CHAPTER V SINGLE-PHASE INDUCTION MOTOR 60. As more fully discussed in the chapters on the single-phase induction motor, in " Theoretical Elements of Electrical Engineer- ing" and " Theory and Calculation of Alternating-current Phenomena," the single-phase induction motor has inherently, no torque at standstill, that is, when used without special device to produce such torque by converting the motor into an unsym- metrical ployphase motor, etc. The magnetic flux at standstill is a single-phase alternating flux of constant direction, and the line of polarization of the armature or secondary currents, that is, the resultant m.m.f. of the armature currents, coincides with the axis of magnetic flux impressed by the primary circuit. When revolving, however, even at low speeds, torque appears in the single-phase induction motor, due to the axis of armature polarization being shifted against the axis of primary impressed magnetic flux, by the rotation. That is, the armature currents, lagging behind the magnetic flux which induces them, reach their maximum later than the magnetic flux, thus at a time when their conductors have already moved a distance or an angle away from coincidence with the inducing magnetic flux. That is, if the armature currents lag ~ = 90° beyond the primary main flux, and reach their maximum 90° in time behind the magnetic flux, at the slip, s, and thus speed (1 — s), they reach their maxi- mum in the position (1 — s) ~ = 90 (1 — s) electrical degrees behind the direction of the main magnetic flux. A component of the armature currents then magnetizes in the direction at right angles (electrically) to the main magnetic flux, and the armature currents thus produce a quadrature magnetic flux, increasing from zero at standstill, to a maximum at synchronism, and approximately proportional to the quadrature component of the armature polarization, P: P sin (1 — s) • 93 ill ELECTRICAL APPARATUS The torque of the single-phase motor then is produced by the action of the quadrature flux on the energy currents induced by the main flux, and thus is proportional to the quadrature flux. At synchronism, the quadrature magnetic flux produced by the armature currents becomes equal to the main magnetic flux produced by the impressed single-phase voltage (approximately, in reality it is less by the impedance drop of the exciting current in the armature conductors) and the magnetic disposition of the single-phase induction motor thus becomes at synchronism iden- tical with that of the polyphase induction motor, and approxi- mately so near synchronism. The magnetic field of the single-phase induction motor thus may be said to change from a single-phase alternating field at standstill, over an unsymmetrical rotating field at intermediate speeds, to a uniformly rotating field at full speed. At synchronism, the total volt-ampere excitation of the single- phase motor thus is the same as in the polyphase motor at the same induced voltage, and decreases to half this value at stand- still, where only one of the two quadrature components of magnetic flux exists. The primary impedance of the motor is that of the circuits used. The secondary impedance varies from the joint impedance of all phases, at synchronism, to twice this value at standstill, since at synchronism all the secondary circuits correspond to the one primary circuit, while at stand- still only their component parallel with the primary circuit corres ponds. 61. Hereby the single-phase motor constants are derived from the constants of the same motor structure as polyphase motor. Let, in a polyphase motor: Y = g — jb = primary exciting admittance; 2o = To + Jin = primary self-inductive im- pedance; Z\ = fi + jxi = secondary self-inductive im- pedance (reduced to the pri- mary by the ratio of turns, in the usual manner}; the characteristic constant of the motor then is: & - y (z„ + zx). (i) The total, or resultant admittance respectively impedance of SINGLE-PHASE INDUCTION MOTOR 95 the motor, that is, the joint admittance respectively impedance of all the phases, then is: In a three-phase motor: 7a = 3 Y, Zo° = H Z,, , (2) Zi° = H Zv In a quarter-phase motor: Y° - 2 Y, ] Zo° - H Zo, (3) Z,° = M Z,. 1 In the same motor, as single-phase motor, it is then: at syn- chronism: 8 = 0: Y' = F°, Z'0 = 2 Z0°, | (4) Z'! = Zx°, hence the characteristic constant : t>'0 - r (z'o + z',) - r°(2Z0,, + Z10)) (5) at standstill : * = 1 : r = H y«, Z'o = 2 Zo°, Z'i = 2 Z,«, (6) hence, the characteristic constant: t>', = Y° (Zo° + Zj0) (7) approximately, that is, assuming linear variation of the constants with the speed or slip, it is then: at slip, s: Y' = F°(l -|), Lt o = 2 #0j Z'i = Zt» (1 + »). J This gives, in a three-phase motor: F'=3F(1- *), Z'o - % z\ Z.x = i + i Zl. (8) (9) 96 ELECTRICAL APPARATUS In 8 quarter-phase motor: Y' =27(1-3, Z 0 = Zoj (10) Thus the characteristic constant, #', of the single-phase motor is higher, that is, the motor inferior in its performance than the polyphase motor; but the quarter-phase motor makes just as good — or poor — a single-phase motor as the three-phase motor. 