Theory and Calculation of Electrical Apparatus (1917) — part 13 of 21

The disadvantage of this type of field construction is the high flux leakage between the field poles, which tends to impair the regulation in alternators, and makes commutation more difficult for direct-current machines, It offers, however, the advantage

INDUCTOR MACHINES

287

of simplicity and material economy in machines of small and moderate size, of many poles, as for instance in small very low- speed synchronous motors, etc.

165. In its structural appearance, inductor machines often have a considerable similarity with reaction machines. The characteristic difference between the two types, however, is, that in the reaction machine voltage is induced by the pulsation of the magnetic flux by pulsating reluctance of the magnetic circuit of the machine. The magnetic pulsation in the reaction machine thus extends throughout the entire magnetic circuit of the machine, and if direct-current excitation were used, the voltage would be induced in the exciting circuit also. In the inductor machine, however, the total magnetic flux does not pulsate, but is constant, and no voltage is induced in the direct- current exciting circuit. Induction is produced in the armature by shifting the— constant — magnetic flux locally from armature coil to armature coil. The important problem of inductor alternator design — and in general of the design of magneto com- mutation apparatus — is to have the shifting of the magnetic flux from path to path so that the total reluctance and thus the total magnetic flux does not vary, otherwise excessive eddy- current losses would result in the magnetic structure.

It is interesting to note, that the number of inductor teeth is one-half the number of poles. An inductor with p projections thus gives twice as many cycles per revolution, thus as syn- chronous motor would run at half the speed of a standard syn- chronous machine of p poles.

As the result hereof, in starting polyphase synchronous machines by impressing polyphase voltage on the armature and using the hysteresis and the induced currents in the field poles, for producing the torque of starting- and acceleration, there frequently appears at half synchronism a tendency to drop into step with the field structure as inductor. This results in an increased torque when approaching, and a reduced torque when passing beyond half synchronism, thus produces a drop in the torque curve and is liable to produce difficulty in passing beyond half speed in starting. In extreme cases, it may result even in a negative torque when passing half synchronism, and make the machine non-self-starting, or at least require a considerable increase of voltage to get beyond half synchronism, over that required to start from rest.

CHAPTER XVIII

SURGING OF SYNCHRONOUS MOTORS

166. In the theory of the synchronous motor the assumption is made that the mechanical output of the motor equals the power developed by it. This is the case only if the motor runs at constant speed. If, however, it accelerates, the power input is greater ; if it decelerates, less than the power output, by the power stored in and returned by the momentum. Obviously, the motor can neither constantly accelerate nor decelerate, without breaking out of synchronism.

If, for instance, at a certain moment the power produced by the motor exceeds the mechanical load (as in the moment of throwing off a part of the load), the excess power is consumed by the momentum as acceleration, causing an increase of speed. The result thereof is that the phase of the counter e.m.f., e , is not constant, but its vector, e, moves backward to earlier time, or counter-clockwise, at a rate depending upon the momentum. Thereby the current changes and the power developed changes and decreases. As soon as the power produced equals the load, the acceleration ceases, but the vector, e, still being in motion, due to the increased speed, further reduces the power, causing a retardation and thereby a decrease of speed, at a rate depend- ing upon the mechanical momentum. In this manner a periodic variation of the phase relation between e and e 0 , and correspond- ing variation of speed and current occurs, of an amplitude and period depending upon the circuit conditions and the mechanical momentum.

If the amplitude of this pulsation has a positive decrement, that is, is decreasing, the motor assumes after a while a constant position of e regarding e 0j that is, its speed becomes uniform. If, however, the decrement of the pulsation is negative, an infinitely small pulsation will continuously increase in amplitude, until the motor is thrown out of step, or the decrement becomes zero, by the power consumed by forces opposing the pulsation, as anti-surging devices, or by the periodic pulsation of the syn- chronous reactance, etc. If the decrement is zero, a pulsation

SURGING OF SYNCHRONOUS MOTORS

289

started once will continue indefinitely at constant amplitude. This phenomenon, a surging by what may be called electro- mechanical resonance, must be taken into consideration in a complete theory of the synchronous motor.

167. Let:

Eq = e 0 = impressed e.m.f. assumed as zero vector.

E = e (cos P — j sin (3) = e.m.f. consumed by counter e.m.f. of motor, where:

p = phase angle between Eo and E .

