Theory and Calculation of Electrical Apparatus (1917) — part 14 of 21

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

Zi = n^'jsx!. (29)

The power output of the motor, per circuit, is

P = [F, h]

e J ^ lH ± il (r , _ jsXl)

[ZZoS + ZZi + ZoZtf {1 J l> ’

(30)

where the brackets [ ] denote the absolute value of the term in- cluded by it, and the small letters, e 0 , z, etc., the absolute values of the vectors, E 0 , Z, etc.

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

Mechanical power output:

P = eoVs (1 - s) r x

[ZZ 0 s + ZZ X + ZoZi] 2 '

This is the power output at the armature conductors, hence in- cludes friction and windage.

The torque of the motor is :

D

1 — s

e Q 2 z 2 riS

[ZZoS + ZZi + ZoZi] 2 ^ [ZZqs + ZZ\ + ZoZi] 2

(32)

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

D =

eo 2 z 2 Tis

[ZZoS + zz x + z 0 zj 2 *

\

(33)

i

ALTERNATING-CURRENT MOTORS

311

The power input of the motor per circuit is: p 0 = Wo, /o]

. * r i Za + z 1 1

0 I/’ZZoS + ZZx + ZoZxJ

= P ' 0 - jPo j

where :

P'o = true power,

Po y = reactive or “ wattless power / 7 Q — VPV + Po j2 = volt-ampere input.

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

P = IE', 1 1]

-hs

• e ' ,s D-gJ

e 2 &Sri . e 2 s 2 Sx i

• 1 2 3 * 1 2

^

Power input: Po = [Po, /o]

9 [ZZos + ZZi + ZoZi Zs -j- Zi~|

~ 6 L ZZi ’ ZZx J

9 r Zo (Zs + Zi) . ^ Z$ + Zf|

• e |_- " I- •*> 2Z X J

■ *’ Pz 2 ‘“] I l 2 " 1 1 + m ‘ r , + i ] 1

• «* t^w-T ( i z - « tS |Z - 11 + $ lz ■ 1] 1

= e 2 I (r ° ~ 3Xo) + z? ( ‘ ri ~i SXl ^ + P { - r ~ JX) ]

= io 2 (ro — jzo) + t'i 2 0 - jxi'j + too 2 (r — jx). (36)

And since:

ri S + s 'Sri

• = n = — + r 1,

s s s

312

ELECTRICAL APPARATUS

and :

ipSri s ’

it is :

P o — (fo 2 ro + ij 2 ri ioo 2 r + P) — j {io 2 Xo + i\ 2 X\ + ioo 2 x). (37) Where:

io 2 r 0 = primary resistance loss, ii 2 ri = secondary resistance loss, im 2 r — core loss (and ecldy-current loss), P — output,

io 2 x 0 = primary reactive volt-amperes, ii 2 Xi = secondary reactive volt-amperes, ioo 2 x = magnetizing volt-amperes.

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

(38)

Where X 0 and \i are small quantities, and X = \o + \i is the “characteristic constant” of the induction motor theory, it is: Primary current:

E o

s + Xi

E o s + Xi

Z s\q + Xi + XoXi Z s \ o ~h X |

Secondary current:

Eq

h

Exciting current:

E o

Z sX o + Xi + XoXi Z s\ o “t~ Xx

7oo = -

E o

Xi

E o Xi

Z «s*Xo 4“ Xi -j- XoXi Z s \ o -j- Xx E.rn.f. of rotation:

W =

Counter e.rn.f. :

= P O .

sX o + Xi + XoXi * 0 sXo 4" Xi

$ = T-rTrn- = #o.

sXo + Xi XoXi sXo + Xi

(39)

(40)

(41)

(42)

(43)

ALTERNATING-CURRENT MOTORS 323

JlLt*, °heTrvi 7: h :rr Ks - w9 - ™ th «*«>

constants: a P ° lyphase Auction motor of the

eo = 320 volts,

% = 1 -f- 10 j ohms,

hence: - Z x = 0.1 + 0.3 j ohms;

X 0 = X x = 0.0307 - 0.0069 j.

