CHAPTER XIII REACTANCE OF SYNCHRONOUS MACHINES 119. The synchronous machine — ^alternating-current generator, synchronous motor or synchronous condenser — consists of an armature containing one or more electric circuits traversed by alternating currents and synchronously revolving relative to a unidirectional magnetic field, excited by direct current. The armature circuit, like every electric circuit, has a resistance, r, in which power is being dissipated by the current, /, and an in- ductance, L, or reactance, a; = 2 irfL^ which represents the mag- netic flux produced by the current in the armature circuit, and interlinked with this circuit. Thus, if ^^ = voltage induced in the armature circuit by its rotation through the magnetic field — or, as now more usually the case, the rotation of the magnetic field through the armature circuit — the terminal voltage of the armature circuit is ^ = ^o-(r+jx)/. In Fig. 110 is shown diagrammatically the path of the field flux, in two different positions, A with an armature slot standing mid- way between two field poles, B with an armature slot standing opposite the field pole. In Fig. Ill is shown diagrammatically the magnetic flux of armature reactance, that is, the magnetic flux produced by the current in the armature circuit, and interlinked with this circuit, which is represented by the reactance x, for the same two relative positions of field and armature. As seen, field flux and armature flux pass through the same iron structures, thus can not have an independent existence, but actual is only their resultant. • This resultant flux of armature self-in- duction and field excitation is shown in Fig. 112, for the same two positions, A and -B, derived by superpositions of the fluxes in Figs. 110 and 111. As seen, in Fig. 112A, all the lines of magnetic forces are inter- linked with the field circuit, but there is no line of magnetic flux interlinked with the armature circuit only, that is, there is ap- 232 REACTANCE OF SYNCHRONOUS MACHINES 233 parently no self-inductive armature flux, and no true self-inducts ive reactance, x, and the self-inductive armature flux of Fig. Ill thus merely is a mathematical fiction, a theoretical component of the resultant flux, Fig. 112. The effect of the armature current, Fra. 110. in changing flux distribution. Fig. IIOA to Fig. 112^, consists in reducing the field flux, that is, flux in the field core, increasing the leakage flux of the field, that is, the flux which leaks from field pole to field pole, without interlinking the armature circuit, and 234 ELECTRIC CIRCUITS still further decreasiDg the armature flux, that is, the flux issiuDg from the field and interlinking with the armature circuit. In position 1 12B, there is no self-inductive armature flux either, but every line of force, which interlinks with the armature circuit, Fig. in. is produced by and interlinked with the field circuit. The effect of the armature current in this case is to increase the field flux and the flux entering the armature at one side of the pole, and decrease it on the other side of the pole, without changing the total field flux and the leakage flux of the field. Indirectly, a reduction of REACTANCE OF SYNCHRONOUS MACHINES 235 the field flux usually occurs, by magnetic saturation limiting the increase of flux at the strengthened pole comer; but this is a sec- ondary effect. Fio. 112. As seen, in 112A the armature current acts demagnetizing, in 112B distorting on the field flux, and in the intermediary position between A and B, a combination of demagnetization (or magneti- zation, in some positions) and distortion occurs. Thus, it may be said that the armature reactance has no inde- pendent existence, is not due to a fiux produced by and interlinked 236 ELECTRIC CIRCUITS • only with the armature circuit, but it is the electrical representa- tion of the efifect exerted on the field flux by them.m.f. of the arma- ture current. Considering the magnetic disposition, an armature current, which alone would produce the flux, Fig. Ill, in the presence of a field excitation which alone would give the flux, Fig. 110, has the following effect: in Fig. 112A, by the counter m.m.f. of the arma- ture current the resultant m.m.f. and with it the resultant flux are reduced from that due to the m.m.f. of field excitation, to that due to field excitation minus the m.m.f. of the armature current. The difference of the magnetic potential between the field poles is increased: in Fig. IIOA it is the sum of the m.m.f s. of the two air- gaps traversed by the