I. General 3. The most important class of alternating-current apparatus consists of the synchronous machines. They comprise the alternating-current generators, single-phase and polyphase, the synchronous motors, the phase compensators, the phase con- verters, the phase balancers, the synchronous boosters and the exciters of induction generators, that is, synchronous machines producing wattless lagging or leading currents, and the con- verters. Since the latter combine features of the commutating machines with those of the synchronous machines they will be considered separately. In the synchronous machines the terminal voltage and the generated e.m.f. are in synchronism with, that is, of the same frequency as, the speed of rotation. These machines consist of an armature, in which e.m.f. is generated by the rotation relatively to a magnetic field, and a continuous magnetic field, excited either by direct current, or by the reaction of displaced phase armature currents, or by per- manent magnetism. The formula for the e.m.f. generated in synchronous machines, commonly called alternators, is E = S2irn3> = 4. where n is the number of armature turns in series interlinked with the magnetic flux <1> (in megalines per pole), / the frequency of rotation (in hundreds of cycles per second), E the e.m.f. gen- erated in the armature turns. This formula assumes a sine wave of e.m.f. If the e.m.f. wave differs from sine shape, the e.m.f. is E = 4.447/n, 2 -\/2 where y = form factor of the wave, or — - times ratio of effect- 7T ive to mean value of wave, that is, the ratio ,of the effective value of the generated e.m.f. to that of a sine wave generated by the same magnetic flux at the same frequency. 126 SYNCHRONOUS MACHINES 127 The form factor 7 depends upon the wave shape of the gener- ated e.m.f. The wave shape of e.m.f. generated in a single con- ductor on the armature surface is identical with that of the dis- tribution of magnetic flux at the armature surface and will be discussed more fully in the chapter on commutating machines. The wave of total e.m.f. is the sum of the waves of e.m.f. in the individual conductors, added in their proper phase relation, as corresponding to their relative positions on the armature surface. 4. In a Y or star-connected three-phase machine, if EQ = e.m.f. per circuit, or Y or star e.m.f., E = E0 \/3 is the e.m.f. between terminals or A (delta) or ring e.m.f., since two e.m.fs. displaced by 60 degrees are connected in series between terminals (V3 = 2 cos 30°). In a A-connected three-phase machine, the e.m.f. per circuit is the e.m.f. between the terminals, or A e.m.f. In a F-connected three-phase machine, the current per circuit is the current issuing from each terminal, or the line current, or Y current. In a A-connected three-pHase machine, if J0 = current per circuit, or A current, the current issuing from each terminal, or the line or F current, is / = /o V3. Thus in a three-phase system, A current and e.m.f., and F current and e.m.f. (or ring and start current and e.m.f. respect- ively), are to be distinguished. They stand in the proportion 1 - V3. As a rule, when speaking of current and of e.m.f. in a three- phase system, under current the F current or current per line, and under e.m.f. the A e.m.f. or e.m.f. between lines is understood. 5. While the voltage wave of a single conductor has the same shape as the distribution of the magnetic flux at the armature circumference and so may differ considerably from a sine, that is, contain pronounced higher harmonics, the terminal voltage is the resultant of the waves of many conductors, and, especially with a distributed armature winding, shows the higher harmonics in a much reduced degree; that is, the resultant is nearer sine shape, and some harmonics may be entirely eliminated in the terminal voltage wave, though they may appear in the voltage wave of a single conductor. Thus, for instance, in a three-phase F-connected machine, the voltage per circuit, or F voltage, may contain a third harmonic and multiples thereof, while in the 128 ELEMENTS OF ELECTRICAL ENGINEERING voltage between the terminals this third harmonic is eliminated. The voltage between the terminals is the resultant of two Y voltages, displaced from each other by 60 degrees. Sixty de- grees for the fundamental, however, is 3 X 60° = 180°, or oppo- sition for the third harmonic; that is, the third harmonics in those two Y voltages, which combine to the delta or terminal voltage, are opposite, and so neutralize each other. Even in a single turn, harmonics existing in the magnetic field and thus in the single conductor can be eliminated by fractional pitch. Thus, if the pitch of the armature turn is not 180 de- grees, but less by -> the e.m.fs. generated in the two conductors n of a single turn are not exactly in phase, but differ by - of a half fl> wave for the fundamental, and thus a whole half wave for the nth harmonic, so that their nth harmonics are in opposition and thus cancel. Fractional pitch winding of a "pitch deficiency" of - thus eliminates the nth harmonic; for instance, with 80 per ^ • . cent, pitch, the fifth harmonic cannot exist. In this manner higher harmonics of the e.m.f. wave can be reduced or entirely eliminated, though in general, with a dis- tributed winding, the wave shape is sufficiently close to sine shape without special precaution being taken in the design.