in. Single -phase Induction Motor •1. INTRODUCTION 146. In the polyphase motor a number of secondary coils displaced in position from each other are acted upon by a num- ber of primary coils displaced in position and excited by e.m.fs. displaced in phase from each other by the same angle as the dis- placement of position of the coils. In the single-phase induction motor a system of secondary circuits is acted upon by one primary coil (or system of primary coils connected in series or in parallel) excited by a single alter- nating current. A number of secondary circuits displaced in position must be used so as to offer to the primary circuit a short-circuited sec- ondary in any position of the armature. If only one secondary coil is used, the motor is a synchronous induction motor and belongs to the class of reaction machines. A single-phase induction motor will not start from rest, but when started in either direction will accelerate with increasing torque and approach synchronism. When running at or very near synchronism, the magnetic field of the single-phase induction motor is practically identical with that of a polyphase motor, that is, can be represented by the theory of the rotating field. Thus, in a turn wound under angle r to the primary winding of the single-phase induction motor, at synchronism an e.m.f. is generated equal to that generated in a turn of the primary winding, but differing therefrom by angle 6 = T in time phase. In a polyphase motor the magnetic flux in any direction is due to the resultant m.m.f. of primary and of secondary currents, in the same way as in a transformer. The same is the case in the direction of the axis of the exciting coil of the single-phase induc- tion motor. In the direction at right angles to the axis of the exciting coil, however, the magnetic flux is due to the m.m.f. of INDUCTION MACHINES 327 the secondary currents alone, no primary e.m.f. acting in this direction. Consequently, in the polyphase motor running synchronously, that is, doing no work whatever, the secondary becomes current- less, and the primary current is the exciting current of the motor only. In the single-phase induction motor, even when running light, the secondary still carries the exciting current of the mag- netic flux in quadrature with the axis of the primary exciting coil. Since, this flux has essentially the same intensity as the flux in the direction of the axis of the primary exciting coil, the current in the armature of the single-phase induction motor run- ning light, and therefore also the primary current corresponding thereto, has the same m.m.f., that is, the same intensity, as the primary exciting current, and the total primary current of the single-phase induction motor running light is thus twice the exciting current, that is, it is the exciting current of the main magnetic flux plus the current producing in the secondary the exciting current of the cross magnetic flux. In reality it is slightly less, especially in small motors, due to the drop of voltage in the self-inductive impedance and the drop of quadrature mag- netic flux below the impressed primary magnetic flux caused thereby. In the secondary at synchronism this secondary exciting current is a current of twice the primary frequency; at any other speed it is of a frequency equal to speed (in cycles) plus synchronism. Thus, if in a quarter-phase motor running light one phase is open-circuited, the current in the other phase doubles. If in the three-phase motor two phases are open-circuited, the current in the third phase trebles, since the resultant m.m.f. of a three- phase machine is 1.5 times that of one phase. In consequence thereof, the total volt-ampere input of the motor remains the same and at the same magnetic density, or the same impressed e.m.f., all induction motors, single-phase as well as polyphase, consume approximately the same volt-ampere input, and the same power input for excitation, and give the same distribution of magnetic flux. 146. Since the maximum . output of a single-phase motor at the same impressed e.m.f. is considerably less than that of a poly- phase motor, it follows therefrom that the relative exciting cur- rent in the single-phase motor must be larger. The cause of this cross magnetization in the single-phase indue- 328 ELEMENTS OF ELECTRICAL ENGINEERING tion motor near synchronism is that the secondary armature currents lag 90 deg. behind the magnetism, and are carried by the synchronous rotation 90 deg. in space before reaching their maximum, thus giving the same magnetic effect as a quarter- phase e.m.f. impressed upon the primary system in quadrature position with the main coil. Hence they can be eliminated by impressing a magnetizing quadrature e.m.f. upon an auxiliary motor circuit, as is done in the monocyclic motor. Below synchronism, the secondary currents are carried less than 90 deg., and thus the cross magnetization due to them is correspondingly reduced, and becomes zero at standstill. The torque is proportional to the power component of the armature currents times the intensity of magnetic flux in quad- rature position thereto. In the single-phase induction motor, the armature power currents I'\ = ea\ can exist only coaxially with the primary coil, since this is the only position in which corresponding pri- mary currents can exist. The magnetic flux in quadrature posi- tion is proportional to the component of e carried in quadrature, or approximately to (1 — s) e, and the torque is thus D = (1 - s) el' = (1 - s) e2alf thus decreases much faster with decreasing speed, and becomes zero at standstill. The power is then P = (1 - sYel' = (1 - s)262a!. Since in the