Theory and Calculation of Electrical Apparatus (1917) — part 7 of 21

negative on the unstable branch of the motor curve, and this rate of change of the torque, with change of current, expressed as fraction of the current, is :

k _ Id?.

it may be called the stability coefficient of the motor.

If k 8 is positive, an increase of i, caused by an increase of slip, s, that is, by a decrease of speed, increases the torque, D, and thereby checks the decrease of speed, and inversely, that is, the motor is stable.

'If, however, k 8 is negative, an increase of i causes a decrease of D, thereby a decrease of speed, and thus further increase of i and decrease of D; that is, the motor slows down with increas- ing rapidity, or inversely, with a decrease of i, accelerates with increasing rapidity, that is, is unstable.

For the motor used as illustration in the preceding, of the constants e = 110 volts; Y = 0.01 - 0.1 j; Z 0 = 0.1 + 0.3 j, Z i = 0.1 + 0.3 j, the stability curve is shown, together with speed, current, and torque, in Fig. 54, as function of the output. As seen, the stability coefficient, k 8 , is very high for light-load, decreases first rapidly and then slowly, until an output of 7000 watts is approached, and then rapidly drops below zero ; that is, the motor becomes unstable and drops out of step, and speed, torque, and current change abruptly, as indicated by the arrows in Fig. 54.

The stability coefficient, k 8} characterizes the behavior of the motor regarding its load-carrying capacity. Obviously, if the terminal voltage of the motor is not constant, but drops with the load, as discussed in 1, a different stability coefficient results; which intersects the zero line at a different and lower torque.

86 . If the induction motor is supplied with constant terminal voltage from a generator of close inherent voltage regulation

INDUCTION-MOTOR REGULATION

139

and of a size very large compared with the motor, over a supply circuit of negligible impedance, so that a sudden change of motor current can not produce even a momentary tendency of change of the terminal voltage of the motor, the stability curve, k SJ of Fig. 54 gives the performance of the motor. If, however,

at a change of load and thus of motor current the regulation of the supply voltage to constancy at the motor terminals re- quires a finite time, even if this time is very short, the maximum output of the motor is reduced thereby, the more so the more rapidly the motor speed can change.

Assuming the voltage control at the motor terminals effected

140

ELECTRICAL APPARATUS

by hand regulation of the generator or the potential regulator in the circuit supplying the motor, or by any other method which is slower than the rate at which the motor speed can adjust itself to a change of load, then, even if the supply voltage at the motor terminals is kept constant, for a momentary fluctuation of motor speed and current, the supply voltage momentarily^ varies, and with regard to its stability the motor corresponds not to the condition of constant supply voltage but to a supply voltage which varies with the current, hence the limit of stability is reached at a lower value of motor torque.

At constant slip, $, the motor torque, D, is proportional to the square of the impressed e.m.f., e 2 . If by a variation of slip caused by a fluctuation of load the motor current, i, varies by di, if the terminal voltage, e, remains constant the motor torque, D,

varies by the fraction k 8 = or the stability coefficient of

the motor. If, however, by the variation of current, di, the impressed e.m.f., e, of the motor varies, the motor torque, D , being proportional to e 2 , still further changes, proportional to

the change e 2 , that is, by the fraction k r = J- —— = ~ ^4 and the

total change of motor torque resultant from a change, di. of the current, i, thus is k 0 — k 8 + k r .

Hence, if a momentary fluctuation of current causes a momen- tary fluctuation of voltage, the stability coefficient of the motor is changed from k 3 to fc 0 = k 8 + k r , and as k r is negative, the voltage, e, decreases with increase of current, i, the stability coefficient of the system is reduced by the effect of voltage regu- lation of the supply, k r , and k r thus can be called the regulation coefficient of the system.

2 de

k r ~ e Wi ^ 1US re P resen ^ s the change of torque produced by

the momentary voltage change resulting from a current change di in the system; hence, is essentially a characteristic of the supply system and its regulation, but depends upon the motor de

only in so far as depends upon the power-factor of the load.

In Fig. 54 is shown the regulation coefficient, k r , of the supply system of the motor, at 110 volts maintained constant at the motor terminals, and an impedance, Z = 0.16 + 0.8 j, between motor terminals and supply e.m.f. As seen, the regulation coefficient of the system drops from a maximum of about 0.03,

INDUCTION-MOTOR REGULATION

141

at no-load, down to about 0.01, and remains constant at this latter value, over a very wide range.

The resultant stability coefficient, or stability coefficient of the system of motor and supply, ko = k 8 + k r , as shown in Fig. 54, thus drops from very high values at light-load down to zero at the load at which the curves, k 3 and 7c r , in Fig. 54 intersect, or at 5800 kw., and there become negative; that is, the motor drops out of step, although still far below its maximum torque point, as indicated by the arrows in Fig. 54.

