CHAPTER XXXVII 
QUARTER-PHASE SYSTEM 

310. In a three- wire quarter-phase system, or quarter-phase 
system with common return-wire of both phases, let the two 
outside terminals and wires be denoted by 1 and 2, the middle 
wire or common return by 0. 

It is then, 

El = E = e.m.f. between 0 and 1 in the generator. 

Ei = jE = e.m.f. between 0 and 2 in the generator. 
Let 1 1 and 1 2 = currents in 1 and in 2, 

7o = current in 0, 

Z] and Z2 = impedances of lines 1 and 2, 

Zo = impedance of line 0, 

Yi and F2 = admittances of circuits 0 to 1, and 0 to 2, 

/'i and /'2 = currents in circuits 0 to 1, and 0 to 2, 

E\ and E'2 = potential differences at circuit 0 to 1, and 
0 to 2. 
it is then, Ii -\- h -\- h = 0, 

or, lo = — {Ii + ^2); 

that is, 7o is common return of Ii and /j. 


(1) 


Further, we have: 

E \ = E — IiZ\ -\- IoZq = E — I]{Z\ -f- Zo) — I2ZQ 
E\ = jE - /2Z0 + /oZo = jE - h{Z2 + Zo) - ZiZi 

and /i = YiE'i 

I2 ^^ J 2-C' 2 

h = - IYiE\ + F2^'2) 

Substituting (3) in (2), and expanding, 

J,, _ J, 1 + F2Z2 + Y2Z0 (1 - j) 

.' . (H- FiZo -f FiZi) (1 -f F2Z0 -t- F2Z2) - F1F2Z02 

J,, _,j, l-fFxZx-fFxZo(l-f-i) 

. ' ~ ^"rn + Y,Z, -f FiZi) (1 + F2Z0 -h F2Z2) - FiF2Zo-^ 

462 


(2) 


(3) 


(4) 


QUARTER-PHASE SYSTEM 463 

Hence, the two e.m.fs. at the end of the Hne are unequal in 
magnitude, and not in quadrature any more. 
311. Special Cases: 

A. Balanced System 
Zi = Z2 = Z 

Fi = F2 = Y. 
Substituting these values in (4) , gives : 

1 + \/2 - i 

1 + V - yz 

^/^ ^ ^ V2 

• ' • 1 + v^ (1 + V2) FZ + (1 + V2) F2Z2 
^ 1 + (1.707 + 0.707 i) YZ 


1 + 3.414 YZ + 2.414 F^Z^ 

1 + \/2 + 7 
1 + ^ ^ ZJ YZ 

E', = jE ^^ 


= jE 


■ 1 + V2(l + V2) FZ + (1 + V2) Y^Z'^ 
1 + (1.707 + 0.707 i) YZ 


1 + 3.414 YZ + 2.414 Y^Z^ 


(5) 


Hence, the balanced quarter-phase system with common re- 
turn is unbalanced with regard to voltage and phase relation, 
or in other words, even if in a quarter-phase system with common 
return both branches or phases are loaded equally, with a load 
of the same phase displacement, nevertheless the system becomes 
unbalanced, and the two e.m.fs. at the end of the hne are neither 
equal in magnitude, nor in quadrature with each other. 

B. One Branch Loaded, One Unloaded 
Zi = Z2 = Z, 
Z -^• 

(a) Fi = 0, F2 = F, 

{b) Fi = Y, Y, = 0. 


464 ALTERNATING-CURRENT PHENOMENA 

Substituting these values in (4), gives: 


(a) 


(b) 


1 + YZ 


E\ = E 


1 + V2 - i 
V2 


i + rz^ + ^"2 


V2 


= ^ 1 


= ^ I 1 


1 + V2 + 


V2 
YZ 


2.414 + 


1.414 


E'2 = jE 


1 


1 + YZ 
= jE 


1 + \/2 

V~2 
1 


E\ = E 


1 + 1.707 YZ 
1 


1 -\-YZ 


1 + \/2 

V2 
1 


1 +FZ 


1 + 1.707 yz 

1 + V2 + i 


E'2 = jE 


V2 


l + FZ^ + ^2 


V2 


=iJ^ 1 + 


A- \/2 

1 + V2 + -Y^ 

f 'I 1 

=i£; I 1 + — 


2.414 


1.414 


YZ 


(6) 


(7) 


These two e.m.fs. are unequal, and not in quadrature with each 
other. 

