Theory and Calculation of Electrical Apparatus (1917) — part 17 of 21

However, while these commutated currents in their resultant

SINGLE-PHASE COMMUTATOR MOTORS 379

give the effect of the frequency of slip, they actually consist of sections of waves of full frequency, that is, meet the full station- ary impedance in the rotor secondary, and not the very much lower impedance of the low-frequency currents in the ordinary induction motor.

If, therefore, the brushes on the commutator are set so that the inserted voltage is in phase with the voltage generated in the secondary, the power-factor of the motor is very poor. Shifting the brushes, by a phase displacement between the generated and the inserted voltage, the secondary currents can be made to lead, and thereby compensate for the lag due to self-inductance and unity power-factor produced. This, however, is the case only at one definite load, and at all other loads either overcompensa- tion or undercompensation takes place, resulting in poor power- factor, either lagging or leading. Such a polyphase adjustable- speed motor thus requires shifting of the brushes with the load or other adjustment, to maintain reasonable power-factor, and for this reason has not been used.

(c) Power-factor compensation. The production of an alter- nating magnetic flux requires wattless or reactive volt-amperes, which are proportional to the frequency. Exciting an induction motor not by the stationary primary but by the revolving sec- ondary, which has the much lower frequency of slip, reduces the volt-amperes excitation in the proportion of full frequency to frequency of slip, that is, to practically nothing. This can be done by feeding the exciting current into the secondary by commuta- tor. If -the secondary contains no other winding but that con- nected to the commutator, the motor gives a poor power-factor. If, however, in addition to the exciting winding, fed by the com- mutator, a permanently short-circuited winding is used, as a squirrel-cage winding, the exciting impedance of the former is reduced to practically nothing by the short-circuit winding coin- cident with it, and so by overexcitation unity power-factor or even leading current can be produced. The presence of the short- circuited winding, however, excludes this method from speed control, and such a motor (Heyland motor) runs near synchron- ism just as the ordinary induction motor, differing merely by the power-factor. Regarding hereto see Chapter on “Induction Motors with Secondary Excitation.”

This method of excitation by feeding the alternating current through a commutator into the rotor has been used very success-'

380

ELECTRICAL APPARATUS

fully abroad in the so-called “ compensated repulsion motor” of Winter-Eichberg. This motor differs from the ordinary repul- sion motor merely by the field coil, F, in Fig. 183 being replaced by a set of exciting brushes, G, in Fig. 184, at right angles to the main brushes of the armature, that is, located so that the m.m.f. of the current between the brushes, G, magnetizes in the same

direction as the field coils, F, in Fig. 183. Usually the exciting brushes are supplied by a transformer or autotransformer, so as to vary the excitation and thereby the speed.

This arrangement then lowers the e.m.f. of self -inductance of field excitation of the motor from that corresponding to full fre-

Fig. 184. — Winter-Eichberg motor.

quency in the ordinary repulsion motor to that of the frequency of slip, hence to a negative value above synchronism; so that hereby a compensation for lagging current can be produced above synchronism, and unity power-factor or even leading currents produced.

381

SINGLE-PHASE COMMUTATOR MOTORS

211. Theoretical Investigation. — In its most general form, the single-phase commutator motor, as represented by Fig. 185, comprises: two armature or rotor circuits in quadrature with each other, the main, or energy , and the exciting circuit of the armature where such exists, which by a multisegmental commu- tator are connected to two sets of brushes in quadrature position with each other. These give rise to two short-circuits, also in quadrature position with each other and caused respectively by the main and by the exciting brushes. • Two stator circuits, the

l 2

field, or exciting, and the cross, or compensating circuit, also in quadrature with each other, and in line respectively with the exciting and the main armature circuit.

These circuits may be separate, or may be parts or components of the same circuit. They may be massed together in a single slot of the magnetic structure, or may be distributed over the whole periphery, as frequently done with the armature windings, and then as their effective number of turns must be considered their vector resultant, that is:

7 r

where n' = actual number of turns in series between the arma- ture brushes, and distributed over the whole periphery, that is, an arc of 180° electrical. Or the windings of the circuit may be distributed only over an arc of the periphery of angle, co, as frequently the case with the compensating winding distributed In the pole face of pole arc, co; or with fractional-pitch armature windings of pitch, oo. In this case, the effective number of turns

2 r . co n = -n sm - oo 2

is:

382

ELECTRICAL APPARATUS

where n f with a fractional-pitch armature winding is the number of series turns in the pitch angle, o, that is:

, n

n = — n,

• x

ft" being the number of turns in series between the brushes, since in the space (x — «) outside of the pitch angle the armature conductors neutralize each other, that is, conductors carrying current in opposite direction are superposed upon each other. See fractional-pitch windings, chapter "Commutating Machine," "Theoretical Elements of Electrical Engineering."

212 . Let :

E o, Jo, Z 0 = impressed voltage, current and self-inductive impedance of the magnetizing or exciter circuit of stator (field coils), reduced to the rotor energy circuit by the ratio of effective turns, Co,

Ely Ii, Zi = impressed voltage, current and self-inductive im- pedance of the rotor energy circuit (or circuit at right angles to Jo),

E 2 , 1 2, Z2 = impressed voltage, current and self-inductive im- pedance of the stator compensating circuit (or circuit parallel to Ji) reduced to the rotor circuit by the ratio of effective turns, c 2 .

