CHAPTER IV INDUCTION MOTOR WITH SECONDARY EXCITATION 38. While in the typical synchronous machine and eommu- tating machine the magnetic field is excited by a direct current, characteristic of the induction machine is, that the magnetic field is excited by an alternating current derived from the alter- nating supply voltage, just as in the alternating-current trans- former. As the alternating magnetizing current is a wattless reactive current, the result is, that the alternating-current input into the induction motor is always lagging, the more so, the larger a part of the total current is given by the magnetizing current. To secure good power-factor in an induction motor, the magnetizing current, that i«, the current which produces the magnetic field flux, must be kept as small as possible. This means as small an air gap between stator and rotor as mechanic- ally permissible, and as large a number of primary turns per pole, that is, as large a pole pitch, as economically permissible. In motors, in which the speed — compared to the motor out- put—is not too low, good constants can be secured. This, however, is not possible in motors, in which the speed is very low, that is, the number of poles large compared with the out- put, and the pole pitch thus must for economical reasons be kept small — as for instance a 100-hp. 60-cycle motor for 90 revolu- tions, that is, 80 poles— or where the requirement of an exutMrVV momentary overload capacity has to be met, etc. In such motors of necessity the exciting current or current at no-load — which is practically all magnetizing current — is a very large part of full-load current, and while fair efficiencies may nevertheless be secured, power-factor and apparent efficiency necessarily are very low. As illustration is shown in Fig. 20 the load curve of a typical 100-hp. 60-cycle 80-polar induction motor (90 revolutions per minute) of the constants: Impressed voltage: ea = 500. Primary exciting admittance: Ya = 0.02 — 0.6 j. Primary self-inductive impedance: Zu = 0.1 + 0.3j. Secondary self-inductive impedance: Zi = 0.1 + 0.3 j. INDUCTION MOTOR 53 As seen, at full-load of 75 kw. output, the efficiency is 80 per cent., which is fair for a slow-speed motor. But the power-factor is 55 per cent., the apparent efficiency only 44 per cent., and the exciting current is 75 per cent, of full- load current. This motor-load curve may be compared with that of a typical induction motor, of exciting admittance: Y0 = 0.01 -O.lj, given on page 234 of "Theory and Calculation of Alternating- current Phenomena" 5th edition, and page 319 of "Theoretical - LOW 8PEE0 1 1DUCTI0N MOTOR l\ '-i*i Y.-.02-.SJ Z,-.l+.3j -'I-. 1 — i- m v. / :> -350 J- > PS j / 1 i 1 1 i 1 I 0 1 0 1 0 1 Fio. 20. — Low-epecd induction motor, load c : the Elements of Electrical Engineering," 4th edition, difference. 39. In the synchronous machine usually the stator, in com- mutating machines the rotor is the armature, that is, the element to -which electrical power is supplied, and in which electrical power is converted into the mechanical power output of the motor. The rotor of the typical synchronous machine, and the stator of the com mutating machine are the held, that is, in them no electric power is consumed by conversion into mechanical work, but their purpose is to produce the magnetic field flux, through which the armature rotates. In the induction machine, it is usually the stator, which is the 54 ELECTRICAL APPARATUS primary, that is, which receives electric power and converts it into mechanical power, and the primary or stator of the induc- tion machine thus corresponds to the armature of the synchro- nous or commutating machine. In the secondary or rotor of the induction machine, low-frequency currents — of the frequency of slip — are induced by the primary, but the magnetic field flux is produced by the exciting current which traverses the primary or armature or stator. Thus the induction machine may be considered as a machine in which the magnetic field is produced by the armature reaction, and corresponds to a synchronous machine, in which the field coils are short-circuited and the field produced by armature reaction by lagging currents in the armature. As the rotor or secondary of the induction machine corresponds structurally to the field of the synchronous or commutating machine, field excitation thus can be given to the induction machine by passing a current through the rotor or secondary and thereby more or less relieving the primary of its function of giv- ing the field excitation. Thus in a slow-speed induction motor, of very high exciting current and correspondingly poor constants, by passing an exciting current of suitable value through the rotor or secondary, the primary can be made non-inductive, or even leading current produced, or — with a lesaer exciting current in the rotor — at least the power-factor increased. Various such methods of secondary excitation have been pro- posed, and to some extent used. 1. Passing a direct current through the rotor for excitation. In this case, as the frequency of the secondary currents is the frequency of slip, with a direct current, the frequency is zero, that is, the motor becomes a synchronous motor. 2. Excitation through commutator, by the alternating supply current, either in shunt or in series to the armature. At the supply frequency,/, and slip, s, the frequency of rotation and thus of commutation is (I — s) /, and the full frequency cur- rents supplied to the commutator thus give in the rotor the effective frequency,/ — (1 — s) / = sf, that is, the frequency of slip, thus are suitable as exciting currents. 