value, thus reaches a lower positive maximum, and thus grad- ually, at a rate depending upon the resistance of the circuit, the - waves of magnetic flux , and thereby current 7, approach their final permanent or symmetrical cycles.

100. In the preceding, the assumption has been made that the magnetic flux, ®, or the flux density, ®, is proportional to the current, or in other words, that the inductance, L, is con-

stant. If the magnetic circuit interlinked with the electric circuit contains iron, and especially if it is an iron-clad or closed magnetic circuit, as that of a transformer, the current is not _ proportional to the magnetic flux or magnetic flux density, but increases for high values of flux density more than proportional, that is, the flux density in the iron reaches a finite limiting value. In the case illustrated above, the current corresponding to double the normal maximum magnetic flux, ®,, or flux density, @,, may be many times greater than twice the normal maximum current, J,. For instance, if the maximum permanent current is J, = 4.5 amperes, the maximum permanent flux density, ®, = 10,000, and the circuit closed, as above, at that point of the e.m.f. wave where the flux density should have its negative maximum, — ®, = — 10,000, but the actual flux density is 0, during the first half wave of e.m-f., the flux density, when neglecting the resistance of the electric circuit, should rise from 0 to 2@, = 20,000, and at this high value of saturation the corresponding current maximum would be, by the magnetic cycle, Fig. 43, 200 amperes, that is, not twice but 44.5 times the normal value. With such excessive values of current, the e.m.f. consumed by resistance would be in general considerable, and the e.m.f. consumed by inductance, and therefore the variation of magnetic flux density, considerably decreased, that is, the maximum magnetic flux density would not rise to 20,000, but remain considerably below this value. The maximum current, however, would be still very much greater than twice the normal maximum. That is, in an iron-clad circuit, in starting, the transient term of current may rise to values very much higher than in air magnetic circuits. While in the latter it is limited to twice the normal value, in the iron-clad circuit, if the magnetic flux density reaches into the range of magnetic saturation, very much higher values of transient current are found. Due to the far greater effect of the resistance with such

—— a MAGNETIC SATURATION AND HYSTERESIS 181 excessive values of current, the transient term of current during the first half waves decreases at a more rapid rate; due to the lack of proportionality between current and magnetic flux density, the transient term does not follow the exponential law any more.

101. In an iron-clad magnetic circuit, the current is not only not proportional to the magnetic flux density, but the same magnetic flux density can be produced by different currents, or with the same current the flux density can have very different values, depending on the point of the hysteresis cycle. Therefere the magnetic flux density for zero current may equal zero, or, on the decreasing branch of the hysteresis cycle, Fig. 43, may be

• 7600, or, on the increasing branch, — 7600. Thus, when closing the electric circuit energizing an iron-clad magnetic circuit, as a transformer, at the moment of zero current, the

magnetic flux density may not be zero, but may still have a high ' value, as remanent magnetism. For instance, closing the circuit at the point of the e.m.f. wave where the permanent wave of magnetic flux density would have its negative maximum value, — ®, = — 10,000, the actual density at this moment may ' be @®, = + 7600, the remanent magnetism of the cycle. During the first half wave of impressed e.m.f. the variation of flux density by 2 ,, as required to generate the counter e.m.f., when _ neglecting the resistance, would bring the positive maximum of _ flux density up to 8, + 2, = 27,600, requiring 1880 amperes maximum current, or 420 times the normal current. Obviously, no such rise could occur, since the resistance of the circuit would consume a considerable part of the e.m.f., and so lower the flux density by reducing the e.m.f. consumed by inductance.

It is obvious, however, that excessive values of transient

current may occur in transformers and other iron-clad magnetic

• direuits. ,

102. When disconnecting a transformer, its current becomes zero, that is, the magnetic flux density is left at the value of the remanent magnetism + @,, and during the period of rest more or less decreases spontaneously towards zero. Hence, in con- necting a transformer into circuit its flux density may be any- where between + @, and — @,. The maximum magnetic flux density during the first half cycle of impressed e.m.f. therefore is produced if the circuit is closed at the moment where the per-

182 TRANSIENT PHENOMENA manent value of the flux density should be a maximum, + &,, and the actual density in this moment is the remanent magnetism in opposite direction, + @,, and the maximum value of density which could occur then is + (®, + 2@,). If therefore the maximum magnetic flux density @, in the transformer is such that @, + 2 @, is still below saturation, the transient term of current cannot reach abnormal values. At ® = 16,000, the flux density is about at the bend of the saturation curve, and the current still moderate. Estimating ®, = 0.75 @, as approx- imate value, ®@, + 28, = 16,000 thus gives @ = 5800, or 37,500 lines of magnetic flux per square inch. °

In 125-cycle transformers, ®, is below 5800 or not much above, for reasons of heating, and this phenomenon of excessive tran- sient currents in starting thus does not appear. At 60 cycles, @, is usually above this value, and under unfavorable conditions considerable transient current may be observed. However, for @, = 0, the limit is @, = 8000, or 51,600 lines per square inch; and since in 60-cycle transformers the flux density rarely exceeds this value to a great extent, and in starting the remanent magnetism is rarely very high, this phenomenon of an excessive transient current is not very marked. At 25 cycles, however, higher densities are used and the transient starting current may then reach formidable values.

