CHAPTER XX SINGLE-PHASE COMMUTATOR MOTORS I. General 189. Alternating-current commutating machines have so far become ef 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-cums! 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-cur rent 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- iiatiiig-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- erics the e.in.f. in the revolving armature conductors, which gives the motor output; the armature reaction, that is, the mag- net k Mux produced by the armature current, distorts and weakens the field, and requires a shifting of the brushes to avoid Bparldag due to the short-circuit current under the commutator brushes, and where the brushes can not l>e 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 333 in the field conductors an e.m.f. of self-inductance, which is not useful but wattless, and therefore harmful in lowering the power- factor, hence must be kept as low as possible. This e.m.f. of self-inductance of the field, e0, is proportional to the field strength, $, to the number of field turns, n0, and to the frequency, /, of the impressed e.m.f. : eo = 2 ir/no* 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, nh and to the fre- quency of rotation of the armature, /<>: e = 2ir/on1 10"8. (2) This later e.m.f., e, is in phase with the magnetic flux, $, 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, e0, is wattless, or in quadrature with the current, and the angle of lag of the motor current thus is given by: tan 6 = -^ (3) 6 -r it where ir = voltage consumed by the motor resistance. Or ap- proximately, since ir is small compared with e (except at very low speed) : tan 6 = -> (4) e and, substituting herein (1) and (2): tan 6 - { -°- (5) Small angle of lag and therewith good power-factor therefore require high values of /0 and n\ and low values of / and n0. High /o 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 ni and low n0 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 motor speed — 750 revolutions with a four-pole 25-cyclc motor — an armature reac- 334 ELECTRICAL APPARATUS tion, n,, equal to the field excitation, n0, would then give tan 6 = 1, 9 = 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, rti, is large compared with the field turns, n0, Let fflo = field reluctance, that is, reluctance of the magnetic field circuit, and n0 tan 0 - J ' (9) =mJ + ^ and this is a minimum; that is, the power-factor a maximum, for: ^-{tanfl} = 0, or: *o = ^ (ID and the maximum power-factor of the motor is then given by: tan 0o = / -> • (12) h \/b Therefore the greater b is the higher the power-factor that can be reached by proportioning field and armature so that Tli 1 no * y/b 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, 6 = 4, gives q = 2, that is, twice as many armature turns as field turns; rti = 2 n0. The angle of lag in this case is, by (12), at synchronism:/© = /, tan 0O = 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 moderate speed, from a straight series motor, even if such a design as discussed above were not inoperative, due to rate distortion and therefore destructive sparking. Thus it becomes necessary in the single-phase com mutator motor to reduce the magnetic flux of armature reaction, thai is, increase the effective magnetic reluctance of the armature fur beyond the value of the true magnetic reluctance. This is m- complished by the compensating winding devised by Eirkemeyer, by surrounding the armature with a stationary winding chist-ly adjacent and parallel to the armature winding, and energized by a current in opposite direction to the armature currem. ;imi ti the same m.m.f., that is, the same number of ampere-turns, the armature winding. s F N. ( / \ rf \ 1 f » e M / > C v / n SI y pha. commutator n 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, 0, is either connected in aeriea - Imt 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 i| ener- gized by the supply current, and the armature short-circuited as SINGLE-PHASE COMMUTATOR MOTORS 337 secondary. The first rase Rives the eonduetively compensated series motor, the second case the inductively compensated series motor, the third case the repulsion motor. In the first case, by giving the compensating winding more turns than the armature, overcompensation, by giving it lesB 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, *i, 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. The compensating winding, or the "cross field," thus fulfils the twofold purpose of reducing the armature self-inductance to that of the leakage flux, and of neutralizing the armature reac- tion and thereby permitting the use of very high armature ampere-turns. The main purpose of the compensating winding thus is to de- crease the armature self-inductance; that is, increase the effect- ive armature reluctance and thereby its ratio to the field reluc- tance, b, and thus permit the use of a much higher ratio, q = ', before maximum power-factor is reached, and thereby a higher power-factor. Even with compensating winding, with increasing q, ultimately a point is reached where the armature self-inductance equals the field self-inductance, and beyond this the power-factor again decreases. It becomes possible, however, by the use of the com- pensating winding, to reach, with a mechanically good design, values of 6 as high as 16 to 20. Assuming b = 16 gives, substituted in (11) and (12): s-4; 338 ELECTRICAL APPARATUS that is, four times as many armature turns as field turns, i*j 4 no and : tan ,. - fe hence, at synchronism: fa = / : tan 0O = 0.5, or 89 per cent, power-factor. At double synchronism, which about represents maximum motor speed at 25 cycles: /o = 2/ : tan 80 = 0.25, or 98 per cent, power-factor; that is, very good power-factors can be reached in the single- phase commutator motor by the use of a compensating winding, far higher than are possible with the same air gap in polyphase induction motors. III. Field Winding and Compensating Winding 192. The purpose of the field winding is to produce the maxi- mum magnetic flux, $, with the minimum number of turns, n,. This requires as large a magnetic section, especially at the air gap, as possible. Hence, a massed field winding with definite polar projections of as great pole arc as feasible, as shown in Fig. 157, gives a better power-factor than a distributed field winding. The compensating winding must be as closely adjacent, to the armature winding as possible, so as to give minimum teoksfj flux between armature conductors and compensating conductors, and therefore is a distributed winding, located in the field poll faces, as shown in Fig. 1,57. The armature winding is distributed over the whole timuO; feretice of the armature, but the compensating winding only in the field pole faces. With the same ampere-turns in armature and compensating winding, their resultant ampere-turns are equal and opposite, and therefore neutralize, but locally the two windings do not neutralize, due to the difference in the distribu- tion curves of their m.m.fs. The m.m.f. of the field winding is constant over the pole faces, and from one pole corner to the next pole corner reverses in direction, as shown diagninniui i. ■:! . by F in Fig. 158, which is the development of Fig. 157. The m.m.f. of the armature is a maximum at the brushes, midway between the field poles, as shown by A in Fig. 158, and from there decreases to zero in the center of the field pole. The m.m.f. of SINGLE-PHASE COMMUTATOR MOTORS 339 the compensating winding, however, is constant in the space from pole corner to pole corner, as shown by C in Fig. 158, and since the total m.m.f. of the compensating winding equals that of the armature, the armature m.m.f. is higher at the brushes, the compensating m.m.f. higher in front of the field poles, as shown by curve R in Fig. 158, which is the difference between A and C; that is, with complete compensation of the resultant armature and compensating winding, locally undercompensation exists at the brushes, overcompensation in front of the field Fio. 158. — Distribution of m.m.f. in compensated motor. poles. The local undercompensated armature reaction at the brushes generates an e.m.f. in the coil short-circuited under the brush, and therewith a short-circuit current of commutation and sparking. In the conductively compensated motor, this can be avoided by overcompensation, that is, raising the flat top of the compensating m.m.f. to the maximum armature m.m.f., but this results in a lowering of the power-factor, due to the self- inductive flux of overcompensation, and therefore is undesirable. 193. To get complete -compensation even locally requires the compensating winding to give the same distribution curve as the armature winding, or inversely. The former is accomplished by distributing the compensating winding around the entire cir- cumference of the armature, as shown in Fig. 159. This, how- ever, results in bringing the field coils further away from the armature surface, aftd so increases the magnetic stray flux of the field winding, that is, the magnetic flux, which passes through the field coils, and there produces a reactive voltage of self-in- 340 ELECTRICAL APPARATUS ductance, but does not pass through the armature conductor? and so does no work; that is, it lowers the power factor, just overcompensation would do. The distribution curve of the armature winding can, however, W made equal to that of the compen- sating winding, and therewith local complete compensation secured, by using a fractional pitch armature winding of a pitch equal to the pole arc. In this case, in the space be- tween the pole corners, the current* are in opposite direction in the upper and the lower layer of con- ductors in each armature slot, shown in Fig. 160, ami thus DeutmUlB magnetically; that is, the armature reaction extends only over the spacr of the armature circumference covered by the pole arc, where it is neutralized by the compensating winding in the pole face. To produce complete compensation even locally, without im- pairing tbe~power-factor, therefore, requires a fractional-pitch Fio. 159.— Completely distributed compensating winding. armature winding, of a pitch equal to the field pole arc, or s< equivalent arrangement. Historically] the first compensated single-phase commutfttov motors, built about 20 years ago, were Prof. Elihu Thomeea^ repulsion motors. In these the field winding and coin pen sating SINGLE-PHASE COMMUTATOR MOTORS 341 winding were massed together in a single coil, as shown diagram- matically in Fig. 161. Repulsion motors are still occasionally built in which field and compensating coils are combined in a single distributed winding, as shown in Fig. 162. Soon after the first repulsion motor, conductively and inductively compensated series motors were built by Eickemeyer, with a massed field winding and a separate compensating winding, or cross coil, either as single coil or turn or distributed in a number of coils or turns, as shown diagrammatically in Fig. 163, and by W. Stanley. QC \±is Fig. 162. — Repulsion motor with Fig. 163. — Eickemeyer inductively distributed winding. compensated series motor. For reversible motors, separate field coils and compensating coils are always used, the former as massed, the latter as dis- tributed winding, since in reversing the direction of rotation either the field winding alone must be reversed or armature and compensating winding are reversed while the field winding re- mains unchanged. IV. Types of Varying-speed Single-phase Commutator Motors 194. The armature and compensating windings are in induc- tive relations to each other. In the single-phase commutator motor with series characteristic, armature and compensating windings therefore can be connected in series with each other, or the supply voltage impressed upon the one, the other closed upon itself as secondary circuit, or a part of the supply voltage im- pressed upon the one, and another part upon the other circuit, and in either of these cases the field winding may be connected in series either to the compensating winding or to the armature winding. This gives the motor types, denoting the armature by 342 ELECTRICAL APPARATUS (D (4) (2) (6) (3) (6) (7) Fio. 164. — Types of alternating-current coramutating motors. SINGLE-PHASE COMMUTATOR MOTORS 343 * • A, the compensating winding by C, and the field winding by F, shown in Fig. 164. Primary Secondary A+F • • • Series motor. A + C + F • • • Conductively compensated series motor. (1) A +F C Inductively compensated series motor. (2) A C + F Inductively compensated series motor with second- ary excitation, or inverted repulsion motor. (3) C + F A Repulsion motor. (4) C A +F Repulsion motor with sec- ondary excitation. (5) A+F,C ■■■} • • • j Series repulsion motors. A, C + F (6) (7) Since in all these motor types all three circuits are connected directly or inductively in series with each other, they all have the same general characteristics as the direct-current series motor; that is, a speed which increases with a decrease of load, and a torque per ampere input which increases with increase of current, and therefore with decrease of speed, and the different motor types differ from each other only by their commutation as affected by the presence or absence of a magnetic flux at the brushes, and indirectly thereby in their efficiency as affected by commutation losses. In the conductively compensated series motor, by the choice of the ratio of armature and compensating turns, overcompensa- tion, complete compensation, or undercompensation can be pro- duced. In all the other types, armature and compensating windings are in inductive relation, and the compensation there- fore approximately complete. A second series of motors of the same varying speed charac- teristics results by replacing the stationary field coils by arma- ture excitation, that is, introducing the current, either directly or by transformer, into the armature by means of a second set of brushes at right angles to the main brushes. Such motors are used to some extent abroad. They have the disadvantage of 344 ELECTRICAL APPARA TU8 Fig. 1(55.- req Hiring two sots of brushes, but the advantage that their power-factor can be controlled and above synchronism even lending current produced. Fig. ll>5 shows diagrammatical!