The value found for the low-resistance motor, t — 5, is how- ever not feasible, as it gives: ei = e 2 = 2.23 eo, and in a quarter- phase motor designed for impressed voltage, e 0 , the impressed voltage, 2.23 e 0 , would be far above saturation. Thus the motor would have to be operated at lower supply voltage single-phase, and then give lower t , though the same value of v = 3.16. At ei = e 2 = eo, the impressed voltage of the single-phase circuit would be about 45 per cent, of e 0 , and then it would be: t = 1.

Thus, in the low-resistance motor, it would be preferable to operate the two motor circuits in series, but shunted by the two different capacities producing true quarter-phase relation.

Series Connection

71 . The calculation of the single-phase starting of a motor with two coils in quadrature position, shunted by two impedances

Yi

Fig. 38. — Diagram of phase-splitting device with series connection of motor

circuits

of different power-factor, as shown diagrammatically in Fig. 38, can be carried out in the same way as that of parallel connection, except that it is more convenient in series connection to use the term “ admittance” instead of impedance.

That is, let the effective admittance per motor coil equal :

Y = ~2 = g-jb,

110

ELECTRICAL APPARATUS

and the two motor coils be shunted respectively by the admit- tances :

Z'~ 9 '~ I (52)

Y 2 = g 2 - jb 2y J

it is then:

Y ^ry- t Y + y 2

the current consumed by the motor, and :

Ti I i n I

Y + Yi

and Eo =

F+ Y 2

the voltages across the two motor circuits.

The phase difference between E x and E 2 thus is given by

m (cos cj) + j sin <j>)

Y+Y* Y + Y i

and herefrom follows t, q and v.

As instance consider a motor of effective admittance per cir- cuit :

Y = g - jb = 1 - 3 j,

with the two circuits connected in series between single-phase mains of voltage, e 0 , and one circuit shunted by a non-inductive resistance of conductance, g } .

What value of g i gives maximum starting torque, and what is this torque?

It is:

(53): / =

1 — _| L

g + gi— jb g - jb

eo(g ~ jb) .

< 54);

_ eo (g - jb )(g + g l -Jb) _ .

2g + Q\ - 2/6 ’ m

— t°Sl ± 9 l ~ fo) •

2g + g t — 2 jb

(55) : m (cos <f> + j sin <t>) = — - — — W — •

g — Jb g 2 A- o 2

tan 0

sm <t> =

, = gi6

• • 6 2 ’

£]6

<h 2 & 2 + [? (0 + »,) + W

SINGLE-PHASE INDUCTION MOTOR

111

and thus:

t

6162 sin <f>

gib

eo

[(2 g + gi) 2 + 4 6 2 ]

gib f

(2g + 0 1 ) 2 + 46 2 and for maximum, t :

thus:

dt

dgi

= 0,

gi = 2 V g 2 '~+ b 2 = 2 y = 6.32,

or, substituting back: (59): t =

= 0.18.

(59)

(60)

(61)

4 (jr + y)

As in single-phase operation, the voltage, e 0 , is impressed upon the two quadrature coils in series, each coil receives only about e 0

V2

Comparing then the single-phase starting torque with that

e 0

of a quarter-phase motor of impressed voltage, — it is:

V2

2 ^ = 0.36.

The reader is advised to study the possibilities of capacity and reactance (inductive or capacity) shunting the two motor coils, the values giving maximum torque, those giving true quarter-phase relation, and the torque and apparent torque efficiencies secured thereby.

B. INDUCTIVE DEVICES External Inductive Devices

72 , Inductively divided circuit : in its simplest form, as shown diagrammatically in Fig. 39, the motor contains two circuits at right angles, of the same admittance.

The one circuit (1) is in series with the one, the other (2) with the other of two coils wound on the same magnetic circuit, M. By proportioning the number of turns, and n 2 , of the two coils, which thus are interlinked inductively with each other on the external magnetic circuit, M , a considerable phase displacement

112

ELECTRICAL APPARATUS

between the motor coils, and thus starting torque can be pro- duced, especially with a high-resistance armature, that is, a motor with starting rheostat.

A full discussion and calculation of this device is contained in the paper on the “Single-phase Induction Motor,” page 63, A. I. E. E. Transactions, 1898.

Fig.

Internal Inductive Devices

The exciting system of the motor consists of a stationary pri- mary coil and a stationary secondary coil, short-circuited upon itself (or closed through an impedance), both acting upon the revolving secondary.

The stationary secondary can either cover a part of the pole face excited by the primary coil, and is then called a “shading coil,” or it has the same pitch as the primary, but is angularly displaced therefrom in space, by less than 90° (usually 45° or 60°), and then has been called accelerating coil.

