Theory and Calculation of Alternating Current Phenomena (1900) — part 18 of 19

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 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 preferable, 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 different 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 un- balanced at unequal distribution of load ; the three-wire quarter-phase system is unbalanced in voltage and phase, even at equal distribution of load.

APPENDICES.

APPENDIX I.

ALGEBRA OF COMPLEX IMAGINARY QUANTITIES.

INTRODUCTION.

296. 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 ex- pressing 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.

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

a, a + 1, a + 2, a + 3 . . . . Counting by a given number of integers :

b integers introduces the operation of addition, as multiple counting :

a + b = c. It is, a + b = b + a,

490 APPENDIX 7.

that is, the terms of addition, or addenda, are interchange- able.

Multiple addition of the same terms :

a -+- a -- a -+- . . . + a = c

b equal numbers introduces the operation of multiplication :

a x b = c. It is, a X b = b X a,

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

Multiple multiplication of the same factors :

aX aX aX . . • X a = c

b equal numbers introduces the operation of involution :

Since ab is not equal to #",

the terms of involution are not interchangeable.

298. The reverse operation of addition introduces the operation of subtraction :

If a + 6 = f,

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 subtraction under any circumstances, the system of abso- lute numbers has to be expanded by the introduction of the negative number:

_ « = (_ 1) X «, .where (- 1)

is the negative unit.

Thereby the system of numbers is subdivided in the

COMPLEX IMAGINARY QUANTITIES. 491

positive and negative numbers, and the operation of sub- traction 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 - (- b) = c + b, thus: (-l)X (-!) = !;

that is, the negative unit is defined by, (—I)2 = 1.

299. The reverse operation of multiplication introduces the operation of division :

If a X b = c, then - = a.

b

In the system of integral numbers this operation can only be carried out, if b 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 infraction:

:©.

where - is the integer fraction, and is defined by :

T- x b = 1.

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

the quantities a and b are not reversible.

Thus V^ = <z, the evolution,

= b, the logarithmation.

The operation of evolution of terms c, which are not •complete powers, makes a further expansion of the system

492 APPENDIX I.

of numbers necessary, by the introduction of the irrational number (endless decimal fraction), as for instance :

V2 = 1.414213.

301. The operation of evolution of negative quantities c with even exponents b, as for instance

2/ -

makes a further expansion of the system of numbers neces- sary, by the introduction of the imaginary unit.

-V^l Thus -x/^ = -v/^T x •#*.

where : V— 1 is denoted by/.

Thus, the imaginary unity is defined by :

f = _ 1.

By addition and subtraction of real and imaginary units, compound numbers are derived of the form :

which are denoted as complex imaginary mimbers.

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.

302. Thus, starting from the absolute integral num- bers 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 neces- sary, into

Positive and negative numbers,

Integral numbers and fractions,

Rational and irrational numbers,

COMPLEX IMAGINARY QUANTITIES. 493

Real and imaginary numbers and complex imaginary numbers.

Therewith closes the field of algebra, and all the alge- braic operations and their reverse operations can be carried out irrespective of the values of terms entering the opera- tion.

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 b can be integers or fractions, positive or negative, rational or irrational.

ALGEBRAIC OPERATIONS WITH COMPLEX IMAGINARY QUANTITIES.

303. Definition of imaginary unit:

f2 = - 1. Complex imaginary number:

Substituting :

a = r cos (3 b = r sin (3, it is A = r (cos /3 -f / sin /?),

where r = a2 -- \

a

r = vector, /3 = amplitude of complex imaginary number A.

Substituting :

eJft 4- c-JP

H

cos

sin/? =

494 APPENDIX I.

it is A = reJP,

where c = lim (l + -}"= yJT _ 1 „=» V n) o~lx2X3x

is the basis of the natural logarithms. Conjugate numbers :

a -- j b = r (cos ft -- j sin ft) = reJ'P •and a — jb = r (cos [—/?]+> sin [— /?]) it is

Associate numbers: a + jb = r (cos ft +/ sin /3) =

and b +

ja = r ( cos 1 ^ — (3\ -f j sin \7-

it is

(a+jb)(b+ja)=j(a*+P)

If

a+jb = a' +jb',

it is

a = af

If

a +J/= 0 ;

it is

a = 0,

304. Addition and Subtraction :

Multiplication :

(a +jb) (a' +jb') = (aa1 - b b') +j(ab' + b a') or r (cos ^3 + / sin ft) X r' (cos /? + / sin ftf) = r r' (cos [£ -p

^]+ysin[/3 + ^]); or re J* X r'^'07 = rr'ef& + M.

