Theory and Calculation of Electrical Apparatus (1917) — part 19 of 21

the problem is to design the magnetic circuit of the converter so as to produce the maximum third harmonic, the minimum fifth and seventh harmonics.

If q = interpolar space, thus (1 - q) = pole arc, as fraction of pitch, the wave shape of the voltage generated between the point, a , of a full-pitch distributed winding — as generally used for commutating machines — and the neutral, or the induced Y voltage of the system is a triangle with the top cut off for dis- tance q, as shown in Fig. 209, when neglecting magnetic spread at the pole corners.

If then e 0 = voltage generated per armature turn while in front of the field pole (which is proportional to the magnetic den- sity in the air gap), m = series turns from brush to brush, the maximum voltage of the wave shown in Fig. 209 is:

E 0 = me o (1 - q) ;

developed into a Fourier series, this gives, as the equation of the voltage wave a, Fig. 188:

« C0£ A".- 1) <?*■

8 E o 2

6 = (1 -g°)T 2 T (2n - 1)* cos (2n “ V °>

REGULATING POLE CONVERTERS

435

\

i

I

i

or, substituting for E 0 , and denoting:

. 8 me o A - r->

ir i

a 2n - 1 V cos 2 qir

e = A + (2»- ~ l)« ' Cos(2n ~ 1)6

= A | cos q g cos 0 + ^ cos 3 q ^ cos 3 0 + j^r cos 5 5 ^ cos 5 0

• cos 7g| cos 70 + |

Thus the third harmonic is a positive maximum for q = 0, or 100 per cent, pole arc, and a negative maximum for ? = M, or 33.3 per cent, pole arc.

For maximum direct voltage, q should therefore be made as small, that is, the pole arc as large, as commutation permits. In general, the minimum permissible value of q is about 0.15 to 0.20.

The fifth harmonic vanishes for q = 0.20 and q = 0.60, and the seventh harmonic for q = 0.143, 0.429, and 0.714.

For small values of q, the sum of the fifth and seventh har- monics is a minimum for about q = 0.18, or 82 per cent, pole arc. Then for q = 0.18, or 82 per cent, pole arc:

ei = A {0.960 cos 6 + 0.0736 cos 3 6 + 0.0062 cos 5 $

— 0.0081 cos 7 6 + . . . }

= 0.960 A {cos 6 + 0.0766 cos 3 6 + 0.0065 cos 5 6

— 0.0084 cos 7 6+ . . . • j j

that is, the third harmonic is less than 8 per cent., so that not much voltage rise can be produced in this manner, while the fifth and seventh harmonics together are only 1.3 per cent., thus negligible.

236. Better results are given by reversing or at least lowering the flux in the center of the field pole. Thus, dividing the pole face into three equal sections, the middle section, of 27 per cent, pole arc, gives the voltage curve, q = 0.73, thus:

e 2 = A {0.411 cos 6 - 0.1062 cos 3 6 + 0.0342 cos 5 6

—0.0035 cos 7 d . . . }

= 0.411 A {cos 6 — 0.258 cos 3 6 + 0.083 cos 5 6

— 0.0085 cos 7 0 . . .}•

The voltage curves given by reducing the pole center to one-

43 G

ELECTRICAL APPARATUS

half intensity, to zero, reversing it to half intensity, to full in- tensity, and to such intensity that the fundamental disappears, then are given by:

Center part of pole density

(1) full, e = ei = 0.960 -A {cos 0+0.077 cos 3 0

+0.0065 cos 5 0-0.0084 cos 7 0. . .}

(2) 0.5, 0 = 61 — 0.5 eo =0.755 A {cos 0+0.168 cos 3 0

-0.0144 cos 5 0-0.0085 cos 7 0. . .}

(3) 0, e = ei-e 2 =0.549 A {cos 0+0.328 cos 3 0

— 0.053 cos 5 0 — 0.084 cos 7 0. . .}

(4) -0.5 e = ei — 1.5 e 2 = 0.344 A {cos 0 +0.680 cos 3 0

-0.131 cos 5 0-0.0084 cos 7 0. . . }

(5) - full, 0 = 01-2 c 2 =0.138 A {cos 0+2.07 cos 3 0

-0.45 cos 5 0-0.008 cos 7 0. . .}

(6) -1.17, 0 = 0 x -2.34 0 2 = O.322 A {cos 3 0-0.227 cos 5 0. . .}.