Thus, at constant voltage maintained at the motor terminals by some regulating mechanism which is slower in its action than the retardation of a motor-speed change by its mechanical momentum, the motor behaves up to 5800 watts output in exactly the same manner as if its terminals were connected directly to an unlimited source of constant voltage supply, but at this point, where the slip is only 7 per cent, in the present instance, the motor suddenly drops out of step without previous warning, and comes to a standstill, while at inherently constant terminal voltage the motor would continue to operate up to 7000 watts output, and drop out of step at 8250 synchronous watts torque at 10 per cent. slip.

By this phenomenon the maximum torque of the motor thus is reduced from 8250 to 6300 synchronous watts, or by nearly 25 per cent.

87 . If the voltage regulation of the supply system is more rapid than the speed change of the motor as retarded by the momentum of motor and load, the regulation coefficient of the system as regards to the motor obviously is zero, and the motor thus gives the normal maximum output and torque. If the regulation of the supply voltage, that is, the recovery of the terminal voltage of the motor with a change of current, occurs at about the same rate as the speed of the motor can change with a change of load, then the maximum output as limited by the stability coefficient of the system is intermediate between the minimum value of 6300 synchronous watts and its normal value of 8250 synchronous watts. The more rapid the recovery of the voltage and the larger the momentum of motor and load, the less is the motor output impaired by this phenomenon of instability. Thus, the loss of stability is greatest with hand regulation, less with automatic control by potential regulator, the more so the more rapidly the regulator works ; it is very little

142

ELECTRICAL APPARATUS

with compounded alternators, and absent where the motor 1

terminal voltage remains constant without any control by prac- tically unlimited generator capacity and absence of voltage drop between generator and motor.

Comparing the stability coefficient, k s , of the motor load and the stability coefficient, k 0 , of the entire system under the assumed conditions of operation of Fig. 54, it is seen that the former intersects the zero line very steeply, that is, the stability remains high until very close to the maximum torque point, and the motor thus can be loaded up close to its maximum torque without impairment of stability. The curve, ko, however, intersects the zero Tine under a sharp angle, that is, long before the limit of stability is reached in this case the stability of the system has dropped so close to zero that the motor may drop out of step by some momentary pulsation. Thus, in the case of instability due to the regulation of the system, the maximum output point, as found by test, is not definite and sharply defined, but the stability !

gradually decreases to zero, and during this decrease the motor f

drops out at some point. Experimentally the difference between the dropping out by approach to the limits of stability of the motor proper and that of the system of supply is very marked by the indefiniteness of the latter. {

In testing induction motors it thus is necessary to guard i

against this phenomenon by raising the voltage beyond normal • before every increase of load, and then gradually decrease the |

voltages again to normal. f

A serious reduction of the overload capacity of the motor, due {

to the regulation of the system, obviously occurs only at very I

high impedance of the supply circuit; with moderate impedance \

the curve, k r is much lower, and the intersection between k r and k a occurs still on the steep part of k 8) and the output thus is not !

materially decreased, but merely the stability somewhat reduced when approaching maximum output.

This phenomenon of the impairment of stability of the induc- tion motor by the regulation of the supply voltage is of prac- tical importance, as similar phenomena occur in many instances.

Thus, with synchronous motors and converters the regulation of the supply system exerts a similar effect on the overload capacity, and reduces the maximum output so that the motor \

drops out of step, or starts surging, due to the approach to the j

stability limit of the entire system. In this case, with syn- gr

INDUCTION-MOTOR REGULATION

143

chronous motors and converters, increase of their field excita- tion frequently restores their steadiness by producing leading currents and thereby increasing the power-carrying capacity of the supply system, while with surging caused by instability of the synchronous motor the leading currents produced by increase of field excitation increase the surging, and lowering the field excitation tends toward steadiness.

CHAPTER VII

HIGHER HARMONICS IN INDUCTION MOTORS

88. The usual theory and calculation of induction motors, as discussed in “ Theoretical Elements of Electrical Engineer- ing” and in “Theory and Calculation of Alternating-current Phenomena,” is based on the assumption of the sine wave. That is, it is assumed that the voltage impressed upon the motor per phase, and therefore the magnetic flux and the current, are sine waves, and it is further assumed, that the distribution of the winding on the circumference of the armature or primary, is sinusoidal in space. While in most cases this is sufficiently the case, it is not always so, and especially the space or air-gap distribution of the magnetic flux may sufficiently differ from sine shape, to exert an appreciable effect on the torque at lower speeds, and require consideration where motor action and braking action with considerable power is required throughout the entire range of speed.