Division :

Expansion of complex imaginary fraction, for rationaliza- tion of denominator or numerator, by multiplication with the conjugate quantity :

COMPLEX IMAGINARY QUANTITIES. 495*"

a+jb = (a+jb}(a' -jb'} = (aar+ bb'} +j (b a' - ab'} -jb'} , *" + *"

(a! -f j b'} (a — jb} (a a' + b b'} +j(ab' — b a') '

or, _ r ^_p ^ _ ^ .

r'

or>

r

involution :

(a +jbY = {r (cos

evolution :

-v/^- (cos /8 + y sin

305. Roots of the Unit : =+l, -1;

</I=+i, -i, +y, -y;

' +i+y +i-y -i +y

V2 V2 V2

-i-y

V2 '

306. Rotation :

In the complex imaginary plane, multiplication with

9 * 2-n-

VI = cos — +y sin — = e

means rotation, in positive direction, by 1 / n of a revolution,

496 APPENDIX I.

multiplication with (—1) means reversal, or rotation by 180°, multiplication with (+y ) means positive rotation by 90°, multiplication with (— /) means negative rotation by 90°.

307. Complex imaginary plane :

While the positive and negative numbers can be rep- resented by the points of a line, the complex imaginary numbers are represented by the points of a plane, with the horizontal axis A' O A as real axis, the vertical axis Br O B as imaginary axis. Thus all

the positive real numbers are represented by the points of half

axis OA towards the right ; the negative real numbers are represented by the points of half

axis OA' towards the left ; the positive imaginary numbers are represented by the points of

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

half axis OB' downwards ; the complex imaginary numbers are represented by the points

outside of the coordinate axes.

APPENDIX II.

OSCILLATING CURRENTS.

INTRODUCTION.

308. An electric current varying periodically between constant maximum and minimum values, — that is, in equal time intervals repeating the same values, — is called an alternating current if the arithmetic mean value equals zero ; and is called a pulsating current if the arithmetic mean value differs from zero.

Assuming the wave as a sine curve, or replacing it by the equivalent sine wave, the alternating current is charac- terized by the period or the time of one complete cyclic change, and the amplitude or the maximum value of the current. Period and amplitude are constant in the alter- nating current.

A very important class are the currents of constant period, but geometrically varying amplitude ; that is, cur- rents in which the amplitude of each following wave bears to that of the preceding wave a constant ratio. Such currents consist of a series of waves of constant length, decreasing in amplitude, that is in strength, in constant proportion. They are called oscillating currents in analogy with mechanical oscillations, — for instance of the pendu- lum,— in which the amplitude of the vibration decreases in constant proportion.

Since the amplitude of the oscillating current varies, constantly decreasing, the oscillating current differs from

497

498

APPENDIX II.

the alternating current in so far that it starts at a definite time, and gradually dies out, reaching zero value theoreti- cally at infinite time, practically in a very short time, short even in comparison with the time of one alternating half- wave. Characteristic constants of the oscillating current are the period T or frequency N = 1/7", the first ampli- tude and the ratio of any two successive amplitudes, the latter being called the decrement of the wave. The oscil- lating current will thus be represented by the product of

V

^ !

I"**'

\

^

-.

\

/

S

r~~

--

__

1

\

180

/

3W

\

MO

^

^-1

raT

X

—

—

TWO

— J

j»W8Q

\

/

\

.

---

^.

•^-i

\

/

_^

—

->T=-

Vy

/.\

-'

-~

t

0

en

atin .

g E 135 cc

M.F

X

" [

E

=5

^ stf>

1

4afs

2°

a periodic function, and a function decreasing in geometric proportion with the time. The latter is the exponential function Af~gt.

309. Thus, the general expression of the oscillating current is

/= ^/-0'COS (2-rrNt — S),

since A'-** = A' A-'* = U~bt.

Where e = basis of natural logarithms, the current may be expressed

7= i(.~bt cos (2-n-JVf— «) = ze-a* cos (<#> - £),

where <#> = %-nNt; that is, the period is represented by a complete revolution.

OSCILLATING CURRENTS.

499

In the same way an oscillating electromotive force will be represented by

E = etra* cos O — 5).