3. Concatenation with a synchronous motor. If a low-frequency synchronous machine is mounted on the induction-motor shaft, and its armature connected into the indue- INDUCTION MOTOR 55 tion-inotor secondary, the synchronous machine feeds low-fre- quency exciting currents into the induction machine, and thereby permits controlling it by using suitable voltage and phase. If the induction machine has n times as many poles as the synchronous machine, the frequency of rotation of the synchro- nous machine is thai of the induction machine, or How- n n ever, the frequency generated by the synchronous machine must be the frequency of the induction-machine secondary currents, that is, the frequency of slip s. Hence; 1 -8 or: 1 * JT+T that is, the concatenated couple its synchronous, that is, runs at constant speed at all loads, but not at synchronous speed, but at constant slip — ■r^r 4. Concatenation with a low-frequency commutating machine. If a commutating machine is mounted on the induction-motor shaft, and connected in series into the induction-motor secondary, the commutating machine generates an alternating voltage of the frequency of the currents which excite its field, and if the field is excited in scries or shunt with the armature, in the circuit of the induction machine secondary, it generates voltage at the frequency of slip, whatever the latter may be. That is, the induction motor remains asynchronous, increases in slip with increase of load. 5. Excitation by a condenser in the secondary circuit of the induction motor. As the magnetizing current required by the induction motor is a reactive, that is, wattless lagging current, it does not require a generator for its production, but any apparatus consuming lead- ing, that is, generating lagging currents, such as a condenser, can be used to supply the magnetizing current. 40, However, condenser, or synchronous or commutating machine, etc., in the secondary of the induction motor do not merely give the magnetizing current and thereby permit power- factor control, but they may, depending on their design or appli- cation, change the characteristics of the induction machine, as regards to speed and speed regulation, the capacity, etc. 56 ELECTRICAL APPARATUS If by synchronous or com mutating machine a voltage is inserted into the secondary of the induction machine, this vol- tage may be constant, or varied with the speed, the load, the slip, etc., and thereby give various motor characteristics. Further- more, such voltage may be inserted at any phase relation from zero to 300°. If this voltage is inserted 90° behind the secondary current, it makes this current leading or magnetizing and so in- creases the power-factor. If, however, the voltage is inserted in phase with the secondary induced voltage of the induction machine, it has no effect on (he power-factor, but merely lowers the speed of the motor if in phase, raises it if in opposition to the secondary induced voltage of the induction machine, and hereby permits speed control, if derived from a commutating machine. For instance, by a voltage in phase with and proportional to the secondary current, the drop of speed of the motor can be increased and series-motor characteristics secured, in the same manner as by the insertion of resistance in the induction-motor secondary. The difference however is, that resistance in the induct ion- motor secondary reduces the efficiency in the same proportion as it lowers the speed, and thus is inefficient for speed control. The insertion of an e.m.f., however, while lowering the speed, docs not lower the efficiency, as the power corresponding to the lowered speed is taken up by the inserted voltage and returned as output of the synchronous or commutating machine. Or, by inserting a voltage proportional to the load and in opposition to the induced secondary voltage, the motor speed can be maintained constant, or increased with the load, etc. If then a voltage is inserted by a commutating machine in the induction-motor secondary, which is displaced in phase by angle a from the secondary induced voltage, a component of this vol- tage: sin a, acts magnetizing or demagnetizing, the other com- ponent: cos a, acts increasing or decreasing the speed, and thus various efferts can be produced. As the current consumed by a condenser is proportional to the frequency, while that passing through an inductive reactance is inverse proportional to the frequency, when using a condenser in the secondary circuit of the induction motor, its effective im- pedance at the varying frequency of slip is: Z,' = n+j («i- 7)' where xt is the capacity reactance at full frequency. INDUCTION MOTOR 57 For s — 0, Zj* = o° , that is, the motor has no power at or near synchronism. For: 8Xi = 0, o or it is: Zf = rh and the current taken by the motor is a maximum. The power output thus is a maximum not when approaching synchronism, as in the typical induction motor, but at a speed depending on the slip, So hi and by varying the capacity reactance, x2, various values of reson- ance slip, So, thus can be produced, and thereby speed control of the motor secured. However, for most purposes, this is uneco- nomical, due to the very large values of capacity required. Induction Motor Converted to Synchronous 41. If, when an induction motor has reached full speed, a direct current is sent through its secondary circuit, unless heavily loaded and of high secondary resistance and thus great slip, it drops into synchronism and runs as synchronous motor. The starting operations of such an induction motor in conver- sion to synchronous motor thus are (Fig. 21) : First step: secondary closed through resistance: A. Second step: resistance partly cut out: B. Third step: resistance all cut out: C. Fourth step: direct current passed through the secondary : D. In this case, for the last or synchronous-motor step, usually the direct-current supply will be connected between one phase and the other two phases, the latter remaining short-circuited to each other, as shown in Fig. 21, D. This arrangement retains a short-circuit in the rotor — now the field — in quadrature with the excitation, which acts as damper against hunting (Danielson motor). 