Let :

Z = r + jx, and z = /V 2 + x 2

= impedance of circuit between

Eo and E, and

, x

tan a — •

r

The current, in the system is:

e ° ~~ e CQS § + r + jx

~ {[e 0 cos a — e cos (a + $)]

— j [e 0 sin a — e sin ( a + p)]} (1)

The power developed by the synchronous motor is:

P 0 = [El] 1 = 6 {[cos p [e 0 cos a — e cos (a + P ) J z

• sin p [e 0 sin a — e sin (a + £)] ] c

= {[e 0 cos (a — P) — e cos a]}. (2)

z

If, now, a pulsation of the synchronous motor occurs, resulting in a change of the phase relation, p, between the counter e.m.f.,. e, and the impressed e.m.f., eo (the latter being of constant fre- quency, thus constant phase), by an angle, 5, where 5 is a periodic function of time, of a frequency very low compared with the impressed frequency, then the phase angle of the counter e.m.f., e, is p + 5; and the counter e.m.f. is:

E = e { cos (p + 8) — j sin (P + 5) } ,

Jo =

eo - E Z

19

290

ELECTRICAL APPARATUS

hence the current:

I = ” { [eo cos a — e cos (a + f3 + 5)] z

— j[e 0 sin a — e sin (a + 0 + 5)]}

= Io + IT Sin 2 ! sin (“ + Z 3 + |) + i cos (“ + i 8 + |) } (3)

the power:

P = - {eo cos (a — 0 — 5) — e cos a}

2 !

D . 2ee 0 . <5 . / d <$\

= + — sm ^ sin - )8 - g j • (4)

Let now :

= mean velocity (linear, at radius of gyration) of syn- chronous machine;

5 = slip, or decrease of velocity, as fraction of v 0 , where s is a (periodic) function of time; hence v ~ vo (1 — s) = actual velocity, at time, £.

During the time element, dt, the position of the synchronous motor armature regarding the impressed e.m.f., e 0 , and thereby the phase angle, 0 + 5, of e, changes by:

where:

and

Let:

d<5 = 2 wfsdt

= $d0, (5)

« = 271 #,

/ = frequency of impressed e.m.f., c 0 .

m = mass of revolving machine elements, and Mo = mvo 2 = mean mechanical momentum, reduced to joules or watt-seconds; then the momentum at time, t , and velocity = t> 0 (1 — s) is:

M = % mv o 2 (1 “ $) 2 >

and the change of momentum during the time element, dt, is:

dM , ds

‘(1 - «) s .

SURGING OF SYNCHRONOUS MOTORS

291

hence, for small values of s:

Since:

and from (5) :

it is:

dM

dt

= — mv o 2

= - 2M 0

ds dd dd dt ds dd . dd dt’

dd

It

= 2 7rf

s

ds

dd

dS

dd’

d 2 J

dd 2

dM

dt

4 TrfMo

d?8

dd'

( 6 )

(7)

Since, as discussed, the change of momentum equals the dif- ference between produced and consumed power, the excess of power being converted into momentum, it' is :

P

( 8 )

and, substituting (4) and (7) into (8) and rearranging:

ee 0

z

sin ~ sin(« - 0 - |) + 27 r/M 0 -^ = 0.

(9)

Assuming 8 as a small angle, that is, considering only small oscillations, it is:

. 8 8

sm ^ ?

sin [a - /S - = sin (a - 0) ;

hence, substituted in (18):

ee 0

8 sin (a — 0) + 4 rfM 0

d 2 8 dd 2

0 ,

and, substituting:-

a =

it is:

ee 0 sin (a — (3) 4 irfzMo

• , d 2 8 aS + dd* = °-

(10)

(ii)

( 12 )

292

ELECTRICAL APPARATUS

This differential equation is integrated by :

8 = Ae C0 ,

which, substituted in (12) gives:

aAe Ce + ACh ce = 0, a + C 2 = 0,

C ± V- a.

168. 1. If a < 0, it is:

8 = J Ll 6 + m0 + A 2 €~ W ",

where :

( 13 )

(14)

m

= V— ( l = yj—

eeo sin^ (/3 — a) itfzMo

Since in this case, e +m0 is continually increasing, the syn- chronous motor is unstable. That is, without oscillation, the synchronous motor drops out of step, if ($ > a.

2. If a > 0, it is, denoting:

. / , lee o sin (a - 0)

n - + Va ” + S —Tfifa '

8 = Aie+ Jn ° + A 2 e- jn0 ,

or, substituting for e +jn0 and e +jn0 the trigonometric functions: 8 = (Ax + A 2 ) cos nd + j (Ax — A 2 ) sin n0,

or,

8 = B cos {nd + y).

(15)

That is, the synchronous motor is in stable equilibrium, when oscillating with a constant amplitude B , depending upon the initial conditions of oscillation, and a period, which for small oscillations gives the frequency of oscillation:

f _ „ f _ If™ osin (a - fi)

/o " n/ “ V 4 «.¥o (16)

As instance, let :

60 = 2200 volts. Z = 1+4 j ohms, or, z = 4.12; a = 76°.

And let the machine, a 16-polar, 60-cycle, 400-kw., revolving- field, synchronous motor, have the radius of gyration of 20 in., a weight of the revolving part of 6000 lb.

