Theory and Calculation of Electrical Apparatus (1917) — part 5 of 21

Z i = n + jx x = secondary self-inductive im- pedance of the induction motor, reduced to full

frequency.

Z 2 = r 2 + jx 2 = self-inductive impedance of the commutating machine, reduced to full frequency.

Z° = pi + jp 2 = effective impedance repre- senting the voltage in-

duced in the commutating machine, reduced to full frequency.

The total secondary impedance, at slip, s, then is:

Z s = (rj + jsx i) + (r 2 + jsx*) + (1 - s) {p x + jp 2 )

= [n + r 2 + (1 - s) + j[s ( xi + x 2 ) + (1 - s) p 2 ] (8)

and, if the mutual inductive voltage of the induction motor -is chosen as base line, e, in the customary manner, the secondary current is :

where:

h *= = Oi- ja 2 )e,

„ _ 5 [n + r 2 + (1 - s) Pi]

1 —

m

„ - s I s (Sl + Xi) + (1 ~ S) p 2 ]

(9)

( 10 )

INDUCTION MOTOR

83

and:

m = [ri + r 2 + (1 — 5 ) pi] 2 + [s (x x + x 2 ) + (1 — 5 ) p 2 ] 2 .

The remaining calculation is the same as on page 318 of “Theoretical Elements of Electrical Engineering,” 4th edition.

As an instance, consider the concatenation of a low-frequency commutating machine to the low-speed induction 'motor, Fig. 20. The constants then are :

60 = 500;

Y 0 = 0.02 - 0.6 j;

Zq = 0.1 -f- 0.3 j ;

Zi = 0.1 +0.3 j;

Z 2 = 0.02 + 0.3 j;

Z° = - 0.2 j.

Fig. 31. — Load curves of high-excitation induction motor concatenated with commutating machine as reactive anti-inductive impedance.

That is, the commutating machine is adjusted to give only reactive lagging voltage, for power-factor compensation.

It then is :

Z* = 0.12 + j [0.6 5 - 0.2 (1 - s)l The load curves of this motor couple are shown in Fig. 31. As

Impressed voltage: Exciting admittance : Impedances:

84

ELECTRICAL APPARATUS

seen, power-factor and apparent efficiency rise to high values, and even the efficiency is higher than in the straight induction motor. However, at light-load the power-factor and thus the apparent efficiency falls off, very much in the same manner as in the con- catenation with a synchronous motor.

It is interesting to note the relatively great drop of speed at light-load, while at heavier load the speed remains more nearly constant. This is a general characteristic of anti-inductive im- pedance in the induction-motor secondary, and shared by the use of an electrostatic condenser in the secondary.

For comparison, on Fig. 28 the curve of apparent efficiency of this motor couple is shown as CC .

Induction Motor with Condenser in Secondary Circuit

55. As a condenser consumes leading, that is, produces lagging reactive current, it can be used to supply the lagging component of current of the induction motor and thereby improve the power-factor.

Shunted across the motor terminals, the condenser consumes a constant current, at constant impressed voltage and frequency, and as the lagging component of induction-motor current in- creases with the load, the characteristics of the combination of motor and shunted condenser thus change from leading current at no-load, over unity power-factor to lagging current at overload. As the condenser is an external apparatus, the characteristics of the induction motor proper obviously are not changed by a shunted condenser.

As illustration is shown, in Fig. 32, the slow-speed induction motor Fig. 20, shunted by a condenser of 125 kva. per phase. Fig. 32 gives efficiency, 77 , power-factor, p, and apparent efficiency, 7 , of the combination of motor and condenser, assuming an efficiency of the condenser of 99.5 per cent., that is, 0.5 per cent, loss in the condenser, or Z = 0.0025 — 0.5 j, that is, a condenser just neutralizing the magnetizing current.

However, when using a condenser in shunt, it must be realized that the current consumed by the condenser is proportional to the frequency, and therefore, if the wave of impressed voltage is greatly distorted, that is, contains considerable higher harmonics — especially harmonics of high order — the condenser may produce considerable higher-frequency currents, and thus by distortion

INDUCTION MOTOR

85

of the current wave lower the power-factor, so that in extreme cases the shunted condenser may actually lower the power- factor. However, with the usual commercial, voltage wave shapes, this is rarely to be expected.

In single-phase induction motors, the condenser may be used in a tertiary circuit, that is, a circuit located on the same member (usually the stator) as the primary circuit, but displaced in posi-

Fig. 32. — Load curves of high-excitation induction motor with shunted

condenser.

tion therefrom, and energized by induction from the secondary. By locating the tertiary circuit in mutual induction also with the primary, it can be used for starting the single-phase motor, and is more fully discussed in Chapter V.

