LECTURE IV. SINGLE-ENERGY TRANSIENTS IN ALTERNATING- CURRENT CIRCUITS. 17. Whenever the conditions of an electric circuit are changed in such a manner as to require a change of stored energy, a transi- tion period appears, during which the stored energy adjusts itself from the condition existing before the change to the condition after the change. The currents in the circuit during the transition period can be considered as consisting of the superposition of the permanent current, corresponding to the conditions after the change, and a transient current, which connects the current value before the change with that brought about by the, change. That is, if ii = current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and 12 = current which should exist at the moment of change in accordance mth the circuit condition after the change, then the actual current ii can be considered as consisting of a part or component 12, and a component ii — 12 = iq. The former, 2*2, is permanent, as result- ing from the established circuit condition. The current compo- nent U, however, is not produced by any power supply, but is a remnant of the previous circuit condition, that is, a transient, and therefore gradually decreases in the manner as discussed in para- graph 13, that is, with a duration T = — - The permanent current 12 may be continuous, or alternating, or may be a changing current, as a transient of long duration, etc. The same reasoning applies to the voltage, magnetic flux, etc. Thus, let, in an alternating-current circuit traversed by current i'l, in Fig. 15 A, the conditions be changed, at the moment t = 0, so as to produce the current iV The instantaneous value of the current ii at the moment t = 0 can be considered as consisting of the instantaneous value of the permanent current (2, shown dotted, and the transient io = i\ — 22. The latter gradually dies down, with the duration T = — , on the usual exponential tran- 30 I SINGLE-ENERGY TRANSIENTS. 31 sient, shown dotted in Fig. 15. Adding the transient current io to the permanent current 22 gives the total current during the transition period, which is shown in drawn line in Fig. 15. As seen, the transient is due to the difference between the instantaneous value of the current ii which exists, and that of the current 2*2 which should exist at the moment of change, and Fig. 15. — Single-energy Transient of Alternating-current Circuit. thus is the larger, the greater the difference between the two currents, the previous and the after current. It thus disappears if the change occurs at the moment when the two currents ii and 2*2 are equal, as shown in Fig. 155, and is a maximum, if the change occurs at the moment when the two currents ii and 12 have the greatest difference, as shown in Fig. 15C, that is, at a point one-quarter period or 90 degrees distant from the intersec- tion of ii and 12. 32 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If the current ii is zero, we get the starting of the alternating current in an inductive circuit, as shown in Figs. 16, A, B,C. The starting transient is zero, if the circuit is closed at the moment when the permanent current would be zero (Fig. 165), and is a maximum when closing the circuit at the maximum point of the permanent-current wave (Fig. 16C). The permanent current and the transient components are shown dotted in Fig. 16, and the resultant or actual current in drawn lines. Fig. 16. — Single-energy Starting Transient of Alternating-current Circuit. 1 8. Applying the preceding to the starting of a balanced three-phase system, we see, in Fig. 17 A, that in general the three transients 2*1°, {2°, and 13° of the three three-phase currents ii, 22, i^ are different, and thus also the shape of the three resultant currents during the transition period. Starting at the moment of zero current of one phase, ii, Fig. 175, there is no transient for this current, while the transients of the other two currents, 2*2 and 23, are equal and opposite, and near their maximum value. Starting, in Fig. 17 C, at the maximum value of one current 23, we have the maximum value of transient for this current 2*3^ while the transients of the two other currents, 21 and 2*2, are equal, have SINGLE-ENERGY TRANSIENTS. 33 half the value of iz^, and are opposite in direction thereto. In any case, the three transients must be distributed on both sides of the zero line. This is obvious: if ii, iV, and is are the instan- taneous values of the permanent three-phase currents, in Fig. 17, the initial values of their transients are: —ii, — zV, —4'. .1 '>?jQc>r>: — .1 '' i, '■ X Fig. 17. — Single-energy Starting Transient of Three-phase Circuit. Since the sum of the three three-phase currents at every moment is zero, the sum of the initial values of the three transient currents also is zero. Since the three transient curves ii^, 12^, 4° are pro- portional to each other fas exponential curves of the same dura- tion T = —\ and the sum of their initial values is zero, it follows 84 ELECTRIC DISCHARGES, WAVES AND IMPULSES. that the sum of their instantaneous