It is interesting to note that in the last case the fundamental frequency disappears and the machine is a generator of triple frequency, that is, produces or consumes a frequency equal to three times synchronous frequency. In this case the seventh harmonic also disappears, and only the fifth is appreciable, but could be greatly reduced by a different kind of pole arc. From above table follows:

(1) (2) (3) (4) (5) (6) normal

Maximum funda- mental alter- 0.960 0.755 0.549 0.344 0.138 0 0.960

nating volts

Direct volts 1.033 0.883 0.743 0.578 0.423 0.322 0.960

237. It is seen that a considerable increase of direct voltage

beyond the normal ratio involves a sacrifice of output, due to the decrease or reversal of a part of the magnetic flux, whereby the air-gap section is not fully utilized. Thus it is not advisable to go too far in this direction.

By the superposition of the third harmonic upon the funda- mental wave of the Y voltage, in a converter with three sections per pole, thus an increase of direct voltage over its normal voltage can be produced by lowering the excitation of the middle section and raising that of the outside sections of the field pole, and also inversely a decrease of the direct voltage below its normal value by raising the excitation of the middle section

REGULATING POLE CONVERTERS

437

and decreasing that of the outside sections of the field poles; that is, in the latter case making the magnetic flux - distribution at the armature periphery peaked, in the former case by making the flux distribution flat-topped or even double-peaked.

Armature Reaction and Commutation

238. In such a split-pole converter let p equal ratio of direct voltage to that voltage which it would have, with the same alternating impressed voltage, at normal voltage ratio, where V > 1 represents an overnormal, p < la subnormal direct voltage. The direct current, and thereby the direct-current armature reaction, then is changed from the value which it

would have at normal voltage ratio, by the factor ™, as the

product of direct volts and amperes must be the same as at normal voltage ratio, being equal to the alternating power input minus losses.

With unity power-factor, the direct-current armature reac- tion, in a converter of normal voltage ratio is equal and opposite, and thus neutralized by the alternating-current armature reac- tion, $Fq, and at a change of voltage ratio from normal, by factor

p, and thus change of direct current by factor - * The direct current armature reaction thus is:

V

hence, leaves an uncompensated resultant.

As the alternating-current armature reaction at unity power- factor is in quadrature with the magnetic flux, and the direct- current armature reaction in line with the brushes, and with this type of converter the brushes stand at the magnetic neutral, that is, at right angles to the magnetic flux, the two armature reactions are in the same direction in opposition with each other, and thus leave the resultant, in the direction of the commutator brushes:

9F' = - fF 0

The converter thus has an armature reaction proportional to the deviation of the voltage ratio from normal.

239. If p > 1, or overnormal direct voltage, the armature

438

ELECTRICAL APPARATUS

reaction is negative, or motor reaction, and the magnetic flux produced by it at the commutator brushes thus a commutating flux. If p < 1, or subnormal direct voltage, the armature reaction is positive, that is, the same as in a direct-current gen- erator, but less in intensity, and thus the magnetic flux of arma- ture reaction tends to impair commutation. In a direct-current generator, by shifting the brushes to the edge of the field poles, the field flux is used as reversing flux to give commutation. In this converter, however, decrease of direct voltage is produced by lowering the outside sections of the field poles, and the edge of the field may not have a sufficient flux density to give commuta-

Fig. 210. — Three-section pole for variable-ratio converter.

tion, with a considerable decrease of voltage below normal, and thus a separate commutating pole is required. Preferably this type of converter should be used only for raising the voltage, for lowering the voltage the other type, which operates by a shift of the resultant flux, and so gives a component of the main field flux as commutating flux, should be used, or a combination of both types.

With a polar construction consisting of three sections, this can be done by having the middle section at low, the outside sections at high excitation for maximum voltage, and, to de- crease the voltage, raise the excitation of the center section, but instead of lowering both outside sections, leave the section in the direction of the armature rotation unchanged, while lowering the other outside section twice as much, and thus produce, in addition to the change of wave shape, a shift of the flux, as represented by the scheme Fig. 210.

Magnetic Density

Pole section ... 1 2 3 1' 2' 3'

Max. voltage . .+(B 0 +(B -(B 0 ~(B

J r%(& — %(B — — (B

<£+(B —

0 +CB +(B 0 -CB — (B

Min. voltage . . . +M® -1J6®

REGULATING POLE CONVERTERS

439

i

s

i

Where the required voltage range above normal is not greater than can be produced by the third harmonic of a large pole arc with uniform density, this combination of voltage regulation by both methods can be carried out with two sections of the field poles, of which the one (toward which the armature moves) is greater than the other, as shown in Fig. 211, and the variation then is as follows:

Magnetic Density

Pole section 1 2 1' 2'

Max. voltage + (B + (B — (B — (B

4- K® + l K& - K® -134B

o IK® o - 134 b

Min. voltage — 34B + 134® + 34 B — 1%®

Fig. 211. — Two-section pole for variable -ratio converter.