103. Since the relation between the current, 7, and the mag-

; netic flux density, @, is empirically given by the magnetic cycle

of the material, and cannot be expressed with sufficient accuracy by a mathematical equation, the problem of determining the transient starting current of a transformer cannot be solved in general, but must be investigated in the individual case by constructing the curves of current and magnetic flux density.

Let the normal magnetic cycle of a transformer be represented by the dotted curve in Figs. 43 and 44; the characteristic points ° are: the maximum values, + ®, = + 10,000; the remanent values, + ®, = + 7600, and the maximum exciting current, tm = + 4.5 amp.

; At very high values of flux density an appreciable part of the total magnetic flux @ may be carried through space, outside of the iron, depending on the construction of the transformer. — The most convenient way of dealing with such a case is to resolve the magnetic flux density, @, in the iron into the “metallic

MAGNETIC SATURATION AND HYSTERESIS 183 | PTT yt tt ttt tt eet att tT tt tt tt tert tT ry 1 loops 800 290 |_| | a ea | ott tet t tt tt pte | SCO eee | SCLC ee Tae | dee — ee al | IT Leer tT ey [TMA Tt nn 27 2S . ra wt Feta tT TT tT TT EE ae 40 / ieee ae 2? Hee o EL VAF I Le | me | | |

• eet te “tel Tg EE ET TE EE Tt 6 tt} tt “3 EA ty | tt ft wld t LETTE Tt ttt tt tt Fig. 43. Magnetic cycle of a transformer starting with low stray field. 28 | SERRE ERPERC OPA mM 0 gp 0 a0 I 40 | SCC EEE EE eee wa tt ttt tt leet tT tt Tt | SCC EEE Eee wt i tye tt tt tt unt tt beer TT et i Laer tt TT tt -wttti va tit tit tT ft a -ti“agy lt tt ttt To - x tt Ae i | tT TE TT TT TT ‘ 8 ae AP Cee .aP CZs ote tid bk bk em oe my | ait tT EE TE TE TT TT ETT atta TTT TT TT TT TT at TTT TTT TT TT rT fl TTT TET TE TT TTT wi dt] TT ETT TTT Te Fig. 44. Magnetic cycle of a transformer starting with high stray field.

184 TRANSIENT PHENOMENA flux density,” @’ = @ — 3X, which reaches a finite limiting value, and the «lensity in space, 3. The total magnetic flux then consists of the flux carried by the molecules of the iron, 2’ = A’e’, where A’ is the section of the iron circuit, and the space flux, &” = A’’3c, where A” is the total section interlinked with the electric circuit, including iron as well as other space. = 8 + 0” = A’@’ + AVR. If then A” = kA’, that is, the total space inside of the coil is k times the space filled by the iron, we have . ® = A’ (@’ + kx), or the total magnetic flux even in a case where considerable stray field exists, that is, magnetic flux can pass also outside of Cot COCCOECCCEIe , Phish ed if Ty a AAS oH CeCe 16 beat 4-4 +++ 1 Ft +t 400 A |_| 4 J 1} —-} 4 35 3” \ CoA x ¢ BOCA HAT ae ie a pelt a -15 SS SenH Baur : 4 100 (EEEEEBECEE PH 100 200 «6300 «6400 «6500 «6600 «670! «6800 «6900 =—:1000 Degrees Fig. 45. Starting current of a transformer. Low stray field. the iron, can be calculated by considering only the iron section as carrying magnetic flux, but using as curve of magnetic flux density not the usual curve, B= @' + x, but a curve derived therefrom, @ = B + ka, where k = ratio of total section to iron section. This, for instance, is the usual method of calculating the m.m.f. consumed in the armature teeth of commutating machines at very high saturations.

MAGNETIC SATURATION AND HYSTERESIS 185

In investigating the transient transformer starting current,

the magnetic density curve thus is corrected for the stray field.

Figs. 43 and 45 correspond to k = 3, or a total effective air

section equal to three times the iron section, that is, @ = @’ + 3 x.

Figs. 44 and 46 correspond to k = 235, or a section of stray field equal to 24 times the iron section, that is, ® = @’ + 25%. SE tT tT iy TT tT tT te aethaN boc FH 3 att N| | Tt tt tT TT of a pet © wtf | IM | YN Hi -| _s 1s Hf is Fife wt AL TT Whe - HH is af) p— te (tt Th Ty wet Tt yt Ta Tay TT AZ Al Tt TT TAT Tt AA Neh TT TTY OPN CeCe tYELIM INA YAN IAT Te VARTA ZR LOLS CES”

PTET TT EE TET ET TT 100 200 300 Desrves 500 600 700 Fig. 46. Starting current of a transformer. High stray field.

104. At very high values of current the resistance consumes a considerable voltage, and thus reduces the e.m.f. generated by the magnetic flux, and thereby the maximum magnetic flux and transient current. The resistance, which comes into con- sideration here, is the total resistance of the transformer primary circuit plus leads and supply lines, back to the point where the voltage is kept constant, as generator, busbars, or supply main.

