VIII. Concatenation of Induction Motors 160. In the secondary of the induction motor an e.m.f. is generated of the frequency of slip. Thus connecting the sec- ondary circuit of the induction motor to the primary of a second induction motor, the latter is fed by a frequency equal to the slip of the first motor, and reaches its synchronism at the frequency of slip of the first motor, the first motor then acting as frequency converter for the second motor. If, then, two equal induction motors are rigidly connected together and thus caused to revolve at the same speed, the speed of the second motor, which is the slip s of the first motor at no load, equals the speed of the first motor: s = 1 — s, and thus s = 0.5. That is, a pair of induction motors connected this way in tandem or in concatenation, that is, " chain connection/' as commonly called, or in cascade, as called abroad, tends to ap- proach s = 0.5, or half synchronism, at no load, slipping below this speed under load; that is, concatenation of two motors re- duces their synchronous speed to one-half, and thus offers as means to operate induction motors at one-half speed. In general, if a number of induction machines are connected INDUCTION MACHINES 357 in tandem, that is, the secondary of each motor feeding the primary of the next motor, the secondary of this last motor being short-circuited, the sum of the speeds of all motors tends toward synchronism, and with all motors connected together so as to revolve at the same speed the system operates at - synchronous speed, when n = number of motors. If the two induction motors on the same shaft have a different number of poles, they synchronize at some other speed below synchronism, or if con- nected differentially, they synchronize at some speed above synchronism. Assuming the ratio of turns of primary and secondary as 1 : 1, with two equal induction motors in concatenation at standstill, the frequency and the e.m.f. 'impressed upon the second motor, neglecting the drop of e.m.f. in the internal impedance of the first motor, equal those of the first motor. With increasing speed, the frequency and the e.m.f. impressed upon the second motor decrease proportionally to each other, and thus the magnetic flux and the magnetic density in the second motor, and its ex- citing current, remain constant and equal to those of the first motor, neglecting internal losses; that is, when connected in con- catenation the magnetic density, current input, etc., and thus the torque developed by the second motor, are approximately equal to those of the first motor, being less because of the internal losses in the first motor. Hence, the motors in concatenation share the work in approxi- mately equal portions, and the second motor utilizes the power which without the use of a second motor at less than one-half synchronous speed would have to be wasted in the secondary resistance; that is, theoretically concatenation doubles the torque and output for a given current, or power input into the motor system. In reality the gain is somewhat less, due to the second motor not being quite equal to a non-inductive resistance for the secondary of the first motor, and due to the drop of voltage in the internal impedance of the first motor, etc. At one-half synchronism, that is, the limiting speed of the con- catenated couple, the current input in the first motor equals its exciting current plus the transformed exciting current of the second motor, that is, equals twice the exciting current. 161. Henee, comparing the concatenated couple with a single motor, the primary exciting admittance is doubled. The total 358 ELEMENTS OF ELECTRICAL ENGINEERING impedance, primary plus secondary, is that of both motors, that is, doubled also, and the characteristic constant of the con- catenated couple is thus four times that of a single motor, but the speed reduced to one-half. FIG. 192. — Comparison of concatenated motors with a single motor of double the number of poles. Comparing the concatenated couple with a single motor re- wound for twice the number of poles, that is, one-half speed also, such rewinding does not change the self-inductive impe- INDUCTION MACHINES 359 dance, but quadruples the exciting admittance, since one-half as many turns per pole have to produce the same flux in one-half the pole arc, that is, with twice the density. Thus the character- istic constant is increased fourfold also. It follows herefrom that the characteristic constant of the concatenated couple is that of one motor rewound for twice the number of poles. The slip under load, however, is less in the concatenated couple than in the motor with twice the number of poles, being due to only one-quarter the internal impedance, the secondary impedance of the second motor only, and thus the efficiency is slightly higher. 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1-0.2-0.3-0.4-0.5-0.0-0.7 FIG. 193. — Concatenation of induction motors, speed curves. Two motors coupled in concatenation are in the range from standstill to one-half synchronism approximately equivalent to one motor of twice the admittance, three times the primary impedance, and the same secondary impedance as each of the two motors, or more nearly 2.8 times the primary and 1.2 times the secondary impedance of one motor. Such motor is called the equivalent motor. 162. The calculation of the characteristic curve of the concate- nated motor system is similar to, but more complex than, that of the single motor. Starting from the generated e.m.f. e of the second motor, reduced to full frequency, we work up to the im- 360 ELEMENTS OF ELECTRICAL ENGINEERING pressed e.m.f. of the first motor e0, by taking due consideration of the proper frequencies of the different circuits. Herefor the reader must be referred to " Theory and Calculation of Electrical Apparatus." The load curves of the pair of three-phase motors of the same constants as the motor in Figs. 176 and 177 are given in Fig. 192, the complete speed curve in Fig. 193. Fig. 192 shows the load curve of the total couple, of the two individual motors, and of the equivalent motor. As seen from the speed curve, the torque from standstill to one-half synchronism has the same shape as the torque curve of a single motor between standstill and synchronism. At one-half synchronism the torque reverses and becomes negative. It reverses again at about two-thirds synchronism, and is positive -0000 -4000 -2000 0 -2000 -4000 -000« , \ Vz ,-0.1- -O.Sj Y - 3.01 - O.tj £ "~"x \ g a z A ^* 2. CO \\ .^ ^" V ""/ / 1.0 < .9 0 8 0 7 0 6 0 x, .5 0 ^ 4 0 3 0 2 0 1 0.( FIG. 194. — Concatenation of induction motors speed curve with resistance in the secondary circuit. between about two-thirds synchronism and synchronism, zero at synchronism, and negative beyond synchronism. Thus, with a concatenated couple, two ranges of positive torque and power as induction motor exist, one from standstill to half synchronism, the other from about two-thirds synchro- nism to synchronism. In the ranges from one-half synchronism to about two-thirds synchronism, and beyond synchronism, the torque is negative, that is, the couple acts as generator. The insertion of resistance in the secondary of the second motor has in the range from standstill to half synchronism the same effect asin a single induction motor, that is, shifts the maxi- mum torque point toward lower speed without changing its value. Beyond half synchronism, however, resistance in the INDUCTION MACHINES 361 secondary lengthens the generator part of the curve, and makes the second motor part of the curve more or less disappear, as seen in Fig. 194, which gives the speed curves of the same motor as Fig. 193, with resistance in circuit in the secondary of the second motor. The main advantages of concatenation are obviously the abil- ity of operating at two different speeds, the increased torque and power efficiency below half speed, and the generator or braking action between half speed and synchronism, and such concatena- tion is therefore used to some extent in three-phase railway motor equipments, while for stationary motors usually a change of the number of poles by reconnecting the primary winding through a suitable switch is preferred where several speeds are desired, as it requires only one motor.