Such an oscillating electromotive force for the values e = 5, a = .1435 or «- 2™ = .4, £ = 0,

is represented in rectangular coordinates in Fig. 207, and in polar coordinates in Fig. 208. As seen from Fig. 207, the oscillating wave in rectangular coordinates is tangent to the two exponential curves,

Fig. 208.

310. In polar coordinates, the oscillating wave is repre- sented in Fig. 208 by a spiral curve passing the zero point twice per period, and tangent to the exponential spiral,

The latter is called the envelope of a system O.L oscillat- ing waves of which one is shown separately, with the same constants as Figs. 207 and 208, in Fig. 209. Its character-

500

APPENDIX II.

istic feature is : The angle which any concentric circle makes with the curve y — ee~a<t>, is

tan a =

which is, therefore, constant ; or, in other words : " The envelope of the oscillating current is the exponential spiral, which is characterized by a constant angle of intersection

Fig. 209.

Fig. 210.

with all concentric circles or all radii vectores." The oscil- lating current wave is the product of the sine wave and the exponential or loxodromic spiral.

311. In Fig. 210 let j/ = e€~a<t> represent the expo-' nential spiral ;

let z = e cos (<£ — a)

represent the sine wave ; and let E = ef.-** cos (<£ — w)

represent the oscillating wave.

We have then

tan y3 =

Ed* _ — sin (<£ — w) — a cos

COS (<£ — oi) = — {tan (<^> — £) + a} ;

— to)

OSCILLATING CURRENTS. 501

that is, while the slope of the sine wave, z = e cos (<£ — w), is represented by

tan y = — tan (<£ — w), the slope of the exponential spiral y = ei'0* is

tan a = — a = constant. That of the oscillating wave E = ?e~a cos (<£ — to) is

tan /3 = — {tan (<£ — w) + a} .

Hence, it is increased over that of the alternating sine wave by the constant a. The ratio of the amplitudes of two consequent periods is

A is called the numerical decrement of the oscillating wave, a the exponential decrement of the oscillating wave, a the angular decrement of the oscillating wave. The oscillating wave can be represented by the equation

£ = ec-**™" cos ($ — 5).

In the instance represented by Figs. 181 and 182> we have A = .4, a = .1435, a = 8.2°.

Impedance and Admittance.

312. In complex imaginary quantities, the alternating wave * = e cos (* - ffl)

is represented by the symbol

E = e (cos w --j sin w) = <?x --jez .

By an extension of the meaning of this symbolic ex- pression, the oscillating wave E = ee~a<t> cos (<f> — w) can be expressed by the symbol

E = e (cos w --j sin w) dec a = (e± --j'e^) dec a, where a = tan a is the exponential decrement, a the angular decrement, e~27ra the numerical decrement.

502 APPENDIX II.

Inductance.

313. Let r = resistance, L = inductance, and x = 2 IT N L = reactance.

In a circuit excited by the oscillating current,

/= /£-«* cos (<£ — w) = /(cos to +y sin w) dec a =

(i --Jz) dec a, where /i = / cos w, /2 = / sin £>, a = tan a.

We have then,

The electromotive force consumed by the resistance r of the circuit ^

The electromotive force consumed by the inductance L of the circuit,

Ef**L—~*iNI&t = *—.

dt d<$> d<$>

Hence Ex = — xif.~a^> (sin (<J> — fy -- a cos (<£ — w)}

xi(.~a^ . ,. „ , N

= sin (^> — w -f- a).

COS a

Thus, in symbolic expression,

£x = - °^—{— sin (w — a) +/ cos (w — a)} dec a

COS a

= — x i (a -f y ) (cos w + 7 sin a>) dec a ; that is, Ex = — x I (a +/') dec a .

Hence the apparent reactance of the oscillating current circuit is, in symbolic expression,

X = x (a +y') dec a.

Hence it contains an energy component ax, and the impedance is

Z = (r — X) dec a = {r — x (a +/')} dec a = (r — ax —jx) dec a.

Capacity.