62. The calculation of the performance curves of the single- phase motor from its constants, then, is the same as that of the polyphase motor, except that : In the expression of torque and of power, the term (1 — *) is added, which results from the decreasing quadrature flux, and it thus is: Torque: T = T(l -*) = (1 - *) a*-. (11) Power: P* =P(1 -*) «(l-*)*aif*. (12) However, these expressions are approximate only, as they assume a variation of the quadrature flux proportional to the speed. 63. As the single-phase induction motor is not inherently self-starting, starting devices are required. Such are: (a) Mechanical starting. As in starting a single-phase induction motor it is not neces- sary, as in a synchronous motor, to bring it up to full speed, but the motor begins to develop appreciable torque already at low speed, it is quite feasible to start small induction motors by hand, by a pull on the belt, etc.. especially at light-load and if«of high- resistance armature. (b) By converting the motor in starting into a shunt or series motor. This has the great objection of requiring a commutator, and a cwuttutating-machine rotor winding instead of the common iftd«c*iQ«i-n*otor squirrel-cage winding. Also, as series motor, tl* KthiKty exists in the starting connection, of running away; SINGLE-PHASE INDUCTION MOTOR 97 as shunt motor, sparking is still more severe. Thus this method is used to a limited extent only. (c) By shifting the axis of armature or secondary polarization against the axis of inducing magnetism. This requires a secondary system, which is electrically un- symmetrical with regards to the primary system, and thus, since the secondary is movable with regards to the primary, requires means of changing the secondary circuit, that is, commutator brushes short-circuiting secondary coils in the position of effective torque, and open-circuiting them in the position of opposing torque. Thus this method leads to the various forms of repulsion motors, of series and of shunt characteristic. It has the serious objection of requiring a commutator and a corresponding armature winding; though the limitation is not quite as great as with the series or shunt motor, since in the re- pulsion motors the armature current is an induced secondary current, and the armature thus independent of the primary system regards current, voltage and number of turns. (d) By shifting the axis of magnetism, that is producing a magnetic flux displaced in phase and in position from that in- ducing the armature currents, in other words, a quadrature magnetic flux, such as at speed is being produced by the rotation. This method does not impose any limitation on stator and rotor design, requires no commutator and thus is the method almost universally employed. It thus may be considered somewhat more in detail. The infinite variety of arrangements proposed for producing a quadrature or starting flux can be grouped into three classes: A. Phase-splitting Devices. — The primary system of the single- phase induction motor is composed of two or more circuits displaced from each other in position around the armature circumference, and combined with impedances of different in- ductance factors so as to produce a phase displacement between them. The motor circuits may be connected in series, and shunted by the impedance, or they may be connected in shunt with each other, but in series with their respective impedance, or they may be connected with each other by transformation, etc. B. Inductive Devices. — The motor is excited by two or more circuits which are in inductive relation with each other so as to produce a phase displacement. 98 ELECTRICAL APPARATUS This inductive relation may be established outside of the motor by an external phase-splitting device, or may take place in the motor proper. C. Monocyclic Devices. — An essentially reactive quadrature voltage is produced outside of the motor, and used to energize a cross-magnetic circuit in the motor, either directly through a separate motor coil, or after combination with the main voltage to a system of voltages of approximate three-phase or quarter- phase relation. D. Phase Converter. — By a separate external phase converter— usually of the induction-machine type — the single-phase supply is converted into a polyphase system. Such phase converter niay be connected in shunt to the motor, or may be connected in series thereto. This arrangement requires an auxiliary machine, running idle, however. It therefore is less convenient, but has the advantage of being capable of giving full polyphase torque and output to the motor, and thus would be specially suitable for railroading. 