CHAPTER XIX 
ALTERNATING- CURRENT MOTORS IN GENERAL 

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

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

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

The main types of synchronous motors are as follows: 

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

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

3. One member excited by alternating current, the other of 
different magnetic reluctance iii different direction! 
construction) — reaction motors. 

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


ALTERNATING-CURRENT MOTORS 301 

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

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

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

The main types of these motors are: 

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

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

lotors with shunt-motor characteristic. 
:?. lioth members connected, through a commutator, directly 


302 


ELECTRICAL APPARATUS 


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

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

Synchronous motors. 

Induction motors. 

Commutator motors. 

There are, however, numerous intermediate forms, which 
belong in several classes, as the synchronous-induction motor, 
the c o oipe n sat ed-in due lion motor, etc. 

172. An alternating current, /, in an electric circuit produces 
a magnetic flux, 41, interlinked with this circuit. Considering 
equivalent sine waves of / and *, 4> lags behind / by the angle 
of hysteretic lag, a. This magnetic flux, $, generates an e.m.f., 
5 = 2 tt/;i<I>, where / = frequency, n = number of turns of 
electric circuit. This generated e.m.f., E, lags 90° behind the 
magnetic flux, *, hence consumes an e.m.f. 90° ahead of ♦, 
or 90—ci degrees ahead of /. This may be resolved in a reactive 
component: E = 2x/ft* eos a = 2 t/LI = xl, the o.m.f, con- 
sumed by self-induction, and power component: E" = 2r/n* 
sin a = 2irfHI = r"I = e.m.f. consumed by hysteresis (eddj 
currents, etc.), and is, therefore, in vector representation denoted 
by: 

E' = jxf and E" = f>% 
where: 

x = 2 irfL — reactance, 
and 

L = inductance, 
r" = effective hysteretic resistance. 

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

E = (r+jx)I = Z{, 
.where : 

Z = r + jx = impedance, in vector denotation, 
z = Vr* + i* = impedance, in absolute terms. 

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


ALTERNATING-CURRENT MOTORS 303 

cuit in two parts — the self-inductance, S, which refers to that 
part of the magnetic flux produced by the current in one circuit 
which is interlinked only with this circuit but not with the other 
circuit, and the mutual inductance, M , which refers to that part 
of the magnetic flux interlinked also with the second circuit. 
The desirability of this separation results from the different char- 
acter of the two components: The self-inductive reactance gen- 
erates a reactive e.m.f. and thereby causes a lag of the current, 
while the mutual inductive reactance transfers power into the 
second circuit, hence generally does the useful work of the ap- 
paratus. This" leads to the distinction between the self-inductive 
impedance, Z0 = r0 + jx0, and the mutual inductive impedance, 
Z = r + jx. 

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

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

r0 is the coefficient of power consumption by ohmic resistance, 
hysteresis and eddy currents of the self-inductive flux — effective 
resistance. 

x0 is the coefficient of e.m.f. consumed by the self-inductive or 
leakage flux — self-inductive reactance. 

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

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

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

# = Zol + Zh l (1) 

If one of the circuits rotates relatively to the other, then in 
addition to the e.m.f. of self-inductive impedance : Z0/, and the 
e.m.f. of mutual-inductive impedance or e.m.f. of alternation: 
ZJy an e.m.f. is consumed by rotation. This e.m.f. is in phase 
with the flux through which the coil rotates — that is, the flux 
parallel to the plane of the coil — and proportional to the speed — 


301 


EL Ei TRIl 'A L APPA RA TVS 


that, is, the frequency of rotation — while the e.m.f. of alternation 
is 90° ahead of the flux alternating through the coil— thai is, Uw 

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

g = z0i + XI - jSZ'l. 
Applying now these considerations to the alternating-current 
motor, we assume all circuits reduced to the same number of 
turns— that is, selecting one circuit, of n effective turns, as start- 
ing point, if n, = number of effective turns of any other circuit, 
all the e.m.fs. of the latter circuit arc divided, the currents multi- 
plied with the ratio, -> the impedances divided, the admittances 

multiplied with I -) . This reduction of the constants of all 
circuits to the same number of effective turns is convenient by 
eliminating constant factors from the equations, and so permit- 


ting a direct comparison. 


When speaking, therefore, in (he fol- 
lowing of the impedance, etc., of the 
different circuits, we always refer to 
their reduced values, as it is cus- 
tomary in induction-motor designing 
practice, and has been done in pre- 
ceding theoretical investigations. 
173. Let, then, in Fig. 147: 
Pn, f«, Zn = impressed voltage, 
current and self-inductive impedance 
respectively of a stationary circuit, 
F,c. 147. Pu h, Z> = impressed voltage, 

current and self-inductive impedance 
respectively of a rotating circuit, 

r = space angle between the axes of the two circuits, 
Z = mutual inductive, or exciting impedance in the direction 
mI the axis (if the stationary coil, 

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

Z" - mutual inductive or exciting impedance in the direction 
at right angles to the axis of (he rotating coil, 


ALTERNATING-CURRENT MOTORS 305 

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

It is then : 

E.m.f. consumed by self-inductive impedance, Z0/o. 

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

$o - Zo/o + Z (/o + /i COS r). (3) 

In the rotating circuit, it is: 

E.m.f. consumed by self-inductive impedance, Zi/i. 

