CHAPTER XXIII SYNCHRONIZING ALTERNATORS 203. All alternators, when brought to synchronism with each other, operate in parallel more or less satisfactorily. This is due to the reversibility of the alternating-current machine; that is, its ability to operate as synchronous motor. In consequence thereof, if the driving power of one of several parallel-operating generators is withdrawn, this generator will keep revolving in synchronism as a synchronous motor; and the power with which it tends to' remain in synchronism is the maximum power which it can furnish as synchronous motor under the conditions of running, 204. The principal and foremost condition of parallel opera- tion of alternators is equality of frequency; that is, the trans- mission of power from the prime movers to the alternators must be such as to allow them to run at the same frequency without slippage or excessive strains on the belts or transmission devices. Rigid mechanical connection of the alternators cannot be con- sidered as synchronizing, since it allows no flexibility or phase adjustment between the alternators, but makes them essentially one machine. If connected in parallel, a difference in the field- excitation, and thus the generated e.m.f. of the machines, may cause large cross-current, since it cannot be taken care of by phase adjustment of the machines. Thus rigid mechanical connection is not desirable for parallel operation of alternators. 205. The second important condition of parallel operation is uniformity of speed; that is, constancy of frequency. If, for instance, two alternators are driven by independent single- cylinder engines, and the cranks of the engines happen to be crossed, the one engine will pull, while the other is near the dead- point, and conversely. Consequently, alternately the one alter- nator will tend to speed up and the other slow down, then the other speed up and the first slow down. This effect, if not taken care of by fly-wheel capacity, causes a "hunting" or surging 292 SYNCHRONIZING ALTERNATORS 293 action; that is, a fluctuation of the voltage with the period of the engine revolution, due to the alternating transfer of the load from one engine to the other, which may even become so excessive as to throw the machines out of step, especially when by an ap- proximate coincidence of the period of engine impulses (or a multiple thereof), with the natural period of oscillation of the revolving structure, the effect is made cumulative. This diffi- culty as a rule does not exist with turbine or water-wheel driving, but is specially severe with gas-engine drive, and special pre- cautions are then often taken, by the use of a short-circuited squirrel cage winding in the field pole faces. 206. In synchronizing alternators, we have to distinguish the phenomena taking place when throwing the machines in parallel or out of parallel, and the phenomena when running in synchronism. When connecting alternators in parallel, they are first brought approximately to the same frequency and same voltage; and then, at the moment of approximate equality of phase, as shown by a phase-lamp or other device, they are thrown in parallel. Equality of voltage is less important with moderate size alter- nators than equality of frequency, and perfect equality of phase is usually of importance only in avoiding an instantaneous flickering of the light of lamps connected to the system. When two alter- nators are thrown together, currents exist between the machines, which accelerate the one and retard the other machine until equal frequency and proper phase relation are reached. With modern ironclad alternators, this interchange of mechan- ical power is usually, even without very careful adjustment before synchronizing, sufficiently limited not to