flux (plus the m.m.f. consumed in the arma- ture iron, which may be neglected as small) ; in Fig. 112^1 it is the sum of the m.m.fs. of the two air-gaps traversed by the flux (which is slightly smaller than in Fig. 110 A, due to the reduced flux) plus the counter m.m.f. of the armature. The increased magnetic potential difference causes an increased magnetic leak- age flux between the field poles, and thereby still further reduces the armature flux and the voltage induced by it. In Fig. 112j&, the m.m.f. of the armature current adds itself to the m.m.f. of field excitation on one side, and thereby increases the flux, and it subtracts on the other side and decreases the flux, and thereby causes an unsymmetrical flux distribution, that is, a field distortion. 120. Both representations of the effect of armature current are used, that by a nominal magnetic flux. Fig. Ill , which gives rise to a nominal reactance, the "synchronous reactance of the arma- ture circuit,' ' and that by considering the direct magnetizing action of the armature current, as "armature reaction," and both have their advantages and disadvantages. The introduction of a synchronous reactance^ a;©, and correspond- ing thereto of a nominal induced e,m,f,y cq, is most convenient in electrical calculations, but it must be kept in mind, that neither Co nor Xq have any actual existence, correspond to actual magnetic fluxes, and for instance, when calculating efficiency and losses, the core loss of the machine does not correspond to eo, but corresponds to the actual or resultant magnetic flux. Fig. 112. Also, in deal- ing with transients involving the dissipation of the magnetic energy stored in the machine, the magnetic energy of the result- ant field, Fig. 112, comes into consideration, and not the — ^much REACTANCE OF SYNCHRONOUS MACHINES 237 larger — energy, which the fields corresponding to e© and Xo would have. Thus the short-circuit transient of a heavily loaded ma- chine is essentially the same as that of the same machine at no- load, with the same terminal voltage, although in the former the field excitation and the nominal induced voltage may be very much larger. The use of the term armature reaction in dealing with the effect of load on the synchronous machine is usually more convenient and useful in design of the machine, but less so in the calculation dealing with the machine as part of an electric circuit. Either has the disadvantage that its terms, synchronous react- ance or armature reaction, are not homogeneous, as the different parts of the reactance field. Fig. Ill, which make up the difference between Fig. 112 and Fig. 110, are very different in their action, especially in their behavior at sudden changes of circuit conditions. 121. Considering the magnetic flux of the armature current. Fig. lllA, which is represented by the synchronous reactance, Xo. A part of this magnetic flux (lines a in Fig. 111-4) interlinks with the armature circuit only, that is, is true self-inductive or leakage flux. Another part, however, (6) interlinks with the field also, and thus is mutual inductive flux of the armature cir- cuit on the field circuit. In a polyphase machine, the resultant armature flux, that is, the resultant of the fluxes. Fig. Ill, of all phases, revolves synchronously at (approximately) constant in- tensity, as a rotating field of armature reaction, and, therefore, is stationary with regard to the synchronously revolving field, F, Hence, the mutual inductive flux of the armature on the field, though an alternating flux, exerts no induction on the field circuit, is indeed a unidirectional or constant flux with regards to the field circuit. Therefore, under stationary conditions of load, no difference exists between the self-inductive and the mutual in- ductive flux of the armature circuit, and both are comprised in the synchronous reactance, Xq, If, however, the armature current changes, as by an increase of load, then with increasing armature current, the armature flux, a and 6, Fig. Ill, also increases, a, being interlinked with the armature current only, increases simul- taneously with it, that is, the armature current can not increase without simultaneously increasing its self-inductive flux, a. The mutual inductive flux, 6, however, interlinks with the field circuit, and this circuit is closed through the exciter, that is, is a closed secondary circuit with regards to the armature circuit as primary, 238 ELECTRIC CIRCUITS and the change of flux, b, thus induces in the field circuit an e.m.f. and causes a current