The momentum then is M 0 = 850,000 joules. •

Deriving the angles, /3, corresponding to given values of output, P, and excitation, e, from the polar diagram, or from the symbolic

SURGING OF SYNCHRONOUS MOTORS

293

representation, and substituting in (16), gives the frequency of oscillation :

P

e

P

e

0 :

1600 volts; p = - 2°;/ 0 = 2.17 cycles,

2180 volts

• 3°

or 130 periods per 2.50 cycles,

minute.

2800 volts

. + 5°

or 150 periods per 2.85 cycles,

minute.

400 kw.

1600 volts; )3

= 33°; /„

or 169 periods per

= 1.90 cycles,

minute.

2180 volts

21"

or 114 periods per 2.31 cycles,

minute.

2800 volts

22"

or 139 periods per 2.61 cycles,

minute.

or 154 periods per

minute.

As seen, the frequency of oscillation does not vary much with the load and with the excitation. It slightly decreases with increase of load, and it increases with increase of excitation.

In this instance, only the momentum of the motor has been considered, as would be the case for instance in a synchronous converter.

In a direct-connected motor-generator set, assuming the momentum of the direct-current-generator armature equal to 60 per cent, of the momentum of the synchronous motor, the total momentum is M o = 1,360,000 joules, hence, at no-load:

P = 0,

e = 1600 volts; jfo = 1.72 cycles, or 103 periods per minute.

1.98 cycles, or 119 periods per minute. 1.23 cycles, or 134 periods per minute.

. 169 . In the preceding discussion of the surging of synchronous machines, the assumption has been made that the mechanical power consumed by the load is constant, and that no damping or anti-surging devices were used.

The mechanical power consumed by the load varies, however, more or less with the speed, approximately proportional to the speed if the motor directly drives mechanical apparatus, as pumps, etc., and at a higher power of the speed if driving direct- current generators, or as synchronous converter, especially

294

ELECTRICAL APPARATUS

when in parallel with other direct-current generators. Assum- ing, then, in the general case the mechanical power consumed by the load to vary, within the narrow range of speed variation con- sidered during the oscillation, at the pth power of the speed, in the preceding equation instead of P o is to be substituted, Po(l - s) p = PoU - ps).

If anti-surging devices are used, and even without these in machines in which eddy currents can be produced by the oscilla- tion of slip, in solid field poles, etc., a torque is produced more or less proportional to the deviation of speed from synchronism. This power assumes the form, Pi = c 2 s , where c is a function of the conductivity of the eddy-current circuit and the intensity of the magnetic field of the machine, c 2 is the power which would be required to drive the magnetic field of the motor through the circuits of the anti-surging device at full frequency, if the same relative proportions could be retained at full fre- quency as at the frequency of slip, s. That is, Pi is the power produced by the motor as induction machine at slip $. In- stead of P, the power generated by the motor, in the preced- ing equations the value, P + Pi, has to be substituted, then:

The equation (8) assumes the form:

P+Pl-Po(l - J») =

or:

(P - Po) - (Pi + vPoS ) = (17)

or, substituting (7) and (4) :

2 e e °- sin | sin [a - 0 - |] + (c 2 + pPo) + 4 irfMo = 0;

and, for small values of 5 :

, , _ , dd , dH ai + 2b di + de= ~

_ (19)

4:7TfzM 0

h - c * + P P ° C201

b ~ 8 irfMo ' (20)

Of these two terms b represents the consumption, a the oscilla- tion of energy by the pulsation of phase angle, /3. 6 and a thus

SURGING OF SYNCHRONOUS MOTORS 295

have a similar relation as resistance and reactance in alternating- current circuits, or in the discharge of condensers. • a is the same term as in paragraph 167.

Differential equation (19) is integrated by:

6 = ( 21 )

which, substituted in (19), gives:

aAe cd + 2 bCAe CQ + C 2 Ae ce = 0, a -f- 2 bC -j- C 2 = 0, which equation has the two roots:

Ci = — b + 's/b 2 — a,

C 2 = —b — \Jb 2 — a. (22)

1. If a < 0, or negative, that is 0 > a, C i is positive and C 2 negative, and the term with C x is continuously increasing, that is, the synchronous motor is unstable, and, without oscillation, drifts out of step.

2. If 0 < a < b 2 , or a positive, and b 2 larger than a (that is, the energy-consuming term very large), C\ and C 2 are both negative, and, by substituting, + *\A 2 — a = g, it is:

Ci = - Q>-0), C 2 = — ( b + g );

lienee:

8 = Aie “ &~o)e + C6 + ^. . (23)

That is, the motor steadies down to its mean position logarith- mically, or without any oscillation.

b 2 > a,

hence:

(c 2 + vPa) 2 eeo sin (a - 0) f .

