Theory and Calculation of Alternating Current Phenomena (1900) — part 11 of 19

At half synchronism the torque of the concatenated couple becomes zero, and above half synchronism the sec- ond motor runs beyond its impressed frequency ; that is, becomes generator. In this case, due to the reversal of current in the secondary of the first motor, its torque becomes negative also, that is the concatenated couple becomes induction generator above half synchronism. At about two-thirds synchronism, with low resistance armature, the torque of the couple becomes zero again, and once more positive between about two-thirds synchronism and full syn- chronism, and negative once more beyond full synchronism. With high resistance in the secondary of the second motor, the second range of positive torque, below full synchronism, disappears, more or less.

173. The calculation of a concatenated couple of in- duction motors is as follows,

Let

N = frequency of main circuit,

s = slip of the first motor from synchronism.

the frequency induced in the secondary of the first motor and thus impressed upon the primary of the second motor is, s N.

The^peed of the first motor is (1 — s) N, thus the slip of the second motor, or the frequency induced in its sec- ondary, is

INDUCTION MOTOR. 277

Let

e = counter E.M.F. induced in the secondary of the sec- ond motor, reduced to full frequency.

Z0 = r0 — jxQ = primary self-inductive impedance.

Z^ = i\ —jxv = secondary self-inductance impedance.

Y — g +jb = primary exciting admittance of each mo- tor, all reduced to full frequency and to the primary by the ratio of turns.

We then have,

Second motor, secondary induced E.M.F.,

(/-!)

secondary current,

where,

(2s-l)r1

i ~ r*+ (2J-1)2^12 z ~ r*+ (2s-

primary exciting current,

4 = * (g +JI>} thus, total primary current,

72 = 7, + 70 = e ( where,

primary induced E.M.F.,

se primary impedance voltage,

ft (ro — >^o) thus, primary impressed E.M.F.,

£3 = se + 72 (r0 -jsx0) = e (^ where,

First motor, secondary current,

278 ALTERNATING-CURRENT PHENOMENA.

secondary induced E.M.F.,

£9 = where,

primary induced E.M.F.,

EI = - where,

s primary exciting current,

total primary current, where,

primary impedance voltage,

|(>o ~>

thus, primary impressed E.M.F., £0 = E, + S(r0 ->0 where,

^i =/i + ^o5i + *b£a

or, absolute,

<-„ and,

V V + V

Substituting now this value of ^ in the preceding gives the values of the currents and E.M.F.'s in the different circuits of the motor series.

• At s = 0 these terms/i and/s become indefinite, and thus at and very near synchronism have to be derived by substituting the complete expressions fory^ andy"2.

INDUCTION MOTOR. 279

In the second motor, the torque is,

T2 = [,/J = ^ hence, its power output,

/»,= (!- s) r2 = (1 - s) <?ai The power input is,

hence, the efficiency,

PS (1 - s) fa,

the power factor,

etc.

In the first motor, the torque is,

the power output,

PI = 71 (1 - j)

= ^ (1 - ,) (/^ -h/A)

the power input,

P1 =

Thus, the efficiency,

^ (1 - Q (/A +/A)

• ^2) - (^ + the power factor of the whole system,

280 ALTERNATING-CURRENT PHENOMENA.

the power factor of the first motor,

the total efficiency of the system,

etc.

f /ff. 724. Concatenation of Induction Motors. Speed Curves. Z=.1— .3/ K=.01 + .l>

174. As instance are given in Fig. 124, the curves of total torque, of torque of the second motor, and of current, for the range of slip from s = + 1.5 to s = — .7 for a pair of induction motors in concatenation, of the constants :

Z0 = Z, = .1 - .Bj

As seen, there are two ranges of positive torque for the whole system, one below half synchronism, and one from about two-thirds to full synchronism, and two ranges of

INDUCTION MOTOR.

281

negative torque, or generator action of the motor, from half to two-third synchronism, and above full synchronism.

With higher resistance in the secondary of the second motor, the second range of positive torque of the system disappears more or less, and the torque curves become as shown in Fig. 125.

001

| | CATENATION jOF IN

SUCTION MOTORS.

L

j SPEED CURVES |z=.|— .3,j Y4=.OI

H-.l

it

rag

RE!

. IN S

;COND

kRY 0

' SECO

NO MC

TOR.

|

H 8000

6000

4000

\

2000

1

—

—

—

""-s.