Ez, Iz, Zi = impressed voltage, current and self-inductive im- pedance of the exciting circuit of the rotor, or circuit parallel to J 0 ,

J 4 , Z4 — current and self-inductive . impedance of the short- circuit under the brushes, Ji, reduced to the rotor circuit,

J 5 , Z 5 = current and self-inductive impedance of the short- circuit under the brushes, 1 3 , reduced to the rotor circuit,

Z = mutual impedance of field excitation, that is, in the direc- tion of Jo,. Js, J4,

Z r = mutual impedance of armature reaction, that is, in the direction of I 1} J 2 , Js-

Z T usually either equals Z, or is smaller than Z.

J 4 and 1 5 are very small, Z 4 and Z 5 very large quantities.

Let S = speed, as fraction of synchronism.

Using then the general equations 7 Chapter XIX, which apply to any alternating-current circuit revolving with speed, S, through a magnetic field energized by alternating-current circuits, gives for the six circuits of the general single-phase commutator motor the six equations:

SINGLE-PHASE COMMUTATOR MOTORS 383

Eo = ZqIq + Z (J o + J3 — A), (1)

Ei = Zih + Z ' (J 1 + J5 — h) — jSZ (J 0 + J3 — J4)/, (2)

E2 = ^2/2 + (J 2 — Ji — /s), (3)

= ^1/3 + Z (J3 + J 0 — J4) — jSZ (J 2 — Ji — J5), (4)

o = Z4J4 + Z (J4 — Jo — J3) — jSZ (Ji + J5 — J2), (5)

0 = ^5/5 + ^ (J5 + Ji — J2) — jSZ (J 0 + J3 — J4)* (6)

These six equations contain ten variables :

Joj Jl, J2, J 3j J4, J5, #0, J/l, E2, Ezj

and so leave four independent variables, that is, four conditions, which may be chosen.

Properly choosing these four conditions, and substituting them into the six equations (1) to (6), so determines all ten variables. That is, the equations of practically all single-phase commutator motors are contained as special cases in above equations, and derived therefrom, by substituting the four conditions, which characterize the motor.

Let then, in the following, the reduction factors to the arma- ture circuit, or the ratio of effective turns of a circuit, i, to the effective turns of the armature circuit, be represented by c*. That is,

number of effective turns of circuit, i

f*. — , z •

. number of effective turns of armature circuit ;

and if Ei, / t , Zi are voltage, current and impedance of circuit, i, reduced to the armature circuit, then the actual voltage, current and impedance of circuit, i , are:

cJSi, ”, Ci 2 Zi.

Ci

213 . The different forms of single-phase commutator motors, of series characteristic are, as shown diagrammatically in Fig. 186:

1. Series motor:

e = cojEJo + j??i; /o = C0J1; J2 = 0; J3 = 0.

2. Conductively compensated series motor (Eickemeyer motor) :

e = cqEo + $1 + c 2 # 2 ; Jo = C0I1] J 2 = C2I1] 1 3 — 0 .

3. Inductively compensated series motor (Eickemeyer motor) :

C = CqEq + j@i] $2 = 0 ; Jo = CqIi] J 3 = Q,

384

ELECTRICAL APPARATUS

4. Inverted repulsion motor, or series motor with secondary excitation :

e — Ei] cqEq -(- C2E2 ~ 0; C2/0 — C0I2] lz — 0.

5. Repulsion motor (Thomson motor) :

6 — CqEq * 4 " C2E2] Ei = 0 ] C2I0 ^ C0I2] Iz ~ 0 .

6. Repulsion motor with secondary excitation:

e — C2E2] CqEq -f* El — 0] Io = Coll] Iz — 0.

7. Series repulsion motor with secondary excitation :

61 = CoEq + Ei] 62 == E2] Jo = Coh] Jz ~ 0.

8. Series repulsion motor with primary excitation (Alexander- sen motor) :

61 = El] 62 — CqEq + C2$2] C2I0 = C0I2] Iz — 0 .

9. Compensated repulsion motor (Winter and Eichberg motor) :

e = C2EI2 + CzE'z] $1 = 0 ; Jo = 0 ; C3/2 = C2/3.

SINGLE-PHASE COMMUTATOR MOTORS 385

10. Rotor-excited series motor with conductive compensation:

e = E 1 + c 2 E 2 + c 3 #3; 1 2 = C 2 I 1 ; U = c 3 /i ; J 0 = 0.

11. Rotor-excited series motor with inductive compensation:

e = Ei + CzElz] E 2 = 0; / 0 = 0; J 3 = c 3 /i.

Numerous other combinations can be made and have been proposed.

All of these motors have series characteristics, that is, a speed increasing with decrease of load.

(1) to (8) contain only one set of brushes on the armature; (9) to (11) two sets of brushes in quadrature.

Motors with shunt characteristic, that is, a speed which does not vary greatly with the load, and reaches such a definite limiting value at no-load that the motor can be considered a constant-speed motor, can also be derived from the above equations. For instance:

Compensated shunt motor (Fig. 187) :

$ i = 0; C 2 U 2 = Cz $ 3 = e; / 0 — 0.

In general, a series characteristic results, if the field-exciting circuit and the armature energy circuit are connected in series with each other directly or inductively, or related to each other so that the currents in the two circuits are more or less propor- tional to each other. Shunt characteristic results, if the voltage impressed upon the armature energy circuit, and the field excita- tion, or rather the magnetic field flux, whether produced or in- duced by the internal reactions of the motor, are constant, or, more generally, proportional to each other.

Repulsion Motor

As illustration of the application of these general equations, paragraph 212, may be considered the theory of the repplsion motor (5) , in Fig. 186.