E.m.f. consumed by mutual-inductive impedance or " e.m.f. of 
alternation": Z' (/i + /0 cos r). (4) 

' E.m.f. of rotation, — jSZ"lo sin t. (5) 

Hence the impressed e.m.f. : 

#i = ZJi + Z' (/! + U cos r) - jSZ"/o sin r. (6) 

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

Z is the reciprocal of the "exciting admittance," Y of the in- 
duction-motor theory. 

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

l$k, hy Zk = impressed e.m.f., current and self-inductive im- 
pedance respectively of any circuit, fc. 

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

t*» = space angle between the axes of coils k and i, and 

S = speed, as fraction of synchronism, or "frequency of 
rotation." 

It is then, in a coil, i: 

${ = ZJi + Zi $* /* cos Tu1 - jSZ" >* h sin rk\ (7) 

i i 

where: 

Ziji = e.m.f. of self-inductive impedance; (8) 

n 

Z*^ /* cos r*1 = e.m.f. of alternation; (9) 


n 


E'i = - jSZiiSjkJk sin tV = e.m.f. of rotation; (10) 

i 

which latter = 0 in a stationary coil, in which 5 = 0. 

20 


306 ELECTRICAL APPARATUS 

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

i 

- - S £ tfZ«J* /* sin r*S /J1, (11) 

11 

and herefrom the torque, in synchronous watts: 

D - ^ - - J? ljZui h sin r*S UK (12) 

o i i 

The power input, in vector denotation, is: 


(13) 


Po = F [Ei9 h] 
i 

= £ [Eit hy + J? [ft, /j/ 
= Po1 + jPo>; 

and therefore: 

Po1 = true power input; 

P</ = wattless volt-ampere input; 

Q = VPo1 + Po* = apparent, or volt-ampere 
input; 

D . = efficiency; 
*o 

n = apparent efficiency; 
iTi = torque efficiency; 
-~ = apparent torque efficiency; 

y 

-Jr = power-factor. 

From the n circuits, i = 1, 2 . . . n, thus result n linear 
equations, with 2 n complex variables, /< and #». 

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

Impressed e.m.fs. /?t may be given. 

Or circuits closed upon themselves #» = 0. 

Or circuits connected in parallel c^i = c*#*, where c, and c* 


ALTERNATING-CURRENT MOTORS 307 

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

Or circuits connected in series: -■* = --» etc. 

Ci ck 

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

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

1. POLYPHASE INDUCTION MOTOR 

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

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

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

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

#o and — j$o, /o and — j'/o, Z0 = impressed e.m.f., currents 
and self-inductive impedance respectively of the primary system. 

0, /i and —jl\9 Z\ = impressed e.m.f., currents and self-in- 
ductive impedance respectively of the secondary system, reduced 
to the primary. Z = mutual-inductive impedance between 
primary and secondary, constant in all directions. 


308 


ELECTRICAL APPARATUS 


S = speed; s = 1 — S — slip, as fraction of synchronism. 
The equation of the primary circuit is then, by (7) : 

E* = ZoJ0 + Z (/, - /,). (14) 

The equation of the secondary circuit: 

0 - ZJi + Z (/i - /.) + jSZ (jfx - jh), (15) 


from (15) follows: 


Zo (1 - S) 


= /< 


Zs 


Z(l-S) + Zl *VZ& + Zi 


(16) 


SU+ 


8 


Fio. 148. 


I. 


I. 


*jl 


and, substituted in (14): 
Primary current: 

/o = Eo 
Secondary current: 


Za + Zi 


ZZoS + ZZi + ZoZi 


r - w - _Zs _ 

/ 1 *° ZZos + ZZX + Z^Zi 


Zi 


Exciting current: 

/oo = h - h = Eo zzliTzzT+zifi 

E.m.f. of rotation: 

E' = jSZ (jh - jh) = sz (/. - /t). 

ZZx 


- SEi 


ZZoS + ZZi + ZoZi 


= (l-8)E0-r?yT- j-' 


ZZx 


ZZoS + ZZX + ZoZi 


(17) 


(18) 


(19) 


(20) 


ALTERNATING-CURRENT MOTORS 309 


It is, 

at synchronism ; s 

- 0: 

h 

Eo 
~ Z'+'Zo 

u 

= 0; 

/oo 

" fa 

V 

2?oZ £10 
Z + Zo t , Zo 

1 + z 

At standstill: 

8 • 

- i; 

u> 

£» (Z + zo . 

ZZo H~ ZZi -f- ZoZi 

* 

h 

_ 2?oZ 

ZZo "h ZZi + ZoZi 

EqZi 

ZZo + ZZi + ZoZi 

#' = 0. 

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

# = #o — Zo/o, (21) 

or: 

#o = # + Zo/o, (22) 

it is, substituted : 
Counter e.m.f. : 

v = ^° zZoT+zz\ + ZoZV (23) 

hence: 

Primary impressed e.m.f.: 

« „ ZZos + Zi + ZZoZi ,0 .v 

#o = # 22 ' '**' 

E.m.f. of rotation: 

#' = #S = # (1 - s). (25) 

Secondary current: 

h = Jj- (26) 


Primary current: 


/o " ^~zzT' z; + z (27) 


310 ELECTRICAL APPARATUS 

Exciting current: 

/oo = § = $Y. (28) 

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

176. Since the frequency of the secondary currents is the fre- 
quency of slip, hence varies with the speed, S = 1 — 8, the sec- 
ondary self-inductive reactance also varies with the speed, and 
so the impedance: 

Zi = n + J8Xl. (29) 

The power output of the motor, per circuit, is: 

P = [£', /i] 

ri ""v'z" r-<7^-i2 (ri ~ J**i)» (3°) 


[ZZos + ZZl + ZoZx]2 

where the brackets [ ] denote the absolute value of the term in- 
cluded by it, and the small letters, c0, z, etc., the absolute values 
of the vectors, #o, Z, etc. 