POLYPHASE INDUCTION MOTOR 320 VOLTS

500—100

450— 90 l 400— 80 i~j 350— 70 — 300— CO -1-1U250- 50

V J — 150— 30 — V-l — 100 — 20

°.l 0.2 0.3

U = 320jl°.30 s - ( s + o.l) j}

(1-03 -f 1.63 s) - j (o.lf - 5.99 s) amp '

D = 2048 (1 — s)

(1.03 -f- 1.63 s) 2 + (0.11 -~ 5 .W^yt synchronous kw.

p = (1 - s) D

tan 6" = — ~ 5.99 s

1.03 + 1.63 ^

tan 6'= i± °_: 1 .

10.3 s ’

• _ cos ( 0 ' ~ n = power-factor.

the power fulp^ “ n 7^ the curr ®‘- ! '

effioiency, , ““ D ’ “» P^er-factor, p, the

Fig. 149.

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-factor 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

Ii

Fig. 150. — Single-phase induction motor.

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:

E 0 , Jo, Z o = impressed e.m.f., current and self-inductive im- pedance, respectively, of the primary circuit,

J 1 , Zi = current and self-inductive impedance, respectively, of the secondary energy circuit,

/ 2 , Z i = current and self-inductive impedance, respectively, of the secondary magnetizing circuit,

Z = mutual-inductive impedance,

S = speed,

and let s 0 = 1 — S 2 (where s 0 is not the slip).

It is then, by equation (7) :

ALTERNATING-CURRENT MOTORS Primary circuit:

Eq = ZqIq + Z (/o — /l).

Secondary energy circuit:

0 = Ziji + Z (/i — /o) — jSZI 2 . Secondary magnetizing circuit:

0 = ZJ 2 + ZI 2 ~jSZ C/o - /i); hence, from (45) and (46) :

r __ r Z (ZSp + Z j)

• 1 i0 Z 2 s o + 2 ZZ,. + Zi 2 ’

jr _ i -or ZZl

U + J'S/o Z2so + 2 ZZl + Zi 2 ’ and, substituted in (44) :

Primary current:

r _ tp Z*s 0 + 2ZZ 1 + Z 1 *

IQ = lio *

Secondary energy current:

T T? ( ZSo + Zi)

/l = -&0 ^

Secondary magnetizing current:

T | -qTTT ^1

h r JoAo

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

7 7

#1 = - jSZI 2 = •

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

E\ = - jSZ (/„ -/:) = - jSfo ^

where :

JC = Z 0 (Z 2 5 0 + 2ZZx + Zi 2 ) + ZZi (Z + Z x ). It is, at synchronism, S = 1, $o = 0:

r jjt _ _ 2 Z + Zi

• 0 Z 0 (2Z + Zi) + Z (Z + ZO’

^ = Z 0 (2 Z + ZO + Z(Z + ZO’

/ 2 = + jBo 2o (2 Z + Zi) + Z (Z + Zi)"

315

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

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, s Q = 1, it is:

T — 7? Z + Z i _ .

/0 *° ZZq -f- ZZi + ZoZi’

T __ T 7

• 1 ZZ 0 + ZZ 1 + Z 0 Z7

h = o.

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 :

E = Eq — Zoh,

and substituting for Z 0 from (49), it is:

Primary impressed e.m.f.:

„ ^Zo (Zs 0 + 2 ZZ + Z x 2 ) + ZZ X (Z + Zi) • 0 * ~ ZZi (Z + Zi)

Primary current:

Jo = E circ Ji = e Tj

Z 2 sq 2 ZZi -J“ Zi 2

ZZi (Z + Zi)

Secondary energy circuit:

Zs 0 -f* Zi

s 0 E_ ^ Z 1 (Z + Zi) Zx ^ Z + Zi

*'• - wrhr

Secondary magnetizing circuit:

h = + E\ = jSE.