214 . Assuming in the following the armature of the repulsion motor as short-circuited upon itself, and applying to the motor the equations (1) to (6), the four conditions characteristic of the repulsion motor are:

386

ELECTRICAL APPARATUS

1. Armature short-circuited upon itself. Hence:

Ei = 0.

2. Field circuit and cross-circuit in series with each other con- nected to a source of impressed voltage, e. Hence, assuming the compensating circuit or cross-circuit of the same number of effective turns as the rotor circuit, or, c 2 = 1 :

Co#o + E% — c.

Herefrom follows:

3. I o = Col 2.-

4. No armature excitation used, but only one set of commu- tator brushes ; hence :

/a = 0,

and therefore:

h = 0.

Substituting these four conditions in the six equations (1) to (6), gives the three repulsion motor equations:

Primary circuit:

Z2/2 + Z' (/ 2 — I) + Co 2 Zo / 2 + cqZ (c 0 j 2 — l a) = e; ( 7 ) • Secondary circuit:

Zih + z' (Jx - h) - jSZ (coh — / 4 ) = 0; (8)

Brush short-circuit:

Z4/4 + Z (I 4 — coh) - jSZ'C/a — /») = 0; (9)

Substituting now the abbreviations:

( 10 )

( 11 )

( 12 )

(13)

where Xi and X 4 , especially the former, are small quantities. From (9) then follows:

J 4 = X 4 {I 2 (c<) ~ jSA) + jSJiA } ;

(14)

SINGLE-PHASE COMMUTATOR MOTORS

387

from (8) follows, by substituting (14) and rearranging:

h = I

jSco - \s(s +&?)

1 + - X 4

(15)

/2 1+Xi-\ 4 S 2

and, substituting (15) in (14), gives:

r r ( Co - jSA) (1 + Xx ~ X 4 ,S 2 ) + jSA - S 2 c 0 - X 4 j,S {SA + jc 0 ) / 4 — X 4 /2 - 1 + Xi — X 4 S 2

or, canceling terms of secondary order in the numerator:

t , T cod -s 2 )

• 4 " X</2 1 + X 4 - X 4 S 2

Equation (7) gives, substituting (10) and rearranging:

1 2 {Zi d - Z 1 ) — I\Z' I4C0Z = e.

Substituting (15) and (16) herein, and rearranging, gives: Primary Current:

e (1 + Xi - S 2 X 4 ) h = JR ’

where:

K ” [A 3 — jSco) +-Xi {A .xd - A) X 4 (S-A 3 S 2 Co4-co 2 jSc^) , (19) and:

(16)

(17)

(18)

A, =

Z 3.

or, since approximately:

A 3 = co 2 ,

( 20 )

( 21 )

it is:

K = (A 3 — jSco) + Xi (co 2 + A) — X 4 Co (co — jS). (22)

Substituting (18), (19), (20) in (15) and (16), gives: Secondary Current:

j > Sco

h

e 1+'

\ i s(s+ l 2)

ZK

Brush Short-circuit Current:

h —

X 4 eco (1 — S 2 ) ZK

(23)

(24)

388

ELECTRICAL APPARATUS

As seen, for S = 1, or at synchronism, 1 4 = 0, that is, the short-circuit current under the commutator brushes of the re- pulsion motor disappears at synchronism, as was to be expected, since the armature coils revolve synchronously in a rotating field.

215. The e.m.f. of rotation, that is, the e.m.f. generated in the rotor by its rotation through the magnetic field, which e.m.f., with the current in the respective circuit, produces the torque and so gives the power developed by the motor, is:

Main circuit :

E'i = jSZ (co/ 2 - A). (25)

Brush short-circuit :

E\ = jSZ' (h - h). (26)

Substituting (18), (23), (24) into (25) and (26), and rearrang- ing, gives:

Main Circuit E.m.f. of Rotation:

l+Xi-X*}. . (27)

Brush Short-circuit E.m.f. of Rotation:

Sr

E\ = {Sco + j\iA - coX 4 ) ; (28)

or, neglecting smaller terms:

, _ S 2 c 0 e F Ji ~ K

(29)

The Power produced by the main armature circuit is: Px = Wi, Jl]\

hence, substituting (22) and (27) :

. 1 1 , jSco

Pi = p— ! - {1 + Xx - x 4 }, — L-

Let:

-x.s(s + f))

" 1

ZK

Kvv)

m = [ZK]

(31)

be the absolute value of the complex product, ZK, and :

j = |

Xx = X'x - j " i

X4 = X' 4 + j"t ,

• )

(32)

SINGLE-PHASE COMMUTATOR MOTORS 389

it is, substituting (31), (32) in (30), and expanding:

Pi = — °f 2 {[* (1 - Scoa") - rSc 0 a'} + (1 - Sc 0 r ") [x (X'x - X' 4 )

KYb

• r(X"x + X" 4 )] — 'Scoa' [r (X'i - X' 4 ) + z(X"x + X" 4 )]

• x (' 4 S 2 - ' 4 Scoa" + " 4 Sc 0 a') + r ( ' 4 Sc 0 a '

• " 4 S 2 + " 4 Sc 0 a")}, (33)

after canceling terms of secondary order.

