CHAPTER XV. INDUCTION MOTOB. 140. A specialization of the general alternating-current transformer is the induction motor. It differs from the sta- tionary alternating-current transformer in so far as the two sets of electric circuits — the primary or excited, and the secondary or induced, circuits — are movable with regard to each other ; and that in general a number of primary and a number of secondary circuits are used, angularly displaced around the periphery of the motor, and containing E.M.Fs. displaced in phase by the same angle. This multi-circuit arrangement has the object always to retain secondary cir- cuits in inductive relation to primary circuits, in spite of their relative motion. The result of the relative motion between primary and secondary is, that the E.M.Fs. induced in the secondary or the motor armature are not of the same frequency as the E.M.F. impressed upon the primary, but of a frequency which is the difference between the impressed frequency and the frequency of rotation, or equal to the ** slip," that is, the difference between synchronism and speed (in cycles). Hence, if N = frequency of main or primary E.M.F., and s = percentage slip ; sN = frequency of armature or secondary E.M.F., and (1 — s) J\r= frequency of rotation of armature. In its reaction upon the primary circuit, however, the armature current is of the same frequency as the primary f 208 ALTERNATING-CURRENT PHENOMENA. [§141 current, since it is carried around mechanically, with such a frequency as always to have the same phase relation, in the same position, with regard to the primary current. 141. Let the primary system consist of/ equal circuits, displaced angularly in space by 1 // of a period, that is, 1// of the width of two poles, and excited by/ E.M.Fs. displaced in phase by 1// of a period; that is, in other words, let the field circuits consist of a symmetrical /-phase system. Analogously, let the armature or secondary circuits con- sist of a symmetrical /^phase system. Let n = number of primary turns per circuit or phase ; «, = number of secondary turns per circuit or phase ; a = — ^ = ratio of total primary turns to total secondary turns or ratio of transformation. Since the number of secondary circuits and number of turns of the secondary circuits, in the induction motor — like in the stationary transformer — is entirely' unessential, it is preferable to reduce all secondary quantities to the primary system, by the ratio of transformation, a ; thus if Ex = secondary E.M.F. per circuit, Ey = a E^ = secondary E.M.F. per circuit reduced to primary system ; if Ii = secondary current per circuit, /^ = _L a = secondary current per circuit reduced to primary system ; if r/ = secondary resistance per circuit, r, = a^ r^ = secondary resistance per circuit reduced to pri- mary system ; if Xx =- secondary reactance per circuit, Xi = a^ Xi^ = secondary reactance per circuit reduced to pri- mary system ; I 142] INDUCTION MOTOR. 209 if 0/ = secondary impedance per circuit, z^ = a^ z^ = secondary impedance per circuit reduced to pri- mary system ; that is, the number of secondary circuits and of turns per secondary circuit is assumed the same as in the primary system. In the following discussion, as secondary quantities ex- clusively, the values reduced to the primary system shall be used, so that, to derive the true secondary values, these quantities have* to be reduced backwards again by the factor np «iA 142. Let * = total maximum flux of the magnetic field per motor pole. It is then E = V2ir«iV*10"® = effective E.M.F. induced by the mag- netic field per primary circuit. Counting the time from the moment where the rising magnetic flux of mutual induction * (flux interlinked with both electric circuits, primary and secondary) passes through zero, in complex quantities, the magnetic flux is denoted by and the primary induced E.M.F., j5 = — ^; where e = V2 TTfiN^ 10~* may be considered as the " Active E.M.F. of the motor." Since the secondary frequency is s Ny the secondary induced E.M.F. (reduced to primary system) is -^1 = — se. 210 AL TERN A TING-CURRENT PHENOMENA, [ § 142 Let lo = exciting current, or current passing through the motor, per primary circuit, when doing no work (at synchronism), and Y ^=^g -\-jb = primary admittance per circuit = — . c It is thus ge = magnetic energy current, ge^ = loss of power by hysteresis (and eddy currents) per primary coil. Hence pgt^= total loss of energy by hysteresis and eddys, as calculated according to Chapter X. b^= magnetizing current, and n be = effective M.M.F. per primary circuit ; hence . i- nbe = total effective M.M.F. ; and -£_ nbe = total maximum M.M.F., as resultant of the M.M.Fs. of "^^ the /-phases, combined by the parallelogram of M.M.Fs.