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When the lamps are not equally distributed, the problem is much more difficult to solve. In the apparently simple case of a single lamp connected at W, a large portion of its current will flow directly from s to W, but a considerable fraction of it will take the path Ja Y W, and also the path J:ZW. Since there is a flow of current from s to A, the latter must be of lower potential than the former; hence a small amount of current will take the course J j AY; and if the network were extended beyond A, some current would follow still more indirect routes. Current would also pass through the remainder of the network shown in Fig. 75, as well as through the portion represented in Fig. 76 ; in fact, a single lamp connected at any point of a network would cause current to flow in every section except a few that might happen to have no potential difference between their ends. With a A number of lamps irregularly located, the conditions become even more complex.
It might be supposed that a solution of the problem Y could be obtained by comparing the resistances of the various paths. For example,
Fig. 76. Portion of Network Represented
in Fig. 75. the course Ja Y W has three times the resistance of J W, hence the current in the former should be one-third as much as in the latter. But this is not true, because J j A a is in parallel with J a. If we attempt to allow for this by calculating the joint resistance, the case is further complicated by the fact that the section J j carries current that flows via X as well as through A, and so on. A correct method would consist in applying Kirchhoff's Laws, which are as follows : 1. The algebraic sum of the currents in all the conductors that meet
at any point is zero. 2. The algebraic sum of all the products of the currents and resis
tances in conductors forming a closed loop equals the algebraic
sum of all the E.M.F.s in the loop. In the networks under consideration there is usually no E.M.F. working within each loop, as, for example, the loop J a Y W; therefore we may simplify the second law as follows:
The algebraic sum of all the products of the currents and resistances
in conductors forming a closed loop is sero.
The application of these principles to simple cases has already been given on pages 35 and 49; but it would be difficult to apply them to the extensive and complicated networks used in practice, particularly when the lamps are unequally distributed. Nevertheless, methods for making such calculations have been given by Herzog and Stark, * Herzog, † Coltri, $ Muellendorff, ş and others.
Electrical Model of Network. — In the pioneer work of Edison in 1882, the designing of the underground network of conductors was aided by constructing models in which the conductors were represented in miniature by copper wires. If the model is correct in scale, it is possible to determine from it the distribution of current and drop with various amounts and positions of load. This plan can be followed in any case, but would ordinarily be considered too much trouble, although it might often effect a considerable saving in, or better arrangement of, the copper.
Mechanical Model of Network. – Another method of solving this problem was devised by H. Helberger || of Munich, and consists in employing a mechanical model in which the conductors are represented in length and position by horizontal strings stretched with a certain force corresponding to the cross-section of the conductor, the load being applied in the form of weights that are hung upon the strings, and are proportional to the current consumed at the various points. The amount that the strings are depressed indicates the drop in voltage, being usually limited to a certain value in a given case. The points at which the strings are supported correspond to the feeding-points, being raised or lowered with respect to each other a distance proportional to the difference in the electrical pressures with which they are fed.
Actual Design of Network. — These methods for determining the size of the mains in a network are not much used in America, although they are applied quite generally in Germany. Experience in this country has shown that it is sufficient to employ a few standard sizes of mains. In New York City, for example, each of the three-wire mains has a cross-section of 350,000 circular mils in the central and heavily loaded portions of the network, and a crosssection of 200,000 circular mils in the outlying or less heavily loaded districts, and in some cases conductors of 150,000 circular mils are large enough. It is not found necessary to specially determine the size of each individual main or section thereof; the same size being used throughout a large district, and having been selected with reference to the general rather than local conditions.
* Elektrotechnische Zeitschrift, 1890, p. 221, and Electrical World, vol. xv., p. 300. † Elektrotechnische Zeitschrift, 1893, p. 10. | Ibid., 1893, p. 425. $ Ibid., 1894, pp. 67 and 236.
|| German Patent No. 68918, Class 21, April 5, 1892. See also Western Elec- . trician, April 27, 1897.
The justification for this apparently crude practice is, first, the simplicity of laying and maintaining a network composed of only two or three standard sizes of mains, larger or smaller sizes being either too clumsy or too weak mechanically ; second, it is practically impossible to predetermine the current that a main will carry, the demand upon it being often much greater or less than was expected; third, an excess of copper in one portion of a network tends to help other portions that are more heavily loaded, and conversely a small section of main acts as a weak link in the chain.
It is important to appreciate this interdependence of the parts of a network of conductors, as it constitutes its chief advantage. In the mechanical model already referred to, it is evident that all of the strings would aid each other in supporting a weight hung at any point upon them. The electrical analogue acts in a similar manner.
As already stated, it is customary to construct networks with certain sizes of mains which have been found by experience to be suitable for towns having a certain density of population, character of service, etc. In cases where this very empirical method cannot be followed, or when it is desired to check it by calculation, a careful plan of the given district should be made, and lines representing the proposed mains are then drawn. A main (2- or 3-wire as the case may be) is run through each street, or two mains may be laid, one on each side, in order to reduce the trouble of making house connections. In the case of unimportant streets in which there are no customers, the mains may be omitted or put in later. Where the mains intersect they are connected, all the + conductors (of which there are usually four or eight) being brought together, the same for the – conductors, and also for the † conductors in a three-wire system. The connection of four + wires and four – wires in a two-wire system is represented at J and N in Fig. 75. Having thus laid out the entire network, certain feedingpoints are then chosen. There is no absolute rule for determining their position, but they should be located to give as nearly uniform voltage as possible throughout the system. They should be arranged so that the feeders can be run conveniently to them and connected to the network, and should be nearer together where the load is great, and vice versa. In case it is subsequently found that they are too far apart, others may be added without disturbing the feeders and mains already laid. In this way increase of load upon the system may be provided for at any time. It is also possible to reënforce the mains by laying others parallel to them, but practically the same result is obtained when additional feeders are put in. If, for example, the average distance between feeding-points is reduced to one-half, the average current on a given main will also be one-half as great as before, and since it flows only one-half as far, the drop would be one-quarter as much. Extending a network of mains in any direction will also tend to help the conductors already laid, because it provides more paths for the current, as explained on page 104.
The Edison system of underground conductors originally adopted for network distribution and still very generally employed for the purpose will be described in the chapter on Underground Conductors. Other methods of constructing such networks will be given under the same heading.
PRINCIPLES OF ALTERNATING CURRENTS.
Introduction. -- The various principles and facts concerning direct current distribution set forth in the preceding chapters, apply also to alternating current systems. But in addition to the simple phenomena due to resistance, which occur in the former case, there are certain additional factors that must be considered in connection with alternating current transmission. The flow of a direct current, which is steady, is entirely determined by the ohmic resistance of the various parts of the circuit ; and if all these resistances are known the distribution of potential and current can be determined exactly. The flow of an alternating current depends not only upon the resistance, but also upon any inductance (self or mutual inductance) or capacity that may be contained in or connected with the circuit. These two factors have absolutely no effect upon a direct current after a steady flow has been established, which usually requires only a small fraction of a second. But in an alternating circuit either or both of them may be far more important than resistance, and in some cases may entirely control the action of the current, the effect of resistance being insignificant.
Since alternating current problems involve a consideration of three factors, they are usually more complicated and difficult to solve than those relating to direct currents. Nevertheless, by an extension of the principles and methods already explained, it will be found that alternating current systems can be designed correctly and without great difficulty.
Practically the only reason for employing alternating currents in electric lighting is to enable the cost of the conductors to be reduced by using high voltages and transformers. It has already been shown that the cross-section of a wire needed to convey a