~ 1 6^/MT > ' ' 2

is the condition under which no oscillation can occur.

As seen, the left side of (24) contains only mechanical, the right side only electrical terms.

3. a > b 2 .

In this case, /b 2 — a is imaginary, and, substituting:

0 = Va — 6 2 ,

it is:

C x = - 1 b + jg,

C 2 “ b jg,

296

ELECTRICAL APPARATUS

hence:

S = 6~ be [A 1 e +1 '° e + A

and, substituting the trigonometric for the exponential functions, gives ultimately :

5 = Be~ ie cos (g$ + y). (25)

That is, the motor steadies down with an oscillation of period :

/o = gf

= / / ee o sin (« — ft) _ (c 2 + yPoP /ofi!

\ 4 7T2Mo 64 T 2 M o 2 ’ 1 ;

and decrement or attenuation constant:

b = T^ 1 ' <“>

170 . It follows, however, that under the conditions considered, a cumulative surging, or an oscillation with continuously increas- ing amplitude, can not occur, but that a synchronous motor, when displaced in phase from its mean position, returns thereto either aperiodically, if b 2 > a, or with an oscillation of vanishing amplitude, if b 2 < a. At the worst, it may oscillate with constant amplitude, if b = 0.

Cumulative surging can, therefore, occur only if in the differ- ential equation ( 19) :

aS + 2b fe + W 2 = 0> (28)

the coefficient, 6, is negative.

Since c 2 , representing the induction motor torque of the damp- ing device, etc., is positive, and pP 0 is also positive {p being

the exponent of power variation with speed), this presupposes

__/ Z 2

the existence of a third and negative term, ~ in b:

8 wfMo

h = c 2 + yP o - h 2 8tt/M 0

This negative term represents a power:

P 2 = —h 2 s;

that is, a retarding torque during slow speed, or increasing and accelerating torque during high speed, or decreasing p.

The source, of this torque may be found external to the motor, or internal, in its magnetic circuit.

SURGING OF SYNCHRONOUS MOTORS

297

External sources of negative, P 2 , may be, for instance, the magnetic field of a self-exciting, direct-current generator, driven by the synchronous motor. With decrease of speed, this field decreases, due to the decrease of generated voltage, and increases with increase of speed. This change of field strength, however, lags behind the exciting voltage and thus speed, that is, during decrease of speed the output is greater than during increase of speed. If this direct-current generator is the exciter of the synchronous motor, the effect may be intensified.

The change of power input into the synchronous motor, with change of speed, may cause the governor to act on the prime mover driving the generator, which supplies power to the motor, and the lag of the governor behind the change of output gives a pulsation of the generator frequency, of c 0 , which acts like a negative power, P 2 . The pulsation of impressed voltage, caused by the pulsation of ft may give rise to a negative, P 2 , also.

An internal cause of a negative term, P 2 , is found in the lag of the synchronous motor field behind the resultant m.m.f. In the preceding discussion, e is the “nominal generated e.m.f.” of the synchronous machine, corresponding to the field excita- tion. The actual magnetic flux of the machine, however, does not correspond to e, and thus to the field excitation, but corre- sponds to the resultant m.m.f. of field excitation and armature reaction, which latter varies in intensity and in phase during the oscillation of /3 . Hence, while e is constant, the magnetic flux is not constant, but pulsates with the oscillations of the machine. This pulsation of the magnetic flux lags behind the pulsation of m.m.f., and thereby gives rise to a term in b in equation (28). If P 0 , ft e, e 0 , Z are such that a retardation of the motor increases the magnetizing, or decreases the demagnetizing force of the armature reaction, a negative term, P 2 , appears, otherwise a positive term.

P 2 in this case is the energy consumed by the magnetic cycle of the machine at full frequency, assuming the cycle at full fre- quency as the same as at frequency of slip, s.

Or inversely, e may be said to pulsate, due to the pulsation of armature reaction, with the same frequency as ft but with a phase, which may either be lagging or leading. Lagging of the pulsation of e causes a negative, leading a positive, P 2 .

P 2 , therefore, represents the power due to the pulsation of e

298

ELECTRICAL APPARATUS

caused by the pulsation of the armature reaction, as discussed in “Theory and Calculation of Alternating-Current Phenomena.”

Any appliance increasing the area of the magnetic cycle of pulsation, as short-circuits around the field poles, therefore, increases the steadiness of a steady and increases the unsteadi- ness of an unsteady synchronous motor.

In self-exciting synchronous converters, the pulsation of e is intensified by the pulsation of direct-current voltage caused thereby, and hence of excitation.

Introducing now the term, P 2 = into the differential

equations of paragraph 169, gives the additional cases:

b < 6 , or negative, that is :

c 2 + p P o — h 2 SirfMo

< 0 ..

