Theory and Calculation of Electric Circuits — part 11 of 15

if z = total reactance, with coil (1) as primary, and (2) as secondary, and z’ = total reactance, with coil (2) as primary, and (1) as secondary, then it is:

With coil (1) as primary and (2) as secondary,

15

226 ELECTRIC CIRCUITS , : Primary reactance,

• 7% ,__"., s to e+e” hata! ‘ Secondary reactance, - a | ma te” ate With coil (2) as primary and (1) as secondary, °* F F . FUNG) | ‘fo | ~ | FRI bi {ll Mi rh nA lal . | : : a 2 . . — ered), = ; ¢ Fig. 108. Primary reactance, ° : : 1 = ve _v. oC ate let Secondary reactance, n= ty = 2. tata e+e!

116. By test, the two total reactances, x and x’, can be derived by considering, that in Fig. 107 at the moments, f and d, the total flux is leakage flux, as more fully shown in Fig. 105f and 105d, and the flux measured from f, gives the reactance, z, measured from d, gives the reactance, d. :

REACTANCE OF INDUCTION APPARATUS 227

Assuming we connect primary coil and secondary coil in series with each other, but in opposition, into an alternating-current circuit, as shown in Fig. 109, and vary the number of primary and secondary turns, until the voltage, ¢:, across the secondary coil, s, becomes equal to rx. Then no flux passes through the secondary coil, that is, the condition, Fig. 107/, exists, and the , voltage, é, across the primary coil, p, gives the total reactance,

2, for p as primary, — €o? = 1? (ro* + 27). Varying now the number of turns so that the voltage across the primary coil equals its resistance drop, é& = rot, then the P & . C7) e Fig. 109. voltage across the secondary coil, s, gives the total reactance, 2, for s as primary, 2 e;? = 27 (r,;? + 2’ ).

It would rarely be possible to vary the turns of the two coils, pands. However, if we short-circuit s and pass an alternating current through p, then at the very low resultant magnetic flux and thus resultant m.m-f., primary and secondary current are practically in opposition and of the same m.mf., and the mag- netic flux in the secondary coil is that giving the resistance drop Txt, that is, e’; = r: 7, is the true primary voltage in the secondary, and the voltage across the primary terminals thus is that giving primary resistance drop, roto, total self-inductive reactance, Xto, and the secondary induced voltage, ri4:. Thus,

€o® = (roto + rit )? + 27X93,

228 ELECTRIC CIRCUITS or, since iy practically equals zo, , eo? = to? [(ro + 71)? + 24], and inversely, impressing a voltage upon coil, s, and short-cir- cuiting the coil p, gives the leakage reactance, 2’, for s as primary, | e? = 77 [(ro + 171)? + 2 )I. Thus, the so-called “impedance test” of the transformer gives the total leakage reactance 2» + 2, for that coil as primary, which is used as such in the impedance test. . Where an appreciable difference of the total leakage flux is | expected when using the one coil as primary, as when using the | other coil, the impedance tests should be made with that coil as primary, which is intended as such. Since, however, the two leakage fluxes are usually approximately equal, it is immaterial ' which coil is used as primary in the impedance test, and gener- ally that coil is used, which gives a more convenient voltage and a current for testing. Magnetic Circuits of Induction Motor —— 117. In general, when dealing with a closed secondary winding, as an induction-motor squirrel-cage, we consider as the mutual inductive voltage, E, the voltage induced by the mutual magnetic flux, &, that is, the magnetic flux due to the resultant of the pri- mary and the secondary m.m.f. This voltage, #, then is con- sumed in the closed secondary winding by the resistance, :f1, ~ and the reactance, jz:f1, thus giving, FE = (ri + jai) (i. © The reactance voltage, jz:{1, is consumed by a, self-inductive flux, ,', that is, a magnetic leakage flux produced by the second- ary current and interlinked with the secondary circuit, and the actual or resultant magnetic flux interlinked with the secondary circuit, that is, the magnetic flux, which passes beyond the second- ary conductor through the armature core, thus is the vector dif- ference, $; = @ — ,?’, and the actual voltage induced in the second- ary circuit by the resultant magnetic flux interlinked with it thus ; is, E, = E — jxif;. This voltage is consumed by the resistance of the secondary circuit, #,; = 7r,J,, and the voltage consumed by self-induction, jf, is no part of £, but as stated, is due to the self-inductive flux, $,’, which vectorially subtracts from the mutual magnetic flux, , and thereby leaves the flux, $1’, which induces £. ee Bee