Since the imaginary term of power seems to have no physical 
meaning, it is: 
Mechanical power output: 

p _c0Vs(l - s)rx ( . 

[ZZos + ZZx + ZoZtf K } 

This is the power output at the armature conductors, hence in- 
cludes friction and windage. 
The torque of the motor is: 


D = 


1 - 8 


eJ&iS • _ eoVsis* _ ,o2\ 

[ZZos + ZZi + ZoZif2 J [ZZos + ZZX + ZoZtf ^ ; 

The imaginary component of torque seems to represent the 
radial force or thrust acting between stator and rotor. Omitting 
this we have: 

T\ _ ^o z__^}s (1*1} 

~ [ZZos + ZZx + z&W* 


where: 


ALTERNATING-CURRENT MOTORS 311 

The power input of the motor per circuit is: 

Po = [#o, /o] 

= *°2 L1' ZZ08 +*ZZl + ZoZj (34) 

= P'o - jPoj 

P'o = true power, 

PJ = reactive or "wattless power," 

Q = a/PV + iV* = volt-ampere input. 

Herefrom follows power-factor, efficiency, etc. 
Introducing the parameter: #, or absolute e, we have: 
Power output: 

- [* a 


= -— * - jii'Szi. (35) 


Power input: 
Po = [#o, /o] 


t [ZZos + ZZt + ZoZi Za_+ Zt 

" c L zzi ' zii . 

Z0 (Z« + Zi) . , Za + Z, 


trZo(Z8 + ZQ Z« + Z,1 

~ C L ZZt " + *' ZZ, ~J 

-«-[^*n[wi+<n.i+i]} 

. rZs + Zn* / , . . e*s , . . , e1 . . , 

- e \rzzv\ l ( ° " J o) + ^» (ri ~ •,*Cl) + 2« (r " Ja:) 

= *V (r, - jxo) + ii* (~ - j*,) + too* (r - j*). (36) 

And since: 


r, 5 + s Su , 

= — n - - + r„ 


312 ELECTRICAL APPARATUS 

and: 

it is: 

P o = (*o2r« + i,2r! + iooV + P) - j (tV*o + *V*i + ioo*x). (37) 
Where: 

2*o2r0 = primary resistance loss, 

i\2ri = secondary resistance loss, 

tooV = core loss (and eddy-current loss), 

P = output, 

io2Xo = primary reactive volt-amperes, 

ii2X\ = secondary reactive volt-amperes, 

ioo2x = magnetizing volt-amperes. 

176. Introducing into the equations, (16), (17), (18), (19), (23) 
the terms: 


z - *°' 


(38) 


Where Xo and Xi are small quantities, and X = Xo + Xi is the 
"characteristic constant" of the induction motor theory, it is: 
Primary current: 

j = &± « + Xt _ _ E0 s + Xi . 

/0 Z sX0 + Xi +"X0X1 2«Xo + X * J 

Secondary current: 

/. = E° * = Eo ! (40) 

41 Z sXo + Xi + XoXi Z 5X0 + Xi v ' 

Exciting current: 

r _ -Bo Xi __ Eo Xi . 

/0° " Z sXo + X~i + XoXi ~~ Z 5X0 + Xi' l l) 

E.m.f . of rotation : 

W = QoS .— ,-^ r .-. = VoS -\ - - (42) 

sXo + Xi + X0X1 5X0 + Xi ' 

Counter e.m.f.: 

sXo + Xi + X0X1 sXo + Xi v ' 


ALTERNATING-CURRENT MOTORS 


313 


177. As an example are shown, in Fig. 149, with the speed 
as abscissae, the curves of a polyphase induction motor of the 
constants: 

e0 = 320 volts, 

Z = 1 + 10j ohms, 

Z0 = Zl = 0.1 + 0.3 j ohms; 
hence : 

X0 = Xi = 0.0307 - 0.0069 j. 


P&D 
140 
180 
120 
110 

POLYPHASE INDUCTION MOTOR 
320 VOLTS 

I 

I 

rr\i\ . 

100 
90 
80 
TO 

^"^ 

•~**£. 

1 WW 

—450- 

^s 

4 

>y 

S,*^ 

\ i 

4UU 

850 

D 

P 

J 

P- 

i i 
\ i 
\ | 

sno 

00 
CO 
40 

^ 

'1^ 

N 

> \i 

0UV 

250 

V 

.— — 5^ 

\ 

OA/t 

•_ _,- — ^» 

\ 

V4|— 150- 

\ 11 1AA 

30 

— "»? 

20 

10 

ft 

—^^^^ 

M ^ft 

*1 WJ 

100 
90 

80 
70 
60 
CO 
40 
80 
20 
—I 10 


0.1 


0.2 0.8 


0.4 0.5 0.6 
Fi«. 149. 


0.7 


0.8 0.9 


1.0 


It is: 
/o = 


320{ 10.30 s - (« + 0.1)i| 
(103 + 1.63*) - j(0.11 - 5.99«)amp* 

D = (1.03 + 1.63^ + (0.11 - 5.99 s)' *y™h™<>™ kw- 

P = (1 - s) D 

0.11 - 5.99 s 


tan 6" = 


1.03 + 1.63 s 


tan*'= * + 01; 
10.3 s ' 

cos (0' — 6") = power-factor. 

