CHAPTER XVI. AIiTEBNATINGh-CURRENT OSNEBATOR. 159. In the alternating-current generator, E.M.F. is induced in the armature conductors by their relative motion through a constant or approximately constant magnetic field. When yielding current, two distinctly different M.M.Fs. are acting upon the alternator armature — the M.M.F. of the field due to the field-exciting spools, and the M.M.F. of the armature current. The former is constant, or approx- imately so, while the latter is alternating, and in synchro- nous motion relatively to the former ; hence, fixed in space relative to the field M.M.F., or uni-directional, but pulsating in a single-phase alternator. In the polyphase alternator, when evenly loaded or balanced, the resultant M.M.F. of the armature current is more or less constant. The E.M.F. induced in the armature is due to the mag- netic flux passing through and interlinked with the arma- ture conductors. This flux is produced by the resultant of both M.M.Fs., that of the field, and that of the armature. On open circuit, the M.M.F. of the armature is zero, and the E.M.F. of the armature is diie to the M.M.F. of the field coils only. In this case the E.M.F. is, in general, a maximum at the moment when the armature coil faces the position midway between adjacent field coils, as shown in Fig. 110, and thus incloses no magnetism. The E.M.F. wave in this case is, in general, symmetrical. An exception from this statement may take place only in those types of alternators where the magnetic reluctance of the armature is different in different directions ; thereby, 1 160] AL TERN A TING-CURRENT GENERA TOR. 235 during the synchronous rotation of the armature, a pulsa- tion of the magnetic flux passing through it is produced. This pulsation of the magnetic flux induces E.M.F. in the field spools, and thereby makes the field current pulsating also. Thus, we have, in this case, even on open circuit, no Fig. 110. rotation through a constant magnetic field, but rotation through a pulsating field, which makes the E.M.F. wave unsymmetrical, and shifts the maximum point from its the- oretical position midway between the field poles. In gen- eral this secondary reaction can be neglected, and the field M.M.F. be assumed as constant. 160. The relative position of the armature M.M.F. with respect to the field M.M.F. depends upon the phase rela- tion existing in the electric circuit. Thus, if there is no displacement of phase between current and E.M.F., the current reaches its maximum at the same moment as the E.M.F. ; or, in the position of the armature shown in Fig. 110, midway between the field poles. In this case the arma- ture current tends neither to magnetize nor demagnetize the field, but merely distorts it ; that is, demagnetizes the trail- ing-pole corner, a, and magnetizes the leading-pole corner, b. A change of the total flux, and thereby of the resultant E.M.F., will take place in this case only when the magnetic densities are so near to saturation that the rise of density at the leading-pole corner will be less than the decrease of 236 AL TERNA TING-CURRENT PHENOMENA. [§ 160 density at the trailing-pole corner. Since the internal self- inductance of the alternator alone causes a certain lag of the current behind the induced E.M.F., this condition of no displacement can exist only in a circuit with external nega- tive reactance, as capacity, etc. If the armature current lags, it reaches the maximum later than the E.M.F. ; that is, in a position where the armature coil partly faces the following-field pole, as shown in diagram in Fig. 111. Since the armature current flows Fiq. Ill, in opposite direction to the current in the following-field pole (in a generator), the armature in this case will tend ta demagnetize the field. If, however, the armature current leads, — that is, reaches its maximum while the armature coil still partly faces the Fig. 112. preceding-field pole, as shown in diagram Fig. 112, — it tends to magnetize this field