Hence, denoting:

gives :

4. If:

b i =

A 2 — c 2 — P P o 8 irfMo ’

bi 2 > a, g = + \ // bp — a,

8 = A 1 e + ^ +f)e + A 2 e + (b ^ f)e .

(31)

(32)

(33)

That is, without oscillation, the motor drifts out of step, in unstable equilibrium.

5. If: a > bi 2 , g = /a — &i 2 ,

8 = Be + bld cos ( gd + 5).

(34)

That is, the motor oscillates, with constantly increasing am- plitude, until it drops out of step. This is the typical case of cumulative surging by electro-mechanical resonance.

The problem of surging of synchronous machines, and its elimination, thus resolves into the investigation of the coefficient:

c 2 + pP o — h 2 WrfMo ’

(35)

while the frequency of surging, where such exists, is given by:

/ 0 = ff ee ° sin ( a ~~ ft) __ t c 2 + pP o - ft 2 ) 2 ^

Case (4), steady drifting out of step, has only rarely been observed.

The. avoidance of surging thus requires:

SURGING OF SYNCHRONOUS MOTORS 299

1. An elimination of the term /i 2 , or reduction as far as possible.

2. A sufficiently large term, c 2 , or

3. A sufficiently large term, pP 0 .

(1) refers to the design of the synchronous machine and the system on which it operates. (2) leads to the use of electro- magnetic anti-surging devices, as an induction motor winding in the field poles, short-circuits between the poles, or around the poles, and (3) leads to flexible connection to a load or a mo- mentum, as flexible connection with a flywheel, or belt drive of the load.

The conditions of steadiness are:

0 > ol,

c 2 -T pP o — h 2 > 0,

and if :

(c 2 + pP o — h 2 ) 2 eeo sin (a — /3)

16 TrfM o z

no oscillation at all occurs, otherwise an oscillation with decreas- ing amplitude.

As seen, cumulative oscillation, that is, hunting or surging, can occur only, if there is a source of power supply converting into low-frequency pulsating power, and the mechanism of con- version is a lag of some effect — in the magnetic field of the machine, or external — which causes the forces restoring the machine into step, to be greater than the forces which oppose the deviation from the position in step corresponding to the load. For further discussion of the phenomenon of cumulative surging, and of cumulative oscillations in general, see Chapter XI of “Theory and Calculation of Electric Circuits.” %

CHAPTER XIX

ALTERNATING-CURRENT MOTORS IN GENERAL

171. The starting point of the theory of the polyphase and single-phase induction motor usually is the general alternating- current transformer. Coming, however, to the commutator motors, this method becomes less suitable, and the following more general method preferable.

In its general form the alternating-current motor consists of one or more stationary electric circuits magnetically related to one or more rotating electric circuits. These circuits can be excited by alternating currents, or some by alternating, others by direct current, or closed upon themselves, etc., and connec- tion can be made to the rotating member either by collector rings — that is, to fixed points of the windings — or by commutator — that is, to fixed points in space.

The alternating-current motors can be subdivided into two classes — those in which the electric and magnetic relations between stationary and moving members do not vary with their relative positions, and those in which they vary with the relative positions of stator and rotor. In the latter a cycle of rotation exists, and therefrom the tendency of the motor results to lock at a speed giving a definite ratio between the frequency of rotation and the frequency of impressed e.m.f. Such motors, therefore, are synchronous motors.

The main types of synchronous motors are as follows:

1. One member supplied with alternating and the other with direct current — polyphase or single-phase synchronous motors.

2. One member excited by alternating current, the other con- taining a single circuit closed upon itself — synchronous induction motors.

3. One member excited by alternating current, the other of different magnetic reluctance in different directions (as polar construction) — reaction motors.

4. One member excited by alternating current, the other by alternating current of different frequency or different direction of rotation — general alternating-current transformer or fre- quency converter and synchronous-induction generator.

QAA

ALTERNATING-CURRENT MOTORS

301

(1) is the synchronous motor of the electrical industry. (2) and (3) are used occasionally to produce synchronous rotation without direct-current excitation, and of very great steadiness of the rate of rotation, where weight efficiency and power- factor are of secondary importance. (4) is used to some extent as frequency converter or alternating-current generator.

(2) and (3) are occasionally observed in induction machines, and in the starting of synchronous motors, as a tendency to lock at some intermediate, occasionally low, speed. That is, in starting, the motor does not accelerate up to full speed, but the acceleration stops .at some intermediate speed, frequently half speed, and to carry the motor beyond this speed, the im- pressed voltage may have to be raised or even external power applied. The appearance of such “dead points ” in the speed curve is due to a mechanical defect — as eccentricity of the rotor — or faulty electrical design: an improper distribution of primary and secondary windings causes a periodic variation of the mutual inductive reactance and so of the effective primary inductive reactance, (2) or the use of sharply defined and im- properly arranged teeth in both elements causes a periodic magnetic lock (opening and closing of the magnetic circuit, (3) and so a tendency to synchronize at the speed corresponding to this cycle.

