Theory and Calculation of Electrical Apparatus (1917) — part 3 of 21

18. In an induction motor, a high-resistance low-reactance secondary is produced by the use of an external non-inductive resistance in the secondary, or in a motor with squirrel-cage secondary, by small bars of high-resistance material located close to the periphery of the rotor. Such a motor has a great slip of speed under load, therefore poor efficiency and poor speed regu- lation, but it has a high starting torque and torque at low and intermediate speed. With a low resistance fairly high-reactance secondary, the slip of speed under load is small, therefore effi- ciency and speed regulation good, but the starting torque and torque at low and intermediate speeds is low, and the current in starting and at low speed is large. To combine good start- ing with good running characteristics, a non-inductive resistance is used in the secondary, which is cut out during acceleration. This, however, involves a complication, which is undesirable in many cases, such as in ship propulsion, etc. By arranging then two squirrel cages, one high-resistance low-reactance one, consisting of high-resistance bars close to the rotor surface, and one of low-resistance bars, located deeper in the armature iron, that is, inside of the first squirrel cage, and thus of higher reactance, a “double squirrel-cage induction motor” is derived, which to some extent combines the characteristics of the high- resistance and the low-resistance secondary. That is, at start- ing and low speed, the frequency of the magnetic flux in the arma- ture, and therefore the voltage induced in the secondary winding is high, and the high-resistance squirrel cage thus carries con- siderable current, gives good torque and torque efficiency, while the low-resistance squirrel cage is ineffective, due to its high reactance at the high armature frequency. At speeds near synchronism, the secondary frequency, being that of slip, is low, and the secondary induced voltage correspondingly low. The high-resistance squirrel cage thus carries little current and gives little torque. In the low-resistance squirrel cage, due to its low reactance at the low frequency of slip, in spite of the relatively

28

ELECTRICAL APPARATUS

low induced e.m.f., considerable current is produced, which is effective in producing torque. Such double squirrel-cage induc- tion motor thus gives a torque curve, which to some extent is a superposition of the torque curve of the high-resistance and that of the low-resistance squirrel cage, has two maxima, one at low speed, and another near synchronism, therefore gives a fairly good torque and torque efficiency over the entire speed range from standstill to full speed, that is, combines the good features of both types. Where a very high starting torque requires locating the first torque maximum near standstill, and large size and high efficiency brings the second torque maximum very close to synchronism, the drop of torque between the two maxima may be considerable. This is still more the case, when the motor is required to reverse at full speed and full power, that is, a very high torque is required at full speed backward, or at or near slip 5 = 2. In this case, a triple squirrel cage may be used, that is, three squirrel cages inside of each other: the outermost, of high resistance and low reactance, gives maximum torque below standstill, at backward rotation; the second squirrel cage, of medium resistance and medium reactance, gives its maximum torque at moderate speed; and the innermost squirrel cage, of low resistance and high reactance, gives its torque at full speed, near synchronism.

Mechanically, the rotor iron may be slotted down to the inner- most squirrel cage, so as to avoid the excessive reactance of a closed magnetic circuit, that is, have the magnetic leakage flux or self-inductive flux pass an air gap.

19 . In the calculation of the standard induction motor, it is usual to start with the mutual magnetic flux, <£, or rather with the voltage induced by this flux, the mutual inductive voltage E — e, as it is most convenient, with the mutual inductive voltage, e , as starting point, to pass to the secondary current by the self-inductive impedance, to the primary current and primary impressed voltage by the primary self-inductive impedance and exciting admittance.

In the calculation of multiple squirrel-cage induction motors, it is preferable to introduce the true induced voltage, that is, the voltage induced by the resultant magnetic flux interlinked with the various circuits, which is the resultant of the mutual and the self-inductive magnetic flux of the respective circuit. This permits starting with the innermost squirrel cage, and

INDUCTION MOTOR

29

gradually building up to the primary circuit, hereof is, that the current in every secondary

with the true induced voltage of this circuit, and is i x =

where r x is the resistance of the circuit. As e x is the voltage induced by the resultant of the mutual magnetic flux coming from the primary winding, and the self-inductive flux corre- sponding to the iiXj of the secondary, the reactance, Xi, does not enter any more in the equation of the current, and e\ is the voltage due to the magnetic flux which passes beyond the cir- cuit in which is induced. In the usual induction-motor theory, the mutual magnetic flux, <£, induces a voltage, E , which produces a current, and this current produces a self-inductive flux, giving rise to a counter e.m.f. of self-induction I\X X , which sub- tracts from E. However, the self-inductive flux, interlinks with the same conductors, with which the mutual flux, $, inter- links* and the actual or resultant flux interlinkage thus is <3>i = <t> — #'i, and this produces the true induced voltage e x = E — Ijx i, from which the multiple squirrel-cage calculation starts. 1

Double Squirrel-cage Induction Motor

20. Let, in a double squirrel-cage induction motor:

Ei = true induced voltage in inner squirrel cage, reduced to full frequency,

/ 2 = current, and

Z<z = r 2 + jx 2 = self-inductive impedance at full frequency, reduced to the primary circuit.