* If (R = reluctance of magnetic circuit per pole, as dis- cussed in Chapter X., it is V2 nbe = (R<^. Thus, from the hysteretic loss, and the reluctance, the constants, g and b, and thus the admittance, K are derived. Let r = resistance per primary circuit ; z = reactance per primary circuit ; thus, Z '=^ r — y .r = impedance per primary circuit ; * Complete discussion hereof, see Chapter XXIII. § 143J INDUCTION MOTOR, 211 ri = resistance per secondary circuit reduced to primary sys- tem; Xi = reactance per secondary circuit reduced to primary system, at full frequency, N\ hence, 5Xx=^ reactance per secondary circuit at slip s ; and Zx = n —Jsxi = secondary internal impedance. 143. It is now. Primary induced E.M.F., E = — e. Secondary induced KM.F., El == ^ se. Hence, Secondary current, /. =§ = - s e Zi n —jsxx Component of primary current, corresponding thereto, Ji — — yj — n — JSXx ' Primary exciting current, hence, Total primary current. = e S + (^+y^)| ; E.M.F. consumed by primary impedance, E, = ZI ^e(r^jx) \ '-r- + {g + jb) \ ; 212 AL TERNA TING-CURRENT PHENOMENA, [8 143 E.^J.F. required to overcome the primary induced £.M.F.y — E =^ e\ hence, Primary terminal voltage, Eo = e-\- E, = ^ i 1 + '/'' " :^y + (^ - A) (^ + >^) j > We get thus, in an induction motor, at slip s and active E.M.F. e, it is Primary terminal voltage, ^o = ^ i 1 + -'^''"".•^''^^ + ir -jx) {g+jb) \ ; Primary current, [ ri - jsxi ) or, in complex expression. Primary terminal voltage, Primary current. To eliminate r, we divide, and get. Primary current, at slip j, and impressed E.M.F., Eq : Zi + sZ+ZZ,y "' or, f ^ s+ (n- jsx{) (g + rb) ^^ (n - /> -^i) + -f ('' - A) + C'' - » ('•i - j^-^i) {i + J^) Neglecting, in the denominator, the small quantity ZZiK, it is §144] INDUCTION MOTOR. 218 (n - j'sx^) + s{r- jx) ^ {s + r^^ + jJT i^ ) + j {r^g - jj:i^) ^ (ri + sr) —js{x^ + x) or, expanded, {r^ + sr)^ + s^(x, + xy Hence,displacement of phase between current and E.M.F., {sr^ + s^r) + r^^g+sr^{rg-x^)+^x^{xg+x^+^^ Neglecting the exciting current, /^, altogether, that is, setting F = 0, it is, \r, + sry + s\x + x,y ^ s_Eo . (r\ + sr) - s {x + x^) ' . « s (x '\- x^ tan Wo = — ^^ — - — — . r, + ^r 144. In graphic representation, the induction motor diagram appears as follows : — F\g, 105. 214 AL TERNA TING-CURRENT PHENOMENA. [§ 14l> ^ Denoting the magnetism by the vertical vector O^ in Fig. 105, the M.M.F. in ampere-turns per circuit is repre- sented by vector OFy leading the magnetism (7* by the angle of hysteretic advance a. The E.M.F. induced in the secondary is proportional to the slip s, and represented by OE^ at the amplitude of 180°. Dividing OE^ by R in the proportion of r^-s-jjTj, and connecting ^ with the middle b of the upper arc of the circle OE^ , this line inter- sects the lower arc of the circle at the point /j r^ . Thus, Ol^r^ is the E.M.F. consumed by the secondary resistance, and OlyX^ equal and parallel to E^I^r^ is the E.M.F. con- sumed by the secondary reactance. The angle, E^OI^f\ = CO J is the angle of secondary lag. The secondary M.M.F. OG^ is in the direction of the vector Ol^r^, Completing the parallelogram of M.M.Fs. with OF as diagonal and OG^ as one side, gives the primary M.M.F. OG as other side. The primary current and the E.M.F. consumed by the primary resistance, represented by Olr, is in line with OG, the E.M.F. consumed by the pri- mary reactance 90° ahead of OGy and represented by OIx^ and their resultant Ola is the E.M.F. consumed by the primary impedance. The E.M.F. induced in the primary circuit is OE^y and the E.M.F. required to overcome this counter E.M.F. is OE equal and opposite to OE^, Com- bining OE with OIs gives the primary terminal voltage represented by vector OE^^ and the angle of primary lag, ^,(9(7 = ^,. 