Fig. 149 gives, with the speed S as abscissae: the current, J; 
the power output, P; the torque, D; the power-factor, p; the 
efficiency, rj. 


314 


ELECTRICAL APPARATUS 


The curves show the well-known characteristics of the poly- 
phase induction motor: approximate constancy of speed at all 

loads, and good efficiency and power- fact or within this narrow- 
speed range, but poor constants at all other speeds. 

1. SINGLE-PHASE INDUCTION MOTOR 
178. In the single-phase induction motor one primary circuit 
acts upon a system of closed secondary circuits which are dis- 
placed from each other in position on the secondary member. 

Let the secondary be assumed as two-phase, that is, containing 
or reduced to two circuits closed upon themselves at right angles 


Fio. 150. — SiiiKle-phosp induction n 


to each other. While it then offers a resultant closed secondary 
circuit to the primary circuit in any position, the electrical dis- 
position of the secondary is not symmetrical, but the directions 
parallel with the primary circuit and at right angles thereto are 
to be distinguished. The former may be called the secondary 
energy circuit, the latter the secondary magnetizing circuit, since 
in the former direction power is transferred from the primary to 
the secondary circuit, while in the latter direction the secondary 
circuit can act magnetizing only. 

Let, in the diagram Fig. 150: 

Ea, Ja, Zn = impressed e.m.f., current and self-inductive im- 
pedance, respectively, of the primary circuit, 

l\, Z\ = current and self-inductive impedance, respectively, 
of the secondary energy circuit, 

/), Zi = current and self-inductive impedance, respectively, 
of the secondary magnetizing circuit, 

Z = mutual -inductive impedance, 

S m speed, 
and let s0 = 1 - S2 (where s0 is not the slip). 

It is then, by equation (7) : 


ALTERNATING-CURRENT MOTORS 315 

Primary circuit: 

E0 = Zolo + Z(h~ /i). ' (44) 

Secondary energy circuit: 

0 = ZXU + Z (/i - /„) - jSZU- (45) 

Secondary magnetizing circuit: 

0 = ZJ, + ZU ~ )SZ (/. - /,) ; (46) 

hence, from (45) and (46) : 

/l ~ /o zuT+2Zz; + zs' (47) 

h = + jS/. ^^2 iz, + Z? ' (48) 

and, substituted in (44) : 
Primary current: 

h = #o ^ (49) 

Secondary energy current: 

U = *. *<*» + *>. (50) 

Secondary magnetizing current: 

/. = + jSE0 ^ (51) 

E.m.f. of rotation of secondary energy circuit: 

#i = - jSZh = S'#0 Z^- (52) 

E.m.f. of rotation of secondary magnetizing circuit: 

E't = - jsz (/. - /o = - is^o ZZl (^+ Zl) ; (53) 

where : 

X = Z„ (Z*So + 2 ZZ, + Z,») + ZZl (Z + Z,). (54) 

It is, at synchronism, S = 1, s0 = 0: 

, _ f 2Z + Zt . 

*° *' Z«(2Z + Zi)+Z(Z+Zi)' 

Jl = ^° Z0 (2 Z + ZO +~Z(Z +' Zi) ' 

/s = + j#o z7(2Z + Z,y + zlz + Z",")" 


316 ELECTRICAL APPARATUS 

Hence, at synchronism, the secondary current of the single- 
phase induction motor does not become zero, as in the polyphase 
motor, but both components of secondary current become equal. 

At standstill, S = 0, s0 = 1, it is: 

/0 ^° ZZo + ZZX + ZoZi' 

?l = ^° zz7+~ zzV+lz'oZi' 
U = 0. 

That is, primary and secondary current corresponding thereto 
' have the same values as in the polyphase induction motor, as 
was to be expected. 

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

and substituting for /0 from (49), it is: 
Primary impressed e.m.f.: 

_ „ Z0 (Z2s0 + 2 ZZX + Z?) + ZZX (Z + Z,) , 

*°-* zz^z + zo ' (o5) 

Primary current: 

r „ Z'so + 2 ZZ, + Z,! riwv 

h~v zzaz + ZiT " m 

Secondary energy circuit: 

_ p ZsojH Z, _ s0E Ss£_ , _> 

71 " * z7(z + z.) z, ~l~ z + z, v " 

£'. = S*V g-Z z (58) 

Secondary magnetizing circuit : 

/.-+JZ + z; (59) 

Vt-JW-' ^ (60) 

And: 

/o - /i = ^ (61) 

These equations differ from the equations of the polyphase 

induction motor by containing the term s0 = (1 — S2), instead 

SE 
of s = (1 — S), and by the appearance of the terms, y~r^~ and 

S2E 
~-, «■» of frequency (1 + S), in the secondary circuit. 


(62) 


ALTERNATING-CURRENT MOTORS 317 

The power output of the motor is: 

P - [Eu /J + [Et, h] 

= ^^-{[ZZu Zs0 + ZJ - [Z, (Z + Z,), ZJ} 

[jq 

and the torque, in synchronous watts: 

D"s~~~~m • m 

From these equations it follows that at synchronism tor- 
que and power of the single-phase induction motor are already 
negative. 

Torque and power become zero for: 

SoZ2 — Zi2 = 0, 

hence: 


->/-e) 


that is, very slightly below synchronism. 

Let z = 10, Zi = 0.316, it is, S = 0.9995. 

In the single-phase induction motor, the torque contains the 
speed S as factor, and thus becomes zero at standstill. 