But the values in case (a) are different from the values in case 
(b). 

That means: 

The two phases of a three-wire, quarter-phase system are 


QUARTER-PHASE SYSTEM 465 

unsymmetrical, and the leading phase, 1, reacts upon the lagging 
phase, 2, in a different manner than 2 reacts upon 1. 

It is thus undesirable to use a three-wire, quarter-phase system, 
except in cases where the line impedances, Z, are negligible. 

In all other cases, the four-wire, quarter-phase system is pref- 
erable, which essentially consists of two independent single-phase 
circuits, and is treated as such. 

Obviously, even in such an independent quarter-phase system, 
at unequal distribution of load, unbalancing effects may take 
place. 

If one of the branches or phases is loaded differently from the 
other, the drop of voltage and the shift of the phase will be differ- 
ent from that in the other branch; and thus the e.m.fs. at the end 
of the lines will be neither equal in magnitude, nor in quadrature 
with each other. 

With both branches, however, loaded equally, the system 
remains balanced in voltage and phase, just like the three-phase 
system under the same conditions. 

Thus the four-wire, quarter-phase system and the three-phase 
system are balanced with regard to voltage and phase at equal 
distribution of load, but are liable to become unbalanced at 
unequal distribution of load; the three-wire, quarter-phase 
system is unbalanced in voltage and phase, even at equal dis- 
tribution of load. 


30 


APPENDIX 


ALGEBRA OF COMPLEX IMAGINARY QUANTITIES 
("See Engineering Mathematics") 

INTRODUCTION 

312. The system of numbers, of which the science of algebra 
treats, finds its ultimate origin in experience. Directly derived 
from experience, however, are only the absolute integral numbers; 
fractions, for instance, are not directly derived from experience, 
but are abstractions expressing relations between different 
classes of quantities. Thus, for instance, if a quantity is divided 
in two parts, from one quantity two quantities are derived, and 
denoting these latter as halves expresses a relation, namely, that 
two of the new kinds of quantities are derived from, or can be 
combined to one of the old quantities. 

313. Directly derived from experience is the operation of 
counting or of numeration, 

a, o+l,a + 2, a + 3 . . . . 

Counting by a given number of integers, 

1 + 1 + 1 . . . . + 1 

a -\ ^ — = c, 

b integers 

introduces the operation of addition, as multiple counting, 

a -\- h = c. 
It is 

a -\- b = h -{- a; 

that is, the terms of addition, or addenda, are interchangeable. 
Multiple addition of the same terms, 

a -\- a -\- a -{- . . . +a 


b equal numbers 

introduces the operation of multiplication, 

a X b = c. 
466 


= c. 


APPENDIX 467 

It is 

a X h = b X a, 

that is, the terms of multipUcation, or factors, are inter- 
changeable. 

Multiple multiplication of the same factors, 

a X a X a X . . . X a 
= c 

b equal numbers 

introduces the operation of involution, 

a* = c. 
Since 

a'' is not equal to 6", 

the terms of involution are not interchangeable. 

314. The reverse operation of addition introduces the opera- 
tion of subtraction. 
If 

a + 6 = c, 
it is 

c — b = a. 

This operation cannot be carried out in the system of absolute 
numbers, if 

b > c. 

Thus, to make it possible to carry out the operation of sub- 
traction under any circumstances, the system of absolute num- 
bers has to be expanded by the introduction of the negative 
number, 

- a = (- 1) X a, 
where 

(— 1) is the negative unit. 

Thereby the system of numbers is subdivided in the positive 
and negative numbers, and the operation of subtraction possible 
for all values of subtrahend and minuend. From the definition 
of addition as multiple numeration, and subtraction as its inverse 
operation, it follows: 

c - (- 6) = c + 6, 
thus: 

(- 1) X (- 1) = 1; 
that is, the negative unit is defined by (— 1)^ = 1. 


468 ALTERNATING-CURRENT PHENOMENA 

315. The reverse operation of multiplication introduces the 
operation of division. 

If 

a y. h = c, 

it is 

c 

^ = a. 