145. Thus far the diagram is essentially the same as the diagram of the stationary alternating-current trans- former. Regarding dependence upon the slip of the motor, the locus of the different quantities for different values of the slip s is determined thus : It is El = s E/ OA -^ /l/*, = jE*! -5- IlSXy^ § 145] INDUCTION MOTOR, 215 OA = -^ ^-^ — 5 = ' * i- = -i- ^1 = constant jf J «r .X'j Yj s x^ x^ That is, /j r^ lies on a half-circle with —1 -fi^^' as diameter. Fig. loe. That means G^ lies on a half-circle g-^ in Fig. 106 with OA as diameter. In consequence hereof, G^ lies on half- circle ^^ with /^^ equal and parallel to OA as diameter. Thus /r lies on a half-circle with DH as diameter, which circle is perspective to the circle FBj and fx lies on a half- circle with /K as diameter, and /s on a half-circle with LJV as diameter, which circle is derived by the combination of the circles /r and /x. 216 AL TERNA TING-CURRENT PHENOMENA, [§ 146 The primary terminal voltage E^ lies thus on a half- * circle eo equal to the half -circle Izy and having to point E the same relative position as the half-circle Iz has to point 0. This diagram corresponds to constant intensity of the maximum magnetism, (?*. If the primary impressed voltage E^ is kept constant, the circle ^^ of the primary impressed voltage changes to an arc with O as center, and all the cor- responding points of the other circles have to be reduced in accordance herewith, thus giving as locus of the other quantities curves of higher order which most conveniently are constructed point for point by reduction from the circle of the locus in Fig. 106. Torque and Power, *4 146. The torque developed per pole by an CiCctric motor equals the product of effective magnetism, <^/V2, times effective armature M.M.F., /^/V2, times the sine of the angle between both, / = ^ sin {i>F,). If n^ = number of turns, /j = current, per circuit, with /-armature circuits, the total maximum current polarization, or M.M.F. of the armature, is Hence the torque per pole, 2 V2 l( li = the number of poles of the motor, the total torque of the motor is. §148] INDUCTION MOTOR, 217 The secondary induced E.M.F., E^^ lags 90® behind the inducing magnetism, hence reaches a maximum displaced in space by 90° from the position of maximum magnetization. Thus, if the secondary current, /, , lags behind its E.M.F., E^, by angle, cS^, the space displacement between armature current and field magnetism is ^ (/, *) = 90<'+ CO, hence, sin (<^ /,) = cos wj It is, however, cos 0*1 = n Vr,2 + ^ x^ , es thus, = — . substituting these values in the equation of the torque, it is or, in practical (C.G.S.) units, _ dp sr^e^ The Torque of the Induction Motor. At the slip s, the frequency N, and the number of poles d, the linear speed at unit radius is V = — ^ (1 - ^) ; hence the output of the motor, P= TS or, substituted, p^ P^\^s(\ -s) 218 AL TERN A TING-CURRENT PHENOMENA, [ § 147 The Power of the Induction Motor. 147. We can arrive at the same results in a different way: By the counter E.M.F. e of the primary circuit with current I = I^-^- 1^ the power is consumed, el = el^ + el^. The power el^ is that consumed by the primary hysteresis and eddys. The power e I^ disappears in the primary circuit by being transmitted to the secondary system. Thus the total power impressed upon the secondary system, per circuit, is P, ^el. Of this power a part, E^I^, is consumed in the secondary circuit by resistance. The remainder, P' = /i (^ - E,), disappears as electrical power altogether ; hence, by the law of conservation of energy, must reappear as some other form of energy, in this case as mechanical power, or as the • output of the motor (included mechanical and secondary magnetic friction). Thus the mechanical output per motor circuit is Substituting, It is £i==se; r se A = : — ; r,^ + ^x,^ hence, since the imaginary part has no meaning as power, r,e^ s(l - s) . n + ^ ^ 1 §148] INDUCTION MOTOR. 