Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients — part 2 of 7

SINGLE-ENERGY TRANSIENTS. 25 The magnetic flux is Bp = 8 X 108, and with 4 n total turns the total number of magnetic interlinkages thus is 4n@p = 32 X 10, hence the inductance —8 L= £n€o 107° = 32n henrys. to 7) . The field excitation is nyo = 6000 ampere turns, . hence 6000 r= v0 : hence .32 X 6000 L= a henrys, and LT 1920 | T =~ = 509 = 3:84 sec. That is, the stored magnetic energy could maintain full field excitation for nearly 4 seconds. It is interesting to note that the duration of the field discharge does not depend on the voltage, current, or size of the machine, but merely on, first, the magnetic flux and m.m.f., — which determine the stored magnetic energy, —- and, second, on the excitation power, which determines the rate of energy dissipation. 15. Assume now that in the moment where the transient be- gins the resistance of the coil in Fig. 10 is increased, that is, the . ae | rs CUTIE) | ; é a r : CHP) CHT eReaaoEl ; Fig. 12. — Magnetic Single-energy Transient. coil is not short-circuited upon itself, but its circuit closed by a resistance 7’. Such would, for instance, be the case in Fig. 12, when opening the switch S.

THE ELECTRIC FIELD. ; 7. Let, in Fig. 7, a generator G transmit electric power over line A into a receiving circuit M. While power flows through A the conductors A, power is con- sumed in these conductors by G M conversion into heat, repre- A sented by vr. This, however, . Vig. 7. is not all, but in the space surrounding the conductor cer- tain phenomena occur: magnetic and electrostatic forces appear. 4? . ‘ \ / 7 ‘ 1 vd _ 4 \ / / Pas d . ‘ ‘ t , ’ w. 4 a ‘ 1 } ‘ ao “i. ~ { nN \ ‘ ‘ ' / . “* _ SO ao a“ : Nee .. ~s, ‘, po m j ; x _ . ween?

• --" ~ © ~~ wee , . -—" a a i *. ** ~- : of ‘ ‘ i ra _ , “Ss ~ “ a ‘ ’ i \ ‘ _ eA , va / ; ( ' . ‘ . “a ra / ' \ \ _ ‘ 4 ’ \ ‘N / ' . \ ‘
  ‘ ! , 1 ‘ /
  Fig. 8. — Electric Field of Conductor. The conductor is surrounded by a magnetic field, or a magnetic flux, which is measured by the number of lines of magnetic force ®. With a single conductor, the lines of magnetic force are concentric ’ circles, as shown in Fig. 8. By the return conductor, the circles 10 ;

LECTURE III. SINGLE-ENERGY TRANSIENTS IN CONTINUOUS- CURRENT CIRCUITS.

• 13. The simplest electrical transients are those in circuits in

which energy can be stored in one form only, as in this case the . change of stored energy can consist only of an increase or decrease; . but no surge or oscillation between several forms of energy can - exist. Such circuits are most of the low- and medium-voltage os _ circuits, — 220 volts, 600 volts, and 2200 volts. In them the eapac- - ity is small, due to the limited extent of the circuit, resulting from

the low voltage, and at the low voltage the dielectric energy thus _ .

. is negligible, that is, the circuit stores appreciable energy only by the magnetic field. A circuit of considerable capacity, but negligible inductance, if —

of high resistance, would also give one form of energy storage only, in the dielectric field. The usual high-voltage capacity circuit, as

that of an electrostatic machine, while of very small inductance,

also is of very small resistance, and the momentary discharge

currents may be very consider-

able, so that ir spite of the very $,

small inductance, considerable : Mn

magnetic-energy storage may oc- CUTTS

cur; that is, the system is one é, ‘ cig L

storing energy in two forms, and bo qq . oscillations appear, as in the dis- : ini

charge of the Leyden jar. Fie. 10.—M sg: .