58 ELECTRICAL APPARATUS In the synchronous motor, Fig. 21, D, produced from the induc- tion motor, Fig. 21, C, it is: Let: l'"» = 8 — jk = primiiry exciting admittance of the induction machine, Z0 = r« -f jxn = primary self-inductive impe- dance, Z\ = t\ + jxt = secondary self-inductive im- pedance. Fio. 21. — Sturtiiig of induction motor and synchronous. The secondary resistance, r,, is that of t lie field exciting winding, thus does not further come into consideration in calculating the motor curves, except in the efficiency, as i|V| is the loss of power in the field, if i\ = field exciting current. Xl is of little further importance, as the frequency is zero. It represents t he magnetic leakage between the synchronous motor poles. r0 is the armature resistance and xa the armature self-inductive reactance of the synchronous machine. However, x0 is net the synchronous impedance, which enters the equation of the synchronous machine, but is only the self- inductive part of il, or the true armature self-induct ancc. The IXDTCTIOX MOTQSt mutual inductive part of the synchronous hapedance. or Ik* effective reactance of anaatare reaction x\ is not contained in x*. The effective reactance of anaaxure reaction of the synchro- nous machine, x*f represents the field excitaiiou consumed by the armature m.m.f., and is the voltage corresponding to this field excitation, divided bj the armature current which consumes this field excitation. 6, the exciting snsoeptance, is the magnetizing armature current, divided bj the voltage induced by it, thus, x\ the effect- ive reactance of synchronous-motor armature reaction, is the reciprocal of the exciting acceptance of the induction machine. The total or synchronous reactance of the induction machine as synchronous motor thus is: * - x« + x' .1 = x. + r The exciting conductance, g, represents the loss by hysteresis, etc., in the iron of the machine. As synchronous machine, this loss is supplied by the mechanical power, and not electrically, and the hysteresis loss in the induction machine as synchronous motor thus is: e*g. We thus have: The induction motor of the constants, per phase: Exciting admittance: 70 = g — jb, Primary self-inductive impedance: Z0 ■= r0 + jx<>, Secondary self-inductive impedance: Zx = r\ + jxi, by passing direct current through the secondary or rotor, be- comes a synchronous motor of the constants, per phase: Armature resistance: r0, Synchronous impedance: x = Xo + r* (1) Total power consumed in field excitation : P = 2 t»r„ (2) where i = field exciting current. Power consumed by hysteresis: P - e*g. (3) it is then: or: 60 ELECTRICAL APPARATUS 42. Let, in a synchronous motor: E0 = impressed voltage, E = counter e.m.f., or nominal induced voltage, Z — r + jx = synchronous impedance, / = i\ — 3H = current, #o = $ + ZJ = # + (n'i + xi2) + j (xt\ - n2), (4) $ = $q — Zf = #o - (n'i + xz2) ~ j (xii - ri2), (5) or, reduced to absolute values, and choosing: g = e = real axis in equation (4), $o = e0 = real axis in equation (5), eo2 = (e + ri\ + xU)2 + (xii — ri2)2 [e = real axis], (6) 02 = (^o — rii + xi2)2 + (xii — ri>2)2 [e0 = real axis]. (7) Equations (6) and (7) are the two forms of the fundamental equation of the synchronous motor, in the form most convenient for the calculation of load and speed curves. In (7), i\ is the energy component, and i2 the reactive com- ponent of the current with respect to the impressed voltage, but not with respect to the induced voltage; in (6), t\ is the energy component and i2 the reactive component of the current with respect to the induced voltage, but not with respect to the impressed voltage. The condition of motor operation at unity power-factor is: i2 = 0 in equation (7). Thus: e2 = (6o ~ rtf + xW (8) at no-load, for i\ = 0, this gives: e = eo, as was to be expected- Equation (8) gives the variation of the induced voltage and thus of the field excitation, required to maintain unity power- factor at all loads, that is, currents, ix. From (8) follows: re0 ± \/z2e2 — xV ,nx '* = - - -o - • W zl INDUCTION MOTOR 61 Thus, the minimum possible value of the counter e.m.f., e, is given by equating the square root to zero, as: x e = - e<>. z For a given value of the counter e.m.f., e, that is, constant field excitation, it is, from (7) : xe0 , /e* 7. re0\* , . or, if the synchronous impedance, x, is very large compared with r, and thus, approximately : z = x: ii = ei± 4i ~ ^ (11) The maximum value, which the energy current, t'i, can have, at a given counter e.m.f., e, is given by equating the square root to zero, as: t, = |- (12) For: ij = 0, or at no-load, it is, by (11): eo ± e ti = ___. Equations (9) and (12) give two values of the currents i\ and *2, of which one is very large, corresponds to the upper or unstable part of the synchronous motor-power characteristics shown on page 325 of "Theory and Calculation of Alternating- current Phenomena," 5th edition. 