Ei = true induced voltage in outer squirrel cage, reduced to full frequency,

1 1 = current, and

Z\ = r x + jx i = self-inductive impedance at full frequency, reduced to primary circuit.

E = voltage induced in secondary and primary circuits by mutual magnetic flux,

Eo = voltage impressed upon primary, fo = primary current,

Z 0 = to + jx o = primary self -inductive impedance, and Y q = g — jb = primary exciting admittance.

1 See "‘Electric Circuits”, Chapter XIL Reactance of Induction Apparatus.

021* 3 i 3

Nn

• ^3

circuit is in phase

33 2 , 1 \

30

ELECTRICAL APPARATUS

The leakage reactance, x 2 , of the inner squirrel cage is that due to the flux produced by the current in the inner squirrel cage, which passes between the two squirrel cages, and does not in- clude the reactance due to the flux resulting from the current, 1 2 , which passes beyond the outer squirrel cage, as the latter is mutual reactance between the two squirrel cages, and thus meets the reactance, xi.

It is then, at slip s:

11

(i)

II

• IS-

(2)

/ 0 = h + h + YoE.

(3)

Ei = E 2 jx 2 1 2 .

(4)

E = Ei + jx 1 (Ji + I 2 ).

(5)

Eq — E - f- ZqIq.

(6)

The leakage flux of the outer squirrel cage is produced by the m.m.f. of the currents of both squirrel cages, h + I 2 , and the reactance voltage of this squirrel cage, in (5), thus iajxi (/ 1 + J 2 ).

As seen, the difference between E * and E 2 is the voltage in- duced by the flux which leaks between the two squirrel cages, in the path of the reactance, x 2 , or the reactance voltage, x 2 l 2 \ the difference between E and #i is the voltage induced by the rotor flux leaking outside of the outer squirrel cage. This has the m.m.f. /i + J 2 , and the reactance x h thus is the reactance voltage £i (Ji + h)- The difference between $ 0 and # is the voltage consumed by the primary impedance: Z 0 /o. (4) and (5) are the

voltages reduced to full frequency; the actual voltages are ,$ times as high, but since all three terms in these equations are induced voltages, the s cancels.

21 . From the equations (1) to (6) follows:

. (7)

— (fli + ja 2 ),

( 9 )

INDUCTION MOTOR

31

where:

a i

1 -

S Z X 1X2

r x r 2

'%i + Xi

ri r 2

• — ) rJ

a 2 = $

thus the exciting current :

Iqq — YoE

= $2 (g - jb) (a,! + ja 2 ) — E 2 (b 1 + jb 2 ) }

where:

bj = a,ig + a 2 b b 2 = a 2 g — a,jb

and the total primary current is (3) :

h = {r 2 + r, ( x +

where:

l r 2 Ti ~ £2 (Cl + jc 2 ),

S , S 1 ,

Ci = f- b\

r 2 n

s 2 x 2 ,

C 2 = — — h b 2

T\T 2

(10).

( 11 )

( 12 )

(13)

(14)

and the primary impressed voltage (6) :

$0 == $2 {(ii + jo , 2 + (r 0 + i^o) (ci + jc 2 ) }

where :

= E 2 (di + jd 2 )j

(15)

hence, absolute:

dj = a x + r 0 c 3 — a; 0 C 2 1 d 2 = a 2 + roC 2 + X 0 C 1 )

(16)

_

62 VdT r +T 2 2

(17)

io = c 2 a/ Cl 2 + C 2 2 -

(18)

22. The torque of the two squirrel cages is given by the product of current and induced voltage in phase with it, as:

n 2 = /# 2 , hr

= (19)

D, = /#!, /,/'

sei s

ri

Sfi 2 2

( 20 )

32

ELECTRICAL APPARATUS

hence, the total torque:

D — Z>2 + Dh

(21)

and the power output 1

P = (1 - 5 ) D .

(22)

(Herefrom subtracts the friction loss, to give the net power output.)