; . g. 10. — Magnetie Single-energy Let, as represented in Fig. 10, Transient. .

a continuous voltage eo be im-

pressed upon a wire coil of resistance r and inductance L (but

negligible capacity). A current 7% = 7 flows through the coil and

a magnetic field Sy) 10-8 = “ts interlinks with the coil. Assuming

now that the voltage ey is suddenly withdrawn, without changing

19 . -

20 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . ; the constants of the coil circuit, as for instance by short- circuiting the terminals of the coil, as indicated at A. With no voltage impressed upon the coil, and thus no power supplied to it, current 2 and magnetic flux ® of the coil must finally be zero. However, since the magnetic flux represents stored energy, it cannot instantly vanish, but the magnetic flux must gradually decrease from its initial value ®o, by the dissipation of its stored energy in the resistance of the coil circuit as 7?r._ Plotting, there- fore, the magnetic flux of the coil as function of the time, in Fig. 11A, the flux is constant and denoted by 9 up to the moment of 2, ! | ; | A | 2 | ! I . in IS 1 oS Ld. B | ND i! | YS ™ . | ‘\ | | aN es | , &, ; . c e ———T.- o } t ty Fig. 11. — Characteristics of Magnetic Single-energy Transient. time where the short circuit is applied, as indicated by the dotted line tp. From ¢) on the magnetic flux decreases, as shown by-curve &. Since the magnetic flux is proportional to the current, the latter must follow a curve proportional to @, as shown in Fig. 11B. The impressed voltage is shown in Fig. 11C as a dotted line; it is €y up to to, and drops to 0 at t9. However, since after ¢) a current - i flows, an e.m.f. must exist in the circuit, proportional to the current. e=7.

, SINGLE-ENERGY TRANSIENTS. 21 This is the e.m.f. induced by the decrease of magnetic flux $, and is therefore proportional to the rate of decrease of #, that is, to a In the first moment of short circuit, the magnetic flux & still has full value $0, and the current 7 thus also full value 7. Hence, at the first moment of short circuit, the induced e.m.f. e must be equal to é, that is, the magnetic flux must begin to decrease at such rate as to induce full voltage eo, as shown in Fig. 11C. The three curves ®, 2, and e are proportional to each other, and as € is proportional to the rate of change of , 6 must be propor- tional to its own rate of change, and thus alsoz and e. That is, the transients of magnetic flux, current, and voltage follow the law of proportionality, hence are simple exponential functions, as seen in Lecture I: - o= Poe — 6(4— bo), t= Ue" (tt) | (1) é= ene ~ 4), ®, i, and e decrease most rapidly at first, and then slower and slower, but can theoretically never become zero, though prac- tically they become negligible in a finite time. | The voltage e is induced by the rate of change of the magnetism, and equals tlie decrease of the number of lines of magnetic force, divided by the time during which this decrease occurs, multiplied by the number of turns 7 of the coil. The induced voltage e times the time during which it is induced thus equals 7 times the decrease of the magnetic flux, and the total induced voltage, . that is, the area of the induced-voltage curve, Fig. 11C, thus equals n times the total decrease of magnetic flux, that is, equals . the initial current 7) times the inductance L: _ . Let = nbol0-§ = Lig. (2) Whatever, therefore, may be the rate of decrease, or the shape of the curves of %, z, and e, the total area of the voltage curve must be the same, and equal to n®y = Lip. If then the current 7 would continue to decrease at its initial rate, as shown dotted in Fig. 11B (as could be caused, for instance, by a gradual increase of the resistance of the coil circuit), the . induced voltage would retain its initial value eo up to the moment of time t = t) +7, where the current has fallen to zero, as

SINGLE-ENERGY TRANSIENTS. 27 The duration of the transient now is , L te TE that is, shorter in the same proportion as the resistance, and thereby the induced voltage is higher. If r’ =oo, that is, no resistance is in shunt to the coil, but the circuit is simply opened, if the opening were instantaneous, it ‘would be: é9’ =o; that is, an infinite voltage would be induced. That is, the insulation of the coil would be punctured and the circuit closed in this inanner. The more rapid, thus, the opening of an inductive circuit, the higher is the induced voltage, and the greater the danger of break- down. Hence it is not safe to have too rapid circuit-opening devices on inductive circuits. To some extent the circuit protects itself by an are following the blades of the circuit-opening switch, and thereby retarding the cir- cuit opening. The more rapid the mechanical opening of the switch, the higher the induced voltage, and further, therefore, the are follows the switch blades and maintains the eircuit. 16. Similar transients as discussed above occur when closing a circuit upon an impressed voltage, or changing the voltage, or the current, or the resistance or inductance of the circuit. A discus- sion of the infinite variety of possible combinations obviously would be impossible. However, they can all be reduced to the same simple case discussed above, by considering that several currents, voltages, magnetic fluxes, etc., in the same circuit add algebraically, without interfering with each other (assuming, as done here, that magnetic saturation is not approached). If an e.m.f. e: produces a current 7, in a circuit, and an e.m.f. é produces in the same circuit a current i, then the e.m.f. e: + e produces the current 7; + %, as is obvious. If now the voltage e: + é, and thus also the current 7, + 7%, con- sists of a permanent term, e; and 2, and a transient term, é and 7%, ' the transient terms é2, 72 follow the same curves, when combined ’ with the permanent terms ¢, 21, as they would when alone in the circuit (the case above discussed). ‘Thus, the preceding discus- ; sion applies to all magnetic transients, by separating the transient from the permanent term, investigating it separately, and then adding it to the permanent term.