219 and the total power of the motor, At the linear speed, V = — — (1 - s) a at unit radius the torque is dp r, e^ s T = ^irN(r^^-^x^ In the foregoing, we found or, approximately, expanded, . = ^. __^^^Z^^__ ; or, eliminating imaginary quantities : Substituting this value in the equations of torque and of power, it is, d pr^E^s torque: r ^^^^^^^^ ^^y^^,^^^^ ^y^\ power : P = — -^ — - ^ , \. — ^2 • Maximum Torque, 148. The torque of the induction motor is a maximum for that value of slip j, where ^ = 0, ds ' 220 AL TERNA TING-CURRENT PHENOMENA. [§ 14S for, ^ j (n + xr)' + ^(^. + ^)M _Q. expanded, this gives, - ^ + '^ + (^i + ^)' = 0» or, J/ = Vr2 + (Ari + Ary Substituting this in the equation of torque, we get the value of maximum torque. Tt = SnJ\r\r+ Vr»+ {x^ + xy\ That is, independent of the secondary resistance, r^ . The power corresponding hereto is, by substitution of s^ in/>, 2^r^+ (X, + xy\^r''+(x, + xy + r\' This power is not the maximum output of the motor, but already below the maximum output. The maximum output is found at a lesser slip, or higher speed, while at the maximum torque point the output is already on the decrease, due to the decrease of speed. With increasing slip, or decreasing speed, the torque of the induction motor increases ; or inversely, with increasing load, the speed of the motor decreases, and thereby the torque increases, so as to carry the load down to the slip Sf , corresponding to the maximum torque. At this point of load and slip the torque begins to decrease again ; that is, as soon as with increasing load, and thus increasing slip, the motor passes the maximum torque point j<, it "falls out of step," and comes to a standstill. Inversely, the torque of the motor, when starting from rest, will increase with increasing speed, until the maximum §149] INDUCTION MOTOR, 221 torque point is reached. From there towards synchronism the torque decreases again. In consequence hereof, the part of the torque-speed curve below the maximum torque point is in general unstable, and can be observed only by loading the motor with an apparatus, whose countertorque increases with the speed faster than the torque of the induction motor. In general, the maximum torque point, j^, is between synchronism and standstill, rather nearer to synchronism. Only in motors of very large armature resistance, that is low efficiency, s^ > 1, that is, the maximum torque falls below standstill, and the torque constantly increases from synchronism down to standstill. It is evident that the position of the maximum torque point, Sfy can be varied by varying the resistance of the secondary circuit, or the motor armature. Since the slip at the maximum torque point, s^ , is directly proportional to the armature resistance, r,, it follows that very constant speed and high efficiency will bring the maximum torque point near synchronism, and give small starting torque, while good starting torque means a maximum torque point at low speed ; that is, a motor with poor speed regulation and low efficiency. Thus, to combine high efficiency and close speed regu- lation with large starting torque, the armature resistance has to be varied during the operation of the motor, and the motor started with high armature resistance, and with in- creasing speed this armature resistance cut out as far as. possible. 149. If X, = 1, it is ri = Vr^ + (xi + xy. In this case the motor starts with maximum torque, and when overloaded does not drop out of step, but gradually slows down more and more, until it comes to rest. < 222 AL TERN A TING-CURRENT PHENOMENA, [§150 If, s, > 1, it is n > Vr^ + (jci + x)\ In this case, the maximum torque point is reached only by driving the motor backwards, as countertorque. As seen above, the maximum torque, t^, is entirely inde- pendent of the armature resistance, and the same is the current corresponding thereto, independent of the armature resistance. Only the speed of the motor depends upon the armature resistance. Hence the insertion of resistance into the motor arma- ture does not change the maximum torque, and the current corresponding thereto, but merely lowers the speed at which the maximum torque is reached. The effect of resistance inserted into the induction motor is merely to consume the E.M.F., which otherwise would find its mechanical equivalent in an increased speed, analogous as resistance in the armature circuit of a continu- ous-current shunt motor. Further discussion on the effect of armature resistance is found under " Starting Torque." Maximum Power, 150. The power of an induction motor is a maximum for that slip, j^, where .