Heating and Rating

240 . The distribution of current in the armature conductors of the variable-ratio converter, the wave form of the actual or differential current in the conductors, and the effect of the wattless current thereon, are determined in the same manner as in the standard converter, and from them are calculated the local heating in the individual armature turns and the mean armature heating.

In an n-phase converter of normal voltage ratio, let E 0 = direct voltage; Jo = direct current; E° = alternating voltage between adjacent collector rings (ring voltage), and 1° = alter- nating current between adjacent collector rings (ring current); then, as seen in the preceding:

Eo sin ■

and as by the law of conservation of energy, the output must equal the input, when neglecting losses:

TO = h Vl, (2)

440

ELECTRICAL APPARATUS

where 1° is the power component of the current corresponding to the direct-current output.

The voltage ratio of a converter can be varied:

(a) By the superposition of a third harmonic upon the star voltage, or diametrical voltage, which does not appear in the ring voltage, or voltage between the collector rings of the converter.

(b) By shifting the direction of the magnetic flux.

(a) can be used for raising the direct voltage as well as for lowering it, but is used almost always for the former purpose, since when using this method for lowering the direct voltage commutation is impaired.

( b ) can be used only for lowering the direct voltage.

It is possible, by proportioning the relative amounts by which the two methods contribute to the regulation of the voltage, to maintain a proper commutating field at the brushes for all loads and voltages. Where, however, this is not done, the brushes are shifted to the edge of the next field pole, and into the fringe of its field, thus deriving the commutating field.

241 . In such a variable-ratio converter let, then, t = intensity of the third harmonic, or rather of that component of it which is in line with the direct-current brushes, and ■ thus does the voltage regulation, as fraction of the fundamental wave, t is chosen as positive if the third harmonic increases the maximum of the fundamental wave (wide pole arc) and thus raises the direct voltage, and negative when lowering the maximum of the fundamental and therewith the direct voltage (narrow pole arc).

pi — loss of power in the converter, which is supplied by the current (friction and core loss) as fraction of the alternating input (assumed as 4 per cent, in the numerical example).

n = angle of brush shift on the commutator, counted positive in the direction of rotation.

0i = angle of time lag of the alternating current (thus negative for lead).

t a = angle of shift of the resultant field from the position at right angles to the mechanical neutral (or middle between the pole corners of main poles and auxiliary poles), counted positive in the direction opposite to the direction of armature rotation, that is, positive in that direction in which the field flux has been shifted to get good commutation, as discussed in the preceding article.

REGULATING POLE CONVERTERS

441

Due to the third harmonic, i, and the angle of shift of the field flux, T aj the voltage ratio differs from the normal by the factor:

(1 4“ 0 COS T a ,

and the ring voltage of the converter thus is:

hence, by (1):

E =

E°

(1 -f- t) COS T d

E o sin

E =

n

a/2 (1 + t) COSTa

( 3 )

( 4 )

and the power component of the ring current corresponding to the direct-current output thus is, when neglecting losses, from ( 2 ):

V == J° (1 + t) COS T a

— Io COS Tg m ^

. 7T

n sm - n

Due to the loss, pi, in the converter, this current is increased by (1 4- pi) in a direct converter, or decreased by the factor (1 — pi) in an inverted converter.

The power component of the alternating current thus is:

h =/'(!+ pi)

■/2 (l+<) (1+Pl) COSTa

;

. 7 r

n sm

n

( 6 )

where pi may be considered as negative in an inverted converter.

With the angle of lag 0i, the reactive component of the current is:

1 2 = /i tan 0i, and the total alternating ring current is:

I =

h

COS 01

_ 7oV2(l+0 (1+p,) COS Tq

( 7 )

442

ELECTRICAL APPARATUS

or, introducing for simplicity the abbreviation:

k = (1+0 (1 + Pi) COS Tg (8)

cos 6i ’ K

it is:

I = (9)

. 7 r

n sin ~ n

242. Let, in Fig. 212, A r OA represent the center line of the magnetic field structure.

The resultant magnetic field flux, 0$, then leads OA by angle $0A = r a .

The resultant m.m.f. of the alternating po wer current, Ii,is 0/i,

Fig. 212. — Diagram of variable ratio converter.

at right angles to 0$, and the resultant m.m.f. of the alternating reactive current, 1 2 , is 01 2 , in opposition to 0<t>, while the total alternating current, J, is 01, lagging by angle di behind Oh.