A condenser may also be used in the secondary of the induction motor. That is, the secondary, circuit is closed through a con- denser in each phase. As the current consumed by a condenser is proportional to the frequency, and the frequency in the secondary circuit varies, decreasing toward zero at synchronism, the cur- rent consumed by the condenser, and thus the secondary current of the motor tends toward zero when approaching synchronism,

86

ELECTRICAL APPARATUS

and peculiar speed characteristics result herefrom in such a motor. At a certain slip, s, the condenser current just balances all the reactive lagging currents of the induction motor, resonance may thus be said to exist, and a very large current flows into the motor, and correspondingly large power is produced. Above this “ resonance speed/' however, the current and thus the power rapidly fall off, and so also below the resonance speed.

It must be realized, however, that the frequency of the sec- ondary is the frequency of slip, and is very low at speed, thus a very great condenser capacity is required, far greater than would be sufficient for compensation by shunting the condenser across the primary terminals. In view of the low frequency and low voltage of the secondary circuit, the electrostatic condenser generally is at a disadvantage for this use, but the electrolytic condenser, that is, the polarization cell, appears better adapted.

56. Let then, in an induction motor, of impressed voltage, e 0 :

Y q = g — jb — exciting admittance;

Z 0 = r 0 + jx o = primary self-inductive impe- dance;

Zi — ri + jx i = secondary self-inductive im- pedance at full frequency;

and let the secondary circuit be closed through a condenser of capacity reactance, at full frequency:

Z o ~ r 2 jx*

where r 2 , representing the energy loss in the condenser, usually is very small and can be neglected in the electrostatic condenser, so that:

Z 2 = - jx 2 .

The inductive reactance, xi, is proportional to the frequency, that is, the slip, s, and the capacity reactance, inverse propor- tional thereto, and the total impedance of the secondary circuit, at slip, s, thus is:

2> = r 1 +j(sx 1 -^), (1)

thus the secondary current :

= e (oi — ja 2 ),

( 2 )

INDUCTION MOTOR

87

where:

and:

o i

r i — , m

( 3 )

All the further calculations of the motor characteristics now are the same as in the straight induction motor.

As instance is shown the low-speed motor, Fig. 20, of constants :

Co — 500 )

F 0 = 0.02 - 0.6 j; Zo = 0.1 + 0.3 j; Z\ = 0.1 + 0.3 j ;

with the secondary closed by a condenser of capacity impedance:

thus giving:

F 2 = - 0.012 j,

Z‘ = 0.1 + 0.3 j (s - 9 ^)-

Fig. 33 shows the load curves of this motor with condenser in the secondary. As seen, power-factor and apparent effi- ciency are high at load, but fall off at light-load, being similar in character as with a commutating machine concatenated to the induction machine, or with the secondary excited by direct current, that is, with conversion of the induction into a synchro- nous motor.

Interesting is the speed characteristic: at very light-load the speed drops off rapidly, but then remains nearly stationary over a wide range of load, at 10 per cent. slip. It may thus be said, that the motor tends to run at a nearly constant speed of 90 per cent, of synchronous speed.

The apparent efficiency of this motor combination is plotted once more in Fig. 28, for comparison with those of the other motors, and marked by C.

Different values of secondary capacity give different operating speeds of the motor: a lower capacity, that is, higher capacity

88

ELECTRICAL APPARATUS

reactance, 3 2 , gives a greater slip, s, that is, lower operating speed, and inversely, as was discussed in Chapter I.

57. It is interesting to compare, in Fig. 28, the various methods of secondary excitation of the induction motor, in their effect in improving the power-factor and thus the apparent efficiency of a motor of high exciting current and thus low power-factor, such as a slow-speed motor.

The apparent efficiency characteristics fall into three groups:

Fig. 33. — Load curves of high-excitation induction motor with condensers in

secondary circuits.

1. Low apparent efficiency at all loads: the straight slow- speed induction motor, marked by I.

2. High apparent efficiency at all loads :

The synchronous motor with unity power-factor excitation, So-

Concatenation to synchronous motor with unity power-factor excitation, CSq.

Concatenation to synchronous motor with constant excitation, CS.

These three curves are practically identical, except at great overloads.

3. Low apparent efficiency at light-loads, high apparent

INDUCTION MOTOR

89

efficiency at load, that is, curves starting from (1) and rising up to (2).

Hereto belong: The synchronous motor at constant excita- tion, marked by S.

Concatenation to a commutating machine, CC.

Induction motor with condenser in secondary circuit, C.

These three curves are very similar, the points calculated for the three different motor types falling within the narrow range between the two limit curves drawn in Fig. 28.

