### Network Analysis and Synthesis: A Modern Systems Theory Approach (Dover Books on Engineering)

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Some other important differences between the classical and modern approaches can be quickly summarized:. The classical approach to synthesis usually relies on the application of ingeniously contrived algorithms to achieve syntheses, with the number of variations on the basic synthesis structures often being severely circumscribed. The modern approach to synthesis, on the other hand, usually relies on solution, without the aid of too many tricks or clever technical artifices, of a well-motivated and easily formulated problem.

At the same time, the modern approach frequently allows the straightforward generation of an infinity of solutions to the synthesis problem. The modern approach to network analysis is ideally suited to implementation on a digital computer. Time-domain integration of state-space differential equations is generally more easily achieved than operations involving computation of Laplace transforms and inverse Laplace transforms.

The modern approach emphasizes the algebraic content of network descriptions and the solution of synthesis problems by matrix algebra. The classical approach is more concerned with using the tools of complex variable analysis. The modern approach is not better than the classical approach in every way. For example, it can be argued that the intuitional pictures provided by Bode diagrams and pole-zero diagrams in the classical approach tend to be lost in the modern approach.

Some, however, would challenge this argument on the grounds that the modern approach subsumes the classical approach. The modern approach too has yet to live up to all its promise. Above we listed problems to which the modern approach could logically be applied; some of these problems have yet to be solved. Accordingly, at this stage of development of the modern system theory approach to network analysis and synthesis, we believe the classical and modern approaches are best seen as being complementary. The fact that this book contains so little of the classical approach may then be queried; but the answer to this query is provided by the extensive array of books on network analysis and synthesis, e.

The classical approach to network analysis and synthesis has been developed for many years. Virtually all the analysis problems have been solved, and a falling off in research on synthesis problems suggests that the majority of those synthesis problems that are solvable may have been solved. Much of practical benefit has been forthcoming, but there do remain practical problems that have yet to succumb. As we noted above, the modern approach has not yet solved all the problems that it might be expected to solve; particularly is it the case that it has solved few practical problems, although in isolated cases, as in the case of active synthesis discussed in Chapter 13, there have been spectacular results.

We attribute the present lack of other tangible results to the fact that relatively little research has been devoted to the modern approach; compared with modern control theory, modern network theory is in its infancy, or, at latest, its early adolescence. We must wait to see the payoffs that maturity will bring. Besides the introductory Part I, the book falls into five parts. Part II is aimed at providing background in two areas, the first being m -port networks and means for describing them, and the second being state-space equations and their relation with transfer-function matrices.

In Chapter 2, which discusses m- port networks, we deal with classes of circuit elements, such network properties as passivity, losslessness, and reciprocity, the immittance, hybrid and scattering-matrix descriptions of a network, as well as some important network interconnections.

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In Chapter 3 we discuss the description of lumped systems by state-space equations, solution of state-space equations, such properties as controllability, observability, and stability, and the relation of state descriptions to transfer-function-matrix descriptions. Part III, consisting of one long chapter, discusses network analysis via state-space procedures. We discuss three procedures for analysis of passive networks, of increasing degree of complexity and generality, as well as analysis of active networks.

This material is presented without significant use of network topology.

## Network Analysis and Synthesis: A Modern Systems Theory Approach

Part IV is concerned with translating into state-space terms the notions of passivity and reciprocity. Chapter 5 discusses a basic result of modern system theory, which we term the positive real lemma. It is of fundamental importance in areas such as nonlinear system stability and optimal control, as well as in network theory; Chapter 6 is concerned with developing procedures for solving equations that appear in the positive real lemma.

Chapter 7 covers two matters; one is the bounded real lemma, a first cousin to the positive real lemma, and the other is the state-space description of the reciprocity property, first introduced in Chapter 2. Part V is concerned with passive network synthesis and relies heavily on the positive real lemma material of Part IV. Chapter 8 introduces the general approaches to synthesis and disposes of some essential preliminaries.