43. Denoting, in equation (5) : V = «' - je", (13) and again choosing J5?0 = eo, as the real axis, (5) becomes: ef — je" = («o — rii — xi2) - j (xii - n2), (14) and the electric power input into the motor then is : Po = /#o, //' = e-oiu (15) the power output at the armature conductor is : = jttij+e'tt, 62 ELECTRICAL APPARATUS hence by (14): " -Pi = U (e0 — ri\ — xi2) + it (xit — n2), (16) expanded, this gives: Pi = cot'i - r (tV + it2) = Po - n2, (17) where: i = total current. That is, the power out- put at the armature conductors is the power input minus the t*r loss. The current in the field is: to - eb, (18) hence, the i2r loss in the field; of resistance, ri. iVn - «Wri. (19) The hysteresis loss in the induction motor of mutual induced voltage, e, is: e2g, or approximately: P' = eoV (20) in the synchronous motor, the nominal induced voltage, e, does not correspond to any flux, but may be very much higher, than corresponds to the magnetic flux, which gives the hysteresis loss, as it includes the effect of armature reaction, and the hys- teresis loss thus is more nearly represented by e02g (20). The difference, however, is that in the synchronous motor the hys- teresis loss is supplied by the mechanical power, and not the electric power, as in the induction motor. The net mechanical output of the motor thus is: P = Pv - toVi - P' = Po— i2r — t'oV] — e2g = e0ii — i2r — 6*ft*ri — e2g, (21) and herefrom follow efficiency, power-factor and apparent efficiency. 44. Considering, as instance, a typical good induction motor, of the constants: Co = 500 volts; 7o = 0.01 -O.lj; Z0 = 0.1 + 0.3j; Zi = 0.lJ+j0.3j. INDUCTION MOTOR 63 The load curves of this motor, as induction motor, calculated in the customary way, are given in Fig. 22. Converted into a synchronous motor, it gives the constants: Synchronous impedance (1): Z - r+jx = 0.1 + 10.3 j. Fig. 23 gives the load characteristics of the motor, with the power output as abscissae, with the direct-current excitation, and thereby the counter e.m.f., c, varied with the load, so as to maintain unity power-factor. The calculation is made in tabular form, by calculating for various successive values of the energy current (here also the total current) t'i, input, the counter e.m.f., c, by equation (8): e* . (500 - 0.1 t,)« + 100.61 if, the power input, which also is the volt-ampere input, the power- factor being unity, is: Po = e0ii = 500 i\. From e follow the losses, by (17), (19) and (20): in armature resistance: 0.1 ii2; in field resistance: 0.001 e2; hysteresis loss: 2.5 kw.; and thus the power output: P = 500 ii - 2.5 - 0.1 ii2 - 0.001 e2 and herefrom the efficiency. Fig. 23 gives the total current as t, the nominal induced voltage as e, and the apparent efficiency which here is the true efficiency, as y. As seen, the nominal induced voltage has to be varied very greatly with the load, indeed, almost proportional thereto. That is, to maintain unity power-factor in this motor, the field excita- tion has to be increased almost proportional to the load. It is interesting to investigate what load characteristics are given by operating at constant field excitation, that is, constant nominal induced voltage, e, as this would usually represent the operating conditions. ELECTRICAL APPARATUS IN UCT ON MOTOR '™ Yo-.OI-.U Z,=.l+.3j -*~ -100. a"j y =~ — jC no 1 ro .,„ m m ■n > J i 0 i 1 I 1 1 i - B f > 1 J 1 0 120 t 0 t 1.1 <■■ Fm. 22.— Load c a of standard induction it 7 INDUCTION MOTOR ' UNITY POWER FACTOR 0 a- 500 Zo-.1t .3j Y0-.01-.|J Z|-.1 +.3i fZ -.1 + 10. 3j) SYNCHRONOUS ;.;. •/ ... .... ... , '■■ -'■; M ^0 1 — X ICHJ .-' -' ~- -- ■m / M „„, IV 7 / -BO -MO 500 i > ' ■■' 10 im • / iff u < 0 | --- in 1 1 ) 0 i) a so so i » 110 ISO I. 0 1 a i j i 0 ] u i I* INDUCTION MOTOR Figs. 24 and 25 thus give the load characteristics of the motor, at constant field excitation, corresponding to: in Fig. 24: in Fig. 25: -- 2e0; - 5 e„. For different values of the energy current, ij, from zero up to the maximum value possible under the given field excitation, INDUCTION MOTOR CONSTANT DIRECT CURRENT EXCITATION e0= 500 . Z0 -.1 + -3J. — - a - .1 + 10.3;) SYNCHRONOUS * ' > V > */ ■m t / f— / "."- 7 7 t a i I t __ — - — ■— so L 3 0 t 0 ■ ... Fia. 24.— Load c as given by equation (12), the reactive current, is, is calculated by equation (11): Fig. 24: u = 48.5 - V9410 - u*; Fig. 25: i, = 48.5 - V58,800 - t,». The total current then is: i = Vfi'T t^i the volt-ampere input: the power input: Q = e0i; P* = eQii, 68 ELECTRICAL APPARATUS the power output given by (21), and herefrom efficiency ij, power-factor p and apparent efficient, 7, calculated and plotted. Figs. 24 and 25 give, with the power output as al.iscissjc, the total current input, efficiency, power-factor and apparent efficiency. As seen from Firs. 24 and 25, the constants of the motor as synchronous motor with constant excitation, are very bad ; the no-load current is nearly equal to full-load current, and power- 1 1 II 1 1 II INDUCTION MOTOR CONSTANT DIRECT CURRENT EXCITATION *= 5e, n or: * = ^h <» that is, the concatenated couple runs at constant slip, s = — — - - > thus constant speed, 1 — s = — .-— of synchronism. (2) n •+- 1 72 ELECTRICAL APPARATUS Thus the machine couple has synchronous- mo tor character- istics, and runs at a speed corresponding to synchronous speed of a motor having the sum of the induction-motor and syn- chronous-motor poles as number of poles. If m = 1, that isj the synchronous motor has the same number of poles as the induction motor, s = 0.5, 1 - * - 0.5, that is, the concatenated couple operates at half synchronous speed, and shares approximately equally in the power output. If the induction motor has 76 poles, the synchronous motor four poles, n = 19, and: a = 0.05, 1 - s = 0.95, that is, the couple runs at 95 per cent, of the synchronous speed of a 76-polar machine, llius at synchronous speed of an SO-polar machine, and thus can be substituted for an 80-polar induction motor. In this case, the synchronous motor gives about 5 per cent., the induction motor 95 per cent, of the output; the synchronous motor thus is a small machine, which could be con- sidered ms a .synchronous exciter of the induction machine. 