}- surli a motor, as designed by Winter- Eichberg-Latour, the so-called compensated repulsion motor. In this case componsatei! meant: compensated for power-factor. The voltage which can be used in the motor armature is limited by the commutator: the voltage per commutator segment is limited by the problem of sparkless commutation, the number of commutator segments Frew brush to brush is limited mechanical consideration of commutator speed and width of segments. In those motet types in which the supply cur- rent traverses the armature, the supply voltage is thus limited to values even lower than in the direct-current motor, while in the repulsion motor (4 and 5), in which the armature is the secondary circuit, the armature voltage is independent of the supply voltage, so can be chosen to suit the requirement! i ■: commutation, while the motor can be built for any supply voltage for which the stator can economically 1m? insulated. Alternating-current motors as well as direct-current scries motors can be controlled by series parallel connection of two or more motors. Further control, as in starting, with direct -current motors is carried out by rheostat, while with alternating-current motors potential conlrol, that is, a change of supply voltage by transformer or autotransformer, offers a more efficient method of control. By changing from one motor type to another motor type, potential control can bo, used in alternating-current motors without any change of supply voltage, by appropriately choosing the ratio of turns of primary and secondary circuit. For in- stance, with an armature wound for half the voltage and thus twice the current as the compensating winding (ratio of turns - = 2) , a change of connection from tvpc 3 to type 2, or from type 5 to type 4, results in doubling the field current and there- SINGLE-PHASE COMMUTATOR MOTORS 345 with the field strength. A change of distribution of voltage be- tween the two circuits, in types 6 and 7, with A and C wound for different voltages, gives the same effect as a change of supply voltage, and therefore is used for motor control. 196. In those motor types in which a transformation of power occurs between compensating winding, C, and armature winding, A, a transformer flux exists in the direction of the brushes, that is, at right angles to the field flux. In general, therefore, the single-phase commutator motor contains two magnetic fluxes in quadrature position with each other, the main flux or field flux, A', in the direction of the axis of the field coils, or at right angles to the armature brushes, and the quadrature flux, or transformer flux, or commu taring flux, *j, in line with the armature brushes, or in the direction of the axis of the compensating winding, that is, at right angles (electrical) with the field flux. The field flux, *, depends upon and is in phase with the field current, except as far as it is modified by the magnetic action of the short-circuit current in the armature coil under the commu- tator brushes. In the conductively compensated series motor, 1, the quad- rature flux is zero at complete compensation, and in the direc- tion of the armature reaction with undercompensation, in oppo- sition to the armature reaction at overcompensation, but in either ease in phase with the current and so approximately with the field. In the other motor types, whatever quadrature flux exists is not in phase with the main flux, but as transformer flux is due to the resultant m.ui.f. of primary and secondary circuit. In a transformer with non-inductive or nearly non-inductive secondary circuit, the magnetic flux is nearly 90° in time phase behind the primary current, a little over 90° ahead of the sec- ondary current, as shown in transformer diagram, Tig. 166. In a transformer with inductive secondary, the magnetic flux is less than 90" liehind the primary current, more than 90° ahead of the secondary current, the more so the higher is the inductivity of the secondary circuit, as shown by the transformer diagram, Fig. 166. Herefrom it follows that: In the inductively compensated series motor, 2, the quad- rature flux is very small and practically negligible, as very little voltage is consumed in the low impedance of the secondary cir- cuit, C; whatever flux there is, lags behind the main flux. 346 ELECTRICAL APPARATUS In the inductively compensated series ipotor with secondary excitation, or inverted repulsion motor, 3, the quadrature flux, $1, is quite large, as a considerable voltage is required for the field excitation, especially at moderate speeds and therefore high currents, and this flux, $i, lags behind the field flux, $, but this lag is very much less than 90°, since the secondary circuit is •-J* Fig. 166. — Transformer diagram, inductive and non-inductive load. highly inductive; the motor field thus corresponding to the con- ditions of the transformer diagram, Fig. 166. As result hereof, the commutation of this type of motor is very good, flux, $i, having the proper phase and intensity required for a commu- tating flux, as will be seen later, but the power-factor is poor. In the repulsion motor, 4, the quadrature flux is very consid- erable, since all the voltage consumed by the rotation of the armature is induced in it by transformation from the compen- SINGLE-PHASE COMMUTATOR MOTORS 347 sating winding, and this quadrature flux, *i, laps nearly 90° be- hind the main flux, *, since the secondary circuit is nearly non- inductive, especially at speed. In the repulsion motor with secondary excitation, 5, the quad- rature flux, *i, is also very large, and practically constant, corre- sponding to the impressed e.m.f., but lags considerably less than 90° behind the main flux, $, the secondary circuit being induct- ive, since it contains the field coil, F. The lag of the flux, *i, increases with increasing speed, since with increasing speed the e.m.f. of rotation of the armature increases, the e.m.f. of self- inductance of the field decreases, due to the decrease of current, and the circuit thus becomes less inductive. The series repulsion motors 6 and 7, give the same phase rela- tion of the quadrature flux, $i, as the repulsion motors, 5 and 6, but the intensity of the quadrature flux, $i, is the less the smaller the part of the supply voltage which is impressed upon the com- pensating winding. V. Commutation 196. In the commutator motor, the current in each armature coil or turn reverses during its passage under the brush. In the armature coil, while short-circuited by the commutator brush, the current must die out to zero and then increase again to its original value in opposite direction. The resistance of the arma- ture coil and brush contact accelerates, the self-inductance re- tards the dying out of the current, and the former thus assists, tin1 latter impairs commutation. If an e.m.f. is generated in the armature coil by its rotation while short-circuited by the commutator brush, this e.m.f. opposes commutation, that is, retards the dying out of the current, if due to the magnetic flux of armature reaction, and assists commutation by reversing the armature current, if due to the magnetic flux of overcompensa- tion, that is, a magnetic flux in opposition to the armature reaction. Therefore, in the direct-current commutator motor with high field strength and low armature reaction, that is, of negligible magnetic flux of armature reaction, fair commutation is produced with the brushes set midway between the field poles — -that is, in the position where the armature coil which is being commu- tated encloses the full field flux and therefore cuts no flux and has no generated e.m.f. — by using high-resistance carbon brushes, 348 ELECTRICAL APPARATUS as the resistance of the brush contact, increasing when the arma- ture coil begins to leave the brush, tends to reverse the current. Such "resistance commutation" obviously can not be perfect; perfect commutation, however, is produced by impressing upon the motor armature at right angles to the main field, thai is, UD the position of the commutator brushes, a magnetic field oppo- site to that of the armature reaction and proportional to the armature current. Such a field is produced by overcompensa- tion or by the use of a commutating pole or interpole. As seen in the foregoing, in the direct-current motor t he counter e.m.f. of self-inductance of commutation opposes the reversal of current in the armature coil under the commutator brush, and this can be mitigated in its effect by the use of high-resistance brushes, and overcome by the commutating field of overcompen- sation. In addition hereto, however, in the alternating-current commutator motor an e.m.f. is generated in the coil short-cir- cuited under the brush, by the alternation of the magnetic flux, and this e.m.f., which does not exist in the direct-current motor, makes the problem of commutation of the alternating-current motor far more difficult. In the position of commutation no e.m.f. is generated in the armature coil by its rotation through the magnetic field, as in this position the coil encloses the maxi- mum field flux; but as this magnetic flux is alternating, in this position the e.m.f. generated by the alternation of the flux en- closed by the coil is a maximum. This "e.m.f. of alternation*1 lags in time 90° behind the magnetic flux which generates it, h proportional to the magnetic flux and to the frequency, but is independent of the speed, hence exists also at standstill, while the "e.m.f. of rotation" — which is a maximum in the position of the armature coil midway between the brushes, or parallel to the field flux — is in phase with the field flux and proportional thereto and to the speed, but independent of the frequency. In the alternating-current commutator motor, no position therefore exists in which the armature coil is free from a generated e.m.f., but in the position parallel to the field, or midway between tin brushes, the e.m.f. of rotation, in phase with the field flux, is a maximum, while the e.m.f. of alternation is zero, and in the posi- tion under the commutator brush, or enclosing the total field flux, the e.m.f. of alternation, in electrical space quadrature with the field flux, is a maximum, the e.m.f. of rotation absent, while in any other position of the armature coil its generated e.m.f. has SINGLE-PHASE COMMUTATOR MOTORS 349 a component due to the rotation — a power e.m.f. — and a com- ponent due to the alternation — a reactive e.m.f. The armature coils of an alternating-current commutator motor, therefore, are the seat of a system of polyphase e.m.f s., and at synchronism the polyphase e.m.fs. generated in all armature coils are equal, above synchronism the e.m.f. of rotation is greater, while below synchronism the e.m.f. of alternation is greater, and in the latter case the brushes thus stand at that point of the com- mutator where the voltage between commutator segments is a maximum. This e.m.f. of alternation, short-circuited by the armature coil in the position of commutation, if not controlled, causes a short-circuit current of excessive value, and therewith destructive sparking; hence, in the alternating-current commuta- tor motor it is necessary to provide means to control the short- circuit current under the commutator brushes, which results from the alternating character of the magnetic flux, and which docs not exist in the direct-current motor; that is, in the alternating- current motor the armature coil under the brush is in the posi- tion of a short-circuited secondary, with the field coil as primary of a transformer; and as in a transformer primary and secondary ampere-turns are approximately equal, if n0 = number of field turns per pole and i = field current, the current in a single arma- ture turn, when short-circuited by the commutator brush, tends to become io = n0i, that is, many times full-load current; and as this current is in opposition, approximately, to the field cur- rent, it would demagnetize the field; that is, the motor field vanishes, or drops far down, and the motor thus loses its torque. Especially is this the case at the moment of starting; at speed, the short-circuit current is somewhat reduced by the self-induc- tance of the armature turn. That is, during the short time during which the armature turn or coil is short-circuited by the brush the short-circuit current can not rise to its full value, if the speed is considerable, but it is still sufficient to cause destruc- tive sparking. 197. The character of the commutation of the motor, and therefore its operativeness, thus essentially depends upon the value and the phase of the short-circuit currents under the com- mutator brushes. An excessive short-eimiit current, gives de- structive sparking by high-current density under the brushes and arcing at the edge of the brushes due to the great and sud- den change of current in the armature coil when leaving the 350 ELECTRICAL APPARATUS brush. But even with a moderate short-circuit current, the sparking at the commutator may be destructive and the motor therefore inoperative, if the phase of the short-circuit current greatly differs from that of the current in the armature coil after it leaves the brush, and so a considerable and sudden change of VI LTS n - / ' I ' a n ' 0 1 AMP a i PKI SQ. H. 0 1GO1SO2O02202W26O29O3W Fin. 167.— E.m.f. consumed at contact of copper brush. current must take place at the moment when the armature coil leaves the brush. That is, perfect commutation occurs, if the short-circuit current in the armature coil under the commutator brush at the moment when the coil leaves the brush has the same value and the same phase as the main-armature current in __v_ LT_=. 0 I 2 | 1 fly 0 .ft 0 HJ 0 I 0 1 0 1 0 i B 1 0 i ■0 . 168.