As first approximation follows herefrom:

Sc 0 e 2 x

1 - Sc ,

• Scoa" - -Sc W) m 2 \ x /

Se 2 a;{l - Sc 0 (a" + ^ «') } Co (1 + S 2 )

hence a maximum for the speed S, given by:

dPi _ n

-V i+c “’(“”+i“’) :

and equal to :

Pi 0 =

1 + Co 2 {a" + ~ a

Co \ a " + x a '

2 — Co (a" + ^ a'J | - (36)

The complete expression of the power of the main circuit is, from (33):

Pi = [l - Sc 0 (o"+ V)] ~bo - biS-btS 2 }, (37)

where b 0y b h &Vare functions of X"i, X' 4 , X" 4 , as derived by rearranging (33).

The Power produced in the brush short-circuit is:

P 4 = [E,

390

ELECTRICAL APPARATUS

hence., substituting (24) and (28) :

r S 2 c 0 e \ 4 ec 0 (1 - S 2 )! 1

-[

]

K ’ ZK

x „

m 2

(38)

hence positive, or assisting, below synchronism, retarding above synchronism.

The total Power , or Output of the motor then is:

P = Pi + P 4

or:

P =

Power Output Sc 0 e 2 X

| J^l Scq (a." + ^ a ') ] ko+$ [eo ” V4 ) — frij

• S 2 6 2 - (S 3 c 0 (x" 4 + r - X'*) ) ; (39)

or, approximately:

P =

hence:

Torque:

Sc 0 e 2 x 1 1 _ Sc() + L ^

Se 2 x{ 1 -

• Sc 0 (a" + - a') }

l

\ x / I

(40)

(41)

Co (1 + $ 2 )

p

D = S’

given in synchronous watts.

The power input into the motor, and the volt-ampere input,

sue , if:

h = i ’ 2 - j*"*, ]

and:

(42)

i 2 = V*V + tV, J

given by:

Power Input:

Po = ei' 2 ,

(43)

SINGLE-PHASE COMMUTATOR MOTORS 391

Volt-ampere Input:

*0

o

II

K>

(44)

Power-factor:

i**

ii

(45)

Efficiency:

P

(46)

y p" >

£ 0

Apparent Efficiency :

P

(47)

Vv =

• an

216. While excessive values of the short-circuit current under the commutator brushes, 1 4 , give bad commutation, due to ex- cessive current densities under the brushes, the best commuta- tion corresponds not to the minimum value of / 4 — as the zero value at synchronism in the repulsion motor — but to that value of h for which the sudden change of current in the armature coil is a minimum, at the moment where the coil leaves the com- mutator brush.

/ 4 is the short-current in the armature coil during commuta- tion, reduced to the armature circuit, I 1 , by the ratio of effective turns:

__ short-circuite d turns under brushes , ,

Ui total effective armature turns

The actual current in the short-circuited coils during commuta- tion then, is:

U = k (49)

C4

or, if we denote :

where A 4 is a fairly large quantity, and substitute (24), it is:

,, A 4 ec 0 (1 — <S 2 ) , u

1 4 7 ZK . fbl)

Before an armature coil passes under the commutator brushes, it carries the current, —Jy] while under the brushes, it carries the current, /'*; and after leaving the brushes, it carries the cur- rent, +/i.

392

ELECTRICAL APPARATUS

While passing under the commutator brushes, the. current in the armature coils must change from, — I h to l, or by:

I'o = /' 4 + /i. (52)

In the moment of leaving the commutator brushes, the cur- rent in the armature coils must change from, J\ to + /i, or by:

I 0 -h~ l. (53)

The value, V 0 , or the current change in the armature coils while entering commutation, is of less importance, since during this change the armature coils are short-circuited by the brushes.

Of fundamental importance for the commutation is the value, T g , of the current change in the armature coils while leaving the commutator brushes, since this change has to be brought about by the resistance of the brush contact while the coil approaches the edge of the brush, and if considerable, can not be completed thereby, but the current, I g , passes as arc beyond the edge of the brushes.

Essential for good commutation, therefore, is that the current, I g , should be zero or a minimum, and the study of the commu- tation of the single-phase commutator thus resolves itself largely into an investigation of the commutation current , I Q , or its abso- lute value, i 0 .

The ratio of the commutation current, i Q , to the main armature current, i, can be called the commutation constant:

k = )■ ' (54)

f'l

For good commutation, this ratio should be small or zero.

The product of the commutation current, i g , and the speed, S, is proportional to the voltage induced by the break of this cur- rent, or the voltage which maintains the arc at the edge of the commutator brushes, if sufficiently high, and may be called the commutation voltage:

Cc — Sig. (55)

In the repulsion motor, it is, substituting (23) and (51) in (53), and dropping the term with X 4 , as of secondary order:

Commutation Current :

e {l -Aicoil- £ 2 ))

} 0 = jo? '

( 56 )

SINGLE-PHASE COMMUTATOR MOTORS

393

Commutation Constant:

4 1 - A 4 c 0 (1 - S 2 )

= 1 -

1 A

Ajcoj 1 ~ S 2 )

! , jSCo

Or, denoting:

Ai — ol\ ja" 4 ; (58)

substituting (32) and expanding:

•o "

4 _ e {1 — Co [<Sa" + (1 -S 2 ) ^-jcotd- S 2 ) a" 4 - So 7 ] } (59)

h

and, absolute:

(l-Sc 0 a") + j£coc/

h = -a/TT 1 Co [So" + (1 - S 2 )V 4 ] } 2 + Co 2 f(l ~ &) a" i — Six ' } 2

, = [[ l-c 0 [£<" + (! - S 2 ) a ' 4 ] } 2 + Co 2 |(1 - S) « "« - Sa!\ 2

\ (1 - Scoa") 2 + SWa' 2

(61)

Perfect commutation, or = 0 , would require from equation (58):

l-Co[So"-+(l-5*)a , J = 0,1 ,„ 9)

.(1 - S 2 ) a " 4 - So' = 0 ; (t>/;

, _ 1 - c 0 Sa" 1

“ 4 Co (1 — S 2 )’ ,

u Sa! _ , (63)

^ 4 1 > g 2 1 a 4 ‘

This condition can usually not be fulfilled.