values must be zero at any moment, and therefore the sum of the instantaneous values of the resultant currents (shown in drawn line) must be zero at any moment, not only during the permanent condition, but also dur- ing the transition period existing before the permanent condi- tion is reached. It is interesting to apply this to the resultant magnetic field produced by three equal three-phase magnetizing coils placed under equal angles, that is, to the starting of the three-phase rotating magnetic field, or in general any polyphase rotating magnetic field. Fig. 18. — Construction of Starting Transient of Rotating Field. As is well known, three equal magnetizing coils, placed under equal angles and excited by three-phase currents, produce a result- ant magnetic field which is constant in intensity, but revolves synchronously in space, and thus can be represented by a concen- tric circle a, Fig. 18. This, however, applies only to the permanent condition. In the moment of start, all the three currents are zero, and their resultant magnetic field thus also zero, as shown above. Since the magnetic field represents stored energy and thus cannot be produced instantly, a transient must appear in the building up of the rotating field. This can be studied by considering separately SINGLE-ENERGY TRANSIENTS. 35 the permanent and the transient components of the three currents, as is done in the preceding. Let ii, iV, is be the instantaneous values of the permanent currents at the moment of closing the circuit, t = 0. Combined, these would give the resultant field OAo in Fig. 18. The three transient currents in this moment are ii^ =—ii, 12^ = —ii', iz^ =—iz, and combined these give a resultant field OBo, equal and opposite to OAq in Fig. 18. The permanent field rotates synchronously on the concentric circle a; the transient field OB remains constant in the direction OBq, since all three transient components of current decrease in propor- tion to each other. It decreases, however, with the decrease of the transient current, that is, shrinks together on the line BqO. The resultant or actual field thus is the combination of the per- manent fields, shown as OAi OA2, • . . , and the transient fields, shown as OBi, OB2, etc., and derived thereby by the parallelo- gram law, as shown in Fig. 18, as OCi, OC2, etc. In this diagram, BiCi, B2C2, etc., are equal to OAi, OA2, etc., that is, to the radius of the permanent circle a. That is, while the rotating field in permanent condition is represented by the concentric circle a, the resultant field during the transient or starting period is repre- sented by a succession of arcs of circles c, the centers of which move from Bq in the moment of start, on the line BqO toward 0, and can be constructed hereby by drawing from the successive points Bo, Bi, B2, which correspond to successive moments of time 0, h, t2 . . . , radii BiCi, B2C2, etc., under the angles, that is, in the direction corresponding to the time 0, ^1, t2, etc. This is done in Fig. 19, and thereby the transient of the rotating field is constructed. Fig. 19, — Starting Transient of Rotating Field: Polar Form. WAVES AND IMPULSES. From this polar diagram of the rotating field, in Fig. 19, values OC can now be taken, corresponding to successive moments of time, and plotted in rectangular coordinates, as done in Fig. 20. As seen, the rotating field builds up from zero at the moment of closing the circuit, and reaches the final value by a series of oscil- lations ; that is, it first reaches beyond the permanent value, then drops below it, rises again beyond it, etc. 4 cycles Fig, 20. — Starting Transient of Rotating Field: Rectangular Form. "^e have here an oscillatory transient, produced in a system with only one form of stored energy (magnetic energy), by the combination of several simple exponential transients. How- ever, it must be considered that, while energy can be stored in one form only, as magnetic energy, it can be stored in three electric circuits, and a transfer of stored magnetic energy between the three electric circuits, and therewith a surge, thus can occur. It is interesting to note that the rotating-field transient is independent of the point of the wave at which the circuit is closed. That is, while the individual transients of the three three-phase currents vary in shape with the point of the wave at which they start, as shown in Fig. 17, their polyphase resultant always has the same oscillating approach to a uniform rotating field, of duration T = — - r The maximum value, which the magnetic field during the transi- tion period can reach, is limited to less than double the final value, as is obvious from the construction of the field. Fig. 19. It is evident herefrom, however, that in apparatus containing rotating fields, as induction motors, polyphase sjaichronous machines, etc., the resultant field may under transient conditions reach nearly double value, and if then it reaches far above magnetic saturation, excessive momentary currents may appear, similar as in starting transformers of high magnetic density. In polyphase rotary SINGLE-ENERGY TRANSIENTS. 