endanger the machines mechanically, since the cross-currents, and thus the interchange of power, are limited by self-induction and armature reaction. In machines of very low armature-reaction, that is, machines of "very good constant-potential regulation," much greater care has to be exerted in the adjustment to equality of frequency, voltage, and phase, or the interchange of current may become so large as to destroy the machine by the mechanical shock; and sometimes the machines are so sensitive in this respect that it is difficult to operate them in parallel. The same applies in getting out of step. 207. When running in synchronism, nearly all types of ma- chines will operate satisfactorily; a medium amount of armature 294 ALTERNATING-CURRENT PHENOMENA reaction is preferable, however, such as is given by modern alter- nators— not too high to reduce the synchronizing power too much, nor too low to make the machine unsafe in case of accident, such as falling out of step, etc. If the armature reaction is very low, an accident — such as a short-circuit, falling out of step, opening of the field circuit, etc. — may destroy the machine. If the armature reaction is very high, the driving power has to be adjusted very carefully to constancy, since the synchronizing power of the alternators is too weak to hold them in step and carry them over irregularities of the driving-power. 208. Series operation of alternators is possible only by rigid mechanical connection, or by some means whereby the machines, with regard to their synchronizing power, act essentially in par- loXUiJL mnrmnmnwv Fig. 143. allel; as, for instance, by the arrangement shown in Fig. 143, where the two alternators, A\, A^, are connected in series, but interlinked by the two coils of a transformer, T, of which the one is connected across the terminals of one alternator and the other across the terminals of the other alternator in such a way that, when operating in series, the coils of the transformer will be with- out current. In this case, by interchange of power through the transformers, the series connection will be maintained stable. 209. In two parallel operating alternators, as shown in Fig. 144, let the voltage at the common busbars be assumed as zero line, or real axis of coordinates of the complex representation; and let e = difference of potential at the common busbars of the two alternators; SYNCHRONIZING ALTERNATORS 295 Z = r -^ jx = impedance of the external circuit; Y = 9 ~ jb == admittance of the external circuit; hence, the current in the external circuit is e I = r -\-jx = e(g - jh). Let El = ei -\- je'i = ai(cos di + j sin di) = generated e.m.f. of first machine; E2 = 62 -{- je'i ^ a2(cos 02. + j sin ^2) = generated e.m.f. of second machine; /i ^ ii — ji'i = current of the first machine; Jg = 12 — ji'2 = current of the second machine; Zi == ri + jxi = internal impedance, and Yi = Qi — jh\ = inter- nal admittance of the first machine; ^2 = ^2 + jxi = internal impedance, and Y2 = ^2 — i&2 = inter- nal admittance of the second machine. Fig. 144. Then, er + e'r = al^• 62^ + e'2^ --^ a2^; Ex = e ■\- hZi, or ei -\- je\ = (e + iiVi + i'lXi) -f j(iiXi — i'\r^; E2 = e -{- I2Z2, or 62 + je'2 = (e + ^'2^2 + ^'2a;2) + j(i2X2 — ^''2^2) ; / = /i 4- h, or eg — jeb = (h + 22) - j{i'i + i'2). This gives the equations: ei = e + iiri + i'lXi; 62 = e -\- 22r2 + ^'2X2; 296 ALTERNATING-CURRENT PHENOMENA e'l = iiXi — i'lVi', e'l = iiXi — i'^Ti; eg = ii + i2\ eh = i'l + i'2; 62- + €2^ = 02^ or eight equations with nine variables, ei, e'l, e^, e'2, ii, i'\, ii, i'2, e. Combining these equations by twos, eiri + e'lXi = eri + iiZi^; e2r2 + ^'2X2 = er2 + ^'222^; substituting in ii + ^2 = eg, we have eiQi + e'l&i + ^2^2 + e'2&2 = e(gi + 02 + g); and analogously, 6161 — e'lgi + 62^2 — ^'2^2 = e(bi + 62 + &): dividing, g + ffi + g2 ^ eiffi + e2g2 + e'lbi + e'2?