64. If: *o = main magnetic flux of single-phase motor, that is, magnetic flux produced by the impressed single-phase voltage, and 4> = auxiliary magnetic flux produced by starting device, and if u> = space angle between the two fluxes, in electrical degrees, and * = time angle between the two fluxes, then the torque of the motor is proportional to: T - a**0 sin u sin tf>) (13) in the same motor as polyphase motor, with the magnetic flux, #o, the torque is: Tn = a*,1; (14) thus the torque ratio of the starting device is; . T * . I = y = ^- am w sin , or, if: (15) = quadrature flux produced by the startiug device, that is, SINGLE-PHASE INDUCTION MOTOR 99 component of the auxiliary flux, in quadrature to the main flux, $o, in time and in space, it is: Single-phase motor starting torque: T = afc'So, (16) and starting-torque ratio: t - £-• (17) As the magnetic fluxes are proportional to the impressed vol- tages, in coils having the same number of turns, it is: starting torque of single-phase induction motor: T = be0e sin a> sin 0 = 6e0e', (18) and, starting-torque ratio: < = -8inw sin 6 e0 e_ where: (19) eo = impressed single-phase voltage, e = voltage impressed upon the auxiliary or starting winding, reduced to the same number of turns as the main winding, and e' = quadrature component, in time and in space, of this voltage, e, and the comparison is made with the torque of a quarter-phase motor of impressed voltage, eo, and the same number of turns. Or, if by phase-splitting, monocyclic device, etc., two voltages, ei and e2, are impressed upon the two windings of a single-phase induction motor, it is: Starting torque : T = be\e% sin a> sin (20) and, starting-torque ratio: t = -^y sin co sin 0, (21) eo where eo is the voltage impressed upon a quarter-phase motor, with which the single-phase motor torque is compared, and all 100 ELECTRICAL APPARATUS these voltages, ej, e^, e0, are reduced to the same number of turns of the circuits, as customary. If then : Q = volt-amperes input of the single-phase motor with starting device, and Qo = volt-amperes input of the same motor with polyphase supply, , = i (22) is the volt-ampere ratio, and thus: v = - (23) q is the ratio of the apparent starting-torque efficiency of the single-phase motor with starting device, to that of the same motor as polyphase motor, v may thus be called the apparent torque efficiency of the single-phase motor-starting device. In the same manner the apparent power efficiency of the start- ing device would result by using the power input instead of the volt-ampere input. 66. With a starting device producing a quadrature voltage, e', t = e' (24) is the ratio of the quadrature voltage to the main voltage, and also is the starting-torque ratio. The quadrature flux: e' = te0 (25) requires an exciting current, equal to t times that of the main voltage in the motor without starting device, the exciting current at standstill is: e0i'= 2 and in the motor with starting device giving voltage ratio, /, the total exciting current at standstill thus is: 'a" U + O SINGLE-PHASE IXDUCTION MOTOR 101 and thus, the exciting admittance: r' = y2°(i + 0; (27) in the same manner, the secondary impedance at standstill is: ZS = W (28) and thus: in the single-phase induction motor with starting device pro- ducing at standstill the ratio of quadrature voltage to main voltage : t = eo the constants are, at slip, s: Z'Q = 2 Zo°, Zi __ * ' s 7 o 1 + 8t (29) However, these expressions (29) are approximate only, as they assume linear variation with s, and furthermore, they apply only under the condition, that the effect of the starting device does not vary with the speed of the motor, that is, that the voltage ratio, t y does not depend on the effective impedance of the motor. This is the case only with a few starting devices, while many depend upon the effective impedance of the motor, and thus with the great change of the effective impedance of the motor with increasing speed, the conditions entirely change, so that no general equations can be given for the motor constants. 