In the system of integral numbers this operation can only be 
carried out if 6 is a factor of c. 

To make it possible to carry out the operation of division 
under any circumstances, the system of integral numbers has to 
be expanded by the introduction of the fraction, 


c 
b 
1 . 


= ^ X © 


where r is the integer fraction, and is defined by 


il) 


Xh = 1. 


316. The reverse operation of involution introduces two new 
operations, since in the involution, 


the quantities a and h are not reversible. 
Thus 

-\A3 = a, the evolution, 
loga c = b, the logarithmation. 

The operation of evolution of terms, c, which are not complete 
powers, makes a further expansion of the system of numbers 
necessary, by the introduction of the irrational number (endless 
decimal fraction), as for instance, 

V^ = 1.414213. . . 

317. The operation of evolution of negative quantities, c, with 
even exponents, 6, as for instance, 

\/— a, 

makes a further expansion of the system of numbers necessary, 
by the introduction of the imaginary unit 


APPENDIX 469 

Thus 

2 / 2 , 2 y— 

, V— a=V— IX va, 

where : 

■y/— 1 is denoted by j. 

Thus, the imaginary unit, j, is defined by 

j' = - 1. 

By addition and subtraction of real and imaginary units, com- 
pound numbers are derived of the form, 

a + jb, 

which are denoted as complex imaginary numbers, or general 
numbers. 

No further system of numbers is introduced by the operation 
of evolution. 

The operation of logarithmation introduces the irrational and 
imaginary and complex imaginary numbers also, but no further 
system of numbers. 

318. Thus, starting from the absolute integral numbers of 
experience, by the two conditions: 

1st. Possibility of carrying out the algebraic operations and 
their reverse operations under all conditions, 

2d. Permanence of the laws of calculation, 
the expansion of the system of numbers has become necessary, 
into 

positive and negative numbers, 

integral numbers and fractions, 

rational and irrational numbers, 

real and imaginary numbers and complex imaginary numbers. 

Therewith closes the field of algebra, and all the algebraic 
operations and their reverse operations can be carried out ir- 
respective of the values of terms entering the operation. 

Thus within the range of algebra no further extension of the 
system of numbers is necessary or possible, and the most general 
number is 

a + jb, 

where a and 6 can be integers or fractions, positive or negative, 
rational or irrational. 

Any attempt to extend the system of numbers beyond the 
complex quantity, leads to numbers, in which the factors of a 
product are not interchangeable, in which one factor of a product 


470 ALTERNATING-CURRENT PHENOMENA 

may be zero without the product being zero, etc., and which thus 
cannot be treated by the usual methods of algebra, that is, are 
extra-algebraic numbers. Such for instance are the double fre- 
quency vector products of Chapter XV. 

Algebraic Operations with General Numbers 
319. Definition of imaginary unit: 

h = - 1. 

Complex imaginary number: 

A = a -\- jb. 

Substituting : 


it is 
where 


a = r cos /3, 
b — r sin ^, 

r(cos /S + i sin /3), 


r = Va^-\- b^ 

b 

tan 8 = -■> 
a 

r = vector, 

j3 = amplitude of general number, A. 

Substituting: 

cos/3 = 2 ' 

sin ^ = — 2j— ' 
it is 

A = re^^, 

where e = jirn^ (l +]) == S j x 2 X 3 X . . • X fc' 

is the basis of the natural logarithms. 
Conjugate numbers are called: 

a -\- jb = r(cos /3 + j sin /3) = re"'^, 

and a - jb = r(cos [- /3]+ j sin [- /3]) = r(cos /3 - j sin /3) = re ^^ 

it is 

(o + j6)(a - jb) = a2 + 62 = ^2, 


APPENDIX 471 

Associate numbers are called: 

a -\- jb = r (cos /? + j sin /S) = re-'^, 


and 

it is 

If 

it is 

If 

it is 


b+3a = r (cos g - /?] + i sin [| - ^]) = re^S"^)' 

(a + i6)(& + ja) = i(a^ + h') = jr'- 

a -{- jb = a' -\- jb', 

a = a' , 
b = b\ 

a-\-jb = 0; 

a = 0, 
6=0. 

320. Addition and Subtraction: 

(a + jb) ± {a' + jb') = (a + a') + j (6 + 6'). 