— E.m.f. consumed ft t of higli-i earhon brush. the coil after leaving the brush. The commutation of such a motor therefore is essentially characterized by the difference between the main-armature current after, and the short-circuit current before leaving the brush. The investigation of the short- circuit current under the commutator brushes therefore is of SINGLE-PHASE COMMUTATOR MOTORS 351 fundamental importance in the study of the alternating-current commutator motor, and the control of this short-circuit current the main problem of alternating-current commutator motor design. Various means have been proposed and tried to mitigate or eliminate the harmful effect of this short-circuit current, as high resistance or high reactance introduced into the armature coil during commutation, or an opposing e.m.f . either from the out- side, or by a commutating field. High-resistance brush contact, produced by the use of very narrow carbon brushes of high resistivity, while greatly improv- ing the commutation and limiting the short-circuit current so that it does not seriously demagnetize the field and thus cause the motor to lose its torque, is not sufficient, for the reason that the resistance of the brush contact is not high enough and also is not constant. The brush contact resistance is not of the nature of an ohmic resistance, but more of the nature of a counter e.m.f.; that is, for large currents the potential drop at the brushes becomes approximately constant, as seen from the volt-ampere characteristics of different brushes given in Figs. 167 and 168. Fig. 167 gives the voltage consumed by the brush contact of a copper brush, with the current density as abscissae, while Fig. 168 gives the voltage consumed by a high-resistance carbon brush, with the current density in the brush as absciss®. It is seen that such a resistance, which decreases approximately in- versely proportional to the increase of current, fails in limiting the current just at the moment where it is most required, that s, at high currents. Commutator Leads 198. Good results have been reached by the use of metallic resistances in the leads between the armature and the commuta- tor. As shown diagrammatically in Fig. 169, each commutator segment connects to the armature, A , by a high non-inductive resistance, CBy and thus two such resistances are always in the circuit of the armature coil short-circuited under the brush, but also one or two in series with the armature main circuit, from brush to brush. While considerable power may therefore l>c consumed in these high-resistance leads, neverthelebs the effi- ciency of the motor is greatly increased by their use; that is, the reduction in the loss of power at the commutator by the reduction 352 ELECTRICAL APPARATUS of the short-circuit current, usually is far greater than the mfltt of power in the resistance leads. To have any apprecial <[•■ i-ffoci , the resistance of the commutator lead must lie far higher thau that of the armature coil to which it connects. Of the e.m.f. of rotation, that is, the useful generated e.m.f., the armature re- sistance consumes only a very small part, a few per cent. only. The e.m.f. of alternation is of the same magnitude as the e.in.f. of rotation — higher below, lower above synchronism. With B short-circuit current equal to full-load current, the resistance of Fee. 160, — Commutation with resistance leads. the short-circuit coil would consume only a small part of the e.m.f. of alternation, and to consume the total e.m.f. the short- circuit current therefore would have to lie about as many times larger than the normal armature current as the useful generated e.m.f. of the motor is larger than the resistance drop in the arma- ture. Long before this value of short-circuit current is reached the magnetic field would have disappeared by the demagnetuui| force of the short-circuit current, that is, the motor would have lost its torque. To limit the short-circuit current under the brush to a value not very greatly exceeding full-load current, thus requires a re- sistance of the lead, many times greater than that of (he animt un- coil. The i-r in the lead, and thus the heat produced in it, then, is many times greater than that in the armature coil. The space available for the resistance lead is, however, less than that avail- able for the armature coil. It is obvious herefrom that it is not feasible to build these resistance leads so that each lead can dissipate continuously, or even for any appreciable time, without rapid si -If-drst ruction, the heat produced in it while in circuit. When the motor is revolving, even very slowly, thiaiaatH DM essary, since each resistance lead is only a very short tmn- in SINGLE-PHASE COMMUTATOR MOTORS circuit, during the moment when the armature c to it are short-circuited by the brushes; that is, if t i connecting ■ number of armature turns from brush to brush, the lead is only - of the time in circuit, and though excessive current densities in mate* rials of high resistivity are used, the heating is moderate. In starting the motor, however, if it does not start instantly, the current continues to flow through the same resistance leads, and thus they are overheated and destroyed if the motor does not start promptly. Hence care has to be taken not to have such motors stalled for any appreciable time with voltage on. The most serious objection to the use of high- re si stance leads, therefore, is their liability to self-destruction by heating if the motor fails to start immediately, as for instance in a railway motor when putting the voltage on the motor before the brakes are released, as is done when starting on a steep up-grade to keep the train from starting to run back. Thus the advantages of resistance commutator leads are the improvement in commutation resulting from the reduced short- circuit current, and the ahsence o fa serious demagnetizing effect on the field at the moment of starting, which would result from an excessive short-circuit current under the brush, and such leads are therefore extensively used ; their disadvantage, however, is that when they are used the motor must be sure to start im- mediately by the application of voltage, otherwise they are liable to l>e destroyed. It is obvious that even with high -resistance commutator leads the commutation of the motor can not be as good as that of the motor on direct-current supply; that is, such an alternating- current motor inherently is more or less inferior in commutation to the direct-current motor, and to compensate for this effect far more favorable constants must be chosen in the ruotui design than permissible with a direct-current motor, that is, a lower voltage per commutator segment and lower magnetic flux per pole, hence a lower supply voltage on the armature, and thus B larger armature current and therewith a larger commutator, etc. The insertion of reactance instead of resistance in the leads connecting the commutator segments with the armature nib of the single-phase motor also has Ix-i-ri proposed And UWd f'ir limiting the short-circuit current, under I lie commutator brush. Reactance has the advantage over resistance, that the voltage 864 ELECTRICAL APPABA TVS consumed by it is wattless and therefore produces no scrum- heating and reactive leads of low resistance thus are not liable to self-destruction by heating if the motor fails to start im- mediately. On account of the limited apace available in the railway motor considerable difficulty, however, is found in designing sufficiently high reactances which du not saturate and thus decrease nt larger currents. At speed, reactance in the armature coils is very objectionable in retarding the reversal of current, and indeed one of the most important problems in the design of commu.tating machines give the armature coils the lowest possible reactance. There- fore, the insertion of reactance in the motor leads tnterfffW seriously with the commutation of the motor at speed, and ihn- requires the use of a suitable commutating or reversing flux, thai is, a magnetic field at the commutator brushes of sufficient strength to reverse the current, against the self-inductance of the armature coil, by means of an e.m.f. generated in the armature coil by its rotation. This commutating flux thus must he m phase with the main current, that is, a flux of overcompensation. Reactive leads require the use of a commutating flux of over- compensation to give fair commutation at speed. Counter E.m.fs. in Commutated Coil 199. Theoretically, the correct way of eliminating the de- structive effect of the short-circuit current under the tutor brush resulting from the e.m.f. of alternation of the main flux would be to neutralize the e.m.f. of alternation by an equal but opposite e.m.f. inserted into the armature coil or generated therein. Practically, however, at least with most motor typos, considerable difficulty is met in producing such a neutralizing e.m.f. of the proper intensity as well as phase. Since the alter- nating current has not only an intensity but also a phase displace- ment, with an alternating-current motor the production of com mutating flux or commutating voltage is more difficult than with direct-current motors in which the intensity is the only v;n t.iM. By introducing an external e.m.f. into the short-circuited under the brush it is rml possible entirely to neutralise itfl BJB ' of alternation, hut simply to reduce it to one-half. Several such arrangements were developed in the early days by Ekkettoyar, SINGLE-PHASE COMMUTATOR MOTORS 355 for instance the arrangement shown in Fig. 170, which represents the development of a commutator. The commutator consists of alternate live segments, S, and dead segments, S', that is, seg- ments not connected to armature coils, and shown shaded in Fig. 170. Two sets of brushes on the commutator, the one, Bt, \MMS\m Fig. 170. — Commutation with external e.m.f, ahead in position from the other, Bt, by one commutator seg- ment, and connected to the first by a coil, N, containing an e.m.f. equal in phase, but half in intensity, and opposite, to the e.m.f. of alternation of the armature coil; that is, if the armature coil contains a single turn, coil A' is a half turn located in the main Fio. 171. — Commutation by external e field space; if the armature coil, A, contains m turns, '„ turns in the main field space are used in coil, N. The dead segments, S', are cut between the brushes, Bx and /Jj, so as not to short-circuit between the brushes. In this manner, during the motion of the brush over the com- 356 ELECTRICAL APPARATUS mutator, as shown by Fig. 171 in its successive steps, in position: 1. There is current through brush, B\\ 2. There is current through both brushes, Si and B«, and the armature coil, A, is closed by the counter e.m.f. of coil, .V, that is, the difference, A — JV, is short-circuited; 3. There is current through brush B3; 4. There is current through both brushes, B, and Bt, and the coil, JV, is short-circuited; 5. The current enters again by brush Bi; thus alternately the coil, JV, of half the voltage of the armature coil, A, or the difference between A and JV is short-circuited, that is, the short-circuit current reduced to one-half. Complete elimination of the short-circuit current can be pro- duced by generating in the armature coil an opposing e.m.f. This e.m.f. of neutralization, however, can not be generated by the alternation of the magnetic flux through the coil, as this would require a flux equal but opposite to the full field flux travers- ing the coil, and thus destroy the main field of the motor. The neutralizing e.m.f., therefore, must be generated by the rotetifin of the armature through the commutating field, and thus can occur only at speed; that is, neutralization of the short-circuit current is possible only when the motor is revolving, but not while at rest. 200. The e.m.f. of alternation in the armature coil short-cir- cuited under the commutator brush is proportional to the main field, *, to the frequency, /, and is in quadrature with the main field, being generated by its rate of change; hence, it can be rep- resented by eo -2r/*10-*/. (17) The e.m.f., e,, generated by the rotation of the armature coil through a commutating field, *', is, however, in phase with the field which produces it; and since d must be equal and in phase with e0 to neutralize it, the commutating field, *', therefore, must be in phase with e0, hence in quadrature with *; that is, the com- mutating field, *', of the motor must be in quadrature witfa tin main'field, *, to generate a neutralizing voltage, e,, of the proper phase to oppose the e.m.f. of alternation in the short-circuited coil. This e.m.f., ei, is proportional to its generating field. *', and to the speed, or frequency of rotation, f„, hence is: ei =2t/„*'10-s, ,lv. SINGLE-PHASE COMMUTATOR MOTORS nil Si = p„ it then follow* that: *' = j-t /.' (19) ] I 1 I that is, the commutating field of the single-phase motor must be in quadrature behind and proportional to the main field, pro- portional to the frequency and inversely proportional to the speed; hence, at synchronism, /» = /, the commutation field equals the main field in intensity, and, being displaced therefrom 1 quadrature both in time and in space, the motor thus must have a uniform rotating field, just as the induction motor. Above synchronism, fa > f, the commutating field, *', is less than the main field; below synchronism, however, /& < /, the commutating field must be greater than the main field to give complete compensation. It obviously is not feasible to increase the commutating field much beyond the main field, u this would require an increase of the iron section of the motor beyond that required to do the work, that is, to carry the main field flux. At standstill *' should be infinitely large, that is, compensation is not possible. Hence, by the use of a commutating field in time and space quadrature, in the single-phase motor the short-circuit current under the commutator brushes resulting from the e.m.f. of alter- nation can be entirely eliminated at and above synchronism, and more or less reduced below synchronism, the more the nearer the speed is to synchronism, but no effect can be produced at standstill. In such a motor either some further method, as re- sistance leads, must, be used to take care of the short-circuit cur- rent at standstill, or the motor designed so that its commutator can carry the short-circuit current for the small fraction of time when the motor is at. standstill or running at very low speed. The main field, *,of the series motor is approximately inversely proportional to the speed, /0, since the product of speed and field strength, /0*, is proportional to the e.m.f. of rotation, or useful e.m.f. of the motor, hence, neglecting losses anil phase displace- ments, to the impressed e.m.f., that is, constant. Substituting . — a. n>tior» &.. — main field at synchronism, into 358 ELECTRICAL APPARATUS that is, the commutating field is inversely proportional to the square of the speed; for instance, at double synchronism it should be one-quarter as high as at synchronism, etc. 201. Of the quadrature field, ', only that part is needed for commutation which enters and leaves the armature at the posi- tion of the brushes; that is, instead of producing a quadrature field, ', in accordance with equation (20), and distributed