T _ , • M

j z +

And:

h~h =

E

(55)

(56)

(57)

(58)

(59)

(60)

(61)

These equations differ from the equations of the polyphase induction motor by containing the term s 0 = (1 — S 2 ), instead

SE

of s = (1 — S ), and by the appearance of the terms, 7,-.- „ and . Z ~r Zi

S 2 jE7

^ fre 9 uenc y (1 + &), in the secondary circuit.

ALTERNATING-CURRENT MOTORS

317

The power output of the motor is :

P = [E h h\ + [# 2 , h)

S*e n z

= -jgp{[ZZx, Zso + Z l ] - [Zx (Z + Zx), Zx]}

_ S 2 eo 2 z 2 r x (soz 2 — z x 2 )

m -

and the torque, in synchronous watts:

n _ P _ St^z 2 r x ( s 0 z 2 — Zi 2 )

S [if] 2

(62)

(63)

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 :

hence:

(64)

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:

j _ p Zsq + 2 Zi

Z (ZqSo + Z i) -f- 2 ZqZi

(65)

j n Zsq + Z i

41 Z {ZoSo + Z x ) + 2 ZoZi

(66)

h- + jSE 0 z ^ oSo + + 2 Z 0 Zi

(67)

77

^ = ^1(2030 + 5 + 2^

(68)

77 • Qf 77 ZZl

• 2 ~ 3 -° Z (ZoSo + Zl) + 2 ZoZi

(69)

p £ 2 eoVriSo

lZ(Z oSo + Zi) +2Z 0 Zy

(70)

„ Sea 2 z 2 riSa

~ [Z (ZoSo + Zx) + 2 ZoZJ 2

(71)

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.

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

e 0 = 400 volts,

Z = 1 + 10 j ohms,

and:

hence :

Zq = Z i = 0.1 + 0.3 j ohms;

N

I o = 400 -g? amp.;

N = (so + 0.2) + j (10 s 0 + 0.6 - 0.6 S);

K = (0.1 + 0.3 j) AH- (1 + 10 j) (0.1 +j) (0.3-0.3 5);

r\ 1616 Ssq . .

D synchronous lew.

ALTERNATING-CURRENT MOTORS

319

Fig. 151 gives, with the speed, S, as abscissae: the current, I 0 , the power output, P, the torque, D , the power-factor, p, the efficiency, rj.

3. POLYPHASE SHUNT MOTOR

4. 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

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

Let Eo and — jEo = e.m.fs. impressed upon the stator, Ei and — jEi = e.m.fs. impressed upon the rotor, 6 0 = phase angle be- tween e.m.f., Eq and Eh and 0i = position angle between the stator and rotor circuits. The e.m.fs., Eo and —jE 0 , produce the same rotating e.m.f. as two e.m.fs. of equal intensity, but dis- placed in phase and in position by angle 0o from E 0 and jE o, and instead of considering a displacement of phase, do, and a dis- placement of position, 6i, between stator and rotor circuits, we can, therefore, assume zero-phase displacement and displacement in position by angle Bo + d± = Q. Phase displacement between stator and rotor e.m.fs. is, therefore, equivalent to a shift of brushes, hence gives no additional feature beyond those pro- duced by a shift of the commutator brushes.

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:

E o and — jE 0} I 0 and —jl 0 , Z 0 = impressed e.m.fs., currents and self-inductive impedance respectively of the stator circuits, cE o and —jcE 0 , /i and —jh, Z x = 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 :

E () = Zolo + Z(/o — 1 1 cos 6 - jl ! sin 6). (72)

Rotor:

cEq — Zih -f- Z (I i — I o cos 6 A" jl o sin 6) —

jSZ ( — jh + lo sin 6 A jh cos 0)

Substituting:

a = cos 0 — j sin 0 ,

8 = cos 6 + j sin 0,

it is:

a 8 = 1 ,

and:

Eo = Zo/o + Z (/o — 5/i), cE 0 = ZJx + Z (Jx - cr/o) + jSZ 07 1 - ier/o)

= Zi/i + sZ (/i — cr/o).