Multiplication: 

(a + i6)(a' + jV) = (aa' - 66') + j (ab' + 6a') ;^ 

or r (cos /3 + j sin /3) X r'(cos /S' + i sin ^') = rr' (cos [/3 + /3'] 

+ jsin[^ + /3']); 
or re^'^X r'e^'^' = rr'e^^ + ^'^ 

Division: 

Expansion of complex imaginary fraction, for rationalization 
of denomxinator or numerator, by multiplication with the con- 
jugate quantity: 

a+i6 _ (g + 3b') {a' - jb') ^ jaa' + 660 + j jba' - ab') 

a' -\-jb' ~ {a' -\-jb') {a' - jb') ~ a'^ + b'^ 

_ (a+jb)(a-jb) ^ a' + ¥ . 

~ (a' + jb') {a - jb) {aa' + 66') + j (ab' - ba') ' 

or, 

r (cos /3 + ?" sin /3) r , ,^ /m , • • ro an\ 

-rr g-T*^. ■ "l' = "7 (cos i3 - &']-^ J sm )3 - /3' ) ; 

r (cos /3 + J sm jS ) r 

or, 

r'e>^' r' 


472 ALTERNATING-CURRENT PHENOMENA 

involution: 

(a + j6)" = {r(cos /3 + j sin /3) }" = {re'^}" 
= r"(cos n^ + i sin n^) = rV"^; 
evolution: 

y/a + j& = Vr (cos iS + j sin /S) = \/re^ 

= Vr (^cos - + J sm ^) = -s/Ve ". 

321. i?oois of the Unit: 

vT = +1, - 1; 

VI = + 1, 2" ' 2 ' 

vT = +1, - 1, +i, -i; 

V2 V2 \/2 \/2 

n^ 27rA; , . . 2Trk ?z^ , 

VI = cos --— + J sin — — = e n , A; = 0, 1, 2 . . . . n - 1. 

322. Rotation: 

In the complex imaginary plane, multiplication with 

nr- 27r , . . 27r 2^' 

V 1 = cos f 7 sm ~- ^ e n 

n ^ n 

means rotation, in positive direction, by - of a revolution, 

n ' 

multiphcation with (- 1) means reversal, or rotation by 180°, 
multiplication with (+ j) means positive rotation by 90°, 
multiphcation with ( - j) means negative rotation by 90°. 

323. Complex Imaginary Plane: 

While the positive and negative numbers can be represented 
by the points of a line, the complex imaginary numbers or general 
numbers are represented by the points of a plane, with the hori- 
zontal axis, AVA, as real axis, the vertical axis, BVB, as im- 
aginary axis. Thus all 

the positive real numbers are represented by the points of half- 
axis OA toward the right; 

the negative real numbers are represented by the points of half- 
axis OA' toward the left; 


APPENDIX 473 

the positive imaginary numbers are represented by the points of 

half-axis OB upward; 
the negative imaginary numbers are represented by the points of 

half-axis OB' downward; 
the complex imaginary or general numbers are represented by the 

points outside of the coordinate axes. 


INDEX 


Absolute values of complex quanti- 
ties, 37 
Actual generated e.m.f., alternator, 

272 
Admittance, 55 

of dielectric, 154 
due to eddy currents, 137 
to hysteresis, 129 
Admittivity of dielectric circuit, 160 
Air-gap in magnetic circuit, 119, 132 
Ambiguity of vectors, 39 
Amplitude, 6, 20 

Apparent capacity of distorted wave, 
386 
efficiency of induction motor, 

234 
impedance of transformer, 201 
torque efficiency of induction 
motor, 234 
Arc causing harmonics, 353 

as pulsating resistance, 352 
volt-ampere characteristic, 354 
wave constrtiction, 355 
Armature reaction of alternator, 260, 

272 
Average value of wave, 11 

Balanced polyphase system, 397 
Balance factor of polyphase system, 

406 
Brush discharge, 112 

Cable, topographical characteristic, 

42 
Capacity, 4, 9 

of line, 174 
Choking coil, 96 

Circuit characteristic of line and 
cable, 44 

dielectric and dynamic, 159 

factor of general wave, 383 
Coefficient of eddy currents, 138 

of hysteresis, 123 
Combination of sine waves, 31 


Compensation for lagging currents 

by condensance, 72 
Condensance in symbolic expression, 

36 
Condenser as reactance and suscep- 
tance, 96 
with distorted wave, 384 
motor on distorted wave, 392 
motor, single-phase induction, 