around the armature periphery in the same manner as the main field, ♦, but in quadrature position thereto, a local commutating field may be used at the brushes, and produced by a commutating pole or commutating coil, as shown diagrammatically in Fig. 172 Fig. 172. — Commutation with commutating poles. as K\ and K. The excitation of this commutating coil, A', then would have to be such as to give a magnetic air-gap density ehind it, the mag- netic flux of the commutating poles, K, can be produced by ener- gizing these poles by an e.m.f. e, which is varied with the speed of the motor, by equation: e = * (£) -, whore e„ is its proper value at synchronism. (22) SINGLE-PHASE COMMUTATOR MOTORS 359 Since (B' lags 90° behind its supply voltage, e, and also lags 90° behind (B, by equation (2), and so behind the supply current and, approximately, the supply e.m.f. of the motor, the voltage, e, required for the excitation of the commutating poles is approxi- mately in phase with the supply voltage of the motor; that is, a part thereof can be used, and is varied with the speed of the motor. Perfect commutation, however, requires not merely the elimi-' nation of the short-circuit current under the brush, but requires a reversal of the load current in the armature coil during its passage under the commutator brush. To reverse the current, an e.m.f. is required proportional but opposite to the current and therefore with the main field; hence, to produce a reversing e.m.f. in the armature coil under the commutator brush a second com- mutating field is required, in phase with the main field and ap- proximately proportional thereto. The commutating field required by a single-phase commutator motor to give perfect commutation thus consists of a component in quadrature with the main field, or the neutralizing component, which eliminates the short-circuit current under the brush, and a component in phase with the main field, or the reversing com- ponent, which reverses the main current in the armature coil under the brush; and the resultant commutating field thus must lag behind the main field, and so approximately behind the sup- ply voltage, by somewhat less than 90°, and have an intensity varying approximately inversely proportional to the square of the speed of the motor. Of the different motor types discussed under IV, the series motors, 1 and 2, have no quadrature field, and therefore can be made to commutate satisfactorily only by the use of commutator leads, or by the addition of separate commutating poles. The inverted repulsion motor, 3, has a quadrature field, which de- creases with increase of speed, and therefore gives a better com- mutation than the series motors, though not perfect, as the quad- rature field does not have quite the right intensity. The repulsion motors, 4 and 5, have a quadrature field, lag- ging nearly 90° behind the main field, and thus give good com- mutation at those speeds at which the quadrature field has the right intensity for commutation. However, in the repulsion motor with secondary excitation, 5, the quadrature field is con- stant and independent of the speed, as constant supply voltage 360 ELECTRICAL APPARATUS is impressed upim the commutating winding, C, which produces the quadrature field, and in the direct repulsion motor, 4, the quadrature field increases with the speed, as the voltage consumed by the main field F decreases, and that left for the compensating winding, C, thus increases with the speed, while to give proper commutating flux it should decrease with the square of the speed. It thus follows that the commutation of the repulsion motors improves with increase of speed, up to that speed where the quadrature field is just right for commutating field — which is about at synchronism — but above this speed the commutation rapily becomes poorer, due to the quadrature field being far in excess of that required for commutating. In the series repulsion motors, 6 and 7, a quadrature field also exfsts, just as in the repulsion motors, but this quadrature field depends upon that part of the total voltage which is impressed upon the commutating winding, C, and thus can be varied by varying the distribution of supply voltage between the two cir- cuits; hence, in this type of motor, the commutating flux can be maintained through all (higher) speeds by impressing the total voltage upon the compensating circuit and short-circuiting the armature circuit for all speeds up to that at which the required commutating flux has decreased to the quadrature, flux given by the motor, and from this speed upward only a part of the supply voltage, inversely proportional (approximately) to the square of the speed, is impressed upon the compensating circuit, the rest shifted over to the armature circuit. The difference between 6 and 7 is that in 6 the armature circuit is more inductive, and the quadrature flux therefore lags less behind the main flux than in 7, and by thus using more or less of the field coil in the arma- ture circuit its inductivity can be varied, and therewith the phase displacement of the quadrature flux against, the main flux adjusted from nearly 90° lag to considerably less lag, hence not only the proper intensity but also the exact phase of the required commutating flux produced. As seen herefrom, the difference between the different motor types of IV is essentially found in their different actions regarding commutation. It follows herefrom that by the selection of the motor-type quadrature fluxes, *i, can be impressed upon the motor, as com- mutating flux, of intensities and phase displacements against the main flux, *, varying over a considerable range. The main SINGLE-PHASE COMMUTATOR MOTORS 361 advantage of the series-repulsion motor type is the possibility which this type affords, of securing the proper commutating field at all speeds down to that where the speed is too low to induce sufficient voltage of neutralization at the highest available commutating flux. VI. Motor Characteristics 202. The single-phase commutator motor of varying speed or series characteristic comprises three circuits, the armature, the compensating winding, and the field winding, which are connected in series with each other, directly or indirectly. The impressed e.m.f. or supply voltage of the motor then con- sists of the components: 1. The e.m.f. of rotation, eh or voltage generated in the arma- ture conductors by their rotation through the magnetic field, $. This voltage is in phase with the field, $>, and therefore approxi- mately with the current, i, that is, is power e.m.f., and is the voltage which does the useful work of the motor. It is propor- tional to the speed or frequency of rotation,/o, to the field strength, $, and to the number of effective armature turns, tii. «i = 2ir/0n1 10"8. (23) The number of effective armature turns, nif with a distributed winding, is the projection of all the turns on their resultant direc- tion. With a full-pitch winding of n series turns from brush to brush, the effective number of turns thus is: ♦* 2 fii = m [avg cos] \ « m. (24) With a fractional-pitch winding of the pitch of r degrees, the effective number of turns is: fix = m [avg cos] / « m sin * (2/5) 2. The e.m.f. of alternation of the field, e«, tliat is, the voltage generated in the field turns by the alternation of the magnetic flux, 4>, produced by them and thus enclosed by them. This vol- tage is in quadrature with the field flux, 4>, and thus approxi- mately with the current [, is proportional to tin* frequency of the 362 ELECTRICAL APPARATUS impressed voltage, /, to the field strength, 4>, and to the number of field turns, n„. «o = 2jirfn0* 10~8. (26) 3. The impedance voltage of the motor: e' = IZ (27) and: Z = r + jx, where r = total effective resistance of field coils, armature with commutator and brushes, and compensating winding, x = total self-inductive reactance, that is, reactance of the leakage flux of armature and compensating winding — or the stray flux passing locally between the armature and the compensating conductors — plus the self-inductive reactance of the field, that is, the reac- tance due to the stray field or flux passing between field coils and armature. In addition hereto, x comprises the reactance due to the quad- rature magnetic flux of incomplete compensation or overcom- pensation, that is, the voltage generated by the quadrature flux, $', in the difference between armature and compensating con- ductors, ni — n2 or n% — n\. Therefore the total supply voltage, Ey of the motor is: E = ei + e0 + e' = 2 irforii* 10-8 + 2jirfnx* lO"8 + (r + jx) /. (28) Let, then, R = magnetic reluctance of field circuit, thus j $ = ~tr = the magnetic field flux, when assuming this flux as in phase with the excitation /, and denoting: as the effective reactance of field inductance, corresponding to the e.m.f. of alternation: S = y = ratio of speed to frequency, or speed f as fraction of synchronism, Tit c = = ratio of effective armature turns to n° field turns; (31) SINGLE-PHASE COMMUTATOR MOTORS 363 substituting (30) and (31) in (28): # = cSxol + jxol + (r + jx) I = [(r + cSxo) + j(x + x0)] I; (32) or: 1 = (r + cSxo) + 7 (x + *o)' (33) and, in absolute values: t = , -_ — . • (34) V(r + cSxo)2 + (x + x0)2 The power-factor is given by: tan * - 7T& (35) The useful work of the motor is done by the e.m.f. of rotation: #i = cSxof, and, since this e.m.f., #i, is in phase with the current, /, the useful work, or the motor output (inclusive friction, etc.), is: p = EJ = cSxoi2 cSxoe2 (r + cSxQ)2 + (x + xo)2 and the torque of the motor is : p D = -~ = cxoi2 (36) cxtf2 (r + cSxo)2 + (x + xo)2 For instance, let: e = 200 volts, c = — = 4, wo (37) then: Z =± r+jx = 0.02 + 0.06 j, x0 = 0.08; . 10,000 1 " vu + i^2+49amp-' ♦ a 1 + 16 S cot 0 = — =- > p = 32^0005 , (1 + 16 Sj2 "+ 49 ' n_ 32,000 , (1 + 16 S)2 + 49 Syn* kW* 364 ELECTRICAL APPARATUS 203. The behavior of the motor at different speeds is l*^t shown by plotting i, p = cos 8, P and D as ordinates with the speed, 8, as abscissae, as shown in Fig. 173. In railway practice, by a survival of the practice of former times, usually the constants are plotted with the current, /, as abscissae, as shown in Fig. 174, though obviously this arrange- ment does not as well illustrate the behavior of the motor. Graphically, by starting with the current, /, as zero axis,0/, the motor diagram is plotted in Fig. 175. £ ^ II. P tl "S \ .. -1 \ -, V \ ri. m A \ \ s „ TV / \ V j» / )f \ P -„ . / \ s! mi ■-- m / "7 1 / 0 mi in -JL a t 8- . „ 1 l'i . 1 L5 1 . 1 wit anc aim C tern EOi I the I flux by Fia. 173. — Bingle-phase commutator-motor spoori characteristics. he voltage consumed by the resistance, r, is OE, — ir. in pi l 01; the voltage consumed by the reactance, x, is OE, = 90° ahead of 01. OE, and OE, combine to the voltage c ed by the motor impedance, OE' — iz. ombining OE' = iz, OE\ — eit and OE0 = c0 thus gives linal voltage, OE = e, of the motor, and the phase an = e. l this diagram, and in the preceding approximate calculat magnetic flux, *, has been assumed in phase with the curren l reality, however, the equivalent sine wave of magn $, lags behind the equivalent sine wave of exciting curren he angle of hysteresis lag, and still further by the po ase tx, on- the file, on, ,/- etic ,/. wcr SINGLE-PHASE COMMUTATOR MOTORS 365 consumed by eddy currents, and, especially in the commutator motor, by the power consumed in the short-circuit current under the brushes, and the vector,0*,therefore is behind the current vector, 01, by an angle a, which is small in a motor in which the short-circuit current under the brushes is eliminated and the eddy currents are negligible, but may reach considerable values in the motor of poor commutation. n„ " r " I \ " g / .r. " \ ■£/«•' u -^ -, ». f- " V? 7 d.. ~~"S_^ -l X m Z i C5^ ' j H: -/ V ^ \--T 7 K , V \ 04 DZ ^ \A-»- 7 s \"Z . ^ i- \ ■ M 30 3OT aio MOWI ^Twu W KOI* 1500 i: MlN Fio. 174. — Single-phase commutator-motor current characteristics. Assuming then, in Fig. 176, 0* lagging behind 01 by angle a, OEi is in phase with 0*, hence lagging behind 01; that is, the e.m.f. of rotation is not entirely a power e.m.f., but contains a wattless lagging component. The e.m.f. of alternation, OE0, is 90° ahead of O*, hence less than 90° ahead of OI, and therefore contains a power component representing the power consumed by hysteresis, eddy currents, and the short-circuit current under the brushes. Completing now the diagram, it is seen that Hie phase angle, 9, is reduced, that is, the power-factor of the motor increased by 366 ELECTRICAL APPARATUS the increased loss of power, but is far greater than corresponding thereto. It is the result of the lag of the e.m.f. of rotation, which produces a lagging e.m.f. component partially compensating for the leading e.m.f. consumed by self -inductance, a lag of the e.m.f. being equivalent to a lead of the current. Fig. 175. — Single-phase commutator-motor vector diagram. As the result of this feature of a lag of the magnetic flux, $, by producing a lagging e.m.f. of rotation and thus compensating for the lag of current by self-inductance, single-phase motors having poor commutation usually have better power-factors, and Fig. 176. — Single-phase commutator-motor diagram with phase displace- ment between flux and current. improvement in commutation, by eliminating or reducing the short-circuit current under the brush, usually causes a slight de- crease in the power-factor, by bringing the magnetic flux, , more nearly in phase with the current, /. 