TOPOLOGICAL ANALYSIS OF ACTIVE NETWORKS

The concept of the “natural frequencies” of a network arises from the consideration of its transient behavior. They are conventionally given as the zeros of the determinant of the loop-impedance matrix or the cutset-admittance matrix. Since in general the zeros of these two determinants are different, this definition involves some ambiguity. One of the classical problems is to count the number of natural frequencies of a network by inspection. An early solution to determine this number called the degrees of freedom was given by Guillemin (1931), applicable only to RLC networks that do not contain any all-capacitor or all-inductor loops. The term originates from the study of mechanical systems, in which we can attach significance to the word “position” or “configuration”. Reza (1955) gave the solution for networks containing only two types of elements and adopted the term “order of complexity”, which we shall follow in this book. The complete solution for RLC networks was obtained independently by Bryant (1959, 1960), Bers (1959), and Seshu and Reed (1961). The extension to active networks was recent and has been considered by many workers [see, for example, Chen (1972a)]. The main difficulty .lies in the fact that, unlike the case for RLC networks, topology of the network alone is not sufficient; network parameters must also be involved, which complicates the problem considerably. However, various upper bounds on the order of complexity of a general network are available. In the first part of the chapter, we shall present a unified summary on many of the known results.

The studies and designs of electronic circuits, signals, and systems are based on a variety of models, which consist of interconnections of idealized physical elements such as inductors, capacitors, resistors, and generators. A physical inductor should be thought of as a coil of wire with series and/or shunt resistance and even capacitance, in addition to inductance, for its complete representation. Similar statements can be made for any physical component. Although such a procedure is physically justifiable, it is extremely inconvenient from a theoretical point of view; for, using this procedure, the equations of a simple network will become very complicated. All of the techniques of network synthesis that have been developed will be useless, and a great many procedures for designing practical networks such as filters, interstage networks, wave shaping networks etc. will be hopelessly complicated. No one is going to give them up merely to avoid the use of idealized elements. Thus, for all purposes, both in analysis and synthesis, we will continue to use models composed of idealized physical elements. A consequence of this idealization is that we cannot always assume the existence and uniqueness of network solutions. It is of great interest, therefore, to determine the conditions under which unique solutions can be obtained. These conditions are especially useful in computer-aided network analysis when a numerical solution does not converge. They help distinguish those cases where a network does not possess a unique solution from those where the fault lies with the integration technique. We note that even when a network response is unstable, a numerical solution exists. Thus when a numerical solution does not converge, it is important to distinguish network instability, divergence due to improper numerical integration, and divergence due to the lack of existence of a unique solution. For these reasons, we shall also present a unified summary of various existence conditions.