The general equations of the electric circuit — III: Variation of constants r, L, C, and g, and its effects

Abstract

In the usual theory of transients the assumption is made, that resistance, inductance, capacity and conductance are constant. This however, is not correct, and as the result thereof, it was not possible to theoretically investigate, and numerically calculate the dissipation of high-frequency disturbances, the flattening of the wave fronts of impulses, the rounding off of steep waves, etc., with the time and the distance of travel, and therefrom to determine the distance, to which the danger from such disturbances extends, and to investigate the conditions of line construction, which limit the danger zone of such phenomena to the smallest local extent. In the following, two of the foremost causes of change of the line constants with the equivalent frequency are investigated, the unequal current distribution in the conductor, and the electric radiation from the conductor, and shown, that within the range of frequencies which may be met in industrial circuits, the effective resistance of the conductor, and its attenuation constant, may increase more than a million fold. Equations of the line constants as function of the equivalent frequency are derived, and applications thereof made to a few problems: (1) The laws of conduction of high-frequency currents, such as produced by lightning discharges and similar disturbances, and the conclusions resulting therefrom on the nature of the conductor. (2) The decay of high-frequency sine waves in transmission lines. (3) The attenuation of rectangular waves. (4) The flattening of the wave front of steep impulses. It is shown that in high-frequency conduction the section of the conductor is of little importance, but the circumference is the determining factor, except at very high frequencies, where size, shape, and material — within certain limits — becomes of secondary importance. The inductance and the capacity of the conductor remain constant up to very high frequencies, but the effective resistance may increase many thousand fold. At high frequencies, not all the energy consumed by the effective resistance of the conductor is converted into heat, but the major part may be radiated into space and thereby affect other circuits. The conditions, which cause a flattening of steep wave fronts and impulses, and a rounding of irregular waves, are investigated, and it is shown, that within the range of distances in which danger may result from a steep wave front or high frequency, the steepness of the wave front decreases with the square root of the distance traveled by the wave; that size and material of the conductor have little influence, but the distance between conductor and return conductor is the main factor in flattening the wave front and thereby limiting the danger zone. For convenience, the theoretical part has been separated and placed in an appendix, giving in the text the discussion, with numerous tables derived from the theoretical equations, and curves illustrating these tables. The investigation is not general, but rather limited to special applications, as the field is too new to permit fargoing generalization.