204. Inversely, by increasing the lag of the magnetic flux, , the phase angle can bo decrejised .and the power-factor improved. Such a shift of the magnetic flux, 4>, behind the supply current, ?, can be produced by dividing the current, i, into components, i' SINGLE-PHASE COMMUTATOR MOTORS 367 and i", and using the lagging component for field excitation. This is done most conveniently by shunting the field by a non- inductive resistance. Let r0 be the non-inductive resistance in shunt with the field winding, of reactance, xQ + x\f where X\ is Fig. 177. — Single-phase commutator-motor improvement of power-factor by introduction of lagging e.m.f. of rotation. that part of the self-inductive reactance, x, due to the field coils. The current, i', in the field is lagging 90° behind the current, i", in a non-inductive resistance, and the two currents have the •# ratio .,-, = — -7- — ; hence, dividing the total current, 01 , in this * Xo T ^1 proportion into the two quadrature components, OF and 01" > Fig. 178. — Single-phase commutator motor. Unity power-factor produced by lagging e.m.f. of rotation. in Fig. 177, gives the magnetic flux, 0$, in phase with 01', and so lagging behind 01, and then the e.m.f. of rotation is OEh the e.m.f. of alternation OE0) and combining 0Eh OE0) and OE' 368 ELECTRICAL APPARATUS gives the impressed e.ra.f., OE, nearer in phase to 01 than with 0* in phase with 01. In this manner, if the e.m.fa, of self-inductance arc not CM large, unity power-factor can be produced, as shown in Fig. 178. Let 01 = total current, OE' = impedance voltage of the motor, OE = impressed e.m.f. or supply voltage, and assumed in phase with 01. OE then must be the resultant of OE' and of OEi, the voltage of rotation plus that of alternation, and resolv- ing therefore 0E% into two components, 0E% and 0Ea, in quadra- ture with each other, and proportional respectively to the e.m.f. of rotation and the e.m.f. of alternation, gives the magnetic flux, 0*, in phase with the e.m.f. of rotation, 0EU and the component of current in the field, 01', and in the non-inductive resistance-, 01", in phase and in quadrature respectively with 04>, which combined make up the total current. The projection of the e.m.f. of rotation 0E\ on 01 then is the power component of the e.m.f., which does the work of the motor, and the quadra- ture projection of, 0Et, is the compensating component of the e.m.f. of rotation, which neutralizes the wattless component of the e.m.f. of self-inductance. Obviously such a compensation involves some loss of power in the non-inductive resistance, ra, shunting the field coils, and as the power-factor of the motor usually is sufficiently high, such compensation is rarely needed. In motors in which some of the circuits are connected induct ively in series with the others the diagram is essentially thesame, except SINGLE-PHASE COMMUTATOR MOTORS 369 that a phase displacement exists between the secondary and the primary current. The secondary current, Ii, of the transformer lags behind the primary current, Jo, slightly less than 180° ; that is, considered in opposite direction, the secondary current leads the primary by a small angle, 0o, and in the motors with secondary excitation the field flux, 4>, being in phase with the field current, 1 1 (or lagging by angle a behind it), thus leads the primary current, Jo, by angle 0O (or angle do — a). As a lag of the mag- netic flux $ increases, and a lead thus decreases the power-factor, motors with secondary field excitation usually have a slightly Fig. 180. — Single-phase commutator motor with secondary excitation power-factor improved by shunting field winding with non-inductive circuit. lower power-factor than motors with primary field excitation, and therefore, where desired, the power-factor may be improved by shunting the field with a non-inductive resistance, r0. Thus for instance, if, in Fig. 179, 01 0 = primary current, 01 \ = sec- ondary current, OEi, in phase with 01 1, is the e.m.f. of rotation, in the case of the secondary field excitation, and OEo, in quadra- ture ahead of 01 1, is the e.m.f. of alternation, while OE' is the impedance voltage, and OEi, OEo and OE' combined give the supply voltage, OE, and EOI = 0 the angle of lag. Shunting the field by a non-inductive resistance, r0, and thus resolving the secondary current OI\ into the components OI\ in the field and 01" \ in the non-inductive resistance, gives the dia- gram Fig. 180, where a = I'iO$ = angle of lag of magnetic field. 24 370 ELECTRICAL APPARATUS 205. The action of the commutator in an alternating-eurreri motor, in permitting compensation for phase displacement iwl thus allowing a control of the power-factor, is very imn. ■-m-.- and important, and can also be used in other types of machines, as induction motors am! alternators, by supplying these machines with a commutator for phase control. A lag of the current is the same as a lead of the e.m.f., and in- versely a leading current inserted into a circuit has the same ef- fect as a lagging e.m.f. inserted. The commutator, however produces an e.m.f. in phase with the current. Inciting the field l>y a lagging current in the field, a lagging e.m.f. of rotation is produced which is equivalent to a leading current. As it is easy to produce a lagging current by self-inductance, the commutator thus affords an easy means of producing the equivalent of a leading current. Therefore, the alternating-current commutator is one of the important methods of compensating for lagging: currents. Other methods are the use of electrostatic or electro- lytic condensers and of overexcited synchronous machines. Based on this principle, a number of designs of induction motors and other apparatus have been developed, using Qm commutator for neutralizing the lagging magnetizing current and the lag caused by self-inductance, and thereby produdng unity power-factor or even leading currents. So far, however, none of them has come into extended use. This feature, however, explains the very high power-factor* feasible in single-phase commutator motors even with COQndtf able air gaps, far larger than feasible in induction motors. VII. Efficiency and Losses 206. The losses in single-phase commutator motors ate BOB ' tially the same as in other types of machines: (a) Friction losses— air friction or windage, lwaring friction and commutator brush friction, and also ■ :■ . mechanical transmission losses. (6) Core losses, as hysteresis and eddy currents. These an1 of two classes — the alternating core hiss, due to the alternation of the magnetic flux in the main field, quadrature field, and arma- ture and the rotating core loss, due to the rotation of the arma- ture; through the magnetic field. The former depends upon the frequency, the latter upon the speed. (cj Commutation losses, as the power consumed by the slum- SINGLE-PHASE COMMUTATOR MOTORS 371 circuit current under the brush, by arcing and sparking, where such exists. (d) ihr losses in the motor circuits — the field coils, the compen- sating winding, the armature and the brush contact resistance. (e) Load losses, mainly represented by an effective resistance, that is, an increase of the total effective resistance of the motor beyond the ohmic resistance. Driving the motor by mechanical power and with no voltage on the motor gives the friction and the windage losses, exclusive of commutator friction, if the brushes are lifted off the commu- tator, inclusive, if the brushes are on the commutator. Ener- gizing now the field by an alternating current of the rated fre- quency, with the commutator brushes off, adds the core losses to the friction losses; the increase of the driving power theto measures the rotating core loss, while a wattmeter in the field exciting circuit measures the alternating core loss. Thus the alternating core loss is supplied by the impressed electric power, the rotating core loss by the mechanical driving power. Putting now the brushes down on the commutator adds the commutation losses. The ohmic resistance gives the i2r losses, and the difference between the ohmic resistance and the effective resistance, calcu- lated from wattmeter readings with alternating current in the motor circuits at rest and with the field unexcited, represents the load losses. However, the different losses so derived have to be corrected for their mutual effect. For instance, the commutation losses are increased by the current in the armature; the load losses are less with the field excited than without, etc. ; so that this method of separately determining the losses can give only an estimate of their general magnitude, but the exact determination of the effi- ciency is best carried out by measuring electric input and me- chanical output. VIII. Discussion of Motor Types 207. Varying-speed single-phase commutator motors can be divided into two classes, namely, compensated series motors and repulsion motors. In the former, the main supply current is through the armature, while in the latter the armature is closed upon itself as secondary circuit, with the compensating winding 372 ELECTRICAL APPARATUS as primary or supply circuit. As the result hereof the repulsion motors contain a transformer flux, in quadrature position to the main flux, and lagging behind it, while in the series motors no such lagging quadrature flux exists, but in quadrature position to the main flux, the flux either is zero — complete compensation ■ — or in phase with the main flux — over- or undercompensation. A. Compensated Series Motors Series motors give the best power-factors, with the exreptioii of those motors in which by increasing the lag of the field flux a compensation for power-factor is produced, as discussed in V. The commutation of the series motor, however, is equally poor at all speeds, due to the absence of any eommutating flux, and with the exception of very small sizes such motors therefore are inoperative without the use of either resistance leads or eom- mutating poles. With high-resistance leads, however, fair opera- tion is secured, though obviously not of the same class with that of the direct-current motor; with eommutating poles or coils producing a local quadrature flux at the brushes good results have been produced abroad. Of the two types of compensation, conductive compensation. 1, with the compensating winding connected in series with the armature, and inductive compensation, 2, with the compensated winding short-circuited upon itself, inductive compensation nec- essarily is always complete or practically complete compensa- tion, while with conductive compensation a reversing flux can be produced at the brushes by overcompensation, and the com- mutation thus somewhat improved, especially at speed, at (fat sacrifice, however, of the power-factor, which is lowered by the increased self-inductance of the compensating winding. On the short-circuit current under the brushes, due to the e.m.f. of alter- nation, such overcompensation obviously has no helpful effect. Inductive compensation has the advantage that the compen- sating winding is not connected with the supply circuit, can I* made of very low voltage, or even of individually short-circuited turns, and therefore larger conductors and less insulation used, which results in an economy of spaee, and therewith an infix Hfld output for the same size of motor. Therefore inductive compttf- satiou is preferable where it can be used. It is not permissible, however, in motors which are required to operate also on direct current, since with direct-current supply no induction takes place .1IXGLB-PHASE COMMUTATOR MOTORS 373 and therefore the compensation fails, and with the high ratio of armature turns to field turns, without compensation, the field distortion is altogether too large to give satisfactory commutation, except in small motors. The inductively compensated series motor with secondary ex- citation, or inverted repulsion motor, 3, takes an intermediary position between the series motors and the repulsion motors; it is a series motor in so far as the armature is in the main supply circuit, but magnetically it has repulsion-motor characteristics, that is, contains a lagging quadrature flux. As the field exci- tation consumes considerable voltage, when supplied from the compensating winding as secondary circuit, considerable voltage must he generated in this winding, thus giving a corresponding transformer flux. With increasing speed and therewith decreas- ing current, the voltage consumed by the field coils decreases, and therewith the transformer flux which generates this voltage. Therefore, the inverted repulsion motor contains a transformer flux which has approximately the intensity and the phase re- quired for commutation; it lags behind the main flux, but less than 90°, thus contains a component in phase with the main flux, as reversing flux, and decreases with increase of speed. Therefore, the commutation of the inverted repulsion motor is very good, far superior to the ordinary series motor, and it can be operated without resistance leads; it has, however, the serious objection of a poor power-factor, resulting from the lead of the field flux against the armature current, due to the secondary ex- citation, as discussed in V. To make such a motor satisfactory in power-factor requires a non-inductive shunt across the field, and thereby a waste of power. For this reason it has not come into commercial use. B. Repulsion Motors 208. Repulsion motors are characterized by a lagging quadra- ture flux, which transfers the power from the compensating wind- ing to the armature. At standstill, and at very low speeds, re- pulsion motors and series motors are equally unsatisfactory in commutation; while, however, in the series motors the commu- tation remains bad (except when using commutating devices), in the repulsion motors with increasing speed the commutation rapidly improves, and becomes perfect near synchronism. As the result hereof, under average conditions a much inferior com- 374 ELECTRICAL APPARATUS mutation can be allowed in repulsion motors at. very low speeds than in series motors, since in the former the period of poor commutation lasts only a very short time. While, therefore, series motors can not be satisfactorily operated Hit hoot resist a m.