single-phase motor only one primary circuit but a multiplicity of secondary circuits exist, all secondary circuits are to be considered as corresponding to the same primary cir- cuit, and thus the joint impedance of all secondary circuits must be used as the secondary impedance, at least at or near syn- chronism. Thus, if the armature has a quarter-phase winding of impedance Zi per circuit, the resultant secondary impedance is r? sr; if it contains a three-phase winding of impedance Zi per a 17 circuit, the resultant secondary impedance is •=- In consequence hereof the resultant secondary impedance of a single-phase motor is less in comparison with the primary im- pedance than in the polyphase motor. Since the drop of speed under load depends upon the secondary resistance, in the single- INDUCTION MACHINES 329 phase induction motor the drop in speed at load is generally less than in the polyphase motor; that is, the single-phase induction motor has a greater constancy of speed than the polyphase induction motor, but just as the polyphase induction motor, it can never reach complete synchronism, but slips below synchro- nism, approximately in proportion to the speed. The further calculation of the single-phase induction motor is identical with that of the polyphase induction motor, as given in the previous chapter. Often no special motors are used for single-phase circuits, but polyphase motors adapted thereto. An induction motor with only one primary winding could not be started by a phase- splitting device, and would necessarily be started by external means. A polyphase motor, as for instance a three-phase motor operating single-phase, by having two of its terminals connected to the single-phase mains, is just as satisfactory a single-phase motor as one built with only one primary winding. The only difference is that in the latter case a part of the circumference of the primary structure is left without winding, while in the polyphase motor this part contains windings also, which, how- ever, are not used, or are not effective when running as single- phase motor, but are necessary when starting by means of displaced e.m.fs. Thus, in a three-phase motor operating from single-phase mains, in starting, the third terminal is connected to a phase-displacing device, giving to the motor the cross mag- netization in quadrature to the axis of the primary coil, which at speed is produced by the rotation of the secondary currents, and which is necessary for producing the torque by its action upon the secondary power currents. Thus the investigation of the single-phase induction motor resolves itself into the investigation of the polyphase motor operating on single-phase circuits. 2. LOAD AND SPEED CURVES 147. Comparing thus a three-phase motor of exciting admit- tance per circuit Y = g — jb and self-inductive impedances ZQ = rQ + jxQ and Zi = TI + jxi per circuit with the same motor operating as single-phase motor from one pair of termi- nals, the single-phase exciting admittance is Y' = 3 Y (so as to give, the same volt-amperes excitation 3 eF), the primary 330 ELEMENTS OF ELECTRICAL ENGINEERING self-inductive impedance is the same, ZQ = r0 + jxo', the sec- ondary self-inductive impedance single-phase, however, is only y Z'i = -5-, since all three secondary circuits correspond to the same primary circuit, and thus the total impedance single-phase 17 is Z' = ZQ + -TT, while that of the three-phase motor is £J = ZlQ "T" Zl\. Assuming approximately Z0 = Zi, we have Thus, in absolute value, s' = % 2, and T' = 2T; that is, the characteristic constant of a motor running single- phase is twice what it is running three-phase, or polyphase in 1000 2000 3000 4000 5000 7000 3000 9000 FIG. 181. — Three-phase induction motor on single-phase circuit, load curves. general; hence, the ratio of exciting current to current at stand- still, or of waste flux to useful flux, is doubled by changing from polyphase to single-phase. This explains the inferiority of the single-phase motor com- pared with the polyphase motor. As a rule, an average polyphase motor makes a poor single- phase motor, and a good single-phase motor must be an excellent polyphase motor. INDUCTION MACHINES 331 As instances are shown in Figs. 181 and 182 the load curves and speed curves of the three-phase motor of which the curves of one circuit are given in Figs. 176 and 177, having the following constants : eo = 110 Three-phase Y = 0.01 -O.lj, ZQ = 0.1 + 0.3 j, Zi = 0.1 + 0.3,7, Thus, 7 = 6.36. Single-phase Y = 0.03 - 0.3 j, ZQ = 0.1 + 0.3 J, Zl = 0.033 +. 0.1 j, Thus, 7 = 12.72. It is of interest to compare Fig. 181 with Fig. 176 and to note the lesser drop of speed (due to the relatively lower secondary SLIP, S"= FIG. 182. — Three-phase induction motor on single-phase circuit, s curves. resistance) and lower power-factor and efficiencies, especially at light load. The maximum output is reduced from 3 X 7000 = 21,000 in the three-phase motor to 9100 watts in the single-phase motor. Since, however, the internal losses are less in the single-phase motor, it can be operated at from 25 to 30 per cent, higher mag- netic density than the same motor polyphase, and in this case its output is from two-thirds to three-quarters that of the poly- phase motor. 