The shading coil, as shown diagrammatically in Fig. 40, is the simplest of all the single-phase induction motor-starting devices, and therefore very extensively used, though it gives only a small starting torque, and that at a low apparent starting- torque efficiency. It is almost exclusively used in very small motors which require little starting torque, such as fan motors, and thus industrially constitutes the most important single- phase induction motor-starting device.

73 . Let, all the quantities being reduced to the primary num- ber of turns and frequency, as customary in induction machines :

Z o = r 0 + jx o = primary self-inductive impedance,

Y = g — jb = primary exciting admittance of unshaded poles (assuming total pole unshaded),

SINGLE-PHASE INDUCTION MOTOR

113

Y f = g f — jV = primary exciting admittance of shaded poles (assuming total pole shaded).

If the reluctivity of the shaded portion of the pole is the same as that of the unshaded, then Y' — Y ; in general, if

b = ratio of reluctivity of shaded to unshaded portion of

pole,

Y' = bY,

b either = 1 , or, sometimes, b > 1 , if the air gap under the shaded portion of the pole is made larger than that under the unshaded portion.

Yi = gi — jbi = self-inductive admittance of the revolving secondary or armature,

Y % — g<i —jbz — self-inductive admittance of the stationary secondary or shading coil, inclusive its exter- nal circuit, where such exists.

Z 0 , Y i and Y 2 thus refer to the self-inductive impedances, in which the energy component is due to effective resistance, and Y and Y' refer to the mutual inductive impedances, in which the energy component is due to hysteresis and eddy currents.

a = shaded portion of pole, as fraction of total pole; thus

(1 — a) = unshaded portion of pole.

If:

Co = impressed single-phase voltage,

= voltage induced by flux in unshaded portion of pole,

Eo = voltage induced by flux in shaded portion of pole,

I o = primary current, it- is then :

Co — E\ + E 2 +- Z 0 /o. (62)

The secondary current in the armature under the unshaded portion of the pole is :

h = Vi Yi. (63) '

The primary exciting current of the unshaded portion of the pole :

h

h

/«, - m

• • ( 66 )

thus :

[4 ELECTRICAL APPARATUS

The secondary current under the shaded portion of the pole is:

V 1 = EzYl (66)

The current in the shading coil is:

h = E 2 Y 2 . (67)

The primary exciting current of the shaded portion of the pole

Jo = /' 1 + Ioo + h = V*[Yi + -Y+ y 2 J ; (69)

from (65) and (69) follows:

Fi + £ Y + Y 2 a

1 — a

ra(cos<£ + j sin <£),

and this gives the angle, <j > , of phase displacement between the two component voltages, $ i and $ 2 .

. If, as usual, 5 = 1, and if a = 0.5, that is, half the pole is shaded, it is:

Ei _ Fi + 2y + y 2 ( ,

2 “ Y 1 + 2Y K

74 . Assuming now, as first approximation, Z 0 - 0, that is, neglecting the impedance drop in the single-phase primary coil — which obviously has no influence on the phase difference between the component voltages, and the ratio of their values, that is, on the approximation of the devices to polyphase relation — then it is:

$1 + $2 = 00; ( 72 )

thus, from (70) :

Ei — eo

y i + ~ y + y 2

a

■■ +i ’(« + ii«)+ y =

E2 = Co

1 — a

2T ' + Y ( l + r~al + y '

( 73 )

SINGLE-PHASE INDUCTION MOTOR

115

or, for:

b = 1; a = 0.5;

_ F, + 2 Y + f 2

• 1 2 Fx + 4 F + F 2 ’

F = Xi + 2r 2 Fx + 4 F + F 2 ’

(74)

and the primary current, or single-phase supply current is, by substituting (73) into (65) :

or, for:

(p + rh K^ + y.)

2 Fx+ F(- + — — ) + F 2 \a 1 — a!

/ o = e 0

6 = 1; a = 0.5:

(Fx + 2F)(Fx + 2F+F 2 ) 2 F x + 4 F + F 2

(75)

(76)

and herefrom follows, by reducing to absolute values, the torque, torque ratio, volt-ampere input, apparent torque efficiency, etc. Or, denoting:

y. + t y ~ - y>,

Y + ^Y + V 2 = Y',

Q /

(77)

it is:

(70):

E Y f

v = v” = m ( cos ^ + j sin 0 ) ;

JJj o / 0

(78)

(73):

p _ <*F' , '

• 1 yo 1|_ y/

*, _ e 0 F°

• 2 yo | y/j

(79)

(75):

II.