18: ELECTRIC DISCHARGES, WAVES AND IMPULSES. . TABLE II. Magnetic Circuit. | Dielectric Circuit. , Electric Circuit. Magnetic flux (magnetic | Dielectric flux (dielectric Electric current: current): current): @=lines of magnetic | W=lines of dielectric t= electric cur- force. force. ; rent. - Magnetomotive force: Electromotive force: Voltage: F = ni ampere turns. e = volts. e = volts. Permeance: M=-— 2... ; Ank Permittance or capacity: | Conductance: Inductance: C = ~ tarads. g =~ mhos. 2 é : é L= 22 19-8 = 219-8 F i henry. Réluctance: (Elastance): Resistance: _F 1 _e _@ | Res CF 7 = = ohms. Magnetic energy: Dielectric energy: Electric power:. —' aL FP ston ~~? _ p=ri? = ge? = ei | wa =a 10 joules. w= > = gy Joules. watts. ; Magnetic density: Dielectric density: Electric-current density: g=" =u JClines per em? Do =xKlines percm? [= 4 = 7G am- | A A A perespercm?. Magnetizing force: Dielectric gradient: Electric gradient: . f= . ampere turns per) G= ; volts per cm. G= j volts percm. em. Magnetic-field intensity:| Dielectric-field inten- sity: H = .4af. K= _& 9p ; , = .27f. 4p? ‘ _ Permeability: Permittivity or specific) Conductivity: capacity:

• 8. = 2. = = mho- w= a r= ¥ @ mho-cm. Reluctivity: (Elastivity ?): ' | Resistivity: . _f. 1_K. 1G P=@ « D a “m Specific magnetic energy:| Specific dielectric energy:| Specific power: uy = Au? 18 19-8 = Usp = #G so ED rose Do = pl? = G=GI 2 2 4m 2 watts per cm? BB 10" joules per em?. 2av’KD joules per cm’. ~

SINGLE-ENERGY -TRANSIENTS. 31 , sient, shown dotted in Fig. 15. Adding the transient current 7o ; to the permanent current 72 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. 7, which exists, and that of the current 7 which should exist at the moment of change, and A Lt oN Vaan ? oe 0 ares Ia ae B. . i, ro) ig lof \ 7 ae 54 ‘ N77 “ 2 ‘ s \ if oe’ 1, eet ; ) . 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 2, and 7, are equal, as shown in Fig. 15B, and is a maximum, if the change occurs at the moment when the two currents 7, and % - 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 Y and to. . ;

; SINGLE-ENERGY TRANSIENTS. 33 _ half the value of 73°, 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 2)’, 7’, and 73’ are the instan- taneous values of the permanent three-phase currents, in Fig. 17, the initial values of their transients are: —7', —w’, —1%3’. A | . i° ~ nie —_—— a . 3) ~ 7m a we x by Ls igi : —— oe | el oe “ Ze i} = — i i : i, 2 3 21 cs pase B ZN EE INN ee OO-e ahr See” . -Q . e 1, 1 1 i, 1, ; i; - 2K x i} LS FX = fl Ss SC OOS | i, 1, 1s 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 a also is zero. Since the three transient curves 71°, 72°, 73° are pro- portional to each other (as exponential curves of the same dura- tion T = 2), and the sum of their initial values is zero, it follows