66. Equations (18) to (23) permit a simple calculation of the starting torque, torque ratio and torque efficiency of the single- phase induction motor with starting device, by comparison with the same motor as polyphase motor, by means of the calculation of the voltages, e'y eh e2, etc., and this calculation is simply that of a compound alternating-current circuit, containing the induc- tion motor as an effective impedance. That is, since the only determining factor in the starting torque is the voltage impressed upon the motor, the internal reactions of the motor do not come into consideration, but the motor merely acts as an effective impedance. Or in other words, the consideration of the internal 102 ELECTRICAL APPARATUS reaction of the motor is eliminated by the comparison with the polyphase motor. In calculating the effective impedance of the motor at stand- still, we consider the same as an alternating-current transformer, and use the equivalent circuit of the transformer, as discussed in Chapter XVII of "Theory and Calculation of Alternating- current Phenomena." That is, the induction motor is con- sidered as two impedances, Za and Z(, connected in series to the -PFL jTRRT it of the induction n impressed voltage, with a shunt of the admittance, Ya, between the two impedances, as shown in Fig. 35. The effective impedance then is: approximately, this is: = ZQ + Zx. (:; + j sin 0) = ^^z" ' ^ Denoting the absolute values of the voltages and currents by small letters, it is: T = beiei sin ; (36) in the motor as quarter-phase motor, with voltage, e0} impressed per circuit, it is: To = 6e02, (37) hence, the torque ratio: t = ei6?8m. (38) The current per circuit, in the machine as quarter-phase motor, is: to = -> (39) z hence the volt-amperes: Qo = 2e0*'o, (40) while the volt-amperes of the single-phase motor, inclusive start- ing impedances, are: Q = eoi, (41) thus: and, the apparent torque efficiency of the starting device: q CqIZ 68. As an instance, consider the motor of effective impedance: Z =r+jx = 0.1 +0.3J, thus: z = 0.316, SINGLE-PHASE INDUCTION MOTOR 105 and assume, as the simplest case, a resistance, a = 0.3, inserted in series to the one motor circuit. That is : Zx = 0, ' (44) Zo = a. It is then: (32):/= <:-. =„, f« -; h= e° e" r+jx 0.1 + 0.3 j " r + a + jx 0.4 + 0.3 J = e0(l-3j), = eo (1.6 - 1.2 j); (33) : / - «0 (2.6 - 4.2 j), i = 4.94 e0; r+jx 0.1 + 0.3 j (34): E> = e0, U = *o f + a +- = e0 Q 4 + Q 3 , ei = eo, e2 = 0.632 eo) /«^x / , . • • ,\ r+jx 0.1 + 0.3.7 (35): m (cos * + j sin *) = r + -+ .f = Q 4 + Mj = 0.52 + 0.36 j, tan * = 0.52' sin <^> = 0.57; (38): t = 0.36; (43) : v = 0.46. Thus this arrangement gives 46 per cent., or nearly half as much starting torque per volt-ampere taken from the supply circuit, as the motor would give as polyphase motor. However, as polyphase motor with low-resistance secondary, the starting torque per volt-ampere input is low. With a high-resistance motor armature, which on polyphase supply gives a good apparent starting-torque efficiency, v would be much lower, due to the lower angle, . In this case, however, a reactance, +ja, would give fairly good starting-torque efficiency . In the same manner the effect of reactance or capacity inserted into one of the two motor coils can be calculated. As instances are given, in Fig. 37, the apparent torque efficiency, v, of the single-phase induction-motor starting device consisting of the insertion, in one of the two parallel motor circuits, of various amounts of reactance, inductive or positive, and capacity 166 ELECTRICAL APPARATUS or negative, for a low secondary resistance motor of impedance: Z - 0.1 +0.3; and a high resistance armature, of the motor impedance: Z = 0.3 + 0.1 j resistance inserted into the one motor circuit, has the same effect .ft .r. z= 1+1 n 1 + 1 o- +,» + S 4 i + 1 + l | 1 I * 1 c PAC TV +.* IN DUC AHC E If ESIS TAN 2L) -Kfi / +.S +1( + i? / +1.4 H ( + 1 s •E*1 + 1 m Pin in t inv 6 the circ 90° anc A 37.— Apparent starting-torque eflutenoei of phase-splitting de parallel cumieition uf motor cireuits. lie first motor, as positive reactance in the second motor, rsely. K Higher values of starting-torque efficiency are aecurec use of capacity in the one, and inductance in the other m nit. It is obvious that by resistance and inductance al phase displacement between the two component curre thus true quarter-phase relation, can not be reached. s resistance consumes energy, the use of resistance is justi and by tor ne, its;, Bed SINGLE-PHASE INDUCTION MOTOR 107 only due to its simplicity and cheapness, where moderate start- ing torques are sufficient, and thus the starting-torque efficiency less important. For producing high starting torque with high starting-torque efficiency, thus, only capacity and inductance would come into consideration. Assume, then, that the one impedance is a capacity: X2 = — fc, or: Z2 = — jk, (45) while the other, xi, may be an inductance or also a capacity, what- ever may be desired: Zi = +jx1}. (46) where X\ is negative for a capacity. It is, then : (35) : m (cos + 3 sin ) = r + j (xi + x) [r2 - (xi + x)(k - x)] + jrxik ( ?. r -j(k - x) ' r2+ (fc- x)* " K } True quadrature relation of the voltages, e\ and e%, or angle, = s' requires: cos = 0, thus: (xx + x) (k - x) = r2 (48) and the two voltages, e\ and 62, are equal, that is, a true quarter- phase system of voltages is produced, if in (34): [Z + ZJ = [Z + Z2], where the [ ] denote the absolute values. This gives: r* + (*i + xY = r* + (k - x)\ or: X\ + x = k — x, (49) hence, by (48) : Xi + x = k — x = r, k = r + x>\ (50) Xt = r — x. t Thus, if x > r, or in a low-resistance motor, the second reactance, Xif also must be a capacity. 