Synchronous machines have been discussed elsewhere. Here shall be considered only that type of motor in which the electric and magnetic relations between the stator and rotor do not vary with their relative positions, and the torque is, therefore, not limited to a definite synchronous speed. This requires that the rotor when connected to the outside circuit be connected through a commutator, and when closed upon itself, several closed cir- cuits exist, displaced in position from each other so as to offer a resultant closed circuit in any direction.

The main types of these motors are:

1. One member supplied with polyphase or single-phase alter- nating voltage, the other containing several circuits closed upon themselves — polyphase and single-phase induction machines.

2. One member supplied with polyphase or single-phase alter- nating voltage, the other connected by a commutator to an alternating voltage — compensated induction motors, commutator motors with shunt-motor characteristic.

3. Both members connected, through a commutator, directly

302

ELECTRICAL APPARATUS

or inductively, in series with each other, to an alternating vol- tage — alternating-current motors with series-motor characteristic.

Herefrom then follow three main classes of alternating-current motors :

Synchronous motors.

Induction motors.

Commutator motors.

There are, however, numerous intermediate forms, which belong in several classes, as the synchronous-induction motor, the compensated-induetion motor, etc.

172. An alternating current, I, in an electric circuit produces a magnetic flux, $>, interlinked with this circuit. Considering equivalent sine waves of I and <£, lags behind I by the angle of hysteretic lag, a. This magnetic flux, $, generates an e.m.f., E = 2 7 rfn$, where / = frequency, n = number of turns of electric circuit. This generated e.m.f., E, lags 90° behind the magnetic flux, $, hence consumes an e.m.f. 90° ahead of <f>, or 90 — a degrees ahead of I. This may be resolved in a reactive component: E* = 2 7 r fn$ cos a — 2 tJLI = xl, the e.m.f. con- sumed by self-induction, and power component: E" = 2 rfn$> sin a — 2 7 r fHI = r"I = e.m.f. consumed by hysteresis (eddy currents, etc.), and is, therefore, in vector representation denoted by:

E' = jxl and E" = r"I,

where :

and

x = 2 7 rfL — reactance,

L — inductance,

r" = effective hysteretic resistance.

The ohmic resistance of the circuit, /, consumes an e.m.f. PI, in phase with the current, and the total or effective resistance of the circuit is, therefore, r == r f + and the total e.m.f. consumed by the circuit, or the impressed e.m.f., is:

E = (t* + jx ) I = ZI ,

where :

Z = r + jx = impedance, in vector denotation, z == Vr 2 + x 2 ~ impedance, in absolute terms.

If an electric circuit is in inductive relation to another electric circuit, it is advisable to separate the inductance, L, of the cir-

ALTERNATING-CURRENT MOTORS-

303

cuit in two parts — the self-inductance, S y which refers to that part of the magnetic flux produced by the current in one circuit which is interlinked only with this circuit but not with the other circuit, and the mutual inductance, M , which refers to that part of the magnetic flux interlinked also with the second circuit. The desirability of this separation results from the different char- acter of the two components: The self-inductive reactance gen- erates a reactive e.m.f. and thereby causes a lag of the current, while the mutual inductive reactance transfers power into the second circuit, hence generally does the useful work of the ap- paratus. This leads to the distinction between the self -inductive impedance, Z 0 = Tq + jx o, and the mutual inductive impedance, Z — r + jx.

The same separation of the total inductive reactance into self- inductive reactance and mutual inductive reactance, represented respectively by the self-inductive or “leakage ” impedance, and the mutual inductive or “exciting” impedance has been made in the theory of the transformer and the induction machine. In those, the mutual inductive reactance has been represented, not by the mutual inductive impedance, Z, but by its reciprocal

value, the exciting admittance: Y = It is then:

r 0 is the coefficient of power consumption by ohmic resistance, hysteresis and eddy currents of the self-inductive flux — effective resistance.

x 0 is the coefficient of e.m.f. consumed by the self-inductive or leakage flux — self-inductive reactance.

r is the coefficient of power consumption by hysteresis and eddy currents due to the mutual magnetic flux (hence contains no ohmic resistance component).

x is the coefficient of e.m.f. consumed by the mutual magnetic flux.