Herefrom follows:

r _ rr (5 + 5c) Z + Zi *° ~ ‘ °sZZo + ZZ X + ZoZ/ r _ rr (crS + c) Z + cZ\

41 " ‘ °sZZ 0 ff- ZZi + Z0Z1 for c = 0 , this gives: •

T p gZ + Z 1

•° * 0 sZZ 0 + ZZi + Z0Z1

^ = <r ^°sZ^FZZ 1 + ZoZi

(73)

(74)

(75)

(76)

(77)

(78)

(79)

ALTERNA TING-C URRENT MOTORS 321

that is, the polyphase induction-motor equations, <r = cos S + L

j sin 0 = l r 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 :

W 1 = —jSZ (— jli + h sin 6 + jl 0 cos 0 ) = SZ (cr/o — /i) ;

hence :

jpr arp Z (<rZ i — cZo)

^ 1 • 0 sZZ 0 + ZZi + Z-oZV

The power output of the motor is:

(80)

P = [JSi, U

• 5zz; + zfftw K " z ' - tZ>) z - ( " + c) z + eZ - i; (8,)

which, suppressing terms of secondary order, gives: p _ Se o 2 3 2 {$(ri+c(a;osin0— r 0 cos 0))+c(riCOS 0+#i sin 0~-cr u )}

[sZZo + ZZi + Z(iZi] * ~ *

(82)

for 5c = o, this gives:

p _ Seo 2 z 2 sri

[sZZ 0 + ZZ X + ZoZJ 2 ’

the same value as for the polyphase induction motor.

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

for the slip

So

P = o,

ri cos 0 + xi sin 0 — ctq Ti + c (x 0 sin 0 — ro cos 0)

(83)

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 s 0 . (84)

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

322

ELECTRICAL APPARATUS

motor, but depends upon the brush angle, 6 , and the ratio, c, of rotor -r- stator impressed voltage.

At this no-load speed, S 0 , the armature current, h, 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, h = o, at no-load, that is, at slip, s 0 .

2. No-load speed equals synchronism, s 0 = o..

3. The armature or rotor current (79):

~r Tj 7 cr$Z + C [Z + Zi)

• 1 *° sZZo + ZZ X + ZoZi

becomes zero, if:

Z

c = - ffS zn;

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

c = —as = —s (cos 6 — j sin 6) ;

hence, resolved:

c = — $ cos 6, o = s sin 6;

hence:

c = —s.

(85)

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, 1

S + c — Si j

(86)

J et SlZ + Zy

U ■ 0 sZZ 0 4- ZZy 4- ZoZ/

(87)

J v SlZ

• 1 • 0 sZZo 4- ZZy + Z 0 Zi’

(88)

p _ /Se 0 2 z 2 si (ri - cr 0 )

[sZZo -j- ZZ\ -f- ZqZi] 2

(89)

These equations of J 0 and h 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, $ 1 .

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, $ 0 = o, or the motor out- put becomes zero at synchronism, just as the ordinary induction motor, if, in equation (83) :

hence:

or. substituting:

7*1 cos 6 + xi sin 6 — cr 0 = o;

c =

ri cos 8 + x i sin 0 .