REACTANCE OF INDUCTION APPARATUS 229 In other words: In any closed secondary circuit, as a squirrel-cage of an induc- tion motor, the true induced e.m.f. in the circuit, that is, the e.m.f. induced by the actual magnetic flux interlinked with the circuit, is the resistance drop of the circuit, #, = rif. This is true whether there is one or any number of closed sec- ; “ondary circuits—or squirrel-cages in an induction motor. Ineach , the current, J, is B , where r; is the resistance of the circuit, and E, the voltage induced by the flux which passes through the cir- ; cuit. The F, of the different squirrel-cages then would differ from each other by the voltage induced by the leakage flux which passes between them, and which is represented by the self- inductive reactance of the next squirrel-cage: ., BE’, = By — je’: where [’; = a is the current in the inner squirrel-cage of voltage, E’,, and resistance, ri, and x’, ]’1, is the reactance of the flux between the two squirrel-cages. , The mutual magnetic flux and the mutual induced e.m.f. of the common induction motor theory thus are mathematical fictions . and not physical realities. The advantage of. the introduction of the mutual magnetic flux, $, and the mutual induced voltage, #, in the induction-motor theory, is the ease and convenience of passing therefrom to the secondary as well as the primary circuit. Where, however, a number of secondary circuits exist, as in a multiple squirrel-cage, it is preferable to start from the innermost magnetic flux, that is, the magnetic flux passing through the innermost squirrel-cage, and the voltage induced by it in the latter, which is the resistance drop of this squirrel-cage. In the same manner, in a primary circuit, the actual or total magnetic flux interlinked with the circuit, ®o, is that due to the impressed voltage, Eo, minus the resistance drop, Tolo, B’o = Eo — Tofo. Of this magnetic flux, ®o, a part, ®’9, passes as primary leak- age flux between primary and secondary, without reaching the secondary, and is represented by the primary reactance voltage, | ' jtofo, and the remainder—usually the major part—is impressed upon the secondary circuit as mutual magnetic flux, = $o — ©’o, corresponding to the mutual inductive voltage, E = E’y — jxofo. The mutual magnetic flux, $, then is impressed upon the second- | { |

| 230 ELECTRIC CIRCUITS ary, and as stated above, a part of it, the secondary leakage flux, %’,, is shunted across outside of the secondary circuit, the re- mainder, —’ = $ — $’,, passes through the secondary circuit and . corresponds to 71/1. 118, Applying this to the polyphase induction motor with single squirrel-cage secondary. Let Yo =g — jo = primary exciting admittance; . . Zo = To + jt = primary self-inductive impedance; ; 41 = 1 + jai = secondary self-inductive impedance at full frequency, reduced to the primary. Let E, = the true induced voltage in the secondary, at full . frequency, corresponding to the magnetic flux in the armature core. The secondary current then is hh = 1 Ti . . The mutual inductive voltage at full frequency, E=Eitjufi . = » 821) = (1 +52) Thus the exciting current, Too = YoR _¢, _ ¢ . 821 = (9 jp) (1 +3 =) Bs = (91 — jqs)E1, . where ; = 80% ; a=9+ ri .Q2 = b- en . and the total current, ; fo=11+T00 | = B12 +0 — Ja] Mh 1 — JQ2 ft» . hence, the primary impressed voltage, Ey =E + Zolo . 8% . 8 . = Bi {1 +i + (ro + jo) [= +4 — ja}, = Fi (C1 + jes),

REACTANCE OF INDUCTION APPARATUS 231 . where

: 8 To ,

c= 1+ ro(= +a) + tage = Lt 37? + regi + toms : 1 TL 8x c= + 2e(2 + qn) — ran = SED + ceo — rege choosing now the impressed voltage as zero vector, . Eo = gives = —°o__ Ae oy jer or, absolute, e = the torque of the motor is

• D = / E 4) vf i/ 1 = Set _ _8e08 T. — -Ti(ex? + C23)’ the power, P= 8e,7(1 — 8) _ 8(1 — s)éo? a T1(¢1? + c2*)
• the volt-ampere input, Q = eoto : ete.

As seen, this method is if anything, rather less convenient than the conventional method, which starts with the mutual inductive voltage £. .

It becomes materially more advantageous, however, when dealing with double and triple squirrel-cage structures, as it permits starting with the innermost squirrel-cage, and gradually building up toward the primary circuit. See “Multiple Squirrel- cage Induction Motor,” “Theory and Calculation of Electrical Apparatus.”

CHAPTER XIII REACTANCE OF SYNCHRONOUS MACHINES 119. The synchronous machine—alternating-current generator, ; synchronous motor or synchronous condenser—consists of an armature containing one or more electric circuits traversed by . ‘ alternating currents and synchronously revolving relative to a unidirectional magnetic field, excited by direct current. The armature circuit, like every electric circuit, has a resistance, r, in which power is being dissipated by the current, J, and an in- ductance, L, or reactance, = 2 xfL, which represents the mag- . netic flux produced by the current in the armature circuit, and interlinked with this circuit. Thus, if fy) = voltage induced in the armature circuit by its rotation through the magnetic field—

. or, a8 now more usually the case, the rotation of the magnetic field through the armature circuit—the terminal voltage of the armature circuit is

E = Ey — (r+ jz) f.