22 ELECTRIC DISCHARGES, WAVES AND IMPULSES. | : shown dotted in Fig. 11C. The area of this new voltage curve would be éo7’, and since it is the same as that of the curve e, as seen above, it follows that the area of the voltage curve e¢ is Let = eoT = oof | (3) = ToT’, and, combining (2) and (3), to cancels, and we get the value of T: L T= 7 (4) That is, the initial decrease of current, and therefore of mag- netic flux and of induced ‘voltage, is such that if the decrease continued at the same rate, the current, flux, and voltage would become zero after the time 7 = a. The total induced voltage, that is, voltage times time, and therefore also the total current and magnetic flux during the transient, are such that, when maintained at their initial value, they would last for the time T = . | Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient we may determine the time in which the transient decreases to half, or to one-tenth, etc., of its initial value. It is preferable, however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is, the duration of a transient is considered as the time T = . This time T has frequently been called the “ time constant ”’ of the circuit. The higher the inductance Z, the longer the transient lasts, obviously, since the stored energy which the transient dissipates is proportional to L. The higher the resistance 7, the shorter is the duration of the transient, since in the higher resistance the stored energy is more _ rapidly dissipated. . Using the time constant 7 = az as unit of length for the abscissa, and the initial value as unit of the ordinates, all exponential transients have the same shape, and can thereby be constructed

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 shown 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 2. 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 2; con- sumes a voltage x42 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. .

24 ELECTRIC DISCHARGES, WAVES AND IMPULSES, _t _t | | = Doe~ * = Doe T= Doe L t rt 1 = ine # = ine T =ige 4, (6) t rt € = eet = ee T = ege 4, The same equations may be derived directly by the integration of the differential equation: di . L Zi +ri=0, © (7) where Lg is the inductance voltage, 77 the resistance voltage, and their sum equals zero, as the coil is short-circuited. Equation (7) transposed gives | dior | TATE dt, hence 7 logi =— zt + logC, 7, t=Ce 4, and, as for ¢ = 0: 7 = ip, it is: C = 1; hence a4 ; t = toe L . . 14. Usually single-energy transients last an appreciable time, and thereby become of engineering importance, only in highly inductive circuits, as motor fields, magnets, etc. To get an idea on the duration of such magnetic transients, consider a motor field: . A 4-polar motor has*8 ml. (megalines) of magnetic flux per pole, produced by 6000 ampere turns m.m.f. per pole, and dissi- . pates normally 500 watts in the field excitation.. ; That is, if 7) = field-exciting current, n = number of field turns per pole, r = resistance, and L = inductance of the field-exciting circuit, it is ; , to’r = 500, hence 8 500, 10" ,

SINGLE-ENERGY TRANSIENTS. 89 _ Thus it is: . momentary short-circuit current open-circuit field flux* _ permanent short-circuit current short-circuit field fux armature reaction plusself-induction _ synchronous reactance _ Zo. ; ~ gelf-induction ~~ self-inductivereactance 2

20. Let &, = field flux of a three-phase alternator (or, in general, , polyphase alternator) at open circuit, and this alternator be short- circuited at the time ¢= 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. 214. If m = ratio

armature reaction plus self-induction _ 2» armature self-induction By? it is By = mo, and the initial value of the field flux consists of the permanent part $o, and the transient part ®’ = 6; —dy = (m—1) $. This is a rather slow transient, frequently of a duration of a second or more.

The armature currents 2, %, t3 are proportional to the field flux ® which produces them, and thus gradually decrease, from initial values, which are as many times higher than the final values as 4, is higher than po, or m times, and are represented in Fig. 21B.

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 7) at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current 7% combines with the armature reaction F’9, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux @p. During the transition period the field flux @ is higher than po, 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.

26 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

The transients of magnetic flux, current, and voltage are shown as A, B, and C in Fig. 18.

The magnetic flux and therewith the current decrease from the initial values $9 and 7» at the moment t» of opening the switch S, on curves which must be steeper than those in Fig. 11, since the current passes through a greater resistance, r+ 7’, and thereby dissipates the stored magnetic energy at a greater rate.

Lc . ! | A ! a | | 7 . B | a 1 |
!
1 eR i ~ £0 e 1 Cc -T-— t, t Fig. 13. — Characteristics of Magnetic Single-energy Transient.