The m.m.f. of direct-current armature reaction is in the direc- tion of the brushes, thus lagging by angle r b behind the position OB, where BOA = 90°, and given by Oh.

The angle by which the direct-current m.m.f., 0/o,lags in space behind the total alternating m.m.f., 01, thus is, by Fig. 212:

To = Qi — T a — T b . (10)

If the alternating m.m.f. in a converter coincides with the direct-current m.m.f., the alternating current and the direct cur- rent are in phase with each other in the armature coil midway

REGULATING POLE CONVERTERS

443

between adjacent collector rings, and the current heating thus a minimum in this coil.

Due to the lag in space, by angle to, of the. direct-current m.m.f. behind the alternating current m.m.f., the reversal of the direct current is reached in time before the reversal of the alter- nating current in the armature coil; that is, the alternating current lags behind the direct current by angle, do = to, in the

Fig. 213. — Alternating and direct current in a coil midway between adjacent collector leads.

armature coil midway between adjacent collector leads, as shown by Fig. 213, and in an armature coil displaced by angle, t, from the middle position between adjacent collector leads the alternating current thus lags behind the direct current by angle (r + do), where r is counted positive in the direction of armature rotation (Fig. 214).

Fig. 214. — Alternating and direct current in a coil at the angle r from the

middle position.

The alternating current in armature coil, t, thus can be ex- pressed by:

i = TV 2 sin (0 — r — do); hence, substituting (9):

% = sin (6 — r — do),

. 7 r

n.sm~ n

and as the direct current in this armature coil is and opposite

( 11 )

( 12 )

444 ELECTRICAL APPARATUS

to the alternating current, i, the resultant current in the arma- ture coil, r, is:

. Jo H - l - 2

hi 4 k , a _ a s

2 . T

i n sin -- I n

sin (6 — r — 0 O ) — 1

and the ratio of heating, of the resultant current, i 0 , compared with the current, of the same machine as direct-current gen- erator of the same output, thus is :

io 2 f 4 k . ( v 1 ] 2

7sr*-j— • (»)

( 2 ) [ nfem 7i I

Averaging (14) over one half wave gives the relative heating of the armature coil, r, as:

y T = - r~rv 2 de = 1 P ( ---- - sin (6 - T - e 0 ) - 1 1 *de. (15)

-J (g) H sin n (

Integrated, this gives:

16 k cos (r + 0 O )

243 . Herefrom follows the local heating in any armature coil, r, in the coils adjacent to the leads by substituting r = ± - ,

71 /

and also follows the average armature heating by averaging

y T fromr — — - to r = +~*

T n n

The average armature heating of the n-phase converter there- fore is :

2 n f + -

= - J . ^

or, integrated:

16 h cos 60

( 17 )

REGULATING POLE CONVERTERS

445

This is the same expression as found for the average armature heating of a converter of normal voltage ratio, when operating with an angle of lag, 0 O , of the alternating current, where k denotes the ratio of 'the total alternating current to the alternating power current corresponding to the direct-current output.

In an ri-phase variable ratio converter (split-pole converter), the average armature heating thus is given by:

where

r =

8/c 2

n 2 sin 2 - n

1 -

16 fc cos i

h = (1 + Q (1 -h pi) cos T q

tv ~ COS 01 ’

Oq = di — r a — Tb] (10)

(18)

and t = ratio of third harmonic to fundamental alternating voltage wave; pi = ratio of loss to output; 6 1 = angle of lag of alternating current; r a — angle of shift of the resultant mag- netic field in opposition to the armature rotation, and = angle of shift of the brushes in the direction of the armature rotation.

244. For a three-phase converter , equation (18) gives [n = 3) :

p = _ f i - 1.6217c cos 0o |

= 1.185 fc 2 -f 1 - 1.621 k cos So- J For a six-phase converter, equation (18) gives (n = 6) :

Q &2 \

T = — — hi — 1.621 k cos 0o

f

= 0.889 fc 2 + 1 - 1.621 k cos 9 0 . \

(19)

( 20 )

Fot a converter of normal voltage ratio:

t = 0, r a = 0,

using no brush shift :

n — 0;

when neglecting the losses :

446

ELECTRICAL APPARATUS

and equations (19) Three-phase :

Six-phase:

The equation (18) is the most general equation of the relative heating of the synchronous converter, including phase displace- ment, 0i, losses, pz, shift of brushes, n, shift of the resultant mag- netic flux, r a , and the third harmonic, t.

