XV. Fluctuating Cross Currents in Parallel Operation 27. In alternators operated from independent prime movers, it is not sufficient that the average frequency corresponding to the average speed of the prime movers be the same, but still more important that the frequency be the same at any instant, that is, that the frequency (and thus the speed of the prime mover) be constant. In rotary prime movers, as turbines or electric motors, this is usually the case; but with reciprocating machines, as steam engines, the torque and thus the speed of rotation rises and falls periodically during each revolution, with the frequency of the engine impulses. The alternator con- nected with the engine will thus not have uniform frequency, but a frequency which pulsates, that is, rises and falls. The amplitude of this pulsation depends upon the design of the engine, the momentum of its fly-wheel, and the action of the engine governor. If two alternators directly connected to equal steam engines are synchronized so that the moments of maximum frequency coincide, there will be no energy cross currents between the machines, but the frequency of the whole system rises and falls periodically. In this case the engines are said to be synchronized. The parallel operation of the alternators is satisfactory in this case provided that the pulsations of engine speeds are of the same size and duration; but apparatus requiring constant fre- quency, as synchronous motors and rotary converters, when operated from such a system, will give a reduced maximum out- put, due to periodic cross currents between the generators of fluctuating frequency and the synchronous motors of constant frequency, and in an extreme case the voltage of the whole sys- tem will be caused to fluctuate periodically. Even with small fluctuations of engine speed the unsteadiness of current due to this source is noticeable in synchronous motors and synchronous converters. If the alternators happen to be synchronized in such a position that the moment of maximum speed of the one coincides with the moment of minimum speed of the other, alternately the one and then the other alternator will run ahead, and thus there 156 ELEMENTS OF ELECTRICAL ENGINEERING will be a pulsating power cross current between the alternators, transferring power from the leading to the lagging machine, that is, alternately from the one to the other, and inversely, with the frequency of the engine impulses. These pulsating cross currents are the most undesirable, since they tend to make the voltage fluctuate and to tear the alternators out of synchro- nism with each other, especially when the conditions are favorable to a cumulative increase of this effect by what may be called mechanical resonance (hunting) of the engine governors, etc. They depend upon the synchronous impedance of the alternators and upon their phase difference, that is, the number of poles and the fluctuation of speed, and are specially objectionable when operating synchronous apparatus in the system. 28. Thus, for instance, if two 80-pole alternators are directly connected to single-cylinder engines of 1 per cent, speed varia- tion per revolution, twice during each revolution the speed will rise, and fall twice; and consequently the speed of each alternator will be above average speed during a quarter revolution. Since the maximum speed is 1/2 per cent, above average, the mean speed during the quarter revolution of high speed is 1/4 per cent, above average speed, and by passing over 20 poles the armature of the machine will during this time run ahead of its mean posi- tion by 1/4 per cent, of 20 or 1/20 pole, that is, 180/20 = 9 elec- trical space degrees. If the armature of the other alternator at this moment is behind its average position by 9 electrical space degrees, the phase displacement between the alternator e.m.fs. is 18 electrical time degrees; that is, the alternator e.m.fs. are represented by OEi and OEZ in Fig. 71, and when running in parallel the e.m.f. OEf = E\E^ is short circuited through the synchronous impedance of the two alternators. . Since E' = OE\ = 2 EI sin 9 deg. the maximum cross current is ffisin9deg. 0.156 ffi 1 = = = U.loo 1 o, 20 20 ET where IQ = -- = short-circuit current of the alternator at full- 20 load excitation. Thus, if the short-circuit current of the alter- nator is only twice full-load current, the cross current is 31.2 per cent, of full-load current. If the short-circuit current is 6 times full-load current, the cross current is 93.6 per cent, of full-load current or practically equal to full-load current. Thus SYNCHRONOUS MACHINES 157 the smaller the armature reaction, or the better the regulation, the larger are the pulsating cross currents between the alternators, due to the inequality of the rate of rotation of the prime movers. Hence for satisfactory parallel operation of alternators connected to steam engines, a certain amount of armature reaction is de- sirable and very close regulation undesirable. By the transfer of energy between the machines the pulsations of frequency, and thus the cross currents, are reduced somewhat. Very high armature reaction is objectionable also, since it reduces the synchronizing power, that is, the tendency of the machines to hold each other in step, by reducing the energy transfer be- tween the machines. As seen herefrom, the problem of parallel operation of alternators is al- most entirely a problem of the regulation of their prime movers, especially steam A ^^ engines. With alternators driven by gas engines, the problem of parallel operation is made more difficult by the more jerky nature of the gas-engine ^ 73._Phase displacement between impulse. In such machines, alternators to be synchronized, therefore, squirrel-cage wind- ings in the field-pole faces are commonly used, to assist synchron- izing by the currents induced in this short-circuited winding, on the principle of the induction machine. From Fig. 73 it is seen that the e.m.f. OEr or EiE2, which causes the cross current between two alternators in parallel con- nection, if their e.m.fs. OEi and OE% are out of phase, is approxi- mately in quadrature with the e.m.fs. OE\ and OE2 of the machines, if these latter two e.m.fs. are equal to each other. The cross current between the machines lags behind the e.m.f. producing it, OE* ', by the angle co, where tan w = — , and XQ = 7*0 reactance, r0 = effective resistance of alternator armature. The energy component of this cross current, or component in phase with OEfj is thus in quadrature with the machine voltages OEi and OE2, that is, transfers no power between them. The power transfer or equalization of load between the two machines takes place by the wattless or reactive component of cross current, E' 158 ELEMENTS OF ELECTRICAL ENGINEERING that is, the component which is in quadrature with OE', and thus in phase with one and in opposition with the other of the machine e.m.fs. OEi and OE^. 29. Hence, machines without reactance would have no syn- chronizing power, or could not be operated in parallel. The theoretical maximum synchronizing power exists if the reactance equals the resistance: XQ = r0. This condition, however, cannot be realized, and if realized would give a dangerously high syn- chronizing power and cross current. In practice, XQ is always very much greater than ro, and the cross current thus practically in quadrature with OE', that is, in phase (or opposition) with the machine voltages, and is consequently an energy-transfer current. If, however, alternators are operated in parallel over a circuit of appreciable resistance, as two stations at some distance from each other are synchronized, especially if the resistance between the stations is non-inductive, as underground cables, with alter- nators of very low reactance, as turbo alternators, the synchro- nizing power may be insufficient. In this case, reactance has to be inserted between the stations, to lag the cross current and thereby make it a power-transferring or synchronizing current. If, however, the machine voltages OEi and OEZ are different in value but approximately in phase with each other, the voltage causing cross currents, E\Ez , is in phase with the machine vol- tages and the crosscurrents thus in quadrature with the machine voltages OEi and OE%, and hence do not transfer energy, but are wattless. In one machine the cross current is a lagging or demagnetizing, and in the other a leading or magnetizing, current. Hence two kinds of cross currents may exist in parallel opera- tion of alternators — currents transferring power between the machines, due to phase displacement between their e.m.fs., and wattless currents transferring magnetization between the ma- chines, due to a difference of their induced e.m.fs. In compound-wound alternators, that is, alternators in which the field excitation is increased with the load by means of a series field excited by the rectified alternating current, it is al- most, but not quite, as necessary as in direct-current machines, when operating in parallel, to connect all the series fields in paral- lel by equalizers of negligible resistance, for the same reason — to insure proper division of current between machines. SYNCHRONOUS MACHINES 159