48. Let: #n = e'fl + je"o = voltage impressed upon in- duction motor. Ei m e'i + je"i = voltage induced in induc- tion motor, by mutual magnetic flux, reduced to full frequency. #? = e'i + je"? = nominal induced voltage of synchronous motor, re- duced to full frequency. /n ■ i'i — jf'"o • primary current in induc- tion motor. /l = i'i — ji"i = secondary current of in- duction motor and cur- rent in synchronous motor. Denoting by Z' the impedance, Z, at frequency, s, it is: Total impedance of secondary circuit, at frequency, s: Z' = Zt + zt- = ('-. + r0 + j-{xx + xt), (3) INDUCTION MOTOR 73 and the equations are: in primary circuit: in secondary circuit: and, current: /o = /i + r^i. From (6) follows: /! = /,- r#„ and, substituting (7) into (5) : s#i = 8fa + Z'lo - Z'YEh sEt + Z'to substituting (8) into (4) gives: sE2 + (Z< + sZo + Z'ZoY) /o hence: #.= and, transposed: 8 + Z'Y or: rrzh = ^° ~'(r+ z«f + Zo) /o- i + =-r Denoting: ri + r2 * = r> X\ + Xt = x7 and: = r'+;V = _/ i „•_/ r/f | it is, substituting into (9; and CIO;: £. (1 + ZT; = Et + ,Y) U, 1 + Z'Y ~ E" M + Z'Y + Z") I* (4) (5) (0) (7) (8) E0 (i + f r) - *, + [* + 2.(1 + * r) ] h, (9) (10) fllj 74 ELECTRICAL APPARATUS Denoting: E* =V = e'+je" (14) 1 + ZY as a voltage which is proportional to the nominal induced voltage of the synchronous motor, and : r+^y + Zo = z = r+ix (15) and substituting (14) and (15) into (13), gives: # = #„ - Z/0. (16) This is the standard synchronous-motor equation, with im- pressed voltage, #o, current, /0, synchronous impedance, Z, and nominal induced voltage, $. Choosing the impressed voltage, $q = e0 as base line, and substituting into (16), gives: e' + je" - (co - ri'o - xi"o) - j (xt'o - ri"o), (17) and, absolute: e2 = (e0 — r0i'o — s0i"o)2 + teot'o — r0i"o)2. (18) From this equation (18) the load and speed curves of the concatenated couple can now be calculated in the same manner as in any synchronous motor. That is, the concatenated couple, of induction and synchronous motor, can be replaced by an equivalent synchronous motor of the constants, e, eo, Z and /0. 49. The power output of the synchronous machine is: P2 = //„ «&/', where: /a+jb, c+jd/' denotes the effective component of the double-frequency prod- uct: (ac + bd); see "Theory and ^Calculation of Alternating- current Phenomena," Chapter XVI, 5th edition. The power output of the induction machine is: Pi = IK (i - *) £i/', (20) thus, the total power output of the concatenated couple: P = P -4- P = /fi, «Ei + (1 - «)£,/'; (2D INDUCTION MOTOR 75 substituting (7) into (21): P = //0 - r#„ s#2 + (1 - «)&/'; (22) from (8) follows: s#2 = #i (« + Z-7) - Z'/o, and substituting this into (22), gives: P - //0 - r^i, #, (1 + Z'Y) - Z'lo/'; (23) from (4) follows: 1$\ = #o — Zo/o, and substituting this into (23) gives: p = fu (i + ZoF) - r#o, #o (i + z*Y) - /o (Z- + Z„ + ZoZ'Y)/'. (24) • Equation (24) gives the power output, as function of impressed voltage, -Fo, and supply current, /o. The power input into the concatenated couple is given by: Po = /*., /o/', (25) or, choosing #0 = e0 as base line: Po = eoi'o. (26) The apparent power, or volt-ampere input ift given by: Q = e0io, (27) where: to = V'i'o* + i"V is the total primary current. From P, Po and Q now follow efficiency, power-factor and apparent efficiency. 60. As an instance may be considered the power-factor control of the slow-speed 80-polar induction motor of Fig. 20, by a small synchronous motor concatenated into its secondary circuit. Impressed voltage: e« = 500 volts. Choosing a four-polar synchronous motor, the induct ion machine would have to be redesigned with 70 poles, giving: n = 19, ' * = 0.05, 70 ELECTRICAL APPARATUS With the same rotor diameter of the induction machine, the pole pitch would be increased inverse proportional to the number of poles, and the exciting susceptance decreased with the square thereof, thus giving the constants: Y0 = g -jb = 0.02 - 0.54 j; Z0 = r0+jxo = 0.1 + 0.3j; Zi = ri+jxt = 0.1 + 0.3./. Assuming as synchronous motor synchronous impedance, reduced to full frequency: Z2 = r2 + jx2 = 0.02 + 0.2 j this gives, for s = 0.05: Z' = (ri + r2) +js (xi + x2) = 0.12 + 0.025 j, and : Z' = r' +jx' = Z's = 2.4 + 0.5 j, Z = r+jx = 0.84 + 1.4 j, and from (14): E2 E = 1.32- 1.29 j' C2 e ~ 1.84 thus: "339 = (50° ~ °'84 *'° ~ 1-4 *"o)* + (1'4 ?'° ~ °<84 '"',)!' (28) and the power output : P = //0(0.836 + 0.048 j) - (10 - 270 j), (508 - 32 j) - /„ (0.241 + 0.326 j)/'. (29) 51. Fig. 29 shows the load curves of the concatenated couple, under the condition that the synchronous-motor excitation and thus its nominal induced voltage, e2, is varied so as to maintain unity power-factor at all loads, that is: t"o = 0; this gives from equation (28) : ~|^ = (500 - 0.84t'0)s + 1.96 tV, INDUCTION MOTOR LO.V SPEED INDUCTION M OTOR EXCITED FOR UNITY POWER FACTOR e„ - 500 z0=.i +■ M ¥,-.02 -.54 j 2i = .l + -3j g - .05 Z, = .02 + .2/ SYNCHRONOUS . ■rr nv- „,. / ,„, . / / JW tm m ., • S / 1 / ' S f h -^ I 1 1 I 0 1 1 0 1 1 1 ] 0 1 0 1 0 II 0 [ t 1 0 1 'J 1 :> 1 0 1 LOW SPEED INDUCTION MOTOR WITH LOW FREQUENCY SYNCHRONOUS IN SECONDARY CONSTANT EXCITATION. « - 1.7C0 eD= 500 Z0 = .1 +.3) 8 ■ .OS Z,- .02 + .2J SYNCHRONOUS * r V ; l-n HI „.( J* j 1NI gg 0 0 I u I 0 " » a H 110 1 0 1 ] -- Fio. 30.