37 apparatus, however, these momentary starting currents usually are far more limited than in transformers, by the higher stray field (self-inductive reactance), etc., of the apparatus, resulting from the air gap in the magnetic circuit. 19. As instance of the use of the single-energy transient in engineering calculations may be considered the investigation of the momentary short-circuit phenomena of synchronous alter- nators. In alternators, especially high-speed high-power ma- chines as turboalternators, the momentary short-circuit current may be many times greater than the final or permanent short- circuit current, and this excess current usually decreases fairly slowly, lasting for many cycles. At the same time, a big cur- rent rush occurs in the field. This excess field current shows curious pulsations, of single and of double frequency, and in the beginning the armature currents also show unsymmetrical shapes. Some oscillograms of three-phase, quarter-phase, and single-phase short circuits of turboalternators are shov/n in Figs. 25 to 28. By considering the transients of energy storage, these rather complex-appearing phenomena can be easily understood, and pre- determined from the constants of the machine with reasonable exactness. In an alternator, the voltage under load is affected by armature reaction and armature self-induction. Under permanent condi- tion, both usually act in the same way, reducing the voltage at noninductive and still much more at inductive load, and increasing it at antiinductive load; and both are usually combined in one quantity, the synchronous reactance Xq. In the transients result- ing from circuit changes, as short circuits, the self-inductive armature reactance and the magnetic armature reaction act very differently:* the former is instantaneous in its effect, while the latter requires time. The self-inductive armature reactance Xi con- sumes a voltage Xii by the magnetic flux surrounding the armature conductors, which results from the m.m.f. of the armature cur- rent, and therefore requires a component of the magnetic-field flux for its production. As the armature magnetic flux and the current which produces it must be simultaneous (the former being an integral part of the phenomenon of current flow, as seen in Lecture II), it thus follows that the armature reactance appears together * So also in their effect on synchronous operation, in hunting, etc. 38 ELECTRIC DISCHARGES, WAVES AND IMPULSES. with the armature current, that is, is instantaneous. The arma- ture reaction, however, is the m.m.f. of the armature current in its reaction on the m.m.f. of the field-exciting current. That is, that part X2 = a;o — Xi of the synchronous reactance which corresponds to the armature reaction is not a true reactance at all, consumes no voltage, but represents the consumption of field ampere turns by the m.m.f. of the armature current, and the corresponding change of field flux. Since, however, the field flux represents stored magnetic energy, it cannot change instantly, and the arma- ture reaction thus does not appear instantaneously with the arma- ture current, but shows a transient which is determined essentially by the constants of the field circuit, that is, is the counterpart of the field transient of the machine. If then an alternator is short-circuited, in the first moment only the true self-inductive part Xi of the synchronous reactance exists, and the armature current thus is ^l = — , where eo is the induced Xi e.m.f., that is, the voltage corresponding to the magnetic-field excitation flux existing before the short circuit. Gradually the armature reaction lowers the field flux, in the manner as repre- sented by the synchronous reactance Xq, and the short-circuit cur- rent decreases to the value ?"o = — • The ratio of the momentary short-circuit current to the perma- nent short-circuit current thus is, approximately, the ratio -^ = — y to Xi that is, synchronous reactance to self-inductive reactance, or armature reaction plus armature self-induction, to armature self-induction. In machines of relatively low self-induction and high armature reaction, the momentary short-circuit cur- rent thus may be many times the permanent short-circuit current. The field flux remaining at short circuit is that giving the volt- age consumed by the armature self-induction, while the decrease of field flux between open circuit and short circuit corresponds to the armature reaction. The ratio of the open-circuit field flux to the short-circuit field flux thus is the ratio of armature reaction plus self-induction, to the self-induction; or of the synchronous reactance to the self-inductive reactance: — • Xi SINGLE-ENERGY TRANSIENTS. 