JLet then :

e = ei cos 4> + ez cos (3 (j> — a 3 ) + 05 cos (5 <j> — as) + 07 cos (7$ — a 7 ) + e$ cos (9 (f> — ao) + . . . (1)

be the voltage impressed upon one phase of the induction motor. If the motor is a quarter-phase motor, the voltage of the

second motor phase, which lags 90° or ~ behind the first motor phase, is:

3 7T

T

a 3

• 02 COS (& 0— — a&J

9 7T ~2

a 9

■)+•••

e' = CiCOs(<£ — + e 3 cos ( 3 <£-

• e 7 cos (j (j> ~ — a 7 ^ + 09 cos (9 <j>

= ei cos + 03 cos (3 4 > - a 3 + + 05 cos ^5 ^ — as —

• e7 cos ^7 <£ — 0:7 + + eg cos ^9 0 — ag — ^ + . . . (2)

The magnetic flux produced by these two voltages thus con- sists of q, series of component fluxes, corresponding respectively

144

HIGHER HARMONICS

145

to the successive components. The secondary currents induced by these component fluxes, and the torque produced by the secondary currents, thus show the same components.

Thus the motor torque consists of the sum of a series of components:

The main or fundamental torque of the motor, given by the usual sine-wave theory of the induction motor, and due to the fundamental voltage wave:

eicos <j) filCOS (<f> —

(3)

is shown as T\ in Fig. 55, of the usual shape, increasing from standstill, with increasing speed, up to a maximum torque, and then decreasing again to zero at synchronism.

The third harmonics of the voltage waves are:

e 3 cos (3 </> — <* 3 ), e 3 cos (^3 <f> — a 3 + '-J

(4)

As seen, these also constitute a quarter-phase system of voltage, but the second wave, which is lagging in the funda- mental, is 90° leading in the third harmonic, or in other words, the third harmonic gives a backward rotation of the poles with triple frequency. It thus produces a torque in opposite direc- tion to the fundamental, and would reach its synchronism, that is, zero torque, at one-third of synchronism in negative direction, or at the speed S, = — given in fraction of synchronous speed. For backward rotation above one-third synchronism, this triple harmonic then gives an induction generator torque, and the complete torque curve given by the third harmonics thus is as shown by curve Tz of Fig. 55.

The fifth harmonics:

escos (5 <t> — 0 : 5 ), e £ cos (5 <f > 5 — a 8 —

(5)

give again phase rotation in the same direction as the funda- mental, that is, motor torque, and assist the fundamental. But synchronism is reached at one-fifth of the synchronous speed of the fundamental, or at: 8 = and above this speed, the

10

146

ELECTRICAL APPARATUS

fifth harmonic becomes induction generator, due to oversyn- chronous rotation, and retards. Its torque curve is shown as Ts in Fig. 55.

The seventh harmonic again gives negative torque, due to backward phase rotation of the phases, and reaches synchronism at S = — H? that one-seventh speed in backward rotation, as shown by curve Ti in Fig. 55.

Fig. 55. — Quarter-phase induction motor, component harmonics and resultant torque.

The ninth harmonic again gives positive motor torque up to its synchronism, S = and above this negative induction generator torque, etc.

We then have the effects of the various harmonics on the

Quarter-phase Induction Motor

Order of harmonics

1

3

5

7

9

11

13

Phase rotation

—

—

Synchronous speed: S =

+1

-H

+M

-Hi

+H 3

Torque positive up to : S « . . . otherwise negative.

+1

+H

+H 8

HIGHER HARMONICS

147

Adding now the torque curves of the various voltage harmonics, Tj, T 5 , T 7 , to the fundamental torque curve, T u of the induction motor, gives the resultant torque curve, T .

As seen from Pig.' 55, if the voltage harmonics are consider- able, the torque curve of the motor at lower speeds, forward and backward, that is, when used as brake, is rather irregular, showing depressions or “dead points.”

89. Assume now, the general voltage wave (1) is one of the three-phase voltages, and is impressed upon one of the phases of a three-phase induction motor. The second and third

2 7T 4 7T

phase then is lagging by and -y respectively behind the first phase (1) :

e' = Ci cos -+ e 3 cos (3 — azj

• cos ^5 0 — — a^j + e 7 cos ^7 4

14 7T \ 3 -

• c 9 cos ^9 0 + . •

= Ci cos (<j> — + e 3 cos (3 <t> — a 3 )

• c 5 cos ^5 </> — <25 + + ^7 cos (7 4> — a 7 —

• eg cos (9 </> — ag) + * - ' = ei cos ^ 4- e 3 cos (3 <f> — a 3 )

• e s cos (?><t> — as + 4- «7 cos (j <t> — on —

4- Co cos (9 <f> — ao) "4 .

(6)

Thus the voltage components of different frequency, impressed upon the three motor phases, are:

n cos 0

es cos

Cfi cos

67 COS

c® cos

(3 <f> — aa)

(50 — as)

(70 — a?)

