Search

A knowledge of the impedance characteristics of loaded lines is of considerable importance in telephone engineering, and particularly in the engineering of telephone repeaters. The first half of the present paper deals with the impedance of non-dissipative loaded lines as a function of the frequency and the line constants, by means of description accompanied by equations transformed to the most suitable forms and by graphs of those equations; and it outlines qualitatively the nature of the modifications produced by dissipation. The characteristics are correlated with those of the corresponding smooth line. The somewhat complicated effects produced by the presence of distributed inductance are investigated rather fully. In the absence of distributed inductance a loaded line would have only one transmitting band, extending from zero frequency to the critical frequency. Actually, however, every line — even a cable — has some distributed inductance; and the effect of distributed inductance, besides altering the nominal impedance and the critical frequency, is to introduce into the attenuating range above the critical frequency a series of relatively narrow transmitting bands — here termed the “minor transmitting bands” — spaced at relatively wide intervals. The paper is concerned primarily with the impedance in the first or major transmitting band; but it investigates the minor transmitting bands sufficiently to determine how they depend on the distributed inductance, and to derive general formulas and graphical methods for finding their locations and widths — an investigation involving rather extensive analysis. The latter half of the paper describes various networks devised for simulating and for compensating the impedance of loaded lines; it furnishes design-formulas and supplementary design-methods for all of the networks depicted; and outlines a considerable number of applications pertaining to lines and to repeaters.

The first section of this paper is devoted to the formal mathematical theory of the propagation of periodic currents over a system of parallel wires energized at its physical terminals only. The theory developed is essentially a generalization of the classical theory of transmission over a single wire (with ground return) or over a balanced metallic circuit. The solution here given furnishes the fundamental formulas and a good deal of information regarding what takes place in a system of parallel wires; for actual calculations, however, the method of treatment is not so well adapted as that developed in the remaining sections of the paper. The second section deals analytically with the problem of propagation over a line or a circuit exposed throughout its length to an arbitrary impressed field of force. The resulting solution is immediately applicable to problems of crosstalk and interference, and to the theory of the wave antenna. The last two sections are devoted to the development and application of a more physical or synthetic method of treatment, based on the substitution of ‘equivalent electromotive forces’ for the arbitrary impressed field. This synthetic treatment, which permits of an intuitive or physical grasp of the various problems, has been found quite useful in dealing with crosstalk and interference, and also with the wave antenna, The method is illustrated (in the last section) by application to two representative problems of a diverse nature.

Part I of this paper contributes methods, theorems, formulas and graphs to meet a previously unfilled need in dealing with certain types of two-dimensional probability problems — especially those relating to alternating current transmission systems and networks, in which the variables occur naturally in complex form and thus are two-dimensional. The paper is concerned particularly with “normal” probability functions (distribution functions) in two dimensions, which are analogous to the familiar “ normal” probability functions in one-dimensional probability problems. It supplies a comprehensive set of graphs for the probability that a “normal” complex chance-variable deviates from its mean value by an amount whose magnitude (absolute value) exceeds any stated value; in other words, the probability that the chance-variable lies without any specified circle centered at the mean value in the plane of its “scatter-diagram,” that is, in the complex plane of the chance-variable. It gives a comprehensive treatment of the distribution-parameters of the “normal” complex chance-variable, and convenient formulas for the necessary evaluation of these parameters. For use in various portions of the paper, as well as for various possible outside uses, it supplies a considerable number of formulas and theorems on “mean values” (“expected values”) of complex chance-variables.

This paper deals mainly with various ‘distribution functions’ and ‘cumulative distribution functions’ pertaining to the modulus and to the angle of the ‘normal’ complex variate, for the case where the mean value of this variate is zero. Also, for auxiliary uses chiefly, the distribution function pertaining to the reciprocal of the modulus is included. For all of these various probability functions the paper derives convenient general formulas, and for four of the functions it supplies comprehensive sets of curves; furthur, it gives a table of computed values of the cumulative distribution function for the modulus, serving to verify the values computed by a different method in an earlier paper by the same author.1

THE transmission of alternating currents over any transmission line between specified terminal impedances depends only on the propagation constant and the characteristic impedance of the line (at the particular frequency contemplated). In this sense, then, the properties of transmission lines may be classed broadly as propagation characteristics and impedance characteristics. In telephony we are primarily concerned with the dependence of these characteristics on the frequency, over the telephonic frequency range.