In Fig. 110 is shown diagrammatically the path of the field flux, in two different positions, A with an armature slot standing mid- way between two field poles, B with an armature slot standing opposite the field pole.

In Fig. 111 is shown diagrammatically the magnetic flux of armature reactance, that is, the magnetic flux produced by the current in the armature circuit, and interlinked with this circuit, which is represented by the reactance x, for the same two relative positions of field and armature.

As seen, field flux and armature flux pass through the same iron structures, thus can not have an independent existence, but actual is only their resultant. This resultant flux of armature self-in- . duction and field excitation is shown in Fig. 112, for the same two

positions, A and B, derived by superpositions of the fluxes in Figs. 110 and 111.

As seen, in Fig. 112A, all the lines of magnetic forces are inter- linked with the field circuit, but there is no line of magnetic flux

interlinked with the armature circuit only, that is, there is ap- 232

REACTANCE OF SYNCHRONOUS MACHINES — 233 parently no self-inductive armature flux, and no true self-induct- ive reactance, x, and the self-inductive armature flux of Fig. 111 thus merely is a mathematical fiction, a theoretical component of the resultant flux, Fig. 112. The effect of the armature current, | . ARMATURE | (| TT REGRERRREE A ==) | rf C= | A (A B Fro. 110. | in changing flux distribution, Fig. 110A to Fig. 112A, consists in | reducing the field flux, that is, flux in the field core, increasing the leakage flux of the field, that is, the flux which leaks from field pole to field pole, without interlinking the armature circuit, and |

234 ELECTRIC CIRCUITS . . still further decreasing the armature flux, that is, the flux issuing from the field and interlinking with the armature circuit. In position 112B, there is no self-inductive armature flux either, but every line of force, which interlinks with the armature circuit, . * ARMATURE am . a A \ ARMATURE (a . os oe to B

1. OES 1 C FIELD | . "We. 111. . is produced by and interlinked with the field circuit. The effect of the armature current in this case is to increase the field fluxand — the flux entering the armature at one side of the pole, and decrease it on the other side of the pole, without changing the total field flux and the leakage flux of the field. Indirectly, a reduction of

REACTANCE OF SYNCHRONOUS MACHINES — 235 the field flux usually occurs, by magnetic saturation limiting the increase of flux at the strengthened pole corner; but this is a sec- ondary effect.

ARMATURE ._ (a moe acer A ; em — —) ARMATUR’ = H | Wf ANI “(én B Fia. 112. . As seen, in 112A the armature current acts demagnetizing, in 112B distorting on the field flux, and in the intermediary position ~ between A and B, a combination of demagnetization (or magneti- zation, in some positions) and distortion occurs. Thus, it may be said that the armature reactance has no inde- pendent existence, is not due to a flux produced by and interlinked

236 ELECTRIC CIRCUITS

only with the armature circuit, but it is the electrical representa- tion of the effect exerted on the field flux by the m.m.f. of the arma- ture current.

Considering the magnetic disposition, an armature current,

which alone would produce the flux, Fig. 111, in the presence of a field excitation which alone would give the flux, Fig. 110, has the following effect: in Fig. 112A, by the counter m.m.f. of the arma- ture current the resultant m.m.f. and with it the resultant flux are | reduced from that due to the m.m-f. of field excitation, to that due to field excitation minus the m.m.f. of the armature current. The difference of the magnetic potential between the field poles is increased: in Fig. 110A it is the sum of the m.m.fs. of the two air- gaps traversed by the flux (plus the m.m.f. consumed in the arma- . ture iron, which may be neglected as small); in Fig. 112A it is the sum of the m.m.fs. of the two air-gaps traversed by the flux (which is slightly smaller than in Fig. 110A, due to the reduced flux) plus the counter m.m.f. of the armature. The increased magnetic potential difference causes an increased magnetic leak- age flux between the field poles,and thereby still further reduces the armature flux and the voltage induced by it.

In Fig. 112B, the m.m.f. of the armature current adds itself to the m.m.f. of field excitation on one side, and thereby increases the flux, and it subtracts on the other side and decreases the flux, and thereby causes an unsymmetrical flux distribution, that is, a field distortion.

120. Both representations of the effect of armature current are used, that by a nominal magnetic flux, Fig. 111 , which gives rise to a nominal reactance, the “synchronous reactance of the arma- ture circuit,’”’ and that by considering the direct magnetizing action of the armature current, as “armature reaction,’ and both have their advantages and disadvantages.

