Flowgraph Techniques for Closed Systems

A two-port with both generator and load constitutes a closed system; it can be defined as a system with only interdependent variables or as a system containing no independent variables. The flowgraph describes the relationships between variables in a closed system as determined by the topology of its components. The corresponding closed flowgraph represents a zero-port network from which N-port networks can be generated as subsets by truncation of the closed system. A constraint governing any closed system expressing the properties of its cut-sets and tie-sets is referred to as the topology equation. The topology equation can be computed from the sum of loops in the closed flowgraph, and forms the key for the derivation of properties of closed systems and derived N-port subsets. Oscillators analyzed as zero-ports are discussed in detail to illustrate the rationale for the flowgraph approach to closed systems. To exploit the closed system model conceptually and computationally for N-ports, an approach is suggested and illustrated for N = 1 by an analysis of transfer functions and for N = 2 by a sensitivity analysis of stochastic networks.