The impressed voltage é) is withdrawn at the moment é, and a voltage thus induced from this moment onward, of such value as to produce the current 7 through the resistance r+ 7’. In the first moment, fo, the current is still 7), and the induced voltage thus must be ;

oS to (r +7’), while the impressed voltage, before to, was €9 = W003

hence the induced voltage eo’ is greater than the impressed volt- _ age éo, in the same ratio as the resistance of the discharge circuit r+ r’ is greater than the resistance of the coil r through which the impressed voltage sends the current

eo ttn

€o r

SINGLE-ENERGY TRANSIENTS. 43 in Fig. 21C 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. 228, 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. 21C by a series of oscillations, as indicated in Fig. 212. ‘ Since the resultant m.m.f. of the machine, which produces the flux, is the difference of the field excitation, Fig. 21D and the armature reaction, then if the armature reaction shows an initial os- cillation, in Fig. 21Z, 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 field current, as shown in Fig. 22C, as an oscillation cf 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. 22C 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 a 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 ; Lo Po ~ (m —1) Tro = Isec., the duration of the armature transient is . T= L _ .l sec. r

• And assuming that the armature reaction is 5 times the armature

28 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

The same reasoning also applies to the transient resulting from several forms of energy storage (provided that the law of propor- tionality of 7, e, &, etc., applies), and makes it possible, in inves- tigating the phenomena during the transition period of energy readjustment, to separate the permanent and the transient tern, and discuss them separately.

re 2, ‘ . H ‘p A} | Beene Poe 1, : B : ! \ a en bee ac cael fy H £5 ' € Cc an os Fig. 14. — Single-energy Starting Transient of Magnetic Circuit.

For instance, in the coil shown in Fig. 10, let the short circuit A be opened, that is, the voltage e) be impressed upon the coil. At the moment of time, ¢, when this is done, current 7, magnetic flux , and voltage e on the coil are zero. In final condition, after the transient has passed, the values 7%, @o, ¢) are reached. We may then, as discussed above, separate the transient from the perma- nent term, and consider that at the time ¢y the coil has a permanent current 2%», permanent flux $9, permanent voltage eo, and in addi- .

| 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 t= 0 ml 2 3 4 Seconds a | A! 7 2 A RATATaatan TATRA ai | i pots 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-

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 7; = current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and % = current which should exist at the moment of change in accordance with the circuit condition after the change, then the actual current 1, can be considered as consisting of a part or component %, and a component 7; — 72 = zo. The former, %, is permanent, as result- ing from the established circuit condition. The current compo- nent 2%, 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 18, that is, with a duration T = = = GL

The permanent current 72 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, in Fig. 15.A, the conditions be changed, at the moment é = 0, so as to produce the current 7%. The instantaneous value of the . current 7, at the moment ¢ = 0 can be considered as consisting of the instantaneous value of the permanent current 7%, shown dotted, and the transient 7) = 7; — 7%. The latter gradually dies down, with the duration T = Z, on the usual exponential tran-

30

SINGLE-ENERGY TRANSIENTS, 49 . frequeney, and as the result an increase of voltage and a distor- tion of the quadrature phase oecurs, as shown in the oscillogram Fig. 25.

Various momentary short-circuit phenomena are illustrated by the oscillograms Figs. 26 to 28. :

Figs. 26A and 26B show the momentary three-phase short cir- cuit of a 4-polar 25-eycle 1500-kw. steam turbine alternator. The eshte tet aa Ua hoececloa ties ceed igi Ea ee RR DR ey Gate ies Pe ae SE SE eae \ " \ . ee CAAA AA BS 1 esr 8 non 4p fo f fifty Lippi

lle dt ~ Life ded? be : Ye Ok PUMA Ve WN on’ 8 a, wu VV fe 8

saforg A(\ APA NAR EN re pede ae, oar es eg

Zeta Pe ET Sy a Rai ae Se eo Pe gee

PRS EE SE p ghee Seen ee a oe RS!

RR Bec Se Bp SER ee ig ig sete, SSRN eu eat be

FOSS ti San 00 PRR AN Se he BES ee ot

OB RS ie SRE Ae et a

sede EEE AL Gee Sir eat gt NE a et tap egy MRS Qt UE 3 TREE gis ie

ELEN Bagh ore eve ee ares ee go Bese

“yatte 4 VV [PA ad . . ‘~ aati Be ESS BE

Fig. 26.4. — ep9399, — Symmetrical. Shot Sy erate he SD ROE gig in ee ey bee Serre ens. ain ® AY TAREE RAE RASASRAALAL EA SAAS

• _ ¢ ou uw ' Oe WA ee ee 7 o- ve

Soke ge OEE A PES OE in gre Soo

PEER a GP AE RE Sy She DRT REET Page Rr

Tyee EE Deter bg yd Ee haba gat Cage Pt pe EE GS

EE GE he hg Se Oe eae

TE SEIS SO NSIC SOY age SR MARE Sn PT Be a

Shee joes sPeB Gary eas ae re ane Ea oe ae

TLL Bek ge oe RET ER NG oth eR Re SOR cee SS

Fig. 268. —cp9397, — Asymmetrical.