The power input is:

P 0 = /Eo, h/'

— 62 2 (cidi + € 262)9 (23)

and the volt-ampere input :

Q = 60 ^* 0 *

Po

Herefrom then follows the power-factor ~q> the torque effi- ciency JJ, the apparent torque efficiency Jr, the power efficiency Jtq W

P P

p- and the apparent power efficiency q-

23 . As illustrations are shown, in Figs. 14 and 15, the speed curves and the load curves of a double squirrel-cage induction motor, of the constants:

eo = 110 volts;

Zo = 0.1 + 0.3 j;

Zj = 0.5 + 0.2 j;

Z 2 = 0.08 + 0.4 j;

Y 0 = 0.01 - 0.1 j;

the speed curves for the range from s = 0 to $ = 2, that is, from synchronism to backward rotation at synchronous speed. The total torque as well as the two individual torques are shown on the speed curve. These curves are derived by calculating, for the values of $:

5 = 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3,

0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0,

REVERSE

SYNCHRONISM

INDUCTION MOTOR

33

r*

1 . 0 - 9 — 8 -. 7 -.6 -. 6-4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Pig. 14.— Speed curves of double squirrel-cage induction motor.

Fis. 15.— Load curves of double squirrel-cage induction motor.

h

1 1

3

SYNCHRONISM

34

ELECTRICAL APPARATUS

the values:

and:

ai

a 2

S 2 X 1X2

nr 2

bi = eng + a 2 b,

b 2 “ d 2 g ciib}

c.i+i+b,,

s 2 x 2 . ,

C — j- b 2,

r*ir 2

di = fli + ^oCi — XoC 2) d 2 *= a 2 + ^ 0^2 + #oCi,

2 __ £ 0 ^

62 di 2 + 4 2 ’

«2,

io = 62 V Ci 2 + C 2 2

D 2 =

Di =

r 2

se 2 2

n

7 ) = D 1 4 " 7 ) 2j P = (1 — s) D,

Po = ^ 2 2 (Cldl + ^2^2),

Q 5=8 Co^Q,

P^ P P P Po Po’Po’Q’ Q’ Q'

Triple Squirrel-cage Induction Motor

24, Let:

$ = flux, E = voltage, I = current, and Z = r + jx = self- inductive impedance, at full frequency and reduced to primary circuit, and let the quantities of the innermost squirrel cage be denoted by index 3, those of the middle squirrel cage by 2, of the outer squirrel cage by 1, of the primary circuit by 0, and the mutual inductive quantities without index.

Also let : Y 0 = g — jb = primary exciting admittance.

It is then, at slip s:

current in .the innermost squirrel cage:

INDUCTION MOTOR

35

current in the middle squirrel cage :

current in the outer squirrel cage :

primary current:

/o = /3 + I 2 + h + Yo$- The voltages are related by :

$2 = Uz + fazlz)

#1 = #2 + jx 2 (I2 + / 3),

^ + JXi (/ 1 + /2 + /s),

#0 = # + 2o/o,

(2)

(3)

(4)

(5)

( 6 )

(7)

( 8 )

where x 3 is the reactance due to the flux leakage between the third and the second squirrel cage; x 2 the reactance of the leak- age flux between second and first squirrel cage; x 2 the reactance of the first squirrel cage and x 0 that of the primary circuit, that is, x } + xo corresponds to the total leakage flux between primary and outer most squirrel cage.

$ 3 , U 2 and are the true induced voltages in the three squirrel cages, j JjJ the mutual inductive voltage between primary and secondary, and $0 the primary impressed voltage 26. From equations (1) to (8) then follows:

#2 = #3 (l + j 1 ( 9 )

fi = ~ #3 (a 1 + JCI2),

( 10 )

(ID

( 12 )

(13)

where:

36

ELECTRICAL APPARATUS

E = E 3 \ai-+ jas + j ^ (<Ji + ja 2) + i ~ . ■ .

'1 r 2 \ T3

/., . . sa; 2 \ , .saiil

, E» ( U - ^ - £SS) + ,(«, + «! + S!i + !S!)

I \ r 3 r 2 r 3 / \ ri r 2 r 3 / J

= ^3 (61 + J&2), where :

—

S 2 3i£3

r*i

r 2 r 3

. , , SXidx ,

62 = a 2 4 1

ri r 2 rz

thus the exciting current:

ho — YoJE

= Ez (bi + jb 2 ) ( g — Ez ( Cj + jc 2 ),

where:

Cj = + b 2 bj

c 2 = % ~ & 1 &,

and the total primary current, by (4) :

(14)

(15)

jb)

(16)

(17)

/° = #3 + ^ 3 ) + ^ + Cl +^ 2 }

= Ez (di + jd 2 ),

(18)

where :

A « 7 <*i + £ + - + ci 1

ri r 2 r 3 1

i 5 , $ 2 x 3 , 1

$2 — ~~ 0 2 + + Ci 1

r i r 2 r 3

(19)

where :

Zoh — Ez (d\ + jdf) (ro + jx o)

= Ez (fl + jfs),

( 20 )

fl = Mi — M 2 |

/a = 7* odz + Xodi j

( 21 )

thus, the primary impressed voltage, by ( 8 ) :

where:

•5'o — (&i + jbi + fi + j/ 2 )

= -Fa (s/i + j^),

( 22 )

ffi = bi+fi 1

g 3 = i>2 •+ f 3 j

(23)

INDUCTION MOTOR

37

hence, absolute:

Co

1

(24)

VW + 02 2

io = C3 ■%/ di 2 -f- dz 2 ,

(25)

L . sW

62 = 63 V 1 +

(20)

6] == C3 /(2i 2 -f- &2 2 *

(27)

26 . The torque

of the innermost squirrel cage thus is:

j) _ «».