(9 0 — «a)

Pj cos

e& cos

ei cos

l 2sr\

es cos

/ 2t\

/ 2 A

e® coa

(*-t)

(3 <(> — m)

(9 0 — as)

ei cos

et, cos

ei cos

/ 4tt\

e% cos

/ 4 t\

( 4 7r\

eo cos

(*-t)

(3 0 — oca )

V 5 *~“ t+ T/

(9 0 — as)

Fundamental....

.3d

5th

1

7th

9th

148

ELECTRICAL APPARATUS

As seen, in this case of the three-phase motor, the third harmonics have no phase rotation, but are in phase with each other, or single-phase voltages. The fifth harmonic gives backward phase rotation, and thus negative torque, while the seventh harmonic has the same phase rotation, as the funda- mental, thus adds its torque up to its synchronous speed, S =

• and above this gives negative or generator torque. The ninth harmonic again is single-phase.

Fig. 56 shows the fundamental torque, T i, the higher harmonics

Fig. 56. — Three-phase induction motor, component harmonics and resultant torque.

of torque, Ts and T 7, and the resultant torque, T . As seen, the distortion of the torque curve is materially less, due to the absence, in Fig. 56, of the third harmonic torque.

However, while the third harmonic (and its multiples) in the three-phase system of voltages are in phase, thus give no phase rotation, they may give torque, as a single-phase induction motor has torque, at speed, though at standstill the torque is zero.

Fig. 57 B shows diagrammatically, as T , the development of the air-gap distribution of a true three-phase winding, such as used in synchronous converters, etc. Each phase 1, 2, 3, covers

one-third of the pitch of a pair of poles or

2 T

of the upper layer,

HIGHER HARMONICS

149

and its return, 1', 2', 3', covers another third of the circumference of two poles, in the lower layer of the armature winding, 180° away from 1, 2, 3. However, this type of true three-phase wind- ing is practically never used in induction or synchronous machines, but the type of winding is used, which is shown as S, in Fig. 57 C. This is in reality a six-phase winding: each of the three

Fig. 57 . — Current distribution at air gap of induction motor, fundamental

and harmonics.

phases, 1, 2, 3, covers only one-sixth of the pitch of a pair of

7T

poles, or ^ or 60°, and between the successive phases is placed

the opposite phase, connected in the reverse direction. Thus the return conductors of phases 1, 2, 3 of the upper layer, are shown in the lower layer as 1', 2', 3'; in the upper layer, above 1', 2', 3', is placed again the phase 1, 2,* 3, but connected in the reverse direction, and indicated as 1 0) 2 0 , 3o. As 1 0 is connected in the reverse direction to 1, and V is the return of 1, lo is in

150

ELECTRICAL APPARATUS

phase with 1', and the return of 1 0 : l'o, is in the lower layer, in phase with, and beneath 1. Thus the phase rotation is: 1,-3, 2, -1, 3, -2, 1, etc.

For comparison, Fig. 57 A shows the usual quarter-phase winding, Q , of the same general type as the winding, Fig. 57 C .

If then the three third harmonics of 1, 2 and 3 are in phase with each other, for these third harmonics the true three-phase winding, T , gives the phase diagram shown as T z in Fig. 57 D. As seen, the current flows in one direction, single-phase, through- out the entire upper layer, and in the opposite direction in the lower layer, and thus its magnetizing action neutralizes, that is, there can be no third harmonic flux in the true three-phase winding.

The third harmonic diagram of the customary six-phase ar- rangement of three-phase winding, S, is shown as S z in Fig. 57 E. As seen, in this case alternately the single-phase third har-

7 r

monic current flows in one direction for 60° or g , and in the

7 r

opposite direction for the next g. In other words, a single-phase

m.m.f. and single-phase flux exists, of three times as many poles as the fundamental flux.

Thus, with the usual three-phase induction-motor winding, a third harmonic in the voltage wave produces a single-phase triple harmonic flux of three times the number of motor poles, and this gives a single-phase motor-torque curve, that is, a torque which, starting with zero at standstill, increases to a maximum in positive direction or assisting, and then decreases again to zero at its synchronous speed, and above this, becomes negative as single-phase induction-generator torque. Triple frequency with three times the number of poles gives a synchronous speed of £ — That is, the third harmonic in a three-phase vol-

tage may give a single-phase motor torque with a synchronous speed of one-ninth that of the fundamental torque, and in either direction, as shown as T% in dotted lines, in Fig. 56.

.As usually the third harmonic is absent in three-phase vol- tages, such a triple harmonic single-phase torque, as shown dotted in Fig. 56, is of rare occurrence: it could occur only in a four-wire three-phase system, that is, system containing the three phase-wires and the neutral.

90. All the torque components produced by the higher har- monics of the voltage wave have the same number of motor poles

vy

Fig. 58. — Current and flux distribution in induction-motor air gap, with different types of windings

many motor poles), but a lower synchronous speed, due to their higher frequency.