The e.m.f. consumed by the circuit is then:

E = Zoi + ZL (1)

If one of the circuits rotates relatively to the other, then in addition to the e.m.f. of self-inductive impedance: Z 0 /, and the e.m.f. of mutual-inductive impedance or e.m.f. of alternation: ZJ, an e.m.f. is consumed by rotation. This e.m.f. is in phase with the flux through which the coil rotates — that is, the flux parallel to the plane of the coil — and proportional to the speed —

304

ELECTRICAL APPARATUS

that is, the frequency of rotation — while the e.m.f. of alternation is 90° ahead of the flux alternating through the coil — that is, the flux parallel to the axis of the coil — and proportional to the fre- quency. If, therefore, Z f is the impedance corresponding to the former flux, the e.m.f. of rotation is — jSZ'I , where S is the ratio of frequency of rotation to frequency of alternation, or the speed expressed in fractions of synchronous speed. The total e.m.f. consumed in the circuit is thus:

E = ZqI + ZI — jSZ'I. (2)

Applying now these considerations to the alternating-current motor, we assume all circuits reduced to the same number of turns — that is, selecting one circuit, of n effective turns, as start- ing point, if rii = number of effective turns of any other circuit, all the e.m.fs. of the latter circuit are divided, the currents multi-

plied with the ratio, the impedances divided, the admittances

multiplied with (~^j \

This reduction of the constants of all

Io

circuits to the same number of effective turns is convenient by eliminating constant factors from the equations, and so permit- ting a direct comparison. When speaking, therefore, in the fol- lowing of the impedance, etc., of the different circuits, we always refer to their reduced values, as it is cus- tomary in induction-motor designing practice, and has been done in pre- ceding theoretical investigations.

173 . Let, then, in Fig. 147:

Eo, h, Z 0 = impressed voltage, current and self-inductive impedance respectively of a stationary circuit, E h 1 1 , Z i = impressed voltage, current and self-inductive impedance respectively of a rotating circuit, t = space angle between the axes of the two circuits,

Z = mutual inductive, or exciting impedance in the direction of the axis of the stationary coil,

Z f = mutual inductive, or exciting impedance in the direction of the axis of the rotating coil,

Z " = mutual inductive or exciting impedance in the direction at right angles to the axis of the rotating coil.

ALTERNATING-CURRENT MOTORS

305

S = speed, as fraction of synchronism, that is, ratio of fre- quency of rotation to frequency of alternation.

It is then :

E.m.f. consumed by self-inductive impedance, Z Q I 0 .

E.m.f. consumed by mutual-inductive impedance, Z (J 0 + / 1 cos r) since the m.m.f. acting in the direction of the axis of the stationary coil is the resultant of both currents. Hence :

Eq = ZqIq + Z (/ o + h cos r). (3)

In the rotating circuit, it is:

E.m.f. consumed by self-inductive impedance, Z\I.

E.m.f. consumed by mutual-inductive impedance or “e.m.f. of alter nation” : Z r (/i + / 0 cos r). (4)

E.m.f. of rotation, — jSZ''Io sin r. (5)

Hence the impressed e.m.f. :

Ei = Zih + Z' {Ji + U cos r) -jSZ"1 0 sin r. (6)

In a structure with' uniformly distributed winding, as used in induction motors, etc., ZJ = Z" — Z , that is, the exciting im- pedance is the same in all directions.

Z is the reciprocal of the “exciting admittance/’ Y of the in- duction-motor theory.

In the most general case, of a motor containing n circuits, of which some are revolving, some stationary, if:

E k) I k , Z k = impressed e.m.f., current and self-inductive im- pedance respectively of any circuit, h.

Z\ and Z u = exciting impedance parallel and at right angles respectively to the axis of a circuit, i,

Tk l = space angle between the axes of coils h and i , and S = speed, as fraction of synchronism, or “frequency of rotation.”

It is then, in a coil, i:

Ei = ZiU + Zjy u cos Tk* — jSZ u )k I k sin r k \

i x

where :

ZiU = e.m.f. of self-inductive impedance;

n

Z^Ik cos Th 1 = e.m.f. of alternation; x #

n

E'i = — jSZ H )k l k sin Tk i = e.m.f. of rotation;

which latter = 0 in a stationary coil, in which S = 0.

(7)

( 8 ) (9)

( 10 )

306

ELECTRICAL APPARATUS

The power output of the motor is the sum of the powers of all the e.m.fs. of rotation, hence, in vector denotation:

P = UY

i

= - Sj[ [jZ il )kI h sin T k \ Ji] 1 ,

1 i

and herefrom the torque, in synchronous watts :

D = ~ = - F [jZ u 5 h Sin n.\ /J l .

O 1 1

The power input, in vector denotation, is:

Po = )£ [Ei, Iil i

= ii\ i + £[%<> w

= Po 1 +iP 0 ? ';

and therefore:

(ID

( 12 )

(13)

P 0 l = true power input;

P 0 7 ‘ = wattless volt-ampere input;

Q — VPo l2 + P/ = apparent, or volt-ampere input;

= efficiency;

Po

P

Q

D

Po 1

D

Q

Po 1

Q

= apparent efficiency;

= torque efficiency;

• = apparent torque efficiency; = power-factor.