The commutation is best for that speed, £, when the commu- tation current, i 0} is a minimum, that is:

= o- I

dS u ’

hence: i (64)

^,{(1 — c 0 [iSq; ,, + (1— *S 2 )a , 4 ]) 2 -)-Co 2 ((l — S 2 )a" 4 — <Sa') 2 } =0

394

ELECTRICAL APPARATUS

This gives a cubic equation in S, of which one root, 0 < Si < 1, represents a minimum.

The relative commutation, that is, relative to the current con- sumed by the motor, is best for the value of speed, S 2 , where the commutation factor, k, is a minimum, that is:

fs = °- < 65 >

217. The power output of the repulsion motor becomes zero at the approximate speed given by substituting P = 0 in the approximate equation (40), as:

’ $o —

Co («"+'-«')

x

( 66 )

and above this speed, the power, P, is negative, that is, the repulsion motor consumes power, acting as brake.

This value, So, however, is considerably reduced by using the complete equations (39), that is, considering the effect of the short-circuit current under the brushes, etc.

For S < 0, P < 0; that is, the power is negative, and the machine a generator, when driven backward, or, what amounts to the same electrically, when reversing either the field-circuit, I o, or the primary energy circuit, / 2 . In this case, the machine then is a repulsion generator .

The equations of the repulsion generator are derived from those of the repulsion motor , given heretofore, by reversing the sign of S.

The power, P 4 , of the short-circuit current under the brushes reverses at synchronism, and becomes negative above synchron- ism. The explanation is: This short-circuit current, / 4 , and a corresponding component of the main current, /i, are two cur- rents produced in quadrature in an armature or secondary, short- circuited in two directions at right angles with each other, and so offering a short-circuited secondary to the single-phase pri- mary, in any direction, that is, constituting a single-phase in- duction motor. The short-circuit current under the brushes so superimposes in the repulsion motor, upon the repulsion-motor torque, a single-phase induction-motor torque, which is positive below synchronism, zero at synchronism, and negative above synchronism, as induction-generator torque. It thereby lowers

SINGLE-PHASE COMMUTATOR MOTORS 395

the speed, So, at which the total torque vanishes, and reduces the power-factor and efficiency.

218 . As an example are shown in Mg. 188 the characteristic curves of a repulsion motor, with the speed, S, as abscissae, for the constants :

Impressed voltage: e = 500 volts.

Exciting impedance, main field: Z = 0.25 +3 j ohms.

cross field: Z' = 0.25 + 2.5 j ohms.

0 0.2 0.4 0.G 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

SPEED .

Fig. 188 .

Self-inductive impedance, main field: Z 0 = 0.1 + 0.3 j ohms.

cross field: Z 2 = 0.025 + 0.075johms. armature: Z\ = 0.025 + 0. 075 j ohms, brush short-circuit: Z 4 = 7.5 + 10 j ohms.

Reduction factor, main field: c 0 = 0.4.

brush short-circuit: c 4 = 0.04.

Hence:

Zz = 0.08 + 0.60 j ohms.

A = 0.835 — 0.014 j.

4 = a' + ja" = 1.20 + 0.02 j.

Xi = 0.031 - 0.007 j.

X 4 = 0.179 + 0.087 j.

A. 4 = 4.475 + 2.175 j.

A 3 = 0.202 - 0.010 j.

396 ELECTRICAL APPARATUS

Then, substituting in the preceding equations:

K = (0.204 - 0.035 S) - j (0.031 + 0.328 S),

ZK = (0.144 + 0.975 8) +j (0.604 - 0.187 S).

Primary or Supply Current:

r 500 {(1.031 - 0.179 S 2 ) - j (0.007 + 0.087 8 2 )} h ~ ZK

Secondary or Armature Current :

r 500 {(1 + 0.048 S - 0.179 S 2 ) + j 0.4 S - 0.087 S 2 )}

11 ~ ' ZK "

Brush Short-circuit Current:

r 500 (1 - S 2 ) (0.072 - 0.035 j)

and absolute:

40 (1 - 8 2 )

2-4

m

Commutation Factor:

, _ /( 1.508 8 2 - 0.673) 2 ”TT0.718 - 0.4 S - 0.704 S*)*

\ (0.697 +0AS - 0.014 ) 2

Main E.m.f. of Rotation:

„ 500 S (4.052 + 0.792 j)

L 1 ~ ZK '

Commutation E.m.f. of Rotation:

500 8 2 (0.4- 4.8 j)

E 4 _ - z gr~ - - •

Power of Main Armature Circuit:

p = (4.052 - 0.122 S - 0.657 8 2 ), in kw.

m 2 .

Power of Brush Short-circuit:

49.2 S 2 (1 - S 2 ) 4 m 2

Total Power Output:

, in kw.

P = ^ 0.075 s _ g2 _ 1Q7 g3) _

m 2

Torque :

oko

D = (4.052 + 0.075 8 - 0.657 S 2 - 0.197 S 3 ),

m 2

SINGLE-PHASE COMMUTATOR MOTORS 397

These curves are derived by calculating numerical values in tabular form, for S = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4.