39 Thus it is: momentary short-circuit current _ open-circuit field flux * _ permanent short-circuit current short-circuit field flux armature reaction plus self-induction _ synchronous reactance _ Xq self-induction self-inductive reactance Xi 20. Let $1 = field flux of a three-phase alternator (or, in general, polyphase alternator) at open circuit, and this alternator be short- circuited at the time t = 0. The field flux then gradually dies down, by the dissipation of its energy in the field circuit, to the short-circuit field flux $o, as indicated by the curve $ in Fig. 21A. If m = ratio armature reaction plus self-induction _ Xq armature self-induction Xi' it is $1 = m$o, and the initial value of the field flux consists of the permanent part $o, and the transient part ^' = $i — $o = (m— 1) $0- This is a rather slow transient, frequently of a duration of a second or more. The armature currents ii, 12, iz are proportional to the held flux $ which produces them, and thus gradually decrease, from initial values, which are as many times higher than the final values as $1 is higher than $0, or m times, and are represented in Fig. 2\B. The resultant m.m.f. of the armature currents, or the armature reaction, is proportional to the currents, and thus follows the same field transient, as shown by F in Fig. 21C The field-exciting current is 2*0 at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current 2*0 combines with the armature reaction Fq, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux $o- During the transition period the held flux $ is higher than $0, and the resultant m.m.f. must therefore be higher in the same proportion. Since it is the dif- ference between the field current and the armature reaction F, and the latter is proportional to , the field current thus must also be * If the machine were open-circuited before the short circuit, otherwise the field flux existing before the short circuit. It herefrom follows that the momentary short-circuit current essentially depends on the field flux, and thereby the voltage of the machine, before the short circuit, but is practically independent of the load on the machine before the short circuit and the field excitation corresponding to this load. 40 ELECTRIC DISCHARGES, WAVES AND IMPULSES. proportional to $. Thus, as it is ?; = io at $0, during the transition period it is i = — Iq. Hence, the field-exciting current traverses $0 the same transient, from an initial value iq to the normal value U, as the field flux ^ and the armature currents. '^l ^' ^'^^ ^ A *0 % To ^ Fig. 21. — Construction of Momentary Short Circuit Characteristic of Poly- phase Alternator. Thus, at the moment of short circuit a sudden rise of field current must occur, to maintain the field flux at the initial value $1 against the demagnetizing armature reaction. In other words, the field flux $ decreases at such a rate as to induce in the field circuit the e.m.f. required to raise the field current in the propor- tion m, from Iq to Iq , and maintain it at the values corresponding to the transient ?*, Fig. 2 ID. As seen, the transients ^I^; I'l, 2*2, H', F; i are proportional to each other, and are a field transient. If the field, excited by current iq SINGLE-ENERGY TRANSIENTS. 41 at impressed voltage eo, were short-circuited upon itself, in the first moment the current in the field would still be ^o, and there- fore the voltage eo would have to be induced by the decrease of magnetic flux ; and the duration of the field transient, as discussed in Lecture III, would be To = — • To The field current in Fig. 21 D, of the alternator short-circuit transient, starts with the value io^ = rnio, and if eo is the e.m.f. supplied in the field-exciting circuit from a source of constant voltage supply, as the exciter, to produce the current iq', the voltage eo' = 7?ieo must be acting in the field-exciting circuit; that is, in addition to the constant exciter voltage eo, a voltage (?7^ — 1) eo must be induced in the field circuit by the transient of the mag- netic flux. As a transient of duration — induces the voltage eo, to induce the voltage (m — l)eo the duration of the transient must be T — -^0 (m — 1) To where Lq = inductance, Tq = total resistance of field-exciting cir- cuit (inclusive of external resistance). The short-circuit transient of an alternator thus usually is of shorter duration than the short-circuit transient of its field, the more so, the greater m, that is, the larger the ratio of momentary to permanent short-circuit current. In Fig. 21 the decrease of the transient is shown greatly exagger- ated compared with the frequency of the armature currents, and Fig. 22 shows the curves more nearly in their actual proportions. The preceding would represent the short-circuit phenomena, if there were no armature transient. However, the armature cir- cuit contains inductance also, that is, stores magnetic energy, and thereby gives rise to a transient, of duration T = —, where L = inductance, r = resistance of armature circuit. The armature transient usually is very much shorter in duration than the field transient. The armature currents thus do not instantly assume their symmetrical alternating values, but if in Fig. 215, ii, iV, is are the instantaneous values of the armature currents in the moment of start, t = 0, three transients are superposed upon these, and 42 ELECTRIC DISCHARGES, WAVES