From the n circuits, i = 1, 2 . . . n, thus result n linear equations, with 2 n complex variables, li and Ei.

Hence n further conditions must be given to determine the variables. These obviously are the conditions of operation of the n circuits.

Impressed e.m.fs. Ei may be given.

Or circuits closed upon themselves E% = 0.

Or circuits connected in parallel dEi = c k E k , where c* and c k

ALTERNATING-CURRENT MOTORS

307

are the reduction factors of the circuits to equal number of effective turns, as discussed before.

Or circuits connected in series: — = — > etc.

Ci- Ck

When a rotating circuit is connected through a commutator, the frequency of the current in this circuit obviously is the same as the impressed frequency. Where, however, a rotating circuit is permanently closed upon itself, its frequency may differ from the impressed frequency, as, for instance, in the polyphase in- duction motor it is the frequency of slip, s = 1 — S, and the self-inductive reactance of the circuit, therefore, is sx ; though in its reaction upon the stationary system the rotating system nec- essarily is always of full frequency.

As an illustration of this method, its application to the theory of some motor types shall be considered, especially such motors as have either found an extended industrial application, or have at least been seriously considered.

1. POLYPHASE INDUCTION MOTOR

2. In the polyphase induction motor a number of primary circuits, displaced in position from each other, are excited by polyphase e.m.fs. displaced in phase from each other by a phase angle equal to the position angle of the coils. A number of sec- ondary circuits are closed upon themselves. The primary usu- ally is the stator, the secondary the rotor.

In this case the secondary system always offers a resultant closed circuit in the direction of the axis of each primary coil, irrespective of its position.

Let us assume two primary circuits in quadrature as simplest form, and the secondary system reduced to the same number of phases and the same number of turns per phase as the primary system. With three or more primary phases the method of procedure and the resultant equations are essentially the same.

Let, in the motor shown diagrammatically in Fig. 148:

E o and —jE 0 , 7 0 and — j7 0 , Z 0 = impressed e.m.f., currents and self-inductive impedance respectively of the primary system.

0, 1 1 and —jh, Z i = impressed e.m.f., currents and self-in- ductive impedance respectively of the secondary system, reduced to the primary. Z = mutual-inductive impedance between primary and secondary, constant in all directions.

308

ELECTRICAL APPARATUS

S = speed; s = 1 — S = slip, as fraction of synchronism.

The equation of the primary circuit is then, by (7)

Eo = ZqIo + Z ( 7 o — /i)- ( 14 )

The equation of the secondary circuit:

0 = Z l I l + Z (/i - /o) + jSZ (jh - i/o), (15)

from (15) follows:

Fig. 14S.

and, substituted in (14): Primary current:

/ _ jp r/jH + 7,1

•° iJ0 ZZ 08 .+ ZZi + ZoZi

Secondary current:

t — w

• 1 _ ■ o 'ZZ 0 s + ZZ 1 + Z 0 Z 1 Exciting current:

Too = h - 1 1 = ^0 z^~+izr+~z&i

E.m.f. of rotation:

E' = jSZ {jh - jh) = SZ (h - l O-

• ^

~ • 0 ZZ 0 s + ZZ, + ZoZ,

‘ /1 „N r.t ZZ 1

a \ TT i

“ • 0 ^o«'TzzTTToZl

(17)

(18)

(19)

( 20 )

ALTERNA TING-C URRENT MOTORS

309

It is, at synchronism; s = 0:

At standstill:

T E o . /o Z~+Z a ’ h = 0 ;

loo — / o',

E' =

E 0 Z Z + Z 0

Eq

1 +

Zo

Z

« = l;

r _ E 0 (Z + Z 1 )

•° ZZ 0 + ZZ! + ZoZS ■

T _ jBo-Z

• 1 ~ ZZ 0 + ZZi + ZoZi’

J SoZl_ . .

ZZo ZZi -j- Zo^i ;

Introducing as parameter the counter e.m.f., or e.m.f . of mutual induction:

E = Eo — ZojTo, or:

#0 = # + Zo/o,

it is, substituted:

Counter e.m.f.:

F = * Ml •

ZZqs -f- ZZ i -j" ZqZi

hence:

Primary impressed e.m.f. :

E o = E

ZZos + Z x + ZZoZi

z7z x

E.m.f. of rotation:

Secondary current

Primary current:

U

F/ = ES = F (1 - 5 ).

^ Zs + Z\ _ Es , E

• ~zzT ~ 27 + 2

(21)

( 22 )

( 23 )

( 24 ) .

( 25 )

( 26 )

( 27 )

310

ELECTRICAL APPARATUS

Exciting current:

loo = | = EY. (28)

/J

These are the equations from which the transformer theory of the polyphase induction motor starts.