148. The preceding discussion of the single-phase induction motor is approximate, and correct only at or near synchronism, 332 ELEMENTS OF ELECTRICAL ENGINEERING where the magnetic field is practically a uniformly rotating field of constant intensity, that is, the quadrature flux produced by the armature magnetization equal to the main magnetic flux produced by the impressed e.m.f. If an accurate calculation of the motor at intermediate speed and at standstill is required, the changes of effective exciting admittance and of secondary impedance, due to the decrease of the quadrature flux, have to be considered. At synchronism the total exciting admittance gives the m.m.f. of main flux and auxiliary flux, while at standstill the quad- rature flux has disappeared or decreased to that given by the starting device, and thus the total exciting admittance has de- 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 183. — Three-phase induction motor on single-phase circuit, torque curves. creased to one-half of its synchronous value, or one-half plus the exciting admittance of the starting flux. The effective secondary impedance at synchronism is the joint impedance of all secondary circuits; at standstill, however, only the joint impedance of the projections of the secondary coils on the direction of the main flux, that is, twice as large as at syn- chronism. In other words, from standstill to synchronism the effective secondary impedance gradually decreases to one-half its standstill value at synchronism. For fuller discussion hereof the reader must be referred to my second paper on the Single-phase Induction Motor, Transactions A. I. E. E., 1900, page 37. The torque in Fig. 182 obviously slopes toward zero at stand- INDUCTION MACHINES 333 still. The effect of resistance inserted in the secondary of the single-phase motor is similar to that in the polyphase motor in so far as an increase of resistance lowers the speed at which the maximum torque takes place. While, however, in the poly- phase motor the maximum torque remains the same, and merely shifts toward lower speed with the increase of resistance, in the single-phase motor the maximum torque decreases proportionally to the speed at which the maximum torque point occurs, due to the factor (1 — s) entering the equation of the torque, D = e2^ (1 - s). Thus, in Fig. 183 are given the values of torque of the single- phase motor for the same conditions and the same motor of which the speed curves polyphase are given in Fig. 179. The maximum value of torque which can be reached at any speed lies on the tangent drawn from the origin onto the torque curve for 7*1 = 0.1 or short-circuited secondary. At low speeds the torque of the single-phase motor is greatly increased by the insertion of secondary resistance, just as in the polyphase motor. 3. STARTING DEVICES OF SINGLE-PHASE MOTORS 149. At standstill, the single-phase induction motor has no starting torque, since the line of polarization due to the second- ary currents coincides with the axis of magnetic flux impressed by the primary circuit. Only when revolving is torque pro- duced, due to the axis of secondary polarization being shifted by the rotation, against the axis of magnetism, until at or near synchronism it is in quadrature therewith, and the magnetic disposition thus the same as that of the polyphase induction motor. Leaving out of consideration starting by mechanical means and starting by converting the motor into a series or shunt motor, that is, by passing the alternating current by means of commutator and brushes through both elements of the motor, the following methods of starting single-phase motors are left: 1st. Shifting of the axis of armature or secondary polarization against the axis of generating magnetism. 2d. Shifting the axis of magnetism, that is, producing a mag- netic flux displaced in position from the flux producing the arma- ture currents. 334 ELEMENTS OF ELECTRICAL ENGINEERING The first method requires a secondary system which is unsym- metrical in regard to the primary, and thus, since the secondary is movable, requires means of changing the secondary circuit, such as 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 repulsion motor, which is a commutator motor also. With the commutatorless induction motor, or motor with permanently closed armature circuits, all starting devices con- sist in establishing an auxiliary magnetic flux in phase with the secondary currents in time, and in quadrature with the line of secondary polarization in space. They consist in producing a component of magnetic flux in quadrature in space with the primary magnetic flux producing the secondary currents, and in phase with the latter, that is, in time quadrature with the primary magnetic flux. Thus, if Fp = polarization due to the secondary currents, a = auxiliary magnetic flux, 6 = phase displacement in time between 3>a and 3>p, and T = phase displacement in space between ^a and Fp, the torque is D = Fp$a sin T cos 6. In general the starting torque, apparent torque efficiency, etc., of the single-phase induction motor with any of these de- vices are given in per cent, of the corresponding values of the same motor with polyphase magnetic flux, that is, with a mag- netic system consisting of two equal magnetic fluxes in quad- rature in time and space. 150. The infinite variety of arrangements proposed for start- ing single-phase induction motors can be grouped into three classes. 