Assuming then at full load of 7,, = 50 amperes effective in the ” transformer, a resistance drop of 8 per cent, or the voltage con- sumed by the resistance, as e, = 0.08 of the impressed e.m.f.

Let now the remanent magnetic flux density be ®, = + 7600, and the eircuit be closed at the moment # = 0, where the flux

186 TRANSIENT PHENOMENA

density should be @ = —®, = — 10,000; then the impressed

e.m.f. is given by d

= — i = F— 1

e E sin 0 E =, (cos 8). (1) It is, however,

1B .

e- AS + Gi, () where A and C are constants; that is, the impressed e.m.f., ¢, is consumed by the self-inductance, or the e.m.f. generated by the changing magnetic density, which is proportional to and by the voltage consumed by the resistance, which is proportional to the current 7.

Combining (1) and (2) gives d® . d cos 0 A wt Ci=E a (3) However, at full load, we have = = effective impresed e.m.f. and 7, = 50 amperes = effective current; hence . Cin = 50C = e.m.f. consumed by resistance, | and since this equals e, = 0.08 of impressed e.m.f., €, Cin = v2 or 50C = 0.08 E , V2 Cc €, 0.08 s=>s = 5 = 0. . 4) or E~ inv 50 V5 0.00113 ¢ | From (3) follows | E Ci ; = — -= 5) d i d cos 0 A d6 ( and 2 _E f Cc fi Sra “A FoI - FG 5 1; ;

MAGNETIC SATURATION AND HYSTERESIS 187 hence, for z = 0, or negligible resistance drop, that is, permanent condition,

E 8, = 47 10,000. (6) Multiplying (4) and (6) gives C e,®, , — =~ = 113, A inv? @ and substituting (6) and (7) in (5) gives . 6B, = 8 _ ——<——— d® od cos 0-2 inva do = 10,000 d cos @ — 11.3id8. (8) Changing now from differential to difference, that is, replacing, as approximation, d by A, gives e,® = 8 -i—2: : AG pA cosd —7 inV5 AG | = 10,000 A cos @ — 11.3 1A0. (9) Assuming now Ad = 10° = 0.175 (10) gives for the increment of magnetic flux density during 10° change of angle the value A® = 10,000 A cos @ — 21 (11) and @ = B’ + AB =_8’ + 10,000 A cos 6 — 21. (12) From equation (12) the instantaneous values of magnetic flux density ®&, and therefrom, by the magnetic cycles, Figs. 43 and 44, respectively, the values of current 7 are calculated, by starting, for 0 = 0, with the remanent density ®’ = ®, = 7600, adding thereto the change of cosine, 10,000 A cos 0, which gives a value ®, = ® + 10,000 1 cos 6, taking the corresponding value of 7 from the hysteresis cycle, Figs. 43 and 44, subtracting 27 from ®,, and then correcting 7 for the value corresponding to B@=68, — 21. . The quantity 27% is appreciable only during the range of the curve where 7 is very large. . . ’

°188 TRANSIENT PHENOMENA 105. The following table is given to illustrate the beginning of the calculation of the curve for low stray field. STARTING CURRENT OF A TRANSFORMER. i = , 4Q.=- | Q= . D= ° cos @ oe | BAB, | in 25 x10" i) 0 +1.00 Lene e ee eee 7.6 0 ce 10 0.98 +0.2 2 Oe 20 0.94 0.4 8.2 190 [occ 30 0.87 0.7 8.9 2.9 |ooccceeelie eee ce eel . 40 0.77 1.0 9.9 3.8 |i fee | ' 50 0.64 1.3 11.2 5.20 |.....eee eles, 60 0.50 1.4 12.6 73 |... dl! 70 0.34 1.6 14.2 wo |... Ll i 80 | +0.17 1.7 15.9 27 Seeeeeee eeeeeene "90 0 | 47 17.6 70 0.1 17.5, 100 | —0.17 | 1.7 19.2 138 0.3 18.9 | 110 0.34 1.7 20.6 220 0.45 20.15 120 0.50 | 1.6 21.75 270 0.55 21.2 130 0.64 1.4 22.6 450 0.9 21.7 140 0.77 1.3 | 23.0 510 1.0 22.0 150 0.87 1.0 23.0 510 1.0 22.0 160 0.94 0.7 22.7 440 0.9 21.8 170 0.98 0.4 22.2 350 0.7 21.5 180 1.00 +0.2 21.7 250 0.5 21.2 190 0.98 -0.2 21.0 200 0.4 20.6 | 200 0.94 0.4 20.2 180 0.35 19.85 210 0.87 0.7 | 19.15 | 130 0.25 18.9 | i 920 0.77 1.0 17.9 76 0.15 17.75 | 1 930 0.64 1.3 16.45 40 0.2 16.25 | 240 0.50 1.4 14.85 14 0.05 14.8 250 0.34 1.6 13.2 7 eeeeeeeee: Dnnnnnen | 260 | —0.17 1.7 11.5 3.2 |oo.e fw 270 0 1.7 98 » -10 |... ey 280 | +0.17 1.7 8.1 aH Lp 290 0.34 1.7 6.4 -1.3 |... fl: 300 0.50 1.6 4.8 -19 | P 310 0.64 1.4 3.4 —22 Joo... pe! 320 0.77 1.3 2.1 —255 [owl 330 0.87 1.0 Ll =2.6 fof 340 0.94 0.7 4 2.75 |.........f lo... 350 0.98 0.4 0 ~28 |e. 360 | +1.00 -0.2 —.2 2:8 | ope. 370 0.98 +0.2 0 -13 |... et 380 0.94 0.4 | +0.4 -05 |... eee. 390 0.87 0.7 1.1 $0.3 [occ pee! 400 0.77 1.0 2.1 10 |........e pec eel 410 0.64 1.3 3.5 1.7 oer errr | 420 0.50 1.4 4.9 2.2 |... eee. | 430 0.34 1.6 6.5 2.7 |... eee 440 | +0.17 1.7 8.2 3.3 |.........,. 00... 450 0 1.7 9.9 43 |. eee. 460 | —0.17 1.7 1.606 (6.2 Pe. 470 =| —0.34 1.7 3.30 1 95 |... pee. 480 | —0.50 1.6 14.9 16.5 beeen es e A