-e leads (or commutating poles), in repulsion motors peristanofl leads are not necessary and not used, and the excessive current density under the brushes in the moment of starting permitted, as it lasts too short a time to cause damage to the commutator. As the transformer field of the repulsion motor is approximately constant, while the proper commutating field should decrease with the square of the speed, above synchronism the transformer field is too large for commutation, and at speeds considerably above synchronism — 50 per cent, and more — the repulsion motor becomes inoperative because of excessive sparking. At syn- chronism, the magnetic field of the repulsion motor is a rotating field, like that of the polyphase induction motor. Where, therefore, speeds far above synchronism are required, the repulsion motor can not be used; but where synchronous speed is not much exceeded the repulsion motor is preferred be- cause of its superior commutation. Thus when using a commu- tator as auxiliary device for starting single- phase induction motors the repulsion-motor type is used. For high frequencies. as 60 cycles, where peripheral speed forbids synchronism being greatly exceeded, the repulsion motor is the type to be considers! Repulsion motors also may be built with primary and str- ondary excitation. The latter usually gives a lietter commuta- tion, because of the lesser lag of the transformer flux, and i here- with a greater in-phase component, that is, greater reversing flux, especially at high speeds. Secondary excitation, however, gives a slightly lower power-factor. A combination of the repulsion-motor and series-motor types is the series repulsion motor, 6 and 7. In this only a part of the supply voltage is impressed upon the conqx-nsating winding and thus transformed to the armature, while the rest of the sup- ply voltage is impressed directly upon the armature, just as in the series motor. As result thereof the transformer flux of tin1 series repulsion motor is less than that of the repulsion motor. in the same proportion in which the voltage impressed upon the compensating winding is less than the total supply voltage. Such a motor, therefore, reaches equality of the transformer flux with the commutating flux, and gives perfect commutation at a SINGLE-PHASE COMMUTATOR MOTORS 375 higher speed than the repulsion motor, that is, above synchron- ism. With-the total supply voltage impressed upon the compen- sating winding, the transformer flux equals the commutating flux at synchronism. At n times synchronous speed the com- mutating flux should be — 2 of what it is at synchronism, and by IV impressing —^ of the supply voltage upon the compensating wind- IV ing, the rest on the armature, the transformer flux is reduced to —j of its value, that is, made equal to the required commuta- ting flux at n times synchronism. In the series repulsion motor, by thus gradually shifting the supply voltage from the compensating winding to the armature and thereby reducing the transformer flux, it can be maintained equal to the required commutating flux at all speeds from syn- chronism upward; that is, the series repulsion motor arrange- ment permits maintaining the perfect commutation, which the repulsion motor has near synchronism, for all higher speeds. With regard to construction, no essential difference exists be- tween the different motor types, and any of the types can be operated equally well on direct current by connecting all three circuits in series. In general, the motor types having primary and secondary circuits, as the repulsion and the series repulsion motors, give a greater flexibility, as they permit winding the circuits for different voltages, that is, introducing a ratio of trans- formation between primary and secondary circuit. Shifting one motor element from primary to secondary, or inversely, then gives the equivalent of a change of voltage or change of turns, Thus a repulsion motor in which the stator is wound for a higher voltage, that is, with more turns, than the rotor or armature, when connecting all the circuits in series for direct-current opera- tion, gives a direct-current motor having a greater field excita- tion compared with the armature reaction, that is, the stronger field which is desirable for direct-current operating but not per- missible with alternating current. 209. In general, tthe constructve differences between motor types are mainly differences in connection of the three circuits. For instacne, let F = field circuit, A = armature circuit, C = compensating circuit, T = supply transformer, R = resistance used in starting and at very low speeds. Connecting, in Fig. 181, the armature, A, between field F and compensate" Ending, C. 376 ELECTRICAL APPARATUS With switch 0 open the starting resistance is in circuit . dofBO| switch 0 short-circuits the starting resistance and gives the run- ning conditions of the motor. With all the other switches open the motor is a conductively compensated series motor. Fia. ] raged to operate Closing 1 gives the inductively compensated series motor. Closing 2 gives the repulsion motor with primary excitation. Closing 3 gives the repulsion motor with secondary excitation. Closing 4 or 5 or 6 or 7 gives the successive speed steps of the scries repulsion motor with armature excitation. opanta Connecting, in Fig. 182, the field, F, between armature, .1 , ud compensating winding, C, the resistance, R, is again controlled by switch 0. All other switches open gives the conductively compensated series motor. SINGLE-PHASE COMMUTATOR MOTORS 377 Switch 1 closed gives the inductively compensated series motor. Snitch 2 closed gives the inductively compensated series motor with secondary excitation, or inverted repulsion motor. Switch 3 closed gives the repulsion motor with primary excitation. Switches 4 to 7 give the different speed steps of the series re- pulsion motor with primary excitation. Opening the connection at x and closing at y (as shown in dotted tine), the steps 3 to 7 give respectively the repulsion motor with secondary excitation and the successive steps of the series repulsion motor with armature excitation. Still further combinations can be produced in this manner, as for instance, in Fig. 181, by closing 2 and 4, but leaving 0 open, the field, F, is connected across a constant- potential supply, in series with resistance, R, while the armature also receives con- stant voltage, and the motor then approaches a finite speed, that is, has shunt motor characteristic, and in starting, the main field, F, and the quadrature field, AC, are displaced in phase, so give a rotating or polyphase field (unsymmetrical). To discuss all these motor types with their in some instances very interesting characteristics obviously is not feasible. In general, they can all be classified under series motor, repulsion motor, shunt motor, and polyphase induction motor, and com- binations thereof. IX. Other Commutator Motors 210. Single-phase commutator motors have been developed as varying-spced motors for railway service. In other directions commutators have been applied to alternating-current motors and such motors developed : (a) For limited speed, or of the shunt-motor type, that is, motors of similar characteristic as the single-phase railway motor, except that the speed does not indefinitely increase with decreasing load but approaches a finite no-load value. Several types of such motors have been developed, as stationary motors for elevators, variable-speed machinery, etc., usually of the single-phase type. By impressing constant voltage upon the field the magnetic field flux is constant, and the speed thus reaches a finite limiting value at which the e.in.f. of rotation of the armature through 378 ELECTRICAL APPARA TVS the constant field flux consumes the impressed voltagi armature. By changing the voltage supply to the field different speeds can be produced, that is, an adjustable-speed motor. The main problem in the design of such motors is to get the field excitation in phase with the armature current and thus pro- duce a good power-factor. (b) Adjustable-speed polyphase induction motors. In the secondary of the polyphase induction motor an e.m.f. is gener- ated which, at constant impressed e.m.f. and therefore apprffld- mately constant flux, is proportional to the slip from synchron- ism. With short-circuited secondary the motor closely ap- proaches synchronism. Inserting resistance into the secondary reduces the speed by the voltage consumed in the secondary. As this is proportional to the current and thus to the load, the speed control of the polyphase induction motor by resistance in the secondary gives a speed which varies with the load, just M the speed control of a direct-current motor by resistance in tin- armature circuit ; hence, the speed is not constant, and the opera- tion at lower speeds inefficient. Inserting, however, a con&taitf voltage into the secondary of the induction motor the speed is decreased if this voltage is in opposition, and is increased if this voltage is in the same direction as the secondary generated e.m.f., and in this manner a speed control can be produced. If c = voltage inserted into the secondary, as fraction of the voltage which would be induced in it at full frequency by the rotating field, then the polyphase induction motor approaches at no-load and runs at load near to the speed (1 — c) or (1 + c) times syn- chronism, depending upon the direction of the inserted voltage. Such a voltage inserted into the induction-motor secondary must, however, have the frequency of the motor secondary cur- rents, that is, of slip, and therefore can be derived from the full- frequency supply circuit only by a commutator revolving with the secondary, If cf is the frequency of slip, then (1 — c)f is the frequency of rotation, and thus the frequency of commuter tion, and at frequency, /, impressed upon the commutator the effective frequency of the eommutated current is/ — (1 — c)/ = cf, or the frequency of slip, as required. Thus the commutator affords a means of inserting voltage into the secondary of induction motors and thus varying its spetd. However, while these eommutated 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_mJ. of the current between the brushes, G, magnetizes in the same D _± Fig. 1*3. — Plain repulsion motor. direction as the field coils. F, in Fig. 1S3. Usually the exciting brushes are supplied by a transformer or autotransfonner. so as to vary the excitation and thereby the speed. This arrangement then lowers the e.mJ. of self-inductance of field excitation of the motor from that corresponding to full fre- C_ qoency in the ordinary repulsion motor to that cc the frequency of sop. hence to a negative value above syririmrJsn: so that hereby a compensation for lagging current ;*r he produced above synchronism, and unity power-dacKc cc even leading currents produced. SINGLE-PHASE COMMUTATOR MOTORS 381 211. 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 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: 2 , n = -n 7T 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, w, as frequently the case with the compensating winding distributed in the pole face of pole arc, w ; or with fractional-pitch armature windings of pitch, w. In this case, the effective number of turns is: 2 . . a) n = - n sin « ELECTRICAL APPARATUS where n' with a fractional-pitch armature winding i: of series turns in the pitch angle, w, that is: n" being the number of turns in series between the brushes, dun in the spaed (*■ — w) outside of the pitch angle the armature conductors neutralize each other, that is, conductors curryine current in opposite direction arc superposed upon each other. See fractional-pitch windings, chapter "Commutating Machine," "Theoretical Elements of Electrical Engineering." 212. Let: Bo, /o, Za = 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, Cn, Ei, I,, Zi = impressed voltage, current and self-inductive im- pedance of the rotor energy circuit (or circuit at right angles to /„), Et, It, Zt = impressed voltage, current and self-inductive im- pedance of the stator compensating circuit (or circuit parallel to /l) reduced to the rotor circuit by the ratio of effective turn*, - ■.. fia, t», Z\ = impressed voltage, current'and self-inductive im- pedance of the exciting circuit of the rotor, or circuit parafld to/„, It, Zt = current and self-inductive impedance of the short- circuit under the brushes, /,, reduced to the rotor cireuit, h, /... = current and self-inductive impedance of the short- circuit under the brushes, /B, reduced to the rotor circuit. Z = mutual impedance of field excitation, that is, in the direc- tion of h, /,, /,, Z' = mutual impedance of armature reaction, that, is, in the direction of /,, I,, /&. Z' usually either equals Z, or is smaller than Z. Ii and Ia are very small, Z, and Z& very large quantities. Let S = speed, as fraction of synchronism. Using then the general equations "Chapter XIX, which ftpplji to any alternating-current circuit revolving with speed, S, bhnmgjb a magnetic field energized by alternating-current circuits, gives for the six circuits of the general single-phase commutator motor the six equations: SIXGLE-PUASE COMMUTATOR MOTORS 383 G o = ZJb + Z (/ E , = Z,/, + Z' (/ & = Z-/- + Z' (/ £* - Zl/, + Z (/; o = ZJ4 + Z (/ o = Z>h + Z' (/. + /i - W, (1) + A - W - J'SZ (/• + /. - /«>, (2) - /i - W, (3) + /o - A> - jSZ v /« - /» - h\ (4) - /o - /,) - jSZ tf. + /* - f,\ (5) + /i - /») - JSZ (h + /. - M. 16) These six equations contain ten variables: /o, /it /•* lit /i, Is, £o, £l, £2, £Y 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, t , to the effective turns of the armature circuit, be represented by 0*. That is, number of effective turns of circuit, i number of effective turns of armature circuit* and if #,, /,, Z» are voltage, current and impedance of circuit, i, reduced to the armature circuit, then the actual voltage, current and impedance of circuit, i, are: /. d$i} * cc Zi. 213. The different forms of single-phase commutator motors, of series characteristic are, as shown diagrammatically in Fig. 186: 1. Series motor: e = c0#o + -Pi; h = Colx] h = 0; /3 = 0. 2. Conduetively compensated series motor (Eickcmeyer motor) : e = c0#o + #i + c2#2; h = co/i; /2 = c2/i; h = 0. 3. Inductively compensated series motor (Eickcmeyer motor) : e = coEq + Pi; #2 = 0; /o = c0/i; /3 = 0. 384 ELECTRICAL APPARATUS 4. Inverted repulsion motor, or series motor with secondary excitation : e = #1; cqE0 + c2E2 = 0; c2/o = c0/2; It = 0. 5. Repulsion motor (Thomson motor) : e = c0#o + c2#2; #1 = 0; C2I0 = coA; It = 0. 6. Repulsion motor with secondary excitation: c = c2#2; co#> + #1 = 0; lo = co/i; It = 0. Fig. 186. 7. Series repulsion motor with secondary excitation : ei = co#o + #i;.e2 = #2; h = c0/i; /s = 0. 8. Series repulsion motor with primary excitation (Alexander- sen motor) : ei = #1; e2 = c0#o + c2#2; c2/0 = c0/2; Js = 0. 9. Compensated repulsion motor (Winter and Eichberg motor) : e = C2E2 + cz$z; gi = 0; /0 = 0; C3/2 = c2/3. SINGLE-PHASE COMMUTATOR MOTORS 385 /m Fig. 187. 10. Rotor-excited series motor with conductive compensation : e = & + cj#2 + c8#3; U = c2/i; h = ci/ij /o = 0. 11. Rotor-excited series motor with inductive compensation: . e = & + c,#3; ft - 0; /o = 0; /, - c3/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) : #1 = 0; caft = c8#3 = e; /o = 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. ReptUtsion Motor As illustration of the application of these general equations, paragraph 212, may be considered the theory of the repulsion motor (5), in Fig. 180. 