The introduction of a synchronous reactance, %, and correspond- ing thereto of a nominal induced e.m.f., eo, is most convenient in electrical calculations, but it must be kept in mind, that neither € hor 2» have any actual existence, correspond to actual magnetic fluxes, and for instance, when calculating efficiency and losses, the core loss of the machine does not correspond to eo, but corresponds to the actual or resultant magnetic flux, Fig. 112. Also, in deal- ing with transients involving the dissipation of the magnetic energy stored in the machine, the magnetic energy of the result- ant field, Fig. 112, comes into consideration, and not the—much

REACTANCE OF SYNCHRONOUS MACHINES 237 larger—energy, which the fields corresponding to é) and 2 would have. Thus the short-circuit transient of a heavily loaded ma- chine is essentially the same as that of the same machine at no- load, with the same terminal voltage, although in the former the field excitation and the nominal induced voltage may be very much larger.

The use of the term armature reaction in dealing with the effect of load on the synchronous machine is usually. more convenient and useful in design of the machine, but less so in the calculation dealing with the machine as part of an electric circuit. ’ Hither has the disadvantage that its terms, synchronous react- ance or armature reaction, are not homogeneous, as the different parts of the reactance field, Fig. 111, which make up the difference ~ between Fig. 112 and Fig. 110, are very different in their action, especially in their behavior at sudden changes of circuit conditions.

121. Considering the magnetic flux of the armature current, Fig. 111A, which is represented by the synchronous reactance, 2o.

A part of this magnetic flux (lines a in Fig. 111A) interlinks with the armature circuit only, that is, is true self-inductive or leakage flux. Another part, however, (6) interlinks with the field also, and thus is mutual inductive flux of the armature cir- cuit on the field circuit. In a polyphase machine, the resultant armature flux, that is, the resultant of the fluxes, Fig. 111, of all phases, revolves synchronously at (approximately) constant in- tensity, as a rotating field of armature reaction, and, therefore, is stationary with regard to the synchronously revolving field, F.

. Hence, the mutual inductive flux of the armature on the field, though an alternating flux, exerts no induction on the field circuit, is indeed a unidirectional or constant flux with regards to the field circuit. Therefore, under stationary conditions of load, no difference exists between the self-inductive and the mutual in- ductive flux of the armature circuit, and both are comprised in the synchronous reactance, 2. If, however, the armature current changes, as by an increase of load, then with increasing armature current, the armature flux, a and b, Fig. 111, also increases. a, . being interlinked with the armature current only, increases simul- taneously with it, that is, the armature current can not increase. : without simultaneously increasing its self-inductive flux,a. The mutual inductive flux, b, however, interlinks with the field circuit, and this circuit is closed through the exciter, that is, is a closed secondary circuit with regards to the armature circuit as primary,

, 238 ELECTRIC CIRCUITS and the change of flux, b, thus induces in the field circuit an e.m.f. and causes a current which retards the change of this flux com- ponent, b. Or, in other words, an increase of armature current tends to increase its mutual magnetic flux, b, and thereby to de- crease the field flux. This decrease of field flux induces in the field circuit an e.m.f., which adds itself to the voltage impressed upon the field, thereby increases the field current and maintains the . field flux against the demagnetizing action of the armature cur- . rent, causing it to decrease only gradually. Inversely, a decrease 7 of armature current gives a simultaneous decrease of the self- inductive part of the flux, a in Fig. 111, but a gradual decrease of the mutual inductive part, b, and corresponding gradual increase of the resultant field flux, by inducing a transient voltage in the field, in opposition to the exciter voltage, and thereby decreasing the field current.

Every sudden increase of the armature current thus gives an equal sudden drop of terminal voltage due to the self-inductive flux, a, produced by it (and the resistance drop in the armature circuit), an equally sudden increase of the field current, and then a gradual further drop of the terminal voltage by the gradual ap- pearance of the mutual flux, b, and corresponding gradual decrease of field current to nominal. The reverse is the case at a sudden decrease of armature current.

The extreme case hereof is found in the momentary short-cir- cuit currents of alternators,! which with some types of machines ' may momentarily equal many times the value of the permanent short-circuit current. However, this phenomenon is not limited to short-circuit conditions only, but every change of current in an alternator causes a momentary overshooting, the more so, the greater and more sudden the change is.