Regarding the speed characteristics, two types exist : the motors So, S, CSo and CS are synchronous, the motors I, CC and C are asynchronous.

In their efficiencies, there is little difference between the different motors, as is to be expected, and the efficiency curves are almost the same up to the overloads where the motor begins to drop out of step, and the efficiency thus decreases.

Induction Motor with Commutator

58. Let, in an induction motor, the turns of the secondary winding be brought out to a commutator. Then by means of brushes be.aring on this commutator, currents can be sent into the secondary winding from an outside source of voltage.

Let then, in Fig. 34, the full-frequency three-phase currents supplied to the three commutator brushes of such a motor be shown as A. The current in a secondary coil of the motor, supplied from the currents, A , through the commutator, then is shown as B. Fig. 34 corresponds to a slip, s = %. As seen from Fig. 34, the commutated three-phase current, B, gives a resultant effect, which is a low-frequency wave, shown dotted in Fig. 34 B , and which has the frequency of slip, s, or, in other words, the commutated current, B , can be resolved into a current of fre- quency, a, and a higher harmonic of irregular wave shape.

Thus, the effect of low-frequency currents, of the frequency of slip, can be produced in the induction-motor secondary by impressing full frequency upon it through commutator and brushes.

The secondary circuit, through commutator and brushes, can be connected to the supply source either in series to the primary,

90

ELECTRICAL APPARATUS

or in shunt thereto, and thus gives series-motor characteristics, or shunt-motor characteristics.

In either case, two independent variables exist, the value of the voltage impressed upon the commutator, and its phase, and the phase of the voltage supplied to the secondary circuit may be varied, either by varying the phase of the impressed voltage by a suitable transformer, or by shifting the brushes on the commutator and thereby the relative position of the brushes with regards to the stator, which has the same effect.

However, with such a commutator motor, while the resultant magnetic effect of the secondary currents is of the low frequency

Fig. 34. — Commutated full-frequency current in induction motor secondary.

of slip, the actual current in each secondary coil is of full fre- quency, as a section or piece of a full-frequency wave, and thus it meets in the secondary the full-frequency reactance. That is, the secondary reactance at slip, s, is not : Z* = r x + jsx h but is : Z 8 = ri + jx i, in other words is very much larger than in the motor with short-circuited secondary.

Therefore, such motors with commutator always require power-factor compensation, by shifting the brushes or choosing the impressed voltage so as to be anti-inductive.

Of the voltage supplied to the secondary through commutator and brushes, a component in phase with the induced voltage lowers the speed, a component in opposition raises the speed, and by varying the commutator supply voltage, speed control of such an induction motor can be produced in the same manner and of the same character, as produced in a direct-current motor

INDUCTION MOTOR

91

by varying the field excitation. Good constants can be secured, if in addition to the energy component of impressed voltage, used for speed control, a suitable anti-inductive wattless component is used.

However, this type of motor in reality is not an induction motor any more, but a shunt motor or series motor, and is more fully discussed in Chapter XIX, on “ General Alternating-current Motors.”

59. Suppose, however, that in addition to the secondary wind- ing connected to commutator and brushes, a short-circuited squirrel-cage winding is used on the secondary. Instead of this, the commutator segments may be shunted by resistance, which gives the same effect, or merely a squirrel-cage winding used, and on one side an end ring of very high resistance em- ployed, and the brushes bear on this end ring, which thus acts as commutator.

In either case, the motor is an induction motor, and has the essential characteristics of the induction motor, that is, a slip, s, from synchronism, which increases with the load; however, through the commutator an exciting current can be fed into the motor from a full-frequency voltage supply, and in this case, the current supplied over the commutator does not meet the full- frequency reactance, Xi, of the secondary, but only the low-fre- quency reactance, sx i, especially if the commutated winding is in the same slots with the squirrel-cage winding: the short-circuited squirrel-cage winding acts as a short-circuited secondary to the high-frequency pulsation of the commutated current, and there- fore makes the circuit non-inductive for these high-frequency pulsations, or practically so. That is, in the short-circuited con- ductors, local currents are induced equal and opposite to the high-frequency component of the commutated current, and the total resultant of the currents in each slot thus is only the low- frequency current.

Such short-circuited squirrel cage in addition to the commu- tated winding, makes the use of a Commutator practicable for power-factor control in the induction motor. It forbids, how- ever, the use of the commutator for speed control, as due to the short-circuited winding, the motor must run at the slip, s, corre- sponding to the load as induction motor. The voltage impressed upon the commutator, and its phase relation, or the brush posi- tion, thus must be chosen so as to give only magnetizing, but

92

ELECTRICAL APPARATUS

no speed changing effects, and this leaves only one degree of freedom.