7*0

XjL

n

— tan o£i,

(90)

(91)

where «i is the phase angle of the rotor impedance, it is:

c = — cos (ai — 9), r o

324

ELECTRICAL APPARATUS

COS (oil

= pc,

Z 1

2l COS (ai — 0)

Since r 0 is usually very much smaller than z i, if c is not very large, it is:

cos (an — 0) = o;

nence:

0 = 90° - «!. (94)

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

Hence:

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

At given rotor-impressed e.m.f., cEq , 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 + Z\ + 5cZ

sZZo + ZZi + ZoZi ■

can be resolved into a component:

= sZZq + ZZi + ZoZi (95)

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

= sZZ 0 + ZZt +~ZoZi Resolving /" o, it assumes the form :

I" o = Eo8c (A i — • jA<I)

= c { Ai cos 9 + A 2 sin 0) + j (Ai sin 0 — A 2 cos 0) }. (97)

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

i ' o = c (A i cos 0 + A% sin 0), (98)

ALTERNATING-CURRENT MOTORS

325

and a wattless or reactive component:

i” o = +jc (Ai sin 0 — A 2 cos 6); (99)

where :

r 0 = i'o - ji" 0. (100)

The reactive component, i" 0 , is zero, if :

A i sin 6 — A 2 cos 0 = o ;

hence:

Ao

tan 0 X = +

Ai

In this case, that is, with brush angle, 0i, the secondary im- pressed voltage, ci7, 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, i' 0 , is zero, if:

Ai cos 0 + A 2 sin 0 = o, (103)

hence :

tan 0 2 = - (104)

■^2

In this case, that is, with brush angle, 0 2 , the secondary im- pressed voltage, cEj does not change power or speed, but pro- duces wattless lagging or leading current. That is, with the brush position, 0 2 , the polyphase shunt motor can be made to produce lagging or leading currents, by 'varying the voltage im- pressed upon the secondary, cljJ, just as a synchronous motor can be made to produce lagging or leading currents by varying its field excitation, and plotting the stator current, 1 0 , 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, 0 1 and 0 2 , 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, 6 h and a change of phase angle, in the brush position, 0 2 , at right angles to 0 X .

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

c$ cos (0i — 0) ;

( 101 )

( 102 )

326 ELECTRICAL APPARATUS

and a change of phase angle corresponding to a voltage.

cE cos (02 — 0)>

= cE sin (0i — 0),

and by choosing then such a position, 0, that the wattless current produced by the component in phase with 0 2 , is equal and op- posite to the wattless lagging current of the motor proper, I 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 o - less current, that is, at no-load and with brush position, 0 2 , it is:

hence, from (78) :

or, approximately:

s = 0;

1 0 = #0

ScZ + Zi

zrTzTzT

Z -h Za

that is, primary exciting current:

ScZ

z[ (z + z5’

or, approximately, neglecting Z o against Z .

T" — / o —

EoC (cos 0 +1 sin 6 1 , ' n + ]xi

^ { ( ri cos 6 + xi sin 6) - j (x x cos 0-n sin 0) },

and, since the power component vanishes:

n cos 0 + xi sin 0 = 0,

„ r x

tan 62 = — r"

or:

( 109 )

ALTERNATING-CURRENT MOTORS

327

Substituting (.109) in (108) gives:

and:

— (*i cos — n sin 0 2 )

.

■ E 0 c .

3 „ ’

2l

( 110 )

( 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

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 :

Eo and —jE 0 , Jo and —jjo, Z 0 = 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,

Ei and —jE u I h and — jji, Z i = impressed e.m.fs. currents and self-inductive impedance of rotor circuits,

Z = mutual-inductance impedance,

S = speed; and, s = 1 — S = slip,

0 — brush angle,

c = ratio of effective stator turns to rotor turns.

If, then:

E and —jE = impressed e.m.fs., J and —jj = currents of motor, it is:

Ji = /, (112)

Jo = cj, (113)

cEo + Ei = E) (114)

and, stator, by equation (7) :

Eo = ZoJo + Z{Iq — 1 1 cos 6 — jh sin 6)] (115)

rotor:'

Ei ~ Z\Ji + Z (/i — Jo cos 6 + jjo sin 6 ) — jSZ (— jji + Jo

sin 6 + ji 0 cos 6); (116)

and, e.m.f. of rotation:

E'i = - jSZ jji + Jo sin 6 + jh cos 6). (117)