122. That part of the synchronous reactance, 2», which is due to : the magnetic lines, a, in Fig. 111, is a true self-inductive reactance, 2, and is instantaneous, but that part of 2 representing the flux lines, b, is mutual inductive reactance with the field circuit, 2’, and is not instantaneous, but comes into play gradually, and when- ever dealing with rapid changes of circuit conditions, the syn- chronous reactance, 2, thus must be divided into a true or self- inductive reactance, z, and a mutual inductive reactance, 2’:

t% =x2+2,! 1 See “Theory and Calculation of Transient Phenomena.”

| . ‘ | : | REACTANCE OF SYNCHRONOUS MACHINES — 239 The change of the flux disposition, caused by a current in the armature circuit, from that of Fig. 110 to that of Fig. 112, thus is | simultaneous with the armature current and instantaneous with a sudden change of armature current only as far as it does not in- volve any change of the flux through the field winding, but the | change of the flux through the field coils is only gradual. Thus | the flux change in the armature core can be instantaneous, but that in the field is gradual. This difference between self-inductive and mutual inductive reactance, or between instantaneous and gradual flux change, comes into consideration only in transients, and then very fre- quently the instantaneous or self-inductive effect is represented by a self-inductive reactance, x, the gradual or mutual inductive ‘effect by an armature reaction, ; : The relation between self-inductive component, z, and mutual inductive component, x’, varies from about 2 + 1 in the unitooth- high frequency alternators of old, to about 1 + 20in some ofthe ¢ earlier turbo-alternators. In those synchronous machines, which contain a squirrel-cage induction-motor winding in the field faces, for starting as motors, or as protection against hunting, or to equalize the armature reaction in single-phase machines, all the armature reactance flux, which interlinks with the squirrel-cage conductors (as the flux, c, | in Fig. 111B), also is mutual inductive flux, and such machines thus have a higher ratio of mutual inductive to self-inductive 7 | armature reactance, that is, show a greater overshooting of cur- rent at sudden changing of load, and larger momentary short- circuit currents. The mutual flux of armature reactance induces in the field cir- cuit only under transient conditions, but under permanent cir- _ cuit conditions the mutual inductance of the armature on the . field has no inducing action, but is merely demagnetizing, and the distinction between self-inductive and mutual inductive react- ance thus is unnecessary, and both combine in the synchronous reactance. In this respect, the synchronous machine differs from the transformer; in the latter, self-inductance and mutual .° inductance are always distinct in their action. 123. In permanent conditions of the circuit, the armature re- actance of the synchronous machine is the synchronous react- ance, % = x + 2’; at the instance of a sudden change of circuit conditions, the mutual inductive reactance, 2’, is still non-exist-

240 ELECTRIC CIRCUITS

ing, and only the self-inductive reactance, z, comes into play. Intermediate between the instantaneous effect and the permanent conditions, for a time up to one or more sec., the effective reactance changes, from z to 2%, and this may be considered as a transient reactance.

' During this period, mutual induction betweef armature cir- cuit and field circuit occurs, and the phenomena in the synchron- ous machine thus are affected by the constants of the field circuit outside of the machine. That is, resistance and inductance of the field circuit appear, by mutual induction, as part of the armature circuit of the synchronous machine, just as resistance and react- ance of the secondary circuit of a transformer appear, trans- formed by the ratio of turns, as resistance and reactance in the pri- mary, in their effect on the primary current and its phase relation.

Thus in the synchronous machine, a high non-inductive re- sistance inserted into the field circuit (with an increase of the exciter voltage to give the same field current) while without effect on the permanent current and on the instantaneous current in the moment of a sudden current change, reduces the duration of the transient armature current; an inductance inserted into the field circuit lengthens the duration of the transient and changes its shape.

The duration of the transient reactance of the synchronous machine is about of the same magnitude as the period of hunting of synchronous machines—which varies from a fraction of a second to over one sec. The reactance, which limits the current fluctations in hunting synchronous machines, thus is neither the synchronous reactance, 2%, nor the true self-inductive reactance, =, but is an intermediate transient reactance; the current change is sufficiently slow that the mutual induction between synchronous machine armature and field has already come into play and the field begun to follow, but is too rapid for the complete develop- ment of the synchronous reactance. .

124. In the polyphase machine on balanced load, the mutual inductive component of the armature reactance has no inductive effect on the field, as its resultant is unidirectional with regard to the field flux. In the single-phase machine, however (or polyphase machine on unbalanced load), such inductive effect exists, as a permanent pulsation of double frequency. The mutual inductive flux of the armature circuit on the field circuit is alternating, and the field circuit, revolving synchronously