| MAGNETIC SATURATION AND HYSTERESIS 189 °

The first column gives angle @,

The second column gives cos 8,

The third column gives A®, = 10 A cos 9, in kilolines per

/ §q. em.,

The fourth column gives ®, = @ + 1@,,

. The fifth column gives 1, ; . The sixth column gives D, = 27 x 107°, and

The seventh column gives ® = 8, — D,,

1 in the fifth column being chosen, by trial, so as to corre- spond, on the hysteresis cycles, not to @,, but to ® = @,— D,.

These values are recorded as magnetic cycles on Figs. 43-and 44, and as waves of flux density, current, etc., in Figs. 45 and 46.

The maximum values of successive half waves are:

: { . A. Low Stray Field. B. High Stray Field. k= 3 | k= 25 |{——________—- r) 0.3 tos | 6 B tom 0 7.6 o | 0 7.6 0 145° 22.0 | 510 || 160° 24.6 | 230 360° | — 2 | —2.8 360° | +2.0 | —2.4 [ ¢ 530° 18.6 | 120 530° 20.7 | 117 720°; —2.2 | —2.9 720° | — .4 | —2.7 gor | 180 | 92 Ij... fo. pole. | 1080° | —2.8 | —3.0 iccseefeceec ee fee cere 1260° 17.4 66 Sepeeeenl eeeeeens neneeent C7 . 1440° | -3.1 | -3.0 |].........[....000) ee |__1620° 16.9 _ 50 J it +10.0 +4.5 oo +10.0 +4.5

As seen, the maximum value of current during the first cycle,

510, is more than one hundred times the final value 4.5, and more

| than 7 times the maximum value of the full-load current, 50/2

| = 70.7 amperes, and the transient current falls below full-load

. current only in the fourth cycle. That is, the excessive value of transient current in an ironclad circuit lasts for a considerable number of cycles.

In the presence of iron in the magnetic field of electric circuits, transient terms of current may thus occur which are very large compared with the transient terms in ironless reactors, which do not follow the exponential curve, can usually not be calculated

;

190 TRANSIENT PHENOMENA . by general equations, but require numerical investigation by the use of the magnetic cycles of the iron.

These transient terms lead to excessive current values only if the normal magnetic flux density exceeds half the saturation value of the iron, and so are most noticeable in 25-cycle circuits.

Fig. 47. Starting current of a 26-cycle transformer.

As illustration is shown, in Fig. 47, an oscillogram of the starting current of a 25-cycle transformer having a resistance in the supply circuit somewhat smaller than in the above instance, thus causing a still longer duration of the transient term of excessive current.

CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD.

106. The resultant of n, equal m.m.fs. equally displaced from each other in space angle and in time-phase is constant in intensity, and revolves at constant synchronous velocity. When acting upon a magnetic circuit of constant reluctance in all directions, such a polyphase system of m.m.fs. produces a revolving magnetic flux, or a rotating field. (‘‘Theory and Calculation of Alternating Current Phenomeha,” 4th edition, . Chapter XXXIII, paragraph 368.) That is, if n, equal mag-

. . 360 netizing coils are arranged under equal space angles of ~~

Pp electrical degrees, and connected to a symmetrical n, phase system, that is, to n, equal e.m.fs. displaced in time-phase by’ “ degrees, the resultant m.m.f. of these n, coils is a constant

P and uniformly revolving m.m.f., of intensity , = 2s, where § is the maximum value (hence S the effective value] of the m.m.f. of each coil.

In starting, that is, when connecting such a system of mag-_ netizing coils to a polyphase system of e.m.fs., a transient term appears, as the resultant magnetic flux first has to rise to its constant value. This transient term of the rotating field is the resultant of the transient terms of the currents and therefore the m.m.fs. of the individual coils.