(80)

T = Ae x e 2 sin <t>,

Q = eoioi

and for a quarter-phase motor, with voltage

e 0

V2

impressed per

116

ELECTRICAL APPARATUS

circuit, neglecting the primary impedance, z 0 , to be comparable with the shaded-coil single-phase motor, it is:

thus:

;° = ^ (F+Fj))

Qq —

2e 0 io

V2

T a

° = ef/Y + Y ,/,

<1

io V" 2 2

. 2eie 2 • ,

t = -v- sin <f>, Co“

V

t u

<2

75 . As instances are given in the following table the compo- nent voltages, e'i and e 2 , the phase angle, <£, between them, the primary current, io, the torque ratio, t , and the apparent starting- torque efficiency, v, for the shaded-pole motor with the constants:

Impressed voltage : = 100;

Primary exciting admittance : F = 0.001 — 0.01 j. b = 1, that is, uniform air gap. a = 0.5, that is, half the pole is shaded.

And for the three motor armatures :

Low resistance: Fi = 0.01 — 0.03 j,

Medium resistance: Y x = 0.02 — 0.02 j 7

High resistance: Yi — 0.03 — 0.01 j;

and for the three kinds of shading coils :

Low resistance: F 2 .= 0.01 — 0.03 j,

Medium resistance: F 2 = 0.02 — 0.02 j,

High resistance: F 2 = 0.03 — 0.01 j .

As seen from this table, the phase angle, <£, and thus the start- ing torque, t , are greatest with the combination of low-resistance armature and high-resistance shading coil, and of high-resistance armature with low-resistance' shading coil; but in the first case the torque is in opposite direction — accelerating coil — from what

SINGLE-PHASE INDUCTION MOTOR

117

it is in the second case — lagging coil. In either case, the torque efficiency is low, that is, the device is not suitable to produce high starting-torque efficiencies, but its foremost advantage is the extreme simplicity.

The voltage due to the shaded portion of the pole, c 2 , is less than that due to the unshaded portion, ei, and thus a somewhat higher torque may be produced by shading more than half of the pole: a > 0.5.

A larger air gap: b > 1, under the shaded portion of the pole, or an external non-inductive resistance inserted into the shad- ing coil, under certain conditions increases the torque somewhat — at a sacrifice of power-factor — particularly with high-resistance armature and low-resistance shading coil.

e 0 = 100 volts; a = 0.5; b = 1; Y - 0.001 - 0.01 j.

Y i:

Yt

s: ei:

e*:

<t>'

to:

v :

V

o

r-H

X

?

o

X

per cent.

per cent.

1 - 3 j 1

3 j 38.3

61.8

• 1.9

1.97

• 1.56

• 4.07

2

—

2 j 40.3

60.2

• 11.0

2.07

• 9.28

+23.00

3

—

1 j 42.0

59.8

+21.5

2.17

• 18.36

+43.70

2 — 2 j l

—

3 j 37.2

62.9

• 4.3

1.70

• 3.52

• 9.65

2

2 j 38.5

61.7

• 6.2

1.76

• 5.12

+13.60

3

—

1 j 39.2

62.0

• 17.3

1.80

• 14.44

+37.40

3 - 1 j 1

—

3 j 37.6

63.0

-11.9

1.66

• 9.76

-25.80

2

—

2 j 37.8

62.5

• 0.8

1.66

• 0.66

• 1.75

3

1 j 37.4

63.0

• 10.3

1.64

• 8.44

+22.60

Monocyclic Starting Device

76 . The monocyclic starting device consists in producing ex- ternally to the motor a system of polyphase voltages with single- phase flow of energy, and impressing it upon the motor, which is wound as polyphase motor.

If across the single-phase mains of voltage, e, two impedances of different inductance factors, Z\ and Z 2 , are connected in series, as shown diagrammatically in Fig. 41, the two voltages, 1 $ i and E 2 , across these two impedances are displaced in phase from each other, thus forming with the main voltage a voltage triangle. The altitude of this triangle, or the voltage, $q, between the com-

118

ELECTRICAL APPARATUS

mon connection of the two impedances, and a point inside of the main voltage, e (its middle, if the two impedances are equal), is a voltage in quadrature with the main voltage, and is a teazer voltage or quadrature voltage of the monocyclic system, e, E h E 2 > that is, it is of limited energy and drops if power is taken off from it. (See Chapter XIV.)

Let then, in a three-phase wound motor, oper- ated single-phase with monocyclic starting device, and shown diagrammatically in Fig. 42:

e = voltage impressed between single-phase

F ig. 41 . lines,

Monocyclic J = current in single-phase lines,

nang e. y = effective admittance per motor circuit,

Y h Ei and /' i, and Y 2} E 2 and /' 2 = admittance, voltage and current respectively, in the two impedances of the mono- cyclic starting device,

r

Fig. 42. — Three-phase motor with monocyclic starting device.

J\ y Ji and lz = currents in the three motor circuits.

