Principles of Electrical Transmission Lines in Power and Communication: The Commonwealth and International Library: Applied Electricity and Electronics Division

By J. H. Gridley and P. Hammond

Principles of Electrical Transmission Lines in Power and Communication is a preliminary study in the transmission of electricity, which particularly discusses principles common to all electrical transmission links, whether their functions be communication or bulk power transfer. This book explains the propagation on loss-free lines I and II and introduces the finite loss-free lines. The sinusoidal excitation of dissipative lines I and II is then examined, and the occurrence of standing waves and quarter-wave is then discussed. This text also looks into topics on frequencies. This book will be invaluable to students and experts in the field of electronics and related disciplines.

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 28, 2014
• ISBN: 9781483186030

Book preview

problems.

CHAPTER 1

Propagation on Loss-free Lines I (The Infinite Loss-free Line)

Publisher Summary

The use of electricity in communication or power requires that as much as possible of the electrical energy generated in a source is transferred to a load. The physical appearances of systems to affect this transfer differ according to the nature of the undertaking; there is a little in common between the high-voltage lines of a major power system and the coaxial cable between aerial and television set. Certain principles underlie the operation of both systems and such principles are to be established. The connection of a source to a load forms a complete circuit. This circuit affects the desired energy transference, but might also dissipate an appreciable amount of energy as heat and store energy in magnetic and electric fields. The circuit must be regarded as having resistance, inductance, and capacitance. It might be that by far the greater part of the energy dissipation or storage is localized, in which case the appropriate parameter can be regarded as concentrated and the familiar concept of a network of lumped parameters follows. This chapter discusses the problems that arise when a finite number of lumped elements no longer suffice to describe a process of electrical energy transference.

1.1 The General Transmission-line Problem

The use of electricity in communication or power requires that as much as possible of the electrical energy generated in a source be transferred to a load. The physical appearances of systems to effect this transfer differ greatly according to the nature of the undertaking; there is at first sight little in common between the high-voltage lines of a major power system and the coaxial cable between aerial and television set. But certain principles underlie the operation of both systems and, in what follows, such principles are to be established.

The connection of a source to a load forms a complete circuit. This circuit effects the desired energy transference, but may also dissipate an appreciable amount of energy as heat and store energy in magnetic and electric fields. The circuit must then be regarded as having resistance, inductance and capacitance. It may be that by far the greater part of the energy dissipation or storage is localized, in which case the appropriate parameter can be regarded as concentrated and the familiar concept of a network of lumped parameters follows. The elementary principles on which the analysis of such networks is based will be assumed; we shall be concerned with problems which arise when a finite number of lumped elements no longer suffices to describe a process of electrical energy transference, and now consider how such a situation can arise.

Suppose a source and load are connected by conducting wires between appropriate terminals. Whatever degree of accuracy is required in analysis it will always be possible to find a distance between source and load such that for this and shorter distances the effect of the connecting wires on the current flow can be neglected. The precise distance will depend on the degree of accuracy demanded, the nature of the source e.m.f.—e.g. whether constant or alternating—the physical nature and disposition of the conductors and the quality of the insulation. Now suppose the distance between source and load to be increased beyond this length. In many practical instances the first noticeable effects are those due to the insertion between source and load of the resistance of the connecting wires, and to the increase in circuit inductance by the increased magnetic flux within the loop formed by source, connecting wires and load. So far as a calculation of the electrical conditions at source and load is concerned the increased resistance and inductance can be assumed concentrated at a point in the connecting wires, and the desired accuracy regained. But as the distance between source and load is increased still further, a more complicated situation arises. The source and load currents will differ by reason of current flow through the conductance and capacitance between conducting wires. Again, the length at which this effect becomes important will depend on the nature of the wires and their disposition, the insulation and the rate at which the source p.d. changes with time. Clearly, we will at some stage be forced to consider a system in which both potential and current at any instant vary appreciably with distance along the connections between source and load. It is at this stage that the connections form a transmission line in the sense that we shall use the term.

A generalized theory of transmission should result in the construction of expressions describing the distribution of potential and current over a transmission line and its terminals, the expressions being applicable whatever the characteristics of the line and the nature of its excitation. Such expressions would, however, by their generality, be so complex as to frustrate any desire to gain understanding which inspired their derivation. Having posed the general problem we shall simplify it as much as possible at first, and consider later the further complications which are important in practice.