AND IMPULSES. start with the values —ii, — iV, —is. The resultant armature currents are derived by the addition of these armature transients upon the permanent armature currents, in the manner as dis- cussed in paragraph 18, except that in the present case even the permanent armature currents ii, 4, is are slow transients. In Fig. 22B are shown the three armature short-circuit currents, in their actual shape as resultant from the armature transient and the field transient. The field transient (or rather its begin- ning) is shown as Fig. 22 A. Fig, 22B gives the three armature Fig. 22. — Momentary Short Circuit Characteristic of Three-phase Alternator. currents for the case where the circuit is closed at the moment when ii should be maximum ; ^l then shows the maximum transient, and 12 and iz transients in opposite direction, of half amplitude. These armature transients rapidly disappear, and the three currents become symmetrical, and gradually decrease with the field tran- sient to the final value indicated in the figure. The resultant m.m.f. of three three-phase currents, or the arma- ture reaction, is constant if the currents are constant, and as the currents decrease with the field transient, the resultant armature reaction decreases in the same proportion as the field, as is shown SINGLE-ENERGY TRANSIENTS. 43 in Fig. 21 C by F. During the initial part of the short circuit, however, while the armature transient is appreciable and the armature currents thus unsymmetrical, as seen in Fig. 22B, their resultant polyphase m.m.f. also shows a transient, the transient of the rotating magnetic field discussed in paragraph 18. That is, it approaches the curve F of Fig. 21 C by a series of oscillations, as indicated in Fig. 21E. Since the resultant m.m.f. of the machine, which produces the flux, is the difference of the field excitation. Fig. 2 ID and the armature reaction, then if the armature reaction shows an initial os- cillation, in Fig. 21 E, the field-exciting current must give the same oscillation, since its m.m.f. minus the armature reaction gives the resultant field excitation corresponding to flux $. The starting transient of the polyphase armature reaction thus appears in the j&eld current, as shown in Fig. 22C, as an oscillation of full machine frequency. As the mutual induction between armature and field circuit is not perfect, the transient pulsation of armature reaction appears with reduced amplitude in the field current, and this reduction is the greater, the poorer the mutual inductance, that is, the more distant the field winding is from the armature wind- ing. In Fig. 22(7 a damping of 20 per cent is assumed, which corresponds to fairly good mutual inductance between field and armature, as met in turboalternators. If the field-exciting circuit contains inductance outside of the alternator field, as is always the case to a slight extent, the pul- sations of the field current. Fig. 22C, are slightly reduced and delayed in phase; and with considerable inductance intentionally inserted into the field circuit, the effect of this inductance would require consideration. From the constants of the alternator, the momentary short- circuit characteristics can now be constructed. Assuming that the duration of the field transient is = 1 sec, {7n — 1) ro the duration of the armature transient is T =- = A sec. r And assuming that the armature reaction is 5 times the armature 44 ELECTRIC DISCHARGES, WAVES AND IMPULSES. self-induction, that is, the synchronous reactance is 6 times the self- inductive reactance, — = m = 6. The frequency is 25 cycles. If $1 is the initial or open-circuit flux of the machine, the short- ^1 1 circuit flux is $o = — = ;^ ^i, and the field transient is a tran- m o sient of duration 1 sec, connecting $i and $o, Fig. 22A, repre- sented by the expression _± «J> = $0 + ($1 — ^o)e ^o. The permanent armature currents ii, i^, is then are currents starting with the values m —, and decreasing to the final short- Xo circuit current — , on the field transient of duration To. To these Xo currents are added the armature transients, of duration T, which start with initial values equal but opposite in sign to the initial values of the permanent (or rather slowly transient) armature currents, as discussed in paragraph 18, and thereby give the asym- metrical resultant currents. Fig. 225. The field current i gives the same slow transient as the flux , starting with ^o' = mio, and tapering to the final value io. Upon this is superimposed the initial full-frequency pulsation of the armature reaction. The transient of the rotating field, of duration T = .1 sec, is constructed as in paragraph 18, and for its instan- taneous values the percentage deviation of the resultant field from its permanent value is calculated. Assuming 20 per cent damping in the reaction on the field excitation, the instantaneous values of the slow field transient (that is, of the current (^ ~ z'o), since I'o is the permanent component) then are increased or de- creased by 80 per cent of the percentage variation of the transient field of armature reaction from uniformity, and thereby the field curve. Fig. 22(7, is derived. Here the correction for the external field inductance is to be applied, if considerable. Since the transient of the armature reaction does not depend on the point of the wave where the short circuit occurs, it follows that the phenomena at the short circuit of a polyphase alternator are always the same, that is, independent of the point of the wave at which the short circuit occurs, with the exception of the initial wave shape of the armature currents, which individually depend SINGLE-ENERGY TRANSIENTS. 