\

I

0

M

\

\

-2000

\

X

^

-4000

£

/

f

-60C(

./

-8000

1

0

9

s

.

6

j

4

3

2

j

„

Fig. 125. Concatenation of Induction Motors. Speed Curves.

SINGLE-PHASE INDUCTION MOTOR.

175. The magnetic circuit of the induction motor at or near synchronism consists of two magnetic fluxes super- imposed upon each other in quadrature, in time, and in position. In the polyphase motor these fluxes are produced by E.M.Fs. displaced in phase. In the monocyclic motor one of the fluxes is due to the primary energy circuit, the other to the primary exciting circuit. In the single-phase

282 AL TERN A TING-CURRENT PHENOMENA.

motor the one flux is produced by the primary circuit, the other by the currents induced in the secondary or armature, which are carried into quadrature position by the rotation of the armature. In consequence thereof, while in all these motors the magnetic distribution is the same at or near syn- chronism, and can be represented by a rotating field of uniform intensity and uniform velocity, it remains such in polyphase and monocyclic motors ; but in the single-phase motor, with increasing slip, — that is, decreasing speed, — the quadrature field decreases, since the induced armature currents are not carried to complete quadrature position ; and thus only a component available for producing the quadrature flux. Hence, approximately, the quadrature flux of a single-phase motor can be considered as proportional to its speed ; that is, it is zero at standstill.

Since the torque of the motor is proportional to the product of secondary current times magnetic flux in quad- rature, it follows that the torque of the single-phase motor is equal to that of the same motor under the same condition of operation on a polyphase circuit, multiplied with the speed ; hence equal to zero at standstill.

Thus, while single-phase induction motors are quite sat- isfactory at or near synchronism, their torque decreases proportionally to the speed, and becomes zero at standstill. That is, they are not self-starting, but some starting device has to be used.

Such a starting device may either be mechanical or elec- trical. All the electrical starting devices essentially consist in impressing upon the motor at standstill a magnetic quad- rature flux. This may be produced either by some outside E.M.F., as in the monocyclic starting device, or by displa- cing the circuits of two or more primary coils from each other, either by mutual induction between the coils, — that is, by using one as secondary to the other, — or by impe- dances of different inductance factors connected with the different primary coils.

INDUCTION MOTOR. 283

176. The starting-devices of .the single-phase induc- tion motor by producing a quadrature magnetic flux can be subdivided into three classes :

177. Phase-Splitting Devices. Two or more primary circuits are used, displaced in position from each other, and either in series or in shunt with each other, or in any other way related, as by transformation. The impedances of these circuits are made different from each other as much as possible, to produce a phase displacement between them. This can be done either by inserting external impedances into the circuits, as a condenser and a reactive coil, or by making the internal impedances of the motor circuits differ- ent, as by making one coil of high and the other of low resistance.

178. Inductive Devices. The different primary circuits of the motor are inductively related to each other in such a way as to produce a phase displacement between them. The inductive relation can be outside of the motor or inside, by having the one coil induced by the other ; and in this latter case the current in the induced coil may be made leading, accelerating coil, or lagging, shading coil.

179. Monocyclic Devices. External to the motor an essentially wattless E.M.F. is produced in quadrature with the main E.M.F. and impressed upon the motor, either directly or after combination with the single-phase main E.M.F. Such wattless quadrature E.M.F. can be produced by the common connection of two impedances of different power factor, as an inductance and a resistance, or an in- ductance and a condensance connected in series across the mains.

The investigation of these starting-devices offers a very instructive application of the symbolic method of investiga- tion of alternating-current phenomena, and a study thereof is thus recommended to the reader.*

» See paper on the Single-phase Induction Motor, A.I.E.E. Transactions, 1898.

284 ALTERNATING-CURRENT PHENOMENA.

177. As a rule, no special motors are built for single- phase operation, but polyphase motors used in single-phase circuits, since for starting the polyphase primary winding is required, the single primary coil motor obviously not allow- ing the application of phase-displacing devices for produ- cing the starting quadrature flux.