The foremost disadvantage of this method of secondary excita- tion of an induction motor, by a commutated winding in addi- tion to the short-circuited squirrel cage, is that secondary excita- tion is advantageous for power-factor control especially in slow-speed motors of very many poles, and in such, the commuta- tor becomes very undesirable, due to the large number of poles. With such motors, it therefore is preferable to separate the commutator, placing it on a small commutating machine of a few poles, and concatenating this with the induction motor. In motors of only a small number of poles, in which a commutator would be less objectionable, power-factor compensation is rarely needed. This is the foremost reason that this type of motor (the Heyland motor) has found no greater application.

CHAPTER V

SINGLE-PHASE INDUCTION MOTOR

60 . As more fully discussed in the chapters on the single-phase induction motor, in “ Theoretical Elements of Electrical Engineer- ing’ 7 and “ Theory and Calculation of Alternating-current Phenomena, ” the single-phase induction motor has inherently, no torque at standstill, that is, when used without special device to produce such torque by converting the motor into an unsym- metrical ployphase motor, etc. The magnetic flux at standstill is a single-phase alternating flux of constant direction, and the line of polarization of the armature or secondary currents, that is, the resultant m.m.f. of the armature currents, coincides with the axis of magnetic flux impressed by the primary circuit. When revolving, however, even at low speeds, torque appears in the single-phase induction motor, due to the axis of armatures polarization being shifted against the axis of primary impressed magnetic flux, by the rotation. That is, the armature currents, lagging behind the magnetic flux which induces them, reach their maximum later than the magnetic flux, thus at a time when their conductors have already moved a distance or an angle 4 , away from coincidence with the inducing magnetic flux. That is,

TT

if the armature currents lag ^ = 90° beyond the primary mam

flux, and reach their maximum 90° in time behind the magnetic flux, at the slip, s, and thus speed (1 — s), they reach their maxi-

7T

mum in the position (1 — s) ^ = 90 (1 — s) electrical degrees

behind the direction of the main magnetic flux. A component of the armature currents then magnetizes in the direction at right angles (electrically) to the main magnetic flux, and the armature currents thus produce a quadrature magnetic flux, increasing from zero at standstill, to a maximum at synchronism, and approximately proportional to the quadrature component of the armature polarization, P:

P sin (1 - s) |-

93

94

ELECTRICAL APPARATUS

The torque of the single-phase motor then is produced by the action of the quadrature flux on the energy currents induced by the main flux, and thus is proportional to the quadrature flux.

At synchronism, the quadrature magnetic flux produced by the armature currents becomes equal to the main magnetic flux produced by the impressed single-phase voltage (approximately, in reality it is less by the impedance drop of the exciting current in the armature conductors) and the magnetic disposition of the single-phase induction motor thus becomes at synchronism iden- tical with that of the polyphase induction motor, and approxi- mately so near synchronism.

The magnetic field of the single-phase induction motor thus may be said to change from a single-phase alternating field at standstill, over an unsymmetrical rotating field at intermediate speeds, to a uniformly rotating field at full speed.

At synchronism, the total volt-ampere excitation of the single- phase motor thus is the same as in the polyphase motor at the same induced voltage, and decreases to half this value at stand- still, where only one of the two quadrature components of magnetic flux exists. The primary impedance of the motor is that of the circuits used. The secondary impedance varies from the joint impedance of all phases, at synchronism, to twice this value at standstill, since at synchronism all the secondary circuits correspond to the one primary circuit, while at stand- still only their component parallel with the primary circuit corresponds.

61. Hereby the single-phase motor constants are derived from the constants of the same motor structure as polyphase motor.

Let, in a polyphase motor:

Y = g — jb = primary exciting admittance;

Zo = tq + jx o = primary self-inductive im- pedance;

Z i = n + jx i = secondary self-inductive im- pedance (reduced to the pri- mary by the ratio of turns, in the usual manner);

the characteristic constant of the motor then is :

• = Y (Zo + ZO. (1)

The total, or resultant admittance respectively impedance of

SINGLE-PHASE INDUCTION MOTOR

95

the motor, that is, the joint admittance respectively impedance of all the phases, then is :

In a three-phase motor:

Y Q 3 y

£o° = HZo, (2)

£i° * HZ*.

In a quarter-phase motor:

Y° = 2 Y

Z o° = M K (3)

Zi° = H ^i.