1)3 -17’

(28)

that of the middle squirrel cage :

D, = — ;

r 2

(20)

and that of the outer squirrel cage:

Dr =

(30)

Ti

the total torque of the triple squirrel-cage motor thus is:

D = + D 2 +

(31)

and the power:

P = (1 - s) D,

(32)

the power input is:

Po = /Vo, ur

= e3 2 {d,g\ + d^g/),

(33)

and the volt-ampere input:

Q = e 0 io. • (34)

P

Herefrom then follows the power-factor the torque effi- ciency Jr, apparent torque efficiency 7^ power efficiency 4 ■* 0 0

and apparent power efficiency g*

27 . As illustrations are shown, in Figs. 16 and 17, the speed and the load curves of a triple squirrel-cage motor with the constants:

eo = HO volts;

Z 0 = 0.1 + 0.3 j; Zi = 0.8 + 0.1 j; Z 2 = 0.2 + 0.3 j; Zi = 0.05 + 0.8 j; Y 0 = 0.01 - 0.1 j;

Pjg. 17 —Load curves of triple squirrel-cage induction motor.

INDUCTION MOTOR

39

the speed curves are shown from 5 = 0 to s = 2, and on them, the individual torques of the three squirrel cages are shown in addition to the total torque.

These numerical values are derived by calculating, for the values of $:

s = 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.20, 0.30, 0.40, 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, 1.8, 20,

the values:

a 1

s 2 x 2 x 3

r 2 r 3

b} — 0,1 — b 2 == o> 2 ■+

SX jCl 2 ri

SXiCh

n

$ 2 X\X 2

r 2 r 3

sx 1

SXi — , Tz

eP

^3,

eo 2

gP + g 2 2 ’

io — Sz a / di 2 + d 2 2 y

0\ = big + b 2 b, c 2 = b 2 g + bib,

ei 2 = e 3 2 (a! 2 + a 2 2 ).

di

_ Sdi s 6 *

• Cj,

d 3

__

T\ ^ r 2 ” r r 3

r 3 ’

d 2

• £?j 4 . i

"■ + C 2 ,

d 2

°Vt|

Si-

ll

Ti r 2 r 3

r 3

r 2

fi

= r 0 di — xo d 2}

Di

_

n 9

f 2

= r 0 d 2 + Xodi,

D

= Di + D 2 + D 3 ,

9i

= 5i + jfi,

P

= (i - *) A

92

= 52+ f 2)

Po

= e 3 2 (di< 7 i + d 2 g 2 ),

Q

= Co A

and

P D P D P 0 Po’ Po’ Q’ Q’ Q'

CHAPTER III

CONCATENATION

Cascade or Tandem Control of Induction Motors

28. If of two induction motors the secondary of the first motor is connected to the primary of the second motor, the second machine operates as a motor with the voltage and frequency impressed upon it by the secondary of the first machine. The first machine acts as general alternating-current transformer or frequency converter (see Chapter XII), changing a part of the primary impressed power into secondary electrical power for the supply of the second machine, and a part into mechanical work.

The frequency of the secondary voltage of the first motor, and thus the frequency impressed upon the second motor, is the fre- quency of slip below synchronism, s. The frequency of the secondary of the second motor is the difference between its im- pressed frequency, s, and its speed. Thus, if both motors are connected together mechanically, to turn at the same speed, 1 — and have the same number of poles, the secondary fre- quency of the second motor is 2 5 — 1, hence equal to zero at $ = 0.5. That is, the second motor reaches its synchronism at half speed. At this speed, its torque becomes zero, the power component of its primary current, and thus the power com- ponent of the secondary current of the first motor, and thus also the torque of the first motor becomes zero. That is, a system of two concatenated equal motors, with short-circuited secondary of the second motor, approaches half synchronism at no-load, in the same manner as a single induction motor approaches synchronism. With increasing load, the slip below half syn- chronism increases.

In reality, at half synchronism, s = 0.5, there is a slight torque produced by the first motor, as the hysteresis energy current of the second motor comes from the secondary of the first motor, and therein, as energy current, produces a small torque.

More generally, any pair of induction motors connected in concatenation divides the speed so that the sum of their two

CONCATENATION

41

respective speeds approaches synchronism at no-load; or, still more generally, any number of concatenated induction motors run at such speeds that the sum of their speeds approaches synchronism at no-load.