— Load n 78 ELECTRICAL APPARATUS P = /(0.8361 i\ - 10) + j (0.048 i'0 4- 270), (508 = 0.241 1'0) - j (32 + 0.326 *'»/' = (0.836 i'» - 10) (508 - 0.241 i'0) - (0.048 j"'d + 270) (32 4- 0.326 i\). As seen from the curve, e2, of the nominal induced voltage, the synchronous motor has to be overexcited at all loads. However, ei first decreases, reaches a minimum and then increases again, thus is fairly constant over a wide range of load, so that with this type of motor, constant excitation should give good results. Fig. 30 then shows the load curves of the concatenated couple for constant excitation, on overexcitation of the synchronous motor of 70 per cent., or c2 = 850 volts. (It must be kept in mind, that ei is the voltage reduced to full frequency and turn ratio 1 :1 in the induction machine: At the slip, s = 0.05, the actual voltage of the synchronous motor would h« set = 12.5 volts, even if the numher of secondary turns of the induction motor equals that of the primary turns, and if, as usual, the induction motor is wound for less turns in the secondary than in the primary, the actual voltage at the synchronous motor terminals is still lower.) As seen from Fig. 30: the power-factor is practically unity over the entire range of load, from less than one-tenth load up to the maximum output point, and the current input into the motor thus is practically proportional to the load. The load curves of this concatenated couple thus arc superior to those, which can be produced in a synchronous motor at con- stant excitation. For comparison, the curve of apparent efficiency, from Fig. 30, is plotted as CS in Fig. 28. It merges indisttnguishably into the unity power-factor curve, So, except at its maximum output point. Induction Motor Concatenated with Commutating Machine 62. While the alternating-current commulating machine, MpB* cially of the polyphase type, is rather poor at higher frequencies, ii becomes better at lower frequencies, and at the extremely low ' frequency of the induction-motor secondary, it is practically as INDUCTION MOTOR 79 good as the direct-current commutating machine, and thus can be used to insert low-frequency voltage into the induction-motor secondary. With series excitation, the voltage of the commutating machine is approximately proportional to the secondary current, and the speed characteristic of the induction motor remains essentially the same: a speed decreasing from synchronism at no-load, by a slip, sf which increases with the load. With shunt excitation, the voltage of the commutating machine is approximately constant, and the concatenated couple thus tends toward a speed differing from synchronism. In either case, however, the slip, s, is not constant and independ- ent of the load, and the motor couple not synchronous, as when using a synchronous machine as second motor, but the motor couple is asynchronous, decreasing in speed with increase of load. The phase relation of the voltage produced by the commutating machine, with regards to the secondary current which traverses it, depends on the relation of the commutator brush position with regards to the field excitation of the respective phases, and thereby can be made anything between 0 and 2t, that is, the voltage inserted by the commutating machine can be energy voltage in phase — reducing the speed — or in opposition to the induction-motor induced voltage — increasing the speed; or it may be a reactive voltage, lagging and thereby supplying the induction-motor magnetizing current, or leading and thereby still further lowering the power-factor. Or the commutating machine voltage may be partly in phase — modifying the speed — and partly in quadrature — modifying the power-factor. Thus the commutating machine in the induction-motor secondary can be used for power-factor control or for speed control or for both. It is interesting to note that the use of the commutating ma- chine in the induction- motor secondary gives two independent variables: the value of the voltage, and its phase relation to the current of its circuit, and the motor couple thus has two degrees of freedom. With the use of a synchronous machine in the induction-motor secondary this is not the case; only the voltage of the synchronous machine can be controlled, but its phase adjusts itself to the phase relation of the secondary circuit, and the synchronous-motor couple thus has only one degree of free- dom. The reason is: with a synchronous motor concatenated to 80 ELECTRICAL APPARATUS the induction machine, the phase of the synchronous machine is fixed in space, by the synchronous-motor poles, thus has a fixed relation with regards to the induction-motor primary system. As, however, the induction-motor secondary has no fixed position relation with regards to the primary, but can have any position slip, the synchronous-motor voltage has no fixed position with regards to the induction-motor secondary voltage and current, thus can assume any position, depending on the relation in the secondary circuit. Thus if we assume that the synchronous- motor field were shifted in space by « position degrees (electrical): this would shift the phase of the synchronous-motor voltage by a degrees, and the induction-motor secondary would slip in posi- tion by the same angle, thus keep the same phase relation with regards to the synchronous-motor voltage. In the couple with a commutating machine as secondary motor, however, the posi- tion of the brushes fixes the relation between commutating- machine voltage and secondary current, and thereby imposes I definite phase relation in the secondary circuit, irrespective of the relations between secondary and primary, and no change of relative position between primary and secondary can change this phase relation of the commutating machine. Thus the commutating machine in the secondary of the induc- tion machine permits a far greater variation of condition! of operation, and thereby gives a far greater variety of speed and load curves of such concatenated couple, than is given by the use of a synchronous motor in the induction-motor secondary. 