249, 257 
synchronous, 339 
Conductance of circuit with induc- 
tive line, 84 
direct current, 55 
due to eddy currents, 137 
effective, 111 

due to hysteresis, 126 
parallel and series connection, 
54 
Conductivity, dielectric, 153 
of dielectric circuit, 160 
Constant current from constant po- 
tential, 76 
synchronous motor, 337 
potential constant current trans- 
formation, 76 
Consumed voltage, by resistance, re- 
actance, impedance, 23 
Control of voltage by shunted sus- 

ceptance, 89 
Corona, 112, 161 

of line, 174 
Counter e.m.f. of impedance, react- 
a,nce, resistance, self-induc- 
tion, 23 
of synchronous motor, 24, 315 
Crank diagram, 19 

and polar diagram, comparison, 
51 
Critical voltage of corona, 166 
Cross currents in alternators, 293 
Cross flux, magnetic of transformer, 

187 
Cycle, magnetic or hysteresis, 114 


475 


476 


tstntK 


Delta connection of thrt^^pha^; *iy«i- 
tern, 416 
current in thrfoptia**-. *jyjjttrfjf», 

417 
delta transformation, 425 
Y transformation, 425 
vohage in three-pliase 9>'9teui, 
417 
I>emagnetizing effect of eddy cur- 
rents, 142 
Diametrical connection of trans- 
formers, six -phase, 429 
Dielectric circuit, 159 
density, 152 
field, 1.50 

hysteresis, 112, 1.50 
strength. 161 
Direct-current system, erhciency, 

441 
Displacement current. 152 
Disruptive gradient. 165 
Distortion by magnetic field, resist- 
ance and reactance pulsa- 
tion. a42 
of magnetizing current. 117 
of wave, see Harmonics 
by hysteresis, 116 
Distributed capacity. 168 
Double delta connections of trans- 
formers to sis-phase. 428 
frequency power and torque 
with distorted wave, 381 
quantities, 180 
peak wave. 370 
T connections of transformers 

to six -phase, 430 
^ connection of transformers to 
six-phase, 429 
Drop of voltage in line, 25 
Dynamic circuit, 159 

Eddy currents, 112 

admittance, 137 

coefficient, 138 

conductance, 137 

in conductor, 144 

loss with distorted wave, 377 
of power, 136 
Effective circuit constants. 168 


.2, f». &. HI 

valiK? at wav*-. It 
in i^olar dia|tmm. «^ 
Kffi«i««MO^ iff drruit wiith indutrtive 

induction motor. 234 

Kl'-' ' trtf 

K.j: 123 

Energy distance oJ dielectric field, 
165 
flow in polyph&K 9>-9lem. 406 
and torque as component of 
double frequency vector, 
186 
Epoch. 6 

Equivalent circuit of transformer, 

202 

sine wave in polar diagram, 53 

single-phase circuit oi poh"phase 

system, 448 

Excitation of induction generator, 

238 

Exciter of induction generator. 238 

Exciting admittance of induction 

motor. 211 

current of induction motor, 211 

single-phase induction motor, 

247 
transformer, 189 

Field characteristic of alternator. 265 
Fifth harmonic. 370 
Five-wire system, efficiency, 466 
Flat top wave. 370 

zero wave, 370 
Foucault ctirrenta, 113 
Four-phase system, 397 

wire systems, efficiency. 466 
Frequency, 6 

General w.*ve, symlxilism, 379 
Generator, induction, 237 

HijrnioiHrs. 7 

caused h\ arr. .liW 


"-^AkWMtftkdSv^ 


^i^v 


■t-«j, .-.>< 


X ^ 


.*■% i ^ v-w. 


-fwt Tjt-^^' STT" 


»e 


'<;--v\ -yS 


leai^^^i^ -j^ 


^■«r. KS? 


H I iiftp^ii*f^ 


174 


11 


miwacvt^ 2P 
pal.ijihaM epibeBk. 397 


—^t, 117 

: .....: of inchicm-e line. 

