Active Network Analysis

The following sections are included:

• PREFACE

• CONTENTS

Over the past two decades, we have witnessed a rapid development of solid-state technology with its apparently unending proliferation of new devices. Presently available solid-state devices such as the transistor, the tunnel diode, the Zener diode, and the varactor diode have already replaced the old vacuum tube in most practical network applications. Moreover, the emerging field of integrated circuit technology threatens to push these relatively recent inventions into obsolescence. In order to understand fully the network properties and limitations of solid-state devices and to be able to cope with the applications of the new devices yet to come, it has become increasingly necessary to emphasize the fundamentals of active network theory that will transcend the advent of new devices and design tools.

The purpose of this chapter is to introduce many fundamental concepts used in the study of linear active networks. We first introduce the concepts of portwise linearity and time invariance. Then we define passivity in terms of the universally encountered physical quantities time and energy, and show that causality is a consequence of linearity and passivity. This is followed by a brief review of the general characterizations of w-port networks in the frequency domain. The translation of the time-domain passivity criteria into the equivalent frequencydomain passivity conditions is taken up next. Finally, we introduce the discretefrequency concepts of passivity. The significance of passivity in the study of active networks is that passivity is the formal negation of activity.

In the preceding chapter, networks were characterized by their port behaviors. Fundamental to the concept of a port is the assumption that the instantaneous current entering one terminal of the port is always equal to the instantaneous current leaving the other terminal of the port. However, we recognize that upon the interconnection of networks, this port constraint may be violated. Thus, it is sometimes desirable and more advantageous to consider n-terminal networks, as depicted in Fig. 2.1.

In this chapter, we discuss a useful description of the external behavior of a multiterminal network in terms of the indefinite-admittance matrix and demonstrate how it can be employed effectively for the computation of network functions. Specifically, we derive formulas expressing the network functions in terms of the first-order and the second-order cofactors of the elements of the indefiniteadmittance matrix. The significance of this approach is that the indefiniteadmittance matrix can usually be written down directly from the network by inspection.

Since in the remainder of this book we deal exclusively with linear, lumped, and time-invariant networks, the adjectives linear, lumped, and time-invariant are omitted in the discussion unless they are used for emphasis.

In Chap. 1 we introduced many fundamental concepts related to linear, time-invariant n-port networks. Some of the results, although very general, are difficult to apply. In Chap. 2 we discussed a useful description of the external behavior of a multiterminal network in terms of the indefinite-admittance matrix, and demonstrated how it can be employed effectively for the computation of network functions.

In practical applications, the most useful class of n-port or n-terminal networks is that of two-port or three-terminal networks. Many active devices of practical importance such as transistors are naturally subsumed in this class. In this chapter, we consider the specialization of the general passivity condition for n-port networks in terms of the more immediately useful two-port parameters. We introduce various types of power gains, sensitivity, and the notion of absolute stability as opposed to potential instability. Llewellyn's conditions for absolute stability and the optimum terminations for absolutely stable two-port networks at a single frequency will be derived.

In the preceding chapter, we demonstrated that by introducing physical feedback loops externally to an active device, we can produce a particular change in the performance of the network. Specifically, we showed that a three-terminal device can be unilateralized by a lossless reciprocal imbedding. In this and following chapters, we shall study the subject of feedback in detail and demonstrate that feedback may be employed to make the gain of an amplifier less sensitive to variations in the parameters of the active components, to control its transmission and driving-point properties, to reduce the effects of noise and nonlinear distortion, and to affect the stability or instability of the network.

We first discuss the conventional treatment of feedback amplifiers, which is based on the ideal feedback model, and analyze several simple feedback networks. We then present Bode's feedback theory in detail. Bode's theory is based on the concepts of return difference and null return difference and is applicable to both simple and complicated feedback amplifiers, where the analysis by conventional method for the latter breaks down. We show that return difference is a generalization of the concept of the feedback factor of the ideal feedback model and can be interpreted physically as the returned voltage. The relationships between the network functions and return difference and null return difference are derived and are employed to simplify the calculation of driving-point impedance of an active network.

In the preceding chapter, we studied the ideal feedback model and demonstrated by several practical examples how to calculate the transfer functions μ(s) and β(s) of the basic amplifier and the feedback network of a given feedback configuration. We introduced Bode’s feedback theory, which is based on the concepts of return difference and null return difference. We showed that the return difference is a generalization of the concept of the feedback factor of the ideal feedback model and can be interpreted physically as the difference between the 1-V excitation and the returned voltage. We demonstrated that return difference and null return difference are closely related to network functions and can therefore be employed to simplify the calculation of driving-point impedance of an active network, thereby observing the effects of feedback on amplifier impedance and gain.