Neglecting quantities of secondary order, it is, approximately : 

h = £o zJZ^ + Zl)+2Zai (65) 

/. - + jSE, z (z^-zj+YzX (67) 

^ = S^0Z(Z„So + Z1)+2Z.Zl' (68) 

zz 

. ^ = " jS** Z (Zoso + zi)+2 ZoZi (69) 

P = SWz^sp , 

[z TZoso + zo + 2 ZoZj*' uu; 

n = _Se0Vri80 /71x 

[Z (ZoSo + Z\j + 2 ZoZi]2' u i} 

This theory of the single-phase induction motor differs from 
that based on the transformer feature of the motor, in that it 
represents more exactly the phenomena taking place at inter- 


318 


ELECTRICAL APPARATUS 


mediate speeds, which are only approximated by the transformer 
theory of the single-phase induction motor. 

For studying the action of the motor at intermediate and at 
low speed, as for instance, when investigating the performance 
of a starting device, in bringing the motor up to speed, that is, 
during acceleration, this method so is more suited. An applica- 
tion to the "condenser motor," that is, a single-phase induction 
motor using a condenser in a stationary tertiary circuit (under 
an angle, usually 60°, with the primary circuit) is given in the 
paper on "Alternating-Current Motors," A. I. E. E. Transac- 
tions, 1904. 


P&D 


Fig. 151. 

180. As example are shown, in Fig. 151, with the speed as 
abscissae, the curves of a single-phase induction motor, having 
the constants: 

e0 = 400 volts, 

Z = 1 + 10 j ohms, 


and: 


hence: 


Z0 = Zi = 0.1 + 0.3 j ohms; 


N 


Io = 400 j* amp. ; 

AT = (s0 + 0.2) + j(10s0 + 0.6 - 0.6 S); 

K = (0.1+0.3j)AT+(l + 10j)(0.1+j)(0.3-0.3S); 


D = 


1616 Ss0 

[K\ 


synchronous kw. 


ALTERXATIXG-CURREXT MOTORS 


319 


Fig. 151 gives, with the speed, S, as abscissa*: the current, 7«, 
the power output, P, the torque, D, the power-factor, p, the 
efficiency, y. 

3. POLYPHASE SHUNT MOTOR 

18L Since the characteristics of the polyphase motor do not 
depend upon the number of phases, here, as in the preceding, a 
two-phase system may be assumed: a two-phase stator winding 
acting upon a two-phase rotor winding, that is, a closed-coil 
rotor winding connected to the commutator in the same manner 
as in direct-current machines, but with two sets of brushes in 
quadrature position excited by a two-phase system of the same 
frequency. Mechanically the three-phase system here has the 
advantage of requiring only three sets of brushes instead of four 


\ jl<* 


Fio. 152. 


as with the two-phase system, but otherwise the general form 
of the equations and conclusions are not different. 

Let #o and — j#0 = e.m.fs. impressed upon the stator, #i and 
— jfli = e.m.fs. impressed upon the rotor, 0O « phase angle be- 
tween e.m.f., #o and #i, and 0i ■» position angle lwtween the 
stator and rotor circuits. The e.m.fH., #o and — j#0, produce the 
same rotating e.m.f. as two e.m.fH. of equal intensity, but dis- 
placed in phase and in position by angle 0O from #», and jf/l,,, 
and instead of considering a displacement of phase, 0,h arid a dis- 
placement of position, 0i, between stator and rotor circuits, we 
can, therefore, assume zero-phase displacement and diMplacemeut 
in position by angle 0O + 0i = 0. Phase diMplaecmcnf l*etween 
stator and rotor e.m.fH. is, therefore, equivalent to n fluff of 
brushes, hence gives no additional feature beyond those pro- 
duced by a shift of the commutator bru«he*. 


320 ELECTRICAL APPARATUS 

Without losing in generality of the problem, we can, therefore, 
assume the stator e.m.fs. in phase with the rotor e.m.fs., and the 
polyphase shunt motor can thus be represented diagrammatically 
by Fig. 152. 

182. Let, in the polyphase shunt motor, shown two-phase in 
diagram, Fig. 152: 

#o and — j#o, /o and — j/o, Z0 = impressed e.m.fs., currents 
and self-inductive impedance respectively of the stator circuits, 

c$o and — jc#0, /i and — j/i, Z\ = impressed e.m.fs., currents 
and self-inductive impedance respectively of the rotor circuits, 
reduced to the stator circuits by the ratio of effective turns, c, 

Z = mutual-inductive impedance, 

S = speed; hence s = 1 — S = slip, 

0 = position angle between stator and rotor circuits, or 
"brush angle." 

It is then : 
Stator: 

#o = Zo/o + Z(h- /i cos 0 - jh sin 0). (72) 

Rotor: 

cGo = ZJX + Z (/i - /o cos 6 + jlo sin 6) - 

jSZ ( - j/i + h sin 6 + j[o cos 0). (73) 
Substituting: 

a = cos 6 — j sin 0, 
b = cos 0 + j sin 0, 
it is: 

ad = 1, (75) 

and : 

£o = Zo/o + Z (/o - 5/0, (76) 

c#o = Z,U + Z(fx- alo) + jSZ 07i " Wo) 

= Z,/i + sZ (/i - cr/o). (77) 

Herefrom follows: 

(« + «c) Z + Z, 

' ° _ *"> 7zz»~+ zzT+ ZoZi (78) 

' ' " *0 izz7+"zzr+~z^' ( ' 9) 

for c = o, this gives: 

, _ „ *Z_+ Z\ 

/0 " *" sZZ* + ZZ\ + ZoZi 

j - v sZ ' 

*l-'r«0szz0-+zzl + z0zi' 


(74) 


ALTERNATING-CURRENT MOTORS 321 

that is, the polyphase induction-motor equations, a = cos 0 + 

j sin 0 = 1» representing the displacement of position between 
stator and rotor currents. 