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

E

1 = (c*Zo + Z{) + 2T(1 + c 2 - 2 c cos 6) + SZ(ca - 1) ’ (118) where :

a = cos 6 — j sin 6, (119)

and:

F , = SZE (ax — L)

' 1 (c 2 Z 0 + ZJ = Z (1 + c -2c cose) +SZ(ccr -1)]’

ALTERNATING-CURRENT MOTORS

329

and the power output:

P = Wi, h ]'

Se 2 { c (r cos 6 + # sin 0) — r}

[(c 2 Z 0 + ZO + Z (1 + c 2 - 2 c cos 6) + SZ ( co- - l)] 2

( 121 )

The characteristics of this motor entirely vary with a change

g e 2 r ( x — 1 )

of the brush angle, 6 . It is, for 6 = 0: P = > hence

very small, while for 6 — 90°: P =

Se*(xc — r)

[Kp *

hence consider-

able. Some brush angles give positive P: motor, others negative, P, generator.

In such a motor, by choosing 6 and c appropriately, unity power-factor or leading current as well as lagging current can be produced.

That is, by varying c and 6 , 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 6 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:

hence: #

I P

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.

e = 640 volts,

Z = 1 + 10 j ohms,

Zq = Z\ — 0.1 + 0.3 j ohms, c = 1,

6 = 37°; (sin 6 — 0.6; cos 6 = 0.8); 640

(0.6 + 5.8 S) + j (4.6 - 2.6 S) amp '’ 4673 S

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

CHAPTER XX

SINGLE-PHASE COMMUTATOR MOTORS I. General

189. Alternating-current commutating machines have so far become of industrial importance mainly as motors of the series or varying-speed type, for single-phase railroading, and as con- stant-speed motors or adjustable-speed motors, where efficient acceleration under heavy torque is necessary. As generators, they would be of advantage for the generation of very low fre- quency, since in this case synchronous machines are uneconom- ical, due to their very low speed, resultant from the low frequency.

The direction of rotation of a direct-current motor, whether shunt or series motor, remains the same at a reversal of the im- pressed e.m.f., as in this case the current in the armature circuit and the current in the field circuit and so the field magnetism both reverse. Theoretically, a direct-current motor therefore could be operated on an alternating impressed e.m.f. provided that the magnetic circuit of the motor is laminated, so as to fol- low the alternations of magnetism without serious loss of power, and that precautions are taken to have the field reverse simul- taneously with the armature. If the reversal of field magnetism should occur later than the reversal of armature current, during the time after the armature current has reversed, but before the field has reversed, the motor torque would be in opposite direc- tion and thus subtract; that is, the field magnetism of the alter- nating-current motor must be in phase with the armature cur- rent, or nearly so. This is inherently the case with the series type of motor, in which the same current traverses field coils and armature windings.

Since in the alternating-current transformer the primary and secondary currents and the primary voltage and the secondary voltage are proportional to each other, the different circuits of the alternating-current commutator motor may be connected with each other directly (in shunt or in series, according to the type of the motor) or inductively, with the interposition of a

331

332

ELECTRICAL APPARATUS

transformer, and for this purpose either a separate transformer may be. used or the transformer feature embodied in the motor, as in the so-called repulsion type of motors. This gives to the alternating-current commutator motor a far greater variety of connections than possessed by the direct-current motor.

While in its general principle of operation the alternating- current commutator motor is identical with the direct-current motor, in the relative proportioning of the parts a great differ- ence exists. In the direct-current motor, voltage is consumed by the counter e.m.f. of rotation, which represents the power output of the motor, and by the resistance, which represents the power loss. In addition thereto, in the alternating-current motor voltage is consumed by the inductance, which is wattless or reactive and therefore causes a lag of current behind the vol- tage, that is, a lowering of the power-factor. While in the direct- current motor good design requires the combination of a strong field and a relatively weak armature, so as to reduce the armature reaction on the field to a minimum, in the design of the alter- nating-current motor considerations of power-factor predominate ; that is, to secure low self-inductance and therewith a high power- factor, the combination of a strong armature and a weak field is required, and necessitates the use of methods to eliminate the harmful effects of high armature reaction.