which retards the change of this flux com- ponent, 6. Or, in other words, an increase of armature current tends to increase its mutual magnetic flux, 6, and thereby to de- crease the field flux. This decrease of field flux induces in the field circuit an e.m.f., which adds itself to the voltage impressed upon the field, thereby increases the field current and maintains the field flux against the demagnetizing action of the armature cur- rent, causing it to decrease only gradually. Inversely, a decrease of armature current gives a simultaneous decrease of the self- inductive part of the flux, a in Fig. Ill, but a gradual decrease of the mutual inductive part, 6, and corresponding gradual increase of the resultant field flux, by inducing a transient voltage in the field, in opposition to the exciter voltage, and thereby decreasing the field current. Every sudden increase of the armature current thus gives an equal sudden drop of terminal voltage due to the self-inductive flux, a, produced by it (and the resistance drop in the armatm*e circuit), an equally sudden increase of the field current, and then a gradual further drop of the terminal voltage by the gradual ap- pearance of the mutual flux, &, and corresponding gradual decrease of field current to nominal. The reverse is the case at a sudden decrease of armature current. The extreme case hereof is fbund in the momentary short-cir- cuit currents of alternators,^ which with some types of machines may momentarily equal many times the value of the permanent short-circuit current. However, this phenomenon is not limited to short-circuit conditions only, but every change of ciurent in an alternator causes a momentary overshooting, the more so, the greater and more sudden the change is. 122. That part of the synchronous reactance, Xoj which is due to the magnetic lines, a, in Fig. Ill, is a true self-inductive reactance, X, and is instantaneous, but that part of Xi representing the flux lines, 6, is mutual inductive reactance with the field circuit, x', and is not instantaneous, but comes into play gradually, and when- ever dealing with rapid changes of circuit conditions, the syn- chronous reactance, xo, thus must be divided into a true or self- inductive reactance, x, and a mutual inductive reactance, x': Xq ^ X "t~ X» *See "Theory and Calculation of Transient Phenomena." REACTANCE OF SYNCHRONOUS MACHINES 239 The change of the flux disposition, caused by a current in the armature circuit, from that of Fig. 110 to that of Fig. 112, thus is simultaneous with the armature current and instantaneous with a sudden change of armature ciu'rent only as far as it does not in- volve any change of the flux through the field winding, but the change of the flux through the field coils is only gradual. Thus the flux change in the armature core can be instantaneous, but that in the field is gradual. This difference between self-inductive and mutual inductive reactance, or between instantaneous and gradual flux change, comes into consideration only in transients, and then very fre- quently the instantaneous or self-inductive effect is represented by a self-inductive reactance, x, the gradual or mutual inductive effect by an armatiu'e reaction. The relation between self-inductive component, x, and mutual inductive component, x\ varies from about 2 -?- 1 in the unitooth- high frequency alternators of old, to about 1 -5- 20 in some of the earlier turbo-alternators. In those synchronous machines, which contain a squirrel-cage induction-motor winding in the field faces, for starting as motors, or as protection against himting, or to equaUze the armature reaction in single-phase machines, all the armature reactance flux, which interlinks with the squirrel-cage conductors (as the flux, c, in Fig. 11 IB), also is mutual inductive flux, and such machines thus have a higher ratio of mutual inductive to self-inductive armature reactance, that is, show a greater overshooting of cur- rent at sudden changing of load, and larger momentary short- circuit currents. The mutual flux of armatiwe reactance induces in the field cir- cuit only under transient conditions, but imder permanent cir- cuit conditions the mutual inductance of the armature on the field has no inducing action, but is merely demagnetizing, and the distinction between self-inductive and mutual inductive react- ance thus is unnecessary, and both combine in the synchronous reactance. In this respect, the synchronous machine differs from the transformer; in the latter, self-inductance and mutual inductance are always distinct in their action. 