Linear Time-Varying Circuits --- Matrix Manipulations, Power Relations, and Some Bounds on Stability

Linear time-varying circuits of special kinds have been designed and analyzed with notable success. On the other hand, theoretical techniques suitable for more general linear time-varying circuits have been developing much more slowly. The development of more general techniques can be approached in various ways. One can seek to specialize the pure mathematics of linear differential equations, in order to discover the properties of those equations which can actually correspond to physical circuits. Alternatively, one can seek to apply the classical analysis of general dynamical systems. As still another alternative, one can seek to generalize, for time-varying circuits, concepts, principles, and techniques which have long been applied to fixed circuits. This paper illustrates the circuit theory approach. After formulating matrix (vector) differential equations corresponding to circuits of linear, time-varying, two-terminal components, it describes some general methods of manipulation. These apply to combinations of time-varying matrices and the differentiation operator, and are time-varying counterparts of manipulations applied to constant matrices in the theory of fixed circuits. Thereafter, the paper uses the manipulations to derive formulas for power, and some bounds on stability. The power equations are conventional and reflect the well known fact 2575