Eo and / o = voltage and current of the quadrature circuit from the common connection of the two impedances, to the motor.

SINGLE-PHASE INDUCTION MOTOR

119

It is then, counting the voltages and currents in the indicated by the arrows of Fig. 42 :

direction

/o = I'l - = U •

substituting:

l'i = BiY h ' /' 2 = ? 2 f 2 ,

-h;

(81)

U = E 2 Y, h =BiY,

gives:

(82)

BiY\ — BOY 2 = ( Bi —

thus:

BO Y,

Ei Y 2 + Y . .

E~ = TT+Y ~ m ( cos ^ + J sm 4)- (83)

This gives the phase angle, <j>, between the voltages, Ei and B 2 , of the monocyclic triangle. Since :

it is, by (83) :

Bi + Bi —

f 2 + f

Yi + F 2 + 2 Y’

Y: + Y

Yi + F 2 + 2 Y’

and the quadrature voltage:

Bo = (Bt - BO ■

_e Y 1 -Y 2 2 7: + F 2 + 2 Y’

(84)

(85)

(86)

and the total current input into the motor, inclusive starting device:

/ = /T + h + h = BiYi + BOY + eY

= e

= e

f(Y } ± Y)(Y 2 + Y) i t Y 1 + Y 2 + 2Y + * ] FiF 2 -f- 2 F(Fj + F 2 ) + 3 F 2 Fi + F 2 + 2 F

(87)

As with the balanced three-phase motor, the quadrature com- ponent of voltage numerically is ^ V3 , it is, when, denoting by:

120 ELECTRICAL APPARATUS

Eo 1 ' the numerical value of the imaginary term of E a ; the torque ratio is:

. _ 2 L a ‘

~ eVz

The volt-ampere ratio is:

i

9 = 3Ta’

thus the apparent starting-torque efficiency:

t

v ~

Q . etc.

77. Three cases have become of special importance:

(а) The resistance-reactance monocyclic starting device ; where one of the two impedances, Z x and Z h is a resistance, the other an inductance. This is the simplest and cheapest arrangement, gives good starting torque, though a fairly high current consump- tion and therefore low starting-torque efficiency, and is therefore very extensively used for starting single-phase induction motors. After starting, the monocyclic device is cut out and the power consumption due to the resistance, and depreciation of the power- factor due to the inductance, thereby avoided.

This device is discussed on page 333 of “Theoretical Elements of Electrical Engineering” and page 253 of “Theory and Calcu- lation of Alternating-current Phenomena.”

(б) The “condenser in the tertiary circuit,” which may be considered as a monocyclic starting device, in which one of the two impedances is a capacity, the other one is infinity. The capacity usually is made so as to approximately balance the mag- netizing current of the motor, is left in circuit after starting, as it does not interfere with the operation, does not consume, power, and compensates for the lagging current of the motor, so that the motor has practically unity power-factor for all loads. This motor gives a moderate starting torque, but with very good start- ing-torque efficiency, and therefore is the most satisfactory single- phase induction motor, where very high starting torque is not needed. It was extensively used some years ago, but went out of use due to the trouble with the condensers of these early days, and it is therefore again coming into use, with the development of the last years, of a satisfactory condenser.

( 88 )

(89)

(90)

SINGLE-PHASE INDUCTION MOTOR

121

The condenser motor is discussed on page 249 of “Theory and Calculation of Alternating-current Phenomena.” ■

(c) The condenser-inductance monocyclic starting device. By suitable values of capacity and inductance, a balanced three- phase triangle can be produced, and thereby a starting torque equal to that of the motor on three-phase voltage supply, with an apparent starting-torque efficiency superior to that of the three-phase motor.

Assuming thus :

Y i = +jbi = capacity, .

Y 2 = —jb 2 = inductance, ^

Y = g — jb.

If the voltage triangle, e, It! i, E 2 , is a balanced three-phase tri- angle, it is :

Bx = 2 a -j Vs),

B2 = 2 (i + j Vs)-

(92)

{Substituting (91) and (92) into (83), and expanding gives:

(b 2 — bi 2 b) /3 ~ j (b 2 + bi — 2 g /3) = 0;

thus:

hence:

thus, if:

the second reactance, Z 2 , must be a capacity also; if

b < g VS,

only the first reactance, Z i, is a capacity, but the second is an inductance.

78 * Considering, as an instance, a low-resistance motor, and a high-resistance motor :

(a) (b)

b 2 -

b 2 + bi

bi -f- 2 b = 0,

• 2 0 VS = 0 ;

i>i = g Vs + b,

b 2 = gVS - b;

(93)

b > g VS,

Y = g-jb = l- Sj,

Y = g— jb = 3— j,

122

ELECTRICAL APPARATUS

it is:

bi — 4.732, capacity, = 6.196, capacity,

5 2 = —1.268, capacity, 62 = 4.196, inductance.

