If the value of l, 200,000, of amperes, 100, and of volts, 2,000, for the transmission above considered are substituted in the formula for total weight, just found, the result is pounds = (3.03 (200,000)2 × 100 × 11) ÷ (1,000,000 × 2,000), which reduced to pounds = 66,660, the weight of copper wire necessary for the transmission with either continuous, single-phase or two-phase current. With three-phase current the weight of copper in the line for this transmission will be 75 per cent of the 66,660 pounds just found. One or more two-wire circuits may be employed for the continuous current or for the single-phase transmission, and if one such circuit is used the weight for each of the two wires is obviously 33,660 pounds. For a two-phase transmission two or more circuits of two wires each will be used, and in the case of two circuits, if all four of the wires are of equal cross section, as would usually be the case, the total weight of each is 16,830 pounds. If the transmission is made with one three-phase circuit, the weight of each of the three wires is 16,830 pounds, and their combined weight, 50,490 pounds of copper. In each of these transmission lines the length of a single conductor in one direction is 100,000 feet, or one-half of the length of the wires in a single two-wire circuit. For the two-wire line the calculated weight of each conductor amounts to 66,660 ÷ 200 = 333.3 pounds per 1,000 feet. For a two-phase four-wire line and also for a three-phase three-wire line, the weight of each conductor is 16,830 ÷ 100 = 168.3 pounds per 1,000 feet. On inspection of a table of weights for bare copper wires it may be seen that a No. 1-0 B. & S. gauge wire has a weight of 320 pounds per 1,000 feet, and being much the nearest size to the calculated weight of 333 pounds should be selected for the two-wire circuit. It may also be seen that a No. 3 wire, with a weight of 159 pounds per 1,000 feet, is the size that comes nearest to the calculated weight of 168 pounds, and should therefore be employed in the three-wire and the four-wire circuits, for two- and three-phase transmissions. Either a continuous-current, single-phase, two-phase, or three-phase transmission line may of course be split up into as many circuits as desired, and these circuits may or may not be designed to carry equal portions of the entire power. In either case the combined weights of the several circuits should equal those above found, the conditions as to power, loss, and length of line remaining constant. It will be noted that the formulæ for the calculation of the circular mils and for the weight of the conductors in the transmission line lead to the selection of the same sizes of wires, as they obviously should do.
Several laws governing the relations of volts lost, length and weight of line conductors, may be readily deduced from the above formulæ. Evidently the circular mils and weight of line conductors vary inversely with the number of volts lost in them when carrying a given current, so that doubling this number of volts reduces the circular mils and weight of conductors by one-half. If the length of the line changes, the circular mils of the required conductors change directly with it, but the weight of these conductors varies as the square of their length. Thus, if the length of the line conductors is doubled, the cross-section in circular mils of each conductor is also doubled, and each conductor is therefore four times as heavy as before for the same current and loss in volts. Should the length of the conductors and also the number of volts lost in them be varied at the same rate, the circular mils of each conductor remain constant, and its weight increases directly with the distance of transmission. Thus, with the same size of line wire, both the number of volts lost and the total weight are twice as great for a 100- as for a fifty-mile transmission. If the total weight of conductors is to be held constant, then the number of volts lost therein must vary as the square of their length, and their circular mils must vary inversely as the length. So that if the length of a transmission line is doubled, the circular mils for conductors of constant weight are divided by two, and the volts lost are four times as great as before. Each of these rules assumes that the watts and percentage of loss in the line are constant.