As seen from Fig. 188, the power-factor, p, rises rapidly, reach- ing fairly high values at comparatively low speeds, and remains near its maximum of 90 per cent, over a wide range of speed. The efficiency, rj, follows a similar curve, with 90 per cent, maxi- mum near synchronism. The power, P , reaches a maximum of 192 kw. at 60 per cent, of synchronism — 450 revolutions with a four-pole 25-cycle motor — is 143 kw. at synchronism, and van- ishes, together with the torque, D, at double synchronism. The torque at synchronism corresponds to 143 kw., the starting torque to 657 synchronous kw.

The commutation factor, fc, starts with 1.18 at standstill, the same value which the same motor would have as series motor, but rapidly decreases, and reaches a minimum of 0.23 at 70 per cent, of synchronism, and then rises again to 1.00 at synchron- ism, and very high values above synchronism. That is, the commutation of the repulsion is fair already at very low speeds, becomes very good somewhat below synchronism, but poor at speeds considerably above synchronism: this agrees with the ex- perience on such motors.

In the study of the commutation, the short-circuit current under the commutator brushes has been assumed as secondary alternating current. This is completely the case only at stand- still, but at speed, due to the limited duration of the short-circuit current in each armature coil — the time of passage of the coij. under the brush — an exponential term superimposes upon the alternating, and so modifies the short-circuit current and thereby the commutation factor, the more, the higher the speed, and greater thereby the exponential term is. The determination of this exponential term is beyond the scope of the present work, but requires the methods of evaluation of transient or momentary electric phenomena, as discussed in “Theory and Calculation of Transient Electric Phenomena and Oscillations.”

B. Series Repulsion Motor

219 . As further illustration of the application of these funda- mental equations of the single-phase commutator motor, (1) to (6), a motor may be investigated, in which the four independent constants are chosen as follows:

398

ELECTRICAL APPARATUS

1. Armature and field connected in series with each other. That is :

E\ 4 “ — JjJ — 6\ }

(67)

where :

Co = reduction factor of field winding to armature; that is,

. „ . field turns

ratio of effective -

' armature turns It follows herefrom:

I o = coh.

( 68 )

2. The e.m.f. impressed upon the compensating winding is given, and is in phase with the e.m.f., e h which is impressed upon field plus armature :

= e s . (69)

That is, JjJ % is supplied by the same transformer or compensator as ci, in series or in shunt therewith.

3. No rotor-exciting circuit is used:

h = 0, (70)

and therefore:

4. No rotor-exciting brushes, or brushes in quadrature posi- tion with the main-armature brushes, are used, and so:

It = 0 ,

(71)

that is, the armature carries only one set of brushes, which give the short-circuit current, 74*

• Since the compensating circuit, e 2? is an independent circuit, it can be assumed as of the same number of effective turns as the armature, that is, e 2 is the e.m.f. impressed upon the com- pensating circuit, reduced to the armature circuit. (The actual e.m.f. impressed upon the compensating circuit thus would be:

C2C2,

, . compensating turns \

where c 2 - ratio effective- 7 , -

armature turns J

220. Substituting (68) into (1), (2), (3), and (5), and (1) and (2) into (67), gives the three motor equations:

ex = Zih + Z' U 1 - U) ~ jSZ [cdi ~

Id (

(72)

• CtfZn Ji -f CaZ (coin — J. 1),

1

&7. ~ Z 21 2 + Z' (/ 2 _ 10,

(73)

0 = l,n +z(} i - Coh) - jSZ' (Ji -

■h).

(74)

SINGLE-PHASE COMMUTATOR MOTORS 399

Substituting now:

Y' = = quadrature, or transformer exciting

L admittance,

i£i = X2 = x ' 2 ■— j" 2,

Z + ^4 = X4 = X,4+iX%

Z'

_ 4 = <*' — = impedance ratio of the

" two quadrature fluxes,

^1 4“ Co 2 (JZo + Z) = Z 3 ,

= A 3 = 0/3 + ja" 3 ,

and:

6 = 61 + e 2 , j

t = 62 I

e 1

Adding (72) and (73), and rearranging, gives:

6 = z«h + h (Zz - jScoZ) - nz (Co - jS ) ; or:

= ~ “ /* ( c o ~ jS).

From (73) follows:

e^P 1 — 1 1 (1 + X 2 ) — / 1 , or:

/1 = /*> (1 4” X 2 ) — • ctlf ' , | and : \

/*2 =: /1 (1 — X 2 ) 4“ ’ • j

From (74) follows:

0 = h (Z + Z A ) - h (coZ 4 - jSZ') + jSZ’U.

(75)

(76)

(77)

(78)

(79)

Since / 4 is a small current, small terms, as X 2 , can be neglected in its evaluation. That is, when substituting (78) in (79), X 2 can be dropped :

or:

U = U - etY', h = h + etY',

approximately.

(80)

400

ELECTRICAL APPARATUS

Hence, (80) substituted in (79) gives:

0 = U (Z + Z 4 ) - c 0 Zh + jSet,

or:

Hence:

0 = ( 4 - Col i + ^- < -

A 4 //

h = X 4 jco/l - J -|-j

and actual value of short-circuit current :

I\ = bU {/i }’

where :

b = C °, a fairly large quantity, and

C 4

c 4 = reduction factor of brush short-circuit to armature circuit.

The commutation current then is:

L = h- l\

= h (1 - b\ 4 ) +

jSetbXj

CoZ

Substituting (81) ;md (80) into (77), gives:

r _ 6

L ~ r.