coil, since the armature current flows in the same direction with the exciting current of the pre- ceding-field spools. ^ § 161] AL TERNA TING-CURRENT GENERA TOR. 237 Thus, with a leading current, the armature reaction of the alternator strengthens the field, and thereby, at con- stant-field excitation, increases the voltage ; with lagging current it weakens the fi,eld, and thereby decreases the vol- tage in a generator. Obviously, the opposite holds for a synchronous motor, in which the direction of rotation is opposite ; and thus a lagging current tends to magnetize, a leading current to demagnetize, the field. 161. The E.M.F. induced in the armature by the re- sultant magnetic flux, produced by the resultant M.M.F. of the field and of the armature, is not the terminal voltage of the machine; the terminal voltage is the resultant of this induced E.M.F. and the E.M.F. of self-inductance and the E.M.F. representing the energy loss by resistance in the alternator armature. That is, in other words, the armature current not only opposes or assists the field M.M.F. in cre- ating the resultant magnetic flux, but sends a second mag- netic flux in a local circuit through the armature, which flux does not pass through the field spools, and is called the magnetic flux of armature self -inductance. Thus we have to distinguish in an alternator between: armature reaction, or the magnetizing action of. the arma- ture upon the field, and armature self-inductance, or the E.M.F. induced in the armature conductors by the current flowing therein. This E.M.F. of self-inductance is (if the magnetic reluctance, and consequently the reactance, of the armature circuit is assumed as constant) in quadrature behind the armature current, and will thus combine with the induced E.M.F. in the proper phase relation. This means that, if the armature current lags, the E.M.F. of self-inductance will be more than 90° behind the induced E.M.F., and therefore in partial opposition, and will reduce the terminal voltage. On the other hand, if the armature current leads, the E.M.F. of self-inductance will be less than 90** behind the induced E.M.F., or in partial conjunc- 288 AL TERNA TIKG-CURRENT PHENOMENA. [ §§ 162, 163 tion therewith, and increase the terminal voltage. This means that the E.M.F. of self-inductance increases the ter- minal voltage with a leading, and decreases it with a lagging current, or, in other words, acts in the same manner as the armature reaction. For this reason both actions can be combined in one, and represented by what is called the synchronous reactance of the alternator. In the following, we shall represent the total reaction of the armature of the alternator by the one term, synchronous reactance. While this is not exact, as stated above, since the reactance should be resolved into the magnetic reaction due to the magnetizing action of the armature current, and the electric reaction due to the self-induction of the armature current, it is in general sufficiently near for prac- tical purposes, and well suited to explain the phenomena taking place in the alternator under the various conditions of load. 162. This synchronous reactance, x, is frequently not constant, but is pulsating, owing to the synchronously vary- ing reluctance of the armature magnetic circuit, and the field magnetic circuit ; it may, however, be considered in what follows as constant; that is, the E.M.Fs. induced thereby may be represented by their equivalent sine waves. A specific discussion of the distortions of the wave shape due to the pulsation of the synchronous reactance is found in Chapter XX. The synchronous reactance, x, is not a true reactance in the ordinary sense of the word, but an equivalent or effective reactance. 