This shows the polyphase induction motor as a special case of 
the polyphase shunt motor, for c = o. 

The e.m.fs. of rotation are: 

£'i = -jSZ (- jh + h sin 0 + j/o cos 0) 

- SZ (*h- I i)i 
hence : 

&l ^'iZZl+ZZx + ZtZt' (80) 

The power output of the motor is: 
P - [£., 7.1 

= m. + zzl + za# l(*Zl " cZo) z> (ff8 + c) z + cZ°]> (81) 

which, suppressing terms of secondary order, gives: 

p _ <Se0V { g(r i + c (x0 sin 0 — r0 cos 0) ) + c (r i cos 0 + xt sin 0 — cr0) } 

~ [sZZo + ZZi + ZoZfr ' 

(82) 

for Sc = o, this gives: 

p Seohhri 

[sZZo + ZZ! + ZoZJ'1' 

the same value as for the polyphase induction motor. 

In general, the power output, as given by equation (82), be- 
comes zero: 

p = °' 

for the slip: 

rx cos 0 + Xi sin 0 - cr0 ,QON 

fi + c (x0 sin 0 — r0 cos 0) 

183. It follows herefrom, that the speed of the polyphase 
shunt motor is limited to a definite value, just as that of a direct- 
current shunt motor, or alternating-current induction motor. 
In other words, the polyphase shunt motor is a constant-speed 
motor, approaching with decreasing load, and reaching at no- 
load a definite speed : 

So = 1 - so. (84) 

The no-load speed, S0, of the polyphase shunt motor is, how- 
ever, in general not synchronous speed, as that of the induction 

21 


322 ELECTRICAL APPARATUS 

motor, but depends upon the brush angle, 0, and the ratio, c, of 
rotor -J- stator impressed voltage. 

At this no-load speed, So, the armature current, i\, of the 
polyphase shunt motor is in general not equal to zero, as it is 
in the polyphase induction motor. 

Two cases are therefore of special interest: , 

1. Armature current, I\ = o, at no-load, that is, at slip, *o. 

2. No-load speed equals synchronism, s0 = o 
1. The armature or rotor current (79): 

T _ F vsZ + c(Z + Zi) 
41 ** sZZo + ZZi + ZoZi 
becomes zero, if: 

Z 

c = — as 


Z + Zx 

or, since Zx is small compared with Z, approximately: 

c = — as = — s (cos 0 — j sin 6); 

hence, resolved: 

c = — s cos 6, 
o = s sin 6; 
hence: 

.' : -.. ) « 

That is, the rotor current can become zero only if the brushes 
are set in line with the stator circuit or without shift, and in this 
case the rotor current, and therewith the output of the motor, 
becomes zero at the slip, s = — c. 

Hence such a motor gives a characteristic curve very similar 
to that of the polyphase induction motor, except that the stator 
tends not toward synchronism but toward a definite speed equal 
to (1 + c) times synchronism. 

The speed of such a polyphase motor with commutator can, 
therefore, be varied from synchronism by the insertion of an 
e.m.f. in the rotor circuit, and the percentage of variation is the 
same as the ratio of the impressed rotor e.m.f. to the impressed 
stator e.m.f. A rotor e.m.f., in opposition to the stator e.m.f. 
reduces, in phase with the stator e.m.f., increases the free-run- 
ning speed of the motor. In the former case the rotor impressed 
e.m.f. is in opposition to the rotor current, that is, the rotor 
returns power to the system in the proportion in which the speed 


ALTERNATING-CURRENT MOTORS 323 

is reduced, and the speed variation, therefore, occurs without 
loss of efficiency, and is similar in its character to the speed con- 
trol of a direct-current shunt motor by varying the ratio between 
the e.m.f . impressed upon the armature and that impressed upon 
the field. 

Substituting in the equations: 


6 = 0, 

8 + C = S\ 


(86) 


it is: 


h ~ V° sZZQ + ZZl + ZoZ/ (87) 

h = ®» sZZo + ZZX + ZoZ/ (88) 

p _ Se<?z*8i (ri - cr0) ( 

r [sZZo + ZZX + ZoZi]2' K ] 

These equations of 70 and I\ are the same as the polyphase 
induction-motor equations, except that the slip from synchron- 
ism, s, of the induction motor, is, in the numerator, replaced by 
the slip from the no-load speed, «i. 

Insertion of voltages into the armature of an induction motor 
in phase with the primary impressed voltages, and by a com- 
mutator, so gives a speed control of the induction motor without 
sacrifice of efficiency, with a sacrifice, however, of the power- 
factor, as can be shown from equation (87). 

184. 2. The no-load speed of the polyphase shunt motor is in 
synchronism, that is, the no-load slip, s0 = o, or the motor out- 
put becomes zero at synchronism, just as the ordinary induction 
motor, if, in equation (83) : 

t\ cos 6 + Xi sin 6 — cr0 = o; 
hence : 

e = n«" !±* "Lf; (90) 

or, substituting: 

= tan «i, (91) 

* where ai is the phase angle of the rotor impedance, it is: 

c = -1 cos (c*i — 0), 
r0 " 


324 ELECTRICAL APPARATUS 

or: 

cos (ttl - 6) = - c, (92) 

or: 

c = !i_^_(«j_^). (93) 

To 

Since r0 is usually very much smaller than zX) if c is not very 
large, it is: 

cos (c*i — 6) = o; 
hence : 

0 = 90° - «i. (94) 

That is, if the brush angle, 0, is complementary to the phase 
angle of the self-inductive rotor impedance, a\, the motor tends 
toward approximate synchronism at no-load. 