S3 

non-md'aCTive cirruit and in- 
ductive line, SI 


476 


INDEX 


Delta connection of three-phase sys- 
tem, 416 
current in three-phase system, 

417 
delta transformation, 425 

Y transformation, 425 
voltage in three-phase system, 

417 
Demagnetizing effect of eddy cur- 
rents, 142 
Diametrical connection of trans- 
formers, six-phase, 429 
Dielectric circuit, 159 
density, 152 
field, 150 

hysteresis, 112, 150 
strength, 161 
Direct-current system, efficiency, 

441 
Displacement current, 152 
Disruptive gradient, 165 
Distortion by magnetic field, resist- 
ance and reactance pulsa- 
tion, 342 
of magnetizing current, 117 
of wave, see Harmonics 
by hysteresis, 116 
Distributed capacity, 168 
Double delta connections of trans- 
formers to six-phase, 428 
frequency power and torque 
with distorted wave, 381 
quantities, 180 
peak wave, 370 

T connections of transformers 
to six-phase, 430 

Y connection of transformers to 

six-phase, 429 
Drop of voltage in line, 25 
Dynamic circuit, 159 

Eddy currents, 112 

admittance, 137 

coefficient, 138 

conductance, 137 

in conductor, 144 

loss with distorted wave, 377 
of power, 136 
Effective circuit constants, 168 


Effective circuit conductance, 111 
power, 180 
reactance, 112 
resistance, 2, 5, 9, 111 
susceptance, 112 
value of wave, 11 
in polar diagram, 53 
Efficiency of circuit with inductive 
line, 88, 95 
induction motor, 234 
Electrostatic, see Dielectric 
E.m.f. of self-induction, 123 
Energy distance of dielectric field, 
165 
flow in polyphase system, 406 
and torque as component of 
double frequency vector, 
186 
Epoch, 6 

Equivalent circuit of transformer, 

202 

sine wave in polar diagram, 53 

single-phase circuit of polyphase 

system, 448 

Excitation of induction generator, 

238 

Exciter of induction generator, 238 

Exciting admittance of induction 

motor, 211 

current of induction motor, 211 

single-phase induction motor, 

247 
transformer, 189 

Field characteristic of alternator, 265 
Fifth harmonic, 370 
Five-wire system, efficiency, 466 
Flat top wave, 370 

zero wave, 370 
Foucault currents, 113 
Four-phase system, 397 

wire systems, efficiency, 466 
Frequency, 6 

General wave, symbolism, 379 
Generator, induction, 237 

Harmonics, 7 

caused by arc, 353 


INDEX 


477 


Harmonics of current, 341 

by hysteresis, 116, 358 

by three-phase transformer, 363 

of voltage, 341 
Hedgehog transformer, 189 
Hemisymmetrical polyphase sys- 
tem, 404 
Higher harmonics, see Harmonics 
Hysteresis, admittance, 129 

advance of phase, 122, 130 

coefficient, 123 

conductance, 126 

cycle, 115 

unsymmetrical, 135 

dielectric, 150 

dielectric and magnetic, 112 

in line, 174 

loss, 122 

with distorted wave, 377 

power current, 117 
voltage, 123 

Imaginary power, 186 
Impedance, 2, 9 

apparent, of transformer, 201 
of induction motor, 211 
in series with circuit, 69 
series and parallel connections, 

55, 59 
in symbolic expression, 35 
synchronous, of alternator, 277 
Independent polyphase system, 397 
Inductance, 3, 9 

factor of general wave, 382 
Induction generator, 237 

machine as inductive reactance, 

96 
motor, 208 

on distorted wave, 392 
Inductive devices, starting single- 
phase induction motor, 246 
line, maximum power, 82 
Inductor alternator, unsymmetrical 

magnetic cycle, 135 
Influence, electrostatic, from line, 

174 
Instantaneous value, 11 
Intensity of wave, 20 
Interlinked polyphase system, 397 


Inverted three-phase system, 398, 
408, 413 
efficiency, 466 
Ironclad circuit, 119, 131 

wave shape distortion, 358, 361 
Iron wire and eddy currents, 140 
unequal current distribution, 
147 

j as distinguishing index, 32 

as imaginary unit, 33 
Joule's law, 1, 5 

Kirchhoff's laws, direct current, 1 
in crank diagram, 22, 60 
in polar diagram, 49 
in symbolic expression, 34 