314. Let r = resistance, C = capacity, and xc = 1 /2-n-JVC = capacity reactance. In a circuit excited by the oscillating

OSCILLATING CURRENTS. 503

current /, the electromotive force consumed by the capacity Cis

or, by substitution,

Ex = x I * e~a* cos (<£

{sin (<£ — w) — a COS (<£ — oi

2

(1 + 02) COS a

hence, in symbolic expression,

sin (</> — u> — a) ;

= 2 (« + /) (cos w +y sin w) dec a ;

hence,

that is, the apparent capacity reactance of the oscillating circuit is, in symbolic expression,

dec

315. We have then:

In an oscillating current circuit of resistance r, induc- tive reactance x, and capacity reactance xc , with an expo- nential decrement a, the apparent impedance, in symbolic expression, is :

*'

1 +a2/ V 1 +**

= ra — jxa;

504 APPENDIX 77.

and, absolute,

Admittance. 316. Let /=/e-a*cos^_£)==current<

Then from the preceding discussion, the electromotive force consumed by resistance r, inductive reactance x, and capa- city reactance xc , is

cos $ — r — ax — a*e — sin (<£ —

= iza(.~a^ cos (<£ — w + 8), where tan 8 = i_^ ,

a r — ax — —. -Xf

substituting & + 8 for G, and ^ = /^a we have

cos <> —

I = — e~a* cos (<#> — w — 8)

,1 \ cos 8 / i ~\ i sin 8 . / , = e e. a<p \ cos (9 — to ) -j sin (9 —

hence in complex quantities,

E = e (cos u> --j sin oi) dec a,

• sin

OSCILLATING CURRENTS. 505

or, substituting,

r — ax — I =E

I- dec a.

317. Thus in complex quantities, for oscillating cur- rents, we have : conductance,

susceptance,

admittance, in absolute values,

/ o i To 1

in symbolic expression,

Y=g+J»

1 + a2/ \ 1 + a2 '

Since the impedance is

Z = ir — ax — we have

506 APPENDIX II.

that is, the same relations as in the complex quantities in alternating-current circuits, except that in the present case all the constants ra , xa , za , g, z, y, depend upon the dec- rement a.

Circuits of Zero Impedance,

318. In an oscillating-current circuit of decrement a, of resistance r, inductive reactance x, and capacity reactance xc, the impedance was represented in symbolic expression by

-jxa =

! + «»

or numerically by

Thus the inductive reactance x, as well as the capacity reactance xc, do not represent wattless electromotive forces as in an alternating-current circuit, but introduce energy components of negative sign

a

— ax — - - x : 1 + a2

that means,

" In an oscillating-current circuit, the counter electro- motive force of self-induction is not in quadrature behind the current, but lags less than 90°, or a quarter period; and the charging current of a condenser is less than 90°, or a quarter period, ahead of the impressed electromotive force."

319. In consequence of the existence of negative en- ergy components of reactance in an oscillating-current cir- cuit, a phenomenon can exist which has no analogy in an alternating-current circuit ; that is, under certain conditions the total impedance of the oscillating-current circuit can equal zero :

In this case we have

r - ax

0 ; x -- ^— = 0,

• — c 1 + a2 1 + fla

OSCILLATING CURRENTS. 507

substituting in this equation

x = 2 TT NL • xc = and expanding, we have a

That is,

" If in an oscillating-current circuit, the decrement 1

and the frequency N = r/4iraL, the total impedance of the circuit is zero ; that is, the oscillating current, when started once, will continue without external energy being impressed upon the circuit."

320. The physical meaning of this is : " If upon an electric circuit a certain amount of energy is impressed and then the circuit left to itself, the current in the circuit will become oscillating, and the oscillations assume the fre- quency N = r/4:7raL, and the decrement

1

That is, the oscillating currents are the phenomena by which an electric circuit of disturbed equilibrium returns to equilibrium.

This feature shows the origin of the oscillating currents, and the means to produce such currents by disturbing the equilibrium of the electric circuit ; for instance, by the discharge of a condenser, by make and break of the circuit, by sudden electrostatic charge, as lightning, etc. Obviously, the most important oscillating currents are

508 APPENDIX II.

those flowing in a circuit of zero impedance, representing oscillating discharges of the circuit. Lightning strokes usually belong to this class.

Oscillating Discharges.

321. The condition of an oscillating discharge is

Z = 0, that is,

~ ~ / .1 r

2aL 2Z~ ~1'

If r = 0, that is, in a circuit without resistance, we have a = 0, Af = 1 / 2 TT VZT ; that is, the currents are alter- nating with no decrement, and the frequency is that of resonance.