Firstly, we shall throughout the study restrict consideration to uniform lines, i.e. lines for which any section is identical with any other section of equal length. If a uniform distribution of potential difference were applied between conductors of such a transmission line, open-circuited at both ends, the capacitance so measurable would tend to increase directly with line length. This is because as the line becomes longer the electrostatic field tends towards axial uniformity, the effect of the ends of the line becoming progressively less important. The line may be regarded as having a certain capacitance per unit length and, by an extension of the same argument, a certain inductance per unit length. For our first discussion we shall also assume the line to be free from power loss, i.e. to have zero resistance along the conductors and zero conductance between them.

This far we have simplified the problem by idealizing the line. Now we shall simplify the system as a whole. Clearly, the electrical conditions on the line depend both on the characteristics of the line and the constraints imposed by source and load. Now if it be accepted that energy transference cannot be instantaneous it follows that changes in load impedance, for instance, cannot affect the source current or p.d. until a finite time has elapsed. In fact, if the load be assumed infinitely distant, its characteristics need not be considered at all in any analysis of the electrical conditions at finite times from the instant of source application.

It is natural to consider next what simplification can be made in the source characteristics. To minimize assumptions we shall characterize the source only by the p.d. which it applies between line conductors, that is, regard the source as of zero impedance.

To crystallize the discussion then, we imagine a source of p.d. to be applied at one end of an infinite loss-free line, and it is our purpose to discover what response the line has to the application of this p.d., that is what potential and current distributions exist along the line and what relationship there may be between the distributions.

1.2 The Infinite Loss-free Line: Solution of Intrinsic Equations

Suppose that at some instant potential and current are distributed along a loss-free line of infinite extent, of inductance per unit length L and capacitance per unit length C. The distributions will be functions both of distance and time since, in general, both current and p.d. will vary with time at any point and with distance from the source at any instant. Using x as distance from the point at which the driving p.d. is applied, and t the time elapsed since the beginning of the application, we may write for the p.d. and current distributions

Now consider two points on the line at distances x and x + Δx. The currents at these points will differ by the capacitance current flowing between conductors over the section Δx. Over any incremental section of length δx , the partial differentiation signifying that it is the variation of p.d. with time at any point that specifies the capacitance current at that point. Then the total capacitance current over the section Δx is

Now if Δx is made to tend to zero, we have in the limit

and therefore

or

(1.1)

Similarly, the excess of p.d. between conductors at x over that between them at x + Δx is given by the inductive drop over the section Δx, i.e.

and by an argument parallel with that used for the current

(1.2)

Equations (1.1) and (1.2) are the intrinsic equations of the line. Whatever time variation the applied p.d. may have, these differential equations must be satisfied by reason of the physical laws governing the transmission. Since two equations connecting v and i have been derived, elimination of either variable is possible. Differentiating (1.1) with respect to time and (1.2) with respect to distance gives

(1.3)

and a similar elimination yields

(1.4)

Now eqns. (1.3) and (1.4) are partial differential equations and, in general, such equations are very difficult to solve. But from a practical viewpoint there is a saving feature; if a supposed solution be obtained by any method whatever, mathematically elegant or not, the validity of the solution can be checked by substitution in the original equation. As we are not concerned with refinements in the mathematics we shall seek a solution by the simple expedient of regarding ∂/∂t as an operator and presuming it to obey the normal rules of algebra. Writing p as ∂/∂t, for compactness, we obtain from the voltage equation (1.3)

(1.5)

Now this is a well-known equation with the solution

For convenience we shall write

and stress the fact that v is a function of distance and time by now writing it in full as v = V(x, t). Then the solution to (1.5) becomes

Now if V(x, t), the p.d. between conductors of the transmission line, is to vanish as x tends to infinity, we must put B = 0. Further, by putting x = 0 we obtain

the function V(0, t) being the time variation of the p.d. applied at the origin. Then the solution to eqn. (1.5) satisfying the constraints we impose becomes

or, expanding the exponential function,

The right-hand side should be recognized as a Taylor series for the value of a function at a point in terms of its value at a neighbouring point,† giving

(1.6)

A parallel argument shows the current solution to be

(1.7)

We now consider how these solutions are to be interpreted.

1.3 Travelling Waves

Neither eqn. (1.6) nor eqn. (1.7) gives the potential and current as explicit functions of place or time; this they could hardly do, since we have not stated explicitly the time variation of the line excitation. But in this very generality lies the great importance of the solutions, for they predict something about the response of the transmission line to any arbitrary excitation. Together, the solutions show that whatever may be the values of p.d. and current at distance x and time t, these values are identical with these imposed at the origin at time t–x/c. Any p.d. or current appearing at the origin at a certain time appears at distance x from the origin after a time delay x/c. Our solutions show that potential and current travel from the point of excitation with velocity c = 1/√(LC), and that the time variation of potential and current at any point is identical with that at the origin, apart from the appropriate