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former costs about 25 per cent more for 60 cycles and about 60 per cent more for 25 cycles than for the high frequency of 125
In the early history of alternating current lighting, the generators were belt-driven and ran at about 1000 r.p.m. Consequently 133 cycles could be obtained with a 16-pole machine. At present large direct-connected alternators running at about 100 r.p.m. are generally installed, and would require 160 poles to give the same frequency. This would make a complicated and expensive construction, so that 60 cycles, requiring 72 poles, would be much more practical. Another objection to high frequency is the fact that inductance or capacity effects are greater. The drop in voltage due to the former and the charging current due to the latter are both directly proportional to the frequency, and the tangent of the angle of lag is also proportional to it. For example, the voltage drop on a No. 0 A.W.G. wire due to its resistance and reactance (p. 116) is only one-half as much at 25 as it is at 125 cycles. Tables showing this difference will be given later in the present chapter. Since the drop is greater it follows that the regulation is poorer on high frequency circuits. The greater wattless currents cause greater heating in generators, lines, transformers, etc. Still another disadvantage of high frequency is the fact that it renders more difficult the parallel operation of generators and rotary converters, as well as the running of motors.
The disadvantages of low frequency, besides the higher cost of transformers already noted, are the difficulties involved in operating arc and incandescent lamps. It is not yet practicable to run the former below 40 cycles and the latter below 25 cycles per second, and even at those values the results are not very satisfactory. Low-voltage or large candle-power incandescent lamps flicker less than the standard 110 volt 16 c.p. lamps, but the practice is determined by the latter. The 220 volt lamp would be still more sensitive to the waves of current, on account of its thinner filament.
In conclusion, it may be said that for lighting alone at moderate distances a frequency of 125 or 133 may be adopted ; but even in such cases it would probably be wiser to choose 60 cycles, in order to permit the operation of motors and the extension of the system to greater distances. For supplying power as well as light 60 cycles are very satisfactory where the circuits are not too long. To transmit energy to great distances, a low frequency, such as 25 or 30 cycles, is suitable. The same is true of long underground submarine cables where the capacity effects would be great. A low frequency of 25 is generally selected also for the simple transmission of energy between generating and distributing stations, where the energy is converted into direct current before it is used, so that the frequency makes no difference so far as the lights are concerned. This question, which is almost always a serious one, is often complicated by the fact that a certain frequency has already been adopted in the original plant, so that there is a great temptation to adhere to it in making additions. In many cases it may be necessary to do so; but frequently it would be wiser and cheaper in the end if the old-fashioned apparatus were sold, even at a great sacrifice, and a new plant designed and installed in accordance with the best practice.
Relative Weights of Copper for Various Systems. — This question is exceedingly important, but belongs more to long-distance power transmission than to electric lighting. Nevertheless, the problem often enters directly or indirectly in electric light engineering, and it will be well to consider briefly the principles involved, and the methods of calculation that are employed.
A comparison between the weights of copper required for the different direct-current systems was given on page 87. In attempting to apply similar reasoning to the alternating current, the difficulty arises that the voltage ordinarily measured is not the maximum value; and since the insulation is subjected to the strain of the latter, the relative figures obtained depend upon which basis of comparison is adopted. For long-distance transmission the highest voltage that is practicable under the circumstances would ordinarily be chosen, but for local distribution the effective pressure would determine the question.
The most important systems of transmission and distribution are represented in Fig. 180, and the relative weights of copper required are given in terms of the common two-wire circuit taken as 100. For equal effective values of E.M.F. the direct and alternating currents demand the same weight of copper ; but if equal maximum values are considered, the latter requires twice as much copper, distances, power in watts, percentage of drop, etc., being equivalent. This is easily understood when we remember that the
maximum value of an alternating E.M.F. is V2 = 1.41 times its effective rating, consequently an alternating E.M.F. of 100 volts would have the same maximum as a direct E.M.F. of 141 volts.