In the same motor, as single-phase motor , it is then: at .syn- chronism: $ = 0*

Y' = Y°j

Z'o = 2 Z 0 °, } (4)

Z'x = Z,»,

hence the characteristic constant:

*'o = 7' (Z'o + Z)

= r° (2 Zo° + Zi°),

at standstill : s = 1 :

7' = M7°,

Z'o = 2 Zo°,

Z'i = 2 Zx°,

hence, the characteristic constant :

0'i = 7° (Zo° + Zi°)

(5)

( 6 )

(7)

approximately, that is, assuming linear variation of the constants with the speed or slip, it is then : at slip, s :

7' = 7° (1 - |),

Z'o = 2 Z 0 ,

Z'i = Zx° (1 + s).

( 8 )

This gives, in a three-phase motor:

7' =37(1- |),

z'o = % z®,

rj, _ 1 + « «•

( 9 )

96

ELECTRICAL APPARATUS

In a quarter-phase motor:

Y' =27(1-1), Z'o = Zo,

( 10 )

Thus the characteristic constant, of the single-phase motor is higher, that is, the motor inferior in its performance than the polyphase motor; but the quarter-phase motor makes just as good — or poor — a single-phase motor as the three-phase motor.

62. The calculation of the performance curves of the single- phase motor from its constants, then, is the same as that of the polyphase motor, except that:

In the expression of torque and of power, the term (1 — $) is added, which results from the decreasing quadrature flux, and it thus is:

Torque:

r = t a - s)

= (1 - $) che 2 . ( 11 )

Power:

P' = p (1 - s )

= (1 - s) 2 aie 2 . (12)

However, these expressions are approximate only, as they assume a variation of the quadrature flux proportional to the speed.

63. As the single-phase induction motor is not inherently self-starting, starting devices are required. Such are:

(а) Mechanical starting.

As in starting a single-phase induction motor it is not neces- sary, as in a synchronous motor, to bring it up to full speed, but the motor begins to develop appreciable torque already at low speed, it is quite feasible to start small induction motors by hand, by a pull on the belt, etc., especially at light-load and if of high- resistance armature.

(б) By c’onverting the motor in starting into a shunt or series motor.

'This has the great objection of requiring a commutator, and a commutating-machine rotor winding instead of the common induction-motor squirrel-cage winding. Also, as series motor, the liability exists in the starting connection, of running away;

SINGLE-PHASE INDUCTION MOTOR

97

as shunt motor, sparking is still more severe. Thus this method is used to a limited extent only.

(c) By shifting the axis of armature or secondary polarization against the axis of inducing magnetism.

This requires a secondary system, which is electrically un- symmetrical with regards to the primary system, and thus, since the secondary is movable with regards to the primary, requires means of changing the secondary circuit, that is, commutator brushes short-circuiting secondary coils in the position of effective torque, and open-circuiting them in the position of opposing torque.

Thus this method leads to the various forms of repulsion motors, of series and of shunt characteristic.

It has the serious objection of requiring a commutator and a corresponding armature winding; though the limitation is not quite as great as with the series or shunt motor, since in the re- pulsion motors the armature current is an induced secondary current, and the armature thus independent of the primary system regards current, voltage and number of turns.

(d) By shifting the axis of magnetism, that is producing a magnetic flux displaced in phase and in position from that in- ducing the armature currents, in other words, a quadrature magnetic flux, such as at speed is being produced by the rotation.

This method does not impose any limitation on stator and rotor design, requires no commutator and thus is the method almost universally employed.

It thus may be considered somewhat more in detail.

The infinite variety of arrangements proposed for producing a quadrature or starting flux can be grouped into three classes :

A. Phase-splitting Devices . — The primary system of the single- phase induction motor is composed of two or more circuits displaced from each • other in position around the armature circumference, and combined with impedances of different in- ductance factors so as to produce a phase displacement between them.

The motor circuits may be connected in series, and shunted by the impedance, or they may be connected in shunt with each other, but in series with their respective impedance, or they may be connected with each other by transformation, etc.

B. Inductive Devices . — The motor is excited by two or more circuits which are in inductive relation with each other so as to produce a phase displacement.

7

98

ELECTRICAL APPARATUS

This inductive relation may be established outside of the motor by an external phase-splitting device, or may take place in the motor proper.

C . Monocyclic Devices. — An essentially reactive quadrature voltage is produced outside of the motor, and used to energize a cross-magnetic circuit in the motor, either directly through a separate motor coil, or after combination with the main voltage to a system of voltages of approximate three-phase or quarter- phase relation.

D. Phase Converter. — By a separate external phase converter — usually of the induction-machine type — the single-phase supply is converted into a polyphase system.

Such phase converter may be connected in shunt to the motor, or may be connected in series thereto.

This arrangement requires an auxiliary machine, running idle, however. It therefore is less convenient, but has the advantage of being capable of giving full polyphase torque and output to the motor, and thus would be specially suitable for railroading.