While in a converter of standard or normal ratio the armature heating is a minimum for unity power-factor, this is not in gen- eral the case, but the heating may be considerably less at same lagging current, more at leading current, than at unity power- factor, and inversely.

245 . It is interesting therefore to determine under which con- ditions of phase displacement the armature heating is a minimum so as to use these conditions as far as possible and avoid con- ditions differing very greatly therefrom, as in the latter case the armature heating may become excessive.

Substituting for k and 0 O from equations (8) and (10) into equation (18) gives:

r = i + 8 (1 + t y (1 + yiY cos 2 r a

7T

n 2 sin 2 - cos 2 0j n

_ lb (1 + t) (1 + yi) cos r a cos (0i - r a - T b ) .

7 r 2 cos 0i ' \ )

Substituting:

and (20) assume the form:

r =

1.185 COS 2 01

0.889 COS 2 01

• 0.621.

• 0.621.

U * T

• sm - = m, (20)

7 r n

which is a constant of the converter type, and is for a three- phase converter, m 3 = 0.744; for a six-phase converter, m Q = 0.955; and rearranging, gives:

r = 1 , 8 (1 +Q 2 (1 Pi) 2 cos 2 T a

^ 7T 2 m 2

2 (1 + 0 (1 + Pz) COS T a COS (r a +- Tb)

7T

REGULATING POLE CONVERTERS

447

• 8 . (l+ytt + PO’oo^T.

7 r 2 m 2

— ^ (1 + 0 (1 + Pi) cos T a sin (r a + n) tan $ x . (21)

T is a minimum for the value, 0 h of the phase displacement given by :

dF

d tan 0 i

= 0 ,

and this gives, differentiated:

tan 0 2

m 2 sin (j a + n)

(1 + t) (1 + pi) cos r a

( 22 )

Equation ( 22 ) gives the phase angle, 0 2 , for which, at given r a , n , t and pi, the armature heating becomes a minimum.

Neglecting the losses, pi , if the brushes are not shifted, n = 0 , and no third harmonic exists, t = 0 :

tan 6 ' 2 = m 2 tan r a ,

where m 2 = 0.544 for a three-phase, 0.912 for a six-phase converter.

For a six-phase converter it thus is approximately 0' 2 = r a , that is, the heating of the armature is a minimum if the alter- nating current lags by the same angle (or nearly the same angle) as the magnetic flux is shifted for voltage regulation.

From equation (22) it follows that energy losses in the con- verter reduce the lag, 0 2 , required for minimum heating; brush shift increases the required lag; a third harmonic, t , decreases the required lag if additional, and increases it if subtractive.

Substituting (22) into (21) gives the minimum armature heat- ing of the converter, which can be produced by choosing the proper phase angle, 0 2 , for the alternating current. It is then, after some transpositions:

_ 1+ 8 ir (l 4-1) O+ff) cos „ T_ + 1)(1 + Vt)

7r 2 l L m J

cos r n cos (r„ + T-ft) — m 1 sin s (r ? + n ) }

_ x - - [ (l+Od+^cosc. _ cos {r> + n) J’) (23)

The term To contains the constants £, pi , r a , n only in the square under the bracket and thus becomes a minimum if this

448

ELECTRICAL APPARATUS

square vanishes, that is, if between the quantities t., pi } r a , n such relations exist that:

(1 + t) (1 + pi) cos r-a

COS (ja + Tb) = 0.

(24)

246 . Of the quantities t, pi , r a , r 6 ; p t and n are determined by the machine design, t and r a , however,- are equivalent to each other, that is, the voltage regulation can be accomplished, either by the flux shift, r a , or by the third harmonic, t, or by both, and in the latter case can be divided between r a and t so as to give any desired relations between them.

Equation (24) gives:

m 2 cos (r g + n)

(1 + pi COS To)

(25)

and by choosing the third harmonic, t, as function of the angle of flux shift T aj by equation (25), the converter heating becomes a minimum, and is:

(26)

hence:

r 0 ° = 0.551 for a three-phase converter, (27)

To 0 = 0.261 for a six-phase converter. (28)

Substituting (25) into (22) gives:

hence :

tan 02 = tan (r a + r b ) ;

6% = T a + Th)

(29)

or, in other words, the converter gives minimum heating To 0 if the angle of lag, d 2} equals the sum of the angle of flux shift, r a , and of brush shift, r&.

