Introduction

Networks are frequently encountered in electronics, control and communication systems in which an input signal is impressed at one pair of terminals and an output signal is taken from another pair of terminals; such networks are called two-terminal pair networks or, more frequently twoport networks.

The resistance voltage divider, first introduced in section 2.2, is an example of an elementary two-port network, and many other examples have occurred in the intervening chapters. In particular, it was demonstrated in section 7.3.2 that a non-linear device such as the transistor could, for small-signal conditions, be modelled by linear, two-port network. The concepts relating to such networks are therefore of considerable generality, and in this chapter we examine the theory of two-ports in greater detail and introduce further applications.

The theory contained in this chapter is concerned with the functional relationships among the voltage and current variables at the two ports of the network as defined in fig. 8.1. If the variables are steady-state a.c. quantities, then these relationships are formulated in terms of impedances or admittances. These same relationships will, of course, apply if the variables are transformed variables; in this case impedances and admittances are generalized functions of complex frequency as defined in chapter 6. In the case of purely resistive networks the functional relationships among variables are identical in form for instantaneous and d.c. quantities as well as for a.c. quantities. It should also be noted that the same relationships apply for incremental (small-signal) a.c. quantities as defined in section 7.3.1.

Introduction

By making the assumption that all of the elements in a circuit are linear, the analysis is greatly simplified. Although all real circuits are nonlinear to some degree, in most cases a linear treatment gives sufficiently accurate results, and even for circuits containing highly non-linear elements, methods can often be devised for dealing with them on a linear basis. It is for these reasons that the study of linear circuit theory is of paramount importance in electrical engineering science.

The theorems and techniques of linear circuit analysis presented in this chapter, while being of general usefulness and validity, are developed in the context of d.c. circuits. The advantages of this approach are twofold: firstly, the theory can be developed on the simplest possible basis and in terms which will be familiar to most students. Secondly, the study of d.c. circuit theory is of great practical importance in its own right since it arises in many branches of power and electronic systems analysis.

D.C. linear circuits comprise assemblies of linear lumped resistances together with ideal direct voltage and current sources. The theory appertaining to such idealized circuits is concerned with real situations since many types of source found in practice, a battery for example, can be represented to a good approximation by an ideal source in combination with a lumped resistance.

A typical voltage-current characteristic, or load characteristic, for a practical voltage source is shown in fig. 2.1.(b).

Introduction

In chapters 3 and 4, circuits were considered in which the alternating energy sources possess the same frequency but, in general, different voltages and internal impedances, and arbitrary phase relationships. Each voltage source in such circuits may be thought of as being generated by the interaction between a stationary coil of wire and a properly shaped rotating magnetic field, as shown schematically in fig. 5.1 (a). Now, suppose that instead of a single coil (generally referred to as a winding) there are n windings symmetrically disposed on the stator of the machine. If the windings are identical, their impedances are equal and the amplitudes of their induced voltages are equal. The voltages will all be of the same frequency, determined by the angular speed of the rotating magnet, and the phase relations among them will be fixed. The phase difference between the voltages of two successive windings will be 2π/n radians or 360/n degrees. A machine constructed in this fashion is an n-phase generator.

The majority of power systems throughout the world utilize the three-phase generator, shown schematically in fig. 5.1 (b). In fig. 5.2 are shown (a) a phasor diagram for the three-phase generator, and (b) graphs of voltage v. time for the three phases. The three voltages differ in phase by 360/3 = 120°.

The generators in a power system are connected to a series of step-up and step-down transformers that provide voltage levels appropriate for the efficient transmission, distribution and consumption of the power generated.

This book provides an introduction to electrical circuits that will serve as a foundation for courses in electronics, communications and power systems at first degree level. The first three chapters will be found particularly suitable as prerequisite reading for the companion volume in this series; Analogue and digital electronics for engineers by H. Ahmed and P.J. Spreadbury. Engineering and science students not intending to specialise in electrical subjects will find in this book most of the circuit theory required for a first degree.

The level of presentation presupposes that students will have encountered the basic ideas of electromagnetism and electrical circuits, including the laws of Faraday, Ohm and Kirchhoff. These ideas are reviewed in chapter 1. Mathematical skills are assumed to extend to the solution of firstorder differential equations, and to the elements of complex algebra. Courses in mathematics taken concurrently with those in electrical subjects during the earlier part of a degree course would be expected to fill in progressively the additional mathematical background required; the subject matter has been arranged with this in mind. Sections which may give rise to mathematical difficulties on a first reading, or which may be too specialised for the general student's requirements, are indicated by an obelus (†).

A traditional approach to the development of electrical circuit theory is adopted: the concept of linearity, and the circuit theorems and analytical techniques which stem from this concept, are all presented in chapter 2 within the context of d.c. circuits.

Introduction

For the sinusoidal steady state, one can calculate the total power supplied to a circuit consisting of linear elements by adding directly the power absorbed by each individual resistive element in the circuit. However, it is often more convenient to express power in terms of the voltage across and the current supplied to the input terminals of a circuit whose detailed configuration is unknown or is of no interest.

In all electric power distribution networks voltage and frequency are maintained substantially constant. A given load will draw a current whose amplitude and phase (relative to the power line voltage) depend upon the load impedance. On the other hand in electronic and telecommunication networks, signal power rather than voltage is fixed and we are concerned more with arranging source and load conditions to achieve maximum power transfer from one part of a circuit to another.

For the above reasons the treatment of power in electrical circuits depends to a marked extent on the type of circuit under consideration. In this chapter we develop general methods for determining the power and total energy supplied to, or dissipated within, a circuit. We also consider one of the most important components involved in the utilization and transmission of power; namely, the transformer.

Average power

Consider a network or load as shown in fig. 4.1, supplied at voltage V(r.m.s. magnitude V) and drawing current I (r.m.s. magnitude I). If the network contains reactive elements, voltage and current will differ in phase by some angle φ.