Torque harmonics may also occur, having the fundamental

152

ELECTRICAL APPARATUS

, 'Vf:

frequency, but higher number of pairs of poles than the funda- mental, and thus lower synchronous speeds, due to the deviation of the space distribution of the motor winding from sine.

The fundamental motor torque, Ti, of Figs. 55 and 56, is given by a sine wave of voltage and thus of flux, if the winding of each phase is distributed around the circumference of the motor air gap in a sinusoidal manner, as shown as F under “Sine,” in Fig. 58, and the flux distribution of each phase around the circum- ference of the air gap is sinusoidal also, as shown as <£ under “Sine,” in Fig. 58. «

This, however, is never the case, but the winding is always distributed in a non-sinusoidal manner.

The space distribution of magnetizing force and thus of flux of each phase, along the circumference of the motor air gap, thus can in the general case be represented by a trigonometric series, with co as space angle, in electrical degrees, that is, counting a pair of poles as 2ir or 360°. It is then:

The distribution of the conductors of one phase, in the motor air gap :

F = Fo { cos co 4* & 3 cos 3 co + a 5 cos 5 co + cos 7 co

-f- &9 cos 9 co ~f* . . . } ) (8)

here the assumption is made, that all the harmonics are in phase, that is, the magnetic distribution symmetrical. This is prac- tically always the case, and if it were not, it would simply add phase angle, a m , to the harmonics, the same as in paragraphs 88 and 89, but would make no change in the result, as the component torque harmonics are independent of the phase relations between the harmonic and the fundamental, as seen below.

In a quarter-phase motor, the second phase is located 90°

7T

or co = g displaced in space, from the first phase, and thus represented by the expression :

F l = Fo | cos (w — + «3 cos ^3 co + a 5 cos ^5 co —

-f- a 7 cos ^7 co + a 9 cos (§ co — + . . . J

= Fo | cos (w — -f a 3 cos (s co + '-j +a 5 cos ^5 co —

4~ a 7 cos ^7 co -f- -f- a 9 cos ^9 co — 4- . . . j • (9)

s.

[

HIGHER HARMONICS

153

Such a general or non-sinusoidal space distribution of magnetiz- ing force and thus of magnetic flux, as represented by F and F', can be considered as the superposition of a series of sinusoidal magnetizing forces and magnetic fluxes:

cos co dz cos 3 co as cos

8 *+f

0,7 cos 7 co cos 9 co ]

5 ( w ~i)

a 5 cos ^5 co

( 10 )

The first component:

COS co,

c°s(co -|),

gives the fundamental torque of the motor, as calculated in the customary manner, and represented by T\ in Figs. 55 and 56.

The second component of space distribution of magnetizing force:

a 3 cos 3 o), az cos ^3 o) + 0 >

gives a distribution, which makes three times as many cycles in the motor-gap circumference, than (10), that is, corresponds to a motor of three times as many poles. This component of space distribution of magnetizing force would thus, with the fundamental voltage and current wave, give a torque curve reaching synchronism as one-third speed; with the third harmonic of the voltage wave, (1 1) would reach synchronism at one-ninth, with, the fifth harmonic of the voltage wave at one-fifteenth of the normal synchronous speed.

In (11), the sign of the second term is reversed from that in (10), that is, in (11), the space rotation is backward from that of (10). In other words, (11) gives a synchronous speed of S = — with the fundamental or full-frequency voltage wave.

The third component of space distribution :

as cos 5 w,

<Z 5 cos (s 03 ““ l) 9

( 12 )

gives a motor of five times as many poles as (10), but with same space rotation as (10), and this component thus would give a torque, reaching synchronism at S — +3^5-

154

ELECTRICAL APPARATUS

In the same manner, the seventh space harmonic gives S = — the ninth space harmonic S = + %, etc.

91 . As seen, the component torque curves of the harmonics of the space distribution of magnetizing force and magnetic flux in the motor air gap, have the same characteristics as the 1 i component torque due to the time harmonics of the impressed

voltage wave, and thus are represented by the same torque •}’ diagrams:

Fig. 55 for a quarter-phase motor, .

Fig. 56 for a three-phase motor.

/ Here again, we see that the three-phase motor is less liable

/ to irregularities in the torque curve, caused by higher harmonics,

than the quarter-phase motor is.

Two classes of harmonics thus may occur in the induction

• motor, and give component torques of lower synchronous speed :

i Time harmonics, that is, harmonics of the voltage wave,

which are of higher frequency, but the same number of motor poles, and

Space harmonics, that is, harmonics in the air-gap distribu- tion, which are of fundamental frequency, but of a higher number l of motor poles.

