CHAPTER XXXIII 
EFFICIENCY OF SYSTEMS 

294. 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 carry the current 
are a sufficient^ large item to require consideration, when decid- 
ing 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 trans- 
mitted is proportional to the product of current and e.m.f., at a 
given power, the current will vary inversely proportionally 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 
current, that is, proportional to the square of the e.m.f.; and 
since the amount of copper is inversely proportional to the resist- 
ance, other things being equal, the amount of copper varies in- 
versely 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 quarter-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 different 
potentials in the different circuits constituting the system, and 
thus the comparison can be made either — 

1st. On the basis of the maximum potential difference between 
any two conductors of the system ; or 

2nd, On the basis of the maximum potential difference between 
any conductor of the system and the ground ; or 

3rd. On the basis of the minimum potential difference in the 
system, or the potential difference per circuit or phase of the 
system. 

431 


432 ALTERNATING-CURRENT PHENOMENA 

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 incan- 
descent 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 apparatus 
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 com- 
parison is on the basis of equality of the maximum difference 
of potential; that is, equal maximum dielectric strain on the 
insulation. 

In this case, the comparison voltage may be either the poten- 
tial difference between any two conductors of the system, or 
it may be the potential difference between any conductor of 
the system and the ground, depending on the character of the 
circuit. 

The dielectric stress is from conductor to conductor, or be- 
tween any two conductors, in a system which is insulated from 
the ground, as is mostly the case in medium voltage overhead 
transmissions, and frequently in underground cables. 

In an ungrounded cable system, in which all the conductors 
are enclosed in the same cable, the insulation stress is mainly 
from conductor to conductor, and this therefore is the basis of 
comparison. But even in an underground cable system with 
grounded neutral, as very commonly used, a direct path exists 
from conductor to conductor inside of the cables, for a disrup- 
tive voltage, and the comparison of systems, therefore, has to be 
made, in this case, on the basis of maximum potential difference 
between conductors as well as between conductor and ground. 

In an ungrounded overhead system, the disruptive stress is 
from conductor to ground and back from ground to conductor. 
If the system is of considerable extent — as is the case where high 
voltages of serious disruptive strength have to be considered — • 
the neutral of the system is maintained at approximate ground 
potential by the capacity of the system, and the normal voltage 
stress from conductor to ground therefore is that from conductor 
to neutral, that is, the same as in a system with grounded neutral, 
and the basis of comparison then is the voltage from line to 
ground, and not between lines. Since, however, one conductor 


EFFICIENCY OF SYSTEMS 433 

of the system may temporarily ground, if it is required to main- 
tain operation even with one conductor of the system grounded, 
the voltage between conductors must be- the basis of comparison, 
since with one conductor grounded, the disruptive stress between 
the other conductors and ground is- the potential difference be- 
tween the conductors of the system. 

In an overhead system with grounded neutral, frequently used 
for transmission systems of very high voltage, or in general in a 
grounded system, the disruptive stress is that due to the potential 
difference between conductor and ground or neutral, and this 
then is the basis of comparison. 

In moderate-potential power circuits, in considering the danger 
to life from live wires entering buildings or otherwise accessible, 
the comparison on the basis of maximum potential also appears 
appropriate. 

Thus the comparison of different systems of long-distance 
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-poten- 
tial 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. 

295. 1st. Comparison on the basis of equality of the minimum 
difference of potential, in low-potential lighting circuits: 

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, 2 ih\ 

Using, however, a three-wire system: the potential between 
outside wires and neutral being given equal to e, the potential 
between the outside wires is equal to 2 e, that is, the distribution 
takes place at twice the potential, or only one-fourth 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, 
however, the neutral wire is made of the same cross-section as 
each of the outside wires, three-eighths as much copper as in the 
two-wire system is needed; if the neutral wire is one-half the 
cross-section of each of the outside wires, five-sixteenths as much 
copper is needed. Obviously, a single-phase, five-wire system 
will be a system of distribution at the potential, 4 e, and there- 
fore require only one-sixteenth of the copper of the single-phase 
system in the outside wires; and if each of the three neutral 

28 


434 ALTERNATING-CURRENT PHENOMENA 

wires is of one-half the cross-section of the outside wires, seven- 
sixty-fourths or 10.93 per cent, of the copper. 