63. Assuming the polyphase low-frequency commutating machine is series-excited, that is, the field coils (and compensat- ing coils, where used) in series with the armature. Assuming also that magnetic saturation is not reached within the range of its use. The induced voltage of the commutating machine then is proportional to the secondary current and to the speed. Thus: et = pix (1) is the commutating-machine voltage at full synchronous speed, where tj is the secondary current and p a constant depending on the design. At the slip, s, and thus the speed (1 — s), the comnmUting machine voltage thus is: (1 -s)e, = (1 - «) pii. ..L'i INDUCTION MOTOR 81 As this voltage may have any phase relation with regards to the current, t'i, we can put: £2 = (Pi + jpt)h (3) where: P = Vpi2 + p22 (4) and: tan w = — (5) Pi is the angle of brush shift of the commutating machine. (Pi + JPt) ls of the nature and dimension of an impedance, and we thus can put : Z° = Pi + m (6) as the effective impedance representing the commutating machine. At the speed (1 — s), the commutating machine is represented by the effective impedance: (1 - s) Z° = (1 - 8) Pl + j (1 - 8) P2. (7) It must be understood, however, that in the effective impedance of the commutating machine, Z° = pi + jp>, Pi as well as p% may be negative as well as positive. That is, the energy component of the effective impedance, or the effective resistance, plf of the commutating machine, may be negative, representing power supply. This simply means, that the commutator brushes are set so as to make the commutating machine an electric generator, while it is a motor, if pi is positive. If pi = 0, the commutating machine is a producer of wattless or reactive power, inductive for positive, anti-inductive for negative, p2. The calculation of an induction motor concatenated with a commutating machine thus becomes identical with that of the straight induction motor with short-circuited secondary, except that in place of the secondary inductive impedance of the induc- tion motor is substituted the total impedance of the secondary circuit, consisting of: 1. The secondary self-inductive impedance of the induction machine. 82 ELECTRICAL APPARATUS 2. The self-inductive impedance of the commutating machine comprising resistance and reactance of armature and of field, and compensating winding, where such exists. 3. The effective impedance representing the commutating machine. It must be considered, however, that in (1) and (2) the re- sistance is constant, the reactance proportional to the slip, s, while (3) is proportional to the speed (1 — s). 64. Let: Yo = g— jb = primary exciting admittance of the induction motor. Z0 =- r0 + jxo = primary self-inductive im- pedance of the induction motor. Z\ = rx + jx\ = secondary self-inductive im- pedance of the induction motor, reduced to full frequency. Zi — r2 + jx% = self-inductive impedance of the commutating machine, reduced to full frequency. Z° = pi + jpi = effective impedance repre- senting the voltage in- duced in the commutating machine, reduced to full frequency. The total secondary impedance, at slip, «, then is: Z$ = (r, + jsxi) + (r2 + jsxi) + (1 - «) (pi + jp2) = [r, + r2 + (1 -» pj + j\s (xi + xt) + (1 - s) p2] (8) and, if the mutual inductive voltage of the induction motor is chosen as base line, e, in the customary manner, the secondary current is: h = 7% = (ai-ja2)e, (9) where: * [r\ + r2 + (1 - s) pi] ii\ — m 8[s(xi + X2) + (1 -s)p2] a2 = m (10) INDUCTION MOTOR m - [p, + rt + (1 - s) p,]* + [s (Xi + x,) + (1 - ) Pd*. The remaining calculation is the same as on page 318 of " Theoretical Elements of Electrical Engineering," 4th edition. As an instance, consider the concatenation of a low-frequency commutating machine to the low-speed induction motor, Fig. 20. The constants then are: Impressed voltage: Exciting admittance: Impedances: ee = 500; YQ = 0.02 - 0.6 j Z* = 0.1 +0.3 j Zv = 0.1 +0.3j" Z, = 0.02 + 0.3 j" Z" = - 0.2 j. LOW SPEED 1 DUCT 0!, MOTOR SERIES EXCITED FOR ANTI-INDUCTIVE REACTIVE V0LTA3E P,+ Jp.-i-.2j ..,. Y,,".02-.8i Zi-.l+.3i ASYMCHRONO JB ■ -» ft * » m zoo m s <* __ *7 -*. •> > i. s M rn T_ / / 100 H / / A o - 1 I 1 4 1 1 t i a ■ a 1 o i 01 i.l 1 0 1 n l n u n i 0" That is, the commutating machine is adjusted to give only reactive lagging voltage, for power-factor compensation. It then is: Z> = 0.12 + j [0.6 s - 0.2 (1 - s)]. The load curves of this motor couple are shown in Fig. 31. As 84 ELECTRICAL APPARATUS Bent, power-factor and apparent efficiency rise to high values, and even the efficiency is higher than in the straight induction motor. However, at light-load the power-factor and thus the apparent efficiency falls off, very much in the same manner as in the con- catenation with a synchronous motor. It is interesting to note the relatively great drop of speed at light-load, while at heavier load the speed remains more nearly constant. This is a general characteristic of anti-inductive im- pedance in the induction-motor secondary, and shared by the use of an electrostatic condenser in the secondary. For comparison, on Fig. 28 the curve of apparent efficiency of this motor couple is shown as CC. Induction Motor with Condenser in Secondary Circuit 66. As a condenser consumes leading, that is, produces lagging reactive current, it can be used to supply the lagging component of current of the induction motor and thereby improve the power-factor. Shunted across the motor terminals, the condenser consumes a constant current, at constant impressed voltage and frequency, Mini as the lagging component of induction-motor current in- creases with the load, the characteristics of the combination of motor and shunted condenser thus change from leading current at no-load, over unity power-factor to lagging current at overload. As the condenser is an external apparatus, the characteristics of the induction motor proper obviously are not changed by a shunted condenser. As illustration is shown, in Fig. 32, the slow-speed induction motor Fig. 20, shunted by a condenser of 125 