It is, by (86) and (92)

Etf = 2 (-®2 T -®l) == rj

thus:

t = 1, as was to be expected.

/a = e(g - jb),

U = e Vg 2 + b 2 = 3.16 e;

it is, however,

by (87):

J = e(3g -jb);

thus:

i =

4.243 e,

i = 9.06 e,

and by (89) :

Q =

0.448,

q = 0.956,

thus:

v =

2.232,

v = 1.046.

Further discussion of the various single-phase induction motor- starting devices, and also a discussion of the acceleration of the motor with the starting device, and the interference or non-inter- ference of the starting device with the quadrature flux and thus torque produced in the motor by the rotation of the armature, is given in a paper on the “Single-phase Induction Motor,” A. I. E. E. Transactions , 1898, page 35, and a supplementary paper on “Notes on Single-phase Induction Motors,” A. I. E. E. Trans- actions , 1900, page 25.

CHAPTER VI

INDUCTION-MOTOR REGULATION AND STABILITY

1. VOLTAGE REGULATION AND OUTPUT

2. Load and speed curves of induction motors are usually calculated and plotted for constant-supply voltage at the motor terminals. In practice, however, this condition usually is only approximately fulfilled, and due to the drop of voltage in the step-down transformers feeding the motor, in the secondary and the primary supply lines, etc., the voltage at the motor terminals drops more or less with increase of load. Thus, if the voltage at the primary terminals of the motor transformer is constant, and such as to give the rated motor voltage at full-load, at no- load the voltage at the motor terminals is higher, but at overload lower by the voltage drop in the internal impedance of the trans- formers. If the voltage is kept constant in. the center of distri- bution, the drop of voltage in the line adds itself to the imped- ance drop in the transformers, and the motor supply voltage thus varies still more between no-load and overload.

With a drop of voltage in the supply circuit between the point of constant potential and the motor terminals, assuming the cir- cuit such as to give the rated motor voltage at full-load, the voltage at no-load and thus the exciting current is higher, the voltage at overload and thus the maximum output and maximum torque of the motor, and also the motor impedance current, that is, current consumed by the motor at standstill, and thereby the starting torque of the motor, are lower than on a constant-poten- tial supply. Hereby then the margin of overload capacity of the motor is reduced, and the characteristic constant of the motor, or the ratio of exciting current to short-circuit current, is in- creased, that is, the motor • characteristic made inferior to that given at constant voltage supply, the more so the higher the voltage drop in the supply circuit.

Assuming then a three-phase motor having, the following con- stants: primary exciting admittance, Y = 0.01 — 0.1 j) primary self-inductive impedance, Z Q = 0.1 + 0.3 j; secondary self-induc-

124

ELECTRICAL APPARATUS

tive impedance, Zi = 0.1 + 0.3 j; supply voltage, *e 0 =110 volts, and rated output, 5000 watts per phase.

Assuming this motor to be operated: *

1. By transformers of about 2 per cent, resistance and 4 per cent, reactance voltage, that is, transformers of good regulation, with constant voltage at the transformer terminals.

2. By transformers of about 2 per cent, resistance and 15 per cent, reactance voltage, that is, very poorly regulating trans- formers, at constant supply voltage at the transformer primaries.

3. With constant voltage at the generator terminals, and about 8 per cent, resistance, 40 per cent, reactance voltage in line and transformers between generator and motor.

This gives, in complex quantities, the impedance between the motor terminals and the constant voltage supply:

1. Z = 0.04 + 0.08 j,

2. Z = 0.04 + 0.3 j,

3. Z = 0.16 + 0.8 j.

It is assumed that the constant supply voltage is such as to give 110 volts at the motor terminals at full-load.

The load and speed curves of the motor, when operating under these conditions, that is, with the impedance, Z, in series between . the motor terminals and the constant voltage supply, e h then can be calculated from the motor characteristics at constant termi- nal voltage, e 0 , as follows :

At slip, s, and constant terminal voltage, e Q , the current in the motor is i 0} its power-factor p = cos 0. The effective or equiva- lent impedance of the motor at this slip then is z° = and, in

complex quantities, Z° = ~ (cos 6 + j sin 0), and the total im-

'Z'O

pedance, including that of transformers and line, thus is:

Zx = Z° + Z = (Jcos0 + r) + jgsin0 + x), or, in absolute values :

-=M™ d+r Y + (t sinf?+a: ) 2 ’

and, at the supply voltage, ei, the current thus Is:

INDUCTION-MOTOR REGULATION

125

and the voltage at the motor terminals is:

z°

t ft * **

e o = zH x = -- e.