Hence: 

At given brush angle, 0, a value of secondary impressed e.m.f., 
c#o, exists, which makes the motor tend to synchronize at no- 
load (93), and, 

At given rotor-impressed e.m.f., c#0, a brush angle, 0, exists, 
which makes the motor synchronize at no-load (92). 

185. 3. In the general equations of the polyphase shunt motor, 
the stator current, equation (78) : 

sZ + Zi +^bcZ 
/o " *° sZZo + ZZ\ + ZoZi 

can be resolved into a component: 

/"° = ^ IZZ* + ZZ, + ZoZy (95) 

which does not contain c, and is the same value as the primary 
current of the polyphase induction motor, and a component: 

r° = $* 'aZzT+zzx'+ZoZ'i (96) 

Resolving /"o, it assumes the form: 

/"o«£o*c(j1, -jA2) 

= c { A\ cos 0 + A2 sin 0) + j (Ai sin 0 — A2 cos 0) }. (97) 

This second component of primary current, 7"0, which is pro- 
duced by the insertion of the voltage, c#, into the secondary cir- 
cuit, so contains a power component: 

z'o = c (Ai cos 0 + A2 sin 0), (98) 


ALTERNATING-CURRENT MOTORS 325 

and a wattless or reactive component: 

t"o = +jc (A i sin 0 - A 2 cos 0); (99) 

where : 

/"o = t'o - j*"o. (100) 

The reactive component, i"o, is zero, if: 

Ai sin 0 — At cos 0 = o; (101) 


hence: 


tan 0! = + ^2- (102) 


In this case, that is, with brush angle, 0i, the secondary im- 
pressed voltage, c#, does not change the reactive current, but 
adds or subtracts, depending on the sign of c, energy, and so 
raises or lowers the speed of the motor: case (1). 

The power component, t'o, is zero, if: 

Ax cos 0 + A2 sin 0 = o, (103) 

hence: 

tan 02 = - 41' (104) 

In this case, that is, with brush angle, 02, the secondary im- 
pressed voltage, cE, does not change power or speed, but pro- 
duces wattless lagging or leading current. That is, with the 
brush position, 02, the polyphase shunt motor can be made to 
produce lagging or leading currents, by varying the voltage im- 
pressed upon the secondary, c$, just* as a synchronous motor 
can be made to produce lagging or leading currents by varying 
its field excitation, and plotting the stator current, /o, of such a 
polyphase shunt motor, gives the same V-shaped phase charac- 
teristics as known for the synchronous motor. 

These two phase angles or brush positions, 0i and 02, are in 
quadrature with each other. 

There result then two distinct phenomena from the insertion 
of a voltage by commutator, into an induction-motor armature : 
a change of speed, in the brush position, 0i, and a change of phase 
angle, in the brush position, 02, at right angles to 6\. 

For any intermediate brush position, 0, a change of speed so 
results corresponding to a voltage: 

c$ cos (0i - 0) ; 


326 ELECTRICAL APPARATUS 

and a change of phase angle corresponding to a voltage: 

c$ cos (02 - 0), 
= c& sin (0i - 0), 

and by choosing then such a position, 0, that the wattless current 
produced by the component in phase with 0*, is equal and op- 
posite to the wattless lagging current of the motor proper, /'o, 
the polyphase shunt motor can be made to operate at unity 
power-factor at all speeds (except very low speeds) and loads. 
This, however, requires shifting the brushes with every change 
of load or speed. 

When using the polyphase shunt motor as generator of watt- 
less current, that is, at no-load and with brush position, 02, it is: 

s = 0; 
hence, from (78) : 

'• - e-zrtrxr (105) 

''• - zfz. <l06> 


or, approximately: 

7'o = 


Eq 


z 

that is, primary exciting current: 


(108) 


'"• - * znrho)' (107) 

or, approximately, neglecting Z0 against Z: 

„ Eo&c 
1 ° - "Z'x 

_ EpC (cos 0 + j sin 0) 

~" ri + jxi 

= -° ■ { (ri cos 0 + Xi sin 0) — j (Xi cos 0 — n sin 0) ) , 

Zl 

and, since the power component vanishes: 

rx cos 0 + X\ sin 0 = 0, 
or: 

tan 02 = - r-- (109) 

X\ 


ALTERNATING-CURRENT MOTORS 

Substituting (109) in (108) gives: 

/"o = ■-!- (^1 cos 02 — t\ sin 02) 


327 


= -J 


m 

. EoC, 


Zl 


(HO) 


and: 


T Eo . EoC 

I'- T~' I,' 


- * (i - 1 c - f.) ) 


(111) 


186. In the exact predetermination of the characteristics of 
such a motor, the effect of the short-circuit current under the 
brushes has to be taken into consideration, however. When a 
commutator is used, by the passage of the brushes from segment 
to segment coils are short-circuited. Therefore, in addition to 
the circuits considered above, a closed circuit on the rotor has 
to be introduced in the equations for every set of brushes. Re- 
duced to the stator circuit by the ratio of turns, the self-inductive 
impedance of the short-circuit under the brushes is very high, 
the current, therefore, small, but still sufficient to noticeably af- 
fect the motor characteristics, at least at certain speeds. Since, 
however, this phenomenon will be considered in the chapters on 
the single-phase motors, it may be omitted here. 