123. In permanent conditions of the circuit, the armature re- actance of the synchronous machine is the synchronous react- ance, xo =^ X + x'; Sit the instance of a sudden change of circuit conditions, the mutual inductive reactance, x', is still non-exist- 240 ELECTRIC CIRCUITS ing, and only the self-inductive reactance, x, comes into play. Intermediate between the instantaneous effect and the permanent conditions, for a time up to one or more sec, the effective reactance changes, from x to Xo, and this may be considered as a transterd reactance. During this period, mutual induction between armature cir- cuit and field circuit occurs, and the phenomena in the synchron- ous machine thus are affected by the constants of the field circuit outside of the machine. That is, resistance and inductance of the field circuit appear, by mutual induction, as part of the armature circuit of the synchronous machine, just as resistance and react- ance of the secondary circuit of a transformer appear, trans- formed by the ratio of turns, as resistance and reactance in the pri- mary, in their effect on the primary current and its phase relation. Thus in the synchronous machine, a high non-inductive re- sistance inserted into the field circuit (with an increase of the exciter voltage to give the same field current) while without effect on the permanent current and on the instantaneous current in the moment of a sudden current change, reduces the duration of the transient armature current; an inductance inserted into the field circuit lengthens the duration of the transient and changes its shape. The duration of the transient reactance of the synchronous machine is about of the same magnitude as the period of hunting of synchronous machines — which varies from a fraction of a second to over one sec. The reactance, which limits the current fluctations in hunting synchronous machines, thus is neither the synchronous reactance, Xo,nor the true self-inductive reactance, x, but is an intermediate transient reactance; the current change is sufficiently slow that the mutual induction between synchronous machine armature and field has already come into play and the field begun to follow, but is too rapid for the complete develop- ment of the synchronous reactance. 124. In the polyphase machine on balanced load, the mutual inductive component of the armature reactance has no inductive effect on the field, as its resultant is imidirectionaJ with regard to the field flux. In the single-phase machine, however (or polyphase machine on imbalanced load), such inductive effect exists, as a permanent pulsation of double frequency. The mutual inductive flux of the armature circuit on the field circuit is alternating, and the field circuit, revolving synchronously REACTANCE OF SYNCHRONOUS MACHINES 241 through this alternating flux, thus has an e.m.f. of double fre- quency induced in it, which produces a double-frequency current in the field circuit, superimposed on the direct ciu-rent from the exciter. The field flux of the single-phase alternator (or poly- phase alternator at imbalanced load) thus pulsates with double frequency, and, by being carried synchronously through the armature circuits, this double-frequency pulsation of flux in- duces a triple-frequency harmonic in the armature. Thus, single-phase alternators, and polyphase alternators at unbalanced load, contain more or less of a third harmonic in their voltage wave, which is induced by the double-frequency pulsation of the field flux, resulting from the pulsating armature reaction, or mutual armature reactance, x\ The statement, that three-phase alternators contain no third harmonics in their terminal voltages, since such harmonics neu- tralize each other, is correct only for balanced load, but at un- balanced load, three-phase alternators may have pronounced third harmonics in their terminal voltage, and on single-phase short-circuit, the not short-circuited phase of a three-phase alternator may contain a third harmonic far in excess of the fundamental. 