feedback structures.