1 —JSt \ 4 (Co —J§)_ - <Xa

or, denoting:

Z -4s — j/Sco — X|Co (co — i<S) + X 2 H

it is:

K = A 3 — iSc 0 — X 4 Co (co — jS ) + XoA,

r e { 1 — jSt\i(c<i — jS ) — 1 X 2 }

/; “ JZfT

It is, approximately:

hence:

A 3 = ^ = co 2 ,

x 2 = 0,

iv = Co (1 — c 0 X 4 ) (co — jS), j = e jl — jStKjjco - jg)j

c 0 2 (1 — c 0 X 4 ) (c 0 — i<S) ” c 0 ^ (1 — c 0 X 4 ) (co — jS

(81)

(82)

(83)

(84)

(85)

( 86 )

(87)

(88)

SINGLE-PHASE COMMUTATOR MOTORS 401

/■„ — e f jSt\ 4 (co — jS ) (1 + i) X2 } , et (1 — X 2 )

' ' “ ' ZK + z 1

l 2 — _ e jStX-je te_

c,Z (1 - coX 4 ) (c 0 - jS) c 0 Z (1 - coX 4 ) + Z r

Armature, or Secondary Current:

j ^ { 1 jSt \ 4 (co jTS) £X 2 }

• 1 “ ~ ' ZK ’

approximately:

h =

CoZ (1 — C0X4) 1 CO — jS Brush Short-circuit Current:

Jet ~ jSt \ 4 [ '

^ 2T(1 - c 0 X 4 ) ic„ - jS

Jq ~~ (1 + X4 — C0X4)

approximately :

h = y

cX 4

f 1

Z (1 — C0X4) l Co — jS Commutation Current:

e f 1 — \J)

c 0 z(r-^cou) + jStXil{b ~ 1}

• 6X4 (1 — Co)] j 1

approximately :

7 C ( 1 111 X 4 & I •

lo = CoZ (1 - C0X4) 1 Co -38 + ' jStXib

(89)

Substituting now (85) respectively (87), (88) into (78), (81) (84), and into:

E\ = *SZ (co/x - / 4 ),1

E\ = jSZ' (/1 - h),

gives the

Equations of the Series Repulsion Motor:

IC = A 3 - jSca - X 4 Co (c 0 - jS) + XzA, approximately:

K = Co (1 — C0X4) (co — jS).

Inducing, or Compensator Current: t = «'{ 1 - jSi

approximately:

(90)

(91)

(92)

(93)

( 94 )

2G

402

ELECTRICAL APPARATUS

Main E.m.f. of Rotation:

E' i =

approximately :

W 1 =

jSe [JL ~X 4 i — c 0 X 4 lc 0 —

• jSt\p (1 — Co) | >

jSe(l-\ t)

(1 — c 0 X 4 ) (co — jS)

Quadrature E.m.f. of Rotation:

W 4 = + jSte.

Power Output:

P = Pi + P 4

= W 1 , J1] 1 - 4 - /4] 1 .

Power Input:

Po = [61, /1] 1 + [c 2 , Id 1 .

Volt-ampere Input:

P = CiZi + C 2 ^2

= c{ (1 — t) ii + Hi 1 ,

( 95 )

(96)

(98)

where the small letters, ii and ?‘ 2 , denote the absolute values of the currents, h and / 2 .

When ii and i 2 are derived from the same compensator or transformer (or are in shunt with each other, as branches of the same circuit, if Ci = c 2 ), as usually the case, in the primary cir- cuit the current corresponds not to the sum, {(1 — t) ii + ti 2 ) of the secondary currents, but to their resultant, [(1 — t) h + tf 2] 1 , and if the currents, h and I 2 , are out of phase with each other, as is more or less the case, the absolute value of their resultant is less than the sum of the absolute values of the components. The volt-ampere input, reduced to the primary source of power, then is:

P ao = e[(l -t)h + (99)

and:

P < P

P

From these equations then follows the torque: D = ^r, the

A3

P

power-factor, p = 5—, etc.

These equations (90) to (99) contain two terms, one with, and

one without t = — , and so, for the purpose of investigating the e

SINGLE-PHASE COMMUTATOR MOTORS 403

effect of the distribution of voltage, e , between the circuits, e x and e 2) they can be arranged in the form: F = K x + tK 2 .

For:

t = 0 ,

that is, all the voltage impressed upon the armature circuit, and the compensating circuit short-circuited, these equations are those of the inductively compensated series motor.

For:

t = 1 ,

that is, all the voltage impressed upon the compensating or in- ducing circuit, and the armature circuit closed in short-circuit, that is, the armature energizing the field, the equations are those of the repulsion motor with secondary excitation.

For:

t > 1,

a reverse voltage is impressed upon the armature circuit.

Study of Commutation

221 . Tiie commutation of the alternating-current commutator motor mainly depends upon :

(a) The short-circuit current under the commutator brush,

j? 4 .

which has the actual value: l\ = — ' High short-circuit current

causes arcing under the brushes, and glowing, by high current density:

( b ) The commutation current, that is, the current change in the armature coil in the moment of leaving the brush short-cir- cuit, I Q = 1 1 — I. This current, and the e.m.f. produced by it, SI g , produce sparking at the edge of the commutator brushes, and is destructive, if considerable.

(a) Short-circuit Current under Brushes Using the approximate equation ($3), the actual value of the

short-circuit current under the brushes is:

r 4 =

e\J)

where :

c 0 Z (1 — C 0 X 4 ) 1 Co — jS

• jSt ;

( 100 )

b = ™ or t = reduction factor of short-circuit under brushes, C4 0

404

ELECTRICAL APPARATUS

to field circuit, that is :

ft — number of field turns

~ number of effective short-circuit turns'

( 101 )

hence a large quantity.