REACTANCE OF SYNCHRONOUS MACHINES 241 through this alternating flux, thus has an e.m.f. of double fre- ; quency induced in it, which produces a double-frequency current ~ in the field circuit, superimposed on the direct current from the exciter. The field flux of the single-phase alternator (or poly- phase alternator at unbalanced load) thus pulsates with double frequency, and, by being carried synchronously through the armature circuits, this double-frequency pulsation of flux in- duces a triple-frequency harmonic in the armature. Thus, single-phase alternators, and polyphase alternators at unbalanced load, contain more or less of a third harmonio _ in their voltage wave, which is induced by the double-frequency pulsation of the field flux, resulting from the pulsating armature : reaction, or mutual armature reactance, 2’. The statement, that three-phase alternators contain no third harmonics in their terminal voltages, since such harmonics neu- tralize each other, is correct only for balanced load, but at un- balanced load, three-phase alternators may have pronounced third harmonics in their terminal voltage, and on single-phase short-circuit, the not short-circuited phase of a three-phase . alternator may contain a third harmonic far in excess of the | fundamental. | 125. Let in a Y-connected three-phase synchronous machine, | the magnetic flux per field pole be &. If this flux is distributed | sinusoidally around the circumference of the armature, at any time, ¢, represented by angle, ¢ = 2x ft, the magnetic flux enclosed . by an armature turn is , | ® = Gy cos > when counting the time from the moment of maximum flux. The voltage induced in an armature circuit of n turns then is . . a = n 22 = coysin ¢ where c= 2afn If, however, the flux distribution around the armature circum- ference is not sinusoidal, it nevertheless can, as a periodic func- tion, be expressed by © = &, [cos ¢ + a2 cos 2(¢ — as) + a3 cos 3(¢ — as) + acos4(6-—a) +... ] and the voltage induced in one armature conductor, by the | 16 |

242 ELECTRIC CIRCUITS synchronous rotation through this flux, is d®& . . . a af&o [sin @ + 2 ae sin 2(¢ — az) + 3 a; sin 3(¢—as) + : 4a,sin4(¢d—a,) +... ] hence, the voltage induced in one full-pitch armature turn, or in . two armature conductors displaced from each other on the arma- ture surface by one pole pitch or an odd multiple thereof, e=2 xf®,[sin +3 a; sin 3(¢—as) +5 a, sin 5(¢—as)+ . . . J that is, the even harmonics cancel. The voltage induced in one armature circuit of n effective series turns then is . €1 = CP [sin @ + bs sin 3(@ — as) + bs sin 5(@ —as) + . . . J where bs=3 a3, b5=5 a, etc., if all the n turns are massed together, and are less, if the armature turns are distributed, due to the overlapping of the harmonics, and partial cancellation caused thereby. As known, by causing proper pitch of the turn, or proper pitch of the arc covered by any phase, any harmonic can be entirely eliminated. . The second and third phase of the three-phase machine then . would have the voltage, €z = CBo [sin (¢ - 120°) + bs sin 3(¢ —asr-— 120°) + bs sin 5(@ — ag — 120°) +... J ; = cp [sin (p — 120°) + bz sin 3(¢ — as) + bs sin . (Bld — as] + 120°) + . - 4] €3 = cy [sin (6 — 240°) + bs sin 3(@ — as) + bs sin (5[@ — as] + 240°)] + . . .] As seen, the third harmonics are all three in phase with each other; the fifth harmonics are in three-phase relation, .but with backward rotation; the seventh harmonics are again in three- phase relation, like the fundamentals, the ninth harmonics in ; phase, etc. . The terminal voltages of the machine then are _— EB, =@€3— é: = V/3 cPo [cos — b, cos 5.(¢ — as) + b;cos 7 (@¢—a7)—+.. | and corresponding thereto E. = e: — eg and Es = é: — é:, differ- ing from E, merely by substituting ¢ — 120° and ¢ — 240° for ¢

. REACTANCE OF SYNCHRONOUS MACHINES 243 As seen, the third harmonic eliminates in the terminal voltages , of the three-phase machine, regardless of the flux distribution, ‘ provided that the flux is constarit in intensity, that is, the load conditions balanced. 126. Assuming, however, that the load on the three-phase : machine is unbalanced, causing a double-frequency pulsation of the magnetic flux, yp (1 + a cos 2 4), assuming for simplicity sinusoidal distribution of magnetic flux. , ; The flux interlinked with a full-pitch armature turn then is & = &,(1 + acos 2 ¢) cos (¢ — a) | | = % [cos (# — a) + 5 c08(¢ + a) +5 008 84 — a) | | | and the voltage induced in an armature circuit of n effective turns, | = nt? = ca, 4 —~a)+2 4 _ a= mF = cb 7,| 008 (¢ a) + 5008 (> + a) + 5008 (3p a) | | | = c&[sin ( — a) + Gain (6+ a) + Sain (8 ¢—a)| | or, if the magnetic flux maximum coincides with the voltage | maximum of the first phase, « = 0, . | 1 = | (1 + 5) cin ¢ + 54ein3 6]. . In the second phase, the flux is the same, ®y (1 + a cos 2 ¢), | but the flux interlinkage 120° later, thus, : | & = & (1 + a cos 2 ¢) cos (¢ — a — 120%), | and the voltage of the second phase thus is derived from that of the first phase, by substituting a + 120° for a, | ex = co [sin (@ — a — 120°) + § sin (p + a + 120°) + 32 sin (3 — a — 120°)| and the third phase, | es = co | sin ( — a — 240°) + 2 ain’ (¢ + a + 240°) + 52 sin (3 — a — 240°)|