With mechanical connection between the two motors, con- catenation thus offers a means of operating two equal motors at full efficiency at half speed in tandem, as well as at full speed, in parallel, and thereby gives the same advantage as does series parallel control with direct-current motors.

With two motors of different number of poles, rigidly con- nected together, concatenation allows three speeds: that of the one motor alone, that of the other motor alone, and the speed of concatenation of both motors. Such concatenation of two motors of different numbers of poles, has the disadvantage that at the two highest speeds only one motor is used, the other idle, and the apparatus economy thus inferior. However, with certain ratios of the fmmber of poles, it is possible to wind one and the s 4 ame motor structure so as to give at the same time two different numbers of poles: For instance, a four-polar and an eight- polar winding; and in this case, one and the same motor struc- ture can be used either as four-polar motor, with the one winding, or as eight-polar motor, with the other winding, or in concatena- tion of the two windings, corresponding to a twelve-polar speed. Such “ internally concatenated 7 J motors thus give three different speeds at full apparatus economy. The only limitation is, that only certain speeds and speed ratios.can economically be produced by internal concatenation.

29. At half synchronism, the torque of the concatenated couple of two equal motors becomes zero. Above half synchronism, the second motor runs beyond its impressed frequency, that is, becomes a generator. In this case, due to the reversal of current in the secondary of the first motor (this current now being out- flowing or generator current with regards to the second motor) its torque becomes negative also, that is, the concatenated couple becomes an induction generator above half synchronism. When approaching full synchronism, the generator torque of the second motor, at least if its armature is of low resistance, becomes very small, as this machine is operating very far above its synchronous speed. With regards to the first motor, it thus begins to act merely as an. impedance in the secondary circuit, that is, the first machine becomes a motor again. Thus, somewhere between

42

ELECTRICAL APPARATUS

half synchronism and synchronism, the torque of the first motor becomes zero, while the second motor still has a small negative or generator torque. A little above this speed, the torque of the concatenated couple becomes zero — about at two-thirds syn- chronism with a couple of low-resistance motors — and above this, the concatenated couple again gives a positive or motor torque — though the second motor still returns a small negative torque — and again approaches zero at full synchronism. Above full synchronism, the concatenated couple once more becomes generator, but practically only the first motor contributes to the generator torque above and the motor torque below full syn- chronism. Thus, while a concatenated couple of induction motors has two operative motor speeds, half synchronism and full synchronism, the latter is uneconomical, as the second motor holds back, and in the second or full synchronism speed range, it is more economical to cut out the second motor altogether, by short-circuiting the secondary terminals of the first motoi*.

With resistance in the secondary of the second motor, the maximum torque point of the second motor above half syn- chronism is shifted to higher speeds, nearer to full synchronism, and thus the speed between half and full synchronism, at which the concatenated couple loses its generator torque and again becomes motor, is shifted closer to full synchronism, and the motor torque in the second speed range, below full synchronism, is greatly reduced or even disappears. That is, with high resist- ance in the secondary of the second motor, the concatenated couple becomes generator or brake at half synchronism, and remains so at all higher speeds, merely loses its braking torque when approaching full synchronism, and regaining it again beyond full synchronism.

The speed torque curves of the concatenated couple, shown m Fig. 18, with low-resistance armature, and in Fig. 19, with high resistance in the armature or secondary of the second motor, illustrate this.

30 . The numerical calculation of a couple of concatenated induction motors (rigidly connected together on the same shaft or the equivalent) can be carried out as follows:

Let:

n — number of pairs of poles of the first motor, n f = number of pairs of poles of the second motor,

CONCATENATION

43

71 .

a = — = ratio of poles, (1)

71

f = supply frequency.

Pull synchronous speed of the first motor then is:

• • n 1 ®

of the second motor:

S'. - 4 - ®

At slip 5 and thus speed ratio (1 — s) of the first motor, its speed is:

S-(l-0'So- (1 -«)£ (4)

71

and the frequency of its secondary circuit, and thus the frequency of the primary circuit of the second motor :

synchronous speed of the second motor at this frequency is:

sS'o = s ; n

the speed of the second motor, however, is the same as that of the first motor, JS,

hence, the slip of speed of the second motor below its synchronous speed, is :

and the slip of frequency thus is:

s' = n r (\ — = a — a (1 — s),

W 71 I

s' = s (1 + a) — a. (5)

This slip of the second motor, s', becomes zero, that is, the couple reaches the synchronism of concatenation, for:

a

1 -f 'CL

So =

(6)

44 ELECTRICAL APPARATUS

The speed in this ease is:

-So 0 = (1 - so) ■ ■ (7)

Tb

f

n (1 +a)

31. If:

a = 1,

that is, two equal motors, as for instance two four-polar motors n = n' = 4,

it is:

So = 0.5,

CO / _ /,

50 — 2n ~ 8

while at full synchronism :

If:

it is:

So

-M’

n 4

a = 2 , ft = 4, n' = 8,

2

*° = r

-So 0 = /- = />’

in 12

that is, corresponding to -a twelve-polar motor. While:

if:

a = 0.5, n = 8, »' = 4,

so

-So 0

/

12

it is:

CONCATENATION

45

r

that is, corresponding to a twelve-polar motor again. That is, as regards to the speed of the concatenated couple, it is immaterial in which order the two motors are concatenated.

