CHAPTER XXX. EFFICIENCY OF SYSTEMS. 288. In electric power transmission and distribution, wherever the place of consumption of the electric energy is distant from the place of production, the conductors which transfer the current are a sufficiently large item to require consideration, when deciding which system and •what potential is to be used. In general, in transmitting a given amount of power at a given loss over a given distance, other things being equal, the amount of copper required in the conductors is inversely proportional to the square of the potential used. Since the total power transmitted is proportional to the product of current and E.M.F., at a given power, the current will vary inversely proportional to the E.M.F., and therefore, since the loss is proportional to the product of current- square and resistance, to give the same loss the resistance must vary inversely proportional to the square of the cur- rent, that is, proportional to the square of the E.M.F. ; and since the amount of copper is inversely proportional to the resistance, other things being equal, the amount of copper varies inversely proportional to the square of the E.M.F. used. This holds for any system. Therefore to compare the different systems, as two-wire single-phase, single-phase three-wire, three-phase and quar- ter-phase, equality of the potential must be assumed. Some systems, however, as for instance, the Edison three-wire system, or the inverted three-phase system, have EFFICIENCY OF SYSTEMS. 409 different potentials in the different circuits constituting the system, and thus the comparison can be made either — 1st. On the basis of equality of the maximum potential difference in the system ; or 2d. On the basis of the minimum potential difference in the system, or the potential difference per circuit or phase of the system. In low potential circuits, as secondary networks, where the potential is not limited by the insulation strain, but by the potential of the apparatus connected into the system, as incandescent lamps, the proper basis of comparison is equality of the potential per branch of the system, or per phase. On the other hand, in long distance transmissions where the potential is not restricted by any consideration of ap- paratus suitable for a certain maximum potential only, but where the limitation of potential depends upon the problem of insulating the conductors against disruptive discharge, the proper comparison is on the basis of equality of the maximum difference of potential in the system ; that is, •equal maximum dielectric strain on the insulation. The same consideration holds in moderate potential power circuits, in considering the danger to life from live wires entering human habitations. Thus the comparison of different systems of long-dis- tance transmission at high potential or power distribution for motors is to be made on the basis of equality of the maximum difference of potential existing in the system. The comparison of low potential distribution circuits for lighting on the basis of equality of the minimum difference of potential between any pair of wires connected to the receiving apparatus. 289. 1st. Comparison on the basis of equality of the minimum difference of potential, in low potential lighting circuits : 4TO ALTERNATING-CURRENT PHENOMENA. In the single-phase alternating-current circuit, if e — E.M.F., i = current, r— resistance per line, the total power is = ei, the loss of power 2z'V. Using, however, a three-wire system, the potential be- tween outside wires and neutral being given = e, the potential between the outside wires is == 2 e, that is, the dis- tribution takes place at twice the potential, or only -'• the copper is needed to transmit the same power at the same loss, if, as it is theoretically possible, the neutral wire has no cross-section. If therefore the neutral wire is made of the same cross-section with each of the outside wires, | of the copper of the two- wire system is needed ; if the neutral wire is £ the cross-section of each of the outside wires, T% of the copper is needed. Obviously, a single-phase five-wire system will be a system of distribution at the potential 4 e, and therefore require only TV °f the copper of the single- phase system in the outside wires ; and if each of the three neutral wires is of i the cross-section of the outside wires, /? = 10.93 per cent of the copper. Coming now to the three-phase system with the poten- tial e between the lines as delta potential, if i = the current per line or Y current, the current from line to line or delta current = ^ / VB ; and since three branches are used, the total power is 3 e i\ / V3 == e z'x V3. Hence if the same power has to be transmitted by the three-phase system as with the single-phase system, the three-phase line current must be z'i = i / V3 where i — single-phase current, r = single-phase resistance per line, at equal power and loss; hence if 1\ = resistance of each of the three wires, the loss per wire is i? rt = iz rt /.3, and the total loss is z2 1\, while in the single-phase system it is 2 t*r. Hence, to get the same loss, it must be : rv = 2 r, that is, each of the three three- phase lines has twice the resistance — that is, half the cop- per of each of the two single-phase lines ; or in other words, the three-phase system requires three-fourths of the copper of the single-phase system of the same potential. EFFICIENCY OF SYSTEMS. 