Zl

If e 0 is the voltage required at the motor terminals at full-load, and i 0 ° the current, Zx° the total impedance at full-load, it is:

Fra. 43. — Induction-motor load curves corresponding to 110 volts at motor terminals at 5000 watts load.

hence, the required constant supply voltage is :

ex = Zi°i 0 °,

and the speed and torque curves of the motor under this condi- tion then are derived from those at constant supply voltage, e 0 ,

by multiplying all voltages and currents by the factor that

is, by the ratio of the actual terminal voltage to the full-load terminal voltage, and the torque and power by multiplying with

126

ELECTRICAL APPARATUS

the square of this ratio, while the power-factors and the efficien- cies obviously remain unchanged.

In this manner, in the three cases assumed in the preceding, the load curves are calculated, and are plotted in Figs. 43, 44, and 45.

80 . It is seen that, even with transformers of good regulation, Fig. 43, the maximum torque and the maximum power are ap-

Fig. 44. — Induction-motor load curves corresponding to 110 volts at motor terminals at 5000 watts load.

preciably reduced. The values corresponding to constant termi- nal voltage are shown, for the part of the curves near maximum torque and maximum power, in Figs. 43, 44, and 45.

In Figs. 46, 47, 48, and 49 are given the speed-torque curves of the motor, for constant terminal voltage, Z = 0, and the three cases above discussed; in Fig. 46 for short-circuited secondaries, or running condition; in Fig. 47 for 0.15 ohm; in Fig. 48 for 0.5 ohm; and in Fig. 49 for 1.5 ohms additional re- sistance inserted in the armature. As seen, the line and trans- former impedance very appreciably lowers the torque, and

INDUCTION-MOTOR REGULATION

127

Fig. 45. — Induction-motor load curves corresponding to 110 volts at motor terminals at 500 watts load.

Fig. 46. — Induction-motor speed torque characteristics with short-circuited

secondary.

128

ELECTRICAL APPARATUS

especially the starting torque, which, with short-circuited arma- ture, in the case 3 drops to about one-third the value given at constant supply voltage.

Fig. 47. — Induction-motor speed torque characteristics with a resistance of 0.15 ohm in secondary circuit.

Fig. 48. — Induction-motor speed torque characteristics with a resistance of 0.5 ohm in secondary circuit.

It is interesting to note that in Fig. 48, with a secondary resistance giving maximum torque in starting, at constant ter-

INDUCTION-MOTOR REGULATION

129

minal voltage, with high impedance in the supply, the starting torque drops so much that the maximum torque is shifted to about half synchronism.

In induction motors, especially at overloads and in starting, it therefore is important to have as low impedance as pos- sible between the point of constant voltage and the motor terminals.

Slip Fraction of Synchronism

Fig. 49. — Induction-motor speed current characteristics with a resistance of 1.5 ohms in secondary circuit.

In Table I the numerical values of maximum power, maxi- mum torque, starting torque, exciting current and starting current are given for above motor, at constant terminal voltage and for the three values of impedance in the supply lines, for such supply voltage as to give the rated motor voltage of 110 volts at full load and for 110 volts supply, voltage. In the first case, maximum power and torque drop down to their full-load values with the highest line impedance, and far below full-load values in the latter case.

130

ELECTRICAL APPARATUS

INDUCTION-MOTOR REGULATION

131

2. FREQUENCY PULSATION

3. If the frequency of the voltage supply pulsates with sufficient rapidity that the motor speed can not appreciably follow the pulsations of frequency, the motor current and torque also pulsate; that is, if the frequency pulsates by the fraction, p, above and below the normal, at the average slip, s, the actual slip pulsates between $ + p and s — p, and motor current and

Fig. 50. — Effect of Frequency Pulsation on Induction Motor.

torque pulsate between the values corresponding to the slips, $ + V and $ — p. If then the average slip s < p, at minimum frequency, the actual slip, s — p, becomes negative; that is, the motor momentarily generates and returns energy.

As instance are shown, in Fig. 50, the values of current and of torque for maximum and minimum frequency, and for the average frequency, for p = 0.025, that is, 2.5 per cent, pulsa- tion of frequency from the average. As seen, the pulsation of current is moderate until synchronism is approached, but be-

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ELECTRICAL APPARATUS

comes very large near synchronism, and from slip, s = 0.025, up to synchronism the average current remains practically con- stant, thus at synchronism is very much higher than the current at constant frequency. The average torque also drops some- what below the torque corresponding to constant frequency, as shown in the upper part of Fig. 50.