4. POLYPHASE SERIES MOTOR 

187. If in a polyphase commutator motor the rotor circuits 
are connected in series to the stator circuits, entirely different 


Fig. 153. 

characteristics result, and the motor no more tends to synchronize 
nor approaches a definite speed at no-load, as a shunt motor, but 
with decreasing load the speed increases indefinitely. In short, 


328 ELECTRICAL APPARATUS 

the motor has similar characteristics as the direct-current series 
motor. 

In this case we may assume the stator reduced to the rotor by 
the ratio of effective turns. 

Let then, in the motor shown diagrammatically in Fig. 153: 

#o and —j$o, lo and — j/o, Z0 = impressed e.m.fs., currents 
and self-inductive impedance of stator circuits, assumed as two- 
phase, and reduced to the rotor circuits by the ratio of effective 
turns, c, 

#i and — j$\, /i, and — jfi, Z\ = impressed e.m.fs. currents 
and self-inductive impedance of rotor circuits, 

Z = mutual-inductance impedance, 

5 = speed; and, s = 1 — S = slip, 

6 = brush angle, 

c = ratio of effective stator turns to rotor turns. 

If, then : 
P and — j$ = impressed e.m.fs., / and — jj = currents of 
motor, it is: 

/i = /, (112) 

h = c/, (113) 

c#o + #i = E; (114) 

and, stator, by equation (7) : 

#o = Zoh + Z(f0 - h cos 6 - jlx sin 6); (115) 

rotor: 

#, = Zi/i + Z (A - U cos 6 + jfo sin 6) - jSZ (- jfl + /0 

sin0 + j/ocos0); (116) 

and, e.m.f. of rotation: 

Q\ = - jSZ (- jfi + /o sin 6 + jfi cos 0). (117) 

Substituting (112), (113) in (115), (116), (117), and (115), (116) 
in (114) gives: 

(c2Zo + Z,j + Z(1 + c2-2ccos0) + SZ(cc - i)'vllo; 

where : 

a = cos0 - jsin 0, (119) 

and: 

-, _ _ SZE(ce-l) . 

* ' " (c*Z0~+ Zx) = Z (1 + c* - 2 c cos 0) + SZ (c<r - 1)] ' 

(120) 


ALTERNATING-CURRENT MOTORS 


329 


and the power output: 

P = U?\ hY 

Se2 { c (r cos 0 + x sin 0) — r\ 


[{c*Z0 + Zi) + Z (1 + c2 - 2 c cos 0) + SZ (or - 1)]* 

(121) 

The characteristics of this motor entirely vary with a change 

Se2r(x— 1) 
of the brush angle, 0. It is, for 0 = 0: P = — rj^i > hence 


-TW_ 

TMl 

110 S* 

x. 

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R 

jr\. 

RRA 

no 

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\ 
\ In 

1)0 

" y^' 

OW 

-460— 

too 

X) 

JO 

a 

-' 

<o 

^lK 

-360— 
-300— 

ro 

/ 

^r « 

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* ^ 

JO 

f 

— ^- 

\ 1 * 

so 

/ 

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i\ 1 

3EBW 

AAA 

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100 

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D&P 
& 

240 %: 
220 110 

200 100 

180 90 

100 80 

140 70 

120 60 

100 60 

80 40 

60 80 

40 20 

20 10 


.2 .4 .6 


.8 1.0 1.2 1.4 

Fig. 154. 


1.6 


1.8 2.0 


Ss2(xc — * t) 
very small, while for 0 = 90°: P = r~p — , hence consider- 
able. Some brush angles give positive P: motor, others negative, 
P, generator. 

In such a motor, by choosing 0 and c appropriately, unity 
power-factor or leading current as well as lagging current can be 
produced. 

That is, by varying c and 0, the power output and therefore 
the speed, as well, as the phase angle of the supply current or 
the power-factor can be varied, and the machine used to produce 
lagging as well as leading current, similarly as the polyphase 
shunt motor or the synchronous motor. Or, the motor can be 
operated at constant unity power-factor at all loads and speeds 
(except very low speeds), but in this case requires changing the 


330 ELECTRICAL APPARATUS 

brush angle, 0, and the ratio, c, with the change of load and speed. 
Such a change of the ratio, c, of rotor -f- stator turns can be pro- 
duced by feeding the rotor (or stator) through a transformer of 
variable ratio of transformation, connected with its primary cir- 
cuit in series to the stator (or. rotor). 

188. As example is shown in Fig. 154, with the speed as 
abscissae, and values from standstill to over double synchronous 
speed, the characteristic curves of a polyphase series motor of 
the constants: 

e = 640 volts, 
Z = 1 + 10 j ohms, 
Z0 = Zx = 0.1 + 0.3 j ohms, 

c = l, 

$ = 370; (ain 6 = 0.6; cos 6 = 0.8); 

hence: 

. 640 

(0.6 + 5.8 S) + j (4.6 - 2.6 S) amp'' 

P = 4_673 S kw 

(0.6 + 5.8 S)2 + (4.6 - 2.6 S)* 

As seen, the motor characteristics are similar to those of the 
direct-current series motor: very high torque in starting and at 
low speed, and a speed which increases indefinitely with the de- 
crease of load. That is, the curves are entirely different from 
those of the induction motors shown in the preceding. The 
power-factor is very high, much higher than in induction motors, 
and becomes unity at the speed S = 1.77, or about one and three- 
quarter synchronous speed.