Active Network Analysis

The following sections are included:

• PREFACE TO FIRST EDITION
• PREFACE TO SECOND EDITION
• 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…

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 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 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…

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 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…

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 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…

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 manyworkers [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…

In Chapters 4 and 5, we showed that Bode's concept of return difference plays an important role in the design of feedback amplifiers. Among the many important properties, the return difference is a generalization of the concept of the feedback factors of ideal feedback model, the sensitivity function of the amplifier is closely related to the return difference, and the return difference is basic to the study of the stability of the feedback system and to the determination of its transmission and driving-point properties…

The indefinite-admittancematrix was introduced in Chapter 2, and its applications to the feedback amplifier theory were taken up in Chapters 4 and 5. We indicated that Bode's concept of return difference plays an important role in the design of feedback amplifiers. Among themany important properties, the return difference is a generalization of the concept of the feedback factors of ideal feedback model, the sensitivity function of the amplifier is closely related to the return difference, and the return difference is basic to the study of the stability of the feedback system and to the determination of its transmission and driving-point properties. We showed that the return difference and the null return difference can elegantly and compactly expressed in terms of the first- and second-order cofactors of the elements of the indefinite-admittancematrix. However, the dual concept, the indefinite-impedance matrix is rarely considered. Perhaps this is due to the fact that measuring the branch voltage is easier than measuring the branch current. However, advances in integrated op-amp circuits have made it possible to measure the branch current on line without opening any branch…

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