471 Introducing, however, a fourth or neutral wire into the three-phase system, and connecting the lamps between the neutral wire and the three outside wires — that is, in Y con- nection— the potential between the outside wires or delta potential will be = e X V3, since the Y potential = e, and the potential of the system is raised thereby from e to e V3 ; that is, only J as much copper is required in the out- side wires as before — that is \ as much copper as in the single-phase two-wire system. Making the neutral of the same cross-section as the outside wires, requires \ more copper, or \ = 33.3 per cent of the copper of the single- phase system ; making the neutral of half cross-section, requires \ more, or ^ = 29.17 per cent of the copper of the single-phase system. The system, however, now is a four-wire system. The independent quarter-phase system with four wires is identical in efficiency to the two-wire single-phase sys- tem, since it is nothing but two independent single-phase systems in quadrature. The four-wire quarter-phase system can be used as two independent Edison three-wire systems also, deriving there- from the same saving by doubling the potential between the outside wires, and has in this case the advantage, that by interlinkage, the same neutral wire can be used for both phases, and thus one of the neutral wires saved. In this case the quarter-phase system with common neu- tral of full cross-section requires -fo = 31.25 per cent, the quarter-phase system with common neutral of one-half cross- section requires ^ = 28.125 per cent, of the copper of the two-wire single-phase system. In this case, however, the system is a five-wire system, and as such far inferior to the five-wire single-phase system. Coming now to the quarter-phase system with common return and potential e per branch, denoting the current in the outside wires by z'2, the current in the central wire is *a V2 ; and if the same current density is chosen for all 472 ALTERNATING-CURRENT PHENOMENA. three wires, as the condition of maximum efficiency, and the resistance of each outside wire denoted by rz, the re- sistance of the central wire = r2/V2, and the loss of power per outside wire is z'22 r2 , in the central wire 2 z'22 r2 / V2 = z'22 r2 V2 ; hence the total loss of power is 2 z'22 r2 + z'22 r2 V2 = z'22 r2 (2 -f V2). The power transmitted per branch is z'2 ^, hence the total power 2 z'2 e. To transmit the same power as by a single-phase system of power, e z, it must be z2 = z'/2; hence the loss, *2;a(2 + ^ . Since this loss shall be the same as the loss 2z'2r in the single- phase system, it must be 2 r = - — — r2 , or r2 = ~ . . 2 -}- V 2 ° 4- V^ Therefore each of the outside wires must be — — times o as large as each single-phase wire, the central wire V2 times larger ; hence the copper required for the quarter- phase system with common return bears to the copper required for the single-phase system the relation : 2 (2 + V2) (2 + V5) V2 . 9 3 + 2V2 ^~ ~T~ ~T~~ per cent of the copper of the single-phase system. Hence the quarter-phase system with common return saves 2 per cent more copper than the three-phase system, but is inferior to the single-phase three-wire system. The inverted three-phase system, consisting of two E.M.Fs. e at 60° displacement, and three equal currents /8 in the three lines of equal resistance r3, gives the out- put 2^z'3, that is, compared with the single-phase system, /8 = z'/2. The loss in the three lines is 3 z'32 r3 = | z2 rs. Hence, to give the same loss 2 z'2 r as the single-phase sys- tem, it must be rs = f r, that is, each of the three wires must have f of the copper cross-section of the wire in the two-wire single-phase system ; or in other words, the in- verted three-phase system requires ^ of the copper of the two-wire single-phase system. EFFICIENCY OF SYSTEMS. 473 We get thus the result, If a given power has to be transmitted at a given loss, and a given minimum potential, as for instance 110 volts for lighting, the amount of copper necessary is : 2 WIRES : Single-phase system, 100.0 3 WIRES : Edison three-wire single-phase sys- tem, neutral full section, 37.5 Edison three-wire single-phase sys- tem, neutral half-section, 31.25 Inverted three-phase system, 56.25 Quarter-phase system with common return, 72.9 Three-phase system, 75.0 4 WIRES : Three-phase, with neutral wire full section, 33.3 Three-phase, with neutral wire half- section, 29.17 Independent quarter-phase system, 100.0 5 WIRES : Edison five-wire, single-phase system, full neutral, 15.625 Edison five-wire, single-phase system, half-neutral, 10.93 Four-wire, quarter-phase, with com- mon neutral full section, 31.25 Four-wire, quarter-phase, with com- mon neutral half-section, 28.125 We see herefrom, that in distribution for lighting — that is, with the same minimum potential, and with the same number of wires — the single-phase system is superior to any polyphase system. The continuous-current system is equivalent in this' comparison to the single-phase alternating-current system of the same effective potential, since the comparison is made on the basis of effective potential, and the power depends upon the effective potential also. 474 AL TERNA TING-CURRENT PHENOMENA. 