107. If, then, § = nJ = maximum value of m.m.f. of each coil, where » = number of turns, and J = maximum value of current, and « = space-phase angle of the coil, the instantaneous value of the m.m.f. of the coil, under permanent conditions, is

J’ = £ cos (6 — 1), (1) 191 a

| 192 TRANSIENT PHENOMENA , and if the time @ is counted from the moment of closing the circuit, the transient term is, by Chapter IV, | ar) | f” = —£e * cost, (2) | | where Z=r— ji. | The complete value of m.m.f. of one coil is | f=f +f = {eos 0 — 2) — 2° cos s}. (3! | In an n,-phase system, successive e.m.fs.-and therefore currents | . 1 . Qi are displaced from each other by "7 of a period, or an angle —. | P P : and the m.m.f. of coil, 7, thus is Qn: -£ 22. i=F cos (0 ~7,— “4 i —€ * cos (= +i). if) Ny n, The resultant of n, such m.m.fs. acting together in the same -direction would be np f 5 np Ix if, = i cos(a — = ~ =* 3) | Bhs m | _t— Iz — Fe z >: cos (= + — i) = 0; (5) 1 Np that is, the sum of the instantaneous values of the permanent | terms as well as the transient terms of all the phases of a sym- .metrical polyphase system equals zero. In the polyphase field, however, these m.m.fs. (4) do not act | in the same direction, but in directions displaced from each 2a . . other by a space angle -— equal to the time angle of their phase p displacement. 108. The component of the m.m-f., f,, acting in the direction (0, — 7), thus is 223 fl = ficos(o,—2- "= i), (6 np

TRANSIENT TERM OF THE ROTATING FIELD 193 and the sum of the components of all the n, m.m.fs., in the direction (0, — 1), that is, the component of the resultant m.m.f. of the polyphase field, in the direction (0, — 1), is

np

f= Dk 1 a Np Qn Lf =§ > feos (9 - + ~ i) —eé * cos (s+ iI, 4 Np Np Qn. cos (0, -;-— i). (7) Ny Transformed, this gives f= ESS c0s(0 +0, - 27 -2 4S oo (9 — @.) 2 2 o " Np 1 * Os ° tf)" -te ™ t ~. Di cos 0, — « z > cos (#,-27- = il, T T Np ) .. 42. and as the sums containing nt equal zero, we have 'p f= } eos (0 = 0) ~ «cox, |, (8) and for 0 = ©, that is as permanent term, this gives Ny Sy = 5 F cos (0 — 9,); (9) hence, a maximum, and equal to we g, that is, constant, for 4, = 96, that is, uniform synchronous rotation. That is, the resultant of a polyphase system of m.m.fs., in permanent con- dition, rotates at constant intensity and constant synchronous velocity. Before permanent condition is reached, however, the resultant m.m.f. in the direction 6, = 0, that is, in the direction of the synchronously rotating vector, in which in permanent condition

194 TRANSIENT PHENOMENA the m.m.f. is maximum and constant, is given during the transient period, from equation (8), by

fates J1 — 6° # cos of (10) that is, it is not constant but periodically varying. -

As example is shown, in Fig. 48, the resultant m.m.f. f, in the direction of the synchronously revolving vector, 6, = 0, for the

ALLL Eye] ae yseeiaG Sak beg Pfeeal ag ° | | J | HH i. | At \ in am Naar eeeae ype ee ode RS GY 0 S | | | | | | roy | eC ee et | | | | | | | iz : 90 180 270 360 450 540 630 720 S10 900 990 108) 1170 1260 1350 1440 Fig. 48. Transient term of polyphase magnetomotive force. constants n, = 3, or a three-phase system; F = 667, and Z =r — jz = 0.32 — 47; hence, fy = 1000 (1 — «~"* cos 8), with 0 as abscissas, showing the gradual oscillatory approach to constancy.

109. The direction, 0, = 9, is, however, not the direction in which the resultant m.m.f. in equation (8) is a maximum, but the maximum is given by

af . db, = 0, (11) this gives sin (0 — 0,) +e * sind, = 0, . (12) cos @ —«¢. ae —e€ =z hence, cot 6, = wna (13) that is, the resultant maximum m.m.f. of the polyphase system does not revolve synchronously, in the starting condition, but revolves with a varying velocity, alternately running ahead and

| | TRANSIENT TERM OF THE ROTATING FIELD 195 dropping behind the position of uniform synchronous rotation, by equation (13), and only for 6 = , equation (12) becomes cot 6, =-cot 0, or 6, = 4, that is, uniform synchronous fotation. The speed of rotation of the maximum m.m/f. is given from equation (12) by differentiation as dd), dé S= ig ~~ dQ’ dé, _ where Q =sin (0 —0,) te * sind,; r -5e, cos (0 — 0.) —-e ” sin 4, . hence, S = > (14) cos (0 — 6,) —¢ * cos 0, or approximately, , 1 fe #" sind, ; | 8 = ze . (15) | l1—e * cos, For 6 = o, equation (14) becomes S = 1, or uniform syn- chronous rotation, but during the starting period the speed alternates between, below and above synchronism. From (13) follows —le6@ cos6—e«e 7 cos 6. = ——g-—— | and (16) . sin 0 sin a, =a” where -fe\2 _! -276. R ~V(coso—e :*) +sin'g =V1—2e * cosote * (17) |