It follows herefrom that, regardless of the losses, pi , of the brush shift, r&, and of the amount of voltage regulation required, that is, at normal voltage ratio as well as any other ratio, the same minimum converter heating IV can be secured by dividing the voltage regulation between the angle of flux shift, r a , and the third harmonic, t, in the manner as given by equation (25), and operating at a phase angle between alternating current and voltage equal to the sum of the angles of flux shift, r a , and of brush shift n ; that is, the heating of the split-pole converter can be made the same as that of the standard converter of normal voltage ratio,

REGULATING POLE CONVERTERS

44 9

Choosing = 0.04, or 4 per cent, loss of current, equation (25) gives, for the three-phase and for the six-phase converter:

(a) no brush shift (n = 0) :

h° = 0.467, 1 (30)

1 6° = 0.123; )

that is, in the three-phase converter this would require a third harmonic of 46.7 per cent., which is hardly feasible; in the six- phase converter it requires a third harmonic of 12.3 per cent., which is quite feasible.

( b ) 20° brush shift (n = 20) :

1 - 0.533

1 - 0.877

COS (r„ + Tb)

,

COS T a

COS (t a ~h Tb) . COS T a ’

(31)

for r a = 0, or no flux shift, this gives:

Since

COS (Ta + Tfc) COS Ta

U 00 = 0.500, 1 U 00 = 0.176. j

(32)

< 1 for brush shift in the direction of

armature rotation, it follows that shifting the brushes increases the third harmonic required to carry out the voltage regulation without increase of converter heating, and thus is undesirable.

It is seen that the third harmonic, t, does not change much with the flux shift, r a , but remains approximately constant, and positive, that is, voltage raising.

It follows herefrom that the most economical arrangement regarding converter heating is to use in the six-phase converter a third harmonic of about 17 to 18 per cent, for raising the vol- tage (that is, a very large pole arc), and then do the regulation by shifting the flux, by the angle, r a , without greatly reducing the third harmonic; that is, keep a wide pole arc excited.

As in a three-phase converter the required third harmonic is impracticably high, it follows that for variable voltage ratio the six-phase converter is preferable, because its armature heating can be maintained nearer the theoretical minimum by propor- tioning t and r a .

29

CHAPTER XXII UNIPOLAR MACHINES Homopolar or Acyclic Machines

247 . If a conductor, C, revolves around one pole of a stationary magnet shown as NS in Fig. 215, a continuous voltage is induced in the conductor by its cutting of the lines of magnetic force of the pole, JV, and this voltage can be supplied to an external cir- cuit, D, by stationary brushes, Bi and B 2} bearing on the ends of the revolving conductor, C.

The voltage is :

e = 10- 8 ,

where / is the number of revolutions per second, 4> the magnetic flux of the magnet, cut by the conductor, C.

Fig. 215. — Diagrammatic illustration of unipolar machine with two high- speed collectors.

Such a machine is called a unipolar machine, as the conductor during its rotation traverses the same polarity, in distinction of bipolar or multipolar machines, in which the conductor during each revolution passes two or many poles. A more correct name is homopolar machine, signifying uniformity of polarity, or acyclic machine, signifying absence of any cyclic change: in all other electromagnetic machines, the voltage induced in a con- ductor changes cyclically, and the voltage in each turn is alter- nating, thus having a frequency, even if the terminal voltage and current at the commutator* are continuous.

450

UNIPOLAR MACHINES

45 1

By bringing the conductor, C, over the end of the magnet close to the shaft, as shown in Fig. 216, the peripheral speed of motion of brash, B 2 , on its collector ring can be reduced. However, at least one brush, B i, in Fig. 216, must bear on a collector ring (not shown in Figs. 215 and 216) at full conductor speed, because the total magnetic flux cut by the conductor, C, must passthrough this collector ring on which B i bears. Thus an essential char- acteristic of the unipolar machine is collection of the current from the periphery of the revolving conductor, at its maximum speed. It is the unsolved problem of satisfactory current collection from high-speed collector rings, at speeds of two or more miles per

Fig. 216. — Diagrammatic illustration of unipolar machine with one high- speed collector.

minute,, which has stood in the way of the commercial intro- duction of unipolar machines.

Electromagnetic induction is due to the relative motion of con- ductor and magnetic field, and every electromagnetic device is thus reversible with regards to stationary and rotary elements. However, the hope of eliminating high-speed collector rings in the unipolar machine, by having the conductor standstill and the magnet revolve, is a fallacy: in Figs. 215 and 216, the con- ductor, C, revolves, and the magnet, NS, and the external circuit, D, stands still. The mechanical reversal thus would be, to have the conductor, C, stand still, and the magnet, NS, and the external circuit revolve, and this would leave high-speed current collection.