Since at or near synchronism, at the same impressed E.M.F. — that is, the same magnetic density — the total voltamperes excitation of the single-phase induction motor must be the same as of the same motor on polyphase circuit, it follows that by operating a quarter-phase motor from single-phase circuit on one primary coil, its primary excit- ing admittance is doubled. Operating a three-phase motor single-phase on one circuit its primary exciting admittance is trebled. The self-inductive primary impedance is the same single-phase as polyphase, but the secondary impe- dance reduced to the primary is lowered, since in single- phase operation all secondary circuits correspond to the one primary circuit used. Thus the secondary impedance in a quarter-phase motor running single-phase is reduced to one-half, in a three-phase motor running single-phase re- duced to one-third. In consequence thereof the slip of speed in a single-phase induction motor is usually less than in a polyphase motor ; but the exciting current is consider- ably greater, and thus the power factor and the efficiency are lower.

The preceding considerations obviously apply only when running so near synchronism that the magnetic field of the single-phase motor can be assumed as uniform, that is the cross magnetizing flux produced by the armature as equal to the main magnetic flux.

When investigating the action of the single-phase motor at lower speeds and at standstill, the falling off of the mag- netic quadrature flux produced by the armature current, the change of secondary impedance, and where a starting device is used the effect of the magnetic field produced by the starting device, have to be considered.

INDUCTION MOTOR. 285

The exciting current of the single-phase motor consists of the primary exciting current or current producing the main magnetic flux, and represented by a constant admit- tance F,,1, the primary exciting admittance of the motor, and' the secondary exciting current, that is that component of primary current corresponding to the secondary current which gives the excitation for the quadrature magnetic flux. This latter magnetic flux is equal to the main magnetic flux 3>0 at synchronism, and falls off with decreasing speed to zero at standstill, if no starting device is used or to 4^ = /<£0 at standstill if by a starting device a quadrature magnetic flux is impressed upon the motor, and at standstill t = ratio- of quadrature or starting magnetic flux to main magnetic flux.

Thus the secondary exciting current can be represented by an admittance Y* which changes from equality with the primary exciting admittance Y^ at synchronism, to Y* = 0, respectively to Y^ — t Y^ at standstill. Assuming thus that the starting device is such that its action is not impaired by the change of speed, at slip s the secondary exciting admit- tance can be represented by :

Y* = [!-(!-/) j] Fo1

The secondary impedance of the motor at synchronism is the joint impedance of all the secondary circuits, since all secondary circuits correspond to the same primary circuit,

hence = -^ with a three-phase secondary, and = -^ with a

two-phase secondary with impedance Z1 per circuit.

At standstill, however, the secondary circuits correspond to the primary circuit only with their projection in the direc- tion of the primary flux, and thus as resultant only one-half of the secondary circuits are effective, so that the secondary impedance at standstill is equal to 2 Zl / 3 with a three-phase, and equal to Z^ with a two-phase secondary. Thus the effective secondary impedance of the single-phase motor

286 ALTERNATING-CURRENT PHENOMENA.

changes with the speed and can at the slip s be represented

by Zf = - -- -^ — - in a three-phase motor, and Z{ = - - <p — -1

in a two-phase motor, with the impedance Z^ per secondary circuit.

In the single-phase motor without starting device, due to the falling off of the quadrature flux, the torque at slip s is :

T = a^ (I - s)

In a single-phase motor with a starting device which at standstill produces a ratio of magnetic fluxes t, the torque at standstill is ;

TQ = /7I

where 7^ = total torque of the same motor on polyphase circuit.

. Thus denoting the value —~ = v &f

the single-phase motor torque at standstill is :

and the single-phase motor torque at slip s is : T = of [1 - (1 - v) s]

178. In the single-phase motor considerably more advantage is gained by compensating for the wattless mag- netizing component of current by capacity than in the polyphase motor, where this wattless current is relatively small. The use of shunted capacity, however, has the dis- advantage of requiring a wave of impressed E.M.F. very close to sine shape ; since even with a moderate variation from sine shape the wattless charging current of the con- denser of higher frequency may lower the power factor more than the compensation for the wattless component of the fundamental wave raises it, as will be seen in the chap- ter on General Alternating Current Waves.

Thus the most satisfactory application of the condenser in the single-phase motor is not in shunt to the primary

INDUCTION MOTOR. 287

circuit, but in a tertiary circuit ; that is, in a circuit stationary with regard to the primary impressed circuit, but induced by the revolving secondary circuit.