1. Phase-splitting Devices. The primary system is composed of two or more circuits displaced from each other in position, and combined with impedances of different inductance factors so as to produce a phase displacement between them. When using two motor circuits, they can either be connected in series between the single-phase* mains, and shunted with impedances of different inductance factors, as, for instance, a INDUCTION MACHINES 335 condensance and an inductance, or they can be connected in shunt between the single-phase mains but in series with impe- dances of different inductance factors. Obviously the impe- dances used for displacing the phase of the exciting coils can either be external or internal, as represented by high-resistance winding in one coil of the motor, etc. In this class belongs the use of the transformer as a phase- splitting device by inserting a transformer primary in series with one motor circuit in the main line and connecting the other motor circuit to the secondary of the transformer, or by feeding one of the motor circuits directly from the mains and the other from the secondary of a transformer connected across the mains with its primary. In either case it is, respectively, the internal impedance, or internal admittance, of the transformer which is combined with one of the motor circuits for displacing its phase, and thus this arrangement becomes most effective by using transformers of high internal impedance or admittance as con- stant power transformers or open magnetic circuit transformers. 2. Inductive Devices. The motor is excited by the combina- tion of two or more circuits which are in inductive relation to each other. This mutual induction between the motor circuits can take place either outside of the motor in a separate phase- splitting device or in the motor proper. In the first case the simplest form is the divided circuit whose branches are inductively related to each other by passing around the same magnetic circuit external to the motor. In the second case the simplest form is the combination of a primary exciting coil and a short-circuited secondary coil on the primary member of the motor, or a secondary coil closed by an impedance. In this class belong the shading coil and the accelerating coil. 3. Monocyclic Starting Devices. An essentially wattless e.m.f. of displaced phase is produced outside of the motor, and used to energize a cross magnetic circuit of the motor, either directly by a special teaser coil on the motor, or indirectly by combining this wattless e.m.f. with the main e.m.f. and thereby deriving a system of e.m.fs. of approximately three-phase or any other relation. In this case the primary system of the motor is supplied essentially by a polyphase system of e.m.fs. with a single-phase flow of energy, a system which I have called "monocyclic." 336 ELEMENTS OF ELECTRICAL ENGINEERING The wattless quadrature e.m.f. is generally produced by con- necting two impedances of different inductance factors in series between the single-phase mains, and joining the connection between the two impedances to the third terminal of a three- phase induction motor, which is connected with its other two terminals to the single-phase lines, as shown diagrammatically in Fig. 184, for a conductance a and an inductive susceptance -jo,. This starting device, when using an inductance and a conden- sance of proper size, can be made to give an apparent starting torque efficiency superior to that of the polyphase induction motor. Usually a resistance and an inductance are used, which, though not giving the same starting torque efficiency as available by the use of a condensance, have the advantage of greater simplicity and cheapness. After starting, the impedances are disconnected. For a complete discussion and theoretical investigation of the FIG. 184. — Connections for starting single-phase motor. different starting devices, the reader must be referred to the paper on the single-phase induction motor, American Institute of Electrical Engineers' Transactions, February, 1898." 151. The use of the resistance-inductance, or monocyclic, starting device with three-phase wound induction motor will be discussed somewhat more explicitly as the only method not us- ing condensers which has found extensive commercial application. It gives relatively the best starting torque and torque efficiencies. In Fig. 184, M represents a three-phase induction motor of which two terminals, 1 and 2, are connected to single-phase mains and the terminal 3 to the common connection of a conduct- ance a (that is, a resistance - j and an equal susceptance — ja (thus a reactance H — ) connected in series across the mains. Let Y = g — jb = total admittance of motor between termi- INDUCTION MACHINES 337 nals 1 and 2 while at rest. We then have HY = total admit- tance from terminal 3 to terminals 1 and 2, regardless of whether the motor is delta- or F-wound. If e = e.m.f. in the single-phase mains and E = difference of potential across conductance a of the starting device, then we have the current in a as /i = Ea, and the e.m.f. across — ja as e — E' thus, the current in — ja is li = - ja (e - E), and the current in the cross magnetizing motor circuit from 3 to 1, 2 is /o = /i - /2 = Ea + ja (e - E). The e.m.f. ^0 of the cross magnetizing circuit is, as may be seen from the diagram of e.m.fs., which form a triangle with e, E and e — E as sides, Eo = E - (e - E) = 2 E - e, and since !» = H YEQ, we have Ea + ja (e - E) = % Y (2E - e). This expression solved for E becomes € which from the foregoing value of EQ gives 3eaQ'+l) . ~3a-3ja-8F' or, substituting Y-g- jb, expanding, and multiplying both numerator and denominator by (3 a -80) +j(3a - 86), gives EQ = ea and the imaginary component thereof, or e.m.f. in quadrature to e in time and in space, is 338 ELEMENTS OF ELECTRICAL ENGINEERING In the same motor on a three-phase circuit this quadrature e.m.f. is the altitude of the equilateral triangle with e as sides, thus = — je — 7r-, and since the starting torque of the motor is pro- portional to this quadrature e.m.f., the relative starting torque of the monocyclic starting device, or the ratio of starting torque of the motor with monocyclic starting device to that of the same motor on three-phase circuit, is /)/ = • EG!_ : 2a 2a- %(