As the varying-speed single-phase commutator motor has found an extensive use as railway motor, this type of motor will as an instance be treated in the following, and the other types discussed in the concluding paragraphs.

II. Power-factor

190. In the commutating machine the magnetic field flux gen- erates the e.m.f. in the revolving armature conductors, which gives the motor output; the armature reaction, that is, the mag- netic flux produced by the armature current, distorts and weakens the field, and requires a shifting of the brushes to avoid sparking due to the short-circuit current under the commutator brushes, and where the brushes can not be shifted, as in a reversible motor, this necessitates the use of a strong field and weak armature to keep down the magnetic flux at the brushes. In the alternating- current motor the magnetic field flux generates in the armature conductors by their rotation the e.m.f. which does the work of the motor, but, as the field flux is alternating, it also generates

SINGLE-PHASE COMMUTATOR MOTORS

838

in the field conductors an e.m.f. of self-inductance, which is not useful but wattless, and therefore harmful in lowering the ppwer- factor, hence must be kept as low as possible. 1

This e.m.f. of self-inductance of the field, e 0 , is proportional to the field strength, <$, to the number of field turns, n 0 , and to the frequency, /, of the impressed e.m.f.:

e 0 = 2 7r/n 0 $ 10~ 8 , (1)

while the useful e.m.f. generated by the field in the armature conductors, or “e.m.f. of rotation,” e, is proportional to the field strength, $>, to the number of armature turns, 7ii, and to the fre- quency of rotation of the armature, / 0 :

e - 2irf 0 n 1 $ 10~ 8 . (2)

This later e.m.f., e, is in phase with the magnetic flux, <3>, and so with the current, i, in the series motor, that is, is a power e.m.f., while the e.m.f. of self-inductance, e 0 , is wattless, or in quadrature with the current, and the angle of lag of the motor current thus is given by:

.tan 6 =

eo

e + ir

(3)

where ir = voltage consumed by the motor resistance. Or ap- proximately, since ir is small compared with e (except at very low speed) :

tan d — (4)

e

and, substituting herein (1) and (2):

tan 0 = > U °- (5)

/o n i

Small angle of lag and therewith good power-factor therefore require high values of f 0 and n\ and low values of / a*nd n 0 .

High fo requires high motor speeds and as large number of poles as possible. Low / means low impressed frequency; there- fore 25 cycles is generally the highest frequency considered for large commutating motors.

High Wi and low n 0 means high armature reaction and low field excitation, that is, just the opposite conditions from that required for good commutator-motor design.

Assuming synchronism, /o = /, as average Uiotor speed — 750 revolutions with a four-pole 25-cycle motor — an armature reac-

334

ELECTRICAL APPARATUS

tion, rti, equal to the field excitation, n 0 , would then give tan 6 = i ? $ = 45° ; or 70.7 per cent, power-factor; that is, with an armature reaction beyond the limits of good motor design, the

power-factor is still too low for use.

The armature, however, also has a self-inductance; that is, the magnetic flux produced by the armature cur- rent as shown diagrammatically in Fig. 155 generates a reactive e.m.f. in the armature conductors, which again lowers the power-factor. While this armature self-inductance is low with small number of armature turns, it becomes considerable when the num- ber of armature turns, n h is large P^ tr J bx i tlon of compared with the field turns, n 0 . ture reaction. Let (Ro = field reluctance, that

is, reluctance of the magnetic

field circuit, and (Ri ^ = the armature reluctance, that is,

ratio of reluctances of the armature and the field mag-

netic circuit; then, neglecting magnetic saturation, the field flux is*

the armature flux is:

$ _

(Hi i/Ro no "

and the e.m.f. of self-inductance of the armature circuit is: 6i = 2 7r/ni# 1 10“ 8

= 2tt/ ^g$10- 8 ; l r n 0

( 6 )

(7)

hence, the total e.m.f. of self-inductance of the motor, or wattless e.m.f., by (1) and (7) is:

ed + ei = 2ir/<£10~ 8

( 8 )

SINGLE-PHASE COMMUTATOR MOTORS 335

and the angle of lag, 9 , is given by:

eo + £i

tan 9 =

f n 0 2 + feni 2 .

