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maximum drop is the controlling consideration in designing electrical conductors, particularly for electric lighting.

Case III. Cylindrical Conductors. Anti-Parallel System. — In this case the mains are fed from opposite ends as already described in connection with Fig. 17. It is evident that this arrangement differs from the two preceding in the fact that no lamp receives the full voltage delivered to the mains, because V0 is at one end of one main, and v0 is at the opposite end of the other.

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Fig.25. Fig.26.

Figs. 25 and- 26. Anti-Parallel Distribution. Cases III. and IV.

A study of Fig. 25 shows that the variation in pressure between the ends of any element dx for both mains is equal to the difference between the drop in one main and that in the other, whereas in the two previous cases it was the sum of the two drops, hence —

d («„ - «') = Rdx (/' - /'), (15)

i and — *' being the currents in the respective mains at the point x, and having the following values : —

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This equation is also that of a parabola, but its axis is perpendicular to the mains at their middle point. When x = 0 or x = L, u0 u' =0, showing that at each end the lamps receive the same voltage. To locate the maximum difference in pressure between the lamps —

that is, the greatest drop is at the center of the mains, and has the value RI0L / 4 obtained by substituting (18) in (17). But it should be carefully noted that this represents the difference between the voltage of the middle lamp and that of either end lamp. For the latter the pressure is less than the difference of potential between the feeding-points (= V0 — i,0) by the quantity RI0L / 2, which is the total drop in either one of the mains. Hence the middle lamp receives a voltage which is less than that supplied to the feeding-points by an amount —

This value is only three-quarters as large as the maximum drop in Case I., which was found to be RI0L ; and the greatest difference between the voltage of lamps is only one-quarter as much, or RIJ. / 4, the weight of copper being the same.

Case IV. Tapering Conductors. Anti-Parallel System. — The plan of feeding from the opposite ends of the mains may be applied to tapering conductors with even greater advantage than in the case of cylindrical conductors. By applying the equations in Cases II. and III. to this arrangement, shown in Fig. 26, the following expression is obtained : —

d (u0 — »i) = (ri r,»,) dx.

rand r', as well as i and — i', being respectively the resistances and currents in the two mains at the point x. Hence by a train of reasoning similar to that in the previous cases, r = p j S, and r' = p j S' ; but pi j S and pi' / S' are constants for each main, by hypothesis, and are equal to each other, hence —

d ("°~ ^ = 0. (20)

7(0 u' = a constant, and

V -v, = V„~v0-R0I0L. (21)

In other words, there is no difference in the voltage supplied to the various lamps, the pressure at any lamp being the difference in potential between the feeding-points less the quantity RJ^L, which latter is therefore the maximum drop for all of the lamps. This is the same value as in Case I., but the amount of copper is only one-half as great; hence the maximum drop is one-half as much for the same weight of copper, and all lamps receive the same voltage.

Drop in Voltage with Irregular Distribution of Lamps. — In

the various cases heretofore considered (Figs. 13 to 26 inclusive), the lamps were assumed to be uniformly distributed on the mains. This represents not only ideal conditions, but also applies fairly well to actual practice at full load; that is, when the maximum number of lamps are lighted. In fact, the circuits should be carefully designed to approximate this condition as closely as practicable in most cases. When only a fraction of the lamps are turned on, it is evident that they may be very irregularly distributed. This would give rise to an almost infinite number of special problems corresponding to the possible arrangements that might be made; but there are certain general facts that apply to such cases.

If in Fig. 13 it be assumed that only the last or right-hand group of lamps is connected, the drop would be equal to the current multiplied by the total resistance of both mains, or 10 x -08 = .8 volt. Hence the potential difference supplied to this group of lamps would be the pressure at the feeding-points minus the drop, that is, 111 — .8 = 110.2 volts. If now the middle group of lamps be turned on also, the potential difference which they receive would be 111 — 20 x .04 = 110.2 volts, and the pressure at the last group becomes 111 - (20 x .04 + 10 x .04) = 109.8 volts. Thus the various groups may be lighted successively, and it will be found that —

1. The addition of each group reduces the pressure for all of

those already connected.

2. The maximum drop occurs when all of the lamps are con

nected.

3. The greatest difference between the voltage of any two

lamps will usually exist when all are turned on.

The first statement might be contradicted on the ground that the pressure at the first group of lamps connected directly to the feeding-points would remain the same whether the others were lighted or not. Theoretically this is true; but practically there would be some drop on the mains even for this group, unless it were connected exactly at the feeding-points; and there would always be a drop on the feeders when any lamps were turned on, unless it is overcome by some of the special methods of feeder regulation which will be described later.

The same statements apply to Fig. 16, in which the portions of the mains on each side of the feeding-points may be considered as corresponding to the whole mains in Fig. 13. Even though all the lamps on one side were connected, and only one on the other side, the total drop and the difference between the voltage of lamps would be no greater than for the full-load conditions represented in the diagram.

Similar reasoning is applicable to the arrangements shown in Figs. 17 and 18; in fact, any two groups of lamps would have the same pressure in the case of the former, and any number less than all would give no greater total or difference in drop than the full load of lamps. If the first three groups were lighted, and only a single lamp out of the last group was turned on, the latter might receive a potential about three volts higher than that of the others. This is greater than the maximum difference when the circuit is fully loaded, which is only two volts. Hence it appears that when one end of a pair of anti-parallel mains is heavily loaded, and there are very few lamps in circuit at the other end, the difference between the voltage of lamps at the two ends is greater than when the full load is turned on. Consequently this is an exception to statement 3 above. But even in this case, the maximum drop and the average drop are less with a fractional load.

In Fig. 27 the curves AB and CD represent the potentials on two cylindrical mains, which are fed according to the anti-parallel method at A and D respectively, being fully and uniformly loaded. The drop between A and E is greater than between C and J, because the average value of the current is greater for the former, as will be seen by comparing Fig. 17. Hence the pressure supplied to the middle lamps EJ is less than that at the end lamps AC, as already explained in connection with Figs. 17 and 25. If now all the lamps on the right-hand half of the mains be disconnected, there will be no drop between E and F, and the fall of potential from // to D will be constant, and will be represented by the straight line HD. It is evident from an inspection of this diagram that FD EH > BD EJ, or in other words, there is a greater difference between the voltage at FD and EH when half the lamps are thrown off. Hence anti-parallel mains, when very unequally loaded, may show an exception to statement 3 (page 47) as already explained. It is also apparent, however, that the maximum as well as average drop are smaller with any load less than the full amount.

The substitution of tapering mains for cylindrical ones makes the pressure still more uniform for fractional loads, since the drop is more nearly equal for the different portions of the conductors.

When lamps are irregularly connected to a closed loop or ring arrangement of mains, the problem becomes somewhat more diffi

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Fig. 27. Effect of Unequal Distribution of Lamps on Anti-Parallel System.

cult, since there are two paths for the current.* Such a case is represented in Fig. 28, in which a pair of ring mains are supplied at the feeding-points, with a voltage V, the total value of the current being I. The resistance of a semicircular portion of each main is R. Two equal lamps are assumed to be connected as shown, the current in each being \ I. Let x be the current that flows downward from the + feeding-point, then the current in the upper half of mains is / — x, and their combined resistance is 2 R, hence, —

V -V2 = 2R(I-x) or Rf-?-^ = Rx, (22) and V -Vt = Rx, (23) also V, -K2 = ^^-(^or Vx - V^ + ^f = Rx. (24)

* This matter will be discussed further under the head of " Networks" of conductors.

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