Coming now to the three-phase system with the potential, 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 = — ~ • 

Vs 

and since three branches are used, the total power is — -pz = ei] \/3- 

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 ii = —-^; where i = single-phase current, 

r = single-phase resistance per line, at equal power and loss: 
hence if ri = resistance of each of the three wires, the loss per 

wire is z'lVi = -^, and the total loss is ihi, while in the single- 
phase system it is 2 ih. Hence, to get the same loss, it must be: 
ri = 2 r, that is, each of the three three-phase lines has twice the 
resistance — that is, half the copper of each of the two single- 
phase lines; or in other words, the three-phase system requires 
three-fourths as much copper as the single-phase system of the 
same potential. 

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 connection — the 
potential between the outside wires or delta potential will be 
= e X a/Sj since the Y potential = e, and the potential of the 
system is raised thereby from e to e\/S ; that is, only one-third 
as much copper is required in the outside wires as before — that 
is one-fourth as much copper as in the single-phase two-wire sys- 
tem. Making the neutral of the same cross-section as the out- 
side wires, requires one-third more copper, or | = 33.3 per cent, 
of the copper of the single-sphase sytem; making the neutral 
of half cross-section, requires one-sixth 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 system, since 
it is nothing but two independent single-phase systems in quad- 
rature. 

The four-wire, quarter-phase system can be used as two inde- 


EFFICIENCY OF SYSTEMS 435 

pendent Edison three-wire systems also, deriving therefrom the 
same saving by doubhng the potential between the outside wires, 
and has in this case the advantage, that by interlinkagc, 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 neutral of 
full cross-section requires fV or 31.25 per cent., the quarter-phase 
system with common neutral of one-half cross-section requires 
"V or 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 in copper efficiency 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 ?2, the current in the central wire is ?2V'2; and if the 
same current density is chosen for all three wires, as the condition 
of maximum efficiency, and the resistance of each outside wire 

denoted by r^, the resistance of the central wire = — p^, and the 

2 z^2r^ 
loss of power per outside wire is i^^r^, in the central wire — -^ 

= ^V?'2\/2; hence the total loss of power is 2 ^2^2 -|- i-ihi\^ 
= ^ 2^2(2 + -\/2). The power transmitted per branch is i^e, 
hence the total power, 2 i^e. To transmit the same power as by 

a single-phase system of power, ei, it must be ?2 = -k', hence the 
loss, J . Since this loss shall be the same as the loss, 

2 f^r, in the single-phase system, it must be 2 r = -^ r2, 

8r 

or r2 = 7=* Therefore each of the outside wires must be 

2 + \/2 

2 + v'2 

^ times as large as each single-phase wire, the central 

wire \/^ 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(2-1- x/2) (2+\/2)\/2 

8 "^ 8 " 

per cent, of the copper of the single-phase system. 


436 ALTERNATING-CURRENT PHENOMENA 

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, consistmg of two e.m.fs. e at 

60° displacement, and three equal currents iz in the three lines 

of equal resistance rs, gives the output 2 eis, that is, compared 

{ 
with the single-phase system, is = -^- The loss in the three lines 

is 3 z'sVs = I ihs. Hence, to give the same loss, 2 ih, as the 
single-phase system, it must be 7'3 = f r, that is, each of the three 
wires must have three-eighths of the copper cross-section of the 
wire in the two-wire single-phase system; or in other words, the 
inverted three-phase system requires nine-sixteenths of the cop- 
per of the two-wire single-phase system. 

Thus 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 system, 

neutral full section, 37 . 5 

Edison three-wire single-phase system, 

neutral half-section. 
Inverted three-phase system, 
Quarter-phase system with common re- 
turn. 
Three-phase system, 

4 Wires : Three-phase, with neutral-wire full sec- 

tion, 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 


31 

.25 

56 

.25 

72, 

.9 

75, 

,0 

EFFICIENCY OF SYSTEMS 437 

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. 

