CHAPTER XIX INDUCTION GENERATORS 173. In the foregoing, the range of speed from 5 = 1, stand- still, to s = 0, synchronism, has been discussed. In this range the motor does mechanical work. It consumes mechanical power, that is, acts as generator or as brake outside of this range. For s > 1, backward driving. Pi becomes negative, repre- senting consumption of power, while D remains positive; hence, since the direction of rotation has changed, represents con- sumption of power also. All this power is consumed in the motor, w^hich thus acts as brake. For s < 0, or negative, Pi and D become negative, and the machine becomes an electric generator, converting mechanical into electric energy. The calculation of the induction generator at constant fre- quency, that is, at a speed increasing with the load by the negative slip, S\, is the same as that of the induction motor except that Si has negative values, and the load curves for the machine shown as motor in Fig. 122, are shown in Fig. 125 for negative slip Si as induction generator. Again, a maximum torque point and a maximum output point are found, and the torque and power increase from zero at synchronism up to a maximum point, and then decrease again, while the current constantly increases. 174. The induction generator differs essentially from the ordinary synchronous alternator in so far as the induction generator has a definite power-factor, while the synchronous alternator has not. That is, in the synchronous alternator the phase relation between current and terminal voltage entirely depends upon the condition of the external circuit. The in- duction generator, however, can operate only if the phase relation of current and e.m.f., that is, the power-factor required by the external circuit, exactly coincides with the internal power-factor of the induction generator. This requires that 237 238 ALTERNATING-CURRENT PHENOMENA the power-factor either of the external circuit or of the induction generator varies with the voltage, so as to permit the generator and the external circuit to adjust themselves to equality of power-factor. Beyond magnetic saturation the power-factor decreases; that is, the lead of current increases in the induction machine. Thus, when connected to an external circuit of constant power- factor the induction generator will either not generate at all, if its power-factor is lower than that of the external circuit, or, if its power-factor is higher than that of the external circuit, the / AMPS-l 'ER QENtI / '>«YNCHHONOU XWATTS-n,^, 140 l^ DUG ION 3ENE RATOR 1 130 Z=.l+. AD-C.„_. -. i • • I COOO 120 no vol T6 C 3N61 ANT FRE 3UEr CY 2> no BPEED j _<: / .■WKI inn — EFF iciEr cy / y — =- =^ -^^ / 90 ,/^ ^ ,-- V— / >'■ 4000 80 / > ^^ .^^ / ,A y r "^ ^> 70 / > / 4. ^ ^\ ;x)oo GO / t » , / ^ 1^ __^- <•'' so 1 1 / y 2000 40 / 4 / ^ ^ 30 1 1 v' ^ ^ 1000 20 1 1 / ^ 10 '/ -\K JO -3( oo ■^ X) El EOT -WOO RICAL CUlTPUt.P,, -5000 1 -€000 WATTS -7000 -» 100 -« 00 -10 KX) Fig. 125. voltage will rise until by magnetic saturation in the induction generator its power-factor has fallen to equality with that of the external circuit. This, however, requires magnetic satura- tion in the induction generator, in some part of the magnetic circuit, as for instance in the armature teeth. To operate below saturation — that is, at constant internal power-factor — the induction generator requires an external circuit with leading current, whose power-factor varies with the voltage, as a circuit containing synchronous motors or syn- chronous converters. In such a circuit, the voltage of the induction generator remains just as much below the counter e.m.f. of the synchronous motor as is necessary to give the INDUCTION GENERATORS 239 required leading exciting current of the induction generator, and the synchronous motor can thus to a certain extent be called the exciter of the induction generator. When operating self-exciting, that is, shunt-wound, con- verters from the induction generator, below saturation of both the converter and the induction generator, the conditions are unstable also, and the voltage of one of the two machines must rise beyond saturation of its magnetic field. When operating in parallel with synchronous alternating cur- rent generators, the induction generator obviously takes its leading exciting current from the synchronous alternator, which thus carries a lagging wattless current. 