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 ((>), the four conditions characteristic of the repulsion motor are: 25 386 ELECTRICAL APPARATUS 1. Armature short-circuited upon itself. Hence: 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% = 1 : Cq$0 + #2 = e. Herefrom follows: 3. io == CqI 2. 4. No armature excitation used, but only one set of commu- tator brushes; hence: /•-0f and therefore: /6 = 0. Substituting these four conditions in the six equations (1) to (6), gives the three repulsion motor equations: Primary circuit: Z2/2 + Z' (h - /1) + Co2Zo/2 + CoZ (co/2 - h) = e; (7) Secondary circuit: Zxh + Z' (h - h) - jSZ (co/2 - I a) = 0; (8) Brush short-circuit: Z4/4 + Z(h- coh) - jSZ'ih - /,) = 0; (9) Substituting now the abbreviations: Z2 + co2 (Z0 + Z) = Z8, (10) (ID (12) Z' z "A, zx Z' = K z\ z , + z = X4; (13) where Xi and X«, especially the former, are small quantities. From (9) then follows: h = X4 1/, (c, - jSA) + jShA } ; (14) SINGLE-PHASE COMMUTATOR MOTORS 387 from (8) follows, by substituting (14) and rearranging: ' ' - '• 1 + x, - x,s< and, substituting (15) in (14), gives: t _x r (co-i5A)(l + X,-X4-S*)+jSA-S»co-Xij,S(-SA + jiCo) /« - Wi ! q: Xi - ^ » or, canceling terms of secondary order in the numerator: '• - k'''TTT^?> (18) where: # = (i43-iSco)+X1(A3+4)-X4(SJi43-S2Co+co2-jSco), (19) and: a, = |8; (20) or, since approximately: Az = Co2, (21) it is: X = (4, - jSco) + Xi (c«2 + A) - X4C0 (Co - j/S). (22) Substituting (18), (19), (20) in (15) and (16), gives: Secondary Current: .{.+*»- **(* + fl } '' ZK Brush Short-circuit Current: (23) /4 = x««oU-.s*). (24) 388 ELECTRICAL APPARATUS As seen, for S = 1, or at synchronism, J \ = 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 rotationf 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: Q\ = jSZ (coh ~ M. (25) Brush short-circuit: V< = jSZ' (/i - /,). (26) Substituting (18), (23), (24) into (25) and (26), and rearrang- ing, gives: Main Circuit E.m.f. of Rotation: £'. =^0fll+Xi-M- (27) Brush Short-circuit E.m.f. of Rotation: $\ = ™{Sco+ j\iA - c0X4} ; (28) or, neglecting smaller terms: £\ = -£*• (29) The Power produced by the main armature circuit is: Pi = [#'., /.is hence, substituting (22) and (27) : i%e |i + x, - M, J ±-gg4 4li] • p Let: (30) m = [ZK] (31) be the absolute value of the complex product, ZK, and: 1 A = cl + ja" X, = X', - j\'\ X4 = X'« + jX"4 (32) SINGLE-PHASE COMMUTATOR MOTORS 389 it is, substituting (31), (32) in (30), and expanding: Pi = ^jr U«U - Scoa") - rSc*a'] + (1 - Sc*r") (x(X\ - X'4) - r(X"i + X"4)] - Scoa'[r(\\ - X'„) + jr(X", + X"4)| - x (X'«iS* - X'iScoa" + X"4»Sc0a') -f r (k\ScW -X"4) m2 \ x / **{i-*.(^+^)} TO' S#X jl - Sc»(«" + ^a') = C0(l +Sl) hence a maximum for the speed S, given by: dS u' (34) or: So = ^l+ Co2 (a" + £ a') 7 - c« (a" + ^ «'). (35) and equal to: e*z Pi0 = 1 2 {^l+a«(^ + ^),-^(^/ + ^«#)|. (36) The complete expression of the power of the main circuit ia, from (33) : Pi - ***£-{ [l - &* (\, can be called the commutation constant: k - For good commutation, this ratio should l>e small or zero. The product of the commutation current, t„ and the speed, S, is proportional to the voltage induced by the break of ih<- mt- rent, or the voltage which maintains the arc at the edge of the commutator brushes, if sufficiently high, and may lie called the commutation vettage: C = Si,. (55) In the repulsion motor, it is, substituting (23) and [51 En and dropping the term with X«, as of secondary order: Commutation Current: jSc0 A,c0(l - S») SINGLE-PHASE COMMUTATOR MOTORS 393 Commutation Constant: I 1 +j~° - ami - S1) 0 /'l 1 + jSco = 1 - A A, at which the total torque vanishes, and reduces the power-factor and efficiency. 218. As an example are shown in Fig. 188 the characteristic curves of a repulsion motor, with the speed, S, as abscissa?, 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. i i i e -5oo volts L25+3J Z,-aolS*0.075i as+i» z, -0.02s + ot07w z- »k. 1 mu \d -O.04 bill X , ■n ;-£i ■ ; " /A N > * v * 1 / ^* s , ^ ■; / N '■'/ - Self-inductive impedance, main field: Z0 = 0.1 + 0.3 j ohms, cross field: Z» = 0.025 + 0.075johms. armature: Zx = 0.025 4- 0.075 j ohms, brush short-circuit: Z* = 7.5 + 10 j ohms. Reduction factor, main field: c0 — 0.4. brush short-circuit: cA = 0.04, Hence: Z, = 0.08 + 0.60 j ohms. A =0.835 - 0.014 j. j - a'+ja"- 1.20 + 0.02 j. Xi = 0.031 - 0.007;. X, = 0.179 + 0.087 j. At = 4.475 + 2.175 j. A» = 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 S) + j (0.604 - 0.187 S). Primary or Supply Current: . _ 500 { (1.031 - 0.179 S*) - j (0.007 + 0.087 S*) \ /2~ " ZK Secondary or Armature Current : T 500 {(1 + 0.048 5 -0.1 79 S*) +j 0.4 S - 0.087 S*)) II --------- 2K — : Brush Short-circuit Current: 500 (1 - Ss) (0.072 - 0.035 j) U= . ZK ' and absolute: 40 (1 - S*) lx = Ttt Commutation Factor: . = /(L508 Sl - 0.673)2 + (0.718 - 6.4 S - 0.704S*)* V (0.697 + 0.4 S- 0.014 )* Main E.m.f. of Rotation: 500 5 (4.052 + 0.792 j) El = z_ Commutation E.m.f. of Rotation : „, 500 S* (0.4 - 4.8 j) E<= - ZK - • Power of Main Armature Circuit: P = 250 S (4 052 _ Q 122 s _ Q 65? S2) Jn kw m2 Power of Brush Short-circuit: n 49.2 S2 (1 - S2) . . P4 = zlt~ — y in kw. Total Power Output: m2 p = ?™ ? (4.052 + 0.075 S - 0.657 S2 - 0.197 S8). m2 Torque : D = -°~ (4.052 + 0.075 S - 0.657 S2 - 0.197 S8), m2 etc. 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, 17, 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 coil 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 fuither 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: #1 + Co#o = # = «ii (67) where: Co = reduction factor of field winding to armature; that is, . „ field turns ratio of effective r — — : armature turns It follows herefrom: /o = coli. (68) 2. The e.m.f. impressed upon the compensating winding is given, and is in phase with the e.m.f., ei, which is impressed upon field plus armature: #2 = e2. (69) That is, #2 is supplied by the same transformer or compensator as 6i, in series or in shunt therewith. 3. No rotor-exciting circuit is used: h = 0, (70) and therefore: 0 4. No rotor-exciting brushes, or brushes in quadrature posi- tion with the main-armature brushes, are used, and so: U - 0, (71) that is, the armature carries only one set of brushes, which give the short-circuit current, J\. Since the compensating circuit, e2, is an independent circuit, it can be assumed as of the same number of effective turns as the armature, that is, e2 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: «. , • compensating turns \ c2Cs, where c, = ratio effective — mature turn8 ) 220. Substituting (68) into (1), (2), (3), and (5), and (1) and (2) into (67), gives the three motor equations: (72) e, = Zih + Z' (/, - /,) - jSZ (co/, - /«) + c0*Zo /i + CoZ (c0/o — h), ei = ZtJt + Z' (Jt - /,), (73) 0 = Z4i + Z(I<- Co/,) - jSZ' (/, - /,). (74) SIXGLE-PHASE COMMUTATOR MOTORS 399 Substituting now: 1 Y* = ~ = quadrature, or transformer exciting | it — Xa — X • — jX 3 z + z< Z' = X4 = X'i+jV'4 and ~ = A = a' — jV = impedance ratio of the two quadrature (tuxes, Zi + c' From '74 foBow« 0 = /< Z ~ Z. - / tJ? ~ ,,>/', - ^Z'/, > > '»/ / 7* i'4, in its eratafcT^oi:. That !•?>, wh*n *>uU*ututiiAg '7^ jij '70,, // can be drr>pj^o : or: ^UUfoxiu^wly 'Vh 400 ELECTRICAL APPARATUS Hence, (80) substituted in (79) gives: 0 = U (Z + Z*) ~ CoZh + jSet, or: /■ jSet Hence : and actual value of short-circuit current: r. -».{/.-£). where: Co b = "°, a fairly large quantity, and C\ = reduction factor of brush short-circuit to armature circuit. The commutation current then is: /. - /l - /'4 »/l(l -6X4) + jSetb\< cQZ Substituting (81) and (80) into (77), gives: /i = or, denoting: e 1 ^JSt\i (c0 - j-S)_— jf_X»_ Z A8 - j«Sco ~ XiCo (c0 - jS) + \2A it is: K = A* - jSco - X4c0 (c0 - j/S) + X2A, . _ ejl -jSt\i(co -jS) -t\i\ (1 - Zk It is, approximately : A3 = Z: — /._2 hence : z = x, = 0, 7C = c0(l — C0X4) (c0 -j e\J) where: Co 1 b = — , or t = reduction factor of short-circuit under brushes, C4 0 ' 104 ELECTRICAL APPARATUS to field circuit, that is: , _ number of field turns number of effective short-circuit turns' hence a large quantity. The absolute value of the short-circuit current, therefore, h: call - cXJ (c* + 8*) hence a minimum for that value of I, where: / = coJ + SMI - ( (c,,1 + S2))* = mi = 1 - ( (c1 + Ss) = 0, hence, t -- 1 " a,' + S>' 8 " >f" * That is, t = — = j,- v" — j gives minimum short-circuit cur- rent at speed, S, and inversely, speed 5 = »/- — e#*, gives minimum short-circuit current at voltage ratio, (. For ( =■ 1, or the repulsion motor with secondary excitation, the short-circuit current is minimum at speed, S = y/\ — ca'. or somewhat below synchronism, and is j'» = - , while in ihe 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, f\, however, is of far less importance than the commutation current, /,. (6) Commutation Current 222. While the value, J'„ = [\ + ft, or the current change in the armature coils while entering commutation, is of niiuor 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: h = h- /'. (103) is zero or a minimum. SINGLE-PHASE COMMUTATOR MOTORS 405 Using the approximate equation of the commutation current (94), it is: /. - c0Z(l — c0X4) I Co — jS e [ l " X?* + jStkJb c0Z(l - c0X4)(co -jS) and, denoting: X4 = X\ + j\"<, it is, expanded : e ;0-r { 1 - X46 + jS (co - jS) t\Ab] ; (104) U-ZTi -m HI - X'lfr + Stb(S\\ - C0X"4)] CoZ(l-c0X4)(co+iS) - j [\"z[l - C0X4]-y/Co2 + fi[2 \[1 - X'4fc + Sft(flX#4 - c0X"4)]2 + [X"4fc - Stb(c0\\ + SX"4)]2, (106) where [1 — c0X4] denotes the absolute value of (1 — c0X4). 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'4fc + Stb (SX'4 - c0X"4) = 0, X"46 - Stb (c0X'4 + SX"4) = 0. (107) From these two equations are calculated the two values, the speed, S, and the voltage ratio, t, as: hence: on — h = Soto = C» (6X4* - X'4) x" 4 x'V c0W(6X4* - X'4) ' X"4 606X4* (108) For instance, if: Z =0.25 + 3j, Z< = 5 + 2.5 j: ELECTRICAL APPARATUS 4 Z-VZi u' ' Co = 0.4, C4 - 0.04; hence: b - 10; and herefrom S, k - 2.02, - 0.197 - 0.248 j = X', + j\"„ that is, at about double synchronism, for es = 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, Su, at which, if the voltage ratio has the proper value, (o, the commutation current, i„, vanishes, and the commutation i.- perfect. At any other speed some commutation current is left, regardless of the value of the voltage ratio, (. With the two voltages, e\ and et, in phase with each other, t lie 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 et and e», that is, if ei 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, ej out of phase with the total voltage, e, hence denoting it. by: #s = et (cos 0s - j sin (?,), (109) the voltage ratio, (, now also is a complex quantity, and expressed by: T = ^ = t (cob 0i - j sin $i) = t' - jt". (110) Substituting (110) in (105), and rearranging, gives: '• " cza-cx.Xc-.M I[1 - v'fc + m w' - **"•> + Xt"b ta,X'( + SX",)| - j|K",6 - St'b (c„X', + SX",i + S!"c.(SX'. - e„X",)]]; (HI) and this expression vanishes, if: 1 - X',6 + Sfb (SX\ - c,X",) + .Sf'o ta,X', + .SX",) - 0, I X'Vi - Sl'b (e.X', + SX",) + Sf'b (SX', - oX",) - 0:| SINGLE-PHASE COMMUTATOR MOTORS 407 and herefrom follows: , = SbA2 - S\\ + c0X"4 S6X42(S2 + Co2) Co2 + S2 1 - S\ 4 — CqX 4 „ = Co6X42 - CqX/4-iSX//4 _ 1 S6X420S2+co2) or approximately: V = t" = SfcX42 Co Cq\\ + SX"4 ] Co2 + S2 \ S Sb\A2 } (112) Co2 + S2' Co (113) S (co2 + S2) t" = 0 substituted in equation (112) gives S = So, the value recorded in equation (108). It follows herefrom, that with increasing speed, S, tf and still more t", decrease rapidly. For S = 0, V and t" become infinite. That is, at standstill, it is not possible by this method to produce zero commutation current. The phase angle, 02, of the voltage ratio, T = t' — jt", is given by: , „ _ ^ _ C06X42 - Cp\'A - SX"4. tan 02 - t, - Sfex^2 _ sx,^ + CqVV rearranged, this gives: Co sin 02 + S cos 02 &X42 — W . (114) and, denoting: Co sin 62 — S sin 02 S f — = tan a, Co ' X" (115) (116) where a may be called the "speed angle," it is, substituted in (115): 6X42 - X\ tan (02 + e approxi- mately in phase with e for all except low speeds. At low speeds, it must lag, the more, the lower the speed. Its absolute value is very large at low speeds, but decreases rapidly with increasing speed, to very low values. For instance, let, as before: X, - 0.304 - 0.248 j, c„ = 0.4, fl, = 0 for (123) A 4 where 6 is the phase angle of X4, and therefore: J/ = J4 " !' I (124) X 4 =* X4 cos 6, I and also to introduce, as before, the speed angle (116): S tan a = ■> Co (125) v - \/c«2»; , hence: ,S - 7*111*, I (126) Ct, «■ 7 com 0, J 410 ELECTRICAL APPARATUS Substituting these trigonometric values into the expression (121) of the voltage ratio for minimum commutation current, it is: 1_ __ sin (a — 6) r Sbqx< <127) Substituting (117) into (106) and expanding gives a relatively simple value, since most terms eliminate: Jg = e {[cos2 (a — 6) + b\ (sin a sin (a — 6) — cos 6)] + j [ sin (a — 5) cos (a — 6) — 6X4 (sin a cos (cr — 6) — sin 6)]} **o = (128) (129) (130) CoZ(l - C0X4, (c0 +JS) and the absolute value: e ( cos ( Co V62X42 - 1. (133) SINGLE-PHASE COMMUTATOR MOTORS 411 That is: The commutation current, i0i can be made to vanish at any speed, S, at given impedance factor, X4, by choosing the phase angle of the impedance of the short-circuited coil, 5, or the resist- ance component, X', provided that X4 is sufficiently small, or the speed, S, sufficiently high, to conform with equations (133). From (132) follows as the minimum value of speed, S, at which the commutation current can be made to vanish, at given X4: Si = Co Vfe2x42 - 1, and: v -1- hence: X". = ^ - I • For high values of speed, S, it is, approximately: cos ( Z= 0.25+3 j Zi=0.025+0.075j Z' 0.;S+2_5J Z,-0.O:5 4-0.O75j ZrO.1 + 0.3J Z,= 7.S + 1D) n n l\" I - --» *H: ", 1 J*£ n ^ ! H ; n K / ... a « h f M n # - IN u i. MM / < C- BOO VOLTS -f = 0.5 / I 0.25»3j Z, -0 025+0075. > D ^ '1 .. _ / -- »E / n. •• * . / j f P i •" \ ;t-- a H ' / -~~ / SPEED Fig. 191. — Scries repulsion motor. SINGLE-PHASE COMMUTATOR MOTORS 413 Self-inductive impedance arma- ture: Zi= 0.025 + 0.075 j ohms, Self-inductive impedance, brush short-circuit: ZA= 7.5 + lOjohms, Reduction factor, main field: c„ = 0.4, brush short-circuit c* = 0.04; that is, the same constants as used in the repulsion motor, Fig. 188. Curves are plotted for the voltage ratios; t = 0: inductively compensated series motor, Fig. 189. t = 0.2: series repulsion motor, high-speed, Fig. 190. ( = 0.5: series repulsion motor, medium-speed, Fig. 191. ( = 1.0: repulsion motor with secondary excitation, low-speed, Fig. 192. e soo ■ Z'"0.!i*2.Sj Z,-0-025t0.07Sj I 3 sv C. - O.J C, 0.04 iir ]v ... — "n - "", ;> *' .._ — r — . . — IT , ' " IsV ■+. s IUL 7~, * V « •±>i / / / \ Mill ■ / -v SPEED Fig. 192— Rcpulsio secondary excitation. is, from above constants Z3 -Zi + c»(Z»+Z) - 0.08 + 0.60 j. i.