296. Comparison on the Basis of Equality of the Maximum 
Difference of Potential between any two Conductors of the System, 
in Long-distance Transmission, Power Distribution, etc. 

Wherever the potential is so high as to bring the question 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 poten- 
tial in the system. 

Hence in this case, since the maximum potential is fixed, noth- 
ing is gained by three- or five-wire, Edison systems. 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 potential; 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 consists of two 
single-phase systems. 

In a quarter-phase system with common return, the potential 
between the outside wires is V^ times the potential 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 "~f^, hence the 

current n = —7^, if i equals the current of the single-phase sys- 
V 2 _ 

tem of equal power, and ii\^2 = i will be the current in the 
central wire. 

Hence, if r^ = resistance per outside wire, —y^- = resistance of 
central wire, and the total loss in the syst.em is 


438 ALTERNATING-CURRENT PHENOMENA 
Since in the single-phase system, the loss = 2 z'^r, it is 

4r 


Ti 


2 + V2 


2 + \/2 

That is, each of the outside wires has to contain -. times 

' 4 

as much copper as each of the single-phase wires. The central 

2 + V 2 ,- 
wires have to contain — ^r \/2 times as much copper; hence 

. 2 (2 -f- V^) 2 + V 2 /- . 
the total system contams — -7 1- - — -7 v 2 tunes as 

much copper as each of the single-phase wires; that is, -^ 

times the copper of the single-phase system. 
Or, in other words, 

A quarter-phasesystem with common .eturn requires =^-^^ 

= 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 an alternating sj^stem is 
V2 times that of a continuous-current circuit of equal effective 
potential, the alternating circuit of effective potential, e, com- 
pares with the continuous-current circuit of potential e \/2, 
which latter requires only half the copper of the alternating 
system. 

This comparison of the alternating with the continuous-cur- 
rent system is not proper, however, since the continuous-current 
voltage may introduce, besides the electrostatic strain, an elec- 
trolytic strain on the dielectric which does not exist in the alter- 
nating system, and thus may make the action of the continuous- 
current voltage on the insulation more severe than that of an 
equal alternating voltage. Besides, self-induction having no 
effect on a steady current, continuous-current circuits as a rule 
have a self-induction far in excess of any alternating circuit. 
During changes of current, as make and break, and changes of 
load, especially rapid changes, there may consequently be gen- 
erated in these circuits e.m.fs. far exceeding their normal poten- 
tials. Inversely, however, with alternating voltages, dielectric 
hysteresis, etc., may cause heating and thereby lower the 
disruptive strength. At the voltages which came under con- 
sideration, the continuous current is usually excluded to begin 
with. 


EFFICIENCY OF SYSTEMS 439 

Thus we get: 

If a given power is to be transmitted at a given loss, and a 
given maxi?num 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 prac- 
tically excluded from long-distance transmission. 

297. In a different way the same comparative results be- 
tween single-phase, three-phase, and quarter-phase systems 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 circuits of e.m.f., 

—j^, 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., ^' and 

no return. Thus, the relative amount of copper in the two sys- 
tems being inversely proportional to the square of e.m.f., bears 

the relation ( ) : (-) =3:4; that is, the three-phase sys- 
tem requires 75 per cent, of the copper of the single-phase system. 
The quarter-phase system with four equal wires requires the 
same copper as the single-phase system, since it consists of two 
single-phase circuits. Replacing two of the four quarter-phase 
wires by one wire of the same cross-section as each of the wires 
replaced thereby, the current in this wire is \/2 times as large 
as in the other wires, hence, the loss is twice as large — that is, 
the same as in the two wires replaced by this common wire, or 
the total loss is not changed — while 25 per cent, of the copper is 
saved, and the system requires only 75 per cent, of the copper of 
the single-phase system, but produces \/2 times as high a poten- 


440 ALTERNATING-CURRENT PHENOMENA 

tial between the outside wires. Hence, to give the same maxi- 
mum potential, the e.m.fs. of the system have to be reduced by 
V'2 , that is, the amount of copper doubled, and thus the quarter- 
phase system with common return of the same cross-section as 
the outside wires requires 150 per cent, of the copper of the single- 
phase system. In this case, however, the current density in the 
middle wire is higher, thus the copper not used most economically, 
and transferring a part of the copper from the outside wires to 
the middle wire, to bring all three wires to the same current den- 
sity, reduces the loss, and thereby reduces the amount of copper 
at a given loss, to 145.7 per cent, of that of a single-phase system. 