Compound harmonics, that is, higher space harmonics of higher time harmonics, theoretically exist, but their torque necessarily is already so small, that they can be neglected, except where they are intentionally produced in the design.

We thus get the two classes of harmonics, and their characteristics:

Order of harmonic

1

3

5

7

9

11

13

15

17

Quarter-phase motor :

Phase rotation

!

•

j

Synchronous speed

• 1

-X

+X

— X

+X

-Mi

• M8

— Ms

• Mr

Timefl { Frequeilcy

/

3/

5/

7/

9/

11/

13/

15/

17/

1 No. of poles

V

V

V

V

V

p

V

P

P

Space ff( Fre<ll r cy

S

f

f

/

f

f

f

/

f

1 No. of poles

V

3 P

5 p

7 p

9 p

u p

13 p

15 p

17 p

Three-phase motor:

Phase rotation

0

0

4-

0

Synchronous speed

+1

(±50

-X

-Mi

4-Ms

(±M«)

-Mt

Time W Fre9uenoy

/

3/

5/

7 f

9/

11/

13/

15/

17/

1 No. of poles

V

(3 p)

P

P

(3j>)

V

P

(3 p)

P

Space fl|S requ “ cy

f

f

f

f j

/

f

f

/

f

1 No. of poles

V

0

5 p

7 P

0

u p

13 p

0

17 p

HIGHER HARMONICS

155

92. The space harmonics usually are more important than the time harmonics, as the space distribution of the winding in the motor usually materially differs from sinusoidal, while the devia- tion of the voltage wave from sine shape in modern electric power- supply systems is small, and the time harmonics thus usually negligible.

The space harmonics can easily be calculated from the dis- tribution of the winding around the periphery of the motor air gap. (See “ Engineering Mathematics,” the chapter on the trigonometric series.)

A number of the more common winding arrangements are shown in Fig. 58, in development. The arrangement of the conductors of one phase is shown to the left, under F, and the wave shape of the m.m.f. and thus the magnetic flux produced by it is shown under <$> to the right. The pitch of a turn of the winding is indicated under F.

Fig. 58 shows:

Full-pitch quarter-phase winding: Q — 0.

Full-pitch six-phase winding: $ — 0.

This is the three-phase winding almost always used in induction and synchronous machines.

Full-pitch three-phase winding: T — 0.

This is the true three-phase winding, as used in closed-circuit armatures, as synchronous converters, but of little importance in induction and synchronous motors.

% and 3 d 2 -piteh quarter-phase windings:

Q - Hi Q - Hi Q - H-

% and Yi - pitch six-phase windings:

8 -Hi 8 -Hi S-H-

%-piteh true three-phase windings: T — H-

As seen, the pitch deficiency, p, is denoted by the index.

Denoting the winding, F, on the left side of Fig. 58, by the Fourier series:

F = Fo (cos a? + a 3 cos 3 w + a 5 cos 5 co + a 7 cos 7 a) + . . .). (13) It is, in general:

• 4 P

Fodn = ~ I

F cos noodo).

If, then:

p = pitch deficiency, q = number of phases

156 ELECTRICAL APPARATUS

(four with quarter-phase, Q, six with six-phase, S } three with three-phase, T);

any fractional pitch winding then consists of the superposition of two layers :

and

From O3 = 0toco = ~-f- ~ q 2

from co = 0too) = --

a 2

and the integral (14) become:

Fo(l n =

Zjl.EE

4f[ ft 2

tt Vrr 2

A 2 A 2 ■

I cos ncodoj + I cos ncoe?o> f

e+

ftx \q

SF . nx

= — sin — cos _ nr q 2

2 /

pnx

• smn €~ ?):

as for: n = l;a n = 1, it is, substituted in (15):

8£

X

F o

. x nx ’

sin - cos ~ q 2

(15)

(16)

hence, substituting (16) into (15) :

dn —

• nr vmr sm — cos — — g 2

• X VTT

sin - cos ~ q 2

(17)

For full-pitch winding:

It is, from (17):

V = 0 .

sm -

nir

Cln

0 _

sm -

(18)

HIGHER HARMONICS

157

and for a fractional-pitch winding of pitch deficiency, p, it thus is :

2

a n = a n ° ( 19 )

pr

cos 2

93. By substituting the values: q — 4, 6, 3 and p — 0, }£, into equation (If), we get the coefficients a n of the trigonometric series:

F = Fo { cos -f- as cos 3 w + a 5 cos 5 co -f- &7 cos 7 co -f- ♦ * • } >

( 20 )

which represents the current distribution per phase through the air gap of the induction machine, shown by the diagrams F of Fig. 58.

The corresponding flux distribution, <3>, in Fig. 58, expressed by a trignometric series :

<£ = $o {sin co + &3 sin 3 co +■ 65 sin 5 co + 67 sin 7 co + . . . }

( 21 )

could be calculated in the same manner, from the constructive characteristics of <3> in Fig. 58.