175. To generate constant frequency, the speed of the in- duction generator must increase with the load. Inversely, when driven at constant speed, with increasing load on the induction generator, the frequency of the current generated thereby decreases. Thus, when calculating the characteristic curves of the constant-speed induction generator, due regard has to be taken of the decrease of frequency with increase of load, or what may be called the slip of frequency, s. Let, in an induction generator, Yo = go — jho = primary exciting admittance, Zo = To -\- jxo = primary self-inductive impedance, Zi = ri + jxi = secondary self-inductive impedance, reduced to primary, all these quantities being reduced to the frequency of synchronism with the speed of the machine, /. Let e = generated em.f., reduced to full frequency. s = slip of frequency, thus: (1 — s) / = frequency generated by machine. We then have the secondary generated e.m.f., se: thus, the secondary current. li = Z — r^^ = e{ai - ja2), where, , S'Xi ai = —^—, — ^ — ^ and 02 = the primary exciting current, ioo = EY^ = e {go - j6o), 240 ALTERNATING-CURRENT PHENOMENA thus, the total primary current, /o = /] + /oo ^ e{bi - J62), where, &]=«! + go and 62= 02 + i>o; the primary impedance voltage, E' = /o(ro+j[l - s]xo); the primary generated e.m.f. is, e(l - s). Thus, primary terminal voltage, ^0 = e(l - s) - 7o(ro +i[l - &]:ro) = e(ci - jcz), whei'e, Ci = 1 — 5 — ro6i — (1 — s)a;o62 and Ca = (1 — s)xo6i — ro62, hence, the absolute value is. Bo = e\/ci2 + C2S and, Co VCI^ + 02^ Thus, the secondary current. Ji = — 7 „ „ > ■'1 = ^o\"^ /~^> 9 -^ ^ ~ " v /- 2 _i_ ^2 ' V Ci^ — C2^ >' ^1 + ^2 the primary current, eo{bi — jbi) h = /&7 the primary terminal voltage, „ eo (ci - JC2) . ILo — — . — > V Ci^ -1- C2^ the torque and mechanical power input, D^P, = [eUY = f2' 2; INDUCTION GENERATORS the electrical output, Po = Po' - jPo' = [Eoh] = [EJoV - j[EohV = C^2 ^ ^^2 \ (^iC] + h.Co) - j (62C1 - 61C2) the volt-ampere output, PaQ ~ Colo = eo- 2 v^7T67^ Cl^ + C2 2 ' 241 ~" / / INDUC TION GENERATOR LOAD CURVES 1 | / ^=1 + 110 vo .3 J TS pONS TANT Di-.lj SPEED / / / / 1 / / ' i EFFICIEN CY ■iUfcN CY 7^ / / J*- lt0 2-- ■* =3 =« :^ ^ ~^ ^ «, t' ^ y •■=^ ^ / ^ ^ :^'— ■ 2 / / ^ y' / 1 .'^'•^ ^ ^ 1 / ^ 1 / 1 / ? 10 w 2r no se ELECTRICAL 00 4C00 output] 6000 1 9 , *A' 00 TS 7Cf K) W M « M _12 )00 Fig. 126. the efficiency. P^ _ hiCx + &2C2 the power-factor, cos t* = &1C1 + 62C2 or. /V ^ 62C1 — &1C2 Po^ ~ &1C1 -f- 62C2 In Fig. 126 is plotted the load characteristic of a constant- speed induction generator, at constant terminal voltage eo = HO, 16 242 ALTERNATING-CURRENT PHENOMENA and the constants: Yq - 0.01 - 0.1 i; Zq = 0.1 + 0.3 j, and Zi = 0.1 + 0.3i. 176. As an example may be considered a power transmission from an induction generator of constants Yq, Zq, Zi, over a line of impedance, Z = r + jx, into a synchronous motor of synchronous impedance, Z2 = r2 -\- JX2, operating at constant- field excitation. Let Co = counter e.m.f. or nominal generated e.m.f. of syn- chronous motor at full frequency; that is, frequency of synchro- nism with the speed of the induction generator. By the preced- ing paragraph the primary current of the induction generator was, 7o = e(bi - J62) ; the primary terminal voltage, Eq = e(ci - JC2) ; thus, terminal voltage at synchronous motor terminals, E', = Eo- h(r +j[l - s]x) ^ e{di - jdn), where, rfi = Ci — rhi — (1 — s) xho and ^2 = C2 + (1 — s) xbi — rh^', the counter e.m.f. of the synchronous motor, E2 = E,' - h (r2 + i [1 - s]x2) where, A'l ^ di — r2hi — (1 — s) .'C2&2 and /c2 = c?2 + (1 — s) 0:261 — r262, or the absolute value E2= e^/ki^ -\- k2^, since, however, E2^ Co (1 - s), we have, eo(l — s) e = Thus, the current, J _ eo(l - s) (61 -jbj), ' Vk7-+k? ^ INDUCTION GENERATORS the terminal voltage at induction generator, eo (1 — s) (ci — JC2) Ef\ Vk^ki' 243 1 TERMING, S^ \ <0L r ARwi*'a % ^o'^ :^ V s, 100 INDUCTION GENERATOR AND 6P ;ed OR FR ;QUE^ OY s aJoo SYNCHRONOUS MOTOR INDUCTION 9ENEHATION ,-'' -J '■(:^C OR — - — —-. ~s^^^ '^ -. 80 Z= .1+.3J Y=.OI-|j |at fIll frequency ,.