108 ELECTRICAL APPARATUS 70. Thus, let: in a low-resistance motor: Z = r+jx = 0.1 + 0.3.?, k = 0.4, xi = - 0.2, Z2 0.4 i, Zx = -0.2j, that is, both reactances are capacities. (34) : ex = e2 = 2.23 e0, * = 5, that is, the torque is five times as great as on true quarter-phase supply. 41 0.1 + o.i / i2 ai-o-ij' / = 10 e0 = i, that is, non-inductive, or unity power-factor. to = y = 3.166o, g = 1.58, v = 3.16, that is, the apparent starting-torque efficiency, or starting torque per volt-ampere input, of the single-phase induction motor with starting devices consisting of two capacities giving a true quarter- phase system, is 3.16 as high as that of the same motor on a quarter-phase voltage supply, and the circuit is non-inductive in starting, while on quarter-phase supply, it has the power- factor 31.6 per cent, in starting. In a high-resistance motor: Z = 0.3 + 0.1 i, it is: k = 0.4, xx = 0.2, Z2 = -0.4j, Z2 = +0.2 j, that is, the one reactance is a capacity, the other an inductance. ei = e2 = 0.743 e0) t = 0.555, i = 3.33 6o, to =3.16 eo, q = 0.527, v = 1.055, SINGLE-PHASE INDUCTION MOTOR 109 that is, the starting-torque efficiency is a little higher than with quarter-phase supply. In other words: This high-resistance motor gives 5.5 per cent, more torque per volt-ampere input, with unity power-factor, on single-phase supply, than it gives on quarter-phase supply with 95 per cent, power-factor. The value found for the low-resistance motor, t = 5, is how- ever not feasible, as it gives: ex = 62 = 2.23 e0, and in a quarter- phase motor designed for impressed voltage, e0, the impressed voltage, 2.23 eo, would be far above saturation. Thus the motor would have to be operated at lower supply voltage single-phase, and then give lower t, though the same value of v = 3.16. At e\ = ej = e0, the impressed voltage of the single-phase circuit would be about 45 per cent, of e0, and then it would be: t = 1. Thus, in the low-resistance motor, it would be preferable to operate the two motor circuits in series, but shunted by the two different capacities producing true quarter-phase relation. Series Connection 71. The calculation of the single-phase starting of a motor with two coils in quadrature position, shunted by two impedances Fia. 38. — Diagram of phase-splitting device with series connection of motor circuits of different power-factor, as shown diagrammatically in Fig. 38, can be carried out in the same way as that of parallel connection, except that it is more convenient in series connection to use the term " admittance" instead of impedance. That is, let the effective admittance per motor coil equal: y = v = (J - A 110 ELECTRICAL APPARATUS and the two motor coils be shunted respectively by the admit- tances: Yi = gi - jbu Y2 = 02 — j&2, it is then: (52) / = 1 6° =— , (53) ir~ + 2 Y +Yi ' Y +Y the current consumed by the motor, and : & = Y~+Ti and ^2 = Y + Y2' (54) the voltages across the two motor circuits. The phase difference between E\ and E2 thus is given by Y + Y* m (cos 4> + j sin 4>) = y^Y ' ^ and herefrom follows t, q and v. As instance consider a motor of effective admittance per cir- cuit: Y = g-jb = l-3j, with the two circuits connected in series between single-phase mains of voltage, e<>, and one circuit shunted by a non-inductive resistance of conductance, gim What value of g\ gives maximum starting torque, and what is this torque? It is: (53}' ' " 1 , __J_ " 2g + gx - 2j6 ~ (5b} ff + flfi — jb 0 - J& (54). *-— -—^ *__.___, (57) (55) : m (cos * + j sin 0) - *-±-«L^ = [^+_^)^Hl^; hence: gi& tan 0 = pfa + gO + fc2 sin * = , g'_. — - (58) VViW + [flf (ff + 91) + 6s]1 SIXGLE-PHASE IS'DUCTIOX MOTOR 111 and thus: '2» + * _ 9*> 1(2? -f g,)* gib 9i)* + *& + 46*] and for thus: maximum. 1: • St = 2 V»f +6* = 2 y = 6.32, or, substituting back: (59): t = ,, . „ = 0.18. (59) (60) (61) ■» \ir ~r yj As in single-phase operation, the voltage, e0, is impressed upon the two quadrature coils in series, each coil receives only about —v=. Comparing then the single-phase starting torque with that of a quarter-phase motor of impressed voltage, —.