III. Armature Reaction 8. The magnetic flux in the field of an alternator under load is produced by the resultant m.m.f. of the field exciting current and of the armature current. It depends upon the phase rela- tion of the armature current. The e.m.f. generated by the field exciting current or the nominal generated e.m.f. reaches a maxi- mum when the armature coil faces the position midway between FIG. 48. — Model for study of armature reaction. Armature coils in position of maximum current. the field poles, as shown in Fig. 48, A and A'. Thus, if the armature current is in phase with the nominal generated e.m.f., it reaches its maximum in the same position A, A' of armature coil as the nominal generated e.m.f., and thus magnetizes the preceding, demagnetizes the following magnet pole (in the di- rection of rotation) in an. alternating-current generator A] magnetizes the following and demagnetizes the preceding mag- net pole in a synchronous motor A' (since in a generator the rotation is against, in a synchronous motor with the magnetic attractions and repulsions between field and armature). In this case the armature current neither magnetizes nor demag- netizes the field as a whole, but magnetizes the one side, demag- SYNCHRONOUS MACHINES 131 netizes the other side of each field pole, and thus merely distorts the magnetic field. 9. If the armature current lags behind the nominal generated e.m.f., it reaches its maximum in a position where the armature coil already faces the next magnetic pole, as shown in Fig. 48, B and Br, and thus demagnetizes the field in a generator B, magnetizes it in a synchronous motor Bf. If the armature current leads the nominal generated e.m.f., it reaches its maximum in an earlier position, while the arma- ture coil still partly faces the pre- ceding magnet pole, as shown in Fig. 48, C and C", and thus mag- netizes the field in a generator, Fig. 48, C, and demagnetizes it in a syn- chronous motor C'. With non-inductive load, or with the current in phase with the ter- minal voltage of an alternating- current generator, the current lags behind the nominal generated e.m.f., due to armature reaction and self- inductance, and thus partly de- magnetizes; that is, the voltage is lower under load than at no load with the same field excitation. In other words, lagging current demag- netizes and leading current magne- tizes the field of an alternating-cur- rent generator, while the opposite is the case with a synchronous motor. 10. In Fig. 49 let OF = F = resultant m.m.f. of field exci- tation and armature current (the m.m.f. of the field excita- tion being alternating with regard to the armature coil, due to its rotation) and 02 the lag of the current / behind the virtual e.m.f. E<2, generated by the resultant m.m.f. The virtual e.m.f. E2 lags in time 90 degrees behind the result- ant flux of OF, and is thus represented by OE2 in Fig. 47, and the m.m.f. of the armature current Fa by OFa, lagging by angle 02 behind OE2. The resultant m.m.f. OF is the diagonal of the parallelogram with the component m.m.fs. OFa = armature m.m.f. and OFQ = total impressed m.m.f. or field excitation, as FIG. 49. — Diagram of m.m.fs. in loaded generator. 132 ELEMENTS OF ELECTRICAL ENGINEERING sides, and from this construction OFQ is found. OF0 is thus the position of the field pole with regard to the armature. It is trigonometrically, If I = current per armature turn in amperes effective, n = number of turns per pole in a single-phase alternator, the arma- ture reaction is Fa = nl ampere-turns effective, and is pulsating between zero and nl \/2. In a quarter-phase alternator with n turns per pole and phase in series and I amperes effective per turn, the armature reaction per phase is nl ampere-turns effective and nl \/2 ampere-turns maximum. The two phases magnetize in quad- rature, in phase and in space. Thus, at the time t, correspond- ing to angle 6 after the maximum of the first phase, the m.m.f. in the direction by angle 6 behind the direction of the magnetiza- tion of the first phase is nl \/2 cos2 6. The m.m.f. of the second phase is nl \/2 sin2 0; thus the total m.m.f. or the armature reaction Fa = nl \/2, and is constant in intensity, but revolves synchronously with regard to the armature; that is, it is station- ary with regard to the field. In a three-phaser of n turns in series per pole and phase and / amperes effective per turn, the m.m.f. of each phase is nl \/2 ampere-turns maximum; thus at angle 6 in positon and angle 0 in time behind the maximum of one phase; The m.m.f. of this phase is nl \/2 cos2 0. The m.m.f. of the second phase is nl V2 cos2 (0 + 120) = n/V2 ( - 0.5 cos 0 - 0.5 \/3 sin 0)*. The m.m.f. of the third phase is nl V2 cos2 (0 + 240) = nl \/2 ( - 0.5 cos 0 + 0.5 \/3 sin 0)2. Thus the total m.m.f. or armature reaction, Fa = nl \/2 (cos2 0 + 0.25 cos2 0 + 0.75 sin2 0 + 0.25 cos2 0 + 0.75 sin2 0) = l.Snl \/2, constant in intensity, but revolving synchronously with regard to the armature, that is, stationary with regard to the field. These values of armature reaction correspond strictly only to the case where all conductors of the same phase are massed SYNCHRONOUS MACHINES 133 together in one slot. If the conductors of each phase are dis- tributed over a greater part of the armature surface, the values of armature reaction have to be multiplied by the average cosine of the total angle of spread of each phase. 11. The single-phase machine thus differs from the poly- phase machines: in the latter, on balanced load, the armature reaction is constant, while in the single-phase machine the armature reaction and thereby the resultant m.m.f. of field and armature is pulsating. The pulsation of the resultant m.m.f. of the single-phase machine causes a pulsation of its magnetic field under load, of double frequency, which generates a third harmonic of e.m.f. in the armature conductors. In machines of high armature reaction, as steam-turbine-driven single-phase alternators, the pulsation of the magnetic field may be sufficient to cause serious energy losses and heating by eddy currents, and thus has to be checked. This is usually done by a squirrel- cage induction machine winding in the field pole faces, or by short-circuited conductors laid in the pole faces in electrical space quadrature to the field coils. In these conductors, secondary currents Ei'_ of double frequency are produced which equalize the resultant m.m.f. of the machine.