For the same number of watts the amperes of the direct current
100 are 117 as great, so that the drop in volts is in the same proportion with equal resistances. But the percentage of drop is only 100 100 1 141 * 141 = zas
as large for the current, or one-half as much copper
would be required for the same percentage of drop. This is simply a particular case under the general law that weight of copper is inversely proportional to the squares of the voltages, other things being equal.
The single-phase, three-wire (Fig. 150) requires 374 per cent as much copper as the two-wire system, and by making the neutral one-half the cross-section of either of the outside conductors the copper is reduced to 314 per cent. These percentages are in the same proportion as for direct-current systems, the figures for which were explained on pages 72 and 87. The above ratios assume that the lamp voltages are equal in all cases; but it is evident that this will give twice the total voltage for the three-wire circuit, since it practically involves the placing of two lamps in series. For the same total voltage, the three-wire would require 50 per cent more copper than the two-wire system, since the former would have three conductors instead of two, and everything else would be the same.
Two-phase four-wire require the same amount of copper as the ordinary single-phase two-wire circuits, the former being equivalent to two single-phase systems. With the two-phase three-wire system (p. 142) the case is not so simple, but may be determined as follows : Assume the voltage V between either outside wire and the common return wire to be the same as in a single-phase circuit. The total power transmitted is VI for the latter, where I is the current, and to transmit the same power by the two-phase system the power must be 2 Vi, in which i is the current in either outside wire, and is equal to 1 • 2. The current in the common conductor is i V2, consequently to have the same current density, which is the condition of maximum efficiency, its cross-section must be V2 times that of either of the others, and its resistance is r • V2 in which r is the ohmic resistance of one of the outer wires. The loss of power for each of the latter is i’r, and for the middle wire it is 2 i?• V2 = į?r V2, hence the total loss in the three wires is 2 i'r + og V2 = ; or (2 + V2) = 1%, (2 + V2) = 4, since i = 1 - 2 The loss in the equivalent single-phase circuit is 21°R, in which R is the resistance of one of the conductors, and this must be the same as the loss for the two-phase system; hence
8R 2 12R = 12v (2 + V2)
2 + V2
(2 + V2) V2 = 1.457 com
Therefore each outer wire must be (2 + V2) + 8 times, and the middle wire V2 (2 + 12) = 8 times as large as each single-phase conductor. It follows that the weight of copper in the two-phase, three-wire conductors is ? (2 + V2) (2 + V2) V2 were conductors iS e t
8 pared with 2 for the single-phase, two-wire system, or in other words is 72.9 per cent as great at the same minimum voltage. If this comparison is made on the basis of maximum voltage, it is necessary to make the potential difference between the outer conductors equal to that in the single-phase, two-wire circuit, hence the voltage between either outside wire and the common return is V • V2, and the current in each branch i = I = V2, so that the power in both branches is 2 (V = V2)(I V2) = VI, which is the same as that in the single-phase system. The current in the common return is i; V2 = 1, and its resistance should be rı + V2 if r, is the resistance of each outer wire. The total loss is
This must be equal to 212R, the loss in the single-phase, hence 12r. (2 + V2) = 212R, or ri = 2 + 12
2 + V2 That is, each outside wire requires
times, and the middle wire 14+ V2) V2
- times as much copper as each of the single-phase conductors.
Hence the total system demands 2(2 + V2), (2 + V2) V2.
a t - - = 2.914 compared with 2 for the single-phase, or 1.457 times as much copper; that is, the two-phase three-wire system requires 45.7 per cent more copper than the twophase at the same maximum voltage.
Considering a three-phase, three-wire system (p. 145), having a voltage V between the lines measured as A potential, the current in each line, or Y current, is in, and the current from line to line, or A current, is iz + V3. Hence the total power in all three branches is 3 Vi, · V3 = Vi, V3, and if this is to equal VI, the power of the single-phase circuit, then VI = Vi, V3, or i, =; V3.