-* it is: t = 0.36. The reader is advised to study the possibilities of capacity and reactance (inductive or capacity) shunting the two motor coils, the values giving maximum torque, those giving true quarter-phase relation, and the torque and apparent torque efficiencies secured thereby. B. INDUCTIVE DEVICES External Inductive Devices 72. Inductively divided circuit: in its simplest form, as shown diagrammatically in Fig. 39, the motor contains two circuits at right angles, of the same admittance. The one circuit (1) is in series with the one, the other (2) with the other of two coils wound on the same magnetic circuit, M. By proportioning the number of turns, n\ and n2, of the two coils, which thus are interlinked inductively with each other on the external magnetic circuit, M, a considerable phase displacement 112 ELECTRICAL APPARATUS between the motor coils, and thus starting torque can be pro- duced, especially with a high-resistance armature, that is, a motor with starting rheostat. A full discussion and calculation of this device is contained in the paper on the " Single-phase Induction Motor," page 63, A. I. E. E. Transactions, 1898. g^flgg -*&>°* ^MATURE FlO. 39. -External inductive device. Fiu. 40. — Diagram of shading coil. Internal Inductive Devices The exciting system of the motor consists of a stationary pri- mary coil and a stationary secondary coil, short-circuited upon itself (or closed through an impedance), both acting upon the revolving secondary. The stationary secondary can either cover a part of the pole face excited by the primary coil, and is then called a "shading coil," or it has the same pitch as the primary, but is angularly displaced therefrom in space, by less than 90° (usually 45° or 60°), and then has been called accelerating coil. The shading coil, as shown diagrammatically in Fig. 40, is the simplest of all the single-phase induction motor-starting devices, and therefore very extensively used, though it gives only a small starting torque, and that at a low apparent starting- torque efficiency. It is almost exclusively used in very small motors which require little starting torque, such as fan motors, and thus industrially constitutes the most important single- phase induction motor-starting device. 73. Let, all the quantities being reduced to the primary num- ber of turns and frequency, as customary in induction machines: Z0 = r<> + jxo = primary self-inductive impedance, y = g — jb = primary exciting admittance of unshaded poles (assuming total pole unshaded), SINGLE-PHASE INDUCTION MOTOR 113 Y' = g' — jb' = primary exciting admittance of shaded poles (assuming total pole shaded). If the reluctivity of the shaded portion of the pole is the same as that of the unshaded, then Y' = Y; in general, if b = ratio of reluctivity of shaded to unshaded portion of pole, Y' = bY, b either = 1, or, sometimes, b > 1, if the air gap under the shaded portion of the pole is made larger than that under the unshaded portion. Yi = gi — jbi = self-inductive admittance of the revolving secondary or armature, Y* = 02 — jb2 = self-inductive admittance of the stationary secondary or shading coil, inclusive its exter- nal circuit, where such exists. Z0, Yi and Y2 thus refer to the self-inductive impedances, in which the energy component is due to effective resistance, and Y and Y' refer to the mutual inductive impedances, in which the energy component is due to hysteresis and eddy currents. a = shaded portion of pqje, as fraction of total pole; thus (1 — a) = unshaded portion of pole. If: eo = impressed single-phase voltage, $i = voltage induced by flux in unshaded portion of pole, $2 = voltage induced by flux in shaded portion of pole, /o = primary current, it is then : e0 = #i + #2 + Zo/o. (62) The secondary current in the armature under the unshaded portion of the pole is: /i = #iVY (03) The primary exciting current of the unshaded portion of the pole : /„„ = flJa, (64) thus: h = ft + f«. - & { r, + , } „!• («5) 114 ELECTRICAL APPARATUS The secondary current under the shaded portion of the pole is: /'i = frYi. (66) The current in the shading coil is: h - #2^2. (67) The primary exciting current of the shaded portion of the pole is: / 00 = EjbY thus: /o = /'i + ho + u = & Yi + -y + Yt (68) (69) from (65) and (69) follows: W* y, + - y + Yt a Fi + = m (cos + j sin 0), (70) I- a and this gives the angle, 4>> of phase displacement between the two component voltages, $1 and $2- If, as usual, 6=1, and if fc = 0.5, that is, half the pole.is shaded, it is: Ei wBYl + 2Y+Y* $2 (71) YS+2Y 74. Assuming now, as first approximation, Z0 = 0, that is, neglecting the impedance drop in the single-phase primary coil — which obviously has no influence on the phase difference between the component voltages, and the ratio of their values, that is, on the approximation of the devices to polyphase relation — then it is: Pi + Pt = e0] (72) thus, from (70) : Pi = e0 Yt + ^-Y+Y* a 2Yl + Y(l + vl-a) + Yi Yt + Et = e0 - 1 - a 2Yl+Y(l + T±- ) +Yt' \a 1 — at (73) SINGLE-PHASE INDUCTION MOTOR 115 or, for: b = 1; a = 0.5; JK, + 2F + F, ** 2 F,"+ 4 F +~F,' „ F, + 2 F ** 2Fl + 4F+F*' (74) and the primary current, or single-phase supply current is, by substituting (73) into (65) : Y \ t„ . b /o = Co (75) or, for: b = \;a = 0.5: . _c (F, + 2 YHY, + 2 F - /0 " e° 2 F! + 4 F + F, + F2) (76) and herefrom follows, by reducing to absolute values, the torque, torque ratio, volt-ampere input, apparent torque efficiency, etc. Or, denoting: y' + T--a = r°' Y + ^Y+Yt=Y', (77) it is: (70): I = ~- = m (cos + j sin ) ; / 0 (78) (73): (75): E, = /.