Modeling Piecewise-Linear Systems by Topological Techniques

Flowgraph techniques are extended to systems with piecewise-linear characteristics by developing criteria for construction of an optimum model from related subregions in which linearity holds. This requires the synthesis of several known techniques and results in a wide range of useful applications including: 1) devices with nonlinear characteristics which may be considered as linear over certain subregions; 2) networks whose response to changes in applied signal frequency or magnitude may be approximated by piecewise-linear asymptotes; 3) systems processing two or more signals simultaneously with different transfer or immitance characteristics for each signal; 4) circuits approximated by different equivalent circuits depending on the numerical values of critical parameters. Representative examples will illustrate these and similar applications. Procedures are presented to provide a logical, orderly, and effective approach to construct a model, to determine figures of merit, and to optimize the model for a prescribed region of operation or for a desired range of parameters.     

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.