Whether the magnet, NS, stands still or revolves, is immaterial in any case, and the question, whether the lines of force of the magnet are stationary or revolve, if the magnet revolves around its axis, is meaningless. If, with revolving conductor, C, and stationary external circuit, D , the lines of force of. the magnet are assumed as stationary, the induction is in C, and the return circuit in D ; if the lines of force are assumed as revolving, the

452

ELECTRICAL APPARATUS

induction is in D, and C is the return, but the voltage in the cir- cuit, CD, is the same. If, then, C and D both stand still, either there is no induction in either, or, assuming the lines of magnetic force to revolve, equal and opposite voltages are induced in C and D, and the voltage in circuit, CD, is zero just the same. However, the question whether the lines of force of a revolving magnet rotate or not, is meaningless for this reason : the lines of force are a pictorial representation of the magnetic field in space. The magnetic field at any point is characterized by an intensity and a-direction, and as long as intensity and direction at any point are constant or stationary, the magnetic field is constant or sta- tionary. This is the case in Figs. 215 and 216, regardless whether the magnet revolves around its axis or not, and the rotation of the magnet thus has no effect whatsoever on the induction phe- nomena. The magnetic field is stationary at any point of space outside of the magnet, and it is also stationary at any point of space inside of the magnet, even if the magnet revolves, and at the same time it is stationary also with regards to any element of the revolving magnet. Using then the pictorial representation of the lines of magnetic force, we can assume these lines of force as stationary in space, or as revolving with the rotating magnet, whatever best suits the convenience of the problem at hand: but whichever assumption we make, makes no difference on the solu- tion of the problem, if we reason correctly from the assumption.

248 . As in the unipolar machine each conductor (correspond- ing to a half turn of the bipolar or multipolar machine) requires a separate high-speed collector ring, many attempts have been made (and are still being made) to design a coil- wound unipolar machine, that is, a machine connecting a number of peripheral conductors in series, without going through collector rings. This is an impossibility, and unipolar induction, that is, continues induction of a unidirectional voltage, is possible only in an open conductor, but not in a coil or turn, as the voltage electromagnetically induced in a coil or turn must always be an alternating voltage.

The fundamental law of electromagnetic induction is, that the induced voltage is proportional to the rate of cutting of the con- ductor through the lines of force of the magnetic field. Applying this to a closed circuit or turn: every line of magnetic force cut by a turn must either go from the outside to the inside, or from the inside to the outside of the turn. This means: the voltage

UNIPOLAR MACHINES

45 3

induced in a turn is proportional (or equal, in absolute units) to the rate of change of the number of line's of magnetic force en- closed by the turn, and a decrease of the lines of force enclosed by the turn, induces a voltage opposite to that induced by an increase. As the number of lines of force enclosed by a turn can not perpetually increase (or decrease), it follows, that a voltage can not be induced perpetually in the same direction in a turn. Every increase of lines of force enclosed by the turn, inducing

Fig. 217. — Mechanical an- Fig. 218. — Mechanical analogy*of alogy of bipolar induction. unipolar induction.

a voltage in it, must sometime later be followed by an equal decrease of the lines of force enclosed by the turn, which induces an equal voltage in opposite direction. Thus, averaged over a sufficiently long time, the total voltage induced in a turn must always be zero, that is, the voltage, if periodical, must be alter- nating, regardless how the electromagnetic induction takes place, whether the turn is stationary or moving, as a part of a machine, transformer, reactor or any other electromagnetic induction device. Thus continuous-voltage induction in a closed turn is impossible, and the coil- wound unipolar machine thus a fallacy. Continuous induction in the unipolar machine is pos- sible only because the circuit is not a closed one, but consists of a conductor or half turn, sliding over the other half turn. Mechan- ically the relation can be illustrated by Figs. 217 and 218. If in Fig. 217 the carriage, C, moves along the straight track of finite length — a closed turn of finite area — the area, A, in front of C decreases, that B behind the carriage, C, increases, but this decrease and increase can not go on indefinitely, but at some time C reaches the end of the track, A has decreased to zero, B is a

454

ELECTRICAL APPARATUS

Fig. 219. — Drum type of unipolar machine with sta- tionary magnet core, section.

maximum, and any further change can only be an increase of A and decrease of B , by a motion of C in opposite direction, repre- senting induction of a reverse voltage. On the endless circular

track, Fig. 218, however, the carriage, C, can continuously move in the same direction, continuously reduce the area, A , in front and increase that of B behind C, corresponding to con- tinuous induction in the same direc- tion, .in the unipolar machine.