= -v.* coF' F° + F' e0F° ft + F'» ] c0F0l^_ •' F° + F'' r = Aetfi sin #, Q = e0io; (79) (80) e0 and for a quarter-phase motor, with voltage —y impressed per V2 116 ELECTRICAL APPARATUS circuit, neglecting the primary impedance, z0, to be comparable with the shaded-coil single-phase motor, it is: ;«= e° AY+Yt), V2 q0 = 2ef* = eovr + y./, V2 T. = A *', thus fo\/2 to 9 = i\72 . 2 6i62 . . *- eo2 sm«, t; = • (7 75. As instances are given in the following table the compo- nent voltages, ei and 62, the phase angle, 4>} between them, the primary current, i0| the torque ratio, t, and the apparent starting- torque efficiency, vy for the shaded-pole motor with the constants: Impressed voltage: e0 = 100; Primary exciting admittance: Y = 0.001 — 0.01 j. 6 = 1, that is, uniform air gap. a = 0.5, that is, half the pole is shaded. And for the three motor armatures : Low resistance: Yx = 0.01 — 0.03 j, Medium resistance: Yx = 0.02 — 0.02 j, High resistance: F, = 0.03 - 0.01 j; and for the three kinds of shading coils: Low resistance: Y2 = 0.01 — 0.03 j, Medium resistance: V2 = 0.02 — 0.02 j, High resistance: ^2 = 0.03 - 0.01 j. As seen from this table, the phase angle, 0, and thus the start- ing torque, t} are greatest with the combination of low-resistance armature and high-resistance shading coil, and of high-resistance armature with low-resistance shading coil; but in the first case the torque is in opposite direction — accelerating coil — from what SIXGLE-PHASE IXDUCTIOX MOTOR 117 it is in the second case — lagging coil. In either case, the torque efficiency is low, that is, the device is not suitable to produce high starting-torque efficiencies, but its foremost advantage is the extreme simplicity. The voltage due to the shaded portion of the pole, €•, is less than that due to the unshaded portion, *i, and thus a somewhat higher torque may be produced by shading more than half of the pole: a > 0.5. A larger air gap: b > 1, under the shaded portion of the pole, or an external non-inductive resistance inserted into the shad- ing coil, under certain conditions increases the torque somewhat — at a sacrifice of power-factor — particularly with high-resistance armature and low-resistance shading coil. Co = 100 volts; a = 0.5; b = 1; Y - 0.001 - 0.01 j. Yii Yti er. e*: : i0: /,: v: X 10~2 X 10~* per cent, per cent. . 1 - 3j 1 - Sj 38.3 61.8 +1.9 1.97 + 1.5!) +4.07 2-2./40.3 60.2 +11.0 2.07 +9.28 +23. (K) . 3 - \j 42.0 59.8 +21.5 2.17 +18.36 +43.70 2 -2>1 -3J 37.2 62.9 -4.3 1.70 -3.52 -.9.65 2-2J38.5 61.7 +6.2 1.76 +5.12 +13.60 3 -I; 39.2 62.0 +17.3 1.80 +14.44 +37.40 . 3 - \j 1 - 3j 37.6 63.0 -11.9 1.66 -9,76 -25.80 2 - 2j 37.8 62.5 - 0.8 1.66 - 0.66 ' - 1.75 3 - Ij 37.4 63.0 +10.3 1.64 +8.44 +22.60 Monocyclic Starting Device 76. The monocyclic starting device consists in producing ex- ternally to the motor a system of polyphase voltages with single- phase flow of energy, and impressing it upon the motor, which is wound as polyphase motor. If across the single-phase mains of voltage, e, two impedances of different inductance factors, Z\ and Z2, are connected in series, as shown diagrammatically in Fig. 41, the two voltages, I$y and #2, across these two impedances are displaced in phase from each other, thus forming with the main voltage a voltage triangle. The altitude of this triangle, or the voltage, #0, between the com- 118 ELECTRICAL APPARATUS mon connection of the two impedances, and a point inside of the main voltage, e (its middle, if the two impedances are equal), is a voltage in quadrature with the main voltage, and is a teazer voltage or quadrature voltage of the monocyclic system, e, E\, Es, that is, it is of limited energy and drops if power is taken off from it. (See Chapter XIV.) Let then, in a three-phase wound motor, oper- ated single-phase with monocyclic starting device, and shown diagrammatically in Fig. 42: ■ — voltage impressed 1 jet. ween single-phase lines, / = current in single-phase lines, Y = effective admittance per motor circuit, I'i, Ei and I',, and Y2, $ 2 and f'2 = admittance, voltage and current