125. Let in a F-connected three-phase synchronous machine, the magnetic flux per field pole be $o. If this flux is distributed sinusoidally around the circumference of the armature, at any time, ^, represented by angle, = 2Tr ft, the magnetic flux enclosed by an armature turn is $ = $0 cos when counting the time from the moment of maximum flux. The voltage induced in an armature circuit of n turns then is ei = n-^r = c^o sin at where c = 2 7r/n If, however, the flux distribution around the armature circum- ference is not sinusoidal, it nevertheless can, as a periodic func- tion, be expressed by ^ = $0 [cos + a2 cos 2{ — a2) + aa cos 3(<^ — az) + a^ cos 4{(t> — a4) + . . . ] and the voltage induced in one armature conductor, by the 16 242 ELECTRIC CIRCUITS synchronous rotation through this flux, is -jr = tt/^o [sin <^ + 2 02 sin 2{ — a2) + 3 az sin 3(0— aa) + 4 a4 sin 4(<^ — ai) + . . . ] hence, the voltage induced in one full-pitch armature turn, or in two armature conductors displaced from each other on the arma- ture surface by one pole pitch or an odd multiple thereof, 6 = 2 7r/$o[sin <^+3 as sin 3(<^ — as) +5 as sin 5(<^— 05)+ . . . ] that is, the even harmonics cancel. The voltage induced in one armature circuit of n effective series turns then is 61 = C$0 [sin <^ + 63 sin 3(<^ — as) + 65 sin 5(<^ — as) + ... ] where 63 = 3 as, 65 = 5 a^j etc., if all the n turns are massed together, and are less, if the armature turns are distributed, due to the overlapping of the harmonics, and partial cancellation caused thereby. As known, by causing proper pitch of the turn, or proper pitch of the arc covered by any phase, any harmonic can be entirely eliminated. The second and third phase of the three-phase machine then would have the voltage, 62 = C$0 [sin (<^ - 120°) + 63 sin 3(<^ - as - 120°) + 65 sin 5(0 - ag - 120°) + . . . ] = C$0 [sin { - 120°) + 63 sin 3(<^ - as) + 65 sin (5[« - aj + 120°) + . . .] 63 = C$0 [sin {4> - 240°) + 63 sin 3(<^ - as) + 65 sin (5[« - ad + 240°)] + . . .] As seen, the third harmonics are all three in phase with each other; the fifth harmonics are in three-phase relation, but with backward rotation; the seventh harmonics are again in three- phase relation, like the fundamentals, the ninth harmonics in phase, etc. The terminal voltages of the machine then are J5?i = Cs — C2 = \/3 C$0 [cos <^ — 65 cos 5 (<^ — ag) + 67 cos 7 {4> — ai) h . • •] and corresponding thereto E2 = Ci — Cs and -Bs = C2 — Ci, differ- ing from El merely by substituting — 120° and — 240° for f REACTANCE OF SYNCHRONOUS MACHINES 243 As seen, the third harmonic eliminates in the terminal voltages of the three-phase machine, regardless of the flux distribution, provided that the flux is constant in intensity, that is, the load conditions balanced. 126. Assuming, however, that the load on the three-phase machine is unbalanced, causing a double-frequency pulsation of the magnetic flux, ^o{l + a cos 2 )y assuming for simplicity sinusoidal distribution of magnetic flux. The flux interlinked with a full-pitch armature turn then is $ = $o(l- + a cos 2 ) cos ((f) — a) = ^0 cos (<^ — a) + ^ COS { + a) + ^ COS (3 <^ — a) and the voltage induced in an armature circuit of n effective turns, ei = n-TT = C$0 ^1 cos (-a) + ^cos ( + a) + ^cos (3 <^ - a)\ = C$0 sin (<^ — a) + o sin ( + a) + -y sin (3 <^ — or) or, if the magnetic flux maximum coincides with the voltage maximum of the first phase, a = 0, ei = c$o[(l + 2) sin + -2-sin3<^J. In the second phase, the flux is the same, $0 (1 + a cos 2 ), but the flux interlinkage 120° later, thus, $ = $0 (1 + a cos 2 ) cos { - a - 120°), and the voltage of the second phase thus is derived from that of the first phase, by substituting a + 120° for a, 62 = C$0 [sin ( - a- 120°) + I sin (<^ + a + 120°) + ^ sin (3 (^ - a - 120°)] and the third phase, es = C$0 [sin ( - a - 240°) + | sin (<^ + a + 240°) + 244 ELECTRIC CIRCUITS the terminal voltages thus are, ^1 = 63 — ^2 = \/3 c^o cos ( — a) — ^ cos ( + a) + -^ cos (3 * - a) J and in the same manner, the other two phases, ^2 = V3 C$0 [cos (<^ - a - 120°) - I cos (<^ + « + 120°) - ^ cos (3 « - a - 120°)] E% = V3 C$0 [cos (<^ - Of - 240°) - I cos (<^ + or + 240°) - ^ cos (3 « - « - 240°)] . For a = 0, this gives J5;i = V3 c$o[ (l - ^) cos <^ + ^ cos 3 «] ^2 = V3 c$o[ (l - I) cos ((^ - 120°) - ?| cos (3 « - 120°)] Ez^ y/Z C$0 [ (l - I) cos ((^ - 240°) - ^ cos (3 « - 240°)] . As seen, all three phases have pronounced third harmonics, and the third harmonic of the loaded phase, E\^ is opposite to that of the unloaded phases. If a = 1, which corresponds about to short-circuit conditions, as it makes the minimum value of $o equal zero, then the quadra- ture phase of the short-cixcuited phase, £i, becomes 3 C$0/ . , , ' n A\ that is, the third harmonic becomes as large as the fundamental. Thus, on unbalanced load, such as on single-phase short-circuit, triple harmonics appear in the terminal voltages of a three-phase generator, though at balanced loads the three-phase terminal voltage can contain no third harmonics. SECTION III