45 on the point of the wave at which the phenomenon begins, but not so in their resultant effect. 21. The conditions with a single-phase short circuit are differ- ent, since the single-phase armature reaction is pulsating, vary- ing between zero and double its average value, with double the machine frequency. The slow field transient and its effects are the same as shown in Fig. 21, A to D. However, the pulsating armature reaction produces a corre- sponding pulsation in the field circuit. This pulsation is of double Fig. 23. — Symmetrical Momentary Single-phase Short Circuit of Alternator. frequency, and is not transient, but equally exists in the final short- circuit current. Furthermore, the armature transient is not constant in its reaction on the field, but varies with the point of the wave at which the short circuit starts. Assume that the short circuit starts at that point of the wave where the permanent (or rather slowly transient) armature current should be zero: then no armature transient exists, and the armature current is symmetrical from the beginning, and shows the slow transient of the field, as shown in Fig. 23, where A i:6 ELECTRIC DISCHARGES, WAVES AND IMPULSES. is the field transient $ (the same as in Fig. 22 A) and B the arma- ture current, decreasing from an initial value, which is m times the final value, on the field transient. Assume then that the mutual induction between field and armature is such that 60 per cent of the pulsation of armature reaction appears in the field current. Forty per cent damping for the double-frequency reaction would about correspond to the 20 per cent damping assumed for the transient full-frequency pulsa- tion of the polyphase machine. The transient field current thus pulsates by 60 per cent around the slow field transient, as shown b}^ Fig. 23 C; passing a maximum for every maximum of armature Fig. 24. — Asymmetrical Momentary Single-phase Short Circuit of Alternator. current, and thus maximum of armature reaction, and a minimum for every zero value of armature current, and thus armature reac- tion. Such single-phase short-circuit transients have occasionally been recorded by the oscillograph, as shown in Fig. 27. Usually, how- ever, the circuit is closed at a point of the wave where the perma- nent armature current would not be zero, and an armature transient appears, with an initial value equal, but opposite to, the initial value of the permanent armature current. This is shown in Fig. 24 for the case of closing the circuit at the moment where the SINGLE-ENERGY TRANSIENTS. 47 armature current should be a maximum, and its transient thus a maximum. The field transient $ is the same as before. The armature current shows the initial asymmetry resulting from the armature transient, and superimposed on the slow field transient. On the field current, which, due to the single-phase armature reaction, shows a permanent double-frequency pulsation, is now superimposed the transient full-frequency pulsation resultant from the transient armature reaction, as discussed in paragraph 20. Every second peak of the permanent double-frequency pulsation then coincides with a peak of the transient full-frequency pulsa- tion, and is thereby increased, while the intermediate peak of the double-frequency pulsation coincides with a minimum of the full- frequency pulsation, and is thereby reduced. The result is that successive waves of the double-frequency pulsation of the field current are unequal in amplitude, and high and low peaks alter- nate. The difference between successive double-frequency waves is a maximum in the beginning, and gradually decreases, due to the decrease of the transient full-frequenc}^ pulsation, and finally the double-frequency pulsation becomes symmetrical, as shown in Fig. 24C. In the particular instance of Fig. 24, the double-frequency and the full-frequency peaks coincide, and the minima of the field- current curve thus are symmetrical. If the circuit were closed at another point of the wave, the double-frequency minima would become unequal, and the maxima more nearly equal, as is easily seen. While the field-exciting current is pulsating in a manner deter- mined by the full-frequency transient and double-frequency per- manent armature reaction, the potential difference across the field winding may pulsate less, if little or no external resistance or inductance is present, or may pulsate so as to be nearly alter- nating and many times higher than the exciter voltage, if consid- erable external resistance or inductance is present; and therefore it is not characteristic of the phenomenon, but may become impor- tant by its disruptive effects, if reaching very high values of voltage. With a single-phase short circuit on a polyphase machine, the double-frequency pulsation of the field resulting from the single- phase armature reaction induces in the machine phase, which is in quadrature to the short-circuited phase, an e.m.f. which con- tains the frequencies /(2 ±1), that is, full frequency and triple 48 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ;±5 o a o 51 CO bJD O .a a b O go o S U I I ^ It ^^ faC ^ SINGLE-ENERGY TRANSIENTS. 