In this case the condenser is supplied with an E.M.F. transformed twice, from primary to secondary, and from secondary to tertiary, through multitooth structures in a uniformly revolving field, and thus a very close approxi- mation to sine wave produced at the condenser, irrespective of the wave shape of primary impressed E.M.F.

With the condenser connected into a tertiary circuit of a single-phase induction motor, the wattless magnetizing current of the motor is supplied by the condenser in a separate circuit, and the primary coil carries the energy cur- rent only, and thus the efficiency of the motor is essentially increased.

The tertiary circuit may be at right angles to the pri- mary, or under any other angle. Usually it is applied on an angle of 60°, so as to secure a mutual induction between tertiary and primary for starting, which produces in start- ing in the condenser a leading current, and gives the quad- rature magnetic flux required.

179. The most convenient way to secure this arrange- ment is the use of a three-phase motor which with two of its terminals 1-2, is connected to the single-phase mains, and with terminals 1 and 3 to a condenser.

Let YQ = g0 --jb0 = primary exciting admittance of the motor per delta circuit.

Z0 = r0 — jxQ = primary self-inductive impedance per delta circuit.

Z^ = i\ —jx^ = secondary self-inductive impedance per delta circuit reduced to primary.

Let

Ys = gs — jb9 = admittance of the condenser connected between terminals 1 and 3.

288 ALTERNATING-CURRENT PHENOMENA.

If then, as single-phase motor,

/ = ratio of auxiliary quadrature flux to main flux in starting,

h = ratio of E.M.F. induced in condenser circuit to

E.M.F. induced in main circuit in starting, starting torque

It is single-phase

Fo1 = 1.5 Y0 = 1.5 (£•„ +/£0) = primary exciting admit-

tance, Y? = 1.5 Y0 [1 - (1 - 0 s]

= 1.5 (g0 +/£<)) [1 — (1 — 0 J] = secondary exciting

admittance at slip s.

Z0l = ?^° = 2fo~^*o) = primary self-inductive impe- o o

dance.

Zxi = £L±^ Zi = ^L + ^ (ri -jsxj = secondary self- o o

inductive impedance.

Z,1 = ^ = 2 (r° ~ ***> = tertiary self-inductive impe- o o

dance of motor. Thus,

Y4 = -^r - T- = total admittance of tertiary circuit.

Since the E.M.F. induced in the tertiary circuit decreases from e at synchronism to he at standstill, the effective ter- tiary admittance or admittance reduced to an induced E.M.F. e is at slip s

Y? = [!-(!-*) s] Y4 Let then,

e = counter E.M.F. of primary circuit, s = slip.

INDUCTION MOTOR. 289

We have, secondary load current

3se

(1 + s) (r, -jsx,) secondary exciting current

secondary condenser current thus, total secondary current primary exciting current

thus, total primary current

/o = 71 + /o1 = /, + /, +

= ' (*i + A) primary impressed E.M.F.

thus, main counter E.M.F.

or,

and, absolute

V^2 + c* hence, primary current

T_slW + %

J* - e° v f* + ^

290 ALTERNATING-CURRENT PHENOMENA.

voltampere input,

Qo = **!» power input

*t — Oo — O 2 , 2

6j T '2

torque at slip .$•

2^= r1 [i - (i - v) s]

and, power output

and herefrom in the usual manner the efficiency, apparent efficiency, torque efficiency, apparent torque efficiency, and power factor.

The derivation o.* the constants /, //, v, which have to be determined before calculating the motor, is as follows :

Let <?0 = single-phase impressed E.M.F.,

Y — total stationary admittance of motor per delta cir-

cuit, Ez = E.M.F. at condenser terminals in starting.

In the circuit between the single-phase mains from ter- minal 1 over terminal 3 to 2, the admittances Y + Y8, and Y, are connected in series, and have the respective E.M.Fs. E^ and e0 - Ey It is thus,

Y+ Ys+ Y=e0-£t+£s,

since with the same current passing through both circuits, the impressed E.M.Fs. are inverse proportional to the re- spective admittances.

Thus,

INDUCTION MOTOR. 291

and quadrature E.M.F.

hence thus

Since in the three-phase E.M.F. triangle, the altitude corresponding to the quadrature magnetic flux = — y= , and

the quadrature and main fluxes are equal, in the single-phase motor the ratio of quadrature to main flux is

/ = — 2 = 1.155 Aa

V3

From /, v is derived as shown in the preceding.