64. If:

$o = main magnetic flux of single-phase motor, that is, magnetic flux produced by the impressed single-phase voltage, and

= auxiliary magnetic flux produced by starting 'device, and if

o) = space angle between the two fluxes, in electrical degrees, and <j> = time angle between the two fluxes,

then the torque of the motor is proportional to :

T = a<f><i>o sin co sin <£; (13)

in the same motor as polyphase motor, with the magnetic flux, $o, the torque is :

To = a$o 2 ; (14)

thus the torque ratio of the starting device is :

T $

t ==_.=; s i n w s i n ^ (15)

1 0 <±>0

or, if :

= quadrature flux produced by the starting device, that is,

, i * 1 u

SINGLE-PHASE INDUCTION^QT^.^ 99

component of the auxiliary flux, in quadrature to'tSSteg^t'fltix 4 ; ^>o 7 in time and in space, it is:

Single-phase motor starting torque :

T = o, (16)

and starting-torque ratio:

t =

11 .

$o

(17)

As the magnetic fluxes are proportional to the impressed vol- tages, in coils having the same number of turns, it is: starting torque of single-phase induction motor :

T = beoe sin co sin 0 = beoe',

(18)

and, starting-torque ratio :

t = — sm w sin <t> eo

= 1, eo

(19)

where :

e 0 = impressed single-phase voltage, e = voltage impressed upon the auxiliary or starting winding, reduced to the same number of turns as the main winding, and

e' = quadrature component, in time and in space, of this voltage, e,

and the comparison is made with the torque of a quarter-phase motor of impressed voltage, e 0 , and the same number of turns.

Or, if by phase-splitting, monocyclic device, etc., two voltages, ex and e 2 , are impressed upon the two windings of a single-phase induction motor, it is:

Starting torque:

T = be ieo sin co sin <t> (20)

and, starting-torque ratio:

t = sin co sin 0, (21)

e 0 2

where eo is the voltage impressed upon a quarter-phase motor, with which the single-phase motor torque is compared, and all

100

ELECTRICAL APPARATUS

these voltages, e h e 2 , e a , are reduced to the same number of turns of the circuits, as customary.

If then:

Q — volt-amperes input of the single-phase motor with starting device, and Q 0 = volt-amperes input of the same motor with polyphase supply,

Q

3 = 0 0

is the volt-ampere ratio, and thus:

t

v — - Q

is the ratio of the apparent starting-torque efficiency of the single-phase motor with starting device, to that of the same motor as polyphase motor, v may thus be called the apparent torque efficiency of the single-phase motor-starting device.

In the same manner the apparent power efficiency of the start- ing device would result by using the power input instead of the volt-ampere input.

65 . With a starting device producing a quadrature voltage, e ! ,

t = y (24)

Co

is the ratio of the quadrature voltage to the main voltage, and also is the starting-torque ratio.

The quadrature flux :

e' = te o (25)

(22)

(23)

requires an exciting current, equal to t times that of the main voltage in the motor without starting device, the exciting current at standstill is : ‘

e<,Y' =

e 0 Y°

2

and in the motor with starting device giving voltage ratio, t, the total exciting current at standstill thus is :

e 0 F°

2

(1 + t)

sf ixun rius motor,

1(H

.tii'l ' In* I'xritiiiK ailmittimi’f ;

i * 1 1 \ tr,

W)

ii, th« • mu miuiinr, tin* m-niidary iutprilattro at alaini, still is;

I

2 Z j fl

I »■ /

( 2K j

aft* I tini'H.

in ilit* m%h* %\uw* liifluH mu iMutor wit li starting <l<vi<r pro- iluriiiK ii! tin rut it# nf voting** to main

villi (ft Rr ,

r

t

ft *

lip” at

*} */ m

m f*U, |

1 U y „

1 f */ '

2

12!))

Howcvi’r, tlnr I'xpmMotiK (29) hit approximate only, hh they town on* linear variation with », nn<i furthermore, t hoy apply only under the witn lit ion, that tin* effect of the ntnrting d«vit>»! dorat not vary with tin ji«d of tin motor, Unit in, that tins voltaga ratio, t, do not d|nd oil tin effective imjiedaiice of tin motor. Thin j» t hr ran** only with n few Marling device*, while many depend upon tin* effective impedance of tin* motor, and lima with tin- grant change of tin* effective imja'daticti of tin*, motor

102

ELECTRICAL APPARATUS

reaction of the motor is eliminated by the comparison with the polyphase motor.