-5 - 0.202 - 0.010}. *-5 - 0.835 - 0.014 j. --* - 0.031 - 0.007 j. x, - -z- ' Q + Z, - 0.179 + 0.087 j. b-* - 10. ~ 414 ELECTKICAL APPARATUS Hence, substituting into the preceding equations: (90) ZK = Z, - jSeoZ - X,CoZ (e, - jS) + X3Z' = (0,160 + 0.975 S) +j {0.590 - 0.187 S), (92) /, = £, - ^ US^ic-jS) + U\ = IR + ^iC-°-031 + 0035's-|>i:'' s - j" ( - 0.007 + 0.072 S + 0.087 #') | , (91) /, = /, (0.969 + 0.007 j) + et (0.010 - 0.096 j), (93) a (0.072 +0.035;)+ 3d [ (0.016 - 0.072 6')-j0.045+0.035g) | h = ZK 226. Ah seen, these four curves are very similar to each other and to those of the repulsion motor, with the exception of the commutation current, i,, and commutation factor, k = -?- The commutation factor of the compensated series motor, that is, the ratio of current change in the armature coil while leaving the brushes, to total armature current, is constant in the series motor, at all speeds. In the series repulsion motors, the commutation factor, h, starts with the same value at standstill, as the series motor, but decreases with increasing speed, thus giving a superior commutation to that of the series motor, reaches a minimum, and then increases again. Beyond the minimum commutation factor, the efficiency, power-factor, torque and out- put of the motor first slowly, then rapidly decrease, due to the rapid increase of the commutation losses. These higher values, however, are of little practical value, since the commutation is bad. The higher the voltage ratio, (, that is, the more voltage is impressed upon the compensating circuit, and the less upon the armature circuit, the lower is the speed at which the commuta- tion factor is a minimum, and the commutation so good or perfect. That is, with ( = 1 , or the repulsion motor with secondary ex- citation, the commutation is best at 70 per cent, of synchronism, and gets poor above synchronism. With t = 0.5, or a series repulsion motor with half the voltage on the compensating, half on the armature circuit, the commutation is best just above syn- chronism, with the motor constants chosen in this instance, and SINGLE-PHASE COMMUTATOR MOTORS 415 gets poor at speeds above 150 per cent, of synchronism. With t = 0.2, or only 20 per cent, of the voltage on the compensating circuit, the commutation gets perfect at double synchronism. Best commutation thus is secured by shifting the supply vol- tage with increasing speed from the compensating to the arma- ture circuit. t > 1, or a reverse voltage, — ei, impressed upon the armature circuit, so still further improves the commutation at very low speeds. For high values of t, however, the power-factor of the motor falls off somewhat. The impedance of the short-circuited armature coils, chosen in the preceding example: ZK = 7.5 + 10 j, corresponds to fairly high resistance and inductive reactance in the commutator leads, as frequently used in such motors. 227. As a further example are shown in Fig. 193 and Fig. 194 curves of a motor with low-resistance and low-reactance com- mutator leads, and high number of armature turns, that is, low reduction factor of field to armature circuit, of the constants: Z4 = 4 + 2j; hence : X4 = 0.373 + 0.267 j, and: Co = 0.3, d = 0.03, the other constants being the same as before. Fig. 193 shows, with the speed as abscissae, the current, torque, power output, power-factor, efficiency and commutation current, i0f under such a condition of operation, that at low speeds t = 1.0, that is, the motor is a repulsion motor with secondary excita- tion, and above the speed at which t = 1.0 gives best commuta- tion (90 per cent, of synchronism in this example), t is gradually decreased, so as to maintain ig a minimum, that is, to maintain best commutation. As seen, at 10 per cent, above synchronism, ig drops below t, that is, the commutation of the motor becomes superior to that of a good direct-current motor. Fig. 194 then shows the commutation factors, fc = -?> of the ELECTRICAL APPARATUS \ i i 1 1 i i i i i i 1 1 -» e SQO VOLTS Z =0.25 +3 j OHM Z,-0.O26 + 0.079 J0"»t \ ,- m so m c. 0.3 n i \p mt / -r . i goo EM , >•> ■v MS H V \ p -= ,' V .. -=^= .-v f J"* VI 1 1 ' / H LK. ^ i ' 5svhv \7 / / Z-0J5 Lfl < / / 3.00 '" SA 7 z. oo;5io.075yo-»i 4-IjOHM V\ II! b, / £:?:" s-' a. 00 - \ V * 1.7B \ \ ■ .v !\ \ / '^f V / \ sj /" v s. - ... \ '- — ■ \ / .... 4- " -- ~A ^ O.ffl 0.1O 0.60 0.80 l.Of SINGLE-PHASE COMMUTATOR MOTORS 417 different motors, all under the assumption of the same constants: Z = 0.25 + Sj, Z' = 0.25 +2.5j, Z0 = 0. 1 4- 0.3 j, Z2 = 0.025 + 0.075 j, Zi = 0.025 + 0.075 j, Z4 = 4 + 2 j, Co = 0.3, c4 = 0.03. Curve I gives the commutation factor of the motor as induct- ively compensated series motor (t = 0), as constant, k = 3.82, that is, the current change at leaving the brushes is 3.82 times the main current. Such condition, under continued operation, would give destructive sparking. Curve II shows the series repulsion motor, with 20 per cent, of the voltage on the compensating winding, t = 0.2; and Curve III with half the voltage on the compensating winding, t = 0.5. Curve IV corresponds to t = 1, or all the voltage on the com- pensating winding, and the armature circuit closed upon itself: repulsion motor with secondary excitation. Curve V corresponds to t = 2, or full voltage in reverse direction impressed upon the armature, double voltage on the compen- sating winding. Curve VI gives the minimum commutation factor, as derived by varying t with the speed, in the manner discussed before. For further comparison are given, for the same motor constants: Curve VII, the plain repulsion motor, showing its good com- mutation below synchronism, and poor commutation above synchronism; and Curve VIII, an overcompensated series motor, that is, con- ductively compensated series motor, in which the compensating winding contains 20 per cent, more ampere-turns than the arma- ture, so giving 20 per cent, overcompensation. As seen, overcompensation does not appreciably improve commutation at low speeds, and spoils it at higher speeds. Fig. 194 also gives the two components of the compensating e.m.f., E2, which are required to give perfect commutation, or zero commutation current: 27 410 ELECTRICAL APPARATUS Substituting these trigonometric values into the expression (121) of the voltage ratio for minimum commutation current, it is: # _ _1 __ sin (a — b) 1 ~ .2 01 x (127) q* Sbq\t v Substituting (117) into (106) and expanding gives a relatively simple value, since most terms eliminate: Jg = e {[cos2 (a — 6) + b\ (sin a sin (a — 6) — cos 6)] + j [ sin (a — 6) cos (a — 5) — 6X4 (sin c0b S > c0 V&2X42 - 1. (133) SINGLE-PHASE COMMUTATOR MOTORS 411 That is: The commutation current, i0} can be made to vanish at any speed, Sy at given impedance factor, \4, by choosing the phase angle of the impedance of the short-circuited coil, 5, or the resist- ance component, X', provided that X4 is sufficiently small, or the speed, Sf sufficiently high, to conform with equations (133). From (132) follows as the minimum value of speed, S, at which the commutation current can be made to vanish, at given X4: Si = Co V62X4* - 1, and: x« ~b' hence: x"< - Vx*' - I- For high values of speed, S, it is, approximately: cos ( • - ... -•... r-'STfiiiT 'irr^r^ i: ;. -.-- * * - ..'•"• *:;<- iiicnc-r *4\t- -nft--:. ■- . i j- !"!.> ••XDi»nt*:iw:^i --rni : * • . ■ . - .• --irr'UTt**: v ■ r:t* iTnn:' - • -- :- v.:; i> r: :i^-rr::t";n^- '■-r '!i;i»»*!>:iTii«r:. ■ r.u* - - r • -. !•.«*.. 1* .'4. -J.*1! .« ■ .". ^ if: : "»-:inn"i *•■ ••::*r . :•■• r* »i "rrtiiMi-!'.! i!:^r:.»:; v^ %%il - :<:■:»■*".«•!] »i ■•iiiiiim:;!":. • ■ •-.'iir'1'* i 'fi:i!ni;'; • -:.' i: :: -i::u-" "■• - -je " >• t:**:::i: I2*»- ■•irr-rr ■-■\" .>*:•?- i.;rr'-x::iiar.-»i . • . " ■ rrv i. ::: 'hi- t •■-- ■ " ..- • '::..'■ ■ r" ■i.'i-ri; * " .' ""•" ! i i iT -»i .]"- ■■'.r ■ • • SINGLE-PHASE COMMUTATOR MOTORS 421 Choosing the e.m.f., E2f impressed upon the compensating winding in phase with, and its magnetic flux, therefore in quad- rature (approximately), behind the main field, gives a com- mutation in the repulsion and the series repulsion motor which is better than that calculated from paragraphs 221 to 224, for all speeds up to the speed of best commutation, but becomes in- ferior for speeds above this. Hence the commutation of the repulsion motor and of the series repulsion motor, when con- sidering the self-induction of commutation, is superior to the calculated values below, inferior above the critical speed, that is, the speed of minimum commutation current. The com- mutation of the overcompensated series motor is superior to the values calculated in the preceding, though not of the same magnitude as in the motors with quadrature commutating flux. It also follows that an increase of the inductive reactance of the armature coil increases the exponential and decreases the alter- nating term of e.m.f. and therewith the current in the short- circuited coil, and therefore requires a commutating flux earlier in phase than that required by an armature coil of lower reac- tance, hence improves the commutation of the series repulsion and the repulsion motor at low speeds, and spoils it at high speeds, as seen from the phase angles of the commutating flux calculated in paragraphs 221 to 224. Causing the armature current to lag, by inserting external inductive reactance into the armature circuit, has the same effect as leading commutating flux: it improves commutation at low, impairs it at high speeds. In consequence hereof the com- mutation of the repulsion motor with secondary excitation — in which the inductive reactance of the main field circuit is in the armature circuit — is usually superior, at moderate speeds, to that of the repulsion motor with primary excitation, except at very low speeds, where the angle of lag of the armature cur- rent is very large. 420 ELECTRICAL APPARATUS change of current in the armature coil when passing under the brush, superimposes upon the e.m.f. generated in the short- circuited coil, and so on the short-circuit current under the brush, and modifies it. the more, the higher the speed, that is, I lie [juicier the current change. Tiiis exponential term of e.m.f. generated in the armature coil short-circuited by the commutator brush, is the so-called "e.m.f. of self-induction of commutation." It exists in direct-current motors as well as in alternating-current motors, and is controlled by overcompensation, that is, hj i oommutating field in phase with the main field, and approxi- mately proportional to the armature current. The investigation of the exponential term of generated e.m.f. and of short-circuit current, the change of the commutation current and commutation factor brought about thereby IBd the study of the conimutating field required to control this exponential term leads into the theory of transient phenomena. that is, phenomena temporarily occurring during and immedi- ately after a change of circuit condition.' The general conclusions are: The control of the e.m.f. of self-induction of commutsti I the single-phase commutator motor requires a COmmutatlDf field, that is, a field in quadrature position in space to the mam field, approximately proportional to the armature current Ittd in phase with the armature current, hence approximately in phase with the main field. Since the conimutating field required to control, in the arma- ture coil under the commutator brush, the e.m.f. of alternation of the main field, is approximately in quadrature behind I he main field — and usually larger than the field controlling the e.m.f. of self-induction of commutation — it follows thai Mm total conimutating field, or the quadrature flux required to give best commutation, must be ahead of the values derived in paragraphs 221 to 224. As the field required by the e.m.f. of alternation in the -)i"ii- circuited coil was found to lag for speeds below the speed of brsl commutation, and to lead above this speed, from the poatMl in quadrature behind the main field, the total GOmmutatiag field must lead this field controlling the e.m.f. of alternation, and it follows: 'See "Theory ami Calculations of Transient Electric Phenomena and Oscillations," Sections I and II, SINGLE-PHASE COMMUTATOR MOTORS 421 Choosing the e.m.f., E2, impressed upon the compensating winding in phase with, and its magnetic flux, therefore in quad- rature (approximately), behind the main field, gives a com- mutation in the repulsion and the series repulsion motor which is better than that calculated from paragraphs 221 to 224, for all speeds up to the speed of best commutation, but becomes in- ferior for speeds above this. Hence the commutation of the repulsion motor and of the series repulsion motor, when con- sidering the self-induction of commutation, is superior to the calculated values below, inferior above the critical speed, that is, the speed of minimum commutation current. The com- mutation of the overcompensated series motor is superior to the values calculated in the preceding, though not of the same magnitude as in the motors with quadrature commutating flux. It also follows that an increase of the inductive reactance of the armature coil increases the exponential and decreases the alter- nating term of e.m.f. and therewith the current in the short- circuited coil, and therefore requires a commutating flux earlier in phase than that required by an armature coil of lower reac- tance, hence improves the commutation of the series repulsion and the repulsion motor at low speeds, and spoils it at high speeds, as seen from the phase angles of the commutating flux calculated in paragraphs 221 to 224. Causing the armature current to lag, by inserting external inductive reactance into the armature circuit, has the same effect as leading commutating flux: it improves commutation at low, impairs it at high speeds. In consequence hereof the com- mutation of the repulsion motor with secondary excitation — in which the inductive reactance of the main field circuit is in the armature circuit — is usually superior, at moderate speeds, to that of the repulsion motor with primary excitation, except at very low speeds, where the angle of lag of the armature cur- rent is very large.