32. It is then, in a concatenated motor couple of pole ratio:

n'

a = — -> n

if:

s = slip of first motor below full synchronism.

The primary circuit of the first motor is of full frequency.

The secondary circuit of the first motor is of frequency 5.

The primary circuit of the second motor is of frequency s.

The secondary circuit of the second motor is of frequency s' = s (1 + a) — a.

Synchronism of concatenation is reached at :

a

So i~- y — ~ *

1 -I- a

Let thus :

e 0 = voltage impressed of first motor primary;

Yq = g — jb = exciting admittance of first motor;" 1

Y'o = g' — jb' = exciting admittance of second motor;

Zo — To + jx 0 = self-inductive impedance of first motor primary;

Z'o = r'o + jx ' 0 = self-inductive impedance of second motor primary;

Zi = r i + jxi = self-inductive impedance of first motor second- ary;

Z'\ = r'i + jx' 1 — self-inductive impedance of second motor secondary.

Assuming all these quantities reduced to the same number of

turns per circuit, and to full frequency, as usual.

If:

e = counter e.m.f . generated in the second motor by its mutual

magnetic flux, reduced to full frequency.

It is then:

secondary current of second motor:

/'x =

[s (1 + a) — a] e

r'i + js'x'i r'i + j [$ (1 + a) — a] x\

e{a x - ja 2 ), (8)

46

ELECTRICAL APPARATUS

where :

a i

CL 2

r\ [s (1 + a) — a]

m

x'i [s (1 + a) — a] 2

m

( 9 )

( 10 )

(ID

m — r'i 2 + z'i 2 (s (1 + a) — a) 2 ; exciting current of second motor:

V oo = el" = e(g' -jb'),

hence, primary current of second motor, and also secondary current of first motor:

V o = h = + /'oo

= e (6i - (12)

where :

h = «i + fif',

&2 = C /<2 f /,

the impedance of the circuit comprising the primary of the 4 second, and the secondary of the first motor, is:

Z — Z* + Z'q 2 = (?*i + r'o) + js (aq + x'q), (14)

hence, the counter e.m.f., or induced voltage in the secondary of the first motor, of frequency is :

= se + l\Z,

hence, reduced to full frequency:

(13)

= e +

hZ

where :

= e (ci +jc 2 ), .

Cj = 1 H — — ~bi -f- (&] + %' o) 62

C 2 = (zi + z'o) bi —

Ti + r' 0

(15)

(16)

33. The primary exciting current of the first motor is:

/oo = $iY

= e (di — jd 2 ),

where :

di = oigr + c 2 b d 2 ^ Cj.5 — c 2 q

(17)

(18)

CONCATENATION

47

thus, the total primary current of the first motor, or supply current:

where:

/

0 = II + loo = e(fi- jf 2 ),

(19)

fi — bi + dA f^ — h 2 d 2 \

( 20 )

and the primary impressed voltage of the first motor,

or supply

voltage:

Eo — Ei -f- Zola

= e (fifi + jgt),

(21)

where:

gi = Ci + r 0 fi + aso/sl

fif2 = c 2 + Xofi — rofi J

(22)

and, absolute:

eo = e Vgi 2 + g 2 2 , .

(23)

thus:

1

1 555 «i.t

II

(24)

Substituting now this value of e in the preceding, gives the values of the currents and voltages in the different circuits.

34. It thus is, supply current :

io = e a//i 2 + f‘ 2

eo

/7 T + /2 2 .

{7i 2 + g* 2 ’

'power input:

volt-ampere input:

p o = /Eo, ur

= e 2 (/igri - / 2 g 2 ) ,

_ „ — f »

_ eo

Q — eoio,

and herefrom power-factor, etc. The torque of the second motor is :

r = /e x h/'

= e 2 ai.

The torque of the first motor is :

Ti = /E h ur

= e 2 {cxfi - c 2 / 2 ),

48

ELECTRICAL APPARATUS

hence, the total torque of the concatenated couple:

T = T +.T X = c 2 (a, + cji - c 2 / 2 ), and herefrom the power output :

P * (1 — s) T,

thus the torque and power efficiencies and apparent efficiencies, etc.

35. As instances are calculated, and shown in Fig. 18, the speed

Fig. 18 . — Speed torque curves of concatenated couple with low resistance

secondary.

torque curves of the concatenated couple of two equal motors: a = 1, of the constants: Co = 110 volts.