3. LOAD AND STABILITY

4. At constant voltage and constant frequency the torque of the polyphase induction motor is a maximum at some definite speed and decreases with increase of speed over that correspond- ing to the maximum torque, to zero at synchronism; it also de- creases with decrease of speed from that at the maximum torque I

point, to a minimum at standstill, the starting torque. This maximum torque point shifts toward lower speed with increase of the resistance in the secondary circuit, and the starting torque thereby increases. Without additional resistance inserted in the secondary circuit the maximum torque point, however, lies at fairly high speed not very far below synchronism, 10 to 20 per cent, below synchronism with smaller motors of good effi- ciency. Any value of torque between the starting torque and the maximum torque is reached at two different speeds. Thus in a three-phase motor having the following constants: impressed e.m.f., eo = 110 volts; exciting admittance, Y = 0.01 — 0.1 j; primary impedance, Zq = 0.1 + 0.3 j, and secondary impedance,

Zi = 0.1 + 0.3 j } the torque of 5.5 synchronous kw. is reached at 54 per cent, of synchronism and also at the speed of 94 per cent, of synchronism, as seen in Fig. 51.

When connected to a load requiring a constant torque, irre- #

spective of the speed, as when pumping water against a constant head by reciprocating pumps, the motor thus could carry the load at two different speeds, the two points of intersection of the horizontal line, L, in Fig. 51, which represents the torque con- sumed by the load, and the motor-torque curve, Z). Of these two points, d and c, the lower one, d, represents unstable con- ditions of operation; that is, the motor can not operate at this speed, but either stops or runs up to the higher speed point, c , at which stability is reached. At the lower speed, d, a momen- tary decrease of speed, as by a small pulsation of voltage, load, etc., decreases the motor torque, Z), below the torque, L, required by the load, thus causes the motor to slow down, but in doing

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133

so its torque further decreases, and it slows down still more, loses more torque, etc., until it comes to a standstill. Inversely, a momentary increase of speed increases the motor torque, D, beyond the torque, L, consumed by the load, and thereby causes an acceleration, that is, an increase of speed. This increase of speed, however, increases the motor torque and thereby the speed still further, and so on, and the motor increases in speed up to the point, c, where the motor torque, D, again becomes

I'xg. 51. — Speed-torque characteristics of induction motor and load for determination of the stability point.

equal to the torque consumed by the load. A momentary in- crease of speed beyond c decreases the motor torque, D, and thus limits itself, and inversely a momentary decrease of speed below c increases the motor torque, D, beyond L, thus accelerates and recovers the speed; that is, at c the motor speed is stable.

With a load, requiring constant torque the induction motor thus is unstable at speeds below that of the maximum torque point, but stable above it; that is, the motor curve consists of two branches, an unstable branch, from standstill, t , to the maxi-

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ELECTRICAL APPARATUS

mum torque point, m, and a stable branch, from the maximum torque point, m, to synchronism.

83. It must be realized, however, that this instability of the lower branch of the induction-motor speed curve is a function of the nature of the load, and as described above applies only to a load requiring a constant torque, L. Such a load the motor could not start (except by increasing the motor torque at low speeds by resistance in the secondary), but when brought up to a speed above d would carry the load at speed, c, in Fig. 51.

If, however, the load on the motor is such as to require a torque which increases with the square of the speed, as shown by curve, C , in Fig. 51, that is, consists of a constant part p (friction of bearings, etc.) and a quadratic part, as when driving a ship’s propeller or driving a centrifugal pump, then the induc- tion motor is stable over the entire range of speed, from standstill to synchronism. The motor then starts, with the load repre- sented by curve C } and runs up to speed, c. At a higher load, represented by curve B , the motor runs up to speed, b , and with excessive overload, curve A , the motor would run up to low | speed, point a, only, but no overload of such nature would stop

the motor, but merely reduce its speed, and inversely, it would always start, but at excessive overloads run at low speed only. Thus in this case no unstable branch of the motor curve exists, fjj but it is stable over the entire range.

With a load requiring a torque which increases proportionally to the speed, as shown by C in Fig. 52, that is, which consists of a constant part, p, and a part proportional to the speed, as when driving a direct-current generator at constant excitation, connected to a constant resistance as load — as a lighting sys- tem — the motor always starts, regardless of the load — provided that the constant part of the torque, p, is less than the starting torque. With moderate load, C, the motor runs up to a speed, c, near synchronism. With very heavy load, A, the motor starts, but runs up to a low speed only. Especially interesting is the case of an intermediary load as represented by line B in Fig. 52. B intersects the motor-torque curve, D, in three points, bij b 2 , bs; that is, three speeds exist at which the motor gives the torque required by the load: 24 per cent., 60 per cent., and 88 per cent, of synchronism. The speeds hi and b z are stable, the speed 6 2 unstable. Thus, with this load the motor starts from standstill, but does not run up to a speed near synchronism, but