244 ELECTRIC CIRCUITS the terminal voltages thus are, , Ey = es — es = V/3 ch [ cos (# — a) — 5 008 (¢ +a)+ “ cos (3 ¢ — a) and in the same manner, the other two phases, By = V3 co [cos ($ — a — 120°) — $ cos (6 +. + 120°) — 3a ; ° . =F cos (3 ¢ — a — 120°) Ey = V3 c&o| 0s (¢ — a — 240°) — $ cos (# + a+ 240°) — 52 cos (3 ¢ —a- 240°) |. For a = 0, this gives E, = V3 es] (1 _ 5) cos } + 32 e083 ¢] BE, = V3 ea (1 _ 5) cos (¢ — 120°) — ae cos (3 ¢ — 120°)| ‘ a ° 3a ) Ey = V3 co] (1 — §) cos (6 — 240°) — “ cos (3 — 240°]. As seen, all three phases have pronounced third harmonics, and the third harmonic of the loaded phase, E,, is opposite to that of the unloaded phases. If a = 1, which corresponds about to short-circuit conditions, as it makes the minimum value of &y equal zero, then the quadra- . ture phase of the short-circuited phase, £1, becomes a4 = $ oP (sin ot sin 3 $); that is, the third harmonic becomes as large as the fundamental. Thus, on unbalanced load, such as on single-phase short-circuit, triple harmonics appear in the terminal voltages of a three-phase . generator, though at balanced loads the three-phase terminal voltage can contain no third harmonics. | | |

SECTION III CHAPTER XIV CONSTANT-POTENTIAL CONSTANT-CURRENT TRANS- FORMATION

127. The generation of alternating-current electric power prac- tically always takes place at constant voltage. For some pur- poses, however, as for operating series arc circuits, and to a lim- ited extent also for electric furnaces, a constant, or approximately constant alternating current is required. While constant alter- .

’ nating-current arcs have largely come out of use and their place taken by constant direct-current luminous arc circuits, or incan- descent lamps, the constant direct current is usually derived by rectification of constant alternating-current supply circuits.

Such constant alternating currents are usually produced from constant-voltage supply circuits by means of constant or variable inductive reactances, and may be produced by the combination of inductive and condensive reactances; and the investigation of different methods of producing constant alternating current from constant alternating voltage, or inversely, constitutes a good application of the terms “impedance,” admittance,’ etc., and — offers a large number of problems or examples for the symbolic

method of dealing with alternating-current phenomena.

Even outside of arc lighting, such combinations of inductance and capacity which tend toward constant-voltage constant-cur- rent transformation are of considerable importance as a possible

| source of danger to the system. In a constant-current circuit, the load is taken off by short-circuiting, while open-circuiting . causes the voltage to rise to the maximum value permitted by the power of the generating source. Hence, where the circuit constants, with a-constant-voltage supply source, are such as to approach constant-voltage constant-current transformation, as is for instance the case in very long transmission lines, open-circuit- ing may lead to dangerous or even destructive voltage rise. 128. With an inductive reactance inserted in series to an alter- 245 |

246 ELECTRIC CIRCUITS nating-current non-inductive circuit, at constant-supply voltage, . _ the current in this circuit is approximately constant, as long as the resistance of the circuit issmall compared with the series inductive reactance. ; Let . Eo = é& = constant impressed alternating voltage; r = resistance of non-inductive receiver circuit; 2%» = inductive reactance inserted in series with this circuit. The impedance of this circuit then is Z=r + jo, . and, absolute, z= Vita, and thus the current, ‘ .

• 2 _ _ P= FH jt @) and the absolute value is . eo Co i Vata ®) the phase angle of the supply circuit is given by tan 0) = - (3) . and the power factor, cos 8) = r, (4) © 2 If in this case, r is small compared with 2p, it is | i-2—__. (5) 0 i@ or, expanded by the binomial theorem, yy (ty ty Bt 1 — = {1+(F)'| =1 Dat + Bast tee Vit) hence, -

. eo r? 3 rt :

i ae taitggint..- ; (6) that is, for small values of r, the current, 7, is approximately constant, and is

. €o ~=>- Zo

CONSTANT-CURRENT TRANSFORMATION _ 247 ~ For small values of r, the power-factor ; cos @ = } is very low, however. . Allowing a variation of current of 10 per cent. from short- circuit or no-load, r = 0, to full-load, or r = 1, it is, substituted ‘ in (2): " No-toad current:

