# What is a Network?

With the advent of the highspeed digital computer, digital networks are being designed to do filtering and control tasks previously performed by electrical networks. Instead of using elements such as resistors, inductors, and capacitors, these digital networks are made up of multipliers, summers, and delay elements. Also, the input and output of digital networks are binary-coded sequences of numbers rather than continuous functions of time. Nevertheless, there is a one-to-one correspondence between electrical and digital networks in the sense that the theory applicable to the analysis and design of one type of network can be translated to apply to the other.

Photo by Jade Colley

## Network Theory

Network theory is primarily the study of the behavior and the design of electrical networks formed by the interconnections of electrical components. Since about 1950, however, much of the theory has also been applied to flow problems, such as the flow of materials in transportation networks or the flow of information in communication networks.

Network analysis and network synthesis form the two parts of network theory. In analysis the problem is to determine the voltages and currents in specified elements of an existing network by applying a set of straightforward rules. The network may contain voltage and current sources and elements such as resistors, inductors, capacitors, gyrators, and transformers, each of which is a model of physical components. In the inverse problem of network synthesis, a performance or behavior characteristic is specified, and the problem is to design a network that achieves the desired performance. The performance is often given in terms of a mathematical function like an impedance. The concept of impedance is one of the most important and fundamental concepts of network theory.

## Applications

Network theory has applications throughout electrical engineering. The theory is also applied in analogous fields such as acoustics, optics, and mechanics. Simply put, network theory makes it possible to analyze exceedingly complex mechanical, acoustical, and electrical networks without becoming bogged down in a maze of detail. The networks that distribute electric power to the home are designed by network theory. It plays a major role in the design of controls for the automation of industrial processes. Carrier telephony - the transmission of many messages on the same pair of wires-would be impossible without the frequency-selective networks known as filters. In addition, network concepts such as input-output, "black box," and feedback have yielded rich rewards in such fields as biology and economics.

## Historical Background

Network theory is a blend of physics, mathematics, and engineering, and it has benefited from contributions in each of these fields. The work of Gustav Robert Kirch-hoff on circuits* and combinatorial topology (more commonly called linear graphs) was a contribution of the first magnitude. The characterization of network elements by a volt-ampere relationship is based on the work of Georg S. Ohm, Michael Faraday, and James Clerk Maxwell.

Many network concepts, such as the normal modes of a system and the characterization of electric and magnetic energy by positive definite quadratic forms, are taken from classical dynamics. In fact, the analysis of networks is analogous to the theory of small vibrations in dissipative mechanical systems. Hence the work in dynamics of physicists and mathematicians such as Sir William R. Hamilton and Joseph L. Lagrange helped to develop network theory. In the 20th century fundamental contributions to network analysis were also made by George A. Campbell, a research mathematician at the Bell Telephone Laboratories.

Network synthesis, in contrast to analysis; is a relatively modern development. Its birth date may be given as 1924 with the publication of the celebrated Reactance Theorem by Ronald M. Foster, who was then at Bell Telephone Laboratories. This paper signaled a revolutionary change in network design. An essentially cut-and-try procedure for the design of networks was replaced by an exact synthesis procedure that satisfied a set of necessary and sufficient conditions. Then followed synthesis contributions by many other research workers, including Yosiro Oono in Japan, Wilhelm Cauer and Hans Piloty in Germany, and Edward L. Norton, Ernst A. Guillemin, Sidney Darlington, Aaron Fialkow, Raoul Bott, and Richard J. DufHn in the United States. The similar results of Piloty and Darlington, arrived at independently and published almost simultaneously, made possible the modern design of precision niters, which are of crucial importance in carrier telephony, radar, radio, and television.

The preceding contributions were in the area of the synthesis of passive networks, which roughly are networks that cannot deliver more power than is furnished to them by external sources. Networks that can amplify power are designated as active networks. They include networks containing transistors in feedback configurations, where a portion of the output is fed back to the input to achieve better sensitivity, noise, and other performance characteristics. Fundamental work in this area was done by Hendrik W. Bode and Harry Nyquist at the Bell Telephone Laboratories in the 1920's and 1930's. Nyquist's work dealt with stability criteria for feedback networks. Other contributions to stability theory were based on the work of Adolf Hurwitz in Germany, E. J. Routh in England, and Aleksandr M. Lyapunov in the USSR.