This thesis presents a method based on Graph Theory concepts that allows us to solve the following aspects regarding the DC general problem.

• Assess the uniquenes of a nonlinear resistive circuit.
• Establish an upper bound on the number of the DC operating points.
• Associate a specific topology-pattern for each DC operating point.
• Determine the stability of each DC operating point with base on the resulting topology-pattern.

This work considers networks containing BJTs, MOSFETs, positive linear resistors, independent voltage and current sources, nonlinear resistors, capacitors and inductors.

The basic idea consists of representing the network to be analyzed by its topological equivalent in cactus graphs. Starting from this representation, the topology is studied by performing graph operations on the original dead graph, and the main objective is to determine if within of the network topology any or some positive feedback structures are present. If none positive feedback structure is found embedded in the network topology, then the uniqueness is guaranteed.

Furthermore, a classification of the feedback structures is achieved. Simple feedback structures are equal to flip-flop structures within the circuit. Composed feedback structures are also a matter of study. The interconnexion beetween feedback structures is also an issue of interest of this thesis.

Once all positive feedback structures have been determined, a series of graph operations are carried out in order to find in a systematic form the number of DC operating points, as well as the associated topology of each operating point, and the stability of each DC operating point is also assessed.

The present work has as conceptual basis the following works:

• The work of Nielsen and Willson
• The work of Nishi and Chua
• The work of Fossepréz and Hasler
• The work of Sarmiento

Last modified November 3th, 2004.

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