The absolute value of the short-circuit current, therefore, is:

eU'Wc l + Sjjl - f (c of + GqZ [1 — C0X4] (Co 2 + S 2 )

( 102 )

hence a minimum for that value of t, where:

/ = c 0 2 + S 2 (1 — t (c 0 2 + S 2 )) 2 = minimum, or = 1 — t (co 2 + S 2 ) = 0, hence,

t = __ 1 _.

Co 2 + S*’

and:

s - 4 -

That is, t = ~~ = cr v — * gives minimum short-circuit cur-

, e S2 + Co 2

rent at speed, S, and inversely, speed S = c 0 2 , gives

minimum short-circuit current at voltage ratio, t.

For t = 1, or the repulsion motor with secondary excitation, the short-circuit current is minimum at speed, S = /l — c 0 2 , or

somewhat below synchronism, and is i\ = “~° e , while in the re- pulsion motor with primary excitation, the short-circuit current is a minimum, and equals zero, at synchronism S = 1.

The lower the voltage ratio, t — the higher is the speed, S,

at which the short-circuit current reaches a minimum.

The short-circuit current, /' 4 , however, is of far less importance than the commutation current, J 0 .

(b) Commutation Current

222 . While the value, /' g — I\ + / 1 , or the current change in the armature coils while entering commutation, is of minor im- portance, of foremost importance for good commutation is that the current change in the armature coils, when leaving the short- circuit under the brushes :

I g = h - ( 103 )

is zero or a minimum.

SINGLE-PHASE COMMUTATOR MOTORS 405

Using the approximate equation of the commutation current (94), it is:

Ia = c 0 Z( 1 - c 0 X 4 ) I c7-JS + jStXib !

= cqZ (1 - c 0 X 4 ) (co - JSj^ 1 ~ X “ & + < - Co “ jS) tXib ^ ’’ < ' 104 ' ) and, denoting:

X 4 = X' 4 + iX" 4 ,

it is, expanded:

h = 7n , -.QV {[1 - XV) + ^ (SX' 4 - c„X" 4 )]

CvZ (1 - c 0 X 4 ) (co 4- jS)

• j[W\b - Stb (C 0 V 4 + SW)]} ; ■ (105)

hence, absolute:

' C 0 z[l - C0X4J -y / Co 2 + & __

V[1 - yj3 + StbiSVi — CoX ,, 4 )] 2 4- [X ;, 4 fe - StbicaW + SX"7)P,

(106)

where [1 — c 0 X 4 ] denotes the absolute value of (1 — c 0 X 4 ).

The commutation current is zero, if either S = 00 , that is, infinite speed, which is obvious but of no practical interest, or the parenthesis in (105) vanishes.

Since this parenthesis is complex, it vanishes when both of its terms vanish. This gives the two equations:

1 — X' 4 t> + Stb (S' 4 — CoX'V = 0,1

'\b - Stb (coX' 4 + SX" 4 ) = O.j 1 ;

From these two equations are calculated the two values, the speed, S, and the voltage ratio, t, as :

a _ c o (&X 4 2 — X' 4 )

£0 = — ,

A 4

. _ XV .

to .c 0 2 6 X 4 W-X' 4 ) j (108)

hence :

Soto =

For instance, if :

Z = 0.25 + 3 j, Z 4 = 5 + 2.5 j:

406

ELECTRICAL APPARATUS

hence :

X 4 =

Z

z + z 4

Co

hence :

and herefrom:

c 4

b

So

to

0.307 + 0.248 j = V 4 + j',

0.4,

0.04;

10 ;

2 . 02 ,

0.197

that is, at about double synchronism, for e 2 = te = 0.197 e, or about 20 per cent, of e, the commutation current vanishes.

In general, there is thus in the series repulsion motor only one speed, So, at which, if the voltage ratio has the proper value, to, the commutation current, i g , vanishes, and the commutation is perfect. At any other speed some commutation current is left, regardless of the value of the yoltage ratio, t.

With the two voltages, e\ and e 2) in phase with each other, the commutation current can not be made to vanish at any desired speed, S.

223. It remains to be seen, therefore, whether by a phase dis- placement between e\ and e 2 , that is, if e 2 is chosen out of phase with the total voltage, e, the commutation current can be made to vanish at any speed, S, by properly choosing the value of the voltage ratio, and the phase difference.

Assuming, then, e 2 out of phase with the total voltage, e, hence denoting it by:

E 2 = e 2 (cos 62 — j sin 0 2 ), (109)

the voltage ratio, t , now also is a complex quantity, and expressed by:

T — — = t (cos 0 2 — j sin 0 2 ) = t' — jt". (110)

€>

Substituting (110) in (105), and rearranging, gives:

= coZ (1 — C0X4) (co-jS) " X ' 4& + Si>b ( ' SX ' i ~ CoX,,4)

• St"b (C0V4 + S')] - - St'b (C0V4 + SX"„)

• St'^iSW - coV' 4 )]}; (111)

and this expression vanishes, if :

1 - X\6 + St'b (SW - c 0 X" 4 ) + St"b (coV 4 + SW) = 0, 1 X" 4 b - St'b (C0V4 + SX" 4 ) + St"b (SX' 4 - CoX" 4 ) = 0:j

SINGLE-PHASE COMMUTATOR MOTORS 407

and herefrom follows :