290. Comparison on the Basis of Equality of the Maximum Difference of Potential in the System, in Long- Distance Transmission, Power Distribution, etc. Wherever the potential is so high as to bring the ques- tion of the strain on the insulation into consideration, or in other cases, to approach the danger limit to life, the proper comparison of different systems is on the basis of equality of maximum potential in the system. Hence in this case, since the maximum potential is fixed, nothing is gained by three- or five-wire Edison sys- tems. Thus, such systems do not come into consideration. The comparison of the three-phase system with the single-phase system remains the same, since the three- phase system has the same maximum as minimum poten- tial ; that is : The three-phase system requires three-fourths of the copper of the single-phase system to transmit the same power at the same loss over the same distance. The four-wire quarter-phase system requires the same amount of copper as the single-phase system, since it con- sists of two single-phase systems. In a quarter-phase system with common return, the potential between the outside wire is V2 times the poten- tial per branch, hence to get the same maximum strain on the insulation — that is, the same potential e between the outside wires as -in the single-phase system — the potential per branch will be ej V2, hence the current z'4 = t/ V2, if i equals the current of the single-phase system of equal power, and t\ V2 = i will be the current in the central wire. Hence, if r± = resistance per outside wire, r± / V2 = resistance of central wire, and the total loss in the sys- tem is : , (2 + V2) = EFFICIENCY OF SYSTEMS. 475 Since in the single-phase system, the loss = 2 i 2 r, it is : 2 + ~v/2 That is, each of the outside wires has to contain — — - - 4 times as much copper as each of the single-phase wires. 2 x V2 /- The central wires have to contain - - V 2 times as ^ (^ -4- ~v/2^ much copper ; hence the total system contains 2 +V2 — T - V2 times as much copper as each of the single- 3 + 2 ~\/2 phase wires ; that is, - — times the copper of the 4 single-phase system. Or, in other words, A quarter-phase system with common return requires 3 + 2 A/2 — == 1.457 times as much copper as a single-phase system of the same maximum potential, same power, and same loss. Since the comparison is made on the basis of equal maximum potential, and the maximum potential of alter- nating system is A/2 times that of a continuous-current circuit of equal effective potential, the alternating circuit of effective potential e compares with the continuous-cur- rent circuit of potential e A/2, which latter requires only half the copper of the alternating system. This comparison of the alternating with the continuous- current system is not proper however, since the continuous- current potential introduces, besides the electrostatic strain, an electrolytic strain on the dielectric which does not exist in the alternating system, and thus makes the action of the continuous-current potential on the insulation more severe than that of an equal alternating potential. Besides, self- induction having no effect on a steady current, continuous current circuits as a rule have a self-induction far in excess 476 ALTERNATING-CURRENT PHENOMENA. of any alternating circuit. During changes of current, as make and break, and changes of load, especially rapid changes, there are consequently induced in these circuits E.M.F.'s far exceeding their normal potentials. At the voltages which came under consideration, the continuous current is excluded to begin with. Thus we get : If a given power is to be transmitted at a given loss, and a given maximum difference of potential in the system, that is, with the same strain on the insulation, the amount of copper required is : 2 WIRES : Single-phase system, 100.0 [Continuous-current system, 50.0] 3 WIRES : Three-phase system, 75.0 Quarter-phase system, with common return, 145.7 4 WIRES : Independent Quarter-phase system, 100.0 Hence the quarter-phase system with common return is practically excluded from long-distance transmission. 291 . In a different way the same comparative results between single-phase, three-phase, and quarter-phase sys- tems can be derived by resolving the systems into their single-phase branches. The three-phase system of E.M.F. e between the lines can be considered as consisting of three single-phase cir- cuits of E.M.F. ^/V3, and no return. The single-phase system of E.M.F. e between lines as consisting of two single-phase circuits of E.M.F.