Chapters 9 and 10 cover impedance synthesis and reciprocal impedance synthesis, respectively. Chapter 11 deals with scattering-matrix synthesis, and Chapter 12 with transfer-function synthesis. Part VI comprises one chapter and deals with active RC synthesis, i. As with the earlier part of the book, state-space methods alone are considered. In this part our main concern is to lay groundwork for the real meat of the book, which occurs in later parts. In particular, we introduce the notion of multiport networks, and we review the notion of state-space equations and their connection with transfer-function matrices.

Almost certainly, the reader will have had exposure to many of the concepts touched upon in this part, and, accordingly, the material is presented in a reasonably terse fashion. The main aim of this chapter is to define what is meant by a network and especially to define the subclass of networks that we shall be interested in from the viewpoint of synthesis. This requires us to list the the types of permitted circuit elements that can appear in the networks of interest, and to note the existence of various network descriptions, principally port descriptions by transfer-function matrices.

We shall define the important notions of passivity, losslessness , and reciprocity for circuit elements and for networks.

At the same time, we shall relate these notions to properties of port descriptions of networks. Section 2. An axiomatic introduction to these concepts may be found in [1]. This material can be found in various texts, e. In Section 2. Next, some commonly encountered methods of connecting networks to form a more complex network are discussed in Section 2.

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In Sections 2. Specializations of the bounded real and positive real properties are discussed, viz. Much of the material from Section 2. We shall offer various port descriptions of a network as alternatives to a network description consisting of a list of elements and a scheme for interconnecting the elements. We shall translate constraints on individual components of a network into constraints on the port descriptions of the network.

An m-port network is a physical device consisting of a collection of circuit elements or components that are connected according to some scheme. Associated with the m- port network are access points called terminals, which are paired to form ports.

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At each port of the network it is possible to connect other circuit elements, the port of another network, or some kind of exciting device, itself possessing two terminals. In general, there will be a voltage across each terminal pair, and a current leaving one terminal of the pair making up a port must equal the current entering the other terminal of the pair.

A simplified representation is shown in Fig. The physical structure of the m port will generally constrain, often in a very general way, the two vector variables v and i , and conversely the constraints on v and i serve to completely describe the externally observable behavior of the m port. Given a network with a list of terminals rather than ports, it is not permissible to combine pairs of terminals and call the pairs ports unless under all operating conditions the current entering one terminal of the pair equals that leaving the other terminal.

Given a network with a list of terminals rather than ports, if one properly constructs ports by selecting terminal pairs and appropriately constraining the excitations, it is quite permissible to include one terminal in two port pairs see Fig. Circuit elements of interest here are listed in Fig. An additional circuit element, a generalization of the two-port transformer, will be defined shortly.

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Interconnections of these elements provide the networks to which we shall devote most attention. Note that each element may also be viewed as a simple network in its own right. For example, the simple one-port network of Fig.

The only possibly unfamiliar element in the list of circuit elements is the gyrator. At audio frequencies the gyrator is very much an idealized sort of component, since it may only be constructed from resistor, capacitor, and transistor or other active components, in the sense that a two-port network using these types of elements may be built having a performance very close to that of the ideal component of Fig. In contrast, at microwave frequencies gyrators can be constructed that do not involve the use of active elements and thus of an external power supply for their operation.

They do however require a permanent magnetic field. In the sequel we shall be concerned almost exclusively with circuit elements with constant element values, and, in the case of resistor, inductor, and capacitor components, with nonnegative element values. The class of all such elements including transformer and gyrator elements will be termed linear, lumped, time invariant, and passive. The term linear arises from the fact that the port variables are constrained by a linear relation; the word lumped arises because the port variables are constrained either via a memoryless transformation or an ordinary differential equation as opposed to a partial differential equation or an ordinary differential equation with delay ; the term time invariant arises because the element values are constant.

## Network Analysis and Synthesis: A Modern Systems Theory Approach by Brian D.O. Anderson

The reason for use of the term passive is not quite so transparent. Passivity of a component is defined as follows:. Suppose that a component contains no stored energy at some arbitrary time t 0. Then the total energy delivered to the component from any generating source connected to the component, computed over any time interval [ t 0, T ], is always nonnegative.