Y = Y' = 0.01 - 0.1 j;

Zo = Z'q = 0.1 -f- 0.3 j}

z x = Z\ = 0.1 + 0.3 j.

Fig. 18 also shows, separately, the torque of the second motor, and the supply current.

Fig. 19 shows the speed torque curves of the same concate- nated couple with an additional resistance r = 0.5 inserted into the secondary of the second motor.

The load curves of the same motor, Fig. 18, for concatenated running, and also separately the load curves of either motor,

CONCATENATION

49

are given on page 358 of “Theoretical Elements of Electrical Engineering.”

36 . It is possible in concatenation of two motors of different number of poles, to use one and the same magnetic structure for both motors. Suppose the stator is wound with an n-polar primary, receiving the supply voltage, and at the same time with an n' polar short-circuited secondary winding. The rotor is wound with an n-polar winding as secondary to the n-polar primary winding, but this n-polar secondary winding is not short-circuited, but connected to the terminals of a second

Fig. 19. — Speed-torque curves of concatenated couple with resistance in

second secondary.

n'-polar winding, also located on the rotor. This latter thus receives the secondary current from the n-polar winding and acts as n'-polar primary to the short-circuited stator winding as secondary. This gives an n-polar motor concatenated to an n'-polar, and the magnetic structure simultaneously carries an n-polar and an n'-polar magnetic field. With this arrangement of “ internal concatenation,” it is essential to choose the number of poles, n and n', so that the two rotating fields do not interfere with each other, that is, the n'-polar field does not induce in the n-polar winding, nor the n-polar field in the n'-polar winding. This is the case if the one field has twice as many poles as the other, for instance a four-polar and an eight-polar field.

If such a fractional-pitch winding is used, that the coil pitch is suited for an n-polar as well as an n'-polar winding, then the same winding can be used for both sets of poles. In the stator, the equipotential points of a 2 p-polar winding are points of opposite polarity of a p-polar winding, and thus, by connecting together the equipotential points of a 2 p-polar primary winding,

4

50

ELECTRICAL APPARATUS

this winding becomes at the same time a p-polar short-circuited winding. On the rotor, in some slots, the secondary current of the n-polar and the primary current of the n'-polar winding flow in the same direction, in other slots flow in opposite direction, thus neutralize in the latter, and the turns can be omitted in concatenation — but would be put in for use of the structure as single motor of n, or of n r poles, where such is desired. Thus, on the rotor one single winding also is sufficient, and this arrange- ment of internal concatenation with single stator and single rotor winding thus is more efficient than the use of two separate motors, and gives somewhat better constants, as the self-inductive im- pedance of the rotor is less, due to the omission of one-third of the turns in which the currents neutralize (Hunt motor).

The disadvantage of this arrangement of internal concatenation with single stator and rotor winding is the limitation of the avail- able speeds, as it is adapted only to 4 -f- 8 -r- 12 poles and multiples thereof, thus to speed ratios of 1 the last

being the concatenated speed.

Such internally concatenated motors may be used advantage- ously sometime as constant-speed motors, that is, always run- ning in concatenation, for very slow-speed motors of very large number of poles.

37 . Theoretically, any number of motors may be concatenated. It is rarely economical, however, to go beyond two motors in concatenation, as with the increasing number of motors, the constants of the concatenated system rapidly become poorer.

If:

Y 0 = g -jb,

Z 0 = n + jx o,

Zl — Ti + jX i,

are the constants of a motor, and we denote:

Z — Zq + Zl = (Tq + Ti) + j ( Xq + Xi)

= r + jx

then the characteristic constant of this motor — which char- acterizes its performance — is :

= yz;

if now two such motors are concatenated, the exciting admittance of the concatenated couple is (approximately) :

CONCATENATION

51

as the first motor carries the exciting current of the second motor.

The total self-inductive impedance of the couple is that of both motors in series:

Z' = 2Z;

thus the characteristic constant of the concatenated couple is:

= y'z'

= 4 yz = 4

that is, four times as high as in a single motor; in other words, the performance characteristics, as power-factor, etc., are very much inferior to those of a single motor.

With three motors in concatenation, the constants of the system of three motors are:

Y" = 3 F,

Z" = 3 Z,

thus the characteristic constant :

= y"z n = 9 yz = 9 tf,

or nine times higher than in a single motor. In other words, the characteristic constant increases with the square of the number of motors in concatenation, and thus concatenation of more than two motors would be permissible only with motors of very good constants.

The calculation of a concatenated system of three or more motors is carried out in the same manner as that of two motors, by starting with the secondary circuit of the last motor, and building up toward the primary circuit of the first motor.