196 "TRANSIENT PHENOMENA 110. The maximum value of the resultant m.m.f., at time- phase 6, and thus of direction 0, as given by equation (13) or (16), (17), is derived by substituting (16), (17) into (8), as: _ ™® Im 5 FR

• 5 Vi ~2e*"cosO te #, (18) hence is not constant, but pulsates periodically, with gradually decreasing amplitude of pulsation, around the mean value 25. For 6 = 0, or at the moment of start, it is, by (13), —n@ . , ©o86—e 7 0 coe = Sind 0 hence, differentiating numerator and denominator, —sin 6 + re ‘ r cot 6) = ——— og and tan 0,/= = ; that is, the position of maximum resultant m.m.f. starts from angle 0,’ ahead of the permanent position, where 6,’ is the time- phase angle of the electric magnetizing circuit. The initial value of the resultant m.m.f., for 0 = 0, is fm = 0, that is, the revolving m.m.f. starts from zero. Substituting (16) in (15) gives the speed as function of time 1—« 2° (cos 0— ~sind) S= Se (19) 1+ «2° 2e =" cos for 0 = 0 this gives the starting speed of the rotating field So= o° or, indefinite; ‘ |

TRANSIENT TERM OF THE ROTATING FIELD 197 hence, after differentiating numerator and denominator twice, this value becomes definite.

1 So = 2 ? (20) that is, the rotating field starts at half speed. As illustration are shown, in Fig. 49, the maximum value of the resultant polyphase m.m.f., fn, and its displacement in |Intensity/, and| position |( 6) -0 | wah} f Choo] v (cus 0-€ | ~9°2 9 we sin? 0 mi NE eee | t | 5 L [ 8 CONCEP SECTS CEPRCOEE PAT ott NET TT HHH wel NE Wor Aaa =e Pegg! 5S 2 OY EEE wf LECCE htt Efe Ss] se a|=| He {el Sow oe SF oe Boar oe .

Fig. 49. Start of rotating field. . position from that of uniform synchronous rotation, 8,— 9, for - the same constants as before, namely: n, = 3; ¥ = 667, and Z=r — jx = 0.32 — 47; hence,

fn = 1000 VI = Be cos TF OM,

with the time-phase angle @ as abscissas, for the first three cycles. 111. As seen, the resultant maximum m.m.f. of the poly- phase system, under the assumed condition, starting at zero in the moment of closing the three-phase circuit, rises rapidly —within 60 time-degrees — to its normal value, overreaches and exceeds it by 78 per cent, then drops down again below normal, by 60 per cent, rises 47 per cent above normal, drops 37 per cent below normal, rises 28 per cent above normal, and thus by a series of oscillations approaches the normal value. The maximum value of the resultant m.m.f. starts in position

198 TRANSIENT PHENOMENA

85 time-degrees ahead, in the direction of rotation, but has in half a period dropped back to the normal position, that is, the position of uniform synchronous rotation, then drops still fur- ther back to the maximum of 40 deg., runs ahead to 34 deg., drops 23 deg. behind, etc.

It is interesting to note that the transient term of the rotat- ‘ing field, as given by equations (10), (13), (18), does not contain the phase angle, that is, does not depend upon the point of the wave, 0 = t, at which the circuit is closed, while in all preced- ing investigations the transient term depended upon the point of the wave at which the circuit was closed, and that this tran- sient term is oscillatory. In the preceding chapter, in circuits containing only resistance and inductance, the transient term has always been gradual or logarithmic, and oscillatory phenom- ena occurred only in the presence of capacity in addition to in-

ductance. In the rotating field, or the polyphase m.m.f., we thus have a case where an oscillatory transient term occurs, in a circuit containing only resistance and inductance but not capacity, and where this transient term is independent of the point of the wave at which the circuits were closed, that is, is always the same, regardless of the moment of start of the phe- nomenon.

The transient term of the polyphase m.m.f. thus is independ- ent of the moment of start, and oscillatory in character, with an amplitude of oscillation depending only on the reactance factor, =, of the circuit.