Lag ill alternator, demagnetizing, 
260 
of current, 21 

in synchronous motor, magnet- 
izing, 261 
Laminated iron and eddy currents, 

138 
Lead in alternator, magnetizing, 260 
of current, 21 

by synchronous condenser, 339 
in synchronous motor, demag- 
netizing, 261 
Leakage, 112, 151 ' 

currents through dielectric, 152 

in transformer, 189 
of line, 174 

reactance of transformer, 187 
Line capacity, 169 
phase control, 99 
power factor control, 99 
topographic characteristic, 43 
Load curves of synchronous motor, 
333 

Magnetic cycle, 114 
hysteresis, 112 
Magnetizing current, 117 
Maximum output of inductive line, 
83 
non-inductive circuit and in- 
ductive line, 81 


478 


INDEX 


Maximum power of induction mo- 
tor, 222 
torque of induction motor, 219 
Mean value of wave, 12 
Metering of polyphase systems, 442 
M.m.f., rotating, of polyphase sys- 
tem, 401 
Molecular friction, 112 
Monocyclic connection of trans- 
formers, 428 
devices, starting single-phase 

induction motor, 246 
system, 409 
Multiple phase control, 108 
Mutual inductance, 174 
induction, 147 
inductive reactance of line, 174 

Neutral voltage of three-phase trans- 
former, 367 

Nominal generated e.m.f. of alter- 
nator, 263, 276, 282 

Non-inductive circuit and inductive 
line, 79, 81 

Ohm's law, 1 

Open delta transformation, 427 
Oscillating waves, 175 
Output, see Power 

of circuit with inductive line, 
82, 95 

in phase control, 104 

Parallel connection of admittances, 
59 
of resistances and conduct- 
ances, 54 
operation of alternators, 292 
Parallelogram of sine waves, 22 

in polar diagram, 48 
Peaked waves, 370 
Peaks of voltage by wave distortion, 

360, 367 
Permittivity, 152 

of dielectric circuit, 160 
Phase, 6, 20 

advance angle by hysteresis, 
122, 1.30 


Phase characteristic of synchronous 
motor, 328 
control, 97 

multiple, 108 
difference in transformer, 29 
splitting devices starting single- 
phase induction motor, 246 
Polar coordinates of alternating 
waves, 46 
and crank diagram, comparison, 
51 
Polarization, 4 

cell as condensive reactance, 96 
Polycyclic systems, 409 
Polygone of sine waves, 22 

in polar diagram, 48 
Polyphase and constituent single- 
phase circuit, 448 
Power, see Output 

characteristics of polyphase sys- 
tems, 409 
components of current and volt- 
age, 168 
consumption by corona, 165 
as double frequency vector, 180 
factor of arc, 356 

correction by synchronous 

condenser, 339 
of dielectric circuit, 152 
of general wave, 382 
of induction motor, 234 
phase control, 99 
of general wave, 381 
of induction motor, 216, 222 
loss in dielectric, 157 
of sine wave, 22 
vector denotation, 179 
of wave in polar diagram, 49 
Primary admittance of transformer, 
197 
impedance of transformer, 198 
Pulsating magnetic circuit, 135 

wave, 11 
Pulsation of magnetic circuit, react- 
ance and resistance, 342 

Quadrature components of alterna- 
tor armature reaction and 
reactance, 282 


INDEX 


479 


Quadrature flux of single-phase in- 
duction motor, 245 
Quarter-phase system, 398 
efficiency, 466 
three-phase transformation, 423 
Quintuple harmonic, see Fifth har- 
monic 

Radiation from line, 174 
Ratio of transformer, 197 
Reactance, 2, 9 
effective, 112 
in phase control, 103 
in series with circuit, 63 
in symbolic expression, 35 
synchronous, of alternator, 277 
Reactive component of current and 
voltage, 168 
power, 180 

with general wave, 382 
Rectangular components, 31 
Reduction of polyphase system to 

single-phase circuit, 448 
Regulation of circuit with inductive 
line, 82, 86 
curve of alternator, 290 
Resistance, effective, 2, 5, 9, 111 
of line, 174 
parallel and series connection, 