It can, however, be derived immediately from the consideration, that $ is the summation, that is, the integral of F :

$ = fFdco (22)

and herefrom follows:

bn = ~ (23)

and this gives the coefficients, b n , of the series,

In the following tables are given the coefficients a n and b n , for the winding arrangements of Fig. 58, up to the twenty-first harmonic.

As seen, some of the lower harmonics are very considerable thus may exert an appreciable effect on the motor torque at low speeds, especially in the quarter-phase motor. *

Space Harmonics op Motor Winding

158

ELECTRICAL APPARATUS

21

-0.0476

-Mi

-0.0023

-0.0952

— Mi

-0.0045

0

0

-0.0349

-0.0017

0

0

+0.0476

+0.0023

-0.0698

-0.0033

0

0

+0.0952

+0.0045

0

0

0.0952

Mi

19

+0.0526

• M9 +0.0028- -0.0526 — Ms -0.0028 +0.0526

+0.0028

+0.0141

+0.0008

-0.0526

-0.0028

-0.0526

-0.0028

-0.0141

-0.0008

+0.0526

+0.0028

+0.0526

+0.0028

-0.0526

-0.0028

0.0526.

M 9

17

+0.0588 +H 7 +0.0035 +0.0588

• M7 +0.0035 -0.0588 — M> -0.0035 -0.0158 -0.0009 -0.0588 -0.0035 +0.0588 +0.0035 -0.0158 -0.0009 -0.0588 -0.0035 +0.0588 +0.0035 +0.0588 +0.0035 0.0588 M 7

-0.0667

-Ms

-0.0044

+0.1333

+Ms

+0.0089

0

0

+0.0488

+0.0033

0

0

-0.0667

-0.0044

-0.0976

+0.0065

0

0

+0.1333

+0.0089

0

0

0.1333

Hi

13

-0.0769 -Ms -0.0059 +0.0769

• K 3 +0.0059 +0.0769
• M 2 +0.0059 +0.0769 +0.0059 -0.0769 -0.0059 +0.0769 +0.0059 -0.0769 +0.0059 +0.0769 +0.0059 -0.0769 -0.0059 +0.0769 +0.0059 0.0769 Ms

tH

+0.0909

• Hi +0.00S2 -0.0909

-Mi

-0.0082

-0.0909

-Ml

-0.0082

-0.0909

-0.0083

+0.0909

+0.0082

-0.0909

-0.0082

+0.0909

+0.0083

-0.0909

— 0.00S2 +0.0909 +0.0082 -0.0909

— 0.00S2 0.0909 Mi

Cl

+0.1111

+M

+0.0123

-0.2222

-H

-0.0247

0

0

-0.0313 -0.0090 * 0

0

+0.1111

+0.0123

+0.1626

+0.0181

0

0

-0.2222

-0.0247

0

0

0.2222

b-

-0.1429

— H

-0.0204

-0.1429

-M

-0.0204

+0.1429

+K

+0.0204

+0.0383

+0.0055

+0.1429

+0.0204

-0.1429

-0.0204

+0.0383

+0.0055

+0.1429

+0.0204

-0.1429

-0.0204

-0.1429

-0.0204

0 . 1429

M

»o

-0.2000

-H

-0.0400

+0.2000

+M

+0.0400

-0.2000

-M

-0.0400

-0.0536

-0.0107

+0.200

+0.040

+0.2000

+0.0400

+0.0536

+0.0107

-0.2000

-0.0400

-0.2000

-0.0400

+0.2000

+0.0400

0.2000

H

CO

CO i-f 1> (N *> — V — ' CO CO i-l b- b- <M

CO tHO <M ^ rH CO r-f 00 <N CD CN

CO H to N ^ CO M H 00 O to Oi t*

CO 1-IO CM (NO M H H O M CO

2 0 rA 0 0^0 ® ° 00 0c> 0000 00 00 00 0

S++++++ ++ II++ II 6 ™

n

c

11 n 11 11 11 11 ii 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 „

«: e t e tt e tteeeeetjciitfrtctfec

G -c. e »o e ■© c* -o q e -o a ,0 0 .0 a -0 a * o-o

s.

s,

2? M 0 SM V 0 \M

££ O O O r>K rK r-f\ -K cN,

tsi

any

4

6

3

4

4

4

6

6

6

3

max.