: | | CHAPTER XIV. | SHORT-CIRCUIT CURRENTS OF ALTERNATORS, 112. The short-circuit current of an alternator is limited by . armature reaction and armature self-inductance; that is, the current in the armature represents a m.m.f. which with lagging current, as at short circuit, is demagnetizing or opposing the impressed m.m.f. of field excitation, and by combining therewith to a resultant m.m.f. reduces the magnetic flux from that corre- sponding to the field excitation to that corresponding to the resultant of field excitation and armature reaction, and thus reduces the generated e.m.f. from the nominal generated e.m.f., e,, to the virtual generated e.mf., e,, The armature current also produces a local magnetic flux in the armature iron and pole- faces which does not interlink with the field coils, but is a true self-inductive flux, and therefore is represented by a reactance z,. Combined with the effective resistance, r,, of the armature winding, this gives the self-inductive impedance Z, = r, — jz,, or 2,= Vr? +2. Vectorially subtracted from the virtual generated e.m.f., e,, the voltage consumed by the armature current in the self-inductive impedance Z, then gives the ter- minal voltage, e. At short circuit, the virtual generated e.m.f., e,, is consumed by the armature self-inductive impedance, z,. As the effective armature resistance, r,, is very small compared with its self- inductive reactance, x,, it can be neglected compared thereto, and the short-circuit current of the alternator, in permanent condition, thus is

• & a =—= Ly As shown in Chapter XXII, “Theory and Calculation of Alternating Current Phenomena,” the armature reaction can be represented by an equivalent, or effective reactance, z,, and the self-inductive reactance, z,, and the effective reactance of 199

200 TRANSIENT PHENOMENA armature reaction, z,, combine to form the synchronous react- ance, 4, = 2, + 2,, and the short-circuit current of the alterna- tor, in permanent condition, therefore can be expressed by i= %, | Xo . where e, = nominal generated e.m.f.

. 113. The effective reactance of armature reaction, z,, differs, however, essentially from the true self-inductive reactance, z,, in that z, is instantaneous in its action, while the effective reactance of armature reaction, z,, requires an appreciable time to develop: x, represents the change of the magnetic field flux produced by the armature m.m.f. The field flux, however, can- not change instantaneously, as it interlinks with the field exciting coil, and any change of the field flux generates an e.m.f. in the field coils, changing the field current so as to retard the change of the field flux. Hence, at the first moment after a change of armature current, the current change meets only the reactance, z,, but not the reactance z, Thus, when suddenly short-cir- cuiting an alternator from open circuit, in the moment before the short circuit, the field flux is that corresponding to the impressed m.m.f. of field excitation and the voltage in the arma- ture, i.e., the nominal generated e.m.f., e, (corrected for mag- netic saturation). At the moment of short circuit, a counter

'. m.m.f., that. of the armature reaction of the short-circuit current, is opposed to the impressed m.m.f. of the field excitation, ~ and the magnetic flux, therefore, begins to decrease at such a rate that the e.m.f. generated in the field coils by the decrease of field flux increases the field current and therewith the m.m.f. so that when combined with the armature reaction it gives a resultant m.m.f. producing the instantaneous value of field flux. Immediately after short circuit, while the field flux still has full value, that is, before it has appreciably decreased, the field m.m.f. thus must have increased by a value equal to the counter m.m.f. of armature reaction. As the field is still practically unchanged, the generated e.m.f. is the nominal generated voltage, e,, and the short-circuit current is pu’, aq,

SHORT-CIRCUIT CURRENTS OF ALTERNATORS 201 and from this value gradually dies down, with a decrease of the field flux and of the generated e.m.f., to

1-4 %, z, Ty Hence, approximately, when short-circuiting an alternator, in the first moment the short-circuit current is i = &, qT, while the field current has increased from its normal value 2, to the value ix Field excitation + Armature reaction , ° Field excitation , gradually the armature current decreases to ; ; t= say = fo, ‘ q, a) and the field current again to the normal value 7,. Therefore, the momentary short-circuit current of an alternator bears to the permanent short-circuit current the ratio ML ate i m4 , that is, Armature self-inductance + Armature reaction Armature self-inductance

In machines of high self-inductance and low armature reaction, as uni-tooth high frequency alternators, this increase of the momentary short-circuit current over the permanent. short- circuit current is moderate, but may reach enormous values in machines of low self-inductance and high armature reaction, as large low frequency turbo alternators.

114. Superimposed upon this transient term, resulting from the gradual adjustment of the field flux to a change of m.m.f., is the transient term of armature reaction. In a polyphase alternator, the resultant m.m.f. of the armature in permanent conditions is constant in intensity and revolves with regard to the armature at uniform synchronous speed, hence is stationary

202 TRANSIENT PHENOMENA with regard to the field. In the first moment, however, the resultant armature m.m.f. is changing in intensity and in velocity, approaching its constant value by a series of oscillations, as discussed in Chapter XIII. Hence, with regard to the field, the transient term of armature reaction is pulsating in intensity and oscillating in position, and therefore generates in the field coils Wi ° ol — . Armature Current Fig. 60. Three-phase short-cireuit current of a turbo-alternator.

an e.m.f. and causes a corresponding pulsation in the field current and field terminal voltage, of the same frequency as the armature current, as shown by the oscillogram of such a three-phase short-circuit, in Fig. 50. This pulsation of field current is independent of the point in the wave, at which the short-circuit occurs,.and dies out gradually, with the dying out of the transient term of the rotating m.m.f.