54 
in series with circuit, 60 
in starting induction motor, 

224 
in symbolic expression, 35 
Resolution of sine waves, 31 
Resonance of condenser with dis- 
torted wave, 387 
by harmonics, 373 
Ring connection of polyphase sys- 
tem, 416 
current in polyphase system, 

417 
voltage in polyphase system, 
417 
Rise of voltage of circuit by shunted 

susceptance, 94 
Rotating field of symmetrical poly- 
phase system, 401 
Ruhmkorff coil, 7 


Saturation, magnetic, induction gen- 
erator, 238 
Saw-tooth wave, 370 
Screening effect of eddy currents, 142 
Secondary impedance of trans- 
former, 198 
Self-excitation of induction genera- 
tor, 238 
Self-inductance, 174 
Self-inductive reactance of alterna- 
tor, 261 
of transformer, 187 
voltage, 123 
Series connection of impedances, 55, 
59 
of resistances and conduct- 
ances, 54 
impedance in circuit, 69 
operation of alternators, 294 
reactance in circuit, 63 
resistance in circuit, 60 
Sharp zero wave, 370 
Short circuit of alternator, 273, 288 
Shunted condensance and lagging 

current, 72 
Silent discharge from line, 174 
Single-phase cable, topographical 
characteristic, 42 
circuit equivalent to polyphase 

system, 448 
efficiency, 466 
induction motor, 245 
system, 398 
Slip of induction motor, 208 
Spheres, dielectric field, 164 
Stability of induction motor, 238 
Star connection of polyphase system, 
415 
current in polyphase system, 

417 
voltage in polyphase system, 
417 
Starting devices of single-phase in- 
duction motor, 245 
torque of induction motor, 223 
single-phase induction motor, 
252 
Susceptance, 55 

of circuit with inductive line, 82 


480 


INDEX 


Susceptance, effective, 112 
Susceptivity, dielectric, 153, 160 
Symbolic expression of power, 181 
Symmetrical polyphase system, 396 
Synchronizing power of alternators, 

294 
Synchronous condenser, 339 

converter for phase control, 98 
impedance of alternator, 277 
machine as shunted susceptance, 

96 
motor, fundamental equation, 
316 
for phase control, 98 
supplied by distorted wave, 
389 
reactance of alternator, 262, 272 
watts as torque, 233 

T connection of transformers, 427 
Terminal voltage of alternator, 263 
Tertiary circuit with condenser, 
single-phase induction mo- 
tor, 249 
Third harmonic, 369 

in three-phase system, 364 
Three-phase line, topographic char- 
acteristic, 43 
quarter-phase transformation, 

423 
system, 397 
efficiency, 466 
voltage drop, 41 
transformer, wave distortion, 
363 
Three-wire single-phase system, effi- 
ciency, 466 
Time constant, 3 

and crank diagram, comparison, 

51 
diagram of alternating wave, 48 
Topographic characteristic of cable 

and line, 42 
Torque as double frequency vector, 
185 
efficiency of induction motor, 
234 


Torque of induction motor, 216, 219, 
223 
single-phase induction motor, 
248, 252 

Transformation by two transform- 
ers, of polyphase systems, 
422 

Transformer, 187 
diagram, 26, 30 
equivalent circuit, 202 

Transmission line, see Line 

Treble peak wave, 370 

Triple harmonic, see Third harmonic 

True power of generator wave, sym- 
bolic, 382 

Unbalanced polyphase system, 397 
quarter-phase system, 463 
three-phase system, 461 

Unequal current distribution in con- 
ductor, 144 

Unsymmetrical hysteresis cycle, 135 
polyphase system, 396 

V connection of transformers on 

three-phase system, 427 
Vector power, 179 
Virtual generated e.m.f. of alter- 
nator, 272 
Voltage of circuit with inductive 
line, 82, 86 
control by shunted susceptance, 
89 
by synchronous condenser, 
339 
peaks by wave distortion, 360, 

367 
phase control, 99 

Y connection of three-phase system, 

416 
current in three-phase system, 

• 417 
Delta transformation, 426 
voltage in three-phase system, 

417 
Y transformation, 426 


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