CHAPTER IV

INDUCTION MOTOR WITH SECONDARY EXCITATION

38. While in the typical synchronous machine and commu- tating machine the magnetic field is excited by a direct current, characteristic of the induction machine is, that the magnetic field is excited by an alternating current derived from the alter- nating supply voltage, just as in the alternating-current trans- former. As the alternating magnetizing current is a wattless reactive current, the result is, that the alternating-current input into the induction motor is always lagging, the more so, the larger a part of the total current is given by the magnetizing current. To secure good' power-factor in an induction motor, the magnetizing current, that is, the current which produces the magnetic field flux, must be kept as small as possible. This means as small an air gap between stator and rotor as mechanic- ally permissible, and as large a number of primary turns per pole, that is, as large a pole pitch, as economically permissible.

In motors, in which the speed — compared to the motor out- put — is not too low, good constants can be secured.. This, however, is not possible in motors, in which the speed is very low, that is, the number of poles large compared with the out- put, and the pole pitch thus must for economical reasons be kept small — as for instance a 100-hp. 60-cycle motor for 90 revolu- tions, that is, 80 poles — or where the requirement of an excessive momentary overload capacity has to be met, etc. In such motors of necessity the exciting current or current at no-load — which is practically all magnetizing current — is a very large part of full-load current, and while fair efficiencies may nevertheless be secured, power-factor and apparent efficiency necessarily are very low.

As illustration is shown in Pig. 20 the load curve of a typical 100-hp. 60-cycle 80-polar induction motor (90 revolutions per minute) of the constants ;

Impressed voltage:

Primary exciting admittance: Primary self-inductive impedance : Secondary self-inductive impedance:.

e 0 = 500.

F 0 = 0.02 - 0.6 j. Zo = 0.1 + 0.3 j. Z x = 0.1 + 0.3 j.

INDUCTION MOTOR

53

As seen, at full-load of 75 kw. output, the efficiency is 80 per cent., which is fair for a slow-speed motor.

But the power-factor is 55 per cent., the apparent efficiency only 44 per cent., and the exciting current is 75 per cent, of full- load current.

This motor-load curve may be compared with that of a typical induction motor, of exciting admittance :

Yo = 0.01 - 0.1 j,

given on page 234 of “Theory and Calculation of Alternating- current Phenomena” 5th edition, and page 319 of “Theoretical

Fig. 20. — Low-speed induction motor, load curves.

Elements of Electrical Engineering,” 4th edition, to. see the difference.

39. In the synchronous machine usually the stator, in com- mutating machines the rotor is the armature, that is, the element to which electrical power is supplied, and in which electrical power is converted into the mechanical power output of the motor. The rotor of the typical synchronous machine, and the stator of the commutating machine are the field, that is, in them no electric power is consumed by conversion into mechanical work, but their purpose is to produce the magnetic field flux, through which the armature rotates.

In the induction machine, it is usually the stator, which is the

54

ELECTRICAL APPARATUS

primary, that is, which receives electric power and converts it into mechanical power, and the primary or stator of the induc- tion machine thus corresponds to the armature of. the synchro- nous or commutating machine. In the secondary or rotor of the induction machine, low-frequency currents — of the frequency of slip — are induced by the primary, but the magnetic field flux is produced by the exciting current which traverses the primary or armature or stator. Thus the induction machine may be considered as a machine in which the magnetic field is produced by the armature reaction, and corresponds to a synchronous machine, in which the field coils are short-circuited and the field produced by armature reaction by lagging currents in the armature.

As the rotor or secondary of the induction machine corresponds structurally to the field of the synchronous or commutating machine, field excitation thus can be given to the induction machine by passing a current through the rotor or secondary and thereby more or less relieving the primary of its function of giv- ing the field excitation.

Thus in a slow-speed induction motor, of very high exciting current and correspondingly poor constants, by passing an exciting current of suitable value through the rotor or secondary, the primary can be made non-inductive, or even leading current produced, or — with a lesser exciting current in the rotor — at least the power-factor increased.

Various such methods of secondary excitation have been pro- posed, and to some extent used.

1. Passing a direct current through the rotor for excitation.

In this case, as the frequency of the secondary currents is the

frequency of slip, with a direct current, the frequency is zero, that is, the motor becomes a synchronous motor.

2. Excitation through commutator, by the alternating supply current, either in shunt or in series to the armature.

At the supply frequency, /, and slip, s, the frequency of rotation and thus of commutation is (1 — s) /, and the full frequency cur- rents supplied to the commutator thus give in the rotor the effective frequency, / — (1 — s) f = $f, that is, the frequency of slip, thus are suitable as exciting currents.

3. Concatenation with a synchronous motor.

If a low-frequency synchronous machine is mounted on the induction-motor shaft, and its armature connected into the indue-

INDUCTION MOTOR

55

tion-motor secondary, the synchronous machine feeds low-fre- quency exciting currents into the induction machine, and thereby permits controlling it by using suitable voltage and phase;