In a single-phase alternator, the armature reaction is alter- nating with regard to the armature, hence pulsating, with double. frequency, with regard to the field, varying between zero and its

, SHORT-CIRCUIT CURRENTS OF ALTERNATORS 203 maximum value, and therefore generates in the field coils a double frequency e.m.f., producing a pulsation of field current ~ of double frequency. This double-frequency pulsation of the field current and voltage at single-phase short-circuit is pro- portional to the armature current, and does not disappear with the disappearance of the transient term, but persists also after the permanent condition of short-circuit has been reached, Armature current

. ‘Field current OO Fig. 51. Single-phase short-circuit current of a three-phase turbo-alternator. merely decreasing with the «lecrease of the armature current. It is shown in the oscillogram of a single-phase short-circuit on a three-phase alternator, Fig. 51. Superimposed on this double frequency pulsation is a single- frequency pulsation due to the transient term of the armature current, that is, the same as on polyphase short-circuit. With single-phase short-circuit, however, this normal frequency pul- ; sation of the field depends on the point of the wave at which the short-circuit occurs, and is zero, if the circuit is closed at the moment when the short-circuit current is zero, as in Fig. 51, and a maximum when the short-circuit starts at the maximum point of the current wave. As this normal frequency pulsation . gradually disappears, it causes the successive waves of the | double frequency pulsation to be unequal in size at the - | beginning of the transient term, and ‘gradually become equal, as shown in the oscillogram, Fig. 52. The calculation of the transient term of the short-circuit current of alternators thus involves the transient term of the

204 TRANSIENT PHENOMENA armature and the field current, as determined by the self- inductance of armature and of field circuit, and the mutual inductance between the armature circuits and the field circuit, and the impressed or generated voltage; therefore is rather complicated; but a simpler approximate calculation can be 0 0 Armature current ; 602 amp. Field : current 2.5 amp. ee Fig. 52. Single-phase short-circuit current of a three-phase turbo-alternator. given by considering that the duration of the transient term is short compared with that of the armature reaction on the field.

(A) Polyphase alternator.

115. Let n, = number of phases; @ = 22ft = time-phase angle; n,= number of field turns in series per pole; n, = number of armature turns in series per pole; Z,= r,— jx, = self-inductive impedance of field circuit; Z, = r, — jx, = self-inductive impe- dlance of armature circuit; p = permeance of field magnetic cir- cuit; a = 2xfn, 10-* = induction coefficient of armature; E, = exciter voltage; 7, = . = field exciting current, in permanent

0 condition; 7, = field exciting current at time 0; i,° = field exciting current immediately after short-circuit; 7 = armature current at time 0, and k, = = “t = transformation ratio of field ee |

SHORT-CIRCUIT CURRENTS OF ALTERNATORS 205 to resultant armature. Counting the time angle @ from the moment of short circuit, 6 = 0, and letting #” = time-phase angle of one of the generator circuits at the moment of short circuit, we have, ¥, = n,/, = field excitation, in permanent or stationary con-

dition, (1) ®, = pF, = pn,/, = magnetic flux corresponding thereto, and e. = aps oo Apel g. (2) = nominal generated voltage, maximum value, at 0 = 0. 0 Hence, p=% = CPMe 7, (3) q, qT, = momentary short-circuit current at time 0 = 0, and 0 Men = MeaPMiNele 4 FY =5 nJ° Dr, (4) . = resultant armature reaction thereof. Assume this armature reaction as opposite to the field excita- tion, Fy = Nelo (5) as is the case at short circuit. . The resultant m.m.f. of the magnetic circuit at the moment of short-circuit is gg? = Fo _— F,°. . (6) At this moment, however, the field flux is still ©,, and the result- ant m.m.f. is given by (1) as -F° = F, = Nel,. (7) Substituting (4), (5), (7) in (6) gives ., Npapn,n,l, ely = Mle — ne 2, + Pm hence, 1,0 = —, I,. (8) ; 1

206 TRANSIENT PHENOMENA Writing t= en, (9) we have io = at I,; (10) 1 that is, at the moment of short circuit the field exciting current rises from J, to 7,°, and then gradually dies down again to I, at — 269 a rate depending on the field impedance Z,, that is, bye ™” , as discussed in preceding chapters. Hence, it can be represented b y “ny j= nthe 7, (11) qT, The resultant armature m.m.f., or armature reaction, is myn T° 2 ? thus the magnetic flux which would be produced by it is . pn n,]° 2 ? and therefore the voltage generated by this flux is apnyn 3 hence, apn,n a a _ Voltage corresponding to the m.m.f. of armature current, a Armature current , that is, x, is the equivalent or effective reactance of armature reaction. In equations (10) and (11) the external self-inductance of the field circuit, that is, the reactance of the field circuit outside of the machine field winding, has been neglected. This would

SHORT-CIRCUIT CURRENTS OF ALTERNATORS 207 introduce a negative transient term in (11), thus giving equation (11) the approximate form L,+ 2 a en’) i, = tN Ey (12) aq, where x, = self-inductive reactance of the field circuit outside of alternator field coils.

The more complete expression requires consideration when zr, is very large, as when an external reactive coil is inserted in the field circuit.

In reality, z, is a mutual inductive reactance, and z, can be represented approximately by a corresponding increase of z,.

116. If 7 = maximum value of armature current, we have