(Figs. 13, 15, and 16). In Fig. 17, for instance, the difference of potential between the feeding-points + and must be 116 volts,

in order that the end group of lamps A shall receive 111 volts, since there is a drop of

in the upper main. Similar reasoning applies to the group A in Fig. 18, the drop being 3 + 2 = 5 volts. This necessity for supplying a considerably higher voltage at the feeding-points of the mains is disadvantageous in two respects. First, it involves a loss of power in watts equal to the extra pressure multiplied by the total current; and second, it may allow great variations in poten

Fig. 19.

Closed Ring; 2,000 ft. No. 8; 100 lbs. Copper; 115.23 Volts Between FeedingPoints; 1.95 Volts Max. Difference Between Lamps; 5.53 Volts Average Drop.

tial to occur when a large number of lamps are thrown on or off the circuit. For example, if all the lamps except one were put out, the remaining one would receive practically the full pressure of 116 volts. This may be overcome by reducing the voltage of the feeders when lamps are disconnected, either by automatic or hand regulation, employing some of the methods described later; but it is evidently simpler to maintain the same pressure at the feedingpoints. On the other hand, the drop in the feeders themselves must be overcome by raising the voltage at the generating plant when the current carried by them increases. In such cases it may not involve very much additional trouble to regulate for the drop in the mains as well as for that in the feeders.

A further extension of the principles shown in Figs. 17 and 18 is indicated in Fig. 19, in which five groups of lamps are connected across the mains, which form complete circles, being fed at diametrically opposite points. In this case, 2,000 feet of No. 8 wire, weighing 100 lbs., is used, instead of 1,600 feet, as in the previous examples. A similar arrangement is shown in Fig. 20; but the lamps are assumed to be divided into four groups, of 25 lamps each. All the lamps receive exactly the same voltage, 1,600 feet of No. 10 wire, weighing only 50 lbs., being required. This exact equality in voltage is due to this being a special case, in which the lamps happen to be symmetrically placed with respect to the feeding-points. In Fig. 17, for example, the second and fourth groups

Fig. 20.

Closed Square; 1,600 ft. No. 10; 50.3 lbs. Copper; No Difference Between
Lamps; 117.5 Volts Between Feeding-Points; 7.5 Volts Average Drop.

of lamps have exactly the same voltage, since they are equally distant from the feeders. The pressure at the feeding-points is 117.5 volts in Fig. 20, being higher than in any of the other cases.

Individual Conductors. The most certain way to obtain a constant voltage in parallel distribution is to provide each lamp or group of lamps with its own particular conductors. One arrangement of this kind is illustrated in Fig. 21, five groups of lamps, each taking 10 amperes and placed 200 feet apart, being assumed, as in the previous examples. The feeding-points, marked and -, are supposed to be located at some distance from the lamps, as shown. The pair of conductors that supply each group are so proportioned in size and length that the drop has an equal value for all of the groups. This condition will be secured if the cross

sections of the various conductors are respectively proportional to their lengths. For example, a conductor twice as long as another should have double the cross-section, so that the resistance of the two will be equal. If the currents are not the same for the different conductors, the cross-sections should be further modified in proportion to the currents. In other words, for all of

il

the pairs of conductors, the fraction should have the same value,

a

i being the current in amperes, the total length of both conductors, and a the cross-section.

It is not apparent what advantages this plan of using individual wires has over the arrangements already described, the weight of copper being even greater than that in Fig. 18, for example. The

Fig. 21. Individual Conductors; Unequal Lengths; 186 lbs. Copper; No Difference of Voltage Between Lamps; 115 Volts Between Feeding-Points; 5 Volts Drop.

In

answer is to be found in the fact that the groups of lamps in Fig. 21 are not only equal in potential when all are burning, but they are also independent of one another, the turning on or off of one or more groups not affecting the others, provided that the voltage at the feeding-points and be kept constant. + the preceding cases, the throwing off of some lamps would vary the pressure of all the others. In fact, it was pointed out that disconnecting every lamp but one would raise its potential practically the whole amount of the drop, which was five or six volts in some instances. It was also stated that the remedy for this variation consists in regulating the pressure at the feeding-points. Thus it appears that it is necessary to maintain a constant voltage at the feeding-points, with some arrangements of conductors, and a variable voltage with others. These questions will be considered later under feeder regulation.

Fig. 22 represents another example of individual conductors, but in this case each group of lamps is supplied through the same total length of conductor; i.e., 800 feet of No. 8 wire, having 0.5 ohm resistance. Consequently the drop is five volts for all, since each group takes 10 amperes. The advantage of this plan over that shown in Fig. 17, which it somewhat resembles, is the freedom from interference already explained. It should be noted, however, that in either Fig. 21 or 22 the turning off of a portion of the lamps in one particular cluster would affect the remaining ones in that group. In order to secure complete independence of operation for every lamp in a system, it would be necessary to provide each one with its own individual wires. This is practically out of the question in almost all cases; but it can be approximated more or less closely, the tendency in the best practice being to subdivide the circuits and reduce the number of lamps on each, as far as economy and simplicity will reasonably allow.

Fig. 22. Individual Conductors; Equal Total Lengths; 200 lbs. Copper; No Difference of Voltage Between Lamps; 115 Volts Between Feeding-Points; 5 Volts Drop.

Calculations of Drop, Weight, etc., of Mains. The examples already given (Figs. 13 to 22) show the results obtained by different arrangements of mains and feeding-points in parallel distribution. These cases having been treated concretely with definite sizes of wire, voltages, currents, etc., bring out the facts clearly, and are intelligible to those who may not possess special mathematical knowledge. It will be well, however, to discuss these important problems in a more general way before dismissing them. For this purpose the following symbols may be adopted:

L is the length of each main in any desired units;

1, the length of each section of main (i.e. between adjacent lamps); I and, the currents in the two mains at the feeding-points;

i and i', the currents in the two mains at the point ;

V and 7%, the potentials on the two mains at the feeding-points; uo, the potential difference between the two feeding-points, or between one feeding-point and the opposite point on the other main ;

=

V'vu', the potential difference between the two mains at any point distant a units from + feeding-point;

D, the fall of potential or "drop" at any given lamp with respect to the difference of potential between the two feeding-points;

C, the current consumed by each lamp;

N, the number of lamps;

R, the resistance of each main per unit of length.

Considering first the ordinary parallel circuit represented in Fig. 13, the drop on both mains from the first to the second lamp (or group of lamps) is 2 RIC (N − 1), and the total drop from the feeding-points to the last lamp

In this equation there are N-1 terms, having an average

value of ; hence we have —

N
2

DRIC —.
RIC (N2 – N),

(1)

and the resistance per unit of length which will give this maximum drop D, is

D
IC (N2 - N)

(2)

In the case illustrated in Fig. 16 the current divides, one-half flowing in each direction, so that it is only necessary to substitute for N in the above formulæ, or

N

2

(3)

(4)

In the case of anti-parallel distribution (Fig. 17) the drop to any lamp, say the xth from the + feeding-point, is R/C [(N − 1) a] on the + main, and R/C [(N-1) + (N − 2) main. x+1)] on the Hence the total drop on

+ (N − 2) +...

both mains is the sum of these values which is

D = R 1C (N2 + 2 Nx − 3 N + 2 x − 2 x2).

2

(5)

From this equation it is evident that the drop depends upon x. Differentiating (5), we find that D is a maximum when