>2. b + bi + 62 ~ eifei + 6262 — e'lfifi - e'igz' substituting g = y cos a £"1=01 cos di €2 = a2 cos 62 b = ^ sin a e'l = ai sin 0i e'2 = ^2 sin 02 gives g -\- gi + gz _ aiVi cos (^i - gQ + ^2^2 cos {(X2 - 62) & + &i + 62 ~ CLiVx sin (ai — 0i) + «22/2 sin (a2 — ^2) as the equation between the phase displacement angles, di and 62, in parallel operation. The power supplied to the external circuit is, V ^ e^g, of which that supplied by the first machine is, Pi = eii] by the second machine. Pi = eii. The total electrical power of both machines is, P = Pi + P2, SYNCHRONIZING ALTERNATORS 297 of which that of the first machine is, Pi = eiii - e'li'i] and that of the second machine, P2 = Ciii — e'li'i. The difference of output of the two machines is, AP = Pi - P2 = e{ix - i2); denoting di + ^2 di — 62 . "T~ = ' ~^~ = ^' AP —r may be called the synchronizing power of the machines, Ao or the power which is transferred from one machine to the other by a change of the relative phase angle. 210. Special Case. — Two equal alternators of equal excitation. ai = a2 = a, Z\ = Z2 = Zf). Substituting this in the eight initial equations, these assume the form, ei = e 4- iiTo -\- i'iXo, 62 = e -\- iiro + i'iXo, e'l = iiXo — i'ir'oj e'2 = iiXQ — ?2ro, eg = ii -\- ^2, eh = i'l + i'2, e^2 _|_ g^'2 = g^^ + 62'" = a2. Combining these equations by twos, 61 + 62 = 2 e + e (rosr + XqV), e'i-\- e'2 = e (xog — rob); e\ = a cos d\, e'l = a sin di, 62 = a cos 02, e'2 = a sin 02, a (cos 01 + cos ^2) = e (2 -\- Tog -\- a:o&), a (sin 01 + sin ^2) = e (xog — nh) ; substituting we have 298 ALTERNATING-CURRENT PHENOMENA expanding and substituting ^1 + ^2 gives hence That is, and cos or, 5 = a cos € cos 5 = e ( 1 H ^ j ' Xog--rob, a sin € cos 5 — e ^ , a;og - rpb tan € = ?r— ; n — "^ — constant. 2 -j-rog -j- xob 01 -\- di = constant; ^ - a\V + 2—) + 1—2^; ' a cos 5 at no-phase displacement between the alternators, or, 5 - 2 " ^' we have a e = From the eight initial equations we get, by combination, eiro + e'ia;o = eoro + iiiro^ + a^o^), eo^o + e'2a:o = Coro + i^iro^ + Xo"); subtracted and expanded. To (ei — 62) + rco (e'l — e'2) . 2i — t2 = 5 » or, since ei — 62 = a (cos 01 — cos ^2) = —2a sin e sin 5, e\ — e' 1 — a (sin 0i — sin ^2) = 2 a cos e sin 5, SYNCHRONIZING ALTERNATORS 299 we have 2 a sin 5 , . , ll — 12 = 2 (^0 ^^^ ^ ~ '"O ^^^ ^t = 2 ayo sin 5 sin (a — e), where tan a = — • The difference of output of the two alternators is AP = Pi - Ps = e (zi - 12) ; hence, substituting, „ 2 ae sin 5 , . , AP = o Xo cos e — ro sin e } ; substituting, a cos 5 e = sin € = Xog — rob 2 ^ res' + Xob 1 + cos e we have, 2a2 sm 8 cos 5 Xo (1 M 2 ) "■ ^^ \ 2 / J AP = expanding, or 2I /1 _L !M+^\'_L /£o^_ILlo6\2 AP = a^ sin 2 6 a^o + -7^- a^ sin 2 5 { ^0 + 2 j Vo^ yo^ + ?{/o + bbo-\-l if ^p 2a2 cos 2 5 1 60 + 2 J^o^ A5 ijo" + {/^o + 6feo + i y' AP 300 ALTERNATING-CURRENT PHENOMENA Hence, the transfer of power between the alternators, AP, is a maximum, if 5 = 45°; or 0i — 02 = 90°; that is, when the alter- nators are in quadrature. AP . The synchronizing power, — , is a maximum if 5 =0; that is, the alternators are in phase with each other. 211. As an instance, curves may be plotted for, a = 2500, ^ = ro + jxo = 1 + 10 j; or Fo = go - jbo - 0.01 - 0.1 j, with the angle, 6 = — ^ — , as abscissas, giving the value of terminal voltage, e; the value of current in the external circuit, ? = ey; the value of interchange of current between the alternators ii - h; the value of interchange of power between the alternators, AP = Pi - P2; the value of synchronizing power, — r-' For the condition of external circuit, g = 0, 6 = 0, y = 0, 0.05, 0, 0.05, 0.08, 0, 0.08, 0.03, + 0.04, 0.05, 0.03, - 0.04, 0.05.