298. Comparison on the basis of equality of the maximum differ- 
ence of potential between any conductor of the system, and the ground, 
in long-distance, three-phase transmissions with grounded neutral, 
single-phase sy sterns with ground return, etc. 

A system may be grounded by grounding its neutral point, 
for the purpose of maintaining constant-potential difference be- 
tween the conductors and ground, without carrying any current 
through the ground, or the ground may be used as return con- 
ductor. In either case the system can be considered as consist- 
ing of and resolved into as many single-phase systems with 
ground return, as there are overhead conductors, and with, zero 
resistance in the ground. 

It immediately follows herefrom, that the copper efficiency of 
such a system is the same as that of a single-phase system with 
ground return, of the same voltage as exists between conductor 
and ground of the system under consideration. If then all the 
overhead conductors have the same potential difference against 
ground, as is the case in a three-phase or quarter-phase system 
with grounded neutral, a single-phase system with grounded 
neutral, or quarter-phase system with common ground return of 
both phases, the copper efficiency is the same. That is: 

All grounded systems, whether with grounded neutral or with 
ground return, have the same copper efficiency, provided that 
all the overhead conductors have the same potential difference 
against ground. 

Hence: 

The three-phase system with grounded neutral has no supe- 
riority over the single-phase or the quarter-phase system with 
grounded neutral, in copper efficiency. The advantage of the 
three-phase system — which causes its practically universal use — 


EFFICIENCY OF SYSTEMS 441 

over the single-phase system is the greater usefulness of polyphase 
power, the advantage over the quarter-phase system is the use 
of three conductors, against four with the quarter-phase system. 

No saving in copper results from the use of the ground (of 
zero resistance) as return circuit, but a single-phase or quarter- 
phase system with ground return, at equal dielectric strain on 
the insulation, requires the same amount of copper as a system 
with grounded neutral, but has a greater self-induction, due to 
the greater distance between conductor and return conductor or 
ground, and has the objection of establishing current through 
the ground and so disturbing neighboring circuits, by electro- 
magnetic and electrostatic induction. 

The apparent saving in copper, in the single-phase system, by 
replacing one of the conductors by the ground as return, there- 
fore is a fallacy. By doing so, the potential difference of the other 
conductors against ground becomes twice what it would be with 
two conductors and grounded neutral, and at the same potential 
difference between conductors. That is, the single-phase system 
with ground return requires the same insulation as a single-phase 
system with grounded neutral, of twice the voltage, and then re- 
quires the same copper. A saving results only in the number of 
insulators required, etc. Only where the amount of power is so 
small that mechanical strength, and not power loss, determines 
the size of the conductor, a saving results by replacing one of the 
conductors by the ground. 

The high-tension, direct-current system, whether insulated, or 
with grounded neutral, or with ground return, appears equal in 
copper efficiency to a single-phase system of the same character 
(insulated, or with grounded neutral, or with ground return) and 
of the same effective voltage, that is, with a sine wave of a maxi- 
mum voltage \/2 times that of the direct current. Due to the 
different character of unidirectional electric stress of the direct- 
current system, from the alternating stress, a general comparison 
of the system by a numerical factor appears hardly feasible. It 
is, however, claimed that usually the insulation stress with per- 
fectly uniform continuous voltage is less than that of an alter- 
nating voltage of the same maximum value, so that continuous- 
current high-voltage transmission would offer advantages, if it 
were not for the difficulty of generating and utilizing very high 
continuous voltages, which with alternating voltages is overcome 
by the interposition of the stationary transformer.