49 frequency, and as the result an increase of voltage and a distor- tion of the quadrature phase occurs, as shown in the oscillogram Fig. 25. Various momentary short-circuit phenomena are illustrated by the oscillograms Figs. 26 to 28. Figs. 26A and 2QB show the momentary three-phase short cir- cuit of a 4-polar 25-cycle 1500-kw. steam turbine alternator. The Fig. 26 A. — CD9399. — Symmetrical. Fig. 265. — CD9397. — Asymmetrical. Momentary Three-phase Short Circuit of 1500-Kw. 2300- Volt Three-phase Alternator (atb-4-1500-1800). Oscillograms of Armatm-e Current and Field Current. lower curve gives the transient of the field-exciting current, the upper curve that of one of the armature currents, — in Fig. 26 A that current which should be near zero, in Fig. 26B that which should be near its maximum value at the moment where the short circuit starts. Fig. 27 shows the single-phase short circuit of a pair of machines in which the short circuit occurred at the moment in which the armature short-circuit current should be zero; the armature cur- 50 ELECTRIC DISCHARGES, TrAT'^>S AND IMPULSES. rent wave, therefore, is s}^nmetrical, and the field current shows only the double-frequency pulsation. Only a few half -waves were recorded before the circuit breaker opened the short circuit. Fig. 27. — CD5128. — Symmetrical. Momentary Single-phase Short Circuit of Alternator. Oscillogram of Armature Current, Armature Voltage, and Field Current. (Circuit breaker opens.) Fig. 28. — cd656o. — Asymmetrical. Momentary Single-phase Short Circuit of 5000-Kw. 11,000-Volt Three-phase Alternator (atb-6-5000-500) . Oscillogram of Armature Current and Field Current. Fig. 28 shows the single-phase short circuit of a 6-polar oOOO-kw. 11,000-volt steam turbine alternator, which occurred at a point of the wave where the armature current should be not far from its maximum. The transient armature current, therefore, starts un- SINGLE-ENERGY TRANSIENTS. 51 symmetrical, and the double-frequency pulsation of the field cur- rent shows during the first few cycles the alternate high and low peaks resulting from the full-frequency transient pulsation of the rotating magnetic field of armature reaction. The irregular initial decrease of the armature current and the sudden change of its wave shape are due to the transient of the current trans- former, through which the armature current was recorded. Fig. 25 shows a single-phase short circuit of a quarter-phase alternator; the upper wave is the voltage of the phase which is not short-circuited, and shows the increase and distortion resulting from the double-frequency pulsation of the armature reaction. While the synchronous reactance Xq can be predetermined with fair accuracy, the self -inductive Xi is not such a definite quantity. It includes a transient component. The armature magnetic cir- cuit is in mutual inductive relation with the field-exciting circuit. At constant alternating current in the armature, the resultant of the armature m.m.f's. and e.m.f's. is constant with regard to the field, and the mutual inductance thus does not come into play. During a transient, however, the armature conditions change, and the self-inductance of the exciting circuit is partly transformed into the armature circuit by the ratio of field turns to armature turns, giving rise to a transient effective component of armature self-induction, which depends on the relative rate of change of the armature and the field, and thereby is a maximum in the beginning, and gradually decreases to zero in stationary conditions. This tends to lower the maximum values of the field transients and to increase the duration of the armature tran- sients. This effect is materially affected by the amount of resist- ance and reactance in the exciting circuit outside of the field winding. There also exists a mutual inductance between the armature circuits of the three-phase machine, which results in an energy transfer between the phases, during the armature transient. The instantaneous power of the momentary short-circuit current, and with it the forces acting on driving shaft and prime mover, are proportional to the short-circuit current, being short- circuit current times magnetic field flux. The forces exerted be- tween the armature conductors — which tend to tear and strip the end windings, etc. — are proportional to the square of the short-circuit current.