^ ds = 0; or, smce p^ pr^E^sjl — s) ds\ -f (1 - -f) i ' expanded, this gives j> = ri + V(ri + r)»+(^i + ^)'' §150] INDUCTION MOTOR, 223 substituted in /*, we get the maximum power, P ^ ' " 2 \(r, + r) + V(n + rf + {x, + xY\ This result has a simple physical meaning : (rj + r) = -^ is the total resistance of the motor, primary plus secondary (the latter reduced to the primary), (x^ + x) is the total reactance, and thus V(ri + rf + (x^^ + x)^ = Z is the total impedance of the motor. Hence it is the maximum output of the induction motor, at the slip, ip — The same value has been derived in Chapter IX., as the maximum power which can be transmitted into a non- inductive receiver circuit over a line of resistance P, and impedance Z, or as the maximum output of a generator, or of a stationary transformer. Hence : T/ie maximum output of an induction motor is expressed by the same fonnula as the maximum output of a generator, or of a stationary trafisformer, or the maximum output which can be trafismitted over an inductive line into a fwn-inductive receiver circuit. The torque corresponding to the maximum output P^ is, ^ SwJVZ(P + ZJ' This is not the maximum torque, but the maximum torque, r^ , takes place at a lower speed, that is, greater slip. St = n _ 9 ^r^+{x,+ xy smce, — ' > ' : Vr^ + (x, + xy r,+ V(^H^)» + (^j + ;c)« that is, Sf > Sp, r 224 AL TERNA TING-CURRENT PHENOMENA, [§ 151 It is obvious from these equations, that, to reach as large an output as possible, R and Z should be as small as possi- ble ; that is, the resistances rj + r, and the impedances, Zy and thus the reactances, x^ + jt, should be small. Since ri + r is usually small compared with x^ + ;r, it follows, that the problem of induction motor design consists in con- structing the motor so as to give the minimum possible reactances, x^ + x. Starting Torque, 151. In the moment of starting an induction motor,. »' IS . = 1 ; hence, starting current : /=: (ri -M) + (r -jx) + (/-I -jxi) (r - jx) (^ +jb) or, expanded, with the neglection of the last term in the denominator, as insignificant : l{r,+r)+g{r,lr,+r-]+x,lx^+x']) + b{rx,-xr,)'\ + T _ j\(x^+x)+b{r^\_r^+r']+x^lx,+x'])-g(rx^-xr;)^ .^ . (r,+ry+(x,+xf and, displacement of phase, or angle of lag, - ^ C ^i + x)-\-b (ri [ri + r ] + - ^1 [-^1 + x']) - g {rx^ - xr^) tan 01, (''i + r) +g{r, [ri + r'] + x^ [.Vi + x'\) + b (rxy - xr^) Neglecting the exciting current, ^ = = ^, these equa- tions assume the form : J ^ (n + ^) +J(^i + ^ ) jj; ^ r^o . (ri + ry+(x, + xy ' {r, + r)^j(x, + x)' or, eliminating imaginary quantities, ^(n + ry+(x, + xy ^' and tan w^ = — ; — . n + r § 152] INDUCTION MOTOR, 226 That means, that in starting the induction motor without additional resistance in the armature circuit, — in which case oTj + ;r is large compared with r^ + r, and the total impe- dance, Z, small, — the motor takes excessive and greatly lagging cunents. The starting torque is dp r, Eo^ T« = _ dp Eq^ rj_ That is, the starting torque is proportional to the armature resistance, and inverse proportional to the square of the total impedance of the motor. It is obvious thus, that, to secure large starting torque, the impedance should be as small, and the armature resis- tance as large, as possible. The former condition is the condition of large maximum output and good efficiency and speed regulation ; the latter condition, however, means inefficiency and poor regulation, and thus cannot properly be fulfilled by the internal resistance of the motor, but only by an additional resistance which is short-circuited while the motor is in operation. 152. Since, necessarily, it is T rr,H _' sri + r,V s + rig P- — > T = or, inversely : s - s = _ 4 TT A'r, T dpF.;' 166, 157] INDUCTION MOTOR, 229 that is : Near synchronism^ the slip, s, of an induction motor, or its drop in speed, is proportional to the armature resistance ri and to the power, P, or torque. Induction Generator. 156. In the foregoing, the range of speed from J = 1, standstill, to j = 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> \, backwards driving, P becomes negative, representing consumption of po>yer, while s remains posi- tive ; hence, since the direction of rotation has changed, represents consumption of power also. All this power is consumed in the motor, which thus acts as brake. For, i" < 0, or negative, P and t become negative, and the machine becomes an electric generator, converting me- chanical into electric energy. Substituting in this case : jj = — j, where k^ is the acceleration, or the slip of the machine above synchronism, we derive the equations of the induction generator, which are the same as those of the induction motor, except that the sign before the " slip " is reversed. 