(9)

/o

or, denoting the ratio of armature turns to field turns by

wi n 0

•Ui+H’ ■ (io)

and this is a minimum; that is, the power-factor a maximum, for:

d

<1 = tan 6

dq

{tan 9} = 0,

or:

90 = VT

(ii)

and the maximum power-factor of the motor is then given by:

tan do

_ / A

( 12 )

/ o •%/&

Therefore the greater b is the higher the power-factor that can be reached by proportioning field and armature so that

Ui 1

w 0 /6

Since b is the ratio of armature reluctance to field reluctance, good power-factor thus requires as high an armature reluctance and as low a field reluctance as possible; that is, as good a mag- netic field circuit and poor magnetic armature circuit as feasible. This leads to the use of the smallest air gaps between field and armature which are mechanically permissible. With an air gap of 0.10 to 0.15 in. as the smallest safe value in railway work, b can not well be made larger than about 4.

Assuming, then, 5 = 4, gives q = 2, that is, twice as many armature turns as field turns; n x = 2 n Q .

The angle of lag in this case is, by (12), at synchronism: / 0 = /,

tan 6q = 1,

giving a power-factor of 70.7 per cent.

It follows herefrom that it is not possible, with a mechanically

336

ELECTRICAL APPARATUS

safe construction, at 25 cycles to get a good power-factor at moderate speed, from a straight series motor, even if such a design as discussed above were not inoperative, due to excessive distortion and therefore destructive sparking.

Thus it becomes necessary in the single-phase commutator motor to reduce the magnetic flux of armature reaction, that is, increase the effective magnetic reluctance of the armature far beyond the value of the true magnetic reluctance. This is ac- complished by the compensating winding devised by Eickemeyer, by surrounding the armature with a stationary winding closely adjacent and parallel to the armature winding, and energized by a current in opposite direction to the armature current, and of the same m.m.f., that is, the same number of ampere-turns, as the armature winding.

Fig. 157. — Massed field winding and distributed compensating winding.

191 . Every single-phase commutator motor thus comprises a' field winding, F, an armature winding, A, and a compensating winding, C, usually located in the pole faces of the field,- as shown in Figs. 156 and 157. -

The compensating winding, C, is either connected in series (but in reversed direction) with the armature winding, and then has the same number of effective turns, or it is short-circuited upon itself, thus acting as a short-circuited secondary with the arma- ture winding as primary, or the compensating winding is ener- gized by the supply current, and the armature short-circuited as

SINGLE-PHASE COMMUTATOR MOTORS 337

secondary. The first case gives the conductively compensated series motor, the second case the inductively compensated series motor, the third ease the repulsion motor.

In the first case, by giving the compensating winding more turns than the armature, overcompensation, by giving it less turns, undercompensation, is produced. In the second case always complete (or practically complete) compensation results, irrespective of the number of turns of the winding, as primary and secondary currents of a transformer always are opposite in direction, and of the same m.m.f. (approximately), and in the third case a somewhat less complete compensation.

With a compensating winding, C, of equal and opposite m.m.f . to the armature winding, A, the resultant armature reaction is zero, and the field distortion, therefore, disappears; that is, the ratio of the armature turns to field turns has no direct effect on the commutation, but high armature turns and low field turns can be used. The armature self-inductance is reduced from that corresponding to the armature magnetic flux, $ 1} in Fig. 155 to that corresponding to the magnetic leakage flux, that is, the magnetic flux passing between armature turns and compensating turns, or the “ slot inductance,” which is small, especially if rela- tively shallow armature slots and compensating slots are used.