If 4 H r2 C - 1 < 0, that is, r > 2 V2T/T, a and N become imaginary ; that is, the discharge ceases to be os- cillatory. An electrical discharge assumes an oscillating nature only, if r < 2 V/, / C. In the case r = 2 VZ, / C we have « = oo , ./V = 0 ; that is, the current dies out without oscillation.

From the foregoing we have seen that oscillating dis- charges, — as for instance the phenomena taking place if a condenser charged to a given potential is discharged through a given circuit, or if lightning strikes the line circuit, — are denned by the equation : Z = 0 dec a.

Since

/ = (/V+y/a) dec a, Er = Ir dec a,

Ex = -x I (a +/) dec a, Exc= ^L/(- a +/) dec a,

we have r-aX--^—Xc = ^

I + a?

hence, by substitution,

Exc= x /(— a +/) dec a.

OSCILLATING CURRENTS. 50 £'

The two constants, t\ and z'2, of the discharge, are deter- mined by the initial conditions, that is, the electromotive force and the current at the time t = 0.

322. Let a condenser of capacity C be discharged through a circuit of resistance r and inductance L. Let e = electromotive force at the condenser in the moment of closing the circuit, that is, at the time t — 0 or <£ = 0. A.t this moment the current is zero ; that is,

7=//2, /1==0. Since Exe= •*/(— a +/) dec a = e at <f> = 0,

we have x /2 Vl + a2 = e or /2 = = .

x V 1 + a2 Substituting this, we have,

I —j — e dec a, Er =je r dec a,

x Vl + a2 x Vl + az

Ex = e (1 -ja) dec a, ^c= e (1 +/ «) dec a,

Vl + «8 Vl + a2

the equations of the oscillating discharge of a condense of initial voltage e.

Since x = 2 *• N L,

1

we have

x =

hence, by substitution,

l

— dec a,

.510 APPENDIX II.

E - ef\fC

-f^r-, — — rr~ / r

47TZ

the final equations of the oscillating discharge, in symbolic expression.

Oscillating Current Transformer.

323. As an instance of the application of the symbolic method of analyzing the phenomena caused by oscillating currents, the transformation of such currents may be inves- tigated. If an oscillating current is produced in a circuit including the primary of a transformer, oscillating currents will also flow in the secondary of this transformer. In a transformer let the ratio of secondary to primary turns be/. Let the secondary be closed by a circuit of total resistance, i= r{ -- TJ", where 1\ = external, 1' = internal, resistance. The total inductance Ll = Z/ -f /,/', where Z/ = external, Zj" = internal, inductance ; total capacity, Cv Then the total admittance of the secondary circuit is

) dec a =

where xl= 2irJVLl= inductive reactance: xcl = \l1-jrNC ' = capacity reactance. Let rQ = effecive hysteretic resistance, Z = inductance ; hence, x^ = Z-n-N LQ = reactance ; hence,

admittance

of the primary exciting circuit of the transformer ; that is, the admittance of the primary circuit at open secondary circuit.

As discussed elsewhere, a transformer can be considered as consisting of the secondary circuit supplied by the im- pressed electromotive force over leads, whose impedance is

OSCILLATING CURRENTS. 511

equal to the sum of primary and secondary transformer im- pedance, and which are shunted by the exciting circuit, out- side of the secondary, but inside of the primary impedance. Let r = resistance ; L = inductance ; C = capacity ;

hence' x = 2 TT NL = inductive reactance,

xc = 1 / 2 TT N C = capacity reactance of the total primary circuit, including the primary coil of the transformer. If EI = EI dec a denotes the electromotive force induced in the secondary of the transformer by the mutual magnetic flux ; that is, by the oscillating magnetism interlinked with the primary and secondary coil, we have Iv = E^ Yl dec a = secondary current.

Hence, // = / 7X dec a = pEJ Yl dec a = primary load current, or component of primary current corresponding to

secondary current. Also, 70 = - 2j/ F0 dec a = primary

/ ' exciting current ; hence, the total primary current is

/= // + 70 = £-'{Fo +/2 Y,} dec a.

E' E' = -^-i- dec a = induced primary electromotive force.

/ Hence the total primary electromotive force is

E = (£' + /Z) dec a = £L (1 + Z F0 +/2Z Y,} dec a. P

In an oscillating discharge the total primary electro- motive force E = 0 ; that is,

or, the substitution

a

1 +

(r0 - ax0) -.

. 0.