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 de- crease again, while the current constantly increases. 157. The induction generator differs from the standard alternating-current generator essentially, in so far as it has no definite frequency of its own, but can operate at any frequency above that corresponding to its speed. But it can generate electric energy only when in circuit with an alternating-current apparatus of definite frequency, as an 230 ALrE/iNAT/NG-CUK/tEAT PHENOMENA. [S ld& alternator or synchronous motor. That is, the induction generator requires a "frequency setter" for its operation. When operating in parallel with standard alternators, the phase relation of the current issuing from the induction generator mainly depends — besides ujxin the slip — upon the self-induction of the induction generator, and can be varied thereby. Hence the induction generator can be used to control the phase relation in an alternating-current circuit. When connected in series in a circuit, the E.M.Iv of the induction generator is approximately proportional to the current. Thus it can be used as booster, to add voltage to a line in proportion to the current passing therein. Example. 168. As an instance are shown, in Fig. 107, charac- teristic curves of a 20 horse-power three-phase induction a> u «1 m BO lUlPHD - Jt j«( !i te m .. -»\ ^"11!' f= on ^\ 1... - T L .-' 1 1 V ' \ .' /• \ S \ y V •/ \ ^ < N / \ i^ ' ^/ \ ^ , \ -1 ■ / * V '', \ "-I ■^ 7 \ "-. \ 13 ,' .- 1--. , — - ~^ ^...^ L\ " W \ ; ia E" w ' 70 ,J^ ft » i«i «.' «, I...' I.J II..' w it :; ™,l±1. M S ' nf. T07. AiMtf <3Kvaet*rM/M 0/ /n INDUCTION MOTOR. 20 H.P ™-P »a ndu !t;= jr. 110 VoH 1. a 30 ..^ otb ., OC cl.. kv "■ ■^.u Vr z' N > / \ ? n / / N 1; / / A s / / > , 3 uf / / -k. ^t. ■\ \^ V \ 10 / / ■,^ M // \ 1*0 ' ' ^ J!0 J £0 1 liW Ami ,r« n ffiO a a 3» fTg, ros. Cumnt ClmrBcterltlla o/ IndaetlM Motor. motor, of 900 revolutions synchronous speed, 8 poles, fre- quency of 60 cycles. The impressed E.M.F. is 110 volts between lines, and the motor star-connected, hence the E.M.F, impressed per circuit : 110 v5 The constants of the motor are : I'rimary admittance, K Primary impedance, Secondary impedanc r = .1 -I- .4 Z =.03 -.01 Z, = .02 — .086/ 232 ALTERNATTNG-CURRENT PHENOMENA. [§158 In Fig. 107 is shown, with the speed in per cent of synchronism, as abscissae, the torque in kilogrammetres, as ordinates, in drawn lines, for the values of armature resistance : ri a .02 : short circuit of armature, full speed. /-J = ,045 : .025 ohms additional resistance. rx = .16 : .16 ohms additional resistance, maximum starting torque. ri => .75 : .73 ohms additional resistance, same starting torque as n — .045. ,2o'Kn.4-P'l-. f \ a 1 llni^uctipn .... / \ t _J\0 Valti. 9,00 !■• on. / ' » 8C e. i 8 Di ^.^m _, -^ ■• N ^ - '/ 7 z 5j . '■- ' ■d » -n, m -■ w- ""^ ' "° ' > 1 t 1 - i' J - mi * ■• , \ -11 - ■"" 11 ■?^ vr --' V 1 Bi uk fg auJ 1 M- / Ab »• hn hrc >ni> n & »- Y -J _ J _} _ L _Lk n». 100. Spt»d Otarairtirlit On the same Figure is shown the current per line, in dotted lines, with the verticals or torque as abscissa, and the horizontals or amperes as ordinates. To the same torque always corresponds the same current, no matter what the speeil be. On Fig. lOS is shown, with the current input per line as abscisssc, the torque in kilogrammetres and the output §168] INDUCTION MOTOR, 238 in kilowatts as ordinates in drawn lines, and the speed and the magnetism, in per cent of their synchronous values, as ordinates in dotted lines, for the armature resistance r^ = .02, or short circuit. In Fig. 109 is shown, with the speed, in per cent of synchronism, as abscissae, the torque in drawn line, and the output in dotted line, for the value of armature resis- tance r, = .045, for the whole range of speed from 120 per cent backwards speed to 220 per cent beyond synchronism, showing the two maxima, the motor maximum at J = .25, and the generator maximum at j = — 25. 234 AL TERN A TING-CURRENT PHENOMENA. [§ 159