Equations for Active Networks: Existence of Unique Solutions

Conditions are presented which are necessary and sufficient to ensure that the element-variable equations for networks of passive elements, independent sources, and dependent sources controlled by admittance currents and voltages have unique solutions. These require that the networks satisfy two types of constraints. The first is a restriction upon the source locations. The second requires that the transmittance relationships between dependent and controlling variables be independent of similar relationships imposed by the network in which the dependent sources are imbedded. These conditions are also necessary and sufficient when voltage sources controlled by open-circuit voltages or current sources controlled by short-circuit currents are considered. The conditions are sufficient but not necessary for networks containing voltage sources controlled by short-circuit currents or current sources controlled by open-circuit voltages.