In the present chapter, we continue our study of feedback amplifier theory. We show that feedback may be employed to make the gain of an amplifier less sensitive to variations in the parameters of the active components and to reduce the effects of noise and nonlinear distortion, and to affect the stability of the network. The concepts of return difference, null return difference, and sensitivity function will be generalized by introducing the general reference value, which is very useful in measurement situations. Since the zeros of the return difference are also the natural frequencies of the network, they are essential for the stability study. To this end, we present three procedures for the physical measurements of return difference. This is especially important in view of the fact that it is difficult to get an accurate formulation of the equivalent network, which, to a greater or lesser extent, is an idealization of the physical reality. The measurement of return difference provides an experimental verification that the system is stable and will remain so under certain prescribed conditions. Finally, we discuss the invariance of return difference and null return difference under different formulations of network equations.

In the preceding two chapters, we demonstrated that the application of negative feedback in amplifiers tends to make the overall gain less sensitive to variations in parameters, reduce noise and nonlinear distortion, and control the input and output impedances. These improvements are all affected by the same factor, which is the normal value of the return difference. However, the price that we paid in achieving these is the net reduction of the overall gain. In addition, we are faced with the stability problem in that, for sufficient amount of feedback, at some frequency the amplifier tends to oscillate and becomes unstable. The objective of this chapter is to discuss various stability criteria and to investigate several approaches to the stabilization of feedback amplifiers.

As indicated in Sec. 5.8, the zeros of the network determinant are called the natural frequencies. A network is stable if all of its natural frequencies are restricted to the open left half of the complex frequency s-plane (LHS). If the network determinant is known, its roots can be readily computed explicitly with the aid of a computer if necessary, and the stability problem can then be settled directly. However, for a physical network, there remains the difficulty of getting an accurate formulation of the network determinant itself. Even if we have the network determinant, the roots alone do not tell us the degree of stability when the feedback amplifier is stable, nor do they provide us with any information as to how to stabilize an unstable amplifier. These limitations are overcome by applying the Nyquist criterion to the return difference, which gives precisely the same information about the stability of a feedback amplifier as does the network determinant itself. Furthermore, the return difference can be measured physically, meaning that we can include all the parasitic effects in the stability study. The discussion of this chapter is confined to single-loop feedback amplifiers. Multipleloop feedback amplifiers are presented in the following chapter.

We first introduce the concepts of a single-loop feedback amplifier and its stability, and then review briefly the Routh-Hurwitz criterion. This is followed by a discussion of the Nyquist stability criterion and the Bode plot. The root-locus technique and the notion of root sensitivity are taken up next. The relationship between gain and phase shift are elaborated. Finally, we discuss means of stabilizing a feedback amplifier and present Bode's design theory.

In the preceding three chapters, we studied the theory of single-loop feedback amplifiers. The concept of feedback was introduced in terms of return difference. We found that return difference plays an important role in the study of amplifier stability, its sensitivity to the variations of the parameters, and the determination of its transfer and driving-point impedances. The fact that return difference can be measured experimentally for many practical amplifiers indicates that we can include all the parasitic effects in the stability study, and that stability problem can be reduced to a Nyquist plot.

In this chapter, we study multiple-loop feedback amplifiers, which contain a multiplicity of inputs, outputs, and feedback loops. We first review briefly the rules of the matrix signal-flow graph, and then generalize the concept of return difference for a controlled source to the notion of return difference matrix for a multiplicity of controlled sources. For measurement situations, we introduce the null return difference matrix and discuss its physical significance. In particular, we show that the determinant of the overall transfer matrix can be expressed explicitly in terms of the determinants of the return difference and the null return difference matrices, thus generalizing Blackman's formula for the input impedance. This is followed by the derivations of the generalized feedback formulas and the formulation of the multiple-loop feedback theory in terms of the hybrid matrix. The problem of multiparameter sensitivity together with its relation to the return difference matrix is discussed. Finally, we develop formulas for computing multiparameter sensitivity functions.

In this chapter, we formulate the network equations in the time-domain as a system of first-order differential equations that govern the dynamic behavior of a network. The advantages of representing the network equations in this form are numerous. First of all, such a system has been widely studied in mathematics and its solution, both analytical and numerical, is known and readily available. Secondly, the representation can easily and naturally be extended to time-varying and nonlinear networks. In fact, nearly all time-varying and nonlinear networks are characterized by this approach. Finally, the first-order differential equations are easily programmed for a digital computer or simulated on an analog computer.

We begin the chapter by first presenting procedures for the systematic formulation of network equations in the form of a system of first-order differential equations known as the state equations, and then discuss the number of dynamically independent state variables required in the formulation of these state equations. We shall demonstrate how this number or its upper bounds can be determined from network topology alone. Finally, we express the familiar feedback matrices in terms of the coefficient parameter matrices of the state equations, and discuss their physical significance. In this chapter, we consider only linear lumped networks that may be passive or active, reciprocal or nonreciprocal, and time-invariant or time-varying.

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.

The following sections are included:

• APPENDIX I HERMITIAN FORMS

• APPENDIX II CONVERSION CHART FOR TWO-PORT PARAMETERS

• APPENDIX III OUTLINE OF A DERIVATION OF EQ. (7.224)

• SYMBOL INDEX

• SUBJECT INDEX