In calculating the effective impedance of the motor at stand- still, we consider the same as an alternating-current transformer, and use the equivalent circuit of the transformer, as discussed in Chapter XVII of “ Theory and Calculation of Alternating- current Phenomena.” That is, the induction motor is con- sidered as two impedances, Z 0 and Z i, connected in series to the

Fig. 35. — The equivalent circuit of the induction motor.

impressed voltage, with a shunt of the admittance, Yo, between the two impedances, as shown in Fig. 35.

The effective impedance then is:

Z = Zo +

z; + Yo

= Zo + J +~ z \y o ;

(30)

approximately, this is :

Z a — Z 0 + Z. (31)

This approximation (31), is very close, if Z\ is highly inductive, as a short-circuited low-resistance squirrel cage, but ceases to be a satisfactory approximation if the secondary is of high resistance, for instance, contains a starting rheostat.

As instances are given in the following the correct values of the effective impedance, Z, from equation (30), the approximate value (31), and their difference, for a three-phase motor without starting resistance, with a small resistance, with the resistance giving maximum torque at standstill, and a high resistance:

SINGLE-PHASE INDUCTION MOTOR

103

0.01 - 0.1 j 0.1 + 0.3 j 0.1 + 0.3 i 0.195 + 0.592; 0.2 + 0.6; — 0.005 - 0.008;

0.25 + 0.3 ; 0.336 + 0.596 ; 0.35 + 0.6 ; - 0.014 - 0.004 ;

0.6 +0.3 ; 0.661 +0.620; 0.7 + 0.6; - 0.039 + 0.020 ;'

1.6 + 0.3; 1.552 + 0.804/ 1.7 + 0.6; - 0.148 + 0.204;

A. PHASE-SPLITTING DEVICES Parallel Connection

67. Let the motor contain two primary circuits at right angles (electrically) in space with each other, and of equal effective impedance:

Z = r + jx.

These two motor circuits are connected in parallel with each

p I0 . 30.— Diagram of phase-splitting device with parallel connection of.

motor circuits.

other between the same single-phase mains of voltage, eo, but the first motor circuit contains in series the impedance

Zx = Ti+ jx i,

the second motor circuit the impedance:

Z 2 = r 2 + jx 2 ,

as shown diagrammatically in Fig. 36.

The two motor currents then are :

( 33 )

104

ELECTRICAL APPARATUS

Ex = 7]Z and E 2 = / 2 Z

Z Z

^0 /"7 1 rr f ^0 7 , i

J z + z x

'Z + Z 2

and the phase angle between E 1 and ^2 is given by:

m (cos </> + j sin <£) ==

Z + Z 1 Z + Z2

Denoting the absolute values of the voltages and currents by small letters, it is :

T = be 162 sin <£; (36)

in the motor as quarter-phase motor, with voltage, eo, impressed per circuit, it is :

To = be o 2 , (37)

hence, the torque ratio:

, Cie 2 • ,

t = - v sin 0. Co"

The current per circuit, in the machine as quarter-phase motor,

hence the volt-amperes:

• - e °

^0 = >

2:

Qo = 2 eoio,

while the volt-amperes of the single-phase motor, inclusive start- ing impedances, are:

Q = e 0 i, (41)

thus:

i iz

q ~ 2i 0 ~ 2e 0 ^

and, the apparent torque efficiency of the starting device:

_ t 2e,e 2 sin <f>

68 . As an instance, consider the naotor of effective impedance: Z = r + jx = 0.1 + 0.3 j,

z = 0.316,

SINGLE-PHASE INDUCTION MOTOR

105

and assume, as the simplest case, a resistance, a = 0.3, inserted in series to the one motor circuit. That is :

=

0 ,

(44)

z 2 =

a.

It is

then:

/'qo'i * 7

_ €q

e 0 T _

60

r +jx

~ 0.1 ■

+• 0.3 j 12 r +

a + jx 0.4

• 0.3 j

= e 0 (1 —

3 j),

= e 0 (1

•6 - 1.2 j);

(33):

I

= e 0 (2.6 — 4.2 j),

i

• 4.94'e 0 ;

(34):

El =

-5*2 :

r + jx r + a ■+ jx

„ 0.1 + 0.3 j

0 0.4 + 0.3/

e\ = 6o,

62 =

= 0.632 ec;

(35):

m (cos cj)

• j sin

,\ _ r +jx

J r + a + jx ”

0.1 -1- 0.3 j 0.4 + 0.3 j

= 0.52 + 0.36 j,

, x 0.36 tan 4> = g j2 >

sin <f> = 0.57;

(38): t = 0.36;

(43) : v = 0.46.

Thus this arrangement gives 46 per cent., or nearly half as much starting torque per volt-ampere taken from the supply circuit, as the motor would give as polyphase motor.