kva. per phase. Fig. 32 gives efficiency, i?, power-factor, p, and apparent efficiency, 7, of the combination of motor and condenser, assuming an efficiency of the condenser of 99.5 per cent., thai is, 0,5 pet cent loss in the condenser, or Z = 0.0025 - 0.5 j, that is, a condenser just neutralizing i he magnetising current. However, when using a condenser in shunt, it must be realized that the current consumed by the condenser is proportional to the frequency, and therefore, if the wave of impressed voltage is greatly distorted, that is, contains considerable higher harmonics — especially harmonics of high order — the condenser may produce i uderable higher-frequency currents, and thus by distortion INDUCTION MOTOR 85 of the current wave lower the power-factor, so that in extreme cases the shunted condenser may actually lower the power- factor. However, with the usual commercial voltage wave shapes, this is rarely to be expected. In single-phase induction motors, the condenser may be used in a tertiary circuit, that is, a circuit located on the same member (usually the stator) as the primary circuit, but displaced in posi- LOW SPEEC INDUCTION MOTOR e„ = 50O Zo = .1 + .3i Yo=.02-.6j Z,«.1 + .3J Zt -.002S-.SJ p * M 1 1 7~ / -ML i. _SL m m i i I I 0 4 1 ) C i) ■ J u 1 Xl 1 o l 0 >.-. Fio. 32.— Load c tion therefrom, and energized by induction from the secondary. By locating the tertiary circuit in mutual induction also with the primary, it can be used for starting the single-phase motor, and is more fully discussed in Chapter V. A condenser may also be used in the secondary of the induction motor. That is, the secondary circuit is closed through a con- denser in each phase. As the current consumed by a condenser is proportional to the frequency, and the frequency in the secondary circuit varies, decreasing toward zero at synchronism, the cur- rent consumed by the condenser, and thus the secondary current of the motor tends toward zero when approaching synchronism, 86 ELECTRICAL APPARATUS and peculiar speed characteristics result herefrom in such a motor. At a certain slip, s, the condenser current just balances all the reactive lagging currents of the induction motor, resonance may thus be said to exist, and a very large current flows into the motor, and correspondingly large power is produced. Above this "resonance speed," however, the current and thus the power rapidly fall off, and so also below the resonance speed. It must be realized, however, that the frequency of the sec- ondary is the frequency of slip, and is very low at speed, thus a very great condenser capacity is required, far greater than would be sufficient for compensation by shunting the condenser across the primary terminals. In view of the low frequency and low voltage of the secondary circuit, the electrostatic condenser generally is at a disadvantage for this use, but the electrolytic condenser, that is, the polarization cell, appears better adapted. 56. Let then, in an induction motor, of impressed voltage, e0: Ya = g — jb — exciting admittance; Z» — H + Jx<> = primary self-inductive impe- dance; Z\ = Ti + jxi = secondary self-inductive im- pedance at full frequency; and let the secondary circuit be closed through a condenser of capacity reactance, at full frequency: Z* — U — j*h where r%, representing the energy loss in the condenser, usually is very small and can lie neglected in the electrostatic condenser, so that: Zt= - jxj. The inductive reactance, Xt, is proportional to the frequency, that is, the slip, s, and the capacity reactance, x:, inverse propor- tional thereto, and the total impedance of the secondary circuit, at slip, j*, thus is: Z--r, +;(»!, -»), (I) Ihus tlir- secondary current: ;, - " - < <«i - >=), (2) INDUCTION MOTOR 87 where: Oi = —i m o2 = (- - ?) m and: m = ri2 + (sxi + yj • (3) All the further calculations of the motor characteristics now are the same as in the straight induction motor. As instance is shown the low-speed motor, Fig. 20, of constants: e0 = 500; Y0 = 0.02 - 0.6 j ; Z0 = 0.1 + 0.3j; Zi = 0.1 + 0.3 j; with the secondary closed by a condenser of capacity impedance: Z, = - 0.012 j, thus giving: 0.04> Z' = 0.1 + 0.3j(s-^p) Fig. 33 shows the load curves of this motor with condenser in the secondary. As seen, power-factor and apparent effi- ciency are high at load, but fall off at light-load, being similar in character as with a commutating machine concatenated to the induction machine, or with the secondary excited by direct current, that is, with conversion of the induction into a synchro- nous motor. Interesting is the speed characteristic: at very light-load the speed drops off rapidly, but then remains nearly stationary over a wide range of load, at 10 per cent. slip. It may thus be said, that the motor tends to run at a nearly constant speed of 90 per cent, of synchronous speed. The apparent efficiency of this motor combination is plotted once more in Fig. 28, for comparison with those of the other motors, and marked by C. Different values of secondary capacity give different operating speeds of the motor: a lower capacity, that is, higher capacity 88 ELECTRICAL APPARATJJB •eactance, xt, gives a greater slip, s, that is, lower operating peed, and inversely, as was discussed in Chapter I. 67. It is interesting to compare, in Fig. 28, the various met hods >f secondary excitation of the induction motor, in their effect in niproving the power-factor and thus the apparent efficiency of v motor of high exciting current and thus low power-factor, >mli is a slow-speed motor. The apparent efficiency characteristics fall into three groups; — LOW SPEED INDUCTION MOTOR WITH CONDENSER IN SECONDARY CIRCUIT ea = 500 Zo-.t +.3i Y„=,02 -.6) Z, = .1 +.3i 2, = - .012) ASYNCHRONOUS \ J / 1 * — ~ / -T?