• _ € . ‘0 To Pt Ett tT Ty tT Ae eo er kT RDA ect TTT et | wt tt ok ING TT ECCS Lt tL tert AIA. ptt ett TT tt | NALS Ltr ti tt tt tt tt AAG Ei td | dered PET tN ia. 113. Full-load current: - i= aes = 0.9%. Hence, —-__ = 0.9%, Vrit + x0? Zo . and therefore, . 71 = 0.485 20, and the power-factor, from (4), is 0.437. That is, even allowing as large a variation of current, 7, as 10 per cent., the maximum power-factor only reaches 43.7 per cent., when producing constant-current regulation by series ' inductance reactance. | ' : |

248 ELECTRIC CIRCUITS As illustrations are shown, in Fig. 113, for the constants: éo = 6600 volts applied e.m.f.; a = 792 ohms series reactance; the current: ; = —8600_ V7? + 792? 8.33 . = amp.; r v1 + (75a) and the power-factor: cos @ = ——2—__ = —". Vr? $792? r\2 792,/1 + (a5) with the voltage at the secondary terminals: e=ri as abscissas.

129. Ifthe receiver circuit is inductive, that is, contains, in addition to the resistance, r, an inductive reactance, x, and if this reactance is proportional to the resistance,

x = kr, as is commonly the case in arc circuits, due to the inductive reactance of the regulating mechanism of the arc lamp (the effective resistance, r, and the inductive reactance, z, in this case are both proportional to the number of lamps, hence pro- portional to each other), it is: total impedance: Z=r+j(mt+2) =r+ j (mo +h); or the absolute value is , z= V rt + (ao + 2)? = Vr? + (20 + kr)?; thus, the current —_— €0 ; [= r+ j(to + kr) (7) and the absolute value is | | jo ely | Vr? + (zo + kr)? = 20 Qhr | r2(1 + k)’ 1+— +————. Zo Loz

CONSTANT-CURRENT TRANSFORMATION 249 and the power-factor: r r cos 9 = - = ———————————: 9 "8 VEE Go + ker)? © By the binomial theorem, it is i re a ee eee ee Xo Lo* , Hence, the current ' ~_ Gof, kr _ r(2—k) i= S(1 pape tne... | ay . that is, the expression of the current, 7 (10), contains the ratio, z, in the first power, with & as coefficient, and if therefore k is not very small, that is, the inductive reactance, x = kr, a : very small fraction of the resistance, 7, the current, 7, is not °

• even approximately constant, but begins to fall off immediately, even at small values of r. Assuming, for instance, _ : k = 0.4. That is, the inductive reactance, 2, of the receiver circuit equals 40 per cent. of its resistance, r, and the power-factor of the receiver circuit accordingly is ; . cos § = ro a . -i 1 + k? = 93 per cent.; it is, substituted in (8), €0 I= eS r 1 toy) (75) + (1 +047)" | As illustrations are shown, in the same Fig. 113, for the constants: | éo = 6600 volts supply e.m.f.; 2%» = 792 ohms series reactance; the current: | . 8.33 ; Sa amp. . \ (a5) + (1 + O-4zG5) This current is shown by dotted line. In this case, in an inductive circuit, the current, 7, has decreased !

250 ELECTRIC CIRCUITS by 10 per cent. below the no-load or short-circuit value of 8.33 amp. that is, has fallen to 7.5 amp., at the resistance r = 187 ohms, or at the voltage of the receiving circuit, e=tVr4+axet=anvil+F = 1.077 nr = 1500 volts; while, in the case of a non-inductive load, the current has fallen off to 7.5 amp., or by 10 per cent. at the resistance. r = 395 ohms, or at the voltage of the receiving circuit: e = 2950 volts.

130. As seen, a moderate constant-current regulation can be produced in a non-inductive circuit, by a constant series inductive reactance, at a considerable sacrifice, however, of the power-factor, while in an inductive receiver circuit, the con-

. stant-current regulation is not even approximate.

To produce constant alternating current, from a constant-

• potential supply, by a series inductive reactance, over a wide range of load and without too great a _

sacrifice of power-factor, therefore re-

[\ my quires a variation of the series inductive

reactance with the load. That is, with

/ \ increasing load, or increasing resistance

fy S| of the receiver circuit, the series induc-

ts tive reactance has to be decreased, so as

to maintain the total impedance of the

° Fie. 114. circuit, and thereby the current, constant.

; In constant-current apparatus, as trans-

formers from constant potential to constant current, or regula-

tors, this variation of series inductive reactance with the load

is usually accomplished automatically by the mechanical motion

caused by the mechanical force exerted by the magnetic field of

the current, upon the conductor in which the current exists.