For further discussion on the Theory and Calculation of the Single-phase Induction Motor, see American Institute Electrical Engineers Transactions, January, 1900.

SYNCHRONOUS INDUCTION MOTOR.

180. The induction motor discussed in the foregoing consists of one or a number of primary circuits acting upon a movable armature which comprises a number of closed secondary circuits displaced from each other in space so as to offer a resultant circuit in any direction. In consequence thereof the motor can be considered as a transformer, having to each primary circuit a corresponding secondary circuit, — a secondary coil, moving out of the field of the primary coil, being replaced by another secondary coil moving into the field.

In such a motor the torque is zero at synchronism, posi- tive below, and negative above, synchronism.

If, however, the movable armature contains one closed circuit only, it offers a closed secondary circuit only in the direction of the axis of the armature coil, but no secondary circuit at right angles therewith. That is, with the rotati .n

292 ALTERNATING-CURRENT PHENOMENA.

of the armature the secondary circuit, corresponding to a primary circuit, varies from short circuit at coincidence of the axis of the armature coil with the axis of the primary coil, to open circuit in quadrature therewith, with the periodicity of the armature speed. That is, the apparent admittance of the primary circuit varies periodically from open-circuit admittance to the short-circuited transformer admittance.

At synchronism such a motor represents an electric cir- cuit of an admittance varying with twice the periodicity of the primary frequency, since twice per period the axis of the armature coil and that of the primary coil coincide. A vary- ing admittance is obviously identical in effect with a varying reluctance, which will be discussed in the chapter on reac- tion machines. That is, the induction motor with one •closed armature circuit is, at synchronism, nothing but a reaction machine, and consequently gives zero torque at synchronism if the maxima and minima of the periodically varying admittance coincide with the maximum and zero values of the primary circuit, but gives a definite torque if they are displaced therefrom. This torque may be positive or negative according to the phase displacement between admittance and primary circuit ; that is, the lag or lead of the maximum admittance with regard to the primary maximum. Hence an induction motor with single-armature circuit at synchronism acts either as motor or as alternat- ing-current generator according to the relative position of the armature circuit to the primary circuit. Thus it can be called a synchronous induction motor or synchronous in- duction generator, since it is an induction machine giving torque at synchronism.

Power factor and apparent efficiency of the synchron- ous induction motor as reaction' machine are very low. Hence it is of practical application only in cases where a small amount of power is required at synchronous rotation, and continuous current for field excitation is not available.

INDUCTION MOTOR. 293

The current induced in the armature of the synchronous induction motor is of double the frequency impressed upon the primary.

Below and above synchronism the ordinary induction motor, or induction generator, torque is superimposed upon the synchronous induction machine torque. Since with the frequency of slip the relative position of primary and of secondary coil changes, the synchronous induction machine torque alternates periodically with the frequency of slip. That is, upon the constant positive or negative torque be- low or above synchronism an alternating torque of the fre- quency of slip is superimposed, and thus the resultant torque pulsating with a positive mean value below, a nega- tive mean value above, synchronism.

When started from rest, a synchronous induction motor will accelerate like an ordinary single-phase induction mo- tor, but not only approach synchronism, as the latter does, but run up to complete synchronism under load. When approaching synchronism it makes definite beats with the frequency of slip, which disappear when synchronism is reached.

THE HYSTERESIS MOTOR.

181. In a revolving magnetic field, a circular iron disk, or iron cylinder of uniform magnetic reluctance in the direction of the revolving field, is set in rotation, even if subdivided so as to preclude the induction of eddy currents. This rotation is due to the effect of hysteresis of the revolv- ing disks or cyclinder, and such a motor may thus be called a hysteresis motor.

Let / be the iron disk exposed to a rotating magnetic field or resultant M.M.F. The axis of resultant magneti- zation in the disk / does not coincide with -the axis of the rotating field, but lags behind the- latter, thus producing a couple. That is, the component of magnetism in a direction of the rotating disk, /, ahead of the axis of rotating M.M.F., is rising, thus below, and in a direction behind the axis

294 AL TERN A TING-CURRENT PHENOMENA.

of rotating M.M.F. decreasing; that is, above proportion- ality with the M.M.F., in consequence of the lag of magnet- ism in the hysteresis loop, and thus the axis of resultant magnetism in the iron disk, /, does not coincide with the axis of rotating M.M.F., but is shifted backwards by an angle, a, which is the angle of hysteretic lead in Chapter X., § 79.

