15° C. A correction formula, for a greater degree of accuracy, is, however, given in Appendix I. Tables and curves I to VI inclusive are of general application, dealing with wires from 1 mm. up to 1 cm. in diameter, and also all interaxial distances up to 250 cms., for the successive frequencies of 40, 60, 80, 100, 120 and 140 ~. Tables and curves VII and VIII apply more particularly to American practice, dealing with all sizes of A. W. G. wires from No. 000 to No. 11 inclusive, and all interaxial distances up to 100 inches for the two frequencies 120 and 140~. Factors for intermediate sizes or frequencies can readily be found by interpolation. For instance, suppose that the impedance factor is required for a pair of No. 6 A. w. G. wires suspended at an interaxial distance of 40 inches at 15° C, and at a simply harmonic frequency of 130 ~. Curves VII and VIII give ~ No. 6 at 140 Taking the mean value for the midway frequency, we have No. 6 at 1301.30. Direct calculation yields in fact 1.300 as the factor at 130 ~. The evidence upon which these tables and curves are based is not merely theoretical. Observations made upon voltage drops on overhead wires in actual alternating current systems have corroborated the results. A particular set of observations is also added in Appendix I, together with a simple development of the formula from which the tables have been computed to the fourth significant digit. The same tables and curves will of course apply to secondary circuits or interior wiring, and even if the pairs of wires are twisted together into double cords, without serious error, the only exception being to wires laid in, or close to iron pipes. An Single copper wires suspended overhead at a mean elevation h, with ground return, have an impedance as though that ground return were replaced by an insulated copper wire of equal size vertically beneath, and at a distance h below the surface. elevation h, thus represents a loop interaxial distance of 2h, and the imaginary wire below ground is the "image" of the suspended wire, so called from the optical analogy with respect to the surface midway between. Heaviside has pointed out' that this result is based upon the assumption that the return current 1. "Reprint of Electrical Papers," vol. i. p. 101. through the ground spreads uniformly through a thin film over the surface, and does not penetrate the soil. As a fact, the current cannot be confined to this 'superficial layer, and the real equivalent loop must be considerably wider than the hypothetical image requires. However, as the real limits can scarcely be assigned, the image impedance factors may be justly regarded as minima, while the increase in the factor is usually very slow with moderate elevations, even for large changes in the interaxial dis tance, so that the error is not so great as might be deemed on first apprehension. The minimum impedance factor for a wire say 20 feet above ground, is therefore, its loop impedance factor at 40 feet between axes. The curves indeed could not conveniently be extended, at their existing scale of construction, to include overhead wires, but the tables have probably sufficient range for practical purposes in this respect. We may next examine how far these tabulated impedance factors are liable to be affected by waiving the restrictions hitherto maintained concerning the type of current waves, and the nonferric nature of the inductances. Impedance factors for conductors remote from iron, while independent of the current strength, depend immediately upon the current wave character. In practice, alternating current circuits contain motors or transformers, in which iron plays a prominent part; and owing to the hysteresis in this iron, the current waves deviate from simple sinusoids, particularly at light loads, even when the E. M. F. supplied by the generator is simply harmonic. Of all possible current wave types, however, the simply harmonic or sinusoidal has the lowest impedance factor, and the entries in the tables are consequently minimum values; but there appears to be no limit to the possible augmentation of these factors under appropriate conditions, as an example will indicate. Let the current waves be of the saw-tooth type [Fig. 7], the rising and falling lines intersecting the zero line z o at equal angles but in opposite directions. Here the time occupied in ascent from, or descent to, the zero line, is just one quarter period. The impedance factor for such currents is 48 n2 12 where n is the frequency and the ratio of inductance to resis TABLE VII. Impedance factors for pairs of parallel copper wires carrying simple harmonic currents at 120 and 140~ 15° C. No. 10, A. W. Gauge. No. 9, A. W. Gauge. No. 8, A. W. Gauge. No. 7, A. W. Gauge. No. 6. A. W. Gauge. No. 5, A. W. Gauge. 1.134 1.008 1.011 1.274 1.013 1.018 1.431 1.021 1.028 1.928 1.037 1.606 1.033 1.045 1.804 1.052 1.070 2.025 1.082 1.109 2.274 1.127 1.169 I 050 2.164 1.057 1.058 1.077 2.430 1.000 1.120 2.728 1.140 1.186 3.241 1.104 1.139 3.638 1.061 1.214 1.070 4.329 1.080 5.772 1.108 4 861 1.125 1 167 5.457 1.193 I 255 6.482 1.141 1.188 7.276 1.217 1.286 1.025 1.033 15.29 1.039 1.052 17.17 1.060 1.081 11.44 1.053 1.071 25.7 1.064 1.086 7.214 1.099 1.133 1.116 10.82 8. 101 1.154 1.205 9.095 1.236 1.311 1.155 12.15 1.179 1.237 13.64 1.273 1.358 1.128 1.171 16.21 1.197 1.261 18.19 1.300 1.393 1.146 1.195 24.30 1.225 1.296 27.28 1.340 1.443 1.160 1.212 32.41 1.245 1.322 36.38 1.369 1.480 tance in any length of conductor, or in other words its time. constant. The tabulated factors for simple harmonic currents we know to be so that the saw-tooth factors are the greater. If we file down the points of the teeth as shown in Fig. 8, the current will rise from zero to a maximum, along the straight line o A in less than period, and the factor diminishes, until this time of ascent is reduced to of a period [indicated in Fig. 8], when the factor has its minimum value for this character of wave, viz.: After this, as we shorten the period of ascent, the factor rises. Neither wave amplitude nor strength of current can of course affect this consideration. When asc. and des. each occ. period, factor = and finally, when the ascending time is reduced to zero, so that the rise is absolutely perpendicular, as shown in Fig. 9, the factor becomes indefinitely great, and the drop on such a wire would no longer be finite with a finite current, which is equivalent to the statement that it is practically impossible to create currents of this strictly rectangular type. Since theory assigns no limits, we are compelled to accept the terms that existing practice may impose. Fig. 10 shows a wave form taken from Professor Ryan's paper on Transformers1 read before this Institute in January, 1890. It was the wave of primary current supplied to an idle transformer by an alternator 1. TRANSACTIONS, vol. vii, p. 1. |