respectively, in the two impedances of the mono- cyclic starting device, Fio. 42. — Three-phase motor with monocyclic starting device. /i, /] and /a = currents in the three motor circuits. E ,, and /« = voltage and current of the quadrature circuit from the common connection of the two impedances, to the motor. SINGLE-PHASE INDUCTION MOTOR 119 It is then, counting the voltages and currents in the direction indicated by the arrows of Fig. 42: substituting: gives: thus: /o = J'i — I't = /* — h'y /'* = EtYt, I* = V>Y, h = W, l$\Yi — jpjl't = (#* ~ jFO F, Ai Yt + Y (81) (82) Zt Yt+Y = m (cos 4> + j sin 0). (83) This gives the phase angle, , between the voltages, #1 and #», of the monocyclic triangle. Since: it is, by (83) : Yt+Y (84) F, + F, + 2 F Ki + r Ki+ F, + 2F' (85) and the quadrature voltage: c Fi - F, ~2F,+ F, + 2F (86) and the total current input into the motor, inclusive starting device: / = /'. + h + h = ViYx + £,F + eY ,(Yl+Y)(Yi+ F) = e + Y Yi+Y, + 2Y = 0 Y\Yt + 2 F(K, +_F2) + 37* " "7!+ FS_+"2F" (87) As with the balanced three-phase motor, the quadrature com- ponent of voltage numerically is « Vo, it is, when denoting by: 120 ELECTRICAL APPARATUS Ee' the numerical value of the imaginary term of #0; the torque ratio is: <=2B;.'- (^ The volt-arnpere ratio is: 8 = 3i. Si, rims I he apparent starting-torque efficiency: (89) (90) 77. Three eases have become of special importance: (a) The resistance-reactance monocyclic starting device; where one of the two impedances, Z, and Zt, is a resistance, the other an inductance. This is the simplest and cheapest arrangement, gives good starting torque, though a fairly high current consump- tion and therefore low starting-torque efficiency, and is therefore very extensively used for starling single-phase induction motors. After starting, the monocyclic device is cut out and the power consumption due to the resistance, and depreciation of the power- factor due to the inductance, thereby avoided. This device is discussed on page 333 of "Theoretical Elements of Electrical Engineering" and page 253 of "Theory and Calcu- lation of Alternating-current Phenomena." (fe) The "condenser in the tertiary circuit," which may be considered as a monocyclic starting device, in which one of the two impedances is a capacity, the other one is infinity. The capacity usually is made so as to approximately balance the mag- netizing current of the motor, is left in circuit after starting, as it does not interfere with the operation, does not consume power, and compensates for the lagging current of the motor, so that the motor has practically unity power-factor for all loads. This motor gives a moderate starting torque, but with very good start- ing-torque efficiency, and therefore is the most satisfactory single- phase induction motor, where very high starting torque is not needed. It was extensively used some years ago, but went out of use due to the trouble with the condensers of these early days, and it is therefore again coming into use, with the development of the last years, of a satisfactory condenser. (92) SINGLE-PHASE INDUCTION MOTOR 121 The condenser motor is discussed on page 249 of " Theory and Calculation of Alternating-current Phenomena.,, (c) The condenser-inductance monocyclic starting device. By suitable values of capacity and inductance, a balanced three- phase triangle can be produced, and thereby a starting torque equal to that of the motor on three-phase voltage supply, with an apparent starting-torque efficiency superior to that of the three-phase motor. Assuming thus: 1\ = +jbi = capacity, 1 . }r2 = — j62 = inductance, J Y = g - jb. If the voltage triangle, e, Eu #2, is a balanced three-phase tri- angle, it is : tfi«£(l -jV3), Substituting (91) and (92) into (83), and expanding gives: (6, - 6i + 26) a/3 - j (62 + 6i - 2g y/S) = 0; thus : 62 - 6, + 2 b = 0, 62 + 61 - 2(7 V3 = 0; hence : bi = gy/S + b, \ 62 = gy/3 ~ b) thus, if: b > g V3, the second reactance, Z2, must be a capacity also; if b= |(#2- Ex) =|V3; thus: t = 1, as was to be expected, /s = e(g - jb), is = e vV + b2 = 3.16 6; it is, however, by (87) : I = e(3g-jb); thus: i = 4.243 e, i = 9.06 e, and by (89) : q = 0.448, g = 0.956, thus: v = 2.232, v = 1.046. Further discussion of the various single-phase induction motor- starting devices, and also a discussion of the acceleration of the motor with the starting device, and the interference or non-inter- ference of the starting device with the quadrature flux and thus torque produced in the motor by the rotation of the armature, is given in a paper on the "Single-phase Induction Motor," A. I. E. E. Transactions, 1898, page 35, and a supplementary paper on "Notes on Single-phase Induction Motors," A. I. E. E. Trans- actions, 1900, page 25.