J 4 4 ^ H TH H

02 Q? CQ

CHAPTER VIII

SYNCHRONIZING INDUCTION MOTORS

94. Occasionally two or more induction motors are operated in parallel on the same load, as for instance in three-phase rail- roading, or when securing several speeds by concatenation. In this case the secondaries of the induction motors may be connected in multiple and a single rheostat used for starting and speed control. Thus, when using two motors in concatena- tion for speeds from standstill to half synchronism, from half synchronism to full speed, the motors may also be operated on a single rheostat by connecting their secondaries in parallel. As in parallel connection the frequency of the secondaries must be the same, and the secondary frequency equals the slip, it follows that the motors in this case must operate at the same slip, that is, at the same frequency of rotation, or in synchronism with each other. If the connection of the induction motors to the load is such that they can not operate in exact step with each other, obviously separate resistances must be used in the motor secondaries, so as to allow different slips. When rigidly connect- ing the two motors with each other, it is essential to take care that the motor secondaries have exactly the same relative posi- tion to their primaries so as to be in phase with each other, just as would be necessary when operating two alternators in parallel with each other when rigidly connected to the same shaft or when driven by synchronous motors from the same supply. As in the induction-motor secondary an e.m.f. of definite fre- quency, that of slip, is generated by its rotation through the revolving motor field, the induction-motor secondary is an alternating-current* generator, which is short-circuited at speed and loaded by the starting rheostat during acceleration, and the problem of operating two induction motors with their secondaries connected in parallel on the same external resistance is thus the same as that of operating two alternators in parallel. In general, therefore, it is undesirable to rigidly connect induction-motor secondaries mechanically if they are electrically connected in parallel, but it is preferable to have their mechanical connection

160

ELECTRICAL APPARATUS

sufficiently flexible, as by belting, etc., so that the motors can drop into exact step with each other and maintain step by their synchronizing power.

It is of interest, then, to examine the synchronizing power of two induction motors which are connected in multiple with their secondaries on the same rheostat and operated from the same primary impressed voltage.

95. Assume two equal induction motors with their primaries connected to the same voltage supply and with their secondaries connected in multiple with each other to a common resistance, r, and neglecting for simplicity the exciting current and the vol- tage drop in the impedance of the motor primaries as not mate- rially affecting the synchronizing power.

Let Zi = ri + jx i = secondary self-inductive impedance at full frequency; s = slip of the two motors, as fraction of syn- chronism; e 0 = absolute value of impressed voltage and thus, when neglecting the primary impedance, of the voltage generated in the primary by the rotating field.

If then the two motor secondaries are out of phase with each other by angle 2 r, and the secondary of the motor 1 is behind in the direction of rotation and the secondary of the motor 2 ahead of the average position by angle r, then :

$1 = se 0 (cos r + j sin r) = secondary generated

l^m.f. of the first motor, (1)

#2 = se o (cos r — j sin r) = secondary generated

e.m.f. of the second motor. (2)

And if h = current coming from the first, jT 2 = current coming from the second motor secondary, the total current, or current in the external resistance, r, is :

/ = h + h; (3)

it is then, in the circuit comprising the first motor secondary and the rheostat, r,

• /i Z - Jr = 0, (4)

in the circuit comprising the second motor secondary and the rheostat, r,

$ 2 — hZ — {r = 0, (5)

where

z « Ti + jsx i;

SYNCHRONIZING INDUCTION MOTORS 161

substituting (3) into (4) and (5) and rearranging gives:

E, - I, (Z + r) . - I,r = 0,

E 2 — hr — h (Z + r) = 0.

These two equations added and subtracted give :

hence,

and

E, + Eo - (h + I 2 ) (Z + 2 r) = 0, jB7i — £7 2 — (/l — 1 2 ) Z — 0,’

II + I2 =

h-h =

E \ -{- £^2 Z + 2r

Ei — E 2

Substituting for convenience the abbreviations, 1

Z + 2r 1

= Y — g — jb, = Yi = gi - jbi

( 6 )

(7)

into equations (6) and substituting (1) and (2) into (6), gives:

h + h = 2 se 0 F cos r,

/1 — I 2 = + 2jse 0 Yi sin t ;

hence,

/ 2 1 = seo {Y cost T jTi sin t)

( 8 )

(9)

is the current in the secondary circuit of the motor, and there- fore also the primary load current, that is, the primary current corresponding to the secondary current, and thus, when neg- lecting the exciting current also the primary motor current, where the upper sign corresponds to the first, or lagging, the lower sign to the second, .'or leading, motor.

Substituting in (9) for Y and Ft gives:

1 2 1 = se 0 { (g cos r ±bi sin r) — j ( b cos t T g 1 sin r) } , (10)

the primary e.m.f. corresponding hereto is:

jpV = e 0 {cost + j sinr}, (11)

where again the upper sign corresponds to the first, the lower to the second motor.

The power consumed by the current, l%\ with the e.m.i, j^ 1 .

162

ELECTRICAL APPARATUS

is the sum of the products of the horizontal components, and of the vertical components, that is, of the real components and of the imaginary components of these two quantities (as a horizontal component of one does not represent any power with a vertical component of the other quantity, being in quadrature therewith).

p 2 > = | whX