Assessing the uniqueness of the DC solutions by tearing of cactus graphs via detection of hinging structures

Cactus graphs are used to determine whether or not a circuit may posses multiple DC solutions. This work introduces a tearing method for cactus graphs that can be used to decompose the initial cactus into smaller cactus subgraphs which are hinging structures. The method is focussed on circuits containing independent current and voltage sources, positive linear resistors, and BJTs, but it can be easily modified in order to cope with devices that can be modelled by equivalent circuits composed by coupled branches. Besides, the concept of path-matrices is introduced in order to determine if the whole graph of the circuit can be split into subgraphs. In a further step, the subgraphs can be analysed separately in order to detect multiple DC operating points.