The above principles and formulæ apply to the design of transmission lines for either continuous or alternating currents, but where the alternating current is employed certain additional factors should be considered. One of these factors is inductance, by which is meant the counter-electromotive force that is always present and opposed to the regular voltage in an alternating current circuit. One effect of inductance is to cut down the voltage at that end of the line where the power is delivered to a sub-station, just as is also done by the ohmic resistance of the line conductors. Between the loss of voltage due to line resistance and the loss due to inductance there is the very important difference that the former represents an actual conversion of electrical energy into heat, while the latter is simply the loss of pressure without any material decrease in the amount of energy. While the loss of energy in a transmission line depends directly on its resistance, the loss of pressure due to inductance depends on the diameter of conductors without regard to their resistance, on the length of the circuit, the distance between the conductors, and on the frequency or number of cycles per second through which the current passes. As a result of these facts, it is not desirable or even practicable to use inductance as a factor in the calculation of the resistance or weight of a transmission line. On transmission lines, as ordinarily constructed, the loss of voltage due to inductance generally ranges between 25 and 100 per cent of the number of volts lost at full load because of the resistance of the conductors. This loss through inductance may be lowered by reducing the diameter of individual wires, though the resistance of all the circuits concerned in the transmission remains the same, by bringing the wires nearer together and by adopting smaller frequencies. In practice the volts lost through inductance are compensated for by operating generators or transformers in the power-plant at a voltage that insures the delivery of energy in the receiving-station at the required pressure. Thus, in a certain case, it may be desirable to transmit energy with a maximum loss of ten per cent in the line at full load, due to the resistance of the conductors, when the effective voltage at the generator end of the line is 10,000, so that the pressure at the receiving-station will be 9,000 volts. If it appears that the loss of pressure due to inductance on this line will be 1,000 volts, then the generators should be operated at 11,000 volts, which will provide for the loss of 1,000 volts by inductance, leave an effective voltage of 10,000 on the line, and allow the delivery of energy at the sub-station with a pressure of 9,000 volts, when there is a ten-percent loss of power due to the line resistance.
Inductance not only sets up a counter-electromotive force in the line, which reduces the voltage delivered to it by generators or transformers, but also causes a larger current to flow in the line than is indicated by the division of the number of watts delivered to it by the virtual voltage of delivery. The amount of current increase depends on both the inductance of the line itself and also on the character of its connected apparatus. In a system with a mixed load of lamps and motors there is quite certain to be some inductance, but it is very hard to predetermine its exact amount. Experience with such systems shows, however, that the increase of line current due to inductance is often not above five and usually less than ten per cent of the current that would flow if there were no inductance. To provide for the flow of this additional current, due to inductance, without an increase of the loss in volts because of ohmic resistance, the cross section of the line conductors must be enlarged by a percentage equal to that of the additional current. This means that in an ordinary case of a transmission with either single, two, or three-phase alternating current, the circular mils of each line wire, as computed with the formulæ above given, should be increased by five to ten per cent. Such increase in the cross section of wires of course carries with it a like rise in the total weight of the conductors for the transmission. If wire of the cross section computed with the formulæ is employed for the alternating current transmission, inductance in an ordinary case will raise the assumed line loss of power by five to ten per cent of what it would be if no inductance existed. Thus, with conductors calculated by the formulæ for a power loss of ten per cent at full load, inductance in an ordinary case would raise this loss to somewhere between 10.5 and eleven per cent. As a rule it may therefore be said that inductance will seldom increase the weight of line conductors, or the loss of power therein, by more than ten per cent.
When an alternating current flows along a conductor its density is not uniform in all parts of each cross section, but the current density is least at the centre of the conductor and increases toward the outside surface. This unequal distribution of the alternating current over each cross section of a conductor through which it is passing increases with the diameter or thickness of the conductor and with the frequency of the alternating current. By reason of this action the ohmic resistance of any conductor is somewhat greater for an alternating than for a continuous current, because the full cross section of the conductor cannot be utilized with the former current. Fortunately, the practical importance of this unequal distribution of alternating current over each cross section of its conductor is usually slight, so far as the sizes of wires for transmission lines are concerned, because the usual frequencies of current and diameters of conductors concerned are not great enough to give the effect mentioned a large numerical value. Thus, sixty cycles per second is the highest frequency commonly employed for the current on transmission lines. With a 4-0 wire, and the current frequency named, the increase in the ohmic resistance for alternating over that for continuous current does not reach one-half of one per cent.
Having calculated the circular mils of weight of a transmission line by the foregoing formulæ, it appears that the only material increase of this weight required by the use of alternating current is that due to inductance. This increase cannot be calculated exactly beforehand because of the uncertain elements in future loads, but experience shows that it is seldom more than ten per cent of the calculated size or weight of conductors.
CHAPTER XVII.
SELECTION OF TRANSMISSION CIRCUITS.
Maximum power, voltage, loss, and weight of conductors having been fixed for a transmission line, the number of circuits that shall make up the line, and the relations of these circuits to each other, remain to be determined.