512 APPENDIX II.

Substituting in this equation, ^r=2 it N C, xc = ~L/'2 etc., we get a complex imaginary equation with the two constants a and N. Separating this equation in the real and the imaginary parts, we derive two equations, from which the two constants a and N of the discharge are calculated.

324. If the exciting current of the transformer is neg- ligible, — that is, if YQ = 0, the equation becomes essentially simplified, —

I a \ . I x \

(r — a x xc 1 — j ( x — I

1+/2v 1 + *8 i v Ljt^l=0;

that is,

or, combined, —

(r, -2aXl) +/2 (r-2 ax) = 0,

Substituting for xlt x, xel, xei we have

+/aZ)

i+/V / 4(A+/ +/2Z) V (n +/V)2 (Ci

!} dec a,

7 =pEi YI dec a, /! = ^/ F! dec a,

the equations of the oscillating-current transformer, with E{ as parameter.

INDEX.

PAGE

Addition 494. 498

Admittance, conductance, suscep-

tance, Chap. vn. ... 52

definition 53

parallel connection ... 57 primary exciting, of trans- former 204

of induction motor . . . 240 Advance of phase, hysteretic . .115 Algebra of complex imaginary

Quantities, App. I. . . . 489 Alternating current generator,

Chap, xvii 297

transformer, xiv 193

motor, commutator, Chap.

xx 354

motor, synchronous, Chap.

xix 321

Alternating wave, definition . . 11 general ..."... . 7 Alternators, Chap. xvii. . . . 297 parallel operation, Chap.

xvin 311

series operation 313

synchronizing, Chap. xvin. . 311 synchronizing power in paral- lel operation 317

Ambiguity of vectors .... 43 Amplitude of alternating wave . 7 Angle of brush displacement in

repulsion motor .... 361 Apparent total impedance of

transformer 208

Arc, distortion of wave shape by 394

power factor of 395

Arithmetic mean value, or average

value of alternating wave 11 Armature reaction of alternators

and synchronous motors . 297 51

Armature reaction of alternators, as affecting parallel opera- tion 313

self-induction of alternators

and synchronous motors . 300 slots, number of, affecting wave shape 384

Associate numbers 494

Asynchronous, see induction . .

Average value, or mean value of

alternating wave .... 11

Balance, complete, of lagging currents by shunted con-

densance 74

Balanced and unbalanced poly- phase systems, Chap.

xxvii 440

Balanced polyphase system . . 431 quarter-phase system . . . 484 three-phase system . . . 481 Balance factor of polyphase sys- tem 441

of lagging currents by shun- ted condensance ... 75 Biphase, see quarter-phase . .

Cables, as distributed capacity . 158 with resistance and capacity topographic circuit charac- teristic 47

Calculation of magnetic circuit

containing iron . . . • 125 of constant frequency induc- tion generator .... 269 of frequency converter . . 232 of induction motor . . . 262 of single-phase induction mo- tor . . 287

514

INDEX.

Calculation of transmission lines,

Chap, ix 83

Capacity and inductance, dis- tributed, Chap. xin. . . 158 as source of reactance . . 6 in shunt, compensating for

lagging currents .... 72 intensifying higher harmon- ics 402

see condenser and conden-

sance. Chain connection of induction

motors, or concatenation . 274 Characteristic circuit of cable with resistance and capa- city 48

circuit of transmission line with resistance, inductance, capacity, and leakage . . 49 curves of transmission lines . 172 field of alternator .... 304 power of polyphase systems 447 Circuit characteristic of cable with resistance and capa- city 48

characteristic of transmission line with resistance, induc- tance, capacity and leakage 49 factor of distorted wave . . 415 with series impedance . . 68 with series reactance ... 61 with series resistance ... 58 Circuits containing resistance, in- ductance, and capacity,

Chap, vin 58

Coefficient of hysteresis . . 116 Combination of alternating sine waves by parallelogram or polygon of vectors ... 21 of double frequency vectors,

as power 163

of sine waves by rectangular

components 35

of sine waves in symbolic

representation .... 38 Commutator motor, Chap. xx. 354

Compensation for lagging cur- rents by shunted conden-

sance 72

Complete diagram of transmis- sion line in space . . .192 Complex imaginary number . . 492 imaginary quantities, algebra

of, App. i 489

imaginary quantities, as sym- bolic representation of al- ternating waves .... 37