The induced magnetism gives with the resultant M.M.F. a mechanical couple, —

T= mF& sin a,

where

F= resultant M.M.F.,

<£ = resultant magnetism,

a = angle of hysteretic advance of phase,

m = a constant.

The apparent or voltampere input of the motor is, — Q = mF®.

Thus the apparent torque efficiency, —

T

2 = sma,

and the power of the motor is, —

P = (1 — s) T= (1 — s) m F<$> sin a, where

s = slip as fraction of synchronism.

The apparent efficiency is, —

P

• = (!_*) sin a.

Since in a magnetic circuit containing an air gap the angle a is extremely small, a- few degrees only, it follows that the apparent efficiency of the hysteresis motor is ex- tremely low, the motor consequently unsuitable for produ- cing larger amounts of mechanical work.

INDUCTION MOTOR. 295

From the equation of torque it follows, however, that at constant impressed E.M.F., or current, — that inconstant F, — the torque is constant and independent of the speed ; and therefore such a motor arrangement is suitable, and occasionally used as alternating-current meter.

The same result can be reached from a different point of view. In such a magnetic system, comprising a mov- able iron disk, /, of uniform magnetic reluctance in a revolving field, the magnetic reluctance — and thus the dis- tribution of magnetism — is obviously independent of the speed, and consequently the current and energy expenditure of the impressed M.M.F. independent of the speed also. If, now, —

V '= volume of iron of the movable part, B = magnetic density, and 77 = coefficient of hysteresis,

the energy expended by hysteresis in the movable disk, /, is per cycle, —

IV, = V^B™,

hence, if N= frequency, the energy supplied by the M.M.F. to the rotating iron disk in the hysteretic loop of the

M.M.F. is, —

P =

At the slip, s N, that is, the speed (1 — s) N, the energy xpended by hysteresis in the rotating disk is, however, —

Hence, in the transfer from the stationary to the revolv- ing member the magnetic energy, —

has disappeared, and thus reappears as mechanical work, and the torque is, —

'-p^iprW'

that is, independent of the speed.

296 AL TERNA TING-CURRENT PHENOMENA.

Since, as seen in Chapter X., sin a is the ratio of the energy of the hysteretic loop to the total apparent energy, in voltampere, of the magnetic cycle, it follows that the apparent efficiency of such a motor can never exceed the value (1 — s) sin a, or a fraction of the primary hysteretic energy.

The primary hysteretic energy of an induction motor, as represented by its conductance, g, being a part of the loss in the motor, and thus a very small part of its output only, it follows that the output of a hysteresis motor is a very small fraction only of the output which the same magnetic structure could give with secondary short-circuited winding, as regular induction motor.

As secondary effect, however, the rotary effort of the magnetic structure as hysteresis motor appears more or less in all induction motors, although usually it is so small as to be neglected.

If in the hysteresis motor the rotary iron structure has not uniform reluctance in all directions — but is, for in- stance, bar-shaped or shuttle-shaped — on the hysteresis motor effect is superimposed the effect of varying magnetic reluctance, which tends to accelerate the motor to syn- chronism, and maintain it therein, as shall be more fully investigated under " Reaction Machine " in Chapter XX.

ALTERNATING-CURRENT GENERATOR. 297

CHAPTER XVII.

ALTERNATING-CURRENT GENERATOR.

182. In the alternating-current generator, E.M.F. is induced in the armature conductors by their relative motion through a constant or approximately constant magnetic field.

When yielding current, two distinctly different M.M.Fs. are acting upon the alternator armature — the M.M.F. of the field due to the field-exciting 'spools, and the M.M.F. of the armature current. The former is constant, or approx- imately so, while the latter is alternating, and in synchro- nous motion relatively to the former ; hence, fixed in space relative to the field M.M.F., or uni-directional, but pulsating in a single-phase alternator. In the polyphase alternator, when evenly loaded or balanced, the resultant M.M.F. of the armature current is more or less constant.

The E.M.F. induced in the armature is due to the mag- netic flux passing through and interlinked with the arma- ture conductors. This flux is produced by the resultant of both M.M.Fs., that of the field, and that of the armature.