Momentary Three-phase Short. Cireuit. of 1500-Kw. 2300-Volt Three-phase Alternator (arp—4-1500-1800). Oscillograms of Armature Current and Field Current. lower curve gives the transient of the field-exeiting current, the upper eurve that of one of the armature currents, —in Fig. 26A that current whieh should be near zero, in Vig. 26B that which should be near its maxinuin yalue at the moment where the short eircuit starts.

Fig. 27 shows the single-phase short circuit of a pair of machines in which the short. circuit oeeurred at the moment in which the amature short-circuit. current should be zero; the armature cur-

32 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . If the current 2; 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. 16B), 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. Loos vo ,
Ayn ff | Fig. 16. — Single-energy Starting Transient of Alternating-current Circuit. . 18. Applying the preceding to the starting of a balanced three-phase system, we see, in Fig. 17A, that in general the three transients 7;°, 72°, and 73° of the three three-phase currents 1%, 72, %3 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, i, Fig. 17B, there is no transient for . this current, while the transients of the other two currents, % and 7%, are equal and opposite, and near their maximum value. Starting, in Fig. 17C, at the maximum value of one current 2, we have the maximum value of transient for this current 73°, while the transients of the two other currents, 2; and 7%, are equal, have

SINGLE-ENERGY TRANSIENTS. 51 symmetrical, and the double-frequency .puisution ofthe field cur- rent shows during the first few cycles the alternate high arid low peaks resulting from the full-frequency transiewt, 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 +rans- 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 2% can be predetermined with fair accuracy, the self-inductive 2, 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 doves 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. _

34 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. :

/ NY Sy a ‘ / a . i / x. 1 / a ; Pane A, : B) B, B,B, O a 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 TRANSIENT OF IRONCLAD CIRCUIT. 58 approximate step-by-step method, as illustrated for the starting transient of an alternating-current transformer in ‘“‘ Transient Elec- . tric Phenomena and Oscillations,” Section I, Chapter XII. Such methods are very cumbersoine and applicable only to numerical instances.

An approximate calculation, giving an idea of the shape of the transient of the ironclad magnetic circuit, can be made by neglect- ing the difference between the rising and decreasing magnetic characteristic, and using the approximation of the magnetic char- acteristic given by Fréhlich’s formula:

ae . aa at dt’ (1) which is usually represented in the form given by Kennelly: ae p= Ba atox; (2) that is, the reluctivity is a linear funetion of the field intensity. It gives a fair approximation for higher magnetic densities.

This formula is based on the fairly rational assumption that the permeability of the iron is proportional to its remaining magnetiza- bility. That is, the magnetic-flux density ® consists of a compo- nent i, the field intensity, which is the flux density in space, and 2 component ®’ = @ — K, which is the additional flux density carried by the iron. @’ is frequently called the ‘“ metallic-flux density.” With increasing 3, @’ reaches a finite limiting value, which in iron is about

®.,’ = 20,000 lines per cm?. *

At any density @’, the remaining magnetizability then is @,,’—@’, and, assuming the (netallic) permeability as proportional hereto, gives

n= c(@,’ — @’), and, substituting GB’ = ae"? gives 1 cB, 3’ OF Ty oe”

• See “On the Law of Hysteresis,” Part II, A.I.E.E. Transactions, 1892,

page 621. ;

36 ELECTRIC DISCHARGES, 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 coérdinates, 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.

| ; Fig. 20. — Starting Transient of Rotating Field: Rectangular Form.

We have here an oscillatory transient, produced in a system with only one form of stored energy (magnetic energy), by the conibination of several simple exponential transients. - How- ever, it must be considered that, while energy can he 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 = Z.

_ The maximum value, which the magnetic field during the transi- tion period ean 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 synchronous 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