249 . In the industrial design of a unipolar machine, naturally a closed magnetic circuit would be used, and the form, Fig. 216, would be exe- cuted as shown in length section in Fig. 219. N is the same pole as in Fig. 216, but the magnetic return circuit is shown by S, concentrically surrounding N. C is the cylindrical con- ductor, revolving in the cylindrical gap be- tween N and S. Bi and B 2 are the two sets of brushes bearing on the collector rings at the end of the conductor, (7, and F is the field exciting winding.

The construction, Fig. 219, has the me- chanical disadvantage of a relatively light structure, C, revolving at high speed between two stationary structures, N and S. As it is immaterial whether the magnet is stationary or revolving, usually the inner core, N, is re- volved with the conductor, as shown in Figs. 221 and 222. This shortens the gap between N and S, but introduces an aux- iliary gap, G . Fig. 221 has the disadvantage of a magnetic end thrust, and thus the con- struction, Fig. 222, is generally used, or its duplication, shown in Fig. 223.

The disk type of unipolar machine, shown in section in Fig. 220, has been frequently proposed in former times, but is economically inferior to the construction of Figs. 221, 222 and 223. The limitation of the unipolar machine is the high collector speed. In Fig. 220, the .average conductor speed is less than the collector speed, and the latter thus relatively

Fig. 220.—” Disc type of unipolar ma- chine, section.

high'

spee<

Hi

G

Fig. 221. — Drum type of unipolar Fig. 222. — Drum type of unipolar machine with revolving magnet core machine with revolving magnet core and auxiliary end gap, section. and auxiliary cylinder gap, section.

derived in the unipolar machine by connecting a number of con- ductors in series. In this case, every series conductor obviously

Fig. 223. — Double drum type of unipolar machine, section.

requires a separate pair of collector rings. This is shown in Figs. 224 and 225, the cross-section and length section of the rotor of

Fig. 224. — Multi- Fig. 225. — Multi-conductor unipolar machine, conductor unipolar length section,

machine, cross-sec- tion.

a four-circuit unipolar. As seen in Fig. 224, the cylindrical con- ductor is slotted into eight sections, and diametrically opposite

456

ELECTRICAL APPARATUS

sections, 1 and 1', 2 and 2', 3 and 3', 4 and 4', are connected in multiple (to equalize the flux distribution) between four pairs of collector rings, shown in Fig. 225 as 1 and li, 2 and 2i, 3 and 3i, 4 and 4i. The latter are connected in series. This machine, Figs. 224 and 225, thus could also be used as a three-wire or five-wire machine, or as a direct-current converter, by bringing out intermediary connections, from the collector rings 2, 3, 4.

250. As each conductor of the unipolar machine requires a separate pair of collector rings, with a reasonably moderate number of collector rings, unipolar machines of medium capacity are suited for low voltages only, such as for electrolytic machines, and have been built for this purpose to a limited extent, but in general it has been found more economical by series connection of the electrolytic cells to permit the use of higher voltages, and then employ standard machines.

For commercial voltages, 250 or 600, to keep the number of collector rings reasonably moderate, unipolar machines require very large magnetic fluxes — that is, large units of capacity — and very high peripheral speeds. The latter requirement made this machine type unsuitable during the days of the slow-speed direct- connected steam engine, but when the high-speed steam turbine arrived, the study -of the design of high-powered steam-turbine- driven unipolars was undertaken, and a number of such machines built and installed.

In the huge turbo-alternators of today, the largest loss is the core loss: hysteresis and eddies in the iron, which often is more than all the other losses together. Theoretically, the unipolar machine has no core loss, as the magnetic flux does not change anywhere, and solid steel thus is used throughout — and has to be used, due to the shape of the magnetic circuit. However, with the enormous magnetic fluxes of these machines, in solid iron, the least variation of the magnetic circuit, such as caused by small unequalities of the air gap, by the reaction of the arma- ture currents, etc., causes enormous core losses, mostly eddies, and while theoretically the unipolar has no core loss, designing experience has shown, that it is a very difficult problem to keep the core loss in such machines down to reasonable values. Fur- thermore, in and at the collector rings, the magnetic reaction of the armature currents is alternating or pulsating. Thus* in Figs. 224 and 225, the point of entrance of the current from the arma- ture conductors into the collector rings revolves with the rotation

UNIPOLAR MACHINES

457

of the machine, and from this point flows through the collector ring, distributing between the next brushes. While this circular flow of current in the collector ring represents effectively a frac- tion of a turn only, with thousands of amperes of current it represents thousands of ampere-turns m.m.f., causing high losses, which in spite of careful distribution of the brushes to equalize the current flow in the collector rings, can not be entirely eliminated.