On open circuit, the M.M.F. of the armature is zero, and the E.M.F. of the armature is due to the M.M.F. of the field coils only. In this case the E.M.F. is, in general, a maximum at the moment when the armature coil faces the position midway between adjacent field coils, as shown in Fig. 126, and thus incloses no magnetism. The E.M.F. wave in this case is, in general, symmetrical.

An exception from this statement may take place only in those types of alternators where the magnetic reluctance of the armature is different in different directions ; thereby,

298 AL TERNA TING-CURRENT PHENOMENA.

during the synchronous rotation of the armature, a pulsa- tion of the magnetic flux passing through it is produced. This pulsation of the magnetic flux induces E.M.F. in the field spools, and thereby makes the field current pulsating also. Thus, we havet in this case, even on open circuit, no

Fig. 126.

rotation through a constant magnetic field, but rotation through a pulsating field, which makes the E.M.F. wave unsymmetrical, and shifts the maximum point from its the- oretical position midway between the field poles. In gen- eral this secondary reaction can be neglected, and the field M.M.F. be assumed as constant.

The relative position of the armature M.M.F. with re- spect to the field M.M.F. depends upon the phase rela- tion existing in the electric circuit. Thus, if there is no displacement of phase between current and E.M.F., the current reaches its maximum at the same moment as the E.M.F. ; or, in the position of the armature shown in Fig. 126, midway between the field poles. In this case the arma- ture current tends neither to magnetize nor demagnetize the field, but merely distorts it ; that is, demagnetizes the trail- ing-pole corner, a, and magnetizes the leading-pole corner, b. A change of the total flux, and thereby of the resultant E.M.F., will take place in this case only when the magnetic densities are so near to saturation that the rise of density at the leading-pole corner will be less than the decrease of

AL TERN A TING-CURRENT GENERA TOR.

299

density at the trailing-pole corner. Since the internal self- inductance of the alternator itself causes a certain lag of the current behind the induced E.M.F., this condition of no displacement can exist only in a circuit with external nega- tive reactance, as capacity, etc.

If the armature current lags, it reaches the maximum later than the E.M.F. ; that is, in a position where the armature coil partly faces the following-field pole, as shown in diagram in Fig. 127. Since the armature current flows

Fig. 127.

in opposite direction to the current in the following-field pole (in a generator), the armature in this case will tend to demagnetize the field.

If, however, the armature current leads, — that is, reaches its maximum while the armature coil still partly faces the

Fig. 128.

preceding-field pole, as shown in diagram Fig. 128, — it tends to magnetize this field coil, since the armature current flows in the same direction with the exciting current of the pre- ceding-field spools.

300 ALTERNA TING-CURRENT PHENOMENA.

Thus, with a leading current, the armature reaction of the alternator strengthens the field, and thereby, at con- stant-field excitation, increases the voltage ; with lagging current it weakens the field, and thereby decreases the vol- tage in a generator. Obviously, the opposite holds for a synchronous motor, in which the armature current flows in the opposite direction ; and thus a lagging current tends to magnetize, a leading current to demagnetize, the field.

183. The E.M.F. induced in the armature by the re- sultant magnetic flux, produced by the resultant M.M.F. of the field and of the armature, is not the terminal voltage of the machine ; the terminal voltage is the resultant of this induced E.M.F. and the E.M.F. of self-inductance and the E.M.F. representing the energy loss by resistance in the alternator armature. That is, in other words, the armature current not only opposes or assists the field M.M.F. in cre- ating the resultant magnetic flux, but sends a second mag- netic flux in a local circuit through the armature, which flux does not pass through the field spools, and is called the magnetic flux of armature self-inductance.

Thus we have to distinguish in an alternator between armature reaction, or the magnetizing action of the arma- ture upon the field, and armature self-inductance, or the E.M.F. induced in the armature conductors by the current flowing therein. This E.M.F. of self-inductance is (if the magnetic reluctance, and consequently the reactance, of the armature circuit is assumed as constant) in quadrature behind the armature current, and will thus combine with the induced E.M.F. in the proper phase relation. Obvi- ously the E.M.F. of self-inductance and the induced E.M.F. do not in reality combine, but their respective magnetic fluxes combine in the armature core, where they pass through the same structure. These component E.M.Fs. are there- fore mathematical fictions, but their resultant is real. This means that, if the armature current lags, the E.M.F. of self-

ALTERNATING-CURRENT GENERATOR. 301