163. Let E^ = induced E.M.F. of the alternator, or the E.M.F. induced in the armature coils by their rotation through the constant magnetic field produced by the cur- rent in the field spools, or the open circuit voltage of the alternator. § 164 J AL TERN A TING-CUKREXT GENERA TOR. 239 Then e^ = -s/^icnN Ml^'^-, where // = total number of turns in series on the armature, N = frequency, J/ = total magnetic ffux per field pole. Let Xo = synchronous reactance, To = internal resistance of alternator ; then Zo = ^o — j x^ = internal impedance. If the circuit of the alternator is closed by the external impedance, and cinrcnt 7 = Zo + Z (,-,+ r)->(a', + ^)' or, j^ Eo . and, tcnninal voltage^ E = IZ = Eo- IZo Eo(r —jx) (ro-\r r) -j{Xo+ xY or. ^^ ^oVr'l_+.v! = ^, V(r, + rf 4- {x, + xf 1 /■« . O ^o ^ T" •^o •^ I ^o "f~ ^o V ^-^ r^ + x' '^ r^ + x^ or, expanded in a series, ^ = ^o J ^ - r^ + .r^" + 2 (r« + ^») =^ • • • p As shown, the terminal voltage varies with the condi- tions of the external circuit. 164. As an instance, in Figs. 118-118, at constant induced, the E.M.F., Eo = 2500 ; 240 ALTERNATINC-CURKENT PHENOMENA. [S164 and the values of the internal impedance, Z, = r„ ~jx^ = 1 - 10/ With the current / as abscissa:, the terminal voltages E as ordinates in drawn line, and the kilowatts output, = P r, in dotted lines, the kilovolt-amperes output, = I E, in. dash- 1 ,' '■ N ,_^ / ' \ •^ ■^ / \ \ S /' ^^ \ ! ^ ', ■■ \ ', .'. N. 1 \ \ °° ,» \ \ / p ELD CHA lACl ERIS Vic \ \ t.= r''= 't\ ■lOj. / \ A,. \ fiq. Its. Fitu m HoH-liKluctliv loaA dotted lines, we have, for the following conditions of external circuit : In Fig. 113, non-inductive external circuit, j: = 0. In Fig. 114, inductive external circuit, of the condition, r/.r = + .75, with a power factor, .6. In Fig. 115, inductive external circuit, of the condition, r = 0, with a power factor, 0, S1641 ALTERNATING-CURRENT GENERATOR. 241 In Fig. 116, external circuit with leading current, of the condi- tion, r jx = — .75, with a power factor, .6. In Fig. 117, external circuit with leading current, of the condi- tion, r = 0, with a power factor, 0. 1 1 \ 1 FIELD CHARA trasoo.^-MOj, 4 CTERI8T □ - 76lor eo< P. s \ \ \ ^ / \ N %% / '\ \ ss -V \ \ »' V --' \ \ I r- \ \ / Y V \ <• ■ ^., A N s^.. N ^^ S ^. \ / s \ / s \ t s s \ 1 \\ 1 ' -^, «» IW FlaUChttt. ocOr tfc P/ i4/t(»ia((ir, en WaW«M Intuctiv UaM. MOO ' s -^-^ \ y \ £ _ ^' ^ y F EL o^ar'a cterIst/ =* ,-25 01, Z "W j.E- -.75 re S . ^ - y /' / / // UN V / yk ,■ ,,' v' / t ' ^ \ VD / ^ ^ /\ E.= 26^0. i,'i oj ) s. \ y ^ ^ \ / ■^ s / \\ 1 ^ / / \ ^ ^1// ,^ ^/ Ui^ im"t' " fin 'y\ " " / \ ^ V/ ■N / / ■00 volts, for the currents, /= 60, 100, 150, 200, 250 amperes, the terminal voltages, E, as ordinates, with the inductance factor of the external circuit, s abscissx. Vr'-t-;. 166. If the internal impedance is negligible compared with the external impedance, then, approximatel)-. il6Q] ALTERNATING-CURRENT GENERATOR. 245 d2.ba.|z. -10 ' 7 7 "^ ~ ^^ (J=o.J„. // m ,. oJ V n ^ '■.""> , ^ ^ ..,200 , ^ 4 " \ as _^ _— ^. r 7 ;6, Ji S- 1^ — ■r' >^\ /7 - 1- L^ \y/ / --71 1 ^\A \0 "^ ^ ' / " 1* pll / / / / rt .-* 5 Sh SI " — ^ ^ -i^ "i / / Fig. f 10. Htgulatha ef Alttrmter en Vartous LuaOt. that is, i7« alternator with small internal resistance and syn- chronous reactance tends to regulate for constant terminal voltage. Kvery alternator tlocs this near open circuit, especially on non-inductive load. Kven if the synchronous reactance, r^, is not quite neg- ligible, this regulation takes place, to a certain extent, on non-inductive circuit, since for = 0, £ = - /1 + 2-^-^ + i^ and thus the expression of the terminal voltage, E, contains the synchronous reactance, x„, only as a term of second order in the denominator. On inductive circuit, however, x„ appears in the denom- inator as a term of first order, and therefore constant poten- tial regulation does not take place as well. 246 ALTERXATING-CURRENT PIIEXOMEXA. [§ 167 With a non-inductive external circuit, if the synchronous reactance, jr^, of the alternator is very large compared with the external resistance, r, Eq 1 current -^ = ^0 \M^* _^. ^0 approximately, or constant ; or, if the external circuit con- tains the reactance, jr, j_ E, 1 Xq -\- X I f r^ A- r \* ^0 v^' -