VH. Frequency Converter or General Alternating-current Transformer 159. The e.m.fs. generated in the secondary of the induction machine are of the frequency of slip, that is, synchronism minus speed, thus of lower frequency than the impressed e.m.f. in the range from standstill to double synchronism; of higher frequency outside of this range. ' Thus, by opening the secondary circuits of the induction machine and connecting them to an external or consumer's cir- cuit, the induction machine can be used to transform from one frequency to another, as frequency converter. It lowers the frequency with the secondary running at a speed between standstill and double synchronism, and raises the fre- quency with the secondary either driven backward or above double synchronism. Obviously, the frequency converter can at the same time change the e.m.f. by using a suitable number of primary and secondary turns, and can change the phases of the system by having a secondary wound for a different number of phases from the primary, as, for instance, convert from three phase 6000 volts 25 cycles to quarter phase 2500 volts 62.5 cycles. Thus, a frequency converter can be called a "general alter- nating-current transformer." For its theoretical discussion and calculation, see " Theory and Calculation of Alternating-current Phenomena." The action and the equations of the general alternating-current INDUCTION MACHINES 355 transformer are essentially those of the stationary alternating- current transformer, except that the ratio of secondary to primary generated e.m.f . is not the ratio of turns but the ratio of. the product of turns and frequency, while the ratio of secondary current and primary load current (that is, total primary current minus primary exciting current) is the inverse ratio of turns. The ratio of the products of generated e.m.f. and current, that is, the ratio of electric power generated in the secondary to electric power consumed in the primary (less excitation), is thus not unity but is the ratio of secondary to primary frequency. Hence, when lowering the frequency with the secondary re- volving at a speed between standstill and synchronism, the secondary output is less than- the primary input, and the differ- ence is transformed into mechanical work; that is, the machine acts at the same time as induction motor, and when used in this manner is usually connected to a synchronous or induction gen- erator feeding preferably into the secondary circuit (to avoid double transformation of its output) or to a synchronous con- verter, which transforms the mechanical power of the frequency converter into electrical power. When raising the frequency by backward rotation, the sec- ondary output is greater than the primary input (or rather the electric power generated in the secondary greater than the pri- mary power consumed by the generated e.m.f.), and the differ- ence is to be supplied by mechanical power by driving the fre- quency changer backward by synchronous or induction motor, preferably connected to the primary circuit, or by any other motor device. Above synchronism the ratio of secondary output to primary input becomes negative; that is, the induction machine generates power in the primary as well as in the secondary, the primary power at the impressed frequency, the secondary power at the frequency of slip, and thus requires mechanical driving power. The secondary power and frequency are less than the primary below double synchronism, more above double synchronism, and are equal at double synchronism, so that at double syn- chronism the primary and secondary may be connected in multi- ple or in series and the machine is then a double synchronous alternator further discussed in the "Theory and Calculation of Electrical Apparatus." As far as its transformer action is concerned, the frequency 356 ELEMENTS OF ELECTRICAL ENGINEERING converter is an open magnetic circuit transformer, that is, a trans- former of relatively high magnetizing current. It combines therewith, however, the action of an induction motor or generator. Excluding the case of over-synchronous rotation, it is approxi- mately (that is, neglecting internal losses) electrical input -r- electrical output -f- mechanical output = primary frequency -f- secondary frequency -f- speed or primary minus secondary fre- quency; that is, the mechanical output is negative when increas- ing the frequency by backward rotation. Such frequency converters are to a certain extent in com- mercial use, and have the advantage over the motor-generator plant of requiring an amount of apparatus equal only to the out- put, while the motor-generator set requires machinery equal to twice the output. An application of the frequency converter when lowering the frequency is made in concatenation or tandem control of induc- tion machines, as described in the next section. In this case the first motor, or all the motors except the last of the series are in reality frequency converters.