INDUCTION-MOTOR REGULATION

135

accelerates only to speed b h and keeps revolving at this low speed (and a correspondingly very large current). If, however* the load is taken off and the motor allowed to run up to syn- chronism or near to it, and the load then put on, the motor slows down only to speed b 3 , and carries the load at this high speed; hence, the motor can revolve continuously at two different speeds, bi and 63, and either of these speeds is stable; that is, a momen- tary increase of speed decreases the motor torque below that

Fig. 52. — Speed torque characteristics of induction motor and load for determination of the stability point.

required by the load, and thus limits itself, and inversely a de- crease of motor speed increases its torque beyond that correspond- ing to the load, and thus restores the speed. At the intermediary speed, hi, the conditions are unstable, and a momentary increase of speed causes the motor to accelerate up to speed b 3 , a momen- tary decrease of speed from 62 causes the motor to slow down to speed Z>i, where it becomes stable again. In the speed range between b% and bz the motor thus accelerates up to 63, in the speed range between b 2 and bj it slows down to 61.

For this character of load, the induction-motor speed curve, D, thus has two stable branches, a lower one, from standstill, t , to the point n, and an upper one, from point m to synchronism,

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ELECTRICAL APPARATUS

where m and n are the points of contact of the tangents from the required starting torque, p, on to the motor curve, D; these two stable branches are separated by the unstable branch, from n to m, on which the motor can not operate.

84. The question of stability of motor speed thus is a func- tion not only of the motor-speed curve but also of the character of the load in its relation to the motor-speed curve, and if the change of motor torque with the change of speed is less than the change of the torque required by the load, the condition is stable,

otherwise it is unstable; that is, it must be ^ ^ to give

stability, where L is the torque required by the load at speed, S .

Fig. 53. — Speed-torque characteristic of single-phase induction motor.

Occasionally on polyphase induction motors on a load as repre- sented in Fig. 52 this phenomenon is observed in the form that the motor can start the load but can not bring it up to speed. More frequently, however, it is observed on single- phase induction motors in which the maximum torque is nearer to synchronism, with some forms of starting devices which de- crease in their effect with increasing speed and thus give motor- speed characteristics of forms similar to Fig. 53. With a torque-speed curve as shown in Fig. 53, even at a load requiring constant torque, three speed points may exist of which the middle one is unstable. In polyphase synchronous motors and converters, when starting by alternating current, that is, as

INDUCTION-MOTOR REGULATION

137

induction machines, the phenomenon is frequently observed that the machine starts at moderate voltage, but does not run up to synchronism, but stops at an intermediary speed, in the neighbor- hood of half speed, and a considerable increase of voltage, and thereby of motor torque, is required to bring the machine beyond the dead point, or rather “dead range,” of speed and make it run up to synchronism. In this case, however, the phenomenon is complicated by the effects due to varying magnetic reluctance (magnetic locking), inductor machine effect, etc.

Instability of such character as here described occurs in elec- tric circuits in many instances, of which the most typical is the electric arc in a constant-potential supply. It occurs whenever the effect produced by any cause increases the cause and thereby becomes cumulative. When dealing with energy, obviously the effect must always be in opposition to the cause (Lena’s Law), as result of the law of conservation of energy. When dealing with other phenomena, however, as the speed-torque relation or the volt-ampere relation, etc., instability due to the effect assisting the cause, intensifying it, and thus becoming cumulative, may exist, and frequently does exist, and causes either indefinite increase or decrease, or surging or hunting, as more fully discussed in Chapters X and XI, of “Theory and Calculation of Electric Circuits.”

4. GENERATOR REGULATION AND STABILITY

5. If the voltage at the induction-motor terminals decreases with increase of load, the maximum torque and output are de- creased the more the greater the drop of voltage. But even if the voltage at the induction motor terminals is maintained con- stant, the maximum torque and’ power may be reduced essen- tially, in a manner depending on the rapidity with which the voltage* regulation at changes of load is effected by the generator or potential regulator, which maintains constancy of voltage, and the rapidity with which the motor speed can change, that is, the mechanical momentum of the motor and its load.

This instability of the motor, produced by the generator regulation, may be discussed for the case of a load requiring constant torque at all loads, though the corresponding pheno- menon may exist at all classes of load, as discussed under 3, and may occur even with a load proportional to the square of the speed, as ship propellors.

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ELECTRICAL APPARATUS

The torque curve of the induction motor at constant terminal voltage consists of two branches, a stable branch, from the maximum torque point to synchronism, and an unstable branch, that is, a branch at which the motor can not operate on a load requiring constant torque, from standstill to maximum torque. With increasing slip, s, the current, i, in the motor increases. If

then D = torque of the motor, ~^r is positive on the stable,