To reduce the capacity of overhead wires, the distance between them and from the ground should be increased. But even in this case the reduction is small compared with the increase in distance. Assume a horizontal wire inch in diameter, and one mile long, to be strung 30 feet above the earth, and another wire of the same size and length to be strung 60 feet above the earth. From (64) the capacities in the two cases will be respectively: 38.83 × 10- 9 log 7200 x = 10.1 × 10-9 farad = .0101 microfarad, and 38.83 × 10 log 14400 9 9 9.3 × 10 farad = .0093 microfarad. The difference between the two values is only about 8 per cent, although one wire is twice as high as the other. The capacity with respect to each other of two parallel overhead wires 3 feet apart, each being inch in diameter and one mile long, is found from (67) to be 19.42 × 10-9 log 360 = 7.6 × 10-9 farad = .0076 microfarad. Increasing the distance between the wires to 6 feet, the capacity becomes 19.42 × 10 log 720 9 = 6.8 x 10' farad = .0068 microfarad. In this case the capacity is reduced 10 per cent by doubling the distance between the wires. From these examples it is evident that this way to diminish capacity is hardly economical where the cost of construction is greatly affected by the height and distance apart of wires, as is the case in pole lines. The method of balancing the reactance of capacity and inductance, already set forth on page 133, can be applied to reducing the effect of capacity in electrical circuits. CHAPTER VIII. PRINCIPLES OF ALTERNATING POLYPHASE CURRENTS. THE advantages of two- and three-phase, or other polyphase systems, apply solely to the operation of motors. In fact, such currents are positively disadvantageous for supplying arc or incandescent lamps. Consequently this subject comes under the head of electric power rather than electric lighting. However, electric lamps are often used upon the same circuits with polyphase motors, and in many cases energy is transmitted over long distances by polyphase currents, to be converted into direct currents for local distribution to lamps; so it is necessary in the present volume to consider the principles of polyphase systems, and the methods of operating lamps upon them. A two-phase current may be regarded as, and in most cases actually consists of, two distinct single-phase currents, flowing in That is, they differ them, their only relation being that of time. in phase. This condition is shown in Fig. 110, in which the curve ABCD represents one alternating current, and EFGHI represents another, the difference in phase being 90°, the maximum value G of the second occurring 90° behind the maximum point A of the first, and so on for other corresponding points. If there is no lag of either current, the same curves can be taken to represent the two E.M.F.s, and with the same lag for both currents they would still be 90° apart in phase. If the lags were not equal, then the phase relation would be altered correspondingly. The two E.M.F.s or currents might have different maximum values, or different wave forms, but in practice they are usually made as nearly alike as possible. It is evident also that the difference in phase might be made anything between 0° and 360°; but it is almost always designed to be 90°, or one-quarter of a period, and for that reason is often called a quarter-phase current. Two-phase currents may be generated by two separate alternators, but in order to preserve the phase relation it would be necessary to have their shafts coupled or posi tively connected together. In practice, a two-phase current is usually generated by two separate windings upon one armature, the machine having the same general form as a single-phase alternator. The two circuits may be kept entirely separate, as in Fig. 111, lamps L being connected to each, in which case four wires are required. In order to save one wire it is possible to use a common R return conductor for both circuits, as in Fig. 112, the dotted portion of one wire, D, being eliminated by connecting across to Cat M and N. For long lines this is economical, but the interconnection of the circuits increases the chance of trouble from grounds or short circuits. It is also a fact that the current in the conductor C will be the resultant of the two currents, differing by 90° in phase. From the principle P shown in Fig. 93, the value of this resultant is found in Fig. 113 to be OR = √2 OP = 1.41 x OP the two-phase currents being represented by the components OP and OQ at right o angles to each other. Consequently the resultant current in C is 1.41 times that flowing in either B or E in Fig. 112 and the cross-section of the wire C should be 41 per cent greater. Q Resultant of Two-Phase Currents. Fig. 113. A three-phase current consists of three alternating currents, differing in phase, as indicated in Fig. 114. One current is represented by the curve JKL, another by the curve MNO, and the third by the curve PQR, the maxima points J, M, and Q (or other corresponding points) being 120° apart in the ideal case, and ap proximately so in practice. These three currents might be carried in three entirely separate circuits requiring six wires, being analogous to the two-phase, four-wire system in Fig. 111; or one common return conductor may be used, thereby saving two wires, and reducing the total number to four, as shown in Fig. 115. The armature windings and their phase relation are represented dia grammatically by the coils MA, MB, and MC, the three main conductors by AE, BG, and CF, the common conductor being indicated by the dotted line MN. The lamps L, L, L, are connected across between the common point N and the three main conductors. If the three circuits are balanced (i. e., have equal currents) the common conductor MN will carry no current, and may be dispensed with. This is a most interesting and important feature of Fig. 115. Three-Phase Circuits with Y Connection. the three-phase system. The simplest way to understand it is to consider that each wire acts as the return conductor for the other two. In other words, the algebraic sum of the three currents meeting at the common point N is equal to zero; consequently Kirchhoff's law is fulfilled. 60 GO This fact is shown in Fig. 114, the algebraic sum of the ordinates of the three curves being equal to zero at any point. For example, at SR the ordinate of curve MNO is zero, and the ordinates of the other two are equal in value, but opposite in sign. At TK the sum of the two positive ordinates of the curves MN and PQ are equal to the negative ordinate of the other curve JKL, because XT = sin 30° and XK = sin 90° = 1, and so on for other points. The same principle is proved in Fig. 116, in which a balanced three-phase current is represented by three equal vectors at 120° with respect to each other. Two of these currents, OT and OU, are equivalent to their resultant OR, which is equal and opposite to the third current OS; consequently the resultant of all three currents is zero. In the operation of motors the three currents are usually equal, all three wires being connected to each machine, so that the fourth R Fig. 116. Resultant Current, = |