However, as polyphase motor with low-resistance secondary, the starting torque per volt-ampere input is low.

With a high-resistance motor armature, which on polyphase supply gives a good apparent starting-torque efficiency, v would be much lower, due to the lower angle, <£. In this case, however, a reactance, +ja , would give fairly good starting-torque efficiency.

In the same manner the effect of reactance or capacity inserted into one of the two motor coils can be calculated.

As instances are given, in Fig. 37, the apparent torque efficiency, v, of the single-phase induction-motor starting device consisting 6f the insertion, in one of the two parallel motor circuits, of various amounts of reactance, inductive or positive, and capacity

106

ELECTRICAL APPARATUS

or negative, for-a low secondary resistance motor of impedance: Z = 0.1 + 0.3 j

and a high resistance armature, of the motor impedance :

Z = 0.3 + 0.1 j

resistance inserted into the one motor circuit, has the same effect

Fig. 37. — Apparent starting-torque efficiencies of phase-splitting device, parallel connection of motor circuits.

in the first motor, as positive reactance in the second motor, and inversely.

69 . Higher values of starting-torque efficiency are secured by the use of capacity in the one, and inductance in the other motor circuit. It is obvious that by resistance and inductance alone, 90° phase displacement between the two component currents, and thus true quarter-phase relation, can not be reached.

As resistance consumes energy, the use of resistance is justified

SINGLE-PHASE INDUCTION MOTOR

107

only due to its simplicity and cheapness, where moderate start- ing torques are sufficient, and thus the starting-torque efficiency less important. For producing high starting torque with high starting-torque efficiency, thus, only capacity and inductance would come into consideration.

Assume, then, that the one impedance is a capacity:

x 2 — — ]c, or: Z 2 = — jk, (45)

while the other, an, may be an inductance or also a capacity, what- ever may be desired :

= +jx u (46)

where xi is negative for a capacity.

It is, then:

(35) : m (cos <j> + j sin 0) =

r + j (an + x) [r 2 - ( gi + x)(k — a?)] + jrxik ( . r — j (k — x) r 2 + (fc — a;) 2 ^

True quadrature relation of the voltages, e\ and e 2 , or angle,

, T

<t> = requires:

cos 0 = 0,

thus:

(an + x) (Jc — x) — r 2 (48)

and the two voltages, e\ and 62, are equal, that is, a true quarter- phase system of voltages is produced, if in

(34): [Z + ZJ = [Z + Zt],

where the [ ] denote the absolute values.

This gives:

r 2 + (an + a;) 2 = r 2 + (k — x) 2 ,

or:

Xi + x — k — x,

(49)

hence, by (48) :

Xi + x = k — x = r,

k = r + x, }

(50)

an = r — x. j

Thus, if x > r, or in a low-resistance motor, the second reactance, Xi, also must be a capacity.

108

ELECTRICAL APPARATUS

70. Thus, let: in a low-resistance motor :

^ = r +-j x = 0.1 + 0.3 j, k = 0.4, Xi — — 0.2,

Zi = - 0.4 j, Zi = - 0.2 j,

that is, both reactances are capacities.

(34) : ei = e 2 = 2.23 e 0 ,

4 = 5,

that is, the torque is five times as great as on true quarter-phase supply.

T e 0 __ j _ eo

• 1 “ 0.1-FO.lj’ 42 - 0.1 o.lj ’

J = 10 eo = i f

that is, non-inductive, or unity power-factor.

to = = 3.16 6 o,

q = 1.58, v = 3.16,

that is, the apparent starting-torque efficiency, or starting torque per volt-ampere input, of the single-phase induction motor with starting devices consisting of two capacities giving a true quarter- phase system, is 3.16 as high as that of the same motor on a quarter-phase voltage supply, and the circuit is non-inductive in starting, while on quarter-phase supply, it has the power- factor 31.6 per cent, in starting.

In a high-resistance motor :

Z - 0.3 + 0.1 j,

it is:

k = 0.4, Xi = 0.2,

Z 2 = — 0.4 j, Z 2 = + 0.2 j,

that is, the one reactance is a capacity, the other an inductance.

6 \ — 62 ^ 0.743 & 0 f t = 0.555, i = 3.33 e 0 , io =3.16 eo, q = 0.527, v = 1.055,

SINGLE-PHASE INDUCTION MOTOR

109

that is, the starting-torque efficiency is a little higher than with quarter-phase supply. In other words:

This high-resistance motor gives 5.5 per cent, more torque per volt-ampere input, with unity power-factor, on single-phase supply, than it gives on quarter-phase supply with 95 per cent, power-factor.