quantity Chap, v 33

Compounding curve of frequency

converter 232

Concatenated couple of induction

motors, calculation . . . 276 Concatenation of induction mo- tors 274

Condensance in shunt, compen- sating for lagging currents 72 in symbolic representation . 40 or capacity reactance ... 6 see capacity and condenser Condensers, distortion of wave

shape by 393

see capacity and condensance with distorted wave . . . 419 with single-phase induction

motor 286

Conductance, effective, definition 104 in alternating current cir- cuits, definition .... 54 in continuous current cir- cuits 52

of receiver circuit, affecting

output of inductive line . 89 parallel connection ... 52 see resistance

Conjugate numbers 494

Constant current — constant po- tential transformation . . 76 current, constant potential transformation by trans- mission line 181

potential, constant current transformation . . 76

INDEX.

515

Constant potential, constant cur- rent transformation by transmission line .... 181

rotating M M.F 436

Constants, characteristic, of in- duction motor .... 262 Continuous current system, distri- bution efficiency .... 473 Control, by change of phase, of

transmission line, Chap. ix. 83 of receiver circuit by shunted

susceptance 96

Converter of frequency, Chap.

xv 219

Counter E.M.F. constant in syn- chronous motor .... 349

of impedance 25

of inductance 25

of resistance .25

of self-induction ..... 24 Counting or numeration . . . 489 Cross-flux, magnetic, of trans- former 193

of transformer, use for con- stant power or constant current regulation . . . 194 Current, minimum, in synchro- nous motor 345

waves, alternating, distorted

by hysteresis 109

Cycle, or complete period ... 10

Decrement of oscillating wave . 501 Delta connection of three-phase

system 453

current in three phase system 455 potential of three-phase sys- tem 455

Y connection of three-phase

transformation .... 463 Demagnetizing effect of armature reaction of alternators and synchronous motors . . 298 effect of eddy currents . . 136 Dielectric and electrostatic phe- nomena . . 144

Dielectic and electrostatic hyste- resis 145

Diphase, see quarter-phase. Discharge, oscillating .... 508 Displacement angle of repulsion

motor 361

of phase, maximum, in syn- chronous motor .... 347 Distorted wave, circuit factor . 415 wave, decreasing hysteresis

loss 407

wave, increasing hysteresis

loss 407

wave of condenser .... 419 wave of synchronous motor . 422 wave, some different shapes . 401 wave, symbolic representa- tion, Chap. xxiv. . . . 410 wave, in induction motor . . 426 Distortion of alternating wave . 9 of wave shape and eddy cur- rents 408

of wave shape, and insulation

strength 409

of wave shape and its causes,

Chap, xxn 383

of wave shape by hysteresis . 109 of wave shape, effect of,

Chap, xxin 398

of wave shape, increasing ef- fective value 405

Distributed capacity, inductance, resistance, and leakage,

Chap, xni 158

Distribution efficiency of systems. 468 Divided circuit, equivalent to

transformer 209

Division 491,494

Double delta connection of three- phase — six-phase transfor- mation 465

frequency quantities, as pow- er, Chap, xii 150

frequency values of distorted wave, symbolic representa- tion . . 413

516

INDEX.

Double peaked wave 399

saw-tooth wave 399

T connection of three-phase

— six-phase transforma- tion 466

Y connection of three-phase

— six-phase transforma- tion 466

.Eddy currents, unaffected by

wave-shape distortion . . 408 demagnetizing or screening

effect 136

in conductor, and unequal

current distribution . . . 139 Eddy or Foucault currents, Chap.

xi 129

Effective reactance and suscep-

tance, definition .... 105 resistance and conductance,

definition 104

resistance and reactance,

Chap, x 104

to maximum value .... 14 value of alternating wave . 11 value of alternating wave,

definition 14

value of general alternating

wave 15

Effects of higher harmonics,

Chap, xxin 398

Efficiency, maximum, of induc- tive line 93

Efficiency of systems, Chap. xxx. 468 Electro-magnetic induction, law

of, Chap. Ill 16

.Electrostatic and dielectric phe- nomena 144

hysteresis 145

Energy component of self-induc- tion 372

flow of, in polyphase system, 441 Epoch of alternating wave ... 7 Equations, fundamental, of alter- nating current transformer,

208, 225

Eauations, fundamental, of gen- eral alternating current transformer, or frequency

converter 224

of induction motor . . 226, 242 of synchronous motor . . . 339 of transmission line . . . 169 Equations, general, of apparatus, see equations,fundamental. Equivalence of transformer with